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Volume of Abstracts

Non-classical Modal and Predicate Logics 2011

Petr Cintula, Shier Ju, and Martin Vítaeditors

Guangzhou, 5–9 December 2011

Conference chairs:

• Petr Cintula, Academy of Sciences of the Czech Republic• Shier Ju, Department of Philosophy and Institute of Logic and Cognition,

Sun Yat-sen University, P.R. China

Program committee:

• Greg Restall, University of Melbourne• Hu Liu, Sun Yat-sen University• Itala M. L. D’Ottaviano, State University of Campinas• Libor Běhounek, Academy of Sciences of the Czech Republic• Michael Rathjen, University of Leeds• Rosalie Iemhoff, Department of Philosophy, Utrecht University• Steve Awodey, Carnegie Mellon University• Yde Venema, University of Amsterdam

Organizing committee:

• Yuping Shen, Institute of Logic and Cognition, Sun Yat-sen University• Minghui Xiong, Institute of Logic and Cognition, Sun Yat-sen University

Organized by: Institute of Logic and Cognition, Sun Yat-sen UniversityInstitute of Computer Science, Academy of Sciences of the Czech Republic

Sponsored by: National Natural Science Foundation of China (Grant No. 61173019)

Published by F solutions, Prague in 2011

ISBN: 987-80-87465-00-4

Non-classical Modal and Predicate Logics Guangzhou, 5–9 December 2011

Welcome to Non-classical Modal and Predicate Logics 2011!

Modalities and predicates have since ancient time been central notions in logic. In the20th century, various systems of non-classical logics have emerged, with applications inmany disciplines like Computer Science, Linguistics, Mathematics, and Philosophy. Thisgave rise to the questions of non-classical treatment of quantification and modalities andtheir accommodation in these non-classical settings. In response, various modal and pred-icate variants of non-classical logics have been introduced and studied in the past decades.

Although there are many good conferences on (mainly propositional) non-classical log-ics, this conference is solely dedicated to modal and predicate non-classical logics. Its aimis to bring together researchers from various branches of non-classical logics, not only topresent recent advances in their particular fields, but also to identify common problemsand methods and foster the exchange of ideas between researchers from separate fields.

This conference is a follow-up event of the workshop: Non-Classical Mathematics 2009which took place in Hejnice, Czech Republic at 18–22 June 2009. Non-Classical Math-ematics is formally understood as the study of any part of mathematics that is, or canin principle be, formalized in some logic other than classical. The scope of the presentconference has broadened to include also the study of necessary logical prerequisites forNCM: the predicate and modal non-classical logics.

This volume contains (extended) abstracts of all 6 invited and 25 contributed talks.There is a planned post-conference special issue of the Logic Journal of IGPL (guest)edited by Petr Cintula, Rosalie Iemhoff, and Shier Ju. The issue is not restricted to thecontributions of the present conference, but will be open to all interested contributors.

We thank all the members of the program and organizing committees for their effort inpreparing the conference and all the participants for their contributions and presentations,and we wish everybody a pleasant time in Guangzhou and a fruitful conference.

Petr Cintula and Shier JuConference chairs

i

Contents

1. Invited talks

Admissible Semantics for Quantified Modal Logics 1Robert Goldblatt

Substructural Aspects of Negative Translations 2Hiroakira Ono

Equality in Modal Predicate Logics 3Valentin Shehtman

Independence Friendly Logic 4Gabriel Sandu

From IF Logic to Partial Logic 7Xuefeng Wen

Floyd-Hoare Logic for Quantum Programs 8Mingsheng Ying

2. Contributed talks

Completions of Basic Algebras and Löb Algebras 11Majid Alizadeh and Hiroakira Ono

Fuzzy Set Theories with Naïve Comprehension 17Libor Běhounek

Quantified Conditional Logics are Fragments of HOL 22Christoph Benzmüller and Valerio Genovese

A Librationist Quest for Strength 33Frode Bjordal

First-Order Hybrid Logic: Introduction and Survey 35Torben Braüner

Algebraic Axiomatization of Tense Intuitionistic Logic 42Ivan Chajda

A Bilattice-based GQT Framework for Interrogatives and Interrogative In-ferences 47

Ka-fat Chow

Fuzzy Functions and Their Representations in Fuzzy Class Theory 52Martina Daňková

A Game Based Approach to Quantification in Łukasiewicz Logic 58Christian G. Fermüller

Multivalued Dependence Logic and Independence Logic 64Pietro Galliani

On Very True Operators on Pocrims 70Radomír Halaš and Michal Botur

Lateral Completions of BCK- and Pseudo-BCK-Algebras 79Jan Kühr

Substructural Epistemic Modalities 82Ondrej Majer

Partial First–Order Logical Semantics Based on Approximations of Sets 85Tamás Mihálydeák

Fuzzy Type Theories: What They Are For 91Vilém Novák

Modal Twist-Structures over Residuated Lattices 95Hiroakira Ono and Umberto Rivieccio

Elementary Submodels of Kripke Models 99Tomasz Połacik

A Partial Characterization of Axiomatizability of First-Order Logics of Lin-ear Kripke Frames over Constant Domains 102

Norbert Preining

The Role of Modal Logic in Formalizing Analogical Reasoning 108Dagmar Provijn

An Expansion of First-order Belnap-Dunn Logic 111Katsuhiko Sano and Hitoshi Omori

From Theory Revision to Logic: A Formal Link between the Logic of Com-parative Similarity CSL and Distance Based Revision 120

Camilla Schwind

Many-Valued Logics for Dynamical Semantics of the Atomic Self-ReferenceStatements 128

Vladimir Stepanov

Abduction of Generalizations in a Modal Adaptive Framework 133Frederik Van De Putte and Tjerk Gauderis

Handling Unnamed Abnormal Objects in (Mathematical) Adaptive LogicTheories 136

Peter Verdée

On the Crispness of ω and an Arithmetic with a Bi-Simulation in a Con-structive Naive Set Theory 140

Shunsuke Yatabe

Invited talks

Admissible Semantics for Quantified Modal LogicsRobert Goldblatt∗

This talk will provide an introduction to the contents of my recently published book:

Quantifiers, Propositions and Identity: Admissible Semantics for QuantifiedModal and Substructural Logics, Cambridge University Press and the Associa-tion for Symbolic Logic, July 2011. www.cambridge.org/9781107010529

The term admissible semantics refers to the use of possible-worlds models in which thereis a restriction on which sets of worlds are admissible as propositions. Such models havepreviously proven effective in characterising propositional modal logics that are incompletefor their Kripke frame semantics.

There are axiomatically defined systems of quantified modal logic that cannot be char-acterised by the kind of possible-worlds models introduced by Kripke, even though thepropositional fragments of those logics are characterised by their Kripke frames. Here weshow that this failure of completeness under Kripke semantics to lift from the propositionalto the quantificational level can be overcome by developing a suitable notion of admissiblemodel for quantified modal logics, leading to semantic characterisations of such logics ingeneral. This requires a new interpretation of universal and existential quantifiers thattakes into account the admissibility of propositions. The talk will explain the motivationfor this interpretation.

The book works out this theory for systems with quantifiers ranging over actual ob-jects, and over all possibilia; as well as for logics with existence and identity predicatesand definite descriptions. New light is shed on the celebrated Barcan Formula, whose rolebecomes that of legitimising the Kripkean interpretation of quantification. Commutativityof quantifiers is analysed, as is the relationship between substitution of formulas for pred-icate letters and the admissibility of propositional functions. The final chapter developsa new admissible cover system semantics for propositional and quantified relevant logic,adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory.

∗School of Mathematics, Statistics and Operations Research, Victoria University, Wellington, NewZealand, [email protected]

1

Substructural Aspects Of Negative TranslationsHiroakira Ono∗

Both algebraic approach and proof-theoretic approach to Glivenko theorems for substruc-tural propositional logics were developed in [2] and [3], respectively. Following the latter,we will discuss here Glivenko theorems and negative translations in substructural predicatelogics.

A classical result on the Glivenko theorem for predicate logics over intuitionistic logicrelative to classical predicate logic (QCl) was obtained in [4] and [1], in which the “doublenegation shift scheme” (DNS) plays an important role. We show first the Glivenko theoremrelative to QCl for an extension G of intuitionistic linear predicate logic QFLe with (DNS).It is shown moreover that the logic G is the weakest logic over QFLe for which the Glivenkotheorem holds.

Using the similar technique, we study also various negative translations of substruc-tural predicate logics. After introducing a negative translation, called “extended Kurodatranslation”, we show that a formula is provable in QCl iff its translation is provable in anextension N of QFLe. The logic N is properly weaker than G, and moreover N is the weakestamong such logics for which the extended Kuroda translation works. Other standard neg-ative translations, including Kolmogorov translation and Gödel-Gentzen translation, areshown to be equivalent to extended Kuroda translation over QFLe. This gives us a uniformview of Glivenko theorems and negative translations from substructural perspective. Thisis a joint work with Hadi Farahani (Shahid Beheshti University, Tehran).

References

[1] Dov Gabbay. Applications of trees to intermediate logics. Journal of Symbolic Logic37:135–138, 1972.

[2] Nikolaos Galatos and Hiroakira Ono. Glivenko theorems for substructural logics overFL Journal Symbolic Logic 71(4):1353–1384, 2006.

[3] Hiroakira Ono. Glivenko theorems revisited Annals of Pure and Applied Logic161:246–250, 2009.

[4] Toshio Umezawa. On some properties of intermediate logics. Proceedings of the JapanAcademy 35:575–577, 1959.

∗Research center for Integrated Science, Japan Advanced Institute of Science and Technology, Asahidai,Nomi, Ishikawa, 923-1292, Japan, [email protected]

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Equality in Modal Predicate LogicsValentin Shehtman∗

In this talk we consider polymodal predicate logics defined in the usual way, as sets ofmodal first-order formulas containing all classical validities, the axioms of polymodal Kand closed under the standard rules (Modus Ponens, Necessitation, Generalization, andPredicate Substitution). Logics with equality should also contain the classical equalityaxioms.

Every logic L without equality has the minimal extension with equality L=. Anotherimportant extension (in the 1-modal case) is

L=c := L= + ∀x∀y(x = y→ (x = y)).

The general problem on correlation of properties between L, L=, and L=c seems quitedifficult. There two main questions:

(i) Conservativity: are the logics L= and L=c conservative over L?

(ii) Transfer of completeness (in different semantics) from L to L= and L=c.

Some results of that kind can be found in the book [1]. In particular:

1. L= is conservative over L for every conically expressive L.

2. Kripke sheaf completeness transfers from L to L= and L=c for every conically expres-sive L.

3. Strong Kripke sheaf completeness always transfers from L to L=.

In the talk we will discuss these and similar further results.

References

[1] D. Gabbay, V. Shehtman, D. Skvortsov. Quantification in Nonclassical Logic, Vol-ume 1. Elsevier, 2009.

∗Institute for Information Transmission Problems, Russian Academy of Sciences,[email protected]

3

Independence Friendly LogicGabriel Sandu∗

1 Game-theoretical semantics (GTS) for first-orderlogic (FOL)

GTS was introduced by Jaakko Hintikka. With each first-order formula φ model M whichinterprets the nonlogical symbols of φ and assignment s in the universe M of M whosedomain includes the free variables of φ, a semantical game G(M, s, φ) is associated whichis played by two players, Eloise and Abelard: intuitively Eloise tries to show that φ is truein M (relative to the assignment s), and her opponent Abelard tries to show that φ is falsein M (relative to the assignment s). At the beginning of the game, the position is (φ, s).Eloise has the role of verifier and Abelard that of falsifier. After each move, the playersreach a position (ψ, r), where ψ is a subformula of φ and r is an assignment whose domainincludes the free variables of ψ, and a specification of the roles of the players. Here are therules of the game.

1. The position is (t1 = t2, r) and the verifier is p ∈ Eloise, Abelard. If M, r |= t1 = t2,then p wins; otherwise the opponent of p wins.

2. The position is (R(t1, ..., tn), r) and the verifier is p. If M, r |= R(t1, ..., tn), then pwins; otherwise the opponent of p wins.

3. The position is (¬ψ, r) and the verifier is p. The game switches to the position (ψ, r)with the opponent of p as the verifier.

