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 [email protected] 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

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r1 @ @ @rb

⊥ c

r−1

3N

ar

@

@ @r ⊥

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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

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ra @ @

@r ⊥

@ @r

r1 @ @ @rb ar @ @ @ @ @r @r r ⊥ @ @ @ @ @r @r [email protected] d @ @r−1

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cr

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@r

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−1

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A1 r1

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

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@

@r ⊥

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rc r

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r @ r @

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A7

⊥

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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 @ [email protected] B @ @ B @ B L∗ gf1 = L∗ gf2r L∗ l [email protected] = L∗ l f2 BrP P L [email protected] [email protected] L Pr r ∗ L L [email protected] 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 ∇.

90

6 Negatively Equivalent Logics

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.

92

6 Negatively Equivalent Logics

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

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.

40

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

42

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.

44

4 The Class of Extensions of Minimal Logic

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

45

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 = ⊥.

46

4 The Class of Extensions of Minimal Logic

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

48

4 The Class of Extensions of Minimal Logic

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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4 The Class of Extensions of Minimal Logic

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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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,

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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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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)

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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

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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 [email protected] 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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5 Adequate Algebraic Semantics for Extensions of Minimal Logic

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 [email protected] 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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5 Adequate Algebraic Semantics for Extensions of Minimal Logic

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 @ [email protected] B @ @ B @ B L∗ gf1 = L∗ gf2r L∗ l [email protected] = L∗ l f2 BrP P L [email protected] [email protected] L Pr r ∗ L L [email protected] 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

82

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);

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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

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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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6 Negatively Equivalent Logics

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

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