Logic and Implication: An Introduction to the General Algebraic Study of Non-classical Logics (Trends in Logic, 57) 3030856747, 9783030856748

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Trends in Logic 57

Petr Cintula Carles Noguera

Logic and Implication An Introduction to the General Algebraic Study of Non-classical Logics

Trends in Logic Volume 57

TRENDS IN LOGIC Studia Logica Library VOLUME 57 Editor-in-Chief Heinrich Wansing, Department of Philosophy, Ruhr University Bochum Bochum, Germany Editorial Board Arnon Avron, Department of Computer Science, University of Tel Aviv, Tel Aviv, Israel Katalin Bimbó, Department of Philosophy, University of Alberta, Edmonton, AB, Canada Giovanna Corsi, Department of Philosophy, University of Bologna, Bologna, Italy Janusz Czelakowski, Institute of Mathematics and Informatics, University of Opole Opole, Poland Roberto Giuntini, Department of Philosophy, University of Cagliari, Cagliari, Italy Rajeev Goré, Australian National University, Canberra, ACT, Australia Andreas Herzig, IRIT, University of Toulouse, Toulouse, France Wesley Holliday, UC Berkeley, Lafayette, CA, USA Andrzej Indrzejczak, Department of Logic, University of Lódz, Lódz, Poland Daniele Mundici, Mathematics and Computer Science, University of Florence, Firenze, Italy Sergei Odintsov, Sobolev Institute of Mathematics, Novosibirsk, Russia Ewa Orlowska, Institute of Telecommunications, Warsaw, Poland Peter Schroeder-Heister, Wilhelm-Schickard-Institut, Universität Tübingen, Tübingen, Baden-Württemberg, Germany Yde Venema, ILLC, Universiteit van Amsterdam Amsterdam, Noord-Holland, The Netherlands Andreas Weiermann, Vakgroep Zuivere Wiskunde en Computeralgebra, University of Ghent Ghent, Belgium Frank Wolter, Department of Computing, University of Liverpool, Liverpool, UK Ming Xu, Department of Philosophy, Wuhan University, Wuhan, China Jacek Malinowski, Institute of Philosophy and Sociology, Polish Academy of Sciences Warszawa, Poland Assistant Editor Daniel Skurt, Ruhr-University Bochum, Bochum, Germany Founding Editor Ryszard Wojcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences Warsaw, Poland The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. Volumes of Trends in Logic may range from highly focused studies to presentations that make a subject accessible to a broader scientific community or offer new perspectives for research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. This thematic spectrum is also reflected in the editorial board of Trends in Logic. Volumes may be devoted to specific logical systems, particular methods and techniques, fundamental concepts, challenging open problems, different approaches to logical consequence, combinations of logics, classes of algebras or other structures, or interconnections between various logic-related domains. This book series is indexed in SCOPUS. Authors interested in proposing a completed book or a manuscript in progress or in conception can contact either [email protected] or one of the Editors of the Series.

More information about this series at http://www.springer.com/series/6645

Petr Cintula Carles Noguera •

Logic and Implication An Introduction to the General Algebraic Study of Non-classical Logics

123

Petr Cintula Institute of Computer Science Czech Academy of Sciences Prague, Czech Republic

Carles Noguera Institute of Information Theory and Automation Czech Academy of Sciences Prague, Czech Republic

ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends in Logic ISBN 978-3-030-85674-8 ISBN 978-3-030-85675-5 (eBook) https://doi.org/10.1007/978-3-030-85675-5 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Marta and my parents (PC) To Monika, Greta, and my parents (CN)

Preface

This research monograph presents our approach to the general algebraic study of non-classical logics. More precisely, it offers a systematic study of weakly implicative logics, a class covering a vast part of the landscape of non-classical logics studied in the literature, including the prominent families of substructural, fuzzy, relevant, and modal logics. Its main method is the abstract mathematical study of the relation between logics and their algebraic semantics, concentrating mainly on the role of particular connectives in this relationship. Needless to say, neither the algebraic method, nor the mentioned prominent families of logics are our invention. What makes this book unique is its focus on implication as the main connective, right balance between abstraction and usability of the presented approach, and its self-contained and didactic presentation which allows it to also serve as a textbook for: • abstract algebraic logic, • substructural and fuzzy logics, and • the study of the role of implication and disjunction in (non-classical) logics. What has led us to writing this book? We are researchers who have been studying substructural and fuzzy logics for years using mainly the tools of algebraic logic. Soon enough in this endeavor, we observed the existence of a great deal of repetition in the papers published on this topic. It was common to encounter articles that studied slightly different logics by repeating the same definitions and essentially obtaining the same results by means of analogous proofs. We felt it as an unnecessary ballast that was delaying the development of such logics and obscuring the reasons behind the main results. This was an area of science screaming for systematization through the development and application of uniform, general, and abstract methods. The need for such a systematization project brought us together. Stemming from our background education, abstract algebraic logic presented itself as the ideal toolbox to rely on. It was a general theory applicable to all non-classical logics and it provided an abstract insight into the fundamental (meta)logical properties at play. However, the existing works in that area (most prominently the excellent monographs [41, 100, 134, 135] and surveys [136, 137]) did not readily give us the desired answers. vii

viii

Preface

Despite their many merits, these texts lived at a level of abstraction a little too far detached from our intended field of application. They were indeed great sources of knowledge and inspiration, but there was still a lot of work to be done in order to bring the theory closer to the characteristic particularities of substructural and fuzzy logics. Namely, we identified a number of properties codified in the logical connectives (mainly in implication and disjunction) that make these families of logics unique and interesting, and we observed that these properties could be (needed to be!) studied with the methods of abstract algebraic logic. These considerations led to an extensive series of papers [27, 68, 74, 75, 78, 81, 84, 85, 87, 89–93, 129, 170, 212, 213, 303] in which we have developed these ideas at different levels of generality and abstraction. Our first attempt at systematizing this bulk of research was a chapter published in 2011 in the Handbook of Mathematical Fuzzy Logic [88]. Naturally, the focus of that chapter was on fuzzy logics, it was intended for researchers in that area, and the limited space did not allow for a truly self-contained text or any extensive didactic aspirations. This book is a much more ambitious project, whose goal is to present a matured up-to-date theory which is powerful, general yet readily accessible, and keeps the right balance between abstraction and usability. Furthermore, we want to do that in a reasonably self-contained and didactic manner, starting from very elementary notions and building brick-by-brick a rather involved theory with a substantial number of results of a moderate level of difficulty and abstraction. Simply put, our goal was to create a book which—as students, researchers, and teachers—we needed and would have liked to read a long time ago. Who is this book intended for? We intend to reach a fairly wide audience: • students and scholars looking for an introduction to a general theory of non-classical logics and their algebraic semantics; • experts in non-classical logics looking for particular results on their favorite logics; • readers with background in mathematics, philosophy, computer science, or related areas, and an interest in formal reasoning systems that are sensitive to a number of intriguing phenomena (vagueness, graded predicates, constructivity, relevance, non-commutativity, non-associativity, resource-awareness, etc.); and • teachers of logic and related fields, who may use parts of the book as supporting material for their courses on topics related to (abstract) algebraic logic, nonclassical, substructural, and fuzzy logics. In spite of our best efforts for simplicity and development from first principles, we do not believe that our text can be taken as a first course in logic. Although we will explicitly define all the involved notions and introduce many particular logics (including the classical one) as examples for the theory, some familiarity with classical logic will certainly be necessary. Furthermore, the reader should be capable of understanding abstract definitions, symbolic notation, and mathematical proofs. Besides that, no other specific previous knowledge is required.

Preface

ix

How should this book be read? We recommend a sequential reading starting from the introduction (where we give the driving motivations of the book and justify its main design choices) and following the subsequent chapters in order. The second and third chapters develop the foundations of our approach on which the remainder of the book rests. Their content is based on deeply intertwined topics that are presented in the necessary logical order; hence skipping an important chunk of the text is likely to make these and the following chapters nearly incomprehensible. The subsequent chapters, from the fourth one onwards, focus on advanced topics, and while these topics are to a large extent independent, many particular results and examples of one chapter are used in the subsequent ones to illustrate and deepen their results, so a sequential reading is highly recommended as well. It is worth stressing that, although this book is about non-classical logics, our metatheory is classical, i.e. we see these logics as objects to be studied within standard mathematics governed by classical (meta)logic. For the reader’s convenience the Appendix contains the necessary elementary notions and facts from set theory (with axiom of choice), lattice theory, and universal algebra. Even readers uninitiated in these areas can start reading the book right away without studying this appendix, and consult it only whenever they encounter some difficulty along the main text. For a maximally beneficial study, the reader is expected to inspect carefully the proofs of the results. Although sometimes they may be routine checking of some properties, more often than not they illustrate the usage of the introduced notions or techniques and offer the best way towards mastering them. Likewise, we strongly encourage the reader to solve the proposed exercises (indicated throughout the text and listed at the end of each chapter) in order to check one’s understanding and progress. We also advise readers to check http://hdl.handle.net/11104/0309152, where we will maintain an updated list of typos and other errata. Prague, June 2021

Petr Cintula Carles Noguera

Acknowledgements

First and foremost we are indebted to our teachers, for their inspiration, patience, and infectious enthusiasm that led us to the area of (abstract) algebraic logic and mathematical fuzzy logic, specially Petr Hájek, Francesc Esteva, Joan Gispert, Franco Montagna, Lluís Godo, Ramon Jansana, Josep Maria Font, Janusz Czelakowski, Jon Michael Dunn, Antoni Torrens, and Ventura Verdú. In no smaller degree we are also indebted to a broad international community of researchers on non-classical logics and related topics who have inspired and guided us through numerous encounters and discussions over the years. This book is the result of years of efforts that go far beyond the work of its two authors. Most importantly, it includes (or it is influenced by) results achieved in cooperation with Guillermo Badia, Libor Běhounek, Marta Bílková, Karel Chvalovský, Pilar Dellunde, Denisa Diaconescu, Francesc Esteva, Àngel García-Cerdaña, Joan Gispert, Lluís Godo, Petr Hájek, Rostislav Horčík, Tomáš Lávička, George Metcalfe, Franco Montagna, and San-min Wang. Moreover, while writing the book we had the privilege of being surrounded by a team of researchers in Prague and elsewhere who have devoted their time to reading various drafts of (parts of) the book and have helped us improving it a great deal: Clint van Alten, Guillermo Badia, Marta Bílková, Diego Castaño, Valeria Castaño, Josep Maria Font, Nikolaos Galatos, Berta Grimau, Tomáš Lávička, George Metcalfe, Tommaso Moraschini, Francesco Paoli, Adam Přenosil, Luca Reggio, Igor Sedlár, Michał Stronkowski, Andrew Tedder, Jamie Wannenburg, Kentaro Yamamoto, and Zhiguang Zhao. Last, but not least, we are grateful to the two anonymous referees for their careful reading and helpful comments, to the editor of the series Heinrich Wansing and the Springer staff for their efficient support. Finally, we acknowledge the support of RVO 67985807 and of the Czech Science Foundation projects GA13-14654S and GA18-00113S.

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Weakly implicative logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Logics as mathematical objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hilbert-style proof systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Some prominent non-classical logics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Classical metalogical properties and their variants . . . . . . . . . . . . . . . 2.5 Logical matrices and semantical consequence . . . . . . . . . . . . . . . . . . . 2.6 The first completeness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Leibniz congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Weakly implicative logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Algebraically implicative logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 History and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 16 21 27 36 43 49 54 60 69 77 82

3

Completeness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.1 Three kinds of completeness and natural extensions . . . . . . . . . . . . . . 88 3.2 Homomorphisms and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3 Submatrices and conservative expansions . . . . . . . . . . . . . . . . . . . . . . 97 3.4 Direct products and Leibniz operator . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.5 Structure of the sets of theories and filters . . . . . . . . . . . . . . . . . . . . . . 109 3.6 Subdirect products and irreducible matrices . . . . . . . . . . . . . . . . . . . . 121 3.7 Filtered products, ultraproducts, and finitarity . . . . . . . . . . . . . . . . . . . 130 3.8 Completeness and description of classes of reduced matrices . . . . . . 138 3.9 History and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

xiii

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Contents

4

On lattice and residuated connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.1 Lattice connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.2 Residuated connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3 Prominent truth-constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.4 Lambek logic and the logic SL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.5 Axiomatization of LL, SL, and their fragments . . . . . . . . . . . . . . . . . . 184 4.6 Substructural logics and prominent extensions of SL . . . . . . . . . . . . . 194 4.7 Strongly separable axiomatic systems for extensions of SLaE . . . . . . 210 4.8 Implicational deduction theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.9 Strongly (MP)–bDT-based axiomatic systems . . . . . . . . . . . . . . . . . . . 230 4.10 Proof by cases property for generalized disjunctions . . . . . . . . . . . . . . 239 4.11 History and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

5

Generalized disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.1 A hierarchy of disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5.2 Characterizations of proof by cases properties via 5-forms . . . . . . . . 266 5.3 Generalized disjunctions and properties of the lattice of filters . . . . . 273 5.4 5-prime theories and 5-prime extension property . . . . . . . . . . . . . . . . 279 5.5 Pair extension property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 5.6 Completeness theorems and localization of prime matrices . . . . . . . . 299 5.7 History and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

6

Semilinear logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 6.1 Basic definitions, examples, and characterization . . . . . . . . . . . . . . . . 313 6.2 Semilinearity and properties of disjunction . . . . . . . . . . . . . . . . . . . . . 321 6.3 Substructural semilinear logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6.4 Completeness w.r.t. densely ordered chains . . . . . . . . . . . . . . . . . . . . . 336 6.5 History and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7

First-order predicate logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.1 Predicate languages and their interpretations . . . . . . . . . . . . . . . . . . . . 351 7.2 Semantically defined predicate logics . . . . . . . . . . . . . . . . . . . . . . . . . . 360 7.3 Predicate logics over L and axiomatization of the minimal one . . . . . 371 7.4 Axiomatization of RFSI-based (witnessed) predicate logics . . . . . . . 382 7.5 Predicate substructural logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 7.6 History and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Contents

A

xv

Basic mathematical notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 A.1 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 A.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 A.3 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 A.4 Varieties and (generalized) quasivarieties . . . . . . . . . . . . . . . . . . . . . . . 429 A.5 Modal, Heyting, G-, and MV-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 432

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 List of axioms and rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

List of Tables

4.1 4.2 4.3 4.4

The language LSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Consecutions defining principal substructural logics . . . . . . . . . . . . . . 197 Notable classes of SL-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Strongly (MP)–bDT-based logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

5.1

Strong p-disjunctions in prominent substructural logics . . . . . . . . . . . . 263

6.1 6.2

Axiomatization of semilinear extensions of notable logics . . . . . . . . . . 324 Axiomatization of Lℓ for prominent substructural logics . . . . . . . . . . . 331

xvii

List of Figures

2.1 2.2

Prominent axiomatic extensions of the logic BCI . . . . . . . . . . . . . . . . . 28 Extended Leibniz hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.1

Hierarchy of infinitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Strongly separable presentation of the Lambek logic LL . . . . . . . . . . . 186 Separable presentation of the logic SL . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Prominent extensions of the logic SL . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Strongly separable presentation of the logic SLaE . . . . . . . . . . . . . . . . . 210 Strongly separable presentation of the logic SLaEio = FLew . . . . . . . . . 215 Strongly separable presentation of the logic SLaECio = IL . . . . . . . . . . 216 Strong (MP)–bDTSLa -based presentation of the logic SLa . . . . . . . . . . 232 Strong (MP)–bDT-based presentation of the logic SL . . . . . . . . . . . . . 235 Prominent extensions of the logics SL and SLo . . . . . . . . . . . . . . . . . . . 247

5.1

Hierarchy of disjunctional logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6.1 6.2

Strongly separable presentation of the logic MTL = FLℓew . . . . . . . . . . 332 Strongly separable presentation of the logic G = ILℓ . . . . . . . . . . . . . . 334

A.1 Examples of Hasse diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 A.2 Examples of prominent non-distributive lattices . . . . . . . . . . . . . . . . . . 421

xix

Acronyms

CIPEP

Completely intersection-prime extension property: A logic has the CIPEP if the completely intersection-prime theories form a basis of the closure system of its theories.

IPEP

Intersection-prime extension property: A logic has the IPEP if the intersection-prime theories form a basis of the closure system of its theories.

5PEP

5-prime extension property: A logic has the 5PEP if the 5-prime theories form a basis of the closure system of its theories.

LEP

Linear extension property: A weakly implicative logic has the LEP if its linear theories form a basis of the closure systems of its theories.

DEP

Dense extension property: A weakly implicative logic has the DEP if for every set Γ ∪ {𝜑} of formulas with infinitely many unused variables such that Γ 0 𝜑, there is a dense theory 𝑇 0 ⊇ 𝑇 such that 𝜑 ∉ 𝑇 0.

DP

Density property: A weakly implicative logic has the DP w.r.t. a protodisjunction 5 if, for any set of formulas Γ ∪ {𝜑, 𝜓, 𝜒} and any variable 𝑝 not occurring in Γ∪{𝜑, 𝜓, 𝜒}, the following metarule is valid: Γ I (𝜑 ⇒ 𝑝) 5 ( 𝑝 ⇒ 𝜓) 5 𝜒 . Γ I (𝜑 ⇒ 𝜓) 5 𝜒

SLP

Semilinearity property: A weakly implicative logic has the SLP if the following metarule is valid: Γ, 𝜑 ⇒ 𝜓 I 𝜒 Γ, 𝜓 ⇒ 𝜑 I 𝜒 . ΓI𝜒 xxi

xxii

sPCP

Acronyms

Strong proof by cases property: A logic has the sPCP w.r.t. a protodisjunction 5 if the following metarule is valid: Γ, Φ I 𝜒 Γ, Ψ I 𝜒 . Γ, Φ 5 Ψ I 𝜒

PCP

Proof by cases property: A logic has the PCP w.r.t. a protodisjunction 5 if the following metarule is valid: Γ, 𝜑 I 𝜒 Γ, 𝜓 I 𝜒 . Γ, 𝜑 5 𝜓 I 𝜒

wPCP

Weak proof by cases property: A logic has the wPCP w.r.t. a protodisjunction 5 if the following metarule is valid: 𝜑I𝜒 𝜓I𝜒 . 𝜑5𝜓 I 𝜒

SKC

Strong K-completeness: A logic L has the SKC if it coincides with K , i.e. each consecution is valid in L iff it is valid in K.

FSKC

Finite strong K-completeness: A logic L has the FSKC if each finitary consecution is valid in L iff it is valid in K.

KC

K-completeness: A logic L has the KC if each formula is valid in L iff it is valid in K.

RSI-model

Relatively subdirectly irreducible model/algebra: a matrix (resp. algebra) from Mod∗ (L) (resp. Alg∗ (L)) which is isomorphic to some element of each of its subdirect representations by elements of Mod∗ (L) (resp. Alg∗ (L)).

RFSI-model

Relatively finitely subdirectly irreducible model/algebra: a matrix (resp. algebra) from Mod∗ (L) (resp. Alg∗ (L)) which is isomorphic to some element of each of its non-empty finite subdirect representations by elements of Mod∗ (L) (resp. Alg∗ (L)).

RSI-completeness

Relatively subdirectly irreducible completeness: A logic L is RSI-complete if it coincides with Mod∗ (L)RSI .

RFSI-completeness

Relatively finitely subdirectly irreducible completeness: A logic L is RFSI-complete if it coincides with Mod∗ (L)RFSI .

Chapter 1

Introduction

Since it was established in the Western tradition by Aristotle in the fourth century BC, logic has been concerned with correct reasoning, that is, with the study of the valid ways by which one can infer a proposition (the conclusion) from a set of previously given propositions (the premises). It is a formal science because it concentrates on inference patterns that are valid solely by virtue of their form, not of their content. A very successful account of valid reasoning is the one known as classical logic, which has been proposed as the cornerstone of all science, rationality, rigorous knowledge, and reliable communication. It is a study of logical inference that stems from the seminal Aristotelian syllogistic presented in the Prior Analytics and from the analysis of propositions developed by the Stoics since the third century BC. Later, it was widely used and developed by medieval logicians, and finally obtained its contemporary presentation in the nineteenth century by the founders of modern mathematical logic (Augustus De Morgan, George Boole, Gottlob Frege, and others). Thanks to their works, today we know classical logic as a well-developed mathematical machinery that allows one to determine the validity of, allegedly, any given argument that can be analyzed in terms of the usual propositional connectives and quantifiers. Propositional classical logic is usually syntactically given in a language with the following logical connectives: negation ¬, implication →, disjunction ∨, conjunction ∧, and equivalence ↔. Semantically, it operates on the basic assumption of bivalence, i.e. each well-formed meaningful proposition must always be either true or false, and any other possibility is excluded. This is implemented by giving a mathematical semantics to all sentences by means of the two-valued Boolean algebra 2: an algebraic structure with two values—1 which stands for true and 0 for false—and operations for the connectives (¬2 for negation, →2 for implication, ∨2 for disjunction, ∧2 for conjunction, and ↔2 for equivalence) defined as: 𝑎 ¬2 𝑎 1 0 0 1

𝑎 1 1 0 0

𝑏 1 0 1 0

𝑎 →2 𝑏 1 0 1 1

𝑎 ∨2 𝑏 1 1 1 0

𝑎 ∧2 𝑏 1 0 0 0

© Springer Nature Switzerland AG 2021 P. Cintula and C. Noguera, Logic and Implication, Trends in Logic 57, https://doi.org/10.1007/978-3-030-85675-5_1

𝑎 ↔2 𝑏 1 0 0 1 1

2

1 Introduction

The semantics of all formulas is established in terms of evaluations1 into 2, i.e. mappings from the set of all formulas to {0, 1} that use the algebraic operations above to interpret connectives (e.g. 𝑒(𝜑 → 𝜓) = 𝑒(𝜑) →2 𝑒(𝜓), where 𝑒 is an evaluation). This gives a very specific character to the classical semantics with several features worth commenting on: Truth-functionality: The value of any complex sentence is computed using only the values of its constituent parts. Material implication: In particular, the value of an implicational formula 𝜑 → 𝜓 depends only on the value of 𝜑 (its antecedent) and the value of 𝜓 (its consequent), disregarding any mutual relations their meanings might have. Preservation of truth: Material implication captures a certain intuition of transmission or preservation of truth. Indeed, if an implication is true, the truth of its antecedent must imply the truth of its consequent; in other words, a true implication can never have a true antecedent and a false consequent. Order of truth: As a simple reformulation of the previous observation, if we think that the two truth-values are ordered (0 < 1), we have that in a true implication the value of the antecedent is less than or equal to the value of the consequent. Notable tautologies: The following formulas are tautologies (i.e. true under any evaluation): Double negation elimination Law of non-contradiction Law of excluded middle Contraction Weakening

¬¬𝜑 → 𝜑 ¬(𝜑 ∧ ¬𝜑) 𝜑 ∨ ¬𝜑 𝜑→𝜑∧𝜑 𝜑 → (𝜓 → 𝜑).

Using evaluations one can define the fundamental logical notion of semantical consequence. That is, if Γ is a set of formulas and 𝜑 is a formula, we say that 𝜑 (the conclusion) is a semantical consequence of Γ (the premises) if any possible evaluation of formulas that makes all the formulas in Γ true (i.e. equal to 1), must also make 𝜑 true; equivalently: there is no evaluation under which all formulas in Γ are true and 𝜑 is false. In symbols, we write: Γ 2 𝜑. Again, as in the case of implication, this definition follows the idea of preservation of truth: if Γ 2 𝜑, the truth of Γ must be preserved in the truth of 𝜑. In this way, classical logic manages to capture one possible precise notion of correct reasoning, that is: an argument that uses Γ as the set of premises and obtains 𝜑 as the conclusion is correct precisely when Γ 2 𝜑. Moreover, besides mathematical semantics, propositional classical logic has also been given a wealth of proof systems; that is, formal calculi allowing us to derive conclusions from sets of premises, based purely on symbolic manipulation of formal sentences, regardless of any semantical interpretation. Among them, Hilbert-style axiomatic systems are very elementary and have a strong theoretical interest, while 1 Some sources use the term ‘valuation’ instead. Since this is a book devoted to both propositional and predicate logic, we will also need to introduce mappings that interpret first-order variables in a domain of individuals. To distinguish them, we have opted to use ‘evaluation’ for the propositional semantics and ‘valuation’ for the predicate semantics in an admittedly arbitrary choice.

1 Introduction

3

they may be difficult to use when looking for particular derivations of some given formulas. Other kinds of systems are more versatile for practical purposes (e.g. natural deduction, sequent and hypersequent calculi, tableaux, resolution, etc.) and have given rise to algorithms for automated theorem proving, a necessary prerequisite for applications of logic in problems of computer science and artificial intelligence. Hilbert-style calculi consist of a set of axioms (formulas that are stipulated to be always true and can be used at any moment) and a set of inference rules (that allow one to derive new formulas from those that one already has). For example, a typical rule to capture the formal behavior of implication is the well-known modus ponens: 𝜑, 𝜑 → 𝜓 I 𝜓. According to this rule, whenever we prove a formula 𝜑 and a formula 𝜑 → 𝜓, we can automatically derive the consequent 𝜓, just by virtue of the kind of symbols (i.e. implication) that are being manipulated, regardless of any semantical interpretation. By introducing a sufficient number of such formal rules, we have the necessary means to obtain a notion of formal proof in the Hilbert-style system, construed as a finite sequence of formulas. The starting point of any proof, besides the axioms, is a set of premises that are taken as hypotheses in a certain context. Additional elements are then conclusions of inference rules whose premises already appear in the proof. Given a set Γ of formulas and a formula 𝜑, we write Γ `CL 𝜑 (CL standing for classical logic) to signify that, when taking Γ as premises, there exists a formal proof in which 𝜑 is the last obtained formula, i.e. the conclusion. Other kinds of proof systems give rise to their corresponding notions of formal proofs. To stress the opposition with the semantical rendering of correct reasoning, we may say that proof systems are purely syntactical devices, i.e. inference machines that pay attention only to the syntactical form of the expressions they manipulate. If proof systems are to make sense and be maximally useful from the theoretical and the applied viewpoints, they should be: Sound: If Γ `CL 𝜑, then Γ 2 𝜑. That is, the proof system is not too strong to formally prove claims which are not correct. Complete: If Γ 2 𝜑, then Γ `CL 𝜑. That is, the proof system is strong enough to obtain formal proofs of all correct claims. Thus a sound and complete (syntactical) proof system is equivalent to—it captures exactly—the (semantical) classical notion of correct reasoning. We are now about to introduce a wealth of non-classical logics described semantically (via classes of models) and syntactically (via proof systems). One of the main topics of this book is the study of under which conditions a given syntactical and semantical presentation coincide, i.e. describe the same logic. Such results are historically called completeness theorems, despite of the fact that they are formulated as equivalence (‘if and only if’) statements, not as a single implication as the name and the classical terminology would suggest. Therefore, from now on, we use in this context the term ‘complete’ in the sense of ‘sound and complete’. As we almost exclusively work in the context when soundness is already established, there is no danger of confusion.

4

1 Introduction

Beyond classical logic There have been a number of claims that classical logic, for all its merits, is not capable of providing a satisfactory explanation of correct reasoning in all possible forms and contexts. Many have pointed at important shortcomings of an analysis that requires the strong assumptions we have listed before, including the oftentimes too limiting restriction to bivalent semantics, or the validity of the law of excluded middle which forces any proposition to be such that either itself or its negation is true (in a given interpretation). Such restrictions confine the classical logician to a narrow set of connectives with several obvious limitations, including among others the following: • The built-in truth-functionality imposes a strong restriction on expressive power. Indeed, it excludes intensional (i.e. non-truth-functional) contexts such as those given by modalities: necessity, possibility, propositional attitudes, etc. • The bivalent connectives can have a rather unnatural behavior, as in the case of material implication, which takes as indisputably true any implication that has a tautology in the consequent or a contradiction in the antecedent. • Bivalence, the law of excluded middle, and contraction cause very serious problems when confronted with some critical logical puzzles, such as self-reference paradoxes or the sorites paradox in the analysis of vagueness. • The law of excluded middle and double negation elimination allow for nonconstructive proofs by contradiction in mathematical reasoning, which has been seen as a commitment to a Platonist view of mathematics (a view that takes mathematical objects as abstract entities with independent existence). The discussion about the limits of classical logic has given rise during the last century to a plethora of alternative non-classical logical systems, based on a wide range of different motivations (to which we have only slightly hinted). Some of them have been proposed to amend alleged deficiencies of classical logic in certain reasoning contexts (e.g. intuitionistic logic for constructive mathematical reasoning, or relevant logics when material implication is not considered adequate). Other non-classical logics have been developed as useful technical devices for a finer analysis of reasoning (e.g. fuzzy logics for reasoning with graded predicates or paraconsistent logics for reasoning in the presence of contradictions) or to model other phenomena (e.g. linear logics for computational tasks or Lambek logic for linguistic analysis). Finally, many others have been defined and studied out of sheer mathematical curiosity. Formally speaking, some of these alternative non-classical propositional logics are expansions of classical logic with new syntactical devices (such as modalities) that ensure a higher expressive power, while others invalidate some problematic classical truths. Let us briefly introduce three classes of such logical systems (which will play an important role in this book), stressing the main aspects in which they deviate from the classical paradigm and exemplifying the difference in the semantics of implication.

1 Introduction

5

Many-valued logics. The twentieth century witnessed a proliferation of logical systems which, though still truth-functional, deviate from classical logic by having an intended algebraic semantics with more than two truth-values (for a historical account see e.g. [141]). Prominent examples are 3-valued systems like Kleene’s logic of indeterminacy and Priest’s logic of paradox, 4-valued systems like Dunn–Belnap’s logic, the 𝑛-valued systems of Łukasiewicz and Post, and even infinitely-valued systems such as Łukasiewicz logic Ł [218] or Gödel–Dummett logic G [113]. Let us illustrate these many-valued semantics by taking a look at the definition of two operations intended as interpretations of the implication connective in Łukasiewicz and Gödel–Dummett logics (for values 𝑎, 𝑏 ∈ [0, 1]): ( 1 if 𝑎 ≤ 𝑏 Ł 𝑎→ 𝑏= 1−𝑎+𝑏 otherwise ( 𝑎 →G 𝑏 =

1

if 𝑎 ≤ 𝑏

𝑏

otherwise.

These examples showcase a typical feature of many-valued logics: the multiple values in the semantics do not form an arbitrary chaotic set, but they follow a prescribed order, in this case the standard order of the real numbers in [0, 1]. Then, we may say that, if 𝑎 ≤ 𝑏, then 𝑏 accounts for propositions that are at least as true as those that are given the value 𝑎. The greatest value of the set, the number 1, is then taken as representing full truth. This allows us to argue that, despite the complexity of the many-valued semantics, the behavior of implication still bears a strong resemblance to some aspects of classical implication. That is to say, the two mentioned examples still follow the guiding idea of truth preservation, which now can be formulated as: if an implication is fully true (i.e. takes value 1), its consequent cannot be less true than its antecedent. As in the case of classical logic, the algebraic (this time many-valued) operations for all connectives present in the language of a many-valued logic give rise to evaluations, i.e. mappings assigning to each formula, in a structure-preserving way, an element of the set of truth-values. Evaluations are then essential for extending the classical definition of semantical consequence. For instance, one defines Γ Ł 𝜑 as: each evaluation in the [0, 1]-valued semantics that gives value 1 to all formulas in Γ must also give value 1 to 𝜑 (and analogously for Γ G 𝜑). It is, hence, preservation of the notion of full truth given by the value 1. Similar truth-preserving definitions can be given in general for any many-valued logic, hence giving a multitude of alternative semantical accounts of correct reasoning. On the other hand, many-valued logics also enjoy the classical repertoire of proof systems by which they are endowed with a syntactical notion of inference. Naturally, a fundamental result in the mathematical study of each many-valued logic is the corresponding completeness theorem that guarantees the perfect link between an intended notion of semantical consequence and a given syntactical proof system.

6

1 Introduction

More recently, the field of algebraic logic has developed a paradigm in which most systems of non-classical logics can be seen as many-valued logics, because they are given a semantics in terms of algebras with more than two truth-values. From this point of view, many-valued logics encompass wide well-studied families of logical systems such as relevance logics, intuitionistic logic and its extensions, and even the family of substructural logics on which we will comment next (see e.g. [143]). Substructural logics. Classical logic can be presented, among other syntactical options, by a proof system based on sequents introduced by Gentzen [150, 151]. It is constituted by • logical rules governing the behavior of connectives and • structural rules which do not refer to any particular connective. Logics lacking some of these structural rules (most importantly, those known as exchange, weakening, contraction, and associativity) are studied in the literature under the name substructural logics. As explained in the monographs [256, 273, 283] they form a huge class of non-classical logics including: relevant logics (amenable to deal with the aforementioned unintuitive behavior of classical material implication), linear logics (introduced in theoretical computer science as resource-aware systems) or Lambek calculi (introduced in linguistics to deal with grammatical categories in formal and natural languages). Although they have been proposed in terms of largely unrelated motivations and have been the subject of many independent studies, in recent years an increasing number of authors have followed the systematic unifying approach of algebraic logic that allows one to see substructural logics as a specific kind of many-valued logics. Indeed, a long stream of purely algebraic papers has concentrated on the algebraic semantics of substructural logics, based on residuated lattices, and have resulted in a deep knowledge of these logics (see e.g. the monograph [143]). One of the advantages of residuated lattices is the presence of their lattice order, which allows us to keep the aforementioned idea of values ordered according to their truth, that is, for elements 𝑎 and 𝑏 of a residuated lattice A with order ≤, we say that 𝑏 is at least as true as 𝑎 whenever 𝑎 ≤ 𝑏. Moreover, the propositional language of substructural logics typically includes a conjunction connective & and constant symbol 1¯ respectively interpreted in A by a binary operation &A and its neutral element 1¯ A. The latter can be used to define the following set of designated elements: 𝐹 = {𝑎 ∈ 𝐴 | 𝑎 ≥ 1¯ A } which accounts for the full truth that has to be preserved in semantical consequences (hence, contrary to the previously seen examples of many-valued logics, now there may be many truth-values which are considered fully true and 1¯ A is just the least of them): Γ A 𝜑

if and only if

for each evaluation 𝑣 in A, if 𝑣(𝛾) ∈ 𝐹 for each 𝛾 ∈ Γ, then 𝑣(𝜑) ∈ 𝐹.

1 Introduction

7

Focusing again on the behavior of implications, let us point out that the semantical counterpart of an implication → in A is a binary operation →A satisfying the following residuation property with respect to the operation &A for any elements 𝑎, 𝑏, 𝑐 ∈ 𝐴:2 𝑎 &A 𝑐 ≤ 𝑏

𝑐 ≤ 𝑎 →A 𝑏.

if and only if

Taking 𝑐 = 1¯ A , we obtain a crucial relation between the lattice order, the implication operation, and the set of designated elements: 𝑎≤𝑏

if and only if

𝑎 →A 𝑏 ∈ 𝐹,

which captures the idea that an implication is fully true exactly when the consequent is truer than the antecedent. Structural rules have natural interpretations, in residuated lattices, as properties of &A : in particular, exchange makes it commutative, weakening identifies its neutral element 1¯ A with the greatest element of the lattice order ≤, and contraction (together with weakening) makes it idempotent. Fuzzy logics. In mathematics, all terms are assumed to be precise and well-defined, in the sense that every property is expected to yield a perfect division between the objects which satisfy it and those which do not. That is why classical logic is well-tailored to model reasoning in usual mathematical practice. However, as soon as one considers non-mathematical contexts, one immediately has to deal with vague predicates (such as old, tall, or warm) for which it is not possible to establish such a clear division. Fuzzy logics have been proposed as non-classical systems for dealing with vagueness [287]. They are based on two main principles: • The truth of vague propositions is a matter of degree. • Degrees of truth must be comparable. Inside the family of many-valued logics one can easily find good systems that satisfy these principles. The most typical examples are [0, 1]-valued logics (like Ł and G). Historically, fuzzy logic emerged from fuzzy set theory (first proposed in 1965 by Lotfi Zadeh [312]). Such theory became extremely popular in computer science and engineering giving rise to a whole area of research with uncountable applications (see e.g. [275, 278]), but it lacked a focus on the usual aspects studied by logicians, e.g. formal language, semantics, proof systems, analysis of arguments, etc. To remedy this shortcoming, at the beginning of the 1990s, based on earlier works [159, 243, 244, 258–260, 292, 293] the Czech mathematician Petr Hájek became the leader of a tour de force to provide solid logical foundations for fuzzy logic. In his approach, that soon was followed by numerous researchers in mathematical logic, fuzzy logics were taken as non-classical systems with a many-valued semantics that reflects the principle of comparability of degrees of truth. He achieved this by restricting the behavior of implication with the following prelinearity axiom: (𝜑 → 𝜓) ∨ (𝜓 → 𝜑). 2 Since we do not assume &A to be commutative, the expression below is, strictly speaking, only a half of the residuation property; see Chapter 4 for more details.

8

1 Introduction

Hájek’s monograph [160] studied fuzzy logics with the tools of algebraic logic and gave birth to a whole new subdiscipline of mathematical logic, called mathematical fuzzy logic (MFL), specialized in the study of this family of many-valued logics. Interestingly enough, many fuzzy logics have been identified as a particular kind of (axiomatic extensions of) substructural logics whose algebraic semantics is generated by linearly ordered algebras [124] (coherently with the principle that all degrees of truth must be comparable). The last two decades have witnessed a great development of MFL (see it presented in the handbook series [79, 82, 83]) and a proliferation of fuzzy logics with diverse properties but always keeping completeness with respect to linearly ordered algebras. Abstract algebraic logic The growing multiplicity of logics certainly calls for a uniform general treatment. A natural candidate for such a general theory is algebraic logic, the branch of mathematical logic that studies logical systems by giving them a semantics based on some algebraic structures. As mentioned above, this branch has served as a unifying approach to deal with the increasingly populated landscape of non-classical logics and has developed a variety of techniques which have been fruitfully applied to many families of logics, including those we have just seen. In the last four decades, algebraic logic has evolved into a more abstract discipline, abstract algebraic logic (AAL), which aims at understanding the various ways in which a logical system (in an arbitrary language) can be endowed with an algebraic semantics. The pioneering works in this area are those from the Polish logic school [216, 269, 294, 310]. Later, the theory was thoroughly developed and systematized mostly by Willem J. Blok, Janusz Czelakowski, Josep Maria Font, Ramon Jansana, Don L. Pigozzi, and James G. Raftery [41, 100, 135, 136, 267]. By understanding the deep connection between logics and their algebraic semantics, AAL has provided a corpus of results that allows one to study properties of the logical systems by means of (equivalent) algebraic properties of their semantics. Also, AAL has led to a finer analysis of the role of the connectives of classical logic, identifying their essential properties, and thus suggesting possible generalizations of these connectives (in non-classical logics) still retaining the essential function(s) they play in classical logic. Notable examples of this approach are the extensive studies on equivalence (or biconditional) connectives in the works we have just mentioned. Indeed, the Lindenbaum–Tarski proof of completeness of the classical propositional calculus, based on the fact that the equivalence connective defines a congruence on the algebra of formulas, has been extended to broad classes of logics by considering a suitable generalized notion of equivalence. That is, equivalence can be taken as a (possibly parameterized, possibly infinite) set of formulas in two variables satisfying certain simple properties. This approach gave rise to the important class of protoalgebraic logics [40] characterized by the presence of a generalized equivalence. Similarly, there have been works focusing on suitable notions of conjunction (e.g. [196]), disjunction (e.g. [89, 100]), negation (e.g. [266]), and implication (e.g. [42,99,195] and, especially, Helena Rasiowa’s theory of implicative logics [269]).

1 Introduction

9

Logics and implication In logical consequence the truth of a set of premises is ‘transmitted’ to a conclusion. We have seen that both in classical and in many non-classical logics this idea guides the definition of both the semantics (algebraic operations) and the inference rule (modus ponens) of implication. In many logics, including classical, the relation between logical consequence and implication is very straightforward thanks to the deduction theorem: 𝜑`𝜓

if and only if

` 𝜑 → 𝜓,

that is, the implication connective internalizes logical consequence, which is one of the reasons why implication may be seen as the main logical connective. Moreover, algebras of fuzzy logics (and, more generally, of substructural logics) have an order relation naturally determined by the implication operation. In the most well-known setting, this can be described in the following way: 𝑎≤𝑏

if and only if

𝑎 →A 𝑏 = 1¯ A .

More generally, as we have seen in substructural logics, one may need to consider not just the value 1¯ A , but the set 𝐹 of designated elements in the algebra, which allows one to define the order in terms of the implication as 𝑎≤𝑏

if and only if

𝑎 →A 𝑏 ∈ 𝐹.

This easy, and very well-known, observation inspired the first author of this book to start developing in 2005 a general framework for all logics with this property. Generalizing Rasiowa’s implicative logics (which still were restricted to logics with a semantics with a greatest degree of truth which, moreover, is the only designated value), the paper [74] introduced the class of weakly implicative logics as those with a binary connective ⇒ that enjoys what we consider the minimal basic requirements any implication connective should satisfy (these requirements have been also considered by other authors; see the last but one paragraph of Section 2.10 for more details): Identity Modus ponens Transitivity Congruence

𝜑⇒𝜑 𝜑, 𝜑 ⇒ 𝜓 I 𝜓 𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜒 I 𝜑 ⇒ 𝜒 𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜑 I 𝑐(𝛼1 , . . . 𝛼𝑖 , 𝜑, . . . 𝛼𝑛 ) ⇒ 𝑐(𝛼1 , . . . 𝛼𝑖 , 𝜓, . . . 𝛼𝑛 ) for each 𝑛-ary connective and each 0 ≤ 𝑖 < 𝑛.

These conditions are enough to guarantee that the connective ⇒ defines an order relation in the algebraic semantics in the way we have just seen. Moreover, the framework allows one to capture a precise mathematical rendering of the informal notion of fuzzy logics as those that are complete with respect to their linearly ordered (by the order given by the implication) algebraic models. These systems and their abstract theory have been studied as semilinear logics by the authors of the present book as an AAL-style foundation of MFL in [88]. This approach has been later extended and developed in detail in a series of papers [87,92,93] as the theory of weakly p-implicational logics, in which implications are taken as connectives defined by (possibly infinite and parameterized) sets of

10

1 Introduction

formulas. The requirements on such generalized connectives are indeed very weak and encompass a very broad class of logics. Actually, p-implicational logics turn out to be an alternative presentation of the protoalgebraic logics, a fundamental class of logics deeply studied in the core theory of AAL. However, the stress on generalized implications has allowed us to focus better on certain aspects of these logics. Interestingly enough, most of the advantages of this implication-based general approach are already available at the level of weakly implicative logics. Indeed, this class already contains most of the prominent non-classical logics studied in the literature, since they almost always have a reasonable implication connective. In particular, weakly implicative logics provide a good framework to study fuzzy, substructural and many-valued logics. A critical point of any general theory is the choice of its level of generality. At the time of its publication, Rasiowa’s monograph on implicative logics [269] had struck an excellent level of generality, broad enough to cover most of the research being done at the time, and yet not too far as to become too abstract and difficult to understand and use. The subsequent development of non-classical logics, however, made it obsolete. Many important logics, studied in theoretical papers and sometimes used in applications to other areas, did not fit anymore in the class of implicative logics with its rather narrow defining restrictions. The present book intends to remedy this shortcoming by presenting the theory of weakly implicative logics as a new framework that can have today the same advantages that Rasiowa’s class used to have.

Content and structure of the book After this introductory chapter, the book is structured in six additional chapters and one Appendix. The latter presents the basic preliminary mathematical notions and results used throughout the book. It starts with elementary notions of basic set theory, orders, closure systems, and lattices; after that it presents the basics of universal algebra and is concluded by recalling some notable classes of algebras related to non-classical logics (modal, Heyting, Gödel, and MV-algebras). However, even readers uninitiated in these areas can start reading the main text without caring for this Appendix, and go there only whenever they encounter some difficulty. As said before, we expect a sequential reading of the subsequent chapters in order. Let us briefly describe their content. Chapter 2. This chapter is the real beginning of our story and hence it is devoted to presenting its main object of study: the class of weakly implicative logics. We start by introducing basic syntactical notions (variables, connectives, formulas, Hilbert-style proof systems, etc.) and giving a purely syntactical definition of logics as mathematical objects (namely, as structural consequence relations). After testing the definition with three extreme, mostly uninteresting examples, we immediately present some of the most important logics that one can find in the literature and that will accompany the reader throughout the book: classical logic, intuitionistic logic, Łukasiewicz logic (both its finitary and infinitary version), Gödel-Dummett logic, and the implicational logics known as BCI, A→ , and BCK. We conclude the

1 Introduction

11

syntactical part of the chapter with a brief study of important metalogical properties of these logics such as (variants of) the deduction theorem and the proof by cases property; these properties are later, in Chapters 4 and 5, studied in a much more abstract setting. Next, we start introducing basic semantical notions. The fundamental one is that of a logical matrix, which is an arbitrary algebra equipped with a set of designated truth-values, called the filter of the matrix. Any logic can be assigned a class of logical matrices, whose members we call models of the logic, which can be shown to provide a complete semantics, albeit a very uninformative one with a very loose connection to the logic in question. In order to obtain a more meaningful semantics, we introduce the notion of Leibniz congruence of a given matrix, which allows us to define, for any given logic, a class of reduced models which provide a more refined complete semantics and are more tightly related to the logic in question. Finally, we give a purely syntactical definition of the class of weakly implicative logics and show that these are exactly the logics where the Leibniz congruence admits a simple description using the implication connective. This connective can also be used to define another fundamental notion of the book: the matrix order in the reduced models of weakly implicative logics. The chapter is concluded by introducing and exploring the class algebraically implicative logics, in which one can disregard logical matrices and work just with algebras instead. Chapter 3. This chapter presents the foundations of the theory of logical matrices with a special focus on the question of which classes of matrices provide a complete semantics for a given logic. We identify three kinds of completeness based on how we restrict the cardinality of the sets of premises: we distinguish strong completeness, where there is no restriction, finite strong completeness, where we restrict ourselves to finite sets of premises, and weak completeness, where we disregard premises altogether and speak about theorems and tautologies only. Throughout the chapter we introduce increasingly complex model-theoretic constructions on (classes of) matrices and use them to characterize not only the completeness properties but also other important notions. We start by introducing submatrices, homomorphisms and direct products of matrices. We use these tools to improve our understanding of the Leibniz congruence and obtain an important semantical characterization of the notion of conservative expansions and of the class of algebraically implicative logics. Our next tools are the subdirect products and subdirectly irreducible matrices. We show that each finitary logic is strongly complete with respect to such matrices, in particular recovering the completeness of classical logic with respect to the two-element Boolean algebra. Finally, the most complex construction we use are the filtered products, in particular countably-filtered products and ultraproducts, which we use to give a purely semantical characterization of finitary logics and completeness properties. Chapter 4. Clearly, not only implication, but also other logical connectives are crucial for the theory and the applications of particular logics. Hence, this chapter is devoted to the study of two groups of important connectives: lattice (∧, ∨, >, ⊥) and ¯ 0): ¯ residuated connectives (&, , 1,

12

1 Introduction

• The lattice connectives allow us to express properties of the matrix order in the models of the logic in question: ∧ and ∨ are binary connectives whose intended interpretation is, respectively, the infimum and the supremum of this order, whereas > and ⊥ are truth-constants that are intended to be interpreted by respectively the greatest and the least element of the order. • The residuated conjunction & is a binary connective intended to work as a means to aggregate premises in chains of nested implications. This conjunction is not assumed, in general, to be commutative (neither associative nor idempotent). In logics in which & is commutative, we can switch the order of premises in nested implications. Otherwise, we can add the binary connective (co-implication) which allows us to do the switch at the price of replacing the original inner implication by . Algebraically, the relation between &, →, and is described by the residuation property: 𝑥 ≤ 𝑦 →A 𝑧

iff

𝑦 &A 𝑥 ≤ 𝑧

iff

𝑦≤𝑥

A

𝑧.

The truth-constant 1¯ is intended to stand for the protounit, i.e. the minimum element of the filter of reduced models, and in certain logics it becomes the unit, i.e. the neutral element of &. Finally, the truth-constant 0¯ is introduced without any intended algebraic interpretation, just as a device used in the literature to define negation connectives. We start by exploring the logical and algebraic properties of these connectives. In the style of this book, we introduce them in terms of Hilbert-style rules which enforce the expected semantical behavior and study consequences of their presence in a logic. After that, we introduce and study two minimal logics containing certain collections of these connectives: Lambek logic LL (the least weakly implicative logic where all the residuated connectives have the minimal intended behavior) and the logic SL (the least expansion of LL where also the lattice connectives have the minimal intended behavior and furthermore the truth constant 1¯ behaves as the unit). We axiomatize them by means of strongly separable Hilbert-style calculi and describe their classes of reduced models and show that they admit regular completions. These two logics also serve as bases for our study of substructural logics. Indeed, we define them as the class of weakly implicative logics expanding the implicative fragment of LL and then we build a hierarchy of substructural logics expanding SL, which contains the most prominent members of this family. The rest of the chapter is devoted to the study of deduction theorems in substructural logics, capitalizing on the presence of implication and residuated conjunction. We introduce a general notion of implicational deduction theorem and provide a characterization in terms of the existence of a presentation that only has one binary rule (modus ponens) and unary rules of a certain simple form. As interesting byproducts, for any logic enjoying such deduction theorem, we obtain a description of filter generation (in its reduced models) and a construction technique for a new form of generalized disjunction connective (given by a set of formulas) with the proof by cases property.

1 Introduction

13

Chapter 5. The general kind of disjunctions enjoying the proof by cases property obtained for substructural logics in the previous chapter (given by a set of formulas instead of a single disjunction connective) motivates the abstract study of generalized disjunctions that we present in the fifth chapter. First, we introduce several forms of proof by cases property and corresponding generalized disjunctions, which we usually denote as 5, yielding a hierarchy of logics that we illustrate and separate with suitable examples. We continue by providing several (groups of) characterizations of generalized disjunctions: • The first characterization is based on the notion of 5-form of a rule, that is the rule obtained from the original rule by disjuncting its premises and conclusion with an arbitrary formula. This notion allows us to characterize logics with a strong p-disjunction 5 as those closed under the formation of 5-forms of its rules. This characterization is then used to study the preservation of the proof by cases property in expansions and to prove its transfer to the general matrix semantics in terms of generated filters. • The second group of characterizations is based on various generalized distributivity properties of the lattice of filters. • The third characterization is based on 5-prime filters, a generalization of the well-known notion of prime filter for classical and intuitionistic logic and its relation to the intersection-prime filters. Finally, we use generalized disjunctions in order to: (1) obtain some axiomatizations of interesting extensions of a given logic, (2) introduce a symmetric notion of consequence relation with a disjunctive reading on the right-hand side, (3) improve some of the characterizations of completeness properties seen in Chapter 3, and, finally, (4) study completeness with respect to finite matrices and matrices with 5-prime filters. Chapter 6. This chapter focuses on the other family that motivated the general study of logics with implication: semilinear logics, defined as logics complete with respect to linearly ordered reduced matrices. We start by formulating and proving useful characterizations of semilinear logics in terms of linear filters, a syntactical metarule akin to the proof by cases property, and the coincidence of finitely subdirectly irreducible and linearly ordered reduced matrices. We use these characterizations to show which of the examples of weakly implicative logics considered in the previous chapters are actually semilinear logics and prove which of them are not semilinear with respect to any possible implication. Then we study the problem of, given an arbitrary weakly implicative logic (in particular, given a substructural logic), finding its least semilinear extension. We also use the presence of disjunction and the results obtained in the previous chapter to prove better characterizations of semilinearity leading to axiomatizations of the least semilinear extension of a given logic. Finally, we focus on completeness with respect to the subclass of linear models in which the order is dense and discuss its relation with completeness with respect to reduced matrices defined over the rational and the real unit intervals.

14

1 Introduction

Chapter 7. The last chapter gives a short introduction to the study of first-order predicate logics built over weakly implicative logics. We follow a semantics-first approach in which we start from semantically defined predicate logics and then propose suitable Hilbert-style axiomatizations and prove corresponding completeness theorems by following non-trivial generalizations of Henkin’s proof of completeness of classical first-order logic. More precisely, to introduce a general semantics of predicate models we utilize the fact that any reduced model of a given weakly implicative logic is ordered and define the truth-value of a universal (resp. existentially) quantified formula as the infimum (resp. supremum) of the truth-values of its instances. Then, for any given class of reduced models of the logic in question, we define a corresponding consequence relation on predicate formulas. We focus on three particular meaningful logics: the predicate logic of all reduced models, the subdirectly irreducible ones, and the restriction of the latter to witnessed predicate models in which quantifiers are actually computed as minima and maxima. We propose axiomatic systems for these three logics and, under certain conditions, prove the corresponding completeness theorems. While the first completeness result is relatively straightforward and works in absolute generality, the other two require non-trivial modifications of Henkin’s proof by making use of a suitable disjunction connective which needs to be added as a requirement for the propositional logic. As a by-product, we also obtain, for certain logics, a form of Skolemization. The relatively modest assumptions on the propositional side allow for a wide generalization of previous approaches and help to illuminate the ‘essentially first-order’ steps in the classical Henkin’s proof.

Chapter 2

Weakly implicative logics

This chapter is the real beginning of our story and hence its main aim is to present the principal mathematical object of study in this book: the class of weakly implicative logics. And being this a mathematical logic story, it necessarily has to start by introducing the usual characters: a formal language, a syntactical notion of consequence (proof systems) and a semantical one (preservation of truth), and completeness theorems binding them together by showing that both kinds of consequence are actually the same. We start by introducing the most basic syntactical notions (variables, connectives, formulas), giving a purely syntactical definition of logic as a mathematical object, and studying the general properties of the class of all logics. We also introduce the crucial notion of theory (a set of formulas that contains all its logical consequences) and show how it connects logics with the important mathematical notion of closure system. After that, we study Hilbert-style proof systems, a very flexible and universal syntactical way of presenting logics and useful tool for the study of their metalogical properties. We use these systems to refine the description of the lattice of all logics. After testing the basic definitions on three extreme, mostly uninteresting, examples, we immediately introduce some of the most important logics that one can find in the literature and that will accompany the reader throughout the book: • classical logic CL, intuitionistic logic IL, Łukasiewicz logic Ł, and Gödel–Dummett logic G; • logics in languages with implication only: BCI, A→ , BCK, and FBCK, and the implicational fragments of the prominent logics mentioned before; • local and global variants of the basic modal logics K, T, K4, and S4; and • an infinitary extension of Łukasiewicz logic Ł∞ . We also use these examples to demonstrate different variants of basic metalogical properties such as the deduction theorem and the proof by cases. After that, we explain the basic semantical notions. In particular, we introduce a very general semantics based on the notion of logical matrix, a pair consisting of (a) an algebra whose domain provides a set of truth-values and whose operations are used to interpret the logical connectives and (b) the set of designated elements, called © Springer Nature Switzerland AG 2021 P. Cintula and C. Noguera, Logic and Implication, Trends in Logic 57, https://doi.org/10.1007/978-3-030-85675-5_2

15

16

2 Weakly implicative logics

the filter, used to define the notion of semantical consequence w.r.t. this matrix as preservation of truth. Each matrix that provides a sound semantics for a given logic is called a model of the logic. In the next section, we show that the class of all models of a given logic actually provides a complete semantics by proving our first completeness theorem. However, this first completeness theorem is still rather weak, because the class of matrices is too wide; in particular, the involved algebras may not be intended for the logic in question. On the other hand, we can use this semantics to show the impossibility of certain syntactical derivations and, in particular, differentiate all the given examples of logics. In order to obtain a more meaningful semantics, we introduce the notion of Leibniz congruence, which identifies elements of a matrix indistinguishable by logical means and allows us to prove completeness of any logic with respect to the class of its reduced models, i.e. models where equivalent elements have been already identified. The reduced models provide a much more meaningful semantics; indeed, we will see that the algebras of such models are intimately related to the logic in question. After this general theory, we finally introduce the class of weakly implicative logics as logics built over countable sets of formulas and possessing a binary connective which satisfies some minimal syntactical requirements one expects from an implication and, more importantly, allows for a simple definition of the Leibniz congruence. We show that almost all the logics in the families considered in this book (which cover the vast majority of non-classical propositional logics studied in the literature) are actually weakly implicative and, thus, the restriction to such class simplifies not only proofs but also the formulations of many definitions and theorems throughout the book, while sacrificing only very little of their scope of applicability. In the last section of this chapter, we introduce algebraically implicative logics in which one can disregard logical matrices and work just with algebras instead, thanks to an equational description of the set of designated elements.

2.1 Logics as mathematical objects The three most basic syntactical elements that we need to formally define in order to introduce logics as mathematical objects are variables, propositional languages, and the sets of formulas they determine: • Propositional variables are the members of an arbitrary infinite set Var. • A propositional language is a function L : 𝐶L → N, where 𝐶L is a set (disjoint with Var) of symbols called connectives, which gives for each connective its arity. • The set FmL of formulas in a language L with variables Var is the least set 𝐹 ⊇ Var such that, for each 𝑛-ary connective 𝑐 of L and 𝜑1 , . . . , 𝜑 𝑛 ∈ 𝐹, we have 𝑐(𝜑1 , . . . , 𝜑 𝑛 ) ∈ 𝐹. Unless said otherwise, we assume that from now on we work with a fixed arbitrary infinite set of variables Var, whose elements we usually denote by lower-case Latin letters 𝑝, 𝑞, 𝑟, . . .

2.1 Logics as mathematical objects

17

Formulas are typically denoted by lower-case Greek letters 𝜑, 𝜓, 𝜒, . . . and sets of formulas by upper-case ones Γ, Δ, Σ, . . . When writing particular formulas, we follow the usual convention and use infix instead of prefix notation for binary connectives, i.e. we write 𝜑 → 𝜓 instead of →(𝜑, 𝜓).1 Also sometimes in longer formulas we may use square brackets to visualize better their structure. Finally, for variables 𝑝 1 , . . . , 𝑝 𝑛 ∈ Var, we write 𝜒( 𝑝 1 , . . . , 𝑝 𝑛 ) to signify that the propositional variables occurring in the formula 𝜒 are among 𝑝 1 , . . . , 𝑝 𝑛 . Furthermore, by 𝜒(𝜑1 , . . . , 𝜑 𝑛 ) we denote the formula resulting from 𝜒 by simultaneous replacement of each variable 𝑝 𝑖 by the formula 𝜑𝑖 . Note that propositional languages are the same mathematical objects as algebraic languages (cf. Definition A.3.1). However, following a well-established tradition, in this logical context we speak about connectives instead of operations and call connectives of arity 0 truth-constants. Formulas in a language L coincide with L-terms (see Definition A.4.1), and thus FmL is the domain of the term L-algebra FmL . This identification of logical and algebraic language greatly simplifies many upcoming notions throughout the whole book. Therefore, from now on, we speak about languages only. Example 2.1.1 The language LCL of classical logic (shared by other important logics such as intuitionistic logic, Łukasiewicz logic, and Gödel–Dummett logic; see Example 2.3.7) consists of a truth-constant ⊥ (falsum) and three binary connectives: → (implication), ∧ (conjunction), and ∨ (disjunction). Besides these primitive connectives, we may also introduce three definable ones: binary ↔ (equivalence), unary ¬ (negation), and nullary > (verum), i.e. ‘𝜑 ↔ 𝜓’ is just an abbreviation for the more complex expression ‘(𝜑 → 𝜓) ∧ (𝜓 → 𝜑)’; ‘¬𝜑’ stands for ‘𝜑 → ⊥’, and ‘>’ stands for ‘⊥ → ⊥’ (see Proposition 2.3.5 for formal details).2 When writing formulas in this language, we follow the usual convention about omitting brackets where → and ↔ have the lowest and ¬ has the highest priority; e.g. we write ‘¬𝜑 ∧ 𝜓 → 𝜑’ instead of ‘((¬𝜑) ∧ 𝜓) → 𝜑’. Throughout the book we will encounter many other languages with more or fewer connectives than LCL . The simplest one, denoted as L→ , is the sublanguage of LCL containing only the binary connective →. Definition 2.1.2 (Substitution) A substitution in a language L is an endomorphism of the algebra FmL , i.e. a mapping 𝜎 : FmL → FmL , such that for each 𝑛-ary connective 𝑐 of L and all 𝜑1 , . . . , 𝜑 𝑛 ∈ FmL , 𝜎(𝑐(𝜑1 , . . . , 𝜑 𝑛 )) = 𝑐(𝜎(𝜑1 ), . . . , 𝜎(𝜑 𝑛 )). 1 Note that the expression ‘𝜑 → 𝜓’ is, technically speaking, not a formula, but the name of the formula resulting from combining the formulas denoted by ‘𝜑’ and ‘𝜓’ via the connective →. 2 Of course, in classical logic even the primitive connectives are definable using the remaining ones (see Definition 2.3.4), e.g. ‘𝜑 ∨ 𝜓’ could stand for ‘¬𝜑 → 𝜓’. However, this definability is not true in intuitionistic logic [222].

18

2 Weakly implicative logics

We often write simply ‘𝜎 𝜒’ instead of ‘𝜎( 𝜒)’. Note that, for a substitution 𝜎 and a formula 𝜒( 𝑝 1 , . . . , 𝑝 𝑛 ), we have 𝜎 𝜒 = 𝜒(𝜎 𝑝 1 , . . . , 𝜎 𝑝 𝑛 ). Since substitutions are mappings whose domain is an absolutely free L-algebra, they are fully determined by their values on the free generators (propositional variables), i.e. for each mapping 𝜎 : Var → FmL , there is a unique substitution 𝜎 such that 𝜎𝑣 = 𝜎𝑣 for each 𝑣 ∈ Var. Now we can present the central notion of this book (and of the discipline): the definition of logic as a mathematical object, a special kind of consequence relation (see the Appendix), i.e, a binary relation between sets of formulas and formulas. As customary, we write ‘Γ `L 𝜑’ to signify that the set of formulas Γ and the formula 𝜑 are in the relation L. Definition 2.1.3 (Logic) We say that a relation L between sets of formulas and formulas in a language L is a logic in L when it satisfies the following conditions for each Γ ∪ Δ ∪ {𝜑} ⊆ FmL : • • • •

{𝜑} `L 𝜑. If Γ `L 𝜑 and Γ ⊆ Δ, then Δ `L 𝜑. If Δ `L 𝜓 for each 𝜓 ∈ Γ and Γ `L 𝜑, then Δ `L 𝜑. If Γ `L 𝜑, then 𝜎[Γ] `L 𝜎(𝜑) for each substitution 𝜎.

(Reflexivity) (Monotonicity) (Cut) (Structurality)

A logic L is said to be finitary if furthermore • If Γ `L 𝜑, then there is a finite Γ0 ⊆ Γ such that Γ0 `L 𝜑.

(Finitarity)

As a matter of convention we say that L is infinitary if it is not finitary. Observe that the Reflexivity condition implies that any logic is non-empty and, thanks to the Cut condition, the Reflexivity and Monotonicity conditions could be equivalently replaced by the following variant of Reflexivity: • If 𝜑 ∈ Γ, then Γ `L 𝜑. Also note that, in finitary logics, the Cut condition could be equivalently replaced by a, in general weaker, variant: • If Γ `L 𝜑 and Γ ∪ {𝜑} `L 𝜒, then Γ `L 𝜒. When the logic and/or its language are known from the context, we omit the parameters L or L in all the notations indexed by them. Moreover, we use the following notational conventions which allow us e.g. to write the Cut condition as “Δ ` Γ and Γ ` 𝜑 implies Δ ` 𝜑” (note that this statement is valid even for Γ = ∅): Γ, Δ ` 𝜑 Γ, 𝜓 ` 𝜑 `𝜑 Γ`Δ Γ a` Δ

stands for stands for stands for stands for stands for

Γ∪Δ` 𝜑 Γ ∪ {𝜓} ` 𝜑 ∅`𝜑 Γ ` 𝜒 for each 𝜒 ∈ Δ Γ ` Δ and Δ ` Γ.

Logics, as all binary relations, can be identified with sets of pairs of corresponding objects. In our case, these pairs are heterogenous (they have a set of formulas in the first place and a formula in the second place) and they are interesting objects in their own right. Furthermore, it is very convenient for us to see logics as (special kinds) of sets of pairs. Therefore, we define:

2.1 Logics as mathematical objects

19

Definition 2.1.4 (Consecution) A consecution in a language L is a pair hΓ, 𝜑i, where Γ ∪ {𝜑} ⊆ FmL . The elements of Γ are called premises and 𝜑 is called the conclusion of consecution hΓ, 𝜑i. A consecution is finitary if it has finitely many premises. To simplify matters, we formally identify a formula 𝜑 with the consecution {∅, 𝜑} and instead of ‘hΓ, 𝜑i’ we write ‘Γ I 𝜑’. Let us stress the difference between the expression ‘Γ I 𝜑’ which denotes a consecution and ‘Γ `L 𝜑’ which (according to our conventions) is the statement ‘Γ I 𝜑 ∈ L’, which we will read as ‘the consecution Γ I 𝜑 is valid in the logic L’. As logics can be now seen as sets of consecutions, we can speak about their (set-theoretic) intersections, unions, inclusions, the least or the largest logic, etc. We say that a logic L2 is an extension of a logic L1 in the same language whenever L1 ⊆ L2 and we say that the extension is proper if furthermore L1 ≠ L2 . Clearly, the extension relation is an order on the set of all logics in a given language, actually, a complete lattice order (we leave the proof of this fact as an exercise for the reader). Proposition 2.1.5 The extension relation is a complete lattice order on the set LogL of all logics in a language L with the corresponding complete lattice LogL = hLogL , ∧, ∨, MinL , IncL i, where L∧M=L∩M Ù L∨M= {N ∈ LogL | N ⊇ L ∪ M} MinL = {Γ I 𝛿 | 𝛿 ∈ Γ} IncL = {Γ I 𝛿 | Γ ∪ {𝛿} ⊆ FmL }. The logics MinL and IncL , called the minimum and the inconsistent logic, are the first two particular, yet very trivial, examples of logics we meet. There is another similar logic called almost inconsistent logic AIncL defined as the set of all consecutions with a non-empty set of premises (we leave the checking of the defining conditions to the reader). Finally, let us note that these three logics are obviously finitary. The difference between the inconsistent and the almost inconsistent logic can be nicely illustrated by means of the syntactical notion of theorem, which captures the idea of an absolute truth valid in the logic without a need for any further premises. Definition 2.1.6 (Theorem) Let L be a logic. The formulas 𝜑 such that `L 𝜑 are called theorems of L. The set of all theorems of L is denoted as Thm(L). Clearly, the logics Min and AInc have no theorems. Actually, it is easy to see that AInc is the greatest element of the complete sublattice of LogL given by logics without theorems, while Min is the least element of this lattice. It is also interesting to observe that, thanks to the Monotonicity condition, Inc is the only logic where all formulas are theorems. Another crucial syntactical notion we will need throughout this book is that of a deductively closed set of formulas which is usually called simply a theory. Definition 2.1.7 (Theory) A theory of a logic L is a set of formulas 𝑇 ⊆ FmL such that: if 𝑇 `L 𝜑, then 𝜑 ∈ 𝑇. We denote by Th(L) the set of all theories of L.

20

2 Weakly implicative logics

Example 2.1.8 It is easy to check that Th(Min) = P (FmL ), Th(AInc) = {FmL , ∅}, and Th(Inc) = {FmL }. Theories are usually denoted by upper case Latin letters 𝑇, 𝑆, 𝑅, . . . It is obvious that the set of all formulas FmL is a theory of any logic L and so is the set Thm(L) of its theorems. Indeed, assume that Thm(L) ` 𝜑; since ∅ ` Thm(L), we can use Cut to obtain ∅ ` 𝜑, i.e. 𝜑 ∈ Thm(L). Let us now explore the structure of the set Th(L). We can show that it is a closure system (see the Appendix) and, hence, it can be seen as the domain of a complete lattice. Proposition 2.1.9 Let L be a logic in a language L. The set Th(L) is a closure system and the function ThL : P (FmL ) → P (FmL ) defined as ThL (Γ) = {𝜑 ∈ FmL | Γ `L 𝜑} is its associated closure operator assigning to each set of formulas its generated theory. Furthermore, ThL is structural, i.e. it satisfies the following additional property: • 𝜎[ThL (Γ)] ⊆ ThL (𝜎[Γ]), for each substitution 𝜎.

(Structurality)

Finally, the following are equivalent (cf. Definition A.1.17): • L is finitary. • ThL is finitary. • Th(L) is inductive. Proof The first claim is straightforward (we know that FmL ∈ Th(L) and the set of all theories is clearly closed under arbitrary intersections). To Ñ prove that ThL is the associated closure operator, we have to show that ThL (Γ) = {𝑇 ∈ Th(L) | Γ ⊆ 𝑇 }. For this, it is enough to prove that ThL (Γ) is the least theory containing Γ (exercise for the reader). The Structurality condition of ThL clearly follows from the corresponding property of L. Finally, the characterization of finitarity follows from Schmidt Theorem A.1.18. As a convenient notational simplification, given Γ ∪ Δ ∪ {𝛼} ⊆ FmL , we will write ‘ThL (Γ, Σ)’ instead of ‘ThL (Γ ∪ Σ)’, and ‘ThL (Γ, 𝛼)’ instead of ‘ThL (Γ ∪ {𝛼})’. Corollary 2.1.10 Let L be a logic. The set Th(L) is the domain of a complete lattice Th(L) = hTh(L), ∧, ∨, Thm(L), FmL i, where 𝑇 ∧𝑆=𝑇 ∩𝑆

𝑇 ∨ 𝑆 = ThL (𝑇, 𝑆).

We have seen that the set of all logics in a given language is the domain of a complete lattice LogL . It is easy to see that, the set of all closure operators over FmL can also be given a complete lattice order defined as 𝐶 ≤ 𝐷 iff 𝐶 (Γ) ⊆ 𝐷 (Γ) for each Γ ⊆ FmL .

2.2 Hilbert-style proof systems

21

Clearly, structural closure operators form a complete sublattice, denoted by SCOL . The next theorem shows that these two lattices are isomorphic via the mappings Th ( ·) : LogL → SCOL

` ( ·) : SCOL → LogL ,

where for each L ∈ LogL and 𝐶 ∈ SCOL , the structural closure operator ThL is defined as above and the logic `𝐶 is defined as {Γ I 𝜑 | 𝜑 ∈ 𝐶 (Γ)}. Theorem 2.1.11 The mappings Th ( ·) and ` ( ·) are mutually inverse isomorphisms between the complete lattices of (finitary) logics in a given language L and that of structural (finitary) closure operators over FmL . Proof In the previous proposition, we have seen that ThL is a structural closure operator (and it is finitary whenever L is finitary). Clearly, if L ⊆ L0, we obtain ThL ≤ ThL0 . Next, we prove that `𝐶 is a logic. Reflexivity is obvious. To establish Cut assume that Δ `𝐶 Γ and Γ `𝐶 𝜑, i.e. Γ ⊆ 𝐶 (Δ) and 𝜑 ⊆ 𝐶 (Γ). Applying Monotonicity and Idempotency of 𝐶, we obtain 𝜑 ∈ 𝐶 (Γ) ⊆ 𝐶 (𝐶 (Δ)) = 𝐶 (Δ), i.e, Δ `𝐶 𝜑. Finally, we show that Structurality is satisfied (finitarity is analogous): if Γ `𝐶 𝜑, 𝜑 ∈ 𝐶 (Γ) and so 𝜎(𝜑) ∈ 𝜎[𝐶 (Γ)] ⊆ 𝐶 (𝜎[Γ]), i.e. 𝜎[Γ] `𝐶 𝜎(𝜑). From the definition it is also clear that, if 𝐶 ≤ 𝐶 0, we obtain `𝐶 ⊆ `𝐶 0 . To complete the proof, we only need to show that ` ( ·) and Th ( ·) are mutually inverse which easily follows if we observe that: • Γ `L 𝜑 iff 𝜑 ∈ ThL (Γ) iff Γ `ThL 𝜑. • 𝜑 ∈ 𝐶 (Γ) iff Γ `𝐶 𝜑 iff 𝜑 ∈ Th`𝐶 (Γ).



Note that the mappings Th ( ·) and ` ( ·) actually give a Galois connection (see the Appendix), i.e. ThL ≤ 𝐶 iff L ≤ `𝐶 for each logic L and each closure operator 𝐶.

2.2 Hilbert-style proof systems Axiomatic Hilbert-style proof systems are an essential device in mathematical logic used to define particular logics by purely syntactical means. We introduce them as the same kind of objects as logics, i.e. as sets of consecutions. In this way, we obtain two equivalent syntactical methods to describe a logic from an axiomatic system (as we will see in Lemma 2.2.7): an external one as the least logic3 containing it or an internal one using the notion of (formal) proof. Definition 2.2.1 (Axiomatic system) An axiomatic system AS in a language L is a set of consecutions closed under arbitrary substitutions. The elements of AS of the form Γ I 𝜑 are called axioms if Γ = ∅, finitary inference rules if Γ is a finite set, and infinitary inference rules otherwise. An axiomatic system is said to be finitary if all its inference rules are finitary, and it is called infinitary otherwise. 3 When we use order-related superlative or comparative adjectives such as ‘least’ or ‘larger’ when speaking about logics we always refer to the order given by inclusion.

22

2 Weakly implicative logics

Axioms play the role of formulas that do not require any justification and can be used at any step of a formal proof (note that we can see them as formulas thanks to our convention of identifying the consecution ∅ I 𝜑 with the formula 𝜑). Inference rules are the syntactical devices used for the construction of formal proofs by deriving a formula from a set of previously proved formulas. These intuitions will be made precise in the next definition. Let us first give an example of an axiomatic system. Due to our requirement that axiomatic systems have to be closed under substitutions (which simplifies our definition of formal proof and formulations and proofs of numerous upcoming statements), we usually present them using axiom/rule schemata. For example, we say that an axiomatic system AS contains the rule 𝜑, 𝜑 → 𝜓 I 𝜓, instead of formally saying either that, (1) for each 𝜑, 𝜓 ∈ FmL , the consecution 𝜑, 𝜑 → 𝜓 I 𝜓 is an inference rule of AS, or (2) that all substitutional instances of 𝑝, 𝑝 → 𝑞 I 𝑞 are inference rules of AS. Before we present the first of many axiomatic systems we will meet in this book, let us establish one important convention. For ease of reference, each axiom/rule is presented with a name (usually that traditionally used in the literature) and a label we use to refer to it in this book (mainly in formal proofs, see below). The labels are usually derived from the names of the axioms/rules. They always occur between round brackets. Moreover, those for axioms are always in lower case letters (even if the name in question contains a proper noun); labels for rules start with a capital letter (and may contain more capital letters if the name consists of more than one word). Later, see e.g. Example 2.2.4, we will meet pairs of rule–axiom that are known under the same name (usually, when the validity of the axiom entails the validity of the rule). In important occurrences we make the distinction explicit; however, in most cases we rely on this convention to disambiguate what is meant. Example 2.2.2 A particular finitary axiomatic system in the language L→ (which consists of a single binary connective →), called BCK,4 consists of the following four consecutions (three axioms and one inference rule):5 (sf) (e) (w) (MP)

(𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)) (𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)) 𝜑 → (𝜓 → 𝜑) 𝜑, 𝜑 → 𝜓 I 𝜓

suffixing exchange weakening modus ponens.

Now we are ready to introduce the notion of formal proof in a given axiomatic system. Since in this book we also deal with infinitary logics and their axiomatic systems, we need a definition of proof that can handle infinitary inference rules 4 The name of this axiomatic system and its corresponding logic BCK (see Lemma 2.2.7) comes from the original historical names of the axioms proposed by Meredith (see e.g. [264]) to formalize the properties of Schönfinkel’s combinators [281]. 5 Recall that while, strictly speaking, axioms are consecutions with empty sets of premises, by our previous convention we identify them with formulas and thus omit the symbol I.

2.2 Hilbert-style proof systems

23

too. Therefore, we introduce the notion of tree-proof, which represents formal proofs using potentially infinite trees (there are other options existing in the literature, e.g. ordinal sequences). However, for didactic reasons, first we restrict to finitary axiomatic systems in which we can use the familiar notion of finitary proof, which represents proofs as finite sequences. Later, in Proposition 2.2.6, we show that in finitary axiomatic systems there is a one-one correspondence between these two notions of proof. Furthermore, for infinitary axiomatic systems, we only define one notion of proof (tree-proof), and hence we can safely omit the adjective finitary or tree- and use the same symbol to denote the existence of a formal proof. Definition 2.2.3 (Finitary proof) Let AS be a finitary axiomatic system in a language L. A finitary proof of a formula 𝜑 from a set of formulas Γ in AS is a finite sequence of formulas ending with 𝜑 such that, for each element 𝜓 in the sequence, • 𝜓 is an axiom of AS, or • 𝜓 is an element of Γ, or • there is a set Δ ≠ ∅ of formulas appearing earlier in the sequence such that Δ I 𝜓 ∈ AS. We write ‘Γ `A S 𝜑’ if there is a proof of 𝜑 from Γ in AS; in this case, we also say that Γ I 𝜑 is a derivable consecution in AS (if Γ ≠ ∅, we speak about derivable rules and, if Γ = ∅, we say that 𝜑 is a derivable/provable formula in AS). Analogously to what we have said about the elements of axiomatic systems, formulas in a proof can also be seen as schemata. Indeed, if we take an arbitrary substitution 𝜎 and replace each element 𝜓 in a proof of 𝜑 from Γ by the formula 𝜎𝜓, we obtain a proof of 𝜎𝜑 from 𝜎[Γ]. Again, it can be seen in two (subtly) different ways: (1) as a proof of the schema using the axiom schemata as such, or (2) as a proof of the particular formula using suitable instances of the axiom schemata. Let us illustrate it with an example. Example 2.2.4 Recall the axiomatic system BCK that we have presented in Example 2.2.2. Let 𝜑 be an arbitrary formula. We show that the formula 𝜑 → 𝜑, known as identity and labeled as (id) in this book, is derivable in BCK by writing a formal proof: a) 𝜑 → ((𝜑 → (𝜓 → 𝜑)) → 𝜑)

(w)

b) [𝜑 → ((𝜑 → (𝜓 → 𝜑)) → 𝜑)] → [(𝜑 → (𝜓 → 𝜑)) → (𝜑 → 𝜑)] c) (𝜑 → (𝜓 → 𝜑)) → (𝜑 → 𝜑) d) 𝜑 → (𝜓 → 𝜑)

(w)

e) 𝜑 → 𝜑

c, d, and (MP)

Another example of a derivable formula in BCK is (pf)

(𝜑 → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓))

(e)

a, b, and (MP)

prefixing.

24

2 Weakly implicative logics

Its formal proof in BCK is simple: a) ( 𝜒 → 𝜑) → ((𝜑 → 𝜓) → ( 𝜒 → 𝜓))

(sf)

b) [( 𝜒 → 𝜑) → ((𝜑 → 𝜓) → ( 𝜒 → 𝜓))] → [(𝜑 → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓))] c) (𝜑 → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓))

(e)

a, b, and (MP)

Later we will see other axiomatic systems in which suffixing and prefixing are not axioms or even derivable formulas, but in which still one can obtain corresponding (weaker) derivable rules. Let us mention three such rules: (Sf) (Pf) (T)

𝜑 → 𝜓 I (𝜓 → 𝜒) → (𝜑 → 𝜒) 𝜑 → 𝜓 I ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓) 𝜑 → 𝜓, 𝜓 → 𝜒 I 𝜑 → 𝜒

suffixing prefixing transitivity.

The first two rules are obtained from the suffixing/prefixing axioms using modus ponens (indeed, their formal proofs would consist of writing the premise, the corresponding axiom, and the conclusion obtained by modus ponens). In a slight abuse of language, we use the same name for the axiom and for its corresponding rule, however, following our convention, we use different labels to distinguish them. The third rule is derivable from any of the two rules by using modus ponens again. Later in the book we also use the rules corresponding to axioms of exchange and weakening in formal proofs: (E) (W)

𝜑 → (𝜓 → 𝜒) I 𝜓 → (𝜑 → 𝜒) 𝜑I𝜓→𝜑

exchange weakening.

Next, we introduce a notion of proof for all axiomatic systems and show that in the finitary case both notions coincide (for an example of an essentially non-finitary proof see Example 5.2.7). Definition 2.2.5 (Tree-proof) Let AS be an axiomatic system in a language L. A tree-proof of a formula 𝜑 from a set of formulas Γ in AS is a tree with no infinite branches and with all nodes labeled by formulas such that, for each node 𝑛 and its label 𝜓, we have • If 𝑛 is the root, then 𝜓 = 𝜑. • If 𝑛 is a leaf, then 𝜓 is an axiom of AS or an element of Γ. • If 𝑛 is not a leaf, then there is a rule Δ I 𝜓 in AS such that Δ is the set of labels of predecessors of 𝑛. We write ‘Γ `A S 𝜑’ and say that it is a derivable consecution in AS if there is a tree-proof of 𝜑 from Γ in AS. Notice that an inference rule {𝜓1 , 𝜓2 , . . . } I 𝜑 gives a way to construct a treeproof of a formula 𝜑 from a set of formulas Γ if we know the tree-proofs of formulas 𝜓1 , 𝜓2 , . . . from Γ; indeed, we just glue them together in a single tree using the rule

2.2 Hilbert-style proof systems

25

{𝜓1 , 𝜓2 , . . . } I 𝜑. In contrast, a metarule ‘from Γ1 ` 𝜓1 , Γ2 ` 𝜓2 , . . . , obtain Σ ` 𝜑’ only tells us that if there are proofs of formulas 𝜓1 , 𝜓2 , . . . from sets of formulas Γ1 , Γ2 , . . . , then there is a proof of a formula 𝜑 from a set of formulas Σ as well, though it gives no hint to its construction. We could say that rules are inferences between formulas, whereas metarules (also called Gentzen-style rules) are, in fact, inferences between consecutions carried out at the metalevel. We will see examples of important metarules later in this text; the first one will be the classical deduction theorem (Proposition 2.4.1). Proposition 2.2.6 Let AS be a finitary axiomatic system in a language L and Γ ∪ {𝜑} a set of formulas. Then, there is a finitary proof of 𝜑 from Γ in AS iff there is a tree-proof of 𝜑 from Γ. Proof From right to left: first observe that tree-proofs in a finitary axiomatic system are always finite. Indeed, they have no infinite branches and, due to finitarity, they are finitely branching and so, by König’s Lemma (see A.1.8) they have to be finite. Consider any tree-proof of 𝜑 from Γ in AS and list the labels of its nodes in a suitable way (starting by leaves in an arbitrary order, then their successors again in arbitrary order, and so on, ending by the root); this gives us the desired finitary proof. To justify the converse direction, one can simply show by induction on the length of the proof that, for each 𝜒 occurring in the finitary proof of 𝜑 from Γ, there is a tree-proof of 𝜒 from Γ, thus in particular there is a tree-proof of 𝜑 (in the induction step we use the gluing construction described above; we leave the details as an exercise for the reader).  Each tree-proof can be seen as a well-founded relation between its nodes with leaves as minimal elements and the root as a maximum (because there are no infinite branches any non-empty set of nodes has a minimal element). Thus, in both finitary and infinitary cases, we can prove facts about formulas by induction on the complexity of their formal proofs and prove a claim about all nodes of a tree (thus in particular about the root), by (1) proving the fact for all leaves and (2) showing that if the fact holds for all predecessors of a node, it holds for the node as well. Note that if in the last case in the definition of tree-proof we dropped the assumption that 𝑛 is not a leaf, it would subsume the case that 𝑛 is a leaf labeled by an axiom; we will sometimes use this observation to simplify the induction proofs. Let us now illustrate both observations in the proof of a simple, yet crucial lemma, proving the promised result that the least logic containing an axiomatic system can be described using the notion of formal proof. Lemma 2.2.7 Let AS be an axiomatic system in a language L. Then, `A S is the least logic in L containing AS. Proof The reader can easily check that `A S is a logic and AS ⊆ `A S . We prove that, for each logic L, if AS ⊆ L, then `A S ⊆ L. Assume that Γ `A S 𝜑, i.e. there is a proof of 𝜑 from Γ. By induction on the complexity of the proof, we show that for each node 𝑛 we have Γ `L 𝜓, where 𝜓 is the label of 𝑛; and hence, in particular, we have Γ `L 𝜑 (as 𝜑 is the label of the root of the proof). Indeed, if 𝜓 ∈ Γ the claim is trivial,

26

2 Weakly implicative logics

otherwise we have Δ I 𝜓 ∈ AS, where Δ is the set of labels of predecessors of 𝑛. From the induction assumption we have Γ `L Δ and, from the assumption AS ⊆ L, we have Δ `L 𝜓; Cut completes the proof.  Now that we know that each axiomatic system gives rise to a logic (described both internally and externally), a natural related question is whether for a given logic L we can find an axiomatic system whose logic coincides with L. Formally we define: Definition 2.2.8 (Presentation) Let AS be an axiomatic system in a language L, and L a logic in L. We say that AS is an axiomatic system for (or a presentation of) L if L = `A S .

Example 2.2.9 Note that the empty set is a presentation of the minimum logic, the set {I 𝜑 | 𝜑 ∈ FmL } of the inconsistent logic, and {𝜑 I 𝜓 | 𝜑, 𝜓 ∈ FmL } of the almost inconsistent one. As a first application of Lemma 2.2.7 we obtain that any logic understood as an axiomatic system is its own presentation.6 Furthermore, we also see that if the logic is finitary, then the set of all its finitary consecutions is actually one of its finitary presentations. In fact, thanks to Proposition 2.2.6, we know even more. Proposition 2.2.10 Each logic has a presentation. Furthermore, a logic is finitary iff it has a finitary presentation. Let us present two obvious, yet very useful corollaries of Lemma 2.2.7. The first one tells us how to recognize when a logic extends another one and, as two logics are equal iff they extend each other, it also gives us a nice method to establish the equality of two logics. The second corollary tells us that, in formal proofs in a given axiomatic system, we can also use consecutions that have been previously derived in the system (see e.g. Example 2.3.1). Corollary 2.2.11 Let L1 be a logic and AS one of its presentations. Then, a logic L2 extends L1 iff AS ⊆ L2 . Corollary 2.2.12 Let AS be an axiomatic system and C a set of derivable consecutions in AS. Then, AS ∪ C and AS are presentations of the same logic. Let us conclude this section by exploring the axiomatizations of finitary logics and their position in the lattice of all logics a bit more. First, we formulate yet another corollary of Lemma 2.2.7 which tells us how to axiomatize the join of an arbitrary set of logics (its proof is straightforward and left as an exercise for the reader). Corollary 2.2.13 Let C beÐa set of logics in a language L and forÔeach L ∈ C take a presentation ASL . Then, {ASL | L ∈ C} is a presentation of C. 6 Using this observation the following three claims are equivalent: Γ I 𝜑 is derivable in a presentation A S of L iff it is valid in L iff it is derivable in L. We use these three statements synonymously.

2.3 Some prominent non-classical logics

27

Thus, in particular, the join of an arbitrary set of finitary logics is finitary. There is no general analog of the above corollary for meets (we will see a partial solution in Chapter 5), but we can easily prove that the meet (i.e. the intersection) of two finitary logics is finitary. Indeed, if Γ `L1 ∩L2 𝜑, then there are finite sets Γ𝑖 ⊆ Γ such that Γ𝑖 `L𝑖 𝜑, for 𝑖 = 1, 2, and so Γ1 ∪ Γ2 `L1 ∩L2 𝜑. Note, however, that this reasoning would not work for the intersection of infinitely many finitary logics (as we will see later in Example 2.6.7). Now it is clear that the set FLog of finitary logics is the domain of a sublattice of Log and, while it need not be a complete sublattice, it is a complete ∨-sub-semilattice. Therefore, for each logic L, there is the greatest finitary logic contained in L, which we naturally call the finitary companion of L and denote by F C(L). Note that F C(L) = L iff L is finitary. In the next proposition, we give an axiomatic system and a more explicit description of this logic. Proposition 2.2.14 Let L be a logic. Then, the set of all finitary consecutions valid in L is one of the finitary presentations of F C(L) and Γ ` F C (L) 𝜑 iff there is a finite subset Γ0 ⊆ Γ such that Γ0 `L 𝜑. Proof Let us denote the proposed axiomatic system as AS and observe that we obviously have AS ⊆ L and that `A S is finitary. Therefore, we obtain `A S ⊆ F C(L) and the right-to-left direction of the second claim. Conversely, from Γ ` F C (L) 𝜑 and finitarity of F C(L), we obtain a finite subset Γ0 ⊆ Γ for which Γ0 ` F C (L) 𝜑 and so Γ0 `L 𝜑 (as F C(L) ⊆ L) and Γ `A S 𝜑. 

2.3 Some prominent non-classical logics By Lemma 2.2.7, we know that the axiomatic system BCK (Example 2.2.2) actually defines a finitary logic; this logic is known as BCK. The goal of this section is to present additional examples of prominent non-classical logics that we will use later to illustrate the upcoming notions. First, we introduce logics in the purely implicational language L→ , then logics in the classical language LCL , and finally we introduce logics in the language L extending LCL with an additional unary connective . In order to understand the relationship between logics in different languages, we introduce the relation of (conservative) expansion (cf. the relation of extension between logics of the same language) and make precise the notion of defined connective. Let us start by considering the following formulas (again we list them together with their traditional names): (id) (c) (waj) (lin) (abe)

𝜑→𝜑 (𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓) ((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑) ((𝜑 → 𝜓) → 𝜒) → (((𝜓 → 𝜑) → 𝜒) → 𝜒) ((𝜑 → 𝜓) → 𝜓) → 𝜑

identity contraction Wajsberg axiom linearity Abelian axiom.

28

2 Weakly implicative logics

Fig. 2.1 Prominent axiomatic extensions of the logic BCI.

The first logic we introduce is BCI, whose presentation is BCK with the axiom (w) replaced by the axiom (id) (from Example 2.2.4 we know that the formula (id) is derivable in BCK, thus, thanks to Corollary 2.2.11, BCK extends BCI). Next, we define the logic A→ as the extension of BCI axiomatized by adding the axiom (abe) to its presentation. Finally, we define a few additional extensions of BCK: Logic FBCK Ł→ IL→ G→ CL→

Presentation BCK extended by BCK extended by BCK extended by BCK extended by BCK extended by

(lin) (lin) and (waj) (c) (c) and (lin) (c), (lin), and (waj)

The mutual extension relationship of these logics is depicted in Figure 2.1. In Example 2.6.4 we show that they are pairwise different and that CL→ is not an extension of A→ ; therefore A→ is an example of the so-called contraclassical logics. In Corollary 3.5.20, the name ‘CL→ ’ is justified by proving that this logic is indeed the implicational fragment (see Definition 2.3.3) of classical logic; see Example 2.3.7 for an explanation of the naming of the remaining logics we have just introduced. Let us show the most common presentation of CL→ and demonstrate the application of, firstly, Corollary 2.2.11 for proving equality of two logics and, secondly, Corollary 2.2.12 to simplify formal proofs (cf. e.g. our use of the derived rules (T), (Sf), and (Pf) in the second formal proof).

2.3 Some prominent non-classical logics

29

Example 2.3.1 The following axiomatic system is a presentation of CL→ : (sf)

(𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))

suffixing

(e)

(𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))

exchange

(w)

𝜑 → (𝜓 → 𝜑)

weakening

(c)

(𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)

contraction

(p)

((𝜑 → 𝜓) → 𝜑) → 𝜑

Peirce’s law

(MP)

𝜑, 𝜑 → 𝜓 I 𝜓

modus ponens.

To prove this claim, we first show that Peirce’s law is derivable in CL→ : a) (𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)

(c)

b) [(𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)] → [((𝜑 → 𝜓) → 𝜑) → 𝜑] c) ((𝜑 → 𝜓) → 𝜑) → 𝜑

(waj) a, b, and (MP)

Next, we show that the linearity and Wajsberg axioms are derivable in the logic given by the proposed axiomatic system (note that it includes BCK so we can use the inference rules (T), (Pf), and (Sf); see Example 2.2.2 and Corollary 2.2.12). Let us start with linearity: a) (𝜓 → 𝜒) → [( 𝜒 → 𝜑) → (𝜓 → 𝜑)]

(sf)

b) [( 𝜒 → 𝜑) → (𝜓 → 𝜑)] → [((𝜓 → 𝜑) → 𝜒) → (( 𝜒 → 𝜑) → 𝜒)] c) (𝜓 → 𝜒) → [((𝜓 → 𝜑) → 𝜒) → (( 𝜒 → 𝜑) → 𝜒)] d) (( 𝜒 → 𝜑) → 𝜒) → 𝜒 e) [((𝜓 → 𝜑) → 𝜒) → (( 𝜒 → 𝜑) → 𝜒)] → [((𝜓 → 𝜑) → 𝜒) → 𝜒] f) (𝜓 → 𝜒) → [((𝜓 → 𝜑) → 𝜒) → 𝜒] g) 𝜓 → (𝜑 → 𝜓) h) ((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒) i) ((𝜑 → 𝜓) → 𝜒) → [((𝜓 → 𝜑) → 𝜒) → 𝜒]

(sf)

a, b, and (T) (p) d and (Pf) c, e, and (T) (w) g and (Sf) f, h, and (T)

Finally, we deal with the Wajsberg axiom (note that now we can also use the formulas (pf) and (lin) as axioms in our proof): a) (𝜓 → 𝜑) → (𝜓 → 𝜑) b) 𝜓 → ((𝜓 → 𝜑) → 𝜑) c) [((𝜑 → 𝜓) → 𝜓) → 𝜓] → (waj)

(id) a and (E) b and (Pf)

d) (𝜑 → 𝜓) → [((𝜑 → 𝜓) → 𝜓) → 𝜓]

(id) and (E)

e) (𝜑 → 𝜓) → (waj)

c, d, and (T)

f) ((𝜑 → 𝜓) → 𝜑) → 𝜑

(p)

g) 𝜑 → ((𝜓 → 𝜑) → 𝜑)

(w)

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2 Weakly implicative logics

h) ((𝜑 → 𝜓) → 𝜑) → ((𝜓 → 𝜑) → 𝜑)

f, g, and (T)

i) [((𝜑 → 𝜓) → 𝜓) → ((𝜑 → 𝜓) → 𝜑)] → (waj)

h and (Pf)

j) (𝜓 → 𝜑) → [((𝜑 → 𝜓) → 𝜓) → ((𝜑 → 𝜓) → 𝜑)]

(pf)

k) (𝜓 → 𝜑) → (waj) l) (waj)

i, j, and (T) e, k, (lin), and (MP) twice

Let us note that, strictly speaking, the previous sequence of formulas is not a proof but a ‘set of instructions’ to produce one. Indeed, before the formula d we should add first this instance of the axiom (id): (𝜑 → 𝜓) → 𝜓) → ((𝜑 → 𝜓) → 𝜓), and before the formula l we should write this instance of the axiom (lin): ((𝜑 → 𝜓) → (waj)) → [((𝜓 → 𝜑) → (waj)) → (waj)] and its consequence by e and (MP): ((𝜓 → 𝜑) → (waj)) → (waj). We use such simplifications from now on without mention. Observe the peculiar nature of the axiom (abe); unlike all the other formulas we have met so far, it is not valid in classical logic. After we establish the completeness theorem, we could easily demonstrate it by taking the evaluation that sends 𝜑 to 0 and 𝜓 to 1; but already now we can prove that adding it to a presentation of classical logic results in a presentation of the inconsistent logic. Actually, the next example shows even more. Example 2.3.2 The axiomatic system AS consisting of consecutions (id), (abe), (MP), and (W) is a presentation of the inconsistent logic Inc. Indeed, the claim is demonstrated by the following simple formal proof of any formula 𝜑: a) 𝜑 → 𝜑 b) (𝜑 → (𝜑 → 𝜑)) → (𝜑 → 𝜑) c) ((𝜑 → (𝜑 → 𝜑)) → (𝜑 → 𝜑)) → 𝜑 d) 𝜑

(id) a and W (abe) b, c, and (MP)

Before we introduce particular examples of logics in richer languages, let us prepare a few useful relevant notions and formalize the notion of a definable connective. Definition 2.3.3 (Expansion, extension, fragment) Let L1 ⊆ L2 be two languages, L𝑖 a logic in L𝑖 , and S a set of consecutions in L2 . We say that the logic L2 is • the expansion of L1 by S if it is the weakest logic in L2 containing L1 and S, i.e. it is axiomatized by all substitutional instances in L2 of consecutions from S ∪ AS, for any presentation AS of L1 .

2.3 Some prominent non-classical logics

31

• an expansion of L1 if L1 ⊆ L2 , i.e. it is the expansion of L1 by a set of consecutions. • an axiomatic expansion of L1 if it is an expansion by a set of formulas (i.e. consecutions of the form ∅ I 𝜑). • a conservative expansion of L1 or, equivalently, L1 is the L1 -fragment of L2 , if L2 is an expansion of L1 and for each consecution Γ I 𝜑 in L1 we have that Γ `L2 𝜑 entails Γ `L1 𝜑. Note that if the languages L1 and L2 are equal, then the notion of expansion coincides with the notion of extension introduced in Section 2.1; therefore we also speak about (axiomatic) extensions of a given logic by a set of consecutions (axioms). Note that no logic has a proper conservative extension. Oftentimes, we describe a fragment by pointing at the connectives that are disregarded, that is, for any L0 ⊆ L, we refer to the (L \ L0 )-fragment as the L0 -free fragment. Note that any sublanguage of a given language is fully determined by its connectives and so, if the original language is clear from the context, we can simply speak about e.g. {∧, ∨}-fragments or {∧, ∨}-free fragments. Recall that in Example 2.1.1 we introduced definable connectives without actually expanding the language; e.g. ↔ is not a real connective and the expression ‘𝜑 ↔ 𝜓’ is simply an abbreviation for ‘(𝜑 → 𝜓) ∧ (𝜓 → 𝜑)’. Let us now explore how this approach is related to: (a) considering the logic as actually formulated in an expanded language and (b) the notion of definability, in a given logic, of a connective already present in a language using the remaining ones. Let us start with some useful notation: • Given a language L and an 𝑛-ary connective 𝑑 not in L, let L 𝑑 be the expansion of L with 𝑑. • If, furthermore, 𝛿 is an L-formula with 𝑛 variables, we inductively define a mapping 𝑑 𝛿 : FmL 𝑑 → FmL as 𝑑 𝛿 (𝑣) = 𝑣 𝑑 𝛿 (𝑐(𝜑1 , . . . , 𝜑 𝑘 )) = 𝑐(𝑑 𝛿 (𝜑1 ), . . . , 𝑑 𝛿 (𝜑 𝑘 )) 𝑑 𝛿 (𝑑 (𝜑1 , . . . , 𝜑 𝑛 )) = 𝛿(𝑑 𝛿 (𝜑1 ), . . . , 𝑑 𝛿 (𝜑 𝑛 )).

for each 𝑣 ∈ Var for each 𝑘-ary 𝑐 ∈ 𝐶 L

The function 𝑑 𝛿 is the formalization of the intuitive notion of replacing a definable connective 𝑑 by its definition (note that 𝑑 𝛿 ( 𝜒) = 𝜒 for each 𝜒 ∈ FmL ). This leads to a notion of definability of a connective in a given logic by saying that it does not matter (for the logic in question) if we formulate a claim using the connective or its definition. Definition 2.3.4 Let L be a language, 𝑑 an 𝑛-ary connective not in L, and L a logic in L 𝑑 . We say that 𝑑 is definable in L if there is an L-formula 𝛿 of 𝑛-variables such that for each Γ ∪ {𝜑} ⊆ FmL 𝑑 , Γ `L 𝜑

iff

𝑑 𝛿 [Γ] `L 𝑑 𝛿 (𝜑).

Later we will show an easy way to check the definability of a connective in weakly implicative logics.

32

2 Weakly implicative logics

The next proposition ties both notions of definability together by showing that we can see each logic with a defined connective as its unique conservative expansion in the bigger language where the new connective is definable using its original definition. Proposition 2.3.5 Let L be a logic in L, 𝛿 an L-formula with 𝑛 variables, and 𝑑 an 𝑛-ary connective not in L. Then, the set L𝑑/ 𝛿 of consecutions in L 𝑑 defined as Γ `L𝑑/ 𝛿 𝜑

𝑑 𝛿 [Γ] `L 𝑑 𝛿 (𝜑)

iff

is the only conservative expansion of L in L 𝑑 where 𝑑 is definable by 𝛿. Proof First, we need to show that L𝑑/ 𝛿 is indeed a logic. Reflexivity and Cut are straightforward; we prove Structurality: consider an L 𝑑 -substitution 𝜎 and define an L-substitution 𝜎 0 as 𝜎 0 (𝑣) = 𝑑 𝛿 (𝜎𝑣). We show that for each 𝜒 ∈ FmL 𝑑 we have 𝜎 0 (𝑑 𝛿 ( 𝜒)) = 𝑑 𝛿 (𝜎 𝜒) by induction on the complexity of 𝜒. If 𝜒 is a variable, the claim is simple; if 𝜒 = 𝑐(𝜑1 , . . . , 𝜑 𝑘 ) for some 𝑘-ary connective 𝑐 of L, then 𝜎 0 (𝑑 𝛿 ( 𝜒)) = 𝜎 0 𝑐(𝑑 𝛿 (𝜑1 ), . . . , 𝑑 𝛿 (𝜑 𝑘 )) = 𝑐(𝜎 0 𝑑 𝛿 (𝜑1 ), . . . , 𝜎 0 𝑑 𝛿 (𝜑 𝑘 )) = 𝑐(𝑑 𝛿 (𝜎𝜑1 ), . . . , 𝑑 𝛿 (𝜎𝜑 𝑘 )) = 𝑑 𝛿 (𝑐(𝜎𝜑1 , . . . , 𝜎𝜑 𝑘 )) = 𝑑 𝛿 (𝜎 𝜒), and, finally, if 𝜒 = 𝑑 (𝜑1 , . . . , 𝜑 𝑛 ), then 𝜎 0 (𝑑 𝛿 ( 𝜒)) = 𝜎 0 𝛿(𝑑 𝛿 (𝜑1 ), . . . , 𝑑 𝛿 (𝜑 𝑛 )) = 𝛿(𝜎 0 𝑑 𝛿 (𝜑1 ), . . . , 𝜎 0 𝑑 𝛿 (𝜑 𝑛 )) = 𝛿(𝑑 𝛿 (𝜎𝜑1 ), . . . , 𝑑 𝛿 (𝜎𝜑 𝑛 )) = 𝑑 𝛿 (𝑑 (𝜎𝜑1 , . . . , 𝜎𝜑 𝑛 )) = 𝑑 𝛿 (𝜎 𝜒). Therefore, if Γ `L𝑑/ 𝛿 𝜑, then 𝑑 𝛿 [Γ] `L 𝑑 𝛿 (𝜑) and so 𝜎 0 [𝑑 𝛿 [Γ]] `L 𝜎 0 𝑑 𝛿 (𝜑) by Structurality of L. Thus, 𝑑 𝛿 [𝜎[Γ]] `L 𝑑 𝛿 (𝜎𝜑) and hence 𝜎[Γ] `L𝑑/ 𝛿 𝜎𝜑. The facts that L𝑑/ 𝛿 is a conservative expansion of L and 𝑑 is definable in L𝑑/ 𝛿 are easy to verify. Indeed, for any Γ ∪ {𝜑} ⊆ FmL and Δ ∪ {𝜓} ⊆ FmL 𝑑 , we have (recall that 𝑑 𝛿 ( 𝜒) = 𝜒 for each 𝜒 ∈ FmL ): Γ `L 𝜑

iff

𝑑 𝛿 [Γ] `L 𝑑 𝛿 (𝜑)

iff

Γ `L𝑑/ 𝛿 𝜑

Δ `L𝑑/ 𝛿 𝜓

iff

𝑑 𝛿 [Δ] `L 𝑑 𝛿 (𝜓)

iff

𝑑 𝛿 [Δ] `L𝑑/ 𝛿 𝑑 𝛿 (𝜓).

Finally, if L0 is a conservative expansion of L where 𝑑 is definable by 𝛿, then Γ `L𝑑/ 𝛿 𝜑

iff

𝑑 𝛿 [Γ] `L 𝑑 𝛿 (𝜑)

iff

𝑑 𝛿 [Γ] `L0 𝑑 𝛿 (𝜑)

iff

Γ `L0 𝜑.



Remark 2.3.6 Note that the definability of 𝑑 in L by 𝛿 can be formulated as L = L0𝑑/ 𝛿 , where L0 is the 𝑑-free fragment of L. Also note that L is finitary iff L𝑑/ 𝛿 is finitary. Finally, let us note that if AS is a presentation of L, then the set AS 𝑑/ 𝛿 = {Γ I 𝜑 | Γ ∪ {𝜑} ⊆ FmL 𝑑 and 𝑑 𝛿 [Γ] I 𝑑 𝛿 (𝜑) ∈ AS} ∪ {𝜑 I 𝑑 𝛿 (𝜑) | 𝜑 ∈ FmL 𝑑 } is a presentation of L𝑑/ 𝛿 . We can show that AS 𝑑/ 𝛿 is closed under substitutions in the same way as we have shown it for L𝑑/ 𝛿 in the proof of the previous proposition.

2.3 Some prominent non-classical logics

33

Clearly, AS 𝑑/ 𝛿 ⊆ L𝑑/ 𝛿 and so ` A S 𝑑/ 𝛿 ⊆ L𝑑/ 𝛿 . Conversely, assume that Γ `L𝑑/ 𝛿 𝜑, i.e. 𝑑 𝛿 [Γ] `L 𝑑 𝛿 (𝜑) and consider a proof of 𝑑 𝛿 (𝜑) from 𝑑 𝛿 [Γ] and we show that for each 𝜒 labeling a node 𝑛 in that proof Γ ` A S 𝑑/ 𝛿 𝜒. For 𝜒 ∈ 𝑑 𝛿 [Γ] the claim is trivial, otherwise there is a consecution 𝑑 𝛿 [Δ] I 𝑑 𝛿 ( 𝜒) = Δ I 𝜒 ∈ AS where Δ is the set of labels of predecessors of 𝑛 (note that Δ could be empty if 𝑛 is a leaf). Therefore, Δ I 𝜒 ∈ AS 𝑑/ 𝛿 and by the induction assumption also Γ ` A S 𝑑/ 𝛿 Δ and thus Cut completes the proof. Let us now present some prominent (axiomatic) expansions of the logics introduced above in the simple implicational language L→ to the classical language LCL which additionally features binary connectives ∧ and ∨ and the truth-constant ⊥. We start by introducing, for each logic L extending BCI, its expansion Llat enforcing the expected lattice-like properties of ∧, ∨, and ⊥.7 Example 2.3.7 Let L be an extension of BCI. Let us denote by Llat its expansion by the following consecutions: (⊥) (lb1 ) (lb2 ) (inf) (ub1 ) (ub2 ) (sup) (Adj)

⊥→𝜑 𝜑∧𝜓 → 𝜑 𝜑∧𝜓 →𝜓 ( 𝜒 → 𝜑) ∧ ( 𝜒 → 𝜓) → ( 𝜒 → 𝜑 ∧ 𝜓) 𝜑 → 𝜑∨𝜓 𝜓 → 𝜑∨𝜓 (𝜑 → 𝜒) ∧ (𝜓 → 𝜒) → (𝜑 ∨ 𝜓 → 𝜒) 𝜑, 𝜓 I 𝜑 ∧ 𝜓

ex falso quodlibet lower bound lower bound infimality upper bound upper bound supremality adjunction.

Let us denote Llat , for L being IL→ , CL→ , Ł→ , and G→ , as, respectively, IL, CL, Ł, and G, and call them intuitionistic logic, classical logic, Łukasiewicz logic, and Gödel–Dummett logic. In many cases we know that Llat is a conservative expansion of L. We prove it for CL→ in Corollary 3.5.20, for BCI, BCK, and IL→ in Corollary 4.7.5, and for FBCK and G→ in Theorem 6.3.8 (it is also known to be true for Ł→ as well; it was proved implicitly in [39, Propositions 2.2. and 4.2.]). However, it is not true for A→ . To prove it, let us first recall that we have considered a connective > defined in LCL as ⊥ → ⊥ and that using this definition we can easily see (using the axioms (⊥) and (E)) that the following formula is a theorem of BCIlat : (>)

𝜑→>

verum ex quolibet.

Now we can use instances of the axioms (>) and (abe): (𝜑 → >) → > and ((𝜑 → >) → >) → 𝜑 to see that Alat → is actually the inconsistent logic (we have obtained even more: the inconsistent logic in a language containing → and > is the only expansion of A→ where the rule (>) is valid). 7 Let us note that the logics BCIlat and BCKlat are not the weakest expansions where ∧ and ∨ behave as infimum and supremum. We have opted for this axiomatization for simplicity. Later, in Section 4.1, we provide a general study of lattice-like connectives and their interplay with implication.

34

2 Weakly implicative logics

Later, in Example 4.6.14, we define the Abelian logic A as a natural expansion of A→ in a language LCL without the truth-constants. In that example, we also justify the notation A→ by showing that, while A is not a conservative expansion of A→ in the sense of our definition, in the language L→ it proves the same theorems as A→ (and thus, due to Example 2.6.4, is not the inconsistent logic). Consider the following three additional related consecutions: (Sup) (Inf) (adj)

𝜑 → 𝜒, 𝜓 → 𝜒 I 𝜑 ∨ 𝜓 → 𝜒 𝜒 → 𝜑, 𝜒 → 𝜓 I 𝜒 → 𝜑 ∧ 𝜓 𝜑 → (𝜓 → 𝜑 ∧ 𝜓)

supremality infimality adjunction.

The first two consecutions are obviously derived rules obtained from the supremality/infimality axioms using the adjunction rule and modus ponens whereas the latter is an axiomatic form of the adjunction rule. As before, in a slight abuse of language, we use the same name for the axiom and for its corresponding rule, however, following our convention, we use different labels to distinguish them. We can easily show that (adj) is a theorem of any extension of BCKlat : a) 𝜑 → (𝜓 → 𝜑) (w) b) 𝜑 → (𝜓 → 𝜓) c) 𝜑 → (𝜓 → 𝜑) ∧ (𝜓 → 𝜓)

(w), (e), and (MP) a, b, and (Inf)

d) (𝜓 → 𝜑) ∧ (𝜓 → 𝜓) → (𝜓 → 𝜑 ∧ 𝜓)

(inf)

e) 𝜑 → (𝜓 → 𝜑 ∧ 𝜓)

c, d, and (T) Llat

Therefore, for any logic L over BCK, we could axiomatize using the axiom (adj) instead of the rule (Adj) and so Llat is an axiomatic expansion of L. On the other hand, in Example 2.6.5 we show that BCIlat is not an axiomatic expansion of BCI. Let us also note that adding the axiom (adj) to BCIlat makes the formula (w) derivable; indeed we can write the following formal proof: a) 𝜑 ∧ 𝜓 → 𝜑 (lb1 ) b)

(𝜓 → 𝜑 ∧ 𝜓) → (𝜓 → 𝜑)

a and (Pf)

c)

(𝜑 → (𝜓 → 𝜑 ∧ 𝜓)) → (𝜑 → (𝜓 → 𝜑))

b and (Pf)

d) 𝜑 → (𝜓 → 𝜑)

c, (adj), and (MP)

Remark 2.3.8 Several remarks about the logics we have just defined and their presentations are in order: • Classical and intuitionistic logic have numerous other presentations, some of them more common in the literature than the one used here. • Classical logic can be introduced with many different sets of primitive connectives. This is related to the interdefinability of Boolean operations (see the Appendix). Later in Corollary 3.5.20 we will prove completeness of this axiomatic system w.r.t. the classical two-valued semantics, showing that it is indeed the classical logic.

2.3 Some prominent non-classical logics

35

• Gödel–Dummett logic is usually presented in a language where ∨ is a defined connective (cf. Example 2.8.14): 𝜑 ∨ 𝜓 = ((𝜑 → 𝜓) → 𝜓) ∧ ((𝜓 → 𝜑) → 𝜑). Furthermore, it is often presented as the axiomatic extension of IL by the axiom of prelinearity (cf. Example 2.4.7): (𝜑 → 𝜓) ∨ (𝜓 → 𝜑). • Łukasiewicz logic is usually presented in a language where ∧ and ∨ are defined connectives (cf. Example 2.8.14): 𝜑 ∨ 𝜓 = (𝜑 → 𝜓) → 𝜓

𝜑 ∧ 𝜓 = ¬(¬𝜑 ∨ ¬𝜓)

with an axiomatic system consisting of modus ponens and axioms (sf), (w), (waj), and the axiom of contraposition: (¬𝜑 → ¬𝜓) → (𝜓 → 𝜑). Also, the following two connectives are usually defined in Łukasiewicz logic: 𝜑 & 𝜓 = ¬(𝜑 → ¬𝜓)

𝜑 ⊕ 𝜓 = ¬𝜑 → 𝜓.

Next, we give an example of a prominent extension of Łukasiewicz logic Ł by an infinitary rule. The fact that the resulting logic is indeed infinitary is demonstrated in Example 2.6.6 and later in Example 3.8.9 we show that Ł is its finitary companion. Example 2.3.9 The logic Ł∞ is defined as the extension of Łukasiewicz logic Ł by the following infinitary rule: (Ł∞ )

{¬𝜑 → 𝜑 & . 𝑛. . & 𝜑 | 𝑛 ≥ 1} I 𝜑

Hay rule.

The last group of examples of logics we introduce in this chapter are the wellknown classical modal logics, i.e. expansions of classical logic CL in the language including an additional unary connective . We consider both the global and the local variants of these logics, which have very different properties from the point of view of this book. Example 2.3.10 Take the language L obtained by adding a unary connective  to LCL . Consider the following consecutions: (d ) (t ) (4 ) (-nec)

(𝜑 → 𝜓) → (𝜑 → 𝜓) 𝜑 → 𝜑 𝜑 → 𝜑 𝜑 I 𝜑

distribution reflexivity transitivity necessitation.

36

2 Weakly implicative logics

The basic global modal logic K is the expansion of CL by the axiom (d ) and the rule (-nec). We also define the following axiomatic extensions of K (also considered as global modal logics): T: axiomatic extension of K by (t ) K4: axiomatic extension of K by (4 ) S4: axiomatic extension of T by (4 ). The local8 variants of these logics are defined as the axiomatic expansions of CL by all the theorems of the corresponding global modal logic and denoted using the prefix 𝑙, i.e. 𝑙K, 𝑙T, 𝑙K4, 𝑙S4. Alternatively, these logics can be axiomatized by taking as axioms all the formulas . 𝑛. .𝜑 for each 𝑛 ≥ 0 and each axiom 𝜑 of the corresponding global logic, and modus ponens as the only inference rule. In Example 2.6.8 we will show that the introduced modal logics are pairwise different.

2.4 Classical metalogical properties and their variants In this section we study two important metalogical properties of the logics defined in the previous section, namely the deduction theorem and the proof by cases property. We will see that both these properties can be formulated in various ways, depending on the logic in question. Later, in Chapter 4 and Chapter 5 respectively, we will analyze them in a much higher level of generality. It is nonetheless useful to present those different formulations and their proofs at this point as an illustration of certain syntactical notions of this chapter. Deduction theorems can be used in (at least) two different ways. The first one works best in logics with the classical deduction theorem, where it allows us to establish the derivability of an implication by proving its consequent from its antecedent (taken as an additional premise). Moreover, in Example 2.4.7, we will see that even non-classical variants of the deduction theorem can be used to establish the derivability of certain implicational formulas. The second usage of deduction theorems transforms derivations from finite sets of premises into derivations of theorems and it is more important from a theoretical point of view, because it shows that finitary logics with the deduction theorem are completely determined by their set of theorems. Since we are dealing with deduction theorems as metalogical properties that any particular logic may or may not have, it should not be surprising that, instead of simply proving them for certain logics, we strive for characterizations that determine which logics enjoy each form of deduction theorem. We start with the simplest case, the classical deduction theorem, and show that it holds only in expansions of IL→ . 8 The terminology global/local for modal logics originates from properties of their relational semantics, which is not a topic of this book, since we concentrate on matricial and algebraic semantics.

2.4 Classical metalogical properties and their variants

37

Proposition 2.4.1 (Classical deduction theorem) Let L be a finitary logic in a language L ⊇ L→ . Then, L is an axiomatic expansion of IL→ if and only if L enjoys the classical deduction theorem, i.e. for any set of formulas Γ ∪ {𝜑, 𝜓} ⊆ FmL , Γ, 𝜑 `L 𝜓

iff

Γ `L 𝜑 → 𝜓.

Proof First, assume that L is an axiomatic expansion of IL→ . Then, L has an axiomatic system AS where modus ponens is the only inference rule. The right-to-left direction of the deduction theorem is then a simple consequence of modus ponens. To prove the converse direction, assume that 𝛼1 , . . . , 𝛼𝑛 = 𝜓 is a finitary proof in AS of 𝜓 from premises Γ ∪ {𝜑}. We show that for each 𝑖 we have Γ `L 𝜑 → 𝛼𝑖 . If 𝛼𝑖 is 𝜑, the claim follows from the fact that in all these logics 𝜑 → 𝜑 is a theorem (see Example 2.2.4). If 𝛼𝑖 is an element of Γ or an instance of an axiom, we know that Γ `L 𝛼𝑖 and so the claim follows using the derived rule (W). Finally, if 𝛼𝑖 is a consequence of modus ponens, then, there have to be indices 𝑘, 𝑙 < 𝑖 such that 𝛼𝑙 = 𝛼 𝑘 → 𝛼𝑖 . Thus, due to the induction assumption, we have Γ `L 𝜑 → 𝛼 𝑘 and Γ `L 𝜑 → (𝛼 𝑘 → 𝛼𝑖 ). Using (E), we obtain Γ `L 𝛼 𝑘 → (𝜑 → 𝛼𝑖 ). Then, by (T), we obtain Γ `L 𝜑 → (𝜑 → 𝛼𝑖 ). The axiom (c) and modus ponens complete the proof. Now assume that the logic L enjoys the deduction theorem. Observe that, from its right-to-left direction and the fact that 𝜑 → 𝜓 `L 𝜑 → 𝜓, we obtain that modus ponens in valid in L. Thus, in particular, we know that 𝜑, 𝜓, 𝜑 → (𝜓 → 𝜒) `L 𝜒 (by using modus ponens twice). Therefore, using the left-to-right direction of the deduction theorem, we obtain: 𝜓, 𝜑 → (𝜓 → 𝜒) `L 𝜑 → 𝜒. Using it twice more we obtain that L proves the axiom (e). Analogously, we can show that L proves all the remaining axioms of IL→ and hence it is an expansion of IL→ . To conclude the proof, we need to show that L is an axiomatic expansion of IL→ . Consider the following axiomatic system: AS 0 = {𝜑, 𝜑 → 𝜓 I 𝜓 | 𝜑, 𝜓 ∈ FmL } ∪ {𝜑 | `L 𝜑}. Clearly, `A S0 ⊆ `L . Assume that Γ `L 𝜑; then, due to the finitarity of L and repeated use of the deduction theorem, we have a set {𝜓1 , . . . , 𝜓 𝑛 } ⊆ Γ such that `L 𝜓1 → (𝜓2 → · · · (𝜓 𝑛 → 𝜑) · · · ). Thus, obviously, Γ `A S0 𝜑.