4. The position is (ψ ∨ θ, r) and the verifier is p. The player p chooses χ ∈ ψ, θ andthe play switches to the position (χ, r) with p as the verifier.

5. The position is (ψ ∧ θ, r) and the verifier is p. The opponent of p chooses χ ∈ ψ, θand the play switches to the position (χ, r) with p as the verifier.

6. The position is (∃xψ, r) and the verifier is p. Player p chooses a ∈ M and the nextposition is (ψ, r(x/a)) with p as the verifier.

7. The position is (∀xψ, r) and the verifier is p. The opponent of p chooses a ∈M andthe next position is (ψ, r(x/a)) with p as the verifier.

∗Department of Philosophy, University of Helsinki, Helsinki, Finland, [email protected]

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A strategy for a player p in G(M, s, φ) is any sequence σ of functions f , one for eachoperator where p is to move. The strategy σ is winning if p wins every possible play whenhe or she follows σ.

Game theoretical truth, M, s |=+ φ, and game-theoretical falsity, M, s |=− φ, aredefined by:

• M, s |=+ φ iff there is a winning strategy for Eloise in G(M, s, φ)

• M, s |=− φ iff there is a winning strategy for Abelard in G(M, s, φ)

The above semantical games can be rephrased as what in classical game theory is knownas two player, win-loss extensive finite games with perfect information. For such games,the principle of determinacy, known as Zermelo Theorem, states that

• For every semantical game G(M, s, φ) there is a winning straegy either for Eloise orfor Abelard.

In other words, the classical principle of bivalence for first-order logic is equivalent to theprinciple of determinacy of certain games.

2 Game-theoretical semantics for IF logic

Hintikka and Sandu (1989) introduce so-called IF (independence-friendly) languages, whichcontain quantifiers of the form

(∃x/W ) (∀x/W )

where W is a finite set of variables. The intended interpretation of (∃x/W ) is: the exis-tential quantifier ∃x is independent of the quantifiers which bind the variables in W. Hereare a couple of examples:

∀x(∃y/x)x = y∀x∀z(x = z ∨ (∃y/x)x = y)

IF formulas are interpreted by semantical games of imperfect information. They areexactly as the semantical games of perfect information sketched above, except for therequirement of the uniformity of strategies. This notion will be explained in the full paper.For the moment suffice it to say that in the case of the first example, a strategy for Eloise isa function defined in the relevant model which is constant: for every choice of an individualby Abelard, the strategy has to give Eloise the same choice.

Imperfect information has several consequences:

• The principle of determinacy of games fails. Thus there are formulas which areneither true nor false.

• Signaling (imperfect recall): a player may fail to know a move of his opponent atsome stage of the game, but may come to know it later on. This phenomena allowsIF logic to express certain forms of communication.

• Gain in expressive power.

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3 Equilibrium semantics for IF logic

Following a suggestion by Aitaj, we interpret formulas of IF logic by two player, win-lossstrategic finite games. M. Sevenster was the first one to explore such an interpretation.The details are contained in A. Mann, G. Sandu, and M. Sevenster, Independence-FriendlyLogic: A Game-theoretic Approach, Cambridge University Press, 2011. The methodologywe follow is the following:

• We convert semantical games (extensive games) into strategic games.

• We switch from pure strategies to mixed strategies.

• Instead of considering the existence of winning strategies for either one of the players,we consider strategies of the two players being in equilibrium.

• Von Neumann’s Minimax theorem shows that every strategic IF game has an equi-librium in mixed strategies.

• Moreover, any two equilibria in such a strategic game return the same expected utilityto the two players.

• We identify the semantic value of an IF formula on a given model with the expectedutility returned to Eloise by the corresponding equilibria.

• The result is a multi-valued semantics.

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From IF Logic to Partial LogicXuefeng Wen∗

We associate the semantic game with chance moves conceived by Blinov with Blamey’spartial logic. We give some equivalent alternatives to the semantic game, some of whichare with a third player, borrowing the idea of introducing the pseudo-player called Nature ingame theory. We observe that IF propositional logic proposed by Sandu and Pietarinen canbe equivalently translated to partial logic, which implies that imperfect information maynot be necessary for IF propositional logic. We also show that independent quantifiers withperfect recall can be regarded as dependent quantifiers of indeterminate sequence, usingthe interjunction connective in partial logic. Finally, we generalize the 2-player semanticgame for partial logic and give a muti-player semantic game for multi-valued partial logic.

∗Institute of Logic and Cognition and Department of Philosophy, Sun Yat-sen University, Guangzhou,510275, China, [email protected]

7

Floyd-Hoare Logic for Quantum ProgramsMingsheng Ying∗

Floyd-Hoare logic is a foundation of axiomatic semantics of classical programs and it pro-vides effective proof techniques for reasoning about correctness of classical programs. Tooffer similar techniques for quantum program verification and to build a logical foundationof programming methodology for quantum computers, we develop a full-fledged Floyd-Hoare logic for both partial and total correctness of quantum programs. It is proved thatthis logic is (relatively) complete by exploiting the power of weakest preconditions andweakest liberal preconditions for quantum programs.

∗Centre for Quantum Computation and Intelligent Systems, P.O. Box 123, Broadway NSW 2007, Aus-tralia, [email protected]

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Contributed talks

Completions of Basic Algebras and Löb AlgebrasMajid Alizadeh∗ Hiroakira Ono†

Abstract

Closure under various completions are studied for the variety basic algebras and thevariety of Löb algebras. Basic algebras arise from (Visser’s) basic logic, and the classof Löb algebras is an important subclass of basic algebras. Canonical extensions, idealcompletions, MacNeille completions and complete ideal completions of these algebrasare discussed.

Keywords: Basic algebra, Löb algebra, Completion

1 Introduction

Basic predicate logic is a predicate logic with intuitionistic language whose semantics isgiven by Kripke frames with transitive accessibility relations. The propositional part ofthe logic was first introduced by Visser [9], called basic propositional logic BPL, andwas studied further by Ardeshir and Ruitenburg [4]. Basic predicate logic BQL which isthe predicate extension of BPL was first introduced by Ruitenburg in [8]. The originalmotivation of this work is to get algebraic semantics suitable for basic predicate logic, buthere we will focus on completions of basic algebras. We will first give a brief overview ofcanonical extensions and ideal completions of basic algebras. An interesting subclass ofbasic algebras is the class of Löb algebras, and both of these two classes of algebras areknown to form varieties, which are denoted by BA and LA, respectively. We show thatBA is closed under both canonical extensions and ideal completions. On the other hand,LA is not closed under any completion. Next we consider both MacNeille completions andcomplete ideal completions. These completions are known to be regular, i.e. embeddingsassociated with them preserve existing infinite joins and meets. It is shown that BA isnot closed under MacNeille completions, while the class of basic algebras with join infinitedistributivity is closed under complete ideal completion.

2 Canonical Extensions and Ideal Completions

We give a brief overview of canonical extensions of basic algebras and then introduce idealcompletion of these algebras.

∗School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran,P. O. Box 14155-6455, Tehran, Iran, [email protected]

†Research center for Integrated Science, Japan Advanced Institute of Science and Technology, Asahidai,Nomi, Ishikawa, 923-1292, Japan, [email protected]

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Definition 2.1. A basic algebra B = ⟨B,∧,∨,→, 0, 1⟩ is an algebra with constants 0 and1, and binary functions ∧, ∨, and →, such that

1. ⟨B,∧,∨, 0, 1⟩ forms a bounded distributive lattice,

2. the function → satisfies the following identities:

a→ b ∧ c = (a→ b) ∧ (a→ c);

b ∨ c→ a = (b→ a) ∧ (c→ a);

a→ a = 1;

a 6 1→ a;

(a→ b) ∧ (b→ c) 6 a→ c.

Note that the relation 6 is the inequality induced by the lattice, and therefore they can bereplaced by indentities. A basic algebra is called a Löb algebra when (1→ x)→ x = 1→ xholds for all x ∈ B. Sometimes, 1 → x is denoted by x. Clearly, both the class of basicalgebras and the class of Löb algebras form varieties. One of the characteristic feature ofbasic algebras is that the law of residuation x 6 y → z ⇐⇒ x ∧ y 6 z does not hold ingeneral. In fact, a basic algebra satisfies x 6 y → z =⇒ x ∧ y 6 z if and only if it is aHeyting algebra. A basic algebra is complete if it is complete as a lattice. A subvariety ofbasic algebras is closed under a given completion method if the completion of every algebrain the variety by that method belongs also to the variety.

Canonical extensions of basic algebras, introduced by Ardeshir [4], are briefly describedas follows. For a basic algebra B, let W =WB be the set of all prime filters of B. Define abinary relation ≺ on W as follows: F ≺ F

′ iff b ∈ F ′ whenever a→ b ∈ F and a ∈ F ′ . Asubset X of W is called an upset of W if and only if conditions F ∈ X and F ⊆ G implyG ∈ X for each F,G ∈ W . Then the algebra ⟨Up(W ),∩,∪,→, ∅,W ⟩, which is called thecanonical extension of B, is shown to be a basic algebra, where Up(W ) is the set of allupsets of W and → is defined by;

X → Y = F ∈W : ∀G ≻ F (if G ∈ X then G ∈ Y ).

Let us define a mapping f from B to Up(W ) by

f(a) = P | P is a prime filter and a ∈ P, for each a ∈ B.

Proposition 2.2. Every basic algebra is embedded into its canonical extension by themapping f .

On the other hand, it was shown by the first author that the variety of Löb algebrasLA is not closed under canonical extensions. But as shown later, we show a much strongerresult on LA. Next we consider ideal completions of basic algebras. A subset I ⊆ B is anideal on a basic algebra B if (1) 0 ∈ I, (2) if a, b ∈ I, then a ∨ b ∈ I and (3) if a 6 b andb ∈ I, then a ∈ I. For each subset S of B, (S] denotes the ideal generated by S, i.e. thesmallest ideal containing S. As usual we use the symbol (a] for the ideal generated by thesingleton set a. For ideals I and J we define I ∧J = I ∩J , and also for a set Γ of ideals,∨Γ is defined by (∪Γ], in particular I ∨ J = (I ∪ J ]. Moreover, for ideals I and J , let

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I →∗ J = x ∈ B : for every i in I there is a j in J such that x 6 i→ j.

Then we can show that I →∗ J is also an ideal.

Theorem 2.3. Let B be a basic algebra. Then I(B) = ⟨I(B),∧,∨,→∗, (0], (1]⟩ is a com-plete basic algebra, where I(B) be the set of all ideals of B. Moreover, the map h definedby h(x) = (x] for each x ∈ B is an embedding of B into the algebra I(B).

The complete algebra I(B) is called the ideal completion of B. It can be shown thatthe basic algebra I(B) is a Heyting algebra if and only if B is a Heyting algebra. Let usrecall that a lattice L is join infinite distributive (JID), if for every a and every existingjoin

∨t bt, the join

∨t(a ∧ bt) exists and the equality a ∧ (

∨t bt) =

∨t(a ∧ bt) holds.

Proposition 2.4. For every basic algebra B, its ideal completion is join infinite distribu-tive.

It is well-known that a complete distributive lattice is a Heyting algebra if and only ifit is join infinite distributive. In fact, the Heyting implication can be defined in complete,join infinite distributive lattice as follows. For every a, b

a → b =∨c : a ∧ c 6 b

From Theorem 2.3 and 2.4 it follows that the ideal completion of every basic algebra formsa Heyting algebra, but the Heyting implication is different from the implication of basicalgebras in general (see the following example).

Example Let B be a bounded distributive lattice with the universe B = 0, a, b, 1 suchthat a ∧ b = 0, a ∨ b = 1. We define an implication on B as follows: 1 → a = b → a = a,1 → b = a → b = 1, x → 0 = a, for x > 0 and x → y = 1, for x 6 y. Then ⟨B,→⟩ is abasic algebra. Since the ideal completion of B is a finite distributive lattice, it is a Heytingalgebra in which the Heyting implication →∗ satisfies B →∗ (b] = (b]. But the implicationof basic algebra satisfies B →∗ (b] = B.

The following result is shown in [2].

Lemma 2.5. In every Löb algebra, x(= 1→ x) = x if and only if x = 1 for any elementx.

By using this lemma, we have the following.

