Constructive Negations and Paraconsistency

Constructive Negations and Paraconsistency TRENDS IN LOGIC Studia Logica Library VOLUME 26 Managing Editor Ryszard Wó...

Author: Odintsov S.P.

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Constructive Negations and Paraconsistency

TRENDS IN LOGIC Studia Logica Library VOLUME 26 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics “Ulisse Dini”, University of Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany

SCOPE OF THE SERIES

Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica – that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.

Volume Editor Heinrich Wansing

The titles published in this series are listed at the end of this volume.

Sergei P. Odintsov

Constructive Negations and Paraconsistency

123

Sergei P. Odintsov Russian Academy of Sciences Siberian Branch Sobolev Institute of Mathematics Koptyug Ave. 4 Novosibirsk Russia

ISBN 978-1-4020-6866-9

e-ISBN 978-1-4020-6867-6

Library of Congress Control Number: 2007940855 © 2008 Springer Science+Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

Contents 1 Introduction

I

1

Reductio ad Absurdum

2 Minimal Logic. Preliminary 2.1 Deﬁnition of Basic Logics 2.2 Algebraic Semantics . . . 2.3 Kripke Semantics . . . . .

13 Remarks 15 . . . . . . . . . . . . . . . . . . . . 15 . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . . . 28

3 Logic of Classical Refutability 31 3.1 Maximality Property of Le . . . . . . . . . . . . . . . . . . . 32 3.2 Isomorphs of Le . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 The Class of Extensions of Minimal Logic 4.1 Extensions of Le . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Intuitionistic and Negative Counterparts for Extensions of Le . . . . . . . . . . . . . . . . . 4.2 Intuitionistic and Negative Counterparts for Extensions of Minimal Logic . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Negative Counterparts as Logics of Contradictions 4.3 Three Dimensions of Par . . . . . . . . . . . . . . . . . . .

. .

45

. . . . . .

48 52 53

5 Adequate Algebraic Semantics for Extensions of Minimal Logic 5.1 Glivenko’s Logic . . . . . . . . . . . . . . . . . 5.2 Representation of j-Algebras . . . . . . . . . . 5.3 Segerberg’s Logics and their Semantics . . . . . 5.4 Kripke Semantics for Paraconsistent Extensions

. . . .

57 57 59 62 78

. . . . . . . . . . . . of Lj

. . . .

. . . .

41 . . 41

. . . .

v

vi

Contents

6 Negatively Equivalent Logics 6.1 Deﬁnitions and Simple Properties . . . . . . . . . . 6.2 Logics Negatively Equivalent to Intermediate Ones 6.3 Abstract Classes of Negative Equivalence . . . . . 6.4 The Structure of Jhn+ up to Negative Equivalence

. . . .

. . . .

. . . .

. . . .

. . . .

81 81 84 88 91

7 Absurdity as Unary Operator 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Le and L ukasiewicz’s Modal Logic . . . . . . . . . . . 7.3 Paradox of Minimal Logic and Generalized Absurdity 7.4 A- and C -Presentations . . . . . . . . . . . . . . . . . 7.4.1 Deﬁnitions and First Results . . . . . . . . . . 7.4.2 Logic CLuN . . . . . . . . . . . . . . . . . . . 7.4.3 Sette’s Logic P1 . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

101 101 104 108 113 113 119 123

II

. . . .

Strong Negation

8 Semantical Study of Paraconsistent 8.1 Preliminaries . . . . . . . . . . . . 8.2 Fidel’s Semantics . . . . . . . . . . 8.3 Twist-structures . . . . . . . . . . 8.3.1 Embedding of N3 into N4 8.4 N4-Lattices . . . . . . . . . . . . . 8.5 The Variety of N4-Lattices . . . . 8.6 The Logic N4⊥ and N4⊥ -Lattices

129 Nelson’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

131 133 135 138 142 145 147 155

159 9 N4⊥ -Lattices 9.1 Structure of N4⊥ -Lattices . . . . . . . . . . . . . . . . . . . . 161 9.2 Homomorphisms and Subdirectly Irreducible N4⊥ -Lattices . 167 10 The 10.1 10.2 10.3 10.4 10.5

Class of N4⊥ -Extensions EN4⊥ and Int+ . . . . . . . . . . . . . . The Lattice Structure of EN4⊥ . . . . . Explosive and Normal Counterparts . . The Structure of EN4C and EN4⊥ C . . Some Transfer Theorems for the Class of

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N4⊥ -Extensions

. . . . .

. . . . .

177 177 185 195 201 211

11 Conclusion

223

Bibliography

227

Index

237

Chapter 1

Introduction The title of this book mentions the concepts of paraconsistency and constructive logic. However, the presented material belongs to the ﬁeld of paraconsistency, not to constructive logic. At the level of metatheory, the classical methods are used. We will consider two concepts of negation: the negation as reduction to absurdity and the strong negation. Both concepts were developed in the setting of constrictive logic, which explains our choice of the title of the book. The paraconsistent logics are those, which admit inconsistent but non-trivial theories, i.e., the logics which allow one to make inferences in a non-trivial fashion from an inconsistent set of hypotheses. Logics in which all inconsistent theories are trivial are called explosive. The indicated property of paraconsistent logics yields the possibility to apply them in diﬀerent situations, where we encounter phenomena relevant (to some extent) to the logical notion of inconsistency. Examples of these situations are (see [86]): information in a computer data base; various scientiﬁc theories; constitutions and other legal documents; descriptions of ﬁctional (and other non-existent) objects; descriptions of counterfactual situations; etc. The mentioned survey by G. Priest [86] may also be recommended for a ﬁrst acquaintance with paraconsistent logic. The study of the paraconsistency phenomenon may be based on diﬀerent philosophical presuppositions (see, e.g., [87]). At this point, we emphasize only one fundamental aspect of investigations in the ﬁeld of paraconsistency. It was noted by D. Nelson in [65, p. 209]: “In both the intuitionistic and the classical logic all contradictions are equivalent. This makes it impossible to consider such entities at all in mathematics. It is not clear to me that such a radical position regarding contradiction is necessary.” Rejecting the principle “a contradiction implies everything”(ex contradictione quodlibet) the paraconsistent logic allows one 1

2

1 Introduction

to study the phenomenon of contradiction itself. Namely this formal logical aspect of paraconsistency will be at the centre of attention in this book. We now turn to constructive logic. Constructive logic is the logic of constructive mathematics, logic oriented on dealing with the universe of constructive mathematical objects. The common feature of diﬀerent variants of constructive mathematics is the rejection of the concept of actual inﬁnity and admitting only the existence of objects constructed on the base of the concept of potential inﬁnity. In any case, passing to constructive logic from the classical one changes the sense of logical connectives. For example, Markov [60] deﬁnes the constructive disjunction as follows: “The constructive understanding of the existence of a mathematical object corresponds to the constructive understanding of the disjunction of sentences of the form “P or Q”. Such a sentence is considered as accepted if at least one of the sentences P , Q was accepted as true.” Of course, this understanding of disjunction does not allow one to accept the law of excluded middle and leads to the rejection of classical logic. In the setting of constructive logic, there are two basic approaches to the concept of negation and they are considered in our investigation. Since the Brouwer works, the negation of statement P , ¬P , is understood as an abbreviation of the statement “assumption P leads to a contradiction”. Note that this concept agrees well with paraconsistency. The above understanding of negation does not assume the principle “contradiction implies everything” (ex contradictione quodlibet) responsible for the trivialization of inconsistent theories. The ﬁrst formalization of intuitionistic logic suggested by A.N. Kolmogorov [44] in 1925 was paraconsistent. In this work, A.N. Kolmogorov reasonably noted that ex contradictione quodlibet (in the form ¬p → (p → q)) has appeared only in the formal presentation of classical logic and does not occur in practical mathematical reasoning. However, A. Heyting was sure that using ex contradictione quodlibet is admissible in intuitionistic reasoning and he added the axiom ¬p → (p → q) to his variant Li of intuitionistic logic [35]. Note that adding ex contradictione quodlibet creates some problems with interpretation of Li as calculus of problems [45]. One cannot consider the implication P → Q as the problem of reducing the problem Q to the problem P . In Li, the implication P → Q means that the problem Q can be reduced to the problem P or the problem P is meaningless. This diﬃculty was known to A. Heyting, but he did not considered this as a serious problem. According to A. Heyting [36, p. 106], “. . . it (ex contradictione quodlibet — S.O.) adds to the precision of the deﬁnition of implication” and “I shall interpret implication in this wider sense.”

1 Introduction

3

Only in 1937 I. Johansson [41] questioned the using of ex contradictione quodlibet in constructive reasoning and suggested the system, which we denote by Lj. Axiomatics for Lj can be obtained by deleting ex contradictione quodlibet from the standard list of axioms for intuitionistic logic, more exactly, Li = Lj + {¬p → (p → q)}. In [41], Johansson proved that many important properties of negation provable in the Heyting logic Li can be proved also in the system Lj. Since that the logic has the name “Johansson’s logic” or “minimal logic”(see the title of Johansson’s article). Note that, in fact, Johansson came back to the Kolmogorov’s variant of intuitionistic logic. More exactly, the implication-negation fragment of Lj coincides with the propositional fragment of the system from [44]. Kolmogorov considered the ﬁrst-order logic, but in the language with only two propositional connectives, implication and negation. Unfortunately, the logic Lj was for a long time on the borderline of studies in the ﬁeld of paraconsistency, which was traditionally motivated by the following “paraconsistent paradox” of Lj. Although Lj is not explosive, admits non-trivial inconsistent theories, we can prove in Lj for any formulas ϕ and ψ that ϕ, ¬ϕ Lj ¬ψ. This means that the negation makes no sense in inconsistent Lj-theories, because all negated formulas are provable in them. In this way, inconsistent Lj-theories are positive. It should be noted that studies in the ﬁeld of paraconsistency were directed during a long period to searching for “the most natural system” of paraconsistent logic, which is maximally close to classical logic (cf. [39, p. 147]). The above paradox obviously shows that Lj cannot play the role of such logic. However, recently more attention has been paid to the study of paraconsistent analogs of well-known logical systems. In this respect, Johansson’s logic Lj is worthy of attention as a paraconsistent analog of intuitionistic logic Li. Turning to the second main approach to negation in constructive logic, the concept of strong negation. Note that the strong negation is namely a proper constructive negation. As happens with most fundamental logical concepts, the concept of strong negation was developed independently by many authors and with different motivations. Constructive logic with strong negation was suggested for the ﬁrst time by D. Nelson in 1949 [64]. The truth of a negation of statement in intuitionistic and minimal logic can be stated only indirectly, via reducing a negated sentence to an absurdity. As a consequence of this, the negation in these logics has the following feature, unsatisﬁable from the

4

1 Introduction

constructive point of view. When the negation of a conjunction ¬(ϕ ∧ ψ) is provable, it does not follow in the general case that either ¬ϕ, or ¬ψ is provable. In the mentioned work, D. Nelson suggested a new constructive interpretation of the negation connective based on the idea that the falseness of atomic formulas can be seen directly, which leads to parallel constructive procedures reducing the truth and falseness of complex statements to the truth and falseness of their components. As a result, D. Nelson obtained a logical system possessing the property: if ∼ (ϕ ∧ ψ),

then ∼ ϕ or ∼ ψ,

where ∼ denotes the negation connective and the derivability in Nelson’s system. Now, the above property is traditionally considered as a characteristic property of constructive negation, and Nelson-type negations are called strong. One year later constructive logic with strong negation was considered by A.A. Markov [59]. The propositional variant of Nelson’s logic was studied by N.N. Vorobiev [114, 115, 116]. Independently, Gentzen-style calculus equivalent to Nelson’s system was developed by F. von Kutschera [49]. A system closely related to strong negation systems also arose in the work by J.P. Cleave [18], who constructed the predicate calculus adequate for the algebra of inexact sets by S. K¨orner [46]. The paraconsistent variant of Nelson’s system was studied independently by R. Routley (later R. Sylvan) in the propositional case in [96], by Lopez-Escobar in [51] and by Nelson himself [1], both in the ﬁrst-order case. It should be noted that the term “strong negation” is connected not with the idea of direct falsiﬁcation, but with comparing strong and intuitionistic negations in the explosive variant of Nelson’s logic [64]. In this logic, one can deﬁne an intuitionistic negation ¬ via a strong negation as follows ¬ϕ := ϕ →∼ ϕ, and prove the implication ∼ ϕ → ¬ϕ showing that the negation ∼ is stronger than the intuitionistic one. In the paraconsistent version of Nelson’s logic, one cannot deﬁne the intuitionistic negation and the above comparison loses its meaning, but traditionally the name “strong negation” is used also in this case. We now say a few words about denotation of logics under consideration. There is no generally accepted convention. Nelson used the denotation N and N − for his system of strong negation and for its paraconsistent variant (see [64, 1]), respectively. In Dunn’s systematization [26], these systems receive the denotation N and BN1 , respectively. We will follow another tradition (see, e.g., [120]) and denote explosive Nelson’s logic by N3 and paraconsistent Nelson’s logic by N4. This choice is motivated by the Kripkestyle semantics for these logics. Kripke semantics for N3 was developed by R. Thomason [107] and R. Routley [96]. As in the case of intuitionistic logics,

1 Introduction

5

N3-frames are partial orderings. But since veriﬁcation and falsiﬁcation are treated in N3 independently, N3-models have two valuations, v + for veriﬁcation and v − for falsiﬁcation, with the additional restriction that v + (p) ∩ v − (p) = ∅, i.e., no atomic statement can be true and false in the same world simultaneously. Omitting the latter restriction we obtain a semantics for N4. It is not hard to check (see [93]) that from the pair (v + , v − ) one can pass to one many-valued valuation, which is three-valued (true, false, neither) in case of N3 and four-valued (true, false, neither, both) in case of N4. Of course, the logic N4 is more attractive for applications, because it allows one to work with inconsistent information. A view of N4 as a logic convenient for information representation and processing is reﬂected in a series of books (see [40, 117, 118]). Also, N4 has proved useful for solving some well-known philosophical logic paradoxes [119, 121]. At the same time, the attention paid to this logic is incomparable with that for N3. In particular, semantic investigations of N4 were restricted mainly to the case of Kripke-style semantics. There was no speciﬁc information about the class of N4-extension, except for the information about its proper subclass, the class of N3-extensions. It should be noted that the latter was thoroughly studied (see [33, 47, 99, 100, 101]). Thus, we have two explosive logics Li and N3, and their paraconsistent analogs Lj and N4. It will be shown that Li can be faithfully embedded into Lj, whereas N3 is faithfully embedded into N4. In this way, refusing the explosion axiom does not lead to a decrease in the expressive power of a logic. Here arises the question: which new expressive possibilities have the logics Lj and N4 as compared to the explosive logics Li and respectively N3, and how regularly this family of new possibilities is structured? In this book we give answers to these question by studying the lattices of extensions of the logics Lj and N4. Studying the lattices of extensions of diﬀerent logics such as, e.g., the intuitionistic logic Li (see, e.g., [16]), the normal modal logic K4 [30, 31], etc., plays an extremely important part in the development of modern nonclassical logic. In the ﬁrst part of the book (Chapters 2–7) we concentrate on the study of the class of extensions of Johansson’s logic. This was the ﬁrst attempt to systematically study the lattice of extensions for a paraconsistent logic. We will see that there is one important feature, which distinguishes the class of Lj-extensions from the classes of extensions of the explosive logics Li and K4. The class Jhn of non-trivial extensions of minimal logic has a non-trivial and interesting global structure (it is three-dimensional in some sense), which allows one to reduce its description (to some extent) to the well-studied classes of intermediate and positive logics.

6

1 Introduction

More exactly, the class Jhn is the disjunctive union of three classes: the class Int of intermediate logics, which are explosive; the class Neg of negative logics, i.e., logics with degenerate negation containing the scheme ¬p; and the class Par of properly paraconsistent extensions of Lj containing logics which do not belong to the ﬁrst two classes. So we have Jhn = Int ∪ Neg ∪ Par. Note that negative logics are deﬁnition equivalent to positive ones. For any L ∈ Par, one can deﬁne its intuitionistic counterpart Lint (negative counterpart Lneg ) as the least logic from the class Int (respectively, from the class Neg) containing L. There are strong translations (i.e., translations preserving the consequence relation) of logics Lint and Lneg to the original paraconsistent L. The logic Lint may also be obtained by adding ex contradictione quodlibet to L. In this way, the above-mentioned translation of Lint shows that the usual explosive reasoning can be modelled in a paraconsistent logic. On the other hand, as was noted above, the important advantage of paraconsistent logics is that they allow one to distinguish contradictions: diﬀerent contradictions are not equivalent in them. In case of Lj-extensions, the structure of contradictions in the paraconsistent logic L can be presented as a formal system, and namely the logic Lneg plays this part. The strong translation of Lneg in L can be done via the contradiction operator C(ϕ) := ϕ ∧ ¬ϕ. Due to this fact, the logic Lneg can really be treated as the logic of contradictions of the logic L. We conclude our study of the class Jhn with an eﬀort to describe the structure of Jhn up to the negative equivalence. Two logics L1 , L2 ∈ Jhn are said to be negatively equivalent if they have the same negative consequence relation, i.e., X L1 ¬ϕ iﬀ X L2 ¬ϕ for an arbitrary set of formulas X and any formula ϕ. The negative equivalence of logics from Lj is equivalent to the fact that they have the same family of inconsistent sets of formulas. From the constructive point of view, these facts mean that negatively equivalent logics have essentially the same concepts of negation and of contradiction. Concluding the ﬁrst part of the book, we suggest a way to overcome the above mentioned paradox of minimal logic. It can be done via introducing the unary operator of absurdity A(ϕ) instead of the constant ⊥ and deﬁning the negation as the reduction to this generalized absurdity: ¬ϕ := ϕ → A(ϕ). The idea of such a deﬁnition arose from comparing the contradiction operator in the logic Le of classical refutability [22] with the necessity operator

1 Introduction

7

in L ukasiewicz’s modal logic L [52, 53]. For the ﬁrst time, a similar interconnection between Le and L was noted by Porte [84, 85]. We prove that one of the modal paradoxes of L exactly corresponds to the fact that the absurdity operator is constant, i.e., is like in Le. Moreover, it turns out that negation in several well-known paraconsistent logics can be deﬁned in this way. For example, in the logic CLuN of Batens [5, 6] and in Sette’s maximal paraconsistent logic P 1 [102, 88], the negation can be presented as the reduction to a unary absurdity operator. In the second part of the book we study the lattice of extensions of paraconsistent Nelson’s logic. This investigation was motivated not only by the interest in Nelson’s logic as an alternative formalization of intuitionistic logic, but also by the desire to prove whether is it possible to apply to this new object the approach developed in the ﬁrst part of our work? The answer to this question is positive, although we discovered essential diﬀerences in the structures of lattices of extensions of minimal logic and paraconsistent Nelson’s logic. In connection with the paraconsistent Nelson logic there also arises a question: in which language should this logic be considered? The explosive N3 is usually considered in the language ∨, ∧, →, ∼, ¬ with symbols for two negations, strong ∼ and intuitionistic ¬. As was noted above, the intuitionistic negation is superﬂuous in this case, because it can be deﬁned via the strong one. If we pass to the paraconsistent N4, the interpretation of ¬ is not clear and it looks natural to consider the language with only the negation symbol ∼. This variant of the paraconsistent Nelson logic will be denoted N4. However, it turns out that the presence of intuitionistic negation is natural and desirable. The conservative extension of N4 in the language ∨, ∧, →, ∼, ⊥ obtained by spreading N4-axioms to the new language and adding axioms ⊥ → p and p →∼ ⊥ for the new constant is denoted N4⊥ . The intuitionistic negation is deﬁned in N4⊥ in the usual way, ¬ϕ := ϕ → ⊥. To study the class EN4 (EN4⊥ ) of extensions of Nelson’s logic N4 (N4⊥ ) we need adequate algebraic semantics. This means that we have to describe the variety of algebras determining N4 (N4⊥ ) such that there is a dual isomorphism between the lattice of subvarieties of this variety and the lattice of N4(N4⊥ )-extensions. For explosive N3, the algebraic semantics is provided by N -lattices, which are well studied [90, 28, 29, 33, 99, 100, 110]. The N4-lattices introduced in [72] provide this kind of semantics for N4. The algebraic semantics for N4⊥ is provided by N4⊥ -lattices, a natural modiﬁcation of N4-lattices. An interesting peculiarity of N4(and N4⊥ )-lattices is that they have a non-trivial ﬁlter of distinguished values.

8

1 Introduction

The advantage of the language with intuitionistic negation becomes obvious, when we start the investigation of the class of N4⊥ -extensions. Its structure diﬀers essentially from that of Jhn. First of all, unlike Jhn containing the subclass Neg of contradictory logics, N4⊥ does not admit contradictory extensions. Despite its paraconsistency the logic N4⊥ admits only local contradictions, adding any contradiction as a scheme to N4⊥ results in a trivial logic. However, the class EN4⊥ decomposes into subclasses of explosive logics, normal logics, and logics of general form. This decomposition reﬂects the local structure of contradictions inside N4⊥ -models and is very similar to the decomposition of Jhn into subclasses of intermediate, negative and properly paraconsistent logics. Note that the negative equivalence relation, which played an important role in the study of extensions of minimal logic, degenerates if we pass to N4(N4⊥ )-extensions. Two extensions of N4 (N4⊥ ) are negatively equivalent if and only if they are equal. We shall now describe more precisely the structure of the book. Chapter 2 contains deﬁnitions of the most important logics from the class Jhn and necessary information concerning algebraic and Kripke-style semantics for Lj-extensions. Chapter 3 is devoted to the logic of classical refutability, the maximal paraconsistent extension of Lj playing the key role in the studying the class of Lj-extensions. In Chapter 4, we investigate the logic Le = Lj + {⊥ ∨ (⊥ → p)} and prove that the class of its extensions coincides with the class of all possible intersections of intermediate and negative logics. Moreover, any logic L extending Le has a unique presentation as an intersection of intermediate logic L1 and negative logic L2 . The logic L1 (resp., L2 ) will be taken as intuitionistic (resp., negative) counterpart of L. The notions of intuitionistic and negative counterparts allow a generalization to the class of all Lj-extensions and it turns out that the class Par of properly paraconsistent Lj-extensions decomposes into a disjoint union of classes Spec(L1 , L2 ) consisting of all logics having L1 and L2 as its intuitionistic and negative counterparts, respectively. Each of the classes Spec(L1 , L2 ) forms an interval in the lattice Par with the upper point L1 ∩ L2 . In this way, studying the structure of Jhn reduces to the investigation of intervals of the form Spec(L1 , L2 ). The next chapter will be devoted to constructing an adequate algebraic semantics, in fact, a suitable presentation of j-algebras, which is convenient to determine the location of diﬀerent logics inside the intervals Spec(L1 , L2 ). The eﬀectiveness of the obtained presentation will be demonstrated via its application to numerous extensions of Lj considered by K. Segerberg [98]. We also provide several facts concerning Kripke semantics for Lj-extensions.

1 Introduction

9

In Chapter 6, we introduce the negative equivalence of logics (see above), which we denote as ≡neg , and by modifying the technique of Jankov’s formulas prove that the quotient lattice Spec(L1 , L2 )/ ≡neg is isomorphic to the interval Spec(Lk, L2 ). We also prove that every interval Spec(L1 , L2 ) contains inﬁnitely many classes of negative equivalence and that there is a continuum of negative equivalence classes in Jhn. The last chapter of the ﬁrst part of the book, Chapter 7, will be devoted to studying absurdity as a unary operator. Chapter 8 starts the second part of the book, devoted to strong negation. In the ﬁrst section, we deﬁne two variants of paraconsistent Nelson’s logic. The logic N4 is determined in the language ∨, ∧, →, ∼ , where ∼ is a symbol for strong negation, whereas the logic N4⊥ is a logic in the language ∨, ∧, →, ∼, ⊥ with an additional constant ⊥. Moreover, N4⊥ is a conservative extension of N4 as well as of intuitionistic logic. The explosive logic N3 is obtained by adding to N4 the explosion axiom ∼ p → (p → q). Notice that by putting ⊥ :=∼ (p0 → p0 ) one can prove in N3 the additional axioms of N4⊥ . In the second section, the logic N4 is characterized via Fidel structures [29]. This is direct generalization of M. Fudel’s result for N3 obtained in [29]. Fidel structures are implicative lattices augmented with a family of unary predicates. In the third section, we describe a semantics for N4 with the help of twist-structures over implicative lattices (see [28, 110]). The completeness result will follow from the equivalence of Fidel structures and twist-structures, also established in this section. A twist-structure is an algebraic structure deﬁned over the Cartesian square of an implicative lattice, the operations of this structure agrees with the operations of the underlying implicative lattice on the ﬁrst component and are “twisted” on the second component. Further, in Section 4 of this chapter, we prove that the class of algebras isomorphic to twist-structures admits a lattice theoretical deﬁnition. We distinguish the class of N4-lattices, prove that any twist-structure is an N4-lattice and that any N4-lattice A is isomorphic to a twist-structure over A , the implicative lattice deﬁned as quotient of A wrt to a congruence of a special form. These results imply that N4 is characterized by N4-lattices. In the next section, it is proved that N4-lattices form a variety VN4 such that the lattice EN4 of N4-extensions is dually isomorphic to the lattice of subvarieties of VN4 . In the last section of Chapter 8, we transfer all these results to the logic N4⊥ and the lattice of its extensions EN4⊥ . In this case, the twist-structures are deﬁned over Heyting algebras and for any N4⊥ -lattice A, the quotient

10

1 Introduction

A is also a Heyting algebra. We call A the basic Heyting algebra of an N4⊥ -lattice A. In Chapter 9, we develop the origins of the algebraic theory of N4⊥ lattices necessary to study the lattice of extensions of the logic N4⊥ . In particular, N4⊥ -lattices are represented in the form of Heyting algebras with distinguished ﬁlter and ideal. We deﬁne a pair of adjoint functors between categories of N4⊥ -lattices and of Heyting algebras. We prove that if a homomorphism of basic algebras can be lifted to N4⊥ -lattices, it can be done in a unique way. It is shown that congruences on an N4⊥ -lattice are in one-to-one correspondence with implicative ﬁlters and that the lattices of congruences of an N4⊥ -lattices and of its basic algebra are isomorphic. As a consequence, we describe subdirectly irreducible N4⊥ -lattices as lattices with subdirectly irreducible basic algebra. Finally, in terms of the above-mentioned representation, we formulate an embeddability criterion and describe the quotients of N4⊥ -lattices. In the last chapter, we study the structure of the lattice of N4⊥ -extensions and show that it is similar to the structure of the class of Lj-extensions. Although the distinctions of the structures of these two classes of logics are also essential. The ﬁrst of these distinctions is that N4⊥ has no contradictory extensions, whereas minimal logic has the subclass of inconsistent extensions isomorphic to the class of extensions of positive logic. We investigate the interrelations between a logic L extending N4⊥ and its intuitionistic fragment. In the lattice EN4⊥ , we distinguish the subclasses Exp of explosive logics, Nor of normal logics, and Gen of logics of general form, which play the roles similar to that of classes Int, Neg, and Par in the lattice of extensions of minimal logic. The interrelations between classes Exp, Nor and Gen are investigated with the help of notions of explosive and normal counterparts for logics in Gen. Finally, we give some ﬁrst applications of the developed theory of the lattice of N4⊥ -extensions. First, we completely describe the lattice of extensions of the logic N4⊥ C obtained by adding the Dummett linearity axiom to N4⊥ . We prove that all extensions of N4⊥ C are ﬁnitely axiomatized and decidable and that given a formula, one can eﬀectively determine which of the N4⊥ C-extensions is axiomatized by this formula. Second, we describe tabular, pretabular logics and logics with Graig’s interpolation property in the lattice of N4⊥ -extensions. Regarding the authorship of the presented results, this book contains mainly the investigations of the author, previously published in a series of articles [66–81]. Chapter 2 and Section 8.1 have a preliminary character and here we do not carefully trace the authorship of the presented results. Except

1 Introduction

11

for Chapter 2 and Section 8.1, we give explicit references to all results quoted from other authors. Acknowledgments. I am deeply indebted to Professors L.L. Maksimova and K.F. Samokhvalov for our fruitful discussions, which inspired, in fact, the beginning of this investigation. The investigations presented in the ﬁrst part of the book were carried out during my stay in Toru´ n, at the Logic Department of Nicholas Copernicus University. I am very grateful to Prof. Jerzy Perzanowski, the head of this department, for the invitation, hospitality and helpful criticism. I want to acknowledge my deep indebtedness to the Alexander von Humboldt Foundation for granting the research fellowship at Dresden University of Technology and the return fellowship. The investigations presented in the second part of the book were carried out during this period. Finally, I am especially grateful to Prof. Heinrich Wansing, my academic host in Dresden, for the very fruitful collaboration.

Chapter 2

Minimal Logic. Preliminary Remarks 2.1

Deﬁnition of Basic Logics

A propositional language L is a ﬁnite set of logical connectives of diﬀerent arities, L = {f1n1 , . . . , fknk }. A propositional constant is a connective of arity 0. Given a set of propositional variables, we deﬁne formulas of the language L via the standard inductive deﬁnition. In the ﬁrst part of the book we will consider logics and deductive systems formulated in the following propositional languages: the language of positive logic L+ := {∧2 , ∨2 , →2 }, the language L⊥ := L+ ∪ {⊥0 } extending L+ with the constant ⊥ for “absurdity”, and the language L¬ := L+ ∪ {¬1 } with the symbol ¬ for negation. Extensions of minimal logic admit equivalent formulations in the languages L⊥ and L¬ . If ϕ is a formula in some propositional language and p1 , . . . , pn are propositional variables, the denotation ϕ(p1 , . . . , pn ) means that all propositional variables of ϕ are from the list p1 , . . . , pn . By a logic we mean a set of formulas closed under the rules of substitution and modus ponens: ϕ(p1 , . . . , pn ) ϕ(ψ1 , . . . , ψn )

and

ϕ ϕ→ψ . ψ

If ϕ(ψ1 , . . . , ψn ) is obtained from ϕ(p1 , . . . , pn ) by the substitution rule, we say that it is a particular case or a substitution instance of ϕ. A deductive system is a collection of axioms and inference rules. A theorem of a deductive system is a formula provable in this system. We will usually deﬁne logics as 15

16

2 Minimal Logic. Preliminary Remarks

sets of theorems of Hilbert style deductive systems with only the inference rules of substitution and modus ponens. Therefore, to deﬁne a logic it is enough to list its axioms. For a logic L and a set of formulas X, L + X denotes the least logic containing L and all formulas of X. The symbol + also denotes the operation of taking the least upper bound in the lattice of logics. With any logic L, we associate in a standard way an inference relation L . For a set of formulas X and a formula ϕ, the relation X L ϕ means that ϕ can be obtained from elements of X and tautologies of L in a ﬁnite number of steps by using the rule of modus ponens. A set X is said to be non-trivial wrt L if X L ϕ for some ϕ. Let Li be a logic in a propositional language Li , i = 1, 2, and L1 ⊆ L2 . We say that L2 is a conservative extension of L1 if L1 ⊆ L2 and for any formula ϕ in the language L1 , ϕ ∈ L1 ⇐⇒ ϕ ∈ L2 . In this case we say also that L1 is an L1 -fragment of L2 . In what follows by a positive fragment we mean an L+ -fragment. Denote by F ∗ the trivial logic, i.e., the set of all formulas in the language L∗ , ∗ ∈ {+, ⊥, ¬}. We now deﬁne several important logics. In the choice of denotation we follow the book [93] by W. Rautenberg. Positive logic Lp is the least logic in the language L+ containing the following axioms: 1. p → (q → p) 2. (p → (q → r)) → ((p → q) → (p → r)) 3. (p ∧ q) → p 4. (p ∧ q) → q 5. (p → q) → ((p → r) → (p → (q ∧ r))) 6. p → (p ∨ q) 7. q → (p ∨ q) 8. (p → r) → ((q → r) → ((p ∨ q) → r))

2.1 Deﬁnition of Basic Logics

17

Positive logic satisﬁes Deduction Theorem: X ∪ {ϕ} Lp ψ ⇐⇒ X Lp ϕ → ψ. To prove this theorem we need axioms 1 and 2 of positive logic and the fact that modus ponens is the only inference rule. All logics considered in the book satisfy these conditions, therefore, Deduction Theorem remains true for all logics considered below. Classical positive logic Lk+ also is a logic in the language L+ and can be axiomatized modulo Lp by either of the following two axioms: P. ((p → q) → p) → p (Peirce law) E. p ∨ (p → q) (extended law of excluded middle) The version Lj⊥ of minimal logic (or Johansson’s logic) in the language L⊥ can be deﬁned as a logic axiomatized by the axioms 1–8 above. The equivalent version of minimal logic Lj¬ in the language L¬ with the negation symbol can be axiomatized by the axioms 1–8 and the following axiom: A. (p → q) → ((p → ¬q) → ¬p) (reductio ad absurdum) To make precise the statement on the equivalence of two versions of minimal logic, we deﬁne the translations θ from the language L¬ to L⊥ and ρ from L⊥ to L¬ as follows. For any ϕ ∈ F ¬ , let θ(ϕ) be a formula in the language L⊥ obtained from ϕ by replacing each subformula of the form ¬ψ by the subformula ψ → ⊥. For any ϕ ∈ F ⊥ , we denote by ρ(ϕ) a formula in the language L¬ obtained from ϕ by replacing every occurrence of ⊥ by the subformula ¬(p → p), where p is some ﬁxed propositional variable. For a set of formulas X ⊆ F ¬ , denote by θ(X) the set {θ(ϕ) | ϕ ∈ X}. Respectively, for X ⊆ F ⊥ , put ρ(X) := {ρ(ϕ) | ϕ ∈ X}. Proposition 2.1.1 The following statements hold. 1. For an arbitrary set of formulas X ⊆ F ¬ and for any formula ϕ ∈ F ¬ , X Lj¬ ϕ if and only if θ(X) Lj⊥ θ(ϕ). Moreover, Lj¬ ϕ ↔ ρθ(ϕ) for any formula ϕ. 2. For an arbitrary set of formulas X ⊆ F ⊥ and for any formula ϕ ∈ F ⊥ , X Lj⊥ ϕ if and only if ρ(X) Lj¬ ρ(ϕ). Moreover, Lj⊥ ϕ ↔ θρ(ϕ) for any formula ϕ.

18

2 Minimal Logic. Preliminary Remarks

Thus, the translations deﬁned above preserve all deductive properties and the subsequent application of two translations results in a formula equivalent to the original one. Due to these facts we pass freely in the following from the language L⊥ to the language L¬ and vice versa. We will omit the superscripts in the denotation of minimal logic and will not explicitly indicate with which version of minimal logic or of its extension we are dealing at the time. And we write F for either F ⊥ or F ¬ . Intuitionistic logic Li and minimal negative logic Ln can be axiomatized modulo minimal logic in the language L⊥ as follows: Li = Lj + {⊥ → p}, Ln = Lj + {⊥}; and in the language L¬ as follows: Li = Lj + {¬p → (p → q)}, Ln = Lj + {¬p}. Classical logic Lk, logic of classical refutability Le, and maximal negative logic Lmn can be axiomatized modulo intuitionistic logic Li, minimal logic Lj, and negative logic Ln respectively, via either the Peirce law P or the extended law of excluded middle E. Lk = Li+{p∨(p → q)}, Le = Lj+{p∨(p → q)}, Lmn = Ln+{p∨(p → q)}. The positive fragments of Lk, Le, and Lmn coincide with classical positive logic, whereas the positive fragments of Li, Lj, and Ln coincide with positive logic. Lk+ = Lk ∩ F + = Le ∩ F + = Lmn ∩ F + , Lp = Li ∩ F + = Lj ∩ F + = Ln ∩ F + . All logics introduced above except for positive and classical positive logics are extensions of minimal logic. The class of all non-trivial extensions of the logic Lj we denote by Jhn, the class of all extensions by Jhn+ . Clearly, the class of logics Jhn+ forms a lattice, where L1 + L2 is the least upper bound of logics L1 and L2 , and the intersection L1 ∩ L2 is the greatest lower bound. For an arbitrary logic L, the lattice of its extensions with lattice operations + and ∩ we denote as EL. Notice that EL is a complete lattice. in EL, the intersection For any family {Li | i ∈ I} of logics i∈I Li is a logic and it extends L. Obviously, i∈I Li is the greatest logic contained in all logics Li . For this reason, in EL, there also exists the sum of logics Σi∈I Li , i.e., the least logic containing all logics Li , i ∈ I. Recall several important formulas provable in Lp and Lj.

2.1 Deﬁnition of Basic Logics

19

Proposition 2.1.2 The following formulas are provable in Lp: 1. p → p

(the identity law).

2. (p ∨ q) ↔ (q ∨ p), (p ∧ q) ↔ (q ∧ p) (the commutativity of ∨ and ∧). 3. (p ∨ (q ∨ r)) ↔ ((p ∨ q) ∨ r), (p ∧ (q ∧ r)) ↔ ((p ∧ q) ∧ r) (the associativity of ∨ and ∧). 4. (p ∨ (q ∧ r)) ↔ ((p ∨ q) ∧ (p ∨ r)), (p ∧ (q ∨ r)) ↔ ((p ∧ q) ∨ (p ∧ r)) (the distributivity laws). 5. (p → (q → r)) ↔ (q → (p → r))

(the permutation law).

6. (p → (p → q)) ↔ (p → q) 7. (p → (q → r)) ↔ ((p ∧ q) → r)

(the contraction law). (import and export of the premiss).

8. ((p → q) ∨ r) → ((p ∨ r) → (q ∨ r)). 9. ((p → q) ∧ r) ↔ ((p ∧ r) → (q ∧ r)) ∧ r. 2 Proposition 2.1.3 The following formulas are provable in Lj: 1. ¬¬(p ∨ ¬p), 2. (p → ¬q) → (q → ¬p), 3. (p → q) → (¬q → ¬p), 4. (¬p ∨ ¬q) → ¬(p ∧ q), 5. ¬(p ∨ q) ↔ (¬p ∧ ¬q), 6. p → ¬¬p, 7. ¬(p ∧ ¬p), 8. (p ∨ q) → ¬(¬p ∧ ¬q), 9. (p ∧ q) → ¬(¬p ∨ ¬q), 10. (p → q) → ¬(p ∧ ¬q)

20

2 Minimal Logic. Preliminary Remarks

For the proof of this and the previous proposition the reader may consult one or another traditional textbook in classical logic and observe that the standard proofs of formulas listed in these propositions use only axioms of Lj or Lp respectively. It is also not hard to prove all these formulas directly or with the help of Deduction Theorem. 2 The next proposition gives some information on the results of adjoining diﬀerent new axioms to Lj. Proposition 2.1.4 [98] 1. The equality Lk = Lj + {ϕ} holds, where ϕ is one of the following formulas: (a) ¬¬p → p, (b) (¬p → q) → (¬q → p), (c) (¬p → ¬q) → (q → p), (d) ¬(¬p ∧ ¬q) → (p ∨ q), (e) ¬(¬p ∨ ¬q) → (p ∧ q), (f ) ¬(p ∧ ¬q) → (p → q). 2. Lk = Lj + {p ∨ ¬p} = Lj + {(p → q) → (¬p ∨ q)}. 3. Lj + {p ∨ ¬p} = Lj + {¬p ∨ ¬¬p} = Lj + {¬(p ∧ q) → (¬p ∨ ¬q)}. 4. Li = Lj + {(¬p ∨ q) → (p → q)}. The next proposition shows how to construct axioms of an intersection of logics. In fact, we repeat the proof of Miura’s result [63] for intersections of superintuitionistic logics (see also [16, p. 111]). We call the formula ϕ(p1 , . . . , pm ) ∨ ψ(pn+1 , . . . , pn+m ) the repeatedless disjunction of the formulas ϕ(p1 , . . . , pn ) and ψ(p1 , . . . , pm ) and denote it by ϕ∨ψ. Proposition 2.1.5 Let L ∈ {Lp, Lj}, L1 = L + {ϕi | i ∈ I}, and L2 := L + {ψj | j ∈ J}. Then L1 ∩ L2 = L + {ϕi ∨ψj | i ∈ I, j ∈ J}. Theorem and the Proof. Suppose χ ∈ L1 ∩ L2 . By Deduction properties of ∧ (see Proposition 2.1.2) we have i∈I ϕi → χ ∈ L and j∈J ψj → χ ∈ L, where I ⊆ I, J ⊆ J, I and J are ﬁnite, and every ϕi and ψj are

2.2 Algebraic Semantics

21

substitution instances of ϕi and ψj respectively. Using axiom 8 of positive logic and the distributivity laws, we then obtain (ϕi ∨ ψj ) → χ ∈ L, i∈I ,j∈J

from which χ ∈ L + {ϕi ∨ψj | i ∈ I, j ∈ J}. Conversely, assume that χ ∈ L + {ϕi ∨ψj | i ∈ I, j ∈ J}. Then χ is derivable in L from some ﬁnite set of substitution instances ϕi ∨ψj of axioms of this logic. Using axioms 6 and 7 of positive logic we can also derive χ from the set of ϕi as well as from the set of ψj . Consequently, χ ∈ L1 ∩ L2 . 2 Proposition 2.1.6 The lattices ELp and ELj are distributive. Moreover, the intersection distributes with the inﬁnite sum in these lattices. Proof. Let L ∈ {Lp, Lj}. We prove only that L ∩ Σi∈I Li = Σi∈I (L ∩ Li ), L, Li ∈ EL. Assume L = L + Γ and Li = L + Δi for i ∈ I. L ∩ Σi∈I Li

= = = = =

(L + Γ) ∩ (L + i∈I Δi ) L+ {ϕ∨ψ | ϕ ∈ Γ, ψ ∈ i∈I Δi } L + i∈I {ϕ∨ψ | ϕ ∈ Γ, ψ ∈ Δi } Σi∈I (L + {ϕ∨ψ | ϕ ∈ Γ, ψ ∈ Δi }) Σi∈I (L + Γ) ∩ (L + Δi ) 2

2.2

Algebraic Semantics

In this section, we give necessary deﬁnitions and facts concerning the algebraic semantics for propositional logics. Detailed information can be found in [92, 93]. Let L = {f1n1 , . . . , fknk } be a propositional language. An algebra A =

A, f1A , . . . , fkA of the language L is a set, where the connectives of L are interpreted as operations of respective arities, fiA : Ani −→ A. The set A is the universe of A and is denoted |A|. We write a ∈ A instead of a ∈ |A|. If A1 , . . . , An are algebras of the same language, then the direct product A1 × . . . × An is deﬁned as an algebra whose universe is the direct product

22

2 Minimal Logic. Preliminary Remarks

of universes |A1 | × . . . × |An | and the operations are componentwise. Note that πi : |A1 | × . . . × |An | −→ |Ai |, the projection onto the ith coordinate, determines an epimorphism of A1 × . . . × An onto Ai . By A → B we denote that the algebra A is embeddable into B, i.e., that there exists a monomorphism h : A → B. If K is a class of algebras, we denote by I(K) the class of algebras isomorphic to algebras from K, by H(K) the class of homomorphic images of algebras from K, by S(K) the class of algebras embeddable into algebras from K, and, ﬁnally, Up(K) denotes the class of algebras isomorphic to ultraproducts [14] of algebras from K. A matrix is, as usual, a pair M = A, DA , where A is an algebra and D A ⊆ |A| is the set of distinguished elements of this matrix. In cases where D A = {1} is one-element, we write A, 1 instead of A, {1} and identify, in fact, a matrix with an algebra in a language with an additional constant 1. A valuation in an algebra A is deﬁned as a mapping from the set of propositional variables into |A|. A valuation extends to the set of all formulas in a homomorphic way. A formula ϕ is said to be true on a matrix M = A, DA , M |= ϕ, if for any A-valuation v, v(ϕ) ∈ DA . An identity ϕ = ψ is true on an algebra A, A |= ϕ = ψ, if v(ϕ) = v(ψ) for any A-valuation v. The set L(M) := {ϕ | M |= ϕ} is the logic of a matrix M and the set Eq(A) := {ϕ = ψ | A |= ϕ = ψ} is the equational theory of an algebra A. For a class Kof matrices (algebras), we deﬁne L(K) := {L(M) | M ∈ K} (Eq(K) := {Eq(A) | A ∈ K}). The direct product of matrices M1 = A1 , DA1 , . . . , Mn = An , DAn is deﬁned as M1 × . . . × Mn = A1 × . . . × An , DA1 × . . . × DAn . Since the operations on the direct product are componentwise, we have L(M1 × . . . × Mn ) = LM1 ∩ . . . ∩ LMn . In this part of the book we deal mainly with matrices having one distinguished element. Let A be an algebra of the language L+ ∪ {1} (L⊥ ∪ {1}, L¬ ∪ {1}). Note that A |= ϕ is equivalent to ϕ = 1 ∈ Eq(A). Elements of LA are called identities of A in this case. An algebra A is a model for a logic L if L ⊆ LA. If L = LA, we say that A is a characteristic model for L. It is clear that the class of models for some logic L forms a variety. Write A |= L if A is a model of L. Denote M od(L) := {A | A |= L}.

2.2 Algebraic Semantics

23

Proposition 2.2.1 [93] Every Lj-extension has a characteristic model. The reader is expected to be familiar with lattices and with distributive lattices. If A = A, ∧, ∨ is a lattice, then the lattice ordering ≤A is deﬁned by the condition a ≤A b ⇐⇒ a ∧ b = a. If a, b ∈ A and a ≤A b, we denote by [a, b] an interval wrt the lattice ordering with end points a and b, i.e., [a, b] := {c ∈ A | a ≤A c ≤A b}. In what follows we omit the lower index in the denotation of the ordering if it does not lead to a confusion. For a ≤ b and c ∈ [a, b], an element d is said to be a complement of c in the interval [a, b] if c ∨ d = b and c ∧ d = a. Recall that if the lattice A is distributive and c ∈ A has a complement in [a, b] ⊆ A, then this complement is unique. An algebra A = A, ∧, ∨, →, 1 is called an implicative lattice if A, ∧, ∨, 1 is a lattice with the greatest element 1 and such that for any a, b ∈ A there exists a supremum {x | a ∧ x ≤ b} equal to the value of the implication (or relative pseudo-complement) operation a → b. Here ≤ denotes the lattice ordering of A. All implicative lattices form a variety and the logic of this variety is Lp [92]. Proposition 2.2.2 [92] Let A be an implicative lattice and a, b ∈ A. Then the following holds. 1) a → b = 1 iﬀ a ≤ b; 2) a = b iﬀ a → b = 1 and b → a = 1; 3) b ≤ a → b; 4) a ∧ (a → b) = a ∧ b. 2 By a j-algebra we mean an algebra A = A, ∧, ∨, →, ⊥, 1 of the language L⊥ ∪ {1} such that A, ∧, ∨, →, 1 is an implicative lattice and the constant ⊥ is interpreted as an arbitrary element of the universe A. Minimal logic Lj corresponds to the variety of j-algebras [92], which we denote as Vj . Equivalently, we can deﬁne j-algebras in the language L¬ ∪{1} as implicative lattices with the negation operation satisfying the property a → ¬b = b → ¬a. These equivalent deﬁnitions are related as follows: ¬a = a → ⊥, ⊥ = ¬1. A Heyting algebra is a j-algebra with the least element ⊥. Intuitionistic logic Li is the logic of the variety Vi of Heyting algebras [92].

24

2 Minimal Logic. Preliminary Remarks

A negative algebra is a j-algebra with ⊥ = 1. Obviously, negative algebras are distinguished in the variety of j-algebras via the identity ⊥. Therefore, minimal negative logic Ln is the logic of the variety Vn of negative j-algebras. An arbitrary variety V of implicative lattices, j-algebras, Heyting algebras, or negative algebras determines a logic LV extending Lp, Lj, Li, or respectively Ln. More exactly, let Sub(V ) denote the lattice of subvarieties of an arbitrary variety V . For a logic L ∈ Jhn+ , deﬁne a variety of j-algebras V (L) := {A | A ∈ Vj , A |= L}, and for a variety V ∈ Sub(Vj ), deﬁne a logic L(V ) := {ϕ | A |= ϕ for all A ∈ V }. It is clear that for any L ∈ Jhn+ and V ∈ Sub(Vj ) we have V (L) ∈ Sub(Vj ) and L(V ) ∈ Jhn+ . Moreover, the following statement holds. Theorem 2.2.3 The mappings V : Jhn+ → Sub(Vj ) and L : Sub(Vj ) → Jhn+ are mutually inverse dual lattice isomorphisms. The restrictions V Sub(Vi ) and L ELi are mutually inverse dual isomorphisms of the lattices ELi and Sub(Vi ), whereas V Sub(Vn ) and L ELn are mutually inverse dual isomorphisms of the lattices ELn and Sub(Vn ). 2 By a dual isomorphism of lattices A1 and A2 we mean an isomorphism from the lattice A1 onto the lattice (A2 )op with the inverse ordering. For a j-algebra A and a Heyting algebra B we denote by A⊕B the direct sum of these algebras. It is a j-algebra in which the unit element of A is identiﬁed with the zero of B, and for any a ∈ A and b ∈ B, we have a ≤ b. Recall that a non-empty subset F of an implicative lattice (a j-algebra) A is a ﬁlter on A if it satisﬁes the following conditions: 1) for any x, y ∈ F , x∧y ∈ F ; 2) for any x ∈ F and y ∈ A, if x ≤ y, then y ∈ F . Denote by F(A) the set of all ﬁlters on A. If X ⊆ A, then X denotes a ﬁlter generated by the set X, i.e., the least ﬁlter on A containing X. It is clear that

X = {a ∈ A | b1 ∧ . . . ∧ bn ≤ a for some b1 , . . . , bn ∈ X}. Instead of {a} we write a .

2.2 Algebraic Semantics

25

For a Heyting algebra A, denote by Fd (A) its ﬁlter of dense elements and by R(A) the Boolean algebra of its regular elements. Recall that Fd (A) = {a ∈ A | ¬¬a = 1} = {a ∨ ¬a | a ∈ A}, R(A) = {a ∈ A | a ∨ ¬a = 1} = {a ∈ A | ¬¬a = a}, and R(A) ∼ = A/Fd (A). Let A be an implicative lattice (a j-algebra). For any congruence θ on A, we deﬁne a ﬁlter Fθ := {a ∈ A | aθ1A }. For any ﬁlter F on A, deﬁne a congruence θF := {(a, b) | a → b, b → a ∈ F }. It is clear that θ = θFθ and F = FθF . We have thus deﬁned a one-to-one correspondence between the set of congruences and the set of ﬁlters on the implicative lattice (j-algebra) A. Notice that for an identity congruence IdA , FIdA = {1A }. We deﬁne a subdirectly irreducible algebra A as an algebra, which has minimal non-identity congruence (comp. [14]). Taking into account the above correspondence between ﬁlters and congruences on implicative lattices and j-algebras, we arrive at the following statement. Proposition 2.2.4 An implicative lattice (a j-algebra) is subdirectly irreducible if and only if {1A } = {F | F is a ﬁlter on A, F = {1A }}. 2 An element A of an implicative lattice (a j-algebra) A is called an opremum, if A = 1 and for any a ∈ A, the inequality a = 1 implies a ≤ A . Proposition 2.2.5 An implicative lattice (a j-algebra) A is subdirectly irreducible iﬀ it has an opremum. 2 For Heyting algebras, a similar result was stated by C.M. McKay [62]. It can be transferred to implicative lattices and j-algebras in a trivial way. Due to the well-known Birkhoﬀ theorem [14], any algebra A is isomorphic to a subdirect product of subdirectly irreducible algebras (being homomorphic images of A). This immediately implies that every variety is completely determined by its subdirectly irreducible algebras. Let M odf si (L) denote the set of ﬁnitely generated subdirectly irreducible models of a logic L. In view of the correspondence between logics and varieties, we have the following

26

2 Minimal Logic. Preliminary Remarks

Proposition 2.2.6 Let L1 and L2 be logics extending Lp (Lj). We have L1 = L2 if and only if M odf si (L1 ) = M odf si (L2 ). 2 We call a Peirce algebra an implicative lattice satisfying the identity P (or, equivalently, E). Let 2P = {0, 1}, ∧, ∨, →, 1 be a two-element Peirce algebra. Proposition 2.2.7 L2P = Lk+ . Proof. It is clear that Lk+ ⊆ L2P . To prove the inverse inclusion we show that there is only one subdirectly irreducible Peirce algebra, 2P . Let A be a Peirce algebra with more than two elements. We show that for any 1 = a ∈ A there exists a ﬁlter Fa = {1} on A with a ∈ Fa . Take 1 = a ∈ A. There is also a b ∈ A with 1 = b = a. If b ≤ a, then a ∈ b . Assuming b ≤ a, we consider an element a → b. Since a = b, we have a ∧ a ≤ b and a ∧ 1 ≤ b. By deﬁnition of relative pseudo-complement we conclude a ≤ a → b and a → b = 1, i.e., a ∈ a → b and a → b = {1}. We have thus constructed a collection {Fa | a ∈ A} of ﬁlters on A such that Fa = {1}, a∈A

and Fa = {1} for all a ∈ A. By Proposition 2.2.4 this means that A is not subdirectly irreducible. 2 Let 2 = {0, 1}, ∨, ∧, →, 0, 1 be a two-element Heyting algebra, which is a characteristic model for classical logic, L2 = Lk. By 2 = {0, 1}, ∨, ∧, →, 1, 1 we denote a two-element negative algebra. Proposition 2.2.8 L2 = Lmn. Proof. Obviously, ⊥ and ((p → q) → p) → p are identities of 2 , and so L2 ⊇ Lmn. The inverse inclusion can be stated similar to Proposition 2.2.7. 2 Proposition 2.2.9 [93] The logic Lj has exactly two maximal non-trivial extensions, Lk and Lmn. Every non-trivial Lj-extension is contained in one of them. Proof. Consider an arbitrary non-trivial extension L of Lj and its characteristic model A, L = LA. Obviously, A is not one-element. If ⊥A = 1A ,

2.2 Algebraic Semantics

27

then for every a ∈ A, a = 1, the set {a, ⊥A } is the universe of a subalgebra isomorphic to 2 . Consequently, LA ⊆ L2 . If ⊥A = 1A , then the subalgebra with the universe {⊥A , 1A } is isomorphic to 2, whence LA ⊆ Lk. 2 Recall several facts from the universal algebra. A variety V is called congruence distributive if for any algebra A ∈ V, the lattice Con(A) of congruences of algebra A is distributive. A variety V is called congruence permutable if for any algebra A ∈ V, the congruences are permutable wrt composition. In this case the join of two congruences coincide with their composition θ1 ∨ θ1 = θ1 ◦ θ2 for any θ1 , θ2 ∈ Con(A). An arithmetic variety is a variety, which is congruence permutable and congruence distributive. According to Pixley’s theorem (see [14]) a variety V is arithmetic if and only if there exists a term m(x, y, z) such that the identities m(x, y, x) = m(x, y, y) = m(y, y, x) = x hold in V. Proposition 2.2.10 The variety of j-algebras is arithmetic. Proof. In case of j-algebras, as well as in case of Heyting algebras (see [14]), we can use the term m(x, y, z) := ((x → y) → z) ∧ ((z → y) → x) ∧ (x ∨ z) to establish that the varieties of j-algebras and Heyting algebras are arithmetic. The veriﬁcation is straightforward. 2 Let us consider an ω-generated free j-algebra Aω and its congruence lattice Con(Aω ), which is distributive by the last proposition. Moreover, congruences of Con(Aω ) are permutable wrt the composition. Elements of Aω can be identiﬁed with classes of equivalence of formulas wrt Lj, |Aω | = {[ϕ] | ϕ ∈ F}, where [ϕ] := {ψ | ϕ ↔ ψ ∈ Lj}. With any L ∈ Jhn+ we associate the congruence θL := {([ϕ0 ], [ϕ1 ]) | ϕ0 ↔ ϕ1 ∈ L}. Clearly, the mapping L → θL is one-to-one and preserves the ordering. Consequently, to prove that it is a lattice embedding it is enough to check that for any L0 , L1 ∈ Jhn+ , the congruences θL0 ∧θL1 and θL0 ∨θL1 also have

28

2 Minimal Logic. Preliminary Remarks

the form θL for a suitable L. Observe that θL is closed under substitution, i.e., if [ϕ0 ]θL [ϕ1 ], then [ϕ0 (ψ1 , . . . , ψn )]θL [ϕ1 (ψ1 , . . . , ψn )] for any ψ1 , . . . , ψn . It can easily be seen that if θ ∈ Con(Aω ) is closed under substitution, then Lθ = {ϕ | [ϕ]θ1}, where 1 is the class of Lj-tautologies, is a logic from Jhn+ and θ = θLθ . In this way, it is enough to check that θL0 ∧ θL1 and θL0 ∨ θL1 are closed under substitution. We consider only the non-trivial case of θL0 ∨ θL1 . Since Aω is congruence permutable, θL0 ∨ θL1 = θL0 ◦ θL1 . So, [ϕ0 ]θL0 ∨ θL1 [ϕ1 ] if and only if there is a formula ψ such that [ϕ0 ]θL0 [ψ] and [ψ]θL1 [ϕ1 ]. This immediately implies that θL0 ∨ θL1 is closed under substitution. So, the set of congruences of the form θL forms a lattice. It is easy to see that L → θL is an order isomorphism of the lattice Jhn+ and the lattice of congruences of the form θL . If two lattices are isomorphic as orders, they are isomorphic as lattices too. We have thus proved in an algebraic way the distributivity of Jhn+ . Corollary 2.2.11 The lattice Jhn+ is distributive.

2.3

Kripke Semantics

In conclusion of this chapter we say a few words on the Kripke-style semantics for minimal logic and its extensions. A j-frame is a triple W = W, , Q , where W is a set of possible worlds, is an accessibility relation such that W, is an ordinary Kripke frame for intuitionistic logic, i.e., a partially ordered set, and Q ⊆ W is a cone (upward closed set) with respect to , which we call the cone of abnormal worlds. Worlds lying out of Q are called normal. A valuation v of a j-frame W is a mapping from the set of propositional variables to the set of cones of the ordering W, . A model μ = W, v is a pair consisting of a j-frame and its valuation. Say in this case that μ is a model on W. The forcing relation between models and formulas is deﬁned in exactly the same way as for ordinary Kripke frames. The only exception is the case of constant ⊥. More precisely, we deﬁne the relation μ |=x ϕ, where μ = W, v is a model, W = W, , Q , x ∈ W , and ϕ is a formula, by induction on the structure of formulas as follows. For a propositional variable p, put μ |=x p

⇐⇒

x ∈ v(p).

2.3 Kripke Semantics

29

And further, μ |=x ϕ ∧ ψ

⇐⇒

μ |=x ϕ and μ |=x ψ;

μ |=x ϕ ∨ ψ

⇐⇒

μ |=x ϕ or μ |=x ψ;

μ |=x ϕ → ψ

⇐⇒

∀y ∈ W (x y ⇒ (μ |=y ϕ ⇒ μ |=y ψ)).

We did not consider yet the case of constant ⊥, and we put μ |=x ⊥

⇐⇒

x ∈ Q.

In particular, for a negated formula ¬ϕ considered as an abbreviation for ϕ → ⊥, we have μ |=x ¬ϕ

⇐⇒

∀y ∈ W (x y ⇒ (μ |=y ϕ ⇒ y ∈ Q)).

Read μ |=x ϕ as “a formula ϕ is true at a world (or at a point) x in a model μ”. A formula ϕ is true on a model μ = W, v , μ |= ϕ, if μ |=x ϕ holds for all x ∈ W . A formula ϕ is true on a j-frame W, W |= ϕ, if it is true on a model W, v for an arbitrary valuation v of the j-frame W. A formula ϕ is valid on the class K of Kripke j-frames if W |= ϕ for any j-frame W ∈ K. Let W = W, , Q be a j-frame and let K ⊆ W be a cone wrt . We deﬁne a j-frame W K in the following way: W K := K, K , QK , where K := ∩K 2 , QK := Q ∩ K. If μ = W, v is a model on W, then μK :=

W K , v K , where v K (p) := v(p) ∩ K for all propositional variables p. If [x] ↑:= {y ∈ W | x y} is a cone generated by x, we write W x and μx instead of W [x]↑ and μ[x]↑ respectively. Lemma 2.3.1 Let W = W, , Q be an arbitrary j-frame, μ a model on W, and K ⊆ W a cone. For any x ∈ K and an arbitrary formula ϕ, we have μ |=x ϕ ⇐⇒ μK |=x ϕ. In particular, W |= ϕ =⇒ W K |= ϕ. We say that a j-frame W is a model for a logic L ∈ Jhn, W |= L, if W |= ϕ for all ϕ ∈ L. For a class of j-frames K we put LK := {ϕ | ∀W ∈ K (W |= ϕ)}. A logic L ∈ Jhn is characterized by a class of j-frames K if L = LK.