Note that, in particular, we have obtained the classical deduction theorem for CL, G, IL, and the local modal logics introduced in Example 2.3.10. Later, in Example 2.6.8, we will see that the global variants of these modal logics are not axiomatic expansions of IL→ and, thus, do not enjoy the classical deduction theorem. However, we can prove an alternative formulation of the theorem for these logics instead. As an illustration, we present it for global modal logics expanding S4 where it is syntactically only slightly different from the classical case. We refer the reader to Section 4.8, where we prove more complex variants of the deduction theorem for weaker modal logics (see Examples 4.8.8 and 4.8.17).

38

2 Weakly implicative logics

Proposition 2.4.2 (Deduction theorem for S4) Let L be a finitary logic expanding CL in a language L ⊇ L . Then, L is an axiomatic expansion of S4 if and only if L enjoys the S4-deduction theorem, i.e. for any set of formulas Γ ∪ {𝜑, 𝜓} ⊆ FmL , Γ, 𝜑 `L 𝜓

iff

Γ `L 𝜑 → 𝜓.

Proof The proof is similar to the proof of the previous proposition, thus we only point out the differences. It is easy to see that if L is an axiomatic expansion of S4, the right-to-left direction of the claim follows from modus ponens and necessitation rules. The converse direction is again proved by induction where the induction hypothesis now is Γ `L 𝜑 → 𝛼𝑖 . The cases in which 𝛼𝑖 is an axiom or an element of Γ or it is a result of modus ponens are handled as before. The case of 𝛼𝑖 = 𝜑 follows from axiom (t ); finally we deal with the case when 𝛼𝑖 = 𝛼 𝑗 is a result of an application of necessitation to 𝛼 𝑗 . The induction assumption gives us 𝜑 → 𝛼 𝑗 ; thus, using necessitation and the axiom (d ), we obtain 𝜑 → 𝛼 𝑗 and the axiom (4 ) completes the proof. To show that a logic with the S4-deduction theorem extends S4, we first observe that necessitation follows from ` 𝜑 → 𝜑. The axioms (d ), (t ), and (4 ) then follow from the obvious derived rules of L: 𝜑 → 𝜓, 𝜑 ` 𝜓, 𝜑 ` 𝜑, and 𝜑 ` 𝜑 respectively.  In Example 2.6.4, we will show that other logics, such as BCK or Łukasiewicz logic, are not expansions of IL→ ; thus, Proposition 2.4.1 (among others) implies that they cannot enjoy the classical deduction theorem. However, we still can prove another kind of deduction theorem for these logics, which has been called local because its form is not uniform for all formulas but it depends on the formulas used in each case. Otherwise, when the form of the deduction theorem is the same for all formulas we call it global.9 Actually, we prove that all axiomatic expansions of BCI enjoy this particular form of the deduction theorem (note that the classical deduction theorem follows from the local one using the axioms (w) and (c)) and in Example 2.6.5 we will use this fact to show that BCIlat is not an axiomatic expansion of BCI. Again, in Section 4.8 we will see an alternative formulation and proof of local deduction theorems for a wider family of logics in richer languages. Proposition 2.4.3 (Local deduction theorem) Let L be an axiomatic expansion of BCI. Then, L enjoys the local deduction theorem, i.e. for any set of formulas Γ ∪ {𝜑, 𝜓} ⊆ FmL , Γ, 𝜑 `L 𝜓 iff there is a natural number 𝑘 s.t. Γ `L 𝜑 → (𝜑 → . . . (𝜑 → 𝜓) . . .). | {z } 𝑘 times

The number 𝑘 can be taken as the number of occurrences of 𝜑 in the leaves of a tree-proof of 𝜓 from Γ, 𝜑 in a presentation AS of L where modus ponens is the only inference rule. 9 Do not confuse this use of the adjectives global/local with the unrelated terminology of modal logics.

2.4 Classical metalogical properties and their variants

39

Proof The right-to-left direction of the equivalence is a simple consequence of modus ponens. To prove the converse direction we need to formulate and prove two auxiliary claims. To formulate them, we need to introduce some notation: for a finite sequence of formulas 𝛼 = h𝜑1 , . . . , 𝜑 𝑛 i, we write 𝛼 → 𝜑 instead of 𝜑1 → (𝜑2 → . . . (𝜑 𝑛 → 𝜑) . . . ) (by convention, we set 𝛼 → 𝜑 = 𝜑 if 𝛼 = hi). We prove that for any finite sequence 𝛼 of formulas and formulas 𝜑, 𝜓 the following are derivable rules in BCI (and, thus, in any expansion): C1 C2

𝛼 → 𝜑, 𝜑 → 𝜓 I 𝛼 → 𝜓 𝛼 → (𝜑 → 𝜓) I 𝜑 → (𝛼 → 𝜓).

We prove these two claims by induction on the length of 𝛼. If 𝛼 = hi or 𝛼 = h𝜒i, the proofs are trivial. Assume, therefore, that 𝛼 = h𝜒1 , . . . , 𝜒𝑚 , 𝜒i, and let us use 𝛽 for h𝜒1 , . . . , 𝜒𝑚 i. Then, we can easily prove C1: a) 𝛽 → ( 𝜒 → 𝜑) the first premise b) 𝜑 → 𝜓

the second premise

c)

( 𝜒 → 𝜑) → ( 𝜒 → 𝜓)

d)

𝛽 → ( 𝜒 → 𝜓)

b and (Pf) a, c, and the induction assumption

The proof of C2 is also simple: a) 𝛽 → ( 𝜒 → (𝜑 → 𝜓))

the premise

b)

( 𝜒 → (𝜑 → 𝜓)) → (𝜑 → ( 𝜒 → 𝜓))

(e)

c)

𝛽 → (𝜑 → ( 𝜒 → 𝜓))

a, b, and C1

d) 𝜑 → (𝛽 → ( 𝜒 → 𝜓))

c and the induction assumption

Now we are ready to finish the proof of the proposition: consider a tree-proof of 𝜓 from premises Γ ∪ {𝜑} in the presentation AS. We show that, for each node 𝑛 of the proof labeled by the formula 𝜒𝑛 , Γ ` 𝜑 → (𝜑 → . . . (𝜑 → 𝜒𝑛 ) . . .), | {z } 𝑘𝑛 times

where 𝑘 𝑛 is the number of occurrences of 𝜑 in the subtree of this proof with root 𝑛. Clearly, taking 𝑛 as the root of the whole proof completes the proof. Note that without loss of generality we can assume that 𝜑 ∉ Γ. If 𝑛 is a leaf, then either: • 𝜒𝑛 = 𝜑, in which case 𝑘 𝑛 = 1 and, indeed, we have: Γ ` 𝜑 → 𝜑, or • 𝜒𝑛 ≠ 𝜑, in which case 𝑘 𝑛 = 0 and, indeed, we have: Γ ` 𝜒𝑛 because 𝜒𝑛 is either an axiom or an element of Γ. If 𝑛 is not a leaf, it must have predecessors 𝑙 and 𝑟 such that 𝜒𝑟 = 𝜒𝑙 → 𝜒𝑛 . Thus, by the induction assumption, Γ ` 𝜑 → (𝜑 → . . . (𝜑 → 𝜒𝑙 ) . . .) | {z } 𝑘𝑙 times

Γ ` 𝜑 → (𝜑 → . . . (𝜑 → ( 𝜒𝑙 → 𝜒𝑛 ) . . .). | {z } 𝑘𝑟 times

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2 Weakly implicative logics

Thus, using C2, we know that Γ ` 𝜒𝑙 → (𝜑 → (𝜑 → . . . (𝜑 → 𝜒𝑛 ) . . .) | {z } 𝑘𝑟 times

and so using C1 we obtain Γ ` 𝜑 → (𝜑 → . . . (𝜑 → (𝜑 → (𝜑 → . . . (𝜑 → 𝜒𝑛 ) . . .) | {z } | {z } 𝑘𝑙 times

𝑘𝑟 times

Since, obviously, 𝑘 𝑛 = 𝑘 𝑙 + 𝑘 𝑟 , the proof is done.



Remark 2.4.4 Note that, unlike the previous two variants of the deduction theorem, we have not formulated Theorem 2.4.3 as a characterization of a class of logics. Actually, one could show that the converse direction is not true. The proper equivalent form of the deduction theorem has a rather complicated form, in which the equivalence is not hard to prove. Indeed, it is easy to show that a logic L with an axiomatic system AS where modus ponens is the only inference rule is an axiomatic expansion of BCI iff for any tree-proof of 𝜓 from Γ ∪ {𝜑1 , 𝜑2 , 𝜑3 } in AS in which the leaf 𝜑𝑖 occurs 𝑘 𝑖 times, we have Γ `L 𝜑1 → ( 𝜑1 → . . . ( 𝜑1 → ( 𝜑2 → ( 𝜑2 → . . . ( 𝜑2 → ( 𝜑3 → ( 𝜑3 → . . . ( 𝜑3 → 𝜓) . . .). | {z } | {z } | {z } 𝑘1 times

𝑘2 times

𝑘3 times

Next, we turn our attention to the proof by cases property, which is again a metalogical property that captures the usual classical mathematical reasoning in which the proof of a conclusion is obtained from separate proofs of the same conclusion from premises divided into possible cases. As in the case of deduction theorems, it has two main uses: it can help establish the derivability of certain formulas (see Example 2.4.7) and it has deep consequences for the general theory; indeed, the whole Chapter 5 is dedicated to the study of (variants of) this property and its consequences.10 Proposition 2.4.5 (Classical proof by cases property) Let L be an axiomatic expansion of BCKlat . Then, for any set of formulas Γ ∪ {𝜑, 𝜓} ⊆ FmL , ThL (Γ, 𝜑 ∨ 𝜓) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓). Proof The left-to-right direction is a simple consequence of the axioms (ub1 ) and (ub2 ). To prove the converse direction, we first observe that, using the rule (Sup), we can easily establish the following claim for any set of formulas Γ ∪ {𝛼, 𝛽, 𝛿}: C1 Γ `L 𝛼 → 𝛿 and Γ `L 𝛽 → 𝛿 imply Γ `L 𝛼 ∨ 𝛽 → 𝛿. 10 Note that we could formulate this property analogously to the deduction theorem, i.e. as Γ, 𝜑 ∨ 𝜓 `L 𝜒

iff

Γ, 𝜑 `L 𝜒 and Γ, 𝜓 `L 𝜒.

However, the fact that the right-hand sides of all derivations are the same allows us to use a more compact formulation.

2.4 Classical metalogical properties and their variants

41

Now assume that Γ, 𝜑 `L 𝜒 and Γ, 𝜓 `L 𝜒. Note that if our logic L expands IL, we can use the classical deduction theorem and the claim C1 to complete the proof. Unfortunately, in general we can only use the local deduction theorem established in Proposition 2.4.3 (recall that any axiomatic expansion of BCKlat is indeed an axiomatic expansion of BCI; see Example 2.3.7). In order to use this weaker deduction theorem, we need a more general version of the claim C1; in particular, we need to prove the following claim for any set of formulas Γ ∪ {𝛼, 𝛽, 𝛿} and any natural numbers 𝑘 and 𝑚: C2 Γ `L 𝛼 → (𝛼 → . . . (𝛼 → 𝛿) . . .) and Γ `L 𝛽 → (𝛽 → . . . (𝛽 → 𝛿) . . .) | {z } | {z } 𝑘 times

𝑚 times

implies Γ, 𝛼 ∨ 𝛽 `L 𝛿. We prove the claim C2 by induction on 𝑘 + 𝑚. If 𝑘 = 0 or 𝑚 = 0, the claim trivially follows (note also that for 𝑘 = 𝑚 = 1 it is actually claim C1). Using the claim C1 from the proof of Proposition 2.4.3 and the axiom (w), we obtain the following from our second premise: Γ `L 𝛽 → (𝛽 → . . . (𝛽 → (𝛼 → 𝛿) . . .). | {z } 𝑚 times

We can formulate the first premise in a syntactically equivalent way Γ `L 𝛼 → (𝛼 → . . . (𝛼 → (𝛼 → 𝛿) . . .) | {z } 𝑘−1 times

and we can use the induction hypothesis to obtain Γ, 𝛼 ∨ 𝛽 `L 𝛼 → 𝛿. Analogously, we can obtain Γ, 𝛼 ∨ 𝛽 `L 𝛽 → 𝛿 and the claim C1 completes the proof.  Observe that we have proved the proof by cases property, in particular, for CL, G, IL, Ł, and local modal logics. On the other hand, it is easy to see that if a logic L expanding a global modal logic K enjoys the classical proof by cases property, then `L 𝜑 → 𝜑. Indeed, from 𝜑 `L 𝜑, we obtain 𝜑 `L ¬𝜑 ∨ 𝜑 using the axiom (ub2 ). Since, using (ub1 ), we have ¬𝜑 `L ¬𝜑 ∨ 𝜑, we can use the proof by cases property to obtain 𝜑 ∨ ¬𝜑 `L ¬𝜑 ∨ 𝜑; therefore, since L expands classical logic, this amounts to saying that `L 𝜑 → 𝜑. Later, in Example 2.6.8, we will see that the formula 𝜑 → 𝜑 is not a theorem of any of our global modal logics, and so these logics do not enjoy the proof by cases property. Also, we can easily observe that, by a small modification of the proof, we obtain an alternative formulation of the property that works for axiomatic expansions of S4 (we leave the proof as an exercise for the reader). We refer the reader to Section 4.10, where we prove more complex variants of the proof by cases property for weaker (modal) logics (see Example 4.10.4). Proposition 2.4.6 (Proof by cases for S4) Let L be an axiomatic expansion of S4. Then, for any set of formulas Γ ∪ {𝜑, 𝜓} ⊆ FmL , ThL (Γ, 𝜑 ∨ 𝜓) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓).

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2 Weakly implicative logics

Let us conclude this section by presenting two applications of the proof by cases property and the local deduction theorem. Example 2.4.7 We can show that FBCKlat is the extension of BCKlat by the axiom: (p∨ )

(𝜑 → 𝜓) ∨ (𝜓 → 𝜑)

prelinearity.

First, observe that FBCKlat proves the prelinearity axiom: a) (𝜑 → 𝜓) → (p∨ ) b) (𝜓 → 𝜑) → (p∨ ) c) (p∨ )

(ub1 ) (ub2 )

a, b, (lin), and (MP) twice

The proof is completed by showing that the extension of BCKlat by the axiom (p∨ ) proves (lin). It is easy to see that this logic satisfies the premises of Propositions 2.4.3 and 2.4.5. Thus, thanks to the proof by cases property, it suffices to show that 𝜑 → 𝜓 ` (lin) and 𝜓 → 𝜑 ` (lin). Observe that there is a proof of 𝜒 from premises 𝜓 → 𝜑 and (𝜓 → 𝜑) → 𝜒 in which (𝜓 → 𝜑) → 𝜒 is used exactly once. Therefore, using the local deduction theorem, we obtain 𝜓 → 𝜑 ` ((𝜓 → 𝜑) → 𝜒) → 𝜒 and thus, using the axiom (w), we obtain 𝜓 → 𝜑 ` (lin). Interchanging 𝜑 and 𝜓 yields 𝜑 → 𝜓 ` ((𝜓 → 𝜑) → 𝜒) → (((𝜑 → 𝜓) → 𝜒) → 𝜒) and the rule (E) completes the proof. Example 2.4.8 We can show that classical logic CL is the extension of BCKlat by the following axiom: (lem)

𝜑 ∨ ¬𝜑

law of excluded middle.

Again, we need to give two formal proofs in two different axiomatic systems. One is the proof of the well-known fact that classical logic proves the law of excluded middle. Recall that from the previous example we know that CL proves the axiom (p∨ ); thus, thanks to the proof by cases property of CL, it suffices to notice that 𝜑 → ¬𝜑 `CL (lem) (which follows from the contraction axiom (c) and (ub2 ) using that ¬𝜑 = 𝜑 → ⊥) and ¬𝜑 → 𝜑 `CL (lem) (which follows from Peirce’s law (p) and (ub1 )). To prove the second direction, recall that, thanks to Example 2.3.1, we only need to prove that the extension of BCKlat by the axiom (lem) proves the contraction axiom (c) and Peirce’s law (p). First, observe that this logic enjoys the proof by cases property due to Proposition 2.4.5. Thus, it suffices to prove these two axioms from a premise 𝜑 or ¬𝜑 respectively. • 𝜑 ` (c): an easy application of the axiom (id) and the rule (E). • ¬𝜑 ` (c): using the rule (T) and the axiom (⊥), we obtain ¬𝜑 ` (𝜑 → 𝜓) and the claim follows using the rule (W). • 𝜑 ` (p): a direct application of the rule (W). • ¬𝜑 ` (p): starting again with ¬𝜑 ` 𝜑 → 𝜓 and observing that ` (𝜑 → 𝜓) → (p) thanks to the axiom (id) and the rule (E).

2.5 Logical matrices and semantical consequence

43

2.5 Logical matrices and semantical consequence In this section, we introduce the basic semantical notions that are necessary for the general theory and illustrate them in particular examples of logics that have been defined in the previous sections. Let us fix a language L and a set of variables Var. Logics in this language are given a semantical interpretation by means of the notion of a logical matrix, which is a pair formed by an L-algebra (which interprets the formulas capitalizing on the fact that L is actually an algebraic language) and a (logical) filter, a subset of designated elements in the domain of the algebra which gives a notion of truth that allows us to define consequence as preservation of truth:11 Definition 2.5.1 (Logical matrix) An L-matrix is a pair A = hA, 𝐹i where A is an L-algebra called the algebraic reduct of A, and 𝐹 is a subset of 𝐴 called the filter of A. The elements of 𝐹 are called designated elements of A. An L-matrix A = hA, 𝐹i is said to be a trivial matrix if 𝐹 = 𝐴, and it is called a Lindenbaum matrix if A = FmL . For simplicity, we usually call these structures just matrices. The cardinality of a matrix is defined as the cardinality of the domain of its underlying algebra. Note that matrices can also be seen as first-order structures for a language (without equality) with one unary predicate symbol interpreted by the filter and function symbols interpreted by operations of its algebraic reduct (in Chapter 3 we will exploit this relationship as a guiding principle in defining operations on logical matrices such as products, submatrices, homomorphic images, etc.). Definition 2.5.2 (Evaluation) Let A be an L-algebra. An A-evaluation is a homomorphism from FmL to A, i.e. a mapping 𝑒 : FmL → 𝐴 such that for each 𝑛-ary connective 𝑐 of L and each 𝑛-tuple of formulas 𝜑1 , . . . , 𝜑 𝑛 , 𝑒(𝑐(𝜑1 , . . . , 𝜑 𝑛 )) = 𝑐A (𝑒(𝜑1 ), . . . , 𝑒(𝜑 𝑛 )). Note that FmL -evaluations are endomorphisms of FmL , i.e. they coincide with substitutions in the language L. Also, as in the case of substitutions, since an A-evaluation is a homomorphism from an absolutely free L-algebra, it is fully determined by its values on the free generators (propositional variables), i.e. for each mapping 𝑒 : Var → 𝐴, there is a unique A-evaluation 𝑒 such that 𝑒(𝑣) = 𝑒(𝑣) for each 𝑣 ∈ Var. Since evaluations are homomorphisms, the value assigned to a formula 𝜑 by an evaluation 𝑒 is actually fully determined by the values 𝑒 assigns to variables occurring in 𝜑. Therefore, for a formula 𝜑 built from variables 𝑝 1 , . . . , 𝑝 𝑛 , an algebra A, elements 𝑎 1 , . . . , 𝑎 𝑛 ∈ 𝐴 and an A-evaluation 𝑒 such that 𝑒( 𝑝 𝑖 ) = 𝑎 𝑖 , we write 𝜑A (𝑎 1 , . . . , 𝑎 𝑛 ) instead of 𝑒(𝜑( 𝑝 1 , . . . , 𝑝 𝑛 )). 11 The term ‘filter’ refers to the fact that, in matrices for prominent logics, the algebras have lattice reducts in which logical filters are indeed filters in the lattice-theoretic sense; see Example 2.5.9.

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2 Weakly implicative logics

We denote by 𝑒 𝑝=𝑎 the evaluation obtained from 𝑒 by assigning the element 𝑎 ∈ 𝐴 to the variable 𝑝 and leaving the values of the remaining variables unchanged. Although the notion of evaluation is defined for algebras, when convenient we also speak about hA, 𝐹i-evaluations instead of A-evaluations. Now we have all the necessary elements to introduce the semantical notion of logical consequence, understood as preservation of truth. Definition 2.5.3 (Semantical consequence) A formula 𝜑 is a semantical consequence of a set Γ of formulas w.r.t. a matrix A = hA, 𝐹i, Γ A 𝜑, in symbols, if for each A-evaluation 𝑒, we have 𝑒(𝜑) ∈ 𝐹 whenever 𝑒[Γ] ⊆ 𝐹. We also extend the notion of semantical consequence to any class K of matrices in the obvious way: Ù K = A . A∈K

Note that, while only the designated elements of a given matrix are explicitly mentioned in the definition of consequence, all the elements of the algebra play their role because they determine the set of evaluations over which we quantify. Clearly, K is a set of consecutions and it is easy to show that it is indeed a logic (see the next proposition for details). Therefore, we have that Γ K 𝜑 iff the consecution Γ I 𝜑 is valid in K ; by an abuse of language, sometimes we will simply say that Γ I 𝜑 is valid in K. A more interesting question is whether it is a finitary logic. In the next proposition we use methods from elementary general topology to prove one sufficient condition: K is finitary whenever K is a finite class of finite matrices.12 Readers unfamiliar with topology may skip this proof because, after building enough structural theory of logical matrices, in Theorem 3.7.7 we will present a self-contained proof of a necessary and sufficient condition for finitarity of K (a condition obviously valid for a finite class of finite matrices). Proposition 2.5.4 Let K be a class of L-matrices. Then, K is a logic in L. Furthermore, if K is a finite class of finite matrices, then K is finitary. Proof Let us first notice that it suffices to prove the claim for K = {hA, 𝐹i} because the set of all (finitary) logics in a given language is closed under (finite) intersections. To prove the first claim, we need to check the three conditions in the definition of logic. Monotonicity is obvious. To show Cut, fix an A-evaluation 𝑒 such that 𝑒[Δ] ⊆ 𝐹. Then, clearly, 𝑒(𝜓) ∈ 𝐹 for each 𝜓 ∈ Γ, i.e. 𝑒[Γ] ⊆ 𝐹, and so 𝑒(𝜑) ∈ 𝐹. To show Structurality, fix hA, 𝐹i and 𝑒 as before and assume that 𝑒(𝜎[Γ]) ⊆ 𝐹. Since 𝑒 0 = 𝑒 ◦ 𝜎 is an A-evaluation and 𝑒 0 [Γ] ⊆ 𝐹, we obtain 𝑒(𝜎(𝜑)) = 𝑒 0 (𝜑) ∈ 𝐹. To prove the second claim, assume that Γ0 2 hA,𝐹 i 𝜑 for each finite Γ0 ⊆ Γ and we want to show that Γ 2 hA,𝐹 i 𝜑. 12 As we will see later, this condition is clearly not necessary: e.g. due to the completeness Theorem 2.6.3, we will know that any finitary logic L is equal to Mod(L) , where Mod(L) is an infinite class of matrices. Furthermore, in Example 5.6.11 we will see that Łukasiewicz logic, while finitary, is not equal to K for any finite class of finite matrices. Let us also note that Example 2.6.7 shows that K need not be finitary if K is an infinite set of finite matrices.

2.5 Logical matrices and semantical consequence

45

Let us consider the finite set 𝐴 endowed with the discrete topology and its power 𝐴Var with the product (i.e. weak) topology. Both spaces are compact (the first one trivially and the second one due to Tychonoff theorem [241, Theorem 37.3]). Clearly, each evaluation 𝑒 can be identified with an element of 𝐴Var and vice versa. For each formula 𝜓, we define a mapping 𝐻 𝜓 : 𝐴Var → 𝐴 as 𝐻 𝜓 (𝑒) = 𝑒(𝜓). It can be shown that these mappings are continuous, thus (𝐻 𝜓 ) −1 [𝐹] is a closed set and so is the set (𝐻 𝜓 ) −1 [𝐹] ∩ (𝐻 𝜑 ) −1 [ 𝐴 \ 𝐹] (i.e. the set of evaluations 𝑒 for which 𝑒(𝜓) ∈ 𝐹, but 𝑒(𝜑) ∉ 𝐹). Let us now consider the system of closed sets {(𝐻 𝜓 ) −1 [𝐹] ∩ (𝐻 𝜑 ) −1 [ 𝐴 \ 𝐹] | 𝜓 ∈ Γ}. This is clearly a centered system (i.e. the intersection of any subsystem given by a finite set Γ0 is non-empty, because it contains any evaluation which witnesses that Γ0 2 hA,𝐹 i 𝜑). Thus, due to the compactness of 𝐴Var, the intersection of the whole system is non-empty and the proof is done (because, for any element 𝑒 of this intersection, we have 𝑒[Γ] ⊆ 𝐹 and 𝑒(𝜑) ∉ 𝐹).  One could ask what matrices naturally correspond to a given logic. We can look at it from two perspectives. First, given a logic L and a matrix A, we can ask whether L is sound w.r.t. the semantics given by A (i.e. whether L ⊆ A ); such matrices are called models of L. Second, given L and an algebra A, we can ask which possible sets of elements of A could be seen as designated elements of some model of L with algebraic reduct A. Formally speaking, we define: Definition 2.5.5 (Model and logical filter) Let L be a logic in L and A = hA, 𝐹i an L-matrix. If L ⊆ A , then we say that • A is a model of L (or an L-model), A ∈ Mod(L) in symbols, and • 𝐹 is an L-filter on A, 𝐹 ∈ FiL (A) in symbols. It is easy to see that the mappings Mod(·) and  ( ·) between logics in a language L and classes of L-matrices give rise to an antitone Galois connection (see the Appendix), i.e. they satisfy the following conditions: • If K ⊆ M, then M ⊆ K . • If L1 ⊆ L2 , then Mod(L2 ) ⊆ Mod(L1 ). • L ⊆ K iff K ⊆ Mod(L). Thus, in particular if L1 ⊆ L2 , then for each A we have FiL2 (A) ⊆ FiL1 (A) and Mod(L) = Mod(Mod(L) )

K = Mod(K ) .

We will use these facts from now on without explicit mention. The following proposition and lemma are useful for recognizing the models and filters of a given logic. The proposition shows that it suffices to check the defining condition for the rules of any presentation (one direction is obvious, the second one follows from Lemma 2.2.7). The lemma assumes that we already have a model of a given logic and shows how to obtain other models using homomorphisms of their underlying algebras (in Chapter 3 we will see its relation to homomorphisms of matrices). Proposition 2.5.6 For each presentation AS of a logic L, we have: A ∈ Mod(L) iff AS ⊆ A .

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2 Weakly implicative logics

Lemma 2.5.7 Let L be a logic in L and 𝑔 : A → B be a homomorphism of L-algebras A, B. Then, 1. hA, 𝑔 −1 [𝐺]i ∈ Mod(L), whenever hB, 𝐺i ∈ Mod(L). 2. hB, 𝑔[𝐹]i ∈ Mod(L), whenever hA, 𝐹i ∈ Mod(L) and 𝑔 is surjective and 𝑔(𝑥) ∈ 𝑔[𝐹] implies 𝑥 ∈ 𝐹. Proof The first claim is straightforward. Indeed, assume that Γ `L 𝜑 and 𝑒[Γ] ⊆ 𝑔 −1 [𝐺] for some A-evaluation 𝑒. Thus, 𝑔[𝑒[Γ]] ⊆ 𝐺 which, since 𝑔 ◦ 𝑒 is a B-evaluation and hB, 𝐺i ∈ Mod(L), implies that 𝑔(𝑒(𝜑)) ∈ 𝐺, i.e. 𝑒(𝜑) ∈ 𝑔 −1 [𝐺]. For the second claim, assume that Γ `L 𝜓 and for a B-evaluation 𝑓 it is the case that 𝑓 [Γ] ⊆ 𝑔[𝐹]. Let us define an A-evaluation 𝑒 by setting 𝑒(𝑣) = 𝑎 for some 𝑎 such that 𝑔(𝑎) = 𝑓 (𝑣) (such an 𝑎 has to exist because 𝑔 is surjective). Next, we show by induction that 𝑓 (𝜑) = 𝑔(𝑒(𝜑)). The base case is trivial. Let us assume that 𝜑 = 𝑐(𝜑1 , . . . , 𝜑 𝑛 ) and observe that 𝑓 (𝑐(𝜑1 , . . . , 𝜑 𝑛 )) = 𝑐B ( 𝑓 (𝜑1 ), . . . , 𝑓 (𝜑 𝑛 )) = 𝑐B (𝑔(𝑒(𝜑1 )), . . . , 𝑔(𝑒(𝜑 𝑛 ))) = 𝑔(𝑐A (𝑒(𝜑1 ), . . . , 𝑒(𝜑 𝑛 ))) = 𝑔(𝑒(𝑐(𝜑1 , . . . , 𝜑 𝑛 ))). From 𝑔[𝑒[Γ]] = 𝑓 [Γ] ⊆ 𝑔[𝐹], we get 𝑒[Γ] ⊆ 𝐹 (due to the additional assumption of 𝑔). Thus, 𝑒(𝜓) ∈ 𝐹 and so 𝑓 (𝜓) = 𝑔(𝑒(𝜓)) ∈ 𝑔[𝐹].  The next proposition tells us that the set of logical filters of a given logic over a given algebra is a closure system (and, thus, the domain of a complete lattice) and its associated closure operator can be internally described by a generalization of the notion of proof introduced in Definition 2.2.5. To this end, for each axiomatic system AS and algebra A, we define the set 𝑉AA S = {h𝑒[Γ], 𝑒(𝜓)i | 𝑒 is an A-evaluation and Γ I 𝜓 ∈ AS}. Note that, if A = FmL , then indeed 𝑉AA S = AS (because FmL -evaluations are L-substitutions and AS is closed under arbitrary substitutions). Proposition 2.5.8 Let L be a logic in a language L and A an L-algebra. Then, FiL (A) is a closure system and the domain of a complete lattice FiAL = hFiL (A), ∧, ∨, FiAL (∅), 𝐴i, where FiAL is the closure operator associated to FiL (A), 𝐹 ∧ 𝐺 = 𝐹 ∩ 𝐺, and 𝐹 ∨ 𝐺 = FiAL (𝐹, 𝐺).13 Furthermore, for any presentation AS of L and each set 𝑋 ∪ {𝑎} ⊆ 𝐴, we have that 𝑎 ∈ FiAL (𝑋) iff there is a tree with no infinite branches (called a proof of 𝑎 from 𝑋 in A) labeled by elements of 𝐴 such that for each node 𝑛 and its label 𝑥 we have: • If 𝑛 is a root, then 𝑥 = 𝑎. • If 𝑛 is a leaf, then h∅, 𝑥i ∈ 𝑉AA S or 𝑥 ∈ 𝑋. • If 𝑛 is not a leaf, and 𝑍 is the set of labels of its predecessors, then h𝑍, 𝑥i ∈ 𝑉AA S . 13 Analogously to the case of generated theories, given 𝑋 ∪𝑌 ∪ {𝑥 } ⊆ 𝐴, we will write ‘FiAL (𝑋 , 𝑌 )’ instead of ‘FiAL (𝑋 ∪ 𝑌 )’, and ‘FiAL (𝑋 , 𝑥)’ instead of ‘FiAL (𝑋 ∪ {𝑥 })’.

2.5 Logical matrices and semantical consequence

47

Proof Let 𝐷 (𝑋) be the set of elements of 𝐴 for which there exists a proof from 𝑋. We can easily show that AS ⊆  hA,𝐷 (𝑋 ) i . Indeed, consider Γ I 𝜑 ∈ AS and an A-evaluation ℎ such that ℎ[Γ] ⊆ 𝐷 (𝑋). Then, for each 𝑥 ∈ ℎ[Γ], there is a proof from 𝑋 and, since hℎ[Γ], ℎ[𝜑]i ∈ 𝑉AA S , we can connect these proofs so that they form a proof of ℎ(𝜑). Thus, 𝐷 (𝑋) ∈ FiL (A) and, since 𝑋 ⊆ 𝐷 (𝑋), we obtain FiAL (𝑋) ⊆ 𝐷 (𝑋). To prove the converse direction, consider 𝑥 ∈ 𝐷 (𝑋) and some proof of 𝑥 from 𝑋 and notice that, for each 𝑦 appearing in this proof, we can prove inductively that 𝑦 ∈ FiAL (𝑋) (because Fi(𝑋) is closed under all the rules of L, in particular those in AS).  Let us now describe models of particular logics and logical filters on particular algebras. Observe that for any L-algebra A, we have FiMin (A) = P ( 𝐴), FiAInc (A) = { 𝐴, ∅}, and FiInc (A) = { 𝐴}, thus every matrix is a model of Min and only matrices with total (or empty) filters are models of the (almost) inconsistent logic. The next example shows that, when we focus on particular logics, whose intended algebras (this notion will be defined properly in Definition 2.9.1) have a lattice reduct, logical filters on these algebras are actually (special kinds of) lattice filters. See the Appendix for the definitions of the algebras; note that Boolean and MV-algebras are defined there (following the literature) in algebraic languages other than LCL ; however, in the Appendix we also show how the operations corresponding to connectives of LCL can be defined in those algebraic languages (and vice versa in the case of MV-algebras). Of course, logical filters need not be always lattice filters; see Example 2.9.13 for a characterization of logical filters on intended algebras of logics in the language L→ . Example 2.5.9 The following table collects several characterizations of logical filters on intended algebras of prominent logics. Logic L CL IL G Ł 𝑙K K

A class of algebras K Boolean algebras Heyting algebras Gödel algebras MV-algebras modal algebras modal algebras

L-filters on the algebras of K lattice filters lattice filters lattice filters lattice filters 𝐹 s.t. 𝑎, 𝑎 → 𝑏 ∈ 𝐹 implies 𝑏 ∈ 𝐹 lattice filters lattice filters 𝐹 s.t. 𝑎 ∈ 𝐹 implies 𝑎 ∈ 𝐹

Let us prove it. In the case of IL, assume that A is a Heyting algebra and 𝐹 ∈ FiIL (A). Since 𝑝 → 𝑝 is a theorem of IL (as shown in Example 2.2.4) we have that for any A-evaluation 𝑒, 𝑒( 𝑝 → 𝑝) ∈ 𝐹, and hence 𝑒( 𝑝 → 𝑝) = 𝑒( 𝑝) → 𝑒( 𝑝) = > ∈ 𝐹. If 𝑎 ∈ 𝐹 and 𝑎 ≤ 𝑏, then 𝑎 → 𝑏 = > ∈ 𝐹 and, since 𝑝, 𝑝 → 𝑞 `IL 𝑞 (modus ponens) taking an A-evaluation 𝑒 such that 𝑒( 𝑝) = 𝑎 and 𝑒(𝑞) = 𝑏, we obtain 𝑏 ∈ 𝐹. Using the axiom 𝑝 → (𝑞 → 𝑝 ∧ 𝑞) and modus ponens it is clear that 𝑝, 𝑞 `IL 𝑝 ∧ 𝑞, hence (using again the same evaluation) if 𝑎, 𝑏 ∈ 𝐹 then 𝑎 ∧ 𝑏 ∈ 𝐹.

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2 Weakly implicative logics

Now assume that 𝐹 is a lattice filter on A and take any 𝑎, 𝑏 ∈ 𝐴. We have 𝑎 ∧𝑏 ≤ 𝑎 ∧𝑏 so, by residuation, 𝑎 ≤ 𝑏 → 𝑎 ∧𝑏 and hence 𝑎 → (𝑏 → 𝑎 ∧𝑏) = > ∈ 𝐹, which means that  hA,𝐹 i 𝜑 → (𝜓 → 𝜑 ∧ 𝜓). For all the remaining axioms one can show, using properties of Heyting algebras, that their value under any evaluation is always equal to >. Next, we observe that 𝜑, 𝜑 → 𝜓  hA,𝐹 i 𝜓: indeed, if 𝑎, 𝑎 → 𝑏 ∈ 𝐹, then using 𝑎 ∧ (𝑎 → 𝑏) ≤ 𝑏, we also have 𝑏 ∈ 𝐹. Therefore, using Proposition 2.5.6 we obtain 𝐹 ∈ FiIL (A). In the case of Boolean and G-algebras, the implications from left to right are already subsumed in the previous proof (because these algebras are Heyting algebras and obviously FiCL (A) ⊆ FiIL (A) and FiG (A) ⊆ FiIL (A)). To prove the converse ones, assume that 𝐹 is a lattice filter on A; thus 𝐹 ∈ FiIL (A) by the first claim (again thanks to the fact that Boolean and G-algebras are also Heyting algebras). To finish the proof, observe the following: • In any Boolean algebra A, for any given 𝑎, 𝑏 ∈ 𝐴, we have: (𝑎 → 𝑏) → 𝑎 = (𝑎 ∧ ¬𝑏) ∨ 𝑎 ≤ 𝑎 ∨ 𝑎 = 𝑎, so ((𝑎 → 𝑏) → 𝑎) → 𝑎 = > and hence  hA,𝐹 i (p) and so 𝐹 ∈ FiCL (A). • Recall that the axiom (lin) in the definition of the Gödel–Dummett logic G can be replaced by the axiom (p∨ ) (see Example 2.4.7) and that in any G-algebra A, for any given 𝑎, 𝑏 ∈ 𝐴, we have (𝑎 → 𝑏) ∨ (𝑏 → 𝑎) = 1. Thus,  hA,𝐹 i (p∨ ) and so 𝐹 ∈ FiG (A). Finally, we deal with the modal logics K and 𝑙K. The left-to-right direction is easy: in both cases we know that 𝐹 ∈ FiCL (A) (as these logics expand CL) and in the global case we also know that 𝑝  hA,𝐹 i 𝑝, thus the claim follows. The converse direction is more complex: • The global case: directly from the assumption we know that 𝜑  hA,𝐹 i 𝜑. From the case for classical logic we also know that 𝜑, 𝜑 → 𝜓  hA,𝐹 i 𝜓 and that  hA,𝐹 i 𝜑 for each axiom 𝜑 of classical logic. Therefore, what remains to prove is  hA,𝐹 i (d ). Let us first observe that the operation  is monotonic: indeed, if 𝑎 = 𝑎 ∧ 𝑏 and to 𝑎 = (𝑎 ∧ 𝑏) = 𝑎 ∧ 𝑏. Next, recall that in any Boolean algebra we have (𝑎 → 𝑏) ∧ 𝑎 ≤ 𝑏 and so (𝑎 → 𝑏) ∧ 𝑎 ≤ 𝑏, which implies (𝑎 → 𝑏) → (𝑎 → 𝑏) = > ∈ 𝐹. • The local case: from the classical case we know that 𝜑, 𝜑 → 𝜓  hA,𝐹 i 𝜓. Therefore, we only need to show that  hA,𝐹 i 𝜑 for each theorem 𝜑 of K. Notice that from the previous case we know that {>} ∈ FiK (A) (because > = >). Thus, for each A-evaluation 𝑒, we have 𝑒(𝜑) = > ∈ 𝐹. The case of Łukasiewicz logic is left as an exercise for the reader. The following example illustrates the fact that the notion of model introduced so far is still very weak. Indeed, it yields models over algebras that are not intended for a particular logic: for example, one can have models for classical logic over a Heyting algebra which is not a Boolean algebra.

2.6 The first completeness theorem

49

Example 2.5.10 Let A be a Heyting algebra and 𝐹 ⊆ 𝐴. Then hA, 𝐹i ∈ Mod(CL) iff 𝐹 is a lattice filter and for each 𝑎 ∈ 𝐴 we have 𝑎 ∨ ¬𝑎 ∈ 𝐹. Thus, in particular, we immediately obtain that h[0, 1] G , (0, 1]i ∈ Mod(CL), where [0, 1] G is the standard G-algebra (see Example A.5.7), which is clearly not a Boolean algebra. To prove the claim first notice that from Example 2.4.8 follows that CL is axiomatized relative to IL using the law of excluded middle 𝜑 ∨ ¬𝜑. Therefore, due to Proposition 2.5.6, we have that hA, 𝐹i ∈ Mod(CL) iff hA, 𝐹i ∈ Mod(IL) and for each A-evaluation 𝑒 we have 𝑒(𝜑 ∨ ¬𝜑) ∈ 𝐹. The rest follows from the previous example.

2.6 The first completeness theorem We start this section by characterizing the Lindenbaum matrices of a given logic by showing that the filters on the absolutely free algebra coincide with the theories of the logic and show that this is all we need to know in order to obtain a first completeness theorem for any logic. Proposition 2.6.1 For any logic L in a language L, we have FiL (FmL ) = Th(L). Proof Let Γ ∈ FiL (FmL ), i.e. if Δ `L 𝜑, then for each FmL -evaluation 𝑒 we have 𝑒(𝜑) ∈ Γ whenever 𝑒[Δ] ⊆ Γ. Therefore, in the particular case where Δ = Γ and the evaluation 𝑒 is the identity, from Γ `L 𝜑 we obtain 𝜑 = 𝑒(𝜑) ∈ Γ, i.e. Γ ∈ Th(L). Next assume that 𝑇 ∈ Th(L), Δ `L 𝜑, and 𝑒 is an FmL -evaluation such that 𝑒[Δ] ⊆ 𝑇. Thus, by Reflexivity, 𝑇 `L 𝑒[Δ]. Since, by Structurality, we have 𝑒[Δ] `L 𝑒(𝜑), we obtain 𝑇 `L 𝑒(𝜑) by Cut. Since 𝑇 is a theory, we have 𝑒(𝜑) ∈ 𝑇. As we will see in several instances in this book, many properties of a logic can be expressed as properties of its lattice of theories. Since, as we have just proved, theories coincide with logical filters over the algebra of formulas, it makes sense to ask whether those properties can be transferred from the lattice of theories to lattices of filters over arbitrary algebras. As an example, we are ready to prove the first of these transfer theorems: Theorem 2.6.2 (Transfer of finitarity) Given a logic L in a language L, the following are equivalent: 1. L is finitary. 2. FiAL is a finitary closure operator for any L-algebra A. 3. FiL (A) is an inductive closure system for any L-algebra A. Proof The equivalence of the last two claims is well known, see e.g. Theorem A.1.18. It is clear that 2 implies 1 by taking A = FmL (cf. Proposition 2.1.9). Let us see that 1 Ð implies 3. Take an upwards directed set F ⊆ FiL (A) and define 𝐹 = F . We need to show that 𝐹 ∈ FiL (A). Assume that Γ `L 𝜑 and 𝑒 is an A-evaluation such that

50

2 Weakly implicative logics

𝑒[Γ] ⊆ 𝐹. Since L is finitary, there is a finite set Γ0 ⊆ Γ such that Γ0 `L 𝜑. Then, since F is upwards directed, there has to be an 𝐹0 ∈ F such that 𝑒[Γ0 ] ⊆ 𝐹0 and so 𝑒(𝜑) ∈ 𝐹0 ⊆ 𝐹.  The description of theories as logical filters also gives us the first completeness theorem of the book: for each logic the class of all its models is one of its possible complete semantics. Theorem 2.6.3 (Completeness w.r.t. all models) Let L be a logic. Then, L = Mod(L) , i.e. for each set Γ of formulas and each formula 𝜑, Γ `L 𝜑

iff

Γ Mod(L) 𝜑.

Proof The implication from left to right is obvious. For the reverse direction assume that Γ Mod(L) 𝜑. Since hFmL , ThL (Γ)i ∈ Mod(L) and the identity 𝑖 is an FmL evaluation such that 𝑖[Γ] ⊆ ThL (Γ) we have 𝜑 = 𝑖(𝜑) ∈ ThL (Γ), i.e. Γ `L 𝜑.  Note that we have actually proved a stronger claim: the completeness w.r.t. Lindenbaum matrices; formally speaking Γ `L 𝜑

iff

Γ  { hFmL ,𝑇 i | 𝑇 ∈ Th(L) } 𝜑.

Part of the appeal of such completeness result is, arguably, its absolute generality and its immediate proof from a few elementary notions of the general theory. Also, the liberal definition of the used semantics (which accepts all models) makes it very suitable for its primary function: to establish that certain consecutions are not derivable. Indeed, one has the whole class of models as the search space for finding counterexamples. On the other hand, the completeness w.r.t. Lindenbaum matrices can be seen as a useful cardinality restriction that ensures that counterexamples can always be found inside the bound given by the cardinality of the set of formulas. Nevertheless, the broadness of the notion of model (even if we consider only Lindenbaum matrices) poses also a serious problem for this completeness result. Such a semantics does not restrict at all the class of algebras one can use, which undermines its meaningfulness for the logic in question. Indeed, the trivial matrix over any algebra is a model of any logic and, furthermore, any logic (other than Inc) has non-trivial Lindenbaum models over the absolutely free algebra, which depends only on the language and bears no other relation whatsoever to the logic (recall also Example 2.5.10). In the next section, we will remedy this drawback by defining, for a given logic L, the class of its reduced models. These models still provide a complete semantics of L with the advantage that their algebraic reducts are more tightly related to the logic L (see Section 2.9 for more details). But, before that, let us introduce several particular models that will be important throughout the book. First, we use them to prove, by means of the first completeness theorem, that all the logics we have presented in the previous section are pairwise different. We also use them to show that the logic Ł∞ introduced in Example 2.3.9 is not finitary, that the intersections of infinitely many finitary logics need not be finitary, and that global modal logics do not enjoy the classical proof by cases property.

2.6 The first completeness theorem

51

Example 2.6.4 The logics depicted in Figure 2.1 and their expansions to the language LCL introduced in Example 2.3.7 are pairwise different and CL is not an expansion of A→ . The last claim follows from 2 h2, {1}i (abe) and it is easy to see that, to prove the rest, it suffices to show the following five claims: • The formula (w) is not a theorem of the logic A→ : Consider the L→ -algebra Z over the domain consisting of integers where 𝑎 →Z 𝑏 = 𝑏 − 𝑎. It is easy to check that hZ, {0}i ∈ Mod(A→ ). We show that 2 hZ, {0}i (w): consider the Z-evaluation 𝑒( 𝑝) = 0 and 𝑒(𝑞) = 1 and compute 𝑒( 𝑝 → (𝑞 → 𝑝)) = 0 → (1 → 0) = 0 → −1 = −1. • The formula (w) is not a theorem of the logic BCIlat : Take the LCL -algebra A over the three-element domain {>, 𝑡, ⊥} where ⊥A = ⊥ and the operations are →A ⊥ 𝑡 >

⊥ > ⊥ ⊥

𝑡 > 𝑡 ⊥

∧A ⊥ 𝑡 >

> > > >

⊥ ⊥ ⊥ ⊥

𝑡 ⊥ 𝑡 𝑡

> ⊥ 𝑡 >

∨A ⊥ 𝑡 >

⊥ ⊥ 𝑡 >

𝑡 𝑡 𝑡 >

> > > >

Note that ∧A and ∨A are, respectively, the inf and sup operations for the order ⊥ < 𝑡 < >. It is easy to check that hA, {𝑡, >}i ∈ Mod(BCIlat ). We show that 2 hA, {𝑡 ,>}i (w) by considering the A-evaluation 𝑒( 𝑝) = 𝑡 and 𝑒(𝑞) = >: 𝑒( 𝑝 → (𝑞 → 𝑝)) = 𝑡 → (> → 𝑡) = 𝑡 → ⊥ = ⊥. • The formula (c) is not a theorem of the Łukasiewicz logic Ł: From Example 2.5.9 we know that Ł∞ = h[0, 1] Ł , {1}i ∈ Mod(Ł) (the MV-algebra [0, 1] Ł is defined in Example A.5.10); then, for the evaluation 𝑒(𝑞) = 0 and 𝑒( 𝑝) = 0.5, 𝑒(( 𝑝 → ( 𝑝 → 𝑞)) → ( 𝑝 → 𝑞)) = (0.5 → 0.5) → 0.5 = 1 → 0.5 = 0.5 ≠ 1. • The formula (lin) is not a theorem of intuitionistic logic: Consider the LCL -algebra B over the domain {⊥, 𝑎, 𝑏, 𝑐, >}, where the lattice operations are induced by the order ⊥ < 𝑎, 𝑏 < 𝑐 < >, 𝑎 and 𝑏 are incomparable, and →B is defined as →B ⊥ 𝑎 𝑏 𝑐 >

⊥ > 𝑏 𝑎 ⊥ ⊥

𝑎 𝑏 𝑐 > > > > > > 𝑏 > > 𝑎 > > > 𝑎 𝑏 > > 𝑎 𝑏 𝑐 >

Clearly, B is a Heyting algebra and so, due to Example 2.5.9, hB, {>}i ∈ Mod(IL). We show that 2 hB, {>}i (lin) using the evaluation 𝑒( 𝑝) = 𝑎, 𝑒(𝑞) = 𝑏, and 𝑒(𝑟) = 𝑐: 𝑒((( 𝑝 → 𝑞) → 𝑟) → (((𝑞 → 𝑝) → 𝑟) → 𝑟)) = > → (> → 𝑐) = 𝑐 ≠ >.

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2 Weakly implicative logics

• The formula (waj) is not a theorem of the logic G: Recall the algebra [0, 1] G (Example A.5.7). By Example 2.5.9, we have G∞ = h[0, 1] G , {1}i ∈ Mod(G); take the evaluation 𝑒(𝑞) = 0 and 𝑒( 𝑝) = 0.5 and compute 𝑒((( 𝑝 → 𝑞) → 𝑞) → ((𝑞 → 𝑝) → 𝑝)) = 1 → 0.5 = 0.5 ≠ 1. Recall that in Section 2.3 we have shown that the extension of BCIlat by the axiom (adj) proves (w); thus the previous example shows that (adj) is not derivable in BCIlat . This, however, does not directly entail that BCIlat is not an axiomatic expansion of BCI, as in principle a more complex axiom(s) could be used instead of the rule (Adj); the next example shows that it is not the case. Example 2.6.5 BCIlat is not an axiomatic expansion of BCI. Indeed, if it were the case, then, thanks to the local deduction theorem 2.4.3, we would know that there is a natural number 𝑘 such that 𝑞 `BCIlat 𝑝 → ( 𝑝 → . . . ( 𝑝 → 𝑝 ∧ 𝑞) . . .). | {z } 𝑘 times

We use the matrix hA, {𝑡, >}i from the previous example to demonstrate that no such 𝑘 exists; indeed, for 𝑘 = 0 consider the A-evaluation 𝑒(𝑞) = 𝑡 and 𝑒( 𝑝) = ⊥, and for the other cases take 𝑒(𝑞) = 𝑡 and 𝑒( 𝑝) = >. The next two examples show three extensions of Łukasiewicz logic that are actually infinitary: the first is the logic Ł∞ introduced in Example 2.3.9, the second is the logic Ł∞ (later in Example 3.5.24 we will show that they are actually the same logic), and the third is semantically defined as the logic of all finite MV-chains. The second example also demonstrates that the intersection of infinitely many finitary logics need not be finitary. Example 2.6.6 Recall the logic Ł∞ introduced in Example 2.3.9 and its infinitary inference rule (Ł∞ )

{¬𝜑 → 𝜑 & . 𝑛. . & 𝜑 | 𝑛 ≥ 1} I 𝜑

Hay rule.

We can now show that Ł∞ is indeed infinitary, and hence, it is a proper extension of Łukasiewicz logic. One can check that, using the operations defined in Proposition A.5.9, Ł∞ ∈ Mod(Ł∞ ). However, for each positive 𝑘 ∈ N, {¬𝜑 → 𝜑 𝑛 | 1 ≤ 𝑛 < 𝑘 } 2Ł∞ 𝜑, where by 𝜑 𝑛 we denote 𝜑 & . 𝑛. . & 𝜑. Indeed, it suffices to take the evaluation 𝑘 1 𝑒(𝜑) = 𝑘+1 and note that 𝑒(𝜑) 𝑛 = 𝑘−𝑛 𝑘+1 ≥ 𝑘+1 = 𝑒(¬𝜑) for 𝑛 < 𝑘. Observe that we have also shown that the semantically given logic Ł∞ is infinitary.

2.6 The first completeness theorem

53

We can also produce an example of a model of Ł in which the rule (Ł∞ ) fails (and so it is not a model of Ł∞ ), thus directly demonstrating the already established fact, that the logics Ł and Ł∞ are different. Consider the MV-algebra C = h𝐶, ⊕C , ¬C , 0¯ C i (which we know can be seen as an LCL -algebra) with the domain 𝐶 = {h0, 𝑖i | 𝑖 ∈ N} ∪ {h1, −𝑖i | 𝑖 ∈ N} and the operations defined as: 0¯ C = h0, 0i ¬C h𝑥, 𝑖i = h1 − 𝑥, −𝑖i   h1, 0i     C h𝑥, 𝑖i ⊕ h𝑦, 𝑗i = h1, 0i     h𝑥 + 𝑦, 𝑖 + 𝑗i 

if 𝑥 + 𝑦 = 2 if 𝑥 + 𝑦 = 1 and 𝑖 + 𝑗 ≥ 0 otherwise.

It is easy to see that C is an MV-algebra, called the Chang algebra, and that   h0, 0i     C h𝑥, 𝑖i & h𝑦, 𝑗i = h0, 0i     h𝑥 + 𝑦 − 1, 𝑖 + 𝑗i 

if 𝑥 + 𝑦 = 0 if 𝑥 + 𝑦 = 1 and 𝑖 + 𝑗 ≤ 0 otherwise.

Using the characterization in Example 2.5.9, we obtain that {h1, 0i} ∈ FiŁ (C). Now we consider the matrix C = hC, {h1, 0i}i and show that {¬𝜑 → 𝜑 𝑛 | 𝑛 ≥ 1} 2C 𝜑. Indeed, we could just take the evaluation 𝑒(𝜑) = h1, −1i, and compute by induction that h1, −1i 𝑛 = h1, −𝑛i and so 𝑒(¬𝜑 → 𝜑 𝑛 ) = h1, −1i ⊕ h1, −𝑛i = h1, 0i. Example 2.6.7 Recall that, for each natural 𝑛 ≥ 2, we denote by MV𝑛 the subalgebra 1 of [0, 1] Ł with the 𝑛-element domain {0, 𝑛−1 , . . . , 1} and consider the matrix Ł𝑛 = hMV𝑛 , {1}i. Thanks to Proposition 2.5.4, we know that Ł𝑛 is a finitary logic and, using claim 1 of Lemma 2.5.7 with 𝑔 being the identity mapping, we know that Ł𝑛 ∈ Mod(Ł∞ ). Thus, we have Ł∞ ⊆  {Ł𝑛 | 𝑛 ≥2} . We show that the right-hand-side logic is strictly stronger than Ł∞ and that it is not finitary. Consider the rule 0 (Ł∞ )

{( 𝑝 𝑖 → 𝑝 𝑖+1 ) 𝑖 (𝑖+1) → 𝑞 | 𝑖 > 0} I 𝑞.

First, we show that this rule is valid in Ł𝑛 . Recall that, for any Ł∞ -evaluation (and, thus, in particular, for any Ł𝑛 -evaluation), we have 𝑒(𝜑 𝑛 ) = max{0, 𝑛𝑒(𝜑) − 𝑛 + 1} and hence: 𝑒(𝜑 𝑛 ) = 1 iff 𝑒(𝜑) = 1. Thus, if there was an Ł𝑛 -evaluation 𝑒 such that 𝑒(𝑞) < 1 and 𝑒(( 𝑝 𝑖 → 𝑝 𝑖+1 ) 𝑖 (𝑖+1) → 𝑞) = 1 for each 𝑖, we would know that, for each 𝑖, we have 𝑒( 𝑝 𝑖 → 𝑝 𝑖+1 ) < 1, i.e. 𝑒( 𝑝 𝑖 ) > 𝑒( 𝑝 𝑖+1 ). Thus, there would be an infinite decreasing chain in MV𝑛 , a contradiction. On the other hand, we can use the Ł∞ -evaluation 𝑒(𝑞) = 0 and 𝑒( 𝑝 𝑖 ) = 1𝑖 to show 0 that (Ł∞ ) is not derivable in Ł∞ . Indeed, we can easily compute

54

2 Weakly implicative logics

𝑒(( 𝑝 𝑖 → 𝑝 𝑖+1 ) 𝑖 (𝑖+1) ) = max{0, 𝑖(𝑖 + 1)(1 −

1 1 + ) − 𝑖(𝑖 + 1) + 1} = 0. 𝑖 𝑖+1

Finally, to prove that the logic  {Ł𝑛 | 𝑛 ≥2} is not finitary, it suffices to note that the Ł∞ -evaluation 𝑒 we used above is also an Ł 𝑛+1 -evaluation and so {( 𝑝 𝑖 → 𝑝 𝑖+1 ) 𝑖 (𝑖+1) → 𝑞 | 𝑛 ≥ 𝑖 > 0} 2 {Ł𝑛 | 𝑛 ≥2} 𝑞.

Next, we prove the well-known fact that the modal logics introduced in Example 2.3.10 are pairwise different (in particular, the global and local variants differ) and in none of them 𝜑 → 𝜑 is a theorem (which, as we argued before, implies that global modal logics do not enjoy the classical proof by cases property). Example 2.6.8 Consider the Boolean algebra 4 on the domain {⊥, 𝑎, 𝑏, >} and expand it to 4 0 with an operator  defined as 𝑥 = > if 𝑥 = > and ⊥ otherwise. It is easy to see that the resulting algebra is a modal algebra. The lattice filter 𝐹 = {𝑎, >} is a logical filter for 𝑙K, but not for K (see Example 2.5.9). Since for each element 𝑥 in the algebra 𝑥 → 𝑥 = > and 𝑥 → 𝑥 = >, we have that {>} is a logical filter for S4. Therefore, h4 0, 𝐹i is a model of 𝑙S4 but not a model of K. Thus: 𝑙K ≠ K, 𝑙T ≠ T, 𝑙K4 ≠ K4, and 𝑙S4 ≠ S4. Also, observe that 𝑎 → 𝑎 = 𝑎 → ⊥ = 𝑏 ∉ {>} which implies that 𝜑 → 𝜑 is not a theorem of S4, and hence it cannot be a theorem of any of the defined (neither global nor local) modal logics. In order to show the remaining differences, it is enough to find a matrix on a modal algebra which verifies (t ) but not (4 ), and another one with the symmetric property. We present an 8-valued modal algebra A as a candidate for the former and leave elaboration of the details and finding the latter as an exercise for the reader (in the latter case an algebra over 4 can be found). The algebra A is built over the Boolean powerset algebra of the set {𝑎, 𝑏, 𝑐} with A given by the table: 𝑥 A 𝑥

∅ {𝑎} {𝑏} {𝑐} {𝑎, 𝑏} {𝑎, 𝑐} {𝑏, 𝑐} {𝑎, 𝑏, 𝑐} ∅ ∅ ∅ {𝑐} {𝑎} {𝑐} {𝑏, 𝑐} {𝑎, 𝑏, 𝑐}

2.7 Leibniz congruence and a second completeness theorem The goal of this section is to obtain a finer complete semantics for all logics. We have seen that the class of all models of a given logic is really wide, which is good for finding counterexamples to the validity of particular consecutions, but has some important drawbacks. First, it makes it a much harder task to establish the derivability of certain rules semantically (by proving them for all matrices and all evaluations); second, it is not suitable for most of the metamathematical results of the general theory we are going to build in this book.

2.7 Leibniz congruence

55

This section is devoted to a particular way of addressing this problem that will prove to be very fruitful. Imagine that we are trying to show that Γ 0L 𝜑 for a given logic L and a given set Γ ∪ {𝜑} of formulas. The first completeness theorem ensures that, if the consecution is indeed not valid in L, then this will be witnessed by some matrix A = hA, 𝐹i ∈ Mod(L). However, there is no guarantee that this model will be optimal in any way. Typically, we may pick an unnecessarily big matrix (e.g. the Lindenbaum matrix) in the sense that it is likely to contain many redundant elements that are indistinguishable from the logical point of view using the properties expressible with the apparatus at our disposal in A. More precisely, there might be many pairs of elements 𝑎, 𝑏 such that, for each formula 𝜒, each propositional variable 𝑝, and each A-evaluation 𝑒, we have: 𝑒 𝑝=𝑎 ( 𝜒) ∈ 𝐹

iff

𝑒 𝑝=𝑏 ( 𝜒) ∈ 𝐹,

i.e. 𝑎 and 𝑏 are indiscernible from the point of view of the matrix because they satisfy exactly the same properties expressible in the language. It is not hard to show that this binary relation of indistinguishability is actually a congruence on A (see Theorem 2.7.1). Alluding to Leibniz’s principle of identity of indiscernibles,14 this equivalence relation has been aptly called the Leibniz congruence of a matrix A = hA, 𝐹i (denoted by Ω(A) or ΩA (𝐹)15), because it identifies indiscernible elements. Recall that congruences provide a natural way of producing smaller algebras (and, hence, smaller matrices) by identifying the congruent elements. We only have to make sure that the resulting matrix is a model of L that allows to obtain a counterexample to Γ `L 𝜑. Algebraically speaking, we construct the quotient algebra of A (see Definition A.3.10), which can be seen as its image under the canonical surjective mapping which assigns to each element the set of its indistinguishable elements. Recall that in Lemma 2.5.7 we have proved that 𝑔(A) = h𝑔[A], 𝑔[𝐹]i ∈ Mod(L) for any surjective homomorphism 𝑔 such that 𝑔(𝑥) ∈ 𝑔[𝐹] implies 𝑥 ∈ 𝐹. Furthermore, it is easy to see that, for any A-evaluation 𝑒 providing a counterexample to Γ `L 𝜑, the mapping 𝑔 ◦ 𝑒 is a 𝑔(A)-evaluation doing the same job (for details see Lemma 2.7.5). Now we only have to show that the additional condition required by Lemma 2.5.7 is satisfied by the canonical mapping given by the Leibniz congruence Ω(A). Actually, it is easy to see that it is true for an arbitrary congruence 𝜃 on A iff we have 𝑏 ∈ 𝐹 whenever h𝑎, 𝑏i ∈ 𝜃 and 𝑎 ∈ 𝐹. In this case, we will call 𝜃 a logical congruence on the matrix A = hA, 𝐹i (alternatively, we say that 𝜃 is compatible with 𝐹). We show not only that the Leibniz congruence is a logical congruence on A (and so we can use it to produce smaller counterexamples as indicated above), but that it is actually the largest logical congruence. This will allow us to show that the Leibniz congruence of the resulting counterexample is the identity, i.e. the counterexample is already as small as possible and cannot be further improved by the same technique. 14 The identity of indiscernibles is a metaphysical principle that establishes that there cannot exist two separate objects with exactly the same properties. 15 The reason for the notation ΩA (𝐹 ) is that in Section 3.4 we introduce the function ΩA , known as the Leibniz operator, that assigns to each filter 𝐹 on A the Leibniz congruence of the matrix hA, 𝐹 i.

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2 Weakly implicative logics

Theorem 2.7.1 Let A = hA, 𝐹i be a matrix. The relation Ω(A) defined as h𝑎, 𝑏i ∈ Ω(A)

if and only if

for each formula 𝜒 and each A-evaluation 𝑒 it is the case that 𝑒 𝑝=𝑎 ( 𝜒) ∈ 𝐹 iff 𝑒 𝑝=𝑏 ( 𝜒) ∈ 𝐹

is the largest logical congruence on A. Proof The relation Ω(A) is clearly an equivalence. We prove the remaining claims: • To show that Ω(A) is a congruence consider, without loss of generality, a binary connective ◦, elements 𝑎, 𝑏, 𝑐 ∈ 𝐴 such that h𝑎, 𝑏i ∈ Ω(A), a formula 𝜒, and an A-evaluation 𝑒. Let 𝑞 be a variable different from 𝑝 and not occurring in 𝜒 and 𝜒 0 be the formula resulting from 𝜒 by substituting 𝑝 by 𝑝 ◦ 𝑞. Observe that 𝑒 𝑝=𝑎◦A 𝑐 ( 𝜒) = (𝑒 𝑞=𝑐 ) 𝑝=𝑎 ( 𝜒 0) and, thus, 𝑒 𝑝=𝑎◦A 𝑐 ( 𝜒) ∈ 𝐹 iff (𝑒 𝑞=𝑐 ) 𝑝=𝑎 ( 𝜒 0) ∈ 𝐹 iff (𝑒 𝑞=𝑐 ) 𝑝=𝑏 ( 𝜒 0) ∈ 𝐹 iff 𝑒 𝑝=𝑏◦A 𝑐 ( 𝜒) ∈ 𝐹 (the central equivalence follows from the assumption that h𝑎, 𝑏i ∈ Ω(A)). • To show that Ω(A) is a logical congruence, it suffices to consider its definition for 𝜒 = 𝑝. • In order to show that Ω(A) is the largest logical congruence, consider an arbitrary logical congruence 𝜃 on A and h𝑎, 𝑏i ∈ 𝜃. Thus, for each formula 𝜒 and each A-evaluation 𝑒, we have h𝑒 𝑝=𝑎 ( 𝜒), 𝑒 𝑝=𝑏 ( 𝜒)i ∈ 𝜃. Hence, by compatibility, 𝑒 𝑝=𝑎 ( 𝜒) ∈ 𝐹 iff 𝑒 𝑝=𝑏 ( 𝜒) ∈ 𝐹.  Let us define the class of matrices that remain unchanged when trying to produce smaller models using Leibniz congruences, namely matrices whose Leibniz congruence is the identity and hence logical indiscernibility implies equality. Definition 2.7.2 (Reduced matrix) A matrix A is said to be reduced if Ω(A) is the identity relation IdA . The class of all the reduced models of a logic L is denoted by Mod∗ (L). Observe that, for each algebra A, we have Ω(hA, ∅i) = Ω(hA, 𝐴i) = 𝐴2 . Thus, a trivial L-matrix (i.e. a matrix where 𝐹 = 𝐴) is reduced iff A is the trivial (one element) L-algebra TrL ; we will call it the trivial reduced matrix. Analogously hA, ∅i is reduced iff A = TrL . The situation becomes, of course, more interesting in non-trivial matrices. Next, we give examples of reduced matrices over Heyting, modal, or MV-algebras. Later, in Proposition 2.8.20, we will see that there are no other reduced matrices over these algebras. Example 2.7.3 The matrix hA, {>A }i is reduced, for A being a Heyting, modal or, MV-algebra. We prove it by showing that the identity is the only logical congruence on hA, {>}i. Indeed, take any logical congruence 𝜃 and elements 𝑎, 𝑏 ∈ 𝐴 such that h𝑎, 𝑏i ∈ 𝜃. Since 𝜃 is a congruence, we obtain h𝑎 → 𝑎, 𝑎 → 𝑏i ∈ 𝜃 and h𝑏 → 𝑎, 𝑏 → 𝑏i ∈ 𝜃 and, since 𝑎 → 𝑎 = 𝑏 → 𝑏 = > and 𝜃 is logical, we obtain 𝑎 → 𝑏 = 𝑏 → 𝑎 = > which is known (see the Appendix) in these algebras to imply that 𝑎 = 𝑏.

2.7 Leibniz congruence

57

The reader might be tempted to think that, for each logic L and each algebra A, there is at most one 𝐹 such that hA, 𝐹i ∈ Mod∗ (L) or, even, that 𝐹 has to be always a singleton. The almost inconsistent logic in a language L and the trivial algebra TrL give counterexamples to both of these claims; as an exercise the reader can check that the minimum logic and any simple algebra (see the Appendix) with at least three elements also provide counterexamples. Let us present another example in a prominent logic (see yet another one in Example 2.8.10). Example 2.7.4 In the case of the logic BCI, we define an algebra M over the finite domain {⊥, 1, 𝑎, >} whose only operation is given by the following table: →M ⊥ 1 𝑎 >

⊥ > ⊥ ⊥ ⊥

1 > 1 ⊥ ⊥

𝑎 > 𝑎 1 ⊥

> > > > >

It is a simple exercise to check that we have FiBCI (M) = {{1, >}, {𝑎, 1, >}, 𝑀 } and ΩM ({1, >}) = ΩM ({𝑎, 1, >}) = IdM , i.e. we have two different reduced models over the same algebra, given by non-singleton filters. The upcoming lemma formalizes the informal idea of producing smaller models for a logic L by means of logical congruences. It shows that the resulting matrix is indeed a model of L, defines the same semantical consequence (thus, it provides counterexamples for the same derivations), and if the congruence is Leibniz, it is reduced (i.e. cannot be made smaller by repeated use of the technique). The second completeness theorem w.r.t. the class Mod∗ (L) then follows as an immediate corollary. We extend the algebraic quotient construction to matrices. For a congruence 𝜃 we denote by 𝑎/𝜃 the equivalence class of 𝑎 with respect to 𝜃, and by A/𝜃 the quotient algebra with domain 𝐴/𝜃 = {𝑎/𝜃 | 𝑎 ∈ 𝐴} and operations 𝑓 A/𝜃 (𝑎 1 /𝜃, . . . , 𝑎 𝑛 /𝜃) = 𝑓 A (𝑎 1 , . . . , 𝑎 𝑛 )/𝜃. For a matrix A = hA, 𝐹i, we define 𝐹/𝜃 = {𝑎/𝜃 | 𝑎 ∈ 𝐹} and denote the matrix hA/𝜃, 𝐹/𝜃i by A/𝜃. In order to lighten the notation, we write A∗ for A/Ω(A). Lemma 2.7.5 Let A be a matrix and 𝜃 a logical congruence on A. Then, A = A/𝜃 . Furthermore, if A ∈ Mod(L) for a logic L, then A∗ ∈ Mod∗ (L). Proof Recall the canonical surjective mapping 𝑔 : A → A/𝜃 defined as 𝑔(𝑎) = 𝑎/𝜃. We show that 𝑔(𝑎) ∈ 𝑔[𝐹] iff 𝑎 ∈ 𝐹, for every 𝑎 ∈ 𝐴. The converse direction is trivial; we show the direct one: the assumption gives us 𝑎/𝜃 = 𝑏/𝜃 for some 𝑏 ∈ 𝐹. Thus, h𝑎, 𝑏i ∈ 𝜃 and, since 𝜃 is a logical congruence, we obtain 𝑎 ∈ 𝐹.

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2 Weakly implicative logics

Recall also that A ∈ Mod(A ) and so we can use Lemma 2.5.7 for the canonical mapping to obtain that A/𝜃 = h𝑔(A), 𝑔[𝐹]i ∈ Mod(A ), i.e. A ⊆ A/𝜃 . To prove the converse direction, assume that Γ 2A 𝜑 and let 𝑒 be an A-evaluation witnessing this fact. Then, clearly, the A/𝜃-evaluation 𝑒 0 (𝜑) = 𝑔(𝑒(𝜑)) does the same job (because, for any 𝜒, we clearly have 𝑒( 𝜒) ∈ 𝐹 iff 𝑒 0 ( 𝜒) ∈ 𝑔[𝐹] = 𝐹/𝜃). To complete the proof, it suffices to show that A/Ω(A) is reduced. Assume that h𝑎/Ω(A), 𝑏/Ω(A)i ∈ Ω(A∗ ). If we show that also h𝑎, 𝑏i ∈ Ω(A), then we obtain 𝑎/Ω(A) = 𝑏/Ω(A) and the proof is done. Consider a formula 𝜒 and an A-evaluation 𝑒. The following chain of equivalent statements ends the proof: 𝑒 𝑝=𝑎 ( 𝜒) ∈ 𝐹 iff 𝑔(𝑒 𝑝=𝑎 ( 𝜒)) ∈ 𝑔[𝐹] iff (𝑔 ◦ 𝑒) 𝑝=𝑎/Ω(A) ( 𝜒) ∈ 𝑔[𝐹] iff (𝑔 ◦ 𝑒) 𝑝=𝑏/Ω(A) ( 𝜒) ∈ 𝑔[𝐹] iff 𝑔(𝑒 𝑝=𝑏 ( 𝜒)) ∈ 𝑔[𝐹] iff 𝑒 𝑝=𝑏 ( 𝜒) ∈ 𝐹.



In Chapter 3 we will define a notion of isomorphism of matrices under which we obtain the expected result that if a matrix A is reduced, then it is isomorphic to A∗ and thus in particular, for any matrix A, A∗ is isomorphic to (A∗ ) ∗ ; see Corollary 3.2.6. Theorem 2.7.6 (Completeness w.r.t. reduced models) Let L be a logic. Then, for any set Γ ∪ {𝜑} of formulas, Γ `L 𝜑

iff

Γ Mod∗ (L) 𝜑.

The second completeness theorem is a clear improvement of the first one in the sense that the obtained complete matrix semantics is based on a restricted class of algebras (which are quotients by Leibniz congruences). In Section 2.9, we will see that these algebras have a much tighter connection to the logic in question. Recall that in the proof of the first completeness theorem we actually obtained a stronger statement: the completeness w.r.t. Lindenbaum matrices (matrices with algebraic reduct FmL ). Similarly, we can strengthen the second completeness theorem w.r.t. reductions of Lindenbaum matrices by their Leibniz congruences; these reduced models are known as Lindenbaum–Tarski matrices. Formally, we can write it as: Γ `L 𝜑

iff

Γ  { hFmL ,𝑇 i∗ | 𝑇 ∈ Th(L) } 𝜑.

The Lindenbaum–Tarski matrices are traditionally based on algebras of classes of ‘equivalent’ formulas modulo the theory; this equivalence can be given by a single binary connective (e.g. in classical logic), or a pair of implications (e.g. in purely implicational logics such as BCK), or even a rather complex set of (defined) binary connectives (e.g. in the case of certain local modal logics; see Example 2.8.6). To cover even such situations we consider arbitrary sets of formulas in two variables 𝑝 and 𝑞, suggestively denoted as ⇔ (e.g. ⇔ = {𝑝 ↔ 𝑞} or ⇔ = {𝑝 → 𝑞, 𝑞 → 𝑝}), and define 𝜑 ⇔ 𝜓 = {𝛿(𝜑, 𝜓) | 𝛿( 𝑝, 𝑞) ∈ ⇔}.

2.7 Leibniz congruence

59

Using the convention, it is well known that in most of the logics we have seen so far (see the next section for a precise formulation) we have for each 𝑇 ∈ Th(L): 𝜑/ΩFmL (𝑇) = {𝜓 ∈ FmL | 𝜑 ⇔ 𝜓 ⊆ 𝑇 }. The next theorem gives a syntactical characterization of the sets of formulas which can be used to obtain this description of Lindenbaum–Tarski matrices. It can also be seen as another example of a transfer theorem, as we show that a certain property of Lindenbaum matrices can be translated to all matrices. Theorem 2.7.7 Let L be a logic in a language L and ⇔ a set of formulas in two variables. Then, the following are equivalent: 1. The following conditions are satisfied: (R⇔ ) (S⇔ ) (T⇔ ) (MP⇔ ) (Cng⇔ )

`L 𝜑 ⇔ 𝜑 𝜑 ⇔ 𝜓 `L 𝜓 ⇔ 𝜑 𝜑 ⇔ 𝜓, 𝜓 ⇔ 𝜒 `L 𝜑 ⇔ 𝜒 𝜑, 𝜑 ⇔ 𝜓 `L 𝜓 𝜑 ⇔ 𝜓 `L 𝑐( 𝜒1 , . . . , 𝜒𝑖 , 𝜑, . . . , 𝜒𝑛 ) ⇔ 𝑐( 𝜒1 , . . . , 𝜒𝑖 , 𝜓, . . . , 𝜒𝑛 ) for each h𝑐, 𝑛i ∈ L and each 0 ≤ 𝑖 < 𝑛.

2. For each hA, 𝐹i ∈ Mod(L), we have h𝑎, 𝑏i ∈ ΩA (𝐹)

iff

𝑎 ⇔A 𝑏 ⊆ 𝐹.16

3. For each hA, 𝐹i ∈ Mod∗ (L), we have 𝑎=𝑏

iff

𝑎 ⇔A 𝑏 ⊆ 𝐹.

4. For each theory 𝑇 ∈ Th(L), we have h𝜑, 𝜓i ∈ ΩFmL (𝑇)

iff

𝜑 ⇔ 𝜓 ⊆ 𝑇.

In this case, ⇔ is called an equivalence set for L. Proof To prove that 1 implies 2, we define a relation 𝜃 as h𝑎, 𝑏i ∈ 𝜃 iff 𝑎 ⇔A 𝑏 ⊆ 𝐹 and prove that 𝜃 is the largest logical congruence on hA, 𝐹i and so it is the Leibniz congruence thanks to Theorem 2.7.1. Obviously, 𝜃 is symmetric due to (S⇔ ), reflexive due to (R⇔ ), and transitive due to (T⇔ ); therefore, due to (Cng⇔ ) and (MP⇔ ), it is a logical congruence. To see that it is the largest one, assume that 𝜃 0 is an arbitrary logical congruence on hA, 𝐹i and h𝑎, 𝑏i ∈ 𝜃 0. Consider any 𝛿( 𝑝, 𝑞) ∈ ⇔. Since 𝜃 0 is a congruence, we have h𝛿A (𝑎, 𝑎), 𝛿A (𝑎, 𝑏)i ∈ 𝜃 0. Observing that 𝛿A (𝑎, 𝑎) ∈ 𝐹 (due to (R⇔ )) and 𝜃 0 is logical, entails that 𝛿A (𝑎, 𝑏) ∈ 𝐹. Thus, we have shown that 𝑎 ⇔A 𝑏 ⊆ 𝐹, i.e. h𝑎, 𝑏i ∈ 𝜃. It is easy to see that 2 implies 3, and 2 implies 4. To conclude the proof it suffices to prove that 3 implies 1, and 4 implies 1, which are immediate consequences of the completeness theorem w.r.t. reduced (resp. Lindenbaum) matrices.  16 By 𝑎 ⇔A 𝑏 we denote the set { 𝛿 A (𝑎, 𝑏) | 𝛿 ( 𝑝, 𝑞) ∈ ⇔}.

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It is easy to see that, for any two equivalence sets ⇔ and ⇔0 of a logic L, we have 𝜑 ⇔ 𝜓 `L 𝜑 ⇔0 𝜓. Therefore, if a finitary logic has a finite equivalence set, then any of its equivalence sets contains a finite subset which is an equivalence set as well. It easy to see that any set of formulas can serve as an equivalence set for the inconsistent logic Inc. Perhaps surprisingly, the empty set can serve as an equivalence set for the almost inconsistent logic AInc and there is no equivalence set for the minimum logic Min. In the next section, we will see that the set {𝑝 → 𝑞, 𝑞 → 𝑝} can serve as an equivalence set for almost all the other logics that we have explicitly introduced in this chapter. The only exception are local modal logics, where more complex equivalence sets are necessary (see Example 2.8.6). Let us conclude this section by proving a proposition which will play an important role in the development of the structural theory of logical matrices. Given a matrix h𝐴, 𝐹i ∈ Mod(L), we denote by [𝐹, 𝐴] the interval {𝐺 ∈ FiL (A) | 𝐹 ⊆ 𝐺} in the lattice of L-filters over A. Proposition 2.7.8 Let L be a logic such that there is a set ⇔ of formulas in two variables such that (R⇔ ) (MP⇔ )

`L 𝜑 ⇔ 𝜑 𝜑, 𝜑 ⇔ 𝜓 `L 𝜓.

Furthermore, let ℎ : A → B be a surjective homomorphism of L-algebras and 𝐺 ∈ FiL (B). Then, the mapping 𝒉 : [ℎ−1 [𝐺], 𝐴] → [𝐺, 𝐵] defined as 𝒉(𝐻) = ℎ[𝐻] is a lattice isomorphism. Proof Let us first observe that, for each 𝐻 ∈ [ℎ−1 [𝐺], 𝐴] and ℎ(𝑥) ∈ ℎ[𝐻], we have 𝑥 ∈ 𝐻. Indeed, the assumption gives us ℎ(𝑥) = ℎ(𝑦) for some 𝑦 ∈ 𝐻. Therefore, due to (R⇔ ), we have ℎ[𝑦 ⇔A 𝑥] = ℎ(𝑦) ⇔B ℎ(𝑥) ⊆ 𝐺, and so 𝑦 ⇔A 𝑥 ⊆ ℎ−1 [𝐺] ⊆ 𝐻. Thus, as 𝑦 ∈ 𝐻, we obtain 𝑥 ∈ 𝐻 due to (MP⇔ ). Then, we use the second claim of Lemma 2.5.7 to obtain that 𝒉(𝐻) ∈ FiL (B) which, together with the obvious fact that 𝒉 is order-preserving, give us 𝒉[ℎ−1 [𝐺], 𝐴] ⊆ [𝐺, 𝐵]. The surjectivity of 𝒉 follows from the fact that, for each 𝐻 ∈ [𝐺, 𝐵], we have ℎ−1 [𝐻] ∈ [ℎ−1 [𝐺], 𝐴] (due to the first claim of Lemma 2.5.7) and 𝒉(ℎ−1 [𝐻]) = 𝐻 (thanks to the surjectivity of ℎ). To complete the proof, we only need to show that 𝒉 is order-reflecting: assume that 𝒉(𝐻) ⊆ 𝒉(𝑆) for some 𝐻, 𝑆 ∈ [ℎ−1 [𝐺], 𝐴]. Consider 𝑥 ∈ 𝐻; then ℎ(𝑥) ∈ ℎ[𝐻] ⊆ ℎ[𝑆] and so 𝑥 ∈ 𝑆, due to the observation at the beginning of the proof. 

2.8 Weakly implicative logics This section presents the main subject studied in this book: the class of weakly implicative logics. We define them as the largest possible class of logics with an implication connective ⇒ that satisfies the minimal requirements an implication should obey to deserve its name, namely: identity, modus ponens, transitivity, and one additional condition which allows the set {𝑝 ⇒ 𝑞, 𝑞 ⇒ 𝑝}, which we call the

2.8 Weakly implicative logics

61

symmetrization of ⇒ and denote by ⇔, to serve as an equivalence set17 in the sense of Theorem 2.7.7 (and, in particular, satisfying also the conditions of Proposition 2.7.8). In order to simplify the formulation of many of the upcoming theorems (and their proofs), we include one additional technical cardinality assumption. Definition 2.8.1 (Weak implication) Let L be a logic in a language L. We say that a binary connective ⇒ (primitive or definable) is a weak implication in L if the following consecutions are valid in L: (id)

𝜑⇒𝜑

(MP)

𝜑, 𝜑 ⇒ 𝜓 I 𝜓

(T)

𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜒 I 𝜑 ⇒ 𝜒

(sCng)

𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜑 I 𝑐( 𝜒1 , . . . , 𝜒𝑖 , 𝜑, . . . , 𝜒𝑛 ) ⇒ 𝑐( 𝜒1 , . . . , 𝜒𝑖 , 𝜓, . . . , 𝜒𝑛 ) for each h𝑐, 𝑛i ∈ L and each 0 ≤ 𝑖 < 𝑛.

Definition 2.8.2 (Weakly implicative logic) Let L be a logic in a countable language L over a countable set of variables. We say that L is a weakly implicative logic if it has a weak implication. Note that if a logic has no theorems, it cannot have any weak implication (as the identity consecution would not be valid for any connective); thus we know that the trivial logics Min and AInc are not weakly implicative; but there are also more interesting logics which cannot be weakly implicative for this reason, such as the conjunction-disjunction-fragment of classical logic CL∧∨ (indeed, consider a Boolean algebra, the filter {>} and the evaluation that sends all variables to ⊥; such evaluation maps all formulas to ⊥ and, hence, the logic has no theorems). On the other hand, any binary connective is a weak implication in the logic Inc. Observe that, clearly, if a connective is a weak implication in a logic, then it remains so in each extension thereof. Let us use this fact to identify more interesting examples of weakly implicative logics. The implication connective in BCI obviously satisfies the conditions of identity, modus ponens, and transitivity as derived rules, so we only have to deal with symmetrized congruence, i.e. we have to prove that 𝜑 → 𝜓, 𝜓 → 𝜑 `BCI (𝜑 → 𝜒) → (𝜓 → 𝜒) 𝜑 → 𝜓, 𝜓 → 𝜑 `BCI ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓). It is easy to see that the first rule follows from the axiom of suffixing (sf) and modus ponens. To prove the second, it suffices to observe that prefixing (pf) is a theorem of BCI (the same proof as in BCK in Example 2.2.4 would work). Example 2.8.3 The logic BCI and all of its extensions, notably all logics depicted in Figure 2.1, are weakly implicative with a weak implication →. 17 Note the difference between the meaning of symbols ↔ and ⇔: the former is a defined connective in the classical language LCL , whereas the latter denotes a set of two formulas.

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2 Weakly implicative logics

Let us turn our attention to logics in richer propositional languages (such as LCL and L ). Observe that expansions of a logic with a weak implication do not necessarily preserve the condition (sCng) and, therefore, it needs to be checked for the new connectives. Lemma 2.8.4 Let L be a logic in a language L with a weak implication ⇒ and L0 an expansion in the language L 0 ⊇ L. The following are equivalent: 1. ⇒ is a weak implication in L0. 2. 𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜑 `L0 𝑐( 𝜒1 , . . . , 𝜒𝑖 , 𝜑, . . . , 𝜒𝑛 ) ⇒ 𝑐( 𝜒1 , . . . , 𝜒𝑖 , 𝜓, . . . , 𝜒𝑛 ) for each h𝑐, 𝑛i ∈ L 0 \ L and each 0 ≤ 𝑖 < 𝑛.

Example 2.8.5 The connective → is a weak implication in BCIlat and, thus, this logic and all of its extensions (and in particular, the prominent logics IL, CL, G, Ł, and Ł∞ , defined in Examples 2.3.7 and 2.3.9) are weakly implicative. To prove this claim, we need to show that 𝜑 → 𝜓, 𝜓 → 𝜑 `BCIlat 𝜑 ∨ 𝜒 → 𝜓 ∨ 𝜒 𝜑 → 𝜓, 𝜓 → 𝜑 `BCIlat 𝜑 ∧ 𝜒 → 𝜓 ∧ 𝜒

𝜑 → 𝜓, 𝜓 → 𝜑 `BCIlat 𝜒 ∨ 𝜑 → 𝜒 ∨ 𝜓 𝜑 → 𝜓, 𝜓 → 𝜑 `BCIlat 𝜒 ∧ 𝜑 → 𝜒 ∧ 𝜓.

We show the first claim; the others are analogous and left as an exercise for the reader. a) 𝜒 → 𝜓 ∨ 𝜒

(ub2 )

b) 𝜓 → 𝜓 ∨ 𝜒

(ub1 )

c) 𝜑 → 𝜓 ∨ 𝜒

premise 𝜑 → 𝜓, b, and (T)

d) 𝜑 ∨ 𝜒 → 𝜓 ∨ 𝜒

a, c, and (Sup)

The reader can also easily check that the equivalence connective ↔ (defined as (𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) is also a weak implication in all extensions of BCIlat , though it differs substantially from → in logical behavior; for instance, we have 𝜑 ` 𝜓 → 𝜑 but not 𝜑 ` 𝜓 ↔ 𝜑. Example 2.8.6 Using the axiom (d ) and the rule (-nec), one can easily check that → is a weak implication in the global modal logics from Example 2.3.10. On the other hand, it is not the case for local modal logics. Indeed, let L be any such logic and assume that > → 𝜑, 𝜑 → > `L > → 𝜑. Since L expands CL, we know that `L 𝜑 → > 𝜑 `L > → 𝜑 `L >. Thus, also, `L > and so 𝜑 `L 𝜑, i.e. L is equal to its global variant, which we already know to be false (see Example 2.6.8). There is an obvious question: could we define some other connective in the language of local modal logics that would behave as a weak implication? In sufficiently strong systems, the answer is positive; indeed, the intended implication of these logics (called strict implication) is a weak implication: • 𝑙K4 and its extensions have the weak implication ( 𝑝 → 𝑞) ∧ ( 𝑝 → 𝑞). • 𝑙S4 and its extensions have the weak implication ( 𝑝 → 𝑞).

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63

Hence, local modal logics extending 𝑙K4 are weakly implicative. Both these facts are easy to prove and are left as an exercise for the reader. Regarding 𝑙K and 𝑙T, one can check that they have an infinite equivalence set in the sense of Theorem 2.7.7: 𝑝 ⇔ 𝑞 = {𝑛 ( 𝑝 → 𝑞), 𝑛 (𝑞 → 𝑝) | 𝑛 ≥ 0}, where 0 𝑝 = 𝑝 and 𝑛+1 𝑝 = 𝑛 𝑝. Indeed, all the syntactical properties are easy to check; the only non-trivial one (Cng⇔ ) follows from the fact that for 𝑛 we have: 𝑛+1 ( 𝑝 → 𝑞) `𝑙K 𝑛 (𝑝 → 𝑞). Therefore, neither 𝑙K nor 𝑙T is weakly implicative. Indeed, if ⇒ was a weak implication in any of these logics, then its symmetrization ⇔ would be a finite equivalence set. Then, since these logics are finitary, there would also be a finite subset of {𝑛 ( 𝑝 → 𝑞), 𝑛 (𝑞 → 𝑝) | 𝑛 ≥ 0} behaving like an equivalence set. Checking that this is impossible is left as an exercise for the reader. Using again that the symmetrization ⇔ of any weak implication ⇒ is an equivalence set of a given logic, we immediately obtain the following two corollaries of Theorem 2.7.7 (note that one could also write a simple syntactical proof; we leave the elaboration of the details as an exercise for the reader). Corollary 2.8.7 (Substitution property) Let L be a logic with a weak implication ⇒, let 𝜑, 𝜓, and 𝜒 be formulas, and let 𝜒ˆ be a formula obtained from 𝜒 by replacing some occurrences of 𝜑 in 𝜒 by 𝜓. Then, 𝜑 ⇔ 𝜓 `L 𝜒 ⇔ 𝜒. ˆ Corollary 2.8.8 Let ⇒ and ⇒0 be weak implications in a logic L. Then, 𝜑 ⇔ 𝜓 a`L 𝜑 ⇔0 𝜓. We also know that ⇔ can be used to define the Leibniz congruence in matrices of weakly implicative logics. Let us now prove a stronger claim. Given a binary connective ⇒ and a matrix A = hA, 𝐹i, we define the following binary relation: 𝑎 ≤A⇒ 𝑏

iff

𝑎 ⇒A 𝑏 ∈ 𝐹.

The next theorem justifies calling ≤A⇒ a matrix (pre)order for a model A of a weakly implicative logic. These (pre)orders will play a crucial role in Chapters 6 and 7. Theorem 2.8.9 Let L be a logic and ⇒ a binary (primitive or definable) connective. Then, the following are equivalent: 1. ⇒ is a weak implication in L. 2. ≤A⇒ is a preorder for each A = hA, 𝐹i ∈ Mod(L), 𝐹 is an upper set w.r.t. ≤A⇒ , and the symmetrization of ≤A⇒ is the Leibniz congruence of A. 3. ≤A⇒ is an order for each A = hA, 𝐹i ∈ Mod∗ (L) and 𝐹 is an upper set w.r.t. ≤A⇒ .

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Proof To prove ‘1 implies 2’ we observe that the fact that ≤A⇒ is a preorder follows from (id) and (T); the fact that 𝐹 is an upper set w.r.t. ≤A⇒ follows from (MP), and the rest follows from Theorem 2.7.7 because ⇔ is an equivalence set of L. The proof of ‘2 implies 3’ is straightforward and the last implication is a simple consequence of the completeness theorem w.r.t. reduced matrices.  Note that claim 2 implies that a model of a weakly implicative logic is reduced iff the matrix preorder is an order. Example 2.8.10 Recall, from Example 2.6.4, that hZ, {0}i ∈ Mod(A→ ), where Z is the L→ -algebra over the domain of integers where 𝑎 →Z 𝑏 = 𝑏 − 𝑎. Let us denote this matrix as Z0 and note that we also have Z+ = hZ, {𝑎 | 𝑎 ≥ 0}i ∈ Mod(A→ ). A simple observation reveals that ≤Z→0 is the discrete order and ≤Z→+ is the usual order of integers. Therefore, thanks to the previous observation, we know that the matrices Z0 and Z+ are both reduced. The following proposition will be often used to semantically produce examples of weakly implicative logics; in particular to show that ⇒ is a weak implication in a logic A . Its proof is analogous to the proof of the implication ‘2 implies 3’ of Theorem 2.8.9 (the fact that A is reduced then follows from the remark before the previous example). Proposition 2.8.11 Let ⇒ be a binary (primitive or definable) connective and A a matrix such that ≤A⇒ is an order and 𝐹 is an upper set w.r.t. ≤A⇒ . Then, ⇒ is a weak implication in A and A ∈ Mod∗ (A ). The following example shows that, in a reduced model A of BCIlat , ≤A→ coincides with the lattice order given by ∧A (or ∨A respectively). We leave the proof of this particular fact to the reader, and give a proof of a more general statement in Section 4.1 (cf. Proposition 4.1.5). Example 2.8.12 Let A be a reduced model of BCIlat (or any of its weakly implicative expansions w.r.t. →). Then, ≤A→ is the lattice order with supremum ∨A , infimum ∧A , and the least and greatest elements are ⊥A and >A . As we have seen, there could be several different weak implications in a given logic. This poses a problem as many notions (such as matrix preorder defined above or linear filter defined in Chapter 6) are parameterized by a chosen weak implication. For the sake of simpler notation, we display the parameter only in rare cases in which we need to speak about notions corresponding to several different weak implications. Furthermore, we use the following conventions: • In generic situations, when we postulate that L is a weakly implicative logic, we will assume that ⇒ is one of its weak implications and all notions are defined relative to this implication.

2.8 Weakly implicative logics

65

• Analogously, when speaking about a specific logic which we have already identified as weakly implicative by singling out its natural weak implication, we assume that all notions are defined relative to this implication. An interesting application of the substitution property is the promised characterization of definability of connectives. First, we deal with the definability of a connective in the language of a given logic by its other connectives (cf. Definition 2.3.4). Note that it entails that the definability of a connective in a logic is preserved in its expansions. Proposition 2.8.13 Let L be a weakly implicative logic in a language L. Then, a connective 𝑑 is definable by a formula 𝛿 iff `L 𝑑 (𝜑1 , . . . , 𝜑 𝑛 ) ⇔ 𝛿(𝜑1 , . . . , 𝜑 𝑛 ). Proof Recall that 𝑑 𝛿 (𝜑) denotes the result of replacing the connective 𝑑 by 𝛿 in 𝜑. In particular, 𝑑 𝛿 (𝑑 (𝜑1 , . . . , 𝜑 𝑛 ) ⇒ 𝛿(𝜑1 , . . . , 𝜑 𝑛 )) = 𝛿(𝑑 𝛿 (𝜑1 ), . . . , 𝑑 𝛿 (𝜑 𝑛 )) ⇒ 𝛿(𝑑 𝛿 (𝜑1 ), . . . , 𝑑 𝛿 (𝜑 𝑛 )). Thus, `L 𝑑 𝛿 (𝑑 (𝜑1 , . . . , 𝜑 𝑛 ) ⇒ 𝛿(𝜑1 , . . . , 𝜑 𝑛 )) and so, by Definition 2.3.4 of definability, we have `L 𝑑 (𝜑1 , . . . , 𝜑 𝑛 ) ⇒ 𝛿(𝜑1 , . . . , 𝜑 𝑛 ). The proof of the converse implication is analogous. To prove the converse direction of the claim, observe that using the substitution property of ⇔ we can show by induction that `L 𝜒 ⇔ 𝑑 𝛿 ( 𝜒). Thus, 𝜒 `L 𝑑 𝛿 ( 𝜒) and 𝑑 𝛿 ( 𝜒) `L 𝜒 (by (MP)) and so obviously Γ `L 𝜑

iff

𝑑 𝛿 [Γ] `L 𝑑 𝛿 (𝜑).



The following examples use the proposition to prove (un)definability of connectives in non-classical logics mentioned in the comments after Example 2.3.7. Example 2.8.14 First we show that ∨ is definable in any expansion of FBCKlat : `FBCKlat 𝜑 ∨ 𝜓 ⇔ ((𝜑 → 𝜓) → 𝜓) ∧ ((𝜓 → 𝜑) → 𝜑). We prove both implications; the first one is relatively straightforward: a) 𝜑 → ((𝜑 → 𝜓) → 𝜓) (id) and (E) b) 𝜓 → ((𝜑 → 𝜓) → 𝜓) c) 𝜑 ∨ 𝜓 → ((𝜑 → 𝜓) → 𝜓)

(w) a, b, and (Sup)

d) 𝜑 → ((𝜓 → 𝜑) → 𝜑)

(w)

e) 𝜓 → ((𝜓 → 𝜑) → 𝜑)

(id) and (E)

f) 𝜑 ∨ 𝜓 → ((𝜓 → 𝜑) → 𝜑) g) 𝜑 ∨ 𝜓 → ((𝜑 → 𝜓) → 𝜓) ∧ ((𝜓 → 𝜑) → 𝜑)

d, e, and (Sup) c, f, and (Inf)

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2 Weakly implicative logics

To prove the converse implication, we first observe that there is a proof of 𝜓 where the premises 𝜑 → 𝜓 and (𝜑 → 𝜓) → 𝜓 are used exactly once. Thus, due to the local deduction theorem 2.4.3, we obtain 𝜑 → 𝜓 `FBCKlat ((𝜑 → 𝜓) → 𝜓) → 𝜑 and hence, using the axioms (lb1 ) and (ub1 ) together with transitivity, we have 𝜑 → 𝜓 `FBCKlat ((𝜑 → 𝜓) → 𝜓) ∧ ((𝜓 → 𝜑) → 𝜑) → 𝜑 ∨ 𝜓. The proof of the same fact only using the premise 𝜓 → 𝜑 is analogous and so we can use the classical proof by cases property together with the prelinearity axiom (𝜑 → 𝜓) ∨ (𝜓 → 𝜑) to obtain the required claim. It is easy to see that, thanks to the Wajsberg axiom, in (expansions of) Łukasiewicz logic we can simplify the definition of ∨; indeed, we can easily prove that `Ł 𝜑 ∨ 𝜓 ⇔ ((𝜑 → 𝜓) → 𝜓). Finally, we leave as an exercise for the reader the proof of the fact that the lattice conjunction is definable in Łukasiewicz logic in the following way: `Ł 𝜑 ∧ 𝜓 ⇔ ¬(¬𝜑 ∨ ¬𝜓). Example 2.8.15 Consider the equivalence connective ↔ in classical logic. As an exercise, the reader should check that ↔ is a weak implication interderivable with its symmetrization. We show that the classical implication → is not definable in CL using only ↔. Indeed, if it were, there would be a formula 𝛿( 𝑝, 𝑞) composed of equivalences such that `CL (𝜑 → 𝜓) ↔ 𝛿(𝜑, 𝜓). Let us assume that 𝛿 is the shortest formula with this property. If we show that 𝛿 cannot contain any subformula of the form 𝑝 ↔ 𝑝 or 𝑞 ↔ 𝑞, then the proof is done, because by repeated use of symmetry of ↔ and the Substitution Property we would obtain that `CL 𝛿(𝜑, 𝜓) ↔ 𝛿(𝜓, 𝜑) and so `CL (𝜑 → 𝜓) ↔ (𝜓 → 𝜑), which is clearly not the case. Assume e.g. that 𝑝 ↔ 𝑝 is a subformula of 𝛿. Then, either 𝛿 = 𝑝 ↔ 𝑝 or there is a formula 𝛼 such that ( 𝑝 ↔ 𝑝) ↔ 𝛼 is a subformula of 𝛿. The first case is impossible because it would entail that `CL (𝜑 → 𝜓) ↔ (𝜑 ↔ 𝜑). The second one is also impossible because `CL [( 𝑝 ↔ 𝑝) ↔ 𝛼] ↔ 𝛼 and so the formula 𝛿 0 resulting from 𝛿 by replacing its subformula ( 𝑝 ↔ 𝑝) ↔ 𝛼 by 𝛼 would be equivalent to 𝛿 and, thus, it would be a strictly shorter definition of →, a contradiction. Next, we deal with logics in languages enlarged with definable connectives. Recall that in Proposition 2.3.5 we showed that each logic L in a language L has a unique conservative expansion, denoted as L𝑑/ 𝛿 , where a connective 𝑑 ∉ L is definable by an L-formula 𝛿. The next proposition gives a simple presentation of L𝑑/ 𝛿 . Corollary 2.8.16 Let L be a weakly implicative logic in a language L, 𝛿 an Lformula with 𝑛 variables, and 𝑑 an 𝑛-ary connective not in L. Then, L𝑑/ 𝛿 is the axiomatic expansion of L by the axiom `L 𝑑 (𝜑1 , . . . , 𝜑 𝑛 ) ⇔ 𝛿(𝜑1 , . . . , 𝜑 𝑛 ).

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67

Proof Let L0 be the axiomatic expansion of L with the mentioned axiom. We only need to show that it is a conservative expansion of L where 𝑑 is definable by 𝛿. The definability immediately follows from Proposition 2.8.13. Regarding conservativity, we proceed semantically: assume that Γ 0L 𝜑 and let A = hA, 𝐹i ∈ Mod∗ (L) be a reduced matrix witnessing this fact. Let us define an L 𝑑 -algebra A𝑑 as the expansion of A with an operation 𝑑 defined as 𝑑

𝑑 A (𝑥1 , . . . , 𝑥 𝑛 ) = 𝛿A (𝑥1 , . . . , 𝑥 𝑛 ). It is easy to see that hA𝑑 , 𝐹i ∈ Mod∗ (L0) and so Γ 0L0 𝜑.



Note that, in certain situations, we may want to speak about definability of an 𝑛-ary connective 𝑑 by a formula 𝛿 with more than 𝑛 variables; this is in cases in which the additional variables intuitively ‘do not matter’. Let us illustrate it with the definability of a constant using a formula with one variable (a situation we will encounter at the end of the section); the reader can formulate a general definition and prove the corresponding results as an exercise. For a constant 𝑑 and a formula 𝛿 with one variable 𝑝, we redefine the 𝑑-clause in the definition of the function 𝑑 𝛿 as 𝑑 𝛿 (𝑑) = 𝛿. The definition of definability remains the same (cf. Definition 2.3.4) and it is clearly equivalent to proving `L 𝑑 ⇔ 𝛿 (just inspect the proof of Proposition 2.8.13). On the other hand, the analog of Corollary 2.8.16 is not so straightforward for two reasons: first, we do not have a readily available reformulation of Proposition 2.3.5 (the problem would be with proving Structurality of L𝑑/ 𝛿 ) and, second, the semantic reasoning in its proof would not go through in the same way. To find a solution, we observe that if a constant 𝑑 is definable by a formula 𝛿( 𝑝), then `L 𝛿( 𝑝) ⇔ 𝛿(𝑞) for each variable 𝑞 ≠ 𝑝. With this observation we can easily prove the following: Corollary 2.8.17 Let L be a weakly implicative logic in a language L, 𝑑 a truthconstant not in L, and 𝛿 an L-formula with one variable 𝑝 such that `L 𝛿( 𝑝) ⇔ 𝛿(𝑞) for a variable 𝑞 ≠ 𝑝. Then, the axiomatic expansion of L by the axioms 𝑑⇒𝛿

𝛿⇒𝑑

is the unique conservative expansion of L in L 𝑑 where 𝑑 is definable by 𝛿. Proof Let us denote the defined logic as L𝑑/ 𝛿 . As argued before, 𝑑 is definable in L𝑑/ 𝛿 by 𝛿 and it is the least logic in L 𝑑 expanding L where it is the case. Thus, it only remains to prove that L𝑑/ 𝛿 expands L conservatively. We proceed semantically as in the proof of Corollary 2.8.16; we only notice that thanks to the fact that A 𝛿( 𝑝) ⇔ 𝛿(𝑞) and Theorem 2.8.9 we know that there is an 𝑎 ∈ 𝐴 such that, for 𝑑 each 𝑏 ∈ 𝐴, we have 𝑎 = 𝛿A (𝑏). Thus, if we define 𝑑 A = 𝑎, we obtain  hA𝑑 ,𝐹 i 𝑑 ⇔ 𝛿 and so the claim follows.  We conclude this section by introducing a particularly well-behaved subclass of weakly implicative logics.

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2 Weakly implicative logics

Definition 2.8.18 (Rasiowa-implicative logics) A logic L is Rasiowa-implicative if it has a weak implication ⇒ such that the following consecution is valid in L: (W)

𝜑I𝜓⇒𝜑

weakening.

We extend our previous convention, by assuming that whenever we say that a logic is Rasiowa-implicative, then ⇒ is one of its weak implications satisfying (W). It is easy to see that any expansion of BCK in which → remains a weak implication (i.e. most of the logics we have introduced so far) is Rasiowa-implicative. Next, we show an example of a weakly implicative logic that is not Rasiowa-implicative (more examples will come later after a useful characterization of Rasiowa-implicative logics in Proposition 2.8.20). Example 2.8.19 Recall that the equivalence fragment of classical logic CL↔ is weakly implicative (see Example 2.8.15). We show that it is not Rasiowa-implicative. Consider the matrix E = hh{0, 1}, ↔i, {1}i whose algebraic part is just the twoelement domain with the classical equivalence operation: 0 ↔ 0 = 1 ↔ 1 = 1 and 0 ↔ 1 = 1 ↔ 0 = 0. It is easy to see that E ∈ Mod(CL↔ ) (because it is a reduct of the Boolean algebra 2). Therefore, ↔ does not satisfy the condition of Rasiowa-implicative logics (because 𝜑 0E 𝜓 ↔ 𝜑). However, CL↔ could still be Rasiowa-implicative with another weak implication ⇒. But then such a connective would have to satisfy the following properties: • 0 ⇒ 0 = 1 ⇒ 1 = 1 (because E 𝑝 ⇒ 𝑝). • 1 ⇒ 0 = 0 (because 𝑝, 𝑝 ⇒ 𝑞 E 𝑞). • 0 ⇒ 1 = 1 (because 𝑝 E 𝑞 ⇒ 𝑝). Therefore, ⇒ would behave as the classical implication, which is undefinable in the pure equivalence fragment (as seen in Example 2.8.15). Clearly, in any Rasiowa-implicative logic we have ` ( 𝑝 ⇒ 𝑝) ⇔ (𝑞 ⇒ 𝑞) and so, due to Corollary 2.8.17, we can see the formula 𝑝 ⇒ 𝑝 as a definition of a truth-constant, which we denote by >. We can then easily prove that `L 𝜑 ⇒ >

`L >

𝜑 `L > ⇒ 𝜑

> ⇒ 𝜑 `L 𝜑.

It is easy to see that any reduced matrix hA, 𝐹i of a Rasiowa-implicative logic L has exactly one element in its filter (and 𝐹 = {>A } if we see L in the language with >). Actually, we can prove even more and obtain a useful characterization: Proposition 2.8.20 A logic L is Rasiowa-implicative iff it has a weak implication ⇒ and for each hA, 𝐹i ∈ Mod∗ (L) we have 𝐹 = {𝑡} where 𝑡 is the maximum of 𝐴 w.r.t. the matrix order ≤⇒ . hA,𝐹 i Proof Let us take an arbitrary 𝑎 ∈ 𝐴 and define 𝑡 = 𝑎 ⇒A 𝑎. Then, 𝑡 ∈ 𝐹, and so for any 𝑏 ∈ 𝐴 we have 𝑏 ⇒A 𝑡 ∈ 𝐹, and if 𝑏 ∈ 𝐹, then also 𝑡 ⇒A 𝑏 ∈ 𝐹. Thus, 𝑡 is the maximum of 𝐴 w.r.t. ≤ hA,𝐹 i and 𝐹 = {𝑡} (due to Theorem 2.8.9). To prove the converse direction, we use the second completeness theorem w.r.t. reduced matrices.

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69

Recall that Theorem 2.7.7 gives an easy way to determine the Leibniz congruence in a given matrix of a weakly implicative logic. Thus, if we find a reduced matrix with non-singleton filter, the previous proposition entails that the logic in question is not Rasiowa-implicative. Example 2.8.21 The following weakly implicative logics are not Rasiowa-implicative: • A→ (and, thus, also BCI): it suffices to recall that in Example 2.8.10 we have presented the matrix Z+ = hZ, {𝑎 | 𝑎 ≥ 0}i ∈ Mod∗ (A→ ). • BCIlat : it suffices to show that the matrix hA, {𝑡, >}i ∈ Mod(BCIlat ) introduced in Example 2.6.4 is reduced. We leave it as an exercise for the reader. • 𝑙S4 and 𝑙K4: it suffices to show that the matrix h4 0, {𝑎, >}i ∈ Mod(lS4) introduced in Example 2.6.8 is reduced. Recall that, thanks to Example 2.8.6, we know that h𝑥, 𝑦i ∈ Ω40 ({𝑎, >})

iff

{(𝑥 → 𝑦), (𝑦 → 𝑥)} ⊆ {𝑎, >}

and as 𝑥 = > if 𝑥 = > and ⊥ otherwise, it is easy to show that h𝑥, 𝑦i ∈ Ω40 ({𝑎, >}) iff 𝑥 = 𝑦.

2.9 Algebraically implicative logics The semantics for propositional logics so far considered (in the general case but also when restricted to weakly implicative logics) has been based on the notion of logical matrix and essentially used both of its components: an algebra to interpret connectives as algebraic operations and a set of designated truth-values (the filter) to interpret a certain notion of truth that allows us to define logical consequence. However, algebraic logic typically tries to reduce the whole semantics of a given logic to just a class of algebras, whenever possible, i.e. to give a logic a purely algebraic semantics. The main goal of this section is to show that a natural candidate for this class of algebras is the class of algebraic reducts of reduced matrices of the logic in question. Let us start with a definition and a collection of examples. Definition 2.9.1 Let L be a logic. We denote by Alg∗ (L) the class of algebraic reducts of reduced models of L; i.e. A ∈ Alg∗ (L) if there is an 𝐹 ⊆ 𝐴 such that hA, 𝐹i ∈ Mod∗ (L). If we recall that FiAInc (A) = { 𝐴, ∅} and FiInc (A) = { 𝐴}, we can immediately observe that Alg∗ (AInc) = Alg∗ (Inc) only contains (isomorphic copies of) the trivial algebra Tr. For the remaining trivial logic, it is clear that Alg∗ (Min) is the class of all the algebras that have a subset whose Leibniz congruence is the identity. We leave as an exercise for the reader showing that the class Alg∗ (Min) does not contain all algebras.

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Recall that in Example 2.5.9 we have characterized logical filters on intended algebras of prominent logics. Now we can show that these algebras are exactly the L-algebras of these logics. Example 2.9.2 The following table collects descriptions of Alg∗ (L) for certain prominent logics: Logic L CL IL G Ł 𝑙K K

Alg∗ (L) Boolean algebras Heyting algebras Gödel algebras MV-algebras modal algebras modal algebras

Thanks to Examples 2.7.3 and 2.5.9, we know that these algebras are indeed elements of Alg∗ (L). We prove the converse inclusion for classical logic; it is easy to see that the proof works for the other logics as well. Take A ∈ Alg∗ (CL). Since A is is the algebraic reduct of a reduced model, we know that `CL 𝜑 ⇔ 𝜓 implies that, for each A-evaluation 𝑒, we have 𝑒(𝜑) = 𝑒(𝜓). Thus, to check that A satisfies all the defining equations of Boolean algebras, we just need to prove the corresponding equivalences in CL. As an example, given arbitrary 𝑎, 𝑏 ∈ 𝐴, let us check that 𝑎 ∧ 𝑏 = 𝑏 ∧ 𝑎. We know that `CL 𝑝 ∧ 𝑞 ⇔ 𝑞 ∧ 𝑝 for any variables 𝑝 and 𝑞. Thus, taking an A-evaluation 𝑒 such that 𝑒( 𝑝) = 𝑎 and 𝑒(𝑞) = 𝑏 completes the proof. The other cases are shown analogously. Note that the class Alg∗ (𝑙K) coincides with Alg∗ (K) (this is not surprising as these logics have the same theorems). This entails that simply specifying a class of algebras does not paint a full semantical picture of a given logic. On the other hand, at least in certain logics, only very little has to be added to obtain the desired purely algebraic semantics. Indeed, if we consider an algebra from Alg∗ (L) for a Rasiowa-implicative logic L with the constant >, we can see (thanks to Proposition 2.8.20) not only that there is exactly one filter which turns it into a reduced matrix, but this filter can be described using purely algebraic notions, that is to say, in terms of equations: 𝑎 ∈ 𝐹 iff 𝑎 = >A . This fact is all we need to present a purely algebraic semantics of L using the equational consequence over Alg∗ (L) (see the Appendix): indeed, for each Γ ∪ {𝜑}, we obviously have Γ `L 𝜑 iff {𝛾 ≈ > | 𝛾 ∈ Γ} Alg∗ (L) 𝜑 ≈ >. Thus, roughly speaking, the class of algebras Alg∗ (L) together with a uniform algebraic definition of reduced matrices over these algebras gives a complete semantics of a Rasiowa-implicative logic L. This yields two natural related questions:

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71

• Can we generalize this semantical picture to a wider class of logics? • Can we do the converse, i.e. use a logic L to describe the equational consequence over Alg∗ (L)? The answer to the second question is surprisingly simple: we can do it in all weakly implicative logics (note that actually we could have formulated it for all logics with an equivalence set in the sense of Theorem 2.7.7). Proposition 2.9.3 Let L be a weakly implicative logic and Π ∪ {𝜑 ≈ 𝜓} a set of equations. Then, Ø Π Alg∗ (L) 𝜑 ≈ 𝜓 iff {𝛼 ⇔ 𝛽 | 𝛼 ≈ 𝛽 ∈ Π} `L 𝜑 ⇔ 𝜓. Proof We show the proof of one implication; the proof of the converse one is similar. Ð Assume Π Alg∗ (L) 𝜑 ≈ 𝜓. To check that {𝛼 ⇔ 𝛽 | 𝛼 ≈ 𝛽 ∈ Π} `L 𝜑 ⇔ 𝜓 it is enough (due Ð to the completeness theorem 2.7.6) to check the equivalent semantical statement {𝛼 ⇔ 𝛽 | 𝛼 ≈ 𝛽 ∈ Π} Mod∗ (L) 𝜑 ⇔ 𝜓. Take any hA, 𝐹i ∈ Mod∗ (L) and an A-evaluation 𝑒 satisfying the premises, i.e. for every 𝛼 ≈ 𝛽 ∈ Π we have 𝑒(𝛼) ⇔A 𝑒(𝛽) ⊆ 𝐹, and hence (since the matrix is reduced) 𝑒(𝛼) = 𝑒(𝛽). By the assumption (using that A ∈ Alg∗ (L)), we know that 𝑒(𝜑) = 𝑒(𝜓) and, thus, 𝑒(𝜑) ⇔A 𝑒(𝜓) ⊆ 𝐹.  A natural answer to the first question is also simple: all we need is a uniform equational description of the filters of all reduced models of a given logic. As we are aiming for the class of all weakly implicative logics that enjoy positive answers to both questions, we give the following definition: Definition 2.9.4 (Algebraically implicative logic) We say that a weakly implicative logic L is algebraically implicative if there is a set E of equations in one variable such that, for each A = hA, 𝐹i ∈ Mod∗ (L) and each 𝑎 ∈ 𝐴, 𝑎∈𝐹

if and only if

𝜇A (𝑎) = 𝜈 A (𝑎) for every 𝜇 ≈ 𝜈 ∈ E.

In this case, Alg∗ (L) is called the equivalent algebraic semantics, its elements are called L-algebras, and E is called the set of defining equations. It is clear that every Rasiowa-implicative logic is algebraically implicative with E = {𝑥 ≈ >}, if there is the constant >, or with E = {𝑥 ≈ 𝑥 ⇒ 𝑥} in general. The next theorem gives several characterizations of algebraically implicative logics. In particular, it shows that it is indeed the desired generalization of Rasiowaimplicative logics, because they are the logics that enjoy both translations (from equational consequence to logic and vice versa). Moreover, the translations come accompanied with additional properties (in claims 2 and 3) that ensure that they are essentially mutually inverse, i.e. if we apply the translation to an equation and then we translate the resulting formula we obtain an equation that is equivalent to the initial one in the sense of equational consequence (the second condition in claim 2), and analogously if we start with a formula (the condition (Alg) in claim 3).

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Theorem 2.9.5 (Characterizations of algebraically implicative logics) Given any weakly implicative logic L, the following are equivalent:18 1. L is algebraically implicative, i.e. there is a set E of equations in one variable such that, for each A = hA, 𝐹i ∈ Mod∗ (L) and each 𝑎 ∈ 𝐴, 𝑎∈𝐹

𝜇A (𝑎) = 𝜈 A (𝑎) for every 𝜇 ≈ 𝜈 ∈ E.

if and only if

2. There is a set of equations E in one variable such that, for every Γ ∪ {𝜑} ⊆ FmL , Γ `L 𝜑

iff

E [Γ] Alg∗ (L) E (𝜑)

and also 𝑝 ≈ 𝑞 Alg∗ (L) E [ 𝑝 ⇔ 𝑞]

E [ 𝑝 ⇔ 𝑞] Alg∗ (L) 𝑝 ≈ 𝑞.