Theorem 2.6. The variety LA of Löb algebras is not closed under any completions.

Proof. We show that there exists a Löb algebra which has no complete extension. Let B bea Löb algebra for which n+10 = n0 for any n ∈ ω. (The existence of such a Löb algebrais known.) In the product algebra Bω, let cn be (0, · · · , 0︸︷︷︸

n−times

, 0,0,20, · · · ) for each n ∈ ω.

Note that in Bω, cn is equal to cn−1 for all but a finite number of coordinates. (Note thatcn is defined coordinate-wise). Define the equivalence relation ∼ on Bω by; two elements

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are equivalent iff they coincide in all but a finite number of coordinates. Clearly ∼ is acongruence relation on Bω. Since LA is a variety, the quotient algebra Bω/∼ is also a Löbalgebra. Denote the equivalence class of cn by cn. We can show that cn = cn = cn−1

for all n > 1. Now suppose that Bω/∼ has a complete extension D with an embedding f .Then in D

∧n>0 f(cn) exists and is not to equal f(1). For, f(1) is the greatest element

of D and cn = 1. Then we have (∧f(cn+1)) 6 f(cn+1) = f(cn+1) = f(cn) for all

n > 0. So (∧f(cn)) 6

∧f(cn). Since the converse inequality holds always, we have

(∧f(cn)) =

∧f(cn). Then by the above lemma

∧n>0 f(cn) = f(1). But this is a

contradiction.

3 MacNeille Completions and Complete Ideal Completions

It is well-known that the MacNeille completion of a distributive lattice preserves existinginfinite joins and meets but does not necessarily preserve distributivity. But MacNeillecompletions of Heyting algebras are still Heyting algebras. The MacNeille completion of agiven lattice is briefly described as follows. Let A be a lattice and B be any subset of A.L(B) (U(B)) denotes the set of all lower bounds (upper bounds, respectively) of B. B iscalled a normal ideal of A if B = LU(B) holds. Then the collection of all normal ideals ofA is the MacNeille completion of A. Different from the variety of Heyting algebras, thefollowing holds.

Theorem 3.1. The variety of basic algebras is not closed under MacNeille completions.

Proof. It is known that the variety of bounded distributive lattices is not closed underMacNeille completions (see e.g. [5]). Let L be an arbitrary bounded distributive lattice.Define → by x → y = 1 for all x, y ∈ L. Then ⟨L,→⟩ is shown to be a basic algebra,indeed a Löb algebra. Now let us take in particular a distributive lattice whose MacNeillecompletion is not distributive for L. Then no matter how implication is defined, it can notbe a basic algebra.

A basic algebra is called an L1-algebra if it satisfies 1 → 0 = 1. Note that in everybasic algebra B, 1→ 0 = 1 if and only if a→ b = 1 for all a, b. In [3] it was shown that theminimal subvarieties of the variety of basic algebras are only two, i.e. the variety of Booleanalgebras and the variety of L1-algebras. Since non-degenerate Boolean algebra cannot bea Löb algebra, the latter is the single minimal variety of the variety of Löb algebras. ForMacNeille completions, we can strengthen the result in Theorem 2.6 in the following way.

Theorem 3.2. (1) Every subvariety of basic algebras containing the two elements L1-algebra is not closed under MacNeille completions.(2) In particular, every subvariety of Löb algebras is not closed under MacNeille comple-tions.

Proof. It was shown [3] that the two elements L1-algebra generates the variety of all L1-algebras. Then similarly to Theorem 3.1 one can (1). The statement (2) is an immediateconsequence of (1).

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Let us introduce the definitions of complete ideal completions (or Crawley completions)for basic algebras. (See [7] for complete ideal completions of residuated lattices.) An subsetI of a basic algebra B is a complete ideal iff (1) it is downward closed and (2) if aj ∈ Ifor each j ∈ S and

∨j aj exists, then

∨j aj ∈ I. Ic(B) is denoted the set of all complete

ideals of B. For every subset X of a basic algebra, there exists the smallest completeideal Ic(X) including X. In particular, we denote Ic(

∪Γ) by

∨c Γ. Also, we defineI →c J = Ic(I →∗ J) for I, J ∈ Ic(B). Note that in the variety of Heyting algebras,MacNeille completions and complete ideal completions coincide.

Let us recall that an embedding of an algebra B to a complete algebra B∗ is said to beregular if it preserves all existing infinite joins and meets in B. In this case the completealgebra B∗ is called a regular completion of B. Let B and B∗ be join infinite distributivebasic algebras and B∗ is a regular completion of B with a regular embedding f . ThenB∗ is called a join dense completion of B if every element a in B∗ can be expressed asa =

∨f(x)| f(x) 6 a and x ∈ B.

Proposition 3.3. Let B be a join infinite distributive basic algebra and X ⊆ B. ThenIc(X) = y | y =

∨i xi for existing

∨i xi such that xi ∈ (X] for each i.

We show that the class of join infinite distributive basic algebras is closed under com-plete ideal completions.

Theorem 3.4. Let B be a join infinite distributive basic algebra. Then

Ic(B) = ⟨Ic(B),∧,∨c,→c, (0], (1]⟩

is a complete, join infinite distributive basic algebra and the embedding f : B −→ Ic(B);f(x) = (x] is regular.

Proof. Clearly Ic(B) is a bounded lattice. we show that it is also join infinite distributive.Obviously

∨cJ∈Γ(I ∧J) 6 I ∧

∨c Γ, for every complete ideal I and every subset Γ of Ic(B).Let x be an element of I∧

∨c Γ, then x ∈ I and x =∨y | y 6 x and y ∈ J for some J ∈ Γ

by the previous proposition. Therefore x ∈ Ic(∪J∈Γ(I ∧J)) =∨cJ∈Γ(I ∧J). One can easily

show that Ic(B) is a basic algebra and f is a regular embedding.

Similarly to [7], we can show the following

Theorem 3.5. (1) Let B be a join infinite distributive basic algebra. Then for everysubset U and V of B, Ic(U) ∩ Ic(V ) = Ic(U ∩ V ).(2) For each basic algebra B, B is join infinite distributive if and only if Ic(B) is.

We end the paper by showing that the complete ideal completion is minimum in thefollowing sense.

Theorem 3.6. Let B be a join infinite distributive basic algebra. Then every join densecompletion of B is lattice isomorphic to Ic(B).

Proof. Let f and g be the embedding of B into Ic(B), and into a join dense completionB∗ of B, respectively. For each complete ideal I we define h(I) =

∨x∈I g(x). Then h is

the required isomorphism from Ic(B) to B∗.

15


References

[1] Alizadeh, M., Ardeshir, M.: Amalgamation property for the class of basic algebras andsome of its natural subclasses. Archive for Mathematical Logic. 45, 913-930 (2006)

[2] Alizadeh, M., Ardeshir, M.: On Löb algebras. Mathematical Logic Quarterly. 52, 95-105(2006)

[3] Alizadeh, M., Ardeshir, M.: On Löb algebras II. To appear in Logic Journal of IGPL.

[4] Ardeshir, M., Ruitenburg, M.: basic propositional calculus I. Mathematical Logic Quar-terly. 44, 317-343 (1998)

[5] Harding, J.: Any lattice can be regularly embedded into the MacNeille completion ofa distributive lattice. Houston J. Math. 19, 39-44 (1993)

[6] Harding, J., Bezhanishvili, G.: MacNeille completions of Heyting algebras. Houston J.Math. 30, 937-950 (2004)

[7] Ono, H., Crawley completions of residuated lattices and algebraic completeness ofsubstructural logics, to appear in Studia Logica.

[8] Ruitenburg, W.: Basic predicate calculus. Notre Dame Journal of Formal Logic. 39,18-46 (1998)

[9] Visser, V.: A propositional logic with explicit fixed points. Studia Logica. 40, 155-175(1981)

16

Fuzzy Set Theories with Naïve ComprehensionLibor Běhounek∗

As notoriously known, the unrestricted (or naïve) comprehension schema is inconsistentover classical logic due to Russell’s paradox. Nevertheless, the schema has been provedconsistent over several contraction-free substructural logics of varying strength, includingthe logic BCK [10] and variants of linear logic [16, 3]. Since deductive fuzzy logics, too,belong to the family of contraction-free substructural logics, it is natural to ask whethernaïve comprehension is consistent over (at least some) deductive fuzzy logics.

The conjecture that naïve comprehension is consistent over (infinite-valued) Łukasiewiczlogic Ł, which later turned out to be one of the prominent deductive fuzzy logics, was firstproposed in 1957 by Skolem [13], who moreover conjectured that a large part of math-ematics could be formalized in naïve set theory over Ł. After several partial results bySkolem [14, 15], Chang [4], and Fenstad [6], a full consistency proof was presented in 1979by White [18]. The theory has further been studied by Hájek [8, 9] and Yatabe [19].However, in 2009, Terui [pers. comm.] found what appears to be a serious gap in White’sconsistency proof. Currently, therefore, the consistency status of naïve set theory overŁukasiewicz logic is still unknown. Nevertheless, Terui has found a consistency proof fornaïve set theory over the weaker fuzzy logic IMTL.1 It has therefore become important toinvestigate which of the results by Hájek and Yatabe for naïve set theory over Łukasiewiczlogic remain valid over IMTL (or even more basic fuzzy logic MTL).2

In this paper, I shall briefly introduce the aforementioned naïve fuzzy set theories andsketch their basic properties; in particular, I shall summarize main results by Hájek andYatabe, show some results that can be adapted for (I)MTL, and indicate the directions forfuture work.

Unrestricted comprehension over the logics Ł and (I)MTL Deductive fuzzy log-ics are a family of intuitionistic substructural logics, i.e., the logics of residuated lattices,characterized by the validity of the prelinearity axiom (φ → ψ) ∨ (ψ → φ). Equivalentlythey can be delimited as the logics of linearly ordered residuated lattices; the best known

∗Institute of Computer Science, Academy of Sciences of the Czech Republic, and Faculty of ComputerScience, Technical University of Vienna. Supported by grant No. P103/10/P234 of the Czech ScienceFoundation. Email: [email protected]

1Terui, pers. comm., March 2011. To my knowledge, the proof has not yet been published and verifiedby the mathematical community.

2It can be easily shown that unrestricted comprehension is inconsistent in fuzzy logics that do notadmit fixed points of negation (such as SMTL, ΠMTL, SBL, Π, or G), or are n-contractive, i.e., validateφn−1 → φn, for n ≥ 1 (including the fuzzy logics CnMTL, SnMTL, NM, and WNM). Of course, nofinitely valued operator (such as the fuzzy connective ∆, defined in linear residuated lattices as ∆x = 1 ifx = 1 and ∆x = 0 otherwise) is compatible with naïve comprehension, either, due to an adapted versionof Russell’s paradox.

17


of them furthermore satisfy the structural rules of exchange and weakening, and so arecharacterized as the logics of commutative integral bounded linear residuated lattices. Theweakest of such logics is the logic MTL, which arises by adding prelinearity to the calculusLJ of intuitionistic logic without the rule of contraction. The logic IMTL extends MTL bythe axiom of double negation, ¬¬φ→ φ. The well-known infinite-valued Łukasiewicz logicŁ further extends IMTL by the axiom of divisibility, φ ∧ ψ → φ & (φ → ψ). Note thatlike in all contraction-free substructural logics, two different conjunction connectives arepresent in the language: the lattice conjunction ∧, and the residuated conjunction &.The two equivalence connectives of contraction-free logics, (φ → ψ) & (ψ → φ) and(φ→ ψ) ∧ (ψ → φ), turn out to be equivalent in deductive fuzzy logics.

The first-order extensions of these logics interpret the quantifiers ∀ and ∃ respectivelyas the infimum and supremum in the residuated lattice of truth values. They can beaxiomatized by Rasiowa’s axioms for quantifiers [11] plus the axiom (∀x)(φ ∨ ψ(x)) →φ ∨ (∀x)ψ(x) ensuring the completeness w.r.t. models over linearly ordered residuatedlattices. For more information on these logics see [7, 5].

The naïve set theory CL over a deductive fuzzy logic L is a first-order theory in L withthe only primitive predicate ∈, axiomatized by the schema of unrestricted comprehension(∃z)(∀x)(x ∈ z ↔ φ) for all formulae φ not containing free occurrences of z. Equivalently,the theory can be introduced with the comprehension terms x | φ for all formulae φ inthe language and the comprehension axioms y ∈ x | φ(x) ↔ φ(y).