30

2 Minimal Logic. Preliminary Remarks

We call a j-frame W = W, , Q normal if Q = ∅, i.e., if all worlds of this frame are normal. It is clear that normal j-frames can be identiﬁed with ordinary Kripke frames for intuitionistic logic. We call a j-frame W =

W, , Q abnormal if Q = W , i.e., if all worlds are abnormal. Finally, a j-frame W = W, , Q will be called identical if the accessibility relation coincides with the identity relation on W , = IdW . Proposition 2.3.2 [98] 1. Minimal logic Lj is characterized by the class of all j-frames. 2. Intuitionistic logic Li is characterized by the class of all normal j-frames. 3. Minimal negative logic Ln is characterized by the class of all abnormal j-frames. 4. Logic of classical refutability Le is characterized by the class of all identical j-frames. 5. Classical logic Lk is characterized by the class of all identical normal j-frames. 6. Maximal negative logic Lmn is characterized by the class of all identical abnormal j-frames. Further, we deﬁne several special classes of j-frames. Let W = W, , Q be a j-frame. We say that W is separated if ∀x, y ∈ W ((x ∈ Q ∧ y ∈ Q) ⇒ x y). And we say that W is closed if ∀x, y ∈ W ((x ∈ Q ∧ y ∈ Q) ⇒ ¬(x y)). Denote by Sep the class of all separated j-frames and by Cl the class of all closed j-frames. Proposition 2.3.3 [98] The logic Lj+{(p → ⊥)∨(⊥ → p)} is characterized by the class Sep, and the logic Lj + {⊥ ∨ (⊥ → p)} is characterized by Cl. A j-frame W = W, , Q is called dense if ∀x ∈ W (x ∈ Q ⇒ ∃y x∀z y(z ∈ Q)). The class of all dense j-frames is denoted by Den. Proposition 2.3.4 [124] The logic Lj + {¬¬(⊥ → p)} is characterized by the class Den.

Chapter 3

Logic of Classical Refutability1 We start the investigation of the class of Lj-extensions with the logic of classical refutability Le = Lj + {((p → q) → p) → p}. This important logic arose in the work of diﬀerent authors and with diﬀerent motivations. It was introduced for the ﬁrst time in the P. Bernays review [10] of H.B. Curry’s articles [20, 21]. P. Bernays observed that one can obtain a new logical system, namely Le, by extending axiom schemes of classical positive logic Lk+ to the language with additional constant ⊥ in the same way as one can obtain minimal logic Lj by extending axiom schemes of positive logic to the language L⊥ . Two years later, the system equivalent to Le was introduced in S. Kanger’s work [42]. Kanger’s reason for deﬁning Le is that “. . . it constitutes a weakened classical calculus in the same sense as the minimal calculus is a weakened intuitionistic calculus”. Further, this logic was studied by S. Kripke [48], who stated, in particular, the equation Le = Lk ∩ Lmn. The name “logic of classical refutability” was suggested in the H.B. Curry monograph [22]. In [22, Ch.6, Sec.A], one can ﬁnd the discussion of this name. In [98], K. Segerberg studied the Kripke-style semantics for numerous extensions of minimal logic, and among them for Le [98, p.46]. J. Porte [84, 85] investigated interrelations between logic of classical refutability and L ukasiewicz’s modal logic [52, 53]. His results will play an essential part in Chapter 7, where we will study the generalized version of negation as reduction to absurdity. In [85], it was stated, in particular, that Le is a four-valued logic. The same four-element matrix for Le will be introduced in Section 3.1 in a diﬀerent way, as a simplest characteristic model for Le. 1

Parts of this chapter were originally published in [67, 68].

31

32

3 Logic of Classical Refutability

Another time, the logic Le arose under the name of Carnot’s logic CAR in the work [19] by N.C.A. da Costa and J.-Y. B´eziau to explicate some ideas of Lasare Carnot. The equality CAR = Le was stated by I. Urbas [108]. The author studies in [66] also led to the system Le, and it arises here in a rather unexpected way, from the investigation of the constructivity concept suggested by K.F. Samokhvalov [97]. In [66], it was proved that Le coincides with the logic of all exactly constructive systems in the sense of K.F. Samokhvalov. We also established in [66] the maximality property for Le. Adjoining to Le a new classical tautology gives classical logic, and adjoining a new maximal negative tautology results in maximal negative logic. In this respect, Le is similar to the logic P 1 suggested by A. Sette [102], the ﬁrst example of maximal paraconsistent logic. The maximality property of Le will be presented in Section 3.1. In the conclusion to this section we show that the class Jhn of all nontrivial extensions of minimal logic is divided into three intervals: the interval of well-known intermediate logics lying between the intuitionistic and the classical logics; the interval of negative logics lying between minimal and maximal negative logics and the interval of properly paraconsistent logics, which all lie between minimal logic and Le. The results of Section 3.2 were inspired by A. Karpenko article [43], where isomorphs of classical logic into three-valued Bochvar’s logic B3 were considered. It turns out that in Le one can naturally deﬁne one isomorph of classical logic and two diﬀerent isomorphs of maximal negative logic. Starting from these isomorphs, we deﬁne translations of Lmn and Lk into Le. In the next chapter, these translations will be generalized to translations of arbitrary negative and intermediate logics into properly paraconsistent extensions of minimal logic, which allow one to deﬁne the notions of intuitionistic and negative counterparts of a paraconsistent Lj-extension.

3.1

Maximality Property of Le

According to Proposition 2.1.5 the intersection of logics Lmn = Lj + {⊥, p∨ (p → q)} and Lk = Lj + {⊥ → p, p ∨ (p → q)} is axiomatized as follows: Lk ∩ Lmn = Lj + {p ∨ (p → q), ⊥ ∨ (⊥ → p)}. The second formula is a substitution instance of the ﬁrst one and we have thus proved the following statement, announced for the ﬁrst time by S. Kripke in 1959 [48].

3.1 Maximality Property of Le

33

Proposition 3.1.1 Le = Lk ∩ Lmn. This representation of Le allows one to make the following observations. Lemma 3.1.2

1. The following formulas are in Le:

p ∨ ¬p, ¬(p ∧ ¬p), ¬(p ∧ q) ↔ (¬p ∨ ¬q), ¬(p ∨ q) ↔ (¬p ∧ ¬q). 2. Le does not contain ¬¬p → p and ¬p → (p → q). Proof. 1. These formulas are classical tautologies, which can be easily deduced in maximal negative logic using the scheme ¬ϕ. 2. Assuming Le ¬¬p → p we have Lmn ¬¬p → p. But Lmn ¬¬p, hence Lmn p. The substitution rule implies that any formula is provable in Lmn, a contradiction. If Lmn ¬p → (p → q), then Lmn p → q. Substituting ¬p for p in the latter formula we again have Lmn q. 2 Now we consider models for Le. We call A = A, ∨, ∧, →, ⊥, 1 a Peirce-Johansson algebra (pj-algebra) if A, ∨, ∧, →, 1 is a Peirce algebra and the constant ⊥ is interpreted as an arbitrary element of the universe A. These algebras provide a semantics for the logic Le. Recall that Peirce algebras provide algebraic semantics for Lk+ and that Le can be considered as an expansion of Lk+ to the language L⊥ . These facts immediately imply the following Proposition 3.1.3 A j-algebra A = A, ∨, ∧, →, ⊥, 1 is a model for Le if and only if A is a pj-algebra. 2 List some simple properties of pj-algebras. Proposition 3.1.4 Let A = A, ∧, ∨, →, ⊥, 1 be a pj-algebra. 1. The interval [⊥, 1]A forms a subalgebra of A, which is a Boolean algebra. 2. For any a ∈ A, if a ≤ ⊥, then ¬a = 1. 3. If ⊥ = 1 and [⊥, 1]A = A, then A contains an element incomparable with ⊥.

34

3 Logic of Classical Refutability

Proof. All statements of the proposition can easily be deduced from the deﬁnition of the relative pseudo-complement. Consider, for example, the last statement. 3. There exists an element a under ⊥ by assumption. Then ⊥ → a is incomparable with ⊥ in view of the equality ⊥ ∨ (⊥ → a) = 1. 2 Proposition 3.1.5 Let A be a pj-algebra. Then either A is a model for Lmn, or A is a model for Lk, or LA ⊆ Le, i.e., A is a characteristic model for Le. Proof. Let A be a pj-algebra. If ⊥ = ¬1 = 1, then ¬a = 1 for any a ∈ A by Item 2 of the previous proposition. This means that A is a model for Lmn. Assume ⊥ = 1, then [⊥, 1]A is a non-trivial Boolean algebra. If [⊥, 1]A = A, then A is a model for classical logic. Finally, assume that ⊥ = 1 and [⊥, 1]A = A. In this case, A contains a subalgebra isomorphic to 2, whence, LA ⊆ L2 = Lk. Consider the ﬁlter ⊥ and the corresponding quotient algebra. Since ¬a ≥ 0 for any a ∈ A, the algebra A/ ⊥ satisﬁes the identity ¬p = 1. Therefore, L(A/ ⊥ ) ⊇ Ln. Moreover, ((p → q) → p) → p is an identity of A/ ⊥ as a quotient of A. Consequently, L(A/ ⊥ ) = Lmn, and we have LA ⊆ Lk ∩ Lmn = Le. 2 Consider the lattice 4 = {0, 1, −1, a}, ≤ , where −1 ≤ a ≤ 1, −1 ≤ 0 ≤ 1, and the elements a and 0 are incomparable. It is a Peirce algebra. Interpreting ⊥ as 0 turns it into a pj-algebra. By the previous proposition we obtain. Corollary 3.1.6 4 is a characteristic model for Le. Proof. Indeed, 4 is neither a Boolean algebra, nor a negative algebra. 2 is the simplest among characteristic models Remark. The j-algebra for Le. Propositions 3.1.4 and 3.1.5 easily imply that any characteristic model for Le must contain at least four elements. Indeed, the unity element diﬀers from ⊥, there is a third element under ⊥, and there is a fourth element incomparable with ⊥. Now we are ready to prove the maximality property for Le. 4

3.2 Isomorphs of Le

35

Theorem 3.1.7 Let ϕ ∈ Le. There are three possible cases: 1. Le + {ϕ} is trivial; 2. Le + {ϕ} = Lmn; 3. Le + {ϕ} = Lk. Proof. Assume that Le+ {ϕ} is non-trivial and A is its characteristic model. The inclusion LA ⊆ Le fails, since ϕ is not in Le. By Proposition 3.1.5 we have either LA = Lk or LA = Lmn. 2 We now make an important observation on the structure of the class Jhn of all nontrivial extensions of Lj. Let Int := {L | L ∈ Jhn, ⊥ → p ∈ L} be the class of all intermediate logics; let Neg := {L | L ∈ Jhn, ¬p ∈ L} be the class of all negative logics, i.e., Neg consists of logics with a degenerate negation. Finally, let Par := Jhn \ (Int ∪ Neg) be the class of all properly paraconsistent logics. Obviously, the class Jhn is a disjoint union of the classes Int, Neg, and Par. It is well known that L ∈ Int if and only if Li ⊆ L ⊆ Lk. It turns out that the other two classes also form intervals in the lattice Jhn+ . Proposition 3.1.8 Let L ∈ Jhn+ . Then the following equivalences hold: 1. L ∈ Neg if and only if Ln ⊆ L ⊆ Lmn; 2. L ∈ Par if and only if Lj ⊆ L ⊆ Le. Proof. 1. If L ∈ Neg, then Ln is contained in L by deﬁnition. At the same time, adding the axiom ¬¬p → p to L leads to a trivialization, therefore, L cannot be extended to Lk, consequently, L ⊆ Lmn by Proposition 2.2.9. 2. Let L ∈ Par, and let A be a characteristic model for L. Since L ∈ Neg, the inequality ⊥ = 1 holds in A, hence {⊥, 1} is a nontrivial Boolean algebra and a subalgebra of A. Consequently, L ⊆ Lk. Further, L ∈ Int, therefore, the quotient A/ ⊥ is nontrivial and has the greatest element ⊥, hence it contains a two-element subalgebra isomorphic to 2 . The latter means that L ⊆ Lmn. Thus, L ⊆ Le = Lk ∩ Lmn. 2

3.2

Isomorphs of Le

The term “isomorph” was used in the ﬁrst monograph on multi-valued logics [94], and now it looks a bit archaic. Let L1 and L2 be logics and let L2 be given via its logical matrix. Due to N. Rescher [94], an isomorph of the logic

36

3 Logic of Classical Refutability

L1 in the logic L2 is a deﬁnition of a matrix for L1 in the matrix for L2 with the help of term operations. We can deﬁne a translation of L1 into L2 whenever some isomorph of L1 in L2 is given. The relations between interdeﬁnability of logical matrices and mutual translations of logics was studied in detail in the works by P. Wojtylak [122, 123]. However, it will be quite enough for our goal to use the old notion of the isomorph. The interest of the author in isomorphs, which can be deﬁned inside logic of classical refutability, is connected with the talk of A. Karpenko at the First World Congress on Paraconsistency (see [43]), in which he considered isomorphs of classical logic in three-valued Bochvar’s logic B3 given via so-called “internal” and “external” connectives. He also tried to argue that diﬀerent isomorphs of classical logic in the given many-valued logic determine the paraconsistent structure of this logic. The logic Le as well as all other extensions of minimal logic takes a borderline position among paraconsistent logics, and this fact naturally gives rise to the question of isomorphs, which can be deﬁned inside the logic Le. It turns out that there is only one natural isomorph of classical logic in Le. At the same time, there are two isomorphs of maximal negative logic. As we will see in the next chapter, the translations of Lk and Lmn deﬁned by these isomorphs can be used to deﬁne translations of intermediate and negative logics into arbitrary paraconsistent extensions of Lj. In this sense, the studying of isomorphs of the logic Le plays the key role in the investigation of the class of Lj-extensions. Consider the 4-element matrix 4 = {1, 0, a, −1}, ∨, ∧, →, ¬, {1} for Le and deﬁne the mapping ε(x) := ¬¬x. x 1 a 0 −1

ε(x) 1 1 0 0

The operations ∨ε , ∧ε , →ε , and ¬ε are deﬁned as follows: x ∗ε y := ε(x ∗ y) = ε(x) ∗ ε(y), ∗ ∈ {∨, ∧, →}, ¬ε x := ¬ε(x), These operations determine an isomorph of classical logic in Le, which we denote Lkε . The fact that Lkε really is an isomorph of Lk can easily be veriﬁed by considering the truth tables for the above operations.

3.2 Isomorphs of Le

37

∧ε 1 a 0 −1

1 1 1 0 0

a 1 1 0 0

0 0 0 0 0

−1 0 0 0 0

∨ε 1 a 0 −1

1 1 1 1 1

→ε 1 a 0 −1

1 1 1 1 1

a 1 1 1 1

0 0 0 1 1

−1 0 0 1 1

¬ε 1 a 0 −1

0 0 1 1

a 1 1 1 1

0 1 1 0 0

−1 1 1 0 0

As we can see, rows and columns corresponding to elements 1 and a are identical. The same holds for elements 0 and −1. Thus, identifying these pairs of elements we obtain two-valued truth tables for operations of classical logic. It is not hard to check with the help of the above truth tables that the mapping ε deﬁnes an epimorphism from 4 onto the two-element Boolean algebra 2ε with the universe {1, 0}. It is also clear that 2ε is a subalgebra of 4 . Lemma 3.2.1 The mapping ε : 4 → 2ε , where 2ε is a subalgebra of 4 with the universe {1, 0}, is an epimorphism. 2 We now consider the mapping δ(x) := ¬¬x → x, which acts on the set of truth-values as follows. x 1 a 0 −1

δ(x) 1 a 1 a

As above, deﬁne the operations ∨δ , ∧δ , →δ , ¬δ : x ∗δ y := δ(x ∗ y) = δ(x) ∗ δ(y), ∗ ∈ {∨, ∧, →}, ¬δ x := ¬δ(x).

38

3 Logic of Classical Refutability

The truth tables of these operations look as follows. ∧δ 1 a 0 −1

1 1 a 1 a

a a a a a

0 1 a 1 a

−1 a a a a

∨δ 1 a 0 −1

1 1 1 1 1

→δ 1 a 0 −1

1 1 1 1 1

a a 1 a 1

0 1 1 1 1

−1 a 1 a 1

¬δ 1 a 0 −1

1 1 1 1

a 1 a 1 a

0 1 1 1 1

−1 1 a 1 a

As we can see, the pairs of elements 1 and 0, a and −1 are indiscernible with respect to the introduced operations. Identifying these elements, we obtain truth tables of the two-element negative algebra 2δ with the universe {1, a}. The algebra 2δ is a characteristic model for maximal negative logic Lmn. This fact allows one to conclude that the introduced operations deﬁne an isomorph of Lmn into Le, which we denote Lmnδ . Moreover, one can check that the mapping δ preserves the operations of 4 . Lemma 3.2.2 The mapping δ : 4 → 2δ , where 2δ is a two-element negative algebra with the universe {1, a}, is an epimorphism. 2 Note that 2δ is not a subalgebra of 4 , though it is an implicative sublattice of 4 . Finally, we deﬁne the mapping τ (x) := x ∧ ⊥ (its action on the truth values of Le is in the table below) x 1 a 0 −1

τ (x) 0 −1 0 −1

and the operations x ∨τ y := τ (x ∨ y) = τ (x) ∨ τ (y), x ∧τ y := τ (x ∧ y) = τ (x) ∧ τ (y),

3.2 Isomorphs of Le

39

x →τ y := τ (x → y) = τ (τ (x) → τ (y)), ¬τ x := τ (¬x). Consider the truth tables of these operations. ∧τ 1 a 0 −1

1 0 −1 0 −1

→τ 1 a 0 −1

1 0 0 0 0

a −1 −1 −1 −1 a −1 0 −1 0

0 0 −1 0 −1 0 0 0 0 0

−1 −1 0 −1 0

−1 −1 −1 −1 −1

∨τ 1 a 0 −1

1 0 0 0 0

¬τ 1 a 0 −1

0 0 0 0

a 0 −1 0 −1

0 0 0 0 0

−1 0 −1 0 −1

Again, we see that the pairs of elements 1 and 0, a and −1 are indiscernible with respect to the introduced operations and that their identiﬁcation yields the truth tables of the two-element negative algebra 2τ :=

{0, −1}, ∨τ , ∧τ , →τ , ¬τ , where 0 plays the part of a unit element and the negation ¬⊥ is identically equal to 0, the conjunction and disjunction operations are induced by the respective operations of the algebra 4 , whereas the implication →τ is deﬁned as x →τ y := (x → y) ∧ ⊥. Since L2τ = Lmn, we conclude that the operations ∨τ , ∧τ , →τ , and ¬τ deﬁne an isomorph of Lmn into Lk with a new distinguished value 0, i.e., the set of tautologies of the matrix 4τ = {1, 0, a, −1}, ∨τ , ∧τ , →τ , ¬τ , {0} with the only distinguished value 0 coincides with Lmn. This isomorph is denoted as Lmnτ . Again we note the following fact Lemma 3.2.3 The mapping τ : 4 → 2τ , where 2τ is a two-element negative algebra with the universe {0, −1} and unit element 0, is an epimorphism. 2 The isomorphs deﬁned above lead to the following translations of classical and maximal negative logics into Le.

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3 Logic of Classical Refutability

Proposition 3.2.4 For any formula ϕ, the following equivalences hold: 1. Lk ϕ ⇐⇒ Le ¬¬ϕ; 2. Lmn ϕ ⇐⇒ Le ¬¬ϕ → ϕ; 3. Lmn ϕ ⇐⇒ Le ⊥ → (ϕ ∧ ⊥) ⇐⇒ Le ⊥ → ϕ. Proof. 1. Assume Le ¬¬ϕ. In this case Lk ¬¬ϕ since Lk extends Le. In Lk, any formula is equivalent to its double negation, whence Lk ϕ. Let us prove the inverse implication. Suppose that a formula ϕ = ϕ (p1 , . . . , pn ) is such that Lk ϕ, but Le ¬¬ϕ. In this case, there is a 4 -valuation v such that v(¬¬ϕ) = 1. Due to Lemma 3.2.1 the double negation preserves the operations of 4 , and so we have v(¬¬ϕ(p1 , . . . , pn )) = v(ϕ(¬¬p1 , . . . , ¬¬pn )). Let v1 be a 2ε -valuation with the property v1 (p1 ) := v(¬¬p1 ), . . . , v1 (pn ) := v(¬¬pn ). In view of the last equality, we have v1 (ϕ(p1 , . . . , pn )) = v(¬¬ϕ(p1 , . . . , pn )) = 1. The latter inequality means that ϕ is not provable in Lk. 2. We can prove this item in the same way as was done above, using Lemma 3.2.2 instead of Lemma 3.2.1. We can also reduce this item to the next one. Indeed, Le ¬¬ϕ ↔ ϕ ∨ ⊥, whence Le ¬¬ϕ → ϕ ⇐⇒ Le (ϕ ∨ ⊥) → ϕ ⇐⇒ Le ⊥ → ϕ. 3. Let Le ⊥ → ϕ. Then Lmn ⊥ → ϕ since Lmn extends Le. In view of ⊥ ∈ Lmn we immediately obtain Lmn ϕ. To prove the inverse implication we consider a formula ϕ such that Le ⊥ → ϕ and a 4 -valuation v such that v(⊥ → ϕ) = v(⊥ → (ϕ ∧ ⊥)) = 1. The latter means that v(ϕ ∧ ⊥) = −1. Let ϕ = ϕ(p1 , . . . , pn ). Due to epimorphism properties of τ (x) = x ∧ ⊥ (see Lemma 3.2.3) we obtain v(ϕ(p1 , . . . , pn ) ∧ ⊥) = v(ϕ(p1 ∧ ⊥, . . . , pn ∧ ⊥)). Consider a 2τ -valuation v1 such that v1 (p1 ) := v(p1 ∧ ⊥), . . . , v1 (pn ) := v(pn ∧ ⊥). Then v1 (ϕ) = v(ϕ ∧ ⊥) = −1, which refutes the provability Lmn ϕ. 2

Chapter 4

The Class of Extensions of Minimal Logic1 In this chapter, we assign to every properly paraconsistent extension L of minimal logic an intermediate logic Lint and negative logic Lneg called intuitionistic and negative counterparts of L, respectively. It will be proved that the negative counterpart Lneg explicates the structure of contradictions of paraconsistent logic L. We show that both counterparts Lint and Lneg are faithfully embedded into the original logic L. Finally, we investigate a question: to what extent is a logic L ∈ Par determined by its counterparts? As a ﬁrst step, we study paraconsistent extensions of the logic Le := Li ∩ Ln = Lj + {⊥ ∨ (⊥ → p)}. The class of extensions of this logic has a nice property that every logic L ∈ ELe ∩ Par is uniquely determined by its intuitionistic and negative counterparts.

4.1

Extensions of Le

In this section, we state that properly paraconsistent extensions of Le are exactly intersections of two logics, one of which is intermediate and the other is negative. Prior to this, we study the algebraic semantics for logics extending Le . 1

Parts of this chapter were originally published in [70] (Nicholas Copernicus University Press, Poland) and in [76] (Elsevier, UK). Reprinted here by permission of the publishers.

41

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4 The Class of Extensions of Minimal Logic

For an implicative lattice A = A, ∨, ∧, →, 1 and a ∈ A, we put Aa := {b ∈ A | b ≥ a} and Aa := {b ∈ A | b ≤ a}. The set Aa is obviously closed under the operations of A and we can deﬁne an implicative sublattice Aa of A, with the universe Aa . Except for the case a = 1, the set Aa forms a sublattice but not an implicative sublattice of A, because Aa is not closed under the implication (a → a = 1). However, the operation x →a y := (x → y) ∧ a turns Aa into an implicative lattice with unit element a. Denote this implicative lattice by Aa . If A is a j-algebra and a = ⊥, Aa can be treated as a Heyting algebra and Aa as a negative one. In the following we call Heyting algebra A⊥ an upper algebra of A. Negative algebra A⊥ is a lower algebra of A. Recall one well-known fact from the theory of distributive lattices. Let A be a distributive lattice, a an arbitrary element of A, and let sublattices Aa and Aa be deﬁned as above. The mappings ε(x) := x ∨ a and τ (x) := x ∧ a are epimorphisms of A onto Aa and Aa respectively. The mapping λ(x) := (x ∨ a, x ∧ a) gives an embedding of A into the direct product of lattices Aa and Aa . These facts do not generally hold for implicative lattices. As before, the mapping τ is an epimorphism of implicative lattices. But ε : A → Aa and λ : A → Aa × Aa are an epimorphism and an embedding of implicative lattices only if some additional condition is imposed on A. More precisely, the following assertions take place. Proposition 4.1.1 For an implicative lattice A and a ∈ A, the mapping τ : A → Aa , τ (x) = x ∧ a, is an epimorphism of implicative lattices. Proof. It follows from the deﬁnition of implication in Aa and the identity (x → y) ∧ z = ((x ∧ z) → (y ∧ z)) ∧ z satisﬁed in all implicative lattices. The latter fact follows from Item 9 of Proposition 2.1.2. 2 Proposition 4.1.2 Let A be an implicative lattice and a ∈ A. The following three conditions are equivalent. 1. For all x, y ∈ A, we have (x ∨ a) → (y ∨ a) ≤ (x → y) ∨ a. 2. The mapping ε : A → Aa given by the rule ε(x) = x ∨ a is an epimorphism of implicative lattices.

4.1 Extensions of Le

43

3. The mapping λ : A → Aa × Aa given by the rule λ(x) = (x ∨ a, x ∧ a) is an isomorphism of A onto a direct product of implicative lattices Aa × Aa . Proof. 1 ⇒ 2. Check that ε preserves the implication, i.e., that the equality (x → y) ∨ a = (x ∨ a) → (y ∨ a) holds. We have Lp ((p → q) ∨ r) → ((p ∨ r) → (q ∨ r)) by Item 8 of Proposition 2.1.2, whence the inequality (x → y)∨a ≤ (x∨a) → (y ∨ a) is valid in any implicative lattice. The inverse inequality holds by assumption. 2 ⇒ 3. It follows easily by assumption that λ is a homomorphism of A into Aa × Aa . We prove the injectivity of λ. Take an element b ∈ A, it is a complement of a in the interval [b ∧ a, b ∨ a]. Assuming λ(c) = λ(b) for some c ∈ A yields that c is a complement of a in the same interval [b ∧ a, b ∨ a]. We have b = c, since complements are unique in distributive lattices. Thus, it remains to prove that λ maps A onto Aa × Aa . For x ∈ Aa and y ∈ Aa , we set z := (a → y) ∧ x. The direct computation shows that z ∧ a = y and z ∨ a = ((a → y) ∨ a) ∧ x. Further, (a → y) ∨ a = (a ∨ a) → (y ∨ a) = a → a = 1 in view of the assumption that ε is a homomorphism, whence z ∨ a = x. We have thereby proved λ(z) = (x, y). 3 ⇒ 1. Obviously, 3 implies 2. Therefore, the desired inequality follows from the fact that ε preserves the implication. 2 Let us consider the following formulas P. ((p → q) → p) → p E. p ∨ (p → q) D. ((p ∨ r) → (q ∨ r)) → ((p → q) ∨ r) We have Lk+ = Lp + {P} = Lp + {E} = Lp + {D}, where Lk+ is the positive fragment of classical logic. It is well-known that Lk+ is axiomatized relative to positive logic by the Peirce law P or by the extended law of excluded middle E. It can be veriﬁed directly that D is true on the 2-element Peirce algebra 2P . On the other hand, substituting p for r in D, we immediately obtain Lp + D E. We have thus obtained that D axiomatizes Lk+ modulo Lp. Combining this fact and Proposition 4.1.2 yields a characterization of Peirce algebras in terms of mappings ε and λ.

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Proposition 4.1.3 Let A be an implicative lattice. The following conditions are equivalent. 1. A is a Peirce algebra. 2. For any a ∈ A, the mapping εa (x) = x ∨ a deﬁnes an epimorphism of A onto Aa . 3. For any a ∈ A, the mapping λa (x) = (x ∨ a, x ∧ a) deﬁnes an isomorphism of A and Aa × Aa . 2 We now turn to the subsystem Le of Le, which can be axiomatized relative to Lj by each of the following substitution instances of E and D: E . ⊥ ∨ (⊥ → p). D . ((p ∨ ⊥) → (q ∨ ⊥)) → ((p → q) ∨ ⊥). The equality Lj + {E } = Lj + {D } can be checked as follows. On the one hand, E follows from the instance of D obtained by replacing p for ⊥. On the other hand, D is equivalent in Lj to (p → (q ∨ ⊥)) → ((p → q) ∨ ⊥), and the latter formula follows in Lj from ⊥ ∨ (⊥ → q). Indeed, ⊥ implies ⊥, and formulas ⊥ → q and p → (q ∨ ⊥) imply p → q. Note a curious fact that the instance of the Peirce law P∗ = ((p → ⊥) → p) → p = (¬p → p) → p, which is known as the Clavius law, is not equivalent to the above formulas relative to Lj. Indeed, Lj P∗ ↔ (p ∨ ¬p) as a particular case of the equivalence P ↔ E, and the logics Lj + (p ∨ ¬p) and Le are incomparable in the lattice of Lj-extensions. To prove the latter assertion, consider the 3-element linearly ordered Heyting algebra 3 and 3-element j-algebra 3 with the universe {−1, ⊥, 1}, −1 ≤ ⊥ ≤ 1. It can be checked directly that 3 × 2 |= p ∨ ¬p, 3 × 2 |= E , 3 |= E , 3 |= p ∨ ¬p. Consider the algebraic semantics for Le . Proposition 4.1.4 Let A be a j-algebra. A is a model for Le if and only if one of the following equivalent conditions holds. 1. The mapping ε(x) = x ∨ ⊥ deﬁnes an epimorphism of the j-algebra A onto the Heyting algebra A⊥ .

4.1 Extensions of Le

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2. The mapping λ(a) = (a ∨ ⊥, a ∧ ⊥) determines an isomorphism of j-algebras A and A⊥ × A⊥ . 3. For any a, b ∈ A with a ≤ ⊥ ≤ b, ⊥ has a complement in the interval [a, b]. Proof. The inclusion Le ⊆ LA is equivalent to the fact that D is an identity of A, which is equivalent in its own right to Item 1 of Proposition 4.1.2 for a = ⊥. In this way, Proposition 4.1.2 implies that each of the conditions 1, 2 characterizes models for Le . Proving Proposition 4.1.2 we established, in fact, that 2 implies 3. Now we check the inverse implication, which completes the proof. Condition 3 means exactly that an embedding of distributive lattices λ : A → A⊥ × A⊥ is “onto”, i.e., that λ is an isomorphism of distributive lattices A and A⊥ × A⊥ . The implication is deﬁned in terms of the ordering preserved by λ, consequently, λ also preserves the implication. 2 Corollary 4.1.5 Let L ∈ Jhn. Then Le ⊆ L ⊆ Le if and only if L = L1 ∩ L2 , where L1 ∈ Int and L2 ∈ Neg. Proof. Let L be an intersection of intermediate and negative logics L1 and L2 . Then Li ⊆ L1 and Ln ⊆ L2 , whence Le = Li ∩ Ln ⊆ L. It is clear that the L is neither intermediate nor negative, therefore, L ∈ Par and L ⊆ Le. Conversely, let Le ⊆ L ⊆ Le and let A be a characteristic model for L. By the above proposition A is presented as a direct product of Heyting algebra A⊥ and negative algebra A⊥ , hence, L = LA = LA⊥ ∩ LA⊥ . It remains to note that LA⊥ is an intermediate logic and LA⊥ is a negative one. 2

4.1.1

Intuitionistic and Negative Counterparts for Extensions of Le

First we state one more property of models for Le . Letting A be a j-algebra, Le ⊆ LA, put CA (⊥) := {a ∈ A | a ∨ ⊥ = 1} and decompose A into a direct product A⊥ × A⊥ . Then CA (⊥) = {(1, b) | b ∈ A⊥ }. Indeed, for a = (x, y) ∈ A⊥ × A⊥ , we have 1 = a ∨ ⊥ ⇐⇒ (1, 1) = (x, y) ∨ (0, 1) = (x, 1) ⇐⇒ x = 1. It follows immediately that the set CA (⊥) is closed under ∨, ∧, and →. We will consider CA (⊥) as a negative algebra with operations induced from A and 1 = ⊥.

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Proposition 4.1.6 Let a j-algebra A be a model for Le . Then CA (⊥) ∼ = A⊥ and the mapping δ(x) = ⊥ → x deﬁnes an epimorphism of the j-algebra A onto the negative algebra CA (⊥). Proof. Again, we need a presentation of A as a direct product of Heyting and negative algebras. The isomorphism CA (⊥) ∼ = A⊥ follows from the above equality CA (⊥) = {(1, b) | b ∈ A⊥ }. Check that δ is an epimorphism. For a = (b, c) ∈ A⊥ × A⊥ , we have δ(a) = δ(b, c) = (0, 1) → (b, c) = (0 → b, 1 → c) = (1, c). Consequently, for ∗ ∈ {∨, ∧, →} and for any (a, b), (c, d) ∈ A⊥ × A⊥ , δ((a, b) ∗ (c, d)) = δ((a ∗ c, b ∗ d)) = (1, b ∗ d) = (1, b) ∗ (1, d) = δ(a, b) ∗ δ(c, d). It remains to note that δ(⊥) = 1 and δ(a) = a for any a ∈ CA (⊥).

2 We are now ready to deﬁne translations of intermediate and negative logics into properly paraconsistent extensions of Le , which are similar to translations of classical logic and maximal negative logic into Le deﬁned at the end of the previous chapter. Theorem 4.1.7 Let L extend Le , L ∈ Par, and let A be a characteristic model for L. Set L1 = LA⊥ and L2 = LA⊥ . Then for an arbitrary formula ϕ, the following equivalences hold. 1. L1 ϕ ⇐⇒ L ϕ ∨ ⊥. 2. L2 ϕ ⇐⇒ L ⊥ → ϕ. Proof. 1) Assume L1 ϕ and for an A-valuation v, compute the value v(ϕ ∨ ⊥). By Proposition 4.1.2, ε : A → A⊥ is an epimorphism, from which we have v(ϕ ∨ ⊥) = εv(ϕ). Here εv denotes an A⊥ -valuation obtained as a composition of v and ε. By deﬁnition L1 = LA⊥ , whence εv(ϕ) = 1. We have thus proved that v(ϕ ∨ ⊥) = 1 for any A-valuation v, i.e., L ϕ ∨ ⊥. Conversely, let L ϕ ∨ ⊥. Every A⊥ -valuation v can be treated as an A-valuation with the property εv = v. As above, we have v(ϕ) = εv(ϕ) = v(ϕ ∨ ⊥) = 1, which immediately implies L1 ϕ. 2) This proof is similar to the previous one with ε replaced by δ. 2 ⊥ As follows from the theorem, the logics L1 := LA and L2 := LA⊥ do not depend on the choice of a characteristic model A for the logic L extending Le . Indeed, L1 = {ϕ | L ϕ ∨ ⊥}, L2 = {ϕ | L ⊥ → ϕ}.

4.1 Extensions of Le

47

It is clear that L1 ∈ Int and L2 ∈ Neg. We call the logics L1 and L2 deﬁned as above intuitionistic and negative counterparts of L ⊇ Le and denote them Lint and Lneg respectively. Since A ∼ = A⊥ × A⊥ , we have L = Lint ∩ Lneg . Let, on the contrary, L = L1 ∩ L2 , where L1 ∈ Int and L2 ∈ Neg. For a suitable Heyting algebra B and for some negative algebra C, we have L1 = LB and L2 = LC. The direct product A = B × C is a characteristic model for L since L(B × C) = LB ∩ LC = L1 ∩ L2 = L. Moreover, B ∼ = A⊥ and C ∼ = A⊥ , consequently, L1 = Lint and L2 = Lneg . In this way, we arrive at the following statement. Proposition 4.1.8 The mapping L → (Lint , Lneg ) deﬁnes a lattice isomorphism between [Le , Le] and the direct product Int × Neg. The inverse mapping is given by the rule (L1 , L2 ) → L1 ∩ L2 . Proof. In fact, it was stated above that the mapping under consideration is a bijection. According to Theorem 4.1.7 it preserves an ordering. It remains to note that an order isomorphism of two lattices is a lattice isomorphism too. 2 We can now describe the class of models for L ⊇ Le as follows. Proposition 4.1.9 Let L ⊇ Le . A j-algebra A is a model for L if and only if A ∼ = A⊥ × A⊥ , A⊥ |= Lint , and A⊥ |= Lneg . Proof. Let A |= L. According to Proposition 4.1.4 the condition L ⊇ Le implies A ∼ = A⊥ × A⊥ . Denote L := LA. Then A⊥ |= Lint and A⊥ |= Lneg by Theorem 4.1.7. In view of the previous proposition Lint ⊆ Lint and Lneg ⊆ Lneg , whence A⊥ |= Lint and A⊥ |= Lneg . Conversely, let A ∼ = A⊥ × A⊥ , A⊥ |= Lint , and A⊥ |= Lneg . Then the direct product A is a model for the intersection Lint ∩ Lneg . But L ⊇ Le , hence, L = Lint ∩ Lneg by Corollary 4.1.5. 2 Thus, the class of properly paraconsistent extensions of Le is completely described in terms of intermediate and negative logics. It should be emphasized that the mapping deﬁned in Proposition 4.1.8 has an essentially eﬀective character. Theorem 4.1.7 allows one to eﬀectively reconstruct intuitionistic and negative counterparts from the given paraconsistent L, whereas the L itself is simply an intersection of its counterparts, i.e., a formula is proved in L if and only if it is proved in both Lint and Lneg . However, the interval [Le , Le] constitutes a relatively small part of the class Par of all properly paraconsistent extensions of Lj. There are many

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interesting logics, which do not belong to this interval. One of them is the Glivenko logic treated in the beginning of the next chapter.

4.2

Intuitionistic and Negative Counterparts for Extensions of Minimal Logic

As the ﬁrst stage in studying the whole class Par we deﬁne intuitionistic and negative counterparts for an arbitrary extension of minimal logic. For extensions of Le , our deﬁnitions will be equivalent to those of the previous section. We deﬁne the following translation In(⊥) = ⊥, In(p) = p ∨ ⊥, In(ϕ ∗ ψ) = In(ϕ) ∗ In(ψ), where p is a propositional variable, ϕ and ψ arbitrary formulas, and ∗ ∈ {∨, ∧, →}. In other words, if ϕ = ϕ(p0 , . . . , pn ), then In(ϕ) = ϕ(p0 ∨ ⊥, . . . , pn ∨ ⊥). For L ∈ Jhn+ , deﬁne Lint := {ϕ | L In(ϕ)}, Lneg := {ϕ | L ⊥ → ϕ}. It can easily be seen that Lint and Lneg are logics. Moreover, Li ⊆ Lint since ⊥ → (p ∨ ⊥) ∈ Lj, and Ln ⊆ Lneg in view of ⊥ → ⊥ ∈ Lj. We call Lint and Lneg intuitionistic and negative counterparts of the logic L respectively. Notice that this deﬁnition of negative counterpart is exactly the same as the deﬁnition of negative counterparts for Le -extensions given in the previous section. As for Lint , using formula D we can easily prove in Le the equivalence (ϕ ∨ ⊥) ↔ In(ϕ) for any formula ϕ. Therefore, if L extends Le , Lint coincides with the intuitionistic counterpart deﬁned in the previous section. List some simple properties of the notions introduced above. Proposition 4.2.1 1. For any L ⊇ Lj, we have Lint ∈ Int ∪ {F}, Lneg ∈ Neg ∪ {F}, and L ⊆ Lint ∩ Lneg . The last inclusion is not proper if and only if L extends Le . 2. L ∈ Int if and only if L = F, L = Lint , and Lneg = F. 3. L ∈ Neg if and only if L = F, L = Lneg , and Lint = F.

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49

4. If Lj ⊆ L1 ⊆ L2 , then L1int ⊆ L2int and L1neg ⊆ L2neg . 5. If L ⊆ L1 ∈ Int, then Lint ⊆ L1 . 6. If L ⊆ L1 ∈ Neg, then Lneg ⊆ L1 . Proof. We only prove the last two assertions. 5. If L In(ϕ), then also L1 In(ϕ). Since L1 is intermediate, we have L1 In(ϕ) → ϕ, and so L1 ϕ, which implies the desired inclusion. 6. Again, from L1 ⊥ → ϕ we conclude L1 ϕ since ⊥ belongs to any negative logic. 2 We have thus proved, in particular, that Lint is the least intermediate logic containing L, and Lneg is the least negative logic with the same property. It can easily be seen that the mappings (−)int : Jhn+ → Int and (−)neg : Jhn+ → Neg can be deﬁned as follows. For any L ∈ Jhn+ , put Lint := L + {⊥ → p} = L + Li and Lneg := L + {⊥} = L + Ln. Proposition 4.2.2 The mappings (−)int and (−)neg are lattice epimorphisms. Proof. This fact easily follows from the distributivity of Jhn+ (Proposition 2.1.6). 2 Further, we prove that upper and lower algebras associated with a given j-algebra are semantic analogs of intuitionistic and negative counterparts. Proposition 4.2.3 For any j-algebra A and formula ϕ, the following equivalences hold. 1. A |= In(ϕ) ⇐⇒ A⊥ |= ϕ. 2. A |= ⊥ → ϕ ⇐⇒ A⊥ |= ϕ. Proof. 1. Assume A⊥ |= ϕ and prove A |= In(ϕ). For an A-valuation v, deﬁne an A⊥ -valuation v by the rule v (p) := v(p) ∨ ⊥. Then it follows easily that v(In(ϕ)) = v (ϕ), which immediately implies the desired conclusion. Conversely, let A |= In(ϕ). For any A⊥ -valuation v, we have v = v , in particular, v(In(ϕ)) = v(ϕ), which completes the proof. 2. We use the mapping τ (x) = x ∧ ⊥, which is an epimorphism of A onto A⊥ by Proposition 4.1.1. Note also that ⊥ → ϕ is equivalent to ⊥ → (ϕ∧ ⊥) in Lj.

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Assuming A⊥ |= ϕ we take an A-valuation v and consider the composition τ v, which is an A⊥ -valuation. By epimorphism properties of τ we have v(ϕ ∧ ⊥) = τ v(ϕ). But τ v(ϕ) = ⊥ by assumption, which yields v(⊥ → (ϕ ∧ ⊥)) = 1. Thus, A |= ⊥ → (ϕ ∧ ⊥) by an arbitrary choice of v. Now, we let A |= ⊥ → (ϕ ∧ ⊥). Clearly, v = τ v for any A⊥ -valuation v. By assumption v(⊥) ≤ v(ϕ ∧ ⊥) = τ v(ϕ) = v(ϕ). The greatest element of A⊥ is ⊥, whence v(ϕ) = ⊥. In this way, A⊥ |= ϕ. 2 Corollary 4.2.4 Let L ∈ Jhn+ . 1. If A is a model for L, then A⊥ |= Lint and A⊥ |= Lneg . 2. If A is a characteristic model for L, then LA⊥ = Lint and LA⊥ = Lneg . 2 Consider classes of logics with given intuitionistic and negative counterparts. For L1 ∈ Int and L2 ∈ Neg, we deﬁne Spec(L1 , L2 ) := {L ⊇ Lj | Lint = L1 , Lneg = L2 }. It is clear that for any pair of intermediate and negative logics, (L1 , L2 ), the set Spec(L1 , L2 ) is non-empty. It contains at least the intersection L1 ∩ L2 . Moreover, in view of Item 1 of Proposition 4.2.1 L1 ∩ L2 is the greatest element of Spec(L1 , L2 ). It turns out this set also contains the least element and forms an interval in the lattice of Lj-extensions. Let L1 ∗ L2 := Lj + {In(ϕ), ⊥ → ψ | ϕ ∈ L1 , ψ ∈ L2 }, where L1 ∈ Int and L2 ∈ Neg. Proposition 4.2.5 Let L1 ∈ Int and L2 ∈ Neg. Then Spec(L1 , L2 ) = [L1 ∗ L2 , L1 ∩ L2 ]. Proof. Let L∗ := L1 ∗ L2 . It follows from deﬁnition that L1 ⊆ L∗int and L2 ⊆ L∗neg . On the other hand, for any L ∈ Spec(L1 , L2 ), we have L∗ ⊆ L. Indeed, L contains all axioms of L∗ . As noted above, L1 ∩ L2 is the greatest element

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51

of Spec(L1 , L2 ), whence, by Item 4 of Proposition 4.2.1 the logic L∗ and all logics intermediate between L∗ and L1 ∩ L2 also belongs to Spec(L1 , L2 ). 2 The logic L1 ∗ L2 , the least element of the interval Spec(L1 , L2 ), we call a free combination of logics L1 and L2 . This name is justiﬁed by the next proposition saying that models for L1 ∗ L2 are all j-algebras such that their upper and lower algebras are models for L1 and L2 , respectively. Proposition 4.2.6 Let L1 ∈ Int and L2 ∈ Neg. For any j-algebra A, we have A |= L1 ∗ L2 if and only if A⊥ |= L1 and A⊥ |= L2 . Proof. This statement easily follows from the deﬁnition of free combination and Corollary 4.2.4. 2 The next proposition allows one to write axioms for L1 ∗ L2 relative to Lj given an axiomatics of L1 relative to Li and of L2 relative to Ln. Proposition 4.2.7 Let L1 ∈ Int, L1 = Li + {ϕi | i ∈ I} and L2 ∈ Neg, L2 = Ln + {ψj | j ∈ J}. Then L1 ∗ L2 = Lj + {In(ϕi ), ⊥ → ψj | i ∈ I, j ∈ J}. Proof. Denote the right-hand side of the last equality by D. The inclusion D ⊆ L1 ∗ L2 is trivial. To state the inverse inclusion we show that L1 ⊆ Dint and L2 ⊆ Dneg . We argue for L2 ⊆ Dneg . Note that Ln = Ljneg , i.e., Ln ϕ if and only if Lj ⊥ → ϕ. Assume ψ ∈ L2 , then Ln (ψj 1 ∧ . . . ∧ ψj n ) → ψ for suitable particular cases ψj 1 , . . . , ψj n of axioms ψj1 , . . . , ψjn , j1 , . . . , jn ∈ J. Whence Lj ⊥ → ((ψj 1 ∧ . . . ∧ ψj n ) → ψ). The last formula implies in Lj ((⊥ → ψj 1 ) ∧ . . . ∧ (⊥ → ψj n )) → (⊥ → ψ), from which we infer ⊥ → ψ ∈ D. Consequently, L2 ⊆ Dneg . The remaining inclusion follows in the same way from the equality Li = Ljint . 2 As we can see from Proposition 4.2.5, the class of properly paraconsistent Lj-extensions decomposes into a union of disjoint intervals Par = {Spec(L1 , L2 ) | L1 ∈ Int, L2 ∈ Neg}.

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Fr X XX XXX XXX XX X rX Lmn $ ' ' rLk $ XX XXX XXX XX Le X r $ ' PP PP PPrLk ∩ Ln r Li ∩ Lmn A A Neg Int A A A rLk ∗ Lmn H A HH H A H Ln P HrLk ∗ Ln& ∗ Lmn r A r Li% rLi % & QPP S A Q PP PPS A Q P Q SPP A Q S PPA r Q Q Le Q S Q S Q S Q Par % S Q r & Lj Figure 4.1

It is interesting that the upper points of these intervals also form an interval in Jhn+ , [Le , Le]. Figure 4.1 illustrates the structure of the class Jhn+ . In this way, the investigation of the class of Lj-extensions is reduced to the problem: what is the structure of the interval Spec(L1 , L2 ) for the given intermediate logic L1 and negative logic L2 ? This problem will be treated in the subsequent sections but ﬁrst we make an observation on the nature of the negative counterpart Lneg of a paraconsistent logic L.

4.2.1

Negative Counterparts as Logics of Contradictions

We deﬁne a contradiction operator C(ϕ) := ϕ∧ ¬ϕ and extend this operator to sets of formulas as follows. Put C(∅) := {⊥} and C(X) := {C(ϕ) | ϕ ∈ X} for X = ∅. The contradiction operator is trivial in all intermediate logics. If the law ex contadictione quodlibet holds, we have C(ϕ) ↔ ⊥ for any formula ϕ. Rejecting ex contadictione quodlibet we obtain the possibility to distinguish contradictions constructed with the help of diﬀerent formulas. In particular, for L ∈ Par we have L C(ϕ) ↔ ⊥ if and only if ϕ ∈ Lneg . Moreover, it turns out that relative to deducibility properties, the behavior

4.3 Three Dimensions of Par

53

of formulas in the negative counterpart Lneg is completely similar to the behavior of contradictions constructed with the help of these formulas in the original logic L. More precisely, the following fact takes place. Proposition 4.2.8 Let L ∈ Par. For an arbitrary set of formulas X and for any formula ϕ, the following equivalence holds: X Lneg ϕ ⇐⇒ C(X) L C(ϕ). The proof of this proposition is an easy exercise on the deducibility in minimal logic. 2 We have thus proved that the contradiction operator deﬁnes a strong translation of the negative counterpart Lneg in a paraconsistent logic L ∈ Par. This fact allows one to consider the negative counterpart Lneg as a logic of contradictions associated with a given paraconsistent logic L.

4.3

Three Dimensions of Par

We can see now that the class Par has a three-dimensional structure. The position of a logic L in this class is determined by its intuitionistic counterpart Lint , which represents reasoning in L under the additional assumption of inconsistency, or of impossibility of contradictions, and by its structure of contradictions explicated in the negative counterpart Lneg . When an explosive pattern of reasoning and a structure of contradictions are ﬁxed, we have a variety of possibilities for combining them presented by the interval of logics Spec(Lint , Lneg ). The place of L in this interval can be considered as its third coordinate in Par, the sense of which is not quite clear yet. It becomes clearer in the next chapter. Now we turn to the question of a scale for this third coordinate. Unlike ﬁrst and second coordinates having absolute scales, Int and Neg respectively, the intervals Spec(I, N ) are mutually disjoint for diﬀerent pairs of logic I ∈ Int and N ∈ Neg. However, one can ﬁnd natural interrelations between these scales, i.e., between the intervals of the form Spec(I, N ) for various I ∈ Int and N ∈ Neg. Consider two pairs of logics P1 = (I1 , N1 ) and P2 = (I2 , N2 ), where I1 , I2 ∈ Int, N1 , N2 ∈ Neg. P1 ≤ P2 means that I1 ⊆ I2 and N1 ⊆ N2 . We write also Spec(P1 ) for Spec(I1 , N1 ). Let P1 = (I1 , N1 ) and P2 = (I2 , N2 ) be such that P1 ≤ P2 . Mappings rP2 ,P1 : Spec(P2 ) → Par and eP1 ,P2 : Spec(P1 ) → Par are deﬁned as follows rP2 ,P1 (L) := L ∩ (I1 ∩ N1 ), eP1 ,P2 (L) := L + (I2 ∗ N2 ).

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4 The Class of Extensions of Minimal Logic

Proposition 4.3.1 Let pairs of logics P1 and P2 be such that P1 ≤ P2 . The following facts hold. 1. For any L ∈ Spec(P2 ), we have eP1 ,P2 rP2 ,P1 (L) = L. 2. For any L ∈ Spec(P1 ), we have rP2 ,P1 eP1 ,P2 (L) = L + rP2 ,P1 (I2 ∗ N2 ) 3. eP1 ,P2 is a lattice epimorphism from Spec(P1 ) onto Spec(P2 ). 4. rP2 ,P1 is a lattice monomorphism from Spec(P2 ) into Spec(P1 ) and rP2 ,P1 (P2 ) = [rP2 ,P1 (I2 ∗ N2 ), I1 ∩ N1 ]. 5. For any P3 such that P2 ≤ P3 , we have eP1 ,P2 eP2 ,P3 = eP1 ,P3 , rP3 ,P2 rP2 ,P1 = rP3 ,P1 . Proof. 1. We calculate eP1 ,P2 rP2 ,P1 (L) = (L ∩ (I1 ∩ N1 )) + (I2 ∗ N2 ) = (L + (I2 ∗ N2 )) ∩ ((I1 ∩ N1 ) + (I2 ∗ N2 )). By Proposition 4.2.5 I2 ∗N2 is the least point of Spec(P2 ), therefore, we have L + (I2 ∗ N2 ) = L. Further, we need one lemma. Lemma 4.3.2 For any L ∈ Spec(I, N ), I ∩ N = L + Le . Proof. (Le )int equals to Li, the least logic in Int, and (Le )neg = Ln, which is the least logic in Neg. Now, it follows from Proposition 4.2.2 that L + Le has the same counterparts as L. By Corollary 4.1.5 L + Le coincides with the greatest point of Spec(I, N ). 2 Using this lemma and the obvious relation I1 ∗ N1 ⊆ I2 ∗ N2 we obtain (I1 ∩ N1 ) + (I2 ∗ N2 ) = ((I1 ∗ N1 ) + Le ) + (I2 ∗ N2 ) = I2 ∗ N2 + Le = I2 ∩ N2 . And ﬁnally, eP1 ,P2 rP2 ,P1 (L) = L ∩ (I2 ∩ N2 ) = L. 2. The direct computation shows rP2 ,P1 eP1 ,P2 (L) = (L + (I2 ∗ N2 )) ∩ (I1 ∩ N1 ) = (L ∩ (I1 ∩ N1 )) + ((I2 ∗ N2 ) ∩ (I1 ∩ N1 )) = L + rP2 ,P1 (I2 ∗ N2 ).

4.3 Three Dimensions of Par

55

3. It follows from the distributivity of Jhn+ that eP1 ,P2 is a lattice homomorphism. Let L ∈ Spec(P1 ) and L := eP1 ,P2 (L) = L + (I2 ∗ N2 ). By Proposition 4.2.2 (L )int = Lint + (I2 ∗ N2 )int = I1 + I2 = I2 . In the same way, (L )neg = N2 . Consequently, L ∈ Spec(P2 ). The fact that eP1 ,P2 is an epimorphism follows from Item 1. 4. As above, we use Proposition 4.2.2 to check that rP2 ,P1 maps Spec(P2 ) into Spec(P1 ). This is a homomorphism due to the distributivity of Jhn+ . If rP2 ,P1 (L1 ) = rP2 ,P1 (L2 ), then applying the formula of Item 1 we obtain L1 = L2 . Thus, rP2 ,P1 is a monomorphism. The equality rP2 ,P1 (P2 ) = [rP2 ,P1 (I2 ∗ N2 ), I1 ∩ N1 ] follows from Item 2. 5. This item follows from the obvious relations I2 ∗ N2 ⊆ I3 ∗ N3 and I1 ∩ N1 ⊆ I2 ∩ N2 . 2 The above proposition shows, in particular, that any interval Spec(I, N ) is isomorphic to an upper subinterval of Spec(Li, Ln). In this way, the latter interval can be considered as a scale for the third dimension of the class Par. Extending intuitionistic and negative counterparts, we restrict simultaneously the part of the scale that can be used to construct a logic with given counterparts. It is also worth noticing the following consequence of the last proposition. Corollary 4.3.3 Let P1 = (I1 , N1 ) and P2 = (I2 , N2 ) be pairs of logics such that P1 ≤ P2 . For any logics L1 , L2 ∈ Spec(P2 ), L1 = L2 , there is a formula ϕ ∈ I1 ∩ N1 such that ϕ ∈ (L1 \ L2 ) ∪ (L2 \ L1 ). Proof. Let L1 , L2 ∈ Spec(P2 ). If L1 = L2 , but these logics are not distinguished by a formula ϕ ∈ I1 ∩ N1 , then rP2 ,P1 (L1 ) = rP2 ,P1 (L2 ). By Item 4 of the previous proposition rP2 ,P1 is a monomorphism, whence L1 = L2 , a contradiction. 2 In particular, any two logics from Spec(I, N ) can be distinguished via a formula from Le = Li∩Ln. Moreover, any logic from the interval Spec(I, N ) can be axiomatized by formulas from Le modulo the least logic of the interval I ∗ N . Indeed, for any L ∈ Spec(I, N ) we have by Item 1 of Proposition 4.3.1 L = (L ∩ Le ) + (I ∗ N ). In this way, any possible combination of intuitionistic and negative logics can be determined by corollaries of the formula ⊥ ∨ (⊥ → p). For further investigations of the class Par, we need semantic considerations.

Chapter 5

Adequate Algebraic Semantics for Extensions of Minimal Logic1 The goal of this chapter is to ﬁnd a representation of j-algebras, convenient for working with logics lying inside the intervals Spec(L1 , L2 ). We have to understand the structure of an arbitrary j-algebra A with given upper algebra A⊥ and lower algebra A⊥ . The semantic characterization of Glivenko’s logic considered in Section 5.1 prompts the solution to this problem. The desired representation is described in Section 5.2. In Section 5.3 with the help of the obtained representation we characterize the Segerberg logics and demonstrate its eﬀectiveness in this way. Finally, in Section 5.4 we consider the Kripke semantics and deﬁne for j-frames analogs of upper and lower algebras associated with a j-algebra.

5.1

Glivenko’s Logic

Consider the following substitution instance of the Peirce law: P . ((⊥ → p) → ⊥) → ⊥ = ¬¬(⊥ → p). We call the logic Lg := Lj + {P } Glivenko’s logic. It was mentioned in [98, p. 46] that Glivenko’s logic is the weakest one in which ¬¬ϕ is derivable whenever ϕ is derivable in classical logic. Unfortunately, this work contains neither the proof of this assertion, nor any further reference. In this section, 1

Parts of this chapter were originally published in [69].

57

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5 Adequate Algebraic Semantics for Extensions of Minimal Logic

we present a natural algebraic proof of this statement. We also show that Lg is a proper subsystem of Le . Proposition 5.1.1 1. Let A be a j-algebra. Then A is a model for Lg if and only if ⊥ ∨ (⊥ → a) ∈ Fd (A⊥ ) for any a ∈ A. 2. Let A be a model for Lg and ∇ := Fd (A⊥ ). Then the mapping π(a) = (a ∨ ⊥)/∇ deﬁnes an epimorphism of A onto A⊥ /∇. Proof. 1. This item immediately follows from the deﬁnition of Glivenko’s logic and the fact that ¬(a ∨ ⊥) = a ∨ ⊥ → ⊥ = a → ⊥ = ¬a for any j-algebra A and a ∈ A. The last equality implies, in particular, Fd (A⊥ ) = {a ∈ A | ¬¬a = 1}. 2. In fact, we need only check that π preserves the implication, i.e., (a → b) ∨ ⊥/∇ = (a ∨ ⊥) → (b ∨ ⊥)/∇. We have ((a → b)∨ ⊥) → ((a∨ ⊥) → (b∨ ⊥)) = 1 ∈ ∇, since the corresponding formula is provable in Lj (see Item 8 of Proposition 2.1.2). Further, it can be veriﬁed directly that Lj (⊥ → q) → (((p ∨ ⊥) → (q ∨ ⊥)) → ((p → q) ∨ ⊥)). In view of Lj (p → q) → (¬¬p → ¬¬q) we obtain Lj ¬¬(⊥ → q) → ¬¬(((p ∨ ⊥) → (q ∨ ⊥)) → ((p → q) ∨ ⊥)), i.e., Lg ¬¬(((p ∨ ⊥) → (q ∨ ⊥)) → ((p → q) ∨ ⊥)). By assumption A is a model for Lg, consequently, ((a ∨ ⊥) → (b ∨ ⊥)) → ((a → b) ∨ ⊥) ∈ ∇, which completes the proof. 2 Theorem 5.1.2 (Generalized Glivenko’s Theorem.) For every logic L ∈ Jhn, the following conditions are equivalent. 1. For any ϕ, Lk ϕ ⇐⇒ L ¬¬ϕ. 2. L ⊇ Lg and L ∈ Neg.