3. There is a set of equations E in one variable satisfying the following condition: Ð (Alg) 𝑝 a`L {𝜇( 𝑝) ⇔ 𝜈( 𝑝) | 𝜇 ≈ 𝜈 ∈ E}. 4. For every hA, 𝐹i ∈ Mod∗ (L), 𝐹 is the least L-filter on A. In the first three items, the sets E can be taken to be the same. Proof We first prove the equivalence of the first three claims and then we attach the last one. 1→2: The first condition is easily checked by using the completeness theorem. To prove 𝑝 ≈ 𝑞 Alg∗ (L) E [ 𝑝 ⇔ 𝑞], take hA, 𝐹i ∈ Mod∗ (L) and an evaluation 𝑒 on A such that 𝑒( 𝑝) = 𝑒(𝑞). Then, 𝑒( 𝑝) ⇒A 𝑒(𝑞) ∈ 𝐹 (by (id)) and so 𝜇A (𝑒( 𝑝) ⇒A 𝑒(𝑞)) = 𝜈 A (𝑒( 𝑝) ⇒A 𝑒(𝑞)) for every 𝜇 ≈ 𝜈 ∈ E, and the same for the reverse implication. To prove E [ 𝑝 ⇔ 𝑞] Alg∗ (L) 𝑝 ≈ 𝑞, take hA, 𝐹i ∈ Mod∗ (L) and an evaluation 𝑒 on A satisfying the equations in the premises. Then, 𝑒( 𝑝) ⇒A 𝑒(𝑞), 𝑒(𝑞) ⇒A 𝑒( 𝑝) ∈ 𝐹, i.e. h𝑒( 𝑝), 𝑒(𝑞)i ∈ ΩA (𝐹) and, since the matrix is reduced, 𝑒( 𝑝) = 𝑒(𝑞). 2→3: Let us the denote the set of formulas on the right-hand side as Γ. First note that, by the second part of condition 2, we have 𝜇( 𝑝) ≈ 𝜈( 𝑝) Alg∗ (L) E (𝜇( 𝑝) ⇔ 𝜈( 𝑝)) and vice versa for each 𝜇 ≈ 𝜈 ∈ E. Thus, we also have E ( 𝑝) Alg∗ (L) E [Γ] and vice versa. Therefore, due to the first part of the condition 2, we have 𝑝 a`L Γ. 3→1: It follows immediately from the completeness theorem and the definition of algebraically implicative logic. 3→4: Consider a filter 𝐺 ∈ FiL (A) and 𝑎 ∈ 𝐹. Then, from (Alg) and the fact that hA, 𝐹i is reduced, we know that 𝜇A (𝑎) = 𝜈 A (𝑎) for every 𝜇 ≈ 𝜈 ∈ E. Thus also Ð {𝜇A (𝑎) ⇔ 𝜈 A (𝑎) | 𝜇 ≈ 𝜈 ∈ E} ⊆ 𝐺 (as  hA,𝐺 i 𝑝 ⇒ 𝑝) and so, by (Alg) again, we have 𝑎 ∈ 𝐺. 18 Given a set E of equations in one variable and a formula 𝜑, we denote by E ( 𝜑) the set of equations in which the variable has been uniformly substituted by 𝜑. Given a set Γ of formulas, we Ð denote by E [Γ] the set { E (𝛾) | 𝛾 ∈ Γ} of equations. Analogously to the notation for logics, for equational consequence K we write Π K Σ whenever Π K 𝛼 ≈ 𝛽 for each 𝛼 ≈ 𝛽 ∈ Σ.

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73

4→3:ÐWe start by proving that, for each pair of theories 𝑇 and 𝑇 0 such that 𝑇 0 = ThL ( {𝛼 ⇔ 𝛽 | 𝛼 ⇔ 𝛽 ⊆ 𝑇 }), we have 𝑇 = 𝑇 0. Obviously, not only 𝑇 0 ⊆ 𝑇 but also for each 𝛼, 𝛽 we have 𝛼 ⇔ 𝛽 ⊆ 𝑇 0 iff 𝛼 ⇔ 𝛽 ⊆ 𝑇. Thus, ΩFmL (𝑇 0) = ΩFmL (𝑇) and so 𝑇 0/ΩFmL (𝑇) = 𝑇 0/ΩFmL (𝑇 0) ∈ F𝑖 L (FmL /ΩFmL (𝑇 0)) = F𝑖L (FmL /ΩFmL (𝑇)). Thus, from our assumption 𝑇/ΩFmL (𝑇) ⊆ 𝑇 0/ΩFmL (𝑇) and so we can use Proposition 2.7.8 for the canonical mapping of the congruence ΩFmL (𝑇) and for the filter 𝑇/ΩFmL (𝑇) to obtain 𝑇 ⊆ 𝑇 0. Let us use the just proved claim for 𝑇 = ThL ( 𝑝) to obtain Ø 𝑝 a` {𝛼 ⇔ 𝛽 | 𝑝 ` 𝛼 ⇔ 𝛽} and thus, applying the substitution 𝜎 which maps all variables to 𝑝, we obtain Ø 𝑝 a` {𝜎(𝛼) ⇔ 𝜎(𝛽) | 𝑝 ` 𝛼 ⇔ 𝛽} which is nothing else but the condition (Alg) for E ( 𝑝) = {𝜎(𝛼) ≈ 𝜎(𝛽) | 𝑝 ` 𝛼 ⇔ 𝛽}.



Note that the third condition implies that any extension an algebraically implicative logic (and even any expansion in which ⇒ is still a weak implication) remains algebraically implicative. Also note that the last condition can be formulated as: for each hA, 𝐹i ∈ Mod∗ (L) we have [𝐹, 𝐴] = {𝐺 ∈ FiL (A) | 𝐹 ⊆ 𝐺} = FiL (A). Observe that in algebraically implicative logics for each A ∈ Alg∗ (L) there is a unique filter that gives a reduced model over A. This simple observation ensures the soundness of the upcoming definition and can be used to easily show that certain logics are not algebraically implicative. Definition 2.9.6 Let L be an algebraically implicative logic, ⇒ one of its weak implications, and A ∈ Alg∗ (L). We denote by 𝐹A the unique filter such that hA, 𝐹A i ∈ Mod∗ (L). The matrix order ≤⇒ is called the intrinsic order on A w.r.t. ⇒.19 hA,𝐹 i A

Example 2.9.7 The logics A→ and BCI, and the local modal logics 𝑙S4 and 𝑙K4 are not algebraically implicative. Indeed, to prove the first two claims just recall that, by Example 2.8.10, we know that there is an algebra Z ∈ Alg∗ (A→ ) with two different filters 𝐹, 𝐺 such that hZ, 𝐹i, hZ, 𝐺i ∈ Mod∗ (A→ ). Regarding modal logics, recall that in Example 2.8.21 we have shown that h4 0, {𝑎, >}i ∈ Mod∗ (lS4) but we also know that {>} ∈ FilS4 (4 0), i.e. {𝑎, >} is not the least filter. The defining equations are an essential part of the semantical picture. Indeed, later in Example 4.6.13 we will present an algebraically implicative logic asA→ , a proper extension of A→ whose equivalent algebraic semantics coincides with Alg∗ (A→ ). 19 Again, following our general convention, we omit the reference to a particular implication when clear from the context.

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Furthermore, in Example 4.6.14 we will present two different natural algebraically implicative logics (expanding A→ and asA→ ) with the same equivalent algebraic semantics. However, this situation can also be illustrated by the following ad hoc example. Example 2.9.8 Consider the three-valued MV-algebra MV3 and use it to define two logics in LCL : Ł3 =  hMV3 , {1}i and J3 =  hMV3 , { 1 ,1}i . 2 Since Ł3 is clearly Rasiowa-implicative, it is algebraically implicative with the ¯ It is easy to show that J3 is also algebraically implicative defining set E = {𝑥 ≈ 1}. ¯ with the defining set {1 ≈ ¬𝑥 → 𝑥}; it suffices to check the validity of the condition (Alg) for hMV3 , { 12 , 1}i. Since this matrix is clearly reduced, we have an example of an algebraically implicative logic which is not Rasiowa-implicative and thus Ł3 ≠ J3 . On the other hand, we can use Corollary 3.8.3 to prove that Alg∗ (Ł3 ) = Alg∗ (J3 ). We have already seen that Rasiowa-implicative logics have very simple sets of defining equations. The next example shows that in algebraically implicative logics without weakening the sets can be more complex. Example 2.9.9 The logic BCIlat introduced in Example 2.3.7 is algebraically implicative with defining equations E = {(𝑥 → 𝑥) ∧ 𝑥 ≈ 𝑥 → 𝑥}. To show that this is indeed the case, we prove the condition (Alg) for this set of defining equations: 𝑝 a`L ( 𝑝 → 𝑝) ∧ 𝑝 ⇔ ( 𝑝 → 𝑝). The left-to-right direction: first observe that 𝑝 `BCIlat ( 𝑝 → 𝑝) ∧ 𝑝 → ( 𝑝 → 𝑝) (due to the axiom (lb1 )). Now we show that 𝑝 `BCIlat ( 𝑝 → 𝑝) → ( 𝑝 → 𝑝) ∧ 𝑝 by writing a proof from the premise 𝑝: a) ( 𝑝 → 𝑝) → ( 𝑝 → 𝑝) b) 𝑝 → (( 𝑝 → 𝑝) → 𝑝) c) ( 𝑝 → 𝑝) → 𝑝 d) ( 𝑝 → 𝑝) → ( 𝑝 → 𝑝) ∧ 𝑝

(id) a and (E) b, premise 𝑝, and (MP) a, c, and (Inf)

The right-to-left direction: clearly from ( 𝑝 → 𝑝) ⇔ ( 𝑝 → 𝑝) ∧ 𝑝 we obtain ( 𝑝 → 𝑝) ∧ 𝑝 (due to the axiom (id)) and thus also 𝑝 (due to the axiom (lb2 ) and (MP)). The previous two examples together with the already established facts that (a) there exist logics that are not weakly implicative, (b) the logic BCI is weakly implicative but not algebraically, and (c) the logics BCIlat and J3 are algebraically implicative but not Rasiowa-implicative, allow us to observe that the four classes of logics that we are studying are indeed different. Proposition 2.9.10 The class of all logics and the classes of weakly, algebraically, and Rasiowa-implicative logics are pairwise different.

2.9 Algebraically implicative logics

75

As another illustrative example, recall that the equivalence fragment of classical logic CL↔ is weakly implicative with the implication ↔. In Example 2.8.19 we have seen that it is not Rasiowa-implicative, but we can easily observe that it is algebraically implicative with the set of equations E ( 𝑝) = {𝑝 ≈ 𝑝 ↔ 𝑝}. The next proposition shows that the defining equations of an algebraically implicative logic allow us to build, from its presentation, a description of the class of its algebras based on (quasi)equations (and generalized quasiequations, if the logic is not finitary); see the Appendix for the necessary algebraic notions. Proposition 2.9.11 Let L be an algebraically implicative logic with defining equations E and a presentation AS. Then, A ∈ Alg∗ (L) iff the following (generalized) quasiequations hold in A for each 𝛼 ≈ 𝛽 ∈ E:20 𝛼(𝜑) ≈ 𝛽(𝜑) Ó E [Γ] → 𝛼(𝜑) ≈ 𝛽(𝜑) Ó E [𝑥 ⇔ 𝑦] → 𝑥 ≈ 𝑦.

for each axiom 𝜑 for each rule Γ I 𝜑

Proof Due to claim 2 of Theorem 2.9.5, we know that each algebra A ∈ Alg∗ (L) satisfies these (generalized) quasiequations (the last one follows from the fact that A is the algebraic reduct of a reduced matrix). Conversely, let us define 𝐹 = {𝑎 | 𝜇A (𝑎) = 𝜈 A (𝑎) for each 𝜇 ≈ 𝜈 ∈ E} and take the matrix A = hA, 𝐹i. It is easy to see that AS ⊆ A (due to the first two conditions) and, since the last condition can be read as ‘if 𝑎 ⇒ 𝑏 ∈ 𝐹 and 𝑏 ⇒ 𝑎 ∈ 𝐹, then 𝑎 = 𝑏’, we obtain that A ∈ Alg∗ (L).  Corollary 2.9.12 If L is an algebraically implicative logic, then Alg∗ (L) is a generalized quasivariety. Furthermore, L is finitary if and only if Alg∗ (L) is a quasivariety and the set E can be taken to be finite. Proof The first claim is a direct consequence of the previous proposition. Next, assume that L is finitary. Using claim 3 of Theorem 2.9.5, one can see (exercise for the reader) that the set E can be taken to be finite. Therefore, the previous proposition gives us the set of quasiequations to axiomatize Alg∗ (L). Conversely, we use claim 2 of Theorem 2.9.5: assume that Γ `L 𝜑, then E [Γ] Alg∗ (L) E (𝜑). Thus, for each 𝛼 ≈ 𝛽 ∈ E, E [Γ] Alg∗ (L) 𝛼(𝜑) ≈ 𝛽(𝜑). Since Alg∗ (L) is a quasivariety, its equational consequence is finitary, and thus there is a finite set of formulas Γ0 ⊆ Γ such that E [Γ0] Alg∗ (L) 𝛼(𝜑) ≈ 𝛽(𝜑). Now, using that E is finite, we have finitely many of such finite subsets of Γ; let Γ0 ⊆ Γ be their union. We have that E [Γ0 ] Alg∗ (L) E (𝜑) and so Γ0 `L 𝜑.  Note that when applying Proposition 2.9.11 to a Rasiowa implicative logic L without the constant > we should use the defining equation 𝑥 ≈ 𝑥 ⇒ 𝑥, which translates any formula 𝜑 into the equation 𝜑 ≈ 𝜑 ⇒ 𝜑. However, we know that in any L-algebra A we have A 𝜑 ⇒ 𝜑 ≈ 𝑥 ⇒ 𝑥, i.e. we could translate 𝜑 using the simpler but equivalent equation 𝜑 ≈ 𝑥 ⇒ 𝑥; let us illustrate it on the case BCK-algebras. 20 Of course, it suffices to take for each scheme of axioms/rules the one which allows us to obtain the others as substitutional instances.

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Example 2.9.13 The quasivariety21 of BCK-algebras is defined using the following quasiequations: • • • • •

(𝑥 → 𝑦) → ((𝑦 → 𝑧) → (𝑥 → 𝑧)) ≈ 𝑥 → 𝑥 (𝑥 → (𝑦 → 𝑧)) → (𝑦 → (𝑥 → 𝑧)) ≈ 𝑥 → 𝑥 𝑥 → (𝑦 → 𝑥) ≈ 𝑥 → 𝑥 If 𝑥 ≈ 𝑥 → 𝑥 and 𝑥 → 𝑦 ≈ 𝑥 → 𝑥, then 𝑦 ≈ 𝑥 → 𝑥 If 𝑥 → 𝑦 ≈ 𝑥 → 𝑥 and 𝑦 → 𝑥 ≈ 𝑥 → 𝑥, then 𝑥 ≈ 𝑦.

Notice that, for any BCK-algebra A, the BCK-filters on A are subsets 𝐹 containing 𝑎 → 𝑎 for each 𝑎 ∈ 𝐴 and closed under modus ponens (i.e. if 𝑎 ∈ 𝐹 and 𝑎 → 𝑏 ∈ 𝐹, then 𝑏 ∈ 𝐹). As an exercise, the reader can characterize Alg∗ (L) and L-filters for prominent extensions of BCK introduced in this chapter. Consider an arbitrary algebraically implicative logic L. Due to the previous proposition and corollary, we know that for each of its extensions S, the class Alg∗ (S) is a generalized subquasivariety of Alg∗ (L). Furthermore, Alg∗ (S) is a subquasivariety (or relative subvariety) of Alg∗ (L) whenever S is finitary (resp. axiomatic) extension of L. We could also easily prove a dual claim: given a description of some relative generalized subquasivariety K of Alg∗ (L), we could use Proposition 2.9.3 to present an axiomatization of the corresponding extension LK of L whose equivalent algebraic semantics is K (thanks to Theorem 2.9.5). Putting it together, it is not hard to prove (we leave the details as an exercise for the reader) that there is a dual order isomorphism between extensions of L and generalized subquasivarieties of Alg∗ (L) (which specializes to isomorphisms between: (1) finitary extensions and subquasivarieties and (2) axiomatic extensions and subvarieties). Theorem 2.9.14 Let L be an algebraically implicative logic in a language L with defining equations E. Ó 1. Let { Π𝑖 → 𝛼𝑖 ≈ 𝛽𝑖 | 𝑖 ∈ 𝐼} be a set of (generalized) quasiequations that, relative to Alg∗ (L), axiomatizes a (generalized) subquasivariety K ⊆ Alg∗ (L). Then, the extension LK of L by the consecutions Ð • {𝛿 ⇔ 𝜀 | 𝛿 ≈ 𝜀 ∈ Π𝑖 } I 𝛼𝑖 ⇔ 𝛽𝑖 for each 𝑖 ∈ 𝐼 is an algebraically implicative logic with defining equations E and Alg∗ (LK ) = K. 2. Let S be the extension of L by {Γ𝑖 I 𝜑𝑖 | 𝑖 ∈ 𝐼}. Then, S is also algebraically implicative, Alg∗ (S) is axiomatized relative to Alg∗ (L) by the (generalized) quasiequations Ó • E [Γ𝑖 ] → 𝛼 ≈ 𝛽 for each 𝑖 ∈ 𝐼 and each 𝛼 ≈ 𝛽 ∈ E [𝜑𝑖 ], and LAlg∗ (S) = S. 21 In [311] it is shown that it is actually a proper quasivariety, i.e. it is not a variety.

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77

Moreover, the mappings L ( ·) and Alg∗ (·) are mutually inverse and give dual order isomorphisms between • the extensions of L and generalized subquasivarieties of K, • finitary extensions of L and relative subquasivarieties of K, and • axiomatic extensions of L and relative subvarieties of K.

Example 2.9.15 Consider the logic Ł and its infinitary proper extension Ł∞ . We know that both logics are algebraically implicative with defining equation E = {𝑥 ≈ >}. Moreover, we know that Alg∗ (Ł) = MV. Now, thanks to the previous theorem, we obtain that Alg∗ (Ł∞ ) is a proper generalized subquasivariety of MV. Example 2.9.16 We leave it as an exercise for the reader to show that Alg∗ (BCKlat ) is the variety defined using the following equations: (𝑥 → 𝑦) → ((𝑦 → 𝑧) → (𝑥 → 𝑧)) ≈ > 𝑥→>≈> 𝑥→𝑥∨𝑦≈> 𝑥∨𝑦 ≈ 𝑦∨𝑥 𝑥∨𝑥 ≈𝑥 (𝑥 ∨ 𝑦) ∨ 𝑧 ≈ 𝑥 ∨ (𝑦 ∨ 𝑧) 𝑥 ∨ ((𝑥 → 𝑦) → 𝑦) ≈ ((𝑥 → 𝑦) → 𝑦)

>→𝑥≈𝑥 ⊥→𝑥≈> 𝑥∧𝑦→𝑥≈> 𝑥∧𝑦 ≈ 𝑦∧𝑥 𝑥∧𝑥 ≈𝑥 (𝑥 ∧ 𝑦) ∧ 𝑧 ≈ 𝑥 ∧ (𝑦 ∧ 𝑧) 𝑥 ∧ ((𝑥 → 𝑦) → 𝑦) ≈ 𝑥.

Therefore, for any axiomatic extension L of BCKlat , the class Alg∗ (L) is a variety (cf. Example 2.9.2). Example 2.9.17 Recall that Alg∗ (K) is the class MA of modal algebras. From the previous theorem, we immediately know that the class Alg∗ (T) of the global modal logic T (resp. K4 or S4) is the subvariety of MA axiomatized by the identity 𝑥 → 𝑥 ≈ > (resp. 𝑥 → 𝑥 ≈ > or both identities).

2.10 History and further reading This chapter can be seen as an introduction to the basic notions of (abstract) algebraic logic. We have presented the first seven sections in a very general fashion, while the remaining two, for didactic reasons, have already been restricted to the main framework of this book: weakly implicative logics. Algebraic logic has been developed since the inception of mathematical logic in the nineteenth century with the pioneering algebraic studies of classical logic by Augustus De Morgan [238], George Boole [50], and Charles Sanders Peirce [261], which were later systematized by Ernst Schröder in [282]. In parallel to these works, classical logic and the foundations of mathematics were also studied from the point of view of formal proof systems, with the essential contributions of Gottlob Frege [139],

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Bertrand Russell and Alfred North Whitehead [308], David Hilbert and Wilhelm Ackermann [185] (who employed the kind of axiomatic systems we have chosen for our presentation), and Gerhard Gentzen [150, 151], among many others. The twentieth century saw a progressive proliferation of logical systems and their algebras that—following a variety of motivations of mathematical, philosophical, and application-driven nature—somehow deviated from the paradigm of classical propositional logic either by increasing its expressivity (e.g. by adding modalities) or by challenging some of the classical properties. Let us list a few historical notes (in chronological order) regarding the examples we have seen in the chapter: 1918

The modern study of modal logics started with Clarence Irving Lewis [215] with the aim of freeing logic from the paradoxes of material implication by replacing it by the strict implication ( 𝑝 → 𝑞).

1920

The first published work on many-valued logics was [217] by Jan Łukasiewicz, in which he introduced the logic of the matrix Ł3 to deal with the problem of future contingents.

1930

The logic Ł—and its infinitely-valued semantics on [0, 1]—was defined by Łukasiewicz and Alfred Tarski in [218], as a mathematically interesting extension of Łukasiewicz’s finitely-valued systems. They conjectured the completeness of the axiomatic system with respect to the semantics.

1930

Intuitionistic logic was defined by Arend Heyting in [184], as a formalization of reasoning in intuitionistic mathematics.

1932

Kurt Gödel in [154] used finite-valued matrices (which can be seen as substructures of the G-matrix G∞ ) to prove that IL is not finitely-valued.

1949

Leon Henkin studied, among other systems, the logics CL→ and IL→ in [180].

1951

Tarski and Bjarni Jónsson [201, 202] introduced modal algebras as Boolean algebras with additional operators and gave a generalization of Stone’s representation [288].

1956

Carew Arthur Meredith started the study of the logic BCI and its extensions (as reported in the second edition of [264]) as purely implicational logics formalizing some properties of the combinators introduced by Moses Schönfinkel [281] and developed by Haskell Curry [96].

1958

A restricted form of completeness of Łukasiewicz logic w.r.t. the matrix Ł∞ was proved by Alan Rose and J. Barkley Rosser in [274] using a syntactical approach.

1958

Chen Chung Chang, in [63], introduced MV-algebras (and, in particular, the Chang algebra C).

1959

Building on the work of Gödel [154], Michael Dummett in [113] introduced the logic G as the extension of IL by the axiom of prelinearity (this axiom and logic had appeared implicitly already in a 1913’s paper of Thoralf Skolem [285]) and proved its completeness with respect to any infinitely-valued linearly ordered G-matrix (in our terminology).

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79

1959

Building on his previous work, Chen Chung Chang, in [64] gave a semantical proof of Rose and Rosser result on completeness of Łukasiewicz logic.

1963

Louise Schmir Hay in the very last line of her paper on predicate Łukasiewicz logic [177] introduced the infinitary rule (Ł∞ ) and thus implicitly proved full completeness of the logic Ł∞ w.r.t. the matrix Ł∞ .

1966

Yasuyuki Imai and Kiyoshi Iseki introduced BCI-algebras in [193].

1978

Iseki and Shotaro Tanaka introduced BCK-algebras in [194].

1979

Robert Meyer and John Slaney presented the contraclassical Abelian logic (whose relation to logic A→ we explore in Example 4.6.14) in a meeting of the Australasian Association for Logic, though their work was only published ten years later in [225] (in 1989 Ettore Casari also published an independent discovery of the same logic in [56]; see also [53, 257]).

With the new multiplicity of propositional logics, algebraic logic evolved into the study of classes of algebras providing semantics for non-classical logics, with a heavy use of tools from universal algebra. Indeed, the latter discipline in its modern setting was founded and systematically developed by Garrett Birkhoff as the study of arbitrary classes of algebraic structures, following the footsteps of Whitehead [307]. In particular, he introduced many notions and results that we have included in the Appendix and in this chapter (and will be the basis for many of the upcoming results) such as equational class and consequence, subdirect representation, and subdirectly irreducible algebras [37]. Other tools particularly important for algebraic logic were created by the Russian school of universal algebra, especially in the works of Anatolii Mal’cev [220]. The basic connection between a propositional logic and a class of algebras was first obtained for classical logic by means of the Lindenbaum–Tarski method [295, 296]. Such result could be analogously obtained in other, non-classical, logics. Capitalizing on this fact, abstract algebraic logic (AAL) developed a general study, of increasing abstraction, of logics (seen as structural consequence operators) and their algebraic counterparts. Consequence operators were introduced by Tarski in [294], the condition of Structurality (invariance under substitutions) was added by Jerzy Łoś and Roman Suszko in [216] (and they used it to prove Proposition 2.2.10), and Ryszard Wójcicki introduced the general matrix semantics and reduced matrix models in [309] and implicitly obtained the corresponding completeness theorems 2.6.3 and 2.7.6. In 1974 Helena Rasiowa focused on the particularly well-behaved class of logical systems that we have called Rasiowa-implicative logics (she referred to them simply as implicative logics) and developed their theory in the landmark monograph [269]. The core modern theory of AAL, however, did not start developing until the 1980s, in the works of Willem J. Blok, Janusz Czelakowski, Don L. Pigozzi, Ramon Jansana, Josep Maria Font and others, with the identification of the crucial role of the Leibniz operator that assigns to each filter its Leibniz congruence. The properties of this operator have given rise to the Leibniz hierarchy consisting of the following classes of logics depicted in Figure 2.2 (in Roman font): protoalgebraic logics, (finitely)

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equivalential logics, and (finitely/regularly/weakly algebraizable logics [40, 41, 100]. These classes have also been characterized in terms of the existence and properties of equivalence sets (cf. Theorem 2.7.7): • The existence of a (finite) equivalence set ⇔ characterizes (finitely) equivalential logics. • If furthermore ⇔ satisfies the consecutions (Alg) from Theorem 2.9.5 for some set of equations (or equivalently, if there is a set of equations defining filters of reduced models), we obtain (finitely) algebraizable logics • If ⇔ obeys the consecutions 𝜑, 𝜓 ` 𝜑 ⇔ 𝜓 (or equivalently, if all reduced models have singleton filters), we obtain regularly (finitely) algebraizable logics. • Finally, if we allowed an even more general notion of equivalence set (possibly involving parameters), we would obtain the classes of protoalgebraic, weakly algebraizable, and regularly weakly algebraizable logics (see Section 3.9 for a precise definition of the class of protoalgebraic logics using the Leibniz operator). The Leibniz hierarchy can be expanded with other classes of logics such as order-algebraizable logics [45, 237, 267] or according to the definability properties of their truth predicates [265]. Moreover, an alternative classification of logics in AAL is the Frege hierarchy [101, 236]. Readers interested in the core modern theory of AAL are referred to the survey [136] and the comprehensive monographs [100, 134, 135] and references thereof.22 In particular, this literature contains a wealth of bridge theorems that connect logical properties (such as the deduction theorems that we have seen) with equivalent algebraic properties of their semantics. Finally, readers interested in further levels of generalization and abstraction of the AAL-approach may consult e.g. [3, 4, 80, 147, 234, 235]. The class of weakly implicative logics was introduced in [74] by the first author of this book as a generalization of Rasiowa-implicative logics. Unbeknownst to the author, the conditions required for weak implications had been singled out independently by different authors before. On the one hand, Richard Sylvan (née Francis Richard Routley) and Meyer already in their 1975 paper [277] argued on page 70 that these conditions are in a certain sense minimal sufficient properties for a connective to be a reasonable implication. They noted that equivalence is a special kind of weak implication and argued that what distinguishes implication is its interplay with other connectives, which is something that we study in detail in Chapter 4. On the other hand, in 1980 Jacek K. Kabziński, in an attempt to capture natural notions of (pre)ordered matrix models by syntactical means, also defined the notion of weakly implicative logics, with different notation and terminology, in [203], building on previous works by Suszko [290, 291]. Regarding our terminological choices, let us note that the term ‘consecution’, taken from [5], was already used in [74]; also, in the terminology of [115], matrices for weakly implicative logics coincide with the class of the prestandard matrices while ordered matrices coincide with standard matrices. 22 It is worth noting that some of these monographs (e.g. [100]) use the term ‘L-algebras’ for the members of Alg∗ (L) for an arbitrary logic; others (e.g. [134] however use it for the members of a wider class of algebras which however, in protoalgebraic logics, coincides with Alg∗ (L).

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Fig. 2.2 Extended Leibniz hierarchy.

Weakly implicative logics have been generalized by the authors of this book in [87,92,93] to weakly p-implicational logics, in a pure AAL fashion, by considering a generalized notion of implication which, instead of being a binary connective, can be defined by sets of (possibly parameterized) formulas. This broader class of logics is actually an alternative presentation of protoalgebraic logics; weakly implicative logics are a subclass of finitely equivalential logics, and algebraically implicative logics are exactly algebraizable weakly implicative logics. In this general setting, we could also consider the defining conditions of the Leibniz hierarchy together with a generalized form of the consecution (W) (defining Rasiowa-implicative logics); the resulting extended Leibniz hierarchy is depicted in Figure 2.2. It is worth mentioning that the results in the last two sections of this chapter (and many results in the subsequent chapters), which have been presented for weakly implicative logics, actually hold for some of the mentioned wider classes of logics studied by modern AAL. We have, however, decided to formulate these results for weakly implicative logics because, as opposed to the pure AAL setting, this offers a finer analysis of the notion of implication and related concepts and, unlike the setting of weakly p-implicational logics, weakly implicative logics offer simplicity while leaving only few interesting logics outside. See the history sections of the subsequent chapters for hints about possible generalizations of their results to wider classes of logics.

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2.11 Exercises Exercise 2.1 Prove Proposition 2.1.5. Exercise 2.2 Check that AIncL is a logic. Exercise 2.3 Given a logic L and a set of formulas Γ, prove that ThL (Γ) is the least theory containing Γ. Exercise 2.4 Finish the proof of Proposition 2.2.6. Exercise 2.5 In the proof of Lemma 2.2.7, check that `A S is a logic. Exercise 2.6 Prove Corollary 2.2.13. Exercise 2.7 Prove Proposition 2.4.6. Exercise 2.8 In Example 2.5.9, prove the result in the case of Łukasiewicz logic. Exercise 2.9 In Example 2.6.4, show that the list of provided examples is sufficient (and necessary) to prove that the axiomatic extensions of BCI introduced in Section 2.1 (and their expansions to the language LCL introduced in Example 2.3.7) are pairwise different. Exercise 2.10 Finish the details in Example 2.6.8. Exercise 2.11 Check that, based on a simple algebra with at least three elements, one can define two different reduced models for the minimum logic, and one of them with a non-singleton filter. Exercise 2.12 In Example 2.7.4, check that FiBCI (M) = {{1, >}, {𝑎, 1, >}, 𝑀 } and ΩM ({1, >}) = ΩM ({𝑎, 1, >}) = IdM . Exercise 2.13 Provide the missing formal proofs in Example 2.8.5. Exercise 2.14 In Example 2.8.6, show that 𝑙K4 and 𝑙S4 are weakly implicative. Hint: use the classical deduction theorem. Also, for the logics 𝑙K and 𝑙T, check that {𝑛 ( 𝑝 → 𝑞), 𝑛 (𝑞 → 𝑝) | 𝑛 ≥ 0} is an equivalence set, and does not have any finite subset behaving like an equivalence set. Exercise 2.15 Give syntactical proofs of Corollaries 2.8.7 and 2.8.8. Exercise 2.16 Check the details in Example 2.8.12. Exercise 2.17 In Exercise 2.8.14, give a formal proof of `Ł 𝜑 ∧ 𝜓 ⇔ ¬(¬𝜑 ∨ ¬𝜓). Exercise 2.18 In Example 2.8.15, check that the equivalence connective ↔ in classical logic is a weak implication interderivable with its symmetrization. Exercise 2.19 Formulate a general definition of definability of an 𝑛-ary connective 𝑑 by a formula 𝛿 with more than 𝑛 variables and prove the corresponding results.

2.11 Exercises

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Exercise 2.20 In Example 2.8.21, check that hA, {𝑡, >}i is reduced. Exercise 2.21 Show that Alg∗ (Min) does not contain all algebras. Hint: take an algebra A with three elements in a language with only one binary connective ∗ such that, for each 𝑎, 𝑏 ∈ 𝐴, 𝑎 ∗A 𝑏 = 𝑎. Exercise 2.22 In the proof of Corollary 2.9.12, show that the set E can be taken to be finite. Hint: use claim 3 of Theorem 2.9.5. Exercise 2.23 Complete Example 2.9.13 by characterizing Alg∗ (L) and L-filters for prominent extensions of BCK introduced in this chapter. Exercise 2.24 Given an algebraically implicative logic L, prove that there is a dual order isomorphism between extensions of L and generalized subquasivarieties of Alg∗ (L) (which specializes to isomorphisms between: (1) finitary extensions and subquasivarieties and (2) axiomatic extensions and subvarieties).

Chapter 3

Completeness properties

The previous chapter has set the stage with the presentation of the two usual faces of logic: syntax and semantics. We have studied both of them for arbitrary logics and, more intensively, for the well-behaved class of weakly implicative logics that is the main object of study in this book. Thus, we have seen logics as consequence relations given by syntactical proof systems or given by truth-preserving semantics (matricial semantics in general, which particularizes to algebraic semantics in the case of algebraically implicative logics). All of these notions have been bountifully illustrated by particular examples of propositional logics. Most importantly, we have shown the intimate link between the syntactical and semantical faces through general completeness theorems with respect to the semantics of all matrix models (Theorem 2.6.3) or all reduced matrix models (Theorem 2.7.6). However, as we have argued, such completeness theorems pay a dear price for their absolutely general scope of applicability: they are bound to general semantics that, as is clearly visible in prominent particular examples, fall too far from the intended semantics of many logical systems. Even in the specialization of the second general completeness theorem to the tame class of algebraically implicative logics (Theorem 2.9.5) we end up with the too wide algebraic semantics Alg∗ (L). For instance, this class gives us all Boolean algebras for classical logic CL, the class of G-algebras for the Gödel-Dummett logic G, or the class of MV-algebras for the Łukasiewicz logic Ł, which surpass by far the intended algebraic models that one would expect: the two-element algebra 2 for CL; the set of finite G-chains or just the standard G-algebra [0, 1] G for G; the set of finite MV-chains or just the standard MV-algebra [0, 1] Ł for Ł. The main goal of this chapter is to remedy the deficiencies of the general completeness theorems seen so far. To this end, given an arbitrary logic L, we allow these theorems to refer to any subclass K ⊆ Mod∗ (L) (disregarding, from now on, the rather meaningless general class Mod(L)). Such free-ranging variability not only lets us obtain, in particular, completeness results w.r.t. the intended semantics for the mentioned prominent examples, but also allows for general meaningful choices of the class K (such as Lindenbaum–Tarski matrices associated to certain kinds of theories). Moreover, to account for the different levels of strength of completeness © Springer Nature Switzerland AG 2021 P. Cintula and C. Noguera, Logic and Implication, Trends in Logic 57, https://doi.org/10.1007/978-3-030-85675-5_3

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results typically seen in prominent examples, we define three kinds of completeness properties for a given class of matrices K ⊆ Mod∗ (L) (and analogously for a class of L-algebras, when L is algebraically implicative): • Strong K-completeness (SKC): L coincides with K , i.e. each consecution is valid in L iff it is valid in K. • Strong finite K-completeness (FSKC): each finitary consecution is valid in L iff it is valid in K. • K-completeness (KC): each formula is valid in L iff it is valid in K. The chapter is mostly a general study of these three properties1 for which we present a wealth of characterization theorems.2 These results have two sources of inspiration (for precise historical and bibliographical remarks, see the last section): 1. The literature of many-valued logics has concentrated, to a large extent, on proving completeness of certain axiomatic systems with respect to particular intended algebraic models. Typically, one has an algebraically implicative logic L, its equivalent algebraic semantics Alg∗ (L), and a certain intended semantics A ⊆ Alg∗ (L). Very often, the strong completeness w.r.t. A is obtained through a two-step proof: first, showing that L is complete with respect to some intrinsically defined class of algebras B ⊆ Alg∗ (L) (such as the finitely subdirectly irreducible members of Alg∗ (L)) and, second, show that each algebra of B can be embedded into an algebra of A; finite strong completeness w.r.t. A is obtained similarly by proving in the second step that each algebra of B can be partially embedded into A. In this way one may prove, for instance, that G is strongly complete with respect to {[0, 1] G } or that Ł is finitely strongly complete with respect to {[0, 1] Ł }. Interestingly, such (partial or total) embeddability properties later were discovered to be, not only sufficient for, but actually equivalent to the completeness properties. 2. The areas of universal algebra and model theory have focused on the study of algebraic and first-order structures by means, among others, of general constructions that can be applied to arbitrary structures, including: homomorphisms, quotients, substructures, direct products, subdirect products, filtered products, and 1 There is yet another kind of completeness property sometimes studied in the literature: “a set of formulas Γ is consistent in L (i.e, Γ 0L 𝜑 for some formula 𝜑) iff it has a model over K (i.e. there is a matrix hA, 𝐹 i ∈ K and an A-evaluation 𝑒 such that 𝑒 [Γ] ⊆ 𝐹 )”. Clearly, the SKC implies the left-to-right direction of this property (to get the right-to-left direction, it suffices to have a formula 𝜒, typically a constant ⊥, such that we never have 𝑒 ( 𝜒) ∈ 𝐹 for any hA, 𝐹 i ∈ K and any A-evaluation 𝑒). In classical logic (for K = { h2, {1}i }) the converse statement is true as well, a fact which is known to be based on the following very strong logical property: “Γ 0L 𝜑 iff Γ ∪ {¬𝜑 } is consistent”. As this property (or a suitable variant) is true in very few logics we consider in this book, we have decided not to explore it in more detail. 2 Additional general results on completeness theorems will be given in subsequent chapters. Namely, in Chapter 4 we study completeness theorem of substructural logics with respect to completely ordered matrices; in Section 5.6 we use the presence of a suitable disjunction connective to strengthen or generalize several results of this chapter and study completeness properties with respect to finite matrices; in Section 6.4 we study the completeness properties of semilinear logics with respect to densely ordered models; and, finally, in Chapter 7 we provide predicate variants of basic completeness theorems.

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ultraproducts. These constructions and their corresponding class operators allow us to formulate, in a very compact way, important facts regarding axiomatizability, preservation of syntactical conditions, generation of classes of structures, characterization of elementary equivalence, etc. In particular, the mentioned properties of embeddability and partial embeddability between classes of structures can be expressed in general in terms of these operators. Abstract algebraic logic has greatly benefited from these developments and has expressed completeness properties of propositional logics in terms of generation of the class of (matricial or algebraic) models by means of the mentioned class operators. The characterizations of completeness theorems we offer in this chapter come in three levels of generality: (1) for arbitrary logics, when we can afford to present the results without an unreasonable effort, (2) for logics over countable sets of formulas, when the cardinality restriction allows for stronger characterizations in terms of countable models, and (3) for weakly implicative logics, when the presence of implication and the stronger properties of the Leibniz operator simplify substantially the proofs and/or offer stronger results. The presentation is almost completely self-contained: the chapter includes all the necessary definitions extending the notions and constructions mentioned above from algebras to matrices (the corresponding elementary algebraic definitions are collected in Section A.3 of the Appendix). Therefore, the text should be accessible to readers familiar with neither universal algebra nor model theory; readers already acquainted with these subjects can quickly skim through the standard definitions and elementary facts. The chapter is structured in terms of the mentioned class operators. That is, most sections are devoted to specific algebraic constructions and the properties of their corresponding class operators. Characterizations of (three kinds of) completeness properties are proved on the fly, as soon as the necessary theoretical elements are available. These characterizations, for a given logic L and a class K ⊆ Mod∗ (L), can be roughly classified as • localization (more or less tight) of the whole class Mod∗ (L) as a subclass of a class generated from K by means of certain operators;3 • localization (more or less tight) of the set of Lindenbaum–Tarski models; • variations of the above characterizations in which the cardinality of the localized models is restricted; and • variations of the above characterizations in which, instead of localization, we prove an equality of the class of models with that generated from K. Moreover, most sections come with valuable theoretical byproducts that enrich the presented knowledge of non-classical logics: completeness properties in natural extensions, complete semantics for fragments and characterization of conservative expansions, additional properties of the Leibniz operator, a study of the structure of the lattice of filters and theories of arbitrary logics, subdirect representation of finitary logics, and characterization of finitarity. The results are illustrated by examples introduced in the previous chapter. 3 Cf. e.g. the first instance of this result, Theorem 3.4.6, where we prove that a weakly implicative logic enjoys the KC if and only if Mod∗ (L) ⊆ HSP(K).

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3.1 Three kinds of completeness and natural extensions We start by formally introducing the three types of completeness properties mentioned in the introduction. Definition 3.1.1 Let L be a logic in a language L and K a class of L-matrices. We say that L enjoys the Strong K-Completeness, SKC for short, if for every set of formulas Γ ∪ {𝜑} ⊆ FmL , Γ `L 𝜑 if and only if Γ K 𝜑. We say that L enjoys the Finite Strong K-Completeness, FSKC for short, if the above equivalence holds for finite sets Γ and, finally, we say that L enjoys the K-Completeness, KC for short, if it holds for Γ = ∅. Recall that, for a given logic L, we denote by Thm(L) the set of its theorems (i.e. the set {𝜑 | `L 𝜑}) and by F C(L) its finitary companion (i.e. the largest finitary logic contained in L). Then, we can write the three above defined properties in a more compact way: SKC FSKC KC

⇔ ⇔ ⇔

L = K F C(L) = F C(K ) Thm(L) = Thm(K ).

It is obvious that the SKC implies the FSKC, and the FSKC implies the KC. Let us show that these implications cannot be reversed in general. To this end, we need the following straightforward proposition. Proposition 3.1.2 A finitary logic has the SKC iff it has the FSKC and K is finitary. Example 3.1.3 Recall that Łukasiewicz logic Ł is finitary whereas, for K being {Ł∞ } or {Ł𝑛 | 𝑛 ≥ 2}, the logics K are not (thanks to Examples 2.6.6 and 2.6.7). Therefore, the previous proposition tells us that it does not enjoy the SKC for any of these classes. Later, in Example 3.8.9, we will see that Ł enjoys the FSKC for both of these classes. Example 3.1.4 Consider the set K = {hFmLCL , Thm(Ł)i}. Clearly, Łukasiewicz logic enjoys the KC. Let us show that it does not enjoy the FSKC. Consider the consecution 𝑝 ↔ ¬𝑝 I ⊥ and note that, on one hand, 𝑝 ↔ ¬𝑝 0Ł ⊥ (just consider the Ł∞ -evaluation 𝑒 such that 𝑒( 𝑝) = 12 ). On the other hand, notice that there is no FmLCL -evaluation, i.e. no LCL -substitution, 𝜎 such that we have 𝜎( 𝑝 ↔ ¬𝑝) ∈ Thm(Ł) (indeed, it would entail that `Ł 𝜎( 𝑝) ↔ ¬ 𝜎( 𝑝) which is easily disproved by any 2-evaluation) and so indeed 𝑝 ↔ ¬𝑝 K ⊥.4 4 This is not an arbitrary example; clearly, any logic L in a language L enjoys the KC for K = { hFmL , Thm(L) i }. However, the question whether L enjoys the FSKC is equivalent to the question whether L is structurally complete. There is a vast literature on the topic with a plethora of examples of logics which are not structurally complete, see e.g. [31, 248, 263, 279]. Let us also note that the reasons for the failure of structural completeness are often not as trivial as in our example, in which we employed a non-unifiable set of premises, i.e. there is no substitution turning all the premises into theorems of the logic in question.

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Later, in Example 4.6.12, we show that the logic A→ enjoys the KC but not the FSKC for K being either {Z+ } or {Z0 } (recall Example 2.8.10). In the previous chapter, we have seen that each logic L enjoys the SKC when K is the class of all (reduced) matrix models of L or the class of Lindenbaum(–Tarski) matrices given by theories of L (see Theorems 2.6.3 and 2.7.6). From the previous chapter we also know the following two important facts (see Definition 2.5.5 and Lemma 2.7.5): • L ⊆ K iff K ⊆ Mod(L). • K =  {A∗ | A∈K} . Therefore, from now on we will only study the three completeness properties of a logic L w.r.t. classes of its reduced models, i.e. we have K ⊆ Mod∗ (L) as the standing assumption (unless said otherwise). Recall that, given an algebraically implicative logic L, each L-algebra A gives rise to a unique reduced L-model hA, 𝐹A i. Therefore, we can formulate the following definition. Definition 3.1.5 Let L be an algebraically implicative logic and A ⊆ Alg∗ (L). We say that L has the A-completeness, AC in symbols, if it has the KC for the class K = {hA, 𝐹A i | A ∈ A}. We define (finite) strong A-completeness analogously. One of the goals of this chapter is to show that completeness properties of a given logic w.r.t. a class K allow us to describe the class of its reduced models by applying model-theoretic operations to the class K. In other words, we will show that a logic L has a completeness property with respect to K if and only if Mod∗ (L) is generated from K using suitable operators. Recall that each logic is built using a set of formulas FmL of a fixed cardinality, which is the maximum of the cardinality of L and the cardinality of its (always infinite) set Var of variables (recall that, by definition, all of these sets are countable in the case of weakly implicative logics). On the other hand, matrices can be of arbitrary cardinality. Therefore, if we want to describe matrices using logical methods, we will need the means to construct an ‘arbitrarily big’ variant of a given logic. Since it is clear that we need to keep the set of connectives intact (to avoid changing the logic too much), we will have to increase instead the cardinality of the set of variables. To this end, we fix a set of variables Var 0 ⊇ Var and denote by FmL0 the set of formulas in a language L and variables Var 0. We will consider FmL0 -substitutions and logics over FmL0 , and 0 0 denote by Var K the semantical consequence over the set FmL with respect to K. Definition 3.1.6 Let L be a logic in a language L and a set of variables Var. Then, 0 for any set Var 0 ⊇ Var, the logic LVar over FmL0 defined semantically as 0

0

LVar = Var Mod(L) is called the natural extension of L to the set of variables Var 0. The following proposition describes the expected strong links between natural extensions and their starting logics.

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Proposition 3.1.7 Let L be a logic in a countable language L over a countable set 0 of variables Var. Take Var 0 ⊇ Var, the corresponding natural extension LVar , a class K ⊆ Mod∗ (L), and Γ ∪ {𝜑} ⊆ FmL0 . 0

1. For any presentation AS of L, the following set is a presentation of LVar : AS 0 = {𝜎[Δ] I 𝜎(𝜓) | Δ I 𝜓 ∈ AS and 𝜎 is an FmL0 -substitution}. 0

2. Mod(L) = Mod(LVar ). 0 3. LVar is a conservative expansion of L and if Γ `LVar0 𝜑, then there is a countable set Γ0 ⊆ Γ such that Γ0 `LVar0 𝜑. 4. Γ `LVar0 𝜑 iff there is a countable set Γ0 ⊆ Γ and a bijection 𝜏 : Var 0 → Var 0 such that 𝜏[Γ0 ∪ {𝜑}] ⊆ FmL and 𝜏[Γ0 ] `L 𝜏(𝜑).5 0 5. If 0LVar0 𝜑 and L has the KC, then 2Var K 𝜑. 0

6. If Γ is finite, Γ 0LVar0 𝜑, and L has the FSKC, then Γ 2Var K 𝜑.

0

7. If Γ is countable, Γ 0LVar0 𝜑, and L has the SKC, then Γ 2Var K 𝜑. Proof We prove the first two claims at once. Let us denote by L0 the logic axiomatized 0 by AS 0. Clearly, AS 0 ⊆ Var (indeed, for each A ∈ Mod(L), each A-evaluation Mod(L) 0 0 𝑒 : FmL → 𝐴, and each FmL -substitution 𝜎, the mapping 𝑒¯ : FmL0 → 𝐴 defined as 0 𝑒( ¯ 𝜒) = 𝑒(𝜎( 𝜒)) is an A-evaluation) and so Mod(L) ⊆ Mod(L0) and L0 ⊆ LVar . 0 0 Conversely, we first notice that obviously L ⊆ L , thus Mod(L) ⊇ Mod(L ) and 0 0 0 so LVar = Var = Var = L0. Therefore, we have: Mod(L) = Mod(L0) = Mod(L) Mod(L0 ) 0

Mod(LVar ). 0 To prove the third claim, we first observe that LVar is clearly an extension of L and, if Γ `LVar0 𝜑, then there is a proof of 𝜑 from Γ in AS 0. To show the conservativity, assume that Γ `LVar0 𝜑 for some Γ ∪ {𝜑} ⊆ FmL and consider the substitution 𝜎 sending each variable from Var to itself and the rest to a fixed 𝑝 ∈ Var. Take any proof of 𝜑 from Γ in AS 0 and observe that the same tree with labels 𝜓 replaced by 𝜎𝜓 is a proof of 𝜑 from Γ in AS. Finally, as obviously any rule in AS 0 has countably many premises, the set of formulas Γ0 ⊆ Γ of labels of leaves not labeled by axioms is countable and Γ0 `LVar0 𝜑. Before we continue with the proof of the fourth claim, let us note that for each countable set Δ ⊆ FmL0 , the set of variables occurring in its formulas is countable, and so it can be embedded into Var. This embedding can be clearly extended to a bijection 𝜏Δ : Var 0 → Var 0 such that 𝜏Δ [Δ] ⊆ FmL . Now the left-to-right direction of the fourth claim is easy: the existence of the set Γ0 is guaranteed by the third claim and so we can set 𝜏 = 𝜏Γ0 ∪{ 𝜑 } (indeed the fact that 0 𝜏[Γ0 ] `L 𝜏(𝜑) follows from the Structurality of LVar and its conservativity over L). To prove the converse, first observe that we have 𝜏[Γ0 ] `LVar0 𝜏(𝜑) and that the function 𝜏 −1 can be seen as an FmL0 -substitution as well. Therefore, by Structurality, we obtain 𝜏 −1 [𝜏[Γ0 ]] `LVar0 𝜏 −1 (𝜏(𝜑)) and, as 𝜏 −1 ◦ 𝜏 = Id, the claim follows. 5 We use the same symbol for the bijection on Var0 and the FmL0 -substitution it induces.

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We prove the final three claims at once. Assume that Γ 0LVar0 𝜑 and Γ is of an appropriate cardinality. Then, since Γ ∪ {𝜑} is countable, as in the proof of the fourth claim we obtain that 𝜏Γ0 ∪{ 𝜑 } [Γ] 0L 𝜏Γ0 ∪{ 𝜑 } (𝜑). The appropriate completeness property gives us hB, 𝐺i ∈ K and a B-evaluation 𝑒 such that 𝑒[𝜏Γ0 ∪{ 𝜑 } [Γ]] ⊆ 𝐺 and 0 𝑒(𝜏Γ0 ∪{ 𝜑 } (𝜑)) ∉ 𝐺. Considering the evaluation 𝑒 ◦ 𝜏Γ0 ∪{ 𝜑 } we obtain Γ 2Var K 𝜑.  Note that any weak implication of a given logic is still a weak implication in all its natural extensions; but, due to the cardinality restriction, a natural extension of a weakly implicative logic need not be weakly implicative. Remark 3.1.8 We can actually prove that, given a logic L in a countable language over 0 a countable set of variables Var, LVar is the only logic over FmL0 satisfying claim 3 of the previous proposition. Indeed, consider any such logic L0 and consider the set AS 0 = {Γ I 𝜑 | Γ `L0 𝜑 and Γ is countable}. This set is clearly an axiomatic system for L0 . Let us show that, seeing L as its own presentation, we obtain AS 0 = {𝜎[Δ] I 𝜎(𝜓) | Δ I 𝜓 ∈ L and 𝜎 is an FmL0 -substitution}. Then, by claim 1 of the previous proposition, the proof will be done. One direction is simple: take 𝜎[Δ] I 𝜎(𝜓) such that Δ `L 𝜓 and 𝜎 is an FmL0 -substitution. Then, Δ is countable and 𝜎[Δ] `L0 𝜎(𝜓). Conversely, assume that Γ is countable and Γ `L0 𝜑. Thus, due to Structurality, 𝜏Γ0 ∪{ 𝜑 } [Γ] `L0 𝜏Γ0 ∪{ 𝜑 } (𝜑) and by conservativity also 𝜏Γ0 ∪{ 𝜑 } [Γ] `L 𝜏Γ0 ∪{ 𝜑 } (𝜑); setting 𝜎 = 𝜏Γ−10 ∪{ 𝜑 } completes the proof. Claim 3 is sometimes used as a definition of a class of logics called natural extensions of L; but as we have just seen, our cardinality restrictions give us unicity 0 of LVar which justifies calling it the natural extension of L to Var 0. Remark 3.1.9 The restriction to countable Γs in the last point of the previous proposition cannot be omitted. Indeed, the Gödel-Dummett logic G enjoys the SKC for K = {G∞ } (see Example 3.6.18) and we leave as an exercise for the reader to show that for any set of variables Var 0 of cardinality bigger than the continuum and the set Γ = {( 𝑝 ↔ 𝑞) → 𝑟 | for each 𝑝, 𝑞 ∈ Var 0 \ {𝑟} such that 𝑝 ≠ 𝑞}, we have Γ 0GVar0 𝑟 0 and Γ Var 𝑟. G∞

3.2 Homomorphisms and congruences In this section we start introducing some basic notions on matrix theory (they expand the corresponding elementary algebraic notions, which can be found in Section A.3). Observe that an L-matrix hA, 𝐹i can be regarded as a first-order structure in the equality-free predicate language with functional symbols from L and a unique unary predicate symbol, with domain 𝐴, functionals interpreted as the operations of A, and the predicate interpreted by 𝐹. From this perspective, the following definition of homomorphism is not surprising.

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Definition 3.2.1 (Homomorphism) Let A = hA, 𝐹i and B = hB, 𝐺i be two Lmatrices. We say that an algebraic homomorphism 𝑔 : A → B is a matrix homomorphism from A to B if 𝑔[𝐹] ⊆ 𝐺. A matrix homomorphism 𝑔 : A → B is • a strict homomorphism if 𝑔[ 𝐴 \ 𝐹] ⊆ 𝐵 \ 𝐺 (i.e. 𝑎 ∈ 𝐹 iff 𝑔(𝑎) ∈ 𝐺). • an embedding if it is strict and injective. • an isomorphism if it is a surjective embedding. It should be noted that a bijective matrix homomorphism is not necessarily an isomorphism (because it need not be strict and, so, its inverse need not be a matrix homomorphism; see Remark 3.2.4). Remark 3.2.2 Consider an algebraic homomorphism 𝑔 : A → B. Then, for any sets 𝐹 ⊆ 𝐴 and 𝐺 ⊆ 𝐵, 𝑔 is a strict matrix homomorphism from hA, 𝑔 −1 [𝐺]i to hB, 𝐺i and a (non-necessarily strict) matrix homomorphism from hA, 𝐹i to hB, 𝑔[𝐹]i. It is easy to see that, for any given algebraically implicative logic L, a homomorphism/embedding of L-algebras A and B is also a homomorphism/embedding of the corresponding matrices hA, 𝐹A i and hB, 𝐹B i. We leave the proof of this fact as an exercise for the reader (note that, in the case of embedding, one has to prove also the strictness condition). From now on, we usually omit the prefix ‘matrix’ or ‘algebraic’ and speak simply about homomorphisms as it is usually clear from the context whether we speak about matrices or algebras. If there is a (strict) surjective homomorphism 𝑔 : A → B, then we say that B is a (strict) homomorphic image of A and that A is a (strict) homomorphic preimage of B. If, moreover, 𝑔 is an isomorphism, we say that A and B are isomorphic (in that case, note that they are a strict image and a strict preimage of each other). As in the case of algebras, it is easy to see that the five binary relations we have just defined on matrices are reflexive (the identity mapping is an isomorphism), transitive (we only have to check the additional fact that a composition of two (strict) homomorphisms is a (strict) homomorphism), and, in the case of isomorphism, also symmetric. Recall that in the proof of Lemma 2.7.5 we have seen that, for any logical congruence 𝜃 on a matrix A, the matrix A/𝜃 is a strict homomorphic image of A via the canonical mapping and that A = A/𝜃 . Thus, in particular, if A is a model of L, so is A/𝜃. We will explore the relation between homomorphisms and congruences in general at the end of this section. Now we reformulate Lemma 2.5.7 in a very compact way as a claim about closure of Mod(L) (for the proof just observe that, for any strict surjective homomorphism ℎ : hA, 𝐹i → hB, 𝐺i, we have 𝐺 = ℎ[𝐹] and 𝐹 = ℎ−1 [𝐺]). Proposition 3.2.3 Let L be a logic. Then, Mod(L) is closed under strict homomorphic images and strict homomorphic preimages (in particular, under isomorphisms). Remark 3.2.4 Let us observe that the class Mod(L) need not be closed under nonstrict homomorphic (pre-)images. Indeed, any matrix A = hA, 𝐹i can be seen as both a homomorphic image of the matrix hA, ∅i and homomorphic preimage of the matrix

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hA, 𝐴i via the identity. Since the matrix hA, ∅i is a model of the almost inconsistent logic and hA, 𝐴i ∈ Mod(Inc) is a model of the inconsistent logic, the same would hold for the matrix A, a contradiction. For a less trivial example for the failure of closure under preimages, recall that in Example 2.5.10 we have shown that h[0, 1] G , (0, 1]i ∈ Mod(CL). It is easy to see that the identity is a bijective homomorphism from h[0, 1] G , {1}i to h[0, 1] G , (0, 1]i and that h[0, 1] G , {1}i ∉ Mod(CL) (indeed, for any 𝑥 ∈ (0, 1), we have 𝑥 ∨ (𝑥 → 0) = 𝑥 ≠ 1 and so 2 h[0,1] G , {1}i (lem)). As an exercise, the reader can check the following non-trivial example for the failure of closure under images: the identity homomorphism from h[0, 1] Ł , {1}i ∈ Mod(Ł) to h[0, 1] Ł , [ 12 , 1]i ∉ Mod(Ł). Let us now explore the closure properties of the class Mod∗ (L). It is obviously not closed under strict homomorphic preimages: indeed any matrix A is a strict preimage of its reduction A∗ via the canonical mapping, so any non-reduced matrix is a counterexample. In Corollary 3.2.6 we prove that Mod∗ (L) is closed under strict homomorphic images for a trivial reason: a strict homomorphic image of a reduced matrix A is isomorphic to A. To prove this claim, we prepare a very useful lemma. Lemma 3.2.5 Let A and B be L-matrices and ℎ : A → B a strict homomorphism. Then, for each 𝑎, 𝑏 ∈ 𝐴 such that hℎ(𝑎), ℎ(𝑏)i ∈ Ω(B), we have h𝑎, 𝑏i ∈ Ω(A). If, furthermore, ℎ is surjective, then the converse implication holds as well. Proof Let 𝐹 and 𝐺 be, respectively, the filters of A and B. Recall that h𝑎, 𝑏i ∈ Ω(A) iff for each formula 𝜒 and each A-evaluation 𝑒 it is the case that 𝑒 𝑝=𝑎 ( 𝜒) ∈ 𝐹 iff 𝑒 𝑝=𝑏 ( 𝜒) ∈ 𝐹. Observe that ℎ ◦ 𝑒 is a B-evaluation and we have: 𝑒 𝑝=𝑎 ( 𝜒) ∈ 𝐹 iff ℎ(𝑒 𝑝=𝑎 ( 𝜒)) ∈ 𝐺 iff (ℎ ◦ 𝑒) 𝑝=ℎ (𝑎) ( 𝜒) ∈ 𝐺 iff (ℎ ◦ 𝑒) 𝑝=ℎ (𝑏) ( 𝜒) ∈ 𝐺 iff ℎ(𝑒 𝑝=𝑏 ( 𝜒)) ∈ 𝐺 iff 𝑒 𝑝=𝑏 ( 𝜒) ∈ 𝐹 (the third equivalence follows from our assumption that hℎ(𝑎), ℎ(𝑏)i ∈ Ω(B); the remaining ones are straightforward). The proof of the second claim is similar; just notice that, as ℎ is surjective, for each B-evaluation 𝑒, there has to be an A-evaluation 𝑒 0 such that 𝑒 = ℎ ◦ 𝑒 0.  Corollary 3.2.6 Let A and B be L-matrices and assume that A is reduced. Then, any strict homomorphism ℎ : A → B is an embedding. Thus, every strict homomorphic image of A is reduced and isomorphic to A. Therefore, the class Mod∗ (L) is closed under strict homomorphic images. Proof Consider elements 𝑎, 𝑏 ∈ 𝐴 such that ℎ(𝑎) = ℎ(𝑏). Thus, hℎ(𝑎), ℎ(𝑏)i ∈ Ω(B) which, by Lemma 3.2.5, gives h𝑎, 𝑏i ∈ Ω(A) and so 𝑎 = 𝑏. Thus, if ℎ is also surjective, it is an isomorphism. Let us show that B is reduced: assume that h𝑎, 𝑏i ∈ Ω(B) for some 𝑎, 𝑏 ∈ 𝐵; as h𝑎, 𝑏i = hℎ(ℎ−1 (𝑎)), ℎ(ℎ−1 (𝑏))i, we obtain hℎ−1 (𝑎), ℎ−1 (𝑏)i ∈ Ω(A) which entails ℎ−1 (𝑎) = ℎ−1 (𝑏) and so 𝑎 = 𝑏.  Recall that the canonical mapping from A to A∗ is a strict surjective homomorphism; thus, this corollary has given us the promised proof of the fact that A∗ is isomorphic to (A∗ ) ∗ and even to A, if A is reduced (cf. Lemma 2.7.5). Let us now introduce a bunch of operators which can be understood as the closure of a class of matrices under the constructions defined above.

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Definition 3.2.7 (Homomorphism related class operators) Let K be a class of L-matrices. We define the following class operators: H(K) = HS (K) = H−1 (K) = H−1 S (K) = I(K) =

the class of homomorphic images of matrices from K the class of strict homomorphic images of matrices from K the class of homomorphic preimages of matrices from K the class of strict homomorphic preimages of matrices from K the class of matrices isomorphic to some matrix from K.

In subsequent sections we will introduce several additional operators. We will often combine them; in order to lighten the notation we omit the brackets and write e.g. ‘HS H(K)’ instead of ‘HS (H(K))’. Let us now formulate the basic properties of these operators and their compositions; including, among others, that they are indeed closure operators. These properties can be seen as schematic compact descriptions of some of the results proved before. We will freely use them from now on, without explicit reference to this result (similarly, for properties of other basic class operators introduced later). Proposition 3.2.8 • Let X be one of the operators I, H, HS , H−1 , or H−1 S . Then, X is monotonic, increasing and idempotent, i.e. for each pair of classes of L-matrices K ⊆ K 0, X(K) ⊆ X(K 0)

K ⊆ X(K) = XX(K).

• For each class K of L-matrices K, we have K ⊆ I(K) ⊆ HS (K) ⊆ H(K). If all matrices in K are reduced, then also: HS (K) = I(K). • For any logic L, we have H−1 S HS (Mod(L)) = Mod(L)

HS (Mod∗ (L)) = I(Mod∗ (L)) = Mod∗ (L).

Our next goal is to show that the reduced models of a logic are essentially its Lindenbaum–Tarski matrices. First, we need to prepare one additional interesting corollary of Lemma 3.2.5. Corollary 3.2.9 Let A and B be L-matrices and ℎ : A → B a strict surjective homomorphism. Then, the mapping 𝑔 : A∗ → B∗ defined as 𝑔(𝑎/Ω(A)) = ℎ(𝑎)/Ω(B) is an isomorphism. Proof We first show that 𝑔 is a well-defined one-one mapping. We know that 𝑎/Ω(A) = 𝑏/Ω(A) iff h𝑎, 𝑏i ∈ Ω(A) iff (due to Lemma 3.2.5) hℎ(𝑎), ℎ(𝑏)i ∈ Ω(B) iff ℎ(𝑎)/Ω(B) = ℎ(𝑏)/Ω(B) iff 𝑔(𝑎/Ω(A)) = 𝑔(𝑏/Ω(A)). It is easy to see that 𝑔 is a surjective algebraic embedding of A/Ω(A) into B/Ω(B). Thus, it only remains to show that it is a strict matrix homomorphism. Assume that 𝐹 and 𝐺 are the filters of A and B respectively and observe that we have this chain of equivalences: 𝑎/Ω(A) ∈ 𝐹/Ω(A) iff 𝑎 ∈ 𝐹 iff ℎ(𝑎) ∈ 𝐺 iff 𝑔(𝑎/Ω(A)) = ℎ(𝑎)/Ω(B) ∈ 𝐺/Ω(B). 

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Second, we have to deal with the problem that matrices can be of arbitrary cardinality whereas the size of Lindenbaum–Tarski matrices is limited by that of the set of formulas (recall that, by definition, it is countable if L is weakly implicative). Exactly for this purpose we have introduced natural extensions in the previous section; 0 recall that by LVar we denote the natural extension of a logic L to Var 0. Given a class of matrices K and an infinite cardinal 𝜅, we denote by K 𝜅 the class of elements of K of cardinality not greater than 𝜅. Theorem 3.2.10 Let L be a logic over FmL , 𝜅 a cardinal greater than or equal to the cardinality of FmL . Then, for any set Var 0 ⊇ Var of cardinality 𝜅, 0

Mod∗ (L) 𝜅 = I({hFmL0 , 𝑇i ∗ | 𝑇 is an LVar -theory}). In particular, if L is weakly implicative, then Mod∗ (L) 𝜔 = I({hFmL , 𝑇i ∗ | 𝑇 is an L-theory}). Proof Due to the cardinality restriction and Proposition 3.1.7, we know that, for 0 0 each LVar -theory 𝑇, we have hFmL0 , 𝑇i ∗ ∈ Mod∗ (LVar ) 𝜅 = Mod∗ (L) 𝜅 ; thus, one inclusion follows using the properties of I. To prove the converse direction, con0 sider a matrix hA, 𝐹i ∈ Mod∗ (L) 𝜅 = Mod∗ (LVar ) 𝜅 and notice that there is a 0 surjective mapping 𝑔 : Var → 𝐴, which gives rise first to a surjective algebraic homomorphism 𝑔 : FmL0 → A and then to a strict surjective matrix homomorphism 𝑔 : hFmL0 , 𝑔 −1 [𝐹]i → hA, 𝐹i. Firstly we observe that, thanks to Proposition 3.1.7, we 0 know that hA, 𝐹i ∈ Mod∗ (LVar ) 𝜅 and so, by Proposition 3.2.8 and 2.6.1, we know 0 that 𝑔 −1 [𝐹] is an LVar -theory. Secondly, we use Corollary 3.2.9 to show that hA, 𝐹i ∗ is isomorphic to hFmL0 , 𝑔 −1 [𝐹]i ∗ and thus, by Corollary 3.2.6, so is hA, 𝐹i.  Now we prove a strengthening of Lemma 3.2.5 and Corollary 3.2.9 to weakly implicative logics. The key difference is that, to prove the converse direction of Lemma 3.2.5, ℎ need not be surjective. Actually, we prove even more: we show that homomorphisms between matrices of a given weakly implicative logic respect the matrix preorders and the strict ones even reflect it. Lemma 3.2.11 Let L be a weakly implicative logic, A, B ∈ Mod(L), and ℎ : A → B a homomorphism. Then, for each 𝑎, 𝑏 ∈ 𝐴, • if 𝑎 ≤A 𝑏, then ℎ(𝑎) ≤B ℎ(𝑏). • if h𝑎, 𝑏i ∈ Ω(A), then hℎ(𝑎), ℎ(𝑏)i ∈ Ω(B). If, furthermore, ℎ is strict, then converse implications hold as well. Proof Let 𝐹 and 𝐺 be respectively the filters of A and B. We prove it for the matrix preorder (the claim for Leibniz congruences immediately follows). Consider the following chain of implications and equivalences: 𝑎 ≤A 𝑏

iff

𝑎 ⇒A 𝑏 ∈ 𝐹

implies iff

ℎ(𝑎) ⇒B ℎ(𝑏) = ℎ[𝑎 ⇒A 𝑏] ∈ 𝐺

ℎ(𝑎) ≤B ℎ(𝑏).

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Both equivalences hold by definition and the implication is trivial because ℎ is a homomorphism. To prove the final claim, just observe that the strictness of ℎ allows us to replace the implication by an equivalence.  Clearly, this lemma allows us to prove an analog of Corollary 3.2.9 without the surjectivity assumption. Corollary 3.2.12 Let L be a weakly implicative logic, take A, B ∈ Mod(L), and let ℎ : A → B be a strict homomorphism. Then, the mapping 𝑔 : A∗ → B∗ given as 𝑔(𝑎/Ω(A)) = ℎ(𝑎)/Ω(B) is well defined and it is an embedding. We conclude this section by exploring the promised relationship between matrix homomorphisms and logical congruences. Recall that each algebraic homomorphism ℎ : A → B defines a congruence relation ker ℎ on A, known as the kernel of ℎ, as h𝑥, 𝑦i ∈ ker ℎ iff ℎ(𝑥) = ℎ(𝑦) (cf. Proposition A.3.12). Proposition 3.2.13 Let ℎ : hA, 𝐹i → hB, 𝐺i be a homomorphism. If ℎ is strict, then ker ℎ is a logical congruence on hA, 𝐹i. Furthermore, if 𝐺 = ℎ[𝐹], then the converse direction is valid as well. Thus, in particular, a surjective homomorphism ℎ : hA, 𝐹i → hB, 𝐺i is strict iff 𝐺 = ℎ[𝐹] and ker ℎ is a logical congruence on hA, 𝐹i. Proof Assume that 𝑥 ∈ 𝐹 and h𝑥, 𝑦i ∈ ker ℎ. Then, ℎ(𝑦) = ℎ(𝑥) ∈ 𝐺 and so, by strictness, 𝑦 ∈ 𝐹. Conversely, assume that ℎ(𝑥) ∈ 𝐺 = ℎ[𝐹], i.e. ℎ(𝑥) = ℎ(𝑦) for some 𝑦 ∈ 𝐹. Therefore, h𝑥, 𝑦i ∈ ker ℎ and, since ker ℎ is a logical congruence, we obtain 𝑥 ∈ 𝐹.  Note that this proposition yields an alternative proof of the first part of Corollary 3.2.6. Indeed, if hA, 𝐹i is reduced, then ker ℎ is the identity and so ℎ has to be an embedding. Corollary 3.2.14 Let ℎ : A → B be a strict surjective homomorphism. Then, the matrices B and A/ker ℎ are isomorphic. Therefore, A = A/ker ℎ = B . Thus, for each K, we have K = HS (K) . Proof We show that the mapping 𝑔(𝑎/ker ℎ) = ℎ(𝑎) is an isomorphism. We first show that 𝑔 is a well-defined one-one mapping. We know that 𝑎/ker ℎ = 𝑏/ker ℎ iff ℎ(𝑎) = ℎ(𝑏) iff 𝑔(𝑎/ker ℎ) = 𝑔(𝑏/ker ℎ). Clearly, 𝑔 is a surjective algebraic embedding of A/ker ℎ into B. Thus, it only remains to show that it is a strict matrix homomorphism. Assume that 𝐹 and 𝐺 are the filters of A and B respectively and observe that we have the following chain of equivalences: 𝑎/ker ℎ ∈ 𝐹/ker ℎ iff 𝑎 ∈ 𝐹 iff 𝑔(𝑎/ker ℎ) = ℎ(𝑎) ∈ 𝐺. The second part follows from Lemma 2.7.5 and the obvious observation that isomorphic matrices induce the same consequence relation. 

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Finally, recall that when a logic L and an algebra C are fixed or clear from the context, given a filter 𝐻 ∈ FiL (C), we denote by [𝐻, 𝐶] the interval {𝐻 0 ∈ FiL (C) | 𝐻 ⊆ 𝐻 0 } and note that this interval can be seen as a sublattice of FiL (C). Clearly, we can write the following reformulation of Proposition 2.7.8 (the proof is trivial, because 𝐹 = ℎ−1 [𝐺] and any weakly implicative logic satisfies the premises of the original proposition). Proposition 3.2.15 Let L be a weakly implicative logic and ℎ : hA, 𝐹i → hB, 𝐺i be a strict surjective homomorphism. Then, the mapping 𝒉 : [𝐹, 𝐴] → [𝐺, 𝐵] defined as 𝒉(𝐻) = ℎ[𝐻] is a lattice isomorphism.

3.3 Submatrices and conservative expansions In this section we introduce submatrices, explore their expected relation with homomorphisms and use them to characterize the notion of conservative expansion. Definition 3.3.1 (Submatrix) An L-matrix A = hA, 𝐹i is a submatrix of an L-matrix B = hB, 𝐺i if A is a subalgebra of B and 𝐹 = 𝐴 ∩ 𝐺. Given a class K of L matrices, let us denote by S(K) the class of all submatrices of matrices from K. Clearly, for each matrix hB, 𝐺i and each subalgebra A of B, the matrix hA, 𝐺 ∩ 𝐴i is a submatrix of hB, 𝐺i. It is easy to see that, for each algebraic homomorphism ℎ : A → B, ℎ[ 𝐴] is the domain of a subalgebra of B; let us denote this algebra by ℎ(A). On the other hand, for an arbitrary set 𝐹 ⊆ 𝐴, the matrix hℎ(A), ℎ[𝐹]i need not be a submatrix of hB, 𝐺i, even if ℎ is a matrix homomorphism ℎ : hA, 𝐹i → hB, 𝐺i. Indeed, consider any pair of matrices hA, 𝐹i and hA, 𝐺i such that 𝐹 ( 𝐺 and the identity mapping id; clearly id is a matrix homomorphism but id[𝐹] ≠ id[ 𝐴] ∩ 𝐺. The next proposition shows that strictness is needed for this claim to hold. Let us denote the matrix hℎ(A), ℎ[𝐹]i by ℎ(hA, 𝐹i). Proposition 3.3.2 Let ℎ : A → B be a matrix homomorphism. 1. If ℎ is strict, then ℎ(A) is a submatrix of B. 2. ℎ is an embedding iff ℎ is one-one and ℎ(A) is a submatrix of B. Proof Let 𝐹 and 𝐺 be, respectively, the filters of A and B. The first claim is easy: 𝑦 ∈ ℎ[𝐹] iff 𝑦 = ℎ(𝑥) for some 𝑥 ∈ 𝐹 iff (due to the strictness of ℎ) 𝑦 = ℎ(𝑥) for some ℎ(𝑥) ∈ 𝐺 iff 𝑦 ∈ ℎ[ 𝐴] ∩ 𝐺. To prove the second claim, observe that we also have that ℎ(𝑥) ∈ 𝐺 implies ℎ(𝑥) ∈ ℎ[ 𝐴] ∩ 𝐺 = ℎ[𝐹] and so, if ℎ is one-one, we get 𝑥 ∈ 𝐹, i.e. ℎ is an embedding of matrices.  Therefore, it is easy to see that A is a submatrix of B iff the identity mapping on 𝐴 is an embedding of A into B. Furthermore, any embedding ℎ : A → B can be seen as an isomorphism between A and a submatrix ℎ(A) of B. These two facts can be schematically written as: A ∈ IS(K)

iff there is a B ∈ K and an embedding ℎ : A → B.

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Note that, if A is reduced, thanks to Corollary 3.2.6 we can formulate this equivalence with strict homomorphism instead of embedding. The previous observation justifies the terminology introduced by the following definition. Definition 3.3.3 A class K of L-matrices is embeddable into a class K 0 of L-matrices whenever we have K ⊆ IS(K 0). Next, we prove a strengthening of Corollary 3.2.14 which, among other things, implies that closing a class under submatrices does not change the induced consequence relation. Proposition 3.3.4 Let K be a class of L-matrices. Then, K = S(K) = HS S(K) . Proof Clearly, K ⊇ S(K) . To prove the converse, assume that Γ 2A 𝜑 for some submatrix A = hA, 𝐹i of a matrix B = hB, 𝐺i ∈ K. Observe that any A-evaluation 𝑒 can be seen as a B-evaluation and 𝑒( 𝜒) ∈ 𝐹 iff 𝑒( 𝜒) ∈ 𝐺 ∩ 𝐴 iff 𝑒( 𝜒) ∈ 𝐺.  This proposition allows us to prove the validity of the principal method of establishing the SKC of a given logic L provided that we already know that it enjoys SK 0C for some other class K 0 ⊆ Mod∗ (L): all we have to do is to show that K 0 is embeddable into K. Indeed, we know that K = HS S(K) ⊆ IS(K) ⊆ K0 = `L . The next result shows that, in order to obtain the FSKC, it suffices to prove a weaker claim: instead of an embedding of the whole matrix, it is enough to have a partial embedding for each finite subset. Let us first define this notion formally. Definition 3.3.5 Let K and K 0 be classes of matrices in the same language L. We say that K is partially embeddable into K 0 if for each hA, 𝐹i ∈ K and each finite subset 𝑋 of 𝐴 there is a hB, 𝐺i ∈ K 0 and a partial embedding 𝑓 : 𝑋 → 𝐵, i.e. a one-one mapping such that, for each 𝑥, 𝑥 1 , . . . , 𝑥 𝑛 ∈ 𝑋 and each 𝑛-ary 𝜆 ∈ L such that 𝜆A (𝑥 1 , . . . , 𝑥 𝑛 ) ∈ 𝑋, we have 𝑥 ∈ 𝐹 iff 𝑓 (𝑥) ∈ 𝐺 A

𝑓 (𝜆 (𝑥1 , . . . , 𝑥 𝑛 )) = 𝜆B ( 𝑓 (𝑥 1 ), . . . , 𝑓 (𝑥 𝑛 )). Lemma 3.3.6 Let L be a logic and K, K 0 ⊆ Mod∗ (L). If L has (F)SK 0C and K 0 is (partially) embeddable into K, then L has the (F)SKC. Proof The claim for the SKC was already proved above. To prove the other one, suppose that for some finite set Γ of formulas we have Γ 0L 𝜑. Then, there is an hA, 𝐹i ∈ K 0 and an A-evaluation 𝑒 witnessing this fact. Let Γ0 be the set of all subformulas of formulas from Γ ∪ {𝜑} and define the finite set 𝑋 = 𝑒[Γ0] ⊆ 𝐴. By the hypothesis, there exist hB, 𝐺i ∈ K and a partial embedding 𝑓 : 𝑋 → 𝐵.

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Finally, take any B-evaluation 𝑒 0 such that, for each atom 𝑝 ∈ Γ0, we have = 𝑓 (𝑒( 𝑝)). Note that it is easy to prove by induction that 𝑒 0 (𝜓) = 𝑓 (𝑒(𝜓)) for each 𝜓 ∈ Γ0. Therefore, 𝑒[Γ] ⊆ 𝐹 implies 𝑓 [𝑒[Γ]] ⊆ 𝑓 [𝐹], i.e. 𝑒 0 [Γ] ⊆ 𝐺 and, analogously, 𝑒 0 (𝜑) ∉ 𝐺.  𝑒 0 ( 𝑝)

Proposition 3.3.7 Let L be a logic. Then, Mod(L) is closed under submatrices. Furthermore, if L is weakly implicative, then also Mod∗ (L) is closed under submatrices. Proof Take A ∈ Mod(L) and observe that L ⊆ A = S(A) ⊆ B for each submatrix B of A, i.e. B ∈ Mod(L). Next, assume that A is reduced and h𝑎, 𝑏i ∈ Ω(B) for some 𝑎, 𝑏 ∈ 𝐵. Then, due to the fact that identity is an embedding of B into A, Lemma 3.2.11 gives us h𝑎, 𝑏i ∈ Ω(A) and, since A is reduced, 𝑎 = 𝑏.  Note that we could prove that Mod∗ (L) is closed under submatrices in weakly implicative logics due to the fact that in those logics we can define the Leibniz congruence using the equivalence set (cf. Theorem 2.7.7) ⇔ = {𝑝 ⇒ 𝑞, 𝑞 ⇒ 𝑝}, i.e. we have h𝑎, 𝑏i ∈ ΩA (𝐹) iff 𝑎 ⇔A 𝑏 ⊆ 𝐹 (actually, the presence of any other equivalence set would be sufficient). The next example shows that, in general, Mod∗ (L) need not be closed under submatrices. Example 3.3.8 Let A be an algebra with domain {𝑎, 𝑏, 𝑐} and one binary operation ◦ defined as 𝑎 ◦ 𝑐 = 𝑐 and 𝑥 ◦ 𝑦 = 𝑥 otherwise; let B be its subalgebra with domain {𝑏, 𝑐}. It is easy to see that the matrix A = hA, {𝑏, 𝑐}i is reduced (indeed, any non-identical logical congruence would have to identify 𝑏 and 𝑐; but we have 𝑎 ◦ 𝑏 = 𝑎, and 𝑎 ◦ 𝑐 = 𝑐, forcing 𝑎 and 𝑐 to be congruent) while the submatrix B = hB, {𝑏, 𝑐}i is not reduced (the total congruence is logical). Thus, B is a counterexample to S(Mod∗ (A )) ⊆ Mod∗ (A ). Remark 3.3.9 Assume that L is an algebraically implicative logic. We leave as an exercise for the reader to show that, for each B ∈ Alg∗ (L) and each subalgebra A ⊆ B, we have that A ∈ Alg∗ (L), 𝐹A = 𝐹B ∩ 𝐴, and hA, 𝐹A i is a submatrix of hB, 𝐹B i. Weakly implicative logics, because their languages are by definition countable, are always complete w.r.t. countable reduced models. Let us prove a generalization of this fact for arbitrary cardinalities. Theorem 3.3.10 Let L be a logic over a set of formulas of cardinality 𝜅 with the SKC for a class K ⊆ Mod∗ (L). Then, L has the SK 0C for the class K 0 = (S(K)) 𝜅 . Proof First, observe that K 0 ⊆ Mod(L) (and, if L is weakly implicative, then even K 0 ⊆ Mod∗ (L)); therefore `L ⊆ K0 . To prove the converse, assume that Γ 0L 𝜑. Therefore, there is a matrix hA, 𝐹i ∈ K and an A-evaluation 𝑒 such that 𝑒[Γ] ⊆ 𝐹 but 𝑒(𝜑) ∉ 𝐹. Consider the submatrix B = h𝑒(FmL ), 𝑒[FmL ] ∩ 𝐹i of hA, 𝐹i. Obviously, B ∈ (S(K)) 𝜅 and 𝑒 can be seen as a B-evaluation. 

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Next, we show that S has the properties of a closure operator, describe its interaction with (strict) homomorphic images, and give schematic compact descriptions of some of the results proved so far. Proposition 3.3.11 • The operator S is monotonic, increasing and idempotent, i.e. for each pair of classes of L-matrices K ⊆ K 0, we have S(K) ⊆ S(K 0)

K ⊆ S(K) = SS(K).

• For each class K of L-matrices, we have SH(K) ⊆ HS(K)

SHS (K) ⊆ HS S(K).

• For any logic L, we have S(Mod(L)) = Mod(L). • For any weakly implicative logic L and any class K ⊆ Mod∗ (L), we have S(Mod∗ (L)) = Mod∗ (L)

IS(K) = HS S(K).

Proof We only prove the second claim and leave the rest as an exercise for the reader. Take an arbitrary element of SHS (K), that is, we have hA, 𝐹i ∈ K, a strict surjective homomorphism 𝑓 : hA, 𝐹i → hB, 𝐺i, and a submatrix hC, 𝐺 ∩ 𝐶i of hB, 𝐺i. We have to argue that hC, 𝐺 ∩ 𝐶i ∈ HS S(K). We leave as an exercise for the reader to show that 𝑓 −1 [𝐶] is the domain of a subalgebra of A denoted by 𝑓 −1 (C) and that h 𝑓 −1 (C), 𝑓 −1 [𝐺 ∩ 𝐶]i is a submatrix of hA, 𝐹i. Then, clearly, 𝑓 can be seen as a strict surjective homomorphism 𝑓 : h 𝑓 −1 (C), 𝑓 −1 [𝐺 ∩ 𝐶]i → hC, 𝐺 ∩ 𝐶i. The proof of the analogous claim for HS runs parallel, without assuming 𝑓 to be strict and taking the submatrix h 𝑓 −1 (C), 𝑓 −1 [𝐺 ∩ 𝐶] ∩ 𝐹i instead.  Let us show an application of Theorem 3.2.10 and prove a characterization of conservative expansions (cf. Definition 2.3.3). Recall that the reduct of an algebra to a smaller language is obtained by simply omitting the interpretation of the discarded operations. In the same way, we define reducts of matrices. Given a class K of L-matrices and a language L 0 ⊆ L, we denote by K L0 the class of L 0-reducts of matrices of K. It is easy to see that, if a logic L enjoys any of our three kinds of completeness w.r.t. a class K, then its fragment L0 in a language L 0 enjoys the same kind of completeness w.r.t. the class K L0 . The next theorem can be seen as the converse of this claim. Theorem 3.3.12 Let L1 be a weakly implicative logic in a language L1 and L2 a weakly implicative expansion in a language L2 . Then, the following are equivalent: 1. L2 is a conservative expansion of L1 . 2. Mod∗ (L1 ) = S(Mod∗ (L2 ) L1 ). 3. L1 has the SKC for a class K ⊆ S(Mod∗ (L2 ) L1 ).

3.3 Submatrices and conservative expansions

101

Proof The proof that 1 implies 2 is the most involved one. First, observe that clearly Mod∗ (L2 ) L1 ⊆ Mod∗ (L1 ); thus, also, S(Mod∗ (L2 ) L1 ) ⊆ Mod∗ (L1 ), thanks to Proposition 3.3.7. To prove the converse direction, let us take a matrix A ∈ Mod∗ (L1 ) and a set Var 0 of an infinite cardinality greater or equal to that of A. Thanks to 0 Theorem 3.2.10, we know that A is isomorphic to hFmL0 1 , 𝑇i ∗ for some LVar 1 -theory 𝑇 and so it suffices to prove that hFmL0 1 , 𝑇i ∗ ∈ S(Mod∗ (L2 ) L1 ). 0 Let us denote by 𝑇 0 the LVar 2 -theory generated by 𝑇 and by C the L1 -reduct of hFmL0 2 , 𝑇 0i. We prove that the identity mapping ℎ : hFmL0 1 , 𝑇i → C is a strict homomorphism. The only complication is caused by the additional variables; indeed 0 to show that 𝜑 ∈ 𝑇 iff ℎ(𝜑) = 𝜑 ∈ 𝑇 0 we need to prove that LVar is a conservative 2 0 expansion of LVar . 1 0 0 Assume that Γ `Var L2 𝜒 for some Γ ∪ { 𝜒} ⊆ FmL1 . By claim 4 of Proposition 3.1.7 used for L2 , there is a countable set Γ0 ⊆ Γ and a bijection 𝜏 on Var 0 (which we can see as an FmL0 2 -substitution) such that 𝜏[Γ0 ∪ { 𝜒}] ⊆ Fm L2 and 𝜏[Γ0 ] `L2 𝜏(𝜑). Thus, also 𝜏[Γ0 ∪ { 𝜒}] ⊆ Fm L1 and so, by conservativity of L2 over L1 , we have 0 𝜏[Γ0 ] `L1 𝜏(𝜑). Using claim 4 of Proposition 3.1.7 for L1 , we obtain Γ `Var L1 𝜒. To complete the proof of this implication, notice that obviously C ∈ Mod(L1 ) and so we can use Corollary 3.2.12 to obtain an embedding from hFmL0 1 , 𝑇i ∗ into C∗ ; i.e. we know that hFmL0 1 , 𝑇i ∗ ∈ IS(C∗ ) ⊆ SI(C∗ ) ⊆ S(Mod∗ (L2 ) L1 ). The proof that 2 implies 3 is trivial. We prove the last implication. Assume that, for some Γ ∪ {𝜑} ∈ FmL1 , we have Γ 0L1 𝜑; by the SKC, there is a reduced L1 -matrix A ∈ K witnessing this fact. Due to the assumption, we know that A is a submatrix of the L1 -reduct of some B ∈ Mod∗ (L2 ). Since, clearly, any A-evaluation can be seen as a B-evaluation, we know that Γ 0L2 𝜑.  Let us show a simple application of this theorem for modal logics. Later, in Corollary 3.5.20, we will see a more involved application for the →-fragment of classical logic and, in the next chapter, it will be the cornerstone of the characterization of strongly separable axiomatic systems (see Proposition 4.5.2) which provide systematic axiomatizations of numerous fragments of prominent logics (cf. Theorems 4.5.5, 4.7.1, 4.5.7, and 6.3.8, and Corollary 4.7.3. Example 3.3.13 Both modal logics 𝑙K and K are conservative expansions of CL. Indeed, since CL and K are Rasiowa-implicative logics, we know from the previous chapter that the matrices in Mod∗ (CL) and Mod∗ (K) are of the form hA, {>}i, where A is a Boolean algebra or a modal algebra respectively (see Example 2.9.2). As modal algebras are Boolean algebras with an additional operator  satisfying certain properties, we obtain Mod∗ (CL) ⊇ S(Mod∗ (K) LCL ). Thanks to the fact that any Boolean algebra with the operator  defined as 𝑥 = 𝑥 is a modal algebra, we obtain the converse direction.6 For 𝑙K we cannot use the theorem above, because it is not weakly implicative. However, as CL ⊆ 𝑙K ⊆ K, the claim follows.

6 Note that we cannot use yet the third claim for K = {2} because we will obtain the strong completeness of CL w.r.t. 2 in Corollary 3.5.20 only.

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3.4 Direct products and Leibniz operator The next matricial construction we want to study is that of direct products. We use them to obtain the first characterization of a completeness property (namely, KC) and for the study of the Leibniz operator. Definition 3.4.1

 (Direct product) The product (or direct product) of a system X = hA𝑖 , 𝐹𝑖 i 𝑖 ∈𝐼 of L-matrices is the L-matrix Ö

X=

Ö 𝑖 ∈𝐼

hA𝑖 , 𝐹𝑖 i =

Ö

A𝑖 ,

𝑖 ∈𝐼

Ö

 𝐹𝑖 .

𝑖 ∈𝐼

Given a class K of L-matrices, we denote by P(K) the class of all direct products of systems of matrices from K. Observe that the product of an empty system is the trivial reduced matrix. It is easy Î to see that the projection mapping 𝜋 𝑗 : 𝑖 ∈𝐼 𝐴𝑖 → 𝐴 𝑗 is a surjective homomorphism of the product matrix onto its 𝑗th component. Unlike the previous class operators, P is not a closure operator. Indeed, it is clearly monotonic but, while the product of a singleton system hAi can be identified with A (identifying elements with 1-tuples), assuming the same for products of products would stretch it a bit too far; so we prove it for a combined operator IP. First we prove a result similar to Proposition 3.3.4, now showing that the consequence relation given by a class K of matrices and that given by its closure under products coincide. Proposition 3.4.2 Let K be a class of L-matrices. Then, K = P(K) = HS SP(K) . Proof Clearly, K ⊇ P(K) . To prove the converse, assume that Γ 2A 𝜑 for some hA, 𝐺i ∈ P(K); let us denote the index set of the product as 𝐼. Therefore, there is an A-evaluation 𝑒 such that 𝑒[Γ] ∈ 𝐺 and 𝑒(𝜑) ∉ 𝐺, i.e. there is an 𝑖 ∈ 𝐼 such that 𝑒(𝜑) ∉ 𝐹𝑖 . As clearly the mapping 𝜋𝑖 ◦ 𝑒 can be seen as an A𝑖 -evaluation, the proof is done.  Now we can prove some important properties of P and note that they entail that actually IP is a closure operator (an immediate consequence of the first two claims and properties of I). Proposition 3.4.3 • For each pair of classes of L-matrices K ⊆ K 0, we have P(K) ⊆ P(K 0)

K ⊆ P(K)

PP(K) ⊆ IP(K).

• Let X be any of the operators S, H, HS , or I. Then, for each class K of L-matrices, we have PX(K) ⊆ XP(K).

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103

• For any logic L, we have P(Mod(L)) = Mod(L). • For any weakly implicative logic L, we have P(Mod∗ (L)) = Mod∗ (L). Proof The only non-straightforward inclusion of the first claim is the last one (note that the central one is due to our identification of elements and their 1-tuples): assume that we have a set of sets
I, define an index set 𝐽 = {h𝐼, 𝑖i | 𝐼 ∈ Î I and Î 𝑖 ∈ 𝐼}, and note that ℎ h𝑎 𝑖𝐼 i𝑖 ∈𝐼 𝐼 ∈I = h𝑎 𝑖𝐼 i h𝐼 ,𝑖 i ∈𝐽 is an isomorphism from 𝐼 ∈I 𝑖 ∈𝐼 A𝑖𝐼 to Î 𝐼 h𝐼 ,𝑖 i ∈𝐽 A𝑖 . Î The first part of the second claim is easy. Indeed, assume that A = 𝑖 ∈𝐼 A𝑖 and that, for each 𝑖 ∈ 𝐼, A𝑖 is a submatrix of A𝑖0 ∈ÎK. Then, it is a simple exercise for the reader to check Î that A is a submatrix of 𝑖 ∈𝐼 A𝑖0 . The second part is similar: Assume that A = 𝑖 ∈𝐼 A𝑖 and for each 𝑖 ∈ 𝐼 we know that there is a matrix A𝑖0 ∈ K and a (strict) surjective homomorphism ℎ𝑖 : A𝑖0 → A𝑖 . Then, clearly, the Î mapping ℎ : 𝑖 ∈𝐼 A𝑖0 → A defined as ℎ(𝑥) = hℎ𝑖 (𝜋𝑖 (𝑥))i𝑖 ∈𝐼 is a (strict) surjective homomorphism. Obviously, if all the ℎ𝑖 are one-one, then so is ℎ. The third claim follows from Proposition 3.4.2 and the first claim. To prove the last claim, it suffices to observe that h𝑎, 𝑏i ∈ ΩA (𝐹) iff 𝑎 ⇔A 𝑏 ⊆ 𝐹 iff for each 𝑖 ∈ 𝐼: 𝜋𝑖 (𝑎) ⇔A𝑖 𝜋𝑖 (𝑏) ⊆ 𝐹𝑖 iff h𝜋𝑖 (𝑎), 𝜋𝑖 (𝑏)i ∈ ΩAi (𝐹𝑖 ).  Example 3.4.4 Mod∗ (L) is not closed under products in general. Let A be the reduct of 2 only with the operation ∧. Let L be the logic  hA, {1}i . As an exercise, the reader can check that hA, {1}i ∈ Mod∗ (L), while the product hA, {1}i × hA, {1}i is not reduced, and hence is not in Mod∗ (L). Remark 3.4.5 Assume that L is an algebraically implicative logic. We leave as an exercise for the reader to show that, for a system hA𝑖 i𝑖 ∈𝐼 of L-algebras, we have Î A ∈ Alg∗ (L) and 𝑖 ∈𝐼 𝑖 Ö 𝐹Î𝑖∈𝐼 A𝑖 = 𝐹A𝑖 . 𝑖 ∈𝐼

Now we are ready to prove the first characterization of a completeness property w.r.t. a class K using a description of Mod∗ (L) in terms of class operators applied to K. We do it for weakly implicative logics and for the weakest of the three properties (KC). Therefore, it should not be surprising that the characterizing condition is rather weak. Indeed, we prove that KC is equivalent to Mod∗ (L) ⊆ HSP(K), which can be equivalently written as H(Mod∗ (L)) = HSP(K); i.e. instead of describing the class of reduced matrices we are really describing the class of their homomorphic images (which, as seen in Remark 3.2.4, could be a strictly bigger class). Later, in Section 3.8, we will see a similar characterization of FSKC and SKC using a better localization of Mod∗ (L).

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Theorem 3.4.6 Let L be a weakly implicative logic and K ⊆ Mod∗ (L). Then, L has the KC if and only if Mod∗ (L) ⊆ HSP(K). Proof To prove the right-to-left direction, assume that 0L 𝜑. Then, there is a reduced matrix A = hA, 𝐹i ∈ Mod∗ (L) and an A-evaluation 𝑒 such that 𝑒(𝜑) ∉ 𝐹. Since A ∈ HSP(K), we know that there exists a submatrix B = hB, 𝐺i of the direct product of a system of matrices hB𝑖 = hB𝑖 , 𝐺 𝑖 ii𝑖 ∈𝐼 from K and a surjective homomorphism ℎ : B → A. Thus, there has to be a B-evaluation 𝑓 such that ℎ ◦ 𝑓 = 𝑒 and 𝑓 (𝜑) ∉ 𝐺 (because otherwise 𝑒(𝜑) = ℎ( 𝑓 (𝜑)) ∈ 𝐹). Therefore, there has to be an index 𝑗 ∈ 𝐼 such that (𝜋 𝑗 ◦ 𝑓 )(𝜑) ∉ 𝐺 𝑗 , that is, we have a B 𝑗 -evaluation 𝜋 𝑗 ◦ 𝑓 demonstrating that 2K 𝜑. To prove the converse direction, we use the notion of natural extension of a given logic introduced in Section 3.1. Let us fix a matrix A ∈ Mod∗ (L) (note that we can assume that A is non-trivial because the trivial reduced matrix obviously lies in HSP(K)) and set Var 0 = Var ∪ {𝑣 𝑎 | 𝑎 ∈ 𝐴} (of course, assume that none of the 𝑣 𝑎 s are in Var). We denote by FmA the set of formulas using only the new variables. For any 𝜑 ∈ FmA , we denote by 𝜑A the value of 𝜑 under the A-evaluation that maps 𝑣 𝑎 to 𝑎 for each 𝑎 ∈ 𝐴 (analogously, we define ΓA = {𝜑A | 𝜑 ∈ Γ}). Observe that, for each formula 𝜒 ∈ FmA such that 𝜒A ∉ 𝐹, we have 0LVar0 𝜒 (just 0 observe that, by the second point of Proposition 3.1.7, A ∈ Mod∗ (LVar ) and consider the A-evaluation 𝑒(𝑣 𝑎 ) = 𝑎). Thus, by the KC of L and Proposition 3.1.7, we have a matrix A 𝜒 = hA 𝜒 , 𝐹𝜒 i ∈ K such that 2A 𝜒 𝜒;7 let 𝑒 𝜒 be an A 𝜒 -evaluation witnessing it. Next, we take the index set 𝐼 = { 𝜒 ∈ FmA | 𝜒A ∉ 𝐹} and define B = hB, 𝐺i as the direct product of hA 𝜒 i 𝜒 ∈𝐼 . We denote by 𝜑 the element h𝑒 𝜒 (𝜑)i 𝜒 ∈𝐼 and consider the subset 𝐵 0 of 𝐵 defined as 𝐵 0 = {𝜑 | 𝜑 ∈ FmA }. Note that 𝐵 0 is a subuniverse of B; indeed, consider for example a binary connective ◦ and formulas 𝜑 and 𝜓 and observe that 𝜑 ◦B 𝜓 = (h𝑒 𝜒 (𝜑)i 𝜒 ∈𝐼 ◦B h𝑒 𝜒 (𝜓)i 𝜒 ∈𝐼 = h𝑒 𝜒 (𝜑) ◦A 𝜒 𝑒 𝜒 (𝜓)i 𝜒 ∈𝐼 = h𝑒 𝜒 (𝜑 ◦ 𝜓)i 𝜒 ∈𝐼 = 𝜑 ◦ 𝜓. Therefore, we can consider the submatrix B0 = hB0, 𝐺 0i of B with domain 𝐵 0 and filter 𝐺 0 = 𝐺 ∩ 𝐵 0 and define a mapping 𝑓 : 𝐵 0 → 𝐴 as 𝑓 (𝜑) = 𝜑A . 0

7 We write simply  instead of, more systematically correct, Var because there is no danger of confusion here.

3.4 Direct products and Leibniz operator

105

The proof is concluded by showing that 𝑓 is a homomorphism of B0 onto A: •

𝑓 is well defined: Assume that 𝜑A ≠ 𝜓 A ; since hA, 𝐹i is reduced, we must have 𝜑A ⇒A 𝜓 A ∉ 𝐹 or 𝜓 A ⇒A 𝜑A ∉ 𝐹. Without loss of generality we can assume the former. Then, 𝜑 ⇒ 𝜓 ∈ 𝐼, and so 𝑒 𝜑⇒𝜓 (𝜑 ⇒ 𝜓) ∉ 𝐹𝜑⇒𝜓 . Thus, 𝑒 𝜑⇒𝜓 (𝜑) ≠ 𝑒 𝜑⇒𝜓 (𝜓) and so 𝜑 ≠ 𝜓. • 𝑓 is an algebraic homomorphism: Consider e.g. a binary connective ◦ and formulas 𝜑, 𝜓, and observe that 0

𝑓 (𝜑 ◦B 𝜓) = 𝑓 (𝜑 ◦ 𝜓) = (𝜑 ◦ 𝜓) A = 𝜑A ◦A 𝜓 A = 𝑓 (𝜑) ◦A 𝑓 (𝜓). • •

𝑓 is an onto mapping: This directly follows from 𝑓 (𝑣 𝑎 ) = 𝑣 A𝑎 = 𝑎. 𝑓 is a matrix homomorphism: Observe that if 𝑓 (𝜑) = 𝜑A ∉ 𝐹, then 𝜑 ∈ 𝐼 and 𝑒 𝜑 (𝜑) ∉ 𝐹𝜑 and, thus, 𝜑 = h𝑒 𝜒 (𝜑)i 𝜒 ∈𝐼 ∉ 𝐺 0. 

Remark 3.4.7 Interestingly enough, it is not possible to extend Theorem 3.4.6 to a characterization of the KC for all (not necessarily weakly implicative) logics. This can be seen via a rather trivial example. Indeed, take CL∧∨ , the {∧, ∨}-fragment of classical logic. We know that it is not weakly implicative because it has no theorems. Let A be the matrix given by the trivial algebra and the empty filter and take K = {A}. Clearly, A ∈ Mod∗ (CL∧∨ ) and CL∧∨ has the KC, whereas Mod∗ (CL∧∨ ) * HSP(K) (because the latter contains only matrices over the trivial algebra, while Mod∗ (CL∧∨ ) is known [135] to contain matrices over non-trivial distributive lattices, e.g. the lattice reduct of the two-valued Boolean matrix). Remark 3.4.8 Note that the previous theorem (and many similar upcoming results) characterizes the KC in terms of the localization of another class of matrices, namely Mod∗ (L), w.r.t. which the logic L enjoys completeness. We might as well have used an arbitrary class K 0 and proved statements such as: “Assume that L has the K 0C. Then, L has the KC iff K 0 ⊆ HSP(K)” (one direction is an obvious corollary of the previous theorem; the other one could be proved by a simple modification of its proof). This formulation would simplify proving KC, as we would only need to show the containment of K 0 in HSP(K). However, instead of duplicating the results, we leave the task of drawing these corollaries to the reader. Furthermore, we will see that if K 0 is one of the ‘intrinsic’ classes of a logic, we obtain tighter localizations (e.g. later in Theorem 3.6.14 we will prove that any finitary logic enjoys the KC for a class of matrices we will denote by Mod∗ (L)RFSI and then in Theorem 5.6.3 we show that a finitary weakly implicative logic with a suitable disjunction connective has the KC iff Mod∗ (L)RFSI ⊆ HSPU (K)). If the logic in question is algebraically implicative, we can use Remarks 3.2.2, 3.3.9, and 3.4.5 to obtain the following purely algebraic characterization of KC: Corollary 3.4.9 Let L be an algebraically implicative logic and A ⊆ Alg∗ (L). Then, L has the AC if and only if Alg∗ (L) ⊆ HSP(A).

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Similar corollaries can be formulated for many upcoming completeness results; however, we will only explicitly state and prove them for the most important cases. Recall that, for most of the algebraically implicative logics we have introduced in the previous chapter, Alg∗ (L) is a variety. In this case, we can reformulate the previous result as ‘L has the AC iff A generates the variety Alg∗ (L)’. Therefore, we can use known algebraic facts to establish AC for several prominent logics and classes A generating Alg∗ (L) (e.g. A = {2} for classical logic or A = {[0, 1] Ł } for Łukasiewicz logic). Later we will prove stronger claims (e.g. SAC for the former and FSAC for the latter) and so we postpone it for now. Later in this chapter (and also in some subsequent ones), we will see numerous results similar to the one we have just proved. As with all characterizations, we are looking for two different kinds of equivalent conditions: those which are easy to check to establish completeness (e.g. in Corollary 3.6.17 we will see that, in certain logics, in order to obtain the SKC it suffices to check that a certain small intrinsically defined class of matrices is embeddable into K) and those providing the tightest description of the largest possible class of models of the logic in question (e.g. in Theorem 3.8.2 we prove that, in certain logics, SKC implies Mod∗ (L) = ISP𝜔 (K); see Section 3.7 for the definition of 𝜔-filtered products). Therefore, it is usually impossible to compare the strength of two different characterizations. Since different situations require different characterizations, we will formulate and prove numerous variants. The following theorem together with its corollary provide the first characterizations of the strongest completeness property, the SKC. Unlike the previous results concerning the KC, here we do not provide a localization of the whole class Mod∗ (L), but only the Lindenbaum–Tarski matrices (resp., in the light of Theorem 3.2.10, of matrices of Mod∗ (L) 𝜅 ). On the other hand, note that (at least when L is weakly implicative) we manage to localize them inside Mod∗ (L) (since, unlike in the case of H, Mod∗ (L) is closed under ISP). To extend this characterization to the whole class Mod∗ (L) we need to introduce additional matrix construction techniques (see Section 3.7 for details). Theorem 3.4.10 Let L be a logic and K ⊆ Mod∗ (L). L has the SKC if and only if {hFmL , 𝑇i ∗ | 𝑇 is a theory} ⊆ HS SP(K). Furthermore, if L is weakly implicative, we can replace HS by I. Proof For the left-to-right direction, consider a theory 𝑇. If 𝑇 = FmL , then hFmL , 𝑇i ∗ is the trivial reduced matrix, which is contained in HS SP(K) as the product of the empty family. Due to the SKC, for each 𝜒 ∉ 𝑇, there is an A 𝜒 = h𝐴 𝜒 , 𝐹𝜒 i ∈ K such that 𝑇 2A 𝜒 𝜒. Let us denote by 𝑒 𝜒 an A 𝜒 -evaluation witnessing this fact. Next, we take the index set 𝐼 = { 𝜒 ∈ FmL | 𝜒 ∉ 𝑇 } and define B = hB, 𝐺i as the direct product of hA 𝜒 i 𝜒 ∈𝐼 .

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107

We show that the mapping 𝑓 : hFmL , 𝑇i → hB, 𝐺i defined as 𝑓 (𝜑) = h𝑒 𝜒 (𝜑)i 𝜒 ∈𝐼 is a strict homomorphism. Clearly, it is an algebraic homomorphism from FmL to B. Strictness is also easy: 𝜑 ∈ 𝑇 iff for each 𝜒 ∈ 𝐼 we have 𝑒 𝜒 (𝜑) ∈ 𝐹𝜒 iff 𝑓 (𝜑) ∈ 𝐺. By Proposition 3.3.2, we know that the matrix C = 𝑓 (hFmL , 𝑇i) is a submatrix of B and so C∗ ∈ HS SP(K). Furthermore, we can see 𝑓 : hFmL , 𝑇i → C as a strict surjective homomorphism and so, due to Corollary 3.2.9, the matrices hFmL , 𝑇i ∗ and C∗ are isomorphic and the claim easily follows. For the proof of the converse implication, we just write the following chain of (in)equalities (the last equality follows from Proposition 3.4.2): `L =  { hFmL ,𝑇 i∗ |𝑇

is a theory}

⊇ HS SP(K) = K .

To prove the last claim, recall that if L is weakly implicative, then all matrices in SP(K) are reduced and so the claim follows due to Proposition 3.2.8.  Corollary 3.4.11 Let L be a logic over a set of formulas of cardinality 𝜅 and K ⊆ Mod∗ (L). Then, L has the SKC if only if Mod∗ (L) 𝜅 ⊆ HS SP(K). Furthermore, if L is weakly implicative, we can replace HS by I. Our next goal is to study the system FiL (A) of filters over A and show its tight relation with the set ConAlg∗ (L) (A) of relative congruences on A. Recall that FiL (A) is a domain of a complete lattice in which the meet is the (set-theoretic) intersection (and, hence, its lattice order is given by inclusion). First, we formulate (and leave its proof as an exercise for the reader) a corollary of Propositions 3.4.3 and 3.3.7 and Theorem A.3.13 which tells us that ConAlg∗ (L) (A) has the same property. Corollary 3.4.12 Let L be a weakly implicative logic in a language L. Then, Alg∗ (L) is closed under subalgebras and direct products. Hence, the set ConAlg∗ (L) (A) of relative congruences of an L-algebra A is the domain of a complete lattice in which meet is the (set-theoretic) intersection. Let us now focus on the function assigning to each filter over a given algebra the Leibniz congruence of the resulting matrix. Definition 3.4.13 Let A be an L-algebra. We define the Leibniz operator associated to A, ΩA : P ( 𝐴) → Con(A), as ΩA (𝐹) = Ω(hA, 𝐹i). Lemma 3.4.14 Let L be a logic in a language L and A an L-algebra. Then, ΩA restricted to FiL (A) is a surjective mapping onto ConAlg∗ (L) (A).

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Proof Observe that, for each 𝐹 ∈ FiL (A), we know that (due to Lemma 2.7.5) hA/ΩA (𝐹), 𝐹/ΩA (𝐹)i ∈ Mod∗ (L) and so A/ΩA (𝐹) ∈ Alg∗ (L). Thus, we only have to prove that the mapping is onto. First, we observe that, for each congruence 𝜃 such that A/𝜃 ∈ Alg∗ (L), we know that hA/𝜃, 𝐹i ∈ Mod∗ (L) for some filter 𝐹 ∈ F𝑖L (A/𝜃). Then, the canonical mapping ℎ from A onto A/𝜃 is a surjective matrix homomorphism hA, ℎ−1 [𝐹]i → hA/𝜃, 𝐹i and so ℎ−1 [𝐹] ∈ FiL (A). If we show that ΩA (ℎ−1 [𝐹]) = 𝜃 the proof is done: h𝑎, 𝑏i ∈ 𝜃 iff ℎ(𝑎) = ℎ(𝑏) iff hℎ(𝑎), ℎ(𝑏)i ∈ ΩA/𝜽 (𝐹) iff h𝑎, 𝑏i ∈ ΩA (ℎ−1 [𝐹]) (the last equivalence is due to Lemma 3.2.5).  Therefore, whenever a logic L is known from the context, we may speak about the Leibniz operator associated to A as a mapping with domain FiL (A). We can show that this mapping linking filters and congruences has interesting lattice-theoretic properties and, in the case of algebraically implicative logics, it is actually an isomorphism. Proposition 3.4.15 Let L be a weakly implicative logic in a language L and A an L-algebra. Then, for every X ⊆ FiL (A), we have Ñ Ñ 1. ΩA ( Ô X) = Ô𝐹 ∈X ΩA (𝐹). Ð 2. ΩA ( X) = 𝐹 ∈X ΩA (𝐹), whenever X is upwards directed and X ∈ FiL (A).8 Thus, ΩA is a surjective order-preserving mapping from FiL (A) onto ConAlg∗ (L) (A). Proof We have seen that ΩA is indeed a surjective mapping. We show that it Ñ preserves Ñ arbitrary meets (and so it is also order-preserving): h𝑥, 𝑦i ∈ ΩA ( X) iff 𝑥 ⇔ 𝑦 ⊆ X iff for each 𝐹 Ñ ∈ X we have 𝑥 ⇔ 𝑦 ⊆ 𝐹 iff for each 𝐹 ∈ X we have h𝑥, 𝑦i ∈ ΩA (𝐹) iff h𝑥, 𝑦i ∈ 𝐹 ∈X ΩA (𝐹). Ô Ô Ð Now we prove the second claim: h𝑥, 𝑦i ∈ ΩA ( X) iff 𝑥 ⇔ 𝑦 ⊆ X = X iff there are 𝐺, 𝐻 ∈ X such that 𝑥 ⇔ 𝑦 ⊆ 𝐺 ∪ 𝐻 iff there is an 𝐹 ∈ X such that Ð 𝑥 ⇔ 𝑦 ⊆ 𝐹 iff there is an 𝐹 ∈ X such that h𝑥, 𝑦i ∈ ΩA (𝐹) iff h𝑥, 𝑦i ∈ 𝐹 ∈X ΩA (𝐹). Ð Thus, among other things, we have shown that 𝐹 ∈X ΩA (𝐹) ∈ ConAlg∗ (L) (A) and so the claim follows.  Observe the crucial use of the weak implication in the previous proof: while the first claim could be proved in any logic with an arbitrary equivalence set ⇔ (i.e. a set of formulas in two variables defining the Leibniz congruence; cf. Theorem 2.7.7), to prove the second claim we need to know that our logic has a finite equivalence set. Theorem 3.4.16 A weakly implicative logic L in a language L is algebraically implicative iff ΩA is a lattice isomorphism for each L-algebra A. Proof Notice that to prove the left-to-right direction it suffices to show that ΩA is oneone (as it commutes with meets). Suppose ΩA (𝐹) = ΩA (𝐺) for some 𝐹, 𝐺 ∈ FiL (A). By claim 4 of Theorem 2.9.5, we know that 𝐹/ΩA (𝐹) and 𝐺/ΩA (𝐺) are the least L-filters on algebras A/ΩA (𝐹) and A/ΩA (𝐺), which coincide, and so their filters also coincide. Then, for any 𝑎 ∈ 𝐹, we have 𝑎/ΩA (𝐺) = 𝑎/ΩA (𝐹) ∈ 𝐹/ΩA (𝐹) = 𝐺/ΩA (𝐺), which gives 𝑎 ∈ 𝐺. Thus, 𝐹 ⊆ 𝐺 and, by the symmetry, also 𝐹 = 𝐺. 8 This property is known in lattice theory as continuity.

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Converse direction: consider a matrix hA, 𝐹i ∈ Mod∗ (L). Since ΩA (𝐹) = IdA and ΩA is an isomorphism, we know that 𝐹 is the least filter in FiL (A). Thus, claim 4 of Theorem 2.9.5 completes the proof.  An algebra A is simple iff Con(A) = {IdA , 𝐴2 } (see the Appendix). Clearly, for any A ∈ Alg∗ (L), we always have Con(A) ⊇ ConAlg∗ (L) (A) ⊇ {IdA , 𝐴2 } and moreover, in the case that Alg∗ (L) is closed under homomorphic images (which, whenever L is algebraically implicative, is equivalent to Alg∗ (L) being a variety), we also have Con(A) = ConAlg∗ (L) (A). Corollary 3.4.17 Let L be an algebraically implicative logic and A ∈ Alg∗ (L). If A is simple, then FiL (A) = {𝐹A , 𝐴}. The converse direction holds whenever Alg∗ (L) is a variety.

Example 3.4.18 We know (see the Appendix) that, for L being Ł, IL, G, CL, or K, Alg∗ (L) is a variety. Therefore, the same holds for any axiomatic extension of any of these logics (cf. Theorem 2.9.14) and so, for these logics, we have that A ∈ Alg∗ (L) is simple iff FiL (A) = {𝐹A , 𝐴}. Example 3.4.19 Now we can easily prove that if an MV-algebra can be embedded into the algebra [0, 1] Ł , then it is simple (we know that the converse direction is true as well, Proposition A.5.11). Clearly, due to Proposition 3.2.15, it suffices to prove it for any subalgebra A of [0, 1] Ł . Consider a set 𝐹 ∈ FiŁ (A) such that there is an ¯ Recall that, due to Example 2.5.9 we know that 𝐹 is a lattice element 𝑎 ∈ 𝐹 \ {1}. filter such that 𝑥, 𝑥 →A 𝑦 ∈ 𝐹 implies 𝑦 ∈ 𝐹. Therefore, if we prove that 0 ∈ 𝐹, the proof is done; we leave the elaboration of the details as an exercise for the reader.

3.5 Structure of the sets of theories and filters Recall that theories of any given logic L and L-filters over any given algebra form closure systems. There is a useful notion of basis of a closure system that can be defined in several equivalent ways. One of these definitions allows us to describe each closed set as the intersection of a family of elements of the basis, while another one shows that each closed set and each element not contained in it can be separated by an element of the basis. Formally, we define: Definition 3.5.1 (Basis) A basis of a closure system C over a set 𝐴 is any B ⊆ C satisfying any of the following equivalent conditions: Ñ 1. For every 𝑇 ∈ C, thereÑis a D ⊆ B such that 𝑇 = D. 2. For every 𝑇 ∈ C, 𝑇 = {𝐵 ∈ B | 𝑇 ⊆ 𝐵}. 3. For every 𝑇 ∈ C and 𝑎 ∈ 𝐴 \ 𝑇, there is an 𝑆 ∈ B such that 𝑇 ⊆ 𝑆 and 𝑎 ∉ 𝑆.

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3 Completeness properties

Clearly, any set containing a basis is itself a basis too; also each closure system is a basis for itself. Also note that all of the conditions above could have been equivalently written for 𝑇 ∈ C \ { 𝐴}. This observation leads to the following, purely technical, notation which will allow for a compact formulation of certain results later on: for any set B ⊆ C, we denote by B + its extension by {𝐴}. Clearly, B is a basis of C iff B + is. We know that each logic L is complete w.r.t. its Lindenbaum–Tarski matrices, i.e. matrices given by the theories of the logic. The next completeness theorem tells us that, instead of using all theories, we can restrict ourselves to those from a basis of Th(L). Theorem 3.5.2 Let L be a logic and B a basis of the closure system Th(L). Then, L has the SKC for K = {hFmL , 𝑇i ∗ | 𝑇 ∈ B}. Proof Assume that Γ 0L 𝜑. Then, there is a theory 𝑇 ∈ B such that 𝑇 ⊇ ThL (Γ) and 𝜑 ∉ 𝑇, and so Γ 2 hFmL ,𝑇 i 𝜑 (just consider the identity evaluation). Therefore, by Lemma 2.7.5, Γ 2 hFmL ,𝑇 i∗ 𝜑.  Therefore, a basis for a system of theories that can be defined in a natural way always yields an interesting completeness theorem. We define several classes of theories that could serve as basis of Th(L). Each of these definitions satisfies, to various degrees, the following three desiderata: • It is applicable to arbitrary closure systems or, at least, to closure systems of filters over arbitrary algebras. • It defines a class of closed sets which is actually a basis for most of the closure systems in which the definition can be applied (or at least the class of such closure systems should be interesting on its own). • The algebras of reduced matrices with filters from the defined class are related to some important algebraic notions or constructions. We start by defining, for each closure system, the classes of meet-irreducible and finitely meet-irreducible closed sets. We will see that these two definitions are applicable to all closure systems, often define a basis (e.g. in all inductive ones), and are related to the notion of subdirectly irreducible algebras (see the next section for details). Then, we define the class of maximally consistent closed sets and see that, while this definition is still universal and related to the important class of simple algebras, maximally consistent closed sets form a basis only in rather special cases (see Corollary 3.5.17). Finally, in order to relate our abstract work to more familiar classes of theories studied for prominent logics (and to generate examples useful to illustrate our subsequent results), we restrict our attention to systems of theories of expansions of BCIlat and define prime, linear, and complete theories. Later, in Chapters 5 and 6, we define abstract notions of prime and linear filters for wide classes of logics and see that they indeed are related to important algebraic notions. We will also see (for logics over BCIlat in this section, and in general later) that quite often these theories do form a basis, because they coincide with finitely meet-irreducible theories (precise formulations will follow soon). Therefore, while from a purely theoretical point of

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111

view, the class of meet-irreducible theories is more interesting (it is a smaller class and, hence, it yields stronger completeness theorems), we will see that the finitely meet-irreducible ones play a more important role for logics. Definition 3.5.3 (Meet-irreducible closed sets) An element 𝑋 of a closure system C over a set 𝐴 is Ñ • meet-irreducible if for each set Y ⊆ C such that 𝑋 = 𝑌 ∈Y 𝑌 , there is a 𝑌 ∈ Y such that 𝑋 = 𝑌 . • finitely meet-irreducible if for each finite non-empty set Y ⊆ C such that 𝑋 = Ñ 𝑌 ∈Y 𝑌 , there is a 𝑌 ∈ Y such that 𝑋 = 𝑌 . Note that set 𝐴 itself is finitely meet-irreducible, but not meet-irreducible (indeed, it is the intersection of the empty set). Also, observe that the finite meet-irreducibility of a set 𝑋 can be equivalently defined by the following condition: For each 𝑌1 , 𝑌2 ∈ C such that 𝑋 = 𝑌1 ∩ 𝑌2 , we have 𝑋 = 𝑌1 or 𝑋 = 𝑌2 . Sometimes, especially when speaking about theories of a logic, we use the terms intersection-prime and completely intersection-prime respectively instead of finitely meet-irreducible and meet-irreducible.9 Given a logic L, we denote the sets of these theories, respectively, as IntPrimeTh(L) and CIntPrimeTh(L). Obviously, we have CIntPrimeTh(L) ⊆ IntPrimeTh(L) and, due to the fact that FmL ∈ IntPrimeTh(L) \ CIntPrimeTh(L), we can never have the converse inclusion. However, in Proposition 3.5.19, we will see that in certain logics we have CIntPrimeTh(L) + = IntPrimeTh(L). Before we prove the promised result that meet-irreducible closed sets form a basis of any inductive closure system, we give an alternative, useful and logically relevant description of these sets and show that being (finitely) meet-irreducible is preserved under strict surjective homomorphisms (a simple consequence of Proposition 3.2.15); cf. an analogous claim for logics with a suitable disjunction connective in Proposition 5.4.5. Proposition 3.5.4 A closed set 𝑋 of a closure system C over a set 𝐴 is meet-irreducible iff it is maximal w.r.t. an element of 𝐴, i.e. there is an 𝑎 ∈ 𝐴 such that 𝑋 is maximal in the set {𝑌 ∈ C | 𝑎 ∉ 𝑌 } w.r.t. the order given by inclusion. Proof To prove this proposition we only need two simple observations: Ñ • 𝑋 is meet-irreducible iff {𝑋 0 ∈ C | 𝑋 0 ) 𝑋 } ) 𝑋. Ñ • 𝑋 is maximal w.r.t. 𝑎 iff 𝑎 ∈ {𝑋 0 ∈ C | 𝑋 0 ) 𝑋 } \ 𝑋.



