Logics of Variable Inclusion (Trends in Logic, 59) [1st ed. 2022] 9783031042966, 9783031042973, 3031042964

Table of contents :
Contents
Acknowledgements
Chapter 1 Logic, analyticity, and significance
1.1 Logic and analyticity
1.1.1 Informational explications
1.1.2 Semantic explications
1.1.3 Syntactic explications
1.2 Logic and significance
1.2.1 The family of Kleene logics
1.2.2 Weak Kleene logics: B3 and PWK
1.2.3 The interpretation of the third value in B3
1.2.4 The interpretation of the third value in PWK
1.2.5 Other logics with infectious values
1.3 From significance to analyticity via variable inclusion
1.3.1 Syntactic characterisations of B3 and PWK
1.3.2 Pure variable inclusion companions of classical logic
1.3.3 Extensions of B3 and PWK
1.4 Logics of variable inclusion: A general frame work
Chapter 2 Płonka sums and regular varieties
2.1 Semilattice direct systems and Płonka sums
2.2 The Płonka decomposition theorem
2.3 Regular varieties
2.3.1 τ-semilattices
2.3.2 Subdirectly irreducible algebras
2.3.3 Subvarieties and equational bases
2.3.4 An example: Bisemilattices
2.4 Generalised involutive bisemilattices
2.4.1 Definition and elementary properties
2.4.2 The structure of the Boolean subalgebras
2.4.3 Characterising Boolean algebras and semilattices
2.4.4 The Płonka sum representation
Chapter 3 Dualities for regular varieties
3.1 Background
3.1.1 Basic notions
3.1.2 The Stone duality
3.1.3 The Priestley duality
3.2 Semilattice systems
3.3 Duality
3.3.1 Other dualities
3.4 Dual spaces
3.4.1 Left normal bands and GR spaces
3.4.2 GR spaces with involution
3.5 A topological counterpart of Płonka sums
Chapter 4 An interlude: Abstract Algebraic Logic
Chapter 5 Logics of left variable inclusion
5.1 Płonka sums of matrices and l-direct systems
5.1.1 General results
5.1.2 Left partition functions
5.2 Hilbert-style axiomatisations
5.3 Suszko reduced models of Ll
5.4 Some well-behaved cases
5.4.1 Equivalential logics
5.4.2 Logics with antitheorems
5.5 Classification in the Leibniz hierarchy
Chapter 6 Logics of right variable inclusion
6.1 Płonka sums of matrices and r-direct systems
6.1.1 General results
6.1.2 Right partition functions
6.2 Hilbert-style axiomatisations
6.3 The algebraic counterpart
6.3.1 Logics without antitheorems
6.3.2 Logics with antitheorems
6.4 Leibniz reduced models
6.5 Suszko reduced models
6.5.1 Truth-equational logics
6.5.2 Two well-behaved cases
Chapter 7 Paraconsistent Weak Kleene Logic
7.1 Abstract Algebraic Logic properties
7.1.1 Basic properties
7.1.2 Deductive filters and matrix models
7.1.3 Suszko reduced models
7.2 Hilbert-style calculi
7.3 Sequent calculi
7.3.1 Systems with linguistic restrictions
7.3.2 Systems without linguistic restrictions
7.4 Other proof-theoretic presentations
7.4.1 Natural deduction calculi
7.4.2 Tableaux
Chapter 8 Conclusions and open problems
8.1 Open problems
8.1.1 Universal Algebra
8.1.2 Abstract Algebraic Logic
8.1.3 Proof Theory
8.1.4 Duality Theory
Bibliography

Citation preview

Trends in Logic 59

Stefano Bonzio Francesco Paoli Michele Pra Baldi

Logics of Variable Inclusion

Trends in Logic Volume 59

TRENDS IN LOGIC Studia Logica Library VOLUME 59 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.

Stefano Bonzio Francesco Paoli Michele Pra Baldi •

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Logics of Variable Inclusion

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Stefano Bonzio Department of Mathematics and Computer Science University of Cagliari Cagliari, Italy

Francesco Paoli Department of Pedagogy, Psychology, Philosophy University of Cagliari Cagliari, Italy

Michele Pra Baldi Spanish National Council of Research (IIIA-CSIC) Artificial Intelligence Research Institute Barcelona, Spain

ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends in Logic ISBN 978-3-031-04296-6 ISBN 978-3-031-04297-3 (eBook) https://doi.org/10.1007/978-3-031-04297-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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

Contents 1 Logic, analyticity, and significance 1.1 Logic and analyticity . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Informational explications . . . . . . . . . . . . . . . . 1.1.2 Semantic explications . . . . . . . . . . . . . . . . . . 1.1.3 Syntactic explications . . . . . . . . . . . . . . . . . . 1.2 Logic and significance . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The family of Kleene logics . . . . . . . . . . . . . . . 1.2.2 Weak Kleene logics: B3 and PWK . . . . . . . . . . . 1.2.3 The interpretation of the third value in B3 . . . . . . 1.2.4 The interpretation of the third value in PWK . . . . . 1.2.5 Other logics with infectious values . . . . . . . . . . . 1.3 From significance to analyticity via variable inclusion . . . . 1.3.1 Syntactic characterisations of B3 and PWK . . . . . . 1.3.2 Pure variable inclusion companions of classical logic 1.3.3 Extensions of B3 and PWK . . . . . . . . . . . . . . . 1.4 Logics of variable inclusion: A general framework . . . . . .

1 1 3 3 5 7 7 9 11 14 16 19 19 21 24 28

2 Płonka sums and regular varieties 2.1 Semilattice direct systems and Płonka sums . . . . . . . 2.2 The Płonka decomposition theorem . . . . . . . . . . . . 2.3 Regular varieties . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 τ-semilattices . . . . . . . . . . . . . . . . . . . . . 2.3.2 Subdirectly irreducible algebras . . . . . . . . . . 2.3.3 Subvarieties and equational bases . . . . . . . . . 2.3.4 An example: Bisemilattices . . . . . . . . . . . . . 2.4 Generalised involutive bisemilattices . . . . . . . . . . . . 2.4.1 Definition and elementary properties . . . . . . . 2.4.2 The structure of the Boolean subalgebras . . . . . 2.4.3 Characterising Boolean algebras and semilattices 2.4.4 The Płonka sum representation . . . . . . . . . . .

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vi 3 Dualities for regular varieties 3.1 Background . . . . . . . . . . . . . . . . . . 3.1.1 Basic notions . . . . . . . . . . . . . 3.1.2 The Stone duality . . . . . . . . . . . 3.1.3 The Priestley duality . . . . . . . . . 3.2 Semilattice systems . . . . . . . . . . . . . . 3.3 Duality . . . . . . . . . . . . . . . . . . . . . 3.3.1 Other dualities . . . . . . . . . . . . 3.4 Dual spaces . . . . . . . . . . . . . . . . . . 3.4.1 Left normal bands and GR spaces . 3.4.2 GR spaces with involution . . . . . 3.5 A topological counterpart of Płonka sums .

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4 An interlude: Abstract Algebraic Logic 5 Logics of left variable inclusion 5.1 Płonka sums of matrices and l-direct systems . 5.1.1 General results . . . . . . . . . . . . . . 5.1.2 Left partition functions . . . . . . . . . 5.2 Hilbert-style axiomatisations . . . . . . . . . . 5.3 Suszko reduced models . . . . . . . . . . . . . 5.4 Some well-behaved cases . . . . . . . . . . . . . 5.4.1 Equivalential logics . . . . . . . . . . . . 5.4.2 Logics with antitheorems . . . . . . . . 5.5 Classification in the Leibniz hierarchy . . . . . 6 Logics of right variable inclusion 6.1 Płonka sums of matrices and r-direct systems 6.1.1 General results . . . . . . . . . . . . . . 6.1.2 Right partition functions . . . . . . . . . 6.2 Hilbert-style axiomatisations . . . . . . . . . . 6.3 The algebraic counterpart . . . . . . . . . . . . 6.3.1 Logics without antitheorems . . . . . . 6.3.2 Logics with antitheorems . . . . . . . . 6.4 Leibniz reduced models . . . . . . . . . . . . . 6.5 Suszko reduced models . . . . . . . . . . . . . 6.5.1 Truth-equational logics . . . . . . . . . 6.5.2 Two well-behaved cases . . . . . . . . .

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7 Paraconsistent Weak Kleene Logic 7.1 Abstract Algebraic Logic properties . . . . . . 7.1.1 Basic properties . . . . . . . . . . . . . . 7.1.2 Deductive filters and matrix models . . 7.1.3 Suszko reduced models . . . . . . . . . 7.2 Hilbert-style calculi . . . . . . . . . . . . . . . . 7.3 Sequent calculi . . . . . . . . . . . . . . . . . . 7.3.1 Systems with linguistic restrictions . . 7.3.2 Systems without linguistic restrictions 7.4 Other proof-theoretic presentations . . . . . . 7.4.1 Natural deduction calculi . . . . . . . . 7.4.2 Tableaux . . . . . . . . . . . . . . . . . .

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8 Conclusions and open problems 8.1 Open problems . . . . . . . . . . 8.1.1 Universal Algebra . . . . 8.1.2 Abstract Algebraic Logic 8.1.3 Proof Theory . . . . . . . 8.1.4 Duality Theory . . . . . .

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Acknowledgements We are greatly indebted towards the colleagues who have been discovering and developing together with us the connection between logics of significance and variable inclusion, regular varieties and their topological duals, and Płonka sums of algebras and logical matrices that is at the centre of this volume: Jos´e Gil F´erez, Andrea Loi, Tommaso Moraschini, and Luisa Peruzzi. The ideas in this book are as much theirs as ours. There is a rapidly growing research community whose interests revolve around the topics of this book. Luckily enough, all of its members are eager and stimulating discussants who believe in the value of scientific collaboration. Having the opportunity to share the ideas expounded here with some of them was a priceless asset, which yielded a number of crucial suggestions and pointers to the literature. We thank in particular Sankha Basu, Massimiliano Carrara, Roberto Ciuni, Ines Corbalan, ´ Vincenzo Fano, Thomas Macaulay Ferguson, Josep Maria Font, Ludovico Fusco, Pierluigi Graziani, Lloyd Humberstone, Hitoshi Omori, Damian Szmuc, as well as our colleagues Davide Fazio, Antonio Ledda, Adam Prenosil, and Gavin St John. Our special and heartfelt gratitude goes to Anna Romanowska, who carefully read portions of a first draft and suggested numerous improvements. We thank two anonymous reviewers of this book series for providing insightful comments and suggestions, as well as for spotting several typos and inaccuracies. The final outcome is in much better shape thanks to their efforts. We would like to express our gratitude towards Roberto Giuntini and Heinrich Wansing, for their encouragement and their help. The completion of this project would have been much more difficult, and much slower, if not for their kind assistance. We gratefully acknowledge the support of the following funding sources: • Fondazione di Sardegna, within the project “Resource sensitive reasoning and logic”, Cagliari, CUP: F72F20000410007;

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ACKNOWLEDGEMENTS • Italian Ministry for University, Education and Research (MIUR), within the projects PRIN 2017: “Theory and applications of resource sensitive logics”, CUP: 20173WKCM5, and PRIN 2017: “From models to decisions”; • European Research Council, within the project “PhilPharm”, grant No. 639276; • European Union and Generalitat de Catalunya, within the programme “Beatriu de Pinos”, co-funded Marie Curie fellowship, EU Horizon 2020 research and innovation programme under the MSCA grant agreement No. 801370. • INDAM GNSAGA, Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni.

On September 17, 2020, while this book was in the making, Jerzy Płonka – whose trailblazing work inspired so many of the developments accounted for in the following pages – passed away at the age of 90. Although we did not have the privilege to meet him personally, this volume is dedicated to his memory.

Chapter 1 Logic, analyticity, and significance 1.1 Logic and analyticity According to a dominant tradition in modern and contemporary philosophy, logic is the paradigmatic example of a discipline consisting of analytic sentences, whose truth or otherwise can be known simply by an analysis of their meanings, and of the meaning of their constituents. This doctrine, championed by Rudolf Carnap in his Logical Syntax of Language [49, § 34] and endorsed by innumerable other authors, has not gone unchallenged. Most famously, Quine questioned the very legitimacy of the analytic-synthetic distinction [196]; other voices, less drastically, have emphasised certain aspects of the notion of analyticity (like open texture [211] or vagueness [43]) that are at least likely to blur the distinction in specific cases; finally, it has been claimed that contemporary first-order logic should be considered synthetic in Kant’s sense [115] or that the methods of logic are continuous with those of science [201, 117]. Still, a vast majority of participants in this debate agree that a failure to display the marks of analyticity is, for a sentence, a reductio of its logical character. What do we mean, then, when we say that logic is analytic? To shed some light on the issue, it pays to go back to the original source of the whole discussion. In his Critique of Pure Reason, Immanuel Kant characterises as follows analytic and synthetic judgments: In all judgments, in which the relationship of a subject to a predicate is thought [...], this relationship is possible in two ways. Either the predicate B belongs to the subject A as something which is contained in the concept A (in a hidden way); or B lies entirely apart from that concept A, though it indeed stands in connection with it. In the first case I term the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_1

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CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE judgment analytical, in the other, synthetical. [...] We could also term the first one explicative, the other expansionary, because the former adds nothing to the concept of the subject through the predicate, but rather only lays it out through dissection into its partial concepts which were already thought in it (though in an indistinct way). The synthetical judgment, on the other hand, adds a predicate to the concept of the subject, a predicate which was not thought in the concept at all and could not have been drawn out of it through any dissection of it. (B10-11)

Thus, in Kant’s own perspective, analyticity is a property of judgments – which are not exactly the same as sentences as we understand them today. On top of that, Kant only considers judgments of the subject-predicate form, a fact (openly deplored by Quine [196]) that renders even more problematic a direct paraphrase of Kant’s notion in contemporary terms. However, we are not straying too much from the spirit of Kant’s definition if we extend this concept to arguments: in this sense, an argument with premisses ϕ1 , ..., ϕn and conclusion ψ is analytic if the truth of ψ can be known simply by an analysis of the meanings of ϕ1 , ..., ϕn . If we follow this path, the thesis that logic is analytic boils down to the claim that logically valid arguments amount to a species of analytic arguments; thus understood, analyticity is a feature of consequence. The early debate on whether logical consequence is analytic took place at a time when classical logic CL had barely any competitors. Nowadays, the supremacy of CL is constantly being undercut by a myriad of nonclassical challengers. Against this backdrop, it makes sense to wonder whether the view that logic is analytic supports CL against its nonclassical rivals. There are indeed prima facie reasons to suspect that it does not. The argument with premisses “Snow is white” and “Snow is not white” and with conclusion “PSG will win the Champions’ League this year”, an instance of the rule of Ex Absurdo Quodlibet: ϕ, ¬ ϕ ` ψ, is classically valid. It can be plausibly argued, though, that by a sheer analysis of the meanings contained in these particular premisses we can’t buttress any conclusion that is about football. Whatever notion of analyticity may appeal to those who claim that CL is analytic, it must be somewhat remote from an immediate, literal construal of the term. We now turn to examining one such notion.

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1.1.1 Informational explications A possible approach to analyticity as meaning containment is via the construct of information content. Wittgenstein, in his Tractatus Logico-Philosophicus [227, 5.14], alludes to the uninformative character of logically valid entailments when he remarks that “If a proposition follows from another, then the latter says more than the former, the former less than the latter [...] A tautology follows from all propositions: it says nothing”. In 1953, Yehoshua Bar-Hillel and Rudolf Carnap [12] propose an influential account of semantic information that echoes similar ideas. According to Bar-Hillel and Carnap, the information content of a sentence ϕ is the set of possible worlds that ϕ rules out. A certain conclusion analytically follows from a given premiss in case the information content of the former is included in that of the latter. Thus, a conjunction ϕ ∧ ψ analytically entails each of its conjuncts ϕ and ψ, since, under this definition, the information content, say, of ϕ is clearly included in the information content of ϕ ∧ ψ – any possible world that is ruled out by ϕ is automatically ruled out by ϕ ∧ ψ. Similarly, any disjunction is analytically entailed by each of its disjuncts. We can take a further step and define the information content of a set of sentences – on the plausible assumption that a finite set of sentences should have the same information content as the conjunction of its members, the obvious choice is to equate the content of Γ with the union of the contents of each ϕ ∈ Γ. More generally, all classically valid entailments abide by the criterion that the information content of their conclusion is included in the content of their set of premisses. In particular, whenever Γ is an inconsistent set of premisses, its content coincides by definition with the set of all possible worlds, a circumstance that vindicates the Ex Absurdo Quodlibet. The Bar Hillel-Carnap perspective is tailored to suit CL together with all its paradoxes of entailment. Yet, as an explication of meaning, it leaves some room for dissatisfaction. For example, we could find it implausible that all tautologies (or, for that matter, all contradictions) have the same meaning and thus co-entail one another. Thereby, other philosophers and logicians have explored different avenues.

1.1.2 Semantic explications Some of the most intriguing suggestions in this regard are due to logicians who gravitated around the Philosophy Department at Harvard in the mid- to late 1920s. The department was then under the sway of the charismatic figure of C.I. Lewis, who famously defended an account of

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entailment in terms of strict implication. We must keep in mind, for the following considerations, that the idea that a logic is primarily a consequence relation did not enjoy back then the same currency as it does nowadays, whereas it was common to regard logics as determined by a set of validities. Therefore, logicians tended to express their views on the structure of logical consequence by means of properties of certain entailment connectives, like strict implication. Analyticity requirements, as a result, were by and large stated in terms of connections between the antecedent and the consequent of an entailment formula, rather than in terms of connections between the premisses and the conclusion of an instance of valid consequence. Interestingly, the informal glosses of the entailment relation provided by Lewis, particularly in his early writings, suggest a more radical departure from classical logic than his calculi actually allow. For example, while in his Survey of Symbolic Logic [144, p. 231] (and elsewhere) Lewis claims that “the ’implication’ and ’equivalence’ of ordinary logic are relations of intension or meaning”, he also accepts that a contradiction entails (strictly implies) an arbitrary conclusion, as well as similar “paradoxes of strict implication”. Typically, his students or younger colleagues would agree with his view of implication in broad strokes, but would translate it into more extreme revisions of CL. Such was the attitude of Charles A. Baylis and of Everett J. Nelson, who obtained their PhDs from Harvard in 1926 and in 1929, respectively. Baylis [15] advances one of the earliest and neatest characterisations of entailment as based on meaning inclusion: Proposition A is subsumed under or implies proposition B if and only if the intensional meaning of B is identical with a part of the intensional meaning of A (p. 397). Baylis, however, does not offer much in the way of a formal clarification of this idea. Nelson [156], on the other hand, is far more specific. In his view, a sentence ϕ entails a sentence ψ just in case ϕ is inconsistent with ¬ψ. Although this is nominally the same definition of entailment one finds in Lewis, the two authors mean very different things by the notion of consistency. For Lewis, two sentences are consistent when they are compossible: hence ϕ and ψ might be inconsistent simply because one of them is impossible. According to Nelson, consistency is an inherently relational notion of conflict of meanings [150]. Therefore, he assumes that each and every sentence (even an impossible one) is consistent with itself and inconsistent with its own negation; moreover, if ϕ entails ψ, then ϕ is consistent with ψ. It is then natural for Nelson to endorse for

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his entailment connective (here represented by an arrow) the principles ( ϕ → ψ) → ¬( ϕ → ¬ψ) (Boethius’ law) and ¬( ϕ → ¬ ϕ) (Aristotle’s law), two laws that fail to qualify as classical tautologies. The calculus he presents both in his PhD thesis and in [156] is free from the paradoxes of strict implication, a feature viewed by Nelson as a considerable advantage over Lewis’s systems. Clearly, in order to obtain a consistent logic, Nelson has to forgo much of CL – and not all these sacrifices are exactly palatable. For example, in his Survey Lewis had shown that a certain variant of the Ex Absurdo Quodlibet is derivable if we assume the laws of Conjunctive Simplification and the rule of Antilogism: ϕ ∧ ψ → ϕ, ϕ ∧ ψ → ψ, ϕ ∧ ψ → χ ` ϕ ∧ ¬χ → ¬ψ. Nelson’s way out of this conundrum, in [156], is to give up Conjunctive Simplification (an approach he would disavow in later work in favour of a restriction to Antilogism). And it wouldn’t be long before some commentators noticed that this much is patently at odds with an explication of logical entailment as analytic. Korner, for example [136], remarks that ¨ if there is an inference that is typically analytic, this is the inference from ϕ ∧ ψ to ϕ: how are we to deny that the intensional meaning of a conjunction contains as a part the intensional meanings of its conjuncts? It can even be argued, as Korner does, that Conjunctive Simplification provides ¨ the common pattern of all analytic inferences. Korner proposes his own solution to this problem, which in the fol¨ lowing decades is amply discussed in the tradition of the so-called connexive logics. We will not further dwell on this topic, referring the interested reader e.g. to [159].

1.1.3 Syntactic explications An important alternative to Nelson’s approach is put forth by another talent from Harvard’s youth academy in the early 1930’s. William T. Parry obtains his PhD in 1931 with a thesis devoted to a concept that is labelled, rather tellingly, analytic implication (see [167], but also [168]). For it, he proposes an axiom system. In [167] he says: The postulates given for [analytic implication] could also be interpreted in terms of an [...] intensional relation i.e., a relation the applicability of which is determined by the meanings

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CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE of the terms (logical or non-logical) involved in the propositions in question. [...] This could be called “analytic implication”, in the sense in which it can be said, e.g., that “This is red” analytically implies “This is colored”. [...] My contention is that (structurally) analytic implication is the same as real implication (pp. 168-169).

Parry is more explicit than Nelson about what he takes to be the proper formal analogue of this informal notion. Indeed, he claims that valid entailments should satisfy a Proscriptive Principle which is to all intents and purposes a variable inclusion requirement, to the effect that: No formula is valid that has [analytic implication] as a main relation and has a variable [...] which occurs in the consequent but not in the antecedent (pp. 170-171). Many truth-functional tautologies violate this criterion – likewise, many classically valid inference rules fail to preserve it. One such example is given by the laws of Addition: ϕ → ϕ ∨ ψ, ψ → ϕ ∨ ψ. Another example is the already mentioned principle of Antilogism. As a matter of fact, Parry blames either Addition or Antilogism as the stumbling blocks in Lewis’s alleged “proofs” of the paradoxes of strict implication. At first sight, Parry’s angle is not that different from Nelson’s: just a distinct technical explication of the idea of analytic entailment in terms of meaning inclusion. Indeed, critics of Parry, like the relevant logicians Anderson and Belnap [3], have found the Proscriptive Principle questionable precisely if advanced as an analysis of meaning inclusion: Surely Kant would have regarded “All brothers are siblings” as an analytic truth, and if sibling is defined in the natural way, we have [...] a case of p → p ∨ q (i.e., brothers are either brothers or sisters). [...] there may be some doubt [...] as to whether p → p ∨ q is correct in the Kantian sense. We think it is (p. 155). Recently, however, Thomas Ferguson [83, 86] has convincingly argued that Anderson and Belnap, like other detractors of Parry, are criticising a strawman. With his Proscriptive Principle, Parry does not have in mind the semantic notion of meaning containment, but an irreducibly syntactic

1.2. LOGIC AND SIGNIFICANCE

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concept that is by no means supposed to act as a surrogate of it. The inference from ϕ to ϕ ∨ ψ, for example, may fail because ϕ ∨ ψ represents an ill-formed sentence of some sort – say, within a larger theory T, a sentence that is built using descriptive vocabulary outside the subtheory T 0 to which the entailment belongs. What is at stake is a concept that is at the same time more general and less metaphysically loaded than meaning: the notion of significance, which will take centre stage in the next section. Parry’s logic and its variants have been studied in [78, 90, 131, 142, 218], although the list is certainly not exhaustive. In particular, Richard Epstein has independently re-introduced a version of Parry’s system under the label of dependence logic: see e.g. [50, 79, 80, 82, 128, 133].

1.2 Logic and significance So far we have been viewing logical consequence as a form of analytic entailment. In this section, instead, we examine logical consequence in terms of truth preservation (or of preservation of other properties) in the presence of possibly non-significant sentences. A probe of the somewhat surprising connections between these two notions will be reserved for the section that follows.

1.2.1 The family of Kleene logics In his Introduction to Metamathematics [132, § 64], S.C. Kleene distinguishes between two senses of the propositional connectives when we are reasoning about partial recursive predicates. If we compute the value of one such predicate P as applied to the argument x, the computation may terminate and return a classical value (0, “false”, or 1, “true”) if P is defined for x, or else return no value at all. In the latter case, it may be convenient to say that Px has value n (“undefined”). In such a 3-valued setting, how are we to calculate the values of sentential compounds built by means of the connectives of negation, disjunction and conjunction? Kleene intends to respect two minimal constraints: • The table of each connective should be exactly the classical one, when all the arguments have classical values. • The table of each connective should be regular, meaning that whenever a column (row) contains a classical value in the row (column) of n, that column (row) should contain that classical value everywhere.

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It is readily seen that the following tables, called by Kleene strong for this reason, maximise the number of classical entries among those that comply with the above criteria:

¬ 0 n 1

1 n 0

∧ 0 n 1

0 0 0 0

n 0 n n

1 0 n 1

∨ 0 n 1

0 0 n 1

n n n 1

1 1 1 1

At the opposite end of the spectrum, we have the following weak tables, where the undefined value n occurs as often as possible:

¬ 0 n 1

1 n 0

∧ 0 n 1

0 0 n 0

n n n n

1 0 n 1

∨ 0 n 1

0 0 n 1

n n n n

1 1 n 1

Kleene, who incidentally develops interesting reflections on the computational meanings of the connectives in both settings, calls these tables “logics”, although this label is improper by present-day terminological standards. In order to obtain logics from these algebras (which we call SK and WK, respectively) we have to turn them into logical matrices, via an appropriate selection of designated values. It is pretty clear that Kleene views 1 only as the value that is to be preserved in inferences, although he does not render this choice fully explicit. Mainly for this reason, the 3-valued logic based on the strong tables, with the single designated value 1, is known today as strong Kleene logic, and denoted as K3 . Its applications are innumerable: it has been used in artificial intelligence as a model of partial information [1] and nonmonotonic reasoning [222], and in philosophy as a bedrock logic for Kripke’s theory of truth and other related proposals [89] – but the list could go on to fill some more lines. Another popular choice consists in keeping the strong tables but including n, along with 1, as a designated value. This policy yields the Logic of Paradox LP, which has been fervently supported by Graham Priest in the context of a dialetheic approach to the truth-theoretic and set-theoretic paradoxes, and has enjoyed an enduring popularity that made it the object of intense study both on the prooftheoretic and on the semantic level [182, 183, 184]. Although logics based on the weak tables do not even remotely compare to K3 or LP in terms of impact, they are far more interesting for the purposes of the present volume. In this subsection and in the next we will simply go over their definitions and some of their elementary

1.2. LOGIC AND SIGNIFICANCE

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formal properties, reserving an assessment of their philosophical motivations for the subsections that follow. In this case, too, we have two reasonable choices of designated values: either 1 only – which gives Paracomplete Weak Kleene logic B3 – or 1 together with n – and then we have Paraconsistent Weak Kleene logic, PWK. The introduction of both logics actually predates Kleene’s book, since B3 and PWK are linguistic fragments of logics respectively introduced by Bochvar in 1938 [30] for dealing with the set-theoretic and semantic paradoxes, and by Halld´en in 1949 [111] as a tool for reasoning in the presence of nonsensical sentences (see also Prior [191]). While the full systems by Bochvar and Halld´en will not be further investigated in what follows, the role of B3 and PWK, and of their generalisations, will be crucial in this book.

1.2.2 Weak Kleene logics: B3 and PWK Let us now switch to a slightly more formal tone in order to give precise definitions of these logics. To do so, we need to help ourselves to some basic tools that will be fully spelt out only later in the book, especially in Chapter 4. Thus, let us fix a propositional language, or similarity type, containing the binary connectives of conjunction (∧) and disjunction (∨) and the unary connective of negation (¬). For reasons that will be explained in due course, we call this type τ2 . The algebras SK and WK from the previous section are clearly algebras of type τ2 . In general, if A is an algebra of type τ, by an A-valuation we mean a homomorphism from the formula algebra Fmτ of type τ to the algebra A: a mapping v : Fmτ → A that respects the logical connectives. When no confusion is likely to arise, by a valuation we mean a WK-valuation, and by a Boolean valuation a B2 -valuation, where B2 is the 2-element Boolean algebra. Since B2 is a subalgebra of WK, Boolean valuations are in particular valuations. It is easy to see that valuations have a property that has been variously called contamination principle [57], principle of component homogeneity [108], principle of infectiousness [86] or doctrine of the predominance of the atheoretical element [5]: if v is a valuation and ϕ ∈ Fmτ2 , then v( ϕ) = n if and only if there is a propositional variable x occurring in ϕ such that v( x ) = n. This property is meant to formally mirror the “infectiousness“ of nonsensical sentences, which can never be a part of a sentential context that is meaningful as a whole, and therefore tallies with the initial motivations for B3 and PWK given by Bochvar and Halld´en (see the next subsections for a fuller discussion). By extension, elements like n are called infectious as well.

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We now introduce the logics B3 and PWK as the consequence relations respectively determined by the matrices hWK, {1}i and hWK, {1, n}i. Definition 1.2.1. The logic B3 = `B3 is the consequence relation of type τ2 defined as follows for every Γ ∪ { ϕ} ⊆ Fmτ2 : Γ `B3 ϕ iff, for every valuation v, v[Γ] ⊆ {1} implies v( ϕ) = 1. The logic PWK = `PWK is the consequence relation of type τ2 defined as follows for every Γ ∪ { ϕ} ⊆ Fmτ2 : Γ `PWK ϕ iff, for every valuation v, v[Γ] ⊆ {1, n} implies v( ϕ) ∈ {1, n}. Observe that classical logic CL can be formulated as a logic of type τ2 , whence it is natural to wonder what are the exact relationships between B3 , PWK and CL in terms of deductive strength. Clearly, both B3 and PWK are subclassical: if Γ 0CL ϕ, then there exists a Boolean valuation v such that v[Γ] = 1 and v( ϕ) = 0; but Boolean valuations are in particular valuations, and this suffices to conclude that Γ 0B3 ϕ and Γ 0PWK ϕ. However, each one of these logics actually coincides with CL with respect to a restricted class of entailments. In what follows, given a logic L of type τ and a formula ϕ ∈ Fmτ , we write `L ϕ as an abbreviation for ∅ `L ϕ, and say that ϕ is a theorem of L; dually, we say that Γ ⊂ Fmτ is an antitheorem of L if σ[Γ] `L ϕ for every substitution σ and formula ϕ. Lemma 1.2.2. We have that: 1. B3 has no theorem, while it has the same antitheorems as CL. 2. PWK has no antitheorem, while it has the same theorems as CL. Proof. (1) Let ϕ ∈ Fmτ2 . Choose a valuation v such that v( x ) = n for any variable x occurring in ϕ. Then v( ϕ) = n, whence 0B3 ϕ. Also, clearly any antitheorem of B3 is an antitheorem of CL. Conversely, if Γ 6= ∅ is not an antitheorem of B3 , there exist a substitution σ, a formula ϕ and a valuation v such that v[σ [Γ]] ⊆ {1} but v( ϕ) ∈ {0, n}. If v( ϕ) = 0, v is Boolean and hence Γ is not an antitheorem of CL. If v( ϕ) = n, there exists a variable x that occurs in ϕ but not in σ (Γ) and such that v( x ) = n. Consider the Boolean valuation w such that w( x ) = 0 and w(z) = v(z) for any variable z 6= x. Since x does not occur in σ[Γ], w[σ[Γ]] ⊆ {1} but w( x ) = 0, so σ[Γ] 0B3 x and once again Γ is not an antitheorem of CL. (2) Let Γ ⊂ Fmτ . Let x, y be distinct variables and σ be such that σ(z) = x, for every z that occurs in Γ. Consider a valuation v such that

1.2. LOGIC AND SIGNIFICANCE

11

v(y) = 0, v( x ) = n and is otherwise arbitrary. Hence v[σ [Γ]] ⊆ {n} but v(y) = 0, and Γ cannot be an antitheorem of PWK. For the nontrivial part of the remaining claim, if 0PWK ϕ, then v( ϕ) = 0 for some valuation v, which must perforce be Boolean – hence 0CL ϕ.  It is not hard to see (in some cases it can be directly inferred from the previous lemma) that in either B3 or PWK some of the most distinctive theorems, or inference rules, of CL fail. B3 , having no theorem, does not validate in particular the principle of Excluded Middle: it is a paracomplete logic. Dually, PWK has no antitheorems and in particular it fails the Ex Absurdo Quodlibet: it is a paraconsistent logic. Both logics have more peculiar failures as well: B3 does not validate (a rule version of) Addition, a respect in which it resembles Parry’s logic. Dually, PWK invalidates (a rule version of) Conjunctive Simplification, a feature that places it in the same ballpark as Nelson’s logic. Finally, we note that PWK fails Disjunctive Syllogism, i.e. Modus Ponens for the material implication ¬ ϕ ∨ ψ – which, for the sake of concision, is henceforth called Modus Ponens simpliciter. We collect all these observations in the next two lemmas. Lemma 1.2.3. None of the following principles hold in B3 : • (EM, Excluded Middle) ` ϕ ∨ ¬ ϕ; • (DA, Addition) ϕ ` ϕ ∨ ψ, ψ ` ϕ ∨ ψ. Lemma 1.2.4. None of the following principles hold in PWK: • (EAQ, Ex Absurdo Quodlibet) ϕ, ¬ ϕ ` ψ; • (CS, Conjunctive Simplification) ϕ ∧ ψ ` ϕ, ϕ ∧ ψ ` ψ; • (MP, Modus Ponens) ϕ, ¬ ϕ ∨ ψ ` ψ.

1.2.3 The interpretation of the third value in B3 So much for the formal details. It is now fair to ask: do weak Kleene logics make philosophical sense? How should we construe logics whose semantics allows for a nonclassical, infectious truth value? Compare the situation with that of strong Kleene logics. Getting our heads around those systems is a bit easier. In K3 , the third value n can be conveniently read as a truth value gap: “neither true nor false”. In turn, this gap can be viewed either as intrinsic to the nature of some particular sentences, like paradoxical sentences, or as an appropriate semantic evaluation for sentences whose truth or falsity is as yet undetermined. Notice,

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indeed, that K3 is the implication-free fragment of 3-valued Łukasiewicz logic, a system that was originally introduced for dealing with future contingents – sentences about contingent events in the future (like Aristotle’s celebrated “There will be a sea battle tomorrow”). Łukasiewicz went 3-valued because he believed that upholding bivalence when reasoning about future contingents entailed a commitment to determinism he was uneasy about. For this interpretation of n, the strong Kleene tables make a decent match for at least some of our intuitions: for example, if ϕ is true today and ψ is about an event possibly happening tomorrow, it stands to reason to tag ϕ ∨ ψ as true, for we already have our true disjunct today irrespective of whether ψ comes out true or false tomorrow. Similarly, it is natural to view ϕ ∧ ψ as false when, say, ϕ is false and ψ is undetermined. In a dual fashion, it is commonplace to read the value n in LP as a truth value glut: “both true and false”. If you think of designated values as values that are “at least true”, designating n is perfectly justified. Such a construal fits Priest’s account of dialetheias (i.e., paradoxical sentences like the Liar, sentences about borderline cases of vague predicates, etc.) as “true contradictions”. In weak Kleene logics, however, n may or may not be designated, but is always infectious. The above readings are at odds with this feature. So, are these logics any good for philosophy? Maybe so. Think of n as a value you would assign to meaningless or nonsensical sentences. It is not far-fetched to maintain that nonsensicality infectiously spreads over any sentential compound, for you can’t make sense of a complex sentence unless you have understood all its subsentences. Since meaningless sentences are clearly pathological, this reading justifies not only the weak tables, but also the particular design choice you have in B3 , where n is not designated. Indeed, this is the interpretation Bochvar [30] had in mind when he first introduced his logic, of which B3 is a fragment. Bochvar wanted to get rid of the set-theoretic and semantic paradoxes by considering them nonsensical. When you reason about the paradoxes, he claimed, you should simply view as nonsensical every sentence in which they are embedded, and continue to aim at truth preservation unblinkingly. Now, Bochvar was aware of Lemma 1.2.2, and the fact that B3 is a theoremless logic (an unusual circumstance back in the 1930s) annoyed him. To overcome this drawback, he expanded his language with a unary modal operator T with classical outputs, evaluated as follows: v( Tϕ) = 1 if v( ϕ) = 1, v( Tϕ) = 0 otherwise. With the help of A, Bochvar defined a new family of external connectives which, unlike the internal connectives of B3 , could only output Boolean values. Unfortunately, in this expansion of B3 , paradoxes resurface [224, § 1.6]: for example, it is enough to define

1.2. LOGIC AND SIGNIFICANCE

13

the Russell set as

{ x | ¬ T ( x ∈ x )} and trouble is around the corner. Of course, if we are not bothered by the fact that B3 has no theorems, we could simply blame Bochvar’s rash attempt to expand his language and choose to stick to the original system. Still, two further objections could be made to the nonsensicality interpretation of the third value: • Is this reading too partial? Beside meaninglessness, there are other philosophically relevant features that propagate infectiously, like empirical unverifiability or the presence of category mistakes. Could nonsensicality be just an instance of a broader phenomenon? • Is this reading even legitimate? It is debatable whether meaningless sentences can permissibly occur in sentential combinations. When I use a conjunction, say, am I not simply presupposing that the conjuncts are meaningful? These two allegations have been respectively discussed by Ferguson [83, 86] and Beall [16]. Ferguson proposes an interpretation of B3 that he directly reconnects to his reading of Parry’s logic (see § 1.1.3). He observes that paradoxical sentences are just one possible instance of propositional constructs exhibiting some form of non-significance. Take empirically unverifiable sentences, that according to some methodologies of science (in primis, logical positivism) are a paragon of lack of significance. In a way, they are not meaningless – competent speakers can make perfect sense of assertions concerning, say, moral values. None the less, empirical unverifiability seems to have an infectious character. At different levels, the same property is shared by ill-formed sentences (“Caesar is an and”) and sentences containing category mistakes (“Caesar is a prime number”). Ferguson also remarks that the B3 disjunction has a strong conjunctive flavour: from what we can read off the weak Kleene tables, for ϕ ∨ ψ to be true (i.e., for it to have the unique designated value of B3 ) two simultaneous requirements need to be satisfied: (a) at least one of ϕ, ψ must be true; (b) both ϕ and ψ must be significant. Unlike the case of CL, a possible algorithm that examines ϕ ∨ ψ and stops, returning the value “true” after finding a true disjunct, would not necessarily return the correct value for the disjunction. A parallel “significance check” must be carried out for both disjuncts. This perspective may assuage some qualms concerning the failure of Addition in B3 . Beall, on the other hand, suggests a more drastic departure from the nonsensicality reading. A sentence has value n when it is off-topic, for

CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE

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example in the context of a given theory. This also brings about a reappraisal of the Boolean values: 1 means now “true and on-topic”, 0 means “false and on-topic”. Thus, B3 consequence becomes a form of on-topic truth preservation, a concept that might play a role in an analysis of compartmentalised reasoning in scientific theories.1 In fairness to Beall, we clarify that he does not consider his construal as alternative to Bochvar’s one, but only as complementary to it. This said, it undisputably points to a quite different direction. Beall’s interpretation is essentially 2-valued – it only classifies sentences along two orthogonal degrees of freedom, true vs false and on-topic vs off-topic. The third value n is neither a genuine truth value, nor a proper truth value gap. Clearly, Bochvar wouldn’t have been on the same page. And this is perhaps a good place to at least mention an approach to logics of significance that will not be further explored in this book, advanced by Goddard and Routley in their monograph [108] and in a series of other works (see [219, 220] for two short, excellent accounts). Goddard and Routley endorse the weak Kleene tables as a basis for their project of a logic of significance. The third value is viewed as an actual gap: non-significant sentences do not have a truth value, the 3-valued tables are just a computationally handy device for keeping track of the propagation of non-significance in sentential combinations. Goddard and Routley, however, do not go on to embrace B3 , which they dislike since it is theoremless. Although they consider different options in their quest for significance logics, their system of choice is one whose language allows for two sorts of variables: ordinary variables, and variables that can only be mapped to Boolean values. Defining logical consequence in terms of truth preservation over formulas built with Boolean variables only clearly validates all CL-valid entailments; it is just as evident that the resulting system is not substitution-invariant.

1.2.4 The interpretation of the third value in PWK Interpreting the value n in PWK involves an additional problem. In PWK, n is both infectious and designated – two features that don’t seem to 1 An

interesting variation on Beall’s approach has been recently developed by Song, Omori, and Tojo [214]. They question Beall’s tenet that all off-topic sentences should be assigned the same semantic value, on the plausible assumption that one should distinguishing between true and false sentences whether or not they are topic-relevant to the theory at issue. This leads the authors of [214] to introducing a 4-valued “noninfectious” semantics for B3 and PWK, based on characteristic matrices that will be briefly discussed below, in Example 6.1.10 and Remark 7.1.4.

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15

sit well with each other. Suppose for a second that we want to stick to the nonsensicality interpretation, as Halld´en did in his [111]. As we have repeatedly observed, it is plausible to ascribe nonsensical sentences a contaminating behaviour, but why on earth should we assign these sentences a designated value? Halld´en had his own motivations for this choice, explained in [111, pp. 44-45]. Retaining the option of 1 as the sole designated value, indeed, would have guaranteed the unrestricted validity of Modus Ponens (Disjunctive Syllogism). In turn, this would have licensed, for an arbitrary sentence ϕ, the inference from the undeniable claims ϕ ∨ ¬ ϕ and ¬( ϕ ∨ ¬ ϕ) ∨ + ϕ to + ϕ (where + ϕ expresses the fact that ϕ is meaningful), wrecking the entire project of a logic of meaninglessness. By taking n on board as a designated value, Halld´en could salvage the classical tautologies while avoiding the commitment to the contention that everything is meaningful. Be it as it may, there is no denying that Halld´en’s approach is at odds with a common reading of designated values in many-valued logics. These values are often conflated with the “truth-like” values, the ones we assign to sentences we’d be ready to assert. Such a view, labelled Assertion-Designation Harmony (ADH) by Ciuni and Carrara [57], is well expressed by Brady and Routley in an oft-quoted passage where they condemn a certain variant2 of PWK [40]: [Designating meaninglessness along with truth] destroys the philosophical point of meaninglessness as a value to be assigned to non-significant sentences; one does not want to be committed to sometimes asserting logical nonsense (p. 219). Well before Brady and Routley wrote these lines, Halld´en himself was mindful of the issue. In his [111, p. 48], he drops a rather obscure remark, which has been variously interpreted (see e.g. [83, 219, 57]), to the effect that “a formula is to be taken as asserting something only about those values of which it can meaningfully assert something”. As Ciuni and Carrara correctly observe in [57], Halld´en seems to bite the bullet here and refuse to assert sentences that assume the third value. If so, his stance is plainly inconsistent with ADH, a popular and widespread assumption among many-valued logicians. Since B3 does not imply the same problem, the two weak Kleene logics are not on an equal footing here. 2 More

precisely, the logic they deplore was introduced by Segerberg [210]. This logic has been expanded and applied in [32] to provide a logical model of ignorance.

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CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE

Or perhaps they are. In the 2-valued setting, CL is usually characterised as the logic of forward truth preservation: if all the premisses of a CL-valid entailment are true, so is the conclusion. By parity of reasoning, though, it can be equally characterised as the logic of backward falsity preservation: if the conclusion of a CL-valid entailment is false, so is at least one of the premisses. If we generalise this picture to a manyvalued context, one might advance the suggestion that ADH is part of the story, but not the whole story. In particular, there is no reason to assign it a primacy over the symmetric principle of Denial-Undesignation Harmony (DUH): rather than treating designated values as those that correspond to sentences we would be ready to assert (and hence that must be preserved forwards in inferences), we could view undesignated values as those that correspond to sentences we’d be ready to deny (and hence that must be preserved backwards). In this perspective, B3 and the logics favoured by Goddard and Routley are as problematic as PWK. The latter is at loggerheads with ADH, the former with DUH, for the idea that we should “sometimes deny logical nonsense” is certainly incompatible with Goddard and Routley’s, and probably also with Bochvar’s, views. A different reading of the value n in PWK is contemplated by Omori and Szmuc [158], who provide a plurivalent semantics for weak Kleene logics along the lines of the semantics given by Graham Priest for BelnapDunn logic BD and its extensions [187]. Omitting details, we confine ourselves to observing that this semantic treatment leads Omori and Szmuc to consider n as a proper truth value gap, and to regard PWK conjunction as a honest-to-goodness conjunction that satisfies an evaluation clause closely resembling the classical one. This makes a sharp contrast to Roberto Ciuni’s view of PWK conjunction as a “disjunction in disguise”: somehow dualising Ferguson’s reading of B3 disjunction, Ciuni observes that for ϕ ∨ ψ to have a designated value in PWK, either one of two alternative requirements needs to be satisfied: (a) both ϕ, ψ are true; (b) either ϕ or ψ must be non-significant [55].3

1.2.5 Other logics with infectious values The matrices for B3 and PWK are both obtained by adding to the matrix for CL an infectious value, which is undesignated in the former case and designated in the latter. What if we perform this trick starting from a 3 In the same paper [55] and elsewhere [56], Ciuni also advances a cotenability interpretation of PWK conjunction, fashioned after the usual interpretation of intensional conjunction in relevant logics, another connective for which Conjunctive Simplification fails. However, in his own opinion, this suggestion is in need of further refinement.

1.2. LOGIC AND SIGNIFICANCE

17

different matrix? And, more importantly, would there be any motivation to do so beyond the mere pursuit of generality? In the late 1970s, Harry Deutsch (see e.g. [73]) added to the matrix for LP an infectious, undesignated value, obtaining a logic that we will later learn to call the right variable inclusion companion of LP, noted LPr . Deutsch, however, dubs (an equivalent presentation of) this logic SFDE , because it is a fragment – the first degree entailment fragment – of a logic he calls S. The justification he provides for SFDE brings us back to the themes we have been entertaining at the beginning of this chapter. Recall that Parry, with his logic of analytic implication, aims at dodging the paradoxes of irrelevance that plague Lewis’s systems of strict implication. However, Parry’s system still admits validities that clash with this motivation. Take for example the entailment

( ϕ ∧ ¬ ϕ) ∨ ( ϕ ∧ ψ) → ψ. Here, a contradiction embedded in a disjunctive antecedent entails an arbitrary, possibly unrelated formula simply because the variables contained in the latter get a free ride in the other disjunct. Keeping in mind that Parry, like many of his peers, ascribed to his entailment connective properties he deemed characteristic of logical consequence, we conclude that consequence a´ la Parry is not beyond reproach from the viewpoint of relevance. Thus, Deutsch tries to build a logic that is analytic and relevant at the same time, devoid of such “hidden” paradoxes of entailment. Later on, Carlos Oller [157] independently rediscovers the same logic with similar purposes. An intriguing interpretation of SFDE has been suggested by Fitting [93] in terms of pooling expert opinion on a given subject. Fitting’s approach, in fact, is more general and applies to several different logics. The semantic value of a formula ϕ is an ordered pair h P, N i consisting of the set P of experts from some group E that accept ϕ and the set N of experts from E that reject ϕ. In real-life contexts where experts weigh up pros and cons on an issue, we have a typically 4-valued situation. Indeed, some experts will only find reasons for accepting ϕ or for rejecting it, but some experts might find reasons for doing both things or for doing neither: hence, possibly, P ∩ N 6= ∅ and P ∪ N 6= E. Formally, given a nonempty set E, a Fitting valuation is a map v from Fmτ2 to ℘( E)2 such that, if v( ϕ) = h P1 , N1 i and v(ψ) = h P2 , N2 i, then: • v( ϕ ∧ ψ) = h P1 ∩ P2 , N1 ∪ N2 i; • v( ϕ ∨ ψ) = h P1 ∪ P2 , N1 ∩ N2 i;

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18

• v(¬ ϕ) = h N1 , P1 i. This yields a semantics for Belnap-Dunn logic BD. Confining ourselves for simplicity to the case of a single premiss, let us define ϕ ` E ψ if for every Fitting valuation v, where v( ϕ) = h P1 , N1 i and v(ψ) = h P2 , N2 i, P1 ⊆ P2 and N2 ⊆ N1 ; then ϕ ` E ψ iff ϕ ` BD ψ (see also [6]). However, one might want to consider a different possibility for evaluating conjunctions and disjunctions, taking into account only those experts who have actually expressed an opinion on the conjuncts (or disjuncts). This choice gives rise to a new consequence relation `∗E , identical to ` E except that Fitting valuations are replaced in its definition by these new valuations. Omitting details once more, Ferguson [86, Ch. 6] observes the rather surprising fact that ϕ `∗E ψ iff ϕ `SFDE ψ. Another possible application of SFDE has been explored by Da R´e, Pailos and Szmuc [69]. Traditional 3-valued approaches to truth theory conflate all the semantical anomalies – paradoxical sentences like the Liar, but also hypodoxical sentences like the Truth-teller, “This very sentence is true“– into the same cauldron of sentences that are assigned the third, nonclassical value. Depending on one’s personal taste, such a value is interpreted as a gap or as a glut. Yet, some authors (e.g. [226]) have objected that whatever reasons we might have for viewing the Liar as a gap push for considering the Truth-teller as a glut, and vice versa. The implication is that we should embrace a 4-valued logic with two different nonclassical values, one for paradoxical and one for hypodoxical sentences. The authors of [69] go on to wonder how these values should interact with the Boolean values and with each other in sentential compounds. Of the several options they examine, SFDE corresponds to a design choice to the effect that the gappy value (whether we decide to assign it to paradoxical or to hypodoxical sentences) is infectious, while the glutty value is not. So much for SFDE .4 By now, however, we bet that a plethora of 4valued logics with infectious values will be looming ahead in the reader’s imagination. Why don’t we take, for example, the matrix for K3 and add to it an infectious, designated value? At your service: this is the logic dSFDE [216, 58], which we will encounter again later on under the name of K3 l , the left variable inclusion companion of K3 . Damian Szmuc [217] has proposed an interesting epistemic interpretation of this logic that dualises Fitting’s interpretation of SFDE , and thus clarifies the relationships between these logics. Else, we might want to add the extra infectious value not to one of the two strong, but to one of the two weak Kleene 4A

more mathematical motivation for SFDE , concerning its relationship with the construction of power matrices [124], will be examined in our concluding chapter.

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19

logics B3 or PWK. The logics HYB1 = PWKr and HYB2 = B3 l were first considered by Szmuc [216, 58] and parlayed in [59, 60] into a hierarchical account of local vs global computer errors. ∗ ∗ Finally, there are the 5-valued logics SFDE [68, 186] and dSFDE [216, 58], where an infectious value (undesignated in the former case, designated in the latter) is added to the matrix for BD. Ciuni et al. [58] argue that these logics fare better than the other members of this family when assessed by relevance criteria. From a different viewpoint, Ferguson [86, Ch. 5] ∗ proposes SFDE as a model for the inferences of an artificial reasoner that is sometimes unable to retrieve the semantic value assigned to a variable.

1.3 From significance to analyticity via variable inclusion In the first two sections of this chapter, we have reviewed two approaches to logical consequence. First, we have focussed on consequence as analytic containment: a conclusion ϕ follows from certain premisses in Γ in case the meaning of ϕ is analytically included in the meanings of the sentences in Γ. Here, the sentences of concern to the logician are supposed to be just the significant sentences. Afterwards, we have switched to a seemingly unrelated perspective, where the presence of non-significant sentences in reasoning is no longer ruled out, but consequence is more traditionally interpreted as truth (or non-falsity) preservation. We now turn to highlighting an unexpected, but robust, connection between these approaches.

1.3.1 Syntactic characterisations of B3 and PWK In 1986, Alasdair Urquhart [224] rather nonchalantly notices a property of B3 . The valid entailments of this logic – with the exception of its antitheorems – can be obtained by filtering classically valid entailments through a variable inclusion strainer. Henceforth, if τ is a propositional language and ϕ is a τ-formula, we denote by Var ( ϕ) the set of propositional variables occurring in ϕ. More generally, if Γ ⊆ Fmτ , we stipulate that Var (Γ) =

[ γ∈Γ

Var (γ).

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CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE

Theorem 1.3.1. [Urquhart] For any Γ ∪ { ϕ} ⊆ Fmτ2 , the following are equivalent: 1. Γ `B3 ϕ; 2. Γ `CL ϕ and either: (a) Γ is an antitheorem of CL, or: (b) Var ( ϕ) ⊆ Var (Γ). Proof. Let B2 be the 2-element Boolean algebra. Observe that the matrix hB2 , {1}i is a submatrix of hWK, {1}i, whence if Γ `B3 ϕ, then Γ `CL ϕ. Now, suppose that Γ is not an antitheorem of CL (hence, that it is a classically satisfiable set of formulas) and that x ∈ Var ( ϕ) \ Var (Γ). Then there is a valuation v such that v[Γ] = {1} and v( x ) = n, whence v( ϕ) = n as well. Thus it is not the case that Γ `B3 ϕ. Conversely, if Γ is an antitheorem of CL, Γ `B3 ϕ holds by Lemma 1.2.2.(1). If, on the other hand, Γ `CL ϕ, Var ( ϕ) ⊆ Var (Γ) and v is a valuation such that v[Γ] = {1}, then v( ϕ) ∈ {0, 1} and so, necessarily, v( ϕ) = 1.  Theorem 1.3.1 immediately opens new prospects for B3 . Independently of its merits as a model of reasoning in the presence of nonsignificant sentences, we are alerted to the possibility that B3 might be also appropriate to describe consequence as analytic containment, at least if we are ready to follow Parry’s syntactic approach in terms of variable inclusion. Also, taking into account that the behaviour of PWK is in many ways dual with respect to B3 , the same result naturally suggests that PWK could be characterised relatively to CL in terms of a reverse variable inclusion constraint. This suspicion is well-founded, as the next theorem, due to Ciuni and Carrara [56], confirms. Theorem 1.3.2. [Ciuni-Carrara] For any Γ ∪ { ϕ} ⊆ Fmτ2 , the following are equivalent: 1. Γ `PWK ϕ; 2. There exists ∆ ⊆ Γ such that ∆ `CL ϕ and Var (∆) ⊆ Var ( ϕ). Proof. Suppose that Γ `PWK ϕ. If `CL ϕ, then the choice ∆ = ∅ will do the job. Otherwise, Γ is nonempty by Lemma 1.2.2.(2). Let then ∆ be the set of all formulas in Γ whose variables are among the variables in ϕ. Since 0CL ϕ, there is a Boolean valuation v such that v( ϕ) = 0. Consider now a valuation u such that u( x ) = v( x ) if x ∈ Var ( ϕ), u(y) = n if y∈ / Var ( ϕ). Hence, u( ϕ) = v( ϕ) = 0, and since Γ `PWK ϕ, there must be a formula ψ ∈ Γ such that u(ψ) = 0. If ψ contained some variable not in

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21

Var ( ϕ), then u(ψ) = n. So, Var (ψ) ⊆ Var ( ϕ), and therefore ψ ∈ ∆ and v(ψ) = u(ψ) = 0. This proves that ∆ `CL ϕ, since v was arbitrary except for falsifying ϕ. For the converse implication, let us suppose that there exists ∆ ⊆ Γ such that Var (∆) ⊆ Var ( ϕ) and ∆ `CL ϕ. Assume moreover that v is a valuation such that v( ϕ) = 0. Because of the structure of the weak Kleene tables, we have that v[Var ( ϕ)] ⊆ {0, 1}, and therefore we can consider a Boolean valuation u such that u( x ) = v( x ) for every x ∈ Var ( ϕ), whence u( ϕ) = v( ϕ) = 0. Since ∆ `CL ϕ, if ∆ is empty we obtain a contradiction; otherwise, there is ψ ∈ ∆ such that u(ψ) = 0. Since Var (ψ) ⊆ Var (∆) ⊆ Var ( ϕ), v(ψ) = u(ψ) = 0. That is, ψ ∈ ∆ ⊆ Γ is such that v(ψ) = 0. This proves that Γ `PWK ϕ.  In the jargon we will become familiar with in the following chapters, therefore, B3 is the right variable inclusion companion of CL, noted CLr , while PWK is its left variable inclusion companion, in symbols CLl . As much as it is tempting to use B3 and PWK to model the concept of analytic consequence and its dual in light of Theorems 1.3.1 and 1.3.2, an objection is to be expected. Let us start with B3 . The underlying intuition is clear: if ϕ classically follows from Γ, B3 only accepts this entailment if ϕ is analytically contained in Γ, as formally specified by the variable inclusion constraint Var ( ϕ) ⊆ Var (Γ). Theorem 1.3.1, however, provides for an important exception: if Γ is an antitheorem of CL (hence also of B3 ), the variable inclusion proviso need not apply. As we have seen, for arbitrary variables x, y, we have that x, ¬ x `B3 y. Yet, recall that the validity of EAQ in CL was one of the reasons for deeming CL inadequate as an explication of analytic containment. Trading B3 in for CL does not seem to have improved the situation. Dually, if ϕ is a theorem of CL (hence also of PWK), the reverse variable inclusion proviso in Theorem 1.3.2 does not apply and we have validities like x `PWK y ∨ ¬y, just as controversial if we are driven by motivations concerning relevance and content inclusion. Shouldn’t we get rid of these exceptions altogether?