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59

Proof. 1 ⇒ 2. This implication is trivial. 2 ⇒ 1. Since L ∈ Neg, the logic L is contained in Lk. If L ¬¬ϕ, then Lk ¬¬ϕ, and so Lk ϕ. Assume Lk ϕ. Let A |= L and v be an A-valuation. Consider an A⊥ /Fd (A⊥ )-valuation v such that v (p) = v(p ∨ ⊥)/Fd (A⊥ ). Using the assumption Lg ⊆ L and Proposition 5.1.1 we obtain v (ϕ) = v(ϕ∨⊥)/Fd (A⊥ ). On the other hand, A⊥ /Fd (A⊥ ) is a Boolean algebra and Lk ϕ, therefore, v (ϕ) = 1. In this way, v(ϕ ∨ ⊥) ∈ Fd (A⊥ ), i.e., v(¬¬(ϕ ∨ ⊥)) = v(¬¬ϕ) = 1. Since A and v are arbitrary, we obtain L ¬¬ϕ. 2 Let us prove that Glivenko’s logic does not belong to the class of Le extensions. To this end it will be enough to show that Glivenko’s logic has models diﬀerent from direct products of Heyting and negative algebras. Proposition 5.1.3 Let A be a model for Le , and B a Heyting algebra. Then A ⊕ B is a model for Lg. Proof. It follows from two facts. For all a ∈ A ⊕ B, we have ⊥ ∨ (⊥ → a) ∈ B. All elements of B are dense in (A ⊕ B)⊥ . Corollary 5.1.4 The inclusion Lg ⊂ Le is proper. Proof. Indeed, according to Proposition 4.1.4 the algebra A ⊕ B is not a model for Le if B is a non-trivial Heyting algebra. But this is a model of Glivenko’s logic by the previous proposition. 2

5.2

Representation of j -Algebras

In this section we give a convenient representation of j-algebras, which allows one to describe classes of models for logics lying inside intervals of the form [L1 ∗ L2 , L1 ∩ L2 ], where L1 ∈ Int and L2 ∈ Neg. We know that an intersection L1 ∩ L2 of intermediate and negative logics is characterized by the class of all direct products of the form A×B, where A is a Heyting algebra being a model for the logic L1 and B is a negative algebra modelling L2 . Indeed, due to Corollary 4.1.5 the intersection L1 ∩ L2 extends Le and each model A for L1 ∩ L2 is isomorphic to the direct product A⊥ × A⊥ by Proposition 4.1.4. It remains to note that by Proposition 4.1.8 we have L1 = (L1 ∩ L2 )int and L2 = (L1 ∩ L2 )neg . Thus, Proposition 4.1.9 implies A⊥ |= L1 and A⊥ |= L2 .

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At the same time, a free combination of logics, L1 ∗ L2 , is characterized by the class of all j-algebras A such that the upper algebra A⊥ is a model for L1 and the lower algebra A⊥ models L2 (see Proposition 4.2.6). At this point the following question arises. If a Heyting algebra B and a negative algebra C are given, what is the diﬀerence between an arbitrary j-algebra A with the condition A⊥ ∼ = B and A⊥ ∼ = C and the direct product of algebras B × C? Proposition 5.1.1 allows us to assume that elements of the form ⊥ ∨ (⊥ → a), where a ∈ A⊥ , play a special part in the structure of A. Proposition 5.2.1 Let A be a j-algebra and a mapping fA : A⊥ → A⊥ is given by the rule fA (x) = ⊥ ∨ (⊥ → x). Then the following two conditions are met. 1. The mapping fA : A⊥ → A⊥ is a semilattice homomorphism preserving the meet ∧ and the greatest element, fA (⊥) = 1; 2. The embedding λ⊥ : A → A⊥ × A⊥ , where λ⊥ (x) = (x ∨ ⊥, x ∧ ⊥), has the following image λ⊥ (A) = {(x, y) | x ≤ fA (y), x ∈ A⊥ , y ∈ A⊥ }. Proof. 1. For brevity, we omit the lower index in denotation fA . We have f (⊥) = ⊥ ∨ (⊥ → ⊥) = 1. Further, f (y1 ) ∧ f (y2 ) = (⊥ ∨ (⊥ → y1 )) ∧ (⊥ ∨ (⊥ → y2 )) = = ⊥ ∨ ((⊥ → y1 ) ∧ (⊥ → y2 )) = ⊥ ∨ (⊥ → (y1 ∧ y2 )) = f (y1 ∧ y2 ). We have thus veriﬁed that f is a semilattice homomorphism preserving the meet and the unit element. 2. If a ∈ A, then (a ∨ ⊥, a ∧ ⊥) ∈ λ⊥ (A) and the inequality a ∨ ⊥ ≤ ⊥ ∨ (⊥ → (a ∧ ⊥)) holds. The latter can be checked, for example, by proving in Lj the formula p ∨ ⊥ → ⊥ ∨ (⊥ → p ∧ ⊥). Thus, the inclusion λ⊥ (A) ⊆ {(x, y) | x ≤ f (y), x ∈ A⊥ , y ∈ A⊥ } is proved. Now, let x, y ∈ A, x ≥ ⊥, y ≤ ⊥, and x ≤ ⊥ ∨ (⊥ → y). We show that there is an element a ∈ A such that x = a ∨ ⊥ and y = a ∧ ⊥. Put a := x ∧ (⊥ → y), then a ∨ ⊥ = (⊥ ∨ x) ∧ (⊥ ∨ (⊥ → y)) = x ∧ (⊥ ∨ (⊥ → y)) = x,

5.2 Representation of j-Algebras

61

moreover, a∧ ⊥ = x∧ (⊥ → y)∧ ⊥ = ⊥ ∧ (⊥ → y) = y. The inverse inclusion is also checked. 2 As we can see from the above proposition, every j-algebra A determines a triple (A⊥ , A⊥ , fA ) consisting of a Heyting algebra, a negative algebra and a semilattice homomorphism. Now, let us take a triple (B, C, f : C → B), where B is an arbitrary Heyting algebra, C a negative algebra and f a semilattice homomorphism from C to B preserving the meet and the greatest element. Starting from this triple we try to construct a j-algebra A, the upper and lower algebras of which are isomorphic to B and C respectively, and the mapping fA is induced in a natural way by the homomorphism f . Deﬁne a lattice B ×f C as a sublattice of the direct product B × C with the universe |B ×f C| := {(x, y) | x ∈ B, y ∈ C, x ≤ f (y)}. This is really a sublattice of B × C, because f preserves the meet and, hence, the ordering, which easily implies the relation f (y1 ) ∨ f (y2 ) ≤ f (y1 ∨ y2 ). From the latter immediately follows that the set |B ×f C| is closed under componentwise lattice operations on the direct product of lattices. As we can see from the proposition below, this lattice can be considered as a j-algebra. Proposition 5.2.2 Let B, C, f , and A := B ×f C be as above. The lattice A has a natural structure of j-algebra, where the relative pseudo-complement operation is given by the rule (x1 , y1 ) → (x2 , y2 ) = ((x1 → x2 ) ∧ f (y1 → y2 ), y1 → y2 ), 1A = (1B , ⊥C ), and ⊥A = (⊥B , ⊥C ). Moreover, B ∼ = A⊥ , C ∼ = A⊥ , and these isomorphisms are given by the rules x → (x, ⊥C ), x ∈ B, and y → (⊥B , y), y ∈ C, respectively. Finally, for all y ∈ C, we have (f (y), ⊥C ) = ⊥A ∨ (⊥A → (⊥B , y)) = fA ((⊥B , y)). Proof. First, we check that the relative pseudo-complement is well deﬁned. Let b1 , b2 ∈ B, c1 , c2 ∈ C, b1 ≤ f (c1 ), and b2 ≤ f (c2 ). The element (b1 , c1 ) → (b2 , c2 ), if it is deﬁned, is the greatest among all elements (x, y) such that x ≤ f (y) and (b1 , c1 ) ∧ (x, y) ≤ (b2 , c2 ). This is equivalent to relations x ≤ (b1 → b2 ) ∧ f (y) and y ≤ c1 → c2 . Taking into account that f preserves the ordering, we immediately obtain that the element ((b1 → b2 ) ∧ f (c1 → c2 ), c1 → c2 ) is the desired relative pseudo-complement.

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5 Adequate Algebraic Semantics for Extensions of Minimal Logic

All other relations, except the last, are trivial. Check the last relation using the obtained formula for relative pseudo-complement. We have ⊥A ∨ (⊥A → (⊥B , y)) = (⊥B , ⊥C ) ∨ ((⊥B , ⊥C ) → (⊥B , y)) = = (⊥B , ⊥C ) ∨ (1B ∧ f (⊥C → y), ⊥C → y) = (⊥B , ⊥C ) ∨ (f (y), y) = (f (y), ⊥C ). 2 As we can see from the above considerations, to deﬁne a class of jalgebras characterizing some extension L of minimal logic, we must choose a class of Heyting algebras and a class of negative algebras isomorphic to upper and lower algebras respectively, associated with models for L. In this way, we ﬁx intuitionistic and negative counterparts of the logic L. Moreover, to determine the place of L inside the interval [Lint ∗ Lneg , Lint ∩ Lneg ], we have to distinguish in one or another way the class of admissible homomorphisms from negative algebras into Heyting ones. If no restrictions are imposed on the class of homomorphisms, we obtain a free combination of intermediate and negative logics characterized by the selected classes of Heyting and negative algebras (see Proposition 4.2.6). If we admit only homomorphisms identically equal to the unit element, we obtain the intersection Lint ∩ Lneg . Indeed, a j-algebra A ×f B coincides with the direct product A × B if and only if f (y) = 1 for all y ∈ B.

5.3

Segerberg’s Logics and their Semantics

It is interesting to consider logics diﬀerent from intersections and free combinations of intermediate and negative logics, i.e., logics lying inside intervals of the form Spec(L1 , L2 ). In this section, using the representation for j-algebras obtained above, we describe an algebraic semantics for logics studied previously by K. Segerberg [98], who characterized these logics in terms of Kripke semantics. Except for Lj K. Segerberg [98] considered logics obtained by adding to Lj one or several axioms from the list below. I. ⊥ → p K. ¬p ∨ ¬¬p X. p ∨ ¬p L. (p → q) ∨ (q → p) E. p ∨ (p → q)

5.3 Segerberg’s Logics and their Semantics

63

L . ¬p ∨ (⊥ → p)(= (p → ⊥) ∨ (⊥ → p)) E . ⊥ ∨ (⊥ → p) Q. ⊥ LN . (p → q ∨ ⊥) ∨ (q → p ∨ ⊥) LQ 1 . ⊥ → (p → q) ∨ (q → p) LQ 2 . (⊥ → (p → q)) ∨ (⊥ → (q → p)) EQ 1 . ⊥ → p ∨ (p → q) EQ 2 . (⊥ → p) ∨ (⊥ → (p → q)) P . ¬¬(⊥ → p)(= ((⊥ → p) → ⊥) → ⊥) We may combine these axioms, which gives rise to a large number of new logics. Some of these logics have traditional denotation, for example Li = Lj + {I}, and others have not. Due to this fact, we need some notational conventions. If some logic is obtained from the logic already having a denotation, say L, by adding some axiom denoted by a capital letter, say X, then the denotation of this new logic will be obtained by joining the corresponding small letter to the existing denotation, Lx := L+{X}. Of course, in this way one logic may obtain diﬀerent denotations. According to this convention we have, for example, Lji = Li, Lje = Le, Ljq = Ln, Lix = Ljix = Lk, and ﬁnally, Ljp = Lg. We shall say a few words on how the above list of axioms arises. The Kripke semantics for extensions of Lj was described in Chapter 2. Recall that any j-frame is divided into two parts consisting of normal worlds and of abnormal worlds. The ﬁrst axiom I distinguish the class of j-frames in which all worlds are normal. The next two axioms are the well-known law of excluded middle X and weak law of excluded middle K. These axioms impose some restrictions on the accessibility relation only in the normal part of a j-frame [98]. It must be identical in case of X and directed in case of K. The Dummett linearity axiom L and the extended law of excluded middle E deﬁne properties of accessibility relation in the whole frame. A j-frame satisfying L is linear, whereas in a j-frame satisfying E the accessibility relation is identical. The next two axioms, L and E , are particular cases of L and E, respectively. They do not impose any restrictions on either the normal

64

5 Adequate Algebraic Semantics for Extensions of Minimal Logic (rLk ( (((( ( ( r ((((Le r((( ( ( ( ( Lje x r ((( ( (rLil ( Ljx (((( ( ( r (((( Lje l r((( N Lje l ((r (((( Ljl ( ( ( (((( ( ( r ( ((rLik (((( LjlN l ((( ( r((( ( ( ((( LjlN (r ((( ( ( Lje k (((( (((r( ( ((rLi ( Ljkl ((( ( r((( ( ( ((( Ljk (r ((( ( ( (((( Lje ((r( ( ( ( ( r (( Ljl

Lj Figure 5.1 or abnormal part of a j-frame, but they deﬁne the way in which the cone of abnormal worlds is situated in the whole frame (see Proposition 2.3.3). The interrelations between logics obtained by joining to Lj one or several axioms reviewed up to this point are presented in Figure 5.1. Note that this diagram (as well as the diagram presented in Figure 5.2 below) respects only the ordering, but not the lattice structure of Jhn+ . All logics presented at the diagram are distinct, and a logic L1 is contained in a logic L2 if and only if there is a path leading from L1 to L2 , which at every point is either rising or horizontal and directed to the right. To explain the explicit irregularities of the above diagram K. Segerberg put some new axioms into consideration, which are not as natural at ﬁrst glance as the axioms considered up to this point. “But as long as we cannot account for the irregularities in the above diagram, we cannot claim to understand the situation fully” [98, p. 41]. As we can see from the above the axiom X can be considered as a relativization of the axiom E to the normal part of a j-frame. Indeed, the axiom E imposes the condition to be identical on the accessibility relation, whereas X imposes essentially the same condition “to be identical” but on the accessibility relation restricted to the normal part of a j-frame. The next six axioms in the list are the axiom Q distinguishing the class of abnormal

5.3 Segerberg’s Logics and their Semantics

65 r

Lnl r

Lmn Ln r

Le u r u r r Ljx u u u r Ljl u u u r r r r r u r r LjlN u Leu r r r r Ljl u r r Ljk u u r r Lj

LjlQ 2

uLk

uLil

uLik

uLi

LjeQ 2

Figure 5.2 j-frames and relativizations of axioms E and L to the normal or to the abnormal part of a j-frame. The axiom LN is a restriction of L to normal Q worlds. The axioms LQ 1 and L2 are variants of relativization of L to the abnormal part of a j-frame. Relativizing E to abnormal words K. Segerberg Q also suggests two variants, EQ 1 and E2 . The last axiom in the list, P , is similar to E and L because it restricts only the way of combination of normal and abnormal parts of a j-frame (see Q Proposition 2.3.4). This axiom, as well as axioms LQ 1 and E1 lie out of the main line of considerations in [98]. Q If we exclude from the above list the axioms P , LQ 1 , and E1 , the logics that can be constructed via adjoining to Lj the other axioms from the list form the beautiful diagram presented in Figure 5.2. The logics of Figure 5.1 are depicted in this diagram by bigger circles. The way, in which these logics are situated in Figure 5.2, explains the irregularities of the previous

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diagram. Only a few points on the diagram are endowed with the names of corresponding logics. The other logics are obtained via a combination of axioms of explicitly designated logics and one can easily reconstruct which logic corresponds to one or another point on the diagram. For example, the non-designated logics lying on the horizontal line ended with Lik are the Q following: Ljke , Ljke lQ 2 , Ljke e2 (from left to right). We also note the Q equalities Ljl = Ljl lN lQ 2 and Le = Lje xe2 . As we will see the equality Q Q Ljl = Ljl lN lQ 1 does not hold. So using axiom L1 instead of L2 results in a diagram of logics, which is not as regular as that of Figure 5.2. This explains the choice of K. Segerberg between variants of relativization of the axiom L to abnormal worlds. In this diagram there are only four intermediate logics, namely, the logics lying on the vertical line from Li to Lk. The three negative logics on the diagram are those lying on the horizontal line from Ln to Lmn. All other logics on the diagram belong to the class Par. They form a three-dimensional ﬁgure, the dimensions of which, as we can see later, correspond to the three parameters, which determine the position of a paraconsistent logic in the class Par. To better explain this correspondence we turn to the algebraic semantics of Segerberg’s logics. Recall that a Stone algebra is a Heyting algebra satisfying the identity K. Let A be a Heyting (negative) algebra. We call A a Heyting (negative) l-algebra if A |= (p → q) ∨ (q → p). Proposition 5.3.1 Let A be an arbitrary j-algebra. The following equivalences hold. 1. A |= Ljk if and only if A⊥ is a Stone algebra. 2. A |= Ljx if and only if A⊥ is a Boolean algebra. 3. A |= Ljl if and only if fA (A⊥ ) ⊆ R(A⊥ ). 4. A |= Ljl if and only if A⊥ and A⊥ are l-algebras, fA (A⊥ ) ⊆ R(A⊥ ), and for all y1 , y2 ∈ A⊥ , we have fA (y1 → y2 ) ∨ fA (y2 → y1 ) = 1. 5. A |= Lg if and only if fA (A⊥ ) ⊆ Fd (A⊥ ). 6. A |= LjlN if and only if A⊥ is a Heyting l-algebra.

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7. A |= LjlQ 1 if and only if A⊥ is a negative l-algebra. 8. A |= LjlQ 2 if and only if A⊥ is a negative l-algebra and for all y1 , y2 ∈ A⊥ , we have fA (y1 → y2 ) ∨ fA (y2 → y1 ) = 1. 9. A |= LjeQ 1 if and only if A⊥ is a negative Peirce algebra. 10. A |= LjeQ 2 if and only if A⊥ is a negative Peirce algebra and for all y1 , y2 ∈ A⊥ , we have fA (y1 ) ∨ fA (y1 → y2 ) = 1. Proof. 1. Let A |= Ljk. We represent A in the form A⊥ ×fA A⊥ , take an arbitrary element (x, y) ∈ A, and compute ((x, y) → (⊥, ⊥)) ∨ (((x, y) → (⊥, ⊥)) → (⊥, ⊥)) = (x → ⊥, y → ⊥)∨ ∨((x → ⊥, y → ⊥) → (⊥, ⊥)) = (¬x, ⊥) ∨ ((¬x, ⊥) → (⊥, ⊥)) = = (¬x, ⊥) ∨ (¬¬x, ⊥) = (¬x ∨ ¬¬x, ⊥) = (1, ⊥). The latter identity is satisﬁed if and only if the identity ¬x ∨ ¬¬x = 1 is true on A⊥ , i.e., if and only if A⊥ is a Stone algebra. 2. This item can also be proved via a direct computation. 3. Let (x, y) ∈ A⊥ ×fA A⊥ . The direct computation shows ((x, y) → (⊥, ⊥)) ∨ ((⊥, ⊥) → (x, y)) = ((x → ⊥) ∨ f (y), ⊥). Here after, we omit the lower index in the denotation fA if it does not lead to confusion. As we can see, L is an identity of A if and only if for all x ∈ A⊥ , y ∈ A⊥ , x ≤ f (y), the equality (x → ⊥) ∨ f (y) = 1A⊥ holds. In particular, we have (f (y) → ⊥)∨f (y) = ¬f (y)∨f (y) = 1 , i.e., each element of the form f (y) is regular. The inverse implication immediately follows from the above and the fact that the implication is descending with respect to the ﬁrst argument. Indeed, if for some y ∈ A⊥ we have (f (y) → ⊥)∨f (y) = 1A⊥ , then for all x ∈ A⊥ , x ≤ f (y), we also have (x → ⊥) ∨ f (y) = 1A⊥ . 4. Assume that A |= Ljl. In this case, the upper algebra A⊥ is a Heyting l-algebra as a subalgebra of A. The inclusion fA (A⊥ ) ⊆ R(A⊥ ) holds by Item 3, because L is a substitution instance of L. Further, recall that the implication →⊥ of A⊥ is deﬁned via the implication → of A as x →⊥ y = (x → y) ∧ ⊥. Calculate (x →⊥ y) ∨ (y →⊥ x) = ((x → y) ∧ ⊥) ∨ ((y → x) ∧ ⊥) = ((x → y) ∨ (y → x)) ∧ ⊥ = 1 ∧ ⊥ = ⊥.

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Thus, A⊥ is a negative l-algebra. To check the last of the conditions listed in this item take arbitrary elements y1 , y2 ∈ A⊥ and represent them in the form (⊥, y1 ) and (⊥, y2 ). We have (1, ⊥) = ((⊥, y1 ) → (⊥, y2 )) ∨ ((⊥, y2 ) → (⊥, y1 )) = (f (y1 → y2 ) ∨ f (y2 → y1 ), (y1 → y2 ) ∨ (y2 → y1 )), in particular, f (y1 → y2 ) ∨ f (y2 → y1 ) = 1. Prove the inverse implication. Let A⊥ and A⊥ be l-algebras, and let fA (A⊥ ) ⊆ R(A⊥ ) and for all y1 , y2 ∈ A⊥ , we have fA (y1 → y2 ) ∨ fA (y2 → y1 ) = 1. Take (x1 , y1 ), (x2 , y2 ) ∈ A and using the formula for implication calculate ((x1 , y2 ) → (x2 , y2 )) ∨ ((x2 , y2 ) → (x1 , y1 )) = (h, (y1 → y2 ) ∨ (y2 → y1 )). The second component of the last pair equals ⊥ = 1A⊥ , because A⊥ is a negative l-algebra, whereas the ﬁrst component has the following form: h = ((x1 → x2 ) ∨ (x2 → x1 )) ∧ (f (y1 → y2 ) ∨ (x2 → x1 ))∧ ((x1 → x2 ) ∨ f (y2 → y1 )) ∧ (f (y1 → y2 ) ∨ f (y2 → y1 )). From our assumptions we immediately infer that ﬁrst and last conjunctive terms of the last expression are equal to the unit element. In this way, we obtain that the satisﬁability of L on A is equivalent to the condition: for all (x1 , y1 ), (x2 , y2 ) ∈ A, (x1 → x2 ) ∨ f (y2 → y1 ) = 1A⊥ . Taking into account the facts that the implication is descending in the ﬁrst argument and ascending in the second and that x ≤ f (y) for all (x, y) ∈ A, we obtain the chain of inequalities (x1 → x2 ) ∨ f (y2 → y1 ) ≥ (x1 → ⊥) ∨ f (⊥ → y1 ) ≥ (f (y1 ) → ⊥) ∨ f (y1 ) = ¬f (y1 ) ∨ f (y1 ) = 1. The latter equality holds due to the condition that every element of the form f (y) is regular. Items 5–10 can be checked via direct computation. 2 Corollary 5.3.2

1. Ljk = Lik ∗ Ln.

2. Ljx = Lk ∗ Ln.

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3. For all L1 ∈ Int and L2 ∈ Neg, we have (L1 ∗ L2 )p , (L1 ∗ L2 )l ∈ Spec(L1 , L2 ) and the following equality holds (L1 ∗ L2 )p + (L1 ∗ L2 )l = L1 ∩ L2 . In particular, Le = Lg + Ljl . 4. For every L1 ∈ Int and L2 ∈ Neg such that L1 = Lk, the logics (L1 ∗ L2 )p and (L1 ∗ L2 )l are diﬀerent from the endpoints of the interval Spec(L1 , L2 ). At the same time, if L1 = Lk, we have (Lk ∗ L2 )p = Lk ∩ L2 and (Lk ∗ L2 )l = Lk ∗ L2 . 5. LjlN = Lil ∗ Ln. 6. LjlQ 1 = Li ∗ Lnl. Q Q 7. LjlQ 2 ∈ Spec(Li, Lnl), Ljl2 = Li ∗ Lnl, Ljl2 = Li ∩ Lnl.

8. LjeQ 1 = Li ∗ Lmn. Q Q 9. LjeQ 2 ∈ Spec(Li, Lmn), Lje2 = Li ∗ Lmn, Lje2 = Li ∩ Lmn.

10. The logic Ljl is a proper extension of (Lil ∗ Lnl)l = Ljl lN lQ 1. Proof. Items 1,2,5,6,8 easily follow from Propositions 4.2.3 and 4.2.7 and suitable items of the last proposition. 3. By Item 3 of Proposition 5.3.1 all elements of the form ⊥∨(⊥ → a) are regular in models of the logic (L1 ∗L2 )l . On the other hand, in models of the logic (L1 ∗ L2 )p all elements of this form are dense, as follows from Item 5 of Proposition 5.3.1. Thus, in models of the least upper bound of logics (L1 ∗ L2 )p and (L1 ∗L2 )l elements of the form ⊥∨(⊥ → a) are regular and dense simultaneously, i.e., they are all equal to the unit element. Consequently, models of the considered least upper bound are exactly direct products of the form B × C, where B |= L1 and C |= L2 , whence we immediately obtain the desired equality by Proposition 4.1.4 and Corollary 4.1.5. 4. The assertion of this item is true due to the fact that for any Heyting algebra A the following three conditions are equivalent: A is a Boolean algebra; the unit element is the only dense element of A; all elements of A are regular. 7. By Item 8 of Proposition 5.3.1 the logic LjlQ 2 belongs to Spec(Li, Lnl). Consider a model A for the free combination Li ∗ Lnl structured as follows. An upper algebra A⊥ is arbitrary; a lower algebra A⊥ is a 4-element negative

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Peirce algebra with universe {⊥, a, b, 0}, where 0 ≤ a ≤ ⊥, 0 ≤ b ≤ ⊥, and elements a and b are incomparable; f (⊥) = 1, f (x) = ⊥ for x = ⊥. Calculate f (a → b) ∨ f (b → a) = f (b) ∨ f (a) = ⊥ ∨ ⊥ = ⊥, which proves that LjlQ 2 diﬀers from Li ∗ Lnl. We now point out a model for LjlQ 2 diﬀerent from the direct product of Heyting and negative algebras. This will prove that LjlQ 2 is not equal to the intersection of logics Li and Lnl. Let B and C be Heyting and negative l-algebras respectively, which are isomorphic as implicative lattices, and let f : C → B be an arbitrary lattice isomorphism. It is not hard to check that B ×f C is the desired model of LjlQ 2. Q 9. The fact that Lje2 belongs to the interval Spec(Li, Lmn) follows from Item 10 of Proposition 5.3.1. Examples of j-algebras showing that LjeQ 2 diﬀers from the endpoints of the indicated interval can be constructed in a way similar to that of Item 7. 10. This item can also be proved in a way similar to Item 7. As a counterexample showing that the indicated extension is proper we may take the j-algebra A from Item 7 with the additional restriction that A⊥ is a Heyting l-algebra. 2 Now we have enough information about j-algebras modelling Segerberg’s axioms and we can come back to the analysis of Figure 5.2. We denote by N eg the line passing trough the logics Ln and Lmn and by Int the line passing through the logics Li and Lk. Recall that logics lying on the line Int (N eg) form the intersection of the class D of logics presented in Figure 5.2 with the class Int (respectively, with the class Neg), D ∩ Int = Int and D ∩ Neg = N eg. These lines play a part of the coordinates for the threedimensional part of Figure 5.2, which we denote by P ar, P ar = D ∩ Par. For any logic L ∈ P ar we can naturally deﬁne its projections I(L) and N (L) to the lines Int and N eg respectively. For example, I(Lj) = Li, N (Lj) = Ln, I(Ljl) = Lil, N (Ljl) = Lnl, I(Ljx) = Lk, N (Ljx) = Ln. Using Proposition 5.3.1 and Corollary 5.3.2 we can easily check that for all logics L ∈ P ar the equalities I(L) = Lint and N (L) = Lneg

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take place. Thus, for any line L on the diagram, which is parallel to the line (Lj, Le ), the logics of this line have ﬁxed intuitionistic and negative counterparts, say L1 and L2 . And so we have L = D ∩ Spec(L1 , L2 ). We stated in this way that the three dimensions of the part P ar of Figure 5.2 exactly correspond to the three parameters determining a position of a logic in the class Par. One coordinate of a logic L is its intuitionistic counterpart Lint ∈ Int, the second coordinate is its negative counterpart Lneg ∈ Neg, and the third coordinate corresponds to a position of L inside the interval Spec(Lint , Lneg ), which is determined in turn by the class of admissible semilattice homomorphisms from models of Lneg to models of Lint . At this point we note one obvious defect of Figure 5.2. Let us consider the planes in the part P ar of the ﬁgure parallel to the plane with points Lj, Ljk, and LjlQ 1 . There are three such planes. We denote by Pj the plane containing the point Lj, by Pl the plane containing the point Ljl, and, ﬁnally, by Pe the plane containing the point Le. If we follow the geometrical analogues sketched above, we would expect that all logics belonging to one of the planes Pj, Pl, Pe will deﬁne the same class of admissible homomorphisms. But this holds only for the plane Pe. For any logic L ∈ Pe we have ⊥ ∨ (⊥ → p) ∈ L, and so L = Lint ∩ Lneg is the greatest point of the interval Spec(Lint , Lneg ), which is determined by the class of homomorphisms identically equal to the unit element. Let us consider the plane Pj. Elements of this plane are the least points in the sets P ar ∩ Spec(L1 , L2 ), where L1 ∈ {Li, Lik, Lil} and L2 ∈ {Ln, Lln, Lmn}. As we know from Proposition 4.2.5 the least point of Spec(L1 , L2 ) is the free combination L1 ∗ L2 of logics L1 and L2 . Moreover, for free combinations all semilattice homomorphisms from models of negative counterpart to models of intuitionistic counterpart are admissible. However, only three points of Pj, namely, the logics Lj, Ljk, and LjlN are free combinations of their intuitionistic and negative counterparts (see Items 1 and 5 of CorolQ lary 5.3.2). Logics LjlQ 2 and Lje2 are proper extensions of free combinations Li∗Lnl and Li∗Lmn respectively, as it follows from Items 7 and 9 of Corollary 5.3.2. Regarding the remaining four logics in Pj, we can easily modify the proofs of Items 7 and 9 of Corollary 5.3.2 to show that the restrictions, Q which axioms LQ 2 and E2 impose on the class of admissible semilattice homomorphisms remain non-trivial, even if the intuitionistic counterpart of a logic satisﬁes axioms K or LN (see also Propositions 5.3.4 and 5.3.5 below). In case of the plane Pl we have a similar situation. Only the logics in the leftmost vertical line have the class of admissible semilattice homomorphisms with a range contained in the set of regular elements of an upper

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algebra (see Item 3 of Proposition 5.3.1). The other logics are characterized by narrower classes of admissible homomorphisms (see Propositions 5.3.4 and 5.3.5). The indicated defect can easily be overcome if we replace the axioms Q Q Q LQ 2 and E2 by L1 and E1 respectively. As follows from Items 7 and 9 of Proposition 5.3.1, these axioms do not impose any restrictions on the class of admissible homomorphisms and restrict only the class of lower algebras. These axioms can thus be considered as an adequate relativization of the axioms L and E to the negative counterpart of a logic. After the abovementioned replacement and deleting axiom L, we obtain a diagram of logics having exactly the same conﬁguration as that of Figure 5.2. Q As we have seen above, the axioms LQ 2 and E2 impose restrictions on the classes of lower algebras of their models and simultaneously on the classes of admissible homomorphisms from the lower algebras of their models to the upper ones. We can separate these restrictions. As follows from PropoQ sition 5.3.1 axioms LQ 1 and E1 restrict the classes of lower algebras in the Q same way as axioms LQ 2 and E2 respectively, and have no inﬂuence on the classes of admissible homomorphisms. On the other hand, as follows from the next proposition, the axioms F1 . ⊥ ∨ (⊥ → (p → q)) ∨ (⊥ → (q → p))(= ⊥ ∨ LQ 2) F2 . ⊥ ∨ (⊥ → p) ∨ (⊥ → (p → q))(= ⊥ ∨ EQ 2) will restrict the classes of admissible homomorphisms in the same way as Q was done by axioms LQ 2 and E2 respectively, and will not change the classes of lower algebras. Proposition 5.3.3 Let A be an arbitrary j-algebra. The following equivalences hold. 1. A |= Ljf1 if and only if we have fA (y1 → y2 ) ∨ fA (y2 → y1 ) = 1 for all y1 , y2 ∈ A⊥ . 2. A |= Ljf2 if and only if we have fA (y1 ) ∨ fA (y1 → y2 ) = 1 for all y1 , y2 ∈ A⊥ . This statement can be proved via a direct computation. It is clear that Q Q N Q = LjlQ 1 f1 , Lje2 = Lje1 f2 , and Ljl = Ljl l l1 f1 . Let us consider the class D1 consisting of logics which can be obtained by adjoining to Lj some subset of the following set of axioms LjlQ 2

Q {I, Q, K, X, L, E, L , E , P , LN , LQ 1 , E1 , F1 , F2 }.

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r L∗ e = L1 ∩ L2 C@ C @ C @ @ C @ C @ C @ C @ rL∗ l f2 r L∗ gf2 CX X XrXX @ r A ∗\ @rL∗ l A r L∗ g L gf\1 L∗ l f1 XXX @ A \ @ Ar \ @ \ L∗ f2 @ \ r @ \ @ L∗ f1 @ @ @ @r

L∗

Figure 5.3 In this way, we take into account all properties involved in Segerberg’s axioms. Obviously, D ⊆ D1 . At the same time, D satisﬁes the condition that for any L1 ∈ Int ∩ D and L2 ∈ Neg ∩ D the intersection Spec(L1 , L2 ) ∩ D is linearly ordered. In case of D1 , this condition fails as we can see from the propositions below. Proposition 5.3.4 Let L1 ∈ {Li, Lik, Lil}, L2 ∈ {Ln, Lnl}, and let L∗ := L1 ∗ L2 . The set of logics Spec(L1 , L2 ) ∩ D1 forms an upper semilattice, shown on Figure 5.3. In the course of proving this and subsequent propositions, we will construct various j-algebras to check the interrelations between diﬀerent logics. The following Heyting and negative algebras will play the part of breaks in our constructions: 2 and 2 are two-element Heyting and negative algebras; 3H and 3N are three-element Heyting and negative algebras, the elements of which are linearly ordered; ﬁnally, 4H and 4N are four-element Heyting and negative algebras respectively, whose implicative lattices are Peirce algebras. For any Heyting algebra B, negative algebra C, and for any j-algebra constructed from them B ×f C, we will identify an element b of B (c of C) with the corresponding element (b, ⊥) of the upper algebra (B ×f C)⊥ ((⊥, c) of the lower algebra (B ×f C)⊥ ).

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r

1

r⊥

2

r

⊥

r−1

2

r

1

r

r

r

a

r⊥

3H

r1 @ @ @rb

⊥ c

r−1

3N

ar

@

@ @r ⊥

4H

r⊥ @ @ @rd

cr

@

@ @r −1

4N

Figure 5.4 Proof (of Proposition 5.3.4.) First of all, we note that due to our assumption L1 = Lk. This fact together with Items 3 and 4 of Corollary 5.3.2, implies that logics L∗ g and L∗ l are diﬀerent from the endpoints of the interval Spec(L1 , L2 ) and the least upper bound of these logics coincides with the greatest point of the interval, L∗ g + L∗ l = L∗ e , which means, in particular, that L∗ g and L∗ l are incomparable. Let us consider the logics L∗ f1 and L∗ f2 . Take an arbitrary model A for L∗ f2 . Due to Proposition 5.3.3 we have fA (y1 ) ∨ fA (y1 → y2 ) = 1 for all y1 , y2 ∈ A⊥ . Since y1 ≤ y2 → y1 , we have fA (y1 ) ≤ fA (y2 → y1 ), and also fA (y2 → y1 ) ∨ fA (y1 → y2 ) = 1 for all y1 , y2 ∈ A⊥ . In view of Proposition 5.3.3 the latter means that A is a model for L∗ f1 , and we have the inclusion L∗ f1 ⊆ L∗ f2 . Let us consider a j-algebra A1 := 3H ×f1 3N , where f1 : 3N → 3H is a uniquely deﬁned implicative lattice isomorphism (see Figure 5.5, in which the structures of algebras constructed in this and the next proposition are presented). For any y1 , y2 ∈ (A1 )⊥ we have f1 (y1 → y2 ) ∨ f1 (y2 → y1 ) = (f1 (y1 ) → f1 (y2 )) ∨ (f1 (y2 ) → f1 (y1 )) = 1 since 3H |= (p → q) ∨ (q → p). Thus, A1 |= L∗ f1 . Now we take the elements −1, c ∈ 3N . It is clear that f1 (−1) = ⊥ and that f1 (c) = a (see Figure 5.4). We have f1 (c) ∨ f1 (c → −1) = f1 (c) ∨ f1 (−1) = a ∨ ⊥ = a = 1. This means that A1 is not a model for L∗ f2 , and so the inclusion L∗ f1 ⊂ L∗ f2 is proper. Consider j-algebras A2 := 2 ×f2 4N , where f2 (⊥) = 1 and f2 (x) = ⊥ for x < ⊥, and A3 := 4H ×f3 4N , where f3 is an implicative lattice isomorphism between 4N and 4H . As in Items 7 and 9 of Corollary 5.3.2 we can show that A2 is a model of L∗ , but is not a model of L∗ f1 , respectively, that A3 is a model of L∗ f2 , but is not a model of L∗ e . We have thus proved that L∗ f1 and L∗ f2 are diﬀerent from the endpoints of the interval Spec(L1 , L2 ).

5.3 Segerberg’s Logics and their Semantics r

r @

r

1

ra @ @

@r ⊥

@ @r

r1 @ @ @rb ar @ @ @ @ @r @r r ⊥ @ @ @ @ @r @r c@ d @ @r−1

1

r⊥ @ @ @r d

cr

@ @

@r

c

75

−1

A2

r−1

A3

A1 r1

r1

r1

r1

ra @ @ @r ⊥ r r @ @ @ @ @ @r r d @ c @ @r

ra @

r⊥

ra @

−1 A4

r @ @

@

@r ⊥

@r

rc r

−1

A5

−1

r r

1

A6

r @ r @

@ @r

@ @r

−1

@ @r⊥

c

A7

⊥

r−1

A8 Figure 5.5

Now we check that each of the logics L∗ f1 and L∗ f2 is incomparable with either of the logics L∗ g or L∗ l . The j-algebras A1 and A3 are models of L∗ f1 and L∗ f2 respectively, but theirs are not models of L∗ g, which implies that L∗ g is not contained in either of the logics L∗ f1 or L∗ f2 . Deﬁne j-algebra A4 as 3H ×f4 4N , where f4 (⊥) = 1 and f4 (x) = a for x < ⊥. A4 is a model for L∗ g, since the element a is dense in 3H , but it is not a model for L∗ f1 , in which case it is not also a model for L∗ f2 . Indeed, for c, d ∈ 4N , we have f4 (c → d) ∨ f4 (d → c) = f4 (d) ∨ f4 (c) = a ∨ a = a. We have thus proved that L∗ f1 and L∗ f2 are incomparable with L∗ g.

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The algebra A2 provides a counterexample, demonstrating that either of the logics L∗ f1 or L∗ f2 is not contained in L∗ l . To state that the inverse inclusions also fail we consider a j-algebra A5 := 3H ×f5 2 , where f5 (−1) = a. This is not a model for L∗ l since a is not regular. At the same time, the direct calculation shows that A5 |= L∗ f2 . In this way, L∗ l is not contained in L∗ f2 , moreover, it is not contained in L∗ f1 . The above facts on incomparability of logics imply, in particular, that L∗ gfi is a proper extension of L∗ fi and of L∗ g, i = 1, 2, and that L∗ l fi is a proper extension of L∗ fi and of L∗ l , i = 1, 2. So, it remains to verify that the inclusions L∗ gf1 ⊆ L∗ gf2 ⊆ L∗ e and L∗ l f1 ⊆ L∗ l f2 ⊆ L∗ e are proper. A j-algebra A6 := 3N ⊕ 2 (∼ = 2 ×f6 3N , where f6 (x) = ⊥ for x < ⊥) ∗ ∗ shows that L l f1 ⊆ L l f2 is a proper inclusion. It is a model for L∗ lf1 since for any y1 , y2 ∈ 3N , either y1 → y2 = ⊥ or y2 → y1 = ⊥. On the other hand, f6 (c) ∨ f6 (c → −1) = ⊥ ∨ ⊥ = ⊥. Note that A3 is a model for L∗ l f2 diﬀerent from the direct product of 4H and 4N . This proves that L∗ e is a proper extension of L∗ l f2 . Finally, consider the algebras A7 := 3H ×f7 3N , where f7 (x) = a for x < ⊥, and A5 deﬁned above. The ﬁrst of these algebras is a counterexample showing that the inclusion L∗ gf1 ⊆ L∗ gf2 is proper. The second algebra can be used to check that L∗ e is a proper extension of L∗ gf2 . 2 Proposition 5.3.5 Let L1 ∈ {Li, Lik, Lil} and L∗ := L1 ∗Lmn. The set of logics Spec(L1 , Lmn) ∩ D1 forms an upper semilattice shown on the semilattice diagram in Figure 5.6. rL∗ e = L1 ∩ Lmn B @ B@ B @ @ B @ B L∗ gf1 = L∗ gf2r L∗ l f1@ = L∗ l f2 BrP P L PP@ P@ L Pr r ∗ L L g@ L∗ l @ L @ Lr @ L∗ f1 = L∗ f2 @ @ @ @r

L∗ Figure 5.6

5.3 Segerberg’s Logics and their Semantics

77

Proof. As in the previous proposition, we have the assumption L1 = Lk, which implies that logics L∗ g and L∗ l are diﬀerent from the endpoints of the interval Spec(L1 , L2 ), are incomparable and their upper bound coincides with the greatest point of the interval. We argue to prove the equality L∗ f1 = L∗ f2 . The inclusion L∗ f1 ⊆ L∗ f2 was stated above. Let us check the inverse inclusion. Take an arbitrary model A of L∗ f1 , which means that fA (x → y) ∨ fA (y → x) = 1 for all x, y ∈ A⊥ . By assumption A⊥ satisﬁes the Peirce law, and so for any x, y ∈ A⊥ , we have x = (x → y) → x. On the other hand, in any j-algebra we have the identity x → y = x → (x → y). In this way, for any x, y ∈ A⊥ , we have fA (x) ∨ fA (x → y) = fA ((x → y) → x) ∨ fA (x → (x → y)) = 1, which proves the desired equality. The lower algebras of j-algebras A2 , A3 , A4 , and A5 deﬁned in Proposition 5.3.4 are models for Lmn and so these algebras can be used in the following reasoning. In particular, j-algebras A2 and A3 can be used to check that the logic L∗ f1 lies inside the interval Spec(L1 , Lmn). With the help of A4 and A8 := 2 ⊕ 2 we can show that the logics L∗ f1 and L∗ g are incomparable. A4 is a model for L∗ g, but not for L∗ f1 . Conversely, A8 is a model for L∗ f1 , but not for L∗ g. In a similar way, one can use algebras A2 and A5 to check that logics ∗ L f1 and L∗ l are incomparable. We are left to check that the following inclusions are proper: L∗ f1 g ⊆ L∗ e and L∗ f1 l ⊆ L∗ e . The suitable counterexamples are provided by algebras A5 and A3 , respectively. 2 We have not yet considered the case when the intuitionistic counterpart coincides with the classical logic. It turns out that only in this case sets of the form Spec(L1 , L2 ) ∩ D1 are linearly ordered with respect to inclusion. Proposition 5.3.6 Let L2 ∈ {Ln, Lnl, Lmn}, and let L∗ := Lk ∗ L2 . The sets of logics Spec(Lk, L2 ) ∩ D1 have the structure presented in Figure 5.7. Proof. First, consider the case L2 ∈ {Ln, Lnl}. Algebras A6 , A8 , and A2 can be used to verify that the inclusions L∗ ⊂ L∗ f1 , L∗ f1 ⊂ L∗ f2 , and respectively L∗ f2 ⊂ L∗ e are proper. In case L2 = Lmn we may again use A8 and A2 to check the corresponding relations between logics. 2

78

5 Adequate Algebraic Semantics for Extensions of Minimal Logic rL∗ e = L∗ g

rL∗ f

2

rL∗ e = L∗ g

rL∗ f

1

rL∗ f = L∗ f 1 2

rL∗ = L∗ l

r

L2 ∈ {Ln, Lnl}

L∗ = L∗ l

L2 = Lmn Figure 5.7

5.4

Kripke Semantics for Paraconsistent Extensions of Lj2

In this section we deﬁne analogs of upper and lower algebras associated with a given j-algebra for j-frames. For an arbitrary j-frame W = W, , Q we deﬁne the following frames W (+) := W \ Q, ∩(W \ Q)2 , ∅ , W (−) := Q, ∩Q2 , Q . It is obvious that W (+) is a model for intuitionistic logic and W (−) is a model for minimal negative logic. Remark. For any j-frame W and any formula ϕ, the translation In(ϕ) is true on j-frame W (−) , W (−) |= In(ϕ). This fact can be checked via an easy induction on the structure of formulas. Lemma 5.4.1 Let W be an arbitrary j-frame, v a valuation of W (+) , and let v be a valuation of W such that for any propositional variable p we have 2

The content of this section was originally published in [76] (Elsevier, UK). Reprinted here by permission of the publisher.

5.4 Kripke Semantics for Paraconsistent Extensions of Lj

79

v(p) = v (p)∩ (W \Q). Then for any formula ϕ and for an arbitrary element x ∈ W \ Q the following equivalence holds

W, v |=x In(ϕ) ⇐⇒ W (+) , v |=x ϕ. Proof. Let μ := W, v and μ(+) := W (+) , v . We argue by induction on the structure of formulas. The case of constant ⊥ is trivial. For an arbitrary propositional variable p and x ∈ W \ Q we have μ |=x p ∨ ⊥ if and only if either x ∈ v (p) or x ∈ Q. The second alternative is impossible by assumption. Thus we have x ∈ v (p) and x ∈ W \ Q, i.e., x ∈ v(p). The latter is equivalent to μ(+) |=x p. Now, we assume that for formulas ϕ and ψ and for all x ∈ W \ Q the equivalences μ |=x In(ϕ) ⇐⇒ μ(+) |=x ϕ and μ |=x In(ψ) ⇐⇒ μ(+) |=x ψ hold. Prove that the desired equivalence takes place for the implication ϕ → ψ. Let μ |=x In(ϕ → ψ)(= In(ϕ) → In(ψ)) for some x ∈ W \ Q. This means that for all y ∈ W , the relations x y and μ |=y In(ϕ) imply μ |=y In(ψ). In view of the assumed equivalences, we have ∀y ∈ W \ Q(x y ⇒ (μ(+) |=y ϕ ⇒ μ(+) |=y ψ)), and so μ(+) |=x ϕ → ψ. Conversely, let μ(+) |=x ϕ → ψ for some x ∈ W \ Q. By assumption for all y ∈ W \ Q such that x y, if μ |=y In(ϕ), then μ |=y In(ψ). If y ∈ Q, then μ |=y In(ϕ) and μ |=y In(ψ). Thus, for all y ∈ W such that x y, we have μ |=y ϕ ⇒ μ |=y ψ, which means that μ |=x In(ϕ → ψ). The cases of disjunction and conjunction are trivial. The next proposition demonstrates that frames considered as analog of upper and lower algebras.

W (+)

and

W (−)

2 can be

Proposition 5.4.2 For a j-frame W and a formula ϕ, the following equivalences hold W |= In(ϕ) ⇐⇒ W (+) |= ϕ, W |= ⊥ → ϕ ⇐⇒ W (−) |= ϕ.

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5 Adequate Algebraic Semantics for Extensions of Minimal Logic

Proof. The ﬁrst equivalence immediately follows from the previous Lemma. If W |= ⊥ → ϕ, then for any valuation v of W, ϕ is true in all abnormal worlds of the model W, v , which means by Lemma 2.3.1 that W (−) , v Q |= ϕ. Any valuation v of W (−) can be considered as a valuation of W, in which case v = v Q . Thus, for all valuations v of W (−) , we have W (−) , v |= ϕ, i.e., W (−) |= ϕ. Conversely, the assumption W (−) |= ϕ implies that for any valuation v of W, W (−) , v Q |= ϕ. In view of Lemma 2.3.1, the latter means that for all valuations v of W, ϕ is true at any abnormal world of W, v , which implies, in turn, W, v |= ⊥ → ϕ. 2 The following fact immediately follows from the deﬁnition of intuitionistic and negative counterparts and from the last proposition. Corollary 5.4.3 Let L ∈ Jhn and W |= L. Then W (+) |= Lint and W (−) |= Lneg . For a class of j-frames K, we deﬁne

2

K(+) := {W (+) | W ∈ K}, K(−) := {W (−) | W ∈ K}. Proposition 5.4.4 Let K be a class of j-frames and let L = LK. Then Lint = LK(+) and Lneg = LK(−) . Proof. The inclusion Lint ⊆ LK(+) follows from Corollary 5.4.3. We argue for the inverse inclusion. Take a ϕ ∈ Lint , in which case In(ϕ) ∈ L. Consequently, there exist a frame W ∈ K, its valuation v, and an element x ∈ W such that W, v |=x In(ϕ). As was remarked above, a formula of the form In(ψ) is true in any model at any abnormal element, therefore, x ∈ Q. Whence, by Lemma 5.4.1 we have W (+) |= ϕ. Now we turn to the second equality. Again, we have to prove only the inclusion LK(−) ⊆ Lneg since the inverse inclusion follows from Corollary 5.4.3. Let ϕ ∈ Lneg , i.e., L ⊥ → ϕ. Consider a j-frame W ∈ K such that W |= ⊥ → ϕ. From the last relation we obtain by Proposition 5.4.2 W (−) |= ϕ, i.e., ϕ ∈ LK(−) .

Chapter 6

Negatively Equivalent Logics1 In the following, by negative formulas we mean formulas of the form ¬ϕ. The well-known Glivenko theorem implies, in particular, that in intuitionistic and in classical logic the same negative formulas are provable. This means that intuitionistic and classical logic, as well as all intermediate logic have, in a sense, the same negation. Generalizing this relation between logics we deﬁne negatively equivalent logics as logics where the same negative formulas are inferable from the same sets of hypotheses. From the constructive point of view we need negation to refute formulas on the basis of one or another set of hypotheses, therefore, negatively equivalent logics have essentially the same negation. Unlike the class of intermediate logics, the relation of negative equivalence is non-trivial on the class Jhn+ and in this chapter we obtain several interesting results on the structure of negative equivalence classes. Simultaneously, we prove the results on cardinality of intervals of the form Spec(L1 , L2 ).

6.1

Deﬁnitions and Simple Properties

Let L1 and L2 be logics in Jhn+ . We say that L1 is negatively lesser than L2 , and write L1 ≤neg L2 , if for any set of formulas X and formula ϕ, the following implication holds: X L1 ¬ϕ =⇒ X L2 ¬ϕ. 1

Parts of this chapter were originally published in [75] (Springer, Netherlands). Reprinted here by permission of the publisher.

81

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6 Negatively Equivalent Logics

In other words, one logic is negatively lesser that the other if passing from one to the other preserves the negative consequence relation, i.e., the consequence relation of the form X ¬ϕ, in which the conclusion is negative. As we can see from the proposition below, the condition of preserving the negative consequence relation can be replaced by that of preserving the class of inconsistent sets of formulas. However, the equivalence proved in this proposition is typical for the class Jhn+ , because in this class the negation is deﬁned via the constant “absurdity”, whereas the absurdity ⊥ can be deﬁned as a negation of tautology. Proposition 6.1.1 For any L1 , L2 ∈ Jhn+ , the following conditions are equivalent. 1. L1 ≤neg L2 . 2. For an arbitrary set of formulas X, if X L1 ⊥, then X L2 ⊥. Proof. 1) ⇒ 2) If X L1 ⊥, then X L1 ¬ϕ for any formula ϕ. By the assumption that L1 ≤neg L2 , we have X L2 ¬ϕ for any ϕ. Take an L2 tautology ψ, then X L2 ψ, ¬ψ, whence X L2 ⊥. 2) ⇒ 1) Let X L1 ¬ϕ. Then X ∪ {ϕ} L1 ⊥. In this case, we have X ∪ {ϕ} L2 ⊥ by assumption, consequently, X L2 ϕ → ⊥ by Deduction Theorem, i.e., X L2 ¬ϕ. 2 The deﬁnition of ≤neg can also be re-worded as follows. Proposition 6.1.2 For any L1 , L2 ∈ Jhn+ , the relation L1 ≤neg L2 holds if and only if for any formula ϕ, the following implication is true: L1 ϕ =⇒ L2 ¬¬ϕ. Proof. Let L1 ≤neg L2 . If L1 ϕ, then {¬ϕ} L1 ⊥. By the last proposition we have {¬ϕ} L2 ⊥, from which we immediately obtain L2 ¬¬ϕ. Now we assume that the right-hand side of the desired equivalence holds. Let X L1 ⊥. This means that for some formulas ϕ1 , . . . , ϕn ∈ X, L1 (ϕ1 ∧. . .∧ϕn ) → ⊥. According to our assumption L2 ¬¬¬(ϕ1 ∧. . .∧ϕn ). In view of Lj ¬¬¬p ↔ ¬p we obtain L2 ¬(ϕ1 ∧. . .∧ϕn ), which immediately implies that X L2 ⊥. Again, Proposition 6.1.1 allows one to conclude that L1 is negatively lesser than L2 . 2 If one of the logics is ﬁnitely axiomatizable relative to the other, the last statement can be simpliﬁed as follows.

6.1 Deﬁnitions and Simple Properties

83

Proposition 6.1.3 Let L1 , L2 ∈ Jhn+ and L2 = L1 + {ϕ1 , . . . , ϕn }. Then L2 ≤neg L1 if and only if ¬¬ϕ1 , . . . , ¬¬ϕn ∈ L1 . Proof. We consider only the non-trivial implication. Let ¬¬ϕ1 , . . . , ¬¬ϕn ∈ L1 . Take an arbitrary set of formulas X with X L2 ⊥, then X ∪{ψ1 , . . . , ψk } is inconsistent in L1 , or equivalently, X L1 ¬(ψ1 ∧ . . . ∧ ψk ), where ψ1 , . . . , ψk are substitution instances of formulas from the list ϕ1 , . . ., ϕn . We have L1 ¬¬ψ1 , . . . , ¬¬ψk by assumption. Consider an arbitrary model A |= L1 and an A-valuation v. The elements v(ψ1 ), . . . , v(ψk ) are dense in A⊥ . Consequently, the element v(ψ1 ∧ . . . ∧ ψk ) is also dense, in particular, v(¬(ψ1 ∧ . . . ∧ ψk )) = ⊥A . Let formulas θ1 , . . . , θm ∈ X be such that L1 (θ1 ∧ . . . ∧ θm ) → ¬(ψ1 ∧ . . . ∧ ψk ). In view of the above considerations, for any model A |= L1 and any Avaluation v, we have v(θ1 ∧ . . . ∧ θm ) ≤ ⊥A . Consequently, we have L1 (θ1 ∧ . . . ∧ θm ) → ⊥, which means that X is inconsistent in L1 . 2 Deﬁne the relation ≡neg as an intersection of ≤neg and its inverse relation: ≡neg :=≤neg ∩(≤neg )−1 . One can easily prove Lemma 6.1.4

1. The relation ≤neg is a preordering.

2. The relation ≡neg is an equivalence. 2 In view of this lemma, logics L1 , L2 ∈ Jhn+ with L1 ≡neg L2 will be called negatively equivalent. From Propositions 6.1.1 and 6.1.2 we immediately obtain Corollary 6.1.5 For any L1 , L2 ∈ Jhn+ , the following conditions are equivalent. 1. L1 ≡neg L2 . 2. An arbitrary set of formulas X is inconsistent in L1 if and only if it is inconsistent in L2 .

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6 Negatively Equivalent Logics

3. For any formula ϕ, the following implications hold: (L1 ϕ =⇒ L2 ¬¬ϕ) and (L2 ϕ =⇒ L1 ¬¬ϕ). Remark. Notice that the inclusion L1 ⊆ L2 implies L1 ≤neg L2 . Therefore, the logics L1 and L2 from Proposition 6.1.3 will be negatively equivalent. Remark. Any two negative logics are negatively equivalent due to the following fact. In an arbitrary negative logic any set of formulas in inconsistent since the absurdity ⊥ belongs to the set of logical tautologies. Any two intermediate logics are also negatively equivalent, which easily follows from Proposition 6.1.3 and Glivenko’s theorem. It is well known that negation in intuitionistic logic is not constructive, from the deducibility Li ¬(ϕ ∧ ψ) does not follow, in a general case, that either of the formulas ¬ϕ or ¬ψ is provable in Li. We have just noted that negation in an arbitrary intermediate logic is close in some sense (negatively equivalent) to classical negation. This fact can be considered as a generalization of Glivenko’s theorem and also emphasizes the non-constructive character of negation in intermediate logics.

6.2

Logics Negatively Equivalent to Intermediate Ones

In this section we consider the question: which logics from the class Jhn are negatively equivalent to intermediate ones? More exactly, let some logics L1 ∈ Int and L2 ∈ Neg be ﬁxed. Which logics L ∈ Par having L1 and L2 as intuitionistic and negative counterparts respectively, are negatively equivalent to intermediate logic L1 and so to an arbitrary intermediate logic? In other words, to which extent can one weaken the law ex contradictione quodlibet while preserving the negative equivalence? Proposition 6.2.1 Let L1 ∈ Int, L2 ∈ Neg, and L ∈ Spec(L1 , L2 ). The equivalence L ≡neg L1 holds if and only if (L1 ∗ L2 )p ⊆ L. Proof. Recall that L1 = L + {⊥ → p}. According to Proposition 6.1.3 L ≡neg L1 whenever L ¬¬(⊥ → p). By deﬁnition (L1 ∗ L2 )p = L1 ∗ L2 + {¬¬(⊥ → p)}. 2 Call G(L1 , L2 ) := (L1 ∗ L2 )p the relativized Glivenko’s logic wrt L1 and L2 . From the last fact we easily infer the following strengthening of Generalized Glivenko’s Theorem (Theorem 5.1.2).

6.2 Logics Negatively Equivalent to Intermediate Ones

85

Corollary 6.2.2 Glivenko’s logic Lg is the least logic in Jhn, which is negatively equivalent to Lk. 2 As we can see from Item 4 of Corollary 5.3.2, the interval [Lk∗L2 , Lk∩L2 ] contains a unique paraconsistent logic negatively equivalent to Lk, namely Lk ∩ L2 . Note that this logic is axiomatized modulo the least logic Lk ∗ L2 of the interval Spec(Lk, L2 ) via the axiom ⊥ ∨ (⊥ → p) having essentially a non-constructive character. At the same time, if L1 = Lk, there is a proper subinterval [G(L1 , L2 ), L1 ∩ L2 ] consisting of logics negatively equivalent to intermediate logics. It turns out that the disjunction property can be transferred from an intuitionistic counterpart to the relativized Glivenko’s logic. This fact was established by M. Stukacheva [104]. Recall that a logic L has the disjunction property if ϕ ∨ ψ ∈ L implies ϕ ∈ L or ψ ∈ L. Let L ∈ Jhn. By induction on the length of formula ϕ we deﬁne an expression |L ϕ (“Kleene’s slash”, see [13]) as follows (further on, instead of “|L ϕ and L ϕ” we write L ϕ): |L ϕ |L ϕ ∧ ψ |L ϕ ∨ ψ |L ϕ → ψ

:= := := :=

L ϕ, where ϕ is an atomic formula; |L ϕ and |L ψ; L ϕ or L ψ; (L ϕ ⇒ |L ψ).