Proposition 3.5.5 Let L be a logic, hA, 𝐹i, hB, 𝐺i ∈ Mod(L), and a mapping ℎ : hA, 𝐹i → hB, 𝐺i a strict surjective homomorphism. If 𝐹 is (finitely) meetirreducible, then so is 𝐺. If, furthermore, L is a weakly implicative logic, then the converse implication holds as well. 9 Later, in Proposition 3.5.21 (and also in a more abstract setting of Section 5.4), we will see how these notions are related with the perhaps more common definition of prime element using disjunction.

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Proof First note that we have 𝐹 = ℎ−1 [𝐺]. Assume that 𝐺 is not (finitely) meetÑ irreducible, i.e. there is a system of filters G ⊆ FiL (B) such that 𝐺 = G and, for each 𝐻 ∈ G, there is an ℎ(𝑥 𝐻 ) ∈ 𝐻 \ 𝐺. Thus, thanks to the strictness of ℎ, we know Ñ that 𝑥 𝐻 ∈ ℎ−1 [𝐻] \ ℎ−1 [𝐺] = ℎ−1 [𝐻] \ 𝐹. Thus, if we prove that 𝐹 = 𝐻 ∈ G ℎ−1 [𝐻], Ñ the proof is done. Clearly,Ñ we have 𝐹 ⊆ 𝐻 ∈ G ℎ−1 [𝐻]. Assume thatÑthe inclusion is strict, i.e. there is an 𝑥 ∈ ( 𝐻 ∈ G ℎ−1 [𝐻]) \ 𝐹; then clearly ℎ(𝑥) ∈ ( 𝐻 ∈ G 𝐻) \ 𝐺, a contradiction. To prove the second claim, we use Proposition 3.2.15 which tells us that the mapping 𝒉 : [𝐹, 𝐴] → [𝐺, 𝐵] defined as 𝒉(𝐻) = ℎ[𝐻] is a lattice isomorphism (recall that we denote by [𝐹, 𝐴] the interval of L-filters containing 𝐹).  Lemma 3.5.6 (Abstract Lindenbaum Lemma) Let C be an inductive closure system. Then, meet-irreducible closed sets form a basis of C. Proof The proof is an easy application of Zorn’s Lemma (Lemma A.1.10): we show that for each set 𝑋 ∈ C and 𝑎 ∉ 𝑋, the set A = {𝑆 ∈ C | 𝑋 ⊆ 𝑆 and 𝑎 ∉ 𝑆} has a maximal element (which by the previous proposition is meet-irreducible).ÐClearly, A is non-empty (𝑋 ∈ A) and for any chain {𝑆𝑖 | 𝑖 ∈ 𝐼} ⊆ A, we have 𝑎 ∉ 𝑖 ∈𝐼 𝑆𝑖 ∈ C Ð (by the inductivity assumption). Therefore, 𝑖 ∈𝐼 𝑆𝑖 ∈ A and so it is an upper bound of that chain in A. Consequently, by Zorn’s Lemma, A has a maximal element.  Recall that each finitary logic can be seen as a finitary closure operator and so its closure system of theories is inductive (Proposition 2.1.9). From Theorem 2.6.2, we actually know that the same holds for closure systems of all L-filters over any algebra. Therefore, we obtain the following corollary of the abstract Lindenbaum Lemma. Corollary 3.5.7 Let L be a finitary logic and A an L-algebra. Then, the class of meetirreducible filters forms a basis of FiL (A). In particular, the class of meet-irreducible theories forms a basis of Th(L) and so L has the SKC for the set K = {hFmL , 𝑇i ∗ | 𝑇 is meet-irreducible}. As we will see later, both the abstract Lindenbaum Lemma and its corollary above, proved for finitary logics only, are crucial for many results (see e.g. the upcoming Theorem 3.5.9) and are actually not restricted to finitary logics. Therefore, we introduce the following two properties of logics (for reasons that will be apparent later, here we prefer to use the logical terminology). Definition 3.5.8 We say that a logic L has the (completely) intersection-prime extension property, IPEP or CIPEP resp. for short, if the class of its (completely) intersection-prime theories forms a basis of Th(L). Clearly, the CIPEP implies the IPEP, and any finitary logic has the CIPEP. Recall that in Example 2.6.6 we have introduced two infinitary logics: the logic given by the standard Łukasiewicz matrix Ł∞ and that given by all finite matrices Ł𝑛 . In Example 3.6.16 we will show that both logics have the CIPEP. Hence, this property defines a natural proper extension of the class of finitary logics.

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The next theorem provides another characterization of the SKC, similar to that of Theorem 3.4.10, in terms of localization of Lindenbaum–Tarski matrices, but now using only those given by meet-irreducible theories and providing a tighter localization (note that we need not speak about products). Theorem 3.5.9 Let L be a logic and K ⊆ Mod∗ (L). If L has the SKC, then {hFmL , 𝑇i ∗ | 𝑇 is a meet-irreducible theory} ⊆ HS S(K). The converse implication is valid as well whenever L has the CIPEP. Finally, if L is weakly implicative, we can replace HS by I. Proof For the left-to-right direction, consider a matrix hFmL , 𝑇i ∗ , where 𝑇 is a meetirreducible theory. Due to the SKC, for each 𝜑 ∉ 𝑇 there is an A 𝜑 = h𝐴 𝜑 , 𝐹𝜑 i ∈ K such that 𝑇 2A 𝜑 𝜑. Let us denote by 𝑒 𝜑 an A 𝜑 -evaluation witnessing this fact and observe. It is easy to see that Ù 𝑇= 𝑒 −1 𝜑 [𝐹𝜑 ]. 𝜑∉𝑇

Thus, due to the meet-irreducibility of 𝑇, there has to be a formula 𝜓 ∉ 𝑇 such that 𝑇 = 𝑒 −1 𝜓 [𝐹𝜓 ]. Therefore, 𝑒 𝜓 is a strict homomorphism from hFmL , 𝑇i into A 𝜑 and we can see it as a strict homomorphism from hFmL , 𝑇i onto the submatrix h𝑒 𝜓 (FmL ), 𝑒 𝜓 [𝑇]i of A 𝜑 . Thus, by Corollary 3.2.9, we know that hFmL , 𝑇i ∗ is isomorphic to h𝑒 𝜓 (FmL ), 𝑒 𝜓 [𝑇]i ∗ , a strict homomorphic image of a submatrix of a matrix from K. For the proof of the converse implication just notice that K = HS S(K) ⊆  { hFmL ,𝑇 i∗ |𝑇

is a meet-irreducible theory}

= `L ,

where the last equality follows from Theorem 3.5.2.



Remark 3.5.10 Note that the weaker assumption of IPEP would allow us to obtain the SKC from {hFmL , 𝑇i ∗ | 𝑇 is a finitely meet-irreducible theory} ⊆ HS S(K). The formulation and the proof of corresponding converse implication is more complicated: not only do we need some additional assumption on the logic L, but we also need to expand the class K by adding the trivial reduced matrix (we call the resulting class K+ ) to account for the fact that the inconsistent theory is finitely meet-irreducible and, thus, the left hand-side class now contains the trivial reduced matrix which cannot be a strict homomorphic image of any non-trivial one (by virtue of Corollary 3.2.6). In Theorem 5.6.5 we will improve this result by showing, in the framework of finitary weakly implicative logics with a suitable disjunction, that the SKC is equivalent to {hFmL , 𝑇i ∗ | 𝑇 is a finitely meet-irreducible theory} ⊆ IS(K+ ).

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As illustrated by several examples in the previous chapter, it is very common to introduce logics as axiomatic extensions of other systems. Hence, an important point of the general theory we are presenting is whether relevant metalogical properties are preserved under axiomatic extensions. This is obviously true for finitarity; the next lemma proves it for the IPEP and the CIPEP. Lemma 3.5.11 Let L0 be an axiomatic extension of L. If L has the IPEP (resp. the CIPEP), then also L0 has it. Proof First note that, obviously, Th(L0) ⊆ Th(L) and we show that every Ltheory 𝑇 ⊇ Thm(L0) is an L0-theory. Indeed, if 𝑇 `L0 𝜑, then there is a proof of 𝜑 from 𝑇 that may use the additional axioms of L0, but, as these additional axioms are in 𝑇, we actually have a proof of 𝜑 from 𝑇 in L, and so 𝜑 ∈ 𝑇. Thus, Th(L0) = {𝑆 ∈ Th(L) | Thm(L0) ⊆ 𝑆}. Therefore, for each 𝑇 ∈ Th(L0), 𝑇 is (completely) intersection-prime in Th(L) iff 𝑇 is (completely) intersection-prime in Th(L0). Now consider an L0-theory 𝑇. Since 𝑇 is also an L-theory, the (C)IPEP Ñ gives us a system 𝐼 of (completely) intersection-prime L-theories such that 𝑇 = 𝑆 ∈𝐼 𝑆. For any 𝑆 ∈ 𝐼, we have 𝑆 ⊇ 𝑇 ⊇ Thm(L0), and hence we know that theories in 𝐼 are actually L0-theories which are (completely) intersection-prime in Th(L0).  Now we are ready to explore the relation between meet-irreducible closed sets and maximal ones. Recall that in Proposition 3.5.4 we showed that a closed set is meet-irreducible iff it is a maximal closed set w.r.t. an element. This leads us to a natural definition of a special kind of meet-irreducible set: Definition 3.5.12 (Maximally consistent closed sets) An element 𝑋 of a closure system C over a set 𝐴 is maximally consistent if it is a maximal element of C \ { 𝐴}. Clearly, any maximally consistent set 𝑋 is maximal w.r.t. any element from 𝐴 \ 𝑋. If C = Th(L), we speak about maximally consistent theories and denote the set as MaxConTh(L). Clearly, we always have MaxConTh(L) ⊆ CIntPrimeTh(L) ⊆ IntPrimeTh(L). Remark 3.5.13 Assume that 𝐴 has an inconsistent element, i.e. an element ⊥ ∈ 𝐴 such that, for each closed set 𝐶, we have 𝐶 = 𝐴 iff ⊥ ∈ 𝐶 (which is the case e.g. in Th(L) for expansions of BCIlat due to the axiom (⊥) and modus ponens). Then, we can obviously characterize maximally consistent sets simply as those that are maximal w.r.t. ⊥. In this case, we can also use the abstract Lindenbaum Lemma to show that, if C is inductive, then every consistent closed set can be extended to a maximally consistent one. However, this does not entail in general that maximally consistent sets form a basis of C (see Example 3.5.18). Let us now explore the promised algebraic relevance of maximally consistent closed sets and simple algebras. As these results are based on Corollary 3.4.17, they are formulated for an algebraically implicative logic L and, at some point, we will need the assumption that Alg∗ (L) is a variety.

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Proposition 3.5.14 Let L be an algebraically implicative logic, A a non-trivial member of Alg∗ (L), and 𝐹 ∈ FiL (A). • If A/ΩA (𝐹) is simple, then 𝐹 is maximally consistent in FiL (A). • If A is simple, then 𝐹A is maximally consistent in FiL (A). If Alg∗ (L) is a variety, the converse implications of both claims hold as well. Proof We prove only the first claim; the second one can be seen as a special case. Let us denote the algebra A/ΩA (𝐹) as B, the filter 𝐹/ΩA (𝐹) as 𝐺, and note that hB, 𝐺i ∈ Mod∗ (L). Recall that by [𝐹, 𝐴] we denote the interval in the lattice FiL (A) consisting of filters containing 𝐹 and, due to Proposition 3.2.15, it is isomorphic to the interval [𝐺, 𝐵] which, as hB, 𝐺i is reduced, is equal to FiL (B). Clearly, 𝐹 is maximally consistent iff [𝐹, 𝐴] = {𝐹, 𝐴} and 𝐹 ≠ 𝐴 iff FiL (B) = {𝐺, 𝐵} and 𝐺 ≠ 𝐵. Thanks to Corollary 3.4.17, we know that the last claim holds whenever B is simple and non-trivial, and the converse is true as well in the case that Alg∗ (L) is a variety. This proposition has two interesting corollaries. The first one gives us a useful characterization of simple algebras which we use to obtain a well known algebraic fact (Example 3.5.16). Corollary 3.5.15 Let L be an algebraically implicative logic such that Alg∗ (L) is a variety, A ∈ Alg∗ (L), and 𝜅 an infinite cardinal bigger than or equal to the cardinality of 𝐴. Then, A is a non-trivial simple algebra iff hA, 𝐹A i is isomorphic to hFmL0 , 𝑇i ∗ 0 for some maximally consistent LVar -theory 𝑇 and some superset Var 0 of Var of cardinality 𝜅. Proof By Theorem 3.2.10, we know that hA, 𝐹A i is a strict homomorphic image of 0 hFmL0 , 𝑇i for some LVar -theory 𝑇. Therefore, FiL (A) and the interval [𝑇, FmL0 ] of 0 FiL (FmL ) are isomorphic, due to Proposition 3.2.15. Since A is simple and non-trivial, 𝑇 is maximally consistent. The proof of the converse direction is analogous. 

Example 3.5.16 The only non-trivial simple Heyting algebra up to isomorphism is the Boolean algebra 2. We first show a related claim: • If 𝑇 is a maximally consistent theory of an axiomatic expansion L of IL→ , then the algebra FmL /ΩFmL (𝑇) has only two elements. Clearly, L is Rasiowa-implicative and so we know that, for any two elements 𝜑, 𝜓 ∈ 𝑇, we have 𝑇 `L 𝜑 ⇔ 𝜓 and so 𝜑/ΩFmL (𝑇) = 𝜓/ΩFmL (𝑇). Next, consider formulas 𝜑, 𝜓 ∉ 𝑇. From the maximal consistency of 𝑇, we know that 𝑇, 𝜓 `L 𝜑 and 𝑇, 𝜑 `L 𝜓. Thus, by the classical deduction theorem 2.4.1, 𝑇 `L 𝜑 ⇔ 𝜓 and so 𝜑/ΩFmL (𝑇) = 𝜓/ΩFmL (𝑇). Finally, consider 𝜑 ∈ 𝑇 and 𝜓 ∉ 𝑇. Then, 𝑇 0L 𝜑 → 𝜓, i.e. 𝜑/ΩFmL (𝑇) ≠ 𝜓/ΩFmL (𝑇). Since Heyting algebras form a variety, we can use the previous corollary and so we know that any non-trivial simple countable Heyting-algebra has two values, i.e. it has to be the Boolean algebra 2. We leave the extension to higher cardinalities as an exercise for the reader.

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The second corollary is now an obvious consequence of Theorem 3.5.2. Its main interest is that the resulting completeness property is easy to refute in most logics, thus showing that indeed MaxConTh(L) is a basis of Th(L) only in special cases. Corollary 3.5.17 Let L be an algebraically implicative logic such that Alg∗ (L) is a variety and MaxConTh(L) forms a basis of Th(L). Then, L has the SKC for K = {hA, 𝐹A i | A is a simple (countable) L-algebra}.

Example 3.5.18 We show that the maximally consistent theories of intuitionistic logic do not form a basis of Th(IL). Indeed, in the previous example we have established that the only non-trivial simple Heyting algebra is 2 and so the previous corollary would entail e.g. the validity of (lem) in IL. We can also use this corollary to demonstrate that the maximally consistent theories of Łukasiewicz logic Ł do not form a basis of Th(Ł). Indeed, in Example 2.6.6 we have shown that the infinitary Hay rule (Ł∞ ) (introduced in Example 2.3.9) is valid in |=Ł∞ but not in Ł. Due to the fact that every non-trivial simple MV-algebra A can be embedded into the standard MV-algebra [0, 1] Ł (see Proposition A.5.11), we know that (Ł∞ ) is valid in hA, {>}i, a contradiction.10 Next, we show that in any axiomatic expansion L of IL→ the maximally consistent theories coincide with (completely) intersection-prime theories and provide a basis of the closure system of theories Th(L) if and only if L expands CL→ . This simple proposition allows us to prove two results that we had promised before: firstly, that CL is indeed the classical two-valued Boolean logic and, secondly, that CL→ is its →-fragment. Proposition 3.5.19 Let L be an axiomatic expansion of IL→ . Then, the following are equivalent: 1. 2. 3. 4.

L expands CL→ . MaxConTh(L) + = CIntPrimeTh(L) + = IntPrimeTh(L). MaxConTh(L) = CIntPrimeTh(L). MaxConTh(L) forms a basis of Th(L).

Proof To prove the first implication, assume that a theory 𝑇 is consistent but not maximally consistent, i.e. there are formulas 𝜑 and 𝜓 such that 𝜑 ∉ 𝑇 and 𝑇, 𝜑 0L 𝜓, and we show that 𝑇 is not intersection-prime as it equals the intersection of the theories generated by 𝑇 ∪ {𝜑} and 𝑇 ∪ {𝜑 → 𝜓} (clearly, 𝜑 → 𝜓 ∉ 𝑇, due to the deduction theorem). 10 Note that we could prove this fact without referring to the embeddability of simple algebras into the standard one; indeed as a consequence of Corollary 4.8.10 we obtain that, for any element 𝑎 ≠ > of a simple MV-algebra A there is an 𝑛 such that 𝑎& 𝑛-times . . . &𝑎 = 0 (otherwise, FiAL ( {𝑎 }) = {𝑥 ∈ 𝐴 | 𝑥 ≥ 𝑎& 𝑚-times . . . &𝑎 for some 𝑚} ≠ 𝐴, a contradiction with the simplicity of 𝐹 ), which immediately implies the validity of the Hay rule (Ł∞ ) in hA, {>}i.

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Assume that 𝑇, 𝜑 `L 𝜒 and 𝑇, 𝜑 → 𝜓 `L 𝜒. Due to the deduction theorem, we obtain 𝑇 `L 𝜑 → 𝜒 and 𝑇 `L (𝜑 → 𝜓) → 𝜒. Using the rule (Sf) twice on the first premise we obtain 𝑇 `L ((𝜑 → 𝜓) → 𝜒) → (( 𝜒 → 𝜓) → 𝜒). Thus, by modus ponens, also 𝑇 `L ( 𝜒 → 𝜓) → 𝜒 and so by Peirce’s law 𝑇 `L 𝜒. The next two implications are simple (the second one is due to Corollary 3.5.7). To prove the last implication, it suffices to observe that if maximally consistent theories form a basis of Th(L), then the set of theorems of L equals the intersection of all such theories and, if we show that any such a theory contains the formula ((𝜑 → 𝜓) → 𝜑) → 𝜑, the proof is done (thanks to Example 2.3.1, we know that CL→ is the axiomatic extension of IL→ by Peirce’s law). The elaboration of the details is left as an exercise for the reader.  Corollary 3.5.20 • CL→ enjoys the SKC for K = {h2→ , {1}i}. • CL enjoys the SKC for K = {h2, {1}i}. • CL→ is the →-fragment of CL. Proof Assume that Γ 0CL→ 𝜑 so that we have a maximally consistent theory 𝑇 containing Γ such that 𝜑 ∉ 𝑇, and consider the Lindenbaum–Tarski matrix A = hFmL→ , 𝑇i ∗ . If we prove that this matrix is isomorphic to h2→ , {1}i, the proof is done. In Example 3.5.16 we have established that A has indeed exactly two values, represented e.g. as 𝜑/ΩFmL→ (𝑇) and (𝜑 → 𝜑)/ΩFmL→ (𝑇). We leave the rest of the proof of this claim as an exercise for the reader. The proof of the second claim is analogous (or we could use Example 3.5.16 which also tells us that 2 is the only non-trivial simple Boolean algebra) and the final claim follows from the first one and Theorem 3.3.12.  Next we introduce three other kinds of theories considered in the literature. We restrict ourselves to logics expanding BCIlat (in order to have all the necessary connectives, in subsequent chapters we expand these definitions to much wider classes of logics) and say that a theory 𝑇 is11 • complete if for each formula 𝜑 we have that 𝜑 ∈ 𝑇 or ¬𝜑 ∈ 𝑇. • prime if for each pair 𝜑, 𝜓 of formulas we have that 𝜑 ∨ 𝜓 ∈ 𝑇 implies that 𝜑 ∈ 𝑇 or 𝜓 ∈ 𝑇. • linear if for each pair 𝜑, 𝜓 of formulas we have that 𝜑 → 𝜓 ∈ 𝑇 or 𝜓 → 𝜑 ∈ 𝑇. We denote the sets of these theories, respectively, as: ComplTh(L), PrimeTh(L), and LinTh(L). Note that the theory FmL belongs to all of these sets. We will see that intersection-prime theories often can be simply described as prime theories (in axiomatic expansions of BCKlat ) or linear theories (in axiomatic expansions of FBCKlat ). In contrast, completely intersection-prime theories are more elusive and are usually a proper subset of intersection-prime theories, only coinciding in very special cases. Let us first prove some positive inclusion/equality results and then we show the separation of these classes of theories. 11 Observe that the defining conditions for prime, complete, and maximally consistent theories are analogous to the usual equivalent definitions of ultrafilters in Boolean algebras.

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Proposition 3.5.21 Let L be an expansion of BCIlat . Then, PrimeTh(L) ⊆ IntPrimeTh(L). If furthermore L has the IPEP, then the following are equivalent: 1. PrimeTh(L) = IntPrimeTh(L). 2. PrimeTh(L) forms a basis of Th(L). 3. L has the classical proof by cases property: ThL (Γ, 𝜑 ∨ 𝜓) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓). Proof Assume that a theory 𝑇 is not intersection-prime, i.e. 𝑇 = 𝑇1 ∩ 𝑇2 for some 𝑇𝑖 ) 𝑇; consider formulas 𝜑𝑖 ∈ 𝑇𝑖 \ 𝑇 and note that, thanks to the axioms (ub1 ) and (ub2 ), we have 𝜑1 ∨ 𝜑2 ∈ 𝑇1 ∩ 𝑇2 = 𝑇, i.e. 𝑇 is not prime. The first implication is just the IPEP. To prove the second implication, observe that ThL (Γ, 𝜑 ∨ 𝜓) ⊆ ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) follows from the axioms (ub1 ) and (ub2 ). To prove the converse, assume that Γ, 𝜑 ∨ 𝜓 0L 𝜒. Then, there is a prime theory 𝑇 such that Γ ∪ {𝜑 ∨ 𝜓} ⊆ 𝑇 and 𝜒 ∉ 𝑇. Since 𝑇 is prime, we know that 𝜑 ∈ 𝑇 or 𝜓 ∈ 𝑇. If 𝜑 ∈ 𝑇, then we have Γ, 𝜑 0L 𝜒; the other case is analogous. To prove the final implication, assume that a theory 𝑇 is not prime, i.e. there are formulas 𝜑, 𝜓 ∉ 𝑇 such that 𝜑 ∨ 𝜓 ∈ 𝑇; we show that 𝑇 is the intersection of the theories generated by 𝑇 ∪ {𝜑} and 𝑇 ∪ {𝜓}. Indeed, take a formula 𝜒 in both theories. Then, by the classical proof by cases property, 𝜒 is in the theory generated by 𝑇 ∪ {𝜑 ∨ 𝜓} which is equal to 𝑇.  Proposition 3.5.22 Let L be an expansion of BCKlat . The sets of theories defined above obey the inclusions depicted in the following diagram: ComplTh(L) ⊆

LinTh(L)

⊆ PrimeTh(L) ⊆

⊆

MaxConTh(L) + ⊆ CIntPrimeTh(L) + ⊆ IntPrimeTh(L) Furthermore, if L is an axiomatic expansion of BCKlat , then PrimeTh(L) = IntPrimeTh(L) and, if it is an axiomatic expansion of IL, then also MaxConTh(L) + = ComplTh(L). Finally, for axiomatic expansions of BCKlat , we also have: 1. The following are equivalent: a. LinTh(L) = IntPrimeTh(L). b. LinTh(L) forms a basis of Th(L). c. L expands FBCKlat .

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2. The following are equivalent: a. All six sets of theories coincide b. ComplTh(L) = CIntPrimeTh(L) + . c. ComplTh(L) forms a basis of Th(L). d. L expands CL. Proof First, observe that from the previous proposition, we know that PrimeTh(L) ⊆ IntPrimeTh(L) and that we even have equality for axiomatic expansions of BCKlat , since they enjoy the classical proof by cases property (Proposition 2.4.5). Next, we show that LinTh(L) ⊆ PrimeTh(L). Assume that 𝑇 is linear and we have 𝜑 ∨ 𝜓 ∈ 𝑇. Without loss of generality, we can assume that 𝜑 → 𝜓 ∈ 𝑇. Thus, also, 𝜑 ∨ 𝜓 → 𝜓 ∈ 𝑇 (thanks to (id) and (Sup)) and so 𝜓 ∈ 𝑇 due to modus ponens. We know that MaxConTh(L) ⊆ CIntPrimeTh(L) ⊆ IntPrimeTh(L) and so it remains to show that consistent complete theories are linear and maximally consistent. Linearity: clearly, if 𝜑 ∈ 𝑇, then 𝜓 → 𝜑 ∈ 𝑇 (thanks to the axiom (w)) and, if ¬𝜑 ∈ 𝑇, then 𝜑 → 𝜓 ∈ 𝑇 (due to the axiom (⊥) and transitivity). Maximal consistency: if 𝜑 ∉ 𝑇, then ¬𝜑 = 𝜑 → ⊥ ∈ 𝑇 and so 𝑇, 𝜑 `L ⊥ by modus ponens, i.e. 𝑇 is maximal w.r.t. ⊥. Next, suppose that L is an axiomatic expansion of IL, take a maximally consistent theory 𝑇 of L, and assume that 𝜑 ∉ 𝑇; then 𝑇, 𝜑 `L ⊥ and so, by the deduction theorem 2.4.1, we obtain 𝑇 `L ¬𝜑. Finally, we suppose that L is an axiomatic expansion of BCKlat and prove the two numbered claims: 1. The first implication is simple: L clearly has the IPEP and so LinTh(L) forms a basis of Th(L). If linear theories form a basis of Th(L), then the set of theorems of L equals the intersection of all such theories and, since any linear theory obviously contains the formula (𝜑 → 𝜓) ∨ (𝜓 → 𝜑) (the axiom (p∨ ) of prelinearity), (p∨ ) is a theorem of L. Thus, due to Example 2.4.7, we know that L expands FBCKlat . Finally, by the same example, any expansion of FBCKlat proves (p∨ ), which clearly implies that any (intersection) prime theory is linear. 2. The first implication is trivial and the next two are proved as above. Recalling that, thanks to Example 2.4.8, we know that if L proves the law of excluded middle, then it expands CL. To prove the final implication, notice that, in any logic with the law of excluded middle, any (intersection) prime theory is clearly complete and so all the classes collapse.  Let us now give some counterexamples to the converse inclusions of some claims proved in the previous propositions. Example 3.5.23 • We know that S4 does not enjoy the classical proof by cases property (see comments right after Proposition 2.4.5) and so, by Proposition 3.5.21, we cannot have PrimeTh(S4) ⊇ IntPrimeTh(S4). We leave as an exercise for the reader to show the same claim for BCIlat .

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• Consider a maximal theory 𝑇 ∈ Th(IL) w.r.t. a formula 𝑝 ∨ ¬𝑝 (which we know is not a theorem of intuitionistic logic and so such a theory has to exist due to the abstract Lindenbaum Lemma 3.5.6). Then, 𝑇 is completely intersection-prime, due to Proposition 3.5.4, but cannot be complete as both 𝑝 ∈ 𝑇 and ¬𝑝 ∈ 𝑇 would imply that 𝑝 ∨ ¬𝑝 ∈ 𝑇. • We can obtain an example in IL of a non-linear completely intersection-prime theory in a similar fashion, if we start with the formula (𝑞 → 𝑝) ∨ ( 𝑝 → 𝑞). • Intuitionistic logic is well known to satisfy the following property12 (which is false in classical and in many other logics): for any formulas 𝜑 and 𝜓, if `IL 𝜑 ∨ 𝜓, then `IL 𝜑 or `IL 𝜓. Therefore, the set Thm(IL) of its theorems is prime and, hence, intersection-prime. But, obviously, it is the intersection of the following infinite system of strictly bigger theories {ThIL (𝑣) | 𝑣 ∈ Var} (indeed, for 𝜑 ∉ Thm(IL) and an atom 𝑣 not occurring in 𝜑, we have 𝑣 0IL 𝜑). • Let us give another example of an intersection-prime (but not completely) theory of IL. Due to Example 2.5.9, we know that G∞ = h[0, 1] G , {1}i ∈ Mod(IL). Consider any evaluation 𝑒, such that 𝑒(𝑣 𝑖 ) = 1 − 1𝑖 for each 𝑖 ≥ 1. Using Lemma 2.5.7, Proposition 2.6.1 and Example 2.5.9 we know that 𝑇𝑖 = 𝑒 −1 [1 − 1𝑖 , 1] is a theory for each 𝑖 ≥ 1; define 𝑇0 = 𝑒 −1 {1}. Clearly, for each 𝜑 and 𝜓, we have: 𝜑 ∨ 𝜓 ∈ 𝑇0 iff 𝑒(𝜑) ∨ 𝑒(𝜓) = 1 iff 𝑒(𝜑) = 1 or 𝑒(𝜓) = 1, i.e. 𝑇0 is prime, but we can easily Ñ observe that 𝑇0 = 𝑖 ≥1 𝑇𝑖 , while for each 𝑖 ≥ 1, 𝑣 𝑖 ∈ 𝑇𝑖 \ 𝑇0 , i.e. 𝑇0 is not completely intersection-prime.

In the next example, we use Proposition 3.5.22 to show the strong completeness of the infinitary logic Ł∞ with respect to the matrix Ł∞ = h[0, 1] Ł , {1}i. We will prove this result via an intermediary claim about the finitary Łukasiewicz logic Ł13 which is interesting on its own and can be seen as an unusual form of the strong completeness theorem of Ł with respect to Ł∞ (we know that the usual form does not hold, cf. Example 2.6.6; later in Example 3.8.9 we will prove the usual finite strong completeness of Ł with respect to Ł∞ ). Example 3.5.24 For any set Γ ∪ {𝜑} of formulas in LCL , the following three claims are equivalent: 1. Γ Ł∞ 𝜑. 2. Γ `Ł ¬𝜑 → 𝜑 𝑛 for each 𝑛 ≥ 1. 3. Γ `Ł∞ 𝜑. The only complicated part of the proof is to show that 1 implies 2. Indeed, the fact that 2 implies 3 is obvious, due to the Hay rule (Ł∞ ), and the fact that 3 implies 1 is just the soundness of the logic Ł∞ w.r.t. Ł∞ which we have already established in Example 2.6.6. 12 Established already by Gödel in [154], it has been called the disjunction property. A study of this property in a wide family of substructural logics can be found in [190]. 13 Later, in Example 5.5.16, we give an alternative direct proof of this completeness result.

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Assume that Γ 0Ł ¬𝜑 → 𝜑 𝑛 for some 𝑛 and note that Γ ∪ {𝜑 𝑛 → ¬𝜑} 0Ł ⊥ (indeed, we know that there is an intersection-prime theory 𝑇0 expanding Γ not containing ¬𝜑 → 𝜑 𝑛 ; due to Proposition 3.5.22 we know that 𝑇0 is linear and so 𝜑 𝑛 → ¬𝜑 ∈ 𝑇0 ). Consider a maximal consistent theory 𝑇 containing Γ ∪ {𝜑 𝑛 → ¬𝜑}. Therefore, by Corollary 3.5.15, FmL /ΩFmL (𝑇) is a non-trivial simple MV-algebra which, due to Proposition A.5.11, can be embedded into [0, 1] Ł and thus, in particular, Γ ∪ {𝜑 𝑛 → ¬𝜑} 2Ł∞ ⊥, which easily entails that Γ 2Ł∞ 𝜑. Note an interesting corollary of the equivalence we have just proved: if Γ `Ł∞ 𝜑, then there is a proof of 𝜑 from Γ where the rule (Ł∞ ) is used only once: in the last step of the proof (of course, there could also be a proof which does not use it at all). As a final result for this section, we observe the relation between linear theories and linearity of matrix preorders in their Lindenbaum–Tarski matrices and show that logics L in which linear theories form a basis of Th(L) are exactly those that are strongly complete w.r.t. their linearly ordered models. Let us first observe that the notions of ⇒-linear theory and ⇒-linearly ordered matrix can be defined for an arbitrary weak implication ⇒ (again, following our general convention, we omit the reference to a particular implication when fixed or clear from the context). Logics enjoying the completeness property w.r.t. linearly ordered matrices will be called semilinear and studied in detail in Chapter 6. In virtue of Proposition 3.5.22, we know that semilinear logics include at least all axiomatic expansions of the logic FBCKlat . Corollary 3.5.25 Let L be a weakly implicative logic in a language L. Then, a theory 𝑇 is linear iff ≤ hFmL ,𝑇 i∗ is a linear order. Furthermore, linear theories form a basis of Th(L) iff L enjoys the SKC for K = {A ∈ Mod∗ (L) | ≤A is linear}. Proof The first claim is an obvious consequence of the definition of matrix preorder, Lemma 3.2.11, and Theorem 2.8.9. The left-to-right direction of the second claim follows from Theorem 3.5.2. To prove the converse direction, assume that 𝜒 ∉ 𝑇. Therefore, 𝑇 0L 𝜒 and there is an A = hA, 𝐹i ∈ Mod∗ (L) such that ≤A is linear and an A-evaluation 𝑒 such that 𝑒[𝑇] ⊆ 𝐹 and 𝑒( 𝜒) ∉ 𝐹. We define 𝑇 0 = 𝑒 −1 [𝐹]. Obviously, 𝑇 ⊆ 𝑇 0 ∈ Th(L) and 𝜒 ∉ 𝑇 0. Since ≤A is a linear order, we know that for each 𝜑, 𝜓 we have: 𝑒(𝜑) ≤A 𝑒(𝜓) or 𝑒(𝜓) ≤A 𝑒(𝜑) Thus, either 𝑒(𝜑 ⇒ 𝜓) ∈ 𝐹 or 𝑒(𝜓 ⇒ 𝜑) ∈ 𝐹, i.e. 𝜑 ⇒ 𝜓 ∈ 𝑇 0 or 𝜓 ⇒ 𝜑 ∈ 𝑇 0. 

3.6 Subdirect products and irreducible matrices In this section, we introduce subdirect products, which are a particular kind of submatrices of direct products. We use this construction to define two special classes of matrices, subdirectly irreducible and finitely subdirectly irreducible matrices, and

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show that they can be characterized in terms of the (finite) meet-irreducibility of their filters and, in weakly implicative logics, the countable ones are exactly (up to isomorphism) the Lindenbaum–Tarski matrices of (finitely) meet-irreducible theories (cf. the analogous claim for simple algebras in Proposition 3.5.14). Thus, we obtain two additional classes of matrices (besides Mod(L) and Mod∗ (L)) that provide complete semantics for a very wide class of logics including all finitary ones. Definition 3.6.1 (Subdirect product) We say that a matrix A is a subdirect product of a family X = hA𝑖 i𝑖 ∈𝐼 of L-matrices if A is a submatrix of the direct product of X and, for each 𝑖 ∈ 𝐼, the restriction of the projection 𝜋𝑖 on 𝐴 is surjective, i.e. 𝜋𝑖 [ 𝐴] = 𝐴𝑖 . Given a class K of L-matrices, we denote by PSd (K) the class of all subdirect products of systems of matrices from K. Note that the condition 𝜋𝑖 [ 𝐴] = 𝐴𝑖 implies that all A𝑖 s are homomorphic images of A; thus, in particular, they cannot have a bigger cardinality than A and, if K is closed under homomorphic images and A is in K, then each A𝑖 is also in K. The following proposition summarizes expected properties of subdirect products and its proof is left as an exercise for the reader. Observe that the proposition entails that, as before, IPSd is a closure operator. Proposition 3.6.2 • For each pair of classes of L-matrices K ⊆ K 0, we have PSd (K) ⊆ PSd (K 0)

K ⊆ PSd (K)

PSd PSd (K) ⊆ IPSd (K).

• For each class K of L-matrices, we have P(K) ⊆ PSd (K) ⊆ SP(K)

PSd I(K) ⊆ IPSd (K).

• For any logic L, we have PSd (Mod(L)) = Mod(L). • For any weakly implicative logic L, we have PSd (Mod∗ (L)) = Mod∗ (L). In Example 3.4.4 we have shown that Mod∗ (L) need not be closed under products, so in general it is not closed under subdirect products either. Next, we define the notion of a subdirect representation of a matrix by a family of matrices. Basically, it says that the matrix is isomorphic to a subdirect product of the family. In accordance with the literature, we use a formulation that helps simplifying some of the upcoming results. Recall that, for a matrix A = hA, 𝐹i and an embedding ℎ : A → B, the matrix ℎ(A) = hℎ(A), ℎ[𝐹]i is a submatrix of B. Definition 3.6.3 We say that an embedding 𝛼 of a matrix A into a direct product of Î system X of matrices is a subdirect representation of A, 𝛼 : A ↩→ X in symbols, if 𝛼(A) is a subdirect product of X.

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123

Next, we show that possible subdirect representations of a matrix are intimately related to the possible expressions of its filter as intersections of families of filters. Proposition 3.6.4 Let L be a logic and A = hA, 𝐹i ∈ Mod∗ (L). Ñ 1. For each set of filters G ⊆ FiL (A) such that 𝐹 = G, there is a subdirect representation Ö 𝛼 : A ↩→ hA, 𝐺i ∗ . 𝐺∈G

Moreover, for each 𝐺 ∈ G, we have 𝐹 = 𝐺 iff 𝜋𝐺 ◦ 𝛼 is an isomorphism. 2. For each subdirect representation Ö 𝛼 : A ↩→ hA𝑖 , 𝐹𝑖 i 𝑖 ∈𝐼

by matrices Mod∗ (L), we have 𝐹¯𝑖 = (𝜋𝑖 ◦ 𝛼) −1 [𝐹𝑖 ] ∈ FiL (A) for each 𝑖 ∈ 𝐼 Ñ from ¯ and 𝐹 = 𝑖 ∈𝐼 𝐹𝑖 . Moreover, for each 𝑖 ∈ 𝐼, we have 𝐹 = 𝐹¯𝑖 iff 𝜋𝑖 ◦ 𝛼 is an isomorphism. Proof To prove the first claim, we start by defining 𝛼(𝑎) = h𝑎/ΩA (𝐺)i𝐺 ∈ G and proving that it is a strict homomorphism. It is clearly an algebraic homomorphism and we have: 𝑎 ∈ 𝐹 iff 𝑎 ∈ 𝐺 for each 𝐺 ∈ G iff 𝑎/ΩA (𝐺) ∈ 𝐺/ΩA (𝐺) for each 𝐺 ∈ G iff Î 𝛼(𝑎) ∈ 𝐺 ∈ G 𝐺/ΩA (𝐺). Since A is reduced, we know that 𝛼 is an embedding thanks to Corollary 3.2.6. Finally, observe that, for each 𝐺 ∈ G, 𝜋𝐺 [𝛼[ 𝐴]] = 𝐴/ΩA (𝐺) and so 𝛼 is indeed a subdirect representation of A. If 𝜋𝐺 ◦ 𝛼 is an isomorphism, then we have: 𝑎 ∈ 𝐹 iff 𝜋𝐺 (𝛼(𝑎)) ∈ 𝐺/ΩA (𝐺) iff 𝑎/ΩA (𝐺) ∈ 𝐺/ΩA (𝐺) iff 𝑎 ∈ 𝐺. To show the converse, assume 𝐹 = 𝐺 for some 𝐺 ∈ G. Then, 𝜋𝐺 ◦ 𝛼 is an isomorphism because it is actually the canonical mapping. The second claim is simpler. Indeed, due to Lemma 2.5.7, we know that 𝐹¯𝑖 = (𝜋𝑖 ◦ 𝛼) −1 [𝐹𝑖 ] ∈ FiL (A). For each 𝑎 ∈ 𝐴, we know that Ñ 𝑎 ∈ 𝐹 iff 𝜋𝑖 (𝛼(𝑎)) ∈ 𝐹𝑖 for each 𝑖 ∈ 𝐼 iff 𝑎 ∈ 𝐹¯𝑖 for each 𝑖 ∈ 𝐼; therefore 𝐹 = 𝑖 ∈𝐼 𝐹¯𝑖 . Next, assume that 𝐹¯𝑖 = 𝐹. As before, to show that 𝜋𝑖 ◦ 𝛼 is an isomorphism it suffices to show that it is strict: 𝑎 ∈ 𝐹 iff 𝑎 ∈ 𝐹¯𝑖 iff 𝜋𝑖 (𝛼(𝑎)) ∈ 𝐹𝑖 . Conversely, we just observe that 𝑎 ∈ 𝐹 iff 𝜋𝑖 (𝛼(𝑎)) ∈ 𝐹𝑖 iff 𝑎 ∈ 𝐹¯𝑖 .  Definition 3.6.5 (Subdirect irreducibility) Let K be a class of reduced matrices. A matrix A ∈ K is subdirectly irreducible relative to K, A ∈ KRSI in symbols, if for each subdirect representation 𝛼 of A by a system hA𝑖 i𝑖 ∈𝐼 of matrices from K, there is an 𝑖 ∈ 𝐼 such that 𝜋𝑖 ◦ 𝛼 is an isomorphism. We say that A ∈ K is finitely subdirectly irreducible relative to K, A ∈ KRFSI in symbols, if the above claim holds for all subdirect representations by finite non-empty systems of matrices from K. Obviously, we have KRSI ⊆ KRFSI . Recall that the definition of subdirect product implies that none of the matrices in X can be of higher cardinality than their subdirect products; therefore, if a subdirect product is (isomorphic to) the trivial reduced matrix, then either it is a product of the empty family or each member of that family is trivial.

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3 Completeness properties

Therefore, the trivial reduced matrix is finitely subdirectly irreducible relative to any class and it is never subdirectly irreducible. Next, we prove the promised characterization of Mod∗ (L)RSI and Mod∗ (L)RFSI using the notion of (finitely) meet-irreducibility of their filters. Theorem 3.6.6 (Characterization of RSI and RFSI reduced models) Let L be a logic and A = hA, 𝐹i ∈ Mod∗ (L). 1. A ∈ Mod∗ (L)RSI if and only if 𝐹 is meet-irreducible in FiL (A). 2. A ∈ Mod∗ (L)RFSI if and only if 𝐹 is finitely meet-irreducible in FiL (A). Proof We prove the first claim; the second one is analogous. Let Ñ us assume that A ∈ Mod∗ (L)RSI and there is a family G ⊆ FiL (A) such that 𝐹 = Î 𝐺 ∈ G 𝐺. Then, due to Proposition 3.6.4, there is a subdirect representation 𝛼 : A ↩→ 𝐺 ∈ G hA, 𝐺i ∗ . As we have hA, 𝐺i ∗ ∈ Mod∗ (L) for each 𝐺 ∈ G and A is subdirectly irreducible relative to Mod∗ (L), there has to be a 𝐺 ∈ G such that 𝜋𝐺 ◦ 𝛼 is an isomorphism and so 𝐹 = 𝐺. Conversely, assume that Î 𝐹 is meet-irreducible in FiL (A) and there is a subdirect Ñ representation 𝛼 : A ↩→ 𝑖 ∈𝐼 hA𝑖 , 𝐹𝑖 i. By Proposition 3.6.4, we know that 𝐹 = 𝐹¯𝑖 for 𝐹¯𝑖 = (𝜋𝑖 ◦ 𝛼) −1 [𝐹𝑖 ] ∈ FiL (A). Thus, by the meet-irreducibility of 𝐹, we have 𝐹 = 𝐹¯𝑖 for some 𝑖 ∈ 𝐼 and so 𝜋𝑖 ◦ 𝛼 is an isomorphism.  Remark 3.6.7 Assume that L is an algebraically implicative logic. We leave as an exercise for the reader to show that any subdirect product of a system of L-algebras is itself an L-algebra. Corollary 3.6.8 Let L be an algebraically implicative logic and A ∈ Mod∗ (L). 1. A ∈ Mod∗ (L)RSI if and only if A ∈ Alg∗ (L) RSI . 2. A ∈ Mod∗ (L)RFSI if and only if A ∈ Alg∗ (L) RFSI . 3. If A is simple, then A ∈ Mod∗ (L)RSI . Proof The first claim is proved by this chain of equivalences: A ∈ Mod∗ (L)RSI iff 𝐹A is meet-irreducible in FiL (A) iff IdA is meet-irreducible in ConAlg∗ (L) (A) iff A ∈ Alg∗ (L) RSI (the first equivalence is by the previous theorem, the second is given by Theorem 3.4.16, the third follows from Proposition A.3.15). The second claim is proved analogously and the last one is a consequence of Corollary 3.4.17. 

Example 3.6.9 Let us show that we can describe the RSI and RFSI-algebras of certain prominent algebraically implicative logics as follows (and so we also describe their RSI and RFSI-models) Alg∗ (CL) RSI Alg∗ (CL) RFSI Alg∗ (G) RSI Alg∗ (G) RFSI Alg∗ (IL) RSI Alg∗ (MV) RFSI

= = = = = =

BARSI BARFSI GRSI GRFSI HARSI MVRFSI

= I(2) = I(2, TrLCL ) = {A ∈ G | A has a unique coatom} = {A ∈ G | A is linearly ordered} = {A ∈ HA | A has a unique coatom} = {A ∈ MV | A is linearly ordered}.

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125

From Corollary A.3.16, we know that a Heyting algebra A is subdirectly irreducible iff ConHA (A) \ {ΔA } has a least element. The lattice ConHA (A) is isomorphic to FiIL (A), due to Theorem 3.4.16, and as FiIL (A) is the set of lattice filters on A, we know that FiIL (A) \ {{>}} has a least element iff A has a unique coatom (the least filter is the principal filter given by the unique coatom). Since BA and G are subvarieties of HA, the remaining claims for RSI-algebras follow. The description of RFSI-algebras is left as an exercise for the reader. Note that, thanks to the fact that in linearly ordered algebras any coatom has to be unique, we can equivalently write Alg∗ (G) RSI = {A ∈ G | A is linearly ordered and has a coatom}. Finally, let us note that h[0, 1] G , {1}i ∈ Alg∗ (G) RFSI \ Alg∗ (G) RSI . Next, we prove two consequences of Theorem 3.6.6. The first one characterizes matrices whose reducts are in Mod∗ (L)RSI (or Mod∗ (L)RSI resp.). The second one builds an analogy with a known result from universal algebra by showing that reduced models of finitary weakly implicative logics can be represented as subdirect products of their subdirectly irreducible models. Corollary 3.6.10 Let L be a logic and A = hA, 𝐹i ∈ Mod(L). 1. If 𝐹 is meet-irreducible in FiL (A), then A∗ ∈ Mod∗ (L)RSI . 2. If 𝐹 is finitely meet-irreducible in FiL (A), then A∗ ∈ Mod∗ (L)RFSI . If, furthermore, L is weakly implicative, then the converse implications are true as well. Proof Let ℎ be the canonical mapping ℎ : A → A∗ . Then, thanks to Proposition 3.5.5, we know that, if 𝐹 is (finitely) meet-irreducible in FiL (A), then ℎ[𝐹] is (finitely) meetirreducible in F𝑖 L (A∗ ) and in weakly implicative logics the reverse implication holds as well. To complete the proof, observe that, using the characterization provided by Theorem 3.6.6), we know that the latter claim is equivalent to A∗ ∈ Mod∗ (L)R(F)SI . Theorem 3.6.11 (Subdirect representation) If L is a finitary weakly implicative logic, then Mod∗ (L) = IPSd (Mod∗ (L)RSI ). Proof One inclusion is easy because Mod∗ (L) is closed under PSd (Proposition 3.6.2). To prove the converse inclusion, consider A = hA, 𝐹i ∈ Mod∗ (L). By Corollary 3.5.7, Ñ there exists a family G of meet-irreducible filters such that 𝐹Î= G. Thus, by Proposition 3.6.4, there is a subdirect representation 𝛼 : A ↩→ 𝐺 ∈ G hA, 𝐺i ∗ and, by Corollary 3.6.10, the matrices hA, 𝐺i ∗ are subdirectly irreducible relative to Mod∗ (L).  Recall that in Theorem 3.2.10 we have shown that, for any cardinal 𝜅 greater than or equal to the cardinality of FmL and for any set Var 0 ⊇ Var of cardinality 𝜅, 0

Mod∗ (L) 𝜅 = I({hFmL0 , 𝑇i ∗ | 𝑇 is an LVar -theory}).

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3 Completeness properties

Therefore, we can use Corollary 3.6.10 to show that the RFSI-models of a given logic can be seen as its Lindenbaum–Tarski matrices of intersection-prime theories (and analogously for RSI-models and completely intersection-prime theories). Recall that in the previous section we have argued that intersection-prime theories are more interesting from the logical point of view than the completely intersection-prime ones. Therefore, the same argument can be used for a logical preference for RFSI-models over the RSI ones. Proposition 3.6.12 Let L be a weakly implicative logic over FmL , 𝜅 a cardinal greater than or equal to the cardinality of FmL , and a set Var 0 ⊇ Var of cardinality 𝜅. Then, 0

𝜅 Mod∗ (L)RFSI = I({hFmL0 , 𝑇i ∗ | 𝑇 is an intersection-prime LVar -theory}) 0

𝜅 Mod∗ (L)RSI = I({hFmL0 , 𝑇i ∗ | 𝑇 is a completely intersection-prime LVar -theory}).

Proof Note that, thanks to Theorem 3.2.10, we know that for any hA, 𝐹i ∈ Mod∗ (L) 𝜅 0 there is an LVar -theory 𝑇 such that hA, 𝐹i is isomorphic to hFmL0 , 𝑇i ∗ . Therefore, there is a strict surjective homomorphism ℎ : hFmL0 , 𝑇i → hA, 𝐹i and, as hFmL0 , 𝑇i, hA, 𝐹i ∈ Mod(L), we know by Proposition 3.5.5 that 𝑇 is (finitely) meetirreducible iff 𝐹 is (finitely) meet-irreducible which, by Theorem 3.6.6, is equivalent 𝜅 to hA, 𝐹i ∈ Mod∗ (L)R(F)SI .  Note that a part of the previous corollary holds for all (not necessarily weakly implicative) logics; namely, we can show that Mod∗ (L)RFSI ⊇ I({hFmL0 , 𝑇i ∗ | 𝑇 is an intersection-prime L-theory}) Mod∗ (L)RSI ⊇ I({hFmL0 , 𝑇i ∗ | 𝑇 is a completely intersection-prime L-theory}). The following two general completeness theorems are simple consequences of this fact and Theorem 3.5.2 where we have seen that, given a basis of the system of all theories, the logic is strongly complete w.r.t. the corresponding class of Lindenbaum– Tarski matrices. Note that the assumption of (C)IPEP can be replaced by the simpler (yet stronger) condition of finitarity (Corollary 3.5.7). Also note that, as in the previous two general completeness theorems (Theorems 2.6.3 and 2.7.6), if the cardinality of the set of formulas is countable, we could restrict ourselves to countable matrices. Theorem 3.6.13 (Completeness w.r.t. RSI reduced models) Let L be a logic satisfying the CIPEP. Then, L enjoys the strong completeness with respect to the class Mod∗ (L)RSI , i.e. for any set Γ ∪ {𝜑} of formulas, Γ `L 𝜑

iff

Γ Mod∗ (L)RSI 𝜑.

If L is a logic over a countable set of formulas, then it enjoys strong completeness 𝜔 . with respect to the class Mod∗ (L)RSI

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127

Fig. 3.1 Hierarchy of infinitary logics.

Theorem 3.6.14 (Completeness w.r.t. RFSI reduced models) Let L be a logic satisfying the IPEP. Then, L enjoys the strong completeness with respect to the class Mod∗ (L)RFSI , i.e. for any set Γ ∪ {𝜑} of formulas, Γ `L 𝜑

iff

Γ Mod∗ (L)RFSI 𝜑.

If L is a logic over a countable set of formulas, then it enjoys strong completeness 𝜔 . with respect to the class Mod∗ (L)RFSI Later we will see that strong completeness w.r.t. (finitely) subdirectly irreducible matrices is an interesting property on its own; let us call the logics that enjoy it RSI-complete (resp. RFSI-complete). The previous two theorems can be compactly formulated in the following way: • Each logic with the IPEP is RFSI-complete. • Each logic with the CIPEP is RSI-complete. Therefore, we have obtained a hierarchy (depicted in Figure 3.1) of classes of logics extending that of finitary logics; in Example 3.6.16 we describe a logic with the CIPEP (and, therefore, with the IPEP) which is not finitary; in Example 5.4.10 we will present a logic which is not RFSI-complete; the strictness of the remaining inclusions is shown, by means of two rather involved counterexamples, in the paper [212].14 In order to provide the promised Example 3.6.16, we prepare the next proposition which tells us that, if Mod∗ (L)RSI is closed under submatrices, then RSI-completeness and the CIPEP coincide (and analogous statements for RFSI-completeness and the IPEP). We formulate it in a slightly more general way: Proposition 3.6.15 Let L be a weakly implicative logic with the SKC. • If S(K) ⊆ Mod∗ (L)RFSI , then L has the IPEP. • If S(K) ⊆ Mod∗ (L)RSI , then L has the CIPEP. 14 This is achieved by producing a rather natural example of a non-RSI-complete logic with the IPEP (which, interestingly enough, is equivalential), and an ad hoc example of an RSI-complete logic without the IPEP (which is even Rasiowa-implicative).

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Proof We prove the first claim; the second one is analogous. Assume that 𝜑 ∉ 𝑇. Then, by the SKC, we have a matrix hA, 𝐹i ∈ K and an A-evaluation 𝑒 such that 𝑒[𝑇] ⊆ 𝐹 but 𝑒(𝜑) ∉ 𝐹. Consider the submatrix B = h𝑒[FmL ], 𝑒[FmL ] ∩ 𝐹i of hA, 𝐹i. From the assumption, we know that B ∈ Mod∗ (L)RFSI , i.e. due to Theorem 3.6.6, 𝑒[FmL ] ∩ 𝐹 is finitely meet-irreducible. Consider the theory 𝑇 0 = 𝑒 −1 [𝑒[FmL ] ∩ 𝐹]. Clearly, 𝑇 0 ⊇ 𝑇, 𝜑 ∉ 𝑇 0, and, as L is weakly implicative and 𝑒 can be seen as a strict surjective homomorphism from hFmL , 𝑇 0i onto B, we can use Proposition 3.5.5 to conclude that 𝑇 0 is intersection-prime.  Example 3.6.16 Recall that the logics Ł∞ and  {Ł𝑛 | 𝑛 ≥2} are not finitary (Examples 2.6.6 and 2.6.7). We show that the former one has the CIPEP and leave the analogous proof for the latter one as an exercise for the reader. Since Ł∞ is weakly implicative and Ł∞ ∈ Mod∗ (Ł∞ ), we know that S(Ł∞ ) ⊆ Mod∗ (Ł∞ ). Thanks to Example 3.4.19, we know that every subalgebra A of [0, 1] Ł is simple and so, by Corollary 3.6.8, we know S(Ł∞ ) ⊆ Mod∗ (Ł∞ )RSI and so, by the previous Proposition 3.6.15, the logic Ł∞ has the CIPEP. We conclude this section by showing that, under certain conditions, we can obtain a characterization of the completeness properties using the localization of (finitely) subdirectly irreducible models. Let us first recall that in Theorem 3.5.9 we have characterized the SKC using the localization of Lindenbaum–Tarski matrices of completely intersection-prime theories. Since, due to Proposition 3.6.12, these are exactly the countable RSI reduced models, we easily obtain the following corollary. Corollary 3.6.17 Let L be a weakly implicative logic. If L has the SKC, then 𝜔 ⊆ IS(K). Mod∗ (L)RSI

The converse implication is valid as well whenever L has the CIPEP.

Example 3.6.18 The Gödel–Dummett logic G (Example 2.3.7) is finitary and so, in virtue of Corollary 3.5.7, it has the CIPEP. Therefore, we can use the previous corollary to prove that it has the SKC but not the SK 0C for K = {h[0, 1] G , {1}i}

and

K 0 = {hG𝑛 , {1}i | 𝑛 ≥ 2}

(see Example A.5.7 for the definitions of these special G-algebras). We show that 𝜔 ⊆ IS(K) and Mod∗ (G) 𝜔 * IS(K 0 ). By Example 3.6.9, we also know Mod∗ (G)RSI RSI 𝜔 , then A is a countable linearly ordered G-algebra. that if hA, {>}i ∈ Mod∗ (G)RSI Let us consider any such algebra A. Then, ≤ hA, {>}i is a countable bounded linear order, and so it can be embedded into [0, 1]. To prove that this order-embedding is an embedding of the corresponding G-algebras, it suffices to show that for each 𝑎, 𝑏 ∈ 𝐴, ( > if 𝑎 ≤ hA, {>}i 𝑏 𝑎 →A 𝑏 = 𝑏 otherwise.

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The elaboration of the details is left as an exercise for the reader. Next, to 𝜔 * IS(K 0 ), it clearly suffices to find a countable infinite matrix prove Mod∗ (G)RSI 𝜔 . Again, we leave the elaboration of the details as an exercise hA, {>}i ∈ Mod∗ (G)RSI for the reader. Interestingly enough, we can use Lemma 3.3.6 to prove that Gödel–Dummett logic G (Example 2.3.7) has the FSK 0C. Clearly, in order to show that K is partially embeddable into K 0, it suffices to observe that any finite subset of [0, 1] extended by {0, 1} is the domain of a subalgebra of [0, 1] G which has to be actually isomorphic to G𝑛 for some 𝑛. Note that the converse direction of the previous corollary can also be proved using Lemma 3.3.6. Indeed, from Theorem 3.6.13, we know that if a logic L has the 𝜔 , and the condition CIPEP, then it enjoys the SK 0C for the class K 0 = Mod∗ (L)RSI ∗ 𝜔 Mod (L)RSI ⊆ IS(K) is nothing else than the embeddability of K 0 into K. By the same reasoning, we can prove the SKC in several other classes of logics and, therefore, using Lemma 3.3.6 for partial embeddings, we obtain sufficient conditions also for the FSKC in those classes of logics. In order to be able to claim that these conditions are, in certain situations, also necessary, we formulate them for the class K+ rather than just K which would be clearly sufficient as well (recall that K+ is K expanded with the trivial reduced matrix and note that, obviously, the (F)SK+ C implies the (F)SKC). Proposition 3.6.19 Let L be a logic and K ⊆ Mod∗ (L) satisfying any of the following conditions: 1. L is RFSI-complete and Mod∗ (L)RFSI is (partially) embeddable into K+ . 2. L is RSI-complete and Mod∗ (L)RSI is (partially) embeddable into K. 𝜔 3. L is a logic over a countable set of formulas with the IPEP and Mod∗ (L)RFSI is + (partially) embeddable into K . 𝜔 is 4. L is a logic over a countable set of formulas with the CIPEP and Mod∗ (L)RSI (partially) embeddable into K. Then, L has the (F)SKC. 𝜔 is In Corollary 3.6.17 we have shown that the condition saying that Mod∗ (L)RSI embeddable into K is actually necessary for the SKC. Later, in Theorem 5.6.5, we 𝜔 will show that the analogous condition for Mod∗ (L)RFSI is necessary in logics with well-behaved disjunctions. On the other hand, it is obvious, by cardinality reasons, that the embeddability of Mod∗ (L)R(F)SI into K (+) cannot be a necessary condition for the SKC in general; indeed, we know that any weakly implicative logic is complete w.r.t. its countable models and some of these logics have uncountable R(F)SI-models, e.g. intuitionistic logic. Finally, let us note that in Theorem 3.8.5 we will prove that, in finitary logics with finite language, the FSKC implies that the class Mod∗ (L)RFSI is partially embeddable into K+ , i.e. all four conditions involving partial embeddability are necessary.

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3 Completeness properties

3.7 Filtered products, ultraproducts, and finitarity Let us continue exploiting the fact that L-matrices hA, 𝐹i can be seen as first-order structures to define a notion of filtered product. Given a system Î X = hA𝑖 i𝑖 ∈𝐼 of L-algebras and a filter F on 𝐼, we know that the relation ∼ F on X defined as 𝑎 ∼F 𝑏

iff

{𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎) = 𝜋𝑖 (𝑏)} ∈ F

Î Î is a congruence on X and we denote the corresponding factor algebra as X/F Î instead of X/∼ F and analogously for its elements and their sets. Note that, for a system h𝐹𝑖 i𝑖 ∈𝐼 , where 𝐹𝑖 ⊆ 𝐴𝑖 , we have Ö 𝑎/F ∈ 𝐹𝑖 /F iff {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎) ∈ 𝐹𝑖 } ∈ F . 𝑖 ∈𝐼

 Definition 3.7.1 (Filtered product) Given a family X = hA𝑖 , 𝐹𝑖 i 𝑖 ∈𝐼 of L-matrices and a filter F on subsets of 𝐼, we define the filtered product of X w.r.t. F as Ö  Ö Ö Ö X/F = hA𝑖 , 𝐹𝑖 i/F = A𝑖 /F , 𝐹𝑖 /F . 𝑖 ∈𝐼

𝑖 ∈𝐼

𝑖 ∈𝐼

Î

If F is an ultrafilter, we call X/F an ultraproduct and, if F is closed under countable intersections, we call it a countably-filtered product. Given a class K of L-matrices, we denote by PF (K) the class of all filtered products of systems of matrices from K. Analogously, we use the notation PU (K) and P𝜔 (K) for classes of ultraproducts and countably-filtered products of matrices from K. We start by proving a variant of Proposition 3.4.2 for countably-filtered products, which will entail that Mod(L) is closed under P𝜔 . Interestingly enough, for ultraproducts and general filtered products the situation is more involved: we will actually show that such closure is equivalent with the finitarity of the logic in question (Theorem 3.7.7). Proposition 3.7.2 Let L be a countable language and K a class of L-matrices. Then, K = P𝜔 (K) = HS SP𝜔 (K) . Proof We will treat all three semantically defined logics in the statement of this proposition as logics over a countable set of propositional variables, i.e. logics over a countable set of formulas. Let us start by proving that K ⊆ P𝜔 (K). Indeed, for any matrix A1 ∈ K, we can consider the index set 𝐼 = {1} and take the filter F = {{1}} on 𝐼 which is trivially closed under countable intersections. With a slight abuse of language, we can identify an element 𝑎 ∈ 𝐴1 with {h𝑎i} = h𝑎i/F and see A1 as a filtered product of hA𝑖 i𝑖 ∈𝐼 w.r.t. F . Therefore, we obtain K ⊇ P𝜔 (K) .

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131

To prove the converse, we assume Γ KÎ𝜑 and show that, for each A ∈ P𝜔 (K), we have Γ A 𝜑. We know that A = hA, 𝐹i = 𝑖 ∈𝐼 hAi , 𝐹𝑖 i/F for some non-empty index set 𝐼, matrices hAi , 𝐹𝑖 i ∈ K, and a filter F on 𝐼 closed under countable intersections. Let 𝑒 be an A-evaluation such that 𝑒[Γ] ⊆ 𝐹. Thus, for each 𝜓 ∈ Γ, 𝑆 𝜓 = {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑒(𝜓)) ∈ 𝐹𝑖 } ∈ F . As F is a filter closed under countable intersections and we have the cardinality restriction, we know that Ù 𝑆 𝜓 = {𝑖 ∈ 𝐼 | for each 𝜓 ∈ Γ we have 𝜋𝑖 (𝑒(𝜓)) ∈ 𝐹𝑖 } ∈ F . 𝜓 ∈Γ

Since 𝜋𝑖 (𝑒(𝜓)) is an A𝑖 -evaluation and Γ K 𝜑, we have 𝜋𝑖 (𝑒(𝜑)) ∈ 𝐹𝑖 for each Ñ 𝑖 ∈ 𝜓 ∈Γ 𝑆 𝜓 and so 𝑒(𝜑) ∈ 𝐹; indeed, we also know that Ù 𝑆 𝜓 ⊆ {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑒(𝜑)) ∈ 𝐹𝑖 } ∈ F .  𝜓 ∈Γ

Let us summarize some basic properties of P𝜔 , PF , or PU ; note that, as in the case of (subdirect) products, we obtain that IX is a closure operator for each X ∈ {P𝜔 , PF , PU }. Proposition 3.7.3 • Let X be one of the operators P𝜔 , PF , or PU . Then, for each pair of classes of L-matrices K ⊆ K 0, we have15 XI(K) ⊆ IX(K)

X(K) ⊆ X(K 0)

K ⊆ X(K)

XX(K) ⊆ IX(K).

• For each class K of L-matrices K, we have P(K) ⊆ P𝜔 (K) ⊆ PF (K) ⊆ HP(K) PU (K) ⊆ PF (K) ISPPU (K) = ISPF (K). • For any weakly implicative logic L, all matrices in PF (Mod∗ (L)) are reduced. • For any logic L over a countable set of formulas, we have P𝜔 (Mod(L)) = Mod(L). • For any weakly implicative logic L, we have P𝜔 (Mod∗ (L)) = Mod∗ (L). 15 As in the case of products we have conflate certain objects in order to be able to see a matrix as its own (countably-)filtered product (or ultraproduct). Consider the sets 𝐼 = {𝑖 } and F = { {𝑖 } } and a matrix A𝑖 . Clearly, F is an ultrafilter on 𝐼 and it is closed under countable intersections; thus, to obtain the claim, it suffices to identify each element 𝑎 ∈ 𝐴𝑖 with the singleton { h𝑎i }.

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Proof We leave the proof of the first inequality of the first claim as an exercise for the reader; the second one is trivial; the third one was proved in the proof of the previous proposition (we only have to observe that the filter F defined there is clearly an ultrafilter); the proof of the last one is not complicated per se but it is very technical. Consider a set of sets ℑ, filter F on ℑ, filters F𝐼 on 𝐼 for each 𝐼 ∈ ℑ and matrices hAIi , 𝐹𝑖𝐼 i ∈ K for each 𝐼 ∈ ℑ and 𝑖 ∈ 𝐼, we define Ö hA𝐼 , 𝐹 𝐼 i = hA𝑖𝐼 , 𝐹𝑖𝐼 i 𝑖 ∈𝐼

hA /F𝐼 , 𝐹 /F𝐼 i =

Ö

hA, 𝐹i =

Ö

𝐼

𝐼

hAIi , 𝐹𝑖𝐼 i/F𝐼

𝑖 ∈𝐼

hA𝐼 /F𝐼 , 𝐹 𝐼 /F𝐼 i/F

𝐼 ∈ℑ

𝐽 = {h𝐼, 𝑖i | 𝐼 ∈ ℑ and 𝑖 ∈ 𝐼} G = {𝑋 ⊆ 𝐽 | {𝐼 ∈ ℑ | {𝑖 | h𝐼, 𝑖i ∈ 𝑋 } ∈ F𝐼 } ∈ F } Ö hB, 𝐺i = hA𝑖𝐼 , 𝐹𝑖𝐼 i/G. h𝐼 ,𝑖 i ∈𝐽

We start by proving that G is a filter on 𝐽 and that it is closed under countable intersections whenever F and all F𝐼 s are. Clearly, it is an upper set; we prove that it is closed under (countable) intersections. Let us first simplify the notation a bit by writing 𝑋 𝐼 for {𝑖 | h𝐼, 𝑖i ∈ 𝑋 } which allows us to formulate 𝑋 ∈ G equivalently as {𝐼 ∈ ℑ | 𝑋 𝐼 ∈ F𝐼 } ∈ F . Consider a finite (or countable) set X ⊆ G. As F is a filter (closed under countable intersections), we can show that Ù {𝐼 ∈ ℑ | 𝑋 𝐼 ∈ F𝐼 for each 𝑋 ∈ X} = {𝐼 ∈ ℑ | 𝑋 𝐼 ∈ F𝐼 } ∈ F . 𝑋 ∈X

Since each F𝐼 is a filter (closed under countable intersections), we obtain Ù {𝐼 ∈ ℑ | ( X) 𝐼 ∈ F𝐼 } = {𝐼 ∈ ℑ | 𝑋 𝐼 ∈ F𝐼 for each 𝑋 ∈ X}. Ñ Therefore, we obtain X ∈ G as required. Next, we show that, whenever all involved filters are ultrafilters, then G is an ultrafilter as well by writing the following simple chain of equivalent statements: 𝑋 ∉ G iff {𝐼 ∈ ℑ | {𝑖 | h𝐼, 𝑖i ∈ 𝑋 } ∈ F𝐼 } ∉ F iff {𝐼 ∈ ℑ | {𝑖 | h𝐼, 𝑖i ∈ 𝑋 } ∉ F𝐼 } ∈ F iff {𝐼 ∈ ℑ | {𝑖 | h𝐼, 𝑖i ∉ 𝑋 } ∈ F𝐼 } ∈ F iff {𝐼 ∈ ℑ | {𝑖 | h𝐼, 𝑖i ∈ 𝐽 \ 𝑋 } ∈ F𝐼 } ∈ F iff 𝐽 \ 𝑋 ∈ G.

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133

To complete the proof, we show that the mapping 𝑓 : 𝐴 → 𝐵 defined as 𝑓 (h𝑎 𝐼 /F𝐼 i𝐼 ∈ℑ /F ) = h𝜋𝑖 (𝑎 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G is an isomorphism of hA, 𝐹i and hB, 𝐺i. • We start by showing that 𝑓 is a soundly defined one-one mapping: h𝑎 𝐼 /F𝐼 i𝐼 ∈ℑ /F = h𝑏 𝐼 /F𝐼 i𝐼 ∈ℑ /F iff {𝐼 ∈ ℑ | 𝑎 𝐼 /F𝐼 = 𝑏 𝐼 /F𝐼 } ∈ F iff {𝐼 ∈ ℑ | {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎 𝐼 ) = 𝜋𝑖 (𝑏 𝐼 )} ∈ FI } ∈ F iff {h𝐼, 𝑖i ∈ 𝐽 | 𝜋𝑖 (𝑎 𝐼 ) = 𝜋𝑖 (𝑏 𝐼 )} ∈ G iff h𝜋𝑖 (𝑎 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G = h𝜋𝑖 (𝑏 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G iff 𝑓 (h𝑎 𝐼 /F𝐼 i𝐼 ∈ℑ /F ) = 𝑓 (h𝑏 𝐼 /F𝐼 i𝐼 ∈ℑ /F ). • To show that 𝑓 is onto, consider an element h𝑎 h𝐼 ,𝑖 i i h𝐼 ,𝑖 i ∈𝐽 /G ∈ 𝐵 and note that it is equal to 𝑓 (hh𝑎 h𝐼 ,𝑖 i i𝑖 ∈𝐼 /F𝐼 i𝐼 ∈ℑ /F ). • Next, we show that 𝑓 is an algebraic homomorphism. Without loss of generality, we show it for a binary connective ◦ only: 𝜑(h𝑎 𝐼 /F𝐼 i𝐼 ∈ℑ /F ◦A h𝑏 𝐼 /F𝐼 i𝐼 ∈ℑ /F ) = 𝑓 (h𝑎 𝐼 /F𝐼 ◦A𝐼 /F𝐼 𝑏 𝐼 /F𝐼 i𝐼 ∈ℑ /F ) = 𝑓 (h(𝑎 𝐼 ◦A𝐼 𝑏 𝐼 )/F𝐼 i𝐼 ∈ℑ /F ) = h𝜋𝑖 (𝑎 𝐼 ◦A𝐼 𝑏 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G 𝐼

= h𝜋𝑖 (𝑎 𝐼 ) ◦A𝑖 𝜋𝑖 (𝑏 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G = h𝜋𝑖 (𝑎 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G ◦B h𝜋𝑖 (𝑏 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G = 𝑓 (h𝑎 𝐼 /F𝐼 i𝐼 ∈ℑ /F ) ◦B 𝑓 (h𝑏 𝐼 /F𝐼 i𝐼 ∈ℑ /F ). • It remains to show that 𝑓 is a strict matrix homomorphism: h𝑎 𝐼 /F𝐼 i𝐼 ∈ℑ /F ∈ 𝐹 iff {𝐼 ∈ ℑ | 𝑎 𝐼 /F𝐼 ∈ 𝐹 𝐼 /F𝐼 } ∈ F iff {𝐼 ∈ ℑ | {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎 𝐼 ) ∈ 𝐹𝑖𝐼 } ∈ FI } ∈ F iff {h𝐼, 𝑖i ∈ 𝐽 | 𝜋𝑖 (𝑎 𝐼 ) ∈ 𝐹𝑖𝐼 } ∈ G iff h𝜋𝑖 (𝑎 𝐼 )i h𝐼 ,𝑖 i ∈𝐽 /G ∈ 𝐺 iff 𝑓 (h𝑎 𝐼 /F𝐼 i𝐼 ∈ℑ /F ) ∈ 𝐺. Next, we prove the second claim; clearly the only non-straightforward inequality to Î prove is PF (K) ⊆ ISPPU (K). Take a matrix hA, 𝐹i ∈ PF (K), i.e. hA, 𝐹i = F on 𝐼. 𝑖 ∈𝐼 hAi , 𝐹𝑖 i/F for some index set 𝐼, matrices hAi , 𝐹𝑖 i from K, and a filterÑ We know that, for each filter 𝐹, there is a set 𝔘 of ultrafilters such that 𝐹 = 𝔘 (cf. Lemma A.2.20). Let us define the matrices Ö Ö hA U , 𝐹U i = hA𝑖 , 𝐹𝑖 i/U hB, 𝐺i = hA U , 𝐹U i 𝑖 ∈U

U ∈𝔘

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3 Completeness properties

and a mapping 𝑓 : 𝐴 → 𝐵 as 𝑓 (𝑎/F ) = h𝑎/Ui U ∈𝔘 . If we show that 𝑓 is an embedding of hA, 𝐹i into hB, 𝐺i, the proof is done. • To prove the soundness of the definition of 𝑓 together with the fact that it is one-one, we write the following chain of equivalences: 𝑎/F = 𝑏/F iff {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎) = 𝜋𝑖 (𝑏)} ∈ F iff {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎) = 𝜋𝑖 (𝑏)} ∈ U for each U ∈ 𝔘 iff 𝑎/U = 𝑏/U for each U ∈ 𝔘 iff 𝑓 (𝑎/F ) = 𝑓 (𝑏/F ). • Next, we show that 𝑓 is an algebraic homomorphism. Without loss of generality, we show it for a binary connective ◦: 𝑓 (𝑎/F ◦A 𝑏/F ) = 𝑓 (h𝜋𝑖 (𝑎) ◦A𝑖 𝜋𝑖 (𝑏)i𝑖 ∈𝐼 /F ) = 𝑓 (h𝜋𝑖 (𝑎 ◦A𝑖 𝑏)i𝑖 ∈𝐼 /F )

 = h𝜋𝑖 (𝑎 ◦A𝑖 𝑏)i𝑖 ∈𝐼 /U) U ∈𝔘

 = h𝜋𝑖 (𝑎) ◦A𝑖 𝜋𝑖 (𝑏)i𝑖 ∈𝐼 /U) U ∈𝔘

 = 𝑎/U ◦AU 𝑏/U U ∈𝔘 = 𝑓 (𝑎/F ) ◦B 𝑓 (𝑏/F ). • It remains to show that 𝑓 is a strict matrix homomorphism: 𝑎/F ∈ 𝐹 iff {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎) ∈ 𝐹𝑖 } ∈ F iff {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑎) ∈ 𝐹𝑖 } ∈ U for each U ∈ 𝔘 iff 𝑎/U ∈ 𝐹U for each U ∈ 𝔘 iff 𝑓 (𝑎/F ) ∈ 𝐺. Î To prove the third claim, assume that A = hA, 𝐹i = 𝑖 ∈𝐼 hAi , 𝐹𝑖 i/F for some hA𝑖 , 𝐹𝑖 i ∈ Mod∗ (L) and 𝑥 ≠ 𝑦 are elements of 𝐴; then {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑥) = 𝜋𝑖 (𝑦)} ∉ F . Since we clearly have {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑥) = 𝜋𝑖 (𝑦)} = {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑥) ⇒A𝑖 𝜋𝑖 (𝑦) ∈ 𝐹𝑖 and 𝜋𝑖 (𝑥) ⇒A𝑖 𝜋𝑖 (𝑦) ∈ 𝐹𝑖 } = {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑥 ⇒A 𝑦) ∈ 𝐹𝑖 } ∩ {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑦 ⇒A 𝑥) ∈ 𝐹𝑖 } and F is a lattice filter, we obtain {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑥 ⇒A 𝑦) ∈ 𝐹𝑖 } ∉ F or {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑦 ⇒A 𝑥) ∈ 𝐹𝑖 } ∉ F , i.e. 𝑥 A 𝑦 or 𝑦 A 𝑥 and so h𝑥, 𝑦i ∉ ΩA (𝐹). The last two claims then follow from Proposition 3.7.2.



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135

Recall that, if L is a weakly implicative logic, then Mod∗ (L) is closed under products. In general this is not true because, while any product of models of L is in Mod(L), it need not be reduced (see Example 3.4.4). Therefore, in general, Mod∗ (L) is not closed under PF either. Unlike the case of products, restricting to weakly implicative logics is not sufficient to obtain this closure property. Thanks to the third claim of the previous proposition, we know that each matrix in PF (Mod∗ (L)) is reduced but the problem is that it need not be a model of L in the first place (unless it is a countably filtered product; cf. the fourth claim). As we have already hinted, the missing condition to prove this claim is the finitarity of L. In order to provide the promised characterization of finitarity in terms of closure under filtered products or ultraproducts, we first need a lemma and a proposition. The lemma shows a sufficient condition for finitarity of a semantically-given logic, which generalizes Proposition 2.5.4 (because any finite class of finite matrices is closed under ultraproducts). The proposition can be seen as a variant of Proposition 3.7.2 and it tells us that the semantical consequence given by general filtered products (or ultraproducts) of a certain class K coincides with the finitary companion of the consequence given by K itself. Lemma 3.7.4 Let K be a class of L-matrices such that PU (K) ⊆ I(K). Then, the logic K is finitary. Proof Take a set Γ ∪ {𝜑} such that Δ 2K 𝜑 for each finite Δ ⊆ Γ, i.e. there is a matrix AΔ = hAΔ , 𝐹Δ i ∈ K and an AΔ -evaluation 𝑒 Δ witnessing this. Let 𝐼 be the set of all finite subsets of Γ and, for each 𝜓 ∈ Γ, take the set 𝜓 ∗ = {Δ ∈ 𝐼 | 𝜓 ∈ Δ}. The system {𝜓 ∗ | 𝜓 ∈ Γ} clearly satisfies the finite intersection property, hence Î there is an ultrafilter U on 𝐼 containing it (see Lemma A.2.20). Let A = hA, 𝐹i = Δ∈𝐼 AΔ /U and consider an A-evaluation 𝑒( 𝜒) = h𝑒 Δ ( 𝜒)iΔ∈𝐼 /U. Then, for each 𝜓 ∈ Γ, 𝜓 ∗ ⊆ {Δ ∈ 𝐼 | 𝑒 Δ (𝜓) ∈ 𝐹Δ } ∈ U

and

{Δ ∈ 𝐼 | 𝑒 Δ (𝜑) ∈ 𝐹Δ } = ∅ ∉ U.

Thus, we have 𝑒[Γ] ⊆ 𝐹 and 𝑒(𝜑) ∉ 𝐹 and, as A ∈ I(K), we obtain Γ 2K 𝜑.



Proposition 3.7.5 Let K be a class of L-matrices. Then, F C(K ) = PF (K) = PU (K) = HS SPPU (K) = HS SPF (K) . Proof We prove the first equality; the proof that F C(K ) = PU (K) is completely analogous and the remaining equalities/inclusions follow from Propositions 3.4.2 and 3.7.3. Clearly, K ⊇ PF (K) . Next, observe that PU PF (K) ⊆ PF PF (K) ⊆ IPF (K), and so by the previous lemma PF (K) is finitary. Therefore, F C(K ) ⊇ PF (K) . To prove the converse direction, assume that Γ I 𝜑 ∈ F C(K ); this implies the existence of a finite set Γ0 such that Γ0 K 𝜑. Clearly, the proof is completed if we show that for any A ∈ PF (K) we have Γ0 A 𝜑.

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This can be done exactly as in the proof of Proposition 3.7.2; the crucial point is that, thanks to the finiteness of Γ0 , it suffices to know that F is closed under finite intersections, which any filter is.  This proposition allows us to obtain seemingly weak embeddability conditions which (in certain contexts) imply the FSKC (it can be seen as a variant of Proposition 3.6.19). Later, in Theorem 3.8.5 and Proposition 5.6.1, we will see that in certain classes of weakly implicative logics the converse implication holds as well. Proposition 3.7.6 Let L be a logic and K ⊆ Mod∗ (L) satisfying any of the following conditions: 1. L is RFSI-complete and Mod∗ (L)RFSI is embeddable into PU (K+ ). 2. L is RSI-complete and Mod∗ (L)RSI is embeddable into PU (K). 𝜔 3. L is a logic over a countable set of formulas with the IPEP and Mod∗ (L)RFSI is + embeddable into PU (K ). 𝜔 is 4. L is a logic over a countable set of formulas with the CIPEP and Mod∗ (L)RSI embeddable into PU (K). Then, L has the FSKC. Proof Denote by K 0 the class of matrices mentioned on any chosen row and note that, under the conditions of that row, we know that L enjoys the SK 0C. Thus, we obtain the first equality in the following chain (the second one is the assumption, the fourth one is due to Proposition 3.7.5 and the remaining ones are trivial): L = K0 ⊇ ISPU (K+ ) ⊇ HS SPF (K+ ) = F C(K+ ) = F C(K ). Therefore, F C(K ) ⊆ F C(L) and the converse direction is a straightforward consequence of L ⊆ K .  Theorem 3.7.7 Let L be a logic. Then, the following are equivalent: 1. L is finitary. 2. Mod(L) is closed under filtered products, i.e. PF (Mod(L)) = Mod(L). 3. Mod(L) is closed under ultraproducts, i.e. PU (Mod(L)) = Mod(L). Furthermore, if L is a weakly implicative logic, we can add other equivalent conditions: 4. 5. 6. 7.

Mod∗ (L) is closed under filtered products, i.e. PF (Mod∗ (L)) = Mod∗ (L). Mod∗ (L) is closed under ultraproducts, i.e. PU (Mod∗ (L)) = Mod∗ (L). There is a class K of reduced matrices s.t. L has the SKC and PF (K) ⊆ Mod∗ (L). There is a class K of reduced matrices s.t. L has the SKC and PU (K) ⊆ Mod∗ (L).

Proof To prove the equivalence of the first five claims, observe that the implications 2 → 3 and 4 → 5 are trivial. Also, using Proposition 3.7.5, we can prove that 1 implies 2 and also 4 (in virtue of the third claim of Proposition 3.7.3). Finally, using Proposition 3.7.5, we can prove that 1 is implied by both 3 and 5 (by setting K = Mod(L), resp. K = Mod∗ (L), we clearly have PU (K) ⊆ I(K)).

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Adding the final two claims is also easy: the implications 4 → 6 and 6 → 7 are trivial and, to prove the final one, 7 → 1, we notice that we obviously have L = K ⊇ PU (K) ⊇ Mod∗ (L) = L and so L = PU (K) . Since PU PU (K) ⊆ IPU (K), Lemma 3.7.4 completes the proof. Remark 3.7.8 Assume that L is an algebraically implicative logic, hA𝑖 i𝑖 ∈𝐼 a system of L-algebras, and F a filter over 𝐼. Furthermore, assume that L is finitary or F is closed Î under countable intersections. As an exercise, the reader can prove that A = 𝑖 ∈𝐼 A𝑖 /F ∈ Alg∗ (L) and Ö 𝐹A = 𝐹A𝑖 /F . 𝑖 ∈𝐼

We conclude this section with an auxiliary lemma that will play an important role in the next section, where we use it to prove the converse of Proposition 3.6.19. It is a folklore result in model theory (an easy consequence of Łoś’s theorem) which we prefer to prove for the sake of self-containment. Lemma 3.7.9 Let K be a class of matrices in a finite language. Then, PU (K) is partially embeddable into K. Î Proof Consider a matrix hB, 𝐺i ∈ PU (K), i.e. hB, 𝐺i = 𝑖 ∈𝐼 hA𝑖 , 𝐺 𝑖 i/U for some index set 𝐼, matrices hA𝑖 , 𝐺 𝑖 i ∈ K, and an ultrafilter U on 𝐼. Then, take a finite set 𝑋 ⊆ 𝐵 and for each 𝑖 ∈ 𝐼, consider mappings ℎ𝑖 : 𝑋 → 𝐴𝑖 such that for each 𝑥 ∈ 𝑋 we have 𝑥 = hℎ𝑖 (𝑥)i𝑖 ∈𝐼 /U. We show that there is a 𝑗 ∈ 𝐼 such that ℎ 𝑗 is a partial embedding of 𝑋 into A 𝑗 . First note that for each pair of elements of 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦 we know that {𝑖 ∈ 𝐼 | ℎ𝑖 (𝑥) = ℎ𝑖 (𝑦)} ∉ U and, since U is an ultrafilter, we also have 𝑆 𝑥,𝑦 = {𝑖 ∈ 𝐼 | ℎ𝑖 (𝑥) ≠ ℎ𝑖 (𝑦)} ∈ U. Next, consider an 𝑛-ary connective 𝜆 and elements 𝑥1 , . . . , 𝑥 𝑛 , 𝜆B (𝑥1 , . . . , 𝑥 𝑛 ) ∈ 𝑋. From the fact that h𝜆Ai (ℎ𝑖 (𝑥1 ), . . . , ℎ𝑖 (𝑥 𝑛 ))i𝑖 ∈𝐼 /U = 𝜆B (hℎ𝑖 (𝑥1 )i𝑖 ∈𝐼 /U, . . . , hℎ𝑖 (𝑥 𝑛 )i𝑖 ∈𝐼 /U) = 𝜆B (𝑥1 , . . . , 𝑥 𝑛 ) = hℎ𝑖 (𝜆B (𝑥1 , . . . , 𝑥 𝑛 ))i𝑖 ∈𝐼 /U (the first equality is the definition of 𝜆B , the latter two are due to the definition of mappings ℎ𝑖 ) we know that 𝑆𝜆,𝑥1 ,...,𝑥𝑛 = {𝑖 ∈ 𝐼 | 𝜆Ai (ℎ𝑖 (𝑥1 ), . . . , ℎ𝑖 (𝑥 𝑛 )) = ℎ𝑖 (𝜆B (𝑥1 , . . . , 𝑥 𝑛 ))} ∈ U.

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Next, consider any 𝑥 ∈ 𝑋. If 𝑥 ∈ 𝐺, then we know that 𝑆 +𝑥 = {𝑖 ∈ 𝐼 | ℎ𝑖 (𝑥) ∈ 𝐺 𝑖 } ∈ U. If 𝑥 ∉ 𝐺, then (because U is an ultrafilter) we know that 𝑆 −𝑥 = {𝑖 ∈ 𝐼 | ℎ𝑖 (𝑥) ∉ 𝐺 𝑖 } ∈ U. Since the set 𝑋 and the language are finite, the intersection of all the sets defined above is non-empty and, clearly, for any 𝑗 in this intersection ℎ 𝑗 is a partial embedding. 

3.8 Completeness and description of classes of reduced matrices We start this section by using (countably) filtered products to characterize the SKC and the FSKC of a given (almost) arbitrary logic L using localizations of the class Mod∗ (L). Recall than in Theorem 3.4.6 we have characterized the KC through a localization of Mod∗ (L) using the matrices in K (namely, L has KC iff Mod∗ (L) ⊆ HSP(K)). Now we prove a characterization of the FSKC using a tighter localization: Mod∗ (L) ⊆ HS SPF (K) (clearly, we have HS SPF (K) ⊆ HSHP(K) ⊆ HHSP(K) ⊆ HSP(K)). We also prove a characterization of the SKC, for logics over a countable set of formulas, by an even tighter localization: Mod∗ (L) ⊆ HS SP𝜔 (K). The cardinality restriction can be avoided if we consider filtered products w.r.t. filters closed under intersections of the cardinality of the set of formulas. As we are ultimately interested in weakly implicative logics, which by definition we have restricted to countable sets of formulas, we are not going to elaborate this any further. Proposition 3.8.1 Let L be a logic and K ⊆ Mod∗ (L). Then, L has the FSKC if and only if Mod∗ (L) ⊆ HS SPF (K). If, furthermore, L is a logic over a countable set of formulas, then L has the SKC if and only if Mod∗ (L) ⊆ HS SP𝜔 (K). Proof Let us first prove the right-to-left directions of both claims. Although in both cases we could give a direct proof analogous to the corresponding direction in Theorem 3.4.6, we can simply use already established facts about matrix operators. The second claim is straightforward: from Proposition 3.7.2 we know that K = HS SP𝜔 (K) ⊆ Mod∗ (L) = L. The proof of the right-to-left direction of the first claim is analogous; we only use Proposition 3.7.5 to obtain F C(K ) = HS SPF (K) ⊆ Mod∗ (L) = L. Therefore, F C(K ) ⊆ F C(L) and the converse direction is a straightforward consequence of L ⊆ K .

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139

Now we prove the left-to-right direction of the second claim. Consider A = hA, 𝐹i ∈ Mod∗ (L) and observe the claim is obviously true for the trivial reduced matrix. Thus, we can assume that A is non-trivial and proceed analogously to the proof of the corresponding direction in Theorem 3.4.6. Again, we set Var 0 = Var ∪ {𝑣 𝑎 | 𝑎 ∈ 𝐴}, denote by FmA the set of formulas using only the new variables, for any 𝜑 ∈ FmA , denote by 𝜑A the value of 𝜑 under the A-evaluation that maps 𝑣 𝑎 to 𝑎 for each 𝑎 ∈ 𝐴, and analogously we define ΓA = {𝜑A | 𝜑 ∈ Γ}). To prove our claim, we need to construct a family of matrices hA𝑖 i𝑖 ∈𝐼 from K, a filter F on 𝐼 closed under countable intersections, a submatrix of the filtered product of that family and a strict surjective homomorphism of A to this submatrix. Let us start by defining the following sets: CS = {Γ ⊆ FmA | Γ is countable and ΓA ⊆ 𝐹} 𝐼 = CS × { 𝜒 ∈ FmA | 𝜒A ∉ 𝐹} Γ∗ = {hΓ0, 𝜒i ∈ 𝐼 | Γ ⊆ Γ0 }

for each Γ ∈ CS.

It is easy to observe that the set F = {𝐻 ⊆ 𝐼 | 𝐻 ⊇ Γ∗ for some Γ ∈ CS} is a proper filter on 𝐼. Clearly, it is closed under supersets, ∅ ∉ F (because from the non-triviality of A we know that each Γ∗ ≠ ∅), and, for any countable H ⊆F ∗ and so ( Ð ∗ and each 𝐻 ∈ H , there is a Γ ∈ CS such that 𝐻 ⊇ Γ 𝐻 𝐻 ∈H Γ 𝐻 ) = 𝐻 Ñ ∗ ⊆ Ñ Γ 𝐻 ∈ F (since CS is clearly closed under countable unions). 𝐻 ∈H 𝐻 𝐻 ∈H Note that, for each 𝑖 = hΓ, 𝜒i ∈ 𝐼, we have Γ 0LVar0 𝜒 (due to the second claim of 0 Proposition 3.1.7, we know that A ∈ Mod(LVar )). Thus, by the SKC and the seventh claim of Proposition 3.1.7, we have a matrix A𝑖 = hA𝑖 , 𝐹𝑖 i ∈ K such that Γ 2A𝑖 𝜒; let 𝑒 𝑖 be an A𝑖 -evaluation witnessing it. Let B = hB, 𝐺i be the filtered product of {A𝑖 | 𝑖 ∈ 𝐼} w.r.t. F . Recall that we know that: 𝑔/F ∈ 𝐺 iff {𝑖 ∈ 𝐼 | 𝜋𝑖 (𝑔) ∈ 𝐹𝑖 } ∈ F . Let us denote the element h𝑒 𝑖 (𝜑)i𝑖 ∈𝐼 /F as 𝜑, define a set 𝐵 0 ⊆ 𝐵 as 𝐵 0 = {𝜑 | 𝜑 ∈ FmA }, and prove that 𝐵 0 is a subuniverse of B. Indeed, consider for example a binary connective ◦ and formulas 𝜑, 𝜓; then we can easily show that 𝜑 ◦B 𝜓 = h𝑒 𝑖 (𝜑)i𝑖 ∈𝐼 /F ◦B h𝑒 𝑖 (𝜓)i𝑖 ∈𝐼 /F = h𝑒 𝑖 (𝜑) ◦A𝑖 𝑒 𝑖 (𝜓)i𝑖 ∈𝐼 /F = h𝑒 𝑖 (𝜑 ◦ 𝜓)i𝑖 ∈𝐼 /F = 𝜑 ◦ 𝜓.

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Therefore, we can consider the submatrix B0 = hB0, 𝐺 0i of B with domain 𝐵 0 and the proof will be done if we prove that the mapping 𝑓 : 𝐵 0 → 𝐴 defined as 𝑓 (𝜑) = 𝜑A is a strict homomorphism from B0 onto A. • First we show that 𝑓 is well defined. Assume 𝜑 = 𝜓 and 𝜑A ≠ 𝜓 A . Since hA, 𝐹i is reduced, then, due to Theorem 2.7.1, there is a 𝛿( 𝑝, 𝑞 1 , . . . , 𝑞 𝑛 ) ∈ FmL and elements 𝑎 1 , . . . , 𝑎 𝑛 ∈ 𝐴 such that 𝛿A (𝜑A , 𝑎 1 , . . . , 𝑎 𝑛 ) ∈ 𝐹 and 𝛿A (𝜓 A , 𝑎 1 , . . . , 𝑎 𝑛 ) ∉ 𝐹. Define the following two formulas from FmA : 𝜒 𝜑 = 𝛿(𝜑, 𝑣 𝑎1 , . . . , 𝑣 𝑎𝑛 ) and 𝜒 𝜓 = 𝛿(𝜓, 𝑣 𝑎1 , . . . , 𝑣 𝑎𝑛 ) and observe that we have removed 𝜒A𝜑 = 𝛿A (𝜑A , 𝑎 1 , . . . , 𝑎 𝑛 ) ∈ 𝐹

and

𝜒A𝜓 = 𝛿A (𝜓 A , 𝑎 1 , . . . , 𝑎 𝑛 ) ∉ 𝐹.

Because 𝜑 = 𝜓, there is a Γ ∈ CS such that Γ∗ ⊆ {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) = 𝑒 𝑖 (𝜓)} ∈ F . Thus, for the index 𝑗 = hΓ ∪ { 𝜒 𝜑 }, 𝜒 𝜓 i ∈ Γ∗ , we know not only that 𝑒 𝑗 ( 𝜒 𝜑 ) ∈ 𝐹 𝑗 and 𝑒 𝑗 ( 𝜒 𝜓 ) ∉ 𝐹 𝑗 but also that 𝑒 𝑗 ( 𝜒 𝜑 ) = 𝛿A 𝑗 (𝑒 𝑗 (𝜑), 𝑒 𝑗 (𝑣 𝑎1 ), . . . , 𝑒 𝑗 (𝑣 𝑎𝑛 )) = 𝛿A 𝑗 (𝑒 𝑗 (𝜓), 𝑒 𝑗 (𝑣 𝑎1 ), . . . , 𝑒 𝑗 (𝑣 𝑎𝑛 )) = 𝑒 𝑗 ( 𝜒 𝜓 ), a contradiction. • Next, we show that 𝑓 is a surjective algebraic homomorphism; let us show it, without loss of generality, for a binary connective ◦: 0

𝑓 (𝜑 ◦B 𝜓) = 𝑓 (𝜑 ◦ 𝜓) = (𝜑 ◦ 𝜓) A = 𝜑A ◦A 𝜓 A = 𝑓 (𝜑) ◦A 𝑓 (𝜓). The surjectivity is straightforward as, for each 𝑎 ∈ 𝐴, we have: 𝑓 (𝑣 𝑎 ) = 𝑣 A𝑎 = 𝑎. • Finally, we have to show that 𝑓 is a strict matrix homomorphism. First assume that 𝑓 (𝜑) ∈ 𝐹. Then, clearly, {𝜑}∗ ⊆ {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) ∈ 𝐹𝑖 } which implies that {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) ∈ 𝐹𝑖 } ∈ F and so 𝜑 ∈ 𝐺 0. Second, assume that 𝑓 (𝜑) ∉ 𝐹. Then, for each Γ ∈ CS, we have: 𝑒 hΓ, 𝜑 i (𝜑) ∉ 𝐹hΓ, 𝜑 i . Hence, for no Γ can we have Γ∗ ⊆ {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) ∈ 𝐹𝑖 } and, therefore, we obtain that {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) ∈ 𝐹𝑖 } ∉ F which implies 𝜑 ∉ 𝐺 0. The proof of the left-to-right direction of the first claim is analogous. The only difference is that, instead of the set CS, we use the set FS = {Γ ⊆ FmA | Γ is finite and ΓA ⊆ 𝐹} and observe that F is now a filter on 𝐼 (because FS is closed under finite unions); note that we need not use any cardinality restriction here.  Now we are ready to formulate and prove, for weakly implicative logics, two final theorems summarizing and improving the characterizations of the SKC and the FSKC presented earlier in the chapter. We will also accompany both theorems with their respective immediate corollaries for algebraically implicative logics (recall the series of Remarks 3.2.2, 3.3.9, 3.4.5, 3.6.7, and 3.7.8, which tell us that matricial class operators correspond to the algebraic ones in a straightforward way, and Corollary 3.6.8, which says the same for R(F)SI matrices/algebras).

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141

Theorem 3.8.2 Let L be a weakly implicative logic and K ⊆ Mod∗ (L). Then, the following are equivalent: 1. L has the SKC. 2. Mod∗ (L) = ISP𝜔 (K). 3. Mod∗ (L) 𝜔 ⊆ ISP(K). If furthermore L has the CIPEP, then we can add the following equivalent condition: 𝜔 ⊆ IS(K). 4. Mod∗ (L)RSI

The implication 1 → 4 is always true. Proof Recall that, as L is weakly implicative, we have P𝜔 (K) ⊆ Mod∗ (L) and so the equivalence of the first two claims is a simple corollary of Proposition 3.8.1. The equivalence of the first and the third claim was proved in Corollary 3.4.11. The remaining claims were proved in Corollary 3.6.17.  Corollary 3.8.3 Let L be an algebraically implicative logic and A ⊆ Alg∗ (L). Then, the following are equivalent: 1. L has the SAC. 2. Alg∗ (L) = GQ(A). 3. Alg∗ (L) 𝜔 ⊆ ISP(A). If, furthermore, L has the CIPEP, then we can add the following equivalent condition: 𝜔 4. Alg∗ (L) RSI ⊆ IS(A).

The implication 1 → 4 is always true. This corollary allows us to use algebraic knowledge to prove the SAC for certain logics and classes of algebras. Also, dually, we can use the fact that we have proved or disproved the SAC to obtain certain purely algebraic facts. Example 3.8.4 Recall that in Corollary 3.5.20 and Example 3.6.18, we have established the strong completeness of classical and Gödel–Dummett logic w.r.t. their intended algebras 2 and [0, 1] G . Since the classes of G-algebras and Boolean algebras are varieties, we immediately obtain the following equalities: BA = V(2) = Q(2) = GQ(2). G = V([0, 1] G ) = Q([0, 1] G ) = GQ([0, 1] G ). Also note that in Theorem 3.6.14 we have proved the SAC of Łukasiewicz logic for the class A of linearly ordered MV-algebras (see Example 3.6.9). However, due to Examples 2.6.6 and 2.6.7, we know that Łukasiewicz logic does not enjoy strong completeness w.r.t. the standard MV-algebra or w.r.t. the set of finite linearly ordered MV-algebras; furthermore, thanks to Example 2.6.7 we know that the logic of finite linearly ordered MV-algebras strictly subsumes that of the standard MV-algebra.

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Therefore, we obtain the following equalities: MV = GQ({A ∈ MV | A is linearly ordered}) ) GQ({[0, 1] Ł }) ) GQ({MV𝑛 | 𝑛 ≥ 2}).

Next, we turn our attention to the FSKC, for which the situation is a bit more complicated. Indeed, roughly speaking, when we move from the SKC to the FSKC, the localizations of prominent classes turn out to be based not directly on K but on the class of its ultraproducts (we formalize it in Corollary 3.8.10). This will however allow us to localize not only the class of all RSI-models16 but even the class of all RFSI-models (which, as we have argued before, is more important from the logical point of view) which in turn will allow us to prove the promised converse direction of Proposition 3.6.19. Let us recall that we denote by K+ the extension of K by the trivial matrix. Theorem 3.8.5 Let L be a weakly implicative logic and K ⊆ Mod∗ (L). Then, the following are equivalent: 1. L has the FSKC. 2. Mod∗ (L) ⊆ ISPPU (K). Furthermore, if L is finitary, then we can add the following conditions: 3. Mod∗ (L) = ISPPU (K). 4. Mod∗ (L)RFSI is embeddable into PU (K+ ), i.e. Mod∗ (L)RFSI ⊆ ISPU (K+ ). 𝜔 is embeddable into P (K), i.e. Mod∗ (L) 𝜔 ⊆ ISP (K). 5. Mod∗ (L)RSI U U RSI If, moreover, the language of L is finite, we can add the following conditions: 6. Mod∗ (L)RFSI is partially embeddable into K+ . 𝜔 is partially embeddable into K. 7. Mod∗ (L)RSI Proof The equivalence of the first three claims is relatively easy and left as an exercise for the reader. The implication ‘1 implies 4’ is rather involved; let us observe that the remaining ones were already established before or are easy. To prove ‘4 implies 5’ and ‘6 implies 7’ just observe that only reduced matrices embeddable into a trivial one have to be trivial, the trivial reduced matrix is never subdirectly irreducible, and PU (K+ ) = (PU (K)) + . As all finitary logics enjoy the CIPEP (Corollary 3.5.7), we obtain ‘5 implies 1’ and ‘7 implies 1’ from Propositions 3.7.6 and Proposition 3.6.19. Thus, after we prove ‘1 implies 4’ we will know, among others, the equivalence of the first five claims and, to finish the proof, it will suffice to observe that the fact that ‘4 implies 6’ follows directly from Lemma 3.7.9. 16 Later, in Theorem 5.6.5, we prove that, in the presence of a suitable disjunction, a logic L has the 𝜔 SKC iff Mod∗ (L) RFSI ⊆ IS(K+ ).

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Let us now prove ‘1 implies 4’. We proceed as in the proof of Proposition 3.8.1; however here we can directly construct the desired embedding. We start by fixing a matrix A = hA, 𝐹i ∈ Mod∗ (L)RFSI (clearly we can assume that A is non-trivial; otherwise it would be isomorphic with the trivial matrix in K+ ) and an A-evaluation 0 𝑒 of formulas from FmLVar∪Var such that, for each 𝑎 ∈ 𝐴, 𝑒(𝑣 𝑎 ) = 𝑎 and, for each 𝑣 ∈ Var, 𝑒(𝑣) = 𝑥 for an arbitrary fixed 𝑥 ∈ 𝐴. Clearly, 𝑒 is a strict surjective 0 homomorphism from hFmLVar∪Var , 𝑒 −1 [𝐹]i to A, and hence, due to Proposition 3.5.5, 𝑒 −1 [𝐹] is finitely meet-irreducible. Next, we define the index set 𝐼 0 = {Δ ⊆ FmA | Δ is finite and ΔA ⊆ 𝐴 \ 𝐹}. Due to the assumption that A is not trivial, we know that 𝐼 0 ≠ ∅. Since 𝑒 −1 [𝐹] is finitely 0 meet-irreducible, for each Δ = {𝛿1 , . . . , 𝛿 𝑛 } ∈ 𝐼 0, there has to be a 𝛿Δ ∈ FmLVar∪Var such that 𝛿Δ ∉ 𝑒 −1 [𝐹] and 𝑒 −1 [𝐹], 𝛿𝑖 `LVar0 𝛿Δ for each 𝑖. Due to the finitarity of 0 LVar (which we know due to the first claim of Proposition 3.1.7), there has to be a finite set ΓΔ ⊆ 𝑒 −1 [𝐹] such that ΓΔ , 𝛿𝑖 `LVar0 𝛿Δ for each 𝑖. Consider the substitution 𝜎 such that 𝜎(𝑣 𝑎 ) = 𝑣 𝑎 for each 𝑎 ∈ 𝐴 and 𝜎(𝑣) = 𝑣 𝑥 for each 𝑣 ∈ Var. Then, 𝜎(𝛿Δ ) A = 𝑒(𝛿Δ ) ∉ 𝐹 and 𝜎[ΓΔ ], 𝛿𝑖 `LVar0 𝜎(𝛿Δ ). Now we define the sets 𝐼 = {hΓ, Δi | Δ ∈ 𝐼 0 and Γ is finite, FmA ⊇ Γ ⊇ 𝜎[ΓΔ ], and ΓA ⊆ 𝐹} hΓ, Δi ∗ = {hΓ0, Δ0i ∈ 𝐼 | Γ ⊆ Γ0 and Δ ⊆ Δ0 }

for each hΓ, Δi ∈ 𝐼.

It is easy to observe that the set {hΓ, Δi ∗ | hΓ, Δi ∈ 𝐼} has the finite intersection property, so there is an ultrafilter U containing it; indeed, we can easily observe that Ø Ø Ù h Γ 𝑗 ∪ 𝜎[ΓÐ 𝑗≤𝑛 Δ 𝑗 ], Δ𝑗i ∈ hΓ 𝑗 , Δ 𝑗 i ∗ . 𝑗 ≤𝑛

𝑗 ≤𝑛

𝑗 ≤𝑛

Note that, for each 𝑖 = hΓ, Δi ∈ 𝐼, we have Γ 0LVar0 𝜎(𝛿Δ ). By the FSKC and Proposition 3.1.7, we have a matrix A𝑖 = hA𝑖 , 𝐹𝑖 i ∈ K such that Γ 2A𝑖 𝜎(𝛿Δ ). Let 𝑒 𝑖 be an A𝑖 -evaluation witnessing it and note that 𝑒 𝑖 [Δ] ⊆ 𝐴𝑖 \ 𝐹𝑖 (because 𝜎[ΓΔ ], 𝛿𝑖 `LVar0 𝜎(𝛿Δ ) and Γ ⊇ 𝜎[ΓΔ ]). Let B = hB, 𝐺i be the ultraproduct of {A𝑖 | 𝑖 ∈ 𝐼} given by the ultrafilter U. We define a mapping 𝑓 : 𝐴 → 𝐵 as 𝑓 (𝜑A ) = h𝑒 𝑖 (𝜑)i𝑖 ∈𝐼 /U. We denote by 𝜑 the element h𝑒 𝑖 (𝜑)i𝑖 ∈𝐼 /U. Recall that 𝜑 ∈ 𝐺 ⇐⇒ {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) ∈ 𝐹𝑖 } ∈ U. The proof will be concluded by showing that 𝑓 is an embedding. We start by showing that 𝜑A ∈ 𝐹 iff 𝑓 (𝜑A ) ∈ 𝐺

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First, assume that 𝜑A ∈ 𝐹. Then, for any Δ ∈ 𝐼 0, we have h{𝜑} ∪ ΓΔ , Δi ∈ 𝐼. Therefore, h{𝜑} ∪ ΓΔ , Δi ∗ ∈ U and so 𝑓 (𝜑A ) = 𝜑 ∈ 𝐺. Conversely, if 𝜑A ∉ 𝐹, then hΓ 𝜑 , {𝜑}i ∈ 𝐼 and due to the construction of 𝛿Δ we know that for each 𝑖 ∈ hΓ 𝜑 , {𝜑}i ∗ we have 𝑒 𝑖 (𝜑) ∉ 𝐹𝑖 and so {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) ∉ 𝐹𝑖 } ∈ U. Since U is a proper filter, this entails {𝑖 ∈ 𝐼 | 𝑒 𝑖 (𝜑) ∈ 𝐹𝑖 } ∉ U, i.e. 𝜑 ∉ 𝐺. Using this fact it is easy to show that 𝑓 is well defined and one-one: 𝜑A = 𝜓 A

iff

(𝜑 ⇔ 𝜓) A ⊆ 𝐹

iff

{𝜑 → 𝜓, 𝜓 → 𝜑} = 𝑓 [(𝜑 ⇔ 𝜓) A ] ⊆ 𝐺

iff

𝜑 = 𝜓.

The rest of the proof is left as an exercise for the reader.



Remark 3.8.6 Later, in Proposition 5.6.1, we will show that in logics with a suitable disjunction we could prove that 1 implies 4 (and thus vicariously also 6) without assuming the finitarity. Moreover, in Theorem 5.6.3, we will show that, under the same conditions, the KC implies a looser localization of Mod∗ (L)RFSI as a subset of HSPU (K). Corollary 3.8.7 Let L be an algebraically implicative logic and A ⊆ Alg∗ (L). Then, the following are equivalent: 1. L has the FSAC. 2. Alg∗ (L) ⊆ Q(A). Furthermore, if L is finitary, then we can add the following conditions: 3. Alg∗ (L) = Q(A). 4. Alg∗ (L) RFSI is embeddable into PU (A+ ), i.e. Alg∗ (L) RFSI ⊆ ISPU (A+ ). 𝜔 𝜔 5. Alg∗ (L) RSI is embeddable into PU (A), i.e. Alg∗ (L) RSI ⊆ ISPU (A). If, moreover, the language of L is finite, we can add the following conditions: 6. Alg∗ (L) RFSI is partially embeddable into A+ . 𝜔 7. Alg∗ (L) RSI is partially embeddable into A.

Example 3.8.8 The assumption of finiteness of the language cannot be omitted. Consider the language L resulting from LCL by adding an infinite countable set 𝐶 = {𝑐 𝑛 | 𝑛 ∈ N} of new 0-ary connectives. Let G𝐶 be the conservative expansion of Gödel–Dummett logic G (see Example 2.3.7) in this language with no additional axioms or rules. Clearly, G𝐶 is a finitary weakly implicative logic and Alg∗ (G𝐶 ) consists of the algebras from Alg∗ (G) with infinitely many constants arbitrarily interpreted. Recall the G-algebra [0, 1] G and consider a class A ⊆ Alg∗ (G𝐶 ) of algebras expanding [0, 1] G in which all constants, except for a finite number, are interpreted as 1. We prove that G𝐶 enjoys FSAC: Consider any finite set Γ ∪ {𝜑} such that Γ 0G𝐶 𝜑.

3.8 Completeness and description of classes of reduced matrices

145

Then, also Γ 0G𝐶 𝜑, where we understand the new constants just as propositional variables. Thus, by Example 3.6.18 and Proposition 3.1.7, there is a [0, 1] G -evaluation 𝑒 such that 𝑒[Γ] ⊆ {1} and also 𝑒(𝜑) < 1. We construct a G𝐶 -algebra B resulting from [0, 1] G by setting 𝑐B𝑛 = 𝑒(𝑐 𝑛 ) for every 𝑐 𝑛 occurring in Γ ∪ {𝜑} and 𝑐B𝑛 = 1 otherwise. Notice that 𝑒 can be viewed as a B-evaluation and, since B ∈ A (because Γ ∪ {𝜑} contains only finitely many constants) we obtain Γ 2A 𝜑. Now we show that the condition 7 of the previous corollary does not hold: ¯ ∪ { 1 | 𝑛 ≥ 1} and its consider the subalgebra A of [0, 1] G with domain {0} 𝑛 expansion A0 obtained by setting 𝑐A = 0 for each 𝑐 ∈ 𝐶. It is easy to see that 𝜔 A0 ∈ Alg∗ (G𝐶 ) RSI (because FiG (A) = FiG𝐶 (A0 ) and due to Example 3.6.9 we know 𝜔 ∗ that A ∈ Alg (G) RSI ) and no finite subset 𝑋 of A0 containing 0 can be partially embedded into any algebra B ∈ A (as we would have 𝑐A0 ∈ 𝑋 for each 𝑐 and the condition of partial embeddability would entail that 𝑐B = ⊥B = 0 for each 𝑐). Example 3.8.9 Using Example 3.6.18 we obtain G = V({G𝑛 | 𝑛 ≥ 2}) = Q({G𝑛 | 𝑛 ≥ 2}) ) GQ({G𝑛 | 𝑛 ≥ 2}). As regards to the Łukasiewicz logic Ł, we can easily observe that it satisfies all the hypotheses of the previous corollary. Using the algebraic fact that the class Alg∗ (Ł) = MV is a variety and it is generated as a quasivariety both by the algebra [0, 1] Ł and by all its finite subalgebras (see Theorem A.5.12), we obtain that Ł enjoys the finite strong completeness w.r.t. any of these two classes of algebras. Note that this fact implies that Ł is the finitary companion of the infinitary logic Ł∞ . Indeed, from Γ `Ł∞ 𝜑 for some finite Γ, we obtain Γ  h[0,1] Ł , {1}i 𝜑 (because, from Example 2.6.6, we know that h[0, 1] Ł , {1}i ∈ Mod∗ (Ł∞ )) and thus, by the completeness theorem just established, we obtain Γ `Ł 𝜑. Furthermore, in Example 3.6.9 we have seen that Alg∗ (Ł) RFSI is exactly the class of linearly ordered MV-algebras. Therefore, the previous corollary entails a well-known result of the theory of MV-algebras (see e.g. [214, Theorem 5.6.1]): each linearly ordered MV-algebra can be embedded into an ultrapower of {[0, 1] Ł }. We conclude this chapter by the promised formalization of the relationship between the SKC and the FSKC via ultraproducts. Corollary 3.8.10 Let L be a finitary weakly implicative logic and K ⊆ Mod∗ (L). Then, L has the FSKC if and only if it has the SPU (K)C. Proof To prove the left-to-right direction, it suffices to prove that Mod∗ (L) = ISP𝜔 PU (K) and use Theorem 3.8.2 (note that, in order to do that, we need to know that PU (K) ⊆ Mod∗ (L), which is the case as L is finitary due to Theorem 3.7.7). This equality is proved by the following simple chain of inclusions/equations (the first one is Theorem 3.8.5, the remaining ones are simple properties of class operators): Mod∗ (L) = ISPPU (K) ⊆ ISP𝜔 PU (K) ⊆ ISP𝜔 PU (Mod∗ (L)) ⊆ Mod∗ (L). The converse direction is left as an exercise for the reader.



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3 Completeness properties

3.9 History and further reading As mentioned in the introduction, the work on completeness theorems presented in this chapter stems from the developments in the following two main groups of bibliographical sources. Algebraic study of many-valued logics: The literature of many-valued logics is rich in proofs of completeness of axiomatic systems with respect to particular intended algebraic models (see e.g. [141] and references therein). A majority of these logics are algebraically implicative and their completeness has been typically obtained by showing (partial) embeddability into a distinguished class of algebras. Early important examples are the completeness w.r.t. the corresponding standard algebra of Łukasiewicz logic (proved by Chen Chung Chang in [63, 64]) and of the logic G (proved implicitly by Michael Dummett in [113]). Starting from the 1990s, there is a plethora of works containing this kind of completeness proofs for families of substructural and fuzzy logics, e.g.: fuzzy logics based on continuous (or left-continuous) t-norms [1,2,72,160,171,189,198,199,230], fuzzy logics on expanded languages [122, 126, 128, 161], substructural (fuzzy) logics [143, 187, 223], or non-associative substructural logics [51, 84, 85]. In the context of mathematical fuzzy logic, the three forms of completeness were first explicitly defined and shown to be different in [121]; the first characterizations in terms of (partial) embeddings were proved for axiomatic expansions of the fuzzy logic MTL in [78] and later generalized by the authors of this book to algebraically implicative semilinear logics in [88]. Universal algebra, model theory, and abstract algebraic logic: These disciplines have developed a general study of, respectively, algebraic, first-order, and matricial structures which, in particular, heavily employs the constructions used in this chapter. Starting from the works of Garrett Birkhoff [37] and Anatolii Mal’cev [220], universal algebra has offered a deep understanding of the operators applied to arbitrary classes of algebras, evolving into an algebraic treatment of universal Horn logic (for its equalityfree counterpart see [43, 108]). Bjarni Jónsson [200] used the class operators to study (finitely) subdirectly irreducible algebras in congruence distributive varieties proving a localization result (known as Jónsson’s Lemma) which can be seen as the original form of the more general localizations of (R)FSIs presented in Theorem 3.8.5 and Corollary 3.8.7. Model theory has developed analogous tools to study the more general notion of first-order structure (see e.g. the comprehensive monographs [66, 186]). Abstract algebraic logic has developed a theory in the intermediate level of generality given by logical matrices. Indeed, Ryszard Wójcicki introduced in 1973 the matrix semantics for propositional logics in [309] and in 1975 Stephen L. Bloom, realizing that matrices are actually first-order structures for an equality-free language with only one monadic predicate, started studying them by model-theoretic means [46]. Following Bloom’s proposal, abstract algebraic logic underwent a substantial development for over two decades as the theory of matrices (and some generalizations thereof [135]) and their connections to propositional logics, specially for well-behaved classes of logics such as the rather general class of protoalgebraic logics.