Equality in naïve fuzzy set theories Like in many other set theories with full com-prehension, the extensional equality (co-extensionality) x ≈ y ≡df (∀u)(x ∈ u ↔ y ∈ u)differs in naïve fuzzy set theories from the intensional (“Leibniz”) equality x = y ≡df

(∀u)(x ∈ u ↔ y ∈ u). In CŁ, the Leibniz identity = can be shown to be bivalent (i.e.,x = y ∨ x = y), while ≈ does have intermediate instances. Both = and ≈ are (fuzzy)equivalence relations, but only = ensures intersubstitutivity (x = y → (φ(x) ↔ φ(y)));consequently, the relation = is stronger than ≈ (i.e., x = y → x ≈ y). Assuming theconverse (i.e., the extensionality of sets) is inconsistent with CŁ; in fact, as shown by Há-jek [9], in CŁ there are infinitely many Leibniz-different sets co-extensional with the emptyset ∅ =df x | ⊥.

Natural numbers in CŁ Basic set operations (such as ∪,∩, . . . ) are defined by theobvious comprehension terms. Due to the crispness of =, the usual definitions of singletons,pairs, ordered pairs (following Kuratowski), and set-successors x ∪ y behave well in CŁ.Like in other naïve set theories over contraction-free substructural logics (e.g., [3, 16])natural numbers can in CŁ be defined more elegantly than in ZF-style set theories: bythe Fixed Point Theorem for comprehension terms (for each formula φ(x, . . . , z) there isa comprehension term ζ such that ζ ≈ x | φ(x, . . . , ζ)), sets can be (extensionally)specified by referring to themselves; natural numbers can thus be defined as successorsof natural numbers or 0, i.e., by a fixed-point comprehension term ω ≈ n | n = ∅ ∨(∃m ∈ ω)(n = m). (Note that there are infinitely many such fixed points that arenon-co-extensional.) Restall [12], Hájek [8], and Yatabe [19] showed the ω-inconsistency of

18


slightly different variants of the theory. In [9] Hájek showed the essential incompletenessand essential undecidability of a rather weak arithmetic defined over ω in CŁ.

Equality in CMTL and CIMTL Many theorems mentioned in the previous paragraphsrely on the bivalence of Leibniz identity in CŁ. Hájek’s [8] proof of the bivalence of =employs the fact that in CŁ, the &-idempotence of the truth value of x = y already impliesthe bivalence of =. This property, however, fails in IMTL, which can have non-trivialidempotent truth values. Still, some important properties of singletons, pairs, orderedpairs, and set successors can be recovered by more cautious proofs that only utilize theidempotence of x = y (which is provable even in CMTL). For example, the followingproperties can still be proved in CMTL:

a = b ↔ a = b a, b = c, d ↔ (a=c ∧ b=d) ∨ (a=d ∧ b=c)a, b ⊆ c ↔ a = b = c a, b ≈ a ↔ a = b

⟨a, b⟩ = ⟨c, d⟩ ↔ a = c ∧ b = d y ≈ y ∪ x ↔ x ∈ y

etc., where x ⊆ y ≡df (∀u)(u ∈ x → u ∈ y) and ⟨x, y⟩ =df x, x, y. Moreover, it canbe proved in CMTL that non-trivial idempotents (and so the non-bivalent values of x = y)can only be ‘large’ in the algebra of truth values. In particular, for rn =df x | (x /∈ x)n,where φn ≡df &n

i=1 φ, CMTL proves x = y ∨ (rn /∈ rn → x = y); the truth valuesof rn /∈ rn form a strictly increasing sequence and are all nilpotent, as CMTL ⊢ (rn /∈rn)

2n ↔ ⊥; therefore, non-trivial idempotents can only be larger than all truth values ofrn /∈ rn (including the fixed point r1 /∈ r1 of negation). Several results on CŁ (such asthe inconsistence of extensionality) can be recovered by means of these theorems alreadyin CMTL. However, it is an open problem whether the bivalence of = can be proved inCMTL, or at least whether some of the more advanced results of [8, 9] can be reconstructedwithout the bivalence of =.

Fuzzy set theory in CMTL Besides having features common in naïve set theories (suchas the existence of universal set, the Fixed Point Theorem, the non-extensionality andnon-well-foundedness of sets, etc.), CMTL moreover proves many usual properties of fuzzysets. In particular, a large part of elementary fuzzy set theory follows from propositionaltheorems of the underlying logic, as shown by the following metatheorem (for its strongervariant in higher-order MTL see [2]):

For a propositional MTL-formula φ(p1, . . . , pn), define the following elementary setoperations and relations:

opφ(x1, . . . , xn) =df q | φ(q ∈ x1, . . . , q ∈ xn)rel∀φ(x1, . . . , xn) ≡df (∀q)φ(q ∈ x1, . . . , q ∈ xn)rel∃φ(x1, . . . , xn) ≡df (∃q)φ(q ∈ x1, . . . , q ∈ xn)

Thus, e.g., x∩ y = opp1&p2(x, y) and x ⊆ y = rel∀p1→p2(x, y). Let φi, φ′i, ψi,j , ψ

′i,j be propo-

sitional formulae of MTL. If MTL ⊢ &ki=1 φi(ψi,1, . . . , ψi,ni) →

∧k′

i=1 φ′i(ψ

′i,1, . . . , ψ

′i,n′

i),

then:

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1. CMTL ⊢&ki=1 rel

∀φi(opψi,1

(x), . . . , opψi,ni(x))→∧k′

i=1 rel∀φ′i(opψ′

i,1(x), . . . , opψ′i,n′

i

(x)), and

2. CMTL ⊢(&k−1i=1 rel∀φi

(opψi,1(x), . . . , opψi,ni

(x))→rel∃φk

(opψk,1(x), . . . , opψk,nk

(x)))→

∧k′

i=1 rel∃φ′i(opψ′

i,1(x), . . . , opψ′i,n′

i

(x)).

(The theorem also works with∨

instead of∧

in the consequent, and the first claim alsowith

∧instead of & in the antecedent.) The usual laws of fuzzy set theory, such as

(x ⊆ z) ∧ (y ⊆ z) → x ∪ y ⊆ z, are thus corollaries of the corresponding propositionaltheorems (in this case, (p→ r) ∧ (q → r)→ (p ∨ q → r)).

A problem with employing CMTL as a foundational basis of fuzzy set theory lies inthe fact that various concepts of fuzzy set theory (such as the kernel of the fuzzy set, i.e.,the set of its 1-elements) are not generally available in CMTL on pain of contradiction. Apossible remedy can be to conservatively extend CMTL with classes, in a manner similar tothe extension of classical ZF to NBG. An extension of this kind, allowing the connective∆ in class terms, is sketched in [1]. The motivation for this move (admittedly alien tothe idea of universal comprehension) is to enable referring to fuzzy collections of sets thatare anyway present in all models of the theory (such as the bivalent kernel of ω). Theresulting theory shows interesting similarities to and differences from non-standard settheories with classes, such as Vopěnka’s Alternative Set Theory AST of [17]: for instance,due to Yatabe’s overspill result of [19], standard natural numbers form a proper class inreal-valued CŁ (just like in AST), but (unlike AST or NBG) not all sets are classes (thoughevery set is co-extensional with a class; a class is proper if it is not an extension of a set).It remains to be seen whether the underlying universe of sets provides a sufficiently richground structure for a mathematically non-trivial class theory over CMTL.

References

[1] L. Běhounek. Extending Cantor–Łukasiewicz set theory with classes. In P. Cintula,E. P. Klement, and L. N. Stout, editors, Lattice-Valued Logic and Its Applications.Abstracts of the 31st Linz Seminar on Fuzzy Set Theory, pages 14–19, Linz, 2010.

[2] L. Běhounek and P. Cintula. Fuzzy class theory. Fuzzy Sets and Systems, 154(1):34–55,2005.

[3] A. Cantini. The undecidability of Grisin’s set theory. Studia Logica, 74:345–368, 2003.

[4] C. C. Chang. The axiom of comprehension in infinite valued logic. MathematicaScandinavica, 13:9–30, 1963.

[5] F. Esteva and L. Godo. Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124(3):271–288, 2001.

[6] J. E. Fenstad. On the consistency of the axiom of comprehension in the Łukasiewiczinfinite valued logic. Mathematica Scandinavica, 14:65–74, 1964.

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[7] P. Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer,Dordercht, 1998.

[8] P. Hájek. On arithmetic in Cantor–Łukasiewicz fuzzy set theory. Archive for Mathe-matical Logic, 44(6):763–782, 2005.

[9] P. Hájek. On equality and natural numbers in Cantor–Łukasiewicz set theory. Toappear in the Logic Journal of the IGPL, doi:10.1093/jigpal/jzq019, 2010.

[10] U. Petersen. Logic without contraction as based on inclusion and unrestricted ab-straction. Studia Logica, 64:365–403, 2000.

[11] H. Rasiowa. An Algebraic Approach to Non-Classical Logics. North-Holland, Amster-dam, 1974.

[12] G. Restall. Arithmetic and truth in Łukasiewicz’s infinitely valued logic. Logique etAnalyse, 36:25–38, 1995.

[13] T. Skolem. Bemerkungen zum Komprehensionsaxiom. Zeitschrift für mathematischeLogik und Grundlagen der Mathematik, 3:1–17, 1957.

[14] T. Skolem. Investigations on comprehension axiom without negation in the definingpropositional functions. Notre Dame Journal of Formal Logic, 1:13–22, 1960.

[15] T. Skolem. A set theory based on a certain 3-valued logic. Mathematica Scandinavica,8:71–80, 1960.

[16] K. Terui. Light affine set theory: A naive set theory of polynomial time. Studia Logica,77(1):9–40, 2004.

[17] P. Vopěnka. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.

[18] R. B. White. The consistency of the axiom of comprehension in the infinite-valuedpredicate logic of Łukasiewicz. Journal of Philosophical Logic, 8:509–534, 1979.

[19] S. Yatabe. Distinguishing non-standard natural numbers in a set theory withinŁukasiewicz logic. Archive for Mathematical Logic, 46:281–287, 2007.

21

Quantified Conditional Logics are Fragments ofHOL

Christoph Benzmüller∗ Valerio Genovese†

1 Introduction

A semantic embedding of propositional conditional logic in classical higher-order logic HOL(Church’s type theory) has been presented in [3]. This embedding exploits the naturalcorrespondence between selection function semantics for conditional logics [10] and HOL.In fact, selection function semantics can be seen as an higher-order extension of well-knownKripke semantics for modal logic and cannot be naturally embedded into first-order logic.

In this paper we extend the embedding in [3] to also include quantification over propo-sitions and individuals. This embedding of quantified conditional logic in HOL is soundand complete.

2 Quantified Conditional Logics

We extend propositional conditional logics with quantification over propositional variablesand over individuals of a first-order domain. Below, we only consider constant domains,i.e., every possible world has the same domain.

Let IV be a set of first-order (individual) variables, PV a set of propositional variables,and SYM a set of predicate symbols of any arity. Formulas of quantified conditional logicare given by the following grammar (where Xi ∈ IV, P ∈ PV, k ∈ SYM):

φ,ψ ::= P | k(X1, . . . , Xn) | ¬φ | φ ∨ ψ | ∀X.φ | ∀P.φ | φ⇒ ψ

From the selected set of primitive connectives, other logical connectives can be in-troduced as abbreviations: for example, φ ∧ ψ, φ → ψ (material implication), and ∃X.φabbreviate ¬(¬φ∨¬ψ), ¬φ∨ψ and ¬∀X.¬φ etc. Syntactically, quantified conditional logicscan be seen as a generalization of quantified multimodal logic where the index of modality⇒ is a formula of the same language. For instance, in (φ⇒ ψ)⇒ δ the subformula φ⇒ ψis the index of the second occurrence of ⇒.

Regarding semantics, many different formalizations have been proposed (see [8]), herewe focus on the selection function semantics [6], which is based on possible world structuresand has been successfully used in [9] to develop proof methods for some conditional logics.We adapt selection function semantics for quantified conditional logics.

