UNIVERSIDADE TÉCNICA DE LISBOA

INSTITUTO SUPERIOR TÉCNICO

Combining Logics

Carlos Manuel Costa Lourenço Caleiro

(Mestre)

Dissertação para Obtenção do Grau de Doutor em Matemática

Orientador: Professor Doutor Amílcar Sernadas

    DOCUMENTO PROVISÓRIO

1 de Agosto de 2000

Combinação de Lógicas

Nome: Carlos Manuel Costa Lourenço Caleiro

Doutoramento em: Matemática

Orientador: Doutor Amílcar dos Santos Costa Sernadas

Provas concluídas em:

Resumo: O mecanismo de sincronização é definido e aplicado a várias noções concretas (enriquecidas ou não) de sistema lógico e de lógica, baseadas no conceito primário de sistema de consequência. Os sistemas de consequência enriquecidos consideram duas consequências explícitas, local e global, relacionadas mas distintas, em lugar de apenas uma operação de consequência sobre uma linguagem não estruturada. As noções semânticas de sistema lógico (enriquecido) consideradas incluem matrizes lógicas e uma variante em que os valores de verdade designados são substituídos por uma estrutura mais rica. As noções dedutivas consideram conjuntos de inferências finitárias, possivelmente distinguindo inferências locais de globais. O mecanismo de fibrilação é definido e aplicado a noções concretas de apresentação de sistema lógico e lógica enriquecidas, baseadas no conceito primário de sistema de consequência estrutural. A estruturalidade corresponde ao fecho para substituições sobre uma linguagem gerada por uma assinatura. A noção semântica de apresentação de sistema lógico enriquecido considerada requer a interpretação homomórfica de fórmulas sobre uma álgebra de valores de verdade. Dedutivamente, as inferências são substituídas por regras esquemáticas. Em ambos os contextos e para os dois mecanismos, são discutidos alguns exemplos de representação e aplicação. Para além do estudo detalhado da relação entre sincronização e fibrilação, são obtidos vários resultados de preservação de correcção e completude para ambos os mecanismos.

Palavras-chave: consequência lógica, sincronização, estruturalidade, semântica algébrica, fibrilação, completude.

Combining Logics

Abstract: The mechanism of synchronization is defined and applied to several concrete notions of (layered or not) logic system and logic, based on the prime concept of consequence system. Layered consequence systems explicitly consider related but distinct local and global consequences, instead of a simple consequence operation over an unstructured language. The model-theoretic notions of (layered) logic system considered include logical matrices and an enhanced version that replaces the set of designated truth-values by a richer internal structure. Proof-theoretic notions consider sets of finitary inferences, possibly with distinguished local and global inferences. The fibring mechanism is defined and applied to concrete notions of presentation of layered logic system and logic, based on the prime concept of structural consequence system. Structurality amounts to substitution closure over a signature generated language. The model-theoretic notion of layered logic system presentation considered further requires homomorphic sentence interpretation over an algebra of truth-values. Proof-theoretically, plain inferences are replaced by schematic inference rules. For both settings and constructions, a few meaningful representation and application examples are discussed. Besides a detailed study of the relationship between synchronization and fibring, several soundness and completeness preservation results for both mechanisms are obtained.

Keywords: logical consequence, synchronization, structurality, algebraic semantics, fibring, completeness.

Preface

My interest in the topic of logic combination traces back to 1996, soon after I had finished my MSc in Applied Mathematics. By then, the whole research background of the Logic and Computation Group of the Department of Mathematics of Instituto Superior Técnico, of which I am proud that I have been a member since 1992, was aimed at the logical foundations of software systems. Translations between logic frameworks, problems of multiple onthologies and their combination were, of course, frequent. Thus, I took it as a natural decision to embrace this subject as the theme for my PhD thesis after a suggestion made by my supervisor, Amílcar Sernadas. His original proposal, back in 1996, had two distinct aims. On the one hand, it focused on the categorial study and characterization, in an institution-like setting, of emerging combination mechanisms such as synchronization and fibring. On the other hand, it aimed at putting computability requirements into play, via the notion of an enumeration operator, and at studying their impact in the combination mechanisms.

Of course, along these 4 years, this ambitious plan has suffered quite a few changes. As we got deep into the subject, it became clear that much of what we could do would depend, first of all, on choosing appropriate notions of logic at the right level of abstraction. Thus, as we explored this field, computability issues were put aside. It would make little sense to try and complicate the problem even further before we reached a full and detailed understanding of its pure form.

The work presented in this dissertation has decisively been influenced by the work of many people. First of all, of course, Amílcar Sernadas and Cristina Sernadas, with whom I have coauthored a few papers on this subject. Apart from these, I should stress as other great sources of inspiration the work of people like Patrick Blackburn, Rod Burstall, Maura Cerioli, José Fiadeiro, Marcelo Finger, Dov Gabbay, Joseph Goguen, José Meseguer, Till Mossakowski, Maarten de Rijke, Andrzej Tarlecki and Alberto Zanardo. The direct or indirect contact with the work of these researchers has been a decisive drive to my understanding of this field. The book Theory of Logical Calculi by Riszard Wójcicki, [Wój88], deserves a special reference for having been the Oracle that helped answer many of my doubts, and for leading me in the track of the pioneering work of Alfred Tarski, Jan Lukasiewicz, Adolf Lindenbaum and other Gods of Logic.

I cannot forget the oportunities I had of exposing my work and learning about other people’s deeds, provided to me by the several national and international projects of the Logic and Computation Group. I can trace this interchange back to the ESPRIT BRAs COMPASS and IS-CORE, and the JNICT project FAC3. These have provided the means for my first scientific contacts. More recently, I should stress the support of the ESPRIT WGs FIREworks and ASPIRE. I should specially mention Hans-Dieter Ehrich, the chairman of ASPIRE, and the TMR Programme, for having invited me, and supported my participation, in the Dagstuhl Seminar, where I presented some ideas about combined logics for multi-agent systems. I should also mention the support by the PRAXIS XXI projects SitCalc and, even more strongly, LogComp and ACL. Not only does the work I have developed fit in with several of the tasks of these two projects, but most of the material I have included was also presented in their Meetings. They have also funded my participation in the AMAST’97 Conference, where I had the chance to present an embryo of the work in Chapters 2 and 3, and in WADT’98, where I presented related work on parameterization of logics. Finally, a word is also due to FCT and the organizer of our local Seminar on Logic and Computation, Amílcar Sernadas, for having provided me with yet more opportunities of disseminating the contents of this thesis and, most of all, for having invited so many interesting and valuable researchers in this and other topics.

Acknowledgments

To my supervisor, Amílcar Sernadas, for his guidance, support and criticism throughout the years, to whom I also owe a set of fundamental reference points concerning modern science, research and academia.

To Cristina Sernadas, a close follower of my work, for her advice, support and willingness, and, most of all, for the enthusiasm that we share about this research topic.

To Jaime Ramos, a kind of older brother in arms, for everything one might expect from an older brother, specially his patience with the youngster.

To Paula Gouveia, Pedro Resende and Miguel Dionísio, for so many big and small reasons, but, essentially, for being there whenever I needed either their friendship or their expertise.

To Antónia Lopes, a good friend and former member of the Group, for all her help and support.

To all other current and former members of the Logic and Computation for the warm and friendly environment they have always provided.

To all colleagues with whom I have discussed my work, and who helped improve it one way or another. Among them, I should emphasize Alberto Zanardo, Joseph Goguen, Andrzej Tarlecki, Till Mossakowski, José Fiadeiro, Walter Carnielli, Marcelo Coniglio, Răzvan Diaconescu, Petros Stefaneas, Pierre-Yves Schöbbens and Mark Ryan.

Last but not least, my deepest gratitude goes to my family and friends. Above all, I thank my parents, Manuel and Feliciana, and my wife, Joana. Their love gives meaning to my life, and I dedicate this dissertation to the three of them.

The following entities have sponsored the work contained in this dissertation:

• Fundação para a Ciência e Tecnologia (FCT) through the PRAXIS XXI projects PCEX/P/MAT/46/96 (ACL), 2/2.1/TIT/1658/95 (LogComp), 2/2.1/MAT/262/94 (SitCalc) and PRAXIS/P/MAT/10002/1998 (ProbLog);

• Centro de Matemática Aplicada - Instituto Superior Técnico (CMA);

• European Community (EC) through the ESPRIT Working Groups 22704 (ASPIRE) and 23531 (FIREworks). vi

Contents

1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Logical consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.2 Layered logical consequence . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.3 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.4 Structural logical consequence . . . . . . . . . . . . . . . . . . . . . . 6 1.4.5 Fibring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Claim of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Logical consequence 11 2.1 Closure and theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Logic systems and logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Models: satisfaction and valuation systems . . . . . . . . . . . . . . 29 2.2.2 Deduction: inference systems . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Layered logical consequence 53 3.1 Local and global reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Layered logic systems and layered logics . . . . . . . . . . . . . . . . . . . . 62

3.2.1 Models: layered valuation systems . . . . . . . . . . . . . . . . . . . 64 3.2.2 Deduction: layered inference systems . . . . . . . . . . . . . . . . . . 68

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Synchronization 79

4.1 General definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1.1 Free synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1.2 Constrained synchronization . . . . . . . . . . . . . . . . . . . . . . 82

4.2 Applying the mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2.1 Satisfaction and (layered) valuation systems . . . . . . . . . . . . . . 88

4.2.2 (Layered) inference systems . . . . . . . . . . . . . . . . . . . . . . . 91 4.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4 Transference results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4.1 Soundness preservation . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.2 Completeness preservation . . . . . . . . . . . . . . . . . . . . . . . . 96

4.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Structural logical consequence 105 5.1 Schematic reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Logic system presentations and logic presentations . . . . . . . . . . . . . . 120

5.2.1 Models: interpretation systems . . . . . . . . . . . . . . . . . . . . . 122 5.2.2 Deduction: deductive systems . . . . . . . . . . . . . . . . . . . . . . 126

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 Fibring 135

6.1 General definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.1.1 Free fibring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.1.2 Constrained fibring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2 Applying the mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2.1 Interpretation systems . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2.2 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.4 Comparison with synchronization . . . . . . . . . . . . . . . . . . . . . . . . 153

6.4.1 Synchronization is fibring . . . . . . . . . . . . . . . . . . . . . . . . 153 6.4.2 Fibring via synchronization . . . . . . . . . . . . . . . . . . . . . . . 157

6.5 Transference results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.5.1 Soundness preservation . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.5.2 Completeness preservation . . . . . . . . . . . . . . . . . . . . . . . . 166

6.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7 Conclusion 181 7.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2 Limitations, outlook and future work . . . . . . . . . . . . . . . . . . . . . . 182

Bibliography 185

Appendix 193

A.1 One problem and two ways of avoiding it . . . . . . . . . . . . . . . . . . . 194 A.1.1 Lifting the source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 A.1.2 Lifting the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

A.2 F -pushout: a solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 A.2.1 Lifting the source yields an F -pushout . . . . . . . . . . . . . . . . . 198 A.2.2 Lifting the result yields an F -pushout . . . . . . . . . . . . . . . . . 199

A.3 Preservation of F -pushouts . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Chapter 1

Introduction

There is an urging need to study more and more complex systems. This fact is one of the major triggers of research in many areas of science, and particularly in computer science. The challenge is to identify and explore abstractions that may help manage this task in a modular and integrated way. Typically, a complex problem is divided in simpler subproblems that can be more easily studied. However, even if all the subproblems are solvable, it is in general a nontrivial task to find a combination of the partial solutions that may lead to a solution of the original problem. This happens because different aspects of the same system may be of quite disparate nature and require quite heterogeneous analytical tools, and all these differences may interact in rather involved ways. In spite of that, as it seems, the problem lies from the beginning on the abstractions underlying the subdivision of the problem to be solved.

1.1 Background

The reasons just exposed are certainly a good justification for the growing interest in mechanisms for combining logics [BdR97a]. However, logic combination as a discipline is still far from having a decisive practical impact. Despite some seminal body of work within the general context of institutions [GB84, GB86, FS88, Mes89, GB92, KM95, SSV95, Mos96, Tar96, AF96, CM97, MTP97, MTP98, Paw98], with intended applications in the area of software engineering, most of the main results in the area have been achieved in the context of modal logics [Tho84, KW91, FG92, FG96, Wol96, Gab96a, Gab96b, GS98, MA98, Kur99, MR99, Gab99, ZSS99], often with quite different motivations.

The unifying concept of trio [BdR97b] as an abstract combination mechanism, where the two logics to be combined are linked together by a bridge controlling the way information flows across them, is certainly appealing. However, for the moment, only a few instances of this general idea have emerged in their own right. Among them, we should emphasize synchronization, parameterization and fibring.

Synchronization, parameterization and fibring are syntax-driven combination mechanisms, in the sense that they envisage particular constructions at the syntactic level. For synchronization one assumes that sentences of logic are fixed unrelated blocks. That is to say that there is no such thing as the common notion of subsentence. The syntax of a synchronization of two logics is therefore obtained by summing the sentences of both, and by eventually requiring that some sentences be shared. It borrows its designation from a similar construction in process theory [WN95], as oposed to interleaving, a terminology sometimes also used to refer to fibring. In fact, for fibring, one assumes that sentences of logic are built by applying constructors from a given signature. The syntax of a fibring of two logics is therefore obtained by applying constructors from both signatures, and by eventually requiring that some constructors be shared. Parameterization, in its turn, is a hybrid between synchronization and fibring, where the two logics involved take different roles. Indeed, while one language is assumed to be structured, the other one is not. The latter shall serve as a parameter for the construction. Thus, the corresponding parameterized language is obtained by applying constructors of the first logic, possibly also to whole sentences of the parameter.

Since synchronization does not take advantage of structure, it is often considered awkward and even uninteresting [MTP97]. Still, the idea of synchronization is present in the very basic intuitions provided by Gabbay to explain fibred semantics [Gab96a, Gab99]. It is also the basis of the “hide and unpack” technique on which the mechanism of parameterization relies [FG92, FG96, BdR97b]. In fact, as hinted in [CSS99], parameterization (of which temporalization is just an example) can be seen as either a restricted form of fibring or as a particular case of synchronization.

Although synchronization is the underlying combination mechanism of institutions, the first general results that we know of it can be found in [SSC97a, SSC97b]. In [SSC97a] synchronizaton was introduced, uniformly, both at model and proof-theoretic levels, and shown to preserve soundness. The results in [SSC97b] concern preservation of model existence and completeness through synchronization of logics with a classical negation, possibly restricted by means of mixed inferences.

Parameterization was first proposed in [FG92]. Ever since, due to its naturality, a strong body of results has been obtained [FG92, FG96, BdR97b, CSS99], which includes soundess, completeness and decidability issues. Thus, it is no surprise that the mechanism has succeeded in finding its way into several practical problems, among which we cite [WZ00].

Fibring as a combination mechanism was first proposed by Gabbay [Gab96a, Gab99]. In its most general form, however, it is too wide a concept for us to give a rigorous account of it. Even at the simplest propositional based level, although it is well understood from a proof-theoretic perspective, fibred semantics is a hard problem. Several examples, namely within modal logics, have always been easily identified with instances of fibring. Still, the first general definition of a fibred model in the context of propositional modal logics is [SSC99]. Nevertheless, the subject of fibred semantics is far from closed. Even more so if we think about multi-sorted logical languages with terms, variables, quantification and binding operators. The most paradigmatic example of this situation is probably first-order modal logic [Tho84, BdR97a]. An attempt to analyse its semantics at the light of the fibring methodology can be found in [SSCM00]. Moreover, the resemblances and differences of fibring with relation to the parchment approach for combining institutions are also quite interesting. Parchments [GB86, MTP97, MTP98, Paw98] appeared as an add-on to the abstract model theory provided by institutions, following the ideas of [Bar74], aimed precisely at achieving the necessary level of abstraction for more sophisticated forms of combination than just synchronization. Their relationship to fibring was briefly sketched in [SSCM00]. Several general results of fibred modal logics are well known, specially concerning modal products. Among the positive and negative results of preservation of decidability, completeness and interpolation we can refer [KW91, Wol96, Gab96a, Gab96b, GS98, MA98, Kur99, MR99, Gab99]. It has also been noted that the simple fibring of propositional classical and intuitionistic logics collapses onto just classical logic [dCH96, Gab96b]. A general completeness preservation theorem for fibred logics with general semantics in the sense of [vB85, HC96] can be found in [ZSS99]. Other relevant work related to fibring is concerned with algebraizable logics [JKE96], tableaux systems [Gab99] and labelled deduction systems [RSSV00].

1.2 Aims

The general goal of the present work is to provide a rigorous insight into this striving research subject. Of course, such a broad title as “Combining Logics” should not be misinterpreted. In fact, we are far from claiming that our work is everything one has ever wanted to know about combination of logics. The subject, by its nature, is too complex to be given a simple answer. First of all, the difficulties arise once one even tries to give a rigorous definition of what logic really is [Gab94]. However, for sure, one cannot study logic combination without first detaching a working universe of logics. Secondly, even if one overcomes this step, there is still a wide choice of possible ways to combine the remaining logics. Clearly, this variety is closely related to the way different logics may relate to each other, and depends, in an essential way, on the amount of structure used to describe a specific system of logic.

Indeed, we shall concentrate just on synchronization and fibring. One good reason is that they are specific of two, quite distinct, levels of abstraction used to describe logics. Moreover, these mechanisms are well referenced in the literature and numerous examples and variants of them have appeared in practice, even before combination of logics established itself as an independent discipline. Last, but not least, although they are so different, synchronization and fibring seem to be related. Besides providing a thorough study of each of the mechanisms per se, we also aim at clarifying the extent of this relationship.

With relation to the two mechanisms, our first aim is to set them up, as abstractly as possible, in their natural setting. By natural, herein, we mean that we shall require just the minimal set of conditions under which the mechanism makes sense. For synchronization, the natural level of abstraction is that of consequence operations [Tar56, Sco74] over sets of unstructured sentences. For fibring, however, much more is neeeded. Indeed we shall consider structural consequence operations over signature generated sets of sentences. Structurality, here, stands for closure under substitutions in the sense of [BS73]. In either case, following other similar treatments, e.g., [FS88, FJ94], finitariness shall be an optional feature used to characterize proof-theoretic content. Given these abstract characterizations, then, we aim at obtaining not only an almost automatic way of applying the mechanisms in any suitable context, but also a good measure of how each available notion of logic is fit for the mechanism. One may even argue that such a general characterization opens the way for synchronizing or fibring logics defined in quite different contexts, e.g., one given a proof-theoretic description and the other a model-theoretic one. Of course, these abstract characterizations must be compatible with previous definitions of the mechanisms within particular notions of logic fulfilling the minimal requirements, such as [SSC97a, SSC97b] as far as synchronization is concerned, and [SSC99, ZSS99] regarding fibring.

As one gets into the details of fibred semantics, it is easy to perceive that its natural setting is universal algebra. For the one reason that the provisos a logic should satisfy in order to be fibred with another are, according to Gabbay [Gab96a, Gab99], essentially that its syntax be a free algebra and its models be homomorphisms with origin in its syntax. Hence, we shall focus on purely algebraic semantics, in a spirit similar to that of algebraic treatments of logic [Cze81, BP89, AKNS94, NA94, ANS97]. Still, by doing so, we are bound to lose a notion that plays an apparently fundamental role in the whole idea of fibring within the context of modal logics: the world. It is well known that Kripke models present powerset algebras (see, e.g., [HC96]), whose values are precisely sets of worlds. However, in an abstract algebra we have no such thing as a world unless we resort to some kind of Stone-like representation technique [Joh82]. Nevertheless, we note that the essential concept of fibred semantics is the internal structure of models, although not necessarily the notion of world. Fortunately, we can still capture the essentials of local versus global reasoning, in a worldless context, by requiring that interpretation structures be endowed with some internal logical structure. As we shall see, our proposal shall be general enough to encompass all sorts of usual ordered or lattice structures. For the sake of symmetry between the two mechanisms, whose characterization shall be given in quite similar terms although on necessarily distinct grounds, we shall also extend the mechanism of synchronization to this local/global setting.

Finally, we shall dwell on transference results for the two mechanisms, i.e., we shall be concerned with preservation of soundness and completeness. In both cases, it is expected that at least soundness is preserved. Indeed, we feel that preservation of soundness should come as an almost immediate consequence of the definitions of the two mechanisms. Completeness is another subject, although difficulties might be guessed at once from the characterization of the mechanisms. Again, for both synchronization and fibring, we shall try to use similar techniques, reminiscent of [ZSS99]. Note that such completeness preservation results for synchronization, in particular, shall be completely unrelated to those presented in [SSC97b].

1.3 Methodology

Category theory is well known to be a very useful tool, particularly in computer science [Gog91]. Moreover, it has been claimed for long that combination, in general terms, is a process intimately related to the way our subject objects relate to each other. Thus, as advocated in [GG78], we shall always look for the right universal property that characterizes our combination mechanisms. Still, we shall not follow the principle of combining via colimits too strictly. Of course, the colimit of every class of diagrams in a category can be seen as a particular form of combination. However, we shall not be interested in all possible forms of combination. In fact, we shall characterize unconstrained combinations using categorial sums, and constrained combinations via F -pushouts, a special categorial construction that mixes the idea of pushout with that of cocartesian lifting. This construction was suggested to us by Joseph Goguen himself, during a visit to our Group in 1999, as an alternative to the two step construction involving sums, pushouts and cocartesian liftings used in previous work [SSC97a, SSC97b, SSC99, ZSS99]. All the details about F -pushouts and their relationship to pushouts and cocartesian liftings can be found in the Appendix.

Of course, prior to any attempt at combination, we shall have to establish all our logic universes categorially. Our approach, following our previous work on the subject [SSC97a, SSC97b, SSC99], shall be to study the basic categories of systems of logic first, and then lift up the results to whole logics, obtained via indexing by means of a simple Grothendieck construction. This technique is reminiscent of early work in the field of institutions [GB86, TBG91]. As expected, however, this effort shall pay back later on in several distinct ways. First, it allows us to formulate the abstract notion of a category of logic systems and logic system presentations, and their corresponding abstract notions of logic and logic presentation, on the basis of the more elementary categories of consequence systems and structural consequence systems. Secondly, it sets the grounds to characterize with precision the universal property of each combination mechanism. This categorial characterization can then be used to classify different notions of logic system and logic system presentation with respect to its being suitable or even exact for the mechanism, in a rigorous way. Suitability is a kind of minimal condition for the mechanism to make sense. Exactness, in its turn, distinguishes only those frameworks where the construction matches precisely its abstract definition. Finally, it shall make many of the proofs simpler and more concise, not to mention the whole body of category-theoretic results and constructions that we can use and apply for free.

This dissertaton is by no means self-contained regarding category theory. However, all the concepts, constructions and results that we use can be found in introductory textbooks as [Mac71, AHS90, BW90, Bor94].

1.4 Outline

The dissertation is organized in 6 more Chapters and an Appendix, as detailed below.

1.4.1 Logical consequence

Chapter 2 introduces the basic concepts underlying the notion of logic that we shall be interested in. Most of the material presented is a rephrasal, in the categorial setting, of well-known concepts and results that can be found in textbooks on logic, namely [Wój88, Cle91, MP96] and universal algebra, mainly [Bir67]. Of course, we share the view of many other modern treatments of logic [Tar56, Sco74, Avr91, FHV92] that consider logical consequence, rather than theoremhood, the central aspect of logic. Hence, we start with a detailed study of the category of consequence systems and their induced theory spaces. The abstract notion of a category of logic systems is then introduced, corresponding, in general terms, to a functor with codomain in the category of consequence systems. The abstract notion of a logic based on a category of logic systems is then obtained by Grothendieck construction. We shall see that our definitions encompass both model-theoretic universes, like satisfaction systems (in the tradition of institutions [GB86]) and valuation systems (following the idea of logical matrices by Lukasiewicz and Tarski [Wój88]), and prooftheoretic ones, like inference systems (after the ideas of Hilbert-like deduction systems [Wój88]). In particular, we shall draw close attention to examples like institutions (based on satisfaction systems) and π-institutions (based on consequence systems). To close the Chapter, we shall show how a collection of well-known systems of logic can be represented in our setting.

1.4.2 Layered logical consequence

Chapter 3 presents a generalization of the basic notions discussed in the previous Chapter, that is, we explore the distinction between the local and global modes of reasoning typical, for instance, of modal and first order logics [FHV92]. The relationship between local and global reasoning and their theory spaces is then explored, and the category of layered consequence systems is introduced and studied. As in Chapter 2, we shall then develop an abstract notion of layered logic system based on layered consequence systems, and explore both a model-theoretic example, layered valuation systems (by enriching the internal structure of the interpretation structures used in valuation systems), and a proof-theoretic one, layered inference systems (by distinguishing between local and global inferences, similarly to [SSC99, ZSS99]). We shall also repeat the process of indexing to obtain corresponding general notions of layered logic. Again, we shall provide the representation of a collection of well-known systems of logic in this layered setting. In particular, some attention will be dedicated to two cases, classical and intuitionistic propositional logics, where local and global reasoning are simply indistinguishable.

1.4.3 Synchronization

Chapter 4 is the first one dealing directly with combining logics. We start by defining the simplest free version of synchronization, in general terms, at its natural level of abstraction: consequence systems. Free synchronization of layered consequence systems is then also obtained by using a straighforward generalization of the underlying categorial construction, a coproduct. We then move towards constrained synchronization, and prove that the construction we propose indeed coincides with previous definitions, namely by filling in the gap between synchronization on models and on sentences as can be found in [SSC97a]. The constrained mechanism is also easily extended to the layered setting, and in both cases characterized as a syntax-driven F -pushout. These categorial characterizations of synchronization are then used to classify each category of (layered) logic systems according to its suitability and exactness for the construction. On these premises, we shall then explore synchronization in our working categories of (layered) logic systems. In fact, they are all easily shown to be suitable, although only the ones with proof-theoretic content are exact. After an illustrating example that combines temporal with equational reasoning, this analysis will then be used to establish our first transference results for synchronization: preservation of soundness for all three relevant cases involving our categories of logic systems, namely inference/satisfaction, inference/valuation and layered inference/layered valuation systems. However, due to the non-exactness of the model-theoretic categories, completeness is in general not preserved. Yet, we shall be able to establish completeness preservation results, in all three cases, by requiring that certain fullness conditions are satisfied by the systems to be synchronized. These fullness conditions amount precisely to guaranteeing that we are left with enough models for completeness to be preserved.

1.4.4 Structural logical consequence

Chapter 5 goes a necessary step down in the level of abstraction considered so far, to pave the way towards the study of fibring in the subsequent Chapter. Herein we shall focus attention on propositional based languages freely generated from an unsorted signature of constructors, pretty much as it was done in [SSC99, ZSS99]. Moreover, the essential feature of structurality, in the sense of [BS73], is brought to the setting by considering also metavariables and metavariable substitutions. We shall introduce structural presentations of consequence and layered consequence systems and study their corresponding categories, and the way they relate to the non-structural version of previous Chapters. Also in analogy to the abstract notions of category of logic systems and layered logic systems, we shall now introduce categories of logic system presentations and layered logic system presentations. The general indexed notions are now called (layered) logic presentations. The meaning of presentations, herein, is explained by the advantage we take of using metavariables. Then, we shall study in detail two categories of layered logic system presentations: interpretation systems and deductive systems. The former, of model-theoretic nature, is a fully algebraic version of those used in [SSC99, ZSS99] and capitalizes on the abstractions also used in layered valuation systems. The latter brings schema rules into the proof-theoretic foreground of layered inference systems, where an implicit closure for substitutions is assumed [SSC99, ZSS99]. We shall close with a few illustrating examples.

1.4.5 Fibring

Chapter 6 deals with the mechanism of fibring, and constitutes the core of the dissertation. To start with, fibring is a combination mechanism with its own tradition and by all ways and means considered much more interesting than synchronization. Furthermore, for the generality of the ideas underlying the intuitions behind fibring, many of its particular forms have deserved close attention [Gab99]. Although with a special focus on the layered case as far as preservation results are concerned, namely for completeness, we shall first define fibring in abstract terms in the context of structural consequence systems. The corresponding set-up in the layered setting will then be carried on in an analogous way. As we did for synchronization, for the sake of simplicity, we shall first tackle the unconstrained case. In both its layered and non-layered versions the construction is characterized as a categorial sum. For the constrained case, similarly to what was done in [SSC99, ZSS99], we shall consider sharing of constructors. We shall define constrained fibring of structural (layered) consequence systems, and show that it is equivalent to enriching free fibring with a set of bridging schema rules for the shared constructors. The categorial characterization of fibring as a signature-driven F -pushout will then be used to classify each category of (layered) logic system presentations according to its suitability and exactness for the construction. Then, we shall explore fibring in our working categories of (layered) logic system presentations, namely interpretation systems and deductive systems. Both are easily shown to be suitable, although only deductive systems are exact for fibring. We shall dedicate particular attention to interpretation systems, due to their generalized form of semantics when compared to previous work. For the sake of illustration, and also for validation, we shall provide and analyze two well known examples: the building up of multi-modal logics by the fibring of modal logic systems, and the fibred combination of classical and intuitionistic systems. Before transference results are tackled, we also explore in detail the relationship between synchronization and fibring. Despite the apparent richness of the fibring mechanism, we shall manage to prove that fibring can be characterized by using only constrained synchronization. The fact that synchronization can be seen as a special case of fibring was already quite clear. As in the case of synchronization, soundness preservation through fibring is obtained as a consequence of the exactness of the category of deductive systems. Again, however, completeness preservation is harder to achieve. Still, under certain fullness conditions, we shall be able to prove the completeness of fibred systems. For more sophisticated forms of fullness, for instance considering only elementary or partially ordered interpretation structures, we shall see that congruence regarding logical equivalence and sharing of implication also play an important role, similarly to what happens in [ZSS99].

1.4.6 Conclusions

Chapter 6 summarizes the main contributions of this dissertation, draws a comparison with other relevant work, and points out several possible paths to be followed in future work. In particular, we spend a few lines discussing the relationship of fibring with the parchment approach to combining institutions [MTP97, MTP98, Paw98]. Special attention is also dedicated to the realm of algebraic logic [Cze81, BP89, AKNS94, FJ94], for the obvious similarities with the techniques we have considered in our completeness preservation results for fibring. Another topic of interest is the further explorations and applications of fibring, namely within alternative abstract semantic frameworks like algebraic non truth-functional semantics [CMSS00], sheaf-theoretic semantics [CC97] and possible-translations [Cd97, CLM99, Mar99]. Another very interesting line of research to be pursued is the development of a meta-logic of logic combination. Some first steps on this subject can be found in [CC00]. The connection with [JKE96] also seems to be worth investigating. We shall conclude by discussing possible applications to the development of logics of distributed agents, namely by returning to the general idea of trios [BdR97b] and on the basis of some lessons learnt in the past [ECSD98, EC00, Cal00].

1.4.7 Appendix

The Appendix introduces a new categorial construction called F -pushout, as it was suggested to us by Joseph Goguen to smooth the gap between pushouts and cocartesian lifting of coequalizers. The principle of combining via colimits has a long standing tradition, namely in computer science [GG78]. However, it often has to be given a loose interpretation. See, for instance, the techniques for combining theories in institutitons [BG77], where combination is just syntax driven but then uplifted by using an adjoint functor. To cope with the difficulties found in providing suitable simple adjoints, in the context of concurrent process theory, Winskel [WN95] and many others after him have used an alternative technique that uses cocartesian liftings. Previous work on combination of logics has used the latter approach to characterize constrained combinations [SSC97a, SSC99, ZSS99]. However, F -pushouts turn out to be a much nicer construction and still provide the desired results. The relationship of F -pushouts with both pushouts and cocartesian liftings is analyzed in detail. Moreover, several preservation results used in the thesis are formulated and proved.

1.5 Claim of contributions

Given the context and aims of the dissertation, we would like to stress those that we think are its main contributions:

• the identification of consequence systems and structural consequence systems, respectively, as the natural settings for defining and studying synchronization and fibring;

• the smooth lifting of both combination mechanisms to any suitable concrete notion of logic and logic presentation based, respectively, on consequence systems and structural consequence systems;

• the explicit distinction between local and global logical consequence, namely the way it was achieved at the model-theoretic level with no (explicit or implicit) reference to the notion of world;

• the characterization of fibring of model-theoretic systems of logic with purely (worldless) algebraic semantics;

• the clarification of the relationship between synchronization and fibring;

• the identification of several sufficient conditions for preservation of completeness by both synchronization and fibring.

Logical consequence

This Chapter has two distinct aims. The first one is to detach the main properties of the universe of logics we shall be dealing with in the subsequent Chapters. Rather than engaging in a more or less philosophical discussion on the nature of logic (refer to [Gab94] for a thorough discussion of this subject), we shall make precise what characteristics a logic should enjoy to be captured by our subsequent results. In doing this, of course, some (a lot of, indeed) more or less well-accepted systems of logic shall be simply ruled out. Still, the restrictions to be made shall be broad enough to encompass many interesting logics. Moreover, we do not advocate a preference for model or proof-theoretic approaches to logic. We shall consider both frameworks, and put due emphasis on the central concept of consequence, as an abstraction of both semantic entailment and deducibility. The second aim is to establish all our working universes categorially, as justified in our introductory section. Consequently, not only the concepts per se but also their relationships shall be analyzed, as deep as necessary, by using the tools of category theory. Furthermore, the setting-up of a system of logic as an indexed family of logic systems, and the clarification of this abstraction, shall also be given in categorial terms via a simple Grothendieck construction.

We shall also present a few categories of logic systems, namely satisfaction systems and inference systems. As it shall be made clear, these lie in the core of the most traditional model and proof-theoretic approaches to logic [Acz94]. We shall end it with an illustration of the concepts introduced by means of a short list of examples that includes some of the most widespread systems of logic: classical and intuitionistic logic, modal logic, temporal logic, and also Lukasiewicz’s three-valued logic, all propositional based, as well as equational and predicate logics.

Most of the concepts we shall introduce are ubiquitous in many areas of mathematics and logic. Most of this material can be found, probably in disguise, in texts on algebra [Bir67, EM85] and logic [Men79, Wój88, Cle91, Gol92, FHMV95, MP96, HC96]. Still, we shall borrow some of the designations to be used, e.g., consequence system and theory space, and also a few results from previous work, namely [SSC97a].

This Chapter is organized as follows. First, we shall introduce the categories of consequence systems and theory spaces and study their relationship, as well as their corresponding finitary versions. In this context we shall spend a few lines characterizing the important notion of logical equivalence. Some emphasis will also be given to the presentation of consequence by means of closure for a set of inferences. Then, we shall define the abstract notion of category of logic systems as a functor to the category of consequence system, and via indexing, the abstract notion of logic. We shall prove that not only consequence systems themselves, but also satisfaction systems and valuation systems on the model-theoretic side, and inference systems on the proof-theoretic side, constitute categories of logic systems. Moreover, we shall show that logics based on consequence systems correspond to π-institutions [FS88] and that logics based on satisfaction systems correspond to institutions [GB84]. We shall also explore a little further these categories of logic systems, namely as regards the important notions of soundness and completeness. We shall close the section with a collection of examples of well known systems of logic, for the sake of illustration, but also because they shall be useful in the sequel.

2.1 Closure and theories

As mentioned in the introductory Chapter, we share the view of many modern treatments of logic that consider consequence, rather than just theoremhood, the central aspect of logic [Wój88, Avr91, Cle91, FHV92, Hac94]. Theoremhood is a concept that is easily derivable from consequence. The contrary, however, is usually not the case. Of course, some systems of logic enjoy sufficiently strong deduction-like theorems that tend to fade this distinction. However, the essential role of consequence really comes to play in the context of systems of relevance logic, some of them designed precisely to capture patterns of reasoning that cannot be explained by just theoremhood. We refer to [Dun86] for a comprehensive exposition on this subject.

The language of a logic is an essential part but, of course, it needs to be given some meaning. In the approach we are going to follow, the essential meaning of a logical language is provided by a consequence operation, in the sense of [Tar56, Sco74].

First a bit of notation: given a set L, we use ℘(L) to denote the set of all subsets of L, and ℘_{fin}(L) to denote the set of all finite subsets of L.

Definition 2.1.1. A consequence system is a pair csy = ⟨L, ▷⟩ where:

• L is a set (of sentences);

• ^{▷} : ℘(L) → ℘(L) is a function (consequence operation),

such that the following three conditions hold, for every Φ, Ψ ⊆ L:

• Φ ⊆ Φ^{▷} (extensiveness);

• Φ^{▷} ⊆ (Φ ∪ Ψ)^{▷} (monotonicity);

• (Φ^{▷})^{▷} ⊆ Φ^{▷} (idempotence).

Given a sentence φ ∈ L and a set of sentences Φ ⊆ L, φ is said to be a consequence of Φ, a fact that we denote by Φ ▷ ψ, if φ ∈ Φ^{▷}. If Φ = ∅, we say that φ is a theorem, and simply write ▷φ. If Φ^{▷} = L then Φ is said to be inconsistent, otherwise Φ is said to be consistent.

Obviously, monotonicity is equivalent to the fact that Φ^{▷} ⊆ Ψ^{▷} whenever Φ ⊆ Ψ. Due to extensiveness, it is also immediate that idempotence states precisely that it is always the case that (Φ^{▷})^{▷} = Φ^{▷}.

As defined above, a consequence system is a generalization of the notion of topological space as provided by the Kuratowski closure axioms (see for instance [MP96]). Just two conditions are dropped, since they are not desirable in logical terms: if L is the set of sentences of some logic, ∅^{▷} = ∅ (groundedness) is clearly undesirable since ∅^{▷} is meant to denote the set of theorems of the logic, and (Φ ∪ Ψ)^{▷} ⊆ Φ^{▷} ∪ Ψ^{▷} (additivity) is not considered because an essential feature of logic is the interplay between its sentences.

With this approach, given a consequence system, we can smoothly enlarge its set of axioms and obtain a corresponding consequence system, as follows:

Definition 2.1.2. Let csy = ⟨L, ▷⟩ be a consequence system and Φ ⊆ L. The consequence system induced by Φ on csy is given by csy(Φ) = ⟨L, ▷_{Φ}⟩ where, for every Ψ ⊆ L:

Ψ^{▷_{Φ}} = (Φ ∪ Ψ)^{▷}.

Proposition 2.1.3. Let csy be a consequence system over language L and Φ ⊆ L. Then, csy(Φ) is a consequence system.

Proof. Assume that csy = ⟨L, ▷⟩. We need to prove that ▷_{Φ} as defined in 2.1.2 is indeed a consequence operation on L. Clearly, Ψ ⊆ Ψ^{▷} ⊆ (Φ ∪ Ψ)^{▷} = Ψ^{▷_{Φ}}, just by using the extensiveness and monotonicity of ▷. To show the monotonicity of ▷_{Φ}, assume that Ψ_{1} ⊆ Ψ_{2}. Then, easily, (Φ ∪ Ψ_{1}) ⊆ (Φ ∪ Ψ_{2}) and Ψ_{1}^{▷_{Φ}} = (Φ ∪ Ψ_{1})^{▷} ⊆ (Φ ∪ Ψ_{2})^{▷} = Ψ_{2}^{▷_{Φ}}, just by using the monotonicity of ▷. To prove the idempotence of ▷_{Φ} just note that Φ ⊆ Φ^{▷} ⊆ (Φ ∪ Ψ)^{▷} = Ψ^{▷_{Φ}}, by means of extensiveness and monotonicity of ▷. Therefore, (Φ ∪ (Φ ∪ Ψ)^{▷}) ⊆ (Φ ∪ Ψ)^{▷} and we can now use monotonicity and idempotence of ▷ to conclude that (Φ ∪ (Φ ∪ Ψ)^{▷})^{▷} ⊆ ((Φ ∪ Ψ)^{▷})^{▷} ⊆ (Φ ∪ Ψ)^{▷}. That is, (Ψ^{▷_{Φ}})^{▷_{Φ}} ⊆ Ψ^{▷_{Φ}} and ▷_{Φ} is a consequence operation. ⊓ ⊔

Probably, the most common way of defining a consequence operation is by providing a set of inferences [Wój88]. We shall take a brief look at this case not only for the sake of illustration, but most importantly because it shall be very useful to us.

Definition 2.1.4. An inference over a set L is a pair i = ⟨Prem(i), Conc(i)⟩ where:

• Prem(i) ⊆ L (the set of premises);

• Conc(i) ∈ L (the conclusion).

We denote the set of all inferences over L by Infer(L).

Definition 2.1.5. Let L be a set and I ⊆ Infer(L). The step operator Ω_{I} : ℘(L) → ℘(L) induced by I is such that, for every Φ ⊆ L,

Ω_{I}(Φ) = (Φ ∪ {Conc(i) : i ∈ I, Prem(i) ⊆ Φ}).

The consequence operation ▷_{I} over L generated by I is such that, for every Φ ⊆ L, Φ^{▷_{I}} is the least fixed point of Ω_{I} that contains Φ.

In other words, Φ^{▷_{I}} is the least superset of Φ that is closed for application of all the inferences in I. Of course, this definition needs to be clarified. We need to guarantee that such a set exists, and futhermore that the construction indeed yields a consequence operation.

Proposition 2.1.6. Let L be a set and I ⊆ Infer(L). Then, ▷_{I} is indeed a consequence operation.

Proof. We start by noting that the step operator is order-preserving, i.e., Ω_{I}(Φ) ⊆ Ω_{I}(Ψ) whenever Φ ⊆ Ψ. Trivially, if i ∈ I and Prem(i) ⊆ Φ then also Prem(i) ⊆ Ψ. Thus, by a famous theorem of Tarski [Tar55], it follows that for every Φ ⊆ L, there is a least fixed point of Ω_{I} that contains Φ given by (⋂_{Ψ∈I(Φ)} Ψ), where I(Φ) = {Ψ ⊆ L : (Φ ∪ Ω_{I}(Ψ)) ⊆ Ψ}. Thus, we are considering Φ^{▷_{I}} = (⋂_{Ψ∈I(Φ)} Ψ).

Now we can easily prove that ▷_{I} is a consequence operation. Extensiveness is immediate from the definition. Being Φ^{▷_{I}} the least fixed point of Ω_{I} that contains Φ, necessarily, Φ ⊆ Φ^{▷_{I}}. If Φ ⊆ Ψ then, using extensiveness, Φ ⊆ Ψ ⊆ Ψ^{▷_{I}}. That is, Ψ^{▷_{I}} is a fixed point of Ω_{I} and contains Φ. Thus, Φ^{▷_{I}} ⊆ Ψ^{▷_{I}} and monotonicity follows. Idempotence is a trivial consequence of Φ^{▷_{I}} being a fixed point of Ω_{I}. ⊓ ⊔

For convenience, we shall use the notation csy_{I} = ⟨L, ▷_{I}⟩.

Of course, in the end, a consequence operation induces a space of closed sets. In logic these correspond to theories, with their intrinsic properties.

Definition 2.1.7. A theory space is a complete lattice tsp = ⟨Th, ≤⟩, i.e., a partial order where every set T ⊆ Th has a least upper-bound (or join) ⋁ T (and thus also a greatest lower-bound (or meet) ⋀ T).

Definition 2.1.8. A theory of a consequence system csy = ⟨L, ▷⟩ is a set of sentences Φ ⊆ L such that Φ^{▷} = Φ. We denote by Th(csy) the set of all theories of c.

It is a trivial consequence of idempotence that Th(csy) = {Φ^{▷} : Φ ⊆ L}.

Proposition 2.1.9. For every consequence system csy, the pair tsp_{csy} = ⟨Th(csy), ⊆⟩ is a theory space.

Proof. It is a trivial fact that ⊆ establishes a partial order on Th(csy).

The meet ⋀ T of a set T ⊆ Th(csy) of theories is just given by their intersection, i.e., ⋀ T = (⋂_{Φ∈T} Φ). This is indeed a theory: by extensiveness, it suffices to prove that (⋂_{Φ∈T} Φ)^{▷} ⊆ (⋂_{Φ∈T} Φ). Clearly, for every Ψ ∈ T, (⋂_{Φ∈T} Φ) ⊆ Ψ. By monotonicity, thus, (⋂_{Φ∈T} Φ)^{▷} ⊆ Ψ^{▷}. But Ψ is a theory of c, so Ψ^{▷} = Ψ. It easily follows that (⋂_{Φ∈T} Φ)^{▷} ⊆ (⋂_{Φ∈T} Φ).

Now, (⋂_{Φ∈T} Φ) is obviously a lower-bound of T. Let Υ ∈ Th(c) also be a lower-bound of T, that is, Υ ⊆ Φ for every Φ ∈ T. Immediately, Υ ⊆ (⋂_{Φ∈T} Φ), and therefore (⋂_{Φ∈T} Φ) is the greatest lower-bound of T. ⊓ ⊔

Of course, top and bottom theories exist: the top theory corresponds to L itself (the inconsistent theory), while the bottom is ∅^{▷} (the set of theorems). Obviously, also joins exist in tsp_{csy}. Given T ⊆ Th(csy), its join ⋁ T corresponds to the meet of the set of all upper-bounds of T, that is, ⋀{Ψ ∈ Th(csy) : (⋃_{Φ∈T} Φ) ⊆ Ψ}. It is quite straightforward to conclude that it amounts to ⋁ T = (⋃_{Φ∈T} Φ)^{▷}, as the following lemma confirms.

Lemma 2.1.10. Let csy = ⟨L, ▷⟩ be a consequence system and {Φ_{k}}_{k∈K} a family of subsets of L, i.e., Φ_{k} ⊆ L for every k ∈ K. Then, ⋁{Φ_{k}^{▷} : k ∈ K} = (⋃_{k∈K} Φ_{k})^{▷}.

Proof. Obviously, (⋃_{k∈K} Φ_{k})^{▷} is a theory of csy. Moreover, for each j ∈ K, Φ_{j} ⊆ (⋃_{k∈K} Φ_{k}). Thus, by the monotonicity of ▷ it follows that Φ_{j}^{▷} ⊆ (⋃_{k∈K} Φ_{k})^{▷}, and (⋃_{k∈K} Φ_{k})^{▷} is an upper-bound of {Φ_{k}^{▷} : k ∈ K}.

Let now Υ ∈ Th(csy) be any upper-bound of {Φ_{k}^{▷} : k ∈ K}. By extensiveness of ▷, we then have that each Φ_{j} ⊆ Φ_{j}^{▷} ⊆ Υ. Hence, (⋃_{k∈K} Φ_{k}) ⊆ Υ and the monotonicity and idempotence of ▷ together guarantee that (⋃_{k∈K} Φ_{k})^{▷} ⊆ Υ. ⊓ ⊔

The categorial treatment of consequence systems considers a notion of morphism that also embodies the spirit of topological spaces. Although they are here introduced in a theory independent form, indeed, morphisms correspond precisely to continuous functions: the inverse images of theories must be theories.

Definition 2.1.11. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems. A consequence systems morphism h : csy_{1} → csy_{2} is a function h : L_{1} → L_{2} such that the following condition holds, for every Φ ⊆ L_{1}:

• h(Φ^{▷_{1}}) ⊆ h(Φ)^{▷_{2}}.

The following is a well known and often useful lemma.

Lemma 2.1.12. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems and h : csy_{1} → csy_{2} a consequence systems morphism. Then, h(Φ^{▷_{1}})^{▷_{2}} = h(Φ)^{▷_{2}} for every Φ ⊆ L_{1}.

Proof. Clearly, by extensiveness of ▷_{1}, Φ ⊆ Φ^{▷_{1}}. Therefore, h(Φ) ⊆ h(Φ^{▷_{1}}) and the monotonicity of ▷_{2} implies that h(Φ)^{▷_{2}} ⊆ h(Φ^{▷_{1}})^{▷_{2}}.

On the other hand, since h is a morphism, h(Φ^{▷_{1}}) ⊆ h(Φ)^{▷_{2}}. By the monotonicity of ▷_{2} we have that h(Φ^{▷_{1}})^{▷_{2}} ⊆ (h(Φ)^{▷_{2}})^{▷_{2}}. Now, idempotence of ▷_{2} implies that (h(Φ)^{▷_{2}})^{▷_{2}} ⊆ h(Φ)^{▷_{2}}. Thus, h(Φ^{▷_{1}})^{▷_{2}} ⊆ h(Φ)^{▷_{2}}. ⊓ ⊔

Clearly the converse of this lemma also holds, thus yielding an alternative characterization of Csy-morphisms.

Definition 2.1.13. Consequence systems and their morphisms constitute the category Csy, with identity id_{⟨L,▷⟩} the identity function on L, and (g ∘ f) the usual composition of functions.

Proposition 2.1.14. Csy is indeed a category.

Proof. Due to the well-known properties of compositions of functions, we just prove that id_{⟨L,▷⟩} : ⟨L, ▷⟩ → ⟨L, ▷⟩ and (g ∘ f) : ⟨L_{1}, ▷_{1}⟩ → ⟨L_{3}, ▷_{3}⟩ are indeed Csy-morphisms, provided that f : ⟨L_{1}, ▷_{1}⟩ → ⟨L_{2}, ▷_{2}⟩ and g : ⟨L_{2}, ▷_{2}⟩ → ⟨L_{3}, ▷_{3}⟩ are.

With respect to identity, trivially, id_{⟨L,▷⟩}(Φ^{▷}) = Φ^{▷} = id_{⟨L,▷⟩}(Φ)^{▷}.

Now, let Φ ⊆ L_{1}. Using the structurality condition of the morphism f, f(Φ^{▷_{1}}) ⊆ f(Φ)^{▷_{2}}. Therefore, g(f(Φ^{▷_{1}})) ⊆ g(f(Φ)^{▷_{2}}). Now, using the structurality condition of the morphism g, g(f(Φ)^{▷_{2}}) ⊆ g(f(Φ))^{▷_{3}}. Thus, g(f(Φ^{▷_{1}})) ⊆ g(f(Φ))^{▷_{3}} and (g ∘ f) is a Csymorphism. ⊓ ⊔

Proposition 2.1.15. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems. A function h : L_{1} → L_{2} constitutes a Csy-morphism h : csy_{1} → csy_{2} if and only if h^{−1}(Φ) ∈ Th(csy_{1}), for every Φ ∈ Th(csy_{2}).

Proof. Assume that h : csy_{1} → csy_{2} is a Csy-morphism, and let Φ ⊆ L_{2} be a theory of csy_{2}, that is, Φ^{▷_{2}} = Φ. Since, by extensiveness of ▷_{1}, h^{−1}(Φ) ⊆ h^{−1}(Φ)^{▷_{1}}, proving that h^{−1}(Φ) is indeed a theory amounts to showing the opposite inclusion. By assumption, h is a morphism and h(h^{−1}(Φ)^{▷_{1}}) ⊆ h(h^{−1}(Φ))^{▷_{2}}. Easily, h(h^{−1}(Φ)) ⊆ Φ, and by monotonicity of ▷_{2}, h(h^{−1}(Φ))^{▷_{2}} ⊆ Φ^{▷_{2}}. Noting that Φ is a theory, it follows that h(h^{−1}(Φ)^{▷_{1}}) ⊆ Φ, and consequently h^{−1}(Φ)^{▷_{1}} ⊆ h^{−1}(Φ).

Assume now that h^{−1}(Φ) ∈ Th(csy_{1}) for every Φ ∈ Th(csy_{2}), and let Ψ ⊆ L_{1}. Clearly, Ψ ⊆ h^{−1}(h(Ψ)). On the other hand, by extensiveness of ▷_{2}, h(Ψ) ⊆ h(Ψ)^{▷_{2}}, and thus h^{−1}(h(Ψ)) ⊆ h^{−1}(h(Ψ)^{▷_{2}}). Therefore Ψ ⊆ h^{−1}(h(Ψ)^{▷_{2}}), and by monotonicity of ▷_{1}, Ψ^{▷_{1}} ⊆ h^{−1}(h(Ψ)^{▷_{2}})^{▷_{1}}. Using the idempotence of ▷_{2}, h(Ψ)^{▷_{2}} is a theory of csy_{2} and from the assumption it follows that h^{−1}(h(Ψ)^{▷_{2}}) is a theory of csy_{1}, i.e., h^{−1}(h(Ψ)^{▷_{2}})^{▷_{1}} = h^{−1}(h(Ψ)^{▷_{2}}). So, Ψ^{▷_{1}} ⊆ h^{−1}(h(Ψ)^{▷_{2}}) and consequently h(Ψ^{▷_{1}}) ⊆ h(Ψ)^{▷_{2}}. ⊓ ⊔

In fact, Csy is a very special concrete category over Set.

Definition 2.1.16. The maps

• Lang_{Csy}(⟨L, ▷⟩) = L;

• Lang_{Csy}(h : ⟨L_{1}, ▷_{1}⟩ → ⟨L_{2}, ▷_{2}⟩) = h : L_{1} → L_{2},

constitute the functor Lang_{Csy} : Csy → Set.

Proposition 2.1.17. Lang_{Csy} : Csy → Set is a topological functor.

Proof. It is a trivial task to note that Lang_{Csy} is indeed a functor.

We start by noting that Lang_{Csy} is faithful. In fact, consequence system morphisms are just functions satisfying an additional condition.

We now show that given any family {csy_{k} = ⟨L_{k}, ▷_{k}⟩}_{k∈K} of consequence systems, any source {h_{k} : L → L_{k}}_{k∈K} in Set has a unique Lang_{Csy}-initial lift {h_{k} : csy → csy_{k}}_{k∈K} in Csy.

First of all, define csy = ⟨L, ▷⟩ where the consequence operation ▷ in L is defined by Φ^{▷} = (⋂_{k∈K} h_{k}^{−1}(h_{k}(Φ)^{▷_{k}})). Extensiveness, monotonicity and idempotence of ▷ are straightforward. Also, each h_{j} : c → c_{j} is a Csy-morphism. Given Ψ ∈ Th(csy_{j}), h_{j}^{−1}(Ψ)^{▷} = (⋂_{k∈K} h_{k}^{−1}(h_{k}(h_{j}^{−1}(Ψ))^{▷_{k}})) ⊆ h_{j}^{−1}(h_{j}(h_{j}^{−1}(Ψ))^{▷_{j}}). So, h_{j}^{−1}(Ψ)^{▷} ⊆ h_{j}^{−1}(Ψ^{▷_{j}}). But Ψ is ▷_{j}-closed, thus h_{j}^{−1}(Ψ)^{▷} ⊆ h_{j}^{−1}(Ψ). The extensiveness of ▷ allows us to conclude that h_{j}^{−1}(Ψ) ∈ Th(csy).

To prove that {h_{k} : csy → csy_{k}}_{k∈K} is Lang_{Csy}-initial, let {g_{k} : csy′ → csy_{k}}_{k∈K} be any other source in Csy, with csy′ = ⟨L′, ▷′⟩, and f : L′ → L a function such that each g_{k} = (h_{k} ∘ f). All we need to prove is that f : csy′ → csy is a Csy-morphism. Indeed, for any Φ ⊆ L′, g_{k}(Φ^{▷′}) ⊆ g_{k}(Φ)^{▷_{k}} because g_{k} : csy′ → csy_{k} is a Csy-morphism for every k ∈ K. Thus, h_{k}(f(Φ^{▷′})) ⊆ h_{k}(f(Φ))^{▷_{k}} or, equivalently, f(Φ^{▷′}) ⊆ h_{k}^{−1}(h_{k}(f(Φ))^{▷_{k}}). So, by definition of csy, f(Φ^{▷′}) ⊆ f(Φ)^{▷}.

The uniqueness of this lift follows immediately from the faithfulness of Lang_{Csy}. ⊓ ⊔

Topological categories like Csy enjoy many nice properties. In particular, not only is the underlying functor faithful, but, furthermore, it is always both an adjoint and a coadjoint, a fibration and a cofibration. Moreover, a topological category is always cocomplete whenever the base category is. We refer the reader to [AHS90] for all these and other properties of topological functors. We are specially interested in another property of topological categories: fibre-completeness. In this case, it corresponds to saying that the set of all consequence operations over a given set of sentences forms a complete lattice under the following definition of order.

Definition 2.1.18. Let L be a set and ⟨L, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems. Then, ▷_{1} ⊑_{L} ▷_{2} provided that Φ^{▷_{1}} ⊆ Φ^{▷_{2}}, for every Φ ⊆ L.

Please note that ▷_{1} ⊑_{L} ▷_{2} is precisely equivalent to id_{L} : ⟨L, ▷_{1}⟩ → ⟨L, ▷_{2}⟩ being a Csymorphism. In fact, ⊑_{L} bears a very strong relationship with the notion of consequence systems morphism, as given by the proof of Csy being topological.

Indeed, even for csy_{1} = ⟨L_{1}, ▷_{1}⟩ and csy_{2} = ⟨L_{2}, ▷_{2}⟩ possibly with L_{1} ≠ L_{2}, we have that h : csy_{1} → csy_{2} is a Csy-morphism if and only if the domain ⟨L, ▷′_{1}⟩ of the unique Lang_{Csy}-initial lift of {h : L_{1} → L_{2}} given {csy_{2}} is such that ▷_{1} ⊑_{L_{1}} ▷′_{1}.

Corollary 2.1.19. Csy is fibre-complete, i.e., each fibre Csy_{L} is a complete lattice (under ⊑_{L}).

Clearly ▷_{1} ⊑_{L} ▷_{2} if and only if id_{L} : ⟨L, ▷_{1}⟩ → ⟨L, ▷_{2}⟩ is a Csy-morphism. Thus, the meet of a family of consequence operations {▷_{k}}_{k∈K} over L is given by the corresponding unique lifting of \{id_{L} : L → L\}_{k∈K}, i.e., ⊓_{L}\{▷_{k} : k ∈ K\} = \underline{▷} with each Φ\underline{▷} = (⋂_{k∈K} Φ^{▷_{k}}) for every Φ ⊆ L.

Obviously, also joins exist. It is not difficult to see that the join ⊔_{L}{▷_{k} : k ∈ K} = \overline{▷} where each Φ\overline{▷} is the least common theory to all consequences ▷_{k} that contains Φ. In fact, for each k ∈ K, let I_{k} = \{⟨Φ, ψ⟩ : Φ ▷_{k} ψ\}. Then, indeed, Φ\overline{▷} is the least superset of Φ that is closed for all inferences in I = (⋃_{k∈K} I_{k}). Clearly, in this case, Ω_{I}(Φ) = (Φ ∪ (⋃_{k∈K} Φ^{▷_{k}})). To see that ⟨L, \overline{▷}⟩ is an upper-bound of \{▷_{k}\}_{k∈K}, just note that each Φ^{▷_{k}} ⊆ Ω_{I}(Φ) ⊆ Ω_{I}(Φ\overline{▷}) = Φ\overline{▷}. Let now ▷′ be an upper-bound of \{▷_{k}\}_{k∈K}. Then, Ω_{I}(Φ^{▷′}) = Φ^{▷′} ∪ (⋃_{k∈K}(Φ^{▷′})^{▷_{k}}) ⊆ Φ^{▷′} ∪ (Φ^{▷′})^{▷′} = Φ^{▷′}. Since Φ ⊆ Φ^{▷′} it follows that Φ\overline{▷} ⊆ Φ^{▷′}.

Easily, the top of the complete lattice is the consequence operation ⊤ such that Φ^{⊤} = L, and the bottom is the consequence operation ⊥ such that Φ^{⊥} = Φ, for every Φ ⊆ L.

The definition of a consequence system embodies the spirit of the original proposal by Tarski [Tar56], in spite of dropping one of the requirements, as it also happens in other works, as for instance [FS88, FJ94]. It is the property of finitariness. By requiring finitariness from scratch we would certainly be excluding a lot of the closure operations one usually finds in logics, namely most of those induced by semantic means. Still, we should stress the important role of finitariness for closures obtained by deductive means.

Definition 2.1.20. A consequence system ⟨L, ▷⟩ is said to be finitary if the following condition holds, for every Φ ⊆ L:

• Φ^{▷} = (⋃_{Ψ∈℘_{fin}(Φ)} Ψ^{▷}).

Clearly, the above definition requires that an element is in the closure of a set if and only if it is in the closure of some of its finite subsets. It is quite easy to give examples of consequence systems that are not finitary.

Definition 2.1.21. Csy_{fin} is the full subcategory of Csy whose objects are precisely the finitary consequence systems.

The finitary part of a consequence system can always be extracted.

Proposition 2.1.22. Let csy = ⟨L, ▷⟩ be a consequence system. The greatest finitary consequence operation contained in ▷ (in the sense of ⊑_{L}) is ▷_{fin} such that, for each Φ ⊆ L:

• Φ^{▷_{fin}} = (⋃_{Ψ∈℘_{fin}(Φ)} Ψ^{▷}).

Proof. To start with, we show that ▷_{fin} is indeed a consequence operation. Let Φ ⊆ L. For every φ ∈ Φ, {φ} is finite and {φ} ▷ φ. Therefore, Φ ⊆ Φ^{▷_{fin}} and we prove extensionality.

If also Φ′ ⊆ L, and Ψ ∈ ℘_{fin}(Φ) it is immediate that also Ψ ∈ ℘_{fin}(Φ ∪ Φ′). Therefore, Φ^{▷_{fin}} ⊆ (Φ ∪ Φ′)^{▷_{fin}} and monotonicity follows.

With respect to idempotence, assume that (Φ^{▷_{fin}}) ▷_{fin}υ, i.e., Ψ ▷ υ for some Ψ ∈ ℘_{fin}(Φ^{▷_{fin}}). Let Ψ = {ψ_{1}, …, ψ_{n}} for some n ∈ IN. For every k ∈ {1, …, n}, since Φ ▷_{fin}ψ_{k}, there must exist Ψ_{k} ∈ ℘_{fin}(Φ) such that Ψ_{k} ▷ ψ_{k}. Clearly, (Ψ_{1} ∪ … ∪ Ψ_{n}) is still finite, and thus the monotonicity of ▷ implies that Ψ ⊆ (Ψ_{1}^{▷} ∪ … ∪ Ψ_{n}^{▷}) ⊆ (Ψ_{1} ∪ … ∪ Ψ_{n})^{▷}. Immediately, using the monotonicity and idempotence of ▷, (Ψ_{1}∪…∪Ψ_{n}) ▷ υ and therefore Φ▷_{fin}υ.

We now prove that ▷_{fin} is indeed finitary. If Φ ▷_{fin}υ, by definition, Ψ ▷ υ for some Ψ ∈ ℘_{fin}(Φ). Trivially, Ψ ∈ ℘_{fin}(Ψ), and therefore, Ψ ▷_{fin}υ completing the proof.

Clearly, ▷_{fin} ⊑_{L} ▷. Just note that Ψ^{▷} ⊆ Φ^{▷} for every Ψ ∈ ℘_{fin}(Φ), by just the monotonicity of ▷. Let now ▷′ be any finitary consequence operation on L such that ▷′ ⊑_{L} ▷. Clearly, Ψ^{▷′} ⊆ Ψ^{▷} ⊆ Φ^{▷_{fin}}. Thus, Φ^{▷′} = (⋃_{Ψ∈℘_{fin}(Φ)} Ψ^{▷′}) ⊆ Φ^{▷_{fin}} and ▷′ ⊑_{L} ▷_{fin}. ⊓ ⊔

For convenience, we use the notation csy_{fin} = ⟨L, ▷_{fin}⟩. This construction can be easily extended to a functor from Csy to Csy_{fin} that is adjoint to the obvious inclusion functor I : Csy_{fin} → Csy. We are, in fact, in the presence of a coreflection.

Proposition 2.1.23. Csy_{fin} is a coreflective subcategory of Csy.

Proof. Let csy = ⟨L, ▷⟩ be a consequence system. We shall prove that id_{L} : csy_{fin} → csy is a coreflection arrow. For the purpose, let h : csy′ → csy be any Csy-morphism, with csy′ = ⟨L′, ▷′⟩ finitary. It suffices to show that h : csy′ → csy_{fin} is also a morphism.

Let Φ ⊆ L′ and assume that Φ ▷′ υ. Since csy′ is finitary, there exists Ψ ∈ ℘_{fin}(Φ) such that Ψ ▷′ υ. Given that h : csy′ → csy is a morphism, it follows that h(Ψ) ▷ h(υ). But clearly h(Ψ) ∈ ℘_{fin}(h(Φ)) and so, by definition of csy_{fin}, h(Φ) ▷_{fin} h(υ). ⊓ ⊔

Corollary 2.1.24. The maps

• Fin(csy) = csy_{fin};

• Fin(h : csy_{1} → csy_{2}) = h : (csy_{1})_{fin} → (csy_{2})_{fin},

constitute a functor Fin : Csy → Csy_{fin}. Moreover, Fin is adjoint to I .

The category Csy_{fin} is also topological over Set. Obviously, we shall use the notation Lang_{Csy_{fin}} = (Lang_{Csy} ∘ I ).

Corollary 2.1.25. The functor Lang_{Csy_{fin}} : Csy_{fin} → Set is topological.

Clearly, Csy_{fin} is a concrete subcategory of Csy. Moreover, Fin : Csy → Csy_{fin} is such that, for every finitary consequence system csy, csy_{fin} = csy. In fact, for every Φ ⊆ L, if Φ ▷ υ, by finitariness of csy, there exists Ψ ∈ ℘_{fin}(Φ) such that Ψ ▷ υ. Therefore, Φ ▷_{fin}υ and, using also 2.1.22, we can conclude that csy_{fin} = csy. Let {csy_{k} = ⟨L_{k}, ▷_{k}⟩}_{k∈K} be a family of finitary consequence systems. Clearly, it can also be seen as a family of consequence systems. Given any source {h_{k} : L → L_{k}}_{i∈I}, let {h_{k} : csy → csy_{k}}_{k∈K} be its unique Lang_{Csy}-initial lift. That is, csy = ⟨L, ▷⟩ and Φ^{▷} = (⋂_{k∈K} h_{k}^{−1}(h_{k}(Φ)^{▷_{k}})) as defined in 2.1.17. If we apply Fin, to this Csy source we obtain the Csy_{fin} source {h_{k} : csy_{fin} → csy_{k}}_{k∈K}. From 2.1.17 it easily follows that it constitutes the desired unique Lang_{Csy_{fin}}-initial lifting.

Thus, also the fibres of Csy_{fin} form complete lattices. However, they are not complete sub-lattices of Csy-fibres.

Corollary 2.1.26. Csy_{fin} is fibre-complete, i.e., each fibre (Csy_{fin})_{L} is a complete lattice (under the restriction of ⊑_{L} to finitary consequence systems).

The meet of a family of finitary consequence operations {▷_{k}}_{k∈K} on L is given by the corresponding unique Lang_{Csy_{fin}}-initial lifting of {id_{L} : L → L}_{k∈K}, i.e., ⊓_{L}^{fin}{▷_{k} : k ∈ K\} = \underline{▷}_{fin} with Φ^{\underline{▷}^{fin}} = ⋃_{Ψ∈℘_{fin}(Φ)}(⋂_{k∈K} Ψ^{▷_{k}}), for every Φ ⊆ L.

Although meets in Csy of families of finitary consequence operations may not be finitary, the same does not happen with joins. In fact, ⊔_{L}^{fin}{▷_{k} : k ∈ K} = ⊔_{L}{▷_{k} : k ∈ K}. Clearly, it suffices to prove that ⊔_{L}\{▷_{k} : k ∈ K\} = \overline{▷} as given by 2.1.19 is indeed finitary, once we know that each ▷_{k} is. We start by showing that, in this case, the map Ω_{I} as given by 2.1.19 is continuous on ⟨℘(L), ⊆⟩. Recall that Φ\overline{▷} was seen to correspond to the least fixed point of Ω_{I} that contains Φ, where Ω_{I}(Φ) = (Φ ∪ (⋃_{k∈K} Φ^{▷_{k}})). Since Ω_{I} is known to be order-preserving, it suffices to show that Ω_{I}(⋃_{Φ∈X} Φ) ⊆ (⋃_{Φ∈X} Ω_{I}(Φ)) for every directed X ⊆ ℘(L). Recall that X is directed provided that every Y ∈ ℘_{fin}(X) has an upper-bound in X. Clearly, if φ ∈ Ω_{I}(⋃_{Φ∈X} Φ) = (⋃_{k∈K}(⋃_{Φ∈X} Φ)^{▷_{k}}), then (⋃_{Φ∈X} Φ) ▷_{j} φ for some j ∈ K. Moreover, since ▷_{j} is finitary, Ψ ▷_{j} φ for some Ψ ∈ ℘_{fin}(⋃_{Φ∈X} Φ), and since Ψ is finite, there exist Φ_{1}, …, Φ_{n} ∈ X such that Ψ ⊆ (Φ_{1} ∪ … ∪ Φ_{n}). But X is directed so there exists Υ ∈ X such that Ψ ⊆ (Φ_{1} ∪ … ∪ Φ_{n}) ⊆ Υ. Therefore, φ ∈ Ψ^{▷_{j}} ⊆ Υ^{▷_{j}} ⊆ (⋃_{Φ∈X} Φ^{▷_{j}}) ⊆ (⋃_{Φ∈X} Ω_{I}(Φ)). Thus, by a famous theorem of Kleene (see for instance [LNS82]), the least fixed point of Ω_{I} that contains Φ can be expressed by (⋃_{n∈IN} Ω_{I}^{n}(Φ)), where:

• Ω_{I}^{0}(Φ) = Φ;

• Ω_{I}^{n+1}(Φ) = Ω_{I}(Ω_{I}^{n}(Φ)).

Thus, \overline{▷} is such that Φ\overline{▷} = (⋃_{n∈IN} Ω_{I}^{n}(Φ)).

We can now show, by induction on n ∈ IN, that {Ω_{I}^{n}(Ψ) : Ψ ∈ ℘_{fin}(Φ)} is directed and Ω_{I}^{n}(Φ) = (⋃_{Ψ∈℘_{fin}(Φ)} Ω_{I}^{n}(Ψ)).

• Base case: n = 0.

Clearly Ω_{I}^{0} is the identity. And trivially, ℘_{fin}(Φ) is directed and Φ = (⋃_{Ψ∈℘_{fin}(Φ)} Ψ).

• Step: n > 0 and the induction hypothesis asserts that {Ω_{I}^{n−1}(Ψ) : Ψ ∈ ℘_{fin}(Φ)} is directed and Ω_{I}^{n−1}(Φ) = (⋃_{Ψ∈℘_{fin}(Φ)} Ω_{I}^{n−1}(Ψ)). For every Ψ_{1}, …, Ψ_{k} ∈ ℘_{fin}(Φ), by induction hypothesis, there exists Ψ ∈ ℘_{fin}(Φ) such that Ω_{I}^{n−1}(Ψ_{1}), …, Ω_{I}^{n−1}(Ψ_{k}) ⊆ Ω_{I}^{n−1}(Ψ). Thus, since Ω_{I} is order-preserving, Ω_{I}^{n}(Ψ_{1}), …, Ω_{I}^{n}(Ψ_{k}) ⊆ Ω_{I}^{n}(Ψ) and {Ω_{I}^{n}(Ψ) : Ψ ∈ ℘_{fin}(Φ)} is directed.

On the other hand, by induction hypothesis, Ω_{I}^{n}(Φ) = Ω_{I}(⋃_{Ψ∈℘_{fin}(Φ)} Ω_{I}^{n−1}(Ψ)). Moreover, {Ω_{I}^{n−1}(Ψ) : Ψ ∈ ℘_{fin}(Φ)} is directed. So, by continuity of Ω_{I}, Ω_{I}^{n}(Φ) = (⋃_{Ψ∈℘_{fin}(Φ)} Ω_{I}(Ω_{I}^{n−1}(Ψ))) = (⋃_{Ψ∈℘_{fin}(Φ)} Ω_{I}^{n}(Ψ)).

Now, easily, Φ\overline{▷} = (⋃_{n∈IN} Ω_{I}^{n}(Φ)) = (⋃_{n∈IN,Ψ∈℘_{fin}(Φ)} Ω_{I}^{n}(Ψ)) = (⋃_{Ψ∈℘_{fin}(Φ)} Ψ\overline{▷}). The same is to say that \overline{▷}_{fin} = \overline{▷}.

Top and bottom of the complete lattice coincide with those for fibers in Csy. Both are easily seen to be finitary.

Now, let us take a closer look at theory spaces. They correspond, in fact, to complete join semi-lattices [Bir67]. That is, in general, only joins are preserved by theory space morphisms.

Definition 2.1.27. Let tsp_{k} = ⟨Th_{k}, ≤_{k}⟩ for k ∈ {1, 2} be theory spaces. A theory spaces morphism h : tsp_{1} → tsp_{2} is a function h : Th_{1} → Th_{2} such that h(⋁_{1} T) = ⋁_{2} h(T), for every T ⊆ Th_{1}.

We prove a straightforward but useful property of Tsp-morphisms.

Proposition 2.1.28. Let tsp_{k} = ⟨Th_{k}, ≤_{k}⟩ for k ∈ {1, 2} be theory spaces and h : tsp_{1} → tsp_{2} a Tsp-morphism. Then h is order-preserving, i.e., for every Φ, Ψ ∈ Th_{1}, if Φ ≤_{1} Ψ then h(Φ) ≤_{2} h(Ψ).

Proof. Clearly, if Φ ≤_{1} Ψ then ⋁_{1}{Φ, Ψ} = Ψ. Therefore, since h preserves joins, h(Ψ) = h(⋁_{1}{Φ, Ψ}) = ⋁_{2}{h(Φ), h(Ψ)}. Consequently, h(Φ) ≤_{2} h(Ψ). ⊓ ⊔

Definition 2.1.29. Theory spaces and their morphisms constitute the category Tsp, with identity id_{⟨Th,≤⟩} the identity function on Th, and (g∘f) the usual composition of functions.

Proposition 2.1.30. Tsp is indeed a category.

Proof. Due to the well known properties of compositions of functions, we just prove that id_{⟨Th,≤⟩} : ⟨Th, ≤⟩ → ⟨Th, ≤⟩ and (g ∘ f) : ⟨Th_{1}, ≤_{1}⟩ → ⟨Th_{3}, ≤_{3}⟩ are indeed Tspmorphisms, provided that f : ⟨Th_{1}, ≤_{1}⟩ → ⟨Th_{2}, ≤_{2}⟩ and g : ⟨Th_{2}, ≤_{2}⟩ → ⟨Th_{3}, ≤_{3}⟩ are.

With respect to identity, trivially, idTh(⋁ T) = ⋁ T = ⋁ idTh(T).

Now, let T ⊆ Th_{1}. Since f is a morphism, f(⋁_{1} T) = ⋁_{2} f(T). Thus, g(f(⋁_{1} T)) = g(⋁_{2} f(T)). But g is also a morphism, so g(⋁_{2} f(T)) = ⋁_{3} g(f(T)) and g(f(⋁_{1} T)) = ⋁_{3} g(f(T)). ⊓ ⊔

Expectedly, the definition of the space of theories induced by a consequence system can be extended to a functor.

Definition 2.1.31. The maps • Th(csy) = tsp_{csy};

• Th(h : csy_{1} → csy_{2}) : tsp_{csy_{1}} → tsp_{csy_{2}}, with Th(h)(Φ) = h(Φ)^{▷_{2}} if csy_{2} = ⟨L_{2}, ▷_{2}⟩, for every Φ ∈ Th(csy_{1}),

consitute the functor Th : Csy → Tsp.

Proposition 2.1.32. Th : Csy → Tsp is indeed a functor.

Proof. At the light of 2.1.9, tsp_{csy} is a theory space whenever csy is a consequence system. We are left to prove that if h : csy_{1} → csy_{2} is a Csy-morphism then the map Th(h) : tsp_{csy_{1}} → tsp_{csy_{2}} defined as above is well defined and preserves joins. For the purpose, let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2}.

Undoubtedly, h(Φ)^{▷_{2}} ∈ Th(csy_{2}) for every Φ ⊆ L_{1}, and Th(h) is well defined.

Let now T ⊆ Th(csy_{1}). We have to show that Th(h)(⋁_{1} T) = ⋁_{2} Th(h)(T). That is, h((⋃_{Φ∈T} Φ)^{▷_{1}})^{▷_{2}} = ⋁_{2}{h(Φ)^{▷_{2}} : Φ ∈ T }. Since h is a Csy-morphism, we can use 2.1.12 to conclude that h((⋃_{Φ∈T} Φ)^{▷_{1}})^{▷_{2}} = h((⋃_{Φ∈T} Φ))^{▷_{2}} = (⋃_{Φ∈T} h(Φ))^{▷_{2}}. But now, we can use 2.1.10 to conclude that (⋃_{Φ∈T} h(Φ))^{▷_{2}} = ⋁_{2}{h(Φ)^{▷_{2}} : Φ ∈ T }. ⊓ ⊔

This functor establishes a reflection between Csy and Tsp. Indeed, Th is coadjoint with a natural isomorphic counit.

Proposition 2.1.33. Th : Csy → Tsp is coadjoint with counit ϵ_{tsp} : Th(csy_{tsp}) → tsp defined by ϵ_{tsp}(T) = ⋁ T, where csy_{tsp} = ⟨Th, ▷_{tsp}⟩ and T^{▷_{tsp}} = {Φ ∈ Th : Φ ≤ ⋁ T } for each theory space tsp = ⟨Th, ≤⟩. Moreover, ϵ_{tsp} is an isomorphism.

Proof. It is straightforward from the definition of a complete lattice and the properties of joins (that always exist in a complete lattice), that ▷_{tsp} is well defined and indeed a consequence operation on Th. Let T ⊆ Th. Extensiveness is immediate from the fact that, if Φ ∈ T, then Φ ≤ ⋁ T. If also T′ ⊆ Th, then T ⊆ T ∪ T′, ⋁(T ∪ T′) is an upper-bound of T, and monotonicity follows. Idempotence is a consequence of the obvious fact that ⋁{Φ ∈ Th : Φ ≤ ⋁ T } ≤ ⋁ T.

The fact that ϵ_{tsp} is indeed a Tsp-morphism, i.e., preserves joins, is a consequence of the fact that, for every T ⊆ ℘(Th), ⋁{⋁ T : T ∈ T } = ⋁(⋃_{T ∈T} T). We should note that the theories of csy_{tsp} are precisely the principal ideals of tsp, i.e., the sets T ⊆ Th such that Φ ∈ T whenever Φ ≤ ⋁ T.

With respect to the universal property, let csy = ⟨L, ▷⟩ be a consequence system and h : Th(csy) → tsp a Tsp-morphism. We need to show that there is a unique Csy-morphism g : csy → csy_{tsp} such that h = (ϵ_{tsp} ∘Th(g)). Note that if g is set to be a morphism, Φ ⊆ L and Φ ▷ ψ, then we must have g(Φ) ▷_{tsp} g(ψ), or equivalently, g(ψ) ≤ ⋁ g(Φ). On the other hand, if Φ ∈ Th(csy), by definition of Th(g), ▷_{tsp} and ϵ_{tsp}, ϵ_{tsp}(Th(g)(Φ)) = ⋁{Ψ ∈ Th : Ψ ≤ ⋁ g(Φ)} = ⋁ g(Φ), thus simplifying the commutativity condition to ⋁ g(Φ) = h(Φ). We define g such that, for every φ ∈ L, g(φ) = h({φ}^{▷}).

If Φ ▷ ψ then, easily, {ψ}^{▷} ⊆ Φ^{▷}. By 2.1.28, since h is order-preserving, it follows that g(ψ) = h({ψ}^{▷}) ≤ h(Φ^{▷}). But clearly, ⋁{{φ}^{▷} : φ ∈ Φ} = Φ^{▷}, and so we have that g(ψ) ≤ h(⋁{{φ}^{▷} : φ ∈ Φ}). Given that h preserves joins, equivalently, we have that g(ψ) ≤ ⋁{h({φ}^{▷}) : φ ∈ Φ} = ⋁{g(φ) : φ ∈ Φ} = ⋁ g(Φ). We conclude that g is indeed a Csy-morphism.

If Φ ∈ Th(csy) then ⋁ g(Φ) = ⋁{g(φ) : φ ∈ Φ} = ⋁{h({φ}^{▷}) : φ ∈ Φ} = h(Φ^{▷}), as we have seen just above. But Φ^{▷} = Φ, so ⋁ g(Φ) = h(Φ) and we conclude that h = (ϵ_{tsp} ∘ Th(g)).

The uniqueness of g follows from the following argument concerning an arbitrary sentence φ ∈ L: since g must be a morphism, if {φ} ▷ ψ then it must be the case that g(ψ) ≤ ⋁{g(φ)} = g(φ); moreover, extensiveness of ▷ guarantees that {φ} ▷ φ, and therefore ⋁{g(ψ) : ψ ∈ {φ}^{▷}} = g(φ); thus, the only way to force the commutativity condition is to have g(φ) = h({φ}^{▷}).

We are now left with proving that ϵ_{tsp} : Th(csy_{tsp}) → tsp is an isomorphism. This is a well known result about principal ideals. We define ϵ_{tsp}^{−1} : Th → Th(csy_{tsp}) by letting ϵ_{tsp}^{−1}(Φ) = {Ψ ∈ Th : Ψ ≤ Φ} for every Φ ∈ Th. Very easily, we can see that ϵ_{tsp}(ϵ_{tsp}^{−1}(Φ)) = ⋁({Ψ ∈ Th : Ψ ≤ Φ}) = Φ. On the other hand, if T ∈ Th(csy_{tsp}), ϵ_{tsp}^{−1}(ϵ_{tsp}(T)) = {Ψ ∈ Th : Ψ ≤ ⋁ T } = T^{▷_{tsp}} = T. We just need to verify that ϵ_{tsp}^{−1} : tsp → Th(csy_{tsp}) is indeed a Tsp-morphism. Let T ⊆ Th. We have that ϵ_{tsp}^{−1}(⋁ T) = {Ψ ∈ Th : Ψ ≤ ⋁ T } = T^{▷_{tsp}}. The property ⋁{{φ}^{▷} : φ ∈ Φ} = Φ^{▷} now with respect to ▷_{tsp}, allows us to conclude that ϵ_{tsp}^{−1}(⋁ T) = ⋁{{Φ}^{▷_{tsp}} : Φ ∈ T }. But ⋁{Φ} = Φ and {Φ}^{▷_{tsp}} = {Ψ ∈ Th : Ψ ≤ Φ} = ϵ_{tsp}^{−1}(Φ). Therefore ϵ_{tsp}^{−1}(⋁ T) = ⋁ ϵ_{tsp}^{−1}(T), ϵ_{tsp}^{−1} is a Tsp-morphism, and ϵ_{tsp} is an isomorphism. ⊓ ⊔

Corollary 2.1.34. The maps

• Csy_{Tsp}(tsp) = csy_{tsp};

• Csy_{Tsp}(h : tsp_{1} → tsp_{2}) = h : csy_{tsp_{1}} → csy_{tsp_{2}},

constitute a functor Csy_{Tsp} : Tsp → Csy. Moreover, Csy_{Tsp} is adjoint to Th.

Trivially, using the proof of coadjointness in 2.1.33. The unit η of the adjunction is such that, for every consequence system csy = ⟨L, ▷⟩ and sentence φ ∈ L, η_{csy} : csy → Csy_{Tsp}(tsp_{csy}) is defined by η_{csy}(φ) = {φ}^{▷}.

A parallel but weaker result could be established between Csy_{fin} and the full subcategory of Tsp whose objects are precisely the algebraic lattices [Bir67]. A coadjoint could again be obtained by restricting the functor Th to Csy_{fin}. However, a necessary change in the counit, obtained by application of Fin, prevents it from being again a reflection. We do not go deeper into this subject here, since it is not of mainstream importance to our results. Further details can be found in [SSV95].

Another issue that we are interested in is the equivalence of consequence systems. In the course of our work we shall have to compare consequence systems with respect to their expressive power. The strongest sense in which two consequence systems can be said to be equivalent is in the presence of an isomorphism.

Proposition 2.1.35. Two consequence systems csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} are isomorphic if and only if there exists a bijection h : L_{1} → L_{2} that satisfies the following condition:

• h(Φ^{▷_{1}}) = h(Φ)^{▷_{2}}, for every Φ ⊆ L_{1}.

Proof. Suppose that g : csy_{1} → csy_{2} is an isomorphism in Csy. That is, there exists a morphism f : csy_{2} → csy_{1} such that (f ∘ g) = id_{L_{1}} and (g ∘ f) = id_{L_{2}}. Clearly, g : L_{1} → L_{2} must be a bijection and f = g^{−1}. Let Φ ⊆ L_{1}. Since g is a morphism, g(Φ^{▷_{1}}) ⊆ g(Φ)^{▷_{2}}. On the other hand, also f = g^{−1} is a morphism and g(Φ) ⊆ L_{2}, thus g^{−1}(g(Φ)^{▷_{2}}) ⊆ g^{−1}(g(Φ))^{▷_{1}} = Φ^{▷_{1}}, i.e., g(Φ)^{▷_{2}} ⊆ g(Φ^{▷_{1}}).

Assume now that h : L_{1} → L_{2} is a bijection and h(Φ^{▷_{1}}) = h(Φ)^{▷_{2}}, for every Φ ⊆ L_{1}. Trivially, h : csy_{1} → csy_{2} is a Csy-morphism. To prove that it is an isomorphism it suffices to show that also h^{−1} : csy_{2} → csy_{1} is a morphism. Let Ψ ⊆ L_{2}. Clearly, h^{−1}(Ψ) ⊆ L_{1} and, by assumption, h(h^{−1}(Ψ)^{▷_{1}}) = h(h^{−1}(Ψ))^{▷_{2}} = Ψ^{▷_{2}}. Equivalently, h^{−1}(Ψ)^{▷_{1}} = h^{−1}(Ψ^{▷_{2}}) and, in particular, h^{−1}(Ψ^{▷_{2}}) ⊆ h^{−1}(Ψ)^{▷_{1}}. ⊓ ⊔

Isomorphism is, however, a very strong requirement. Many times, consequence systems differ just on the amount of equivalent sentences they possess, while still enclosing essentially the same closure properties. An extreme solution would be to require isomorphism of the corresponding theory spaces. The theory space of a consequence system is clearly a good measure of its expressive power.

Proposition 2.1.36. Two theory spaces tsp_{k} = ⟨Th_{k}, ≤_{k}⟩ for k ∈ {1, 2} are isomorphic if and only if there exists a bijection h : Th_{1} → Th_{2} that satisfies the following condition:

• Φ ≤_{1} Ψ if and only if h(Φ) ≤_{2} h(Ψ), for every Φ, Ψ ∈ Th_{1}.

Proof. Suppose that g : tsp_{1} → tsp_{2} is an isomorphism in Tsp. That is, there exists a morphism f : tsp_{2} → tsp_{1} such that (f ∘ g) = id_{Th_{1}} and (g ∘ f) = id_{Th_{2}}. Clearly, g : Th_{1} → Th_{2} must be a bijection and f = g^{−1}. If Φ ≤_{1} Ψ, since g is a morphism, 2.1.28 implies that g(Φ) ≤_{2} g(Ψ). On the other hand, since f is also a morphism, if g(Φ) ≤_{2} g(Ψ) 2.1.28 now implies that Φ = g^{−1}(g(Φ)) ≤_{1} g^{−1}(g(Ψ)) = Ψ.

Assume now that h : Th_{1} → Th_{2} is a bijection, and Φ ≤_{1} Ψ if and only if h(Φ) ≤_{2} h(Ψ), for every Φ, Ψ ∈ Th_{1}. We first show that h : tsp_{1} → tsp_{2} is a Tsp-morphism. For the purpose, let T ⊆ Th_{1}. If Φ ∈ T, obviously Φ ≤_{1} ⋁_{1} T. Therefore, h(Φ) ≤_{2} h(⋁_{1} T) and consequently ⋁_{2} h(T) ≤_{2} h(⋁_{1} T). On the other hand, clearly, h(Φ) ≤_{2} ⋁_{2} h(T). Since h is a bijection, we can rephrase this to h(Φ) ≤_{2} h(h^{−1}(⋁_{2} h(T))). Therefore, Φ ≤_{1} h^{−1}(⋁_{2} h(T)) and we conclude that ⋁_{1} T ≤_{1} h^{−1}(⋁_{2} h(T)). Thus, h(⋁_{1} T) ≤_{2} ⋁_{2} h(T) and h is indeed a Tsp-morphism. Analogously, we can show that h^{−1} : tsp_{2} → tsp_{1} is also a morphism. ⊓ ⊔

But there is still a strong drawback with this notion of equivalence: theory spaces and their morphisms are not guided by syntax. That is, they do not reflect the finer expressibility power of isolated sentences.

Consider these two, very simple, consequence systems csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2}:

• L_{1} = {a, b} and ▷_{1} is the identity;

• L_{2} = {a, b, a&b} and ▷_{2} extends ▷_{1} by letting Φ^{▷_{2}} = L_{2} if a&b ∈ Φ.

Easily, Th(csy_{1}) = {∅, {a}, {b}, {a, b}} and Th(csy_{2}) = {∅, {a}, {b}, {a, b, a&b}} are isomorphic. However, csy_{1} and csy_{2} must not be considered equivalent, in our terms. In particular, in csy_{2}, a&b can be seen as an inconsistent sentence since {a&b}^{▷_{2}} = L_{2}. In csy_{1}, clearly, there is no such sentence.

What we would like to have is some notion that ensures isomorphism of theory spaces while still keeping sentences as first class citizens.

Definition 2.1.37. Two consequence systems csy_{k} for k ∈ {1, 2} are said to be equivalent if there exist Csy-morphisms f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} such that Th(f) and Th(g) establish an isomorphism of tsp_{csy_{1}} and tsp_{csy_{2}} with Th(f) = Th(g)^{−1}. We denote the fact that csy_{1} and csy_{2} are equivalent by csy_{1} ≡ csy_{2}.

Trivially, isomorphism of consequence systems is a very special case of equivalence. Furthermore, as expected, equivalent consequence systems have isomorphic theory spaces. We also omit the straightforward proof of the fact that ≡ establishes an equivalence relation on consequence systems.

We shall now pave the way to an alternative characterization of equivalence in terms of the internal logical relation between the sentences of each consequence system.

Definition 2.1.38. Let csy = ⟨L, ▷⟩ be a consequence system and Φ ⊆ L. Two sentences ψ, υ ∈ L are said to be equivalent in csy modulo-Φ whenever both (Φ ∪ {υ}) ▷ ψ and (Φ ∪ {ψ}) ▷ υ. We denote this fact by ψ ≡_{csy}^{Φ} υ. If Φ = ∅, we simply say that ψ and υ are equivalent in csy, and write ψ ≡_{csy} υ instead of ψ ≡_{csy}^{∅} υ.

This definition trivially yields a collection of equivalence relations ≡_{csy}^{Φ} on sentences. It is straightforward to verify that ≡_{csy}^{Φ} corresponds precisely to logical equivalence in the consequence system induced by Φ on csy as in 2.1.2, i.e., ≡_{csy}^{Φ}=≡_{csy(Φ)}.

A special interesting case is what we shall call an elementary consequence system.

Definition 2.1.39. A consequence system csy = ⟨L, ▷⟩ is said to be elementary if ≡_{csy}= Δ_{L}, i.e., {φ} ▷ ψ and {ψ} ▷ φ together imply φ = ψ.

Quotients of consequence systems with respect to these equivalence relations could be shown to establish a reflection on Csy. Although we shall not fully explore this construction we shall still present it here. Its usefulness shall be made clear later on, but it obviously yields an elementary system.

Definition 2.1.40. Let csy = ⟨L, ▷⟩ be a consequence system and Φ ⊆ L. The quotient consequence system induced by Φ on csy is given by csy/_{≡_{csy}^{Φ}} = ⟨L/_{≡_{csy}^{Φ}}, ▷_{Φ}^{≡_{csy}}⟩ where L/_{≡_{csy}^{Φ}} is the quotient set of L by the equivalence relation, and

{[ψ]_{≡_{csy}^{Φ}} : ψ ∈ Ψ}^{▷_{Φ}^{≡csy}} = {[υ]_{≡_{csy}^{Φ}} : (Φ ∪ Ψ) ▷ υ}.

Of course, we have used [υ]_{≡_{csy}^{Φ}} to denote the ≡_{csy}^{Φ}-equivalence class of υ. Please note also that csy/_{≡_{csy}^{Φ}} = csy(Φ)/_{≡_{csy(Φ)}}.

Proposition 2.1.41. Let csy = ⟨L, ▷⟩ be a consequence system and Φ ⊆ L. Then, csy/_{≡_{csy}^{Φ}} is an elementary consequence system.

Proof. We start by showing that ▷_{Φ}^{≡_{csy}}, as in 2.1.40, is well defined. Just consider Ψ_{1}, Ψ_{2} ⊆ L such that {[ψ_{1}]_{≡_{csy}^{Φ}} : ψ_{1} ∈ Ψ_{1}} ⊆ {[ψ_{2}]_{≡_{csy}^{Φ}} : ψ_{2} ∈ Ψ_{2}}. We first prove that, in that case, Ψ_{1} ⊆ (Φ∪Ψ_{2})^{▷} and therefore the properties of ▷ imply that (Φ∪Ψ_{1})^{▷} ⊆ (Φ∪Ψ_{2})^{▷}. Indeed, let ψ_{1} ∈ Ψ_{1}. Easily, there exists ψ_{2} ∈ Ψ_{2} such that ψ_{1} ≡_{csy}^{Φ} ψ_{2}. Thus, (Φ ∪ {ψ_{2}})▷ψ_{1} and we are done by the monotonicity of ▷. Assume now that {[ψ_{1}]_{≡_{csy}^{Φ}} : ψ_{1} ∈ Ψ_{1}} = {[ψ_{2}]_{≡_{csy}^{Φ}} : ψ_{2} ∈ Ψ_{2}}. Clearly, then, (Φ ∪ Ψ_{1})^{▷} = (Φ ∪ Ψ_{2})^{▷} and ▷_{Φ}^{≡_{csy}} is properly defined.

Next, we check that ▷_{Φ}^{≡_{csy}} indeed defines a consequence operation on L/_{≡_{csy}^{Φ}}. Clearly, by extensiveness of ▷, if ψ ∈ Ψ we have that (Φ ∪ Ψ) ▷ ψ. Thus, {[ψ]_{≡_{csy}^{Φ}} : ψ ∈ Ψ} ▷_{Φ}^{≡_{csy}} [ψ]_{≡_{inf}^{Φ}}

and the extensiveness of ▷_{Φ}^{≡_{csy}} follows. Consider now Ψ_{1}, Ψ_{2} ⊆ L such that {[ψ_{1}]_{≡_{csy}^{Φ}} : ψ_{1} ∈ Ψ_{1}} ⊆ {[ψ_{2}]_{≡_{csy}^{Φ}} : ψ_{2} ∈ Ψ_{2}}, and assume that {[ψ_{1}]_{≡_{csy}^{Φ}} : ψ_{1} ∈ Ψ_{1}} ▷_{Φ}^{≡_{csy}} [υ]_{≡_{csy}^{Φ}}. This means that (Φ ∪ Ψ_{1})▷υ. Since we already know that (Φ ∪ Ψ_{1})^{▷} ⊆ (Φ ∪ Ψ_{2})^{▷} the monotonicity of ▷_{Φ}^{≡_{csy}} easily follows. As for idempotence of ▷_{Φ}^{≡_{csy}}, assume that {[ψ]_{≡_{csy}^{Φ}} :

ψ ∈ Ψ}^{▷_{Φ}^{≡csy}} ▷_{Φ}^{≡_{csy}} [υ]_{≡_{csy}^{Φ}}. By definition, this means precisely that υ ∈ (Φ ∪ (Φ ∪ Ψ)^{▷})^{▷}. However, by the extensiveness of ▷, Φ ⊆ (Φ ∪ Ψ)^{▷} and thus υ ∈ ((Φ ∪ Ψ)^{▷})^{▷} = (Φ ∪ Ψ)^{▷}), just by using the idempotence of ▷. Therefore, {[ψ]_{≡_{csy}^{Φ}} : ψ ∈ Ψ} ▷_{Φ}^{≡_{csy}} [υ]_{≡_{csy}^{Φ}} and we are done.

We now know that csy/_{≡_{csy}^{Φ}} is a consequence system. We have however to check that it is elementary. In fact, suppose that {[ψ_{1}]_{≡_{csy}^{Φ}}} ▷_{Φ}^{≡_{csy}} [ψ_{2}]_{≡_{csy}^{Φ}} and also {[ψ_{2}]_{≡_{csy}^{Φ}}} ▷_{Φ}^{≡_{csy}} [ψ_{1}]_{≡_{csy}^{Φ}}. Then we must have that (Φ ∪ {ψ_{1}}) ▷ ψ_{2} and (Φ ∪ {ψ_{2}}) ▷ ψ_{1}. Thus, ψ_{1} ≡_{csy}^{Φ} ψ_{2} and so [ψ_{1}]_{≡_{csy}^{Φ}} = [ψ_{2}]_{≡_{csy}^{Φ}}. ⊓ ⊔

In our quest for a nicer characterization of equivalence of consequence systems, we shall now prove that equivalent sentences are precisely those that are indistinguishable with respect to theories.

Proposition 2.1.42. Let csy = ⟨L, ▷⟩ be a consequence system and ψ, υ ∈ L. The following conditions are equivalent:

• ψ ≡_{csy} υ;

• {ψ}^{▷} = {υ}^{▷};

• for every Φ ⊆ L, ψ ≡_{csy}^{Φ} υ;

• for every theory Φ ∈ Th(csy), ψ ∈ Φ if and only if υ ∈ Φ.