3.9 History and further reading

147

In his landmark monograph [100] published in 2001, Janusz Czelakowski provided a comprehensive study of protoalgebraic logics that, among many other topics, covered completeness properties as well. Indeed, it contains (in the first two chapters) several results involving SKC and KC. In particular, it includes consequences of the completeness properties: (a) generation of the whole class of reduced models from K using certain model-theoretic operators, and (b) localization of RFSI-models in classes generated by K. On a few occasions, these theorems are obtained as a characterization of the completeness properties. Finally, let us mention our earlier paper [93], where we offered a systematic presentation of existing results on completeness theorems, including some newly considered variations for the FSKC and often proved as characterizations (that is, with optimal sets of assumptions); this paper is the basis of this chapter and contains the majority of its results on completeness theorems (which sometimes we have particularized to weakly implicative logics). This chapter also contains numerous results on the structure of (classes of) logical matrices and their model-theoretic properties, and on the properties of the Leibniz operator. Most of these results are standard elements of AAL and can be found e.g. in the monographs [100, 134]; the interested reader may also consult these monographs for further references and historical remarks. It is worth mentioning that, even though we have proved some of these results for weakly implicative logics, they are known to hold in (and even provide characterizations for) wider classes of logics studied in AAL as constituents of the Leibniz hierarchy depicted in Figure 2.2; see the end of Section 2.10 for syntactic definitions of these classes. Indeed, it is known that a logic is • protoalgebraic iff the Leibniz operator (restricted to its filters) is monotonic iff the class of its reduced models is closed under subdirect products (cf. Propositions 3.6.2 and 3.4.15). • equivalential iff the Leibniz operator (restricted to its filters) is monotonic and commutes with inverse substitutions iff the class of its reduced models is closed under submatrices and products. • finitely equivalential iff the Leibniz operator (restricted to its filters) is continuous iff (if, furthermore, the logic in question is finitary) the class of its reduced models is closed under submatrices, products, and ultraproducts (cf. Proposition 3.4.15 and Theorem 3.7.7). In contrast, we have also obtained results that cannot be much generalized in the AAL framework, e.g. the proof of the third claim of Proposition 3.7.3 is based on the finite nature of the description of Leibniz congruence we have in weakly implicative logics; that is why this result cannot be generalized to more general classes of the Leibniz hierarchy beyond finitely equivalential logics.

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3 Completeness properties

3.10 Exercises Exercise 3.1 In Remark 3.1.9, given a set of variables Var 0 of cardinality bigger than the continuum and Γ = {( 𝑝 ↔ 𝑞) → 𝑟 | for each 𝑝, 𝑞 ∈ Var 0 \ {𝑟 } such that 𝑝 ≠ 𝑞}, 0 show that Γ 0GVar0 𝑟 and Γ Var 𝑟. G∞ Exercise 3.2 In Remark 3.2.2, given algebraically implicative logic L, show that a homomorphism/embedding of L-algebras A and B is also a homomorphism/embedding of the corresponding matrices hA, 𝐹A i and hB, 𝐹B i. Exercise 3.3 In Remark 3.2.4, show that the class Mod(L) is not closed, in general, under homomorphic images by considering the identity homomorphism from h[0, 1] Ł , {1}i ∈ Mod(Ł) to h[0, 1] Ł , [ 12 , 1]i ∉ Mod(Ł). Exercise 3.4 In Example 3.3.8, prove that the class Mod∗ (L) is not closed, in general, under submatrices by checking that hA, 𝐹i ∈ Mod∗ (L), while the submatrix hB, 𝐹i ∉ Mod∗ (L). Exercise 3.5 In Remark 3.3.9, given an algebraically implicative logic L, B ∈ Alg∗ (L), and a subalgebra A ⊆ B, show that A ∈ Alg∗ (L), 𝐹A = 𝐹B ∩ 𝐴, and hA, 𝐹A i is a submatrix of hB, 𝐹B i. Exercise 3.6 Complete the proof of Proposition 3.3.11. Î Exercise 3.7 In the proof of Proposition 3.4.3, take A = 𝑖 ∈𝐼 A𝑖 and suppose Î that, for each 𝑖 ∈ 𝐼, A𝑖 is a submatrix of A𝑖0 ∈ K. Check that A is a submatrix of 𝑖 ∈𝐼 A𝑖0 . Exercise 3.8 In Example 3.4.4, prove that the class Mod∗ (L) is not closed, in general, under products. If A is the reduct of 2 only with the operation ∧, check that hA, {1}i ∈ Mod∗ (L), while the product hA, {1}i × hA, {1}i ∉ Mod∗ (L). Exercise 3.9 In Remark 3.4.5, given L and a system Î an algebraically implicative logicÎ hA𝑖 i𝑖 ∈𝐼 of L-algebras, show that 𝑖 ∈𝐼 A𝑖 ∈ Alg∗ (L) and 𝐹Î𝑖∈𝐼 A𝑖 = 𝑖 ∈𝐼 𝐹A𝑖 . Exercise 3.10 Prove Corollary 3.4.12. Exercise 3.11 In Example 3.4.19, take a subalgebra A of [0, 1] Ł and 𝐹 ∈ FiŁ (A) ¯ Prove that 0 ∈ 𝐹. Hint: recall the definition such that there is an element 𝑎 ∈ 𝐹 \ {1}. of the operation & in MV-algebras and show that 𝑥, 𝑦 ∈ 𝐹 implies that 𝑥 &A 𝑦 ∈ 𝐹 and that there is an 𝑛 such that 𝑎 & 𝑎 & . . . 𝑎 = 0. | {z } 𝑛 times

Exercise 3.12 In Example 3.5.16, finish proving that the only non-trivial simple Heyting algebra (up to isomorphism) is the Boolean algebra 2 by extending the reasoning to higher cardinalities. Exercise 3.13 In Proposition 3.5.19, complete the proof of the last implication.

3.10 Exercises

149

Exercise 3.14 Complete the proof of Corollary 3.5.20. Hint: show that 𝑇 proves formulas corresponding, in an obvious way, to the rows of the truth table for →2 . Exercise 3.15 Show that IntPrimeTh(BCIlat ) * PrimeTh(BCIlat ). Hint: consider Example 4.4.11. Exercise 3.16 Prove Proposition 3.6.2. Hint: the proof of the first claim is a simple extension of the proof of analogous statement for products. Exercise 3.17 In Remark 3.6.7, given an algebraically implicative logic L, show that any subdirect product of a system of L-algebras is itself an L-algebra. Exercise 3.18 In Example 3.6.9, check that the description of RFSI-algebras is correct. Hint: prove, mimicking the proof of the relevant part of Proposition 3.5.22, that for any logic L extending FBCK, LCL -algebra A, and filter 𝐹 ∈ FiL (A) we have: 𝐹 is finitely meet-irreducible iff 𝑎 → 𝑏 ∈ 𝐹 or 𝑏 → 𝑎 ∈ 𝐹 for each 𝑎, 𝑏 ∈ 𝐴. Exercise 3.19 In Example 3.6.16, prove that  {Ł𝑛 | 𝑛≥2} has the CIPEP. Exercise 3.20 In Example 3.6.18, show that the order-embedding is an embedding of 𝜔 * G-algebras. Hint: use the fact that 𝑎 ∧ 𝑏 = 𝑎 ∧ (𝑎 → 𝑏). Also, prove Mod∗ (G)RSI IS(K) 0. Hint: consider the G-algebra with domain { 𝑛1 | 𝑛 ≥ 1}. Exercise 3.21 In Proposition 3.7.3, if X is any of the operators P𝜔 , PF , or PU , show that XI(K) ⊆ IX(K). Exercise 3.22 In Remark 3.7.8, given an algebraically implicative logic L with a set E of defining equations, a system of L-algebras A𝑖 𝑖 ∈𝐼 , and F a filter over 𝐼, and assumingÎthat L is finitary or that F is closed Î under countable intersections, prove that A = 𝑖 ∈𝐼 A𝑖 /F ∈ Alg∗ (L) and 𝐹A = 𝑖 ∈𝐼 𝐹A𝑖 /F . Exercise 3.23 Prove the equivalence of the first three claims (assuming finitarity for the last implication) in Theorem 3.8.5. Exercise 3.24 Complete the details at the end of the proof of Theorem 3.8.5. Exercise 3.25 Complete the proof of Corollary 3.8.10 by showing that for any logic L, if L = PU (K) , then L enjoys the FSKC (note that the premise is, in general, not equivalent to SPU (K)C of L, because we need not have PU (K) ⊆ Mod∗ (L)).

Chapter 4

On lattice and residuated connectives

The most important logical connective in our approach is implication. This is explicitly stressed in the title of this book and our reasons have been elaborated in the previous chapters. We have introduced the class of weakly implicative logics, along with several prominent examples, and have proved a number of results in which the implication connective plays indeed an essential role. However, there are other connectives present in many important particular logics which make a substantial contribution to the expressivity, the modeling power, and the metalogical properties of these logics. Let us focus, in particular, on the following groups of properties: Order-theoretic properties: We have already seen that, in weakly implicative logics, the implication connective ⇒ defines a preorder relation in all matrix models, which is actually an order in reduced models. Moreover, in algebras of expansions of BCIlat (e.g. in IL, CL, Ł, or modal logics), we have seen that this binary relation is actually a bounded lattice order. Indeed, the algebras in the Alg∗ (L) of these logics are expansions of bounded lattices (satisfying certain additional conditions), where the connectives ∧, ∨, >, and ⊥ are respectively interpreted as the lattice meet, lattice join, the maximum element, and the minimum element. Therefore, not only implication but also these other connectives are intrinsically related to the description of the order-theoretic properties. Adjunction: An additional property that is usually expected from a conjunction ∧, as seen in several examples, is expressed by the adjunction rule (Adj): 𝜑, 𝜓 I 𝜑∧𝜓. Semantically, adjunction forces filters on an algebra A (of a matrix model of the logic) to be closed under the operation ∧A . Proof by cases property: This property of a connective ∨ is useful for obtaining formal derivations in many logics. It ensures the existence of a proof (in a given context) of a formula 𝜒 from 𝜑 ∨ 𝜓, as soon as we know that 𝜒 is provable from 𝜑 and also from 𝜓. Thanks to Proposition 2.4.5, we have this property, in particular, for CL, G, IL, Ł, and local modal logics. Also we have argued that it does not hold for global modal logics (although in Proposition 2.4.6 we have shown how it can be recovered in another variant that also involves the connective ; we devote the whole Chapter 5 to the abstract study of such disjunctive connectives). © Springer Nature Switzerland AG 2021 P. Cintula and C. Noguera, Logic and Implication, Trends in Logic 57, https://doi.org/10.1007/978-3-030-85675-5_4

151

152

4 On lattice and residuated connectives

Deduction theorems: In Proposition 2.4.1, we have obtained, for axiomatic expansions of IL→ , the deduction theorem in its classical form, which tells us that proving (in a given context) 𝜓 from 𝜑 is equivalent to proving the formula 𝜑 → 𝜓. So, in particular, we have this useful metalogical property of implication for CL, G, IL, and local modal logics (again, for global modal logics, we need to involve also the connective , as seen in Proposition 2.4.2). However, for weaker logics the property needs to be modified; e.g. for axiomatic expansions of BCI we have shown in Proposition 2.4.3 that 𝜓 is provable from 𝜑 (in a given context) if and only if there is a natural number 𝑘 ≥ 0 such that the formula 𝜑 → (𝜑 → . . . (𝜑 → 𝜓) . . .) with 𝑘 nested implications is provable. Thus, these logics still retain a deduction theorem purely expressed in terms of implication, albeit now with two obvious drawbacks: (1) the number 𝑘 depends on the involved formulas, (2) it partly loses the appeal of having just one implication that somehow internalizes the notion of provability. The latter can be overcome in certain expansions of BCI, namely those with a connective & that allows one to aggregate premises of nested implications (the order of premises in & is purely conventional; see Remark 4.2.6 for more details): 𝜑 → (𝜓 → 𝜒) a` 𝜓 & 𝜑 → 𝜒. Clearly, using these rules, one can rewrite the local deduction theorem in terms of the provability of 𝜑 & (𝜑 & (. . . & 𝜑) . . .) → 𝜓, where the conjunction & is repeated 𝑘 times (for some natural number 𝑘). We have already met two illustrative examples of such connectives: the connective & defined in Łukasiewicz logic as ¬(𝜑 → ¬𝜓) and the connective ∧ in intuitionistic or classical logic. We know that the latter is idempotent (i.e. 𝜑 ∧ 𝜑 ↔ 𝜑 is a theorem) and thus we obtain the classical deduction theorem for IL and CL, whereas the former is not, hence giving only a local version of deduction theorem for Ł. Residuation property: Not only does the connective & obeying the rules mentioned above need not be idempotent, in many weakly implicative logics it is not even associative or commutative. Actually, it is easy to see that & is commutative in a given logic iff the rule (E) is valid. This rule allows us to switch the order of premises in nested implications; moreover, in many logics where (E) fails, we have an additional binary connective which allows us to do the switch at the price of replacing the original inner implication by , i.e. 𝜑 ⇒ (𝜓 ⇒ 𝜒) a` 𝜓 ⇒ (𝜑

𝜒).

Combining these rules with those for &, we obtain the following: 𝜑 ⇒ (𝜓 ⇒ 𝜒) a` 𝜓 & 𝜑 ⇒ 𝜒 a` 𝜓 ⇒ (𝜑

𝜒).

The algebraic rendering of these rules is called residuation property and it is the cornerstone of the study of substructural logics, a prominent family of non-classical logics that will receive a lot of attention in the present chapter. Also, the residuation property motivates the terminology of calling & a residuated conjunction, ⇒ its left residuum, and its right residuum.

4 On lattice and residuated connectives

153

Filter definability: Recall that in reduced matrices of weakly implicative logics the filter is an upper set with respect to the matrix order. An interesting question is when is this filter principal; i.e. when does the matrix have a least designated element? In this case, it may be useful to have a (definable) truth-constant 1¯ (called in this book protounit, for reasons that will be apparent soon) to be interpreted as this least designated element in each matrix. This behavior can be easily described by a pair of rules known as push and pop: 𝜑 a` 1¯ ⇒ 𝜑 Then, one can define the filter as the set of elements of the algebra bigger than ¯ Clearly, if the logic is Rasiowaor equal to the interpretation of the protounit 1. implicative, then any of its theorems can play the role of the protounit. Perhaps less obvious is the observation that, if the logic also has a residuated conjunction &, then the protounit 1¯ is its right-unit As we shall see, it need not be the left-unit (recall that & need not be commutative) but, perhaps surprisingly, any right-unit of & has to be the protounit and thus also the left-unit. This chapter is dedicated to the study of weakly implicative logics that, besides their usual implication connective ⇒, also enjoy the presence of additional connectives with (some of) the mentioned properties. In the first third of the chapter, we focus on individual connectives, which we split into two families: lattice and residuated connectives (see Section 4.11 for the historical origins of this taxonomy). In Section 4.1, we start with the binary connectives ∧ and ∨, which we call lattice protoconjunction/protodisjunction of a given weakly implicative logic L, whenever they are interpreted as the infimum/supremum of the underlying matrix order in any reduced model.1 We drop the prefix ‘proto’ if they satisfy the additional properties mentioned above: adjunction and proof by cases, respectively. Then, in Section 4.2, we perform a similar study of the binary connectives & and , which we call residuated conjunction and co-implication whenever they satisfy the residuation property: 𝑥 ≤A 𝑦 ⇒A 𝑧

iff

𝑦 &A 𝑥 ≤ A 𝑧

iff

𝑦 ≤A 𝑥

A

𝑧.

In Section 4.3 we focus on the truth-constants >, ⊥, and 1¯ which we call top, bottom, and protounit whenever their interpretations are the greatest/least elements of the underlying matrix order and the least element of the filter. We also introduce a stronger version of protounit called unit and explore their interplay with the connectives introduced before. Terminologically speaking, we consider top and bottom as lattice connectives and (proto)unit as a residuated connective. Finally, we also consider the residuated truth-constant 0¯ which, in general, has no intended algebraic interpretation and it is simply used in substructural logics to define negation (actually two, in ¯ which, principle different, negation connectives: ¬𝜑 = 𝜑 ⇒ 0¯ and ∼𝜑 = 𝜑 0) however, will not be studied in this book. 1 Actually, following the spirit of this book, we use definitions in terms of the Hilbert-style rules and prove the equivalence with the expected semantical behavior later.

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4 On lattice and residuated connectives

The second third of the chapter is focused on the study of a wide class of weakly implicative logics with (some of) these additional connectives. We start, in Section 4.4, by introducing two important particular logics: • the logic LL (known as Lambek Logic) as the least weakly implicative logic with all the residuated connectives and • the logic SL as the extension of LL with all the lattice connectives in which, moreover, 1¯ is required to be the unit. We describe the semantical counterparts of these two logics in terms of residuated ordered protounital groupoids and prove that SL is an algebraically implicative (though not Rasiowa-implicative) logic, while LL is not. Furthermore, we show that the reduced models of both of these logics admit regular completions, i.e. they can be embedded (preserving all the existing suprema and infima) into models where the matrix order is complete. In Section 4.5, we axiomatize LL and SL by means of Hilbert-style calculi which are strongly separable for certain sublanguages (in the case of LL for all sublanguages containing the implication), i.e. calculi whose axioms and rules involving connectives from a given sublanguage axiomatize the corresponding fragment of the logic in question. In Section 4.6, we select some basic implicational properties of LL to define our rendition of substructural logics, encompassing virtually all the logics that have been studied in the literature under this moniker. We explore this family of logics in further detail by describing the extensions of SL obtained by combinations of structural rules (see the resulting hierarchy in Figures 4.3 and 4.9) and the corresponding classes of algebras for these logics. Finally, in Section 4.7 we obtain strongly separable presentations for extensions of the substructural logic SLaE , which allows us to obtain several promised conservativity results (e.g. we can show that the logic BCI is indeed the implicational fragment BCIlat and similar results). The final third of the chapter builds on the mentioned motivation of residuated conjunction as a means to recover a form of deduction theorem using only one implication. Section 4.8 offers a quite detailed study of a general notion of implicational deduction theorem, which is exemplified in many logics encountered so far in the text and characterized in terms of the existence of a presentation that only has one binary rule (modus ponens) and unary rules of a certain simple form. As an interesting by-product of this study, we obtain a description of filter generation in the algebraic counterpart. In Section 4.9, we show that the main logics studied in this chapter can indeed be presented in the way mentioned above and, furthermore, the needed unary rules obey certain additional properties. Finally, Section 4.10 uses the previous results to obtain a construction technique for a generalized form of disjunction connective (given not by one, but by a set of formulas) with the proof by cases property, which will be studied in full generality in the next chapter. Therefore, we will obtain that the prominent substructural logics studied in this chapter not only have implicational deduction theorems, proof by cases properties, and a nice description of generated filters, but also we will have prepared the ground for Chapter 6 where we will use these results to easily describe their semilinear extensions.

4.1 Lattice connectives

155

4.1 Lattice connectives In this section, we consider two binary connectives ∧ and ∨ whose intended interpretation, in a reduced matrix A of a given weakly implicative logic L, is respectively the infimum and supremum with respect to the matrix order ≤A⇒ given by a weak implication ⇒ of L.2 Therefore, strictly speaking, the meaning of these connectives depends on the used implication; however, we follow our general convention and do not mention explicitly the implication when already known or fixed by the context. As in the case of the language LCL , while writing formulas featuring these connectives, they have a higher binding power than ⇒, i.e. we write ‘𝜑 ∧ 𝜓 ⇒ 𝜑 ∨ 𝜓’ instead of ‘(𝜑 ∧ 𝜓) ⇒ (𝜑 ∨ 𝜓)’. Recall that in Example 2.3.7 we have introduced consecutions which ensured the intended interpretation of the connectives ∧ and ∨ in the logic BCIlat. In a more general setting, however, we replace the axioms (sup) and (inf) by their corresponding rules. While in many logics these rules entail the corresponding axioms (see Corollary 4.2.11 and Proposition 4.3.12), it is not the case in general (actually, already over BCK, as we show in Example 4.1.8). Thus, we consider the following consecutions:3 (lb1 ) (lb2 ) (ub1 ) (ub2 ) (Adj) (Inf) (Sup)

𝜑∧𝜓 ⇒ 𝜑 𝜑∧𝜓 ⇒𝜓 𝜑 ⇒ 𝜑∨𝜓 𝜓 ⇒ 𝜑∨𝜓 𝜑, 𝜓 I 𝜑 ∧ 𝜓 𝜒 ⇒ 𝜑, 𝜒 ⇒ 𝜓 I 𝜒 ⇒ 𝜑 ∧ 𝜓 𝜑 ⇒ 𝜒, 𝜓 ⇒ 𝜒 I 𝜑 ∨ 𝜓 ⇒ 𝜒

lower bound lower bound upper bound upper bound adjunction infimality supremality.

As before, we use the same name (but a different symbol) for a rule and its corresponding axiomatic form, i.e. we have the rule of infimality (Inf) and the axiom of infimality (inf). Also, to simplify upcoming formulations, we use the denotation (ub) when we mean both formulas (ub1 ) and (ub2 ) or when, mainly in formal proofs, it is clear from context which one we need; we use the same convention for (lb) and numerous upcoming pairs of consecutions. One can immediately notice a certain asymmetry between the rules for ∧ and ∨. Indeed, the rule of adjunction does not seem to be necessary to enforce the expected interpretation of ∧ as the infimum and there is no counterpart of this rule for ∨. On the other hand, this rule seems to be crucial for any reasonable conjunction connective as it says that if two formulas are provable so is their conjunction. Therefore, we speak about lattice protoconjunctions and lattice conjunctions to mark this difference. The situation in the case of disjunction is analogous: recall that, in Section 2.4, we have 2 See Example 4.6.14 for a possible alternative interpretation of these connectives. 3 It is interesting to note that, in all the formulas appearing in these consecutions, the implication ⇒ only appears as the principal connective, i.e. it is never nested; cf. the rules describing the behavior of residuated connectives in the next section where it is not the case.

156

4 On lattice and residuated connectives

seen that in many logics the connective ∨ satisfies an important metalogical property known as proof by cases property (Proposition 2.4.5); but in Example 2.6.8 we have shown that in global modal logics this property fails even though the connective ∨ obviously satisfies (ub) and (Sup). As the adjunction rule seems to be crucial for a reasonable notion of conjunction, in Chapter 5 we will see that the proof by cases property is crucial for any reasonable disjunction. Definition 4.1.1 Let L be a weakly implicative logic. A (possibly definable) binary connective • ∧ is a lattice protoconjunction in L if the consecutions (lb) and (Inf) are valid. • ∨ is a lattice protodisjunction in L if the consecutions (ub) and (Sup) are valid. Furthermore, we say that a lattice • protoconjunction ∧ is a lattice conjunction in L if the rule (Adj) is valid. • protodisjunction ∨ is a lattice disjunction in L if it satisfies the proof by cases property: PCP If Γ, 𝜑 `L 𝜒 and Γ, 𝜓 `L 𝜒, then Γ, 𝜑 ∨ 𝜓 `L 𝜒. Example 4.1.2 Consider any logic L expanding BCIlat . Clearly, ∧ is a lattice conjunction and ∨ is a lattice protodisjunction. If, furthermore, L is an axiomatic expansion of BCKlat , then ∨ is a lattice disjunction (see Proposition 2.4.5). As we mentioned above, ∨ is a lattice protodisjunction in global modal logics but not a lattice disjunction. It is easy to see that if L is a Rasiowa-implicative logic, then any lattice protoconjunction is a lattice conjunction. Indeed, from the premises 𝜑 and 𝜓, we can prove: a) ( 𝜒 ⇒ 𝜒) ⇒ 𝜑 assumption 𝜑 and (W) b)

( 𝜒 ⇒ 𝜒) ⇒ 𝜓

c)

( 𝜒 ⇒ 𝜒) ⇒ 𝜑 ∧ 𝜓

assumption 𝜓 and (W) a, b, and (Inf)

d) 𝜑 ∧ 𝜓

c, (id), and (MP)

However, this fact does not hold in general as demonstrated by the next example. Example 4.1.3 Consider the matrix A with domain {⊥, 𝑎, 𝑏}, filter {𝑎, 𝑏}, and connectives ⇒ and ∧ interpreted as: ⇒A ⊥ 𝑎 𝑏

⊥ 𝑎 ⊥ ⊥

𝑎 𝑎 𝑎 ⊥

𝑏 𝑏 ⊥ 𝑏

∧A ⊥ 𝑎 𝑏

⊥ ⊥ ⊥ ⊥

𝑎 ⊥ 𝑎 ⊥

𝑏 ⊥ ⊥ 𝑏

As an exercise, the reader should prove that ≤A⇒ is a matrix order which can be schematically depicted as ⊥ < 𝑎, 𝑏 and, thus, A is a weakly implicative logic with a weak implication ⇒, and ∧ is a lattice protoconjunction but not a lattice conjunction.

4.1 Lattice connectives

157

Remark 4.1.4 One can also introduce an abstract notion of (not necessarily lattice) conjunction, defined as any connective u such that 𝜑, 𝜓 a` 𝜑 u 𝜓. It is easy to describe a logic with an (additional) conjunction which is not a lattice protoconjunction (consider e.g. the expansion of classical logic given by the 4-valued Boolean algebra B4 expanded with a connective u defined as: 𝑥 u 𝑦 = > if 𝑥 = 𝑦 = > and ⊥ otherwise). Note however that, for any two conjunctions u and u0, we have 𝜑 u 𝜓 a` 𝜑 u0 𝜓. Also note that any conjunction u and any weak implication ⇒ allow us to define in a natural way an equivalence connective: 𝜑 ⇔u 𝜓 = (𝜑 ⇒ 𝜓) u (𝜓 ⇒ 𝜑). Moreover, if u0 and ⇒0 are respectively a different conjunction and a weak implication, we have 𝜑 ⇔u 𝜓 a` {𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜑} a` 𝜑 ⇔u0 0 𝜓. Recalling our standard notation 𝜑 ⇔ 𝜓 = {𝜑 ⇒ 𝜓, 𝜓 ⇒ 𝜑}, it is easy to see why we actually have no need to use connectives of the form ⇔u . We will not study these abstract conjunctions any further as they have little relevance for the theory we are building in this book. In contrast, the abstract notion of disjunction has much more relevance and hence we dedicate the whole Chapter 5 to its study. Let us just mention now that in S4 the connective ∨0 defined as 𝜑 ∨0 𝜓 = 𝜑 ∨ 𝜓 enjoys the proof by cases property (see Proposition 2.4.6). Thus, ∨0 can reasonably be called a disjunction, although it is not a lattice disjunction (as it would entail `S4 𝜑 → 𝜑 which is not the case; see Example 2.6.8). The next proposition justifies the names of our connectives by formalizing their expected relation with the order on any reduced matrix of a weakly implicative logic. Proposition 4.1.5 Let L be a weakly implicative logic with binary connectives ∧ and ∨ in its language. Assume that L has the SKC for a class K ⊆ Mod∗ (L). Then, the properties 1–4 below are equivalent and so are the properties 10–40. 1) ∧ is a lattice protoconjunction in L .

10) ∨ is a lattice protodisjunction in L .

2) For every A = hA, 𝐹i ∈ Mod∗ (L) and every 𝑥, 𝑦 ∈ 𝐴, we have

20) For every A = hA, 𝐹i ∈ Mod∗ (L) and every 𝑥, 𝑦 ∈ 𝐴, we have

𝑥 ∧A 𝑦 = inf ≤A {𝑥, 𝑦}.

𝑥 ∨A 𝑦 = sup ≤A {𝑥, 𝑦}.

3) For every A = hA, 𝐹i ∈ K and every 𝑥, 𝑦 ∈ 𝐴, we have 𝑥 ∧A 𝑦 = inf ≤A {𝑥, 𝑦}. 4) The following consecutions are valid in L: (i∧ ) (c∧ ) (a∧,1 ) (a∧,2 ) (Df1∧ ) (Df2∧ )

I 𝜑∧𝜑⇒𝜑 I 𝜑∧𝜓 ⇒𝜓∧𝜑 I 𝜑 ∧ (𝜓 ∧ 𝜒) ⇒ (𝜑 ∧ 𝜓) ∧ 𝜒 I (𝜑 ∧ 𝜓) ∧ 𝜒 ⇒ 𝜑 ∧ (𝜓 ∧ 𝜒) 𝜑 ⇒𝜓 I 𝜑 ⇒ 𝜑∧𝜓 𝜑 ⇒ 𝜑 ∧ 𝜓 I 𝜑 ⇒ 𝜓.

30) For every A = hA, 𝐹i ∈ K and every 𝑥, 𝑦 ∈ 𝐴, we have 𝑥 ∨A 𝑦 = sup ≤A {𝑥, 𝑦}. 40) The following consecutions are valid in L: (i∨ ) (c∨ ) (a∨,1 ) (a∨,2 ) (Df1∨ ) (Df2∨ )

I 𝜑∨𝜑⇒𝜑 I 𝜑∨𝜓 ⇒𝜓∨𝜑 I 𝜑 ∨ (𝜓 ∨ 𝜒) ⇒ (𝜑 ∨ 𝜓) ∨ 𝜒 I (𝜑 ∨ 𝜓) ∨ 𝜒 ⇒ 𝜑 ∨ (𝜓 ∨ 𝜒) 𝜑 ⇒𝜓 I 𝜑∨𝜓 ⇒𝜓 𝜑 ∨ 𝜓 ⇒ 𝜓 I 𝜑 ⇒ 𝜓.

158

4 On lattice and residuated connectives

Proof We prove the equivalence of 1–4; the second quadruplet is analogous. The implications ‘1 implies 2’, ‘2 implies 3’, and ‘3 implies 4’ are easy to prove and left as an exercise for the reader. To prove the remaining implication ‘4 implies 1’, we need to show the validity of the consecutions (lb) and (Inf). First, observe that from (Df1∧ ) and (id) together with (i∧ ) we obtain that `L 𝜓 ⇔ 𝜓 ∧ 𝜓 and so, using the substitution property of ⇔, we obtain `L 𝜑 ∧ 𝜓 ⇔ 𝜑 ∧ (𝜓 ∧ 𝜓). Thus, by (a∧ ) and transitivity, we obtain `L 𝜑 ∧ 𝜓 ⇒ (𝜑 ∧ 𝜓) ∧ 𝜓 which, using the (Df ∧ ), entails `L 𝜑 ∧ 𝜓 ⇒ 𝜓, i.e. (lb2 ) is valid and so is (lb1 ) due to (c∧ ). To prove (Inf), we start by applying the rule (Df ∧ ) to the premises 𝜒 ⇒ 𝜑 and 𝜒 ⇒ 𝜓 to obtain `L 𝜒 ⇒ 𝜒 ∧ 𝜑 and `L 𝜒 ⇒ 𝜒 ∧ 𝜓. From the former fact and the theorem (lb), we obtain `L 𝜒 ⇔ 𝜒 ∧ 𝜑 and so `L 𝜒 ∧ 𝜓 ⇔ ( 𝜒 ∧ 𝜑) ∧ 𝜓 (using the substitution property of ⇔) which, using the latter fact and the transitivity, entails that `L 𝜒 ⇒ ( 𝜒 ∧ 𝜑) ∧ 𝜓. Using the theorems (a∧ ) and transitivity, we obtain 𝜒 ⇒ 𝜒 ∧ (𝜑 ∧ 𝜓), and so the rule (Df ∧ ) completes the proof.  This proposition has several interesting consequences. The first one justifies speaking about the lattice protodisjunction / protoconjunction of a given logic. Indeed, according to the following proposition, these connectives are intrinsic. The proof is simple and left as an exercise for the reader (an interested reader can also find a direct syntactical proof). Note that this proposition implies that, if a logic has a lattice protodisjunction which is not a disjunction, then no lattice disjunction is definable in that logic (and analogously for conjunctions). Thus, in particular, we know that no lattice disjunction is definable in the global modal logic S4. Proposition 4.1.6 Let L be a weakly implicative logic and ◦, ◦0 be two lattice protodisjunctions or two lattice protoconjunctions in L. Then, `L 𝜑 ◦ 𝜓 ⇔ 𝜑 ◦0 𝜓. The second consequence uses the consecutions (Df ∧ ) and (Df ∨ ), together with Corollary 2.8.8, to establish the interderivability of two different weak implications of a given logic in the presence of certain lattice connectives. Corollary 4.1.7 Let L be a logic with weak implications ⇒ and ⇒0 and a connective which is a lattice protoconjunction (or a lattice protodisjunction) for both ⇒ and ⇒0 in L. Then, 𝜑 ⇒ 𝜓 a`L 𝜑 ⇒0 𝜓. We can also use Proposition 4.1.5 to provide the promised example of a logic with a lattice (proto)conjunction not satisfying the axiomatic form of the rule (Inf). Example 4.1.8 Consider a matrix A with domain {⊥, 𝑎, 𝑏, >}, filter 𝐹 = {>}, and connectives → and ∧ interpreted as: →A ⊥ 𝑎 𝑏 >

⊥ > ⊥ ⊥ ⊥

𝑎 > > 𝑎 𝑎

𝑏 > > > 𝑏 > > > 𝑏 >

∧A ⊥ 𝑎 𝑏 >

⊥ ⊥ ⊥ ⊥ ⊥

𝑎 ⊥ 𝑎 ⊥ 𝑎

𝑏 > ⊥ ⊥ ⊥ 𝑎 𝑏 𝑏 𝑏 >

4.1 Lattice connectives

159

Observe that the binary relation defined by → is an order which can be schematically described as ⊥ < 𝑎, 𝑏 < >. Therefore, by Proposition 2.8.11, A is a weakly implicative logic; we leave as an exercise for the reader to show that it is an expansion of BCK. Also, for each 𝑥, 𝑦 ∈ 𝐴, 𝑥 ∧ 𝑦 = inf ≤A {𝑥, 𝑦}. Due to Proposition 4.1.5, ∧ is a protoconjunction (actually, it is a conjunction). However, (𝑎 → 𝑎) ∧ (𝑎 → 𝑏) = > ∧ 𝑏 = 𝑏 whereas 𝑎 → 𝑎 ∧ 𝑏 = 𝑎 → ⊥ = ⊥. This example also shows that BCKlat is not the weakest logic in LCL with (proto)conjunction expanding BCK. Let us now look at the role of the lattice connectives when checking that a logic is algebraically implicative. Recall that in Example 2.9.9 we have proved that the logic BCIlat (and so all of its weakly implicative expansions) is algebraically implicative with defining equations E = {(𝑥 ⇒ 𝑥) ∧ 𝑥 ≈ 𝑥 ⇒ 𝑥}. The next proposition identifies the essential ingredients of that proof: the presence of a lattice protoconjunction or protodisjunction and the existence of a formula 𝜒 with no other variable than 𝑝 such that `L 𝜒 and 𝑝 `L 𝜒 ⇒ 𝑝. In the mentioned example, the formula 𝑝 → 𝑝 has the required properties (in BCI); in Proposition 4.3.12, we will see that the constant 1¯ called the (proto)unit of an implication is an even simpler option; and in Proposition 4.2.13 we will see that, in certain logics weaker than BCI, we can use the formula 𝑝 𝑝, where is a connective called dual co-implication. Proposition 4.1.9 Let L be a weakly implicative logic such that the following two conditions are met: • There is a formula 𝜒 with no other variable than 𝑝 such that `L 𝜒 and 𝑝 `L 𝜒 ⇒ 𝑝. • L has a lattice protoconjunction or protodisjunction. Then, L is an algebraically implicative logic. Proof We show that, depending on the second assumption, 𝑝 ∧ 𝜒 ≈ 𝜒 or 𝑝 ∨ 𝜒 ≈ 𝑝 is a defining equation. In the first case, we need to show that 𝑝 a`L 𝑝 ∧ 𝜒 ⇔ 𝜒. One direction is easy: due to (lb) we have 𝑝 `L 𝑝 ∧ 𝜒 ⇒ 𝜒 and also 𝑝 `L 𝜒 ⇒ 𝑝 ∧ 𝜒 (due to 𝑝 `L 𝜒 ⇒ 𝑝, `L 𝜒 ⇒ 𝜒, and (Inf)). The second direction: clearly, 𝑝 ∧ 𝜒 ⇔ 𝜒 `L 𝑝 ∧ 𝜒 (due to `L 𝜒) and so 𝑝 ∧ 𝜒 ⇔ 𝜒 `L 𝑝 (due to (lb)). In the second case, we need to show that 𝑝 a`L 𝑝 ∨ 𝜒 ⇔ 𝑝. One direction is easy: due to (ub), we have 𝑝 `L 𝑝 ⇒ 𝑝 ∨ 𝜒 and also 𝑝 `L 𝑝 ∨ 𝜒 ⇒ 𝑝 (due to 𝑝 `L 𝜒 ⇒ 𝑝, `L 𝑝 ⇒ 𝑝, and (Sup)). The second direction: clearly, 𝑝 ∨ 𝜒 ⇔ 𝑝 `L 𝜒 ⇒ 𝑝 (due to (ub) and transitivity) and so 𝑝 ∨ 𝜒 ⇔ 𝑝 `L 𝑝 (due to `L 𝜒).  It is easy to see that a lattice protoconjunction ∧ in a logic L is a lattice conjunction iff for every hA, 𝐹i ∈ Mod∗ (L) and every 𝑥, 𝑦 ∈ 𝐹, we have 𝑥 ∧A 𝑦 ∈ 𝐹. From Theorem 5.4.7, we will obtain, for RFSI-complete logics, a more complex analogous result for (proto)disjunctions: a lattice protodisjunction ∨ is a lattice disjunction iff for each hA, 𝐹i ∈ Mod∗ (L)RFSI and each 𝑥 ∨A 𝑦 ∈ 𝐹, we have 𝑥 ∈ 𝐹 or 𝑦 ∈ 𝐹 (cf. the notion of prime theory in expansions of BCKlat or prime filter in Boolean algebras; later in Chapter 5 we elaborate this connection in details).

160

4 On lattice and residuated connectives

4.2 Residuated connectives In this section, we focus on connectives of a different nature. As explained in the introduction, our goal is to consider two connectives which will allow us to work with nested implications of the form 𝜑 ⇒ (𝜓 ⇒ 𝜒): the residuated conjunction & (also known, in certain contexts, as fusion or multiplicative/strong conjunction) whose role can be described as ‘aggregation of premises’ and co-implication (also known, in certain contexts, as left/right residuum) which allows for switching the order of premises at the price of replacing the internal implication by . Interestingly enough, although a co-implication quite often is a weak implication that even defines the same order as the underlying implication (see Proposition 4.2.4 and Corollary 4.3.11), in general it need not be a weak implication (see Example 4.2.17). Therefore, as in the case of lattice connectives, we distinguish two kinds of co-implication. However, we do not use the prefix proto to distinguish the weaker one but rather the adjective dual to single out the stronger notion. The expected behavior of these connectives is therefore described by the following consecutions:4 (Res1 ) (Res2 ) (E ,1 ) (E ,2 ) (Symm1 ) (Symm2 )

𝜑 ⇒ (𝜓 ⇒ 𝜒) I 𝜓 & 𝜑 ⇒ 𝜒 𝜓 & 𝜑 ⇒ 𝜒 I 𝜑 ⇒ (𝜓 ⇒ 𝜒) 𝜑 ⇒ (𝜓 ⇒ 𝜒) I 𝜓 ⇒ (𝜑 𝜒) 𝜑 ⇒ (𝜓 𝜒) I 𝜓 ⇒ (𝜑 ⇒ 𝜒) 𝜑⇒𝜓I𝜑 𝜓 𝜑 𝜓I𝜑⇒𝜓

residuation residuation -exchange -exchange symmetry symmetry.

Definition 4.2.1 Let L be a weakly implicative logic. A (possibly definable) binary connective • & is a residuated conjunction in L if the consecutions (Res) are valid. • is a co-implication in L if the consecutions (E ) are valid. Furthermore, we say that a co-implication (Symm) are valid.

is dual if additionally the consecutions

It is easy to see that the residuated conjunction is indeed a conjunction in the sense of Remark 4.1.4, but it is a lattice (proto)conjunction only in very strong logics (see Corollary 4.6.9). Let us observe that, as in the case of lattice connectives, we can easily prove that residuated connectives are intrinsic and can be easily semantically characterized (cf. Propositions 4.1.6 and 4.1.5); we leave the necessary proofs as an exercise for the reader. Note that, analogously to the case of lattice connectives, the first proposition implies that no dual co-implication is definable in a logic with a non-dual co-implication. Proposition 4.2.2 Let L be a weakly implicative logic and ◦, ◦0 be two residuated conjunctions or two co-implications in L. Then, `L 𝜑 ◦ 𝜓 ⇔ 𝜑 ◦0 𝜓. 4 As before, we use (X) to denote either the pair of consecutions (X1 ) and (X2 ) or any one of them, when it is clear from the context which one we need.

4.2 Residuated connectives

161

Proposition 4.2.3 Let L be a weakly implicative logic with the binary connectives & and in its language. Assume that L has the SKC for a class K ⊆ Mod∗ (L). Then, the properties 1–3 below are equivalent and so are the properties 10–30. 1) & is a residuated conjunction in L. 10) is a co-implication in L. 2) For every A ∈ Mod∗ (L) and every 𝑥, 𝑦 ∈ 𝐴 we have 𝑥 ≤A 𝑦 ⇒A 𝑧

iff

20) For every A ∈ Mod∗ (L) and every 𝑥, 𝑦 ∈ 𝐴 we have

𝑦 &A 𝑥 ≤A 𝑧.

𝑥 ≤A 𝑦 ⇒A 𝑧

iff

𝑦 ≤A 𝑥

A

𝑧.

A

𝑧.

30) For every A ∈ K and every 𝑥, 𝑦 ∈ 𝐴 we have

3) For every A ∈ K and every 𝑥, 𝑦 ∈ 𝐴 we have 𝑥 ≤A 𝑦 ⇒A 𝑧

iff

𝑦 &A 𝑥 ≤A 𝑧.

𝑥 ≤A 𝑦 ⇒A 𝑧

iff

𝑦 ≤A 𝑥

∗

Furthermore, is a dual co-implication iff for every A ∈ Mod (L) (or just A ∈ K) we have ≤A⇒ = ≤A and for every 𝑥, 𝑦 ∈ 𝐴 we have 𝑥 ≤ A 𝑦 ⇒A 𝑧

iff

𝑦 ≤A 𝑥

A

𝑧.

It is obvious that any dual co-implication of ⇒ is a weak implication and ⇒ is its dual implication, i.e. we can prove that 𝜑

(𝜓

𝜒) a`L 𝜓

(𝜑 ⇒ 𝜒).

The next proposition solves the problem of determining whether ⇒ and are actually the same, i.e. when a weak implication can be seen as its own (dual) coimplication. It is easy to show that it happens if and only if the residuated conjunction is commutative, which can be equivalently expressed with the rule (E) (which we have already encountered in Chapter 2). We leave the proof as an exercise for the reader (cf. Proposition 4.6.7 for a formal proof of a related statement). Proposition 4.2.4 Let L be a weakly implicative logic. Then, the following are equivalent: 1. 2. 3. 4.

⇒ is the dual co-implication. ⇒ is the co-implication. 𝜑 ⇒ (𝜓 ⇒ 𝜒) `L 𝜓 ⇒ (𝜑 ⇒ 𝜒). L has a co-implication and `L (𝜑 ⇒ 𝜓) ⇔ (𝜑

𝜓).

If, moreover, L has a residuated conjunction &, we can add the following condition: 4. `L 𝜑 & 𝜓 ⇒ 𝜓 & 𝜑. Example 4.2.5 In certain logics, a weakly implication is its own co-implication (obviously a dual one). Examples of such logics are all weakly implicative expansions of BCI. However, in many prominent logics that we will meet later in this chapter these connectives differ; e.g. the logics LL, SL and SLa have a weak implication → and a distinct co-implication (which is dual in the latter two cases only; see Examples 4.4.6 and 4.6.20).

162

4 On lattice and residuated connectives

Furthermore, the lattice conjunction ∧ is a residuated conjunction in any expansion of intuitionistic logic and the connective & defined in Proposition A.5.9 is a residuated conjunction in any expansion of Łukasiewicz logic. On the other hand, ∧ is not a residuated conjunction in Łukasiewicz logic (just consider the matrix Ł∞ ∈ Mod∗ (Ł) introduced in Example 2.6.4). Finally, in Example 4.6.11 we will show that the residuated conjunction is definable in the logic A→ . Remark 4.2.6 Let us fix a weakly implicative logic L with a weak implication ⇒. Observe that & is a residuated conjunction of ⇒ iff the connective &0 defined as 𝜑 &𝑡 𝜓 = 𝜓 & 𝜑 satisfies the following transposed version of (Res): 𝜑 ⇒ (𝜓 ⇒ 𝜒) a`L 𝜑 &𝑡 𝜓 ⇒ 𝜒. Let us call this connective dual residuated conjunction (only for the purposes of this remark and additional considerations in the history section). Therefore, the order of arguments in the rules (Res) is purely conventional: a logic has a residuated conjunction iff it has a dual residuated conjunction. In the presence of the dual co-implication, the above symmetry is even more pronounced (cf. Definition 4.6.15 and Theorem 4.6.16), but the residuated conjunction can help us distinguish these two otherwise (deductively) indistinguishable connectives. Indeed, the following four claims are clearly equivalent: & is a residuated conjunction of ⇒, i.e. 𝜑 ⇒ (𝜓 ⇒ 𝜒) a`L 𝜓 & 𝜑 ⇒ 𝜒 𝑡

& is a dual residuated conjunction of ⇒, i.e. 𝜑 ⇒ (𝜓 ⇒ 𝜒) a`L 𝜑 &𝑡 𝜓 ⇒ 𝜒 &𝑡 is a residuated conjunction of

, i.e. 𝜑

(𝜓

𝜒) a`L 𝜓 &𝑡 𝜑

𝜒

& is a dual residuated conjunction of

, i.e. 𝜑

(𝜓

𝜒) a`L 𝜑 & 𝜓

𝜒.

Thus, if we take the residuated conjunction as the principal connective, we can disambiguate the situation by calling the weak implication satisfying the rule (Res) the left residuum of & and the one satisfying its transposed version the right residuum of & (see Section 4.11 for the semantical origins of this terminology). Thus, in our setting, ⇒ is the left residuum and its dual co-implication is the right residuum. Although one may study the residuated connectives separately, for simplicity of formulation of definitions and results we will usually assume that we have both of them in a given logic and we leave to the reader, as an exercise, the task of finding minimal sufficient conditions for the validity of particular properties. Let us start by providing an abstract formulation of the residuation property and an obvious reformulation of the previous proposition. Definition 4.2.7 We say that hh𝐴, &, ⇒, i, ≤i is a residuated ordered groupoid if h𝐴, &, ⇒, i is an algebra with three binary operations, h𝐴, ≤i is an ordered set and the residuation property is satisfied, i.e. for each 𝑥, 𝑦, 𝑧 ∈ 𝐴, 𝑥≤𝑦⇒𝑧

iff

𝑦&𝑥 ≤ 𝑧

iff

𝑦≤𝑥

𝑧.

4.2 Residuated connectives

163

Proposition 4.2.8 Let L be a weakly implicative logic with the binary connectives & and in its language. Assume that L has the SKC for a class K ⊆ Mod∗ (L). Then, the following are equivalent: 1. & is a residuated conjunction and is a co-implication. 2. For every A ∈ Mod∗ (L), the structure hh𝐴, &A , ⇒A , A i, ≤A i is a residuated ordered groupoid. 3. For every A ∈ K, the structure hh𝐴, &A , ⇒A , A i, ≤A i is a residuated ordered groupoid. Thanks to this proposition, we know that proving general properties of residuated ordered groupoids will tell us a lot about logics with residuated conjunction and coimplication; e.g. the next proposition tells us that &A is monotonic in both arguments w.r.t. ≤A , and so on. Proposition 4.2.9 Let hh𝐴, &, ⇒,

i, ≤i be a residuated ordered groupoid. Then:

1. & is monotonic in both arguments w.r.t. ≤. 2. ⇒ and are antitone w.r.t. ≤ in the first argument and monotonic in the second one. 3. For every 𝑥, 𝑦, 𝑧 ∈ 𝐴, we have: 𝑥 ⇒ 𝑦 = max{𝑧 | 𝑥 & 𝑧 ≤ 𝑦} = max{𝑧 | 𝑥 ≤ 𝑧 𝑦} 𝑥 𝑦 = max{𝑧 | 𝑧 & 𝑥 ≤ 𝑦} = max{𝑧 | 𝑥 ≤ 𝑧 ⇒ 𝑦} 𝑥 & 𝑦 = min{𝑧 | 𝑦 ≤ 𝑥 ⇒ 𝑧} = min{𝑧 | 𝑥 ≤ 𝑦 𝑧}. 4. For every 𝑧 ∈ 𝐴 and 𝑋, 𝑌 ⊆ 𝐴 such that the supremum of 𝑋 and the infimum of 𝑌 exist, we have Ü Ü Ü Ü (𝑥 & 𝑧) = 𝑋&𝑧 (𝑧 & 𝑥) = 𝑧 & 𝑋 𝑥 ∈𝑋

Û

𝑥 ∈𝑋

(𝑥 ⇒ 𝑧) =

Ü

𝑋⇒𝑧

𝑥 ∈𝑋

Û 𝑥 ∈𝑋

Û

(𝑧 ⇒ 𝑦) = 𝑧 ⇒

Û

𝑌

(𝑧

Û

𝑌.

𝑦 ∈𝑌

(𝑥

𝑧) =

Ü

𝑋

𝑧

Û

𝑦) = 𝑧

𝑦 ∈𝑌

Proof Claims 1 and 2: We prove the antitonicity of in the first argument; the other proofs are analogous and left as an exercise for the reader. Let us assume that 𝑦 0 ≤ 𝑦 and observe that, from 𝑦 𝑧≤𝑦 𝑧, we obtain 𝑦 ≤ (𝑦 𝑧) ⇒ 𝑧 (due to the residuation property) and so 𝑦 0 ≤ (𝑦 𝑧) ⇒ 𝑧 and using residuation again completes the proof. Claim 3: Here we only prove the case for ⇒ and leave the rest as an exercise for the reader. Note that the second equality is a direct consequence of the residuation property; to prove the first one, let us denote the set {𝑧 | 𝑥 & 𝑧 ≤ 𝑦} as 𝑋. First, we observe that 𝑥 ⇒ 𝑦 ∈ 𝑋: indeed, 𝑥 & (𝑥 ⇒ 𝑦) ≤ 𝑦 follows immediately from the residuation property. Second, we use the residuation property again to obtain that, if 𝑧 ∈ 𝑋, then 𝑧 ≤ 𝑥 ⇒ 𝑦.

164

4 On lattice and residuated connectives

Claim 4: The proofs of these equations are analogous; we show the proof of two of them to demonstrate the need for the presence of additional operations and leave the proof of the remaining ones as an exercise for the reader. The first one we prove Ô Ô Ô is 𝑥 ∈𝑋 (𝑧 & 𝑥) = 𝑧 & 𝑋. Note that, for each 𝑥 ∈ 𝑋, we have 𝑥 ≤ 𝑋 and so, Ô Ô by the monotonicity of &, also 𝑧 & 𝑥 ≤ 𝑧 & 𝑋, i.e. 𝑧 & 𝑋 is an upper bound of {𝑧 & 𝑥 | 𝑥 ∈ 𝑋 }. Consider an arbitrary upper bound 𝑡 of this Ô set, i.e. 𝑧 & 𝑥 ≤ 𝑡 for each 𝑥 ∈ 𝑋. Then, by residuation, also 𝑥 ≤ 𝑧 ⇒ 𝑡 and so 𝑋 ≤ 𝑧 ⇒ 𝑡 and, by Ô residuation once more, also 𝑧 & 𝑋 ≤ 𝑡.Ó Ô As the second example we show that 𝑥 ∈𝑋 (𝑥 ⇒ 𝑧) = ( 𝑋 ⇒ 𝑧). We start by observing that, due to antitonicity of ⇒ in the first argument, we obtain for each 𝑥 ∈ 𝑋 Ô Ô that ( 𝑋 ⇒ 𝑧) ≤ (𝑥 ⇒ 𝑧), i.e, ( 𝑋 ⇒ 𝑧) is a lower bound of {𝑥 ⇒ 𝑧 | 𝑥 ∈ 𝑋 }. Consider an arbitrary lower bound Ô 𝑡, i.e. 𝑡 ≤ 𝑥 ⇒ 𝑧 for each 𝑥 ∈Ô 𝑋. Then also 𝑥≤𝑡 𝑧, for each 𝑥 ∈ 𝑋, and so 𝑋 ≤ 𝑡 𝑧 which implies 𝑡 ≤ 𝑋 ⇒ 𝑧.  The next two corollaries of the previous two propositions show that the presence of residuated connectives has an influence on the underlying weak implication and on the lattice connectives (if present).5 The subsequent proposition then shows that so does the fact whether the co-implication is dual or not. These facts will play a crucial role in Section 4.5, especially in our definition of substructural logics (cf. Definition 4.6.1). Corollary 4.2.10 Let L be a logic with a residuated conjunction & and a coimplication . Then, the following consecutions are valid: (Sf) (Pf) (Sf ) (Pf ) (adj& ) (Mon& 1) (Mon& 2)

𝜑 ⇒ 𝜓 I (𝜓 ⇒ 𝜒) ⇒ (𝜑 ⇒ 𝜒) 𝜑 ⇒ 𝜓 I ( 𝜒 ⇒ 𝜑) ⇒ ( 𝜒 ⇒ 𝜓) 𝜑 ⇒ 𝜓 I (𝜓 𝜒) ⇒ (𝜑 𝜒) 𝜑 ⇒ 𝜓 I (𝜒 𝜑) ⇒ ( 𝜒 𝜓) 𝜓 ⇒ (𝜑 ⇒ 𝜑 & 𝜓) 𝜑 ⇒𝜓 I 𝜒&𝜑 ⇒ 𝜒&𝜓 𝜑⇒𝜓 I 𝜑&𝜒⇒𝜓&𝜒

suffixing prefixing -suffixing -prefixing &-adjunction &-monotonicity &-monotonicity.

Corollary 4.2.11 Let L be a logic with a residuated conjunction &, co-implication , lattice protoconjunction ∧, and lattice protodisjunction ∨. Then, the following consecutions are valid: (Mon∧1 ) (Mon∧2 ) (Mon∨1 ) (Mon∨1 ) (inf) (sup)

𝜑⇒𝜓 I 𝜑∧𝜒⇒𝜓∧𝜒 𝜑 ⇒𝜓 I 𝜑∧𝜑 ⇒ 𝜒∧𝜓 𝜑⇒𝜓 I 𝜑∨𝜒⇒𝜓∨𝜒 𝜑⇒𝜓 I 𝜑∨𝜒⇒𝜓∨𝜒 ( 𝜒 ⇒ 𝜑) ∧ ( 𝜒 ⇒ 𝜓) ⇒ ( 𝜒 ⇒ 𝜑 ∧ 𝜓) (𝜑 ⇒ 𝜒) ∧ (𝜓 ⇒ 𝜒) ⇒ (𝜑 ∨ 𝜓 ⇒ 𝜒)

∧-monotonicity ∧-monotonicity ∨-monotonicity ∨-monotonicity infimality supremality.

5 Again, for simplicity, we assume the presence of both residuated (and in the second case even lattice) connectives at once and leave as an exercise for the reader the task of finding minimal sufficient conditions for the validity of each particular claim.

4.2 Residuated connectives

165

Proposition 4.2.12 Let L be a logic with a co-implication

.

• The rule (Symm1 ) is valid in L iff so is the formula (id )

𝜑

𝜑

-identity.

• The rule (Symm2 ) is valid in L iff so is the rule (As)

𝜑 I (𝜑 ⇒ 𝜓) ⇒ 𝜓

assertion.

Proof The first claim: the validity of (id ) is a direct consequence of (Symm1 ) and (id) and the converse claim follows using (MP) on the following instance of (Pf ): 𝜑 ⇒ 𝜓 `L (𝜑 𝜑) ⇒ (𝜑 𝜓). The second claim: for one direction apply (E ) on (𝜑 ⇒ 𝜓) ⇒ (𝜑 ⇒ 𝜓) to obtain 𝜑 ⇒ ((𝜑 ⇒ 𝜓) 𝜓) and so the claim follows by (MP) and (Symm2 ). The converse direction easily follows from the following instance of (As): 𝜑 𝜓 I ((𝜑 𝜓) ⇒ (𝜑 𝜓)) ⇒ (𝜑 𝜓).  Proposition 4.2.13 Let L be a weakly implicative logic with a dual co-implication and lattice protoconjunction or protodisjunction. Then, L is algebraically implicative. Proof We prove the claim by using Proposition 4.1.9: clearly, it suffices to show that that there is a formula 𝜒 with no other variable than 𝑝 such that 𝑝 `L 𝜒 ⇒ 𝑝 and `L 𝜒. Clearly, due to the validity of the consecution (As), we can set 𝜒 = 𝑝 ⇒ 𝑝. 

Example 4.2.14 We can use Corollary 4.2.11 to show that no residuated conjunction of → is definable in the logic BCI. Consider the least logic expanding BCI with a lattice protoconjunction and denote it as BCI∧ . Due to Example 4.1.8, we know that the formula (inf) is not provable in BCI∧ , but as → is its own dual co-implication, no residuated conjunction of → could be definable in BCI∧ and thus consequently neither in BCI. Even more, it implies that BCI∧ cannot be a fragment of any weakly implicative logic with a weak implication → which would have a residuated conjunction. Interestingly enough, BCI is a fragment of such logic (see Theorem 4.7.1). We conclude this section by presenting a lemma which provides a useful tool for defining residuated ordered groupoids. It tells us, roughly speaking, that any completely ordered groupoid where the groupoid operation distributes over arbitrary joins in both arguments is residuated. Lemma 4.2.15 Let ≤ be a complete lattice order on 𝐴. Then, hh𝐴, &, ⇒, i, ≤i is a residuated ordered groupoid if and only if the following conditions are met: • For each 𝑧 ∈ 𝐴 and 𝑋 ⊆ 𝐴, we have Ü Ü (𝑧 & 𝑥) = 𝑧 & 𝑋 𝑥 ∈𝑋

Ü 𝑥 ∈𝑋

(𝑥 & 𝑧) =

Ü

𝑋 & 𝑧.

166

4 On lattice and residuated connectives

• For each 𝑥, 𝑦 ∈ 𝐴, we have 𝑥 ⇒ 𝑦 = max{𝑧 | 𝑥 & 𝑧 ≤ 𝑦} 𝑥 𝑦 = max{𝑧 | 𝑧 & 𝑥 ≤ 𝑦}. Proof One implication follows directly from Proposition 4.2.9. To prove the converse one, we have to prove the residuation property. Let us show the first equivalence; the proof of the other one is analogous and left as an exercise for the reader. Let us denote the set {𝑧 | 𝑥 & 𝑧 ≤ 𝑦} as 𝑋 and notice that, using the distributivity of & Ô over , we can easily prove that & is monotonic Ô in both arguments and so 𝑋 is a ≤-downset. Furthermore, we can also get that 𝑋 ∈ 𝑋: Ü Ü 𝑥& 𝑋= {𝑥 & 𝑧 | 𝑥 & 𝑧 ≤ 𝑦} ≤ 𝑦. Ô Therefore, 𝑥 ⇒ 𝑦 = 𝑋 and so 𝑋 = {𝑧 | 𝑧 ≤ 𝑥 ⇒ 𝑦}. Thus, we have indeed proved that 𝑥 & 𝑧 ≤ 𝑦 iff 𝑧 ≤ 𝑥 ⇒ 𝑦.  Remark 4.2.16 Recall that in Proposition 4.2.9 we proved additional distributivity equations and the fact that ⇒ fully determines & and (resp. determines & and ⇒). Therefore, it would be easy to formulate and prove two additional versions of the above lemma. Indeed, we could replace the second condition by the distributivity of ⇒ (or respectively) over joins in the second argument and meets in the first one and the third condition by analogous statements defining & and (resp. ⇒). The details are left to the reader as an exercise. The next example shows two logics with residuated conjunction and non-dual co-implication (thus, no dual co-implication can be defined in those logics); actually we show even a bit more: the rules (Symm1 ) and (Symm2 ) are independent. Example 4.2.17 Consider the set 𝐴 = {⊥, 𝑎, >} and define the algebras A𝑖 = h𝐴, &𝑖 , ⇒𝑖 , 𝑖 i, for 𝑖 = 1 or 2, by setting: &1 ⊥ 𝑎 >

⊥ ⊥ ⊥ ⊥

𝑎 ⊥ 𝑎 >

> ⊥ 𝑎 >

⇒1 ⊥ 𝑎 >

⊥ > ⊥ ⊥

𝑎 > > ⊥

> > > >

&2 ⊥ 𝑎 >

⊥ ⊥ ⊥ ⊥

𝑎 ⊥ ⊥ ⊥

> ⊥ 𝑎 >

⇒2 ⊥ 𝑎 >

⊥ > 𝑎 𝑎

𝑎 > > > > > 𝑎 >

1

⊥ 𝑎 > 2

⊥ 𝑎 >

⊥ > ⊥ ⊥

𝑎 > 𝑎 𝑎

> > > >

⊥ > > ⊥

𝑎 > > 𝑎

> > > >

It is easy to see that, by considering the usual linear order ≤ schematically depicted as ⊥ < 𝑎 < >, both the structures hA1 , ≤i and hA2 , ≤i are residuated ordered groupoids (either by direct proof or using the previous lemma). It is also easy to see that, for the matrices A𝑖 = hA𝑖 , {>}i, we obtain ≤ = ≤A⇒𝑖 𝑖 . Therefore, the matrices A𝑖 are reduced and, by Proposition 2.8.11, A𝑖 is a logic with weak implication ⇒𝑖 and, due to Proposition 4.2.8, &𝑖 is a residuated conjunction and 𝑖 a co-implication.

4.3 Prominent truth-constants

167

But, clearly, neither 1 nor 2 are dual (they are not even weak implications because in one case we do not have 𝑎 ≤A11 𝑎 and in the second case we have both ⊥ ≤A22 𝑎 and 𝑎 ≤A22 ⊥) and, furthermore, we know that • the theorem (id ) and the rule (Symm1 ) are not valid in A1 but the rules (Symm2 ) and (As) are. • the rules (Symm2 ) and (As) are not valid in A2 but the theorem (id ) and the rule (Symm1 ) are. In the subsequent sections we will also make use of the matrix B = hA1 , {𝑎, >}i. Note that we still have ≤ = ≤B⇒1 and thus B is reduced, ⇒1 is a weak implication in B and &1 is its residuated conjunction and 1 its co-implication.

4.3 Prominent truth-constants In this section, we focus on truth-constants and, as before, we distinguish between lattice (order) and residuated truth-constants. We start with two lattice truth-constants: top > and bottom ⊥. Their intended interpretation, given a weakly implicative logic L and a matrix A ∈ Mod∗ (L), are the largest/least elements with respect to the matrix order ≤A (of course, modulo the chosen weak implication). Notice that, if L has lattice protoconjunction/disjunction ∧/∨, this behavior can be equivalently characterized by requiring that the interpretation of >/⊥ is the unit of ∧A /∨A (i.e. for each 𝑎 ∈ 𝐴 we have >A ∧A 𝑎 = 𝑎 A ∧A >A = 𝑎, and analogously for ⊥ and ∨.) ¯ in a logic Next, we consider two residuated truth-constants. The first one, unit 1, with a residuated conjunction & is intended to be interpreted as the unit of &A . Interestingly enough, this behavior can be described by using only the implication (see Proposition 4.3.8) and has the following profound consequence: in a given reduced model hA, 𝐹i of a logic with unit, 1¯ A is the least element of 𝐹. The existence of a truth-constant with this property is interesting on its own (e.g. in Section 4.4 it allows us to describe the semantics of prominent substructural logics and in Chapter 7 it ensures the validity of the rule of ∀-generalization in predicate logics). We call this truth-constant the protounit and show that 1¯ A is always the right unit of &A (Proposition 4.3.6) but not necessarily the left unit (see Example 4.3.3).6 Furthermore, in Example 4.3.7 we will see an example of a logic with a constant that denotes the right unit of its residuated conjunction but is not a protounit. ¯ only has the role of defining Finally, the second residuated truth-constant, 0, ¯ and also ∼𝜑 = 𝜑 negation connectives (as ¬𝜑 = 𝜑 ⇒ 0, 0¯ in logics with a co-implication different from ⇒). However, although negation is certainly a crucial logical notion, it plays almost no role in the theory built in this book. There 6 Note that a protounit is a right but not necessarily left unit because of our formulation of the residuation rules. If we had decided to formulate them in the transposed way (cf. Remark 4.2.6), it would be the other way around.

168

4 On lattice and residuated connectives

¯ in many are no generally agreed ‘governing’ rules that determine the behavior of 0: logics it is identified with the bottom ⊥ (see Remark 4.6.19), in many other logics it can be any non-designated element different from bottom, and there are even certain logics (e.g. the logics A→ and A, see Examples 4.6.11 and 4.6.14) in which it is identified with the unit (i.e. a designated element). The expected behavior of the truth-constants outlined above is determined by the following consecutions: (⊥) (>) (Push) (Pop) (push) (pop)

⊥⇒𝜑 𝜑⇒> 𝜑 I 1¯ ⇒ 𝜑 1¯ ⇒ 𝜑 I 𝜑 𝜑 ⇒ ( 1¯ ⇒ 𝜑) ( 1¯ ⇒ 𝜑) ⇒ 𝜑

ex falso quodlibet verum ex quolibet push pop push pop.

Definition 4.3.1 Let L be a weakly implicative logic. A (possibly definable) truthconstant • • • •

⊥ is the bottom in L if the consecution (⊥) is valid. > is the top in L if the consecution (>) is valid. 1¯ is the protounit in L if the consecutions (Push) and (Pop) are valid. 1¯ is the unit in L if the consecutions (push) and (pop) are valid.

Before we give the first examples of the truth-constants we have just defined, let us recall that in Corollary 2.8.17 we have shown that a truth-constant 𝑑 can be defined in a logic L by a formula 𝛿 containing at most one variable 𝑝 (under the assumption that `L 𝛿( 𝑝) ⇔ 𝛿(𝑞)).7 Thus, we can identify a huge class of logics with a top that is also the protounit, and show that we can actually use these constants to characterize this class of logics. Proposition 4.3.2 A weakly implicative logic is Rasiowa-implicative iff it has the top and it is also the protounit. Proof To prove the left-to-right direction, see the text after Example 2.8.19 where we show that the formula 𝑝 → 𝑝 is both a defined top and a defined protounit. The converse direction is a direct consequence of (Push) and (>).  Let us now present a few basic examples of notable logics with truth-constants, showing, among others, that the top (protounit) in a Rasiowa-implicative logic need not be the unit and that A→ is a logic with defined unit. Example 4.3.3 Consider any logic L expanding BCIlat . Clearly, ⊥ is the bottom and > is the top but need not be the protounit (otherwise, it would entail that BCIlat is 7 Note that this assumption implies that, in any A ∈ Alg∗ (L), we could meaningfully write 𝑑 A to denote the unique element which is the value of the formula 𝛿 under any A-evaluation.

4.3 Prominent truth-constants

169

Rasiowa-implicative). However, if L expands BCKlat , then > is also the protounit and, in this case, we can easily check (or use Corollary 4.3.11) that it is actually the unit. Of course, not all logics have (proto)units which are also the tops and, by the previous proposition, we know that such logics cannot be Rasiowa-implicative. An interesting example is the logic A→ , where the formula 𝑝 → 𝑝 is a defined unit (see below) and, thanks to Examples 2.3.7 and 2.6.4, we know that it is not the top and that this logic is not a fragment of any logic with top (furthermore, due to the upcoming Proposition 4.3.9, we will see that both claims hold for bottom as well). To show that 𝑝 → 𝑝 is really a defined unit in A→ , we use Corollary 2.8.17, i.e. first we give a formal proof of (𝜓 → 𝜓) → (𝜑 → 𝜑): a) ((𝜑 → 𝜑) → 𝜓) → ((𝜑 → 𝜑) → 𝜓) b) ((𝜑 → 𝜑) → 𝜓) → 𝜓

(id) a, (E), (id), and (MP)

c) (𝜓 → 𝜓) → (((𝜑 → 𝜑) → 𝜓) → 𝜓)

b and (Sf)

d) ((𝜑 → 𝜑) → 𝜓) → 𝜓) → (𝜑 → 𝜑)

(abe)

e) (𝜓 → 𝜓) → (𝜑 → 𝜑)

c, d, and (T)

To complete the proof, note that the step b is actually (pop) and that (push) follows by using the rule (E) on ( 𝑝 → 𝑝) → (𝜑 → 𝜑). Later in this chapter we will see many additional examples of logics with (proto)units which are not the top, e.g. in Theorem 4.7.1 we show that the logics BCI and BCIlat can be conservatively expanded into the logic SLaE . Finally, we can use the previous proposition to produce a class of logics with top/protounit which is not unit. Indeed, consider any Rasiowa-implicative logic together with its weak implication ⇒ and the top/protounit >, any at least threevalued reduced model A of this logic, and any element 𝑎 ∈ 𝐴 \ {>A }. Then, define a new binary connective 𝑥 ⇒0 𝑦 as >A if 𝑥 ≤ 𝑦 and 𝑎 otherwise, and consider the logic L of the matrix A expanded by ⇒0. Clearly, 𝜑 ⇒0 𝜓 a`L 𝜑 ⇒ 𝜓 and so L is a Rasiowa-implicative logic with the weak implication ⇒0 (the only non-trivial fact to check is the congruence rule for ⇒0) and so > is still the top and protounit for ⇒0 (it defines the same order as ⇒). Taking any 𝑥 ∈ 𝐴 \ {𝑎, >A }, we obtain > ⇒0 𝑥 = 𝑎 ≠ 𝑥, i.e. > is not the unit of ⇒0. Particular examples of such logics can be given by the matrices A𝑖0 from Example 4.2.17, taking ⇒𝑖0 = ⇒𝑖 (note that they are of the proper form). Actually, we can use these logics to show even the independence of the axioms (push) and (pop): A01 (pop)

2A01 (push)

A02 (push)

2A02 (pop).

Let us observe that, as in the case of previously considered connectives, we can easily prove that the lattice truth-constants and (proto)units are intrinsic (which justifies speaking about ‘the’ top / bottom / protounit) and can be easily semantically characterized, which shows that our terminology is well chosen indeed. The proofs of the first two propositions are obvious; note that the first one implies that no unit can be defined in a logic which has a protounit that is not the unit.

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4 On lattice and residuated connectives

Proposition 4.3.4 If 𝑐 and 𝑐 0 are two tops, bottoms, or protounits in L, then `L 𝑐 ⇔ 𝑐 0. Proposition 4.3.5 Let L be a weakly implicative logic with truth-constants ⊥ and > in its language. Assume that L has the KC for a class K ⊆ Mod∗ (L). Then, the properties 1–3 below are equivalent and so are the properties 𝑎–𝑐. 1) ⊥ is the bottom in L. 2) ⊥A

𝑎) > is the top in L ∗

= min ≤A 𝐴 for each A ∈ Mod (L). 𝑏) >A = max ≤A 𝐴 for each A ∈ Mod∗ (L).

3) ⊥A = min ≤A 𝐴 for each A ∈ K.

𝑐) >A = max ≤A 𝐴 for each A ∈ K.

Proposition 4.3.6 Let L be a weakly implicative logic with a truth-constant 1¯ in its language. Assume that L has the FSKC for a class K ⊆ Mod∗ (L). Then, the following are equivalent: 1. 1¯ is the protounit in L. 2. 1¯ A = min ≤A 𝐹 for each A ∈ Mod∗ (L). 3. 1¯ A = min ≤A 𝐹 for each A ∈ K. Furthermore, assume that 1¯ is the protounit and & the residuated conjunction in L. Then, 1¯ A is the right unit of &A for each A ∈ Mod∗ (L), i.e. for each 𝑥 ∈ 𝐴, 𝑥 &A 1¯ A = 𝑥. Proof The equivalence of the first three claims is obvious. To complete the proof, notice that from 𝑥 ≤ 𝑥 we obtain 1¯ A ≤ 𝑥 ⇒A 𝑥 using (Push) and, thus by residuation also, 𝑥 &A 1¯ A ≤ 𝑥. To prove the converse inequality, observe that from 𝑥 &A 1¯ A ≤ 𝑥 &A 1¯ A , by residuation, we obtain 1¯ A ≤ 𝑥 ⇒A 𝑥 &A 1¯ A and so the rule (Pop) completes the proof.  Note that, for any weakly implicative logic L with top / protounit, bottom and any model A = hA, 𝐹i ∈ Mod∗ (L), we have: (1) >A , 1¯ A ∈ 𝐹 and (2) ⊥A ∈ 𝐹 iff A is the trivial matrix (i.e. 𝐹 = 𝐴 = {⊥A }). Example 4.3.7 Recall that in Example 4.1.3 we have introduced a reduced matrix A with domain {⊥, 𝑎, 𝑏}, filter {𝑎, 𝑏}, and implication → such that the matrix order ≤A⇒ could be schematically depicted as ⊥ < 𝑎, 𝑏. Thus, by the previous two propositions, neither top nor the protounit are definable in this logic. Second, consider the matrix B0 defined by expanding the matrix B introduced in 0 0 Example 4.2.17 by the constant 1¯ interpreted as 1¯ B = >. It is easy to see that 1¯ B is 0 B the right unit of & but, clearly, 1¯ is not a protounit in B0 . The next proposition gives the corresponding semantical characterizations of the unit. Note that, as in the previous ones, we could add additional conditions demanding satisfaction of the involved semantical properties for matrices from K ⊆ Mod∗ (L), provided that L enjoys the KC.

4.3 Prominent truth-constants

171

Proposition 4.3.8 Let L be a weakly implicative logic with residuated conjunction and a truth-constant 1¯ in its language. Then, the following are equivalent: 1) 1¯ is the unit in L. 2) For each A ∈ Mod∗ (L), 1¯ A is the left-unit of ⇒A , i.e. for each 𝑥 ∈ 𝐴, we have: 1¯ A ⇒A 𝑥 = 𝑥. 3) For each A ∈ Mod∗ (L), 1¯ A is the left-unit of &A , i.e. for each 𝑥 ∈ 𝐴, we have: 1¯ A &A 𝑥 = 𝑥. 4) For each A ∈ Mod∗ (L), 1¯ A is the unit of &A , i.e. for each 𝑥 ∈ 𝐴, we have: 1¯ A &A 𝑥 = 𝑥 &A 1¯ A = 𝑥. Proof The equivalence of the first two claims is obvious. We show the equivalence of the second and third one; actually, we show a bit more: the equivalence of the corresponding inequalities. • By the residuation property, we know that 𝑥 ≤ 1¯ A ⇒A 𝑥 is equivalent to 1¯ A &A 𝑥 ≤ 𝑥. • Assume that 1¯ A ⇒A 𝑥 ≤ 𝑥 holds for each 𝑥. Using residuation on 1¯ A &A 𝑥 ≤ 1¯ A &A 𝑥, we obtain 𝑥 ≤ 1¯ A ⇒A 1¯ A &A 𝑥 and, using the assumption for 𝑥 = 1¯ A &A 𝑥, we obtain 𝑥 ≤ 1¯ A &A 𝑥. • Assume that 𝑥 ≤ 1¯ A &A 𝑥 holds for each 𝑥. Using residuation on 1¯ A ⇒A 𝑥 ≤ 1¯ A ⇒A 𝑥, we obtain 1¯ A &A ( 1¯ A ⇒A 𝑥) ≤ 𝑥 and, using the assumption for 𝑥 = 1¯ A ⇒A 𝑥, we obtain 1¯ A ⇒A 𝑥 ≤ 𝑥. The equivalence with the last claim follows from Proposition 4.3.6.



The rest of this section is dedicated to the study of the interplay of these truthconstants with each other and with other connectives. First, we focus on the interplay of truth-constants with co-implications. The first proposition deals with the interdefinability of lattice truth-constants and the second one extends Proposition 4.2.12 by ¯ showing how the symmetry rules can be described by means of properties of 1. Proposition 4.3.9 In any weakly implicative logic with co-implication bottom ⊥, the constant > is definable as ⊥ ⇒ ⊥.

, top >, and

Proof The proof is an easy consequence of Proposition 2.8.13: it suffices to observe that, from the axiom (>), we obtain (⊥ ⇒ ⊥) ⇒ > and, from the axiom (⊥) in the form ⊥ ⇒ (> ⊥), we obtain > ⇒ (⊥ ⇒ ⊥) using the rule (E ).  Proposition 4.3.10 Let L be a logic with a co-implication . Then, the properties 1–4 below are equivalent and so are the properties 10–40 (modulo the presence of additional connectives): 1) (Symm1 ) is valid in L.

10) (Symm2 ) is valid in L.

2) (id ) is valid in L.

20) (As) is valid in L.

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4 On lattice and residuated connectives

¯ we can add If there is a protounit 1, 30) (pop) is valid in L.

3) (push) is valid in L.

If there is also a residuated conjunction &, we can add 4) For each A ∈ Mod∗ (L) and each

40) For each A ∈ Mod∗ (L) and each

𝑥 ∈ 𝐴, we have: 1¯ A &A 𝑥 ≤A⇒ 𝑥

𝑥 ∈ 𝐴, we have: 𝑥 ≤A⇒ 1¯ A &A 𝑥.

Proof The equivalence of the first two points was proved in Proposition 4.2.12 and it is easy to observe that the equivalence of the last two points was established in the proof of Proposition 4.3.8. So it remains to prove two pairs of implications. To show that the 2s imply the 3s observe that from 𝜑 𝜑 using the rule (Push) we obtain 1¯ ⇒ (𝜑 𝜑) and so (E ) gives us the theorem (push). The theorem (pop) easily follows from the following instance of (As): 1¯ I ( 1¯ ⇒ 𝜑) ⇒ 𝜑. Finally, we show that the 4s imply the 1s. Let us recall that from Proposition 4.3.5 we know that 1¯ A = min ≤A⇒ 𝐹 for any A = hA, 𝐹i ∈ Mod∗ (L) and so we have the following chain of equivalences: 𝑥 ≤A 𝑦

iff

𝑥

𝑦∈𝐹

iff

1¯ ≤A⇒ 𝑥

𝑦

iff

1¯ & 𝑥 ≤A⇒ 𝑦.

Therefore, if 1¯ & 𝑥 ≤ 𝑥, we know that 𝑥 ≤A⇒ 𝑦 implies 𝑥 ≤A 𝑦, i.e. the rule (Symm1 ) is valid in L. Clearly, if we have 𝑥 ≤A⇒ 1 & 𝑥, then 𝑥 ≤A 𝑦 implies 𝑥 ≤A⇒ 𝑦.  The following corollary is a compact formulation of the previous proposition and adds one obvious observation about co-implication (the proof is left as an exercise for the reader). Note that, in particular, it tells us that, in any logic where ⇒ is its own co-implication, any protounit is the unit. Corollary 4.3.11 Let L be a weakly implicative logic with a co-implication and a ¯ Then, 1¯ is the unit iff protounit 1. is dual. Furthermore, if 1¯ is the unit in L, then it is a left unit of as well. Finally, we study the interplay of protounits with lattice connectives. First, we prove an analog of Proposition 4.2.13 and, then, we show two halves of certain distributivity laws which will be important later. Proposition 4.3.12 Let L be a weakly implicative logic with the protounit. Then, any lattice protoconjunction in L is a lattice conjunction. Furthermore, if L has a lattice protoconjunction or protodisjunction, then it is an algebraically implicative logic. Proof To prove the first claim, just observe that, due to the rule (Push), we know that 𝜑 `L 1¯ ⇒ 𝜑 and 𝜑 `L 1¯ ⇒ 𝜓 and so we obtain 𝜑, 𝜓 `L 1¯ ⇒ 𝜑 ∧ 𝜓 using the rule (Sup). The proof is then done using the rule (Pop). We prove the second claim by using Proposition 4.1.9: clearly, it suffices to observe that obviously the formula 𝜒 = 1¯ is the required formula with no other variable than 𝑝 such that 𝑝 `L 𝜒 ⇒ 𝑝 and `L 𝜒. 

4.4 Lambek logic and the logic SL

173

Table 4.1 The language LSL . Symbol → & 0¯ 1¯ ∧ ∨ ⊥ >

Arity 2 2 2 0 0 2 2 0 0

Name implication residuated conjunction dual co-implication unit lattice protoconjunction lattice protodisjunction bottom top

Alternative names left residuum fusion, multiplicative conjunction right residuum multiplicative falsity multiplicative truth additive/weak/lattice conjunction additive/weak/lattice disjunction additive/lattice falsity, falsum additive/lattice truth, verum

In sublanguage LLL , LCL , L→ LLL LLL LLL LLL LCL LCL LCL

Let us note that an analog of the first claim for protodisjunction is not true even if 1¯ is the unit; see Example 4.4.11 for details. ¯ and Proposition 4.3.13 Let L be a logic with residuated conjunction &, protounit 1, lattice (proto)conjunction ∧. Then, the following consecutions are valid in L: ¯

1 ) (distr⇒ 1¯ ) (distr&

¯ (𝜑 ⇒ 𝜓) ∧ 1¯ ⇒ (𝜑 ∧ 1¯ ⇒ 𝜓 ∧ 1) ¯ & (𝜓 ∧ 1) ¯ ⇒ (𝜑 & 𝜓) ∧ 1¯ (𝜑 ∧ 1)

¯ 1-⇒-distributivity ¯ 1-&-distributivity.

Proof Consider A = hA, 𝐹i ∈ Mod∗ (L), and 𝑥, 𝑦 ∈ 𝐴. For the proof of the first claim, ¯ & (𝑦 ∧ 1) ¯ ≤ 𝑥 & 𝑦 thanks notice that from 𝑥 ∧ 1¯ ≤ 𝑥 and 𝑦 ∧ 1¯ ≤ 𝑦, we obtain (𝑥 ∧ 1) ¯ ¯ ≤ 1¯ & 1¯ ≤ 1¯ to the monotonicity of &; analogously, we show that (𝑥 ∧ 1) & (𝑦 ∧ 1) (the last inequality is due to Proposition 4.3.6). To prove the second claim, we use residuation to obtain 𝑥 & (𝑥 ⇒ 𝑦) ≤ 𝑦 and ¯ applying the previous claim and residuation again so (𝑥 & (𝑥 ⇒ 𝑦)) ∧ 1¯ ≤ 𝑦 ∧ 1; completes the proof. 

4.4 Lambek logic and the logic SL The goal of this section is to identify the minimal logics containing certain natural collections of connectives behaving according to their expected properties as studied in the previous sections. Obviously, there are many degrees of freedom to fulfill this goal: we have introduced many connectives, in some cases even in two variants (namely (proto)conjunction/disjunction, (proto)units, and (dual) co-implications). The selection we make is guided mainly by mathematical interest, elegance of presentation, importance for the theory we build in this book, and the existing body of work in the literature. For these reasons we will distinguish whether 1¯ is a unit or just a protounit (or equivalently whether the co-implication is dual or not, cf. Corollary 4.3.11) and furthermore we will not expect ∨ to be a lattice disjunction but

174

4 On lattice and residuated connectives

just a protodisjunction8 (due to our selection of languages and Proposition 4.3.12 any lattice protoconjunction will be automatically a conjunction). The first logic we consider is called Lambek logic, denoted LL, which is intended to be the least logic of residuated connectives, i.e. the least logic with residuated con¯ As we have seen in Corollary 4.2.10, junction &, co-implication , and protounit 1. the presence of these connectives has consequences for the underlying weak implication. Therefore, we use the symbol → rather than the generic ⇒; i.e. the language LLL of the logic LL consists of three binary connectives →, , and & and two ¯ truth-constants 0¯ and 1.9 The second logic we consider is an expansion of LL to the language LSL that contains also ∧, ∨, >, and ⊥; see Table 4.1 which lists the connectives of LSL together with their arities and (alternative) names and also displays its relation to other languages.10 This logic, SL, is known as bounded non-associative Full Lambek logic and defined as the least expansion of LL where ∧/∨ are lattice protoconjunction/protodisjunction, >/⊥ are top/bottom, and 1¯ is the unit (or equivalently where is dual). While this selection of logics might seem arbitrary at first (in particular, the treatment of (proto)unit and (dual) co-implication, cf. Remark 4.4.2), it is well justified by their historical significance and existing body of literature (see Section 4.11 for details) and by the mathematical content of this and subsequent sections. In particular, we will see that • reduced models/algebras of LL can be easily described using residuated ordered protounital groupoids (see Theorem 4.4.5). • SL-algebras are shown to be the variety of residuated lattice-ordered unital groupoids (see Theorem 4.4.8); • reduced models of both logics can be regularly completed and, thus, these logics are complete w.r.t. completely ordered models (see Theorem 4.4.15 and Corollary 4.4.16). • both LL and SL are substructural logics according to our formal definition (Definition 4.6.1). • we can find a natural presentation of Lambek logic (see Theorem 4.5.5), which is strongly separable for all sublanguages of LLL containing implication, i.e. it gives us a presentation of all these fragments of LL; in particular it proves that its implicational fragment is the least substructural logic. 8 As we have seen before, there are natural strong logics (such as the global modal logic S4) where ∨ is only a lattice protodisjunction (and actually no lattice disjunction is definable), and thus requiring our minimal logics to have a lattice disjunction would be incoherent with their supposed minimality. 9 When writing formulas in a language containing (some of) these connectives, we assume that & has a higher binding power than → and (which have the same power), i.e. we write 𝜑& 𝜓 → ( 𝜒 𝜓) instead of ( 𝜑 & 𝜓) → ( 𝜒 𝜓); if furthermore the language contains the connectives ∨ or ∧, then we assume them to have the same binding power as &. 10 The names are selected based on the roles those connectives play in the logic SL; that is e.g. why 1¯ is called unit instead of protounit. Note in particular, that LSL can be seen as the union of the languages LLL and LCL extended by the truth-constant >. We will see that this constant is definable in SL, cf. Proposition 4.3.9, but it need not be definable in all of its fragments, so we prefer to have it as a primitive connective.

4.4 Lambek logic and the logic SL

175

• we can find two interesting presentations of SL: a natural one which is strongly separable for a restricted family of sublanguages of LSL , and nonetheless gives us axiomatizations of certain important fragments of SL (see Corollary 4.4.17) and another which is strongly (MP)-based (i.e. it has modus ponens as the only rule with two or more premises and its rules with one premise are of a particular form) which allows us to prove a local deduction theorem for this logic and identify a generalized disjunction connective with the proof by cases property. Let us start with the formal definitions of these two logics. Note that these definitions are sound because the set of all logics where all the connectives behave as required contains the inconsistent logic and it is closed under arbitrary intersections. Of course, these definitions can be seen as implicit; later in Figures 4.1 and 4.2 we will present particular axiomatic systems for these logics. Definition 4.4.1 (The logics LL and SL) • The Lambek logic LL is the least logic in the language LLL with weak implication ¯ →, co-implication , residuated conjunction &, and protounit 1. • The logic SL is the least logic in the language LSL with weak implication →, ¯ lattice protoconjunction ∧, co-implication , residuated conjunction &, unit 1, lattice protodisjunction ∨, top >, and bottom ⊥. Remark 4.4.2 Let us stress the different role of 1¯ in both logics. We could naturally define two different related logics by requiring that 1¯ is unit in the definition of Lambek logic and that it is just protounit in the definition of SL. In Corollary 4.4.17 we prove that the former logic is the LLL -fragment of SL and the latter one is a conservative expansion of LL. However, these logics are not important for us in this book, so we prefer not to increase the already quite extensive stock of examples. We know that the defining consecutions of the connectives featuring in LL and SL and all the consecutions listed in Corollaries 4.2.10 and 4.2.11 are respectively valid in LL and SL (with the implication → used instead of the generic ⇒). Furthermore, the following consecutions are also valid in SL (see the corresponding propositions in the previous section): (id )

𝜑

(As)

𝜑 I (𝜑 → 𝜓) → 𝜓

assertion

(Symm1 )

𝜑→𝜓I𝜑

𝜓

symmetry

(Symm2 )

𝜑

𝜓I𝜑→𝜓

symmetry

(r-unit1 )

𝜑 & 1¯ → 𝜑 𝜑 → 𝜑 & 1¯

right unit

1¯ & 𝜑 → 𝜑 𝜑 → 1¯ & 𝜑

left unit

(r-unit2 ) (l-unit1 ) (l-unit2 ) ¯

1 ) (distr→ ¯

1) (distr&

-identity

𝜑

¯ (𝜑 → 𝜓) ∧ 1¯ → (𝜑 ∧ 1¯ → 𝜓 ∧ 1) ¯ & (𝜓 ∧ 1) ¯ → (𝜑 & 𝜓) ∧ 1¯ (𝜑 ∧ 1)

right unit left unit ¯ 1-→-distributivity ¯ 1-&-distributivity.

176

4 On lattice and residuated connectives

Let us explore the semantical properties of LL and SL. Observe that, thanks to Propositions 4.2.13 and 4.3.12, we know that SL is algebraically implicative. Actually, because of the richness of its language, these propositions give us four alternative ¯ defining equations for SL; in the future we will use the simplest one: E = {𝑥 ∧ 1¯ ≈ 1}. Therefore, we could use Proposition 2.9.11 and the axiomatization of SL to obtain a quasiequational base of Alg∗ (SL) (which we know uniquely determines Mod∗ (SL)). Instead, we will focus first on the weaker logic LL and give an ordered-algebraic description of its reduced models and, based on this, we will obtain a natural latticetheoretic description of Alg∗ (SL) and even prove that it is a variety (Theorem 4.4.8). Not only that: we will see that LL is not algebraically implicative and SL is not Rasiowa-implicative (see Example 4.4.10). Our main tool will be the residuated ordered groupoids which we have introduced in Definition 4.2.7; we only need to take care of the residuated truth-constants. Definition 4.4.3 We say that the structure hh𝐴, &, ⇒, ordered protounital groupoid if

¯ 1i, ¯ ≤i is a residuated , 0,

1. hh𝐴, &, ⇒, i, ≤i is a residuated ordered groupoid and 2. 1¯ is a right unit of &, i.e. for each 𝑥 ∈ 𝐴, 𝑥 & 1¯ = 𝑥. ¯ 1i, ¯ ≤i is a residuated ordered unital groupoid if We say that hh𝐴, &, ⇒, , 0, furthermore11 3. 1¯ is a left unit of &, i.e. for each 𝑥 ∈ 𝐴, 1¯ & 𝑥 = 𝑥. We know that, for each LLL -algebra A and each subset 𝐹 ⊆ 𝐴, we can assign a binary relation ≤ → on 𝐴 defined as hA,𝐹 i 𝑥 ≤→ hA,𝐹 i 𝑦

iff

𝑥 →A 𝑦 ∈ 𝐹

and that this relation is an order whenever hA, 𝐹i ∈ Mod∗ (LL). We use the constant 1¯ to introduce a kind of dual construction that assigns to every relation ≤ on A a set 𝐹≤ = {𝑥 ∈ 𝐴 | 1¯ A ≤ 𝑥}. The next lemma shows how these two constructions are intertwined and will allow us to immediately obtain characterizations of Mod∗ (LL) and Alg∗ (LL). Lemma 4.4.4 Let A be an LLL -algebra. 1. If A = hA, 𝐹i ∈ Mod∗ (LL), then hA, ≤A→ i is a residuated ordered protounital groupoid and 𝐹 = 𝐹≤A→ . 11 In the literature of substructural logics, these structures are usually called pointed residuated ordered unital groupoids in order to signify the presence of the constant 0¯ in the signature.

4.4 Lambek logic and the logic SL

177

2. If hA, ≤i is a residuated ordered protounital groupoid, then A ≤ = hA, 𝐹≤ i ∈ Mod∗ (LL)

and

≤ = ≤A→≤ .

Proof To prove the first claim, observe that hA, ≤A→ i is a residuated ordered protounital groupoid, due to Propositions 4.2.8 and 4.3.6, and we have the following chain of equivalences (the first equivalence is due to the validity of the rules (Push) and (Pop) and the other two are just definitions): 𝑥∈𝐹

iff

1¯ A →A 𝑥 ∈ 𝐹

iff

1¯ A ≤A→ 𝑥

iff 𝑥 ∈ 𝐹≤→ . hA,𝐹i

To prove the second claim, we first show that ≤ = ≤A→≤ by writing the following chain of equivalences (the first one is due to residuation property and the fact that 1¯ is the right unit; the other two are just definitions): 𝑥 &A 1¯ A = 𝑥 ≤ 𝑦

iff 1¯ A ≤ 𝑥 →A 𝑦

iff 𝑥 →A 𝑦 ∈ 𝐹≤

iff

𝑥 ≤A→≤ 𝑦.

Therefore, by Propositions 4.2.8 and 4.3.6, we know that & is a residuated conjunction, a co-implication, and 1¯ is the protounit in A≤ and so LL ⊆ A , i.e. we have A ≤ ∈ Mod∗ (LL). Theorem 4.4.5 Let A be an LLL -algebra. • A = hA, 𝐹i ∈ Mod∗ (LL) iff hA, ≤A→ i is a residuated ordered protounital groupoid. • A ∈ Alg∗ (LL) iff there is an order ≤ on 𝐴 such that hA, ≤i is a residuated ordered protounital groupoid. Example 4.4.6 Due to the previous theorem, we know that the matrices A10 and A20 introduced in Example 4.3.3 are models of Lambek logic, but provide counterexamples to the validity of the consecutions (Symm), (l-unit) (push), (pop), and (As). Therefore, Lambek logic is not the LLL -fragment of SL. Actually, already the implicative fragments of these two logics differ. Now we turn our attention to the logic SL. We start with a straightforward consequence of Propositions 4.1.5 and 4.3.5 which formulates the expected relationship between models of LL and SL and bounded lattices. Let us first introduce the following two notations (recall that the former was already used for classes of matrices in Theorem 3.3.12): • Given an LSL -matrix A, we denote by ALLL the LLL -reduct of A. • Given an LLL -matrix B where ≤B is bounded lattice order, we denote by B LSL the expansion of B to the language LSL with lattice operation defined using ≤B . Proposition 4.4.7 If A ∈ Mod∗ (SL), then ≤A is a bounded lattice order, 1¯ A is the unit of &A , ALLL ∈ Mod∗ (LL), and A = (ALLL ) LSL . Conversely, let B ∈ Mod∗ (LL) be a matrix such that ≤B is a bounded lattice order and 1¯ B is the unit of &B . Then, B LSL ∈ Mod∗ (SL) and B = (B LSL )LLL .

178

4 On lattice and residuated connectives

Building on these results, we are ready to formulate the promised characterization of Alg∗ (SL) and Mod∗ (SL). Recall that, in any algebra containing a lattice operation ∧, we can define the relation ≤∧ by 𝑥 ≤∧ 𝑦 iff 𝑥 ∧ 𝑦 = 𝑥. Recall that SL is an algebraically implicative logic; therefore, for each A ∈ Alg∗ (SL), there is a unique filter 𝐹A such ¯ we obtain that hA, 𝐹A i ∈ Mod∗ (SL) and, since the defining equation is 𝑥 ∧ 1¯ ≈ 1, 𝐹A = 𝐹≤∧ . Theorem 4.4.8 Let A be an LSL -algebra. Then, the following are equivalent: 1. A ∈ Alg∗ (SL). 2. hA, 𝐹≤∧ i ∈ Mod∗ (SL). 3. hALLL , ≤∧ i is a residuated ordered unital groupoid and ≤∧ is a bounded lattice order. 4. h𝐴, ∧A , ∨A , ⊥A , >A i is a bounded lattice, h𝐴, &A , 1¯ A i is a unital groupoid and for each 𝑥, 𝑦, 𝑧, ∈ 𝐴, 𝑥 &A 𝑦 ≤ 𝑧

iff

𝑦 ≤ 𝑥 →A 𝑧

iff

𝑥≤𝑦

A

𝑧.

5. h𝐴, ∧A , ∨A , ⊥A , >A i is a bounded lattice, h𝐴, &A , 1¯ A i is a unital groupoid and, for each 𝑥, 𝑦, 𝑧 ∈ 𝐴, 𝑥 & (𝑦 ∨ 𝑧) = (𝑥 & 𝑦) ∨ (𝑥 & 𝑧) (𝑦 ∨ 𝑧) & 𝑥 = (𝑦 & 𝑥) ∨ (𝑧 & 𝑥)

𝑥≤𝑦 ((𝑥 & 𝑦) ∨ 𝑧) 𝑦 ≤ 𝑥 → ((𝑥 & 𝑦) ∨ 𝑧)

(𝑦 𝑥) & 𝑦 ≤ 𝑥 𝑦 & (𝑦 → 𝑥) ≤ 𝑥.

Therefore, Alg∗ (SL) is a variety and SL-algebras are known as bounded residuated lattice-ordered unital groupoids. Proof The equivalence of the first two claims is obvious. The equivalence of the second and third claims follows from Proposition 4.4.7 and Theorem 4.4.5. The fourth claim is trivially equivalent to the third one. To complete the proof, we first observe that all the equations in the fifth claim are valid in any algebra A ∈ Alg∗ (SL) (for the distributivity see Proposition 4.2.9; the remaining ones are direct consequences of the residuation property) and so we have established that 1 implies 5. To complete the proof of 5 implies 4, it suffices to show the residuation condition. We show the first equivalence; the proof of the other one is left as an exercise for the reader. Assume that 𝑥 & 𝑦 ≤ 𝑧; then (𝑥 & 𝑦) ∨ 𝑧 = 𝑧; thus, from 𝑦 ≤ 𝑥 → ((𝑥 & 𝑦) ∨ 𝑧), we obtain the required 𝑦 ≤ 𝑥 → 𝑧. Conversely, from 𝑦 ≤ 𝑥 → 𝑧 and monotonicity of & (which follows from the distributivity of & over joins), we obtain 𝑥 & 𝑦 ≤ 𝑥 & (𝑥 → 𝑧) and so indeed 𝑥 & 𝑦 ≤ 𝑧.  The next lemma provides a useful tool for defining reduced models of LL and SL. It is a direct consequence of Lemmas 4.2.15 and 4.4.4 and Proposition 4.4.7; we leave the elaboration of details as an exercise for the reader. Lemma 4.4.9 Let A be an LLL -algebra and ≤ be a complete lattice order on A. Then, hA, 𝐹≤ i ∈ Mod∗ (LL) if and only if the following conditions are met 1. For each 𝑥 ∈ 𝐴, we have 𝑥 &A 1¯ A = 𝑥.

4.4 Lambek logic and the logic SL

179

2. For each 𝑧 ∈ 𝐴 and 𝑋 ⊆ 𝐴, we have Ü Ü (𝑧 &A 𝑥) = 𝑧 &A 𝑋 𝑥 ∈𝑋

Ü

(𝑥 &A 𝑧) =

Ü

𝑋 &A 𝑧.

𝑥 ∈𝑋

3. For each 𝑥, 𝑦 ∈ 𝐴, we have 𝑥 →A 𝑦 = max{𝑧 | 𝑥 &A 𝑧 ≤ 𝑦} 𝑥 A 𝑦 = max{𝑧 | 𝑧 &A 𝑥 ≤ 𝑦}. Furthermore, hA, 𝐹≤ i LSL ∈ Mod∗ (SL) if and only if conditions 1–3 are satisfied and additionally for each 𝑥 ∈ 𝐴, we have 1¯ A &A 𝑥 = 𝑥. We use this lemma to show that LL is not algebraically implicative, there are no definable lattice protoconjunctions or protodisjunctions in LL, SL is not Rasiowaimplicative, and ∨ is not a lattice disjunction in SL. Example 4.4.10 Consider the LLL -algebra M with domain {⊥, 1, 𝑎, >}, constants 0¯ and 1¯ defined as ⊥ and 1 and operations →, , and & given by the following tables (note that M is an expansion of the L→ -algebra of Example 2.7.4): →M =

M

⊥ 1 𝑎 >

⊥ > ⊥ ⊥ ⊥

1 > 1 ⊥ ⊥

𝑎 > 𝑎 1 ⊥

> > > > >

&M ⊥ 1 𝑎 >

⊥ ⊥ ⊥ ⊥ ⊥

1 ⊥ 1 𝑎 >

𝑎 > ⊥ ⊥ 𝑎 > > > > >

Next, consider two orders ≤1 and ≤2 given schematically as ⊥ } whose lattice reduct is the non-distributive five-element lattice (recall Proposition A.2.11) (i.e. the order can be schematically depicted as ⊥ < 𝑎, 𝑏, 1 < >), constants 0¯ and 1¯ are respectively interpreted as ⊥ and 1, & is defined as 𝑥 & 1 = 1 & 𝑥 = 𝑥 and 𝑥 & 𝑦 = 𝑥 ∧ 𝑦 for 𝑥, 𝑦 ≠ 1; and implications 𝑥 𝑦 = 𝑥 → 𝑦 = max{𝑧 | 𝑧 & 𝑥 ≤ 𝑦}. We leave as an exercise for the reader to check that M3 is indeed a non-distributive ¯ >}. SL-algebra and 𝐹≤∧ = {1,

180

4 On lattice and residuated connectives

Consider the logic L =  hM3 ,𝐹≤ i . Clearly, it has a lattice conjunction and a lattice protodisjunction. Let us show that ∨ is not a lattice disjunction. Assume now that ∨ ¯ we easily get 𝜑 `L (𝜑 ∧ 1) ¯ ∨ 𝜓. Since we also satisfies the PCP. From 𝜑 `L 𝜑 ∧ 1, ¯ ¯ ∨ 𝜓. As clearly have 𝜓 `L (𝜑 ∧ 1) ∨ 𝜓, we can use the PCP to obtain 𝜑 ∨ 𝜓 `L (𝜑 ∧ 1) ¯ 𝑎 ∨ 𝑏 = > but not (𝑎 ∧ 1) ∨ 𝑏 = ⊥ ∨ 𝑏 = 𝑏, we obtain a contradiction. The next example shows that there are reduced models of LL where 1¯ A is the unit of &A but whose matrix order is not a lattice order, and hence they cannot be reducts of any model of SL. Example 4.4.12 Consider the LLL -algebra M with domain {0, 1}, constants 0¯ and 1¯ defined as 0 and 1 and operations →, , and & given by the following tables: →M =

M

0 1

0 1 0

&M 0 1

1 0 1

0 1 0

1 0 1

and take the matrix M = hM, {1}i. It is easy to see that ≤M is the discrete (i.e. non-lattice) order. We leave as an exercise for the reader to prove that we indeed have hM, {1}i ∈ Mod∗ (LL). However, we will be able to prove that each B ∈ Mod∗ (LL) where 1¯ A is the unit of &A is a subreduct of some A ∈ Mod∗ (SL). Therefore, due to Theorem 3.3.12, we will obtain the promised result that the LLL -fragment of SL is the axiomatic extension of LL by the axioms (push) and (pop). In order to prove this claim, we prove a stronger and very useful result: any A ∈ Mod∗ (LL) can be embedded into B ∈ Mod∗ (LL) in which ≤B is a complete order. We will prove similar claims for several logics and they will be a prerequisite for certain other results (especially in Chapter 7 on predicate logics, where the regular variant will be particularly useful). Definition 4.4.13 Let L be a weakly implicative logic and A ∈ Mod∗ (L). We say a matrix A𝑐 ∈ Mod∗ (L) is a completion of A if ≤A→𝑐 is a complete order and there is an embedding 𝑓 : A → A𝑐 . We say that a completion A𝑐 ∈ Mod∗ (L) is regular if there is a regular embedding 𝑓 : A → A𝑐 , i.e. an embedding such that additionally, for each 𝑋, 𝑌 ⊆ 𝐴 for which the infimum of 𝑋 and the supremum of 𝑌 exist in 𝐴, we have Û Û 𝑓( 𝑋) = 𝑓 (𝑥) 𝑥 ∈𝑋

𝑓(

Ü

𝑌) =

Ü

𝑓 (𝑦).

𝑦 ∈𝑌

Finally, we say that a class K ⊆ Mod∗ (L) admits (regular) completions if, for each A ∈ K, there is a (regular) completion A𝑐 ∈ K.

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181

Observe that, if we denote by K𝑐 the class of completely ordered matrices in K, the fact that K admits completions can be equivalently formulated as: K ⊆ IS(K𝑐 ). Recall that Corollary 3.6.17 tells us that a weakly implicative logic L with the CIPEP has the SKC iff 𝜔 Mod∗ (L)RSI ⊆ IS(K). Therefore, we immediately obtain the following theorem.12 Theorem 4.4.14 Let L be a weakly implicative logic with the CIPEP and let K be a 𝜔 ⊆ K ⊆ Mod∗ (L). If K admits completions, class of matrices such that Mod∗ (L)RSI then L enjoys strong completeness w.r.t. the class K𝑐 . Now we are ready to prove the crucial theorem about regular completions of models of LL and SL. The proof is based on the Dedekind–MacNeille completion of orders (see Theorem A.1.13). Theorem 4.4.15 The classes Mod∗ (LL) and Mod∗ (SL) admit regular completions. Proof Let us assume that A = hA, 𝐹i ∈ Mod∗ (LL) and denote its matrix order as ≤. For each 𝑋 ⊆ 𝐴, we define the following sets of lower and upper bounds: 𝑋 𝑙 ={𝑙 | 𝑙 ≤ 𝑥 for each 𝑥 ∈ 𝑋 } 𝑋 𝑢 ={𝑢 | 𝑥 ≤ 𝑢 for each 𝑥 ∈ 𝑋 }. It is well known (see Theorem A.1.13) that the inclusion relation gives a complete order on the set 𝐶 = {𝑋 ⊆ 𝐴 | 𝑋 = (𝑋 𝑢 ) 𝑙 } and that the mapping 𝑓 : h𝐴, ≤i → h𝐶, ⊆i defined as 𝑓 (𝑥) = {𝑥}𝑙 is an orderpreserving and reflecting embedding which preserves all existing suprema and infima from 𝐴. From now on, let us write (𝑥] instead of {𝑥}𝑙 . We define an LLL -algebra C = h𝐶, &C , →C , C , 1¯ C , 0¯ C i by setting 0¯ C = ( 0¯ A ] 1¯ C = ( 1¯ A ] ØØ 𝑋 &C 𝑌 = (𝑥 &A 𝑦] 𝑥 ∈𝑋 𝑦 ∈𝑌

𝑋 →C 𝑌 =

ÙØ

(𝑥 →A 𝑦]

𝑥 ∈𝑋 𝑦 ∈𝑌

𝑋

C

𝑌=

ÙØ

(𝑥

A

𝑦].

𝑥 ∈𝑋 𝑦 ∈𝑌

First, we prove that hC, ⊆i gives rise to a residuated ordered protounital groupoid. Since we already know that ⊆ is an order on 𝐶, we only have to show the following two facts: 𝜔 we obtain even equivalence and, using Theorem 5.6.5, we could 12 Note that for K = Mod∗ (L) RSI 𝜔 . obtain, for certain logics, equivalence for K = Mod∗ (L) RFSI

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4 On lattice and residuated connectives

¯ i.e. there are 𝑥 ∈ 𝑋 and 𝑦 ≤ 1¯ such that • 𝑋 & 1¯ = 𝑋: Assume that 𝑧 ∈ 𝑋 & 1, ¯ 𝑧 ≤ 𝑥 & 𝑦, but then 𝑧 ≤ 𝑥 & 𝑦 ≤ 𝑥 & 1 = 𝑥 and so 𝑧 ∈ 𝑋 (because 𝑋 is a ≤-downset). ¯ Conversely, assume that 𝑥 ∈ 𝑋. Clearly, 𝑥 ≤ 𝑥 & 1¯ and so 𝑥 ∈ 𝑋 & 1. • 𝑋 & 𝑌 ⊆ 𝑍 iff 𝑌 ⊆ 𝑋 → 𝑍 iff 𝑋 ⊆ 𝑌 𝑍: Again we prove the first equivalence; the proof of the second one is analogous. Assume that 𝑋 & 𝑌 ⊆ 𝑍. To prove that 𝑌 ⊆ 𝑋 → 𝑍, we need to show that, for each 𝑦 ∈ 𝑌 and each 𝑥 ∈ 𝑋, there is a 𝑧 ∈ 𝑍 such that 𝑦 ≤ 𝑥 → 𝑧. From the assumption, we know that 𝑥 & 𝑦 ∈ 𝑍 and, as 𝑦 ≤ 𝑥 → 𝑥 & 𝑦, the proof is done. To prove the converse, we need to show that, for each 𝑥 ∈ 𝑋 and for each 𝑦 ∈ 𝑌 , we have 𝑥 & 𝑦 ∈ 𝑍. From the assumption, we know that there is a 𝑧 ∈ 𝑍 such that 𝑦 ≤ 𝑥 → 𝑧, which implies that 𝑥 & 𝑦 ≤ 𝑧 and, as 𝑍 is a ≤-downset, the proof is done. Therefore, due to Lemma 4.4.4, we have that A𝑐 = hC, 𝐹⊆ i ∈ Mod∗ (LL) and ≤A𝑐 = ⊆ is a complete order. Thus, it remains to show that 𝑓 is a matrix embedding. We already know that it is one-one and, as the following chain of equivalences is obviously valid: 𝑥∈𝐹

iff

1¯ ≤ 𝑥

iff

¯ ⊆ 𝑓 (𝑥) 𝑓 ( 1)

iff

1¯ ⊆ 𝑓 (𝑥)

iff

𝑓 (𝑥) ∈ 𝐹⊆ ,

¯ = 1¯ and it suffices to prove that 𝑓 is an algebraic homomorphism. Clearly, 𝑓 ( 1) ¯ ¯ 𝑓 ( 0) = 0, and, thanks to the monotonicity of the connectives in A, we can easily prove the following equations: (𝑎] & (𝑏] = (𝑎 & 𝑏]

(𝑎] → (𝑏] = (𝑎 → 𝑏]

(𝑎]

(𝑏] = (𝑎

𝑏].

Indeed, let us show e.g. the case of implication →; the other two are analogous. First, assume that 𝑧 ∈ (𝑎] → (𝑏], i.e. for each 𝑥 ≤ 𝑎 there is a 𝑦 𝑥 ≤ 𝑏 such that 𝑧 ≤ 𝑥 → 𝑦 𝑥 . Thus, in particular, 𝑧 ≤ 𝑎 → 𝑦 𝑎 and so (by monotonicity of → in the second argument) 𝑧 ≤ 𝑎 → 𝑏. Second, assume that 𝑧 ≤ 𝑎 → 𝑏. Thus, for each 𝑥 ≤ 𝑎, we have (by antitonicity of → in the first argument) that 𝑧 ≤ 𝑥 → 𝑏, i.e. 𝑧 ∈ (𝑎] → (𝑏]. The claim for SL is an easy corollary of the claim for LL and Proposition 4.4.7. Indeed, for any A = hA, 𝐹i ∈ Mod∗ (SL) we take its LLL -reduct, construct its completion and then expand it back to a matrix A𝑐 ∈ (Mod∗ (SL)) 𝑐 . Let us elaborate the details. First, set B = A LLL ∈ Mod∗ (LL) and note that ≤B = ≤A is a bounded lattice order, and 1¯ A = 1¯ B is the unit of &A = &B . Consider the matrix B𝑐 and the embedding 𝑓 : B → B𝑐 . Then, ≤B𝑐 is a complete lattice order (it is a regular 𝑐 𝑐 completion of bounded lattice order) and, if we show that 1¯ B is the unit of &B , then ∗ 𝑐 𝑐 L 𝑐 A = ((A LLL ) ) SL ∈ (Mod (SL)) and, since 𝑓 preserves all existing suprema and infima, we can see it as an embedding 𝑓 : A → A𝑐 . To prove the last missing part, it suffices to show that, for each 𝑋 ∈ 𝐶, we have 𝑋 = 1¯ & 𝑋. For one inequality, assume that 𝑧 ∈ 1¯ & 𝑋, i.e. there are 𝑥 ∈ 𝑋 and 𝑦 ≤ 1¯ such that 𝑧 ≤ 𝑦 & 𝑥, but then 𝑧 ≤ 𝑦 & 𝑥 ≤ 1¯ & 𝑥 = 𝑥 and so 𝑧 ∈ 𝑋 (because 𝑋 is a ≤-downset). For the converse inequality, assume that 𝑥 ∈ 𝑋. Clearly, 𝑥 ≤ 1¯ & 𝑥 and so 𝑥 ∈ 1¯ & 𝑋. 

4.4 Lambek logic and the logic SL

183

Next, we state two easy-to-prove corollaries of the previous theorem. The first one tells us that both LL and SL enjoy strong completeness w.r.t. the class of their completely ordered matrices (direct consequence of Theorem 4.4.14 and the fact that both these logics are finitary) and the second one is the promised conservativeness result describing what would happen it we would switch the requirements for 1¯ in the definition of Lambek logic and SL. For compactness of notation, let us write Mod𝑐 (L) instead of (Mod∗ (L)) 𝑐 to denote the class of all reduced models of L whose matrix order is complete. Corollary 4.4.16 Let L be either LL or SL. Then, L enjoys the strong Mod𝑐 (L)completeness. Corollary 4.4.17 • The LLL -fragment of SL is the least logic in the language LLL with weak ¯ implication →, co-implication , residuated conjunction &, and unit 1. • Lambek logic is the LLL -fragment of the least logic in the language LSL with ¯ weak implication →, co-implication , residuated conjunction &, protounit 1, lattice protoconjunction ∧, lattice protodisjunction ∨, top >, and bottom ⊥. Proof Let us denote these two logics, only for the purposes of this proof, as LL+ and SL− . The proof of both claims is done by Theorem 3.3.12 as soon as we prove that (1) every matrix from Mod∗ (LL+ ) is a subreduct of a matrix from Mod∗ (SL) and (2) that every matrix from Mod∗ (LL) is a subreduct of a matrix from Mod∗ (SL− ). If we prove that Mod∗ (LL+ ) admits (regular) completions, the first claim follows from Proposition 4.4.7. Consider any A ∈ Mod∗ (LL+ ) ⊆ Mod∗ (LL) and its completion A𝑐 guaranteed by Theorem 4.4.15. Inspecting that proof (the last part when we deal with SL), we know that 1¯ is left unit of & (in A𝑐 ) and thus, by Proposition 4.3.8, it is the unit in A𝑐 . Thus, the proof follows by observing that indeed → is a weak implication, a co-implication, and & a residuated conjunction & in A𝑐 , i.e. A𝑐 ∈ Mod∗ (LL+ ). The second claim easily follows from the following variant of the second claim of Proposition 4.4.7: for any matrix B ∈ Mod∗ (LL) such that ≤B is a bounded lattice order, we have B LSL ∈ Mod∗ (SL− ).  It will be useful, from time to time, to refer to the logics LL+ and SL− , which appear in the proof of the previous corollary, by means of a more descriptive notation. Indeed, the first claim of the corollary tells us that we can denote the logic LL+ by SL LLL , as an instance of the systematic notation for fragments introduced in the beginning of the next section. The second claim, together with the upcoming Proposition 4.5.11, justifies denoting the logic SL− by LLlat , because it indeed is the expansion of LL by the same consecutions we used in Example 2.3.7 to define the logic BCIlat as an expansion of BCI.13 Finally, let us observe that all the consecutions listed in Corollary 4.2.11 are valid already in LLlat and the same holds for theorems 1¯ ), and (distr1¯ ). (r-unit), (distr→ & 13 Recall however that, unlike in this case, BCIlat is not the least expansion of BCI where the lattice connectives behave as expected; cf. Example 4.1.8.

184

4 On lattice and residuated connectives

4.5 Axiomatization of LL, SL, and their fragments The goal of this section is to find natural presentations of logics LL and SL as well as some of their important fragments. In the case of Lambek logic we even manage to find a strongly separable presentation (depicted in Figure 4.1), i.e. an axiomatic system such that each fragment of LL containing → is axiomatized by consecutions of this system featuring only connectives of the fragment in question (see Definition 4.5.1 for a formal definition). In the case of SL we manage to replicate this result only partially and find an axiomatic system of SL (depicted in Figure 4.2) where the axiomatizability condition holds for certain fragments only. We start by presenting a general definition of strongly separable axiomatic systems. First, however, we need to prepare some necessary notations and conventions: given any set C of consecutions in a language L0 , we denote by CL the set of elements of C which are consecutions in L ⊆ L0 . As any logic can be seen as a set of consecutions, given a logic L in the language L0 , we can write LL and observe that it is the L-fragment of L. Note that if a connective ⇒ which is primitive or definable in L is a weak implication in L, then same is the case for LL . For simplicity, we only list the names of the connectives in the subscript instead of the full language, e.g. we write L→ instead of LL→ (cf. our already used notation CL→ for denoting the implicational fragment of CL) and, whenever L ) L→ , we omit the implication connective (e.g. we write LL instead of LL→, ). Definition 4.5.1 Let L be a logic in a language L0 . We say that a presentation AS of L is strongly separable for a set X of sublanguages of L0 , if ASL is a presentation of LL for each L ∈ X. If, furthermore, L is a weakly implicative logic in a language L0 with a weak implication ⇒ and AS is strongly separable for all sublanguages of L0 containing ⇒ (as primitive or definable), we say that AS is strongly separable for ⇒.14 Recall that in Theorem 3.3.12 we proved (for weakly implicative logics) that a logic L in a language L is the L-reduct of a logic L0 whenever the models of L are subreducts of models of L0 . We use this fact to prove the next proposition, which is our main tool in proving separability results (recall that given a class K of L0 -matrices and a language L ⊆ L0 , we denote by K L the class of L-reducts of matrices of K). Proposition 4.5.2 Let L be a weakly implicative logic in a language L0 with a weak implication ⇒, X a set of sublanguages of L0 containing ⇒ (as primitive or definable), and AS an axiomatic system. Then, AS is a presentation of L which is strongly separable for X if and only if AS ⊆ L and, for each L ∈ X ∪ {L0 }, Mod∗ (`A SL ) ⊆ S(Mod∗ (L)L ). Proof To prove the left-to-right implication, first notice that, since AS is a presentation of L, we clearly have AS ⊆ L. Next, take a language L ∈ X ∪ {L0 }. 14 Following our standing convention, we omit referring to the weak implication whenever it is clear from the context and speak simply about strongly separable axiomatic systems.