Proposition 6.2.3 [104] Let L1 ∈ Int, L2 ∈ Neg, and L1 has the disjunction property. If L1 ∗L2 ϕ, then |G(L1 ,L2 ) ϕ. Proof. Let G(L1 ,L2 ) ϕ. By induction on the length of proof, we show that |G(L1 ,L2 ) ϕ. In the proof we omit the lower index G(L1 , L2 ). Prove that this statement holds for axioms of G(L1 , L2 ). a) The case of Lj-axioms can be easily veriﬁed; b) For L1 ∗ L2 -axioms of the form ⊥→ ψ, where ψ ∈ L2 , the conclusion is obvious since G(L1 ,L2 ) ⊥; c) By induction of the structure of In(ϕ), ϕ ∈ L1 , prove that In(ϕ) implies |In(ϕ). The basis is obvious. Indeed, since L1 is non-trivial, we have L1 p, i.e., G(L1 ,L2 ) In(p);

86

6 Negatively Equivalent Logics

Let In(ϕ) and In(ψ) be such that In(ϕ) ⇒ |In(ϕ) and In(ψ) ⇒ |In(ψ). If In(ϕ ∧ ψ), then In(ϕ) and In(ψ). By induction hypothesis we have |In(ϕ) and |In(ψ), which means by deﬁnition |In(ϕ) ∧ In(ψ). Recall that In(ϕ) ∧ In(ψ) = In(ϕ ∧ ψ). If In(ϕ) ∨ In(ψ), then In(ϕ) or In(ψ), since L1 satisﬁes the disjunction property. By induction hypothesis In(ϕ) or In(ψ), i.e., |In(ϕ) ∨ In(ψ). Assume In(ϕ) → In(ψ) and In(ϕ), then In(ψ) and |In(ψ) by induction hypothesis. d) It remains to prove | ¬¬(⊥ → p). By deﬁnition we have | ¬¬(⊥ → p) ⇐⇒ (( (⊥ → p) → ⊥ and | (⊥ → p) → ⊥) ⇒ | ⊥). Since ¬¬(⊥ → p) and ⊥, we have (⊥ → p) → ⊥, which means that the right-hand side implication is true. Finally, let ϕ is obtained by modus ponens from ψ ∈ G(L1 , L2 ) and ψ → ϕ ∈ G(L1 , L2 ). We have by induction hypothesis | ψ and | ψ → ϕ. Consequently, ψ implies | ϕ, hence, | ϕ. 2 Proposition 6.2.4 Let L1 ∈ Int, L2 ∈ Neg, and L1 has the disjunction property. Then G(L1 , L2 ) has the disjunction property. Proof. Let G(L1 , L2 ) ϕ ∨ ψ. According to Proposition 6.2.3 we have |G(L1 ,L2 ) ϕ ∨ ψ, consequently, G(L1 ,L2 ) ϕ or G(L1 ,L2 ) ψ. 2 This fact shows that we can resign the law ex contradictione quodlibet preserving not only the class of inconsistent sets of formulas, but also constructive properties of intuitionistic logic. But the disjunction property does not hold in all relativized Glivenko’s logics. In particular, if L1 = Lk, we have G(Lk, L2 ) = Lk ∩ L2 ⊥ ∨ (⊥ → p) (Item 4 of Corollary 5.3.2). Moreover, if L1 does not possess the disjunction property, then G(L1 , L2 ) also does not. Indeed, let ϕ ∨ ψ be a corresponding counterexample, i.e., L1 ϕ ∨ ψ, but neither ϕ nor ψ are provable in L1 . In this case, the formula In(ϕ ∨ ψ)(= In(ϕ) ∨ In(ψ)) will refute the disjunction property for G(L1 , L2 ): G(L1 , L2 ) In(ϕ) ∨ In(ψ), but neither In(ϕ) not In(ψ) are provable in G(L1 , L2 ).

6.2 Logics Negatively Equivalent to Intermediate Ones

87

However, one can point out an interesting weak analog of the disjunction property, which holds in all relativized Glivenko’s logics G(L1 , L2 ) with L1 = Lk. We try to ﬁnd a property that holds in all relativized Glivenko’s logics, independently of constructive properties of intuitionistic counterparts. Therefore, it should be a property that is trivially satisﬁed in all intermediate logics, but becomes non-trivial in paraconsistent extensions of Lj. The property of a logic to be closed under the rule ϕ∨⊥ ϕ can serve as an example of such property. It can be considered as a weak analog of the disjunction property, because as well as in case of the disjunction property we conclude from a deducibility of disjunction to a deducibility of disjunction term. Proposition 6.2.5 Let Lk = L1 ∈ Int, L2 ∈ Neg, and let ϕ be an arbitrary formula. If G(L1 , L2 ) ϕ ∨ ⊥, then G(L1 , L2 ) ϕ. Proof. Let ϕ = ϕ(p1 , . . . , pn ). Assume that G(L1 , L2 ) ϕ ∨ ⊥, but ϕ is not provable in G(L1 , L2 ). This implies, in particular, that ϕ is not provable in L2 . Indeed, if L2 ϕ, then G(L1 , L2 ) ⊥ → ϕ and one can easily infer G(L1 , L2 ) ϕ. Thus, there exists a negative algebra B being a model for L2 , B |= L2 , and elements b1 , . . . , bn ∈ B such that ϕ(b1 , . . . , bn ) = ⊥. By assumption Lk = L1 , hence, there exists a Heyting algebra A with A |= L1 and a non-trivial ﬁlter of dense elements, Fd (A) = {1}. Take an element a ∈ Fd (A), a = 1, and consider a j-algebra A ×f B, where a semilattice homomorphism f is deﬁned as follows: f (⊥) = 1 and f (x) = a for x = ⊥. In this case, for any pair (x, y) ∈ A ×f B, we have x ≤ a when y = ⊥. Moreover, ρf = {a, 1} ⊆ Fd (A), which means that A ×f B is a model for G(L1 , L2 ) (see Proposition 5.3.1). Compute the value of ϕ on the elements (0, b1 ), . . . , (0, bn ) ∈ A ×f B. Taking into account that the mapping (x, y) → y deﬁnes an epimorphism of j-algebras A ×f B → B we have the equality ϕ((0, b1 ), . . . , (0, bn )) = (x, ϕ(b1 , . . . , bn )), where x ≤ a in view of ϕ(b1 , . . . , bn ) = ⊥. Thus, we have ϕ((0, b1 ), . . . , (0, bn )) ∨ (⊥, ⊥) = (x, ⊥) = (1, ⊥), which contradicts our assumption that G(L1 , L2 ) ϕ ∨ ⊥.

2 Remark. It is interesting that in the class of extensions of minimal logic the inference rule ϕ∨⊥ ϕ is equivalent to disjunctive syllogism in the following

88

6 Negatively Equivalent Logics

sense. Let L ∈ Jhn and let Ld be a deductive system with the set of axioms L and the only deductive rule modus ponens. Adding to Ld either of the rules ¬ϕ, ϕ ∨ ψ ϕ∨⊥ or ϕ ψ results with the deductive system having exactly the same consequence relation.

6.3

Abstract Classes of Negative Equivalence

For an arbitrary logic L ∈ Jhn+ , we deﬁne ∇(L) := {ϕ | ¬¬ϕ ∈ L}. We now observe that the set ∇(L) is itself a logic, possibly a trivial one, and point out some simple properties of the operator ∇ : Jhn+ → Jhn+ . Proposition 6.3.1 For an arbitrary L ∈ Jhn+ , the following facts take place. 1. L ⊆ ∇(L). 2. If L1 ∈ Jhn+ and L ⊆ L1 , then ∇(L) ⊆ ∇(L1 ). 3. ∇(L) ∈ Jhn+ . 4. ∇∇(L) = ∇(L). 5. ∇(L) = F if and only if L ∈ Neg ∪ {F}. Proof. 1. This is true because ϕ → ¬¬ϕ ∈ Lj. 2. This item trivially follows from the deﬁnition. 3. Let formulas ϕ and ϕ → ψ belong to ∇(L). Consider a model A |= L and take an arbitrary A-valuation v. By deﬁnition of ∇(L) we have ¬¬ϕ, ¬¬(ϕ → ψ) ∈ L, which means that the values of formulas ϕ ∨ ⊥ and (ϕ → ψ) ∨ ⊥ are dense, v(ϕ ∨ ⊥), v((ϕ → ψ) ∨ ⊥) ∈ Fd (A⊥ ). Calculate v(ϕ ∨ ⊥) ∧ v((ϕ → ψ) ∨ ⊥) = v((ϕ ∧ (ϕ → ψ)) ∨ ⊥) = v((ϕ ∧ ψ) ∨ ⊥) ≤ v(ψ ∨ ⊥) ∈ ∇(A). Thus, A |= ¬¬ψ for an arbitrary model A for L, i.e., L ¬¬ψ, whence ψ ∈ ∇(L). In this way, the set ∇(L) is closed under modus ponens. The

6.3 Abstract Classes of Negative Equivalence

89

fact that it is closed under the substitution rule follows directly from the deﬁnition. We have thus proved that ∇(L) is a logic, the fact that it extends Lj follows from Item 1. 4. First, we note that the object ∇∇(L) is well deﬁned in view of the previous item. The inclusion ∇(L) ⊆ ∇∇(L) follows from Item 1. Take a formula ϕ ∈ ∇∇(L), in this case ¬¬ϕ ∈ ∇(L) and ¬¬¬¬ϕ ∈ L. The last formula is equivalent in Lj to ¬¬ϕ, and so ϕ ∈ ∇(L), which proves the inverse inclusion. 5. If L ∈ Neg ∪ {F}, then ∇(L) = F, because an arbitrary negative formula belongs to L in this case. Assume L ∈ Jhn \ Neg. Then L ⊆ Lk and by Item 2 ∇(L) ⊆ ∇(Lk) = Lk. The last equality is due to the fact that a formula and its double negation are equivalent in Lk. 2 The operator ∇ is closely related to the negative equivalence relation, as we can see from the following Proposition 6.3.2

1. For any L ∈ Jhn+ , we have L ≡neg ∇(L).

2. For any L1 , L2 ∈ Jhn+ , the following equivalence holds L1 ≡neg L2 ⇐⇒ ∇(L1 ) = ∇(L2 ). Proof. 1. It follows from Item 3 of Corollary 6.1.5. 2. Let L1 ≡neg L2 . By deﬁnition ϕ ∈ ∇(L1 ) if and only if ¬¬ϕ ∈ L1 . In virtue of the negative equivalence of L1 and L2 , the last fact is equivalent to ¬¬ϕ ∈ L2 , which is equivalent, in turn, to ϕ ∈ ∇(L2 ). We have thus proved that ∇(L1 ) = ∇(L2 ). To prove the inverse implication assume ∇(L1 ) = ∇(L2 ). If ϕ ∈ L1 , then also ϕ ∈ ∇(L1 ), whence, by assumption ϕ ∈ ∇(L2 ), and so ¬¬ϕ ∈ L2 . In the same way, ϕ ∈ L2 implies ¬¬ϕ ∈ L1 . Applying Item 3 of Corollary 6.1.5 we conclude that L1 and L2 are negatively equivalent. 2 + For a logic L ∈ Jhn , we denote by [L]neg its abstract class with respect to negative equivalence, [L]neg := {L1 ∈ Jhn+ | L1 ≡neg L}. It turns out that each of such abstract classes forms an interval in the lattice Jhn+ , moreover the greatest point of the interval [L]neg can be calculated by ∇.

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Proposition 6.3.3 For any L ∈ Jhn \ Neg, [L]neg = [L , ∇(L)], where L ⊆ Lg. Proof. First we state that the set [L]neg is convex. Letting L1 , L2 ∈ [L]neg we check that the interval [L1 , L2 ] is contained in [L]neg . Take an arbitrary L ∈ [L1 , L2 ], we then have L1 ≤neg L ≤neg L2 . Taking into account L1 ≡neg L2 we immediately obtain L ∈ [L]neg . The logic ∇(L) is the greatest point of [L]neg . Indeed, if L ≡neg L and ϕ ∈ L , then ¬¬ϕ ∈ L, i.e., ϕ ∈ ∇(L), and we have the inclusion L ⊆ ∇(L). To state that [L]neg has the least point it is enough to observe that the intersection of an arbitrary family of logics from the class [L]neg again belongs to this class. One can give a more explicit presentation of the least logic from [L]neg . Put L := Lj + {¬¬ϕ | ϕ ∈ L}. Due to Proposition 6.1.2 every logic negatively equivalent to L must contain L . On the other hand, the logic L itself belongs to [L]neg . Indeed, the relation L ≤neg L follows from an obvious inclusion L ⊆ L, the inverse relation L ≤neg L follows from Proposition 6.1.2. According to Corollary 6.2.2, the logic Lg is the least logic in [Lk]neg , and so it has a presentation Lg = Lj + {¬¬ϕ | ϕ ∈ Lk}. Using this fact and the inclusion L ⊆ Lk we immediately obtain L ⊆ Lg. 2 Logics of the form ∇(L) admit another interesting characterization independent of the operator ∇ and the notion of negative equivalence. We deﬁne ∇-logics as ﬁxed-points of the operator ∇, i.e., we say that a logic L ∈ Jhn is a ∇-logic if ∇(L) = L. In view of Item 4 of Proposition 6.3.1, any logic of the form ∇(L) is a ∇-logic. The ∇-logics have a description, in which again arises the rule ϕ∨⊥ ϕ . Proposition 6.3.4 A logic L ∈ Jhn is a ∇-logic if and only if Lint = Lk and L is closed under the rule ϕ∨⊥ . ϕ

6.4 The Structure of Jhn+ up to Negative Equivalence

91

Proof. Recall that Lj ¬¬(p ∨ ¬p). This means that for every L ∈ Jhn, the formula p ∨ ¬p belongs to ∇(L). It was proved in Item 2 of Corollary 5.3.2 that Lj + {p ∨ ¬p} = Lk ∗ Ln, and so any logic of the form ∇(L) contains Lk ∗ Ln. An inclusion of logics implies the inclusion of respective counterparts (see Proposition 4.2.2), therefore, Lk ⊆ (∇(L))int . We have thus proved that intuitionistic counterparts of ∇-logics are classical. We now observe that every logic ∇(L) is closed under the rule ¬¬ϕ ϕ . This fact easily follows from the idempotentness of ∇. If ¬¬ϕ ∈ ∇(L), then ϕ ∈ ∇∇(L) = ∇(L). According to the lemma below, the double negation ¬¬ϕ is equivalent to ϕ∨⊥ in Lk ∗ Ln, and so in any ∇-logic, which completes the proof of the direct implication. Lemma 6.3.5 ¬¬p ↔ (p ∨ ⊥) ∈ Lk ∗ Ln. Proof. By deﬁnition of free combination we have ¬¬(p ∨ ⊥) ↔ (p ∨ ⊥) = In(¬¬p ↔ p) ∈ Lk ∗ Ln. It remains to note that ¬(p ∨ ⊥) ↔ ¬p ∈ Lj.

2 Prove the inverse implication. The condition Lint = Lk implies the inclusion Lk ∗ Ln ⊆ L, and we apply Lemma 6.3.5 to conclude that L is closed under the rule ¬¬ϕ ϕ . If ϕ ∈ ∇(L), then by deﬁnition ¬¬ϕ ∈ L, and applying the above rule we obtain ϕ ∈ L. 2

6.4

The Structure of Jhn+ up to Negative Equivalence

In this section, we give a characterization of the partial ordering

Jhn+ / ≡neg , neg , where neg :=≤neg / ≡neg . To obtain the main results we apply the technique of Jankov’s formulas suggested by V. A. Jankov [37, 38] and modiﬁed by H. Ono [83] and A. Wro´ nski [125, 126]. Usually, this technique is used for constructing uncountable families of logics. We are interested ﬁrst of all for Jankov’s formulas themselves. In our considerations, they will have the form of negative formulas, which allows one to prove that diﬀerent logics are not negatively equivalent. We recall basic elements of Jankov’s method adopting it for j-algebras.

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A relation X |=A ϕ, where X is a set of formulas, ϕ a formula, and A a j-algebra, means that for any A-valuation v, if v(ψ) = 1 for all ψ ∈ X, then v(ϕ) = 1. If K is a class of j-algebras, then X |=K ϕ means that X |=A ϕ for all A ∈ K. Finally, write X |=L ϕ instead of X |=M od(L) ϕ. Let A = A, ∨, ∧, →, ⊥, 1 be a not more than countable and subdirectly irreducible j-algebra. For each element a ∈ A, a = ⊥, we attach a unique propositional variable pa . Further, for any a ∈ A, we attach a unique atomic formula Za as follows pa , if a = ⊥ Za := ⊥, if a = ⊥. A diagram D(A) of A is the following set of formulas D(A)

:=

{Za∨b ↔ (Za ∨ Zb ) | a, b ∈ A}

∪

∪

{Za∧b ↔ (Za ∧ Zb ) | a, b ∈ A}

∪

∪

{Za→b ↔ (Za → Zb ) | a, b ∈ A}.

Let A be a ﬁnite subdirectly irreducible j-algebra. Then D(A) is a ﬁnite set of formulas and we can deﬁne a Jankov formula of A by J(A) := ( D(A)) → Z A , where ( D(A)) is the conjunction of all formulas in D(A), and A is the opremum of A. It is easy to see that J(A) ∈ LA. Moreover, the following statement holds. Lemma 6.4.1 Let A be a ﬁnite and subdirectly irreducible j-algebra. For each j-algebra B, the following two conditions are equivalent. 1. J(A) ∈ LB. 2. A is embeddable into a quotient algebra of B. Proof. 1 ⇒ 2. Assume B | = J(A). Let v be a B-valuation such that v( D(A)) ≤ v(Z A ). Put a0 := v( D(A)) and consider the quotient B/ a0 . Deﬁne a mapping h : A → B/ a0 by the rule h(a) := v(Za )/ a0 . It follows from v( D(A)) ∈ a0 that h is a homomorphism. Since a0 ≤ v(Z A ), we have h( A ) = 1, i.e., A ∈ Ker(h). This means that h is an embedding. 2 ⇒ 1. Let F be a ﬁlter on B and h : A → B/F be an embedding. Consider a B-valuation v such that v(Za )/F = h(a) for all a ∈ A. Homomorphism properties of h imply that for all ψ ∈ D(A) we have v(ψ) ∈ F ,

6.4 The Structure of Jhn+ up to Negative Equivalence

93

and so v( D(A)) ∈ F . At the same time, h( A ) =1/F since h is an embedding, which implies v(Z A ) ∈ F . In this way, v( D(A)) ≤ v(Z A ), i.e., B |= J(A). 2 In case A is not ﬁnite, we cannot, of course, deﬁne a Jankov formula of A. However, one can prove Lemma 6.4.2 Let A be a countable and subdirectly irreducible j-algebra. For each j-algebra B, the following conditions are equivalent. 1. D(A) |=B Z A . 2. A is embeddable into B. Proof. 1 ⇒ 2. Let v be a B-valuation such that v(ψ) = 1 for all ψ ∈ D(A) and v(Z A ) = 1B . Consider a mapping h : A → B given by the rule h(a) = v(Za ). It follows easily from our assumption and the deﬁnition of D(A) that h is a homomorphism. If h is not a monomorphism, then Ker(h) = {1A } and A ∈ Ker(h), i.e., h( A ) = 1B . The latter conﬂicts with the assumption that v(Z A ) = 1B . 2 ⇒ 1. Assume h embeds A into B. Consider a B-valuation such that v(pa ) = h(a) for a = ⊥A . Naturally, v(⊥) = h(⊥A ) = ⊥B . It is clear that v(ψ) = 1 for all ψ ∈ D(A) and v(Z A ) = 1B , i.e., D(A) |=B Z A . 2 A sequence {Li }i<ω of logics from Jhn is said to be strongly independent if Li ⊆ Σj =i Lj for each i < ω, where Σj =i Lj is the least logic containing all Lj , j = i. The following two facts are natural generalizations of Proposition 1.2 and Lemma 1.4 of [106] to the class of extensions of minimal logic. Proposition 6.4.3 Let {Li }i<ω be a strongly independent sequence of logics from Jhn. For any subsets I and J of ω, Σi∈I Li = Σi∈J Li if and only if I = J. Proof. Prove the non-trivial implication. Assume that I = J and there is k ∈ I \ J. Obviously, Σi∈J Li ⊆ Σi =k Li . Since {Li }i<ω is strongly independent, we have Lk ⊆ Σi =k Li , moreover, Lk ⊆ Σi∈J Li . Therefore, Σi∈I Li = Σi∈J Li . 2

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6 Negatively Equivalent Logics

Proposition 6.4.4 Let {Ai }i<ω be a sequence of subdirectly irreducible j-algebras satisfying the following conditions: (1) each Ai is ﬁnite; (2) for every i, j < ω, i = j implies that Ai cannot be embedded into any quotient algebra of Aj . If a logic L is contained in every LAi (i < ω), the sequence {Li }i<ω of logics deﬁned by Li = L + {J(Ai )} (i < ω) is strongly independent. Proof. It suﬃces to show that J(Ai ) ∈ Σj =i Li for all i. According to Lemma 6.4.1 we have J(Ai ) ∈ LAj for j = i. Consequently, L ∪ {J(Aj ) | j = i} ⊆ LAi , and we conclude Σj =i Lj ⊆ LAi . In view of J(Ai ) ∈ LAi we have J(Ai ) ∈ Σj =i Li . 2 We now turn to studying the partial ordering Jhn+ / ≡neg , neg . First we prove Proposition 6.4.5 Let L ∈ Neg. For any L1 , L2 ∈ Spec(Lk, L), we have the equivalence L1 ≡neg L2 ⇐⇒ L1 = L2 . Proof. We need only prove that diﬀerent logics from the class Spec(Lk, L) are not negatively equivalent. Let logics L1 , L2 ∈ Spec(Lk, L) be such that there is a formula ϕ ∈ L1 \ L2 . Since ϕ ∈ L2 , according to Proposition 2.2.6 it must be refuted at some subdirectly irreducible model of L2 . Let A be a subdirectly irreducible j-algebra such that A |= L2 and A |= ϕ. First of all we note that A is not negative. If A is negative, then A is a model of (L2 )neg = L2 + {⊥} = L. The logics L1 and L2 have a common negative counterpart L and L1 ⊆ L. In this way, we conclude that A |= L1 , which conﬂicts with our assumption that A refutes ϕ. We now consider the upper algebra A⊥ , which is non-trivial due to the last remark. The opremum of A equals to the opremum of A⊥ . Indeed, ⊥A = 1A , and so ⊥A ≤ A . Therefore, Heyting algebra A⊥ is a subdirectly irreducible model of (L1 )int = Lk, whence, it is a Boolean algebra. It is well known that only two-element Boolean algebra is subdirectly irreducible, and so A = ⊥A . We have thus proved the following Lemma 6.4.6 Let L ∈ Neg and L1 ∈ Spec(Lk, L). Every subdirectly irreducible model A of L1 is either negative, in which case A |= L, or has the form A ∼ = B ⊕ 2, where B is a negative algebra modelling L and 2 is a two-element Boolean algebra.

6.4 The Structure of Jhn+ up to Negative Equivalence

95

Assume additionally that A is ﬁnite. Since A is not a model of L1 , it is not embeddable into a quotient algebra of any j-algebra B modelling L1 . According to Lemma 6.4.1, this means that for any model B of L1 , B |= J(A), and so J(A) ∈ L1 . It is clear that J(A) ∈ L2 since J(A) ∈ LA. As noted above, A = ⊥A , whence, J(A) = ( D(A)) → ⊥ = ¬( D(A)). This formula proves that logics L1 and L2 are not negatively equivalent. It remains to consider the case when a j-algebra A refuting ϕ ∈ L1 is inﬁnite. We use Lemma 6.4.2 to conclude that for any j-algebra B with B |= L1 , we have D(A) |=B ⊥, i.e., that D(A) |=L1 ⊥. We may consider j-algebras as models of the ﬁrst-order predicate calculus of the language ∨, ∧, →, ⊥, 1 with identity “=” and classical negation “∼”. Propositional variables will be treated as individual ones, propositional formulas as terms, and valuations as denotations of individual variables. To avoid confusion we will denote the ﬁrst order satisﬁability relation by |=1 . In this case, for any j-algebra B, B-valuation v, and formula ϕ, we have the following equivalences: v(ϕ) = 1 ⇔ B |=1 ϕ = 1[v], B |= ϕ ⇔ B |=1 (ϕ = 1) , where (ψ) denotes the universal closure of a ﬁrst-order formula ψ. The relation D(A) |=L1 ⊥ is equivalent to the statement that there is no j-algebra B and B-valuation v such that B |= L1 , ⊥B = 1B , and v(ϕ) = 1 for all ϕ ∈ D(A). In view of the above equivalences, the latter means exactly that the set of formulas Σ := {(ϕ = 1) | ϕ ∈ L1 } ∪ {ϕ = 1 | ϕ ∈ D(A)} ∪ {∼ (⊥ = 1)} is not satisﬁable. Due to the Malcev local theorem for ﬁrst-order logic, there exists a ﬁnite non-satisﬁable subset X ⊆ Σ. Further, we argue to state that ∼(⊥ = 1) ∈ X. Consider the quotient algebra A/ ⊥ . The set of formulas {ϕ = 1 | ϕ ∈ D(A)} is satisﬁable in A, it will be also satisﬁable in the quotient algebra A/ ⊥ . The algebra A/ ⊥ is negative, and so it is a model for L = (L2 )neg . Since L1 ⊆ L, the sentences from the set {(ϕ = 1) | ϕ ∈ L1 } will be true in A/ ⊥ . We have thus proved that the set of formulas Σ \ {∼(⊥ = 1)} is satisﬁable, from which we conclude that ∼(⊥ = 1) ∈ X.

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6 Negatively Equivalent Logics

Denote by X the following subset of D(A), X := {ϕ | ϕ ∈ D(A), ϕ = 1 ∈ X} Coming back to identities of j-algebras, we can see that X |=B ⊥ for any B with B |= L1 , i.e., X |= L1 ⊥. We claim that the formula ( X ) → ⊥ = ¬( X ), where X is the conjunction of all formulas from X , is true in all subdirectly irreducible models B of L1 , and so belongs to L1 . Take an arbitrary subdirectly irreducible model B of L1 . If Bis negative, then any negated formula is true on B, in particular, B |= ( X ) → ⊥. Assume that B is not negative and v is a B-valuation. Relation X |=L1 ⊥ means that the set of formulas X is not satisﬁable in any non-negative model of L1 , whence v( X ) = 1. Taking into account that B is subdirectly irreducible and ⊥B = B by Lemma 6.4.6 we obtain v( X ) ≤ ⊥B , consequently, v(( X ) → ⊥) = 1. We have thus proved that ¬( X ) ∈ L1 . Recall that X is a subset of D(A), the diagram of the non-negative model in A and ¬( X ) does not belong to L2 . A for L2 , therefore, it is satisﬁable In this way, the formula ¬( X ) proves that L1 and L2 are not negatively equivalent. 2 Proposition 6.4.7 For any logics L ∈ Jhn \ Neg and L1 ∈ Neg such that L1 ⊆ Lneg , there exists a logic L ∈ Spec(Lk, L1 ) such that L ≡neg L. Proof. Consider the logic ∇(L). By Proposition 6.3.2 ∇(L) ≡neg L. Due to Proposition 6.3.4 (∇(L))int = Lk. At the same time, by Proposition 6.3.1 L ⊆ ∇(L) and consequently, Lneg ⊆ ∇(L)neg . Combining the last two facts yields ∇(L) ∈ Spec(Lk, L2 ) for some L2 ⊇ Lneg ⊇ L1 , in particular, Lk ∗ L2 ⊆ ∇(L). Deﬁne a logic L by setting L := Lk ∗ L1 + {¬ϕ | ¬ϕ ∈ L}(= Lk ∗ L1 + {¬ϕ | ¬ϕ ∈ ∇(L)}). It is clear that Lk ∗ L1 ⊆ ∇(L), whence, L ⊆ ∇(L), and so L ≤neg L. Further, let ϕ ∈ L. Then also ¬¬ϕ ∈ L, and ¬¬ϕ ∈ L by deﬁnition. We now apply Proposition 6.1.2 to conclude L ≤neg L . We have thus proved that L and L are negatively equivalent and it remains to check that L belongs to the desired interval Spec(Lk, L1 ).

6.4 The Structure of Jhn+ up to Negative Equivalence

97

Obviously, L1 ⊆ Lneg . If this inclusion is proper, there is a negative algebra A, which is a model of L1 but not of Lneg . It is clear that A models Lk ∗ L1 . All negated formulas are true in A, therefore, A |= L . Moreover, A is negative, whence, A |= Lneg , a contradiction. 2 + Now the ordering Jhn / ≡neg , neg can be described as follows. Proposition 6.4.8

1. The following isomorphism takes place

Jhn+ / ≡neg , neg ∼ = Spec(Lk, Ln) ∪ {F}, ⊆ . 2. For L1 ∈ Int and L2 ∈ Neg, we have

Spec(L1 , L2 )/ ≡neg , neg ∼ = Spec(Lk, L2 ), ⊆ . Proof. 1. Applying Proposition 6.4.7 for the case L1 = Ln, we conclude that for any L ∈ Jhn \ Neg, there is a logic in Spec(Lk, Ln) negatively equivalent to L. Such logic is unique due to Proposition 6.4.5 and we denote it by h(L). For every L ∈ Neg ∪ {F}, put h(L) = F. We have thus deﬁned a mapping h : Jhn+ → Spec(Lk, Ln) ∪ {F}. Obviously, h(L) = L for all L ∈ Spec(Lk, Ln), therefore, h is a mapping onto. Moreover, it satisﬁes the property that L ≡neg h(L) for every L ∈ Jhn+ . Take arbitrary L1 , L2 ∈ Jhn \ Neg such that L1 ≤neg L2 . In view of the envisaged property of h we have h(L1 ) ≤neg h(L2 ). If ϕ ∈ h(L1 ), then ¬¬ϕ ∈ h(L1 ) and by Proposition 6.1.2 ¬¬ϕ ∈ h(L2 ). Due to the same proposition we have h(L1 ) ≤neg h(L1 ) ∩ h(L2 ). The inverse relation h(L1 ) ∩ h(L2 ) ≤neg h(L1 ) follows from the set-theoretical inclusion. Applying Proposition 6.4.5 to h(L1 ) ≡neg h(L1 ) ∩ h(L2 ) we infer h(L1 ) ⊆ h(L2 ). Note also that for any L1 ∈ Jhn and L2 ∈ Neg ∪ {F}, L1 ≤neg L2 and h(L1 ) ⊆ h(L2 ) = F. We have thus proved that h is an epimorphism of Jhn+ , ≤neg onto

Spec(Lk, Ln) ∪ {F}, ⊆ . Passing to the quotient mapping h/ ≡neg of this epimorphism with respect to the negative equivalence we obtain the desired isomorphism of Jhn+ / ≡neg , neg and Spec(Lk, Ln) ∪ {F}, ⊆ . 2. Arguing as in the previous item we construct a mapping h : Spec(L1 , L2 ) → Spec(Lk, L2 ) such that L ≡neg h(L) for any L ∈ Spec(L1 , L2 ). We need only check that this is a mapping onto. Take an arbitrary L ∈ Spec(Lk, L2 ) and put

98

6 Negatively Equivalent Logics r r

1

r

⊥

@

r

B1 ⊕ 2

1 r⊥

r @

@r

@r

B2 ⊕ 2

r

1 r⊥

@ r @r r @ @ r @r @ @ @r

...

B3 ⊕ 2 Figure 6.1 L := L1 ∗ L2 + {¬ϕ | ¬ϕ ∈ L}. As in Proposition 6.4.7 we prove that L ≡neg L. Now we conclude the proof as in the previous item. 2 In conclusion, we make some remarks on the cardinality of intervals Spec(L1 , L2 ). It turns out that any interval of this form is inﬁnite, moreover, inﬁnite up to negative equivalence, and in many cases such an interval has the power of the continuum. We start with the simplest interval Spec(Lk, Lmn). Proposition 6.4.9 The interval Spec(Lk, Lmn) has the following structure: Lk ∗ Lmn ⊆ . . . ⊆ Ln ⊆ . . . ⊆ L1 ⊆ L0 , where L0 = Le = Lk ∩ Lmn; for n > 0, Ln = L(Bn ⊕ 2) = Lk ∗ Lmn + {J(Bn+1 ⊕ 2)}, where Bn is a negative Peirce algebra with n atoms (see Figure 6.1); ﬁnally, Lk ∗ Lmn = L(B ⊕ 2), where B is an arbitrary inﬁnite negative Peirce algebra. Proof. All logics from the interval Spec(Lk, Lmn) have the same negative models, namely, the models of Lmn. Therefore, every logic L in Spec(Lk, Lmn) is determined by the class of its non-negative ﬁnitely generated subdirectly irreducible models and we denote this class by M od+ f si (L). Due to Lemmas 6.4.6 and 6.4.10 every such model model has the form B ⊕ 2, where B is a ﬁnitely generated model of Lmn, i.e., a ﬁnitely generated negative Peirce algebra. Lemma 6.4.10 The algebra B ⊕ 2 is ﬁnitely generated if and only if B is ﬁnitely generated. Proof. Let B := B ⊕ 2 and := B⊕2 . Note that |B | = |B| ∪ {1B } and = 1B = ⊥B .

6.4 The Structure of Jhn+ up to Negative Equivalence

99

If X generates B , then X \ { } generates B. Indeed, if b ∈ B is a value of some term deﬁned on generators, then replacing by c → c, where c ∈ X \ { }, does not change the value of this term. If X generates B, then X ∪ { } generates B . It is a consequence of the following obvious relations. For any a, b ∈ B, we have a ∧B b = a ∧B b, a ∨B b = a ∨B b, if a →B b = 1B , then a →B b = a →B b. 2 The only diﬀerence between Boolean and negative Peirce algebras is in the interpretation of ⊥. In Boolean algebras, ⊥ is interpreted as the least element, whereas in negative Peirce algebras, it is interpreted as the greatest element of an algebra. In this way, all ﬁnitely generated negative Peirce algebras are ﬁnite. And we have a countable chain of diﬀerent (up to isomorphism) ﬁnitely generated negative Peirce algebras {Bn | n ∈ ω}, where Bn is a negative Peirce algebra with n-atoms, i.e., with n minimal elements in the set Bn \ {f } for f denoting the least element of Bn . Clearly, Bn is isomorphically embedded into Bm if and only if n ≤ m. Therefore, the class M od+ f si (L) for L ∈ Spec(Lk, Lmn) is of the form Mα = {Bn ⊕ 2 | n < α}, where 1 ≤ α ≤ ω. It is not hard to check that each of these sets can be realized as a set of non-negative ﬁnitely generated subdirectly irreducible models of a suitable logic Lα from the interval Spec(Lk, Lmn). Indeed, the class M1 = {B0 ⊕ 2} = {2} corresponds to the logic Le = Lk ∩ Lmn, which has a unique non-negative subdirectly irreducible model, 2. For 1 < α < ω, consider a logic Lα := Lk ∗ Lmn + {J(Bα ⊕ 2)}. According to Lemma 6.4.1 the class M od+ f si (Lα ) coincides with the class of algebras of the form Bn ⊕ 2 such that the algebra Bα ⊕ 2 is not embeddable into a quotient algebra of Bn ⊕ 2. Every proper quotient algebra of Bn ⊕ 2 is negative and non-negative algebra can not be embedded into it. The algebra Bm ⊕ 2 is embeddable into Bn ⊕ 2 if and only if Bm is embeddable into Bn . Combining all these facts we obtain M od+ f si (Lα ) = Mα . From the last equality and the fact that Lmn = L(Bn ) for any n we have Lα = L(Bα ⊕ 2). The class Spec(Lk, Lmn) has the least element Lω = Lk ∗ Lmn and, obviously, M od+ f si (Lk ∗ Lmn) = Mω . If we take an arbitrary inﬁnite negative Peirce algebra B, any algebra of the form Bn ⊕ 2 will be embeddable into B ⊕ 2. Therefore, Lω = L(B ⊕ 2).

100

6 Negatively Equivalent Logics

Classes of models Mα , 1 ≤ α ≤ ω, form an ascending chain of type ω + 1 with respect to inclusion, which means that the corresponding logics Lα , 1 ≤ α ≤ ω, form a descending chain of type (ω + 1)∗ . 2 Further, we obtain a suﬃcient condition guaranteeing that Spec(L1 , L2 ) is of the power of the continuum. Proposition 6.4.11 Let L1 ∈ Int and L2 ∈ Neg. Assume that there exists a family {Bi | i < ω} of ﬁnite negative algebras such that Bi |= L2 for all i < ω, and Bi is not embeddable into Bj for i = j. Then |Spec(L1 , L2 )| = 2ω . Proof. In view of Proposition 6.4.8, it is enough to consider the case L1 = Lk. Consider the sequence {Ai }i<ω of j-algebras deﬁned by Ai = Bi ⊕ 2, i < ω. It is a sequence of subdirectly irreducible j-algebras modelling Lk ∗ L2 . By assumption if i = j, Bi is not embeddable into Bj , in which case also Ai is not embeddable into Aj . Further, note that every proper quotient algebra of Aj is negative. This means that the non-negative algebra Ai cannot be embedded into a quotient algebra of Aj for i = j. Deﬁne a sequence of logics {Li }i<ω by Li = Lk ∗ L2 + {J(Ai )}. By Proposition 6.4.4 this sequence is strongly independent and we obtain a continuum of diﬀerent logics of the form Σi∈I Li , I ⊆ ω. Each of these logics belongs to the interval Spec(Lk, L2 ), because all J(Ai ) are negative formulas and belong to L2 . 2 Corollary 6.4.12 |Jhn/ ≡neg | = |Spec(Lk, Ln)| = 2ω . Proof. It is not hard to check that the family {Bi | i ∈ ω} (see Figure 6.2) of negative algebras satisﬁes the condition of the previous proposition. 2 r⊥ r @r r @r @r @r @r @r

B0

r⊥ r @r @r @r r @r @r @r @r @r

B1

r⊥ r @r @r @r @r @r r @r @r @r @r @r

Figure 6.2

B2

...

Chapter 7

Absurdity as Unary Operator1 7.1

Introduction

This chapter ﬁnishes the ﬁrst part of the book devoted to the concept of negation as reduction to absurdity. As was mentioned in Chapter 1, minimal logic lies on the border line of paraconsistency. We have in Lj for arbitrary formulas ϕ and ψ, {ϕ, ¬ϕ} ¬ψ. This means that although inconsistent Lj-theories may be non-trivial, they are trivial with respect to negation. Any negated formula is provable in any inconsistent Lj-theory. Due to this “paraconsistent paradox” of minimal logic it looks natural to ﬁnish the investigation of the class of Lj-extensions with an attempt to overcome this paradox. We try to do it by merging the class of Ljextensions into a more general class of paraconsistent logics and pointing out some special property distinguishing extensions of minimal logic in the latter class. We also suggest that the negation in logics from the abovementioned class should preserve the most essential property of intuitionistic negation, namely, that the negation must be deﬁned as reduction to absurdity. In intuitionistic logic, the negation is characterized by three important features: 1) we assert ¬ϕ if supposing ϕ leads to absurdity; 2) absurdity may be explicated as a propositional constant ⊥; 3) absurdity implies everything, 1

Parts of this chapter were originally published in [73] (Nicholas Copernicus University Press, Poland) and in [81] (Rodopi, Netherlands). Reprinted here by permission of the publishers.

101

102

7 Absurdity as Unary Operator

⊥ → p, or ¬p → (p → q). Note that Item 3 implies, in fact, Item 2. If any contradiction implies everything, all contradictions are equivalent, and so we may use a propositional constant do denote an arbitrary contradictory or absurd statement. In minimal logic Lj, we omit Item 3, which allows one to distinguish non-equivalent contradictions. In an extension L of Lj, the contradiction ϕ ∧ ¬ϕ is equivalent to ⊥ if and only if ϕ is provable in Lneg = L + {⊥}, the negative counterpart of L (see Chapter 4), moreover, it was stated that Lneg may be considered as the logic of contradictions of L. At the same time, Item 2 is preserved in Lj, which results, in particular, in the above-mentioned “paraconsistent paradox”. It looks natural to make the next step and to resign not only Item 3, but also its consequence, Item 2. This gives rise to a question: how should the absurdity be explicated in a logical system in this case? We suggest an answer, which is based on the interrelations between logic of classical refutability Le and L ukasiewicz’s modal logic L . In Chapter 3 it was proved that the simplest matrix for Le has four elements, and so Le can be considered as a four-valued logic as well as L ukasiewicz’s modal logic. In Section 3.2, it was proved that in Le, one can deﬁne one isomorph of classical logic and two diﬀerent isomorphs of maximal negative logic. These isomorphs, on the one hand, correspond to translations of Lk and Lmn into Le playing the key role in studying the structure of the class Jhn (see Section 4.1). On the other hand, the isomorphs were induced by mappings, which can be identiﬁed in a natural way with the modalities of L , which easily implies the fact that Le is deﬁnition equivalent to the positive fragment of L and that Le extended by the classical negation is deﬁnition equivalent to L . Similar interrelations between Le and L were stated previously by J. Porte [84, 85]. For our goals the most essential is the fact that the necessity operator of L may be identiﬁed with the contradiction operator C(ϕ) := ϕ ∧ ¬ϕ of logic of classical refutability. This leads to the idea of considering a contradiction operator C as a primary logical connective and deﬁning a negation as a reduction to this operator, ¬ϕ := ϕ → C(ϕ). In this way, we merge the class Jhn into the class of so-called C-logics and show that logics from Jhn will be distinguished by some property of C, which becomes paradoxical if C is considered as a modal operator. So, more natural modal properties of the contradiction operator result in a more satisﬁable consequence relation from the point of view of paraconsistent logic. Further, it turns out that the same negation may be deﬁned via diﬀerent operators. There is a sense to distinguish, for example, between the contradiction operator C that should satisfy the axiom C(ϕ) → ϕ and the more general absurdity operator, for which a similar axiom does not obey.

7.1 Introduction

103

At the end of this chapter, we show that in such well-known paraconsistent logics as the logic CLuN by D. Batens [6, 7] and Sette’s maximal paraconsistent logic P 1 [88, 102] the negation can be deﬁned via a suitable contradiction or absurdity operator. It is interesting that the negation of CLuN may be deﬁned via an absurdity operator about which nothing is postulated. In this way, CLuN may be considered as a counterpart of Le, which has the same positive fragment and where the negation is deﬁned via the constant ⊥, about which nothing is postulated. Let L1 and L2 be propositional languages, L1 and L2 logics in languages L1 and L2 respectively, and let θ : L1 → L2 and ρ : L2 → L1 be translations from the language L1 into L2 and vice versa, i.e., mappings from the set of L1 -formulas to the set L2 -formulas and vice versa. We say that L1 is faithfully embedded into L2 via θ if for any L1 -formula ϕ, we have the equivalence ϕ ∈ L1 ⇐⇒ θ(ϕ) ∈ L2 . Further, the logic L1 is said to be deﬁnition equivalent to L2 via translations θ and ρ if L1 is faithfully embedded into L2 via θ, L2 is faithfully embedded into L1 via ρ and moreover, for any L1 -formula ϕ and L2 -formula ψ we have L1 ϕ ↔ ρθ(ϕ) and L2 ψ ↔ θρ(ψ). A translation α from L1 into L2 preserves propositional variables if α(p) = p for any propositional variable p and it preserves an n-ary connective ∗ if ∗ ∈ L1 ∩ L2 and α(∗(ϕ1 , . . . , ϕn )) = ∗(αϕ1 , . . . αϕn ) for all L1 -formulas ϕ1 , . . . , ϕn . Remark. If L1 is faithfully embedded into L2 via θ, θ preserves the implication connective and logics L1 and L2 satisfy the Deduction Theorem, then it is not hard to prove that for any set of L1 -formulas X and L1 -formula ϕ, the equivalence holds: X L1 ϕ ⇐⇒ θ(X) L2 θ(ϕ), where θ(X) = {θ(ψ) | ψ ∈ X}. In other words, θ is a strong translation. Taking into account this remark and the above deﬁnitions, Proposition 2.1.1 can be re-worded as follows. Proposition 7.1.1 Logics Lj¬ and Lj⊥ are deﬁnition equivalent via translations θ and ρ. 2

104

7.2

7 Absurdity as Unary Operator

Le and L ukasiewicz’s Modal Logic

Recall that logic of classical refutability Le is characterized by the class of Peirce-Johansson (or pj-)algebras. A pj-algebra is an implicative lattice satisfying the Peirce law and with ⊥ interpreted as an arbitrary element of its universe. This characterization easily implies that any characteristic model of Le must contain at least four elements: unit element 1, 0(= ⊥ = ¬1), some element −1 under 0, because Le is paraconsistent and ﬁnally, element a = 0 → −1 incomparable with 0. It turns out that the four-element lattice 4 with universe {1, 0, a, −1} and ⊥ interpreted as 0 is really a characteristic model for Le (see Corollary 3.1.6). L ukasiewicz’s modal logic L was introduced in [52] (see also [53]). The intention of J. L ukasiewicz was to construct a system of modal logic, which is an extension of classical propositional logic with two interdeﬁnable modal operators of “necessity” L and of “possibility” M , Lp ↔ ¬M ¬p and M p ↔ ¬L¬p. These operators, in turn, should satisfy the conditions Lp → p, p → M p, and p → Lp, M p → p. He used the matrix approach to deﬁne his system. Since the ordinary twoelement matrix 2 for classical logic does not allow one to deﬁne modal operators satisfying the above-mentioned conditions, L ukasiewicz took the four-element matrix 2 × 2, which also deﬁnes classical logic. Note that if we identify the elements of 2 × 2 and the elements of Lemodel 4 in the following way (1, 1) → 1, (1, 0) → a, (0, 1) → 0, and (0, 0) → −1, the binary operations of classical logic and of Le will coincide on 4 , only negations will act diﬀerently on the set of truth values {1, a, 0, −1}. q1 @ 6 @

@ @q0

q a @

@ @ @? q

−1

∼

q1 @ I 6 @ @ -q0 R @

aq

@

@ @

@q

¬

−1

It these diagrams the classical negation ∼ is on the left and the negation ¬ of Le on the right.

7.2 Le and L ukasiewicz’s Modal Logic

105

Modalities L and M and dual modalities W and V are deﬁned on 2 × 2 via the following truth-tables:

x 1 a 0 −1

Lx 0 −1 0 −1

Mx 1 a 1 a

Wx a a −1 −1

Vx 1 1 0 0

It is easy to check that the modalities L and M are interdeﬁnable via classical negation and satisfy L ukasiewicz’s conditions. We deﬁne L ukasiewicz’s modal logic L in the language L+ + {L, ∼} as the set of tautologies of the following algebra := {−1, a, 0, 1}, ∨, ∧, →, ∼, L, 1 . 4L Denote by L + its positive fragment in the language L+ + {L}. Note that the second modality M is deﬁnable through L in positive fragment L + as follows M p ↔ L(p → p) → p. The pairs of modalities L, M and W , V are interdeﬁnable, e.g., as follows W p ↔ p ∧ M (∼(p → p)), Lp ↔ p ∧ V (∼(p → p)). If we use modalities W and V instead of L and M , respectively, we obtain the logic with the same set of tautologies. Taking into account the above identiﬁcation of truth-values of logics L and Le we can easily see that the actions of modalities L, M , and V on the set of truth-values coincide, respectively with the mappings τ , δ, and ε, which are deﬁnable in the logic of classical refutability and which were used in Section 3.2 to construct isomorphs of classical (ε) and maximal negative logic (τ , δ) in Le. On the other hand, the modality W cannot be deﬁned in Le, because algebra 4 has a subalgebra with the universe {1, 0}, i.e., the set {1, 0} is closed under all operators deﬁnable in Le. Thus, we can see that there is a close interrelation between L and Le. J. Porte [84, 85] was the ﬁrst to pay attention to the above-mentioned interrelation but he had L ukasiewicz’s modal logic as a starting point for his considerations. More exactly, in [84], it was proved that the modalities of

106

7 Absurdity as Unary Operator

L can be deﬁned in a system obtained from classical logic by the addition of a constant, Ω in Porte’s denotation, about which nothing is postulated. And conversely, that the modalities of L can be used to deﬁne the constant Ω. In [85], J. Porte observed that his constant Ω is close to the negation of logic of classical refutability and stated that Le is deﬁnition equivalent to the positive part of L . Extend the language L⊥ of Le via a new negation symbol ∼, L⊥,∼ := L⊥ ∪ {∼}, and deﬁne the logic Le∼ as a logic in the language L⊥,∼ with axiom schemes of Le and, additionally, axiom schemes of classical negation for ∼: 1∼ . (p → q) → ((p →∼ q) →∼ p) 2∼ . ∼ p → (p → q) 3∼ . p∨ ∼ p Deﬁne a translation θ : L⊥,∼ → L+ +{L, ∼} from the language of Le∼ to the language of L ukasiewicz’s modal logic and an inverse translation ρ from the language of L to the language of Le∼ in such a way that they preserve propositional variables and all classical connectives and moreover, θ(⊥) = L(p → p), ρ(Lϕ) = ρϕ ∧ ¬ρϕ, where p is a propositional variable and ϕ is an arbitrary formula in the language L+ + {L, ∼}. Deﬁning an algebra 4∼ as an expansion of 4 via the classical negation, i.e., 4∼ := {−1, a, 0, 1}, ∨, ∧, →, ⊥, ∼, 1 , we state the following facts. Proposition 7.2.1 4∼ is a characteristic model for Le∼ . Proof. Obviously, Le∼ has only two subdirectly irreducible models, 2∼ and 2 ∼ , which are expansions of Boolean algebra 2 and negative algebra 2 via an additional Boolean negation ∼. Therefore, Le∼ = L2∼ ∩ L2 ∼ . Taking ∼ into account 4∼ ∼ = 2∼ × 2 we immediately arrive at L4∼ = Le∼ . 2 Proposition 7.2.2 Le∼ is a conservative extension of Le.

7.2 Le and L ukasiewicz’s Modal Logic

107

Proof. If ϕ is a formula in the language L⊥ and Le∼ ϕ, then 4∼ |= ϕ, and also 4 |= ϕ, because the connective ∼ does not occur in ϕ. Since L4 = Le, we have ϕ ∈ Le, which completes the proof. 2 We now give in a slightly modiﬁed version of Porte’s results [84, 85]. are deﬁnition equivalent via θ Proposition 7.2.3 The logics Le∼ and L and ρ, the logic Le is deﬁnition equivalent to the positive fragment of L via the same translations. The proof is by direct veriﬁcation, because all logics involved in this statement are four-valued. 2 The only diﬀerence in the above translations from Porte’s translations is the item ρ(Lϕ) = ρϕ ∧ ¬ρϕ. The original version was ρ(Lϕ) = ρϕ ∧ ⊥, but, obviously, Lj (ϕ ∧ ¬ϕ) ↔ (ϕ ∧ ⊥). Proposition 7.2.3 means, ﬁrst of all, that there is a close connection between L ukasiewicz’s modalities and the paraconsistent negation of logic of classical refutability. From this fact Porte [85, pp. 87–88.] inferred a rather categorical conclusion that “the modalities of the L -system are very far from what everybody calls “possibility” and “necessity” and/or that the weak negation of CR (the logic of classical refutability) is very far from what everybody calls “negation”.” As noted above, L ukasiewicz constructed his system so as to satisfy the minimal list of requirements for modal operators, which gives rise to a long history of critics of L ukasiewicz’s modal logic L , but special discussion of this logic lies outside the scope of the present research. The resent work by Font and Hajek [32] may be recommended to become acquainted with the topic. Regarding the critics of the negation in the logic of classical refutability based on its similarity to modal operators, in recent years such similarity was not considered as something “negative” and was intensively studied. For example, K. Dosen in a series of works [23, 24, 25] treated the negation namely as a modal operator. In fact, the deﬁnition equivalence of L ukasiewicz’s modal logic and logic of classical refutability means that the list of properties of modal operators given in the beginning of this section is not broad enough to distinguish the necessity operator and the operator of contradiction. The interrelation between Le and L stated in Proposition 7.2.3 will be used in the next section to suggest the way to generalize the notion of negation, which allows one to overcome the paradox of minimal logic.

108

7.3

7 Absurdity as Unary Operator

Paradox of Minimal Logic and Generalized Absurdity

Recall that by the paradox of minimal logic we mean the following property of Lj: from any contradiction one can infer in Lj an arbitrary negative formula, i.e., for any ϕ and ψ, we have ϕ, ¬ϕ Lj ¬ψ. Due to this property, the negation makes no sense in inconsistent Lj-theories, because all negative formulas are provable in them. This property is conditioned, on the one hand, by the axiom for implication p → (q → p) (which is sometimes called “the positive implication paradox”) and, on the other hand, by the unrestricted law reductio ad absurdum, (p → q) → ((p → ¬q) → ¬p), saying that if a formula implies a contradiction, one can negate this formula without any restriction on the nature of this contradiction. Indeed, let T be some inconsistent Lj-theory and ϕ, ¬ϕ ∈ T . Then for an arbitrary formula ψ, we can infer in T the implications ψ → ϕ and ψ → ¬ϕ using the positive paradox, and we then infer ¬ψ applying reductio ad absurdum. Of course, one can try to overcome the above mentioned paradox via rejecting the positive implication paradox and passing in this way into the ﬁeld of relevant logic. But we choose another way, leave intuitionistic implication unchanged and consider possible ways of restricting reductio ad absurdum. It is worth noting that this idea has been exploited many times in investigations in the ﬁeld of paraconsistency. For example, such well-known paraconsistent logics as Sette’s P 1 [88, 102] and Da Costa’s C1 [19] have reductio ad absurdum restricted to complex formulas, in case of P 1 , and to formulas “behaving consistently”, in case of C1 . Our approach is based on the correspondence between the necessity operator L of L ukasiewicz’s modal logic and the contradiction operator C, C(ϕ) := ϕ ∧ ¬ϕ, in the logic of classical refutability, which was stated in the previous section. In Le as well as in an arbitrary extension of minimal logic, the negation can be deﬁned via the constant “absurdity”, ¬ϕ := ϕ →⊥, but it can also be deﬁned via the contradiction operator C(ϕ), ¬ϕ := ϕ → C(ϕ). It is clear that Lj ¬ϕ ↔ (ϕ → C(ϕ)).

7.3 Paradox of Minimal Logic and Generalized Absurdity

109

This leads to the idea of deﬁning a negation via the contradiction operator C considered as a primary logical connective, ¬ϕ := ϕ → C(ϕ). It will be shown below that the unrestricted reductio ad absurdum for negation deﬁned as above exactly corresponds to some paradoxical properties of C(ϕ) considered as a modal operator. The results of Section 7.2 allow us to identify Le with a subsystem of L and after such identiﬁcation, we have L C(ϕ) ↔ L(ϕ) and L ¬ϕ ↔ (ϕ → L(ϕ)). Notice that modal operators L and M have the following properties: L L(ϕ ∧ ψ) ↔ L(ϕ) ∧ ψ and L M (ϕ ∨ ψ) ↔ M (ϕ) ∨ ψ. Each of these properties can be inferred from the other modulo classical logic and the relation deﬁning M through L, and both of these properties have a paradoxical nature. Indeed, if we accept the conjunction of two conditions as necessary, it means from the intuitive point of view more than stating that one of these conditions is necessary and the second just takes place. In a similar way, assuming that it is possible that one of the two conditions takes place should be weaker than the alternative of one of the conditions and the possibility of the other. It is interesting that the numerous authors who criticized L ukasiewicz’s modalities did not pay any attention to these paradoxes. For any formula ϕ, we have ϕ ↔ ϕ ∧ (p → p) and ϕ ↔ ϕ∨ ∼ (p → p), from which we infer, using the above paradoxical properties, L L(ϕ) ↔ ϕ ∧ L(p → p) and L M (ϕ) ↔ ϕ ∨ M (∼(p → p)). The ﬁrst of these equivalences corresponds to the relation C(ϕ) ↔ ϕ ∧ ⊥ ↔ ϕ ∧ ¬(p → p) for the contradiction operator in Le. Thus, the paradoxical properties envisaged above allow us to deﬁne the negation via the constant “absurdity”. Now we consider the language LC = ∨, ∧, →, C , i.e., L+ extended by the contradiction operator C, and deﬁne a C-logic as a logic in this language containing axioms of Lp and the formula C(p) → p. We say that C is extensional in a C-logic L if L is closed under the rule ϕ↔ψ . C(ϕ) ↔ C(ψ)

110

7 Absurdity as Unary Operator

Lemma 7.3.1 Let L be a C-logic and C(p ∧ q) ↔ C(p) ∧ q, C(p ∧ q) ↔ p ∧ C(q) ∈ L, then C is extensional in L. Proof. Let L satisfy the conditions of the lemma. Assume ϕ ↔ ψ ∈ L. Using the ﬁrst equivalence and the axiom C(p) → p we have C(ϕ ∧ ψ) ↔ C(ϕ) ∧ ψ ↔ C(ϕ) ∧ ϕ ↔ C(ϕ). In a similar way, we infer from the equivalence C(p ∧ q) ↔ p ∧ C(q) that C(ϕ ∧ ψ) ↔ C(ψ), and, ﬁnally, C(ϕ) ↔ C(ψ). 2 Proposition 7.3.2 Let L be a C-logic and ¬ϕ := ϕ → C(ϕ). Then C(p) ↔ p ∧ ¬p ∈ L,

(7.1)

moreover, (p → q) → ((p → ¬q) → ¬p) ∈ L iﬀ C(p ∧ q) ↔ C(p) ∧ q, C(p ∧ q) ↔ p ∧ C(q) ∈ L. Proof. From the well-known property p∧(p → q) ↔ p∧q of the intuitionistic implication and the axiom C(p) → p, we immediately obtain the equivalence C(p) ↔ p ∧ ¬p, which justiﬁes the name “contradiction operator” for C. Assume that (p → q) → ((p → ¬q) → ¬p) ∈ L and show that in this case the negation may be deﬁned via a constant, ¬ϕ ↔ ϕ → ⊥, where ⊥ := C(p0 → p0 ) for some ﬁxed variable p0 . Substituting the deﬁnition of negation in reductio ad absurdum and exporting the second premise we obtain ((p → q) ∧ (p → (q → C(q)))) → (p → C(p)) Taking into account Lp p → (q ∧ r) ↔ (p → q) ∧ (p → q) we arrive at (p → (q ∧ (q → C(q))) → (p → C(p)). Due to (7.1), the latter is equivalent to (p → C(q)) → (p → C(p)).

(7.2)

7.3 Paradox of Minimal Logic and Generalized Absurdity

111

Substituting the tautology p0 → p0 for q, we obtain L (p → C(p0 → p0 )) → ¬p. To state the inverse implication, we substitute p0 → p0 for p and p for q in (7.2) ((p0 → p0 ) → C(p)) → ((p0 → p0 ) → C(p0 → p0 )), from which we have C(p) → C(p0 → p0 ). Taking into account that the intuitionistic implication is increasing in the second argument, we arrive at ¬p → (p → C(p0 → p0 )) and, ﬁnally, at ¬p ↔ p → ⊥. Further, from this fact and the equivalence C(p) ↔ p ∧ ¬p we infer as in Lj that L C(p) ↔ p ∧ ⊥. This equivalence implies in a trivial fashion the extensionality of C in L, moreover, C(p ∧ q) ↔ (p ∧ q) ∧ ⊥ ↔ (p ∧ ⊥) ∧ q ↔ C(p) ∧ q. The second desired equivalence easily follows from the one just proved by extensionality. We now assume that the equivalences C(p∧q) ↔ C(p)∧q and C(p∧q) ↔ p ∧ C(q) hold in L. By Lemma 7.3.1 the operator C will be extensional in L. Deﬁning ⊥ := C(p0 → p0 ) and using the extensionality of C we obtain C(p) ↔ C((p0 → p0 ) ∧ p) ↔ C(p0 → p0 ) ∧ p ↔ p ∧ ⊥ and, further, ¬p ↔ p → C(p) ↔ p → (p ∧ ⊥) ↔ p → ⊥. The equivalence ¬p ↔ p → ⊥ easily implies the unrestricted version of reductio ad absurdum. 2 As noted at the beginning of this section, the paradox of minimal logic is conditioned by the “positive implication paradox” and the unrestricted version of reductio ad absurdum. In the deﬁnition of C-logics we leave the intuitionistic implication unchanged, therefore, a C-logic meets the paradox of minimal logic if and only if the negation deﬁned via its operator C satisﬁes the unrestricted version of reductio ad absurdum. In this way, we infer from the last proposition

112

7 Absurdity as Unary Operator

Corollary 7.3.3 A C-logic L satisfy for all formulas ϕ and ψ the condition ϕ, ¬ϕ L ¬ψ iﬀ C(p ∧ q) ↔ C(p) ∧ q, C(p ∧ q) ↔ p ∧ C(q) ∈ L. 2 As we can see from this corollary, deleting the paradoxical property of C allows one to avoid the paradox of minimal logic. In other words, if the operator C of contradiction has more natural properties from the viewpoint of modal logic, then the negation deﬁned via C generates more a satisﬁable inference relation from the viewpoint of paraconsistent logic. It was noted above that in minimal logic the negation may be deﬁned in two equivalent ways: via the constant ⊥ and via the non-constant operator C(ϕ) of contradiction. This simple observation leads naturally to the distinction between the contradiction operator and what we will call the absurdity operator, which we denote A(ϕ). In case of Lj, the absurdity operator is constant (A(ϕ) ↔ ⊥). There are diﬀerent intuitions behind these operators. By C(ϕ) we mean a contradiction expressed in terms of ϕ, i.e., a simultaneous stating of ϕ and its negation ¬ϕ, from which follows that C(ϕ) should imply ϕ. By A(ϕ) we mean such a statement that reducing ϕ to it, i.e., proving the implication ϕ → A(ϕ), is enough to negate ϕ. This understanding of A(ϕ) does not assume that A(ϕ) implies ϕ. The formulas A(ϕ) and ϕ may be incomparable, as takes place in case of Lj, where the formulas ϕ and ⊥ are incomparable in the general case. In fact, the contradiction operator C can be considered as a special case of an absurdity operator satisfying an additional assumption that C(ϕ) → ϕ for any ϕ. Moreover, if a negation of some system can be deﬁned in terms of an absurdity operator, such a negation itself can be taken as an absurdity operator, which will produce the same negation. To present these considerations in a precise form, we deﬁne an A-language as the positive language L+ with an additional unary operator A, LA := L+ ∩ {A}, and an A-logic as a logic in the language LA containing axioms of positive logic Lp. Proposition 7.3.4 Let L be an A-logic. Deﬁne the contradiction operator C as C(ϕ) := ϕ ∧ A(ϕ) and the negation ¬ as ¬ϕ := ϕ → A(ϕ). The following formulas are provable in L:

7.4 A- and C -Presentations

113

¬p ↔ p → ¬p

and

¬p ↔ p → C(p);

C(p) ↔ p ∧ ¬p

and

C(p) → p;

C(p) → A(p)

and

A(p) → ¬p.