Proof. Assume that ψ ≡_{csy} υ. By definition {υ} ▷ ψ and {ψ} ▷ υ, i.e., {ψ} ⊆ {υ}^{▷} and {υ} ⊆ {ψ}^{▷}. Thus, by monotonicity and idempotence of ▷, {ψ}^{▷} ⊆ {υ}^{▷} and {υ}^{▷} ⊆ {ψ}^{▷}.

Let now {ψ}^{▷} = {υ}^{▷}. Clearly, {ψ} ⊆ (Φ ∪ {ψ}) and the monotonicity of ▷ implies that (Φ ∪ {ψ}) ▷ υ. Analogously, we have that also (Φ ∪ {υ}) ▷ ψ, and thus ψ ≡_{csy}^{Φ} υ.

Let ψ ≡_{csy}^{Φ} υ for every Φ ⊆ L. Given a theory Υ ∈ Th(csy) assume, without loss of generality, that ψ ∈ Υ. Clearly, ψ ≡_{csy}^{Υ} υ and in particular (Υ∪{ψ}) ▷ υ. But (Υ∪{ψ})^{▷} = Υ^{▷} = Υ, and hence υ ∈ Υ.

Assume now that, for every theory Φ ∈ Th(csy), ψ ∈ Φ if and only if υ ∈ Φ. Clearly, {ψ}^{▷} and {υ}^{▷} are theories, and by extensiveness of ▷, {ψ} ▷ ψ and {υ} ▷ υ. Thus, {ψ} ▷ υ and {υ} ▷ ψ, that is, ψ ≡_{csy} υ. ⊓ ⊔

The following lemma shows that theories are in fact independent, modulo equivalent sentences, of the way they are presented.

Lemma 2.1.43. Let csy = ⟨L, ▷⟩ be a consequence system and Φ, Ψ ⊆ L. Then, Φ^{▷} = Ψ^{▷} whenever the following two conditions are satisfied:

• for every φ ∈ Φ there exists φ′ ∈ Ψ such that φ ≡_{csy} φ′;

• for every ψ ∈ Ψ there exists ψ′ ∈ Φ such that ψ ≡_{csy} ψ′.

Proof. Let us assume that both conditions hold. If φ ∈ Φ then φ′ ∈ Ψ ⊆ Ψ^{▷}, just using the extensiveness of ▷. But φ ≡_{csy} φ′ and therefore 2.1.42 implies that also φ ∈ Ψ^{▷}. Analogously, we can show that Ψ ⊆ Φ^{▷}, and conclude that Φ^{▷} = Ψ^{▷} by using the monotonicity and idempotence of ▷. ⊓ ⊔

Please note that this result could be a little weaker: e.g., the first condition could be safely replaced by the requirement that for every φ ∈ Φ there exists φ′ ∈ Ψ such that φ ≡_{csy}^{Ψ} φ′. However, our formulation is enough to enable a nicer characterization of equivalence of consequence systems.

Proposition 2.1.44. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems. Then, csy_{1} ≡ csy_{2} if and only if there exist Csy-morphisms f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} such that the following two conditions are satisfied:

• φ ≡_{csy_{1}} g(f(φ)) for every φ ∈ L_{1};

• ψ ≡_{csy_{2}} f(g(ψ)) for every ψ ∈ L_{2}.

Proof. Assuming that csy_{1} ≡ csy_{2}, let f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} be Csymorphisms such that Th(f) and Th(g) are isomorphisms, inverse to each other. Thus, given φ ∈ L_{1}, we must have Th(g)(Th(f)({φ}^{▷_{1}})) = {φ}^{▷_{1}}. However, Th(f)({φ}^{▷_{1}}) = f({φ}^{▷_{1}})^{▷_{2}} = {f(φ)}^{▷_{2}}, just using 2.1.12. Similarly, Th(g)({f(φ)}^{▷_{2}}) = g({f(φ)}^{▷_{2}})^{▷_{1}} = {g(f(φ))}^{▷_{1}}, thus yielding {g(f(φ))}^{▷_{1}} = {φ}^{▷_{1}}, or equivalently, φ ≡_{csy_{1}} g(f(φ)). Analogously, we can prove that ψ ≡_{csy_{2}} f(g(ψ)) for every ψ ∈ L_{2}.

Assume now that f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} are Csy-morphisms satisfying the two conditions stated above. If Φ ∈ Th(csy_{1}), then Th(g)(Th(f)(Φ)) = Th(g∘f)(Φ) = g(f(Φ))^{▷_{1}}. But now, trivially, Φ and g(f(Φ)) are in the conditions of 2.1.43, and we have that g(f(Φ))^{▷_{1}} = Φ^{▷_{1}} = Φ. Using an analogous argument, we can conclude that also Th(f)(Th(g)(Ψ)) = Ψ for every Ψ ∈ Th(csy_{2}). ⊓ ⊔

2.2 Logic systems and logics

Having studied consequence systems, we can now state our intended notions of logic system and logic. As can be understood from our previous discussion, consequence plays an essential role. That is, a logic system shall be seen as any mathematical object from which a consequence system can be extracted. Following many modern considerations [GB84, Mes89, Acz94, NA94], a logic shall be seen as the family of all its logic systems.

Definition 2.2.1. A category of logic systems is a pair ⟨A, Cn⟩ where:

• A is a category (of logic systems and their morphisms);

• Cn : A → Csy is a functor.

An object a ∈ |A| is said to be an ⟨A, Cn⟩-based logic system. If, in addition, Cn(a) is a finitary consequence system then a is said to be a finitary logic system. ⟨A, Cn⟩ is further said to be a category of finitary logic systems provided that every ⟨A, Cn⟩-based logic system is finitary.

Proposition 2.2.2. A category of logic systems ⟨A, Cn⟩ is finitary if and only if there exists a functor Cn_{fin} : A → Csy_{fin} such that Cn = (I ∘ Cn_{fin}).

Proof. Trivially, as already noted in 2.1.25, Fin does not change finitary elements. Therefore, just consider Cn_{fin} = (Fin ∘ Cn). ⊓ ⊔

Consequently, in the sequel, we shall sometimes identify a category of finitary logic systems with a functor Cn_{fin} : A → Csy_{fin}.

A logic shall embody the intuition that many particular vocabularies are admissible, each yielding a particular logic system, provided that changes from one vocabluary to another can be smoothly handled, via appropriate morphisms.

Definition 2.2.3. Let ⟨A, Cn⟩ be a category of logic systems and Voc a category (of vocabularies). An ⟨A, Cn⟩-based logic over Voc is a functor Log : Voc → A. If ⟨A, Cn⟩ is a category of finitary logic systems, then Log is also said to be finitary.

Whenever the functor Cn is well understood, we sometimes just refer to A, and call Log an A-based logic.

Please note also that if, for instance, F : A_{1} → A_{2} is a functor and ⟨A_{1}, Cn_{1}⟩ and ⟨A_{2}, Cn_{2}⟩ are categories of logic systems then, every ⟨A_{1}, Cn_{1}⟩-based logic Log : Voc → A_{1} can also be seen as the ⟨A_{2}, Cn_{2}⟩-based logic (F ∘ Log) : Voc → A_{2}.

Definition 2.2.4. Let ⟨A, Cn⟩ be a category of logic systems and Log an ⟨A, Cn⟩-based logic over Voc. The language functor of Log is Lang_{Log} = (Lang_{Csy} ∘ Cn ∘ Log) : Voc → Set.

That is, the language of a logic is the set of sentences corresponding to each possible vocabulary, and the sentence translation function induced by each vocabularies morphism. Clearly, this agrees with the notation Lang_{A} = (Lang_{Csy} ∘ Cn). Trivially, Csy is itself a category of logic systems under its own identity functor.

As we show, logics based on Csy, modulo a few minor presentation details, are known in the literature as π-institutions [FS88].

Definition 2.2.5. A π-institution is a tuple ⟨Voc, Sen, ⊢⟩ where:

• Voc is a category (of vocabularies^{1});

• Sen : Voc → Set is a (sentence) functor; • ⊢ is a |Voc|-indexed family of consequence operations ^{⊢_{V}} : ℘(Sen(V)) → ℘(Sen(V))

1In the literature on institutions this category is usually called Sig and its objects signatures. We have chosen Voc and vocabularies to avoid confusion with the notion of signature we shall introduce in the sequel.

for each vocabulary V ∈ |Voc |,

such that the following structurality condition holds for every Voc-morphism h : V_{1} → V_{2} and set of sentences Φ ⊆ Sen(V_{1}):

• Sen(h)(Φ^{⊢_{V_{1}}}) ⊆ Sen(h)(Φ)^{⊢_{V_{2}}}.

The π-institution is further said to be finitary if ⊢_{V} is a finitary consequence operation, for each vocabulary V ∈ |Voc |.

Note that the condition above means precisely that, for every Voc-morphism h : V_{1} → V_{2}, Sen(h) : ⟨Sen(V_{1}), ⊢_{V_{1}} ⟩ → ⟨Sen(V_{2}), ⊢_{V_{2}} ⟩ is a Csy-morphism. Of course, this is not just a coincidence.

Proposition 2.2.6. π-institutions correspond precisely to Csy-based logics.

Proof. Let ⟨Voc, Sen, ⊢⟩ be a π-institution. We can alternatively see the π-institution as the functor Log : Voc → Csy with Log(V) = ⟨Sen(V), ⊢_{V} ⟩ and Log(h : V_{1} → V_{2}) : Log(V_{1}) → Log(V_{2}) the Csy-morphism such that Log(h) = Sen(h).

On the other hand, let Log : Voc → Csy be a functor. To fix some notation let Log(V) = ⟨L_{V} , ▷_{V} ⟩ for each vocabulary V. Then, ⟨Voc, Sen, ▷⟩ is an institution with:

• Sen(V) = L_{V} and Sen(h : V_{1} → V_{2}) = Log(h).

The constructions are obviously well defined and inverse to each other. ⊓ ⊔

Another trivial example of a category of logic systems is Csy_{fin} under the inclusion functor to Csy.

Proposition 2.2.7. Finitary π-institutions correspond precisely to Csy_{fin}-based logics.

Proof. Similar to 2.2.6. ⊓ ⊔

Logics over a given category of logic systems also constitute a category. Logic morphisms are chosen in order to capture the intuition of a domain logic that is a component of the more more complex codomain.

Definition 2.2.8. Let ⟨A, Cn⟩ be a category of logic systems, and Log_{k} : Voc_{k} → A for k ∈ {1, 2} be ⟨A, Cn⟩-based logics. An ⟨A, Cn⟩-based logics morphism h : Log_{1} → Log_{2} is a pair h = ⟨H, η⟩ where:

• H : Voc_{2} → Voc_{1} is a functor;

• η : (Log_{1} ∘ H) → Log_{2} is a natural transformation.

Definition 2.2.9. Let ⟨A, Cn⟩ be a category of logic systems. ⟨A, Cn⟩-based logics and ⟨A, Cn⟩-based logics morphisms constitute the category ⟨A, Cn⟩-Log, with identity id_{Log} composed of the identity functor on the vocabulary category and the identity natural transformation on Log, and composition (⟨F, τ⟩ ∘ ⟨G, λ⟩) equal to ⟨(G ∘ F), (τ ∘ λF)⟩.

Proposition 2.2.10. ⟨A, Cn⟩-Log is indeed a category.

Proof. ⟨A, Cn⟩-Log is the category Funct(A) of A.3.5, in Appendix. ⊓ ⊔

We are often faced with systems of logic for which we have both a (finitary) prooftheoretic and a model-theoretic formulation. Such systems may or may not enjoy many interesting properties, which we shall be interested in later on.

Definition 2.2.11. Let ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩ be categories of logic systems, the former one being finitary, and cal ∈ |Cal| and sem ∈ |Sem| be such that Ded(cal) = ⟨L, ⊢⟩, Ent(sem) = ⟨L, ⊨⟩ for some set L. The pair ⟨cal, sem⟩ is said to be:

• weakly sound if ∅^{⊢} ⊆ ∅^{⊨};

• (strongly) sound if ⊢ ⊑_{L} ⊨, i.e., Φ^{⊢} ⊆ Φ^{⊨} for every Φ ⊆ L;

• weakly complete if ∅^{⊨} ⊆ ∅^{⊢};

• finitely complete if ⊨_{fin} ⊑_{L} ⊢_{fin}, i.e., Φ^{⊨} ⊆ Φ^{⊢} for every Φ ∈ ℘_{fin}(L);

• (strongly) complete if ⊨ ⊑_{L} ⊢, i.e., Φ^{⊨} ⊆ Φ^{⊢} for every Φ ⊆ L.

In the definition above Cal stands for calculus and Ded for deducibility, while Sem stands for semantics and Ent for entailment. Finite soundness was not defined since it coincides with soundness.

It is a trivial matter to extend these definitions to logics. In the following definitition, P denotes any of the properties defined above.

Definition 2.2.12. Let Voc be a category, and Prf : Voc → Cal and Mod : Voc → Sem be logics based, respectively, on the categories of logic systems ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩, the former being finitary, sharing the same language functor, i.e., Lang_{Prf} = Lang_{Mod}. The pair ⟨Prf , Mod⟩ is said to have property P provided that each pair ⟨Prf (V), Mod(V)⟩ does.

In the above definition, Prf stands for proof-theoretic and Mod for model-theoretic. In this context, we denote Ded(Prf (V)) by ⟨L_{V} , ⊢_{V} ⟩ and Ent(Mod(V)) by ⟨L_{V} , ⊨_{V} ⟩, for each vocabulary V. Please note that the condition that Lang_{Prf} = Lang_{Mod} immediately implies that the language L_{V} is shared by both.

2.2.1 Models: satisfaction and valuation systems

Our definition of a logic is wide enough to encompass many other abstract notions of logic. The next example is based on a common category of model-theoretic logic systems: satisfaction systems. They help us pave the way to the subsequent sections, and furthermore, they are the basis for another very well known abstract notion of logic: institutions [GB84, GB92]. With slight variations, the concept was already present in [SSC97a].

Definition 2.2.13. A satisfaction system is a triple sat = ⟨L, M, ⊩⟩ where:

• L is a set;

• M is a class (of models);

• ⊩ ⊆ (M × L) (the satisfaction relation).

Given a sentence φ ∈ L and a model m ∈ M, m is said to satisfy φ provided that m⊩φ.

In the context of a satisfaction system, as above, we extend the notion of satisfaction by a model to a set of sentences Φ ⊆ L, by requiring that m⊩Φ if and only if m⊩φ for every φ ∈ Φ.

Satisfaction systems morphisms embody the usual contravariance between sentences and models, known as Galois connection [Bir67, AHS90].

Definition 2.2.14. Let sat_{k} = ⟨L_{k}, M_{k}, ⊩_{k}⟩ for k ∈ {1, 2} be satisfaction systems. A satisfaction systems morphism h : sat_{1} → sat_{2} consists of:

• a (translation) function \overline{h} : L_{1} → L_{2};

• a (reduct) map \underline{h} : M_{2} → M_{1},

such that the following (satisfaction) condition holds for every sentence φ ∈ L_{1} and model m ∈ M_{2}:

• m⊩_{2} \overline{h}(φ) if and only if \underline{h}(m)⊩_{1}φ.

Definition 2.2.15. Satisfaction systems and their morphisms constitute the category Sat, with identity id_{⟨L,M, ⊩⟩} such that \overline{id}_{⟨L,M, ⊩⟩} = id_{L}, the identity function on L, and \underline{id}_{⟨L,M, ⊩⟩} = id_{M}, the identity map on M, and composition (g∘f) such that \overline{(g ∘ f)} = (\overline{g}∘\overline{f}) and \underline{(g ∘ f)} = (\underline{f} ∘ \underline{g}).

Proposition 2.2.16. Sat is indeed a category.

Proof. Due to the well known properties of compositions of maps, we just prove that id_{⟨L,M, ⊩⟩} : ⟨L, M, ⊩⟩ → ⟨L, M, ⊩⟩ and (g ∘ f) : ⟨L_{1}, M_{1}, ⊩_{1}⟩ → ⟨L_{3}, M_{3}, ⊩_{3}⟩ are indeed Sat-morphisms, provided that f : ⟨L_{1}, M_{1}, ⊩_{1}⟩ → ⟨L_{2}, M_{2}, ⊩_{2}⟩ and g : ⟨L_{2}, M_{2}, ⊩_{2}⟩ → ⟨L_{3}, M_{3}, ⊩_{3}⟩ are.

With respect to identity, trivially, \overline{id}_{⟨L,M, ⊩⟩}(φ) = φ and \underline{id}_{⟨L,M, ⊩⟩}(m) = m, and the condition holds.

Now, let φ ∈ L_{1}, m ∈ M_{3} and m⊩_{3} \overline{(g ∘ f)}(φ). By definition, \overline{(g ∘ f)}(φ) = \overline{g}(\overline{f}(φ)). Thus, using the satisfaction condition of the morphism g, we have that m⊩_{3} \overline{(g ∘ f)}(φ) if and only if \underline{g}(m)⊩_{2} \overline{f}(φ). Again, using the satisfaction condition of the morphism f, we have that m⊩_{3} \overline{(g ∘ f)}(φ) if and only if \underline{f}(\underline{g}(m))⊩_{1}φ. Therefore the condition holds since, by definition, \underline{(g ∘ f)}(m) = \underline{f}(\underline{g}(m)). ⊓ ⊔

Semantic entailment can be defined in general terms for any given satisfaction system.

Definition 2.2.17. Let sat = ⟨L, M, ⊩⟩ be a satisfaction system. A sentence ψ ∈ L is said to be entailed in sat by a set of sentences Φ ⊆ L, fact that we denote by Φ⊨_{sat} ψ, provided that for every model m ∈ M, m⊩ψ whenever m⊩Φ. The semantic entailment consequence operation ⊨_{sat} induced by sat is then defined, for each set of sentences Φ ⊆ L, by Φ^{⊨_{sat}} = {ψ ∈ L : Φ⊨_{sat} ψ}.

Proposition 2.2.18. Let sat = ⟨L, M, ⊩⟩ be a satisfaction system. Then, csy_{sat} = ⟨L, ⊨_{sat} ⟩ constitutes a consequence system.

Proof. We first prove extensiveness. Let Φ ⊆ L, and m ∈ M be any model such that m⊩Φ. Trivially, Φ⊨_{sat} φ for every φ ∈ Φ.

Let us now turn to monotonicity. Let Φ, Ψ ⊆ L, Φ⊨_{sat} υ, and m ∈ M be any model such that m⊩(Φ ∪ Ψ). Trivially, m⊩Φ, and so, by definition of ⊨_{sat} , m⊩υ. Therefore, (Φ ∪ Ψ)⊨_{sat} υ.

Finally, we show the idempotence of ⊨_{sat} . Let Φ ⊆ L, (Φ^{⊨_{sat}})⊨_{sat} υ, and m ∈ M be any model such that m⊩Φ. By definition of ⊨_{sat} we can conclude that m⊩Φ^{⊨_{sat}}. So, again by definition of ⊨_{sat} , m⊩υ, and Φ⊨_{sat} υ. ⊓ ⊔

The previous construction can be easily extended to a functor.

Definition 2.2.19. The maps

• Ent_{Sat}(sat) = csy_{sat};

• Ent_{Sat}(h : sat_{1} → sat_{2}) = \overline{h},

constitute the functor Ent_{Sat} : Sat → Csy.

Proposition 2.2.20. Ent_{Sat} : Sat → Csy is indeed a functor.

Proof. By definition of the categories Sat and Csy it is trivial to verify that identities and compositions are preserved. By 2.2.18, Ent_{Sat} is well defined on objects. Thus, we are left with proving that \overline{h} : csy_{sat_{1}} → csy_{sat_{2}} is indeed a Csy-morphism whenever h : sat_{1} → sat_{2} is a Sat-morphism. For the purpose, let sat_{k} = ⟨L_{k}, M_{k}, ⊩_{k}⟩ for k ∈ {1, 2}, Φ ⊆ L_{1}, Φ⊨_{sat_{1}} ψ and m ∈ M_{2} be such that m⊩_{2} \overline{h}(Φ). Since h : sat_{1} → sat_{2} is a morphism, we conclude that \underline{h}(m)⊩_{1}Φ. Thus, by definition of ⊨_{sat_{1}} , \underline{h}(m)⊩_{1}ψ. Therefore, again by the satisfaction condition of h, m⊩_{2} \overline{h}(ψ), and \overline{h}(Φ)⊨_{sat_{2}} \overline{h}(ψ). ⊓ ⊔

Corollary 2.2.21. The pair ⟨Sat, Ent_{Sat}⟩ constitutes a category of logic systems.

In the sequel, given a Sat-based logic Log : Voc → Sat, we shall denote Log(V) by ⟨L_{V} , M_{V} , ⊩_{V} ⟩ and its induced entailment consequence by ⊨_{V} , for each vocabulary V. Logics based on Sat, modulo presentation details, are known in the literature as institutions^{2}. This has already been noticed by the creators of the concept [GB86].

Definition 2.2.22. An institution is a tuple ⟨Voc, Sen, Mod, ⊩⟩ where:

• Voc is a category (of vocabularies);

• Sen : Voc → Set is a (sentence) functor;

• Mod : Voc → Cls^{op} is a (model) functor;

• ⊩ is a |Voc |-indexed family of relations ⊩_{V} ⊆ Mod(V)×Sen(V), for each vocabulary V ∈ |Voc |, such that the following satisfaction condition holds for every vocabulary morphism h : V_{1} → V_{2}, sentence φ ∈ Sen(V_{1}) and model m ∈ Mod(V_{2}):

2To be precise, we should say simple institutions since we just consider classes, instead of the categories of models used in fully-fledged institutions. It makes the example easier, and pinpoints to the fact that we shall not need categories of models in the sequel. Of course, to encompass institutions, Sat would have to incorporate this change.

• m⊩_{V_{2}} Sen(h)(φ) if and only if Mod(h)(m)⊩_{V_{1}} φ.

Please note that the condition above means that, for every vocabulary morphism h : V_{1} → V_{2}, Sen(h) and Mod(h) set up a Sat-morphism from ⟨Sen(V_{1}), Mod(V_{1}), ⊩_{V_{1}} ⟩ to ⟨Sen(V_{2}), Mod(V_{2}), ⊩_{V_{2}} ⟩. Again, this is no coincidence.

Proposition 2.2.23. Institutions correspond precisely to Sat-based logics.

Proof. Let ⟨Voc, Sen, Mod, ⊩⟩ be an institution. We can alternatively see the institution as the functor Log : Voc → Sat with Log(V) = ⟨Sen(V), Mod(V), ⊩_{V} ⟩ and Log(h : V_{1} → V_{2}) : Log(V_{1}) → Log(V_{2}) the Sat-morphism such that \overline{Log(h)} = Sen(h) and \underline{Log(h)} = Mod(h).

On the other hand, let Log : Voc → Sat be a functor. Then, ⟨Voc, Sen, Mod, ⊩⟩ is an institution with:

• Sen(V) = L_{V} and Sen(h : V_{1} → V_{2}) = \overline{Log(h)};

• Mod(V) = M_{V} and Mod(h : V_{1} → V_{2}) = \underline{Log(h)}.

The constructions are obviously well defined and inverse to each other. ⊓ ⊔

We do not wish to go deeper into the subject of institutions here, but it is worthwhile noticing that morphisms of Sat-based logics as we have defined them correspond precisely to contravariant institution morphisms as proposed in [GB92]. That is, the category of institutions and institution morphisms is isomorphic to the dual of Sat-Log.

Another interesting property is that the forgetful functor Lang_{Sat} is coadjoint.

Proposition 2.2.24. Lang_{Sat} : Sat → Set is coadjoint. Namely, id_{L} : Lang_{Sat}(sat) → L is a Lang_{Sat}-couniversal arrow for L, where sat = ⟨L, ℘(L), ∋⟩.

Proof. Let ⟨L′, M′, ⊩′⟩ be a satisfaction system and \overline{h} : L′ → L a function. We have to show that there exists a unique map \underline{h} : M′ → ℘(L) such that h : ⟨L′, M′, ⊩′⟩ → sat is a Sat-morphism. Obviously, by definition of sat we must have \underline{h}(m′) = \{φ ∈ L : m′ ⊩′\overline{h}(φ)\}, for every m′ ∈ M′, and the result follows. ⊓ ⊔

We shall now focus our attention on a natural generalization of satisfaction systems. In fact, intuitively, we may think of a model as somehow assigning a meaning to sentences. Then, depending only on its meaning, a sentence is bound to be satisfied or not by the model. This mechanism is also present in satisfaction systems, as we shall see, although in an indirect way. Our interest in this notion resides not only on the fact that we shall be able to build another example of a category of logic systems, but mainly because it adequately paves the way towards the analysis of the internal structure of models and its algebraic treatment that we shall develop in the next Chapters. The L-structures introduced below can be easily recognized as being a form of so-called logical matrices, in the tradition of the work of Tarski and Lukasiewicz [Wój88].

Definition 2.2.25. Let L be a set. An L-structure is a triple ⟨T, [[ ]], D⟩ where:

• T is a set (of possible truth values);

• [[ ]] : L → T (a valuation);

• D ⊆ T (the set of designated values).

We denote by Str(L) the class of all L-structures.

Given a function h : L′ → L and an L-structure ⟨T, [[ ]], D⟩ we denote by ⟨T, [[ ]], D⟩|_{h} the L′-structure ⟨T, ([[ ]] ∘ h), D⟩.

Definition 2.2.26. A valuation system is a triple val = ⟨L, M, S⟩ where:

• L is a set;

• M is a class;

• S : M → Str(L) (endows each model with an L-structure).

For each model m ∈ M, we denote S(m) by ⟨T_{m}, [[ ]]_{m}, D_{m}⟩.

Valuation system morphisms suitably generalize Sat-morphisms.

Definition 2.2.27. Let val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩ for k ∈ {1, 2} be valuation systems. A valuation systems morphism h : val_{1} → val_{2} consists of:

• a (translation) function \overline{h} : L_{1} → L_{2};

• a (reduct) map \underline{h} : M_{2} → M_{1},

such that the following condition holds for every model m ∈ M_{2}:

• S_{1}(\underline{h}(m)) = S_{2}(m)|\overline{h}.

Note that a morphism, as above, implies that T_{\underline{h}(m)} = T_{m}, D_{\underline{h}(m)} = D_{m} and [[φ]]_{\underline{h}(m)} = [[\overline{h}(φ)]]_{m} for every φ ∈ L_{1}.

Definition 2.2.28. Valuation systems and their morphisms constitute the category Val, with identity id_{⟨L,M,S⟩} such that \overline{id}_{⟨L,M,S⟩} = id_{L}, the identity function on L, and \underline{id}_{⟨L,M,S⟩} = id_{M}, the identity map on M, and composition (g ∘ f) such that \overline{(g ∘ f)} = (\overline{g} ∘ \overline{f}) and \underline{(g ∘ f)} = (\underline{f} ∘ \underline{g}).

Proposition 2.2.29. Val is indeed a category.

Proof. Due to the well known properties of compositions of maps, we just prove that id_{⟨L,M,S⟩} : ⟨L, M, S⟩ → ⟨L, M, S⟩ and (g ∘ f) : ⟨L_{1}, M_{1}, S_{1}⟩ → ⟨L_{3}, M_{3}, S_{3}⟩ are indeed Val-morphisms, provided that f : ⟨L_{1}, M_{1}, S_{1}⟩ → ⟨L_{2}, M_{2}, S_{2}⟩ and g : ⟨L_{2}, M_{2}, S_{2}⟩ → ⟨L_{3}, M_{3}, S_{3}⟩ are.

With respect to identity, trivially, \overline{id}_{⟨L,M,S⟩}(φ) = φ and \underline{id}_{⟨L,M,S⟩}(m) = m, and the condition holds.

Now, let m ∈ M_{3}. Using the condition of the morphism g, we have that S_{2}(\underline{g}(m)) = S_{3}(m)|\overline{g}. But \underline{g}(m) ∈ M_{2}, thus, using the condition of the morphism f, we have that S_{1}(\underline{f}(\underline{g}(m))) = (S_{2}(\underline{g}(m)))|\overline{f} = (S_{3}(m)|\overline{g})|\overline{f}. Or equivalently, S_{1}(\underline{(g ∘ f)}(m)) = S_{3}(m)|\overline{(g∘f)} and (g ∘ f) : ⟨L_{1}, M_{1}, S_{1}⟩ → ⟨L_{3}, M_{3}, S_{3}⟩ is a morphism. ⊓ ⊔

We show that Val is a category of logic systems by means of a functor to Sat. A model is set to satisfy a sentence provided that the meaning assigned to the sentence by the model is a designated value.

Definition 2.2.30. The maps

• Sat_{Val}(⟨L, M, S⟩) = ⟨L, M, ⊩_{S} ⟩, where m⊩_{S} φ provided that [[φ]]_{m} ∈ D_{m};

• Sat_{Val}(h : val_{1} → val_{2}) = h,

constitute the functor Sat_{Val} : Val → Sat.

Proposition 2.2.31. Sat_{Val} : Val → Sat is indeed a functor.

Proof. By definition of the categories Val and Sat it is trivial to verify that identities and compositions are preserved. Moreover, it is clear that ⊩_{S} is a well defined relation between sentences and models. We just show that h : Sat_{Val}(val_{1}) → Sat_{Val}(val_{2}) is indeed a Sat-morphism whenever h : val_{1} → val_{2} is a Val-morphism. For the purpose, let val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩ for k ∈ \{1, 2\}, φ ∈ L_{1} and m ∈ M_{2}. Then m⊩_{S_{2}} \overline{h}(φ) if and only if [[\overline{h}(φ)]]_{m} ∈ D_{m} if and only if [[φ]]_{\underline{h}(m)} ∈ D_{\underline{h}(m)} if and only if \underline{h}(m)⊩_{S_{1}} φ. ⊓ ⊔

In the sequel we shall use the notation Ent_{Val} = (Ent_{Sat} ∘ Sat_{Val}).

Corollary 2.2.32. The pair ⟨Val, Ent_{Val}⟩ constitutes a category of logic systems.

In the sequel, given a Val-based logic Log : Voc → Val, we shall denote Log(V) by ⟨L_{V} , M_{V} , S_{V} ⟩ and its induced entailment consequence by ⊨_{V} , for each vocabulary V.

It can be easily shown that Sat and Val are similar, as categories of logic systems. Indeed, a satisfaction system bears with it a two-valued structure with one distinguished value.

Definition 2.2.33. The maps

• 2(⟨L, M, ⊩⟩) = ⟨L, M, S^{⊩}⟩, where S^{⊩}(m) = ⟨{0, 1}, [[ ]]m^{⊩}, {1}⟩ with

[[φ]]m^{⊩} = \{ \begin{matrix}1 & if m⊩φ \\ 0 & otherwise & ; \end{matrix}

• 2(h : sat_{1} → sat_{2}) = h,

constitute the functor 2 : Sat → Val.

Proposition 2.2.34. 2 : Sat → Val is indeed a functor.

Proof. By definition of the categories Sat and Val it is trivial to verify that identities and compositions are preserved. Moreover, it is clear that 2(⟨L, M, ⊩⟩) is a well defined valuation system. We just show that h : 2(sat_{1}) → 2(sat_{2}) is indeed a Val-morphism whenever h : sat_{1} → sat_{2} is a Sat-morphism. For the purpose, let sat_{k} = ⟨L_{k}, M_{k}, ⊩_{k}⟩ for k ∈ \{1, 2\}, and m ∈ M_{2}. Clearly, both S^{⊩_{1}}(\underline{h}(m)) and S^{⊩_{2}}(m) have \{0, 1\} as set of truth values with \{1\} distinguished. Moreover, [[\overline{h}(φ)]]m^{⊩_{2}} = 1 if and only if m⊩_{2} \overline{h}(φ) if and only if \underline{h}(m)⊩_{1}φ if and only if [[φ]]\underline{h}(m)^{⊩_{1}} = 1. ⊓ ⊔

It is quite easy to show that although Sat_{Val} : Val → Sat and 2 : Sat → Val are both adjoint, none is coadjoint (none preserves the initial objects). This means that the functors are unrelated. However, Sat and Val are still quite similar as categories of logics systems. Clearly, (Sat_{Val} ∘ 2) = ID_{Sat} and therefore (Ent_{Val} ∘ 2) = Ent_{Sat}. In addition, Lang_{Val} is still coadjoint.

Proposition 2.2.35. Lang_{Val} : Val → Set is coadjoint. Namely, id_{L} : Lang_{Val}(val) → L is a Lang_{Val}-couniversal arrow for L, where val = ⟨L, Str(L), id_{Str(L)}⟩.

Proof. Let ⟨L′, M′, S′⟩ be a valuation system and \overline{h} : L′ → L a function. We have to show that there exists a unique map \underline{h} : M′ → Str(L) such that h : ⟨L′, M′, S′⟩ → val is a Val-morphism. Obviously, by definition of val we must have \underline{h}(m′) = S′(m′)|\overline{h}, for every m′ ∈ M′, and the result follows. ⊓ ⊔

Moreover, 2 : Sat → Val can be easily shown to be a cofibration.

Proposition 2.2.36. 2 : Sat → Val is a cofibration. Namely, h : sat → Sat_{Val}(val′) is a cocartesian lift of ⟨sat, h : 2(sat) → val′⟩.

Proof. Obviously, h : sat → Sat_{Val}(val′) is a Sat-morphism. Moreover, since h : 2(sat) → val′ is a Val-morphism, val′ is necessarily two-valued and 2(Sat_{Val}(val′)) = val′.

As for the universal property, assume that g : sat → sat′′ is Sat-morphism and f : val′ → 2(sat′′) a Val-morphism such that (f ∘ h) = g. Easily, then, f : Sat_{Val}(val′) → Sat_{Val}(2(sat′′)), Sat_{Val}(2(sat′′)) = sat′′ and f : Sat_{Val}(val′) → sat′′ establishes a Satmorphism. ⊓ ⊔

Furthermore, also Lang_{Sat} : Sat → Set and Lang_{Val} : Val → Set, the corresponding forgetful functors, are cofibrations.

Proposition 2.2.37. Lang_{Val} : Val → Set is a cofibration. Namely, h : ⟨L, M, S⟩ → ⟨L′, M′, S′⟩ is a cocartesian lift of ⟨⟨L, M, S⟩, \overline{h} : L → L′⟩, where:

• M′ is the class of all pairs ⟨m, ϑ⟩ such that ϑ : (L′ \ \overline{h}(L)) → T_{m} is a function and m ∈ M satisfies the following condition, for every φ_{1}, φ_{2} ∈ L:

– if \overline{h}(φ_{1}) = \overline{h}(φ_{2}) then [[φ_{1}]]_{m} = [[φ_{2}]]_{m};

• S′(m, ϑ) = ⟨T_{m}, [[ ]], D_{m}⟩ with:

– [[\overline{h}(φ)]] = [[φ]]_{m}, for every φ ∈ L;

– [[ψ]] = ϑ(ψ), for every ψ ∈ (L′ \ \overline{h}(L));

• \underline{h}(m, ϑ) = m.

Proof. Clearly, ⟨L′, M′, S′⟩ is a valuation sytem. In particular, S′ is well defined due to the conditions imposed to models in M′. Moreover, by definition, h : ⟨L, M, S⟩ → ⟨L′, M′, S′⟩ trivially establishes a Val-morphism.

To prove the universal property, assume that g : ⟨L, M, S⟩ → ⟨L′′, M′′, S′′⟩ is a Valmorphism and \overline{f} : L′ → L′′ a function such that (\overline{f} ∘ \overline{h}) = \overline{g}. We have to prove that there is a unique map \underline{f} : M′′ → M′ such that f : ⟨L′, M′, S′⟩ → ⟨L′′, M′′, S′′⟩ is a Valmorphism and (\underline{h} ∘ \underline{f}) = \underline{g}. Given m′′ ∈ M′′, we first note that if \overline{h}(φ_{1}) = \overline{h}(φ_{2}) then also \overline{g}(φ_{1}) = \overline{g}(φ_{2}) and thus [[φ_{1}]]_{\underline{g}(m′′)} = [[φ_{2}]]_{\underline{g}(m′′)}. Furthermore, if ψ ∈ (L′ \ \overline{h}(L)), we must have [[\overline{f}(ψ)]]_{m′′} = [[ψ]]_{\underline{f}(m′′)}. Thus, necessarily, \underline{f}(m′′) = ⟨\underline{g}(m′′), ([[ ]]_{m′′} ∘ \overline{f})⟩. ⊓ ⊔

Obviously, Lang_{Sat} = (Lang_{Val} ∘ 2) and the corollary below follows.

Corollary 2.2.38. Lang_{Sat} : Sat → Set is a cofibration. Namely, h : ⟨L, M, ⊩⟩ → ⟨L′, M′, ⊩′⟩ is a cocartesian lift of ⟨⟨L, M, ⊩⟩, \overline{h} : L → L′⟩, where:

• M′ is the class of all pairs ⟨m, Υ⟩ such that Υ ⊆ (L′ \ \overline{h}(L)) and m ∈ M satisfies the following condition, for every φ_{1}, φ_{2} ∈ L:

– if \overline{h}(φ_{1}) = \overline{h}(φ_{2}) then m⊩φ_{1} = m⊩φ_{2};

• ⊩′ ⊆ (M′ × L′) is such that:

– ⟨m, Υ⟩⊩′\overline{h}(φ) if and only if m⊩φ, for every φ ∈ L;

– ⟨m, Υ⟩⊩′ψ if and only if ψ ∈ Υ, for every ψ ∈ (L′ \ \overline{h}(L));

• \underline{h}(m, Υ) = m.

These two cofibrations shall be useful later on, for combination purposes.

2.2.2 Deduction: inference systems

We now turn to another example: logics based on finitary inference systems. Such systems are in the heart of most proof-theoretic approaches to logics. Their ubiquity and also their simplicity motivate our choice. In [SSC97a], for instance, such systems are designated by deduction systems.

Definition 2.2.39. An inference i = ⟨Prem(i), Conc(i)⟩ over a set L is said to be finitary if Prem(r) ∈ ℘_{fin}(L). We denote the set of all finitary inferences over L by Infer_{fin}(L).

Definition 2.2.40. An inference system is a pair inf = ⟨L, I⟩ where:

• L is a set;

• I ⊆ Infer_{fin}(L) (the finitary inferences).

Inference systems morphisms are quite trivial.

Definition 2.2.41. Let inf_{k} = ⟨L_{k}, I_{k}⟩ for k ∈ {1, 2} be inference systems. An inference systems morphism h : inf_{1} → inf_{2} is a function h : L_{1} → L_{2} such that the following condition holds, for every inference i ∈ I_{1}:

• h(i) = ⟨h(Prem(i)), h(Conc(i))⟩ ∈ I_{2}.

Definition 2.2.42. Inference systems and their morphisms constitute the category Inf, with identity id_{⟨L,I⟩} the identity function on L, and (g ∘ f) the usual composition of functions.

Proposition 2.2.43. Inf is indeed a category.

Proof. Due to the well known properties of compositions of maps, we just prove that id_{⟨L,I⟩} : ⟨L, I⟩ → ⟨L, I⟩ and (g∘f) : ⟨L_{1}, I_{1}⟩ → ⟨L_{3}, I_{3}⟩ are indeed Inf-morphisms, provided that f : ⟨L_{1}, I_{1}⟩ → ⟨L_{2}, I_{2}⟩ and g : ⟨L_{2}, I_{2}⟩ → ⟨L_{3}, I_{3}⟩ are.

With respect to identity, trivially, id_{L} maps each inference i ∈ I to itself.

Now, let i ∈ I_{1}. Since f : ⟨L_{1}, I_{1}⟩ → ⟨L_{2}, I_{2}⟩ is a morphism, we have f(i) ∈ I_{2}. But g : ⟨L_{2}, I_{2}⟩ → ⟨L_{3}, I_{3}⟩ is also a morphism, and thus g(f(i)) ∈ I_{3}. Therefore, (g ∘ f) : ⟨L_{1}, I_{1}⟩ → ⟨L_{3}, I_{3}⟩ is indeed a morphism. ⊓ ⊔

Deducibility can be defined, in general terms, for any given inference system. Naturally, the deducibility closure of a set is obtained by applying the construction in 2.1.5. Notably, since all the inferences involved have finitely many premises, this consequence turns out to be finitary.

Definition 2.2.44. Let inf = ⟨L, I⟩ be an inference system. The deducibility consequence operation induced by inf is ⊢_{inf} = ▷_{I}, i.e., for every Φ ⊆ L, Φ^{⊢_{inf}} is the least superset of Φ that is closed for all inferences in I.

For convenience, in this case, we use the notation csy_{inf} = ⟨L, ⊢_{inf} ⟩ and Ω_{inf} = Ω_{I}.

Proposition 2.2.45. Let inf = ⟨L, I⟩ be an inference system. Then, csy_{inf} is a finitary consequence system.

Proof. This result is a corollary of 2.1.26. In fact, it easily follows from the construction in 2.1.5 that ⊢_{inf} = ⊔_{L}{▷_{{i}} : i ∈ I}. Thus, it suffices to show that each ▷_{{i}}, the consequence generated by a single inference, is finitary. Let Φ ⊆ L and Φ ▷_{{i}} ψ. If ψ ∈ Φ obviously {ψ}▷_{{i}}ψ and we are done. Otherwise, it must be the case that ψ = Conc(i) and Prem(i) ⊆ Φ. But i is finitary, so, Prem(i) is finite and trivially Prem(i) ▷_{{i}} Conc(i). ⊓ ⊔

This construction can be easily extended to a functor.

Definition 2.2.46. The maps

• Ded_{Inf}(inf ) = csy_{inf} ;

• Ded_{Inf}(h : inf_{1} → inf_{2}) = h,

constitute the functor Ded_{Inf} : Inf → Csy_{fin}.

Proposition 2.2.47. Ded_{Inf} : Inf → Csy_{fin} is indeed a functor.

Proof. By definition of the categories Inf and Csy it is trivial to verify that identities and compositions are preserved. By 2.2.45, Ded_{Inf} is well defined on objects. Thus, we are left with proving that h : csy_{inf_{1}} → csy_{inf_{2}} is indeed a Csy-morphism whenever h : inf_{1} → inf_{2} is an Inf-morphism. For the purpose, let inf_{k} = ⟨L_{k}, I_{k}⟩ for k ∈ {1, 2} and Φ ⊆ L_{1}. We need to prove that h(Φ^{⊢_{inf_{1}}}) ⊆ h(Φ)^{⊢_{inf_{2}}}. Clearly, from 2.2.45, it is enough to show that h(Ω_{inf_{1}}^{n}(Φ)) ⊆ Ω_{inf_{2}}^{n}(h(Φ)) for every n ∈ IN. We proceed by induction on n.

• Base case: n = 0.

Trivially, Ω_{inf_{k}}^{0} is the identity for k ∈ {1, 2} and h(Φ) = h(Φ).

• Step: n > 0 and, by hypothesis, h(Ω_{inf_{1}}^{n−1}(Φ)) ⊆ Ω_{inf_{2}}^{n−1}(h(Φ)).

Clearly, Ω_{inf_{1}}^{n}(Φ) = Ω_{inf_{1}}(Ω_{inf_{1}}^{n−1}(Φ)) = (Ω_{inf_{1}}^{n−1}(Φ) ∪ {Conc(i) : i ∈ I_{1}, Prem(i) ⊆ Ω_{inf_{1}}^{n−1}(Φ)}). By the induction hypothesis, h(Ω_{inf_{1}}^{n−1}(Φ)) ⊆ Ω_{inf_{2}}^{n−1}(h(Φ)) ⊆ Ω_{inf_{2}}^{n}(h(Φ)).

On the other hand, if i ∈ I_{1} and Prem(i) ⊆ Ω_{inf_{1}}^{n−1}(Φ) then h(Prem(i)) ⊆ h(Ω_{inf_{1}}^{n−1}(Φ)). By induction hypothesis, then, h(Prem(i)) ⊆ Ω_{inf_{2}}^{n−1}(h(Φ)). But we know that h : inf_{1} → inf_{2} is an Inf-morphism. So, h(i) ∈ I_{2}, and thus h(Conc(i)) ∈ Ω_{inf_{2}}(Ω_{inf_{2}}^{n−1}(h(Φ))) = Ω_{inf_{2}}^{n}(h(Φ)). We conclude that h(Ω_{inf_{1}}^{n}(Φ)) ⊆ Ω_{inf_{2}}^{n}(h(Φ)).

Therefore, we conclude that h(Φ^{⊢_{inf_{1}}}) ⊆ h(Φ)^{⊢_{inf_{2}}}. ⊓ ⊔

Corollary 2.2.48. The pair ⟨Inf, Ded_{Inf}⟩ constitutes a category of finitary logic systems.

In the sequel, given an Inf-based logic Log : Voc → Inf, we shall denote Log(V) by ⟨L_{V} , I_{V} ⟩ and its induced deducibility consequence by ⊢_{V} , for each vocabulary V.

Our interest in satisfaction and inference system is not just as illustration of the concepts. We shall often consider logics based on them. Moreover, we shall also consider systems of logic that have both satisfaction and inference/valuation system formulations. As it should be clear, soundness is a minimal requirement in this kind of situation. We close this section with a necessary and sufficient condition for soundness in such context.

Definition 2.2.49. Let sat be a satisfaction system. An inference i ∈ Infer(L) is said to be sound with respect to sat if Prem(i)⊨_{sat} Conc(i).

Proposition 2.2.50. Let inf = ⟨L, I⟩ be an inference system and sat a satisfaction system. Then, the pair ⟨inf , sat⟩ is sound if and only if every inference in I is sound with respect to sat.

Proof. Assume that ⟨inf , sat⟩ is sound, and let i ∈ I. Clearly, by definition of ⊢_{inf} , Prem(i)⊢_{inf} Conc(i). Thus, by soundness, Prem(i)⊨_{sat} Conc(i) and i is sound with respect to sat.

Suppose now that every inference i ∈ I is sound with respect to sat, and let Φ ⊆ L. We show, by induction on n ∈ IN, that every Ω_{inf}^{n} (Φ) ⊆ Φ^{⊨_{sat}}.

• Base case: n = 0.

By definition, Ω_{inf}^{0} (Φ) = Φ. And extensiveness of ⊨_{sat} implies that Φ ⊆ Φ^{⊨_{sat}}.

• Step: n > 0 and the induction hypothesis asserts that Ω_{inf}^{n−1}(Φ) ⊆ Φ^{⊨_{sat}}. By definition, Ω_{inf}^{n} (Φ) = Ω_{inf} (Ω_{inf}^{n−1}(Φ)) = (Ω_{inf}^{n−1}(Φ) ∪ {Conc(i) : i ∈ I, Prem(i) ⊆ Ω_{inf}^{n−1}(Φ)}). Given the induction hypothesis, it remains to prove that {Conc(i) : i ∈ I, Prem(i) ⊆ Ω_{inf}^{n−1}(Φ)} ⊆ Φ^{⊨_{s}}. Let i ∈ I be such that Prem(i) ⊆ Ω_{inf}^{n−1}(Φ). Reusing the induction hypothesis we have that Prem(i) ⊆ Φ^{⊨_{sat}}. But i is sound with respect to sat, therefore Prem(i)⊨_{sat} Conc(i), and the monotonicity and idempotence of ⊨_{sat} together imply that Φ⊨_{sat} Conc(i).

Clearly then, Φ^{⊢_{inf}} ⊆ Φ^{⊨_{sat}}. ⊓ ⊔

This characterization can of course be used to study the soundness of Inf-based logics with respect to Sat-based logics over the same vocabulary and with the same language. Analogously, we can characterize soundness of Inf-based logics with respect to Val-based logics.

Definition 2.2.51. Let val be a satisfaction system. An inference i ∈ Infer(L) is said to be sound with respect to val if Prem(i)⊨_{val} Conc(i).

Proposition 2.2.52. Let inf = ⟨L, I⟩ be an inference system and val a satisfaction system. Then, the pair ⟨inf , val⟩ is sound if and only if every inference in I is sound with respect to val.

Proof. Immediate from 2.2.50, since Ent_{Val} = (Ent_{Sat} ∘ Sat_{Val}). ⊓ ⊔

We are also interested in some relevant properties of the forgetful functor Lang_{Inf}. Obviously, Lang_{Inf} = (Lang_{Csy_{fin}} ∘ Ded_{Inf}). To start with, Lang_{Inf} : Inf → Set is a cofibration.

Proposition 2.2.53. Lang_{Inf} : Inf → Set is a cofibration. Namely, h : ⟨L, I⟩ → ⟨L′, h(I)⟩ is a cocartesian lift of ⟨⟨L, I⟩, h : L → L′⟩.

Proof. Trivially, ⟨L′, h(I)⟩ is an inference sytem and h : ⟨L, I⟩ → ⟨L′, h(I)⟩ an Infmorphism. To prove the universal property, assume that g : ⟨L, I⟩ → ⟨L′′, I′′⟩ is an Inf-morphism and f : L′ → L′′ a function such that (f ∘ h) = g. Immediately, we have that f(h(I)) = g(I) ⊆ I′′. Thus, f : ⟨L′, h(I)⟩ → ⟨L′′, I′′⟩ is an Inf-morphism and the result follows. ⊓ ⊔

Recall that in 2.1.25, we have shown that Lang_{Csy_{fin}} is topological. Thus, among other properties, Lang_{Csy_{fin}} is also coadjoint and a cofibration. Thus, although Ded_{Inf} can be shown not to be a cofibration, it still keeps some nice properties.

Proposition 2.2.54. Ded_{Inf} : Inf → Csy_{fin} is coadjoint. Namely, id_{L} : Ded_{Inf}(inf ) → L is a Ded_{Inf}-couniversal arrow for the finitary consequence system ⟨L, ▷⟩, where inf = ⟨L, I⟩ and I = {i ∈ Infer_{fin}(L) : Prem(i)▷Conc(i)}. Moreover, Ded_{Inf} is cocartesian with respect to the cofibrations Lang_{Inf} and Lang_{Csy_{fin}}.

Proof. Trivially, since ▷ is finitary, we have ⊢_{inf} = ▷_{I} = ▷. Immediately, Φ^{▷} ∈ I(Φ). Moreover, if Ψ ∈ I(Φ) then Φ^{▷} ⊆ Ψ. In fact, if Φ ▷ υ, by finitariness, there exists Υ ∈ ℘_{fin}(Φ) such that Υ ▷ υ. But then, ⟨Υ, υ⟩ ∈ I and υ ∈ Φ since (Φ ∪ Ω_{I}(Ψ)) ⊆ Ω_{I}(Ψ).

As for the universal property, let inf′ = ⟨L′, I′⟩ be an inference system and g : ⟨L′, ⊢_{inf′} ⟩ → ⟨L, ▷⟩ be a Csy-morphism. Clearly, we have to prove that g : inf′ → inf is an Inf-morphism. But if i′ ∈ I′ then, obviously, Prem(i′)⊢_{inf′} Conc(i′) and g(Prem(i′)) ▷ g(Conc(i′)).

To prove that Ded_{Inf} is cocartesian, according to 2.2.53, we need to prove that the morphism h : ⟨L, ▷_{I}⟩ → ⟨L′, ▷_{h(I)}⟩ is a Lang_{Csy_{fin}}-cocartesian lift of ⟨⟨L, ▷_{I}⟩, h : L → L′⟩, where I ⊆ Infer_{fin}(L). For the purpose, let g : ⟨L, ▷_{I}⟩ → ⟨L′′, ▷′′⟩ be a Csy_{fin}- morphism and f : L′ → L′′ a function such that (f ∘ h) = g. All we need to show is that f : ⟨L′, ▷_{h(I)}⟩ → ⟨L′′, ▷′′⟩ is a Csy_{fin}-morphism. For the purpose, let Φ ⊆ L′.

Clearly, it is enough to show that f^{−1}(f(Φ)^{▷′′}) ∈ h(I)(Φ). Trivially, Φ ⊆ f^{−1}(f(Φ)) ⊆ f^{−1}(f(Φ)^{▷′′}) just by making use of the extensiveness of ▷′′. Assume now that i ∈ I and h(Prem(i)) ⊆ f^{−1}(f(Φ)^{▷′′}). Immediately, we have f(Φ) ▷′′f(h(Prem(i))) = g(Prem(i)). But Prem(i) ▷_{I} Conc(i), g is a morphism, and so we must have g(Prem(i)) ▷′′ g(Conc(i)). Therefore, it must be the case that f(Φ) ▷′′f(h(Conc(i))) = g(Conc(i)), i.e., h(Conc(i)) ∈ f^{−1}(f(Φ)^{▷′′}). We conclude that Ω_{I}(f^{−1}(f(Φ)^{▷′′})) ⊆ f^{−1}(f(Φ)^{▷′′}). ⊓ ⊔

Furthermore, we have the following corollary. Note that id_{L} : Lang_{Csy_{fin}}(⟨L, ⊤⟩) → L is the obvious Lang_{Csy_{fin}}-couniversal arrow for L.

Corollary 2.2.55. Lang_{Inf} : Inf → Set is coadjoint. Namely, id_{L} : Lang_{Inf}(inf ) → L is a Lang_{Inf}-couniversal arrow for L, where inf = ⟨L, Infer_{fin}(L)⟩.

2.3 Examples

We have already set the focus of our work on propositional based logics. Still, at this level, we shall also dwell on logics with terms such as equational and predicate logic. Although there is no great novelty in the notions we have introduced in this Chapter, it is worth trying to see what some familiar logics look like in this setting. For obvious reasons, we start with classical propositional logic.

Example 2.3.1. Classical propositional logic system. Let P be a set of propositional symbols. The language of CPL over P consists of the set L_{cpl}^{P} of sentences defined inductively by:

• p ∈ L_{cpl}^{P} if p ∈ P;

• (¬ φ), (φ ⇒ ψ), (φ ∧ ψ), (φ ∨ ψ) ∈ L_{cpl}^{P} if φ, ψ ∈ L_{cpl}^{P}.

As for many other interesting logics, there are essentially two ways of defining the intended consequence system csy_{cpl}^{P} = ⟨L_{cpl}^{P}, ▷_{cpl}^{P}⟩ of CPL: either model-theoretically, using the two-valued Boolean algebra, or proof-theoretically, by means of the adequate inference system. In this case, it is well known that exactly the same consequence operation is obtained by both means: soundness and completeness theorems for CPL are very well known (see for instance [Men79]).

The semantics of CPL considers a class of models M_{cpl}^{P} consisting of all possible binary {0, 1}-valuations of the propositional symbols, i.e., M_{cpl}^{P} = {0, 1}^{P}. By convention, 0 stands for falsity and 1 for truth. Each valuation ϑ ∈ M_{cpl}^{P} is then inductively extended to a function ϑ^{∗} : L_{cpl}^{P} → {0, 1} by requiring that:

• ϑ^{∗}(p) = ϑ(p);

• ϑ^{∗}((¬ φ)) = 1 − ϑ^{∗}(φ);

• ϑ^{∗}((φ ⇒ ψ)) = 1 + ϑ^{∗}(φ)ϑ^{∗}(ψ) − ϑ^{∗}(φ);

• ϑ^{∗}((φ ∧ ψ)) = ϑ^{∗}(φ)ϑ^{∗}(ψ);

• ϑ^{∗}((φ ∨ ψ)) = ϑ^{∗}(φ) + ϑ^{∗}(ψ) − ϑ^{∗}(φ)ϑ^{∗}(ψ). The valuation system of CPL over P is val_{cpl}^{P} = ⟨L_{cpl}^{P}, M_{cpl}^{P}, S_{cpl}^{P}⟩ where, for each ϑ ∈ M_{cpl}^{P},

S_{cpl}^{P}(ϑ) = ⟨{0, 1}, ϑ^{∗}, {1}⟩.

The inference system for CPL consists of the following family Ax_{cpl}^{P} of axioms: Ax01. (φ ⇒ (ψ ⇒ φ)); Ax02. ((φ ⇒ (ψ ⇒ υ)) ⇒ ((φ ⇒ ψ) ⇒ (φ ⇒ υ))); Ax03. ((φ ∧ ψ) ⇒ φ); Ax04. ((φ ∧ ψ) ⇒ ψ); Ax05. ((φ ⇒ ψ) ⇒ ((φ ⇒ υ) ⇒ (φ ⇒ (ψ ∧ υ)))); Ax06. (φ ⇒ (φ ∨ ψ)); Ax07. (ψ ⇒ (φ ∨ ψ)); Ax08. ((φ ⇒ υ) ⇒ ((ψ ⇒ υ) ⇒ ((φ ∨ ψ) ⇒ υ))); Ax09. ((φ ⇒ (¬ ψ)) ⇒ (ψ ⇒ (¬ φ))); Ax10. ((¬(φ ⇒ φ)) ⇒ ψ); Ax11. ((¬(¬ φ)) ⇒ φ),

plus the rule of modus ponens:

MP. from φ and (φ ⇒ ψ) conclude ψ, where φ, ψ and υ range over the whole language L_{cpl}^{P}. Thus, the inference system of CPL over P is the pair inf _{cpl}^{P} = ⟨L_{cpl}^{P}, I_{cpl}^{P}⟩, where

I_{cpl}^{P} = ({⟨∅, φ⟩ : φ ∈ Ax_{cpl}^{P}} ∪ {⟨{φ, (φ ⇒ ψ)}, ψ⟩ : φ, ψ ∈ L_{cpl}^{P}}).

We identify ▷_{cpl}^{P} = ⊨_{cpl}^{P} = ⊢_{cpl}^{P}. For the sake of illustration, we shall also prove that the logic system of classical propo-

sitional logic is equivalent, for instance, to its complete expressive fragment with only negation and implication. Example 2.3.2. Complete expressiveness of {¬, ⇒}. Let P be a set of propositional symbols. The language of the {¬, ⇒}-fragment of CPL over P consists of the set L_{cplf}^{P} ⊆ L_{cpl}^{P} of sentences defined inductively just by:

• p ∈ L_{cplf}^{P} if p ∈ P;

• (¬ φ), (φ ⇒ ψ) ∈ L_{cplf}^{P} if φ, ψ ∈ L_{cplf}^{P}.

The valuation system of the {¬, ⇒}-fragment of CPL over P is given by val_{cplf}^{P} = ⟨L_{cplf}^{P} , M_{cplf}^{P} , S_{cplf}^{P} ⟩ where M_{cplf}^{P} = M_{cpl}^{P}, and for each valuation ϑ, S_{cplf}^{P} (ϑ) is now the restriction of ϑ^{∗} to L_{cplf}^{P}.

We now show that val_{cplf}^{P} and val_{cpl}^{P} induce (via Ent_{Val}) equivalent consequence systems, that is ⟨L_{cplf}^{P} , ⊨_{cplf}^{P} ⟩ ≡ ⟨L_{cpl}^{P}, ⊨_{cpl}^{P} ⟩.

Let f : L_{cplf}^{P} → L_{cpl}^{P} be the inclusion of languages, i.e., f(φ) = φ for every φ ∈ L_{cplf}^{P}. On the other hand, let g : L_{cpl}^{P} → L_{cplf}^{P} be defined, inductively, by:

• g(p) = p;

• g((¬ φ)) = (¬ g(φ));

• g((φ ⇒ ψ)) = (g(φ) ⇒ g(ψ));

• g((φ ∧ ψ)) = (¬(g(φ) ⇒ (¬ g(ψ))));

• g((φ ∨ ψ)) = ((¬ g(φ)) ⇒ g(ψ)).

Easily, by using 2.1.44, it turns out that it suffices to show that ϑ^{∗}(g(φ)) = ϑ^{∗}(φ) for every ϑ ∈ {0, 1}^{P} and φ ∈ L_{cpl}^{P}. A simple inductive argument does the trick. In fact, it all amounts to verifying that, for every φ, ψ ∈ L_{cpl}^{P}:

• ϑ^{∗}(g((φ ∧ ψ))) = ϑ^{∗}((¬(g(φ) ⇒ (¬ g(ψ))))) = 1 − (1 + ϑ^{∗}(g(φ))(1 − ϑ^{∗}(g(ψ))) − ϑ^{∗}(g(φ)))

= 1 − (1 + ϑ^{∗}(g(φ)) − ϑ^{∗}(g(φ))ϑ^{∗}(g(ψ)) − ϑ^{∗}(g(φ)))
= 1 − (1 − ϑ^{∗}(g(φ))ϑ^{∗}(g(ψ)))
= ϑ^{∗}(g(φ))ϑ^{∗}(g(ψ))
= ϑ^{∗}((g(φ) ∧ g(ψ)));

• ϑ^{∗}(g((φ ∨ ψ))) = ϑ^{∗}(((¬ g(φ)) ⇒ g(ψ))) = 1 + (1 − ϑ^{∗}(g(φ)))ϑ^{∗}(g(ψ)) − (1 − ϑ^{∗}(g(φ)))

= 1 + ϑ^{∗}(g(ψ)) − ϑ^{∗}(g(φ))ϑ^{∗}(g(ψ)) − 1 + ϑ^{∗}(g(φ))
= ϑ^{∗}(g(φ)) + ϑ^{∗}(g(ψ)) − ϑ^{∗}(g(φ))ϑ^{∗}(g(ψ))
= ϑ^{∗}((g(φ) ∨ g(ψ))).

The inference system inf _{cplf}^{P} = ⟨L_{cplf}^{P} , I_{cplf}^{P} ⟩ for the {¬, ⇒}-fragment of CPL over P is built similarly to inf _{cpl}^{P} but considering only axioms Ax01 , Ax02 and Ax09 -Ax11 .

A proof of the fact that also inf _{cplf}^{P} and inf _{cpl}^{P} induce (now via Ded_{Inf}) equivalent consequence systems, can also be easily obtained. Again according to 2.1.44, and using the same translations f and g, it is straightforward to observe that a proof of ⟨L_{cplf}^{P} , ⊢_{cplf}^{P} ⟩ ≡ ⟨L_{cpl}^{P}, ⊢_{cpl}^{P} ⟩ essentially ammounts to showing that, for every φ ∈ L_{cpl}^{P}, φ and g(φ) are deducible from each other in ⊢_{cpl}^{P}, and that the g translation of axioms Ax03 -Ax08 yields ⊢_{cplf}^{P} theorems. Although quite tedious, all these proofs are quite easy and most of them can be easily found in logic textbooks as [Men79].

We illustrate the construction of logics from logic systems using the same example.

Example 2.3.3. Classical propositional logic. Clearly, the set P of propositional symbols in 2.3.1 has only played the role of a parameter.

Thus, we can easily define CPL as a functor CPL : Set → Csy obtained by CPL = (Ent_{Sat}∘Sat_{Val}∘CPLVal) = (Ded_{Inf}∘CPLInf ), where CPLVal : Set → Val and CPLInf : Set → Inf are defined as follows:

• CPLVal(P) = val_{cpl}^{P} and CPLInf (P) = inf _{cpl}^{P};

• CPLVal(h : P_{1} → P_{2}) with \overline{CPLVal(h)} = CPLInf (h) = ĥ and \underline{CPLVal(h)}(ϑ) = (ϑ ∘ h), where ĥ : L_{cpl}^{P_{1}} → L_{cpl}^{P_{2}} translates the propositional symbols occurring in sentences as follows: – ĥ(p) = h(p);

– ĥ((¬ φ)) = (¬ ĥ(φ));

– ĥ((φ ⇒ ψ)) = (ĥ(φ) ⇒ ĥ(ψ)). In order to check that CPLVal and CPLInf are indeed functors we have to guarantee

that CPLVal(h) : val_{cpl}^{P_{1}} → val_{cpl}^{P_{2}} is indeed a Val-morphism and CPLInf (h) : inf _{cpl}^{P_{1}} → inf _{cpl}^{P_{2}} is indeed an Inf-morphism.

The first fact is a trivial consequence of the fact that (ϑ ∘ h)^{∗}(φ) = ϑ^{∗}(ĥ(φ)) for all φ ∈ L_{cpl}^{P_{1}}. The second just amounts to checking that ĥ(i) ∈ I_{cpl}^{P_{2}} whenever i ∈ I_{cpl}^{P_{1}}. It is straightforward consequence of the definition of ĥ.

Another well known equivalent possibility is to consider the algebraic semantics for CPL, using Boolean algebras. Example 2.3.4. Algebraic semantics for CPL. The algebraic semantics of CPL considers a class of models M_{cpla} consisting of all pairs ⟨B, ϑ⟩ with B = ⟨B, ≤⟩ a Boolean algebra and ϑ : P → B a function. Recall from [Bir67] that B is a Boolean algebra provided that it is a partial order and further satisfies the following properties, for every x, y, z ∈ B:

• both a least upper-bound x ∨ y and a greatest lower-bound x ∧ y exist;

• ∧ distributes over ∨, i.e., x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z);

• top and bottom elements ⊤, ⊥ ∈ B exist, with x ≤ ⊤ and ⊥ ≤ x;

• a complement x′ exists such that x ∨ x′ = ⊤ and x ∧ x′ = ⊥. Given ⟨B, ϑ⟩ ∈ M_{cpla}, ϑ is extended to a function ϑ^{∗} : L_{cpl} → B by requiring that:

• ϑ^{∗}(p) = ϑ(p);

• ϑ^{∗}((¬ φ)) = ϑ^{∗}(φ)′;

• ϑ^{∗}((φ ⇒ ψ)) = ϑ^{∗}(φ)′ ∨ ϑ^{∗}(ψ) = (ϑ^{∗}(φ) ∧ ϑ^{∗}(ψ)′)′.

• ϑ^{∗}((φ ∧ ψ)) = ϑ^{∗}(φ) ∧ ϑ^{∗}(ψ);

• ϑ^{∗}((φ ∨ ψ)) = ϑ^{∗}(φ) ∨ ϑ^{∗}(ψ).

The algebraic valuation system of CPL over P is val_{cpla}^{P} = ⟨L_{cpl}^{P}, M_{cpla}^{P}, S_{cpla}^{P}⟩ where, for each ⟨B, ϑ⟩ ∈ M_{cpla}^{P},

S_{cpla}^{P}(⟨B, ϑ⟩) = ⟨B, ϑ^{∗}, {⊤}⟩.

Note that the usual total order on {0, 1} constitutes precisely the two-valued Boolean algebra we have used in 2.3.1. The above definitions obviously generalize the ones we had previously. However, an application of Birkhoff’s theorem [Bir67] shows that, still, ⊨_{cpla}^{P} = ▷_{cpl}^{P}.

Let us now take a look at propositional intuitionistic logic.

Example 2.3.5. Intuitionistic propositional logic (IPL). Let P be a set of propositional symbols. The language of IPL over P coincides with the language of CPL, that is L_{ipl}^{P} = L_{cpl}^{P}.

As in the classical case, there are two equivalent ways of defining the intended consequence system ⟨L_{ipl}^{P}, ▷_{ipl}^{P}⟩ of IPL. Again, it is well known that exactly the same consequence operation is obtained by both means: soundness and completeness theorems for IPL are very well known since Heyting [Wój88].

The semantics of IPL considers a class of models M_{ipl}^{P} consisting of all pairs ⟨H, ϑ⟩ with H = ⟨H, ≤⟩ a Heyting algebra and ϑ : P → H a function. Recall from [Bir67] that H is a Heyting algebra if it is a bounded lattice with relative pseudo-complementation, that is, provided that it is a partial order and further satisfies the following properties, for every x, y, z ∈ H:

• both a least upper-bound x ∨ y and a greatest lower-bound x ∧ y exist;

• a bottom element ⊥ ∈ B exists, with ⊥ ≤ x;

• a relative pseudo-complement x → y of x to y exists such that (x → y) ∧ x ≤ y, and z ∧ x ≤ y implies z ≤ x → y.

Given ⟨H, ϑ⟩ ∈ M_{ipl}^{P}, ϑ can be extended to a function ϑ^{∗} : L_{ipl} → H by defining:

• ϑ^{∗}(p) = ϑ(p);

• ϑ^{∗}((¬ φ)) = ϑ^{∗}(φ) → ⊥;

• ϑ^{∗}((φ ⇒ ψ)) = ϑ^{∗}(φ) → ϑ^{∗}(ψ);

• ϑ^{∗}((φ ∧ ψ)) = ϑ^{∗}(φ) ∧ ϑ^{∗}(ψ);

• ϑ^{∗}((φ ∨ ψ)) = ϑ^{∗}(φ) ∨ ϑ^{∗}(ψ).

The valuation system of IPL over P is val_{ipl}^{P} = ⟨L_{ipl}^{P}, M_{ipl}^{P}, S_{ipl}^{P}⟩ where, for each ⟨H, ϑ⟩ ∈ M_{ipl},

S_{ipl}^{P}(⟨H, ϑ⟩) = ⟨H, ϑ^{∗}, {⊥ → ⊥}⟩.

Recall that a Boolean algebra B is always a Heyting algebra, with x → y = x′ ∨ y. A Heyting algebra H is Boolean if and only if (x → ⊥) → ⊥ = x for every x ∈ H. Besides ⊤ = ⊥ → ⊥, we can then define x′ = x → ⊥.

The inference system inf _{ipl}^{P} = ⟨L_{ipl}^{P}, I_{ipl}^{P}⟩ of IPL over P is built similarly to inf _{cpl}^{P} in 2.3.1, but leaving out Ax11 .

Soundness and completeness results again allow us to identify ▷_{ipl}^{P} = ⊨_{ipl}^{P} = ⊢_{ipl}^{P}.

Another example we would like to cover is propositional modal logic. Example 2.3.6. Propositional normal modal logic (K). The language of K over a given set P of propositional symbols corresponds to the set of sentences L_{K}^{P} inductively defined as follows:

• p ∈ L_{K}^{P} if p ∈ P;

• (¬ φ), (□φ), (φ ⇒ ψ), (φ ∧ ψ), (φ ∨ ψ) ∈ L_{K}^{P} if φ, ψ ∈ L_{K}^{P}.

The traditional semantic structures of normal modal logic are Kripke models (see for instance [HC96]) m = ⟨W, R, ϑ⟩ over P, where:

• W is a set (of worlds);

• R ⊆ W × W is a relation (accessibility);

• ϑ : P → ℘(W) is a (valuation) function.

Recall that pairs ⟨W, R⟩ are traditionally called Kripke frames. As expected, ϑ(p) represents the set of worlds where p holds, for each p ∈ P. The accessibility relation R is used to give meaning to the modality □. Traditionally, one starts by defining what it means for a sentence to be satisfied at a world w in m, inductively, as follows:

• m⊩_{w}^{P} p if w ∈ ϑ(p);

• m⊩_{w}^{P} ( ¬ φ) if not m⊩_{w}^{P} φ;

• m⊩_{w}^{P} (□φ) if m⊩_{w′}^{P} φ for every w′ ∈ W such that wRw′;

• m⊩_{w}^{P} (φ ⇒ ψ) if not m⊩_{w}^{P} φ or m⊩_{w}^{P} ψ;

• m⊩_{w}^{P} (φ ∧ ψ) if m⊩_{w}^{P} φ and m⊩_{w}^{P} ψ;

• m⊩_{w}^{P} (φ ∨ ψ) if m⊩_{w}^{P} φ or m⊩_{w}^{P} ψ.

One can then define what it means for a sentence φ ∈ L_{K}^{P} to be satisfied by m:

• m⊩_{K}^{P} φ if m⊩_{w}^{P} φ for every w ∈ W.

The satisfaction system sat_{K}^{P} = ⟨L_{K}^{P}, M_{K}^{P}, ⊩_{K}^{P} ⟩ thus defined, with M_{K}^{P} the class of all Kripke models over P, is then used as a way of presenting the consequence system ⟨L_{K}^{P}, ⊨_{K}^{P} ⟩.

The usual inference system inf _{K}^{P} = ⟨L_{K}^{P}, I_{K}^{P}⟩ for K contains, for φ, ψ, υ ∈ L_{K}^{P}, all the axioms and rules Ax01 -Ax11 and MP from 2.3.1, plus the axiom of normality:

□-Norm. ((□(φ ⇒ ψ)) ⇒ ((□φ) ⇒ (□ψ))), and the rule of necessitation:

□-Nec. from φ conclude (□φ).

Please note that most of the “classical” literature on modal logic assumes a perspective which is slightly different from ours. Usually, a modal logic is identified with the set of its theorems rather than with a consequence operation. In that sense, a normal modal logic can be identified with a ⊢_{K}^{P} theory. For instance, while the basic system K coincides with ∅^{⊢_{K}^{P}}, the system S4 corresponds to (⋃_{φ∈L_{K}^{P}}{((□φ) ⇒ φ), ((□φ) ⇒ (□(□φ)))})^{⊢_{K}^{P}}. Usually, also, completeness results are expressed in terms of frames rather than models. Still, a rephrasal of well-known soundness and completeness results allows us to conclude that ⊢_{K}^{P} = ⊨_{K}^{P}. More details on this shall be supplied in example 3.3.3 at the end of the next Chapter. In our setting, for instance, the usual system S4 would not just amount to add axioms corresponding to all instances of T and 4 (the schemas above). Semantically, we would restrict the class of models to just those Kripke models whose accessibility relation is reflexive and transitive. This would still yield a strong completeness result [HC96], but which does not follow directly from the completeness of K as stated above. In fact, there are many models satisfying T and 4 whose accessibility relation is not a preorder. Another example would be the system KL, which encompasses Löb’s axiom L: ((□((□φ)⇒φ))⇒(□φ)). Model theoretically, we would restrict the class of models to just those corresponding to finite trees. This system is well known not to be complete, but only finitely complete. Still, in K, we have all models satisfying the L axioms, which may not correspond to finite trees, and therefore the corresponding semantic theory is obtained by the inference system.

Another system of logic that we would like to discuss is linear temporal logic. Its applications, namely in the area of computer science, are numerous and significant [Pnu77, Ser80, Eme90, MP91, Wol95]. Herein we shall present a simple, discrete and future directed version due to [GPSS80].

Example 2.3.7. Propositional linear temporal logic (PTL). The language of PTL over a given set P of propositional symbols corresponds to the set of sentences L_{ptl}^{P} inductively defined as follows:

• p ∈ L_{ptl}^{P} if p ∈ P;

• (¬ φ), (G φ), (X φ), (φ ⇒ ψ), (φ ∧ ψ), (φ ∨ ψ), (φ U ψ) ∈ L_{ptl}^{P} if φ, ψ ∈ L_{ptl}^{P}.

The simplest semantic structures of discrete temporal logic are based over IN equiped with its usual ordering. That is, M_{ptl}^{P} = ℘(IN)^{P}. Traditionally, one starts by defining what it means for a sentence to be satisfied at an instant n ∈ IN by a model ϑ ∈ M_{ptl}^{P}, inductively, as follows:

• ϑ⊩_{n}^{P} p if n ∈ ϑ(p);

• ϑ⊩_{n}^{P} ( ¬ φ) if not ϑ⊩_{n}^{P} φ;

• ϑ⊩_{n}^{P} ( G φ) if ϑ⊩_{m}^{P}φ for every m ≥ n;

• ϑ⊩_{n}^{P} ( X φ) if ϑ⊩_{n+1}^{P}φ;

• ϑ⊩_{n}^{P} (φ ⇒ ψ) if not ϑ⊩_{n}^{P} φ or ϑ⊩_{n}^{P} ψ; • ϑ⊩_{n}^{P} (φ ∧ ψ) if ϑ⊩_{n}^{P} φ and ϑ⊩_{n}^{P} ψ;

• ϑ⊩_{n}^{P} (φ ∨ ψ) if ϑ⊩_{n}^{P} φ or ϑ⊩_{n}^{P} ψ;

• ϑ⊩_{n}^{P} (φ U ψ) if there exists m ≥ n with ϑ⊩_{m}^{P}ψ and ϑ⊩_{k}^{P} φ for all n ≤ k < m.

One can then define what it means for a sentence φ ∈ L_{ptl}^{P} to be satisfied by ϑ:

• ϑ⊩_{ptl}^{P} φ if ϑ⊩_{n}^{P} φ for every n ∈ IN.

The satisfaction system sat_{ptl}^{P} = ⟨L_{ptl}^{P}, M_{ptl}^{P}, ⊩_{ptl}^{P} ⟩ thus defined is then used as a way of presenting the consequence system ⟨L_{ptl}^{P}, ⊨_{ptl}^{P} ⟩.

The usual inference system inf _{ptl}^{P} = ⟨L_{ptl}^{P}, I_{ptl}^{P}⟩ for PTL contains, for φ, ψ, υ ∈ L_{ptl}^{P}, all the axioms and rules Ax01 -Ax11 and MP of classical logic, plus the following axioms:

G-Norm. ((G(φ ⇒ ψ)) ⇒ ((G φ) ⇒ (G ψ))); X-Norm. ((X(φ ⇒ ψ)) ⇒ ((X φ) ⇒ (X ψ))); X1. ((X(¬ φ)) ⇒ (¬(X φ))); X2. ((¬(X φ)) ⇒ (X(¬ φ))); G1. ((G φ) ⇒ (φ ∧ (X(G φ)))); G2. ((G(φ ⇒ (X φ))) ⇒ (φ ⇒ (G φ))); U1. ((G(¬ ψ)) ⇒ (¬(φ U ψ))); U2. ((φ U ψ) ⇒ (ψ ∨ (φ ∧ (X(φ U ψ))))); U3. ((ψ ∨ (φ ∧ (X(φ U ψ)))) ⇒ (φ U ψ)),

and the two rules of necessitation:

G-Nec. from φ conclude (G φ);

X-Nec. from φ conclude (X φ). It is well known that this inference system is sound and finitely complete with respect to the above semantics [Gol92]. Strong completeness fails because semantic entailment consequence is not finitary. For instance, {(φ ⇒ X^{n} ψ) : n ∈ IN}⊨_{ptl}^{P} (φ ⇒ (G ψ)) but no finite set {(φ⇒X^{n_{1}} ψ), …, (φ⇒X^{n_{m}} ψ)} does. A more detailed analysis shall be postponed to the end of next Chapter. Although X-Nec is usually considered for the sake of symmetry with G (both are modalities), it is in fact well known to be derivable from the rest.

Next we present an example of a paraconsistent logic due to [dC74]. Example 2.3.8. Paraconsistent logic (C_{1}). Let P be a set of propositional symbols. The language of paraconsistent logic C_{1} over P consists of the set L_{c1}^{P} = L_{cpl}^{P} of sentences. As we shall see, in this logic, negation has a very special meaning that prevents contradiction from raising inconsistency. A special abbreviation is often used: φ^{∘} stands for (¬(φ ∧ (¬ φ))) and means essentailly that φ behaves classically.

The semantics of C_{1} considers a class of models M_{c1}^{P} consisting of all non-homorphic bi-valuations ϑ : L_{c1}^{P} → {0, 1} satisfying:

• ϑ((φ ∧ ψ)) = 1 if and only if ϑ(φ) = 1 and ϑ(ψ) = 1;

• ϑ((φ ∨ ψ)) = 1 if and only if ϑ(φ) = 1 or ϑ(ψ) = 1;

• ϑ((φ ⇒ ψ)) = 1 if and only if ϑ(φ) = 0 or ϑ(ψ) = 1;

• if ϑ(φ) = 0 then ϑ((¬ φ)) = 1;

• if ϑ((¬(¬ φ))) = 1 then ϑ(φ) = 1;

• if ϑ(φ^{∘}) = ϑ((ψ ⇒ φ)) = ϑ((ψ ⇒ (¬ φ))) = 1 then ϑ(ψ) = 1;

• if ϑ(φ^{∘}) = φ(ψ^{∘}) = 1 then ϑ((φ ∧ ψ)^{∘}) = ϑ((φ ∨ ψ)^{∘}) = ϑ((φ ⇒ ψ)^{∘}) = 1.

The satisfaction system sat_{c1}^{P} = ⟨L_{c1}^{P} , M_{c1}^{P} , ⊩_{c1}^{P} ⟩ is then defined by ϑ⊩_{c1}^{P} φ provided that ϑ(φ) = 1.

It is known [dCeEA77] that a strong complete inference system for C_{1}, inf _{c1}^{P} = ⟨L_{c1}^{P} , I_{c1}^{P} ⟩, can be obtained by taking Ax01 -Ax08 and Ax11 plus MP from 2.3.1 and adding the following three schemas:

• (φ ∨ (¬ φ));

• (φ^{∘} ⇒ ((ψ ⇒ φ) ⇒ ((ψ ⇒ (¬ φ)) ⇒ (¬ ψ))));

• ((φ^{∘} ∧ ψ^{∘}) ⇒ ((φ ∧ ψ)^{∘} ∧ ((φ ∨ ψ)^{∘} ∧ (φ ⇒ ψ)^{∘}))).

A last example, to illustrate the definition of less “classical” systems, concerns the semantics of Lukasiewicz three-valued logic.

Example 2.3.9. Lukasiewicz three-valued logic (L_{3}). Let P be a set of propositional symbols. The language of Lukasiewicz three-valued logic over P consists of the set L_{l3}^{P} = L_{cpl}^{P} of sentences.

The semantics of L_{3} considers a class of models M_{l3}^{P} consisting of all possible threevalued valuations of the propositional symbols, i.e., M_{l3}^{P} = \{0, \frac{1}{2}, 1\}^{P}. By convention, the value \frac{1}{2} designates uncertainty.

Each valuation ϑ ∈ M_{l3}^{P} is then inductively extended to a function ϑ^{∗} : L_{l3}^{P} → \{0, \frac{1}{2}, 1\} by requiring that:

• ϑ^{∗}(p) = ϑ(p);

• ϑ^{∗}((¬ φ)) = 1 − ϑ^{∗}(φ);

• ϑ^{∗}((φ ⇒ ψ)) = min(1, 1 + ϑ^{∗}(ψ) − ϑ^{∗}(φ));

• ϑ^{∗}((φ ∧ ψ)) = min(ϑ^{∗}(φ), ϑ^{∗}(ψ));

• ϑ^{∗}((φ ∨ ψ)) = max(ϑ^{∗}(φ), ϑ^{∗}(ψ)).

The satisfaction system sat_{l3}^{P} = ⟨L_{l3}^{P} , M_{l3}^{P} , ⊩_{l3}^{P} ⟩ is then defined by ϑ⊩_{l3}^{P} φ provided that ϑ^{∗}(φ) = 1.

It is known that a strong complete inference system for L_{3} can be obtained by taking all theorems of ⊨_{l3}^{P} as axioms and further considering the rule MP. That is inf _{l3}^{P} = ⟨L_{l3}^{P} , I_{l3}^{P} ⟩ with

I_{l3}^{P} = ({⟨∅, φ⟩ : ⊨_{l3}^{P} φ} ∪ {⟨{φ, (φ ⇒ ψ)}, ψ⟩ : φ, ψ ∈ L_{l3}^{P} }).

The fact that ⊢_{l3}^{P} = ⊨_{l3}^{P} is a consequence of the fact that the following deduction-like theorem holds for ⊨_{l3}^{P}:

• (Φ ∪ {φ})⊨_{l3}^{P} ψ if and only if Φ⊨_{l3}^{P} (φ ⇒ (φ ⇒ ψ)).

For more details, we refer the reader to [Wój88].

Let us now take a brief look at two very well known examples of non-propositional based logics.

Example 2.3.10. (Unsorted) equational logic (EQL). Let F be an (unsorted) equational signature, i.e., a IN-indexed family of sets of operation symbols F = {F_{n}}_{n∈IN}, and X a set (of variables). The term language of EQL over F and X consists of the set Term_{F} (X) defined inductively by:

• x ∈ Term_{F} (X) if x ∈ X;

• f(t_{1}, …, t_{n}) ∈ Term_{F} (X) if f ∈ F_{n} and t_{k} ∈ Term_{F} (X) for 1 ≤ k ≤ n.

The language of EQL over F and X is the set L_{eql}^{F}(X) of all equations, as follows:

• (t_{1} = t_{2}), with t_{1}, t_{2} ∈ Term_{F} (X).

The semantics of EQL considers M_{eql}^{F} to be the class of all F -algebras T = ⟨T, ^{T} ⟩, where T is a set and, for every n ∈ IN and f ∈ F_{n}, \underline{f} _{n}^{T} : T^{n} → T.

Traditionally, one starts by defining satisfaction of an equation (t_{1} = t_{2}) by an algebra T given an assignment to X in T , i.e., a function ρ : X → T. We shall use Asn(T , X) to denote the set of all assignments to X in T . Clearly, terms can be freely interpreted given T and ρ. Inductively, [[ ]]_{ρ}^{T} : Term_{F} (X) → T, is defined by:

• [[x]]_{ρ}^{T} = ρ(x);

• [[f(t_{1}, …, t_{n})]]_{ρ}^{T} = \underline{f} _{n}^{T} ([[t_{1}]]_{ρ}^{T} , …, [[t_{n}]]_{ρ}^{T}).

Obviously, T and ρ satisfy an equation (t_{1} = t_{2}) ∈ L_{eql}^{F}(X), denoted T ⊩_{ρ}^{F} (t_{1} = t_{2}) if:

• [[t_{1}]]_{ρ}^{T} = [[t_{2}]]_{ρ}^{T}.

The satisfaction system sat_{eql}^{⟨F,X⟩} = ⟨L_{eql}^{F}(X), M_{eql}^{F}, ⊩_{eql}^{⟨F,X⟩}⟩ is defined by requiring that:

• T ⊩_{eql}^{⟨F,X⟩}(t_{1} = t_{2}) if and only if T ⊩_{ρ}^{F} (t_{1} = t_{2}) for every ρ ∈ Asn(T , X).

The inference system for EQL consists of all the instances of the following rules I_{eql}^{⟨F,X⟩}:

Reflex. ⟨∅, (t = t)⟩;

Symm. ⟨{(t_{1} = t_{2})}, (t_{2} = t_{1})⟩;

Trans. ⟨{(t_{1} = t_{2}), (t_{2} = t_{3})}, (t_{1} = t_{3})⟩;

Congr. ⟨{(s_{i} = r_{i}) : 1 ≤ i ≤ n}, (f(s_{1}, …, s_{n}) = f(r_{1}, …, r_{n}))⟩;

Subst. ⟨{(t_{1} = t_{2})}, (t_{1}σ = t_{2}σ)⟩, where t, t_{1}, t_{2}, t_{3}, s_{1}, …, s_{n}, r_{1}, …, r_{n} range over the whole term language Term_{Σ}(X), n ranges over IN, f over Σ_{n} and σ over the set of all substitution functions from X to Term_{Σ}(X). Obviously, tσ denotes the term obtained from t by uniform substitution of each variable x occuring in t by the term σ(x). Thus, the inference system of EQL over F and X is the pair inf _{eql}^{⟨F,X⟩} = ⟨L_{eql}^{Σ}(X), I_{eql}^{⟨F,X⟩}⟩.

Soundness and completeness results are well known for equational logic [Bir67, EM85]. In particular, completeness can be proved by considering the quotient term algebra on the congruence generated by ⊢_{eql}^{⟨F,X⟩}. Thus, we identify ▷eql^{⟨F,X⟩} = ⊨_{eql}^{⟨F,X⟩} = ⊢_{eql}^{⟨F,X⟩}.

Example 2.3.11. First-order logic (FOL). Let ⟨F, P ⟩ be a first-order signature, i.e., a pair consisting of a IN-indexed family of sets of operation symbols F = {F_{n}}_{n∈IN} and a IN-indexed family of sets of predicate symbols P = {P_{n}}_{n∈IN}, and X a set (of variables). The term language of FOL over ⟨F, P ⟩ and X is precisely Term_{F} (X) as defined above in 2.3.10. The language of FOL over ⟨F, P ⟩ and X is the set L_{fol}^{⟨F,P ⟩}(X) inductively defined as follows:

• p(t_{1}, …, t_{n}) ∈ L_{fol}^{⟨F,P ⟩}(X) if p ∈ P_{n} and t_{k} ∈ Term_{F} (X) for 1 ≤ k ≤ n;

• (¬ φ), (φ ⇒ ψ), (∀x φ), (∃x φ) ∈ L_{fol}^{⟨F,P ⟩}(X) if φ, ψ ∈ L_{fol}^{⟨F,P ⟩}(X) and x ∈ X.

The semantics of FOL considers M_{fol}^{⟨F,P ⟩} to be the class of all ⟨F, P ⟩-structures T = ⟨T, ^{T} ⟩, where T is a set and, for every n ∈ IN, f ∈ F_{n} and p ∈ P_{n}, \underline{f} _{n}^{T} : T^{n} → T and \underline{p} _{n}^{T} ⊆ T^{n}.

Traditionally, one starts by defining satisfaction of a sentence^{3} by a structure T given an assignment to X in T . As before, terms are freely interpreted given T and ρ ∈ Asn(T , X), similarly to 2.3.10. Inductively:

• T ⊩_{ρ}^{⟨F,P ⟩}p(t_{1}, …, t_{n}) if ⟨[[t_{1}]]_{ρ}^{T} , …, [[t_{n}]]_{ρ}^{T} ⟩ ∈ \underline{p} _{n}^{T};

• T ⊩_{ρ}^{⟨F,P ⟩}( ¬ φ) if T ⊩_{ρ̸}^{⟨F,P ⟩}φ;

• T ⊩_{ρ}^{⟨F,P ⟩}(φ ⇒ ψ) if T ⊩_{ρ̸}^{⟨F,P ⟩}φ or T ⊩_{ρ}^{⟨F,P ⟩}ψ;

• T ⊩_{ρ}^{⟨F,P ⟩}(∀x φ) if T ⊩_{ρ′}^{⟨F,P ⟩}φ for every ρ′ that coincides with ρ on X \ {x};

• T ⊩_{ρ}^{⟨F,P ⟩}(∃x φ) if T ⊩_{ρ′}^{⟨F,P ⟩}φ for some ρ′ that coincides with ρ on X \ {x}.

The satisfaction system sat_{fol}^{⟨F,P,X⟩} = ⟨L_{fol}^{⟨F,P ⟩}(X), M_{fol}^{⟨F,P ⟩}, ⊩_{fol}^{⟨F,P,X⟩}⟩ is defined by:

• T ⊩_{fol}^{⟨F,P,X⟩}φ if and only if T ⊩_{ρ}^{⟨F,P ⟩}φ for every assignment ρ to X in T .

The inference system for FOL, I_{fol}^{⟨F,P,X⟩}, consists of all the instances of the axioms and rules Ax01 -Ax11 and MP of classical logic, plus all the appropriate instances of the axioms:

3Note that, in our terminology, all elements of Lfol^{⟨F,P ⟩}(X) are called sentences. Some authors, however, designate them by formulas and reserve the word “sentence” to refer to a closed formula. In this thesis, if that is the case, we shall use “closed sentence” instead.

• ((∀x φ) ⇒ φ_{t}^{x}), if t is a term free for x in φ, where φ_{t}^{x} denotes the sentence obtained by replacing every free occurrence of x in φ by t;

• ((∀x (φ ⇒ ψ)) ⇒ (φ ⇒ (∀x ψ))), if x does not occur free in φ,

and the rule of generalization:

• from φ conclude (∀x φ),

where t ranges over the term language Term_{Σ}(X), x ranges over X, φ and ψ over L_{fol}^{⟨F,P ⟩}(X). The notions of free occurrence of a variable in a sentence and term free for a variable in a sentence are standard and can be found in any textbook on predicate logic [Men79]. Thus, the inference system of EQL over ⟨F, P ⟩ and X is the pair inf _{fol}^{⟨F,P,X⟩} = ⟨L_{fol}^{⟨F,P ⟩}(X), I_{fol}^{⟨F,P,X⟩}⟩.

Soundness and completeness of first-order logic are well known as Gödel’s completeness theorem [Men79]. The typical proof of this result, however, is due to Henkin. More details on this shall be supplied on example 3.3.6 at the end of the next Chapter. Thus, we identify ▷_{fol}^{⟨F,P,X⟩} = ⊨_{fol}^{⟨F,P,X⟩} = ⊢_{fol}^{⟨F,P,X⟩}.

Clearly, we could have as easily developed logics based on the logic systems presented in any of the examples, by mirroring what we did in 2.3.3.

2.4 Remarks

We conclude with a brief summary of what has been achieved in this Chapter. We have introduced consequence system as the basic concept of logic and explored the natural way of obtaining a space of theories out of a consequence operation. In categorial terms this relationship was shown to correspond to an adjunction between the categories Csy and Tsp. The forgetful functor from Csy to Set was also shown to be topological. Among the several interesting consequences of this fact we paid particular attention to the fact that the class of all consequence operations over a fixed set forms a complete lattice. The concept of a finitary consequence system was also explored, and the corresponding full subcategory of Csy shown to be coreflective. As a consequence, the class of all finitary consequence operations over a fixed set also forms a complete lattice, although not a complete sublattice of the previous one. We also explored the notion of logical equivalence, and used it to establish a characterization of equivalence of consequence systems, which consists of the existence of a Csy-based isomorphism between induced theory spaces. Then, we defined a category of logic systems as the source category of any functor to Csy and, by indexing, a corresponding general notion of logic. In particular, we saw that institutions and π-institutions are particular cases of this definition. Moreover, we studied satisfaction systems, valuation systems and inference systems as interesting categories of logic systems, the first two Sat and Val of model-theoretic nature, and the latter Inf of proof-theoretic nature as confirmed by its finitariness. The syntactic forgetful functor corresponding to each of these categories was also shown to constitute both a coadjoint and a cofibration. The Chapter finished with a set of representation examples of well known systems of logic into our setting, namely propositional classic and intuitionistic logics, normal modal logic, linear temporal logic, paraconsistent logic, Lukasiewicz’s three-valued logic and, outside the propositional realm, equational and first-order logics.

Layered logical consequence

In this Chapter we envisage an enrichment of the overall setting developed so far. In fact, we wish to go a step further and explicitly incorporate a new feature into our framework. This further step aims precisely at clarifying a layered dimension of logical consequence that is often neglected. Indeed, as discussed in [FHV92], it is often the case that several related but distinct notions of consequence are available in a single system of logic. Typical well-known examples of this fact are modal and first-order logics. These different shades of logical consequence tend to be assumed and used, in context, but are usually disregarded as an essential a priori feature. However, a sharp distinction between these levels is often a crucial point in the study of logics. Many of what are usually known as deduction theorems and the way they interact with necessitation-like rules are perhaps the most visible outcome of this fact. There are, of course, two main reasons for this state of affairs. The first one is that this question is almost meaningless if we just think about theoremhood. The second one is that even if that is not the case, many usual systems like propositional classical and intuitionistic logics do not raise this distinction. Herein we aim not only at exploring the difference between local and global reasoning, but also the connections between these two layers of consequence, and bring them to the forefront.

A brief outline of this Chapter follows. After making our point as regards the distinction between local and global reasoning, much of what is presented is a remake of the previous Chapter. First, we shall introduce the category of layered consequence systems and study its relationship to consequence systems, both at the local and global levels. This relationship shall set the basis for a deeper study of the category of layered consequence systems. Special attention will be dedicated to the corresponding finitary case. Again, the notion of logical equivalence will be explored and shown to coincide with local equivalence. Abstract notions of category of layered logic systems and layered logic will also be similarly provided. Two special cases of layered logic systems will be carefully detailed: layered valuation systems and layered inference systems. Both are developments of the notions of valuation system and inference system studied before. Layered valuation systems use a generalization of the previous notion of interpretation structure, where its whole internal structure is specified, besides just a set of distinguished truth-values. Layered inference systems incorporate into inference systems an explicit distinction between local and global inferences. Naturally, we will explore these extensions to study a few interesting properties of these layered versions. Layered valuation systems and their underlying layered interpretation structures, in particular, deserve some careful attention. We shall close the Chapter with a collection of well-known examples, now set up in the layered setting. Special focus will be put on modal and first-order logics for their naturalness in this setting, and on classical and intuitionistic propositional logics, where local and global reasoning are shown to collapse.

3.1 Local and global reasoning

Although these facts are well known, let us make the subject clear by reexamining propositional normal modal logic K, as it was introduced in example 2.3.6. We shall start with its semantic aspects, and then also analyse the deduction.

Remark 3.1.1. Local versus global consequence in K. Recall from 2.3.6 that we have defined satisfaction of a modal sentence by a Kripke model in two steps. First, given a Kripke model m = ⟨W, R, ϑ⟩, we have defined satisfaction at a world w ∈ W, inductively, by:

• m⊩_{w}^{P} p if w ∈ ϑ(p);

• m⊩_{w}^{P} ( ¬ φ) if not m⊩_{w}^{P} φ;

• m⊩_{w}^{P} (φ ⇒ ψ) if not m⊩_{w}^{P} φ or m⊩_{w}ψ;

• m⊩_{w}^{P} (□φ) if m⊩_{w′}^{P} φ for every w′ ∈ W such that wRw′.

Then, the satisfaction relation between models and sentences was defined by:

• m⊩_{K}^{P} φ if m⊩_{w}^{P} φ for every w ∈ W.

Let us denote the semantic entailment consequence obtained from ⊩_{K}^{P} by ⊨_{K}^{gP}, where the superscript g stands for “global”.

Clearly, this process completely forgot about the internal structure of Kripke models. To keep track of it, one may define a different consequence ⊨_{K}^{lP}, by considering the notion of satisfaction ⊩_{w}^{P} at a given world of a given model. That is:

• Φ⊨_{K}^{lP} ψ iff m⊩_{w}^{P} ψ whenever m⊩_{w}^{P} Φ, for every m ∈ M_{K}^{P} and w in m.