4.5 Axiomatization of LL, SL, and their fragments

185

From the assumption, we know that ASL is a presentation of LL . Thus, we obtain Mod∗ (`A SL ) = Mod∗ (LL ) = S(Mod∗ (L)L ) (using Theorem 3.3.12 for the last equality, thanks to the fact that ⇒ is in L and so LL remains weakly implicative). To prove the converse implication, we show that ASL is a presentation of LL for any language L ∈ X ∪ {L0 } (thus, in particular, it is indeed a presentation of L). From AS ⊆ L, we obtain ` A SL ⊆ LL . For the reverse inclusion, we observe that Mod∗ (`A SL ) ⊆ S(Mod∗ (L)L ) = Mod∗ (LL ) (using again Theorem 3.3.12) and hence LL ⊆ ` A SL .  Using this proposition for proving strong separability of a certain presentation of LL will involve constructing models of LL, which we know are certain residuated ordered protounital groupoids (cf. Lemma 4.4.4). The following lemma presents a cornerstone of one such construction in a rather general setting.15 Lemma 4.5.3 Let h𝑈, ≤i be an ordered set, 𝑓 an element of 𝑈, and 𝑅 a ternary relation of 𝑈 such that the following conditions hold for each 𝑥, 𝑦, and 𝑧: 1. 2. 3. 4.

If 𝑥 ≤ 𝑥 0 and 𝑅(𝑥 0, 𝑧, 𝑦), then 𝑅(𝑥, 𝑧, 𝑦). If 𝑧 ≤ 𝑧 0 and 𝑅(𝑥, 𝑧 0, 𝑦), then 𝑅(𝑥, 𝑧, 𝑦). If 𝑦 ≤ 𝑦 0 and 𝑅(𝑥, 𝑧, 𝑦), then 𝑅(𝑥, 𝑧, 𝑦 0). 𝑅(𝑥, 𝑓 , 𝑦) iff 𝑥 ≤ 𝑦.

¯ 1i, ¯ ⊆i where 𝑈 ≤ (𝑈) is the set of non-empty Then, the structure hh𝑈 ≤ (𝑈), &, →, , 0, ¯ ≤-uppersets on 𝑈, 0 is an arbitrary element of 𝑈 ≤ (𝑈), and the remaining operations are defined as 1¯ = {𝑢 ∈ 𝑈 | 𝑓 ≤ 𝑢} 𝑋 → 𝑌 = {𝑢 ∈ 𝑈 | for all 𝑥, 𝑦 ∈ 𝑈, if 𝑅(𝑥, 𝑢, 𝑦) and 𝑥 ∈ 𝑋, then 𝑦 ∈ 𝑌 } 𝑋

𝑌 = {𝑢 ∈ 𝑈 | for all 𝑥, 𝑦 ∈ 𝑈, if 𝑅(𝑢, 𝑥, 𝑦) and 𝑥 ∈ 𝑋, then 𝑦 ∈ 𝑌 }

𝑋 & 𝑌 = {𝑢 ∈ 𝑈 | there are 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 such that 𝑅(𝑥, 𝑦, 𝑢)} is a residuated ordered protounital groupoid. Furthermore, 1¯ is the unit of & whenever also the following additional condition holds for each 𝑥 and 𝑦: 5. 𝑅( 𝑓 , 𝑥, 𝑦) iff 𝑥 ≤ 𝑦. Proof First, we have to show that 𝑈 ≤ (𝑈) is closed under the defined operations. We prove the first case in detail and leave the remaining cases as an exercise for the reader. Assume that 𝑢 ≤ 𝑢 0 and 𝑢 ∈ 𝑋 → 𝑌 ; we need to show that 𝑢 0 ∈ 𝑋 → 𝑌 . Consider 𝑥, 𝑦 ∈ 𝑈 such that 𝑅(𝑥, 𝑢 0, 𝑦) and 𝑥 ∈ 𝑋. Thus, from 𝑢 ≤ 𝑢 0 and 𝑅(𝑥, 𝑢 0, 𝑦), we get 𝑅(𝑥, 𝑢, 𝑦) and, therefore, from 𝑢 ∈ 𝑋 → 𝑌 , we obtain 𝑦 ∈ 𝑌 , as required. Next, we prove the residuation property: 𝐶⊆𝐷→𝐸

iff

𝐷 &𝐶 ⊆ 𝐸

iff

𝐷⊆𝐶

𝐸.

15 This construction is tightly related to a well-known ternary relational semantics for substructural logics, related duality theoretic properties of ordered sets with operators and a special kind of completions known as canonical completions (see Section 4.11 for more details).

186

4 On lattice and residuated connectives

(id)

𝜑→𝜑

identity

(MP)

𝜑, 𝜑 → 𝜓 I 𝜓

modus ponens

(Sf)

𝜑 → 𝜓 I (𝜓 → 𝜒) → (𝜑 → 𝜒)

suffixing

(Pf)

𝜑 → 𝜓 I ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓)

prefixing

(Res1 )

𝜑 → (𝜓 → 𝜒) I 𝜓 & 𝜑 → 𝜒

residuation

(Res2 )

𝜓 & 𝜑 → 𝜒 I 𝜑 → (𝜓 → 𝜒)

residuation

(E

,1 )

𝜑 → (𝜓 → 𝜒) I 𝜓 → (𝜑

𝜒)

-exchange

(E

,2 )

𝜑 → (𝜓 𝜒) I 𝜓 → (𝜑 → 𝜒) 𝜑 I 1¯ → 𝜑 1¯ → 𝜑 I 𝜑

-exchange

(Push) (Pop)

push pop

Fig. 4.1 Strongly separable presentation A S LL of the Lambek logic LL.

We proceed by showing three implications. To prove the first one, we show that, from 𝐶 ⊆ 𝐷 → 𝐸 and 𝑢 ∈ 𝐷 & 𝐶, we obtain 𝑢 ∈ 𝐸. From the latter assumption, we know that there are 𝑥 ∈ 𝐷 and 𝑦 ∈ 𝐶 such that 𝑅(𝑥, 𝑦, 𝑢). Thus, from the former assumption, we know that 𝑦 ∈ 𝐷 → 𝐸, which means that for all 𝑥 0, 𝑦 0 ∈ 𝑈, if 𝑅(𝑥 0, 𝑦, 𝑦 0) and 𝑥 0 ∈ 𝐷, then 𝑦 0 ∈ 𝐸. Taking 𝑥 0 = 𝑥 and 𝑦 0 = 𝑢 completes the proof. The second implication: assume that 𝐷 & 𝐶 ⊆ 𝐸 and 𝑢 ∈ 𝐷 and we want to show that 𝑢 ∈ 𝐶 𝐷; i.e. for each 𝑥, 𝑦 ∈ 𝑈 such that 𝑅(𝑢, 𝑥, 𝑦) and 𝑥 ∈ 𝐶 we need to show that 𝑦 ∈ 𝐸. Assume that 𝑥 and 𝑦 are such and note that it implies that 𝑦 ∈ 𝐷 & 𝐶 and so 𝑦 ∈ 𝐸 as required. The final implication: assume that 𝐷 ⊆ 𝐶 𝐸 and 𝑢 ∈ 𝐶 and we want to show that 𝑢 ∈ 𝐷 → 𝐸; i.e. for each 𝑥, 𝑦 ∈ 𝑈 such that 𝑅(𝑥, 𝑢, 𝑦) and 𝑥 ∈ 𝐷, we need to show that 𝑦 ∈ 𝐸. Assume that 𝑥 and 𝑦 are such and note that it implies that 𝑥 ∈𝐶 𝐸. This means that, for all 𝑥 0, 𝑦 0 ∈ 𝑈, if 𝑅(𝑥, 𝑥 0, 𝑦 0) and 𝑥 0 ∈ 𝐶, then 0 𝑦 ∈ 𝐸. Taking 𝑥 0 = 𝑢 and 𝑦 0 = 𝑦 completes the proof. Next, we prove that 1¯ is the right unit of &: assume that 𝑢 ∈ 𝑋, then from 𝑓 ∈ 1¯ and ¯ Conversely, assume that 𝑢 ∈ 𝑋 & 1; ¯ then 𝑅(𝑥, 𝑦, 𝑢) 𝑅(𝑢, 𝑓 , 𝑢), we obtain 𝑢 ∈ 𝑋 & 1. ¯ for some 𝑥 ∈ 𝑋 and 𝑦 ∈ 1. From the latter, we obtain 𝑓 ≤ 𝑦 and so 𝑅(𝑥, 𝑓 , 𝑢) which implies that 𝑥 ≤ 𝑢, and this (because 𝑋 is ≤-upperset) entails that 𝑢 ∈ 𝑋. Finally, we prove that, with the additional assumption, 1¯ is left unit of &: assume that 𝑢 ∈ 𝑋 and, as we know that 𝑓 ∈ 1¯ and 𝑅( 𝑓 , 𝑢, 𝑢), we know that 𝑢 ∈ 1¯ & 𝑋. Conversely, assume that 𝑢 ∈ 1¯ & 𝑋; then 𝑅(𝑥, 𝑦, 𝑢) for some 𝑥 ∈ 1¯ and 𝑦 ∈ 𝑋. From the former, we obtain 𝑓 ≤ 𝑥 and so 𝑅( 𝑓 , 𝑦, 𝑢) which implies that 𝑦 ≤ 𝑢, and this (because 𝑋 is ≤-upperset) entails that 𝑢 ∈ 𝑋.  The following lemma is the final ingredient we need to be able to prove that the axiomatic system AS LL listed in Figure 4.1 is a strongly separable presentation of Lambek logic. Note that, while obviously → is a weak implication in any fragment of LL which contains it, the same fact is not automatically true for the logic ` ( A S LL ) L .

4.5 Axiomatization of LL, SL, and their fragments

187

Lemma 4.5.4 Let L ⊆ LLL be a language containing →. Then, → is a weak implication in ` ( A S LL ) L . Proof First, note that (id) and (MP) are elements of (AS LL )L and that transitivity (T) and the congruence rules for implication follow from both suffixing and prefixing. To prove the congruence axioms for & and , it is clearly sufficient to formally prove (in an appropriate logic) their mono/antitonicity properties (which we know are valid in LL, see Corollary 4.2.11). In particular, we have to show that the following consecutions are derivable in ` ( A S LL )& : (Mon& 1) (Mon& 2)

𝜑 →𝜓 I 𝜒&𝜑 → 𝜒&𝜓 𝜑→𝜓 I 𝜑&𝜒→𝜓&𝜒

&-monotonicity &-monotonicity

and that the following consecutions are valid in ` ( A S LL ) : (Sf ) (Pf )

𝜑 → 𝜓 I (𝜓 𝜑 → 𝜓 I (𝜒

𝜒) → (𝜑 𝜑) → ( 𝜒

𝜒) 𝜓)

-suffixing -prefixing.

To prove the former two claims, we first observe that from 𝜒 & 𝜓 → 𝜒 & 𝜓 using (Res) we obtain the theorem (adj& ): 𝜓 → ( 𝜒 → 𝜒 & 𝜓). The rule (Mon& 1 ) then follows from (adj& ) using transitivity and (Res) again. To prove (Mon& ), we start by 2 a particular instance of the rule (Sf): 𝜑 → 𝜓 ` (𝜓 → 𝜓 & 𝜒) → (𝜑 → 𝜓 & 𝜒) and thanks to transitivity and (adj& ), we obtain 𝜑 → 𝜓 ` 𝜒 → (𝜑 → 𝜓 & 𝜒) and so the rule (Res) completes the proof. To prove the latter two claims, we first observe that, from (𝜓 𝜒) → (𝜓 𝜒), using (E ), we obtain the theorem (as ): 𝜓 → ((𝜓 𝜒) → 𝜒). The rule (Sf ) then follows from (as ) using transitivity and (E ) again. To prove (Pf ), we start by a particular instance of (Pf): 𝜑 → 𝜓 ` (( 𝜒 𝜑) → 𝜑) → (( 𝜒 𝜑) → 𝜓) and use (as ) and transitivity to obtain 𝜑 → 𝜓 ` 𝜑 → (( 𝜒 𝜑) → 𝜓) and so (E ) completes the proof.  Theorem 4.5.5 The axiomatic system AS LL listed in Figure 4.1 is a strongly separable presentation of Lambek logic. Proof We prove the theorem by using Proposition 4.5.2. Let us write (for the purposes of this proof) simply `L instead of ` ( A S LL ) L . The inclusion AS LL ⊆ LL is obviously true, thus all we have to do is to show that Mod∗ (`L ) ⊆ S(Mod∗ (LL)L ) for each language L ⊆ LLL containing →. Consider any such language and a matrix A = hA, 𝐹i ∈ Mod∗ (`L ); we need to show that there is a matrix A0 ∈ Mod∗ (LL) and a strict embedding 𝑓 : A → AL0 . Our trick is to build the matrix A0 using only the implication connective and show that the construction works for all fragments at once. First, observe that thanks to the previous lemma, we know that → is a weak implication in `L ; let us denote by ≤ the matrix order on A given by →. Consider the set 𝑈 ≤ ( 𝐴) of non-empty ≤-upper sets on 𝐴 and let us denote by [𝑎) the set {𝑥 ∈ 𝐴 | 𝑎 ≤ 𝑥}. Note that 𝑈 ≤ ( 𝐴) is ordered by inclusion ⊆ and we have 𝐹 ∈ 𝑈 ≤ ( 𝐴). Let us define a ternary relation 𝑅 on 𝑈 ≤ ( 𝐴) as 𝑅(𝑋, 𝑍, 𝑌 ) iff for each 𝑥, 𝑦 ∈ 𝐴 we have 𝑥 ∈ 𝑋 and 𝑥 → 𝑦 ∈ 𝑍 implies 𝑦 ∈ 𝑌 .

188

4 On lattice and residuated connectives

It is easy to see that the relation 𝑅 satisfies the conditions 1–4 required by Lemma 4.5.3 (the only non-trivial part is condition 4; but it is easy to observe that to prove it suffices to show that 𝑅(𝑋, 𝐹, 𝑋): indeed if 𝑥 ∈ 𝑋 and 𝑥 → 𝑦 ∈ 𝐹, then 𝑥 ≤ 𝑦 and so 𝑦 ∈ 𝑋). Therefore, using Lemma 4.4.4, we know that A0 = hh𝑈 ⊆ (𝑈 ≤ ( 𝐴)), →,

¯ 1i, ¯ 𝐹⊆ i ∈ Mod∗ (LL). , &, 0,

Recall that Lemma 4.5.3 did no postulate any restriction for the possible interpretation of 0¯ in the domain 𝑈 ⊆ (𝑈 ≤ ( 𝐴)); therefore, if we have 0¯ as a truth-constant in L, we can interpret it as {𝑋 ∈ 𝑈 ≤ ( 𝐴) | 0¯ ∈ 𝑋 }; otherwise we can take an arbitrary element of 𝑈 ⊆ (𝑈 ≤ ( 𝐴)). Clearly, in order to complete the proof of this theorem, all we have to do is to prove that the mapping 𝑔 defined as 𝑔(𝑎) = {𝑋 ∈ 𝑈 ≤ ( 𝐴) | 𝑎 ∈ 𝑋 } is the required embedding 𝑔 : A → AL0 . Let us first observe that indeed 𝑔(𝑎) ∈ 𝑈 ⊆ (𝑈 ≤ ( 𝐴)) (it is clearly an ⊆-upper set and [𝑎) ∈ 𝑔(𝑎); note that this also shows that indeed 0¯ ∈ 𝑈 ⊆ (𝑈 ≤ ( 𝐴)) in the case ¯ and so, due to Corollary 3.2.12, it suffices to show that 𝑔 is a that L contains 0) strict homomorphism. The matricial part is easily proved by the following chain of equivalences (recall that 1¯ = {𝑈 ∈ 𝑈 ≤ ( 𝐴) | 𝐹 ⊆ 𝑈}): 𝑎∈𝐹

iff

𝐹 ∈ 𝑔(𝑎)

iff

1¯ ⊆ 𝑔(𝑎)

iff

𝑔(𝑎) ∈ 𝐹⊆ .

Thus, it remains to prove that it is an algebraic homomorphism. We take care of all possible connectives in L. 𝑔(𝑎 → 𝑏) = 𝑔(𝑎) → 𝑔(𝑏): To prove one inclusion, assume that 𝑈 ∈ 𝑔(𝑎 → 𝑏), i.e. 𝑎 → 𝑏 ∈ 𝑈 and we show that 𝑈 ∈ 𝑔(𝑎) → 𝑔(𝑏): consider 𝑋, 𝑌 such that 𝑅(𝑋, 𝑈, 𝑌 ) and 𝑋 ∈ 𝑔(𝑎); thus, from the latter assumption, we know 𝑎 ∈ 𝑋 and, as we also assume that 𝑎 → 𝑏 ∈ 𝑈, the assumption 𝑅(𝑋, 𝑈, 𝑌 ) yields 𝑏 ∈ 𝑌 , i.e. 𝑌 ∈ 𝑔(𝑏) as required. To prove the converse inclusion, assume that 𝑈 ∈ 𝑔(𝑎) → 𝑔(𝑏), i.e. for each 𝑋, 𝑌 ∈ 𝑈 ≤ ( 𝐴), we have 𝑌 ∈ 𝑔(𝑏) whenever 𝑋 ∈ 𝑔(𝑎) and 𝑅(𝑋, 𝑈, 𝑌 ). Let us consider the following two sets: 𝑋 = {𝑥 ∈ 𝐴 | 𝑎 ≤ 𝑥}

and

𝑌 = {𝑦 ∈ 𝐴 | 𝑎 → 𝑦 ∈ 𝑈}.

Clearly, 𝑋 ∈ 𝑔(𝑎) and, if we show that 𝑌 ∈ 𝑈 ≤ ( 𝐴) and 𝑅(𝑋, 𝑈, 𝑌 ), the proof is done: indeed, from 𝑌 ∈ 𝑔(𝑏) we get 𝑏 ∈ 𝑌 , i.e. 𝑎 → 𝑏 ∈ 𝑈 and thus 𝑈 ∈ 𝑔(𝑎 → 𝑏). To prove that 𝑌 ∈ 𝑈 ≤ ( 𝐴), observe that, due to the validity of the rule (Pf), we know that 𝑦 ≤ 𝑦 0 implies 𝑎 → 𝑦 ≤ 𝑎 → 𝑦 0 and because 𝑈 is a ≤-upper set, so is 𝑌 . Finally, we show that 𝑅(𝑋, 𝑈, 𝑌 ): consider 𝑥, 𝑦 ∈ 𝐴 such that 𝑥 ∈ 𝑋 and 𝑥 → 𝑦 ∈ 𝑈. Then 𝑎 ≤ 𝑥 and, due to the validity of the rule (Sf), we know that 𝑥 → 𝑦 ≤ 𝑎 → 𝑦 and, as 𝑈 is a ≤-upper set, we obtain 𝑎 → 𝑦 ∈ 𝑈, i.e. 𝑦 ∈ 𝑌 as required.

4.5 Axiomatization of LL, SL, and their fragments

189

𝑔(𝑎 & 𝑏) = 𝑔(𝑎) & 𝑔(𝑏): First, assume that 𝑈 ∈ 𝑔(𝑎 & 𝑏), i.e. 𝑎 & 𝑏 ∈ 𝑈 and to show that 𝑈 ∈ 𝑔(𝑎) & 𝑔(𝑏) it suffices to show that 𝑅([𝑎), [𝑏), 𝑈): consider 𝑥 ∈ [𝑎) and 𝑥 → 𝑦 ∈ [𝑏), then clearly 𝑎 & 𝑏 ≤ 𝑥 & (𝑥 → 𝑦) and, since due to the rule (Res) we have 𝑥 & (𝑥 → 𝑦) ≤ 𝑦 and 𝑈 is ≤-upperset, we obtain 𝑦 ∈ 𝑈 as required. To prove the converse, assume that 𝑈 ∈ 𝑔(𝑎) & 𝑔(𝑏), i.e. there are 𝑋 ∈ 𝑔(𝑎) and 𝑌 ∈ 𝑔(𝑏) such that 𝑅(𝑋, 𝑌 , 𝑈), i.e. for each 𝑥 ∈ 𝑋 and 𝑥 → 𝑦 ∈ 𝑌 we have 𝑦 ∈ 𝑈. The proof is done by taking 𝑥 = 𝑎 and 𝑦 = 𝑎 & 𝑏 (because, due to the rule (Res), we have 𝑏 ≤ 𝑎 → 𝑎 & 𝑏). 𝑔(𝑎 𝑏) = 𝑔(𝑎) 𝑔(𝑏): The proof is analogous to the previous case. First, assume that 𝑈 ∈ 𝑔(𝑎 𝑏), i.e. 𝑎 𝑏 ∈ 𝑈 and we show that 𝑈 ∈ 𝑔(𝑎) 𝑔(𝑏): consider 𝑋, 𝑌 such that 𝑅(𝑈, 𝑋, 𝑌 ) and 𝑋 ∈ 𝑔(𝑎); thus, from the latter assumption, we know 𝑎 ∈ 𝑋 and, as we also assume that 𝑎 𝑏 ∈ 𝑈, the former assumption yields 𝑏 ∈ 𝑌 (because, due to the rules (E ), we have 𝑎 ≤ (𝑎 𝑏) → 𝑏 and so (𝑎 𝑏) → 𝑏 ∈ 𝑋), i.e. 𝑌 ∈ 𝑔(𝑏) as required. To prove the converse, assume that 𝑈 ∈ 𝑔(𝑎) 𝑔(𝑏), i.e. for each 𝑋, 𝑌 ∈ 𝑈 ( 𝐴), we have 𝑌 ∈ 𝑔(𝑏) whenever 𝑋 ∈ 𝑔(𝑎) and 𝑅(𝑈, 𝑋, 𝑌 ). Let us consider the sets 𝑋 = [𝑎) 𝑌 = {𝑦 | 𝑎

𝑦 ∈ 𝑈}.

Clearly, 𝑋 ∈ 𝑔(𝑎) and, if we show that 𝑌 ∈ 𝑈 ( 𝐴) and 𝑅(𝑈, 𝑋, 𝑌 ), the proof is done. Indeed, from 𝑌 ∈ 𝑔(𝑏), we obtain 𝑏 ∈ 𝑌 , i.e. 𝑎 𝑏 ∈ 𝑈 and thus we clearly have 𝑈 ∈ 𝑔(𝑎 𝑏). To prove 𝑌 ∈ 𝑈 ( 𝐴), observe that, due to the validity of the rule (Pf ), we know that 𝑦 ≤ 𝑦 0 implies 𝑎 𝑦≤𝑎 𝑦 0 and, because 𝑈 is ≤-upper set, so is 𝑌 . Finally, we show that 𝑅(𝑈, 𝑋, 𝑌 ). Consider elements 𝑥, 𝑦 ∈ 𝐴 such that 𝑥 ∈ 𝑈 and 𝑥 → 𝑦 ∈ [𝑎). Then, 𝑎 ≤ 𝑥 → 𝑦 and, due to the validity of the rule (E ), we know that 𝑥 ≤ 𝑎 𝑦 and, as 𝑈 ∈ 𝑈 ( 𝐴), we obtain 𝑎 𝑦 ∈ 𝑈, i.e. 𝑦 ∈ 𝑌 as required. ¯ = 1: ¯ ¯ and, 𝑔( 1) Recall that, in the presence of a protounit, we know that 𝐹 = [ 1) thus, we can write the following chain of equations: ¯ = {𝑋 ∈ 𝑈 ≤ ( 𝐴) | 1¯ ∈ 𝑈} = {𝑋 ∈ 𝑈 ≤ ( 𝐴) | 𝐹 ⊆ 𝑈} = 1. ¯ 𝑔( 1) ¯ = 0: ¯ 𝑔( 0)

This follows directly from the definition of 0¯ in A0.



Inspecting the proof of the previous theorem, one can observe that the matrix AL0 is actually a completion of A (for any language L). Therefore, we obtain the following corollary (the part about the strong completeness follows from Theorem 4.4.14): Corollary 4.5.6 Let L ⊆ LLL be a language containing implication. Then, the class Mod∗ (LLL ) admits completions and LLL enjoys the strong Mod𝑐 (LLL )completeness. Our next goal is to find a presentation of SL which is strongly separable at least for a restricted class of fragments (see Remark 4.5.8 for comments on the reasons for these restrictions).

190

4 On lattice and residuated connectives

(id)

𝜑→𝜑

(id )

𝜑

(As)

𝜑 I (𝜑 → 𝜓) → 𝜓

assertion

(MP)

𝜑, 𝜑 → 𝜓 I 𝜓

modus ponens

(Sf)

𝜑 → 𝜓 I (𝜓 → 𝜒) → (𝜑 → 𝜒)

suffixing

(Pf)

𝜑 → 𝜓 I ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓)

prefixing

(Res1 )

𝜑 → (𝜓 → 𝜒) I 𝜓 & 𝜑 → 𝜒

residuation

(Res2 )

𝜓 & 𝜑 → 𝜒 I 𝜑 → (𝜓 → 𝜒)

residuation

(E

,1 )

𝜑 → (𝜓 → 𝜒) I 𝜓 → (𝜑

𝜒)

-exchange

(E

,2 )

-exchange push

(pop)

𝜑 → (𝜓 𝜒) I 𝜓 → (𝜑 → 𝜒) ¯ 𝜑 → ( 1 → 𝜑) ( 1¯ → 𝜑) → 𝜑

(⊥)

⊥→𝜑

ex falso quodlibet

(>)

𝜑→>

verum ex quolibet

(Adj)

𝜑, 𝜓 I 𝜑 ∧ 𝜓

adjunction

(lb1 )

𝜑∧𝜓 → 𝜑

lower bound

(lb2 )

𝜑∧𝜓 →𝜓

lower bound

(inf)

( 𝜒 → 𝜑) ∧ ( 𝜒 → 𝜓) → ( 𝜒 → 𝜑 ∧ 𝜓)

infimality

(ub1 )

𝜑 → 𝜑∨𝜓

upper bound

(ub2 )

𝜓 → 𝜑∨𝜓

upper bound

(Sup)

𝜑 → 𝜒, 𝜓 → 𝜒 I 𝜑 ∨ 𝜓 → 𝜒

supremality

(push)

identity -identity

𝜑

pop

Fig. 4.2 Separable presentation A S SL of the logic SL.

Theorem 4.5.7 The axiomatic system AS SL listed in Figure 4.2 is a strongly separable presentation of SL for the set X consisting of all sublanguages L of LSL satisfying the following three conditions: • L contains →. ¯ • L contains also or 1. • If L contains ∨, then L ⊇ LLL . Furthermore, for each L ∈ X, Mod∗ (SLL ) admits completions and SLL enjoys the strong Mod𝑐 (SLL )-completeness. Proof The proof is done using Proposition 4.5.2 and a modification of the proof of Theorem 4.5.5. As before, let us write (for the purposes of this proof) simply `L instead of ` ( A S SL ) L . We start by observing that AS SL ⊆ SL and that → is a weak implication in `L for any language L ⊆ LSL (the monotonicity rules for ∧ and ∨ are

4.5 Axiomatization of LL, SL, and their fragments

191

easy to prove from the corresponding rules of AS SL ). Therefore, in particular, ` A S SL is a weakly implicative logic and so AS SL is an axiomatic system for SL (recall, that we have already established the inclusion ` A S SL ⊆ SL). The converse inclusion follows from the definition of SL and the fact that all connectives of LSL behave in ` A S SL as expected). Let us now process all possible languages L allowed by our assumptions. First, assume that L = LLL . Obviously, `LLL is the least logic in the language LLL with weak implication →, co-implication , residuated conjunction &, and unit 1¯ and in Corollary 4.4.17 we have shown that this logic is the LLL -fragment of SL (i.e. indeed `LLL = SLLLL as required) and in the proof of that corollary we have shown that the class of its reduced models admits regular completions. Next, assume that L contains ∨. Thus, from the assumption, we know that L ⊇ LLL and so for any A ∈ Mod∗ (`L ) we know that A LLL ∈ Mod∗ (`LLL ) = Mod∗ (SLLLL ). Because Mod∗ (SLLLL ) admits regular completions, we obtain a matrix B ∈ Mod∗ (SLLLL ) and an embedding 𝑓 : A LLL → B preserving all existing suprema and infima. The proof is complete if we realize that B LSL ∈ Mod∗ (SL) and that 𝑓 can be seen as an embedding 𝑓 : A → (B LSL )L . Next, assume that neither ∨ nor ∧ are in L: Consider the ternary relation 𝑅 and the matrix A0 constructed in the proof of Theorem 4.5.5. If we prove that 𝑅(𝐹, 𝑋, 𝑌 )

iff

𝑋 ⊆𝑌

we can use the final part of Lemma 4.5.3 to obtain that A0 ∈ Mod∗ (SLLLL ) and therefore (A0) LSL ∈ Mod∗ (SL); the fact that the embedding 𝑓 preserves lattice truth-constants possibly present in L is left as an exercise for the reader. To prove one direction of this equivalence, assume that 𝑢 ∈ 𝑋 and 𝑅(𝐹, 𝑋, 𝑌 ). Taking 𝑥 = 1¯ or 𝑥 = 𝑢 𝑢 (depending on which connective is in L), we obtain ¯ or the axiom (id )) and 𝑢 ≤ 𝑥 → 𝑢 (due to the axiom 𝑥 ∈ 𝐹 (due to the theorem 116 (push) or the rule (E )) which implies that 𝑥 → 𝑢 ∈ 𝑋 (as 𝑋 is ≤-upperset) and so 𝑢 ∈ 𝑌 due to the assumption 𝑅(𝐹, 𝑋, 𝑌 ). To prove the converse direction, it clearly suffices to prove that 𝑅(𝐹, 𝑋, 𝑋) and then the monotonicity condition on 𝑅 completes the proof. Assume that 𝑥 ∈ 𝐹 and 𝑥 → 𝑦 ∈ 𝑋, then, due to the rule (As), we obtain 𝑥 → 𝑦 ≤ 𝑦 and so 𝑦 ∈ 𝑋. Finally, assume that ∧ is in L but ∨ is not. Here we have to do a bigger modification of the proof of Theorem 4.5.5 and build the matrix A0 using ∧-filters of ≤ instead of all ≤-uppersets. Let us elaborate it in details. Consider any such language and a matrix A = hA, 𝐹i ∈ Mod∗ (`L ); we need to show that there is a matrix A0 ∈ Mod∗ (SL) and a strict embedding 𝑓 : A → AL0 . We know that → is a weak implication in `L ; let us denote by ≤ the matrix ∧-semilattice order on A given by → (or ∧). Let us define the set Fi ≤ ( 𝐴) = {𝑋 ∈ 𝑈 ≤ ( 𝐴) | 𝑥 ∧A 𝑦 ∈ 𝑋 for each 𝑥, 𝑦 ∈ 𝐴}. Clearly, the restriction of 𝑅 to Fi ≤ ( 𝐴) satisfies the conditions 1–5 required by Lemma 4.5.3 (the last one is proved as in the case for languages without ∧ and ∨). 16 Proved in the usual way using (pop), (MP), and (id).

192

4 On lattice and residuated connectives

Therefore, using Lemma 4.4.4 and Theorem 4.4.8, we know that A0 = hh𝑈 ⊆ (Fi ≤ ( 𝐴)), →,

¯ 1, ¯ ∧, ∨, ⊥, >i, 𝐹⊆ i ∈ Mod∗ (SL) , &, 0,

(where, as before, 0¯ is defined as {𝑋 ∈ Fi ≤ ( 𝐴) | 0¯ ∈ 𝑋 } if 0¯ is in L and arbitrarily otherwise). Clearly, we also know that [𝑎) ∈ Fi ≤ ( 𝐴) and that 𝑔(𝑎) = {𝑋 ∈ Fi ≤ ( 𝐴) | 𝑎 ∈ 𝑋 } is a mapping from A into AL0 and we have to show that it is an embedding. The proof of this fact proceeds along the lines of the proof of Theorem 4.5.5 with only a few alterations. When dealing with implication (resp. co-implication) we have to make sure that for the set 𝑌 defined as {𝑦 ∈ 𝐴 | 𝑎 → 𝑦 ∈ 𝑈} (or as {𝑦 ∈ 𝐴 | 𝑎 𝑦 ∈ 𝑈} resp.) we have 𝑌 ∈ Fi ≤ 𝐴. Clearly, in both cases we have 𝑌 ∈ 𝑈 ≤ ( 𝐴). We only have to show that 𝑌 is closed under ∧. Let us deal with implication first: assume that 𝑥, 𝑦 ∈ 𝑌 , i.e, 𝑎 → 𝑥, 𝑎 → 𝑦 ∈ 𝑈; therefore (𝑎 → 𝑥)∧(𝑎 → 𝑦) ∈ 𝑈 and, due to the validity of (inf), we know that 𝑎 → 𝑥 ∧ 𝑦 ∈ 𝑈, i.e. 𝑥 ∧ 𝑦 ∈ 𝑌 as required. The case of co-implication is analogous; it suffices to show that `L ( 𝜒 𝜑) ∧ ( 𝜒 𝜓) → ( 𝜒 𝜑 ∧ 𝜓). We give a formal proof (let us set 𝛿 = ( 𝜒 𝜑) ∧ ( 𝜒 𝜓)): a) 𝛿 → ( 𝜒 𝜑) (lb) b)

𝜒 → (𝛿 → 𝜑)

a and (E )

c)

𝜒 → (𝛿 → 𝜓)

analogously

d)

𝜒 → (𝛿 → 𝜑) ∧ (𝛿 → 𝜓)

e)

𝜒 → (𝛿 → 𝜑 ∧ 𝜓)

f) 𝛿 → ( 𝜒

b, c, (Adj), (inf), and (MP) d, (inf), and (Pf)

𝜑 ∧ 𝜓)

e and (E )

Finally, we have to prove that 𝑔(𝑎 ∧ 𝑏) = 𝑔(𝑎) ∧ 𝑔(𝑏). As obviously ∧ on 𝑈 ⊆ (Fi ≤ ( 𝐴)) is the simple intersection, the claim easily follows from the fact that each 𝑈 ∈ Fi ≤ ( 𝐴) is closed under ∧.  Remark 4.5.8 Besides the expected condition of having the main implication in the language, the previous theorem requires two additional restrictions on possible fragments. The first one is especially troublesome as it entails that we do not know how to axiomatize SL→ and, in particular, whether it is the extension of LL→ by the assertion rule (As) (which would be implied by the unrestricted formulation of the theorem). It is easy to see that our proof needs the presence of or 1¯ in the language to allow us to apply the last part of Lemma 4.5.3 and ensure that 1¯ is the unit in A0. We can show that if neither nor 1¯ are in the language, our proof method fails. Indeed, consider the implicational reduct of the matrix A1 from Example 4.2.17, i.e. the matrix A = hA, {>}i where 𝐴 = {⊥, 𝑎, >} and →A is given by the following table: →A ⊥ 𝑎 >

⊥ > ⊥ ⊥

𝑎 > > ⊥

> > > >

4.5 Axiomatization of LL, SL, and their fragments

193

It is easy to see that A ∈ Mod∗ (`L→ ) and the matrix preorder ≤A is a linear one which can be schematically depicted as ⊥ < 𝑎 < >. We leave as an exercise for the reader to show that the matrix A0 constructed using our proof method would still be a three-element matrix which is actually an expansion of A with naturally defined lattice connectives and truth-constants; indeed, we would obtain the following equations: 𝑈 ≤ ( 𝐴) = {{>}, {𝑎, >}, 𝐴} 𝑈 ⊆ (𝑈 ≤ ( 𝐴)) = {{ 𝐴}, { 𝐴, {𝑎, >}}, 𝑈 ≤ ( 𝐴)} 𝐹⊆ = {𝑈 ≤ ( 𝐴)}. 0 0 Note that, in particular, we would obtain that 1¯ A = >A = 𝑔(>) = 𝑈 ≤ ( 𝐴) and 0 0 A A therefore also 1¯ → 𝑔(𝑎) = 𝑔(> → 𝑎) = 𝑔(⊥) ≠ 𝑔(𝑎), which is a contradiction with A0 ∈ Mod∗ (SL). Unfortunately, this example only demonstrates the insufficiency of our proof method, not the failure of the conjecture itself. Indeed, as an exercise the reader can check that A is a subreduct of the matrix B ∈ Mod∗ (SL1¯ ) with domain 𝐵 = {⊥, 𝑎, 1, >}, truth-constant 1¯ interpreted as 1 and implication → as

→B ⊥ 𝑎 1 >

⊥ > ⊥ ⊥ ⊥

𝑎 > > 𝑎 ⊥

1 > > 1 1

> > > > >

and thus, by the previous theorem, A is a subreduct of some C ∈ Mod∗ (SL). Therefore, the question whether SL→ is the extension of LL→ by (As) remains open (see Section 4.11 for more comments on this issue). The problem in the case of disjunction is perhaps less surprising due to the already demonstrated different nature of (lattice) conjunctions and disjunctions. On the other hand, our restriction is motivated mainly by the simplicity of the proof. One could consider the weaker condition ‘if ∨ ∈ L, then so is ∧’, but that would require a rather complex modification of the proof which is not warranted by the limited utility of a slightly more general result (again, more details are to be found in Section 4.11; see footnote 37). Remark 4.5.9 Note that SL is the expansion of LL by the axioms (>), (⊥), (lb), and (ub) and the rules (Symm), (Inf), and (Sup). Indeed, we know that all these consecutions are valid in SL and, conversely, all connectives in the thusly defined logic behave as they should and so it has to extend SL which is the least such logic. The axiomatic systems AS SL depicted in Figure 4.2 is more complex in order to obtain the separability result. First, it contains several redundant consecutions: the axiom (id) (a consequence of the axioms (push) and (pop) using the transitivity), the axioms (id ) and (As) (cf. Proposition 4.3.10) and the rule (Adj) (cf. Proposition 4.3.12.) Second, we had to include the axiom (inf) instead of the rule (Adj) (cf. Example 4.1.8) and the rule (Sup) instead of the axiom (sup) (due to its use of the connective ∧).

194

4 On lattice and residuated connectives

Remark 4.5.10 We have formulated the previous theorem for SL seen as a logic in LSL . Recall that from Proposition 4.3.9 we know that > is definable in any fragment of SL containing and ⊥. Thus, we could see SL as a logic in LSL \ {>} where > is a just an abbreviation for the formula ⊥ → ⊥. Indeed, we could formulate the previous corollary for all languages not containing > and the axiomatic system without the axiom (top). However, this would be a clearly inferior result because it would not yield axiomatizations for certain fragments of SL where > has to be considered primitive e.g. SL1,> ¯ . In stronger logics where more connectives are definable (see Corollary 4.6.9) the situation is more complicated; see Remark 4.7.2 for more details. Recall that in Corollary 4.4.17 we have considered two logics resulting from using opposing requirements for 1¯ in the definition of Lambek logic and SL: the logic SLLLL (the least extension of LL where 1¯ is the unit) and LLlat (the conservative expansion of LL by lattice connectives). Inspecting the proof of the previous corollary we can find certain strongly separable presentations of these logics. Moreover, we can prove that the matrices of corresponding fragments admit completions and, thus, these fragments enjoy the strong completeness w.r.t. their completely ordered reduced matrices. In particular, we obtain the following: Proposition 4.5.11 • The axiomatic system listed in Figure 4.1 with the rules (Push) and (Pop) replaced by the axioms (push) and (pop) is a strongly separable presentation of SL LLL for ¯ all the sublanguages of LLL containing → and, additionally, or 1. • The axiomatic system listed in Figure 4.1 expanded by the axioms (lb), (ub), (inf), (sup), (⊥), (>) and the rule (Adj) is a strongly separable presentation of LLlat for all the sublanguages of LSL containing → and extending LLL whenever they contain ∨.

4.6 Substructural logics and prominent extensions of SL This section has two primary goals: (1) to propose a formal definition of a large family of well-behaved logics which will encompass most logics studied in the literature under the moniker substructural logics and (2) to study some of the most prominent examples. We start by stressing that our definition of the family of substructural logics is of a pragmatic nature, rather than motivated by any philosophical or methodological analyses of the involved pretheoretical ideas. Therefore, a concerned reader should see it as a mere convention. Our goal is to give a simple and natural definition of a very broad class of logics that includes the majority of logics that may have been labeled in this manner in the literature (formally or informally) and also other logics to which our methods will conveniently apply (thus, we include logics such as global modal ones, which are usually not considered substructural); see Example 4.6.2. We

4.6 Substructural logics and prominent extensions of SL

195

aim at striking a balance between the generality of the results and the simplicity of their presentation. Indeed, we could achieve a greater level of generality by means of a more complex, and probably less natural, definition or dually simplify our presentation at the price of leaving certain important logics outside of its scope (let us note that there are other formal definitions of the class of substructural logics, usually more specific; see Section 4.11). Definition 4.6.1 A logic L in a language L ⊇ L→ is substructural if → is a weak implication and the following consecutions are valid: (Sf) (Pf)

𝜑 → 𝜓 I (𝜓 → 𝜒) → (𝜑 → 𝜒) 𝜑 → 𝜓 I ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓)

suffixing prefixing.

Example 4.6.2 Almost all the examples of weakly implicative logics introduced up to now are substructural, in particular: • the logic BCI and its expansions where → remains a weak implication; • the logic BCIlat and its extensions, including Łukasiewicz, intuitionistic and classical logics, and the infinitary logic Ł∞ ; • the contraclassical logics (i.e. logics that prove some formula that is not a theorem of CL) A and asA, which we will introduce in upcoming Example 4.6.14; • the global modal logic K and its extensions; and • the logics LL and SL together with all their fragments containing implication and all their expansions where → is a weak implication. It is obvious that the inconsistent logic in any language containing → is the largest substructural logic in that language. In contrast, it is easy to see that the minimum and the almost inconsistent logic, and local modal logics are not substructural logics (as we have seen in Example 2.8.6, 𝑙K and 𝑙T are not even weakly implicative, whereas 𝑙K4 and 𝑙S4 are weakly implicative and expand SL→ but the connective → does not behave as a weak implication in these logics). Thanks to Theorem 4.5.5, we obtain the following description of the least substructural logic and an alternative definition of this class of logics. Proposition 4.6.3 The logic LL→ , i.e. the implicative fragment of Lambek logic, is the least substructural logic (in the language L→ ) and it has the presentation: (id) (MP) (Sf) (Pf)

𝜑→𝜑 𝜑, 𝜑 → 𝜓 I 𝜓 𝜑 → 𝜓 I (𝜓 → 𝜒) → (𝜑 → 𝜒) 𝜑 → 𝜓 I ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓)

identity modus ponens suffixing prefixing.

Therefore, a logic L is substructural iff it expands LL→ and → is a weak implication.

196

4 On lattice and residuated connectives

Our next step is to find an axiomatization for the least substructural logic expanding a given substructural logic L by any of the particular kinds of connectives we have considered in this chapter, with the exceptions of lattice disjunction17 and, of course, ¯ Also we will show that, if the starting logic is the least substructural logic LL→ , of 0.18 we usually obtain a fragment of LL, LLlat , or SL.19 Note that we could additionally use the previous proposition to axiomatize the least substructural logic expanding L where any set of those connectives behaves as prescribed by simply using the collection of the corresponding consecutions. Proposition 4.6.4 Let L be a substructural logic and 𝑐 a connective of LSL other than →. Then, the least substructural logic expanding L where 𝑐 is

expands L by adding

and, if L = LL→ , it coincides with

top bottom protounit unit lattice protodisjunction lattice protoconjunction lattice conjunction residuated conjunction co-implication dual co-implication

(>) (⊥) (Push) and (Pop) (push) and (pop) (ub) and (Sup) (lb) and (Inf) (lb), (inf), and (Adj) (Res) (E ) (E ) and (Symm)

LLlat > LLlat ⊥ LL1¯ SL1¯

LLlat ∧ LL& LL SL .

Proof First of all, we have to prove that → is a weak implication in the expansions of L by the additional connectives. For residuated connectives it follows from Lemma 4.5.4 and for the lattice ones it is implicit in the beginning of the proof of Theorem 4.5.7 (it just suffices to prove the expected monotonicity rules). Next, consider any row of the table and note that for any extension L0 of L we know that 𝑐 is, in L0, a connective of the type specified in its first column iff the consecutions in its second column are valid in L0. Thus, the extension of L by these consecutions is the least logic where 𝑐 is a connective of the required type. The claims about fragments follow from the already established strongly separable axiomatic systems for the corresponding logics; see Theorems 4.5.5 and 4.5.7 and Proposition 4.5.11 (the fact that the rules (Symm) can be replaced by (As) and (id ) was established in Proposition 4.2.12).  17 Recall that, unlike all other cases, the lattice disjunction is defined through the validity of a metarule, called proof by cases property, instead of just a (set of) consecution(s). Therefore, it should not be surprising that the problem of axiomatizing the least substructural logic expanding L with a lattice disjunction is rather non-trivial; we give a solution to this problem in Corollary 5.5.19. 18 As we have assumed nothing about this truth-constant, this logic is simply the expansion of L to the language with 0¯ by the empty set of consecutions. 19 Recall that LLlat is the conservative expansion of LL by lattice connectives or equivalently the logic axiomatized by A S SL (Figure 4.2) with the axioms (push) and (pop) replaced by the corresponding rules.

4.6 Substructural logics and prominent extensions of SL

197

Table 4.2 Consecutions defining principal substructural logics.

(a1 ) (a2 ) (E) (C) (i) (o)

𝜑 & (𝜓 & 𝜒) → (𝜑 & 𝜓) & 𝜒 (𝜑 & 𝜓) & 𝜒 → 𝜑 & (𝜓 & 𝜒) 𝜑 → (𝜓 → 𝜒) I 𝜓 → (𝜑 → 𝜒) 𝜑 → (𝜑 → 𝜓) I 𝜑 → 𝜓 𝜑 → 1¯ 0¯ → 𝜑

re-associate to the left re-associate to the right exchange contraction 1¯ ex quolibet ex 0¯ quodlibet.

Remark 4.6.5 Let us note that, in axiomatizing LLlat ∧ , we could replace the rule (Adj) ¯ 1 with the axiom (distr& ) and the rule (Adj1¯ ): ¯

1) (distr& (Adj1¯ )

¯ & (𝜓 ∧ 1) ¯ → (𝜑 & 𝜓) ∧ 1¯ (𝜑 ∧ 1) ¯ 𝜑 I 𝜑∧1

¯ 1-&-distributivity ¯ 1-adjunction. ¯

1 ) is shown to Indeed, (Adj1¯ ) can be obtained as an instance of (Adj), and (distr& be a theorem of LLlat in Proposition 4.3.13. Conversely, a) 𝜑 ` 𝜑 ∧ 1¯ (Adj1¯ ) ¯ b) 𝜓 ` 𝜓 ∧ 1 (Adj1¯ )

¯ & (𝜓 ∧ 1) ¯ c) 𝜑, 𝜓 ` (𝜑 ∧ 1) d) 𝜑, 𝜓 ` 𝜑 ∧ 𝜓

a, b, and (Adj& ) ¯

1 ), and (lb) c, (distr&

Note that this fact, together with Theorem 4.5.7 and Remark 4.5.9, entails that the logic SL can be axiomatized by the following consecutions: (Pf), (Sf), (E ), (Res), 1¯ ), and (Adj ). (MP), (push), (pop), (⊥), (>), (lb), (ub), (inf), (sup), (distr& 1¯ Now we are ready to introduce the principal family of examples of substructural logics studied in the literature, which will see to play an important role throughout this book. These logics are defined by combinations of the consecutions listed in Table 4.2 (recall our standing convention on the usage of lower/upper case letters in names of axioms/rules). The reader familiar with Gentzen-type sequent calculi for classical or intuitionistic logic can clearly see that these consecutions correspond to the so-called structural rules. Note that in the case of exchange and contraction we consider their rule forms and, instead of the weakening rule, we use the axiom 1¯ ex quolibet. Our selection of defining consecutions is motivated by their mutual interplay (see Proposition 4.6.7) and an effort to avoid clashes with existing notational conventions for substructural logics (see Section 4.11 for more details). Definition 4.6.6 Given any 𝑋 ⊆ {a1 , a2 , E, C, i, o}, we denote by SL𝑋 (𝑋 written as a list without spaces or commas) the extension of SL by the consecutions corresponding to the letters in 𝑋. If both a1 and a2 are in 𝑋, then we denote the pair by the symbol a.

198

4 On lattice and residuated connectives

Fig. 4.3 Prominent extensions of the logic SL.

A part of the resulting family of basic substructural logics is depicted in Figure 4.3. For simplicity, we include only logics satisfying either both or none of the associativity axioms and logics without the axiom (o),20 i.e. logics SL𝑋 , for 𝑋 ⊆ {a, E, C, i}. In Corollary 4.6.9 we will see that if {C, i} ⊆ 𝑋, then SL𝑋 = SLaECi , which explains the inclusion relation between the logics depicted in the figure; on the other hand in Example 4.6.20 we will show that all logics depicted there are mutually different. For a better understanding of the behavior of connectives in this family of logics, let us now consider equivalent formulations of the rules in Table 4.2. The next proposition shows that almost all of them can be described as properties of &. Moreover, it also shows under which conditions the conjunction & is associative. It turns out that both halves of associativity are equivalent to other interesting logical laws, usually resulting from strengthening rules of SL into an axiomatic form. Note, that while we formulate the proposition for SL, the proved equivalences involving only connectives of LLL are valid already in the logic SLLLL ; note, however, that it would not always be the case for LL, cf. Footnote 21. 20 The reasons for not including axiom (o) will be apparent after Remark 4.6.19 where we show that, seen as logics in LSL , SL𝑋o is strictly stronger than SL𝑋 and, if SL𝑋 is strictly stronger than SL𝑌 , then SL𝑋o is strictly stronger than SL𝑌 o ; thus we could produce exactly the same hierarchy of substructural logics with axiom (o); we depict this hierarchy later, in Figure 4.9, where we use the traditional names some of these logics have in the literature.

4.6 Substructural logics and prominent extensions of SL

199

Proposition 4.6.7 The logic SLa1 is the axiomatic extension of SL by any of the following formulas:21 (e

,2 )

(𝜑 → (𝜓

𝜒)) → (𝜓

(𝜑 → 𝜒))

-exchange

(res2 )

(𝜑 & 𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))

residuation

(pf)

(𝜑 → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓))

prefixing.

The logic SLa2 is the axiomatic extension of SL by any of the following formulas: (e

,1 )

(𝜓

(𝜑 → 𝜒)) → (𝜑 → (𝜓

𝜒))

(res1 )

(𝜓 → (𝜑 → 𝜒)) → (𝜑 & 𝜓 → 𝜒)

(sf )

(𝜑

𝜓) → ((𝜓

𝜒) → (𝜑

-exchange residuation

𝜒))

-suffixing.

The logic SLE is the axiomatic extension of SL by any of the following formulas: (symm1 )

(𝜑 → 𝜓) → (𝜑

(symm2 )

(𝜑

(C& )

𝜑&𝜓 →𝜓&𝜑

𝜓)

symmetry

𝜓) → (𝜑 → 𝜓)

symmetry &-commutativity.

The logic SLC is the axiomatic extension of SL by any of the following formulas: (s-i)

𝜑→𝜑&𝜑

square-increasingness

(∧-sup-&)

𝜑∧𝜓 → 𝜑&𝜓

&-supinfimality.

The logic SLi is the (axiomatic) extension of SL by any of the following consecutions: (W)

𝜑I𝜓→𝜑

weakening

(w)

𝜑 → (𝜓 → 𝜑)

weakening

(∧-sub-&)

𝜑&𝜓 → 𝜑∧𝜓

&-subinfimality.

The logic SLaE coincides with SLa1 E , and SLa2 E is the axiomatic extension of SL by the following formula: (e)

(𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))

exchange.

The logic SLa2 C is the axiomatic extension of SLa2 by the following formula: (c)

(𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)

contraction.

Proof All, but the last two, claims of this proposition will be proved by showing (a chain of) implications of the form ‘the extension of SL by the consecution 𝑥 derives the consecution 𝑦’ (𝑥  𝑦 in symbols). 21 We use the name -exchange and symbol (e ,2 ) for simplicity, even though it breaks our standing convention that it should entail the corresponding rule (E ,2 ) just by using modus ponens; clearly we need the additional rule (Symm) to obtain this claim.

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4 On lattice and residuated connectives

SLa1 : We first show the equivalence of (e ,2 ), (res2 ), and (pf). Let us prove (e ,2 )  (res2 ): we apply the rules (E ) and (Res) on (𝜑&𝜓 → 𝜒) → (𝜑&𝜓 → 𝜒) to obtain 𝜓 → (𝜑 → ((𝜑 & 𝜓 → 𝜒) 𝜒)). Thus, by the axiom (e ,2 ) and transitivity, we obtain 𝜓 → ((𝜑 & 𝜓 → 𝜒) (𝜑 → 𝜒)) and the rule (E ) finishes the proof. To prove (res2 )  (pf), we start with ( 𝜒 → 𝜑) → ( 𝜒 → 𝜑), use the rule (Res) to obtain 𝜒 & ( 𝜒 → 𝜑) → 𝜑 and then, by using the rule (Sf), we obtain (𝜑 → 𝜓) → ( 𝜒 & ( 𝜒 → 𝜑) → 𝜓). The axiom (res2 ) in the form ( 𝜒 & ( 𝜒 → 𝜑) → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓))) and transitivity complete the proof. To prove the final implication, (pf)  (e ,2 ), we start with the theorem (as ) in the form 𝜓 → ((𝜓 𝜒) → 𝜒), use the axiom (pf) and transitivity to obtain 𝜓 → ((𝜑 → (𝜓 𝜒)) → (𝜑 → 𝜒)) and the rule (E ) finishes the proof. Next, we show that (res2 )  a1 and a1  (res2 ). To prove the former claim, we start with the theorem (adj& ) in the form 𝜒 → (𝜑&𝜓 → (𝜑&𝜓) & 𝜒) and, using the axiom (res2 ) and transitivity, we obtain 𝜒 → (𝜓 → (𝜑 → (𝜑&𝜓) & 𝜒)); the rule (Res) used twice completes the proof. To prove the latter, we start with ( 𝜒 → 𝜑) → ( 𝜒 → 𝜑) and apply the rules (Res) and (Sf) to obtain (𝜑 → 𝜓) → ( 𝜒 & ( 𝜒 → 𝜑) → 𝜓); the rule (Res), then, gives us ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓) → 𝜓 and therefore, using the axiom a1 , we obtain 𝜒 & (( 𝜒 → 𝜑) & (𝜑 → 𝜓)) → 𝜓 and the rule (Res) used twice completes the proof. SLa2 : We first show the equivalence of a2 , (e ,1 ), and (res1 ). To prove a2  (e ,1 ), we start with (𝜓 (𝜑 → 𝜒)) → (𝜓 (𝜑 → 𝜒)), use the rule (E ) to obtain 𝜓 → ((𝜓 (𝜑 → 𝜒)) → (𝜑 → 𝜒)), and then the rule (Res) twice to obtain 𝜑 & ((𝜓 (𝜑 → 𝜒)) & 𝜓) → 𝜒. Therefore, using the axiom a2 , we obtain (𝜑 & (𝜓 (𝜑 → 𝜒))) & 𝜓 → 𝜒 and so by rules (Res) and (E ) we also have (𝜑 & (𝜓 (𝜑 → 𝜒))) → (𝜓 𝜒) and so the rule (Res) completes the proof of this implication. To prove (e ,1 )  (res1 ), observe that from (𝜓 → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)) and the rule (E ) we obtain 𝜓 → ((𝜓 → (𝜑 → 𝜒)) (𝜑 → 𝜒)). Thus, by (e ,1 ) and transitivity, we also have 𝜓 → (𝜑 → ((𝜓 → (𝜑 → 𝜒)) 𝜒)) and so the rules (Res) and (E ) complete the proof. To prove the final implication, (res1 )  a2 , we start by using the rule (Res) twice on 𝜑 & (𝜓 & 𝜒) → 𝜑 & (𝜓 & 𝜒) to obtain 𝜒 → (𝜓 → (𝜑 → 𝜑 & (𝜓 & 𝜒))); then, by (res1 ) and transitivity, we obtain 𝜒 → (𝜑 & 𝜓 → 𝜑 & (𝜓 & 𝜒)); and, finally, the rule (Res) completes the proof. Next, we show that (e ,1 )  (sf ) and (sf )  (e ,1 ). To prove the former claim, we start with the theorem (as ) in the form 𝜓 → ((𝜓 𝜒) → 𝜒), use the rule (Pf ) to obtain (𝜑 𝜓) → (𝜑 ((𝜓 𝜒) → 𝜒)) and the axiom (e ,1 ) finishes the proof. To prove the latter, we start with (𝜑 → 𝜒) → (𝜑 → 𝜒) and use the rules (E) and (Sf) to obtain (((𝜑 → 𝜒) 𝜒) → (𝜓 𝜒)) → (𝜑 → (𝜓 𝜒)), and so the axiom (sf ) completes the proof. SLE : Again we prove a chain of implications starting with E  (symm1 ). To prove this, just use the rules E and (E ) on (𝜑 → 𝜓) → (𝜑 → 𝜓). Next, we prove that (symm1 )  (symm2 ): starting with the theorem (as ): 𝜑 → ((𝜑 𝜓) → 𝜓) we use the axiom (symm1 ) and transitivity to obtain 𝜑 → ((𝜑 𝜓) 𝜓) and the rule (E ) completes the proof. To prove the next implication, (symm1 )  (C& ), start with the theorem (adj& ) in the form 𝜑 → (𝜓 → 𝜓 & 𝜑), use the rule (E ), the axiom (symm2 ), and transitivity to obtain 𝜓 → (𝜑 → 𝜓 & 𝜑) and then the rule (Res) to

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complete the proof. To prove the final implication, (C& )  E, apply the rule (Res) on the assumption 𝜑 → (𝜓 → 𝜒) to obtain 𝜓 & 𝜑 → 𝜒. The axiom (C& ), together with transitivity and the rule (Res), again completes the proof. SLC : The implications C(s-i), (s-i)C, and (∧-sup-&)(s-i) are straightforward and the proof of the final implication, (s-i)  (∧-sup-&), is also easy: using the axioms (lb) and the rules (Mon& ) we obtain (𝜑 ∧ 𝜓) & (𝜑 ∧ 𝜓) → 𝜑 & 𝜓 and so (s-i) completes the proof. SLi : We first show the equivalence of (i), (W), and (w). The first implication is easy: indeed from (i) we know that 𝜓 → 1¯ and as we always have 𝜑 ` 1¯ → 𝜑 the transitivity completes the proof of this implication. Next, we use (W) to obtain 𝜓 → (𝜑 𝜑) and so (E ) completes the proof of the next implication. The ¯ To prove the rest of this final implication just follows from (w) by taking 𝜑 = 1. claim, observe that, from (W) and (w), we directly obtain 𝜓 → (𝜑 → 𝜑) and 𝜓 → (𝜑 → 𝜓) and so the claim follows using (Res) and (Inf). Conversely, we easily obtain (w) from (∧-sub-&) using (lb) and (Res). The last two claims are easy consequences of the already established characterizations of SLa1 , SLa2 , SLE , and SLC .  Remark 4.6.8 Using the previous proposition and Remark 4.6.5 we can easily observe that the logic SLa can be axiomatized by (pf), (MP), (res), (e ), (Symm), (push), 1¯ ), and (Adj ). (pop), (⊥), (>), (lb), (ub), (inf), (sup), (distr& 1¯ Indeed, all these consecutions are valid in SLa and to prove the converse it suffices to observe that the rules (Sf), (Pf), (res), and (E ) are derivable in the proposed axiomatic system (cf. footnote 21). Later, in Figure 4.7 we will give a more common presentation of this logic which replaces the rules (Symm) with the syntactically simpler rules of product normality, see Theorem 4.9.5. The next corollary speaks about the definability (see Definition 2.3.4) of certain connectives in some of the just defined substructural logics and the resulting collapse of some of them. We leave the proof as an exercise for the reader. Let us first observe that in any substructural logic L with lattice conjunction, the equivalence connective ↔ defined as in LCL , i.e. as (𝜑 → 𝜓) ∧ (𝜓 → 𝜑), is a weak implication and we have 𝜑 ↔ 𝜓 a`L 𝜑 ⇔ 𝜓 (for 𝜑 ⇔ 𝜓 being either {𝜑 → 𝜓, 𝜓 → 𝜑} or {𝜑 ↔ 𝜓, 𝜓 ↔ 𝜑}. Therefore, when speaking about definability, we can use this connective instead of the equivalence set. Corollary 4.6.9 Let L be an expansion of SL. ¯ 1. L expands SLo iff 0¯ and ⊥ coincide in L, i.e. `L ⊥ ↔ 0. 2. L expands SLi iff L is Rasiowa-implicative iff 1¯ and > coincide in L, i.e. ¯ Note that in this case 1¯ can be also viewed as a defined connective (by `L > ↔ 1. 1¯ ↔ (𝜑 → 𝜑)). 3. L expands SLE iff → and coincide in L, i.e. `L (𝜑 → 𝜓) ↔ (𝜑 𝜓). 4. L expands SLCi iff & is the lattice conjunction in L iff ∧ and & coincide in L, i.e. `L 𝜑 ∧ 𝜓 ↔ 𝜑 & 𝜓. Therefore, SLCi = SLaECi and SLCio = SLaECio .

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4 On lattice and residuated connectives

Therefore, any logic extending SLCio can be seen as a logic in the language ¯ 1, ¯ LCL (since, clearly, each of the connectives 0, , and & is definable in SLCio using a formula of LCL ) and, modulo this identification, as an easy consequence of Corollary 4.7.5, we obtain SLCio = SLaECio = IL. Note that the language of weaker logics can also be simplified, in particular in the following two cases for which we will obtain specific results later: ¯ 1, ¯ &}. • The logic SLaE can be seen as a logic in LSL \ { } = LCL ∪ {0, ¯ ¯ • The logic SLaEio can be seen as a logic in LSL \ { , >, 0, 1} = LCL ∪ {&}. Because of its prominence in the literature, we refer to the logic SLaEio as FLew and see it as a logic in the language LCL ∪ {&} (see Section 4.11 for historical explanation of this notation). Remark 4.6.10 Recall the general fact that a logic in a language in which a connective 𝑑 ∈ L is definable (by a formula 𝜑 𝑑 not using 𝑑) can be seen as a logic in the languages L and L \ {𝑑} and, in the latter case, the connective 𝑑 can, but need not, be used as an abbreviation for its defining formula 𝜑 𝑑 . For example, even though > is definable already in SL as ⊥ → ⊥, we consider SL as a logic in the language LSL ; however, intuitionistic logic can be seen as a logic in LCL which includes neither > nor , but while we consider > as an abbreviation for ⊥ → ⊥, we do not see as an abbreviation for → (even though it would make perfect sense as we have seen above). This arbitrariness in seeing connectives as primitive or defined will play a role in the next section. An extreme case of definability is the logic A→ which, while being a logic in the language containing solely the implication connective, can actually be seen as a logic in the language LLL , and the class Alg∗ (A→ ) actually coincides (modulo translations provided by the definability) with the variety of Abelian groups.22 The next series of examples elaborates these connections in order to obtain some interesting facts about A→ and its expansions. Example 4.6.11 The logic A→ can be seen as the extension of (SLaE ) LLL by the axiom (abe). Obviously, we can define as →; by Example 4.3.3, we know that we can define the unit 1¯ as 𝑝 → 𝑝; let us set 0¯ as 𝑝 → 𝑝 as well; and, finally, define 𝜑 & 𝜓 ¯ or, strictly speaking in this case, as ¬(𝜑 → ¬𝜓) (recall that ¬𝜒 is defined as 𝜒 → 0, as 𝜒 → ( 𝜒 → 𝜒), i.e. we need not use any extra variable). Inspecting the relevant parts of Proposition 4.6.7 (and noticing that the lattice connectives play no role in their proof), it is clear that the validity of the residuation rule (Res) is all we have to 22 Let us recall that an algebra h𝐴, ·, −1 , 𝑒i is an Abelian group whenever · is a commutative and associative binary operation with unit 𝑒 and −1 is the inverse operation (i.e. for each 𝑎 ∈ 𝐴, we have 𝑎 · 𝑎−1 = 𝑒). It is well-known that the variety of Abelian groups is generated by the additive group of integers hZ, +, (−1) ·, 0i.

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prove to obtain the claim. To do that, let us first prove two interesting properties of the negation connective (which justify identifying 0¯ with 1¯ in this particular logic): (dne) (cp)

¬¬𝜑 → 𝜑 ¬𝜑 → 𝜓 I ¬𝜓 → 𝜑

double negation elimination contraposition.

The validity of (dne) follows from the axiom (abe), and (cp) follows from (Sf), (dne), and (Pf). The validity of the residuation rule then easily follows: 𝜑 → (𝜓 → 𝜒) a` 𝜑 → (¬𝜒 → ¬𝜓) a` ¬𝜒 → (𝜑 → ¬𝜓) a` ¬(𝜑 → ¬𝜓) → 𝜒. Let us extend this identification of logics to algebras and matrices. Of course, here it works well only if, in the algebra in question, the unary function 𝑥 → 𝑥 is constant (but this is clearly the case when it is an element of Alg∗ (A→ )). Thus, we can see the algebra Z and the matrices Z+ = hZ, {𝑎 | 0 ≤ 𝑎}i and Z0 = hZ, {0}i introduced in Example 2.8.10 (where we have shown that Z+ , Z0 ∈ Mod∗ (A→ )) as LLL -algebra/matrices. To show that promised relation between Alg∗ (A→ ) and Abelian groups, we need one additional straightforward translation: for any LLL -algebra A, we can set 𝑒 = 1¯

𝑥 −1 = 𝑥 → 0¯

𝑥 · 𝑦 = 𝑥 & 𝑦,

and, conversely, for any algebra h𝐴, ·,−1 , 𝑒i, we can define 𝑥&𝑦=𝑥·𝑦

𝑥→𝑦=𝑥

𝑦 = 𝑦 · 𝑥 −1

0¯ = 1¯ = 𝑒.

Now we can easily show that A ∈ Alg∗ (A→ ) iff A is an Abelian group. Indeed, to prove the left-to-right direction, it suffices to use that 𝜑 & ¬𝜑 ⇔ 0¯ is a theorem of A→ ; to prove the converse direction, just check that hA, {𝑒}i ∈ Mod∗ (A→ ), where 𝑒 is the unit of A. Of course, as A→ is not algebraizable, the algebras in Alg∗ (A→ ) do not completely determine Mod∗ (A→ ). For that, we need the notion of ordered Abelian group: we say that a residuated ordered protounital groupoid hA, ≤i is an ordered Abelian group, whenever A is an Abelian group.23 Now we can use Lemma 4.4.4 and Theorem 4.4.5 to obtain the following two facts: • A ∈ Mod∗ (A→ ) iff hA, ≤A→ i is an ordered Abelian group. • hA, ≤i is an ordered Abelian group iff hA, {𝑎 ∈ 𝐴 | 𝑒 ≤ 𝑎}i ∈ Mod∗ (A→ ). Thus, in particular, the matrices Z0 and Z+ correspond to the group of integers with, respectively, the discrete and the natural order. 23 Note that we use the translation mentioned above. It is worth noting that the residuation condition is, in this context, implied by the much simpler monotonicity condition expressible in the original language of Abelian groups: 𝑎 ≤ 𝑏 implies 𝑎 · 𝑐 ≤ 𝑏 · 𝑐.

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4 On lattice and residuated connectives

We can use the just established identification of languages and algebraic facts on Abelian groups to prove that A→ enjoys completeness (but not finite strong completeness) w.r.t. (each of) Z0 and Z+ . Example 4.6.12 • A→ is complete w.r.t. Z0 . Due to Theorem 3.4.6, it suffices to show that for each matrix hA, 𝐹i ∈ Mod∗ (A→ ), we have hA, 𝐹i ∈ HSP(K). As the variety of Abelian groups is generated by Z, we know that A ∈ HSP(Z), i.e. there is a surjective group homomorphism ℎ : C → A, where C is a subalgebra of a product of Z. Let us denote the index set of the product by 𝐼 and note that h0i𝑖 ∈𝐼 is the unit of C and, thus, ℎ(h0i𝑖 ∈𝐼 ) = 0¯ A . Therefore, ℎ is a matrix homomorphism of hC, h0i𝑖 ∈𝐼 i onto hA, 𝐹i and, as we obviously have hC, h0i𝑖 ∈𝐼 i ∈ SP(hZ, {0}i), the proof is done. • A→ is also complete w.r.t. Z+ . Note that for each formula 𝜑 built from variables 𝑝 1 , . . . , 𝑝 𝑛 , there are integers 𝑎 1 , . . . , 𝑎 𝑛 such that, for each Z-evaluation 𝑒, we have 𝑒(𝜑) = 𝑎 1 𝑒( 𝑝 1 ) + · · · + 𝑎 𝑛 𝑒( 𝑝 𝑛 ). Therefore, if 𝜑 is valid in Z+ , in particular we have 𝑎 1 (−𝑎 1 ) + · · · + 𝑎 𝑛 (−𝑎 𝑛 ) ≥ 0 which can only be true if 𝑎 𝑖 = 0 for each 𝑖 ≤ 𝑛; hence 𝜑 is valid in Z0 and thus, due to the previous result, it is a theorem of A→ . • A→ is finite strong complete w.r.t. neither Z0 nor Z+ . Note that the consecution ¬𝜑 → 𝜑 I 𝜑 is valid both in Z0 and Z+ but we can show that it not a valid in A→ using the matrix hZ2 , {0}i, where Z2 is the Abelian group of addition modulo 2 (recall that this group has two elements 0 and 1; unit 0; identity as inverse function, and 𝑎 + 𝑏 = 1 iff 𝑎 ≠ 𝑏) and the Z2 -evaluation 𝑒(𝜑) = 1. Recall that A→ is not algebraically implicative; perhaps surprisingly we can define a proper extension asA→ with the same set of theorems which is algebraically implicative and such that Alg∗ (A→ ) = Alg∗ (asA→ ). Example 4.6.13 Let us define the logic asA→ (where ‘as’ stands for ‘assertional’) as the extension of A→ by the rule 𝜑 → 𝜓 I 𝜓 → 𝜑 (it is a proper extension, as this rule clearly fails e.g. in Z+ ). The new rule implies that A ∈ Mod∗ (asA→ ) iff hA, ≤A→ i is a discretely ordered Abelian group iff A = hA, {𝑒}i, where A is an Abelian group with unit 𝑒. Therefore, asA→ is an algebraically implicative logic with the defining equation 𝑝 ≈ 1¯ (indeed, to show the condition (Alg) of Theorem 2.9.5, it suffices to observe that 𝑝 `asA→ 1¯ ⇔ 𝑝 follows from (Push) and the additional rule of asA→ ) and the variety of Abelian groups as its equivalent algebraic semantics. Due to Corollary 3.4.9 and fact that Z generates the variety of Abelian groups, we know that asA→ enjoys the completeness w.r.t. Z0 (but it does not enjoy the finite strong completeness due to the same example as in the case of A→ ). This, together with the completeness of A→ w.r.t. Z0 , implies that A→ and asA→ have the same theorems.

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The final example in this series dedicated to A→ introduces two of its expansions into the language with lattice connectives which are well studied in the literature: the Abelian logic A and the assertional Abelian logic asA. Both logics are algebraically implicative and have the same equivalent algebraic semantics (but different defining equations). Although A is not a conservative extension of A→ , it does not prove any additional theorems in the LLL , thus justifying our naming conventions. Example 4.6.14 Let us first notice that any ordered Abelian group where the underlying order is a lattice can be seen as an algebra A = h𝐴, ∧, ∨, ·,−1 , 𝑒i with group and lattice reducts h𝐴, ·,−1 , 𝑒i and h𝐴, ∧, ∨i; such algebras are called Abelian ℓ-groups and, as before, we can identify them with algebras in the language LLL expanded by two binary connectives ∧ and ∨. It is known that the variety of Abelian ℓ-groups is generated as a quasivariety by the naturally ordered integers [206] (but not as a generalized quasivariety; as we will see later); let us denote this algebra by Zlat and lat matrices hZlat , {𝑎 | 𝑎 ≥ 0}i and hZlat , {0}i as Zlat + and Z0 . respectively. Let us now define two logics in the language LLL expanded by two binary connectives ∧ and ∨: • Abelian logic A as the expansion of A→ by the rule (Adj) and the axioms (ub), (lb), (inf), and (sup). • Assertional Abelian logic asA as the expansion of asA→ by the congruence rules for ∨ and ∧ and axiom 𝜑 → 𝜓 for each equation 𝜑 ≈ 𝜓 in the definition of lattice. It is easy to see (using Corollary 4.2.11) that A is the least expansion of A→ with lattice conjunction and protodisjunction (later, in Example 5.1.13, we show that this would be true even without the prefix proto). On the other hand, ∨ and ∧ are not lattice protodisjunction/protoconjunction w.r.t. the implication → in asA even though, as we will immediately see, the algebras from Alg∗ (asA) have lattice reducts and ∨ and ∧ are interpreted as supremum and infimum. The problem is that the lattice order they induce differs from the matrix order (which is actually always discrete) and thus e.g. 𝜑 → 𝜑∨𝜓 is not a theorem (note that it also implies that these two logics are incomparable). It can be easily shown that both logics are algebraically implicative with the same equivalent algebraic semantics being the variety of Abelian ℓ-groups but with different defining equations: in the case of A it is 𝑝 ∧ 1¯ ≈ 1¯ and in the case of asA ¯ it is 𝑝 ≈ 1.Thus, using Corollary 3.8.7, we obtain the finite strong completeness of lat A and asA w.r.t. the matrices Zlat + and Z0 respectively.24 24 We could show that the strong completeness fails in both cases. In the case of A we can show it directly: first we observe that {𝜓 → 𝜑 𝑛 | 𝑛 ≥ 1} Zlat 𝜑: if 𝑒 ( 𝜑) < 0, then for some 𝑛 we much + have 𝑒 ( 𝜓 → 𝜑 𝑛 ) = 𝑛𝑒 ( 𝜑) − 𝑒 ( 𝜓) < 0. To prove that it fails in A, we can consider e.g. the Abelian ℓ-group given by the lexicographic second power of Zlat (i.e. the algebra Zlat ×lex Zlat with group reduct Z × Z and lattice operations given by the lexicographic (i.e. linear) order: h𝑎, 𝑏i ≤ h𝑐, 𝑑 i iff 𝑎 < 𝑏 or 𝑎 = 𝑐 and 𝑏 ≤ 𝑑) and the evaluation 𝑒 ( 𝜑) = h0, −1i and 𝑒 ( 𝜓) = h−1, 0i. Indeed, in such case we have 𝑒 ( 𝜓 → 𝜑 𝑛 ) = h0, −𝑛i − h−1, 0i = h1, −𝑛i ≥ h0, 0i. Therefore, by Corollary 3.8.3, we know that Zlat does not generate the variety of Abelian ℓ-groups as a generalized quasivariety and thus, by the same Corollary3.8.3, we also know the failure of strong completeness of asA w.r.t. Zlat 0 .

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4 On lattice and residuated connectives

These results imply that both these logics are not conservative expansions of the starting logics A and asA in the sense of our definition: indeed, we now know that ¬𝜑 → 𝜑 `A 𝜑 and ¬𝜑 → 𝜑 `asA 𝜑 which we have seen in Example 4.6.12 to fail both in A→ and asA→ . Nevertheless, due to all the completeness results established in these examples we know that the conservativeness condition holds when restricted to theorems: indeed, we know that an LLL -formula is a theorem of A iff it is a theorem of A→ iff if it is a theorem asA→ iff it is a theorem of asA. Next, we present an interesting property of substructural logics expanding SL whose axiomatization has certain symmetries. Namely, we prove for these logics that the two implications → and are almost indistinguishable from a proof-theoretic point of view, since they can be interchanged at the cost of reversing all instances of the residuated conjunction & (if present). We use this fact to simplify certain formal proofs of derivability of some consecutions in these substructural logics. Definition 4.6.15 (Mirror image) Given a formula 𝜒, its mirror image 𝜒 𝑚 is obtained by replacing in 𝜒 all occurrences of → by , and vice versa, and, by replacing all subformulas of the form 𝛼 & 𝛽 by 𝛽 & 𝛼. Our definition of mirror image is done for all possible formulas, but of course in some it makes better sense than in others: in particular, if 𝜒 is a formula of LL , then so is 𝜒 𝑚 and the same is true for formulas of LL ,& . We extend the mirror notation to sets of formulas in the usual way: Γ𝑚 = {𝛾 𝑚 | 𝛾 ∈ Γ}. Next, we prove an easy theorem showing which logics are closed under mirror versions of their consecutions and show that the logics SL𝑋 are among them. Theorem 4.6.16 (Mirror image theorem) Let L be a substructural logic expanding SL by a set of consecutions C such that, for each Ψ I 𝜓 ∈ C, we have Ψ𝑚 `L 𝜓 𝑚 . Then, for each set of formulas Γ ∪ {𝜑}, we have Γ `L 𝜑

iff

Γ 𝑚 `L 𝜑 𝑚 .

Proof We show only one direction (the second one immediately follows from the fact that (𝜑 𝑚 ) 𝑚 = 𝜑). Recall the axiomatic system of SL from Theorem 4.5.7; if for each consecution Ψ I 𝜓 of this axiomatic system we prove Ψ𝑚 `L 𝜓 𝑚 , the claim easily follows by induction on the complexity of the proof of 𝜑 from Γ. We leave the proof as an exercise for the reader.  Corollary 4.6.17 (Mirror image theorem for SL𝑋 ) Let 𝑋 ⊆ {a, E, C, i, o}. Then, for each set of formulas Γ ∪ {𝜑}, we have 𝑇 `SL𝑋 𝜑

iff

Γ𝑚 `SL𝑋 𝜑 𝑚.

Proof We use the previous theorem: first observe that due to Remark 4.5.9 we know that SL is an expansion of SL by the axioms (>), (⊥), (lb), and (ub) and the rules (Res), (Inf), and (Sup). The mirror forms of (Res) are easy consequences of the rules (Symm) and, for the remaining consecutions, we can observe that neither & nor

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Table 4.3 Notable classes of SL-algebras.

Logic L SL SLa SLE SLi SLC SLCi

Alg∗ (L) SL-algebras associative SL-algebras commutative SL-algebras integral SL-algebras square-increasing SL-algebras idempotent SL-algebras

defining equation 𝑥 & (𝑦 & 𝑧) ≈ (𝑥 & 𝑦) & 𝑧 𝑥&𝑦≈𝑦&𝑦 1¯ ≈ > 𝑥 ≤ 𝑥&𝑥 𝑥 ≈𝑥&𝑥

appears in their formulas and → appears exactly once as a principal connective. Therefore, we show the validity of their mirror version simply by the rules (Symm). Finally, thanks to Proposition 4.6.7, we know that SL𝑋 is the extension of SL by the consecutions whose mirror forms are valid in SL𝑋 .  Finally, let us deal with semantical aspects of (prominent) substructural logics. Recall that SL is algebraically implicative and, hence, we can use Theorem 2.9.14 to establish the dual isomorphism between subvarieties of SL-algebras and axiomatic extensions of the logic SL. Let us note that, as SL-algebras have lattice reducts, the special equations of the form 𝑡 ≤ 𝑠 for terms 𝑡 and 𝑠 correspond to formulas 𝑡 → 𝑠: indeed 𝑡 ≤ 𝑠 is actually the equation 𝑡 ≈ 𝑡 ∧ 𝑠, which corresponds to the formula 𝑡 ↔ 𝑡 ∧ 𝑠 which (thanks to the validity of (Df ∧ ); see Proposition 4.1.5) is interderivable in SL with 𝑡 → 𝑠. Dually, the formula 𝑡 → 𝑠 corresponds to the equation (𝑡 → 𝑠) ∧ 1¯ ≈ 1¯ which is clearly equivalent with 𝑡 ≤ 𝑠. In Table 4.3 we list subvarieties of SL-algebras which constitute the equivalent algebraic semantics of prominent substructural logics (we can use equations involving & thanks to Proposition 4.6.7 and Corollary 4.6.9). Recall that in Theorem 4.4.15 we have proved that Mod∗ (SL) admits regular completions. Now we show that the same claim holds for all the logics of the form SL𝑋 and thus, thanks to Theorem 4.4.14, these logics enjoy completeness w.r.t. completely ordered matrices. Corollary 4.6.18 Let 𝑋 ⊆ {a1 , a2 , E, C, i, o}. Then, Mod∗ (SL𝑋 ) admits regular completions and SL𝑋 enjoys the strong Mod𝑐 (SL𝑋 )-completeness. Proof Clearly, it suffices to prove the claim for 𝑋 being all possible singletons. Consider the matrix A ∈ Mod∗ (SL𝑋 ) and its completion A𝑐 constructed in the proof of Theorem 4.4.15 together with the embedding 𝑓 : A → A𝑐 . Recall that (A𝑐 ) LLL = hC, 𝐹⊆ i, where 𝐶 = {𝑌 ⊆ 𝐴 | 𝑌 = (𝑌 𝑢 ) 𝑙 } (𝑥] = {𝑎 | 𝑎 ≤A 𝑥} ≤A𝑐 = ⊆

0¯ C = ( 0¯ A ] 1¯ C = ( 1¯ A ] Ð Ð 𝑌 &C 𝑍 = 𝑦 ∈𝑌 𝑧 ∈𝑍 (𝑦 &A 𝑧].

208

4 On lattice and residuated connectives

Let us distinguish all possible cases: o: From 0¯ A ≤A 𝑎 for each 𝑎 ∈ 𝐴, we obtain 0¯ C = {0¯ A } ⊆ 𝑍 for each 𝑍 ∈ 𝐶, i.e. A𝑐 0¯ → 𝜑. i: From 𝑎 ≤A 1¯ A for each 𝑎 ∈ 𝐴, we obtain 1¯ C = 𝐴 ⊇ 𝑍 for each 𝑍 ∈ 𝐶, i.e. ¯ A𝑐 𝜑 → 1. c: From 𝑎 ≤A 𝑎 &A 𝑎 for each 𝑎 ∈ 𝐴, we obtain 𝑧 ∈ (𝑧 &A 𝑧] ⊆ 𝑍 &C 𝑍 for any 𝑍 ∈ 𝐶 and 𝑧 ∈ 𝑍, i.e. A𝑐 𝜑 → 𝜑 & 𝜑. E: From 𝑎 &A 𝑏 = 𝑎 &A 𝑏 for each 𝑎, 𝑏 ∈ 𝐴, we obtain 𝑌 &C 𝑍 = 𝑍 &C 𝑌 for each 𝑌 , 𝑍 ∈ 𝐶, i.e. A𝑐 𝜑 & 𝜓 → 𝜓 & 𝜑. a1 : Clearly, for each 𝑎 ∈ 𝑌 &C (𝑍 &C 𝑉), there have to be 𝑦 ∈ 𝑌 , 𝑧 ∈ 𝑍, 𝑣 ∈ 𝑉 such that there is a 𝑏 ≤ 𝑧 &A 𝑣 and 𝑎 ≤ 𝑦 &A 𝑏. Then, 𝑎 ≤ 𝑦 &A (𝑧 &A 𝑣) ≤ (𝑦 &A 𝑧) &A 𝑣 (the last inequality is due to a1 ) and, since clearly 𝑦 &A 𝑧 ∈ 𝑌 &C 𝑍, we obtain 𝑎 ∈ (𝑌 &C 𝑍) &C 𝑉. Thus, we have shown that 𝑌 &C (𝑍 &C 𝑉) ⊆ (𝑌 &C 𝑍) &C 𝑉 and so A𝑐 𝜑 & (𝜓 & 𝜒) → (𝜑 & 𝜓) & 𝜒. a2 : This case is analogous.  We conclude this section by showing that all the substructural logics depicted in Figure 4.3 and their extensions by the axiom (o), (i.e. the logics SL𝑋 for 𝑋 ⊆ {a, E, C, i, o}) are pairwise different. First, let us deal with the axiom o. Remark 4.6.19 Let us consider any 𝑋 ⊆ {a1 , a2 , E, C, i}. First notice that SL𝑋 and ¯ where 0¯ is defined as SL𝑋 o coincide when seen as logics in the language LSL \ {0}, ⊥. However, when seen as logics in the full language LSL , SL𝑋 o is strictly stronger than SL𝑋 . Indeed, consider the Boolean LCL -algebra 2 and define its expansion 20 to ¯ >, & and the language LSL by interpreting the connectives 1, in the obvious way, 0 but setting 0¯ 2 = 1. Observe that 20 ∈ Alg∗ (SLaECi ) \ Alg∗ (SLo ). Let us now consider any 𝑋, 𝑌 ⊆ {a1 , a2 , E, C, i} such that SL𝑋 is strictly stronger than SL𝑌 . Then, SL𝑋 o is strictly stronger than SL𝑌 o . Indeed, the assumption gives us a matrix A ∈ Mod∗ (SL𝑌 ) \ Mod∗ (SL𝑋 ) and, if we consider the matrix A0 resulting 0 from A by setting 0¯ A = ⊥A , we have A0 ∈ Mod∗ (SL𝑌 o ) \ Mod∗ (SL𝑋 o ), proving the claim.

Example 4.6.20 (The logics SL𝑋 for 𝑋 ⊆ {a, E, C, i, o} are pairwise different.) Thanks to the previous remark, we know that it is enough to show the claim for 𝑋 ⊆ {a, E, C, i}. It is easy to see, after a short contemplation, that it suffices to find the following six matrices: A1 ∈ Mod∗ (SLaEi ) \ Mod∗ (SLC ) A2 ∈ Mod∗ (SLaEC ) \ Mod∗ (SLi ) A3 ∈ Mod∗ (SLai ) \ Mod∗ (SLE ) A4 ∈ Mod∗ (SLEC ) \ Mod∗ (SLa ) A5 ∈ Mod∗ (SLEi ) \ Mod∗ (SLa ) A6 ∈ Mod∗ (SLaC ) \ Mod∗ (SLE ).

4.6 Substructural logics and prominent extensions of SL

209

We have already met the first one: we can set A1 = h[0, 1] Ł , {1}i ∈ Mod(Ł); see Example 2.6.4. We describe the domains, matrix orders, and the interpretation of the unit and the residuated conjunction in the remaining five matrices. We leave the rest as an exercise for the reader. • The matrices A2 and A6 share the domain {⊥, 1, >} and the matrix order schematically depicted as ⊥ < 1 < >, the unit is interpreted as 1¯ A2 = 1¯ A6 = 1, and the operations &A2 and &A6 are defined as &A2 ⊥ 1 >

⊥ ⊥ ⊥ ⊥

&A6 ⊥ 1 >

1 > ⊥ ⊥ 1 > > >

⊥ ⊥ ⊥ >

1 ⊥ 1 >

> ⊥ > >

• The matrix A3 has domain {⊥, 𝑎, 𝑏, >} and the matrix order schematically depicted as ⊥ < 𝑎 < 𝑏 < >, the unit is interpreted as 1¯ A3 = >, and the operation &A3 is defined as &A3 ⊥ 𝑎 𝑏 >

⊥ ⊥ ⊥ ⊥ ⊥

𝑎 ⊥ ⊥ 𝑎 𝑎

𝑏 ⊥ ⊥ 𝑏 𝑏

> ⊥ 𝑎 𝑏 >

• The matrix A4 has domain {⊥, 𝑎, 1, >} and the matrix order schematically depicted as ⊥ < 𝑎 < 1 < >, the unit is interpreted as 1¯ A4 = 1, and the operation &A4 is defined as &A4 ⊥ 𝑎 1 >

⊥ ⊥ ⊥ ⊥ 𝑎

𝑎 ⊥ 𝑎 𝑎 1

1 ⊥ 𝑎 1 >

> 𝑎 1 > >

To show the failure of associativity, note that (⊥ & 𝑎) & > = ⊥ & > = 𝑎 ≠ ⊥ = ⊥ & 1 = ⊥ & (𝑎 & >). • The matrix A5 has domain the real unit interval [0, 1] and its matrix order is the usual order of reals, the unit is interpreted as 1¯ A5 = 1, and the operation &A5 is defined as 𝑥 &A5 𝑦 =

max{𝑥𝑦, 𝑥𝑦 + 𝑥 + 𝑦 − 1} . 2

To show the failure of associativity, note that (0.6 & 0.5) & 0.4 = 0.2 & 0.4 = 0.04 ≠ 0.03 = 0.6 & 0.1 = 0.6 & (0.5 & 0.4).

210

4 On lattice and residuated connectives

(id)

𝜑→𝜑

identity

(sf)

(𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))

suffixing

(e)

(𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))

exchange

(MP)

𝜑, 𝜑 → 𝜓 I 𝜓

modus ponens

(res1 )

(𝜑 → (𝜓 → 𝜒)) → (𝜓 & 𝜑 → 𝜒)

residuation

(res2 )

residuation

(push)

(𝜓 & 𝜑 → 𝜒) → (𝜑 → (𝜓 → 𝜒)) 𝜑 → ( 1¯ → 𝜑)

(pop)

( 1¯ → 𝜑) → 𝜑

pop

(⊥)

⊥→𝜑

ex falso quodlibet

(>)

𝜑→>

verum ex quolibet

(lb1 )

𝜑∧𝜓 → 𝜑

lower bound

(lb2 )

𝜑∧𝜓 →𝜓

lower bound

(inf)

( 𝜒 → 𝜑) ∧ ( 𝜒 → 𝜓) → ( 𝜒 → 𝜑 ∧ 𝜓)

infimality

(Adj)

𝜑, 𝜓 I 𝜑 ∧ 𝜓

adjunction

(ub1 )

𝜑 → 𝜑∨𝜓

upper bound

(ub2 )

𝜓 → 𝜑∨𝜓

upper bound

(Sup)

𝜑 → 𝜒, 𝜓 → 𝜒 I 𝜑 ∨ 𝜓 → 𝜒

supremality

push

Fig. 4.4 Strongly separable presentation A S SLaE of the logic SLaE .

4.7 Strongly separable axiomatic systems for extensions of SLaE The goal of this section is to find a strongly separable axiomatic system for the logic SLaE and some of its notable extensions. Recall that such axiomatic systems (see Definition 4.5.1), when restricted to consecutions featuring only a limited set of connectives, axiomatize the corresponding fragment of the logic in question. Since the connectives and > are definable in SLaE , we can see it as a logic in four possible languages. We prove the result for the most natural one, LSL \ { }, and then remark how the system would have to be adjusted for the remaining three options: LSL \ { , >}, LSL \ {>}, and LSL . Theorem 4.7.1 The axiomatic system depicted in Figure 4.4 is a strongly separable presentation of SLaE seen as a logic in the language LSL \ { }. Proof We will use Proposition 4.5.2. First, observe that all the consecutions of AS SLaE are valid in SLaE . Next, consider any sublanguage L containing → and let us write (for the purposes of this proof) simply `L instead of ` ( A S SLaE ) L . It suffices to show that Mod∗ (`L ) ⊆ S(Mod∗ (SLaE )L ), i.e. for each reduced model hA, 𝐹i of `L , we need to show that there is an SLaE -model A0 = hA0, 𝐹 0i and a strict embedding 𝑓 : A → AL0 .

4.7 Strongly separable axiomatic systems for extensions of SLaE

211

As in the previous proofs of analogous statements for LL and SL, our trick is to build the matrix A0 using only the implication connective and show that the construction works for all fragments at once. We start by noticing that, due to Lemma 4.5.4, → is a weakly implication in `L . Furthermore, note that `L→ equals the logic BCI and recall that in the proof of Theorem 2.4.3 we have defined, for a finite sequence of formulas 𝛼 = h𝜑1 , . . . , 𝜑 𝑛 i, the notation 𝛼 → 𝜑 to stand for 𝜑1 → (𝜑2 → . . . (𝜑 𝑛 → 𝜑) . . . ) (and, by convention, we set 𝛼 → 𝜑 = 𝜑 if 𝛼 = hi). We have also proved that, for any finite sequence 𝛼 of formulas and any formulas 𝜑, 𝜓, the following rules are derivable in BCI and thus also in `L : C1 C2

𝛼 → 𝜑, 𝜑 → 𝜓 I 𝛼 → 𝜓 𝛼 → (𝜑 → 𝜓) I 𝜑 → (𝛼 → 𝜓).

Later in the proof, we will need to use that the following formula is a theorem of `L for any L containing ∧: C3

(𝛼 → 𝜑) ∧ (𝛼 → 𝜓) → (𝛼 → 𝜑 ∧ 𝜓).

As before, we prove this claim by induction on the length of 𝛼. If 𝛼 = hi, the proof is trivial. Assume that 𝛼 = h𝜒, 𝜒1 , . . . , 𝜒𝑚 i and let us use 𝛽 for h𝜒1 , . . . , 𝜒𝑚 i; then, we can easily prove C3: a)

𝜒 → (𝛽 → 𝜑)

the first premise

b)

𝜒 → (𝛽 → 𝜓)

the second premise

c)

𝜒 → (𝛽 → 𝜑) ∧ (𝛽 → 𝜓)

d)

𝜒 → (𝛽 → 𝜑 ∧ 𝜓)

a, b, and (Inf) c, the induction assumption, and transitivity

Let us denote by ≤ the matrix order on A and by 𝐴< 𝜔 the set of all finite sequences of elements of 𝐴. Then, for each 𝑠, 𝑞 ∈ 𝐴< 𝜔 and 𝑦 ∈ 𝐴, we denote by 𝑠 ◦ 𝑞 the concatenation of 𝑠 and 𝑞 and introduce the notation 𝑠 → 𝑦 as before. Furthermore, we define, for each 𝑠 ∈ 𝐴< 𝜔 and each 𝑆 ⊆ 𝐴< 𝜔 , [𝑠] = {𝑦 | 𝑠 → 𝑦 ∈ 𝐹} Ù [𝑆] = [𝑠]. 𝑠 ∈𝑆

Clearly, [{𝑠}] = [𝑠], so we omit the curly brackets; analogously, for 𝑎 ∈ 𝐴 we write [𝑎] instead of [h𝑎i]. Note that [𝑆] is a ≤-upper subset of 𝐴, [hi] = 𝐹 (which equals ¯ if 𝐴 has unit 1), ¯ [∅] = 𝐴 (which equals [⊥] if 𝐴 has a bottom element ⊥), and [ 1] < 𝜔 [ 𝐴 ] equals {>} = [>] if ≤ has a top element > or to the empty set otherwise. Finally, we need to notice that for each 𝑠, 𝑞 ∈ 𝐴< 𝜔 we have, thanks to the repeated use of condition C2, [𝑠 ◦ 𝑞] = [𝑞 ◦ 𝑠]. Let us continue by defining the set 𝐴 0 = {[𝑆] | 𝑆 ⊆ 𝐴< 𝜔 } 0

and noticing that ⊇ is complete lattice order on 𝐴 0 with the least element ⊥A = [∅], 0 0 0 the largest element >A = [ 𝐴< 𝜔 ] and meet ∧A and join ∨A , which can be defined

212

4 On lattice and residuated connectives

for all sets X ⊆ P ( 𝐴 0) as (the second equality in the definition of meet is obviously valid due to the definition of [𝑆]): ÜA0 ÛA0

X=

Ù

X o Ùn o Ùn Ø Ø X= [𝑆] ∈ 𝐴 0 | X ⊆ [𝑆] = [𝑠] ∈ 𝐴 0 | X ⊆ [𝑠] .

Next, we define for each 𝑆, 𝑄 ⊆ 𝐴< 𝜔 0

Ù

[𝑆] &A [𝑄] =

[𝑠 ◦ 𝑞].

𝑠 ∈𝑆,𝑞 ∈𝑄 0

Let us show that &A is well defined. Consider sets 𝑆, 𝑄, 𝑅 ⊆ 𝐴< 𝜔 such that [𝑄] = [𝑅], i.e. for each 𝑎 ∈ 𝐴, we have (∀𝑞 ∈ 𝑄)(𝑞 → 𝑎 ∈ 𝐹)

(∀𝑟 ∈ 𝑅)(𝑟 → 𝑎 ∈ 𝐹).

iff

Then, for any 𝑠 ∈ 𝑆 and 𝑏 ∈ 𝐴, we can use the above equivalence for 𝑎 = 𝑠 → 𝑏 to show that (∀𝑞 ∈ 𝑄)(𝑞 → (𝑠 → 𝑏) ∈ 𝐹)

(∀𝑟 ∈ 𝑅)(𝑟 → (𝑠 → 𝑏) ∈ 𝐹)

iff

which implies that, for each 𝑠 ∈ 𝑆, Ù Ù [𝑠 ◦ 𝑞] = [𝑠 ◦ 𝑟] 𝑞 ∈𝑄

𝑟 ∈𝑅

and so 0

[𝑆] &A [𝑄] =

Ù

[𝑠 ◦ 𝑞] =

𝑠 ∈𝑆,𝑞 ∈𝑄

0

Ù

[𝑠 ◦ 𝑟] = [𝑆] &A [𝑅].

𝑠 ∈𝑆,𝑟 ∈𝑅

0

Since &A is clearly a commutative operation, we also obtain that 0

0

[𝑄] &A [𝑆] = [𝑅] &A [𝑆]. 0

To prove that &A is associative, we first note that 𝑧 ∈ [𝑠 ◦ 𝑞] iff 𝑠 → 𝑧 ∈ [𝑞]. 0 Thus, we have 𝑧 ∈ [𝑆] &A [𝑄] iff for each 𝑠 ∈ 𝑆 we have 𝑠 → 𝑧 ∈ [𝑄] and so, by repeated use of this observation and the fact that 𝑟 → (𝑠 → 𝑧) = 𝑠 → (𝑟 → 𝑧), we obtain the following chain of equivalent statements: • • • • • •

0

0

𝑧 ∈ [𝑆] &A ([𝑄] &A [𝑅]). 0 0 For each 𝑠 ∈ 𝑆, we have 𝑠 → 𝑧 ∈ [𝑄] &A [𝑅] = [𝑅] &A [𝑄]. For each 𝑠 ∈ 𝑆 and 𝑟 ∈ 𝑅, we have 𝑟 → (𝑠 → 𝑧) ∈ [𝑄]. For each 𝑟 ∈ 𝑅 and 𝑠 ∈ 𝑆, we have 𝑠 → (𝑟 → 𝑧) ∈ [𝑄]. 0 For each 𝑟 ∈ 𝑅, we have 𝑟 → 𝑧 ∈ [𝑆] &A [𝑄]. 0 0 0 0 𝑧 ∈ [𝑅] &A ([𝑆] &A [𝑄]) = ([𝑆] &A [𝑄]) &A [𝑅].

4.7 Strongly separable axiomatic systems for extensions of SLaE

213

0 0 Finally, we define 1¯ A = 𝐹 and show that it is the unit of &A . Indeed, Ù 0 [𝑆] & 𝐹 = [𝑆] &A [hi] = [𝑠 ◦ hi] = [𝑆].

𝑠 ∈𝑆

Note that, due to the definition, we trivially have: 0

𝑋 &A

ÜA0

Z=

ÜA0

0

𝑍 ∈Z

𝑋 &A 𝑍.

Thus, if we define (observe that second equality is obvious) Ü 0 0 𝑋 →A 𝑌 = {𝑍 ∈ 𝐴 0 | 𝑋 &A 𝑍 ⊇ 𝑌 } Ü 0 = {[𝑠] ∈ 𝐴 0 | 𝑋 &A [𝑠] ⊇ 𝑌 }, we can use Lemma 4.4.9 and Proposition 4.4.7 to obtain that the matrix 0

0

0

0

0

0

0

A0 = hh𝐴 0, &A , →A , ∨A , ∧A , 1¯ A , ⊥A , >A i, {[𝑆] | 𝐹 ⊇ [𝑆]}i is a model of SLaE . We also need to show that 𝑓 : A → A0 defined as 𝑓 (𝑥) = [𝑥] is a strict embedding. First, we show that it is an algebraic homomorphism. We proceed by the list of connectives that could be in L: • Implication: recall that 𝑎 → [(𝑎 → 𝑏) → 𝑏] ∈ 𝐹. Thus, for each 𝑧 such that 0 𝑏 → 𝑧 ∈ 𝐹, we obtain 𝑎 → [(𝑎 → 𝑏) → 𝑧] ∈ 𝐹, i.e. [𝑎] &A [𝑎 → 𝑏] = 0 [h𝑎, 𝑎 → 𝑏i] ⊇ [𝑏], and so [𝑎 → 𝑏] ⊇ [𝑎] →A [𝑏]. Conversely, assume that 𝑧 ∈ [𝑎 → 𝑏], i.e. (𝑎 → 𝑏) → 𝑧 ∈ 𝐹 and, using 0 the definition of →A , we have to show that 𝑧 ∈ [𝑠] for each [𝑠] ∈ 𝐴 0 such 0 A that [𝑎] & [𝑠] ⊇ [𝑏]. Clearly, for each such 𝑠, we have 𝑏 ∈ [𝑎 ◦ 𝑠], i.e. 𝑠 → (𝑎 → 𝑏) ∈ 𝐹 and so 𝑠 → 𝑧 ∈ 𝐹, i.e. indeed 𝑧 ∈ [𝑠]. 0

• Residuated conjunction is simple: [𝑎 & 𝑏] = [𝑎 ◦ 𝑏] = [𝑎] &A [𝑏]. • Lattice protodisjunction is also rather simple: [𝑎 ∨ 𝑏] = {𝑧 | 𝑎 ∨ 𝑏 → 𝑧 ∈ 𝐹} = 0 {𝑧 | 𝑎 → 𝑧, 𝑏 → 𝑧 ∈ 𝐹} = [𝑎] ∩ [𝑏] = [𝑎] ∨A [𝑏]. Ñ 0 • Lattice conjunction: recall that [𝑎] ∧A [𝑏] = {[𝑠] ∈ 𝐴 0 | [𝑎] ∪ [𝑏] ⊆ [𝑠]}. 0 Thus, as clearly [𝑎] ∪ [𝑏] ⊆ [𝑎 ∧ 𝑏], we obtain [𝑎] ∧A [𝑏] ⊆ [𝑎 ∧ 𝑏]. To prove the converse, we have to show that, for each 𝑧 ∈ [𝑎 ∧ 𝑏], we have 𝑧 ∈ [𝑠] for each [𝑠] ∈ 𝐴 0 such that [𝑎] ∪ [𝑏] ⊆ [𝑠]. Note that we have 𝑎, 𝑏 ∈ [𝑠] and so (𝑠 → 𝑎) ∧ (𝑠 → 𝑏) ∈ 𝐹. Using C3, we know that (𝑠 → 𝑎) ∧ (𝑠 → 𝑏) ≤ 𝑠 → 𝑎 ∧ 𝑏 and so 𝑠 → 𝑎 ∧ 𝑏 ∈ 𝐹 and, by condition C1, we know that 𝑠 → 𝑧 ∈ 𝐹 as required. ¯ = {𝑧 | 1¯ ≤ 𝑧} = 𝐹 = 1¯ A0 . • Unit: [1] 0

• Top: [>] = {𝑧 | > ≤ 𝑧} = {>} = >A . 0

• Bottom: [⊥] = {𝑧 | ⊥ ≤ 𝑧} = 𝐴 = ⊥A . To complete the proof, it suffices to observe that 𝑓 is a strict matrix homomorphism. Indeed, 𝑎 ∈ 𝐹 iff 𝐹 ⊇ [𝑎] = 𝑓 (𝑎) iff 𝑓 (𝑎) ∈ {[𝑆] | 𝐹 ⊇ [𝑆]}. 

214

4 On lattice and residuated connectives

Remark 4.7.2 We have formulated the previous theorem for SLaE seen as a logic in LSL \ { }. This makes good sense even though > is definable because we need ⊥ for its definition and so > is not definable in the, possibly interesting, {→, >}-fragment of SLaE . On the other hand, is always definable simply as → and so there is no need to consider the {→, }-fragment of SLaE . Let us list the strongly separable presentations of SLaE seen as a logic in the other three languages, which are obtained by modifying the axiomatic system from Figure 4.4 in the following ways: • LSL \ { , >}: we remove the axiom (>) (note that in this case we do not have to care about the {→, >}-fragment of SLaE ). • LSL : we add the axioms (𝜑 → 𝜓) → (𝜑 𝜓) and (𝜑 𝜓) → (𝜑 → 𝜓). • LSL \{>}: we remove the axiom (>) and adding the axioms (𝜑 → 𝜓) → (𝜑 𝜓) and (𝜑 𝜓) → (𝜑 → 𝜓). Let us also note that the axiomatic system depicted in Figure 4.4 is a strongly ¯ (with 0¯ defined separable presentation of SLaEo seen in the language LSL \ { , 0} as ⊥) and its extension by the axiom (o) is a strongly separable presentation of SLaEo seen in the language LSL \ { }. The same observation can be made about strongly separable axiomatic systems of other prominent substructural logics (cf. Corollary 4.7.3). Let us now present strongly separable axiomatic systems for two prominent substructural logics, namely the logics FLew and IL. Again, we present the results for these logics seen in their natural languages. For instance, in FLew we disregard not only the constants 1¯ and >, as they are definable as 𝑝 → 𝑝, but also 0¯ which of course is not definable using just implication, but thanks to the axiom (o), 0¯ coincides with the constant ⊥ (thus, 0¯ is definable in the →, ⊥-fragment and ⊥ in ¯ the →, 0-fragment and so these fragments are the same modulo these definitions). The situation in IL is analogous: here the connective & coincides with ∧, and so we disregard the former. Corollary 4.7.3 • The axiomatic system AS FLew depicted in Figure 4.5 is a strongly separable ¯ 1, ¯ >} = LCL ∪ {&}. presentation of FLew seen in the language LSL \ { , 0, • The axiomatic system AS IL depicted in Figure 4.6 is a strongly separable presentation of IL seen in the language LCL . Proof Both of these claims are proved by a simple inspection of the proof of Theorem 4.7.1. Let us first observe that the formula (adj) is derivable in FLew and in IL (an easy consequence of (∧-sub-&) and (adj& )), the formula (id) is derivable in (AS FLew )→ and the rule (Adj) is derivable in (AS FLew )∧ (and, thus, the same is true for the corresponding fragments of ` A S IL ). Therefore, to prove the first claim, it suffices to see that, for each reduced model hA, 𝐹i of ` ( A S FLew ) L , the resulting matrix A0 = hA0, 𝐹 0i is a model of FLew . To do this, we can simply use Proposition 2.8.20 to observe that 𝐹 is a singleton and so 𝐹 0 is also singleton, which immediately makes the axiom (i) valid in A0.

4.7 Strongly separable axiomatic systems for extensions of SLaE

215

(sf)

(𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))

suffixing

(e)

(𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))

exchange

(w)

𝜑 → (𝜓 → 𝜑)

weakening

(MP)

𝜑, 𝜑 → 𝜓 I 𝜓

modus ponens

(res1 )

(𝜑 → (𝜓 → 𝜒)) → (𝜓 & 𝜑 → 𝜒)

residuation

(res2 )

(𝜓 & 𝜑 → 𝜒) → (𝜑 → (𝜓 → 𝜒))

residuation

(⊥)

⊥→𝜑

ex falso quodlibet

(lb1 )

𝜑∧𝜓 → 𝜑

lower bound

(lb2 )

𝜑∧𝜓 →𝜓

lower bound

(inf)

( 𝜒 → 𝜑) ∧ ( 𝜒 → 𝜓) → ( 𝜒 → 𝜑 ∧ 𝜓)

infimality

(adj)

𝜑 → (𝜓 → 𝜑 ∧ 𝜓)

adjunction

(ub1 )

𝜑 → 𝜑∨𝜓

upper bound

(ub2 )

𝜓 → 𝜑∨𝜓

upper bound

(Sup)

𝜑 → 𝜒, 𝜓 → 𝜒 I 𝜑 ∨ 𝜓 → 𝜒

supremality

Fig. 4.5 Strongly separable presentation A S FLew of the logic SLaEio = FLew in LCL ∪ {&}.

To prove the second claim, it suffices to show that, for a reduced model hA, 𝐹i of ` ( A S IL ) L , the resulting matrix A0 = hA0, 𝐹 0i is a model of IL. We do that by proving the validity of the axiom (s-i) of square-increasingness, i.e. Ù 0 [𝑆] ⊇ [𝑆] &A [𝑆] = [𝑠 ◦ 𝑞]. 𝑠 ∈𝑆,𝑞 ∈𝑆 0

Consider 𝑧 ∈ [𝑆] &A [𝑆]. Then, in particular, for each 𝑠, we have 𝑠 ◦ 𝑠 → 𝑧 ∈ 𝐹. If we show that the following rule is derivable in `L→ , the proof is done: C4

𝛼 → (𝛼 → 𝜑) I 𝛼 → 𝜑.

As before, we do it by induction on the length of 𝛼: it is trivial for 𝛼 = hi and, if 𝛼 = h𝜒1 , . . . , 𝜒𝑚 , 𝜒i, we set 𝛽 = h𝜒1 , . . . , 𝜒𝑚 i and write the following formal proof: a)

𝜒 → ( 𝜒 → (𝛽 → (𝛽 → 𝜑)))

b)

𝜒 → (𝛽 → (𝛽 → 𝜑))

a, (c), and (MP)

c)

𝛽 → (𝛽 → ( 𝜒 → 𝜑))

b and repeated use of C2

d)

𝛽 → ( 𝜒 → 𝜑)

the premise and C2 twice

c and the induction assumption



Inspecting the proofs of Theorem 4.7.1 and the previous corollary, one can observe that the matrix AL0 is actually a completion of A (for any language L and any of the three logics SLaE , FLew , and IL). Therefore, we obtain the following corollary (the part about strong completeness follows from Theorem 4.4.14).

216

4 On lattice and residuated connectives

(sf)

(𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))

suffixing

(e)

(𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))

exchange

(w)

𝜑 → (𝜓 → 𝜑)

weakening

(c)

(𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)

contraction

(MP)

𝜑, 𝜑 → 𝜓 I 𝜓

modus ponens

(⊥)

⊥→𝜑

ex falso quodlibet

(lb1 )

𝜑∧𝜓 → 𝜑

lower bound

(lb2 )

𝜑∧𝜓 →𝜓

lower bound

(inf)

( 𝜒 → 𝜑) ∧ ( 𝜒 → 𝜓) → ( 𝜒 → 𝜑 ∧ 𝜓)

infimality

(adj)

𝜑 → (𝜓 → 𝜑 ∧ 𝜓)

adjunction

(ub1 )

𝜑 → 𝜑∨𝜓

upper bound

(ub2 )

𝜓 → 𝜑∨𝜓

upper bound

(Sup)

𝜑 → 𝜒, 𝜓 → 𝜒 I 𝜑 ∨ 𝜓 → 𝜒

supremality

Fig. 4.6 Strongly separable presentation A S IL of the logic SLaECio = IL in LCL .

Corollary 4.7.4 Let L be a logic in L0 such that one of the following three conditions holds: • L = SLaE , seen as a logic in the language L0 = LSL \ { }. ¯ 1, ¯ >} = LCL ∪ {&} • L = FLew , seen as a logic in the language L0 = LSL \ { , 0, • L = IL, seen as a logic in the language L0 = LCL . Then, for any language L ⊆ L0 containing implication, the class Mod∗ (LL ) admits completions and LL enjoys the strong Mod𝑐 (LL )-completeness. Finally, let us observe that Theorem 4.7.1 and its Corollary 4.7.3 allow us to obtain some of the results promised in Chapter 2 about the relationship between purely implicative logics and their lattice expansions. Recall that we introduced several axiomatic extensions of BCI (namely, BCK, FBCK, IL→ , Ł→ , G→ , and CL→ ) and claimed that all these logics are fragments of their corresponding lat variants (recall that ILlat → = IL and the same holds for Ł, G, and CL). In particular, for CL→ we proved it in Corollary 3.5.20; for BCI, BCK, and IL→ we prove it in the following corollary; for the logics FBCK and G→ we will prove it in Theorem 6.3.8 after establishing corresponding strongly separably axiomatic systems. Corollary 4.7.5 • The logics BCI and BCIlat are fragments of SLaE , and thus also BCI is a fragment of BCIlat . • The logics BCK and BCKlat are fragments of FLew and, thus, also BCK is a fragment of BCKlat . • The logic IL→ is a fragment of IL.

4.8 Implicational deduction theorems

217

4.8 Implicational deduction theorems In this section, we focus on another special form of axiomatic systems with a neat syntactical form for prominent substructural logics. Namely, we require these systems to have modus ponens as the only inference rule with more than one premise, and, possibly some unary rules whose premise is a subformula of the conclusion. We have already met some typical examples of such rules, e.g. (Nec ): 𝜑 I 𝜑, (Adj1¯ ): ¯ or (As): 𝜑 I (𝜑 → 𝜓) → 𝜓 and, of course, we have also seen many rules 𝜑 I 𝜑 ∧ 1, that are not of this form, e.g. (Symm): 𝜑 → 𝜓 I 𝜑 𝜓. It turns out that the existence of an axiomatic system of this kind for a given logic is tightly (though not straightforwardly) related to the validity of a form of deduction theorem and proof by cases (cf. Section 2.4) in the logic in question. Since the conclusion of a rule may include more variables than its premise and the premise may occur in the conclusion several times (cf. e.g. the rule (As) for 𝜓 = 𝜑 & 𝜑 → 𝜒), we introduce an auxiliary notation to avoid formalistic complications. We will work with a set of formulas FmL in a fixed language L ⊇ L→ over a fixed set of propositional variables Var. Let ★ ∉ Var be a new symbol which acts as a placeholder for a special kind of substitutions. A ★-formula is built using variables Var ∪ {★} (and connectives from L) and a ★-substitution is a substitution in the extended language such that 𝜎(★) = ★. We denote by 𝛿(𝜑) the ★-formula resulting from replacing all occurrences of ★ in 𝛿 by 𝜑; note that if 𝜑 is a formula in the original set of variables, so is 𝛿(𝜑). Therefore, we can see a ★-formula as a function 𝛿 : FmL → FmL . Definition 4.8.1 (Deduction term) A ★-formula 𝛿 is called a deduction term of a logic L if the following rule is valid in L for each formula 𝜑: (𝛿-nec)

𝜑 I 𝛿(𝜑)

𝛿-necessitation.

Clearly, it would suffice to assume the validity of (𝛿-nec) for any variable not occurring in 𝛿. Now we are ready to introduce the two central notions of this section. Definition 4.8.2 ((MP)(–bDT)-based axiomatic system and logic) Let bDT be a set of ★-formulas closed under ★-substitutions. We say that an axiomatic system AS is (MP)–bDT-based if, besides axioms and instances of modus ponens, it only contains 𝛽-necessitation rules for 𝛽 ∈ bDT. The elements of bDT are called basic deduction terms of AS. A substructural logic L is (MP)–bDT-based if it has an (MP)–bDT-based presentation. Finally, we speak about (MP)-based axiomatic systems/logics instead of (MP)–∅-based ones. Clearly, basic deduction terms of an (MP)–bDT-based axiomatic system are deduction terms of the corresponding logic. Also, if L is an (MP)–bDT-based logic, then it is (MP)–bDT0-based for any set of bDT0 ⊇ bDT of its deduction terms. The adjective basic signifies the obvious desiderata of having the set bDT as small as possible and ideally not containing redundant rules, in particular those directly obtainable from more elementary ones by either iteration or conjunction. Indeed,

218

4 On lattice and residuated connectives

if 𝛿 = 𝛿1 (𝛿2 ) or 𝛿 = 𝛿1 & 𝛿2 (assuming that L is a substructural logic expanding LL& ), then 𝛿-necessitation is easily derivable from 𝛿1 - and 𝛿2 -necessitation rules (the first case it trivial; in the second we use the rule (Adj& ): 𝜑, 𝜓 I 𝜓 & 𝜑). One of the obvious advantages of having the bDT of a given logic L as small as possible is that it simplifies the description of the set of L-filters. Note that, if L is an (MP)-bDT-based logic, then so are all its axiomatic expansions (assuming that we close the set bDT under ★-substitutions of the new expanded language). Example 4.8.3 Clearly, the logic BCI and all of its axiomatic expansions are (MP)based. This includes, among others, the logics IL, FLew and the {∧, ∨}-free fragment of SLaE (this follows from Theorem 4.7.1); let us denote this logic (for future reference mainly in examples of this section) by SL−aE . As a consequence of Corollary 4.8.14, we will see that the full logic SLaE is not ¯ (MP)-based. However, we can easily show that it is (MP)–{★ ∧ 1}-based. Indeed, its axiomatization from Theorem 4.7.1 has two problematic rules: (Sup), which can clearly be replaced by its axiomatic version (sup), and (Adj), which can be replaced 1¯ ) (cf. Remark 4.6.5). by the rule (Adj1¯ ): 𝜑 I 𝜑 ∧ 1¯ and the axiom (distr& The global modal logic K is not (MP)-based (see Proposition 2.4.2), but it is (MP)–{★}-based (and so are all of its axiomatic expansions). As a convention, for certain important logics L, we choose particular natural sets of deduction terms bDTL such that L is (MP)–bDT-based, e.g. we set bDTFLew = {★} and bDTK = {★}. Note that so far we have considered logics with basic deduction terms containing no variables other than ★. However, in Theorems 4.9.5 and 4.9.8, we will see examples of logics where we use more complex basic deduction terms and for which our ★ notation will simplify substantively the formulations of definitions and results. Now we define the general form of (local) deduction theorem that will turn out to be equivalent to the existence of an (MP)(–bDT)-based axiomatic system, for many substructural logics under certain conditions. In Remark 4.8.18, we will discuss an even more general form of deduction theorem that would extend the characterization to a wider class of substructural logics (including, in particular, weak logics without residuated conjunction or protounit). Definition 4.8.4 (Δ-Implicational deduction theorem) Let Δ be a set of ★-formulas closed under ★-substitutions. A substructural logic L has the Δ-implicational deduction theorem, if for each set Γ ∪ {𝜑, 𝜓} of formulas, Γ, 𝜑 `L 𝜓

iff

Γ `L 𝛿(𝜑) → 𝜓 for some 𝛿 ∈ Δ.

We speak about (non-)parameterized implicational deduction theorem, whenever we want to stress the presence (resp. the lack) of variables other than ★ in Δ. Furthermore, we call the theorem global if the set Δ is a singleton, and local otherwise. Finally, if Δ = {★}, we speak about the classical deduction theorem.