∗Freie Universität Berlin, [email protected]†University of Luxembourg, [email protected]

22


An interpretation is a structureM = ⟨S, f,D,Q, I⟩ where, S is a set of possible itemscalled worlds, f : S×2S 7→ 2S is the selection function, D is a non-empty set of individuals(the first-order domain), Q is a non-empty collection of subsets of W (the propositionaldomain), and I is a classical interpretation function where for each n-ary predicate symbolk, I(k,w) ⊆ Dn.

A variable assignment g = (giv, gpv) is a pair of maps where, giv : IV 7→ D maps eachindividual variable in IV to an object in D, and gpv : maps each propositional variable inPV to a set of worlds in Q.

Satisfiability of a formula φ for an interpretation M = ⟨S, f,D,Q, I⟩, a world s ∈ S,and a variable assignment g = (giv, gpv) is denoted as M, g, s |= φ and defined as follows,where [a/Z]g denote the assignment identical to g except that ([a/Z]g)(Z) = a:

M, g, s |= k(X1, . . . , Xn) if and only if ⟨giv(X1), . . . , giv(Xn)⟩ ∈ I(k,w)M, g, s |= P if and only if s ∈ gpv(P )M, g, s |= ¬φ if and only if M, g, s |= φ (that is, not M, g, s |= φ )M, g, s |= φ ∨ ψ if and only if M, g, s |= φ or M, g, s |= ψM, g, s |= ∀X.φ if and only if M, ([d/X]giv, gpv), s |= φ for all d ∈ DM, g, s |= ∀P.φ if and only if M, (giv, [p/P ]gpv), s |= φ for all p ∈ QM, g, s |= φ⇒ ψ if and only if M, g, t |= ψ for all t ∈ S such that t ∈ f(s, [φ])

where [φ] = u |M, g, u |= φAn interpretationM = ⟨S, f,D,Q, I⟩ is a model if for every variable assignment g and

every formula φ, the set of worlds s ∈ S | M, g, s |= φ is a member of Q. As usual, aconditional formula φ is valid in a model M = ⟨S, f,D,Q, I⟩, denoted withM |= φ, if andonly if for all worlds s ∈ S and variable assignments g holds M, g, s |= φ. A formula φ isa valid, denoted |= φ, if and only if it is valid in every model.

f is defined to take [φ] (called the proof set of φ w.r.t. a given modelM) instead of φ.This approach has the consequence of forcing the so-called normality property: given amodel M, if φ and φ′ are equivalent (i.e., they are satisfied in the same set of worlds),then they index the same formulas w.r.t. to the ⇒ modality. The axiomatic counterpartof the normality condition is given by the rule (RCEA)

φ↔ φ′

(RCEA)(φ⇒ ψ)↔ (φ′ ⇒ ψ)

Moreover, it can be easily shown that the above semantics forces also the following rulesto hold:

(φ1 ∧ . . . ∧ φn)↔ ψ(RCK)

(φ0 ⇒ φ1 ∧ . . . ∧ φ0 ⇒ φn)→ (φ0 ⇒ ψ)

φ↔ φ′

(RCEC)(ψ ⇒ φ)↔ (ψ ⇒ φ′)

We refer to CK [6] as the minimal quantified conditional logic closed under rulesRCEA, RCEC and RCK. In what follows, only quantified conditional logics extending CKare considered.

23


3 Classical Higher-Order Logic

HOL is a logic based on simply typed λ-calculus [7, 2]. The set T of simple types in HOL isusually freely generated from a set of basic types o, i using the function type constructor→. Here we instead consider a set of basic type o, i, u, where o denotes the type ofBooleans, and where i and u denote some non-empty domains. Without loss of generality,we will later identify i with a set of worlds and u with a domain of individuals.

Let α, β, o ∈ T . The terms of HOL are defined by the grammar (pα denotes typedconstants and Xα typed variables distinct from pα):

s, t ::= pα | Xα | (λXα.sβ)α→β | (sα→β)β | (¬o→o so)o | (so ∨o→o→o to)o | (Π(α→o)→o sα→o)o

Complex typed terms are constructed via abstraction and application. The primitivelogical connectives are ¬o→o,∨o→o→o and Π(α→o)→o (for each type α). From these, otherlogical connectives can be introduced as abbreviations: for example, ∧ and → abbreviatethe terms λA.λB.¬(¬A ∨ ¬B) and λA.λB.¬A ∨ B, etc. HOL terms of type o are calledformulas. Binder notation ∀Xα.so is used as an abbreviation for (Π(α→o)→o (λXα.so)).Substitution of a term Aα for a variable Xα in a term Bβ is denoted by [A/X]B, where itis assumed that the bound variables of B avoid variable capture. Well known operationsand relations on HOL terms include βη-normalization and βη-equality, denoted by s =βη t.

The following definition of HOL semantics closely follows the standard literature [1, 2].A frame is a collection Dαα∈T of nonempty sets called domains such that Do =

T, F where T represents truth and F falsehood, Di = ∅ and Du = ∅ are chosen arbitrary,and Dα→β are collections of total functions mapping Dα into Dβ .

An interpretation is a tuple ⟨Dαα∈T , I⟩ where Dαα∈T is a frame and where functionI maps each typed constant cα to an appropriate element of Dα, which is called thedenotation of cα. The denotations of ¬,∨ and Π(α→o)→o are always chosen as usual. Avariable assignment ϕ maps variables Xα to elements in Dα.

An interpretation is a Henkin model (general model) if and only if there is a binaryvaluation function V such that V(ϕ, sα) ∈ Dα for each variable assignment ϕ and termsα, and the following conditions are satisfied for all ϕ, variables Xα, constants pα, andterms lα→β, rα, sβ (for α, β ∈ T ): V(ϕ,Xα) = ϕ(Xα), V(ϕ, pα) = I(pα), V(ϕ, (lα→β rα)) =(V(ϕ, lα→β))(V(ϕ, rα)), and V(ϕ, λXα.sβ) represents the function from Dα into Dβ whosevalue for each argument z ∈ Dα is V(ϕ[z/Xα], sβ), where ϕ[z/Xα] is that variable assign-ment such that ϕ[z/Xα](Xα) = z and ϕ[z/Xα]Yβ = ϕYβ when Yβ = Xα.

If an interpretation H = ⟨Dαα∈T , I⟩ is an Henkin model the function V is uniquelydetermined and V(ϕ, sα) ∈ Dα is called the denotation of sα. H is called a standard modelif and only if for all α and β, Dα→β is the set of all functions from Dα into Dβ . It is easyto verify that each standard model is also a Henkin model. A formula A of HOL is validin a Henkin model H if and only if V(ϕ,A) = T for all variable assignments ϕ. In this casewe write H |= A. A is (Henkin) valid, denoted as |= A, if and only if H |= A for all Henkinmodels H.

24


Proposition 1. Let V be the valuation function of Henkin model H. The followingproperties hold for all assignments ϕ, terms so, to, lα, rα, and variables Xα, Vα (for α ∈T ): V(ϕ, (¬so)) = T if and only if V(ϕ, so) = F , V(ϕ, (so ∨ to)) = T if and only ifV(ϕ, so) = T or V(ϕ, so) = T , V(ϕ, (so ∧ to)) = T if and only if V(ϕ, so) = T andV(ϕ, so) = T , V(ϕ, (so → to)) = T if and only if V(ϕ, so) = F or V(ϕ, so) = T ,V(ϕ, (∀Xα.so)) = V(ϕ, (Π(α→o)→o (λXα.so))) = T if and only if for all v ∈ Dα holdsV(ϕ[v/Vα], ((λXα.so) V )) = T , and if lα =βη rα then V(ϕ, lα) = V(ϕ, rα)

4 Embedding Quantified Conditional Logics in HOL

Quantified conditional logic formulas are identified with certain HOL terms (predicates) oftype i→ o. They can be applied to terms of type i, which are assumed to denote possibleworlds.

Definition 2. The mapping ⌊·⌋ translates formulas φ of quantified conditional logic CKinto HOL terms ⌊φ⌋ of type i→ o. The mapping is recursively defined as follows:

⌊P ⌋ = Pi→o

⌊k(X1, . . . , Xn)⌋ = (⌊k⌋⌊X1⌋ . . . ⌊Xn⌋)⌋= (kun→(i→o) X

1u . . . X

nu )

⌊¬φ⌋ = ¬i→o ⌊φ⌋

⌊φ ∨ ψ⌋ = ∨(i→o)→(i→o)→(i→o) ⌊φ⌋⌊ψ⌋⌊φ⇒ ψ⌋ = ⇒(i→o)→(i→o)→(i→o) ⌊φ⌋⌊ψ⌋⌊∀X.φ⌋ = Π(u→(i→o))→(i→o) λXu.⌊φ⌋⌊∀P.φ⌋ = Π((i→o)→(i→o))→(i→o) λPi→o.⌊φ⌋

Pi→o and X1u, . . . , X

nu are HOL variables and kun→(i→o) is a HOL constant. ¬i→o,

∨(i→o)→(i→o)→(i→o), ⇒(i→o)→(i→o)→(i→o), Π(u→(i→o))→(i→o) and Π((i→o)→(i→o))→(i→o) re-alize the quantified conditional logics connectives in HOL. They abbreviate the followingproper HOL terms:

¬(i→o)→(i→o) = λAi→o.λXi.¬(AX)

∨(i→o)→(i→o)→(i→o) = λAi→o.λBi→o.λXi.(AX) ∨ (BX)

⇒(i→o)→(i→o)→(i→o) = λAi→o.λBi→o.λXi.∀Wi.(f X AW )→ (BW )

Π(u→(i→o))→(i→o) = λQu→(i→o).λWi.∀Xu.(QXW )

Π((i→o)→(i→o))→(i→o) = λR(i→o)→(i→o).λWi.∀Pi→o.(RP W )

The constant symbol f in the mapping of ⇒ is of type i→ (i→ o)→ (i→ o). It realizesthe selection function, i.e., its interpretation is chosen appropriately (cf. below).

This mapping induces mappings ⌊IV⌋, ⌊PV⌋ and ⌊SYM⌋ of the sets IV, PV andSYM respectively.

Analyzing the validity of a translated formula ⌊φ⌋ for a world represented by term ticorresponds to evaluating the application (⌊φ⌋ ti). In line with [4], we define vld(i→o)→o =λAi→o.∀Si.(A S). With this definition, validity of a quantified conditional formula φ inCK corresponds to the validity of the corresponding formula (vld ⌊φ⌋) in HOL, and viceversa.

25


5 Soundness and Completeness

To prove the soundness and completeness of the embedding, a mapping from selection func-tion models into Henkin models is employed. This mapping will employ a correspondingmapping of variable assignments for quantified conditional logics into variable assignmentsfor HOL.

Definition 3 (Mapping of Variable Assignments). Let g = (giv : IV −→ D, gpv : PV −→Q) be a variable assignment for a quantified conditional logic. We define the correspondingvariable assignment ⌊g⌋ = (⌊giv⌋ : ⌊IV⌋ −→ D, ⌊gpv⌋ : ⌊PV⌋ −→ Q) for HOL so that⌊g⌋(Xu) = ⌊g⌋(⌊X⌋) = g(X) and ⌊g⌋(Pi→o) = ⌊g⌋(⌊P ⌋) = g(P ) for all Xu ∈ ⌊IV⌋ andPi→o ∈ ⌊PV⌋. Finally, a variable assignment ⌊g⌋ is extended to an assignment for variablesZα of arbitrary type by choosing ⌊g⌋(Zα) = d ∈ Dα arbitrary, if α = u, i→ o.

Definition 4 (Henkin model HM). Given a quantified conditional logic model M =⟨S, f,D,Q, I⟩. The Henkin model HM = ⟨Dαα∈T , I⟩ for M is defined as follows: Di

is chosen as the set of possible worlds S, Du is chosen as the first-order domain D (cf.definition of ⌊giv⌋), Di→o is chosen as the set of sets of possible worlds Q (cf. definitionof ⌊gpv⌋)1, and all other sets Dα→β are chosen as (not necessarily full) sets of functionsfrom Dα to Dβ. For all sets Dα→β the rule that everything denotes must be obeyed, inparticular, we require that the sets Dun→(i→o) and Di→(i→o)→(i→o) contain the elementsIkun→(i→o) and Ifi→(i→o)→(i→o) as characterized below.