Obviously, the superscript l captures the local character of this consequence.

It is well known that these two consequences are related but different. In fact, we have that ⊨_{K}^{lP} ⊑ ⊨_{K}^{gP} and ∅^{⊨_{K}^{lP}} = ∅^{⊨_{K}^{gP}}. These relationships can be easily checked. On the one hand, let Φ⊨_{K}^{lP} ψ and m = ⟨W, R, ϑ⟩ be a Kripke model such that m⊩_{K}^{P} Φ. This means that, for each w ∈ W, one has m⊩_{w}^{P} Φ. So, by definition of ⊨_{K}^{lP}, one also has m⊩_{w}^{P} ψ for each w ∈ W. That is, m⊩_{K}^{P} ψ and Φ⊨_{K}^{gP} ψ. On the other hand, let ⊨_{K}^{gP} φ, m = ⟨W, R, ϑ⟩ be a Kripke model and w ∈ W. By definition of ⊨_{K}^{gP}, one also has m⊩_{K}^{P} φ, i.e., m⊩_{w′}^{P} φ for every w′ ∈ W. Thus, in particular, m⊩_{w}^{P} φ and ⊨_{K}^{lP} φ.

However, in general Φ^{⊨_{K}^{gP}} ⊈ Φ^{⊨_{K}^{lP}}. In fact, a simple argument using the properties of submodels of m generated via R^{∗} (the reflexive and transitive closure of R) is well known to show that Φ^{⊨_{K}^{gP}} = {□^{n}φ : φ ∈ Φ, n ∈ IN}^{⊨_{K}^{lP}}. Inductively on n ∈ IN, we have:

• □^{0}φ = φ; • □^{n+1}φ = (□□^{n}φ).

In fact, it can be precisely stated that Φ⊨_{K}^{lP} ψ if and only if there exist φ_{1}, …, φ_{n} ∈ Φ such that ⊨_{K}^{gP} (φ_{1} ⇒ (… (φ_{n} ⇒ ψ) …)). This result, however, relies heavily on the presence of classical implication, and therefore we shall not take it as typical. More details on this shall be supplied in example 3.3.3 at the end of this chapter.

These two layers also have their impact in deductive terms. Obviously, no differences are to be expected from axioms. We already know that local and global entailment coincide for ∅. The difference arises, however, with the rules MP and □-Nec. It is a trivial task to note that both are sound with respect to global entailment, that is ψ ∈ {φ, (φ ⇒ ψ)}^{⊨_{K}^{gP}} and (□φ) ∈ {φ}^{⊨_{K}^{gP}}. That is, both can be safely used for global reasoning. The same cannot be said at the local level. In fact, modus ponens is still sound with respect to local entailment, i.e., {φ, (φ ⇒ ψ)}⊨_{K}^{lP} ψ. It suffices to note that the only constructor involved (⇒) is interpreted locally. However, the satisfaction of a boxed sentence at a given world may depend on satisfaction at other worlds. Thus, necessitation is obviously not appropriate for local reasoning since we do not have {φ}⊨_{K}^{lP} (□φ).

The obvious counterpart is that the deduction theorem holds only for inferences that do not use □-Nec.

Clearly, the plot would thicken even further [FHV92] if we also considered frame semantics [HC96].

Of course, similar arguments would also apply, for instance, to first-order logic. We shall reexamine this example at the end of this Chapter.

Since there is no reason why local reasoning should not be considered as important as the global one, we should abstract from this distinction and formulate a notion of logic that encompasses both layers. It should be noted that a good reason for this difference to be frequently disregarded is the most traditional view of a logic as a set of theorems instead of a consequence relation. As it must be clear from our example, global and local theorems are exactly the same. Another good reason is that, for instance, classical and intuitionistic propositional logics do not exhibit these two layers (c.f., 3.3.1 and 3.3.2 at the end of this Chapter). Or rather, global and local reasoning coincide there. Of course, we shall not exclude this possibility.

We shall start by introducing the notion of layered consequence system.

Definition 3.1.2. A layered consequence system is a triple csy = ⟨L, ▷^{l}, ▷^{g}⟩ where both csy^{l} = ⟨L, ▷^{l}⟩ and csy^{g} = ⟨L, ▷^{g}⟩ are consequence systems, such that the following two conditions hold:

• ▷^{l} ⊑_{L} ▷^{g};

• ∅^{▷^{l}} = ∅^{▷^{g}}.

The operations ▷^{l} and ▷^{g} are said to be the local and global consequences, respectively. Given a sentence φ ∈ L and a set of sentences Φ ⊆ L, φ is said to be a local consequence of Φ if Φ ▷^{l} ψ. Analogously, φ is said to be a global consequence of Φ if Φ ▷^{g} ψ.

Clearly, a layered consequence system also induces a layered theory space, or rather, it induces both a local and a global theory space that inherit the stated relationship between the two consequence operations.

Lemma 3.1.3. For every layered consequence system csy = ⟨L, ▷^{l}, ▷^{g}⟩, the following condition holds, for every Φ ⊆ L:

• (Φ^{▷^{g}})^{▷^{l}} = (Φ^{▷^{l}})^{▷^{g}} = Φ^{▷^{g}}.

Proof. We start by showing that (Φ^{▷^{g}})^{▷^{l}} = Φ^{▷^{g}}. By the extensiveness of ▷^{l}, we have that Φ^{▷^{g}} ⊆ (Φ^{▷^{g}})^{▷^{l}}. Moreover, ▷^{l} ⊑_{L} ▷^{g}, thus also (Φ^{▷^{g}})^{▷^{l}} ⊆ (Φ^{▷^{g}})^{▷^{g}} = Φ^{▷^{g}}, by using the idempotence of ▷^{g}.

We now show that also (Φ^{▷^{l}})^{▷^{g}} = Φ^{▷^{g}}. By extensiveness of ▷^{l}, we have that Φ ⊆ Φ^{▷^{l}}, and the monotonicity of ▷^{g} then yields Φ^{▷^{g}} ⊆ (Φ^{▷^{l}})^{▷^{g}}. Moreover, ▷^{l} ⊑_{L} ▷^{g}, so Φ^{▷^{l}} ⊆ Φ^{▷^{g}} and again using the monotonicity of ▷^{g} we have that (Φ^{▷^{l}})^{▷^{g}} ⊆ (Φ^{▷^{g}})^{▷^{g}}. The idempotence of ▷^{g} completes the proof. ⊓ ⊔

This lemma has several pleasant corollaries. An easy one relates equivalent local sets with equivalent global sets: if Φ, Ψ ⊆ L and Φ^{▷^{l}} = Ψ^{▷^{l}} then also Φ^{▷^{g}} = Ψ^{▷^{g}}.

Another consequence is that any global theory is also a local theory. However, in general, the inclusion of Th(csy^{g}) in Th(csy^{l}) does not induce a morphism of the corresponding theory spaces. Note that if, for instance, csy^{l} = ⟨L, ⊥⟩ then csy^{g} would have to be additive. Still, for modal logic, it is indeed a morphism due to the nature of necessitation and the way the normality axioms propagate boxes (c.f., 3.3.3 ahead). Still, this inclusion is trivially order-preserving. On the other hand, the function that maps each local theory to its global closure always constitutes a morphism of theory spaces. Moreover, the two together constitute a special kind of Galois connection.

Proposition 3.1.4. Let csy = ⟨L, ▷^{l}, ▷^{g}⟩ be a layered consequence system and consider g : tsp_{csy}_{l} → tsp_{csy}^{g}, defined by g(Φ) = Φ^{▷^{g}} for every Φ ∈ Th(csy^{l}). Then, g constitutes a Tsp-morphism and establishes a Galois reflection between the ordered structures T h(csy^{g}) and T h(csy^{l}).

Proof. We know that ▷^{l} ⊑_{L} ▷^{g}, i.e., id_{L} : csy^{l} → csy^{g} is a Csy-morphism. Hence, g = Th(id_{L}) is indeed a Tsp-morphism, according to 2.1.33. Moreover, from lemma 3.1.3 above, we know that Th(csy^{g}) embeds into Th(csy^{l}). Now, given any Φ ∈ Th(csy^{l}), extensiveness of ▷^{g} ensures that Φ ⊆ g(Φ). Moreover, if for some Ψ ∈ Th(csy^{g}) we have that Φ ⊆ Ψ then, using the monotonicity and extensiveness of ▷^{g}, we can conclude that g(Φ) = Φ^{▷^{g}} ⊆ Ψ^{▷^{g}} = Ψ. ⊓ ⊔

Our category of layered consequence systems is built, as it should be expected, by extending the definitions from the previous chapter.

Definition 3.1.5. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems. A layered consequence systems morphism h : csy_{1} → csy_{2} is a function h : L_{1} → L_{2} such that both h : csy_{1}^{l} → csy_{2}^{l} and h : csy_{1}^{g} → csy_{2}^{g} are Csy-morphisms.

Definition 3.1.6. Layered consequence systems and their morphisms constitute the category lCsy, with identity id_{⟨L,▷^{l},▷^{g}⟩} the identity function on L, and (g ∘ f) the usual composition of functions.

Proposition 3.1.7. lCsy is indeed a category.

Proof. A trivial consequence of the fact that Csy is a category. ⊓ ⊔

In fact, there is an expectedly strong relationship between lCsy and Csy.

Definition 3.1.8. The maps

• lg(⟨L, ▷⟩) = ⟨L, ▷, ▷⟩;

• lg(h : csy_{1} → csy_{2}) = h,

constitute the functor lg : Csy → lCsy.

Proposition 3.1.9. lg : Csy → lCsy is a full embedding.

Proof. That lg is both faithful and injective on objects is immediate from its definition. Fullness of lg follows from the fact that, for all consequence systems csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2}, h : csy_{1} → csy_{2} is a Csy-morphism if and only if h : lg(csy_{1}) → lg(csy_{2}) is an lCsy-morphism. ⊓ ⊔

The full subcategory of lCsy induced by lg has as objects, precisely, all layered consequence systems ⟨L, ▷^{l}, ▷^{g}⟩ such that ▷^{l} = ▷^{g}. We denote this category by lCsy=.

Proposition 3.1.10. Csy is isomorphic to lCsy=. Moreover, lCsy= is both a reflective and coreflective subcategory of lCsy.

Proof. Clearly, every object ⟨L, ▷, ▷⟩ of lCsy= is the lg image of ⟨L, ▷⟩. Thus, we can define lg : Csy → lCsy= as above. This functor is obviously an isomorphism.

For every layered consequence system ⟨L, ▷^{l}, ▷^{g}⟩ we now show that id_{L} : ⟨L, ▷^{l}, ▷^{g}⟩ → ⟨L, ▷^{g}, ▷^{g}⟩ is a reflection arrow. For the purpose, let h : ⟨L, ▷^{l}, ▷^{g}⟩ → ⟨L′, ▷, ▷⟩ be any lCsy-morphism. Since (g ∘ id_{L}) = h implies g = h, it suffices to note that h : ⟨L, ▷^{g}, ▷^{g}⟩ → ⟨L′, ▷, ▷⟩ is also an lCsy-morphism. Obviously, we already know that h : ⟨L, ▷^{g}⟩ → ⟨L′, ▷⟩ is a Csy-morphism, and the result follows.

We now show that, for every layered consequence system ⟨L, ▷^{l}, ▷^{g}⟩, id_{L} : ⟨L, ▷^{l}, ▷^{l}⟩ → ⟨L, ▷^{l}, ▷^{g}⟩ is a coreflection arrow. For the purpose, let h : ⟨L′, ▷, ▷⟩ → ⟨L, ▷^{l}, ▷^{g}⟩ be any lCsy-morphism. Since (id_{L} ∘ g) = h implies g = h, it suffices to note that h : ⟨L′, ▷, ▷⟩ → ⟨L, ▷^{l}, ▷^{l}⟩ is also an lCsy-morphism. Obviously, we already know that h : ⟨L′, ▷⟩ → ⟨L, ▷^{l}⟩ is a Csy-morphism, and the result follows. ⊓ ⊔

As a consequence of the above, the functor lg : Csy → lCsy is both adjoint and coadjoint.

Corollary 3.1.11. The functor Loc : lCsy → Csy defined by

• Loc(csy) = csy^{l};

• Loc(h : csy_{1} → csy_{2}) = h : csy_{1}^{l} → csy_{2}^{l},

is adjoint to lg : Csy → lCsy.

The functor Glob : lCsy → Csy defined by • Glob(csy) = csy^{g};

• Glob(h : csy_{1} → csy_{2}) = h : csy_{1}^{g} → csy_{2}^{g},

is coadjoint to lg : Csy → lCsy.

Notably, Glob is also adjoint. In fact we can even show that it is topological.

Proposition 3.1.12. Glob : lCsy → Csy is a topological functor.

Proof. We have to show that given any family {csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩}_{k∈K} of layered consequence systems, any source {h_{k} : csy^{g} → csy_{k}^{g}}_{k∈K} in Csy has a unique Glob-initial lift {h_{k} : csy → csy_{k}}_{k∈K} in lCsy.

To start with, assume that csy^{g} = ⟨L, ▷^{g}⟩, and let ⟨L, ▷⟩ be the unique Lang_{Csy}-initial lift of {h_{k} : L → L_{k}}_{k∈K} given {csy_{k}^{l}}_{k∈K} in Csy. Recall from 2.1.17 that, for every Φ ⊆ L, we have Φ^{▷} = (⋂_{k∈K} h_{k}^{−1}(h_{k}(Φ)^{▷_{k}^{l}})). Let now csy = ⟨L, ▷^{l}, ▷^{g}⟩ where ▷^{l} = ⊓_{L}{▷, ▷^{g}}, i.e., Φ^{▷^{l}} = (Φ^{▷^{g}} ∩ Φ^{▷}). Since obviously ▷^{l} ⊑_{L} ▷^{g} and each h_{k}(∅^{▷^{g}}) ⊆ ∅^{▷_{k}^{g}}, it immediately follows that ∅^{▷^{g}} ⊆ ∅^{▷} and csy is a layered consequence system.

Clearly {h_{k} : csy → csy_{k}}_{k∈K} is an adequate source in lCsy. Namely, h_{k}(Φ^{▷^{l}}) ⊆ h_{k}(Φ^{▷}) ⊆ h_{k}(Φ)^{▷_{k}^{l}}. To prove its initiality, let {f_{k} : csy′ → csy_{k}}_{k∈K} be any lCsy source, with csy′ = ⟨L′, ▷^{′l}, ▷^{′g}⟩, and let f : csy^{′g} → csy^{g} be such that (h_{k} ∘ f) = f_{k} for every k ∈ K. All we need to prove is that then f : csy′ → csy is indeed an lCsy-morphism. Thus, it suffices to show that f : csy^{′l} → csy^{l} is a Csy-morphism. But we know that f(Φ^{▷^{′l}}) ⊆ f(Φ^{▷^{′g}}) ⊆ f(Φ)^{▷^{g}}. And on the other hand, the initiality of ▷ implies that also f(Φ^{▷^{′l}}) ⊆ f(Φ)^{▷}. Thus, trivially, f(Φ^{▷^{′l}}) ⊆ (f(Φ)^{▷^{g}} ∩ f(Φ)^{▷}) = f(Φ)^{▷^{l}}. ⊓ ⊔

Corollary 3.1.13. Glob : lCsy → Csy is adjoint.

The couniversal pair for each consequence system csy = ⟨L, ▷⟩ is given by ⟨L, ▷_{min}, ▷⟩ and id_{L} : csy → csy, where Φ^{▷_{min}} = (Φ ∪ ∅^{▷}).

However, the apparent symmetry between local and global is now broken down. In fact, in its turn, Loc has quite a different behaviour. It is out of the scope of this study to prove whether or not Loc is coadjoint ^{1}. However, we can easily show that Loc is not topological. Indeed, given any consequence system ⟨L, ▷⟩, let us denote by ▷_{max} the biggest consequence on L such that ⟨L, ▷, ▷_{max}⟩ is a layered consequence system. It is quite easy to see that it amounts to

Φ^{▷_{max}} = \{ \begin{matrix}∅^{▷} & if Φ ⊆ ∅^{▷} & . \\ L & otherwise \end{matrix}

The problem arises in the quite usual situation where there are both theorems and nontheorems, i.e., ∅ ≠ ∅^{▷} ≠ L. Then, this gap can be explored to build a layered consequence system ⟨L, ▷^{l}, ▷^{g}⟩ that fulfils ▷^{l} ⊑_{L} ▷ but not ▷^{g} ⊑_{L} ▷_{max}. It suffices to consider ▷^{l} = ⊥ and ▷^{g} defined as follows:

1It is not hard to see that Loc preserves colimits, and that in fact all colimits exist in lCsy, which makes the task much harder.

Φ^{▷^{g}} = \{ \begin{matrix}∅ & if Φ = ∅ & . \\ L & otherwise \end{matrix}

Easily, (∅^{▷})^{▷^{g}} = L ⊈ (∅^{▷})^{▷_{max}} = ∅^{▷}.

This means, basically, that ▷ can be seen as being global (with the local one varying between ▷_{min} and ▷), but not as being local (in general, we cannot safely set an upperbound on the global side, although we know that ▷ is a lower-bound).

Nevertheless, Loc and Glob inherit the properties that relate their corresponding parts in a layered consequence system. In particular, this relationship extends to a natural transformation.

Proposition 3.1.14. The map

• id(⟨L, ▷^{l}, ▷^{g}⟩) = id_{L} : ⟨L, ▷^{l}⟩ → ⟨L, ▷^{g}⟩,

constitutes a natural transformation id : Loc → Glob.

Proof. Trivial from the definitions of lCsy and the functors Loc and Glob. ⊓ ⊔

Moreover, lCsy is still a topological category over Set.

Definition 3.1.15. The maps

• Lang_{lCsy}(⟨L, ▷^{l}, ▷^{g}⟩) = L;

• Lang_{lCsy}(h : ⟨L_{1}, ▷_{1}^{l}, ▷_{1}^{g}⟩ → ⟨L_{2}, ▷_{2}^{l}, ▷_{2}^{g}⟩) = h : L_{1} → L_{2},

constitute the functor Lang_{lCsy} : lCsy → Set.

Trivially, Lang_{lCsy} = (Lang_{Csy} ∘ Loc) = (Lang_{Csy} ∘ Glob).

Proposition 3.1.16. Lang_{lCsy} : lCsy → Set is a topological functor.

Proof. Immediate, since Lang_{lCsy} = (Lang_{Csy} ∘ Glob) and both Lang_{Csy} and Glob are topological. ⊓ ⊔

Thus, again, the fibres of Lang_{lCsy} form complete lattices under the obvious definition of order.

Definition 3.1.17. Let L be a set and ⟨L, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems. Then, ⟨▷_{1}^{l}, ▷_{1}^{g}⟩ ⊑_{L} ⟨▷_{2}^{l}, ▷_{2}^{g}⟩ if and only if both ▷_{1}^{l} ⊑_{L} ▷_{2}^{l} and ▷_{1}^{g} ⊑_{L} ▷_{2}^{g}.

Please note again that ⟨▷_{1}^{l}, ▷_{1}^{g}⟩ ⊑_{L} ⟨▷_{2}^{l}, ▷_{2}^{g}⟩ is precisely equivalent to id_{L} : ⟨L, ▷_{1}^{l}, ▷_{1}^{g}⟩ → ⟨L, ▷_{2}^{l}, ▷_{2}^{g}⟩ being an lCsy-morphism.

Corollary 3.1.18. Lang_{lCsy} : lCsy → Set is fibre-complete, i.e., each fibre lCsy_{L} is a complete lattice (under ⊑_{L}).

Clearly, the meet of a family of pairs of layered consequence operations {⟨▷_{k}^{l}, ▷_{k}^{g}⟩}_{k∈K} is given by the corresponding unique Lang_{lCsy}-initial lifting of {id_{L} : L → L}_{k∈K}, i.e., ⊓_{L}\{⟨▷_{k}^{l}, ▷_{k}^{g}⟩ : k ∈ K\} = ⟨\underline{▷}^{l}, \underline{▷}^{g}⟩ where \underline{▷}^{l} = ⊓_{L}\{▷_{k}^{l} : k ∈ K\} and \underline{▷}^{g} = ⊓_{L}\{▷_{k}^{g} : k ∈ K\}. Top and bottom are also obtained componentwise. They are, respectively, ⟨⊤, ⊤⟩ and ⟨⊥, ⊥⟩.

As it should be expected, joins of families are a bit different. They are first built globally, as before, and then locally. That is, ⊔_{L}\{⟨▷_{k}^{l}, ▷_{k}^{g}⟩ : k ∈ K\} = ⟨\overline{▷}^{l}, \overline{▷}^{g}⟩ where \overline{▷}^{g} = ⊔_{L}\{▷_{k}^{g} : k ∈ K\}, and \overline{▷}^{l} is then generated by the set of inferences I = ((⋃_{k∈K} I_{k}) ∪ \{⟨∅, φ⟩ : \overline{▷}^{g}φ\}) with each I_{k} = \{⟨Φ, ψ⟩ : Φ ▷_{k}^{l} ψ\}.

We now turn to finitariness in the context of layered consequence systems.

Definition 3.1.19. A layered consequence system csy = ⟨L, ▷^{l}, ▷^{g}⟩ is said to be finitary if both csy^{l} and csy^{g} are finitary consequence systems.

Definition 3.1.20. lCsy_{fin} is the full subcategory of lCsy whose objects are precisely the finitary layered consequence systems.

As in the non-layered case, the finitary part of a layered consequence system can always be extracted.

Proposition 3.1.21. Let csy = ⟨L, ▷^{l}, ▷^{g}⟩ be a layered consequence system. The greatest pair of finitary layered consequence operations contained in ⟨▷^{l}, ▷^{g}⟩ (in the sense of ⊑_{L}) is ⟨▷_{fin}^{l}, ▷_{fin}^{g}⟩.

Proof. Reusing the proof of 2.1.22, it suffices to prove that ⟨L, ▷_{fin}^{l}, ▷_{fin}^{g}⟩ is indeed a layered consequence system. In fact, ▷^{l} ⊑_{L} ▷^{g} is equivalent to id_{L} : csy^{l} → csy^{g} being a Csymorphism. Thus, 2.1.23 implies that id_{L} : csy_{fin}^{l} → csy_{fin}^{g} is a morphism, that is ▷_{fin}^{l} ⊑_{L} ▷_{fin}^{g}. Moreover, by definition, ∅^{▷_{fin}^{l}} = ∅^{▷^{l}} = ∅^{▷^{g}} = ∅^{▷_{fin}^{g}}. ⊓ ⊔

For convenience, we use the notation csy_{fin} = ⟨L, ▷_{fin}⟩. Again, we can extend this construction to a functor from lCsy to lCsy_{fin} that is adjoint to the corresponding inclusion functor lI : lCsy_{fin} → lCsy.

Proposition 3.1.22. lCsy_{fin} is a coreflective subcategory of lCsy.

Proof. Immediate from 2.1.23. ⊓ ⊔

Corollary 3.1.23. The maps

• lFin(csy) = csy_{fin};

• lFin(h : csy_{1} → csy_{2}) = h : csy_{1fin} → csy_{2fin},

constitute a functor lFin : lCsy → lCsy_{fin}. Moreover, lFin is adjoint to lI .

The category lCsy_{fin} is also topological over Set. Obviously, we shall use the notation Lang_{lCsy_{fin}} = (Lang_{lCsy} ∘ lI ).

Corollary 3.1.24. The functor Lang_{lCsy_{fin}} : lCsy_{fin} → Set is topological.

Proof. Immediate from 2.1.25. ⊓ ⊔

Thus, the fibres of lCsy_{fin} form complete lattices similarly to corresponding lCsy fibres.

Corollary 3.1.25. lCsy_{fin} is fibre-complete, i.e., each fibre (lCsy_{fin})_{L} is a complete lattice (under the restriction of ⊑_{L} to finitary layered consequence systems).

Proof. Immediate from 2.1.26. ⊓ ⊔

Meets are taken separately local and globally, by extracting their finitary parts. The result is still layered because ∅ is finite. Top and bottom are also obvious. Joins coincide with the construction for lCsy, since we already know that finitariness is preserved.

The functors Loc and Glob, previously defined, obviously have their counterparts in terms of theory spaces, respectively, (Th ∘ Loc) and (Th ∘ Glob). And the latter obviously establishes a reflection between lCsy and Tsp. Moreover, the natural transformation id : Loc → Glob clearly extends to Th(id) : (Th ∘ Loc) → (Th ∘ Glob).

With respect to equivalence of layered consequence systems, we shall use an obvious generalization of the non-layered case. As we have seen before, global theories are certainly also local theories. Therefore, equivalence with respect to local consequence implies equivalence with respect to global consequence. However, the converse does not hold in general. This means that we shall always use local equivalence in layered consequence systems.

Definition 3.1.26. Two layered consequence systems csy_{k} for k ∈ {1, 2} are said to be equivalent if there exist lCsy-morphisms f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} such that Loc(csy_{1}) ≡ Loc(csy_{2}) with Loc(f) and Loc(g), and Glob(csy_{1}) ≡ Glob(csy_{2}) with Glob(f) and Glob(g). We denote the fact that csy_{1} and csy_{2} are equivalent by csy_{1} ≡ csy_{2}.

That is, we must have both csy_{1}^{l} ≡ csy_{2}^{l} and csy_{1}^{g} ≡ csy_{2}^{g} with the same f and g.

In order to have a nicer characterization of equivalence, let us again have a look at equivalence of sentences.

Definition 3.1.27. Let csy = ⟨L, ▷^{l}, ▷^{g}⟩ be a layered consequence system and Φ ⊆ L. Two sentences ψ, υ ∈ L are said to be equivalent in csy modulo-Φ whenever ψ ≡_{csy^{l}}^{Φ} υ. We denote this fact by ψ ≡_{csy}^{Φ} υ. As before, if Φ = ∅, we simply say that ψ and υ are equivalent in csy, and write ψ ≡_{csy} υ instead of ψ ≡_{csy}^{∅} υ.

Again, we shall prove that equivalent sentences are precisely those that are indistinguishable concerning theories, both locally and globally.

Lemma 3.1.28. Let csy = ⟨L, ▷^{l}, ▷^{g}⟩ be a layered consequence system and Φ, Ψ ⊆ L. Then, Φ^{▷^{l}} = Ψ^{▷^{l}} and Φ^{▷^{g}} = Ψ^{▷^{g}} whenever the following two conditions are satisfied:

• for every φ ∈ Φ there exists φ′ ∈ Ψ such that φ ≡_{csy} φ′;

• for every ψ ∈ Ψ there exists ψ′ ∈ Φ such that ψ ≡_{csy} ψ′.

Proof. Since ≡_{csy} coincides with ≡_{csy}_{l}, we already know from 2.1.43 that Φ^{▷^{l}} = Ψ^{▷^{l}}. Thus, using 3.1.3, we also have that Φ^{▷^{g}} = (Φ^{▷^{l}})^{▷^{g}} = (Ψ^{▷^{l}})^{▷^{g}} = Ψ^{▷^{g}}. ⊓ ⊔

We are now able to formulate a nice characterization of equivalence of layered consequence systems.

Proposition 3.1.29. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems. Then, csy_{1} ≡ csy_{2} if and only if there exist lCsy-morphisms f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} such that the following two conditions are satisfied:

• φ ≡_{csy_{1}} g(f(φ)) for every φ ∈ L_{1};

• ψ ≡_{csy_{2}} f(g(ψ)) for every ψ ∈ L_{2}.

Proof. Assuming that csy_{1} ≡ csy_{2}, let f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} be lCsymorphisms fulfilling the conditions in definition 3.1.26. In particular, Th(Loc(f)) and Th(Loc(g)) are isomorphisms of tsp_{csy}l_{1} and tsp_{csy}l_{2}, with Th(Loc(f)) = Th(Loc(g))^{−1}. Obviously then, Loc(f) = f : csy_{1}^{l} → csy_{2}^{l} and Loc(g) = g : csy_{2}^{l} → csy_{1}^{l} are Csymorphisms and csy_{1}^{l} ≡ csy_{2}^{l}. Therefore, we can use to 2.1.44 to conclude that φ ≡_{csy_{1}} g(f(φ)) and ψ ≡_{csy_{2}} f(g(ψ)), for every φ ∈ L_{1} and ψ ∈ L_{2}, respectively.

Assume now that f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} are lCsy-morphisms satisfying the two conditions stated above. We can use 2.1.44 again, to conclude that Th(Loc(f)) and Th(Loc(g)) are isomorphisms of tsp_{csy}l_{1} and tsp_{csy}l_{2}, with Th(Loc(f)) = Th(Loc(g))^{−1}. We are bound to show that the same happens as regards global consequence. If Φ ∈ Th(csy_{1}^{g}) then Th(Glob(g))(Th(Glob(f))(Φ)) = Th(Glob(g ∘ f))(Φ) = g(f(Φ))^{▷_{1}^{g}}. However, from 3.1.3, we know that g(f(Φ))^{▷_{1}^{g}} = (g(f(Φ))^{▷_{1}^{l}})^{▷_{1}^{g}}. We already know that Φ ∈ Th(csy_{1}^{l}) and thus we use the assumption to conclude that g(f(Φ))^{▷_{1}^{l}} = Φ^{▷_{1}^{l}} = Φ. Therefore, Th(Glob(g))(Th(Glob(f))(Φ)) = Φ^{▷_{1}^{g}} = Φ. Using an analogous argument, we can conclude that also Th(Glob(f))(Th(Glob(g))(Ψ)) = Ψ for every Ψ ∈ Th(csy_{2}^{g}), and therefore csy_{1} ≡ csy_{2}. ⊓ ⊔

3.2 Layered logic systems and layered logics

Having studied layered consequence systems, we can now state our intended notions of layered logic system and layered logic.

Definition 3.2.1. A category of layered logic systems is a pair ⟨A, lCn⟩ where:

• A is a category (of layered logic systems and their morphisms);

• lCn : A → lCsy is a functor.

An object a ∈ |A| is said to be an ⟨A, lCn⟩-based logic system. If, in addition, lCn(a) is a finitary consequence system then a is said to be a finitary layered logic system.

⟨A, lCn⟩ is further said to be a category of finitary layered logic systems provided that every ⟨A, lCn⟩-based layered logic system is finitary.

Proposition 3.2.2. A category of layered logic systems ⟨A, lCn⟩ is finitary if and only if there exists a functor lCn_{fin} : A → lCsy_{fin} such that lCn = (lI ∘ lCn_{fin}).

Proof. Since lFin does not change finitary elements, just consider lCn_{fin} = (lFin ∘lCn). ⊓ ⊔

As a consequence, in the sequel, we shall sometimes identify a category of finitary layered logic systems with a functor lCn_{fin} : A → lCsy_{fin}.

Again, the notion of a layered logic is straightforward.

Definition 3.2.3. Let ⟨A, lCn⟩ be a category of layered logic systems and Voc a category (of vocabularies). An ⟨A, lCn⟩-based layered logic over Voc is a functor Log : Voc → A. If ⟨A, lCn⟩ is a category of finitary logic systems, then Log is also said to be finitary.

Whenever the functor lCn is well understood, we sometimes just refer to A, and call Log an A-based layered logic.

Definition 3.2.4. Let ⟨A, lCn⟩ be a category of layered logic systems and Log an ⟨A, lCn⟩- based logic over Voc. The language functor of Log is Lang_{Log} = (Lang_{lCsy} ∘ lCn ∘ Log) : Voc → Set.

That is, the language of a layered logic is the language of each of its layered consequence systems. As before, we shall also denote (Lang_{lCsy} ∘ lCn) by Lang_{A}.

Trivially, lCsy is itself a category of layered logic systems under its own identity functor. Moreover, Csy can also be seen as a category of layered logic systems by using the lg functor from 3.1.8. Another trivial example of a category of layered logic systems is lCsy_{fin} under the inclusion functor to lCsy.

Repeating the steps we followed in the previous chapter, layered logics over a given category of layered logic system also constitute a category. Again, we reuse the construction from A.3.5.

Definition 3.2.5. Let ⟨A, lCn⟩ be a category of layered logic systems. We denote the category Funct(A) by ⟨A, lCn⟩-Log, and call it the category of ⟨A, lCn⟩-based layered logics.

Notions like soundness and completeness in a proof/model theoretic setting also have their layered versions. As in the previous Chapter, in the following definitions P denotes one of the following properties: weak soundness, soundness, weak completeness, finite completeness, completeness.

Definition 3.2.6. Let ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩ be categories of layered logic systems, the former one being finitary, and cal ∈ |Cal| and sem ∈ |Sem| be such that Ded(cal) = ⟨L, ⊢^{l}, ⊢^{g}⟩, Ent(sem) = ⟨L, ⊨^{l}, ⊨^{g}⟩ for some set L. The pair ⟨cal, sem⟩ is said to have the local property P provided that ⟨cal^{l}, sem^{l}⟩ = ⟨⟨L, ⊢^{l}⟩, ⟨L, ⊨^{l}⟩⟩ does. Analogously, ⟨cal, sem⟩ has the global property P provided that ⟨cal^{g}, sem^{g}⟩ = ⟨⟨L, ⊢^{g}⟩, ⟨L, ⊨^{g}⟩⟩ does. Furthermore, ⟨cal, sem⟩ is said to have the property P provided that it holds both locally and globally.

Definition 3.2.7. Let Voc be a category, and Prf : Voc → Cal and Mod : Voc → Sem be layered logics based, respectively, on the categories of layered logic systems ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩, the former being finitary, sharing the same language functor, i.e., Lang_{Prf} = Lang_{Mod}. The pair ⟨Prf , Mod⟩ is said to have (local/global) property P provided that each pair ⟨Prf (V), Mod(V)⟩ does.

3.2.1 Models: layered valuation systems

The notion we propose for going beyond satisfaction of a sentence by a model and making explicit the internal (local) logical structure of models is a further development of valuation systems. In fact, in a layered interpretation system we shall have to distinguish between the different shades of “grey” truth values, instead of just “black” and “white”. As it was mentioned in the introductory Chapter regarding fully algebraic semantics, we have to assign some richer internal structure to truth values. Namely, as an abstraction which encompasses all usual cases of ordered or lattice structures, we shall endow the whole set of truth values with a closure operation. The role of designated truth values is then played by the values in the closure of ∅, i.e., the “theorem” truth values. Similar techniques are used in [Cle91].

Definition 3.2.8. Let L be a set. An L-layered structure is a triple ⟨T, [[ ]], ⋉⟩ where:

• T is a set (of truth values);

• [[ ]] : L → T (a valuation);

• ⟨T, ⋉⟩ is a consequence system (it fixes an internal logical structure in T).

We denote by lStr(L) the class of all L-layered structures.

In practice, a common particular case happens when the set of truth values in a model is endowed with a binary relation expressing “closeness to truth” (usually a pre-ordering, but not necessarily), and letting the set of designated values D to be represented by the set of all its maximal elements. Please observe again that, in our case, the idea is that the role of the set D in a non-layered L-structure is herein represented by ∅^{⋉}. This clearly generalizes the case where a relation on truth values is known. It suffices to use a simple construction, well known from polarities in relational structures [Bir67].

Given a binary relation ⟨T, ρ⟩, i.e., ρ ⊆ T × T, and V ⊆ T, we shall denote by:

• U_{ρ}(V) = {t ∈ T : vρt for every v ∈ V } (the set of upper-bounds of V);

• L_{ρ}(V) = {t ∈ T : tρv for every v ∈ V } (the set of lower-bounds of V).

Lemma 3.2.9. Let ⟨T, ρ⟩ be a binary relation. Then, ⟨T, ⋉_{ρ}⟩ is a consequence system, where ⋉_{ρ} is defined by:

• V^{⋉_{ρ}} = U_{ρ}(L_{ρ}(V)), for every V ⊆ T.

Proof. We first show that ⋉_{ρ} is extensive. Given t ∈ L_{ρ}(V), easily, tρv for every v ∈ V. Hence, V ⊆ U_{ρ}(L_{ρ}(V)) = V^{⋉_{ρ}}.

Monotonicity follows easily from the fact that both U_{ρ} and L_{ρ} are order-reversing. Indeed, let U ⊆ V. If t ∈ U_{ρ}(V) we have that vρt for every v ∈ V, and therefore uρt for every u ∈ U. We conclude that t ∈ U_{ρ}(U) and hence U_{ρ}(V) ⊆ U_{ρ}(U). Analogously, if t ∈ L_{ρ}(V) we have that tρv for every v ∈ V, and therefore tρu for every u ∈ U. We conclude that t ∈ L_{ρ}(U) and hence L_{ρ}(V) ⊆ L_{ρ}(U). Immediately, then, U^{⋉_{ρ}} = U_{ρ}(L_{ρ}(U)) ⊆ U_{ρ}(L_{ρ}(V)) = V^{⋉_{ρ}}.

To prove idempotence, we shall also capitalize on the fact that U_{ρ} is order-reversing. In fact, let t ∈ L_{ρ}(V) and u ∈ U_{ρ}(L_{ρ}(V)). Easily, tρu and thus t ∈ L_{ρ}(U_{ρ}(L_{ρ}(V))). We conclude that L_{ρ}(V) ⊆ L_{ρ}(U_{ρ}(L_{ρ}(V))). Hence, (V^{⋉_{ρ}})^{⋉_{ρ}} = U_{ρ}(L_{ρ}(U_{ρ}(L_{ρ}(V)))) ⊆ U_{ρ}(L_{ρ}(V)) = V^{⋉_{ρ}}. ⊓ ⊔

Please note that ∅^{⋉ρ} = U_{ρ}(L_{ρ}(∅)) = U_{ρ}(T) = {v ∈ T : tρv for every t ∈ T } is precisely the set of all ρ-maximal elements. For T = {0, 1}, in particular, there is a unique consequence operation such that ∅^{⋉} = {1}. It corresponds precisely to the usual total order on {0, 1}.

Given a function h : L′ → L and an L-layered structure ⟨T, [[ ]], ⋉⟩ we denote by ⟨T, [[ ]], ⋉⟩|_{h} the L′-structure ⟨T, ([[ ]] ∘ h), ⋉⟩.

Definition 3.2.10. A layered valuation system is a triple val = ⟨L, M, S⟩ where:

• L is a set (of sentences);

• M is a class (of models);

• S : M → lStr(L) (endows each model with an L-layered structure).

For each model m ∈ M, we denote S(m) by ⟨T_{m}, [[ ]]_{m}, ⋉_{m}⟩.

Morphisms are the obvious generalization of the non-layered case.

Definition 3.2.11. Let val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩ for k ∈ {1, 2} be layered valuation systems. A layered valuation systems morphism h : val_{1} → val_{2} consists of:

• a (translation) function \overline{h} : L_{1} → L_{2};

• a (reduct) map \underline{h} : M_{2} → M_{1},

such that the following condition holds for every model m ∈ M_{2}:

• S_{1}(\underline{h}(m)) = S_{2}(m)|\overline{h}.

Note that a morphism, as above, implies that T_{\underline{h}(m)} = T_{m}, ⋉_{\underline{h}(m)} = ⋉_{m} and [[φ]]_{\underline{h}(m)} = [[\overline{h}(φ)]]_{m} for each φ ∈ L_{1}.

Definition 3.2.12. Layered valuation systems and their morphisms constitute the category lVal, with identity id_{⟨L,M,S⟩} such that \overline{id}_{⟨L,M,S⟩} = id_{L}, the identity function on L, and \underline{id}_{⟨L,M,S⟩} = id_{M}, the identity map on M, and composition (g∘f) such that \overline{(g ∘ f)} = (\overline{g}∘\overline{f}) and \underline{(g ∘ f)} = (\underline{f} ∘ \underline{g}).

Proposition 3.2.13. lVal is indeed a category.

Proof. Trivial. ⊓ ⊔

We shall now show, in three steps, that lVal is indeed a category of layered logic systems. First, we shall establish a suitable functor from lVal to Val that is bound to define the global consequence resulting from each layered valuation system. Secondly, we shall define the corresponding local consequence operation by exploring the internal consequence of models. Finally, we shall set up the appropriate functor to lCsy by showing that these two consequences fulfill the layered conditions.

Definition 3.2.14. The maps

• D(⟨L, M, S⟩) = ⟨L, M, S^{D}⟩, where S^{D}(m) = ⟨T_{m}, [[ ]]_{m}, ∅^{⋉_{m}}⟩;

• D(h : val_{1} → val_{2}) = h,

constitute the functor D : lVal → Val.

For obvious reasons we shall sometimes use D_{m} to denote ∅^{⋉_{m}}.

Proposition 3.2.15. D : lVal → Val is indeed a functor.

Proof. By definition of the categories lVal and Val it is trivial to verify that identities and compositions are preserved. Moreover, it is clear that S^{D} assigns a well defined Lstructure to each model. The fact that h : D(val_{1}) → D(val_{2}) is indeed a Val-morphism whenever h : val_{1} → val_{2} is an lVal-morphism is immediate. Necessarily, for each model m of val_{2}, we have ⋉_{\underline{h}(m)} = ⋉_{m} and hence also D_{\underline{h}(m)} = D_{m}. ⊓ ⊔

Definition 3.2.16. Let val = ⟨L, M, S⟩ be a layered valuation system. The global entailment consequence system csy_{val}^{g} = ⟨L, ⊨_{val}^{g} ⟩ induced by val is precisely Ent_{Val}(D(s)). Namely, a sentence ψ ∈ L is said to be globally entailed in val by a set of sentences Φ ⊆ L, a fact that we denote by Φ⊨_{val}^{g} ψ, if and only if for every model m ∈ M, [[ψ]]_{m} ∈ D_{m} whenever {[[φ]]_{m} : φ ∈ Φ} ⊆ D_{m}. Moreover, Φ^{⊨_{val}^{g}} = {ψ ∈ L : Φ⊨_{val}^{g} ψ}.

We shall now define local entailment by capitalizing on the internal consequence operation endowed to the set of truth values of each model.

Definition 3.2.17. Let val = ⟨L, M, S⟩ be a layered valuation system. A sentence ψ ∈ L is said to be locally entailed in val by a set of sentences Φ ⊆ L, a fact that we denote by Φ⊨_{val}^{l} ψ, if and only if for every model m ∈ M, {[[φ]]_{m} : φ ∈ Φ}⋉_{m}[[ψ]]_{m}. The local entailment consequence operation ⊨_{val}^{l} induced by val is then defined, for each set of sentences Φ ⊆ L, by Φ^{⊨_{val}^{l}} = {ψ ∈ L : Φ⊨_{val}^{l} ψ}.

Proposition 3.2.18. Let val = ⟨L, M, S⟩ be a layered valuation system. Then, csy_{val}^{l} = ⟨L, ⊨_{val}^{l} ⟩ constitutes a consequence system.

Proof. The fact that ⊨_{val}^{l} is a consequence operation follows immediately from the fact that each ⋉_{m} is also a consequence operation. Easily, ⊨_{val}^{l} is the unique Lang_{Csy}-initial lift of {[[ ]]_{m} : L → T_{m}}_{m∈M} given {⟨T_{m}, ⋉_{m}⟩}_{m∈M}, as defined in 2.1.17. ⊓ ⊔

Let us now establish the functor from lVal to lCsy.

Definition 3.2.19. The maps

• Ent_{lVal}(⟨L, M, S⟩) = ⟨L, ⊨_{⟨L,M,S⟩}^{l}, ⊨_{⟨L,M,S⟩}^{g}⟩;

• Ent_{lVal}(h : val_{1} → val_{2}) = \overline{h},

constitute the functor Ent_{lVal} : lVal → lCsy.

For every layered valuation system val, we shall denote Ent_{lVal}(val) by csy_{val}.

Proposition 3.2.20. Ent_{lVal} : lVal → lCsy is indeed a functor.

Proof. We first prove that given a layered valuation system val = ⟨L, M, S⟩, the triple csy_{val} = ⟨L, ⊨_{val}^{l} , ⊨_{val}^{g} ⟩ is a layered consequence system. To start with, let Φ ⊆ L, Φ⊨_{val}^{l} ψ and m ∈ M be such that {[[φ]]_{m} : φ ∈ Φ} ⊆ D_{m}. Immediately, the monotonicity and idempotence of ⋉_{m} together imply that {[[φ]]_{m} : φ ∈ Φ}^{⋉_{m}} ⊆ D_{m}. However, we know that {[[φ]]_{m} : φ ∈ Φ}⋉_{m}[[ψ]]_{m}. Hence, ⋉_{m}[[ψ]]_{m}, i.e., [[ψ]]_{m} ∈ D_{m}, and we conclude that Φ^{⊨_{val}^{l}} ⊆ Φ^{⊨_{val}^{g}}. Now, if ψ ∈ ∅^{⊨_{val}^{g}}, we know that ⋉_{m}[[ψ]]_{m} for every m ∈ M. Thus, immediately, ⊨_{val}^{l} ψ and we conclude that ∅^{⊨_{val}^{g}} ⊆ ∅^{⊨_{val}^{l}}.

Given that (Glob ∘ Ent_{lVal}) = (Ent_{Val} ∘ D), we can immediately conclude that \overline{h} : csy_{val_{1}}^{g} → csy_{val_{2}}^{g} is indeed a Csy-morphism whenever h : val_{1} → val_{2} is an lValmorphism. It remains to prove that also \overline{h} : csy_{val_{1}}^{l} → csy_{val_{2}}^{l} is a Csy-morphism. For the purpose, let val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩ for k ∈ {1, 2}, Φ⊨_{val_{1}}^{l} ψ and m ∈ M_{2}. Obviously, \underline{h}(m) ∈ M_{1} and therefore we have that \{[[φ]]_{\underline{h}(m)} : φ ∈ Φ\}⋉_{\underline{h}(m)}[[ψ]]_{\underline{h}(m)}. Thus, by the conditions on a Val-morphism, this is equivalent to \{[[\overline{h}(φ)]]_{m} : φ ∈ Φ\}⋉_{m}[[\overline{h}(ψ)]]_{m}. We conclude that \overline{h}(Φ)⊨_{val_{2}}^{l} \overline{h}(ψ). ⊓ ⊔

Corollary 3.2.21. The pair ⟨lVal, Ent_{lVal}⟩ constitutes a category of layered logic systems.

In the sequel, given an lVal-based layered logic Log : Voc → lVal, we shall denote Log(V) by ⟨L_{V} , M_{V} , S_{V} ⟩ and its induced local and global entailment consequences by ⊨_{V}^{l} and ⊨_{V}^{g} , respectively, for each vocabulary V.

Below, we study a few interesting properties of Lang_{lVal}.

Proposition 3.2.22. Lang_{lVal} : lVal → Set is coadjoint. Namely, id_{L} : Lang_{lVal}(val) → L is a Lang_{lVal}-couniversal arrow for L, where val = ⟨L, lStr(L), id_{lStr(L)}⟩.

Proof. Let ⟨L′, M′, S′⟩ be a layered valuation system and \overline{h} : L′ → L a function. We have to show that there exists a unique map \underline{h} : M′ → lStr(L) such that h : ⟨L′, M′, S′⟩ → val is an lVal-morphism. Obviously, by definition of val we must have \underline{h}(m′) = S′(m′)|\overline{h}, for every m′ ∈ M′, and the result follows. ⊓ ⊔

Moreover, Lang_{lVal} can also be easily shown to be a cofibration.

Proposition 3.2.23. Lang_{lVal} : lVal → Set is a cofibration. Namely, h : ⟨L, M, S⟩ → ⟨L′, M′, S′⟩ is a cocartesian lift of ⟨⟨L, M, S⟩, \overline{h} : L → L′⟩, where:

• M′ is the class of all pairs ⟨m, ϑ⟩ such that ϑ : (L′ \ \overline{h}(L)) → T_{m} is a function and m ∈ M satisfies the following condition, for every φ_{1}, φ_{2} ∈ L:

– if \overline{h}(φ_{1}) = \overline{h}(φ_{2}) then [[φ_{1}]]_{m} = [[φ_{2}]]_{m};

• S′(m, ϑ) = ⟨T_{m}, [[ ]], ⋉_{m}⟩ with:

– [[\overline{h}(φ)]] = [[φ]]_{m}, for every φ ∈ L;

– [[ψ]] = ϑ(ψ), for every ψ ∈ (L′ \ \overline{h}(L));

• \underline{h}(m, ϑ) = m.

Proof. Clearly, ⟨L′, M′, S′⟩ is a valuation sytem. In particular, S′ is well defined due to the conditions imposed to models in M′. Moreover, by definition, h : ⟨L, M, S⟩ → ⟨L′, M′, S′⟩ trivially establishes an lVal-morphism.

To prove the universal property, assume that g : ⟨L, M, S⟩ → ⟨L′′, M′′, S′′⟩ is an lValmorphism and \overline{f} : L′ → L′′ a function such that (\overline{f} ∘ \overline{h}) = \overline{g}. We have to prove that there is a unique map \underline{f} : M′′ → M′ such that f : ⟨L′, M′, S′⟩ → ⟨L′′, M′′, S′′⟩ is an lVal-morphism and (\underline{h} ∘ \underline{f}) = \underline{g}. Given m′′ ∈ M′′, we first note that if \overline{h}(φ_{1}) = \overline{h}(φ_{2}) then also \overline{g}(φ_{1}) = \overline{g}(φ_{2}) and thus [[φ_{1}]]_{\underline{g}(m′′)} = [[φ_{2}]]_{\underline{g}(m′′)}. Furthermore, if ψ ∈ (L′ \ \overline{h}(L)), we must have [[\overline{f}(ψ)]]_{m′′} = [[ψ]]_{\underline{f}(m′′)}. Thus, necessarily, \underline{f}(m′′) = ⟨\underline{g}(m′′), ([[ ]]_{m′′} ∘ \overline{f})⟩. ⊓ ⊔

3.2.2 Deduction: layered inference systems

We shall now turn to logics based on layered inference systems. That is, we shall enrich the setting of inference systems by distinguishing local from global inferences. As expected, they constitute a category of finitary layered logic systems. Just recall that a finitary inference over L is a pair i = ⟨Prem(i), Conc(i)⟩ where Prem(i) ∈ ℘_{fin}(L) is the set of premises of i and Conc(i) ∈ L its conclusion. Another reminder for the fact that we denote the set of all finitary inferences over L by Infer_{fin}(L).

Definition 3.2.24. A layered inference system is a tuple inf = ⟨L, I, gI⟩ where (I ∪gI) ⊆ Infer_{fin}(L), such that the following condition holds:

• Prem(i) ≠ ∅, for every i ∈ gI.

In the definition above, gI stands for the set of inferences that are of a strictly global nature, while I denotes the set of all other allowed inferences. This definition is equivalent to other proposals in previous work, namely the Hilbert calculi of [SSC99, ZSS99]. Since local and global consequences must have the same theorems, the requirement that global inferences must have premises is quite natural.

Layered inference systems morphisms are not surprising.

Definition 3.2.25. Let inf_{k} = ⟨L_{k}, I_{k}, gI_{k}⟩ for k ∈ {1, 2} be layered inference systems. A layered inference systems morphism h : inf_{1} → inf_{2} is a function h : L_{1} → L_{2} such that the following conditions hold:

• h(i) ∈ I_{2} for every i ∈ I_{1};

• h(i) ∈ gI_{2} for every i ∈ gI_{1};

Definition 3.2.26. Layered inference systems and their morphisms constitute the category lInf, with identity id_{⟨L,I,gI⟩} the identity function on L, and (g ∘ f) the usual composition of functions.

Proposition 3.2.27. lInf is indeed a category.

Proof. Trivial. ⊓ ⊔

Local and global deducibility can be defined in general terms for any given layered inference system. Indeed, we shall show that lInf is indeed a category of finitary layered logic systems by establishing two functors (a local and a global one) to Inf, and then proving that the pairing of their compositions with Ded_{Inf} : Inf → Csy_{fin} can be extended to a functor to lCsy_{fin}.

Definition 3.2.28. The maps

• Glob_{Inf}(⟨L, I, gI⟩) = ⟨L, (I ∪ gI)⟩;

• Glob_{Inf}(h : inf_{1} → inf_{2}) = h,

constitute the functor Glob_{Inf} : lInf → Inf.

In the sequel, for every layered inference system inf = ⟨L, I, gI⟩, we shall use the notation inf^{g} = ⟨L, (I ∪ gI)⟩.

Proposition 3.2.29. Glob_{Inf} : lInf → Inf is indeed a functor.

Proof. Straightforward, by definition of lInf. ⊓ ⊔

Definition 3.2.30. Let inf = ⟨L, I, gI⟩ be a layered inference system. The global deducibility consequence system csy_{inf}^{g} = ⟨L, ⊢_{inf}^{g} ⟩ induced by inf is precisely Ded_{Inf}(inf^{g}).

Namely, ⊢_{inf}^{g} = ⊢_{inf}_{g}, i.e., Φ^{⊢_{inf}^{g}} is the least superset of Φ that is closed for all inferences in (I ∪ gI).

For convenience, in this case, we use Ω_{inf}^{g} to denote Ω_{(I∪gI)}.

We can now proceed to establish local deducibility, by promoting global theoremhood through to local inferences.

Definition 3.2.31. The maps

• Loc_{Inf}(⟨L, I, gI⟩) = ⟨L, (I ∪ {⟨∅, φ⟩ : ⊢_{⟨L,I,gI⟩}^{g}φ})⟩;

• Loc_{Inf}(h : inf_{1} → inf_{2}) = h,

constitute the functor Loc_{Inf} : lInf → Inf.

In the sequel, for every layered inference system inf = ⟨L, I, gI⟩, we shall use the notation inf^{l} = ⟨L, (I ∪ {⟨∅, φ⟩ : ⊢_{inf}^{g} φ})⟩.

Proposition 3.2.32. Loc_{Inf} : lInf → Inf is indeed a functor.

Proof. We just prove that h : inf _{1}^{l} → inf _{2}^{l} is indeed an lInf-morphism whenever h : inf_{1} → inf_{2} is an lInf-morphism. Indeed, letting inf_{k} = ⟨L_{k}, I_{k}, gI_{k}⟩ for k ∈ {1, 2}, we know that h(I_{1}) ⊆ I_{2}. We just have to note that, additionally, h(∅^{⊢_{inf_{1}}^{g}}) ⊆ ∅^{⊢_{inf_{2}}^{g}}. In fact, h : csy_{inf_{1}}^{g} → csy_{inf_{2}}^{g} is a Csy-morphism and, in particular, h(∅^{⊢_{inf_{1}}^{g}}) ⊆ h(∅)^{⊢_{inf_{2}}^{g}} = ∅^{⊢_{inf_{2}}^{g}}.

⊓ ⊔ Definition 3.2.33. Let inf = ⟨L, I, gI⟩ be a layered inference system. The local deducibility consequence system csy_{inf}^{l} = ⟨L, ⊢_{inf}^{l} ⟩ induced by inf is precisely Der_{Inf}(inf^{l}).

Namely, ⊢_{inf}^{l} = ⊢_{inf^{l}}, i.e., Φ^{⊢_{inf}^{l}} is the least superset of Φ that is closed for all inferences in (I ∪ {⟨∅, φ⟩ : ⊢_{inf}^{g} φ}).

Obviously, equivalently, Φ^{⊢_{inf}^{l}} is the least superset of (Φ ∪ ∅^{⊢_{inf}^{g}}) that is closed for all inferences in I. For convenience, in this case, we use Ω_{inf}_{l} to denote Ω_{(I∪{⟨∅,φ⟩:⊢inf^{g} φ})}.

Definition 3.2.34. The maps

• Ded_{lInf}(⟨L, I, gI⟩) = ⟨L, ⊢_{⟨L,I,gI⟩}^{l}, ⊢_{⟨L,I,gI⟩}^{g}⟩;

• Ded_{lInf}(h : inf_{1} → inf_{2}) = h,

constitute the functor Ded_{lInf} : lInf → lCsy_{fin}.

Proposition 3.2.35. Ded_{lInf} : lInf → lCsy_{fin} is indeed a functor.

Proof. The only non trivial observation is that Ded_{lInf}(inf ) is indeed a layered consequence system, whenever inf is a layered inference system. Thus, we shall prove that for every inf = ⟨L, I, gI⟩ both ⊢_{inf}^{l} ⊑_{L} ⊢_{inf}^{g} and ∅^{⊢_{inf}^{g}} ⊆ ∅^{⊢_{inf}^{l}}. So, we use 3.2.33 and 3.2.30, to show

that Φ^{⊢_{inf}^{l}} ⊆ Φ^{⊢_{inf}^{g}}. Clearly, it suffices to prove that Ω_{inf^{l}}^{n}(Φ) ⊆ Φ^{⊢_{inf}^{g}} for every n ∈ IN. We shall proceed by induction on n.

• Base case: n = 0.

Clearly Ω_{inf^{l}}^{0}(Φ) = Φ ⊆ Φ^{⊢_{inf}^{g}}, just by extensiveness of ⊢_{inf}^{g} .

• Step: n > 0 and the induction hypothesis asserts that Ω_{inf^{l}}^{n−1}(Φ) ⊆ Φ^{⊢_{inf}^{g}}. Clearly, Ω_{inf^{l}}^{n}(Φ) = Ω_{inf}_{l}(Ω_{inf^{l}}^{n−1}(Φ)). Since Ω_{inf}_{l} is order-preserving, we also have that

Ω_{inf^{l}}^{n}(Φ) ⊆ Ω_{inf}_{l}(Φ^{⊢_{inf}^{g}}) = (Φ^{⊢_{inf}^{g}} ∪ ∅^{⊢_{inf}^{g}} ∪ {Conc(i) : i ∈ I, Prem(i) ⊆ Φ^{⊢_{inf}^{g}} }). Ob-

viously, ∅^{⊢_{inf}^{g}} ⊆ Φ^{⊢_{inf}^{g}} by just the monotonicity of ⊢_{inf}^{g} . Moreover, easily, {Conc(i) :

i ∈ I, Prem(i) ⊆ Φ^{⊢_{inf}^{g}} } ⊆ Ω_{inf}^{g}(Φ^{⊢_{inf}^{g}}) = Φ^{⊢_{inf}^{g}}. We conclude that Ω_{inf^{l}}^{n}(Φ) ⊆ Φ^{⊢_{inf}^{g}}.

Regarding theorems, necessarily, ∅^{⊢_{inf}^{l}} = Ω_{inf}_{l}(∅^{⊢_{inf}^{l}}) = (∅^{⊢_{inf}^{l}} ∪ ∅^{⊢_{inf}^{g}} ∪ {Conc(i) : i ∈

I, Prem(i) ⊆ ∅^{⊢_{inf}^{l}} }), and therefore ∅^{⊢_{inf}^{g}} ⊆ ∅^{⊢_{inf}^{l}}. ⊓ ⊔

Corollary 3.2.36. The pair ⟨lInf, Ded_{lInf}⟩ constitutes a category of finitary layered logic systems.

In the sequel, given a lInf-based layered logic Log : Voc → lInf, we shall denote Log(V) by ⟨L_{V} , I_{V} , gI_{V} ⟩ and its induced local and global deducibility consequences by ⊢_{V}^{l} and ⊢_{V}^{g} , respectively, for each vocabulary V.

Again, we shall consider systems of logic that involve both a layered inference system and a layered valuation system.

Definition 3.2.37. Let val = ⟨L, M, S⟩ be a layered valuation system. An inference i ∈ Infer(L) is said to be locally sound with respect to val if Prem(i)⊨_{val}^{l} Conc(i). Analogously, i is said to be globally sound with respect to val if Prem(i)⊨_{val}^{g} Conc(i).

Notice that an immediate consequence is that locally sound inferences are also globally sound.

Proposition 3.2.38. Let inf = ⟨L, I, gI⟩ and val = ⟨L, M, S⟩ be, respectively, layered inference and valuation systems. Then, the pair ⟨inf , val⟩ is sound if and only if every inference in I is locally sound and every inference in gI is globally sound, with respect to val.

Proof. Assume that ⟨inf , val⟩ is sound, and let i ∈ I. Clearly, by definition of ⊢_{inf}^{l} , Prem(i)⊢_{inf}^{l} Conc(i). Thus, by soundness, Prem(i)⊨_{val}^{l} Conc(i) and i is locally sound. Similarly, if i ∈ gI, by definition of ⊢_{inf}^{g} , we have that Prem(i)⊢_{inf}^{g} Conc(i). Hence, by soundness, Prem(i)⊨_{val}^{g} Conc(i) and therefore i is globally sound.

Suppose now that every inference in I is locally sound and every inference in gI is globally sound with respect to val, and let Φ ⊆ L. To show that ⟨inf , val⟩ is globally sound, we can use 2.2.52, provided that we observe that every local rule is also globally sound. This follows immediately from the fact that ⊨_{val}^{l} ⊑_{L} ⊨_{val}^{g}. Therefore, every rule in (I ∪ gI) is globally sound with respect to val, and Φ^{⊢_{inf}^{g}} ⊆ Φ^{⊨_{val}^{g}}.

Locally, we can now show by induction on n ∈ IN, that every Ω_{inf^{l}}^{n}(Φ) ⊆ Φ^{⊨_{val}^{l}}.

• Base case: n = 0.

Easily, Ω_{inf^{l}}^{0}(Φ) = Φ ⊆ Φ^{⊨_{val}^{l}}, just by extensiveness of ⊨_{val}^{l}.

• Step: n > 0 and the induction hypothesis asserts that Ω_{inf^{l}}^{n−1}(Φ) ⊆ Φ^{⊨_{val}^{l}}. By definition, Ω_{inf^{l}}^{n}(Φ) = Ω_{inf}_{l}(Ω_{inf^{l}}^{n−1}(Φ)). Using the hypothesis and the fact that

Ω_{inf}_{l} is order-preserving, then, Ω_{inf^{l}}^{n}(Φ) ⊆ Ω_{inf}_{l}(Φ^{⊨_{val}^{l}}) = (Φ^{⊨_{val}^{l}} ∪ ∅^{⊢_{inf}^{g}} ∪ {Conc(i) :

i ∈ I, Prem(i)) ⊆ Φ^{⊨_{val}^{l}}}. Easily, ∅^{⊢_{inf}^{g}} ⊆ ∅^{⊨_{val}^{g}} = ∅^{⊨_{val}^{l}} ⊆ Φ^{⊨_{val}^{l}}. Moreover, every rule in I is locally sound and therefore {Conc(i) : i ∈ I, Prem(i) ⊆ Φ^{⊨_{val}^{l}}} ⊆ Φ^{⊨_{val}^{l}} by just using the monotonicity and idempotence of ⊨_{val}^{l}.

Thus, also Φ^{⊢_{inf}^{l}} = (⋃_{n∈IN} Ω_{inf^{l}}^{n}(Φ)) ⊆ Φ^{⊨_{val}^{l}}. ⊓ ⊔

Once again, we are interested in the properties of the forgetful functor Lang_{lInf}. Obviously, Lang_{lInf} = (Lang_{lCsy_{fin}} ∘ Ded_{lInf}). To start with, Lang_{lInf} : lInf → Set is a cofibration.

Proposition 3.2.39. Lang_{lInf} : lInf → Set is a cofibration. Namely, h : ⟨L, I, gI⟩ → ⟨L′, h(I), h(gI)⟩ is a cocartesian lift of ⟨⟨L, I, gI⟩, h : L → L′⟩.

Proof. Clearly, the triple ⟨L′, h(I), h(gI)⟩ is a layered inference system and h : ⟨L, I, gI⟩ → ⟨L′, h(I), h(gI)⟩ a lInf-morphism. To prove the universal property, assume that g : ⟨L, I, gI⟩ → ⟨L′′, I′′, gI′′⟩ is a lInf-morphism and f : L′ → L′′ a function such that (f ∘ h) = g. Immediately, we have that f(h(I)) = g(I) ⊆ I′′ and f(h(gI)) = g(gI) ⊆ gI′′. Thus, f : ⟨L′, h(I), h(gI)⟩ → ⟨L′′, I′′, gI′′⟩ is a lInf-morphism and the result follows. ⊓ ⊔

Recall that in 3.1.24, we have shown that Lang_{lCsy_{fin}} is topological. Thus, among other properties, Lang_{lCsy_{fin}} is also coadjoint and a cofibration. Thus, although Ded_{lInf} can be shown not to be a cofibration, it still keeps some nice properties.

Proposition 3.2.40. Ded_{lInf} : lInf → lCsy_{fin} is coadjoint. Namely, id_{L} : Ded_{lInf}(inf ) → L is a Ded_{lInf}-couniversal arrow for the finitary layered consequence system ⟨L, ▷^{l}, ▷^{g}⟩, where inf = ⟨L, I, gI⟩, I = {i ∈ Infer_{fin}(L) : Prem(i) ▷^{l} Conc(i)} and gI = {i ∈ Infer_{fin}(L) : Prem(i) ▷^{g} Conc(i), Prem(i) ≠ ∅}. Furthermore, Ded_{lInf} is cocartesian with respect to the cofibrations Lang_{lInf} and Lang_{lCsy_{fin}}.

Proof. Trivially, since ⟨L, ▷^{l}, ▷^{g}⟩ is finitary, we have ⊢_{inf}^{l} = ▷^{l} and ⊢_{inf}^{g} = ▷^{g}. Easily, we have Φ^{▷^{g}} ∈ (I ∪ gI)(Φ). Moreover, if Ψ ∈ (I ∪ gI)(Φ) then Φ^{▷^{g}} ⊆ Ψ. In fact, let Φ ▷^{g} υ. By finitariness, there exists Υ ∈ ℘_{fin}(Φ) such that Υ ▷^{g} υ. Depending on whether Υ = ∅ or not, we have either ⟨∅, υ⟩ ∈ I or ⟨Υ, υ⟩ ∈ gI. Thus, in any case, ∅ ⊆ Υ ⊆ Φ ⊆ Ψ and Ω_{(I∪gI)}(Ψ) ⊆ Ψ together imply that υ ∈ Ψ. Locally, now, it is also obvious that Φ^{▷^{l}} ∈ (I ∪ {⟨∅, υ⟩ : ⊢_{inf}^{g} υ})(Φ). Let now Ψ ∈ (I ∪ {⟨∅, υ⟩ : ⊢_{inf}^{g} υ})(Φ). If Φ ▷^{l} υ then, by finitariness, there exists Υ ∈ ℘_{fin}(Φ) such that Υ ▷^{l} υ. Thus, ⟨Υ, υ⟩ ∈ I and thus υ ∈ Ψ.

As for the universal property, let inf′ = ⟨L′, I′, gI′⟩ be a layered inference system and g : ⟨L′, ⊢_{inf′}^{l} , ⊢_{inf′}^{g} ⟩ → ⟨L, ▷^{l}, ▷^{g}⟩ be an lCsy-morphism. Clearly, we have to prove that g : inf′ → inf is an lInf-morphism. But if i′ ∈ I′ then, obviously, Prem(i′)⊢_{inf′}^{l} Conc(i′) and g(Prem(i′)) ▷^{l} g(Conc(i′)). Analogously, if i′ ∈ gI′ then, obviously, Prem(i′)⊢_{inf′}^{g} Conc(i′) and g(Prem(i′)) ▷^{g} g(Conc(i′)).

To prove that Ded_{lInf} is cocartesian, according to 3.2.39, we need to show that the morphism h : Ded_{lInf}(⟨L, I, gI⟩) → Ded_{lInf}(⟨L′, h(I), h(gI)⟩) is a Lang_{lCsy_{fin}}-cocartesian lift of ⟨Ded_{lInf}(⟨L, I, gI⟩), h : L → L′⟩, where ⟨L, I, gI⟩ is any layered inference system. Let g : Ded_{lInf}(⟨L, I, gI⟩) → ⟨L′′, ▷^{′′l}, ▷^{′′g}⟩ be an lCsy_{fin}-morphism and f : L′ → L′′ a function such that (f ∘h) = g. All we need to show is that f : Ded_{lInf}(⟨L′, h(I), h(gI)⟩) → ⟨L′′, ▷^{′′l}, ▷^{′′g}⟩ is an lCsy_{fin}-morphism. Let Φ ⊆ L′. Globally, it is enough to show that f^{−1}(f(Φ)^{▷^{′′g}}) ∈ h(I ∪ gI)(Φ). Trivially, Φ ⊆ f^{−1}(f(Φ)) ⊆ f^{−1}(f(Φ)^{▷^{′′g}}) just by making use of the extensiveness of ▷^{′′g}. Assume now that i ∈ (I ∪ gI) and h(Prem(i)) ⊆ f^{−1}(f(Φ)^{▷^{′′g}}). Immediately, we have that f(Φ) ▷^{′′g}f(h(Prem(i))) = g(Prem(i)). But Prem(i)⊢_{⟨L,I,gI⟩}^{g}Conc(i), g is a morphism, and so we must have g(Prem(i))▷^{′′g} g(Conc(i)). Therefore, it must be the case that f(Φ) ▷^{′′g}f(h(Conc(i))) = g(Conc(i)), i.e., h(Conc(i)) ∈ f^{−1}(f(Φ)^{▷^{′′g}}). Locally, it is enough to show that f^{−1}(f(Φ)^{▷^{′′l}}) is a superset of Φ that contains ∅^{⊢_{⟨L′,h(I),h(gI)⟩}^{g}} and is closed for h(I). Trivially, Φ ⊆ f^{−1}(f(Φ)) ⊆ f^{−1}(f(Φ)^{▷^{′′l}}) just by the use of the extensiveness of ▷^{′′l}. Moreover, we have already shown that f(∅^{⊢_{⟨L′,h(I),h(gI)⟩}^{g}}) ⊆ f(∅)^{▷^{′′g}} = ∅^{▷^{′′g}}. Thus, ∅^{⊢_{⟨L′,h(I),h(gI)⟩}^{g}} ⊆ f^{−1}(∅^{▷^{′′g}}) ⊆ f^{−1}(f(Φ)^{▷^{′′l}}). Assume now that i ∈ I and h(Prem(i)) ⊆ f^{−1}(f(Φ)^{▷^{′′l}}). Immediately, we have that f(Φ) ▷^{′′l}f(h(Prem(i))) = g(Prem(i)). But Prem(i)⊢_{⟨L,I,gI⟩}^{l}Conc(i), g is a morphism, and we must have g(Prem(i)) ▷^{′′l} g(Conc(i)). Therefore, we have f(Φ) ▷^{′′l}f(h(Conc(i))) = g(Conc(i)), i.e., h(Conc(i)) ∈ f^{−1}(f(Φ)^{▷^{′′l}}). ⊓ ⊔

Furthermore, we have the following corollary. Note that id_{L} : Lang_{lCsy_{fin}}(⟨L, ⊤, ⊤⟩) → L is the obvious Lang_{lCsy_{fin}}-couniversal arrow for L.

Corollary 3.2.41. Lang_{lInf} : lInf → Set is coadjoint. Namely, id_{L} : Lang_{lInf}(inf ) → L is a Lang_{lInf}-couniversal arrow for L, where inf = ⟨L, Infer_{fin}(L), {i ∈ Infer_{fin}(L) : Prem(i) ≠ ∅}⟩.

3.3 Examples

We shall start by a careful analysis of both classical and intuitionistic propositional logics, i.e., we shall show that their layered presentation brings nothing new and local and global consequences coincide for both.

Example 3.3.1. Layered CPL. Given a set P propositional symbols, the language of CPL over P consists of the set L_{cpl}^{P} as defined in 2.3.1.

The layered valuation system of CPL considers exactly the same class of models M_{cpl}^{P} as in 2.3.1, that is, M_{cpl}^{P} = {0, 1}^{P}. It consists of val_{lcpl}^{P} = ⟨L_{cpl}^{P}, M_{cpl}^{P}, S_{lcpl}^{P}⟩ where, for each ϑ ∈ M_{cpl}^{P}, we have

S_{lcpl}^{P}(ϑ) = ⟨{0, 1}, ϑ^{∗}, ⋉⟩

where ∅^{⋉} = {1}^{⋉} = {1} and {0}^{⋉} = {0, 1}^{⋉} = {0, 1}. As we have noted before, ⋉ is ⋉_{ρ} for ρ = {⟨0, 0⟩, ⟨0, 1⟩, ⟨1, 1⟩}, the usual order on {0, 1}.

The layered inference system for CPL over P is quite trivial. It just has no global inferences. It consists of the triple inf _{lcpl}^{P} = ⟨L_{cpl}^{P}, I_{cpl}^{P}, ∅⟩, where ⟨L_{cpl}^{P}, I_{cpl}^{P}⟩ is the inference system for CPL as defined in 2.3.1.

Trivially, ⊨_{lcpl}^{gP} = ⊨_{cpl}^{P} and ⊢_{lcpl}^{gP} = ⊢_{cpl}^{P}. Moreover, since it is always the case that local consequence is weaker than global consequence, to show that the two coincide, we just need to prove that ⊨_{lcpl}^{gP} ⊑_{Lcpl^{P}} ⊨_{lcpl}^{lP} and ⊢_{lcpl}^{gP} ⊑_{Lcpl^{P}} ⊢_{lcpl}^{lP}. In terms of deduction the result

is quite trivial. With respect to semantic entailment, assume that Φ⊨_{lcpl}^{gP} ψ. By definition, since ∅^{⋉} = {1}, this means that ϑ^{∗}(ψ) = 1 whenever ϑ^{∗}(Φ) ⊆ {1}, for every valuation ϑ. Therefore, if ϑ^{∗}(Φ) ⊆ {1}, we have ϑ^{∗}(Φ)^{⋉} = {1} and easily ϑ^{∗}(Φ) ⋉ ϑ^{∗}(ψ). Otherwise, obviously, ϑ^{∗}(ψ) ∈ ϑ^{∗}(Φ)^{⋉} = {0, 1}. Hence, it is always the case that ϑ^{∗}(Φ) ⋉ ϑ^{∗}(ψ) and we have proved that Φ⊨_{lcpl}^{lP} ψ.

Of course, the same applies to the layered version of the algebraic semantics of CPL presented in 2.3.4. In this case, the result could be shown by capitalizing on the arguments above by using Birkhoff’s famous representation theorem of Boolean algebras in terms of products of the two-valued Boolean algebra [Bir67]. However, we shall not do that. Instead, we shall prove a similar result for intuitionistic propositional logic. Since Boolean algebras are always Heyting algebras, the proof we shall present also works for CPL.

Example 3.3.2. Layered IPL. Given a set P propositional symbols, the language of IPL over P consists of the set L_{ipl}^{P} as defined in 2.3.5.

The layered valuation system of IPL considers exactly the same class of models M_{cpl}^{P} as in 2.3.5, that is, M_{cpl}^{P} is the class of all pairs ⟨H, ϑ⟩ with H = ⟨H, ≤⟩ a Heyting algebra and ϑ : P → H a function. It consists of val_{lipl}^{P} = ⟨L_{ipl}^{P}, M_{ipl}^{P}, S_{lipl}^{P}⟩ where, for each ⟨H, ϑ⟩ ∈ M_{cpl}^{P}, we have

S_{lipl}^{P}(ϑ) = ⟨H, ϑ^{∗}, ⋉_{≤}⟩.

Please note that ∅^{⋉_{≤}} = U_{≤}(L_{≤}(∅)) = U_{≤}(H) = {⊤}.

The layered inference system for IPL over P is again quite trivial, having no global inferences. It consists of the triple inf _{lipl}^{P} = ⟨L_{ipl}^{P}, I_{ipl}^{P}, ∅⟩, where ⟨L_{ipl}^{P}, I_{ipl}^{P}⟩ is the inference system for IPL as defined in 2.3.5.

Trivially, ⊨_{lipl}^{gP} = ⊨_{ipl}^{P} and ⊢_{lipl}^{lP} = ⊢_{lipl}^{gP} = ⊢_{ipl}^{P}. Moreover, since it must be the case that ⊨_{lipl}^{lP} ⊑_{Lipl^{P}} ⊨_{lipl}^{gP}, to show that the two coincide, we just need to prove that ⊨_{lipl}^{gP} ⊑_{Lcpl^{P}} ⊨_{lipl}^{lP}.

From strong completeness, as stated in 2.3.5, we have that ⊨_{lipl}^{gP} = ⊨_{ipl}^{P} = ⊢_{ipl}^{P}. Moreover, since all the axioms are both locally and globally sound, it suffices to show that MP is sound with respect to local entailment, i.e., for every Heyting algebra H, {x, x → y} ⋉_{≤} y for every x, y ∈ H. Clearly, this amounts to showing that y ∈ U_{≤}(L_{≤}({x, x → y})). Thus, let z ∈ H be such that z ≤ x and z ≤ x → y. Therefore, z ≤ x ∧ (x → y) ≤ y, just by definition of meet and the properties of →.

We conclude that ⊢_{ipl}^{P} ⊑_{Lcpl^{P}} ⊨_{lipl}^{lP} and thus ⊨_{lipl}^{lP} = ⊨_{lipl}^{gP}.

We shall finally dwell on our motivating example. Let us see how to adequately set up propositional normal modal logic in terms of a layered logic system.

Example 3.3.3. Layered K. Given a set P propositional symbols, the language of K over P consists of the set L_{K}^{P} as defined in 2.3.6.

The layered valuation system of K considers exactly the same class of models M_{K}^{P} as in 2.3.6, that is, M_{K}^{P} is the class of all Kripke models ⟨W, R, ϑ⟩ over P. It consists of val_{lK}^{P} = ⟨L_{K}^{P} , M_{K}^{P} , S_{lK}^{P} ⟩ where, for each ⟨W, R, ϑ⟩ ∈ M_{K}^{P}, we have

S_{lK}^{P} (⟨W, R, ϑ⟩) = ⟨℘(W), ϑ^{∗}, ⋉_{⊆}⟩

where ϑ^{∗} : L_{K}^{P} → ℘(W) extends ϑ : P → ℘(W) by:

• ϑ^{∗}(p) = ϑ(p);

• ϑ^{∗}((¬ φ)) = (W \ ϑ^{∗}(φ));

• ϑ^{∗}((□φ)) = {w ∈ W : wRw′ implies w′ ∈ ϑ^{∗}(φ)};

• ϑ^{∗}((φ ⇒ ψ)) = ((W \ ϑ^{∗}(φ)) ∪ ϑ^{∗}(ψ));

• ϑ^{∗}((φ ∧ ψ)) = (ϑ^{∗}(φ) ∩ ϑ^{∗}(ψ));

• ϑ^{∗}((φ ∨ ψ)) = (ϑ^{∗}(φ) ∪ ϑ^{∗}(ψ)).

Clearly, we are considering the inclusion ordering ⊆ on ℘(W). Therefore, given X ⊆ ℘(W), we have that X^{⋉_{⊆}} = U_{⊆}(L_{⊆}(X)) = U_{⊆}(℘(⋂_{X∈X} X)) = {Y ⊆ W : (⋂_{X∈X} X) ⊆ Y }. In particular, ∅^{⋉_{⊆}} = {Y ⊆ W : W ⊆ Y } = {W }.

The layered inference system of K over P is now different. It has global inferences corresponding to necessitation. It consists of the triple inf _{lK}^{P} = ⟨L_{K}^{P} , I_{lK}^{P} , gI_{lK}^{P} ⟩, where I_{lK}^{P} contains all the axioms Ax01 -Ax11 of classical logic, the axiom of normality □-Norm and all the instances of the rule MP of modus ponens, while gI_{lK}^{P} just contains all the instances of the rule □-Nec of necessitation. Note that, still, (I_{lK}^{P} ∪ gI_{lK}^{P}) = I_{K}^{P}.

Trivially, for every m = ⟨W, R, ϑ⟩ ∈ M_{K}^{P} and φ ∈ L_{K}^{P}, ϑ^{∗}(φ) = {w ∈ W : m⊩_{w}^{P} φ}. Therefore, m⊩_{K}^{P} φ if and only if ϑ^{∗}(φ) = W, and we conclude that ⊨_{K}^{P} = ⊨_{K}^{gP} = ⊨_{lK}^{gP}. Note that the first of these consequences comes from 2.3.6, the second from remark 3.1.1 at the beginning of this Chapter, and the third is the local consequence induced by val_{lK}^{P}. Moreover, ⊨_{K}^{lP} = ⊨_{lK}^{lP} as we shall show. According to 3.1.1, Φ⊨_{K}^{lP} ψ if and only if m⊩_{w}^{P} ψ whenever m⊩_{w}^{P} Φ, for every model m and world w in m. But this is clearly equivalent to ϑ^{∗}(ψ) ∈ (⋂_{φ∈Φ} ϑ^{∗}(φ)), i.e., {ϑ^{∗}(φ) : φ ∈ Φ} ⋉_{⊆} ϑ^{∗}(ψ). The same is to say that Φ⊨_{lK}^{lP} ψ. It is also immediate that ⊢_{K}^{P} = ⊢_{lK}^{gP}.

The usual formulation of strong completeness for K can be stated, in our terms, precisely as follows:

• if Φ⊨_{lK}^{lP} ψ then there exist φ_{1}, …, φ_{m} ∈ Φ such that ⊢_{lK}^{lP} (φ_{1} ⇒ (… ⇒ (φ_{m} ⇒ ψ) …)).

This fact is usually proved by contraposition, by constructing a canonical model over the maximal consistent extensions of (¬(φ_{1} ⇒(…⇒(φ_{m} ⇒ψ) …))). See, for instance, [HC96]. This result easily implies both local and global (strong) completeness.

• ⊨_{lK}^{lP} ⊑_{LK^{P}} ⊢_{lK}^{lP}.

Let Φ⊨_{lK}^{lP} ψ. Then, ⊢_{lK}^{lP} (φ_{1} ⇒(…⇒(φ_{m} ⇒ψ) …)) for some φ_{1}, …, φ_{m} ∈ Φ. Clearly, it suffices to note that extensiveness implies that Φ⊢_{lK}^{lP} φ_{k} for k ∈ {1, …, m}, monotonicity implies that Φ⊢_{lK}^{lP} (φ_{1} ⇒ (… ⇒ (φ_{m} ⇒ ψ) …)), and therefore Φ⊢_{lK}^{lP} ψ, just by m applications of MP.

• ⊨_{lK}^{gP} ⊑_{LK^{P}} ⊢_{lK}^{gP}.

As noted in 3.1.1, if Φ⊨_{lK}^{gP} ψ then {□^{n}φ : φ ∈ Φ, n ∈ IN}⊨_{lK}^{lP} ψ. Thus, we have that ⊢_{lK}^{lP} (□^{n_{1}}φ_{1} ⇒ (… ⇒ (□^{n_{m}}φ_{m} ⇒ ψ) …)) for some φ_{1}, …, φ_{m} ∈ Φ and n_{1}, …, n_{m} ∈ IN. Clearly, it suffices to note that, for k ∈ {1, …, m}, extensiveness implies that Φ⊢_{lK}^{gP} φ_{k}, and therefore also Φ⊢_{lK}^{gP} □^{n_{k}}φ_{k}, just by n_{k} applications of □-Nec. Moreover, Φ⊢_{lK}^{gP} (□^{n_{1}}φ_{1} ⇒ (… ⇒ (□^{n_{m}}φ_{m} ⇒ ψ) …)) follows from monotonicity and the fact that ∅^{⊢_{lK}^{lP}} = ∅^{⊢_{lK}^{gP}}. Therefore, Φ⊢_{lK}^{gP} ψ, just by m applications of MP.

Clearly, the deduction theorem only holds locally. That is, Φ⊢_{lK}^{lP} (ψ ⇒ υ) if and only if (Φ ∪ {ψ})⊢_{lK}^{lP} υ. A proof of this fact can be trivially adapted from the corresponding proof for CPL, which can be found in any basic textbook on logic (see for instance [Men79]).

Another interesting fact, but very specific of the interaction between classical implication and normality axioms, is the fact that (⋃_{k∈K} Φ_{k})^{⊢_{lK}^{gP}} ⊆ (⋃_{k∈K} Φ_{k})^{⊢_{lK}^{lP}}, for every family {Φ_{k}}_{k∈K} of global theories. It all amounts to observing that □-Nec has just one premise, and the fact that applications of MP never give rise to new unboxed sentences. Given ψ ∈ Φ_{k_{1}} and (ψ ⇒ υ) ∈ Φ_{k_{2}} by MP we would get υ. But we must also have (□ψ) ∈ Φ_{k_{1}} and (□(ψ ⇒ υ)) ∈ Φ_{k_{2}}. Moreover, ((□(ψ ⇒ υ)) ⇒ ((□ψ) ⇒ (□υ))) is a normality axiom, and by using MP twice we also get (□υ).

For its similarity with modal logic, we refrain from detailing the case of PTL. A layered approach to the paraconsistent logic C_{1} is also possible, namely because the deduction theorem holds there. See, for instance, [Mar99]. However, still, local and global consequences would coincide. We postpone some further comments on C_{1} to the end of Chapter 5. Lukasiewicz three-valued system L_{3} is more interesting. We present here one of the possibilities, and refer the reader to [Wój88] for other alternatives.

Example 3.3.4. Layered L_{3}. Given a set P propositional symbols, the language of L_{3} over P consists of the set L_{l3}^{P} as defined in 2.3.9.

The layered valuation system of L3 considers exactly the same class of models M_{l3}^{P} as in 2.3.9, that is, \{0, \frac{1}{2}, 1\}^{P}. It consists of val_{ll3}^{P} = ⟨L_{l3}^{P} , M_{l3}^{P} , S_{ll3}^{P} ⟩ where, for each ϑ ∈ M_{l3}^{P}, we have

S_{ll3}^{P} (ϑ) = ⟨\{0, \frac{1}{2}, 1\}, ϑ^{∗}, ⋉⟩

where ⋉ is ⋉_{ρ} for ρ = \{⟨0, 0⟩, ⟨0, \frac{1}{2}⟩, ⟨0, 1⟩, ⟨\frac{1}{2}, \frac{1}{2}⟩, ⟨\frac{1}{2}, 1⟩, ⟨1, 1⟩\}, the usual order on \{0, \frac{1}{2}, 1\}.

Trivially, ⊨_{ll3}^{gP} = ⊨_{l3}^{P} = ⊢_{l3}^{P}. However, ⊨_{ll3}^{lP} is quite different. In fact, it is strictly weaker than global consequence, i.e., ⊨_{ll3}^{lP} ⊏ ⊨_{ll3}^{gP}. Due to the equality above, it suffices to note that MP is not locally sound. In fact, given p, q ∈ P, it is not true that {p, (p ⇒ q)}⊨_{ll3}^{lP} q. Just consider ϑ such that ϑ(p) = \frac{1}{2} and ϑ(q) = 0.

In fact, it is not very hard to show that Φ⊨_{ll3}^{lP} ψ if and only if there exist φ_{1}, …, φ_{m} ∈ Φ such that ⊨_{ll3}^{gP} ((φ_{1} ∧ (... ∧ (φ_{m−1} ∧ φ_{m})...)) ⇒ ψ). A proof of this fact can be easily obtained as a corollary of the fact that ⊨_{ll3}^{lP} could also have been generated by the valuation system that considers two possible kinds of structures over \{0, \frac{1}{2}, 1\}: those with just 1 distinguished, and those with both 1 and \frac{1}{2} distinguished [Wój88].

Let us now see what equational and predicate logics look like in the layered setting.

Example 3.3.5. Layered EQL. Let F be an (unsorted) equational signature and X a set of variables. The term language Term_{F} (X) and language L_{eql}^{F}(X) of EQL over F and X are defined precisely as in 2.3.10, from where we also fetch much of the notation below.

The layered semantics of EQL considers again, as models, the class M_{eql}^{F} of all F - algebras. The layered valuation system for EQL over F and X is then set to be val_{leql}^{⟨F,X⟩} =

⟨L_{eql}^{F}(X), M_{eql}^{F}, S_{leql}^{⟨F,X⟩}⟩ where, for each F -algebra T , we have

S_{leql}^{⟨F,X⟩}(T ) = ⟨℘(Asn(T , X)), [[ ]]^{T} , ⋉_{⊆}⟩

where [[ ]]^{T} : L_{eql}^{F}(X) → ℘(Asn(T , X)) is defined by:

• [[t_{1} = t_{2}]]^{T} = {ρ ∈ Asn(T , X) : T ⊩_{ρ}^{F} t_{1} = t_{2}}.

Clearly, we are considering the inclusion ordering ⊆ on ℘(Asn(T , X)). Thus, in particular, ∅^{⋉_{⊆}} = Asn(T , X) and ⊨_{eql}^{⟨F,X⟩} = ⊨_{leql}^{g⟨F,X⟩}. Moreover, easily, Φ⊨_{leql}^{l⟨F,X⟩}t_{1} = t_{2} if, for every T ∈ M_{eql}^{F} and ρ ∈ Asn(T , X), T ⊩_{ρ}^{F} Φ implies that T ⊩_{ρ}^{F} t_{1} = t_{2}.

The layered inference system inf _{leql}^{⟨F,X⟩} = ⟨L_{eql}^{F}(X), I_{leql}^{⟨F,X⟩}, gI_{leql}^{⟨F,X⟩}⟩ for EQL distin-

guishes the rule of substitution as global only. That is, we have I_{leql}^{⟨F,X⟩} consisting of

all instances of Reflex, Symm, Trans and Congr, and gI_{leql}^{⟨F,X⟩} consisting of all instances of Subst.

Soundness and completeness results are well known also for layered equational logic [Bir67, EM85], although not exactly proved explicitly as such. In particular, completeness in the local case can be proved by using the quotient term algebra, but now on the congruence generated by local deduction. Clearly, only the global part requires substitution.

Let us now reanalyze first-order logic on the layered setting.

Example 3.3.6. Layered FOL. Let ⟨F, P ⟩ be a first-order signature and X a set of variables, as considered in 2.3.11. The term language Term_{F} (X) and language L_{fol}^{⟨F,P ⟩}(X) of FOL over ⟨F, P ⟩ and X are defined as before.

The layered semantics of FOL considers again, as models, the class M_{fol}^{⟨F,P ⟩} of all ⟨F, P ⟩-structures. The layered valuation system for FOL over ⟨F, P ⟩ and X is then set to be val_{lfol}^{⟨F,P,X⟩} = ⟨L_{fol}^{⟨F,P ⟩}(X), M_{fol}^{⟨F,P ⟩}, S_{lfol}^{⟨F,P,X⟩}⟩ where, for each ⟨F, P ⟩-structure T , we have

S_{lfol}^{⟨F,P,X⟩}(T ) = ⟨℘(Asn(T , X)), [[ ]]^{T} , ⋉_{⊆}⟩

where [[ ]]^{T} : L_{fol}^{⟨F,P ⟩}(X) → ℘(Asn(T , X)) is defined by:

• [[φ]]^{T} = {ρ ∈ Asn(T , X) : T ⊩_{ρ}^{⟨F,P ⟩}φ}.

Clearly, again, we are considering the inclusion ordering ⊆ on ℘(Asn(T , X)). Thus, in particular, ∅^{⋉_{⊆}} = Asn(T , X) and ⊨_{fol}^{⟨F,P,X⟩} = ⊨_{lfol}^{g⟨F,P,X⟩}. Moreover, easily, Φ⊨_{lfol}^{l⟨F,P,X⟩}φ if,

for every T ∈ M_{fol}^{⟨F,P ⟩} and ρ ∈ Asn(T , X), T ⊩_{ρ}^{⟨F,P ⟩}Φ implies that T ⊩_{ρ}^{⟨F,P ⟩}φ.

The layered inference system inf _{lfol}^{⟨F,P,X⟩} = ⟨L_{fol}^{⟨F,P ⟩}(X), I_{lfol}^{⟨F,P,X⟩}, gI_{lfol}^{⟨F,P,X⟩}⟩ for FOL dis-

tinguishes the rule of generalization as global only. That is, we have I_{lfol}^{⟨F,P,X⟩} consisting of

all the inferences of I_{fol}^{⟨F,P,X⟩} except those corresponding to generalization, and gI_{lfol}^{⟨F,P,X⟩} consisting precisely of all pairs ⟨{φ}, (∀x φ)⟩.

As we have already mentioned in 2.3.11, Gödel’s completeness theorem corresponds precisely to soundness and completeness of global reasoning in FOL. Soundness being trivial, let us focus attention on completeness. Although completeness of FOL was first proved by Gödel, the most widespread proof of this result is, instead, the one discovered by Henkin a few years later. See [Hen96] for a detailed explanation. Essentially, it involves proving a global model existence property: “every global consistent set of sentences Φ is satisfied by some first-order structure T ”. This task is accomplished by building a complete consistent extension of Φ plus a set of witness sentences over a signature enlarged with suitable constant symbols [Men79].

However, in fact, this construction also embodies a proof of a local model existence property: “every local consistent set of sentences Φ is satisfied by some first-order structure T and assignment ρ”. Just consider \overline{Φ} to be the set of all sentences in Φ, where free occurrences of variables are uniformly replaced by “new” constant symbols. Easily, Φ is locally satisfiable (by some structure and assignment) if and only if \overline{Φ} is globally satisfiable (by some structure). Moreover, Φ is local consistent if and only if \overline{Φ} is global consistent. This last remark just amounts to the classic result that \overline{φ} is a theorem of FOL if and only if φ is. This shows that the two versions of the model existence property are in fact equivalent.

It is now immediate that local completeness follows from the local version of the model existence property.

3.4 Remarks

We shall briefly summarize the contents of this Chapter. We have recalled the distinction between local and global reasoning, namely in the context of modal logic, and have decided to give it an explicit treatment. Thus, we have introduced layered consequence systems in such a way that they reflect the expected relationship between the global and local layers. The new category lCsy was then compared with Csy, both via local and global forgetful functors. Expectedly, while all properties of the global part carry on smoothly, the same does not happen with relation to the local part. This fact is well expressed by the topologicity of the functor Glob and the non-topologicity of Loc. Yet, the forgetful functor from lCsy to Set could still be shown to be topological, and hence all layered consequence operations over a fixed set still form a complete lattice. Finitariness was also brought into play and shown to behave essentially as in the non-layered case. In particular, finitary layered consequence systems again constitute a coreflective full subcategory of Csy, and the class of all finitary layered consequence operations over a fixed set still form a complete lattice, which is not a complete sub-lattice of the previous. We also explored the notion of logical equivalence in this layered setting and showed that it coincides with logical equivalence at the local level. Then, we defined the abstract notion of category of layered logic systems as the source category of any functor to lCsy, and used indexing to introduce the corresponding general notion of layered logic. Moreover, we studied in some detail two categories of layered logic systems. On the model-theoretic side we enriched the notion of valuation system by adding an explicit internal consequence to the previous notion of interpretation structure. We showed that this procedure abstracts from the usual requirements of an ordered or lattice structure on truth values. This new category lVal of layered valuation systems was then studied, namely by exploring its relationship with Val. Among other things, its syntactic forgetful functor could still be shown to constitute both a coadjoint and a cofibration. Proof-theoretically, we enriched our previous inference systems by explicitly distinguishing between local and global inferences. The new category lInf of layered inference systems was then shown to relate smoothly to Inf. The Chapter finished with a set of representation examples of well-known systems of logic in the layered setting. Special emphasis was put on modal and first-order systems, for their naturalness in this context, and to classical and intuitionistic propositional logics, for being among those special cases where the two layers of reasoning are indistinguishable.

Synchronization

Up to now, it may seem that we have completely neglected our main goal of studying logic combination. But that was unavoidable. As explained before, we firmly believe that there is no point in trying to study the combination of any kinds of systems without knowing exactly what a system is, and how systems relate to each other. That was the role of the two preceding Chapters regarding our first combination mechanism: synchronization. So far, we have proceeded to the accurate detachment of our working categories of logic systems and logics, therefore establishing our synchronization framework. Again, we emphasize our combination methodology: we advocate that combinations should be carried out categorially, and have a universal characterization. In the case of synchronization, of course, we capitalize on previous work. That the natural setting for defining synchronization is the category of consequence systems just follows from its very first characterization in [SSC97a]. The intuition is that if we synchronize two given consequence systems, we obtain a new consequence system that extends the two given systems in a minimal way and respects all the constraints one may require. Of course, if the synchronization is free, we obtain exactly the least consequence operation over the disjoint union of the two languages that extends the two given consequences. In general, this construction is not very interesting in logical terms. However, in its constrained form, it turns out to be quite powerful as we shall show later on in Chapter 6. The idea is that a constraint specifies a set of pairs of formulas to be shared (synchronized) [SSC97a, SSC97b]. In this case, by sharing sentences, we typically have to perform a quotient, both in the obtained language and consequence. As we shall see, the result is similar to adding to the free synchronized consequence a set of bridging inferences corresponding to each pair of shared sentences. This picture is easily seen to be a particular case of the more general form of constraining synchronization used in [SSC97b]. The extension of the mechanism to the layered case can be expected to generalize these ideas in a smooth way.