4.8 Implicational deduction theorems

219

Note that the right-to-left direction of the Δ-implicational deduction theorem is equivalent to the assumption that the elements of Δ are actually deduction terms of L. Indeed, the 𝛿-necessitation follows from the fact that `L 𝛿(𝜑) → 𝛿(𝜑) and conversely Γ, 𝜑 `L 𝜓 follows from 𝛿-necessitation and modus ponens. Therefore, if L has the Δ-implicational deduction theorem, then it has the Δ0-implicational deduction theorem for any set of Δ0 ⊇ Δ of its deduction terms.25 Again, our goal will be to minimize the set Δ, but as we will see already in the next example, the situation here is more complex and in some logics the set Δ will have to contain deductively redundant deduction terms (e.g. in FLew or T). Let us again use a conventional notation ΔL for a chosen prominent set of deduction terms such that L has the Δ-implicational deduction theorem. Example 4.8.5 Recall that in Section 2.4 we have proved several instances of the implicational deduction theorem. First, in Proposition 2.4.1, we have shown that a finitary logic enjoys the classical deduction theorem iff it is an axiomatic expansion of IL→ . In Proposition 2.4.3, we have proved a local deduction theorem for axiomatic expansions of BCI. Under certain conditions, it can also be seen as an implicational deduction theorem. Indeed, we know that the logic SL−aE is an axiomatic expansion of BCI and we can use the residuation rule and the properties of 1¯ to conclude that (any axiomatic expansion of) SL−aE enjoys the ΔSL−aE -implicational deduction theorem for the set ¯ ∪ {★𝑘 | 𝑘 ≥ 1}, ΔSL−aE = {1} ¯ note that in the where 𝜑1 = 𝜑 and 𝜑 𝑛+1 = 𝜑 𝑛 & 𝜑 (we set conventionally 𝜑0 = 1; associative setting the bracketing inside 𝜑 𝑘 does not matter). Thus, in particular, while FLew does not enjoy the classical deduction theorem (see Proposition 2.4.1), it enjoys the ΔFLew -implicational deduction theorem for the set ΔFLew = {★𝑘 | 𝑘 ≥ 1} (we can drop 1¯ thanks to the axiom (w)). Finally, let us recall that in Proposition 2.4.2 we have proved that a finitary logic expanding CL in a language L ⊇ L enjoys the {(★)}-implicational deduction theorem iff it is an axiomatic extension of S4 (let us denote the set {(★)} as ΔS4 ). Therefore, T does not enjoy the {(★)}-implicational deduction theorem. However, later in Example 4.8.8, we will see that it enjoys the ΔT -implicational deduction theorem for the set ΔT = { 𝑘 (𝜑) | 𝑘 ≥ 1}, where we define 0 (𝜑) = 𝜑 and 𝑛+1 (𝜑) = (𝑛 (𝜑)). 25 We can use this observation to notice that the assumption that Δ is closed under ★-substitutions is not essential (we have added it for simplicity of the formulation of upcoming theorems): indeed if L enjoys the Δ-implicational deduction theorem (defined for Δs not necessarily closed under ★-substitutions), then 𝜎 𝛿 is a deduction term of L for each 𝛿 ∈ Δ and each ★-substitution 𝜎 and thus L enjoys the Δ0 -implicational deduction theorem, where Δ0 is a closure of Δ under ★-substitutions.

220

4 On lattice and residuated connectives

Later, in Example 4.8.8, we will show how all these facts are simple corollaries of our general theory. We leave as an exercise for the reader to find a semantical counterexample showing that neither FLew nor T can have the Δ-implicational deduction theorem for any finite set Δ. Let us now link the two notions, just defined, of (MP)–bDT-based axiomatic systems and Δ-implicational deduction theorems. It is easy to observe (see Theorem 4.8.6 for a detailed proof) that, if L is a finitary logic with the Δ-implicational deduction theorem, then it is (MP)–Δ-based. We already know that the question about the converse directions is not so straightforward: i.e. the logics IL and FLew are both (MP)-based, but only IL enjoys classical deduction theorem and FLew enjoys ΔFLew -implicational deduction theorem for a much more complex set ΔFLew only. Other important examples are the logics T and S4: both are (MP)–{★}-based, but only S4 enjoys the {(★)}-implicational deduction theorem while T enjoys ΔT -implicational deduction theorem for the more complex set ΔT only. Clearly, the complications are caused by the fact that there could be many deductively redundant 𝛿-necessitation rules for 𝛿 ∈ Δ, in particular when the deduction terms are composed from more elementary ones by either iteration or conjunction. The next theorem gives the first very general answer to our question and provides a necessary and sufficient condition for an (MP)-bDT-based logic to enjoy the Δ-implicational deduction theorem for some Δ ⊇ bDT. The conditions we use may seem rather ad hoc and not easy to check, but actually they will allow us to establish deduction theorems for several logics (see Example 4.8.8). More importantly, they have interesting theoretical consequences: they allow us to prove the transfer of the deduction theorem and describe logical filters generated by sets in arbitrary algebras. Later, in Theorem 4.8.13 and Corollary 4.8.16, we present more easy-to-check (equivalent and sufficient) conditions when restricted to particular deduction terms. Theorem 4.8.6 (Characterization of the implicational deduction theorem) Let L be a finitary substructural logic and Δ a set of ★-formulas closed under ★-substitutions. Then, the following are equivalent: 1. L has the Δ-implicational deduction theorem. 2. L is (MP)–bDT-based for some set bDT ⊆ Δ and the following four conditions are valid: DT1

There is a 𝛿𝑖 ∈ Δ such that `L 𝛿𝑖 (𝜑) → 𝜑.

DT2

There is a 𝛿 𝑤 ∈ Δ such that 𝜓 `L 𝛿 𝑤 (𝜑) → 𝜓.

DT3

(MP) For each 𝛾, 𝛿 ∈ Δ, there is a 𝛿 𝛾, 𝛿 ∈ Δ such that (MP) 𝛾(𝜑) → (𝛿(𝜑) → 𝜓) `L 𝛿 𝛾, 𝛿 (𝜑) → 𝜓.

4.8 Implicational deduction theorems

DT4

221 (𝛽-nec)

For each 𝜖 ∈ Δ and 𝛽 ∈ bDT, there is a 𝛿 𝜖 (𝛽-nec)

𝜖 (𝜑) → 𝜓 `L 𝛿 𝜖

∈ Δ such that

(𝜑) → 𝛽(𝜓).

3. For each L-algebra A and 𝑋 ∪ {𝑥, 𝑦} ⊆ 𝐴, we have 𝑦 ∈ FiAL (𝑋, 𝑥)

iff

𝛿A (𝑥, 𝑎 1 , . . . , 𝑎 𝑛 ) →A 𝑦 ∈ FiAL (𝑋) for some 𝛿(★, 𝑝 1 , . . . , 𝑝 𝑛 ) ∈ Δ and 𝑎 1 , . . . , 𝑎 𝑛 ∈ 𝐴.

Proof To prove that 1 implies 2, we set bDT = Δ and consider the axiomatic system AS whose axioms are all the theorems of L and whose rules are modus ponens and (𝛿-nec) for each 𝛿 ∈ Δ. Clearly, all these axioms and rules are valid in L (we already know that each 𝛿 ∈ Δ is a deduction term of L). Conversely, assume that Γ `L 𝜓. Due to the finitarity, we have 𝜑1 , . . . , 𝜑 𝑛 `L 𝜓 for some 𝜑1 , . . . , 𝜑 𝑛 ∈ Γ. By repeatedly using the left-to-right direction of the deduction theorem, we obtain `L 𝛿1 (𝜑1 ) → (𝛿2 (𝜑2 ) → (· · · → (𝛿 𝑛 (𝜑 𝑛 ) → 𝜓) . . . ) for some 𝛿1 , . . . , 𝛿 𝑛 ∈ Δ. Therefore, this formula is an axiom of AS and thus, by repeated use of of 𝛿𝑖 necessitation and modus ponens, we obtain Γ ` A S 𝜓. Therefore, L is (MP)–Δ-based. To complete the proof of this implication, it suffices to check that the consecutions DT1–DT4 are indeed valid in L; we prove the last one, the proofs of the remaining ones are similar. Consider 𝜖 ∈ Δ, 𝛽 ∈ bDT, and atoms 𝑝 and 𝑞 not occurring in 𝜖 and 𝛽. Clearly, we have 𝑝, 𝜖 ( 𝑝) → 𝑞 `L 𝛽(𝑞) (because both 𝛽 and 𝜖 are deduction terms of L). Therefore, by the implicational deduction theorem, there is a 𝛿 ∈ Δ such (𝛽-nec) that 𝜖 ( 𝑝) → 𝑞 `L 𝛿( 𝑝) → 𝛽(𝑞). We set 𝛿 𝜖 the instance of 𝛿 with all atoms 𝑝 replaced by ★. Observe that substituting 𝜑 and 𝜓 for 𝑝 and 𝑞 completes the proof. The proof that 2 implies 3 is a bit more complicated (mainly due to the possible presence of additional variables in the deduction terms and the fact that we work on an arbitrary algebra; we leave as an exercise for the reader to find a direct proof of the implication from 2 to 1). Let us recall that, since FiAL (𝑋) and FiAL (𝑋, 𝑥) are filters of the logic L, they are closed under all rules of L (we drop the suband superscripts from now on). We first use this fact to prove the right-to-left direction: assume that 𝛿A (𝑥, 𝑎 1 , . . . , 𝑎 𝑛 ) →A 𝑦 ∈ Fi(𝑋) ⊆ Fi(𝑋, 𝑥). Since, clearly, 𝛿(𝑥, 𝑎 1 , . . . , 𝑎 𝑛 ) ∈ Fi(𝑋, 𝑥) (using the 𝛿-necessitation and any appropriate evaluation), we obtain 𝑦 ∈ Fi(𝑋, 𝑥) (using modus ponens). The converse direction is the complicated one. First, note that given a formula 𝛿(★, 𝑝 1 , . . . , 𝑝 𝑛 ) ∈ Δ and an A-evaluation 𝑒, we can define a function 𝛿𝑒 : 𝐴 → 𝐴 as 𝛿𝑒 (𝑥) = 𝛿A (𝑥, 𝑒( 𝑝 1 ), . . . , 𝑒( 𝑝 𝑛 )) (cf. the case of formulas). Let us denote by ΔA the set of all such functions and note that we can reformulate our goal as: 𝑦 ∈ FiAL (𝑋, 𝑥) implies 𝑓 (𝑥) →A 𝑦 ∈ FiAL (𝑋) for some 𝑓 ∈ ΔA . We use the proof-like description of elements of Fi(𝑋, 𝑥) introduced in Proposition 2.5.8 and show that, for any element 𝑎 which is a label of a node in some proof of 𝑦 from the assumptions 𝑋 ∪ {𝑥}, there is an 𝑓 𝑎 ∈ ΔA such that 𝑓 𝑎 (𝑥) → 𝑎 ∈ Fi(𝑋). Let AS be the presentation given by the claim 2 and recall that we have defined 𝑉AA S = {h𝑒[Γ], 𝑒(𝜓)i | 𝑒 is an A-evaluation and Γ I 𝜓 ∈ AS}.

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4 On lattice and residuated connectives

In our case, we can be more specific and observe that elements of 𝑉AA S with non-empty first constituents are of the form 𝑉𝑅A = {h{𝑥, 𝑥 → 𝑦}, 𝑦i | 𝑥, 𝑦 ∈ 𝐴} ∪ {h{𝑥}, 𝑓 (𝑥)i | 𝑥 ∈ 𝐴 and 𝑓 ∈ bDTA }. Consider any node 𝑛 of the proof labeled by an element 𝑎 and distinguish four cases: • 𝑛 is a leaf and 𝑎 = 𝑥. Then, we can set 𝑓 𝑎 = (𝛿𝑖 )𝑒 for any A-evaluation 𝑒 such that 𝑒( 𝑝) = 𝑥, where 𝑝 is a variable not occurring in 𝛿𝑖 . Indeed, thanks to condition DT1, we know that 𝛿𝑖 ( 𝑝) → 𝑝 and so 𝑓 𝑎 (𝑥) → 𝑥 ∈ Fi(𝑋). • 𝑛 is a leaf and 𝑎 ∈ 𝑋 or h∅, 𝑎i ∈ 𝑉AA S . Then, 𝑎 ∈ Fi(𝑋) and, as in the previous case, if we set 𝑓 𝑎 = (𝛿 𝑤 )𝑒 for some suitable A-evaluation 𝑒, we obtain that 𝑓 𝑎 (𝑥) → 𝑥 ∈ Fi(𝑋), thanks to a suitable instance of condition DT2. • 𝑛 has two predecessors labeled as 𝑏 and 𝑏 → 𝑎 for some 𝑏 ∈ 𝐴. Then, by the induction assumption, 𝑓𝑏 (𝑥) → 𝑏, 𝑓𝑏→𝑎 (𝑥) → (𝑏 → 𝑎) ∈ Fi(𝑋) for some 𝑓𝑏 , 𝑓𝑏→𝑎 ∈ ΔA . Thus, using (Sf), we have (𝑏 → 𝑎) → ( 𝑓𝑏 (𝑥) → 𝑎) ∈ Fi(𝑋) and, by transitivity, also 𝑓𝑏→𝑎 (𝑥) → ( 𝑓𝑏 (𝑥) → 𝑎) ∈ Fi(𝑋). Thanks to the fact that Δ is closed under arbitrary ★-substitutions, we know that there are 𝛾(★, 𝑝 1 , . . . , 𝑝 𝑛 ), 𝛿(★, 𝑞 1 , . . . , 𝑞 𝑛 ) ∈ Δ and an A-evaluation 𝑒 such that – the variables 𝑝 1 , . . . , 𝑝 𝑛 , 𝑞 1 , . . . , 𝑞 𝑛 , 𝑝, 𝑞 are pairwise different, – 𝑓𝑏 = 𝛿𝑒 and 𝑓𝑏→𝑎 = 𝛾𝑒 , and – 𝑒( 𝑝) = 𝑥 and 𝑒(𝑞) = 𝑎. Therefore, 𝑒(𝛾( 𝑝) → (𝛿( 𝑝) → 𝑞)) ∈ Fi(𝑋) and so, due to condition DT3, we (MP) (MP) have 𝑒(𝛿 𝛾, 𝛿 ( 𝑝) → 𝑞) ∈ Fi(𝑋); thus we can take 𝑓 𝑎 = (𝛿 𝛾, 𝛿 ) 𝑒 . • 𝑛 has one predecessor labeled as 𝑏 and 𝑎 = 𝑔(𝑏) for some 𝑔 ∈ bDTA . Then, by induction assumption, 𝑓𝑏 (𝑥) → 𝑏 ∈ Fi(𝑋) for some 𝑓𝑏 ∈ ΔA . As before, we can assume that 𝑓𝑏 = 𝛾𝑒 and 𝑔 = 𝛽𝑒 for some suitable 𝛾 ∈ Δ and 𝛽 ∈ bDT and an (𝛽-nec) A-evaluation 𝑒 and use condition DT4 to justify that we can take 𝑓 𝑎 = (𝛿 𝛾 )𝑒 . The proof that 3 implies 1 is done by taking A = FmL and recalling that Δ is closed under arbitrary ★-substitutions.  Remark 4.8.7 Note that we could prove a variant of the previous theorem for any logic L in a language L ⊇ L→ that validates the axiom (id) and the rule (MP). We would only have to replace the condition DT3 by the (in substructural logics equivalent) condition: DT30

(MP) For each 𝛾, 𝛿 ∈ Δ, there is a 𝛿 𝛾, 𝛿 ∈ Δ such that (MP) 𝛾(𝜑) → 𝜒, 𝛿(𝜑) → ( 𝜒 → 𝜓) `L 𝛿 𝛾, 𝛿 (𝜑) → 𝜓.

We decided to formulate the theorem for substructural logics not only for simplicity but also because the stronger versions of this theorem (Theorem 4.8.13 and Corollary 4.8.16) work only for substructural logics and because, actually, most logics of interest for which we would like to obtain a deduction theorem are substructural.

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The exceptions are local modal logics, which are still (MP)-based and the classical deduction theorem for these logics would be a consequence of a more general formulation of our theorem. Consequently, in the rest of this chapter the only modal logics we will refer to are the global ones.

Example 4.8.8 We can use the previous theorem to obtain all the deduction theorems mentioned in Example 4.8.5 and more. • Axiomatic expansions of IL→ : these logics are (MP)-based and so we can take (MP) bDT = ∅, which makes DT4 void and, setting 𝛿 𝑤 = 𝛿𝑖 = 𝛿★,★ = ★, we have the classical deduction theorem (note that DT3 becomes the rule contraction). • Axiomatic expansions of FLew : again we can take bDT = ∅, which makes DT4 𝑛+𝑚 , we have DT1–DT3 and so we void and, taking 𝛿 𝑤 = 𝛿𝑖 = ★ and 𝛿★(MP) 𝑛 ,★𝑚 = ★ obtain the ΔFLew -implication deduction theorem (note that we use weakening and associativity to prove it). • Axiomatic expansions of SL−aE (the {∧, ∨}-free fragment SLaE ): as before but, due the lack of weakening, we have to set 𝛿 𝑤 (𝜑) = 1¯ and include 1¯ in ΔSL−aE . • T: recall that this logic is (MP)–{(★)}-based and setting 𝛿 𝑤 = 𝛿𝑖 = (★), max{𝑚,𝑛} (★), and 𝛿 (-nec) = 𝑛+1 (★) fulfills the conditions 𝛿(MP) 𝑛 (★),𝑚 (★) =  𝑛 (★) DT1–DT4 (we leave the details as an exercise for the reader) and so T has the ΔT -implicational deduction theorem for ΔT = { 𝑘 (★) | 𝑘 ≥ 1}. • S4: again this logic is (MP)–{(★)}-based and, to obtain its {(★)}-implicational deduction theorem, it suffices to check the conditions DT1–DT4 for 𝛿 𝑤 = 𝛿𝑖 = (MP) (-nec) (★) = 𝛿(★),(★) = (★), and 𝛿(★) = (★). Again, as an exercise the reader should check the necessary conditions. • K4: we leave as an exercise for the reader to show that this logic has both the {★ ∧ (★)}- and {★, (★)}-implicational deduction theorems.

Recall that, if a logic L is (MP)-bDT-based, so is each of its axiomatic expansions (assuming that we close the set bDT under ★-substitutions of the new language). Thanks to the previous theorem, the analogous claim is obviously valid for implicational deduction theorems as well.26 However, as we have seen, in stronger logics we can often consider a restricted set of deduction terms; the next obvious proposition provides a simple characterization telling us under which conditions we can do that. Let us point to the special case in which Δ = Δ0 (which particularizes to the previous observation) or L = L0 (which tells us when we can simplify some already established deduction theorem of a given logic, usually inherited from a weaker logic). We formulate it for axiomatic extensions to avoid the problem with substitutions and leave its proof and generalization to expansions as an exercise for the reader. 26 Note that for axiomatic extensions we could see this even without resorting to the previous theorem by simply considering the additional axioms as elements of the set Γ; but this reasoning would not work in the presence of additional connectives.

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4 On lattice and residuated connectives

Proposition 4.8.9 Let L be a logic with the Δ-implicational deduction theorem and let L0 be an axiomatic extension. Then, L0 has the Δ0-implicational deduction theorem for a set Δ0 ⊆ Δ iff for every 𝜒 ∈ Δ there is a 𝛿 ∈ Δ0 such that `L0 𝛿(𝜑) → 𝜒(𝜑). The previous theorem can also be used to obtain the following algebraic description of the logical filter generated by a set for a wide class of substructural logics. Corollary 4.8.10 (Filter generation) Let L be a finitary substructural algebraically implicative logic expanding LL&,1¯ with the Δ-implicational deduction theorem, A an L-algebra with the intrinsic order ≤, and 𝑋 ⊆ 𝐴. Then, Ø FiAL (𝑋) = {𝑎 ∈ 𝐴 | 𝑧 ≤ 𝑎}, 𝑧 ∈ΔA (𝑋 )

where ΔA (𝑋) is the least set closed under &A which contains the set {𝛿A (𝑧, 𝑎 1 , . . . , 𝑎 𝑛 ) | 𝛿(★, 𝑝 1 , . . . , 𝑝 𝑛 ) ∈ Δ, 𝑧 ∈ 𝑋, and 𝑎 1 , . . . , 𝑎 𝑛 ∈ 𝐴}. Proof One inclusion is straightforward: due to the rule of 𝛿-necessitation we know that 𝛿A (𝑧, 𝑎 1 , . . . , 𝑎 𝑛 ) ∈ Fi(𝑋) for each 𝛿(★, 𝑝 1 , . . . , 𝑝 𝑛 ) ∈ Δ, 𝑎 1 , . . . , 𝑎 𝑛 ∈ 𝐴, and 𝑧 ∈ 𝑋. Therefore, we know that ΔA (𝑋) ⊆ Fi(𝑋) (due to (Adj& ) and modus ponens). Now take 𝑎 ≥ 𝑥 for some 𝑥 ∈ ΔA (𝑋); then, clearly, 𝑥 → 𝑎 ∈ Fi(𝑋) and so 𝑎 ∈ Fi(𝑋). To prove the other inclusion, assume that 𝑎 ∈ Fi(𝑋). Due to the finitarity of L and Theorem 2.6.2, there has to be a finite set {𝑥 1 , . . . , 𝑥 𝑛 } ⊆ 𝑋 such that 𝑎 ∈ Fi(𝑥1 , . . . , 𝑥 𝑛 ). Thus, by the previous theorem, 𝑥 𝑛0 → 𝑎 ∈ Fi(𝑥1 , . . . , 𝑥 𝑛−1 ) for some 𝑥 𝑛0 ∈ ΔA (𝑋). Repeating this for 𝑖 ≤ 𝑛, we obtain {𝑥10 , . . . , 𝑥 𝑛0 } ⊆ ΔA (𝑋) such that 𝑥 10 → (𝑥20 → · · · → (𝑥 𝑛0 → 𝑎) . . .) ∈ Fi(∅). Therefore, by residuation, we know that 𝑥 𝑛0 & (. . . (𝑥20 & 𝑥10 ) . . .) → 𝑎 ∈ Fi(∅). Setting 𝑧 = 𝑥 𝑛0 & (. . . (𝑥20 & 𝑥10 ) . . .) completes the proof (since 𝑎 ≥ 𝑧 ∈ ΔA (𝑋)). 

Example 4.8.11 Let us list a few particular instances of the previous corollary whose validity we know due to Example 4.8.8. Let us assume that L is a substructural logic in a language L, A an L-algebra, and 𝑋 ⊆ 𝐴. If L is an axiomatic expansions of IL→ , then FiAL (𝑋) = {𝑎 ∈ 𝐴 | 𝑥 ≤ 𝑎 for some 𝑥 ∈ 𝑋 }. FLew , then FiAL (𝑋) = {𝑎 ∈ 𝐴 | 𝑥 𝑖𝑘𝑖 & · · · & 𝑥 𝑛𝑘𝑛 ≤ 𝑎 for some 𝑥1 , . . . , 𝑥 𝑛 ∈ 𝑋 and natural numbers 𝑘 1 , . . . , 𝑘 𝑛 }. S4, then FiAL (𝑋) = {𝑎 ∈ 𝐴 | 𝑥 ≤ 𝑎 for some 𝑥 ∈ 𝑋 }.

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Our next goal is to improve Theorem 4.8.6. We start by considering deduction terms that are obtained from more elementary ones by either iteration or conjunction; in order to analyze this situation, we introduce the following notation. Definition 4.8.12 (Iterated and conjuncted deduction terms) Given a set Δ of ★-formulas in L, we define the set Δ∗ of ★-formulas as the smallest set 𝐹 such that • ★ ∈ 𝐹 and • 𝐹 is closed under each 𝛿 ∈ Δ, i.e. 𝛿(𝛾) ∈ 𝐹 for each 𝛿 ∈ Δ and each 𝛾 ∈ 𝐹. Furthermore, if L contains 1¯ and &, we define the set Π(Δ) as the smallest set of ¯ and closed under &.27 ★-formulas containing Δ ∪ {1} It is easy to see that, if a logic L is (MP)–Δ-based (e.g. whenever it has the Δimplicational deduction theorem) and there is a set bDT ⊆ Δ such that Δ ⊆ Π(bDT∗ ), then L is also (MP)–bDT-based.28 Clearly, taking any (MP)–bDT-based logic and Δ = Π(bDT∗ ), conditions DT1– ¯ and DT3 of Theorem 4.8.6 become obviously valid (by setting 𝛿𝑖 = ★, 𝛿 𝑤 = 1, (𝛽-nec) (MP) 𝛿 𝛾, 𝛿 = 𝛾 & 𝛿) and condition DT4 requires us to find certain 𝛿 𝜖 ∈ Π(bDT∗ ) for each 𝜖 ∈ Π(bDT∗ ) and 𝛽 ∈ bDT. In the next theorem, we replace it by an equivalent simpler condition DT6 which will require us to find certain 𝛿1 , 𝛿2 ∈ Π(bDT∗ ) for each 𝜄 ∈ bDT∗ . On the other hand, recall that we have seen many examples where (MP)–bDT-based logics enjoy the Δ-implicational deduction theorem for a proper subset Δ ( Π(bDT∗ ). Therefore, we formulate the following theorem for arbitrary Δ and add condition DT5 which makes sure that the set Δ is big enough (note that for Δ = Π(bDT∗ ) it is trivially satisfied). Theorem 4.8.13 Let L be a finitary substructural logic expanding LL&,1¯ and let bDT and Δ be sets of ★-formulas closed under ★-substitutions such that bDT ⊆ Δ ⊆ Π(bDT∗ ). Then, L has the Δ-implicational deduction theorem iff L is (MP)–bDTbased and the following two conditions are valid: DT5

For each 𝛿 ∈ Π(bDT∗ ), there is a 𝛿Δ ∈ Δ such that `L 𝛿Δ (𝜑) → 𝛿(𝜑).

DT6

For each 𝜄 ∈ bDT∗ \ {★}, there are 𝛿1 , 𝛿2 ∈ Π(bDT∗ ) such that `L 𝛿1 (𝜑 → 𝜓) → (𝛿2 (𝜑) → 𝜄(𝜓)).

27 The fact that our definition entails that 1¯ ∈ Π(bDT∗ ) is purely conventional and included for simplicity of presentation; cf. Remark 4.8.18. 28 Clearly, a set bigger than Π(bDT∗ ) could be used here; in the extreme it could be the set of all deduction terms of the given logic. Another option, which appears in the literature, is the notion ¯ ★} and of confusion of a set bDT defined as the smallest set of ★-formulas containing bDT ∪ {1, closed under & and under each 𝛿 ∈ bDT. However, we will see that, in our setting, we always obtain the Δ-implicational deduction theorem for Δ ⊆ Π(bDT∗ ) for some neatly defined set bDT, so there is no need for this additional layer of complexity.

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4 On lattice and residuated connectives

Proof The left-to-right direction is simple. From (the proof of) Theorem 4.8.6, we know that L is (MP)–Δ-based and so, from the assumption bDT ⊆ Δ ⊆ Π(bDT∗ ), we obtain that L is (MP)–bDT-based (the first subsethood makes sure that the elements of bDT are deduction terms of L and the second one ensures, for each 𝛿 ∈ Π(bDT∗ ) ⊇ Δ, the derivability of 𝛿-necessitation in the (MP)–bDT-based axiomatic system containing as axioms all theorems of L). Finally, we use the left-to-right direction of the Δ-implicational deduction theorem to obtain DT5 and DT6 (in the second case start with 𝜑 → 𝜓, 𝜑 `L 𝜄(𝜓)). To prove the converse implication, we also use Theorem 4.8.6. We start by ¯ ★ ∈ Π(bDT∗ ) and Π(bDT∗ ) is closed under &. Therefore, we can observing that 1, (MP) Δ set 𝛿 𝑤 = 1¯ Δ , 𝛿𝑖 = ★Δ , and 𝛿 𝛾, 𝛿 = (𝛾 & 𝛿) to obtain the conditions DT1–DT3. In order to prove DT4, first we establish three auxiliary claims. DT-AUX1

For each 𝛾 ∈ Π(bDT∗ ), there exists 𝛾 0 ∈ Π(bDT∗ ) such that 𝜑 → 𝜓 `L 𝛾 0 (𝜑) → 𝛾(𝜓).

We prove it by induction on the smallest number 𝑛 of conjuncts from bDT∗ constituting ¯ we can set 1¯ 0 = 1. ¯ If 𝑛 = 1, then the formula 𝛾.29 Clearly, if 𝑛 = 0, i.e. 𝛾 = 1, ∗ 0 𝛾 ∈ bDT . We can use DT6, set 𝛾 = 𝛿2 , and use 𝛿1 -necessitation and modus ponens to prove the claim. Finally, for the induction step, assume that 𝛾 = 𝜈1 & 𝜈2 . Then, by the induction assumption, we have 𝜈10 and 𝜈20 such that 𝜑 → 𝜓 `L 𝜈𝑖0 (𝜑) → 𝜈𝑖 (𝜓); setting 𝛾 0 = 𝜈10 & 𝜈20 and using (Mon& ) completes the proof. DT-AUX2

For each 𝜄 ∈ bDT∗ , there exist 𝛾1 , 𝛾2 ∈ Π(bDT∗ ) such that `L 𝛾1 (𝜑) & 𝛾2 (𝜓) → 𝜄(𝜑 & 𝜓).

To prove this claim, we use DT6 to obtain 𝛿1 , 𝛿2 ∈ Π(bDT∗ ) such that `L 𝛿1 (𝜑 → 𝜑 & 𝜓) → (𝛿2 (𝜑) → 𝜄(𝜑 & 𝜓)). Then, we use DT-AUX1 on the axiom (adj& ) to obtain a 𝛿10 ∈ Π(bDT∗ ) such that `L 𝛿10 (𝜓) → 𝛿1 (𝜑 → 𝜑 & 𝜓). Setting 𝛾1 = 𝛿2 and 𝛾2 = 𝛿10 , the claim follows using (T) and (Res). DT-AUX3

For each 𝛾, 𝛿 ∈ Π(bDT∗ ), there exists 𝛿ˆ ∈ Π(bDT∗ ) such that ˆ `L 𝛿(𝜑) → 𝛾(𝛿(𝜑)).

Note that it suffices to prove the claim for 𝛾 ∈ bDT∗ and then, by the same method as in the proof of DT-AUX1, we extend it to all 𝛾 ∈ Π(bDT∗ ). 29 We need to speak about the smallest number of conjunctions because bDT∗ could contain the ¯ and so 𝛾 need not be a unique conjunction of conjunction of some of its elements (or formula 1) elements of bDT∗ .

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We reason by induction on the smallest number 𝑛 of conjuncts from bDT∗ constituting 𝛿. The cases 𝑛 = 0 and 𝑛 = 1 are trivial by setting 𝛿ˆ = 1¯ and 𝛿ˆ = 𝛾(𝛿). Next, assume that 𝛿 = 𝜂1 & 𝜂2 for some 𝜂1 , 𝜂2 ∈ Π(bDT∗ ) with less conjunctions. Using DT-AUX2, we obtain 𝛾1 , 𝛾2 ∈ Π(bDT∗ ) such that `L 𝛾1 (𝜂1 (𝜑)) & 𝛾2 (𝜂2 (𝜑)) → 𝛾 0 (𝜂1 (𝜑) & 𝜂2 (𝜑)). By the induction assumption, we obtain 𝛿ˆ1 , 𝛿ˆ2 ∈ Π(bDT∗ ) such that `L 𝛿ˆ1 (𝜑) → 𝛾1 (𝜂1 (𝜑))

and

`L 𝛿ˆ2 (𝜑) → 𝛾2 (𝜂2 (𝜑)).

Setting 𝛿ˆ = 𝛿ˆ1 & 𝛿ˆ2 completes the proof using (Mon& ). Now we are finally ready to prove (DT4) for any 𝜖 ∈ Δ ⊆ Π(bDT∗ ) and 𝛽 ∈ bDT. We start by using DT-AUX1 to obtain 𝛽 0 ∈ Π(bDT∗ ) such that 𝜖 (𝜑) → 𝜓 `L 𝛽 0 (𝜖 (𝜑)) → 𝛽(𝜓) and so, by DT-AUX3, we obtain 𝛿ˆ ∈ Π(bDT∗ ) such that ˆ `L 𝛿(𝜑) → 𝛽 0 (𝜖 (𝜑)). (𝛽-nec)

Setting 𝛿 𝜖

(𝜑) = 𝛿ˆΔ and using (T) completes the proof.



{★𝑘

Using the previous theorem for bDT = ∅ and Δ = | 𝑘 ≥ 0} (we set ¯ we directly obtain the following corollary which can be seen as a variant of ★0 = 1), Proposition 2.4.3 (note that condition DT5 is satisfied due to associativity). Corollary 4.8.14 (Local deduction theorem for associative (MP)-based logics) ¯ →}-fragment of SLa . Then, L is Let L be a substructural logic expanding the {&, 1, (MP)-based iff L is finitary and, for each set Γ ∪ {𝜑, 𝜓} of formulas, Γ, 𝜑 `L 𝜓

iff

Γ `L 𝜑 𝑛 → 𝜓 for some 𝑛 ≥ 0.

¯ Recall that from Example 4.8.3 we know that the logic SLaE is (MP)–{★∧ 1}-based; we now can use the previous theorem to describe which implicational deduction theorem it enjoys and use this corollary to show that it is not an (MP)-based logic. Example 4.8.15 The logic SLaE is not (MP)-based. Indeed, otherwise 𝜑 `SLaE 𝜑 ∧ 1¯ would entail the provability of the formula 𝜑 𝑛 → 𝜑 ∧ 1¯ for some 𝑛 which the reader can refute by a simple semantical counterexample as an exercise. ¯ On the other hand, we know that SLaE is (MP)–{★ ∧ 1}-based and we can use Theorem 4.8.13 to show that it has the ΔSLaE -implicational deduction theorem for the set ¯ 𝑘 | 𝑘 ≥ 1}. ΔSLaE = {(★ ∧ 1) Indeed, observe that for any 𝜄 ∈ bDT∗ we have 𝜓 ∧ 1¯ → 𝜄(𝜓) and, thus, we obtain 1¯ ) (for 𝛿 = 𝛿 = ★ ∧ 1) ¯ and DT5 by repeated DT6 by double use of (Pf) and (distr→ 1 2 use of (Mon& ) (the case 𝛿 = 1¯ is trivial).

228

4 On lattice and residuated connectives

These facts are also true for the Abelian logic A (introduced in Example 4.6.14) and are proved in the same way. Note that condition DT6 is rather easy to check and resembles the axiom (K) of 1¯ ) valid already in SL. However, the fact that we modal logics or the axiom (distr→ ∗ have to consider all 𝜄 ∈ bDT is an eyesore: we would prefer to check condition DT6 for basic deduction terms only. The next corollary shows that we can do it at a price: we obtain no longer a characterization, but just a sufficient condition, and we have to find 𝛿1 , 𝛿2 already in bDT∗ (and not in Π(bDT∗ ) like in the case of DT6). Corollary 4.8.16 Let L be a finitary substructural logic expanding LL&,1¯ and let bDT and Δ be sets of ★-formulas closed under ★-substitutions such that bDT ⊆ Δ ⊆ Π(bDT∗ ). Furthermore, assume that L is (MP)–bDT-based and the following two conditions are valid: DT5

For each 𝛿 ∈ Π(bDT∗ ), there is a 𝛿Δ ∈ Δ such that `L 𝛿Δ (𝜑) → 𝛿(𝜑).

DT60

For each 𝛽 ∈ bDT, there are 𝜄, 𝛾 ∈ bDT∗ such that `L 𝛾(𝜑 → 𝜓) → (𝜄(𝜑) → 𝛽(𝜓)).

Then, the logic L has the Δ-implicational deduction theorem. Proof Clearly, due to Theorem 4.8.13, all we have to do is to prove condition DT6. First, we prove a weaker variant of DT-AUX1 from the proof of Theorem 4.8.13. DT-AUX10

For each 𝜄 ∈ bDT∗ , there exists 𝜄0 ∈ bDT∗ such that 𝜑 → 𝜓 `L 𝜄0 (𝜑) → 𝜄(𝜓).

Assume that 𝜄 = 𝛽1 . . . 𝛽𝑛 (we omit the brackets for simplicity). Then, thanks to DT60, there are 𝜄𝑖 , 𝛾𝑖 ∈ bDT∗ such that for 𝜄0 = 𝜄1 . . . 𝜄𝑛 and 𝜒𝑖 = 𝜄𝑖 . . . 𝜄𝑛 (𝜑) → 𝛽𝑖 . . . 𝛽𝑛 (𝜓), we obtain the following series of facts: `L 𝛾𝑛 (𝜑 → 𝜓) → (𝜄𝑛 (𝜑) → 𝛽𝑛 (𝜓)) `L 𝛾𝑛−1 (𝜄𝑛 (𝜑) → 𝛽𝑛 (𝜓)) → (𝜄𝑛−1 𝜄𝑛 (𝜑) → 𝛽𝑛−1 𝛽𝑛 (𝜓)) `L 𝛾𝑛−2 ( 𝜒𝑛−1 ) → 𝜒𝑛−2 .. . `L 𝛾2 ( 𝜒3 ) → 𝜒2 `L 𝛾1 ( 𝜒2 ) → (𝜄0 (𝜑) → 𝜄(𝜓)) Repeated use of 𝛾𝑖 -necessitation and (MP) completes the proof of DT-AUX10. Now we are ready to prove DT6. Consider again 𝜄 = 𝛽1 . . . 𝛽𝑛 and let 𝜄0, 𝛾𝑖 , and 𝜄0 be as before and set 𝜒𝑛+1 = 𝜑 → 𝜓. We show by induction that, for each 𝑖 > 1, there

4.8 Implicational deduction theorems

229

is an 𝜂𝑖 ∈ bDT∗ such that 𝜂𝑖 ( 𝜒𝑖+1 ) → (𝜄0 (𝜑) → 𝜄(𝜓)); therefore, for 𝑖 = 𝑛 we obtain the desired claim. For the base case 𝑖 = 1, we set 𝜂1 = 𝛾1 . Assume that the claim holds for 𝑖 and recall that we know that 𝛾𝑖+1 ( 𝜒𝑖+2 ) → 𝜒𝑖+1 and thus, thanks to DT-AUX10, there is an 𝜂𝑖0 ∈ bDT∗ such that 𝜂𝑖0 (𝛾𝑖+1 ( 𝜒𝑖+2 )) → 𝜂𝑖 ( 𝜒𝑖+1 ). Setting 𝜂𝑖+1 = 𝜂𝑖0 (𝛾𝑖+1 ) and using transitivity together with the induction assumption completes the proof. ¯ or in K for Clearly, condition (DT60) is met in SLaE for bDTSLaE = {★ ∧ 1} bDTK = {(★)}. This observation could simplify the proof of some implicational deduction theorems e.g. for SLaE or T (cf. Examples 4.8.8 and 4.8.15) and allow us to easily prove a new one (e.g. for the logic K in the next straightforward example). Example 4.8.17 K enjoys the ΔK -implicational deduction theorem for the set Û ΔK = Π({(★)}∗ ) = {★} ∪ {  𝑘𝑖 (★) | 𝑛 ≥ 1 and 𝑘 1 , . . . 𝑘 𝑛 ≥ 1}. 𝑖 ≤𝑛

Let us note that we have established deduction theorems for most of the logics introduced in Chapter 2 and for axiomatic expansions of SLaE . The goal of the following section is to deal with weaker substructural logics. Before that, we conclude this section with a remark on a possible generalization of the notion of implicational deduction theorem which, at the price of increased complexity, would cover some additional substructural logics, especially logics in purely implicative languages. Remark 4.8.18 We say that a substructural logic L enjoys the generalized Δ-implicational deduction theorem if for each set Γ ∪ {𝜑, 𝜓} of formulas,30 Γ, 𝜑 `L 𝜓

iff

Γ `L h𝛿1 (𝜑), . . . , 𝛿 𝑛 (𝜑)i → 𝜓 for some 𝑛 ≥ 0 and 𝛿1 , . . . , 𝛿 𝑛 ∈ Δ.

This definition allows us to obtain deduction theorems for logics in languages without ¯ e.g. we know that axiomatic expansions of BCI enjoy the generalized & or 1; {★}-implicational deduction theorem (see Proposition 2.4.3). Let us observe that any logic with the Δ-implicational deduction theorem enjoys the generalized one too. Indeed, the left-to-right direction is obvious and the right-to-left one follows from the fact that such logic is (MP)–Δ-based. Clearly, if L has the residuated conjunction, then the notion above is equivalent to Γ, 𝜑 `L 𝜓

iff

Γ `L 𝜓 or Γ `L 𝛿1 (𝜑) & . . . & 𝛿 𝑛 (𝜑) → 𝜓 for some 𝑛 ≥ 1 and 𝛿1 , . . . , 𝛿 𝑛 ∈ Δ.

Therefore, any substructural logic expanding LL&,1¯ has the generalized Δ-implicational deduction theorem iff it has the Π(Δ)-implicational deduction theorem (the presence of 1¯ in Π(Δ) takes care of the case Γ `L 𝜓). 30 Let us recall our earlier convention of writing, for 𝑛 ≥ 1, h𝜑1 , . . . , 𝜑𝑛 i → 𝜑 for the formula 𝜑1 → ( 𝜑2 → . . . ( 𝜑𝑛 → 𝜑) . . . ) and setting hi → 𝜑 = 𝜑.

230

4 On lattice and residuated connectives

4.9 Strongly (MP)–bDT-based axiomatic systems In Corollary 4.8.16, we have obtained an easy way of proving that an (MP)–bDTbased logic L has the Δ-implicational deduction theorem for some set Δ; in particular, it says that if L satisfies DT60

for each 𝛽 ∈ bDT, there are 𝜄, 𝛾 ∈ bDT∗ such that `L 𝛾(𝜑 → 𝜓) → (𝜄(𝜑) → 𝛽(𝜓)),

then it has the Π(bDT∗ )-implicational deduction theorem. In almost all the particular logics we have seen so far, we can obtain a tighter form of deduction theorem (the exception is the logic K), but using DT60 we could obtain some form of deduction theorem for all these logics almost effortlessly. Indeed, for (MP)-based logics 1¯ ), and for ¯ we have it trivially, for the (MP)–{★ ∧ 1}-based using theorem (distr→ (MP)–{(★)}-based using axiom (K). Note that in both cases we can set 𝜄 = 𝛾 = 𝛽. However, there are important substructural logics for which we have not yet proved any form of deduction theorem, namely the non-commutative and non-associative ones. As we will see, for these logics the necessary sets of deduction terms are rather complex: as in the case of the logic K, we will need to work with conjunctions and iterations, and now will even need to use terms with additional variables. On the other hand, we will be able to prove an even stronger form of DT60 for these sets (though we will not be able to set 𝜄 = 𝛾 = 𝛽, as in the logics seen so far) in which we can assume that 𝜄 is a basic deduction term as well. This seemingly unimportant alteration will play a crucial role later in Chapter 6 (see Theorem 6.3.3). Towards this end, we introduce the notion of strongly (MP)–bDT-based axiomatic system and logic. Definition 4.9.1 (Strongly (MP)(–bDT)-based axiomatic system and logic) An (MP)–bDT-based axiomatic system AS is strong if the following condition holds: SDT

For each 𝛽 ∈ bDT, there are 𝛽 0 ∈ bDT and 𝛾 ∈ bDT∗ such that `L 𝛾(𝜑 → 𝜓) → (𝛽 0 (𝜑) → 𝛽(𝜓)).

A substructural logic L is strongly (MP)–bDT-based if it has a strong (MP)–bDTbased presentation. Note that any (MP)-based logic is trivially strongly (MP)–∅-based. We have seen that FLew is an example of an (MP)-based logic, while SLaE and K are examples of ¯ strongly (MP)–{★ ∧ 1}-based and strongly (MP)–{(★)}-based logics which are not (MP)-based. Due to Corollary 4.8.16, we know that any strongly (MP)–bDT-based logic has the Π(bDT∗ )-implicational deduction theorem. However, as the notion of basic deduction term is purely conventional, we can prove a claim which can be seen as a variant of the missing converse claim (note that the price we pay for obtaining (SDT) is that we have to consider a rather big set of basic deduction terms, which induces many redundant 𝛿-necessitation rules).

4.9 Strongly (MP)–bDT-based axiomatic systems

231

Theorem 4.9.2 Let L be a substructural logic expanding LL&,1¯ and let bDT and Δ be sets of ★-formulas closed under ★-substitutions. • If L is strongly (MP)–bDT-based, then it has the Π(bDT∗ )-implicational deduction theorem. • If L has the Δ-implicational deduction theorem, then it is strongly (MP)–Δ-based. Proof The first claim follows directly from Corollary 4.8.16 and the second one is an easy consequence of Theorem 4.8.13 for Δ = bDT. Indeed, we know that L is strongly (MP)–Δ-based and that, due to DT6, for each 𝛽 ∈ bDT there are 𝛿1 , 𝛿2 ∈ Π(bDT∗ ) such that `L 𝛿1 (𝜑 → 𝜓) → (𝛿2 (𝜑) → 𝜄(𝜓)). Due to DT5, we know that there are 𝛿1Δ , 𝛿2Δ ∈ Δ = bDT such that 𝛿1Δ (𝜑) → 𝛿1 (𝜑) and 𝛿2Δ (𝜑) → 𝛿2 (𝜑); using suffixing and prefixing completes the proof.  Remark 4.9.3 Note that in the proof we show an even stronger version of SDT in which both 𝛽 0 and 𝛾 can be taken basic. However, we disregard it here because in this situation there are many redundant bDTs and we are not able to prove this variant of SDT for the logic SL. It is obvious that if L is strongly (MP)–bDT-based, then so is each of its axiomatic expansions (we only have to take care about closing bDT under the ★-substitutions of the expanded language); but of course, sometimes a smaller set suffices. Therefore, for didactic reasons, we first prove that SLa is strongly (MP)–bDTSLa -based for a set of basic deduction terms bDTSLa , which is rather complex but still much simpler than the set bDTSL . Let us start by introducing the candidates for its basic deduction terms, which in the literature on substructural logics are known as conjugates; the corresponding necessitation rules are known as product normality rules (there could be some terminological discrepancies, see Remark 4.9.6 for details). Definition 4.9.4 (Left and right conjugates) Given a formula 𝛼, we define the left and right conjugates w.r.t. 𝛼 as 𝜆 𝛼 (★) = 𝛼 → ★ & 𝛼

𝜌 𝛼 (★) = 𝛼

𝛼 & ★.

Theorem 4.9.5 The logic SLa is strongly (MP)–bDTSLa -based for the set bDTSLa = {𝜆 𝛼 (★), 𝜌 𝛼 (★), ★ ∧ 1¯ | 𝛼 ∈ FmLSL }. Proof We start by showing that the axiomatic system depicted in Figure 4.7, denoted by AS in this proof, is a presentation of SLa and, thus, we establish that it is obviously (MP)–bDTSLa -based (the set bDTSLa is obviously closed under ★-substitutions). Recall that, in Remark 4.6.8, we have shown that SLa can be axiomatized by (pf), (MP), (res), (e ), (Symm), (push), (pop), (⊥), (>), (lb), (ub), (inf), (sup), 1¯ ), and (Adj ). Thus, to prove that ` (distr& A S ⊆ SLa , it suffices to take care of 1¯ (auxFL ) and (Pn). To show that (auxFL ) is valid in SLa , we start with (Res) to obtain 2 𝜑 & (𝜑 → 𝜓) → 𝜓 and (Pf ) completes the proof; to show the validity of (Pn2 ), we use the theorem (adj& ) and the rules (MP) and (Symm); the remaining two

232

4 On lattice and residuated connectives

(pf)

(𝜑 → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓))

prefixing

(res1 )

(𝜑 → (𝜓 → 𝜒)) → (𝜓 & 𝜑 → 𝜒)

residuation

(res2 )

(𝜓 & 𝜑 → 𝜒) → (𝜑 → (𝜓 → 𝜒))

residuation

(e

(𝜑

,1 )

(e

,2 ) (auxFL 1 ) (auxFL 2 )

(𝜓 → 𝜒)) → (𝜓 → (𝜑

𝜒))

-exchange

(𝜑 → 𝜒))

-exchange

(𝜑 → (𝜓

𝜒)) → (𝜓

(𝜑 → (𝜑

𝜓) & 𝜑) → (𝜑 → 𝜓)

(𝜑

𝜑 & (𝜑 → 𝜓)) → (𝜑

fusion divisions fusion divisions

𝜓)

push

(pop)

𝜑 → ( 1¯ → 𝜑) ( 1¯ → 𝜑) → 𝜑

(⊥)

⊥→𝜑

ex falso quodlibet

(>)

𝜑→>

verum ex quolibet

(ub1 )

𝜑 → 𝜑∨𝜓

upper bound

(ub2 )

𝜓 → 𝜑∨𝜓

upper bound

(sup)

(𝜑 → 𝜒) ∧ (𝜓 → 𝜒) → (𝜑 ∨ 𝜓 → 𝜒)

supremality

(lb1 )

𝜑∧𝜓 → 𝜑

lower bound

(lb2 )

𝜑∧𝜓 →𝜓

lower bound

(inf)

( 𝜒 → 𝜑) ∧ ( 𝜒 → 𝜓) → ( 𝜒 → 𝜑 ∧ 𝜓) ¯ & (𝜓 ∧ 1) ¯ → (𝜑 & 𝜓) ∧ 1¯ (𝜑 ∧ 1)

infimality ¯ 1-&-distributivity

(Adj1¯ )

𝜑, 𝜑 → 𝜓 I 𝜓 𝜑 I 𝜑 ∧ 1¯

modus ponens ¯ 1-adjunction

(Pn1 )

𝜑 I 𝜓 → 𝜑&𝜓

product normality

(Pn2 )

𝜑I𝜓

𝜓&𝜑

product normality

(push)

1¯ ) (distr&

(MP)

pop

Fig. 4.7 Strong (MP)–bDTSLa -based presentation of the logic SLa .

consecutions are mirror images of these two and so they are valid in SLa thanks to Corollary 4.6.17. Conversely, to prove that SLa ⊆ ` A S , it suffices to show the validity of the rules (Symm): using (Pn1 ) and (auxFL 𝜓 and 𝜓 = 𝜑 we obtain 1 ) for 𝜑 = 𝜑 (Symm2 ) and, analogously, we obtain the rule (Symm1 ) from (Pn2 ) and (auxFL 2 ). To complete the proof, i.e. to establish the condition SDT, we prove an even stronger claim: for each 𝛽 ∈ bDTSLa , ` 𝛽(𝜑 → 𝜓) → (𝛽(𝜑) → 𝛽(𝜓)). First, observe that from (adj& ) using (Pf) we get (𝜑 → 𝜓) → (𝜑 → (𝛼 → 𝛼 & 𝜓)) and so, using the axiom (res), we obtain the following theorem of SLa : (mon& 1)

(𝜑 → 𝜓) → (𝛼 & 𝜑 → 𝛼 & 𝜓)

&-monotonicity.

4.9 Strongly (MP)–bDT-based axiomatic systems

233

Now we prove the mentioned stronger claim. First, we note that for 𝛽 = ★ ∧ 1¯ it is 1¯ ) (a known theorem of SL). Next, we prove it for 𝛽 = 𝜆 = 𝛼 → ★ & 𝛼 just (distr→ 𝛼 (recall that we denote by 𝜒 𝑚 the mirror form of 𝜒 and, if 𝜒 is a theorem of SLa , then so is 𝜒 𝑚 ; Corollary 4.6.17): a)

((𝜑 → 𝜓)

((𝜑 → 𝜓) & 𝛼

𝜓)

(mon& 1)

𝜓 & 𝛼)

𝑚

𝜓 & 𝛼)

(as ) 𝑚 , a, and (T) 𝑚

c)

(𝜑 → 𝜓) & 𝛼 → (𝜑 → 𝜓 & 𝛼)

b, (Symm), and (E )

d)

(𝛼 → (𝜑 → 𝜓) & 𝛼) → [𝛼 → (𝜑 → 𝜓 & 𝛼)]

((𝜑 → 𝜓) & 𝛼

b) 𝜑

c and (Pf)

e) 𝜆 𝛼 (𝜑 → 𝜓) → (𝜑 & 𝛼 → 𝜓 & 𝛼)

d, (res), and (T)

f) 𝜆 𝛼 (𝜑 → 𝜓) → [𝜆 𝛼 (𝜑) → 𝜆 𝛼 (𝜓)]

e, (pf), and (T)

Finally, we prove it for 𝛽 = 𝜌 𝛼 = 𝛼

𝛼 & ★: (mon& 1 ) and (E )

a) 𝛼 & 𝜑 → ((𝜑 → 𝜓)

𝛼 & 𝜓)

b) (𝛼

((𝜑 → 𝜓)

𝛼 & 𝜑) ((𝜑 → 𝜓)

[𝛼

a, (Pf ), and (Symm)

𝛼 & 𝜓)]

[𝛼 & (𝜑 → 𝜓)

(res) 𝑚

c)

[𝛼

d)

𝜌 𝛼 (𝜑)

e)

[𝛼 & (𝜑 → 𝜓)

f)

𝜌 𝛼 (𝜑)

𝜌 𝛼 (𝜓)]

d, e, and (T) 𝑚

g)

𝜌 𝛼 (𝜑 → 𝜓) → [𝜌 𝛼 (𝜑) → 𝜌 𝛼 (𝜓)]

e, (Symm), and (E ) 

𝛼 & 𝜓)]

[𝛼 & (𝜑 → 𝜓) 𝛼 & 𝜓]

[𝜌 𝛼 (𝜑 → 𝜓)

𝛼 & 𝜓]

b, c, and (T) 𝑚

𝛼 & 𝜓] [𝜌 𝛼 (𝜑 → 𝜓)

(pf) 𝑚

𝜌 𝛼 (𝜓)]

Remark 4.9.6 In the literature on substructural logics, the names left/right conjugate are usually used for more complex terms, namely 𝜆 0𝜀 = (𝜀 → ★ & 𝜀) ∧ 1¯ and ¯ However, it is easy to see that the logic SLa is strongly 𝜌 0𝜀 = (𝜀 𝜀 & ★) ∧ 1. 0 (MP)–bDTSL -based with the set of basic deduction terms (note that terms 𝜆10¯ , 𝜌10¯ , a and ★ & 1¯ are equivalent): 0 bDTSL = {𝜆 0𝛼 , 𝜌 0𝛼 | 𝛼 ∈ FmLSL }. a 0 The elements of (bDTSL ) ∗ are then known as iterated conjugates. We prefer our a approach because the elements of bDTSLa are indecomposable and so they better illustrate the substructural nature of the failure of (MP)-basedness: ★ & 1¯ takes care of the lack of weakening, while 𝜆 𝛼 and 𝜌 𝛼 take care of the lack of exchange.

Next, we deal with the much more complicated case of SL. Let us start by ¯ the role of basic introducing ★-formulas which will play (together with ★ ∧ 1) deduction terms in SL: 𝛼 𝛿, 𝜀 = 𝛿 & 𝜀 → 𝛿 & (𝜀 & ★)

𝛽 𝛿, 𝜀 = 𝛿 → (𝜀 → (𝜀 & 𝛿) & ★)

𝛼 0𝛿, 𝜀

𝛽 0𝛿, 𝜀 = 𝛿 → (𝜀

= 𝛿 & 𝜀 → (𝛿 & ★) & 𝜀

(𝛿 & 𝜀) & ★)

and exploring some of their elementary properties that we will need in order to prove the main result of this section.

234

4 On lattice and residuated connectives

Later (see Proposition 4.9.10), we show how these terms, and hence the axiomatic systems in which they appear, can be simplified in stronger substructural logics (e.g. in the presence of exchange the prime versions are equivalent to the non-prime ones and associativity allows us to replace them by left conjugates). Lemma 4.9.7 For every ★-formula 𝛾 ∈ {𝛼 𝛿, 𝜀 , 𝛼 0𝛿, 𝜀 , 𝛽 𝛿, 𝜀 , 𝛽 0𝛿, 𝜀 | 𝛿, 𝜀 ∈ FmLSL } and every pair of formulas 𝜑 and 𝜓, we have: 𝜑 `SL 𝛾(𝜑) 𝜑 → 𝜓 `SL 𝛾(𝜑) → 𝛾(𝜓). Proof All the cases are easily proved in a similar way. Let us show the case of 𝛼 𝛿, 𝜀 as an example and leave the remaining cases as an exercise for the reader. To prove the first claim, we get 𝜑 ` 𝜀 → 𝜀 & 𝜑 (due to (adj& ) and (MP)) and so the use of (Mon& ) completes the proof. The second one is a simple consequence of using (Mon& ) twice and (Pf) on the formula 𝜑 → 𝜓.  Theorem 4.9.8 The logic SL is strongly (MP)–bDTSL -based for the set bDTSL = {𝛼 𝛿, 𝜀 , 𝛼 0𝛿, 𝜀 , 𝛽 𝛿, 𝜀 , 𝛽 0𝛿, 𝜀 , ★ ∧ 1¯ | 𝛿, 𝜀 ∈ FmLSL }. Proof Let us denote by AS the axiomatic system depicted in Figure 4.8. We start by showing that AS is indeed a presentation of SL. To prove one direction of the claim, first notice that the validity of the rules (𝛼-nec), (𝛼 0-nec), (𝛽-nec) and (𝛽 0-nec) in SL was established already in Lemma 4.9.7 and all the other consecutions but (auxSL ) are known to be valid in SL (the axiom (adj&, ) follows from (adj& ) 𝑚 using (Symm)). Clearly, (auxSL 1 ) follows from (𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → 𝜒)) by using the rule (Res) twice and, analogously, we obtain (auxSL 2 ) from (𝜑 → (𝜓

𝜒)) → (𝜑 → (𝜓

𝜒))

(we use (E ) before the second use of (Res)). The proof of (auxSL 3 ) is more complex: a) 𝜒 & ( 𝜒 → 𝜑) → 𝜑 b) (𝜑 → 𝜓) → ( 𝜒 & ( 𝜒 → 𝜑) → 𝜓) c) ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓) → 𝜓 d) ( 𝜒 → ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓)) → ( 𝜒 → 𝜓)

(id) and (Res) a and (Sf) b and (Res) c and (Pf)

The proof of (auxSL 4 ) is similar: a) b) c) d)

(𝜒 𝜑) & 𝜒 → 𝜑 (𝜑 → 𝜓) → (( 𝜒 𝜑) & 𝜒 → 𝜓) (( 𝜒 𝜑) & 𝜒) & (𝜑 → 𝜓) → 𝜓 (𝜒 (( 𝜒 𝜑) & 𝜒) & (𝜑 → 𝜓)) → ( 𝜒

𝜓)

(id), (E ), and (Res) a and (Sf) b and (Res) c and (Pf )

4.9 Strongly (MP)–bDT-based axiomatic systems

235

(adj& )

𝜑 → (𝜓 → 𝜓 & 𝜑)

&-adjunction

(adj&, )

&,

(pop)

𝜑 → (𝜓 𝜑 & 𝜓) ¯ 𝜑 → ( 1 → 𝜑) ( 1¯ → 𝜑) → 𝜑

(⊥)

⊥→𝜑

ex falso quodlibet

(>)

𝜑→>

verum ex quolibet

(ub1 )

𝜑 → 𝜑∨𝜓

upper bound

(ub2 )

𝜓 → 𝜑∨𝜓

upper bound

(sup)

(𝜑 → 𝜒) ∧ (𝜓 → 𝜒) → (𝜑 ∨ 𝜓 → 𝜒)

supremality

(lb1 )

𝜑∧𝜓 → 𝜑

lower bound

(lb2 )

𝜑∧𝜓 →𝜓

lower bound

(inf)

( 𝜒 → 𝜑) ∧ ( 𝜒 → 𝜓) → ( 𝜒 → 𝜑 ∧ 𝜓) ¯ & (𝜓 ∧ 1) ¯ → (𝜑 & 𝜓) ∧ 1¯ (𝜑 ∧ 1)

infimality ¯ 1-&-distributivity

(push)

1¯ ) (distr& (auxSL 1 ) (auxSL 2 ) (auxSL 3 ) (auxSL 4 )

(MP)

-adjunction

push pop

𝜓 & (𝜑 & (𝜑 → (𝜓 → 𝜒))) → 𝜒 (𝜑 & (𝜑 → (𝜓

𝜒))) & 𝜓 → 𝜒

( 𝜒 → ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓)) → ( 𝜒 → 𝜓) (𝜒

(( 𝜒

𝜑) & 𝜒) & (𝜑 → 𝜓)) → ( 𝜒

𝜓)

(Adj1¯ )

𝜑, 𝜑 → 𝜓 I 𝜓 𝜑 I 𝜑 ∧ 1¯

modus ponens ¯ 1-adjunction

(𝛼-nec)

𝜑 I 𝛿 & 𝜀 → 𝛿 & (𝜀 & 𝜑)

𝛼-necessitation

(𝛼 0-nec)

𝜑 I 𝛿 & 𝜀 → (𝛿 & 𝜑) & 𝜀

𝛼 0-necessitation

(𝛽-nec)

𝜑 I 𝛿 → (𝜀 → (𝜀 & 𝛿) & 𝜑)

𝛽-necessitation

(𝛽 0-nec)

𝜑 I 𝛿 → (𝜀

𝛽 0-necessitation

(𝛿 & 𝜀) & 𝜑)

Fig. 4.8 Strong (MP)–bDT-based presentation of the logic SL.

Now we are ready to prove the converse direction. Recall that, in Remark 4.6.5, we have shown that SL can be axiomatized by (Pf), (Sf), (E ), (Res), (MP), (push), 1¯ ), and (Adj ). Note that all but the (pop), (⊥), (>), (lb), (ub), (inf), (sup), (distr& 1¯ first four are also elements of AS, so we only have to prove the remaining ones (we will also need to prove some auxiliary claims along the way). (T): 𝜑 → 𝜓, 𝜓 → 𝜒 I 𝜑 → 𝜒 is proved as: a) b) c)

( 𝜒 → 𝜑) → ( 𝜒 → ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓)) 𝜒 → ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓)) 𝜒→𝜓

𝜑 → 𝜓 and (𝛽-nec) 𝜒 → 𝜑, a, and (MP) b, (auxSL 3 ), and (MP)

236

4 On lattice and residuated connectives

(Pf): 𝜑 → 𝜓 I ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓) is proved as: ( 𝜒 → 𝜑) → ( 𝜒 → (( 𝜒 → 𝜑) & 𝜒) & (𝜑 → 𝜓)) ( 𝜒 → 𝜑) → ( 𝜒 → 𝜓)

a) b)

(Pf ): 𝜑 → 𝜓 I ( 𝜒 a) ( 𝜒 b) ( 𝜒

𝜑) → ( 𝜒

𝜑) → ( 𝜒 𝜑) → ( 𝜒

(( 𝜒 𝜓)

𝜑 → 𝜓 and (𝛽-nec) a, (auxSL 3 ), and (T)

𝜓) is proved as: 𝜑 → 𝜓 and (𝛽 0-nec) a, (auxSL 4 ), and (T)

𝜑) & 𝜒) & (𝜑 → 𝜓))

(Res1 ): 𝜑 → (𝜓 → 𝜒) I 𝜓 & 𝜑 → 𝜒 is proved as: a) 𝜓 & 𝜑 → 𝜓 & (𝜑 & (𝜑 → (𝜓 → 𝜒))) b) 𝜓 & 𝜑 → 𝜒 (Res

,1 ):

𝜑 → (𝜓

𝜑 → (𝜓 → 𝜒) and (𝛼-nec) a, (auxSL 1 ), and (T)

𝜒) I 𝜑 & 𝜓 → 𝜒 is proved as:

a) 𝜑 & 𝜓 → (𝜑 & (𝜑 → (𝜓 𝜒))) & 𝜓 b) 𝜑 → (𝜓 𝜒) ` 𝜑 & 𝜓 → 𝜒

a,

(𝛼 0-nec) and (T)

(auxSL 2 ),

(Res2 ): 𝜓 & 𝜑 → 𝜒 I 𝜑 → (𝜓 → 𝜒) is proved as: a) (𝜑 → (𝜓 → 𝜓 & 𝜑)) → (𝜑 → (𝜓 → 𝜒)) b) 𝜑 → (𝜓 → 𝜒) (Res

,2 ):

𝜓 & 𝜑 → 𝜒 I 𝜓 → (𝜑

a) (𝜓 → (𝜑 b) 𝜓 → (𝜑 (E

,1 ):

𝜒) is proved as:

𝜓 & 𝜑)) → (𝜓 → (𝜑 𝜒)

𝜓 → (𝜑 → 𝜒) I 𝜑 → (𝜓

𝜓 & 𝜑 → 𝜒 and (Pf) twice a, (adj& ), and (MP)

𝜒))

𝜓 & 𝜑 → 𝜒, (Pf ), and (Pf) a, (adj&, ), and (MP)

𝜒) is proved as:

a) 𝜑 & 𝜓 → 𝜒 b) 𝜑 → (𝜓 𝜒) (E

,1 ):

𝜑 → (𝜓

𝜓 → (𝜑 → 𝜒) and (Res1 ) a and (Res ,2 ) 𝜒) I 𝜓 → (𝜑 → 𝜒) is proved as:

a) 𝜑 & 𝜓 → 𝜒 b) 𝜓 → (𝜑 → 𝜒)

𝜑 → (𝜓

𝜒) and (Res ,1 ) a and (Res2 )

(Sf): 𝜑 → 𝜓 I (𝜓 → 𝜒) → (𝜑 → 𝜒) is proved as: a) b) c) d)

(𝜓 → 𝜒) → (𝜓 → 𝜓 → ((𝜓 → 𝜒) 𝜑 → ((𝜓 → 𝜒) (𝜓 → 𝜒) → (𝜑 →

𝜒) 𝜒) 𝜒) 𝜒)

(push), (pop), and (T) a and (E 1 ) 𝜑 → 𝜓, b and (T) c and (E 2 )

Before we get to the final part of the proof, we show that the following formulas are theorems of SL: (aux1)

𝛼 𝛿, 𝜑 (𝜑 → 𝜓) → (𝛿 & 𝜑 → 𝛿 & 𝜓)

(aux2)

𝛼 0𝜑, 𝜖 (𝜑 → 𝜓) → (𝜑 & 𝜖 → 𝜓 & 𝜖)

(aux3)

𝛽 𝜒→𝜑,𝜒 (𝜑 → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓)).

4.9 Strongly (MP)–bDT-based axiomatic systems

237

The first one is easy; recall that 𝛼 𝛿, 𝜑 (𝜑 → 𝜓) = (𝛿 & 𝜑 → 𝛿 & (𝜑 & (𝜑 → 𝜓))): a) 𝜑 & (𝜑 → 𝜓) → 𝜓 b) 𝛿 & (𝜑 & (𝜑 → 𝜓)) → 𝛿 & 𝜓 c) (𝛿 & 𝜑 → 𝛿 & (𝜑 & (𝜑 → 𝜓))) → (𝛿 & 𝜑 → 𝛿 & 𝜓)

(id) and (Res) a and (Mon& ) b and (Pf)

The proof of (aux2) is analogous and the proof of (aux3) is a bit more complex (recall that 𝛽 𝜒→𝜑,𝜒 (𝜑 → 𝜓) = ( 𝜒 → 𝜑) → ( 𝜒 → ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓))): a) b) c) d) e) f)

𝜒 & ( 𝜒 → 𝜑) → 𝜑 ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓) → 𝜑 & (𝜑 → 𝜓) 𝜑 & (𝜑 → 𝜓) → 𝜓 ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓) → 𝜓 ( 𝜒 → ( 𝜒 & ( 𝜒 → 𝜑)) & (𝜑 → 𝜓)) → ( 𝜒 → 𝜓) 𝛽 𝜒→𝜑,𝜒 (𝜑 → 𝜓) → (( 𝜒 → 𝜑) → ( 𝜒 → 𝜓))

(id) and (Res) a and (Mon& ) (id) and (Res) b, c, and (T) d and (Pf) e and (Pf)

To complete the proof of the fact that SL is strongly (MP)–bDTSL -based, we have to establish the condition (DT7). We prove a stronger claim: for each 𝛾 ∈ bDTSL there is a 𝛾 0 ∈ bDT∗SL such that ` 𝛾 0 (𝜑 → 𝜓) → (𝛾(𝜑) → 𝛾(𝜓)). ¯

1 ). Next, we prove the claim for ¯ we can set 𝛾 0 = 𝛾 thanks to (distr→ If 𝛾 is ★ ∧ 1,

𝛾 = 𝛼 0𝛿, 𝜀 = 𝛿 & 𝜀 → (𝛿 & ★) & 𝜀. The other cases are proved analogously and are left as an exercise for the reader. First, we use (aux1), (aux2), and (aux3) to obtain the following theorems of SL: 𝛼 𝛿, 𝜑 (𝜑 → 𝜓) → (𝛿 & 𝜑 → 𝛿 & 𝜓) 𝛼 0𝛿&𝜑, 𝜀 (𝛿 & 𝜑 → 𝛿 & 𝜓) → ((𝛿 & 𝜑) & 𝜀 → (𝛿 & 𝜓) & 𝜀) 𝛽 𝛿&𝜀→( 𝛿&𝜑)&𝜀, 𝛿&𝜀 ((𝛿 & 𝜑) & 𝜀 → (𝛿 & 𝜓) & 𝜀) → (𝛼 0𝛿, 𝜀 (𝜑) → 𝛼 0𝛿, 𝜀 (𝜓)). ¯ Then, by Lemma 4.9.7 applied on the first Let us denote 𝛽 𝛿&𝜀→( 𝛿&𝜑)&𝜀, 𝛿&𝜀 as 𝛽. ¯ and once on the second theorem (for 𝛽), ¯ we theorem twice (for 𝛼 0𝛿&𝜑, 𝜀 and for 𝛽) obtain the following theorems of SL: ¯ 0 ¯ 0 𝛽(𝛼 𝛿&𝜑, 𝜀 (𝛼 𝛿, 𝜑 (𝜑 → 𝜓))) → 𝛽(𝛼 𝛿&𝜑, 𝜀 (𝛿 & 𝜑 → 𝛿 & 𝜓)) ¯ 0 ¯ 𝛽(𝛼 (𝛿 & 𝜑 → 𝛿 & 𝜓)) → 𝛽((𝛿 & 𝜑) & 𝜀 → (𝛿 & 𝜓) & 𝜀). 𝛿&𝜑, 𝜀

Thus, using transitivity on the previous three theorems, we obtain 𝛽 𝛿&𝜀→( 𝛿&𝜑)&𝜀, 𝛿&𝜀 (𝛼 0𝛿&𝜑, 𝜀 (𝛼 𝛿, 𝜑 (𝜑 → 𝜓))) → (𝛼 0𝛿, 𝜀 (𝜑) → 𝛼 0𝛿, 𝜀 (𝜓)).



We can easily observe that the consecutions from Figure 4.8 involving only formulas of LLL form a strong (MP)–bDTSLLLL -based presentation of the LLL -fragment of SL and, therefore, we obtain the following proposition.

238

4 On lattice and residuated connectives

Proposition 4.9.9 The LLL -fragment of SL is strongly (MP)–bDTSLLLL -based for the set bDTSLLLL = {𝛼 𝛿, 𝜀 , 𝛼 0𝛿, 𝜀 , 𝛽 𝛿, 𝜀 , 𝛽 0𝛿, 𝜀 , | 𝛿, 𝜀 ∈ FmLLL }. Note that Theorem 4.9.8 tells us that all axiomatic extensions of SL are strongly (MP)–bDTSL -based and thus, due to Theorem 4.9.2, they enjoy the Π(bDT∗SL )implicational deduction theorem. Of course, for particular logics we have already seen (much) simpler sets of basic deduction terms and tighter implicational deduction theorems. Some of these results (especially those simplifying the set of basic deduction terms) can be obtained by observing that, in the presence of certain logical rules, some of the basic deduction terms become equivalent and certain 𝛿-necessitation rules redundant. The simplest case is ★ ∧ 1¯ which can be omitted in all Rasiowa-implicative logics (e.g. in expansions of SLi ). The next easy proposition, left as an exercise for the reader, shows how it works with the more complex basic deduction terms of SL 0 0 (note that we also have `SL 𝛾1, ¯ 1¯ (𝜑) ↔ 𝜑 for each 𝛾 ∈ {𝛼, 𝛼 , 𝛽, 𝛽 }). Proposition 4.9.10 The following formulas are theorems of SLE for each 𝛿 and 𝜀: 𝛼 𝛿, 𝜀 (𝜑) ↔ 𝛼 0𝜀, 𝛿 (𝜑)

𝛽 𝛿, 𝜀 (𝜑) ↔ 𝛽 0𝛿, 𝜀 (𝜑).

The following formulas are theorems of SLa for each 𝛿 and 𝜀: 0 (𝜑) 𝜆 𝜀 (𝜑) ↔ 𝛼1, ¯ 𝜀

0 (𝜑) 𝜌 𝜀 (𝜑) ↔ 𝛽1, ¯ 𝜀

𝜑 → 𝛼 𝛿, 𝜀 (𝜑)

𝜑 → 𝛽 𝛿, 𝜀 (𝜑)

𝜆 𝜀 (𝜑) →

𝛼 0𝛿, 𝜀 (𝜑)

𝜌 𝜀 (𝜑) → 𝛽 0𝛿, 𝜀 (𝜑).

The following formulas are theorems of SLaE for each 𝜀: 𝜑 → 𝜆 𝜀 (𝜑)

𝜑 → 𝜌 𝜀 (𝜑).

¯ Using this proposition (and the already mentioned fact that 1-adjunction is redundant in extensions of SLi ), we can identify, for certain prominent substructural logics L, sets bDTL ⊆ bDTSL such that the logic L is strongly (MP)–bDTL -based. The results are summarized in Table 4.4; note that in certain cases we have obtained alternative proofs to already known facts. Remark 4.9.11 We can also use the second claim of the previous proposition to give an alternative proof of Theorem 4.9.5, i.e. to show that SLa is strongly (MP)–bDTSLa based. It is easy to see that taking as axioms all the theorems of SLa , modus ponens and all product normality rules yields an (MP)–bDTSLa -based presentation of SLa , what remains is to prove the condition SDT for which all we need to show is that for each 𝛾 ∈ bDT∗SL , there is a 𝛾 0 ∈ bDT∗SLa such that `SLa 𝛾 0 (𝜑) → 𝛾(𝜑), which easily follows from the second claim of the previous proposition and Lemma 4.9.7.

4.10 Proof by cases property for generalized disjunctions

239

Table 4.4 Strongly (MP)–bDT-based logics.

Logic L SL SLi SLE

is strongly (MP)–bDTL -based for bDTL being {𝛼 𝛿, 𝜀 , 𝛼 0 , 𝛽 𝛿, 𝜀 , 𝛽 0 , ★ ∧ 1¯ | 𝛿, 𝜀 ∈ FmLSL } 𝛿, 𝜀

𝛿, 𝜀

{𝛼 𝛿, 𝜀 , 𝛼 0𝛿, 𝜀 , 𝛽 𝛿, 𝜀 , 𝛽 0𝛿, 𝜀 | 𝛿, 𝜀 ∈ FmLSL } {𝛼 𝛿, 𝜀 , 𝛽 𝛿, 𝜀 , ★ ∧ 1¯ | 𝛿, 𝜀 ∈ FmLSL }

SLa

{𝛼 𝛿, 𝜀 , 𝛽 𝛿, 𝜀 | 𝛿, 𝜀 ∈ FmLSL } {𝜆 𝜀 , 𝜌 𝜀 , ★ ∧ 1¯ | 𝜀 ∈ FmLSL }

SLaE

¯ {★ ∧ 1}

SLaEi

∅

SLEi

4.10 Proof by cases property for generalized disjunctions Recall that in Proposition 2.4.5 we have used the deduction theorem of the logic BCKlat (which we know that, over FLew , can be seen as the ΔFLew -implicational deduction theorem) to prove a variant of the classical proof by cases property for any axiomatic expansion L of BCKlat , i.e.: for any set of formulas Γ ∪ {𝜑, 𝜓} ⊆ FmL , ThL (Γ, 𝜑 ∨ 𝜓) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓). Note that this can be equivalently expressed by saying that in any axiomatic expansion L of BCKlat the connective ∨ is a lattice disjunction. The goal of this section is to use the Δ-implicational deduction theorem to prove an analog of the proof by cases property, not for the lattice protodisjunction ∨, but for a more complex generalized notion of disjunction. Let us start by an easy-to-prove proposition. Proposition 4.10.1 (Proof by cases property) Let L be a substructural logic with lattice protodisjunction enjoying the Δ-implicational deduction theorem. Then, for each Γ ∪ {𝜑, 𝜓} ⊆ FmL , ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) = ThL (Γ, {𝛾(𝜑) ∨ 𝛿(𝜓) | 𝛾, 𝛿 ∈ Δ}). Proof The right-to-left inclusion is easy: recall that all formulas in Δ are deduction terms of the logic and, thanks to this and the axiom (ub), we obtain for each 𝛾, 𝛿 ∈ Δ that: 𝜑 `L 𝛾(𝜑) ∨𝛿(𝜓) which entails ThL (Γ, 𝜑) ⊇ ThL (Γ, {𝛾(𝜑) ∨𝛿(𝜓) | 𝛾, 𝛿 ∈ Δ0 }). The analogous claim holds for 𝜓 instead of 𝜑 and so the proof of one inclusion follows. The proof of the converse direction is also easy. Indeed, from the assumption that 𝜒 ∈ ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓), we obtain, by the Δ-implicational deduction theorem, two formulas 𝛾, 𝛿 ∈ Δ such that 𝛾(𝜑) → 𝜒, 𝛿(𝜓) → 𝜒 ∈ ThL (Γ) and so (by the rule (Sup)) also 𝛾(𝜑) ∨ 𝛿(𝜓) → 𝜒 ∈ ThL (Γ) and the proof is done. 

240

4 On lattice and residuated connectives

Nevertheless, the generalized proof by cases property sometimes can be proved also for a smaller set Δ0 . Consider e.g. the logic FLew , which does not enjoy the classical deduction theorem but only the ΔFLew -implicational deduction theorem with ΔFLew = {★𝑘 | 𝑘 ≥ 1}. As we will see at the end of this section, for Δ0 = {★}, ThFLew (Γ, 𝜑) ∩ ThFLew (Γ, 𝜓) = ThFLew (Γ, {𝛾(𝜑) ∨ 𝛿(𝜓) | 𝛾, 𝛿 ∈ Δ0 }). Recall that, by Theorem 4.8.13, we know that the Δ-implicational deduction theorem implies that the logic in question is bDT–(MP)-based for any bDT such that bDT ⊆ Δ ⊆ Π(bDT∗ ). Therefore, one might be tempted to assume that we could always take Δ0 = bDT. However, as the next example shows, the situation is more complex. Example 4.10.2 Let L be any of the modal logics K, K4, or T. The following equation fails: ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) = ThL (Γ, 𝜑 ∨ 𝜓). In the case of K, using the equation for 𝜑 = 𝜓 and Γ = ∅, we would obtain 𝜑 `L 𝜑, which can be easily refuted in any, at least four-valued, modal algebra h𝐴, ∧, ∨, ¬, ⊥, >, i, where 𝑎 = > for each 𝑎 ∈ 𝐴 (note that such kind of algebra would be in Alg∗ (K4), so we know that this form of proof by cases property fails even in K4). In the case of T, observe that the equation would entail that 𝜑 ∨ 𝜓 `T 𝜑 ∨ 𝜓. Considering the modal algebra A from Example 2.6.8 and the evaluation 𝑒 (𝜑) = {𝑎, 𝑏} and 𝑒(𝜓) = {𝑏, 𝑐}, we obtain a contradiction. The next theorem shows us that, at least in Rasiowa-implicative logics, we can improve Proposition 4.10.1 by setting Δ0 = bDT∗ (again, we prefer a more general formulation taking care of possible simplifications of the set of deduction terms needed for the deduction theorem). We also prove a variant for logics that need not be Rasiowa-implicative but have a protounit and a lattice (proto)conjunction. Theorem 4.10.3 (Proof by cases property) Let L be a substructural logic expanding SL&,1,∨ ¯ and enjoying the Δ-implicational deduction theorem and let Δ0 be a set of ★-formulas such that Δ0 ⊆ Δ ⊆ Π(Δ0 ). • If L is Rasiowa-implicative, then, for each Γ ∪ {𝜑, 𝜓} ⊆ FmL , we have ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) = ThL (Γ, {𝛾(𝜑) ∨ 𝛿(𝜓) | 𝛾, 𝛿 ∈ Δ0 }). • If L has a lattice (proto)conjunction ∧, then, for each Γ ∪ {𝜑, 𝜓} ⊆ FmL , we have ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) = ThL (Γ, {𝛾(𝜑) ∨ 𝛿(𝜓) | 𝛾, 𝛿 ∈ {𝜂 ∧ 1¯ | 𝜂 ∈ Δ0 }}). Proof The right-to-left inclusions in both claims are easy; in the first case it actually follows directly from Proposition 4.10.1 and to obtain the second one we only have to use the rule (Adj) 1¯ .

4.10 Proof by cases property for generalized disjunctions

241

We prove the converse inclusion for both claims at once by using the following ¯ while in the auxiliary notation: in the second case we denote by 𝜑 0 the formula 𝜑 ∧ 1, 0 first case we simply take 𝜑 = 𝜑. We start by proving that in both cases we have ` (𝜑10 ∨ 𝜓 0) & (𝜑20 ∨ 𝜓 0) → (𝜑1 & 𝜑2 ) 0 ∨ 𝜓 0 . We give a semantical proof. Consider A ∈ Mod∗ (L) and an evaluation 𝑒, set 𝑥𝑖 = 𝑒(𝜑𝑖0 ) and 𝑦 = 𝑒(𝜓 0), and observe that we have 𝑥1 & 𝑦 ≤ 𝑦 𝑦 & 𝑥2 ≤ 𝑦 𝑦&𝑦 ≤ 𝑦 Indeed, we know that 𝑥1 ≤ 1¯ and, therefore, 𝑥1 & 𝑦 ≤ 1¯ & 𝑦 = 𝑦; the other cases are analogous. Thanks to this observation, Proposition 4.2.9, and elementary properties of ∨ we know that (𝑥1 ∨ 𝑦) & (𝑥2 ∨ 𝑦) = (𝑥1 & 𝑥 2 ) ∨ (𝑥 1 & 𝑦) ∨ (𝑦 & 𝑥2 ) ∨ (𝑦 & 𝑦) ≤ (𝑥1 & 𝑥2 ) ∨ 𝑦. This fact already completes the proof for Rasiowa-implicative logics (where 𝜑10 & 𝜑20 = 1¯ ) (which can be written (𝜑1 & 𝜑2 ) 0); in the other case we also use the theorem (distr& 0 0 0 as 𝜑1 & 𝜑2 → (𝜑1 & 𝜑2 ) ) and properties of ∨. Now we are ready to prove the reverse inclusion of the both claims. Note that, thanks to Proposition 4.10.1, it suffices to show that, for each 𝛾, 𝛿 ∈ Δ, {𝛼 0 (𝜑) ∨ 𝛽 0 (𝜓) | 𝛼, 𝛽 ∈ Δ0 } `L 𝛾 0 (𝜑) ∨ 𝛿 0 (𝜓) and observing that `L 𝛾 0 (𝜑) ∨ 𝛿 0 (𝜓) → 𝛾(𝜑) ∨ 𝛿(𝜓) (in Rasiowa-implicative logics this is a trivial claim; in other logics it immediately follows from (lb), (ub), and the rule (Sup)). Clearly, the claim holds if 𝛾 = 1¯ and 𝛿 = 1¯ (as ten we have `L 1¯ → 𝛾 0 (𝜑) ∨ 𝛿 0 (𝜓)). ¯ can be expressed From the assumption, we know that each element 𝜒 ∈ Δ \ {1} as a conjunction of elements of Δ0 ; let us denote by 𝑛 𝜒 the minimal number of elements of Δ0 to do so (i.e. 𝑛 𝜒 = 1 iff 𝜒 ∈ Δ0 ). We prove the claim by induction on 𝑛 𝛾 + 𝑛 𝛿 . The base case 𝑛 = 2 is trivial. For the induction step assume, without loss of generality, that 𝛾 = 𝛾1 & 𝛾2 for some 𝛾𝑖 ∈ Π(Δ). Then, from the induction assumption, we obtain the provability of 𝛾10 (𝜑) ∨ 𝛿 0 (𝜓) and 𝛾20 (𝜑) ∨ 𝛿 0 (𝜓) for the set {𝛼 0 (𝜑) ∨ 𝛽 0 (𝜓) | 𝛼, 𝛽 ∈ Δ0 }. Using the rule (Adj& ) and the claim we established earlier, the proof is completed.  Note that the ∧-free fragment of SLaE in not covered by any of the claims in the theorem and, actually, we do not know whether this logic is bDT–(MP)-based. Let us conclude this section with an example that collects variants of the proof by cases property for prominent logics.

242

4 On lattice and residuated connectives

Example 4.10.4 The following equality ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) = ThL (Γ, {𝛾(𝜑) ∨ 𝛿(𝜓) | 𝛾, 𝛿 ∈ Δ0 }) holds in axiomatic expansions of IL→ FLew SLaE SLai SLa SLi SL S4 K4 T K

for for for for for for for for for for for

Δ0 Δ0 Δ0 Δ0 Δ0 Δ0 Δ0 Δ0 Δ0 Δ0 Δ0

= = = = = = = = = = =

{★} {★} ¯ {★ ∧ 1} {𝜆 𝜀 , 𝜌 𝜀 | 𝜀 ∈ FmLSL }∗ {𝜆 𝜀 , 𝜌 𝜀 , ★ ∧ 1¯ | 𝜀 ∈ FmLSL }∗ {𝛼 𝛿, 𝜀 , 𝛼 0𝛿, 𝜀 , 𝛽 𝛿, 𝜀 , 𝛽 0𝛿, 𝜀 | 𝛿, 𝜀 ∈ FmLSL }∗ {𝛼 𝛿, 𝜀 , 𝛼 0 , 𝛽 𝛿, 𝜀 , 𝛽 0 , ★ ∧ 1¯ | 𝛿, 𝜀 ∈ FmLSL }∗ 𝛿, 𝜀

𝛿, 𝜀

{(★)} {★, (★)} and {★ ∧ (★)} { 𝑘 (★) | 𝑘 ≥ 1} {★} ∪ { 𝑘 (★) | 𝑘 ≥ 1}.

All these claims follow directly from an appropriate application of the previous theorem and a known form of implicational deduction theorem for the logic in question; only in the case of SLaE the last claim of the previous theorem would give ¯ ∧ 1}, ¯ but (★ ∧ 1) ¯ ∧ 1¯ is obviously equivalent to ★ ∧ 1. ¯ us the result for {(★ ∧ 1)

4.11 History and further reading The additional connectives that we have studied in this chapter are very common in (non-classical) propositional logics and a bigger part of the research of these logics is dedicated to the study of these connectives and their interplay with implication and with each other. Before we go into details on particular connectives and related logics, let us draw the reader’s attention to Humberstone’s monumental monograph (and references therein) called The Connectives [192] which, through its 1500 A4 pages, leaves very few stones unturned. Regarding lattice connectives and truth-constants, we have already seen that they are present in classical logic and in most examples of non-classical logics introduced in the Chapter 2 (we refer to Section 2.10 for historical and bibliographical remarks). The general treatment of these connectives in the context of weakly implicative logics presented here is based on our paper [92]. In the next chapter, we will develop a more abstract approach to disjunction connectives (a similar abstract study of conjunction is conducted in [196]).

4.11 History and further reading

243

Residuated connectives (often together with lattice ones) are typically found in substructural logics, a very diverse family of logics, extensively studied in the literature, that stem from different origins and had to wait a long time to receive this all-encompassing label and several definitions attempting at a formal delimitation (see below). Roughly speaking, there have been four main streams of research leading towards the general study of substructural logics: • Relevant logics: Although the first example of relevant logic was introduced by Ivan E. Orlov already in 1928 [112, 255], the study of this family really flourished from the 1960s onwards when these logics were explored as forms of entailment that avoid the paradoxes of classical material implication (see e.g. [5]). • Lambek logic and its variants: Introduced in 1958 by Joachim Lambek [210,211], they were intended to deal with grammatical categories in formal and natural languages (see e.g. [233]).31 • Logics without the contraction rule: With their influential paper published in 1985 [254], Hiroakira Ono and Yuichi Komori started a purely mathematical line of research on non-classical logics without the contraction rule (and, possibly, without exchange and weakening) encompassing, among others, relevant logics, some many-valued logics, BCK, and related systems, and considering their proof theory and their algebraic and relational semantics (see e.g. [143]). • Linear logic and its variants: These systems were introduced first in 1987 by Jean Yves Girard [152] to model processes in computer science that need to keep track of resource usage (see e.g. [298]). It was not until 1990 that all these diverse families of logics were unified under the umbrella term of substructural logics by Kosta Došen (during a conference in Tübingen, see [283]). The rationale for the label was the fact that prototypical examples of these logics can be presented by taking Gentzen-style sequent calculi presentations of classical or intuitionistic logic and removing some of the rules that do not refer to any particular connective, that is, the so-called structural rules: associativity, exchange, weakening, and contraction. Besides this proof-theoretic perspective, these logics have been given uniform semantical treatments by means of algebraic and relational semantics which have not only opened the door to a plethora of other examples (largely transcending the Gentzen-style description) but also have fully justified seeing these logics as a family interesting on its own and worthy of a general/abstract study. Nowadays there are two main streams of research focusing on the in-depth study of the whole family of substructural logics (and numerous streams focusing on its notable subfamilies, such as linear logic and its variants, superintuitionistic logics, fuzzy logics,32 etc.). 31 The accounts of Lambek logic most common in the literature disregard the truth-constants 0¯ and 1¯ and define it as the truth-constant-free fragment of our Lambek logic (cf. Theorem 4.5.5). 32 Fuzzy logics have been identified as particular substructural logics in [124]. Chapter 6 presents an abstract study of fuzzy logics under the formal name of semilinear logics, defined as logics enjoying strong completeness w.r.t. matrices with linear matrix order. See Section 6.5 for the rationale behind this definition and this terminology and for more details and references. Finally, let us point out that, despite its name, linear logic (and its most common variants) is not a semilinear logic.

244

4 On lattice and residuated connectives

Although they do overlap to some extent and share some mathematical methods, they differ in motivations, preferred methodology, terminological and naming conventions (esp. regarding logics), and goals. In order to avoid applying misleading and potentially controversial labels, we refer to them by neutral ad hoc labels: • Stream 𝔄 stems from the work of Hiroakira Ono and Yuichi Komori [254] and focuses on an abstract mathematical study of substructural logics seen as logics of (classes of) residuated lattices.33 The motivation is mainly mathematical and hence this research utilizes a full range of mathematical methods: not only (universal) algebra, but also proof theory, relational semantics, topology, duality theory, etc. The interested reader may enter this research area through the monographs [143, 256]. • Stream ℜ stems from the study of relevant logics and retains its philosophical motivations rooted in the paradoxes of classical material implication. In this tradition, substructural logics are primarily given syntactically via Fitch, Hilbert, Gentzen, or other proof systems. Besides other mathematical methods, this research stream heavily utilizes relational semantics based on generalizations of the Kripke frames semantics used for modal and (super)intuitionistic logics (see e.g. [62]) and the correspondence between axioms/rules for (co-)implication and (dual) combinators studied by combinatory logic (see e.g. [281]). Representative recent references are the monographs [35, 36, 115, 273]. The exposition in this chapter does not follow closely either of these streams. As we were interested in casting substructural logics in our general implicationbased approach, our presentation can be seen as a blend of both streams: the primary objects for us are logics described using their axiomatic systems, but we have also been interested in their mathematical properties and have studied them using mainly algebraic methods. Therefore, some remarks on our particular design choices and their relation to the developments in the mentioned main streams are in order. Regarding the definition of substructural logics, it is clear that Došen’s characterization is an informal one and hence has invited several mathematical precisifications. We have defined them as the expansions of LL→ in which → is a weak implication. Our definition encompasses other major definitions proposed by both streams:34 • Stream 𝔄: Their most prevailing formal definition (see e.g. [143]) identifies substructural logics with the axiomatic extensions of the Full Lambek logic FL, which is the {>, ⊥}-free fragment of SLa .35 Let us, however, stress that they 33 Residuated lattices were already introduced in 1939 by Morgan Ward and Robert P. Dilworth [305] as a tool for the study of ideal theory in rings. In our terminology, their algebras coincide with SLaio -algebras. However, the most prevailing definition in Stream 𝔄 identifies them with {⊥, >}-free subreducts of SLa -algebras (see [143]). It is also quite common, especially in the literature on mathematical fuzzy logic [160], to identify them with FLew -algebras. 34 Let us also mention our previous proposal from [88], where we defined substructural logics in a language L as expansions of the L ∩ LSL -fragment of SL, where → is a weak implication. 35 There is also an informal proposal, in the conclusion of the paper [252] by Ono, to identify substructural logics with fragments with implication of logics expanding FL.

4.11 History and further reading

245

study also logics outside of their formal definition (but within ours), such as the {>, ⊥}-free fragment of SL in [145] or other fragments of these logics, e.g. (the associative variant of) Lambek logic, or purely implicative logics such as BCI or BCK in [44, 133, 254]. • Stream ℜ: Their most prevailing recent formal definition from [273] identifies substructural logics with expansions of the -free fragment of the variant of LL (modulo small syntactical transformations; see below) given by combination of certain logical axioms/rules concerning also additional connectives such as negation, co-implication, modalities, etc. Of course, there are logics studied in this stream that fall outside the scope of this definition; e.g. the historically most prominent relevant logics are cast in a language with lattice connectives (and often ¯ 1, ¯ or &; some of them are even outside of our formal also negation) but without 0, definition (e.g. the logic F+ studied in [277]; see below for details). Another difference between both streams and our approach is the notation of logical connectives. Recall that the central notion tying together both implications and the residuated conjunction is that of residuation property which is algebraically (e.g. in the definition of residuated groupoid) formulated as 𝑥≤𝑦⇒𝑧

iff

𝑦&𝑥 ≤ 𝑧

iff

𝑦≤𝑥

𝑧.

Our notation for the (dual) co-implication is due to Hájek [163]. In stream 𝔄, however, the residuation property is formulated for two connectives \ and /, instead of our ⇒ and , in a way that allows to see it as the description of the result of dividing the inequality 𝑦 & 𝑥 ≤ 𝑧 from the left and from the right: 𝑥 ≤ 𝑦\𝑧

iff

𝑦&𝑥 ≤ 𝑧

iff

𝑦 ≤ 𝑧 / 𝑥.

Note that, to facilitate this intuition, one has to change the order of the arguments in .36 This intuition is the source of the terminology left residuum for ⇒ and right residuum for (though, strictly speaking it should be the operation /). To simplify notation, it is common, in logics validating the exchange rule (E), to use the symbol → for both \ and / (after changing the order of arguments of course). Furthermore, it is also quite common to drop the symbol & when writing the conjunction of two formulas (following the usual practice in group theory). Recall that in Remark 4.2.6 we have explained that the decision between considering residuated conjunctions or their dual variants given by the following transposed version of the residuation rules (Res), 𝜑 → (𝜓 → 𝜒) a` 𝜑 & 𝜓 → 𝜒, is purely conventional. Our decision follows the common practice in stream 𝔄. 36 One also has to assume the following facts which, in turn, are known consequences of the residuation property: (1) monotonicity of & in both arguments; (2) monotonicity of \ in the second one and / in the first one; and (3) the inequalities: 𝑥 ≤ 𝑦 \ ( 𝑦 & 𝑥), 𝑦 ≤ ( 𝑦 & 𝑥) / 𝑥, 𝑦 & ( 𝑦 \ 𝑧) ≤ 𝑧, and (𝑧 / 𝑥) & 𝑥 ≤ 𝑧.

246

4 On lattice and residuated connectives

In stream ℜ, however, the dual residuated conjunction is preferred, i.e. the basic substructural logic from [273] is axiomatized as the -free fragment of LL with the usual residuation rules replaced by their dual variant. This logic is clearly (termwise) equivalent to our presentation of the -free fragment of LL: all we have to do is to translate & as &𝑡 . Furthermore, in stream ℜ it is customary to use the symbol ← instead of our with reversed order of arguments. Note that, as explained in Remark 4.2.6, in strong enough logics where is the dual co-implication (i.e. in logics validating the rules (Symm)) we can, instead of transposing the residuated conjunction, switch the implications. All these considerations lead us to the following simple translation table between our approach and both streams (we also include the usual notation for the residuated truth-constants): our notation 𝜑&𝜓 𝜑→𝜓 𝜑 𝜓 1¯ 0¯

stream 𝔄 𝜑&𝜓 𝜑\𝜓 𝜓/𝜑 1 0

stream ℜ 𝜓&𝜑 𝜑→𝜓 𝜓←𝜑 𝑡 𝑓

stream ℜ in strong enough logics 𝜑&𝜓 𝜓←𝜑 𝜑→𝜓 𝑡 𝑓

Let us conclude our comparison with streams 𝔄 and ℜ by discussing the naming conventions for prominent logical systems. Let us start by mentioning that our logic SL is the conservative expansion of the logic studied in [145] by > and ⊥. However, stream 𝔄 usually takes as the basic logic its associative version, known as full Lambek logic FL, and studies its extensions by structural rules/axioms (analogously to our definition of SL𝑋 for 𝑋 ⊆ {a1 , a2 , E, C, i, o}). Namely, for each 𝑌 ⊆ {e, c, i, o} (note that here we consider the axiomatic versions e and c of the rules E and C), • the extension FL𝑌 of FL by the axioms corresponding to the elements of 𝑌 . • the expansion FL𝑌+ of FL𝑌 by constant ⊥ and the axiom ⊥ → 𝜑. Note that > is definable in FL+ as ⊥ → ⊥ and, more importantly, ⊥ is definable in FLo as simply 0¯ and so we can identify FLo and FL+o . Therefore, for each 𝑋 ⊆ {E, C, i} and 𝑋 0 which results from 𝑋 by replacing the capital letters by the corresponding lower case ones, we have the following mapping between our terminology and that of [143]: • SLa𝑋 = FL+𝑋 0 • SLa𝑋 o = FL𝑋 0 o = FL+𝑋 0 o • FL𝑋 0 is the {⊥, >}-free fragment of SLa𝑋 . Furthermore, there is one additional convention used in the literature on substructural logics: if both i ∈ 𝑋 and o ∈ 𝑋, then the letter w is used instead, so we have the following identifications: FLw = FLio = SLaio FLew = FLeio = SLaEio FLcw = FLcio = SLCio = SLaECio = IL.

4.11 History and further reading

247

Fig. 4.9 Prominent extensions of the logics SL and SLo .

Based on all the mentioned relations, we draw the hierarchy of notable extensions of SL and SLo in Figure 4.9 using the stream 𝔄 names for certain logics (cf. the related Figure 4.3). The naming conventions for logical systems in stream ℜ are a bit more convoluted, mainly due to different language preferences. Historically, relevant logics were cast in a language consisting of implication →, lattice connectives ∧ and ∨, and a negation ¬ taken as a primitive connective (it cannot be defined in the usual way, as the ¯ The negation is usually language does not contain the necessary truth-constant 0). required to obey the axioms of contraposition 𝜑 → 𝜓 I ¬𝜓 → ¬𝜑 and double negation elimination I ¬¬𝜑 → 𝜑 (but it hardly ever obeys the law of excluded middle I 𝜑 ∨ ¬𝜑 and never the law of explosion I 𝜑 ∧ ¬𝜑 → 𝜓) and the lattice connectives to obey the law of distributivity I (𝜑 ∨ 𝜓) ∧ 𝜒 ↔ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒). The difference between the major logical systems was then (as in our case) described mainly using rules and axioms describing various properties of implication. The following four logics are prominent examples of such systems: B TW T R

basic affixing system [276] ticket entailment logic without contraction [5] ticket entailment logic [5] relevant entailment logic [5]

Using the notation L+ to denote the ¬-free fragment (commonly known as positive fragment) and L→ for the implication fragment, we can write the following translations between our respective notations: B→ = TW→ = T→ = R→ =

LL→ B→ + (sf) + (pf) TW→ + (c) T→ + (e) = (SLaEC )→

B+ = TW+ = T+ = R+ =

LLlat ∧,∨ B+ TW+ TW+

+ Distr + (sf) + (pf) + (c) + (e) = (SLaEC )∧,∨ + Distr.

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4 On lattice and residuated connectives

It is worth noting, that in [277], the authors studied the logic F+ which results from the usual axiomatization of B+ (see Proposition 4.5.11) by replacing the rules (Pf) and (Sf) by the congruence axiom for →. Clearly, this logic is still weakly implicative (as the congruence axioms for ∧ and ∨ can be proved anyway), but it is no longer substructural according to our definition. Finally, let us now compare our approach and naming conventions with that of linear logic, one of the most prominent research streams in substructural logic focusing on a particular family of logics. Usually, the language is that of SLE (i.e. with one single implication) plus two unary connectives called exponentials. However, the symbols used for connectives are quite different from ours; to get the identification results mentioned below, we need the following transformation: our notation 𝜑→𝜓 𝜑&𝜓 ¬𝜑 𝜑∧𝜓 𝜑∨𝜓 1¯ > ⊥

linear logic 𝜑(𝜓 𝜑⊗𝜓 𝜑⊥ 𝜑&𝜓 𝜑⊕𝜓 1 > 0

Modulo this translation, the fragment of linear logics without the exponentials (known as multiplicative–additive fragment and denoted as MALL) is the extension of our logic SLaE by the double negation elimination axiom. This literature also studies the affine and intuitionistic variants of linear logic (denoted by using prefixes A and I); using our notation these logics can be describes as MALL = MAALL = MAIALL = MAILL =

(SLaE ) + double negation elimination (SLaEio ) + double negation elimination (SLaEio ) (SLaE ). 

In Sections 4.4, 4.5, 4.6, and 4.7, the proofs of results on axiomatization of fragments, completeness w.r.t. completely ordered models, and strong separability make a heavy use of the notion of completion, that is, embeddability of algebras into complete lattices (we refer the reader to the survey paper [175]). In some cases (namely, Theorem 4.4.15 and Corollaries 4.4.16 and 4.6.18), the involved completions are variants of Dedekind–MacNeille completions (see Theorem A.1.13). They originate in a construction proposed by Holbrook Mann MacNeille in [219] which allows to embed any partially ordered set into a minimal complete lattice and generalizes Richard Dedekind’s construction of the reals from the rational numbers. Such completions are regular, i.e. preserve all existing suprema

4.12 Exercises

249

and infima. The completion results mentioned above for various classes of associative residuated lattices can be found in [143] and for the non-associative lattices in [145]. Finally, let us mention the paper [69] where the authors propose a syntactical hierarchy of formulas and prove that classes of residuated lattices axiomatized by simple enough identities are closed under Dedekind–MacNeille completions. On the other hand, the proofs of Theorems 4.5.5 and 4.5.7 involve another completion construction based on the ternary relation introduced in Lemma 4.5.3. Ternary relations like this one appear often in the literature to provide generalizations of the Kripke-style frame semantics of intuitionistic and modal logics giving rise to a rich duality theory between algebraic and relational semantics of big families of substructural logics; the interested reader can find an introduction to this area in the monograph [36]. Starting from an algebraic (or matricial) model of a given logic, generating its corresponding frame and, from that, the corresponding algebra (known as complex algebra of the frame), one typically obtains a completion of the original model with some good properties known as canonical completion. Moreover, this construction gives rise to a duality with certain topological spaces that generalizes those given by Stone in the case of Boolean algebras [288] and by Tarski and Jónsson for modal algebras [201, 202].37 Finally, in Theorem 4.7.1 and Corollary 4.7.4 we have used yet another completion construction which is essentially the same as the one used in [299]; see also the similar constructions in [44, 81, 133, 254, 268].  The general study of deduction theorems and proof by cases in substructural logics presented in the last three sections is based on our previous works [88, Section 2.5] and [84]. Deduction theorems for FL and its main axiomatic extensions were already known (see e.g. [143, 144]); the novelties of our approach are the use of the various notions of (MP)-based logic and the link with the proof by cases property. The obtained local deduction theorems are particular forms of abstract deduction theorems extensively studied by abstract algebraic logic (see e.g. [42, 99, 195]). Similarly, the obtained forms of the proof by cases are particular cases of the abstract notion of generalized disjunction that we will study in the next chapter (for bibliographical references see Section 5.7).

4.12 Exercises Exercise 4.1 In Example 4.1.3, prove that A is a weakly implicative logic with a weak implication ⇒ and ∧ is a lattice protoconjunction but not a lattice conjunction. Hint: use Proposition 2.8.11. 37 A similar method was used in [145] to prove a stronger version of our Theorem 4.5.7. Indeed, the paper claims that (modulo the inconsequential addition of truth-constants ⊥ and >) the axiomatic system A S SL from Figure 4.2 is a separable presentation of SL for languages with implication involving ∧ whenever they involve ∨. Disregarding the weaker assumption on disjunction, we do now see how their proof could avoid the need for the presence of 1¯ or as explained in Remark 4.5.8.

250

4 On lattice and residuated connectives

Exercise 4.2 Prove the implications ‘1 implies 2’, ‘2 implies 3’, and ‘3 implies 4’ of Proposition 4.1.5. Exercise 4.3 Prove Proposition 4.1.6. Exercise 4.4 In Example 4.1.8, show that A is an expansion of BCK. Exercise 4.5 Show that the residuated connectives (introduced in Definition 4.2.1) are intrinsic by showing an analog of Proposition 4.1.6). Exercise 4.6 Prove Proposition 4.2.3. Exercise 4.7 Prove Proposition 4.2.4. Exercise 4.8 Determine which connectives are necessary for each of the properties in the first two claims of Proposition 4.2.9. Exercise 4.9 Complete the proof of Proposition 4.2.9. Exercise 4.10 In Corollaries 4.2.10 and 4.2.11, determine which connectives are needed for the validity of each claim. Exercise 4.11 Prove Corollaries 4.2.10 and 4.2.11. Exercise 4.12 Complete the proof of Lemma 4.2.15. Exercise 4.13 Elaborate the details in Remark 4.2.16. Exercise 4.14 Prove Proposition 4.3.5. Exercise 4.15 In Proposition 4.3.8, determine which of the equivalent formulations would be valid if only one of the connectives were present. Exercise 4.16 Prove Corollary 4.3.11. Exercise 4.17 Complete the proof of Theorem 4.4.8. Exercise 4.18 Prove Lemma 4.4.9. Exercise 4.19 In Example 4.4.10, show that hM, 𝐹≤𝑖 i, ∈ Mod∗ (LL), hM, 𝐹≤𝑖 i LSL ∈ Mod∗ (LL), and that 𝐹≤1 = {1, >} and 𝐹≤2 = {1, 𝑎, >}. Hint: either prove the residuation directly or use Remark 4.2.16. Exercise 4.20 In Example 4.4.11, check that M3 is indeed a non-distributive SL¯ >}. algebra and 𝐹≤ = {1, Exercise 4.21 Find an example simpler than Example 4.4.11 using BCI-matrices. Exercise 4.22 In Example 4.4.12, prove that hM, {1}i ∈ Mod∗ (LL). Hint: use Theorem 4.4.5. Exercise 4.23 In the proof of Theorem 4.4.15, find an alternative justification of the fact that hC, ⊆i is a residuated ordered unital groupoid using Lemma 4.4.9.

4.12 Exercises

251

Exercise 4.24 Complete the proof of Lemma 4.5.3. Exercise 4.25 In the proof of Theorem 4.5.7, show that the embedding 𝑓 preserves possible lattice truth-constants present in L. Hint: observe that, in the order given by ⊆, { 𝐴} is the least element in 𝑈 ⊆ (𝑈 ≤ ( 𝐴)) and 𝑈 ≤ ( 𝐴) is the largest one. Exercise 4.26 Complete the details of Remark 4.5.8. Exercise 4.27 Prove Proposition 4.5.11. Exercise 4.28 Prove Corollary 4.6.9. Exercise 4.29 Finish the proof of Theorem 4.6.16. Hint: recall that, thanks to rule (Symm), we can change the central implication from to → on both sides of the consequence relation using the consecutions of LL proved in the proof of Theorem 4.5.5. Recall that the rules (Symm) are derivable in any expansion of SL . Exercise 4.30 Elaborate the details in Example 4.6.20 by determining the filters of the matrices and the interpretations of the remaining connectives and checking that the matrices enjoy the required properties. Hint: use Lemma 4.4.9 and Proposition 4.6.7. Exercise 4.31 Show, by a semantical counterexample, that neither FLew nor T have the Δ-implicational deduction theorem for any finite set Δ. Exercise 4.32 In Theorem 4.8.6, find a direct proof for the implication from 2 to 1. Exercise 4.33 Elaborate the details in Example 4.8.8. Exercise 4.34 Prove Proposition 4.8.9 and generalize it to axiomatic expansions. Exercise 4.35 In Example 4.8.15, give a semantical counterexample to the formula ¯ 𝜑 𝑛 → 𝜑 ∧ 1. Exercise 4.36 Finish the proof of Lemma 4.9.7. Exercise 4.37 Finish the proof of Theorem 4.9.8 by taking care of the cases 𝛾 = 𝛼 𝛿, 𝜀 , 𝛾 = 𝛽 𝛿, 𝜀 , and 𝛾 = 𝛽 0𝛿, 𝜀 . Hint: use the theorem (aux4)

𝛽 0𝜒

𝜑,𝜒 (𝜑

→ 𝜓) → (( 𝜒

𝜑) → ( 𝜒

which is proved in the same way as (aux3). Exercise 4.38 Prove Proposition 4.9.10.

𝜓)),

Chapter 5

Generalized disjunctions

In the last section of the previous chapter, we have seen that substructural logics may retain a form of the classical proof by cases property at the price of using a complex disjunction, given possibly by not one, but a set of formulas. In particular, in the cases of SL, SLa , SLi , and SLai , we have used rather complicated sets of infinitely many formulas involving two variables and parameters. The proof by cases property has many interesting consequences and characterizations and, moreover, will play an important role in the next two chapters, where we study the interplay of disjunctions and implications (in particular, disjunctions will be used to provide a powerful characterization of logics complete with respect to linearly ordered models) and use it to axiomatize natural predicate extensions of prominent logics. Therefore, we dedicate this whole chapter to the study of generalized disjunctions. Up to this point, we have formulated (and proved) the proof by cases property as an equation involving generated theories; for an arbitrary set of formulas Γ ∪ {𝜑, 𝜓} of the corresponding language we have shown the following facts: • ThL (Γ, 𝜑 ∨ 𝜓) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓), for axiomatic expansions of BCKlat (Theorem 2.4.5). • ThL (Γ, 𝜑 ∨ 𝜓) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓), for axiomatic expansions of S4 (Theorem 2.4.6). • ThL (Γ, {𝛾(𝜑) ∨ 𝛿(𝜓) | 𝛾, 𝛿 ∈ Δ}) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓), for substructural logics with a lattice protodisjunction enjoying the Δ-implicational deduction theorem (Proposition 4.10.1). • Under certain additional conditions, we can use a much smaller set Δ0 ⊆ Δ in the previous equality (Theorem 4.10.3). Our study of generalized disjunctions starts by identifying the essential form underlying the multiplicity of variants of the proof by cases property mentioned above: ThL (Γ, 𝜑 5 𝜓) = ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓), where 5 is a set of formulas (possibly infinite, possibly with many variables), in which two variables are used to mark the positions where 𝜑 and 𝜓 will appear, and the rest are considered parameters (this notation will be formally introduced later). © Springer Nature Switzerland AG 2021 P. Cintula and C. Noguera, Logic and Implication, Trends in Logic 57, https://doi.org/10.1007/978-3-030-85675-5_5

253

254

5 Generalized disjunctions

In a closer look, one may realize that this general equation involves a rather trivial inclusion and, as expected, a more interesting one, which actually embodies the common mathematical proof method known as proof by cases. Indeed, the trivial inclusion from left to right corresponds to a generalization of the rule form of axioms (ub) which we have already seen when speaking of lattice protodisjunctions and which can be schematically written as 𝜑 I 𝜑5𝜓

𝜑 I 𝜓 5 𝜑.

We say that a set 5 is a p-protodisjunction in a given logic (‘p’ indicating the possible presence of parameters in the set) if these consecutions are valid in the logic. But then, to be coherent with this analysis, we should also write the non-trivial right-to-left inclusion of the equation in terms of consecutions; indeed, we can equivalently express it by the validity of the following metarule: PCP

Γ, 𝜑 I 𝜒 Γ, 𝜓 I 𝜒 . Γ, 𝜑 5 𝜓 I 𝜒

As this metarule corresponds to the interesting inclusion in the equation, it may deservingly be called the proof by cases property and any p-protodisjunction 5 of a given logic L for which the PCP is valid is then called a p-disjunction in L. However, this general notion of p-disjunction is still not powerful enough for the general theory that we are presenting in this book. If we were content with a study of finitary logics, the PCP and its corresponding p-disjunctions would suffice. But, as demonstrated in previous chapters, infinitary logics are an unrenounceable part of our study. Therefore, we consider the following stronger metarule: sPCP

Γ,

Ð

Γ, Φ I 𝜒 Γ, Ψ I 𝜒 , {𝛼 5 𝛽 | 𝛼 ∈ Φ and 𝛽 ∈ Ψ} I 𝜒

in which, instead of single formulas 𝜑 and 𝜓, we allow arbitrary sets Φ and Ψ. We call it the strong proof by cases property and a p-protodisjunction with this property is called a strong p-disjunction. As we will see, sPCP captures a good notion of disjunction for both finitary and infinitary logics; it is equivalent to the PCP in finitary logics (even more: in all logics with the IPEP); but, in general, it allows us to formulate interesting results for all logics that were previously known (in terms of the PCP) for finitary logics only. Another modification of the PCP that will be of interest goes in the opposite direction of considering a weaker property obtained by restricting to Γ = ∅: wPCP

𝜑I𝜒 𝜓I𝜒 , 𝜑5𝜓 I 𝜒

which we call the weak proof by cases property; p-protodisjunctions satisfying it are called weak p-disjunctions. This weaker notion is interesting on its own and will play a role in some useful characterizations.

5 Generalized disjunctions

255

The taxonomy of generalized disjunctions can be further enriched by distinguishing whether 5 does really have parameters or not, whether it is finite, whether it is actually just one formula, and whether it additionally satisfies the lattice properties seen in the previous chapter. All of these distinctions give rise to a hierarchy of disjunctions and corresponding classes of logics that we study in detail and with a wealth of examples in Section 5.1. Section 5.2 is devoted to syntactical characterizations of generalized disjunctions. The key element is the notion of 5-form of a rule. Given a p-protodisjunction 5 in a logic in language L and a consecution Γ I 𝜑, its 5-form is the set (Γ I 𝜑)5 = {Γ 5 𝜒 I 𝛿 | 𝜒 ∈ FmL and 𝛿 ∈ 𝜑 5 𝜒}, i.e. roughly speaking, the rule obtained by doing disjunctions with arbitrary formulas in the premises and in the conclusion of the original rule. We will characterize strong p-disjunctions as those that keep the logic stable under the formation of 5-forms (Theorem 5.2.6). The usefulness of this result will be illustrated by showing that the lattice disjunction ∨ of the infinitary logic Ł∞ has the sPCP. Moreover, the characterization allows us to prove that the sPCP is preserved in axiomatic expansions (Theorem 5.2.8) and the transfer of the sPCP (Theorem 5.2.9). Furthermore, for weakly implicative logics, we also prove the transfer of the PCP (Theorem 5.2.10). In Section 5.3, we aim at characterizations of generalized disjunctions in terms of the lattices of theories and filters. We start with an instrumental characterization of the wPCP by means of substitutions (Theorem 5.3.1) and, eventually obtain the main result of the section (Theorem 5.3.8): a characterization of strongly p-disjunctional logics as those in which the lattice of filters (or the lattice of theories) is a frame or, in the presence of the IPEP, as logics with a distributive lattice of filters (or theories). Section 5.4 considers the generalization of the well-known notion of prime theory which we have already considered for classical, intuitionistic and other lattice-disjunctive logics in Section 3.5. We say that a theory 𝑇 is 5-prime if 𝜑5𝜓 ⊆𝑇

implies that

𝜑 ∈ 𝑇 or 𝜓 ∈ 𝑇

(we define 5-prime filters in the same way). Then, we can define an extension property analogous to the IPEP: a logic L has the 5PEP if 5-prime theories form a basis of the closure system Th(L). We show that under the PCP, intersection-prime theories and 5-prime theories coincide, and also that the 5PEP implies the IPEP and the sPCP, and hence it is the strongest property of generalized disjunctions considered in the chapter. The 5PEP has several interesting characterizations. In particular, in Proposition 5.4.6, we prove that it implies the strong completeness of L w.r.t. the class of its reduced models with 5-prime filters, and that the converse also holds in the parameter-free case. Furthermore, in Theorem 5.4.7, we show that the 5PEP is equivalent to the IPEP plus the sPCP, or to RFSI-completeness plus the sPCP (in the parameter-free case). Finally, we define the logic L5 as the extension of L given by its reduced models with 5-prime filters and use 5 to axiomatize L5 (Proposition 5.4.13) and other interesting extensions of L given by non-negative clauses (Theorem 5.4.19).

256

5 Generalized disjunctions

In Section 5.5, we introduce an auxiliary syntactical formalism, which actually has an independent interest, that allows us to establish and characterize the 5PEP also for infinitary logics under certain conditions. Given two sets of formulas Γ and Ψ in a logic L with a weak p-disjunction, we say that a pair hΓ, Ψi of sets of formulas is an L-consistent pair if Γ does not prove any disjunction of finitely many elements of Ψ, and it is a full L-consistent pair if Γ ∪ Ψ = FmL . The logic L is said to have the pair extension property if each L-consistent pair hΓ, Ψi can be extended into a full L-consistent pair. We prove that the pair extension property holds for finitary logics with a finite disjunction (Theorem 5.5.7) and that a restricted form, which is equivalent to the 5PEP, holds for logics with a weak p-disjunction and a countable presentation (Proposition 5.5.8). Finally, we use the latter result to formulate and prove additional interesting characterizations of generalized disjunctions (Theorems 5.5.13 and 5.5.14). Finally, in Section 5.6, we revisit the characterization results for completeness properties studied in Chapter 3 and obtain some improvements thanks to the expressive power of generalized disjunctions. In particular: (1) we show that the characterization of the FSKC in terms of the embeddability of Mod∗ (L)RFSI into PU (K+ ) seen in Theorem 3.8.5 actually need not assume that L is finitary as long as it has a p-disjunction (Proposition 5.6.1), (2) if L is an RFSI-complete weakly implicative logic with a p-disjunction, the KC is equivalent to Mod∗ (L)RFSI ⊆ HSPU (K) (Theorem 5.6.3), (3) if L is a finitary weakly implicative logic with a lattice disjunction, 𝜔 the SKC is equivalent to Mod∗ (L)RFSI ⊆ IS(K+ ) (Theorem 5.6.5). Furthermore, we characterize completeness properties with respect to the semantics of finite models (Proposition 5.6.8 and Theorem 5.6.12).