1.3.2 Pure variable inclusion companions of classical logic In [166], in the context of a more general investigation, two pure variable inclusion companions of CL have been introduced: a right companion CL pr and a left companion CL pl . Their definitions are as follows: Γ `CL pr ϕ

⇐⇒

Γ `CL ϕ and Var ( ϕ) ⊆ Var (Γ);

22

CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE ( Γ `CL pl ϕ

⇐⇒

there exists ∆ ⊆ Γ, ∆ 6= ∅ such that ∆ `CL ϕ and Var (∆) ⊆ Var ( ϕ).

It is instructive to compare the definition of CL pr with the right-hand side of the statement of Theorem 1.3.1. Here the variable inclusion requirement is in force across the board, unlike in B3 . Clearly, therefore, CL pr is a sublogic of B3 that has neither theorems nor antitheorems. Similarly, if we compare the definition of CL pl with Theorem 1.3.2, we see that the subset ∆ is required to be nonempty. The dual variable inclusion requirement, unlike in PWK, admits of no exceptions. Thus, CL pl is a sublogic of PWK that, again, has neither theorems nor antitheorems. These two logics appear to be promising candidates to address the worries voiced at the end of the previous subsection. Observe that, somehow, we reversed the approach followed so far. We introduced B3 and PWK via their respective characteristic matrices, and then showed that they can be obtained from CL by imposing appropriate variable inclusion conditions. Such linguistic constraints, instead, are part and parcel of the very definitions we have given of CL pr and CL pl . The obvious question now is: what about their matrix semantics? Do these logics have finite characteristic matrices? Here, CL pr and CL pl go their separate ways. While the former has no single characteristic matrix and can only be described as the intersection of two matrix consequence relations (one of which is B3 )5 , the latter is indeed sound and complete with respect to a 5-element matrix. Theorem 1.3.3. We have that: 1. CL pr has no single characteristic matrix. 2. For any Γ ∪ { ϕ} ⊆ Fmτ2 , Γ `CL pr ϕ iff: (a) for every valuation v, v[Γ] ⊆ {1} implies v( ϕ) = 1 and (b) for every valuation v, v[Γ] ⊆ {1, 0} implies v( ϕ) ∈ {1, 0}. Proof. (1) Recall from e.g. [228] that a a logic L =`L of type τ has a single characteristic matrix if and only if Γ ∪ {Γi | i ∈ I } `L ϕ implies Γ `L ϕ for S all Γ ∪ { ϕ}, {Γi | i ∈ I } ⊆ Fmτ such that: (i) Var (Γ ∪ { ϕ}) ∩ Var ( {Γi | i ∈ I }) = ∅; S

(ii) Var (Γi ) ∩ Var (Γ j ) = ∅, for all i 6= j; 5 This is not at all unprecedented, even in the broad family of Kleene logics. The Kleene logic of order K≤ has no single characteristic matrix, but, being the intersection of K3 and LP, can be characterised via two different matrices on SK: see e.g. [123, § 2.1].

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23

(iii) for all i ∈ I: it is not the case that Γi `L ψ, for all ψ. Let Γ = { x }, I = {1}, Γ1 = {y, ¬y}, ϕ = ¬ x. Notice Γ, Γ1 `CL ϕ and Var ( ϕ) ⊆ Var (Γ ∪ Γ1 ), whence Γ, Γ1 `CL pr ϕ. Additionally, (i) Var (Γ ∪ { ϕ}) ∩ Var (Γ1 ) = ∅, (ii) is vacuously satisfied, and (iii) it is not the case that Γ1 `CL pr ψ, for all ψ. Yet, although all these things are true, Γ 0CL pr ϕ. Therefore, CL pr has no single characteristic matrix. (2) From right to left, if Γ 0CL pr ϕ, then either Γ 0CL ϕ or Var ( ϕ) * Var (Γ). Suppose first that Γ 0CL ϕ. Then Γ 0B3 ϕ and thus condition (a) fails. If Var ( ϕ) * Var (Γ), there is x ∈ Var ( ϕ) \Var (Γ). Consider a valuation v s.t. v ( x ) = n, v (y) = 0 for y 6= x. Then v [Γ] ⊆ {1, 0} , v ( ϕ) = n. Hence condition (b) fails. Conversely, we distinguish two cases. If condition (a) fails, then Γ 0B3 ϕ, hence by Theorem 1.3.1 either Γ 0CL ϕ or Γ is a CL-antitheorem such that Var ( ϕ) * Var (Γ). In both cases, Γ 0CL pr ϕ. If condition (b) fails, there is a valuation w s.t. w[Γ] ⊆ {1, 0} , w ( ϕ) = n. Then some variable in ϕ is assigned n by w, while none of the variables in Γ is assigned n by w. It follows that Var ( ϕ) * Var (Γ), whence again Γ 0CL pr ϕ.  This theorem invites an enthralling reading of consequence in CL pr , which assumes different connotations according as we interpret the third value n as “significant“, or, following Beall, as “on-topic“. In the former case, CL pr would be a logic of joint truth and significance preservation. In the latter, it would be a logic of joint truth and topic preservation, as opposed to B3 , which merely requires preservation of on-topic truth. We are not going to dwell on these issues here, referring the reader to [166] for a fuller discussion. Lemma 1.3.4. CL pl has a 5-element characteristic matrix hPK, {1, n}i, where PK is described by the tables below:

0 p n m 1

¬ 1 m n p 0

∧ 0 p n m 1

0 0 0 n 0 0

p 0 p n p 0

n n n n n n

m 0 p n m 1

1 0 0 n 1 1

∨ 0 p n m 1

0 0 0 n 1 1

p 0 p n m 1

n n n n n n

m 1 m n m 1

1 1 1 n 1 1

Proof. Suppose first that Γ `CL pl ϕ, whence there exists a nonempty ∆ ⊆ Γ such that ∆ `CL ϕ and Var (∆) ⊆ Var ( ϕ). Observe that for any PKvaluation, if v( x ) = n for some x occurring in ∆, then v( ϕ) = n, whence ϕ follows from Γ in the matrix hPK, {1, n}i. Hence we can restrict our

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CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE

attention to PK-valuations u such that u( x ) 6= n, for any x ∈ Var (∆). If u[Γ] ⊆ {1, n}, then u[∆] ⊆ {1}. A quick inspection of the tables above reveals that it is impossible that u( ϕ) ∈ {m, p}. If we can rule out the case u( ϕ) = 0 our job is completed, for then u( ϕ) ∈ {1, n}. If u( ϕ) = 0, then consider w such that for any variable y, w(y) = 1 if u(y) = m, w(y) = 0 if u(y) = p, w(y) = u(y) otherwise. It is readily seen that w is a valuation on the Boolean algebra over {0, 1}, that w[∆] ⊆ {1} and that w( ϕ) = 0. This, however, contradicts ∆ `CL ϕ. Suppose now that for any PK-valuation v, if v[∆] ⊆ {1, n}, then v[ ϕ] ∈ {1, n}. Let ∆ = {ψ ∈ Γ | Var (ψ) ⊆ Var ( ϕ)}. If ∆ = ∅, there exists u such that u[∆] ⊆ {n} and u[ ϕ] ∈ {m, p}, against the assumption. So ∆ is nonempty. If it is not the case that ∆ `CL ϕ, there is a Boolean valuation w on the Boolean subalgebra of PK over {0, 1} such that w[∆] ⊆ {1} and w[ ϕ] = 0, a contradiction.  In [166] it is shown that, although CL pl has a single finite characteristic matrix, it can also be characterised, exactly like CL pr , as the intersection of two logics (one of which is PWK), each determined by a single matrix on WK.

1.3.3 Extensions of B3 and PWK CL pr and CL pl are weaker logics than B3 and PWK, respectively. It is tempting to go in the opposite direction and wonder if there are logics strictly in between B3 and CL, or strictly in between PWK and CL. These questions have been tackled in [164] and in [165]; here we simply survey the facts, referring the reader to those papers for their proofs. The unique logic strictly stronger than B3 and strictly weaker than CL is Suszko logic SL (see [146]), defined as follows for any Γ ∪ { ϕ} ⊆ Fmτ2 : Γ `SL ϕ iff Γ `CL ϕ and Γ 6= ∅. It is a theoremless logic that exactly coincides with CL as regards its proper rules, whence there are formulas ψ such that 0SL ψ but ϕ `SL ψ for any formula ϕ. In other words, it is a pseudo-axiomatic logic, a property which confers to it a certain pathological character (see [96, p. 428]). There is also a unique logic strictly stronger than PWK and strictly weaker than CL. This non-paraconsistent logic, dubbed PWKE in [164], can be characterised in three different ways: • axiomatically, as the logic obtained by adding EAQ to PWK;

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25

• via a certain 6-valued matrix; • via a variable inclusion constraint in the style of Theorems 1.3.1 and 1.3.2. To establish these equivalences, we need to use some basic but as yet undefined results and notions about Płonka sums, which will be the object of next chapter. Readers who lack these prerequisites may skip the rest of this section. Let EK be the 6-element algebra of type τ2 described by the following tables (we write down only the tables for ∧ and for ¬, since the table for ∨ can be computed via x ∨ y = ¬(¬ x ∧ ¬y)):

¬ 0 ⊥ b a > 1

1 > a b ⊥ 0

∧ 0 ⊥ b a > 1

0 0 0 0 0 0 0

⊥ b a > 1 0 0 0 0 0 ⊥ ⊥ ⊥ ⊥ 0 ⊥ b ⊥ b 0 ⊥ ⊥ a a 1 ⊥ b a > 1 0 0 1 1 1

Define PWKE as the logic determined by the matrix hEK, {1, >}i. Observe: Lemma 1.3.5. hEK, {1, >}i is isomorphic to each of the following matrices: 1. hWK, {1, n}i × hB2 , {1}i; 2. hB4 ⊕ B2 , {1, >}i. Proof. For (1), consider the mapping f defined by f (>) = h1, 1i, f ( a) = h0, 1i, f (b) = h1, 0i, f (⊥) = h0, 0i, f (1) = hn, 1i, f (0) = hn, 0i. and remark that the designated values of the matrix hEK, {1, >}ii are exactly the ordered pairs of designated values of the matrices hWK, {1, n}i and hB2 , {1}i. For (2), the subalgebra of EK whose universe is {⊥, a, b, >} is isomorphic to the 4-element Boolean algebra B4 , with bottom element ⊥, top element > and atoms a, b. Also, the subalgebra of EK whose universe is {0, 1} is isomorphic to the 2-element Boolean algebra B2 , with bottom element 0 and top element 1. From the tables for the operations, it is clear that the latter fibre is above the former one. 

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CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE

We now provide a Ciuni-Carrara-type description of PWKE . To that effect, somewhere down the road, we will need to use an exercise in the truth-table semantics of classical logic, whose proof we gladly leave to the reader. Lemma 1.3.6. Let Γ ∪ { ϕ} ⊆ Fmτ2 . If ϕ is not a classical tautology and Γ is classically satisfiable, then there exists a B4 -valuation u, where B4 is the 4element Boolean algebra: 1, 1

1, 0

0, 1

0, 0 such that u ( ϕ) ∈ {h0, 0i , h1, 0i} and u (ψ) ∈ {h1, 1i , h1, 0i} for all ψ ∈ Γ. Theorem 1.3.7. The following conditions are equivalent for any Γ ∪ { ϕ} ⊆ Fmτ2 : 1. Γ `PWKE ϕ; 2. There exists ∆ ⊆ Γ such that ∆ `CL ϕ and either (A) Var (∆) ⊆ Var ( ϕ) or (B) ∆ is a CL-antitheorem. Proof. (1) implies (2). Let Γ `PWKE ϕ. If `CL ϕ, then ∆ = ∅ satisfies ∆ `CL ϕ and condition (A). If Γ contains an unsatisfiable subset ∆, then such a ∆ satisfies ∆ `CL ϕ and condition (B). Thus, we will suppose that 0CL ϕ and that Γ does not contain unsatisfiable subsets. We use the isomorphism in Lemma 1.3.5.(2). Since 0CL ϕ and Γ is itself satisfiable, by Lemma 1.3.6 there is a B4 -valuation v such that v ( ϕ) ∈ {⊥, a} and v (ψ) ∈ {>, a} for all ψ ∈ Γ. Since Γ does not contain unsatisfiable subsets, for every Σ ⊆ Γ there is a Boolean valuation wΣ such that wΣ [Σ] ⊆ {1}. Now, let ∆ = {ψ ∈ Γ | Var (ψ) ⊆ Var ( ϕ)} . Define an EK-valuation u as follows:  v ( x ) , if x ∈ Var ( ϕ) ; u (x) = wΓ\∆ ( x ) , otherwise.

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27

Thus, u ( ϕ) = v ( ϕ) ∈ {⊥, a}. Since Γ `PWKE ϕ, there is β ∈ Γ such that u ( β) ∈ { a, b, ⊥, 0}. Now, if Var ( β) ⊆ Var ( ϕ), then u ( β) = v ( β) ∈ { a, b, ⊥, 0} ∩ { a, >} = { a} . Then v, which was arbitrary except for falsifying ϕ, determines a Boolean valuation that also falsifies β. Thus ∆ satisfies ∆ `CL ϕ and condition (A). Finally, suppose that Var ( β) * Var ( ϕ), i.e. there is y ∈ Var ( β) \Var ( ϕ). Consequently, u (y) = wΓ\∆ (y) ∈ {0, 1}. We show by induction on the complexity of β that u ( β) = wΓ\∆ ( β) if Var ( β) * Var ( ϕ). (β = y). Then u ( β) = u (y) = wΓ\∆ (y) = wΓ\∆ ( β). (β = ¬γ). Observe that β and γ contain the same variables. Using the inductive hypothesis, then, u ( β) = u (¬γ) = ¬u (γ) = ¬wΓ\∆ (γ) = wΓ\∆ (¬γ) = wΓ\∆ ( β) . (β = γ ∧ δ). There are two possible cases: either Var (γ) * Var ( ϕ) and Var (δ) * Var ( ϕ), or (w.l.o.g.) Var (γ) * Var ( ϕ) and Var (δ) ⊆ Var ( ϕ). In the former subcase, we can use the inductive hypothesis for both conjuncts: u ( β ) = u ( γ ∧ δ ) = u ( γ ) ∧ u ( δ ) = wΓ\∆ ( γ ) ∧ wΓ\∆ ( δ ) = wΓ\∆ ( γ ∧ δ ) = wΓ\∆ ( β ) . In the latter subcase, since β ∈ Γ\∆, wΓ\∆ ( β) = 1 and thus wΓ\∆ (γ) = 1. Since v ( β) = v (γ ∧ δ) ∈ {>, a}, we have that v (δ) ∈ {>, a}. Then u ( β ) = u ( γ ∧ δ ) = u ( γ ) ∧ u ( δ ) = wΓ\∆ ( γ ) ∧ v ( δ ) = 1 = wΓ\∆ ( γ ∧ δ ) = wΓ\∆ ( β ) . Since disjunction can be defined in terms of the other connectives, our induction is complete. So we have just proved that u ( β) = wΓ\∆ ( β) = 1 because β ∈ Γ\∆; on the other hand, u ( β) ∈ { a, b, ⊥, 0} — a contradiction. (2) implies (1). We use Lemma 1.3.5.(1). Suppose (2) holds. There are two possibilities. Assume first that there exists ∆ ⊆ Γ such that ∆ `CL ϕ and Var (∆) ⊆ Var ( ϕ). Then ∆ `CL ϕ and, by Theorem 1.3.2, ∆ `PWK ϕ. Moreover, any EK-valuation v that assigns members of ∆ values in {1, >} naturally induces a WK-valuation v1 such that v1 [∆] ⊆ {1, n} and a B2 valuation v2 such that v2 [∆] ⊆ {1}. Since ∆ `CL ϕ and ∆ `PWK ϕ, v1 ( ϕ) ∈ {1, n} and v2 ( ϕ) = 1. Hence v( ϕ) ∈ {1, >}. We conclude that ∆ `PWKE ϕ and hence Γ `PWKE ϕ. Next, assume that there exists ∆ ⊆ Γ such that

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CHAPTER 1. LOGIC, ANALYTICITY, AND SIGNIFICANCE

∆ `CL ϕ and ∆ is a CL-antitheorem. But then any EK-valuation v that assigns members of ∆ values in {1, >} would in particular induce a B2 valuation that satisfies ∆, which is impossible. Once again, our conclusion readily follows.  As already mentioned, in [164] it is shown that PWKE is axiomatised relatively to PWK by EAQ, hence it is a non-paraconsistent logic. Also, it is the unique logic between PWK and CL, which is in turn obtained by adding MP to PWKE . Two noteworthy facts strike the eye. In the first place, it could be tempting to conjecture that, exactly as PWK has an extension obtained by adding to it the classical antitheorems, which is however still strictly weaker than CL, something similar should happen if we add to B3 the classical theorems. As implicitly remarked at the outset, however, this is not true. Indeed, B3 , unlike PWK, validates MP, whence the addition of classical theorems yields nothing short of full CL. Secondly, we have just seen that PWKE admits a linguistic characterisation in terms of a variable inclusion requirement. Exceptions must be provided for, but after all neither PWK nor B3 is exempt from such dispensations. However, its defining matrix contains no infectious value. One is led to surmise, then, that the relationship between infectiousness and variable inclusion is not as straightforward as one might think. Indeed, we will see that logics of variable inclusion, as we will call them throughout the present book, arise in the presence of certain conditions that admit of a convenient semantic description, and of which infectiousness is only a limit case. To clarify all this, however, we need a detour through universal algebra.

1.4

Logics of variable inclusion: A general framework

In the late 1960’s, the Polish algebraist Jerzy Płonka [173, 174, 175, 176, 177] devised a powerful construction to represent algebras meeting particular requirements as sums of algebras with stronger properties, in such a way as to enable transfer of important information from the summands onto the sum. This construction, now known as Płonka sum over a semilattice direct system of algebras (although of course this is not the name used by Płonka himself), is, in a sense, a fibration indexed by a join semilattice and whose fibres are the algebras of the system. Thus, one can think of it as a new algebra within which the algebras of the system live separately.

1.4. LOGICS OF VARIABLE INCLUSION: A GENERAL FRAMEWORK 29 It turns out that the technique of Płonka sums works well for regular varieties, i.e. varieties of algebras axiomatised by identities whose lefthand side and whose right-hand side contain exactly the same variables. If a regular variety V satisfies a certain additional condition, its members are representable as Płonka sums over semilattice direct systems of algebras in a variety V 0 , such that V satisfies all and only the regular identities satisfied by V 0 . This and other results will be at the centre of Chapter 2, where we also examine in detail some important examples of regular varieties: distributive bisemilattices, representable as Płonka sums over semilattice direct systems of distributive lattices, and (generalised) involutive bisemilattices, representable as Płonka sums over semilattice direct systems of Boolean algebras. These algebras, as we will see, play a crucial role in the algebraic analysis of the logics B3 and PWK.

Useful information about many varieties of algebras can be retrieved from their topological duals. For example, the topological counterpart of Boolean algebras is provided by Stone spaces, while Priestley spaces are the topological analogue of distributive lattices. Now, suppose that V is a regular variety whose members are representable as Płonka sums over semilattice direct systems of algebras in the variety V 0 , whose topological dual is known. Is it possible to endow V with a topological dual consisting of spaces constructed in some way out of spaces from the topological counterpart of V 0 ? This question is answered in the affirmative in Chapter 3, based, among other sources, on the papers [31, 34, 36]. It is shown that this topological dual can be obtained via a topological construction that dualises the Płonka sum construction.

The identities holding in regular varieties are subject to a syntactic constraint on the sets of variables that occur therein. This circumstance invites a conjecture to the effect that regular varieties are somehow connected with logics of variable inclusion. This connection is explored in detail in Chapters 5 and 6, where we present a general framework for logics of variable inclusion that we developed together with Jos´e Gil F´erez, Tommaso Moraschini, and Luisa Peruzzi in the papers [33, 37, 38, 164]. We associate to an arbitrary logic L a left variable inclusion companion Ll and a right variable inclusion companion Lr following the blueprint provided by

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30

Theorems 1.3.1 and 1.3.2. Thus:6 Γ `Ll ϕ ⇐⇒ there is ∆ ⊆ Γ s.t. Var (∆) ⊆ Var ( ϕ) and ∆ `L ϕ. Γ `Lr ϕ ⇐⇒ either: Γ `L ϕ and Var ( ϕ) ⊆ Var (Γ) or: Γ is an antitheorem of L. A semantic analysis of these logics of variable inclusion is possible by generalising the Płonka sum construction from algebras to logical matrices. By so doing, we can investigate the different classes of models of Ll and Lr of interest for Abstract Algebraic Logic: general models, Leibnizand Suszko-reduced models, etc. It will become apparent that these models can be constructed in appropriate ways as Płonka sums of (general, Leibniz- or Suszko-reduced) models of the logic L. Also, the different “directionality”, so to speak, of the variable inclusion constraints appearing in right and left variable inclusion logics (from the conclusion to the premisses, or vice versa) can be explained by recourse to the different conditions imposed on the filters in the semilattice direct systems of matrices in the left and in the right case, respectively. By way of example, let us point out that: • the defining matrix of B3 , i.e. hWK, {1}i, can be represented as the Płonka sum of the two matrices hB2 , {1}i and h1, ∅i, where 1 is the trivial algebra of type τ2 ; • the defining matrix of PWK, i.e. hWK, {1, n}i, can be represented as the Płonka sum of the two matrices hB2 , {1}i and h1, {1}i; • the defining matrix of CL pl , i.e. hPK, {1, n}i, can be represented as a Płonka sum of the three matrices hB2 , ∅i, hB2 , {1}i and h1, {1}i; • the defining matrix of PWKE , i.e. hEK, {1, >}i, can be represented as a Płonka sum of the two matrices hB4 , {h1, 1i}i and hB2 , {1}i. More generally, while all logics of variable inclusion (in the above sense) can be given a semantics in terms of Płonka sums over nontrivial semilattice direct systems of matrices, logics with infectious values arise out 6 The

following definitions of a left and a right companion of a given logic do not enhance the inherent duality between these notions at its most perspicuous. We could do a better job if we moved to a multiple-conclusion framework, as done e.g. in [57, 58, 60]. Throughout this book, however, we have decided to stick to the more traditional singleconclusion account of consequence relations, so as to help ourselves to the incomparably larger and deeper stock of results available on such single-conclusion relations in the Abstract Algebraic Logic literature.

1.4. LOGICS OF VARIABLE INCLUSION: A GENERAL FRAMEWORK 31 of those semilattice direct systems containing at least one trivial algebra. This confirms the idea, already hinted at, that infectiousness is a special manifestation of a more general phenomenon. Chapter 7 is devoted to what is perhaps the best understood among all logics of variable inclusion, PWK. Using results from the papers [33, 163, 164] as well as other sources, we survey its semantic properties from the viewpoint of Abstract Algebraic Logic (some of which are mere corollaries of the general theory laid out in the preceding chapters, while other ones must be established directly) and its proof theory, in terms of Hilbert systems, natural deduction calculi, sequent calculi, and tableaux calculi. Chapter 8 presents some concluding remarks and a list of open problems. The general perspective that we have just quickly summarised, but that we will unfold in detail in the rest of this volume, subsumes and clarifies the particular examples of logics we have mentioned in this chapter, and many more. Hopefully, it can contribute to shed some light on the profound connections between syntactic variable inclusion requirements and the matrix semantics of certain many-valued logics in the vicinity of B3 and PWK, and thus, on a more conceptual level, on the relationships between analyticity and significance.

Chapter 2 Płonka sums and regular varieties As we have hinted in the previous chapter, the algebraic construction of Płonka sums yields a convenient representation for algebras defined by means of regular identities, namely, those identities where the same variables appear on both sides. Informally, Płonka sums build new algebras out of families of algebras organised into a certain system, called semilattice direct system. This chapter is a precis of the fundamental results concerning this construction – the connection with logics of variable inclusion will be outlined in Chapters 5 and 6, where we will parlay Płonka sums, and the algebras we build thereby, into suitable algebraic semantics for the mentioned logics. The denomination of this construction is a homage that a well-established tradition pays to the Polish mathematician Jerzy Płonka, who first introduced this “method of construction of abstract algebras” [173] in 1967 (see also [174, 175, 179]). There is no shortage of comprehensive surveys on this topic [179, 209]. Our more modest aim is to provide the reader with a self-contained exposition tailored to the needs of applications of Płonka sums to logic. The chapter is organised as follows. We begin by introducing, in Section 2.1, semilattice direct systems of algebras and Płonka sums. Then, in Section 2.2, we define partition functions with an eye to providing the main results from Płonka’s original work. In Section 2.3, we describe regular varieties, in particular regularisations of strongly irregular varieties, which play a prominent role in the theory, as each member of these classes is representable as a Płonka sum. Within this section we introduce τ-semilattices and we deal with some basic structure theory: the description of subdirectly irreducible members of regularisations of strongly irregular varieties and equational bases. Finally, in Section 2.4 we zoom in on the motivating example of generalised involutive bisemilattices, which

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_2

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34

will prove crucial in applications to logic. Hereafter, a running assumption is that all similarity types under consideration will contain at least one operation symbol of arity 2 or greater.

2.1 Semilattice direct systems and Płonka sums Definition 2.1.1. A semilattice direct system of algebras of type τ is a triple A = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i consisting of: (i) a semilattice I = h I, ≤i with join ∨; (ii) a family {Ai }i∈ I of τ-algebras with pairwise disjoint universes; (iii) a homomorphism pij : Ai → A j , for every i, j ∈ I such that i ≤ j. Moreover, pii is the identity map for every i ∈ I, and if i ≤ j ≤ k, then pik = p jk ◦ pij . We sometimes say that the semilattice direct system A = h{Ai }i∈ I , h I, ≤ i, { pij | i ≤ j}i is over the semilattice I = h I, ≤i. In essence, organising a family of algebras {Ai }i∈ I into a semilattice direct system means requiring that the index set I forms a semilattice and that algebras whose indices are comparable with respect to the order be “connected” by homomorphisms, whose “direction” is bottom-up,1 namely from algebras whose indices are lower to algebras whose indices are greater. The nomenclature in Definition 2.1.1 is deliberately chosen to emphasise the presence of an index set equipped with the structure of a semilattice. Systems of this kind are special cases of direct systems (of algebras), whose index sets are universes of directed preorders. Recall that a preorder (or quasiorder) is a reflexive and transitive binary relation . It is said to be directed if for each pair of elements x, y there exists a common upper bound, namely an element z such that x, y  z. Given a semilattice direct system A = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i, it is possible to define as follows a new algebra in the same type as the Ai ’s. Definition 2.1.2. Let τ be a similarity type that does not contain constants, and let A = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i be a semilattice direct system of type τ. The Płonka sum over A, in symbols Pł (A ) or Pł (Ai )i∈ I , is the algebra, also of type τ, such that 1 This

is simply a convention. The same construction can be done top-down, trading in a meet-semilattice for our join-semilattice.

2.1. SEMILATTICE DIRECT SYSTEMS AND PŁONKA SUMS (i) the universe of Pł (A ) is the union

[

35

Ai ;

i∈ I

(ii) for every n-ary basic operation g (with n > 1) in τ, and a1 ∈ A i1 , . . . , a n ∈ A i n , gPł (A) ( a1 , . . . , an ) := gA j ( pi1 j ( a1 ), . . . , pin j ( an )), where j = i1 ∨ · · · ∨ in . In plain words, operations on the elements a1 , . . . , an of the Płonka sum are computed by homomorphically mapping their arguments to the algebra A j , whose index is the join of the indices corresponding to the algebras where the elements a1 , . . . , an live, respectively, and then applying the pre-existing operation gA j to the images of a1 , . . . , an thus obtained. Sometimes we will refer to the algebras in {Ai }i∈ I as the fibres of the Płonka sum Pł (A ). A simple example can help to clarify this notion. Example 2.1.3. Let Li and L j be the two distributive lattices whose respective orders are given by the following Hasse diagrams: n

d Li = b

Lj = m

c

l

a

Let L = h{Li , L j }, I, { pii , pij , p jj }i where I = h{i, j}, ≤i is the linear order with two elements i < j, and the homomorphism pij : Li → L j is depicted in the following diagram via dashed arrows from Li to L j : n d m b

c l a

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES

36

We show how to compute binary operations in Pł (L ) among some elements in order to grasp the content of Definition 2.1.2. As regards elements belonging to the universe of the same algebra (like, for instance b and c) we have: b ∧Pł (L) c = pii (b) ∧Li pii (c) = b ∧Li c = a. In general, any operation whose arguments live in the same algebra restricts to the corresponding operation in such an algebra. On the other hand, when it comes to elements belonging to the universes of different algebras, we have for example: b ∨Pł (L) m = pij (b) ∨L j m = n ∨L j m = n. An alternate perspective on semilattice direct systems and Płonka sums is offered by category theory (see e.g. [206, p. 32] or [209]; see also the next chapter for a quick rehearsal of its main concepts). It is wellknown that we can view any poset P = h P, ≤i as a small category P: the objects of P are simply the elements of P, and given any two objects a, b in P, there is exactly one morphism from a to b if a ≤ b, no morphism from a to b if a  b. A semilattice direct system can then be defined as a triple hI, A, F i, where: • I = h I, ≤i is a semilattice, viewed as a small category; • A is an algebraic category (i.e. a category whose objects are similar algebras, and whose morphisms are algebra homomorphisms); • F is a covariant functor from I to A. If we denote by sij the morphism in I from i to j (whenever it exists), it can be observed that the last condition in the above list is responsible for the fact that F (sii ) = pii = id F(i) and that


pki = F (ski ) = F s ji ◦ skj = F s ji ◦ F skj = p ji ◦ pkj . We observed that any Płonka sum Pł (A ) is an algebra of the same type τ as the algebras organised into the semilattice direct system A. However, Definition 2.1.2 does not encompass the case in which τ contains constants. In such a circumstance, it is necessary to strengthen the definition of a semilattice direct system by requiring that the index set I is a semilattice with bottom element2 i0 (namely i0 ≤ i, for all i ∈ I), also 2 This

idea was introduced by Płonka himself in [178, 177].

2.2. THE PŁONKA DECOMPOSITION THEOREM

37

called a semilattice with zero. It is not difficult to check that the elements realised by the constants in the Płonka sum are exactly the elements realised by the constants in the algebra whose index is i0 . As a matter of convention, from now on we always assume that our semilattice direct systems of algebras with constants will be indexed by elements of a semilattice with zero.

2.2 The Płonka decomposition theorem A natural question arises: when is it the case that an algebra A is isomorphic to the Płonka sum over a certain semilattice direct system B? This problem is successfully addressed by Płonka [173, 175, 176], by introducing and putting to good use the core concept of a partition function, which will serve as the centrepiece of the present section. Definition 2.2.1. Let A be an algebra whose type τ does not contain constants. A function : A2 → A is a partition function in A if the following conditions are satisfied for all a, b, c ∈ A, a1 , ..., an ∈ An and for any operation g ∈ τ of arity n > 1. (PF1) a a = a, (PF2) a (b c) = ( a b) c, (PF3) a (b c) = a (c b), (PF4) gA ( a1 , . . . , an ) b = gA ( a1 b, . . . , an b), (PF5) b gA ( a1 , . . . , an ) = b a1 ... an . Definition 2.2.1 streamlines other, more elaborate definitions of a partition function to be found in the literature (see for instance [174, 173]). Our variant has the advantage of cutting down to a minimum the number of equations, in accordance with the most recent trends [206, 20]. This is also briefly discussed in [179, page 133]. Once again, the reader may notice that Definition 2.2.1 does not allow for the presence of constants in the type τ. In case τ contains constants, then the definition is easily adapted by adding [178] the following condition: (PF6) a cA = a, for all a ∈ A and all constants c in the type τ. The reason for the addition of the above condition will be clarified presently (Remark 2.2.9).

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Lemma 2.2.2. Let be a partition function in the algebra A of type τ. Then, for each g ∈ τ and a1 , . . . , an ∈ A, g( a1 , . . . , an ) ai = g( a1 , . . . , an ), for each 1 ≤ i ≤ n. Proof. gA ( a1 , ..., an ) ai = = = = =

gA ( a1 , ..., an ) gA ( a1 , ..., an ) ai gA ( a1 , ..., an ) a1 ... an ai gA ( a1 , ..., an ) a1 ... an gA ( a1 , ..., an ) gA ( a1 , ..., an ) gA ( a1 , ..., an )

(PF1) (PF5) (PF1-3) (PF5) (PF1) 

If a class K of algebras is such that a partition function is termdefinable in the same way in all members of K, we sometimes say that K has a partition function given by the formula x y. Before proceeding, we list some appropriate examples. Example 2.2.3. Recall that a left normal band (see e.g. [170, Ch. 2] or [120, §§ 4.4-4.6]) is an idempotent semigroup satisfying the identity x · y · z ≈ x · z · y. Left normal bands are important algebras in semigroup theory to which we shall revert in the next chapter. The defining axioms of left normal bands imply that the conditions (PF1)-(PF3) in Definition 2.2.1 are satisfied by the semigroup operation, and the compatibility conditions (PF4)-(PF5) are also satisfied with respect to the same operation, which is the only one in the type. It is not difficult to check that the binary projection operation on the first component is a partition function in every algebra. The examples that follow include varieties of algebras that have a partition function given by a formula defining the left projection. Example 2.2.4. The variety of lattices, and more generally all the varieties of algebras with a lattice reduct, have a partition function given by the formula x y := x ∧ ( x ∨ y). Example 2.2.5. The classes of Heyting algebras, BCK algebras [127] and, more generally, all the classes having a Hilbert algebra reduct [74], have a partition function given by the formula x y := (y → y) → x.

2.2. THE PŁONKA DECOMPOSITION THEOREM

39

Example 2.2.6. The variety of groups has a partition function given by the formula: x y := x + (y − y). Example 2.2.7. A variety V whose type contains (at least) a constant 1 is said to be 1-subtractive (or simply subtractive [225, 2]) if there is a binary formula x → y, such that V satisfies the following identities: (S1) (S2)

x → x ≈ 1; 1 → x ≈ x.

Examples of subtractive varieties include groups, rings, Boolean algebras. Every subtractive variety has a partition function given by the formula x y := (y → y) → x. It is readily seen that this example generalises the two preceding ones. As the terminology suggests, the presence of a partition function is the key to decomposing an algebra into disjoint pieces. The main connection between partition functions and Płonka sums is given by the following result, hereafter occasionally referred to as the Płonka decomposition (or representation) theorem. Theorem 2.2.8. [173, Thm. II] Let A be an algebra of type τ with a partition function . The following conditions hold: 1. There is a set I such that A can be partitioned into a family { Ai }i∈ I , where any two elements a, b ∈ A belong to the same equivalence class Ai exactly when a = a b and b = b a. 2. If τ does not contain constants, each Ai is the universe of a subalgebra Ai of A. 3. The relation ≤ on I given by the condition i ≤ j ⇐⇒ there exist a ∈ Ai , b ∈ A j s.t. b a = b is a partial order and h I, ≤i is a semilattice. 4. For all i, j ∈ I such that i ≤ j and for all b ∈ A j , the map pij : Ai → A j , defined by pij ( x ) = x b is a homomorphism.

40

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES 5. If τ does not contain constants, A = h{Ai }i∈ I , h I, ≤i, { pij : i ≤ j}i is a semilattice direct system of algebras such that Pł (A ) = A.

Proof. 1. Observe that our claim is equivalent to showing that the relation ∼, defined as: a ∼ b if and only if a b = a and b a = b, for every a, b ∈ A, is an equivalence relation over A. Since a a = a by (PF1), ∼ is reflexive. The fact that ∼ is symmetric is obvious. For transitivity, suppose that a ∼ b and b ∼ c, for a, b, c ∈ A, i.e. a b = a, b a = b, b c = b and c b = c. Then a c = a (c b) = a (b c) = a b = a, where we have used (PF3). Analogously, c a = c ( a b) = c (b a) = c b = c. 2. If a1 , ..., an ∈ Ai and gn ∈ τ (n > 1), then for j, k ≤ n we have that a j ak = a j , whence by (PF5) ai gA ( a1 , ..., an ) = ai a1 ... an = ai . Lemma 2.2.2, on the other hand, implies that gA ( a1 , ..., an ) ai = ( a1 , ..., an ). Our claim follows. 3. To show that the definition of ≤ does not depend on the choice of a and b, we prove that if a ∈ Ai and b ∈ A j , i ≤ j iff b a = b. For the nontrivial direction, suppose a c = a, c a = c, b d = b, d b = d, d c = d. Then: gA

b a = b a c = b c a = b c = b d c = b d = b. Since a a = a, for every a ∈ Ai , we have that ≤ is reflexive. Now, suppose that i ≤ j and j ≤ i, i.e. there exists elements a ∈ Ai , b ∈ A j such that b a = b. By our previous remark, a b = a. Thus a ∼ b, whence i = j. For transitivity, suppose that i ≤ j ≤ k, whence there exist elements a ∈ Ai , b ∈ A j and c ∈ Ak such that b a = b and c b = c. Thus: c a = c b a = c b = c, which proves that i ≤ k, i.e. ≤ is a partial order over the index set I.

2.2. THE PŁONKA DECOMPOSITION THEOREM

41

In order to show that I is semilattice-ordered, suppose that i 6= j, and let a ∈ Ai , b ∈ A j . Thus, w.l.o.g., b a ∈ Ak for some k 6= j. We claim that k is the least upper bound of i and j. Indeed, b a a = b a, and b a b = b b a = b a, which shows that i, j ≤ k. Moreover, let i, j ≤ l, for some l ∈ I, whence there exists an element d ∈ Al such that d a = d, d b = d, for some a ∈ Ai , b ∈ A j . Then d b a = d a = d, i.e. k ≤ l. This settles our claim. We denote by x ∨ y the l.u.b. of x and y. 4. First, we prove that each map pij is well defined, i.e. it is independent of the choice of the element b ∈ A j . To this end, let c ∈ A j with c 6= b. Applying (PF1) and (PF3), we get x b = x (b c) = x (c b) = x c. With an eye to showing that pij is also a homomorphism, let a1 , . . . , an be elements in Ai and g any n-ary operation in τ (due to Remark 2.2.9 below we can assume n > 1, as constants live in the fibre whose index is the least element of I). Then pij ( g( a1 , . . . , an )) = g( a1 , . . . , an ) b

= g( a1 b, . . . , an b) = g( pij ( a1 ), . . . , pij ( an )), where, in the second equality, we have used (PF4). 5. In order to show that A is a semilattice direct system, we only need to check the final part of condition (iii) in Definition 2.1.(1) (as (i)-(ii) and the first part of (iii) have been proved above). Let a ∈ Ai , for some i ∈ I. Then pii ( a) = a b = a, for any choice of b ∈ Ai , which ensures that pii is the identity. Moreover, assume that i ≤ j ≤ k. Then, for some elements a ∈ Ai , b ∈ A j , c ∈ Ak , we use (PF2) to obtain: p jk ( pij ( a)) = p jk ( a b) = a b c = a p jk (b) = pik ( a). To prove that A = Pł (A ), it is enough to show that, for arbitrary elements a1 , . . . , an ∈ A and an n-ary operation g ∈ τ, the element g( a1 , . . . , an ) belongs to the same fibre as gPł (A) ( a1 , . . . , an ) (this is a consequence of the definition of operations in Płonka sums). So, let g( a1 , . . . , an ) ∈ Ak .

42

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES

W.l.o.g. we assume that a1 ∈ Ai1 , . . . , an ∈ Ain and set j = i1 ∨ · · · ∨ in ; then gPł (A) ( a1 , . . . , an ) ∈ A j . We claim that j = k. If b ∈ A j , then: b g ( a1 , . . . , a n ) = = = = = =

b b g ( a1 , . . . , a n ) b g ( a1 , . . . , a n ) b b a1 · · · a n b b a1 · · · a n b · · · b b ( a1 b ) · · · ( a n b ) b p i1 j ( a 1 ) · · · p i n j ( a n )

= b,

(PF1) (PF2-PF3) (PF5) (PF1) (PF3) (4) (1)

and this shows (by 3) that k ≤ j. The fact that j ≤ k follows from Lemma 2.2.2. This, in virtue of 3., implies that i1 , . . . , in ≤ k, therefore i1 ∨ · · · ∨ in = j ≤ k. We have proved that j = k and this shows that the operations coincide.  Remark 2.2.9. In case the type τ of A contains constants and the partition function is assumed to satisfy the further condition (PF6), then Theorem 2.2.8.(3) is enforced: I has the structure of a semilattice with least element i0 and all the constants of A belong to the fibre Ai0 . In order to show the latter, suppose that 0 and 1 are two constants in A, then, by (PF6) 0 1 = 1 and 1 0 = 1, which shows that they belong to the same fibre Ai0 . Moreover, let a ∈ Ai for some i ∈ I, then a 0 = a, therefore i0 ≤ i. Whenever an algebra A has a partition function and the above Theorem can be applied, we say that Pł (A ) is the Płonka sum representation of A. Actually, [173, Thm. II] proves more than we have established in Theorem 2.2.8. This paper contains proofs to the effect that: (i) Every representation of an algebra A as a Płonka sum Pł (A ) is obtained by starting with a suitable partition function in A. (ii) There is a bijective correspondence between partition functions in a given algebra and Płonka sum representations of the same algebra. Due to Theorem 2.2.8.(3), binary projections on the first component correspond (as partition functions) to trivial Płonka sum decompositions, namely, decompositions with a single fibre. Nevertheless, there are special classes of algebras whose members always admits a non-trivial representation. These will be discussed in the next section.

2.3. REGULAR VARIETIES

43

2.3 Regular varieties We now recall an important concept in universal algebra, crucially related to the material above. Definition 2.3.1. An identity ϕ ≈ ψ of type τ is regular provided that Var ( ϕ) = Var (ψ). Examples of regular identities are easy to come by: the associativity, the commutativity or the idempotency of a binary operation are all expressed by regular identities. Notably, the Płonka construction preserves all regular identities – on the other hand, it fails to preserve any other identity that is satisfied in all the fibres. This is the content of the following: Theorem 2.3.2. [173, Thm. I] If A is a semilattice direct system of type τ containing at least two algebras, then all regular identities satisfied in all algebras of A are satisfied in the algebra Pł (A ), whereas any other identity is not satisfied in Pł (A ). Proof. Let A = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i. We start by showing that Pł (A ) satisfies all the regular identities holding in {Ai }i∈ I . Let ϕ ≈ ψ be a regular identity, i.e., let Var ( ϕ) = Var (ψ) = { x1 , . . . , xn }. Let a1 , . . . , an be such that for all k ≤ n, ak ∈ Aik . Then ϕPł (A) ( a1 , . . . , an ) ∈ A j and ψPł (A) ( a1 , . . . , an ) ∈ A j , where j = i1 ∨ · · · ∨ in . By assumption Ai |= ϕ ≈ ψ, for all i ∈ I, whereby ϕPł (A) ( a1 , . . . , an ) = ψPł (A) ( a1 , . . . , an ). It follows that Pł (A ) |= ϕ ≈ ψ. For the remaining part, assume that ϕ ≈ ψ is a non-regular identity satisfied in Pł (A ), hence in each Ai . Let Var ( ϕ) ∪ Var (ψ) ⊆ { x1 , . . . , xn , y}, and let w.l.o.g. y ∈ Var (ψ) \ Var ( ϕ). Suppose that a ∈ Ak , b ∈ A j , with j 6= k (the existence of two different algebras in A is granted by our initial assumption). By construction of the Płonka sum Pł (A ), we have that ϕPł (A) ( a, . . . , a, b) ∈ Ak , while ψPł (A) ( a, . . . , a, b) ∈ A j∨k and, since Ak , A j∨k have disjoint universes, ϕPł (A) ( a, . . . , a, b) 6= ψPł (A) ( a, . . . , a, b). It follows that Pł (A ) 6|= ϕ ≈ ψ.  We are now interested in singling out varieties whose members are representable as Płonka sums. To attain this goal, we introduce a few additional concepts. Definition 2.3.3. A variety V is called regular if it satisfies regular identities only.

44

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES

Examples of regular varieties include semigroups, monoids and semilattices. A variety which is not regular is called irregular. Regular and irregular varieties have been amply investigated over time [112, 207]. A taxonomy can be found in [207, p. 231]. For our purposes, we will focus only on strongly irregular varieties. Definition 2.3.4. A variety V of type τ is strongly irregular if there is a τ-formula φ( x, y) such that V |= φ( x, y) ≈ x. In other words, being strongly irregular for a variety V of type τ means that there is a formula of the same type realising the projection operation on the first component in all algebras in the variety. Obviously, any strongly irregular variety is also irregular. All the varieties introduced in the Examples 2.2.4, 2.2.5, 2.2.6 and 2.2.7, as well as all congruence modular varieties [22, p. 5], are strongly irregular. The simplest example of an irregular variety which is not strongly irregular can be found in [176] (see also [205]). It is the variety of groupoids satisfying only the identity x · y ≈ x · z. Given a strongly irregular variety V , it is possible to associate to it a variety R(V ) which satisfies all and only the regular identities holding in V . R(V ) is called the regularisation of V . Elements of the regularisation of a strongly irregular variety can always be represented as Płonka sums. Theorem 2.3.5. [179, Thm. 7.1] Let V be a strongly irregular variety of type τ. Then A ∈ R(V ) iff A is decomposable as a Płonka sum over a semilattice direct system of algebras in V . Proof. By Theorem 2.2.8, in order to prove that an algebra A ∈ R(V ) is decomposable as a Płonka sum it is enough to show that A possesses a partition function. For this, preliminarly observe that, since V is strongly irregular, then there exists a formula φ( x, y) (where x, y actually occur) such that V |= φ( x, y) ≈ x. Note also that an arbitrary B ∈ V , upon defining for all a, b ∈ B, a b := φB ( a, b), satisfies all the equations (PF1)(PF5) in Definition 2.2.1. We give a sketch of (PF4) only. Let g be an arbitrary operation symbol of type τ. Then, for all a1 , . . . , an , b ∈ B, gB ( a1 , . . . , an ) b = gB ( a1 , . . . , an ) = gB ( a1 b, . . . , an b). Since (PF1)-(PF5) are regular identities satisfied by V , then they also hold in R(V ), from which we conclude that φA is a partition function on A. The fact that the fibres of the Płonka sum are algebras in V follows from Theorem 2.2.8.(1). The converse direction follows from Theorem 2.3.2. 

2.3. REGULAR VARIETIES

45

The above theorem does not hold for algebras in the regularisations of arbitrary irregular varieties: if V is an irregular, but not strongly irregular variety, then R (V ) need not consist in Płonka sums of algebras in V , nor even in subalgebras of Płonka sums of algebras in V [109]. To get a suitable representation in this case, we have to replace Płonka sums by more general (and more complicated) constructions, the so-called coherent Lallement sums: see e.g. [204] or [205, §4]. The extant theory of regular varieties is dominated by investigations into regularisations of strongly irregular varieties, like the variety of distributive lattices and of Boolean algebras. We will come back to them in the following sections.

2.3.1

τ-semilattices

It is already clear from the material covered so far that semilattices and semilattices with zero play a fundamental role in the theory of regular varieties. The reason behind this fact is that the variety SL of semilattices has a counterpart in any arbitrary similarity type without constants ([206, p. 31]; if the type contains constants, semilattices with zero must be considered instead). For each type τ, this counterpart is the regularisation of the trivial variety of type τ. In this short subsection, we give a quick proof of these results.

 Definition 2.3.6. Let S = S, ∨S , 0S be a semilattice with zero, and let τ be a similarity type. Sτ is the algebra of type τ whose universe is S and whose operations are so defined: τ

• for any constant c ∈ τ, cS = 0S ; τ

• for any unary f ∈ τ and any a ∈ S, f S ( a) = a; • for any k-ary g ∈ τ (k > 2) and any a1 , ..., ak ∈ S, τ

gS ( a1 , ..., ak ) = a1 ∨S ... ∨S ak . Any such algebra is called a τ-semilattice. We set SLτ = {Sτ | S ∈ SL}. Lemma 2.3.7. SLτ is a variety of type τ. Proof. Let Sτ ∈ SLτ , and let A be a subalgebra of Sτ with universe A. Let h AiS be the subsemilattice generated in S by A. Clearly, A = (h AiS )τ , whence A ∈ SLτ , and SLτ is closed under subalgebras.

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46

To check closure under products, for the sake of simplicity we spell out in detail only the binary case. It suffices to prove that Sτ × Tτ = (S × T)τ . In fact, these two algebras share the same universe S × T. As regards constants and unary operation symbols, D E D τ τE τ τ τ c ( S × T ) = 0S × T = 0S , 0T = c S , c T = c S × T ; D τ E τ τ τ τ f (S×T) (h a, bi) = h a, bi = f S ( a) , f T (b) = f S ×T (h a, bi) . For operation symbols of arity greater than 1, τ

g(S×T) (h a1 , b1 i , ..., h ak , bk i) = h a1 , b1 i ∨S×T ... ∨S×T h ak , bk i  = Da1 ∨S ... ∨S ak , b1 ∨T ... ∨T bk E τ τ = gS ( a1 , ..., ak ) , gT ( a1 , ..., ak )

= gS

τ ×Tτ

(h a1 , b1 i , ..., h ak , bk i) .

Finally, since τ contains at least a binary operation, then S and Sτ have the same congruences, so that Sτ /θ = (S/θ )τ and SLτ is closed under quotients.  Hereafter, we denote by T the trivial variety of a given type τ. Theorem 2.3.8. [206, Prop. 235] Let τ be a similarity type. The following varieties of type τ coincide with one another: (A) R (T ); (B) the class of Płonka sums over a semilattice direct system of trivial algebras of type τ; (C) SLτ . Proof. Notice that T is strongly irregular, because it satisfies e.g. f ( x, y) ≈ x, where f is the binary operation symbol we assumed to exist in τ. Therefore (A) is included in (B) by Theorem 2.3.5.

 It is readily observed that whenever A = {Ai }i∈ I , I, { pij | i ≤I j} , with all of the Ai ’s trivial, P ł(A ) is Iτ . Therefore (B) is included in (C). For the remaining inclusion, we show that SLτ satisfies all regular identities of type τ and fails any other identity. Thus, let ϕ ( x1 , ..., xn ) ≈ ψ ( x1 , ..., xn ) be a regular identity of type τ. We have to prove that it is satisfied in any τ-semilattice. Thus, let Sτ be a τ-semilattice. If ϕ and ψ τ τ contain no variable, then ϕS = ψS = 0S . If ϕ and ψ contain a single

2.3. REGULAR VARIETIES

47 τ

τ

variable x and a ∈ S, then ϕS ( a) = ψS ( a) = a, by Definition 2.3.6 and the idempotency of join. If n > 1 and a1 , ..., an ∈ S, then by the associativity, commutativity and idempotency of join, together with the fact that 0S is the least element of S, τ

τ

ϕS ( a1 , ..., an ) = ψS ( a1 , ..., an ) = a1 ∨S ... ∨S an . This establishes our claim. Let now ϕ ≈ ψ be an irregular identity of type τ. Let Sτ be a nontrivial τ-semilattice such that Sτ  ϕ ≈ ψ, and let a, b ∈ S, with a 6= b. W.l.o.g., a 6= 0S . If ψ does not contain variables and ϕ contains the variables τ τ x1 , ..., xn , then ϕS ( a, ..., a) = a 6= 0S = ψS ( a, ..., a), a contradiction. If ϕ does not contain variables (whence ψ must contain at least one variable), we argue similarly. Otherwise, let Var ( ϕ) ∪ Var (ψ) = { x1 , ..., xn , y}, where w.l.o.g. y ∈ Var ( ϕ) \ Var (ψ). Then τ

τ

a ∨S b = ϕS ( a, ..., a, b) = ψS ( a, ..., a, b) = a, whence b ≤S a. Similarly, a ≤S b and thus a = b, a contradiction again.  Observe that semilattices coincide with h2i-semilattices, and that semilattices with zero can be identified with h2, 0i-semilattices.

2.3.2 Subdirectly irreducible algebras Strongly irregular varieties are amenable to a nice description of their subdirectly irreducible members. This result, due to Lakser, Padmanabhan and Platt [139], needs to be mentioned also for the construction it involves, which will be used several times in the book. Let A be an algebra of a fixed type τ and let ∞ 6∈ A. Let us define a new algebra A∗ whose universe is A∗ = A ∪ {∞} and whose operations, for every g ∈ τ and a1 , . . . , an ∈ A∗ , are defined as follows: ( ∞ if ∞ ∈ { a1 , . . . , an }, A∗ g ( a1 , . . . , a n ) = gA ( a1 , . . . , an ) otherwise. The element ∞ is called an absorbing element in the algebra A∗ . The meaning of this should appear evident: whenever ∞ is an argument of an operation g then the result is ∞, irrespective of any other argument involved. The algebra A∗ constructed above is nothing but the Płonka sum over a semilattice direct system whose index set is the 2-element

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES

48

chain i < j, where Ai = A and A j is the trivial τ-algebra with universe {∞}. The system is completed by the identity automorphisms on Ai and A j and by the unique homomorphism pij : Ai → A j . Example 2.3.9. The 3-element algebra WK = h{0, 1, n}, ∧, ∨, ¬i (see Section 1.2.1), whose operations are defined by the weak Kleene tables:

¬ 0 n 1

1 n 0

∧ 0 n 1

0 0 n 0

n n n n

1 0 n 1

∨ 0 n 1

0 0 n 1

n n n n

1 1 n 1

is obtained applying the construction outlined above, adding to the 2element Boolean algebra B2 the absorbing element n.  For proving our main result on subdirectly irreducibles, we need an appropriate lemma. Lemma 2.3.10. Let A be an algebra such that A ∼ = Pł (A ), where A = h{Ai }i∈ I , I, { pij | i ≤I j}i is a semilattice direct system of algebras. If A is subdirectly irreducible, then the semilattice I is subdirectly irreducible. Proof. For every θ ∈ Con(I) we define a relation θ A ⊆ A2 as follows, for every a ∈ Ai , b ∈ A j :

h a, bi ∈ θ A iff hi, ji ∈ θ and there exists k ∈ I, i, j ≤ k such that pik ( a) = p jk (b). It is routine to check that θ A is a congruence on A. Let D be a set of T congruences on I such that {θ | θ ∈ D } = ∆I . We have to show that ∆I ∈ D. Define D 0 = D ∪ {ψ | ∆I 6= ψ ∈ Con(I), ψ ⊆ θ, for some θ ∈ D }. Clearly, {θ | θ ∈ D 0 } = ∆I . We claim that {θ A | θ ∈ D 0 } = ∆A and, since A is subdirectly irreducible, there exists θ0A ∈ D 0 such that θ0A = ∆A , from which it follows that θ0 = ∆I ∈ D. T In order to show the above claim, let h x, yi ∈ {θ A | θ ∈ D 0 }. Since T {θ | θ ∈ D 0 } = ∆I then x, y belong to the same fibre Ai of Pł (A ), for some i ∈ I. Moreover, for each θ ∈ D 0 , there exists an index k(θ ) ∈ I such that i ≤ k (θ ) and pik(θ ) ( x ) = pik(θ ) (y). We now show that there exists θ 0 ∈ D 0 such that k (θ 0 ) = i, which implies that x = y. Suppose, by T contradiction, that there is no such θ 0 . Since {θ | θ ∈ D 0 } = ∆I , there exist θ1 , θ2 ∈ D 0 with θ1 6= θ2 , whence k (θ1 ) 6= k (θ2 ) and i < k (θ1 ), k(θ2 ). T

T

2.3. REGULAR VARIETIES

49

Therefore i < k (θ2 ) < k (θ1 ) ∨ k (θ2 ). Let Φ be the smallest congruence on I containing the pair hk(θ1 ), k (θ1 ) ∨ k (θ2 )i. As hi, k (θ2 )i ∈ θ2 , we have that Φ ⊆ θ2 , hence Φ ∈ D 0 . Moreover, it is easily verified that, if hz, i i ∈ Φ then either i = z or k (θ1 ) ≤ i. Since the latter is not the case then the Φ-equivalence class of i is the singleton {i }, i.e. i = k (Φ), the desired contradiction which shows the claim.  Theorem 2.3.11. [139, p. 487] Let A be a nontrivial algebra such that A ∼ = Pł (A ), where A = h{Ai }i∈ I , I, { pij | i ≤ j}i is a semilattice direct system of algebras in a variety V . Then A is subdirectly irreducible if and only if one of the following holds: 1. A is subdirectly irreducible in V ; 2. A ∼ = B∗ , with B a subdirectly irreducible algebra in V that has no absorbing elements. Proof. By Lemma 2.3.10 we know that, if A is subdirectly irreducible then also I = h I, ≤i is subdirectly irreducible as semilattice, i.e., I is either trivial or the chain with two elements i, j s.t. i < j. If the former, then we directly get case (1). If the latter, then we define two relations θ1 , θ2 on A as follows: h x, yi ∈ θ1 iff x = y or x, y ∈ A j ,

h x, yi ∈ θ2 iff plj ( x ) = plj (y) for x, y ∈ Al , with l ∈ {i, j}. It is easy to check that θ1 , θ2 are congruences and θ1 ∩ θ2 = ∆; moreover, w.l.o.g. we can assume that θ2 6= ∆. Since A is s.i., then necessarily θ1 = ∆, i.e. | A j |= 1. Therefore we have shown that A ∼ = Ai∗ . In order to conclude our proof we show that A is subdirectly irreducible if and only if Ai is subdirectly irreducible and Ai has no absorbing elements. From left to right, suppose contrapositively that a ∈ Ai is an absorbing element. Define two congruence relations, Φ1 and Φ2 , on A as follows:

h x, yi ∈ Φ1 iff x, y ∈ Ai or x = y; h x, yi ∈ Φ2 iff { x, y} ⊆ { a, ∞} or x = y. Observe that, since a is an absorbing element, then Φ2 6= ∆ and since | A | > 1 we also have Φ1 6= ∆. However, Φ1 ∩ Φ2 = ∆, which shows that A is not subdirectly irreducible. From right to left, assume that Ai has no absorbing element. Then, any congruence on A, distinct from the total relation ∇, is determined by its restriction to Ai . In particular, for any Θ ∈ Con(A), if x 6= y then h x, yi ∈ Θ implies that x, y ∈ Ai . Therefore A is subdirectly irreducible if and only if Ai is. 