The Chapter is organized as follows. First, we shall define free synchronization of consequence systems and, by analogy, of layered consequence systems. Both shall be given a similar categorial characterization, in Csy and lCsy. Then, capitalizing on the abstract notions of logic system and layered logic system, and on this categorial characterization, we shall introduce the notions of suitability and exactness for free synchronization. Suitability will mean that the corresponding construction in a category of (layered) logic systems coincides at the level of syntax with the construction for the corresponding (layered) consequence systems. Exactness is much stronger and will require that the whole construction in a category of (layered) logic systems indeed matches the construction for the corresponding (layered) consequence systems. Regarding constrained synchronization, we shall start by defining it at the level of consequence systems and then show that it indeed corresponds to the addition of corresponding mixed rules to the free construction. Analogously, we shall define constrained synchronization of layered consequence systems, and prove that in both cases the construction corresponds to a syntax-driven F -pushout. The notions of suitability and exactness for synchronization, in general, will then be introduced by suitably generalizing the free case. Next, we shall explore synchronization in our working categories of (layered) logic systems. All the proof-theoretic categories (Inf and lInf) and model-theoretic categories (Sat, Val and lVal) shall be shown to be suitable for synchronization and the corresponding constructions made explicit. However, only Inf and lInf fulfill exactness. We shall illustrate synchronization by means of a simple example involving temporal and equational logics. We shall conclude with the study of two fundamental transference results, soundness and completeness, in all three cases Inf/Sat, Inf/Val and lInf/lVal. Soundness preservation shall appear as a trivial corollary of the suitability for synchronization of all these categories, and moreover of the exactness of Inf and lInf. Regarding completeness preservation, however, due to the non-exactness of the categories Sat, Val and lVal a general result cannot be expected. Still, we shall be able to prove a general result in the presence of fullness, a requirement that guarantees that one does not throw away essential models during the synchronization process.

4.1 General definition

Synchronization is a combination mechanism whose set-up is made at the level of consequence systems, logic systems and logics as presented in Chapter 2. We shall first introduce the construction at this level, and then extend it to the layered setting of Chapter 3.

Intuitively, if we synchronize two given consequence systems, we obtain a new consequence system that extends the two given systems in a minimal way.

4.1.1 Free synchronization

This construction corresponds to the simplest case of synchronization. No constraints being made, the language shall be the disjoint union of the two given languages, and the consequence operation the least possible that still extends the given ones. In other words, given two consequence systems csy_{1} and csy_{2}, their free synchronization corresponds to the least consequence system (csy_{1} ▷◁ csy_{2}) that “contains” both csy_{1} and csy_{2}. Of course, this should be read as meaning that (csy_{1} ▷◁ csy_{2}) is a sum in Csy.

Definition 4.1.1. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems. The free synchronization of csy_{1} and csy_{2} is (csy_{1} ▷◁ csy_{2}) = ⟨(L_{1} ▷◁ L_{2}), ▷⟩, where:

• (L_{1} ▷◁ L_{2}) = ({φ_{1} : φ ∈ L_{1}} ∪ {ψ_{2} : ψ ∈ L_{2}});

• Υ^{▷} is the least superset of Υ that is closed for I_{▷◁}, where:

I_{▷◁} = ({⟨Φ_{1}, φ_{1}⟩ : Φ ▷_{1} φ} ∪ {⟨Ψ_{2}, ψ_{2}⟩ : Ψ ▷_{2} ψ}).

Of course, in Φ_{1} and Ψ_{2}, the subscript is to be applied to each of its elements.

Proposition 4.1.2. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems. Then, {ι_{k} : csy_{k} → (csy_{1} ▷◁ csy_{2})}_{k∈{1,2}}, with each ι_{k}(υ) = υ_{k}, is a sum in Csy.

Proof. In 2.1.17 we have shown that Lang_{Csy} : Csy → Set is topological. Moreover, Set is well known to be cocomplete, and thus also Csy is cocomplete. In particular, since the sink {ι_{k} : L_{k} → (L_{1} ▷◁ L_{2})}_{k∈{1,2}} is a sum in Set, (csy_{1} ▷◁ csy_{2}) = ⟨(L_{1} ▷◁ L_{2}), ▷⟩ is a sum in Csy if ▷ = ⊓_{(L_{1}▷◁L_{2})}{▷′ : both ι_{k} : csy_{k} → ⟨(L_{1} ▷◁ L_{2}), ▷′⟩ are Csy-morphisms}. That is, according to 2.1.19, for every Φ ⊆ L_{1} and Ψ ⊆ L_{2}, (Φ_{1} ∪ Ψ_{2})^{▷} = ({φ_{1} : Φ ▷_{1} φ} ∪ {ψ_{2} : Ψ ▷_{2} ψ}). Hence, the result follows trivially. ⊓ ⊔

It is clear that this combination mechanism is not very flexible. One just puts together the languages and the deductive power of the two systems. But no kind of interaction between them is allowed. Still, it is worth studying, not only as a guiding step towards the much higher level mechanism of fibring, but also because it is possible to obtain interesting combinations by restricting free synchronization. In that case, free synchronization is just the unrestricted version of a broader mechanism. We shall postpone this discussion until a little further on, in order to extend the mechanism of free synchronization also to the layered setting.

Definition 4.1.3. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems. The free synchronization of csy_{1} and csy_{2} is (csy_{1} ▷◁ csy_{2}) = ⟨(L_{1} ▷◁ L_{2}), ▷^{l}, ▷^{g}⟩, where ⟨(L_{1} ▷◁ L_{2}), ▷^{l}⟩ = (csy_{1}^{l} ▷◁ csy_{2}^{l}) and ⟨(L_{1} ▷◁ L_{2}), ▷^{g}⟩ = (csy_{1}^{g} ▷◁ csy_{2}^{g}). That is:

• Υ^{▷^{l}} is the least superset of Υ that is closed for I_{l▷◁}, where:

I_{l▷◁} = ({⟨Φ_{1}, φ_{1}⟩ : Φ ▷_{1}^{l} φ} ∪ {⟨Ψ_{2}, ψ_{2}⟩ : Ψ ▷_{2}^{l} ψ});

• Υ^{▷^{g}} is the least superset of Υ that is closed for I_{g▷◁}, where:

I_{g▷◁} = ({⟨Φ_{1}, φ_{1}⟩ : Φ ▷_{1}^{g} φ} ∪ {⟨Ψ_{2}, ψ_{2}⟩ : Ψ ▷_{2}^{g} ψ}).

Proposition 4.1.4. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems. Then, {ι_{k} : csy_{k} → (csy_{1} ▷◁ csy_{2})}_{k∈{1,2}}, with each ι_{k}(υ) = υ_{k}, is a sum in lCsy.

Proof. In 3.1.16 we have shown that Lang_{lCsy} : lCsy → Set is topological. Thus, lCsy is cocomplete. In particular, (csy_{1} ▷◁ csy_{2}) = ⟨(L_{1} ▷◁ L_{2}), ▷^{l}, ▷^{g}⟩ is a sum in lCsy if ⟨▷^{l}, ▷^{g}⟩ = ⊓_{(L_{1}▷◁L_{2})}{⟨▷^{′l}, ▷^{′g}⟩ : both ι_{k} : csy_{k} → ⟨(L_{1} ▷◁ L_{2}), ▷^{′l}, ▷^{′g}⟩ are lCsy-morphisms}. The result follows easily from 4.1.2 and 3.1.18. ⊓ ⊔

Free synchronization in an arbitrary category A of (layered) logic systems just exploits the construction for consequence system. The same applies to the layered setting. First of all, binary sums must exist in A, and in such a way that the language of a binary sum in A corresponds to a binary sum of the languages. Obviously, the construction is precise if also the consequence operation of a sum in A is a sum of the consequence operations.

Definition 4.1.5. A category of (layered) logic systems ⟨A, (l)Cn⟩ is suitable for free synchronization provided that A has binary sums and Lang_{A} preserves them. Moreover, ⟨A, (l)Cn⟩ is exact for free synchronization if, in addition, also (l)Cn preserves binary sums.

Definition 4.1.6. Let ⟨A, (l)Cn⟩ be a category of (layered) logic systems suitable for free synchronization and a_{1}, a_{2} ∈ |A|. Then, given any sum {h_{k} : a_{k} → (a_{1} ▷◁ a_{2})}_{k∈{1,2}} in A, (a_{1} ▷◁ a_{2}) is said to be a free synchronization of a_{1} and a_{2}.

Please note that Lang_{A}(a_{1} ▷◁ a_{2}) may not coincide with (Lang_{A}(a_{1}) ▷◁ Lang_{A}(a_{2})), as defined in 4.1.1 (4.1.3) with injections ι_{k} as in 4.1.2 (4.1.4). However, by definition of suitability for free synchronization, it must always be the case that the unique function h : (Lang_{A}(a_{1}) ▷◁ Lang_{A}(a_{2})) → Lang_{A}(a_{1} ▷◁ a_{2}) such that (h ∘ ι_{1}) = h_{1} and (h ∘ ι_{2}) = h_{2} is necessarily a bijection. Moreover, from the universal property of sums it follows that h : ((l)Cn(a_{1}) ▷◁ (l)Cn(a_{2})) → (l)Cn(a_{1} ▷◁ a_{2}) is an (l)Csy morphism. Exactness, in its turn, implies that this morphism is furthermore an (l)Csy isomorphism.

Finally, let us consider free synchronization of logics and layered logics. Of course, we capitalize on the constructions at the level of logic systems and layered logic systems.

Definition 4.1.7. Let ⟨A, (l)Cn⟩ be a category of (layered) logic systems suitable for free synchronization, and Log_{k} be ⟨A, (l)Cn⟩-based (layered) logics over Voc_{k} for k ∈ {1, 2}. The free synchronization of Log_{1} and Log_{2} is (Log_{1} ▷◁ Log_{2}) over (Voc_{1} × Voc_{2}) where:

• (Log_{1} ▷◁ Log_{2})(⟨V_{1}, V_{2}⟩) = (Log_{1}(V_{1}) ▷◁ Log_{2}(V_{2}));

• (Log_{1} ▷◁ Log_{2})(⟨h_{1} : V_{1} → V′_{1}, h_{2} : V_{2} → V′_{2}⟩) is the unique A-morphism such that ((Log_{1} ▷◁ Log_{2})(⟨h_{1}, h_{2}⟩) ∘ ι_{k}(V_{1}, V_{2})) = (ι_{k}(V′_{1}, V′_{2}) ∘ Log(h_{k})) for k ∈ {1, 2}, with ι_{k}(X_{1}, X_{2}) : Log_{k}(X_{k}) → (Log_{1}(X_{1}) ▷◁ Log_{2}(X_{2})) the corresponding injection morphisms.

The result below is a corollary of well-established properties of indexed categories and the Grothendieck construction.

Proposition 4.1.8. Let Log_{k} for k ∈ {1, 2} be ⟨A, (l)Cn⟩-based (layered) logics over Voc_{k}, for some category ⟨A, (l)Cn⟩ of (layered) logic systems suitable for free synchronization. Then, {⟨Π_{k}, ι_{k}⟩ : Log_{k} → (Log_{1} ▷◁ Log_{2})}_{k∈{1,2}}, where each Π_{k} : (Voc_{1} × Voc_{2}) → Voc_{k} is the corresponding projection, is a sum in ⟨A, (l)Cn⟩-Log.

Proof. This result is a corollary of A.3.8, in Appendix. The construction is of course a simplification of that in A.3.9, and uses 4.1.2 or 4.1.4. ⊓ ⊔

We should now have a look at whether our working categories of logic systems and layered logic systems are suitable or even exact for free synchronization. However, we postpone this analysis until we study synchronization in its general constrained form.

4.1.2 Constrained synchronization

In the course of our definition of free synchronization, we have recognized that Csy is cocomplete. In fact, we might very well see every small diagram in Csy as a specification of a combination of consequence systems, and the colimit of this diagram as the result of this combination. This is what we have done for free synchronization regarding sums. For the purpose of this thesis, however, we are not interested in all possible kinds of colimits. Still, we would like to go beyond sums. In fact, it is usual that combination constraints are syntactically driven. In that sense, and in the light of the Appendix, we shall also study what we call constrained synchronization. The idea is that the previous free construction is modified by requiring that certain sentences of each language should be shared with sentences of the other one.

Definition 4.1.9. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems. A sharing constraint on csy_{1} and csy_{2} is a source S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} in Set.

Please note that such a source S essentially defines the subset of L_{1} × L_{2} that contains all pairs ⟨s_{1}(υ), s_{2}(υ)⟩ for υ ∈ L_{0}. This set shall be referred to as the set of shared sentences. The intuition is that, then, the sentences s_{1}(υ) and s_{2}(υ) are shared. That is, in the language of the constrained synchronization, we shall not be able to distinguish between s_{1}(υ)_{1} and s_{2}(υ)_{2}.

Conversely, any set L′_{0} ⊆ L_{1} × L_{2} defines the constraint {π_{k} : L′_{0} → L_{k}}_{k∈{1,2}}, where each π_{k} is the corresponding projection.

Definition 4.1.10. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a constraint. The synchronization of csy_{1} and csy_{2} constrained by S is (csy_{1} ▷◁^{S} csy_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), ▷⟩, where:

• (L_{1} ▷◁^{S} L_{2}) = (L_{1} ▷◁ L_{2})/_{≈^{S}} (the quotient set), with:

– ≈^{S} the least equivalence on (L_{1} ▷◁ L_{2}) such that s_{1}(υ)_{1} ≈^{S} s_{2}(υ)_{2} for every υ ∈ L_{0};

• E^{▷} is the least superset of E that is closed for I_{▷◁}^{≈^{S}}, where:

I_{▷◁}^{≈^{S}} = ({⟨[Φ_{1}]_{≈^{S}}, [φ_{1}]_{≈^{S}}⟩ : Φ ▷_{1} φ} ∪ {⟨[Ψ_{2}]_{≈^{S}}, [ψ_{2}]_{≈^{S}}⟩ : Ψ ▷_{2} ψ}).

Of course, we are using [υ]_{≈^{S}} to denote the ≈^{S}-equivalence class of υ. For clarity, we have also used E to denote a set of ≈^{S}-equivalence classes, and we shall use e to denote one equivalence class.

This definition can be made clearer by noting that (csy_{1} ▷◁^{S} csy_{2}) corresponds (up to equivalence) to an enrichment of (csy_{1}▷◁csy_{2}) with inferences that force the sharing represented by S.

Proposition 4.1.11. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a constraint. Then, (csy_{1} ▷◁^{S} csy_{2}) is equivalent to the consequence system csy_{S} = ⟨(L_{1} ▷◁ L_{2}), ▷_{S}⟩, with Υ^{▷_{S}} the least superset of Υ that is closed for I_{▷◁}^{S}, where:

I_{▷◁}^{S} = (I_{▷◁} ∪ {⟨{s_{1}(υ)_{1}}, s_{2}(υ)_{2}⟩ : υ ∈ L_{0}} ∪ {⟨{s_{2}(υ)_{2}}, s_{1}(υ)_{1}⟩ : υ ∈ L_{0}}).

Proof. Let f : (L_{1} ▷◁^{S} L_{2}) → (L_{1} ▷◁ L_{2}) be any function such that [f(e)]_{≈^{S}} = e (if necessary, use the axiom of choice), and g : (L_{1} ▷◁ L_{2}) → (L_{1} ▷◁^{S} L_{2}) be such that g(υ) = [υ]_{≈^{S}}. Assume also that csy = (csy_{1} ▷◁^{S} csy_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), ▷⟩. We start by proving the following lemma:

if Ω_{I_{▷◁}^{S}}(Υ) ⊆ Υ and υ ∈ Υ then [υ]_{≈^{S}} ⊆ Υ.

Easily, the relation ≈ defined on (L_{1} ▷◁ L_{2}) by υ′ ≈ υ′′ if and only if both υ′, υ′′ ∈ Υ or both υ′, υ′′ ∉ Υ, is an equivalence. All we need to check is that ≈^{S}⊆≈. Clearly, it suffices to note that s_{1}(υ)_{1} ≈ s_{2}(υ)_{2} for every υ ∈ L_{0}, since Υ is closed for I_{▷◁}^{S}.

We first prove that f : csy → csy_{S} is a morphism. For the purpose, assume that E ▷ e and Υ ∈ I_{▷◁}^{S}(f(E)). We now show that [Υ]_{≈^{S}} ∈ I_{▷◁}^{≈^{S}}(E). Easily, f(E) ⊆ Υ, and by definition of f we must have E ⊆ [Υ]_{≈^{S}}. Now, let Φ ▷_{1} φ and suppose that [Φ_{1}]_{≈^{S}} ⊆ [Υ]_{≈^{S}}. Using the lemma above, we can conclude that also Φ_{1} ⊆ Υ, and therefore φ_{1} ∈ Υ. Analogously, if Ψ ▷_{2} ψ and [Ψ_{2}]_{≈^{S}} ⊆ [Υ]_{≈^{S}}, we can show that ψ_{2} ∈ Υ. In either case, [φ_{1}]_{≈^{S}}, [ψ_{2}]_{≈^{S}} ∈ [Υ]_{≈^{S}} and [Υ]_{≈^{S}} ∈ I_{▷◁}^{≈^{S}}(E). Thus, e ∈ [Υ]_{≈^{S}} and the lemma again implies that f(e) ∈ Υ. Therefore, f(E) ▷_{S} f(e).

We now prove that g : csy_{S} → csy is also a morphism. For the purpose, assume that Υ ▷_{S} υ and E ∈ I_{▷◁}^{≈^{S}}([Υ]_{≈^{S}}). We show that (⋃_{e∈E} e) ∈ I_{▷◁}^{S}(Υ). Clearly, [Υ]_{≈^{S}} ⊆ E so also Υ ⊆ (⋃_{e∈E} e). Now, if Φ▷_{1}φ and Φ_{1} ⊆ (⋃_{e∈E} e), we also have that [Φ_{1}]_{≈^{S}} ⊆ E. Therefore, [φ_{1}]_{≈^{S}} ∈ E and φ_{1} ∈ (⋃_{e∈E} e). Analogously, we can prove that (⋃_{e∈E} e) is closed for ▷_{2}. Let now υ ∈ L_{0} and suppose that s_{1}(υ)_{1} ∈ (⋃_{e∈E} e). Clearly, [s_{2}(υ)_{2}]_{≈^{S}} = [s_{1}(υ)_{1}]_{≈^{S}} ∈ E, and we conclude that also s_{2}(υ)_{2} ∈ (⋃_{e∈E} e). Analogously, if s_{2}(υ)_{2} ∈ (⋃_{e∈E} e) we can show that also s_{1}(υ)_{1} ∈ (⋃_{e∈E} e). Thus, (⋃_{e∈E} e) ∈ I_{▷◁}^{S}(Υ) and υ ∈ (⋃_{e∈E} e). So, [υ]_{≈^{S}} ∈ E and [Υ]_{≈^{S}} ▷ [υ]_{≈^{S}}.

According to 2.1.44, we further need to show that e ≡_{csy} g(f(e)) and υ ≡_{csy_{S}} f(g(υ)), for every e ∈ (L_{1} ▷◁^{S} L_{2}) and every υ ∈ (L_{1} ▷◁ L_{2}). Easily, e = g(f(e)). On the other hand, υ ≈^{S} f(g(υ)) and the result follows from the lemma above. ⊓ ⊔

In particular, it must be obvious that free synchronization can be obtained as a particular case of this construction, when the set L_{0} that guides the sharing is empty.

Now that constrained synchronization is made clearer, let us provide it with a categorial characterization. It corresponds precisely to a Lang_{Csy}-pushout, as defined in A.2.1 in Appendix.

Proposition 4.1.12. Let csy_{k} = ⟨L_{k}, ▷_{k}⟩ for k ∈ {1, 2} be consequence systems, and

S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a constraint. Then, {ι_{k}^{S} : csy_{k} → (csy_{1} ▷◁^{S} csy_{2})}_{k∈{1,2}}, with

each ι_{k}^{S}(υ) = [υ_{k}]_{≈^{S}}, is a Lang_{Csy}-pushout of {csy_{k}}_{k∈{1,2}} along S.

Proof. Since Set has all coequalizers and we have shown, in 2.1.17, that Lang_{Csy} is topological, we know in particular that Lang_{Csy} is a cofibration. Thus, using A.2.13 and 4.1.2, we just need to show that [ ]_{≈^{S}} : (csy_{1} ▷◁ csy_{2}) → (csy_{1} ▷◁^{S} csy_{2}) is a Lang_{Csy}-cocartesian lift of ⟨(csy_{1} ▷◁ csy_{2}), [ ]_{≈^{S}}⟩, since [ ]_{≈^{S}} : (L_{1} ▷◁ L_{2}) → (L_{1} ▷◁^{S} L_{2}) is a coequalizer in Set of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}.

However, from topologicity, it follows that h : (csy_{1} ▷◁ csy_{2}) → ⟨L′, ▷′⟩ is a Lang_{Csy}- cocartesian lift of ⟨(csy_{1} ▷◁ csy_{2}), h : (L_{1} ▷◁ L_{2}) → L′⟩ provided that the equality ▷′ = ⊓_{L′}{▷′′ : h : (csy_{1} ▷◁ csy_{2}) → ⟨L′, ▷′′⟩ is a Csy-morphism} holds. According to 2.1.19, it easily follows that ▷′ is generated by the set of inferences {⟨h(Φ), h(ψ)⟩ : Φ ▷ ψ}. ⊓ ⊔

Constrained synchronization can be easily extended to the layered setting.

Definition 4.1.13. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems. A sharing constraint on csy_{1} and csy_{2} is a source S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} in Set.

Again sentences φ ∈ L_{1} and ψ ∈ L_{2} are shared under S whenever there exists ϑ ∈ L_{0} such that s_{1}(ϑ) = φ and s_{2}(ϑ) = ψ.

Definition 4.1.14. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a constraint. The synchronization of csy_{1} and csy_{2} constrained by S is (csy_{1} ▷◁^{S} csy_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), ▷^{l}, ▷^{g}⟩, where ⟨(L_{1} ▷◁^{S} L_{2}), ▷^{g}⟩ = (csy_{1}^{g} ▷◁^{S} csy_{2}^{g}), i.e., E^{▷^{g}} is the least superset of E that is closed for I_{g▷◁}^{≈^{S}}, where:

I_{g▷◁}^{≈^{S}} = ({⟨[Φ_{1}]_{≈^{S}}, [φ_{1}]_{≈^{S}}⟩ : Φ ▷_{1}^{g} φ} ∪ {⟨[Ψ_{2}]_{≈^{S}}, [ψ_{2}]_{≈^{S}}⟩ : Ψ ▷_{2}^{g} ψ}),

and E^{▷^{l}} is the least superset of E that is closed for I_{l▷◁}^{≈^{S}}, where:

I_{l▷◁}^{≈^{S}} = ({⟨[Φ_{1}]_{≈^{S}}, [φ_{1}]_{≈^{S}}⟩ : Φ ▷_{1}^{l} φ} ∪ {⟨[Ψ_{2}]_{≈^{S}}, [ψ_{2}]_{≈^{S}}⟩ : Ψ ▷_{2}^{l} ψ} ∪ {⟨∅, e⟩ : ▷^{g}e}).

Following a similar strategy to the one used in the non-layered case, we can better explain the intuition behind this definition by noting that, also in the layered case, (csy_{1} ▷◁^{S} csy_{2}) corresponds (up to equivalence) to an enrichment of (csy_{1}▷◁csy_{2}) with inferences that force the sharing represented by S.

Proposition 4.1.15. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a constraint. Then, (csy_{1} ▷◁^{S} csy_{2}) is equivalent to the layered consequence system csy_{S} = ⟨(L_{1} ▷◁ L_{2}), ▷_{S}^{l}, ▷_{S}^{g}⟩, with Υ^{▷_{S}^{g}} the least superset of Υ that is closed for I_{g▷◁}^{S}, where:

I_{g▷◁}^{S} = (I_{g▷◁} ∪ {⟨{s_{1}(υ)_{1}}, s_{2}(υ)_{2}⟩ : υ ∈ L_{0}} ∪ {⟨{s_{2}(υ)_{2}}, s_{1}(υ)_{1}⟩ : υ ∈ L_{0}}),

and Υ^{▷_{S}^{l}} the least superset of Υ that is closed for I_{l▷◁}^{S}, where:

I_{l▷◁}^{S} = (I_{l▷◁} ∪ {⟨{s_{1}(υ)_{1}}, s_{2}(υ)_{2}⟩ : υ ∈ L_{0}} ∪ {⟨{s_{2}(υ)_{2}}, s_{1}(υ)_{1}⟩ : υ ∈ L_{0}} ∪ {⟨∅, υ⟩ : ▷_{S}^{g}υ}).

Proof. Assume that csy = (csy_{1} ▷◁^{S} csy_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), ▷^{l}, ▷^{g}⟩, and let f : (L_{1} ▷◁^{S} L_{2}) → (L_{1} ▷◁ L_{2}) and g : (L_{1} ▷◁ L_{2}) → (L_{1} ▷◁^{S} L_{2}) be defined as in 4.1.11, i.e., [f(e)]_{≈^{S}} = e for every e ∈ (L_{1} ▷◁^{S} L_{2}), and g(υ) = [υ]_{≈^{S}} for every υ ∈ (L_{1} ▷◁ L_{2}). It immediately follows from definition 4.1.14 above that csy^{g} ≡ csy_{S}^{g} with f and g. That is, f : csy^{g} → csy_{S}^{g} and g : csy_{S}^{g} → csy^{g} are Csy-morphisms such that e ≡_{csy}^{g} g(f(e)) and υ ≡_{csy}_{S}^{g} f(g(υ)).

According to 3.1.29, it remains to be proved that also f : csy^{l} → csy_{S}^{l} and g : csy_{S}^{l} → csy^{l} are Csy-morphisms, and moreover e ≡_{csy}_{l} g(f(e)) and υ ≡_{csy}l_{S} f(g(υ)). The proof follows the sequence of steps already used in 4.1.11. We start by proving the following lemma:

if Ω_{I_{l▷◁}^{S}}(Υ) ⊆ Υ and υ ∈ Υ then [υ]_{≈^{S}} ⊆ Υ.

Easily, the relation ≈ defined on (L_{1} ▷◁ L_{2}) by υ′ ≈ υ′′ if and only if both υ′, υ′′ ∈ Υ or both υ′, υ′′ ∉ Υ, is an equivalence. To check that ≈^{S}⊆≈, it suffices to note that s_{1}(υ)_{1} ≈ s_{2}(υ)_{2} for every υ ∈ L_{0}, since Υ is closed for I_{l▷◁}^{S}.

Let us now show that f : csy^{l} → csy_{S}^{l} is indeed a morphism. For the purpose, assume that E ▷^{l} e and Υ ∈ I_{l▷◁}^{S}(f(E)). We now show that [Υ]_{≈^{S}} ∈ I_{l▷◁}^{≈^{S}}(E). Easily, f(E) ⊆ Υ, and by definition of f we must have E ⊆ [Υ]_{≈^{S}}. Now, let Φ ▷_{1} φ and suppose that [Φ_{1}]_{≈^{S}} ⊆ [Υ]_{≈^{S}}. Using the lemma above, we can conclude that also Φ_{1} ⊆ Υ. Hence, φ_{1} ∈ Υ and therefore [φ_{1}]_{≈^{S}} ∈ [Υ]_{≈^{S}}. Analogously, if Ψ ▷_{2} ψ and [Ψ_{2}]_{≈^{S}} ⊆ [Υ]_{≈^{S}}, we can show that ψ_{2} ∈ Υ. Hence, φ_{1} ∈ Υ and therefore also [ψ_{2}]_{≈^{S}} ∈ [Υ]_{≈^{S}}. If we have ▷^{g}e, since f : csy^{g} → csy_{S}^{g} is a morphism, it follows that ▷_{S}^{g}f(e). Immediately, we have that ⟨∅, f(e)⟩ ∈ I_{l▷◁}^{S}. Hence, f(e) ∈ Υ and e ∈ [Υ]_{≈^{S}}.

We now prove that g : csy_{S}^{l} → csy^{l} is also a morphism. For the purpose, assume that Υ ▷_{S}^{l} υ and E ∈ I_{l▷◁}^{≈^{S}}([Υ]_{≈^{S}}). We show that (⋃_{e∈E} e) ∈ I_{l▷◁}^{S}(Υ). Clearly, [Υ]_{≈^{S}} ⊆ E so also Υ ⊆ (⋃_{e∈E} e). Now, if Φ▷_{1}φ and Φ_{1} ⊆ (⋃_{e∈E} e), we also have that [Φ_{1}]_{≈^{S}} ⊆ E. Therefore, [φ_{1}]_{≈^{S}} ∈ E and φ_{1} ∈ (⋃_{e∈E} e). Analogously, we can prove that (⋃_{e∈E} e) is closed for ▷_{2}. Let now υ ∈ L_{0} and suppose that s_{1}(υ)_{1} ∈ (⋃_{e∈E} e). Clearly, [s_{2}(υ)_{2}]_{≈^{S}} = [s_{1}(υ)_{1}]_{≈^{S}} ∈ E, and we conclude that also s_{2}(υ)_{2} ∈ (⋃_{e∈E} e). Analogously, if s_{2}(υ)_{2} ∈ (⋃_{e∈E} e) we can show that also s_{1}(υ)_{1} ∈ (⋃_{e∈E} e). Thus, (⋃_{e∈E} e) ∈ I_{▷◁}^{S}(Υ) and υ ∈ (⋃_{e∈E} e). So, [υ]_{≈^{S}} ∈ E and [Υ]_{≈^{S}} ▷ [υ]_{≈^{S}}. Finally, assume that ▷_{S}^{g}υ. Since g : csy_{S}^{g} → csy^{g} is a Csy-morphism, it follows that ▷^{g}[υ]_{≈^{S}} and thus ⟨∅, [υ]_{≈^{S}}⟩ ∈ I_{l▷◁}^{≈^{S}}. Therefore, [υ]_{≈^{S}} ∈ E and υ ∈ (⋃_{e∈E} e).

We further need to show that e ≡_{csy}_{l} g(f(e)) and υ ≡_{csy}l_{S} f(g(υ)), for every e ∈ (L_{1} ▷◁^{S} L_{2}) and every υ ∈ (L_{1} ▷◁ L_{2}). Easily, e = g(f(e)). On the other hand, υ ≈^{S} f(g(υ)) and the result follows from the lemma above. ⊓ ⊔

Constrained synchronization of layered consequence systems also corresponds to a Lang_{lCsy}-pushout.

Proposition 4.1.16. Let csy_{k} = ⟨L_{k}, ▷_{k}^{l}, ▷_{k}^{g}⟩ for k ∈ {1, 2} be layered consequence systems, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a constraint. Then, {ι_{k}^{S} : csy_{k} → (csy_{1} ▷◁^{S} csy_{2})}_{k∈{1,2}}, with each ι_{k}^{S}(υ) = [υ_{k}]_{≈^{S}}, is a Lang_{lCsy}-pushout of {csy_{k}}_{k∈{1,2}} along S.

Proof. Since Set has all coequalizers and we have shown, in 3.1.16, that Lang_{lCsy} is topological, we know in particular that Lang_{lCsy} is a cofibration. Thus, using A.2.13 and 4.1.4, we just need to show that [ ]_{≈^{S}} : (csy_{1} ▷◁ csy_{2}) → (csy_{1} ▷◁^{S} csy_{2}) is a Lang_{lCsy}- cocartesian lift of ⟨(csy_{1} ▷◁ csy_{2}), [ ]_{≈^{S}}⟩, since [ ]_{≈^{S}} : (L_{1} ▷◁ L_{2}) → (L_{1} ▷◁^{S} L_{2}) is a coequalizer in Set of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}.

From topologicity, it follows that h : (csy_{1} ▷◁ csy_{2}) → ⟨L′, ▷^{′l}, ▷^{′g}⟩ is a Lang_{lCsy}- cocartesian lift of ⟨(csy_{1} ▷◁ csy_{2}), h : (L_{1} ▷◁ L_{2}) → L′⟩ if ⟨▷^{′l}, ▷^{′g}⟩ = ⊓_{L′}{⟨▷^{′′l}, ▷^{′′g}⟩ : h : (csy_{1} ▷◁ csy_{2}) → ⟨L′, ▷^{′′l}, ▷^{′′g}⟩ is an lCsy-morphism}. And according to 3.1.18 it easily follows that ▷^{′g} is the consequence generated by {⟨h(Φ), h(ψ)⟩ : Φ ▷^{g} ψ}, and ▷^{′l} is the consequence generated by ({⟨h(Φ), h(ψ)⟩ : Φ ▷^{l} ψ} ∪ {⟨∅, υ⟩ : ▷^{′g}υ}). ⊓ ⊔

Constrained synchronization in an arbitrary category A of (layered) logic systems again exploits the construction for consequence system. The same applies to the layered setting. First of all, language oriented pushouts must exist in A, and in such a way that the corresponding language is a pushout of the given languages. Obviously, the construction is precise if also the consequence operation of a language oriented pushout is a language oriented pushout of the given consequence operations.

Definition 4.1.17. A category of (layered) logic systems ⟨A, (l)Cn⟩ is suitable for synchronization provided that all Lang_{A}-pushouts exist and are transformed by Lang_{A} into pushouts in Set. Moreover, ⟨A, (l)Cn⟩ is exact for synchronization if, in addition, also (l)Cn transforms Lang_{A}-pushouts into Lang_{(l)Csy}-pushouts.

Definition 4.1.18. Let ⟨A, (l)Cn⟩ be a category of (layered) logic systems suitable for synchronization, a_{1}, a_{2} ∈ |A| and S a sharing constraint on (l)Cn(a_{1}) and (l)Cn(a_{2}). Then, given any Lang_{A}-pushout {h_{k} : a_{k} → (a_{1} ▷◁^{S} a_{2})}_{k∈{1,2}} of a_{1} and a_{2} along S, (a_{1} ▷◁^{S} a_{2}) is said to be a synchronization of a_{1} and a_{2} constrained by S.

Please note that Lang_{A}(a_{1} ▷◁^{S} a_{2}) may not coincide with (Lang_{A}(a_{1}) ▷◁^{S} Lang_{A}(a_{2})), as defined in 4.1.10 (4.1.14) with pseudo-injections ι_{k}^{S} as in 4.1.12 (4.1.16). However, by definition of suitability for synchronization, it must always be the case that the unique function h : (Lang_{A}(a_{1}) ▷◁^{S} Lang_{A}(a_{2})) → Lang_{A}(a_{1} ▷◁^{S} a_{2}) such that (h ∘ ι_{1}^{S}) = h_{1} and (h∘ι_{2}^{S}) = h_{2} is necessarily a bijection. Moreover, from the underlying universal property, it follows that h : ((l)Cn(a_{1}) ▷◁^{S} (l)Cn(a_{2})) → (l)Cn(a_{1} ▷◁^{S} a_{2}) is an (l)Csy morphism. Once again, exactness further implies that this morphism is indeed an (l)Csy isomorphism.

Moreover, we would like to emphasize the fact that we did not define suitability or exactness for just constrained synchronization. In fact, since Set has an initial object (the empty set), the conditions above easily imply, respectively, suitability and exactness for free synchronization. For details see A.2.3 in the Appendix.

Finally, let us consider constrained synchronizations of logics and layered logics. Of course, we capitalize on the constructions at the level of logic systems and layered logic systems. Clearly, the first thing to do is to provide an enhancement of our previous notion of constraint. For the purpose, recall again the construction in A.3.5.

Definition 4.1.19. Let Log_{k} for k ∈ {1, 2} be (layered) ⟨A, (l)Cn⟩-based logics over Voc_{k}, for some category ⟨A, (l)Cn⟩ of (layered) logic systems suitable for synchronization. A sharing constraint on Log_{1} and Log_{2} is a source S = {⟨S_{k}, s_{k}⟩ : (Lang_{0} : Voc_{0} → Set) → Lang_{Log_{k}}}_{k∈{1,2}} in Funct(Set).

Please note that such a source S essentially defines a restriction to all pairs of vocabularies, by selecting just those pairs ⟨V_{1}, V_{2}⟩ such that S_{1}(V_{1}) = S_{2}(V_{2}) = V_{0}. Given such a pair, clearly, S(V_{1}, V_{2}) = {s_{k}(V_{k}) : Lang_{0}(V_{0}) → Lang_{Log_{k}}(V_{k})}_{k∈{1,2}} is a constraint on Log_{1}(V_{1}) and Log_{2}(V_{2}). Moreover, it also defines a restriction to all pairs of Vocmorphisms allowed between such vocabularies, by selecting just those pairs ⟨h_{1} : V_{1} → V′_{1}, h_{2} : V_{2} → V′_{2}⟩ such that S_{1}(h_{1}) = S_{2}(h_{2}) = h_{0} : V_{0} → V′_{0}. The conditions satisfied by functors and natural transformations guarantee that the following coherence property holds for k ∈ {1, 2}: (Lang_{Log_{k}}(h_{k}) ∘ s_{k}(V_{k})) = (s_{k}(V′_{k}) ∘ Lang_{0}(h_{0})). The same is to say that pairs of shared sentences are preserved across vocabulary translations.

Definition 4.1.20. Let ⟨A, (l)Cn⟩ be a category of (layered) logic systems suitable for synchronization, Log_{k} be ⟨A, (l)Cn⟩-based (layered) logics over Voc_{k} for k ∈ {1, 2}, and S = {⟨S_{k}, s_{k}⟩ : (Lang_{0} : Voc_{0} → Set) → Lang_{Log_{k}}}_{k∈{1,2}} a constraint. The synchronization of Log_{1} and Log_{2} constrained by S is (Log_{1} ▷◁^{S} Log_{2}) over Voc where:

• Voc is the subcategory of (Voc_{1} × Voc_{2}) such that:

– objects ⟨V_{1}, V_{2}⟩ must satisfy S_{1}(V_{1}) = S_{2}(V_{2});

– morphisms ⟨h_{1}, h_{2}⟩ : ⟨V_{1}, V_{2}⟩ → ⟨V′_{1}, V′_{2}⟩ must satisfy S_{1}(h_{1}) = S_{2}(h_{2});

• (Log_{1} ▷◁^{S} Log_{2}) : Voc → A is such that:

– (Log_{1} ▷◁^{S} Log_{2})(⟨V_{1}, V_{2}⟩) = (Log_{1}(V_{1}) ▷◁^{S(V_{1},V_{2})} Log_{2}(V_{2}));

– (Log_{1} ▷◁^{S} Log_{2})(⟨h_{1} : V_{1} → V′_{1}, h_{2} : V_{2} → V′_{2}⟩) is the unique A-morphism such

that ((Log_{1} ▷◁^{S} Log_{2})(⟨h_{1}, h_{2}⟩)∘ι_{k}^{S}(V_{1}, V_{2})) = (ι_{k}^{S}(V′_{1}, V′_{2})∘Log(h_{k})) for k ∈ {1, 2},

with ι_{k}^{S}(X_{1}, X_{2}) : Log_{k}(X_{k}) → (Log_{1}(X_{1}) ▷◁^{S(X_{1},X_{2})} Log_{2}(X_{2})) the corresponding Lang_{A}-pushout morphisms.

Similarly to the free case, we now have the following result.

Proposition 4.1.21. Let Log_{k} for k ∈ {1, 2} be ⟨A, (l)Cn⟩-based (layered) logics over Voc_{k}, for some category ⟨A, (l)Cn⟩ of (layered) logic systems suitable for synchronization, and S = {⟨S_{k}, s_{k}⟩ : (Lang_{0} : Voc_{0} → Set) → Lang_{Log_{k}}}_{k∈{1,2}} a constraint. Then, {⟨Π_{k}^{S}, ι_{k}^{S}⟩ : Log_{k} → (Log_{1} ▷◁^{S} Log_{2})}_{k∈{1,2}}, where each Π_{k}^{S} is the restriction to Voc of the projection Π_{k} : (Voc_{1} × Voc_{2}) → Voc_{k}, is a Funct(Lang_{A})-pushout of Log_{1} and Log_{2} along S.

Proof. This result is a corollary of A.3.9, in Appendix, and uses 4.1.12 or 4.1.16. ⊓ ⊔

4.2 Applying the mechanism

For now, let us take a look at whether our working categories of model-theoretic logic systems and layered logic systems are suitable or even exact for free synchronization. As it turns out, all of them are suitable for synchronization but none is exact. On the way, we also provide characterizations of synchronization in each of them.

4.2.1 Satisfaction and (layered) valuation systems

Let us start with satisfaction systems.

Proposition 4.2.1. Sat is suitable, but not exact for synchronization. Namely, the synchronization of satisfaction systems sat_{k} = ⟨L_{k}, M_{k}, ⊩_{k}⟩, for k ∈ {1, 2}, constrained by

S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} is given by (sat_{1} ▷◁^{S} sat_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), M, ⊩⟩ where:

• M is the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ (M_{1} × M_{2}) such that, for all υ ∈ L_{0}:

– m_{1} ⊩_{1}s_{1}(υ) if and only if m_{2} ⊩_{2}s_{2}(υ);

• ⟨m_{1}, m_{2}⟩⊩e if and only if m_{1} ⊩_{1}φ for every φ_{1} ∈ e and m_{2} ⊩_{2}ψ for every ψ_{2} ∈ e.

Proof. It is well known that Set has all coequalizers. Moreover, we have shown in 2.2.38 that Lang_{Sat} is a cofibration. Hence, according to A.2.13, it suffices to show that binary sums exist in Sat to guarantee that all Lang_{Sat}-pushouts exist.

For the purpose, let (sat_{1} ▷◁ sat_{2}) = ⟨(L_{1} ▷◁ L_{2}), (M_{1} × M_{2}), ⊩_{12}⟩ be such that, for all m_{1} ∈ M_{1}, m_{2} ∈ M_{2}, k ∈ {1, 2} and υ ∈ L_{k}:

• ⟨m_{1}, m_{2}⟩⊩_{12}υ_{k} if and only if m_{k} ⊩_{k}υ.

We are going to show that {h_{k} : sat_{k} → (sat_{1} ▷◁ sat_{2})}_{k∈{1,2}} is a sum in Sat, where each \overline{h}_{k} = ι_{k} : L_{k} → (L_{1} ▷◁ L_{2}) is the corresponding injection function, and each \underline{h}_{k} = π_{k} : (M_{1} × M_{2}) → M_{k} the corresponding projection. Trivially, both are Sat-morphisms. Let now {g_{k} : sat_{k} → sat′}_{k∈{1,2}} be morphisms in Sat, with sat′ = ⟨L′, M′, ⊩′⟩. Obviously, there exists a unique function \overline{f} : (L_{1} ▷◁ L_{2}) → L′ such that (\overline{f} ∘ ι_{k}) = \overline{g}_{k} for k ∈ \{1, 2\}. It is such that \overline{f}(υ_{k}) = \overline{g}_{k}(υ). Moreover, there is a unique map \underline{f} : M′ → (M_{1} × M_{2}) such that (π_{k} ∘ \underline{f}) = \underline{g}_{k} for k ∈ \{1, 2\}. It is such that \underline{f}(m′) = ⟨\underline{g}_{1}(m′), \underline{g}_{2}(m′)⟩. We just need to prove that f : (sat_{1} ▷◁ sat_{2}) → sat′ establishes a Sat-morphism. In fact, for m′ ∈ M′, k ∈ \{1, 2\} and υ ∈ L_{k}, we have that \underline{f}(m′)⊩_{12}υ_{k} if and only if \underline{g}_{k}(m′)⊩_{k}υ iff m⊩′\overline{g}_{k}(υ), and \overline{g}_{k}(υ) = \overline{f}(υ_{k}).

Thus, since [ ]_{≈^{S}} : (L_{1} ▷◁ L_{2}) → (L_{1} ▷◁^{S} L_{2}) is a coequalizer in Set of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}, we obtain (sat_{1} ▷◁^{S} sat_{2}) by Lang_{Sat}-cocartesian lift of ⟨(sat_{1} ▷◁ sat_{2}), [ ]_{≈^{S}}⟩. Furthermore, we have also proved in 2.2.24 that Lang_{Sat} is coadjoint. So, according to A.3.3, we can conclude that Sat is suitable for synchronization.

Exactness, in its turn, can be easily shown to fail, even for free synchronization. In fact Ent_{Sat} does not preserve binary sums. As an example, suppose that there exists a set Ψ ⊆ L_{2} with no model in sat_{2}, i.e., there exists no model m_{2} ∈ M_{2} such that m_{2} ⊩_{2}Ψ. Then, obviously, for every Φ ⊆ L_{1}, it turns out that (Φ_{1} ∪ Ψ_{2}) has no model in (sat_{1} ▷◁ sat_{2}). Therefore, immediately, (Φ_{1}∪Ψ_{2})⊨_{(sat_{1}▷◁sat_{2})} φ_{1} for every φ ∈ L_{1}. However, it might very well be the case that Φ⊨_{s̸at_{1}}φ. ⊓ ⊔

Obviously, (sat_{1} ▷◁ sat_{2}) as shows up in the proof above is a free synchronization of the satisfaction systems sat_{1} and sat_{2}.

Valuation systems are quite similar, although the pairing of models in Val is less straightforward.

Proposition 4.2.2. Val is suitable, but not exact for synchronization. Namely, the synchronization of valuation systems val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩, for k ∈ {1, 2}, constrained by

S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} is given by (val_{1} ▷◁^{S} val_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), M, S⟩ where:

• M is the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ (M_{1} × M_{2}) such that T_{m_{1}} = T_{m_{2}} = T_{⟨m_{1},m_{2}⟩}, D_{m_{1}} = D_{m_{2}} = D_{⟨m_{1},m_{2}⟩} and, for all υ ∈ L_{0}:

– [[s_{1}(υ)]]_{m_{1}} = [[s_{2}(υ)]]_{m_{2}};

• S(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, [[ ]], D_{⟨m_{1},m_{2}⟩}⟩ where, for every e ∈ (L_{1} ▷◁^{S} L_{2}):

– [[e]] = [[υ]]_{m_{k}} for every k ∈ {1, 2} and υ_{k} ∈ e.

Proof. Set has all coequalizers. And again, we have shown in 2.2.37 that Lang_{Val} is a cofibration. Hence, by A.2.13, if we show that binary sums exist in Val we guarantee that all Lang_{Val}-pushouts exist.

For the purpose, let (val_{1} ▷◁ val_{2}) = ⟨(L_{1} ▷◁ L_{2}), M′, S′⟩ be such that:

• M′ is the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ (M_{1} × M_{2}) such that T_{m_{1}} = T_{m_{2}} = T_{⟨m_{1},m_{2}⟩} and D_{m_{1}} = D_{m_{2}} = D_{⟨m_{1},m_{2}⟩};

• S′(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, [[ ]], D_{⟨m_{1},m_{2}⟩}⟩ where, for every k ∈ {1, 2} and υ ∈ L_{k}:

– [[υ_{k}]] = [[υ]]_{m_{k}}.

We are going to show that {h_{k} : val_{k} → (val_{1} ▷◁ val_{2})}_{k∈{1,2}} is a sum in Val, where each \overline{h}_{k} = ι_{k} : L_{k} → (L_{1} ▷◁ L_{2}) is the corresponding injection function, and each \underline{h}_{k} : M′ → M_{k} is the corresponding restriction of the projection π_{k} : (M_{1} × M_{2}) → M_{k}.

Trivially, both are Val-morphisms. Let now {g_{k} : val_{k} → val′′}_{k∈{1,2}} be morphisms in Val, with val′′ = ⟨L′′, M′′, S′′⟩. Obviously, \overline{f}(υ_{k}) = \overline{g}_{k}(υ) defines the unique function \overline{f} : (L_{1} ▷◁ L_{2}) → L′′ such that (\overline{f} ∘ ι_{k}) = \overline{g}_{k} for k ∈ \{1, 2\}. Moreover, for every m′′ ∈ M′′, trivially T_{\underline{g}_{1}(m′′)} = T_{m′′} = T_{\underline{g}_{2}(m′′)} and D_{\underline{g}_{1}(m′′)} = D_{m′′} = D_{\underline{g}_{2}(m′′)} and we conclude that ⟨\underline{g}_{1}(m′′), \underline{g}_{2}(m′′)⟩ ∈ M′. Therefore, \underline{f}(m′′) = ⟨\underline{g}_{1}(m′′), \underline{g}_{2}(m′′)⟩ defines the unique map \underline{f} : M′′ → M′ such that (\underline{h}_{k} ∘ \underline{f}) = \underline{g}_{k} for k ∈ \{1, 2\}. Now, we just need to prove that f : (val_{1} ▷◁ val_{2}) → val′′ establishes a Val-morphism. In fact, for m′′ ∈ M′′, k ∈ {1, 2} and υ ∈ L_{k}, we have that [[υ_{k}]]_{\underline{f}(m′′)} = [[υ]]_{\underline{g}_{k}(m′′)} = [[\overline{g}_{k}(υ)]]_{m′′} = [[\overline{f}(υ_{k})]]_{m′′}.

Thus, since [ ]_{≈^{S}} : (L_{1} ▷◁ L_{2}) → (L_{1} ▷◁^{S} L_{2}) is a coequalizer in Set of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}, we obtain (val_{1} ▷◁^{S} val_{2}) by Lang_{Val}-cocartesian lift of ⟨(val_{1} ▷◁ val_{2}), [ ]_{≈^{S}}⟩. Furthermore, we have also proved in 2.2.35 that Lang_{Val} is coadjoint. So, according to A.3.3, we can conclude that Val is suitable for synchronization.

Again, exactness fails because Ent_{Val} does not preserve binary sums. Instead of using an example similar to the one in 4.2.1, we shall explore another source of non-exactness in Val. Let Φ ⊆ L_{1} and φ′ ∈ L_{1}, and assume that Φ⊨_{v̸al_{1}}φ′. However, suppose that for every model m_{1} ∈ M_{1} such that {[[φ]]_{m_{1}} : φ ∈ Φ} ⊆ D_{m_{1}} and [[φ′]]_{m_{1}} ∉ D_{m_{1}} there is no model m_{2} ∈ M_{2} with T_{m_{1}} = T_{m_{2}} and D_{m_{1}} = D_{m_{2}}. Then, Φ_{1} ⊨_{(val_{1}▷◁val_{2})}φ′_{1}. ⊓ ⊔

Obviously, (val_{1} ▷◁ val_{2}) as defined in the proof above corresponds again to a free synchronization of the valuation systems val_{1} and val_{2}.

Let us move to the layered setting, and analyze layered valuation systems. These are quite similar to valuation systems.

Proposition 4.2.3. lVal is suitable, but not exact for synchronization. Namely, the synchronization of layered valuation systems val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩, for k ∈ {1, 2}, constrained

by S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} is given by (val_{1} ▷◁^{S} val_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), M, S⟩ where:

• M is the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ (M_{1} × M_{2}) such that T_{m_{1}} = T_{m_{2}} = T_{⟨m_{1},m_{2}⟩}, ⋉_{m_{1}} = ⋉_{m_{2}} = ⋉_{⟨m_{1},m_{2}⟩} and, for all υ ∈ L_{0}:

– [[s_{1}(υ)]]_{m_{1}} = [[s_{2}(υ)]]_{m_{2}};

• S(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, [[ ]], ⋉_{⟨m_{1},m_{2}⟩}⟩ where, for every e ∈ (L_{1} ▷◁^{S} L_{2}):

– [[e]] = [[υ]]_{m_{k}} for every k ∈ {1, 2} and υ_{k} ∈ e.

Proof. Again, since Set has all coequalizers and we have shown, in 3.2.23, that Lang_{lVal} is a cofibration, we can use A.2.13 and conclude that all Lang_{lVal}-pushouts exist, once we show that all binary sums exist in lVal.

For the purpose, let (val_{1} ▷◁ val_{2}) = ⟨(L_{1} ▷◁ L_{2}), M′, S′⟩ be such that:

• M′ is the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ (M_{1} × M_{2}) such that T_{m_{1}} = T_{m_{2}} = T_{⟨m_{1},m_{2}⟩} and ⋉_{m_{1}} = ⋉_{m_{2}} = ⋉_{⟨m_{1},m_{2}⟩};

• S′(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, [[ ]], ⋉_{⟨m_{1},m_{2}⟩}⟩ where, for every k ∈ {1, 2} and υ ∈ L_{k}:

– [[υ_{k}]] = [[υ]]_{m_{k}}.

The fact that \{h_{k} : val_{k} → (val_{1} ▷◁ val_{2})\}_{k∈\{1,2\}} is a sum in lVal, where each \overline{h}_{k} = ι_{k} : L_{k} → (L_{1} ▷◁ L_{2}) is the corresponding injection function, and each \underline{h}_{k} : M′ → M_{k} is the corresponding restriction of the projection π_{k} : (M_{1} × M_{2}) → M_{k}, can be given a proof similar to that in 4.2.2. Also analogously, we obtain (val_{1} ▷◁^{S} val_{2}) by Lang_{lVal}-cocartesian lift of ⟨(val_{1} ▷◁ val_{2}), [ ]_{≈^{S}}⟩. Furthermore, we have also proved in 3.2.22 that Lang_{lVal} is coadjoint. Thus, we can use A.3.3 to conclude that lVal is suitable for synchronization.

Once again, exactness fails because Ent_{lVal} does not preserve sums. Instead of reusing the example from 4.2.2 within the context of global entailment, we shall focus on local entailment. Let Φ ⊆ L_{1} and φ ∈ L_{1}, and assume that Φ⊨_{v̸al_{1}}^{l}φ′. However, suppose that for every model m_{1} ∈ M_{1} such that {[[φ]]_{m_{1}} : φ ∈ Φ}⋉_{m̸_{1}}[[φ′]]_{m_{1}} there is no model m_{2} ∈ M_{2} with T_{m_{1}} = T_{m_{2}} and ⋉_{m_{1}} = ⋉_{m_{2}}. Then, obviously, Φ_{1} ⊨_{(val_{1}▷◁val_{2})}^{l}φ′_{1}. ⊓ ⊔

The layered valuation system (val_{1} ▷◁ val_{2}) as defined above obviously corresponds to a free synchronization of val_{1} and val_{2}.

4.2.2 (Layered) inference systems

Let us now analyze the proof-theoretic categories. As it turns out they are not only suitable but also exact for synchronization. Characterizations of synchronization of inference and layered inference systems are also provided.

Proposition 4.2.4. Inf is exact for synchronization. Namely, the synchronization of inference systems inf_{k} = ⟨L_{k}, I_{k}⟩, for k ∈ {1, 2}, constrained by S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} is given by (inf_{1} ▷◁^{S} inf_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), I⟩ where:

I = ( ⋃_{k∈{1,2}}{⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ : i ∈ I_{k}}).

Proof. Again, since Set has all coequalizers and we have shown, in 2.2.53, that Lang_{Inf} is a cofibration, we can use A.2.13 and conclude that all Lang_{Inf}-pushouts exist, once we show that all binary sums exist in Inf.

For the purpose, let (inf_{1} ▷◁ inf_{2}) = ⟨(L_{1} ▷◁ L_{2}), (ι_{1}(I_{1})∪ι_{2}(I_{2}))⟩. We are going to show that {ι_{k} : inf_{k} → (inf_{1} ▷◁ inf_{2})}_{k∈{1,2}} is a sum in Inf, where each ι_{k} : L_{k} → (L_{1} ▷◁ L_{2}) is the corresponding injection function. Trivially, both are Inf-morphisms. Let now {g_{k} : inf_{k} → inf′}_{k∈{1,2}} be morphisms in Inf, with inf′ = ⟨L′, I′⟩. Obviously, f(υ_{k}) = g_{k}(υ) defines the unique function f : (L_{1} ▷◁ L_{2}) → L′ such that (f ∘ ι_{k}) = g_{k} for k ∈ {1, 2}. We just need to prove that f : (inf_{1} ▷◁ inf_{2}) → inf′ establishes an Inf-morphism. In fact, for every k ∈ {1, 2} and i ∈ I_{k}, f(i_{k}) = f(ι_{k}(i)) = g_{k}(i) ∈ I′.

Thus, since [ ]_{≈^{S}} : (L_{1} ▷◁ L_{2}) → (L_{1} ▷◁^{S} L_{2}) is a coequalizer in Set of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}, we obtain (inf_{1} ▷◁^{S} inf_{2}) by Lang_{Inf}-cocartesian lift of ⟨(inf_{1} ▷◁ inf_{2}), [ ]_{≈^{S}}⟩. Furthermore, we have also proved in 2.2.55 that Lang_{Inf} is coadjoint. Thus, we can use A.3.3 to conclude that Inf is suitable for synchronization.

Exactness follows from A.3.2, since we have shown in 2.2.54 that Ded_{Inf} is a cocartesian coadjoint. Note that, from 2.1.25, we know that Lang_{Csy_{fin}} is a cofibration. ⊓ ⊔

Again, of course, the inference system (inf_{1} ▷◁ inf_{2}) as it shows up above corresponds to a free synchronization of inf_{1} and inf_{2}.

Layered inference systems are quite similar.

Proposition 4.2.5. lInf is exact for synchronization. Namely, the synchronization of layered inference systems inf_{k} = ⟨L_{k}, I_{k}, gI_{k}⟩, for k ∈ {1, 2}, constrained by S = {s_{k} :

L_{0} → L_{k}}_{k∈{1,2}} is given by (inf_{1} ▷◁^{S} inf_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), I, gI⟩ where:

I = ( ⋃_{k∈{1,2}}{⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ : i ∈ I_{k}})

and

gI = ( ⋃_{k∈{1,2}}{⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ : i ∈ gI_{k}}).

Proof. Again, since Set has all coequalizers and we have shown, in 3.2.39, that Lang_{lInf} is a cofibration, we can use A.2.13 and conclude that all Lang_{lInf}-pushouts exist, once we show that all binary sums exist in lInf.

For the purpose, let (inf_{1} ▷◁ inf_{2}) = ⟨(L_{1} ▷◁ L_{2}), (ι_{1}(I_{1}) ∪ ι_{2}(I_{2})), (ι_{1}(gI_{1}) ∪ ι_{2}(gI_{2}))⟩. The fact that show that {ι_{k} : inf_{k} → (inf_{1} ▷◁ inf_{2})}_{k∈{1,2}} is a sum in lInf, where each ι_{k} : L_{k} → (L_{1} ▷◁ L_{2}) is the corresponding injection function, can be proved similarly to what was done in 4.2.4. Analogously, we obtain (inf_{1} ▷◁^{S} inf_{2}) by Lang_{lInf}-cocartesian lift of ⟨(inf_{1} ▷◁ inf_{2}), [ ]_{≈^{S}}⟩. Furthermore, we have also proved in 3.2.41 that Lang_{lInf} is coadjoint. Thus, we can use A.3.3 to conclude that lInf is suitable for synchronization.

Exactness follows from A.3.2, since we have shown in 3.2.40 that Ded_{lInf} is a cocartesian coadjoint. Note that, from 3.1.24, we know that Lang_{lCsy_{fin}} is a cofibration. ⊓ ⊔

Again, we can identify (inf_{1} ▷◁ inf_{2}) as a free synchronization of the layered inference systems inf_{1} and inf_{2}.

4.3 An example

We illustrate the synchronization mechanism by means of a meaningful, although simple, example. It consists of a combination of temporal and equational logics suited for dealing, for instance, with the behaviour of a system with a time dependent attribute that takes values over an equationally specified data type. A similar example can be found in [SSC97a].

Example 4.3.1. Equational reasoning within temporal logic. Let us consider the logic systems of propositional temporal logic and equation logic as defined, respectively, in 2.3.7 and 2.3.10. We are interested in taking a set P of propositional symbols and an unsorted equational signature F such that P = {p_{t} : t ∈ Term_{F} (∅)}, where each p_{t} should be read as “the attribute has value t”. Fixed a set of equational variables X, our purpose is to synchronize both the corresponding satisfaction systems and inference systems under the constraint S = ⟨s_{ptl} : Term_{F} (∅)^{2} → L_{ptl}^{P}, s_{eql} : Term_{F} (∅)^{2} → L_{eql}^{F}(X)⟩ where for every pair ⟨t_{1}, t_{2}⟩ ∈ Term_{F} (∅)^{2} we define

s_{1}(⟨t_{1}, t_{2}⟩) = (p_{t_{1}} ⇒ p_{t_{2}}) and s_{1}(⟨t_{1}, t_{2}⟩) = (t_{1} = t_{2}).

Clearly, the idea is that the attribute does not take distinct values at same time.

In fact, if we consider sat_{ptl}^{P} ▷◁^{S} sat_{eql}^{⟨F,X⟩}, not only the language obtained is roughly the sum of the two but where we cannot distinguish between (p_{t_{1}} ⇔ p_{t_{2}}) and (t_{1} = t_{2}) for variable free terms t_{1} and t_{2}, but we shall only have models ⟨ϑ, T ⟩ such that ϑ⊩_{ptl}^{P} (p_{t_{1}} ⇒ p_{t_{2}}) if and only if T ⊩_{eql}^{⟨F,X⟩}(t_{1} = t_{2}). The power of the logic obtained is perhaps better illustrated at the deductive level. So, let us consider the corresponding synchronization inf = inf _{ptl}^{P} ▷◁^{S} inf _{eql}^{⟨F,X⟩} of the inference systems. In particular, it is possible to take advantage of equational reasoning within temporal reasoning to prove. For illustration purposes, assume that f ∈ F_{1}, x ∈ X and u ∈ Term_{F} (∅). We prove that if f is idempotent and, in each time step, the attribute value is obtained by applying f to its previous value, then a value obtained at one instant in time shall be obtained again within two steps. That is, letting

Φ = ({(x = f(f(x)))} ∪ {(p_{t} ⇒ (X p_{f(t)})) : t ∈ Term_{F} (∅)})

we show that

Φ⊢_{inf} (p_{u} ⇒ (G(p_{u} ∨ (X p_{u})))).

1. (p_{u} ⇒ (X p_{f(u)}))  Hypothesis
2. (p_{f(u)} ⇒ (X p_{f(f(u))}))  Hypothesis
3. (X(p_{f(u)} ⇒ (X p_{f(f(u))})))  X-Nec:2
4. ((X(p_{f(u)} ⇒ (X p_{f(f(u))}))) ⇒ ((X p_{f(u)}) ⇒ (X(X p_{f(f(u))}))))  X-Norm
5. ((X p_{f(u)}) ⇒ (X(X p_{f(f(u))})))  MP:3,4
6. (p_{u} ⇒ (X(X p_{f(f(u))})))  Syl:1,5
7. (x = f(f(x)))  Hypothesis
8. (f(f(x)) = x)  Symm:7
9. (f(f(u)) = u) alias (p_{f(f(u))} ⇒ p_{u})  Subst:8
10. (X(p_{f(f(u))} ⇒ p_{u}))  X-Nec:9
11. ((X(p_{f(f(u))} ⇒ p_{u})) ⇒ ((X p_{f(f(u))}) ⇒ (X p_{u})))  X-Norm
12. ((X p_{f(f(u))}) ⇒ (X p_{u}))  MP:10,11
13. (X((X p_{f(f(u))}) ⇒ (X p_{u})))  X-Nec:12
14. ((X((X p_{f(f(u))}) ⇒ (X p_{u}))) ⇒ ((X(X p_{f(f(u))})) ⇒ (X(X p_{u}))))  X-Nec:12
15. ((X(X p_{f(f(u))})) ⇒ (X(X p_{u})))  MP:13,14
16. (p_{u} ⇒ (X(X p_{u})))  Syl:6,15
17. ((X p_{u}) ⇒ (((X p_{u}) ⇒ (X p_{u})) ⇒ (X p_{u})))  Ax01
18. (((X p_{u}) ⇒ (((X p_{u}) ⇒ (X p_{u})) ⇒ (X p_{u})))⇒
(((X p_{u}) ⇒ ((X p_{u}) ⇒ (X p_{u}))) ⇒ ((X p_{u}) ⇒ (X p_{u})))) Ax02
19. ((X p_{u}) ⇒ ((X p_{u}) ⇒ (X p_{u})))  Ax01
20. (((X p_{u}) ⇒ ((X p_{u}) ⇒ (X p_{u}))) ⇒ ((X p_{u}) ⇒ (X p_{u})))  MP:19,18
21. ((X p_{u}) ⇒ (X p_{u}))  MP:17,20
22. (p_{u} ⇒ (p_{u} ∨ (X p_{u})))  Ax06
23. ((X p_{u}) ⇒ (p_{u} ∨ (X p_{u})))  Ax07
24. (X(p_{u} ⇒ (p_{u} ∨ (X p_{u}))))  X-Nec:22
25. ((X(p_{u} ⇒ (p_{u} ∨ (X p_{u})))) ⇒ ((X p_{u}) ⇒ (X(p_{u} ∨ (X p_{u})))))  X-Norm
26. ((X p_{u}) ⇒ (X(p_{u} ∨ (X p_{u}))))  MP:24,25
27. (X((X p_{u}) ⇒ (p_{u} ∨ (X p_{u}))))  X-Nec:23
28. ((X((X p_{u}) ⇒ (p_{u} ∨ (X p_{u})))) ⇒ ((X(X p_{u})) ⇒ (X(p_{u} ∨ (X p_{u}))))) X-Norm
29. ((X(X p_{u})) ⇒ (X(p_{u} ∨ (X p_{u}))))  MP:27,28
30. (p_{u} ⇒ (X(p_{u} ∨ (X p_{u}))))  Syl:16,29

31. ((p_{u} ⇒ (X(p_{u} ∨ (X p_{u})))) ⇒ (((X p_{u}) ⇒ (X(p_{u} ∨ (X p_{u}))))⇒
((p_{u} ∨ (X p_{u})) ⇒ (X(p_{u} ∨ (X p_{u}))))))  Ax08
32. (((X p_{u}) ⇒ (X(p_{u} ∨ (X p_{u})))) ⇒ ((p_{u} ∨ (X p_{u})) ⇒ (X(p_{u} ∨ (X p_{u}))))) MP:30,31
33. ((p_{u} ∨ (X p_{u})) ⇒ (X(p_{u} ∨ (X p_{u}))))  MP:26,32
34. (G((p_{u} ∨ (X p_{u})) ⇒ (X(p_{u} ∨ (X p_{u})))))  G-Nec:33
35. ((G((p_{u} ∨ (X p_{u})) ⇒ (X(p_{u} ∨ (X p_{u})))))⇒
((p_{u} ∨ (X p_{u})) ⇒ (G(p_{u} ∨ (X p_{u})))))  G2
36. ((p_{u} ∨ (X p_{u})) ⇒ (G(p_{u} ∨ (X p_{u}))))  MP:34,35
37. (p_{u} ⇒ (G(p_{u} ∨ (X p_{u}))))  Syl:22,36

In the proof we have freely used Syl, which corresponds to the syllogistic derived rule of classical logic:

Syl. from (φ ⇒ ψ) and (ψ ⇒ υ) conclude (φ ⇒ υ).

Next we present a simple proof of this fact, i.e., {(φ ⇒ ψ), (ψ ⇒ υ)}⊢_{cpl}^{P}(φ ⇒ υ).

1. ((ψ ⇒ υ) ⇒ (φ ⇒ (ψ ⇒ υ)))  Ax01
2. (ψ ⇒ υ)  Hypothesis
3. (φ ⇒ (ψ ⇒ υ))  MP:2,1
4. ((φ ⇒ (ψ ⇒ υ)) ⇒ ((φ ⇒ ψ) ⇒ (φ ⇒ υ))) Ax02
5. ((φ ⇒ ψ) ⇒ (φ ⇒ υ))  MP:3,4
6. (φ ⇒ ψ)  Hypothesis
7. (φ ⇒ υ)  MP:6,5

Further examples of application of this combination mechanism shall appear in Chapter 6, when we analyze the relationship between synchronization and fibring.

4.4 Transference results

As we have stressed from the beginning, the point in having a theory of logic combination relies a lot on obtaining general results of transference of properties. Herein we shall give an account of how two essential properties of logic, soundness and completeness, interact with synchronization. Although we shall only deal explicitly with (layered) logic systems, according to 2.2.12 and 3.2.7, the results we obtain shall be immediately applicable to the corresponding indexed notion of (layered) logic. Let us tackle soundness first.

4.4.1 Soundness preservation

The question to be answered is: are there necessary, or at least sufficient, conditions to be satisfied in order to guarantee that if we are given two sound proof-theoretic systems with respect to two model-theoretic systems, a synchronization of the proof-theoretic components is still sound with respect to the corresponding synchronization of the modeltheoretic ones?

In the setting that we have developed so far, this question can be posed either in the non-layered context of inference/satisfaction systems or inference/valuation systems, or at the level of layered inference/layered valuation systems. In all these cases, however, the answer is the same: soundness is always preserved, both by free and by constrained synchronization. This is an immediate consequence of the fact that all the involved categories of (layered) model-theoretic logic systems are suitable and the proof-theoretic ones are exact, for synchronization. Let us first prove an auxiliary lemma.

Lemma 4.4.1. Let ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩ be categories of (layered) logic systems, the former being finitary and exact for synchronization, and the latter suitable for synchronization. Moreover, let cal_{k} ∈ |Cal| and sem_{k} ∈ |Sem| be such that Lang_{Cal}(cal_{k}) = Lang_{Sem}(sem_{k}), for k ∈ {1, 2}, and S a corresponding sharing constraint. If both pairs ⟨cal_{1}, sem_{1}⟩ and ⟨cal_{2}, sem_{2}⟩ are sound, and {f_{k} : cal_{k} → (cal_{1} ▷◁^{S} cal_{2})}_{k∈{1,2}} and {g_{k} : sem_{k} → (sem_{1} ▷◁^{S} sem_{2})}_{k∈{1,2}} correspond to synchronizations in Cal and Sem constrained by S, then the unique h : Lang_{Cal}(cal_{1} ▷◁^{S} cal_{2}) → Lang_{Sem}(sem_{1} ▷◁^{S} sem_{2}) such that (h ∘ Lang_{Cal}(f_{k})) = Lang_{Sem}(g_{k}), for k ∈ {1, 2}, is a bijection and induces an (l)Csy-morphism h : Ded(cal_{1} ▷◁^{S} cal_{2}) → Ent(sem_{1} ▷◁^{S} sem_{2}).

Proof. Let us start by fixing the notation L_{k} = Lang_{Cal}(cal_{k}) = Lang_{Sem}(sem_{k}) for k ∈ {1, 2}, and assuming that S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}}.

Given that ⟨Cal, Ded⟩ is exact for synchronization, it immediately follows that not only {Ded(f_{k}) : Ded(cal_{k}) → Ded(cal_{1} ▷◁^{S} cal_{2})}_{k∈{1,2}} is a Lang_{(l)Csy}-pushout of Ded(cal_{1}) and Ded(cal_{2}) along S, but also that {Lang_{Cal}(f_{k}) : L_{k} → Lang_{Cal}(cal_{1} ▷◁^{S} cal_{2})}_{k∈{1,2}} is a pushout of S in Set. Moreover, {Lang_{Sem}(g_{k}) : L_{k} → Lang_{Sem}(sem_{1} ▷◁^{S} sem_{2})}_{k∈{1,2}} is also a pushout of S in Set, since ⟨Sem, Ent⟩ is suitable for synchronization. In particular, we obviously have that (Lang_{Sem}(g_{1}) ∘ s_{1}) = (Lang_{Sem}(g_{2}) ∘ s_{2}).

Clearly, the fact that each pair ⟨cal_{k}, sem_{k}⟩ is sound amounts precisely to the fact that id_{L_{k}} : Ded(cal_{k}) → Ent(sem_{k}) is an (l)Csy-morphism. Thus, let us consider the sink {(Ent(g_{k}) ∘ id_{L_{k}}) : Ded(cal_{k}) → Ent(sem_{1} ▷◁^{S} sem_{2})}_{k∈{1,2}}. Easily, then, we have (Lang_{(l)Csy}(Ent(g_{1}) ∘ id_{L_{1}}) ∘ s_{1}) = (Lang_{Sem}(g_{1}) ∘ id_{L_{1}} ∘ s_{1}) = (Lang_{Sem}(g_{1}) ∘ s_{1}) = (Lang_{Sem}(g_{2})∘s_{2}) = (Lang_{Sem}(g_{2})∘id_{L_{2}} ∘s_{2}) = (Lang_{(l)Csy}(Ent(g_{2})∘id_{L_{2}})∘s_{2}). So, using the universal property of the Lang_{(l)Csy}-pushout Ded(cal_{1} ▷◁^{S} cal_{2}), there exists a unique (l)Csy-morphism h : Ded(cal_{1} ▷◁^{S} cal_{2}) → Ent(sem_{1} ▷◁^{S} sem_{2}) such that (h ∘ Ded(f_{k})) = (Ent(g_{k}) ∘ id_{L_{k}}) for k ∈ {1, 2}.

Therefore, we also have that Lang_{(l)Csy}(h ∘ Ded(f_{k})) = Lang_{(l)Csy}(Ent(g_{k}) ∘ id_{L_{k}}), i.e., (h ∘ Lang_{Cal}(f_{k})) = (Lang_{Sem}(g_{k}) ∘ id_{L_{k}}) = Lang_{Sem}(g_{k}). The universal property of the pushout Lang_{Cal}(cal_{1} ▷◁^{S} cal_{2}) clearly imposes that h : Lang_{Cal}(cal_{1} ▷◁^{S} cal_{2}) → Lang_{Sem}(sem_{1} ▷◁^{S} sem_{2}) is the unique such function. Given that Lang_{Sem}(sem_{1} ▷◁^{S} sem_{2}) is also a pushout of S, it is well known that h must be a bijection. ⊓ ⊔

Please note that the finitariness condition is not essential here. However, it encompasses the intuition that ⟨Cal, Ded⟩ is a category of proof-theoretic logic systems.

With all this generality, we can now instantiate this result with our parameters of interest.

Proposition 4.4.2. Let inf_{k} = ⟨L_{k}, I_{k}⟩ and sat_{k} = ⟨L_{k}, M_{k}, ⊩_{k}⟩, for k ∈ {1, 2}, be inference and satisfaction systems, respectively, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a corresponding sharing constraint. Then, the pair ⟨(inf_{1} ▷◁^{S} inf_{2}), (sat_{1} ▷◁^{S} sat_{2})⟩ (as defined in 4.2.4 and 4.2.1) is sound whenever both ⟨inf_{1}, sat_{1}⟩ and ⟨inf_{2}, sat_{2}⟩ are sound.

Proof. We already know, from 2.2.48 and 4.2.4, that Inf is finitary and exact for synchronization. From 4.2.1 we also know that Sat is suitable for synchronization. Moreover, in both 4.2.4 and 4.2.1 the syntactic construction coincides precisely with (L_{1} ▷◁^{S} L_{2}). Therefore, if both ⟨inf_{1}, sat_{1}⟩ and ⟨inf_{2}, sat_{2}⟩ are sound, we can use lemma 4.4.1 and conclude that the unique function h referred by the lemma is precisely id_{(L_{1}\begin{matrix}S \\ ▷◁ \end{matrix}L_{2})} : (L_{1} ▷◁^{S} L_{2}) → (L_{1} ▷◁^{S} L_{2}). Hence, id_{(L_{1}\begin{matrix}S \\ ▷◁ \end{matrix}L_{2})} : (inf_{1} ▷◁^{S} inf_{2}) → (sat_{1} ▷◁^{S} sat_{2}) is guaranteed to be a Csymorphism, or equivalently, ⟨(inf_{1} ▷◁^{S} inf_{2}), (sat_{1} ▷◁^{S} sat_{2})⟩ is sound. ⊓ ⊔

The next results are obtained analogously.

Proposition 4.4.3. Let inf_{1} = ⟨L_{1}, I_{1}⟩ and inf_{2} = ⟨L_{2}, I_{2}⟩ be inference system, val_{1} = ⟨L_{1}, M_{1}, S_{1}⟩ and val_{2} = ⟨L_{2}, M_{2}, S_{2}⟩ be valuation systems, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a corresponding sharing constraint. Then, ⟨(inf_{1} ▷◁^{S} inf_{2}), (val_{1} ▷◁^{S} val_{2})⟩ is sound whenever both ⟨inf_{1}, val_{1}⟩ and ⟨inf_{2}, val_{2}⟩ are sound.

Proof. We already know, from 2.2.48 and 4.2.4, that Inf is finitary and exact for synchronization. From 4.2.2 we also know that Val is suitable for synchronization. Moreover, in both 4.2.4 and 4.2.2 the syntactic construction coincides precisely with (L_{1} ▷◁^{S} L_{2}). Therefore, if both ⟨inf_{1}, val_{1}⟩ and ⟨inf_{2}, val_{2}⟩ are sound, we can use lemma 4.4.1 and conclude that the unique function h referred by the lemma is precisely id_{(L_{1}\begin{matrix}S \\ ▷◁ \end{matrix}L_{2})} : (L_{1} ▷◁^{S} L_{2}) → (L_{1} ▷◁^{S} L_{2}). Hence, id_{(L_{1}\begin{matrix}S \\ ▷◁ \end{matrix}L_{2})} : (inf_{1} ▷◁^{S} inf_{2}) → (val_{1} ▷◁^{S} val_{2}) is guaranteed to be a Csymorphism, or equivalently, ⟨(inf_{1} ▷◁^{S} inf_{2}), (val_{1} ▷◁^{S} val_{2})⟩ is sound. ⊓ ⊔

Proposition 4.4.4. Let inf_{1} = ⟨L_{1}, I_{1}, gI_{1}⟩ and inf_{2} = ⟨L_{2}, I_{2}, gI_{2}⟩ be layered inference system, val_{1} = ⟨L_{1}, M_{1}, S_{1}⟩ and val_{2} = ⟨L_{2}, M_{2}, S_{2}⟩ be layered valuation systems, and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a corresponding sharing constraint. Then, ⟨(inf_{1} ▷◁^{S} inf_{2}), (val_{1} ▷◁^{S} val_{2})⟩ is sound whenever both ⟨inf_{1}, val_{1}⟩ and ⟨inf_{2}, val_{2}⟩ are sound.

Proof. We already know, from 3.2.36 and 4.2.5, that lInf is finitary and exact for synchronization. From 4.2.3 we also know that lVal is suitable for synchronization. Moreover, in both 4.2.5 and 4.2.3 the syntactic construction coincides precisely with (L_{1} ▷◁^{S} L_{2}). Therefore, if both ⟨inf_{1}, val_{1}⟩ and ⟨inf_{2}, val_{2}⟩ are sound, we can use lemma 4.4.1 and conclude that the unique function h referred by the lemma is precisely id_{(L_{1}\begin{matrix}S \\ ▷◁ \end{matrix}L_{2})} : (L_{1} ▷◁^{S} L_{2}) → (L_{1} ▷◁^{S} L_{2}). Hence, id_{(L_{1}\begin{matrix}S \\ ▷◁ \end{matrix}L_{2})} : (inf_{1} ▷◁^{S} inf_{2}) → (val_{1} ▷◁^{S} val_{2}) is guaranteed to be an lCsy-morphism, or equivalently, ⟨(inf_{1} ▷◁^{S} inf_{2}), (val_{1} ▷◁^{S} val_{2})⟩ is sound. ⊓ ⊔

4.4.2 Completeness preservation

General transference results for completeness are necessarily harder to obtain, when compared with those for soundness. As we have seen, soundness transfers, without further provisos, for both free and constrained synchronization as a consequence of just the facts that all our working categories of model-theoretic (layered) logic systems (Sat, Val and lVal) are suitable, and all our working categories of proof-theoretic (layered) logic systems (Inf and lInf) are exact for synchronization. However, the converse does not hold. As we have seen in 4.2.1, 4.2.2 and 4.2.3, it is not hard to find examples showing that Sat, Val and lVal can misbehave. Work reported in [SSC97b] led to a bulk of completeness preservation results for synchronization, namely concerning systems with a classical negation. Herein, however, we shall dwell on more or less orthogonal results. Moreover, the techniques that we are going to use in this context shall also open the way to our subsequent study of the problem of completeness preservation by the more sophisticated combination mechanism of fibring, later on in Chapter 6. Thus, for the moment, our aim is to establish fair sufficient conditions for completeness to transfer through synchronization, either in the context of Inf/Sat, Inf/Val or lInf/lVal pairs. The key idea, in any case, is to guarantee the existence of enough models of a certain kind in our target semantic systems. Let us start with the simplest case of Inf and Sat.

Definition 4.4.5. Let inf = ⟨L, I⟩ be an inference system and sat = ⟨L, M, ⊩⟩ a satisfaction system. The pair ⟨inf , sat⟩ is said to be full if for every Φ ⊆ L there exists m ∈ M such that:

• m⊩φ if and only if Φ⊢_{inf} φ.

Clearly, if ⟨inf , sat⟩ is full and sound, then sat cannot have any other models than those postulated by the definition above.

In this setting, it is easy to show that fullness is preserved by synchronization.

Proposition 4.4.6. For k ∈ {1, 2}, let ⟨inf_{k}, sat_{k}⟩ be a full pair of inference and satisfaction systems, over the same language. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ▷◁^{S} inf_{2}), (sat_{1} ▷◁^{S} sat_{2})⟩ is full.

Proof. Assume that each sat_{k} = ⟨L_{k}, M_{k}, ⊩_{k}⟩ and fix the notation sat = (sat_{1} ▷◁^{S} sat_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), M, ⊩⟩. Assume also that each inf_{k} = ⟨L_{k}, I_{k}⟩ and fix the notation inf = (inf_{1} ▷◁^{S} inf_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), I⟩. Furthermore, let E ⊆ (L_{1} ▷◁^{S} L_{2}).

Define Φ = {φ ∈ L_{1} : [φ_{1}]_{≈^{S}} ∈ E^{⊢_{inf}} } and Ψ = {ψ ∈ L_{2} : [ψ_{2}]_{≈^{S}} ∈ E^{⊢_{inf}} }. Letting {ι_{k}^{S} : L_{k} → (L_{1} ▷◁^{S} L_{2})}_{k∈{1,2}} be the sink corresponding to the synchronization, we have indeed that Φ = (ι_{1}^{S})^{−1}(E^{⊢_{inf}}) and Ψ = (ι_{2}^{S})^{−1}(E^{⊢_{inf}}) are theories of ⊢_{inf_{1}} and ⊢_{inf_{2}} , respectively. This is immediate from the fact that each ι_{k}^{S} : inf_{k} → (inf_{1} ▷◁^{S} inf_{2}) is an Inf-morphism, and hence each ι_{k}^{S} : ⟨L_{k}, ⊢_{inf_{k}} ⟩ → ⟨(L_{1} ▷◁^{S} L_{2}), ⊢_{inf} ⟩ a Csy-morphism. Therefore, since each pair ⟨inf_{k}, sat_{k}⟩ is full, there exist m_{1} ∈ M_{1} and m_{2} ∈ M_{2} such that m_{1} ⊩_{1}φ if and only if φ ∈ Φ, and m_{2} ⊩_{2}ψ if and only if ψ ∈ Ψ. We now prove that ⟨m_{1}, m_{2}⟩ ∈ M and that ⟨m_{1}, m_{2}⟩⊩e′ if and only if E ⊢_{inf} e′.

First of all suppose that S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}}, and let υ ∈ L_{0}. Obviously, s_{1}(υ)_{1} ≈^{S} s_{2}(υ) and we can denote [s_{1}(υ)_{1}]_{≈^{S}} = [s_{2}(υ)_{2}]_{≈^{S}} by e. Thus, clearly, m_{1} ⊩_{1}s_{1}(υ) if and only if E ⊢_{inf} e if and only if m_{2} ⊩_{2}s_{2}(υ), and we can conclude that ⟨m_{1}, m_{2}⟩ is a model of (sat_{1} ▷◁^{S} sat_{2}).

Now, ⟨m_{1}, m_{2}⟩⊩e′ if and only if m_{1} ⊩_{1}φ for every φ_{1} ∈ e′ and m_{2} ⊩_{2}ψ for every ψ_{2} ∈ e′. The same is to say that e′ ⊆ (Φ_{1} ∪ Ψ_{2}), or equivalently, that E ⊢_{inf} e′. ⊓ ⊔

It is a trivial task to verify that fullness implies completeness.

Proposition 4.4.7. Let ⟨inf , sat⟩ be full pair of inference and satisfaction systems, over the same language. Then, the pair ⟨inf , sat⟩ is complete.

Proof. Let sat = ⟨L, M, ⊩⟩ and suppose that Φ⊢_{i̸nf} ψ. From the fullness of ⟨inf , sat⟩ we can guarantee the existence of a model m ∈ M such that m⊩φ if and only if Φ⊢_{inf} φ. Since ⊢_{inf} is extensive, necessarily, m⊩Φ. However, obviously, m⊩ψ̸ and we can conclude that Φ⊨_{s̸at}ψ. ⊓ ⊔

Thus, our completeness preservation result for inference and satisfaction systems follows.

Proposition 4.4.8. For k ∈ {1, 2}, let ⟨inf_{k}, sat_{k}⟩ be a full pair of inference and satisfaction systems over the same language. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ▷◁^{S} inf_{2}), (sat_{1} ▷◁^{S} sat_{2})⟩ is complete.

Proof. Immediate from 4.4.6 and 4.4.7. ⊓ ⊔

Expectedly, a similar result can be obtained in the context of Inf and Val.

Definition 4.4.9. Let inf = ⟨L, I⟩ be an inference system. An L-structure ⟨T, [[ ]], D⟩ is said to be a structure for inf provided that, for every i ∈ I, the following condition holds:

• if [[Prem(i)]] ⊆ D then [[Conc(i)]] ∈ D.

We denote the class of all structures for inf by Str(inf ).

Clearly, this means that the structure is such that all the inferences of inf are sound with respect to int. Making the parallel with the Inf/Sat case, this means precisely that {φ ∈ L : [[φ]] ∈ D} is a ⊢_{inf} -theory.

In what follows we shall consider as fixed an arbitrary property P of the pair ⟨T, D⟩, and for each language L, denote by Str(L, P) the class of all P -structures over L, i.e., L-structures ⟨T, [[ ]], D⟩ such that ⟨T, D⟩ satisfies P.

Fullness with respect to P -structures, now, takes the following form.

Definition 4.4.10. Let inf = ⟨L, I⟩ be an inference system and val = ⟨L, M, S⟩ a valuation system. The pair ⟨inf , val⟩ is said to be full with respect to P if (Str(L, P)∩Str(inf )) ⊆ S(M).

Clearly, if ⟨inf , val⟩ is full with respect to the class of all L-structures, we have that ⟨inf , val⟩ is sound if and only if S(M) = Str(inf ).

In this setting, it is easy to show that fullness with respect to P is preserved by synchronization. We first prove an auxiliary lemma.

Lemma 4.4.11. Let inf_{k} for k ∈ {1, 2} be inference systems, S a corresponding sharing constraint, and {ι_{k}^{S} : inf_{k} → (inf_{1} ▷◁^{S} inf_{2})}_{k∈{1,2}} be the sink corresponding to the synchronization of inf_{1} and inf_{2} constrained by S (as given by 4.2.4). If ⟨T, [[ ]], D⟩ is a structure for (inf_{1} ▷◁^{S} inf_{2}) then ⟨T, [[ ]], D⟩|_{ι_{k}^{S}} is a structure for inf_{k} for every k ∈ {1, 2}.

Proof. Let each inf_{k} = ⟨L_{k}, I_{k}⟩, and denote ⟨T, [[ ]], D⟩|_{ι_{k}^{S}} simply by ⟨T, [[ ]]_{k}, D⟩, i.e., [[ ]]_{k} = ([[ ]]∘ι_{k}^{S}). Recall that ι_{k}^{S} : L_{k} → (L_{1} ▷◁^{S} L_{2}) is the function such that ι_{k}^{S}(υ) = [υ_{k}]_{≈^{S}} for every υ ∈ L_{k}. Trivially, both ⟨T, [[ ]]_{1}, D⟩ and ⟨T, [[ ]]_{2}, D⟩ are P -structures, once they share both the set of truth values and the subset of designated values with ⟨T, [[ ]], D⟩. So, we are left with showing that each ⟨T, [[ ]]_{k}, D⟩ is a structure for inf_{k}.

Let i ∈ I_{k} and suppose that [[Prem(i)]]_{k} ⊆ D, i.e., [[[Prem(i)_{k}]_{≈^{S}}]] ⊆ D. Clearly, we have that ⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ is an inference of (inf_{1} ▷◁^{S} inf_{2}). Thus, since ⟨T, [[ ]], D⟩ is a structure for it, we must also have [[[Conc(i)_{k}]_{≈^{S}}]] ∈ D. But this is equivalent to [[Prem(i)]]_{k} ∈ D and we are done. ⊓ ⊔ Proposition 4.4.12. For k ∈ {1, 2}, let ⟨inf_{k}, val_{k}⟩ be a pair of inference and satisfaction systems over the same language and full with respect to P. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ▷◁^{S} inf_{2}), (val_{1} ▷◁^{S} val_{2})⟩ is full with respect to P.

Proof. Assume that each val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩ and (val_{1} ▷◁^{S} val_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), M, S⟩, and let ⟨T, [[ ]], D⟩ be a P -structure for (inf_{1} ▷◁^{S} inf_{2}). According to 4.4.11, letting {ι_{k}^{S} : L_{k} → (L_{1} ▷◁^{S} L_{2})}_{k∈{1,2}} be the sink corresponding to the synchronization, we have that ⟨T, [[ ]], D⟩|_{ι_{k}^{S}} = ⟨T, ([[ ]] ∘ ι_{k}^{S}), D⟩ is a P -structure for inf_{k}. Therefore, since each pair ⟨inf_{k}, val_{k}⟩ is full with respect to P, there exist models m_{1} ∈ M_{1} and m_{2} ∈ M_{2} such that S_{1}(m_{1}) = ⟨T, ([[ ]]∘ι_{1}^{S}), D⟩ and S_{2}(m_{2}) = ⟨T, ([[ ]]∘ι_{2}^{S}), D⟩. We now prove that ⟨m_{1}, m_{2}⟩ ∈ M and S(⟨m_{1}, m_{2}⟩) = ⟨T, [[ ]], D⟩. Clearly, T_{m_{1}} = T_{m_{2}} = T and D_{m_{1}} = D_{m_{2}} = D. Thus, it suffices to show that m_{1} and m_{2} agree on the interpretation of shared sentences. Suppose that S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}}, and let υ ∈ L_{0}. Obviously, s_{1}(υ)_{1} ≈^{S} s_{2}(υ) and we can denote [s_{1}(υ)_{1}]_{≈^{S}} = [s_{2}(υ)_{2}]_{≈^{S}} by e. Thus, clearly, [[s_{1}(υ)]]_{m_{1}} = [[e]] = [[s_{2}(υ)]]_{m_{2}}, and we can conclude that ⟨m_{1}, m_{2}⟩ is a model of (val_{1} ▷◁^{S} val_{2}). It is straigthforward to verify that S(⟨m_{1}, m_{2}⟩) = ⟨T, [[ ]], D⟩. In fact it suffices to recall that [[[υ_{k}]_{≈^{S}}]] = [[υ]]_{m_{k}}. ⊓ ⊔

Our first completeness preservation result for inference and valuation systems is quite general. We shall admit any possible structure as good, or, in other words, we shall consider P to be universally satisfied, i.e., for every language Str(L, P) = Str(L). In this broad case we can very easily build arbitrary structures for an inference system, as follows:

Proposition 4.4.13. Let inf be an inference system over language L and Φ ⊆ L. Then, the triple

⟨L, id_{L}, Φ^{⊢_{inf}} ⟩

is a structure for inf .

Proof. It is obvious that ⟨L, id_{L}, Φ^{⊢_{inf}} ⟩ is an L-structure. Therefore, we just have to prove that it is indeed a structure for inf . For the purpose, let i ∈ I and suppose that Prem(i) ⊆ Φ^{⊢_{inf}}. Then, easily, Conc(i) ∈ Prem(i)^{⊢_{inf}} ⊆ Φ^{⊢_{inf}}. ⊓ ⊔

As it stands, this straightforward representation of the theory as an interpretation structure is sufficient to guarantee the following completeness theorem.

Proposition 4.4.14. Let ⟨inf , val⟩ be a pair of inference and valuation systems over the same language and full with respect to all structures. Then, the pair ⟨inf , val⟩ is complete.

Proof. Let val = ⟨L, M, S⟩ and suppose that Φ⊢_{i̸nf} ψ. Of course, we need to show that Φ⊨_{v̸al}ψ. So, it is sufficient to find a model m of val such that [[Φ]]_{m} ⊆ D_{m} but [[ψ]]_{m} ∉ D_{m}. Consider the structure ⟨L, id_{L}, Φ^{⊢_{inf}} ⟩ defined from inf as in 4.4.13 above. As we know, it is a structure for inf . Hence, since ⟨inf , val⟩ is full with respect to the class of all L-structures, there exists a model m ∈ M with S(m) = ⟨L, id_{L}, Φ^{⊢_{inf}} ⟩. Trivially, then, Φ ⊆ Φ^{⊢_{inf}} just by using extensiveness while ψ ∉ Φ^{⊢_{inf}}, which concludes the proof. ⊓ ⊔

The preservation result follows easily.

Proposition 4.4.15. For k ∈ {1, 2}, let ⟨inf_{k}, val_{k}⟩ be a pair of inference and valuation systems over the same language and full with respect to all structures. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ⊛^{S} inf_{2}), (val_{1} ⊛^{S} val_{2})⟩ is complete.

Proof. Immediate from 4.4.12 and 4.4.14. ⊓ ⊔

Of course, one may argue about the legitimacy of such semantic structures for inf as those defined in 4.4.13 and used to establish 4.4.14 and 4.4.15. However, we can improve this state of affairs by trying to forbid such general structures. Among several possibilities, we shall choose P to be the class of all two-valued structures.

Definition 4.4.16. The unique two-valued pair of sets is ⟨{0, 1}, {1}⟩.

Necessarily, the following results are closely related with those obtained for satisfaction systems.

Proposition 4.4.17. Let inf be an inference system over language L and Φ ⊆ L. Then, the triple

⟨{0, 1}, [[ ]]_{Φ}, {1}⟩

is a structure for inf , where:

[[φ]]_{Φ} = \{ \begin{matrix}1 & if Φ⊢_{inf} φ \\ 0 & otherwise & . \end{matrix}

Proof. It is obvious that ⟨L, id_{L}, Φ^{⊢_{inf}} ⟩ is a two-valued L-structure. Therefore, we only have to prove that it is indeed a structure for inf . For the purpose, let i ∈ I and suppose that [[Prem(i)]]_{Φ} ⊆ {1}. This means that Prem(i) ⊆ Φ^{⊢_{inf}}.Then, easily, Conc(i) ∈ Prem(i)^{⊢_{inf}} ⊆ Φ^{⊢_{inf}} and [[Conc(i)]]_{Φ} = 1. ⊓ ⊔

Proposition 4.4.18. Let ⟨inf , val⟩ be a pair of inference and valuation systems over the same language and full with respect to all two-valued structures. Then, the pair ⟨inf , val⟩ is complete.

Proof. Let val = ⟨L, M, S⟩ and suppose that Φ⊢_{i̸nf} ψ. Of course, we need to show that Φ⊨_{v̸al}ψ. So, it is sufficient to find a model m of val such that [[Φ]]_{m} ⊆ D_{m} but [[ψ]]_{m} ∉ D_{m}. Consider the structure ⟨{0, 1}, [[ ]]_{Φ}, {1}⟩ defined from inf as in 4.4.17 above. As we know, it is a two-valued structure for inf . Hence, since ⟨inf , val⟩ is full with respect to the class of all two-valued structures, there exists a model m ∈ M with S(m) = ⟨{0, 1}, [[ ]]_{Φ}, {1}⟩. Trivially, then, Φ ⊆ Φ^{⊢_{inf}} just by using extensiveness while ψ ∉ Φ^{⊢_{inf}}, which is equivalent to saying that [[Φ]]_{Φ} ⊆ {1} and [[ψ]]_{Φ} = 0, thus concluding the proof. ⊓ ⊔

Proposition 4.4.19. For k ∈ {1, 2}, let ⟨inf_{k}, val_{k}⟩ be a pair of inference and satisfaction systems over the same language and full with respect to all two-valued structures. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ⊛^{S} inf_{2}), (val_{1} ⊛^{S} val_{2})⟩ is complete.

Proof. Immediate from 4.4.12 and 4.4.28. ⊓ ⊔

We shall now repeat the process, with due adaptations, in the layered context of lInf/lVal.

Definition 4.4.20. Let inf = ⟨L, I, gI⟩ be a layered inference system. An L-layered structure ⟨T, [[ ]], ⋉⟩ is said to be a structure for inf provided that the following conditions hold:

• for every i ∈ I, [[Prem(i)]] ⋉ [[Conc(i)]];

• for every i ∈ gI, if ⋉[[Prem(i)]] then ⋉[[Conc(i)]].

We denote the class of all layered structures for inf by lStr(inf ).

Now, we shall consider fixed an arbitrary property P of consequence systems, and for each language L, denote by lStr(L, P) the class of all P -layered structures over L, i.e., L-layered structures ⟨T, [[ ]], ⋉⟩ such that ⟨T, ⋉⟩ satisfies P.

Definition 4.4.21. Let inf = ⟨L, I, gI⟩ be a layered inference system and val = ⟨L, M, S⟩ a layered valuation system. The pair ⟨inf , val⟩ is said to be full with respect to P if (lStr(L, P) ∩ lStr(inf )) ⊆ S(M).

Once again, it is easy to show that fullness with respect to P is preserved by synchronization. We shall first prove an auxiliary lemma.

Lemma 4.4.22. Let inf_{k} for k ∈ {1, 2} be layered inference systems, S a corresponding sharing constraint, and {ι_{k}^{S} : inf_{k} → (inf_{1} ▷◁^{S} inf_{2})}_{k∈{1,2}} be the sink corresponding to the synchronization of inf_{1} and inf_{2} constrained by S (as given by 4.2.5). If ⟨T, [[ ]], ⋉⟩ is a structure for (inf_{1} ▷◁^{S} inf_{2}) then ⟨T, [[ ]], ⋉⟩|_{ι_{k}^{S}} is a structure for inf_{k} for every k ∈ {1, 2}.