5.1 A hierarchy of disjunctions We start by introducing a few useful conventions that simplify the notation and formulation of the results and their proofs throughout this chapter. Given a set Γ ∪ Δ of formulas, we write Γ I Δ to denote the set of consecutions {Γ I 𝛿 | 𝛿 ∈ Δ} and we say that the set Γ I Δ is valid in a given logic L if all its elements are (i.e. whenever Γ `L Δ). Next, recall that by writing 𝜒( 𝑝 1 , . . . , 𝑝 𝑛 ) we signify that the propositional variables occurring in 𝜒 are among 𝑝 1 , . . . , 𝑝 𝑛 ; let us denote by 𝜒(𝜓1 , . . . , 𝜓 𝑛 ) the formula resulting from 𝜒 by simultaneous replacing each variable 𝑝 𝑖 by the formula 𝜓𝑖 (formally speaking, it is the formula 𝜎 𝜒, where 𝜎 is any substitution such that 𝜎 𝑝 𝑖 = 𝜓𝑖 for each 𝑖 ∈ {1, . . . , 𝑛}); whenever the number of variables/formulas is immaterial or known from the context we use the vector notations 𝜒( 𝑝) and 𝜒(𝜓). Finally, recall that throughout the book we have met numerous metalogical statements, i.e. statements about a logic L that go beyond simply claiming that a particular consecution is valid in L. Some of them are purely syntactic: the most notorious examples are the deduction theorems and the proof by cases properties, and other examples include finitarity, and the properties IPEP and CIPEP. Some metalogical properties mix syntax and semantics (such as the R(F)SI-completeness

5.1 A hierarchy of disjunctions

257

and other completeness theorems or the transfer of finitarity proved in Theorem 2.6.2) and some are purely semantical (e.g. the numerous properties of classes of matrices studied in Chapter 3). A particularly interesting class of metalogical properties are those expressible using metarules: formal expressions which have a set of consecutions in the position of premises and a consecution in the position of conclusion. We are not going to develop a formal theory of these expressions; we just say that, as expected, a metarule is valid in a given logic, if for each substitution, the validity of its premises implies validity of its conclusion. We will depict the validity of a metarule in a logic L using the following notation (which, probably looks familiar to many readers): Γ1 `L 𝜑1

Γ2 `L 𝜑2 Γ `L 𝜑

... .

Let us formally introduce the notation that we will use for generalized disjunctions and the underlying basic notion of p-protodisjunction. Definition 5.1.1 (Protodisjunction and p-protodisjunction) Let L be a logic and 5 a set of formulas in two fixed variables 𝑝 and 𝑞 and (possibly) some additional variables called parameters of 5. For any set of formulas Φ ∪ Ψ ∪ {𝜑, 𝜓}, we define 𝜑 5 𝜓 = {𝛿(𝜑, 𝜓, 𝛼) | 𝛿( 𝑝, 𝑞, 𝑟) ∈ 5 and 𝛼 ∈ FmL } Ø Φ5Ψ= {𝛼 5 𝛽 | 𝛼 ∈ Φ and 𝛽 ∈ Ψ}. Furthermore, we say that 5 is a p-protodisjunction in L whenever it satisfies the following sets of consecutions: (PD1 ) (PD2 )

𝜑 I 𝜑5𝜓 𝜑I𝜓5𝜑

protodisjunction protodisjunction.

If 5 has no parameters, we drop the prefix ‘p-’. Note that, if a connective ∨ is a lattice protodisjunction in the sense of Definition 4.1.1, then the set {𝑝 ∨ 𝑞} is a protodisjunction in the sense of the previous definition; therefore, we say that a binary connective ◦ is a protodisjunction iff the set {𝑝 ◦ 𝑞} is. Note that the notion of (p-)protodisjunction does not have a theoretical interest of its own because, actually, any theorem (or set of theorems) of a given logic would be a protodisjunction in this logic. We introduce it as a useful means to shorten the formulation of many upcoming definitions and results. Also, as mentioned in the introduction of this chapter, p-protodisjunctions capture the following half of the proof by cases property: ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) ⊇ ThL (Γ, 𝜑 5 𝜓). Actually, we have even more: ThL (Γ, Φ) ∩ ThL (Γ, Ψ) ⊇ ThL (Γ, Φ 5 Ψ).

258

5 Generalized disjunctions

Let us formulate the other half of the proof by cases property, the genuine one, in three different versions giving rise to corresponding notions of generalized disjunctions. Definition 5.1.2 Let L be a logic and 5 a (p-)protodisjunction in L. We say that 5 is a weak (p-)disjunction, (p-)disjunction, or strong (p-)disjunction, respectively, if for each set of formulas Γ ∪ Φ ∪ Ψ ∪ {𝜑, 𝜓, 𝜒}, we have 𝜑 `L 𝜒 𝜓 `L 𝜒 wPCP weak proof by cases property 𝜑 5 𝜓 `L 𝜒 PCP

Γ, 𝜑 `L 𝜒 Γ, 𝜓 `L 𝜒 Γ, 𝜑 5 𝜓 `L 𝜒

proof by cases property

Γ, Φ `L 𝜒 Γ, Ψ `L 𝜒 strong proof by cases property. Γ, Φ 5 Ψ `L 𝜒 Finally, we say that a lattice protodisjunction ∨ is a (weak/strong) lattice disjunction whenever {𝑝 ∨ 𝑞} is a (weak/strong) disjunction. sPCP

We will use the informal term generalized disjunctions to refer non-specifically to any of the formal notions just defined. We are going to give a series of examples of various kinds of generalized disjunctions in various logics, demonstrating both the difference between the three kinds of proof by cases properties and the need for the chosen level of generality. Let us first prepare three propositions and one definition which will help us illuminating the landscape of generalized disjunctions. The first one gives their formulations as equations involving generated theories. Proposition 5.1.3 Let L be a logic and 5 a p-protodisjunction. Then, 5 is a • weak p-disjunction iff for each pair of formulas 𝜑 and 𝜓, ThL (𝜑) ∩ ThL (𝜓) = ThL (𝜑 5 𝜓). • p-disjunction iff for each set of formulas Γ ∪ {𝜑, 𝜓}, ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) = ThL (Γ, 𝜑 5 𝜓) iff for each set Γ ∪ Φ ∪ Ψ of formulas where Φ ∪ Ψ is finite, ThL (Γ, Φ) ∩ ThL (Γ, Ψ) = ThL (Γ, Φ 5 Ψ). • strong p-disjunction iff for each set Γ ∪ Φ ∪ Ψ of formulas, ThL (Γ, Φ) ∩ ThL (Γ, Ψ) = ThL (Γ, Φ 5 Ψ) iff for each set of formulas Φ ∪ Ψ, ThL (Φ) ∩ ThL (Ψ) = ThL (Φ 5 Ψ).

5.1 A hierarchy of disjunctions

259

Proof We only prove one particular implication and leave the elaboration of the whole proof of this proposition as an exercise for the reader. Namely, we prove that, if 5 is a p-disjunction, then the following metarule is valid for each set Γ ∪ Φ ∪ Ψ of formulas where Φ ∪ Ψ is finite: Γ, Φ I 𝜒 Γ, Ψ I 𝜒 . Γ, Φ 5 Ψ I 𝜒 Let us call a pair of consecutions Γ, Φ I 𝜒 and Γ, Ψ I 𝜒 valid in L a situation; define the complexity of a situation as a pair h𝑛, 𝑚i where 𝑛 and 𝑚 are respectively the cardinality of Φ \ Ψ and Ψ \ Φ. We will show by induction on 𝑘 = 𝑛 + 𝑚 that, for each situation, we have Γ, Φ 5 Ψ `L 𝜒. First, assume 𝑘 ≤ 2. If 𝑛 = 0, i.e. Φ ⊆ Ψ, we obtain Φ 5 Φ ⊆ Φ5 Ψ and, since Γ, Φ 5 Φ `L Γ ∪ Φ, the proof is done. The proof for 𝑚 = 0 is the same. If 𝑛 = 𝑚 = 1, we use the PCP. For the induction step, consider a situation with complexity h𝑛, 𝑚i, where 𝑛 + 𝑚 > 2. We can assume without loss of generality that 𝑛 ≥ 2. Take a formula 𝜑 ∈ Φ \ Ψ and define Φ10 = Φ \ {𝜑}. We know that Γ, Φ10 , 𝜑 `L 𝜒 and Γ, Ψ `L 𝜒. Thus, we also know that Γ, Φ10 , 𝜑 `L 𝜒 and Γ, Φ10 , Ψ `L 𝜒; the complexity of this situation is h1, 𝑚i and so we can use the induction assumption to obtain Γ, Φ10 , 𝜑 5 Ψ `L 𝜒. Thus, we have the situation Γ, 𝜑 5 Ψ, Φ10 `L 𝜒 and Γ, 𝜑 5 Ψ, Ψ `L 𝜒 (the second claim is trivial) with complexity h𝑛 0, 𝑚 0i, where 𝑛 0 ≤ 𝑛 − 1 and 𝑚 0 ≤ 𝑚, and so, by the induction assumption, we obtain Γ, 𝜑 5 Ψ, Φ10 5 Ψ `L 𝜒 (which is exactly what we wanted).  Note that the characterization of p-disjunctions we have just proved implies that, if L is finitary, then each p-disjunction is actually a strong p-disjunction. In the next proposition we prove that the same claim holds for a larger class of logics, namely, logics with the intersection-prime extension property (recall that a logic L has the IPEP if the intersection-prime theories form a basis of Th(L)). However, as shown in Example 5.4.10, logics with a strong p-disjunction need not have the IPEP. Moreover, in Example 5.1.10 we present a logic with a disjunction which is not strong (thus, this logic cannot have the IPEP). Proposition 5.1.4 Let L be a logic with the IPEP. Then, any p-disjunction in L is actually a strong p-disjunction. Proof Clearly, all we have to prove is the validity of the metarule sPCP. Assume, for a contradiction, that we have Γ, Φ 5 Ψ 0L 𝜒 but Γ, Φ `L 𝜒 and Γ, Ψ `L 𝜒. Due to the IPEP and the first assumption, there has to be an intersection-prime theory 𝑇 such that Γ ∪ (Φ 5 Ψ) ⊆ 𝑇 and 𝜒 ∉ 𝑇. Thus, due to the latter two assumptions, there has to be a 𝜑 ∈ Φ \ 𝑇 and 𝜓 ∈ Ψ \ 𝑇. Using the PCP and the fact that 𝑇 = ThL (𝑇, 𝜑 5 𝜓), we obtain 𝑇 = ThL (𝑇, 𝜑) ∩ ThL (𝑇, 𝜓), a contradiction with the assumption that 𝑇 is intersection-prime.  The next proposition is an analog of Proposition 4.1.6, which showed that any two lattice protodisjunctions are interdefinable. Its proof is a consequence of the suitable proof by cases properties.

260

5 Generalized disjunctions

Proposition 5.1.5 Let L be a logic with a (weak/strong) p-disjunction 5. Then, a p-protodisjunction 5 0 is a (weak/strong) p-disjunction in L iff 𝜑 5 𝜓 a`L 𝜑 5 0 𝜓. Thus, in particular, if L has a (strong) p-disjunction, then each of its weak p-disjunctions is actually a (strong) p-disjunction. Note that the contarpositive reading of the last claim of the previous proposition is particularly useful: it tells us that, if there is a (weak) p-disjunction which is not strong in a given logic, then there is no strong p-disjunction in that logic (and analogously if there is a weak p-disjunction which is not a p-disjunction, then there is no p-disjunction). Thus, in particular, the already mentioned logic with a disjunction which is not strong (see Example 5.1.10) cannot have any strong p-disjunction. This fact allows for a robust classification of logics based on how strong and simple disjunctions they have. Indeed, we will distinguish logics based on the existence of a set 5 satisfying a variant of the proof by cases property, but also based on the structure of the set 5, that is, distinguishing traditional disjunctions defined by a single binary (primitive or definable) connective (including lattice disjunctions), generalized disjunctions defined by a parameter-free set, and the most general case where 5 is allowed to be a parameterized set. Therefore, we define the following 12 classes of logics: Definition 5.1.6 (p-disjunctional logics) We say that a logic L is (strongly/weakly) • • • •

p-disjunctional if it has a (strong/weak) p-disjunction. disjunctional if it has a (strong/weak) disjunction. disjunctive if it has a (strong/weak) disjunction given by a singleton. lattice-disjunctive if it has a (strong/weak) lattice disjunction.

Theorem 5.1.7 The subsumption order of the classes of logics introduced in the previous definition is depicted in Figure 5.1. Furthermore, all these classes are pairwise different and their intersections are infima w.r.t. the depicted subsumption order. Finally, when restricted to logics with the IPEP, the layer of strongly p-disjunctional logics collapses with the layer of p-disjunctional logics, giving a hierarchy of eight pairwise different classes. Proof The intersection property follows from Proposition 5.1.5. The collapse when restricted to logics with the IPEP follows from Proposition 5.1.4. The separation of all the classes (both in the general case and in the IPEP case) is established by the forthcoming series of examples.  Remark 5.1.8 Note that all of the upcoming counterexamples are weakly implicative logics and those demonstrating separations in the IPEP case are actually finitary. On the other hand, note if we have a logic L in a language L 0 ⊇ L, then any 5 ⊆ FmL which is a (weak/strong) disjunction in L is a (weak/strong) disjunction in the L-fragment of L. Therefore, we can easily produce (e.g. using the next two examples) (strongly/weakly) disjunctional logics which are not weakly implicative, the prime example being the conjunction-disjunction fragment of classical logic.

5.1 A hierarchy of disjunctions

261

Fig. 5.1 Hierarchy of disjunctional logics.

Remark 5.1.9 We could have also introduced the classes of (strongly/weakly) finitely (p-)disjunctional logic defined by the presence of a finite (strong/weak)(p-)disjunction. We have decided not to do so in this book, in order to keep the picture (relatively) simple. Let us just note that the logic G→ which is used in Example 5.1.15 to separate strongly disjunctional and weakly disjunctive logics would actually separate strongly finitely disjunctional and weakly disjunctive logics. Also, it can be shown that the global modal logic K is an example of a strongly disjunctional logic which is not weakly finitely disjunctional. Furthermore, the logic IL→ could serve as example of a strongly finitely p-disjunctional logic which is not weakly disjunctional (see Example 5.1.14). Finally, we conjecture that SL is an example of a strongly p-disjunctional logic which is not weakly finitely p-disjunctional. First, we present counterexamples to all right-facing arrows, i.e. we separate disjunctions for their strong/weak variants. It is easy to see that it can be achieved by finding two particular logics. Example 5.1.10 (An infinitary weakly implicative logic which is lattice-disjunctive but not strongly p-disjunctional) Let A be a complete Heyting algebra which is not a dual frame (cf. Example A.2.13 and Lemma A.5.5), i.e. there are elements 𝑎 𝑖 ∈ 𝐴 for 𝑖 ≥ 0 such that Û Û (𝑎 0 ∨ 𝑎 𝑖 )  𝑎 0 ∨ 𝑎𝑖 . 𝑖 ≥1

𝑖 ≥1

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5 Generalized disjunctions

We expand the language of A by constants {𝑐 𝑖 | 𝑖Ó≥ 0} ∪ {𝑐} and define an 0 algebra A0 in this language by setting 𝑐A𝑖 = 𝑎 𝑖 and 𝑐 = 𝑖 ≥1 𝑎 𝑖 . Then, we define the logic K in this language given semantically by the class of matrices K = {hA0, 𝐹i | 𝐹 is a principal lattice filter in A}. Clearly, the logic K is an expansion of intuitionistic logic by constants and, thus, it is weakly implicative and ∨ is a lattice disjunction. In order to prove that it has no strong p-disjunction, it suffices to show that ∨ is not a strong disjunction. We first prove that Û Γ K 𝜑 iff for each A-evaluation 𝑒 holds: 𝑒(𝜓) ≤ 𝑒(𝜑). 𝜓 ∈Γ

Ó One direction: as the principal lattice filter generated by 𝜓 ∈Γ 𝑒(𝜓) clearly contains 𝑒[Γ], it has to contain 𝑒(𝜑) too. The converse direction: just observe that any principal Ó filter containing 𝑒[Γ] has to contain 𝜓 ∈Γ 𝑒(𝜓) and so it contains 𝑒(𝜑). Therefore, 𝑐 0 K 𝑐 0 ∨ 𝑐

and

{𝑐 𝑖 | 𝑖 ≥ 1} K 𝑐 0 ∨ 𝑐

and so, by the metarule sPCP, we would obtain {𝑐 0 ∨ 𝑐 𝑖 | 𝑖 ≥ 1} K 𝑐 0 ∨ 𝑐 which is a contradiction with our assumptions about 𝑐 and 𝑐 𝑖 s. Example 5.1.11 (A finitary weakly implicative logic which is weakly lattice-disjunctive but not p-disjunctional) Consider the LCL -algebra A whose lattice reduct is the bounded non-distributive lattice M3 with the domain {⊥, 𝑎, 𝑏, 𝑐, >}, order ≤, and implication → defined as 𝑥 →A 𝑦 = > if 𝑥 ≤ 𝑦 and ⊥ otherwise. Note that, for each non-trivial lattice filter 𝐹, the matrix preorder ≤ hA,𝐹 i is equal to the order ≤ and so, if we denote by K the class of such matrices, we have that K is a finitary (Proposition 2.5.4) weakly implicative logic (Proposition 2.8.11) and ∨ is a lattice protodisjunction (Proposition 4.1.5). As in the previous example, we could prove that Û Γ K 𝜑 iff for each M3 -evaluation 𝑒 holds: 𝑒(𝜓) ≤ 𝑒(𝜑). 𝜓 ∈Γ

From this it easily follows that ∨ is a weak lattice disjunction (indeed, 𝑒(𝜑) ≤ 𝑒( 𝜒) and 𝑒(𝜓) ≤ 𝑒( 𝜒) implies 𝑒(𝜑 ∨ 𝜓) = 𝑒(𝜑) ∨A 𝑒(𝜓) ≤ 𝑒( 𝜒)) but not a disjunction. Indeed, starting from the obviously valid claims 𝜑, 𝜓 K (𝜑 ∧ 𝜓) ∨ 𝜒 and 𝜒, 𝜓 K (𝜑 ∧ 𝜓) ∨ 𝜒, we obtain using the PCP that 𝜑 ∨ 𝜒, 𝜓 K (𝜑 ∧ 𝜓) ∨ 𝜒 and, thus, also (applying the PCP again) 𝜑 ∨ 𝜒, 𝜓 ∨ 𝜒 K (𝜑 ∧ 𝜓) ∨ 𝜒. Then, we reach a contradiction by observing that 𝑎 ∨ 𝑏 = 𝑐 ∨ 𝑏 = > while (𝑎 ∧ 𝑐) ∨ 𝑏 = ⊥ ∨ 𝑏 = 𝑏. Let us now focus on the structure of the possible p-disjunctions in a given logic. As before, to obtain counterexamples to the left-facing arrows we only need a few well-chosen logics. Note that the existence of these examples justifies the level of generality of disjunction connectives, defined by possibly infinite sets of formulas with parameters.

5.1 A hierarchy of disjunctions

263

Table 5.1 Strong p-disjunctions in prominent substructural logics.

Logic L IL BCKlat FLew SLaE SLai SLa SLi SL S4 K4 T K

strong p-disjunction 5L {𝑝 ∨ 𝑞} {𝑝 ∨ 𝑞} {𝑝 ∨ 𝑞} ¯ ∨ (𝑞 ∧ 1)} ¯ {( 𝑝 ∧ 1) {𝛾( 𝑝) ∨ 𝛿(𝑞) | 𝛾, 𝛿 ∈ {𝜆𝑟 , 𝜌𝑟 | 𝑟 ∈ Var}∗ } {𝛾( 𝑝) ∨ 𝛿(𝑞) | 𝛾, 𝛿 ∈ {𝜆𝑟 , 𝜌𝑟 , ★ ∧ 1¯ | 𝑟 ∈ Var}∗ } {𝛾( 𝑝) ∨ 𝛿(𝑞) | 𝛾, 𝛿 ∈ {𝛼𝑟 ,𝑠 , 𝛼𝑟0 ,𝑠 , 𝛽𝑟 ,𝑠 , 𝛽𝑟0 ,𝑠 | 𝑟, 𝑠 ∈ Var}∗ } {𝛾( 𝑝) ∨ 𝛿(𝑞) | 𝛾, 𝛿 ∈ {𝛼𝑟 ,𝑠 , 𝛼𝑟0 ,𝑠 , 𝛽𝑟 ,𝑠 , 𝛽𝑟0 ,𝑠 , ★ ∧ 1¯ | 𝑟, 𝑠 ∈ Var}∗ } {𝑝 ∨ 𝑞} {( 𝑝 ∧ 𝑝) ∨ (𝑞 ∧ 𝑞)} { 𝑘 𝑝 ∨ 𝑙 𝑞 | 𝑘, 𝑙 ≥ 1} { 𝑘 𝑝 ∨ 𝑙 𝑞 | 𝑘, 𝑙 ≥ 0}

Before we present those three examples, let us recall that we have already identified disjunctions in various logics in the previous chapters and summarized our results in Example 4.10.4. Some of these generalized disjunctions are rather complex, e.g. in the logic SLa we have one (see Table 5.1) expressed using the left and right conjugates 𝜆 and 𝜌 which can be parametrized by any formula; in the case of SL the situation is even more complex. Our present terminology and conventions allow us a certain simplification: we can use variables instead of formulas.1 Let us also note, thanks to Proposition 5.1.4 and the fact that all the logics listed in the next example are finitary, we know that the listed disjunctions are strong. Example 5.1.12 Let L be an axiomatic expansion of a logic in the first column on any row in Table 5.1. Then, the set 5L in the second column of the same row is a strong p-disjunction in L (cf. Theorem 5.2.8). Example 5.1.13 (Abelian logic A is strongly lattice-disjunctive) Let us first note that, thanks to Examples 4.6.11 and 4.6.14, we know that A is an axiomatic expansion of ¯ ∨ (𝑞 ∧ 1)} ¯ is a strong disjunction the ⊥, >-free fragment of SLaE and, thus, {( 𝑝 ∧ 1) in A as well.2 Next, let us recall that in Example 4.6.14 we have also established the completeness of A w.r.t. the matrix Zlat + and, since its underlying lattice is linearly 1 Actually, we can simplify it even further: if we enumerate the variables and define the depth of an iterated deduction term in the obvious way, it suffices to use only the 𝑖th variable on the 𝑖the level; e.g. consider only formulas 𝜌𝑟2 (𝜆𝑟1 ) instead of 𝜌𝑟 (𝜆𝑠 ) for every atoms 𝑟 and 𝑠. 2 We could prove this in two ways: (1) notice that the lattice truth-constants play no role in establishing the results for SLaE in Example 4.10.4 using Theorem 4.10.3; (2) using the observation ¯ ∨ (𝑞 ∧ 1) ¯ } is a strong disjunction in ⊥, >-free fragment from Remark 5.1.8 to establish that { ( 𝑝 ∧ 1)

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5 Generalized disjunctions

ordered and thus distributive, we know that `A (𝜑 ∨ 𝜓) ∧ 𝜒 ↔ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒). Therefore, we can easily show that ¯ ∨ (𝜓 ∧ 1) ¯ 𝜑 ∨ 𝜓 a`A (𝜑 ∧ 1) and, thus, Proposition 5.1.5 completes the proof. Thus, in particular, the logic SL has a rather complicated p-disjunction given by an infinite set with parameters. Sato in [280, Proposition 6.9] showed that there is no finite disjunction in FL (and, thus, in SL); however, the question whether there could be an infinite disjunction remains to be open. We can nonetheless give a simple example of a logic with a strong p-disjunction which is not even weakly disjunctional. Example 5.1.14 (A finitary weakly implicative logic which is strongly p-disjunctional but not weakly disjunctional) Consider the logic IL→ (Section 2.3). We show that 5 = {( 𝑝 → 𝑟) → ((𝑞 → 𝑟) → 𝑟)} is a p-disjunction in IL (and, thus, a strong p-disjunction due to Proposition 5.1.4). We leave the proof of the validity of (the sets of) consecutions (PD), namely that, for each 𝜑, 𝜓, and 𝜒, 𝜑 `IL→ (𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)

𝜓 `IL→ (𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒),

as an exercise for the reader and we show the PCP. Assume that Γ, 𝜑 `IL→ 𝜒 and Γ, 𝜓 `IL→ 𝜒 and so, by the deduction theorem (see Theorem 2.4.1), Γ `IL→ 𝜑 → 𝜒 and Γ `IL→ 𝜓 → 𝜒. Since (𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒) ∈ 𝜑 5 𝜓, we easily obtain that Γ, 𝜑 5 𝜓 `IL→ 𝜒. Next, assume, for a contradiction, that a set 5 0 is a weak disjunction in IL→ . Thus, by Theorem 5.2.8, it is also a weak disjunction in the full intuitionistic logic IL. Recall that the lattice connective ∨ is a disjunction in IL too (due to Proposition 2.4.5) and so, by Proposition 5.1.5, we have 𝑝 5 0 𝑞 a`IL 𝑝 ∨ 𝑞. Using finitarity, the presence of the lattice conjunction ∧ in the language of IL and the deduction theorem of IL (Theorem 2.4.1), we obtain a formula ∨0 of two variables 𝑝, 𝑞 built using only implication and lattice conjunction such that `IL 𝑝 ∨0 𝑞 ↔ 𝑝 ∨𝑞, which contradicts the known fact that no connective of intuitionistic logic is definable from the remaining ones (see e.g. [222]). By a simple modification of the previous example, we could show that any axiomatic extension of BCK weaker that IL→ is an example of strongly p-disjunctional logic which is not weakly disjunctional. Indeed, the negative part is obvious: any weak disjunction in such a logic would be a weak disjunction in IL→ (this can be either directly observed or obtained as a special corollary of Theorem 5.2.8). of SLaE and thus it is so in all of its axiomatic extensions (an easy observation or a consequence of the upcoming Theorem 5.2.8).

5.1 A hierarchy of disjunctions

265

To show the positive part, it suffices to prove that the set 5 = {[ 𝑝 → ( 𝑝 → . . . ( 𝑝 → 𝑟)] → [𝑞 → (𝑞 → . . . (𝑞 → 𝑟)] → 𝑟) | 𝑘, 𝑙 ≥ 1} | {z } | {z } 𝑘 times

𝑙 times

is a p-disjunction in the given logic. We leave the elaboration of details as an exercise for the reader. The next example shows that there are natural logics with strong (finite) disjunction but no weak disjunction given by a singleton. Example 5.1.15 (A finitary weakly implicative logic which is strongly disjunctional but not weakly disjunctive) Recall the logic G→ (Section 2.3). We leave as an exercise for the reader to show that set 5 = {( 𝑝 → 𝑞) → 𝑞, (𝑞 → 𝑝) → 𝑝} is a disjunction in G→ . Assume, for a contradiction, that there is some weak disjunction 5 0 = { 𝜒( 𝑝, 𝑞)} in G→ . By Theorem 5.2.8, we know that 5 0 is a weak disjunction in the logic G as well and, thus, due to Proposition 5.1.5 and the known fact that ∨ is a lattice disjunction in G, we obtain 𝜒(𝜑, 𝜓) a`G 𝜑 ∨ 𝜓. Using the deduction theorem of G, this entails that ∨ is definable in G by the formula 𝜒, i.e. we have `G 𝜑 ∨ 𝜓 ↔ 𝜒(𝜑, 𝜓) and thus, due to the strong completeness of Gödel–Dummett logic G w.r.t. the matrix h[0, 1] G , {1}i established in Example 3.6.18, we know, in particular, that for every 𝑎, 𝑏 ∈ [0, 1) we have 𝜒 [0,1] G (𝑎, 𝑏) = max{𝑎, 𝑏}. We show by an infinite descent argument that no such formula 𝜒 in the language L→ could exist. Clearly, 𝜒 cannot be an atom and so it has to be a formula of the form 𝜒( 𝑝, 𝑞) = 𝛼 → 𝛽. If 𝑎, 𝑏 < 1, we have 𝜒 [0,1] G (𝑎, 𝑏) < 1 and so 𝜒 [0,1] G (𝑎, 𝑏) = 𝛽 [0,1] G (𝑎, 𝑏), i.e. 𝛽( 𝑝, 𝑞) is a strictly shorter formula with the same property. As before, we can use this example to show that any axiomatic extension of FBCK weaker that G→ is an example of a strongly disjunctional but not weakly disjunctive logic. Indeed, the negative part follows from the case of G→ and the positive one could be proven analogously using the more complex deduction theorem of FBCK; later we will see that this is a special case of Proposition 6.3.6. Finally, we have to give an example of a strongly disjunctive logic which is not weakly lattice-disjunctive; such examples are very natural logics which are actually easy to produce. Example 5.1.16 (Finitary weakly implicative logics which are strongly disjunctive but not weakly lattice-disjunctive) Consider the global modal logic K4 or S4, or the substructural logic SLaE . Thanks to Example 5.1.12, we know that these logics indeed are strongly disjunctive and we also already know that ∨ is a not a weak lattice disjunction in any of these logics: for modal logics it was established in the text after Proposition 2.4.5 and for SLaE in Example 4.4.11 (actually, in both cases,

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we speak about the PCP but it is easy to observe that the provided counterexamples disprove already the wPCP; also, in the second case, one has to observe that the constructed algebra is actually an SLaE -algebra). Since ∨ is a lattice protodisjunction in these logics, we know, thanks to Propositions 4.1.6 and 5.1.5, that no weak lattice disjunction exists in these logics.

5.2 Characterizations of proof by cases properties via 5-forms Weak p-disjunctions satisfy some additional properties that any kind of disjunction is expected to satisfy, namely idempotency, commutativity, and associativity (which, however, are also typically satisfied by conjunction connectives, whereas the (PD) and (variants of) proof by cases property are typically satisfied only by disjunction connectives). These properties are formulated in the following straightforward lemma. Lemma 5.2.1 Let L be a logic with a weak p-disjunction 5. Then, the following (sets of) consecutions are valid in L: (I 5 ) (C 5 ) (A 5,1 ) (A 5,2 )

𝜑5𝜑I 𝜑 𝜑5𝜓 I𝜓5𝜑 𝜑 5 (𝜓 5 𝜒) I (𝜑 5 𝜓) 5 𝜒 (𝜑 5 𝜓) 5 𝜒 I 𝜑 5 (𝜓 5 𝜒)

idempotency commutativity associativity associativity.

The properties (C 5 ), (I 5 ), (A 5 ) must be properly read: they respectively give commutativity, idempotency and associativity as regards to membership in the filter of matrix models, but they do not imply that these properties hold for disjunctions of arbitrary elements in the matrix. In symbols, if A = hA, 𝐹i ∈ Mod(L), the validity of (C 5 ) means that, for every 𝑎, 𝑏 ∈ 𝐴, 𝑎 5A 𝑏 ⊆ 𝐹 implies that 𝑏 5A 𝑎 ⊆ 𝐹; but it does not necessarily mean that 𝑎 5A 𝑏 = 𝑏 5A 𝑎, and analogously for the other properties. Example 5.2.2 An element-wise non-commutative strong disjunction. Consider the matrix A in the language LCL extended by a binary connective t whose LCL -reduct is the standard matrix of Gödel–Dummett logic, i.e. h[0, 1] G , {1}i ∈ Mod∗ (G), and t is defined as ( 𝑎 if 𝑏 < 1 A 𝑎t 𝑏= 1 otherwise. Clearly, t is a strong disjunction in A (exercise for the reader), but for any 𝑎 < 𝑏 < 1, we have 𝑎 t 𝑏 = 𝑎 ≠ 𝑏 = 𝑏 t 𝑎. We could also show the independence of the conditions (C 5 ), (I 5 ), (A 5 ) of p-protodisjunctions by several artificial examples, all of them finitary. We leave it as

5.2 Characterizations of proof by cases properties via 5-forms

267

an exercise for the reader and just mention that in Łukasiewicz logic one can introduce a connective ⊕ defined as 𝜑 ⊕ 𝜓 = ¬(¬𝜑 & ¬𝜓) which satisfies the conditions (PD), (C 5 ), and (A 5 ) but not (I 5 ). Recall that in Example 5.1.16 we have seen examples of logics with a lattice protodisjunction ∨ (and, thus, satisfying the consecutions (I∨ ), (C∨ ), and (A∨ )) which is not a weak disjunction. An interesting question is to determine whether there exist easily checkable additional properties that guarantee that a p-protodisjunction 5 satisfying (C 5 ), (I 5 ), and (A 5 ) is actually a (weak/strong) p-disjunction. The answer is surprisingly simple and has rather profound consequences. Let us first deal with the case of weak p-disjunctions. Theorem 5.2.3 (Syntactical characterization of weak p-disjunctions) Let L be a logic with a p-protodisjunction 5. Then, 5 is a weak p-disjunction iff the consecutions (C 5 ) and (I 5 ) are valid in L and whenever 𝜑 `L 𝜒, then, for each 𝛿, 𝜑 5 𝛿 `L 𝜒 5 𝛿. Proof The left-to-right direction is straightforward. Conversely, assume that 𝜑 `L 𝜒 and 𝜓 `L 𝜒. Then, due to our assumption, we have 𝜑 5 𝜓 `L 𝜒 5 𝜓 and 𝜓 5 𝜒 `L 𝜒 5 𝜒. The rest easily follows from the validity of (C 5 ) and (I 5 ).  In hindsight, we could have used this theorem before (e.g. in Example 5.1.16) to show that ∨ is not a weak disjunction in any global modal logic (using the fact that, for a certain formula 𝜒, we do not have 𝜑 ∨ 𝜒 0S4 (𝜑) ∨ 𝜒) or in substructural logics below SLaE (using the fact that, for a certain formula 𝜒, we do not have ¯ ∨ 𝜒). 𝜑 ∨ 𝜒 0SLaE (𝜑 ∧ 1) An analogous characterization can be proven for strong p-disjunctions. However, we will first prepare a certain notion which will allow not only for a compact formulation of the claim but also for an easy-to-check sufficient condition. Definition 5.2.4 (5-form) Let L be a logic with a p-protodisjunction 5 and Γ I 𝜑 a consecution. We call the set Ø (Γ I 𝜑)5 = (Γ 5 𝜒 I 𝜑 5 𝜒) = {Γ 5 𝜒 I 𝛿 | 𝜒 ∈ FmL and 𝛿 ∈ 𝜑 5 𝜒} 𝜒 ∈FmL

the 5-form of the consecution Γ I 𝜑. Lemma 5.2.5 Let 𝑅 be a consecution such that 𝑅 5 ⊆ L. 1. If 5 satisfies (I 5 ), then 𝑅 ∈ L. 2. If 5 satisfies (A 5 ), then (𝑅 5 )5 ⊆ L. Proof The first claim: From the assumption, we know Γ 5 𝜑 `L 𝜑 5 𝜑; (PD) and (I 5 ) complete the proof. To prove the second claim, we start with Γ 5 (𝜓1 5 𝜓2 ) `L 𝜑 5 (𝜓1 5 𝜓2 ); repeated use of (A 5 ) completes the proof.  Note that the first part of this lemma tells us that in the next theorem we could write ‘𝑅 5 ⊆ L iff 𝑅 ∈ L’, instead of ‘𝑅 5 ⊆ L for each 𝑅 ∈ L’.

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5 Generalized disjunctions

Theorem 5.2.6 (Characterizations using 5-forms) Let L be a logic with a presentation AS and a p-protodisjunction 5. Then, the following are equivalent: 1. 5 is a strong p-disjunction. 2. The consecutions (C 5 ) and (I 5 ) are valid in L and 𝑅 5 ⊆ L for each 𝑅 ∈ AS. 3. The consecutions (C 5 ) and (I 5 ) are valid in L and 𝑅 5 ⊆ L for each 𝑅 ∈ L. Proof The proof that 1 implies 2 is easy: consider a consecution Γ I 𝜑 ∈ AS; from Γ `L 𝜑, we obtain Γ `L 𝜑 5 𝜒 using (PD). By (PD), we also obtain 𝜒 `L 𝜑 5 𝜒. Thus, by the sPCP, we obtain Γ 5 𝜒 `L 𝜑 5 𝜒. To prove that 2 implies 3, assume that Γ `L 𝜑 and consider some proof of this fact in the presentation AS. We show Γ 5 𝜒 `L 𝛿 5 𝜒 for each formula 𝜒 and each 𝛿 appearing in the chosen proof of 𝜑 from Γ. If 𝛿 ∈ Γ, the proof is trivial; if it is an axiom, it follows from (PD). Now assume that 𝑅 = Γ0 I 𝛿 is the inference rule used to obtain 𝛿 (we can assume it because axiomatic systems are closed under substitutions). From the induction assumption, we have Γ 5 𝜒 `L Γ0 5 𝜒. Since 𝑅 5 ∈ L, the proof is done. To prove the final implication, assume that Γ, Φ `L 𝜒 and Γ, Ψ `L 𝜒. Then, by the hypothesis, Γ 5 𝜒, Ψ 5 𝜒 `L 𝜒 5 𝜒 and Γ 5 𝜓, Φ 5 𝜓 `L 𝜒 5 𝜓, for each 𝜓 ∈ Ψ, and hence Γ 5 Ψ, Φ 5 Ψ `L 𝜒 5 Ψ. Using (PD), (C 5 ), and (I 5 ),we obtain Γ, Φ 5 Ψ `L 𝜒. Let us use this theorem to prove that ∨ is a strong lattice disjunction in Ł∞ (which will be a crucial step in proving the completeness of this infinitary logic w.r.t. the matrix Ł∞ = h[0, 1] Ł , {1}i). Example 5.2.7 (Ł∞ is a strongly lattice-disjunctive logic) Recall that the logic Ł∞ is the extension of Łukasiewicz logic by the following rule (recall that we set 𝜑0 = 1¯ and 𝜑 𝑛+1 = 𝜑 𝑛 & 𝜑): (Ł∞ )

{¬𝜑 → 𝜑 𝑛 | 𝑛 ≥ 1} I 𝜑

Hay rule.

Due to Proposition 2.4.5, we know that ∨ is a (strong lattice-)disjunction in Ł and so, due to the previous theorem, Ł∞ proves (PD), (I∨ ), (C∨ ), and the ∨-form of modus ponens (the only rule of the Łukasiewicz logic). Thus, to prove the claim, again due to the previous theorem, it suffices to prove the validity of the ∨-form of (Ł∞ ) i.e. {(¬𝜑 → 𝜑 𝑛 ) ∨ 𝜒 | 𝑛 ≥ 1} `Ł∞ 𝜑 ∨ 𝜒. To do so, we first observe that ¬𝜑 → 𝜑 𝑛 `Ł ¬(𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜒) 𝑛

𝜒 `Ł ¬(𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜒) 𝑛 .

We leave finding of the formal proofs as an exercise for the reader. Thus, we can use the sPCP (in Ł) to obtain (¬𝜑 → 𝜑 𝑛 ) ∨ 𝜒 `Ł ¬(𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜒) 𝑛 and so the proof is completed by an obvious instance of (Ł∞ ).

5.2 Characterizations of proof by cases properties via 5-forms

269

As another application of Theorem 5.2.3, we show under which conditions a strong p-disjunction of a given logic remains a strong p-disjunction in an expansion. Clearly, p-protodisjunctions are preserved by expansions. Thus, we have to focus on the strong proof by cases property. This property (and both of its variants) is clearly preserved in axiomatic extensions, but we can prove a much stronger statement. Theorem 5.2.8 (Preservation of sPCP) Let L1 be a logic in a language L1 with a strong p-disjunction 5, and L2 an expansion of L1 in a language L2 ⊇ L1 by a set C of consecutions closed under L1 -substitutions. Then, 5 is a strong p-disjunction in L2 iff 𝑅 5 ⊆ L2 for each 𝑅 ∈ C. In particular, each axiomatic expansion of a strongly p-disjunctional logic is strongly p-disjunctional. Proof The left-to-right direction is a straightforward application of Theorem 5.2.6. For the reverse direction, take a presentation AS of L1 . We know that L2 has a presentation AS 0 = {𝜎[Γ] I 𝜎𝜑 | 𝜎 is an L2 -substitution, Γ I 𝜑 ∈ AS ∪ C}. Thus, we need to prove that, for each Γ I 𝜑 ∈ AS ∪ C and for each L2 -substitution 𝜎, we have (𝜎[Γ] I 𝜎𝜑)5 ⊆ L2 , i.e. for each 𝛿( 𝑝, 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 ) ∈ 5, each sequence 𝛼 of L2 -formulas, and each L2 -formula 𝜒, we have 𝜎[Γ] 5 𝜒 `L2 𝛿(𝜎𝜑, 𝜒, 𝛼). If 5 were parameter-free, the proof would be almost straightforward. In general, however, we have to take cake of the parameters by using three suitably chosen substitutions. Consider any enumeration of the propositional variables such that 𝑝 0 = 𝑞, 𝑝 𝑖 = 𝑟 𝑖 , and L1 -substitutions 𝜌, 𝜌 −1 and L2 -substitution 𝜎 ¯ defined as ( 𝜌 𝑝 𝑖 = 𝑝 𝑖+𝑛+1

−1

𝑝 𝑖−𝑛−1

for 𝑖 > 𝑛

𝑝𝑖

for 𝑖 ≤ 𝑛

𝜌 𝑝𝑖 =

  𝜒    𝜎 ¯ 𝑝 𝑖 = 𝛼𝑖    𝜎( 𝑝 𝑖−𝑛−1 ) 

for 𝑖 = 0 for 0 ≠ 𝑖 ≤ 𝑛 for 𝑖 > 𝑛.

Note that 𝜌 and 𝜌 −1 can also be seen as L2 -substitutions and, for each L2 -formula 𝜓, we have 𝜌 −1 𝜌𝜓 = 𝜓 and 𝜎𝜌𝜓 ¯ = 𝜎𝜓. Clearly, 𝜌[Γ] I 𝜌𝜑 ∈ AS ∪ C and thus, thanks to our assumptions, we obtain 𝜌[Γ] 5 𝑞 `L2 𝛿(𝜌𝜑, 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 ) and consequently also 𝜎[𝜌[Γ] ¯ 5 𝑞] `L2 𝜎𝛿(𝜌𝜑, ¯ 𝑞, 𝑟). Obviously, 𝜎𝛿(𝜌𝜑, ¯ 𝑞, 𝑟) = 𝛿(𝜎𝜑, 𝜒, 𝛼) and so if, we prove 𝜎[𝜌[Γ] ¯ 5 𝑞] ⊆ 𝜎[Γ] 5 𝜒, the proof is done. To show this, it is enough to observe that the formulas in 𝜎[𝜌[Γ] ¯ 5 𝑞] are clearly of the form 𝛿 0 (𝜎𝜓, 𝜒, 𝜎𝛽 ¯ 1 , . . . , 𝜎𝛽 ¯ 𝑘 ) ∈ 5 for some 𝜓 ∈ Γ, 𝛿 0 ( 𝑝, 𝑞, 𝑠1 , . . . , 𝑠 𝑘 ) ∈ 5, and a sequence of L2 -formulas 𝛽1 , . . . , 𝛽 𝑘 .  Let us now prove one more crucial consequence of Theorem 5.2.6. Given a logic L in a language L, a p-protodisjunction 5, an L-algebra A, and a set 𝑋 ∪𝑌 ∪ {𝑎, 𝑏} ⊆ 𝐴, we define 𝑎 5A 𝑏 = {𝛿A (𝑎, 𝑏, 𝑐) | 𝛿( 𝑝, 𝑞, 𝑟) ∈ 5 and 𝑐 ∈ 𝐴} Ø 𝑋 5A 𝑌 = {𝑥 5A 𝑦 | 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 }.

270

5 Generalized disjunctions

Note that the definition of 𝜑 5 𝜓 is a special case of the general definition of 𝑎 5A 𝑏 for A = FmL . Next, recall that we denote by FiAL (𝑋) the L-filter generated by the set 𝑋 and observe that we clearly have FiAL (𝑋) ∩ FiAL (𝑌 ) ⊇ FiAL (𝑋 5A 𝑌 ). We know that the converse inclusion for A = FmL is equivalent to the fact that 5 is a strong p-disjunction. We can use the characterization given in Theorem 5.2.6 to show that in such case the inclusion holds for all algebras, thus proving a property which can be naturally called transferred strong proof by cases property. Theorem 5.2.9 (Transfer of sPCP) Let L be a logic with a strong p-disjunction 5. Then, for each L-algebra A and each 𝑋, 𝑌 ⊆ 𝐴, FiAL (𝑋) ∩ FiAL (𝑌 ) = FiAL (𝑋 5A 𝑌 ). Proof The inclusion Fi(𝑋 5A 𝑌 ) ⊆ Fi(𝑋) ∩ Fi(𝑌 ) follows easily from (PD). Indeed, consider any 𝛿( 𝑝, 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 ) ∈ 5; we know that 𝑝 `L 𝛿( 𝑝, 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 ) and so 𝛿A (𝑥, 𝑦, 𝑎 1 , . . . , 𝑎 𝑛 ) ∈ Fi(𝑥) for each 𝑥, 𝑦, 𝑎 1 , . . . , 𝑎 𝑛 ∈ 𝐴, thus 𝑋 5A 𝑌 ⊆ Fi(𝑋). To prove the converse inclusion, we start by proving the following claim: C

For each 𝑥 ∈ Fi(𝑋) and 𝑦 ∈ 𝐴, we have 𝑥 5A 𝑦 ⊆ Fi(𝑋 5A 𝑦).

Let us fix some presentation AS of L and 𝑥 ∈ Fi(𝑋). Recall that, due to Proposition 2.5.8, we know that there is a tree labeled by elements of 𝐴 such that the root is labeled by 𝑥 and, for each node 𝑛 and its label 𝑧, the following holds: • If 𝑛 is a leaf, then h∅, 𝑧i ∈ 𝑉AA S or 𝑧 ∈ 𝑋. • If 𝑛 is not a leaf, and 𝑍 is the set of labels of its predecessors, then h𝑍, 𝑧i ∈ 𝑉AA S , where 𝑉AA S = {h𝑒[Γ], 𝑒(𝜓)i | 𝑒 is an A-evaluation and Γ I 𝜓 ∈ AS}. Clearly, the proof is done by showing that 𝑧 5A 𝑦 ⊆ Fi(𝑋 5A 𝑦) for each 𝑧 labeling any node of that proof, i.e. for each 𝛿( 𝑝, 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 ) ∈ 5 and each 𝑎 1 , . . . , 𝑎 𝑛 ∈ 𝐴, we have 𝛿A (𝑧, 𝑦, 𝑎 1 , . . . , 𝑎 𝑛 ) ∈ Fi(𝑋 5A 𝑦). The case of leaves is trivial (𝑧 is either an element of 𝑋, in which case it is trivial, or the value of some axiom in some evaluation, in which case it follows using (PD)). Otherwise, there is a non-empty set 𝑍 of labels of the preceding nodes, a consecution Γ I 𝜑 ∈ AS, and an evaluation ℎ, such that 𝑒[Γ] = 𝑍 and 𝑒(𝜑) = 𝑧. Without loss of generality, we could assume that the variables 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 do not occur3 in Γ ∪ {𝜑} and so we can set 𝑒(𝑞) = 𝑦 and 𝑒(𝑟 𝑖 ) = 𝑎 𝑖 for every 𝑖 ∈ {1, . . . , 𝑛}. Thus, 𝑒[Γ 5 𝑞] ⊆ 𝑍 5A 𝑦 ⊆ Fi(𝑋 5A 𝑦) (the last inclusion follows from the induction 3 We could define a new suitable Γ I 𝜑 with the same properties using a Hilbert-hotel-style argument; indeed, consider any enumeration of the variables such that 𝑝0 = 𝑞, 𝑝𝑖 = 𝑟𝑖 , a substitution 𝜎 ( 𝑝𝑖 ) = 𝑝𝑖+𝑛+1 , and an evaluation 𝑒0 such that 𝑒0 ( 𝜎 𝑝) = ℎ ( 𝑝). Then, 𝜎 [Γ] I 𝜎 𝜑 is the needed consecution; indeed, 𝜎 [Γ] I 𝜎 𝜑 ∈ A S, 𝑒0 [ 𝜎 [Γ] ] = 𝑍 , and 𝑒0 ( 𝜎 𝜑) = 𝑧. Note that we have used our assumption that axiomatic systems are closed under substitutions.

5.2 Characterizations of proof by cases properties via 5-forms

271

assumption). From the characterization of the sPCP in Theorem 5.2.6, we know that Γ 5 𝑞 `L 𝛿(𝜑, 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 ) and, therefore, 𝛿A (𝑧, 𝑦, 𝑎 1 , . . . , 𝑎 𝑛 ) = 𝑒(𝛿(𝜑, 𝑞, 𝑟 1 , . . . , 𝑟 𝑛 )) ∈ Fi(𝑋 5A 𝑦). Now we can finally use the claim C to prove that Fi(𝑋) ∩ Fi(𝑌 ) ⊆ Fi(𝑋 5A 𝑌 ). Indeed, from 𝑧 ∈ Fi(𝑋) ∩ Fi(𝑌 ), we obtain the following two inclusions: 𝑧 5A 𝑦 ⊆ Fi(𝑋 5A 𝑦) for each 𝑦 ∈ 𝑌

𝑧 5A 𝑧 ⊆ Fi(𝑌 5A 𝑧).

Due to the validity of (C 5 ) and (I 5 ) and the monotonicity of the operator Fi(), these two inclusions entail the following facts: Ø 𝑌 5A 𝑧 = 𝑦 5A 𝑧 ⊆ Fi(𝑋 5A 𝑌 ) 𝑧 ∈ Fi(𝑌 5A 𝑧). 𝑦 ∈𝑌

As the former claim clearly implies Fi(𝑌 5A 𝑧) ⊆ Fi(𝑋 5A 𝑌 ), the proof is done.  Observe that, as in the syntactical case, the equality proved in this theorem can be equivalently replaced by FiAL (𝑍, 𝑋) ∩ FiAL (𝑍, 𝑌 ) = FiAL (𝑍, 𝑋 5A 𝑌 ). Let us conclude this section by proving, in a certain setting, an analog of the previous theorem for p-disjunctions, i.e. the transferred proof by cases property. Obviously, we get it for logics with the IPEP where any p-disjunction is strong, but we can also prove it in the incomparable setting of weakly implicative logics. Of course,we have to prove it in a rather different way as the proof of the previous theorem was based on the characterization of strong p-disjunctions from Theorem 5.2.6. Theorem 5.2.10 (Transfer of PCP) Let L be a weakly implicative logic with a p-disjunction 5. Then, for each L-algebra A and each 𝑋 ∪ {𝑎, 𝑏} ⊆ 𝐴, FiAL (𝑋, 𝑎) ∩ FiAL (𝑋, 𝑏) = FiAL (𝑋, 𝑎 5A 𝑏). Proof To prove the non-trivial inclusion, we show that, for each 𝑡 ∉ Fi(𝑋, 𝑎 5A 𝑏), we have 𝑡 ∉ Fi(𝑋, 𝑎) or 𝑡 ∉ Fi(𝑋, 𝑏). We use the notion of natural extension of a given logic studied in Section 3.1. Since L is assumed to be weakly implicative, we know that it is a logic in a countable language L over a countable set of variables Var. Let us set Var 0 = Var ∪ {𝑣 𝑎 | 𝑎 ∈ 𝐴} (of course, assume that none of the 𝑣 𝑎 s is in Var). Recall that we denote by FmL0 the 0 new set of formulas and by LVar the natural extension of L to the set of variables Var 0. 0 Var We start by showing that L has the PCP. Note that, due to the possible presence of parameters and since we are working with logics with two different sets of variables, we have to be careful when speaking about 𝜑 5 𝜓; hence, we use the Var0 systematic notation 𝜑 5FmL 𝜓 and 𝜑 5FmL 𝜓. We assume that Γ, 𝜑 `LVar0 𝜒 and Γ, 𝜓 `LVar0 𝜒 and use point 4 of Proposition 3.1.7 to obtain a countable set Γ0 ⊆ Γ and

272

5 Generalized disjunctions

a bijection 𝜏 : Var 0 → Var 0 such that4 𝜏[Γ0 ∪ {𝜑, 𝜓, 𝜒}] ⊆ FmL , 𝜏[Γ0 ], 𝜏𝜑 `L 𝜏 𝜒, and 𝜏[Γ0 ], 𝜏𝜓 `L 𝜏 𝜒 (we use the same symbol for the bijection on Var 0 and the FmL0 -substitution it induces, analogously for its inverse 𝜏 −1 ). Thus, due to the PCP of L, we obtain 𝜏[Γ0 ], 𝜏𝜑 5FmL 𝜏𝜓 `L 𝜏 𝜒. Therefore, in order to be able to use Var0 point 4 of Proposition 3.1.7 again to obtain Γ, 𝜑 5FmL 𝜓 `LVar0 𝜒, all we have to do is to show that Var0 𝜏𝜑 5FmL 𝜏𝜓 ⊆ 𝜏[𝜑 5FmL 𝜓], which is easy because for each 𝛿0 (𝜏𝜑, 𝜏𝜓, 𝛼1 , . . . , 𝛼𝑛 ) ∈ 𝜏𝜑 5FmL 𝜏𝜓, Var0

𝛿0 (𝜏𝜑, 𝜏𝜓, 𝛼1 , . . . , 𝛼𝑛 ) = 𝜏(𝛿0 (𝜑, 𝜓, 𝜏 −1 𝛼1 , . . . , 𝜏 −1 𝛼𝑛 )) ∈ 𝜏[𝜑 5FmL 𝜓]. Now we are ready to proceed to show the transferred proof by cases property. Let us define, for any 𝑛-ary connective 𝑐 of L, the sets: Γ𝑐 = {𝑐(𝑣 𝑧1 , . . . , 𝑣 𝑧𝑛 ) ⇔ 𝑣 𝑐A (𝑧1 ,...,𝑧𝑛 ) | 𝑧1 , . . . , 𝑧 𝑛 ∈ 𝐴} Ø Γ = {𝑣 𝑧 | 𝑧 ∈ Fi(𝑋, 𝑎 5A 𝑏)} ∪ Γ𝑐 . 𝑐 ∈𝐶L 0

Clearly, Γ, 𝑣 𝑎 5 𝑣 𝑏 0 𝑣 𝑡 (because hA, Fi(𝑋, 𝑎 5A 𝑏)i ∈ Mod(L) = Mod(LVar ) and, for the A-evaluation 𝑒(𝑣 𝑧 ) = 𝑧, we obtain 𝑒[Γ ∪ 𝑣 𝑎 5 𝑣 𝑏 ] ⊆ Fi(𝑋, 𝑎 5A 𝑏) and 𝑒(𝑣 𝑡 ) ∉ Fi(𝑋, 𝑎 5A 𝑏)). Thus, by the PCP, we have Γ, 𝑣 𝑎 0 𝑣 𝑡 or Γ, 𝑣 𝑏 0 𝑣 𝑡 . Assume (without loss of generality) the former case and set 𝑇 = ThLVar0 (Γ, 𝑣 𝑎 )

𝜃 = ΩFmVar0 (𝑇) L

0

B = FmVar L /𝜃.

Note that, due the fact that Γ𝑐 ⊆ 𝑇, we know that, for each 𝑧1 , . . . , 𝑧 𝑛 ∈ 𝐴, 𝑣 𝑐A (𝑧1 ,...,𝑧𝑛 ) /𝜃 = 𝑐(𝑣 𝑧1 , . . . , 𝑣 𝑧𝑛 )/𝜃 and thus the mapping ℎ : A → B defined as ℎ(𝑧) = 𝑣 𝑧 /𝜃 is a homomorphism; indeed: ℎ(𝑐A (𝑧1 , . . . , 𝑧 𝑛 )) = 𝑣 𝑐A (𝑧1 ,...,𝑧𝑛 ) /𝜃 = 𝑐(𝑣 𝑧1 , . . . , 𝑣 𝑧𝑛 )/𝜃 = 𝑐B (𝑣 𝑧1 /𝜃, . . . , 𝑣 𝑧𝑛 /𝜃) = 𝑐B (ℎ(𝑧 1 ), . . . , ℎ(𝑧 𝑛 )). Thus, 𝐹 = ℎ−1 [𝑇] ∈ FiLVar0 (A) = FiL (A) and, since clearly 𝑋 ∪ {𝑎} ⊆ 𝐹 and 𝑡 ∉ 𝐹, we have established that 𝑡 ∉ Fi(𝑋, 𝑎).  Let us mention an interesting consequence of the previous two theorems: if L is a logic with a finite disjunction 5 such that L is weakly implicative or 5 is strong, we know that the intersection of finitely generated filters is finitely generated and, thus, such filters are the domain of a sublattice of the lattice of all filters. 4 Strictly speaking, Proposition 3.1.7 directly guarantees the existence of a bijection for each of our two assumptions, but inspecting its proof one can see that we can assume that these two bijections are equal. We leave the elaboration of the details as an exercise for the reader.

5.3 Generalized disjunctions and properties of the lattice of filters

273

5.3 Generalized disjunctions and properties of the lattice of filters The goal of this section is to provide characterizations of generalized disjunctions using various forms of distributivity of the lattice of theories/filters. Let us however start with a theorem of a different nature: a characterization of weakly (p-)disjunctional logics which, among others, identifies a particular weak (p-)disjunction present in all weakly (p-)disjunctional logics. Theorem 5.3.1 (Characterization of weakly (p-)disjunctional logics) Let L be a logic and 𝑝 and 𝑞 two fixed distinct atoms. Then, the properties 1–4 are equivalent and so are the properties 10–40: 1) L is weakly p-disjunctional.

10) L is weakly disjunctional.

2) For each surjective substitution 𝜎 and any pair 𝜑, 𝜓 of formulas,

20) For each substitution 𝜎 and any pair 𝜑, 𝜓 of formulas,

ThL (𝜎𝜑) ∩ ThL (𝜎𝜓) = ThL (𝜎[ThL (𝜑) ∩ ThL (𝜓)]). 3) For each surjective substitution 𝜎,

30) For each substitution 𝜎,

ThL (𝜎 𝑝) ∩ ThL (𝜎𝑞) = ThL (𝜎[ThL ( 𝑝) ∩ ThL (𝑞)]). 4) The set 51 is a p-disjunction: 51 = ThL ( 𝑝) ∩ ThL (𝑞).

40) The set 52 is a disjunction: 52 = 𝜏[ThL ( 𝑝) ∩ ThL (𝑞)], ( 𝑝 if 𝑟 = 𝑝 where 𝜏𝑟 = 𝑞 otherwise.

Proof Let us first notice that the implications from 2 to 3 and from 4 to 1 are trivial and analogously for their primed versions. In order to prove that 1 implies 2 and 10 implies 20, let us consider a (p-)protodisjunction 5 and observe that, if 𝜎 is surjective or 5 is parameter-free, we have 𝜎𝜑 5 𝜎𝜓 = 𝜎[𝜑 5 𝜓]. Indeed, in the second case it is trivial and in the first one we can write this chain of equations: 𝜎𝜑 5 𝜎𝜓 = {𝛿(𝜎𝜑, 𝜎𝜓, 𝛼) | 𝛿( 𝑝, 𝑞, 𝑟) ∈ 5, 𝛼 ∈ FmL } = {𝛿(𝜎𝜑, 𝜎𝜓, 𝜎𝛽) | 𝛿( 𝑝, 𝑞, 𝑟) ∈ 5, 𝛽 ∈ FmL } = 𝜎[𝜑 5 𝜓]. Thus, we can write, in both cases, the following chain of equations (the first and the last one are due to the fact that 5 is a weak p-disjunction, the second was just established, and the third one is an instance of the general fact ThL (𝜎[Γ]) = ThL (𝜎[ThL (Γ)])): ThL (𝜎𝜑) ∩ ThL (𝜎𝜓) = ThL (𝜎𝜑 5 𝜎𝜓) = ThL (𝜎[𝜑 5 𝜓]) = ThL (𝜎[ThL (𝜑 5 𝜓)]) = ThL (𝜎[ThL (𝜑) ∩ ThL (𝜓)]).

274

5 Generalized disjunctions

Let us now prove the final two implications. Clearly, 51 is a p-protodisjunction and 52 is a protodisjunction, and thus we only have to check that they satisfy the wPCP. To this end, let us consider any surjective substitution 𝜎1 such that 𝜎1 𝑝 = 𝜑 and 𝜎1 𝑞 = 𝜓 and define 𝜎2 = 𝜎1 ◦ 𝜏. Note that 𝜎2 is not surjective and we still have 𝜎2 𝑝 = 𝜑 and 𝜎2 𝑞 = 𝜓. Therefore, for 𝑖 being 1 or 2, 𝜎𝑖 [5𝑖 ] = 𝜎1 [ 𝑝 5𝑖 𝑞] ⊆ 𝜑 5𝑖 𝜓 and so we can write the following of chain equations/inclusions: ThL (𝜑) ∩ ThL (𝜓) = ThL (𝜎𝑖 𝑝) ∩ ThL (𝜎𝑖 𝑞) = ThL (𝜎𝑖 [ThL ( 𝑝) ∩ ThL (𝑞)]) ⊆ ThL (𝜑 5𝑖 𝜓).



By virtue of Proposition 5.1.5, we immediately obtain the following characterization of (strongly) p-disjunctional logics (note that we use an alternative but clearly equivalent description of the disjunction in the non-parameterized case). Corollary 5.3.2 A logic is (strongly) p-disjunctional if and only if the set ThL ( 𝑝) ∩ ThL (𝑞) is one of its (strong) p-disjunctions. Furthermore, a logic is (strongly) disjunctional if and only if the following set is one of its (strong) disjunctions: ( 𝑝 for r = p 𝜏[ThL ( 𝑝) ∩ ThL (𝑞)] for 𝜏𝑟 = 𝑞 otherwise. Now we are ready for the core of this section. Recall that we denote by FiL (A) the set of L-filters on an algebra A; FiL (A) is a closure system with associated closure operator FiAL and it is the domain of a complete lattice FiAL = hFiL (A), ∧, ∨, FiAL (∅), 𝐹i, where 𝐹 ∧ 𝐺 = 𝐹 ∩ 𝐺, and 𝐹 ∨ 𝐺 = FiAL (𝐹, 𝐺). Also recall that, for any system X of subsets of 𝐴, Ü Ø FiAL (𝑋) = FiAL ( X). 𝑋 ∈X

Definition 5.3.3 A logic L is filter-distributive if for each L-algebra A, the lattice FiAL is distributive. A logic L is filter-framal if for each L-algebra A, the lattice FiAL is a frame, i.e. for each F ∪ {𝐺} ⊆ FiL (A), Ü Ü 𝐺∩ 𝐹= (𝐺 ∩ 𝐹). 𝐹 ∈F

𝐹 ∈F

We omit the prefix ‘filter-’ whenever the corresponding property holds for A = FmL (i.e. for the lattice of L-theories).

5.3 Generalized disjunctions and properties of the lattice of filters

275

Remark 5.3.4 Recall that, for any algebraically implicative logic L, the lattice of L-filters FiL (A) is isomorphic with the lattice of relative congruences ConAlg∗ (L) (A). Therefore, in all subsequent results, whenever we assumed that the logic L in question is algebraically implicative, we could replace the condition that the logic L is filterdistributive/framal by the equivalent condition that the class Alg∗ (L) is relatively congruence distributive/framal. Congruence distributivity is a prominent algebraic property related to numerous others. It is known to be always true whenever Alg∗ (L) is a variety of algebras with lattice reducts, but it is rather elusive in quasivarieties; see Example 5.3.11 for possible links between these algebraic facts and our results. At the end of this section we show (in Corollary 5.3.9) the transfer of framality in weakly implicative and weakly p-disjunctional logics and (in Corollary 5.3.10) that if such logic enjoys the IPEP, then its distributivity implies framality (and, thus, vicariously we also show the transfer of distributivity in this setting). Note that the fact that distributivity implies framality can actually be proved very straightforwardly if we assume that the logic in question is finitary. Proposition 5.3.5 Every (filter-)distributive finitary logic is (filter-)framal. Proof Clearly, one of the inclusions in the definition of filter-framal logic is valid in all complete lattices; let usÔprove the other one. Ð Assume that 𝑥 ∈ 𝐺 ∩ 𝐹 ∈ F 𝐹, i.e. 𝑥 ∈ 𝐺 and 𝑥 ∈ FiAL ( F ). Due to finitarity, Ð Ð there must be finite sets F 0 ⊆ F and 𝑋 ⊆ F 0 such that 𝑥 ∈ FiAL ( F 0). Therefore, due to filter-distributivity, we have: Ü Ü Ü 𝑥∈𝐺∩ 𝐹0 = (𝐺 ∩ 𝐹) ⊆ (𝐺 ∩ 𝐹).  𝐹 ∈F 𝐹 ∈ F0 𝐹 ∈F In order to explore the relationship between generalized disjunctions and various forms of distributivity, let us start by showing that a logic with a (strong) p-disjunction always satisfies a certain restricted form of distributivity (framality). Note that the left-hand sides of both equations in the upcoming proposition can obviously be written in more compact ways (we use our notation to stress the relation to distributivity/framality): (𝑇 ∨ ThL (𝜑)) ∩ (𝑇 ∨ ThL (𝜓)) = ThL (𝑇, 𝜑) ∩ ThL (𝑇, 𝜓) Ü 𝑇∩ ThL (𝜑) = 𝑇 ∩ ThL (Γ). 𝜑 ∈Γ

Proposition 5.3.6 A logic L is • p-disjunctional iff it is weakly p-disjunctional and, for each theory 𝑇 and any pair 𝜑, 𝜓 of formulas, (𝑇 ∨ ThL (𝜑)) ∩ (𝑇 ∨ ThL (𝜓)) = 𝑇 ∨ (ThL (𝜑) ∩ ThL (𝜓)).

276

5 Generalized disjunctions

• strongly p-disjunctional iff it is weakly p-disjunctional and, for each theory 𝑇 and each set Γ, Ü Ü 𝑇∩ ThL (𝜑) = (𝑇 ∩ ThL (𝜑)). 𝜑 ∈Γ

𝜑 ∈Γ

Proof All implications in both claims are proved by simple chains of equations using the characterizations of generalized disjunctions provided in Proposition 5.1.3. The left-to-right direction of the first claim: ThL (𝑇, 𝜑) ∩ ThL (𝑇, 𝜓) = ThL (𝑇, 𝜑 5 𝜓) = ThL (𝑇) ∨ ThL (𝜑 5 𝜓) = 𝑇 ∨ (ThL (𝜑) ∩ ThL (𝜓)). The left-to-right direction of the second claim: 𝑇 ∩ ThL (Γ) = ThL (𝑇 5 Γ)

Ð

= ThL (

(𝑇 5 𝜑))

𝜑 ∈Γ

= ThL (

Ð 𝜑 ∈Γ

=

Ô

Ð

ThL (𝑇 5 𝜑)) = ThL (

(ThL (𝑇) ∩ ThL (𝜑)))

𝜑 ∈Γ

(𝑇 ∩ ThL (𝜑)).

𝜑 ∈Γ

The converse direction of the first claim: ThL (Γ, 𝜑) ∩ ThL (Γ, 𝜓) = ThL (ThL (Γ), 𝜑) ∩ ThL (ThL (Γ), 𝜓) = ThL (Γ) ∨ (ThL (𝜑) ∩ ThL (𝜓)) = ThL (Γ) ∨ ThL (𝜑 5 𝜓) = ThL (Γ, 𝜑 5 𝜓). The converse direction of the second claim (using Proposition 5.1.3): Ü ThL (Φ) ∩ ThL (Ψ) = (ThL (Φ) ∩ ThL (𝜓)) 𝜓 ∈Ψ

=

Ü

ThL (𝜑) ∩ ThL (𝜓)

𝜑 ∈Φ, 𝜓 ∈Ψ

=

Ü

ThL (𝜑 5 𝜓)

𝜑 ∈Φ, 𝜓 ∈Ψ

= ThL (

Ø

ThL (𝜑 5 𝜓))

𝜑 ∈Φ, 𝜓 ∈Ψ

= ThL (

Ø

𝜑 5 𝜓)

𝜑 ∈Φ, 𝜓 ∈Ψ

= ThL (Φ 5 Ψ).



5.3 Generalized disjunctions and properties of the lattice of filters

277

Interestingly enough, we do not know whether every p-disjunctional logic is distributive. However, we will prove (in Theorem 5.3.8) that every strongly pdisjunctional logic is framal (even filter-framal), and thus, when restricted to logics with the IPEP, the p-disjunctional ones are distributive. Let us first prove a stronger version of the previous proposition for weakly implicative logics where, as will see, we can drop the assumption that the logic in question is weakly p-disjunctional. Proposition 5.3.7 A weakly implicative logic L is • p-disjunctional iff for each theory 𝑇 and each pair 𝜑, 𝜓 of formulas, (𝑇 ∨ ThL (𝜑)) ∩ (𝑇 ∨ ThL (𝜓)) = 𝑇 ∨ (ThL (𝜑) ∩ ThL (𝜓)). • strongly p-disjunctional iff for each theory 𝑇 and set Γ, Ü Ü 𝑇∩ ThL (𝜑) = (𝑇 ∩ ThL (𝜑)). 𝜑 ∈Γ

𝜑 ∈Γ

Proof The left-to-right directions of both claims follow from the previous proposition. To prove the converse direction of the first claim, we use Theorem 5.3.1 to show that any weakly implicative logic satisfying our assumption is weakly p-disjunctional and so the previous proposition completes the proof. Thus, our goal now is to show that, for a fixed pair of variables 𝑝, 𝑞 and each surjective substitution 𝜎, ThL (𝜎 𝑝) ∩ ThL (𝜎𝑞) = ThL (𝜎[ThL ( 𝑝) ∩ ThL (𝑞)]). We start by defining a theory 𝑌 = 𝜎 −1 [ThL (∅)] and observe that the mapping 𝜎 : hFmL , 𝑌 i → hFmL , ThL (∅)i is a strict surjective homomorphism and thus, by Proposition 3.2.15, we know that the mapping 𝝈 defined as 𝝈(𝑇) = 𝜎[𝑇] for each theory 𝑇 ⊇ 𝑌 is a lattice isomorphism between [𝑌 , FmL ] and Th(L) (recall that by [𝑌 , FmL ] we denote the sublattice of Th(L) of theories containing 𝑌 ). Next, we prove that ThL (𝜎[Σ]) = 𝝈(𝑌 ∨ ThL (Σ)) for any set of formulas Σ. The first inclusion follows from the inclusion 𝜎[Σ] ⊆ 𝝈(𝑌 ∨ ThL (Σ)) and the fact that 𝝈(𝑌 ∨ ThL (Σ)) is a theory. In order to prove the second inclusion, assume that 𝜒 ∈ 𝝈(𝑌 ∨ ThL (Σ)) = 𝝈(ThL (𝑌 ∪ Σ)); therefore, we know that 𝜒 = 𝜎𝛿 for some 𝛿 such that 𝑌 , Σ `L 𝛿. Thus, 𝜎[𝑌 ], 𝜎[Σ] `L 𝜎(𝛿) and, since 𝜎[𝑌 ] = ThL (∅), we obtain 𝜎[Σ] `L 𝜒. Now we can finish the proof of the first claim by the following series of equations: ThL (𝜎 𝑝) ∩ ThL (𝜎𝑞) = 𝝈(𝑌 ∨ ThL ( 𝑝)) ∩ 𝝈(𝑌 ∨ ThL (𝑞)) = 𝝈((𝑌 ∨ ThL ( 𝑝)) ∩ (𝑌 ∨ ThL (𝑞))) = 𝝈(𝑌 ∨ (ThL ( 𝑝) ∩ ThL (𝑞))) = ThL (𝜎[ThL ( 𝑝) ∩ ThL (𝑞)]). To prove the reverse implication of the second claim, it suffices to show that our assumption implies the right-hand side of the first claim. Then, we will know that L is p-disjunctional and so we can use the previous proposition to complete the proof.

278

5 Generalized disjunctions

We prove it by the following chain of equations: Ü ThL (𝑇, 𝜑) ∩ ThL (𝑇, 𝜓) = (ThL (𝑇, 𝜑) ∩ ThL ( 𝜒)) 𝜒 ∈𝑇 ∪{𝜓 }

=(

Ü

(ThL (𝑇, 𝜑) ∩ ThL ( 𝜒))) ∨ (ThL (𝑇, 𝜑) ∩ ThL (𝜓))

𝜒 ∈𝑇

=(

Ü

ThL ( 𝜒)) ∨ (ThL (𝑇, 𝜑) ∩ ThL (𝜓))

𝜒 ∈𝑇

= 𝑇 ∨ (ThL (𝜓) ∩ ThL (𝑇, 𝜑)) Ü =𝑇 ∨ (ThL (𝜓) ∩ ThL ( 𝜒)) 𝜒 ∈𝑇 ∪{ 𝜑 }

=𝑇 ∨(

Ü

(ThL (𝜓) ∩ ThL ( 𝜒))) ∨ (ThL (𝜓) ∩ ThL (𝜑))

𝜒 ∈𝑇

= 𝑇 ∨ (ThL (𝜓) ∩ ThL (𝜑)).



Now we are ready to prove the main theorem of this section, draw from it several interesting corollaries, and give an example demonstrating its utility for establishing that a given logic is p-disjunctional or that a given class of algebras is congruence-distributive (cf. Remark 5.3.4). Theorem 5.3.8 Let L be a weakly implicative or weakly p-disjunctional logic. Then, the following properties are equivalent: 1. L is strongly p-disjunctional. 2. L is filter-framal. 3. L is framal. Furthermore, if L has the IPEP, we can add the following equivalent conditions: 4. L is p-disjunctional. 5. L is filter-distributive. 6. L is distributive. Proof To prove that 1 implies 2, we use Theorem 5.2.9, which entails Fi(𝑋) ∩ Fi(𝑌 ) = Fi(𝑋 5 𝑌 ). Therefore, for any set {𝐹} ∪ G of filters, we can write the following chain of equations: Ü Ø Ø 𝐹∩ 𝐺 = Fi(𝐹) ∩ Fi( G) = Fi(𝐹 5 G) 𝐺∈G

= Fi(

Ø 𝐺∈G

= Fi(

Ø 𝐺∈G

(𝐹 5 𝐺)) = Fi(

Ø

Fi(𝐹 5 𝐺))

𝐺∈G

(Fi(𝐹) ∩ Fi(𝐺))) =

Ü 𝐺∈G

(𝐹 ∩ 𝐺).

5.4 5-prime theories and 5-prime extension property

279

To complete the proof of the equivalence of the first three claims, it suffices to observe that every filter-framal logic is obviously framal, and every framal logic satisfies a particular instance of framality which is shown in Propositions 5.3.6 and 5.3.7 to be a sufficient condition for the strong p-disjunctionality of the logic in question. Next, observe that the chain of implications ‘2 implies 5 implies 6 implies 4’ holds without any additional assumptions about L (the first two implications are obvious, the last one follows from Propositions 5.3.6 and 5.3.7). Thus, to complete the proof, we just recall that, due to Proposition 5.1.4, we know that each p-disjunctional logic with the IPEP is strongly p-disjunctional.  Corollary 5.3.9 (Transfer of framality) Let L be a weakly implicative or weakly p-disjunctional logic. If L is framal, then it is filter-framal. Corollary 5.3.10 (Distributivity implies framality) Let L be a weakly implicative logic or weakly p-disjunctional logic. If L has the IPEP and is distributive, then it is filter-framal.

Example 5.3.11 We can use the previous theorem to obtain that the logic SL is strongly p-disjunctional. In Section 4.4, we have established that it is algebraically implicative and the class Alg∗ (SL) is a variety of algebras with lattice reducts and so it is congruence-distributive. Thus, in the light of Remark 5.3.4, we know that SL is filter-distributive and, as SL is finitary, the previous theorem indeed tells us that it is strongly p-disjunctional. Conversely, we can use our results to give an alternative proof of certain known algebraic facts. In Example 5.1.14, we have shown that the logic IL→ is strongly p-disjunctional logic and, in the subsequent text, we have shown the same for the logic BCK. As both of these logics are Rasiowa-implicative, we obtain that Alg∗ (IL→ ) (whose elements are called Hilbert algebras) and Alg∗ (BCK) are congruence distributive classes. Note that the algebras in neither of these classes do not have lattice reducts and, while Alg∗ (IL→ ) is a variety, Alg∗ (BCK) is not (it is known to be a proper quasivariety).

5.4 5-prime theories and 5-prime extension property Recall that in Section 3.5 we have defined, for logics over BCIlat , the notion of prime theory (if, for each pair 𝜑, 𝜓 of formulas, we have that 𝜑 ∨ 𝜓 ∈ 𝑇 implies that 𝜑 ∈ 𝑇 or 𝜓 ∈ 𝑇) and showed in Proposition 3.5.21 that prime theories are always intersection-prime, and the converse direction holds in a logic with the IPEP iff L has the classical proof by cases property iff prime theories for a basis of Th(L). The goal of this section is to study these notions in our abstract setting and, among others, prove a general version of the mentioned results.

280

5 Generalized disjunctions

Definition 5.4.1 (Prime filters and matrices and Prime Extension Property) Let L be a logic in language L with a p-protodisjunction 5 and A an L-algebra. We say that a filter 𝐹 ∈ FiL (A) is 5-prime (as customary, if A = FmL , we speak about 5-prime theories), if for every 𝑎, 𝑏 ∈ 𝐴,5 𝑎 5A 𝑏 ⊆ 𝐹

implies

𝑎 ∈ 𝐹 or 𝑏 ∈ 𝐹.

If, furthermore, the matrix A = hA, 𝐹i is reduced, we say that A is 5-prime and write p A ∈ Mod 5 (L). Finally, we say that L has the 5-prime extension property, 5PEP for short, if 5-prime theories form a basis of the closure system Th(L). By Proposition 5.1.5, we know that, for any two weak disjunctions 5 and 5 0, 5-prime and 5 0-prime filters coincide. Observe, the following two lemmas imply, among others, that, if 5 is a strong p-disjunction in a given logic, then its 5-prime and intersection-prime filters coincide and furthermore that any logic with the 5PEP has the IPEP and 5 is one of its strong p-disjunctions. Lemma 5.4.2 Let L be a logic in a language L with a p-protodisjunction 5. • Every 5-prime filter in an arbitrary L-algebra is intersection-prime and, therefore, p Mod 5 (L) ⊆ Mod∗ (L)RFSI . • If 5 is a p-disjunction, then intersection-prime theories and 5-prime theories coincide. • If 5 is a strong p-disjunction or L is a weakly implicative logic where 5 is a p-disjunction, then in any L-algebra intersection-prime and 5-prime filters p coincide and so Mod 5 (L) = Mod∗ (L)RFSI .6 Proof Note that the claims about RFSI matrices are direct consequences of the characterization provided by Theorem 3.6.6. To prove the first claim, assume that 𝐹 is not intersection-prime; i.e. 𝐹 = 𝐹1 ∩ 𝐹2 for some 𝐹𝑖 ) 𝐹. Let us consider 𝑎 𝑖 ∈ 𝐹𝑖 \ 𝐹. Thus, by (PD), we know that 𝑎 1 5A 𝑎 2 ⊆ 𝐹𝑖 and so 𝑎 1 5A 𝑎 2 ⊆ 𝐹, i.e. 𝐹 is not 5-prime. To prove the second claim, consider any 𝑇 ∈ Th(L) and assume first that 𝑇 is not 5-prime, i.e. there are 𝜑 ∉ 𝑇 and 𝜓 ∉ 𝑇 such that 𝜑 5 𝜓 ⊆ 𝑇. By the PCP, we know that 𝑇 = ThL (𝑇, 𝜑 5 𝜓) = ThL (𝑇, 𝜑) ∩ ThL (𝑇, 𝜓), i.e. 𝑇 is the intersection of two strictly bigger theories. The proof of the final claim is analogous, by using the transferred PCP implied either by Theorem 5.2.9 or by Theorem 5.2.10.  Lemma 5.4.3 Let L be a logic with a p-protodisjunction 5 enjoying the 5PEP. Then, 5 is a strong p-disjunction and L enjoys the IPEP. 5 Note that the fact that 5 a p-protodisjunction entails that the converse implication is always satisfied. 6 Note that we could formulate this claim with the weaker assumption of 5 satisfying the transferred PCP; however, we prefer a simpler formulation which is sufficient for our needs.

5.4 5-prime theories and 5-prime extension property

281

Proof Assume that Φ 5 Ψ 0L 𝜒. Then, using our assumption, there has to be a 5-prime theory 𝑇 ⊇ ThL (Φ 5 Ψ) such that 𝑇 0L 𝜒. First, assume that Φ ⊆ 𝑇. Then, Φ 0L 𝜒 and the proof is done. Otherwise, assume that there is a 𝜑 ∈ Φ \ 𝑇. Since 𝜑 5 𝜓 ⊆ 𝑇 for each 𝜓 ∈ Ψ and 𝑇 is 5-prime, we obtain that Ψ ⊆ 𝑇 and so Ψ 0L 𝜒. The fact that L enjoys the IPEP follows from Lemma 5.4.2.  Let us continue exploring the properties of 5-prime filters. Recall that in Proposition 3.5.5 we showed that, for each strict surjective homomorphism ℎ : hA, 𝐹i → hB, 𝐺i, if 𝐹 is intersection-prime, then so is 𝐺, and in weakly implicative logics we even have an equivalence. We show in Proposition 5.4.5 that the same equivalence holds in all strongly p-disjunctional logics (which we know is a class of logics incomparable with that of weakly implicative logics; cf. Remark 5.1.8). Let us first prove a simple lemma showing that an analog of the tricky implication (the one for which we need to assume that L is weakly implicative) formulated for 5-prime filters is true in a much wider setting. Lemma 5.4.4 Let L be a logic with a (p-)protodisjunction 5 and ℎ : hA, 𝐹i → hB, 𝐺i a strict (surjective) homomorphism for hA, 𝐹i, hB, 𝐺i ∈ Mod(L). If 𝐺 is 5-prime, then so is 𝐹. Proof Assume that 𝐹 = ℎ−1 [𝐺] is not 5-prime, i.e. there are 𝑎, 𝑏 ∉ ℎ−1 [𝐺] and 𝑎 5A 𝑏 ⊆ ℎ−1 [𝐺]. Thus, ℎ(𝑎), ℎ(𝑏) ∉ 𝐺 and ℎ[𝑎 5B 𝑏] ⊆ 𝐺. Using that ℎ is surjective or that 5 has no parameters, we obtain ℎ(𝑎) 5B ℎ(𝑏) = ℎ[𝑎 5A 𝑏] and, thus, 𝐺 is not 5-prime.  Using this lemma together with Propositions 5.4.2 (to know that 5-prime and intersection-prime filters coincide) and 3.5.5 (to obtain the converse implication), we easily obtain the promised variant of Proposition 3.5.5. Proposition 5.4.5 Let L be a logic with a strong p-disjunction 5 and ℎ : hA, 𝐹i → hB, 𝐺i a strict surjective homomorphism for hA, 𝐹i, hB, 𝐺i ∈ Mod(L). Then, 𝐺 is 5-prime (intersection-prime) iff 𝐹 is 5-prime (intersection-prime). Before we state and prove (in Theorem 5.4.7) the promised abstract rendering of Proposition 3.5.21, we need to prepare one final ingredient: the relation between the p 5PEP and strong completeness w.r.t. the class Mod 5 (L) (see also Corollary 6.2.4). Proposition 5.4.6 (5-prime completeness) Let L be a logic with a p-protodisjunction p 5. If L has the 5PEP, then L is strongly complete w.r.t. the class Mod 5 (L). If 5 is parameter-free, the converse implication holds as well. p

Proof From Lemmas 5.4.3 and 5.4.2, we know that L enjoys the IPEP and Mod 5 (L) = Mod∗ (L)RFSI . The former claim and Theorem 3.6.14 imply that L is RFSI-complete and so the latter claim completes the proof. To prove the converse claim, assume that 𝑇 0L 𝜒. Therefore, there is a matrix p hA, 𝐹i ∈ Mod 5 (L) and 𝑒 such that 𝑒[𝑇] ⊆ 𝐹 and 𝑒( 𝜒) ∉ 𝐹. Define 𝑇 0 = 𝑒 −1 [𝐹]. Clearly, 𝑇 0 is a theory, 𝑇 0 ⊇ 𝑇, and 𝜒 ∉ 𝑇 0 and, by Lemma 5.4.4, 𝑇 0 is 5-prime. 

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5 Generalized disjunctions

Theorem 5.4.7 Let 5 be a p-protodisjunction in a logic L. Then, the following are equivalent: 1. L has the 5PEP. 2. L has the IPEP and any of the following (in this context equivalent) conditions holds: a. 5 is a p-disjunction. b. 5 is a strong p-disjunction. c. In every L-algebra, the intersection-prime and the 5-prime filters coincide. d. The intersection-prime and the 5-prime theories coincide. If 5 is parameter-free, we can add the following equivalent claims: 3. L is RFSI-complete and 5 is a strong disjunction. p 4. L is RFSI-complete and Mod 5 (L) = Mod∗ (L)RFSI . If, furthermore, L is weakly implicative, we can add the following equivalent claim: 5. L is RFSI-complete and 5 is a disjunction. Proof We prove the equivalence of 1 and 2a–2d by showing the usual circle of implications: the fact that 1 implies 2a follows from Lemma 5.4.3; 2a implies 2b from Proposition 5.1.4, and 2b implies 2c from Lemma 5.4.2. The facts that 2c implies 2d and 2d implies 1 are trivial. To add the conditions 3 and 4, first recall the known fact that the IPEP implies RFSI-completeness and so 2b trivially implies 3 (in all logics) and ‘3 implies 4’ holds (also in all logics) due to Lemma 5.4.2. The final fact, that 4 implies 1, follows from Proposition 5.4.6. To add the final condition 5, observe that ‘3 implies 5’ is trivial and ‘5 implies 4’ follows from Lemma 5.4.2.  Corollary 5.4.8 (Preservation of 5PEP) Let L0 be an axiomatic extension of L. If L has the 5PEP, then so does L0. Proof Thanks to the previous theorem, we know that L is strongly p-disjunctional and has the IPEP. Due to Lemma 3.5.11 and Theorem 5.2.8, we also know that L0 has the IPEP and is strongly p-disjunctional. Therefore, the previous theorem completes the proof.  Corollary 5.4.9 Let L be a strongly disjunctional logic or a weakly implicative disjunctional logic. Then, L is RFSI-complete iff it has the IPEP. Recall that in Example 5.1.10 we have presented an infinitary weakly implicative logic which is lattice-disjunctive but not strongly p-disjunctional, thus, due to Proposition 5.1.4, this logic cannot have the IPEP and, due to the previous corollary, it is not even RFSI-complete. The next example gives another non-RFSI-complete logic which, however, is strongly disjunctive (thus, it is also the promised example of a logic with strong p-disjunction but without the IPEP).

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Example 5.4.10 (A strongly disjunctive logic which is not RFSI-complete) Consider a language L consisting of a binary connective ∨, a unary connective 𝑠, and two constants 0 and 𝜔. Let us denote by 𝑛 the formula defined inductively as (𝑛 + 1) = 𝑠(𝑛). Let L be a logic in the language L axiomatized by the rules (PD), (C∨ ), (I∨ ), and (A∨ ), the ∨-forms of these rules, and the following rule for each infinite subset 𝐶 of natural numbers: (InfC )

{𝑖 ∨ 𝜓 | 𝑖 ∈ 𝐶} I 𝜓.

First, observe that, thanks to (A∨ ), we know that the rule (InfC ) entails its ∨-form and thus, due to Theorem 5.2.6 together with Lemma 5.2.5, we know that ∨ is a strong disjunction in L. To prove that L is not RFSI-complete, it suffices, thanks to Theorem 5.4.7, to show that it does not enjoy the ∨PEP. Consider the following set 𝐴 of subsets of natural numbers (denoted as 𝜔 here): 𝐴 = {𝜔} ∪ {𝑋 ⊆ 𝜔 | 𝑋 finite and for each 𝑖 ∈ 𝜔 : 2𝑖 ∉ 𝑋 or 2𝑖 + 1 ∉ 𝑋 }. Note that 𝐴 is closed under arbitrary intersections of its element and thus we can see it a domain of a complete join-semilattice with ∨ such that ( 𝑋 ∨𝑌 =

Ù

𝑋 ∪𝑌

if 𝑋 ∪ 𝑌 ∈ 𝐴

𝜔

otherwise (i.e. if 2𝑖, 2𝑖 + 1 ∈ 𝑋 ∪ 𝑌 ).

{𝑍 ∈ 𝐴 | 𝑋 ∪𝑌 ⊆ 𝑍 } =

A

Consider an algebra A = h𝐴, ∨, 𝑠, 0, 𝜔i where 0 = {0}, 𝜔A = 𝜔, and 𝑠A ({𝑖}) = {𝑖 + 1} and 𝑠A (𝑋) = ∅ otherwise (note that in particular 𝑛A = {𝑛}). We show that A = hA, {𝜔}i ∈ Mod(L). Let us check the validity of the rules of L in A : • The validity of the rules (C∨ ), (I∨ ), and (A∨ ) and their ∨-forms is straightforward (recall that the join of any sets from A is 𝜔 if and only if its union contains {2𝑖, 2𝑖 + 1} for some 𝑖). • Next, consider a rule (InfC ) and an A-evaluation 𝑒 such that 𝑒(𝜓) ≠ 𝜔. Therefore, 𝑒(𝜓) is finite and, as 𝐶 is infinite, there is an 𝑚 ∈ 𝐶 such that 𝑚 > max(𝑒(𝜓)) + 1 and so 𝑒(𝑚 ∨ 𝜓) = 𝑒(𝜓) ∪ {𝑚} ≠ 𝜔. Next, observe that, for any A evaluation, we have 𝑒(2𝑖 ∨ 2𝑖 + 1) = 𝜔 and thus for the theory 𝑇 generated by the set of formulas {2𝑖 ∨ 2𝑖 + 1 | 𝑖 ∈ 𝜔}, we know that 𝑇 0L 0. Consider any ∨-prime theory 𝑇 0 ⊇ 𝑇 and note that, for each 𝑖, we have 2𝑖 ∈ 𝑇 0 or 2𝑖 + 1 ∈ 𝑇 0. Thus, there is an infinite set 𝐶 such that {𝑖 ∨ 0 | 𝑖 ∈ 𝐶} ⊆ 𝑇 0 and so, by the rule (InfC ), we obtain 𝑇 0 `L 0. Now, given a logic L with a p-protodisjunction 5, we will focus on the logic L5 defined semantically by the reduced 5-prime models of L and explore some of its properties.

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5 Generalized disjunctions

Definition 5.4.11 (The 5-extension L5 ) Let L be a logic with a p-protodisjunction 5. We define the 5-extension L5 of L semantically as L5 = Modp5 (L) . By Proposition 5.4.6, we know that if L has the 5PEP, then L = L5 . Actually, this equality only says that L is strongly complete w.r.t. its own 5-prime models. The next proposition shows that L5 is the least logic extending L with such property and under certain conditions it is the least logic extending L with the 5PEP or where 5 is a (strong) disjunction. Proposition 5.4.12 Let L be a logic with a p-protodisjunction 5. • L5 is the least logic L0 extending L such that L0 = L50 ; thus, in particular, (L5 )5 = L5 . • If 5 is a protodisjunction, then L5 is the least logic with the 5PEP extending L. • If 5 is a finite protodisjunction and L is finitary, then L5 is finitary and it is the least logic extending L in which 5 is a (strong) disjunction. Proof The fact that L5 extends L is obvious. Consider any logic L0 extending L such that L0 = L50 . Then, clearly, L0 = Modp5 (L0 ) ⊇ Modp5 (L) = L5 . p

It remains to show that (L5 )5 ⊆ L5 . Consider any A ∈ Mod 5 (L) and observe that we can easily prove that L5 = Modp5 (L) ⊆ A . p

p

Therefore, Mod 5 (L) ⊆ Mod 5 (L5 ) and so the claim easily follows. The second claim is a direct consequence of the first one and Proposition 5.4.6. To prove the last claim, first note that, thanks to Theorem 5.4.7 and the second claim, we know that 5 is a disjunction in L5 . Next, consider a logic L0 where 5 is a disjunction. We leave as an exercise for the reader to show that the logic F C(L0 ) (the finitary companion of L0 ) has the 5PEP and, as obviously L ⊆ F C(L0 ) ⊆ L0 , then, due to the second claim, we know that L5 ⊆ F C(L0 ) ⊆ L0 . In particular, we also obtain that L5 ⊆ F C(L5 ) and so L5 is finitary.  The two assumptions of the last claim of the previous proposition (finitarity of L and finiteness of 5) may seem quite restrictive, but they are satisfied in the arguably most interesting case: for a finitary logic L with a lattice protodisjunction, in which case L5 is the least lattice-disjunctive extension of L. Also, under these two assumptions, we can easily axiomatize the logic L5 .7 In Corollary 5.5.18, we will show that we can replace those two assumptions by the assumption that L has a countable presentation (note that these two results are incomparable in strength; see 7 For simplicity, in Proposition 5.4.13 we formulate it with a few additional side-conditions, namely the validity of (C5 ), (I5 ), or (A5 ), but it is easy to see that these conditions could be relaxed at the price of having to add these formulas and their 5-forms to obtain the presentation of L5.

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Remark 5.5.9) and still axiomatize L5 and prove that it is the least logic extending L in which 5 is a strong disjunction. Proposition 5.4.13 (Axiomatization of L5 ) Let L be a finitary logic with a presentation AS and a finite protodisjunction 5 such that the consecutions (C 5 ), (I 5 ), and Ð (A 5 ) are valid in L.Then, the logic L5 is axiomatized by AS ∪ {𝑅 5 | 𝑅 ∈ AS}. Ð Proof Let us denote by Lˆ the logic axiomatized by AS ∪ {𝑅 5 | 𝑅 ∈ AS} (this set is closed under all substitutions because we assume 5 to be parameter-free). Clearly, for each consecution 𝐶 from this axiomatic system, we have 𝐶 5 ⊆ Lˆ (due to Lemma 5.2.5 part 2), hence we can use Theorem 5.2.6 to obtain that 5 is a strong ˆ disjunction in Lˆ which, due to the previous proposition, implies L5 ⊆ L. To prove the converse direction, we use Theorem 5.2.6 and the fact that 5 is a strong disjunction in L5 to obtain that, for any 𝑅 ∈ AS, we have 𝑅 5 ⊆ L5 .  The proof of the following corollary is left as an exercise for the reader. Corollary 5.4.14 Let L0 be a finitary logic with a finite disjunction 5 and L an 5 extension of L0 by a set of finitary Ð 5consecutions C. Then, the L is the extension of L0 by the set of consecutions {𝑅 | 𝑅 ∈ C}. In the rest of this section, we deal with the problem of axiomatizing extensions of a given logic L defined semantically by special classes of matrices from Mod∗ (L) described using so called non-negative clauses; see Remark 5.4.23 for a justification of this terminology and the relation of non-negative clauses (and their corresponding classes of matrices) with certain notions of classical first-order logic. Definition 5.4.15 Let Γ and Ψ be two finite sets of formulas such that Ψ is nonempty. We say that a pair hΓ, Ψi is a non-negative clause; we replace the adjective non-negative by positive if Γ = ∅. Moreover, we say that a set of non-negative clauses H is valid in a matrix A = hA, 𝐹i, written as A |= H , whenever, for each hΓ, Ψi ∈ H and each A-evaluation 𝑒 such that 𝑒[Γ] ⊆ 𝐹, we have 𝑒[Ψ] ∩ 𝐹 ≠ ∅. Thus, given a logic L, each set of non-negative clauses H in its language defines a class of matrices which, in turn, defines the following extension of L:  {A∈Mod∗ (L)

| A|=H } .

Note that a non-negative clause 𝐶 = hΓ, Ψi where Ψ = {𝜑} is actually a finitary consecution, and we have A |= hΓ, Ψi iff Γ A 𝜑. Observe that, to axiomatize the extension of L given by a set of consecutions C, we just add C to any presentation of L; indeed, denoting such extension by L + C, we could formally write this claim as  {A∈Mod∗ (L)

| ΓA 𝜑 for each ΓI 𝜑 ∈ C }

= Mod∗ (L+C) = L + C.

Before we show how to axiomatize extensions of L given by sets of non-negative clauses, let us give several examples of interesting classes of matrices that we can define using these syntactical objects.

286

5 Generalized disjunctions p

Example 5.4.16 The most obvious is the class Mod 5 (L) of 5-prime matrices of a given logic L with a finite protodisjunction 5 defined as p

Mod 5 (L) = {A ∈ Mod∗ (L) | A |= h𝜑 5 𝜓, {𝜑, 𝜓}i}. A more interesting case is that of linearly ordered matrices in a given weakly implicative logic L defined as {A ∈ Mod∗ (L) | ≤A is linear} = {A ∈ Mod∗ (L) | A |= h∅, {𝜑 → 𝜓, 𝜓 → 𝜑}i}. Finally, we show that we can use non-negative (positive) clauses to describe the intersection of two (axiomatic) extensions of a given logic. Let us consider two logics L𝑖 (for 𝑖 = 1, 2) which are extensions of a logic L by sets of finitary consecutions R 𝑖 . Let us further assume that the sets R 𝑖 are closed under arbitrary substitutions. It is easy to see that L1 ∩ L2 = |=Mod∗ (L1 )∪Mod∗ (L2 ) . We show that we can describe the class Mod∗ (L1 ) ∪ Mod∗ (L2 ) using non-negative clauses (note that, if the elements of R 𝑖 are axioms, we can use positive clauses): {A ∈ Mod∗ (L) | A |= hΓ ∪ Δ, {𝜑, 𝜓}i for each Γ I 𝜑 ∈ R 1 , Δ I 𝜓 ∈ R 2 }. One inclusion is trivial. We prove the converse one contrapositively. Consider A ∈ Mod∗ (L) such that A ∉ Mod∗ (L1 ) ∪ Mod∗ (L2 ), i.e. there are Γ𝑖 I 𝜑𝑖 ∈ R 𝑖 and evaluations 𝑒 𝑖 witnessing that Γ𝑖 2A 𝜑𝑖 . Since the R 𝑖 s are closed under arbitrary substitutions, we can assume that the formulas from Γ1 ∪ {𝜑1 } do not share any propositional variable with the formulas from Γ2 ∪ {𝜑2 } and so we can assume that 𝑒 1 = 𝑒 2 . This evaluation also shows that A 6 |= hΓ1 ∪ Γ2 , {𝜑1 , 𝜑2 }i. Our aim is to use properties of generalized disjunctions to find axiomatizations for the extensions of L given by sets of non-negative clauses. As we will see, we can do so in general for sets of positive clauses only, and otherwise we axiomatize only the 5-extension of such logics (which already has interesting consequences of its own, see Corollary 5.4.22). The crucial ingredient we need to prove these axiomatization results is the next expected lemma stating that, in a logic with a p-(proto)disjunction 5, a non-negative clause hΓ, {𝜓1 , . . . , 𝜓 𝑛 }i semantically behaves, in a certain way, as a set of consecutions Γ I 𝜓1 5 (𝜓2 5 (· · · 5 𝜓 𝑛 ) . . .). Remark 5.4.17 In order to simplify the notation, for an arbitrary given finite set Ψ = {𝜓1 , . . . , 𝜓 𝑛 }, we write 5 Ψ instead of 𝜓1 5 (𝜓2 5 (· · · 5 𝜓 𝑛 ) . . .) and set, by the way of convention, 5 {𝜑} = 𝜑 and 5 ∅ as the set of all formulas. This convention is, strictly speaking, not sound; but it is sufficient for our purposes as we will use it only in the context of logics with a p-protodisjunction 5 satisfying the rules (C 5 ) and (A 5 ). For a proper formal definition we would first need to fix an enumeration of all formulas and make a convention that whenever we describe a finite set of formulas by a list, we write these formulas in a strictly increasing sequence according to the

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287

fixed enumeration. Now 5 Ψ is defined soundly and it is easy to see that, if 5 is a p-protodisjunction satisfying the rules (C 5 ) and (A 5 ) in a given logic L, then both the bracketing in the definition of 5 Ψ and the background enumeration are irrelevant for the purposes of determining whether Γ `L 5 Ψ; 5 Ψ `L 𝜑, Γ A 5 Ψ or 5 Ψ A 𝜑 for any A ∈ Mod∗ (L). Finally, let us note that in such a setting, we also know that 5 Ψ0 `L 5 Ψ for each set Ψ0 ⊆ Ψ. Lemma 5.4.18 Let L be a logic with a p-protodisjunction 5 such that (C 5 ) and (A 5 ) are valid in L, A = hA, 𝐹i ∈ Mod(L), and hΓ, Ψi a non-negative clause. If A |= hΓ, Ψi, then Γ A 5 Ψ. If 𝐹 is 5-prime, the reverse implication holds as well. Proof The first claim follows by virtue of (PD); the second one is due to primality: if 5 is a parameter-free singleton, the proof is trivial, and even without the singleton assumption the proof is easy. However, the possible presence of parameters complicates things. Assume that Ψ = {𝜓1 , . . . , 𝜓 𝑛 } and consider an A-evaluation 𝑒 such that 𝑒[Γ] ⊆ 𝐹 and 𝑒[Ψ] ∩ 𝐹 = ∅. We show by finite induction that, for each 𝑘 ≤ 𝑛, there is a 𝛿 𝑘 ∈ 5 {𝜓1 , . . . , 𝜓 𝑘 } such that 𝑒(𝛿 𝑘 ) ∉ 𝐹. The base step is trivial as 5 {𝜓1 } = {𝜓1 }. As 𝐹 is prime, from 𝑒(𝛿 𝑘 ) ∉ 𝐹 and 𝑒(𝜓 𝑘+1 ) ∉ 𝐹, we know that there is a 𝛿( 𝑝, 𝑞, 𝑟) ∈ 5 and 𝑎 ∈ 𝐴 such that 𝛿A (𝑒(𝛿 𝑘 ), 𝑒(𝜓 𝑘+1 ), 𝑎) ∉ 𝐹. Without loss of generality, we can assume that 𝛿 𝑘+1 = 𝛿(𝛿 𝑘 , 𝜓 𝑘+1 , 𝑟) ∈ 5 {𝜓1 , . . . , 𝜓 𝑘+1 } and the variables 𝑟 do not occur in any formula from Γ ∪ {𝜓 𝑘+1 , 𝛿 𝑘 }. Therefore, we can also assume that 𝑒(𝑟) = 𝑎, and so we have 𝛿A (𝑒(𝛿 𝑘 ), 𝑒(𝜓 𝑘+1 ), 𝑎) = 𝑒(𝛿(𝛿 𝑘 , 𝜓 𝑘+1 , 𝑟) = 𝑒(𝛿 𝑘+1 ) ∉ 𝐹. Theorem 5.4.19 Let L be a logic with a p-protodisjunction 5 such that (C 5 ) and (A 5 ) are valid in L and H a set of non-negative clauses. Then, Ø ( {A∈Mod∗ (L) | A|=H } )5 = (L + {Γ I Ψ | hΓ, Ψi ∈ H })5.

5

If, furthermore, L has the 5PEP and all clauses in H are positive, then Ø  {A∈Mod∗ (L) | A |=H } = L + {∅ I Ψ | h∅, Ψi ∈ H }.

5

Proof Let us set L1 = L + previous lemma, we have

Ð

{Γ I

5Ψ

| hΓ, Ψi ∈ H } and observe that, by the

p

Mod 5 (L1 ) ⊆ {A ∈ Mod∗ (L) | A |= H } ⊆ Mod∗ (L1 ) from which we directly obtain L1 ⊆  {A∈Mod∗ (L)

| A|=H }

⊆ L51 .

Thus, if L has the 5PEP and all clauses in H are positive, then L1 has the 5PEP (due to Corollary 5.4.8), and so L1 = L51 (due to Proposition 5.4.6), which completes the proof of the second claim. To prove the first claim, observe that, by the previous chain of inclusions, the definition of L5, and Proposition 5.4.12, we also know that L51 ⊆ ( {A∈Mod∗ (L)

5 | A |=H } )

⊆ (L51 )5 = L51 .



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5 Generalized disjunctions

Recall that in Example 5.4.16 we have shown that certain interesting classes of matrices can be described using non-negative clauses and so, due to the previous theorem, we (almost) immediately obtain the following three corollaries. The first one tells us, given a weakly implicative logic L with the 5PEP, how to axiomatize the logic given by its linearly ordered matrices. In the next chapter, we call this logic the least semilinear expansion of L (see Definition 6.1.14). Corollary 5.4.20 Let L be a weakly implicative logic with the 5PEP. Then, |= {B∈Mod∗ (L) | B is linearly ordered} = L + (𝜑 ⇒ 𝜓) 5 (𝜓 ⇒ 𝜑). The next two corollaries deal with the problem of finding a presentation of the intersection of two given logics, i.e. the description of the meet in the lattice of logics in a given language. Recall that, in the text after Corollary 2.2.13 (where we have shown how to axiomatize the joins in that lattice), we have commented that there is no known method to do it in general. Interestingly enough, the presence of a disjunction allows us to do it in two particular settings (incomparable in strength): the first one involves axiomatic extensions of any logic with the 5PEP and the second one involves finitary extensions of a finitary logic with a finite disjunction. Corollary 5.4.21 Let L be a logic with the 5PEP and let L1 and L2 be axiomatic extensions of L by sets of axioms A1 and A2 , respectively. Then, L1 ∩ L2 is an axiomatic extension of L and Ø L1 ∩ L2 = L + {𝜑 5 𝜓 | 𝜑 ∈ A1 , 𝜓 ∈ A2 }. Therefore, the axiomatic extensions of L form a sublattice of its extensions. Corollary 5.4.22 Let L be a finitary logic with a finite disjunction 5 and let L1 and L2 be extensions of L by sets of finitary consecutions C1 and C2 , respectively such that 5 is a disjunction in L𝑖 . Then, L1 ∩ L2 = L + {Γ 5 𝜒, Ψ 5 𝜒 I 𝜑 5 𝜓 5 𝜒 | Γ I 𝜑 ∈ C1 , Ψ I 𝜓 ∈ C2 }. Proof Using Theorem 5.4.19 and Example 5.4.16, we know that Ø (L1 ∩ L2 )5 = (L + {Γ, Ψ I 𝜑 5 𝜓 | Γ I 𝜑 ∈ C1 , Ψ I 𝜓 ∈ C2 })5 . Note that L1 ∩ L2 is obviously a finitary logic with the disjunction 5 and thus, by Proposition 5.4.12, we know that L1 ∩ L2 = (L1 ∩ L2 )5 and the proof is completed by Corollary 5.4.14.  Remark 5.4.23 Recall that in previous chapters we have used the fact that L-matrices can be seen as classical first-order structures (in the equality-free predicate language P with function symbols from L and a unique unary predicate symbol 𝐷) to in order to define model-theoretic operations on logical matrices such as submatrices, homomorphisms, products, or reduced products.

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By identifying the set of propositional variables Var on one side and the set of object variables on the other side, we obtain that the formulas in L are exactly the terms in P. Then, we can further extend this identification to non-negative (and, in particular, positive) clauses. Indeed, recall that in classical logic, a non-negative clause in P is a P-formula 𝐻 given by two finite sets of P-terms (i.e. sets of formulas in the language L) Γ 𝐻 and Ψ 𝐻 where Ψ 𝐻 is required to be non-empty: Û Ü 𝐻= 𝐷 (𝜑) → 𝐷 (𝜓) 𝜑 ∈Γ 𝐻

𝜓 ∈Ψ 𝐻

Let us show that our definition of validity of non-negative clauses (Definition 5.4.15) is in alignment with the classical notion of validity of formulas in first-order structures. Let us start by recalling that, for each L-matrix hA, 𝐹i, its corresponding first-order structure 𝔐 in the predicate language P has the domain 𝐴, interprets the function symbols using the operations of A, and interprets the unary predicate symbol 𝐷 by the set 𝐹. Clearly, A-evaluations can be identified with 𝔐-valuations, i.e. mappings v : Var → 𝐴. Therefore, for each propositional formula 𝜑 and each A-evaluation 𝑒, we have 𝑒(𝜑) ∈ 𝐹 iff the formula 𝐷 (𝜑) is satisfied in 𝔐 by the 𝔐-valuation 𝑒. Now it is easy to see that a non-negative clause 𝐻 is valid in 𝔐 iff the non-negative clause hΓ 𝐻 , Ψ 𝐻 i is valid in hA, 𝐹i, and vice versa.

5.5 Pair extension property Recall that in Theorem 5.4.7 we have seen that any logic with a (strong) p-disjunction 5 and the IPEP enjoys the 5PEP. An interesting question is whether we may replace the not-so-easy-to-prove IPEP assumption by something more accessible. Of course, the easiest option would be to assume the finitarity of L, but we will see that we can do better: if 5 is a strong p-disjunction in L and L has a countable presentation, then L enjoys the 5PEP and so vicariously also the IPEP. Thus, we obtain a (relatively) easy way of establishing the 5PEP (and the IPEP) for infinitary logics: all we have to do is to find a countable presentation such that the logic in question proves the 5-form of each of its rules (cf. Theorem 5.2.6). In order to prove this result, we introduce a particular framework which is interesting on its own and will be useful also later; in particular, we use it to obtain stronger consequences of the SKC of a given logic in Theorem 5.6.5; for our study of completeness w.r.t. densely ordered chains in Section 6.4; and to construct the Henkin extension of a given predicate theory which is a crucial step in the completeness proof of the predicate logics in Section 7.4. We start by introducing a class of particular binary relations on the sets of formulas. We use them mainly as technical tools; interestingly, at the end of the section, we will show that in certain cases they are examples of the so-called symmetric consequence relations. Let us note that the following definition is sound by virtue of Remark 5.4.17 and, thanks to the fact that any two weak p-disjunctions in a given logic are mutually derivable (see Proposition 5.1.5), it is immaterial with one we actually use.

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Definition 5.5.1 Let L be a weakly p-disjunctional logic. The binary relation L on the sets of formulas is defined as Γ L Ψ iff there is a finite Ψ0 ⊆ Ψ and Γ `L

5 Ψ0.

Remark 5.5.2 Note the fundamental difference between the expressions ‘Γ `L Ψ’ and ‘Γ L Ψ’. The former has a conjunctive reading: it just says that all elements of Ψ are provable from Γ; whereas the latter has a disjunctive reading: it says that some disjunction of finitely many elements of Ψ is provable from Γ (which, if Γ is 5-prime, even implies that some element of Ψ is provable from Γ). Let us also note that Γ L ∅ iff Γ `L FmL . When writing instances of the relation L with more complex right or left hand sides, we use conventions similar to those in play for `, i.e. we write 𝜑, 𝜓 L Ψ, 𝜒 instead of {𝜑, 𝜓} L Ψ ∪ {𝜑} and, as usual, we omit the parameter L whenever it is fixed by the context. The relation L obviously has the following properties that we will freely use in the subsequent proofs (during the course of this section, we will additionally consider various versions of cut and pair extension properties): • Monotonicity on both sides: if Γ L Ψ, then Γ, Γ0 L Ψ, Ψ0. • Reflexivity: Γ, 𝜑 L Ψ, 𝜑. • Right-finitarity: if Γ L Ψ, then Γ L Ψ0 for some finite Ψ0 ⊆ Ψ. The next two definitions and three propositions show the relevance of the relation

L for our study of logics with generalized disjunction. From now on, we will always assume that we work in a weakly p-disjunctional logic (in order to define L ) and, since for any two weak p-disjunctions 5 and 5 0 the classes of 5-prime and 5 0-prime filters coincide, we will speak about prime filters and prime extension property PEP. Let us observe that, if 𝑇 is a prime theory of L, then not only no formula from FmL \ 𝑇 is provable from 𝑇 (in L) but also no finite disjunction of such formulas is provable from 𝑇. Conversely, note that any set of formulas with such property is a prime theory. We can see such tuple h𝑇, FmL \ 𝑇i as a complete consistent description of a certain logical situation (we use the word situation here purposefully vaguely; note, however, that in classical logic it would fully determine an evaluation) and see any such tuple hΓ, Ψi (where not necessarily Γ ∪ Ψ = FmL ) of sets of formulas as a partial consistent description of a certain logical situation. An obvious question then would be under which conditions a partial description can be extended into a complete one. The following formal definition and proposition formalize the reasoning above. Definition 5.5.3 Let L be a weakly p-disjunctional logic. We say that a tuple hΓ, Ψi of sets of formulas is an L-consistent pair if Γ 1L Ψ. Furthermore, we say that an L-consistent pair hΓ, Ψi is full if Γ ∪ Ψ = FmL and that it extends an L-consistent pair hΓ0 , Ψ0 i if Γ0 ⊆ Γ and Ψ0 ⊆ Ψ. Moreover, we say that L has the (right-finite) pair extension property if each L-consistent pair hΓ, Ψi (where Ψ is finite) can be extended into a full L-consistent pair.

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Proposition 5.5.4 Let L be a weakly p-disjunctional logic. • If a tuple hΓ, Ψi is a full consistent pair, then Γ is a prime theory. • If 𝑇 is a prime theory, then the tuple h𝑇, FmL \ 𝑇i is a full consistent pair. • L has the right-finite pair extension property iff it has the PEP. Proof We leave the proof of the first two claims and the left-to-right direction of the third one as an exercise for the reader and prove here the reverse direction only. Consider an L-consistent pair hΓ, Ψi, where Ψ is finite. Therefore, due to the definition of L and the (PD) assumption, we know that there is a 𝛿 ∈ 5 Ψ such that Γ 0L 𝛿. Thus, by PEP, where is a prime theory 𝑇 ⊇ Γ such that 𝛿 ∉ 𝑇. Thanks to the second claim, we know that h𝑇, FmL \ 𝑇i is a full consistent pair and, to finish the proof, we only have to notice that we know that, for each 𝜓 ∈ Ψ, we have 𝜓 `L 𝛿 and so Ψ ⊆ FmL \ 𝑇.  In Example 5.5.10, we show that the infinitary Łukasiewicz logic Ł∞ (which is weakly lattice-disjunctive) enjoys the right-finite pair extension property (and so the PEP and the IPEP) but not the full one. Thus, the pair extension property can be seen as a (in general strictly) stronger version of the PEP. Before we continue studying these two extension properties, let us note that they can be schematically written as: a logic L enjoys the (right-finite) pair extension property iff for each set Γ of formulas and each (finite) set Ψ of formulas, {Γ, FmL \ Σ L Ψ, Σ | Σ ⊆ FmL } . Γ L Ψ This metarule may appear as a rather complicated form of cut property from the theory of sequent calculi, and indeed it is an essential ingredient in defining the so-called symmetric consequence relation.8 Let us now introduce three simpler and more familiar forms of cut property and show their, perhaps surprising, relationship with the underlying generalized disjunction and their, less surprising, relation to pair extension properties. Definition 5.5.5 We say that a weakly p-disjunctional logic L has the • cut property if, for each set Γ of formulas and each finite set Ψ ∪ {𝜑} of formulas, Γ L Ψ, 𝜑 Γ, 𝜑 L Ψ . Γ L Ψ • (right-finite) strong cut property if, for each set Γ ∪ Φ of formulas and each (finite) set Ψ of formulas, {Γ L Ψ, 𝜑 | 𝜑 ∈ Φ} Γ L Ψ

Γ, Φ L Ψ

.

8 According to [115] a relation ⊆ P (FmL ) × P (FmL ) is called symmetric consequence relation if it enjoys monotonicity, reflexivity, and the prime extension property.

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Proposition 5.5.6 Let L be weakly p-disjunctional logic. • L has the cut property iff it is p-disjunctional. • L has the right-finite strong cut property iff it is strongly p-disjunctional. • If L has the (right-finite) pair extension property, then it has the (right-finite) strong cut property. Proof We prove the second claim and observe that the first one could be proved analogously. To prove the right-to-left direction, let us consider any formula 𝜒 ∈ 5 Ψ. From the second premise of cut, we obtain (thanks to (PD)) Γ, Φ `L 𝜒 and, as trivially Γ, 𝜒 `L 𝜒, we can use the sPCP to obtain Γ, Φ 5 𝜒 `L 𝜒. On the other hand, from the first premise of cut, we know that, for each 𝜑 ∈ Φ, we have: Γ `L 𝜑 5 𝜒. Therefore, Γ `L 𝜒 and so Γ L Ψ. The converse direction is a simple application of Theorem 5.2.6. Clearly, (C 5 ) and (I 5 ) are valid in L and, whenever Γ `L 𝜑, we can use the following instance of right-finite strong cut to obtain Γ 5 𝜒 `L 𝜑 5 𝜒 (the first assumption is trivial and the second one follows from Γ `L 𝜑 using the monotonicity L ): {Γ 5 𝜒 L 𝜒, 𝛾 | 𝛾 ∈ Γ} Γ 5 𝜒, Γ L 𝜑, 𝜒 . Γ 5 𝜒 L 𝜑, 𝜒 To prove the third claim, we show that L enjoys the (right-finite) strong cut property contrapositively: assume that Γ 1L Ψ for some set Γ and (finite) set Ψ. Thus, hΓ, Ψi is a consistent pair and, due to the (right-finite) pair extension property, we know that there is a set Σ such that Γ, FmL \ Σ 1L Ψ, Σ. If there is a formula 𝜑 ∈ Φ ∩ Σ, then, due to the monotonicity of L , we must have Γ 1L Ψ, 𝜑. If Φ ∩ Σ = ∅, then Φ ⊆ FmL \ Σ and, due to the monotonicity of L , we must have Γ, Φ 1L Ψ.  Recall that one of the main tasks of this section is to find some (preferably easy to check) sufficient conditions for (strongly) p-disjunctional logics to enjoy the PEP (and so vicariously also the IPEP). Thanks to the previous two propositions, we know that this question can be phrased using the language of symmetric consequence relations as: under which conditions does the logic with right-finite strong cut enjoy the right-finite pair extension property. Of course, this opens the related question of relations between other variants of cut and full pair extension properties. Thanks to the fact that any weakly p-disjunctional logic with the cut property is disjunctional, we can see the following theorem as the first answer to these questions. Note that, regarding the properties of generalized disjunctions, this theorem does not add to our stock of knowledge, because the fact that finitary disjunctional logics enjoy the PEP was established already in Theorem 5.4.7. However, we present this classical result in the theory of symmetric consequence relations because we know that the pair extension property is strictly stronger than the PEP (see Example 5.5.10) and it is instrumental for understanding the mutual relationship between the (meta)logical properties studied in this chapter (see Theorems 5.5.13 and 5.5.14). In Chapter 7, we will generalize it to the setting of predicate logics and use it to obtain completeness results. Theorem 5.5.7 Let L be a finitary logic with a finite disjunction. Then, L has the pair extension property.

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Proof First, we show that L enjoys a limited form of finitarity: if Γ L Ψ for some non-empty Ψ, then Γ `L 5 Ψ0 for some non-empty Ψ0 ⊆ Ψ and, as under our assumptions 5 Ψ0 is a finite set, we can use the finitarity of L to obtain a finite Γ0 ⊆ Γ such that Γ0 L Ψ0. Take any consistent pair hΓ0 , Ψ0 i and note that, without a loss of has to be a formula 𝜑 such that Γ0 0L 𝜑 and so hΓ0 , {𝜑}i is a consistent pair extending hΓ0 , Ψ0 i). Let 𝜅 be the cardinality of FmL and we enumerate all formulas by ordinals 𝜇 < 𝜅. We construct an increasing sequence of consistent pairs hΓ 𝜇 , Ψ 𝜇 i for each 𝜇 < 𝜅 by a transfinite recursion. Let us consider an ordinal 𝜇 < 𝜅 and assume that, for each 𝜈 < 𝜇, we have a consistent pair hΓ𝜈 , Ψ𝜈 i. Then, due to the induction assumption Ð Ð and finitarity, we know h 𝜈