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES

50

Combining Theorem 2.3.11 with Theorem 2.3.5 we obtain the following Corollary 2.3.12. Let V be a strongly irregular variety of type τ. Then the subdirectly irreducible elements in R(V ) are: (i) the subdirectly irreducible members of V ; (ii) the algebras of the form A∗ , where A is nontrivial and subdirectly irreducible in V ; (iii) the 2-element τ-semilattice. Observe that the 2-element τ-semilattice mentioned in (iii) is nothing but T∗ , where T is the trivial τ-algebra (which is subdirectly irreducible).

2.3.3 Subvarieties and equational bases Interestingly enough, the lattices of subvarieties of regularisations of strongly irregular varieties can be conveniently described. The main result along these lines is due to Dudek and Graczynska [77] (see also [22]). We first ´ state without a proof an auxiliary lemma – hereafter, given a variety V , we denote by L[V ] the lattice of its subvarieties.

Lemma 2.3.13. [77, Thms. 1 and 2] 1. If V is an irregular variety, then for any W ∈ L [V ], R (W ) covers W in the lattice L [V ]. 2. If V is a strongly irregular variety, then the mapping given by h (W ) = R (W ) embeds L [V ] into L [ R (V )]. We now are in a position to proceed to the proof of the main result by Dudek and Graczynska. By D2 , we mean the 2-element chain on {0, 1}, ´ understood as a lattice. Theorem 2.3.14. [77, Thm. 3] If V is a strongly irregular variety, then L [ R (V )] is isomorphic to L [V ] × D2 . Proof. Define h : L [V ] × {0, 1} → L [ R (V )] via the following stipulations: h (hU , 0i) = U and h (hU , 1i) = R (U ) .

2.3. REGULAR VARIETIES

51

Clearly, h is well-defined. For injectivity, suppose h (hU , i i) = h (hW , ji). If this variety is irregular, then i = j = 0 and thus U = W . If it is regular, U = W by Lemma 2.3.13.(2). We now check that   h hU , i i ∨L[V ]×D2 hW , ji = h (hU , i i) ∨L[ R(V )] h (hW , ji) .

We distinguish three cases: (A) i = j = 1; (B) i = j = 0; (C) w.l.o.g., i = 0, j = 1. (A)   D E h hU , 1i ∨L[V ]×D2 hW , 1i = h U ∨L[V ] W , 1   = R U ∨L[V ] W

= R (U ) ∨L[ R(V )] R (W ) = h (hU , 1i) ∨L[ R(V )] h (hW , 1i) , where the third equality follows from Lemma 2.3.13.(2). (B)   D E h hU , 0i ∨L[V ]×D2 hW , 0i = h U ∨L[V ] W , 0

= U ∨L[V ] W = U ∨L[ R(V )] W = h (hU , 0i) ∨L[ R(V )] h (hW , 0i) , where, again, the third equality is implied by Lemma 2.3.13.(2). (C)   D E h hU , 0i ∨L[V ]×D2 hW , 1i = h U ∨L[V ] W , 1   = R U ∨L[V ] W

= U ∨L[ R(V )] R (W ) = h (hU , 0i) ∨L[ R(V )] h (hW , 1i) . L[V ] The third equality is justified W  as follows.  By Lemma 2.3.13.(1), U ∨ L[V ] is covered in L [ R (V )] by R U ∨ W . However,

  U ∨L[ R(V )] W ≤ U ∨L[ R(V )] R (W ) ≤ R (U ) ∨L[ R(V )] R (W ) = R U ∨L[V ] W , whence our equality ensues. We are now left with the task of proving that h is onto. However, every W ∈ L [V ] is the image under h of some hW , 0i. Assume now that U ∈

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES

52

L [ R (V )] \ L [V ]. Then there exists an A satisfying only regular identities s.t. U = HSP (A) (A can be taken to be e.g. the free U -algebra with
denumerably many generators). By Theorem 2.3.5, A = Pł {Ai }i∈ I , where each Ai belongs to V . Moreover, the set of identities satisfied by A (hence by U ) is the intersection of the sets of regular identities satisfied by each Ai , which is the same as the set of all regular identities satisfied by ∏ Ai ∈ V . Thus: i∈ I

Y := HSP

∏ Ai

!

≤ V.

i∈ I

Since ∏ Ai and Y satisfy the same regular identities, then for any formui∈ I

las ϕ and ψ of the appropriate type, R (Y )  ϕ ≈ ψ ⇔ Y  ϕ ≈ ψ and ϕ ≈ ψ is regular ⇔ ∏ Ai  ϕ ≈ ψ and ϕ ≈ ψ is regular i∈ I

⇔ A  ϕ ≈ ψ and ϕ ≈ ψ is regular ⇔ Aϕ≈ψ ⇔ U  ϕ ≈ ψ. Thus, R (Y ) = U , meaning that U = h (hY , 1i).



If we are given some variety, it is natural to search appropriate equational bases for its subvarieties. Let us now recall a few very general results that bridge the equational bases of irregular varieties and those of their regularisations. Our starting point is the following basic fact. Theorem 2.3.15. [205, Theorem 7.1] Let V be an irregular variety of a given type τ. Then V is axiomatised by the set of all regular identities satisfied in V and any irregular identity of V . Proof. By Lemma 2.3.13.(1)



The above result is the key to obtaining an equational basis for an irregular variety V out of a given basis for its regularisation. From Theorem 2.3.15 we immediately get: Corollary 2.3.16. Let V be an irregular variety of type τ and let Σ be an equational basis for its regularisation R(V ). Then there exists a binary τ-formula x · y such that Σ ∪ { x · y ≈ x · z} is an equational basis for V .

2.3. REGULAR VARIETIES

53

Notice that, if V is, in particular, a strongly irregular variety, then the form of the above identity can be taken to be x · y ≈ x. The problem of providing an equational basis for R(V ) given a basis for V is more complicated. We confine our analysis to the case of strongly irregular varieties and refer the interested reader directly to [205]. We state without a proof the following: Theorem 2.3.17. [205, Theorem 7.6] Let V be a strongly irregular variety of type τ defined by a set Σ of regular identities together with an identity x · y ≈ x, for a binary τ-formula x · y. Then R(V ) is axiomatised by the identities in Σ and the following: 1. x · x ≈ x; 2. x · (y · z) ≈ ( x · y) · z, 3. x · (y · z) ≈ x · (z · y), 4. g( x1 , . . . , xn ) · y = g( x1 · y, . . . , xn · y), 5. y · g( x1 , . . . , xn ) = y · x1 ·... · xn , for any n-ary operation g ∈ τ. Observe that the conditions in the above theorem coincide with the clauses that define a partition function. It readily follows that, if V has a finite basis, then so does its regularisation. The most well-known examples of regular varieties that are representable as Płonka sums are the regularisations of distributive lattices and of Boolean algebras. It is to these that we now turn.

2.3.4 An example: Bisemilattices Bisemilattices and distributive bisemilattices have been investigated by Płonka [174] and many other algebraists, including Balbes, Kalman, Romanowska and Harding [10, 105, 129, 161, 112]. In particular, distributive bisemilattices served as the motivating example for the construction of Płonka sums. A bisemilattice is an algebra A = h A, ∧, ∨i of type h2, 2i such that both ∧ and ∨ are idempotent, associative and commutative operations. A Birkhoff system is a bisemilattice satisfying the identity x ∧ ( x ∨ y) ≈ x ∨ ( x ∧ y). Finally, a quasilattice is a Birkhoff system that satisfies the quasi-identity x ∨ y ≈ x ⇒ ( x ∧ z) ∨ (y ∧ z) ≈ x ∧ z.

54

CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES

Padmanhaban [161] proved that this quasi-identity can be replaced by identities, and thus quasilattices form a variety. In addition, he showed that such a variety is the regularisation of the variety of lattices. Clearly, both lattices and h2, 2i-semilattices are examples of quasilattices. Any bisemilattice A induces two different partial orders, namely x ≤∧ y iff x ∧ y = x and x ≤∨ y iff x ∨ y = y. These partial orders coincide if and only if A is a lattice, and are dual to each other if and only if A is a h2, 2i-semilattice. Bisemilattices where ∧ (∨ respectively) distributes over ∨ (∧ respectively) will be denoted as distributive bisemilattices. Observe that distributive bisemilattices are quasilattices. The varieties of bisemilattices and of distributive bisemilattices will be respectively denoted by BSL and by DBSL. Example 2.3.18. The paramount example of a distributive bisemilattice is the 3-element algebra 3 = h{0, 1, n}, ∧, ∨i, which is the h2, 2i-reduct of the algebra WK of Example 2.3.9. Recall that the operations are defined as follows:

∧

0

n

1

∨

0

n

1

0

0

n

0

0

0

n

1

n

n

n

n

n

n

n

n

1

0

n

1

1

1

n

1

The two partial orders may be represented by the Hasse diagrams in Figure 2.1. The absorption identity x ≈ x ∨ ( x ∧ y) and its dual are irregular, and it is easy to check that 3 fails them. Indeed, 0 ∨ (0 ∧ n) = n 6= 0.  Although the absorption identities fail to hold in DBSL, appropriate restrictions of this principle are indeed satisfied: Lemma 2.3.19. Every distributive bisemilattice satisfies the following identities: x ∨ y ∨ ( x ∧ y) x ∧ y ∧ ( x ∨ y) x ∨ ( x ∧ y) ∨ (y ∧ z) x ∧ ( x ∨ y) ∧ (y ∨ z)

≈ ≈ ≈ ≈

x ∨ y; x ∧ y; x ∨ ( y ∧ z ); x ∧ ( y ∨ z ).

2.3. REGULAR VARIETIES

55

n • 1 •

• 1

≤∨

0 •

≤∧

• 0 • n

Figure 2.1: The two partial orderings induced in 3 by join (left-hand side), and meet (right-hand side). Proof. If A ∈ DBSL and a, b ∈ A, we have for example that a ∨ b ∨ ( a ∧ b) = ( a ∨ b ∨ a) ∧ ( a ∨ b ∨ b) = ( a ∨ b) ∧ ( a ∨ b) = a ∨ b.



Płonka proved the following: Theorem 2.3.20 ([174, Thm. 3]). Any distributive bisemilattice is isomorphic to the Płonka sum over a semilattice direct system of distributive lattices. Since distributive lattices form a strongly irregular variety, by Theorem 2.3.5 the variety of distributive bisemilattices is the regularisation of the variety of distributive lattices, i.e. it is the variety satisfying exactly the regular identities holding for distributive lattices. By applying Corollary 2.3.12, we directly obtain that DBSL is generated by the 3-element algebra 3. More precisely: Proposition 2.3.21. The only nontrivial subdirectly irreducible distributive bisemilattices are:

(i) 3; (ii) the 2-element distributive lattice D2 ; (iii) the 2-element h2, 2i-semilattice S2 . Since D2 , S2 are subalgebras of 3, DBSL = V (3). Before the general theory of Płonka sums was fully developed, Proposition 2.3.21 was proved by Kalman [129], by introducing an ad-hoc technique.

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2.4 Generalised involutive bisemilattices 2.4.1 Definition and elementary properties Boolean algebras can be defined in a number of different similarity types. Two of the most popular choices are the types τ1 = h2, 2, 1, 0, 0i, with primitive operation symbols ∧, ∨, ¬, 0, and 1, and the type τ2 = h2, 2, 1i, already encountered in Chapter 1, which differs from τ1 in that the constants 0 and 1 are dropped. For the sake of definiteness, we respectively call BA0,1 and BA these varieties, which are obviously term-equivalent because 0 and 1 can be defined in τ2 as x ∧ ¬ x and x ∨ ¬ x, respectively, for x an arbitrary variable. Similarly, we denote by B2 and B0,1 2 the 2-element Boolean algebra in the types τ2 and τ1 , respectively.3 The choice of the type τ1 or τ2 , however, is far from being inconsequential when it comes to regularising Boolean algebras. As we will see, BA0,1 and BA have quite different regularisations, respectively called involutive bisemilattices (first introduced by Płonka in [177], although the name comes from [33]) and generalised involutive bisemilattices (introduced in [164]). These varieties are allotted a whole section here, since, as we will see, they, and some subclasses thereof, will play an important role in the algebraic semantics of variable inclusion logics associated with classical logic. Definition 2.4.1. A generalised involutive bisemilattice is an algebra B = h B, ∧, ∨, ¬i of type τ2 satisfying: I1. x ∨ x ≈ x; I2. x ∨ y ≈ y ∨ x; I3. x ∨ (y ∨ z) ≈ ( x ∨ y) ∨ z; I4. ¬¬ x ≈ x; I5. x ∧ y ≈ ¬(¬ x ∨ ¬y); I6. x ∧ (¬ x ∨ y) ≈ x ∧ y. An involutive bisemilattice is an algebra B = h B, ∧, ∨, ¬, 0, 1i of type τ1 such that h B, ∧, ∨, ¬i is a generalised involutive bisemilattice and the following conditions are satisfied: 3 We will not rigorously observe the same notational distinction as regards the 2element semilattice, which we will denote as S2 in any type, relying on the context for disambiguation.

2.4. GENERALISED INVOLUTIVE BISEMILATTICES

57

I7 0 ∨ x ≈ x; I8 1 ≈ ¬0. Thus, the classes of generalised involutive bisemilattices and involutive bisemilattices are varieties, which we denote by GIB and IBSL, respectively. One can readily see that every generalised involutive bisemilattice has, in particular, the structure of a join semilattice, by virtue of axioms (I1)– (I3); by (I7), in the case of an involutive bisemilattice, this is a join semilattice with zero. Moreover, the h2, 2i-reduct of an arbitrary (generalised) involutive bisemilattice is a bisemilattice, whence the label we have chosen is not a misnomer. Notice that, by virtue of axioms (I5) and (I8), the operations ∧ and 1 are completely determined by ∨, ¬, and 0. In Definition 2.4.1 binary operations are not assumed to distribute over each other, as this property can be derived from the other identities (see Proposition 2.4.7). Example 2.4.2. Boolean algebras (expressed in the type τ2 ) and τ2 -semilattices are generalised involutive bisemilattices. More precisely, Boolean algebras are those members of GIB for which ≤∧ =≤∨ , while τ2 -semilattices – henceforth also called semilattices, with a slight abuse – are those members of GIB for which ≤∧ is dual to ≤∨ . The variety of semilattices will be denoted by SLτ2 . Similarly, Boolean algebras in the type τ1 and τ1 -semilattices (henceforth also called semilattices with zero) are involutive bisemilattices.

Example 2.4.3. Our example of primary interest is the generalised involutive bisemilattice WK introduced in Example 2.3.9. Upon considering the order ≤∨ induced by join in its bisemilattice reduct, it becomes a 3-element chain with n = ¬n as its top element. As already observed, this algebra is isomorphic to the unique Płonka sum over the 2-element chain {i, j} (i < j) such that Ai = B2 and A j = 1, the trivial algebra of type τ2 . We can represent WK, as well as the 2-element Boolean algebra B2 and the 2-element semilattice S2 , by means of the following diagrams (the dashes represent ≤∨ , while the arrows represent negation): a

1 S2 =

B2 = 0

n WK = 1

1=0

0

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Analogous considerations hold for the algebra WK0,1 , which differs from WK for the presence in the type of the constants 0, 1 and is an involutive bisemilattice.  It is not difficult to check that every generalised involutive bisemilattice has also the structure of a meet semilattice, and that the identities x ∨ y ≈ ¬(¬ x ∧ ¬y) x ∨ y ≈ x ∨ (¬ x ∧ y)

(I9) (I10)

are satisfied. Indeed, given B ∈ GIB and a, b ∈ B, a ∨ b = ¬¬(¬¬ a ∨ ¬¬b) = ¬(¬ a ∧ ¬b), by virtue of (I4) and (I5), and a ∨ (¬ a ∧ b) = ¬(¬ a ∧ ¬(¬ a ∧ b)) = ¬(¬ a ∧ ¬(¬¬¬ a ∧ ¬¬b)) = ¬(¬ a ∧ (¬¬ a ∨ ¬b)) = ¬(¬ a ∧ ¬b) = a ∨ b, by virtue of (I4), (I6), and (I9). Notice that (I9) and (I10) are the result of swapping ∨ and ∧ in axioms (I5) and (I6), respectively. Given any property ( P) of type τ1 or τ2 , we call the property ( P0 ) that results from swapping ∨ and ∧, as well as 0 and 1, the dual of ( P). More generally, the following duality principle holds: Proposition 2.4.4. The following holds: 1. If B = h B, ∧, ∨, ¬i ∈ GIB , then also B∂ = h B, ∨, ∧, ¬i ∈ GIB , and moreover the map ¬ : B → B∂ is an isomorphism. Therefore, given any property ( P) of type τ2 , we have that ( P) is true in GIB if and only if ( P0 ) is also such. 2. If B = h B, ∧, ∨, ¬, 0, 1i ∈ IBSL, then also B∂ = h B, ∨, ∧, ¬, 1, 0i ∈ IBSL, and moreover the map ¬ : B → B∂ is an isomorphism. Therefore, given any property ( P) of type τ1 , we have that ( P) is true in IBSL if and only if ( P0 ) is also such. Proposition 2.4.4 is a very useful tool in computations. We will use it with no further mention. Proposition 2.4.5. Every B ∈ GIB satisfies the following identities: 1. x ∨ ( x ∧ y) ≈ x ∨ (y ∧ ¬y); 2. ( x ∧ ¬ x ) ∨ ¬( x ∧ ¬ x ) ≈ x ∨ ¬ x;

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59

3. x ∨ ( x ∧ y) ≈ x ∧ ( x ∨ y); 4. x ∨ (y ∧ z) ≈ x ∨ (( x ∨ y) ∧ z); 5. x ∨ ¬ x ∨ y ≈ y ∨ ¬y ∨ x. Proof. Let a, b, c ∈ B. Then:

(1)

a ∨ ( a ∧ b) = = = =

(2)

( a ∧ ¬ a) ∨ ¬( a ∧ ¬ a) = ( a ∧ ¬ a) ∨ ¬ a ∨ a = a ∨ (¬ a ∧ a) ∨ ¬ a = a ∨ a ∨ ¬a = a ∨ ¬a

(3)

(4)

a ∨ ( a ∧ b) = = = = = = = =

a ∨ (¬ a ∧ a ∧ b) a ∨ (¬ a ∧ ( a ∨ ¬b) ∧ b) a ∨ (¬ a ∧ ¬b ∧ b) a ∨ (b ∧ ¬b)

(I10) (I6) (I4, I6) (I2), (I10)

(I10)

a ∨ (b ∧ ¬b) ( a ∨ (b ∧ ¬b)) ∧ ( a ∨ (b ∧ ¬b)) ( a ∨ (b ∧ ¬b)) ∧ ( a ∨ (b ∧ ¬b) ∨ (b ∧ ¬b)) ( a ∨ (b ∧ ¬b)) ∧ ((b ∧ ¬b) ∨ ¬(b ∧ ¬b)) ( a ∨ (b ∧ ¬b)) ∧ (b ∨ ¬b) (b ∨ ¬b) ∧ ( a ∨ ¬(b ∨ ¬b)) a ∧ (b ∨ ¬b) a ∧ ( a ∨ b)

a ∨ (b ∧ c) = a ∨ (¬ a ∧ b ∧ c) = a ∨ (¬ a ∧ ( a ∨ b) ∧ c) = a ∨ (( a ∨ b) ∧ c)

(I10) (I4, I6) (I10)

(1)

(1) (2) (I6) (1)



(5) Let a, b ∈ B. By two applications of (I10), we obtain: a ∨ ¬ a ∨ b = a ∨ (¬ a ∧ ¬b) ∨ b = a ∨ ¬b ∨ b = b ∨ ¬b ∨ a. Proposition 2.4.6. In every involutive bisemilattice B the following identities are satisfied: 1. x ∨ ¬ x ≈ 1 ∨ x;

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CHAPTER 2. PŁONKA SUMS AND REGULAR VARIETIES 2. 1 ∨ x ≈ 1 ∨ ¬ x.

Proof. By taking y = 0 in (5) of the previous proposition, we obtain x ∨ ¬ x = x ∨ ¬ x ∨ 0 = 0 ∨ ¬0 ∨ x = 1 ∨ x. (2) is a consequence of the symmetry of (1).  Notice that the bisemilattice reducts of generalised involutive bisemilattices are Birkhoff systems by Proposition 2.4.5. We are now able to show that distributivity follows from the axiomatisation in Definition 2.4.1. Proposition 2.4.7. If B = h B, ∧, ∨, ¬i ∈ GIB , then h B, ∧, ∨i is a distributive bisemilattice, that is, the identity x ∨ (y ∧ z) ≈ ( x ∨ y) ∧ ( x ∨ z) and its dual are satisfied. Proof. Let a, b, c ∈ B. Then: a ∨ (b ∧ c) = = = = = = = = = = = = = =

a ∨ (b ∧ c) ∨ (b ∧ c) a ∨ (b ∧ ( a ∨ c)) ∨ (b ∧ c) a ∨ (b ∧ c) ∨ (b ∧ ( a ∨ c)) a ∨ (b ∧ c) ∨ (b ∧ ( a ∨ (b ∧ c))) ( a ∨ (b ∧ c)) ∨ (( a ∨ (b ∧ c)) ∧ b) ( a ∨ (b ∧ c)) ∧ (( a ∨ (b ∧ c)) ∨ b) ( a ∨ (b ∧ c)) ∧ (b ∨ a ∨ (b ∧ c)) ( a ∨ (b ∧ c)) ∧ (b ∨ a ∨ (( a ∨ b) ∧ c)) ( a ∨ (b ∧ c)) ∧ ( a ∨ b) ∧ ( a ∨ b ∨ c) ( a ∨ (( a ∨ b) ∧ c)) ∧ ( a ∨ b) ∧ ( a ∨ b ∨ c) ( a ∨ c) ∧ ( a ∨ b) ∧ ( a ∨ b ∨ c) ( a ∨ c) ∧ ( a ∨ b ∨ (( a ∨ b) ∧ c)) ( a ∨ c) ∧ ( a ∨ b ∨ ( a ∧ c)) ( a ∨ c ) ∧ ( a ∨ ( a ∧ c ) ∨ b ),

Prop. 2.4.5.(4) Prop. 2.4.5.(4) Prop. 2.4.5.(3) Prop. 2.4.5.(4) Prop. 2.4.5.(3) Prop. 2.4.5.(4) Prop. 2.4.5.(4) Prop. 2.4.5.(3) Prop. 2.4.5.(4)

and by Proposition 2.4.5.(3)–(4), ( a ∨ c) ∧ ( a ∨ ( a ∧ c) ∨ b) = ( a ∨ c) ∧ (( a ∧ ( a ∨ c)) ∨ b) = ( a ∨ b) ∧ ( a ∨ c).  In [91] Finn and Grigolia, and independently Brzozowski in [42], introduce, under the same name of De Morgan bisemilattices, another expansion of DBSL by constants 0, 1 and an involutive negation operation that

2.4. GENERALISED INVOLUTIVE BISEMILATTICES

61

obeys the De Morgan laws. It is not difficult to check that De Morgan bisemilattices are nothing but the regularisation of De Morgan algebras; see [11, Ch. 9] for these notions. We already observed that every generalised involutive bisemilattice is equipped with the two partial orderings ≤∨ and ≤∧ inherited by its bisemilattice reduct. Rather unsurprisingly, these two orders are closely related, since ∧ is completely determined by ∨ and ¬. Lemma 2.4.8. Let B ∈ GIB . Then, for every a, b ∈ B, a ≤∨ b ⇐⇒ ¬b ≤∧ ¬ a. Proof. If a, b ∈ B, then using (I4) and (I9) we have: a ≤∨ b

⇐⇒ a∨b = b ⇐⇒ ¬(¬ a ∧ ¬b) = b ⇐⇒ ¬ a ∧ ¬b = ¬b ⇐⇒ ¬b ≤∧ ¬ a. 

Corollary 2.4.9. In every semilattice (with zero), the order ≤∧ is the dual of the order ≤∨ .

2.4.2 The structure of the Boolean subalgebras We now prove that any generalised involutive bisemilattice can be partitioned into a number of Boolean subalgebras, which later will be shown to correspond to the fibres of its Płonka sum representation. In the rest of this chapter we will write ≤ in place of ≤∨ , since the other ordering will not play an important role. Definition 2.4.10. Let B ∈ GIB . The element c ∈ B is said to be positive if ¬c ≤ c, negative if c ≤ ¬c, and fix if ¬c = c. We denote the set of positive elements of B by P(B). Clearly, by Proposition 2.4.4 and De Morgan properties: Lemma 2.4.11. For every B ∈ GIB and every a ∈ B, we have that: 1. a ∨ ¬ a is positive and a ∧ ¬ a is negative; 2. a is fix if and only if a and ¬ a are positive.

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Next, given a positive element c in B ∈ GIB , we analyse the structure of the interval [¬c, c] = { x ∈ B : ¬c ≤ x ≤ c}. It turns out that all such intervals are universes of Boolean algebras, and that they partition B. Lemma 2.4.12. Let B ∈ GIB . For every a ∈ B, there exists a unique c a ∈ P(B) such that a ∈ [¬c a , c a ]. Moreover, c a = a ∨ ¬ a. Proof. Let a ∈ B and consider c a = a ∨ ¬ a. First, we prove that ¬c a ≤ a ≤ c a . Obviously, a ≤ a ∨ ¬ a. For the other inequality, notice that a ∨ ¬c a = a ∨ ¬( a ∨ ¬ a) = a ∨ (¬ a ∧ a) = a ∨ a = a, by virtue of (I9). Consider now a positive c ∈ B such that ¬c ≤ a ≤ c. Then,

¬c a = ¬( a ∨ ¬ a) = ¬ a ∧ a = ¬ a ∧ ( a ∨ ¬c) = ¬ a ∧ ¬c = ¬( a ∨ c) = ¬c, whence it follows that c = c a .



Lemma 2.4.13. Let B ∈ GIB . If c ∈ P(B), we have that: 1. For every a ∈ B, c ∨ a = c ∨ ¬ a. 2. If a ≤ c and ¬c ≤ b, then b ∧ (b ∨ a) = b. Proof. Repeatedly applying Proposition 2.4.5.(5) and (I4), as well as the fact that c is positive, we have that for every a ∈ B, c ∨ a = c ∨ ¬c ∨ a = c ∨ ¬ a ∨ a = c ∨ ¬c ∨ ¬ a = c ∨ ¬ a. For (2), suppose that c is positive, a ≤ c and ¬c ≤ b. It follows that b ∧ (b ∨ a) = (b ∨ ¬c) ∧ (b ∨ a) = b ∨ (¬c ∧ a) = b ∨ ¬(c ∨ ¬ a) = b ∨ ¬(c ∨ a) = b ∨ ¬c = b.  Proposition 2.4.14. Let B ∈ GIB . For every positive c ∈ B, the interval [¬c, c] is the universe of a Boolean algebra C under the restrictions of the operations of B. Proof. Let c be as in the statement. Notice that, for every a ∈ B, c¬a = ¬ a ∨ ¬¬ a = a ∨ ¬ a = c a . Hence, by Lemma 2.4.12, if a ∈ [¬c, c], then c = c a = c¬a , and therefore ¬ a ∈ [¬c¬a , c¬ a ] = [¬c, c]. That is to say, [¬c, c] is closed under negation, and since it is obviously closed under ∨, it is closed under ∧ as well. Since ¬c and c are the bottom and the top of [¬c, c], respectively, C is a bounded generalised involutive bisemilattice, whence all we have to prove is that the absorption laws are satisfied and that for every a ∈ [¬c, c], ¬ a is its complement in C. If we are given a, b ∈ [¬c, c], then a ≤ c and ¬c ≤ b, and thus by virtue of Lemma 2.4.13.(2), b ∨ (b ∧ a) = b. Therefore the absorption laws are satisfied. Finally, by Lemma 2.4.12, we have that if a ∈ [¬c, c], then c = c a = a ∨ ¬ a, and hence ¬c = ¬c a = ¬( a ∨ ¬ a) = a ∧ ¬ a, which proves that ¬ a is the complement of a. 

2.4. GENERALISED INVOLUTIVE BISEMILATTICES

63

In particular, given an involutive bisemilattice B, the interval [0, 1] = { a ∈ B : 0 ≤ a ≤ 1} is a Boolean subuniverse of B. The intervals determined by the other positive elements (if any), however, have a Boolean structure but fail to be subuniverses of B, since they are not closed with respect to the nullary operations. Observe, moreover, that in an involutive bisemilattice a is a positive element iff 1 ≤ a.

2.4.3 Characterising Boolean algebras and semilattices This short subsection provides criteria to determine when a given (generalised) involutive bisemilattice is subsumed under the two special cases of Example 2.4.2, i.e. when it is either a Boolean algebra or a semilattice. Proposition 2.4.15. Let B ∈ IBSL. The following statements are equivalent: 1. B ∈ BA0,1 . 2. 1 is the maximum of B with respect to the ordering ≤. 3. B satisfies x ∨ ¬ x ≈ 1. Proof. If B is a Boolean algebra, then obviously it satisfies x ≤ 1. For the other implication, just notice that (2) is equivalent to B = [0, 1], but [0, 1] is the universe of a Boolean algebra by virtue of Proposition 2.4.14. The second and the third statement are equivalent by Proposition 2.4.6.(1).  Let us list some additional properties of fix elements and positive elements, in order to get a better grasp of the structure of a generalised involutive bisemilattice. In the next Lemma we show that the set of fix elements of a member of GIB is closed upwards with respect to the ordering ≤. Lemma 2.4.16. Let B ∈ GIB . Then, we have: 1. If c is a fix element, then for every a ∈ B, c ∧ a = c ∨ a. 2. If c is a fix element and c ≤ a, then a is also a fix element. Proof. For (1), suppose that c is a fix element of B, and pick an arbitrary a ∈ B. Then, c ∨ a = c ∨ (¬c ∧ a) = c ∨ (c ∧ a) = c ∧ (c ∨ a) = c ∧ (¬c ∨ a) = c ∧ a. Regarding (2), under the given hypothesis we use (1) and obtain a = c ∨ a = c ∨ ¬c ∨ a = c ∨ ¬ a ∨ a = c ∧ ( a ∨ ¬ a) = c ∧ (c ∨ ¬ a) = c ∧ (¬c ∨ ¬ a) = c ∧ ¬ a = ¬c ∧ ¬ a = ¬ a, since a = c ∨ a . 

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64

We enumerate below a number of characterisations of SLτ1 as a subvariety of IBSL, and of SLτ2 as a subvariety of GIB . Proposition 2.4.17. Let B ∈ GIB . The following statements are equivalent: 1. B satisfies x ∧ y ≈ x ∨ y. 2. B satisfies ¬ x ≈ x. 3. B satisfies x ∧ ( x ∨ y) ≈ x ∧ y. Moreover, if B ∈ IBSL, each of the previous conditions is equivalent to: 4. B satisfies 1 ≈ 0. Proof. (1) ⇒ (2) If B satisfies x ∧ y ≈ x ∨ y, then in particular for any a, a ∧ ¬ a is fix, whence by Lemma 2.4.16.(2), a is fix as well. (2) ⇒ (3) By (I6). (3) ⇒ (1) We have that: a∨b = = = =

( a ∨ b) ∧ (b ∨ a) ( a ∧ ( a ∨ b)) ∨ (b ∧ (b ∨ a)) ( a ∧ b) ∨ ( a ∧ b) a∧b

Prop. 2.4.7 

(4) ⇔ (3) If 1 = 0, then for every a, b ∈ B, we have that a ∧ ( a ∨ b) = a ∨ ( a ∧ b) = ( a ∧ 1) ∨ ( a ∧ b) = a ∧ (1 ∨ b) = a ∧ (0 ∨ b) = a ∧ b. For the other direction, using our assumption as well as (I7)-(I8), 1 = 1 ∧ 1 = 1 ∧ (1 ∨ 0) = 1 ∧ 0 = 0.

2.4.4

The Płonka sum representation

In this subsection we prove that GIB is the regularisation of BA by giving a detailed proof of the Płonka sum representation for generalised involutive bisemilattices. Indeed, there are insights to be gained here that will be useful in the sequel, once we come to discuss the connection between generalised involutive bisemilattices and the logic PWK. We will closely follow [33] in presenting such a proof. Lemma 2.4.18. Let B ∈ GIB . Then we have that: 1. If a and b are positive, then also a ∧ b = a ∨ b is positive.

2.4. GENERALISED INVOLUTIVE BISEMILATTICES

65

2. If ¬ a ≤ a ≤ b, then ¬ a ≤ ¬b. Proof. (1) We prove the claim that a ∧ b = a ∨ b, from which the rest of the item easily follows. a∨b = = = = = =

a ∨ ¬b a ∨ (b ∧ ¬b) a ∨ ( a ∧ b) a ∧ ( a ∨ b) a ∧ (¬ a ∨ b) a∧b

Lm. 2.4.13.(i) Prop. 2.4.5.(i) Prop. 2.4.5.(iii) Lm. 2.4.13.(i) I6

(2) If ¬ a ≤ a ≤ b, then clearly a ∈ P(B). By Lemma 2.4.13.(1), b = a ∨ b = a ∨ ¬b, whence ¬b ∨ b = ¬b ∨ a ∨ ¬b = ¬b ∨ a = b, hence b ∈ P(B) as well. By (1) a ∧ b = a ∨ b, whence ¬ a ∨ ¬b = ¬ ( a ∧ b) = ¬ ( a ∨ b) = ¬b.  We now present the main result of this section, according to which every Płonka sum over a semilattice direct system of Boolean algebras is a generalised involutive bisemilattice, and every generalised involutive bisemilattice admits a representation as a Płonka sum of Boolean algebras.4 Theorem 2.4.19. 1. If A = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i is a semilattice direct system of Boolean algebras, then the Płonka sum Pł (A ) is a generalised involutive bisemilattice. 2. Any generalised involutive bisemilattice B is isomorphic to the Płonka sum over the semilattice direct system of Boolean algebras whose semilattice of indices is h P(B), ≤i and the homomorphism pcd : [¬c, c] → [¬d, d], with c ≤ d, is given by pcd ( a) = ¬d ∨ a. Proof. (1) This is a direct consequence of Theorem 2.3.2 and the axiomatisation of generalised involutive bisemilattices. Indeed, all the identities in Definition 2.4.1 are regular identities satisfied in every Boolean algebra, and therefore they are satisfied in Pł (A ) as well. That is, Pł (A ) is a generalised involutive bisemilattice. (2) Let B ∈ GIB . We know that a b = a ∧ ( a ∨ b) is a partition function on its bisemilattice reduct; hence, in order to show that it is 4 For a more refined representation of a class of algebras that includes IBSL in terms of involutorial Płonka sums of algebras satisfying a generalised version of the absorption law, see [76].

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a partition function on B, it suffices to verify that for all a, b ∈ B the following conditions are satisfied:

¬ a b = ¬( a b);

b ¬ a = b a.

For the first one, notice that, by Proposition 2.4.5.(1)-(3), ¬ a b = ¬ a ∨ (¬ a ∧ b) = ¬ a ∨ (b ∧ ¬b) = ¬ a ∨ (¬ a ∧ ¬b) = ¬( a ∧ ( a ∨ b)) = ¬( a b). The remaining one is immediate in view of Definition 2.4.1 and Proposition 2.4.5.(1)-(3). Therefore, by Theorem 2.2.8, B is representable as the Płonka sum over a direct system of generalised involutive bisemilattices satisfying x y ≈ x, and thus such that for all a, b, a ∨ b = b ⇒ a = a ∧ ( a ∨ b) = a ∧ b, and similarly a ≤∧ b implies a ≤∨ b. These Płonka fibres are Boolean by Example 2.4.2. We now show that the semilattice of indices of this direct system is isomorphic to the semilattice h P(B), ∨i of the positive elements of B. For a start, we prove that two elements b, c ∈ B are such that b c = b and c b = c iff they are such that b ∨ ¬b = c ∨ ¬c (and, consequently, iff b ∧ ¬b = c ∧ ¬c). In fact, by Lemmas 2.4.12 and 2.4.13, if b ∧ ¬b ≤ b, c ≤ b ∨ ¬b, then b c = b and c b = c; conversely, if b c = b and c b = c, then by Proposition 2.4.5.(1) b ∨ (c ∧ ¬c) = b and c ∨ (b ∧ ¬b) = c. So b ∈ [c ∧ ¬c, c ∨ ¬c] and c ∈ [b ∧ ¬b, b ∨ ¬b], whence our conclusion follows by Lemma 2.4.12. So, the proof of Theorem 2.2.8 implies that the semilattice of indices in our direct system is isomorphic to the semilattice h P(B), ∨i of the positive elements of B, and by Lemma 2.4.12 each fibre has the form [¬c, c], for c a positive element. Finally, the proof of Theorem 2.2.8 implies further that ϕcd ( a) = a d, and by Proposition 2.4.5.(1), a d = a ∨ ( a ∧ d) = a ∨ (d ∧ ¬d) = ¬d ∨ a.



With only a few slight modifications, which can be safely left to the reader, the argument carries over to the case when constants are present in the type. Thus: Theorem 2.4.20. 1. If A = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i is a semilattice direct system of Boolean algebras (in the type τ1 ), then the Płonka sum Pł (A ) is an involutive bisemilattice. 2. Any involutive bisemilattice B is isomorphic to the Płonka sum over the semilattice direct system of Boolean algebras (in the type τ1 ) whose semilattice of indices is h P(B), ≤, 0i and the homomorphism pcd : [¬c, c] → [¬d, d], with c ≤ d, is given by pcd ( a) = ¬d ∨ a.

2.4. GENERALISED INVOLUTIVE BISEMILATTICES

67

Since BA and BA0,1 are strongly irregular varieties satisfying x y ≈ x, where the partition function x y can be taken to be x ∧ ( x ∨ y), Theorem 2.3.5 immediately yields: Corollary 2.4.21. GIB = R(BA) and IBSL = R(BA0,1 ). Corollary 2.4.21 entitles us to helping ourselves to the general theory of regular varieties developed earlier in the chapter. In particular: Proposition 2.4.22. We have that: 1. The only nontrivial subdirectly irreducible generalised involutive bisemilattices are WK, S2 , and B2 , up to isomorphism. Since B2 is a subalgebra of WK and S2 is a quotient of such, GIB = V (WK). 2. The only nontrivial subdirectly irreducible involutive bisemilattices are 0,1 WK0,1 , S2 , and B0,1 2 , up to isomorphism. Since B2 is a subalgebra of WK0,1 and S2 is a quotient of such, IBSL = V (WK0,1 ). The reader may find a direct constructive proof of Proposition 2.4.22.(2) (which adapts Kalman’s technique [129]) in [33]. Corollary 2.4.23. We have that: 1. The only nontrivial proper subvarieties of GIB are the disjoint varieties BA of Boolean algebras and SLτ2 of τ2 -semilattices. 2. The only nontrivial proper subvarieties of IBSL are the disjoint varieties BA0,1 of Boolean algebras and SLτ1 of τ1 -semilattices. In light of the preceding corollary, any generalised involutive bisemilattice B falls under one of the following three cases, in terms of its Płonka sum representation: 1. B has only one Boolean fibre. In that case, B is a Boolean algebra. 2. All the Boolean fibres of B are trivial. In this case, B satisfies the identity x ∨ y ≈ x ∧ y, and hence it is a semilattice. 3. B has at least two Boolean fibres, not all trivial. These are the proper generalised involutive bisemilattices that are neither Boolean algebras nor semilattices.

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Clearly, x ∧¬ x ≈ 0and x ∨ ¬ x ≈ 1 are not regular identities, whence they fail in R BA0,1 = IBSL. However, the type τ1 is expressive enough to dictate that (the Płonka sum representation of) each member of IBSL contain a designated fibre where the interpretations of 0 and 1 live — namely, the fibre indexed by the bottom element of the semilattice of indices. On the other hand, the Płonka sum representation of a generic member of GIB need not contain any such bottom element. Generalised involutive bisemilattices, thus, are not the τ2 -reducts of involutive bisemilattices. However, they are τ2 -subreducts of such. In fact: Lemma 2.4.24. If A ∈ GIB , then A embeds into the τ2 -reduct of an involutive bisemilattice. Proof. Preliminarly, observe that if I = h I, ≤i is a (join-)semilattice and i0 ∈ / I, then I0 = h I ∪ {i0 } , ≤ ∪ {hi0 , x i | x ∈ I ∪ {i0 }}i is a (join-)semilattice with zero and I is a subsemilattice of it. Now, let A ∈ GIB . By Theorem 2.4.19, A is isomorphic to a Płonka sum Pł (A ) over a semilattice direct system A = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i of Boolean algebras. Consider the triple: D n oE  A0 = {Ai }i∈ I ∪ Ai0 , I0 , pij | i, j ∈ I0 , i ≤I0 j , where Ai0 = B2 , and, for all j ∈ I, pi0 j is the unique homomorphism from B2 to A j . This is again a semilattice direct system of Boolean algebras. Clearly, its fibres can be seen as (Boolean) algebras of type τ1 , whence Pł (A0 ) is also an algebra of type τ1 . By Theorem 2.3.5 and Corollary 2.4.21, Pł (A0 ) is an involutive bisemilattice, and Pł (A ) clearly embeds into the τ2 -reduct of it. 

The lattice of subquasivarieties of GIB has been completely described in [22] (see also [164]) in the context of a more general study of subquasivarieties of regular varieties. There are exactly nine subquasivarieties of GIB , arranged as in the following Hasse diagram:

2.4. GENERALISED INVOLUTIVE BISEMILATTICES

69

GIB

SGIB

N GIB ∨ SLτ2

N GIB

IN GIB ∨ SLτ2

IN GIB

SLτ2 BA T

In the next table, we list relative quasiequational bases w.r.t. GIB and generators for each member of the lattice.

GIB SGIB N GIB ∨ SLτ2 N GIB IN GIB ∨ SLτ2 IN GIB BA SLτ2 T

x ≈ ¬ x & y ≈ ¬y ⇒ x ≈ y x ≈ ¬ x ⇒ y ≈ ¬y x ≈ ¬x ⇒ y ≈ z ¬ x ∨ ¬y ≤∨ x ∧ y ⇒ ¬ x ≤∨ x ¬ x ∨ ¬y ≤∨ x ∧ y ⇒ ¬ x ≤∨ x x ≈ ¬x ⇒ y ≈ z x ∧ ( x ∨ y) ≈ x x∧y ≈ x∨y x≈y

WK, S2 WK EK, S2 EK B2 × S2 , S2 B2 × S2 B2 S2 S1

We finish this chapter by   providing a direct proof of the fact that the axiomatisation of R BA0,1 given by Płonka in [177] is equivalent to the one in Definition 2.4.1. Theorem 2.4.25. An algebra A = h A, ∧, ∨, ¬, 0, 1i of type τ1 is an involutive bisemilattice if and only if it satisfies the following identities:

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R1. x ∨ x ≈ x; R2. x ∨ y ≈ y ∨ x; R3. x ∨ (y ∨ z) ≈ ( x ∨ y) ∨ z; R4. ¬¬ x ≈ x; R5. x ∧ y ≈ ¬(¬ x ∨ ¬y); R6. 0 ∨ x ≈ x; R7. 1 ≈ ¬0. R8. x ∧ ¬ x ≈ x ∧ 0; R9. ( x ∨ y) ∧ z ≈ ( x ∧ z) ∨ (y ∧ z); R10. ( x ∧ y) ∨ z ≈ ( x ∨ z) ∧ (y ∨ z). Proof. We observe first that the identities (R1)-(R7) are common to both axiomatisations. By Proposition 2.4.7, (R9)-(R10) hold in IBSL. (R8) is easily derived from (I1)-(I8): a ∧ 0 = a ∧ (¬ a ∨ 0) = a ∧ ¬ a, where in the former equality we have used (I6) and in the latter (I7). On the other hand, (I6) is derivable from the identities above: a ∧ (¬ a ∨ b) = ( a ∧ ¬ a) ∨ ( a ∧ b) = ( a ∧ 0) ∨ ( a ∧ b) = a ∧ (0 ∨ b) = a ∧ b, where we have used (R8) and (R10). 

Chapter 3 Dualities for regular varieties Mathematicians often try to shed new light on the properties of abstract or unfamiliar structures by somehow linking them to more concrete or better understood objects. Representation theorems are a convenient illustration of this situation – think of the commonplace examples of Cayley’s representation of groups as groups of permutations or of Stone’s representation of Boolean algebras as fields of sets. An even subtler approach to this goal, on the other hand, is provided by category theory. Here, the notions of an equivalence and of a dual equivalence of categories (see Definition 3.1.8 below) are the key not only to interpreting possibly esoteric structures from universal algebra in terms of relatively mundane ones like topological spaces, but also to preserving enough properties in the process so that we can conduct our investigation on whichever side of the fence we find more convenient (usually at the “concrete” end) and then cash in the results, via the (dual) categorical equivalence, on the other side. Regular varieties are no exception to this general picture. It is helpful to view these classes of algebras under a different perspective, provided by a categorical association with appropriate (enriched) topological spaces. It is also instructive to assess the extent to which the algebraic constructions of semilattice direct systems and Płonka sums have topological counterparts, which can be used to decompose such dual spaces into more familiar ones. This is the aim of the present chapter. In Section 3.1 we give a few basic definitions of category theory and quickly review two standard examples of topological dualities, the Stone duality and the Priestley duality. In Section 3.2 we raise the notion of a semilattice direct (and inverse) system, already defined for algebras in Chapter 1, to the level of categories. In Section 3.3 we show that the categories of semilattice direct and inverse systems of dually equivalent

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_3

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categories are dually equivalent, and that strongly irregular varieties admitting topological duals are such that their regularisations also admit dually equivalent topological categories. We also briefly discuss other approaches to dualities for regular varieties. In Section 3.4 we focus on the case studies of distributive and involutive bisemilattices, where the dual spaces provided by the general theory can be given a more perspicuous description. Finally, in Section 3.5 we give an account of a partial attempt to introduce a topological counterpart of Płonka sums.

3.1

Background

3.1.1 Basic notions In this section we only go over the basic notions of category theory which will be used in the present book. Fully comprehensive accounts of the topic are e.g. [148], [130], [143]. Definition 3.1.1. A category is a structure C consisting of: • a (proper or improper) class of objects; • a set of morphisms HomC ( X, Y ), for every pair of objects X, Y; • a composition function ◦ defined, for every object X, Y, Z, as:

◦ : HomC ( X, Y ) × HomC (Y, Z ) → HomC ( X, Z ) h ϕ, ψi 7→ ψ ◦ ϕ, satisfying the following additional requirements: 1. η ◦ (ψ ◦ ϕ) = (η ◦ ψ) ◦ ϕ, for all morphisms η, ψ, ϕ s.t. ◦ is well-defined; 2. for every object X, HomC ( X, X ) contains the identity object, i.e. a morphism id X such that: idY ◦ ϕ = ϕ = ϕ ◦ id X , for every ϕ ∈ HomC ( X, Y ). As the definition of a category may sound very abstract, the following examples can clarify it.

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Example 3.1.2. The category Set has sets as objects, maps as morphisms and ◦ is the usual composition of maps. Example 3.1.3. Given a certain similarity type τ, the category Algτ has τ-algebras as objects, homomorphisms as morphisms and ◦ is the usual composition of homomorphisms. More generally, categories whose objects form a class of similar algebras and whose morphisms are algebra homomorphisms are called algebraic categories. Example 3.1.4. The category Top has topological spaces as objects, continuous functions as morphisms and composition of continuous functions as composition. More generally, categories whose objects are topological spaces and whose morphisms are continuous maps are called topological categories. Definition 3.1.5. Given two categories C and D, a covariant (resp. contravariant) functor from C to D is a pair of applications F = hF1 , F2 i such that, for every object X in C, F1 ( X ) is an object in D and, for every ϕ ∈ HomC ( X, Y ), F2 ( ϕ) ∈ HomD (F ( X ), F (Y )) (resp., F2 ( ϕ) ∈ HomD (F (Y ), F ( X ))) is the induced morphism satisfying: 1. F (id X ) = idF (X ) ; 2. F ( ϕ ◦ ψ) = F ( ϕ) ◦ F (ψ) (resp., F ( ϕ ◦ ψ) = F (ψ) ◦ F ( ϕ)) The following are easy examples of functors between categories. Example 3.1.6. The identity covariant function idC : C → C defined by idC ( X ) = X and idC ( ϕ) = ϕ, for every object X and every morphism ϕ in C, is a covariant function from C to itself. Example 3.1.7. The forgetful functor from the category Alg (Top, respectively) into Set is the covariant functor which associates to every algebra (resp., topological space) its universe, and to every homomorphism f (resp., continuous function) the function f itself. Examples of contravariant functors will be given in the next sections. Definition 3.1.8. Two categories C and D are equivalent provided that there exist two covariant functors, F : C → D and G : D → C such that the compositions G ◦ F and F ◦ G are naturally isomorphic with the identity in the categories C and D, respectively. Whenever the functors considered in the above definition are contravariant (instead of covariant), the two categories C and D are said to be dually equivalent or, briefly, duals.

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3.1.2 The Stone duality Marshall Stone [215] has to be credited with the insight that Boolean algebras, whose study had been driven by purely algebraic or logical motivations (and which would later prove to be crucial in computer science as well), can be understood in even greater depth via their topological duals – compact, totally disconnected Hausdorff spaces, which we call, nowadays, Stone spaces or Boolean spaces. In this subsection we quickly rehearse the object part of this duality, referring the reader to [107] for further details. The duality can be constructed using the 2-element set {0, 1} as dualising object, namely an object living in both categories. In this case, one can see it as the universe of the 2-element Boolean algebra B0,1 2 = h{0, 1}, ∧, ∨, ¬, 0, 1i or as the universe of a topological space, call it 2˜ = h{0, 1}, Tdis i, equipped with the discrete topology. In detail, if B is a Boolean algebra, let b := HomBA (B, B0,1 ) = { f : B → {0, 1} | f is a homomorphism}. B 2 b is a subset of 2B , hence it forms a subspace Clearly the universe of B B b is a totally disconnected, closed Hausdorff ˜ of the product space 2 . B space (because it is a subspace of a Hausdorff space), and hence compact (because it is a closed subset of a space that is compact by Tychonoff’s b is the dual space of the Boolean algebra B (see Theorem). Actually B Theorem 3.1.9). A similar construction can performed on the topological side. Namely, let S be a Stone space, and let e := Hom(S, 2˜ ) = { f : S → 2 | f is continuous}. S Hom(S, 2˜ ) can be turned into a Boolean algebra by defining operations pointwise, for any f , g ∈ Hom(S, 2˜ ) and a ∈ S: f ∧ g( a) := f ( a) ∧ g( a); f ∨ g( a) := f ( a) ∨ g( a);

¬ f ( a) := ¬( f ( a)). The constants are defined as constant functions, i.e. f 0 ( a) = 0 and f 1 ( a) = 1, for any a ∈ S. We have that:

3.1. BACKGROUND

75

Theorem 3.1.9 (Stone). Let S be a Stone Space and B be a Boolean algebra. b˜ and B ∼ b˜ Moreover, the functor F from the Then S is homeomorphic to S = B. algebraic category BA of Boolean algebras to the category ST of Stone spaces with continuous functions, and the functor G from ST to BA, defined as follows: b G(S) = S˜ F (B) = B, are contravariant. b˜ via the map α defined for any b ∈ B Proof. (Sketch.) B is isomorphic to B and f ∈ HomBA (B, B0,1 2 ) by α(b) = eb , with eb ( f ) = f (b). b˜ via the map β defined for any x ∈ S and f ∈ S is homeomorphic to S ˜ Hom(S, 2) by β( x ) = ex , with ex ( f ) = f ( x ). 

3.1.3 The Priestley duality The Stone duality has been extended by Priestley [190, 189] to bounded distributive lattices. Bounded distributive lattices are the ¬-free subreducts of Boolean algebras [110, Thm. 153], but on the topological side we have to enrich the topological structure with an order relation to obtain appropriate duals. Under most respects, Priestley duality follows the very same technique as Stone duality, whence we will be even more cursory in its presentation. The dualising object is again the 2-element set {0, 1}, which can be seen as the universe of the bounded distributive lattice D2 = h{0, 1}, ∧, ∨, 0, 1i on the algebraic side and as a topological space on the other. The topology is however quite different from the previous subsection, as {0, 1} is here taken as the carrier set of an ordered topological space. By an ordered topological space X = h X, ≤, T i we simply mean a topological space h X, T i endowed with a partial order ≤. A subset U ⊆ X is a lower set whenever, if x ∈ U and y ≤ x then y ∈ U. Definition 3.1.10. An ordered topological space X = h X, ≤, T i is a Priestley space if 1. h X, T i is compact;

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CHAPTER 3. DUALITIES FOR REGULAR VARIETIES 2. {h x, yi ∈ X 2 | x ≤ y} is closed; 3. if x 6≤ y then there is an open and closed (clopen) lower set U such that y ∈ U and x 6∈ U.

The 2-element chain 2˜ ≤ = ({0, 1}, ≤, Tdis ), with its natural order (0 < 1) and the discrete topology, can be regarded as a Priestley space. The duality is then constructed as for Boolean algebras. If B is a b := HomDL (B, D2 ) bounded lattice, then the role of the dual space is played by B which is topologised as a subspace of the product space 2˜ B ≤ . In this case HomDL (B, D2 ) stands for the set of distributive lattice homomorphisms from B to D2 . On the other hand, we associate to any Priestley space X a bounded e as follows: distributive lattice X e := Hom(X, 2˜ ≤ ) = { f : X → 2˜ ≤ | f is continuous and preserves ≤} X where operations are defined pointwise. e are a Priestley space and e and X It is not very difficult to check that B a bounded distributive lattice, respectively (for details, see for example [70]). The object part of the duality can be stated as follows: Theorem 3.1.11 (Priestley). Let X be a Priestley space and B a bounded disb˜ and B ∼ b˜ Moreover, the tributive lattice. Then X is homeomorphic to X = B. ˜ are contravariant. b and G(X) = X functors F (B) = B

3.2 Semilattice systems In the previous chapter, we have introduced semilattice direct systems of algebras as the essential ingredients of Płonka sums. Semilattice direct (and inverse) systems can be defined at a more abstract, categorical level (see also the remarks in Section 2.1). Definition 3.2.1. Let C be an arbitrary category. A semilattice direct system in C is a triple X = h{ Xi }i∈ I , h I, ≤i, { pij | i ≤ j}i such that: (i) h I, ≤i is a semilattice. (ii) Xi is an object in C, for each i ∈ I; (iii) pij : Xi → X j is a morphism of C, for each pair i 6 j, satisfying the following further conditions:

3.2. SEMILATTICE SYSTEMS

77

– pii is the identity in Xi ; – p jk ◦ pij = pik , for each i ≤ j ≤ k. We will refer to h I, ≤i and { Xi }i∈ I as the index set and the terms of the direct system, respectively, while we refer to the maps pij as transition morphisms. As usual, the semilattice operation of h I, ≤i is denoted by ∨. Semilattice direct systems consist in obvious generalisations of direct systems in a given category, obtained by assuming the index set to be a semilattice instead of a (directed) pre-ordered set1 . These concepts find applications in several fields of mathematics (see for example [149]). Definition 3.2.2. Given two semilattice direct systems X = h{ Xi }i∈ I , h I, ≤ I i, { pij | i ≤ I j}i, Y = h{Yj } j∈ J , h J, ≤ J i, {qkl | k ≤ J l }i, in the same category C, a morphism from X to Y is a pair h ϕ, { f i }i∈ I i such that: i) ϕ : I → J is a semilattice homomorphism; ii) f i : Xi → Yϕ(i) is a morphism of C (for each i ∈ I), making the diagram in Figure 3.1 commutative for each i, j ∈ I, i ≤ I j.