Proof. Let each inf_{k} = ⟨L_{k}, I_{k}, gI_{k}⟩, and denote ⟨T, [[ ]], ⋉⟩|_{ι_{k}^{S}} simply by ⟨T, [[ ]]_{k}, ⋉⟩. Of course, [[ ]]_{k} = ([[ ]] ∘ ι_{k}^{S}). Recall that ι_{k}^{S} : L_{k} → (L_{1} ▷◁^{S} L_{2}) is such that ι_{k}^{S}(υ) = [υ_{k}]_{≈^{S}} for every υ ∈ L_{k}. Trivially, both ⟨T, [[ ]]_{1}, ⋉⟩ and ⟨T, [[ ]]_{2}, ⋉⟩ are P -structures, once they share both the set of truth values and the internal consequence with ⟨T, [[ ]], ⋉⟩. So, we just need to show that each ⟨T, [[ ]]_{k}, ⋉⟩ is indeed a structure for inf_{k}.

Let i ∈ I_{k}. Clearly, we have that ⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ is a local inference of (inf_{1} ▷◁^{S} inf_{2}). Thus, since ⟨T, [[ ]], ⋉⟩ is a structure for it, we must have [[[Prem(i)_{k}]_{≈^{S}}]] ⋉ [[[Conc(i)_{k}]_{≈^{S}}]]. But this is equivalent to [[Prem(i)]]_{k} ⋉ [[Conc(i)]]_{k}, and we are done.

Let i ∈ gI_{k} and suppose that ⋉[[Prem(i)]]_{k}, i.e., ⋉[[[Prem(i)_{k}]_{≈^{S}}]]. Clearly, we have that ⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ is a global inference of (inf_{1} ▷◁^{S} inf_{2}). Thus, since ⟨T, [[ ]], ⋉⟩ is a structure for it, we must also have ⋉[[[Conc(i)_{k}]_{≈^{S}}]]. Clearly, this is equivalent to ⋉[[Conc(i)]]_{k} thus ending the proof. ⊓ ⊔

Proposition 4.4.23. For k ∈ {1, 2}, let ⟨inf_{k}, val_{k}⟩ be a pair of layered inference and valuation systems over the same language and full with respect to P. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ▷◁^{S} inf_{2}), (val_{1} ▷◁^{S} val_{2})⟩ is full with respect to P.

Proof. Assume that each val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩ and (val_{1} ▷◁^{S} val_{2}) = ⟨(L_{1} ▷◁^{S} L_{2}), M, S⟩, and let ⟨T, [[ ]], ⋉⟩ be a P -structure for (inf_{1} ▷◁^{S} inf_{2}). According to 4.4.22, letting {ι_{k}^{S} : L_{k} → (L_{1} ▷◁^{S} L_{2})}_{k∈{1,2}} be the sink corresponding to the synchronization, we have that ⟨T, [[ ]], ⋉⟩|_{ι_{k}^{S}} = ⟨T, ([[ ]] ∘ ι_{k}^{S}), ⋉⟩ is a P -structure for inf_{k}. Therefore, since each pair ⟨inf_{k}, val_{k}⟩ is full with respect to P, there exist models m_{1} ∈ M_{1} and m_{2} ∈ M_{2} such that S_{1}(m_{1}) = ⟨T, ([[ ]]∘ι_{1}^{S}), ⋉⟩ and S_{2}(m_{2}) = ⟨T, ([[ ]]∘ι_{2}^{S}), ⋉⟩. We now prove that ⟨m_{1}, m_{2}⟩ ∈ M and S(⟨m_{1}, m_{2}⟩) = ⟨T, [[ ]], ⋉⟩. Clearly, T_{m_{1}} = T_{m_{2}} = T and ⋉_{m_{1}} = ⋉_{m_{2}} = ⋉. Thus, it suffices to show that m_{1} and m_{2} agree on the interpretation of shared sentences. Suppose that S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}}, and let υ ∈ L_{0}. Obviously, s_{1}(υ)_{1} ≈^{S} s_{2}(υ) and we can denote [s_{1}(υ)_{1}]_{≈^{S}} = [s_{2}(υ)_{2}]_{≈^{S}} by e. Thus, clearly, [[s_{1}(υ)]]_{m_{1}} = [[e]] = [[s_{2}(υ)]]_{m_{2}}, and we can conclude that ⟨m_{1}, m_{2}⟩ is a model of (val_{1} ▷◁^{S} val_{2}). It is straigthforward to verify that S(⟨m_{1}, m_{2}⟩) = ⟨T, [[ ]], ⋉⟩. In fact it suffices to recall that [[[υ_{k}]_{≈^{S}}]] = [[υ]]_{m_{k}}. ⊓ ⊔

Our first completeness preservation result for layered inference and valuation systems again admits any possible structure as good. That is to say, we shall consider P to be universal, i.e., for every language lStr(L, P) = lStr(L). In this broad case we can very easily build arbitrary structures for a layered inference system, by ellaborating a little bit on what we did in the non-layered case.

Proposition 4.4.24. Let inf = ⟨L, I, gI⟩ be a layered inference system and Φ ⊆ L be ⊢_{inf}^{g} -closed. Then, the triple

⟨L, id_{L}, ⋉_{Φ}⟩

is a structure for inf , where:

Ψ^{⋉_{Φ}} = (Φ ∪ Ψ)^{⊢_{inf}^{l}}.

Proof. It is obvious that ⟨L, id_{L}, ⋉_{Φ}⟩ is an L-layered structure. In particular, ⋉_{Φ} corresponds precisely to the consequence induced by Φ on ⟨L, ⊢_{inf}^{g} ⟩ as in 2.1.2. Therefore, we just have to prove that it is indeed a structure for inf .

For the purpose, let i ∈ I. Easily, we have that Prem(i)⊢_{inf}^{l} Conc(i). Therefore, by monotonicity we also have (Φ ∪ Prem(i))⊢_{inf}^{l} Conc(i), i.e., Prem(i) ⋉_{Φ} Conc(i).

Let now i ∈ gI, and suppose that ⋉_{Φ}Prem(i). This means that Φ⊢_{inf}^{l} Prem(i). But we know that Φ is ⊢_{inf}^{g} -closed, and therefore also ⊢_{inf}^{l} -closed. So, Prem(i) ⊆ Φ and hence,

immediately, Conc(i) ∈ Prem(i)^{⊢_{inf}^{g}} ⊆ Φ^{⊢_{inf}^{g}} = Φ. Trivially, then, ⋉_{Φ}Conc(i). ⊓ ⊔

Proposition 4.4.25. Let ⟨inf , val⟩ be a pair of layered inference and valuation systems over the same language and full with respect to all structures. Then, the pair ⟨inf , val⟩ is complete.

Proof. Let val = ⟨L, M, S⟩. We first prove global completeness. For the purpose, suppose that Ψ⊢_{i̸nf}^{g} φ. Of course, we need to show that Ψ⊨_{v̸al}^{g}φ. So, it is sufficient to find a model

m of val such that ⋉_{m}[[Ψ]]_{m} but ⋉_{m̸}[[φ]]_{m}. Make Φ = Ψ^{⊢_{inf}^{g}}, and consider the structure ⟨L, id_{L}, ⋉_{Φ}⟩ defined from inf as in 4.4.24 above. As we know, it is a structure for inf . Hence, since ⟨inf , val⟩ is full with respect to the class of all L-layered structures, there exists a model m ∈ M with S(m) = ⟨L, id_{L}, ⋉_{Φ}⟩. Clearly, now, ⋉_{Φ}Ψ since Ψ ⊆ Φ = Ψ^{⊢_{inf}^{g}}. However, since Φ must be ⊢_{inf}^{l} -closed and φ ∉ Φ = Ψ^{⊢_{inf}^{g}} we have that ⋉_{Φ̸}φ.

Let us now prove local completeness. Suppose that Ψ⊢_{i̸nf}^{l} φ. We need to show that Ψ⊨_{v̸al}^{l}φ, i.e., that there exists a model m of val such that [[Ψ]]_{m}⋉_{m̸}[[φ]]_{m}. In this case, make Φ = ∅^{⊢_{inf}^{g}}, and consider the structure ⟨L, id_{L}, ⋉_{Φ}⟩ defined from inf as in 4.4.24 above. We know that it is a structure for inf . Since ∅^{⊢_{inf}^{g}} = ∅^{⊢_{inf}^{l}} ⊆ Ψ^{⊢_{inf}^{l}}, we have that ψ ∉ (∅^{⊢_{inf}^{g}} ∪ Ψ)^{⊢_{inf}^{l}} = Ψ^{⊢_{inf}^{l}}. Thus, we can conlude that Ψ⋉_{Φ̸}φ. ⊓ ⊔

Proposition 4.4.26. For k ∈ {1, 2}, let ⟨inf_{k}, val_{k}⟩ be a pair of layered inference and valuation systems over the same language and full with respect to all structures. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ⊛^{S} inf_{2}), (val_{1} ⊛^{S} val_{2})⟩ is complete.

Again, we can refine this result by forbiding such general structures. In fact, in general, they are allowed to have many equivalent truth values, and in particular, several distinguished values. To avoid this situation, we shall now consider P to be the class of all elementary consequence systems. As expected, we shall recall the notion of equivalence as defined in 3.1.27, over the layered consequence system induced by inf . Thus, given Φ, we shall use ≡_{inf}^{Φ} to denote the logical equivalence relation under Φ induced by ⟨L, ⊢_{inf}^{l} , ⊢_{inf}^{g} ⟩. Remember that it coincides with equivalence in the non-layered system ⟨L, ⊢_{inf}^{l} ⟩.

Proposition 4.4.27. Let inf = ⟨L, I, gI⟩ be a layered inference system and Φ ⊆ L be ⊢_{inf}^{g} -closed. Then, the triple

⟨L/_{≡_{inf}^{Φ}} , [ ]_{≡_{inf}^{Φ}} , ⋉_{Φ}^{≡_{inf}} ⟩

is an elementary structure for inf , where:

{[ψ]_{≡_{inf}^{Φ}} : ψ ∈ Ψ}^{⋉_{Φ}^{≡_{inf}}} = {[υ]_{≡_{inf}^{Φ}} (Φ ∪ Ψ)⊢_{inf}^{l} υ}.

Proof. It is obvious that ⟨L/_{≡_{inf}^{Φ}} , [ ]_{≡_{inf}^{Φ}} , ⋉_{Φ}^{≡_{inf}} ⟩ is an L-layered elementary structure.

In particular, ⟨L/_{≡_{inf}^{Φ}} ⋉_{Φ}^{≡_{inf}} ⟩ corresponds precisely to the quotient consequence system

induced by Φ on ⟨L, ⊢_{inf}^{l} ⟩ as in 2.1.40. Therefore, we just have to prove that it is indeed a structure for inf .

If i ∈ I we know that Prem(i)⊢_{inf}^{l} Conc(i). Therefore, using the monotonicity of ⊢_{inf}^{l} it follows that (Φ∪Prem(i))⊢_{inf}^{l} Conc(i), and we conclude that {[ψ]_{≡inf^{Φ}} : ψ ∈ Prem(i)}⋉_{Φ}^{≡_{inf}}

[Conc(i)]_{≡inf^{Φ}} .

Suppose now that i ∈ gI and ⋉_{Φ}^{≡_{inf}} {[ψ]_{≡inf^{Φ}} : ψ ∈ Prem(i)}, i.e., Φ⊢_{inf}^{l} Prem(i). Of

course, we know that Φ is ⊢_{inf}^{g} -closed, and thus also ⊢_{inf}^{l} -closed. Hence, we can conclude that Prem(i) ⊆ Φ and therefore we also have Conc(i) ∈ Φ. Thus, by extensiveness, Φ⊢_{inf}^{l} Conc(i). That is, ⋉_{Φ}^{≡_{inf}} [Conc(i)]_{≡inf^{Φ}} and the result follows. ⊓ ⊔

Proposition 4.4.28. Let ⟨inf , val⟩ be a pair of layered inference and valuation systems over the same language and full with respect to the class of all elementary structures. Then, the pair ⟨inf , val⟩ is complete.

Proof. Let val = ⟨L, M, S⟩. We start by proving that ⟨inf , val⟩ is globally complete. For the purpose, suppose that Ψ⊢_{i̸nf}^{g} υ. Of course, we need to show that Ψ⊨_{v̸al}^{g}υ. So, it is

sufficient to find a model m of val such that ⋉_{m}[[Ψ]]_{m} but ⋉_{m̸}[[υ]]_{m}. Make Φ = Ψ^{⊢_{inf}^{g}}, and consider the structure ⟨L/_{≡_{inf}^{Φ}} , [ ]_{≡_{inf}^{Φ}} , ⋉_{Φ}^{≡_{inf}} ⟩ defined from inf as in 4.4.17 above. Clearly, we obtain an elementary structure for inf . Thus, given that ⟨inf , val⟩ is full with respect to the class of all elementary structures, there exists a model m ∈ M with S(m) = ⟨L/_{≡_{inf}^{Φ}} , [ ]_{≡_{inf}^{Φ}} , ⋉_{Φ}^{≡_{inf}} ⟩. Easily, we have ⋉_{Φ}^{≡_{inf}} [Ψ]_{≡_{inf}^{Φ}} since Ψ ⊆ Φ = Ψ^{⊢_{inf}^{g}}, by

just using the extensiveness of both ⊢_{inf}^{g} and ⊢_{inf}^{l} . Furthermore, ⋉_{Φ̸}^{≡_{inf}} [υ]_{≡_{inf}^{Φ}} since Φ

is ⊢_{inf}^{g} -closed and therefore also ⊢_{inf}^{l} -closed, and υ ∉ Φ = Ψ^{⊢_{inf}^{g}}. Obviously then, by definition of ≡_{inf}^{Φ} , ([υ]_{≡_{inf}^{Φ}} ∩ Φ) = ∅.

Let us turn to local completeness. Suppose, now, that Ψ⊢_{i̸nf}^{l} υ. We need to show that Ψ⊨_{v̸al}^{l}υ. So, it is sufficient to find a model m of val such that [[Ψ]]_{m}⋉_{m̸}[[υ]]_{m}. Make Φ = ∅^{⊢_{inf}^{g}}, and consider the structure ⟨L/_{≡_{inf}^{Φ}} , [ ]_{≡_{inf}^{Φ}} , ⋉_{Φ}^{≡_{inf}} ⟩ defined from inf , again as in 4.4.17 above. Again, we obtain an elementary structure for inf , and since ⟨inf , val⟩ is full with respect to the class of all elementary structures, there exists a model m ∈ M with S(m) = ⟨L/_{≡_{inf}^{Φ}} , [ ]_{≡_{inf}^{Φ}} , ⋉_{Φ}^{≡_{inf}} ⟩. Clearly, we need to show that [Ψ]_{≡_{inf}^{Φ}} ⋉_{Φ̸}^{≡_{inf}} [υ]_{≡_{inf}^{Φ}} .

But this is immediate because υ ∉ Ψ^{⊢_{inf}^{l}} = (Φ ∪ Ψ)^{⊢_{inf}^{l}}, since we have Φ = ∅^{⊢_{inf}^{g}} ⊆ ∅^{⊢_{inf}^{l}}. Therefore, by definition of ≡_{inf}^{Φ} , ([υ]_{≡_{inf}^{Φ}} ∩ Ψ^{⊢_{inf}^{l}}) = ∅. ⊓ ⊔

Proposition 4.4.29. For k ∈ {1, 2}, let ⟨inf_{k}, val_{k}⟩ be a pair of layered inference and valuation systems over the same language and full with respect to all elementary structures. Then, for every corresponding constraint S, the pair ⟨(inf_{1} ⊛^{S} inf_{2}), (val_{1} ⊛^{S} val_{2})⟩ is complete.

Proof. Immediate from 4.4.23 and 4.4.28. ⊓ ⊔

4.5 Remarks

We shall close the Chapter with a brief summary. We have defined free synchronization of consequence systems and also of layered consequence systems and shown that they both correspond to categorial sums. Then, we have introduced the concepts of suitability and exactness for free synchronization of any given category of logic systems or layered logic systems. Next, constrained synchronization was introduced for consequence systems and then also generalized to layered consequence systems, and shown to correspond to a syntax-driven F -pushout. The concepts of suitability and exactness for free synchronization were then lifted to synchronization in its most general form. Then we studied synchronization at each of the categories of logic systems Sat, Val and Inf, and layered logic systems lVal and lInf. All were shown to be suitable for synchronization, but only Inf and lInf satisfy exactness. Given these characterizations, we presented a simple example of application of the mechanism, namely a combination of temporal logic with equational logic. To conclude, we established soundness and completeness preservation results for the various relevant cases, namely Inf/Sat, Inf/Val and lInf/lVal. Soundness preservation was established as a consequence of the exactness for synchronization of the proof-theoretic categories Inf and lInf. Completeness preservation was proved under the general assumption of fullness of the model-theoretic systems being combined with respect to their corresponding proof-theoretic counterparts. The non-exacteness for synchronization of the model-theoretic categories of logic systems and layered logic systems can be identified as the general source of failure of more general completeness preservation results.

Structural logical consequence

This Chapter plays, regarding the mechanism of fibring, essentially the same role played by Chapters 2 and 3 regarding synchronization. As we have already mentioned in the introduction, the first key towards fibring is to consider a logical language as corresponding to some initial algebra, thus being freely generated by application of a certain set of constructors to atoms. In our case, we shall consider signatures that are just ranked alphabets of constructors, which is the main reason for restricting attention, from now on, only to propositional based systems. Going beyond just syntax, schematicity or structurality of consequence [BS73] is the other essential ingredient, as already recognized in [SSC99, ZSS99]. This Chapter is therefore devoted to the study of what happens when we add the structurality dimension to the notions of consequence system and layered consequence system. Note that several properties of structural consequence are picked up from the literature, namely [Wój88], and rephrased in categorial terms. An essential difference from early Chapters, and not only in linguistic terms, is that, based on these categories, we shall develop abstract notions of category of (layered) logic system presentations, instead of just (layered) logic systems, and (layered) logic presentation, instead of just (layered) logics. Indeed, the implicit way in which we shall deal with closure for substitutions using metavariables, as it was already the case in [SSC99, ZSS99], easily justifies the distinction. Moreover, we shall then give little attention to the non-layered case. The main reason is that it does not meet our intuitions about fibring in the context of modal logics, which was why we introduced the two-layered setting in the first place. The main excuse for this omission is that, expectedly, the non-layered case can always be seen as a particular case of the layered one, as shown in Chapter 3. Thus, we shall only dwell on two examples of categories of layered logic system presentations. One of them, of model-theoretic nature, is closely related to layered valuation systems. We call its objects interpretation systems, following the designation already used in [SSC99, ZSS99]. Observe, however, that they are substantially more general in nature than those used in the papers just mentioned. In the first case, interpretation structures were precisely taken to be the powerset algebras associated to Kripke-like models, with an explicit set of worlds. In the second case, interpretation structures were taken to be set algebras associated to general Kripke models [vB85, HC96], of a more general nature, but still with an explicit set of worlds. Note that general models, as considered in modal logic, are cast in the tradition of Henkin’s general semantics for second-order logic [Hen50, Hen96]. In our case, we shall consider fully algebraic interpretation structures, in a way comparable to that of algebraic logic [Cze81, BP89, AKNS94, NA94, ANS97]. As regards the internal logical structure associated to such algebras, we shall use precisely the same trick as we did for layered valuation systems. As we already know, both previously studied cases can be seen as particular cases of the kind of structure we shall be using here. On the model-theoretic side, we shall consider deductive systems, by enriching the framework of layered inference systems with the notion of schema rule and by using metavariables, in a similar way to what was done in [SSC99, ZSS99]. The distinction between local and global rules shall be brought untouched from Chapter 3.

With the due differences in content, the outline of this Chapter is pretty similar to those of Chapters 2 and 3. We shall start by introducing signatures and the details of signature generated languages and schema languages. The difference between the two resides in the use of metavariables. The relationship between the category of signatures and the category of unstructured languages (Set in fact) is briefly explored. Then, we shall introduce the notion of structural consequence over a schematic signature generated language involving metavariables as corresponding to closure for substitutions, that is, every substitution of metavariables by schema sentences must correspond to an endomorphism. The obtained categories of structural consequence systems and structural layered consequence systems, as well as their finitary versions, will then be studied in some detail. After that, we shall define the abstract notion of category of (layered) logic system presentations as a functor to the category of structural (layered) consequence system, and, via indexing, the abstract notion of (layered) logic presentation. Next, we shall introduce interpretation systems and deductive systems and show that they constitute categories of layered logic system presentations. Furthermore, deductive system will be shown to be finitary. We shall also explore a little further these categories of logic systems, namely with respect to the notions of soundness and completeness, adapted to the structural case. We shall close the Chapter with a collection of examples of well-known structural systems of logic. For the sake of illustration we shall also discuss, very briefly, an interesting case that cannot be represented in this way.

5.1 Schematic reasoning

Our approach is based on the perspective that the abstract syntax of sentences corresponds to a free algebra over a given (unsorted) signature.

Definition 5.1.1. A signature Σ is a IN-indexed family of sets {Σ_{n}}_{n∈IN}. For each n ∈ IN, the elements of Σ_{n} are said to be constructors of arity n.

Definition 5.1.2. Let Σ^{k} for k ∈ {1, 2} be signatures. A signatures morphism h : Σ^{1} → Σ^{2} is a IN-indexed family of arity preserving functions {h_{n} : Σ_{n}^{1} → Σ_{n}^{2}}_{n∈IN}.

Definition 5.1.3. Signatures and their morphisms constitute the category Sig, with identity such that (id_{Σ})_{n} corresponds to the identity function on Σ_{n}, and (g ∘ f)_{n} to the usual composition of functions (g_{n} ∘ f_{n}), for each arity n ∈ IN.

Proposition 5.1.4. Sig is indeed a category.

This category is well known to be isomorphic to [IN, Set], i.e., the category of functors from IN (seen as a discrete category) to Set, and natural transformations between them. Or, in other words, IN-indexed sets and index-preserving functions. Isomorphisms in Sig, for instance, are precisely IN-indexed families of bijections. Hence, in particular, it is well known to be cocomplete: the unique initial signature is Σ^{∅}, where Σ_{n}^{∅} = ∅ for every n ∈ IN (because ∅ is the unique initial object of Set); the sum of two signatures Σ^{1} and Σ^{2} is (Σ^{1} + Σ^{2}) with injections ι_{k} : Σ^{k} → (Σ^{1} + Σ^{2}) for k ∈ {1, 2}, where (Σ^{1} + Σ^{2})_{n} is a sum in Set (disjoint union) of Σ_{n}^{1} and Σ_{n}^{2} with injections ι_{k,n} for every n ∈ IN; a coequalizer of two Sig-morphisms f, g : Σ^{1} → Σ^{2} is h : Σ^{2} → Σ, where h_{n} : Σ_{n}^{2} → Σ_{n} is a coequalizer in Set (quotient by the equivalence relation generated by the pair of arrows) of f_{n}, g_{n} : Σ_{n}^{1} → Σ_{n}^{2}. In particular, as in any category with binary sums, we denote by ( + ) : (Sig × Sig) → Sig the sum bifunctor on Sig that is left adjoint to the diagonal functor Δ_{Sig} : Sig → (Sig × Sig). By fixing any signature Σ, we can also consider the endofunctor (Σ + ) : Sig → Sig that sends each signature Σ′ to (Σ + Σ′) and each Sigmorphism h : Σ^{1} → Σ^{2} to (id_{Σ} + h) : (Σ + Σ^{1}) → (Σ + Σ^{2}).

We recall from example 2.3.10 a few notational conventions on algebras. In the sequel, we shall use calligraphic letters do denote algebras over a given signature Σ, and use the corresponding non-calligraphic version to denote its carrier. For instance, T shall denote the carrier of an algebra T . For every arity n ∈ IN, the denotation of a constructor c ∈ Σ_{n} in T shall be given by \underline{c} _{n}^{T} : T^{n} → T. Thus, T = ⟨T, ^{T} ⟩. Moreover, given a Sig-morphism h : Σ^{1} → Σ^{2} and Σ^{2}-algebra T , we shall denote by T |_{h} the reduct Σ^{1}-algebra ⟨T, ( ^{T} ∘h)⟩.

For compositionality purposes, we shall need to use metavariables that can afterwards be uniformly instantiated with sentences. We consider fixed, once and for all, a denumerable set Ξ = {ξ_{n} : n ∈ IN} of metavariable symbols.

Definition 5.1.5. The schema language induced by a signature Σ corresponds to the carrier of the free algebra L_{Σ}(Ξ) generated by Ξ in Alg(Σ). We denote this set by L_{Σ}(Ξ), and refer to its elements as Σ-schema sentences. Analogously, the language induced by a signature Σ corresponds to the carrier of the initial algebra L_{Σ} in Alg(Σ). We denote this set by L_{Σ}, and refer to its elements as Σ-sentences.

Explicitly, L_{Σ} is inductively defined by:

• c ∈ L_{Σ}, if c ∈ Σ_{0};

• c(φ_{1}, …, φ_{n}) ∈ L_{Σ}, if n > 0, c ∈ Σ_{n} and φ_{k} ∈ L_{Σ} for k ∈ {1, …, n}.

Similarly, L_{Σ}(Ξ) is the least set such that:

• ξ ∈ L_{Σ}(Ξ), if ξ ∈ Ξ;

• c ∈ L_{Σ}(Ξ), if c ∈ Σ_{0};

• c(γ_{1}, …, γ_{n}) ∈ L_{Σ}(Ξ), if n > 0, c ∈ Σ_{n} and γ_{k} ∈ L_{Σ}(Ξ) for k ∈ {1, …, n}.

Thus, we can say that Σ presents L_{Σ}, but has a richer schematic language L_{Σ}(Ξ). We use it, for instance, for defining derived constructors. An n-ary derived constructor is simply a schema sentence on the metavariables ξ_{1}, …, ξ_{n}. By abuse of notation, sometimes, we shall write (c_{1} γ) instead of c_{1}(γ) for a unary constructor c_{1}, and (γ c_{2} δ) instead of c_{2}(γ, δ) for a binary constructor c_{2}.

Notice that we shall keep using φ, ψ, υ, … to denote sentences, and thus also Φ, Ψ, Υ, … to denote sets of sentences. Additionally, we shall use γ, δ, θ, … to denote schema sentences and Γ, Δ, Θ … to denote sets of schema sentences. Clearly, L_{Σ} corresponds to L_{Σ}(∅) and thus we can assume that L_{Σ} ⊆ L_{Σ}(Ξ).

The definitions above can of course be extended to functors.

Definition 5.1.6. The maps

• PresL(Σ) = L_{Σ};

• PresL(h : Σ^{1} → Σ^{2}) = ĥ, the function underlying the unique Alg(Σ^{1})-morphism from L_{Σ^{1}} to the reduct L_{Σ^{2}}|_{h},

constitute the functor PresL : Sig → Set.

The maps

• SL(Σ) = L_{Σ}(Ξ);

• SL(h : Σ^{1} → Σ^{2}) = ĥ, the function underlying the unique Alg(Σ^{1})-morphism from L_{Σ^{1}}(Ξ) to the reduct L_{Σ^{2}}(Ξ)|_{h} that extends id_{Ξ},

constitute the functor SL : Sig → Set.

Explicitly, ĥ : L_{Σ^{1}} → L_{Σ^{2}} is inductively defined by:

• ĥ(c) = h_{0}(c);

• ĥ(c(φ_{1}, …, φ_{n})) = h_{n}(c)(ĥ(φ_{1}), …, ĥ(φ_{n})).

Moreover, ĥ : L_{Σ^{1}}(Ξ) → L_{Σ^{2}}(Ξ) further satisfies:

• ĥ(ξ) = ξ.

Proposition 5.1.7. PresL : Sig → Set and SL : Sig → Set are indeed functors.

Proof. Both PresL and SL are clearly well defined. Trivially, also, L_{Σ}|_{id_{Σ}} = L_{Σ} and L_{Σ}(Ξ)|_{id_{Σ}} = L_{Σ}(Ξ), and thus îd_{Σ} = id_{L_{Σ}} and id_{L_{Σ}(Ξ)}, respectively. Moreover, L_{Σ_{3}}|_{(g∘f)} =

(L_{Σ_{3}}|_{g})|_{f}, L_{Σ_{3}}(Ξ)|_{(g∘f)} = (L_{Σ_{3}}(Ξ)|_{g})|_{f} and { \atop (g ∘ f)} = (ĝ ∘ f̂). ⊓ ⊔

Obviously, the metavariables in a schema sentence can be instantiated.

Definition 5.1.8. A substitution over a signature Σ is a function σ : Ξ → L_{Σ}(Ξ). We denote by Sub(Σ) the set of all substitutions over Σ. If σ ∈ Sub(Σ) and σ(ξ) ∈ L_{Σ} for every ξ ∈ Ξ, then σ is said to be a ground substitution. We denote by GSub(Σ) the set of all ground substitutions over Σ.

Definition 5.1.9. Let Σ be a signature and σ ∈ Sub(Σ). The instance of a schema sentence γ ∈ L_{Σ}(Ξ) by σ, denoted by γσ, is the schema sentence σ̃(γ) where σ̃ is the corresponding free extension of σ to schema sentences.

Obviously, σ̃ is the unique Alg(Σ) endomorphism in L_{Σ}(Ξ) that extends σ. In detail, σ̃ is inductively defined by:

• σ̃(ξ) = σ(ξ);

• σ̃(c) = c;

• σ̃(c(γ_{1}, …, γ_{n})) = c(σ̃(γ_{1}), …, σ̃(γ_{n})).

Trivially, the instance of a schema sentence by a ground substitution is a sentence (to be precise, a schema sentence without metavariables).

The language functor PresL (although the same does not apply to its schematic companion SL) has a counterpart that explores the fact that every language can be presented by some signature. In fact, it shows that unstructured languages are just a particular case of signature generated languages. It is enough to consider each sentence as a constructor of arity 0.

Definition 5.1.10. The maps such that

• @(L) is the signature such that @(L)_{0} = L and @(L)_{n} = ∅ for every n > 0;

• @(h : L_{1} → L_{2}) : @(L_{1}) → @(L_{2}), where @(h)_{0} = h and @(h)_{n} is the empty function for every n > 0,

constitute the functor @ : Set → Sig.

The following properties are straightforward.

Proposition 5.1.11. @ : Set → Sig is a full embedding and (PresL ∘ @) = ID_{Set}.

Proof. Trivially, @ is indeed a functor. To check that it is an embedding, let h_{1} : L_{1} → L′_{1} and h_{2} : L_{2} → L′_{2} be any two functions such that @(h_{1}) = @(h_{2}). Immediately, we must also have @(L_{1}) = @(L_{2}) and @(L′_{1}) = @(L′_{2}). This obviously implies that L_{1} = L_{2}, L′_{1} = L′_{2} and h_{1} = h_{2}, thus showing that @ is injective on morphisms.

To prove that @ is full, let L, L′ be sets and h : @(L) → @(L′) be a Sig-morphism. Clearly, for every n > 0, h_{n} must be the empty function from ∅ to ∅. Thus, easily, h_{0} : L → L′ is such that @(h_{0}) = h.

Moreover, trivially, L_{@(L)} = L and { \atop @(h)} = h. ⊓ ⊔

As a corollary, of course, we get that the full subcategory of Sig whose objects are the signatures only with 0-ary constructors is isomorphic to Set.

Having introduced signature generated languages, the obvious step is to consider now consequence in such a context. For the purpose, let Σ be a signature and consider a consequence system csy = ⟨L_{Σ}, ▷⟩. It is quite straightforward to extract from ▷ a consequence operation ▷^{Ξ} over L_{Σ}(Ξ) that represents the schematic reasoning embodied by ▷. In simple terms, a schema sentence δ should follow from a set of schema sentences Γ provided that every possible instantiation of the metavariables in δ yields a sentence that follows via ▷ from the corresponding instantiation of Γ.

Definition 5.1.12. Let Σ be a signature and csy = ⟨L_{Σ}, ▷⟩ a consequence system. The structural consequence system induced by csy is csy^{Ξ} = ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩ where, for every Γ ⊆ L_{Σ}(Ξ) and δ ∈ L_{Σ}(Ξ):

• Γ ▷^{Ξ}δ if and only if (Γσ) ▷ δσ for every σ ∈ GSub(Σ)}.

Below we shall show that this is the greatest schematic consequence that mirrors ▷ under any ground substitution, thus assigning the intended meaning to the metavariables. It intuitively corresponds to a structural completion [Wój88] of ▷.

Proposition 5.1.13. Let Σ be a signature and csy = ⟨L_{Σ}, ▷⟩ a consequence system. Then, csy^{Ξ} is indeed a consequence system. Moreover, ▷^{Ξ} is the biggest consequence operation on L_{Σ}(Ξ) (in the sense of ⊑_{L_{Σ}(Ξ)}) such that σ̃ : csy^{Ξ} → csy is a Csy-morphism for every σ ∈ GSub(Σ).

Proof. This is a corollary of 2.1.17. Indeed, we have that Γ^{▷^{Ξ}} = (⋂_{σ∈GSub(Σ)} σ̃^{−1}(σ̃(Γ)^{▷})), and {σ̃ : csy^{Ξ} → csy}_{σ∈GSub(Σ)} is the unique Lang_{Csy}-initial lift of {σ̃ : L_{Σ}(Ξ) → L_{Σ}}_{σ∈GSub(Σ)} with respect to the family {csy}_{σ∈GSub(Σ)}. ⊓ ⊔

Unfortunately, this definition is not functorial. It is easy to conceive an example where, for instance, a consequence system over a signature generated language has a certain schematic property that is lost by a trivial extension of the signature. Let Σ^{1} and Σ^{2} be signatures such that:

• Σ_{0}^{1} = {a}, Σ_{0}^{2} = {a, b} and, for k ∈ {1, 2}, Σ_{1}^{k} = {f} and Σ_{n}^{k} = ∅ for every n > 1.

Very easily, L_{Σ^{1}} = {f^{n}(a) : n ∈ IN} and L_{Σ^{2}} = {f^{n}(a) : n ∈ IN} ∪ {f^{n}(b) : n ∈ IN}. Consider also ▷_{1} : ℘(L_{Σ^{1}}) → ℘(L_{Σ^{1}}) defined by:

• Φ^{▷_{1}} = {f^{m}(a) : f^{n}(a) ∈ Φ for some n ≥ m},

and ▷_{2} : ℘(L_{Σ^{2}}) → ℘(L_{Σ^{2}}) defined by:

• Φ^{▷_{2}} = {f^{m}(a) : f^{n}(a) ∈ Φ for some n ≥ m} ∪ {f^{n}(b) : f^{n}(b) ∈ Φ}.

It is quite trivial to verify that csy_{1} = ⟨L_{Σ^{1}}, ▷_{1}⟩ and csy_{2} = ⟨L_{Σ^{2}}, ▷_{2}⟩ are consequence systems and that ĥ : csy_{1} → csy_{2} is a Csy-morphism, with h such that:

• h_{0}(a) = a and h_{1}(f) = f.

However, while {f(ξ)} ▷_{1}^{Ξ}ξ, it is easy to see that ξ ∉ {f(ξ)}^{▷_{2}^{Ξ}}. Therefore, obviously, ĥ : csy _{1}^{Ξ} → csy _{2}^{Ξ} is not a Csy-morphism.

Still, this construction embodies the kind of compositional property we should expect from schematic consequence.

Proposition 5.1.14. Let Σ be a signature and csy = ⟨L_{Σ}, ▷⟩ a consequence system. Then, for every σ ∈ Sub(Σ), σ̃ : csy^{Ξ} → csy^{Ξ} is a Csy-morphism.

Proof. Let σ ∈ Sub(Σ). We must show that σ̃ : csy^{Ξ} → csy^{Ξ} is a Csy-morphism. Given σ′ ∈ GSub(Σ), it is trivial that (σ̃′ ∘ σ) ∈ GSub(Σ) and { \atop (σ̃′ ∘ σ)} = (σ̃′ ∘ σ̃). Moreover, from 5.1.13, we know that (σ̃′ ∘ σ̃) : csy^{Ξ} → csy is a Csy-morphism. Therefore, ((Γ^{▷^{Ξ}})σ)σ′ ⊆ ((Γσ)σ′)^{▷}. Thus, (Γ^{▷^{Ξ}})σ ⊆ (⋂_{σ′∈GSub(Σ)} σ̂^{′−1}(σ̂′(Γσ)^{▷})) = (Γσ)^{▷^{Ξ}}. ⊓ ⊔

Note that the condition above means precisely that, for every metavariable substitution σ on Σ and every Γ ⊆ L_{Σ}(Ξ), it must be the case that (Γ^{▷^{Ξ}})σ ⊆ (Γσ)^{▷^{Ξ}}.

Thus, in the context of a signature generated language, of course, we shall require that a consequence operation fulfills this structural property.

Definition 5.1.15. A structural consequence system is a pair csy = ⟨Σ, ▷^{Ξ}⟩ such that csy(Ξ) = ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩ is a consequence system and the following condition holds, for every σ ∈ Sub(Σ):

• σ̃ : csy(Ξ) → csy(Ξ) is a Csy-morphism.

Clearly, a structural consequence operation, besides incorporating the desired consequence at the level of sentences, also prescribes the amount of structurality allowed. Indeed, if we apply the operation on 5.1.13 to such a consequence but considering all possible instantiations instead of just the ground ones, we obtain exactly the same operation.

Definition 5.1.16. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems. A structural consequence systems morphism h : csy_{1} → csy_{2} is a Sig-morphism h : Σ^{1} → Σ^{2} such that ĥ : csy_{1}(Ξ) → csy_{2}(Ξ) is a Csy-morphism.

Definition 5.1.17. Structural consequence systems and their morphisms constitute the category sCsy, with identity id_{⟨Σ,▷^{Ξ}⟩} the identity morphism on Σ, and (g ∘ f) the composition of Sig-morphisms.

Proposition 5.1.18. sCsy is indeed a category.

Proof. Trivial. ⊓ ⊔

Obviously, we can extract, over the (non-schematic) language of any structural consequence system, the consequence system it presents.

Definition 5.1.19. Let csy = ⟨Σ, ▷^{Ξ}⟩ be a structural consequence system. The nonschematic consequence system presented by csy is csy(∅) = ⟨L_{Σ}, ▷⟩ where, for every Φ ⊆ L_{Σ}:

• Φ^{▷} = (L_{Σ} ∩ Φ^{▷^{Ξ}}).

Proposition 5.1.20. For every structural consequence system csy = ⟨Σ, ▷^{Ξ}⟩, csy(∅) is indeed a consequence system.

Proof. We prove that ▷, as defined in 5.1.19, is indeed a consequence operation on L_{Σ}. Since only subsets of L_{Σ} are considered, extensiveness follows immediately from the extensiveness of ▷^{Ξ}. Monotonicity is also straightforward from the monotonicity of ▷^{Ξ}. Concerning idempotence, easily, (L_{Σ} ∩ Φ^{▷^{Ξ}})^{▷^{Ξ}} ⊆ (Φ^{▷^{Ξ}})^{▷^{Ξ}} ⊆ Φ^{▷^{Ξ}}, by just using the monotonicity and idempotence of ▷^{Ξ}. Thus, (Φ^{▷})^{▷} = (L_{Σ}∩(L_{Σ}∩Φ^{▷^{Ξ}})^{▷^{Ξ}}) ⊆ (L_{Σ}∩Φ^{▷^{Ξ}}) = Φ^{▷}. ⊓ ⊔

Indeed, this definition easily extends to a functor. However, another functor from sCsy to Csy can be easily established. Although this second one does not match our intuition of a structural consequence system as being a presentation of a consequence over the language generated by the corresponding signature, it will still be useful.

Definition 5.1.21. The maps

• PresC (csy) = csy(∅);

• PresC (h : csy_{1} → csy_{2}) = ĥ,

constitute the functor PresC : sCsy → Csy.

The maps

• SC (csy) = csy(Ξ);

• SC (h : csy_{1} → csy_{2}) = ĥ,

constitute the functor SC : sCsy → Csy.

By definition of sCsy, it is trivial to verify that SC : sCsy → Csy is indeed a functor. Below we shall show that the same happens with respect to PresC .

Proposition 5.1.22. PresC : sCsy → Csy is indeed a functor.

Proof. We just prove that ĥ : csy_{1}(∅) → csy_{2}(∅) is indeed a Csy-morphism whenever h : csy_{1} → csy_{2} is an sCsy-morphism. Knowing that ĥ : csy_{1}(Ξ) → csy_{2}(Ξ) is a Csymorphism, and letting csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2}, Φ ⊆ L_{Σ^{1}} and ψ ∈ L_{Σ^{1}} be such that Φ ▷_{1}ψ, we immediately have that Φ ▷_{1}^{Ξ}ψ and therefore ĥ(Φ) ▷_{2}^{Ξ}ĥ(ψ). But obviously, ĥ(ψ) ∈ L_{Σ^{2}} and thus ĥ(Φ) ▷_{2}ĥ(ψ). ⊓ ⊔

As expected, however, this relationship between sCsy and Csy is not reciprocal. For any consequence system csy = ⟨L_{Σ}, ▷⟩, recall from 5.1.12 the definition of csy^{Ξ} = ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩. From 5.1.13, it follows that ⟨Σ, ▷^{Ξ}⟩ is a structural consequence system. Moreover, it can be easily checked that PresC (⟨Σ, ▷^{Ξ}⟩) = csy.

On the contrary, however, it can be easily shown that any given structural consequence system csy = ⟨Σ, ▷^{Ξ}⟩ is, in general, smaller than the system obtained by applying the construction in 5.1.12 to csy(∅). This happens because, as we very well know, there are many non-structural ways of presenting structural phenomena. Consider, for instance, a signature Σ such that:

• Σ_{0} = {a}, Σ_{1} = {f} and Σ_{n} = ∅ for every n > 1.

Consider also ▷^{Ξ} : ℘(L_{Σ}(Ξ)) → ℘(L_{Σ}(Ξ)) defined by:

• Γ^{▷^{Ξ}} = ({f^{m}(a) : f^{n}(a) ∈ Γ for some n ≥ m} ∪ (Γ \ L_{Σ})).

It is quite trivial to verify that csy = ⟨Σ, ▷^{Ξ}⟩ is a structural consequence system and that it induces csy(∅) = ⟨L_{Σ}, ▷⟩ with:

• Φ^{▷} = {f^{m}(a) : f^{n}(a) ∈ Φ for some n ≥ m}.

However, by applying the construction in 5.1.12, we shall be able to capture, for instance, the schematic rule that asserts that ξ follows from f(ξ) which is not encompassed by ▷^{Ξ}, i.e., ξ ∉ {f(ξ)}^{▷^{Ξ}}.

Still, the relationship between sCsy and Csy via PresC is kept similar to the one between Sig and Set via PresL. Please just note that given a set L, L_{@(L)}(Ξ) = (L ∪ Ξ).

Definition 5.1.23. The maps such that

• @(⟨L, ▷⟩) = ⟨@(L), @(▷)^{Ξ}⟩, where (Φ ∪ χ)^{@(▷)^{Ξ}} = (Φ^{▷} ∪ χ) for every Φ ⊆ L, χ ⊆ Ξ;

• @(h : csy_{1} → csy_{2}) = @(h) : @(csy_{1}) → @(csy_{2}),

constitute the functor @ : Csy → sCsy.

The following properties are straightforward.

Proposition 5.1.24. @ : Csy → sCsy is a full embedding and (PresC ∘ @) = ID_{Csy}.

Proof. The fact that @ is a functor follows from the definition of @(▷)^{Ξ}. Indeed, it is trivially structural and thus trivially preserved by morphisms. To check that it is an embedding, let h_{1} : ⟨L_{1}, ▷_{1}⟩ → ⟨L′_{1}, ▷′_{1}⟩ and h_{2} : ⟨L_{2}, ▷_{2}⟩ → ⟨L′_{2}, ▷′_{2}⟩ be any two Csymorphisms such that @(h_{1}) = @(h_{2}). Immediately, we must also have @(L_{1}) = @(L_{2}) and @(L′_{1}) = @(L′_{2}), @(▷)_{1}^{Ξ} = @(▷)_{2}^{Ξ} and @(▷)_{1}^{′Ξ} = @(▷)_{2}^{′Ξ}. This obviously implies that L_{1} = L′_{1} and L_{2} = L′_{2}, ▷_{1} = ▷_{2} and ▷′_{1} = ▷′_{2} and h_{1} = h_{2}, thus showing that @ is injective on morphisms.

To prove that @ is full, let csy = ⟨L, ▷⟩, csy′ = ⟨L′, ▷′⟩ be consequence systems and h : @(csy) → @(csy′) an sCsy-morphism. Clearly, h_{0} : csy → csy′ is a morphism and @(h_{0}) = h.

Moreover, trivially, @(csy)(∅) = csy and { \atop @(h)} = h. ⊓ ⊔

This evolution, from an unstructured to a freely generated language, and from a consequence operation to a structural one, still carries along much of the interesting internal structure we have found in the category Csy. In fact, we shall now show that sCsy is topological over Sig. Let us start by fixing the corresponding forgetful functor.

Definition 5.1.25. The maps

• Sgn_{sCsy}(⟨Σ, ▷^{Ξ}⟩) = Σ;

• Sgn_{sCsy}(h : csy_{1} → csy_{2}) = h,

constitute the functor Sgn_{sCsy} : sCsy → Sig.

Proposition 5.1.26. Sgn_{sCsy} : sCsy → Sig is a topological functor.

Proof. Clearly, Sgn_{sCsy} is a faithful functor. We now show that given any family {csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩}_{k∈K} of structural consequence systems, any source {h_{k} : Σ → Σ^{k}}_{k∈K} in Sig has a unique Sgn_{sCsy}-initial lift {h_{k} : csy → csy_{k}}_{k∈K} in sCsy.

Clearly, by applying SC and SL, we obtain a family {csy_{k}(Ξ)}_{k∈K} of consequence systems, and a source {ĥ_{k} : L_{Σ}(Ξ) → L_{Σ^{k}}(Ξ)}_{k∈K} in Set. Thus, from 2.1.17, we know that the corresponding unique Lang_{lCsy}-initial lift in Csy is {ĥ_{k} : ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩ → csy_{k}(Ξ)}_{k∈K}, where Γ^{▷^{Ξ}} = (⋂_{k∈K} ĥ_{k}^{−1}(ĥ_{k}(Γ)^{▷_{k}^{Ξ}})) for every Γ ⊆ L_{Σ}(Ξ). Indeed, the unique Sgn_{sCsy}- initial lift is now given by {h_{k} : csy → csy_{k}}_{k∈K}, where csy = ⟨Σ, ▷^{Ξ}⟩. We just have to show that csy is indeed a structural consequence system. Given σ ∈ Sub(Σ), it easily follows that σ_{k} = (ĥ_{k} ∘ σ) ∈ Sub(Σ^{k}) for every k ∈ K. Moreover, from the definition of each σ_{k}, it trivially follows that (ĥ_{k} ∘ σ̃) = (σ̃_{k} ∘ ĥ_{k}). Thus, for each k ∈ K, ĥ_{k}((Γ^{▷^{Ξ}})σ) = (ĥ_{k}(Γ^{▷^{Ξ}}))σ_{k} ⊆ (ĥ_{k}(Γ)^{▷_{k}^{Ξ}})σ_{k} just by using the definition of ▷^{Ξ}. But every csy_{k} is known to be structural, hence ĥ_{k}((Γ^{▷^{Ξ}})σ) ⊆ ((ĥ_{k}(Γ))σ_{k})^{▷_{k}^{Ξ}} = ĥ_{k}(Γσ)^{▷_{k}^{Ξ}}. Thus, (Γ^{▷^{Ξ}})σ ⊆ ĥ_{k}^{−1}(ĥ_{k}(Γσ)^{▷_{k}^{Ξ}}) for every k ∈ K, and so (Γ^{▷^{Ξ}})σ ⊆ (Γσ)^{▷^{Ξ}}.

To prove that {h_{k} : csy → csy_{k}}_{k∈K} is Sgn_{sCsy}-initial, let {g_{k} : csy′ → csy_{k}}_{k∈K} be any source in sCsy, with csy′ = ⟨Σ′, ▷^{′Ξ}⟩, and f : Σ′ → Σ a Sig-morphism such that each g_{k} = (h_{k} ∘f). All we need to prove is that f : csy′ → csy is an sCsy-morphism. In fact we have {ĝ_{k} : csy′(Ξ) → csy_{k}(Ξ)}_{k∈K} a source in Csy and a function f̂: L_{Σ′}(Ξ) → L_{Σ}(Ξ). Easily, ĝ_{k} = { \atop (h_{k} ∘ f)} = (ĥ_{k}∘f̂), and immediately f̂: csy′(Ξ) → csy(Ξ) is a Csy-morphism.

The uniqueness of this lift follows immediately from the faithfulness of Sgn_{sCsy}. ⊓ ⊔

The Sgn_{sCsy}-fibres of sCsy are again complete lattices. The corresponding ordering in the set of all structural consequence operations over a given signature is to be expected.

Definition 5.1.27. Let Σ be a signature and csy_{k} = ⟨Σ, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems. Then, ▷_{1}^{Ξ} ⊑_{Σ} ▷_{2}^{Ξ} if and only if ▷_{1}^{Ξ} ⊑_{L_{Σ}(Ξ)} ▷_{2}^{Ξ}.

Obviously, this means that ▷_{1}^{Ξ} ⊑_{Σ} ▷_{2}^{Ξ} if and only if Γ^{▷_{1}^{Ξ}} ⊆ Γ^{▷_{2}^{Ξ}}, for every Γ ⊆ L_{Σ}(Ξ). Moreover, please note that since îd_{Σ} = id_{L_{Σ}(Ξ)}, ▷_{1}^{Ξ} ⊑_{Σ} ▷_{2}^{Ξ} is precisely equivalent to id_{Σ} : ⟨Σ, ▷_{1}^{Ξ}⟩ → ⟨Σ, ▷_{2}^{Ξ}⟩ being an sCsy-morphism.

As a corollary, we have that the fibres of sCsy are complete sub-lattices of Csy-fibres.

Corollary 5.1.28. sCsy is fibre-complete, i.e., each fibre sCsy_{Σ} is a complete lattice (under ⊑_{Σ}).

The meet of a family {▷_{k}^{Ξ}}_{k∈K} of structural consequence operations over Σ consists of ⊓_{Σ}\{▷_{k}^{Ξ} : k ∈ K\} = \underline{▷}^{Ξ} = ⊓_{L_{Σ}(Ξ)}\{▷_{k}^{Ξ} : k ∈ K\}, i.e, Γ^{\underline{▷}^{Ξ}} = (⋂_{k∈K} Γ^{▷_{k}^{Ξ}}).

Clearly, also joins exist and coincide with joins in Csy. That is, ⊔_{Σ}{▷_{k}^{Ξ} : k ∈ K} = \overline{▷}^{Ξ} = ⊔_{L_{Σ}(Ξ)}\{▷_{k}^{Ξ} : k ∈ K\}, i.e, Γ^{\overline{▷}^{Ξ}} is the least superset of Γ that is closed for I = (⋃_{k∈K} I_{k}), and each I_{k} = \{⟨Δ, δ⟩ : Δ▷_{k}^{Ξ} δ\}. All we need to prove is that \overline{▷}^{Ξ} as above is still structural. Recall that, in this case, Ω_{I}(Γ) = (Γ∪(⋃_{k∈K} Γ^{▷_{k}^{Ξ}})). In fact, it all amounts to showing that, for every σ ∈ Sub(Σ) and Θ ∈ I(Γσ), there exists Θ′ ∈ I(Γ) such that Θ′σ ⊆ Θ. Thus, let σ ∈ Sub(Σ) and Θ ∈ I(Γσ). Obviously, (Γσ ∪ Ω_{I}(Θ)) ⊆ Θ. Let now Θ′ = σ̃^{−1}(Θ). First of all, since σ̃(Γ) = Γσ ⊆ Θ we have Γ ⊆ σ̃^{−1}(Θ) = Θ′. On the other hand, Ω_{I}(Θ) ⊆ Θ means that Θ^{▷_{k}^{Ξ}} ⊆ Θ for every k ∈ K. That is, Θ is a theory of all ▷_{k}^{Ξ}. Since every csy_{k} is structural, we have that σ̃ : csy_{k}(Ξ) → csy_{k}(Ξ) is a Csy-morphism. Thus, using 2.1.15, Θ′ = σ̃^{−1}(Θ) is also a theory of all csy_{k}, i.e., Θ^{′▷_{k}^{Ξ}} ⊆ Θ′. Therefore Ω_{I}(Θ′) ⊆ Θ′ and Θ′ ∈ I(Γ). Moreover, trivially, Θ′σ = σ̃^{−1}(Θ)σ = σ̃(σ̃^{−1}(Θ)) ⊆ Θ. Thus, it follows that, (Γ^{\overline{▷}^{Ξ}})σ = (⋂_{Θ′∈I(Γ)} Θ′)σ ⊆ (⋂_{Θ′∈I(Γ)} Θ′σ) ⊆ (⋂_{Θ∈I(Γσ)} Θ) = (Γσ)^{\overline{▷}^{Ξ}}.

Top and bottom are again ⊤ and ⊥, both trivially structural.

Finitariness in the structural setting raises no problems. As we shall show below, the finitary part of a structural consequence system is still structural.

Definition 5.1.29. A structural consequence system csy is finitary if csy(Ξ) is.

Definition 5.1.30. sCsy_{fin} is the full subcategory of sCsy whose objects are precisely the finitary structural consequence systems.

Proposition 5.1.31. Let csy = ⟨Σ, ▷^{Ξ}⟩ be a structural consequence system. The greatest finitary structural consequence system contained in ▷^{Ξ} (in the sense of ⊑_{Σ}) is ▷_{fin}^{Ξ}.

Proof. From 2.1.22, it follows that Γ^{▷_{fin}^{Ξ}} = (⋃_{Δ∈℘_{fin}(Γ)} Δ^{▷^{Ξ}}). Clearly, it suffices to prove that ▷_{fin}^{Ξ} is structural. Given σ ∈ Sub(Σ), we have that (Γ^{▷_{fin}^{Ξ}})σ = (⋃_{Δ∈℘_{fin}(Γ)} Δ^{▷^{Ξ}})σ = (⋃_{Δ∈℘_{fin}(Γ)}(Δ^{▷^{Ξ}})σ). But ▷^{Ξ} is known to be structural and thus it follows that (Γ^{▷_{fin}^{Ξ}})σ ⊆ (⋃_{Δ∈℘_{fin}(Γ)}(Δσ)^{▷^{Ξ}}) = (⋃_{Δ∈℘_{fin}(Γσ)} Δ^{▷^{Ξ}}) = (Γσ)^{▷^{Ξ}}. ⊓ ⊔

For convenience, we use the notation csy_{fin} = ⟨Σ, ▷_{fin}^{Ξ}⟩. The construction easily extends to a functor from sCsy to sCsy_{fin} that is adjoint to the obvious inclusion functor sI : sCsy_{fin} → sCsy. We have, in fact, a coreflection.

Proposition 5.1.32. sCsy_{fin} is a coreflective subcategory of sCsy.

Proof. Let csy = ⟨Σ, ▷^{Ξ}⟩ be a structural consequence system. We prove that id_{Σ} : csy_{fin} → csy is a coreflection arrow. Let h : csy′ → csy be any sCsy-morphism, with csy′ = ⟨Σ′, ▷^{′Ξ}⟩ finitary. Clearly, it suffices to show that h : csy′ → csy_{fin} is also a morphism. This follows immediately from the coreflection 2.1.23. ⊓ ⊔

Corollary 5.1.33. The maps

• sFin(csy) = csy_{fin};

• sFin(h : csy_{1} → csy_{2}) = h : csy_{1fin} → csy_{2fin},

constitute a functor sFin : sCsy → sCsy_{fin}. Moreover, sFin is adjoint to sI .

This subcategory sCsy_{fin} is also topological over Sig. Obviously, we shall use the notation Sgn_{sCsy_{fin}} = (Sgn_{sCsy} ∘ sI ).

Corollary 5.1.34. The functor Sgn_{sCsy_{fin}} : sCsy_{fin} → Sig is topological.

Clearly, sFin does not change finitary systems. Let {csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩}_{k∈K} be a family of finitary structural consequence systems. Clearly, we can also see it as a family of structural systems. Given a source {h_{k} : Σ → Σ^{k}}_{k∈K}, {h_{k} : csy → csy_{k}}_{k∈K} is the unique Sgn_{sCsy}-initial lifting, with csy = ⟨Σ, ▷^{Ξ}⟩ and Γ^{▷^{Ξ}} = (⋂_{k∈K} h_{k}^{−1}(h_{k}(Γ)^{▷_{k}^{Ξ}})), as in 5.1.26. If we apply sFin to this sCsy source, we obtain the sCsy_{fin} source {h_{k} : csy_{fin} → csy_{k}}_{k∈K}. Now, 5.1.26 and 2.1.25 imply that it constitutes the desired unique Sgn_{sCsy_{fin}}-initial lifting.

Thus, also the fibres of sCsy_{fin} form complete lattices.

Corollary 5.1.35. sCsy_{fin} is fibre-complete, i.e., each fibre (sCsy_{fin})_{Σ} is a complete lattice (under the restriction of ⊑_{Σ} to finitary structural consequence systems).

The meet of a family of finitary structural consequence operations {▷_{k}^{Ξ}}_{k∈K} over Σ is given by the unique Sgn_{sCsy_{fin}}-initial lifting of {id_{Σ} : Σ → Σ}_{k∈K}, i.e., it corresponds to ⊓_{Σ}^{fin}\{▷_{k}^{Ξ} : k ∈ K\} = \underline{▷}Ξ_{fin} with Γ^{\underline{▷}Ξ_{fin}} = ⋃_{Δ∈℘_{fin}(Γ)}(⋂_{k∈K} Δ^{▷_{k}^{Ξ}}). Joins, top and bottom also coincide with those in Csy.

Let us now add the two-layered dimension to structural consequence systems, according to the principles studied in the previous chapter.

Definition 5.1.36. A structural layered consequence system is a triple csy = ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩ such that csy(Ξ) = ⟨L_{Σ}(Ξ), ▷^{lΞ}, ▷^{gΞ}⟩ is a layered consequence system and both csy^{l} = ⟨Σ, ▷^{lΞ}⟩ and csy^{g} = ⟨Σ, ▷^{gΞ}⟩ are structural consequence systems.

We can still restrict a structural layered consequence system to non-schematic sentences and obtain the layered consequence system presented.

Definition 5.1.37. Let csy = ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩ be a structural layered consequence system. The non-schematic layered consequence system presented by csy is csy(∅) = ⟨L_{Σ}, ▷^{l}, ▷^{g}⟩ where ⟨L_{Σ}, ▷^{l}⟩ = csy^{l}(∅) and ⟨L_{Σ}, ▷^{g}⟩ = csy^{g}(∅).

Proposition 5.1.38. For every structural consequence system csy, csy(∅) is indeed a layered consequence system.

Proof. Let csy = ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩. From 5.1.20 we already now that ▷^{l} and ▷^{g} are consequence operations on L_{Σ}. Since, ▷^{lΞ} ⊑_{L_{Σ}(Ξ)} ▷^{gΞ}, we have that Φ^{▷^{l}} = (L_{Σ} ∩ Φ^{▷^{lΞ}}) ⊆ (L_{Σ} ∩ Φ^{▷^{gΞ}}) = Φ^{▷^{g}}. Thus, we conclude that ▷^{l} ⊑_{L_{Σ}} ▷^{g}. Moreover, ∅^{▷^{lΞ}} = ∅^{▷^{gΞ}} thus ∅^{▷^{l}} = (L_{Σ} ∩ ∅^{▷^{lΞ}}) = (L_{Σ} ∩ ∅^{▷^{gΞ}}) = ∅^{▷^{g}}. ⊓ ⊔

This definition easily extends to a functor, from the appropriate category of structural layered consequence systems to slCsy.

Definition 5.1.39. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems. A structural layered consequence systems morphism h : csy_{1} → csy_{2} is a Sig-morphism h : Σ^{1} → Σ^{2} such that ĥ : csy_{1}(Ξ) → csy_{2}(Ξ) is an lCsy-morphism.

Definition 5.1.40. Structural layered consequence systems and their morphisms constitute the category slCsy, with identity id_{⟨Σ,▷^{lΞ},▷^{gΞ}⟩} the identity morphism on Σ, and (g ∘f) the composition of Sig-morphisms.

Proposition 5.1.41. slCsy is indeed a category.

Proof. Trivial. ⊓ ⊔

Again, along with the functor yielding the layered consequence system presented by each structural system, there is another useful functor from slCsy to lCsy.

Definition 5.1.42. The maps

• PreslC (csy) = csy(∅);

• PreslC (h : csy_{1} → csy_{2}) = ĥ,

constitute the functor PreslC : slCsy → lCsy.

The maps

• SlC (csy) = csy(Ξ);

• SlC (h : csy_{1} → csy_{2}) = ĥ,

constitute the functor SlC : slCsy → lCsy.

As before, it is immediate that SlC : slCsy → lCsy is indeed a functor. We shall show below that the same happens with PreslC .

Proposition 5.1.43. PreslC : slCsy → lCsy is indeed a functor.

Proof. Since csy(∅) is indeed layered, the result follows immediately from 5.1.22. ⊓ ⊔

Moreover, lCsy is still isomorphic to a full subcategory of slCsy.

Definition 5.1.44. The maps such that

• @(⟨L, ▷^{l}, ▷^{g}⟩) = ⟨@(L), @(▷)^{lΞ}, @(▷)^{gΞ}⟩;

• @(h : csy_{1} → csy_{2}) = @(h) : @(csy_{1}) → @(csy_{2}),

constitute the functor @ : lCsy → slCsy.

Proposition 5.1.45. @ : lCsy → slCsy is a full embedding and (PreslC ∘ @) = ID_{lCsy}.

Proof. An immediate consequence of 5.1.24. ⊓ ⊔

As expected, also slCsy is topological over Sig.

Definition 5.1.46. The maps

• Sgn_{slCsy}(⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩) = Σ;

• Sgn_{slCsy}(h : csy_{1} → csy_{2}) = h,

constitute the functor Sgn_{slCsy} : slCsy → Sig.

Proposition 5.1.47. Sgn_{slCsy} : slCsy → Sig is a topological functor.

Proof. Clearly, Sgn_{slCsy} is a faithful functor. We now show that given any family {csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩}_{k∈K} of structural layered consequence systems, any source {h_{k} : Σ → Σ^{k}}_{k∈K} in Sig has a unique Sgn_{sCsy}-initial lift {h_{k} : c → c_{k}}_{k∈K} in slCsy.

Clearly, from {csy_{k}}_{k∈K}, we obtain two families {csy_{k}^{l}}_{k∈K} and {csy_{k}^{g}}_{k∈K} of structural consequence systems. Thus, using 5.1.26 we obtain two Sgn_{sCsy}-initial liftings in sCsy, given by {h_{k} : csy^{l} → csy_{k}^{l}}_{k∈K} and {h_{k} : csy^{g} → csy_{k}^{g}}_{k∈k} where csy^{l} = ⟨Σ, ▷^{lΞ}⟩ and csy^{g} = ⟨Σ, ▷^{gΞ}⟩ are such that Γ^{▷^{lΞ}} = (⋂_{k∈K} ĥ_{k}^{−1}(ĥ_{k}(Γ)^{▷_{k}^{lΞ}})) and Γ^{▷^{gΞ}} = (⋂_{k∈K} ĥ_{k}^{−1}(ĥ_{k}(Γ)^{▷_{k}^{gΞ}})). Indeed, the unique Sgn_{slCsy}-initial lift is now given by {h_{k} : csy → csy_{k}}_{k∈K} where csy = ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩. We just have to show that csy is indeed layered. In fact, both ▷^{lΞ} ⊑_{Σ} ▷^{gΞ} and ∅^{▷^{lΞ}} = ∅^{▷^{gΞ}} follow easily from the fact that each csy_{k} is layered.

To prove that {h_{k} : csy → csy_{k}}_{k∈K} is Sgn_{slCsy}-initial, let {g_{k} : csy′ → csy_{k}}_{k∈K} be any source in slCsy, with csy′ = ⟨Σ′, ▷^{′lΞ}, ▷^{′gΞ}⟩, and f : Σ′ → Σ a Sig-morphism such that each g_{k} = (h_{k} ∘ f). All we need to prove is that f : csy′ → csy is an slCsy-morphism. This follows immediately from the uniqueness of csy^{l} and csy^{g} with respect to the sources {g_{k} : csy^{′l} → csy_{k}^{l}}_{k∈K} and {g_{k} : csy^{′g} → csy_{k}^{g}}_{k∈K}, respectively.

The uniqueness of this lift follows immediately from the faithfulness of Sgn_{slCsy}. ⊓ ⊔

The Sgn_{slCsy}-fibres of slCsy are again complete lattices. The corresponding ordering in the set of all structural consequence operations over a given signature is to be expected.

Definition 5.1.48. Let Σ be a signature and csy_{k} = ⟨Σ, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems. Then, ⟨▷_{1}^{lΞ}, ▷_{1}^{gΞ}⟩ ⊑_{Σ} ⟨▷_{2}^{lΞ}, ▷_{2}^{gΞ}⟩ if both ▷_{1}^{lΞ} ⊑_{Σ} ▷_{2}^{lΞ} and ▷_{1}^{gΞ} ⊑_{Σ} ▷_{2}^{gΞ}.

Again, note that ⟨▷_{1}^{lΞ}, ▷_{1}^{gΞ}⟩ ⊑_{Σ} ⟨▷_{2}^{lΞ}, ▷_{2}^{gΞ}⟩ is precisely equivalent to id_{Σ} : ⟨Σ, ▷_{1}^{lΞ}, ▷_{1}^{gΞ}⟩ → ⟨Σ, ▷_{2}^{lΞ}, ▷_{2}^{gΞ}⟩ being an slCsy-morphism.

Corollary 5.1.49. slCsy is fibre-complete, i.e., each fibre slCsy_{Σ} is a complete lattice (under ⊑_{Σ}).

The meet of a family {⟨▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩}_{k∈K} of pairs of structural layered consequence operations over Σ is given by ⊓_{Σ}\{⟨▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ : k ∈ K\} = ⟨\underline{▷}^{lΞ}, \underline{▷}^{gΞ}⟩, where \underline{▷}^{lΞ} = ⊓_{Σ}\{▷_{k}^{lΞ} : k ∈ K\} and \underline{▷}^{gΞ} = ⊓_{Σ}\{▷_{k}^{gΞ} : k ∈ K\}. Top and bottom are obtained as in the non-structural case.

However, again, joins are more complex. As in lCsy, ⊔_{Σ}{⟨▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ : k ∈ K} = ⟨\overline{▷}^{lΞ}, \overline{▷}^{gΞ}⟩ where \overline{▷}^{gΞ} = ⊔_{Σ}\{▷_{k}^{gΞ} : k ∈ K\}, and \overline{▷}^{lΞ} is then generated by the set of rules I = ((⋃_{k∈K} I_{k}) ∪ \{⟨∅, γ⟩ : \overline{▷}^{gΞ}γ\}) with each I_{k} = \{⟨Γ, δ⟩ : Γ ▷_{k}^{lΞ} δ\}.

Let us now take a look at finitariness in the layered structural setting. Again, no surprises are to be expected.

Definition 5.1.50. A structural layered consequence system csy is said to be finitary if both csy^{l} and csy^{g} are finitary.

Definition 5.1.51. slCsy_{fin} is the full subcategory of slCsy whose objects are precisely the finitary structural layered consequence systems.

The finitary part of a structural layered consequence system is still structural.

Proposition 5.1.52. Let csy = ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩ be a structural layered consequence system. The greatest finitary structural layered consequence system contained in ⟨▷^{lΞ}, ▷^{gΞ}⟩ (in the sense of ⊑_{Σ}) is ⟨▷_{fin}^{lΞ}, ▷_{fin}^{gΞ}⟩.

Proof. From 3.1.21, it follows that ⟨L_{Σ}(Ξ), ▷_{fin}^{lΞ}, ▷_{fin}^{gΞ}⟩ is indeed a layered consequence system. From 5.1.31, it follows that both ▷_{fin}^{lΞ} and ▷_{fin}^{gΞ} are structural. ⊓ ⊔

As before, for convenience, we use the notation csy_{fin} = ⟨Σ, ▷_{fin}^{lΞ}, ▷_{fin}^{gΞ}⟩. The construction easily extends to a functor from slCsy to slCsy_{fin} that is adjoint to the obvious inclusion functor slI : slCsy_{fin} → slCsy. Again, we have a coreflection.

Proposition 5.1.53. slCsy_{fin} is a coreflective subcategory of slCsy.

Proof. Immediate from 5.1.32. ⊓ ⊔

Corollary 5.1.54. The maps

• slFin(csy) = csy_{fin};

• slFin(h : csy_{1} → csy_{2}) = h : csy_{1fin} → csy_{2fin},

constitute a functor slFin : slCsy → slCsy_{fin}. Moreover, slFin is adjoint to slI .

This subcategory slCsy_{fin} is also topological over Sig. Again, we shall use the notation Sgn_{slCsy_{fin}} = (Sgn_{slCsy} ∘ slI ).

Corollary 5.1.55. The functor Sgn_{slCsy_{fin}} : slCsy_{fin} → Sig is topological.

Thus, also the fibres of slCsy_{fin} form complete lattices.

Corollary 5.1.56. slCsy_{fin} is fibre-complete, i.e., each fibre (slCsy_{fin})_{Σ} is a complete lattice (under the restriction of ⊑_{Σ} to finitary structural layered consequence systems).

The meet of a family of finitary structural pairs of layered consequence operations {⟨▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩}_{k∈K} on Σ is given by the corresponding unique Sgn_{slCsy_{fin}}-initial lifting of

\{id_{Σ} : Σ → Σ\}_{k∈K}, i.e., ⊓_{Σ}^{fin}\{⟨▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ : k ∈ K\} = ⟨\underline{▷}lΞ_{fin}, \underline{▷}gΞ_{fin}⟩ with \underline{▷}lΞ_{fin} = ⊓_{Σ}^{fin}\{▷_{k}^{lΞ} : k ∈ K\}

and \underline{▷}gΞ_{fin} = ⊓_{Σ}^{fin}\{▷_{k}^{gΞ} : k ∈ K\}. Joins, top and bottom coincide with those for slCsy.

Following the same script that we have used in Chapters 2 and 3, we shall now define what we mean by an equivalence of structural (possibly layered) consequence systems. A simplistic approach would be to require that the non-structural (ground) systems presented by them (via PresC or PreslC ) should be equivalent. This would be a fair measure of equivalence, but still it would completely neglect structurality. To cope with it, we could instead require that the non-structural (schematic) systems presented by them (via SC or SlC ) should be equivalent. Yet, this alternative would still ignore the way schema sentences are generated from signatures. Thus, putting all these pieces together, we arrive at the following definition.

Definition 5.1.57. Two structural (layered) consequence systems csy_{k} over signature Σ^{k}, for k ∈ {1, 2}, are said to be equivalent if there exist s(l)Csy-morphisms f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} such that S(l)C (csy_{1}) ≡ S(l)C (csy_{2}) with S(l)C (f) and S(l)C (g). We denote the fact that csy_{1} and csy_{2} are equivalent by csy_{1} ≡ csy_{2}.

That is, csy_{1} ≡ csy_{2} means that we must have csy_{1}(Ξ) ≡ csy_{2}(Ξ) with f̂ and ĝ.

Clearly, isomorphism of structural (layered) consequence systems is a particular case of equivalence. As we have done in the non-structural case in Chapters 2 and 3, we shall now dwell on an alternative characterization of equivalence.

Definition 5.1.58. Let csy be a structural (layered) consequence system over signature Σ, and Γ ⊆ L_{Σ}(Ξ). Two schema sentences δ, θ ∈ L_{Σ}(Ξ) are said to be equivalent in csy modulo-Φ whenever δ ≡_{csy(Ξ)}^{Γ} θ. We denote this fact by δ ≡_{csy}^{Γ} θ. If Γ = ∅, we simply say that δ and θ are equivalent in csy, and simply write δ ≡_{csy} θ.

This definition obviously capitalizes on both 2.1.38 and 3.1.27. In the non-layered case, hence, if csy = ⟨Σ, ▷^{Ξ}⟩, δ ≡_{csy}^{Γ} θ means that both (Γ ∪ {δ}) ▷^{Ξ} θ and (Γ ∪ {θ}) ▷^{Ξ} δ. In the layered case, if csy = ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩, δ ≡_{csy}^{Γ} θ means that both (Γ ∪ {δ}) ▷^{lΞ} θ and (Γ ∪ {θ}) ▷^{lΞ} δ.

Proposition 5.1.59. Let csy_{k} over signature Σ^{k}, for k ∈ {1, 2}, be structural (layered) consequence systems. Then, csy_{1} ≡ csy_{2} if and only if there exist s(l)Csy-morphisms f : csy_{1} → csy_{2} and g : csy_{2} → csy_{1} such that the following two conditions are satisfied:

• γ ≡_{csy_{1}} ĝ(f̂(γ)) for every γ ∈ L_{Σ^{1}}(Ξ);

• δ ≡_{csy_{2}} f̂(ĝ(δ)) for every δ ∈ L_{Σ^{2}}(Ξ).

Proof. Clearly, by definition, csy_{1} ≡ csy_{2} with f and g if and only if csy_{1}(Ξ) ≡ csy_{2}(Ξ) with f̂ and ĝ. Thus, the result follows immediately from either 2.1.44 in the non-layered case, or 3.1.29 in the layered case. ⊓ ⊔

5.2 Logic system presentations and logic presentations

Having based the notion of a logic on a given category of logic systems, it is immediate that we should try and develop logic presentations on the basis of a category of logic system presentations. The same applies to the layered case. Below, besides establishing these concepts, we shall also introduce the two main approches we shall use for presenting structural layered logic systems: interpretation systems and deductive systems.

Definition 5.2.1. A category of (layered) logic system presentations is a pair ⟨A, s(l)Cn⟩ where:

• A is a category (of (layered) logic system presentations and their morphisms);

• s(l)Cn : A → s(l)Csy is a functor.

An object a ∈ |A| is said to be an ⟨A, s(l)Cn⟩-based (layered) logic system presentation. If, in addition, s(l)Cn(a) is a finitary structural (layered) consequence system then a is said to be a finitary (layered) logic system presentation.

⟨A, s(l)Cn⟩ is further said to be a category of finitary (layered) logic system presentations provided that every ⟨A, s(l)Cn⟩-based (layered) logic system presentation is finitary.

Proposition 5.2.2. A category of (layered) logic system presentations ⟨A, s(l)Cn⟩ is finitary if and only if there exists a functor s(l)Cn_{fin} : A → s(l)Csy_{fin} such that s(l)Cn = (s(l)I ∘ s(l)Cn_{fin}).

As before we shall sometimes identify a category of finitary (layered) logic system presentations with a functor s(l)Cn_{fin} : A → s(l)Csy_{fin}.

Please note that every category of (layered) logic system presentations ⟨A, s(l)Cn⟩ can be seen as presenting the category of (layered) logic systems ⟨A, (Pres(l)C ∘ s(l)Cn)⟩. In particular, structural (layered) consequence systems obviously present (layered) consequence systems, via Pres(l)C : s(l)Csy → (l)Csy.

Still, obviously, given a category ⟨A, s(l)Cn⟩ of (layered) logic system presentations, A can also be seen as a category of (layered) logic systems via the functor S(l)C : s(l)Csy → (l)Csy. However, ⟨A, (S(l)C ∘s(l)Cn)⟩ is quite different from the category of logic systems presented by ⟨A, s(l)Cn⟩.

Again, the notion of a (layered) logic presentation is straightforward.

Definition 5.2.3. Let ⟨A, s(l)Cn⟩ be a category of (layered) logic system presentations and Voc a category (of vocabularies). An ⟨A, s(l)Cn⟩-based (layered) logic presentation over Voc is a functor Log : Voc → A. If ⟨A, s(l)Cn⟩ is a category of finitary (layered) logic system presentations, then Log is also said to be finitary.

In both cases, whenever the functors sCn and slCn are well understood, we sometimes just refer to A, and call Log an A-based (layered) logic presentation.

Definition 5.2.4. Let ⟨A, s(l)Cn⟩ be a category of (layered) logic system presentations and Log an ⟨A, s(l)Cn⟩-based (layered) logic presentation over Voc. The signature functor of Log is Sgn_{Log} = (Sgn_{s(l)Csy} ∘ s(l)Cn ∘ Log) : Voc → Set.

That is, the signature of a (layered) logic presentation is the signature of each of its (layered) logic system presentations. In the sequel, we shall also use Sgn_{A} to denote (Sgn_{s(l)Csy} ∘ s(l)Cn).

Further repeating the steps followed in the previous chapters, (layered) logic presentations over a given category of (layered) logic system presentations also constitute a category. Recall again the construction in A.3.5.

Definition 5.2.5. Let ⟨A, s(l)Cn⟩ be a category of (layered) logic system presentations. We denote Funct(A) by ⟨A, s(l)Cn⟩-Log, and call it the category of ⟨A, s(l)Cn⟩-based (layered) logic presentations.

Notions like soundness and completeness in a proof/model theoretic setting also have their structural versions. However, we shall require that the whole schematic consequences are sound or complete over the schema languages, instead of just the consequences over the non-schematic languages presented by them. As before, in the following definitions P denotes one of the following properties: weak soundness, soundness, weak completeness, finite completeness, completeness.

Definition 5.2.6. Let ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩ be categories of logic system presentations, the former one being finitary, and cal ∈ |Cal| and sem ∈ |Sem| be such that Ded(cal) = ⟨Σ, ⊢^{Ξ}⟩, Ent(sem) = ⟨Σ, ⊨^{Ξ}⟩ for some signature Σ. The pair ⟨cal, sem⟩ is said to have property P provided that ⟨⟨L_{Σ}(Ξ), ⊢^{Ξ}⟩, ⟨L_{Σ}(Ξ), ⊨^{Ξ}⟩⟩ does.

Definition 5.2.7. Let ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩ be categories of layered logic system presentations, the former one being finitary, and cal ∈ |Cal| and sem ∈ |Sem| be such that Ded(cal) = ⟨Σ, ⊢^{lΞ}, ⊢^{gΞ}⟩, Ent(sem) = ⟨Σ, ⊨^{lΞ}, ⊨^{gΞ}⟩ for some signature Σ. The pair ⟨cal, sem⟩ is said to have the local property P provided that the pair ⟨⟨Σ, ⊢^{lΞ}⟩, ⟨Σ, ⊨^{lΞ}⟩⟩ does. Analogously, ⟨cal, sem⟩ has the global property P if the pair ⟨⟨Σ, ⊢^{gΞ}⟩, ⟨Σ, ⊨^{gΞ}⟩⟩ does. Furthermore, ⟨cal, sem⟩ has the property P provided that it holds both locally and globally.

Definition 5.2.8. Let Voc be a category, and Prf : Voc → Cal and Mod : Voc → Sem be (layered) logic presentations based, respectively, on the categories of (layered) logic system presentations ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩, the former being finitary, sharing the same signature functor, i.e., Sgn_{Prf} = Sgn_{Mod}. The pair ⟨Prf , Mod⟩ is said to have (local/global) property P provided that each pair ⟨Prf (V), Mod(V)⟩ does.

5.2.1 Models: interpretation systems

It is now time to introduce the model-theoretic category of layered logic system presentations we shall use in this dissertation. Being an algebraic approach, it relies on the free generation of sentences from a signature to provide them with a truth-functional algebraic interpretation. Of course, algebras over a signature must be enriched with some logical content. Or, conversely, our valuation structures from the previous chapters must be endowed with a suitable algebraic structure. Although we refrain from doing it here, we could easily simplify all the concepts and results to be given in order to put up a modeltheoretic category of non-layered logic system presentations. As we already know, it would suffice to replace each internal consequence ⋉_{m} on a model, by a set D_{m} of designated values.

Definition 5.2.9. Let Σ be a signature. A Σ-structure is a pair ⟨T , ⋉⟩ where:

• T = ⟨T, ^{T} ⟩ is a Σ-algebra;

• ⟨T, ⋉⟩ is a consequence system.

Obviously, given a Σ-algebra T , we can freely interpret the sentences of L_{Σ} in T. Moreover, if metavariables are assigned values in T, we can even extend it to an interpretation of schema sentences.

Definition 5.2.10. Let Σ be a signature and T a Σ-algebra. An assignment in T is a function α : Ξ → T. We denote by Asg(T ) the set of all assignments in T .

Obviously, a substitution over Σ is no more that an assignment in L_{Σ}(Ξ). Similarly, a ground substitution over Σ is an assignment in L_{Σ}.

Definition 5.2.11. Let Σ be a signature, T a Σ-algebra and α ∈Asg(T ). The interpretation function [[ ]]_{α}^{T} : L_{Σ}(Ξ) → T is the function underlying the unique Alg(Σ)-morphism from L_{Σ}(Ξ) to T that extends α.

Explicitly, we have:

• [[ξ]]_{α}^{T} = α(ξ);

• [[c]]_{α}^{T} = \underline{c}_{0}^{T};

• [[c(γ_{1}, …, γ_{n})]]_{α}^{T} = \underline{c}_{n}^{T} ([[γ_{1}]]_{α}^{T} , …, [[γ_{n}]]_{α}^{T}).

Obviously, the fact that L_{Σ} is initial in Alg(Σ) implies that the unique morphism [[ ]]^{T} : L_{Σ} → T is such that, for every sentence φ ∈ L_{Σ}, [[φ]]_{α}^{T} = [[φ]]^{T}.

Reducts may be extended to Σ-structures in an obvious manner. If ⟨T , ⋉⟩ is a Σ_{2}- structure and h : Σ_{1} → Σ_{2} a Sig-morphism then we denote by ⟨T , ⋉⟩|_{h} the Σ_{1}-structure ⟨T |_{h}, ⋉⟩.

We can now define our interpretation systems.

Definition 5.2.12. An interpretation system is a tuple ⟨Σ, M, S⟩ where:

• Σ is a signature; • M is a class (of models);

• S : M → Str(Σ) is a map (endowing each model with a Σ-structure).

For each model m, we denote its associated Σ-structure S(m) by ⟨T_{m}, ⋉_{m}⟩.

Morphisms are expectedly similar to those of layered valuation systems.

Definition 5.2.13. Let int_{k} = ⟨Σ^{k}, M_{k}, S_{k}⟩ for k ∈ {1, 2} be interpretation systems. An interpretation systems morphism h : int_{1} → int_{2} consists of:

• a signatures morphism \overline{h} : Σ^{1} → Σ^{2};

• a map \underline{h} : M_{2} → M_{1},

such that, for every model m ∈ M_{2}, the following condition holds:

• S_{1}(\underline{h}(m)) = S_{2}(m)|\overline{h}.

Definition 5.2.14. Interpretation systems and their morphisms constitute the category Int, with identity id_{⟨Σ,M,S⟩} such that \overline{id}_{⟨Σ,M,S⟩} = id_{Σ}, the identity morphism on Σ, and \underline{id}_{⟨Σ,M,S⟩} = id_{M}, the identity map on M, and composition (g∘f) such that \overline{(g ∘ f)} = (\overline{g}∘\overline{f}) and \underline{(g ∘ f)} = (\underline{f} ∘ \underline{g}).

Indeed, well known properties of functions and their compositions justify the fact that Int is a category.

Proposition 5.2.15. Int is indeed a category.

Now, we shall prove that Int constitutes a category of layered logic system presentations. Of course, we shall capitalize on what we have already learnt about layered valuation systems. However, we have to face the novelty of structurality. Easily, any Σ-structure ⟨T , ⋉⟩ induces an L_{Σ}-structure ⟨T, [[ ]]^{T} , ⋉⟩. Moreover, given any α ∈Asg(T ), it also induces an L_{Σ}(Ξ)-structure ⟨T, [[ ]]_{α}^{T} , ⋉⟩. Thus, taking into account all possible assignments, every interpretation system shall give rise to a layered valuation system on L_{Σ}(Ξ).

Definition 5.2.16. Let int = ⟨Σ, M, S⟩ be an interpretation system. The layered valuation system induced by int is val_{int}(Ξ) = ⟨L_{Σ}(Ξ), M^{Ξ}, S^{Ξ}⟩ where:

• M^{Ξ} = {⟨m, α⟩ : m ∈ M, α ∈ Asg(T_{m})};

• S^{Ξ}(⟨m, α⟩) = ⟨T_{m}, [[ ]]_{α}^{T_{m}}, ⋉_{m}⟩, for every ⟨m, α⟩ ∈ M^{Ξ}.

In fact we can show that Int is a category of layered logic systems by just extending the above construction to a functor from Int to lVal.

Definition 5.2.17. The maps

• lVal_{Int}(int) = val_{int}(Ξ);

• lVal_{Int}(h : int_{1} → int_{2}) with \overline{lVal_{Int}(h)} = ˆ \overline{h} and \underline{lVal_{Int}(h)}(⟨m, α⟩) = ⟨\underline{h}(m), α⟩, constitute the functor lVal_{Int} : Int → lVal.

Proposition 5.2.18. lVal_{Int} : Int → lVal is indeed a functor.

Proof. lVal_{Int} obviously maps each interpretation system to a layered valuation system. We just prove that lVal_{Int}(h) : val_{int_{1}}(Ξ) → val_{int_{2}}(Ξ) is indeed an lVal-morphism whenever h : int_{1} → int_{2} is an Int-morphism. For the purpose, let int_{k} = ⟨Σ^{k}, M_{k}, S_{k}⟩ for k ∈ \{1, 2\}. Clearly, since \overline{h} : Σ^{1} → Σ^{2} is a Sig-morphism, ˆ \overline{h} : L_{Σ^{1}}(Ξ) → L_{Σ^{2}}(Ξ) is a well defined function. Moreover, \underline{h} : M_{2} → M_{1} is such that S_{1}(\underline{h}(m)) = S_{2}(m)|\overline{h}, for every m ∈ M_{2}, and in particular T_{m}|\overline{h} = T_{\underline{h}(m)} and T_{m} = T_{\underline{h}(m)}. Therefore, given m ∈ M_{2} and α ∈ Asg(T_{m}), \underline{V (h)}(⟨m, α⟩) ∈ M_{1}^{Asg}. Since also ⋉_{m} = ⋉_{\underline{h}(m)}, we are left with showing

that [[ ]]_{α}^{T_{\underline{h}(m)}} = ([[ ]]_{α}^{T_{m}} ∘ ˆ \overline{h}). Indeed, it suffices to note that ([[ ]]_{α}^{T_{m}} ∘ ˆ \overline{h}) extends α, i.e., [[\underline{ˆ\overline{h}(ξ)}]]_{α}^{T_{m}} = [[\underline{ξ}]]_{α}^{T_{m}} = α(ξ). ⊓ ⊔

This definition embodies the mechanisms of global and local entailment induced by an interpretation system, by capitalizing on its induced layered valuation system.

Definition 5.2.19. Let int = ⟨Σ, M, S⟩ be an interpretation system. The local entailment structural consequence system csy_{int}^{lΞ} = ⟨Σ, ⊨_{int}^{lΞ} ⟩ is such that ⟨L_{Σ}(Ξ), ⊨_{int}^{lΞ} ⟩ is the one induced by val_{int}(Ξ). Namely, a schema sentence δ ∈ L_{Σ}(Ξ) is said to be locally entailed in int by a set of schema sentences Γ ⊆ L_{Σ}(Ξ), a fact that we denote by Γ⊨_{int}^{lΞ} δ, if and only if for every model m ∈ M and every assignment α ∈ Asg(T_{m}), {[[γ]]_{α}^{T_{m}} : γ ∈ Γ}⋉_{m}[[δ]]_{α}^{T_{m}}. Moreover, Γ^{⊨_{int}^{lΞ}} = {δ ∈ L_{Σ}(Ξ) : Γ⊨_{int}^{lΞ} δ}.

Definition 5.2.20. Let int = ⟨Σ, M, S⟩ be an interpretation system. The global entailment structural consequence system csy_{int}^{gΞ} = ⟨Σ, ⊨_{int}^{gΞ}⟩ is such that ⟨L_{Σ}(Ξ), ⊨_{int}^{gΞ}⟩ is the one induced by val_{int}(Ξ). Namely, a schema sentence δ ∈ L_{Σ}(Ξ) is said to be globally entailed in int by a set of schema sentences Γ ⊆ L_{Σ}(Ξ), a fact that we denote by Γ⊨_{int}^{gΞ}δ, if and only if for every model m ∈ M and every assignment α ∈ Asg(T_{m}), ⋉_{m}[[δ]]_{α}^{T_{m}} whenever ⋉_{m}[[γ]]_{α}^{T_{m}} for every γ ∈ Γ. Moreover, Γ^{⊨_{int}^{gΞ}} = {δ ∈ L_{Σ}(Ξ) : Γ⊨_{int}^{gΞ}δ}.

Of course, we have to check that these two consequences are indeed structural. For that purpose we need to use the following lemma.

Lemma 5.2.21. Let Σ be a signature, T a Σ-algebra, σ ∈ Sub(Σ) and α ∈ Asg(T ). Then, for every schema sentence γ ∈ L_{Σ}(Ξ), we have:

• [[γσ]]_{α}^{T} = [[γ]]_{([[ ]]_{α}^{T} ∘σ)}^{T}.

Proof. This is a straightforward result from universal algebra. Clearly, ([[ ]]_{α}^{T} ∘σ) is still an assignment in T . Thus, it suffices to note that the Alg(Σ)-morphism ([[ ]]_{α}^{T} ∘ σ̃) extends the assignment. Easily, σ̃(ξ) = σ(ξ), for every ξ ∈ Ξ, and the result follows. ⊓ ⊔

Proposition 5.2.22. Let int be an interpretation system. Then, both csy_{int}^{lΞ} and csy_{int}^{gΞ} are structural consequence systems.

Proof. The result is immediate from lemma 5.2.21. Indeed, assume that int = ⟨Σ, M, S⟩, and let σ ∈ Sub(Σ), m ∈ M and α ∈ Asg(T_{m}).

If Γ⊨_{int}^{lΞ} δ, by definition of ⊨_{int}^{lΞ} , we have {[[γ]]_{([[ ]]_{α}^{Tm}∘σ)}^{T_{m}} : γ ∈ Γ}⋉_{m}[[δ]]_{([[ ]]_{α}^{Tm}∘σ)}^{T_{m}}. Thus,

equivalently, {[[γσ]]_{α}^{T_{m}} : γ ∈ Γ}⋉_{m}[[δσ]]_{α}^{T_{m}}. We conclude that Γσ ⊨_{int}^{lΞ} δσ and ⊨_{int}^{lΞ} is structural.

If Γ⊨_{int}^{gΞ}δ and ⋉_{m}[[γσ]]_{α}^{T_{m}} for every γ ∈ Γ then, equivalently, we have that ⋉_{m}[[γ]]_{([[ ]]_{α}^{Tm}∘σ)}^{T_{m}}

for every γ ∈ Γ. Thus, by definition of ⊨_{int}^{gΞ}, ⋉_{m}[[δ]]_{([[ ]]_{α}^{Tm}∘σ)}^{T_{m}}, or equivalently, ⋉_{m}[[δσ]]_{α}^{T_{m}}.

We conclude that Γσ ⊨_{int}^{gΞ}δσ and ⊨_{int}^{gΞ} is structural. ⊓ ⊔

Now we just need to use the fact that layered valuation systems indeed induce layered consequence systems.

Corollary 5.2.23. Let int = ⟨Σ, M, S⟩ be an interpretation system. Then, the triple csy_{int}^{Ξ} = ⟨Σ, ⊨_{int}^{lΞ} , ⊨_{int}^{gΞ}⟩ constitutes a structural layered consequence system.

Definition 5.2.24. The maps

• Ent_{Int}(int) = csy_{int}^{Ξ};

• Ent_{Int}(h : int_{1} → int_{2}) = \overline{h},

constitute the functor Ent_{Int} : Int → slCsy.

Proposition 5.2.25. Ent_{Int} : Int → slCsy is indeed a functor.

Proof. Immediate from 5.2.18, by definition of slCsy. ⊓ ⊔

Corollary 5.2.26. The pair ⟨Int, Ent_{Int}⟩ constitutes a category of layered logic system presentations.

Easily, one notices that (SlC ∘ Ent_{Int}) = (Ent_{lVal} ∘ lVal_{Int}).

In the sequel, given an Int-based layered logic presentation Log : Voc → Int, we shall denote Log(V) by ⟨Σ^{V} , M_{V} , S_{V} ⟩ and its induced local and global entailment structural consequences by ⊨_{V}^{lΞ} and ⊨_{V}^{gΞ}, respectively, for each vocabulary V.

Below we shall prove some interesting properties of the functor Sgn_{Int} : Int → Sig.

Proposition 5.2.27. Sgn_{Int} : Int → Sig is coadjoint. Namely, id_{Σ} : Sgn_{Int}(int) → Σ is an Sgn_{Int}-couniversal arrow for Σ, where int = ⟨Σ, Str(Σ), id_{Str(Σ)}⟩.

Proof. Let ⟨Σ′, M′, S′⟩ be an interpretation system and \overline{h} : Σ′ → Σ a Sig-morphism. We have to show that there exists a unique map \underline{h} : M′ → Str(Σ) such that h : ⟨Σ′, M′, S′⟩ → int is an Int-morphism. Obviously, by definition of int we must have \underline{h}(m′) = S′(m′)|\overline{h}, for every m′ ∈ M′, and the result follows. ⊓ ⊔

Moreover, Sgn_{Int} can also be easily shown to be a cofibration.

Proposition 5.2.28. Sgn_{Int} : Int → Sig is a cofibration. Namely, h : ⟨Σ, M, S⟩ → ⟨Σ′, M′, S′⟩ is a cocartesian lift of ⟨⟨Σ, M, S⟩, \overline{h} : Σ → Σ′⟩, where:

• M′ is the class of all pairs ⟨m, ∂⟩ with ∂ = \{∂_{n} : (Σ′_{n} \ \overline{h}_{n}(Σ_{n})) → ((T_{m})^{n} → T_{m})}_{n∈IN}, such that m ∈ M satisfies the following condition, for every c^{1}, c^{2} ∈ Σ_{n}:

– if \overline{h}_{n}(c^{1}) = \overline{h}_{n}(c^{2}) then \underline{c^{1}}_{n}^{T_{m}} = \underline{c^{2}}_{n}^{T_{m}};

• S′(m, ∂) = ⟨T_{⟨m,∂⟩}, ⋉_{m}⟩ with T_{⟨m,∂⟩} = T_{m} and:

– \underline{\overline{h}_{n}(c)}_{n}^{T_{⟨m,∂⟩}} = \underline{c}_{n}^{T_{m}}, for every c ∈ Σ_{n};

– \underline{c′}_{n}^{T_{⟨m,∂⟩}} = ∂_{n}(c′), for every c′ ∈ (Σ′_{n} \ \overline{h}_{n}(Σ_{n}));

• \underline{h}(m, ∂) = m.

Proof. Clearly, ⟨Σ′, M′, S′⟩ is an interpretation sytem. In particular, S′ is well defined due to the conditions imposed to models in M′. Moreover, by definition, h : ⟨Σ, M, S⟩ → ⟨Σ′, M′, S′⟩ trivially establishes an Int-morphism.

To prove the universal property, assume that g : ⟨Σ, M, S⟩ → ⟨Σ′′, M′′, S′′⟩ is an Intmorphism and \overline{f} : Σ′ → Σ′′ a Sig-morphism such that (\overline{f} ∘ \overline{h}) = \overline{g}. We have to prove that there is a unique map \underline{f} : M′′ → M′ such that f : ⟨Σ′, M′, S′⟩ → ⟨Σ′′, M′′, S′′⟩ is an Intmorphism and (\underline{h} ∘ \underline{f}) = \underline{g}. Given m′′ ∈ M′′, we first note that if \overline{h}_{n}(c^{1}) = \overline{h}_{n}(c^{2}) then also

\overline{g}_{n}(c^{1}) = \overline{g}_{n}(c^{2}) and thus \underline{c^{1}}_{n}^{T_{\underline{g}(m′′)}} = \underline{c^{2}}_{n}^{T_{\underline{g}(m′′)}}. Furthermore, if c′ ∈ (Σ′_{n} \ \overline{h}_{n}(Σ_{n})), we must

have \underline{\overline{f}_{n}(c′)}_{n}^{T_{m′′}} = \underline{c′}_{n}^{T_{\underline{f}(m′′)}}. Thus, necessarily, \underline{f}(m′′) = ⟨\underline{g}(m′′), \{( _{n}^{T_{m′′}} ∘ \overline{f}_{n})\}_{n∈IN}⟩. ⊓ ⊔

5.2.2 Deduction: deductive systems

Our proof-theoretic approach to layered logic presentations is based on Hilbert-like deductive systems. These are usually presented by means of a set of schematic axioms and rules. For the purpose, we shall make use of metavariables. Moreover, their being a layered framework, we shall again distinguish between local and global inference rules. Although we refrain from doing it here, we could easily drop this distinction and analogously build up a proof-theoretic category of non-layered logic system presentations.