Proof. All the formulas listed in the proposition can easily be inferred from the given deﬁnitions and axioms of Lp. For example, the equivalence ¬p ↔ p → ¬p is an abbreviation for p → A(p) ↔ p → (p → A(p)) and the latter is just a particular case of the contraction law. 2 This proposition shows that C deﬁned in terms of the absurdity operator A really can be considered as a contradiction operator corresponding to the negation deﬁned via A and that there is the whole “interval” of operators, from C to ¬ deﬁning the same negation.

7.4 7.4.1

A- and C -Presentations Deﬁnitions and First Results

Now we want to deﬁne more exactly what a negation in one or another logic can be deﬁned via an absurdity operator means. We restrict ourselves to the class of ¬-logics, which are sets of formulas in the language L¬ containing axioms of positive logic and closed under the rules of substitution and modus ponens. Further, we need several translations between languages L¬ , LC , and LA . Translations θ : L¬ → LC , ρ : LC → L¬ , ξ : L¬ → LA , and ζ : LC → LA preserve propositional variables and all positive connectives and act on other connectives as follows: θ(¬ϕ) := θϕ → C(θϕ); ρ(C(ϕ)) := ρϕ ∧ ¬ρϕ; ξ(¬ϕ) := ξϕ → A(ξϕ); ζ(C(ϕ)) := ζϕ ∧ A(ζϕ). Let L be a ¬-logic. An A-logic L such that ξ deﬁnes a faithful embedding of L into L and is called an A-presentation of L. A C-presentation of L is a C-logic L such that θ faithfully embeds L into L . By an exact A-(C-)presentation of a ¬-logic L we mean its A-(C-) presentation, which is the least among all A-(C-)presentations of L.

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7 Absurdity as Unary Operator

A strong C-presentation of L is a C-logic L deﬁnition equivalent to L via θ and ρ. If L is a C-logic, then its A-presentation is an A-logic L such that ζ faithfully embeds L into L , and its exact A-presentation is the least A-presentation. If L is a C-logic, then its ¬-presentation is a ¬-logic L such that ρ faithfully embeds L into L , and its exact ¬-presentation is the least ¬-presentation. First we state some simple properties of the notions introduced. Proposition 7.4.1 1. Let L be an A-(C-)logic. There exists a unique ¬-logic L such that L is an A-(C-)presentation of L . 2. Let L be an A-logic. There exists a unique C-logic L such that L is an A-presentation of L . Proof. 1. It is enough to consider the case of A-logics. We deﬁne by induction the subset of ¬-formulas in the set of all LA -formulas: 1. a propositional variable is a ¬-formula; 2. if ϕ and ψ are ¬-formulas, then ϕ∨ψ, ϕ∧ψ, and ϕ → ψ are ¬-formulas; 3. if ϕ is a ¬-formula, then ϕ → A(ϕ) is a ¬-formula. For a ¬-formula ϕ, deﬁne a formula ϕ¬ as a result of replacing all subformulas of the form ψ → A(ψ) by subformulas ¬ψ. Note that ϕ = ξ(ϕ¬ ). Consider the intersection S of the logic L with the set of all ¬-formulas. It is clear that S is closed under the substitution of ¬-formulas and the restricted version of modus ponens: if ϕ, ϕ → ψ ∈ S and ψ does not have the form A(ϕ), then ψ ∈ S. Therefore, the set L := {ϕ¬ | ϕ ∈ S}

(7.3)

is a ¬-logic. Check that L is an A-presentation of L . If L ϕ, then ϕ = ψ ¬ for a suitable formula ψ ∈ S ⊆ L. Consequently, ξϕ = ξ(ψ ¬ ) = ψ ∈ L. Conversely, if L ξϕ, then ξϕ ∈ S and (ξϕ)¬ = ϕ, whence, L ϕ. The uniqueness of a ¬-logic L , for which L is an A-presentation, easily follows from the above considerations. Indeed, every such logic L must have the form (7.3). 2. Deﬁne a set of C-formulas by replacing Item 3 in the deﬁnition of ¬-formulas by

7.4 A- and C -Presentations

115

3 . if ϕ is a C-formula, then ϕ ∧ A(ϕ) is a C-formula. The proof of this item is similar to the previous one. The desired C-logic has the form L := {ϕC | ϕ ∈ L and ϕ is a C-formula}, where ϕC is obtained from ϕ by replacing every subformula ψ ∧ A(ψ) by C(ψ). 2 In view of this statement Proposition 7.3.2 can be re-worded as Corollary 7.4.2 Let L be a C-logic. Then C(p ∧ q) ↔ C(p) ∧ q, C(p ∧ q) ↔ p ∧ C(q) ∈ L iﬀ L is a C-presentation for some extension of Lj. 2 Proposition 7.4.3 1. If L is a ¬-logic and L has an A-(C-)presentation, then L has also an exact A-(C-)presentation. 2. If L is a C-logic having an A-presentation, it also has an exact A-presentation. Proof. This statement follows from the next simple lemma. Lemma 7.4.4 Let L0 and L1 be propositional languages and τ : L0 → L1 a translation. Further, let L be a logic in the language L0 and {Li | i ∈ I} a family of logics in the language L1 such thatτ faithfully embeds L into Li for all i ∈ I. Then τ faithfully embeds L into i∈I Li . In this way, if the set of A-(C-)presentations of some logic is non-empty, the intersection of all such presentations gives an exact A-(C-)presentation of that logic. 2 To prove further results we impose the restriction of extensionality on the considered logics. Deﬁne the extensionality of negation and absurdity operators in the same way as for the operator C above. Let L be a ¬-(A-) logic. We say that the negation ¬ (respectively, the absurdity operator A) is extensional in L if the logic L is closed under the rule ϕ↔ψ ϕ↔ψ (respectively ). ¬ϕ ↔ ¬ψ A(ϕ) ↔ A(ψ)

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7 Absurdity as Unary Operator

If L is a ¬-(C-, A-)logic and the operator ¬ (C, A) is extensional in L, we say that L is an extensional ¬-(C-, A-) logic. Remark. Note that in all logics under consideration the positive connectives ∨, ∧, and → are extensional in a natural sense, which easily follows from the fact that ¬-(C-, and A-) logics was deﬁned so as to satisfy all axioms of the positive logic Lp. Proposition 7.4.5 If a ¬-logic L has an A-presentation, then we have L ¬p ↔ p → ¬p. Conversely, if the equivalence ¬p ↔ p → ¬p can be proved in a ¬-logic L and additionally, L is an extensional ¬-logic, then L has an A-presentation. Proof. Let a ¬-logic L has an A-presentation L . By the contraction law for →, we have L (p → A(p)) ↔ (p → (p → A(p))). The last equivalence graphically equals θ(¬p) ↔ θ(p → ¬p) = θ(¬p ↔ p → ¬p). By assumption θ is a faithful embedding of L into L , consequently, L ¬p ↔ p → ¬p. We now assume that L is an extensional ¬-logic and L ¬p ↔ p → ¬p. For a formula ϕ in the language L¬ , deﬁne a formula ϕA as a result of replacing every subformula of the form ¬ψ by A(ψ). Put LA := {ϕA | ϕ ∈ L}. It is clear that LA is an A-logic and the extensionality of ¬ in L immediately implies that A is extensional in LA . We check that ξ is a faithful embedding of L into LA . First of all, we note that for any formula ϕ, L ϕ iﬀ LA ϕA . Therefore, it is enough to prove that LA ϕA ↔ ξ(ϕ) for any ϕ. We use the induction on the structure of formulas. This statement trivially holds for propositional variables. The case of positive connectives easily follows by extensionality. It remains to consider the case ϕ = ¬ψ. By the induction hypothesis we have ψ A ↔ ξ(ψ) in LA .

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117

This fact and the extensionality of A imply LA A(ψ A ) ↔ A(ξ(ψ)). On the other hand, we have A(ξψ) ↔ (ξ(ψ) → A(ξ(ψ))) since the equivalence ¬p ↔ (p → ¬p) holds in L by assumption. The last two facts together imply LA A(ψ A ) ↔ (ξ(ψ) → A(ξ(ψ))), i.e., LA (¬ψ)A ↔ ξ(¬ψ).

2

Proposition 7.4.6 1. If an extensional ¬-logic L has an A-presentation, it has also a C-presentation. 2. If an extensional ¬-logic L has a C-presentation, this C-presentation is unique and it is also an exact and strong C-presentation. Proof. 1. The set C of formulas in the language L¬ we deﬁne as the least set of formulas satisfying the following conditions: 1) propositional variables belong to C; 2) if ϕ ∈ C and ψ ∈ C, then the formulas ϕ ∨ ψ, ϕ ∧ ψ, and ϕ → ψ are in C; 3) if ϕ ∈ C, then ϕ ∧ ¬ϕ ∈ C. Let LC := L ∩ C. The set LC is closed under the rules of modus ponens and substitution of formulas from C, moreover, it contains all formulas of the form (ϕ ∧ ¬ϕ) → ϕ for ϕ in C. Therefore, the set L := {ϕC | ϕ ∈ LC }, where ϕC is obtained from ϕ by substituting a subformula C(ψ) for any subformula of the form ψ ∧ ¬ψ, forms a C-logic. Check that L is the desired C-presentation of L. First we note that the equivalence L ϕ iﬀ L ϕC holds for any ϕ ∈ C. The logic L has an A-presentation by assumption, whence, by Proposition 7.4.5 we have in L the equivalences ¬p ↔ (p → ¬p) ↔ (p → (p ∧ ¬p)). Therefore, in L any formula ϕ is equivalent to a formula ϕ obtained by replacing every subformula ¬ψ of ϕ by ψ → (ψ ∧ ¬ψ). Note that ϕ belongs

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7 Absurdity as Unary Operator

to C, moreover, θ(ϕ) = ϕC . In this way, we arrive at the following chain of equivalences L ϕ ⇐⇒ L ϕ ⇐⇒ L ϕC ⇐⇒ L θϕ, which completes the proof of this item. 2. It follows easily from the considerations of the previous item that any C-presentation of an extensional ¬-logic L coincides with the set {ϕC | ϕ ∈ L ∩ C}. Therefore, the logic L may have at most one C-presentation. The exact C-presentation equals the intersection of all C-presentations, it therefore coincides with the unique C-presentation of L. It remains to prove that a C-presentation L of L is strong, i.e., that ρ is a faithful embedding of L into L and that the translations θ and ρ are mutually inverse. It is clear that for any formula ϕ in the language LC , the formula ρ(ϕ) belongs to C and, conversely, any formula in C has the form ρ(ϕ). Therefore, the above presentation of L immediately implies that ρ is a faithful embedding. Let ϕ be a formula in the language L¬ . The equivalence ϕ ↔ ρθ(ϕ) follows from the fact ρθ(ϕ) graphically equals the formula ϕ deﬁned in the previous item. For a formula ϕ in the language LC , the equivalence ϕ ↔ θρ(ϕ) easily follows from the extensionality of L and the equivalence C(p) ↔ (p ∧ (p → C(p))) which holds in any C-logic. 2 Proposition 7.4.7 Let L be an extensional ¬-logic, L1 an extensional A-presentation of L, and L2 a C-presentation of L. Then L1 is an A-presentation of L2 . Proof. 1. By the previous proposition L2 is a strong C-presentation of L. Therefore, we have for any formula ϕ, L2 ϕ ⇐⇒ L ρϕ ⇐⇒ L1 ξρϕ. Moreover, ξρC(ϕ) = ξ(ρϕ ∧ ¬ρϕ) = ξρϕ ∧ (ξρϕ → A(ξρϕ)). The latter formula is equivalent in L1 to ξρϕ ∧ A(ξρϕ). From this fact and the extensionality of L1 , we obtain by induction L1 ξρϕ ↔ ζϕ for any ϕ. 2

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119

The above results concerning the existence of A- and C-presentations, uniqueness of C-presentations, etc., used essentially the extensionality property. Most of the well-known systems of paraconsistent logic are not extensional and need, therefore, special treatment. In the next sections, we take two basic systems of paraconsistent logic and consider the problem of presenting the negation via absurdity and contradiction operators for these systems.

7.4.2

Logic CLuN

All logics considered in this and the following subsections have the same positive fragment, namely, Lk+ , and admit semantic characterization in terms of {0, 1}-interpretations non-truth-functional for operators of negation, absurdity or contradiction. The latter means that the value of an interpretation ν on a formula of the form ¬ϕ (A(ϕ), C(ϕ)) is not determined uniquely by the value ν(ϕ). For the language L , where # ∈ {¬, A, C}, we deﬁne an interpretation ν as a mapping from the set of L -formulas to the set {0, 1} satisfying for arbitrary ϕ and ψ the following properties: 1ν ) ν(ϕ ∧ ψ) = 1 iﬀ ν(ϕ) = 1 and ν(ψ) = 1; 2ν ) ν(ϕ ∨ ψ) = 1 iﬀ ν(ϕ) = 1 or ν(ψ) = 1; 3ν . ν(ϕ → ψ) = 1 iﬀ ν(ϕ) = 0 or ν(ψ) = 1. The logic CLuN was introduced in [6] (under the name P I). The name CLuN we can understand as classical logic admitting gluts (see also [7]). It is a fairly weak paraconsistent logics, which is in a sense basic for the class of logics considered by D. Batens in [6]. The logic CLuN can be axiomatized modulo classical positive logic via the law of excluded middle, i.e., we have CLuN = Lk+¬ + {p ∨ ¬p}, where Lk+¬ is an expansion of Lk+ to the language L¬ . We denote by U the class of interpretations ν satisfying for any ϕ the property: U . If ν(ϕ) = 0, then ν(¬ϕ) = 1.

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7 Absurdity as Unary Operator

Proposition 7.4.8 [6, p.205] For any formula ϕ, CLuN ϕ iﬀ ν(ϕ) = 1 for all ν ∈ U. We start to consider A- and C-presentations of CLuN with the following observation. Proposition 7.4.9 The logic CLuN has no strong C-presentation. Proof. If CLuN has a strong C-presentation, the equivalence ϕ ↔ ρθϕ holds in CLuN for any ϕ. Take a formula ¬¬p and compute ρθ(¬¬p) = (p → (p ∧ ¬p)) → ((p → (p ∧ ¬p)) ∧ ¬(p → (p ∧ ¬p))) ↔ ↔ (p → ¬p) → ¬(p → (p ∧ ¬p)). Let ν be an interpretation from U such that ν(p) = 0, ν(¬p) = ν(¬¬p) = 1, and ν(¬(p → (p ∧ ¬p))) = 0. One can ﬁnd out such interpretation in U , because ν(p) = 0 implies ν(p → (p ∧ ¬p)) = 1. As we can see, ν(ρθ(¬¬p)) = 0, and so ν(¬¬p ↔ ρθ(¬¬p)) = 0. By the previous proposition we have CLuN ¬¬p ↔ ρθ(¬¬p). 2 However, one can ﬁnd exact A- and C-presentations of CLuN. Deﬁne an A-logic CLuNA as an expansion of Lk+ to the language LA , and let U A be the set of all interpretations of the language LA . We have the following completeness theorem. Proposition 7.4.10 For any ϕ, CLuNA ϕ if and only if ν(ϕ) = 1 for all ν ∈ U A . Proof. The soundness can be checked directly and to prove the completeness we use an easy modiﬁcation of Henkin’s construction. As usual, deﬁne a prime CLuNA -theory as a set of formulas T , which contains CLuNA , is closed under modus ponens and has the disjunction property, i.e., ϕ ∨ ψ ∈ T implies ϕ ∈ T or ψ ∈ T . In a standard way, one can prove the extension lemma. Lemma 7.4.11 Let a set of formulas S and a formula ϕ be such that S CLuNA ϕ. There is then a prime CLuNA -theory T such that S ⊆ T and ϕ ∈ T . 2

7.4 A- and C -Presentations

121

Let CLuNA ϕ0 and T be a prime CLuNA -theory such that ϕ0 ∈ T . Deﬁne an interpretation νT on propositional variables and formulas of the form A(ϕ) as follows: νT (p) = 1 ⇐⇒ p ∈ T, νT (A(ϕ)) = 1 ⇐⇒ A(ϕ) ∈ T. By deﬁnition of interpretation, the mapping νT extends uniquely to the set of all formulas. For any prime CLuNA -theory, the following equivalences hold: ϕ∨ψ ∈T

⇔

ϕ ∈ T or ψ ∈ T ;

ϕ∧ψ ∈T

⇔

ϕ ∈ T and ψ ∈ T ;

ϕ→ψ∈T

⇔

ϕ ∈ T or ψ ∈ T .

The ﬁrst equivalence follows from the disjunction property, the second is obvious, check the last one. If ϕ → ψ ∈ T and ϕ ∈ T , then ψ ∈ T by modus ponens. Conversely, if ϕ ∈ T , then ϕ → ψ ∈ T in view of ϕ ∨ (ϕ → ψ) ∈ CLuNA . If ψ ∈ T , then ϕ → ψ ∈ T by the positive axiom ψ → (ϕ → ψ). From the deﬁnition of νT and the above equivalences we obtain νT (ϕ) = 1 ⇔ ϕ ∈ T, consequently, νT (ϕ0 ) = 0.

2

Proposition 7.4.12 CLuNA is an exact A-presentation of CLuN. Proof. First we check that for any ϕ, if CLuN ϕ, then CLuNA ξϕ. The logics CLuN and CLuNA have the same positive fragment, therefore, it is enough to check that the formula ξ(p ∨ ¬p) = p ∨ (p → A(p)) is provable in CLuNA , which is obviously true. Conversely, assume that for some ϕ0 we have CLuN ϕ0 . In this case, ν(ϕ0 ) = 0 for suitable ν ∈ U. Consider an interpretation ν of the language LA satisfying the following conditions: ν (p) = ν(p) for any propositional variable p; ν (A(ξψ)) = ν(ψ ∧ ¬ψ) for any ψ in the language L¬ . Obviously, we can construct such an interpretation and it belongs to the class U A , which contains all interpretations of the language LA . Prove by induction on the structure of formulas that ν(ψ) = ν (ξψ) for all ψ. This is obviously true for propositional variables. The case of positive

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7 Absurdity as Unary Operator

connectives is trivial, so it remains to consider the case of negation. Assume that ν(ψ) = ν (ξψ) and calculate ν (ξ(¬ψ)) = ν (ξψ → A(ξψ)) = ν (ξψ) → ν (A(ξψ)) = νψ → ν(ψ ∧ ¬ψ) = ν(ψ → (ψ ∧ ¬ψ)) = ν(¬ψ). The last equation is due to the fact that CLuN ¬p ↔ p → (p ∧ ¬p). According to the proven equality ν (ξϕ0 )) = ν(ϕ0 ) = 0, i.e., the formula ξϕ0 is not provable in CLuNA . We have thus proved that CLuNA is an A-presentation of CLuN. The exactness of this presentation immediately follows from the fact that CLuNA is the least A-logic with classical positive fragment. Obviously, any A-presentation of CLuN must have Lk+ as its positive fragment. 2 Similarly to the case of A-presentation we can take the least C-logic with classical positive fragment as a C-presentation of CLuN. Put CLuNC := Lk+C + {C(p) → p}, where Lk+C is an expansion of Lk+ to the language LC . We denote by U C the class of all interpretations ν of the language LC , satisfying for any ϕ the condition: U C . If ν(ϕ) = 0, then ν(C(ϕ)) = 0. Proposition 7.4.13 For any ϕ, CLuNC ϕ if and only if ν(ϕ) = 1 for all ν ∈ U C . Proof is analogous to Proposition 7.4.10. 2 Proposition 7.4.14 CLuNC is an exact C-presentation of CLuN. Proof of this statement can be obtained, in fact, by replacing A by C and ξ by θ in the proof of Proposition 7.4.12. 2 C Due to Proposition 7.4.9 the logic CLuN cannot be a strong Cpresentation, but we can prove that the inverse translation also embeds faithfully CLuNC into CLuN. Proposition 7.4.15 CLuN is an exact ¬-presentation of CLuNC .

7.4 A- and C -Presentations

123

Proof. The fact that CLuNC ϕ implies CLuN ρϕ for any ϕ can be checked directly. Assume that for some formula ϕ0 we have CLuNC ϕ0 and consider an interpretation ν ∈ U C with ν(ϕ0 ) = 0. In U, one can ﬁnd an interpretation ν of the language L¬ such that ν (p) = ν(p) for all propositional variables p and ν (¬ρψ) = ν(ψ → Cψ) for any ψ. Using induction on the structure of formulas, we check that ν ρψ = νψ for any ψ. Clearly, it is enough to consider the case of C-operation. Let ν ρψ = νψ for some ψ. Then ν (ρCψ) = ν (ρψ ∧ ¬ρψ) = ν ρψ ∧ ν (¬ρψ) = νψ ∧ ν(ψ → Cψ) = ν(Cψ). The desired equality is proved, in particular, we have ν (ρϕ0 ) = 0. Consequently, CLuN is a ¬-presentation of CLuNC . Further, any ¬-presentation of CLuNC has positive fragment Lk+ and contains the formula p ∨ ¬p, because this formula is equivalent modulo Lk+ to p ∨ (p → (p ∧ ¬p)) = ρ(p ∨ (p → Cp)). Therefore, CLuN is the least C-presentation of CLuNC and so it is exact. 2

7.4.3

Sette’s Logic P1

We now turn to the ﬁrst example of maximal paraconsistent logic, namely, to the system P 1 suggested by A. Sette in [102]. This is a sublogic of classical logic maximal in a sense that adjoining to P 1 any new classical tautology yields classical logic. The negation of P 1 is not extensional, however, as we will see, P 1 has even a strong C-presentation. Originally P 1 was deﬁned as a three-valued logic with the set of truth-values {T 0 , T 1 , F }, where T 0 and F are, in fact, the classical truth and falsehood, and T 1 is in a sense a second truth, whose negation is also true. Both truths T 0 and T 1 are distinguished values. The matrix for P 1 is {T 0 , T 1 , F }, →, ¬, {T 0 , T 1 } , where operations → and ¬ are deﬁned via the following truth tables: → T0 T1 F

T0 T0 T0 T0

T1 T0 T0 T0

F F F T0

¬ T0 T1 F

F T0 T0

The connectives ∨ and ∧ are introduced via the deﬁnitions: ϕ ∨ ψ := (ϕ → ¬¬ϕ) → (¬ϕ → ψ), ϕ ∧ ψ := (((ϕ → ϕ) → ϕ) → ¬((ψ → ψ) → ψ)) → ¬(ϕ → ¬ψ).

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7 Absurdity as Unary Operator

A. Sette [102] gives the following axiomatics for P 1 : 1. p → (q → p) 2. (p → (q → r)) → ((p → q) → (p → r)) 3. (¬p → ¬q) → ((¬p → ¬¬q) → p)) 4. (p → q) → ¬¬(p → q) The only inference rules are, as usual, substitution and modus ponens. It is clear that P 1 is a sublogic of Lk, moreover, the following theorem holds. Theorem 7.4.16 [102] For any ϕ ∈ Lk, if ϕ is not provable in P 1 , then P 1 + {ϕ} = Lk. 2 In [2], it was stated that P 1 is equivalent to the system β2 introduced in [50]. Axiomatics of β2 includes traditional axioms of classical positive logic and the following axiom schema for negation: 1β . (¬ϕ → ψ) → ((¬ϕ → ¬ψ) → ϕ), where ψ is molecular, i.e., diﬀerent from a propositional variable. From β2 we can easily pass to an axiomatization of P 1 convenient for our goals. Proposition 7.4.17 The logic P 1 is axiomatizable modulo the classical positive logic via the following axiom schemes: 1S ) ¬¬ϕ → ϕ; 2S ) (ϕ → ψ) → ((ϕ → ¬ψ) → ¬ϕ), where ψ is molecular. Proof. On the one hand, we can use truth-tables to check that schemes 1S and 2S hold in P 1 . On the other, the schema 1β easily follows from (¬ϕ → ψ) → ((¬ϕ → ¬ψ) → ¬¬ϕ), a substitution case of 2S , and 1S . 2 Using the maximal property we can easily obtain a semantic characterization of P 1 in terms of {0, 1}-interpretations. Denote by P the class of all interpretations ν of the language L¬ satisfying the properties: 1P . If νϕ = 0, then ν¬ϕ = 1. 2P . If ϕ is molecular and νϕ = 1, then ν¬ϕ = 0.

7.4 A- and C -Presentations

125

Proposition 7.4.18 For any formula ϕ, P 1 ϕ if and only if νϕ = 1 for all ν ∈ P. Proof. Put L∗ := {ϕ | νϕ = 1 for all ν ∈ P}. We have P 1 ⊆ L∗ ⊆ Lk. Indeed, the former inclusion can be veriﬁed directly, the latter follows from the fact that all classical interpretations belong to P. Due to Theorem 7.4.16 one of these inclusions is not proper. In P, there is an interpretation ν such that νp = ν¬p = 1 and ν¬¬p = 0, which means that p → ¬¬p ∈ L∗ , i.e., L∗ = Lk. Therefore, L∗ = P 1 . 2 Having convenient axiomatics and semantics for P 1 , we may turn to its presentations. Lemma 7.4.19 If L is an A-presentation of P 1 , then L contains the following schemes: 1A . A(ϕ → Aϕ) → ϕ; 2A . (ψ ∧ Aψ) → ϕ, where ψ is molecular. Proof. Consider a ξ-translation of 2S , (ϕ → ψ) → ((ϕ → (ψ → Aψ)) → (ϕ → Aϕ). Modulo Lk+ it is equivalent to (ϕ → (ψ ∧ Aψ)) → (ϕ → Aϕ) ↔ ↔ (ϕ ∧ (ϕ → (ψ ∧ Aψ))) → Aϕ ↔ (ϕ ∧ (ψ ∧ Aψ)) → Aϕ ↔ ↔ (ψ ∧ Aψ) → (ϕ → Aϕ). In particular, we have L (ψ ∧ Aψ) → ((ϕ → Aϕ) → A(ϕ → Aϕ)). At the same time, the formula ((ϕ → Aϕ) → A(ϕ → Aϕ)) → ϕ,

(7.4)

a ξ-translation of ¬¬ϕ → ϕ, is provable in L. From the last two formulas we immediately have L (ψ ∧ Aψ) → ϕ, where ψ is molecular.

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7 Absurdity as Unary Operator

Further, the positive axiom A(ϕ → Aϕ) → ((ϕ → Aϕ) → A(ϕ → Aϕ)) together with (7.4) yields the scheme 1A . 2 Denote by P A the A-logic axiomatized modulo Lk+ by schemes 1A and 2A and by P A the class of interpretations ν of the language LA satisfying the condition 1Aν . If ϕ is molecular and νϕ = 1, then ν(Aϕ) = 0. Lemma 7.4.20 If P A ϕ, then νϕ = 1 for all ν ∈ P A . The proof is by direct veriﬁcation. 2 Proposition 7.4.21 The logic P A is an exact A-presentation of P 1 . Proof. Let L∗ := {ϕ | ξϕ ∈ P A }. As it was stated in the proof of Lemma 7.4.19, the ξ-translation of 2S is equivalent in P A to a particular case of 2A . The formula ξ(¬¬ϕ → ϕ) has the following quasi-proof in P A : 1) A(ϕ → Aϕ) → ϕ, an axiom; 2) ((ϕ → Aϕ) → A(ϕ → Aϕ)) → ((ϕ → Aϕ) → ϕ), by monotonicity of → in the second argument from 1); 3) ((ϕ → Aϕ) → ϕ) → ϕ, the Peirce law; 4) ((ϕ → Aϕ) → A(ϕ → Aϕ)) → ϕ (= ξ(¬¬ϕ → ϕ)), by transitivity from 2) and 3). We have thus proved the inclusion P 1 ⊆ L∗ . Let P A ξϕ0 and ν is an arbitrary classical interpretation. Deﬁne an interpretation ν of LA by the conditions ν p = νp for any p and ν (Aϕ) = |1 − νϕ|. It is not hard to see that ν ∈ P A and ν ξϕ = νϕ for any ϕ. By Lemma 7.4.20, νϕ0 = 1. And we have in this way L∗ ⊆ Lk. This inclusion is proper, because the formula ξ(p → ¬¬p) is not in P A . The last fact can be easily checked using Lemma 7.4.20. Only one of the inclusions P 1 ⊆ L∗ ⊆ Lk is proper, whence P 1 = L∗ . We have thus proved that P A is an A-presentation of P 1 , the exactness immediately follows from Lemma 7.4.19. 2 1 We now turn to C-presentations of P .

7.4 A- and C -Presentations

127

Lemma 7.4.22 If L is a C-presentation of P 1 , then it contains the schema: 1C ) Cψ → ϕ, where ψ is molecular. Proof. By Lemma 7.4.19 L (ψ ∧ Cψ) → ϕ for molecular ψ, which is equivalent to 1C in view of the axiom Cp → p of C-logics. 2 + C Let P be a C-logic axiomatized modulo Lk by the axiom scheme 1C . We denote by P C the class of all interpretations ν of LC satisfying the conditions: 1Cν . If νϕ = 0, then νCϕ = 0. 2Cν . If ϕ is molecular and νϕ = 1, then νCϕ = 0. Lemma 7.4.23 If P C ϕ, then νϕ = 1 for all ν ∈ P C .

2

Proposition 7.4.24 The logic P C is an exact and strong C-presentation of P 1 . Proof. The fact that P C is an exact presentation can be proved similarly to Proposition 7.4.21. We can directly check that P 1 is a ¬-presentation of P C . It remains to prove that any formula ϕ is equivalent to ρθϕ in P 1 and that ψ is equivalent to θρψ in P C . This can be easily stated using induction on the structure of formulas and the facts that P 1 (ϕ ↔ ψ) → (¬ϕ ↔ ¬ψ) for molecular ϕ and ψ and that P C (ϕ ↔ ψ) → (C(ϕ) ↔ C(ψ)) for molecular ϕ and ψ. 2 1 C Remark. Note that P is not an exact ¬-presentation of P . The least ¬-presentation of P C can be deﬁned via the class of interpretations ν satisfying the condition νϕ = 1 ⇒ ν¬ϕ = 0 for molecular ϕ. And this logic has no C-presentation.

Chapter 8

Semantical Study of Paraconsistent Nelson’s Logic1 This chapter starts the second part of the book devoted to paraconsistent Nelson’s logic N4 and to the class of its extensions. The natural ﬁrst step in the investigation of the class of N4-extensions is to provide an adequate algebraic semantics for the logic N4, i.e., characterizing N4 via a variety of algebraic systems V such that there exists a natural dual isomorphism between the lattice of N4-extensions and the lattice of subvarieties of V. We consider two variants of Nelson’s paraconsistent logic. The logic N4 is determined in the language ∨, ∧, →, ∼ and its positive fragment coincides with positive logic Lp. The logic, which we denote N4⊥ , is a conservative extension of N4 in the language with an additional symbol ⊥ for the constant “absurdity” and ∨, ∧, →, ⊥ -fragment of N4⊥ coincides with intuitionistic logic Li. In the course of semantic investigations, we work mainly with the logic N4 and we then show how the obtained results can be adapted to N4⊥ . For the logic N3, an explosive variant of Nelson’s logic, the algebraic semantics was introduced in [90], where N3 was characterized via the variety of N -lattices. Later D. Vakarelov [110] and independently M.M. Fidel [28] found a very convenient representation of N -lattices in the form of so-called twist-structures. Finally, M.M. Fidel [29] suggested a characterization of N3 1

Parts of this chapter were originally published in [72].

131

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via Heyting algebras with a special family of unary predicates distinguishing classes of counterexamples for elements of a Heyting algebra. Structures of this kind are called Fidel-structures, or F-structures. In our semantic studies of N4, we go in the opposite direction compared to N3 and start with Fidel-structures (Section 8.2). A semantics of this kind for N4 can be obtained as an immediate generalization of the semantics from [29] and the completeness theorem has an easy and short proof. Then (Section 8.3) we introduce twist-structures for N4 and show that Fidel-structures and twist-structures are mutually deﬁnable, which leads to a characterization of N4 in terms of twist-structures. In Section 8.4, generalizing the notion of N -lattices we deﬁne N4-lattices and state that all twiststructures are N4-lattices and conversely, that any N4-lattice is isomorphic to a twist-structure. With any N4-lattice A, we associate the set DA of its distinguished elements and prove the completeness result with respect to the obtained class of logical matrices. It is essential that the set D A is not one-element in the general case. This fact has the following explanation. In N4, as well as in N3, the usual equivalence ϕ ↔ ψ := (ϕ → ψ) ∧ (ψ → ϕ) does not have the congruence property with respect to negation. It is possible that the negations of equivalent formulas are not equivalent. Only the strong equivalence ϕ ⇔ ψ := (ϕ ↔ ψ) ∧ (∼ ψ ↔∼ ϕ) is a congruence wrt negation. Fortunately, due to the axiom schema ∼ ϕ → (ϕ → ψ), all negations of tautologies are equivalent in N3. Namely this fact allows one to characterize N3 by the class of N -lattices, where the unit element (the greatest element wrt the lattice ordering) is the only distinguished element. Another situation is in case of N4. For example, ∼ (p → p) and ∼ (p → (q → p)) are not equivalent in this logic. The ﬁrst formula is equivalent to p∧ ∼ p, the second to (p∧ ∼ p) ∧ q, and these two contradictions are not equivalent in the paraconsistent logic N4. This complication can be overcome due to the fact that sets of distinguished elements in N4-matrices are deﬁnable. In Section 8.5, we prove that N4-lattices form a variety and establish a dual lattice isomorphism between the lattice of N4-extensions and the lattice of subvarieties of the variety of N4-lattices. To obtain an axiomatization of the variety of N4-lattices the results of A. Pynko [89] are essentially used. The desired axiomatization is obtained as appropriate weakening of an axiomatization of the class of implicative De Morgan lattices introduced in [89]. In the last section of this chapter we show how to deﬁne twist-structures for the logic N4⊥ , and then deﬁne N4⊥ -lattices and establish the dual isomorphism between the lattice of N4⊥ -extensions and the lattice of subvarieties of the variety of N4⊥ -lattices.

8.1 Preliminaries

8.1

133

Preliminaries

In this part of the book, we deal with the propositional languages L := {∨, ∧, →, ∼}, where ∼ is a symbol for strong negation, and L⊥ := L ∪ {⊥} with an additional symbol for the constant “absurdity”. By F or (F or ⊥ ) we denote the trivial logic in the language L (L⊥ ), i.e., the set of formulas of this language. The connectives of equivalence ↔ and of strong equivalence ⇔ are deﬁned as follows: ϕ ↔ ψ := (ϕ → ψ) ∧ (ψ → ϕ), ϕ ⇔ ψ := (ϕ ↔ ψ) ∧ (∼ ψ ↔∼ ϕ). As above, logics will be deﬁned via Hilbert-style deductive systems with only the rules of substitution and modus ponens. In this way, to deﬁne a logic it is enough to give its axioms. Paraconsistent Nelson’s logic N4 is a logic in the language L characterized by the following list of axioms: A1) p → (q → p); A2) (p → (q → r)) → ((p → q) → (p → r)); A3) (p ∧ q) → p; A4) (p ∧ q) → q; A5) (p → q) → ((p → r) → (p → (q ∧ r))); A6) p → (p ∨ q); A7) q → (p ∨ q); A8) (p → r) → ((q → r) → ((p ∨ q) → r)); A9) ∼∼ p ↔ p; A10) ∼ (p ∨ q) ↔ (∼ p∧ ∼ q); A11) ∼ (p ∧ q) ↔ (∼ p∨ ∼ q); A12) ∼ (p → q) ↔ (p∧ ∼ q). To obtain explosive Nelson’s logic N3 we add to the list of N4-axioms, the explosion axiom for strong negation: A13) ∼ p → (p → q).

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The logic N4⊥ is a logic in the language L⊥ determined by the axioms A1–A13 and the two additional axioms for constant ⊥: A14) ⊥ → p and A15) p →∼ ⊥. Due to axiom A14 the intuitionistic negation can be deﬁned in N4⊥ as ¬ϕ := ϕ → ⊥. If we put ⊥ :=∼ (p0 → p0 ), one can prove N3 ⊥ → p, p →∼ ⊥. The ﬁrst formula is equivalent to a particular case of axiom A13 modulo positive logic, the second follows from the equivalence ∼ ⊥ ↔ (p0 → p0 ) obtained in turn from A9. In particular, the intuitionistic negation is deﬁnable in N3. This is why we do not consider two diﬀerent logics N3 and N3⊥ . We say that a formula ϕ of the language L or L⊥ is a negation normal form (nnf) if it contains the symbol ∼ only before atomic formulas. The following translation (·) sends every formula ϕ to a negation normal form, where p ∈ P rop and # ∈ {∨, ∧, →}: p = p ∼∼ ϕ = ϕ

∼p = ∼p ϕ#ψ = ϕ#ψ

∼ (ϕ ∨ ψ) = ∼ ϕ ∧ ∼ ψ ∼ (ϕ ∧ ψ) = ∼ ϕ ∨ ∼ ψ ∼ (ϕ → ψ) = ϕ ∧ ∼ ψ

⊥ = ⊥

Proposition 8.1.1 For any formula ϕ in the language L (L⊥ ), N4(N4⊥ ) ϕ ↔ ϕ. The proof easily follows from strong negation axioms A9–A12. 2 Proposition 8.1.2 The logic N4⊥ is a conservative extension of N4. Proof. Let ϕ be a formula in the language L and ϕ0 , . . . , ϕn = ϕ is a proof ⊥ of ϕ in N4 . Let P be the set of all variables occurring in this proof. Put φ := p∈P p∧ ∼ p. Replacing in the above proof any occurrence of ∼ ⊥ by pi → pi , pi ∈ P , and then any occurrence of ⊥ by φ we obtain a quasi-proof in N4. Indeed, every axiom of the form ψ →∼ ⊥ is obviously replaced by a provable formula. That every axiom ⊥ → ψ is replaced by a provable formula follows from Proposition 8.1.1 and the fact that in positive logic Lp, every formula follows from the conjunction of its variables. 2

8.2 Fidel’s Semantics

135

An important peculiarity of Nelson’s logics N4 and N4⊥ is that the provable equivalence is not a congruence relation. However, axioms A1-A8 constitute the axiomatization of positive logic Lp (see Chapter 2). This means that the provable equivalence is a congruence relation with respect to positive connectives. More exactly, for any formulas ϕ0 , ϕ1 and a positive formula ψ(p) with a propositional parameter, the provability of ϕ0 ↔ ϕ1 in N4 (or N4⊥ ) implies the provability of ψ(ϕ0 ) ↔ ψ(ϕ1 ) in that logic. The provable strong equivalence ⇔ will be a congruence relation in N4 as well as in N4⊥ . Proposition 8.1.3 The logics N4 and N4⊥ are closed under the weak replacement rule ϕ0 ↔ ϕ1 ∼ ϕ0 ↔∼ ϕ1 . ψ(ϕ0 ) ↔ ψ(ϕ1 ) Proof. It follows easily from Proposition 8.1.1. 2

8.2

Fidel’s Semantics

First, we present a semantics for N4 in terms of F-structures (Fidelstructures), which is an immediate generalization of the semantics for N3 developed by M.M. Fidel in [29]. Definition 8.2.1 An F-structure is a tuple A = A, ∨, ∧, →, 1, {Na }a∈A , where A, ∨, ∧, →, 1 is an implicative lattice and {Na }a∈A is a family of sets satisfying the following properties: 1) for any a ∈ A, ∅ = Na ⊆ A; 2) for any a, b ∈ A, a ∈ Na , and b ∈ Nb , the following relations hold a ∨ b ∈ Na∧b , a ∧ b ∈ Na∨b , a ∈ Na ; 3) for any a, b ∈ A and b ∈ Nb , we have a ∧ b ∈ Na→b . We say that an F-structure A is saturated if the equality Na = A holds for all a ∈ A. For F-structures we use the denotation A, {Na }a∈|A| , where A is an implicative lattice and {Na }a∈|A| a family of subsets of |A|.

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With an arbitrary set of formulas Γ non-trivial with respect to N4 we associate an equivalence relation ≡Γ on the set of formulas in a standard way. For any formulas ϕ and ψ, we set ϕ ≡Γ ψ ⇐⇒ Γ N4 ϕ ↔ ψ. The coset of an element ϕ with respect to ≡Γ is denoted by [ϕ]Γ and the family of all cosets by LΓ . In case of N4, the relation ≡Γ is a congruence with respect to all positive connectives, which allows one to deﬁne on the set LΓ operations ∨, ∧ and → as follows: [ϕ]Γ ∗ [ψ]Γ := [ϕ ∗ ψ]Γ , where ∗ ∈ {∨, ∧, →}. Due to the fact that N4 contains all axioms of positive logic, the structure LΓ , ∨, ∧, →, 1 , where 1 denotes the coset of formulas deducible from Γ, 1 = [p → p]Γ , is an implicative lattice. We extend it to an F-structure putting N[ϕ]Γ := {[∼ ψ]Γ | ψ ∈ [ϕ]Γ } for any [ϕ]Γ ∈ LΓ . Later on, we omit the lower index Γ when it does not lead to confusion. Let LΓ := LΓ , ∨, ∧, →, 1, {N[ϕ] }[ϕ]∈LΓ . Lemma 8.2.2 For every set of formulas Γ non-trivial wrt N4, the structure LΓ is an F-structure. Proof. It is enough to check Items 2, 3 of Deﬁnition 8.2.1, which can easily be done with the help of the axioms for strong negation. 2 We say that an F-structure A, {Na }a∈|A| is a substructure of an F-structure B, {Nb }b∈|B| if (1) A is an implicative sublattice of B and (2) for any a ∈ |A|, Na ⊆ Na . Lemma 8.2.3 Any F-structure is a substructure of a saturated F-structure. Proof. Any structure A, {Na }a∈|A| can be embedded into a saturated structure A, {Na }a∈|A| , where A is the same implicative lattice and Na = |A| for all a ∈ A. 2

8.2 Fidel’s Semantics

137

Definition 8.2.4 Let A, {Na }a∈|A| be an F-structure. The mapping v from F or into |A| is said to be a valuation into this structure if the following conditions hold for arbitrary formulas ϕ and ψ: 1) v(∼ p) ∈ Nv(p) for a propositional variable p; 2) v(ϕ ∗ ψ) = v(ϕ) ∗ v(ψ) for ∗ ∈ {∨, ∧, →}; 3) v(∼ (ϕ ∨ ψ)) = v(∼ ϕ) ∧ v(∼ ψ) and v(∼ (ϕ ∧ ψ)) = v(∼ ϕ) ∨ v(∼ ψ); 4) v(∼ (ϕ → ψ)) = v(ϕ) ∧ v(∼ ψ); 5) v(∼∼ ϕ) = v(ϕ). Remark. It can easily be seen from this deﬁnition that every valuation into an F-structure is uniquely determined by its restriction to the set of propositional variables and their negations. The following fact can be checked by induction on the structure of formulas. Lemma 8.2.5 For any F-structure A, {Na }a∈|A| , any valuation v and formula ϕ, we have v(∼ ϕ) ∈ Nv(ϕ) . 2 Now we deﬁne a semantic consequence relation |=F between sets of formulas and formulas. Let Γ be a set of formulas and ϕ be a formula. For an F-structure A, the relation Γ |=A F ϕ holds if and only if for any valuation v into A the condition v(ψ) = 1 for all ψ ∈ Γ implies v(ϕ) = 1. The relation Γ |=F ϕ means that Γ |=A F ϕ for any F-structure A. Lemma 8.2.6 Let B be a substructure of an F-structure A. Then Γ |=A F ϕ ϕ. implies Γ |=B F Proof. Indeed, if v is a valuation into B, it is also a valuation into A.

2 It is not hard to check that the introduced consequence relation is closed under substitution. Lemma 8.2.7 Let Γ(p1 , . . . , pn ) be a set of formulas, ϕ(p1 , . . . , pn ), ψ1 , . . ., ψn be formulas, and A an F-structure. If Γ |=A F ϕ, then Γ(ψ1 , . . . , ψn ) |=A F ϕ(ψ1 , . . . , ψn ). 2

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Theorem 8.2.8 Let Γ be a set of formulas non-trivial with respect to N4 and ϕ a formula. The following conditions are equivalent: 1) Γ N4 ϕ; 2) Γ |=F ϕ; 3) Γ |=A F ϕ for any saturated F-structure A; Γ 4) Γ |=L F ϕ.

Proof. 1) ⇒ 2) The soundness of consequence relation with respect to the introduced semantics can be proved in a standard way via induction on the length of proof. Implications 2) ⇒ 3) and 3) ⇒ 4) are obvious. Γ 4) ⇒ 1) Assume Γ |=L F ϕ and consider a mapping v from the set of propositional variables and their negations to LΓ such that for a variable p, we have v (p) = [p] and v (∼ p) = [∼ p]. This mapping extends in a unique way to a valuation v into LΓ . Using Items 2–5 of Deﬁnition 8.2.4 one can easily check that v(ϕ) = [ϕ] for any formula ϕ. If ψ ∈ Γ, then clearly Γ N4 ψ, i.e., [ψ] = 1. In view of the above property of v, we have v(ψ) = 1 for all ψ ∈ Γ. This implies by assumption that v(ϕ) = [ϕ] = 1, i.e., Γ N4 ϕ. 2 If ϕ is a positive formula, the validity of this formula on an F-structure

A, {Na }a∈|A| is equivalent to its validity on the implicative lattice A. This fact and Theorem 8.2.8 imply Corollary 8.2.9 N4 is a conservative extension of positive logic.

8.3

Twist-structures

Following M.M. Fidel [28] and D. Vakarelov [110] we introduce the notion of twist-structures. Because we need a semantics for the logic N4, where the intuitionistic negation is not deﬁnable, as underlying algebraic systems for twist-structures we take implicative lattices unlike the works [28, 110], where twist-structures were deﬁned over Heyting algebras. The apt term “twiststructure” was introduced in [47]. We deﬁne an algebraic structure on the direct square of the universe of an underlying algebra in such a way that operations are not componentwise, but they are “twisted” in some sense.

8.3 Twist-structures

139

Definition 8.3.1 Let A = A, ∨, ∧, →, 1 be an implicative lattice. 1. The full twist-structure over A is the algebra A = A × A, ∨, ∧, →, ∼ with the twist-operations deﬁned for (a, b), (c, d) ∈ A × A as follows: (a, b) ∨ (c, d) := (a ∨ c, b ∧ d), (a, b) ∧ (c, d) := (a ∧ c, b ∨ d) (a, b) → (c, d) := (a → c, a ∧ d), ∼ (a, b) := (b, a). 2. A twist-structure over A is an arbitrary subalgebra B of the full twiststructure A such that π1 (B) = A (in which case also π2 (B) = A). 3. The class of all twist-structures over A is denoted S (A). It should be noted that our terminology diﬀers from that of [47], where the term “twist-structure” is used for twist-structures over Heyting algebras (see Section 8.6) and with the unverse of the form {(a, b) | a ∧ b = 0}, where 0 is the least element of the underlying algebra. A valuation into a twist-structure B is deﬁned in the usual way as a homomorphism of the algebra of formulas into B. The semantic consequence relation |= over twist-structures is deﬁned as follows. Let Γ be a set of formulas, ϕ a formula, and B a twist-structure. The relation Γ |=B ϕ holds if and only if for any valuation v in B the condition that π1 v(ψ) = 1 for all ψ ∈ Γ implies π1 v(ϕ) = 1. The relation Γ |= ϕ means that Γ |=B ϕ for all twist-structures B. One can easily pass from F-structures to twist-structures and back. Let A∗ = A, {Na }a∈|A| be an F-structure. We associate with A∗ a twiststructure A∗ over A with the following universe |A∗ | = {(a, b) ∈ |A| × |A| | b ∈ Na }. Using the deﬁnitions of twist-operations and of F-structures one can check that for any F-structure A∗ the set |A∗ | is closed under all twist-operations, and so the structure A∗ is well deﬁned. On the other hand, given a twist-structure B ∈ S (A) we deﬁne an F-structure B F = A, {Na }a∈|A| , where Na = {b ∈ A | (a, b) ∈ B} for any a ∈ |A|. The correctness of this deﬁnition can be veriﬁed directly.

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Further, let A be an F-structure and v a valuation into A. We deﬁne a valuation v into A for a propositional variable p as v (p) := (v(p), v(∼ p)). By induction on the structure of formulas we obtain then the following statement. Lemma 8.3.2 For any formula ϕ, v (ϕ) = (v(ϕ), v(∼ ϕ)). 2 Now, let B be a twist-structure and v a valuation. We deﬁne a valuation vF into B F as a unique valuation satisfying the conditions vF (p) = π1 v(p) and vF (∼ p) = π2 v(p) for all propositional variables p. Again, by induction on the structure of formulas one can prove the following lemma. Lemma 8.3.3 For any formula ϕ, vF (ϕ) = π1 v(ϕ).

2

Proposition 8.3.4 1. Let A∗ = A, {Na }a∈|A| be an F-structure and v a valuation into A∗ . Then (A∗ )F = A∗ and (v )F = v. 2. Let B be a twist-structure and v a valuation into B. Then (B F ) = B and (vF ) = v. Proof. 1. Let (A∗ )F = A, {Na }a∈|A| . For any pair of elements a, b ∈ |A|, we have the equivalences b ∈ Na ⇐⇒ (a, b) ∈ A∗ ⇐⇒ b ∈ Na . This proves the equality (A∗ )F = A∗ . Take an arbitrary formula ϕ and calculate (v )F (ϕ) = π1 v (ϕ) = π1 ((v(ϕ), v(∼ ϕ))) = v(ϕ). The second equality in this chain is due to Lemma 8.3.2.

8.3 Twist-structures

141

2. Let B be a twist-structure over A, B F = A, {Na }a∈|A| and let a, b ∈ |A|. The ﬁrst equality of this item follows from the equivalences: (a, b) ∈ (B F ) ⇐⇒ b ∈ Na ⇐⇒ (a, b) ∈ B. For a formula ϕ, we calculate applying Lemma 8.3.3 (vF ) (ϕ) = (vF (ϕ), vF (∼ ϕ)) = = (π1 v(ϕ), π1 v(∼ ϕ)) = (π1 v(ϕ), π2 v(ϕ)) = v(ϕ). 2 Proposition 8.3.5 Let Γ be a set of formulas and ϕ a formula. 1. For any F-structure A, A Γ |=A F ϕ ⇐⇒ Γ |= ϕ.

2. For any twist-structure A, F

A Γ |=A ϕ ⇐⇒ Γ |=F ϕ.

Proof. 1. In view of the previous proposition, any valuation v into A has the for a suitable A -valuation v and any valuation v into A can be form vF for some A-valuation v . These observations easily imply represented as v the desired equivalence. 2. Again, in view of the previous proposition (AF ) = A, which allows one to reduce this item to the previous one. 2 For a set of formulas Γ non-trivial with respect to N4, we deﬁne a Lindenbaum twist-structure L Γ := (LΓ ) , i.e., L Γ = {([ϕ]Γ , [∼ ψ]Γ ) | ψ ∈ [ϕ]Γ }, ∨, ∧, →, ∼ . The next completeness result easily follows from the completeness theorem for Fidel structures, Proposition 8.3.5 and an obvious equivalence: (an F-structure B is saturated) ⇐⇒ (the twist-structure B is full).

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Theorem 8.3.6 Let Γ be a set of formulas, non-trivial with respect to N4, and ϕ a formula. The following conditions are equivalent: 1) Γ N4 ϕ; 2) Γ |= ϕ; 3) Γ |=B ϕ for any full twist-structure B; L

4) Γ |=Γ ϕ. Remark. It was already mentioned that the provable equivalence ⇔ is a congruence in N4, therefore, on the set := {[ϕ]⇔ | ϕ ∈ F or}, L⇔ N4⊥ where [ϕ]⇔ := {ψ | N4 ϕ ⇔ ψ}, one can deﬁne a structure of Lindenbaum algebra in a standard way. Denote this algebra by L⇔ N4 . Note that the mapping [ϕ]⇔ → ([ϕ], [∼ ϕ]) determines an isomorphism of Lindenbaum algebra L⇔ N4 and Lindenbaum twist-structure LN4 .

8.3.1

Embedding of N3 into N42

Having the semantics for N4 in terms of twist-structures we are ready to prove that explosive N3 is faithfully embedded into paraconsistent N4. First we describe twist-structures, which are models of N3. Lemma 8.3.7 Let A be an implicative lattice and B ∈ S (A). We have B |= N3 if and only if A has the least element 0 and for any (a, b) ∈ B, a ∧ b = 0. Proof. As follows from the deﬁnition of twist-operations, the truth of axiom A13 on B is equivalent to the fact that for any (a, b), (c, d) ∈ B, a ∧ b ≤ c. By Deﬁnition 8.3.1 c is an arbitrary element of A, whence for any (a, b) ∈ B, a ∧ b is the least element of A. 2 Definition 8.3.8 Let :=∼ (p0 → p0 ), where p0 is a ﬁxed propositional variable. We deﬁne a transformation (·)∗ of formulas as follows. 1. For a propositional variable p, (p)∗ := p ∨ , (∼ p)∗ := (∼ p ∧ (p → )) ∨ . 2

This section was originally published in [71].

8.3 Twist-structures

143

2. (ϕ # ψ)∗ := ϕ∗ # ψ ∗ , where ϕ and ψ are nnf and # ∈ {∨, ∧, →}. 3. (ϕ)∗ (ϕ)∗ for any formula ϕ not a nnf. Theorem 8.3.9 For any formula ϕ, 1. ϕ ∈ N3 if and only if ϕ∗ ∈ N4. 2. N3 ϕ ↔ ϕ∗ . Proof. 1. First, we prove the direct implication. For any substitution instance ϕ of one or another N4-axiom, ϕ is provable in N4 due to Proposition 8.1.1. The transformation (·)∗ preserves all positive connectives for nnf, therefore, (ϕ)∗ is also provable in N4 as a substitution instance of ϕ. Let ϕ, ϕ → ψ ∈ N3 and ϕ∗ , (ϕ → ψ)∗ ∈ N4. Note that (ϕ → ψ)∗ = (ϕ → ψ)∗ = (ϕ → ∗ ψ)∗ = ϕ∗ → ψ , which immediately implies that ψ ∗ = (ψ)∗ ∈ N4. We have thus proved that the set of formulas, for which the considered implication holds, is closed under modus ponens. In this way, it remains to check that formulas of the form (∼ ϕ → (ϕ → ψ))∗ = (∼ ϕ)∗ → (ϕ∗ → ψ ∗ ) are provable in N4. The latter formula is equivalent in N4 to ((∼ ϕ)∗ ∧ϕ∗ ) → ψ ∗ , therefore, the desired result follows from the next lemma. Lemma 8.3.10 For any formula ϕ, the following holds: a) N4 → ϕ∗ ; b) N4 ↔ ϕ∗ ∧ (∼ ϕ)∗ . Proof. a) This item is true by deﬁnition for propositional variables and their negations. Any formula ϕ∗ can be obtained from formulas of the form p∗ and (∼ p)∗ with the help of positive connectives, which allows one to complete the proof by an easy induction on the structure of formulas. b) We use again an induction on the structure of formulas. For a propositional variable p, we have p∗ ∧ (∼ p)∗ ↔ (p∧ ∼ p ∧ (p → )) ∨ ↔ (∼ p ∧ p ∧ ) ∨ ↔ .

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In case of the conjunction of formulas, we apply the induction hypotheses and Item a) to obtain the following chain of equivalences: (ϕ ∧ ψ)∗ ∧ (∼ (ϕ ∧ ψ))∗ ↔ ϕ∗ ∧ ψ ∗ ∧ (∼ (ϕ ∧ ψ))∗ ↔ ↔ ϕ∗ ∧ ψ ∗ ∧ (∼ ϕ ∨ ∼ ψ)∗ ↔ ϕ∗ ∧ ψ ∗ ∧ ((∼ ϕ)∗ ∨ (∼ ψ)∗ ) ↔ ↔ ((ϕ∗ ∧ (∼ ϕ)∗ ) ∧ ψ ∗ ) ∨ (ϕ∗ ∧ (ψ ∗ ∧ (∼ ψ)∗ )) ↔ ( ∧ ψ ∗ ) ∨ (ϕ∗ ∧ ) ↔ . The case of disjunction can be considered similarly. For the implication, we have (ϕ → ψ)∗ ∧ (∼ (ϕ → ψ))∗ ↔ (ϕ∗ → ψ ∗ ) ∧ (∼ (ϕ → ψ))∗ ↔ ↔ (ϕ∗ → ψ ∗ ) ∧ (ϕ ∧ ∼ ψ)∗ ↔ (ϕ∗ → ψ ∗ ) ∧ ϕ∗ ∧ (∼ ψ)∗ ↔ ↔ ϕ∗ ∧ ψ ∗ ∧ (∼ ψ)∗ ↔ ϕ∗ ∧ ↔ . Thus, it remains to consider the case of negation, which is trivial in view of the graphical equality (∼ ϕ)∗ ∧ (∼∼ ϕ)∗ = (∼ ϕ)∗ ∧ ϕ∗ . 2 We now turn to the inverse implication. Assume N3 ϕ. Then there is a twist-structure B, B |= N3, and its valuation v such that π1 vϕ = 1. By Lemma 8.3.7, B ∈ S (A), where A is an implicative lattice with the least element 0. Since p0 → p0 ∈ N4, we have π1 v(p0 → p0 ) = 1. In view of Lemma 8.3.7 π2 v(p0 → p0 ) = 0. In this way, v() = (0, 1). For any propositional variable p, v(p) = (a, b), we have v(p∗ ) = v(p) ∨ v() = v(p) ∨ (0, 1) = v(p), v((∼ p)∗ ) = (∼ v(p) ∧ (v(p) → (0, 1))) ∨ (0, 1) = = (b, a) ∧ ((a, b) → (0, 1)) = (b ∧ (a → 0), a) = (b, a) = v(∼ p). Here we have taken into account the fact that a ∧ b = 0 is equivalent to b ≤ a → 0. Thus, if ψ is a propositional variable or its negation, we have v(ψ ∗ ) = v(ψ). Due to the fact that (·)∗ preserve all logical connectives for nnf, the relation v(ψ ∗ ) = v(ψ) holds for any nnf ψ. So for any formula ψ, by Item 3 of Deﬁnition 8.3.8 we have v(ψ ∗ ) = v(ψ). By Proposition 8.1.1 we have N4 ψ ↔ ψ, whence π1 v(ψ ∗ ) = π1 v(ψ) for any ψ. In particular, π1 v(ϕ∗ ) = 1, i.e., ϕ∗ ∈ N4. 2. In the proof of the previous item, it was stated that for any formula ϕ, for any twist-structure B, B |= N3, and any B-valuation v, π1 v(ϕ∗ ) = π1 v(ϕ), which implies N3 ϕ ↔ ϕ∗ . 2

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145

We have proved that as well as in case of intuitionistic and minimal logics deleting the explosion axiom from the logic N3 does not decrease the expressive power of the logic. For this reason, we conclude this section with the following problem: to ﬁnd a natural example of an explosive logic such that it is not faithfully embeddable into its paraconsistent analog. An example should be natural in a sense that the logic must be described in the literature and not constructed especially to solve this problem.