Xi

pij

Xj fj

fi

Yϕ(i)

q ϕ (i ) ϕ ( j )

Yϕ( j)

Figure 3.1: The commuting diagram defining morphisms of semilattice direct systems When clear from the context we will often write h ϕ, f i i instead of h ϕ, { f i }i∈ I i. 1 For

[52].

a comprehensive survey of the algebraic approach to directed ordered set, see

CHAPTER 3. DUALITIES FOR REGULAR VARIETIES

78

Given three semilattice direct systems X = h{ Xi }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i, Y = h{Yj } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i, Z = h{ Zk }k∈K , hK, ≤K i, {rkk0 | k ≤K k0 }i, with morphisms h ϕ, f i i : X → Y and hψ, g j i : Y → Z, the composition of morphisms is defined as

h ϕ, f i i ◦ hψ, g j i = hψ ◦ ϕ, g ϕ(i) ◦ f i i.

(3.1)

Lemma 3.2.3. The composition of morphisms between semilattice direct systems is a morphism. Proof. Let h ϕ, f i i : X → Y, hψ, g j i : Y → Z. Then h ϕ, f i i ◦ hψ, g j i, as defined in 3.1, is a semilattice homomorphism, because it is the composition of semilattice homomorphisms. The other claim follows from the commutativity of the following diagram (we omitted the indexes for the maps p, q, r to make the notation less cumbersome): Xi

p...

fi

Xi 0 f i0

Yϕ(i)

q...

g ϕ (i )

Zψ◦ ϕ(i)

Yϕ(i0 )

g ϕ (i 0 )

r...

Zψ◦ ϕ(i0 ) 

Proposition 3.2.4. Let C be an arbitrary category. The structure whose objects are semilattice direct systems in C and morphisms are as in Definition 3.2.2 is a well-defined category. Proof. Left to the reader.



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79

The category of Proposition 3.2.4 will be referred to as Sem-dir-C. Semilattice inverse systems, for an arbitrary category, are defined in an analogous, dual way. Definition 3.2.5. Let C be an arbitrary category. A semilattice inverse system in C is a triple X = h{ Xi }i∈ I , h I, ≤i, { pii0 | i ≤ i0 }i such that (i) h I, ≤i is a semilattice; (ii) for each i ∈ I, Xi is an object in C; (iii) pii0 : Xi0 → Xi is a morphism of C, for each pair i 6 i0 , satisfying the further conditions in Definition 3.2.1(iii). Again, we will refer to h I, ≤i and the Xi as the index set and the terms of the system X , respectively; in this case, however, the maps pii0 will be called bonding morphisms. As already mentioned, the only additional feature that distinguishes a semilattice inverse system from a generic inverse system in C is the requirement that the index set be a semilattice instead of a directed preorder. Definition 3.2.6. Given two semilattice inverse systems

X = h{ Xi }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i, Y = h{Yj } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i, a morphism between X and Y is a pair h ϕ, f j i such that: i) ϕ : J → I is a semilattice homomorphism; ii) for each j ∈ J, f j : X ϕ( j) → Yj is a morphism in C, such that if j ≤ j0 , then the diagram in Figure 3.2 commutes. Notice that the assumption that ϕ : J → I is a semilattice homomorphism implies that whenever j ≤ j0 , ϕ( j) ≤ ϕ( j0 ). Given three systems

X = h{ Xi }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i, Y = h{Yj } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i, Z = h{ Zk }k∈K , hK, ≤K i, {rkk0 | k ≤K k0 }i,

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80

p ϕ( j) ϕ( j0 )

X ϕ( j)

X ϕ( j0 ) f j0

fj

Yj

Yj0

q jj0

Figure 3.2: The commuting diagram defining morphisms of semilattice inverse systems. the composition is defined as:

hφ, f j i ◦ hψ, gk i = hφ ◦ ψ, gk ◦ f φ◦ψ(k) i. It is easily verified that this is a morphism. Moreover, ◦ is associative and semilattice inverse systems form a well-defined category which we will call Sem-inv-C.

3.3

Duality

In this section we aim at showing two main results: 1. The categories of semilattice direct and semilattice inverse systems of dually equivalent categories are dually equivalent. 2. Regularisations of strongly irregular varieties admitting topological duals admit in turn topological duals. We start by showing how to lift a duality between two categories C and D to a duality between the categories Sem-dir-C and Sem-inv-D. In detail, suppose that F and G are the contravariant functors yielding a duality between C and D.

F C

D

G

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81

Then, it is possible to construct two new functors Fe and Ge, as follows:

Fe Sem-dir-C

Sem-inv-D

Ge Fe (X ) := h{F ( Xi )}i∈ I , h I, ≤i, {F ( pii0 ) | i ≤ i0 }i Fe ( ϕ, f i ) := h ϕ, F ( f i )i,

(3.2) (3.3)

where X = h{ Xi }i∈ I , h I, ≤i, { pii0 | i ≤ i0 }i is an object and h ϕ, f i i a morphism in the category Sem-dir-C. Clearly, Ge is constructed using the functor G and the same prescription in (3.2). In the next Lemma we show that Fe and Ge are indeed contravariant functors. Lemma 3.3.1. Let F : C → D and G : D → C be contravariant functors between the categories C and D. Then: 1. Fe is a contravariant functor between Sem-dir-C and Sem-inv-D; 2. Ge is a contravariant functor between Sem-inv-D and Sem-dir-C. Proof. Proof of 1. and 2. are essentially analogous, so we just give the details of 1. (the reader can check that they can be easily adapted to prove 2.). Assume that X = h{ Xi }i∈ I , h I, ≤i, { pii0 | i ≤ i0 }i is an object in Semdir-C. We first show that Fe (X ) is an object in Sem-inv-D, namely it satisfies conditions (i), (ii), (iii) of Definition 3.2.5. Recall that, by (3.2), Fe (X ) := h{F ( Xi )}i∈ I , h I, ≤i, {F ( pii0 ) | i ≤ i0 }i. (i) is clearly satisfied as h I, ≤i is a semilattice; (ii) since Xi (for each i ∈ I) is an object in C and F is a functor, F ( Xi ) is an object in D; (iii) let i ≤ i0 . Then there exist morphisms pii0 : Xi → Xi0 in C such that pii is the identity in Xi and moreover, if i ≤ i0 ≤ i00 then pi0 i00 ◦ pii0 = pii00 . Since F is a contravariant functor, F ( pii0 ) : F ( Xi0 ) → F ( Xi ) is a morphism in D. Moreover, F ( pii ) = F (idC ) = idD and compositions are obviously preserved.

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We have thus shown that Fe (X ) is an object. Now, suppose that h ϕ, f i i is a morphism in Sem-dir-C between two objects X = h{ Xi }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i Y = h{Yj } j∈ J , h J, ≤ J i, { p jj0 | j ≤ J j0 }i We show that Fe ( ϕ, f i ) := h ϕ, F ( f i )i is a morphism from

Fe (Y ) := h{F (Yj )} j∈ J , h J, ≤ J i, {F ( p jj0 ) | j ≤ J j0 }i to

Fe (X ) := h{F ( Xi )}i∈ I , h I, ≤ I i, {F ( pii0 ) | i ≤ I i0 }i

(and get for free that Fe is contravariant). We check that Fe ( ϕ, f i ) satisfies conditions i) and ii) in Definition 3.2.6. i) clearly holds as ϕ : I → J is a semilattice homomorphism from the index set of Fe (X ) to the index set of Fe (Y ). ii) For every i ∈ I, f i : Xi → Yϕ(i) is a morphism in C making the Diagram in Fig. 3.1 commutative. Therefore, F ( f i ) : F (Yϕ(i) ) → F ( Xi ) is a morphism in D. Suppose that i ≤ i0 , for some i, i0 ∈ I. Then pii0 : Xi → Xi0 and F ( pii0 ) : F ( Xi0 ) → F ( Xi ) is the corresponding morphism in D. Since ϕ is a semilattice homomorphism, we have ϕ(i ) ≤ ϕ(i0 ), and Fe (X ) is a semilattice inverse system, then the following diagram commutes:

F (Yϕ(i) )

F ( q ϕ (i ) ϕ (i 0 ) )

F ( fi )

F ( Xi )

This establishes our claim.

F (Yϕ(i0 ) ) F ( f i0 )

F ( pii0 )

F ( Xi 0 )



Notice that the statement of Lemma 3.3.1 is false when considering covariant, instead of contravariant, functors, as shown by the following example.

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Example 3.3.2. Let C be an algebraic category, and let F : C → Set be the forgetful functor. For any object X = h{ Xi }i∈ I , h I, ≤i, { pii0 | i ≤ i0 }i, Fe (X ) is not an object in Sem-inv-Set. Indeed, for any pair of indices such that i ≤ i0 , we have a morphism in C, pii0 : Xi → Xi0 ; since F is covariant, Fe ( pii0 ) = F ( pii0 ) is a function (a morphism in Set) from F ( Xi ) to F ( Xi0 ), hence it does not fulfill condition (iii) in Definition 3.2.5. Theorem 3.3.3. Let C and D be dually equivalent categories. Then Sem-dir-C and Sem-inv-D are dually equivalent. Proof. Let F and G be contravariant functors

F C

D

G such that G ◦ F = idC and F ◦ G = idD . By Lemma 3.3.1, we know that the functors Fe and Ge

Fe Sem-dir-C

Sem-inv-D

Ge defined in (3.2) are contravariant. We only need to check that the compositions Ge ◦ Fe and Fe ◦ Ge are naturally isomorphic with the identities in the categories Sem-dir-C and Sem-inv-D, respectively. Let X = h{ Xi }i∈ I , h I, ≤ i, { pii0 | i ≤ i0 }i be an object in Sem-dir-C. Then e Fe (X )) = G(h{F e G( ( Xi )}i∈ I , h I, ≤i, {F ( pii0 ) | i ≤ i0 }i) = h{G ◦ F ( Xi )}i∈ I , h I, ≤i, {G ◦ F ( pii0 ) | i ≤ i0 }i = h{ Xi }i∈ I , h I, ≤i, { pii0 | i ≤ i0 }i, where the last equality is obtained by the fact G ◦ F is naturally isomorphic with the identity. The claim is analogously verified for Fe ◦ Ge.  In order to introduce the second announced result, firstly observe that strongly irregular varieties and regularisations of a variety can be defined as algebraic categories. We will say that C is a strongly irregular algebraic

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category provided that its objects form a strongly irregular variety. When C is an algebraic category, we will refer to R(C) as the algebraic category whose objects are algebras in the regularisation of the variety to which the objects in C belong. On the other hand, we will say that an algebraic category C is dualisable if it admits a dually equivalent topological category (a category whose objects are topological spaces). Examples of strongly irregular dualisable categories include the categories of Boolean algebras and of distributive lattices in virtue of the dualities recalled in Subsections 3.1.2 and 3.1.3. Theorem 2.3.5 can be then read as: whenever C is a strongly irregular algebraic category, the objects in R(C) are isomorphic to the objects of the category Sem-dir-C. We will show the equivalence extends at the level of categories. Definition 3.3.4. Given two semilattice direct systems A = h{Ai }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i, B = h{B j } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i from an algebraic category C and a homomorphism h : Pł (A ) → Pł (B ), we say that h preserves the Płonka fibres if, for every i ∈ I there exists an index j ∈ J such that h( Ai ) ⊆ Bj . In words, homomorphisms between Płonka sums (of algebras) preserve the fibres if their images do not split an algebra into the fibres of the Płonka sum Pł (B ). Lemma 3.3.5. Let A = h{Ai }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i, B = h{B j } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i be semilattice direct systems of algebras, whose terms belong to an irregular variety V of type τ. Then any homomorphism h : Pł (A ) → Pł (B ) preserves the fibres. Proof. Suppose that V is irregular and, by contradiction, that there exists a homomorphism h : Pł (A ) → Pł (B ) which does not preserve the fibres. This implies that there exist elements a, b ∈ Ai (for some i ∈ I) such that h( a) ∈ Bj and h(b) ∈ Bk with j 6= k. Since V is irregular, there exist formulas ε, η such that Var (η ) r Var (ε) 6= ∅ and V |= ε ≈ η. Let y ∈ Var (η ) r Var (ε) and x1 , . . . , xn the variables actually appearing in both ε and η. Since Ai ∈ V then ε( a, . . . , a) = η ( a, . . . , a, b) and ε(b, . . . , b) = η (b, . . . , b, a) (we have mapped x1 , . . . , xn to a, and y to b, in the first equality and x1 , . . . , xn to b, and y to a, in the second). Observe that, since

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85

h is a homomorphism, then h(ε( a, . . . , a)) = ε(h( a), . . . , h( a)) ∈ Bj and h(η ( a, . . . , a, b)) = η (h( a), . . . , h( a), h(b)) ∈ Bj∨k , from which we deduce that j ∨ k = j. Reasoning analogously, in the second equality, we get ε(h(b), . . . , h(b)) = η (h(b), . . . , h(b), h( a)), from which we deduce that j ∨ k = k, thus j = k, a contradiction.  Lemma 3.3.6. Let A = h{Ai }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i, B = h{B j } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i be semilattice direct systems of an algebraic category C and ( ϕ, f i ) a morphism from A to B. Then h : Pł (A ) → Pł (B ), defined as h( a) := f i ( a), where i ∈ I is the index such that a ∈ Ai , is a morphism in R(C). Proof. Preliminarily observe that, for every i ∈ I, the map h is welldefined, as f i is morphism in C. Since C is an algebraic category (where morphisms are homomorphisms of algebras), we only have to check that h is compatible with all the operations of the Płonka sum. To simplify notation, we set A = Pł (A ), B = Pł (B ), a1 , ..., an ∈ A with i1 , ..., in indexing the algebras to which they belong, g a generic n-ary operation in the type of the considered algebras and, finally, k = i1 ∨ ... ∨ in . Then,

h( gA ( a1 , ..., an )) = h( gAk ( pi1 k ( a1 ), ..., pin k ( an )))

= f k ( gAk ( pi1 k ( a1 ), ..., pin k ( an ))) = gB ϕ(k) ( f k ( pi1 k ( a1 )), ..., f k ( pin k ( an ))) = gB ϕ(k) (q ϕ(i1 ) ϕ(k) ( f i1 ( a1 )), ..., q ϕ(in ) ϕ(k) ( f in ( an )) = gB ( f i1 ( a1 ), ..., f in ( an )) = gB (h( a1 ), ..., h( an )), where the fourth equality is justified by the commutativity of the following diagram (which holds as, by assumption, h ϕ, f i i is a morphism in Sem-dir-C), for every i ∈ {i1 , ..., in }:

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86

pik

Ai

Ak

fi

fk

B ϕ (i )

B ϕ(k)

q ϕ (i ) ϕ ( k )

 Lemma 3.3.7. Let A = h{Ai }i∈ I , h I, ≤ I i, { pii0 | i ≤ I i0 }i, B = h{B j } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i be semilattice direct systems of algebras in a variety V and h : Pł (A ) → Pł (B ) a homomorphism. Let, moreover, ϕh : I → J be a map such that h( Ai ) ⊆ B ϕ (i) . h Then ϕh is a semilattice homomorphism. Proof. Let a1 , . . . , an ∈ i∈ I Ai , with a1 ∈ Ai1 , . . . , an ∈ Ain (i1 , . . . , in ∈ I) and set k = i1 ∨ · · · ∨ in . We want to show that ϕh (k) = ϕh (i1 ) ∨ · · · ∨ ϕh (in ). Set A = Pł (A ) and B = Pł (B ). Consider an n-ary operation symbol f in the type of V . Clearly, h( f A ( a1 , . . . , an )) = f B (h( a1 ), . . . , h( an )). S

By hypothesis, h( a1 ) ∈ B ϕ

h

( i1 ) , . . . , h ( a n )

∈ Bϕ

h

(in ) ,

therefore f B (h( a1 ), . . . , h( an )) ∈

Bj with j = ϕh (i1 ) ∨ · · · ∨ ϕh (in ). On the other hand, f A ( a1 , . . . , an ) ∈ Ak , hence h( f A ( a1 , . . . an )) ∈ B ϕh (k) . This shows that ϕh (k ) = ϕh (i1 ) ∨ · · · ∨ ϕh (in ), i.e. ϕn is a semilattice homomorphism.  Theorem 3.3.8. [31, Thm. 4.5] Let C be a strongly irregular algebraic category. Then the categories R(C) and sem-dir-C are equivalent. Proof. The equivalence is proved via the following functors:

F R (C)

sem-dir-C

G Let A be an object in the category R(C). Since C is strongly irregular, by Theorem 2.3.5, we know that A ∼ = Pł (A ), with A a semilattice direct

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system of algebras in C. F associates to A the semilattice direct system A. Consider a morphism in R(C), h : A → B and set A ∼ = Pł (A ), B ∼ = I I 0 Pł (B ), with A = h{Ai }i∈ I , h I, ≤ i, { pii0 | i ≤ i }i and B = h{B j } j∈ J , h J, ≤ J i, {q jj0 | j ≤ J j0 }i semilattice direct systems of algebras in C. Since C is a strongly irregular variety, we know, by Lemma 3.3.5, that h preserves the Płonka fibres of the direct system A (arising from the Płonka sum representation of A), i.e. h( Ai ) ⊆ Bj , for some j ∈ J. Hence, we can define a map ϕh : I → J satisfying the assumptions of Lemma 3.3.7, which assures that ϕh is a semilattice homomorphism. Moreover, for each i ∈ I, the restriction of h to Ai , h| Ai is a homomorphism of algebras (objects) in C. F associates to the morphism h, the pair h ϕh , h| Ai i. Moreover, it is easily checked that the following diagram is commutative for each i ≤ I i0 (indeed i ≤ I i0 implies ϕh (i ) ≤ J ϕh (i0 )) Ai

pii0

h| A 0

h | Ai

B ϕ h (i )

Ai 0

i

q ϕ h (i ) ϕ h (i 0 )

B ϕ h (i 0 )

Therefore F (h) is a morphism from A to B, showing that F is a covariant functor. On the other hand, G associates to an object A in the category Semdir-C, the Płonka sum Pł (A ) over A, which is an object in R(C) (as C is strongly irregular). Moreover, to each morphism h ϕ, f i i, G associates the map h : Pł (A ) → Pł (B ), defined as h( a) := f i ( a), for each a ∈ Ai and i ∈ I. Lemma 3.3.6 guarantees that h is indeed a morphism in R(C). It is easy to check that the compositions of the two functors are naturally isomorphic with the identities (in both categories).  Theorem 3.3.8, combined with Theorem 3.3.3, provides a duality for the regularisations of all dualisable strongly irregular algebraic categories. Corollary 3.3.9. Let C be a dualisable strongly irregular algebraic category with C∗ as topological dual. Then the categories R(C) and sem-inv-C∗ are dually equivalent.

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It is natural to wonder whether a more concrete description of seminv-C∗ , possibly as a unique topological space, can be provided. This is done in some known cases, such as for distributive bisemilattices [105] and involutive bisemilattices [36]. Moreover, to the best of our knowledge, the construction of Płonka sums has no analogue on the side of the topological representation spaces (a question raised by Gierz and Romanowska [105]) so, in most cases, the class sem-inv-C∗ remains basically a collection of spaces organised into a semilattice inverse system. In the following sections we will show concrete dualities describing the topological dual spaces and an attempt to provide an analogue of Płonka sums for topological spaces.

3.3.1 Other dualities Different techniques, over time, have been employed in the investigation into dualities for regular varieties. The duality result presented above resembles semilattice-based dualities established by Romanowska and Smith in [207, 208], where the authors also lift a duality between two categories to a duality involving the corresponding semilattice representations. There is a substantial difference between their approach and the one we have developped here: Romanowska and Smith consider, on the one side, the semilattice sum of an algebraic category and, on the other, the semilattice representation of the dual spaces, and the duality is obtained by dualising the semilattice of indices. In order to achieve this, they rely on the duality due to Hofmann, Mislove and Stralka [118] for semilattices (see also [70] for details). This means that the semilatttice representation of the dual spaces (of the considered categories) is constructed via compact topological semilattices with 0 which carry the Boolean topology. Brian Davey, the founding father of natural dualities [61], investigated the problem of lifting a natural duality from a strongly irregular variety to its regularisation [71]. Recently, a different approach has been suggested by Ledda [141], who establishes a “Stone-type” duality for varieties of bisemilattices, including distributive bisemilattices and expansions thereof such as De Morgan and (generalised) involutive bisemilattices. To begin with, a Balbes-type representation [10] is given for algebras in the mentioned varieties; then, Dunn-Hartonas dualities [114] are extended to the cases in point. In a few words, the technique introduced by Dunn and Hartonas for (varieties of) lattices consists in “splitting” a lattice into its ∧ and ∨ semilattice reducts,

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89

and dualising each of them.

3.4 Dual spaces The results in the previous section, despite their generality, do not provide a grasp of dual spaces of regular varieties. The aim of this section is to give a full description of the topological dual spaces in two special cases, namely distributive and involutive bisemilattices.

3.4.1 Left normal bands and GR spaces It is possible to construct a duality for distributive bisemilattices using the three-elements algebra 3 (see Example 2.3.18) as dualising object. The result, due to Gierz and Romanowska [105], provides a detailed description of the topological dual space: a compact totally disconnected partially ordered left normal band with constants, here called a GR space in honour of the authors. Recall from Section 2.2 that a left normal band is an idempotent semigroup h A, ∗i satisfying the following identity (which can be understood as a weak form of commutativity): x ∗ ( y ∗ z ) ≈ x ∗ ( z ∗ y ). Notable examples of left normal bands are left zero bands, which are left normal bands satisfying the identity x ∗ y ≈ x. Every left normal band can be equipped with a partial order. Definition 3.4.1. A partially ordered left normal band is a structure A = h A, ∗, ≤i such that: i) h A, ∗i is a left normal band; ii) h A, ≤i is a partially ordered set; iii) if x ≤ y then x ∗ z ≤ y ∗ z and z ∗ x ≤ z ∗ y; iv) x ∗ y ≤ x. In any partially ordered left normal band it is possible to define a second partial order via ∗ and ≤ as follows: a v b iff a ∗ b ≤ b and b ∗ a = b. Next, we add constants to the structure.

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Definition 3.4.2. A partially ordered left normal band with constants is a structure A = h A, ∗, ≤, c0 , c1 , cn i such that h A, ∗, ≤i is a partially ordered left normal band and c0 , c1 and cn are constants satisfying, for all x ∈ A: (1) x ∗ cn = cn ∗ x = cn ; (2) x ∗ c0 = x ∗ c1 = x; (3) c0 v x ≤ c1 and cn ≤ x v cn ; (4) if c0 ∗ x = c1 ∗ x then x = cn . GR spaces are now obtained just by adding a topology. Definition 3.4.3. A GR space is a structure A = h A, ∗, ≤, c0 , c1 , cn , T i, such that h A, ∗, ≤, c0 , c1 , cn i is a partially ordered left normal band with constants and T is a topology making: 1. ∗ a continuous map; 2. h A, ≤, T i a totally order disconnected space. Example 3.4.4. Consider the universe 3 = {0, 1, n} of the algebra 3, equipped with discrete topology, where ≤ ≡ ≤∧ , c0 = 0, c1 = 1, cn = n and ∗ is defined as follows: ( a if b 6= n, a∗b = b otherwise.

This is a GR space. It is readily checked that a ∗ b = a ∨ ( a ∧ b) = a ∧ ( a ∨ b), that h{0, 1, n}, ∗i is a left normal band and that the induced order v coincides with ≤∨ . We call DB the algebraic category of bisemilattices and GR the category with GR spaces as objects and continuous maps preserving ∗, constants and the order as morphisms. As previously mentioned, the duality is obtained by the usual technique in the presence of a dualising object. Theorem 3.4.5. [105, Thm. 7.5] The categories DB and GR are dual to each other under the invertible functor HomDB (−, 3) : DB → GR and its inverse HomGR (−, 3) : GR → DB. For a detailed proof of this result the reader is referred to [105]. Here, we only provide a sketch. Given a bisemilattice S, its dual GR space is Sˆ = HomDB (S, 3), i.e. the space of bisemilattice homomorphisms from S to 3. Analogously, if A is a GR space, then the dual is given

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91

˜ = Hom (A, 3), the bisemilattice of GR-morphisms from A to 3. by A GR The isomorphism between S and S˜ˆ is given by: ˜ˆ x 7→ ε ( x ), ε ( x )( ϕ) = ϕ( x ), ε S : S → S, S S

(3.4)

˜ for every x ∈ S and ϕ ∈ S. ˜ˆ the isomorphism is given by: Analogously, for A and A, ˆ˜ x 7→ δ ( x ), δ ( x )( ϕ) = ϕ( x ), δA : A → A, A A

(3.5)

ˆ for every x ∈ A and ϕ ∈ A. In order to provide a more concrete grasp of how a GR-space looks like, we present a detailed description of the space 3ˆ (the dual of 3). It is the 6-element set 3ˆ = { ϕ0 , ϕ1 , ϕn , id, ϕ2 , ϕ3 } of the endomorphisms of 3. ϕ0 , ϕ1 , ϕn stand for the constants homomorphisms on 0, 1 and n, respectively, id is the identity homomorphism, and ϕ2 , ϕ3 are defined as follows:   n 7−→ n ϕ2 = 0 7−→ 1   1 7−→ 1   n 7−→ n ϕ3 = 0 7−→ 0   1 7−→ 0 One may check that no other map from 3 to itself is a bisemilattice homomorphism. It is readily seen that ϕn ≤ ϕ0 ≤ ϕ1 , ϕ3 ≤ id ≤ ϕ2 and ϕ3 ≤ ϕ0 , ϕ2 ≤ ϕ1 (≤ coincides with ≤∧ induced by the ∧ operation in 3). For the order v (coinciding with ≤∨ ), one has ϕ0 v ϕ1 v ϕn , ϕ3 v id v ϕ2 , ϕ1 v ϕ2 and ϕ0 v ϕ3 . Notice that the two orders, represented by the Hasse diagrams in Figure 3.3, coincide on the subset { ϕ3 , id, ϕ2 }, which then forms a sublattice. The constant elements are cn = ϕn , c0 = ϕ0 and c1 = ϕ1 , while the following table spells out the behaviour of the binary operation ∗ (notice that for the constants: x ∗ ϕn = ϕn ∗ x = ϕn , x ∗ ϕ0 = x ∗ ϕ1 = x, for any x∈b 3).

CHAPTER 3. DUALITIES FOR REGULAR VARIETIES

92 ϕn • ϕ2 •

ϕ1 •

• id

v

• ϕ2

≤

ϕ1 •

ϕ0 •

• ϕ3

• id

ϕ0 •

ϕ3 • ϕn •

Figure 3.3: The Hasse diagrams of the two partial orderings in b 3. ϕ2 ∗ ϕ3 = ϕ2 ϕ0 ∗ ϕ3 = ϕ3 ϕ0 ∗ ϕ2 = ϕ3 ϕ1 ∗ ϕ3 = ϕ2 ϕ1 ∗ ϕ2 = ϕ2 ϕ3 ∗ id = ϕ3 ϕ2 ∗ id = ϕ2 ϕ0 ∗ id = ϕ3 ϕ1 ∗ id = ϕ2

ϕ3 ∗ ϕ2 = ϕ3

id ∗ ϕ3 = id id ∗ ϕ2 = id

It might be useful to look at the algebra b 3˜ = HomGR (b 3, 3). Its elements are maps that preserve the constants, the order ≤ and the operation ∗. Respecting the constants implies that any element Φ ∈ b 3˜ maps ϕn to n, ϕ0 to 0 and ϕ1 to 1. Moreover, preserving the order implies that Φ( ϕ3 ) ≤ Φ(id) ≤ Φ( ϕ2 ). Hence, the following maps are the only elements in b 3˜ :  ϕn 7−→ n      ϕ0 7−→ 0     ϕ 7−→ 1 1 Φn =  ϕ 3 7 −→ n      id 7−→ n    ϕ2 7−→ n

 ϕn 7−→ n      ϕ0 7−→ 0     ϕ 7−→ 1 1 Φ0 =  ϕ 3 7 −→ 0      id 7−→ 0    ϕ2 7−→ 1

 ϕn 7−→ n      ϕ0 7−→ 0     ϕ 7−→ 1 1 Φ1 =  ϕ 3 7 −→ 0      id 7−→ 1    ϕ2 7−→ 1

3.4. DUAL SPACES

93

It can be checked (pointwise) that Φn ≤∧ Φ0 ≤∧ Φ1 and Φ0 ≤∨ Φ1 ≤∨ Φn . It is then immediate to see that b 3˜ is isomorphic to 3.

3.4.2 GR spaces with involution A concrete topological description of the dual space of an involutive bisemilattice can be given by considering GR spaces with an additional unary operation (as introduced in [36]). Definition 3.4.6. A GR space with involution is a GR space G equipped with a continuous map ¬ : G → G such that, for any a ∈ G: G1. ¬(¬ a) = a; G2. ¬( a ∗ b) = ¬ a ∗ ¬b; G3. if a ≤ b then ¬b v ¬ a; G4. ¬c0 = c1 , ¬c1 = c0 and ¬cn = cn ; G5. the space HomGR (G, WK0,1 ) equipped with the unary operation ¬, defined as 0,1 ¬ ϕ( a) := ¬WK ( ϕ(¬G a)) satisfies ϕ ∧ (¬ ϕ ∨ ψ) = ψ ∧ ϕ, where the operations are defined pointwise; G6. there exist ϕ0 , ϕ1 ∈ HomGR (G, WK0,1 ) such that ¬ ϕ0 = ϕ1 and ϕ ∨ ϕ0 = ϕ, for each ϕ ∈ HomGR (G, WK0,1 ). We will refer to the unary operation ¬ as involution, as it resembles the properties of an involutive negation. The Definition of a GR space may sound odd, due to conditions (G5) and (G6) which forces the GRhomomorphism to satisfy some properties typical of involutive bisemilattices (namely (I6), (I7) and (I8) from Definition 2.4.1). However, this is an essential requirement in order to establish the desired duality and, unfortunately, they are unlikely to be improved upon (see Remark 3.4.8). Example 3.4.7. WK0,1 , equipped with the discrete topology, is the canonical example of a GR space with involution. Remark 3.4.8. A natural solution to avoid conditions (G5) and (G6) in Definition 3.4.3 would be having, for every GR space G, two maps ϕ0 , ϕ1 ∈ HomGR (G, WK0,1 ) such that: ϕ0 ( a ) = 0

and

ϕ1 ( a) = 1,

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for every a ∈ A r {c0 , c1 , cn }. However, it is not difficult to show that, in general, ϕ0 and ϕ1 are not morphisms of GR spaces, as witnessed by the following example. Consider the direct system formed by N ∞ , the Alexandroff compactification [213] of N, a trivial GR space {c} and the unique map from N ∞ to {c} (one should also, pedantically, add the identity maps). Upon defining n ∗ m = min(n, max(n, m)), N ∞ turns into a partially ordered left zero band where ≤ is the usual ordering over N and n ≤ ∞, for every n ∈ N. Since, by [105, Theorem 4.8], every partially ordered left normal band is the Płonka sum of partially ordered left zero bands, the introduced Płonka sum over the 2-element semilattice can be turned into a GR space, topologised via the disjoint union topology and by setting c0 = 0, c1 = ∞ and cn = c. It is easy to see that ϕ0 : N ∞ ∪ {c} → 3 is not continuous, since ϕ0−1 ({1}) = {∞} = N ∞ r N is not an open set. Definition 3.4.9. IGR is the category whose objects are GR spaces with involution and morphisms are GR-morphisms preserving the involution. Given a GR space with involution G, we can consider its GR space reduct (simply its involution-free reduct), call it A, which can be associ˜ = Hom (A, 3). Aiming at ated to the dual distributive bisemilattice A GR ˜ turning it into an involutive bisemilattice, we define an involution on A as follows: 0,1 ¬Φ( a) := ¬WK (Φ(¬G a)), ˆ and a ∈ G. for each Φ ∈ A To simplify notation, in the rest of this section we write ¬ in place of 0,1 ¬G and 0 in place of ¬WK . Lemma 3.4.10. Let G be a GR space with involution and A its ¬-free reduct. If ˜ then ¬Φ ∈ A. ˜ Φ∈A Proof. Assuming that Φ is a morphism of GR spaces, we have to verify that also ¬Φ is, i.e., that it is a continuous map, preserving the operation ∗, the constants and the order ≤. Observe that ¬Φ is continuous as it is the composition of continuous maps. Concerning operations and constants, we have: ¬Φ( a ∗ b) = (Φ¬( a ∗ b))0 = (Φ(¬ a ∗ ¬b))0 = (Φ(¬ a) ∗ Φ(¬b))0 = (Φ(¬ a))0 ∗ (Φ(¬b))0 = ¬Φ( a) ∗ ¬Φ(b). ¬Φ(c0 ) = (Φ(¬c0 ))0 = (Φ(c1 ))0 = 10 = 0. Similarly, ¬Φ(c1 ) = (Φ(¬c1 ))0 = (Φ(c0 ))0 = 00 = 1 and ¬Φ(cα ) = (Φ(¬cα ))0 = (Φ(cα ))0 = α0 = α. As for the order, let a ≤ b, but then ¬b v ¬ a. Since Φ preserves both orders, Φ(¬b) ≤∨ Φ(¬ a), thus (Φ(¬ a))0 ≤∧ (Φ(¬ a))0 , i.e. ¬Φ( a) ≤ ¬ Φ ( b ). 

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Proposition 3.4.11. Let G be a GR space with involution and A its ¬-free ˜ = hA, ˜ ¬i is an involutive bisemilattice. reduct. Then G Proof. We have to check that conditions (I1) to (I8) of Definition 2.4.1 hold b Clearly, (I1), (I2) and (I3) hold as A ˆ is a distributive bisemilattice, for G. while (I6), (I7) and (I8) hold by definition. For the remaining ones, let ˆ and a ∈ A. ϕ, ψ ∈ A I4. ¬(¬ ϕ( a)) = ¬ ϕ(¬( a))0 = ϕ(¬¬ a)00 = ϕ( a). I5. ¬( ϕ ∨ ψ)( a) = ( ϕ ∨ ψ(¬ a))0 = ( ϕ(¬ a) ∨ ψ(¬ a))0 = (0 ϕ(¬ a))0 ∧ (ψ(¬ a))0 = ¬ ϕ( a) ∧ ¬ψ( a).  b˜ Proposition 3.4.12. Let G be a GR space with involution. Then G ∼ = G. b˜ where A is the GR space Proof. By Theorem 3.4.5 we have that A ∼ = A, reduct of G. To prove our claim we only have to prove that the isomor˜ preserves phism, given by (3.5), δA ( x )( ϕ) = ϕ( x ), for x ∈ A and ϕ ∈ A, the involution. Indeed

(¬δA ( x ))( ϕ) = (δA ( x )(¬ ϕ))0 = (¬ ϕ( x ))0 = ( ϕ(¬ x ))00 = ϕ(¬ x ).  We proceed analogously, on the algebraic side. Given an arbitrary involutive bisemilattice B, we consider its bisemilattice reduct S = h B, ∧, ∨i, which is distributive by Proposition 2.4.7, and therefore can be associated b = Hom (S, 3). The bisemilattice 3 turns into to the dual GR space, S DBL 0,1 WK just by adding the involution ¬ and the constants 0, 1. This allows b as to define a unary operation ¬ on S

¬ ϕ( x ) := ¬B ( ϕ(¬S ( x ))), b

b is closed with respect to the for any ϕ ∈ Sb and x ∈ B. We prove that S b above-defined operation. Again, to simplify notation, we put ¬S = ¬ and B 0 ¬ =. b then ¬ ϕ ∈ S. b Lemma 3.4.13. If ϕ ∈ S b It suffices to verify that also ¬ ϕ preserves the Proof. Suppose that ϕ ∈ S. operations ∨ and ∧. We give the details for one, as for the other it can be done analogously:

¬ ϕ( x ∨ y) = ( ϕ( x ∨ y)0 )0 = ( ϕ( x 0 ∧ y0 ))0 = ( ϕ( x 0 ) ∧ ϕ(y0 ))0 = = ( ϕ( x 0 ))0 ∨ ( ϕ(y0 ))0 = ¬ ϕ( x ) ∨ ¬ ϕ(y). 

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The reader may wonder whether the above construction is actually needed and if it would be possible to construct a dual space using WK0,1 as dualising object. Unfortunately, this is not the case as witnessed by the following remark. Remark 3.4.14. The role of the dual space cannot be played by the space HomISBL (B, WK0,1 ) of homomorphisms of involutive bisemilattices (namely those maps preserving also involution) with ϕ0 ( x ) = ϕ( x )0 . Indeed, this space is not closed under such involution: ϕ0 ( x ∨ y) = ( ϕ( x ∨ y))0 = ( ϕ( x ) ∨ ϕ(y))0 = ϕ0 ( x ) ∧ ϕ0 (y), which is in general different from ϕ0 ( x ) ∨ ϕ 0 ( y ). Despite the negative observation in the above remark, the object WK0,1 is still dualising in the sense that it belongs both to the algebraic category IBSL of involutive bisemilattices and to its topological counterpart. However, the duality cannot be constructed as for the cases of Boolean algebras and distributive lattices (see Subsections 3.1.2 and 3.1.3). b ¬i is a GR space with involution. b = hS, Proposition 3.4.15. B b is a GR space, thus we only have Proof. By Theorem 3.4.5 we have that S b and to check that ¬ has the properties in Definition 3.4.6. Let ϕ, ψ ∈ S x ∈ S; properties (G1)-(G4) can be easily verified as follows:

¬(¬ ϕ( x )) = ¬( ϕ( x 0 ))0 = ( ϕ( x 00 ))00 = ϕ( x ). ¬( ϕ ∗ ψ)( x ) = ( ϕ ∗ ψ( x 0 ))0 = ( ϕ( x 0 ) ∗ ϕ( x 0 ))0 = ( ϕ( x 0 ))0 ∗ (ψ( x 0 ))0 = ¬ ϕ ( x ) ∗ ¬ ψ ( x ). Let ϕ ≤ ψ, i.e. ϕ( x ) ≤∧ ψ( x ) for each x ∈ S. In particular ϕ( x 0 ) ≤∧ ψ( x 0 ), thus (ψ( x 0 ))0 ≤∨ ( ϕ( x 0 ))0 , i.e. ¬ψ v ¬ ϕ. Let ϕ0 , ϕ1 and ϕn be the constant homomorphisms (of bisemilattices) on 0, 1 and n, respectively. ¬ ϕ0 ( x ) = ( ϕ0 ( x 0 ))0 = 00 = 1 = ϕ1 ( x ); ¬ ϕ1 ( x ) = ( ϕ1 ( x 0 ))0 = 10 = 0 = ϕ0 ( x ); ¬ ϕn ( x ) = ( ϕn ( x 0 ))0 = n0 = n = ϕ n ( x ). b ∼ B. b In order to prove (G5) and (G6), we preliminarily prove that B = b b under the Recall that the bisemilattice reduct S of B is isomorphic to S isomorphism given by (3.4), namely ε S ( x )( ϕ) = ϕ( x ), for every ϕ ∈ Sˆ and x ∈ S. The map ε S is obviously a homomorphism of bisemilattices and b b \ {Φ0 , Φ1 }, where by Φ0 , Φ1 we indicate a bijection from B \ {0, 1} to B

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b b This map can be extended to a bijection from B to the constants in B. b b by setting ε (0) = Φ0 and ε (1) = Φ1 . We have to prove that Φ0 and B, S S b and that ε also preserves b Φ indeed play the role of the constants in B 1

S

involution. We start with the latter task:

(¬ε S ( x ))( ϕ) = (ε S ( x )(¬ ϕ))0 = (¬ ϕ( x ))0 = ( ϕ( x 0 ))00 = ϕ( x 0 ). Regarding the constants, we only need to prove that ¬Φ0 = Φ1 and Ψ ∨ b b Indeed, for any ϕ ∈ B, b one has: Φ0 = Ψ, for each Ψ ∈ B.

¬Φ0 ( ϕ) = ¬ε S (0)( ϕ) = ϕ(00 ) = ϕ(1) = ε S (1)( ϕ) = Φ1 ( ϕ). b b there exists x ∈ I such that Finally, since ε S is onto, then, for any Ψ ∈ B, Ψ = ε S ( x ). Therefore Ψ( ϕ) = ε S ( x )( ϕ) = ε S ( x ∨ 0)( ϕ) = ϕ( x ∨ 0) = ϕ( x ) ∨ ϕ(0) = ε S ( x )( ϕ) ∨ ε S (0)( ϕ) = (Ψ ∨ Φ0 )( ϕ). This concludes the proof.  In order to prove Theorem 3.4.17 we are only left with proving that the functors Homb (−, WK0,1 ) : IBSL → IGR and HomGR (−, WK0,1 ) : IGR → IBSL are contravariant; we consider just the first functor, as for the other the proof runs analogously. Proposition 3.4.16. Any morphism f : B → C of IBSL induces a morphism b → B, b B b where C, b are the dual spaces of C and B, respectively. of IGR f ∗ : C Proof. f ∗ is defined in the usual way, i.e. f ∗ ( jˆ)(i ) = jˆ( f (i )), for each i ∈ B and jˆ ∈ bJ. It suffices to prove that f ∗ preserves the involution, namely f ∗ (¬ jˆ) = ¬ f ∗ ( jˆ), for all j ∈ J, which is immediate as

(¬ f ∗ ( jˆ))(i ) = ¬ jˆ( f (i )) = f ∗ (¬ jˆ)(i ).  All the above facts entail: Theorem 3.4.17. The categories of involutive bisemilattices and GR spaces with involution are dually equivalent. In virtue of Corollary 3.3.9, we have that involutive bisemilattices are also dually equivalent to the category of semilattice inverse systems of Stone spaces, which leads to the following Corollary 3.4.18. The category of semilattice inverse systems of Stone spaces is equivalent to the category of GR spaces with involution.

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Corollary 3.4.18 highlights an interesting as well as unexpected topological properties of Stone spaces. Indeed the category of semilattice inverse systems of Stone spaces can be described by a specific class of topological spaces, namely GR spaces with involution.

3.5

A topological counterpart of Płonka sums

In the previous part of the section, we have seen how to associate a dual topological space to some regular varieties. Interestingly enough, the dual structures such as GR spaces have an algebraic reduct admitting a Płonka sum representation (left normal bands are Płonka sums of left zero bands). An interesting, yet open, question is whether it is possible to design a “topological” version of the Płonka sum, yielding a duality with the algebraic construction. A partial attempt to introduce a topological counterpart is represented by the Płonka product of topological spaces. We recall here the main ingredients and results. The key intuition is that the direct product is (a sort of) “dual” construction with respect to disjoint sum. Since the Płonka sum is obtained out of a disjoint sum of algebras, a specific notion of product can play the role of topological dual of a Płonka sum. When considering a product Πi∈ I Xi of a family of topological spaces { Xi }i∈ I , we will think it topologised with the Tychonoff topology [213]. Also, for the purposes of the present section only, we will not distinguish between a topological space and its underlying set. Definition 3.5.1. Let { Xi }i∈ I be a family of topological spaces, where I = h I, ≤i is a join semilattice. The Płonka product of the family { Xi }i∈ I is a quadruple h X, I, λ, f i such that: (a) X is a space homeomorphic to the product space Πi∈ I Xi via the homeomorphism f : X → Πi∈ I Xi ; (b) λ is a continuous action of I on X, namely λ(i, λ( j, x )) = λ(i ∨ j, x ),

∀i, j ∈ I, ∀ x ∈ X;

(c) The homeomorphism f is compatible with the action λ, i.e. π j (λ f (i, x )) = π j (λ f (i, x 0 )) if πi∨ j ( x ) = πi∨ j ( x 0 ), x, x 0 ∈ Πi∈ I Xi , where λ f is the action of I on Πi∈ I Xi induced by λ and the map f (i.e. λ f (i, x ) = f (λ(i, f −1 ( x ))) and πi : Πi∈ I Xi → Xi is the canonical projection;

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(d) πi (λ f (i, x )) = πi ( x ). Notice that, since h I, ≤i is a semilattice, hence a semigroup, the action is a semigroup action in the usual sense [62]. The following notion of morphism allows one to turn Płonka products into a category. Definition 3.5.2. Given two Płonka products, h X, I, λ, f i and hY, J, µ, gi, a morphism between them is a pair h ϕ, Φi such that: (i) ϕ : J → I is a semilattice homomorphism; (ii) Φ : X → Y is an equivariant continuous function, namely Φ(λ( ϕ( j), x )) = µ( j, Φ( x )), for all x ∈ X and j ∈ J; (iii) π j ◦ g ◦ Φ ◦ f −1 (z) = π j ◦ g ◦ Φ ◦ f −1 (z0 ), if π ϕ( j) (z) = π ϕ( j) (z0 ), for some j ∈ J and z ∈ ∏i∈ I Xi . Theorem 3.5.3. Let C be an arbitrary topological category. Then the category Pl (C) of Płonka products in C is equivalent to the category of semilattice inverse systems in C. The technical details required for the proofs do not provide additional insights; the interested reader may consult [34]. Theorem 3.5.3 yields a new characterisation of the dual space for algebras belonging to regular varieties, which admits a representation as a Płonka sum over a semilattice direct system (Theorem 2.3.5) whose algebraic members are dualisable, strongly irregular varieties. Corollary 3.5.4. Let C be a dualisable strongly irregular algebraic category, with topological dual category C∗ . Then the categories R(C) and Pl (C∗ ) are dually equivalent. The Płonka product consists only of a partial “solution” to problem of finding a topological analogue of Płonka sums of algebras. While the presence of a partition function allows us to decompose an algebra into a disjoint sum, no analogue is known on the topological side. More precisely, it is unknown under what assumptions a topological space can be decomposed into a product (in the sense that it is homeomorphic to a product of spaces), even under strong assumptions, e.g. that the space is a topological or differential manifold.

Chapter 4 An interlude: Abstract Algebraic Logic Since the next three chapters will contain some heavy-duty Abstract Algebraic Logic, we include here some basic notions on the subject to keep this book reasonably self-contained. This chapter is not an Abstract Algebraic Logic primer: for a far more extensive treatment and for the occasional bits of unexplained terminology and notation, the reader is referred to what is by now the standard reference in the discipline, the textbook [96]. We will be working with propositional languages in an arbitrary but fixed similarity type τ, built over a denumerable set Var of propositional variables x, y, z, . . . . Fmτ will denote the algebra of τ-formulas, standardly defined. Recall that, given a set Γ of τ-formulas, we denote by Var (Γ) the set of all variables occurring in members of Γ; when Γ is a singleton, braces will be omitted. A τ-identity is an ordered pair h ϕ, ψi of τ-formulas, usually written ϕ ≈ ψ. A logic of type τ is an ordered pair L = hFmτ , `L i, where `L ⊆ P ( Fmτ ) × Fmτ is a consequence relation that is substitution-invariant, meaning that for every σ ∈ End(Fmτ ) and for every Γ ∪ { ϕ} ⊆ Fmτ , if Γ `L ϕ, then σ [Γ] `L σ ( ϕ). Unless otherwise specified, we identify a logic with its underlying consequence relation. A logic L of type τ is inconsistent if Γ `L ϕ for every Γ ∪ { ϕ} ⊆ Fmτ (equivalently, if ∅ `L x for some variable x), and almost inconsistent if the same condition holds with Γ nonempty. A logic is consistent when it is not inconsistent. A logic L of type τ is finitary when the following holds for all Γ ∪ { ϕ} ⊆ Fmτ : if Γ `L ϕ, then there is a finite ∆ ⊆ Γ s.t. ∆ `L ϕ. Given Γ, ∆ ⊆ Fmτ , we write Γ `L ∆ as shorthand for: Γ `L ϕ, for all ϕ ∈ ∆. We also write `L ϕ in place of ∅ `L ϕ; in such case, we say that ϕ is a theorem of L.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_4

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As we have seen in Chapter 1, an antitheorem of a logic L of type τ, also called a set of inconsistency terms for L ([140], but see also [47, 200]) is a set Γ ⊂ Fmτ such that for every σ ∈ End(Fmτ ) and for every ϕ ∈ Fmτ , we have that σ[Γ] `L ϕ. For example, for any formula ϕ, the set {¬( ϕ → ϕ)} is an antitheorem for all superintuitionistic logics, all axiomatic extensions of the logic MTL [54, 81] including Łukasiewicz logic [53], and all local and global consequences of normal modal logics. Observe that if L has an antitheorem, then L has an antitheorem in the single variable x. If, moreover, L is finitary, then it has a finite set of inconsistency terms in the single variable x. Given a type τ, a τ-inference is an ordered pair hΓ, ϕi, with Γ = { ϕ1 , ..., ϕn } a finite subset of Fmτ and ϕ ∈ Fmτ . A τ-rule is a set R of τ-inferences such that, whenever hΓ, ϕi ∈ R and σ ∈ End(Fmτ ), also hσ[Γ], σ( ϕ)i ∈ R. Rules are often presented schematically and thus written more conveniently either in the form ϕ1 , ..., ϕn . ϕ or in the form ϕ1 , ..., ϕn . ϕ. If we want to stress that a τ-rule of this form belongs to some consequence relation `L , we also write ϕ1 , ..., ϕn `L ϕ. The following τ2 -rules, already introduced in Chapter 1, will be used later: ¬ϕ ∨ ψ ( MP, Modus Ponens) ψ

ϕ ϕ∧ψ ϕ

ϕ∧ψ (CS, Conj. Simplification) ψ

¬ϕ

ϕ ψ

( EAQ, Ex Absurdo Quodlibet)

ϕ ϕ∨ψ ϕ ∨ ¬ϕ

ψ ( DA, Addition) ϕ∨ψ

( EM, Excluded Middle)

A Hilbert-style calculus of type τ is a set H of τ-rules, with its attendant derivability relation `H standardly defined. We say that the calculus H is for the logic L in case `H = `L . A τ-matrix is an ordered pair hA, F i where A is a τ-algebra and F ⊆ A. In this case, A is called the algebraic reduct of hA, F i. A τ-matrix hA, F i is a submatrix of a τ-matrix hB, G i if A is a subalgebra of B and F ⊆ G. The direct product Πi∈ I hAi , Fi i of a class {hAi , Fi i}i∈ I of τ-matrices is the τ-matrix hΠAi , ΠFi ii∈ I . Given a class M of τ-matrices, we denote by S(M), P(M) and PSD (M) its closure under formation of submatrices, direct products, and subdirect products respectively.

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Recall from Chapter 1 that an A-valuation, for A a τ-algebra, is a member of Hom (Fmτ , A). Every τ-matrix hA, F i induces a logic of the same type whose consequence relation is determined as follows: Γ `hA,Fi ϕ iff for every A-valuation v, if v[Γ] ⊆ F, then v( ϕ) ∈ F. For M a class of τ-matrices, we write Γ `M ϕ to mean Γ `hA,Fi ϕ for every hA, F i ∈ M. A τ-matrix hA, F i is a model of the logic L, also of type τ, when `L ⊆`hA,Fi . A logic L is complete w.r.t. a τ-matrix hA, F i when `L =`hA,Fi ; likewise, it is complete w.r.t. a class of τ-matrices M when `L =`M . A set F ⊆ A is a deductive filter of L (or an L-filter) on the τ-algebra A, when the τ-matrix hA, F i is a model of L. The L-filter generated by X ⊆ A on A is denoted by FgLA ( X ). The following lemma is folklore. Lemma 4.0.1. [96, Prop. 2.24] Let hA, F i, hB, G i be two matrices and let h : A → B be a homomorphism from the algebra A to the algebra B such that F = h−1 [ G ]. If G is a L-filter on B then F is a L-filter on A. Let A be a τ-algebra and F ⊆ A. A congruence θ on A is compatible with F when F is a union of θ-cosets. The largest congruence of A that is compatible with F always exists; this congruence is called the Leibniz congruence of F on A, and is denoted by ΩA F. The next lemma characterises the Leibniz congruence of a filter F on an algebra A: Lemma 4.0.2. [96, Thm. 4.23] Let A be a τ-algebra, F ⊆ A, and a, b ∈ A.

h a, bi ∈ ΩA F ⇐⇒ for every unary polynomial ϕ( x, ~c) on A, ϕA ( a, ~c) ∈ F if and only if ϕA (b, ~c) ∈ F. The Suszko congruence of F ⊆ A on A relative to the logic L, noted A e ΩL F, on the other hand, is the intersection of all ΩA G, where G is an L-filter on A that contains F. The next lemma characterises the Suszko congruence of a L-filter F on an algebra A: Lemma 4.0.3. [96, Thm. 5.32] Let L be a logic, A be an algebra of the same type, F ⊆ A, and a, b ∈ A. e A F ⇐⇒ for every unary polynomial ϕ( x, ~c) on A, h a, bi ∈ Ω L FgLA ( F ∪ { ϕA ( a, ~c)}) = FgLA ( F ∪ { ϕA (b, ~c)}).

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e A assigns to each L-filter F on A its Suszko conThe Suszko operator Ω L gruence, and is monotone [96, Lemma 5.37]. By means of the Leibniz and Suszko congruences, we can associate to logics three distinguished classes of models. More precisely, given a logic L of type τ, we set Mod(L) = {hA, F i | hA, F i is a model of L}; Mod∗ (L) = {hA, F i ∈ Mod(L) | ΩA F = ∆A }; e A F = ∆ A }, ModSu (L) = {hA, F i ∈ Mod(L) | Ω L

where ∆A is the identity on A. The above classes of τ-matrices are called, respectively, the classes of models, Leibniz reduced models, and Suszko reduced models of L. A τ-matrix is trivial if it is of the form hA, Ai. Observe that the trivial τ-matrix h1, {1}i, whose algebraic reduct is the trivial algebra 1 of type τ, is a Leibniz reduced model of every logic. Moreover, if L is a logic and hA, Ai ∈ Mod∗ (L) is a trivial τ-matrix, then hA, Ai = h1, {1}i. Let L be a logic of type τ. A translation of type τ is a set E = {γi ( x ) ≈ δi ( x )}i∈ I of τ-identities in a single variable. We may also view a translation E as a function that maps τ-formulas to sets of τ-identities. Thus, we let E( ϕ) stand for the set

{γi ( x/ϕ) ≈ δi ( x/ϕ)}i∈ I , where γi ( x/ϕ) refers to the result of uniformly replacing any occurrences of x in γi by ϕ, and similarly for δi ( x/ϕ). For Γ ⊆ Fmτ , E [Γ] is defined as [ { E(γ) | γ ∈ Γ}. On occasion, if A is a τ-algebra and E is a translation,  we denote by SolEA the set of solutions a ∈ A : A  EA ( a) . Using some of the previously defined notions, we can give a brief survey of the Leibniz hierarchy, where logics are ranked according to their behaviour “with regard to their matrix semantics and their lattices of filters” [96, p. 317]. A logic L of type τ is: • protoalgebraic, if there is a set of τ-formulas ∆( x, y) in two variables, called protoimplication formulas for L, such that `L ∆( x, x ) and x, ∆( x, y) `L y; • equivalential, if there is a set of τ-formulas ∆( x, y) in two variables, called congruence formulas for L, such that for every hA, F i ∈ Mod(L),

h a, bi ∈ ΩA F iff ∆A ( a, b) ⊆ F for all a, b ∈ A;

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• truth-equational, if there is a translation E of type τ such that for all hA, F i ∈ Mod∗ (L), a ∈ F iff a ∈ SolEA . E = E( x ) is called a set of defining equations for L. • algebraisable, if it is both equivalential and truth-equational. Any equivalential logic is in particular protoalgebraic. Also: Theorem 4.0.4 ([96],Theorem 6.106). A logic L of type τ is truth-equational if and only if the Suszko operator is injective over the set of its filters, for any τ-algebra. A finitary algebraisable logic L enjoys an especially tight and fruitful relationship with an attendant class of algebras K, called the equivalent algebraic semantics of L. This relationship can be expressed in different ways — most notably, as an isomorphism of certain expanded lattices of theories of L and of theories of the equational consequence relation of K, or as the presence of mutually inverse substitution-invariant mappings between entailments in L and in the equational consequence relation of K: see [96] for more details. Given a logic L, we set Alg∗ (L) = {A | there is F ⊆ A s.t. hA, F i ∈ Mod∗ (L)}. Alg(L) = {A | there is F ⊆ A s.t. hA, F i ∈ ModSu (L)}. In other words, Alg∗ (L) is the class of the algebraic reducts of Leibniz reduced models of L, Alg(L) is the class of algebraic reducts of Suszko reduced models of L. We have that [96, Thms. 5.70, 5.71, 6.7, 6.15, 6.92]: Proposition 4.0.5. Let L be a logic. 1. ModSu (L) = PSD (Mod∗ (L)) and Alg(L) = PSD (Alg∗ (L)). 2. If L is protoalgebraic, then Mod∗ (L) = ModSu (L) and hence Alg(L) = Alg∗ (L). 3. If a non almost-inconsistent logic is either protoalgebraic, or truth-equational, it has theorems. 4. Alg(L) is closed w.r.t. direct products. Henceforth, we will omit references to types when no danger of ambiguity is present. For example, we will say “matrix”, instead of “τ-matrix”, when the type τ is clear from the context. The following theorem can be inferred from [200, Thm. 3.6], and it discloses fundamental properties of antitheorems for finitary protoalgebraic logics.