Definition 5.2.29. An inference rule over a signature Σ is a finitary inference over L_{Σ}(Ξ). We denote the set of all inference rules over Σ by Rule(Σ).

It should be obvious that an inference over L_{Σ} is a particular case of an inference rule over Σ. But in general inference rules use metavariables and thus a set of inferences can be obtained by uniformly instantiating the metavariables occurring in it.

Definition 5.2.30. A deductive system is a pair dsy = ⟨Σ, R, gR⟩ where Σ is a signature and inf_{dsy} = ⟨L_{Σ}(Ξ), R, gR⟩ is a layered inference system.

Deductive systems morphisms are expectedly simple.

Definition 5.2.31. Let dsy_{k} = ⟨Σ^{k}, R_{k}, gR_{k}⟩ for k ∈ {1, 2} be deductive systems. A deductive systems morphism h : dsy_{1} → dsy_{2} is a Sig-morphism h : Σ^{1} → Σ^{2} such that ĥ : inf_{dsy_{1}} → inf_{dsy_{2}} is an lInf-morphism.

Definition 5.2.32. Deductive systems and their morphisms constitute the category Dsy, with identity and composition as in Sig.

Proposition 5.2.33. Dsy is indeed a category.

Proof. Trivial. ⊓ ⊔

Expectedly, in order to define deducibility in a deductive system, we need to instantiate rules. A deductive system is nothing other than a smart way of presenting an inference system. All we have to do is to take all possible instances of the given inference rules.

Definition 5.2.34. Let Σ be a signature and r ∈ Rule(Σ). Then, r^{Ξ} denotes the set of all instances of r, i.e., r^{Ξ} = {rσ : σ ∈ Sub(Σ)}.

Obviously, rσ denotes the rule ⟨Prem(r)σ, Conc(r)σ⟩.

Definition 5.2.35. Let dsy = ⟨Σ, R, gR⟩ be a deductive system. We denote by inf_{dsy}(Ξ) the layered inference system ⟨L_{Σ}(Ξ), R^{Ξ}, gR^{Ξ}⟩.

This definition can be trivially extended to a functor.

Definition 5.2.36. The maps

• lInf_{Dsy}(dsy) = inf_{dsy}(Ξ);

• lInf_{Dsy}(h : dsy_{1} → dsy_{2}) = ĥ,

constitute the functor lInf_{Dsy} : Dsy → lInf.

Proposition 5.2.37. lInf_{Dsy} : Dsy → lInf is indeed a functor.

Proof. We just prove that lInf_{Dsy}(h) : inf_{dsy_{1}}(Ξ) → inf_{dsy_{2}}(Ξ) is indeed a lInf-morphism whenever h : dsy_{1} → dsy_{2} is a Dsy-morphism. The result is a consequence of the fact

that, for every σ ∈ Sub(Σ^{1}), (ĥ ∘ σ) ∈ Sub(Σ^{2}) and (ĥ ∘ σ̃) = ({ \atop (ĥ ∘ σ)} ∘ ĥ). Indeed, it

suffices to note that ({ \atop (ĥ ∘ σ)} ∘ ĥ)(ξ) = (ĥ ∘ σ)(ĥ(ξ)) = (ĥ ∘ σ)(ξ) = ĥ(σ(ξ)) = ĥ(σ̃(ξ)).

Let dsy_{k} = ⟨Σ^{k}, R_{k}, gR_{k}⟩ for k ∈ {1, 2}, and assume that ĥ(r) ∈ R_{2} for every r ∈ R_{1}. Then, for every σ ∈ Sub(Σ), ĥ(rσ) = ĥ(r)(ĥ ∘ σ) ∈ R_{2}^{Ξ}. A proof of ĥ(gR_{1}^{Ξ}) ⊆ gR_{2}^{Ξ} can be obtained analogously. ⊓ ⊔

As an immediate corollary we have that Dsy is a category of finitary layered logic systems. However, we can get more. We want to show that it constitutes a category of finitary layered logic system presentations.

This definition embodies the mechanisms of global and local deducibility induced by a deductive system.

Definition 5.2.38. Let dsy = ⟨Σ, M, S⟩ be a deductive system. The global deducibility finitary structural consequence system csy_{dsy}^{gΞ} = ⟨Σ, ⊢_{dsy}^{gΞ} ⟩ is such that ⟨L_{Σ}(Ξ), ⊢_{dsy}^{gΞ} ⟩ is the global consequence induced by inf_{dsy}(Ξ). Namely, a schema sentence δ ∈ L_{Σ}(Ξ) is said to be locally deducible in dsy by a set of schema sentences Γ ⊆ L_{Σ}(Ξ) if δ belongs to the least superset of Γ that is closed for (R^{Ξ} ∪ gR^{Ξ}). We denote this fact by Γ⊢_{dsy}^{gΞ} δ.

For convenience, we shall use Ω_{dsy}^{g} to denote Ω_{(R^{Ξ}∪gR^{Ξ})}.

Definition 5.2.39. Let dsy = ⟨Σ, M, S⟩ be a deductive system. The local deducibility finitary structural consequence system csy_{dsy}^{lΞ} = ⟨Σ, ⊢_{dsy}^{lΞ} ⟩ is such that ⟨L_{Σ}(Ξ), ⊢_{dsy}^{lΞ} ⟩ is the local consequence induced by inf_{dsy}(Ξ). Namely, a schema sentence δ ∈ L_{Σ}(Ξ) is said to be locally deducible in dsy by a set of schema sentences Γ ⊆ L_{Σ}(Ξ) if δ belongs to the least superset of Γ that is closed for (R^{Ξ} ∪ {⟨∅, γ⟩ : ⊢_{dsy}^{gΞ} γ}). We denote this fact by Γ⊢_{dsy}^{lΞ} δ.

For convenience, we shall use Ω_{dsy}_{l} to denote Ω_{(R^{Ξ}∪{⟨∅,γ⟩:⊢_{dsy}^{gΞ}γ})}.

Proposition 5.2.40. Let dsy be a deductive system. Then, both csy_{dsy}^{lΞ} and csy_{dsy}^{gΞ} are structural consequence systems.

Proof. Let Γ ⊆ L_{Σ}(Ξ) and σ ∈ Sub(Σ). Clearly, for global deducibility, it suffices to show that Ω_{dsy}^{g}(Γ)σ ⊆ Ω_{dsy}^{g}(Γσ). Thus, assume that r ∈ (R ∪ gR) and Prem(r)σ′ ⊆ Γ. We must find a substitution σ′′ such that Prem(r)σ′′ ⊆ Γσ and Conc(r)σ′σ = Conc(r)σ′′. The obvious candidate is σ′′ = (σ̃ ∘ σ′). Since Prem(r)σ′ ⊆ Γ, then also Prem(r)(σ̃ ∘ σ′) = Prem(r)σ′σ ⊆ Γσ, because { \atop (σ̃ ∘ σ′)} = (σ̃ ∘ σ̃′).

Regarding local deducibility, similarly, we proceed by showing that Ω_{dsy}_{l}(Γ)σ ⊆ Ω_{dsy}_{l}(Γσ).

To start with, we already know that ⊢_{dsy}^{lΞ} is structural. Therefore, ∅^{⊢_{dsy}^{gΞ}}σ ⊆ ∅σ^{⊢_{dsy}^{gΞ}} = ∅^{⊢_{dsy}^{gΞ}}. Assume now that r ∈ R and Prem(r)σ′ ⊆ Γ. Again, Prem(r)σ′σ ⊆ Γσ, and the result follows. ⊓ ⊔

Now, we just need to use the fact that layered valuation systems indeed induce layered consequence systems.

Corollary 5.2.41. Let dsy = ⟨Σ, R, gR⟩ be a deductive system. Then, the triple csy_{dsy}^{Ξ} = ⟨Σ, ⊢_{dsy}^{lΞ} , ⊢_{dsy}^{gΞ} ⟩ constitutes a structural finitary layered consequence system.

Definition 5.2.42. The maps

• Ded_{Dsy}(dsy) = csy_{dsy}^{Ξ};

• Ded_{Dsy}(h : dsy_{1} → dsy_{2}) = ĥ,

constitute the functor Ded_{Dsy} : Dsy → slCsy_{fin}.

Proposition 5.2.43. Ded_{Dsy} : Dsy → slCsy_{fin} is indeed a functor.

Proof. Immediate from 5.2.37, by definition of slCsy. ⊓ ⊔

Corollary 5.2.44. The pair ⟨Dsy, Ded_{Dsy}⟩ constitutes a category of finitary layered logic system presentations.

In the sequel, given a Dsy-based layered logic presentation Log : Voc → Dsy, we shall denote Log(V) by ⟨Σ^{V} , R_{V} , gR_{V} ⟩ and its induced local and global deducibility structural consequences by ⊢_{V}^{lΞ} and ⊢_{V}^{gΞ}, respectively, for each vocabulary V.

As expected, we shall be specially interested in layered logics presented via interpretation systems or deductive systems. These correspond, precisely, to Int and Dsy-based logics, respectively. As before, whenever we are given a system of logic that has both an interpretation and a deductive sytems based formulation, soundness is a desired minimal requirement. We shall next develop a sufficient condition for soundness in this context.

Definition 5.2.45. Let int = ⟨Σ, M, S⟩ be an interpretation system. A inference rule r ∈ Rule(Σ) is said to be locally sound with respect to int if Prem(r)⊨_{int}^{lΞ} Conc(r). Analogously, r is said to be globally sound with respect to int if Prem(r)⊨_{int}^{gΞ}Conc(r).

Clearly, the local/global soundness of a rule r with respect to int corresponds precisely to local/global soundness of r with respect to val_{int}(Ξ), seen as an inference over L_{Σ}(Ξ). Thus we shall be able to reuse the result we have proved in the last chapter.

Proposition 5.2.46. Let dsy = ⟨Σ, R, gR⟩ be a deductive system and int = ⟨Σ, M, S⟩ an interpretation system. Then, the pair ⟨dsy, int⟩ is sound if and only if every inference rule in R is locally sound and every inference rule in gR is globally sound, with respect to int.

Proof. By definition, ⟨dsy, int⟩ is sound if and only if ⟨csy_{dsy}^{Ξ}, csy_{int}^{Ξ}⟩ is sound if and only if ⟨inf_{dsy}(Ξ), val_{int}(Ξ)⟩ is sound. Therefore, the result is an instance of 3.2.38. Just note that the soundness of both R^{Ξ}, locally, and gR^{Ξ}, globally, with respect to val_{int}(Ξ), follows immediately from the soundness of R and gR, respectively, and the structurality of both entailments. ⊓ ⊔

Once again, we are interested in the properties of the forgetful functor Sgn_{Dsy}. Obviously, Sgn_{Dsy} = (Sgn_{slCsy_{fin}} ∘ Ded_{Dsy}). To start with, Sgn_{Dsy} : Dsy → Sig is a cofibration.

Proposition 5.2.47. Sgn_{Dsy} : Dsy → Sig is a cofibration. Namely, h : ⟨Σ, R, gR⟩ → ⟨Σ′, ĥ(R), ĥ(gR)⟩ is a cocartesian lift of ⟨⟨Σ, R, gR⟩, h : Σ → Σ′⟩.

Proof. Clearly, the triple ⟨Σ′, ĥ(R), ĥ(gR)⟩ is a deductive system and h : ⟨Σ, R, gR⟩ → ⟨Σ′, ĥ(R), ĥ(gR)⟩ a Dsy-morphism. To prove the universal property, assume that g : ⟨Σ, R, gR⟩ → ⟨Σ′′, R′′, gR′′⟩ is a Dsy-morphism and f : Σ′ → Σ′′ a Sig-morphism such that (f ∘ h) = g. Immediately, we have that f̂(ĥ(R)) = ĝ(R) ⊆ R′′ and f̂(ĥ(gR)) = ĝ(gR) ⊆ gR′′. Thus, f : ⟨Σ′, ĥ(R), ĥ(gR)⟩ → ⟨Σ′′, R′′, gR′′⟩ is a Dsy-morphism and the result follows. ⊓ ⊔

Recall that in 5.1.55, we have shown that Sgn_{slCsy_{fin}} is topological. Thus, among other properties, Sgn_{slCsy_{fin}} is also coadjoint and a cofibration. Thus, although Ded_{Dsy} can be shown not to be a cofibration, it still keeps some nice properties.

Proposition 5.2.48. Ded_{Dsy} : Dsy → slCsy_{fin} is coadjoint. Namely, we have that id_{Σ} : Ded_{Dsy}(dsy) → Σ is a Ded_{Dsy}-couniversal arrow for the structural finitary layered consequence system ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩, where dsy = ⟨Σ, R, gR⟩ with R = {r ∈ Rule(Σ) : Prem(r) ▷^{lΞ} Conc(r)} and gR = {r ∈ Rule(Σ) : Prem(r) ▷^{gΞ} Conc(r), Prem(r) ≠ ∅}. Furthermore, Ded_{Dsy} is cocartesian with respect to the cofibrations Sgn_{Dsy} and Sgn_{slCsy_{fin}}.

Proof. Trivially, since ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩ is finitary, we have ⊢_{dsy}^{lΞ} = ▷^{lΞ} and ⊢_{dsy}^{gΞ} = ▷^{gΞ}. To start with, from the structurality of ▷^{lΞ} and ▷^{gΞ}, it easily follows that R^{Ξ} = R and gR^{Ξ} = gR. Easily, we have Γ^{▷^{gΞ}} ∈ (R ∪ gR)(Γ). Moreover, if Δ ∈ (R ∪ gR)(Γ) then Γ^{▷^{gΞ}} ⊆ Δ. In fact, let Γ ▷^{gΞ} θ. By finitariness, there exists Θ ∈ ℘_{fin}(Γ) such that Θ ▷^{gΞ} θ. Depending on whether Θ = ∅ or not, we have either ⟨∅, θ⟩ ∈ R or ⟨Θ, θ⟩ ∈ gR. Thus, in any case, ∅ ⊆ Θ ⊆ Γ ⊆ Δ and Ω_{(R∪gR)}(Δ) ⊆ Δ together imply that θ ∈ Δ. Locally, now, it is also obvious that Γ^{▷^{lΞ}} ∈ (R ∪ {⟨∅, θ⟩ : ⊢_{dsy}^{gΞ} θ})(Γ). Let now Δ ∈ (R ∪ {⟨∅, θ⟩ : ⊢_{dsy}^{gΞ} θ})(Γ). If Γ ▷^{lΞ} θ then, by finitariness, there exists Θ ∈ ℘_{fin}(Γ) such that Θ ▷^{lΞ} θ. Thus, ⟨Θ, θ⟩ ∈ R and θ ∈ Δ.

As for the universal property, let dsy′ = ⟨Σ′, R′, gR′⟩ be a deductive system and g : ⟨Σ′′, ⊢_{dsy′}^{lΞ} , ⊢_{dsy′}^{gΞ} ⟩ → ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩ be an slCsy-morphism. Clearly, we have to prove that g : dsy′ → dsy is a Dsy-morphism. But if r′ ∈ R′ then, obviously, Prem(r′)⊢_{dsy′}^{lΞ} Conc(r′) and ĝ(Prem(r′)) ▷^{lΞ} ĝ(Conc(r′)). Analogously, if r′ ∈ gR′, then Prem(r′)⊢_{dsy′}^{gΞ} Conc(r′) and ĝ(Prem(r′)) ▷^{gΞ} ĝ(Conc(r′)).

To prove that Ded_{Dsy} is cocartesian, according to 5.2.47, we need to prove that the morphism h : Ded_{Dsy}(⟨Σ, R, gR⟩) → Ded_{Dsy}(⟨Σ′, ĥ(R), ĥ(gR)⟩) is an Sgn_{slCsy_{fin}}- cocartesian lift of ⟨Ded_{Dsy}(⟨Σ, R, gR⟩), h : Σ → Σ′⟩, where ⟨Σ, R, gR⟩ is any deductive system. For the purpose, let g : Ded_{Dsy}(⟨Σ, R, gR⟩) → ⟨Σ′′, ▷^{′′lΞ}, ▷^{′′gΞ}⟩ be an slCsy_{fin}- morphism and f : Σ′ → Σ′′ a morphism such that (f∘h) = g. All we need to show is that f : Ded_{Dsy}(⟨Σ′, ĥ(R), ĥ(gR)⟩) → ⟨Σ′′, ▷^{′′lΞ}, ▷^{′′gΞ}⟩ is an slCsy_{fin}-morphism. For the purpose, let Γ ⊆ L_{Σ′}(Ξ). Globally, it is enough to show that f̂^{−1}(f̂(Γ)^{▷^{′′gΞ}}) ∈ (ĥ(R) ∪ ĥ(gR))^{Ξ}(Γ). Trivially, Γ ⊆ f^{−1}(f(Γ)) ⊆ f^{−1}(f(Γ)^{▷^{′′gΞ}}) just by making use of the extensiveness of ▷^{′′gΞ}. Assume now that r ∈ (R ∪ gR), σ ∈ Sub(Σ′) and ĥ(Prem(r))σ ⊆ f^{−1}(f(Γ)^{▷^{′′gΞ}}).

Moreover, as noted before, (f̂ ∘ σ) ∈ Sub(Σ′′) and (f̂ ∘ σ̃) = ({ \atop (f̂∘ σ)} ∘ f̂). Immediately, we have that f̂(Γ) ▷^{′′gΞ}f̂(ĥ(Prem(r))σ) = ĝ(Prem(r))(f̂∘ σ). But g is a morphism, and so we must have ĝ(Prem(r)) ▷^{′′gΞ} ĝ(Conc(r)), and by structurality ĝ(Prem(r))(f̂ ∘ σ) ▷^{′′gΞ} ĝ(Conc(r))(f̂∘ σ). Therefore, it must be the case that f̂(Γ) ▷^{′′gΞ}ĝ(Conc(r))(f̂∘ σ), i.e., ĥ(Conc(r))σ ∈ f^{−1}(f(Γ)^{▷^{′′gΞ}}). Locally, it is enough to show that f^{−1}(f(Γ)^{▷^{′′lΞ}}) is

a superset of Γ that contains ∅^{⊢_{⟨Σ′,_{h}(R),_{h}(gR)⟩}^{gΞ}} and is closed for ĥ(R)^{Ξ}. Trivially, Γ ⊆ f^{−1}(f(Γ)) ⊆ f^{−1}(f(Γ)^{▷^{′′lΞ}}) now by using of the extensiveness of ▷^{′′lΞ}. Moreover, we

have already shown that f̂(∅^{⊢_{⟨Σ′,_{h}(R),_{h}(gR)⟩}^{gΞ}}) ⊆ f̂(∅)^{▷^{′′gΞ}} = ∅^{▷^{′′gΞ}}. Thus, ∅^{⊢_{⟨Σ′,_{h}(R),_{h}(gR)⟩}^{gΞ}} ⊆

f̂^{−1}(∅^{▷^{′′gΞ}}) ⊆ f̂^{−1}(f̂(Γ)^{▷^{′′lΞ}}). Assume now that r ∈ R, σ ∈ Sub(Σ′) and ĥ(Prem(r))σ ⊆ f^{−1}(f(Γ)^{▷^{′′lΞ}}). Immediately, we have that f̂(Γ) ▷^{′′lΞ}f̂(ĥ(Prem(r))σ) = ĝ(Prem(r))(f̂∘ σ). But g is a morphism, and so we must have ĝ(Prem(r)) ▷^{′′lΞ} ĝ(Conc(r)), and by structurality ĝ(Prem(r))(f̂ ∘ σ) ▷^{′′lΞ} ĝ(Conc(r))(f̂ ∘ σ). Therefore, it must be the case that f̂(Γ) ▷^{′′lΞ}ĝ(Conc(r))(f̂∘ σ), i.e., ĥ(Conc(r))σ ∈ f^{−1}(f(Γ)^{▷^{′′lΞ}}). ⊓ ⊔

Furthermore, we have the following corollary. Note that id_{Σ} : Sgn_{slCsy_{fin}}(⟨Σ, ⊤, ⊤⟩) → Σ is the obvious Sgn_{slCsy_{fin}}-couniversal arrow for Σ.

Corollary 5.2.49. Sgn_{Dsy} : Dsy → Sig is coadjoint. Namely, id_{Σ} : Sgn_{Dsy}(dsy) → Σ is an Sgn_{Dsy}-couniversal arrow for Σ, where dsy = ⟨Σ, Rule(Σ), {r ∈ Rule(Σ) : Prem(r) ≠ ∅}⟩.

5.3 Examples

Most of the examples we analyzed in the previous sections admit structural presentations, both semantically, using interpretation systems, and proof-theoretically, using deductive systems. However, the semantics of the paraconsistent system C_{1} constitutes an exception.

The cases of equational and first-order logic are left out for the obvious reason that they do not fit our propositional-based setting. We shall start with classical propositional logic.

Example 5.3.1. Structural CPL. Let P be a set of propositional symbols. The signature of CPL over P is Σ_{cpl}^{P} such that:

• Σ_{cpl,0}^{P} = P, Σ_{cpl,1}^{P} = {¬}, Σ_{cpl,2}^{P} = {⇒, ∧, ∨}, and Σ_{cpl,n}^{P} = ∅ if n > 2.

Clearly its induced non-schematic language is precisely L_{cpl}^{P}.

The algebraic interpretation system int_{cpl}^{P} = ⟨Σ_{cpl}^{P}, M_{cpla}^{P}, S_{cpl}^{P}⟩ of CPL considers the same class of models M_{cpla} as in 2.3.4, consisting of all pairs ⟨B, ϑ⟩ with B = ⟨B, ≤⟩ a Boolean algebra and ϑ : P → B a function. For each ⟨B, ϑ⟩ ∈ M_{cpla}^{P},

S_{cpl}^{P}(⟨B, ϑ⟩) = ⟨B, ⋉_{≤}⟩

where:

• \underline{p}_{0}^{B} = ϑ(p);

• \underline{¬}_{1}^{B}(x) = x′;

• \underline{⇒}_{2}^{B}(x, y) = x′ ∨ y;

• \underline{∧}_{2}^{B}(x, y) = x ∧ y;

• \underline{∨}_{2}^{B}(x, y) = x ∨ y.

Clearly, if φ is a non-schematic sentence we have [[φ]]^{B} = ϑ^{∗}(φ) as defined in 2.3.4.

The deductive system dsy_{cpl}^{P} = ⟨Σ_{cpl}^{P}, R_{cpl}^{P}, gR_{cpl}^{P}⟩ is similar to the layered inference system considered in 3.3.1, but now with schema rules instead of inferences. Namely, gR_{cpl}^{P} = ∅ and R_{cpl}^{P} contains schematic versions of each of the axioms Ax01 -Ax11 plus MP. Namely, for instance, Ax01 looks like:

Ax01. ⟨∅, (ξ_{1} ⇒ (ξ_{2} ⇒ ξ_{1}))⟩,

and MP like:

MP. ⟨{ξ_{1}, (ξ_{1} ⇒ ξ_{2})}, ξ_{2}⟩.

Example 5.3.2. Structural IPL. Let P be a set of propositional symbols. The signature of IPL over P is Σ_{ipl}^{P} = Σ_{cpl}^{P}.

The interpretation system int_{ipl}^{P} = ⟨Σ_{ipl}^{P}, M_{ipl}^{P}, S_{ipl}^{P}⟩ of CPL is similar to the layered valuation system defined in 3.3.2. Namely, again, M_{ipl}^{P} consists of all pairs ⟨H, ϑ⟩ with H = ⟨H, ≤⟩ a Heyting algebra and ϑ : P → H a function. For each ⟨H, ϑ⟩ ∈ M_{ipl}^{P},

S_{ipl}^{P}(⟨H, ϑ⟩) = ⟨H, ⋉_{≤}⟩

where:

• \underline{p}_{0}^{H} = ϑ(p);

• \underline{¬}_{1}^{H}(x) = x → ⊥; • \underline{⇒}_{2}^{H}(x, y) = x → y;

• \underline{∧}_{2}^{H}(x, y) = x ∧ y;

• \underline{∨}_{2}^{H}(x, y) = x ∨ y.

Clearly, if φ is a non-schematic sentence we have [[φ]]^{H} = ϑ^{∗}(φ) as defined in 2.3.5.

The deductive system dsy_{ipl}^{P} = ⟨Σ_{ipl}^{P}, R_{ipl}^{P}, gR_{ipl}^{P}⟩ is similar to the layered inference system considered in 3.3.2, but again with schema rules instead of inferences. Namely, gR_{cpl}^{P} = ∅ and R_{cpl}^{P} contains schematic versions of each of the axioms Ax01 -Ax10 plus MP.

Example 5.3.3. Structural K. Given a set P propositional symbols, the signature of K over P is Σ_{K}^{P} such that:

• Σ_{K ,0}^{P} = P, Σ_{K ,1}^{P} = {¬, □}, Σ_{K ,2}^{P} = {⇒, ∧, ∨}, and Σ_{K ,n}^{P} = ∅ if n > 2.

Clearly its induced non-schematic language is precisely L_{K}^{P}.

The interpretation system int_{K}^{P} = ⟨Σ_{K}^{P} , M_{K}^{P} , S_{K}^{P} ⟩ of CPL is similar to the layered valuation system defined in 3.3.3. Namely, again, M_{K}^{P} consists of all Kripke models ⟨W, R, ϑ⟩ with ϑ : P → ℘(W). For each ⟨W, R, ϑ⟩ ∈ M_{K}^{P}, let T = ℘(W) and define

S_{K}^{P} (⟨W, R, ϑ⟩) = ⟨T , ⋉_{⊆}⟩

where:

• \underline{p}_{0}^{T} = ϑ(p);

• \underline{¬}_{1}^{T} (X) = (W \ X);

• \underline{□}_{1}^{T} (X) = \{w ∈ W : wRw′ implies w′ ∈ X\};

• \underline{⇒}_{2}^{T} (X, Y) = ((W \ X) ∪ Y);

• \underline{∧}_{2}^{T} (X, Y) = (X ∩ Y);

• \underline{∨}_{2}^{T} (X, Y) = (X ∪ Y).

Clearly, if φ is a non-schematic sentence we have [[φ]]^{T} = ϑ^{∗}(φ) as defined in 3.3.3.

The deductive system dsy_{K}^{P} = ⟨Σ_{K}^{P} , R_{K}^{P} , gR_{K}^{P} ⟩ is similar to the layered inference system considered in 3.3.3, but again with schema rules instead of inferences. Namely, R_{cpl}^{P} contains schematic versions of each of the axioms Ax01 -Ax11 and □-Norm and MP, and gR_{K}^{P} a schematic version of □-Nec, namely:

□-Nec. ⟨{ξ_{1}}, (□ξ_{1})⟩.

It is well known that normal modal logic can also be given a purely algebraic semantics.

Example 5.3.4. Algebraic semantics for K. The algebraic interpretation system for K over a set P of propositional symbols is int_{aK}^{P} = ⟨Σ_{K}^{P} , M_{aK}^{P} , S_{aK}^{P} ⟩ where:

• M_{aK}^{P} is the class of all pairs ⟨B, ϑ⟩ where B = ⟨B, ≤⟩ is a Boolean algebra with an additional operation □ : B → B such that:

– □(⊤) = ⊤;

– □(x ∧ y) = □(x) ∧ □(y),

and ϑ : P → B a function;

• S_{aK}^{P} (⟨B, ϑ⟩) = ⟨B, ⋉_{≤}⟩, where:

– \underline{p}_{0}^{B} = ϑ(p);

– \underline{¬}_{1}^{B}(x) = x′;

– \underline{□}_{1}^{B}(x) = □(x);

– \underline{⇒}_{2}^{B}(x, y) = x′ ∨ y;

– \underline{∧}_{2}^{B}(x, y) = x ∧ y;

– \underline{∨}_{2}^{B}(x, y) = x ∨ y.

The deductive system for K is well known to be sound and complete with respect to this semantics. Soundness is trivial, and completeness follows from an easy Lindenbaum construction [HC96].

To close this collection of examples, let us briefly comment on the non-representability of the semantics of the paraconsistent system C_{1}.

Example 5.3.5. The case of C_{1}. The signature of C_{1} over a set P of propositional symbols is Σ_{c1}^{P} = Σ_{cpl}^{P}.

As easily, the deductive system dsy_{c1}^{P} = ⟨Σ_{c1}^{P} , R_{c1}^{P} , gR_{c1}^{P} ⟩ of C_{1} has gR_{cpl}^{P} = ∅ and R_{cpl}^{P} containing schematic versions of each of the axioms considered in 2.3.8.

However, the semantics also proposed in 2.3.8 is non-homomorphic. In particular, it encompasses bi-valuations ϑ such that ϑ((¬ φ)) ≠ 1 − ϑ(φ). In fact it is known, see for instance [Mar99], that the system is not congruent, particularly with respect to negation. An homomorphic algebraic semantics for C_{1} is still an open problem, and therefore we do not know how to represent it as an interpretation system.

5.4 Remarks

We shall conclude with a brief summary of this Chapter. We have introduced signatures and signature generated languages and schema languages, and set up these constructions in context of the category Sig of signatures and its relationship to Set. Then, we have brought the notion of structurality into play by making explicit use of metavariable substitutions. The categories sCsy of structural consequence systems and slCsy of structural layered consequence systems, as well as their finitary versions, were then studied in some detail, i.e., the forgetful signature functors from sCsy and slCsy were shown to be topological, and, therefore, the families of structural and layered structural consequence operations over the same signature were shown to form complete lattices. The relationship between the non-layered and layered versions was shown to be similar to the non-structural case. The same applies to the corresponding finitary versions. Then, we defined a category of (layered) logic system presentations as the source category of any functor to s(l)Csy, and, by indexing, a corresponding general notion of (layered) logic presentation. The notions of soundness and completeness were also reshaped in this structural context. Two particular cases of categories of layered logic system presentations were then analyzed in detail: the categories Int of interpretation systems and Dsy of deductive systems. In particular, the corresponding forgetful signature functors were both proved to be both coadjoints and cofibrations. The Chapter closes with some examples of structural logic systems, namely classical, intuitionistic and modal propositional logics, and with a counterexample borrowed from [dC74].

Fibring

Now that we have settled the notions of (layered) logic system presentation and logic presentation on the basis of (layered) structural consequence systems, we can finally focus our attention on fibring. Although in a different context, the path that shall lead us to fibring is completely analogous to our previous treatment of synchronization in Chapter 4. To a great extent, we capitalize on previous work [SSC99, ZSS99], namely in detaching this level of abstraction as the natural one for the study of fibring. The intuition is that if we fibre two given structural consequence systems, we obtain a new structural consequence system that extends the two given structural systems in a minimal way and respects all the constraints one may require. Note that, if it was not for the keyword “structural”, we would certainly be repeating what we said about synchronization. However, structurality makes all the difference. First of all we shall be summing together two signatures instead of two unstructured language. The language of the signature obtained shall therefore include sentences built up by using constructors from both given signatures. As far as consequence is concerned, structurality does the rest. Just note that any possible schematic pattern of reasoning coming from either side can be instantiated with such mixed sentences of the fibred language. Of course, if the fibring is free, we obtain exactly the least structural consequence operation over the disjoint union of the two signatures that extends the two given structural consequences. In its constrained form, the idea is that a constraint specifies a set of pairs of constructors to be shared. As we shall see, the result is similar to adding to the free fibred structural consequence a set of bridging schema rules corresponding to each pair of shared constructors. The extension of the mechanism to the layered case is straightforward.

This Chapter is organized as follows. First, we shall define free fibring of structural consequence systems and, by analogy, of structural layered consequence systems. Both shall be given a similar categorial characterization, in sCsy and slCsy. Then, capitalizing on the abstract notions of logic system presentation and layered logic system presentation, and on this categorial characterization, we shall introduce the notions of suitability and exactness for free fibring. As before, suitability shall mean that the corresponding construction in a category of (layered) logic system presentations coincides at the level of signatures with the construction for the corresponding structural (layered) consequence systems. Exactness is much stronger and shall require that the whole construction in a category of (layered) logic system presentations precisely matches the corresponding construction for the induced structural (layered) consequence systems. Regarding constrained fibring, we shall start by defining it at the level of structural consequence systems and then show that it indeed corresponds to the addition of corresponding mixed rules to the free construction. Analogously, we shall define constrained fibring of structural layered consequence systems, and prove that in both cases the construction corresponds to a signaturedriven F -pushout. The notions of suitability and exactness for fibring, in general, shall then be introduced by suitably generalizing the free case. Next, we shall explore fibring in our working categories of layered logic system presentations. The proof-theoretic category Dsy is shown to be exact for fibring, and the model-theoretic category Int shown to be just suitable. We shall illustrate fibring by means of two examples, namely multi-modal logics as fibred modal-logics, and the interesting case of fibring classical and intuitionistic logic. Then, we dwell on the relationship between fibring and synchronization. The coincidence between the development of the two mechanisms, each on its own basis, and the known relationships studied in Chapter 5 between the structural and non-structural settings shall pave the ground for our investigation. The fact that synchronization can be seen as a special case of fibring was already rather clear, and shall be obtained with no surprise. However, despite the apparent added richness of the fibring mechanism, we shall also manage to simulate fibring using constrained synchronization. We shall conclude with the study of two fundamental transference results, soundness and completeness, in all the relevant case Dsy/Int. Soundness preservation shall appear as a trivial corollary of the suitability for fibring of both categories, and, moreover, of the exactness of Dsy. Regarding completeness preservation, however, due to the non-exactness of Int, a fully general result cannot be expected. Still, under reasonable fullness conditions, we shall able to prove the completeness of fibred systems in three different contexts: arbitrary interpretation structures, elementary structures and partially ordered ones. As we shall see, congruence, logical equivalence, implication and conjunction play an important role in the last two cases, similarly to what happens in [ZSS99].

6.1 General definition

Fibring is a combination mechanism whose set-up is made at the level of structural consequence systems, logic system presentations and logic presentations, as well as of their layered versions, as presented in Chapter 5. Intuitively, if we fibre two given structural consequence system, we obtain a new structural consequence system whose signature contains the signatures of the two given systems and whose structural consequence operation extends the two given structural ones in some minimal way.

6.1.1 Free fibring

This construction corresponds to the simplest case of fibring. No constraints being made, the signature shall be the disjoint union of the two signatures, and the structural consequence operation the least possible one that still extends the structural consequences given. In other words, given two structural consequence systems csy_{1} and csy_{2}, their free fibring corresponds to the least structural consequence system (csy_{1} ⊛ csy_{2}) that contains both csy_{1} and csy_{2}. Of course, this should be read as meaning that (csy_{1}⊛csy_{2}) is precisely a sum in sCsy.

Definition 6.1.1. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems. The free fibring of csy_{1} and csy_{2} is (csy_{1} ⊛ csy_{2}) = ⟨(Σ^{1} ⊛ Σ^{2}), ▷^{Ξ}⟩, where:

• (Σ^{1} ⊛ Σ^{2})_{n} = ({c_{1} : c ∈ Σ_{n}^{1}} ∪ {c_{2} : c ∈ Σ_{n}^{2}}) for every n ∈ IN;

• Θ^{▷^{Ξ}} is the least superset of Θ that is closed for RΞ_{⊛}, with:

R_{⊛} = ({⟨Γ_{1}, γ_{1}⟩ : Γ ▷_{1}^{Ξ} γ} ∪ {⟨Δ_{2}, δ_{2}⟩ : Δ ▷_{2}^{Ξ} δ}).

Of course, above, we are using θ_{k} to denote ι̂_{k}(θ) for every k ∈ {1, 2} and θ ∈ L_{Σ^{k}}(Ξ), where ι_{k} : Σ^{k} → Σ is the corresponding injection morphism. Obviously, θ_{k} is obtained from θ by replacing each constructor c (from Σ^{k}) by c_{k}.

Proposition 6.1.2. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems. Then, {ι_{k} : csy_{k} → (csy_{1} ⊛ csy_{2})}_{k∈{1,2}}, with each ι_{k,n}(c) = c_{k}, is a sum in sCsy.

Proof. In 5.1.26 we have shown that Sgn_{sCsy} : sCsy → Sig is topological. Moreover, Sig is well known to be cocomplete, and thus also sCsy is cocomplete. In particular, since the sink {ι_{k} : Σ^{k} → (Σ^{1} ⊛ Σ^{2})}_{k∈{1,2}} is a sum in Sig, (csy_{1} ⊛ csy_{2}) = ⟨(Σ^{1} ⊛ Σ^{2}), ▷^{Ξ}⟩ is a sum in sCsy if ▷^{Ξ} = ⊓_{(Σ^{1}⊛Σ^{2})}{▷^{′Ξ} : both ι_{k} : csy_{k} → ⟨(Σ^{1} ⊛ Σ^{2}), ▷^{′Ξ}⟩ are sCsy-morphisms}. That is, according to 5.1.28, ▷^{Ξ} is generated by all instances of ι_{1} translations of ▷_{1}^{Ξ} consequences and ι_{2} translations of ▷_{2}^{Ξ} consequences, i.e., by RΞ_{⊛}. ⊓ ⊔

Similarly to synchronization, we extend the mechanism of free fibring also to the layered setting.

Definition 6.1.3. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems. The free fibring of csy_{1} and csy_{2} is (csy_{1} ⊛ csy_{2}) = ⟨(Σ^{1} ⊛ Σ^{2}), ▷^{lΞ}, ▷^{gΞ}⟩, where ⟨(Σ^{1} ⊛ Σ^{2}), ▷^{gΞ}⟩ = (csy_{1}^{g} ⊛ csy_{2}^{g}), i.e., Θ^{▷^{gΞ}} is the least superset of Θ that is closed for R_{g⊛}^{Ξ}, where:

R_{g⊛} = ({⟨Γ_{1}, γ_{1}⟩ : Γ ▷_{1}^{gΞ} γ} ∪ {⟨Δ_{2}, δ_{2}⟩ : Δ ▷_{2}^{gΞ} δ}),

and Θ^{▷^{lΞ}} is the least superset of Θ that is closed for (R_{l⊛}^{Ξ} ∪ {⟨∅, θ⟩ : ▷^{gΞ}θ}), where:

R_{l⊛} = ({⟨Γ_{1}, γ_{1}⟩ : Γ ▷_{1}^{lΞ} γ} ∪ {⟨Δ_{2}, δ_{2}⟩ : Δ ▷_{2}^{lΞ} δ}).

Proposition 6.1.4. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems. Then, {ι_{k} : csy_{k} → (csy_{1} ⊛ csy_{2})}_{k∈{1,2}}, with each ι_{k,n}(c) = c_{k}, is a sum in slCsy.

Proof. From 5.1.47 we know that Sgn_{slCsy} : slCsy → Sig is topological. So, slCsy is cocomplete. In particular, (csy_{1}⊛csy_{2}) = ⟨(Σ^{1}⊛Σ^{2}), ▷^{lΞ}, ▷^{gΞ}⟩ is a sum in slCsy if ⟨▷^{lΞ}, ▷^{gΞ}⟩ = ⊓_{(Σ^{1}⊛Σ^{2})}{⟨▷^{′lΞ}, ▷^{′gΞ}⟩ : both ι_{k} : csy_{k} → ⟨(Σ^{1} ⊛ Σ^{2}), ▷^{′lΞ}, ▷^{′gΞ}⟩ are slCsy-morphisms}. The result follows easily from 6.1.2 and 5.1.49. ⊓ ⊔

Free fibring in an arbitrary category A of (layered) logic system presentations extends smoothly from (layered) structural consequence systems. In abstract terms, everything is absolutely similar to the lifting of free synchronization from consequence systems to logic systems. Thus, expectedly, sums must exist in A, and in such a way that the signature of a sum corresponds to a sum of the signatures. Obviously, the construction is precise if also the structural consequence operation of a sum is a sum of the structural consequence operations.

Definition 6.1.5. A category of (layered) logic system presentations ⟨A, s(l)Cn⟩ is suitable for free fibring provided that A has binary sums and Sgn_{A} preserves them. Moreover, ⟨A, s(l)Cn⟩ is exact for free fibring if, in addition, also s(l)Cn preserves binary sums.

Definition 6.1.6. Let ⟨A, s(l)Cn⟩ be a category of (layered) logic system presentations suitable for free fibring and a_{1}, a_{2} ∈ |A|. Then, given any sum {h_{k} : a_{k} → (a_{1} ⊛ a_{2})}_{k∈{1,2}} in A, (a_{1} ⊛ a_{2}) is said to be a free fibring of a_{1} and a_{2}.

Please note that Sgn_{A}(a_{1} ⊛ a_{2}) may not coincide with (Sgn_{A}(a_{1}) ⊛ Sgn_{A}(a_{2})), as defined in 6.1.1 (6.1.3) with injections ι_{k} as in 6.1.2 (6.1.4). However, by definition of suitability for free fibring, it must always be the case that the unique signature morphism h : (Sgn_{A}(a_{1}) ⊛ Sgn_{A}(a_{2})) → Sgn_{A}(a_{1} ⊛ a_{2}) such that (h ∘ ι_{1}) = h_{1} and (h ∘ ι_{2}) = h_{2} is necessarily an isomorphism of signatures. Moreover, from the universal property of sums it follows that h : (s(l)Cn(a_{1}) ⊛ s(l)Cn(a_{2})) → s(l)Cn(a_{1} ⊛ a_{2}) is an s(l)Csy morphism. Exactness, in its turn, implies that this morphism is, furthermore, an s(l)Csy isomorphism.

Finally, let us consider free fibring of logic and layered logic presentations. Of course, we capitalize on the constructions at the level of logic and layered logic system presentations.

Definition 6.1.7. Let ⟨A, s(l)Cn⟩ be a category of (layered) logic system presentations suitable for free fibring, and Log_{k} be ⟨A, s(l)Cn⟩-based (layered) logic presentations over Voc_{k} for k ∈ {1, 2}. The free fibring of Log_{1} and Log_{2} is (Log_{1} ⊛Log_{2}) over (Voc_{1} ×Voc_{2}) where:

• (Log_{1} ⊛ Log_{2})(⟨V_{1}, V_{2}⟩) = (Log_{1}(V_{1}) ⊛ Log_{2}(V_{2}));

• (Log_{1} ⊛ Log_{2})(⟨h_{1} : V_{1} → V′_{1}, h_{2} : V_{2} → V′_{2}⟩) is the unique A-morphism such that ((Log_{1} ⊛ Log_{2})(⟨h_{1}, h_{2}⟩) ∘ ι_{k}(V_{1}, V_{2})) = (ι_{k}(V′_{1}, V′_{2}) ∘ Log(h_{k})) for k ∈ {1, 2}, with ι_{k}(X_{1}, X_{2}) : Log_{k}(X_{k}) → (Log_{1}(X_{1}) ⊛ Log_{2}(X_{2})) the corresponding injection morphisms.

Proposition 6.1.8. Let Log_{k} for k ∈ {1, 2} be ⟨A, s(l)Cn⟩-based (layered) logic presentations over Voc_{k}, for some category ⟨A, s(l)Cn⟩ of (layered) logic system presentations suitable for free fibring. Then, {⟨Π_{k}, ι_{k}⟩ : Log_{k} → (Log_{1} ⊛ Log_{2})}_{k∈{1,2}}, where each Π_{k} : (Voc_{1} × Voc_{2}) → Voc_{k} is the corresponding projection, is a sum in ⟨A, s(l)Cn⟩- Log.

Proof. Again, this result is a corollary of A.3.8, in Appendix. The construction is of course a simplification of the one in A.3.9, and uses 6.1.2 or 6.1.4. ⊓ ⊔

As with synchronization, we shall postpone the analysis of our working categories of logic system presentations until after the introduction of the more general mechanism of constrained fibring.

6.1.2 Constrained fibring

Constrained fibring, now, modifies the previous free construction by requiring that certain constructors of each of the signatures involved should be shared.

Definition 6.1.9. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems. A sharing constraint on csy_{1} and csy_{2} is a source S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} in Sig.

Such a source S essentially defines, for each n ∈ IN, the subset of Σ_{n}^{1} × Σ_{2}^{2} that contains all pairs ⟨s_{1,n}(c), s_{2,n}(c)⟩ for c ∈ Σ_{n}^{0}. In that case, the constructors s_{1,n}(c) ∈ Σ_{n}^{1} and s_{2,n}(c) ∈ Σ_{n}^{2} are said to be shared under S. The intuition is that then, in the language of the constrained synchronization, we shall not be able to distinguish between s_{1,n}(c)_{1} and s_{2,n}(c)_{2}.

Definition 6.1.10. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} a constraint. The fibring of csy_{1} and csy_{2} constrained by S is (csy_{1} ⊛^{S} csy_{2}) = ⟨(Σ^{1} ⊛^{S} Σ^{2}), ▷^{Ξ}⟩, where:

• (Σ^{1} ⊛^{S} Σ^{2}) is such that, for each n ∈ IN, (Σ^{1} ⊛^{S} Σ^{2})_{n} = (Σ^{1} ⊛ Σ^{2})_{n}/_{≈_{n}^{S}} with:

– ≈_{n}^{S} the least equivalence on (Σ^{1} ⊛ Σ^{2})_{n} with s_{1,n}(c)_{1} ≈_{n}^{S} s_{2,n}(c)_{2} for every c ∈ Σ_{n}^{0};

• E^{▷^{Ξ}} is the least superset of E that is closed for (R≈^{S}_{⊛})^{Ξ}, where:

R≈^{S}_{⊛} = ({⟨[Γ_{1}]_{≈^{S}}, [γ_{1}]_{≈^{S}}⟩ : Γ ▷_{1}^{Ξ} γ} ∪ {⟨[Δ_{2}]_{≈^{S}}, [δ_{2}]_{≈^{S}}⟩ : Δ ▷_{2}^{Ξ} δ}),

and ≈^{S} is the congruence on L_{(Σ^{1}⊛Σ^{2})}(Ξ) generated by the set of all equations

s_{1,n}(c)_{1}(θ_{1}, …, θ_{n}) = s_{2,n}(c)_{2}(θ_{1}, …, θ_{n})

with n ∈ IN, c ∈ Σ_{n}^{0} and θ_{1}, …, θ_{n} ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ).

Please note that {[ ]_{≈_{n}^{S}}}_{n∈IN} : (Σ^{1} ⊛ Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}) is a Sig-morphism, and that the (Σ^{1} ⊛ Σ^{2})-algebra L_{(Σ^{1}S_{⊛}Σ^{2})}(Ξ)|_{{[ ]_{≈^{S}_{n}} }_{n∈IN}} is isomorphic to the quotient algebra

L_{(Σ^{1}⊛Σ^{2})}(Ξ)/_{≈^{S}}. This isomorphism is made clearer in the next proposition. This is the reason why we can safely confuse^{1} [ ]_{≈^{S}} with { \atop \{[ ]_{≈_{n}^{S}}\}}_{n∈IN}.

Again, we can make the idea behind constrained fibring more explicit, by noting that (csy_{1} ⊛^{S} csy_{2}) corresponds (up to equivalence) to an enrichment of (csy_{1}⊛csy_{2}) with inference rules that force the sharing represented by S. The proof is quite similar to the one in 4.1.11.

Proposition 6.1.11. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} a constraint. Then, (csy_{1} ⊛^{S} csy_{2}) is equivalent to the structural consequence system csy_{S} = ⟨(Σ^{1} ⊛ Σ^{2}), ▷_{S}^{Ξ}⟩, with Θ^{▷_{S}^{Ξ}} the least superset of Θ that is closed for (RS_{⊛})^{Ξ}, where:

RS_{⊛} = (R_{⊛} ∪ {⟨{s_{1,n}(c)_{1}(ξ_{1}, …, ξ_{n})}, s_{2,n}(c)_{2}(ξ_{1}, …, ξ_{n})⟩ : n ∈ IN, c ∈ Σ_{n}^{0}}∪

{⟨{s_{2,n}(c)_{2}(ξ_{1}, …, ξ_{n})}, s_{1,n}(c)_{1}(ξ_{1}, …, ξ_{n})⟩ : n ∈ IN, c ∈ Σ_{n}^{0}}).

1To be precise, for θ ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ), while [θ]_{≈^{S}} denotes the set of all L_{(Σ^{1}⊛Σ^{2})}(Ξ) schema sentences that can be obtained from θ by replacing constructors by equivalent ones, { \atop \{[ ]_{≈_{n}^{S}}\}}_{n∈IN}(θ) represents the schema sentence on L_{(Σ^{1}\begin{matrix}S \\ ⊛ \end{matrix}Σ^{2})}(Ξ) obtained from θ by replacing every constructor by its equivalence class.

Proof. Let f : (Σ^{1} ⊛^{S} Σ^{2}) → (Σ^{1} ⊛ Σ^{2}) be any signature morphism such that, for every n ∈ IN, f_{n}([c]_{≈_{n}^{S}})≈_{n}^{S}c (if necessary, use the axiom of choice), and g : (Σ^{1} ⊛Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}) be such that, for every n ∈ IN, g(c) = [c]_{≈_{n}^{S}}. It is straigtforward that we can identify ĝ(θ) with [θ]_{≈^{S}}, for every θ ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ), and also [f̂(e)]_{≈^{S}} with e, for every e ∈ L_{(Σ^{1}S_{⊛}Σ^{2})}(Ξ). Assume also that csy = (csy_{1} ⊛^{S} csy_{2}) = ⟨(Σ^{1} ⊛^{S} Σ^{2}), ▷^{Ξ}⟩. We start by proving the following lemma:

if Ω_{(RS_{⊛})^{Ξ}}(Θ) ⊆ Θ and θ ∈ Θ then [θ]_{≈^{S}} ⊆ Θ.

Easily, the relation ≈ defined on L_{(Σ^{1}⊛Σ^{2})}(Ξ) by θ′ ≈ θ′′ if and only if both θ′, θ′′ ∈ Θ or both θ′, θ′′ ∉ Θ, is an equivalence. All we need to check is that ≈^{S}⊆≈. Clearly, it suffices to note that s_{1,n}(c)_{1}(θ_{1}, …, θ_{n}) ≈ s_{2,n}(c)_{2}(θ_{1}, …, θ_{n}), for every n ∈ IN, c ∈ Σ_{n}^{0} and θ_{1}, …, θ_{n} ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ), since Θ is closed for all instances of RS_{⊛}.

We first prove that f : csy → csy_{S} is an sCsy-morphism. For the purpose, assume that E ▷^{Ξ} e and Θ ∈ (RS_{⊛})^{Ξ}(f̂(E)). We now show that [Θ]_{≈^{S}} ∈ (R≈^{S}_{⊛})^{Ξ}(E). Easily, f̂(E) ⊆ Θ, and therefore we must have E ⊆ [Θ]_{≈^{S}}. Now, let Γ ▷_{1}^{Ξ} γ and σ ∈ Sub(Σ^{1} ⊛^{S} Σ^{2}), and suppose that ([Γ_{1}]_{≈^{S}})σ ⊆ [Θ]_{≈^{S}}. Trivially, we have indeed that [θ]_{≈^{S}}σ = [θσ′]_{≈^{S}} for every σ′ ∈ Sub(Σ^{1}⊛Σ^{2}) such that σ(ξ) = [σ′(ξ)]_{≈^{S}}. Given σ, such a σ′ can be obtained, if necessary by using again the axiom of choice. Thus we have, in fact, [Γ_{1}σ′]_{≈^{S}} ⊆ [Θ]_{≈^{S}}. Using the lemma above, we can conclude that also Γ_{1}σ′ ⊆ Θ, and therefore γ_{1}σ′ ∈ Θ. Analogously, if Δ ▷_{2}^{Ξ} δ and ([Δ_{2}]_{≈^{S}})σ ⊆ [Θ]_{≈^{S}}, we can show that δ_{2}σ′ ∈ Θ. In either case, ([γ_{1}]_{≈^{S}})σ = [γ_{1}σ′]_{≈^{S}}, and ([δ_{2}]_{≈^{S}})σ = [δ_{2}σ′]_{≈^{S}} are in [Θ]_{≈^{S}} and [Θ]_{≈^{S}} ∈ (R≈^{S}_{⊛})^{Ξ}(E). Thus, e ∈ [Θ]_{≈^{S}} and the lemma again implies that f̂(e) ∈ Θ. Therefore, f̂(E) ▷_{S}^{Ξ} f̂(e).

We now prove that g : csy_{S} → csy is also a morphism. For the purpose, assume that Θ▷_{S}^{Ξ}θ and E ∈ (R≈^{S}_{⊛})^{Ξ}([Θ]_{≈^{S}}). We show that (⋃_{e∈E} e) ∈ (RS_{⊛})^{Ξ}(Θ). Clearly, [Θ]_{≈^{S}} ⊆ E so also Θ ⊆ (⋃_{e∈E} e). Now, if Γ ▷_{1}^{Ξ} γ, σ′ ∈ Sub(Σ^{1}⊛Σ^{2}) and Γ_{1}σ′ ⊆ (⋃_{e∈E} e), we can consider σ such that σ(ξ) = [σ′(ξ)]_{≈^{S}}, and conclude that [Γ_{1}σ′]_{≈^{S}} = ([Γ_{1}]_{≈^{S}})σ ⊆ E. Therefore, [γ_{1}σ′]_{≈^{S}} = ([γ_{1}]_{≈^{S}})σ ∈ E and γ_{1}σ′ ∈ (⋃_{e∈E} e). Analogously, we can prove that (⋃_{e∈E} e) is closed for instances of ▷_{2}^{Ξ}. Let now n ∈ IN, c ∈ Σ_{n}^{0} and suppose that s_{1,n}(c)_{1}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e). Clearly, [s_{2,n}(c)_{2}(θ_{1}, …, θ_{n})]_{≈^{S}} = [s_{1,n}(c)_{1}(θ_{1}, …, θ_{n})]_{≈^{S}} ∈ E, and we conclude that also s_{2,n}(c)_{2}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e). Analogously, if s_{2,n}(c)_{2}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e) we can show that also s_{1,n}(c)_{1}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e). Thus, (⋃_{e∈E} e) ∈ (RS_{⊛})^{Ξ}(Θ) and θ ∈ (⋃_{e∈E} e). So, [θ]_{≈^{S}} ∈ E and [Θ]_{≈^{S}} ▷^{Ξ} [θ]_{≈^{S}}.

According to 5.1.59, we further need to show that e ≡_{csy} ĝ(f̂(e)) and θ ≡_{csy_{S}} f̂(ĝ(θ)), for every e ∈ L_{(Σ^{1}S_{⊛}Σ^{2})}(Ξ) and every θ ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ). Easily, e = ĝ(f̂(e)). On the other

hand, θ ≈^{S} f̂(ĝ(θ)) and the result follows from the lemma above. ⊓ ⊔

Let us now provide a categorial characterization for constrained fibring. It corresponds precisely to an Sgn_{sCsy}-pushout.

Proposition 6.1.12. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ for k ∈ {1, 2} be structural consequence systems, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} a constraint. Then, {ι_{k}^{S} : csy_{k} → (csy_{1} ⊛^{S} csy_{2})}_{k∈{1,2}}, with each ι_{k,n}^{S}(c) = [c_{k}]_{≈_{n}^{S}}, is an Sgn_{sCsy}-pushout of {csy_{k}}_{k∈{1,2}} along S.

Proof. Since Sig has all coequalizers and we have shown, in 5.1.26, that Sgn_{sCsy} is topological, we know in particular that Sgn_{sCsy} is a cofibration. Thus, using A.2.13 and 6.1.2, we just need to show that {[ ]_{≈_{n}^{S}}}_{n∈IN} : (csy_{1} ⊛ csy_{2}) → (csy_{1} ⊛^{S} csy_{2}) is an Sgn_{sCsy}- cocartesian lift of ⟨(csy_{1} ⊛ csy_{2}), {[ ]_{≈_{n}^{S}}}_{n∈IN}⟩, since {[ ]_{≈_{n}^{S}}}_{n∈IN} : (Σ^{1} ⊛ Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}) is a coequalizer in Sig of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}.

However, from topologicity, it follows that h : ⟨Σ, ▷^{Ξ}⟩ → ⟨Σ′, ▷^{′Ξ}⟩ is an Sgn_{sCsy}- cocartesian lift of ⟨⟨Σ, ▷^{Ξ}⟩, h : Σ → Σ′⟩ if and only if ▷^{′Ξ} = ⊓_{Σ′}{▷^{′′Ξ} : h : ⟨Σ, ▷^{Ξ}⟩ → ⟨Σ′, ▷^{′′Ξ}⟩ is an sCsy-morphism}. And according to 5.1.28 it easily follows that ▷^{′Ξ} is the consequence generated by the set of all instances of {⟨ĥ(Θ), ĥ(θ)⟩ : Θ ▷^{Ξ} θ}. ⊓ ⊔

Constrained fibring can be now extended to the layered setting.

Definition 6.1.13. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems. A sharing constraint on csy_{1} and csy_{2} is a source S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} in Sig.

Definition 6.1.14. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} a constraint. The fibring of csy_{1} and csy_{2} constrained by S is (csy_{1} ⊛^{S} csy_{2}) = ⟨(Σ^{1} ⊛^{S} Σ^{2}), ▷^{lΞ}, ▷^{gΞ}⟩, where ⟨(Σ^{1} ⊛^{S} Σ^{2}), ▷^{gΞ}⟩ = (csy_{1}^{g} ⊛^{S} csy_{2}^{g}), i.e., E^{▷^{gΞ}} is the least superset of E that is closed for (R_{g⊛}^{≈^{S}})^{Ξ}, where:

R_{g⊛}^{≈^{S}} = ({⟨[Γ_{1}]_{≈^{S}}, [γ_{1}]_{≈^{S}}⟩ : Γ ▷_{1}^{gΞ} φ} ∪ {⟨[Δ_{2}]_{≈^{S}}, [δ_{2}]_{≈^{S}}⟩ : Δ ▷_{2}^{gΞ} δ}),

and E^{▷^{lΞ}} is the least superset of E that is closed for ((R_{l⊛}^{≈^{S}})^{Ξ} ∪ {⟨∅, e⟩ : ▷^{gΞ}e}), where:

R_{l⊛}^{≈^{S}} = ({⟨[Γ_{1}]_{≈^{S}}, [γ_{1}]_{≈^{S}}⟩ : Γ ▷_{1}^{lΞ} γ} ∪ {⟨[Δ_{2}]_{≈^{S}}, [δ_{2}]_{≈^{S}}⟩ : Δ ▷_{2}^{lΞ} δ}).

Following a similar strategy to the one used in the non-layered case, we can better explain the intuition behind this definition by noting that, also in the layered case, (csy_{1} ⊛^{S} csy_{2}) corresponds (up to equivalence) to an enrichment of (csy_{1}⊛csy_{2}) with inference rules that force the sharing represented by S.

Proposition 6.1.15. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} a constraint. Then, (csy_{1} ⊛^{S} csy_{2}) is equivalent to the structural layered consequence system csy_{S} = ⟨(Σ^{1} ⊛ Σ^{2}), ▷_{S}^{lΞ}, ▷_{S}^{gΞ}⟩, with Θ^{▷_{S}^{gΞ}} the least superset of Θ that is closed for (R_{g⊛}^{S})^{Ξ}, where:

R_{g⊛}^{S} = (R_{g⊛} ∪ {⟨{s_{1,n}(c)_{1}(ξ_{1}, …, ξ_{n})}, s_{2,n}(c)_{2}(ξ_{1}, …, ξ_{n})⟩ : n ∈ IN, c ∈ Σ_{n}^{0}}∪
{⟨{s_{2,n}(c)_{2}(ξ_{1}, …, ξ_{n})}, s_{1,n}(c)_{1}(ξ_{1}, …, ξ_{n})⟩ : n ∈ IN, c ∈ Σ_{n}^{0}}),

and Θ^{▷_{S}^{lΞ}} the least superset of Θ that is closed for ((R_{l⊛}^{S})^{Ξ} ∪ {⟨∅, θ⟩ : ▷_{S}^{gΞ}θ}), where:

R_{l⊛}^{S} = (R_{l⊛} ∪ {⟨{s_{1,n}(c)_{1}(ξ_{1}, …, ξ_{n})}, s_{2,n}(c)_{2}(ξ_{1}, …, ξ_{n})⟩ : n ∈ IN, c ∈ Σ_{n}^{0}}∪
{⟨{s_{2,n}(c)_{2}(ξ_{1}, …, ξ_{n})}, s_{1,n}(c)_{1}(ξ_{1}, …, ξ_{n})⟩ : n ∈ IN, c ∈ Σ_{n}^{0}}).

Proof. Assume that csy = (csy_{1} ⊛^{S} csy_{2}) = ⟨(Σ^{1} ⊛^{S} Σ^{2}), ▷^{lΞ}, ▷^{gΞ}⟩, and let f : (Σ^{1} ⊛^{S} Σ^{2}) → (Σ^{1} ⊛ Σ^{2}) and g : (Σ^{1} ⊛ Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}) be defined as in 6.1.11, i.e., for every n ∈ IN and c ∈ (Σ^{1} ⊛ Σ^{2})_{n}, f_{n}([c]_{≈_{n}^{S}})≈_{n}^{S}c and g_{n}(c) = [c]_{≈_{n}^{S}}, and therefore also, ĝ(θ) = [θ]_{≈^{S}} and [f̂(e)]_{≈^{S}} = e, for every θ ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ) and e ∈ L_{(Σ^{1}S_{⊛}Σ^{2})}(Ξ). It immediately follows from definition 6.1.14 above that csy^{g} ≡ csy_{S}^{g} with f and g. That is, f : csy^{g} → csy_{S}^{g} and g : csy_{S}^{g} → csy^{g} are sCsy-morphisms such that e ≡_{csy}^{g} ĝ(f̂(e)) and θ ≡_{csy}_{S}^{g} f̂(ĝ(θ)).

According to 5.1.59, it remains to be proved that also f : csy^{l} → csy_{S}^{l} and g : csy_{S}^{l} → csy^{l} are sCsy-morphisms, and moreover e ≡_{csy}_{l} ĝ(f̂(e)) and θ ≡_{csy}l_{S} f̂(ĝ(θ)). The proof follows the sequence of steps already used in 6.1.11. We start by proving the following lemma:

if Ω_{(R_{l⊛}^{S})^{Ξ}}(Θ) ⊆ Θ and θ ∈ Θ then [θ]_{≈^{S}} ⊆ Θ.

Easily, the relation ≈ defined on L_{(Σ^{1}⊛Σ^{2})}(Ξ) by θ′ ≈ θ′′ if and only if both θ′, θ′′ ∈ Θ or both θ′, θ′′ ∉ Θ, is an equivalence. To check that ≈^{S}⊆≈, it suffices to note that s_{1,n}(c)_{1}(θ_{1}, …, θ_{n}) ≈ s_{2,n}(c)_{2}(θ_{1}, …, θ_{n}), for every n ∈ IN, c ∈ Σ_{n}^{0} and θ_{1}, …, θ_{n} ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ), since Θ is closed for all instances of R_{l⊛}^{S}.

Let us now show that f : csy^{l} → csy_{S}^{l} is indeed an sCsy-morphism. For the purpose, assume that E ▷^{lΞ} e and Θ ∈ ((R_{l⊛}^{S})^{Ξ} ∪ {⟨∅, θ⟩ : ▷_{S}^{gΞ}θ})(f̂(E)). We prove that [Θ]_{≈^{S}} ∈ ((R_{l⊛}^{≈^{S}})^{Ξ} ∪ {⟨∅, e′⟩ : ▷^{gΞ}e′})(E). Easily, f̂(E) ⊆ Θ, and therefore we must have E ⊆ [Θ]_{≈^{S}}. Now, let Γ ▷_{1}^{lΞ} γ and σ ∈ Sub(Σ^{1} ⊛^{S} Σ^{2}), and suppose that ([Γ_{1}]_{≈^{S}})σ ⊆ [Θ]_{≈^{S}}. As in 6.1.11, we can consider σ′ ∈ Sub(Σ^{1}⊛Σ^{2}) σ′ ∈ Sub(Σ^{1}⊛Σ^{2}) such that σ(ξ) = [σ′(ξ)]_{≈^{S}}, and have [Γ_{1}σ′]_{≈^{S}} ⊆ [Θ]_{≈^{S}}. Using the lemma above, we can conclude that also Γ_{1}σ′ ⊆ Θ, and therefore γ_{1}σ′ ∈ Θ. Analogously, if Δ ▷_{2}^{lΞ} δ and ([Δ_{2}]_{≈^{S}})σ ⊆ [Θ]_{≈^{S}}, we can show that δ_{2}σ′ ∈ Θ. In either case, ([γ_{1}]_{≈^{S}})σ = [γ_{1}σ′]_{≈^{S}}, and ([δ_{2}]_{≈^{S}})σ = [δ_{2}σ′]_{≈^{S}} are in [Θ]_{≈^{S}}. Let now ▷^{gΞ}e′. Since f : csy^{g} → csy_{S}^{g} is an sCsy-morphism we have ▷_{S}^{gΞ}f̂(e′). Thus, f̂(e′) ∈ Θ and hence e′ ∈ [Θ]_{≈^{S}}. Therefore, [Θ]_{≈^{S}} ∈ ((R_{l⊛}^{≈^{S}})^{Ξ} ∪ {⟨∅, e′⟩ : ▷^{gΞ}e′})(E). So, e ∈ [Θ]_{≈^{S}} and the lemma again implies that f̂(e) ∈ Θ. Therefore, f̂(E) ▷_{S}^{lΞ} f̂(e).

We now prove that g : csy_{S}^{l} → csy^{l} is also an sCsy-morphism. For the purpose, assume that Θ ▷_{S}^{lΞ} θ and E ∈ ((R_{l⊛}^{≈^{S}})^{Ξ} ∪ {⟨∅, e′⟩ : ▷^{gΞ}e′})([Θ]_{≈^{S}}). We show that (⋃_{e∈E} e) ∈ ((R_{l⊛}^{S})^{Ξ} ∪ {⟨∅, θ′⟩ : ▷_{S}^{gΞ}θ′})(Θ). Clearly, [Θ]_{≈^{S}} ⊆ E so also Θ ⊆ (⋃_{e∈E} e). Now, if Γ ▷_{1}^{lΞ} γ, σ′ ∈ Sub(Σ^{1}⊛Σ^{2}) and Γ_{1}σ′ ⊆ (⋃_{e∈E} e), we can consider σ such that σ(ξ) = [σ′(ξ)]_{≈^{S}}, and conclude that [Γ_{1}σ′]_{≈^{S}} = ([Γ_{1}]_{≈^{S}})σ ⊆ E. Therefore, [γ_{1}σ′]_{≈^{S}} = ([γ_{1}]_{≈^{S}})σ ∈ E and γ_{1}σ′ ∈ (⋃_{e∈E} e). Analogously, we can prove that (⋃_{e∈E} e) is closed for instances of ▷_{2}^{lΞ}. Let now n ∈ IN, c ∈ Σ_{n}^{0} and suppose that s_{1,n}(c)_{1}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e). Clearly, [s_{2,n}(c)_{2}(θ_{1}, …, θ_{n})]_{≈^{S}} = [s_{1,n}(c)_{1}(θ_{1}, …, θ_{n})]_{≈^{S}} ∈ E, and we conclude that also s_{2,n}(c)_{2}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e). Analogously, if s_{2,n}(c)_{2}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e) we can show that also s_{1,n}(c)_{1}(θ_{1}, …, θ_{n}) ∈ (⋃_{e∈E} e). Finally, assume that ▷_{S}^{gΞ}θ′. Since g : csy_{S}^{g} → csy^{g} is a morphism, we must have ▷^{gΞ}[θ′]_{≈^{S}}. Therefore, [θ′]_{≈^{S}} ∈ E and so θ′ ∈ (⋃_{e∈E} e). Thus, (⋃_{e∈E} e) ∈ ((R_{l⊛}^{S})^{Ξ} ∪ {⟨∅, θ′⟩ : ▷_{S}^{gΞ}θ′})(Θ) and θ ∈ (⋃_{e∈E} e). So, [θ]_{≈^{S}} ∈ E and [Θ]_{≈^{S}} ▷^{lΞ} [θ]_{≈^{S}}.

We further need to show that e ≡_{csy}_{l} ĝ(f̂(e)) and θ ≡_{csy}l_{S} f̂(ĝ(θ)), for every e ∈

L_{(Σ^{1}S_{⊛}Σ^{2})}(Ξ) and every θ ∈ L_{(Σ^{1}⊛Σ^{2})}(Ξ). Easily, e = ĝ(f̂(e)). On the other hand, θ ≈^{S} f̂(ĝ(θ)) and the result follows from the lemma above. ⊓ ⊔

As expected, constrained fibring of structural layered consequence systems corresponds precisely to an Sgn_{slCsy}-pushout.

Proposition 6.1.16. Let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{gΞ}⟩ for k ∈ {1, 2} be structural layered consequence systems, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} a constraint. Then, {ι_{k}^{S} : csy_{k} → (csy_{1} ⊛^{S} csy_{2})}_{k∈{1,2}}, with each ι_{k,n}^{S}(c) = [c_{k}]_{≈_{n}^{S}}, is an Sgn_{slCsy}-pushout of {csy_{k}}_{k∈{1,2}} along S.

Proof. Since Sig has all coequalizers and we have shown, in 5.1.47, that Sgn_{slCsy} is topological, we know in particular that Sgn_{slCsy} is a cofibration. Thus, using A.2.13 and 6.1.4, we just need to show that {[ ]_{≈_{n}^{S}}}_{n∈IN} : (csy_{1} ⊛ csy_{2}) → (csy_{1} ⊛^{S} csy_{2}) is an Sgn_{slCsy}- cocartesian lift of ⟨(csy_{1} ⊛ csy_{2}), {[ ]_{≈_{n}^{S}}}_{n∈IN}⟩, since {[ ]_{≈_{n}^{S}}}_{n∈IN} : (Σ^{1} ⊛ Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}) is a coequalizer in Sig of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}.

However, from topologicity, it follows that h : ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩ → ⟨Σ′, ▷^{′lΞ}, ▷^{′gΞ}⟩ is an Sgn_{slCsy}-cocartesian lift of ⟨⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩, h : Σ → Σ′⟩ if ⟨▷^{′lΞ}, ▷^{′gΞ}⟩ = ⊓_{Σ′}{⟨▷^{′′lΞ}, ▷^{′′gΞ}⟩ : h : ⟨Σ, ▷^{lΞ}, ▷^{gΞ}⟩ → ⟨Σ′, ▷^{′′lΞ}, ▷^{′′gΞ}⟩ is an slCsy-morphism}. And according to 5.1.49 it easily follows that ▷^{′gΞ} is the consequence generated by all instances of {⟨ĥ(Θ), ĥ(θ)⟩ : Θ ▷^{gΞ} θ}, and ▷^{′lΞ} is the consequence generated by all instances of ({⟨ĥ(Θ), ĥ(θ)⟩ : Θ ▷^{lΞ} θ} and {⟨∅, θ⟩ : ▷^{′gΞ}θ}). ⊓ ⊔

It must be obvious that, in either case, free fibring can be obtained as a particular case, when the signature Σ^{0} that guides the sharing is empty.

Constrained fibring in an arbitrary category A of (layered) logic system presentations again exploits the constructions for structural (layered) consequence system. Signature oriented pushouts must exist in A and correspond to pushouts of the given languages. Obviously, the construction is precise if also the structural consequence operation of a signature oriented pushout is a signature oriented pushout of the given structural consequence operations.

Definition 6.1.17. A category of (layered) logic system presentations ⟨A, s(l)Cn⟩ is suitable for fibring provided that all Sgn_{A}-pushouts exist and are transformed by Sgn_{A} into pushouts in Sig. Moreover, ⟨A, s(l)Cn⟩ is exact for fibring if, in addition, also s(l)Cn transforms Sgn_{A}-pushouts into Sgn_{s(l)Csy}-pushouts.

Once more, since Sig has an initial object (the empty signature), the conditions above easily imply, respectively, suitability and exactness for free fibring.

Definition 6.1.18. Let ⟨A, s(l)Cn⟩ be a category of (layered) logic system presentations suitable for fibring, a_{1}, a_{2} ∈ |A| and S a sharing constraint on s(l)Cn(a_{1}) and s(l)Cn(a_{2}). Then, given any Sgn_{A}-pushout {h_{k} : a_{k} → (a_{1} ⊛^{S} a_{2})}_{k∈{1,2}} of a_{1} and a_{2} along S, (a_{1} ⊛^{S} a_{2}) is said to be a fibring of a_{1} and a_{2} constrained by S.

Please note again that Sgn_{A}(a_{1} ⊛^{S} a_{2}) may not coincide with (Sgn_{A}(a_{1}) ⊛^{S} Sgn_{A}(a_{2})), as defined in 6.1.10 (6.1.14) with pseudo-injections ι_{k}^{S} as in 6.1.12 (6.1.16). However, the definition of suitability for fibring implies that the unique signature morphism h : (Sgn_{A}(a_{1}) ⊛^{S} Sgn_{A}(a_{2})) → Sgn_{A}(a_{1} ⊛^{S} a_{2}) such that (h ∘ ι_{1}^{S}) = h_{1} and (h ∘ ι_{2}^{S}) = h_{2} is necessarily a signature isomorphism. Moreover, from the underlying universal property, it follows that h : (s(l)Cn(a_{1}) ⊛^{S} s(l)Cn(a_{2})) → s(l)Cn(a_{1} ⊛^{S} a_{2}) is an s(l)Csy morphism. Exactness, of course, further implies that this morphism is indeed an s(l)Csy isomorphism.

Finally, let us consider constrained fibrings of (layered) logic presentations. Of course, we capitalize on the constructions at the level of (layered) structural logic systems. As with synchronization, we shall first upgrade our previous notion of sharing constraint.

Definition 6.1.19. Let Log_{k} for k ∈ {1, 2} be (layered) ⟨A, s(l)Cn⟩-based logic presentations over Voc_{k}, for some category ⟨A, s(l)Cn⟩ of (layered) logic system presentations suitable for fibring. A sharing constraint on Log_{1} and Log_{2} is a source S = {⟨S_{k}, s_{k}⟩ : (Sgn_{0} : Voc_{0} → Sig) → Sgn_{Log_{k}}}_{k∈{1,2}} in Funct(Sig).

Please note that such a source S essentially defines a restriction to all pairs of vocabularies, by selecting just those pairs ⟨V_{1}, V_{2}⟩ such that S_{1}(V_{1}) = S_{2}(V_{2}) = V_{0}. Given such a pair, clearly, S(V_{1}, V_{2}) = {s_{k}(V_{k}) : Sgn_{0}(V_{0}) → Sgn_{Log_{k}}(V_{k})}_{k∈{1,2}} is a constraint on Log_{1}(V_{1}) and Log_{2}(V_{2}). Moreover, it also defines a restriction to all pairs of Vocmorphisms allowed between such vocabularies, by selecting just those pairs ⟨h_{1} : V_{1} → V′_{1}, h_{2} : V_{2} → V′_{2}⟩ such that S_{1}(h_{1}) = S_{2}(h_{2}) = h_{0} : V_{0} → V′_{0}. The conditions satisfied by functors and natural transformations guarantee that the following coherence property holds for k ∈ {1, 2}: (Sgn_{Log_{k}}(h_{k}) ∘ s_{k}(V_{k})) = (s_{k}(V′_{k}) ∘ Sgn_{0}(h_{0})). The same is to say that pairs of shared constructors are preserved across vocabulary translations.

Definition 6.1.20. Let ⟨A, s(l)Cn⟩ be a category of structural (layered) logic systems suitable for fibring, Log_{k} be ⟨A, s(l)Cn⟩-based (layered) logic presentations over Voc_{k} for k ∈ {1, 2}, and S = {⟨S_{k}, s_{k}⟩ : (Sgn_{0} : Voc_{0} → Sig) → Sgn_{Log_{k}}}_{k∈{1,2}} a constraint. The fibring of Log_{1} and Log_{2} constrained by S is (Log_{1} ⊛^{S} Log_{2}) over Voc where:

• Voc is the subcategory of (Voc_{1} × Voc_{2}) such that:

– objects ⟨V_{1}, V_{2}⟩ must satisfy S_{1}(V_{1}) = S_{2}(V_{2});

– morphisms ⟨h_{1}, h_{2}⟩ : ⟨V_{1}, V_{2}⟩ → ⟨V′_{1}, V′_{2}⟩ must satisfy S_{1}(h_{1}) = S_{2}(h_{2});

• (Log_{1} ⊛^{S} Log_{2}) : Voc → A is such that:

– (Log_{1} ⊛^{S} Log_{2})(⟨V_{1}, V_{2}⟩) = (Log_{1}(V_{1}) ⊛^{S(V_{1},V_{2})} Log_{2}(V_{2}));

– (Log_{1} ⊛^{S} Log_{2})(⟨h_{1} : V_{1} → V′_{1}, h_{2} : V_{2} → V′_{2}⟩) is the unique A-morphism such

that ((Log_{1} ⊛^{S} Log_{2})(⟨h_{1}, h_{2}⟩)∘ι_{k}^{S}(V_{1}, V_{2})) = (ι_{k}^{S}(V′_{1}, V′_{2})∘Log(h_{k})) for k ∈ {1, 2},

with ι_{k}^{S}(X_{1}, X_{2}) : Log_{k}(X_{k}) → (Log_{1}(X_{1}) ⊛^{S(X_{1},X_{2})} Log_{2}(X_{2})) the corresponding Sgn_{A}-pushout morphisms.

As expected we have the following result.

Proposition 6.1.21. Let Log_{k} for k ∈ {1, 2} be ⟨A, s(l)Cn⟩-based (layered) logic presentations over Voc_{k}, for some category ⟨A, s(l)Cn⟩ of structural (layered) logic systems suitable for fibring, and S = {⟨S_{k}, s_{k}⟩ : (Sgn_{0} : Voc_{0} → Sig) → Sgn_{Log_{k}}}_{k∈{1,2}} a constraint. Then, {⟨Π_{k}^{S}, ι_{k}^{S}⟩ : Log_{k} → (Log_{1} ⊛^{S} Log_{2})}_{k∈{1,2}}, where each Π_{k}^{S} is the restriction to Voc of the projection Π_{k} : (Voc_{1} × Voc_{2}) → Voc_{k}, is a Funct(Sgn_{A})-pushout of Log_{1} and Log_{2} along S.

Proof. This result is a corollary of A.3.9, in Appendix, and uses 6.1.12 or 6.1.16. ⊓ ⊔

6.2 Applying the mechanism

For now, let us take a look at whether our working model-theoretic category of structural layered logic system presentations is suitable or even exact for fibring. Again, as we shall see, Int is only suitable while Dsy is exact. On the way, we shall also provide a characterization of fibring of interpretation systems.

6.2.1 Interpretation systems

Our model-theoretic category of layered logic system presentations Int is suitable, although not exact, for fibring. Below, we shall analyze fibring of interpretation systems in detail. The simplicity of the construction, namely if compared to that in [SSC99], can be explained by the assumption of implicit closure conditions on the models associated to interpretation systems. Indeed, the intuitions given by Gabbay [Gab96a, Gab99] about fibred semantics can be interpreted as defining a fibred model to be a matching collection of models of each of the two systems being fibred. As first shown in [ZSS99], these conditions can be achieved by assuming that our working interpretation structures satisfy a few closure conditions. Indeed, there, a primary notion of pre-interpretation system was used and shown to generate a closed interpretation system in a natural way, without changing its entailment relations. This simple but meaningful idea opened the way to this simplification of the definition in [SSC99], where these closure conditions were not required a priori but were hidden in the (quite complex) notion of morphism considered. The closure conditions we shall consider herein are a world-free purely algebraic version of those considered in [ZSS99].

Proposition 6.2.1. Int is suitable, but not exact for fibring. Namely, the fibring of interpretation systems int_{k} = ⟨Σ^{k}, M_{k}, S_{k}⟩, for k ∈ {1, 2}, constrained by S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} is given by (int_{1} ⊛^{S} int_{2}) = ⟨(Σ^{1} ⊛^{S} Σ^{2}), M, S⟩ where:

• M is the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ (M_{1} × M_{2}) such that T_{m_{1}} = T_{m_{2}} = T_{⟨m_{1},m_{2}⟩}, ⋉_{m_{1}} = ⋉_{m_{2}} = ⋉_{⟨m_{1},m_{2}⟩} and, for all c ∈ Σ_{n}^{0}:

– \underline{s_{1,n}(c)}_{n}^{T_{m_{1}}} = \underline{s_{2,n}(c)}_{n}^{T_{m_{2}}} ;

• S(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, ⋉_{⟨m_{1},m_{2}⟩}⟩ where, for every e ∈ Σ_{n}:

– \underline{e}_{n}^{T_{⟨m_{1},m_{2}⟩}} = \underline{c}_{n}^{T_{m_{k}}} for every k ∈ \{1, 2\} and c_{k} ∈ e.

Proof. Since Sig has all coequalizers and we have shown, in 5.2.28, that Sgn_{Int} is a cofibration, we can use A.2.13 and conclude that all Sgn_{Int}-pushouts exist, once we show that all binary sums exist in Int.

For the purpose, let (int_{1} ⊛ int_{2}) = ⟨(Σ^{1} ⊛ Σ^{2}), M′, S′⟩ be such that:

• M′ is the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ (M_{1} × M_{2}) such that T_{m_{1}} = T_{m_{2}} = T_{⟨m_{1},m_{2}⟩} and ⋉_{m_{1}} = ⋉_{m_{2}} = ⋉_{⟨m_{1},m_{2}⟩};

• S′(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, ⋉_{⟨m_{1},m_{2}⟩}⟩ where, for every k ∈ {1, 2} and c ∈ Σ_{n}^{k}:

– \underline{c_{k}}_{n}^{T_{⟨m_{1},m_{2}⟩}} = \underline{c}_{n}^{T_{m_{k}}}.

We are going to show that {h_{k} : int_{k} → (int_{1} ⊛ int_{2})}_{k∈{1,2}} is a sum in Int, where each \overline{h}_{k} = ι_{k} : Σ^{k} → (Σ^{1} ⊛ Σ^{2}) is the corresponding injection morphism, and each \underline{h}_{k} : M′ → M_{k} is the corresponding restriction of the projection π_{k} : (M_{1} × M_{2}) → M_{k}. Trivially, both are Int-morphisms. Let now {g_{k} : int_{k} → int′′}_{k∈{1,2}} be morphisms in Int, with int′′ = ⟨Σ′′, M′′, S′′⟩. Obviously, \overline{f}_{n}(c_{k}) = \overline{g}_{k,n}(c) defines the unique Sigmorphism \overline{f} : (Σ^{1} ⊛ Σ^{2}) → Σ′′ such that (\overline{f} ∘ ι_{k}) = \overline{g}_{k} for k ∈ \{1, 2\}. Moreover, for every m′′ ∈ M′′, trivially T_{\underline{g}_{1}(m′′)} = T_{m′′} = T_{\underline{g}_{2}(m′′)} and ⋉_{\underline{g}_{1}(m′′)} = ⋉_{m′′} = ⋉_{\underline{g}_{2}(m′′)} and we conclude that ⟨\underline{g}_{1}(m′′), \underline{g}_{2}(m′′)⟩ ∈ M′. Therefore, \underline{f}(m′′) = ⟨\underline{g}_{1}(m′′), \underline{g}_{2}(m′′)⟩ defines the unique map \underline{f} : M′′ → M′ such that (\underline{h}_{k} ∘ \underline{f}) = \underline{g}_{k} for k ∈ \{1, 2\}. Now, we just need to prove that f : (int_{1} ⊛ int_{2}) → int′′ establishes an Int-morphism. In fact, for m′′ ∈ M′′,

k ∈ \{1, 2\} and c ∈ Σ_{n}^{k}, we have that \underline{c_{k}}_{n}^{T_{\underline{f}(m′′)}} = \underline{c}_{n}^{T_{\underline{g}_{k}(m′′)}} = \underline{\overline{g}_{k,n}(c)}_{n}^{T_{m′′}} = \underline{\overline{f}_{n}(c_{k})}_{n}^{T_{m′′}}.

Thus, since {[ ]_{≈_{n}^{S}}}_{n∈IN} : (Σ^{1} ⊛ Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}) is a coequalizer in Sig of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}, we obtain (int_{1} ⊛^{S} int_{2}) by Sgn_{Int}-cocartesian lift of ⟨(int_{1}⊛int_{2}), {[ ]_{≈_{n}^{S}}}_{n∈IN}⟩. Furthermore, we have also proved in 5.2.27 that Sgn_{Int} is coadjoint. Thus, we can use A.3.3 to conclude that Int is suitable for fibring.

As expected, exactness fails because Ent_{Int} does not preserve sums. Let c ∈ Σ_{1}^{1} and assume that {ξ}⊨_{i̸nt_{1}}^{lΞ}c(ξ). However, suppose that for every model m_{1} ∈ M_{1} for which there exists an assignment α ∈ Asg(T_{m_{1}}) such that \{α(ξ)\}⋉_{m̸_{1}} \underline{c}_{1}^{T_{m_{1}}} (α(ξ)) there is no model m_{2} ∈ M_{2} with T_{m_{1}} = T_{m_{2}} and ⋉_{m_{1}} = ⋉_{m_{2}}. Then, clearly, {ξ}⊨_{(int_{1}⊛int_{2})}^{lΞ}c_{1}(ξ). A similar example could be obtained for global entailment. ⊓ ⊔

Obviously, (int_{1} ⊛ int_{2}) as defined in the proof above is a free fibring of the interpretation systems int_{1} and int_{2}.

As we have already mentioned, in general, the construction presented here lacks the power to gather several models of each of the given systems into a unique fibred model. Thus, one could argue about how legitimate it is to call it a “fibring”. In fact, in [SSC99], although with a very particular case of this algebraic semantics, this problem was overcome categorially by enriching (and complicating) the notion of morphism. In [ZSS99], later on, now with a less particular case of our semantics, the previous solution was much simplified by just requiring that certain closure conditions should hold on the class of structures associated to models. This seems to be not only the most simple but also the most elegant way to deal with the problem of fibred semantics. In particular, it was recognized that one should require closures for isomorphisms and unions. Closure for substructures was also considered but does not play an essential role in the construction. Systems without those properties were, there, referred to as just pre-interpretation systems. In our case, thus, in order to justify our legitimate use of the word “fibring”, we shall require closures for isomorphism and product of interpretation structures. A condition matching the closure for substructures shall not be considered herein, but we shall comment on what such a condition might look like in the concluding Chapter.

To start with, let us set up the obvious category of Σ-structures, for any signature Σ, that we shall consider fixed. We can endow Str(Σ) with a notion of morphism that extends homomorphisms of Σ-algebras by further requiring that it should establish a Csymorphism between the closure operations internal to the structures.

Definition 6.2.2. Let S_{k} = ⟨T_{k}, ⋉_{k}⟩ for k ∈ {1, 2} be Σ-structures. A Σ-structures morphism h : S_{1} → S_{2} is an Alg(Σ)-morphism h : T_{1} → T_{2} such that h : ⟨T_{1}, ⋉_{1}⟩ → ⟨T_{2}, ⋉_{2}⟩ is a Csy-morphism.

Clearly, h : T_{1} → T_{2} is a function such that, for every n ∈ IN, c ∈ Σ_{n} and x_{1}, …, x_{n} ∈ T_{1}, we must have h(\underline{c}_{n}^{T_{1}}(x_{1}, …, x_{n})) = \underline{c}_{n}^{T_{2}}(h(x_{1}) …, h(x_{n})). Moreover, for every X ⊆ T_{1}, h(X^{⋉_{1}}) ⊆ h(X)^{⋉_{2}}.

Definition 6.2.3. Σ-structures and their morphisms constitute the category Str(Σ), with identities and composition as in Alg(Σ) (and Csy).

Proposition 6.2.4. Str(Σ) is indeed a category, for every signature Σ.

We start by characterizing isomorphisms in Str(Σ).

Proposition 6.2.5. Two Σ-structures S_{k} = ⟨T_{k}, ⋉_{k}⟩ for k ∈ {1, 2} are isomorphic if and only if there exists an Alg(Σ)-isomorphism h : T_{1} → T_{2} such that, for every X ⊆ T_{1}:

• h(X^{⋉_{1}}) = h(X)^{⋉_{2}}.

Proof. Immediate from 2.1.35, by noting that Alg(Σ)-isomorphisms are precisely the bijective Alg(Σ)-morphisms. ⊓ ⊔

Let us now move to products.

Proposition 6.2.6. Let K be a set and S_{k} = ⟨T_{k}, ⋉_{k}⟩ for k ∈ K be Σ-structures. Then, the Σ-structure (∏_{k∈K} S_{k}) = ⟨T , ⋉⟩ defined by:

• T = (∏_{k∈K} T_{k}), that is:

– T = (∏_{k∈K} T_{k});

– \underline{c}_{n}^{T}(x_{1}, …, x_{n}) = ⟨\underline{c}_{n}^{T_{k}}(π_{k}(x_{1}), …, π_{k}(x_{n}))⟩_{k∈K}, for every n ∈ IN, c ∈ Σ_{n} and x_{1}, …, x_{n} ∈ T;

• X^{⋉} = (∏_{k∈K} π_{k}(X)^{⋉_{k}}), for every X ⊆ T,

together with the corresponding projections π_{k} : T → T_{k} is a product in Str(Σ).

Proof. It is well known that {π_{k} : T → T_{k}}_{k∈K} is a product in Alg(Σ). Thus, we just need to show that ⟨T, ⋉⟩ is indeed a consequence system, and that given any possible Str(Σ)- source {h_{k} : S′ → S_{k}}_{k∈K}, with S′ = ⟨T′, ⋉′⟩, the unique Alg(Σ)-morphism h : T′ → T such that h_{k} = (π_{k}∘h) for all k ∈ K also establishes a Csy-morphism h : ⟨T′, ⋉′⟩ → ⟨T, ⋉⟩.

Let X ⊆ T. Obviously, X ⊆ (∏_{k∈K} π_{k}(X)). Therefore, the extensiveness of each ⋉_{k} easily implies that X ⊆ X^{⋉}. With respect to monotonicity, assume that X ⊆ Y ⊆ T. Obviously, π_{k}(X) ⊆ π_{k}(Y) and X^{⋉} ⊆ Y^{⋉} by just using the monotonicity of each ⋉_{k}. Assume now that y ∈ (X^{⋉})^{⋉}, i.e., y ∈ (∏_{k∈K} π_{k}(X^{⋉})^{⋉_{k}}). Obviously, this means that π_{j}(y) ∈ π_{j}(X^{⋉})^{⋉_{j}}, for every j ∈ K. That is, for every j ∈ K, π_{j}(y) ∈ π_{j}(∏_{k∈K} π_{k}(X)^{⋉_{k}})^{⋉_{j}}. But clearly π_{j}(∏_{k∈K} π_{k}(X)^{⋉_{k}}) ⊆ π_{j}(X)^{⋉_{j}}, and we can use the monotonicity and idempotence of each ⋉_{j} to conclude that each π_{j}(y) ∈ π_{j}(X)^{⋉_{j}}, and hence y ∈ X^{⋉}.

Assume now that h : T′ → T is the unique Alg(Σ)-morphism such that h_{k} = (π_{k}∘h) for all k ∈ K. Obviously, we must have h(z) = ⟨h_{k}(z)⟩_{k∈K} for every z ∈ T′. Let us now show that h : ⟨T′, ⋉′⟩ → ⟨T, ⋉⟩ is a Csy-morphism. For the purpose, let Z ⊆ T′ and assume that z ∈ Z^{⋉′}. Since each h_{k} is a morphism, we also have that h_{k}(z) ∈ h_{k}(Z)^{⋉_{k}}. Therefore, we have that h(z) = ⟨h_{k}(z)⟩_{k∈K} ∈ (∏_{k∈K} h_{k}(Z)^{⋉_{k}}) = (∏_{k∈K} π_{k}(h(Z))^{⋉_{k}}) = h(Z)^{⋉}. ⊓ ⊔

We can now define what we mean by a closed interpretation system.

Definition 6.2.7. An interpretation system int = ⟨Σ, M, S⟩ is said to be closed for:

isomorphisms if for every m ∈ M and every structure S′ isomorphic to S(m), there exists m′ ∈ M such that S(m′) = S′;

products if for every set-indexed family {m_{k}}_{k∈K} of models in M, there exists m′ ∈ M such that S(m′) = (∏_{k∈K} S(m_{k})).

We further say that int is closed if it is closed for isomorphisms and products.

In fact, this property is too much to expect in general of a given interpretation system. However, this slight drawbak can be easily overcome by providing a mechanism that picks up any interpretation system and gives as result a closed interpretation system, but leaves the original logical properties untouched. That is, the local and global structural entailment operations should be the same for the original system and its closure.

Definition 6.2.8. Let int = ⟨Σ, M, S⟩ be an interpretation system. The closure of int is the interpretation system \overline{int} = ⟨Σ, \overline{M}, \overline{S}⟩ where:

• \overline{M} is the class of all tuples m = ⟨\{m_{k}\}_{k∈K}, b : (∏_{k∈K} T_{m_{k}}) → T ⟩ such that:

– {m_{k}}_{k∈K} is a set-indexed family of models in M;

– b : (∏_{k∈K} T_{m_{k}}) → T is a bijection;

• \overline{S}(m) = ⟨T , ⋉⟩ with:

– \underline{c}_{n}^{T}(b(x_{1}), …, b(x_{n})) = b(⟨\underline{c}_{n}^{T_{m_{k}}} (π_{k}(x_{1}), …, π_{k}(x_{n}))⟩_{k∈K}), for every n ∈ IN, c ∈ Σ_{n} and x_{1}, …, x_{n} ∈ T′;

– b(X)^{⋉} = b((∏_{k∈K} π_{k}(X)^{⋉_{k}})), for every X ⊆ (∏_{k∈K} T_{m_{k}}).

It is a trivial fact that b : (∏_{k∈K} S(m_{k})) → \overline{S}(m) is an Str(Σ)-isomorphism.

Proposition 6.2.9. The closure \overline{int} of any interpretation system int is a closed interpretation system.

Proof. Let int = ⟨Σ, M, S⟩. By definition, \overline{int} = ⟨Σ, \overline{M}, \overline{S}⟩ is clearly also an interpretation system. We shall now prove that \overline{int} is closed for:

isomorphisms if m = ⟨\{m_{k}\}_{k∈K}, b : (∏_{k∈K} T_{m_{k}}) → T ⟩ ∈ \overline{M} and h : \overline{S}(m) → S′ is an Str(Σ)- isomorphism, with S′ = ⟨T′, ⋉′⟩, then m′ = ⟨{m_{k}}_{k∈K}, (h ∘ b) : (∏_{k∈K} T_{m_{k}}) → T′⟩ ∈

\overline{M} and obviously \overline{S}(m′) = S′;

products if J is a set and {m_{j} = ⟨{m_{k_{j}}}_{k_{j}∈K_{j}}, b_{j} : (∏_{k_{j}∈K_{j}} T_{m_{k_{j}}}) → T_{j}⟩}_{j∈J} is a family of models m_{j} ∈ \overline{M}, then m = ⟨\{m_{k}\}_{k∈( ∑_{j∈J} K_{j})}, ⟨b_{j}⟩_{j∈J} : (∏_{k∈( ∑_{j∈J} K_{j})} T_{m_{k_{j}}}) →

(∏_{j∈J} T_{j})⟩ ∈ \overline{M}, and clearly \overline{S}(m) = (∏_{j∈J} \overline{S}(m_{j})).

We conclude that \overline{int} is indeed closed. ⊓ ⊔

It is not difficult to extend this definition to an endofunctor on Int.

Definition 6.2.10. The maps

• Cl(int) = \overline{int};

• Cl(h : ⟨Σ^{1}, M_{1}, S_{1}⟩ → ⟨Σ^{2}, M_{2}, S_{2}⟩) : ⟨Σ^{1}, \overline{M}_{1}, \overline{S}_{1}⟩ → ⟨Σ^{2}, \overline{M}_{2}, \overline{S}_{2}⟩ such that the signature morphism \overline{Cl(h)} = \overline{h} : Σ^{1} → Σ^{2} and the map \underline{Cl(h)} : \overline{M}_{2} → \overline{M}_{1} is defined, for every model m = ⟨\{m_{k}\}_{k∈K}, b : (∏_{k∈K} T_{m_{k}}) → T ⟩ ∈ \overline{M}_{2}, by:

– \underline{Cl(h)}(m) = ⟨\{\underline{h}(m_{k})\}_{k∈K}, b : (∏_{k∈K} T_{\underline{h}(m_{k})}) → T ⟩,

constitute the functor Cl : Int → Int.

Proposition 6.2.11. Cl : Int → Int is indeed a functor.

Proof. We just prove that Cl(h) : \overline{int}_{1} → \overline{int}_{2} is indeed an Int-morphism whenever h : int_{1} → int_{2} is. For the purpose, assume that int_{1} = ⟨Σ^{1}, M_{1}, S_{1}⟩ and int_{2} = ⟨Σ^{2}, M_{2}, S_{2}⟩ and let m = ⟨\{m_{k}\}_{k∈K}, b : (∏_{k∈K} T_{m_{k}}) → T ⟩ ∈ \overline{M}_{2}. Clearly, \underline{Cl(h)} : \overline{M}_{2} → \overline{M}_{1} is well defined. In fact, since S_{1}(\underline{h}(m_{k})) = S_{2}(m_{k})|\overline{h} for every k ∈ K, it follows that (∏_{k∈K} S_{1}(\underline{h}(m_{k}))) = (∏_{k∈K} S_{2}(m_{k}))|\overline{h}. To see that Cl(h) is an Int-morphism, now, just note that b : (∏_{k∈K} S_{2}(m_{k})) → \overline{S}_{2}(m) is an Str(Σ^{2})-isomorphism, and thus b : (∏_{k∈K} S_{2}(m_{k}))|\overline{h} → \overline{S}_{2}(m)|\overline{h} is an Str(Σ^{1})-isomorphism and hence \overline{S}_{1}(\underline{Cl(h)}(m)) = \overline{S}_{2}(m)|\overline{h}. ⊓ ⊔

Now our subject matter is to show that this closure mechanism for interpretation systems leaves their structural entailments untouched. We shall first prove a useful result.