8.4

N4-Lattices

In this section, we give an algebraic deﬁnition of twist-structures. Namely, we describe the class of algebraic systems isomorphic to twist-structures. The deﬁnition below is closely related to that of N -lattices (see [92, 33]). Recall that in case of semantic investigations for N3, the notion of N -lattices was ﬁrst introduced [90] and only later was it proved that N -lattices can be represented as twist-structures [28, 110]. Definition 8.4.1 An algebra A = A, ∨, ∧, →, ∼ is said to be an N4lattice if the following hold. 1. The reduct A, ∨, ∧, ∼ is a De Morgan algebra, i.e., A, ∨, ∧ is a distributive lattice (non-bounded in general case) and the following identities hold: ∼ (p ∨ q) = ∼ p∧ ∼ q and ∼∼ p = p. 2. The relation $, where a $ b denotes (a → b) → (a → b) = a → b, is a preordering on A. 3. The relation ≈, where a ≈ b if and only if a $ b and b $ a, is a congruence relation with respect to ∨, ∧, → and the quotient-algebra A := A, ∨, ∧, → / ≈ is an implicative lattice. 4. For any a, b ∈ A, ∼ (a → b) ≈ a∧ ∼ b. 5. For any a, b ∈ A, a ≤ b if and only if a $ b and ∼ b $∼ a, where ≤ is a lattice ordering on A. First we check that the twist-structures belong to the just deﬁned class of lattices.

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Proposition 8.4.2 Every twist-structure B ∈ S (A) is an N4-lattice, moreover, the following facts are true for all (a, b), (c, d) ∈ |B|: a) (a, b) $ (c, d) if and only if a ≤ c; b) (a, b) ≈ (c, d) if and only if a = c; c) (a, b) ≤ (c, d) if and only if a ≤ c and d ≤ b; d) The mapping [(a, b)]≈ → a determines an isomorphism of implicative lattices B and A. Proof. Check statements a)–d). a) We calculate (a, b) → (c, d) = (a → c, a ∧ d) and (a → c, a ∧ d) → (a → c, a ∧ d) = (1, (a → c) ∧ a ∧ d) = (1, a ∧ c ∧ d). Thus, the inequality (a, b) $ (c, d) is equivalent to the conditions a → c = 1 and a ∧ d = a ∧ c ∧ d. The ﬁrst condition means a ≤ c and implies the second. Item b) immediately follows from a) and Item c) follows from the deﬁnition of twist-operations. d) In view of b) we have [(a, b)]≈ = {(a, c) | (a, c) ∈ |B|}. Moreover, twist-operations are componentwise with respect to the ﬁrst component and π1 (B) = A. These facts easily imply that [(a, b)]≈ → a determines the desired isomorphism. We now turn to the deﬁnition of N4-lattices. The fact that |B|, ∨, ∧, ∼ is a De Morgan algebra easily follows from the deﬁnition of twist-operations. Items 2 and 3 of the deﬁnition follows from b) and d), respectively. Item 4 can be veriﬁed directly, and 5 follows from a) and c). 2 Proposition 8.4.3 Every N4-lattice A is isomorphic to a twist-structure over A . Proof. Consider a mapping h from A to (A ) deﬁned by the rule h(a) := ([a]≈ , [∼ a]≈ ). We claim that h is an isomorphic embedding. Indeed, for any a, b ∈ A, we have h(a∧ b) = ([a∧ b]≈ , [∼ (a∧ b)]≈ ) = ([a]≈ ∧ [b]≈ , [∼ a]≈ ∨ [∼ b]≈ ) = h(a)∧ h(b).

8.5 The Variety of N4-Lattices

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We have used De Morgan’s law and the congruence properties of ≈. That h preserves ∨ can be checked similarly. Consider the case of implication: h(a → b) = ([a]≈ → [b]≈ , [a]≈ ∧ [∼ b]≈ ) = h(a) → h(b). We have again used the congruence properties of ≈ and Item 4 of Deﬁnition 8.4.1. Finally, by identity ∼∼ p = p we have h(∼ a) = ([∼ a]≈ , [∼∼ a]≈ ) = ([∼ a]≈ , [a]≈ ) =∼ h(a). Now assume h(a) = h(b), which means that a ≈ b and ∼ a ≈∼ b. By Item 5 of Deﬁnition 8.4.1 we immediately obtain a = b, which completes the proof. 2 A For an N4-lattice A, deﬁne a set D := {a ∈ |A| | a → a = a}. Let N denote the class of all matrices of the form A, DA , where A is an N4lattice, and let |=N be a consequence relation deﬁned in a natural way by the class N . Lemma 8.4.4 Let A ∈ S (B). Then D A = {(1, b) | (1, b) ∈ A}. Proof. Calculate (a, b) → (a, b) = (1, a∧b). Obviously, the equality (1, a∧b) = (a, b) is equivalent to a = 1. 2 From this lemma and Theorem 8.3.6 we immediately obtain that N4 is characterized by the class of matrices N . Theorem 8.4.5 Let Γ be a set of formulas and ϕ a formula. The following equivalence holds: Γ N4 ϕ ⇐⇒ Γ |=N ϕ.

8.5

The Variety of N4-Lattices

We have thus obtained a characterization of the logic N4 via the class N of matrices. We now show that the underlying algebras of these matrices, namely, N4-lattices, form a variety. The fact that sets of distinguished elements of matrices in N are deﬁnable allows one to state the dual isomorphism between the lattice EN4 of logics extending N4 and the lattice of subvarieties of the variety of N4-lattices. In this section, we will use

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essentially the results of [89], where the class of implicative De Morgan lattices was introduced and it was stated that this class provides an algebraic semantics for Belnap’s four-valued logic in the language ∨, ∧, →, ∼ . Our axiomatization of the class of N4-lattices is obtained by weakening the axiomatization of the class of implicative De Morgan lattices. Definition 8.5.1 Let VN4 be a variety of algebras in the language L satisfying the identities of De Morgan algebras and the following identities: 1N . (p → p) → q = q 2N . (p ∧ q) → r = p → (q → r) 3N . p → (q ∧ r) = (p → q) ∧ (p → r) 4N . (p ∨ q) → r = (p → r) ∧ (q → r) 5N . q ≤ p → q 6N . (p → q) ∧ (q → r) $ p → r 7N . p → p ≤∼ (p → q) → p 8N . q → q ≤ p → (∼ q →∼ (p → q)) 9N . q → q ≤∼ (p → q) →∼ q 10N . p ∧ (p → q) ≤ q∨ ∼ (∼ q →∼ p) 11N . p ≤ (q → q)∨ ∼ (∼ q →∼ p) Here, we consider p ≤ q as an abbreviation of the identity p ∨ q = q and p $ q as an abbreviation for (p → q) → (p → q) = p → q. Remark. Since algebras from VN4 are De Morgan algebras, for any A ∈ VN4 and a, b ∈ A, holds a ≤ b ⇒∼ b ≤∼ a.

(8.1)

Indeed, ∼ a∧ ∼ b =∼ (a ∨ b) =∼ b. As in the case of N4-lattices for an arbitrary algebra A ∈ VN4 , we deﬁne D A := {a ∈ |A| | a → a = a}. First we state some properties of sets DA in algebras of VN4 .

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Lemma 8.5.2 Let A ∈ VN4 . For any a, b ∈ A, the following holds: a) if a ∈ D A , then a → b = b; b) D A = {a → a | a ∈ |A|}; c) if a ∈ DA and a $ b, then b ∈ DA ; d) if a ≤ b, then a $ b; e) if a ∈ D A and a ≤ b, then b ∈ DA ; f ) if a, b ∈ D A , then a ∧ b ∈ DA . g) a $ a ∧ b if and only if a $ b. Proof. a) We have a → b = (a → a) → b = b by identity 1N . b) If a ∈ D A , then by deﬁnition a = a → a. On the other hand, by 1N we have (a → a) → (a → a) = a → a, whence a → a ∈ DA . c) By assumption (a → b) → (a → b) = a → b. On the other hand, a ∈ D A implies by a) a → b = b, whence b → b = b. d) By 2N we have (a → b) → (a → b) = ((a → b) ∧ a) → b. The latter is equal to a → b, because a ≤ b by assumption and b ≤ a → b by identity 5N . e) It follows immediately from c) and d). f) Applying identity 2N and Item a) we calculate (a ∧ b) → (a ∧ b) = a → (b → (a ∧ b)) = b → (a ∧ b) = a ∧ b. g) By deﬁnition a $ a ∧ b means a → (a ∧ b) ∈ DA , which is equivalent to (a → a) ∧ (a → b) ∈ D A . By Items e) and f) the latter is equivalent to a → a ∈ D A and a → b ∈ DA . By b) a → a ∈ DA , which completes the proof. 2 Theorem 8.5.3 For any algebra A = A, ∨, ∧, →, ∼ , the following holds: A ∈ VN4 ⇐⇒ A is an N4-lattice. Proof. If A is an N4-lattice, then by Proposition 8.4.3 A can be isomorphically embedded into B for a suitable implicative lattice B. Checking that the deﬁning identities of VN4 are satisﬁed in A is reduced in this way to a routine calculation with the help of twist-operations and the application of well-known identities of implicative lattices. In case of identity 6N , we will need also Proposition 8.4.2.a).

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We now take an algebra A ∈ VN4 and check successively Items 1–5 of Deﬁnition 8.4.1. 1. A is a De Morgan algebra by deﬁnition. 2. Let us check that the relation $ on A is reﬂexive and transitive. Note that for a, b ∈ A, a $ b if and only if a → b ∈ DA . By Lemma 8.5.2 the relation a $ a holds for any a ∈ A. Take elements a, b, c ∈ A such that a $ b and b $ c. We then have (a → b) ∧ (b → c) ∈ D A by Lemma 8.5.2.f). This fact and identity 6N imply then a $ c. In this way, $ is transitive. 3. Check ﬁrst the congruence properties of the equivalence ≈. Let a, b, c, d ∈ A, a ≈ b and c ≈ d. We have by Lemma 8.5.2.d) b → (b ∨ d) ∈ D A , moreover, a → b ∈ DA by assumption. By Lemma 8.5.2.f) and identity 6N we obtain a → (b∨d) ∈ DA . In a similar way, from c → d ∈ D A we infer c → (b ∨ d) ∈ DA . Further, applying Lemma 8.5.2.f) and identity 4N yields (a ∨ c) → (b ∨ d) = (a → (b ∨ d)) ∧ (c → (b ∨ d)) ∈ DA , i.e., a ∨ c $ b ∨ d. We omit the veriﬁcation of the inverse inequality. From the relations a → b, c → d ∈ D A one can conclude by 6N and Lemma 8.5.2.f) that the implications (a ∧ c) → b and (a ∧ c) → d are in DA . Now we need identity 3N to obtain (a ∧ c) → (b ∧ d) ∈ DA . Finally, consider the case of implication. We have (b → a) ∧ (a → c) $ (b → c) by identity 6N , whence, (b → a) → ((a → c) → (b → c)) ∈ DA by 2N . By assumption, b → a ∈ DA , from which we infer by Lemma 8.5.2.c) (a → c) $ (b → c). Similarly, the inequality (b → c) $ (b → d) can be obtained from (b → c)∧ (c → d) $ (b → d) and the assumption c → d ∈ DA . By transitivity of $ we have (a → c) $ (b → d). Analogously, one can check the inverse inequality. We now show that A := A, ∨, ∧, → / ≈ is an implicative lattice. A is a distributive lattice as a quotient of a distributive lattice. Therefore, it remains to prove that → is a relative pseudo-complement operation on A , i.e., that for any a, b, c ∈ A, the following condition is met: [c]≈ ≤ [a → b]≈ if and only if [a]≈ ∧ [c]≈ ≤ [b]≈ . We can replace it by the condition: c ≈ c ∧ (a → b) if and only if a ∧ c ∧ b ≈ a ∧ c,

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which is equivalent by Lemma 8.5.2.d) to: c $ c ∧ (a → b) if and only if a ∧ c $ a ∧ c ∧ b, and by Lemma 8.5.2.g) to: c $ (a → b) if and only if a ∧ c $ b. The latter immediately follows from the equality c → (a → b) = (a ∧ c) → b, which is true by 2N and the commutativity of ∧. 4. In view of Items b) and e) of Lemma 8.5.2 and 2N , identities 7N –9N can be rewritten as ∼ (p → q) $ p, ∼ (p → q) $∼ q, and p∧ ∼ q $∼ (p → q) respectively. These facts together mean that the identity ∼ (p → q) ≈ p∧ ∼ q holds on A . 5. Let a, b ∈ A. Assume a ≤ b, then the inequality ∼ b ≤∼ a follows by (8.1). By Lemma 8.5.2.d) we have a $ b and ∼ b $∼ a. We now assume that a $ b and ∼ b $∼ a and try to prove a ≤ b. The idea of this proof is taken from the work [89] by A. Pynko, where it was realized in another setting. But as we will see, it works also for VN4 . First we state some new identities in essentially the same way as was done in [89] for the class of implicative De Morgan lattices. Lemma 8.5.4 The following identities hold in VN4 : a) ∼ p →∼ p = p → p; b) ∼ (p → p) ≤ q → q; c) q → q ≤ (p → q) → (p → q). Proof. a) Take an algebra A ∈ VN4 and any a, b ∈ A. From 1N and 8N we obtain b → b ≤ (a → a) → (∼ b →∼ ((a → a) → b)) =∼ b →∼ b. Similarly, one can state the inverse inequality ∼ b →∼ b ≤ b → b. b) By Lemma 8.5.2 we have (a → a) ∧ (b → b) ∈ D A . At the same time, by 5N and a) for any c ∈ DA , holds ∼ c ≤∼ c →∼ c = c → c = c. Hence, ∼ (a → a)∨ ∼ (b → b) ≤ (a → a) ∧ (b → b). Finally, ∼ (a → a) ≤ b → b. c) We calculate (a → b) → (a → b) = (a → b) → ((a ∧ a) → b) = [by 2N ] = (a → b) → (a → (a → b)) = [by 2N ] (a ∧ (a → b)) → (a → b) = [by 2N ]

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= a → ((a → b) → (a → b)) = [by a)] = a → (∼ (a → b) →∼ (a → b)) = [by 5N ] = a → ((∼ b∧ ∼ (a → b)) →∼ (a → b)) = [by 2N ] =∼ (a → b) → (a → (∼ b →∼ (a → b))) ≥ [by 5N ] ≥ a → (∼ b →∼ (a → b)) ≥ [by 8N ] b → b. 2 Due to Item c) of the previous lemma conditions a $ b and ∼ b $∼ a imply b → b ≤ a → b and ∼ a →∼ a ≤∼ b →∼ a. These inequalities together with 10N and (8.1) yield a ∧ (b → b) ≤ a ∧ (a → b) ≤ b∨ ∼ (∼ b →∼ a) ≤ b∨ ∼ (∼ a →∼ a). The inequality ∼ a →∼ a ≤∼ b →∼ a, identity 11N and (8.1) allow us to obtain a ≤ (b → b)∨ ∼ (∼ a →∼ a). From this fact and item b) of Lemma 8.5.4, we infer a ≤ b → b. Substituting in 11N ∼ b for p and ∼ a for q yields ∼ b ≤∼ a →∼ a, and so ∼ (∼ a →∼ a) ≤ b by (8.1). Comparing this with the chain of inequalities above and taking into account a ≤ b → b we obtain a ≤ b. 2 This theorem allows one to state one more completeness result for N4. Theorem 8.5.5 For any formula ϕ, the following equivalence holds: N4 ϕ ⇐⇒ ϕ → ϕ = ϕ ∈ Eq(VN4 ). Proof. Due to Theorem 8.4.5 the fact N4 ϕ means that for any N4-lattice A and valuation v, v(ϕ) ∈ D A . By deﬁnition of DA the latter means exactly that for any A ∈ VN4 , the identity ϕ → ϕ = ϕ holds in A. 2 The latter characterization of the logic N4 is the most attractive, because it can be extended to any extension of N4, moreover, we can establish a dual isomorphism between the lattice EN4 and the lattice Sub(VN4 ) of all subvarieties of VN4 . For any L ∈ EN4, deﬁne a variety V ar(L) := {A | ϕ → ϕ = ϕ ∈ Eq(A) for all ϕ ∈ L}. Clearly, V ar(L) ∈ Sub(VN4 ). Moreover, the logic L is characterized by the variety V ar(L).

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Proposition 8.5.6 For any logic L ∈ EN4 and a formula ϕ, we have the equivalence: ϕ ∈ L ⇐⇒ ϕ → ϕ = ϕ ∈ Eq(V ar(L)). Proof. The direct implication is by deﬁnition of V ar(L). To state the inverse implication consider the canonical twist-structure L L for the logic L ∈ V ar(L). By deﬁnition (see Section 8.3). Prove that L L |L L | = {([ϕ], [∼ ψ]) | ψ ∈ [ϕ]}, where [ϕ] is a coset of ϕ wrt the equivalence ≡L . Taking into account that [ϕ] = [ψ] for ψ ∈ [ϕ] we obtain |L L | = {([ϕ], [∼ ϕ]) | ϕ ∈ F or}. Let ϕ = ϕ(p1 , . . . , pn ) and an L L -valuation v be such that v(pi ) = ([ψi ], [∼ ψi ]), i = 1, . . . , n. Using induction on the structure of formulas, we state that v(ϕ) = ([ϕ(ψ1 , . . . , ψn )], [∼ ϕ(ψ1 , . . . , ψn )]). In this way, if ϕ ∈ L, then for any L L -valuation v we have π1 v(ϕ) = 1, i.e., L L v(ϕ) ∈ D . Consequently, ϕ → ϕ = ϕ ∈ Eq(L L ) and LL ∈ V ar(L). Take a formula ϕ ∈ L and LL -valuation v such that v(p) := ([p], [∼ p]). Then v(ϕ) = ([ϕ], [∼ ϕ]).

By assumption L ϕ, consequently, [ϕ] = 1. This means that v(ϕ) ∈ DLL , i.e., ϕ → ϕ = ϕ ∈ Eq(L L ). 2 Corollary 8.5.7

1. If L1 , L2 ∈ EN4, then L1 ⊆ L2 ⇐⇒ V ar(L2 ) ⊆ V ar(L1 ).

2. The mapping V ar : EN4 → Sub(VN4 ) is one-to-one. Proof. 1. The direct implication immediately follows from the deﬁnition. Assume that Eq(V ar(L1 )) ⊆ Eq(V ar(L2 )) and take ϕ ∈ L1 . Then ϕ → ϕ = ϕ ∈ Eq(V ar(L2 )). By the previous proposition we have ϕ ∈ L2 . 2. This item follows from the previous one. 2 Now for any V ∈ Sub(VN4 ), deﬁne a set of formulas L(V ) := {ϕ | ϕ → ϕ = ϕ ∈ Eq(V )}.

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Proposition 8.5.8 1. For any V ∈ Sub(VN4 ), the set L(V ) is a logic and L(V ) ∈ EN4. 2. The mappings V ar : EN4 → Sub(VN4 ) and L : Sub(VN4 ) → EN4 are mutually inverse. Proof. 1. It is clear that N4 ⊆ L(V ) and that the set L(V ) is closed under substitution. Assume ϕ, ϕ → ψ ∈ L(V ) and take an arbitrary N4-lattice A ∈ V and a valuation v into A. By assumption v(ϕ → ϕ) = v(ϕ) and v((ϕ → ψ) → (ϕ → ψ)) = v(ϕ → ψ), i.e., v(ϕ), v(ϕ) → v(ψ) ∈ DA . Hence, due to Lemma 8.5.2 we have v(ψ) ∈ D A . The latter means that ψ → ψ = ψ ∈ Eq(V ). Thus, L(V ) is closed under modus ponens and L(V ) ∈ EN4. 2. It follows from Proposition 8.5.6 that L(V ar(L)) = L for any L ∈ EN4. Let us compare the varieties V and V ar(L(V )). It follows from the deﬁnition that V ar(L(V )) ⊆ V . To state the inverse inclusion we need the following lemma. Lemma 8.5.9 For any N4-lattice A and formulas ϕ and ψ, we have ϕ = ψ ∈ Eq(A) if and only if (ϕ ⇔ ψ) → (ϕ ⇔ ψ) = ϕ ⇔ ψ ∈ Eq(A). Proof. Without loss of generality we may assume that A is a twist-structure over a suitable implicative lattice B. It is enough to check the equivalence: ϕ = ψ ∈ Eq(A) if and only if v(ϕ ⇔ ψ) ∈ DA for any A-valuation v. Due to Lemma 8.4.4 the condition v(ϕ ⇔ ψ) ∈ D A is equivalent to π1 v(ϕ ⇔ ψ) = 1. Let v be an arbitrary A-valuation. Assume v(ϕ) = (a, b) and v(ψ) = (c, d) for some a, b, c, d ∈ B. The direct computation shows that π1 v(ϕ ⇔ ψ) = (a ↔ c) ∧ (b ↔ d). It is obvious that (a ↔ c) ∧ (b ↔ d) = 1 if and only if a = c and b = d, i.e., v(ϕ) = v(ψ). 2 For brevity we denote the identity ϕ → ϕ = ϕ as ϕN . We now turn to the inclusion V ⊆ V ar(L(V )). Let ϕ = ψ ∈ Eq(V ). By the previous lemma we have (ϕ ⇔ ψ)N ∈ Eq(V ), then ϕ ⇔ ψ ∈ L(V ) and (ϕ ⇔ ψ)N ∈ Eq(V ar(L(V ))). Again by Lemma 8.5.9 ϕ = ψ ∈ Eq(V ar(L(V ))). 2 Theorem 8.5.10 The mapping V ar is a dual lattice isomorphism between EN4 and Sub(VN4 ).

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155

Proof. By Item 2 of Proposition 8.5.8 the mapping V ar is a bijection. Therefore, the fact that V ar is a dual lattice isomorphism follows from Corollary 8.5.7.

8.6

The Logic N4⊥ and N4⊥ -Lattices

First, we consider when twist-structures over implicative lattices are bounded. Lemma 8.6.1 Let A be an implicative lattice and B ∈ S (A). 1. If A has the least element 0, then (0, 1), (1, 0) ∈ B and B is a bounded lattice with zero element (0, 1) and unit element (1, 0). 2. If B is bounded, then A has the least element 0, zero element of B is (0, 1) and unit element is (1, 0). Proof. 1. By deﬁnition of twist-structures there are a, b ∈ A such that (0, a), (1, b) ∈ B, then (0, 1) = (0, a) ∧ (b, 1) ∈ B and (1, 0) =∼ (0, 1) ∈ B. That (0, 1) is the least and (1, 0) the greatest element of B follows from the deﬁnition of twist-operations. 2. Assume that (a, b) is the least element of B and c ∈ A. There is d ∈ A such that (c, d) ∈ B. Then (a, b) ≤ (c, d), whence a ≤ c. Thus, a is the least element of A. 2 This allows us to deﬁne twist-structures over Heyting algebras as follows. Definition 8.6.2 Let A = A, ∨, ∧, →, 0, 1 be a Heyting algebra. 1. A full twist-structure over A is an algebra A = A × A, ∨, ∧, →, ∼, ⊥ such that A × A, ∨, ∧, →, ∼ is the full twist-structure over the implicative lattice A, ∨, ∧, →, 1 and ⊥ = (0, 1). 2. A twist-structure over A is an arbitrary subalgebra B of the full twiststructure A such that π1 (B) = A . 3. The class of all twist-structures over A is denoted S (A).

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8 Semantical Study of Paraconsistent Nelson’s Logic

Definition 8.6.3 An algebra A = A, ∨, ∧, →, ∼, ⊥ is an N4⊥ -lattice if

A, ∨, ∧, →, ∼ is an N4-lattice and ⊥ is the least element of A. In this case A := A, ∨, ∧, →, ⊥ / ≈ is a Heyting algebra, which we call a basic Heyting algebra of A. We can easily prove the equivalence of the last two deﬁnitions. Proposition 8.6.4 Every twist-structure B ∈ S (A) over a Heyting algebra A is an N4⊥ -lattice. Every N4⊥ -lattice A is isomorphic to a twiststructure over A and an isomorphism is given by the rule a → ([a]≈ , [∼ a]≈ ). 2 The mapping i : A → (A ) , i (a) = ([a]≈ , [∼ a]≈ ), is called a canonical embedding of an N4⊥ -lattice into the full twist-structure over the basic Heyting algebra. The validity of formulas in twist-structures over Heyting algebras and in N4⊥ -lattices is deﬁned in the same way as for twist-structures over implicative lattices and for N4-lattices. Note that although an N4⊥ -lattice A has the greatest element, its set of distinguished values is D A = {a → a | a ∈ A} and it is not one-element in the general case. To prove that N4⊥ is complete with respect to the class of twist-structures over Heyting algebras we have only to note that LN4⊥ is a Heyting algebra due to axiom A14. Consequently, the Lindenbaum twist-structure L N4 is a twist-structure over a Heyting algebra. Theorem 8.6.5 For any formula ϕ, N4⊥ ϕ if and only if A |= ϕ for any N4⊥ -lattice A. 2 Taking into account that the action of twist-operations on the ﬁrst component agreed with the operations of the basic Heyting algebra, we easily obtain Corollary 8.6.6 N4⊥ is a conservative extension of intuitionistic logic. 2 Since N4⊥ -lattices are exactly bounded N4-lattices, they also form a variety determined by the identities of de Morgan algebras, the set of identities 1N –11N and the new identity: 12N . ⊥ ≤ p

8.6 The Logic N4⊥ and N4⊥ -Lattices

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The mappings V ar : EN4⊥ → Sub(VN4⊥ ) and L : Sub(VN4⊥ ) → EN4⊥ are deﬁned exactly as for the logic N4 and the variety of N4-lattices. Repeating the reasoning from the previous section we prove Theorem 8.6.7 The mappings V ar and L are mutually inverse dual lattice isomorphisms between EN4⊥ and Sub(VN4⊥ ). 2

Chapter 9

N4⊥-Lattices1 In the previous chapter, we considered two variants of Nelson’s paraconsistent logic, N4 and N4⊥ , which were deﬁned in diﬀerent languages. N4⊥ is a conservative extension of N4 in the language with the additional constant ⊥ allowing us to deﬁne in this logic the intuitionistic negation. The addition of this constant results in an easy modiﬁcation of the semantics. Models of N4 are isomorphic to twist-structures over implicative lattices and due to the fact that implicative lattices do not necessarily have the least element and the lattices modelling N4 do not have the greatest element either. In fact, no constant can be naturally deﬁned in N4. Extending the language with ⊥ we obtain the class of models isomorphic to twist-structures over Heyting algebras, i.e., implicative lattices with the least element 0. A twist-structure over bounded lattice is also bounded, it contains necessarily elements (0, 1) and (1, 0), which are the least and the greatest elements. Thus, the semantics for N4⊥ is given by the class of bounded lattices. It turns out that the introduction of ⊥ has essential consequences for the class of extensions EN4⊥ . As we will see in Chapter 10, adding the constant ⊥ enriches the class of N4⊥ -extensions as compared to EN4, and, which is more important, provides it with a regular structure close to some extent to the structure of the class of extensions of minimal logic studied in the ﬁrst part of the book. This is why we will work mainly with the logic N4⊥ and with N4⊥ -lattices. However, all results in this chapter remains true if we replace “N4⊥ -lattice” with “N4-lattice” and “Heyting algebra” with “implicative lattice”. Our investigation of N4⊥ -lattices is based essentially on the representation of N -lattices obtained by A. Sendlewski [100]. N -lattices are exactly 1

Parts of this chapter were originally published in [74] (Springer, Netherlands). Reprinted here by permission of the publisher.

159

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9 N4⊥ -Lattices

N4⊥ -lattices modelling N3. It is well known (see, [28, 110, 47]) that N -lattices can be represented as twist-structures over Heyting algebras satisfying the condition a∧b = 0 for any (a,b). We call such twist-structures explosive. D. Vakarelov [110] suggested the following intuitive interpretation of an explosive twist-structure: the underlying Heyting algebra A represents a set of assertions of an intuitionistic theory, i.e., it is a Lindenbaum algebra of this theory. Let a and b be assertions. We say that b is a counter-example of a if a∧b is contradictory. For intuitionistic theory, the latter means exactly that a∧b = 0. Thus, an explosive twist-structure is a set of statements of an intuitionistic theory together with their counter-examples. The twist-operations determine how to construct a counter-example of a complex statement from counter-examples of components. For instance, a counter-example of a conjunction of two assertions is a disjunction of counter-examples of these assertions, (a, b) ∧ (c, d) = (a ∧ b, c ∨ d). Any statement can be considered as a counter-example of its counter-example, therefore, ∼ (a, b) = (b, a). Of course, any assertion has a counter-example, therefore, for any twiststructure B over A, π1 (B) = A. To represent arbitrary N4⊥ -lattices we need arbitrary, not only explosive twist-structures over Heyting algebras. Therefore, the condition a ∧ b = 0 should be omitted and the Vakarelov interpretation should be modiﬁed. One of the possible solutions to this problem is as follows: we consider the notion of a counter-example as a primary notion, independent of the notion of contradiction in the underlying intuitionistic theory. As above, we consider a twist-structure B over A as some set of pairs (a, b) consisting of an assertion a ∈ A and its counter-example b. Of course, this set should be closed with respect to twist-operations. The contradictory assertion is now deﬁned as an assertion of the form a ∧ b for (a, b) ∈ B, i.e., as a conjunction of an assertion and its counter-example. A. Sendlewski [100] proved that an explosive twist-structure B over a Heyting algebra A is completely determined by a ﬁlter on A consisting of all elements of the form a ∨ b, where (a, b) ∈ B. On the other hand, for any ﬁlter ∇ over A containing the ﬁlter of dense elements, there exists an explosive twist-structure B over A such that ∇ = {a ∨ b | (a, b) ∈ B}. In what follows, we call ∇ a ﬁlter of completions of the twist-structure B. In this chapter, we generalize Sendlewski’s characterization to arbitrary twist-structures over Heyting algebras. An explosive twist-structure is characterized by conditions on the join and meet of an assertion and its counterexample, i.e., on the join and meet of the ﬁrst and second components of its elements. The join belongs to some ﬁxed ﬁlter of the basic Heyting algebra and the meet is contradictory. The second condition means the same for

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any explosive twist-structure, namely, to be the least element of the basic Heyting algebra. But in case of arbitrary twist-structures, this condition may be varied. This leads to the hypothesis that a twist-structure B over a Heyting algebra A is determined by this algebra, by the set of possible joins of assertions and their counter-examples, which plausibly forms a ﬁlter on A as well as in Sendlewski’s characterization, and by the set of contradictions, i.e., elements of A of the form a ∧ b for (a, b) ∈ B. It turns out that this set forms an ideal on the basic Heyting algebra, which we call an ideal of contradictions. One may assume that there is also some interrelation between the ideal of contradictions and the ﬁlter of completions but as we will see below, it is not the case. We prove the main result of the chapter on the representation of twiststructures in Section 9.1. Moreover, we formulate and prove the embeddability criterium for N4⊥ -lattices in terms of the above representation. In Section 9.2, we deal with homomorphisms of N4⊥ -lattices. In particular, we deﬁne special ﬁlters of the ﬁrst kind (sﬀk) on N4⊥ -lattices and prove that sﬀk are exactly kernels of homomorphisms of N4⊥ -latices. Moreover, it is proved that sﬀk on an N4⊥ -lattice A are in a one-to-one correspondence with ﬁlters on its basic Heyting algebra, which implies the isomorphism of congruence lattices Con(A) ∼ = Con(A ). The latter fact allows one to obtain a characterization of subdirectly irreducible N4⊥ -lattices. Namely, an N4⊥ -lattice is subdirectly irreducible if and only if its basic Heyting algebra is subdirectly irreducible. A similar result on subdirectly irreducible N -lattices was proved in [99] and independently in [17], the proof was based on the Priestley duality. The results of this and the previous chapters constitute the necessary semantic basis for the investigation of the lattice of logics EN4⊥ .

9.1

Structure of N4⊥ -Lattices

In this section, we obtain a characterization of arbitrary twist-structures similar to that of A. Sendlewski [100]. First, we note that the condition a ∧ b = 0 can be replaced by the condition a ∧ b ∈ Δ, where Δ is an arbitrary ideal of the basic Heyting algebra of a twist-structure. Recall that an ideal on a Heyting algebra (implicative lattice) A is a non-empty subset X ⊆ A such that: 1) if a, b ∈ X, then a ∨ b ∈ X; 2) if a ∈ X and b ≤ a, then b ∈ X. The set of all ideals on A is denoted by I(A).

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Proposition 9.1.1 Let A be a Heyting algebra, ∇ be a ﬁlter on A such that ∇ ⊇ Fd (A), and let Δ be an ideal on A. Then there exists a twist-structure T w(A, ∇, Δ) ∈ S (A) with the universe |T w(A, ∇, Δ)| = {(a, b) | a, b ∈ |A|, a ∨ b ∈ ∇, a ∧ b ∈ Δ}. Proof. Check that the set T := |T w(A, ∇, Δ)| is closed under twist operations. Let (a, b), (c, d) ∈ |A| × |A| be such that a ∨ b, c ∨ d ∈ ∇ and a ∧ b, c ∧ d ∈ Δ. For the disjunction (a, b) ∨ (c, d) = (a ∨ c, b ∧ d), we have (a ∨ c) ∨ (b ∧ d) = (a ∨ c ∨ b) ∧ (a ∨ c ∨ d). Both conjunction terms in the latter expression belong to ∇, and so (a ∨ c) ∨ (b ∧ d) ∈ ∇. In a similar way, we have (a ∨ c) ∧ (b ∧ d) = (a ∧ b ∧ d) ∨ (c ∧ b ∧ d) ∈ Δ. Thus, (a, b) ∨ (c, d) ∈ T . The case of conjunction can be considered similarly. Let us consider the implication (a, b) → (c, d) = (a → c, a ∧ d). We have (a → c) ∨ (a ∧ d) = ((a → c) ∨ a) ∧ ((a → c) ∨ d). The element (a → c) ∨ a belongs to ∇, because it is dense. At the same time, c ∨ d ≤ (a → c) ∨ d ∈ ∇. In this way, (a → c) ∨ (a ∧ d) ∈ ∇. Further, (a → c) ∧ (a ∧ d) = a ∧ c ∧ d. This element belongs to Δ by the condition c ∧ d ∈ Δ. We have proved (a, b) → (c, d) ∈ T . It is obvious that ∼ (a, b) = (b, a) ∈ T . 2 It turns out that any twist-structure can be represented in the form described in the last proposition. Proposition 9.1.2 Let A be a Heyting algebra and B ∈ S (A). We deﬁne I(B) := {a∨ ∼ a | a ∈ B}, ∇(B) := π1 (I(B)), Δ(B) := π2 (I(B)). Then Fd (A) ⊆ ∇(B), ∇(B) is a ﬁlter on A, and Δ(B) is an ideal on A. Moreover, B = T w(A, ∇(B), Δ(B)). Proof. First, we obtain a more convenient deﬁnition for I(B) and a representation of B via I(B). Lemma 9.1.3 Let A be a Heyting algebra and B ∈ S (A). Then |B| = {(a, b) | a, b ∈ A, (a ∨ b, a ∧ b) ∈ I(B)}, I(B) = {(a ∨ b, a ∧ b) | (a, b) ∈ B} = {(a, b) | (a, b) ∈ B, a ≥ b}.

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Proof. Let us check the ﬁrst equation. If (a, b) ∈ B, then (a ∨ b, a ∧ b) = (a, b) ∨ (b, a) = (a, b)∨ ∼ (a, b) ∈ I(B). To check the inverse inclusion take an element (a, b) ∈ |A| × |A| such that (a ∨ b, a ∧ b) ∈ I(B) and prove that (a, b) ∈ B. By Deﬁnition 8.6.2 there is an element c ∈ A such that (a, c) ∈ B. We have I(B) ⊆ B by deﬁnition, therefore, (a ∧ b, a ∨ b) =∼ (a ∨ b, a ∧ b) ∈ B. Further, (a, c) → (a ∧ b, a ∨ b) = (a → (a ∧ b), a ∧ (a ∨ b)) = (a → b, a) ∈ B. And ﬁnally, we have (a ∨ b, a ∧ b) ∧ (a → b, a) = ((a ∧ (a → b)) ∨ (b ∧ (a → b)), a) = = ((a ∧ b) ∨ b, a) = (b, a). Thus, the elements (b, a) and (a, b) =∼ (b, a) are in B. The representation for I(B) immediately follows from the facts that for any (a, b) ∈ B, (a, b)∨ ∼ (a, b) = (a ∨ b, a ∧ b) , and (a, b)∨ ∼ (a, b) = (a, b) whenever a ≥ b. 2 This allows us to conclude that a twist-structure B is completely determined by the set I(B). Corollary 9.1.4 Let B1 and B2 be twist-structures over a Heyting algebra A. Assume I(B1 ) = I(B2 ). Then B1 = B2 . 2 We now turn to the proof of the proposition and show that Δ(B) is an ideal of A. We need one more lemma. Lemma 9.1.5 Δ(B) = {a ∈ A | (1, a) ∈ B} = π2 (DB ). Proof. If (1, a) ∈ B, then (1, a)∨ ∼ (1, a) = (1, a) ∨ (a, 1) = (1, a), i.e., (1, a) ∈ I(B) and a ∈ Δ(B). Conversely, let a ∈ Δ(B). This means that there is a d = (b, c) ∈ B such that π2 (d∨ ∼ d) = a, i.e., a = b ∧ c. Further, (b, c) → (b, c) = (1, b ∧ c) = (1, a) ∈ B. 2 Let a, b ∈ Δ(B). In view of the previous lemma (1, a), (1, b) ∈ B, whence (1, a) ∧ (1, b) = (1, a ∨ b) ∈ B, and so a ∨ b ∈ Δ(B). Now assume that a ∈ Δ(B) and b ∈ A, b ≤ a. Let c ∈ A be such that (c, b) ∈ B. We have (1, a) ∨ (c, b) = (1, a ∧ b) = (1, b) ∈ B, which means that b ∈ Δ(B).

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We have thus proved that Δ(B) is an ideal. Let us check that ∇(B) is a ﬁlter on A and Fd (A) ⊆ ∇(B). Let a, b ∈ ∇(B) and elements c, d ∈ Δ(B) are such that (a, c), (b, d) ∈ I(B). In view of Lemma 9.1.3 we have c ≤ a, d ≤ b. The fact that Δ(B) is an ideal and Lemma 9.1.5 imply (1, c ∧ d) ∈ I(B), whence the element (a, c) → (c ∧ d, 1) = (a → (c ∧ d), a) is in B. We have also (a → (c ∧ d), a) ∧ (a, c) = (a ∧ c ∧ d, a ∨ c) = (c ∧ d, a) ∈ B. From this fact and Lemma 9.1.3 we have (a, c ∧ d) ∈ I(B). Similarly, (b, c ∧ d) ∈ I(B). Further, (a, c∧d)∧(b, c∧d) = (a∧b, c∧d) ∈ I(B), from which we conclude a∧b ∈ ∇(B). Assume a ∈ ∇(B), b ∈ A, and a ≤ b. In this case, there exist elements c, d ∈ A such that a ≥ c, (a, c) ∈ I(B), and (b, d) ∈ B. We have then (a∨b, c∧d) = (b, c∧d) ∈ B. Moreover, c∧d ≤ a ≤ b, whence (b, c∧d) ∈ I(B), i.e., b ∈ ∇(B). Thus, ∇(B) is a ﬁlter. Let a, b ∈ A be such that (a, b) ∈ B. Then (a, b) ∨ ((a, b) → (0, 1)) = (a ∨ (a → 0), a ∧ b) ∈ B. We have a ∧ b ≤ a ≤ a ∨ (a → 0), consequently, a ∨ (a → 0) ∈ ∇(B) and we conclude that Fd (A) ⊆ ∇(B). It remains to prove the equality B = T w(A, ∇(B), Δ(B)). As it follows directly from deﬁnitions (a, b) ∈ T w(A, ∇(B), Δ(B)) if and only if (a ∨ b, a ∧ b) ∈ I(B). In view of Lemma 9.1.3, the latter fact means that (a, b) ∈ B. 2 ⊥ This proposition can be generalized to arbitrary N4 -lattices as follows. Corollary 9.1.6 Let A = A, ∨, ∧, →, ∼, ⊥ be an N4⊥ -lattice. We deﬁne ∇l (A) := e ({a∨ ∼ a | a ∈ A}), Δl (A) := e ({a∧ ∼ a | a ∈ A}), where e : A, ∨, ∧, →, ⊥ → A is a canonical epimorphism. Then ∇l (A) is a ﬁlter on A , Fd (A ) ⊆ ∇l (A), and Δl (A) is an ideal on A . Moreover, we have i (A) = T w(A , ∇l (A), Δl (A)), where i : A → (A ) is a canonical embedding, i.e., A∼ = T w(A , ∇l (A), Δl (A)). Proof. Let us consider a twist-structure A := i (A). Clearly, A ∈ S (A ) and it has the form A = T w(A , ∇(A ), Δ(A )) by the previous proposition. In this way, the conclusion of the corollary will follow from the equations ∇l (A) = ∇(A ) and Δl (A) = Δ(A ), which we prove now.

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By Proposition 8.4.3 each element b ∈ A is of the form b = (e (a), e (∼ a)) for some a ∈ A. Thus, b∨ ∼ b = (e (a), e (∼ a)) ∨ (e (∼ a), e (a)) = (e (a∨ ∼ a), e (a∧ ∼ a)). We have in this way the equalities: ∇(A ) = π1 (I(A )) = {e (a∨ ∼ a) | a ∈ A} = ∇l (A), Δ(A ) = π2 (I(A )) = {e (a∧ ∼ a) | a ∈ A} = Δl (A). 2 Remark. If B ∈ S (A) and h : B → A is an isomorphism given by the rule h([(a, b)]≈ ) = a (see Proposition 8.4.2), then h(∇l (B)) = ∇(B) and h(Δl (B)) = Δ(B). For this reason we omit later on the lower index “−l ” in the denotation ∇l (A) and Δl (A). As mentioned, all results in this chapter also hold true for N4-lattices. To formulate the above results for twist-structures over implicative lattices, we have to deﬁne a ﬁlter of dense elements for an arbitrary implicative lattice. Let A be an implicative lattice. We deﬁne Fd (A) := {a ∨ (a → b) | a, b ∈ A} and call Fd (A) a ﬁlter of dense elements of A. If A has the least element, the ﬁlter Fd (A) will coincide with the ﬁlter of dense elements of A considered as a Heyting algebra. This explains why we use the same name and denotation. It turns out that any element of Fd (A) has the form a ∨ (a → b). Proposition 9.1.7 For any implicative lattice A, Fd (A) = {a ∨ (a → b) | a, b ∈ A} an , b1 , . . . ,bn ∈ A Proof. Let a ∈ F d (A). Then one can ﬁnd elements a1 , . . . , such that a ≥ ni=1 ai ∨ (ai → bi ). For the element d = ni=1 ai ∧ ni=1 bi , consider the implicative lattice Ad , a sublattice of A with the universe {a ∈ A | a ≥ d}. It is clear that a ∈ Fd (Ad ) ⊆ Fd (A). On the other hand, Ad has the least element and the ﬁlter Fd (Ad ) coincides with the ﬁlter of dense elements of the Heyting algebra Ad . Therefore, for some c ∈ Ad , a = c ∨ ¬c = c ∨ (c → d). We have thus obtained the desired representation for an arbitrary element of Fd (A). 2 Analogs of Propositions 9.1.1 and 9.1.2 can be proved for twist-structures over implicative lattices in essentially the same way.

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Proposition 9.1.8 Let A be an implicative lattice, ∇ be a ﬁlter on A such that ∇ ⊇ Fd (A), and let Δ be an ideal on A. Then there exists a twiststructure T w(A, ∇, Δ) ∈ S (A) with the universe |T w(A, ∇, Δ)| = {(a, b) | a, b ∈ |A|, a ∨ b ∈ ∇, a ∧ b ∈ Δ}. 2 Proposition 9.1.9 Let A be an implicative lattice and B ∈ S (A). We deﬁne I(B) := {a∨ ∼ a | a ∈ B}, ∇(B) := π1 (I(B)), Δ(B) := π2 (I(B)). Then Fd (A) ⊆ ∇(B), ∇(B) is a ﬁlter on A, and Δ(B) is an ideal on A. Moreover, B = T w(A, ∇(B), Δ(B)). An analog of Corollary 9.1.6 also holds. In the following we will formulate results only for twist-structures over Heyting algebras and for N4⊥ -lattices. At the end of this section, we make some remarks on subalgebras of twist-structures. Proposition 9.1.10 1. Let B ∈ S (A). Then B = A if and only if ∇(B) = Δ(B) = A. 2. Let A0 and A1 be Heyting algebras, ∇i ∈ F(Ai ), Fd (Ai ) ⊆ ∇i , and Δi ∈ I(Ai ), i = 0, 1. Moreover, let A0 ≤ A1 ,∇0 ⊆ ∇1 , and Δ0 ⊆ Δ1 . Then T w(A0 , ∇0 , Δ0 ) ≤ T w(A1 , ∇1 , Δ1 ). 3. Let B ∈ S (A) and C ≤ B. Then C = T w(A1 , ∇, Δ), where A1 ≤ A, ∇ ∈ F(A1 ) and ∇ ⊆ ∇(B), Δ ∈ I(A1 ) and Δ ⊆ Δ(B). 4. Let B ∈ S (A) and A1 ≤ A. There exists a subalgebra C of B such that C ∈ S (A1 ). 5. Let B ∈ S (A) and A1 ≤ A. The greatest subalgebra C of B satisfying the condition C ∈ S (A1 ) has the following form C = T w(A1 , ∇(B) ∩ A1 , Δ(B) ∩ A1 ).

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167

Proof. 1. Let B = A , i.e., |B| = |A| × |A|. Then I(B) = {(a, b) | a ≥ b, a, b ∈ A}. This fact and the deﬁnition of ∇(B) and Δ(B) as projections of I(B) onto the ﬁrst and the second coordinates, respectively, imply ∇(B) = Δ(B) = A. If ∇(B) = Δ(B) = A, then |B| = {(a, b) | a ∨ b ∈ ∇(B), a ∧ b ∈ Δ(B)} = {(a, b) | a, b ∈ A}. 2. This item follows immediately from the deﬁnition of twist-structures T w(Ai , ∇i , Δi ), i = 0, 1. 3. The operations ∨, ∧, and → are componentwise with respect to the ﬁrst coordinate, therefore, A1 := π1 (C) = π2 (C) is a subalgebra of A. In this case C ∈ S (A1 ). Obviously, I(C) = {a∨ ∼ a | a ∈ C} ⊆ I(B), from which we have ∇(C) = π1 (I(C)) ⊆ π1 (I(B)) = ∇(B). Similarly, Δ(C) ⊆ Δ(B). 4. Note that Fd (A1 ) ⊆ Fd (A), therefore, Fd (A1 ) ⊆ ∇(B) ∩ A1 . Moreover, 0 ∈ Δ(B) ∩ A1 = ∅. Since ∇(B) ∩ A1 is obviously a ﬁlter and Δ(B) ∩ A1 an ideal on C, we can consider a twist-structure C = T w(A1 , ∇(B) ∩ A1 , Δ(B) ∩ A1 ). That C ≤ B follows from Item 2. 5. As was noted above, C is a subalgebra of B. By Item 2 it is the greatest subalgebra with the condition C ∈ S (A1 ). 2

9.2

Homomorphisms and Subdirectly Irreducible N4⊥ -Lattices

In Deﬁnition 8.6.2 for any Heyting algebra A, there was deﬁned the N4⊥ lattice A , the full twist-structure over A. At the same time, for any N4⊥ lattice A, there was deﬁned the Heyting algebra A . We extend these mappings to homomorphisms of Heyting algebras and of N4⊥ -lattices. For any Heyting algebra homomorphism h : A → B, the homomorphism of N4⊥ lattices h : A → B is deﬁned by the rule: h (a, b) := (h(a), h(b)). That this is really a homomorphism follows immediately from the deﬁnition of twist-structures. Now let A and B be N4⊥ -lattices. For a homomorphism h : A → B, we deﬁne a mapping h : A → B by the rule h ([a]≈ ) := [h(a)]≈ . The facts that h is a homomorphism and that the relation ≈ is deﬁnable via basic operations of an N4⊥ -lattice guarantee that the above deﬁnition is correct. Due to the congruence properties of ≈, h is a homomorphism of Heyting

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algebras. It turns out that we have deﬁned a functor from the category H of Heyting algebras and their homomorphisms to the category N 4⊥ of N4⊥ -lattices and their homomorphisms and the adjoint functor from N 4⊥ to H. Theorem 9.2.1 2 The mapping (·) is a covariant functor from the category H to the category N 4⊥ . The mapping (·) is a covariant functor from the category N 4⊥ to the category H. The functor (·) is left adjoint to (·) . Proof. One can directly verify that (·) and (·) are covariant functors. We prove only that (·) is left adjoint to (·) . To this end, it will be enough to ⊥ show (see e.g. [12]) that the family of canonical embeddings i B, B ∈ N4 , determines a natural transformation of an identical functor 1N 4⊥ to the composition ((·) ) such that for any B ∈ N 4⊥ , the pair (B , i B ) is a reﬂection of B along (·) . The latter means that for any Heyting algebra A and homomorphism f : B → A there exists a unique homomorphism g : B → A such that f = g ◦ i B. First, take a homomorphism f : B → C of N4⊥ -lattice and check the equality i C ◦ f = (f ) ◦ iC . For any b ∈ B, we have i C ◦ f (b) = ([f (b)]≈C , [∼ f (b)]≈C ) = ([f (b)]≈C , [f (∼ b)]≈C ) = = (f ([b]≈B ), f ([∼ b]≈B )) = (f ) ([b]≈B , [∼ b]≈B ) = (f ) ◦ i C (b). ⊥ We have thus established that the family i B , B ∈ N 4 , determines the desired natural transformation. Further, for a Heyting algebra A and a homomorphism f : B → A we deﬁne a mapping g : B → A by the rule g([a]≈B ) := π1 f (a) for all a ∈ B. This deﬁnition is correct due to the following observation. If a, b ∈ B and a ≈B b, then f (a) ≈A f (b), from which we obtain π1 f (a) = π1 f (b) by Proposition 8.4.2. That g is a Heyting algebra homomorphism follows from the congruence properties of ≈B and the deﬁnition of positive operations on A . For b ∈ B, we calculate

g ◦ i B (b) = (g([b]≈B ), g([∼ b]≈B )) = (π1 f (b), π1 f (∼ b)) = = (π1 f (b), π1 (∼ f (b))) = (π1 f (b), π2 f (b)) = f (b). 2 This is the unique place where we use the special notions from the category theory. For this reason, we do not give all the necessary definitions and ask the reader to consult, e.g., [12]. Note that categorical properties of N3-lattices were studied by A. Sendlewski [100] and R. Cignoli [17].

9.2 Homomorphisms and Subdirectly Irreducible N4⊥ -Lattices

169

On the other hand, for a homomorphism h : B → A, the equality f = h ◦ i B immediately implies h([a]≈B ) := π1 f (a) for a ∈ B. Thus, g is a unique homomorphism with the desired property. 2 For Heyting algebras and N -lattices, interrelations of this kind were proved by V. Goranko [33] and A. Sendlewski [99, 100]. We note also that every homomorphism h of N4⊥ -lattices is uniquely determined by the homomorphism h of basic Heyting algebras. Proposition 9.2.2 Let A and B be N4⊥ -lattices, and let hi : A → B, i = 1, 2, be homomorphisms such that h1 = h2 . Then h1 = h2 . Proof. Let gi := (hi ) , i = 1, 2. Since h1 = h2 , we have g1 = g2 . For any a, b ∈ A, gi ([a]≈A , [b]≈A ) = (hi ([a]≈A ), hi ([a]≈A )) = ([hi (a)]≈B , [hi (b)]≈B ). Further, for a ∈ A, we have −1 i i −1 i i hi (a) = (i B ) ([h (a)]≈B , [∼ h (a)]≈B ) = (iB ) ([h (a)]≈B , [h (∼ a)]≈B ).

In view of the above calculation, we obtain −1 ◦ gi ([a]≈A , [∼ a]≈A ). hi (a) = (i B)

In this way, the equality h1 = h2 follows from g1 = g2 .

2 We now introduce the notion of a special ﬁlter of the ﬁrst kind on an N4⊥ -lattice in exactly the same way as a special ﬁlter of the ﬁrst kind on an N -lattice was deﬁned by H. Rasiowa [92]. Let A be an N4⊥ -lattice. A non-empty subset ∅ = ∇ ⊆ A is called a special ﬁlter of the ﬁrst kind (sﬀk) on A if: 1) a ∈ ∇ and b ∈ ∇ imply a ∧ b ∈ ∇; 2) a ∈ ∇ and a $ b imply b ∈ ∇. It is obvious that the set of all sﬀk on A forms a lattice, which we denote F 1 (A). For a homomorphism h : A → B of N4⊥ -lattices we deﬁne a kernel Ker(h) := h−1 (DB ). As we can see from the following proposition, sﬀk are in one-to-one correspondence with congruences on an N4⊥ -lattice and the least sﬀk on A coincides with D A . Proposition 9.2.3

1. For any N4⊥ -lattice A, DA is an sﬀk.

2. Let h : A → B be an epimorphism of N4⊥ -lattices. Then Ker(h) is an sﬀk. For any a, b ∈ A, h(a) = h(b) if and only if a ⇔ b ∈ Ker(h).

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3. Let A be an N4⊥ -lattice and ∇ an sﬀk on A. Then DA ⊆ ∇. The relation ≈∇ , a ≈∇ b iﬀ a ⇔ b ∈ ∇, is a congruence relation on A and ∇ = Ker(h), where h : A → A/≈∇ is a canonical epimorphism. Proof. 1. This item follows from the lemma below. Lemma 9.2.4 Let A be an N4⊥ -lattice. Then the following holds: 1. If a, b ∈ D A , then a ∧ b ∈ DA . 2. If a ∈ A and b ∈ D A , then a $ b; 3. For any a, b ∈ A, a = b if and only if a ⇔ b ∈ DA . 4. For any a, b ∈ A, a ∧ (a → b) $ b. Proof. Due to Proposition 8.4.3 we can identify A with a twist-structure over A , in which case DA = {(1, a) | a ∈ A } and (a, b) $ (c, d) is equivalent to a ≤ b. The latter two facts easily imply all assertions of the lemma. 2 B 2. By the previous item D is an sﬀk, moreover, h preserves the relation $ as deﬁnable via basic operations. These facts imply that Ker(h) is an sﬀk. By Lemma 9.2.4 the equality h(a) = h(b) is equivalent to h(a ⇔ b) = h(a) ⇔ h(b) ∈ D B , i.e., a ⇔ b ∈ Ker(h). 3. The relation DA ⊆ ∇ follows from Item 2 of Lemma 9.2.4. Recall that the strong equivalence ⇔ has congruence properties in N4⊥ . This means, in particular, that the formula (p ⇔ q) → ((r ⇔ t) → ((p ∨ r) ⇔ (q ∨ t))) is provable in N4⊥ . By Theorem 8.4.5 for any a, b, c, d ∈ A, (a ⇔ b) → ((c ⇔ d) → ((a ∨ c) ⇔ (b ∨ d))) ∈ DA ⊆ ∇. If we assume a ⇔ b, c ⇔ d ∈ ∇, we may conclude by properties of sﬀk that (a ∨ c) ⇔ (b ∨ d) ∈ ∇. Other congruence properties can be checked similarly. Let [a] be a coset of a with respect to ≈∇ . Assume that a ∈ Ker(h), i.e., [a] ∈ D A/≈∇ . This is equivalent to [a → a] = [a], i.e., a → a ≈∇ a or (a → a) ⇔ a ∈ ∇. Since a ≤ b implies a $ b, the relation (a → a) ⇔ a ∈ ∇ implies (a → a) → a ∈ ∇. By a → a ∈ D A ⊆ ∇ and Item 4 of Lemma 9.2.4 we obtain a ∈ ∇. Conversely, let a ∈ ∇. By identity 5N (see Deﬁnition 8.5.1), we have (a → a) → a ∈ ∇. It follows from a → a ∈ DA ⊆ ∇ and 5N that a → (a → a) ∈ ∇. At the same time, ∼ a →∼ (a → a) ≈∼ a → (a∧ ∼ a) ≈∼ a → a.

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Since a ∈ ∇, we have ∼ a → a ∈ ∇ and so, ∼ a →∼ (a → a) ∈ ∇. Finally, ∼ (a → a) →∼ a ≈ (∼ a ∧ a) →∼ a ∈ DA ⊆ ∇. Whence ∼ (a → a) →∼ a ∈ ∇. In this way, (a → a) ⇔ a ∈ ∇ and a ∈ Ker(h). We have thus proved the equality ∇ = Ker(h). 2 Corollary 9.2.5 Let A be an N4⊥ -lattice and ∅ = ∇ ⊆ A. The set ∇ is an sﬀk on A if and only if: 1) a, b ∈ ∇ implies a ∧ b ∈ ∇; 2) a, a → b ∈ ∇ implies b ∈ ∇. Proof. The direct implication follows from Item 4 of Lemma 9.2.4. The inverse implication follows from the fact that DA is the least sﬀk on A. Indeed, if a ∈ ∇ and a $ b, then a → b ∈ D A ⊆ ∇. 2 ⊥ In this way, sﬀk are exactly implicative ﬁlters on N4 -lattices. Corollary 9.2.6 For any N4⊥ -lattice A, F 1 (A) is isomorphic to Con(A) and mutually inverse isomorphisms are deﬁned by the rules: ∇ → ≈∇ for 1 ∇ ∈ F (A) and θ → {a ∈ A | aθ(a → a)} for θ ∈ Con(A). 2 Corollary 9.2.7 Let h : A → B be a homomorphism of N4⊥ -lattices. Then h is a monomorphism if and only if Ker(h) = D A . Proof. Assume h : A → B is a monomorphism and a ∈ DA , i.e., a → a = a, then h(a) → h(a) = h(a). The equality Ker(h) = D A now follows from the fact that D A is the least sﬀk on A. If Ker(h) = DA and h(a) = h(b) for some a, b ∈ A, then a ⇔ b ∈ DA by Item 2 of Proposition 9.2.3 and we conclude that a = b by Item 3 of Lemma 9.2.4. 2 As we can see from Proposition 9.2.2, homomorphisms of N4⊥ -lattices are completely determined by homomorphisms of basic Heyting algebras. This leads to a suggestion that there is a close connection between sﬀk on an N4⊥ -lattice and ﬁlters on the basic Heyting algebra. This is really so and we consider ﬁrst the case of twist-structures.

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9 N4⊥ -Lattices

Proposition 9.2.8 Let B ∈ S (A). 1. For any ∇ ∈ F(A), ∇ := π1−1 (∇) = {(a, b) ∈ |A| × |A| | a ∈ ∇} is an sﬀk on B. 2. For any ∇ ∈ F 1 (B), ∇ := π1 (∇) is a ﬁlter on A. Moreover, ∇ = {(a, b) ∈ |A| × |A| | a ∈ ∇ } = (∇ ) . 3. The lattices F(A) and F 1 (B) are isomorphic and the mappings ∇ → ∇ , ∇ ∈ F 1 (B) determine mutually inverse ∇ , ∇ ∈ F(A) and ∇ → isomorphisms. Proof. Items 1 and 2 can be proved via direct veriﬁcation, whereas Item 3 is an immediate consequence of them. 2 Recall that due to Proposition 8.4.3 and Deﬁnition 8.6.3 any N4⊥ -lattice A is isomorphic to a twist-structure over A and the canonical epimorphism e : A → A corresponds to the projection π1 of the twist-structure onto the ﬁrst coordinate. In this way, we generalize the previous statement to all N4⊥ -lattices. Proposition 9.2.9 Let A be an N4⊥ -lattice. 1. For any ∇ ∈ F(A ), ∇ := (e )−1 (∇) is an sﬀk on A. 2. For any ∇ ∈ F 1 (A), ∇ := e (∇) is a ﬁlter on A . Moreover, ∇ = (∇ ) . 3. The lattices F(A ) and F 1 (A) are isomorphic and the mappings ∇ → ∇ , ∇ ∈ F 1 (A) determine mutually ∇ , ∇ ∈ F(A ) and ∇ → inverse isomorphisms. 2 Taking into account the dualities of congruences and ﬁlters on Heyting algebras and of congruences and sﬀk on N4⊥ -lattices, we obtain the following Corollary 9.2.10 For any N4⊥ -lattice A, Con(A) ∼ = Con(A ).