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Theorem 4.0.6. Let L be a finitary protoalgebraic logic and let Σ( x ) = { x, ψ1 ( x ), . . . , ψn ( x )} be an antitheorem for L. Let moreover F be a L-filter on an algebra A and a ∈ A. Then: A = FgLA ({ a} ∪ F ) ⇐⇒ {ψ1A ( a) . . . ψnA ( a)} ⊆ F. Lemma 4.0.7. Let L be a logic of type τ, and let e, δ ∈ Fmτ . The following are equivalent: 1. Alg(L)  e ≈ δ; 2. ϕ(e, ~z) a`L ϕ(δ, ~z), for every τ-formula ϕ(v, ~z). Proof. See [96, Lm. 5.74(1)] and [96, Thm. 5.76].



In general, the intended algebraic counterpart of a logic L, namely Alg(L), is not a variety. However, there is a variety that is ”naturally” associated to a logic L of type τ, the so-called intrinsic variety of L, defined as e Fmτ ( F0 )), V (L) = V (Fmτ /Ω L where F0 is the smallest L-filter on Fmτ . The following results concerning the intrinsic variety of a logic are well-known and particularly useful in practice (see [96, Thm.5.76]). Theorem 4.0.8. For any logic L of type τ the following classes of algebras coincide: 1. V (L); 2. V (Alg∗ (L)); 3. V (Alg(L)). We conclude by briefly citing a further important property of logics. A logic L of type τ is selfextensional if and only if the relation

{h ϕ, ψi | ϕ `L ψ, ψ `L ϕ} is a congruence on Fmτ .

Chapter 5 Logics of left variable inclusion Regular varieties are defined by regular identities, where the same set of variables occurs on both sides. Therefore, we are not taking a wild guess if we assume that there must be some kinship between regular varieties and logics of variable inclusion, of which we have given a cursory preview in Section 1.4. The aim of the next two chapters is, on the one hand, to fully clarify the nature and extent of this connection; on the other, to lift the algebraic construction of Płonka sums to the level of logical matrices and to relate, in turn, these sums of matrices to logics of variable inclusion. In this chapter, in particular, we focus on logics of left variable inclusion – roughly, logics that impose some constraint on valid entailments to the effect that the variables in the premisses must be contained in the conclusion, too. More precisely, given an arbitrary logic L, its left variable inclusion counterpart Ll is defined by enforcing the following syntactic requirement: Γ `Ll ϕ ⇐⇒ there is ∆ ⊆ Γ s.t. Var (∆) ⊆ Var ( ϕ) and ∆ `L ϕ. The chapter is structured as follows. We begin by generalising the construction of Płonka sums from algebras to logical matrices (Section 5.1). In a nutshell, the connection between logics of left variable inclusion and Płonka sums can be epitomised in the following slogan: The left variable inclusion companion Ll of a logic L is complete w.r.t. the class of Płonka sums of matrix models of L (Corollary 5.1.8). As a matter of fact, a logic of the form Ll is especially well-behaved in case the original logic L has an l-partition function [209] – a not too uncommon feature, by the way. The importance of partition functions is reflected both at a syntactic and at a semantic level. Accordingly, on the one hand we present a general method to transform every Hilbert-style calculus for a finitary logic L with an l-partition function into a Hilbert-style calculus

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_5

107

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for Ll (Theorem 5.2.2). On the other hand, l-partition functions can be exploited to tame the structure of the matrix semantics ModSu (Ll ) of Ll , given by the Suszko reduced models of Ll . In particular, we obtain a full description of ModSu (Ll ) in case L is a finitary equivalential logic with an l-partition function (Theorems 5.4.3 and 5.4.8). Finally, we determine the whereabouts of the logics Ll in the Leibniz hierarchy, depending on the properties of L (Section 5.5).

5.1

Płonka sums of matrices and l-direct systems

5.1.1 General results The definition of a semilattice direct system of algebras (Definition 2.1.1) can be extended as follows to logical matrices: Definition 5.1.1. An l-direct system of matrices of type τ is an ordered pair X = hA, { Fi }i∈ I i such that:

 1. A = {Ai }i∈ I , h I, ≤i, { pij | i ≤ j} is a semilattice direct system of algebras of type τ; 2. for every i ∈ I, Fi ⊆ Ai ; 3. for every i, j ∈ I such that i ≤ j, pij [ Fi ] ⊆ Fj . . It may be expedient to view l-direct systems as the results of replacing, as a fibre of a semilattice direct system, each algebra Ai by the matrix hAi , Fi i. The chosen name, therefore, is no misnomer. Instead of the compact notation hA, { Fi }i∈ I i, we will often use the expanded form

h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i. Given an l-direct system of matrices X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i as above, the matrices in {hAi , Fi i}i∈ I are called the fibres of X. Moreover, we set [ Pł ( X ) := hPł (Ai )i∈ I , Fi i. i∈ I

5.1. PŁONKA SUMS OF MATRICES AND L-DIRECT SYSTEMS

109

The matrix Pł ( X ) is the Płonka sum of the l-direct system of matrices X. Clearly, if all the algebra reducts of the fibres of X have type τ, Pł ( X ) will be a τ-matrix. Given a class M of matrices, we denote by Pł (M) the class of all Płonka sums of l-direct systems of matrices in M. In the rest of the chapter, to simplify notation, we sometimes write Pł in place of Pł (Ai )i∈ I , when the family {Ai }i∈ I is clear from the context. The following observation is a routine computation: Lemma 5.1.2. SPł (M) ⊆ Pł (S(M)) and PPł (M) ⊆ Pł ( P(M)), for every class of matrices M. Definition 5.1.3. Let L be a logic of type τ. The left variable inclusion companion of L is the logic Ll =`Ll ⊆ P ( Fmτ ) × Fmτ defined for every Γ ∪ { ϕ} ⊆ Fmτ as Γ `Ll ϕ ⇐⇒ there is Γ0 ⊆ Γ s.t. Var (Γ0 ) ⊆ Var ( ϕ) and Γ0 `L ϕ. It is immediate to check that Ll is indeed a logic. Throughout this chapter, we will often refer to the left variable inclusion companion of a logic simply as its variable inclusion companion, as we will not examine any different pattern of syntactic sieve on entailments. Example 5.1.4. If L is classical propositional logic CL formulated in the type τ2 without constants, then CLl is the logic PWK, already discussed in Chapter 1 and to be examined more in depth in Chapter 7. It is remarkable that this logic can be equivalently defined syntactically, by imposing to CL the variable inclusion constraint as in Definition 5.1.3, or semantically, as the matrix consequence relation `hWK,{1,n}i (see Section 1.2.2 and Example 2.3.9). This is the content of the Ciuni-Carrara Theorem (Theorem 1.3.2) proved in Chapter 1. It is not difficult to check that the matrix hWK, {1, n}i is the Płonka sum of the matrices hB2 , {1B2 }i and h1, {1}i, where the latter is the trivial matrix on the trivial algebra of type τ2 , the index set is the 2-element semilattice and the homomorphisms are uniquely determined. If classical propositional logic CL0,1 is formulated in the type τ1 with constants, its variable inclusion companion is the logic PWK0,1 , also of type τ1 . Example 5.1.5. If L is Belnap-Dunn logic BD [160], then BDl is the logic ∗ , for which see [216], [58]. If L is strong Kleene logic K (see Chapter dSFDE 3 1), then K3 l is the logic dSFDE , introduced and discussed in [86], [58]. Finally, if L is B3 (see again Chapter 1), then B3 l is Szmuc’s HYB2 , also known as Sw FDE ([216], [69]). For all these logics, see also Section 1.2.5.

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110

In [33], it is shown that an algebraic semantics for PWK can be obtained via Płonka sums of Boolean algebras (see also Chapter 7 below). This result can be generalised to the variable inclusion companion of any logic L. Lemma 5.1.6. Let L be a logic of type τ and let X be an l-direct system of models of L. Then Pł ( X ) is a model of Ll . Proof. Assume that X is as in Definition 5.1.1 and that Γ `Ll ϕ. Consider S a Pł (Ai )i∈ I -valuation v such that v[Γ] ⊆ i∈ I Fi . By the definition of `Ll , there exists ∆ ⊆ Γ such that Var (∆) ⊆ Var ( ϕ) and ∆ `L ϕ. Consider an enumeration { x1 , . . . , xn } of Var ( ϕ). There are i1 , . . . , in ∈ I such that v( x1 ) ∈ Ai1 , . . . , v( xn ) ∈ Ain . We set j := i1 ∨ · · · ∨ in . Now, define an A j -valuation g such that g( xm ) = pim j (v( xm )), for every m ≤ n. (The behaviour of g with respect to variables not in Var ( ϕ) need not concern us.) We claim that g[∆] ⊆ Fj . To prove this, consider an arbitrary formula δ ∈ ∆. Since Var (∆) ⊆ { x1 , . . . , xn }, we can assume that Var (δ) = { xm1 , . . . , xmk } ⊆ { x1 , . . . , xn } for some k ≤ n. Set l := im1 ∨ · · · ∨ imk . From the definition of Pł ( X ) we have that v(δ) = δPł (v( xm1 ), . . . , v( xmk )) = δAl ( pim l (v( xm1 )), . . . , pim l (v( xmk ))). 1

Since v(δ) ∈

S

i∈ I

k

Fi , this implies that

δAl ( pim l (v( xm1 )), . . . , pim l (v( xmk ))) ∈ Fl . 1

k

(5.1)

Now observe that l ≤ j. Therefore there is a homomorphism plj : Al → A j such that plj [ Fl ] ⊆ Fj . Together with (5.1), this implies that g(δ) =δA j ( pim j (v( xm1 )), . . . , pim j (v( xmk ))) 1

k

=δA j ( plj ◦ pim l (v( xm1 )), . . . , plj ◦ pim l (v( xmk ))) 1

k

= plj (δAl ( pim1 l (v( xm1 )), . . . , pim l (v( xmk )))) k

∈ plj [ Fl ] ⊆ Fj . This establishes our claim.

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111

Recall that ∆ `L ϕ. Since hA j , Fj i is a model of L and g[∆] ⊆ Fj , we conclude that g( ϕ) ∈ Fj . But this means that v( ϕ) = ϕPł (v( x1 ), . . . , v( xn ))

= ϕA j ( pi1 j (v( x1 )), . . . , pin j (v( xn ))) = g( ϕ) ∈ Fj ⊆

[

Fi .

i∈ I

Hence we conclude that Pł ( X ) is a model of Ll as desired.



For subsequent use, we lift to matrices the construction of the algebra A∗ from Subsection 2.3.2. Given an arbitrary matrix hA, F i, there is always an l-direct system of matrices over the 2-element semilattice with universe {i, j}, i < j, where i indexes hA, F i and j indexes h1, {1}i. This system is equipped with the identity automorphisms and the unique homomorphism pij : A → 1. The Płonka sum of this system is the matrix hA∗ , F ∪ {1}i. Theorem 5.1.7. Let L be a logic of type τ and M be a class of τ-matrices containing h1, {1}i. If L is complete w.r.t. M, then Ll is complete w.r.t. Pł (M). Proof. By Lemma 5.1.6 it will be enough to show that if Γ 0Ll ϕ, then Γ 0Pł (M) ϕ. To this end, suppose that Γ 0Ll ϕ. Define Γ+ := {γ ∈ Γ | Var (γ) ⊆ Var ( ϕ)} Γ− := {γ ∈ Γ | Var (γ) * Var ( ϕ)}. Clearly Γ = Γ+ ∪ Γ− . Since Γ 0Ll ϕ, we know that Γ+ 0L ϕ. Together with the fact that L is complete w.r.t. M, this implies that there exist a matrix hA, F i ∈ M and an A-valuation v such that v[Γ+ ] ⊆ F and v( ϕ) ∈ / F. Since hA, F i, h1, {1}i ∈ M, we have that hA∗ , F ∪ {1}i ∈ Pł (M). Now, consider the A∗ -valuation g defined for every variable x as follows:  v( x ) if x ∈ Var ( ϕ) g( x ) := 1 otherwise. From the definition of A∗ it follows that: g [ Γ − ] ⊆ {1} ⊆ F ∪ {1}; g(γ) = v(γ) for every γ ∈ Γ+ ∪ { ϕ}. Together with the fact that v[Γ+ ] ⊆ F and v( ϕ) ∈ / F, this implies that g[Γ] = g[Γ+ ∪ Γ− ] ⊆ F ∪ {1} and g( ϕ) ∈ / F ∪ {1}. Hence we conclude that Γ 0Pł (M) ϕ as desired.



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Corollary 5.1.8. Let L be a logic. Its variable inclusion companion Ll is complete w.r.t. any of the following classes of matrices:

Pł (Mod(L));

Pł (Mod∗ (L));

Pł (ModSu (L)).

Proof. Observe that L is complete w.r.t. any of the classes Mod(L), Mod∗ (L), ModSu (L). Moreover any of these classes contains the (trivial) matrix h1, {1}i. Thus we can apply Theorem 5.1.7.  The proof of Theorem 5.1.7 also entails the following result, which considerably simplifies the characterisation of a complete matrix semantics for logics of left variable inclusion: Theorem 5.1.9. Let L be a logic of type τ. If L is complete w.r.t. the class of matrices M, then Ll is complete w.r.t. the class {hA∗ , F ∪ {1}i | hA, F i ∈ M}. The results in [58, Theorem 2] and [60, Theorem 3.1] are similar to Theorem 5.1.9. They are, however, at the same time more general and more specific. They are more general in that they cover the case of multipleconclusion consequence relations, which subsume as fragments logics as understood in this book. They are more specific because they are restricted to singleton classes of matrices. Remark 5.1.10. Observe that, if L is complete w.r.t. the class of matrices M, Ll is complete w.r.t. several different classes of matrices other than the one in Theorem 5.1.9. Indeed, in the above construction we can replace the matrix h1, {1}i, by any trivial matrix hB, Bi, where B is a τ-algebra, and get a completeness result w.r.t. sums of matrices in M and trivial matrices of this kind. This turns out to be particularly significant when the logic L is defined by a single matrix, i.e. when the class M contains exactly one element.

5.1.2 Left partition functions In this subsection we show that the theory of Płonka sums of matrices behaves especially well for logics that have left partition functions. Definition 5.1.11. Let τ be a similarity type. An essentially binary τformula x · y is a left partition function (for short l-partition function) for the logic L of type τ if x `L x · y and ·A is a partition function for every algebra in Alg(L). In this case, we also say that L has the l-partition function x · y.

5.1. PŁONKA SUMS OF MATRICES AND L-DIRECT SYSTEMS

113

Remark 5.1.12. By Lemma 4.0.7, the above definition can be rephrased in purely logical terms, by requiring that x `L x · y and that ϕ(e, ~z ) a`L ϕ(δ, ~z ) for every formula ϕ(v, ~z ), for every identity of the form e ≈ δ in (PF1)-(PF5), see Definition 2.2.1.



Example 5.1.13. Logics with an l-partition function abound in the literature. Indeed, the term x · y := x ∧ ( x ∨ y) is an l-partition function for every logic L such that every algebra in Alg(L) has a lattice term reduct. On the other hand, x · y := (y → y) → x is an l-partition function for all logics L such that every algebra in Alg(L) is 1-subtractive (Example 2.2.7).  Remarkably, the presence of l-partition functions transfers from a logic on to its variable inclusion companion. Lemma 5.1.14. Let L be a logic. The formula x · y is an l-partition function for L if and only if it is an l-partition function for Ll . Proof. By Remark 5.1.12, the fact that · is an l-partition function for L is witnessed by the validity of some inferences ϕ `L ψ such that Var ( ϕ) ⊆ Var (ψ). Hence these inferences also hold in Ll . Another application of Remark 5.1.12 yields that · is an l-partition function for Ll . The other direction follows from the inclusion Ll ⊆ L.  The following result is the generalisation of Theorem 2.2.8 to the setting of logical matrices. Theorem 5.1.15. Let L be a logic of type τ with an l-partition function ·, and let hA, F i be a model of L such that A ∈ Alg(L). Conditions 1., 3. and 4. of Theorem 2.2.8 hold for A. Moreover, if τ does not contain constants, Condition 2. holds and, setting Fi := F ∩ Ai for every i ∈ I, the triple X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i is an l-direct system of matrices such that Pł ( X ) = hA, F i. Proof. Theorem 2.2.8 provides us with all that is needed to prove our claim, except the fact that pij [ Fi ] ⊆ Fj for every i, j ∈ I such that i ≤ j. To this end, consider a ∈ Fi and b ∈ A j with i ≤ j. Since · is an l-partition function for L, we have x `L x · y. Since hA, F i ∈ Mod(L) and a ∈ F, this implies that a · b ∈ F. Observe that a · b ∈ A j and, therefore, that a · b ∈ Fj . Hence pij ( a) = a · b ∈ Fj . 

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What comes next is a partial converse to Lemma 5.1.6 above. Lemma 5.1.16. Let L be a finitary logic whose type τ does not contain constants with l-partition function ·, and let hA, F i ∈ Mod(Ll ), with A ∈ Alg(Ll ). Then, the fibres of hA, F i are models of L. Proof. Let hAi , Fi i be a fibre of hA, F i and Γ `L ϕ, with Γ a finite set. Then consider an Ai -valuation v such that v[Γ] ⊆ Fi . First, suppose Γ = ∅. Then clearly ∅ `Ll ϕ. Since Ai is a subalgebra of A (as τ does not contain constants) and hA, F i is a model of Ll , this implies that v( ϕ) ∈ F ∩ Ai = Fi . If Γ is nonempty, then there are γ1 , . . . , γn ∈ Fmτ such that Γ = {γ1 , . . . , γn }. Let ψ = γ1 · (γ2 · . . . (γn−1 · γn ) . . . ). Since · is an l-partition function, we have x `L x · y. In particular, this implies that ϕ `L ϕ · ψ. Then Γ `L ϕ · ψ. Given that the variable inclusion constraint holds for this inference, we obtain that Γ `Ll ϕ · ψ. Since Ai is a subalgebra of A and hA, F i is a model of Ll , this implies that v( ϕ · ψ) ∈ Ai ∩ F = Fi . However, v( ϕ) and v(ψ) belong to Ai , whereby v( ϕ) = v( ϕ) · v(ψ) = v( ϕ · ψ) and, therefore, v( ϕ) ∈ Fi , as desired.



5.2 Hilbert-style axiomatisations We now present a method for endowing logics of left variable inclusion with Hilbert-style axiomatisations. Definition 5.2.1. Let H be a Hilbert-style calculus whose derivability relation `H is a logic with an l-partition function ·. H l is defined as the Hilbert-style calculus given by the following rules: ∅Bψ γ1 , . . . , γn B ϕ · (γ1 · (γ2 · . . . (γn−1 · γn ) . . . )) ϕBϕ·ψ χ(δ, ~z ) C B χ(e, ~z ) for every (i) ∅ B ψ rule in H; (ii) γ1 , . . . , γn B ϕ rule in H;

(H1) (H2) (H3) (H4)

5.2. HILBERT-STYLE AXIOMATISATIONS

115

(iii) δ ≈ e equation in the definition of partition function, and formula χ(v, ~z ). Theorem 5.2.2. Let H be a Hilbert-style calculus whose type τ does not contain constants. If `H is a logic with an l-partition function ·, then H l is a complete Hilbert-style calculus for `lH . Proof. We begin by showing that `Hl ≤`lH . It will be sufficient to prove that every rule in H l holds in `lH . This is clear for (H1). Moreover, the rules (H3, H4) are valid in `lH , because · is a partition function for `lH by Lemma 5.1.14. It only remains to prove that (H2) holds in `lH . To this end, consider a rule γ1 , . . . , γn B ϕ in H. Clearly we have that γ1 , . . . , γn `H ϕ. Since · is an l-partition function for `H , we have x `H x · y. In particular, if ψ = γ1 · (γ2 · . . . (γn−1 · γn ) . . . ), then ϕ `H ϕ · ψ. Hence we conclude that γ1 , . . . , γn `lH ϕ · ψ, as desired. To prove `lH ≤`Hl , we reason as follows. Pick hA, F i ∈ ModSu (`Hl ). Clearly A ∈ Alg(`Hl ). Moreover, · is an l-partition function in `Hl by Remark 5.1.12 and (H3, H4). Hence we can apply Theorem 5.1.15, obtaining that hA, F i = Pł ( X ), where X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i is the l-direct system of matrices given in the statement of Theorem 5.1.15. Thanks to the rules of H l we can carry out the construction in the proof of Lemma 5.1.16, obtaining that each fibre hAi , Fi i is a model of `H . This observation, together with the fact that hA, F i = Pł ( X ) and Corollary 5.1.8, implies that hA, F i is a model of `lH . Hence we conclude that ModSu (`Hl ) ⊆ Mod(`lH ). This implies our claim.  The proof of the above result establishes the following: Corollary 5.2.3. If L is a finitary logic whose type τ does not contain constants, having an l-partition function, then ModSu (Ll ) ⊆ Pł (Mod(L)). Observe that the Hilbert calculus H l is infinite, as witnessed by condition (H4) in Definition 5.2.1. However, there can be cases in which H l can be reduced to a finite calculus. Recall from [96, p. 230] that the Leibniz congruence is finitisable in a class of τ-matrices when there is a finite set Φ ⊆ Fmτ such that for every matrix hA, F i in the class and every a, b ∈ A it holds that h a, bi ∈ ΩA F if and only if

~ A ( a, ~c) ∈ F ⇐⇒ δA (b, ~c) ∈ F. for all δ ∈ Φ and all ~c ∈ Aδ

(5.2)

CHAPTER 5. LOGICS OF LEFT VARIABLE INCLUSION

116

In other words, the quantifier in Lemma 4.0.2 can be taken to range over a finite set. This, together with Lemma 4.0.7, tells us that if the Leibniz congruence is finitisable in Mod(H l ), then the infinite set of rules determined by (H4) can be reduced to a finite one, therefore leading to a finite Hilbert calculus for Ll . It would be interesting to determine under what conditions the finitisability of the Leibniz congruence transfers from H to Hl . An alternative method for obtaining a Hilbert calculus for Ll out of a corresponding calculus for L consists in keeping the same axioms and attaching to each proper rule ϕ1 , ..., ϕn B ψ the proviso Var ( ϕ1 , ..., ϕn ) ⊆ Var (ψ). This approach sometimes works: as we will see in Chapter 7, a case in point is PWK. It is natural to wonder whether the method can be applied to any logic. Interestingly, Basu and Chakraborty [14] have answered this question in the negative. The derivability relation of such a restricted rules companion of a logic L always defines a sublogic of Ll – yet, this sublogic may be strictly weaker than Ll .

Suszko reduced models of Ll

5.3

In this section we investigate the structure of the Suszko reduced models ModSu (Ll ) of the variable inclusion companion Ll of a logic L. To begin with, we need a technical lemma. Lemma 5.3.1. Let L be a logic with an l-partition function ·, and let X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i be an l-direct system of models of L. Given an upset J ⊆ I, we define for every i ∈ I,  Ai if i ∈ J J Gi := Fi otherwise. Then

S

i∈ I

GiJ is a Ll -filter on Pł (Ai )i∈ I .

Proof. The matrices in {hAi , GiJ i}i∈ I naturally give rise to an l-direct system of matrices, when equipped with the homomorphisms in X. MoreS over, by assumption each hAi , GiJ i is a model of L. Thus i∈ I GiJ is a Ll -filter on Pł (Ai )i∈ I by Lemma 5.1.6.  The following result identifies the Płonka sums of matrices in ModSu (L) that belong to ModSu (Ll ).

5.3. SUSZKO REDUCED MODELS

117

Theorem 5.3.2. Let L be a logic of type τ with an l-partition function ·, and let X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i be an l-direct system of matrices in ModSu (L). The following conditions are equivalent: 1. Pł ( X ) ∈ ModSu (Ll ). 2. For every n, i ∈ I such that hAn , Fn i is trivial and n < i, there exists j ∈ I s.t. n ≤ j, i  j and A j is nontrivial. Proof. (1)⇒(2): Suppose that Pł ( X ) ∈ ModSu (Ll ), and consider n, i ∈ I such that hAn , Fn i is trivial and n < i. Taking into account that hAn , Fn i is trivial and belongs to ModSu (L), it follows that An is the trivial algebra. Then hAn , Fn i = h1, {1}i. Moreover, set a := pni (1). Since n < i, we know that a 6= 1. As Pł ( X ) ∈ ModSu (Ll ), there is a Ll -filter G of Pł (Ai )i∈ I such S that i∈ I Fi ⊆ G and h a, 1i ∈ / ΩPł (Ai )i∈ I G. Thus, by Lemma 4.0.3, there is S a formula ϕ( x, ~z) and elements ~c ∈ i∈ I Ai such that ϕPł ( a, ~c ) ∈ G ⇐⇒ ϕPł (1, ~c ) ∈ / G.

(5.3)

We can assume w.l.o.g. that all the elements in the sequence ~c belong to the same fibre Ak of the Płonka sum Pł (Ai )i∈ I .1 We claim that indeed ϕ(1, ~c ) ∈ / G. Suppose the contrary towards a contradiction. Then ϕ(1, ~c ) ∈ G. First observe that ϕPł ( a, ~c ) = ϕAi∨k ( pi,i∨k ( a), pk,i∨k (~c ))

(5.4)

=ϕ

Ai ∨k

( pi,i∨k ◦ pn,i (1), pk,i∨k (~c ))

(5.5)

=ϕ

Ai ∨k

( pn∨k,i∨k ◦ pn,n∨k (1), pn∨k,i∨k ◦ pk,n∨k (~c ))

(5.6)

= pn∨k,i∨k ( ϕ

An∨k

( pn,n∨k (1), pk,n∨k (~c )))

Pł

= pn∨k,i∨k ( ϕ (1, ~c )) Pł

= pn∨k,i∨k ( ϕ (1, ~c )) · Pł

= ϕ (1, ~c) · ∈ G.

Pł

a

(5.7) (5.8)

Ai ∨k

pi,i∨k ( a)

(5.9) (5.10) (5.11)

The non-obvious equalities above are justified as follows: (5.6) is a consequence of the fact that X is an l-direct system of matrices and that precisely, if ~c = c1 , . . . , cm and c1 ∈ A p1 , . . . , cm ∈ A pm , then we set k := p1 ∨ · · · ∨ pm and replace ci by p pi k (ci ). 1 More

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n ∨ k ≤ i ∨ k (since n ≤ i), (5.9) follows from the fact that x · y is the projection on the first component on the algebra Ai∨k . Condition (5.11) follows from the fact that ϕ(1, ~c ) ∈ G, G is a Ll -filter and, by Lemma 5.1.14 · is an l-partition function for Ll , hence x `Ll x · y. Hence we have that ϕ( a, ~c ), ϕ(1, ~c ) ∈ G, which contradicts (5.3), establishing the claim. From the claim and (5.3) we get that ϕ( a, ~c ) ∈ G and ϕ(1, ~c ) ∈ / G. Set j := n ∨ k and m := k ∨ i. We claim that j is such that: (A) n ≤ j, (B) A j is nontrivial and (C) i  j. We proceed to prove (A, B, C). (A): Since j = n ∨ k, we have that n ≤ j. (B): Observe that ϕPł (1, ~c ) = ϕA j ( pnj (1), pkj (~c )) ∈ A j . Together with ϕ(1, ~c ) ∈ / G, this implies that ϕPł (1, ~c ) ∈ A j r G. On the other hand, since Fn = An , we have that pnj (1) ∈ pnj [ Fn ] ⊆ Fj ⊆ A j ∩ G. Thus both A j ∩ G and A j r G are non-empty. We conclude that A j is nontrivial. (C): Suppose, by way of contradiction, that i ≤ j. In particular, this implies that m = j (indeed, i ≤ j = n ∨ k, thus i ∨ k ≤ n ∨ k, i.e. m ≤ j; on the other hand, since n < i then n ∨ k ≤ i ∨ j, i.e. j ≤ m). Therefore we have that ϕPł (1, ~c ) = ϕA j ( pnj (1), pkj (~c ))

(5.12)

Aj

(5.13)

Aj

= ϕ ( pij ( a), pkj (~c ))

(5.14)

= ϕAm ( pim ( a), pkm (~c ))

(5.15)

= ϕ ( pij ◦ pni (1), pkj (~c ))

Pł

= ϕ ( a, ~c ) ∈ G.

(5.16)

The non-obvious equalities above are justified as follows. (5.13) follows from the fact that i ≤ m = j. (5.14) is a consequence of a = pni (1). (5.15) follows from j = m and (5.16) from m = i ∨ k. This establishes the above equalities, yielding that ϕPł (1, ~c ) ∈ G. But this contradicts the fact that ϕ(1, ~c ) ∈ / G. Hence (A), (B) and (C) hold, establishing our claim. In particular, this implies that j ∈ I satisfies the condition in the statement. (2)⇒(1): By Lemma 5.1.6 we know that Pł ( X ) is a model of Ll . It only remains to be proved that it is Suszko reduced.

5.3. SUSZKO REDUCED MODELS

119

Let thus θ be the Suszko congruence of Pł ( X ). We claim that, in order to prove that θ = ∆, it will be enough to show that it does not identify distinct elements in fibres which are comparable with respect to the order ≤. Suppose indeed that this is the case. Consider two elements a, b ∈ S A = i∈ I Ai , with a 6= b. There exist i, j ∈ I such that a ∈ Ai and b ∈ A j . If i and j are comparable, then by assumption h a, bi ∈ / θ. Suppose instead that i and j are incomparable. Set k := i ∨ j. Clearly we have that i, j < k. In particular, we have that b · b = b ∈ A j and a · b ∈ Ak and, therefore, b · b 6= a · b. Since j and k are comparable, this implies that hb · b, a · bi ∈ / θ. In particular, this means that h a, bi ∈ / θ as well, and our claim is established. Now, take two different elements a, b ∈ A such that a ∈ Ai and b ∈ A j with i ≤ j. We have two cases: either i = j or i < j. First consider the case where i = j, that is a, b ∈ Ai . By assumption, we have that hAi , Fi i ∈ ModSu (L). Therefore we can suppose w.l.o.g. that there is a L-filter Gi on Ai such that Fi ⊆ Gi , some elements ~c ∈ Ai , and a τ-formula ϕ( x, ~z) such that ϕAi ( a, ~c ) ∈ Gi and ϕAi (b, ~c ) ∈ / Gi . Keeping the notation of Lemma 5.3.1, we let J = {l ∈ I | i ≤ l }. So, S G := i∈ I GiJ is a Ll -filter on Pł (Ai )i∈ I . Moreover, observe that ϕPł ( a, ~c) = ϕAi ( a, ~c) ∈ G ϕPł (b, ~c) = ϕAi (b, ~c) ∈ / G. We conclude that h a, bi ∈ / θ. If i < j, then either Ai is trivial or it is not. If Ai is nontrivial, then Fi 6= Ai as hAi , Fi i ∈ ModSu (L). Let now J = {l ∈ I | i < l }. By Lemma S 5.3.1 we know that G := i∈ I GiJ is a Ll -filter on Pł (Ai )i∈ I . Choose an element c ∈ Ai r Fi . We have that c ·Pł a = c ·Ai a = c ∈ Ai r Fi = Ai r GiJ and c ·Pł b ∈ A j = GjJ . Therefore, c · a ∈ / G and c · b ∈ G. Hence we conclude that h a, bi 6∈ θ, as desired. Finally, if Ai is trivial, either Fi = ∅ or Fi = Ai . First suppose that Fi = ∅. Rehearsing the argument in the previous paragraph (taking c := a) we obtain that h a, bi ∈ / θ. If, on the other hand, Fi = Ai , hAi , Fi i is a trivial matrix. Therefore we can apply our assumption, obtaining an element k ∈ I such that Ak is nontrivial, i < k and j  k. Then let S J = {l ∈ I | k ∨ j ≤ l }. By Lemma 5.3.1 we know that G := i∈ I GiJ is a Ll -filter on Pł (Ai )i∈ I . Since Ak is nontrivial and hAk , Fk i ∈ ModSu (L),

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there is c ∈ Ak r Fk . Since k < k ∨ j, we have that c ·Pł a = c ·Ak pik ( a) = c ∈ Ak r Fk = Ak r GkJ c ·Pł b ∈ A j∨k = GjJ∨k . Hence we conclude that c ·Pł a ∈ / G and c ·Pł b ∈ G. But this means that h a, bi ∈ / θ. By our claim, what we established so far suffices for the proof of our theorem.  Theorem 5.3.2 identifies the Suszko reduced models of Ll that can be expressed in terms of Płonka sums of Suszko reduced models of L. It is natural to wonder whether all the Suszko reduced models of Ll are of this kind. The following shows that this does not hold in general: Example 5.3.3. Consider the logic L of type h2, 2i determined by the following class of matrices: M := {hA, F i | A is a distributive lattice and F is an upset}. Let A1 be the 3-element chain with a < b < c and let F1 = {b, c}. Moreover, let A2 be the 4-element distributive lattice with universe {0, d, e, 1}, with 0 as bottom element and d, e as atoms. Let F2 = A2 r {0}. Clearly both hA1 , F1 i and hA2 , F2 i are models of L (as they belong to M). However, it is easy to see that hA1 , F1 i ∈ / ModSu (L). Now, let f : A1 → A2 be any of the two embeddings of A1 into A2 . Clearly these two matrices plus f uniquely determine a direct system X of matrices depicted in the following figure. We denote by hB, G i the Płonka sum Pł ( X ).

1 c b

d

e

a 0 Since hA1 , F1 i and hA2 , F2 i are models of L, by Lemma 5.1.6 hB, G i is a model of Ll . We now show that it is indeed Suszko reduced. Elements

5.3. SUSZKO REDUCED MODELS

121

belonging to the algebra A1 , as for example b and c (any other pair of elements in A1 is distinguished by the identity function), can be distinguished by means of the function ∧B , the filter G and the element e, as follows: b ∧ B e = d ∧ A2 e = 0 6 ∈ G c ∧B e = 1 ∧A2 e = e ∈ G. One can reason similarly (using G as filter) for pairs of elements belonging to A2 . We illustrate the only interesting case: d ∧ B b = d ∧ A2 d = d ∈ G e ∧B b = e ∧A2 d = 0 6∈ G. On the other hand, pairs of elements belonging to different algebras are distinguished by considering the filter H := F1 ∪ A2 on B (the fact that it is a filter is guaranteed by Lemma 5.3.1) , the function ∧B and the element a. Consider, for instance, the elements b and d: b ∧B a = a 6∈ H; d ∧B a = d ∧A2 0 = 0 ∈ H. This is enough to show that hB, G i is Suszko reduced. To conclude the example we need to disprove that hB, G i is a Płonka sum of any Suszko reduced models of L. Suppose that hB, G i is the Płonka sum of a direct system Y of Suszko reduced models

hB1 , G1 i, . . . , hBn , Gn i of L. First observe that n ≤ 2. Suppose the contrary towards a contradiction. Then n > 3. We choose three elements b1 ∈ B1 , b2 ∈ B2 and b3 ∈ B3 . Clearly b1 , b2 and b3 are different. Moreover, for every 1 ≤ i < j ≤ 3 we have that either bi ·B b j 6= bi or b j ·B bi 6= b j , where · indicates the partition function, i.e. x · y := x ∧ ( x ∨ y). It is easy to see that no such three elements exist in B, which is a contradiction. Hence n ≤ 2. We have cases. If n = 1, then hB1 , G1 i = hB, G i. In particular, this implies that hB, G i ∈ ModSu (L) and, therefore, B ∈ Alg(L). By [96, Lm. 5.78] this implies that B is a lattice, which is false. Thus, the only possible case is that n = 2. Now, by [96, Lm. 5.78] again we conclude that B1 and B2 are distributive lattices. Since the only way of partitioning B into two subalgebras that are distributive lattices is {A1 , A2 }, we conclude that w.l.o.g. B1 = A1 and B2 = A2 , i.e. hB, G i cannot be the Płonka sum of any Suszko reduced models of L. 

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5.4 Some well-behaved cases 5.4.1 Equivalential logics It is when L is a finitary equivalential logic that the class of matrices ModSu (Ll ) can be described at its most perspicuous, as we proceed to prove (see Theorem 5.4.3). Lemma 5.4.1. Let L be a finitary equivalential logic with an l-partition function, whose type τ does not contain constants. Then Mod∗ (Ll ) ⊆ Pł (Mod∗ (L)). Proof. Recall from Lemma 5.1.14 that, under the assumptions in our statement, also Ll has an l-partition function. Then consider hA, F i ∈ Mod∗ (Ll ) and let X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i be the direct system of matrices given in Theorem 5.1.15. Thus, Pł ( X ) = hA, F i. Moreover, by Lemma 5.1.16, we know that each fibre of X is a model of L. It only remains to prove that the fibres of X are Leibniz reduced. S We claim that i∈ I ΩAi Fi is a congruence of A. To show this, let Σ( x, y) be a set of congruence formulas for L, which has to exist as the latter is equivalential. Then consider an n-ary basic operation g and elements S a1 , . . . , an , b1 , . . . , bn ∈ A such that h a j , b j i ∈ i∈ I ΩAi Fi , for all 1 ≤ j ≤ n. This implies that there are indices m1 , . . . , mn ∈ I such that a j , b j ∈ Am j , for all j ≤ n, and moreover that h a j , b j i ∈ Ω of congruence formulas for L implies that Σ Pł ( a j , b j ) = Σ

Am j

Am j

Fm j . The fact that Σ is a set

( a j , b j ) ⊆ Fm j .

Set k := m1 ∨ · · · ∨ mn . We have that [

ΣAk ( pm j k ( a j ), pm j k (b j )) ⊆ Fk .

(5.17)

j≤n

Given that Σ is a set of congruence formulas for L, [

Σ( x j , y j ) ` Σ( g(~x ), g(~y)).

(5.18)

j≤n

Together with (5.17) and (5.18), the fact that hAk , Fk i is a model of L implies that

5.4. SOME WELL-BEHAVED CASES

123

ΣAk ( gPł ( a1 , . . . , an ), gPł (b1 , . . . , bn ))

= ΣAk ( gAk ( pm1 k ( a1 ), . . . , pmn k ( an )), gAk ( pm1 k (b1 ), . . . , pmn k (bn ))) ⊆ Fk . Hence

h gPł (~a), gPł (~b)i ∈ ΩAk Fk ⊆

[

ΩAi Fi .

i∈ I

This establishes the claim. Since Fi is a union of cosets for each ΩAi Fi , F is a union of cosets of the S S congruence i∈ I ΩAi Fi . In particular, this implies that i∈ I ΩAi Fi ⊆ ΩA F. Since ΩA = ∆A , we conclude that for all i, ΩAi Fi = ∆Ai . Hence we obtain that hAi , Fi i ∈ Mod∗ (L) for every i ∈ I and, therefore, that

hA, F i = Pł ( X ) ∈ Pł (Mod∗ (L)). We conclude that Mod∗ (Ll ) ⊆ Pł (Mod∗ (L)), as desired.



Corollary 5.4.2. Let L be a finitary equivalential logic with an l-partition function, whose type τ does not contain constants. Then ModSu (Ll ) ⊆ Pł (Mod∗ (L)) = Pł (ModSu (L)). Proof. By Lemma 4.0.5.(2), ModSu (L) = Mod∗ (L), since L is equivalential. Thus it will be enough to prove that ModSu (Ll ) ⊆ Pł (Mod∗ (L)). We have that ModSu (Ll ) = PSD Mod∗ (Ll ) ⊆ SPMod∗ (Ll )

(5.19) (5.20)

⊆ SPP (Mod∗ (L))

(5.21)

⊆ P (SPMod∗ (L)) ł

(5.22)

= P (Mod∗ (L)). ł

(5.23)

ł

These equalities or inclusions are justified as follows: (5.19) obtains by Lemma 4.0.5.(1); (5.21) is a consequence of Lemma 5.4.1, (5.22) follows from Lemma 5.1.2, and (5.23) from the fact that Mod∗ (L) is closed under S and P, by Lemma 4.0.5.(1) and (2). Hence we conclude that ModSu (Ll ) ⊆ Pł (Mod∗ (L)). 

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We are now ready to provide a full characterisation of the Suszko reduced models of the variable inclusion companion of a finitary equivalential logic with an l-partition function. Theorem 5.4.3. Let L be a finitary equivalential logic with an l-partition function, whose type τ does not contain constants, and let hA, F i be a τ-matrix. The following conditions are equivalent: 1. hA, F i ∈ ModSu (Ll ). 2. There exists an l-direct system of matrices X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i ⊆ Mod∗ (L) such that hA, F i ∼ = Pł ( X ) and for every n, i ∈ I such that hAn , Fn i is trivial and n < i, there exists j ∈ I s.t. n ≤ j, i  j and A j is nontrivial. Proof. This is a consequence of Theorem 5.3.2 and Corollary 5.4.2.



Example 5.4.4. Observe that all substructural logics that extend FL [101, 162] are finitary, equivalential, and have an l-partition function. The same holds for all local and global consequences of normal modal logics [24]. As a consequence, the above result provides a description of the Suszko reduced models of the variable inclusion companion of all substructural and modal logics (when the latter are understood as local and global consequences of normal modal logics [23, 51, 137]), formulated in the appropriate type. 

5.4.2 Logics with antitheorems The goal of this section is to show that if L is a logic with antitheorems, then we can further improve on the description of the Suszko reduced models of its variable inclusion companion, as given in Theorems 5.3.2 and 5.4.3. The next result discloses the logical meaning of antitheorems. Algebraic versions of it first appeared in [135] and [48] in the setting of varieties and quasivarieties of algebras respectively. Lemma 5.4.5. Let L be a logic of type τ. The following are equivalent: 1. L has an antitheorem Σ. 2. If hA, F i ∈ Mod(L) is nontrivial, then it has no trivial submatrix.

5.4. SOME WELL-BEHAVED CASES

125

Proof. (1)⇒(2): Suppose that L has an antitheorem Σ. As remarked above, we can assume w.l.o.g. that Σ is in the single variable x. Suppose, in view of a contradiction, that there is a nontrivial matrix hA, F i ∈ Mod(L) with a trivial submatrix hB, Bi. Since hA, F i is nontrivial, there exists an element a ∈ A r F. Consider any A-valuation v such that v( x ) = b and v(y) = a, where b is any element of B. Since Σ = Σ( x ) and hB, Bi is a submatrix of hA, F i, we have that v[Σ] ⊆ B ⊆ F. Together with the fact that Σ `L y, this implies that a = v(y) ∈ F, which is a contradiction. (2)⇒(1): Let Fm( x ) be the set of τ-formulas in variable x only. We show that Fm( x ) is an antitheorem for L. To this end, consider a substitution σ and a formula ψ. It is enough to show that σ [ Fm( x )] `L ψ. Let ϕ := σ( x ). Observe that σ[ Fm( x )] coincides with the universe of the subalgebra SgFmτ ( ϕ) of Fmτ generated by ϕ. For Γ ⊆ Fmτ , let CnL (Γ) = { ϕ ∈ Fmτ | Γ `L ϕ}. Consider the matrices

hA1 , F1 i :=hFmτ , CnL (σ[ Fm( x )])i hA2 , F2 i :=hSgFmτ ( ϕ), σ[ Fm( x )]i. Clearly, hA1 , F1 i is a model of L and hA2 , F2 i a trivial submatrix of hA1 , F1 i. By the assumption, we get that hA1 , F1 i is a trivial matrix, i.e. Fmτ = CnL (σ [ Fm( x )]). Hence we conclude that ψ ∈ Fmτ = CnL (σ [ Fm( x )]). Clearly this implies that σ[ Fm( x )] `L ψ, as desired.



We now obtain a refinement of Theorem 5.3.2 for logics possessing an antitheorem: Theorem 5.4.6. Let L be a logic of type τ with an l-partition function and an antitheorem. For every l-direct system X of matrices in ModSu (L), the following conditions are equivalent: 1. Pł ( X ) ∈ ModSu (Ll ). 2. X contains at most one trivial fibre. Proof. Throughout the proof we fix X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i. First we claim that if a fibre hAn , Fn i of X is trivial, then so is hAk , Fk i, for every k > n. To prove this, consider a trivial fibre hAn , Fn i of X and k > n. Observe that pnk [ An ] = pnk [ Fn ] ⊆ Fk .

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Then h pnk [ An ], pnk [ Fn ]i is a trivial submatrix of hAk , Fk i. Since L has an antitheorem, we can apply Lemma 5.4.5, obtaining that hAk , Fk i is trivial. This establishes the claim. (1)⇒(2): Suppose, in view of a contradiction, that Pł ( X ) ∈ ModSu (Ll ) and that X contains two distinct trivial fibres h1n , {1n }i and h1k , {1k }i (their algebra reducts are trivial, as the fibres of X belong to ModSu (L)). Set hA, F i := Pł ( X ). Observe that, for every formula ϕ( x, ~z ) in which x really occurs, and every tuple ~c ∈ A, we have that ϕA (1n , ~c ), ϕA (1k , ~c ) ∈ F. Indeed, the element ϕ(1n , ~c ) belongs to a fibre hAl , Fl i of X with n ≤ l. By the previous claim, we know that hAl , Fl i is trivial and, therefore, that ϕ(1n , ~c ) ∈ Fl ⊆ F, as desired. A similar argument shows that ϕ(1k , ~c ) ∈ F as well. Hence we have that for every unary polynomial ϕ( x, ~c) on A FgLAl ( F ∪ { ϕA (1n , ~c)}) = FgLAl ( F ∪ { ϕA (1k , ~c)}). ∼ By Lemma 4.0.3 this implies that h1n , 1k i ∈ ΩLA F. Since hA, F i ∈ ModSu (Ll ), this implies that 1n = 1k , which is a contradiction. (2)⇒(1): Suppose that X contains at most one trivial matrix. If X contains no trivial fibre, then by Theorem 5.3.2 Pł ( X ) ∈ ModSu (Ll ). Then consider the case where X contains exactly one trivial fibre. By the claim, this fibre is the top element of h I, ≤i. Again, with an application of Theorem 5.3.2, we conclude that Pł ( X ) ∈ ModSu (Ll ).  The assumption that L has an antitheorem in the above theorem is essential, as shown in the following Example 5.4.7. The statement of Theorem 5.4.6 is in general false for logics without an antitheorem, as witnessed by the following example of a logic L where we have a Suszko reduced model of Ll , which is the Płonka sum of direct systems of Suszko reduced models of L containing two trivial matrices. Let L be the {∧, ∨}-fragment of CL. Moreover, let 1 be the trivial lattice and D2 = h{⊥, >}, ∧, ∨i be the 2-element distributive lattice (with ⊥< >). Consider the l-direct system X of matrices formed by 6 copies of the matrix hD2 , {>}i and two trivial matrices h1, {1}i sketched in the following figure (lines represent lattice order in the Płonka fibres, dotted lines the homomorphisms, and circles, filters in any fibre).

5.4. SOME WELL-BEHAVED CASES

127

•

•

•

•

• •

•

•

•

•

•

•

1

1

Clearly each matrix in X, which contains two trivial matrices, is a Suszko reduced model of L. Moreover, by applying Theorem 5.3.2, one immediately checks that Pł ( X ) ∈ ModSu (Ll ). Theorem 5.4.6 dovetails with Theorem 5.4.3 into a description of the Suszko reduced models of the variable inclusion companion of a finitary equivalential logic with an l-partition function and antitheorems. Theorem 5.4.8. Let L be an equivalential and finitary logic whose type τ does not contain constants, having an l-partition function and antitheorems. Let moreover hA, F i be a τ-matrix. The following conditions are equivalent: 1. hA, F i ∈ ModSu (Ll ). 2. There exists an l-direct system of matrices X ⊆ Mod∗ (L) with at most one trivial fibre such that hA, F i ∼ = Pł ( X ). Proof. This is a combination of Theorems 5.4.6 and 5.4.3.



The above result provides a full description of the Suszko reduced models of the variable inclusion companions of most well-known logics (under the mentioned restriction), including all logics listed in Example 5.4.4.

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5.5 Classification in the Leibniz hierarchy We conclude this chapter by investigating how logics of left variable inclusion can be placed in the Leibniz hierarchy. Theorem 5.5.1. Let L be a logic whose type contains no constants. 1. If L is consistent, then Ll is not protoalgebraic. 2. If L is finitary, algebraisable and has an l-partition function, then Ll is truth-equational. Proof. (1): We reason by contraposition. Suppose that Ll is protoalgebraic. Then there is a set of formulas ∆( x, y) such that ∅ `Ll ∆( x, x ) and x, ∆( x, y) `Ll y. Thus, the definition of Ll implies that there is a subset Σ(y) ⊆ ∆( x, y) such that Σ(y) `L y. Since ∅ `Ll ∆( x, x ), we have that ∅ `Ll Σ(y). From Σ(y) `L y and ∅ `Ll Σ(y) it follows that ∅ `Ll y. By the definition of Ll we conclude that ∅ `L y and, therefore, that L is inconsistent. (2): Suppose that L is finitary, algebraisable and has an l-partition function. In particular, L is truth-equational with set of defining equations E( x ). We will show that E( x ) is a set of defining equations for Ll as well. Let hA, F i ∈ Mod∗ (Ll ). Since L is finitary, equivalential and with an lpartition function, we can apply Lemma 5.4.1 obtaining that there exists an l-direct system of Leibniz reduced models of L X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i such that hA, F i = Pł ( X ). Consider an element a ∈ A. There is i ∈ I such that a ∈ Ai . We have that A  E( a) ⇐⇒ Ai  E( a) ⇐⇒ a ∈ Fi ⇐⇒ a ∈ F.

(5.24)

The above equivalences are justified as follows. The first one follows from the fact that A = Pł (Ai )i∈ I . The second one follows from the fact that hAi , Fi i ∈ Mod∗ (L) and that E( x ) is a set of defining equations for L. The last one follows from the assumption that hA, F i = Pł ( X ). By (5.24) we obtain that for every a ∈ A, A  E( a) ⇐⇒ a ∈ F. Hence we conclude that E( x ) is a set of defining equations for Ll and, therefore, Ll is truth-equational. 

5.5. CLASSIFICATION IN THE LEIBNIZ HIERARCHY

129

In [33, Theorem 48] it is proved that the variety IBSL of involutive bisemilattices is not the equivalent algebraic semantics of any algebraisable logic. Recall from Chapter 1 that the type τ1 of IBSL contains constants. If τ, on the other hand, is a type with no constants, a similar result can be shown to hold in general. Theorem 5.5.2. Let K be a class of similar algebras whose type τ contains no constants, including two trivial algebras and closed under Płonka sums. There is no protoalgebraic logic L such that Alg(L) = K. Proof. Suppose, in view of a contradiction, that there are a class K of similar algebras containing two trivial algebras and closed under Płonka sums, and a protoalgebraic logic L such that Alg(L) = K. Let 1 a , 1b ∈ K be distinct trivial algebras and consider the unique semilattice direct system A over the 2-element chain on { a, b}, with a < b, such that A a = 1 a and Ab = 1b . Clearly, A = Pł (A ) ∈ K. Therefore there is F ⊆ A such that hA, F i ∈ ModSu (L). As Alg(L) contains a nontrivial algebra, it is not difficult to see that x 0L y. Since L is protoalgebraic, there is a set of formulas ∆( x, y) such that ∅ `L ∆( x, x ) and x, ∆( x, y) `L y. Taking into account that x 0L y and x, ∆( x, y) `L y, we conclude that ∆( x, y) 6= ∅. So there exists ϕ( x, y) ∈ ∆( x, y). Since ∅ `L ∆( x, x ), we conclude that ∅ `L ϕ( x, x ). Now, observe that the variable x actually occurs in ϕ( x, x ), since the type τ is without constants. Hence we obtain that ϕA (1a , 1a ) = 1a and ϕA (1b , 1b ) = 1b . Together with the fact that ∅ `L ϕ( x, x ), this implies that A = {1a , 1b } is the smallest L-filter on A. In particular, this implies that A is the unique L-filter on A. Since F is a L-filter on A, we conclude that A = F. Hence hA, Ai is a Suszko reduced model of L. This implies that A is trivial, which is false.  The results in these last sections exemplify the potential of matrix semantics for the investigation of logics of left variable inclusion. A better granularity level can be attained by resorting e.g. to nondeterministic matrices [8]. Caleiro, Marcelino, and Filipe [46] put to good use this method in investigating a generalisation of variable inclusion logics where the validity of entailments require more stringent criteria, e.g. the inclusion of certain subformulas.

Chapter 6 Logics of right variable inclusion In Chapter 5 we associated to an arbitrary logic L its left variable inclusion companion Ll . Here, we focus on another sublogic of L, which we denote by Lr and whose syntactic characterisation is:  either: Γ `L ϕ and Var ( ϕ) ⊆ Var (Γ) r Γ `L ϕ ⇐⇒ or: Γ is an antitheorem of L. We will refer to the logic Lr as the right variable inclusion companion of the logic L. It is worth remarking that logics of right variable inclusion have been less thoroughly investigated than their left variable inclusion counterparts. Indeed, as noticed in Chapter 5, [33] contains a detailed investigation of PWK from the standpoint of Abstract Algebraic Logic, while no comparable enquiry has been carried out either for its paracomplete cousin B3 , or for any other logic of right variable inclusion. In Section 6.1, logics of right variable inclusion are formally introduced. Moreover, by providing the appropriate notion of Płonka sum of logical matrices, we obtain soundness and completeness for an arbitrary finitary logic of the form Lr with respect to Płonka sums of matrix models of L. Then we focus on a specific class of logics, namely, those possessing a binary formula, called an r-partition function (Definition 6.1.15). In Section 6.2, we provide a method for obtaining a Hilbert-style axiomatisation for a finitary logic Lr (Theorem 6.2.3) out of a given axiomatisation for L. It is worth to mention that almost all examples of logics of right variable inclusion, among which B3 , belong to this class and that the rules of the calculi at issue are devoid of syntactic restrictions. In Section 6.3 we investigate the algebraic counterpart of a right variable inclusion logic, while Sections 6.4 and 6.5 revolve on the structure of the Leibniz and Suszko reduced models of Lr , respectively. It turns out

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_6

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that whether a model is (Leibniz or Suszko) reduced hinges on conditions pertaining to the semilattice structure of the system of matrix models involved in the construction of the Płonka sums. In this chapter, all logics will be assumed to be finitary.