Proposition 6.2.12. Let int = ⟨Σ, M, S⟩ be an interpretation system. Then, h : \overline{int} → int, where \overline{h} = id_{Σ} and \underline{h} : M → \overline{M} is defined by \underline{h}(m) = ⟨\{m\}, id_{T_{m}} : T_{m} → T_{m}⟩ for every m ∈ M, is an Int-morphism.

Proof. Trivially, \underline{h} is well defined, since every Σ-structure can be seen as a singular product. Moreover, immediately, \overline{S}(\underline{h}(m)) = S(m) = S(m)|_{id_{Σ}}. ⊓ ⊔

Although we do not dwell on this subject here, please note that the dual functor Cl^{op} : Int^{op} → Int^{op} gives rise to a monad [AHS90], whose set-up uses the result we have just proved.

Proposition 6.2.13. Both local and global structural entailment are invariant under Cl, that is, (Ent_{Int} ∘ Cl) = Ent_{Int}.

Proof. With respect to morphisms, since \overline{Cl(h)} = \overline{h}, this is a trivial result. We are thus left with proving that Ent_{Int}(\overline{int}) = Ent_{Int}(int), for every interpretation system int = ⟨Σ, M, S⟩. For the purpose, let Ent_{Int}(int) = ⟨Σ, ⊨^{lΞ}, ⊨^{gΞ}⟩ and Ent_{Int}(\overline{int}) = ⟨Σ, \overline{⊨}^{lΞ}, \overline{⊨}^{gΞ}⟩.

Using 6.2.12, we can conclude that id_{Σ} : ⟨Σ, \overline{⊨}^{lΞ}, \overline{⊨}^{gΞ}⟩ → ⟨Σ, ⊨^{lΞ}, ⊨^{gΞ}⟩ is an slCsymorphism. In other words, \overline{⊨}^{lΞ} ⊑_{Σ} ⊨^{lΞ} and \overline{⊨}^{gΞ} ⊑_{Σ} ⊨^{gΞ}.

Assume now that Γ⊨^{lΞ}δ and let m = ⟨\{m_{k}\}_{k∈K}, b : (∏_{k∈K} T_{m_{k}}) → T ⟩ ∈ \overline{M} and α ∈ Asg(T_{m}). Clearly, for each k ∈ K, (π_{k} ∘ b^{−1} ∘ α) ∈ Asg(T_{m_{k}}). Thus, since Γ⊨^{lΞ}δ, we necessarily have

{[[γ]]_{(π_{k}∘b^{−1}∘α)}^{T_{m_{k}}} : γ ∈ Γ} ⋉_{m_{k}} [[δ]]_{(π_{k}∘b^{−1}∘α)}^{T_{m_{k}}}.

Therefore, by definition of product, letting (∏_{k∈K} S(m_{k})) = ⟨(∏_{k∈K} T_{m_{k}}), ⋉⟩, we also have

{[[γ]]_{(b^{−1}∘α)}^{( ∏_{k∈K}T_{m_{k}})} : γ ∈ Γ} ⋉ [[δ]]_{(b^{−1}∘α)}^{( ∏_{k∈K}T_{m_{k}})}.

Now, because b : (∏_{k∈K} S(m_{k})) → \overline{S}(m) is an isomorphism, we finally have that

{[[γ]]_{α}^{T_{m}} : γ ∈ Γ} ⋉_{m} [[δ]]_{α}^{T_{m}}.

We conclude that Γ\overline{⊨}^{lΞ}δ, and therefore \overline{⊨}^{lΞ} = ⊨^{lΞ}.

With respect to global entailment, assume that Γ⊨^{gΞ}δ and suppose that, for every γ ∈ Γ, we have ⋉_{m}[[γ]]_{α}^{T_{m}}. Since b : (∏_{k∈K} S(m_{k})) → \overline{S}(m) is an isomorphism, then, we

also have ⋉[[γ]]_{(b^{−1}∘α)}^{( ∏_{k∈K}T_{m_{k}})}. So, for every γ ∈ Γ and k ∈ K, we have ⋉_{m_{k}}[[γ]]_{(π_{k}∘b^{−1}∘α)}^{T_{m_{k}}}. But

since Γ⊨^{gΞ}δ, we also have, for every k ∈ K, ⋉_{m_{k}}[[δ]]_{(π_{k}∘b^{−1}∘α)}^{T_{m_{k}}}. Hence, ⋉[[δ]]_{(b^{−1}∘α)}^{( ∏_{k∈K}T_{m_{k}})}, and

again because b is an isomorphism, we have that ⋉_{m}[[δ]]_{α}^{T_{m}}. We conclude that Γ\overline{⊨}^{gΞ}δ, and therefore \overline{⊨}^{gΞ} = ⊨^{gΞ} and ⟨Σ, ⊨^{lΞ}, ⊨^{gΞ}⟩ = ⟨Σ, \overline{⊨}^{lΞ}, \overline{⊨}^{gΞ}⟩. ⊓ ⊔

For the remainder of the thesis, unless otherwise stated, whenever we consider the fibring of two interpretation systems we shall always assume that they are closed. Of course, if the given interpretation systems are not closed as such, it goes without saying that we apply the construction above to close each of them before they are fibred.

6.2.2 Deductive systems

As for synchronization before our proof-theoretic category of layered logic system presentations is both suitable and exact for fibring. Along the way, we shall also provide a characterization of fibring of deductive systems in similar terms to those in [SSC99, ZSS99].

Proposition 6.2.14. Dsy is suitable and exact for fibring. Namely, the fibring of deductive systems dsy_{k} = ⟨Σ^{k}, R_{k}, gR_{k}⟩, for k ∈ {1, 2}, constrained by S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} is given by (dsy_{1} ⊛^{S} dsy_{2}) = ⟨(Σ^{1} ⊛^{S} Σ^{2}), R, gR⟩ where:

R = ( ⋃_{k∈{1,2}}{⟨[Prem(r)_{k}]_{≈^{S}}, [Conc(r)_{k}]_{≈^{S}}⟩ : r ∈ R_{k}})

and

gR = ( ⋃_{k∈{1,2}}{⟨[Prem(r)_{k}]_{≈^{S}}, [Conc(r)_{k}]_{≈^{S}}⟩ : r ∈ gR_{k}}).

Proof. Again, since Sig has all coequalizers and we have shown, in 5.2.47, that Sgn_{Dsy} is a cofibration, we can use A.2.13 and conclude that all Sgn_{Dsy}-pushouts exist, once we show that all binary sums exist in Dsy.

For the purpose, let (dsy_{1} ⊛ dsy_{2}) = ⟨(Σ^{1} ⊛ Σ^{2}), (ι̂_{1}(R_{1}) ∪ ι̂_{2}(R_{2})), (ι̂_{1}(gR_{1}) ∪ ι̂_{2}(gR_{2}))⟩. We are going to show that {ι_{k} : dsy_{k} → (dsy_{1} ⊛ dsy_{2})}_{k∈{1,2}} is a sum in Dsy, where each ι_{k} : Σ^{k} → (Σ^{1} ⊛ Σ^{2}) is the corresponding injection morphism. Trivially, both are Dsy-morphisms. Let now {g_{k} : dsy_{k} → dsy′}_{k∈{1,2}} be morphisms in Dsy, with dsy′ = ⟨Σ′, R′, gR′⟩. Obviously, f_{n}(c_{k}) = g_{k,n}(c) defines the unique Sig-morphism f : (Σ^{1}⊛Σ^{2}) → Σ′ such that (f ∘ι_{k}) = g_{k} for k ∈ {1, 2}. We just need to prove that f : (dsy_{1}⊛dsy_{2}) → dsy′ establishes a Dsy-morphism. In fact, for every k ∈ {1, 2} and r ∈ R_{k}, f̂(r_{k}) = f̂(ι̂_{k}(r)) = ĝ_{k}(r) ∈ R′. And analogously, if r ∈ gR_{k}, f̂(r_{k}) ∈ gR′.

Thus, given that {[ ]_{≈_{n}^{S}}}_{n∈IN} : (Σ^{1} ⊛ Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}) is a coequalizer in Sig of {(ι_{k} ∘ s_{k})}_{k∈{1,2}}, we obtain (dsy_{1} ⊛^{S} dsy_{2}) by Sgn_{Dsy}-cocartesian lift of ⟨(dsy_{1} ⊛ dsy_{2}), {[ ]_{≈_{n}^{S}}}_{n∈IN}⟩. Furthermore, we have also proved in 5.2.49 that Sgn_{Dsy} is coadjoint. Thus, we can use A.3.3 to conclude that Dsy is suitable for fibring.

Exactness follows from A.3.2, since we have shown in 5.2.48 that Ded_{Dsy} is a cocartesian coadjoint. Note that, from 5.1.55, we know that Sgn_{slCsy_{fin}} is a cofibration. ⊓ ⊔

Obviously, (dsy_{1} ⊛ dsy_{2}), as defined above, is a free fibring of the deductive systems dsy_{1} and dsy_{2}.

6.3 Examples

Next, we illustrate the fibring mechanism by means of two well known examples. The first one shows that multi-modal logics can be obtained as fibred modal logics by sharing the purely propositional constructors.

Example 6.3.1. Multi-modal logic. Let us consider two copies of the structural systems for K presented in 5.3.3. Namely, assuming that they share a set P of propositional symbols let us denote their signatures by Σ_{K}^{P 1} and Σ_{K}^{P 2}. Our purpose is to synchronize both the corresponding interpretation and deductive systems by sharing also all the purely propositional constructors. Thus, taking the propositional signature Σ_{cpl}^{P} as defined in 5.3.1, the desired constraint is S = ⟨s_{1} : Σ_{cpl}^{P} → Σ_{K}^{P 1}, s_{2} : Σ_{cpl}^{P} → Σ_{K}^{P 2}⟩ with:

• s_{1,0}(p) = s_{2,0}(p) = p for each p ∈ P;

• s_{1,1}(¬) = s_{2,1}(¬) = ¬;

• s_{1,2}(⇒) = s_{2,2}(⇒) = ⇒;

• s_{1,2}(∧) = s_{2,2}(∧) = ∧;

• s_{1,2}(∨) = s_{2,2}(∨) = ∨.

It is straightforward that Σ = (Σ_{K}^{P 1} ⊛^{S} Σ_{K}^{P 2}) essentially consists of:

• Σ_{0} = P, Σ_{1} = {¬, □_{1}, □_{2}}, Σ_{2} = {⇒, ∧, ∨}, and Σ_{n} = ∅ if n > 2,

where □_{1} and □_{2} stand, respectively, for the modalitites of the first and second signatures.

Clearly, then, the fibred deductive system (dsy_{K}^{P 1} ⊛^{S} dsy_{K}^{P 2}) can be identified with dsy = ⟨Σ, R, gR⟩ where R contains the schemas of Ax01 -Ax11 and MP as in 5.3.3 plus two distinct normality schemas:

□_{1}-Norm. ⟨∅, ((□_{1}(ξ_{1} ⇒ ξ_{2})) ⇒ ((□_{1}ξ_{1}) ⇒ (□_{1}ξ_{2})))⟩;

□_{2}-Norm. ⟨∅, ((□_{2}(ξ_{1} ⇒ ξ_{2})) ⇒ ((□_{2}ξ_{1}) ⇒ (□_{2}ξ_{2})))⟩,

and gR contains precisely two corresponding necessitation schemas:

□_{1}-Nec. ⟨{ξ_{1}}, (□_{1}ξ_{1})⟩;

□_{2}-Nec. ⟨{ξ_{1}}, (□_{2}ξ_{1})⟩.

In terms of interpretation systems, we have first to recall that the interpretation systems int_{K}^{P 1} = int_{K}^{P 2} to be fibred must first be closed for isomorphisms and products of interpretation structures. In this case, it is straightforward to verify that the product of structures corresponding to Kripke models {⟨W_{k}, R_{k}, ϑ_{k}⟩}_{k∈K} is essentially the same as (isomorphic to) the structure corresponding to the Kripke model obtained by taking the union of all the Kripke models ⟨W_{k}, R_{k}, ϑ_{k}⟩, in the sense used in [ZSS99]. Therefore, the interpretation system (int_{K}^{P 1} ⊛^{S} int_{K}^{P 2}) can be identified (up to isorphism) with the closure \overline{int} (indeed it suffices a closure for isomorphisms) of int = ⟨Σ, M, S⟩, where M is the class of all Kripke models ⟨W, R_{1}, R_{2}, ϑ⟩ with two distinct accessibility relations and, letting T = ℘(W), we have

S(⟨W, R_{1}, R_{2}, ϑ⟩) = ⟨T , ⋉_{⊆}⟩

similarly to 5.3.3 but now with:

• \underline{□_{1}}_{1}^{T} (X) = \{w ∈ W : wR_{1}w′ implies w′ ∈ X\};

• \underline{□_{2}}_{1}^{T} (X) = \{w ∈ W : wR_{2}w′ implies w′ ∈ X\}.

In fact, a fibred model ⟨⟨W_{1}, R_{1}, ϑ_{1}⟩, ⟨W_{2}, R_{2}, ϑ_{1}⟩⟩ would have to be such that an isomorphism h : ⟨℘(W_{1}), ⋉_{⊆}⟩ → ⟨℘(W_{2}), ⋉_{⊆}⟩ of consequence systems existed. Easily, this implies that h : ⟨℘(W_{1}), ⊆⟩ → ⟨℘(W_{2}), ⊆⟩ is an order isomorphism and in particular one can define a bijection h^{∗} : W_{1} → W_{2} by letting h^{∗}(x) be the unique element of h({x}). It can be easily verified that h(X) = {h^{∗}(x) : x ∈ X}. Furthermore, due to the sharing of propositional symbols, we must also have h(ϑ_{1}(p)) = ϑ_{2}(p).

The second example concerns the fibring of classical with intuitionistic propositional systems.

Example 6.3.2. Fibring of classical and intuitionistic logics: the collapse. Let us consider the interpretation systems of classical and intuitionistic logic defined, respectively, in 5.3.1 and 5.3.2. In this case, we shall consider their free fibring. Of course, it might be expected that their fibring constrained by sharing all the constructors would yield classical logic. That is indeed the case, both at the model and the prooftheoretic levels. However, we show that the same happens even if the fibring is free, for their corresponding interpretation systems. Clearly, the signature Σ = (Σ_{cpl}^{P} ⊛ Σ_{ipl}^{P}) contains two copies of each of the propositional constructors. However, if we think of int = (int_{cpl}^{P} ⊛ int_{ipl}^{P}) it is easy to verify that (up to isomorphism) int = ⟨Σ, M, S⟩, where M does only contain Boolean algebra models. Indeed, we start by noting that both Boolean algebras and Heyting algebras are well known to be closed for isomorphisms and products [Bir67]. Therefore, a fibred model would have to be a pair ⟨⟨B, ϑ_{1}⟩, ⟨H, ϑ_{2}⟩⟩ with B = ⟨B, ≤_{1}⟩ a Boolean algebra and H = ⟨H, ≤_{2}⟩ a Heyting algebra such that B = H and ⋉_{≤_{1}} = ⋉_{≤_{2}}. Easily, this turns out to mean precisely that ≤_{1}=≤_{2} and therefore H = B must in fact be a Boolean algebra. The interpretation of the matching classical and intuitionistic constructors would therefore coincide, despite the fact that they were not required to be shared.

This fact has been studied in [dCH96]. Related to this phenomenon, we refer the reader to [MP96] for a general proof of the fact that the presence of classical negation always forces a weaker negation such as the intuitionistic to collapse to the classical case.

6.4 Comparison with synchronization

By now, we can safely say that the two syntax-driven combination mechanisms we have been studying, synchronization and fibring, are quite distinct from each other. However, we have developed both of them in parallel, using the same kind of tools, but in different contexts. Both correspond to the same kind of categorial construction, although on sufficiently different categories. However, it has been shown along Chapter 5 that these categories are related. So, it is not surprising that some kind of relationship between synchronization and fibring pops up naturally. In fact, since we have shown that (l)Csy is isomorphic to a well behaved subcategory of s(l)Csy, we can show that synchronization can be seen as a very particular, unstructured case of fibring. Perhaps more surprising, although intuitively not so hard to grasp, is the possibility of representing the logic system presented by a fibring of presentations as a constrained synchronization of the given presented systems. This result, which we shall prove below, reflects the intuitions provided by Gabbay [Gab96a, Gab99], as it had also been noticed in [Sel98].

6.4.1 Synchronization is fibring

The whole idea of seeing synchronization as a particular case of fibring is to look at each sentence in a language as being a nullary constructor, thus giving rise to a signature with only constants. We have already seen how to do it, namely through the definition of the functors @ : Set → Sig in 5.1.10, and also how to extend it not only to the case of consequence systems, using @ : Csy → sCsy as defined in 5.1.23, but also to the case of layered consequence systems, using @ : lCsy → slCsy as defined in 5.1.44. Recall also the definition of the functors PresL : Sig → Set, PresC : sCsy → Csy and PreslC : slCsy → lCsy.

We shall show that any given synchronization of (layered) consequence systems can be obtained by extracting the (layered) consequence presented by the corresponding fibring of their promotion via @ to the structural context. First we shall prove an auxiliary lemma.

Proposition 6.4.1. Let S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}}. Then, @(L_{1} ▷◁^{S} L_{2}) coincides with (@(L_{1}) ⊛^{@(S)} @(L_{2})), where @(S) = {@(s_{k}) : @(L_{0}) → @(L_{k})}_{k∈{1,2}}. So, in particular,

(L_{1} ▷◁^{S} L_{2}) = PresL(@(L_{1}) ⊛^{@(S)} @(L_{2})).

Proof. By definition of @ : Set → Sig, each @(L_{k}) is such that @(L_{k})_{0} = L_{k} and @(L_{k})_{n} = ∅ for n ≠ 0, and each @(s_{k}) is such that @(s_{k})_{0} = s_{k} and @(s_{k})_{n} is the empty function for n ≠ 0. Thus, recalling the constructions in 4.1.10 and 6.1.10, we have that ≈_{0}^{@(S)}=≈^{S} and ≈_{n}^{@(S)}= ∅ for n ≠ ∅, which implies that (@(L_{1}) ⊛^{@(S)} @(L_{2})) = @(L_{1} ▷◁^{S} L_{2}). Therefore, from 5.1.11, we can conclude that PresL(@(L_{1}) ⊛^{@(S)} @(L_{2})) = (L_{1} ▷◁^{S} L_{2}). ⊓ ⊔ Proposition 6.4.2. Let csy_{k} over language L_{k}, for k ∈ {1, 2}, be consequence systems and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a corresponding sharing constraint. Then, @(csy_{1} ▷◁^{S} csy_{2}) coincides with (@(csy_{1}) ⊛^{@(S)} @(csy_{2})). So, in particular, (csy_{1} ▷◁^{S} csy_{2}) = PresC (@(csy_{1}) ⊛^{@(S)} @(csy_{2})).

Proof. Let each csy_{k} = ⟨L_{k}, ▷_{k}⟩. Using the previous 6.4.1, and recalling the constructions in 4.1.10 and 6.1.10, we can consider that (csy_{1} ▷◁^{S} csy_{2}) = ⟨L, ▷⟩ and (@(csy_{1}) ⊛^{@(S)} @(csy_{2})) = ⟨@(L), ▷^{Ξ}⟩, where L = (L_{1} ▷◁^{S} L_{2}). Since L_{@(L)}(Ξ) = (L ∪ Ξ), all we need to prove is that (E ∪ χ)^{@(▷)^{Ξ}} = (E ∪ χ)^{▷^{Ξ}} for every Φ ⊆ L_{1}, Ψ ⊆ L_{2} and χ ⊆ Ξ with E = ([Φ_{1}]_{≈^{S}} ∪ [Ψ_{2}]_{≈^{S}}). Recall from 5.1.23 that (E ∪ χ)^{@(▷)^{Ξ}} = (E^{▷} ∪ χ), and from 4.1.10 that ▷ is generated by the following set I_{▷◁}^{≈^{S}} of inferences:

({⟨[Φ_{1}]_{≈^{S}}, [φ_{1}]_{≈^{S}}⟩ : Φ ▷_{1} φ} ∪ {⟨[Ψ_{2}]_{≈^{S}}, [ψ_{2}]_{≈^{S}}⟩ : Ψ ▷_{2} ψ}).

On the other hand, recall from 6.1.10 that ▷^{Ξ} is generated by all the instances of the set R≈^{S}_{⊛} of rules:

({⟨[Γ_{1}]_{≈^{S}}, [γ_{1}]_{≈^{S}}⟩ : Γ@(▷_{1})^{Ξ}γ} ∪ {⟨[Δ_{2}]_{≈^{S}}, [δ_{2}]_{≈^{S}}⟩ : Δ@(▷_{2})^{Ξ}δ}).

Due to the definitions of each @(▷_{k})^{Ξ} this clearly ammounts to partitioning R≈^{S}_{⊛} into two kinds of rules, the first being

({⟨([Φ_{1}]_{≈^{S}} ∪ χ), [φ_{1}]_{≈^{S}}⟩ : Φ ▷_{1} φ} ∪ {⟨([Ψ_{2}]_{≈^{S}} ∪ χ), [ψ_{2}]_{≈^{S}}⟩ : Ψ ▷_{2} ψ})

and the second

({⟨([Φ_{1}]_{≈^{S}} ∪ χ), ξ⟩ : ξ ∈ χ} ∪ {⟨([Ψ_{2}]_{≈^{S}} ∪ χ), ξ⟩ : ξ ∈ χ}).

We first prove that (E^{▷} ∪ χ) ⊆ (E ∪ χ)^{▷^{Ξ}}. Clearly, the monotonicity and extensiveness of ▷^{Ξ} imply that E, χ ⊆ (E ∪ χ)^{▷^{Ξ}}. To show that E^{▷} ⊆ (E ∪ χ)^{▷^{Ξ}} it clearly suffices to show that (E ∪χ)^{▷^{Ξ}} is closed for every inference in I_{▷◁}^{≈^{S}}. This is immediate, however, since every inference in I_{▷◁}^{≈^{S}} belongs to the first kind of rules in R≈^{S}_{⊛}. Just take χ = ∅.

Let us now prove (E ∪ χ)^{▷^{Ξ}} ⊆ (E^{▷} ∪ χ). Clearly, since E ⊆ E^{▷}, we just need to show that (E^{▷} ∪ χ) is closed for every instance of a rule in R≈^{S}_{⊛}. If an instance of a rule of the first kind applicable to (E^{▷} ∪ χ), clearly, the corresponding inference from I_{▷◁}^{≈^{S}} is also applicable and yields exactly the same conclusion. So, the conclusion must belong to E^{▷}. Application of instances of rules of the second kind is trivial since their conclusions always belong to the premises.

Therefore, @(L_{1} ▷◁^{S} L_{2}) = (@(L_{1}) ⊛^{@(S)} @(L_{2})) and from 5.1.24 we can conclude that also (csy_{1} ▷◁^{S} csy_{2}) = PresC (@(csy_{1}) ⊛^{@(S)} @(csy_{2})). ⊓ ⊔

The corresponding layered version is quite similar.

Proposition 6.4.3. Let csy_{k} over language L_{k}, for k ∈ {1, 2}, be layered consequence systems and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a corresponding sharing constraint. Then, @(csy_{1} ▷◁^{S} csy_{2}) coincides with (@(csy_{1}) ⊛^{@(S)} @(csy_{2})). Thus, in particular, (csy_{1} ▷◁^{S} csy_{2}) = PreslC (@(csy_{1}) ⊛^{@(S)} @(csy_{2})).

Proof. Immediate, from the definitions of 4.1.14 and 6.1.14, by using 6.4.2 and 5.1.45. ⊓ ⊔

Note that, owing to the exactness of the constructions, we can legitimately expect a similar trade-off within our working categories of proof-theoretic systems, namely by explaining synchronization in lInf by using fibring in Dsy. Let us define the relevant functors. Given a rule r over a signature Σ, we shall denote by \overline{r} the set of all its ground instances, i.e., {rσ : σ ∈ GSub(Σ)}.

Definition 6.4.4. The maps

• PreslInf_{Dsy}(⟨Σ, R, gR⟩) = ⟨L_{Σ}, \overline{R}, \overline{gR}⟩;

• PreslInf_{Dsy}(h : dsy_{1} → dsy_{2}) = ĥ,

constitute the functor PreslInf_{Dsy} : Dsy → lInf.

Proposition 6.4.5. PreslInf_{Dsy} : Dsy → lInf is indeed a functor.

Proof. Similar to 5.2.37. ⊓ ⊔

Definition 6.4.6. The maps such that

• @(⟨L, I, gI⟩) = ⟨@(L), I, gI⟩;

• @(h : inf_{1} → inf_{2}) = @(h) : @(csy_{1}) → @(csy_{2}),

constitute the functor @ : lInf → Dsy.

Proving that it constitutes a full embedding is straightforward.

Proposition 6.4.7. @ : lInf → Dsy is a full embedding and (PreslInf_{Dsy} ∘ @) = ID_{lInf}.

Proof. The fact that @ is a functor is trivial, since { \atop @(h)} = h. To check that it is an embedding, let h_{1} : ⟨L_{1}, I_{1}, gI_{1}⟩ → ⟨L′_{1}, I′_{1}, gI′_{1}⟩ and h_{2} : ⟨L_{2}, I_{2}, gI_{2}⟩ → ⟨L′_{2}, I′_{2}, gI′_{2}⟩ be any two lInf-morphisms such that @(h_{1}) = @(h_{2}). Immediately, we must have @(L_{1}) = @(L_{2}) and @(L′_{1}) = @(L′_{2}), which implies L_{1} = L_{2} and L′_{1} = L′_{2}, and also I_{1} = I_{2}, gI_{1} = gI_{2}, I′_{1} = I′_{2} and gI′_{1} = gI′_{2}. This obviously means that @ is injective on morphisms.

To prove that @ is full, let inf = ⟨L, I, gI⟩, inf′ = ⟨L′, I′, gI′⟩ be layered inference systems and h : @(inf ) → @(inf′) a Dsy-morphism. Clearly, h_{0} : inf → inf′ is a morphism and @(h_{0}) = h.

Trivially, every inference over L yields a rule over @(L) without metavariables. Therefore, \overline{I} = I and \overline{gI} = gI. Moreover, L_{@(L)} = L, and we can conclude that for every inference system inf we have PreslInf_{Dsy}(@(inf )) = inf . Since { \atop @(h)} = h the result follows. ⊓ ⊔

Proposition 6.4.8. Let inf_{k} over language L_{k}, for k ∈ {1, 2}, be layered inference systems and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a corresponding constraint. Then, @(inf_{1} ▷◁^{S} inf_{2}) coincides with (@(inf_{1}) ⊛^{@(S)} @(inf_{2})) and, in particular, (inf_{1} ▷◁^{S} inf_{2}) = PreslInf (@(inf_{1}) ⊛^{@(S)} @(inf_{2})).

Proof. Let each inf_{k} = ⟨L_{k}, I_{k}, gI_{k}⟩. Using 6.4.1, and recalling the constructions in 4.2.5 and 6.2.14, we can consider that (inf_{1} ▷◁^{S} inf_{2}) = ⟨L, I, gI⟩ and (@(inf_{1}) ⊛^{@(S)} @(inf_{2})) =

⟨@(L), R, gR⟩, where L = (L_{1} ▷◁^{S} L_{2}), I = R = (⋃_{k∈{1,2}}{⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ :

i ∈ I_{k}}) and gI = gR = (⋃_{k∈{1,2}}{⟨[Prem(i)_{k}]_{≈^{S}}, [Conc(i)_{k}]_{≈^{S}}⟩ : i ∈ gI_{k}}). So, immediately, @(inf_{1} ▷◁^{S} inf_{2}) = (@(inf_{1}) ⊛^{@(S)} @(inf_{2})). Using 6.4.7 we also obtain that (inf_{1} ▷◁^{S} inf_{2}) = PreslInf (@(inf_{1}) ⊛^{@(S)} @(inf_{2})). ⊓ ⊔

Model-theoretically, although the constructions are not exact, the relationship still holds. In particular, it is still possible to explain synchronization in lVal by means of fibring in Int. Again, we use exactly the same technique. The relevant functors are as follows:

Definition 6.4.9. The maps

• PreslVal_{Int}(⟨Σ, M, S⟩) = ⟨L_{Σ}, M, \overline{S}⟩, with \overline{S}(m) = ⟨T_{m}, [[ ]]^{T_{m}}, ⋉_{m}⟩;

• PreslVal_{Int}(h : int_{1} → int_{2}) such that \overline{lVal_{Int}(h)} = ˆ \overline{h} and \underline{lVal_{Int}(h)} = \underline{h},

constitute the functor PreslVal_{Int} : Int → lVal.

Proposition 6.4.10. PreslVal_{Int} : Int → lVal is indeed a functor.

Proof. Similar to 5.2.18. ⊓ ⊔

Definition 6.4.11. The maps such that

• @(⟨L, M, S⟩) = ⟨@(L), M, S^{@}⟩, with S^{@}(m) = ⟨T_{m}, ⋉_{m}⟩ and \underline{φ} _{0}^{T_{m}} = [[φ]]_{m};

• @(h : val_{1} → val_{2}) : @(val_{1}) → @(val_{2}) such that \overline{@(h)} = @(\overline{h}) and \underline{@(h)} = \underline{h},

constitute the functor @ : lVal → Int.

Proposition 6.4.12. @ : lVal → Int is a full embedding and (PreslVal_{Int} ∘ @) = ID_{lVal}.

Proof. The fact that @ is a functor is straightforward. Each S^{@}(m) as defined above is clearly an @(L)-structure. Moreover, if h : ⟨L_{1}, M_{1}, S_{1}⟩ → ⟨L_{2}, M_{2}, S_{2}⟩ is an lValmorphism then \overline{h} : L_{1} → L_{2} is a function and @(\overline{h}) : @(L_{1}) → @(L_{2}) a signature morphism. Moreover let m ∈ M_{2} and S_{2}(m) = ⟨T_{m}, [[ ]]_{m}, ⋉_{m}⟩. Obviously, S_{1}(\underline{h}(m)) = ⟨T_{m}, ([[ ]]_{m} ∘\overline{h}), ⋉_{m}⟩. Therefore, S_{2}^{@}(m) and S_{1}^{@}(\underline{h}(m)) share the same structure ⟨T_{m}, ⋉_{m}⟩. Furthermore, for every φ ∈ L_{1}, \underline{φ} _{0}^{T_{\underline{h}(m)}} = [[φ]]_{\underline{h}(m)} = [[\overline{h}(φ)]]_{m} = \underline{\overline{h}(φ)} _{0}^{T_{m}}.

To check that it is an embedding, let h_{1} : ⟨L_{1}, M_{1}, S_{1}⟩ → ⟨L′_{1}, M′_{1}, S′_{1}⟩ and h_{2} : ⟨L_{2}, M_{2}, S_{2}⟩ → ⟨L′_{2}, M′_{2}, S′_{2}⟩ be any two lVal-morphisms such that @(h_{1}) = @(h_{2}). Immediately, we must have @(L_{1}) = @(L_{2}) and @(L′_{1}) = @(L′_{2}), which implies L_{1} = L_{2} and L′_{1} = L′_{2}, and also M_{1} = M_{2} and M′_{1} = M′_{2}. Moreover, having S_{1}^{@} = S_{2}^{@} and S_{1}^{′@} = S_{2}^{′@}, it easily follows that S_{1} = S_{2}, S′_{1} = S′_{2}, and @ is injective on morphisms.

To prove that @ is full, let val = ⟨L, M, S⟩, val′ = ⟨L′, M′, S′⟩ be layered valuation systems and h : @(val) → @(val′) an Int-morphism. Clearly, k : val → val′ with \overline{k} = \overline{h}_{0} and \underline{k} = \underline{h} is a morphism and @(k) = h.

Given a layered valuation system val = ⟨L, M, S⟩, since L_{@(L)} = L, to prove that PreslVal_{Int}(@(val)) = val we are left with showing that \overline{S^{@}} = S. Clearly, for every φ ∈ L and m ∈ M, [[φ]]^{T_{m}} = \underline{φ} _{0}^{T_{m}} = [[φ]]_{m}. Morphisms are trivially preserved and therefore (PreslVal_{Int} ∘ @) = ID_{lVal}. ⊓ ⊔ Proposition 6.4.13. Let val_{k} over language L_{k}, for k ∈ {1, 2}, be layered valuation systems and S = {s_{k} : L_{0} → L_{k}}_{k∈{1,2}} a corresponding sharing constraint. Then, @(val_{1} ▷◁^{S} val_{2}) coincides with (@(val_{1}) ⊛^{@(S)} @(val_{2})), and, in particular, (val_{1} ▷◁^{S} val_{2}) = PreslVal(@(val_{1}) ⊛^{@(S)} @(val_{2})).

Proof. Let each val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩. Using 6.4.1, and recalling the constructions in 4.2.3 and 6.2.1, we can consider that (val_{1} ▷◁^{S} val_{2}) = ⟨L, M, S⟩ and (@(inf_{1}) ⊛^{@(S)} @(inf_{2})) = ⟨@(L), M, S′⟩, where L = (L_{1} ▷◁^{S} L_{2}) and M = M′ consist of the class of all pairs ⟨m_{1}, m_{2}⟩ ∈ M_{1} × M_{2} such that T_{m_{1}} = T_{m_{2}} = T_{⟨m_{1},m_{2}⟩}, ⋉_{m_{1}} = ⋉_{m_{2}} = ⋉_{⟨m_{1},m_{2}⟩} and [[s_{1}(υ)]]_{m_{1}} = \underline{@(s_{1})_{0}(υ)} _{0}^{T_{m_{1}}} = \underline{@(s_{2})_{0}(υ)} _{0}^{T_{m_{2}}} = [[s_{2}(υ)]]_{m_{2}} for every υ ∈ L_{0}. Furthermore, S(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, [[ ]]_{⟨m_{1},m_{2}⟩}, ⋉_{⟨m_{1},m_{2}⟩}⟩ and S′(⟨m_{1}, m_{2}⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, ⋉_{⟨m_{1},m_{2}⟩}⟩

are such that \underline{υ} _{0}^{T_{⟨m_{1},m_{2}⟩}} = [[υ]]_{⟨m_{1},m_{2}⟩} for every υ ∈ L, which implies that S′ = S^{@} and @(val_{1} ▷◁^{S} val_{2}) = (@(val_{1}) ⊛^{@(S)} @(val_{2})).

Using 6.4.12 we also obtain that (val_{1} ▷◁^{S} val_{2}) = PreslVal(@(val_{1}) ⊛^{@(S)} @(val_{2})). ⊓ ⊔

6.4.2 Fibring via synchronization

The intuitions behind the forthcoming representation of fibring using synchronization are indeed on the very deep nature of fibring, ever since they were first described by Gabbay in [Gab96a, Gab99]. In fact, every schema sentence of a fibred signature can be seen as a sentence on the side of its principal constructor, but where one is allowed to use, as a whole, a monolith, to represent each of its subsentences whose principal constructor falls on the other signature. Thus, by enriching each of the initially given signatures with as many constant symbols as necessary to represent every possible such monolith and by freely extending the initially given consequence, proof or semantic system presentations, we can achieve a system which is equivalent (even isomorphic) to the system induced by the fibring of the initial presentations by means of a suitable synchronization. For the sake of simplicity, we shall just focus our attention on explaining free fibring using synchronization. Still, with due changes, we could have similarly dealt with the constrained case. The phrasing, however, would be much more cumbersome.

We shall start by showing that the system induced by any given free fibring of structural consequence systems is equivalent to a suitable synchronization of the consequence systems induced by the enrichments of the original ones. First, we shall dwell on a few auxiliary definitions and lemmas. For the remainder of the section we shall fix two signatures Σ^{1} and Σ^{2}, and denote the freely fibred signature (c.f., 6.1.1) (Σ^{1} ⊛ Σ^{2}) just by Σ.

Definition 6.4.14. For every k, j ∈ {1, 2} with j ≠ k, the set of k-monoliths is:

• L_{Σ}^{k}(Ξ) = {c_{j}(θ_{1}, …, θ_{n}) : n ∈ IN, c ∈ Σ_{n}^{j}, θ_{1}, …, θ_{n} ∈ L_{Σ}(Ξ)}.

Clearly, we have that L_{Σ}(Ξ) = (L_{Σ}^{1}(Ξ) ∪ L_{Σ}^{2}(Ξ) ∪ Ξ), the three sets being mutually disjoint.

Definition 6.4.15. For each k ∈ \{1, 2\}, the corresponding enriched signature is \overline{Σ}^{k} defined by:

• \overline{Σ}_{0}^{k} = (Σ_{0}^{k} ∪ \{ θ : θ ∈ L_{Σ}^{k}(Ξ)\});

• \overline{Σ}_{n}^{k} = Σ_{n}^{k} for n ≠ 0.

Note that the boxed constructors above correspond precisely to the monoliths over the other signature. Moreover, it goes without saying that there are obvious signature inclusion morphisms from each signature to its enrichment. The following result introduces two maps that help us relate the fibred sentences with those built in each of the enriched signatures.

Proposition 6.4.16. For each k ∈ {1, 2}, consider the following functions:

• ι_{k} : L\overline{Σ}_{k}(Ξ) → L_{Σ}(Ξ) inductively defined by:

– ι_{k}(ξ) = ξ;

– ι_{k}( θ ) = θ;

– ι_{k}(c(γ_{1}, …, γ_{n})) = c_{k}(ι_{k}(γ_{1}), …, ι_{k}(γ_{n})),

• ȷ_{k} : L_{Σ}(Ξ) → L\overline{Σ}_{k}(Ξ) inductively defined by:

– ȷ_{k}(ξ) = ξ;

– ȷ_{k}(c_{k}(θ_{1}, …, θ_{n})) = c(ȷ_{k}(θ_{1}), …, ȷ_{k}(θ_{n}));

– ȷ_{k}(c_{j}(θ_{1}, …, θ_{n})) = c_{j}(θ_{1}, …, θ_{n}) , where j ∈ {1, 2} and j ≠ k.

Then, (ι_{k} ∘ ȷ_{k}) = id_{L_{Σ}(Ξ)} and (ȷ_{k} ∘ ι_{k}) = id_{L_{\overline{Σ}k} (Ξ)}.

Proof. It is easy to see that both functions are well defined. We shall now prove the results for the case k = 1. Proofs for k = 2 can be obtained similarly.

Let us show, by induction on the structure of θ ∈ L_{Σ}(Ξ), that ι_{1}(ȷ_{1}(θ)) = θ.

• Base case.

If we consider a metavariable ξ, trivially, ι_{1}(ȷ_{1}(ξ)) = ι_{1}(ξ) = ξ. If it is the case of a constant c_{1} (with c ∈ Σ_{0}^{1}), then ι_{1}(ȷ_{1}(c_{1})) = ι_{1}(c) = c_{1}. If, on the other hand, we have a constant c_{2} (with c ∈ Σ_{0}^{2}), then ι_{1}(ȷ_{1}(c_{2})) = ι_{1}( c_{2} ) = c_{2}.

• Step: θ is either of the form c_{1}(θ_{1}, …, θ_{n}) or c_{2}(θ_{1}, …, θ_{n}), with n ≠ 0, and the induction hypothesis asserts that ι_{1}(ȷ_{1}(θ_{1})) = θ_{1}, …, ι_{1}(ȷ_{1}(θ_{n})) = θ_{n}. If we have c_{1}(θ_{1}, …, θ_{n}) (with c ∈ Σ_{n}^{1}) then, it follows that ι_{1}(ȷ_{1}(c_{1}(θ_{1}, …, θ_{n}))) = ι_{1}(c(ȷ_{1}(θ_{1}), …, ȷ_{1}(θ_{n}))) = c_{1}(ι_{1}(ȷ_{1}(θ_{1})), …, ι_{1}(ȷ_{1}(θ_{n}))) = c_{1}(θ_{1}, …, θ_{n}), by just using the induction hypothesis on the last step. Otherwise, if we have c_{2}(θ_{1}, …, θ_{n})

(with c ∈ Σ_{n}^{2}) then, immediately, ι_{1}(ȷ_{1}(c_{2}(θ_{1}, …, θ_{n}))) = ι_{1}( c_{2}(θ_{1}, …, θ_{n}) ) =

c_{2}(θ_{1}, …, θ_{n}).

Therefore, (ι_{1} ∘ ȷ_{1}) = id_{L_{Σ}(Ξ)}.

Finally we show, now by induction on the structure of γ ∈ L\overline{Σ}_{1}(Ξ), that ȷ_{1}(ι_{1}(γ)) = γ.

• Base case.

If we consider a metavariable ξ, trivially, ȷ_{1}(ι_{1}(ξ)) = ȷ_{1}(ξ) = ξ. If it is the case of a constant c ∈ Σ_{0}^{1}, then ȷ_{1}(ι_{1}(c)) = ȷ_{1}(c_{1}) = c. If, on the other hand, we have a boxed constant θ with θ ∈ L_{Σ}^{2}(Ξ), then ȷ_{1}(ι_{1}( θ )) = ȷ_{1}(θ) = θ , by definition of L_{Σ}^{2}(Ξ) and ȷ_{1}.

• Step: γ is of the form c(γ_{1}, …, γ_{n}), with n ≠ 0, and the induction hypothesis asserts that ȷ_{1}(ι_{1}(γ_{1})) = γ_{1}, …, ȷ_{1}(ι_{1}(γ_{n})) = γ_{n}. Then, by just using the induction hypothesis on the last step, ȷ_{1}(ι_{1}(c(γ_{1}, …, γ_{n}))) = ȷ_{1}(c_{1}(ι_{1}(γ_{1}), …, ι_{1}(γ_{n}))) = c(ȷ_{1}(ι_{1}(γ_{1})), …, ȷ_{1}(ι_{1}(γ_{n}))) = c(γ_{1}, …, γ_{n}).

Thus, we also get (ȷ_{1} ∘ ι_{1}) = id_{L_{\overline{Σ}1}(Ξ)}. ⊓ ⊔

Obviously this means that L_{Σ}(Ξ), L\overline{Σ}_{1}(Ξ) and L\overline{Σ}_{2}(Ξ) are all bijective. In particular, the maps ȷ_{1} and ȷ_{2} provide us with a way of representing each fibred schema sentence as a schema sentence over each of the enriched signatures. For instance, a fibred sentence like c_{1}(a_{2}, c_{2}(a_{1})) corresponds to c( a_{2} c_{2}(a_{1}) ) in the first side, and directly to

c_{1}(a_{2}, c_{2}(a_{1})) in the second one. Clearly, the fibred subsentences a_{2}, c_{2}(a_{1}) and a_{1} also have their counterparts on both sides.

In the sequel, we shall consider the constraint S^{⊛} = \{ȷ_{k} : L_{Σ}(Ξ) → L\overline{Σ}_{k}(Ξ)\}_{k∈\{1,2\}} for synchronizing L\overline{Σ}_{1}(Ξ) and L\overline{Σ}_{2}(Ξ). Recall the constructions in 4.1.1 and 4.1.10.

Proposition 6.4.17. The equivalence relation ≈^{S^{⊛}} induced by S^{⊛} on (L\overline{Σ}_{1}(Ξ) ▷◁ L\overline{Σ}_{2}(Ξ)) is such that for every γ, γ′, γ′′ ∈ L\overline{Σ}_{1}(Ξ) and δ, δ′, δ′′ ∈ L\overline{Σ}_{2}(Ξ):

• γ′_{1} ≈^{S^{⊛}} γ′′_{1} if and only if γ′ = γ′′;

• δ′_{2} ≈^{S^{⊛}} δ′′_{2} if and only if δ′ = δ′′;

• γ_{1} ≈^{S^{⊛}} δ_{2} if and only if ι_{1}(γ) = ι_{2}(δ).

Proof. Let L_{▷◁} = (L\overline{Σ}_{1}(Ξ) ▷◁ L\overline{Σ}_{2}(Ξ)). It is straightforward to verify that we have ≈^{S^{⊛}}= ({⟨υ, υ⟩ : υ ∈ L} ∪ {⟨ȷ_{1}(θ)_{1}, ȷ_{2}(θ)_{2}⟩ : θ ∈ L_{Σ}(Ξ)} ∪ {⟨ȷ_{2}(θ)_{2}, ȷ_{1}(θ)_{1}⟩ : θ ∈ L_{Σ}(Ξ)}). ⊓ ⊔

Owing to the result just proved, we can now conclude that we can synchronize the languages of the two enriched signatures under the constraint S^{⊛} and still obtain essentially the free fibred language of the original signatures.

Proposition 6.4.18. Let L_{▷◁}^{S^{⊛}} = (L\overline{Σ}_{1}(Ξ) ▷◁^{S^{⊛}} L\overline{Σ}_{2}(Ξ)). Consider the following functions:

• ι : L_{▷◁}^{S^{⊛}} → L_{Σ}(Ξ) defined by:

– ι([γ_{1}]_{≈^{S^{⊛}}}) = ι_{1}(γ);

– ι([δ_{2}]_{≈^{S^{⊛}}}) = ι_{2}(δ),

• ȷ : L_{Σ}(Ξ) → L_{▷◁}^{S^{⊛}} defined by:

– ȷ(θ) = [ȷ_{1}(θ)_{1}]_{≈^{S^{⊛}}} = [ȷ_{2}(θ)_{2}]_{≈^{S^{⊛}}}.

Then, (ι ∘ ȷ) = id_{L_{Σ}(Ξ)} and (ȷ ∘ ι) = id_{L_{▷◁}^{S^{⊛}}}.

Proof. At the light of 6.4.17 it is immediate that both ι and ȷ are well defined. Easily, ι(ȷ(θ)) = ι([ȷ_{1}(θ)_{1}]_{≈^{S^{⊛}}}) = ι_{1}(ȷ_{1}(θ)) = θ. On the other hand, let e = {ȷ_{1}(θ)_{1}, ȷ_{2}(θ)_{2}} ∈ L_{▷◁}^{S^{⊛}} = [ȷ_{1}(θ)_{1}]_{≈^{S^{⊛}}} = [ȷ_{2}(θ)_{2}]_{≈^{S^{⊛}}}. Then, ȷ(ι(e)) = ȷ(ι_{1}(ȷ_{1}(θ))) = ȷ(θ) = e. ⊓ ⊔

In the sequel we shall reuse the notation L_{▷◁} = (L\overline{Σ}_{1}(Ξ) ▷◁ L\overline{Σ}_{2}(Ξ)) for the freely synchronized language and L_{▷◁}^{S^{⊛}} = (L\overline{Σ}_{1}(Ξ) ▷◁^{S^{⊛}} L\overline{Σ}_{2}(Ξ)) for the constrained one.

We are at last in a position to state and prove our characterization of free fibring via constrained synchronization.

Proposition 6.4.19. For k ∈ {1, 2}, let csy_{k} = ⟨Σ^{k}, ▷_{k}^{Ξ}⟩ be structural consequence systems, and consider the enriched structural consequence systems \overline{csy}_{k} = ⟨\overline{Σ}^{k}, \overline{▷}_{k}^{Ξ}⟩, with \overline{▷}_{k}^{Ξ} the consequence on L\overline{Σ}_{k}(Ξ) generated by all instances of the rules

{⟨Θ, θ⟩ : Θ ▷_{k}^{Ξ} θ}

where Θ and θ range over L_{Σ^{k}}(Ξ). Then, (\overline{csy}_{1}(Ξ) ▷◁^{S^{⊛}} \overline{csy}_{2}(Ξ)) is isomorphic to the consequence system (csy_{1} ⊛ csy_{2})(Ξ).

Proof. Obviously, according to 5.1.28, \overline{▷}_{k}^{Ξ} is the least structural consequence on L\overline{Σ}_{k}(Ξ)

such that the inclusion morphism of Σ^{k} into \overline{Σ}^{k} still yields a morphism from csy_{k} to \overline{csy}_{k}.

From 6.1.1 and 5.1.15, we can conclude that (csy_{1} ⊛ csy_{2})(Ξ) = ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩ where ▷^{Ξ} is generated by all the instances of the rules

({⟨Γ_{1}, γ_{1}⟩ : Γ ▷_{1}^{Ξ} γ} ∪ {⟨Δ_{2}, δ_{2}⟩ : Δ ▷_{2}^{Ξ} δ})

where Γ and γ range over L_{Σ^{1}}(Ξ) and Δ and δ range over L_{Σ^{2}}(Ξ).

From 4.1.10, we can conclude that (\overline{csy}_{1}(Ξ) ▷◁^{S^{⊛}} \overline{csy}_{2}(Ξ)) = ⟨L_{▷◁}^{S^{⊛}}, ▷⟩ where ▷ is generated by the inferences

(\{⟨[Γ_{1}]_{≈^{S^{⊛}}} , [γ_{1}]_{≈^{S^{⊛}}} ⟩ : Γ\overline{▷}_{1}^{Ξ}γ\} ∪ \{⟨[Δ_{2}]_{≈^{S^{⊛}}} , [δ_{2}]_{≈^{S^{⊛}}} ⟩ : Δ\overline{▷}_{2}^{Ξ}δ\})

where Γ and γ now range over L\overline{Σ}_{1}(Ξ) and Δ and δ over L\overline{Σ}_{2}(Ξ).

Let us consider ȷ as defined before. Using 2.1.35, we shall prove that ȷ : ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩ → ⟨L_{▷◁}^{S^{⊛}}, ▷⟩ is an isomorphism in Csy. From 6.4.18 we know that ȷ is a bijection, so we are left with showing that ȷ(Θ^{▷^{Ξ}}) = ȷ(Θ)^{▷}, for every Θ ⊆ L_{Σ}(Ξ). Equivalently, using 2.1.12 and 2.1.15, we shall indeed prove that Θ is ▷^{Ξ}-closed if and only if ȷ(Θ) is ▷-closed.

First assume that ȷ(Θ) is ▷-closed and, without loss of generality, that Γ ▷_{1}^{Ξ} γ and Γ_{1}σ ⊆ Θ for some σ ∈ Sub(Σ). Take σ′ = (ȷ_{1} ∘ σ) ∈ Sub(\overline{Σ}^{1}). Easily, we have that ȷ_{1}(γ′σ) = ȷ_{1}(γ′)σ′ for every γ′ ∈ L_{Σ}(Ξ). Therefore, we have also that ȷ_{1}(Γ_{1})σ′ = ȷ_{1}(Γ_{1}σ) ⊆ ȷ_{1}(Θ). Since Γ ⊆ L_{Σ^{1}}(Ξ), by definition of ȷ_{1}, we have that ȷ_{1}(Γ_{1}) = Γ and Γσ′ ⊆ ȷ_{1}(Θ), which implies ι_{1}(Γσ′) ⊆ Θ and [(Γσ′)_{1}]_{≈^{S^{⊛}}} = [ȷ_{1}(ι_{1}(Γσ′))_{1}]_{≈^{S^{⊛}}} = ȷ(ι_{1}(Γσ′)) ⊆ ȷ(Θ). Moreover, since Γ ▷_{1}^{Ξ} γ we have that (Γ, γ) is a generating rule for \overline{▷}_{1}^{Ξ}. Therefore, Γσ′\overline{▷}_{1}^{Ξ}γσ′ and ⟨[(Γσ′)_{1}]_{≈^{S^{⊛}}} , [(γσ′)_{1}]_{≈^{S^{⊛}}} ⟩ is a generating inference for ▷. Thus, since ȷ(Θ) is ▷-closed it must be the case that [(γσ′)_{1}]_{≈^{S^{⊛}}} ∈ ȷ(Θ). Hence, ι_{1}(γσ′) = ι([(γσ′)_{1}]_{≈^{S^{⊛}}}) ∈ Θ. But γ ∈ L_{Σ^{1}}(Ξ) and therefore ȷ_{1}(γ_{1}) = γ. So, ι_{1}(γσ′) = ι_{1}(ȷ_{1}(γ_{1})σ′) = ι_{1}(ȷ_{1}(γ_{1}σ)) = γ_{1}σ ∈ Θ, which completes the first half of the proof.

Assume now that Θ is ▷^{Ξ}-closed and, without loss of generality, that Δ\overline{▷}_{2}^{Ξ}δ and [Δ_{2}]_{≈^{S^{⊛}}} ⊆ ȷ(Θ). Clearly, this is equivalent to ι_{2}(Δ) = ι([Δ_{2}]_{≈^{S^{⊛}}}) ⊆ Θ. Suppose that ι_{2} : \overline{csy}_{2}(Ξ) → ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩ is a Csy-morphism. Then, it easily follows that ι_{2}(Δ) ▷^{Ξ} ι_{2}(δ) and since Θ is ▷^{Ξ}-closed we immediately have ι_{2}(δ) ∈ Θ. Hence, as we wanted, [δ_{2}]_{≈^{S^{⊛}}} = [ȷ_{2}(ι_{2}(δ))_{2}]_{≈^{S^{⊛}}} = ȷ(ι_{2}(δ)) ∈ ȷ(Θ).

However, we need to prove that ι_{2} : \overline{csy}_{2}(Ξ) → ⟨L_{Σ}(Ξ), ▷^{Ξ}⟩ is indeed a Csy-morphism. So, we have to show that ι_{2}^{−1}(Θ) = ȷ_{2}(Θ) is \overline{▷}_{2}^{Ξ}-closed still assuming that Θ is ▷^{Ξ}-closed. For the purpose, assume now that Δ▷_{2}^{Ξ} δ with Δ and δ now ranging over L_{Σ^{2}}(Ξ), and consider σ ∈ Sub(\overline{Σ}^{2}) such that Δσ ⊆ ȷ_{2}(Θ). Take σ′ = (ι_{2} ∘ σ)Sub(Σ). Easily, ȷ_{2}(θσ′) = ȷ_{2}(θ)σ for every θ ∈ L_{Σ}(Ξ). Since Δ and δ range over L_{Σ^{2}}(Ξ), by definition of ȷ_{2}, we have that ȷ_{2}(Δ_{2}) = Δ and ȷ_{2}(δ_{2}) = δ. Therefore, ȷ_{2}(Δ_{2}σ′) = ȷ_{2}(Δ_{2})σ = Δσ ⊆ ȷ_{2}(Θ) and immediately Δ_{2}σ′ ⊆ Θ. But ⟨Δ_{2}, δ_{2}⟩ is a generating rule for ▷^{Ξ}, and since Θ is ▷^{Ξ}-closed we must also have δ_{2}σ′ ⊆ Θ. So, δσ = ȷ_{2}(δ_{2})σ = ȷ_{2}(δ_{2}σ′) ∈ ȷ_{2}(θ) and we are done. ⊓ ⊔

A similar result for the layered case follows immediately.

Proposition 6.4.20. For k ∈ {1, 2}, let csy_{k} = ⟨Σ^{k}, ▷_{k}^{lΞ}, ▷_{k}^{lΞ}⟩ be structural layered consequence systems, and consider the enriched structural layered consequence systems \overline{csy}_{k} = ⟨\overline{Σ}^{k}, \overline{▷}_{k}^{lΞ}, \overline{▷}_{k}^{gΞ}⟩, with \overline{▷}_{k}^{gΞ} the consequence on L\overline{Σ}_{k}(Ξ) generated by all instances of the rules

{⟨Θ, θ⟩ : Θ ▷_{k}^{gΞ} θ}

and \overline{▷}_{k}^{lΞ} the consequence on L\overline{Σ}_{k}(Ξ) generated by \{⟨∅, θ′⟩ : \overline{▷}_{k}^{gΞ}θ′\} and all instances of the rules

{⟨Θ, θ⟩ : Θ ▷_{k}^{lΞ} θ}

where Θ and θ range over L_{Σ^{k}}(Ξ). Then, (\overline{csy}_{1}(Ξ) ▷◁^{S^{⊛}} \overline{csy}_{2}(Ξ)) is equivalent to the layered consequence system (csy_{1} ⊛ csy_{2})(Ξ).

Proof. Immediate, from the definitions of 4.1.14 and 6.1.14, by using 6.4.19. ⊓ ⊔

Analogous results can be proved to explain Dsy fibring using lInf synchronization and Int fibring using lVal synchronization. The proof-theoretic part might be expected, given the exactness of the construction. However, model-theoretically, the relationship also holds. We start by settling a characterization of isomorphisms in lInf.

Proposition 6.4.21. Two layered inference systems inf_{k} = ⟨L_{k}, I_{k}, gI_{k}⟩ for k ∈ {1, 2} are isomorphic if and only if there exists a bijection h : L_{1} → L_{2} such that h(I_{1}) = I_{2} and h(gI_{1}) = gI_{2}.

Proof. Suppose that f : inf_{1} → inf_{2} is an isomorphism in lInf. That is, there exists a morphism g : inf_{2} → inf_{1} such that (g∘f) = id_{L_{1}} and (f ∘g) = id_{L_{2}}. Obviously, f must be a bijection and g = f^{−1}. Since f is a morphism we have f(I_{1}) ⊆ I_{2} and f(gI_{1}) ⊆ f(gI_{2}). On the other hand, also f^{−1} is a morphism and we have f^{−1}(I_{2}) ⊆ I_{1} and f^{−1}(I_{2}) ⊆ I_{1}. Therefore, we also have I_{2} = f(f^{−1}(I_{2})) ⊆ f(I_{1}) and I_{2} = f(f^{−1}(I_{2})) ⊆ f(I_{1}).

Assume now that h : L_{1} → L_{2} is a bijection with h(I_{1}) = I_{2} and h(gI_{1}) = gI_{2}. Clearly, h : inf_{1} → inf_{2} is a lInf-morphism. We just need to prove that also h^{−1} : inf_{2} → inf_{1} is a lInf-morphism. Easily, h^{−1}(I_{2}) = h^{−1}(h(I_{1})) = I_{1} and h^{−1}(gI_{2}) = h^{−1}(h(gI_{1})) = gI_{1} and we conclude the proof. ⊓ ⊔

Of course the desired property of our proof-theoretic setting follows easily. The enrichments made to the deductive systems just do not need to add further rules. The point is that just by enriching the signature, one is already allowing a much wider ranger of possible instantiations of the existing rules.

Proposition 6.4.22. For k ∈ {1, 2}, let dsy_{k} = ⟨Σ^{k}, R_{k}, gR_{k}⟩ be deductive systems, and consider the enriched deductive systems \overline{dsy}_{k} = ⟨\overline{Σ}^{k}, R_{k}, gR_{k}⟩. Then, the layered inference systems (lInf_{Dsy}(\overline{dsy}_{1}) ▷◁^{S^{⊛}} lInf_{Dsy}(\overline{dsy}_{2})) and lInf_{Dsy}(dsy_{1} ⊛ dsy_{2}) are isomorphic.

Proof. Recall from 5.2.36 the definition of lInf_{Dsy} : Dsy → lInf. From 6.2.14, we can conclude that lInf_{Dsy}(dsy_{1} ⊛ dsy_{2}) = ⟨L_{Σ}(Ξ), I, gI⟩, where

I = ( ⋃_{k∈{1,2}}{r_{k}σ : r ∈ R_{k}, σ ∈ Sub(Σ)})

and

gI = ( ⋃_{k∈{1,2}}{r_{k}σ : r ∈ gR_{k}, σ ∈ Sub(Σ)}).

From 4.2.5, we can conclude that (lInf_{Dsy}(\overline{dsy}_{1}) ▷◁^{S^{⊛}} lInf_{Dsy}(\overline{dsy}_{2})) = ⟨L_{▷◁}^{S^{⊛}}, \overline{I}, \overline{gI}⟩, where

\overline{I} = ( ⋃_{k∈\{1,2\}}\{[(rσ′)_{k}]_{≈^{S^{⊛}}} : r ∈ R_{k}, σ′ ∈ Sub(\overline{Σ}^{k})\})

and

\overline{gI} = ( ⋃_{k∈\{1,2\}}\{[(rσ′)_{k}]_{≈^{S^{⊛}}} : r ∈ gR_{k}, σ′ ∈ Sub(\overline{Σ}^{k})\}).

Again, let us consider ȷ as defined before. Using 6.4.21, we are going to prove that ȷ : ⟨L_{Σ}(Ξ), I, gI⟩ → ⟨L_{▷◁}^{S^{⊛}}, \overline{I}, \overline{gI}⟩ is an isomorphism in lInf. From 6.4.18 we already know that ȷ is a bijection, so we are left with showing that ȷ(I) = \overline{I} and ȷ(gI) = \overline{gI}.

Clearly, that ȷ(I) ⊆ \overline{I} and ȷ(gI) ⊆ \overline{gI} follows immediately from the fact that, for every k ∈ {1, 2}, θ ∈ L_{Σ^{k}}(Ξ) and σ ∈ Sub(Σ), it is the case that ȷ(θ_{k}σ) = [(θσ′)_{k}]_{≈^{S^{@}}} , where σ′ = (ȷ_{k} ∘ σ) ∈ Sub(\overline{Σ}^{k}). In fact, as we have seen before, ȷ_{1}(θ′σ) = ȷ_{1}(θ′)σ′ for every θ′ ∈ L_{Σ}(Ξ). Moreover, since θ ∈ L_{Σ^{k}}(Ξ) we clearly have ȷ_{k}(θ_{k}) = θ. Thus, ȷ(θ_{k}σ) = [ȷ_{k}(θ_{k}σ)_{k}]_{≈^{S^{@}}} and ȷ_{k}(θ_{k}σ) = ȷ_{k}(θ_{k})σ′ = θσ′.

On the other hand, the reverse inclusions \overline{I} ⊆ ȷ(I) and \overline{gI} ⊆ ȷ(gI) also follow from the property above, since for any σ′ ∈ Sub(\overline{Σ}^{k}) we can build σ ∈ Sub(Σ) such that σ′ = (ȷ_{k}∘σ). Just define σ = (ι_{k} ∘ σ′). Easily, (ȷ_{k} ∘ σ) = (ȷ_{k} ∘ ι_{k} ∘ σ′) = σ′. ⊓ ⊔

Isomorphisms in lVal are also very simple.

Proposition 6.4.23. Two layered valuation systems val_{k} = ⟨L_{k}, M_{k}, S_{k}⟩ for k ∈ {1, 2} are isomorphic if and only if there exist bijections \overline{h} : L_{1} → L_{2} and \underline{h} : M_{2} → M_{1} such that h : val_{1} → val_{2} is an lVal-morphism.

Proof. Suppose that f : val_{1} → val_{2} is an isomorphism in lVal. That is, there exists a morphism g : val_{2} → val_{1} such that (\overline{g} ∘ \overline{f}) = id_{L_{1}} and (\overline{f} ∘ \overline{g}) = id_{L_{2}}, and (\underline{f} ∘ \underline{g}) = id_{M_{1}} and (\underline{g} ∘ \underline{f}) = id_{M_{2}}. Obviously, both \overline{f} and \underline{f} are bijections.

Assume now that \overline{h} : L_{1} → L_{2} and \underline{h} : M_{2} → M_{1} are bijections with h : val_{1} → val_{2} an lVal-morphism. Clearly, we need to prove that also h^{−1} : val_{2} → val_{1} is an lValmorphism. If m ∈ M_{1}, obviously, \underline{h}^{−1}(m) ∈ M_{2}. So, since h is a morphism, we have that S_{1}(m) = S_{1}(\underline{h}(\underline{h}^{−1}(m))) = S_{2}(\underline{h}^{−1}(m))|\overline{h}. Therefore, S_{1}(m)|\overline{h}_{−1} = (S_{2}(\underline{h}^{−1}(m))|\overline{h})|\overline{h}_{−1} = S_{2}(\underline{h}^{−1}(m))|_{(\overline{h}∘\overline{h}^{−1})} = S_{2}(\underline{h}^{−1}(m))|_{id_{L_{2}}} = S_{2}(\underline{h}^{−1}(m)) and h^{−1} is indeed a morphism. ⊓ ⊔

In order to obtain the desired result, as expected, the interpretation systems must be enriched in such a way as to allow any possible extension of existing structures with arbitrary values for the newly introduced monoliths.

Proposition 6.4.24. For k ∈ {1, 2}, let int_{k} = ⟨Σ^{k}, M_{k}, S_{k}⟩ be interpretation systems, and consider the enriched interpretation systems \overline{int}_{k} = ⟨\overline{Σ}^{k}, \overline{M}_{k}, \overline{S}_{k}⟩ with

\overline{M}_{k} = \{⟨m, ϑ⟩ : m ∈ M_{k}, ϑ ∈ T_{m}^{L_{Σ}^{k}(Ξ)}\}

and each

\overline{S}_{k}(⟨m, ϑ⟩) = ⟨T_{⟨m,ϑ⟩}, ⋉_{m}⟩

where T_{⟨m,ϑ⟩} = T_{m}, \underline{c} _{n}^{T_{⟨m,ϑ⟩}} = \underline{c} _{n}^{T_{m}} for every n ∈ IN and c ∈ Σ_{n}^{k}, and \underline{θ} _{0}^{T_{⟨m,ϑ⟩}} = ϑ(θ).

Then, the layered valuation systems (lVal_{Int}(\overline{int}_{1}) ▷◁^{S^{⊛}} lVal_{Int}(\overline{int}_{2})) and lVal_{Int}(int_{1}⊛int_{2}) are isomorphic.

Proof. Recall from 5.2.17 the definition of lVal_{Int} : Int → lVal. From 6.2.1, we can conclude that lVal_{Int}(int_{1} ⊛ int_{2}) = ⟨L_{Σ}(Ξ), M, S⟩, where M is the class of all tuples

⟨⟨m_{1}, m_{2}⟩, α⟩

with m_{1} ∈ M_{1} and m_{2} ∈ M_{2}, T_{m_{1}} = T_{m_{2}}, ⋉_{m_{1}} = ⋉_{m_{2}} and α ∈ Asg(T_{m_{1}}) = Asg(T_{m_{1}}). Of course, we define T_{⟨m_{1},m_{2}⟩} = T_{m_{1}} = T_{m_{2}} and ⋉_{⟨m_{1},m_{2}⟩} = ⋉_{m_{1}} = ⋉_{m_{2}} ,and obtain

S(⟨⟨m_{1}, m_{2}⟩, α⟩) = ⟨T_{⟨m_{1},m_{2}⟩}, [[ ]]_{α}^{T_{⟨m_{1},m_{2}⟩}}, ⋉_{⟨m_{1},m_{2}⟩}⟩.

From 4.2.3, we can conclude that (lVal_{Int}(\overline{int}_{1}) ▷◁^{S^{⊛}} lVal_{Int}(\overline{int}_{2})) = ⟨L_{▷◁}^{S^{⊛}}, \overline{M}, \overline{S}⟩, where \overline{M} is the class of all tuples

m = ⟨⟨⟨m_{1}, ϑ_{1}⟩, α_{1}⟩, ⟨⟨m_{2}, ϑ_{2}⟩, α_{2}⟩⟩

with m_{1} ∈ M_{1} and m_{2} ∈ M_{2}, T_{⟨⟨m_{1},ϑ_{1}⟩,α_{1}⟩} = T_{⟨⟨m_{2},ϑ_{2}⟩,α_{2}⟩}, ⋉_{⟨⟨m_{1},ϑ_{1}⟩,α_{1}⟩} = ⋉_{⟨⟨m_{2},ϑ_{2}⟩,α_{2}⟩} and α_{1}, α_{2} ∈ Asg(T_{⟨m_{1},ϑ_{1}⟩}) = Asg(T_{⟨m_{2},ϑ_{2}⟩}), such that, for every θ ∈ L_{Σ}(Ξ), we have

[[ȷ_{1}(θ)]]_{α_{1}}^{T_{⟨m_{1},ϑ_{1}⟩}} = [[ȷ_{2}(θ)]]_{α_{2}}^{T_{⟨m_{2},ϑ_{2}⟩}}.

Of course, we define T_{m} = T_{⟨⟨m_{1},ϑ_{1}⟩,α_{1}⟩} = T_{⟨⟨m_{2},ϑ_{2}⟩,α_{2}⟩} and ⋉_{m} = ⋉_{⟨⟨m_{1},ϑ_{1}⟩,α_{1}⟩} = ⋉_{⟨⟨m_{2},ϑ_{2}⟩,α_{2}⟩}, and obtain

\overline{S}(m) = ⟨T_{m}, [[ ]]_{m}, ⋉_{m}⟩

where

[[e]]_{m} = [[ȷ_{1}(ι(e))]]_{α_{1}}^{T_{⟨m_{1},ϑ_{1}⟩}} = [[ȷ_{2}(ι(e))]]_{α_{2}}^{T_{⟨m_{2},ϑ_{2}⟩}}.

Thus, in particular, ȷ_{1}(ξ) = ȷ_{2}(ξ) = ξ and we must have α_{1}(ξ) = α_{2}(ξ) for every ξ ∈ Ξ. The same is to say that α_{1} = α_{2} = α.

Again, let us consider ȷ as defined before. Using 6.4.23, we just need to find a bijection \underline{h} : \overline{M} → M such that S(\underline{h}(m)) = \overline{S}(m)|_{ȷ}. Given m = ⟨⟨⟨m_{1}, ϑ_{1}⟩, α_{1}⟩, ⟨⟨m_{2}, ϑ_{2}⟩, α_{2}⟩⟩ define \underline{h}(m) = ⟨⟨m_{1}, m_{2}⟩, α⟩. This map is clearly well defined. Indeed, we have that T_{m_{1}} = T_{⟨m_{1},ϑ_{1}⟩} = T_{m} = T_{⟨m_{2},ϑ_{2}⟩} = T_{m_{2}} and ⋉_{m_{1}} = ⋉_{⟨m_{1},ϑ_{1}⟩} = ⋉_{m} = ⋉_{⟨m_{2},ϑ_{2}⟩} = ⋉_{m_{2}}. Thus, ⟨m_{1}, m_{2}⟩ is a fibred model and α = α_{1} = α_{2} ∈ Asg(T_{⟨m_{1},m_{2}⟩}).

Injectivity of \underline{h} follows immediately if we prove that m_{1}, m_{2} and α completely determine ϑ_{1} and ϑ_{2}. In fact, we shall prove that ϑ_{k}(θ) = [[θ]]_{α}^{T_{⟨m_{1},m_{2}⟩}} for every θ ∈ L_{Σ}^{k}(Ξ). Indeed, we already know that [[ȷ_{1}(θ)]]_{α}^{T_{⟨m_{1},ϑ_{1}⟩}} = [[ȷ_{2}(θ)]]_{α}^{T_{⟨m_{2},ϑ_{2}⟩}} for every θ ∈ L_{Σ}(Ξ). In particular,

if θ ∈ L_{Σ}^{k}(Ξ), we have that ϑ_{k}(θ) = [[ȷ_{j}(θ)]]_{α}^{T_{⟨m_{j} ,ϑ_{j} ⟩}} where k, j ∈ {1, 2} and j ≠ k. Thus, it suffices that we prove that [[ȷ_{k}(θ)]]_{α}^{T_{⟨m_{k},ϑ_{k}⟩}} = [[θ]]_{α}^{T_{⟨m_{1},m_{2}⟩}} for every k ∈ {1, 2} and θ ∈ L_{Σ}(Ξ). We proceed by induction on the structure of θ.

• Base case.

For a metavariable ξ, trivially, ȷ_{k}(ξ) = ξ and [[ξ]]_{α}^{T_{⟨m_{k},ϑ_{k}⟩}} = α(ξ) = [[ξ]]_{α}^{T_{⟨m_{1},m_{2}⟩}}.

• Step: θ is either of the form c_{1}(θ_{1}, …, θ_{n}) or c_{2}(θ_{1}, …, θ_{n}), and the induction hypothesis asserts that [[ȷ_{k}(θ_{m})]]_{α}^{T_{⟨m_{k},ϑ_{k}⟩}} = [[θ_{m}]]_{α}^{T_{⟨m_{1},m_{2}⟩}} for m ∈ {1, …, n} and k ∈ {1, 2}.

If we have c_{k}(θ_{1}, …, θ_{n}) (with c ∈ Σ_{n}^{k}) then, it follows that ȷ_{k}(c_{k}(θ_{1}, …, θ_{n})) = c(ȷ_{k}(θ_{1}), …, ȷ_{k}(θ_{n})) and thus we easily obtain that [[c(ȷ_{k}(θ_{1}), …, ȷ_{k}(θ_{n}))]]_{α}^{T_{⟨m_{k},ϑ_{k}⟩}} = \underline{c}_{n}^{T_{m_{k}}} ([[ȷ_{k}(θ_{1})]]_{α}^{T_{⟨m_{k},ϑ_{k}⟩}}, …, [[ȷ_{k}(θ_{n})]]_{α}^{T_{⟨m_{k},ϑ_{k}⟩}}). So, by induction hypothesis, we obtain \underline{c}_{n}^{T_{m_{k}}} ([[θ_{1}]]_{α}^{T_{⟨m_{1},m_{2}⟩}}, …, [[θ_{n}]]_{α}^{T_{⟨m_{1},m_{2}⟩}}) = [[c_{k}(θ_{1}, …, θ_{n})]]_{α}^{T_{⟨m_{1},m_{2}⟩}}.

On the other hand, if we have c_{j}(θ_{1}, …, θ_{n}) (with c ∈ Σ_{n}^{j} and j ≠ k) then, eas-

ily, ȷ_{k}(c_{j}(θ_{1}, …, θ_{n})) = c_{j}(θ_{1}, …, θ_{n}) , and we obtain [[ c_{j}(θ_{1}, …, θ_{n}) ]]_{α}^{T_{⟨m_{k},ϑ_{k}⟩}} =

ϑ_{k}( c_{j}(θ_{1}, …, θ_{n}) ). But we already know that, necessarily, ϑ_{k}( c_{j}(θ_{1}, …, θ_{n}) ) =

[[ȷ_{j}(c_{j}(θ_{1}, …, θ_{n}))]]_{α}^{T_{⟨m_{j} ,ϑ_{j} ⟩}} and ȷ_{j}(c_{j}(θ_{1}, …, θ_{n})) = c(ȷ_{j}(θ_{1}), …, ȷ_{j}(θ_{n})). Thus, we

have that [[c(ȷ_{j}(θ_{1}), …, ȷ_{j}(θ_{n}))]]_{α}^{T_{⟨m_{j} ,ϑ_{j} ⟩}} = \underline{c}_{n}^{T_{m_{j}}} ([[ȷ_{j}(θ_{1})]]_{α}^{T_{⟨m_{j} ,ϑ_{j} ⟩}}, …, [[ȷ_{j}(θ_{n})]]_{α}^{T_{⟨m_{j} ,ϑ_{j} ⟩}}). Using the induction hypothesis for j, we obtain \underline{c}_{n}^{T_{m_{j}}} ([[θ_{1}]]_{α}^{T_{⟨m_{1},m_{2}⟩}}, …, [[θ_{n}]]_{α}^{T_{⟨m_{1},m_{2}⟩}}) = [[c_{j}(θ_{1}, …, θ_{n})]]_{α}^{T_{⟨m_{1},m_{2}⟩}}.

As for the surjectivity of \underline{h}, easily, for every ⟨⟨m_{1}, m_{2}⟩, α⟩ ∈ M, we just need to pick each ϑ_{k}(θ) = [[θ]]_{α}^{T_{⟨m_{1},m_{2}⟩}}.

Finally, we prove that S(\underline{h}(m)) = \overline{S}(m)|_{ȷ}, given m = ⟨⟨⟨m_{1}, ϑ_{1}⟩, α⟩, ⟨⟨m_{2}, ϑ_{2}⟩, α⟩⟩ ∈ \overline{M}. Obviously, we have T_{⟨m_{1},m_{2}⟩} = T_{m} and ⋉_{⟨m_{1},m_{2}⟩} = ⋉_{m}. Thus, we are left with showing that

[[θ]]_{α}^{T_{⟨m_{1},m_{2}⟩}} = [[ȷ(θ)]]_{m} for every θ ∈ L_{Σ}(Ξ). Easily, for instance, [[ȷ(θ)]]_{m} = [[ȷ_{1}(ι(ȷ(θ)))]]_{m} = [[ȷ_{1}(θ)]]_{α}^{T_{⟨m_{1},ϑ_{1}⟩}} = [[θ]]^{T_{⟨m_{1},m_{2}⟩}}. ⊓ ⊔

As we have said before, the above representation of fibring is perhaps the richest illustration of the “hidden” power of synchronization.

6.5 Transference results

As we have said about synchronization, a useful theory of logic combination must encompass general results of preservation of properties. We shall concentrate on soundness and completeness. Again we shall only deal explicitly with layered logic system presentations, but according to 5.2.8 the results obtained shall be immediately applicable to the corresponding indexed notion of layered logic presentation.

6.5.1 Soundness preservation

As it was the case for synchronization, and we have already seen that both synchronization and fibring have been settled by using similar tools, we shall see that soundness is always preserved. The question is posed, of course, just in the context of interpretation/deductive systems. Still, it could similarly be obtained for corresponding non-layered versions, developed as hinted in Chapter 5. In fact, we can again sum it all up in the following lemma, whose proof mimics that of 4.4.1.

Lemma 6.5.1. Let ⟨Cal, Ded⟩ and ⟨Sem, Ent⟩ be categories of (layered) logic system presentations, the former being finitary and exact for fibring, and the latter suitable for fibring. Moreover, let cal_{k} ∈ |Cal| and sem_{k} ∈ |Sem| be such that Sgn_{Cal}(cal_{k}) = Sgn_{Sem}(sem_{k}), for k ∈ {1, 2}, and S a corresponding sharing constraint. If both pairs ⟨cal_{1}, sem_{1}⟩ and ⟨cal_{2}, sem_{2}⟩ are sound, and {f_{k} : cal_{k} → (cal_{1} ⊛^{S} cal_{2})}_{k∈{1,2}} and {g_{k} : sem_{k} → (sem_{1} ⊛^{S} sem_{2})}_{k∈{1,2}} correspond to fibrings in Cal and Sem constrained by S, then the unique h : Sgn_{Cal}(cal_{1} ⊛^{S} cal_{2}) → Sgn_{Sem}(sem_{1} ⊛^{S} sem_{2}) such that (h ∘ Sgn_{Cal}(f_{k})) = Sgn_{Sem}(g_{k}), for k ∈ {1, 2}, is a signature isomorphism and induces an s(l)Csy-morphism h : Ded(cal_{1} ⊛^{S} cal_{2}) → Ent(sem_{1} ⊛^{S} sem_{2}).

Proof. Let us start by fixing the notation Σ^{k} = Sgn_{Cal}(cal_{k}) = Sgn_{Sem}(sem_{k}) for k ∈ {1, 2}, and assuming that S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}}.

Once ⟨Cal, Ded⟩ is exact for fibring, it immediately follows that not only the sink {Ded(f_{k}) : Ded(cal_{k}) → Ded(cal_{1} ⊛^{S} cal_{2})}_{k∈{1,2}} is an Sgn_{s(l)Csy}-pushout of Ded(cal_{1}) and Ded(cal_{2}) along S, but also that {Sgn_{Cal}(f_{k}) : Σ^{k} → Sgn_{Cal}(cal_{1} ⊛^{S} cal_{2})}_{k∈{1,2}} is a pushout of S in Sig. Moreover, {Sgn_{Sem}(g_{k}) : Σ^{k} → Sgn_{Sem}(sem_{1} ⊛^{S} sem_{2})}_{k∈{1,2}} is also a pushout of S in Sig, since ⟨Sem, Ent⟩ is suitable for fibring. In particular, we obviously have that (Sgn_{Sem}(g_{1}) ∘ s_{1}) = (Sgn_{Sem}(g_{2}) ∘ s_{2}).

Clearly, the fact that each pair ⟨cal_{k}, sem_{k}⟩ is sound amounts precisely to the fact that id_{Σ^{k}} : Ded(cal_{k}) → Ent(sem_{k}) is an s(l)Csy-morphism. Thus, let us consider the sink {(Ent(g_{k}) ∘ id_{Σ^{k}}) : Ded(cal_{k}) → Ent(sem_{1} ⊛^{S} sem_{2})}_{k∈{1,2}}. Easily, then, we have (Sgn_{s(l)Csy}(Ent(g_{1}) ∘ id_{Σ^{1}}) ∘ s_{1}) = (Sgn_{Sem}(g_{1}) ∘ id_{Σ^{1}} ∘ s_{1}) = (Sgn_{Sem}(g_{1}) ∘ s_{1}) = (Sgn_{Sem}(g_{2}) ∘ s_{2}) = (Sgn_{Sem}(g_{2}) ∘ id_{Σ^{2}} ∘ s_{2}) = (Sgn_{s(l)Csy}(Ent(g_{2}) ∘ id_{Σ^{2}}) ∘ s_{2}). So, using the universal property of the Sgn_{s(l)Csy}-pushout Ded(cal_{1} ⊛^{S} cal_{2}), there exists a unique s(l)Csy-morphism h : Ded(cal_{1} ⊛^{S} cal_{2}) → Ent(sem_{1} ⊛^{S} sem_{2}) such that (h ∘ Ded(f_{k})) = (Ent(g_{k}) ∘ id_{Σ^{k}}) for k ∈ {1, 2}.

Therefore, we also have that Sgn_{s(l)Csy}(h ∘ Ded(f_{k})) = Sgn_{s(l)Csy}(Ent(g_{k}) ∘ id_{Σ^{k}}), i.e., (h ∘ Sgn_{Cal}(f_{k})) = (Sgn_{Sem}(g_{k}) ∘ id_{Σ^{k}}) = Sgn_{Sem}(g_{k}). The universal property of the pushout Sgn_{Cal}(cal_{1} ⊛^{S} cal_{2}) clearly imposes that h : Sgn_{Cal}(cal_{1} ⊛^{S} cal_{2}) → Sgn_{Sem}(sem_{1} ⊛^{S} sem_{2}) is the unique such morphism. Given that Sgn_{Sem}(sem_{1} ⊛^{S} sem_{2}) is also a pushout of S, it is well known that h must be an isomorphism in Sig. ⊓ ⊔

Please note that, again, the finitariness condition is not essential. It justifies the intuition that ⟨Cal, Ded⟩ is a category of proof-theoretic logic system presentations.

We can now instantiate this result with our parameters of interest, i.e., interpretation systems and deductive systems. Please observe that it applies even if we work with interpretation systems that are not closed (for isomorphisms, products and generated substructures).

Proposition 6.5.2. Let dsy_{k} = ⟨Σ^{k}, R_{k}, gR_{k}⟩ and int_{k} = ⟨Σ^{k}, M_{k}, S_{k}⟩, for k ∈ {1, 2}, be deductive and interpretation systems, respectively, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}} a corresponding sharing constraint. Then, the pair ⟨(dsy_{1} ⊛^{S} dsy_{2}), (int_{1} ⊛^{S} int_{2})⟩ (as defined in 6.2.14 and 6.2.1) is sound whenever both ⟨dsy_{1}, int_{1}⟩ and ⟨dsy_{2}, int_{2}⟩ are sound.

Proof. We already know, from 5.2.44 and 6.2.14, that Dsy is finitary and exact for fibring. From 6.2.1 we also know that Int is suitable for fibring. Moreover, in both 6.2.14 and 6.2.1 the syntactic construction coincides precisely with (Σ^{1} ⊛^{S} Σ^{2}). Therefore, if both ⟨dsy_{1}, int_{1}⟩ and ⟨dsy_{2}, int_{2}⟩ are sound, we can use lemma 6.5.1 and conclude that the unique morphism h referred by the lemma is precisely id_{(Σ^{1}S_{⊛}Σ^{2})} : (Σ^{1} ⊛^{S} Σ^{2}) → (Σ^{1} ⊛^{S} Σ^{2}). Hence, id_{(Σ^{1}S_{⊛}Σ^{2})} : (dsy_{1} ⊛^{S} dsy_{2}) → (int_{1} ⊛^{S} int_{2}) is guaranteed to be an slCsy-morphism, or equivalently, ⟨(dsy_{1} ⊛^{S} dsy_{2}), (int_{1} ⊛^{S} int_{2})⟩ is sound. ⊓ ⊔

6.5.2 Completeness preservation

General transference results for completeness are necessarily harder to obtain, when compared with those for soundness. As we have seen, soundness transfers, without further provisos, for both free and constrained fibring as a consequence of just the facts that Int is suitable and Dsy is exact for fibring. However, the converse does not hold. As we have seen in 6.2.1, it is not hard to find examples showing that Int can misbehave. Concrete examples of non-preservation of completeness through fibring have also been identified [Wol96]. Thus, our aim is to establish sufficient fair conditions for completeness to transfer through fibring. Below, we shall establish three different completeness preservation results. The key idea for all three cases is to guarantee the existence of enough models of a certain kind in our target interpretation systems.

Definition 6.5.3. Let dsy = ⟨Σ, R, gR⟩ be a deductive system. A Σ-structure ⟨T , ⋉⟩ is said to be a structure for dsy provided that, for every assignment α ∈ Asg(T ):

• for each r ∈ R, {[[γ]]_{α}^{T} : γ ∈ Prem(r)} ⋉ [[Conc(r)]]_{α}^{T};

• for each r ∈ gR, if ⋉[[γ]]_{α}^{T} for every γ ∈ Prem(r) then ⋉[[Conc(r)]]_{α}^{T}.

We denote the class of all structures for dsy by Str(dsy).

This means precisely that the structure is such that all the rules of dsy are sound with respect to it.

In what follows we shall consider as fixed an arbitrary property P of consequence systems, and for each signature Σ, denote by Str(Σ, P) the class of all P -structures over Σ, i.e, Σ-structures ⟨T , ⋉⟩ such that ⟨T, ⋉⟩ satisfies P.

Definition 6.5.4. Let dsy = ⟨Σ, R, gR⟩ be a deductive system and int = ⟨Σ, M, S⟩ an interpretation system. The pair ⟨dsy, int⟩ is said to be full with respect to P if (Str(Σ, P)∩ Str(dsy)) ⊆ S(M).

Clearly, if ⟨dsy, int⟩ is full with respect to the class of all Σ-structures, we have that ⟨dsy, int⟩ is sound if and only if S(M) = Str(dsy).

In this setting it is easy to show that fullness with respect to P is preserved by fibring. We first prove an auxiliary lemma.

Lemma 6.5.5. Let dsy_{k} for k ∈ {1, 2} be deductive systems, S a corresponding sharing constraint, and {ι_{k}^{S} : dsy_{k} → (dsy_{1} ⊛^{S} dsy_{2})}_{k∈{1,2}} be the sink corresponding to the fibring of dsy_{1} and dsy_{2} constrained by S (as given by 6.2.14). If ⟨T , ⋉⟩ is a structure for (dsy_{1} ⊛^{S} dsy_{2}) then ⟨T |_{ι_{k}^{S}}, ⋉⟩ is a structure for dsy_{k} for every k ∈ {1, 2}.

Proof. Let each dsy_{k} = ⟨Σ^{k}, R_{k}, gR_{k}⟩, and denote T |_{ι_{k}^{S}} simply by T_{k}. Recall that ι_{k}^{S} : Σ^{k} → (Σ^{1} ⊛^{S} Σ^{2}) is the signature morphism such that ι_{k,n}^{S}(c) = [c_{k}]_{≈_{n}^{S}}, for every n ∈ IN and c ∈ Σ_{n}^{k}. Thus, T_{k} is the Σ^{k}-algebra with the same carrier T as T and such that \underline{c}_{n}^{T_{k}} = \underline{[c_{k}]_{≈_{n}^{S}}} _{n}^{T}. Moreover, easily, Asg(T_{k}) = Asg(T ), and for every α ∈ Asg(T_{k}) and δ ∈ L_{Σ^{k}}(Ξ) we have that [[δ]]_{α}^{T_{k}} = [[[δ_{k}]_{≈^{S}}]]_{α}^{T}.

Let us show that ⟨T_{k}, ⋉⟩ is a structure for dsy_{k}. If r ∈ R_{k} then it follows that ⟨[Prem(r)_{k}]_{≈^{S}}, [Conc(r)_{k}]_{≈^{S}}⟩ is a local rule of (dsy_{1} ⊛^{S} dsy_{2}). Thus, since ⟨T , ⋉⟩ is a structure for it, we have that [[[Prem(r)_{k}]_{≈^{S}}]]_{α}^{T} ⋉ [[[Conc(r)_{k}]_{≈^{S}}]]_{α}^{T}. But this is equivalent to [[Prem(r)]]_{α}^{T_{k}} ⋉ [[Conc(r)]]_{α}^{T_{k}} and we are done.

Similarly, if r ∈ gR_{k} then it follows that ⟨[Prem(r)_{k}]_{≈^{S}}, [Conc(r)_{k}]_{≈^{S}}⟩ is a global rule of (dsy_{1} ⊛^{S} dsy_{2}). Thus, since ⟨T , ⋉⟩ is a structure for it, we have that if ⋉[[[Prem(r)_{k}]_{≈^{S}}]]_{α}^{T} then ⋉[[[Conc(r)_{k}]_{≈^{S}}]]_{α}^{T}. But this is equivalent to having ⋉[[Conc(r)]]_{α}^{T_{k}} whenever we have ⋉[[Prem(r)]]_{α}^{T_{k}} and we are done, again. ⊓ ⊔

Proposition 6.5.6. For k ∈ {1, 2}, let ⟨dsy_{k}, int_{k}⟩ be a pair of deductive and interpretation systems over the same signature and full with respect to P. Then, for every corresponding constraint S, the pair ⟨(dsy_{1} ⊛^{S} dsy_{2}), (int_{1} ⊛^{S} int_{2})⟩ is full with respect to P.

Proof. Assume that each int_{k} = ⟨Σ^{k}, M_{k}, S_{k}⟩ and (int_{1} ⊛^{S} int_{2}) = ⟨(Σ^{1} ⊛^{S} Σ^{2}), M, S⟩, and let ⟨T , ⋉⟩ be a P -structure for (dsy_{1} ⊛^{S} dsy_{2}). According to 6.5.5, letting {ι_{k}^{S} : Σ^{k} → (Σ^{1} ⊛^{S} Σ^{2})}_{k∈{1,2}} be the sink corresponding to the fibring, we have that ⟨T_{k}, ⋉⟩ with T_{k} = T |_{ι_{k}^{S}} is a P -structure for dsy_{k}. Therefore, since each pair ⟨dsy_{k}, int_{k}⟩ is full with respect to P, there exist models m_{1} ∈ M_{1} and m_{2} ∈ M_{2} such that S_{1}(m_{1}) = ⟨T_{1}, ⋉⟩ and S_{2}(m_{2}) = ⟨T_{2}, ⋉⟩. We now prove that ⟨m_{1}, m_{2}⟩ ∈ M and S(⟨m_{1}, m_{2}⟩) = ⟨T , ⋉⟩. Clearly, T_{m_{1}} = T_{m_{2}} = T and ⋉_{m_{1}} = ⋉_{m_{2}} = ⋉. Thus, it suffices to show that T_{1} and T_{2} agree on the interpretation of shared symbols. Thus, assume that S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}}, and let n ∈ IN and c ∈ Σ_{n}^{0}. Obviously, \underline{s_{k,n}(c)}_{n}^{T_{k}} = \underline{[s_{k,n}(c)_{k}]_{≈_{n}^{S}}} _{n}^{T} for k ∈ \{1, 2\}. But we know that s_{1,n}(c)_{1} ≈_{n}^{S} s_{2,n}(c)_{2}, i.e., [s_{1,n}(c)_{1}]_{≈_{n}^{S}} = [s_{2,n}(c)_{2}]_{≈_{n}^{S}} and the result follows. It is straigthforward to verify that S(⟨m_{1}, m_{2}⟩) = ⟨T , ⋉⟩. In fact it suffices to recall that \underline{[c_{k}]_{≈_{n}^{S}}} _{n}^{T} = \underline{c}_{n}^{T_{k}}. ⊓ ⊔

Our first completeness preservation result is quite general. We shall admit any possible structure as good, or in other words, we shall consider P to be satisfied by all consequence systems. So, for every signature Σ, Str(Σ, P) = Str(Σ). In this broad case we can build arbitrary structures for a deductive system, very easily, as follows.

Proposition 6.5.7. Let dsy = ⟨Σ, R, gR⟩ be a deductive system and Γ ⊆ L_{Σ}(Ξ) be ⊢_{dsy}^{gΞ} - closed. Then, the pair

⟨L_{Σ}(Ξ), ⋉_{Γ}⟩

is a structure for dsy, where:

Δ^{⋉_{Γ}} = (Γ ∪ Δ)^{⊢_{dsy}^{lΞ}}, for every Δ ⊆ L_{Σ}(Ξ).

Proof. Obviously, L_{Σ}(Ξ) is the free Σ-algebra generated by Ξ. Moreover, the construction of ⋉_{Γ} as a consequence operation on L_{Σ}(Ξ) is an instance of 2.1.2. Therefore, we have a Σ-structure and it remains to be proved that it is indeed a structure for dsy. Recall that an assignment in L_{Σ}(Ξ) is precisely a substitution over Σ. Furthermore, given σ ∈ Sub(Σ) = Asg(L_{Σ}(Ξ)), [[γ]]_{σ}^{L_{Σ}(Ξ)} = γσ.

If r ∈ R, then, Prem(r)σ⊢_{dsy}^{lΞ} Conc(r)σ just by using rσ ∈ R^{Ξ}. Therefore, using the monotonicity of ⊢_{dsy}^{lΞ} it follows that (Γ ∪ Prem(r)σ)⊢_{dsy}^{lΞ} Conc(r)σ, and we conclude that Prem(r)σ ⋉_{Γ} Conc(r)σ.

Suppose now that r ∈ gR and ⋉_{Γ}Prem(r)σ, i.e., Γ⊢_{dsy}^{lΞ} Prem(r)σ. Of course, we know that Γ is ⊢_{dsy}^{gΞ} -closed, and thus also ⊢_{dsy}^{lΞ} -closed. Hence, we can conclude that Prem(r)σ ⊆ Γ and, by now using rσ ∈ gR^{Ξ} and the ⊢_{dsy}^{gΞ} -closedness of Γ, we also have Conc(r)σ ∈ Γ. Thus, by extensiveness, Γ⊢_{dsy}^{lΞ} Conc(r)σ. That is, ⋉_{Γ}Conc(r)σ and the result follows. ⊓ ⊔

As it stands, this straightforward representation of the deductive system as an interpretation structure is sufficient to guarantee the following completeness theorem.

Proposition 6.5.8. Let ⟨dsy, int⟩ be a pair of deductive and interpretation systems over the same signature and full with respect to all structures. Then, the pair ⟨dsy, int⟩ is complete.

Proof. Let int = ⟨Σ, M, S⟩. We start by proving that ⟨dsy, int⟩ is globally complete. For the purpose, suppose that Δ⊢_{d̸sy}^{gΞ}θ. Of course, we need to show that Δ⊨_{i̸nt}^{gΞ}θ. So, it is sufficient to find a model m of int and an assignment α ∈ Asg(T_{m}) such that ⋉_{m}[[δ]]_{α}^{T_{m}} for every δ ∈ Δ, but ⋉_{m̸}[[θ]]_{α}^{T_{m}}. Make Γ = Δ^{⊢_{dsy}^{gΞ}}, and consider the structure ⟨L_{Σ}(Ξ), ⋉_{Γ}⟩ defined from dsy as in 6.5.7 above. As we know, it is a structure for dsy. Hence, since ⟨dsy, int⟩ is full with respect to the class of all Σ-structures, there exists a model m ∈ M with S(m) = ⟨L_{Σ}(Ξ), ⋉_{Γ}⟩. Let now α = id_{Ξ}. Clearly, [[γ]]_{α}^{L_{Σ}(Ξ)} = γ. Now, easily, we have ⋉_{Γ}Δ since Δ ⊆ Γ = Δ^{⊢_{dsy}^{gΞ}}, by just using the extensivenes of both ⊢_{dsy}^{gΞ} and ⊢_{dsy}^{lΞ} .

Furthermore, ⋉_{Γ̸}θ since Γ is ⊢_{dsy}^{gΞ} -closed and therefore also ⊢_{dsy}^{lΞ} -closed, and θ ∉ Γ = Δ^{⊢_{dsy}^{gΞ}}.

Let us turn to local completeness. Suppose, now, that Δ⊢_{d̸sy}^{lΞ}θ. Of course, we need to show that Δ⊨_{i̸nt}^{lΞ}θ. So, it is sufficient to find a model m of int and an assignment α ∈ Asg(T_{m}) such that {[[δ]]_{α}^{T_{m}} : δ ∈ Δ}⋉_{m̸}[[θ]]_{α}^{T_{m}}. Make Γ = ∅^{⊢_{dsy}^{gΞ}}, and consider the structure ⟨L_{Σ}(Ξ), ⋉_{Γ}⟩ defined from dsy, again as in 6.5.7 above. It is for sure a structure for dsy, and since ⟨dsy, int⟩ is full with respect to the class of all structures, again there exists a model m ∈ M with S(m) = ⟨L_{Σ}(Ξ), ⋉_{Γ}⟩. Once again, take α = id_{Ξ}. Clearly, we need to show that Δ⋉_{Γ̸}θ. But this is immediate because θ ∉ Δ^{⊢_{dsy}^{lΞ}} = (Γ ∪ Δ)^{⊢_{dsy}^{lΞ}}, since we have Γ = ∅^{⊢_{dsy}^{gΞ}} ⊆ ∅^{⊢_{dsy}^{lΞ}}. ⊓ ⊔

Thus, our first completeness preservation result follows easily.