2

The property of an algebra to be subdirectly irreducible is determined by the structure of its congruence lattice. Combining this fact with the previous corollary, we obtain

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Corollary 9.2.11 An N4⊥ -lattice A is subdirectly irreducible if and only if the Heyting algebra A is subdirectly irreducible. 2 The above description of the congruence lattice of an N4⊥ -lattice allows one to prove that the lattice of N4⊥ -extensions is distributive. Proposition 9.2.12 VN4⊥ is an arithmetic variety. Proof. The congruence distributivity of the variety of Heyting algebras and Corollary 9.2.10 allows one to conclude that the variety of N4⊥ -lattices is congruence distributive. Prove that VN4⊥ is congruence permutable. Let A be a Heyting algebra and B ∈ S (A). Consider some congruences θ1 , θ2 ∈ Con(B) determined by sﬀk F1 and F2 , where F1 , F2 ∈ F(A), i.e., ((a, b), (c, d)) ∈ θi iﬀ (a, b) ⇔ (c, d) ∈ Fi , i = 1, 2. The deﬁnition of ﬁlters Fi and calculation with the help of twist-operations allows one to conclude that (a, b) ⇔ (c, d) ∈ Fi is equivalent to (a ↔ c) ∧ (b ↔ d) ∈ Fi . In this way, ((a, b), (c, d)) ∈ θi iﬀ (a, c), (b, d) ∈ θFi , i = 1, 2, where θFi is a congruence on A determined by Fi . This equivalence easily implies that ((a, b), (c, d)) ∈ θ1 ◦ θ2 iﬀ (a, c), (b, d) ∈ θF1 ◦ θF2 . The permutability of congruences on A imply that θ1 and θ2 are now permutable. 2 Proposition 9.2.13 The lattice EN4⊥ is distributive. Proof. This reasoning is similar to Corollary 2.2.11 but contains several peculiarities. Consider a free ω-generated N4⊥ -lattice Aω and its congruence lattice Con(Aω ), which is distributive according to the previous proposition. Moreover, the congruences of Con(Aω ) are permutable wrt composition. In this case, the elements of Aω can be identiﬁed with cosets of formulas wrt the strong equivalence in N4⊥ , |Aω | = {[ϕ]⇔ | ϕ ∈ F},

9 N4⊥ -Lattices

174

where [ϕ]⇔ := {ψ | ϕ ⇔ ψ ∈ N4⊥ }. Each logic L ∈ EN4⊥ corresponds to the congruence θL := {([ϕ0 ]⇔ , [ϕ1 ]⇔ ) | ϕ0 ⇔ ϕ1 ∈ L}. Clearly, the mapping L → θL is one-to-one and preserves the ordering. Check that this is a lattice embedding. First, we verify that for any L0 , L1 ∈ EN4⊥ , the congruences θL0 ∧ θL1 and θL0 ∨ θL1 are of the form θL for a suitable logic L. Note that θL is closed under substitution, i.e., [ϕ0 ]⇔ θL [ϕ1 ]⇔ implies [ϕ0 (ψ1 , . . . , ψn )]⇔ θL [ϕ1 ((ψ1 , . . . , ψn ))]⇔ for any formulas ψ1 , . . . , ψn . Let θ ∈ Con(Aω ) be closed under substitution. Put Lθ := {ϕ | [ϕ]⇔ θ[ϕ → ϕ]⇔ }. Check that Lθ is a logic from EN4⊥ . Obviously, the set Lθ is closed under substitution. If ϕ ∈ N4⊥ , it is not hard to prove that ϕ ⇔ (ϕ → ϕ) ∈ N4⊥ . Therefore, N4⊥ ⊆ Lθ . It remains to check that Lθ is closed under modus ponens. According to Corollary 9.2.6 Fθ = {[ϕ]⇔ | [ϕ]⇔ θ[ϕ → ϕ]⇔ } is an sﬀk on Aω . If ϕ, ϕ → ψ ∈ Lθ , then [ϕ]⇔ , [ϕ → ψ]⇔ = [ϕ]⇔ → [ψ]⇔ ∈ Fθ . From Corollary 9.2.5 we obtain [ψ]⇔ ∈ Fθ , i.e., ψ ∈ Lθ . Check the equality θ = θLθ . It can easily be seen that this equality is equivalent to the following relation ([ϕ]⇔ , [ψ]⇔ ) ∈ θ

⇐⇒

([ϕ ⇔ ψ]⇔ , [(ϕ ⇔ ψ) → (ϕ ⇔ ψ)]⇔ ) ∈ θ

for any ϕ and ψ, which is equivalent in turn to [ϕ ⇔ ψ]⇔ ∈ Fθ

⇐⇒

[ϕ ⇔ ψ]⇔ ⇔ [(ϕ ⇔ ψ) → (ϕ ⇔ ψ)]⇔ ∈ Fθ .

The latter is a particular case of the relation a∈F

⇐⇒

a ⇔ (a → a) ∈ F,

which was established in the proof of Item 3 of Proposition 9.2.3 for any sﬀk F on any N4⊥ -lattice. Thus, we have proved that the congruences of the form θL are exactly the congruences closed under substitution. Check that θL0 ∧ θL1 and θL0 ∨ θL1 are closed under substitution. Consider the non-trivial case of θL0 ∨ θL1 . Since the lattice Aω is congruence permutable, θL0 ∨ θL1 = θL0 ◦ θL1 . Therefore [ϕ0 ]⇔ θL0 ∨ θL1 [ϕ1 ]⇔ iﬀ there is a formula ψ such that [ϕ0 ]⇔ θL0 [ψ]⇔ and [ψ]⇔ θL1 [ϕ1 ]⇔ . This equivalence immediately implies that θL0 ∨ θL1 is closed under substitution. We have proved that the set of congruences θL forms a lattice. It is clear that the mapping L → θL is an order isomorphism of EN4⊥ and the lattice

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175

of congruences of the form θL . If two lattices are isomorphic as orderings, they are isomorphic as lattices too. 2 We now prove the criterion when there exists a homomorphism from one N4⊥ -lattice into the other, which generalizes the embeddability criterion from the previous section. But ﬁrst, we point out some facts on interrelations of homomorphisms h and h . Proposition 9.2.14 Let h : A → B be a homomorphism of N4⊥ -lattices. 1. Ker(h ) = (Ker(h)) and Ker(h) = (Ker(h )) . 2. h is a monomorphism if and only if h is a monomorphism. Proof. 1. By deﬁnition (Ker(h)) = {[a]≈A | h(a) ∈ DB }. The condition h(a) ∈ D B is equivalent to h ([a]≈A ) = [h(a)]≈B = 1B . From this equivalence we immediately obtain Ker(h ) = (Ker(h)) . The second equality immediately follows from the ﬁrst one and Proposition 9.2.9. 2. If h is a monomorphism, then Ker(h) = D A by Corollary 9.2.7. Taking into account that (D A ) = {1A } we infer from the previous item Ker(h ) = {1A }, i.e., h is a monomorphism. The inverse implication is similar. 2 Proposition 9.2.15 Let A and B be N4⊥ -lattices. There exists a homomorphism (monomorphism) h : A → B if and only if there exists a homomorphism (monomorphism) g : A → B such that g(∇(A)) ⊆ ∇(B) and g(Δ(A)) ⊆ Δ(B). Proof. If h : A → B is a homomorphism, then h can be chosen as g. Indeed, for [a∨ ∼ a]≈A ∈ ∇(A), we have h ([a∨ ∼ a]≈A ) = [h(a∨ ∼ a)]≈B = [h(a)∨ ∼ h(a)]≈B ∈ ∇(B). Analogously, h (Δ(A)) ⊆ Δ(B). If h is additionally a monomorphism, then h is a monomorphism by Item 2 of the previous proposition. Assume that there exists a homomorphism g : A → B such that g(∇(A)) ⊆ ∇(B) and g(Δ(A)) ⊆ Δ(B). Let us consider g and check g ◦ i A (A) ⊆ iB (B). Let (a, b) ∈ i A (A). By Corollary 9.1.6 we have a ∨ b ∈ ∇(A) and a ∧ b ∈ Δ(A). Applying to these facts our assumption we obtain

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g(a) ∨ g(b) = g(a ∨ b) ∈ ∇(B) and g(a) ∧ g(b) = g(a ∧ b) ∈ Δ(B). Again, by Corollary 9.1.6 we have g ((a, b)) = (g(a), g(b)) ∈ i B (B). Thus, the desired homomorphism from A to B can be deﬁned as h := −1 ◦ g ◦ i . ) (i B A If g is a monomorphism, then g and h are also monomorphisms. 2 A homomorphic image of a twist-structure has the following presentation. Proposition 9.2.16 Let B be a Heyting algebra, F a ﬁlter on B. Let A ∈ S (B) and A = T w(B, ∇, Δ). Then A/F ∼ = T w(B/F, ∇/F, Δ/F ). Proof. Deﬁne a mapping h : A/F → (B/F ) as follows. For any (a, b) ∈ A, h((a, b)/F ) := (a/F, b/F ). Clearly, h is a homomorphism. Check that this is a monomorphism. The equality (a/F, b/F ) = (c/F, d/F ) is equivalent to (a ↔ c)∧(b ↔ d) ∈ F . At the same time, (a, b)/F = (c, d)/F is equivalent by Proposition 9.2.3 to (a, b) ⇔ (c, d) ∈ F , which is equivalent in turn to (a ↔ c) ∧ (b ↔ d) ∈ F . Thus, h is a monomorphism and it remains to check that h(A/F ) = T w(B/F, ∇/F, Δ/F ), Let A := h(A/F ) = {(a/F, b/F ) | (a, b) ∈ A}. Then ∇(A ) = π1 (I(A )) = π1 ({(a/F, b/F )∨ ∼ (a/F, b/F ) | (a, b) ∈ A}) = π1 ({(a ∨ b/F, a ∧ b/F ) | (a, b) ∈ A}) = {a ∨ b/F | (a, b) ∈ A} = ∇/F. In a similar way, Δ(A ) = π2 (I(A )) = Δ/F .

2

Chapter 10

The Class of N4⊥-Extensions1 In this chapter, we study the structure of the lattice EN4⊥ and discover a deﬁnite similarity to the structure of the lattice Jhn+ studied in the ﬁrst part of the book. It should be noted that diﬀerences in the structure of these two classes of logics are also essential. Moreover, we give ﬁrst applications of the developed theory: the class of extensions of the logic N4⊥ C obtained by adding Dummett’s linearity axiom (p → q) ∨ (q → p) to N4⊥ is completely described; two classical results by L.L. Maksimova, namely, the description of pretabular logics and the description of logics with Craig interpolation property, are transferred from the class of superintuitionistic logics to the class of N4⊥ -extensions.

10.1

EN4⊥ and Int+

We start the investigation of the class EN4⊥ with the question on interrelations between a logic in EN4⊥ and its intuitionistic fragment. We deﬁne a mapping σ from EN4⊥ into the class Int+ of extensions of intuitionistic logic Li so that σ(L) is a ∨, ∧, →, ⊥ -fragment of L. Later on, we call formulas of the language ∨, ∧, →, ⊥ intuitionistic formulas and denote the set of all intuitionistic formulas by F. As in the ﬁrst part of the book, Int denotes the lattice of intermediate logics, i.e., Int+ = Int ∪ {F}. The restriction of σ to the class EN3 was investigated by M. Kracht [47] and A. Sendlewski [101]. 1

Parts of this chapter were originally published in [77, 78, 79] (Springer, Netherlands). Reprinted here by permission of the publisher.

177

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10 The Class of N4⊥ -Extensions

First, we point out that σ(L) is determined by basic Heyting algebras of L-models. For a class K of N4⊥ -lattices, put K := {A | A ∈ K}. Proposition 10.1.1 For any L ∈ EN4⊥ and a class K of N4⊥ -lattices, if L = LK, then σ(L) = LK . Proof. Due to Proposition 8.4.3 it is enough to consider twist-structures isomorphic to elements of K. Let B ∈ IK and B ∈ S (A) for some Heyting algebra A. Assume that A(∼ = B ) is not a model for σ(L). Let ϕ ∈ σ(L) and A-valuation v are such that v(ϕ) = 1. For any propositional variable p, there is an element bp ∈ |A| with (v(p), bp ) ∈ |B|. Let v be a B-valuation such that v (p) = (v(p), bp ) for any p. The twist-operations are componentwise wrt the ﬁrst component, therefore, π1 v (ϕ) = v(ϕ) for any ϕ. Thus, π1 v (ϕ) = 1 and B |= ϕ, which conﬂicts with the assumption B |= σ(L) ⊆ L. We have thus proved the inclusion σ(L) ⊆ LK . To check the inverse inclusion take an intuitionistic formula ϕ ∈ σ(L). Let B ∈ IK, B ∈ S (A) and a B-valuation v be such that π1 v(ϕ) = 1. Then π1 v is an A-valuation proving that A |= ϕ. 2 The following fact was stated in the last proof and it makes sense to distinguish it as a separate lemma. Lemma 10.1.2 Let A be an N4⊥ -lattice and ϕ an intuitionistic formula. Then A |= ϕ ⇐⇒ A |= ϕ. 2 We now study the inverse image σ −1 (L) for any L ∈ Int+ , i.e., the class of all conservative extensions of L in the class EN4⊥ . We show that σ −1 (L) forms an interval in the lattice EN4⊥ and consider the mappings sending L to the end points of the interval σ −1 (L). Let for any L ∈ Int+ , η(L) := N4⊥ + L, i.e., η(L) is obtained by extending language and adding the strong negation axioms to L, and η ◦ (L) = η(L) + {∼ p → (p → q), ¬¬(p∨ ∼ p)}. Logics in EN4⊥ having the form η(L) we call special or s-logics, and logics of the form η ◦ (L) are called normal explosive or ne-logics.

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Prior to proving that η(L) and η ◦ (L) are the end points of the inverse image of L wrt σ, we describe models of η(L) and η ◦ (L) and classes of models generating such logics. Proposition 10.1.3 Let L ∈ Int+ and A be an N4⊥ -lattice. 1. A |= η(L) iﬀ A |= L 2. A |= η ◦ (L) iﬀ A |= L, ∇(A) = Fd (A ) and Δ(A) = {0}. Proof. 1. The direct implication follows by Lemma 10.1.2. The inverse implication follows from the same lemma and the fact that the axioms of N4⊥ hold in any N4⊥ -lattice. 2. If A |= η ◦ (L), then A is a model of N3 and Δ(A) = {0}. The direct computation shows that for a twist-structure A ∈ S (B), the condition A |= ¬¬(p∨ ∼ p) is equivalent to ¬¬(a ∨ b) = 1 for any (a, b) ∈ A, i.e., ∇(A) ⊆ Fd (B). The inclusion Fd (B) ⊆ ∇(A) holds for any twist-structure, which proves the direct implication. We now assume that the right hand conditions of the equivalence are satisﬁed. Due to Lemma 10.1.2 the ﬁrst condition implies A |= L. We have A |=∼ p → (p → q) by Δ(A) = {0}. Finally, in view of the above considerations, ∇(A) = Fd (A ) implies A |= ¬¬(p∨ ∼ p). 2 For a Heyting algebra A, deﬁne A◦ := T w(A, Fd (A), {0}). This is the least twist-structure over A. For any B ∈ S (A), A ◦ ≤ B. Namely, such lattices are up to isomorphism models of logics of the form η ◦ (L). If an N4⊥ -lattice A is isomorphic to a lattice of the form B◦ , we call it a normal N3-lattice. This is really an N3-lattice, since in this case, Δ(A) = {0}. In the following, N4⊥ -lattices isomorphic to full twist-structures will be called special N4⊥ -lattices. Further, we deﬁne translations of η ◦ (L) and η(L) into L. Extend the set P rop of propositional variables adding a new propositional variable p for any p ∈ P rop. This extended set of variables will be denoted P rop. To any formula ϕ in the language L⊥ we assign intuitionistic formulas ϕ and ϕ◦ over P rop deﬁned as follows. If ϕ is a nnf, then ϕ is the result of replacement of every occurrence of ∼ p in ϕ by p. If ϕ is not in nnf, then

and ϕ◦ as follows: ϕ := (ϕ) . We deﬁne formulas ϕ ϕ

:=

p∈var(ϕ)

¬(p ∧ p) ∧

p∈var(ϕ)

→ ϕ . ϕ◦ := ϕ

¬¬(p ∨ p),

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Proposition 10.1.4 Let A be a Heyting algebra and ϕ ∈ F or ⊥ . The following equivalences hold. 1. A |= ϕ if and only if A |= ϕ . ◦ 2. A ◦ |= ϕ if and only if A |= ϕ .

Proof. In the course of the proof we assume that ϕ is a nnf. 1. For any A -valuation v, we deﬁne v : P rop → |A| by the rule v (p) := π1 v(p) and v (p) := π2 v(p). For any v : P rop → |A|, we deﬁne A -valuation v : P rop → |A| × |A| by the rule v (p) := (v(p), v(p)). Note that (v ) = v for any A -valuation v, and (v ) = v for any Avaluation v. We have thus deﬁned a bijection between A - and A-valuations. Taking into account the fact that the action of intuitionistic operations on a twist-structure agrees with their action on the ﬁrst component and the equality π1 v(∼ p) = π2 v(p) we obtain π1 v(ϕ) = v (ϕ ) and π1 v (ϕ) = v(ϕ ) for any nnf ϕ. These relations and the above-stated bijection between A and A-valuations imply the desired equivalence. 2. Let A |= ϕ◦ and v be an A ◦ -valuation. By deﬁnition of A◦ we have π1 v(p) ∧ π2 v(p) = 0 and π1 v(p) ∨ π2 v(p) ∈ Fd (A) for any p ∈ var(ϕ),

= 1, whence v (ϕ ) = 1. By the equivalence from the which implies v (ϕ) previous item v(ϕ) = 1. ◦ Assume now A ◦ |= ϕ and v(ϕ ) = 1 for some A-valuation v. Then

and consider the quotient algebra A/F and v(ϕ)

≤ v(ϕ ). Put F = v(ϕ)

= 1 and v (ϕ ) = 1. The the quotient valuation v := v/F . We have v (ϕ)

= 1 implies that π1 (v ) (p) ∧ π2 (v ) (p) = 0 and π1 (v ) (p) ∨ equality v (ϕ) π2 (v ) (p) ∈ Fd (A) for any p ∈ var(ϕ), i.e., v is an (A/F ) ◦ -valuation. Arguing as in the previous item from v (ϕ ) = 1, we infer (A/F ) ◦ |= ϕ. It /F by Proposition 9.2.16. ∼ (A) remains to note that (A/F ) = ◦ ◦ 2 Corollary 10.1.5 Let L ∈ Int+ and ϕ ∈ F or ⊥ . The following equivalences hold. 1. ϕ ∈ η(L) if and only if ϕ ∈ L. 2. ϕ ∈ η ◦ (L) if and only if ϕ◦ ∈ L.

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181

Proof. 1. Let ϕ ∈ η(L) and A |= L. Then A |= η(L) by Proposition 10.1.3, and A |= ϕ by Item 1 of the last proposition. Conversely, let ϕ ∈ L and A |= η(L). Then A |= L by Proposition 10.1.3, and (A ) |= η(L). Since A → (A ) , we have A |= ϕ. 2. This item follows similarly from Item 2 of Proposition 10.1.3. 2 Corollary 10.1.6 For any L ∈ Int+ , the logics η(L) and η ◦ (L) are conservative extensions of L. Proof. Intuitionistic formulas are ∼-free, therefore, ϕ = ϕ for any such formula. In this way, the fact that η(L) is a conservative extension of L follows from Item 1 of Corollary 10.1.5. For any intuitionistic ϕ, ϕ

∈ η ◦ (L). Thus, the equivalence ϕ ↔ ϕ◦ is provable in η ◦ (L) for any intuitionistic ϕ. Now, Item 2 of Corollary 10.1.5 implies that η ◦ (L) is conservative over L. 2 Let K be a class of Heyting algebras. Put K := {A | A ∈ K} and K◦ := {A ◦ | A ∈ K}. Proposition 10.1.3 states, in fact, that η(L) = L(M od(L)) and η ◦ (L) = L(M od(L)) ◦ . The proposition below generalizes this result. Proposition 10.1.7 Let L ∈ Int+ and L = LK. Then η(L) = LK and η ◦ (L) = LK◦ . Proof. The inclusions η(L) ⊆ LK and η ◦ (L) ⊆ LK◦ follow from Proposition 10.1.3. Let ϕ ∈ η(L). Then ϕ ∈ L and there is A ∈ K such that A |= ϕ . It follows by Item 1 of Proposition 10.1.4 that A |= ϕ, i.e., ϕ ∈ LK . Similarly, we use Item 2 of Proposition 10.1.4 to prove the second equality. 2 Taking into account Proposition 10.1.1, the last proposition can be reworded as follows.

182

10 The Class of N4⊥ -Extensions

Corollary 10.1.8 Let L ∈ EN4⊥ . 1. L is a special logic if and only if L is determined by some family of special N4⊥ -lattices. 2. [33] L is an ne-logic if and only if L is determined by some family of normal N3-lattices. 2 Proposition 10.1.9 For any L ∈ Int+ , σ −1 (L) = [η(L), η ◦ (L)]. Proof. If L1 ∈ σ −1 (L) and an N4⊥ -lattice A is such that A |= L1 , then A |= L by Lemma 10.1.2 and A |= η(L) by Item 1 of Proposition 10.1.3. Thus, η(L) ⊆ L1 . For any A ∈ M od(L1 ), (A ) ◦ embeds into A and belongs to M od(L1 ). Let K = {A | A ∈ M od(L1 )}. Then K◦ ⊆ M od(L1 ). By Proposition 10.1.1, L = LK, and η ◦ (L) = LK◦ by Proposition 10.1.7. We have thus proved L1 ⊆ η ◦ (L). That η(L) and η ◦ (L) belong to σ −1 (L) was stated in Corollary 10.1.6. 2 Remark. The analog of the above proposition does not hold for the lattice of N4-extensions. If σ p (L) is the positive fragment of L ∈ EN4, one can still prove that N4 + L is the least element of the inverse image (σ p )−1 (L), where L is some extension of positive logic. But in the general case, the set (σ p )−1 (L) has no greatest element. This follows from the fact that the family of intermediate logics with the same positive fragment does not necessarily have a greatest element. Following M. Kracht [47] we obtain another useful characterization of special and normal explosive logics. Proposition 10.1.10 Let L ∈ EN4⊥ . 1. L is a special logic if and only if all rules of the form ψ/ψ are admissible in L. 2. L is an ne-logic if and only if L ∈ EN3 and ¬¬(p∨ ∼ p) ∈ L. Proof. 1. Let L be a special logic. According to Corollary 10.1.8 L is determined by some family of special N4⊥ -lattices. The admissibility of rules ψ/ψ now follows from Proposition 10.1.4 and Lemma 10.1.2. Conversely, assume that all rules of the form ψ/ψ are admissible in L.

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Lemma 10.1.11 All rules of the form ψ /ψ are admissible in any L ∈ EN4⊥ . Proof. Let A be an N4⊥ -lattice. If A |= ψ , then A |= ψ by Lemma 10.1.2, and (A ) |= ψ by Proposition 10.1.4. Since A → (A ) , we conclude A |= ψ. 2 Taking into account this lemma and Corollary 10.1.5, we obtain for any ϕ, ϕ ∈ L if and only if ϕ ∈ L if and only if ϕ ∈ η(σ(L)). Thus, L = η(σ(L)) and L is a special logic. 2. If L is an ne-logic, then ∼ p → (p → q) and ¬¬(p∨ ∼ p) belong to L by deﬁnition. Let L ∈ EN3, ¬¬(p∨ ∼ p) ∈ L, and A |= L. In this case, A is an N3-lattice, i.e., Δ(A) = {0}. The validity of ¬¬(p∨ ∼ p) is equivalent to ∇(A) = Fd (A). We have thus proved that any model of L is a normal N3-lattice. By Corollary 10.1.8 L is an ne-logic. 2 Denote N3◦ := N3 + {¬¬(p∨ ∼ p)}. Proposition 10.1.12 1. σ : EN4⊥ → Int+ is a lattice epimorphism commuting with inﬁnite meets and joins. 2. η : Int+ → EN4⊥ is a lattice monomorphism commuting with inﬁnite meets and joins. 3. η ◦ is a lattice isomorphism of Int+ and EN3◦ . Proof. 1. It follows immediately from deﬁnition that σ commutes with inﬁnite meets. Prove that it commutes with inﬁnite joins. The inclusion Σi∈I σ(Li ) ⊆ σ(Σi∈I Li ), where Li ∈ EN4⊥ , i ∈ I, is obvious. Check the inverse inclusion. ◦ Take a model A of Σi∈I σ(Li ). By Proposition 10.1.3 we have A ◦ |= η σ(Li ) for i ∈ I. Since Li ⊆ η ◦ σ(Li ) by Proposition 10.1.9, we obtain A ◦ |= Li for |= Σ L . According to Lemma 10.1.2 A |= σ(Σi∈I Li ), all i ∈ I, and so A i∈I i ◦ which proves the inverse inclusion. 2. That η is one-to-one follows fromCorollary 10.1.6.Check that η commutes with inﬁnite meets. Let L∗ := i∈I Li . Since σ( i∈I η(Li )) is obviously equal to L∗ , the equality η(L∗ ) = i∈I η(Li ) follows from the fact that rules of the i∈I η(Li ) is a special logic. Each of η(Li ) is closed under all form ψ/ψ by Proposition 10.1.10. Obviously, the intersection i∈I η(Li ) is also closed under such rules and it is also a special logic.

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184

3. η ◦ embeds Int into EN3◦ by Corollary 10.1.6. That η ◦ is onto follows from Item 2 of Proposition 10.1.10. 2 We have thus presented the class EN4⊥ as a union of disjoint intervals of the form σ −1 (L): [η(L), η ◦ (L)]. EN4⊥ = L∈Int+

Now, we establish interrelations between these intervals. It turns out that if L1 ⊆ L2 , then σ −1 (L2 ) is embedded into σ −1 (L1 ) as upper subinterval, at the same time, σ −1 (L2 ) is isomorphic to a homomorphic image of σ −1 (L1 ). Let L1 , L2 ∈ Int+ and L1 ⊆ L2 . Mappings rL2 ,L1 : σ −1 (L2 ) → EN4⊥ and eL1 ,L2 : σ −1 (L1 ) → EN4⊥ are deﬁned as follows: rL2 ,L1 (L) = L ∩ η ◦ (L1 ) and eL1 ,L2 (L) = L + η(L2 ) Proposition 10.1.13 Let L1 , L2 ∈ Int+ and L1 ⊆ L2 . The following facts hold. 1. For any L ∈ σ −1 (L2 ), we have eL1 ,L2 rL2 ,L1 (L) = L. 2. For any L ∈ σ −1 (L1 ), we have rL2 ,L1 eL1 ,L2 (L) = L + rL2 ,L1 (η(L2 )). 3. eL1 ,L2 is a lattice epimorphism from σ −1 (L1 ) onto σ −1 (L2 ). 4. rL2 ,L1 is a lattice monomorphism from σ −1 (L2 ) into σ −1 (L1 ) and rL2 ,L1 (σ −1 (L2 )) = [rL2 ,L1 (η(L2 )), η ◦ (L1 )]. 5. For any L3 ∈ Int+ such that L2 ⊆ L3 , we have eL1 ,L2 eL2 ,L3 = eL1 ,L3 and rL3 ,L2 rL2 ,L1 = rL3 ,L1 . Proof. 1. Let L ∈ σ −1 (L2 ). We calculate eL1 ,L2 rL2 ,L1 (L) = (L + η(L2 )) ∩ (η ◦ (L1 ) + η(L2 )). We have L+ η(L2 ) = L since η(L2 ) is the least element of σ −1 (L2 ). Consider L := η ◦ (L1 )+η(L2 ). Due to homomorphism properties of σ we have σ(L ) = L2 . Moreover, L ∈ EN3◦ since η ◦ (L1 ) ⊆ L . By Item 3 of Proposition 10.1.12

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185

we conclude L = η ◦ (L2 ). Finally, L ∩ η ◦ (L2 ) = L since η ◦ (L2 ) is the greatest element of σ −1 (L2 ). 2. Again for L ∈ σ −1 (L1 ), we have rL2 ,L1 eL1 ,L2 (L) = (L ∩ η ◦ (L1 )) + (η(L2 ) ∩ η ◦ (L1 )), where the ﬁrst term is equal to L since η ◦ (L1 ) is the greatest element of σ −1 (L1 ) and the second term is exactly rL2 ,L1 (η(L2 )). 3. That eL1 ,L2 is a lattice homomorphism follows from the distributivity of EN4⊥ . Let L ∈ σ −1 (L1 ). Since σ is a homomorphism, we have σeL1 ,L2 (L) = σ(L) + ση(L2 ) = L1 + L2 = L2 . Thus, the range of eL1 ,L2 is contained in σ −1 (L2 ). Item 1 implies that eL1 ,L2 is onto. 4. As above, we use the distributivity of EN4⊥ and the homomorphism properties of σ to prove that rL2 ,L1 is a lattice homomorphism from σ −1 (L2 ) into σ −1 (L1 ). If rL2 ,L1 (L) = rL2 ,L1 (L ), then applying the formula of Item 1 we obtain L = L . In this way, rL2 ,L1 is a monomorphism. The equality rL2 ,L1 (σ −1 (L2 )) = [rL2 ,L1 (η(L2 )), η ◦ (L1 )] follows from Item 2. 5. This follows immediately from deﬁnitions. 2

10.2

The Lattice Structure of EN4⊥

It was proved in the previous section that the lattice EN4⊥ decomposes into a union of disjoint intervals σ −1 (L), where L ∈ Int+ and that σ −1 (L1 ) is isomorphic to an upper subinterval of σ −1 (L2 ), whenever L2 ⊆ L1 . Since Lk is the greatest point of Int, an isomorphic copy of σ −1 (Lk) is an upper part of any interval of the form σ −1 (L) for L = F. For this reason, we start the study of EN4⊥ with a description of the interval σ −1 (Lk). Consider subdirectly irreducible models of logics in σ −1 (Lk). According to Corollary 9.2.11, any such model is isomorphic to an element of S (A), where A is a subdirectly irreducible model of Lk. Any subdirectly irreducible model of Lk is isomorphic to the two-element Boolean algebra 2. Therefore, we have exactly four subdirectly irreducible models of logics in σ −1 (Lk): 2 = T w(2, {0, 1}, {0, 1}), 2 ◦ = T w(2, {1}, {0}), 2 R = T w(2, {1}, {0, 1}), 2L = T w(2, {0, 1}, {0}).

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186

These lattices deﬁne logics closely related to well-known ﬁnite valued logics. Consider matrices of the form M (A) = A, DA corresponding to the above N4⊥ -lattices. The lattice 2 ◦ is two-element. Its elements (0, 1) and (1, 0) can be identiﬁed with the classical truth-values f and t respectively. The strong negation ∼ coincides with the classical one and DA = {t} in this case. Thus, the matrix corresponding 2 ◦ deﬁnes, in fact, the classical logic := L2 . with an additional negation ∼, such that ∼ p ↔ ¬p. We denote Lk ◦ The matrix M (2 ) has additional truth-values & := (1, 1) and ⊥ := (0, 0). It has two distinguished values, D2 = {t, &}, and the following lattice structure.

tq

@

@

⊥ q@

@q &

@ @q

f

In fact, M (2 ) can be considered as a four-valued Belnap’s matrix ([8, 9]) enriched with the weak implication → and the constant ⊥. We denote B→ 4 := L2 . The logic of 2 R diﬀers from the maximal extension RM3 of the relevant logic RM [3, 4] only by an additional constant ⊥. Indeed, RM3 can be deﬁned via the matrix M = {f, t, &}, ∨, ∧, ⊃, ∼, {t, &} , where f ≤ & ≤ t, ∨ and ∧ are the usual lattice operations, ∼ is an order reversing involution and the implication is deﬁned by the rule: a ⊃ b :=

b, if a ∈ D M t, if a ∈ DM .

(10.1)

2R = {t, &}. The lattice structures of We have |2 R | = {f, t, &} and D M and 2 R are identical, the strong negation is also an order-reversing involution, ﬁnally, the direct computation shows that ⊃ coincides with the implication of 2 R . Thus, L2R is RM3 enriched with the constant ⊥. We denote RM3⊥ := L2 R.

10.2 The Lattice Structure of EN4⊥

187

It was pointed out in [110] that the logic of 2 L is deﬁnition equivalent ukasiewicz. We have |2 to the three-valued logic L 3 of L L | = {f, ⊥, t}. Let us set a ⊃ b := (a → b) ∧ (∼ b →∼ a) and calculate the truth tables for ⊃ and ∼ on {f, ⊥, t}: → f ⊥ t

⊥ t t ⊥

f t ⊥ f

∼ f ⊥ t

t t t t

t ⊥ f

Thus, {f, ⊥, t}, ⊃, ∼, {t} is the well-known matrix for L 3 . All operations can be deﬁned through ⊃ and ∼ as follows: of 2 L ¬a := a ⊃∼ a, a ∨ b := (a ⊃ b) ⊃ b, a ∧ b :=∼ (∼ a∨ ∼ b), a → b := a ⊃ (a ⊃ b). 3. For this reason, we denote L2 L as L , Note that N4⊥ -lattices 2 , 2 ◦ 2R , and 2L have no non-trivial homomorphic images and are embedded one into another as follows

2 @ I

@

@

2 R

2 L

I @

@

@

2 ◦

Thus, the interval σ −1 (Lk) contains exactly ﬁve logics determined by the following sets of subdirectly irreducible N4⊥ -lattices: {2 }, {2 R , 2L }, {2R }, {2L }, {2◦ }.

We have thus proved the following

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188

Proposition 10.2.1 The interval σ −1 (Lk) has the following structure:

Lk q @

RM3⊥

q @ @ @q

@

@q

L 3

RM3⊥ ∩ L 3

q

B→ 4 ◦ In particular, η(Lk) = B→ 4 and η (Lk) = Lk.

2

Taking into account that L ⊆ η ◦ σ(L) for any L ∈ EN4⊥ and that ◦ 1 ) ⊆ η (L2 ) whenever L1 ⊆ L2 , we conclude

η ◦ (L

Corollary 10.2.2 the logic Lk.

1. Any non-trivial extension of N4⊥ is contained in

2. The logic N4⊥ has no contradictory non-trivial extension. 2 We have thus pointed out the ﬁrst essential diﬀerence between the structure of EN4⊥ and the structure of Jhn+ . Minimal logic has the subclass of contradictory extensions isomorphic to the class of extensions of positive logic, whereas in the case of N4⊥ , no contradiction can be added to N4⊥ as a scheme. Further, unlike Jhn, the class EN4⊥ \ {F or ⊥ } of non-trivial and our next step N4⊥ -extensions forms a lattice with the unit element Lk, ⊥ ⊥ is to describe coatoms of the lattice EN4 \ {F or }. We know two examples of coatoms. It follows from Proposition 10.2.1 3 are coatoms in EN4⊥ \ {F or ⊥ }. A further example of that RM3⊥ and L coatoms provides the twist-structure 3 ◦ , where 3 is a three-element linearly ordered Heyting algebra, |3| = {0, 1, 2}, 0 ≤ 1 ≤ 2. Since Fd (3) = {1, 2}, the lattice 3 ◦ has four elements (0, 1), (1, 0), (0, 2), and (2, 0). The lattice structure and the action of strong negation on 3 ◦ are presented in the diagram below.

10.2 The Lattice Structure of EN4⊥

189

(2,q 0)

6 @ @

@ @q(1, 0)

(0, 1) q

@

@ @

@q?

(0, 2) as a lattice, but not as an We can see that 3 ◦ is isomorphic to 2 implicative lattice, the negation also acts on 3 ◦ in another way.

Proposition 10.2.3 The lattice EN4⊥ \ {F or ⊥ } has exactly three coatoms: ⊥ ⊥ 3 , and L3 RM3⊥ , L ◦ . Each element of EN4 \ {F or } diﬀerent from Lk is contained in one of them. Assume L is contained neither in Proof. Let L ∈ EN4⊥ \{F or ⊥ } and L = Lk. ⊥ 3 . By Proposition 10.2.1 σ(L) = Lk in this case. It follows that RM3 nor in L L has a model A = T w(B, ∇, Δ), where B is not a Boolean algebra, in which case ∇ = {1}. Take an element a in B such that a = 0, 1 and a ∈ ∇, and consider a twist-structure A0 := T w({0, a, 1}, {a, 1}, {0}). Clearly, A0 ≤ A and A0 ∼ = 3 ◦ . Thus, 3◦ |= L and L ⊆ L3◦ . ⊥ 3 . On the one hand, the intuitionL3◦ is incomparable with RM3 and L istic fragment of L3◦ is diﬀerent from classical logic. On the other hand, ⊥ . ∼ p → (p → q) ∈ L3 3 ◦ \ RM3 and ¬¬(p∨ ∼ p) ∈ L3◦ \ L 2 Recall that an element x of a lattice A is called a splitting element if there exists an y such that for every element z ∈ A, exactly one of the following conditions holds: z ≤ x or y ≤ z. If x is a splitting element, the corresponding y is called the splitting of A by x and is denoted by A/x. We write A/{x, y} for A/x ∨ A/y. 3 , and L3 Proposition 10.2.4 The logics RM3⊥ , L ◦ are splitting elements ⊥ in the lattice EN4 and the following holds: → EN4⊥ /RM3⊥ = N3, EN4⊥ /L3 = N4⊥ + ¬¬(p∨ ∼ p), EN4⊥ /L3 ◦ = B4

and 3 } = N3◦ . EN4⊥ /{RM3⊥ , L

190

10 The Class of N4⊥ -Extensions

Proof. If L ∈ EN3, then Δ(A) = {0} for any A |= L. Therefore, 2 R is not a model of L and L ⊆ RM3⊥ . If L ∈ EN3, there is A = T w(B, ∇, Δ) such that A |= L and Δ = {0}. Choose an a ∈ Δ such that a = 0 and consider the quotient A0 := A/ a . By Proposition 9.2.16 A0 ∼ = A1 := T w(B/ a , ∇/ a , Δ/ a ). Since a ∈ Δ, Δ/ a = B/ a . Consequently, A1 contains a subalgebra T w({0/ a , 1/ a }, {1/ a }, {0/ a , 1/ a }), which is isomorphic to 2 R. Thus, L ⊆ RM3⊥ and we have proved the equality EN4⊥ /RM3⊥ = N3. If ¬¬(p∨ ∼ p) ∈ L, then every model of L is up to isomorphism of the form T w(B, Fd (B), Δ). Obviously, 2 L does not satisfy this condition since 3. ∇(2L ) = 2. Therefore, L |= 2L , and L is not contained in L If ¬¬(p∨ ∼ p) ∈ L, then L has a model A = T w(B, ∇, Δ) such that ∇= Fd (B). Passing to the quotient A/(Fd (B)) we obtain a model A0 of L such that A0 = T w(B0 , ∇0 , Δ0 ), B0 is a Boolean algebra and ∇0 = {1}. Let a ∈ ∇0 and a = 1. Let a be a Boolean complement of a in B0 . By Proposition 9.2.16 we have A0 / a ∼ = A2 := T w( a , a , Δ1 ). Since a = 1, the twist-structure A2 contains a subalgebra T w({1, a}, {1, a}, {a}) that is isomorphic to 2 L . Therefore, L |= 2L . We have thus proved ⊥ ⊥ EN4 /L3 = N4 + ¬¬(p∨ ∼ p). → ◦ To prove EN4⊥ /L3 ◦ = B4 , we notice that L3◦ = η (L3) according ◦ to Proposition 10.1.7. Thus, if σ(L) ⊆ L3, then L ⊆ η (L) ⊆ η ◦ (L3). It is well known that L3 is the opremum of Int. Consequently, if σ(L) ⊆ L3, then σ(L) = Lk and L extends η(Lk). It remains to notice that η(Lk) = B→ 4 by Proposition 10.2.1. The last equality immediately follows from the ﬁrst and the second. 2 We denote N4N := N4⊥ + {¬¬(p∨ ∼ p)} and distinguish in EN4⊥ the following subclasses: Exp := {L ∈ EN4⊥ |∼ p → (p → q) ∈ L}, Nor := {L ∈ EN4⊥ | ¬¬(p∨ ∼ p) ∈ L}, Gen := EN4⊥ \ (Exp ∪ Nor).

10.2 The Lattice Structure of EN4⊥

191

Let L ∈ EN4⊥ . We say that L is explosive if L ∈ Exp, we call L normal if L ∈ Nor. Finally, if L ∈ Gen, we say that L is a logic of general form. In the proposition below, we collect a series of simple facts on the introduced classes. Proposition 10.2.5 1. Exp ∩ Nor = EN3◦ and L3 ◦ is an opremum in ⊥ (Exp ∩ Nor) \ {F or }. 2. Exp = EN3. 3. Exp \ Nor = [N3, L 3] 4. L ∈ Exp iﬀ L ⊆ RM3⊥ iﬀ for any A ∈ M od(L), Δ(A) = {0}. 5. Nor = EN4N . 6. Nor \ Exp = [N4N , RM3⊥ ] 7. L ∈ Nor iﬀ L ⊆ L 3 iﬀ for any A ∈ M od(L), ∇(A) = Fd (A ). 3 ]. 8. Gen = [N4⊥ , RM3⊥ ∩ L 2 All statements in this proposition easily follow from Propositions 10.2.3 and 10.2.4. Figure 10.1 presents the structure of EN4⊥ . Remark. Let us note the similarity of classes Exp, Nor, Gen and respectively, the classes Int, Neg, Par studied in the ﬁrst part of the book. Both classes Exp and Int consist of explosive logics. The similarity of classes Nor and Neg is not so obvious. The class Neg consists of contradictory logics, which represent the structures of contradictions in all Lj-extensions in the following sense. If L1 ∈ Neg is a negative counterpart of an Lj-extension L, then L1 is embedded into L by the contradiction operator (see Section 4.2.1). According to Corollary 10.2.2 the logic N4⊥ has no contradictory extension and the structures of contradictions cannot be explicated in the same way as for Lj-extensions. However, they can be represented semantically as ideals of contradictions Δ(A). Every N4⊥ -lattice is determined by its basic algebra A , the ﬁlter of completions ∇(A) and the ideal of contradictions Δ(A). By Proposition 10.1.1 basic algebras of L-models for L ∈ EN4⊥ determine the intuitionistic fragment of L. According to the last proposition, the ideal of contradictions is trivial in models of explosive logics and so, a logic from Exp is characterized only by its intuitionistic fragment and by ﬁlters of completions of its models. At the same time, the ﬁlter of completions has a ﬁxed

10 The Class of N4⊥ -Extensions

192 rF or ⊥

rLk X XXX XXX

rL3◦$ ' ⊥ RM 3 r $ '

XX

XX X XXX L Xr 3 ' XXX XX XXX XXX XX XXX Exp ∩ Nor r $ '

$

RM3⊥ ∩ L 3

Nor \ Exp

Exp \ Nor

r & % X X ◦ XXX XXX N3 XX XXX XXXr r % & & HH Gen N3 HH N4N HH HH HH H HH HH Hr & %

%

N4⊥

Figure 10.1 value, the least possible one, in models of normal logics. Thus, a normal logic is characterized by its intuitionistic fragment and by ideals of contradictions of its models. In this sense, the normal logics represent the structures of contradictions in all N4⊥ -extensions and are similar to negative logics in the class of Lj-extensions. It will be shown in the next section that the class Gen is connected to the classes Exp and Nor via the system of counterparts in a similar way to the connection of the class Par with the classes Int and Neg. Our plans for the rest of this section are as follows. First, we consider the restrictions of the operator σ to classes Exp and Nor and point out the perfect analogy with the situation described in Section 10.1. For any L ∈ Int+ , the inverse image of L with respect to the corresponding restriction of σ forms an interval in the class Exp (Nor), and the end points of such interval can be translated into L. Further, we study interrelations of the class Gen and classes Exp and Nor.

10.2 The Lattice Structure of EN4⊥

193

Denote σ 3 := σ Exp and σ n := σ Nor . The mappings η 3 : Int+ → Exp and η n : Int+ → Nor are deﬁned as follows. For every L ∈ Int+ , η 3 (L) := η(L) + {∼ p → (p → q)} and η n (L) := η(L) + {¬¬(p∨ ∼ p)}. Clearly, η ◦ (L) = η 3 (L) + η n (L). Logics in Exp having the form η 3 (L) we call special explosive or se-logics, and logics in Nor of the form η n (L) are called special normal or sn-logics. Models of se- and sn-logics are described in the following. Proposition 10.2.6 Let L ∈ Int+ and A be an N4⊥ -lattice. 1. A |= η 3 (L) iﬀ A |= L and Δ(A) = {0}. 2. A |= η n (L) iﬀ A |= L and ∇(A) = Fd (A ). Proof. It can be proved similarly to Proposition 10.1.3. 2 In the following, we omit the proofs if they can be obtained similarly to the results of the previous section. For a Heyting algebra A, we deﬁne A 3 := T w(A, A, {0}) and An := T w(A, Fd (A), A).

If an N3-lattice A is isomorphic to a lattice of the form B3 , we call it a special N3-lattice. If an N4⊥ -lattice A is isomorphic to a lattice of the form Bn , we call it a special normal N4⊥ -lattice. Translations of η 3 (L) and η n (L) into L can be deﬁned in the following way. The extended set P rop of propositional variables and formula ϕ were deﬁned in the previous section. Put ϕ3 :=

¬(p ∧ p) → ϕ ,

p∈var(ϕ)

ϕn :=

p∈var(ϕ)

¬¬(p ∨ p) → ϕ .

10 The Class of N4⊥ -Extensions

194

Proposition 10.2.7 Let A be a Heyting algebra and ϕ a formula in the language L⊥ . The following equivalences hold. 3 1. A 3 |= ϕ if and only if A |= ϕ . n 2. A n |= ϕ if and only if A |= ϕ .

2 Corollary 10.2.8 Let L ∈ Int+ and ϕ ∈ F or ⊥ . The following equivalences hold. 1. ϕ ∈ η 3 (L) if and only if ϕ3 ∈ L. 2. ϕ ∈ η n (L) if and only if ϕn ∈ L. 2 Let K be a class of Heyting algebras. Put K3 := {A 3 | A ∈ K} and Kn := {An | A ∈ K}.

Proposition 10.2.9 Let L ∈ Int+ and L = LK. Then η 3 (L) = LK3 and η n (L) = LKn . 2 Corollary 10.2.10 Let L ∈ EN4⊥ . 1. L is a special explosive logic if and only if L is determined by some family of special N3-lattices. 2. L is a special normal logic if and only if L is determined by some family of special normal N4⊥ -lattices. 2 Proposition 10.2.11 For any L ∈ Int+ , (σ 3 )−1 (L) = [η 3 (L), η ◦ (L)] and (σ n )−1 (L) = [η n (L), η ◦ (L)]. 2 By analogy with Proposition 10.1.10, special explosive and special normal logics can be characterized via the admissibility of inference rules.

10.3 Explosive and Normal Counterparts

195

Proposition 10.2.12 Let L ∈ EN4⊥ . 1. L is a special explosive logic if and only if L ∈ Exp and all rules of the 3 are admissible in L. form ψ/ψ 2. L is a special normal logic if and only if L ∈ Nor and all rules of the n are admissible in L. form ψ/ψ 2 This characterization allows one to prove an analog of Proposition 10.1.12. The statements on σ 3 and η 3 were established earlier in [47]. Proposition 10.2.13 1. σ 3 : Exp → Int+ and σ n : Nor → Int+ are lattice epimorphisms commuting with inﬁnite meets and joins. 2. η 3 : Int+ → Exp and η n : Int+ → Nor are lattice monomorphisms commuting with inﬁnite meets and joins. 2

10.3

Explosive and Normal Counterparts

The decomposition of EN4⊥ into classes Exp, Nor, and Gen is very similar to the decomposition of the class Jhn of non-trivial extensions of minimal logic into subclasses of intermediate, negative and properly paraconsistent logics. Our next step is to deﬁne explosive and normal counterparts for logics in EN4⊥ , in exactly the same way as we deﬁned intuitionistic and negative counterparts for extensions of minimal logic. The mappings (−)exp : EN4⊥ → Exp, (−)nor : EN4⊥ → Nor, and (−)ne : EN4⊥ → Exp ∩ Nor are deﬁned by the rules: Lexp := L + N3, Lnor := L + N4N , Lne := L + N3◦ , where L ∈ EN4⊥ . Obviously, Lne := Lexp + Lnor . Call the logic Lexp an explosive counterpart of L, Lnor a normal counterpart of L, and Lne a normal explosive counterpart of L. Thus, by deﬁnition the explosive (normal) counterpart of a logic L ∈ EN4⊥ is the least explosive (normal) logic containing L. Notice that N4⊥ has the following counterparts: ⊥ N ⊥ ◦ N4⊥ exp = N3, N4nor = N4 , N4ne = N3 .

Some simple properties of counterparts are collected in the proposition below.

10 The Class of N4⊥ -Extensions

196

Proposition 10.3.1 phisms.

1. (−)exp , (−)not , and (−)ne are lattice epimor-

2. L ∈ Exp iﬀ L = Lexp iﬀ Lnor = Lne . 3. L ∈ Nor iﬀ L = Lnor iﬀ Lexp = Lne . 4. L ∈ EN3◦ iﬀ Lexp = Lnor . 5. σ(L) = σ(Lexp ) = σ(Lnor ) = σ(Lne ) for every L ∈ EN4⊥ . 6. Lne = η ◦ σ(L) for every L ∈ EN4⊥ . Proof. Item 1 follows from the distributivity of the lattice EN4⊥ . Items 2 and 3 hold by the deﬁnition of counterparts. Item 4 follows from the relation N3◦ = N3 + N4N . Homomorphism properties of σ imply Item 5. We prove the last item. By Item 5 σ(Lne ) = σ(L) and Lne ∈ EN3◦ by deﬁnition. Thus, the equality Lne = η ◦ σ(L) follows from the fact that η ◦ is a lattice isomorphism of Int+ and EN3◦ (see Proposition 10.1.12). 2 The semantic characterization of logics N3, N4N , and N3◦ allows one to characterize models of counterparts as follows. Proposition 10.3.2 Let L ∈ EN4⊥ and A be an N4⊥ -lattice. 1. A |= Lexp if and only if A |= L and Δ(A) = {0}. 2. A |= Lnor if and only if A |= L and ∇(A) = Fd (A). 3. A |= Lne if and only if A |= L, Δ(A) = {0}, and ∇(A) = Fd (A). 2 We now study how the counterparts of L can be deﬁned in L. For any formula ϕ, we put ¬(p∧ ∼ p), ϕn := ¬¬(p∨ ∼ p), ϕe := p∈var(ϕ)

p∈var(ϕ)

and ϕexp := ϕe → ϕ, ϕnor := ϕn → ϕ, ϕne := (ϕe ∧ ϕn ) → ϕ. Let A be a twist-structure, A = T w(B, ∇, Δ). We associate with A the following substructures: Aexp = T w(B, ∇, {0}), Anor = T w(B, Fd (B), Δ), Ane = T w(B, Fd (B), {0}).

10.3 Explosive and Normal Counterparts

197

According to Proposition 10.3.2 if A is a model of L, then Aexp , Anor , and Ane are models of explosive, normal and normal explosive counterparts of L, respectively. The validity of formulas on Aexp, Anor , and Ane can be simulated in A as follows. Proposition 10.3.3 Let A ∈ S (B) and ϕ be a formula. 1. Aexp |= ϕ if and only if A |= ϕexp . 2. Anor |= ϕ if and only if A |= ϕnor . 3. Ane |= ϕ if and only if A |= ϕne . Proof. 1. Let A |= ϕexp and v be an Aexp-valuation. Note that v is also an A-valuation, since Aexp ≤ A. For any (a, b) ∈ Aexp , a ∧ b = 0. Consequently, π1 v( p∈var(ϕ) ¬(p∧ ∼ p)) = 1. And since A |= ϕexp , we have π1 v(ϕ) = 1. We proved Aexp |= ϕ. Assume now that Aexp |= ϕ but A |= ϕexp . Let v be an A-valuation such that π1 v(ϕexp ) = 1. Put a := π1 v( p∈var(ϕ) ¬(p∧ ∼ p)) and F := a . Then a ≤ π1 v(ϕ). Consider the quotient structure A/F and the quotient valuation v/F . By Proposition 9.2.16 A/F ∼ = A1 := T w(B/ a , ∇/ a , Δ/ a ). Applying the same proposition we obtain (A1 )exp ∼ = Aexp /F . Clearly, π1 v/ F ( p∈var(ϕ) ¬(p∧ ∼ p)) = 1, whence v/F can be considered as a valuation in (A1 )exp . Since a ≤ π1 v(ϕ), we have π1 v/F (ϕ) = 1. Thus, Aexp /F |= ϕ, which conﬂicts with the assumption Aexp |= ϕ. The proofs of Items 2 and 3 are similar and are based on the following facts. For any A-valuation v, v is an Anor -valuation iﬀ π1 v(¬¬(p∨ ∼ p)) = 1 for all p, and v is an Ane -valuation iﬀ for all p, π1 v(¬(p∧ ∼ p)) = 1 and π1 v(¬¬(p∨ ∼ p)) = 1. 2 Proposition 10.3.4 Let L ∈ EN4⊥ and ϕ be a formula. 1. ϕ ∈ Lexp if and only if ϕexp ∈ L. 2. ϕ ∈ Lnor if and only if ϕnor ∈ L. 3. ϕ ∈ Lne if and only if ϕne ∈ L.

10 The Class of N4⊥ -Extensions

198

Proof. We prove only the ﬁrst item. The proofs of others are similar and are based on the respective items of the last proposition. Assume ϕ ∈ Lexp . Let A be a twist-structure such that L |= A. Then Aexp |= Lexp . Indeed, Aexp |= L as a substructure of A and Δ(Aexp ) = 0. By Item 1 of the previous proposition we have A |= ϕexp . Finally, since every model of L can be represented as a twist-structure, we conclude ϕexp ∈ L. Conversely, let ϕexp ∈ L. Consider a twist-structure A |= Lexp . By Proposition 10.3.2 Δ(A) = 0, i.e., A = Aexp . By Item 1 of the last proposition A |= ϕ if and only if A |= ϕexp , from which we conclude ϕ ∈ Lexp . 2 Let K be a class of twist-structures and τ ∈ {exp, nor, ne}. We put Kτ := {Aτ | A ∈ K}. Proposition 10.3.5 Let K be a class of twist-structures, L = LK, and τ ∈ {exp, nor, ne}. Then Lτ = LKτ . Proof. Consider the case of explosive counterparts. Obviously, Lexp ⊆ LKexp . If ϕ ∈ Lexp , then ϕexp ∈ L by the previous proposition. Consequently, there is A ∈ K such that A |= ϕexp . By Proposition 10.3.3 Aexp |= ϕ. 2 At the end of this section, we consider classes of logics having given explosive and normal logics as explosive and normal counterparts. According to Proposition 10.3.1 explosive and normal counterparts of a logic have the same intuitionistic fragment. Therefore, for given L1 ∈ Exp and L2 ∈ Nor, there is a logic L with Lexp = L1 and Lnor = L2 if and only if σ(L1 ) = σ(L2 ). For L1 ∈ Exp and L2 ∈ Nor with σ(L1 ) = σ(L2 ), we deﬁne the family of logics Spec(L1 , L2 ) := {L ∈ EN4⊥ | Lexp = L1 and Lnor = L2 } and the logic L1 ∗ L2 := N4⊥ + {ϕexp | ϕ ∈ L1 } ∪ {ϕnor | ϕ ∈ L2 }.

First of all, we note that all logics of Spec(L1 , L2 ) have the same intuitionistic fragment and that if one of the logics L1 or L2 belongs to the intersection Exp ∩ Nor, the class Spec(L1 , L2 ) is one-element.

10.3 Explosive and Normal Counterparts

199

Proposition 10.3.6 Let L1 ∈ Exp and L2 ∈ Nor be such that L = σ(L1 ) = σ(L2 ). 1. Spec(L1 , L2 ) ⊆ σ −1 (L). 2. If L2 ∈ Exp, then Spec(L1 , L2 ) = {L1 }. 3. If L1 ∈ Nor, then Spec(L1 , L2 ) = {L2 }. Proof. 1. If L ∈ Spec(L1 , L2 ), then Lexp = L1 and L = σ(Lexp ) = σ(L ) + σ(N3) = σ(L ) + Li = σ(L ). 2. If L2 ∈ Exp∩Nor = EN3◦ , then L2 = η ◦ (L). Since η ◦ (L) is the greatest element of σ −1 (L), we obtain L1 ⊆ L2 . Let L ∈ Spec(L1 , L2 ). We then have L + N4N = L2 ⊇ L1 ⊇ N3. We claim that N3 ⊆ L in this case. Assume that L is not explosive. There then is a twist-structure A |= L with Δ(A) = {0}. By Proposition 10.3.2 Anor |= L2 = Lnor and Δ(Anor ) = Δ(A) = {0}. The latter is impossible, since L2 is explosive. From N3 ⊆ L and L1 = L + N3 we obtain L = L1 . 3. From L1 ∈ Nor we conclude L2 ⊆ L1 and ﬁnish the proof similar to the previous item. 2 In the case when neither of L1 or L2 is normal explosive, the class Spec(L1 , L2 ) forms an interval in the lattice EN4⊥ containing at least two points. Proposition 10.3.7 Let L1 ∈ Exp and L2 ∈ Nor be such that σ(L1 ) = Nor and L2 ∈ Exp, σ(L2 ). Then Spec(L1 , L2 ) = [L1 ∗ L2 , L1 ∩ L2 ]. If L1 ∈ then L1 ∗ L2 = L1 ∩ L2 . Proof. It follows by deﬁnition that L ⊆ Lexp ∩ Lnor for all L ∈ EN4⊥ , i.e., L ⊆ L1 ∩ L2 for all L ∈ Spec(L1 , L2 ). Let us check that L1 ∩ L2 ∈ Spec(L1 , L2 ). We calculate (L1 ∩ L2 )exp = (L1 + N3) ∩ (L2 + N3) = L1 ∩ η ◦ σ(L1 ) = L1 . That (L1 ∩ L2 )nor = L2 , can be checked similarly. We have thus proved that L1 ∩ L2 is the greatest point of Spec(L1 , L2 ). Let us consider the logic L1 ∗ L2 . If L ∈ Spec(L1 , L2 ), then {ϕexp | ϕ ∈ L1 } ∪ {ϕnor | ϕ ∈ L2 } ⊆ L

200

10 The Class of N4⊥ -Extensions

by Proposition 10.3.4, i.e., L1 ∗ L2 ⊆ L. It remains to verify that L1 ∗ L2 belongs to Spec(L1 , L2 ). Let L∗ := L1 ∗ L2 . The inclusions L1 ⊆ L∗exp and L2 ⊆ L∗nor follow from the deﬁnition of L1 ∗ L2 and Proposition 10.3.4. Let us prove the inverse inclusions. By deﬁnition L∗exp = N3 + {ϕexp | ϕ ∈ L1 } ∪ {ϕnor | ϕ ∈ L2 } = L1 + {ϕne | ϕ ∈ L2 }. The last equality is due to the fact for any ϕ, N3 ϕ ↔ ϕexp and N3 ϕnor ↔ ϕne , which follows from ¬(p∧ ∼ p) ∈ N3. By Proposition 10.3.4 a formula of the form ϕne belongs to L∗exp if and only if ϕ ∈ (L∗exp )ne . The equality σ(L1 ) = σ(L2 ) implies L2 ⊆ (L2 )ne = (L1 )ne , whence, ϕ ∈ L2 implies ϕne ∈ L1 . Thus, L∗exp = L1 . That L∗nor = L2 , can be proved similarly. Assume L1 ∈ Nor and L2 ∈ Exp. Prove that in this case (∼ p → (p → q)) ∨ ¬¬(r∨ ∼ r) ∈ (L1 ∩ L2 ) \ (L1 ∗ L2 ). Since ∼ p → (p → q) ∈ L1 and ¬¬(r∨ ∼ r) ∈ L2 , the disjunction of these formulas belongs to the intersection L1 ∩ L2 . To prove (∼ p → (p → q)) ∨ 3 ∗RM3⊥ , ¬¬(r∨ ∼ r) ∈ L1 ∗L2 note that on the one hand, L1 ∗L2 ⊆ B→ 4 =L and on the other hand, (∼ p → (p → q)) ∨ ¬¬(r∨ ∼ r) ∈ B→ 4 . 2 As a consequence, we obtain a semantic characterization of logics of the form L1 ∗ L2 . Corollary 10.3.8 Let L1 ∈ Exp and L2 ∈ Nor be such that σ(L1 ) = σ(L2 ). For every N4⊥ -lattice A holds the equivalence A |= L1 ∗ L2 ⇔ Aexp |= L1 and Anor |= L2 . Proof. The direct implication follows from the fact that L1 ∗ L2 ∈ Spec(L1 , L2 ) stated above. The inverse implication is by the deﬁnition of L1 ∗ L2 . 2

10.4 The Structure of EN4C and EN4⊥ C

10.4

201

The Structure of EN4C and EN4⊥ C

In this section, we completely describe the extensions of the logics N4⊥ C := N4⊥ + {C} and N4C := N4 + {C}, which are obtained by adding to N4⊥ and respectively to N4 Dummett’s linearity axiom C = (p → q) ∨ (q → p). This result is interesting for the following reasons. First, comparison of the structures of EN4⊥ C and EN4C vividly demonstrates how the regular structure of the lattice N4⊥ is collapsed if we delete the intuitionistic negation from the language. In particular, one cannot deﬁne normal logics in the class EN4. Second, it is worth comparing the structures of the lattices EN4⊥ C, EN3C and ELC, where N3C := N3 + {C} and LC is Dummett’s logic obtained by adding the linearity axiom to intuitionistic logic, i.e. LC = Lil in the denotation of the ﬁrst part of the book. Dummett’s logic is the ﬁrst example of superintuitionistic pretabular logic, whose structure of extensions was completely described [27]. Recall that a logic is pretabular if any of its extensions is tabular, i.e., is determined by a ﬁnite algebra, but the logic is not tabular itself. In fact, the article [27] by M.J. Dunn and R. Meyer was published earlier then the notion of pretabular logic was introduced and generated an interest in pretabular logics as logics whose class of extensions admits an exhaustive description. M. Kracht [47] described the lattices of N3C-extensions and proved that although this logic is not pretabular, it preserves the most essential properties of pretabular logics. Namely, all extensions of N3C are ﬁnitely axiomatizable and decidable. Moreover, given a formula ϕ, one can determine which of the N3C-extensions is axiomatized by ϕ. It turns out that the class of N4⊥ C-extensions also satisﬁes all these properties. Moreover, comparing lattices EN4⊥ C and EN3C demonstrates the complexity of the lattice of N4⊥ -extensions. Finally, notice that the description of the lattice N3C by M. Kracht [47] was motivated by M. Gelfond’s question on the structure of the upper part of the lattice EN3. This is connected with the so-called answer set semantics for logic programs. The logic of “here-and-there” with strong negation lying in the upper part of EN3 plays an important role in the study of the answer set semantics. For this reason, it is important to have same information on the surroundings of this logic in the lattice EN3. In the study of paraconsistent answer sets [82] there arises a paraconsistent version of the logic “here-andthere” with strong negation, which belongs to the upper part of EN4⊥ . This naturally leads to the question on the description of the lattice EN4⊥ C.