The interpretation I is constructed as follows: (i) Let kun→(i→o) = ⌊k⌋ for n-ary k ∈ SYM and let Xi

u = ⌊Xi⌋ for Xi ∈ IV, i = 1, . . . , n. We chooseIkun→(i→o) ∈ Dun→(i→o) such that (I kun→(i→o))(⌊g⌋(X1

u), . . . , ⌊g⌋(Xnu ), w) = T for all

worlds w ∈ Di such that M, g, w |= k(X1, . . . , Xn), that is, if ⟨giv(X1), . . . , giv(Xn)⟩ ∈I(k,w). Otherwise we choose (I kun→(i→o))(⌊g⌋(X1

u), . . . , ⌊g⌋(Xnu ), w) = F . (ii) We choose

Ifi→(i→o)→(i→o) ∈ Di→(i→o)→(i→o) such that (Ifi→(i→o)→(i→o))(s, q, t) = T for all worldss, t ∈ Di and q ∈ Di→o with t ∈ f(s, x ∈ S | q(x) = T) in M. Otherwise we choose(Ifi→(i→o)→(i→o))(s, q, t) = F . (iii) For all other constants sα, choose Isα arbitrary.2

It is not hard to verify that HM is a Henkin model.

Lemma 5. Let HM be a Henkin model for a selection function modelM. For all quantifiedconditional logic formulas δ, variable assignments g and worlds s it holds: M, g, s |= δif and only if V(⌊g⌋[s/Si], (⌊δ⌋ Si)) = T

Proof. The proof is by induction on the structure of δ. The cases for δ = P , δ =k(X1, . . . , Xn), δ = (¬φ), δ = (φ ∨ ψ), and δ = (φ ⇒ ψ) are similar to Lemma 1in [3]. The cases for δ = ∀X.φ and δ = ∀P.φ adapt the respective cases from Lemmas 4.3and 4.7 in [5].

Theorem 6 (Soundness and Completeness). |=(vld⌊φ⌋) in HOL if and only if |=φ in CK.1To keep things simple, we identify sets with their characteristic functions.2In fact, we may safely assume that there are no other typed constant symbols given, except for the

symbol fi→(i→o)→(i→o), the symbols , kun→(i→o), and the logical connectives.

26


Proof. (Soundness) The proof is by contraposition. Assume |= φ in CK, that is, thereis a model M = ⟨S, f,D,Q, I⟩, a variable assignment g and a world s ∈ S, such thatM, g, s |= φ. By Lemma 5 we have that V(⌊g⌋[s/Si], (⌊φ⌋ S)) = F in Henkin model HM =⟨Dαα∈T , I⟩ for M. Thus, by Proposition 1, definition of vld and since (∀Si.⌊φ⌋ S) =βη

(vld ⌊φ⌋) we know that V(⌊g⌋, (∀Si.⌊φ⌋ S)) = V(⌊g⌋, (vld ⌊φ⌋)) = F . Hence, HM |=(vld ⌊φ⌋), and thus |= (vld ⌊φ⌋) in HOL.

(Completeness) The proof is again by contraposition. Assume |= (vld ⌊φ⌋) in HOL,that is, there is a Henkin model H = ⟨Dαα∈T , I⟩ and a variable assignment ϕ withV(ϕ, (vld ⌊φ⌋)) = F . Without loss of generality we can assume that Henkin Model H isin fact a Henkin model HM for a corresponding quantified conditional logic modelM andthat Φ = ⌊g⌋ for a corresponding quantified conditional logic variable assignment g. ByProp. 1 and since (vld ⌊φ⌋) =βη (∀Si.⌊φ⌋ S) we have V(⌊g⌋, (∀Si.⌊φ⌋ S)) = F , and hence,by definition of vld, V(⌊g⌋[s/Si], ⌊φ⌋ S) = F for some s ∈ D. By Lemma 5 we thus knowthatM, g, s |= φ, and hence |= φ in CK.

6 Conclusion

We have presented an embedding of quantified conditional logics in HOL. This embeddingenables the uniform application of higher-order automated theorem provers and modelfinders for reasoning about and within quantified conditional logics. In previous work wehave studied related embeddings in HOL, including propositional conditional logics [3] andquantified multimodal logics [5]. First experiments with these embeddings have providedevidence for their practical relevance. Moreover, an independent case study on reasoning inquantified modal logics shows that the embeddings based approach may even outperformspecialist reasoners quantified modal logics [12]. Future work will investigate whether HOLreasoners perform similarly well also for quantified conditional logics. For a first impressionof such studies we refer to the Appendices A and B, where we also present the concreteencoding of our embedding in TPTP THF0 [11] syntax. Unfortunately we are not awareof any other (direct or indirect) prover for quantified conditional logics that could be usedfor comparison.

References

[1] P. B. Andrews. General models and extensionality. J. of Symbolic Logic, 37:395–397,1972.

[2] P. B. Andrews. Church’s type theory. In The Stanford Encyclopedia of Philosophy.2009.

[3] C. Benzmüller, D. Gabbay, V. Genovese, and D. Rispoli. Embedding and au-tomating conditional logics in classical higher-order logic. Technical report, 2011.http://arxiv.org/abs/1106.3685.

[4] C. Benzmüller and L.C. Paulson. Multimodal and intuitionistic logics in simple typetheory. Logic J. of the IGPL, 18:881–892, 2010.

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[5] C. Benzmüller and L.C. Paulson. Quantified multimodal logics in simple type theory.Logica Universalis (Special Issue om Multimodal Logics), 2011. To appear. See alsohttp://arxiv.org/abs/0905.2435.

[6] B.F. Chellas. Modal Logic: An Introduction. Cambridge: Cambridge University Press,1980.

[7] A. Church. A formulation of the simple theory of types. J. of Symbolic Logic, 5:56–68,1940.

[8] D. Nute. Topics in conditional logic. Reidel, Dordrecht, 1980.

[9] N. Olivetti, G.L. Pozzato, and C. Schwind. A sequent calculus and a theorem proverfor standard conditional logics. ACM Trans. Comput. Log., 8(4), 2007.

[10] R. Stalnaker. A theory of conditionals. In N. Rescher, editor, Studies in Logical Theory,American Philosophical Quarterly, Monograph Series no.2, page 98–112. Blackwell,Oxford, 1968.

[11] G. Sutcliffe and C. Benzmüller. Automated reasoning in higher-order logic using theTPTP THF infrastructure. J. of Formalized Reasoning, 3(1):1–27, 2010.

[12] T. Raths und J. Otten. Implementing and evaluating theorem provers for first-ordermodal logics. In M. Giese, editor, Proceedings of FTP 2011. CEUR Workshop Pro-ceedings, 2011.

A The Embedding of Quantified Conditional Logic in HOLin THF0 Syntax

We present an encoding of our embedding of quantified conditional logics in HOL in theTPTP THF0 [11] syntax. Satisfiability of this embedding is shown by the HOL reasonerSatallax3 in only 0.01 seconds.

%---------------------------------------------------------------------

%---- reserved constant for selection function f

thf(f_type,type,(

f: $i > ( $i > $o ) > $i > $o )).

%---- ’not’ in conditional logic

thf(cnot_type,type,(

cnot: ( $i > $o ) > $i > $o )).

thf(cnot_def,definition,

( cnot

= ( ^ [Phi: $i > $o,X: $i] :

~ ( Phi @ X ) ) )).

3http://www.ps.uni-saarland.de/~cebrown/satallax/

28


%---- ’or’ in conditional logic

thf(cor_type,type,(

cor: ( $i > $o ) > ( $i > $o ) > $i > $o )).

thf(cor_def,definition,

( cor

= ( ^ [Phi: $i > $o,Psi: $i > $o,X: $i] :

( ( Phi @ X )

| ( Psi @ X ) ) ) )).

%---- ’true’ in conditional logic

thf(ctrue_type,type,(

ctrue: $i > $o )).

thf(ctrue_def,definition,

( ctrue

= ( ^ [X: $i] : $true ) )).

%---- ’false’ in conditional logic

thf(cfalse_type,type,(

cfalse: $i > $o )).

thf(cfalse_def,definition,

( cfalse

= ( ^ [X: $i] : $false ) )).

%---- ’conditional implication’ in conditional logic

thf(ccond_type,type,(

ccond: ( $i > $o ) > ( $i > $o ) > $i > $o )).

thf(ccond_def,definition,

( ccond

= ( ^ [Phi: $i > $o,Psi: $i > $o,X: $i] :

! [W: $i] :

( ( f @ X @ Phi @ W )

=> ( Psi @ W ) ) ) )).

%---- ’and’ in conditional logic

thf(cand_type,type,(

cand: ( $i > $o ) > ( $i > $o ) > $i > $o )).

thf(cand_def,definition,

( cand

= ( ^ [Phi: $i > $o,Psi: $i > $o,X: $i] :

( ( Phi @ X )

& ( Psi @ X ) ) ) )).

29


%---- ’conditional equivalence’ in conditional logic

thf(ccondequiv_type,type,(

ccondequiv: ( $i > $o ) > ( $i > $o ) > $i > $o )).

thf(ccondequiv_def,definition,

( ccondequiv

= ( ^ [Phi: $i > $o,Psi: $i > $o] :

( cand @ ( ccond @ Phi @ Psi ) @ ( ccond @ Psi @ Phi ) ) ) )).

%---- ’material implication’ in conditional logic

thf(cimpl_type,type,(

cimpl: ( $i > $o ) > ( $i > $o ) > $i > $o )).

thf(cimpl_def,definition,

( cimpl

= ( ^ [Phi: $i > $o,Psi: $i > $o,X: $i] :

( ( Phi @ X )

=> ( Psi @ X ) ) ) )).

%---- ’material equivalence’ in conditional logic

thf(cequiv_type,type,(

cequiv: ( $i > $o ) > ( $i > $o ) > $i > $o )).

thf(cequiv_def,definition,

( cequiv

= ( ^ [Phi: $i > $o,Psi: $i > $o] :

( cand @ ( cimpl @ Phi @ Psi ) @ ( cimpl @ Psi @ Phi ) ) ) )).

%---- ’universal quantification (individuals)’ in conditional logic

thf(cforall_ind_type,type,(

cforall_ind: ( mu > $i > $o ) > $i > $o )).

thf(cforall_ind,definition,

( cforall_ind

= ( ^ [Phi: mu > $i > $o,W: $i] :

! [X: mu] :

( Phi @ X @ W ) ) )).

%---- ’universal quantification (propositions)’ in conditional logic

thf(cforall_prop_type,type,(

cforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o )).

thf(cforall_prop,definition,

( cforall_prop

= ( ^ [Phi: ( $i > $o ) > $i > $o,W: $i] :

! [P: $i > $o] :

( Phi @ P @ W ) ) )).

30


%---- ’existential quantification (individuals)’ in conditional logic

thf(cexists_ind_type,type,(

cexists_ind: ( mu > $i > $o ) > $i > $o )).

thf(cexists_ind,definition,

( cexists_ind

= ( ^ [Phi: mu > $i > $o] :

( cnot

@ ( cforall_ind

@ ^ [X: mu] :

( cnot @ ( Phi @ X ) ) ) ) ) )).

%---- ’existential quantification (propositions)’ in conditional logic

thf(cexists_prop_type,type,(

cexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o )).

thf(cexists_prop,definition,

( cexists_prop

= ( ^ [Phi: ( $i > $o ) > $i > $o] :

( cnot

@ ( cforall_prop

@ ^ [P: $i > $o] :

( cnot @ ( Phi @ P ) ) ) ) ) )).

%---- ’validity’ of a conditional logic formula

thf(valid_type,type,(

valid: ( $i > $o ) > $o )).

thf(valid_def,definition,

( valid

= ( ^ [Phi: $i > $o] :

! [S: $i] :

( Phi @ S ) ) )).

%---------------------------------------------------------------------

B The Barcan Formula and the Converse Barcan Formula

Using the above THF0 encoding, the Barcan formula (∀X.A⇒ B(x))→ (A⇒ ∀X.B(x))can be encoded in THF0 as given below. The HOL provers LEO-II4 and Satallax can bothprove this theorem in 0.01 seconds. This confirms that we our encoding assumes constantdomain structure.

%---------------------------------------------------------------------

include(’CK_axioms.ax’).