Proposition 6.5.9. For k ∈ {1, 2}, let ⟨dsy_{k}, int_{k}⟩ be a pair of deductive and interpretation systems over the same signature and full with respect to all structures. Then, for every corresponding constraint S, the pair ⟨(dsy_{1} ⊛^{S} dsy_{2}), (int_{1} ⊛^{S} int_{2})⟩ is complete.

Of course, it is hard to accept as legitimate semantic structures such structures for dsy as those defined in 6.5.7 and used to establish 6.5.8 and 6.5.9. These structures are certainly too close to deduction. And, of course, it is not surprising that a deductive system is complete with respect to itself when seen as an interpretation structure. However, we can improve this state of affairs by trying to forbid such structures. In fact, in general, these do even lack the property of being congruent with respect to their own internal logical equivalence. As a consequence, we are allowed to have many equivalent truth values, and in particular, several distinguished values. To avoid this situation, we shall now consider P to be the class of all elementary consequence systems. Recall that, in particular, an elementary consequence system has at the most one theorem. Of course, the obvious thing to do now is to apply an adapted Lindenbaum construction to the model built in 6.5.7. That is only possible, of course, when logical equivalence in the deductive system constitutes a congruence on the free algebra of schema sentences. The following situation, which we adapt from [ZSS99], is enough for our purposes. As expected, we shall recall the notion of equivalence as defined in 3.1.26, over the schematic layered consequence system induced by dsy. Thus, given Γ, we shall use ≡_{dsy}^{Γ} to denote the logical equivalence relation under Γ induced by ⟨L_{Σ}(Ξ), ⊢_{dsy}^{lΞ} , ⊢_{dsy}^{gΞ} ⟩. Remember that it coincides with equivalence in the non-layered system ⟨L_{Σ}(Ξ), ⊢_{dsy}^{lΞ} ⟩.

Proposition 6.5.10. Let dsy be a deductive system over signature Σ and Γ ⊆ L_{Σ}(Ξ) be ⊢_{dsy}^{gΞ}-closed, with ≡_{dsy}^{Γ} a congruence on L_{Σ}(Ξ). Then, the pair

⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{Γ}^{≡_{dsy}}⟩

is an elementary structure for dsy, where:

{[δ]_{≡_{dsy}^{Γ}} : δ ∈ Δ}^{⋉_{Γ}^{≡_{dsy}}} = {[θ]_{≡dsy^{Γ}} : (Γ ∪ Δ)⊢_{dsy}^{lΞ}θ}, for every Δ ⊆ L_{Σ}(Ξ).

Proof. Knowing that ≡_{dsy}^{Γ} is a congruence on L_{Σ}(Ξ), we certainly have a quotient Σalgebra L_{Σ}(Ξ)/_{≡dsy^{Γ}} . Moreover, ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{Γ}^{≡_{dsy}}⟩ is certainly an elementary conse-

quence system, since it corresponds to a particular case of 2.1.40. So, we are left with showing that ⟨L_{Σ}(Ξ)/_{≡dsy^{Γ}} , ⋉_{Γ}^{≡_{dsy}}⟩ is a structure for dsy.

In fact, given α ∈ Asg(L_{Σ}(Ξ)/_{≡dsy^{Γ}} ), it easily follows that [[γ]]_{α}^{L_{Σ}(Ξ)/_{≡{Γ \atop dsy}}} = [γσ]_{≡dsy^{Γ}} for

any substitution σ such that [σ(ξ)]_{≡dsy^{Γ}} = α(ξ), for every ξ ∈ Ξ. Such a σ can always be

obtained, if necessary by using the axiom of choice.

If r ∈ R, then, Prem(r)σ⊢_{dsy}^{lΞ} Conc(r)σ just by using rσ ∈ R^{Ξ}. Therefore, using the monotonicity of ⊢_{dsy}^{lΞ} it follows that (Γ ∪ Prem(r)σ)⊢_{dsy}^{lΞ} Conc(r)σ, and we conclude that {[γσ]_{≡dsy^{Γ}} : γ ∈ Prem(r)} ⋉_{Γ}^{≡_{dsy}} [Conc(r)σ]_{≡dsy^{Γ}} .

Suppose now that r ∈ gR and ⋉_{Γ}^{≡_{dsy}}{[γσ]_{≡dsy^{Γ}} : γ ∈ Prem(r)}, i.e., Γ⊢_{dsy}^{lΞ} Prem(r)σ. Of

course, we know that Γ is ⊢_{dsy}^{gΞ} -closed, and thus also ⊢_{dsy}^{lΞ} -closed. Hence, we can conclude that Prem(r)σ ⊆ Γ and, by using now rσ ∈ gR^{Ξ} and the ⊢_{dsy}^{gΞ} -closedness of Γ, we also have Conc(r)σ ∈ Γ. Thus, by extensiveness, Γ⊢_{dsy}^{lΞ} Conc(r)σ. That is, ⋉_{Γ}^{≡_{dsy}}[Conc(r)σ]_{≡dsy^{Γ}} and

the result follows. ⊓ ⊔

The following property of a deductive system obviously guarantees that the contruction above can always be done.

Definition 6.5.11. A deductive system dsy over signature Σ is said to be congruent if for every ⊢_{dsy}^{gΞ}-closed Γ ⊆ L_{Σ}(Ξ), n ∈ IN, c ∈ Σ_{n} and δ_{1}, θ_{1}, …, δ_{n}, θ_{n} ∈ L_{Σ}(Ξ) the following condition holds:

• if δ_{k} ≡_{dsy}^{Γ} θ_{k} for k ∈ {1, …, n} then c(δ_{1}, …, δ_{n}) ≡_{dsy}^{Γ} c(θ_{1}, …, θ_{n}).

Clearly, this is equivalent to requiring that, for every ⊢_{dsy}^{gΞ}-closed Γ, ≡_{dsy}^{Γ} is a congruence on L_{Σ}(Ξ). Fortunately, most usual deductive systems are congruent in this sense. But there are exceptions, as for instance the paraconsistent system of example 2.3.8. However, if we had not required Γ to be ⊢_{dsy}^{gΞ}-closed even normal modal logic would fall out.

We are now in a position to guarantee the following completeness theorem. Its proof is all in all similar to that of our previous 6.5.8, but now it uses interpretation structures built according to 6.5.10.

Proposition 6.5.12. Let ⟨dsy, int⟩ be a pair of deductive and interpretation systems over the same signature and full with respect to the class of all elementary structures. If dsy is congruent then the pair ⟨dsy, int⟩ is complete.

Proof. Let int = ⟨Σ, M, S⟩. We start by proving that ⟨dsy, int⟩ is globally complete. For the purpose, suppose that Δ⊢_{d̸sy}^{gΞ}θ. Of course, we need to show that Δ⊨_{i̸nt}^{gΞ}θ. So, it is sufficient to find a model m of int and an assignment α ∈ Asg(T_{m}) such that ⋉_{m}[[δ]]_{α}^{T_{m}} for every δ ∈ Δ, but ⋉_{m̸}[[θ]]_{α}^{T_{m}}. Make Γ = Δ^{⊢_{dsy}^{gΞ}}, and consider the structure ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{Γ}^{≡_{dsy}}⟩ de-

fined from dsy as in 6.5.10 above. Clearly, because dsy is congruent, we have that ≡_{dsy}^{Γ} is a congruence on L_{Σ}(Ξ), the construction is possible, and we obtain an elementary structure for dsy. Thus, given that ⟨dsy, int⟩ is full with respect to the class of all elementary structures, there exists a model m ∈ M with S(m) = ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{Γ}^{≡_{dsy}}⟩. Let now α be the

assignment on L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}} such that each α(ξ) = [ξ]_{≡_{dsy}^{Γ}}. Clearly, [[γ]]_{α}^{L_{Σ}(Ξ)/_{≡{Γ \atop dsy}}} = [γ]_{≡_{dsy}^{Γ}}.

Now, easily, we have ⋉_{Γ}^{≡_{dsy}}[δ]_{≡_{dsy}^{Γ}}, for every δ ∈ Δ, since Δ ⊆ Γ = Δ^{⊢_{dsy}^{gΞ}}, by just using

the extensiveness of both ⊢_{dsy}^{gΞ} and ⊢_{dsy}^{lΞ} . Furthermore, ⋉_{Γ̸}^{≡_{dsy}}[θ]_{≡_{dsy}^{Γ}} since Γ is ⊢_{dsy}^{gΞ} -closed

and therefore also ⊢_{dsy}^{lΞ} -closed, and θ ∉ Γ = Δ^{⊢_{dsy}^{gΞ}}. Obviously then, by definition of ≡_{dsy}^{Γ}, ([θ]_{≡_{dsy}^{Γ}} ∩ Γ) = ∅.

Let us turn to local completeness. Suppose, now, that Δ⊢_{d̸sy}^{lΞ}θ. Of course, we need to show that Δ⊨_{i̸nt}^{lΞ}θ. So, it is sufficient to find a model m of int and an assignment α ∈ Asg(T_{m}) such that {[[δ]]_{α}^{T_{m}} : δ ∈ Δ}⋉_{m̸}[[θ]]_{α}^{T_{m}}. Make Γ = ∅^{⊢_{dsy}^{gΞ}}, and consider the structure ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{Γ}^{≡_{dsy}}⟩ defined from dsy, again as in 6.5.10 above. Again, the

construction is possible because dsy is congruent and therefore ≡_{dsy}^{Γ} is a congruence on L_{Σ}(Ξ). Thus we obtain an elementary structure for dsy, and since ⟨dsy, int⟩ is full with respect to the class of all elementary structures, again there exists a model m ∈ M with S(m) = ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{Γ}^{≡_{dsy}}⟩. Once again, take α such that each α(ξ) = [ξ]_{≡_{dsy}^{Γ}}. Clearly,

we need to show that {[δ]_{≡_{dsy}^{Γ}} : δ ∈ Δ}⋉_{Γ̸}^{≡_{dsy}}[θ]_{≡_{dsy}^{Γ}}. But this is immediate because θ ∉ Δ^{⊢_{dsy}^{lΞ}} = (Γ ∪ Δ)^{⊢_{dsy}^{lΞ}}, since we have Γ = ∅^{⊢_{dsy}^{gΞ}} ⊆ ∅^{⊢_{dsy}^{lΞ}}. Therefore, by definition of ≡_{dsy}^{Γ},

([θ]_{≡_{dsy}^{Γ}} ∩ Δ^{⊢_{dsy}^{lΞ}}) = ∅. ⊓ ⊔

Notwithstanding the completeness proof above, our second completeness preservation result does not follow immediately. In fact, we have no guarantee that the fibring of two congruent deductive systems is still congruent. Indeed, counterexamples can be easily found (see [ZSS99]). However, as it was also shown in [ZSS99], there is a fair situation in which everything works smoothly. The requirement is that the given deductive systems can suitably internalize logical equivalence. We shall rephrase that result here.

Definition 6.5.13. Let dsy = ⟨Σ, R, gR⟩ be a deductive system. A constructor ⇒ ∈ Σ_{2} is said to be an implication on dsy whenever the following condition holds, for every Γ ⊆ L_{Σ}(Ξ), δ_{1}, δ_{2} ∈ L_{Σ}(Ξ):

• Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}) if and only if (Γ ∪ {δ_{1}})⊢_{dsy}^{lΞ} δ_{2}.

This condition can be easily understood as being equivalent to both the metatheorems of modus ponens (left to right) and deduction (right to left).

It is now easy to verify that back and forth implications coincide with logical equivalence in any deductive system.

Proposition 6.5.14. Let dsy = ⟨Σ, R, gR⟩ be a deductive system with implication, Γ ⊆

L_{Σ}(Ξ) and δ_{1}, δ_{2} ∈ L_{Σ}(Ξ). Then, δ_{1} ≡_{dsy}^{Γ} δ_{2} if and only if (δ_{1} ⇒ δ_{2}), (δ_{2} ⇒ δ_{1}) ∈ Γ^{⊢dsy^{lΞ}} .

Proof. Suppose that δ_{1} ≡_{dsy}^{Γ} δ_{2}, i.e., (Γ∪{δ_{1}})⊢_{dsy}^{lΞ} δ_{2} and (Γ∪{δ_{2}})⊢_{dsy}^{lΞ} δ_{1}. Using deduction on both, then, we obtain Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}) and Γ⊢_{dsy}^{lΞ} (δ_{2} ⇒ δ_{1}).

On the other hand, assume that both Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}) and Γ⊢_{dsy}^{lΞ} (δ_{2} ⇒ δ_{1}). Using modus ponens on both, then, we obtain (Γ∪{δ_{1}})⊢_{dsy}^{lΞ} δ_{2} and (Γ∪{δ_{2}})⊢_{dsy}^{lΞ} δ_{1}. Thus, equivalently, we have that δ_{1} ≡_{dsy}^{Γ} δ_{2}. ⊓ ⊔

But, of course, we need to go further on in order to ensure that equivalence defined via back and forth implication is indeed a congruence.

Definition 6.5.15. A deductive system dsy with implication over signature Σ is said to have substitution by equivalents whenever the following condition holds, for every n ∈ IN, c ∈ Σ_{n} and ξ_{1}, ξ′_{1}, …, ξ_{n}, ξ′_{n} ∈ Ξ:

• {(ξ_{1} ⇒ ξ′_{1}), (ξ′_{1} ⇒ ξ_{1}), …, (ξ_{n} ⇒ ξ′_{n}), (ξ′_{n} ⇒ ξ_{n})}⊢_{dsy}^{gΞ} (c(ξ_{1}, …, ξ_{n}) ⇒ c(ξ′_{1}, …, ξ′_{n})).

Note that many usual deductive systems have implication and substitution by equivalents. In particular, every system including the basic logic of [SBF99] is covered. Normal modal logics also have this property.

Easily, we can now show that in the presence of implication, substitution by equivalents and congruence coincide.

Proposition 6.5.16. Let dsy be a deductive system with implication. Then, dsy has substitution by equivalents if and only if dsy is congruent.

Proof. Let dsy be over signature Σ and assume that Γ ⊆ L_{Σ}(Ξ) is ⊢_{dsy}^{gΞ}-closed , n ∈ IN, c ∈ Σ_{n} and δ_{1}, θ_{1}, …, δ_{n}, θ_{n} ∈ L_{Σ}(Ξ) are such that δ_{k} ≡_{dsy}^{Γ} θ_{k} for k ∈ {1, …, n}. According to 6.5.14, this means that Γ⊢_{dsy}^{lΞ} (δ_{k} ⇒ θ_{k}) and Γ⊢_{dsy}^{lΞ} (θ_{k} ⇒ δ_{k}) for every k ∈ {1, …, n}. However, since Γ is ⊢_{dsy}^{gΞ}-closed it must also be ⊢_{dsy}^{lΞ}-closed, and it immediately follows that {(δ_{1} ⇒ θ_{1}), (δ_{1} ⇒ θ_{1}) …, (δ_{n} ⇒ θ_{n}), (δ_{n} ⇒ θ_{n})} ⊆ Γ. Therefore, if dsy has substitution by equivalents, according to 6.5.15, we can use twice the structurality of ⊢_{dsy}^{gΞ} to immediately obtain, by monotonicity, that (c(δ_{1}, …, δ_{n})⇒c(θ_{1}, …, θ_{n})), (c(θ_{1}, …, θ_{n})⇒c(δ_{1}, …, δ_{n})) ∈ Γ^{⊢_{dsy}^{gΞ}} = Γ. The arguments for each of them are of course symmetric. Therefore, again by 6.5.14, we have that c(δ_{1}, …, δ_{n}) ≡_{dsy}^{Γ} c(θ_{1}, …, θ_{n}).

Now, on the other hand, let us assume that dsy is congruent and consider the ⊢_{dsy}^{gΞ}-

closed set Γ = {(ξ_{1} ⇒ ξ′_{1}), (ξ′_{1} ⇒ ξ_{1}), …, (ξ_{n} ⇒ ξ′_{n}), (ξ′_{n} ⇒ ξ_{n})}^{⊢_{dsy}^{gΞ}}. Obviously, Γ is also ⊢_{dsy}^{lΞ}-closed and according to 6.5.14 we have that ξ_{k} ≡_{dsy}^{Γ} ξ′_{k} for k ∈ {1, …, n}. Thus, by congruence, we also have that c(ξ_{1}, …, ξ_{n}) ≡_{dsy}^{Γ} c(ξ′_{1}, …, ξ_{n}). Therefore, again by 6.5.14, we have that in particular c(ξ_{1}, …, ξ_{n}) ⇒ c(ξ′_{1}, …, ξ_{n})) ∈ Γ. ⊓ ⊔

And now, going via implication, we can finally prove our second completeness preservation result. First, however, we shall present a more suitable characterization of implication.

Proposition 6.5.17. A constructor ⇒ ∈ Σ_{2} is an implication on a deductive system dsy = ⟨Σ, R, gR⟩ if and only if the following conditions hold:

• ⊢_{dsy}^{lΞ} (ξ_{1} ⇒ ξ_{1});

• {(ξ_{1} ⇒ ξ_{2}), ξ_{1}}⊢_{dsy}^{lΞ} ξ_{2};

• {ξ_{2}}⊢_{dsy}^{lΞ} (ξ_{1} ⇒ ξ_{2});

• {(ξ_{r} ⇒ γ_{1}), …, (ξ_{r} ⇒ γ_{n})}⊢_{dsy}^{lΞ} (ξ_{r} ⇒ γ), for every r = ⟨{γ_{1}, …, γ_{n}}, γ⟩ ∈ R,

where we take each ξ_{r} as a metavariable not occuring in r.

Proof. Assume that dsy has implication. We prove, in turn, that all the four conditions above hold. Clearly, by extensiveness, {ξ_{1}}⊢_{dsy}^{lΞ} ξ_{1}. So, from deduction, we obtain ⊢_{dsy}^{lΞ} (ξ_{1} ⇒ ξ_{1}). Again by extensiveness, we have that {(ξ_{1} ⇒ ξ_{2})}⊢_{dsy}^{lΞ} (ξ_{1} ⇒ ξ_{2}). So, now from modus ponens, we obtain {(ξ_{1} ⇒ ξ_{2}), ξ}⊢_{dsy}^{lΞ} ξ_{2}. By extensiveness, once again, we have {ξ_{2}, ξ_{1}}⊢_{dsy}^{lΞ} ξ_{2}. Thus, from deduction, we obtain {ξ_{2}}⊢_{dsy}^{lΞ} (ξ_{1} ⇒ ξ_{2}). Let now r = ⟨{γ_{1}, …, γ_{n}}, γ⟩ ∈ R. By definition, we have that {γ_{1}, …, γ_{n}}⊢_{dsy}^{lΞ} γ. Moreover, for each k ∈ {1, …, n}, we have already shown that {(ξ_{r} ⇒ γ_{k}), ξ_{r}}⊢_{dsy}^{lΞ} γ_{k}. Therefore, from the properties of ⊢_{dsy}^{lΞ} , we can conclude that {(ξ_{r} ⇒ γ_{1}), …, (ξ_{r} ⇒ γ_{k}), ξ_{r}}⊢_{dsy}^{lΞ} γ. Hence, from deduction, we obtain {(ξ_{r} ⇒ γ_{1}), …, (ξ_{r} ⇒ γ_{n})}⊢_{dsy}^{lΞ} (ξ_{r} ⇒ γ).

Suppose, on the other hand, that all four conditions above hold. Clearly, the metatheorem of modus ponens is an immediate consequence of the second condition, the rule of modus ponens. In fact, if Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}), since from the rule of modus ponens plus the structurality of ⊢_{dsy}^{lΞ} we know that {(δ_{1} ⇒ δ_{2}), δ_{1}}⊢_{dsy}^{lΞ} δ_{2}, it easily follows from the properties of ⊢_{dsy}^{lΞ} that (Γ ∪ {δ_{1}})⊢_{dsy}^{lΞ} δ_{2}. To prove the metatheorem of deduction, as-

sume that (Γ ∪ {δ_{1}})⊢_{dsy}^{lΞ} δ_{2}. Recall that (Γ ∪ {δ_{1}})^{⊢_{dsy}^{lΞ}} = (⋃_{n∈IN} Ω_{dsy^{l}}^{n}(Γ ∪ {δ_{1}})), with Ω_{dsy}_{l}(Δ) = (Δ ∪ ∅^{⊢_{dsy}^{gΞ}} ∪ {Conc(r)σ : r ∈ R, σ ∈ Sub(Σ), Prem(r)σ ⊆ Δ}). Therefore, it is enough that we prove, by induction on n ∈ IN, that if δ_{2} ∈ Ω_{dsy^{l}}^{n}(Γ ∪ {δ_{1}}) then Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}).

• Base case: n = 0.

Assume that, δ_{2} ∈ Ω_{dsy^{l}}^{0}(Γ ∪ {δ_{1}}) = (Γ ∪ {δ_{1}}). If δ_{2} ∈ Γ then, by extensiveness, Γ⊢_{dsy}^{lΞ} δ_{2}. Moreover, from the third condition plus the structurality of ⊢_{dsy}^{lΞ} we know that {δ_{2}}⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}). Thus, it easily follows from the properties of ⊢_{dsy}^{lΞ} that Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}). If δ_{2} = δ_{1} then, since from the first condition plus the structurality of ⊢_{dsy}^{lΞ} we know that ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{1}), we easily obtain Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{1}).

• Step: n > 0 and the induction hypothesis asserts that if δ′_{2} ∈ Ω_{dsy^{l}}^{n−1}(Γ ∪ {δ_{1}}) then Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ′_{2}).

Assume that, δ_{2} ∈ Ω_{dsy^{l}}^{n}(Γ∪{δ_{1}}) = Ω_{dsy}_{l}(Ω_{dsy^{l}}^{n−1}(Γ∪{δ_{1}})) = (Ω_{dsy^{l}}^{n−1}(Γ∪{δ_{1}})∪∅^{⊢_{dsy}^{gΞ}} ∪ {Conc(r)σ : r ∈ R, σ ∈ Sub(Σ), Prem(r)σ ⊆ Ω_{dsy^{l}}^{n−1}(Γ∪{δ_{1}})}). If δ_{2} ∈ Ω_{dsy^{l}}^{n−1}(Γ∪{δ_{1}})

then the induction hypothesis asserts that Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}). If δ_{2} ∈ ∅^{⊢_{dsy}^{gΞ}} then, since

∅^{⊢_{dsy}^{gΞ}} = ∅^{⊢_{dsy}^{lΞ}}, we also have that ⊢_{dsy}^{lΞ} δ_{2}. Therefore, since from the third condition plus the structurality of ⊢_{dsy}^{lΞ} we know that {δ_{2}}⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}), we can easily conclude that Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}). Lastly, if δ_{2} = γσ for some r = ⟨{γ_{1}, …, γ_{n}}, γ⟩ ∈ R and σ ∈ Sub(Σ) such that {γ_{1}σ, …, γ_{n}σ} ⊆ Ω_{dsy^{l}}^{n−1}(Γ ∪ {δ_{1}}), we proceed as follows. From the fourth condition we know that {(ξ_{r} ⇒ γ_{1}), …, (ξ_{r} ⇒ γ_{n})}⊢_{dsy}^{lΞ} (ξ_{r} ⇒ γ), where ξ_{r} does not occur in γ_{1}, …, γ_{n}, γ. So, if we consider the substitution σ′ which coincides with σ except that σ′(ξ_{r}) = δ_{1}, we have γ_{k}σ = γ_{k}σ′ for k ∈ {1, …, n}, γσ = γσ′, and the structurality of ⊢_{dsy}^{lΞ} implies that {(δ_{1} ⇒ γ_{1}σ), …, (δ_{1} ⇒ γ_{n}σ)}⊢_{dsy}^{lΞ} (δ_{1} ⇒ γσ). Furthermore, the induction hypothesis guarantees that Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ γ_{k}σ) for k ∈ {1, …, n}. Thus, we easily obtain that Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ γσ).

So, Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{2}) and the metatheorem of deduction follows. ⊓ ⊔

Please note that the last condition needs only be checked for local rules with premises. For axioms, the condition easily follows from the third. These conditions are surely familiar to anyone acquainted with standard proofs of the metatheorem of deduction, e.g., for classical logic.

Proposition 6.5.18. For k ∈ {1, 2}, let dsy_{k} be a congruent deductive system with implication. Then, for every corresponding constraint S sharing the implications, the deductive system (dsy_{1} ⊛^{S} dsy_{2}) is congruent and has implication.

Proof. Let each dsy_{k} = ⟨Σ^{k}, R_{k}, gR_{k}⟩ with ⇒_{k} its implication constructor, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}}. Moreover, let {ι_{k}^{S} : dsy_{k} → (dsy_{1} ⊛^{S} dsy_{2})}_{k∈{1,2}} be the sink corresponding to the fibring of dsy_{1} and dsy_{2} constrained by S (as given by 6.2.14). If S shares ⇒_{1} and ⇒_{2} then, obviously, ι_{1,2}^{S}(⇒_{1}) = ι_{2,2}^{S}(⇒_{2}) = ⇒. Our aim is to prove that ⇒ is an implication in (dsy_{1} ⊛^{S} dsy_{2}) that further satisfies substitution by equivalents.

Obviously, each ι_{k}^{S} : csy_{dsy_{k}}^{Ξ} → csy_{(dsy_{1}S_{⊛}dsy_{2})}^{Ξ} is a morphism of structural consequence

systems. Therefore, each of the first three conditions from 6.5.17 for ⇒ in (dsy_{1} ⊛^{S} dsy_{2}) follows trivially the corresponding condition for either the ⇒_{1} implication in dsy_{1} or the ⇒_{1} implication in dsy_{1}. The fourth condition from 6.5.17 follows imediatelly for ⇒ in (dsy_{1} ⊛^{S} dsy_{2}), but now provided that it holds for both the ⇒_{1} implication in dsy_{1} and the ⇒_{2} implication in dsy_{2}. This is because the local rules of (dsy_{1} ⊛^{S} dsy_{2}) are either ι_{1}^{S} translations of local rules of dsy_{1} or ι_{2}^{S} translations of local rules of dsy_{2}. Thus, we conclude that ⇒ is an implication in (dsy_{1} ⊛^{S} dsy_{2}). According to 6.5.16, it now suffices to prove substitution by equivalents for ⇒ in (dsy_{1} ⊛^{S} dsy_{2}). But again it follows immediately from substitution by equivalents for ⇒_{1} in dsy_{1} and ⇒_{2} in dsy_{2}, because the constructors in (Σ^{1} ⊛^{S} Σ^{2}) are either ι_{1}^{S} translations of constructors of Σ^{1} or ι_{2}^{S} translations of constructors of Σ^{2}. Note that 6.5.16 ensures that we indeed have substitution by equivalents for ⇒_{1} in dsy_{1} and ⇒_{2} in dsy_{2}. ⊓ ⊔

Proposition 6.5.19. For k ∈ {1, 2}, let ⟨dsy_{k}, int_{k}⟩ be a pair of deductive and interpretation systems over the same signature and full with respect to all elementary structures. If both dsy_{1} and dsy_{2} are congruent and have implication then, for every corresponding constraint S sharing the implications, the pair ⟨(dsy_{1} ⊛^{S} dsy_{2}), (int_{1} ⊛^{S} int_{2})⟩ is complete.

Proof. Immediate from 6.5.6, 6.5.12 and 6.5.18. ⊓ ⊔

Note that, in the presence of implication, the construction in 6.5.10 yields exactly one distinguished truth-value. Although we have guaranteed that there is at the most one for deductive systems without axioms, we would have none for Γ = ∅^{⊢_{dsy}^{gΞ};}. With implication we shall always have, at least (ξ ⇒ ξ). Once again, we should stress that this construction is essentially the same as the one in [ZSS99], notwithstanding the fact that our proof does not need to consider maximal consistent sets.

Although this result is already quite significant, and still widely applicable, we would like to explore another issue. As it stands, it is very simple to verify that implication establishes a partial order on the truth values of structures as those defined in 6.5.10 that we have just used. Still, we did not consider the internal closure of the structure as being defined from this partial order. What happens if we do that? That is what we are going to explore next, bearing in mind that a completeness preservation result with respect to such partial structures would be even nicer to achieve. Please recall the construction of a consequence system from a partial order as presented in 3.2.9.

Definition 6.5.20. Let csy = ⟨L, ▷⟩ be a consequence system. Then, csy is said to be a partial order consequence system if there exists a partial order ⟨L, ≤⟩ such that ▷ = ▷_{≤}, i.e., Φ ▷ ψ if and only if ϑ ≤ φ for every φ ∈ Φ implies ϑ ≤ ψ.

Please note that every partial order consequence system is necessarily elementary, due to the antisymmetry of ≤.

Proposition 6.5.21. Let dsy over signature Σ be a deductive system with implication ⇒, and Γ ⊆ L_{Σ}(Ξ) be ⊢_{dsy}^{lΞ}-closed. Then, the relation defined by

[δ_{1}]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ_{2}]_{≡_{dsy}^{Γ}} if and only if (δ_{1} ⇒ δ_{2}) ∈ Γ

defines a partial order ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ≤_{Γ}⟩.

Proof. To see that ≤_{Γ} is well defined, assume that δ_{1} ≡_{dsy}^{Γ} θ_{1}, δ_{2} ≡_{dsy}^{Γ} θ_{2} and (δ_{1}⇒δ_{2}) ∈ Γ. Then, from δ_{1} ≡_{dsy}^{Γ} θ_{1} we have that (Γ ∪ {θ_{1}})⊢_{dsy}^{lΞ} δ_{1} and by the rule of modus ponens from (δ_{1} ⇒ δ_{2}) ∈ Γ we get that (Γ ∪ {θ_{1}})⊢_{dsy}^{lΞ} δ_{2}. But from δ_{2} ≡_{dsy}^{Γ} θ_{2} we also have that (Γ ∪ {δ_{2}})⊢_{dsy}^{lΞ} θ_{2} and therefore also (Γ ∪ {θ_{1}})⊢_{dsy}^{lΞ} θ_{2}. Using deduction and the ⊢_{dsy}^{lΞ}- closedness of Γ we finally conclude that also (θ_{1} ⇒ θ_{2}) ∈ Γ.

By the first condition of 6.5.17, the structurality of ⊢_{dsy}^{lΞ} implies that ⊢_{dsy}^{lΞ}(δ ⇒ δ). Therefore, since Γ is ⊢_{dsy}^{lΞ}-closed we have that (δ ⇒ δ) ∈ Γ, [δ]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ]_{≡_{dsy}^{Γ}} and reflexivity of ≤_{Γ} follows. Suppose now that [δ_{1}]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ_{2}]_{≡_{dsy}^{Γ}} and [δ_{2}]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ_{3}]_{≡_{dsy}^{Γ}}. By definition, this means that (δ_{1} ⇒ δ_{2}), (δ_{2} ⇒ δ_{3}) ∈ Γ. Therefore, by the rule of modus ponens from (δ_{1} ⇒ δ_{2}) ∈ Γ we get that (Γ ∪ {δ_{1}})⊢_{dsy}^{lΞ} δ_{2}, and again by the rule of modus ponens but now from (δ_{2} ⇒δ_{3}) ∈ Γ we conclude that (Γ∪{δ_{1}})⊢_{dsy}^{lΞ} δ_{3}. By using deduction, then, we have Γ⊢_{dsy}^{lΞ} (δ_{1} ⇒ δ_{3}) and therefore (δ_{1} ⇒ δ_{3}) ∈ Γ and ≤_{Γ} is transitive. To prove antisymmetry, let now [δ_{1}]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ_{2}]_{≡_{dsy}^{Γ}} and [δ_{2}]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ_{1}]_{≡_{dsy}^{Γ}}. That is, by definition,

(δ_{1} ⇒ δ_{2}), (δ_{2} ⇒ δ_{1}) ∈ Γ. But then, according to 6.5.14, δ_{1} ≡_{dsy}^{Γ} δ_{2} and we conclude that [δ_{1}]_{≡_{dsy}^{Γ}} = [δ_{2}]_{≡_{dsy}^{Γ}}. ⊓ ⊔

We now adapt slightly the result in 6.5.10.

Proposition 6.5.22. Let dsy be a deductive system with implication, over signature Σ, Γ ⊆ L_{Σ}(Ξ) be ⊢_{dsy}^{gΞ} -closed, with ≡_{dsy}^{Γ} a congruence on L_{Σ}(Ξ). Then, the pair

⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{≤_{Γ}}⟩

is a partial order structure for dsy.

Proof. It is trivial that ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{≤_{Γ}}⟩ is a partial order Σ-structure, since Γ is neces-

sarily ⊢_{dsy}^{lΞ}-closed and we have shown in 6.5.21 that ≤_{Γ} is a partial order on L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}. So, we just need to show that it is a structure for dsy.

Recall from 6.5.10 that given α ∈ Asg(L_{Σ}(Ξ)/_{≡dsy^{Γ}} ), we have [[γ]]_{α}^{L_{Σ}(Ξ)/_{≡{Γ \atop dsy}}} = [γσ]_{≡dsy^{Γ}}

for any substitution σ such that [σ(ξ)]_{≡dsy^{Γ}} = α(ξ), for every ξ ∈ Ξ, and that such a σ can

always be obtained, if necessary by using the axiom of choice.

If r = ⟨{γ_{1}, …, γ_{n}}, γ⟩ ∈ R, then, {γ_{1}σ, …, γ_{n}σ}⊢_{dsy}^{lΞ} γσ just by using rσ ∈ R^{Ξ}. To show that {[γ_{1}σ]_{≡dsy^{Γ}} , …, [γ_{n}σ]_{≡dsy^{Γ}} } ⋉_{≤_{Γ}} [γσ]_{≡dsy^{Γ}} , let δ ∈ L_{Σ}(Ξ) and suppose that

[δ]_{≡dsy^{Γ}} ≤_{Γ} [γ_{k}σ]_{≡dsy^{Γ}} for k ∈ {1, …, n}, i.e., {(δ ⇒ γ_{1}σ), …, (δ ⇒ γ_{n}σ)} ⊆ Γ. But then,

using the last condition of 6.5.17 and the structurality of ⊢_{dsy}^{lΞ} , with σ′ coinciding with σ except that σ′(ξ_{r}) = δ, we easily get that Γ⊢_{dsy}^{lΞ} (δ ⇒ γσ). However, since Γ is necessarily ⊢_{dsy}^{lΞ} -closed, this means that (δ ⇒ γσ) ∈ Γ and thus [δ]_{≡dsy^{Γ}} ≤_{Γ} [γσ]_{≡dsy^{Γ}} .

Suppose now that r = ⟨{γ_{1}, …, γ_{n}}, γ⟩ ∈ gR and ⋉_{≤_{Γ}}[γ_{k}σ]_{≡dsy^{Γ}} for k ∈ {1, …, n}.

This means that, for every δ ∈ L_{Σ}(Ξ) and k ∈ {1, …, n}, we have [δ]_{≡dsy^{Γ}} ≤_{Γ} [γ_{k}σ]_{≡dsy^{Γ}} ,

or equivalently, (δ ⇒ γ_{k}σ) ∈ Γ. In particular, if we consider δ′ = (ξ_{1} ⇒ ξ_{1}) ∈ Γ due to the first condition of 6.5.17 and the necessary ⊢_{dsy}^{lΞ} -closedness of Γ, we obtain that {γ_{1}σ, …, γ_{n}σ} ⊆ Γ. So, by using now rσ ∈ gR^{Ξ} and the ⊢_{dsy}^{gΞ} -closedness of Γ, we also have γσ ∈ Γ. Therefore, now according to the third condition of 6.5.17, we have that Γ⊢_{dsy}^{lΞ} (δ ⇒ γσ) for every δ ∈ L_{Σ}(Ξ). Thus, since Γ is ⊢_{dsy}^{lΞ} -closed, it must be the case that (δ ⇒ γσ) ∈ Γ and [δ]_{≡dsy^{Γ}} ≤_{Γ} [γσ]_{≡dsy^{Γ}} . Given that δ ∈ L_{Σ}(Ξ) is arbitrary, we finally

conclude that ⋉_{≤_{Γ}}[γσ]_{≡dsy^{Γ}} . ⊓ ⊔

We can now obtain the following partial completeness theorem.

Proposition 6.5.23. Let ⟨dsy, int⟩ be a pair of deductive and interpretation systems, over the same signature Σ, full with respect to the class of all partial order structures. If dsy is congruent and has implication, then the pair ⟨dsy, int⟩ is globally complete and locally weak complete.

Proof. Let int = ⟨Σ, M, S⟩. We start by proving that ⟨dsy, int⟩ is globally complete. For the purpose, suppose that Δ⊢_{d̸sy}^{gΞ}θ. Of course, we need to show that Δ⊨_{i̸nt}^{gΞ}θ. So, it is sufficient to find a model m of int and an assignment α ∈ Asg(T_{m}) such that ⋉_{m}[[δ]]_{α}^{T_{m}} for every δ ∈ Δ, but ⋉_{m̸}[[θ]]_{α}^{T_{m}}. Make Γ = Δ^{⊢_{dsy}^{gΞ}}, and consider the structure ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{≤_{Γ}}⟩ defined from dsy as in 6.5.22 above. Besides having implication, since dsy is congruent, we have that ≡_{dsy}^{Γ} is a congruence on L_{Σ}(Ξ), the construction is possible, and we obtain a partial structure for dsy. Thus, given that ⟨dsy, int⟩ is full with respect to the class of all partial order structures, there exists a model m ∈ M with S(m) = ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{≤_{Γ}}⟩. Let now α be the assignment on L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}} such that each α(ξ) = [ξ]_{≡_{dsy}^{Γ}}. Recall that

[[γ]]_{α}^{L_{Σ}(Ξ)/_{≡{Γ \atop dsy}}} = [γ]_{≡_{dsy}^{Γ}}. Now, easily, we have ⋉_{≤_{Γ}}[δ]_{≡_{dsy}^{Γ}}, for every δ ∈ Δ. In fact, since

Δ ⊆ Γ, we can use the third condition of 6.5.17 together with the ⊢_{dsy}^{lΞ} -closedness of Γ to conclude that (γ ⇒ δ) ∈ Γ for every δ ∈ Δ and γ ∈ L_{Σ}(Ξ), i.e., [γ]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ]_{≡_{dsy}^{Γ}}. Furthermore, ⋉_{≰_{Γ}}[θ]_{≡_{dsy}^{Γ}} because there exists γ ∈ L_{Σ}(Ξ) such that [γ]_{≡_{dsy}^{Γ}} ≰_{Γ} [θ]_{≡_{dsy}^{Γ}}, or

equivalently, (γ ⇒ θ) ∉ Γ. Just consider γ = (ξ_{1} ⇒ ξ_{1}). Since Γ must be ⊢_{dsy}^{lΞ} -closed, from the first condition of 6.5.17 we immediately conclude that (ξ_{1} ⇒ ξ_{1}) ∈ Γ. Therefore, if also ((ξ_{1} ⇒ ξ_{1}) ⇒ θ) ∈ Γ we would immediately obtain by modus ponens that θ ∈ Γ = Δ^{⊢_{dsy}^{gΞ}}, which is absurd.

Local weak completeness is just a corollary of global completeness. Clearly, if ⊢_{d̸sy}^{lΞ}θ then also ⊢_{d̸sy}^{gΞ}θ and thus, as we have shown, ⋉_{≰_{Γ}}[θ]_{≡_{dsy}^{Γ}}. ⊓ ⊔

Just note that global completeness always implies local weak completeness, due to the properties of layered systems.

Clearly, this result is not as strong as our previous ones. But in fact, locally, we cannot in general go much beyond weak completeness. Let us try and understand why. If Δ⊢_{d̸sy}^{lΞ}θ two situations have to be distinguished. If also Δ⊢_{d̸sy}^{gΞ}θ we can proceed as above and everything works smoothly. The problem arises, in fact, when it is the case that Δ⊢_{dsy}^{gΞ} θ. Then, since the ⊢_{dsy}^{gΞ} -closed Γ to be considered cannot include θ, Γ must be strictly weaker

than Δ^{⊢_{dsy}^{gΞ}}. However, in general terms now, the task of finding out a schema sentence γ such that (γ ⇒ δ) ∈ Γ for every δ ∈ Δ but (γ ⇒ θ) ∉ Γ would correspond to the existence of some kind of “infinitary” conjunction that would allow us to represent the whole set Δ^{⊢_{dsy}^{gΞ}} \ Γ deductively by a unique sentence γ. Nevertheless, if that was the case, the structure built according to 6.5.22 would indeed coincide with the one built according to 6.5.10. Still, this analysis suggests that we can slightly improve this result if we are in the presence of a deductive system with conjunction.

Definition 6.5.24. A constructor ∧ ∈ Σ_{2} is said to be a conjunction on a deductive system dsy over Σ whenever the following condition holds, for every Γ ⊆ L_{Σ}(Ξ), δ_{1}, δ_{2}, θ ∈ L_{Σ}(Ξ):

• (Γ ∪ {δ_{1}, δ_{2}})⊢_{dsy}^{lΞ} θ if and only if (Γ ∪ {(δ_{1} ∧ δ_{2})})⊢_{dsy}^{lΞ} θ.

Fortunately, most usual deductive systems have conjunction. Again, this includes every system including Basic Logic and normal modal logics.

The following alternative characterization of conjunction is better suited to our purposes.

Proposition 6.5.25. A constructor ∧ ∈ Σ_{2} is a conjunction on a deductive system dsy over Σ if and only if the following conditions hold:

• {ξ_{1}, ξ_{2}}⊢_{dsy}^{lΞ} (ξ_{1} ∧ ξ_{2});

• {(ξ_{1} ∧ ξ_{2})}⊢_{dsy}^{lΞ} ξ_{1};

• {(ξ_{1} ∧ ξ_{2})}⊢_{dsy}^{lΞ} ξ_{2}.

Proof. Assume that dsy has conjunction. We prove that all three conditions above hold. Clearly, by extensiveness, {(ξ_{1} ∧ ξ_{2})}⊢_{dsy}^{lΞ} (ξ_{1} ∧ ξ_{2}). So, using the right to left direction of 6.5.24, we obtain {ξ_{1}, ξ_{2}}⊢_{dsy}^{lΞ} (ξ_{1} ∧ ξ_{2}). Extensiveness, again, guarantees that {ξ_{1} ξ_{2}}⊢_{dsy}^{lΞ} ξ_{1} and {ξ_{1} ξ_{2}}⊢_{dsy}^{lΞ} ξ_{2}. Thus, using now the left to right implication of 6.5.24, we obtain both {(ξ_{1} ∧ ξ_{2})}⊢_{dsy}^{lΞ} ξ_{1} and {(ξ_{1} ∧ ξ_{2})}⊢_{dsy}^{lΞ} ξ_{2}.

Suppose, on the other hand, that the three conditions above hold. First assume that (Γ∪{δ_{1}, δ_{2}})⊢_{dsy}^{lΞ} θ. By monotonicity and structurality of ⊢_{dsy}^{lΞ} , using the last two conditions we obtain (Γ ∪ {(δ_{1} ∧ δ_{2})})⊢_{dsy}^{lΞ} δ_{1}, δ_{2}. Thus, immediately, (Γ ∪ {(δ_{1} ∧ δ_{2})})⊢_{dsy}^{lΞ} θ. Suppose, now, that (Γ∪{(δ_{1} ∧δ_{2})})⊢_{dsy}^{lΞ} θ. By monotonicity and structurality of ⊢_{dsy}^{lΞ} , using the first condition we obtain (Γ ∪ {δ_{1}, δ_{2}})⊢_{dsy}^{lΞ} (δ_{1} ∧ δ_{2}). Immediately, then, (Γ ∪ {δ_{1}, δ_{2}})⊢_{dsy}^{lΞ} θ. ⊓ ⊔

Easily, now we can see that conjunction propagates through fibring.

Proposition 6.5.26. For k ∈ {1, 2}, let dsy_{k} be deductive systems. If at least one of dsy_{1} and dsy_{2} has conjunction then, for every corresponding constraint S, the deductive system (dsy_{1} ⊛^{S} dsy_{2}) has conjunction.

Proof. Let each dsy_{k} be over signature Σ^{k}, and S = {s_{k} : Σ^{0} → Σ^{k}}_{k∈{1,2}}. Moreover, let {ι_{k}^{S} : dsy_{k} → (dsy_{1} ⊛^{S} dsy_{2})}_{k∈{1,2}} be the sink corresponding to the fibring of dsy_{1} and dsy_{2} constrained by S (as given by 6.2.14). Furthermore assume that j ∈ {1, 2} and that ∧_{j} ∈ Σ_{2}^{j} is a conjunction in dsy_{j}. Our aim is to prove that ∧ = ι_{j,2}^{S}(∧_{j}) is a conjunction in (dsy_{1} ⊛^{S} dsy_{2}).

Obviously, ι_{j}^{S} : csy_{dsy_{j}}^{Ξ} → csy_{(dsy_{1}S_{⊛}dsy_{2})}^{Ξ} is a morphism of structural consequence sys-

tems. Therefore, each of the three conditions from 6.5.25 for ∧ in (dsy_{1} ⊛^{S} dsy_{2}) follows trivially from the corresponding conditions for the conjunction ∧_{j} in dsy_{j}. ⊓ ⊔

The following improvement of our previous completeness result can now be established.

Proposition 6.5.27. Let ⟨dsy, int⟩ be a pair of deductive and interpretation systems, over the same signature Σ, full with respect to the class of all partial order structures structures. If dsy is congruent and has implication and conjunction then the pair ⟨dsy, int⟩ is globally complete and locally finite complete.

Proof. Since all the preconditions of 6.5.23 are satisfied, we obviously have global completeness. To show local finite completeness, let int = ⟨Σ, M, S⟩, and suppose that Δ⊢_{d̸sy}^{lΞ}θ where Δ = {δ_{1}, …, δ_{n}} is finite. Of course, we need to show that Δ⊨_{i̸nt}^{lΞ}θ. So, it is sufficient to find a model m of int and an assignment α ∈ Asg(T_{m}) such that {[[δ]]_{α}^{T_{m}} : δ ∈ Δ}⋉_{m̸}[[θ]]_{α}^{T_{m}}. Make Γ = ∅^{⊢_{dsy}^{gΞ}}, and consider the structure ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{≤_{Γ}}⟩ defined from dsy, again as in 6.5.22 above. Obviously, the construction is possible because dsy is congruent and has implication.

Thus we obtain a partial order structure for dsy, and since ⟨dsy, int⟩ is full with respect to the class of all partial order structures, there exists a model m ∈ M with S(m) = ⟨L_{Σ}(Ξ)/_{≡_{dsy}^{Γ}}, ⋉_{≤_{Γ}}⟩. Once again, take α such that each α(ξ) = [ξ]_{≡_{dsy}^{Γ}}. Clearly, we need to show that {[δ_{1}]_{≡_{dsy}^{Γ}}, …, [δ_{n}]_{≡_{dsy}^{Γ}}}⋉_{≰_{Γ}}[θ]_{≡_{dsy}^{Γ}}, i.e., that there exists γ ∈ L_{Σ}(Ξ) such that [γ]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ_{k}]_{≡_{dsy}^{Γ}} for every k ∈ {1, …, n}, but [γ]_{≡_{dsy}^{Γ}} ≰_{Γ} [θ]_{≡_{dsy}^{Γ}}. The obvious candidate is, of course, γ = (δ_{1} ∧ … ∧ δ_{n}). From the properties of conjunction, it is a trivial task to conclude that {(δ_{1} ∧ … ∧ δ_{n})}⊢_{dsy}^{lΞ} δ_{k} or equivalently, using implication, that ⊢_{dsy}^{lΞ} ((δ_{1} ∧ … ∧ δ_{n}) ⇒ δ_{k}), for every k ∈ {1, …, n}. Therefore, since Γ must be ⊢_{dsy}^{lΞ} - closed, each ((δ_{1} ∧ … ∧ δ_{n}) ⇒ δ_{k}) ∈ Γ and thus [(δ_{1} ∧ … ∧ δ_{n})]_{≡_{dsy}^{Γ}} ≤_{Γ} [δ_{k}]_{≡_{dsy}^{Γ}}. The fact that [(δ_{1} ∧ … ∧ δ_{n})]_{≡_{dsy}^{Γ}} ≰_{Γ} [θ]_{≡_{dsy}^{Γ}} can now be easily verified. In fact, if it were the case

that ((δ_{1} ∧ … ∧ δ_{n}) ⇒ θ) ∈ Γ = ∅^{⊢_{dsy}^{gΞ}} = ∅^{⊢_{dsy}^{lΞ}} the properties of conjunction would easily lead to Δ⊢_{dsy}^{lΞ} θ, which is absurd. ⊓ ⊔

Our last completeness preservation result, necessarily weaker than the previous ones, now follows.

Proposition 6.5.28. For k ∈ {1, 2}, let ⟨dsy_{k}, int_{k}⟩ be a pair of deductive and interpretation systems over the same signature and full with respect to all partial order structures. If both dsy_{1} and dsy_{2} are congruent and have implication and at least one of them has conjunction then, for every corresponding constraint S sharing the implications, the pair ⟨(dsy_{1} ⊛^{S} dsy_{2}), (int_{1} ⊛^{S} int_{2})⟩ is complete.

Proof. Immediate from 6.5.6, 6.5.18, 6.5.26 and 6.5.27. ⊓ ⊔

Please note that, together with implication, the partial orders underlying the structures as built in 6.5.22 are furthermore meet-semilattices with top. Given the definition of ≤_{Γ}, just take ⊤_{Γ} = [(ξ_{1} ⇒ ξ_{1})]_{≡_{dsy}^{Γ}} and ([δ_{1}]_{≡dsy^{Γ}} ⊓_{Γ} [δ_{2}]_{≡dsy^{Γ}} ) = [(δ_{1} ∧ δ_{2})]_{≡_{dsy}^{Γ}}. Therefore, the

result above could still be proved under the assumption that each pair ⟨dsy_{k}, int_{k}⟩ is full with respect to all topped meet-semilattice structures.

Although we do not obtain local strong completeness, this result is still fully satisfactory for systems whose local and global components coincide.

6.6 Remarks

We shall conclude this Chapter with a summary of its contents. We have defined free fibring of structural consequence systems and of structural layered consequence systems, and we have shown that they both correspond to categorial sums. Then, we have introduced the concepts of suitability and exactness for free fibring of any given category of logic system presentations or layered logic system presentations. Next, constrained fibring was introduced for structural consequence systems and then also generalized to structural layered consequence systems, and shown to correspond to a signature-driven F -pushout. The concepts of suitability and exactness for free fibring were lifted to fibring in its most general form. Then, we studied fibring at each of our working categories of layered logic system presentations, namely Dsy and Int. Both were shown to be suitable for fibring, but only Dsy exact. Given these characterizations, we have presented two examples to illustrate the mechanism: the fibring of modal logics as a multi-modal logic, and the collapse of the fibring of classical and intuitionistic logics to the classical case. We also established a close relationship between fibring and constrained synchronization. To conclude, we proved soundness and completeness preservation results in the context of Dsy/Int. Soundness preservation appeared as a consequence of the exactness for fibring of the proof-theoretic category Dsy. Several completeness preservation results were then proved, for alternative classes of relevant structures, under the general assumption of fullness of the model-theoretic systems being combined with respect to their corresponding proof-theoretic counterparts. The non-exacteness for fibring of the model-theoretic category Int could again be identified as the general source of failure of more general completeness preservation results.

Conclusion

To conclude this dissertation, we are going to summarize its main contributions and limitations. Of course, although after all we hope to have helped shed a little more light on the misteries of logic combination, still a lot remains to be done. This happens not only at the purely theoretic level of abstract logic but also, perhaps even more, in the field of practical applications. Thus, besides drawing some comparisons with related work, we have also dedicated our attention to these open questions.

7.1 Summary of contributions

As far as contributions are concerned, we should mention four major aspects. First of all, we presented, in Chapters 4 and 6, categorial definitions of synchronization and fibring of logics in those we consider their most natural settings, respectively, consequence systems and structural consequence systems. Furthermore, we defined general notions of logic (system) and logic (system) presentation, where the combination mechanisms can then be explored. Indeed, suitability and exactness for each of the mechanisms do not only provide the means for this exploration but they also draw a sharp distinction between those categories of systems where the mechanisms just make sense, and what is more, those where they can be faithfully applied. This general picture complies, of course, with previous knowledge about synchronization and fibring. The second reference goes to the way we managed to extend both combination mechanisms to the local/global layered setting, namely in semantic terms. Our treatment of fully algebraic fibred semantics, in particular, deserves a special reference for raising the level of abstraction previously achieved. The notion of world, essential in previous related work, was simply abolished. This was possible owing to the world-free way we found to represent the internal logic structure of models, first with layered valuation systems in Chapter 3 and later, in Chapter 5, with interpretation systems. The third contribution is related to the analogy we drew between the apparently unlevel mechanisms of synchronization and fibring. This parallelism shows both in the setting up of their base frameworks in Chapters 2, 3 and 5, and in the way they were defined and studied in Chapters 4 and 6. However, far from being just an exercise of mathematical elegance, we took advantage of this analogy to show that a close relationship between the two mechanisms really exists. After Chapter 6, it became clear that although synchronization is a low level particular case of fibring, it is still interesting enough to simulate the apparent extra power of fibring. Last, but not least, we have to refer to the transference results obtained. Obviously, there is no point in studying combination of logics without having in mind preservation of properties. Soundness being a minimal requirement, completeness is, however, known to be a different and far more difficult subject. Thus, the general preservation results obtained, although restricted by the requirement of fullness conditions, are certainly worth special attention.

7.2 Limitations, outlook and future work

As we have already mentioned, we postulate consequence systems as the general grounds for studying synchronization, as much as structural consequence systems for studying fibring. Such decisions are perhaps arguable, but they nevertheless comply with previous knowledge about the mechanisms, which is a good sign, anyway. Moreover, following our general methodology, they open up the way for applying the mechanisms in an heterogeneous setting, for instance by fibring a model-theoretic presented system with a proof-theoretic one. Of course, the result obtained cannot be expected to be given in more concrete terms than those of structural consequence. The close relationship between synchronization and fibring is also a striking subject. Indeed, the quite general ideas underlying their relationships are not so rare. Similar phenomena were recognized, for instance in [EC00], in a study of temporal logics for distributed systems. Similar ideas also underly the (as far as we know) unpublished paper [Sel98], which we discovered almost by accident on a web exploration, where they are also used to provide characterizations of interleaved languages by means of paired languages, although with motivations coming from the philosophy of language and thus quite distinct from ours. The constructions there, however, very much resemble our definitions of fibring and synchronization.

Regarding limitations, the main aspect we should mention is our focus on propositional based logics, namely as far as fibring is concerned. Indeed, as we start thinking about more sophisticated logical languages, with several sorts, terms, quantification and so on, things get much more complicated. However, in practice, the propositional case is well known to be insufficient. Extensions of this work towards logic systems with terms and quantifiers have yet to be provided. A first attempt can be found in [SSCM00]. Further explorations should perhaps be developed within our algebraic setting, with the study of cylindric algebra [HMT71], and in connection to the work also being developed in this direction in the context of parchments [Paw98]. Another important aspect concerns our completeness preservation results, namely for fibring. To start with, although our results are similar in spirit to those proved in [ZSS99], they are perhaps much less surprising. An old slogan by van Benthem [vB84] argues that the Lindenbaum-like completeness techniques of algebraic semantics are essentially algebra in disguise, as opposed to Henkin’s deepest techniques, which also encompass Stone-like representation. However, the field of algebraic logic has imposed itself and shown to be very useful [Cze81, BP89, AKNS94] owing to its rich modeltheory. Moreover, although some of our results may in part suffer from this handicap, the result about partially ordered structures in particular is already set in the direction of set-theoretic representation. Yet, the fullness conditions used may still be thought of as a little idealistic. However, the established link to algebraic logic may prove fruitful in this particular aspect. Imagine the following picture: take the logics you want to combine, cast them in algebraic terms using the algebraization methodology made available and then use our fibring construction. Of course, this may seem too far from reality. However, we feel that the resemblances between our interpretation structures, congruence and fullness conditions and the so-called sentential logics, equivalential logics, strongly nice and seminice general logics of [Cze81, BP89, AKNS94, FJ94] are worth being explored, via an equational representation of our setting. Another limitation worth mentioning is that we have only studied preservation of soundness and (strong) completeness. Many other interesting properties are certainly worth studying, among them interpolation, decidability, the finite model property, and the exploration of corresponding theories namely with respect to finite axiomatizability. All of these, and others we may have failed to refer, constitute obvious topics for future work.

An interesting question that we have not solved concerns the precise role played by the closure condition for substructures used in [ZSS99]. Of course, the fibred semantics we obtain with and without such condition are not the same. However, it seems that all the intuitions behind fibring are suitably captured without considering it. Thus, in our purely algebraic context we have not considered such a condition. However, it turns out that, in our setting, a simple closure for substructures as subalgebras with internal generated consequence may not be appropriate. It seems much more reasonable to require instead a closure for subdirect products in a way close to the usual in universal algebra [Bir67]. Moreover, although we have not investigated this issue in detail, requiring such a condition may open the way for avoiding that the fibring of classical and intuitionistic logics collapses into the classical case.

Before we explain a collection of other ideas for continuing this line of research, let us first compare our work with the parchment approach to combining logics [MTP97, MTP98, Paw98]. In essence, the single sorted version of the so-called rooms of modeltheoretic parchments are very similar to our Σ-structures, although the local dimension is neglected. Thus, the model-theoretic notions of logic underlying parchments are roughly the same as the global fragment of our interpretation systems. The big difference lies, however, on the notion of morphism. Indeed, our morphisms are comparatively poorer. While we require that, for each model, S^{1}(\underline{h}(m)) = S^{2}(m)|\overline{h}, parchments morphisms are broad enough to encompass the existence of a homomorphism between S^{1}(\underline{h}(m)) and S^{2}(m)|\overline{h}. As a result, structures for the combined language are then obtained by free construction: for any constructor of one of the signatures applied to a value in the other one, a new truth-value is generated. Of course, by applying quotients to these large structures, one may end up obtaining our fibred structures. But that is in general not the case and thus, despite the similarities, parchments aim at a form of combination that is interesting in its own right, but is distinct from fibring.

Computer science applications have brought an increased interest in many heterodox logics, such as many-valued and paraconsistent logics. It turns out that, in many cases, such logics have rather unconventional semantics. So, we are very much interested in extending the fibring methodology to alternative semantic frameworks. The work started in [CMSS00] aims precisely at coping with algebraic non-truth functional semantics, where interpretation of sentences is not homomorphic. Other alternatives already under consideration are the possible-translations and society semantics of [Cd97, CLM99, Mar99] and sheaf-theoretic semantics [CC97]. The latter can indeed be expected to be a firm ground for exploring the geometric aspects of local versus global phenomena. Another very interesting topic is the development of a meta-logic of logic combination, and namely its relationship to [JKE96]. Some first steps on this subject can be found in [CC00].

Last, but not least, we would like to reanalyze some previous work on logics for distributed systems [ECSD98, EC00, Cal00] in the light of logic combination. As hinted there, we hold the hope of using the ideas behind trios [BdR97b] to obtain nice completeness results and tableaux systems for distributed temporal logic by capitalizing on well-known results from linear temporal logic. As we said at the very beginning of this dissertation, the divide and conquer strategy is everywhere in modern science, and general compositionality results, although hard to achieve, are very powerful and useful tools. Since we firmly believe that combining logics is a very relevant abstraction, namely in computer science, we have no doubts that the body of practical applications of this methodology will grow exponentially in the near future.

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Appendix

Categorial combination

There is an already well-acknowledged tradition of applying categorial methods in computer science and logic [Gog91]. The best advertisement one can make of category theory, beyond the essential fact that its language is concise, very rich and at the same time easy to express in pictorial terms, is that, given a certain universe of mathematical objects, we should not only study them as isolated objects, but also learn about the way these objects relate to each other. And we stress that this is, in our opinion, the key issue. The beauty and simplicity of a diagram to express a concept is certainly very appealing, but, essentially, because diagrams of arrows are used. For that purpose, besides studying the structure of such mathematical objects, we must furthermore establish the adequate notion of morphism (arrow) between objects that carries along this structure.

Thus, whenever compositional issues arise in a categorial setting, it is to be expected that the combination of simpler objects to form more complex objects might correspond to a universal categorial construction. This category theoretic self-evidence was brought up quite a long while ago to general systems theory, and computer science in particular, as a recipe for combining systems. The slogan in [GG78] might be summarized by a very simple idea: in your category C of systems, set up a diagram D of your component objects with arrows expressing the desired relationships among them, and obtain your combined complex system by taking the colimit of D in C (if it exists, of course).

[figure: C; D; .; .; .; .; .; .; c′; =; c; =; =]

Figure A.1: A colimit c of D in C and its universal property.

Such constructions, namely sums (for simple combinations) and pushouts (for constrained ones), have been widely used. Sometimes, even when they were not the more adequate. That is often the case with pushouts, a construction that we are particularly interested in, and so we shall concentrate on them.

[figure: c′; c; h; c_{1}; g_{1}; h_{1}; c_{2}; g_{2}; h_{2}; c_{0}; f_{1}; f_{2}; =; =; =]

Figure A.2: A pushout c of f_{1} and f_{2} and its universal property.

A.1 One problem and two ways of avoiding it

In computer science, for instance, the desired relationships between components are often described by purely syntactic means. In general, however, our reasoning applies not only to syntax but also to any other suitable fragment of a system. In general terms, the picture can be drawn as follows: besides the category of systems C, one must also consider a “syntactical” base category B and a (usually forgetful) functor F : C → B. In this setting, a two-source in C with codomain {c_{1}, c_{2}} may not be the best way of describing the relationship between the component objects c_{1} and c_{2} of a complex system. It is likely that, instead, we are given a two-source in B with codomain {F (c_{1}), F (c_{2})} describing the relationship between their “syntactical” fragments.

[figure: C; B; F; ?; F (?); c_{1}; ?_{1}; c_{2}; ?_{2}; F (c_{1}); F (?_{1}); F (c_{2}); F (?_{2}); b_{0}; f_{1}; f_{2}; =]

Figure A.3: The problem: how to combine c_{1} and c_{2} by sharing b_{0} along f_{1} and f_{2}?

The envisaged combined object must, of course, be minimal among those that “extend” both c_{1} and c_{2} while still embodying the shared fragment b_{0} along f_{1} and f_{2}.

Let us therefore consider fixed two categories C and B, a functor F : C → B, two C-objects c_{1} and c_{2}, and a source D = {f_{k} : b_{0} → F (c_{k})}_{k∈{1,2}} in B.

A.1.1 Lifting the source

We first describe the probably most common partial solution to this problem. It applies whenever the source D = {f_{1}, f_{2}} in B can be adequately lifted to C. A usual sufficient condition for this to be possible is that F is an adjoint. The usual pushout construction in C for the two-source thus obtained yields the desired complex object.

Just to freeze notation, we shall recall here the notion of adjunction. To start with, by an F -structured arrow with domain b ∈ |B| we shall mean a pair ⟨f, c⟩ with c ∈ |C| and f : b → F (c) a B-morphism.

Definition A.1.1. An F -universal arrow for b ∈ |B| is an F -structured arrow ⟨η_{b}, c(b)⟩ with domain b such that the following universal property holds:

• for every F -structured arrow ⟨f, c⟩ with domain b there exists a unique C-morphism f^{∗} : c(b) → c such that (F (f^{∗}) ∘ η_{b}) = f.

F is said to be adjoint if every B-object has an F -universal arrow.

[figure: C; B; F; F (c); F (c(b)); F (f^{∗}); b; η_{b}; f; c; c(b); f^{∗}; =]

Figure A.4: An F -universal arrow ⟨η_{b}, c(b)⟩ for b and its universal property.

So, F being an adjoint, we can use the existence of an F -universal arrow for b_{0} to lift f_{1} and f_{2} to f_{1}^{∗} and f_{2}^{∗}, and obtain the desired complex system by building their pushout in C (if it exists).

[figure: C; B; F; c; c_{1}; g_{1}; c_{2}; g_{2}; c(b_{0}); f_{1}^{∗}; f_{2}^{∗}; F (c_{1}); F (c_{2}); b_{0}; f_{1}; f_{2}]

Figure A.5: If possible, lift the source to C and take its pushout to obtain c.

Note that this approach clearly complies to the principle of combining via colimits. Simple but illustrative examples of this situation have abounded in the techniques for combining theories, at least since [BG77].

A.1.2 Lifting the result

There is, however, an alternative approach to the problem. It is not so uncommon that C has sums, and, moreover, F supports an abstraction operation known as a cofibration. In that case, we can build a sum {ι_{k} : c_{k} → (c_{1} + c_{2})}_{k∈{1,2}} of c_{1} and c_{2} in C, obtain a coequalizer of {F (ι_{1}) ∘ f_{1}, F (ι_{2}) ∘ f_{2}} in B (if it exists), and then lift it to C using the fact that F is a cofibration, as explained below.

Just to freeze notation, we shall recall here the notion of cofibration. Similarly, in the sequel, by an F -costructured arrow with codomain b′ ∈ |B| we shall mean a pair ⟨c, f⟩ with c ∈ |C| and f : F (c) → b′ a B-morphism.

Definition A.1.2. An F -cocartesian lift of an F -costructured arrow ⟨c, f⟩ with codomain b′ is a C-morphism f^{#} : c → c′ with F (f^{#}) = f such that the following universal property holds:

• for every C-morphism g : c → c′′ and B-morphism h : b′ → F (c′′) with (h∘f) = F (g) there exists a unique C-morphism h^{•} : c′ → c′′ with (h^{•} ∘ f^{#}) = g such that F (h^{•}) = h.

F is said to be a cofibration if every F -costructured arrow has an F -cocartesian lift.

[figure: C; B; F; F (c′′); b′; h=F (h^{•}); F (c); f; F (g); c′′; c′; h^{•}; c; f^{#}; g; F (f^{#}); ∥; F (c′); =; =; =]

Figure A.6: An F -cocartesian lift of ⟨c, f⟩ with codomain b′ and its universal property.

So, F being a cofibration and C having sums, we can obtain the complex system by a cocartesian lift of ⟨(c_{1}+c_{2}), q⟩, where q : F (c_{1}+c_{2}) → b is the above mentioned coequalizer.

[figure: C; B; F; c; b; (c_{1} + c_{2}); q^{#}; F (c_{1} + c_{2}); q; c_{1}; ι_{1}; (q^{#}∘ι_{1}); c_{2}; ι_{2}; (q^{#}∘ι_{2}); F (c_{1}); (q∘F (ι_{1})); F (ι_{1}); F (c_{2}); (q∘F (ι_{2})); F (ι_{2}); b_{0}; f_{1}; f_{2}]

Figure A.7: If possible, lift a coequalizer of {(F (ι_{1}) ∘ f_{1}), (F (ι_{1}) ∘ f_{2})} to obtain c.

Illustrative examples of this approach can be found, for instance, within categorial studies of models for concurrency [WN95, CSSE92]^{1}.

1Although these often adopt the dual view of building complex systems via limits, instead of colimits.

A.2. F -pushout: a solution 197

A.2 F -pushout: a solution

There certainly seems to be some kind of relationship between these two approaches. The relationship between pushouts, coequalizers and sums is very tight. As it will be made clear, they do lead to precisely the same result under the general conditions for their application. From that standpoint, however, the cofibration approach seems to be simpler, since it does not require the calculation of the two-source in C. However, it has been criticized for not falling into the mainstream approach of (explicitly) building a colimit in C.

In a recent discussion of these issues with Joseph Goguen, (one of the pioneers of the slogan “combine via colimits”), he has pointed out to us that the adequate universal construction in the present situation should be neither a pushout nor a cocartesian lift. Instead, he suggested a new construction, which he called an F -pushout.

Definition A.2.1. An F -pushout of {c_{1}, c_{2}} along D = {f_{1}, f_{2}} is a sink S = {g_{k} : c_{k} → c}_{k∈{1,2}} in C such that (F (g_{1}) ∘ f_{1}) = (F (g_{2}) ∘ f_{2}), and the following universal property holds:

• for every sink S′ = {h_{k} : c_{k} → c′}_{k∈{1,2}} in C with (F (h_{1}) ∘ f_{1}) = (F (h_{2}) ∘ f_{2}), there exists a unique C-morphism h : c → c′ such that (h ∘ S) = S′.

[figure: C; B; F; =; =; c′; F (c′); c; h; F (c); c_{1}; g_{1}; h_{1}; c_{2}; g_{2}; h_{2}; F (g_{1}); F (h_{1}); F (c_{1}); F (g_{2}); F (h_{2}); F (c_{2}); b_{0}; f_{1}; f_{2}; =]

Figure A.8: An F -pushout of {c_{1}, c_{2}} along {f_{1}, f_{2}} and its universal property.

It goes without saying that F -pushouts, as any other well defined universal construction, are unique up to isomorphism. In fact, this new construction is an obvious generalization of the notion of pushout.

Proposition A.2.2. Pushouts in C and ID_{C}-pushouts coincide.

Furthermore, trivially, if B has an initial object, then binary sums in C are a special case of F -pushouts.

Proposition A.2.3. Let b_{0} be initial in B. Then, S = {g_{k} : c_{k} → c}_{k∈{1,2}} is an F - pushout of {c_{1}, c_{2}} along D = {f_{1}, f_{2}} if and only if S is a sum of c_{1} and c_{2} in C.

Our aim, in the next few pages, is to justify the use we make of the simple and intuitively clear notion of F -pushout. In particular, we shall show that under the conditions that allow their application the two before mentioned shortcut approaches correspond precisely to F -pushouts.

A.2.1 Lifting the source yields an F -pushout

Let us start with the first approach (Figure A.5). Under the assumption of adjointness for F, we can not only prove that the construction yields an F -pushout, but also that all F -pushouts can be obtained by applying it.

Proposition A.2.4. Let ⟨c(b_{0}), η_{b_{0}}⟩ be an F -universal arrow for b_{0} and S = {g_{k} : c_{k} → c}_{k∈{1,2}} a pushout in C of D^{∗} = {f_{k}^{∗} : c(b_{0}) → c_{k}}_{k∈{1,2}}. Then, S is an F -pushout of {c_{1}, c_{2}} along D.

Proof. Since S is a pushout in C of D^{∗}, necessarily, (g_{1}∘f_{1}^{∗}) = (g_{2}∘f_{2}^{∗}). Thus, immediately, (F (g_{1}) ∘ f_{1}) = (F (g_{1} ∘ f_{1}^{∗}) ∘ η_{b_{0}}) = (F (g_{2} ∘ f_{2}^{∗}) ∘ η_{b_{0}}) = (F (g_{2}) ∘ f_{2}). Now, let S′ = {h_{k} : c_{k} → c′}_{k∈{1,2}} be a sink in C with (F (h_{1}) ∘ f_{1}) = (F (h_{2}) ∘ f_{2}) = f : b_{0} → F (c′). Clearly, for each k ∈ {1, 2}, (F (h_{k} ∘ f_{k}^{∗}) ∘ η_{b_{0}}) = (F (h_{k}) ∘ f_{k}) = f. Thus, by uniqueness of f^{∗}, (h_{1}∘f_{1}^{∗}) = (h_{2}∘f_{2}^{∗}). The universal property of the pushout S then guarantees the existence of a unique C-morphism h : c → c′ such that (h ∘ S) = S′. We conclude that S is an F -pushout of {c_{1}, c_{2}} along D. ⊓ ⊔

Proposition A.2.5. Let S = {g_{k} : c_{k} → c}_{k∈{1,2}} be an F -pushout of {c_{1}, c_{2}} along D and ⟨c(b_{0}), η_{b_{0}}⟩ an F -universal arrow for b_{0}. Then, S is a pushout in C of D^{∗}.

Proof. Since S is an F -pushout of {c_{1}, c_{2}} along D, necessarily, (F (g_{1}) ∘ f_{1}) = (F (g_{2}) ∘ f_{2}) = g : b_{0} → F (c). Thus, for each k ∈ {1, 2}, (F (g_{k} ∘ f_{k}^{∗}) ∘ η_{b_{0}}) = (F (g_{k}) ∘ f_{k}) = g. Therefore, by uniqueness of g^{∗}, (g_{1} ∘ f_{1}^{∗}) = (g_{2} ∘ f_{2}^{∗}). Let now S′ be such that (h_{1} ∘ f_{1}^{∗}) = (h_{2} ∘ f_{2}^{∗}). Immediately, (F (h_{1}) ∘ f_{1}) = (F (h_{1} ∘ f_{1}^{∗}) ∘ η_{b_{0}}) = (F (h_{2} ∘ f_{2}^{∗}) ∘ η_{b_{0}}) = (F (h_{2}) ∘ f_{2}). The universal property of the F -pushout S then guarantees the existence of a unique Cmorphism h : c → c′ such that (h ∘ S) = S′. We conclude that S is a pushout of D^{∗}. ⊓ ⊔

Corollary A.2.6. If F is adjoint, then pushouts of D^{∗} in C and F -pushouts of {c_{1}, c_{2}} along D coincide.

This shows that the adjointness of F collapses the two constructions. In particular, the existence of all pushouts in C is both a necessary and a sufficient condition for the existence of all F -pushouts.

Obviously, even when an F -universal arrow for b_{0} does not exist, it may still be the case that D can somehow be lifted to C. Notably, in such a situation and provided that F is a faithful functor, the two constructions still collapse.

Proposition A.2.7. Let D′ = {f′_{k} : c_{0} → c_{k}}_{k∈{1,2}} be a source in C such that F (D′) = D, and S = {g_{k} : c_{k} → c}_{k∈{1,2}} a pushout in C of D′. Then, S is an F -pushout of {c_{1}, c_{2}} along D.

Proof. Since S is a pushout in C of D′, necessarily, (g_{1}∘f′_{1}) = (g_{2}∘f′_{2}). Thus, (F (g_{1})∘f_{1}) = (F (g_{1} ∘ f′_{1})) = (F (g_{2} ∘ f′_{2})) = (F (g_{2}) ∘ f_{2}). Now, let S′ = {h_{k} : c_{k} → c′}_{k∈{1,2}} be a sink in C with (F (h_{1}) ∘ f_{1}) = (F (h_{2}) ∘ f_{2}). Clearly, (h_{1} ∘ f′_{1}), (h_{2} ∘ f′_{2}) : c_{0} → c′ are C-morphisms and F (h_{1} ∘ f′_{1}) = (F (h_{1}) ∘ f_{1}) = (F (h_{2}) ∘ f_{2}) = F (h_{2} ∘ f′_{2}). The faithfulness of F thus implies that (h_{1} ∘ f′_{1}) = (h_{2} ∘ f′_{2}). Therefore, the universal property of the pushout S then guarantees the existence of a unique C-morphism h : c → c′ such that (h ∘ S) = S′. We conclude that S is an F -pushout of {c_{1}, c_{2}} along D. ⊓ ⊔ Proposition A.2.8. Let S = {g_{k} : c_{k} → c}_{k∈{1,2}} be an F -pushout of {c_{1}, c_{2}} along D and D′ = {f′_{k} : c_{0} → c_{k}}_{k∈{1,2}} in C such that F (D′) = D. Then, S is a pushout in C of D′.

Proof. Since S is an F -pushout of {c_{1}, c_{2}} along D, necessarily, (F (g_{1})∘f_{1}) = (F (g_{2})∘f_{2}). Clearly, (g_{1} ∘ f′_{1}), (g_{2} ∘ f′_{2}) : c_{0} → c are C-morphisms and F (g_{1} ∘ f′_{1}) = (F (g_{1}) ∘ f_{1}) = (F (g_{2}) ∘ f_{2}) = F (g_{2} ∘ f′_{2}). The faithfulness of F thus implies that (g_{1} ∘ f′_{1}) = (g_{2} ∘ f′_{2}). Let now S′ be such that (h_{1} ∘ f′_{1}) = (h_{2} ∘ f′_{2}). Immediately, (F (h_{1}) ∘ f_{1}) = F (h_{1} ∘ f′_{1}) = F (h_{2} ∘ f′_{2}) = (F (h_{2}) ∘ f_{2}). The universal property of the F -pushout S then guarantees the existence of a unique C-morphism h : c → c′ such that (h ∘ S) = S′. We conclude that S is a pushout of D′. ⊓ ⊔

Corollary A.2.9. If F is faithful functor and D′ = {f′_{k} : c_{0} → c_{k}}_{k∈{1,2}} in C is such that F (D′) = D, then pushouts of D′ in C and F -pushouts of {c_{1}, c_{2}} along F (D′) coincide.

A.2.2 Lifting the result yields an F -pushout

Let us now analyze the second approach (Figure A.7). Under the assumptions that C has sums, we can not only prove that the construction yields an F -pushout, but also that all F -pushouts can be thus obtained provided that F is a cofibration.

In the sequel, we shall consider as fixed a sum {ι_{k} : c_{k} → (c_{1} + c_{2})}_{k∈{1,2}} of c_{1} and c_{2} in C.

Proposition A.2.10. Let q : F (c_{1} +c_{2}) → b be a coequalizer in B of {(F (ι_{1})∘f_{1}), (F (ι_{2})∘ f_{2})}, and q^{#} : (c_{1}+c_{2}) → c an F -cocartesian lift of ⟨(c_{1}+c_{2}), q⟩. Then, {(q^{#}∘ι_{1}), (q^{#}∘ι_{2})} is an F -pushout of {c_{1}, c_{2}} along D.

Proof. Since q^{#} is an F -cocartesian lift of ⟨(c_{1} + c_{2}), q⟩, necessarily, F (q^{#}) = q. Thus, since q is a coequalizer of {(F (ι_{1}) ∘ f_{1}), (F (ι_{2}) ∘ f_{2})}, (F (q^{#} ∘ ι_{1}) ∘ f_{1}) = (q ∘ F (ι_{1}) ∘ f_{1}) = (q ∘ F (ι_{2}) ∘ f_{2}) = (F (q^{#} ∘ ι_{2}) ∘ f_{2}). Let now S′ = {h_{k} : c_{k} → c′}_{k∈{1,2}} be a sink in C with (F (h_{1}) ∘ f_{1}) = (F (h_{2}) ∘ f_{2}). By the universal property of the sum (c_{1} + c_{2}), there exists a unique C-morphism h : (c_{1} + c_{2}) → c′ such that (h ∘ ι_{k}) = h_{k} for k ∈ {1, 2}. Clearly, then, (F (h)∘F (ι_{1})∘f_{1}) = (F (h_{1})∘f_{1}) = (F (h_{2})∘f_{2}) = (F (h)∘F (ι_{2})∘f_{2}). The universal property of the coequalizer q then guarantees the existence of a unique B-morphism t : b → F (c′) such that (t ∘ q) = F (h). And the universal property of the F -cocartesian lift q^{#} then guarantees the existence of a unique C-morphism t^{•} : c → c′ such that (t^{•} ∘ q^{#}) = h and F (t^{•}) = t. Easily, (t^{•} ∘ q^{#} ∘ ι_{k}) = (h ∘ ι_{k}) = h_{k} for k ∈ {1, 2}. To show that t^{•} is the unique, assume that s : c → c′ is a C-morphism such that (s ∘ q^{#} ∘ ι_{k}) = h_{k} for k ∈ {1, 2}. By the uniqueness of h, then, (s ∘ q^{#}) = h. Therefore, (F (s) ∘ q) = F (s ∘ q^{#}) = F (h) and by the uniqueness of t we must have F (s) = t. Finally, by the uniqueness of t^{•}, it must be the case that s = t^{•}. ⊓ ⊔

Proposition A.2.11. Let g : (c_{1}+c_{2}) → c be a C-morphism such that S = {(g∘ι_{1}), (g∘ι_{2})} is an F -pushout of {c_{1}, c_{2}} along D, and F (g) is a coequalizer in B of {(F (ι_{1})∘f_{1}), (F (ι_{2})∘ f_{2})}. Then, g is an F -cocartesian lift of ⟨(c_{1} + c_{2}), F (g)⟩.

Proof. Let r : (c_{1} + c_{2}) → c′ be a C-morphism, and u : F (c) → F (c′) a B-morphism such that (u ∘ F (g)) = F (r). Clearly, since F (g) is a coequalizer of {(F (ι_{1}) ∘ f_{1}), (F (ι_{2}) ∘ f_{2})}, (F (r ∘ ι_{1}) ∘ f_{1}) = (u ∘ F (g) ∘ F (ι_{1}) ∘ f_{1}) = (u ∘ F (g) ∘ F (ι_{2}) ∘ f_{2}) = (F (r ∘ ι_{2}) ∘ f_{2}), and by its universal property, u is the unique such that (u ∘ F (g)) = F (r). Moreover, by the universal property of the F -pushout S, there exists a unique C-morphism v : c → c′ such that (v ∘ g ∘ ι_{k}) = (r ∘ ι_{k}) for k ∈ {1, 2}. Thus, using the universal property of (c_{1} + c_{2}), we conclude that v is the unique such that (v ∘ g) = r. So, (F (v) ∘ F (g)) = F (v ∘ g) = F (r) and thus, by uniqueness of u, we must have that F (v) = u. ⊓ ⊔

Proposition A.2.12. Let F : C → B be a cofibration, and g : (c_{1}+c_{2}) → c a C-morphism such that S = {(g ∘ ι_{1}), (g ∘ ι_{2})} is an F -pushout of {c_{1}, c_{2}} along D. Then, F (g) is a coequalizer in B of {(F (ι_{1}) ∘ f_{1}), (F (ι_{2}) ∘ f_{2})}.

Proof. Since S is an F -pushout of {c_{1}, c_{2}} along D, necessarily, (F (g)∘F (ι_{1})∘f_{1}) = (F (g)∘ F (ι_{2}) ∘ f_{2}). Assume that r : F (c_{1} + c_{2}) → b′ is a B-morphism such that (r ∘ F (ι_{1}) ∘ f_{1}) = (r ∘F (ι_{2})∘f_{2}), and let r^{#} : (c_{1} +c_{2}) → c′ be an F -cocartesian lift of ⟨(c_{1} +c_{2}), r⟩. The fact that F is a cofibration guarantees that such an r^{#} exists. Then, necessarily, F (r^{#}) = r and therefore (F (r^{#} ∘ ι_{1}) ∘ f_{1}) = (r ∘ F (ι_{1}) ∘ f_{1}) = (r ∘ F (ι_{2}) ∘ f_{2}) = (F (r^{#} ∘ ι_{2}) ∘ f_{2}). Thus, by the universal property of the F -pushout S, there exists a unique C-morphism s : c → c′ such that (s ∘ g ∘ ι_{k}) = (r^{#} ∘ ι_{k}) for k ∈ {1, 2}. So, using the universal property of (c_{1} + c_{2}), we conclude that s is the unique such that (s ∘ g) = r^{#}. Moreover, (F (s) ∘ F (g)) = F (s ∘ g) = F (r^{#}) = r. To show that F (s) is unique, assume that u : F (c) → b′ is a B-morphism such that (u ∘ F (g)) = r. Since F is a cofibration, there exists an F -cocartesian lift of ⟨c, u⟩. Let it be u^{#} : c → c′′. Obviously, F (u^{#}) = u and therefore F (u^{#} ∘ g) = (u ∘ F (g)) = r. Then, by the universal property of the cocartesian lifting r^{#}, there exists a unique C-morphism v : c′ → c′′ such that (v ∘ r^{#}) = (u^{#} ∘ g) and F (v) = id_{b′}. On the other hand, easily, (F (u^{#} ∘ g ∘ ι_{1}) ∘ f_{1}) = (u ∘ F (g) ∘ F (ι_{1}) ∘ f_{1}) = (u∘F (g)∘F (ι_{2})∘f_{2}) = (F (u^{#}∘g∘ι_{2})∘f_{2}). But then, (v∘s∘g∘ι_{k}) = (v∘r^{#}∘ι_{k}) = (u^{#}∘g∘ι_{k}) for k ∈ {1, 2}, and the universal property of the F -pushout S guarantees that u^{#} = (v ∘s). Thus, u = F (u^{#}) = F (v ∘ s) = (F (v) ∘ F (s)) = (id_{b′} ∘ F (s)) = F (s). ⊓ ⊔

Clearly, if S is an F -pushout but F (g) fails to be a coequalizer of {(F (ι_{1})∘f_{1}), (F (ι_{2})∘ f_{2})} then, either that coequalizer does not exist or, if it does, it is bound not to have an F -cocartesian lifting.

Corollary A.2.13. If F : C → B is a cofibration, then F -cocartesian liftings to (c_{1} + c_{2}) of coequalizers of {(F (ι_{1}) ∘ f_{1}), (F (ι_{2}) ∘ f_{2})} and F -pushouts of {c_{1}, c_{2}} along D coincide.

This shows that, F being a cofibration, it collapses the two constructions. In particular, the existence of an F -pushout is a necessary and sufficient condition for the existence of the corresponding coequalizer in B.

Of course, a particularly nice case happens when both approaches to the problem can be used. That is, besides the categories C and B having all relevant colimits, it may be the case that F : C → B is both an adjoint and a cofibration. Of course, although most functors are not like this, there are still a lot of them that fulfill both conditions. We shall be specially interested in topological functors [AHS90]. Topological functors abstract away a lot of smooth structural properties that are satisfied by topological spaces. In particular, topological functors are always adjoint and cofibrations.

A.3. Preservation of F -pushouts 201

A.3 Preservation of F -pushouts

In the text of this dissertation, whenever we use F -pushouts, we shall be in a setting where the second approach can be used. Thus, in that setting, we shall be interested in the following problem. Let F : C → B and F′ : C′ → B be cofibrations, both C and C′ have binary sums and B have coequalizers. To what extent can we guarantee that a functor H : C → C′ transforms F -pushouts into F′-pushouts?

To that purpose we shall recall here the notion of cocartesian functor (see, e.g., [Bor94]).

Definition A.3.1. Let F : C → B and F′ : C′ → B be cofibrations. A functor H : C → C′ is cocartesian provided that (F′ ∘ H) = F and, for every F -costructured arrow ⟨c, h⟩, if h^{#} is an F -cocartesian lift of ⟨c, h⟩ then H(h^{#}) is an F′-cocartesian lift of ⟨G(c), h⟩.

Assuming the hypotheses stated above, i.e., F and F′ are cofibrations, C and C′ have binary sums and B has coequalizers, we can now prove the following result.

Proposition A.3.2. If H : C → C′ preserves binary sums and is cocartesian, then F - pushouts are transformed by H into F′-pushouts, i.e, H(S) is an F′-pushout of H(c_{1}) and H(c_{2}) along D whenever S is an F -pushout of c_{1} and c_{2} along D.

Proof. Assume that S = {g_{k} : c_{k} → c}_{k∈{1,2}} is an F -pushout of c_{1} and c_{2} along D = {f_{k} : b_{0} → F (c_{k})}_{k∈{1,2}}. Since C has sums, let {ι_{k} : c_{k} → (c_{1} + c_{2})}_{k∈{1,2}} be a sum of c_{1} and c_{2}, and g : (c_{1} + c_{2}) → c the unique morphism such that (g ∘ ι_{k}) = g_{k} for k ∈ {1, 2}. According to A.2.12, once F is a cofibration, we can conclude that F (g) is a coequalizer of {(F (ι_{k}) ∘ f_{k})}_{k∈{1,2}}. From A.2.11 it then follows that g is an F -cocartesian lift of ⟨(c_{1} + c_{2}), F (g)⟩. But H is known to preserve sums, and thus, {H(ι_{k}) : H(c_{k}) → H(c_{1} + c_{2})}_{k∈{1,2}} is a sum in C′ and H(g) : H(c_{1} + c_{2}) → H(c) is the unique morphism such that H(g) ∘ H(ι_{k}) = H(g_{k}). Moreover, H is cocartesian. Hence, on the one hand, F = (F′ ∘ H) and we have that F′(H(g)) is a coequalizer of {(F′(H(ι_{k})) ∘ f_{k})}_{k∈{1,2}}. On the other hand, H preserves cocartesian lifts, and we have that H(g) is an F′-cocartesian lift of ⟨(c_{1} +c_{2}), F′(H(g))⟩. We can now just use A.2.10 to conclude that H(S) = {H(g_{k}) : H(c_{k}) → H(c)}_{k∈{1,2}} is an F′-pushout of H(c_{1}) and H(c_{2}) along D. ⊓ ⊔

Since identity functors are always trivial cofibrations, we have the following interesting corollary:

Corollary A.3.3. If F : C → B preserves binary sums, then F -pushouts are transformed by F into pushouts in B, i.e., F (S) is a pushout of D whenever S is an F -pushout of c_{1} and c_{2} along D.

As for similar results for the first approach, or even the general approach, it seems that one must consider a notion close to fibred coadjoints (see [Bor94]). This investigation is out of the scope of this thesis.

Another result that we use in the dissertation concerns the preservation of the construction through Grothendieck constructions from indexed categories. Indexed categories (see [TBG91]) are an alternative, although not exactly equivalent, to the notion of (co)fibration. An indexed category is a functor C : Ind^{op} → Cat. Given an indexed category C, the so-called Grothendieck construction allows us to build a flattening of this functor.

Definition A.3.4. Let C : Ind^{op} → Cat be an indexed-category. The flat category C^{♭} is built as follows:

• objects are pairs ⟨i, c⟩ such that i ∈ |Ind| and c ∈ |C(i)|;

• morphisms are pairs ⟨f, g⟩ : ⟨i_{1}, c_{1}⟩ → ⟨i_{2}, c_{2}⟩ such that f : i_{1} → i_{2} is an Indmorphism and g : c_{1} → C(f)(c_{2}) is a C(i_{1})-morphism;

• identities are id_{⟨i,c⟩} = ⟨id_{i}, id_{c}⟩;

• compositions are (⟨h, k⟩ ∘ ⟨f, g⟩) = ⟨(h ∘ f), (C(f)(k) ∘ g)⟩.

We are particularly interested in a certain instance of this construction. It is related to the categories [D, B] of functors from D to B and natural transformations between them. We refer the reader to [AHS90] for further details, in particular for the fact that [D, B] is cocomplete whenever B is.

Definition A.3.5. Let B be a category. The indexed category C_{B} : Cat^{op} → Cat is defined by:

• C_{B}(D) = [D, B]^{op};

• C_{B}(H : D → D′) : [D′, B]^{op} → [D, B]^{op} is such that:

– C_{B}(H)(F : D′ → B) = (F ∘ H) : D → B;

– C_{B}(H)(η : F_{1} → F_{2}) = (ηH) : (F_{1} ∘ H) → (F_{2} ∘ H).

We denote by Funct(B) the category (C_{B}^{♭})^{op}.

Note that objects of Funct(B) are just functors with codomain in B, and morphisms are pairs ⟨H, η⟩ : (F : D → B) → (F′ : D′ → B) such that H : D′ → D is a functor and η : (F ∘ H) → F′ is a natural transformation.

Moreover, it is easy to see that any functor G : B → B′ can be extended to a functor Funct(G) : Funct(B) → Funct(B′).

Definition A.3.6. Given G : B → B′, the functor Funct(G) : Funct(B) → Funct(B′) is defined by:

• Funct(G)(F : D → B) = (G ∘ F) : D → B′;

• Funct(G)(⟨H, η⟩ : (F_{1} : D_{1} → B) → (F_{2} : D_{2} → B)) = ⟨H, Gη⟩ : (G ∘ F_{1}) → (G ∘ F_{2}).

The result we intend to prove is an extension of a well-knownt theorem that states certain sufficient conditions for a Grothendieck category to be complete^{2}. A proof of this result can be found, for instance, in [TBG91].

Theorem A.3.7. Let C : Ind^{op} → Cat be an indexed-category. The flat category C^{♭} is complete whenever the following conditions hold:

2In fact, we shall be interested in colimits in Funct(B). Clearly, they correspond to limits in CB^{♭}.

• Ind is complete;

• C(i) is complete, for every i ∈ |Ind|;

• C(f) : C(i_{2}) → C(i_{1}) preserves limits, for every Ind-morphism f : i_{1} → i_{2}.

Regarding our indexed categories C_{B}, clearly, Cat is complete. Moreover, [D, B] is cocomplete whenever B is. And, furthermore, the functors C_{B}(H)^{op} trivially preserve colimits. All these facts can be found, for instance, in [AHS90]. Thus, we have the following result.

Corollary A.3.8. Let B be a category. If B is cocomplete then Funct(B) is cocomplete.

Finally, we shall state and prove our result about preservation of the construction through the Grothendieck construction.

Proposition A.3.9. Let G : B → B′ be a functor. If all G-pushouts exist then also all Funct(G)-pushouts exist.

Proof. Let {F_{k} : D_{k} → B}_{k∈{1,2}} be objects in Funct(B), and consider any source D = {⟨H_{k}, η_{k}⟩ : (H : D_{0} → B′) → (G ∘ F_{k})}_{k∈{1,2}} in Funct(B′). Then, the sink {⟨J_{k}, λ_{k}⟩ : F_{k} → (F : D → B)}_{k∈{1,2}} is a Funct(G)-pushout of F_{1} and F_{2} along D, where:

• {J_{k} : D → D_{k}}_{k∈{1,2}} is a pullback of {H_{k} : D_{k} → D_{0}}_{k∈{1,2}} in Cat, that is:

– D is the subcategory of (D_{1} × D_{2}) with only those pairs of objects ⟨b_{1}, b_{2}⟩ such that H_{1}(b_{1}) = H_{2}(b_{2}), and only those pairs of arrows ⟨f_{1}, f_{2}⟩ between these objects such that H_{1}(f_{1}) = H_{2}(f_{2});

– each J_{k} is the corresponding restriction to D of the projection functor Π_{k} : (D_{1} × D_{2}) → D_{k};

• {λ_{k}(b_{1}, b_{2}) : F_{k}(b_{k}) → F (b_{1}, b_{2})}_{k∈{1,2}} is a G-pushout of F_{1}(b_{1}) and F_{2}(b_{2}) along {η_{k}(b_{k}) : H(H_{k}(b_{k})) → G(F_{k}(b_{k}))}_{k∈{1,2}}, for each ⟨b_{1}, b_{2}⟩ ∈ |D|;

• F (f_{1}, f_{2}) : F (b_{1}, b_{2}) → F (b′_{1}, b′_{2}) is the unique B-morphism (as given by definition of G-pushout) such that (F (f_{1}, f_{2}) ∘ λ_{k}(b_{1}, b_{2})) = (λ_{k}(b′_{1}, b′_{2}) ∘ F_{k}(h_{k})) for k ∈ {1, 2}, for each D-morphism ⟨f_{1}, f_{2}⟩ : ⟨b_{1}, b_{2}⟩ → ⟨b′_{1}, b′_{2}⟩.

All the constructions are well defined. The verification that F : D → B is a functor is straightforward. It is also immediate to check that each ⟨J_{k}, λ_{k}⟩ : F_{k} → F is indeed a Funct(B)-morphism. Moreover, the definition of D clearly guarantees that (H_{1} ∘ J_{1}) = (H_{2} ∘ J_{2}). By definition of G-pushout, in addition, we have that also (G(λ_{1}(b_{1}, b_{2})) ∘ η_{1}(b_{1})) = (G(λ_{2}(b_{1}, b_{2})) ∘ η_{2}(b_{2})) for every ⟨b_{1}, b_{2}⟩ ∈ |D|.

As for the universal property, assume that a sink {⟨J′_{k}, λ′_{k}⟩ : F_{k} → (F′ : D′ → B)}_{k∈{1,2}} is such that also (H_{1} ∘ J′_{1}) = (H_{2} ∘ J′_{2}), and (Gλ′_{1} ∘ η_{1}J′_{1}) = (Gλ′_{2} ∘ η_{2}J′_{2}). Because D is a pullback, there exists a unique functor J : D′ → D such that (J_{k} ∘ J) = J′_{k} for k ∈ {1, 2}. Moreover, given b′ ∈ |D′|, let J(b′) = ⟨b_{1}, b_{2}⟩. Clearly, by definition of G-pushout, there exists a unique B-morphism λ(b′) : F (b_{1}, b_{2}) → F′(b′) such that (λ(b′) ∘ λ_{k}(b_{1}, b_{2})) = λ′_{k}(b′) for k ∈ {1, 2}. Thus, immediately, ⟨J, λ⟩ : F → F′ is the unique Funct(B)-morphism such that (⟨J, λ⟩ ∘ ⟨J_{k}, λ_{k}⟩) = ⟨J′_{k}, λ′_{k}⟩ for k ∈ {1, 2}. ⊓ ⊔

A.4 Remarks

A small remark is due, here, concerning the relevance of F -pushouts. Clearly, whenever either one of the two alternative solutions to the combination problem is available, F - pushouts are just syntactic-sugar. Everything that we do with them we can also do without them. That is precisely the case in all applications of F -pushouts used in this dissertation. Although very artificial examples can be found, it remains to be seen if F - pushouts can be useful in a situation where none of the other approaches works. We leave this question unanswered, since it falls out of the scope of the dissertation.