6.1 Płonka sums of matrices and r-direct systems 6.1.1 General results Logics of right variable inclusion, sometimes called containment logics [38],1 are defined as follows: Definition 6.1.1. Let L be a logic of type τ. The right variable inclusion companion of L is the relation Lr =`Lr ⊆ P ( Fmτ ) × Fmτ defined for every Γ ∪ { ϕ} ⊆ Fmτ as  either: Γ `L ϕ and Var ( ϕ) ⊆ Var (Γ) Γ `Lr ϕ ⇐⇒ or: Γ is an antitheorem of L. Example 6.1.2. If L is classical propositional logic CL formulated in the type τ2 without constants, then CLr is the logic B3 , already discussed in Chapter 1. The considerations about PWK of Example 5.1.4 also apply to B3 , which can be defined either syntactically, by imposing to CL the variable inclusion constraint as in Definition 6.1.1, or semantically, as the matrix consequence relation `hWK,{1}i (see Section 1.2.2) – namely, by trading in the two designated values of PWK for the single designated value 1. This is the content of the Urquhart Theorem (Theorem 1.3.1) discussed in Chapter 1 (it is also a consequence of Theorem 6.1.8 below). As already noticed in Example 5.1.4, it is not difficult to check that the algebra WK = h{0, 1, n}, ∧, ∨, ¬i is the Płonka sum of the 2-element Boolean algebra in the language τ2 and the trivial Boolean algebra (the index set is the 2-element semilattice). Example 6.1.3. Other logics of right variable inclusion are listed in the following table. More precisely, for each logic among CL, BD, K3 , LP, PWK and B3 , we specify both its left and its right variable inclusion companion, citing the names under which they are known in the literature as well 1 In the present volume we prefer to avoid this alternative terminology, since it is not wholly consistent with other usages of this term in the literature. Ferguson, for example, labels as containment logics systems that lack antitheorems [86].

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as the papers where they were introduced (sometimes independently by more than one author). Many of these logics are discussed in Section 1.2.5. We observe parenthetically that SFDE was not originally introduced as the right variable inclusion companion of LP, but via its characteristic matrix; the equivalence of these descriptions was established in [84]. ∗ Similarly, it is proved in [85] that SFDE (also originally introduced via its 5-element characteristic matrix) is the right variable inclusion companion of BD. Base logic CL BD K3 LP PWK B3

Left companion PWK ∗ dSFDE [216, 58] dSFDE [216, 58] Snfl [216, 18] Sw FDE or HYB2 [216, 69]

Right companion B3 ∗ SFDE [68, 186] Setl [216, 18] SFDE [73, 157] dSw FDE or HYB1 [216, 69, 221]

The gaps in the above table are due to the fact that, for any logic L, Lrr = Lr and Lll = Ll . We also have that L = Ll ∨ Lr [181]. Just like l-direct systems of matrices are the appropriate semantic tool for logics of left variable inclusion (Definition 5.1.1), r-direct systems of matrices work well for logics of right variable inclusion. The differences between these two notions mainly concern the interplay between the homomorphisms of the system and the filters in the matrices. Definition 6.1.4. An r-direct system of matrices is an ordered pair X = hA, { Fi }i∈ I i such that:

 1. A = {Ai }i∈ I , I, { pij | i ≤I j} is a semilattice direct system of algebras of type τ; 2. for every i ∈ I, Fi ⊆ Ai ; 3. I + := {i ∈ I | Fi 6= ∅} is the universe of a subsemilattice of I; 4. for every i, j ∈ I such that i ≤ j, if Fj 6= ∅, then pij−1 [ Fj ] = Fi . Like for l-direct systems, also an r-direct system can be viewed as the result of replacing, as a fibre of a semilattice direct system, each algebra Ai by the matrix hAi , Fi i. Instead of the compact notation hA, { Fi }i∈ I i, we will generally use the expanded form

h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i. The matrices in {hAi , Fi i}i∈ I are called the fibres of X.

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Given an r-direct system of matrices X, we define a new matrix as

Pł ( X ) := hPł (Ai )i∈ I ,

[

Fi i.

i∈ I

We refer to the matrix Pł ( X ) as the Płonka sum of the r-direct system of matrices X. Given a class M of matrices, in this chapter Pł (M) will denote the class of all Płonka sums of r-direct systems of matrices in M. Let h : Fmτ → Pł (Ai )i∈ I be a homomorphism from the formula algebra into a generic Płonka sum of algebras. Then, for any formula ϕ ∈ Fmτ , we set ih ( ϕ) :=

_

{i ∈ I | h( x ) ∈ Ai , x ∈ Var ( ϕ)}.

In other words, ih ( ϕ) denotes the index of the fibre where the formula ϕ is interpreted by the Pł (Ai )i∈ I -valuation h. Also, for any finite set of τ-formulas Γ, we define ih (Γ) :=

_

{ i h ( ϕ ) | ϕ ∈ Γ }.

Observe that, for every Pł ( X )-valuation h, and every finite Γ ∪ { ϕ} ⊆ Fmτ , it is immediate to check that Var ( ϕ) ⊆ Var (Γ) implies ih ( ϕ) ≤ ih (Γ). A set of models of a logic L is said to be nontrivial, if it does not contain trivial matrices. We indicate by Mod+ (L) the set of nontrivial models of L. Lemma 6.1.5. Let X be an r-direct system of nontrivial models of a logic L of type τ. Then Pł ( X ) is a model of Lr . Proof. Let X be an r-direct system of nontrivial models of L. Assume Γ `Lr ϕ. Since Lr is finitary, there exists a finite subset ∆ ⊆ Γ such that ∆ `Lr ϕ. We distinguish the following cases: (a) ∆ = ∆( x ) is an antitheorem of L; (b) ∆ `L ϕ with Var ( ϕ) ⊆ Var (∆). Suppose (a) is the case. Then ∆( x ) `L ψ, for any ψ ∈ Fmτ . Let hAi , Fi i ∈ X. Preliminarily, observe that, for any Ai -valuation v, we have v[∆( x )] 6⊆ Fi (as, otherwise we would have v(ψ) ∈ Fi , for any formula ψ, implying that Fi = Ai , in contradiction with the fact that hAi , Fi i is nontrivial). From this fact, it easily follows that, for any Pł (Ai )i∈ I -valuation h, h[∆( x )] 6⊆ F. Therefore ∆( x ) `Pł (X ) ϕ.

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Suppose (b) is the case, i.e. ∆ `L ϕ with Var ( ϕ) ⊆ Var (∆). Let h be a S Pł (Ai )i∈ I -valuation such that h[∆] ⊆ i∈ I Fi . Since ∆ is a finite set, then we can fix j := ih (∆) and, for any formula δ ∈ ∆, we have h(δ) ∈ Fih (δ) . This implies that each ih (δ) ∈ I + and, as I + forms a subsemilattice of I, we have that j ∈ I + . Now, define an A j -valuation g as g( x ) := pih ( x) j ◦ h( x ), for every x ∈ Var (∆). For any δ ∈ ∆, we have g(δ) = pih (δ) j ◦ h(δ), hence g[∆] ⊆ Fj . From the fact that ∆ `L ϕ and hA j , Fj i ∈ Mod(L), it follows that g( ϕ) ∈ Fj . Setting k := ih ( ϕ), we have k ≤ j and this, together 1 with the observation that Fj 6= ∅, implies p− kj [ Fj ] = Fk by Definition 6.1.4. Moreover, we claim that Fk 6= ∅. Suppose, by contradiction, that 1 Fk = p− kj [ Fj ] = ∅. Then, there exists no a ∈ Ak such that pkj ( a ) ∈ Fj . On the other hand, since Var ( ϕ) ⊆ Var (∆), then g( ϕ) = pkj ◦ h( ϕ) ∈ Fj , a 1 contradiction. From the fact that g( ϕ) ∈ Fj together with p− kj [ Fj ] = Fk , we S conclude h( ϕ) ∈ Fk . This proves that h( ϕ) ∈ Fk ⊆ i∈ I Fi .  Remark 6.1.6. Observe that in Lemma 6.1.5, it is crucial to assume that the models of the logic L are nontrivial, as witnessed by the following example. Let L be a theoremless logic possessing an antitheorem Σ( x ). Let moreover X be the r-direct system of models of L over the 2-element chain, consisting of the two matrices hA, Ai and h1, ∅i with the unique homomorphism p : A → 1 (plus the identity automorphisms). Then Σ( x ) `L y, for an arbitrary variable y, and therefore Σ( x ) `Lr y. However, Pł ( X ) is not a model of the latter inference – consider, for instance, a Pł ( X )-valuation v such that v( x ) = a ∈ A and v(y) = 1. Observe that, if the logic L does not possess an antitheorem, then the following holds: Corollary 6.1.7. Let X be an r-direct system of models of a logic L possessing no antitheorems. Then Pł ( X ) is a model of Lr . Given a logic L which is complete with respect to a class M of matrices, we set M∅ := M ∪ hC, ∅i, for any arbitrary C ∈ Alg(L). Theorem 6.1.8. Let L be a logic of type τ which is complete w.r.t. a class of nontrivial matrices M. Then Lr is complete w.r.t. Pł (M∅ ). Proof. We aim at showing that Lr = `Pł (M∅ ) . (Lr ≤ `Pł (M∅ ) ). Firstly, observe that, using the same argument as in

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Lemma 6.1.5, if Σ( x ) is an antitheorem of L, then Pł (M∅ ) satisfies the rule Σ( x ) `Lr ϕ, for any ϕ ∈ Fmτ . Moreover, if the matrix hC, ∅i is a model of L, then the claim follows from Lemma 6.1.5. We are left with treating the case where hC, ∅i is not a model of L. S Consider a Płonka sum hA, i∈ I Fi i of matrices in M∅ and suppose that Γ `Lr ϕ, with Var ( ϕ) ⊆ Var (Γ). W.l.o.g. we can assume Γ to be finite. S Let h be an A-valuation such that h[Γ] ⊆ i∈ I Fi . Suppose, in view of S a contradiction, that h( ϕ) 6∈ i∈ I Fi . Set ih ( ϕ) = j and ih (Γ) = k; since Var ( ϕ) ⊆ Var (Γ) then j ≤ k. We define an Ak -valuation v as follows: v( x ) := plk ◦ h( x ), where l = ih ( x ). Clearly, v[Γ] = pkk ◦ h[Γ] = h[Γ] ⊆ Fk and v( ϕ) = p jk ◦ h( ϕ) ∈ Ak r Fk , since h( ϕ) ∈ A j r Fj and Fj = f jk−1 [ Fk ] (as we know that Fk 6= ∅). Therefore, we have Γ 6`L ϕ, which is a contradiction. (`Pł (M∅ ) ≤ `Lr ). By contraposition, we prove that Γ 0Lr ϕ implies Γ 0Pł (M∅ ) ϕ. To this end, assume Γ 0Lr ϕ. Firstly, consider the case where Var ( ϕ) ⊆ Var (Γ). It follows that Γ 0L ϕ. Since M is a class of matrices complete for L, then there exists a matrix hAi , Fi i ∈ M and an Ai -valuation h such that h[Γ] ⊆ Fi and h( ϕ) ∈ / Fi . Upon considering the r-direct system S X = hhAi , Fi i, {i }, {idAi }i, h is a Pł ( X )-valuation such that h[Γ] ⊆ i∈ I Fi S and h( ϕ) ∈ / i∈ I Fi , proving that Γ 0Pł (M∅ ) ϕ. The only other case to consider is Var ( ϕ) * Var (Γ). Preliminarily, observe that the assumption Γ 0Lr ϕ implies that Γ is not an antitheorem of L. Therefore, since M is a class of models complete with respect to L, there exist a matrix hB, G i ∈ M and a B-valuation v such that v[Γ] ⊆ G and v( ϕ) ∈ / G. Consider the r-direct system formed by the matrices hB, G i and hC, ∅i, where C ∈ Alg(L) (observe that the choice C = 1 is always appropriate), indexed over the 2-element chain with universe {i, j} with pij any homomorphism from B to C (plus the identity automorphisms pii and p jj ). Denote by B ⊕ C∅ the Płonka sum over the r-direct system just described. For an arbitrary a ∈ C, we define the B ⊕ C∅ -valuation g as follows:  v( x ) if x ∈ Var (Γ), g( x ) := a otherwise. Clearly, g[Γ] = v[Γ] ⊆ G. On the other hand, since Var ( ϕ) * Var (Γ), there exists y ∈ Var ( ϕ) such that y ∈ / Var (Γ). Therefore g(y) = a and, by the construction of B ⊕ C∅ , we have g( ϕ) ∈ C and C ∩ G = ∅. This shows that Γ 0Pł (M∅ ) ϕ, as desired. 

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Observe that Theorem 5.1.9 can be adapted to the case of Lr . Indeed, the proof of Theorem 6.1.8 shows that, given a complete class of matrices M for L, the class {hA∗ , F i | hA, F i ∈ M} is a complete matrix semantics for Lr . Theorem 6.1.8 provides a complete class of matrices for an arbitrary logic of right variable inclusion. This class is obtained performing Płonka sums over an r-direct systems of models of L together with the matrices hC, ∅i for any C ∈ Alg(L). Obviously, it is not generally the case that the matrix hC, ∅i is a model of a logic L. For this reason, it is not always true that Płonka sums over an r-direct systems of models of L provide a complete matrix semantics for Lr . Thus, the right variable inclusion companion of a logic is a logic of “Płonka sums” (of matrices) in a weaker sense than it applies to its left variable inclusion companion. Nonetheless, if h1, ∅i ∈ Mod(L), the correspondence between Lr and Płonka sums is fully recovered. This is actually the case of every theoremless logic, such as Strong Kleene Logic or the conjunction-disjunction fragment of classical logic [100]. On the other hand, a logic which is not complete w.r.t. a complete class of models of L is Bochvar logic B3 . It is indeed easy to check that all the matrices in the class Pł (Mod∗ (CL)) are models of the inference x ` x ∨ y, while we have that x 0B3 x ∨ y. Corollary 6.1.9. A logic Lr of type τ is complete w.r.t. any of the following classes of matrices:

∗ (L) ∪ hC, ∅i), P (ModSu (L) ∪ hC, ∅i), Pł (Mod+ (L) ∪ hC, ∅i), Pł (Mod+ ł + for C ∈ Alg(L). Example 6.1.10. By Corollary 6.1.9, Bochvar logic B3 is complete with respect to the matrix whose algebra reduct is B2 ⊕ B2 , the unique Płonka sum of two copies of the 2-element Boolean algebra over the 2-element chain of indices, and whose unique designated value is the top element of its lower fibre. This fact is the basis of the “non-infectious” semantics for B3 proposed by Song et al. in [214]. Observe that the above-mentioned matrix is not a Leibniz reduced model of B3 , and that its quotient modulo the Leibniz congruence of its filter is precisely hWK, {1}i. Observing that if h1, ∅i ∈ Mod(L) then h1, ∅i ∈ Mod∗ (L), the following holds: Corollary 6.1.11. Let L be a logic of type τ such that h1, ∅i ∈ Mod(L). Then Lr is complete w.r.t. any of the following classes of matrices:

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∗ (L)), P (ModSu (L)). Pł (Mod+ (L)), Pł (Mod+ ł + In case L does not possess antitheorems, then the above corollaries can be restated as follows: Corollary 6.1.12. Let L be a logic of type τ without antitheorems. Then Lr is complete w.r.t. any of the following classes of matrices: P (Mod(L) ∪ hC, ∅i), P (Mod∗ (L) ∪ hC, ∅i), P (ModSu (L) ∪ hC, ∅i), ł

ł

ł

for any C ∈ Alg(L). Corollary 6.1.13. Let L be a logic of type τ without antitheorems such that h1, ∅i ∈ Mod(L). Then Lr is complete w.r.t. any of the following classes of matrices: P (Mod(L)), P (Mod∗ (L)), P (ModSu (L)). ł

ł

ł

We close this subsection with a parenthetical observation, which collects a couple of rather disappointing facts as regards the position of logics of right variable inclusion within the Leibniz hierarchy, and makes a contrast with the analogous result for logics of left variable inclusion (Theorem 5.5.1). Theorem 6.1.14. Let L be a logic of type τ. If τ does not contain constants, then 1. if L is consistent, then Lr is not protoalgebraic; 2. Lr is not truth-equational. Proof. Since τ does not contain constants, Lr is theoremless. Hence, by Proposition 4.0.5.(3), it can be neither protoalgebraic nor truth-equational. 

6.1.2 Right partition functions Definition 6.1.15. Let τ be a similarity type. An essentially binary τformula x ∗ y is a right partition function (for short r-partition function) for the logic L of type τ if (i) x, y `L x ∗ y, (ii) x ∗ y `L x and (iii) ∗A is a partition function for every algebra in Alg(L). In this case, we also say that L has the r-partition function x ∗ y. Remark 6.1.16. By Lemma 4.0.7, the above definition can be rephrased in purely logical terms, by requiring that x, y `L x ∗ y, x ∗ y `L x and that ϕ(ε, ~z ) a `L ϕ(δ, ~z ) for every τ-formula ϕ(v, ~z ), for every identity of the form ε ≈ δ in (PF1)-(PF5) of Definition 2.2.1.



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Notice that the above definition is essentially different from the definition of an l-partition function (Definition 5.1.11). However, in most cases (for instance, all substructural logics, classical and modal logics) the very same formula plays both the role of an r-partition function and of an l-partition function. Example 6.1.17. The formula x ∗ y := x ∧ ( x ∨ y) is an r-partition function for every logic L such that Alg(L) has a lattice reduct. Such examples include all modal and substructural logics [101]. On the other hand, the formula x ∗ y := (y → y) → x is an r-partition function for all the logics L such that every algebra in Alg(L) is 1-subtractive (Example 2.2.7). Remark 6.1.18. It is easily checked that x ∗ y is an r-partition function for L if and only if it is an r-partition function for Lr (cf. Lemma 5.1.14). The next result is an analogue of Theorem 5.1.15 for r-partition functions. Theorem 6.1.19. Let L be a logic of type τ with an r-partition function ∗, and let hA, F i be a model of L such that A ∈ Alg(L). Conditions 1., 3. and 4. in Theorem 2.2.8 hold for A. Moreover, if τ does not contain constants, Condition 2. holds and, setting Fi := F ∩ Ai for every i ∈ I, the triple X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i is an r-direct system of matrices such that Pł ( X ) = hA, F i. Proof. Our claims concerning Conditions (1-4) of Theorem 2.2.8 hold true, by simply observing that ∗ is a partition function for A. For the remaining part, it will be enough to show: (a) for every i, j ∈ I such that i ≤ j, if Fj 6= ∅ then pij−1 [ Fj ] = Fi ; (b) I + is a subsemilattice of I. In order to prove (a), consider i, j ∈ I such that i ≤ j and let Fj be non-empty. Assume, in view of a contradiction, that pij−1 [ Fj ] 6= Fi . This implies that Fi * pij−1 [ Fj ] or that pij−1 [ Fj ] * Fi . In the former case, let a ∈ Fi be such that pij ( a) = c ∈ A j r Fj . As Fj 6= ∅, there exists an element b ∈ Fj . Since ∗ is an r-partition function for L, then x, y `L x ∗ y holds. However, we have that a, b ∈ F while a ∗A b = pij ( a) ∗A j b = c ∗A j b = c ∈ / F, a contradiction. In the latter case, let a ∈ Ai r Fi be such that pij ( a) ∈ Fj . Fix pij ( a) = c. Again, as ∗ is an r-partition function for L

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it holds x ∗ y `L x. This, however, is in contradiction with the fact that a ∗A c = pij ( a) ∗A j c = c ∗A j c = c ∈ F while a ∈ / F. This proves (a). In order to prove (b), consider i, j ∈ I + and let k = i ∨ j, with i, j, k ∈ I. As ∗ is an r-partition function for L, x, y `L x ∗ y. Since i, j ∈ I + , then Fi and Fj are non-empty, therefore there exist two elements a ∈ Fi , b ∈ Fj . We have a ∗A b = pik ( a) ∗Ak p jk (b) ∈ Ak . This, together with the fact that hA, F i ∈ Mod(L) implies a ∗ b ∈ Fk , i.e. Fk 6= ∅. So k ∈ I + and this establishes (b).  Given a logic L with an r-partition function ∗ and a model hA, F i of L such that A ∈ Alg(L), we call (Płonka) fibres of hA, F i the matrices {hAi , Fi i}i∈ I given by the decomposition in Theorem 6.1.19. From now on, when considering a model hA, F i of a logic L with an r-partition function, we will assume that hA, F i = Pł ( X ), for a given r-direct system X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i, often without explicitly mentioning the r-direct system X. Lemma 6.1.20. Let Lr be a logic with an r-partition function ∗, and let the type τ of Lr contain no constants. Let moreover hA, F i ∈ Mod(Lr ), with A ∈ Alg(Lr ). The fibres hAi , Fi i, such that i ∈ I + , are models of L. Proof. Firstly, observe that each algebra Ai is a subalgebra of A, because τ does not contain constants. Let Γ `L ϕ and suppose, by contradiction, that there exists a matrix hA j , Fj i, with j ∈ I + , and an A j -valuation h such that h[Γ] ⊆ Fj and h( ϕ) ∈ / Fj . Observe that Var ( ϕ) * Var (Γ) and, moreover, if L has an antitheorem Σ( x ), then Σ( x ) * Γ, for otherwise Γ `Lr ϕ, which is in contradiction with our assumption that hA, F i ∈ Mod(Lr ). Denote by X the non-empty set of variables occurring in ϕ but not in Γ and, for γ ∈ Γ, let Xγ := {γ ∗ x | x ∈ X } and Γ− γ := Γ r { γ }. r r Since ∗ is an r-partition function for L , we have γ ∗ x `L γ. Therefore γ ∗ x `L γ and Xγ `L γ, which implies Xγ , Γ− γ `L ϕ, for any γ ∈ Γ. As Var ( ϕ) ⊆ Var ( Xγ ) ∪ Var (Γ− ) , we have that X , Γ− γ γ γ `Lr ϕ. Since h(γ), h( ϕ) ∈ A j and x ∈ Var ( ϕ), for every x ∈ X, we have that h(γ ∗ x ) = h(γ), whence h[ Xγ ] = h(γ). Now, for an arbitrary a ∈ A, we define an A-valuation g as follows:  g( x ) :=

h( x ), if x ∈ Var (Γ) ∪ Var ( ϕ); a, otherwise.

− We have g[ Xγ ] = h[ Xγ ] = h(γ) ∈ Fj , g[Γ− γ ] = h [ Γγ ] ∈ Fj and g ( ϕ ) = h( ϕ) 6∈ Fj , a contradiction. 

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6.2 Hilbert-style axiomatisations In this section we show how to provide a sound and complete Hilbertstyle calculus for a logic of right variable inclusion possessing an r-partition function. Interestingly enough, the postulates in the calculi we present are free from linguistic (variable-inclusion) constraints. Throughout this section, we implicitly assume that the logic L possesses an antitheorem. If this is not so, our strategy can be easily adapted (see Remark 6.2.4). Definition 6.2.1. Let H be a Hilbert-style calculus whose derivability relation `H is a logic with an r-partition function ∗ and an antitheorem Σ( x ). Hr is defined as the Hilbert-style calculus given by the following rules: x∗ ϕBϕ x, y B x ∗ y x∗yBx {γ1 , . . . , γn } r {γi }, γi ∗ ϕ B ϕ Σ( x ) B ϕ χ(δ, ~z ) C B χ(ε, ~z )

(P0) (P1) (P2) (P3) (P4) (P5)

for every (i) ∅ B ϕ rule in H; (ii) γ1 , . . . , γn B ϕ rule in H; (iii) δ ≈ ε equation in the definition of partition function, and formula χ(v, ~z ). Lemma 6.2.2. Let L =`Hr be a logic whose type τ contains no constants. Assume moreover that L has an r-partition function ∗ and an antitheorem Σ( x ), and let hA, F i ∈ ModSu (L). Then: 1. hA, F i ∼ = Pł ( X ), where X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i is an r-direct system of matrices; 2. if X contains a trivial matrix, then A = 1. Proof. (1) Since hA, F i ∈ ModSu (L), A ∈ Alg(L). Moreover, observe that ∗ is an r-partition function for L (thanks to conditions (P1), (P2), (P5)). These facts, together with Theorem 6.1.19, imply that hA, F i ∼ = Pł ( X ),

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where X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i is an r-direct system of matrices. (2) Suppose that, for some j ∈ I, hA j , Fj i is a trivial fibre of hA, F i, i.e. Fj = A j . Since Σ( x ) is an antitheorem (for L) and (P4) is a rule of Hr , then, for every i ∈ I, we have Ai = Fi , i.e. each fibre is trivial. Indeed, if there exists a nontrivial fibre hAk , Fk i and an element c ∈ Ak r Fk , then the A-valuation h, defined by h( x ) = a, h(y) = c (for an arbitrary a ∈ A j ) is such that h[Σ( x )] ⊆ F while h(y) ∈ / F, against the fact that Σ( x ) `Hr y. ∼ Moreover, the fact that each fibre is trivial, together with ΩA F = ∆A , immediately implies A = 1.  Theorem 6.2.3. Let H be a Hilbert-style calculus whose type τ does not contain constants. If L =`H is a logic with an r-partition function ∗ and an antitheorem Σ( x ), then Hr is a complete Hilbert-style calculus for Lr . Proof. We show that `Hr =`Lr . (`Hr ≤`Lr ). In order to verify the desired inequality, it is enough to prove that every rule of Hr holds in Lr . This is immediate for (P0), (P1), (P2), (P4) and (P5), as, by Remark 6.1.18, ∗ is an r-partition function for Lr . W.l.o.g., let i = 1 in (P3). By condition (ii) in Definition 6.1.15, we have that γ1 ∗ ϕ `L γ1 , hence (by monotonicity) γ1 ∗ ϕ, γ2 , . . . , γn `L γ1 . Moreover, as H is complete for L and γ1 , . . . , γn B ϕ is a rule in H, we have that γ1 , . . . , γn `L ϕ and therefore, by transitivity, we obtain γ1 ∗ ϕ, γ2 , . . . , γn `L ϕ. Since Var ( ϕ) ⊆ Var (γ1 ∗ ϕ, γ2 , . . . , γn ) we conclude γ1 ∗ ϕ, γ2 , . . . , γn `Lr ϕ, and this proves that (P3) holds in Lr . (`Hr >`Lr ). We have to show that ModSu (`Hr ) ⊆ Mod(Lr ). So let hA, F i ∈ ModSu (`Hr ). By Lemma 6.2.2.(1), we know that hA, F i ∼ = Pł ( X ), where X = h{hAi , Fi i}i∈ I , h I, ≤i, { pij | i ≤ j}i is an r-direct system of matrices. In order to show that hAi , Fi i ∈ Mod(`H ), for each i ∈ I + we adapt the proof strategy of Lemma 6.1.20 to the calculus Hr as follows. Suppose Γ B ϕ is a rule of H, and assume towards a contradiction that for hAi , Fi i (i ∈ I + ) there exists an Ai -valuation h such that h[Γ] ⊆ Fi , while h( ϕ) ∈ Ai r Fi . We distinguish two cases, (a): Γ = ∅, (b): Γ = {γ1 , . . . , γn }. In case (a), by condition (P0), x ∗ ϕ B ϕ holds in Hr . So consider the Avaluation v defined as v( x ) = a ∈ Fi for x ∈ / Var ( ϕ) and v(y) = h(y), for every y ∈ Var ( ϕ). As v( x ∗ ϕ) = v( x ) ∗ v( ϕ) = v( x ) ∗ h( ϕ) = a ∈ Fi and v( ϕ) = h( ϕ) ∈ Ai r Fi , we obtain that v falsifies a rule of Hr , which is a contradiction.

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The strategy for proving the remaining case (b) can be carried out in a very similar way by using condition (P3). Therefore, recalling that H is complete for L, we have proved that hAi , Fi i ∈ Mod(L), for each i ∈ I + . By Lemma 6.2.2.(2), we know that if X contains a trivial matrix hA j , Fj i, then A = 1. So, two cases may arise: (1) A = 1, (2) X contains no trivial fibres. If (1) then clearly hA, F i ∈ {h1, ∅i, h1, {1}i}. As Lr is a theoremless logic, {h1, ∅i, h1, {1}i} ⊆ Mod(Lr ). If (2), then we can apply Lemma 6.1.5, so hA, F i = Pł ( X ) ∈ Mod(Lr ).  Remark 6.2.4. It is easy to check that, if the logic L does not possess antitheorems, then a Hilbert-style calculus for Lr can be defined by simply dropping condition (P4) from Definition 6.2.1. The completeness of Lr with respect to such calculus can be proven by adapting the strategy in the proof of Theorem 6.2.3.

6.3 The algebraic counterpart In this section we describe the structure of the class Alg(Lr ), where Lr is some right variable inclusion logic. Such a characterisation depends on specific properties of the initial logic L. We begin by stating a lemma which can directly be inferred from Corollaries 6.1.12, 6.1.13. Lemma 6.3.1. Let hPł (Ai ), Fi ii∈ I be a r-direct system of matrices with Ai ∈ Alg(L) for every i ∈ I. Suppose, moreover, that hA j , Fj i ∈ Mod(L) for every j ∈ I+. S

(i) If L has no antitheorem, then hPł (Ai ),

S

Fi ii∈ I is a model of Lr ;

(ii) If L has an antitheorem and A j 6= Fj for every j ∈ I + , then hPł (Ai ), is a model of Lr .

S

Fi ii∈ I

Let us now state a result which will be proved as a corollary to Theorem 6.4.2. Corollary 6.3.2. For any logic L, Alg∗ (Lr ) ⊆ Pł (Alg∗ (L)). The following lemma will be applied multiple times in this section. Lemma 6.3.3. Let L be a logic of type τ such that Alg(L) is closed under subalgebras. If τ contains no constants, then Alg(Lr ) ⊆ Pł (Alg(L)).

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Proof. We have Alg(Lr ) = PSD (Alg∗ (Lr )) ⊆ PSD (Pł (Alg∗ (L))) ⊆ SP(P (Alg∗ (L))) ⊆

(6.1)

P (SP(Alg∗ (L)) ⊆

(6.4) (6.5)

ł

ł

Pł (SP(Alg(L) = Pł (Alg(L)).

(6.2) (6.3)

The inclusions or equalities are justified as follows. (6.1) holds by Proposition 4.0.5.(1). (6.2) holds in virtue of Corollary 6.3.2. (6.4) is a consequence of Lemma 5.1.2. Finally, the last equality holds since Alg(L) is closed under subalgebras, and because of Proposition 4.0.5.(4).  Remark 6.3.4. Observe that any equivalential logic L falls under the scope of Lemma 6.3.3, as well as all the non-equivalential logics whose class Alg is a quasivariety. The following lemma establishes how a filter of a logic L can be extended to a Lr -filter by means of pre-images of Płonka homomorphisms. Lemma 6.3.5. Let Lr be a logic of type τ having a r-partition function ∗ and let A ∼ = Pł (Ai )i∈ I with Ai ∈ Alg(L) for each i ∈ I. If Gi 6= Ai is a non-empty L-filter, then [ 1 r hA, ( p− ki [ Gi ])i ∈ Mod(L ), k ≤i 1 Proof. At first observe that, by Lemma 4.0.1, hAk , p− ki [ Gi ]i ∈ Mod(L), for each k ≤ i. since A ∼ = Pł (Ai ), by construction it is immediate to check that [ 1 hA, p− ki [ Gi ]i k ≤i

is isomorphic to a Płonka sum over a r-direct system of matrices. We consider two possibilities: (1) L does not have an antitheorem, (2) Σ( x ) is an antitheorem of L. If (1), our conclusion follows by Lemma 6.3.1. 1 If (2) we need to verify that for each k ≤ i, p− ki [ Gi ] 6 = Ak . Suppose the contrary towards a contradiction. Consider an arbitrary Ak -valuation 1 h. Clearly h[Σ( x )] ⊆ Ak = p− ki [ Gi ] and so pki [ h [ Σ ( x )]] ⊆ Gi . Consider now d ∈ Ai r Gi and an Ai -valuation v such that v( x ) = pki ◦ h( x ) and v(y) = d for all the variables y 6= x. Clearly we have v[Σ( x )] ⊆ Gi and v(y) ∈ / Gi against the assumption that hAi , Gi i is a model of L. This proves

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1 that for each k ≤ i, p− ki ( Gi )) 6 = Ak . So, by Lemma 6.3.1, we conclude that [ 1 r hA, p−  ki ( Gi ))i is an L -model. k ≤i

Before moving on, let us state one last preliminary lemma. Lemma 6.3.6. Let A ∈ Pł (Alg(L)), a ∈ Ai , b ∈ A j with j  i. If for some S Ai 1 unary polynomial function ϕ( x, ~z) and ~c ∈ Ai , F = k≤i p− c))] is ki [FgL ( ϕ ( a, ~ r A a L -filter on A, then h a, bi ∈ / Ω F. Proof. It suffices to note that ϕ(b, ~c) ∈ / F, so the statement follows by Lemma 4.0.2. 

6.3.1 Logics without antitheorems We set off by investigating the structure of the class Alg of a logic without antitheorems. Remark 6.3.7. Observe that, when L has no antitheorem, the assumption that Gi 6= Ai can be safely dropped from the statement of Lemma 6.3.5. It is natural to wonder to what extent the class Alg(Lr ) is expressible in terms of Płonka sums over Alg(L). A sufficient condition for this to happen is stated in the following: Proposition 6.3.8. Let L be a logic of type τ possessing no antitheorem. Then Pł (Alg(L)) ⊆ Alg(Lr ). Proof. Consider a logic L without antitheorems and set A = Pł ({hAi , Fi i}i∈ I ) with Ai ∈ Alg(L) for each i ∈ I. It is immediate to check that hA, ∅i ∈ Mod(Lr ). Let a ∈ Ai , b ∈ A j (for some i, j ∈ I). If i = j, then clearly there exists Fi ⊆ Ai such that hAi , Fi i ∈ ModSu (L), i.e., by Lemma 4.0.3, for some ~c ∈ Ai and unary polynomial ϕ( x, ~c) it holds ϕA ( a, ~c) ∈ Fi A ( b, ~ and ϕ[ c) ∈ / Fi . Lemma 6.3.5 together with Remark 6.3.7 entails that ∼ 1 r G= ( p− ( F / ΩA G, as desired. i )) is a L -filter on A and clearly h a, b i ∈ ki k ≤i

Let now i 6= j; set k = i ∨ j and consider b ∗ c for c = pik ( a) ∈ Ak . [ 1 r By Lemma 6.3.5 and Remark 6.3.7, G = ( p− wk [ Ak ]) is a L -filter on A. w≤k ∼ Since a ∗ a = a ∈ G, while b ∗ a = b ∗ c ∈ / G, we conclude h a, bi ∈ / ΩA G, as desired. 

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Remark 6.3.9. The assumption on the lack of antitheorems is crucial in Proposition 6.3.8, as witnessed by the following example. Clearly the trivial algebra belongs to Alg(CL), while the unique Płonka sum over the chain whose universe is {i, j}, with Ai , A j both trivial, does not belong to Alg(CLr ) = Alg(B3 ). It is indeed immediate to verify that the only CLr -filters on this algebra are the empty and the total filter. A full description of the class Alg(Lr ) can be obtained provided that L possesses no antitheorems and Alg(L) is closed under subalgebras. Theorem 6.3.10. Let L be a logic of type τ, without antitheorems, and such that Alg(L) is closed under subalgebras. If τ does not contain constants, then Pł (Alg(L)) = Alg(Lr ). Proof. Combine Proposition 6.3.8 and Lemma 6.3.3.



Since every equivalential logic L satisfies Alg(L) = S(Alg(L)), we obtain the following Corollary 6.3.11. Let L be an equivalential logic without antitheorems and such that its type τ does not contain constants. Then Alg(Lr ) = Pł (Alg(L)). Examples of equivalential logics with no antitheorems include the implicative fragment of classical logic, as well as every equivalential logic satisfying EAQ.

6.3.2 Logics with antitheorems We now address the study of the class Alg(Lr ) in case L has antitheorems. The following lemma establishes necessary conditions under which Płonka sums of algebras belonging to Alg(L) actually belong to Alg(Lr ). Lemma 6.3.12. Let L be a logic of type τ with antitheorems and an r-partition function ∗, A ∈ Pł (Alg(L)) and suppose that A ∈ Alg(Lr ). Then, in case { ai }, { a j } are universes of distinct trivial fibres Ai , A j (of A), then there exists a nontrivial fibre Ak , with i < k or j < k such that ai ∗ b 6= a j ∗ b, for some b ∈ Ak . Proof. Assume A ∈ Alg(Lr ) and let Ai = { ai }, A j = { a j } be universes of two distinct trivial fibres. Since A ∈ Alg(Lr ), there exists a unary polynomial function ϕ( x, ~z), elements ~c ∈ A and Lr -filter F on A such that ϕA ( ai , ~c) ∈ F and ϕA ( a j , ~c) ∈ / F. Let us set p = i ∨ q, where q is the join of the indexes of fibres of each element of ~c. Observe that Lr has antitheorems, since L also does, and thus | A p | > 1. Indeed, suppose that

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147

A p is trivial, then A p = Fp , thus A = F (otherwise it is immediate to have a counterexample to an antitheorem), against the fact that ϕA ( a j , ~c) ∈ / F. Observe that we also have i < p, as otherwise we would get Fi = Ai , and thus A = F. Let now b ∈ A p . If a j ∗ b ∈ A p , clearly ai ∗ b 6= a j ∗ p, because ϕ( ai , ~c) = ϕ( a j , ~c). If, on the other hand, a j ∗ b ∈ / A p , we obtain ai ∗ b 6= a j ∗ p. This concludes the proof.  The converse of the above result can be proved assuming the logic L is protoalgebraic, as shown by the next proposition. Proposition 6.3.13. Let L be a protoalgebraic logic of type τ with antitheorems and an r-partition function ∗, and A ∈ Pł (Alg(L)). Moreover, assume that, if { ai }, { a j } are the universes of distinct trivial fibres Ai , A j , then there exists a nontrivial fibre Ak , with i < k or j < k such that ai ∗ b 6= a j ∗ b, for some b ∈ Ak . Then A ∈ Alg(Lr ). Proof. Let a, b be two arbitrary distinct elements in A. If a, b belongs to the same fibre Ai , or neither Ai nor A j are trivial, let Fi be a Suszko L[ 1 filter on Ai and assume w.l.o.g. that j 6< i. By Lemma 6.3.5 ( p− wi [ Fi ]) w ≤i

is a Lr -filter, and our conclusion follows by Lemma 6.3.6. So consider a ∈ Ai , b ∈ A j with i 6= j, for some i, j ∈ I and let { x, ψ1 ( x ), . . . , ψn ( x )} be an antitheorem of L. Firstly, suppose that Ai , A j are trivial fibres. Then, by assumption there exists a non-trivial fibre Ak with i < k or j < k such that a ∗ c 6= b ∗ c, for some c ∈ Ak . Without loss of generality consider i < k (the case j < k is analogous). We have ak = a ∗ c = pik ( a) 6= bu = b ∗ c = p ju (b) where u = k ∨ j. Clearly, if k = u, our conclusion directly follows by the fact that Ak ∈ Alg(L) and Lemmas 6.3.5, 6.3.6. So Let k 6= u, which entails k < u. Consider a L-filter Fk on Ak such that Fk 6= Ak , which exists A because Ak is non-trivial. Two cases may arise. If Ak = FgL k ({ ak } ∪ Fk ) then, since L is protoalgebraic and finitary, we can apply Theorem 4.0.6 [ 1 getting {ψ1 ( ak ), . . . , ψn ( ak )} ⊆ Fk . So, by Lemma 6.3.5, K := ( p− wk [ Fk ]) w≤k ∼ is a Lr -filter on A and ψ1 ( ak ) ∈ K while ψ1 (bu ) ∈ / K, i.e. h a, bi ∈ / ΩLAr K. A Otherwise, let Hi = FgL k ({ ak } ∪ Fk ) 6= Ak . A further application of [ 1 r Lemma 6.3.5 guarantees that H := ( p− wk [ Hk ]) is a L -filter and clearly w≤k ∼ ak ∈ H, bu ∈ / H, i.e. h a, bi ∈ / Ω LAr H, as desired. Finally, suppose a ∈ Ai , b ∈ A j with i 6= j and at least one among Ai , A j is non-trivial. Without loss of generality, assume Ai is non-trivial

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and consider a Suszko filter Fi over Ai . Observe that, by Lemma 6.3.5, [ −1 we can claim that if Gi 6= Ai is a L-filter on Ai , then G := ( pwi [ Gi ]) w ≤i A

cannot contain the universe of a trivial fibre. Now, if FgL i ({ a} ∪ Fi ) = Ai , by Theorem 4.0.6 we have {ψ1 ( a) . . . ψn ( a)} ⊆ Fi and clearly ψ1 (b) ∈ / [ −1 ( pwi [ Fi ]). Indeed, since b ∈ A j , we have ψ1 (b) = b, so ψ1 (b) ∈ F if w ≤i

and only if b ∈ Fj , which is impossible by the above claim. This entails

h a, bi ∈ / ΩA

[

Ai 1 ( p− wi [ Fi ]), as desired. The only case left is Gi = FgL ({ a } ∪

w ≤i

Fi ) 6= Ai . In this case, the above claim ensures b ∈ / G. Since a ∈ G, b ∈ / G, we obtain again the desired conclusion that h a, bi ∈ / ΩA G. This concludes the proof.  The following corollary provides a full characterisation of the algebraic counterpart of an equivalential logic with antitheorems. Corollary 6.3.14. Let L be an equivalential logic of type τ with antitheorems and an r-partition function ∗, and let A be an algebra. If τ does not contain constants, the following are equivalent: 1. A ∈ Alg(Lr ); 2. A ∈ Pł (Alg(L)) and if { ai }, { a j } are the universes of trivial fibres Ai , A j , then there exists a nontrivial fibre Ak , with i < k or j < k such that ai ∗ b 6= a j ∗ b, for some b ∈ Ak . Proof. Since in an equivalential logic the class Alg is closed under subalgebras, our statement follows by Lemmas 6.3.3, 6.3.12 and Proposition 6.3.13.  For a logic L possessing antitheorems, a simpler characterization of the class Alg(Lr ) can be given in case every nontrivial member of Alg(L) lacks trivial subalgebras. A quasivariety of algebras with the property that each of its non-trivial members lacks trivial subalgebras is called Koll´ar in [154]. Coherently, we extend this terminology also to an arbitrary class of algebras. More precisely, a class of algebras is Koll´ar if every non-trivial member of the class lacks trivial subalgebras. A full description of the class Alg(Lr ) is provided when L is also protoalgebraic.

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Theorem 6.3.15. Let L be a logic of type τ with antitheorems and an r-partition function such that Alg(L) is Koll´ar, and let A ∈ Pł (Alg(L)). If A ∈ Alg(Lr ), then A has at most one trivial fibre. Proof. Let A ∈ Alg(Lr ) and assume, by contradiction, that there are two trivial fibres Ai , A j in A, with Ai = { a}, A j = {b}. Let k ∈ I with i ∨ j ≤ k; then pik (Ai ) and p jk (A j ) are subuniverses of Ak . Since each nontrivial member of Alg(L) lacks trivial subalgebras then Ak is trivial. This contradicts Lemma 6.3.12, so there exist at most one trivial fibre.  Corollary 6.3.16. Let L be a protoalgebraic logic of type τ with antitheorems and an r-partition function such that Alg(L) is Koll´ar. If A ∈ Pł (Alg(L)), then the following are equivalent: 1. A ∈ Alg(Lr ); 2. A has at most one trivial fibre. Proof. (1) ⇒ (2) follows from Theorem 6.3.15. (2) ⇒ (1) follows from Proposition 6.3.13.



Remark 6.3.17. It follows from the proof of Theorem 6.3.15 that, in case A ∈ Alg(Lr ) contains a (unique) trivial subalgebra then the semilattice of indices of the Płonka sum representation has a top element, which coincides with the index of the trivial algebra.

6.4 Leibniz reduced models Leibniz reduced models of right variable inclusion logics can be conveniently described, and it is the aim of this section to afford such a description. We first examine the Płonka sum representation of the filter of a Leibniz reduced model of a generic right variable inclusion logic Lr . Lemma 6.4.1. Let L be a logic of type τ with an r-partition function. Let also A 6= 1 and A ∈ Alg(Lr ). The following hold: 1. If τ does not contain constants and hA, F i ∈ Mod∗ (Lr ), then I + is a singleton. 2. If τ contains constants and hA, F i ∈ Mod∗ (Lr ) is isomorphic to a r-direct system of matrices whose underlying semilattice is I, then I + = {i }.

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Proof. (1). Let hA, F i ∈ Mod∗ (Lr ). Since Alg∗ (Lr ) ⊆ Alg(Lr ), then, by applying Theorem 6.1.19, the matrix hA, F i is a Płonka sum over a rdirect system X of matrices. Suppose, by contradiction, that I + is not a singleton. Clearly I + 6= ∅; differently, hA, F i is a Płonka sum of matrices with empty filters, any of which cannot be Leibniz reduced as we have assumed that A 6= 1. W.l.o.g. we can consider two elements i, j ∈ I + such that i ≤ j (this is justified by the fact that I + is a semilattice). Since Fi 6= ∅, let a ∈ Fi and pij ( a) = b ∈ Fj . We claim that h a, bi ∈ ΩA F. In order to show this, we use the characterisation provided in Lemma 4.0.2. Let ϕ(v, ~c ) be an arbitrary unary polynomial on A and assume ϕA ( a, ~c ) ∈ F, with ~c ∈ As , for some s ∈ I. Clearly, ϕA ( a, ~c ) ∈ Fk , where k = i ∨ s. Observe that j, k ∈ I + , hence also k ∨ j = p ∈ I + (as I + is a sub-semilattice of I). In particular: ϕA (b, ~c ) =

(6.6)

ϕ ( pij ( a), ~c ) =

(6.7)

( p jp ( pij ( a)), psp (~c )) =

(6.8)

ϕA p ( pkp ( pik ( a)), pkp ( psk (~c ))) =

(6.9)

A

ϕ

Ap

Ak

pkp ( ϕ ( pik ( a), psk (~c )) = A

pkp ( ϕ ( a, ~c )) ∈ Fp .

(6.10) (6.11)

In particular, (6.9) holds as s ∨ j = p; (6.10) by observing that pip = p jp ◦ pij = pkp ◦ pik and s ≤ k ≤ p; (6.11) since ϕA ( a, ~c ) ∈ Fk implies that pkp ( ϕA ( a, ~c)) ∈ Fp . Similarly, assume ϕ(b, ~c ) ∈ F, that is ϕ(b, ~c ) ∈ Fp . Suppose, towards a contradiction that ϕ( a, ~c ) ∈ / F, which means ϕA ( a, ~c ) = ϕAk ( pik ( a), psk (~c )) ∈ / A Fk , whence pkp ( ϕ k ( pik ( a), psk (~c ))) ∈ / Fp . However, pkp ( ϕAk ( pik ( a), psk (~c ))) = ϕA p ( pkp ( pik ( a)), pkp ( psk (~c ))) = ϕA p ( p jp ( pij ( a)), pkp ( psk (~c ))) = ϕA p ( p jp (b), psp (~c )) = ϕA (b, ~c ) ∈ Fp . This is a contradiction, so ϕ( a, ~c) ∈ Fk ⊆ F. This establishes our claim that h a, bi ∈ ΩA F. Therefore a = b, which implies that i = j, i.e. I + does not possess two different elements. Then I + is a singleton. (2) can be proved similarly. 

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151

Recalling from Section 2.3.2 the definition of the algebra A∗ , we can now proceed to prove the following characterisation of the Leibniz reduced models of a right variable inclusion logic. Theorem 6.4.2. Let L be a logic of type τ with an r-partition function ∗, , and let A be a nontrivial member of Alg(Lr ). If τ does not contain constants, the following are equivalent: 1. hA, F i ∈ Mod∗ (Lr ); 2. I + is a singleton {i } and, either A = Ai or A = Ai∗ , with hAi , Fi i ∈ Mod∗ (L). Proof. (1)⇒(2). The fact that the matrix is a Płonka sum over a r-direct system of matrices follows by Theorem 6.1.19, while that I + is a singleton follows by Lemma 6.4.1. Now, the equivalence relation θ which collapses all the fibres j for j  i to a single point is a congruence which is compatible with F. Because the matrix hA, F i is reduced, θ is the identity congruence and therefore either i is the top element of the semilattice I or it is the coatom, with the top fibre having 1 as algebraic reduct. Similarly, the equivalence relation ψ which collapses each a ∈ Ak for k < i with pki ( a) ∈ Ai is also a congruence which is compatible with F. It follows that i is the bottom element of the semilattice. This proves the implication. (2)⇒ (1). Let Pł ( X ) = hA, F i satisfy (2). Since the Płonka sum over an r-direct system of matrices is a model of Lr (by Lemma 6.1.20), hA, F i ∈ Mod(Lr ). Moreover, since hAi , Fi i ∈ Mod∗ (L), for any pair of elements a, b ∈ Ai , there exists a unary polynomial ϕ( x, ~c) on Ai such that ϕA ( a, ~c) ∈ Fi if and only if ϕA (b, ~c) ∈ / Fi . In order to prove (1), we just need to show that hd, 1i ∈ / ΩA F, for an A arbitrary d ∈ Ai . To this end, let e ∈ Fi . Clearly e ∗ d = e ∈ F, while e ∗A 1 = 1 ∈ / F. That is, the unary polynomial e ∗ x witnesses that hd, 1i ∈ / ΩA F. This concludes our proof.  Remark 6.4.3. Observe that Theorem 6.4.2 can be weakened in order to deal with the case in which τ has constants. All is needed is to assume that hA, F i ∈ Mod∗ (Lr ) is isomorphic to an r-direct system of matrices whose underlying semilattice is I. From the above Theorem 6.4.2 Corollary 6.3.2 follows (we have actually already used in the proof Lemma 6.3.3).

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6.5 Suszko reduced models We now turn our attention to the structure of the Suszko reduced models of a right variable inclusion logic. In what follows, given a logic L and hA, G i ∈ ModSu (L), we say that G is a Suszko filter over A. We start by proving that, in the Płonka sum representation of a Suszko reduced model of a right variable inclusion logic, there can be at most one fibre with a nonempty filter. Lemma 6.5.1. Let Lr be a logic of type τ with an r-partition function. The following hold: 1. If τ does not contain constants and hA, F i ∈ ModSu (Lr ) then | I + |≤ 1; 2. If τ contains constants and hA, F i ∈ ModSu (Lr ) is isomorphic to a r-direct system of matrices whose underlying semilattice is I, then | I + |≤ 1 Proof. (1). By [164, Lemma 6] and the definition of direct product of matrices, if {hAw , Fw i}w∈W is a family of Płonka sums of matrices such that | Iw+ |≤ 1 for each w ∈ W, then ∏w∈W hAw , Fw i is a Płonka sum of matrices having at most one fibre with non-empty filter too. By Theorem 6.4.2, every Leibniz reduced model of Lr has at most one fibre with nonempty filter. Therefore, by Proposition 4.0.5.(1), our conclusion follows. (2). Suppose ex absurdo that there exist i, j ∈ I + with i 6= j. Let k = i ∨ j. Since I + is a semilattice, then k ∈ I + . Let a ∈ Fi and b = pik ( a) ∈ Fk . ∼ We claim that h a, bi ∈ ΩA F, whence a contradiction follows. ∼ To show the claim, suppose, again by contradiction, that h a, bi 6∈ ΩA F. Then, there exist an Lr -filter G ⊇ F, a unary polynomial ϕ( x, ~c ) on A and elements ~c ∈ A such that ϕ( a, ~c ) ∈ G ⇐⇒ ϕ(b, ~c ) ∈ / G. Observe that a ∈ Gi , b ∈ Gk (as G ⊇ F). Suppose that ϕ( a, ~c ) ∈ G. W.l.o.g. consider ~c ∈ Aq , hence ϕ( a, ~c ) ∈ Gs , with s = i ∨ q. On the other hand, ϕ(b, ~c ) ∈ A p with p = k ∨ q. Clearly, as i ≤ k, we have s ≤ p and so p = s ∨ k. This, together with the fact that s, k ∈ I + implies that p ∈ I + (as, otherwise, we would have ϕ( a, ~c ), b ∈ G, while ϕ( a, ~c ) ∗ b ∈ / G, against the fact that G is an Lr -filter). In particular, we obtain psp ( ϕ( a, ~c)) ∈ G p . Now, recalling that pip = psp ◦ pis = pks ◦ pik we have

6.5. SUSZKO REDUCED MODELS

153

psp ( ϕ( a, ~c)) psp ( ϕ( pis ( a), pqs (~c ))) ϕ( psp ( pis ( a)), psp ( pqs (~c ))) ϕ( pkp ( pik ( a)), pqp (~c ))

= = = =

ϕ( pkp (b), pqp (~c )) = ϕ(b, ~c ) ∈ G p , a contradiction.



Theorem 6.5.2. Let Lr be a logic of type τ with an r-partition function ∗ and let hA, F i ∈ Mod(Lr ) be such that A ∈ Alg(Lr ), hAi , Fi i ∈ ModSu (L) for every i ∈ I + . Assume, moreover, that, for each j ∈ I, A j ∈ Alg(L) and there exists a Suszko filter Gj on A j such that Fi ⊆ pij−1 [ Gj ]. If τ does not contain constants, the following are equivalent: 1. hA, F i ∈ ModSu (Lr ); 2.

(a) I + = ∅ or (b) I + = {i } is the bottom element in I.

Proof. (1)⇒(2). By Lemma 6.5.1, we have that | I + |≤ 1, that is, either I + = ∅, namely F = ∅, or I + = {i }, i.e. F = Fi . In order to prove (2) we only need to show that if I + = {i } then i is the bottom element in I. We reason by contradiction, so assume that i is not the bottom element of I, i.e. there exists j ∈ I such that i  j. Let a ∈ A j and s = i ∨ j; consider an element b = p js ( a) ∈ As . Since j ∈ / I + , we have that Fj = ∅ and 1 then, by Definition 6.1.4, b ∈ / Fs (since if b ∈ Fs then Fj = p− js [ Fs ] 6 = ∅). Moreover, as hA, F i ∈ ModSu (Lr ), there exist an Lr -filter G ⊇ F and a unary polynomial ϕ(v, ~c) on Ak such that ϕ( a, ~c ) ∈ G ⇐⇒ ϕ(b, ~c ) ∈ / G. W.l.o.g. assume ϕ( a, ~c ) ∈ Gq ⊆ Aq (with q = j ∨ k). Now, as Gi 6= ∅ and Gq 6= ∅, by Theorem 6.1.19, we have G p 6= ∅ (with p = s ∨ k). Observe also that this implies pqp ( ϕ( a, ~c )) ∈ G p . Moreover, by applying the same strategy used in the proof of Lemma 6.5.1 pqp ( ϕ( a, ~c )) = ϕ(b, ~c ) ∈ G p ,

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154

which is a contradiction. The same argument can be applied to the case ϕ(b, ~c ) ∈ G. This proves (2). (2)⇒(1). We have to show that each of the conditions (a) and (b) implies (1). (a)⇒(1). Assume the Płonka decomposition of hA, F i is such that I + = ∅. ∼ Consider a, b ∈ A, with a 6= b. We aim at showing h a, bi ∈ / ΩA F. Consider first the case when a ∈ Ai , b ∈ A j for arbitrary i 6= j. We assume w.l.o.g. that if i, j are comparable then i < j. Now, as A ∈ Alg(L), consider a [i 1 non-empty L-filter Gi 6= Ai . By Lemma 6.3.5, hA, ( p− ki ( Gi ))i is a model k ≤i

of Lr . In particular, as F = ∅, H :=

[

1 r p− ki [ Gi ]) is a L -filter extending F.

k ≤i

Now fix c ∈ Gi . We have that c ∗ a = c ∈ H, while c ∗ b ∈ / H, proving ∼ h a, bi ∈ / ΩA H, i.e., h a, bi ∈ / ΩA F. This shows hA, F i ∈ ModSu (Lr ), as desired. The only case left is a, b ∈ Ai . As Ai ∈ Alg(L) there exists hAi , Gi i ∈ Mod(L) such that h a, bi ∈ / ΩAi Gi , i.e., there exist ~c ∈ Ai and a unary polynomial ϕ(v, ~c) on Ai satisfying ϕ( a, ~c) ∈ Gi if and only if ϕ(b, ~c) ∈ / Gi . Observe this implies Gi 6= Ai , for otherwise ΩAi Gi = Ai × Ai and, by Lemma 6.3.5, this entails hA, H i ∈ Mod(Lr ). So, we obtain ϕ( a, ~c) ∈ H if ∼ and only if ϕ(b, ~c) ∈ / H, proving h a, bi ∈ / ΩA F. (b)⇒(i). Assume that I + = {i } is the bottom element of I and consider ∼ arbitrary a, b ∈ A. Again, we aim at showing h a, bi ∈ / ΩA F. The case a, b ∈ ∼Ai Ai is immediate, as F = Fi and Ω Fi = ∆Ai , for h Ai , Fi i ∈ ModSu (L). So let a ∈ A j , b ∈ Ak assuming w.l.o.g. that if j, k are comparable then j < k. The argument of Lemma 6.3.5, together with the fact that there exists a Suszko filter Gj such that Fi ⊆ pij−1 [ Gj ] for each j > i, imply that hA, K i := hA,

[

1 r p− sj ( G j )i is a model of L and F ⊆ K. Moreover, as

s≤ j

∼ Gj 6= ∅, we can fix c ∈ Gj . Clearly c ∗ a ∈ K and c ∗ b ∈ / K, so h a, bi ∈ / ΩA F, as desired. The only case left is a, b ∈ A j . Again consider hA j , Gj i ∈ ModSu (L) such that Fi ⊆ pij−1 [ Gj ] and let Hj ⊇ Gj be the L-filter on A j such that

h a, bi ∈ / ΩA j Hj . This is to say that there exist a unary polynomial ϕ(v, ~c) on A j such that ϕ( a, ~c) ∈ Hj if and only if ϕ(b, ~c) ∈ / Hj . As Hj ⊇ −1 −1 Gj and Fi ⊆ pij [ Gj ] we have Fi ⊆ pij [ Hj ]. This, as before, implies hA,

[ k≤ j

1 r p− c) ∈ kj [ H j ]i is a model of L and therefore we obtain ϕ ( a, ~

[ k≤ j

1 p− kj [ H j ]

6.5. SUSZKO REDUCED MODELS if and only if ϕ(b, ~c) ∈ /

[

155

∼ 1 p− / ΩA F and conkj [ H j ]. This proves h a, b i ∈

k≤ j

cludes the proof.