4http://www.leoprover.org

31


%---- conjecture statement

thf(a,type,(

a: $i > $o )).

thf(b,type,(

b: mu > $i > $o )).

thf(bf,conjecture,

( valid

@ ( cimpl

@ ( cforall_ind

@ ^ [X: mu] :

( ccond @ a @ ( b @ X ) ) )

@ ( ccond @ a

@ ( cforall_ind

@ ^ [X: mu] :

( b @ X ) ) ) ) )).

%---------------------------------------------------------------------

The converse Barcan formula (A ⇒ ∀X.B(x)) → (∀X.A ⇒ B(x)) can be encodedanalogously. Again, the HOL provers LEO-II and Stallax need only 0.01 seconds to provethis theorem.

%---------------------------------------------------------------------

include(’CK_axioms.ax’).

%---- conjecture statement

thf(a,type,(

a: $i > $o )).

thf(b,type,(

b: mu > $i > $o )).

thf(cbf,conjecture,

( valid

@ ( cimpl

@ ( ccond @ a

@ ( cforall_ind

@ ^ [X: mu] :

( b @ X ) ) )

@ ( cforall_ind

@ ^ [X: mu] :

( ccond @ a @ ( b @ X ) ) ) ) )).

%---------------------------------------------------------------------

32

A Librationist Quest for StrengthFrode Bjordal∗

I will seek to show that the librationist theory of sorts, which I have partly put forth inpublications and talks available from my web page and also remarked upon recently onthe Foundations of Mathematics email list, is strong enough to offer a model of ZF . Themethod made use of here involves making use of the same coding of Gödel numbers inthe object language as well as in the meta language appealed to in the semantic processwhich justifies the non-triviality of the librationist theory. This involves the accomodationof a truth-sort (predicate) T and a bijection sort (relation) E taking us from the sort ofnatural numbers (understood as von Neumann’s omega) in the object language of the puretheory of sorts to the full universe of sorts. The truth *operator* made use of in previoussuperseded accounts becomes superfluos. In setting up the semantic apparatus we rely onthe prescience that the system we build up is strong enough to offer a coding of its ownformulas and terms in an arithmetical fragment. The parametrized coding makes use ofthe fact that distinct variables in the contentual (super-formal or semi-formal) frameworkof librationism, given the semantic set up and the conditions on the universal bijectionE, serve as standard names of sorts. This means that it will hold schematically thatall variables denote a sort, distinct variables denote distinct sorts and that all sorts aredenoted by a variable.

In the semantic Herzbergerian-like semi-inductive process, with a librationist twist inthat we focus upon *one designated* model by means of a presupposed external enumera-tion of sort-constants, one will use merely parts of the coding. In order to isolate a modelof strong theories in the librationist framework we make use of a fixed-point constructiongoing back to Andrea Cantini and Albert Visser which we in our context call manifestationpoints. For A(x, y) any formulas with the free variables shown, we can find a manifestationpoint M such that (x)(xeM ⇔ TTA(x,M)) is a maxim (a theorem whose negation is notalso a theorem) of librationism (the two “T ”s are essential ingredients here). In order toaccomodate relative power sorts we need to have an appeal to the manifestation point itselfin the definition of the relative power-sort defined in the definition e.g. of the least Jensenclosure + this relative power-sort containing e.g. von Neumann’s ordinal omega. In orderto get stronger replacement, we need to invoke the parametrized coding as well as the truthpredicate in order to achieve sufficient strength to model ZF . As things look now, therewill have to be an appeal to bisimulation in order to interpret the isolated non-paradoxicalsort also with extensionality appropriately.

The librationist framework I have developed can be understood as one which involvestechniques which transcend the famous limitative recsults on syntactic modalities and truthpredicates by Montague, Friedman-Sheard and others. This also involves the isolation of a

∗University of Oslo, Faculty of Humanities, Department of Philosophy, Classics, History of Arts andIdea, Postboks 1020, Blindern 0315 Oslo, Norway, [email protected]

33


theory which is non-classical as compared with classical predicate logic in as far as inferencerules are concerned; but all theorems of classical predicate logic are retained and none ofthem are contradicted. Connectives should in this framework be reinterpreted in a libra-tionist manner and perhaps even be given slightly deviant neames: e.g. “negjunction” (−),“adjunction” (&), “veljunction” (V ), “subjunction” (⇒), and “equijunction” (⇔). Inferencerules will in several cases appeal both to the theoremhood and the lack of theoremhood ofa formula. A theorem A is a *maxim* if −A is not also a theorem. Else a theorem is aminor. Modus ponens for ⇒ in the sense that B is a theorem if both A and A ⇒ B aretheorems does not hold universally. But modus maximus, viz. that B is a maxim if bothA and A ⇒ B are maxims, does hold universally. For further such details the reader issent to my writings as available from or directed to from my web page. More details areavailable upon request.

Some will consider my work as one belonging to the paraconsistent group of ideas. Thereis kinship, but I am not so inclined as to characterize it in that way. From the librationistpoints of view, there indeed are paradoxical sentences. E.g., if r is Russell’s sort of alland only those sorts that are not members of themselves, it will be a theorem that rer aswell as -rer. But these sentences should, to my mind, in the librationist framework notbe considered contradictory. They have complementary *significances*: the significance ofa sentence (formula) is the set of ordinals below the closure ordinal of the semi inductiveprocess where it holds. Complementary sentences such as these are always *incompatible*,so their conjunction is *not* a theorem. It is preferable, I submit, to think of the librationistpoint of view as “parasistent”, which means *standing beyond*. “paraconsistent” has adifferent, and to my mind less attractive etymology. For similar reasons, one should notthink of librationism as *diaelethic*, which, as pointed out by e.g. John Burgess, hasa bad etymology. More importantly, I prefer “bialethic” because dialetheism is presentlyprevalently characterized as the point of view that there are true contradictions. Accordingto librationism, contradictions are never true.

34

First-Order Hybrid Logic:Introduction and Survey

Torben Braüner∗

Summary

Hybrid logic is an extension of modal logic which allows us to refer explicitly to pointsof the model in the syntax of formulas. It is easy to justify interest in hybrid logic onapplied grounds, with the usefulness of the additional expressive power. For example,when reasoning about time one often wants to build up a series of assertions about whathappens at a particular instant, and standard modal formalisms do not allow this.

What is less obvious is that the route hybrid logic takes to overcome this problem oftenactually improves the behaviour of the underlying modal formalism. For example, it be-comes far simpler to formulate proof-systems for hybrid logic, and completeness results canbe proved of a generality that is simply not available in modal logic. That is, hybridizationis a systematic way of remedying a number of known deficiencies of modal logic.

First-order hybrid logic is obtained by adding first-order machinery to propositionalhybrid logic, or equivalently, by adding hybrid-logical machinery to first-order modal logic.In this short paper we introduce first-order hybrid logic and we give a survey of work inthe area.

1 Introduction

We shall take first-order modal logic as a starting point. First-order hybrid logic is obtainedby adding to first-order modal logic further expressive power in the form of a new sortof propositional symbol called a nominal, and moreover, by adding so-called satisfactionoperators. It is stipulated that a nominal is true at exactly one world, so in this sensea nominal refers to a world. If a is a nominal and ϕ is an arbitrary formula, then anew formula @aϕ called a satisfaction statement can be formed. It is the part @a ofthe satisfaction statement @aϕ which is called a satisfaction operator. The satisfactionstatement @aϕ expresses that the formula ϕ is true at one particular world, namely theworld at which the nominal a is true. Furthermore, the so-called binders ∀ and ↓ mightbe added. The two binders bind nominals to worlds in two different ways: The binder ↓binds a nominal to the actual world whereas the binder ∀ quantifies over worlds (but it isnot to be confused with the first-order quantifier). The ↓ binder is definable in terms of ∀.In what follows we shall concentrate on the ↓ binder.

∗Programming, Logic and Intelligent Systems Research Group, Roskilde University, P.O. Box 260,DK-4000 Roskilde, Denmark, [email protected]

35


The history of what now is known as hybrid logic goes back to the philosopher ArthurPrior’s work in the 1960s, more precisely, it goes back to what he called four grades oftense-logical involvement. They were presented in the book [16], Chapter XI (also ChapterXI in the new edition [12]). See also [15] Chapter V.6 and Appendix B.3–4. The stagesprogress from what can be regarded as pure first-order earlier-later logic to what can beregarded as pure tense logic; the goal being to be able to consider the tense logic of thefourth stage as encompassing the earlier-later logic of the first stage. In other words, thegoal was to be able to translate the first-order logic of the earlier-later relation into tenselogic. With this in mind, Prior introduced so-called instant-propositions:

What I shall call the third grade of tense-logical involvement consists in treatingthe instant-variables a, b, c, etc. as also representing propositions. ([12], p. 124)

In the context of modal logic, Prior called such propositions possible-world-propositions.Of course, this is what we here call nominals. Prior also introduced the binder ∀ and whatwe here call satisfaction operators. See the handbook chapter [14] for a general accountof Prior’s work. See also the the paper [4]. The term “hybrid logic” was coined in PatrickBlackburn and Jerry Seligman’s paper [7] published in 1995. See the handbook chapter[2] for a detailed introduction to hybrid logic. See the book [8] on hybrid logic and itsproof-theory, also covering first-order hybrid logic.

2 Syntax and semantics

We now extend the formal syntax and semantics of first-order modal logic with hybrid-logical machinery. See the books [10] and [13] as well as the handbook chapters [11] and [9]for the basics of first-order modal logic. We hybridize a common varying domain version offirst-order modal logic. Proof procedures are available for the resulting first-order hybridlogic: A tableau system can be found in [5] whereas natural deduction and axiom systemscan be found in [8]. We shall return to the proof-theory of hybrid logic in the next section.

First the syntax. It is assumed that a countably infinite set of nominals is given.The metavariables a, b, c, . . . range over nominals. We represent ordinary propositionalsymbols by 0-place predicate symbols. We also add satisfaction operators and the binder↓ as mentioned above. We furthermore assume that a set of non-rigid constant symbols isgiven, and we follow [5] in overloading the notation for the satisfaction operator by defininga term to be either a first-order variable or an expression of the form @ai where a is anominal and i is a non-rigid constant symbol. Of course, the term @ai denotes the valueof i at the world where a is true. Such terms are called rigidified constants. The formulasof first-order hybrid logic are defined by the grammar

S ::= P (t1, . . . , tn) | t = u | a | S ∧ S | S → S | ⊥ | S | @aS | ∀xS | ↓aS

where P ranges over n-place predicate symbols, t1, . . . , tn as well as t and u range overterms, a ranges over nominals, and x ranges over ordinary first-order variables. The freenominal occurrences in the formula @aϕ is the occurrence of a together with the freenominal occurrences in ϕ. The free nominal occurrences in ↓ aϕ are the free nominal

36


occurrences in ϕ, except for occurrences of a. Substitution of nominals for nominals isdefined accordingly. Standard substitution of terms for first-order variables is modifiedto take into account that terms might contain nominals (that can be bound). Otherpropositional and first-order connectives such as ∨, ¬, and ∃ are defined in the usual way.We let ϕ be an abbreviation for ¬¬ϕ and we define the existence predicate by lettingE(t) be an abbreviation for ∃y(y = t) where y is a variable distinct from any variableoccuring in t.

Now the semantics. We first define models.

Definition 1. A model for first-order hybrid logic is a tuple (W,R,D, δww∈W , Vww∈W )where

1. W is a non-empty set;

2. R is a binary relation on W ;

3. D is a non-empty set;

4. for each w, δw is a subset of D; and

5. for each w, Vw is a function that to each non-rigid designator assigns an element ofD, and moreover, to each n-place predicate symbol assigns a subset of Dn.

The set δw is called the domain of quantification at the world w. A model has increasingdomains if and only if δw ⊆ δv whenever wRv, and similarly, it has decreasing domains ifand only if δw ⊇ δv whenever wRv.

Given a model M = (W,R,D, δww∈W , Vww∈W ), an assignment is a function thatto each nominal assigns an element of W and to each first-order variable assigns an elementof D. Given an assignment g, each term t is assigned an element tM,g of D as follows: If tis of the form @ai, then tM,g = Vg(a)(i), otherwise t is a variable, in which case tM,g = g(t).Given assignments g′ and g, g′ x∼ g means that g′ agrees with g on all nominals and first-order variables, save possibly on the first-order variable x (and analogously if x is replacedby a nominal a). The relation M, g, w |= ϕ is defined by induction, where w is a world, gis an assignment, and ϕ is a formula of first-order hybrid logic.