10 The Class of N4⊥ -Extensions

202

q Lch2 = Lk q Lch3 q Lch4 pp p q LC

Figure 10.2 Recall that the class ELC of extensions of Dummett’s logic has the following structure. Let chn be a linearly ordered n-element Heyting algebra. Algebras chn are all up to isomorphism ﬁnitely generated subdirectly irreducible models of LC. Obviously, for any n ∈ ω, chn → chn+1 . Therefore, every proper LC-extension has the form Lchn for some n ∈ ω, and ELC is isomorphic to a linear order of type (ω + 1)∗ (see Figure 10.2). Given a logic L, consider the class M odf si (L) of its ﬁnitely generated subdirectly irreducible models. Deﬁne on M odf si (L) the following preordering: A B ⇔ LA ⊆ LB. The following theorem by Jonsson is well known. In a congruence distributive variety, all subdirectly irreducible elements of the subvariety generated by the class K of algebras belong to HSUp(K). If an algebra A is ﬁnite, then HSUp({A}) = HS({A}). The varieties of N4-lattices and N4⊥ -lattices are congruence distributive, therefore, for any ﬁnite N4-(N4⊥ -)lattices A and B, holds the equivalence LA ⊆ LB ⇔ B ∈ HS({A}). As was noted above, all ﬁnitely generated subdirectly irreducible models of Dummett’s logic LC are ﬁnite. According to Corollary 9.2.11, an N4(N4⊥ )lattice A is subdirectly irreducible if and only if A is subdirectly irreducible. The fact that if A is ﬁnitely generated, then A is ﬁnitely generated, follows from the lemma below. Lemma 10.4.1 If an N4(N4⊥ )-lattice A is ﬁnitely generated, then A is ﬁnitely generated.

10.4 The Structure of EN4C and EN4⊥ C

203

Proof. If A is generated by the set {a1 , . . . , an }, then its reduct |A|, ∨, ∧, → ( |A|, ∨, ∧, →, ⊥ ) is generated by the set {a1 , . . . , an , ∼ a1 , . . . , ∼ an }. This follows from the fact that every formula is equivalent in N4(N4⊥ ) to an nnf. By deﬁnition A = |A|, ∨, ∧, → / ≈ ( |A|, ∨, ∧, →, ⊥ / ≈). Consequently, A is generated by the set {[a1 ]≈ , . . . , [an ]≈ , [∼ a1 ]≈ , . . . , [∼ an ]≈ }. 2 Thus, every ﬁnitely generated subdirectly irreducible model of N4C (N4⊥ C) is isomorphic to a twist-structure over chn for suitable n ∈ ω, therefore, all ﬁnitely generated subdirectly irreducible models of N4C and N4⊥ C are ﬁnite. In this section, we consider only extensions of the logics N4C and N4⊥ C, therefore, in the following for A, B ∈ M odf si (L), holds the equivalence: A B ⇔ B ∈ HS({A}). Every extension of a logic L is determined by the class of its ﬁnitely generated subdirectly irreducible models. This class forms a cone in the preordering (M odf si (L), ). Of course, not every cone of (M odf si (L), ) can be represented in the form M odf si (L ) for a suitable L-extension L in the general case. However, we prove that for every cone U of (M odf si (N4⊥ C), ), there exists L ∈ EN4⊥ C such that M odf si (L) = U . Cones of the preordering (M odf si (N4⊥ C), ) are closed under isomorphism. Note that the isomorphism relation, coincides with the equivalence determined by the preordering . Factoring out the preordering (M odf si (N4⊥ C), ) wrt the isomorphism relation, we obtain the partial ordering of isomorphism types of ﬁnitely generated subdirectly irreducible N4⊥ C-models, which we denote (M od∗f si (N4⊥ C), ∗ ). In this way, the description of the lattice EN4⊥ C is reduced to the description of the ordering (M od∗f si (N4⊥ C), ∗ ). Let |chn | = {0, 1, . . . , n − 1}, 0 ≤ 1 ≤ . . . ≤ n − 1. Ideals of chn are exactly non-empty initial segments of chn . The ﬁlter Fd (chn ) equals {1, . . . , n − 1}. Therefore, if A = T w(chn , ∇, Δ), then ∇ = chn or ∇ = Fd (chn ), and Δ = {0, . . . , m} for some m ≤ n − 1. We have thus proved |S (chn )| = 2n. Denote chn (k, +) := T w(chn , chn , {0, . . . , k − 1}), chn (k, −) := T w(chn , Fd (chn ), {0, . . . , k − 1}), where 1 ≤ k ≤ n.

10 The Class of N4⊥ -Extensions

204

chn (1, −) chn (2, −) p p

chn (n − 2, −) chn (n − 1, −) chn (n, −)

q @ @

q @ @

p

q @

q @ @ p p

@

q @

@ @q

chn (1, +)

@q

p

chn (2, +)

@q

chn (n − 2, +)

@q

chn (n − 1, +)

@q

chn (n, +) Figure 10.3

Let A, B ∈ S (chn ). As it follows from Propositions 9.2.8 and 9.2.16 every proper homomorphic image of B is isomorphic to a twist-structure over a proper homomorphic image of chn , i.e., over chm , m < n. In this way, B A if and only if A → B. Thus, according to Proposition 9.1.10 B A if and only if |∇(A)| ≤ |∇(B)| and |Δ(A)| ≤ |Δ(B)|. The structure of S (chn ) ordered by is depicted in Figure 10.3. Proposition 9.2.2 states that a homomorphic image h(A) of an N4⊥ lattice A is uniquely determined by the homomorphic image h (A ) of the basic Heyting algebra. Since every ﬁlter on chn has the form Fm := {m, . . . , n − 1} and chn /Fm ∼ = chm+1 , Proposition 9.2.16 allows one to conclude that quotients of twist-structures over chn have the following form: chn (k, )/(Fm ) ∼ = chm+1 (min{k, m + 1}, ), m ≤ n − 1, ∈ {+, −}. Consider the question of embedding twist-structures of the form chn (k, ). If h : chn1 (k1 , 1 ) → chn2 (k2 , 2 ), then according to Proposition 9.2.15 h embeds chn1 into chn2 and the inclusions h (∇(chn1 (k1 , 1 ))) ⊆ ∇(chn2 (k2 , 2 )), h (Δ(chn1 (k1 , 1 ))) ⊆ Δ(chn2 (k2 , 2 )) hold. Consequently, n1 ≤ n2 and k1 ≤ k2 . From h (0) = 0 and ﬁrst of the above inclusions we conclude that 1 = + implies 2 = +. We set 1 ≤ 2 if 1 = 2 , or 1 = − and 2 = +. It is easy to see that the conditions 1 ≤ 2 , n1 ≤ n2 , and k1 ≤ k2 guarantee that there exists an embedding g : chn1 → chn2 such that

10.4 The Structure of EN4C and EN4⊥ C

205

g({0, . . . , k1 −1}) ⊆ {0, . . . , k2 −1}. Again, by Proposition 9.2.15 we conclude that there exists an embedding h : chn1 (k1 , 1 ) → chn2 (k2 , 2 ) such that h = g. We have thus proved the following statement. Lemma 10.4.2 Let n1 , n2 , k1 , k2 ∈ ω, k1 ≤ n1 , k2 ≤ n2 , and 1 , 2 ∈ {+, −}. 1. chn1 (k1 , 1 ) → chn2 (k2 , 2 ) if and only if n1 ≤ n2 , k1 ≤ k2 , and 1 ≤ 2 . 2. If chn1 (k1 , 1 ) ∈ H(chn2 (k2 , 2 )), then chn1 (k1 , 1 ) → chn2 (k2 , 2 ). 2 Let T := n∈ω S (chn ). Combining Lemma 10.4.2 with the structure of (S (chn ), ) presented in Figure 10.3, we obtain that T is ordered by as depicted in Figure 10.4. Since every ﬁnitely generated subdirectly irreducible model of N4⊥ C is isomorphic to some element of T, we arrive at the following statement. Proposition 10.4.3 The preordering (M od∗f si (N4⊥ C), ∗ ) has the structure depicted in Figure 10.4. qch2 (1, −) @ qch2 (1, +) S (ch2 ) q q @ @ @ q@qq @qch3 (1, +) @ @ @ q @qch4 (1, +) q q q S (ch3 ) @ @ @ q@qq@q @q @ @ q q @q @ q S (ch4 ) q q @ @ q q @q @ q @ q q q @q q q q

Figure 10.4

10 The Class of N4⊥ -Extensions

206

Describe now the cones of (T, ). Let A ∈ T. If A = chn (k, ), we say that A is of type (n, k, ) and write tp(A) = (n, k, ). By tpi (A), i = 1, 2, 3, we denote the i-th coordinate of tp(A). For a cone U of (T, ) and i = 1, 2, we put tp− i (U ) tp+ i (U )

:= :=

max{tpi (A) | A ∈ U and tp3 (A) = −}, max{tpi (A) | A ∈ U and tp3 (A) = +}.

− Clearly, tp+ i (U ) ≤ tpi (U ) for i = 1, 2. Further, we distinguish the following cones:

T− Tk T− k

:= := :=

{A ∈ T | tp3 (A) = −}, {A ∈ T | tp2 (A) ≤ k}, {A ∈ T | tp3 (A) = − and tp2 (A) ≤ k}.

Proposition 10.4.4 Every cone of (T, ) can be represented as U ∪ V , where U is a ﬁnite cone and V is one of the following cones: − − ∅, T, T− , Tk , T− k , T ∪ Tk , Tk ∪ Tk ,

where k < k . − Proof. Let U be a cone in (T, ). If tp− 1 (U ) < ω and tp2 (U ) < ω, it can easily be seen from Figure 10.4 that U is ﬁnite or empty. + If tp+ 1 (U ) = ω and tp2 (U ) = ω, then U = T. − − ⊆ U . If additionally Let tp− 1 (U ) = ω and tp2 (U ) = ω. Then T + + − tp1 (U ) < ω and tp2 (U ) < ω, then U = T ∪ U , where U is ﬁnite. + + If tp+ 1 (U ) = ω and tp2 (U ) < ω, there exists m ≤ tp2 (U ) such that chn (m, +) ∈ U for all n. Let

k := max{m | chn (m, +) ∈ U for all n}. Then U = T− ∪ Tk ∪ U , where U is ﬁnite. − Let tp− 1 (U ) = ω and tp2 (U ) < ω. Put k := max{m | chn (m, −) ∈ U for all n}. Then T− k ⊆U U = T− k ∪ Tk

and arguing as above, we obtain that either U = T− k ∪ U , or ∪ U , k < k, or U = Tk ∪ U , where U is some ﬁnite cone. 2

Theorem 10.4.5 The lattice EN4⊥ C is isomorphic to the lattice of cones of the partial ordering presented in Figure 10.4. All elements of EN4⊥ C are ﬁnitely axiomatized and decidable.

10.4 The Structure of EN4C and EN4⊥ C

207

Proof. In view of Proposition 10.4.3, the lattice EN4⊥ C is embedded into the lattice of cones of the partial order presented in Figure 10.4. We have to prove that this embedding is onto, i.e., that for any cone of (T, ), there is L ∈ EN4⊥ C such that T(L) := M odf si (L) ∩ T = U . Since T contains all up to isomorphism ﬁnitely generated subdirectly irreducible models of N4⊥ C, the equality T(L) = U implies LU = L. First, we note that the empty cone corresponds to the trivial logic F or ⊥ . According to Corollary 10.2.2 every non-trivial extension of N4⊥ is consistent, therefore, every contradiction axiomatizes F or ⊥ over N4⊥ C. Consider p0 ∧ ∼ p0 as a “standard” axiom of the trivial logic. Prove the equality T− = T(N4⊥ C + {¬¬(p∨ ∼ p)}). By deﬁnition the cone T− contains exactly algebras from T satisfying ∇(A) = Fd (A). This condition is equivalent to the validity of ¬¬(p∨ ∼ p) on A, which implies the desired equality. Consider the formulas (pk ∧ ∼ pk ) ↔ (pm ∧ ∼ pm ), Dn := 1≤k<m≤n+1

En :=

pk ↔ pm .

1≤k<m≤n+1

Let A ∈ T. Recall that by deﬁnition Δ(A) = π2 {a∨ ∼ a | a ∈ A}. We have π2 (a∨ ∼ a) = π1 (∼ (a∨ ∼ a)) = π1 (a∧ ∼ a). Thus, Δ(A) = π1 ({a∧ ∼ a | a ∈ A}). In this way, A |= Dn is equivalent to the fact that v(En ) = 1 for any A -valuation such that v(pk ) ∈ Δ(A) for 1 ≤ k ≤ n + 1. Since A is linearly ordered, we immediately obtain that A |= Dn if and only if Δ(A) contains at most n elements. In other words, A |= Dn if and only if tp2 (A) ≤ n. We have thus proved Tn = T(N4⊥ C + {Dn }). − Since T− n = T ∩ Tn , this cone is distinguished in T by the set of axioms {¬¬(p∨ ∼ p), Dn }. For A ∈ T, the condition tp1 (A) ≤ n is equivalent to A = chm , m ≤ n. The class of models with this property is distinguished in M odf si (LC) by the formula En . By Lemma 10.1.2 the validity of an intuitionistic formula ϕ on a twiststructure over a Heyting algebra B is equivalent to B |= ϕ. In particular, A |= En if and only if A |= En . The cone of algebras satisfying the condition tp1 (A) ≤ n corresponds to the logic N4⊥ C + {En }. Note that by Lemma 10.4.2 this cone is generated by chn (n, +), i.e., by the full twiststructure over chn .

208

10 The Class of N4⊥ -Extensions

The cone generated by chn (k, +) consists of algebras chm (s, ) satisfying the conditions m ≤ n and s ≤ k. Therefore, it is distinguished in T by the formulas {En , Dk }. Algebras over chn (k, −) satisfy additionally the condition = −. Therefore, the cone generated by chn (k, −) is distinguished in T by the formulas {En , Dk , ¬¬(p∨ ∼ p)}. Note that all logics corresponding to the cones up to this moment are ﬁnitely axiomatizable. In view of Proposition 10.4.4 every cone of (T, ) can be represented as a ﬁnite union of cones of the forms considered above. The logic of a union of cones is the intersection of logics of cones. If L1 = L + {ϕ1 , . . . , ϕn } and L2 = L + {ψ1 , . . . , ψn }, moreover, the formulas ϕ1 , . . . , ϕn do not contain propositional variables occurring ψ1 , . . . , ψn and vice versa, then L1 ∩ L2 := L + {ϕi ∨ ψj | 1 ≤ i ≤ n, 1 ≤ j ≤ m}. In the case when L, L1 and L2 are extensions of positive or minimal logic, this result was stated in Proposition 2.1.5. Its proof is based on the facts that modus ponens is the only inference rule and that L, L1 and L2 contain axioms of positive logics. Therefore, the result remains true for extensions of N4⊥ . In this way, Proposition 10.4.4 implies that every extension of N4⊥ C is ﬁnitely axiomatizable. We know axiomatics for any logic in EN4⊥ C. Given a logic L ∈ EN4⊥ C with a ﬁnite list of axioms A and some formula ϕ, we enumerate all formulas deducible from A and from A ∪ {ϕ}. At a ﬁnite step we deduce ϕ from A or we deduce from A ∪ {ϕ} an axiom of some other logic not contained in L. Thus, L is decidable. 2 ⊥ − Note that every logic L ∈ EN4 C such that T ⊆ M odf si (L) is not tabular. Consequently, N4⊥ C is not pretabular. Moreover, it has inﬁnitely many non-tabular extensions. It follows from the proof of Theorem 10.4.5 that we can construct such an enumeration L0 , L1 , . . . of all logics from EN4⊥ C that given n one can eﬀectively construct the list of axioms of Ln . Fix some enumeration of this kind. Proposition 10.4.6 There is an algorithm ﬁnding for any formula ϕ a logic Ln ∈ EN4⊥ C such that Ln = N4⊥ C + {ϕ}. Proof. Let ψ0 , ψ1 , . . . be an eﬀective enumeration of all formulas inferable from N4⊥ C + {ϕ}. The desired algorithm works as follows.

10.4 The Structure of EN4C and EN4⊥ C

209

Step n. For all m ≤ n, check whether all axioms of Lm are contained in the list of formulas ψ0 , . . . , ψn . If we have a positive answer for some m0 ≤ n, i.e., we have established the inclusion Lm0 ⊆ N4⊥ C + {ϕ}, we then check whether ϕ ∈ Lm0 . In case of the second positive answer, we have proved Lm0 = N4⊥ C + {ϕ} and the algorithm terminates. Otherwise, we pass on to the next step. Since the list L0 , L1 , . . . contains all logics from EN4⊥ C, and the list ψ0 , ψ1 , . . . all formulas from N4⊥ C + {ϕ}, at some ﬁnite step we meet a number n such that Ln = N4⊥ C + {ϕ}. 2 We have thus proved that given a formula, one can determine which of the extensions of N4⊥ C is axiomatized by this formula. In conclusion to this section, we consider brieﬂy how the situation changes if we pass on to the logic N4C in the restricted language L = ∨, ∧, →, ∼ and to the explosive logic N3C. Consider the logic LpC := Lp + {(p → q) ∨ (q → p)}. Finitely generated subdirectly irreducible models of LpC are ﬁnite linearly ordered implicative lattices, the reducts of Heyting algebras chn to the language

∨, ∧, → . As above, we have Fd (chn ) = {1, . . . , n − 1} (recall that for implicative lattices Fd (A) = {a ∨ (a → b) | a, b ∈ |A|} ). The lattice ELpC as well as ELC is a linear order of type (ω + 1)∗ . We can identify twiststructures from M odf si (N4C) with elements of T. As above chn (k, ) denotes a twist-structure A over n-element linearly ordered implicative lattice with k-element ideal Δ(A) and ∇(A) = A if = +, ∇(A) = Fd (A) if = −. However, the ordering diﬀers in this case (see Figure 10.5). The structures chn (k, ) have the same homomorphic images but the embedding relation changes. In the language ∨, ∧, →, ∼ , the condition h(0) = 0 should not be satisﬁed for a homomorphism h : A → B of N4lattices. Therefore, the additional embeddings chn (k, +) → chn+1 (k + 1, ) are possible. For this reason, the ordering (T, ) has the structure depicted in Figure 10.5. This ordering has fewer cones than the ordering presented in Figure 10.4. For example, in denotation of Proposition 10.4.4 we have Tk ⊆ T− k+1 . This shows that N4-lattices A satisfying the condition ∇(A) = Fd (A) do not form a variety and we cannot deﬁne analogs of normal logics as was done in the class of N4⊥ -extensions. Similarly to Theorem 10.4.5, we can prove the following statement. Theorem 10.4.7 The lattice EN4C is isomorphic to the lattice of cones of the partial order presented in Figure 10.5. All elements of EN4C are ﬁnitely axiomatizable and decidable.

10 The Class of N4⊥ -Extensions

210

qch2 (1, −) @ qch2 (1, +) S (ch2 ) q q!@ ! @!@ !q @qch (1, +) q! @ 3 !q ! !! @ @ ! ! !qq @qch (1, +) q! q! @ 4 S (ch3 ) ! @ @ !! !! @! ! !q @q !qq! @ q! @ ! ! ! ! @ @ q ! ! ! ! ! @ ! @ q q q q S (ch4 ) ! q ! @! @ q ! ! q q @q @ @ q q q @ q q q q

Figure 10.5

ch1 (1, −) q ch2 (1, −) q

@ @

ch3 (1, −) q

p p

p

@q

ch1 (1, +)

@q

ch2 (1, +)

@

@

ch4 (1, −) q

@ @

@q

ch3 (1, +)

@ @

@q

p p

p

ch4 (1, +)

Figure 10.6

Let us consider the logic N3C. Only twist-structures of the form chn (1, ) are models of N3C. Therefore, the isomorphism types of algebras from M odf si (N3C) are ordered wrt as shown in Figure 10.6. According to Theorem 10.4.5, the lattice EN3C is isomorphic to the lattice of cones of the above ordering. It is easy to see that EN3C has the structure depicted in Figure 10.7. This diagram was obtained for the ﬁrst time by M. Kracht [47].

10.5 Some Transfer Theorems for the Class of N4⊥ -Extensions

211

q

Lch1 (1, +) q Lch2 (1, +) q Lch3 (1, −) q Lch4 (1, −) q

@

@

@

@qLch1 (1, +)

@q @

@q @qLch2 (1, +) @ @ p @q @q p p @ @ p q @ p q @qLch3 (1, +) N3Cne p @ @ p @q @q p p @ p @ @q p @qLch4 (1, +) p @ p @q p p @ @q p p

p q

N3C Figure 10.7

10.5

Some Transfer Theorems for the Class of N4⊥ -Extensions

In this section we transfer several results well known for the class of intermediate logics to the class of N4⊥ -extensions. We obtain the characterization of tabular and pretabular logics and logics with the Craig interpolation property in the class EN4⊥ . The corresponding results for Int we mention without proofs. First, we describe tabular and pretabular logics in EN4⊥ , which generalizes the result by A. Sendlewski [99] for EN3. Let E be a lattice of logics and L ∈ E. We say that L is of ﬁnite codimension [93] if there is no inﬁnite descending chain of logics containing L. It is known that Theorem 10.5.1 [93] Let L ∈ Int. L is tabular if and only if L is of ﬁnite codimension in Int. 2 It turns out that an N4⊥ -extension is tabular if and only if its intuitionistic fragment is tabular.

10 The Class of N4⊥ -Extensions

212

Proposition 10.5.2 Let L ∈ EN4⊥ . L is tabular if and only if σ(L) is tabular. Proof. Let L = LA for some ﬁnite N4⊥ -lattice A. Then σ(L) = LA by Proposition 10.1.1. Thus, if L is tabular, σ(L) is tabular too. Assume σ(L) = LB for some ﬁnite Heyting algebra B. Since Heyting algebras are congruence distributive, all ﬁnitely generated subdirectly irreducible σ(L)-models belong to HS(B). Consequently, σ(L) has only a ﬁnite number of ﬁnitely generated subdirectly irreducible models {B1 , . . . , Bn } and all these models are ﬁnite. For every ﬁnitely generated subdirectly irreducible model A of L, A is a subdirectly irreducible model of σ(L) by Corollary 9.2.11, and A is ﬁnitely generated by Lemma 10.4.1. Consequently, every element of M odf si (L) is isomorphic to some A ∈ S (Bi ), i = 1, . . . , n. Thus, M odf si (L) contains only a ﬁnite number of algebras up to isomorphism, which means that L is tabular. 2 This fact and Theorem 10.5.1 yield the description of tabular logics in EN4⊥ . Theorem 10.5.3 Let L ∈ EN4⊥ . L is tabular if and only if L is of ﬁnite codimension in EN4⊥ . Proof. If L is tabular, up to isomorphism it has only a ﬁnite number of ﬁnitely generated subdirectly irreducible models {A1 , . . . , An }. This follows from the fact that the variety of N4⊥ -lattices is congruence distributive and the above-mentioned result by Jonsson. Any extension of L is determined by some subset of {A1 , . . . , An }. Therefore, L has ﬁnitely many extensions, in particular, it has a ﬁnite codimension. Let L have a ﬁnite codimension in EN4⊥ . By Proposition 10.1.12 the mapping η ◦ embeds the lattice Eσ(L) into EL. Therefore, σ(L) also has a ﬁnite codimension. Now the tabularity of σ(L) follows from Theorem 10.5.1 and the tabularity of L from Proposition 10.5.2. 2 In [55], L.L. Maksimova proved that there are exactly three pretabular intermediate logics. Theorem 10.5.4 [55] The class Int contains exactly three pretabular logics: LC LJ LH

:= := :=

Li + {(p → q) ∨ (q → p)}, Li + {p ∨ (p → (q ∨ ¬q))}, Li + {¬p ∨ ¬¬p, p ∨ (p → (q ∨ (q → (r ∨ ¬r))))}. 2

10.5 Some Transfer Theorems for the Class of N4⊥ -Extensions

213

Proposition 10.5.5 Let L ∈ EN4⊥ . L is pretabular if and only if L = Lne and σ(L) is pretabular. Proof. Let L be pretabular. Then σ(L) is pretabular by Proposition 10.5.2. Indeed, every proper extension of σ(L) is of the form σ(L ) for some proper extension L of L. By assumption L is tabular, whence σ(L ) is tabular. At the same time, σ(L) is not tabular since L is not tabular. Assume L = Lne . We have σ(Lne ) = σ(L) and L ⊆ Lne , consequently, Lne is a non-tabular proper extension of L, which conﬂicts with the assumption that L is pretabular. Assume L = Lne and σ(L) is pretabular. Then L ∈ EN3◦ . By Proposition 10.1.12, every proper L-extension is of the form η ◦ (L ) for some proper σ(L)-extension L . By assumption L is tabular, consequently, η ◦ (L ) is tabular by Proposition 10.5.2. Thus, L is pretabular. 2 By Item 6 of Proposition 10.3.1 if L = Lne , then L = η ◦ σ(L). This fact and the last two assertions imply. Theorem 10.5.6 The class EN4⊥ contains exactly three pretabular logics: η ◦ (LC), η ◦ (LJ), and η ◦ (LH). 2 We say that a logic L possesses the Craig interpolation property (CIP ) if ϕ → ψ ∈ L implies that there exists a formula χ such that ϕ → χ ∈ L and χ → ψ ∈ L, and χ has occurrences of common variables of ϕ and ψ only. The algebraic counterpart of this property is as follows. Let K be a class of algebras. We say that K has an amalgamation property if for any algebras A0 , A1 , A2 ∈ K and monomorphisms i1 : A0 → A1 and i2 : A0 → A2 , there exist an algebra A ∈ K and monomorphisms ε1 : A1 → A and ε2 : A2 → A such that ε1 i1 = ε2 i2 , i.e., the diagram A1

i1

@

A0

@ ε1 @@ R

@

i@ 2

@ R @

A

ε2

A2

commutes. The triple (A, ε1 , ε2 ) is called an amalgam of A1 and A2 over A0 .

10 The Class of N4⊥ -Extensions

214

It was proved in [56] that an intermediate logic possesses CIP if and only if the corresponding variety of Heyting algebras has the amalgamation property. The next statement can be obtained by a natural modiﬁcation of the proof from [56]. Proposition 10.5.7 Let L ∈ EN4⊥ . L possesses CIP if and only if the variety V ar(L) has the amalgamation property. Proof. Assume that L possesses CIP and prove the amalgamation property for V ar(L). Let A0 , A1 , A2 ∈ V ar(L) be such that A0 is a common subalgebra of A1 and A2 . Assign to each element a ∈ Ai , i = 0, 1, 2, a propositional variable pia . Assume additionally that p0a = p1a = p2a for a ∈ A0 and all other variables are diﬀerent. Denote by Fi the set of formulas with variables pia . Put F := F0 ∪ F1 ∪ F2 . Let v : F → A1 ∪ A2 be such that v(pia ) = a. For i = 1, 2, denote Ti := {ϕ ∈ Fi | v(ϕ) ∈ DAi }. It is easy to see that Fi ∩ L ⊆ Ti and that the sets Ti are closed under modus ponens. Put T := {ϕ ∈ F | T1 ∪ T2 L ϕ}. Lemma 10.5.8 Let {i, j} = {1, 2}, ϕ ∈ Fi , and ψ ∈ Fj . Then ϕ→ψ∈T

⇐⇒ ∃χ ∈ F0 (v(ϕ) $ v(χ) and v(χ) $ v(ψ)).

Proof. If v(ϕ) $ v(χ) and v(χ) $ v(ψ), then v(ϕ → χ) ∈ DAi and v(χ → ψ) ∈ D Aj . Therefore, ϕ → χ, χ → ψ ∈ T , whence ϕ → ψ ∈ T . Conversely, let ϕ → ψ ∈ T . There are ﬁnite subsets Γi ⊆ Ti and Γj ⊆ Tj such that Γi , Γj L ϕ → ψ. By Deduction Theorem L

Γi → ( Γj → (ϕ → ψ)),

which is equivalent to L

Γi ∧ ϕ → ( Γj → ψ).

such that Γi ∧ ϕ → χ and L According to CIP , there is a χ ∈ F 0 L χ → ( Γj → ψ). We obtain Γi L ϕ → χ and Γj L χ → ψ. Since Γi ⊆ Ti , we have ϕ → χ ∈ Ti . Consequently, v(ϕ → χ) ∈ DAi , i.e., v(ϕ) $ v(χ). Similarly, v(χ) $ v(ψ). 2

10.5 Some Transfer Theorems for the Class of N4⊥ -Extensions

215

Let ψ ∈ Fj be such that T L ψ. Take some ϕ ∈ L∩Fi , then T L ϕ → ψ. By the previous lemma there is a formula χ ∈ F0 such that v(ϕ) $ v(χ) and v(χ) $ v(ψ). It follows from ϕ ∈ L∩Fi that v(ϕ) ∈ DAi . Taking into account the ﬁrst inequality we obtain v(χ) ∈ D Ai , or, equivalently, v(χ → χ) = v(χ). Consequently, we have also v(χ) ∈ D Aj . Hence, v(ψ) ∈ DAj , i.e., ψ ∈ Tj . In this way, we have proved T ∩ Fj = Tj , j = 1, 2. Deﬁne on F the relation ∼ =:= {(ϕ, ψ) | ϕ ⇔ ψ ∈ T }. Since the provable strong equivalence ⇔ has in N4⊥ the congruence properties, ∼ = is a congruence on F. Let ϕ, ψ ∈ Fi . The equality T ∩ Fi = Ti implies ϕ∼ = ψ if and only if v(ϕ) = v(ψ)

(*)

Put A := F/ ∼ =. Clearly, A is an N4⊥ -lattice. Let mappings εi : Ai → A, i = 1, 2, be given by the rules εi (a) := [pia ]∼ = , a ∈ Ai . The mapping εi is a homomorphism since ∼ = is a congruence, and is one-toone in view of (∗). Since p0a = p1a = p2a , we have ε1 (a) = ε2 (a) for a ∈ A0 . Thus, (A, ε1 , ε2 ) is an amalgam of A1 and A2 over A0 . Prove the inverse implication. Assume that V ar(L) has the amalgamation property. We need one more lemma from [56]. Lemma 10.5.9 Let A0 is a common subalgebra in Heyting algebras A1 and A2 . If a ∈ A1 and b ∈ A2 are such that there is no c ∈ A0 with a ≤A1 c and c ≤A2 b, then there are ﬁlters F1 on A1 and F2 on A2 such that a ∈ F1 , b ∈ F2 , and A0 ∩ F1 = A0 ∩ F2 . Proof. We write ≤i for ≤Ai , i = 1, 2. Consider the sets ∇ := {z ∈ A0 | a ≤1 z} and Δ := {z ∈ A0 | z ≤2 b}. By assumption ∇ ∩ Δ = ∅. Consider an ideal on A2 generated by b, i.e., the set I = {x ∈ A2 | x ≤2 b}. Then F2 := A2 \ I is a ﬁlter on A2 such that b ∈ F2 . Put F0 := F2 ∩ A0 and I0 := I ∩ A0 . It is clear that F0 = A0 \ I0 , ∇ ⊆ F0 , and Δ ⊆ I0 . Further, consider a ﬁlter on A1 generated by F0 ∪ {a}, i.e., F1 = {x ∈ A1 | a ∧ z ≤1 x for some z ∈ F0 }.

216

10 The Class of N4⊥ -Extensions

We claim that F1 ∩ I0 = ∅. Indeed, if x ∈ F1 ∩ I0 , then a ∧ z ≤1 x for z ∈ F0 . The latter means that a ≤1 z → x. Since x ∈ I0 ⊆ A0 and z ∈ A0 , we have z → x ∈ ∇ ⊆ F0 . Taking into account z ∈ F0 we obtain x ∈ F0 , which conﬂicts with F0 ∩ I0 = ∅. From F0 ⊆ F1 and F1 ∩ I0 = ∅ we immediately infer that F1 ∩ A0 = F0 , i.e., A0 ∩ F1 = A0 ∩ F2 . 2 We adopt this statement for N4⊥ -lattices. Lemma 10.5.10 Let an N4⊥ -lattice A0 is a common subalgebra in N4⊥ lattices A1 and A2 . If a ∈ A1 and b ∈ A2 are such that there is no c ∈ A0 with a $A1 c and c $A2 b, then there are sﬀk Φ1 on A1 and Φ2 on A2 such that a ∈ Φ1 , b ∈ Φ2 , and A0 ∩ Φ1 = A0 ∩ Φ2 . Proof. Without loss of generality we may assume that Ai ∈ S (Bi ), i = 0, 1, 2, a = (a1 , a2 ) and b = (b1 , b2 ). As we can see from Proposition 8.4.2 the algebras B0 , B1 , B2 and the elements a1 ∈ B1 and b1 ∈ B2 satisfy the conditions of the previous lemma. Let the conclusion of this lemma hold for ﬁlters F1 on B1 and F2 on B2 . Then sﬀk Φ1 := F1 and Φ2 := F2 satisfy all the desired conditions. We check only the equality A0 ∩ Φ1 = A0 ∩ Φ2 . If (a, b) ∈ A0 ∩ Φ1 , then a ∈ B0 ∩ F1 = B0 ∩ F2 , whence (a, b) ∈ F2 . The inverse inclusion can be checked similarly. 2 Let x, y, z be disjoint tuples of variables and let for ϕ(x, y) and ψ(x, z) there is no χ(x) such that ϕ(x, y) → χ(x) ∈ L and χ(x) → ψ(x, z) ∈ L. The relation ϕ → ψ ∈ L is equivalent to ϕ $ ψ ∈ Eq(V ar(L)). Therefore, there is no χ(x) such that V ar(L) |= ϕ(x, y) $ χ(x) $ ψ(x, z). Let Ax be a free algebra in V ar(L) with free generators x and Ax,y,z a free algebra with generators x, y, z. Consider Ax as a subalgebra of Ax,y,z . According to Lemma 10.5.10 there are sﬀk Φ1 and Φ2 on Ax,y,z such that [ϕ(x, y)] ∈ Φ1 , [ψ(x, z)] ∈ Φ2 , and Ax ∩ Φ1 = Ax ∩ Φ2 . Here [φ] denotes the element of the free algebra corresponding to the formula φ. Put A1 := Ax,y,z /Φ1 and A2 := Ax,y,z /Φ2 . Then [ϕ(x, y)]/Φ1 ∈ DA1 and [ψ(x, z)]/Φ2 ∈ DA2 . Recall that for any u, v ∈ Ax , the equality u/Φ1 = v/Φ1 is equivalent to u ⇔ v ∈ Φ1 , which is equivalent in turn to u ⇔ v ∈ Φ2 in view of Ax ∩ Φ1 = Ax ∩ Φ2 . Thus, u/Φ1 = v/Φ1 if and only if u/Φ2 = v/Φ2 . Therefore, there is a natural embedding i2 (u/Φ1 ) := u/Φ2 of N4⊥ -lattice A0 := {u/Φ1 | u ∈ Ax } ⊆ A1 into N4⊥ -lattice A2 .

10.5 Some Transfer Theorems for the Class of N4⊥ -Extensions

217

By the amalgamation property of V ar(L) there are A ∈ V ar(L) and monomorphisms εi : Ai → A, i = 1, 2, such that ε1 A0 = ε2 i2 . Put h([x]) := ε1 ([x]/Φ1 ), h([y]) := ε1 ([y]/Φ1 ), h([z]) := ε2 ([z]/Φ2 ), where h([x]) := h([p1 ]), . . . , h([pn ]) for x = p1 , . . . , pn , etc. Extend this mapping to a homomorphism h : Ax,y,z → A. We obtain h([ϕ(x, y)]) = ε1 ([ϕ(x, y)]/Φ1 ) ∈ DA , h([ψ(x, z)]) = ε2 ([ψ(x, z)]/Φ2 ) ∈ DA . Therefore, V ar(L) |= ϕ(x, y) $ ψ(x, z), i.e., ϕ(x, y) → ψ(x, z) ∈ L.

2

The description of intermediate logics with CIP looks as follows: Theorem 10.5.11 [56] In the class Int, there exists exactly 7 logics with CIP : L1 L2 L3 L4 L5 L6 L7

:= := := := := := :=

Li, Li + {¬p ∨ ¬¬p}, Li + {p ∨ (p → (q ∨ ¬q))}, L3 + {(p → q) ∨ (q → p) ∨ (p ↔ ¬q)}, L3 + {¬p ∨ ¬¬p}, Li + {(p → q) ∨ (q → p)}, Lk. 2

We use the facts presented above to ﬁnd all logics in EN4⊥ possessing CIP . First of all, we notice that the intuitionistic fragment of a logic in EN4⊥ inherits CIP . Proposition 10.5.12 If L ∈ EN4⊥ possesses CIP , then σ(L) possesses CIP . Proof. According to Proposition 10.5.7 the variety V ar(L) has the amalgamation property. We show that the class of σ(L)-models has the amalgamation property too. Let Heyting algebras A0 , A1 , and A2 model σ(L), and i1 : A0 → A1 and i2 : A0 → A2 are embeddings. Consider twist-structures (A0 ) ◦ , (A1 )◦ , and (A2 )◦ . According to Proposition 10.1.3 these twist-structures are models of η ◦ σ(L). Logics L and η ◦ σ(L) have the same intuitionistic fragment, therefore, L ⊆ η ◦ σ(L) by Proposition 10.1.9. Thus, (A0 ) ◦ , (A1 )◦ , and (A2 )◦ model L. Moreover,

10 The Class of N4⊥ -Extensions

218

i 1 : (A0 )◦ → (A1 )◦ and i2 : (A0 )◦ → (A2 )◦ are embeddings according to Proposition 9.2.14. By assumption one can ﬁnd an amalgam (A, ε1 , ε2 ) of (A1 ) ◦ and (A2 )◦ over (A0 )◦ . In particular, ε1 i1 = ε2 i2 . Using functor properties of (−) we obtain (ε1 ) i1 = (ε1 i 1 ) = (ε2 i2 ) = (ε2 ) i2 .

Again by Proposition 9.2.14 (ε1 ) : A1 → A and (ε2 ) : A2 → A are embeddings. Thus, (A , (ε1 ) , (ε2 ) ) is an amalgam of A1 and A2 over A0 . 2 The implication in the above proposition can be reversed for some special logics from the interval σ −1 (L). Proposition 10.5.13 Let L ∈ Int. If L possesses CIP , then the logics η(L), η 3 (L), η n (L), and η ◦ (L) possess CIP . Proof. Recall that for any logic L ∈ σ −1 (L) for any N4⊥ -lattice B modelling L , B is isomorphic to a twist-structure over a Heyting algebra A such that A |= L. Let A0 , A1 , A2 |= L. Consider twist-structures Bi := T w(Ai , ∇i , Δi ), i = 0, 1, 2, and their embeddings i1 : B0 → B1 and i2 : B0 → B2 . Then (i1 ) embeds A0 into A1 , and (i2 ) embeds A0 into A2 . Since L possesses CIP , there is an amalgam (A, ε1 , ε2 ) of A1 and A2 over A0 . Proposition 9.2.15 allows us to lift up monomorphisms ε1 and ε2 so that we obtain the following diagram. B1

i1 B0

@ @ ε1 @ R @

@

i@ 2

@@ R

A

ε 2

B2

The functor properties of (−) guarantee that this diagram commutes. The full twist-structure A is a model of η(L) by Proposition 10.1.3 and we have thus proved that η(L) possesses CIP . If Bi |= η 3 (L), then Δi = {0}, i = 0, 1, 2, by Proposition 10.2.6. In this 3 case, ε i embeds Bi into Aexp . Again by Proposition 10.2.6 Aexp |= η (L), which proves that η 3 (L) possesses CIP .

10.5 Some Transfer Theorems for the Class of N4⊥ -Extensions

219

In a similar way, applying Propositions 10.2.6 and 10.1.3, we prove that η n (L) and η ◦ (L) possess CIP . 2 It turns out that all logic from σ −1 (L) that can possess CIP were listed in the last proposition. Proposition 10.5.14 If L ∈ EN4⊥ and L possesses CIP , then L is of the form η(L ), η 3 (L ), η n (L ), or η ◦ (L ) for some L ∈ Int. Proof. Let L possesses CIP . If L = Lne , then L = η ◦ σ(L). Let L = Lne . Then one of the formulas ¬(p∧ ∼ p) or ¬¬(p∨ ∼ p) does not belong to L. Assume ¬(p∧ ∼ p) ∈ L and ¬¬(p∨ ∼ p) ∈ L and prove that L = η 3 σ(L) in this case. Consider the Lindenbaum twist-structure of L, L L . Recall that |L L | = {([ϕ]L , [∼ ψ]L ) | ψ ∈ [ϕ]L }, where [ϕ]L is a coset of the equivalence ϕ ≡L ψ ⇔ ϕ ↔ ψ ∈ L. Thus, L L is a twist-structure over the Heyting algebra LL of all cosets [ϕ]L . Note that ¬(p∧ ∼ p) ∈ L implies the equality Δ(L L ) = {0}. Put F := {[ϕ]L | ¬(p0 ∨ ∼ p0 ) → ϕ ∈ L}. Obviously, F is a ﬁlter on LL and F is an sﬀk on L L . Consider a quotient . /F algebra L0 := L L Let ([ϕ0 ]L , [ψ0 ]L ), ([ϕ1 ]L , [ψ1 ]L ) ∈ L L . By Proposition 9.2.3 the equality ([ϕ0 ]L , [ψ0 ]L )/F = ([ϕ1 ]L , [ψ1 ]L )/F is equivalent to ([ϕ0 ]L , [ψ0 ]L ) ⇔ ([ϕ1 ]L , [ψ1 ]L ) ∈ F . By deﬁnition of F the latter is equivalent to [ϕ0 ↔ ψ0 ]L , [ϕ1 ↔ ψ1 ]L ∈ F. Thus, an element of L0 can be identiﬁed with a pair of elements of the quotient LL /F = LF , [([ϕ]L , [ψ]L )]F = ([ϕ]F , [ψ]F ). Consider the element a := ([p0 ]F , [∼ p0 ]F ) ∈ L0 . Then a∨ ∼ a = ([p0 ∨ ∼ p0 ]F , [p0 ∧ ∼ p0 ]F ),

220

10 The Class of N4⊥ -Extensions

which means that [p0 ∨ ∼ p0 ]F ∈ Δ(L0 ). By deﬁnition [¬(p0 ∨ ∼ p0 )]L ∈ F , i.e., [¬(p0 ∨ ∼ p0 )]F = ¬[p0 ∨ ∼ p0 ]F = 1. Consequently, [p0 ∨ ∼ p0 ]F = 0. We have thus proved that ∇(L0 ) = LF , i.e., L0 is a special N3-lattice. Let us check the equality L = LL0 . We have L ⊆ LL0 , since L0 is a quotient of L L . Let L0 |= ϕ(q0 , . . . , qn ) and p0 does not occur in ϕ. Then [ϕ(q0 , . . . , qn )]L ∈ F , i.e., ¬(p0 ∨ ∼ p0 ) → ϕ(q0 , . . . , qn ) ∈ L. By CIP we have ¬(p0 ∨ ∼ p0 ) → φ ∈ L and φ → ϕ(q0 , . . . , qn ) ∈ L, where φ is some formula without propositional variables, i.e., constructed from ⊥ and logical connectives. Denote & :=∼ ⊥. Then & ∈ N4⊥ . By an easy induction on the structure of formulas one can prove that φ ↔ ⊥ ∈ N4⊥ or φ ↔ & ∈ N4⊥ . In this way, we obtain ¬(p0 ∨ ∼ p0 ) → ⊥ ∈ L or & → ϕ(q0 , . . . , qn ) ∈ L. In view of our assumption ¬(p0 ∨ ∼ p0 ) → ⊥ ∈ L, consequently, ϕ(q0 , . . . , qn ) ∈ L. We have thus proved that L is determined by the special N3-lattice L0 . By Corollary 10.2.10 L is a special explosive logic, i.e., a logic of the form η 3 (L ). Now, we assume ¬(p∧ ∼ p) ∈ L and ¬¬(p∨ ∼ p) ∈ L and prove that L = η n σ(L). In this case, the Lindenbaum twist-structure L L is normal. Let us consider the ﬁlter F1 := {[ϕ]L | (p0 ∧ ∼ p0 ) → ϕ ∈ L} and the quotient L1 := L L /F1 . The element b = ([p0 ]F1 , [∼ p0 ]F1 ) is such that b∧ ∼ b = ([p0 ∧ ∼ p0 ]F1 , [p0 ∨ ∼ p0 ]F1 ).

Since [p0 ∧ ∼ p0 ]L ∈ F1 , we obtain [p0 ∧ ∼ p0 ]F1 = 1 and Δ(L1 ) = L1 . Thus, L1 is a special normal N4⊥ -lattice. Arguing as above, we check that L = LL1 . Thus, L is determined by a special normal N4⊥ -lattice. By Corollary 10.2.10 L is a special normal logic, i.e., a logic of the form η n (L ). It remains to consider the case when ¬(p∧ ∼ p) ∈ L and ¬¬(p∨ ∼ p) ∈ L. We have ¬((p∧ ∼ p) ∧ ¬(q∨ ∼ q)) ∈ L. Indeed, this is equivalent to (p∧ ∼ p) → ¬¬(q∨ ∼ q) ∈ L. If (p∧ ∼ p) → ¬¬(q∨ ∼ q) belongs to L, by CIP we have (p∧ ∼ p) → ⊥ ∈ L or & → ¬¬(q∨ ∼ q) ∈ L. Both alternatives conﬂict with our assumption. Let us consider the ﬁlter F2 := {[ϕ]L | (p0 ∧ ∼ p0 ) ∧ ¬(p1 ∨ ∼ p1 ) → ϕ ∈ L}

10.5 Some Transfer Theorems for the Class of N4⊥ -Extensions

221

on LL . Arguing as above, we can prove that the quotient L2 := L L /F2 is a characteristic model for L and that ∇(L2 ) = Δ(L2 ) = LF2 , i.e., L2 is a special N4⊥ -lattice. Since L is determined by a special N4⊥ -lattice, it is a special logic, i.e., L = ησ(L). 2 3 n ◦ The logics η(L), η (L), η (L), and η (L) are diﬀerent for any L in Int. This fact and the above propositions imply

Theorem 10.5.15 In EN4⊥ , there are exactly 28 non-trivial logics possessing CIP : η(Li ), η 3 (Li ), η n (Li ), and η ◦ (Li ), i = 1, . . . , 7, where L1 , . . . , L7 are logics from Theorem 10.5.11. 2

Chapter 11

Conclusion Discussing the question “Why is paraconsistency worthy?” J.-Y. B´eziau [11] emphasized that paraconsistent logic is an important contribution to the theory of negation and to modern logic in general. The distinction between triviality and inconsistency made in paraconsistent logic is similar to the distinction between implication and inference relation and allows one to elucidate new features of traditional logical notions. Suppose that the investigations presented in this book also contribute to the general theory of logical systems. For two explosive logics with the same positive fragment and with essentially diﬀerent kinds of negation, we investigated how the lattice of extensions of a logic changes when the explosion axiom is deleted, i.e., if we pass from a logic to its paraconsistent analog. It turns out that in both cases the lattices of extensions extend in a rather regular manner. In the class of extensions of a paraconsistent logic, one can distinguish the subclass of explosive logics, i.e., the class of extensions of the original explosive logic; the subclass of logics, which can be used to represent the structures of contradictions in all extensions of the considered paraconsistent logic (see Remark after Proposition 10.2.5). Finally, all other logics can be obtained via a combination of logics from the above two subclasses. The manner of combination can be explicated via a suitable representation theory for algebras modelling the logics under consideration. Thus, admitting non-trivial inconsistent theories leads to quite natural and interesting mathematical structures arising. Moreover, the above results allow one to consider paraconsistent logic not as an alternative to traditional explosive logic, but as a more general setting for considering explosive systems. It would be natural to continue the study of lattices of extensions of paraconsistent logics for some other concepts of negation. According to the author’s opinion the 223

224

11 Conclusion

most natural candidate for such investigation is the subminimal negation by D. Vakarelov [111, 112]. The subminimal logic has the same positive fragment as intuitionistic logic. Its negation is a natural generalization of minimal negation and can be deﬁned via two constants ⊥ and & such that ⊥ → & holds by the formula ¬p ↔ (p → ⊥) ∧ &. In this case, the constants are deﬁnable through negation as ⊥ := ¬(p → p) and & := ¬¬(p → p). To obtain minimal logic from the subminimal one we have to add the axiom &. As other candidates for such investigations one could consider the negation as impossibility by K. Doˇsen [24, 25], diﬀerent kinds of negations considered by M. Dunn in [26] and ﬁnally, the regular and co-regular negations by D. Vakarelov. At the same time, one can continue the study of the classes of extensions of minimal logic and of Nelson’s logic N4⊥ using the coordinate system depicted in this work. It should be mentioned that there are interesting results on minimal logic extensions that are not reﬂected in this book. L.L. Maksimova [57, 58] studied interpolation and deﬁnability properties in extensions of positive and minimal logic. She constructed several series of logics with Craig interpolation property in the class of minimal logic extensions. However, it is still unknown whether the total number of logics with Craig interpolation property in this class of logics is ﬁnite. M.V. Stukacheva studied the disjunction property in the class of minimal logic extensions [103, 104] and transferred the technique of canonical formulas by M. Zakharyaschev to this class of logics [105]. And the very ﬁnal remark concerns the technique of twist-structures. In the second part of the book we used twist-structures to solve algebraization problems for Nelson’s logics N4 and N4⊥ . The main diﬃculty here was connected with the fact that Nelson’s logics are not closed under the replacement rule. But they are closed under weak forms of replacement rules: ϕ ↔ ψ, ∼ ϕ ↔∼ ψ ϕ↔ψ and , + ↔ χ (ψ) χ(ϕ) ↔ χ(ψ)

χ+ (ϕ)

where χ+ is a ∼-free formula. Namely, these rules allow one to deﬁne the semantics for Nelson’s logics in terms of twist-structures. The ﬁrst of these rules and the fact that ∼-free fragment of N4 (N4⊥ ) is equal to positive (intuitionistic) logic imply that implicative lattices (Heyting algebras) should be taken as basic algebras of twist-structures for N4 (N4⊥ ). The number of premises in the second rule corresponds to the number of components in

11 Conclusion

225

elements of twist-structures. Finally, the strong negation axioms determine the twist-operations. A more general situation, where a technique of this kind could be applied would be as follows. Assume that we have a logic L in a language L, L0 is a sublanguage of L and L0 is an L0 -fragment of L. Further, let the logic L0 be extensional, the class of algebras K0 provides an algebraic semantics for L0 , and the logic L is closed under the rule: ϕ↔ψ , where χ0 is an L0 -formula. ↔ χ0 (ψ)

χ0 (ϕ)

Finally, assume that there are formulas α1 (p), . . . , αn (p) such that L is closed under the following form of replacement rule: α1 (ϕ) ↔ α1 (ψ), . . . , αn (ϕ) ↔ αn (ψ) , χ(ϕ) ↔ χ(ψ) where χ is an arbitrary L-formula. Note that in case of N4 and N4⊥ we have n = 2, α1 (p) = p and α2 (p) =∼ p. In this case, one can try to construct a semantics for the logic L in terms twist-structures over algebras from K0 with the number of components equalling n. Naturally, to deﬁne twist-operations some reduction axioms for connectives from L \ L0 are needed. Possibly, some additional conditions should be imposed on the form of formulas α1 , . . . , αn . It is interesting to look for natural examples of logics, which are closed under rules of the form described above and to deﬁne semantics for such logics in terms of twist-structures. It may also be interesting to develop a general theory of twist-structures.

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Index absurdity operator, 7, 102, 112 algebra, 21 Boolean algebra of regular elements, 25 De Morgan algebra, 145 Heyting l-algebra, 66 Heyting algebra, 23 basic, 156 j-algebra, 23 lower, 42 negative, 24 negative l-algebra, 66 Peirce algebra, 26 Peirce-Johansson algebra, 33 pj-algebra, 33 Stone algebra, 66 subdirectly ireducible, 25 upper, 42 amalgam, 213 amalgamation property, 213 canonical embedding, 156 complement, 23 conservative extension, 16 contradiction operator, 6, 52, 109 counterpart explosive, 195 intuitionistic, 47, 48 negative, 47, 48 normal, 195 normal explosive, 195 Craig interpolation property, 213

Deduction Theorem, 17 deductive system, 15 deﬁnition equivalent, 103 disjunction property, 85 dual lattice isomorphism, 24 Dummett’s linearity axiom, 63, 177 element dense, 25 distinguished, 22 regular, 25 equational theory, 22 extensional operator, 109, 115 F-structure, 135 saturated, 135 faithfully embedded via, 103 ﬁlter, 24 generated by, 24 of completions, 160 of dense elements, 25 special ﬁlter of the ﬁrst kind, 169 sﬀk, 169 fragment L1 -fragment, 16 positive, 16 free combination, 51 Generalized Glivenko’s Theorem, 58 ideal, 161 of contradictions, 161 implicative lattice, 23 237

238

inference relation, 16 interpretation, 119 intuitionistic formula, 177 isomorph, 35 j-frame, 28 abnormal, 30 closed, 30 dense, 30 identical, 30 normal, 30 separated, 30 Jankov formula, 92 Jonsson’s theorem, 202 kernel, 169 law contraction law, 19 distributivity law, 19 ex contradictione quodlibet, 1 extended law of excluded middle, 17 identity law, 19 import and export of the premiss, 19 Peirce law, 17 permutation law, 19 reductio ad absurdum, 17 weak law of excluded middle, 63 logic, 15 A-logic, 112 C-logic, 109 L ukasiewicz’s modal logic, 105 characterized by, 29 classical logic, 18 classical positive logic, 17 explosive, 1, 191 extensional, 116 Glivenko’s logic, 57 relativized, 84

Index

intermediate, 35 intuitionistic logic, 18 Johansson’s logic, 17 logic of classical refutability, 18 maximal negative logic, 18 minimal logic, 17 minimal negative logic, 18 ne-logic, 178 negative, 35 Nelson’s logic explosive, 133 paraconsistent, 133 normal, 191 normal explosive, 178 of ﬁnite codimension, 211 of general form, 191 paraconsistent, 1 positive logic, 16 pretabular, 201 properly paraconsistent, 35 s-logic, 178 se-logic, 193 sn-logic, 193 special, 178 special explosive, 193 special normal, 193 tabular, 201 trivial logic, 16 ¬-logic, 113 matrix, 22 model, 22 characteristic, 22 nnf, 134 N3-lattice normal, 179 special, 193 N4-lattice, 145

Index

N4⊥ -lattice, 156 special, 179 special normal, 193 negation normal form, 134 negatively equivalent, 83 negatively lesser, 81

239

weak replacement rule, 135 world abnormal, 28 normal, 28

2, 26 2 , 26 opremum, 25 2P , 26 4 , 34 particular case, 15 , 105 4L presentation A ⊕ B, 24 A-presentation, 113 A⊥ , 42 exact, 113 A , 139 C-presentation, 113 A 3 , 193 exact, 113 A ◦ , 179 strong, 114 A n , 193 ¬-presentation, 114 A⊥ , 42 exact, 114 A , 145 prime theory, 120 A → B, 22 propositional language, 15 A B, 202 B ×f C, 61 relative pseudo-complement, 23 chn (k, ), 203 repeatedless disjunction, 20 CIP , 213 splitting, 189 CLuN, 119 splitting element, 189 Con(A), 27 strong equivalence, 133 D(A), 92 strongly independent sequence, 93 D A , 147, 148 substitution instance, 15 Δ(B), 162 sum of logics, 18 Δl (B), 164 e , 164 theorem, 15 EL, 18 twist-operations, 139 Eq(A), 22 twist-structure, 139, 155 Eq(K), 22 full, 139, 155 Lindenbaum twist-structure, 141 ≡neg , 83 η(L), 178 η 3 (L), 193 valuation, 22, 28 η ◦ (L), 178 variety η n (L), 193 arithmetic, 27 Exp, 190 congruence distributive, 27 Fd (A), 25 congruence permutable, 27

240

F or, 133 F or ⊥ , 133 G(L1 , L2 ), 84 Gen, 191 H(K), 22 h , 167 h , 167 I(B), 162 I(K), 22 i , 164 Int, 35 J(A), 92 Jhn, 18 Jhn+ , 18 K , 178 Ker(h), 169 L(V ), 153 L1 ∗ L2 , 50, 198 L Γ , 141 Le, 18 Le , 41 ≤neg , 81 Lexp , 195 Lg, 57 Li, 18 Lint , 48 Lj⊥ , 17 Lj¬ , 17 Lk, 18 Lk+ , 17 Lmn, 18 Ln, 18 Lne , 195 Lneg , 48 Lnor , 195 Lp, 16 L , 105

Index

Γ |= ϕ, 139 Γ |=F ϕ, 137 N3, 133 N3◦ , 183 N4, 133 N4N , 190 N4⊥ , 134 N4⊥ C, 201 N4C, 201 ∇(B), 162 ∇ , 172 ∇ , 172 ∇l (B), 164 ∇(L), 88 Neg, 35 Nor, 190 P 1 , 123 Par, 35 ϕ , 179 ϕ3 , 193 ϕ◦ , 179 ϕn , 193 R(A), 25 S(K), 22 S (A), 139 ⇔, 133 σ(L), 177 σ 3 , 193 σ n , 193 Σi∈I Li , 18 Spec(L1 , L2 ), 50, 198 A , 25 T w(A, ∇, Δ), 162 Up(K), 22 VN4 , 148 V ar(L), 152

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