Remark 6.5.3. Observe that the assumption concerning the existence of a specific Suszko filter in Theorem 6.5.2 is fundamental, as witnessed by the following example. Consider the Płonka sum of matrices hA, G i represented in the diagram below. The algebraic reduct is a Płonka sum of two distributive lattices D3 , D2 , namely the 3-element chain with universe { a, b, c} and the 2-element chain with universe {d, e}. Dotted lines are Płonka homomorphisms and circled elements represent the filter G = {b, c}. It is immediate to verify that it is a model of CLr∧,∨ . Moreover, hD3 , {c, b}i, hD2 , ∅i ∈ ModSu (CL∧,∨ ), I + is the bottom of I but ∼ ΩD3 G 6= ∆D3 . However, there is no Suszko filter F on D2 such that c, b are contained in the pre-image of F. c

e

b a

d

6.5.1 Truth-equational logics If the logic L is truth-equational, the characterisation of the Suszko reduced models can be significantly simplified, thanks to Theorem 4.0.4. The following technical lemma highlights the effect of truth-equationality in a Płonka sum over an r-direct system. Lemma 6.5.4. Let L be a truth-equational logic of type τ. Let hA, F i ∈ Mod(Lr ) be isomorphic to an r-direct system of matrices indexed by I, with Ai ∈ Alg(L), for each i ∈ I. If, for some k, j ∈ I, k ≤ j and hA j , Gj i, hAk , Gk i ∈ ModSu (L), 1 then Gk ⊆ p− kj ( G j ). Proof. Let hA, F i ∈ Mod(Lr ) be such that Ai ∈ Alg(L), for each i ∈ I. Consider k ≤ j and let hA j , Gj i, hAk , Gk i ∈ ModSu (L). Observe first that L is truth-equational, therefore Gj , Gk 6= ∅. Hence, our conclusion follows from Definition 6.1.4.  The next result is a refinement of Theorem 6.5.2 that characterises the Suszko reduced models of Lr in case L is a truth-equational logic.

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Theorem 6.5.5. Let L be a truth-equational logic of type τ with a r-partition function. Let moreover hA, F i ∈ Mod(Lr ) be isomorphic to a r-direct system of matrices with A j ∈ Alg(L), for every j ∈ I and hA j , Fj i ∈ ModSu (L) for every j ∈ I + . The following are equivalent: 1. hA, F i ∈ ModSu (Lr ); 2. I + = ∅ or I + = {i } with i the bottom element in I. Proof. (1)⇒(2). The proof is analogous to that of Theorem 6.5.2 (it is immediate to verify that the additional assumption in Theorem 6.5.2 is not used in this direction). (2)⇒(1). We need to show that any of the two conditions (I + = ∅, I + = {i } is the least element in I) implies (1). If I + = ∅ then the argument is the same as in the proof of Theorem 6.5.2. Suppose that I + = {i } is the least element in I, and consider two distinct ∼ elements a, b ∈ A. The case a, b ∈ Ai is immediate, as h a, bi ∈ / ΩA Fi . So, suppose that a ∈ A j , b ∈ Ak (with j 6= k); we can consider a Suszko filter Gj on A j and, by applying Lemma 6.3.5 and Lemma 6.5.4, we obtain that [ 1 r hA, U i := hA, p− sj ( G j )i is a model of L and that Fi = F ⊆ U. If j 6 = k s≤ j

then, as before, we can fix c ∈ Gj and observe that c ∗ a ∈ U while c ∗ b ∈ / Su U. Differently, if j = k, then, from the fact that hA j , Gj i ∈ Mod (L), we deduce that there exists a L-filter Hj ⊇ Gj such that h a, bi ∈ / ΩA j Hj . This is equivalent to the fact that ϕ( a, ~c) ∈ Hj if and only if ϕ(b, ~c) ∈ / Hj , for a unary polynomial ϕ(v, ~c) on A j . Clearly Hj ⊇ Gj implies V := [ 1 r p− sj ( H j ) ⊇ U, so by Lemma 6.3.5, V is a L -filter extending U. As s≤ j

V ∩ A j = Hj we have that ϕ( a, ~c) ∈ V if and only if ϕ(b, ~c) ∈ / V. ∼A Su r In any case, then, h a, bi ∈ / Ω F, i.e. hA, F i ∈ Mod (L ).



Corollary 6.5.6. Let Lr be a the right variable inclusion companion of a logic with a r-partition function. Then Mod∗ (Lr ) ⊂ ModSu (Lr ).

6.5.2

Two well-behaved cases

We conclude this chapter with two representative cases in which a full characterisation of the Suszko reduced models of Lr is available. Corollary 6.5.7. Let L be an algebraisable logic of type τ with antitheorems and a r-partition function and such that Alg(L) is Koll´ar. If τ does not contain constants and hA, F i is a τ-matrix, the following are equivalent:

6.5. SUSZKO REDUCED MODELS

157

1. hA, F i ∈ ModSu (Lr ); 2. A ∼ = Pł (Ai )i∈ I , A j ∈ Alg(L) for each j ∈ I, Pł (Ai )i∈ I has at most one trivial fibre indexed by the top element of I and (a) I + = ∅ or (b) I + = {i } is the bottom element of I and hAi , Fi i ∈ ModSu (L). Proof. The statement follows from Lemma 6.3.3 and Theorems 6.5.2, 6.5.5, upon recalling that an algebraisable logic is both equivalential and truthequational.  Corollary 6.5.8. Let L be a truth-equational logic of type τ without antitheorems, with a r-partition function and such that Alg(L) is closed under subalgebras. If τ does not contain constants and hA, F i is a τ-matrix, the following are equivalent: 1. hA, F i ∈ ModSu (Lr ); 2. A ∼ = Pł (Ai )i∈ I , A j ∈ Alg(L) for each j ∈ I and (a) I + = ∅ or (b) I + = {i } is the bottom element of I and hAi , Fi i ∈ ModSu (L). Proof. The statement follows by Lemma 6.3.3 and Theorems 6.3.10, 6.5.5. 

Chapter 7 Paraconsistent Weak Kleene Logic We close this volume with a focus on one of the most notable and best understood logics of variable inclusion – Paraconsistent Weak Kleene Logic, the left variable inclusion companion of classical logic. We know that this logics comes in two main variants: PWK = CLl , formulated in the constant-free type τ2 , and PWK0,1 = (CL0,1 )l , formulated in the type τ1 with constants. The former incarnation will take centre stage in this chapter, although we will not refrain from giving results on the latter and from commenting the occasional differences between them. We had ample opportunity in Chapter 1 to go over the conceptual underpinnings and the potential applications of this logic. The present chapter, thus, will be entirely devoted to its formal properties, whether syntactic or semantic. We hope thereby to provide a convincing case study that shows what features of individual logics of variable inclusion can be derived from the general theory, and what features need instead to be addressed with a recourse to methods that are tailored to each specific case. We set off in Section 7.1 with a study of the properties of PWK and PWK0,1 from the viewpoint of Abstract Algebraic Logic. We will see that their placement in the Leibniz hierarchy, as well as the characterisation of their Suszko reduced models and their algebra reducts, can be straightforwardly obtained from the results in Chapter 5. Other aspects, like the description of deductive filters and matrix models – especially the Leibniz reduced ones – will benefit from the application of ad hoc methods. In Section 7.2 we present two Hilbert-style axiomatisations. Again, one of them will be mechanically generated via the general algorithm given in Chapter 5, while the other will be extracted from an axiomatisation of CL by imposing a variable inclusion constraint on its sole rule, Modus

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_7

159

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Ponens. In Section 7.3 we present sequent calculi for PWK. One of them coincides with the classical sequent calculus except that appropriate variable inclusion provisos are attached to the operational rules for some of the connectives. The next calculus, on the other hand, is devoid of such restrictions; its flip side, however, is that it includes, in the absence of the Cut rule, (left and right) elimination rules in addition to the customary introduction rules. A final section explores other proof-theoretic formalisms like natural deduction or tableaux systems. Throughout this chapter, by a valuation we mean either a WK- or a WK0,1 -valuation; it will be clear from the context whether the type we are using is τ1 or τ2 .

7.1 Abstract Algebraic Logic properties 7.1.1 Basic properties In this short subsection, we locate the place of both PWK and PWK0,1 in the Leibniz and in the Frege hierarchies. With such an aim in mind, we will have occasion to put to good use the results obtained in Chapter 5. We also give, for later use, a normal form theorem for both logics. Theorem 7.1.1. Both PWK and PWK0,1 are: 1. non-protoalgebraic; 2. truth-equational; 3. non-selfextensional. Proof. (1)-(2). This follows from Theorem 5.5.1 since: a) PWK = CLl and PWK0,1 = (CL0,1 )l ; b) CL and CL0,1 are consistent, finitary, algebraisable logics with an l-partition function. Observe that the set of defining equations for both logics is the singleton set E( x ) = { x ≈ x ∨ ¬ x }, and that for every (generalised) involutive bisemilattice B, SolEB = P(B), the set of positive elements of B. (3). We prove our claim for PWK; an argument requiring no modification also works for PWK0,1 . All we have to show is that the interderivability relation a`PWK is not a congruence of the formula algebra Fmτ2 . In order to do that, notice that ¬ x ∨ x a`PWK ¬y ∨ y, since all instances of the excluded middle are classical theorems, and therefore theorems of PWK by Theorem 1.3.2. However, the valuation v that sends y to 0

7.1. ABSTRACT ALGEBRAIC LOGIC PROPERTIES

161

and all the remaining variables to n is such that v(¬(¬ x ∨ x )) = n and v(¬(¬y ∨ y)) = 0, and therefore ¬(¬ x ∨ x ) 0PWK ¬(¬y ∨ y).  Despite the foregoing result, a certain refinement of the interderivability relation in PWK is a congruence of the formula algebra Fmτ2 . In fact, let Θ = {h ϕ, ψi ∈ Fmτ2 : Var ( ϕ) = Var (ψ)}; then Θ ∩ a`PWK is a congruence of Fmτ2 . To round off this section, we establish a normal form theorem for PWK and PWK0,1 . As usual, a literal is either a variable or the negation of such, and a disjunctive clause is a (finite) disjunction of literals. The Conjunctive Normal Form (CNF) Theorem for CL ensures that every formula is interderivable with a conjunction of disjunctive clauses. By using Theorem 1.3.2, it is not hard to prove that such a theorem holds for PWK and PWK0,1 as well. But this result is of little avail, since in PWK Conjunctive Simplification (CS) fails (Lemma 1.2.4) and then we cannot replicate the classical proof that leads from the CNF theorem to the fact that every formula is interderivable with the set of the disjunctive clauses of its conjunctive normal form. To make some headway, we need to relax the notion of a clause in such a way as to take good care of the variables involved. An elementary contradiction is a formula of the form x ∧ ¬ x, where x is a variable. A clause is a finite disjunction of literals and elementary contradictions. Proposition 7.1.2. (1) Every formula ϕ is interderivable in PWK with a conjunction of clauses ψ1 , . . . , ψn such that, for every i ≤ n, Var (ψi ) = Var ( ϕ). Moreover, ϕ is interderivable in PWK with the set of the clauses of its conjunctive normal form, that is ϕ a`PWK {ψ1 , . . . , ψn }. (2) The same results hold for PWK0,1 , if ϕ is a formula containing variables. Proof. (1) Let γ1 ∧ · · · ∧ γn be a conjunctive normal form of ϕ in CL. Thus, every γi is a finite disjunction of literals. The classical theorem also ensures that the variables of the normal form are the same as the variables of ϕ. For every i ≤ n, let ψi = γi ∨ ( xi1 ∧ ¬ xi1 ) ∨ · · · ∨ ( ximi ∧ ¬ ximi ), where { xi1 , . . . , ximi } is the set of variables of ϕ that are missing in γi . It is obvious that ψi a`CL γi , and therefore ϕ a`CL ψ1 ∧ · · · ∧ ψn . Thus, since Var ( ϕ) = Var (ψ1 ∧ · · · ∧ ψn ), by Theorem 1.3.2 we have that ϕ a`PWK ψ1 ∧ · · · ∧ ψn . Finally, since the variables of every clause are always the same, Theorem 1.3.2 yields that ψ1 ∧ · · · ∧ ψn a`PWK {ψ1 , . . . , ψn }, and we are done with our proof. (2) Clear. 

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7.1.2 Deductive filters and matrix models Our study of matrix models will be centred on PWK, although the same results can be proved in an analogous way when constants are added to the type. For generic matrix models of PWK, we deduce from the general theory that: Theorem 7.1.3. PWK is complete w.r.t. the Płonka sum of any class of τ2 matrices containing hB2 , {1}i and the trivial matrix h1, {1}i; in particular, it is complete w.r.t. hWK, {1, n}i. Proof. By Theorem 5.1.7.



Remark 7.1.4. By Theorem 7.1.3, taking further into account Remark 5.1.10, PWK is complete with respect to the matrix whose algebra reduct is B2 ⊕ B2 , the unique Płonka sum of two copies of the 2-element Boolean algebra over the 2-element chain of indices, and whose designated values are the top element of its lower fibre and both elements in its upper fibre. This fact is the basis of the “non-infectious” semantics for PWK proposed by Song et al. in [214]. Observe that the above-mentioned matrix is not a Leibniz reduced model of PWK, and that its quotient modulo the Leibniz congruence of its filter is precisely hWK, {1, n}i. For Leibniz reduced models, although PWK is not protoalgebraic (Theorem 7.1.1), it still makes sense to characterise the class Alg*(PWK) of their algebra reducts. In order to do so, we need to provide a workable description of the Leibniz congruence of a PWK-filter of an arbitrary algebra of the appropriate similarity type. In the following proposition, we establish such a characterisation, very much in the same spirit as the one in [95] for Belnap and Dunn’s BD. Proposition 7.1.5. Let A be an algebra of type τ2 and F ⊆ A a PWK-filter. Then, for every a, b ∈ A, h a, bi ∈ ΩA F if and only if for every c ∈ A, a ∨ c ∈ F ⇐⇒ b ∨ c ∈ F

and

¬ a ∨ c ∈ F ⇐⇒ ¬b ∨ c ∈ F. (Leib)

Proof. If F is a PWK-filter of an algebra A and h a, bi ∈ ΩA F, then by Lemma 4.0.2 for every unary polynomial ϕ( x, ~c) on A and for every a, b ∈ A, we have that ϕA ( a, ~c) ∈ F if and only if ϕA (b, ~c) ∈ F. In particular, for the formulas x ∨ y and ¬ x ∨ y, we obtain the implications of (Leib). Now, suppose that h a, bi ∈ / ΩA F. W.l.o.g. we can assume that there is a unary polynomial ϕ( x, ~c) on A such that ϕA ( a, ~c) ∈ F and ϕA (b, ~c) ∈ / F. Let ψ1 ∧ · · · ∧ ψn be the conjunctive normal form of ϕ( x, ~y) given by

7.1. ABSTRACT ALGEBRAIC LOGIC PROPERTIES

163

Proposition 7.1.2. It follows that ϕ a`PWK {ψ1 , . . . , ψn }, and therefore ϕA ( a, ~c) ∈ F if and only if for every i, ψiA ( a, ~c) ∈ F, and analogously for b. Therefore, there is a clause, say ψ1 ( x, ~y), such that ψ1A ( a, ~c) ∈ F and ψ1A (b, ~c) ∈ / F. Obviously, the variable x necessarily appears in ψ1 . If the remaining variables in ψ1 are among ~y = y1 , . . . , yk , ψ1 ( x, ~y) coincides with some of the following, up to equivalence:

(i ) x; (iv) x ∨ γ(~y);

(ii ) ¬ x; (v) ¬ x ∨ γ(~y);

(iii ) x ∨ ¬ x; (vi ) x ∨ ¬ x ∨ γ(~y).

However, the cases (iii) and (vi) can be ruled out, since these formulas are always evaluated to positive elements of A, and therefore in both cases we would have that ψ1 (b, ~c) ∈ F. In cases (i) or (ii), one or the other of the implications in (Leib) would fail by considering c = b ∧ ¬b (or c = a ∧ ¬ a). The same would happen in cases (iv) or (v), by considering c = γ(~c). Thus we have seen that if h a, bi ∈ / ΩA F, then one or the other of the implications of (Leib) fails for some c, as was to be shown.  Thanks to this characterisation of the Leibniz congruence, we can restrict our search of members of Alg*(PWK) to generalised involutive bisemilattices. Theorem 7.1.6. Alg*(PWK) ⊆ GIB . Proof. Let hA, F i be a Leibniz reduced matrix model of PWK and let ϕ ≈ ψ be one of the identities (I1)-(I6) in Definition 2.4.1. We need to prove that A  ϕ ≈ ψ. Observe that, in every case, we have ϕ a`CL ψ, and thus ϕ ∨ y a`CL ψ ∨ y and ¬ ϕ ∨ y a`CL ¬ψ ∨ y, where y is a fresh variable not in Var ( ϕ) = Var (ψ). Hence, ϕ ∨ y a`PWK ψ ∨ y and ¬ ϕ ∨ y a`PWK ¬ψ ∨ y by virtue of Theorem 1.3.2, because Var ( ϕ ∨ y) = Var (ψ ∨ y) and Var (¬ ϕ ∨ y) = Var (¬ψ ∨ y). Let v be an A-valuation, and let c be an arbitrary element of A. Consider the A-valuation u such that u( x ) = v( x ) for every variable x 6= y, and u(y) = c. Hence, v( ϕ) ∨ c = u( ϕ ∨ y) ∈ F ⇐⇒ v(ψ) ∨ c = u(ψ ∨ y) ∈ F, and similarly ¬v( ϕ) ∨ c ∈ F ⇐⇒ ¬v(ψ) ∨ c ∈ F. Therefore, by virtue of Proposition 7.1.5, we obtain that hv( ϕ), v(ψ)i ∈ ΩA F = ∆A , since the matrix is reduced, and hence v( ϕ) = v(ψ). Since the A-valuation v was arbitrary, we have A  ϕ ≈ ψ. Thus, A ∈ GIB .  Corollary 7.1.7. The intrinsic variety V (PWK) of PWK is GIB .

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Proof. This is an immediate consequence of Theorem 4.0.8, Theorem 7.1.6, Proposition 2.4.22, and the fact that WK ∈ Alg*(PWK), since these imply that V (PWK) = V (Alg*(PWK)) ⊆ GIB = V (WK) ⊆ V (Alg*(PWK)) = V (PWK). 

We will prove that the inclusion of Theorem 7.1.6 is actually proper. But before that, we need a firmer grasp of the PWK-filters of generalised involutive bisemilattices, which the next Proposition will help us to build. In what follows, recall from Definition 2.4.10 that P(B) is the set of positive elements of B. Proposition 7.1.8. Let B ∈ GIB . F ⊆ B is a PWK-filter of B if and only if: F1. P(B) ⊆ F; F2. a ∈ F, a ≤∨ b ⇒ b ∈ F; F3. a, b ∈ F ⇒ a ∧ b ∈ F. Proof. Suppose that F is a PWK-filter of a generalised involutive bisemilattice B. Let moreover c ∈ P(B) and let v be a B-valuation s.t. v( x ) = c. Since x ∨ ¬ x is a theorem of PWK, c = c ∨ ¬c = v( x ∨ ¬ x ) ∈ F. Suppose now that a ≤∨ b and a ∈ F. Since x `CL x ∨ y, by virtue of Theorem 1.3.2, it follows that x `PWK x ∨ y. Thus, considering a B-valuation v such that v( x ) = a and v(y) = b, since v( x ) = a ∈ F, then also b = a ∨ b = v( x ∨ y) ∈ F. Assume finally that a, b ∈ F. By Theorem 1.3.2 again, we have that x, y `PWK x ∧ y. Thus, for a B-valuation v sending x to a and y to b, we would have a ∧ b = v( x ∧ y) ∈ F. Conversely, suppose that F contains P(B), is closed upwards w.r.t. ≤∨ and is closed under ∧. Suppose that Γ `PWK ϕ and that v[Γ] ⊆ F, for a designated B-valuation v. By Theorem 1.3.2 and the fact that CL is finitary, there exists a finite or empty ∆ ⊆ Γ s.t. ∆ `CL ϕ, with Var (∆) ⊆ Var ( ϕ). If ∆ = ∅, then `CL ϕ, whence by Lemma 1.2.2 `PWK ϕ, and it is readily checked, using Proposition 7.1.2, that v( ϕ) ∈ P(B) ⊆ F. If ∆ = δ1 , ..., δn (1 ≤ n), then `CL ¬(δ1 ∧ ... ∧ δn ) ∨ ϕ, whence `PWK ¬(δ1 ∧ ... ∧ δn ) ∨ ϕ and thus v(¬(δ1 ∧ ... ∧ δn ) ∨ ϕ) ∈ F. Since v(δ1 ∧ ... ∧ δn ) = v(δ1 ) ∧ ... ∧ v(δn ) ∈ F by closure under meets, it will suffice to prove that F is closed w.r.t. the rule

( RMP)

ϕ, ¬ ϕ ∨ ψ , ψ

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165

with Var ( ϕ) ⊆ Var (ψ). Suppose that we have two formulas ϕ and ψ such that v( ϕ) ∈ F and v(¬ ϕ ∨ ψ) ∈ F, and moreover Var ( ϕ) ⊆ Var (ψ), with an eye to showing that v(ψ) ∈ F. By (I6) in Definition 2.4.1, our assumption entails that v( ϕ) ∧ v(ψ) = v( ϕ) ∧ (¬v( ϕ) ∨ v(ψ)) ∈ F. Now, since Var ( ϕ) ⊆ Var (ψ), we have that the equation ( ϕ ∧ ψ) ∨ ψ ≈ ψ, which is obviously valid in every Boolean algebra qua instance of the absorption law, is regular, and therefore valid in every generalised involutive bisemilattice, by virtue of Corollary 2.4.21. So

(v( ϕ) ∧ v(ψ)) ∨ v(ψ) = v(ψ). Thus, v( ϕ) ∧ v(ψ) ≤∨ v(ψ), and since v( ϕ) ∧ v(ψ) ∈ F and F is closed upwards, we also obtain v(ψ) ∈ F, as was to be proved.  Remark 7.1.9. In any generalised involutive bisemilattice B, the set P(B) is the smallest PWK-filter. We can also provide a characterisation of the PWK-filters of B ∈ GIB in terms of its Płonka sum representation. Proposition 7.1.10. Let A ∈ GIB be the Płonka sum over the semilattice direct  system of Boolean algebras A = {Ai }i∈ I , h I, ≤i, { pij | i ≤ j} . For F ⊆ A, the following are equivalent: 1. F is a PWK-filter of A. 2. For every i ∈ I, there is a CL-filter Fi of the Boolean algebra Ai such that X = hA, { Fi }i∈ I i is an l-direct system of matrices and hA, F i is the Płonka sum over X. Proof. We first prove that (1) implies (2). If (1) holds, for i ∈ I let Fi = F ∩ Ai . Suppose ex absurdo that Fi is not a CL-filter of Ai . We distinguish three cases: either (a) the top element of Ai does not belong to Fi ; or (b) it is not upward closed w.r.t. ≤Ai , which is ≤∨ restricted to Ai ; or (c) it is not closed w.r.t. meets. Given the way operations are computed in Płonka sums, these cases respectively entail that F fails either F1, or F2, or F3 in Proposition 7.1.8. Hence F is not a PWK-filter of A, a contradiction. It remains to be shown that for every i, j ∈ I such that i ≤ j, pij [ Fi ] ⊆ Fj . However, let a ∈ Fi = F ∩ Ai . By Proposition 7.1.8 F satisfies F2, whence pij ( a) = a ∨ pij ( a) ∈ F ∩ A j = Fj . Thus, X = hA, { Fi }i∈ I i is an l-direct

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system of matrices and hA, F i is the Płonka sum over X, since clearly S F = i∈ I Fi . For the converse direction, suppose that (2) holds, and let Γ `PWK ϕ. By Theorem 1.3.2, there is ∆ ⊆ Γ such that ∆ `CL ϕ and Var (∆) ⊆ Var ( ϕ). S Let moreover v[Γ] ⊆ F = i∈ I Fi , for a given A-valuation v. If ∆ = ∅, then S ϕ is a classical theorem and thus v( ϕ) ∈ i∈ I Fi = F. Otherwise, we can assume w.l.o.g. that ∆ = {δ1 , ..., δn } and in particular v[∆] ⊆ F. For 1 ≤ k ≤ n, let iv (δk ) be the index of the fibre to which v(δk ) belongs, and let l = iv (δ1 ) ∨ ... ∨ iv (δn ). Since X = hA, { Fi }i∈ I i is an l-direct system of matrices and hA, F i is the Płonka sum over X, for every i, j ∈ I such that i ≤ j we have pij [ Fi ] ⊆ Fj , and hence v(δ1 ∧ ... ∧ δn ) ∈ Fl . But Fl is a classical filter, whereby v( ϕ) ∈ Fl ⊆ F.  We now show that the Leibniz congruence of a PWK-filter F of B ∈ GIB , when restricted to a given fibre Ac of its Płonka representation, is the Leibniz congruence (on Ac ) of the restriction Fc of F to Ac . Lemma 7.1.11. Let B be a generalised involutive bisemilattice, Ac any of its Boolean fibres, F a PWK-filter, and Fc = F ∩ Ac . Then, A2c ∩ ΩB F = ΩAc Fc . Proof. In order to prove that A2c ∩ ΩB F ⊆ ΩAc Fc , suppose that a, b ∈ Ac are such that h a, bi ∈ ΩB F. Hence, for every x ∈ B, a ∨ x ∈ F if and only if b ∨ x ∈ F, according to Proposition 7.1.5. By taking x = ¬ a, we obtain that b ∨ ¬ a ∈ F, since a ∨ ¬ a ∈ F. Analogously, we can see that a ∨ ¬b ∈ F. Now, b ∨ ¬ a, a ∨ ¬b ∈ Ac , whence b ∨ ¬ a, a ∨ ¬b ∈ Fc . This implies that h a, bi ∈ ΩAc Fc , since Fc is a filter of the Boolean algebra Ac . For the other inclusion, consider h a, bi ∈ ΩAc Fc . We will prove that h a, bi ∈ ΩB F by recourse to Proposition 7.1.5. Pick then an arbitrary element x ∈ B. Since Fc is a filter of the Boolean algebra Ac , we have that b ∨ ¬ a, a ∨ ¬b ∈ Fc ⊆ F, and therefore also b ∨ ¬ a ∨ x, a ∨ ¬b ∨ x ∈ F. If we suppose that a ∨ x ∈ F, then

( a ∧ b) ∨ x = ( a ∧ (¬ a ∨ b)) ∨ x = ( a ∨ x ) ∧ (¬ a ∨ b ∨ x ) ∈ F, and since a, b ∈ Ac , we have a ∧ b ≤∨ b, and hence ( a ∧ b) ∨ x ≤∨ b ∨ x, whence we obtain that b ∨ x ∈ F. By symmetry, if b ∨ x ∈ F, then a ∨ x ∈ F. Similarly, if ¬ a ∨ x ∈ F, then (¬ a ∧ ¬b) ∨ x ∈ F, and again, since ¬ a, ¬b ∈ Ac , then ¬ a ∧ ¬b ≤∨ ¬b, whence (¬ a ∧ ¬b) ∨ x ≤∨ ¬b ∨ x, and therefore ¬b ∨ x ∈ F. The other implication follows by symmetry.  Lemma 7.1.12. If hB, F i is a Leibniz reduced model of PWK, then B is a generalised involutive bisemilattice and F is the set P(B) of positive elements of B.

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Proof. If hB, F i is a Leibniz reduced model of PWK, then by Theorem 7.1.6, B is a generalised involutive bisemilattice and F is a PWK-filter, which implies that P(B) ⊆ F. Putting once more to good use Proposition 7.1.5, let us now prove that, whenever a ∈ F, h a, c a i ∈ ΩB F, where c a = a ∨ ¬ a. Since a ∈ F by hypothesis and c a ∈ P(B) ⊆ F, we have that for every c ∈ B, a ∨ c, c a ∨ c ∈ F, whence the first implication of (Leib) is trivial. For the second implication, notice that ¬c a ≤∨ ¬ a, whereby for every c ∈ B, ¬c a ∨ c ∈ F implies ¬ a ∨ c ∈ F. Let us now suppose that for a certain c ∈ B, ¬ a ∨ c ∈ F. Hence,

¬c a ∨ c = (¬ a ∧ a) ∨ c = (¬ a ∨ c) ∧ ( a ∨ c) ∈ F, since the former conjunct is in F by hypothesis, and the latter is in F because a ∈ F. Thus, for all a ∈ F, h a, c a i ∈ ΩB F = ∆B , since hB, F i is reduced, and therefore a = c a ∈ P(B), as we wanted to prove.  We can now finally state the result we have been after. Theorem 7.1.13. B ∈ Alg*(PWK) if and only if B is a generalised involutive bisemilattice and for every a }i ∈ Mod*(PWK). However, B2 × S2 is isomorphic to a subalgebra of EK with universe {>, ⊥, 1, 0} which is not in Alg*(PWK), since hB2 × S2 , {1, >}i is not Leibniz reduced, and {1, >} = P(B2 × S2 ).

7.1.3 Suszko reduced models Given that PWK is not protoalgebraic, it comes as no surprise that the class Alg*(PWK) is a little unwieldy (recall that it is not even a prevariety, as we saw in Example 7.1.14). We can expect to be better off with the class Alg(PWK) of algebra reducts of Suszko reduced models of PWK, which, as argued in [96, p. 279], has to be considered as the proper algebraic counterpart of the non-protoalgebraic logic PWK. Let us first settle an easier task, though. The general theory of logics of left variable inclusion provides us with a ready description of the Suszko reduced models of PWK. Theorem 7.1.15. For hA, F i a τ2 -matrix, the following are equivalent: 1. hA, F i ∈ ModSu (PWK); 2. there exists an l-direct system X of Leibniz reduced models of CL with at most one trivial fibre s.t. hA, F i ∼ = Pł ( X ).

7.1. ABSTRACT ALGEBRAIC LOGIC PROPERTIES Proof. By Theorem 5.4.8.

169 

We immediately obtain from this theorem that Alg(PWK) consists of all generalised involutive bisemilattices with at most one fix element, or equivalently, with at most one trivial fibre. This condition can be expressed by a quasiequation – indeed, as we will observe presently, Alg(PWK) is a proper quasivariety. This is a very unusual situation, as there are few natural examples in the literature of a logic L such that Alg(L) is a quasivariety and not a variety. As noticed by Font [96, p. 279], PWK is also one of the rare natural examples of logics L such that Alg*(L), Alg(L) and V (L) are all pairwise distinct. Corollary 7.1.16. Alg(PWK) is the class SGIB of generalised involutive bisemilattices satisfying the quasiequation

¬ x ≈ x & ¬y ≈ y ⇒ x ≈ y. Proof. By Theorem 7.1.15.



Observe that, unlike the results proven so far in this chapter, we cannot apply Corollary 7.1.16 to PWK0,1 , because Theorem 5.4.8 works for logics whose type contains no constants. If we aim at a description that can be applicable to both logics, we have to proceed in a more piecemeal way. The most we can make of the big picture sketched in Chapter 5 is a description of Suszko reduced models of PWK and PWK0,1 on (generalised) involutive bisemilattices. Until the end of this section, PWK will once again take centre stage, but all the results hereafter hold for PWK0,1 as well, with the obvious modifications. Theorem 7.1.17. If X is an l-direct system of Leibniz reduced models of CL, the following conditions are equivalent: 1. Pł ( X ) ∈ ModSu (PWK). 2. X contains at most one trivial fibre. Proof. By Theorem 5.4.6.



We now improve on this result with a characterisation of Alg(PWK). Theorem 7.1.18. Alg*(PWK) ⊂ Alg(PWK) ⊂ GIB .

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Proof. By Proposition 4.0.5, Alg*(PWK) ⊆ Alg(PWK), and by Theorem 4.0.8 the varieties they generate are the same, namely the intrinsic variety of PWK, which coincides with GIB by Corollary 7.1.7. Therefore Alg*(PWK) ⊆ Alg(PWK) ⊆ GIB . Now, the generalised involutive bisemilattice S2 has only one filter, which is S2 itself, and therefore ∼ Ω S 2 S2 = Ω S 2 S2 = ∇ S 2 , which is not the identity. Hence, S2 ∈ / Alg(PWK). Finally, consider the generalised involutive bisemilattice B = B2 ⊕ B4 given by the following diagram: c 1

¬a

a

¬c 0

hB, P(B)i is not Leibniz reduced, since ΩB P(B) is the congruence that identifies 1 and c, 0 and ¬c, and nothing else. Thus, B ∈ / Alg*(PWK). On the other hand, the set F = {1, ¬c, a, ¬ a, c} is a PWK-filter of B and ΩB F is the congruence identifying all the elements ¬c, a, ¬ a, c, and nothing ∼B else. The Suszko congruence ΩPWK P(B) is included in the intersection of these two congruences, which is the identity. Therefore, hB, P(B)i is Suszko reduced, whence B ∈ Alg(PWK).  This observation yields a new proof of the fact that Alg*(PWK) is not a quasivariety. Indeed, by Proposition 4.0.5, for any logic L, Alg*(L) and Alg(L) generate the same quasivariety, and were it the case that Alg*(PWK) is a quasivariety, we would get Alg*(PWK) = Alg(PWK). Finally, we also observe (as announced earlier in the section) that Alg(PWK) cannot be a variety, since it contains WK but it fails to contain S2 . To end this section, we provide an alternative proof, independent of the general theory, of the fact that Alg(PWK) is the class SGIB of generalised involutive bisemilattices with at most one fix element (see § 2.4.4). Theorem 7.1.19. Let A be an algebra of type τ2 . The following are equivalent: 1. A ∈ Alg(PWK); 2. A ∈ SGIB , i.e. A satisfies the quasiequation

¬ x ≈ x & ¬y ≈ y ⇒ x ≈ y;

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3. there exists an l-direct system X of Leibniz reduced models of CL with at most one trivial fibre s.t. hA, F i ∼ = Pł ( X ); 4. A ∈ ISP(WK). Proof. (1) ⇒ (2). This follows directly from Theorem 7.1.17, since Alg(PWK) ⊆ GIB by Theorem 7.1.18. (2) ⇔ (3). Clear. (2) ⇒ (4). We prove that every nontrivial generalised involutive bisemilattice with at most one fix element is embeddable into a power of WK. Given a semilattice direct system of Boolean algebras T = h{Ai }i∈ I , h I, ≤i, { pij | i ≤ j}i we will prove that the Płonka sum T over T is embeddable into the product ∏ I Ai∗ , where for any given Boolean algebra A, A∗ is defined according to the guidelines given in Section 2.3.2. Indeed, consider the function η : T → ∏ I Ai∗ defined as follows: for every i ∈ I, and every a ∈ Ai , η ( a) = hη ( a) j | j ∈ I i, where for every j ∈ I, ( pij ( a) if i ≤∨ j, η ( a) j = (∗) ∞ otherwise. The function η is injective, because if a, b ∈ Ai , then η ( a) = η (b) implies a = pii ( a) = η ( a)i = η (b)i = pii (b) = b; and if a ∈ Ai and b ∈ A j , with i 6= j, then either i 6≤∨ j or j 6≤∨ i, and therefore η ( a) j = ∞ 6= b = η (b) j or η (b)i = ∞ 6= a = η ( a)i . Moreover, for every i ∈ I, a ∈ Ai , and j ∈ I, we have that η (¬ a) j = ¬η ( a) j , since if i ≤∨ j, then η (¬ a) j = pij (¬ a) = ¬ pij ( a) = ¬η ( a) j , and otherwise η (¬ a) j = ∞ = ¬∞ = ¬η ( a) j . Finally, given i, j ∈ I, a ∈ Ai , b ∈ A j , and k = i ∨ j, we have that for every l ∈ I, if k ≤∨ l, η ( a ∨T b)l = η ( pik ( a) ∨Ak p jk (b))l = pkl ( pik ( a) ∨Ak p jk (b))

= pkl ( pik ( a)) ∨Al pkl ( p jk (b)) = pil ( a) ∨Al p jl (b) ∗

∗

= pil ( a) ∨Al p jl (b) = η ( a)l ∨Al η (b)l . On the other hand, if k 6≤∨ l, then i 6≤∨ l or j 6≤∨ l, or both. Let us assume i 6≤∨ l and j ≤∨ l. We would have: ∗

∗

η ( a ∨T b)l = η ( pik ( a) ∨Ak p jk (b))l = ∞ = ∞ ∨Al p jl (b) = η ( a)l ∨Al η (b)l . The other two cases are analogous. We have proved that η : T → ∏ I Ai∗ is an embedding. In case T does not have any fix element, the Boolean

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algebra Ai is nontrivial, for every i ∈ I. If T has exactly one fix element, and it is not the trivial algebra, then there is m ∈ I such that Am = 1, I has more than one element, and for every i ∈ I, i 6= m, Ai 6= 1, and hence i < m. Thus, we can prove that the function η : T → ∏ I \{m} Ai∗ defined by (∗) is again an embedding. Hence, we have proved that every nontrivial member of SGIB can be embedded in a product of “extended” Boolean algebras of the form A∗ , where A is nontrivial. Now, if A and B are Boolean algebras such that A is embeddable into B, then it is not difficult to see that A∗ is embeddable into B∗ . So, since every nontrivial Boolean algebra A is embeddable into a power of B2κ , with κ > 0, every nontrivial generalised involutive bisemilattice with at most one fix element can be embedded into a product of “extended” nontrivial powers of B2 . All that remains to prove is that for every κ > 0, the algebra (B2κ )∗ is embeddable into a power of WK. The required embedding is given by ρ : (B2κ )∗ → (WK)κ , where for every a ∈ ( B2κ )∗ , ( a if a 6= ∞, ρ( a) = ~n otherwise, where ~n is the sequence constantly equal to n. (iv) ⇒ (i). We have seen that Alg(PWK) is a quasivariety, and since WK ∈ Alg∗ (PWK) ⊆ Alg(PWK), ISP(WK) = Q(WK) ⊆ Alg(PWK). 

7.2 Hilbert-style calculi The proof theory of PWK and PWK0,1 is comparatively well-developped. There is no shortage of proof systems for these logics, ranging from natural deduction calculi to sequent and tableaux calculi. In any case, we deliberately start our survey with the most traditional syntactic presentation a logic can be given – Hilbert-style axiomatic calculi. The first calculus we introduce is a calculus for PWK (but it wouldn’t be hard to convert it into one for PWK0,1 ) with linguistic restrictions on its single inference rule. It is a rather concise system, whose chief merit is to further underscore the connection with classical logic that is at the heart of Theorem 1.3.2. There is a downside as well, which is the need to attach an unelegant variable inclusion proviso to the rule of Modus Ponens. In detail, we introduce a new logic HPWK by means of a Hilbert-style calculus and then show that HPWK exactly coincides with PWK. Throughout the rest of this section, ϕ → ψ is used as shorthand for ¬ ϕ ∨ ψ.

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173

Definition 7.2.1. HPWK =`HPWK is the derivability relation of the axiomatic system HP W K of type τ2 with the following axioms and inference rules: A1. ( ϕ ∨ ϕ) → ϕ; A2. ϕ → ( ϕ ∨ ψ); A3. ( ϕ ∨ ψ) → (ψ ∨ ϕ); A4. ( ϕ → ψ) → ((γ ∨ ϕ) → (γ ∨ ψ)); A5. ( ϕ ∧ ψ) → ¬(¬ ϕ ∨ ¬ψ); A6. ¬(¬ ϕ ∨ ¬ψ) → ( ϕ ∧ ψ); A7. ϕ ∧ ¬ ϕ → ψ; A8. ϕ → ψ ∨ ¬ψ.

( RMP)

ϕ

ϕ→ψ ψ

with the proviso that Var ( ϕ) ⊆ Var (ψ). As a matter of fact, there is nothing special about our choice of the axioms (A1)–(A8). We could have picked any other set of axioms that, together with Modus Ponens (MP), yields a complete Hilbert system for CL, with the caveat that the working language is τ2 (see [116], [121], or [122] concerning the importance of the language when choosing a certain set of axioms for a particular logic). Notice that the only difference between HPWK and CL is the proviso that constrains Modus Ponens (RMP means Restricted Modus Ponens). Because of this inferential restriction, HPWK turns out to be weaker than CL. Nevertheless, we prove in the next proposition that both logics have the same theorems. Proposition 7.2.2. For any ϕ ∈ Fmτ2 , `HPWK ϕ if and only if `CL ϕ. Proof. If `HPWK ϕ, then there is a proof D of ϕ that uses only the axioms (A1)–(A8) and RMP. Since (A1)–(A8) are classical theorems, and RMP is an instance of the usual Modus Ponens MP, D also counts as a proof of ϕ in the Hilbert system CL for CL given by (A1)–(A8) and MP. All we have to prove now is that we can transform any proof D = h ϕ1 , . . . , ϕn i of ϕ in CL into another proof that only uses RMP. We will proceed by induction on the length n of D. If n = 1, then ϕ = ϕ1 is an

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axiom, and there is nothing to prove. Let us now assume that n > 1, and that ϕ = ϕn follows from ϕi and ϕ j = ϕi → ϕ by MP, with i, j < n. By the induction hypothesis, we have proofs of ϕi and ϕ j in HP W K. Let hψ1 , . . . , ψm i be the result of concatenating these two proofs, in such a way that ψs = ϕi and ψm = ϕ j = ϕi → ϕ = ψs → ϕ. This is still a proof in HP W K. Consider a substitution σ fixing all the variables of ϕ and sending all the variables of ψ1 , . . . , ψm that are not contained among the variables of ϕ to some particular variable of ϕ . Thus, hσ (ψ1 ), . . . , σ(ψm )i is still a proof in HP W K. Now, σ(ψm ) = σ(ψs → ϕ) = σ(ψs ) → σ( ϕ) = σ (ψs ) → ϕ, since σ fixes the variables of ϕ. Moreover, Var (σ (ψs )) ⊆ Var ( ϕ), by our choice of σ. Therefore, ϕ follows by an application of RMP to σ(ψs ) and σ(ψm ), and thus hσ (ψ1 ), . . . , σ(ψm ), ϕi is a proof of ϕ in HP W K.  By Lemma 1.2.4, PWK fails Conjunctive Simplification (CS), and in particular ϕ ∧ ( ϕ ∨ ψ) 0PWK ϕ. Nonetheless, we can derive in HPWK certain linguistic restrictions of CS and a further weak form of MP. Proposition 7.2.3. The following rules are derivable in HPWK:

[ RCS1 ]

[ RCS2 ]

ϕ∧ψ ϕ

ϕ∧ψ ψ

provided that Var(ψ) ⊆ Var( ϕ),

provided that Var( ϕ) ⊆ Var(ψ),

[ ADJ ]

[W MP]

ϕ ψ , ϕ∧ψ

ϕ ∧ ( ϕ → ψ) . ϕ∧ψ

Proof. If Var (ψ) ⊆ Var ( ϕ), then, taking into account Proposition 7.2.2,

ϕ∧ψ

.. . ϕ∧ψ → ϕ ( RMP) ϕ

This shows that RCS1 is derivable, and the proof for RCS2 is similar. For ADJ, we have:

ϕ ψ

.. . ϕ → (ψ → ϕ ∧ ψ) ( RMP) ψ → ϕ∧ψ ( RMP) ϕ∧ψ

The derivability of WMP is a straightforward consequence of the fact that ( ϕ ∧ ( ϕ → ψ)) → ( ϕ ∧ ψ) is a theorem of CL and Var ( ϕ ∧ ( ϕ → ψ)) = Var ( ϕ ∧ ψ). 

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Notice that, in the presence of ADJ, WMP is indeed a weak form of MP. The consequent of WMP cannot be replaced by ψ, since this would imply an unrestricted version of CS. Actually, a kind of converse to the previous proposition is available, whence the axioms (A1)–(A8) and rules ADJ, WMP, RCS2 provide an alternate axiomatisation of HPWK. Proposition 7.2.4. The rule RMP is derivable in any Hilbert system including the rules ADJ, WMP and RCS2 . Proof. If Var ( ϕ) ⊆ Var (ψ), RMP is derivable as follows: ϕ ϕ→ψ ( ADJ ) ϕ ∧ ( ϕ → ψ) (W MP) ϕ∧ψ ( RCS2 ) ψ  We are now ready to show that HPWK coincides with PWK. Theorem 7.2.5. HPWK = PWK. Proof. It is easy to check, by direct inspection, that for every axiom ϕ of HPWK and for every valuation v, v( ϕ) ∈ {1, n}. Moreover, if Var ( ϕ) ⊆ Var (ψ) and v is a valuation such that v(ψ) = 0, then v( x ) 6= n for every x ∈ Var (ψ), and hence for every x ∈ Var ( ϕ). Therefore, v( ϕ) ∈ {0, 1}. Thus, if v( ϕ) = 1, then v( ϕ → ψ) = 0. This proves that the rule RMP is sound with respect to `PWK . Therefore, Σ `HPWK ϕ implies Σ `PWK ϕ. For the other direction, suppose that Σ `PWK ϕ. By Theorem 1.3.2, there is a finite or empty subset ∆ ⊆ Σ such that ∆ `CL ϕ and Var (∆) ⊆ Var ( ϕ). If ∆ = ∅, then `HPWK ϕ, by virtue of Proposition 7.2.2, whereby we get Σ `HPWK ϕ. Otherwise, let ∆ = { ϕ1 , . . . , ϕn }. By the Deduction Theorem for CL, `CL ϕ1 → ( ϕ2 → (· · · → ( ϕn → ϕ) · · · )). Thus, `HPWK ϕ1 → ( ϕ2 → (· · · → ( ϕn → ϕ) · · · )), by Proposition 7.2.2, and since Var ( ϕ1 , . . . , ϕn ) ⊆ Var ( ϕ), by several applications of RMP we obtain { ϕ1 , . . . , ϕn } `HPWK ϕ, so Σ `HPWK ϕ.  We can also obtain a Hilbert system with no linguistic restriction on inference rules from the general axiomatisation of left variable inclusion logics in Theorem 5.2.2. Since this particular application presupposes the general theory of left variable inclusion logics, it cannot be adapted to PWK0,1 , unlike the other axiomatisation.

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Definition 7.2.6. HPWK∗ is the derivability relation of the axiomatic system HP W K ∗ of type τ2 that comprises axioms A1-A8 in Definition 7.2.1 and the following inference rules (x · y is short for x ∧ ( x ∨ y)):

( R ∗ 1)

ϕ ϕ→ψ ψ · ( ϕ · ( ϕ → ψ))

( R ∗ 2) ( R∗)

χ(e, ~z ) χ(δ, ~z )

ϕ ϕ·ψ χ(δ, ~z ) χ(e, ~z )

for every identity e ≈ δ in the definition of a partition function (Definition 2.2.1). Theorem 7.2.7. HPWK∗ = PWK. Proof. By Theorem 5.2.2.



7.3 Sequent calculi Hilbert-style calculi are sometimes convenient for metatheoretical enquiries, but far from optimal for proof search – and, more generally, for working within the system. Proof theory practitioners who investigate nonclassical logics usually favour Gentzen-type sequent systems [155, 125], or generalisations thereof, which are friendlier under these (and other) respects. There are essentially two ways to obtain a sequent calculus for PWK out of a corresponding calculus for CL. The first approach, followed by Coniglio and Corbal´an [64], proceeds along the lines of the characterisation result in Theorem 1.3.2, namely by placing appropriate variable inclusion restrictions on some of the classical operational rules for logical connectives (Subsection 7.3.1). The ultimate goal is to obtain a calculus S such that, for every Γ ∪ { ϕ} ⊆ Fmτ2 , Γ finite, we have that Γ `PWK ϕ iff `S Γ ⇒ ϕ – namely, a calculus whose internal consequence relation (in the sense of [7]) coincides with PWK. It can be argued, though, that such restricted rules, rather than shedding light on the operational meaning of the PWK connectives, merely give a formal clothing to a rule of thumb “Just apply classical logic so long as this does not conflict with the prescribed variable-inclusion requirement”. The alternative approach, originally developped in [163] and related in Subsection 7.3.2, circumvents this problem by focussing on a different

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goal: that of finding a calculus S such that, for every Γ ∪ { ϕ} ⊆ Fmτ2 , we have that Γ `PWK ϕ iff ⇒ ϕ is provable in the calculus S supplemented by the additional axioms {⇒ γ | γ ∈ Γ} – namely, a calculus whose external consequence relation coincides with PWK. This much is possible by stripping away the rule of Cut from the classical calculus and restoring some of its deductive power with (left and right) elimination rules for the connectives. We end these introductory remarks with some definitions that will be repeatedly used in this section. Definition 7.3.1. 1. A τ2 -sequent is an ordered pair (Γ, ∆) of finite, possibly empty subsets of Fmτ2 , written Γ ⇒ ∆ for ease of notation. 2. Seq (τ2 ) is the set of all τ2 -sequents. The usual notational and terminological conventions for sequent calculi [155, 125] will be adopted with no particular comment. We take the liberty of borrowing Gentzen’s original label LK for the calculus we present hereafter, with the disclaimer, though, that it does not coincide with the propositional part of Gentzen’s system either in the choice of primitives (Gentzen included implication in addition to the connectives of τ2 ) or in the choice of rules. Gentzen postulated exchange and contraction rules which are here redundant because our sequents, unlike Gentzen’s, are ordered pairs of sets of formulas; moreover, for reasons that will become apparent in the sequel, we choose the operational rules of the connectives – among various equivalent alternatives we have in LK – in such a way that they are invertible. Definition 7.3.2. The sequent calculus LK is the calculus of type τ2 whose axioms and rules are as follows:

( Id)

φ⇒φ

(Cut)

Γ ⇒ ∆, φ φ, Γ ⇒ ∆ Γ⇒∆

(W L )

Γ⇒∆ φ, Γ ⇒ ∆

(WR)

Γ⇒∆ Γ ⇒ ∆, φ

(¬ L)

Γ ⇒ ∆, φ ¬φ, Γ ⇒ ∆

(¬ R)

φ, Γ ⇒ ∆ Γ ⇒ ∆, ¬φ

(∧ L)

φ, ψ, Γ ⇒ ∆ φ ∧ ψ, Γ ⇒ ∆

(∧ R)

Γ ⇒ ∆, φ Γ ⇒ ∆, ψ Γ ⇒ ∆, φ ∧ ψ

(∨ L)

φ, Γ ⇒ ∆ ψ, Γ ⇒ ∆ φ ∨ ψ, Γ ⇒ ∆

(∨ R)

Γ ⇒ ∆, φ, ψ Γ ⇒ ∆, φ ∨ ψ

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Recall that an LK-derivation D of the sequent S from X ⊆ Seq(τ2 ) is a finite tree labelled by members of Seq(τ2 ) such that: 1. S labels the root; 2. For each node labelled S0 , either S0 ∈ X or its child nodes are labelled S1 , ..., Sn and ({S1 , ..., Sn } , S0 ) is an instance of a rule of LK. If there is a LK-derivation D of S from X, then S is said to be LKderivable from X, written X `LK S. S is a theorem of LK iff ∅ `LK S (also written `LK S). Observe that X `LK Γ ⇒ ∆ iff Γ ⇒ ∆ is provable in the calculus obtained from LK by taking the members of X as additional axioms. Analogous definitions of a derivation and of a theorem, mutatis mutandis, apply to all the other sequent calculi hereafter defined. Definition 7.3.3. The sequent calculus LK − is the calculus obtained by removing from LK the rule Cut. Theorem 7.3.4. 1. For every finite subset Γ ∪ { ϕ} of Fmτ2 , we have that `LK Γ ⇒ ϕ iff {⇒ γ | γ ∈ Γ} `LK ⇒ ϕ iff Γ `CL ϕ. 2. (Admissibility of Cut in LK: [103]) For every finite or empty subsets Γ, ∆ of Fmτ2 , `LK Γ ⇒ ∆ iff `LK− Γ ⇒ ∆. The property `LK Γ ⇒ ϕ iff {⇒ γ | γ ∈ Γ} `LK ⇒ ϕ (Item 1 in Theorem 7.3.4), namely, the coincidence between the internal and the external consequence relation of the calculus, is a distinctive property that LK shares with relatively few other systems. In general, the internal consequence relation of a calculus need not be a consequence relation at all – whether because the antecedent of a sequent is not a set of formulas but a multiset, or list, or else because the calculus lacks the structural rules required to ensure reflexivity, monotonicity and transitivity of the relation [75]. Even when it is, though, it need not coincide with the external relation, as we will have occasion to see later in this section. Moreover, although by Item 2 in Theorem 7.3.4 we have that `LK Γ ⇒ ∆ iff `LK− Γ ⇒ ∆, it is crucial to notice that LK − is a different calculus than LK and their respective derivability relations are likewise different. For example, x ⇒ y, y ⇒ z `LK x ⇒ z, while x ⇒ y, y ⇒ z 0LK− x ⇒ z, for distinct propositional variables x, y, z. These two calculi just happen to have the same theorems.

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7.3.1 Systems with linguistic restrictions Marcelo Coniglio and In´es Corbal´an [64] obtain a calculus for PWK by curbing applications of the rules ¬ L and ∧ L in LK via a variable inclusion strainer. In detail, their suggestion is presented in the next definition. Definition 7.3.5. The sequent calculus LK cc is the calculus obtained from LK by replacing ¬ L and ∧ L with the following rules:

(¬ L H )

Γ ⇒ ∆, φ provided that Var (φ) ⊆ Var (∆); ¬φ, Γ ⇒ ∆

(∧ L H )

φ, ψ, Γ ⇒ ∆ provided that Var (φ) ∪ Var (ψ) ⊆ Var (∆). φ ∧ ψ, Γ ⇒ ∆

We now prove that the calculus thus defined is sound with respect to PWK. Theorem 7.3.6. For every finite subset Γ ∪ { ϕ} of Fmτ2 , we have that `LKcc Γ ⇒ ϕ implies Γ `PWK ϕ. Proof. We will establish a more general claim, to wit, that for every finite Γ, ∆ ⊆ Fmτ2 , we have that if `LKcc Γ ⇒ ∆, then every valuation v is such that either v(γ) = 0 for some γ ∈ Γ or v(δ) ∈ {1, n} for some δ ∈ ∆. This much is verified via a standard induction on the length of the assumed derivation of Γ ⇒ ∆ in LK cc . The base step is easy. For every ϕ ∈ Fmτ2 and for every valuation v, either v( ϕ) = 0 or v( ϕ) ∈ {1, n}, so every instance of Id complies with the claim. For the inductive step, we confine ourselves to the rule ¬ L H . Thus, suppose Var ( ϕ) ⊆ Var (∆), and that v is such that either v(γ) = 0 for some γ ∈ Γ or v(δ) ∈ {1, n} for some δ ∈ ∆ ∪ { ϕ}. Assume further that v[Γ ∪ {¬ ϕ}] ⊆ {1, n}. Then either v(δ) ∈ {1, n} for some δ ∈ ∆, or v( ϕ) ∈ {1, n}. If the former, we are finished. If the latter, since v(¬ ϕ) ∈ {1, n}, v( ϕ) ∈ {0, n}. If v( ϕ) = 0 we are done. If v( ϕ) = n, then there is x ∈ Var ( ϕ) ⊆ Var (∆) such that v( x ) = n. Thus, there is δ ∈ ∆ such that v(δ) = n, and we are done in this case as well.  For completeness, we first need a lemma. Lemma 7.3.7. For every finite subset Γ ∪ ∆ of Fmτ2 , if `LK Γ ⇒ ∆ and Var (Γ) ⊆ Var (∆), then `LKcc Γ ⇒ ∆.

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Proof. Assume that Var (Γ) ⊆ Var (∆) and that D is a proof of Γ ⇒ ∆ in LK; we can assume that D is Cut-free by Theorem 7.3.4. D may or may not be a proof in LK cc as well. If it is, we are done. If it is not, somewhere down the line we must have applied either ¬ L or ∧ L disrespecting the linguistic provisos. To exemplify, suppose there are consecutive nodes in D respectively labelled by Π ⇒ Σ, ϕ and by ¬ ϕ, Π ⇒ Σ, with Var ( ϕ) not included in Var (Σ). D is Cut-free, so all the variables contained in the sequents occurring in D also occur in Γ ∪ ∆ – hence, by assumption, in ∆. Now, turn D into a labelled tree E as follows: the underlying trees of D and E are the same, but if Λ ⇒ Θ labels a node in D , the corresponding node in E is labelled by Λ ⇒ Θ, ∆. Next, we prolong each branch of E , terminating with a leaf whose label has the form ϕ ⇒ ϕ, ∆, by a chain of applications of WR, ending in a new leaf labelled by ϕ ⇒ ϕ. The resulting tree F is clearly a proof of Γ ⇒ ∆ in LK. Now, backtrack to the nodes in F that correspond to the nodes labelled by Π ⇒ Σ, ϕ and by ¬ ϕ, Π ⇒ Σ in D . Their labels are Π ⇒ Σ, ∆, ϕ and ¬ ϕ, Π ⇒ Σ, ∆ respectively, so we have now an instance of ¬ L H . Similar considerations apply to all the other applications of rules in D that do not comply with the restrictions. Hence, F is a Cut-free proof of Γ ⇒ ∆ in LK cc .  Theorem 7.3.8. For every finite subset Γ ∪ { ϕ} of Fmτ2 , we have that Γ `PWK ϕ implies `LKcc Γ ⇒ ϕ. Proof. Assume that Γ `PWK ϕ. Then, by Theorem 1.3.2 and Theorem 7.3.4, there is Π ⊆ Γ such that Var (Π) ⊆ Var ( ϕ) and `LK− Π ⇒ ϕ. Using Lemma 7.3.7, we conclude that `LKcc Π ⇒ ϕ (and the Cut rule is not needed in the proof). The required Cut-free proof in LK cc of Γ ⇒ ϕ is then obtained by appling W L as many times as needed.  Remark 7.3.9. It is important to observe that the Cut rule is not used in the proof of either Lemma 7.3.7 or Theorem 7.3.8. Corollary 7.3.10. The Cut rule is admissible in LK cc . We observe that the paper [64] contains calculi with syntactical restrictions not only for PWK, but also for B3 . Coniglio and Corbal´an’s paper has been subsequently generalised by other authors to a number of logics of left and right variable inclusion. To date, one can encounter sequent calculi with variable inclusion restrictions for dSFDE = K3 l and ∗ ∗ SFDE = LPr [217], for dSFDE = BDl and SFDE = BDr [58], for HYB2 = B3 l r and HYB1 = PWK [60], [59], as well as 4-sided calculi for dSFDE and SFDE [69].

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7.3.2 Systems without linguistic restrictions AS an alternative to LK cc , we can resort to an approach that harks back to some classical work in structural proof theory from the 1970’s. Somewhat surprisingly, Jean-Yves Girard (see e.g. the account in [106]) showed that the classical sequent calculus without Cut LK − can be given a 3valued semantics based on the Strong Kleene tables. The intuition behind this result is that, in the absence of Cut, occurrences of a formula on the left-hand side of a sequent do not have the same deductive strength as occurrences of the same formula on the right-hand side, but are actually stronger (dually, in the absence of Id, occurrences on the left-hand side would be weaker: see [119]). Girard shows that a sequent Γ ⇒ ∆ is provable in LK − iff every SK-valuation v S-satisfies Γ ⇒ ∆, i.e. there is δ ∈ ∆ s.t. either v(δ) = 1 or v(δ) = n whenever for all γ ∈ Γ, v(γ) = 1. Moreover, all the rules of LK − (crucially, unlike Cut) preserve S-satisfaction. The main results by Girard were independently obtained by Cobreros, Egr´e, Ripley, and van Rooij in a series of papers (see e.g. [63]). The same authors suggest an attractive philosophical application for LK − , which, if enriched with rules for quantifiers and for an unrestricted disquotational truth predicate, allows for a reconstruction of naive truth theory while at the same time blocking the derivation of the paradoxes. The bottom line, in the authors’ interpretation, is that classical logic is after all compatible with disquotational truth. Some reservations on this plan are expressed in the paper [75]. Leaving aside some doubts that may be raised on the classicality of LK − as opposed to full LK, we just mention here that LK − is only weakly complete for Girard’s semantics. In order to obtain completeness with respect to derivations from arbitrary assumptions (and not only with respect to sequent proofs, i.e. derivations from axioms), one must add to LK − the inverses of the logical rules for negation, conjunction and disjunction1 , obtaining thereby a new calculus labelled LK − S . As a corollary, E it follows that the external relation of this calculus, defined by Γ `LK − ϕ S

iff {⇒ γ | γ ∈ Γ} `LK− ⇒ ϕ, coincides with the consequence relation of S

Priest’s LP, a result that had been proved independently by Pynko [195] and by Barrio et al. [13] (see also [184, p. 78]). However, there is nothing in Girard’s 3-valued semantics that constrains its application to the strong Kleene tables. We will immediately see 1 Recall that the inverse of the one-premiss schematic rule r = ({ S } , S0 ) is the rule r 0 = ({S0 } , S) obtained from r by swapping premiss and conclusion. A two-premiss rule ({S1 , S2 } , S) has two inverses, ({S} , S1 ) and ({S} , S2 ), respectively.

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that all the relevant definitions can be recast in terms of WK-valuations. In particular, given a τ2 -sequent Γ ⇒ ∆, we can stipulate that it is satisfied by one such valuation v in case v (δ) ∈ {1, n} for some δ ∈ ∆ whenever v (γ) = 1 for all γ ∈ Γ. We also require that Γ ⇒ ∆ follows from the premiss-sequents in the set X if and only if it is satisfied (in the above sense) by all valuations that satisfy each member of X. The guiding idea is to tweak the calculus LK − S into a calculus that is complete with respect to this 3-valued semantics on the weak Kleene tables, in such a way as to obtain PWK as its external relation. This task is not as mechanical as it might seem. On the one hand, some of the rules of LK − S (e.g. both inverses of the rules ∧ R) become unsound once we trade the strong semantics for the weak one; on the other hand, the calculus LK − S must be supplemented by some extra rules, because e.g. ϕ, ¬ ϕ ∨ ψ 0LP ϕ ∧ ψ E (hence ϕ, ¬ ϕ ∨ ψ 0LK − ϕ ∧ ψ), while ϕ, ¬ ϕ ∨ ψ `PWK ϕ ∧ ψ. S

The next definition can be found, essentially, in Correia [65] and Szmuc and Ferguson [219]. Ferguson [87] observes that the rules ∧ R1−2 and ∨ L1−2 in Definition 7.3.12 below are sound w.r.t. this semantics and conjectures that completeness can be attained by adding them to LK − . As we will see, a few more rules must be added to prove this conjecture. We now proceed to laying down the formal definitions.

Definition 7.3.11. 1. A valuation v W-satisfies a τ2 -sequent Γ ⇒ ∆ (in symbols, v W Γ ⇒ ∆) iff either there is γ ∈ Γ s.t. v (γ) ∈ {0, n} or there is δ ∈ ∆ s.t. v (δ) ∈ {1, n}.

2. A τ2 -sequent Γ ⇒ ∆ is W-valid (in symbols, W Γ ⇒ ∆) if v W Γ ⇒ ∆ for all valuations v.

3. A τ2 -sequent Γ ⇒ ∆ is a W-consequence of a set of sequents X (in symbols, X W Γ ⇒ ∆) if every valuation that W-satisfies all members of X also satisfies Γ ⇒ ∆.

− Definition 7.3.12. LKW is the sequent calculus of type τ2 with the following set of rules:

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( Id)

183

φ⇒φ

(W L )

Γ⇒∆ φ, Γ ⇒ ∆

(WR)

Γ⇒∆ Γ ⇒ ∆, φ

(¬ L)

Γ ⇒ ∆, φ ¬φ, Γ ⇒ ∆

(¬ R)

φ, Γ ⇒ ∆ Γ ⇒ ∆, ¬φ

(¬ L ∂ )

Γ ⇒ ∆, ¬φ φ, Γ ⇒ ∆

(¬ R ∂ )

¬φ, Γ ⇒ ∆ Γ ⇒ ∆, φ

(∧ L)

φ, ψ, Γ ⇒ ∆ φ ∧ ψ, Γ ⇒ ∆

(∧ R1 )

Γ ⇒ ∆, φ φ, Γ ⇒ ∆, ψ Γ ⇒ ∆, φ ∧ ψ

(∧ R2 )

Γ ⇒ ∆, φ φ, Γ ⇒ ∆, ψ Γ ⇒ ∆, ψ ∧ φ

(∧ L ∂ )

φ ∧ ψ, Γ ⇒ ∆ φ, ψ, Γ ⇒ ∆

(∧ R1∂ )

Γ ⇒ ∆, φ ∧ ψ Γ ⇒ ∆, φ, ψ

(∧ R2∂ )

Γ ⇒ ∆, φ ∧ ψ φ, Γ ⇒ ∆, ψ

(∧ R3∂ )

Γ ⇒ ∆, φ ∧ ψ ψ, Γ ⇒ ∆, φ

(∨ L1 )

φ, Γ ⇒ ∆, ψ ψ, Γ ⇒ ∆ φ ∨ ψ, Γ ⇒ ∆

(∨ L2 )

φ, Γ ⇒ ∆, ψ ψ, Γ ⇒ ∆ ψ ∨ φ, Γ ⇒ ∆

(∨ R)

Γ ⇒ ∆, φ, ψ Γ ⇒ ∆, φ ∨ ψ

(∨ L1∂ )

φ ∨ ψ, Γ ⇒ ∆ φ, ψ, Γ ⇒ ∆

(∨ L2∂ )

φ ∨ ψ, Γ ⇒ ∆ φ, Γ ⇒ ∆, ψ

(∨ L3∂ )

φ ∨ ψ, Γ ⇒ ∆ ψ, Γ ⇒ ∆, φ

(∨ R ∂ )

Γ ⇒ ∆, φ ∨ ψ Γ ⇒ ∆, φ, ψ

− Observe that LKW is fairly atypical as a sequent calculus due to the presence of left and right elimination rules in addition to the customary left and right introduction rules. In the absence of a Cut rule, elimination rules allow one to decompose formulas in derivations from arbitrary assumptions, which would otherwise be impossible. Rules ∧ R1−2 , beside their primary role as right introduction rules for conjunction, also encode part of the deductive power of Cut – notice that cutting the principal formula is harmless here, as it does not interfere with the subformula property.

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− We now list a few easy facts about LKW and its derivability relation.

Lemma 7.3.13. 1. The LK right introduction rule for conjunction and left introduction rule for disjunction: Γ ⇒ ∆, φ Γ ⇒ ∆, ψ Γ ⇒ ∆, φ ∧ ψ φ, Γ ⇒ ∆ ψ, Γ ⇒ ∆ (∨ L) φ ∨ ψ, Γ ⇒ ∆

(∧ R)

− are derivable in LKW .

2. `LK− ⊂`LK− ⊂ `LK . W

3. For any S ∈ Seq (τ2 ), `LK− S iff `LK S. W

− 4. Cut is admissible in LKW .

5. ∧ R1−2 are jointly equivalent to the three-premiss invertible rule

(∧ R3 )

Γ ⇒ ∆, φ, ψ

φ, Γ ⇒ ∆, ψ Γ ⇒ ∆, φ ∧ ψ

ψ, Γ ⇒ ∆, φ

Proof. (1) Given the premisses Γ ⇒ ∆, ϕ and Γ ⇒ ∆, ψ, apply W L to the latter to get ϕ, Γ ⇒ ∆, ψ; an application of ∧ R1 delivers the conclusion Γ ⇒ ∆, ϕ ∧ ψ. For the other rule we argue analogously. (2) By item (1), `LK− ⊆`LK− . For the remaining inclusion, it is not W

− hard to check that all the rules of LKW are derivable in LK. For example, consider ∧ R1 . Assume in LK the premisses Γ ⇒ ∆, ϕ and ϕ, Γ ⇒ ∆, ψ and derive via WR and Cut Γ ⇒ ∆, ψ. From this sequent and Γ ⇒ ∆, ϕ, an application of ∧ R yields Γ ⇒ ∆, ϕ ∧ ψ. Therefore `LK− ⊆ `LK . MoreW over, these inclusions are strict. In fact, given three distinct propositional variables x, y, z, we have that x ⇒ y, y ⇒ z `LK x ⇒ z, while x ⇒ y, y ⇒ z 0LK− x ⇒ z; on the other, ⇒ x ∨ y `LK− ⇒ x, y, while W W ⇒ x ∨ y 0LK− ⇒ x, y. Observe that we are presupposing Theorem 7.3.14 − below in our non-derivability claim for LKW , while the one for LK − can be verified by inspection of the rules. − (3) All the rules of LKW are classically sound, whence `LK− S only if W `LK S. The converse direction follows by item (2). In fact, suppose that `LK S; then `LK− S (by Theorem 7.3.4) and therefore `LK− S. W (4) From item (3).

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(5) Suppose ∧ R3 is given and assume the premisses Γ ⇒ ∆, ϕ and ϕ, Γ ⇒ ∆, ψ of ∧ R1 . Then Γ ⇒ ∆, φ ψ, Γ ⇒ ∆, φ

φ, Γ ⇒ ∆, ψ Γ ⇒ ∆, φ ∧ ψ

Γ ⇒ ∆, φ Γ ⇒ ∆, φ, ψ .

The conclusion of ∧ R2 is obtained similarly. For the other direction, suppose that ∧ R1 and ∧ R2 are given and assume the premisses (i) Γ ⇒ ∆, ϕ, ψ, (ii) ϕ, Γ ⇒ ∆, ψ and (iii) ψ, Γ ⇒ ∆, ϕ of ∧ R3 . Then by ∧ R1 we obtain that: Γ ⇒ ∆, φ, ψ φ, Γ ⇒ ∆, ψ . (∧ R1 ) Γ ⇒ ∆, φ ∧ ψ, ψ Applying ∧ R2 we have that:

(∧ R2 )

Γ ⇒ ∆, φ, ψ ψ, Γ ⇒ ∆, φ . Γ ⇒ ∆, φ ∧ ψ, φ

A final application of the derivable rule ∧ R yields:

(∧ R)

Γ ⇒ ∆, φ ∧ ψ, ψ Γ ⇒ ∆, φ ∧ ψ, φ . Γ ⇒ ∆, φ ∧ ψ 

− Observe that, as a consequence of Lemma 7.3.13.(4), the calculus LKW can be viewed as the result of adding to its fragment consisting of its structural rules and of its left and right introduction rules, all the inverses − of such rules, in the same way as LK − S is obtained out of LK . − We first prove the soundness of LKW with respect to our 3-valued semantics.

Theorem 7.3.14. Let {Πi ⇒ Λi }i≤n ∪ {Γ ⇒ ∆} ⊆ Seq (τ2 ). Then

{Πi ⇒ Λi }i≤n `LK− Γ ⇒ ∆ implies {Πi ⇒ Λi }i≤n W Γ ⇒ ∆. W

− Proof. By induction on the length of the LKW -derivation of Γ ⇒ ∆ from {Πi ⇒ Λi }i≤n . It is enough to check that any valuation W-satisfies the − axiom of LKW and that all rules preserve W-satisfaction by a given valuation. The weakening rules W L, WR are trivial and the disjunction rules are left to the reader. Let us check the remaining rules. (Id) Given any valuation v and any ϕ ∈ Fmτ2 , it is not possible that both v ( ϕ) = 1 and v ( ϕ) = 0.

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(¬ L) Suppose that v W Γ ⇒ ∆, ϕ and, by contradiction, that v [{¬ ϕ} ∪ Γ] = 1, v [∆] = 0. Then v ( ϕ) = 0, whence v [Γ] = 1, v [∆ ∪ { ϕ}] = 0, a contradiction. (¬ R) Similar. (¬ R∂ ) Suppose that v W ¬ ϕ, Γ ⇒ ∆ and that v [Γ] = 1, v [∆ ∪ { ϕ}] = 0. Then v (¬ ϕ) = 1, whence v [{¬ ϕ} ∪ Γ] = 1, v [∆] = 0, a contradiction. (¬ L∂ ) Similar. (∧ L) Suppose that v W ϕ, ψ, Γ ⇒ ∆ and that v [{ ϕ ∧ ψ} ∪ Γ] = 1, v [∆] = 0. Then v ( ϕ) = v (ψ) = 1, whence v [{ ϕ} ∪ {ψ} ∪ Γ] = 1, v [∆] = 0, a contradiction. (∧ R1−2 ) Suppose that v W Γ ⇒ ∆, ϕ, that v W ϕ, Γ ⇒ ∆, ψ and that v [Γ] = 1, v [∆ ∪ { ϕ ∧ ψ}] = 0. Then either v ( ϕ) = 0, or (v ( ϕ) = 1 and v (ψ) = 0). If the former, then v [Γ] = 1, v [∆ ∪ { ϕ}] = 0, a contradiction. If the latter, then v [{ ϕ} ∪ Γ] = 1, v [∆ ∪ {ψ}] = 0, also a contradiction. The rule ∧ R2 is checked similarly. (∧ L∂ ) Suppose that v W ϕ ∧ ψ, Γ ⇒ ∆ and that v [{ ϕ} ∪ {ψ} ∪ Γ] = 1, v [∆] = 0. Then v ( ϕ ∧ ψ) = 1, whence v [{ ϕ ∧ ψ} ∪ Γ] = 1, v [∆] = 0, a contradiction. (∧ R1∂−3 ) Suppose throughout that v W Γ ⇒ ∆, ϕ ∧ ψ. If v [Γ] = 1, v [∆ ∪ { ϕ} ∪ {ψ}] = 0, then v [Γ] = 1, v [∆ ∪ { ϕ ∧ ψ}] = 0, a contradiction. The very same contradiction obtains if v [Γ ∪ {ψ}] = 1, v [∆ ∪ { ϕ}] = 0 or if v [Γ ∪ { ϕ}] = 1, v [∆ ∪ {ψ}] = 0.  Corollary 7.3.15. For any Γ ∪ { ϕ} ⊆ Fmτ2 , {⇒ γ | γ ∈ Γ} `LK− ⇒ ϕ imW plies Γ `PWK ϕ. Proof. By Theorem 7.3.14, {⇒ γ | γ ∈ Γ} `LK− ⇒ ϕ implies {⇒ γ | γ ∈ Γ} W ⇒ W

ϕ, a condition which obtains exactly in case Γ `PWK ϕ.



As usual, completeness is a bit trickier to establish. Actually, we will not prove the converse of Theorem 7.3.14 in full generality, but a weaker condition which is enough to attain the converse of Corollary 7.3.15. For a start, we prove a number of lemmas whose upshot is a sort of “hourglass − − lemma” for LKW : every sequent Γ ⇒ ∆ can be derived from itself in LKW by first decomposing it into atomic sequents whose formulas are exactly the propositional variables occurring in Γ ∪ ∆, and then reassembling Γ ⇒ ∆ out of such atomic sequents. Lemma 7.3.16. Let Γ ⇒ ∆ ∈ Seq (τ2 ) be such that ϕ( x1 , . . . , xn ) ∈ Γ ∪ ∆. − Then there exists a derivation D of Γ ⇒ ∆ from itself in LKW such that:

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1. if ϕ( x1 , . . . , xn ) ∈ ∆, then for every branch Bi ∈ D there exists a node ni ∈ Bi labelled by a sequent of the form Γ, Vi1 ⇒ Vi2 , ∆ r { ϕ( x1 , . . . , xn )} such that Vi1 ∪ Vi2 = { x1 . . . xn }; 2. if ϕ( x1 , . . . , xn ) ∈ Γ, then for every branch Bi ∈ D there exists a node ni ∈ Bi labelled by a sequent of the form Γ r { ϕ( x1 , . . . , xn )}, Vi1 ⇒ Vi2 , ∆ such that Vi1 ∪ Vi2 = { x1 . . . xn }; 3. each branch Bi terminates with leaves labelled by Γ ⇒ ∆. Proof. We prove the lemma by induction on the complexity of ϕ( x1 , . . . , xn ). We only consider the case ϕ( x1 , . . . , xn ) ∈ ∆ (case (1)), as the other case is symmetric. Let Σ = ∆ r { ϕ( x1 , . . . , xn )}. Base. If ϕ( x1 . . . xn ) = x the statement is obviously true. Let ϕ( x1 . . . xn ) = ¬ x. Γ ⇒ ¬ x, Σ Γ, x ⇒ Σ Γ ⇒ ¬ x, Σ The middle node is what we need to prove claim (1) for the single branch in this tree, and, upon recalling that ∆ = Σ ∪ { ϕ( x1 , . . . , xn )}, the derivation complies with claim (3). Let now ϕ( x1 , . . . , xn ) = x ∧ y. The following derivation, where “contraction” steps are made explicit for the sake of clarity, establishes our claim. Γ ⇒ x ∧ y, Σ Γ ⇒ x ∧ y, Σ Γ ⇒ x ∧ y, Σ Γ ⇒ x ∧ y, Σ Γ, x ⇒ y, Σ Γ ⇒ x, y, Σ Γ, y ⇒ x, Σ Γ ⇒ x, y, Σ Γ ⇒ x ∧ y, y, Σ Γ ⇒ x ∧ y, x, Σ Γ, Γ ⇒ x ∧ y, x ∧ y, Σ, Σ Γ ⇒ x ∧ y, Σ The case ϕ( x1 , . . . , xn ) = x ∨ y is left to the reader. Inductive step. We confine ourselves to the case ϕ( x1 , . . . , xn ) = α(~x ) ∧ β(~y). Consider the following derivation of Γ ⇒ α(~x ) ∧ β(~y), Σ from itself: B1 B4 B3 B2 Γ ⇒ α(~x ) ∧ β(~y), Σ Γ ⇒ α(~x ) ∧ β(~y), Σ Γ ⇒ α(~x ) ∧ β(~y), Σ Γ ⇒ α(~x ) ∧ β(~y), Σ Γ, α(~x ) ⇒ β(~y), Σ Γ ⇒ α(~x ), β(~y), Σ Γ, β(~y) ⇒ α(~x ), Σ Γ ⇒ α(~x ), β(~y), Σ Γ ⇒ α(~x ) ∧ β(~y), β(~y), Σ Γ ⇒ α(~x ), α(~x ) ∧ β(~y), Σ Γ, Γ ⇒ α(~x ) ∧ β(~y), α(~x ) ∧ β(~y), Σ, Σ Γ ⇒ α(~x ) ∧ β(~y), Σ

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Now, let us consider the branch B1 and the sequent Γ, α(~x ) ⇒ β(~y), Σ in that branch. Since β(~y) has a strictly lower complexity than α(~x ) ∧ β(~y), by inductive hypothesis there exists a derivation D1 of Γ, α(~x ) ⇒ β(~y), Σ from itself, fulfilling conditions (1)-(3). That is, D1 has the following structure:

Bk1 B11 Γ, α(~x ) ⇒ β(~y), Σ Γ, α(~x ) ⇒ β(~y), Σ .. .. . . 1 2 Γ, α(~x ), Vk1 ⇒ Vk2 , Σ Γ, α(~x ), V1 ⇒ V1 , Σ .. .. . ··· . Γ, α(~x ) ⇒ β(~y), Σ where Vi1 ∪ Vi2 = {~y} for 1 ≤ i ≤ k. On the other hand, since α(~x ) has complexity strictly less than the complexity of α(~x ) ∧ β(~y), there exists a derivation D11 of Γ, α(~x ), V11 ⇒ V12 , Σ from itself verifying conditions (1)-(3): Γ, α(~x ), V11 ⇒ V12 , Σ Γ, α(~x ), V11 ⇒ V12 , Σ .. .. . . Γ, V11 , W11 ⇒ V12 , W12 , Σ Γ, V11 , Wr1 ⇒ Wr2 , V12 , Σ .. .. . ··· . Γ, α(~x ), V11 ⇒ V12 , Σ where Wi1 ∪ Wi2 = {~x } for 1 ≤ i ≤ r. Now, it is possible to combine these two proofs (replacing part of the branch B1 ) and to get the following derivation Γ, α(~x ) ⇒ β(~y), Σ .. . Γ, α(~x ), V11 ⇒ V12 , Σ ... 1 1 Γ, V1 , W1 ⇒ W12 , V12 , Σ

···

Γ, α(~x ) ⇒ β(~y), Σ .. . Γ, α(~x ), V11 ⇒ V12 , Σ ... 1 1 Γ, V1 , Wr ⇒ Wr2 , V12 , Σ

Γ, α(~x ), V11 ⇒ V12 , Σ .. .

··· Γ, α(~x ) ⇒ β(~y), Σ

Bk1 Γ, α(~x ) ⇒ β(~y), Σ .. . Γ, α(~x ), Vk1 ⇒ Vk2 , Σ .. .

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with the property that (Wj1 ∪ V11 ) ∪ (Wj2 ∪ V12 ) = {~x, ~y} for 1 ≤ j ≤ r. By repeating this procedure for every branch, it is possible to construct a deduction enjoying the desired characteristics. This concludes the proof.  Theorem 7.3.17. Let Γ ⇒ ∆ ∈ Seq (τ2 ) be such that Var (Γ) ∪ Var (∆) = − { x1 . . . xn }. Then there exists a derivation D of Γ ⇒ ∆ in LKW such that: 1. for every branch Bi ∈ D there exists a node ni ∈ Bi labelled by a sequent of the form Vi1 ⇒ Vi2 , containing only propositional variables, and such that Vi1 ∪ Vi2 = { x1 . . . xn }. 2. each branch Bi terminates with leaves labelled by Γ ⇒ ∆. Proof. Preliminarly, observe that, since Var (Γ) ∪ Var (∆) = { x1 . . . xn }, either Γ 6= ∅, or ∆ 6= ∅. By applying Lemma 7.3.16.(2) to each formula in Γ and Lemma 7.3.16.(1) to each formula in ∆, for each branch Bi there will be a node ni labelled by a sequent of the form Vi1 ⇒ Vi2 , whose formulas are exactly the variables occurring in Γ ∪ ∆. By Lemma 7.3.16.(3), every branch Bi will terminate with leaves labelled by Γ ⇒ ∆.  In light of the Hilbert-style axiomatisation of PWK given in Definition 7.2.1, whose sole inference rule is the linguistically constrained version RMP of Modus Ponens, it is to be expected that a restricted form of Cut, respecting some variable-inclusion constraint from premisses to − conclusion, be derivable (not only admissible!) in LKW . The next lemma confirms this insight. − Lemma 7.3.18. The following rule is derivable in LKW whenever Var ( ϕ) ⊆ Var (Γ ∪ ∆): Γ, ϕ ⇒ ∆ Γ ⇒ ϕ, ∆ Γ⇒∆

Proof. We prove the statement by induction on the complexity of the Cutformula ϕ. Base. Let ϕ( x1 , . . . , xn ) = x. So there is a formula ϕ( x, ~y) ∈ Γ ∪ ∆, with x actually occurring in it. W.l.o.g. let ϕ( x, ~y) ∈ ∆. By Theorem 7.3.17 − there is a LKW -derivation of the following form: V11 ⇒ V12 .. .

··· Γ⇒∆

Vk1 ⇒ Vk2 .. .

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where each Vi1 ⇒ Vi2 (1 ≤ i ≤ k) is a sequent containing only propositional variables and such that Vi1 ∪ Vi2 = Var (Γ ∪ ∆). This entails that, for each 1 ≤ i ≤ k, the variable x actually occurs in Vi1 ∪ Vi2 . According as x occurs on the left-hand or on the right-hand side, we apply Theorem 7.3.17 in order to obtain the following derivation (here we assume w.l.o.g. x ∈ V11 , Vk2 ) Γ ⇒ x, ∆ Γ, x ⇒ ∆ .. .. . . V11 , x ⇒ V12

Vk1 ⇒ x, Vk2

V11 ⇒ V12 .. .

Vk1 ⇒ Vk2 .. .

··· Γ⇒∆

as desired. Observe that the critical step Vi1 , x ⇒ x, Vi2 Vi1 ⇒ Vi2 where the displayed x may not occur in the sequent’s succedent or antecedent according as i = 1 or i = k, relies on the assumption that x ∈ Var (Γ ∪ ∆). The case ϕ = ¬ x is similar. Let now ϕ( x1 , . . . xn ) = x ∧ y, so x, y ∈ Var (Γ ∪ ∆). We adopt the same strategy as above, applying Theorem 7.3.17 to Γ ⇒ ∆ in order to get a derivation of the form V11 ⇒ V12 .. .

··· Γ⇒∆

Vk1 ⇒ Vk2 .. .

where each Vi1 ⇒ Vi2 (1 ≤ i ≤ k) is a sequent containing only propositional variables and such that Vi1 ∪ Vi2 = Var (Γ ∪ ∆). This entails that the variables x, y actually occur in each node Vi1 ⇒ Vi2 (1 ≤ i ≤ k). There are four possibilities, namely (a) x, y ∈ Vi1 , (b) x ∈ Vi1 , y ∈ Vi2 , (c) y ∈ Vi1 , x ∈ Vi2 , (d) x, y ∈ Vi2 . Thanks to Theorem 7.3.17, in all cases (a)-(d) we can conclude the derivation as follows (here, for the sake of exemplification, we assume that the combinations (a)-(d) hold, from left to right, in the displayed branches):

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Γ, x ∧ y ⇒ ∆ Γ, x, y ⇒ ∆ .. . V11 , x, y ⇒ V12

Γ ⇒ x ∧ y, ∆ Γ, x ⇒ y, ∆ .. . V21 , x ⇒ y, V22

Γ ⇒ x ∧ y, ∆ Γ, y ⇒ x, ∆ .. . Vk1−1 , y ⇒ x, Vk2−1

Γ ⇒ x ∧ y, ∆ Γ ⇒ x, y, ∆ .. . Vk1 ⇒ x, y, Vk2

V11 ⇒ V12 .. .

V21 ⇒ V22 .. .

Vk1−1 ⇒ Vk2−1 .. .

Vk1 ⇒ Vk2 .. .

··· Γ⇒∆

Inductive step. We need a derivation of Γ ⇒ ∆ all of whose branches terminate either with axioms or with the assumption Γ ⇒ ∆, ϕ, or else with the assumption ϕ, Γ ⇒ ∆. We only consider the case ϕ = α( x1 . . . , xn ) ∧ β(y1 , . . . , ym ), where { x1 , . . . , xn , y1 , . . . , ym } ⊆ Var (Γ ∪ ∆). By inductive hypothesis, since { x1 , . . . , xn } ⊆ Var (Γ ∪ ∆) and the complexity of α(~x ) is smaller than the complexity of ϕ, there exists a derivation of Γ ⇒ ∆ all of whose branches terminate either with axioms or with the assumption Γ ⇒ ∆, α(~x ), or else with the assumption α(~x ), Γ ⇒ ∆: Γ ⇒ ∆, α(~x ) .. .

··· Γ⇒∆

α(~x ), Γ ⇒ ∆ .. .

We now apply the inductive hypothesis to all leaves labelled by Γ ⇒ ∆, α(~x ). Since {y1 , . . . , ym } ⊆ Var (Γ ∪ ∆) and the complexity of β(~x ) is smaller than the complexity of ϕ, every such branch can be extended as follows: Γ ⇒ ∆, α(~x ), β(~y) β(~y), Γ .. . ··· Γ ⇒ ∆, α(~x ) .. .

⇒ ∆, α(~x ) .. .

Γ⇒∆

···

α(~x ), Γ ⇒ ∆ .. .

Here, all of the new leaves are labelled either by axioms or by Γ ⇒ → → ∆, α(~x ), β(− y ) or by β(− y ), Γ ⇒ ∆, α(~x ). Symmetrically, we apply the H.I. to all leaves labelled by α(~x ), Γ ⇒ ∆. The corresponding branches can be extended in such a way that the new leaves are labelled either by → → axioms or by α(~x ), Γ ⇒ ∆, β(− y ) or by α(~x ), β(− y ), Γ ⇒ ∆. Now apply to all the leaves that are not labelled by axioms the rules ∧ L∂ or ∧ R1∂−3 , as appropriate. The resulting proof-tree will be such that its leaves are labelled either by axioms or by Γ ⇒ ∆, ϕ, or else by ϕ, Γ ⇒ ∆. This concludes the proof. 

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Theorem 7.3.19. The following are equivalent for any Γ ∪ { ϕ} ⊆ Fmτ2 : 1. {⇒ γ | γ ∈ Γ} `LK− ⇒ ϕ; W

2. Γ `PWK ϕ. Proof. (1)⇒(2) follows from Corollary 7.3.15. (2)⇒(1). Assume that Γ `PWK ϕ. By Theorem 1.3.2, either Γ = ∅ and `CL ϕ (whence our conclusion follows from Theorem 7.3.4 and Lemma 7.3.13.(3)) or there exists a nonempty, and necessarily finite, Γ0 = {γ1 , . . . γm } ⊆ Γ such that Γ0 CL ϕ and Var (Γ0 ) ⊆ Var ( ϕ). As Var (Γ0 ) ⊆ Var ( ϕ), Lemma 7.3.13.(1)-(3) and Lemma 7.3.18 ensure that we can obtain the following derivation:

.. . Γ0 ⇒ ϕ γ1 ∧ · · · ∧ γm ⇒ ϕ

⇒ γ2 ⇒ γ1 ⇒ γ1 ∧ γ2 .. . ⇒ γ1 ∧ · · · ∧ γm−1 ⇒ γ1 ∧ · · · ∧ γm ⇒ γ3

⇒ γm ⇒ϕ

and, therefore, {⇒ γ | γ ∈ Γ} `LK− ⇒ ϕ. W



While we are still in the process of surveying existing sequent systems (broadly conceived) for PWK, let us also mention Fjellstad’s five-sided cal− culus in [94], which, unlike LKW , can be extended to the first-order case and tweaked to a calculus for B3 , and the three-sided calculus generated by the automated system Multlog (https://www.logic.at/multlog/), designed to algorithmically output several types of calculi for finite-valued logics [9]

7.4

Other proof-theoretic presentations

7.4.1 Natural deduction calculi In this very short subsection we survey the existing natural deduction calculi for PWK. In the interests of space, we will refrain from going into the details and we will omit proofs of the soundness and completeness results. Chronologically, the first such calculus was introduced by Petrukhin [171]. Slightly different calculi were suggested by Priest [188] and Belikov [17]. All of these calculi take their cue from earlier calculi for

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logics in the Strong Kleene family [224, 183, 134] that are appropriately tweaked to fit PWK. On the face of it, though, Priest’s system appears more compact and streamlined, whence we proceed to present it for ease of exposition. If ϕ ∈ Forτ2 , let E ( ϕ) = {ψ ∈ Forτ2 | Var ( ϕ) ⊆ Var (ψ)}. We denote by ϕ∗ a generic member of E ( ϕ). The calculus N DP W K , of type τ2 , has the following rules: ϕ ( DN I ) ¬¬ ϕ

¬¬ ϕ ( DNE) ϕ

¬ ϕ ∧ ¬ψ ( DM1) ¬ ( ϕ ∨ ψ)

¬ ( ϕ ∨ ψ) ( DM2) ¬ ϕ ∧ ¬ψ

¬ ϕ ∨ ¬ψ ( DM3) ¬ ( ϕ ∧ ψ)

¬ ( ϕ ∧ ψ) ( DM4) ¬ ϕ ∨ ¬ψ

ϕ ψ (∧ I ) ϕ∧ψ

ϕ ϕ∨ψ

ϕ ∨ ¬ϕ

( EM)

[ ϕ] [ψ] .. .. . . ϕ∨ψ χ χ ϕ ¬ϕ ϕ∧ψ (∨ E) (WExp) χ ϕ† ϕ ∨ ψ†

ψ (∨ I ) ϕ∨ψ

ϕ∧ψ (W ∧ E ) ϕ† ∨ ψ

Observe that the rules WExp and W ∧ E are restricted version of the classical conjunction elimination and negation elimination rules, abiding by the familiar variable inclusion constraint. For Γ ∪ { ϕ} ⊆ Forτ2 , we write Γ `N DP W K ϕ to denote the fact that there is a derivation of ϕ in N DP W K from assumptions in Γ. Theorem 7.4.1. [188, § 7.3] If Γ ∪ { ϕ} ⊆ Forτ2 , then Γ `N DP W K ϕ iff Γ `PWK ϕ. Petrukhin’s calculus in [171] is similar, except that the rules WExp and W ∧ E are parcelled out in a series of different rules according to the various logical forms ϕ† may take. Before closing this subsection, we also mention that Caleiro et al. [46] have introduced an elegant multiple-conclusion natural deduction system for PWK in the style of Shoesmith and Smiley [212].

7.4.2 Tableaux We conclude this chapter by presenting a tableaux system for PWK, modelled after the standard proof-theoretical format for many-valued logics

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to be found in [185] (although earlier versions can be traced back to [67] or [25]), which uses signed formulas — namely, formulas prefixed either by the sign + (read as ”designated”) or by the sign − (read as ”nondesignated”). Priest’s textbook has tableaux calculi both for LP and for K3 , but does not provide one for B3 or PWK. As opposed to the existing calculi for K3 or LP, the distinctive features of the system in this subsection are the presence of semi-branching rules alongside the usual simple rules and branching rules, as well as of a third sign ∗ (which stands for “classical”) alongside the customary signs + (read as ”designated”) and − (read as ”non-designated”). The annotation ∗ is not to be interpreted as a truth value, as it is customary to do in other kinds of labelled proof systems, but essentially as a guarantor that a formula does not take a particular truth value, namely n. Definition 7.4.2. By a signed formula we mean an ordered pair h ϕ, si, where ϕ ∈ Fmτ2 and s ∈ {+, −, ∗}. A member of this last set is called a sign. Hereafter, when referring to signed formulas, angles are omitted whenever there is no danger of confusion. Definition 7.4.3. A PWK-tableau is a labelled tree T whose vertices are labelled by signed formulas. The signed formulas labelling the root of T (and, possibly, some of its immediate successors) are called initial formulas of T . The other formulas occurring in T are called non-initial. Non-initial formulas in T are obtained from predecessor vertices by one of the rules below (where the usual typographical conventions are adopted): ϕ ∧ ψ, + ϕ, + ψ, + ¬ ϕ, − | | ¬ ϕ, + ¬ψ, + ¬ψ, −

ϕ ∨ ψ, + ϕ, + | ψ, +

ϕ ∧ ψ, − ϕ, ∗ ψ, ∗ ϕ, − | ψ, −

ϕ ∨ ψ, − ϕ, − ψ, −

¬ ( ϕ ∧ ψ) , + ¬ ϕ, + | ¬ψ, +

¬ ( ϕ ∨ ψ) , + ϕ, + ψ, + ϕ, − | | ¬ ϕ, + ¬ψ, + ψ, −

¬ ( ϕ ∧ ψ) , − ¬ ϕ, − ¬ψ, −

¬ ( ϕ ∨ ψ) , − ϕ, ∗ ψ, ∗ ¬ ϕ, − | ¬ψ, −

¬¬ ϕ, + ϕ, +

¬¬ ϕ, − ϕ, −

ϕ ∧ ψ, ∗ ϕ, ∗ ψ, ∗

ϕ ∨ ψ, ∗ ϕ, ∗ ψ, ∗

¬ ϕ, ∗ ϕ, ∗

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The rules of Definition 7.4.3 divide into α-rules (simple), β-rules (branching) and γ-rules (semi-branching). In the next table, which provides the corresponding classification, each rule is labelled after the signed formula which constitutes its premiss. α

β

γ

ϕ ∨ ψ, − ϕ ∧ ψ, + ϕ ∧ ψ, − ¬ ( ϕ ∧ ψ) , − ϕ ∨ ψ, + ¬ ( ϕ ∨ ψ) , − ¬¬ ϕ, + ¬ ( ϕ ∧ ψ) , + ¬¬ ϕ, − ¬ ( ϕ ∨ ψ) , + ϕ ∧ ψ, ∗ ϕ ∨ ψ, ∗ ¬ ϕ, ∗ Definition 7.4.4. If ϕ1 , ..., ϕn , ψ ∈ Fmτ2 , we say that the PWK-tableau T is for h{ ϕ1 , ..., ϕn } , ψi iff its root is labelled by h ϕ1 , +i and followed by a chain of vertices respectively labelled by h ϕ2 , +i, ..., h ϕn , +i, hψ, −i. The next definitions will be useful in what follows. Definition 7.4.5. A branch of a PWK-tableau T is called closed iff either (i) for some formula ϕ, it contains both h ϕ, +i and h ϕ, −i; or (ii) for some formula ϕ, it contains both h ϕ, −i and h¬ ϕ, −i; or else (iii) for some formula ϕ, it contains all of h ϕ, +i, h¬ ϕ, +i and h ϕ, ∗i . Closed branches will be decorated by crosses. Definition 7.4.6. A branch B of a PWK-tableau T is called complete iff: i) whenever it contains the premiss of an α-rule, it contains all its conclusion(s); ii) whenever it contains the premiss of a β-rule, it contains at least one of its conclusions; iii) whenever it contains the premiss of a γ-rule, it contains all its conclusions of the form h ϕ, ∗i and at least one of its remaining conclusions. A PWK-tableau T is said to be completed iff all of its branches are complete. Definition 7.4.7. By the notation ϕ1 , ..., ϕn ` T −PWK ψ we mean that there is a completed PWK-tableau T for h{ ϕ1 , ..., ϕn } , ψi whose branches are all closed. We now provide some examples of PWK-tableaux. Example 7.4.8. (De Morgan Laws)

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¬ ( ϕ ∧ ψ) , + ¬ ϕ ∨ ¬ψ, − ¬ ϕ, − ¬ψ, − ¬ ϕ, + ¬ψ, + | × ×

¬ ( ϕ ∨ ψ) , + ¬ ϕ ∧ ¬ψ, − ¬ ϕ, ∗ ¬ψ, ∗ ϕ, ∗ ψ, ∗ ¬ ϕ, − ¬ψ, − ϕ, + ψ, + ϕ, − | ϕ, + ψ, + ϕ, − | | | | ¬ ϕ, + ¬ψ, + ψ, − ¬ ϕ, + ¬ψ, + ψ, − × × × × × ×

To show completeness of the calculus with respect to PWK, we follow the standard technique of Hintikka sets (see e.g. [92]). As a preliminary move, we give two more definitions. Definition 7.4.9. A valuation v is called faithful to the signed formula h ϕ, si iff one of the following conditions hold: • h ϕ, si has the form hψ, +i and v (ψ) ∈ {1, n}; • h ϕ, si has the form h¬ψ, +i and v (ψ) ∈ {0, n}; • h ϕ, si has the form hψ, −i and v (ψ) = 0; • h ϕ, si has the form h¬ψ, −i and v (ψ) = 1; • h ϕ, si has either the form hψ, ∗i or the form h¬ψ, ∗i and v (ψ) ∈ {0, 1}. A set S of signed formulas is called satisfiable iff there is a valuation v which is faithful to every signed formula in S. Definition 7.4.10. A set S of signed formulas is called a Hintikka set iff the following conditions hold: • for every formula ϕ, it is not the case that both h ϕ, +i and h ϕ, −i belong to S; • for every formula ϕ, it is not the case that both h ϕ, −i and h¬ ϕ, −i belong to S; • for every formula ϕ, it is not the case that all of h ϕ, +i, h¬ ϕ, +i and h ϕ, ∗i belong to S;

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• if h ϕ, si is a possible premiss for an α-rule of a PWK-tableau, the conclusion(s) that would result from the application of the rule to h ϕ, si occur in S; • if h ϕ, si is a possible premiss for a β-rule of a PWK-tableau, at least one of the conclusions that would result from the application of the rule to h ϕ, si occur in S; • if h ϕ, si is a possible premiss for a γ-rule of a PWK-tableau, all the conclusions of the form hψ, ∗i that would result from the application of the rule to h ϕ, si, and at least one of the remaining conclusions, occur in S. Theorem 7.4.11. For every ϕ1 , ..., ϕn , ψ ∈ Fmτ2 , the following are equivalent: 1. ϕ1 , ..., ϕn ` T −PWK ψ; 2. ϕ1 , ..., ϕn `PWK ψ. Proof. (1)⇒(2) Just remark that: i) any valuation which is faithful to the premiss of an α-rule is faithful to its conclusion(s) as well; ii) any valuation which is faithful to the premiss of a β-rule is faithful to at least one of its conclusions as well; iii) any valuation which is faithful to the premiss of a γ-rule is faithful to its conclusions of the form hψ, ∗i, as well as to at least one of the remaining conclusions. Reasoning inductively, it follows that if v is faithful to h ϕ1 , +i, ..., h ϕn , +i, hψ, −i, then in any PWK-tableau T there is at least a branch B s.t. v is faithful to all the signed formulas in B . However, if T is a PWK-tableau all of whose branches are closed, no interpretation can be faithful to all the signed formulas in any of its branches. Therefore, whenever v ( ϕ1 ) , ..., v ( ϕn ) ∈ {1, n}, it is also the case that v (ψ) ∈ {1, n}. (2)⇒(1) We prove that every complete open branch of a PWK-tableau is satisfiable. Since the set of signed formulas occurring in a complete open branch of a PWK-tableau is a Hintikka set, it will be enough to show that any Hintikka set S is satisfiable. Construct a valuation v as follows, for any variable x occurring in some signed formula in S (the remaining variables may be assigned arbitrary values): • h x, +i ∈ S, h¬ x, +i ∈ / S: v( x ) = 1; • h x, −i ∈ S, h¬ x, −i ∈ / S: v( x ) = 0; • h x, +i ∈ S, h¬ x, +i ∈ S: v( x ) = n;

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• h¬ x, −i ∈ S, h x, −i ∈ / S: v( x ) = 1; • h¬ x, +i ∈ S, h x, +i ∈ / S: v( x ) = 0; • h x, +i ∈ / S, h x, −i ∈ / S, h¬ x, +i ∈ / S, h¬ x, −i ∈ / S, h x, ∗i ∈ S: v( x ) = 1 (alternatively: v( x ) = 0). In light of Definition 7.4.10, v is well-defined because the cases envisaged above do not clash and exhaust all the alternatives. Moreover, v is faithful to all the signed literals in S. Upon remarking that: 1. a valuation which is faithful to the possible conclusion(s) of an αrule is also faithful to the respective premiss; 2. a valuation which is faithful to at least one of the possible conclusions of a β-rule is also faithful to the respective premiss; 3. a valuation which is faithful to the possible conclusions of the form hψ, ∗i of a γ-rule, and to at least one of the remaining conclusions, is also faithful to the respective premiss; we conclude by induction that v is faithful to all the signed formulas in S. 

Chapter 8 Conclusions and open problems The present book aims at providing readers with a primer on logics of variable inclusion, systems that have attracted more and more attention in the last few years. The main focus of the volume is algebraic, given the prominent role played by the application of Płonka sums to logical matrices. We are not claiming that the algebraic approach is the correct way to address these logics. On the contrary, one of our goals has been to build as many bridges as possible between the algebraic and the syntactic analysis. The novelty and intricacy of the topics covered in this work inevitably led to certain design choices on the inclusion of specific subjects. We deliberately opted to devote a whole chapter to one of the most important examples of a logic of variable inclusion, PWK, but we did not do the same for B3 . This asymmetry is due to the different state of the art regarding these two logics. While PWK has been the object of many detailed articles on its proof theory, algebraic semantics, and philosophical applications, the extant results on B3 are comparatively fewer. Moreover, in accordance with the most recent trend, PWK and B3 are presented in the book as logics in the same language as CL. However, as we briefly saw in Chapter 1, Bochvar and Halld´en approached them as fragments of logics in an expanded language, containing external and internal connectives, given the particular philosophical or foundational uses they had in mind. The treatment of the external expansions of PWK and B3 (see [210] and [91]) has not been included, given their weaker connection with the topic of variable inclusion. We also omitted the seminal investigations on a theory of probability for involutive bisemilattices (see [35]). In any case, we believe that the best way to wind up an introductory book is to look ahead to the many directions that can unfold from the ground we have covered, instead of looking back to the stretch of road

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Bonzio et al., Logics of Variable Inclusion, Trends in Logic 59, https://doi.org/10.1007/978-3-031-04297-3_8

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that’s already behind our shoulders.

8.1 Open problems We present a list of promising directions for future research, organising them into thematic categories, so that different readers can directly jump, if needed, to the motives and topics that befit them best.

8.1.1 Universal Algebra • Congruences of algebras in regular varieties Determining the general structure of congruence lattices of algebras in regular varieties is a very natural task to pursue. In particular, such a description may depend both on the congruence lattices of the algebras that inhabit the fibres of the corresponding Płonka sum decompositions, and on the underlying semilattices of indices. To the best of our knowledge, this problem has been not completely solved even for well-known regular varieties, such as involutive bisemilattices, despite some preliminary results to be found in [22]. A related open problem is to describe factor congruences of an algebra represented as a Płonka sum. In turn, this would lead to new information concerning directly indecomposable members of regular varieties. • Płonka sums of relational structures As such, the notion of a Płonka sum only applies to algebras, not to other kinds of first order structures, which may also include relations. Extending the construction to the general case would be of intrinsic interest. The approach presented in this book goes some way towards undertaking this generalisation, since logical matrices can be seen as first order structures with a unique unary relational symbol in the signature, representing the relation of “belonging to the filter” [96, Sec 4.5]. However, a general model-theoretic study of Płonka sums of first order structures has not yet been developped. • The fine spectrum of a regular variety Given a variety V , the task of determining, for each natural number n, how many members of V have cardinality n is a classical goal which usually relies on combinatorial and algebraic methods. Very little is known in case V is a regular(ised) variety. The problem has

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been solved in [39] for a specific subclass of involutive bisemilattices, those whose underlying semilattice of indices is linearly ordered. It turns out that, building on the algebraic properties of Płonka sums and the relative dual spaces, a combinatorial approach allows for an effective algorithm which takes a natural number n as input, and returns a natural number m, where m is the number of nonisomorphic involutive, linearly ordered bisemilattices of cardinality n. A remarkable achievement would be to generalise this approach to the non-linearly ordered case and, even more, to other regular varieties. • Płonka sums in quasivarieties The theory of Płonka sums is well developped for regularisations of strongly irregular varieties. Switching the attention from varieties to quasivarieties may afford new inroads. The first attempt towards this direction is a work by Bergman and Romanowska [22], which lays the foundations of a theory of the quasi-regularisation of a quasivariety: under this respect, Płonka sums over semilattice systems whose homomorphisms are injective turn out to be particularly relevant. The more recent paper [21] tackles the problems of determining the subquasivarieties of a regular variety, and relates Płonka sums with the well-known construction of Maltsev products. This line of enquiry would be particularly fruitful for the logical applications, as quasivarieties are the key notion in determining the algebraic counterpart of a finitary logic. An investigation of logics connected with subquasivarieties of generalised involutive bisemilattices can be found in [164, 165]. Among others, the subquasivariety IN GIB of GIB , consisting of algebras whose Płonka sum representations have injective homomorphisms and nontrivial fibres, brings about a connection with the theory of states explored in [35]. Indeed, this quasivariety comprises algebras carrying faithful states.

8.1.2 Abstract Algebraic Logic • Gentzen algebraisability An immediate outcome of the algebraic analysis of logics of variable inclusion is that neither the left, nor the right companion of a logic is algebraisable (Theorems 5.5.1 and 6.1.14). Even worse, right variable inclusion logics lie outside the Leibniz hierarchy, while left variable inclusion logics may be at best truth-equational. It is well-

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CHAPTER 8. CONCLUSIONS AND OPEN PROBLEMS known that non-algebraisable logics can nonetheless become algebraisable when considered as Gentzen systems, i.e., as substitutioninvariant consequence relations over a certain set of sequences of formulas. This is the case of the {∧, ∨}-fragment of classical logic. A promising research direction is to study logics of variable inclusion in the framework of Gentzen systems [198]. In the case of PWK, it is shown in [180] that there exists an equivalential Gentzen calculus. However, a systematic treatment of the topic must still come into existence.

• First order versions of logics of variable inclusion Although there is some material on predicate versions of logics of variable inclusion [94, 87, 88], from the point of view of Abstract Algebraic Logic this is a virtually untrodden territory. Building on the fact that classical first order logic has cylindric algebras as an algebraic counterpart, a very natural case study would be to perform the construction of Płonka sums on members of this class of algebras, with an eye to getting insights on the algebraic behaviour of the first order versions of both variable inclusion companions of CL. • Modal expansions of logics of variable inclusion A different suggestion is to investigate logics on languages with modal operators. This can be done in two ways: more traditionally, by considering Płonka sums of matrix models of modal logics; or else, by starting with Płonka sums of matrices over non-modal algebras and defining some intensional operators “externally”. The latter approach is taken in [142], where it is shown that a certain expansion of generalised involutive bisemilattices (with nontrivial fibres) by means of a modality-like intensional implication is the equivalent algebraic semantics of Parry and Dunn’s logic of demodalised analytic implication. • Power matrices Let A be an algebra of type τ. Recall that the complex algebra of A [102, 41] is the algebra ℘(A), also of type τ, whose universe is the powerset ℘( A) and whose operations are so defined, for every n-ary g in τ and any n-tuple A1 , ..., An of subsets of A: g℘(A) ( A1 , ..., An ) = { gA ( a1 , ..., an ) | a1 ∈ A1 , ..., an ∈ An }. Humberstone [124] discusses an interesting class of matrices whose algebraic reducts are complex algebras, and dubs them power ma-

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trices. In detail, the power matrix of the matrix hA, F i is the matrix whose algebraic reduct is ℘(A), and whose filter is { B ⊆ A | B ∩ F 6= ∅}. Humberstone’s focus is especially on the power matrix of the classical matrix hB2 , {1}i, which upon inspection turns out to be none other than the characteristic matrix of the right variable inclusion logic LPr – aka SFDE . It would be interesting to ascertain if this particular connection can be made more general, bridging the theories of power matrices and of variable inclusion logics.

8.1.3 Proof Theory • Finite Hilbert calculi without linguistic restrictions Finding a complete Hilbert-style calculus is a typical goal whenever we investigate a certain logic. In the case of a logic of variable inclusion Lr or Ll , assuming a complete calculus H for the original logic L is known, a general answer concerning how to obtain a complete calculus for Lr or Ll has been given in Sections 5.2 and 6.2. However, as already noticed, such calculi contain infinitely many rule schemata. In the case of PWK a finite Hilbert calculus is available, with the drawback of having syntactic restrictions on its rules. It could be worthwhile to determine if, and under what conditions, a finite Hilbert calculus without linguistic restrictions is available for a logic of variable inclusion. • Sequent calculi and external consequences In Section 7.3.2 we introduced a sequent calculus whose external consequence coincides with PWK. We believe that this perspective, which arises in the study of substructural logics, can be fruitfully applied in a more general and systematic way. On the one hand, it would be interesting to apply to a wider class of left variable inclusion logics the recipe used to obtain the calculus for PWK in Section 7.3.2. On the other, it would be worth investigating if a modification of this approach would work for right variable inclusion logics. The latter research may also help to fill the existing gaps in the literature concerning the proof theory of right variable inclusion logics.

8.1.4 Duality Theory • Description of dual spaces The duality introduced in Chapter 3 for regularised varieties works well when we start from a strongly irregular variety. In such a case,

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CHAPTER 8. CONCLUSIONS AND OPEN PROBLEMS the dual space of an algebra essentially consists in a semilattice inverse system of the dual spaces of the algebras in a Płonka sum representation. It is natural to wonder whether a more intelligible description can be given to the dual space as a unique topological space. This is known for distributive bisemilattices [105] and involutive bisemilattices (as described in Chapter 3). However, it would be desirable to obtain a more elegant description of dual spaces of involutive bisemilattices, as well as to provide, in general, topological descriptions of spaces that are homeomorphic to semilattice inverse systems of duals of certain algebras.

• Topological analogue of the Płonka sum In the duality for (distributive) bisemilattices given by Gierz and Romanowska [105], the authors mention the problem of devising a construction that represents a “topological analogue” of Płonka sums. In Chapter 3, we introduced the Płonka product of topological spaces as a partial answer to the problem. Also assuming that the product constitutes the best construction to look at, the main problem still stands. Indeed, while on the algebraic side it is enough for an algebra to possess a partition function to be split into a sum of disjoint algebras, no conditions are known that guarantee the possibility to split a topological space into a product of spaces (with the use of purely topological techniques).

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