M, g, w |= P (t1, . . . , tn) iff (tM,g1 , . . . , tM,g

n ) ∈ Vw(P )M, g, w |= t = u iff tM,g = uM,g

M, g, w |= a iff w = g(a)M, g, w |= ϕ ∧ ψ iff M, g, w |= ϕ and M, g, w |= ψM, g, w |= ϕ→ ψ iff M, g, w |= ϕ implies M, g, w |= ψ

M, g, w |= ⊥ iff falsumM, g, w |= ϕ iff for any v ∈W such that wRv, M, g, v |= ϕM, g, w |= @aϕ iff M, g, g(a) |= ϕ

M, g, w |= ∀xϕ iff for any g′ x∼ g where g′(x) ∈ δw, M, g′, w |= ϕ

M, g, w |=↓aϕ iff M, g′, w |= ϕ where g′ a∼ g and g′(a) = w

By convention M, g |= ϕ means M, g, w |= ϕ for every world w and M |= ϕ means M, g |= ϕfor every assignment g. A formula ϕ is valid if and only if M |= ϕ for any model M.

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3 Proof-theory

Many modal-logical proof systems lack important proof-theoretic properties and the re-lationships between proof systems for different modal logics are often unclear. This issuccintly expressed in the quotation below.

Compared with the multitude of not only existing but also interesting axiomat-ically presentable normal modal propositional logics, the number of systems forwhich sequent calculus presentations (of some sort) are known is disappoint-ingly small. In contrast to the axiomatic approach, the standard sequent-styleproof theory for normal modal logics fails to be ‘modular’, and the very mech-anism behind the small range of known possible variations is not very clear.([17], p. 128)

It is notable that that hybrid-logical proof-theory remedies this lack of uniformity in modal-logical proof systems. This applies to propositional as well as first-order hybrid logic.Hybrid-logical tableau systems can be found in the papers [3] and [5]. Natural deductionand Gentzen systems can be found in the book [8]. In what follows we briefly describe thenatural deduction systems of [8].

In the propositional case, what is crucial to formulate proof-rules is that hybrid logicallows us to express that a formula ϕ is true at a world a (this is expressed by the formula@aϕ) and that a world a is R-related to a world c (which is expressed by @ac). Usingthis, we can formulate natural deduction introduction and elimination rules1 for the modaloperator .

[@ac]···@cϕ

(I)@aϕ

@aϕ @ac(E)

@cϕ

The introduction rule is equipped with the side-condition that the nominal c does notoccur in @aϕ or in any undischarged assumptions other than the specified occurrencesof @ac. These rules can simply be read off from the modal operator’s truth-condition inthe Kripke semantics.

ϕ is true at a iff for any c such that aRc, ϕ is true at c

The introduction rule can be read off from the right-to-left direction of the truth-conditionand the elimination rule can be read off from the left-to-right direction. The remainingrules for propositional hybrid logic can be found in [8], Chapter 2.

In the first-order case it is crucial to be able to express that a formula ϕ is true at aworld a, which is expressed as in the propositional case, but it is moreover crucial to beable to express that an individual t exists at a world a (which can be done by the formula

1Natural deduction systems have two different kinds of rules for each connective; there are rules whichintroduce a connective (that is, the connective occurs in the conclusion of the rule, but not in the premises)and there are rules which eliminate a connective (vice versa). Natural deduction rules may dischargeassumptions which is indicated by putting brackets [ . . . ] around the assumptions in question.

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@aE(t)). We can then formulate introduction and elimination rules for the first-orderquantifier ∀.

[@aE(z)]···

@aϕ[z/x](∀I1)∗

@a∀xϕ

@a∀xϕ @aE(t)(∀E1)

@aϕ[t/x]

The introduction rule is equipped with the side-condition that the first-order variable zdoes not occur free in @a∀xϕ or in any undischarged assumptions other than the specifiedoccurrences of @aE(z). The expression ϕ[t/x] denotes the formula ϕ where the term t hasbeen substituted for all free occurrences of the variable x. Again there is a correspondanceto the truth-condition of the connective which is introduced and eliminated, namely thefirst-order quantifier.

∀xϕ is true at a iff for any z existing at a, ϕ[z/x] is true at a

The remaining rules for first-order hybrid logic can be found in [8], Chapter 6.Besides allowing the formulation of the above introduction and elimination rules, hybrid

logic also allows the formulation of rules corresponding to first-order conditions on theaccessibility relations and quantifier domains. Thereby natural deduction systems for awide class of hybrid logics can be obtained in a uniform way simply by adding rulesas appropriate. The rules in question correspond to conditions expressed by so-calledgeometric theories. Here is the technical definition: A two-sorted first-order formula isgeometric if it is built out of atomic formulas of the forms R(a, c), E(a, x), a = c, andx = y using only the connectives ⊥, ∧, ∨, and ∃ (where R(a, c) says that a is R-related to cand E(a, x) says that x exists at a). A geometric theory is a finite set of closed two-sortedfirst-order formulas each having the form ∀a x(ϕ → ψ) where the formulas ϕ and ψ aregeometric, a is a list a1, . . . , al of variables ranging over worlds, x is a list x1, . . . , xh ofvariables ranging over individuals, and ∀a x is an abbreviation for ∀a1 . . . ∀al∀x1 . . . ∀xh.Space does not allow going into details with rules corresponding to geometric theories, butwe shall give one example. The first-order formula

∀a∀c∀x((R(a, c) ∧ E(a, x))→ E(c, x))

corresponds to the following rule.

@ac @aE(t)

@cE(t)

The formula, and hence the derivation rule, says that if the world a is R-related to theworld c, and an individual exists at a, then the individual also exists at c, in other words,quantifier domains are increasing.

It should be mentioned that the natural deduction systems described above satisfy therequirements such systems are expected to satisfy, namely normalization such that normalproofs satisfy a version of the subformula property, see [8].

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4 Concluding remarks

As described above, hybrid logic systematically remedies a number of known deficienciesof modal-logical proof-theory. Another notable feature of hybrid logic is that first-orderhybrid logic offers precisely the features needed to prove interpolation theorems2. Whileinterpolation fails in a number of well-known first-order modal logics, their hybridizedcounterparts have this property, see the paper [1] as well as [6]. The first paper gives amodel-theoretic proof of interpolation whereas the second paper gives an algorithm forcalculating interpolants based on a tableau system.

Acknowledgements

The author wish to acknowledge the financial support received from The Danish NaturalScience Research Council as funding for the project HYLOCORE (Hybrid Logic, Compu-tation, and Reasoning Methods, 2009–2012).

References

[1] C. Areces, P. Blackburn, and M. Marx. Repairing the interpolation theorem in quan-tified modal logic. Annals of Pure and Applied Logic, 124:287–299, 2003.

[2] C. Areces and B. ten Cate. Hybrid logics. In P. Blackburn, J. van Benthem, andF. Wolter, editors, Handbook of Modal Logic, pages 821–868. Elsevier, 2007.

[3] P. Blackburn. Internalizing labelled deduction. Journal of Logic and Computation,10:137–168, 2000.

[4] P. Blackburn. Arthur Prior and hybrid logic. Synthese, 150:329–372, 2006. Specialissue edited by T. Braüner, P. Hasle, and P. Øhrstrøm.

[5] P. Blackburn and M. Marx. Tableaux for quantified hybrid logic. In U. Egly andC. Fermüller, editors, Automated Reasoning with Analytic Tableaux and Related Meth-ods, TABLEAUX 2002, volume 2381 of Lecture Notes in Artificial Intelligence, pages38–52. Springer-Verlag, 2002.

[6] P. Blackburn and M. Marx. Constructive interpolation in hybrid logic. Journal ofSymbolic Logic, 68:463–480, 2003.

[7] P. Blackburn and J. Seligman. Hybrid languages. Journal of Logic, Language andInformation, 4:251–272, 1995.

[8] T. Braüner. Hybrid Logic and Its Proof-Theory, volume 37 of Applied Logic Series.Springer, 2011.

2The interpolation theorem for propositional logic says that for any valid formula ϕ → ψ there existsa formula θ containing only the common propositional symbols of ϕ and ψ such that the formulas ϕ → θand θ → ψ are valid. Interpolation theorems for other logics are formulated in an analogous fashion.

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[9] T. Braüner and S. Ghilardi. First-order modal logic. In P. Blackburn, J. van Benthem,and F. Wolter, editors, Handbook of Modal Logic, pages 549–620. Elsevier, 2007.

[10] M. Fitting and R.L. Mendelsohn. First-Order Modal Logic. Kluwer, 1998.

[11] J.W. Garson. Quantification in modal logic. In D.M. Gabbay and F. Guenthner, edi-tors, Handbook of Philosophical Logic, 2nd Edition, volume 3, pages 267–323. KluwerAcademic Publishers, 2001.

[12] P. Hasle, P. Øhrstrøm, T. Braüner, and J. Copeland, editors. Revised and ExpandedEdition of Arthur N. Prior: Papers on Time and Tense. Oxford University Press,2003.

[13] G.E. Hughes and M.J. Cresswell. A New Introduction to Modal Logic. Routledge,1996.

[14] P. Øhrstrøm and P. Hasle. A.N. Prior’s logic. In D.M. Gabbay and J. Woods, editors,Logic and the Modalities in the Twentieth Century. The Handbook of the History ofLogic, volume 6. Elsevier, 2005.

[15] A. Prior. Past, Present and Future. Clarendon/Oxford University Press, 1967.

[16] A. Prior. Papers on Time and Tense. Clarendon/Oxford University Press, 1968.

[17] H. Wansing. Sequent calculi for normal modal propositional logics. Journal of Logicand Computation, 4:125–142, 1994.

41

Algebraic Axiomatization of Tense IntuitionisticLogic

Ivan Chajda∗

Intuitionistic logic was introduced by L. E. J. Brouwer (see e.g. [3]) and his collaboratorA. Heyting (e.g. [11]). Semantic of intuitionistic logic is algebraically axiomatized by theso-called Brouwerian lattice which is a relatively pseudocomplemented lattice or by theso-called Heyting algebra which is a bounded Brouwerian lattice. For our goals, we needwork with a complete lattice which is Brouwerian and hence also a Heyting algebra thuswe will not make distinction between these two algebras.

It is known that propositional logic (classic or non-classic) does not incorporate thedimension of time. To obtain a tense logic we enrich the given propositional logic by newunary operators which are usually denoted by G, H, F and P (see e.g. [4, 7, 8] and [9]).We usually define F and P via G and H as follows: F (x) = ¬G(¬x) and P (x) = ¬H(¬x),where ¬x denotes negation of the proposition x.

It is worth noticing that the operators G and H can be considered as certain kind ofmodal operators which were already studied for intuitionistic calculus by D. Wijesekera[14] and in a general setting by W. B. Ewald [10].

Consider a pair (T ;≤) where T is a non-empty set and ≤ is a reflexive and transitivebinary relation on T . Let t ∈ T and f(x) be a formula of a given propositional logic L.We say that G(f(t)) is valid if for every s ≥ t the formula f(s) is valid in L. Analogously,H(f(t)) is valid if f(s) is valid in L for each s ≤ t. Hence, P (f(t)) is valid if there existss ≤ t such that f(s) is valid in L and F (f(t)) is valid if there exists s ≥ t such that f(s)is valid in L. Thus the unary operator G constitutes an algebraic counterpart of the tenseoperator “it is always going to be the case that” and H constitutes an algebraic counterpartof the tense operator “it has always been the case that”. In other words, G and H can berecognized as tense “for all” quantifiers and P and F as tense existential quantifiers.

Tense operators were introduced for the classical propositional calculus in [4] as opera-tors on the corresponding Boolean algebra satisfying the axioms

G(1) = 1, H(1) = 1

G(x ∧ y) = G(x) ∧G(y), H(x ∧ y) = H(x) ∧H(y)

x ≤ GP (x), x ≤ HF (x).

However, studying non-classical logics, the list of axioms for tense operators has beenenlarged. For example, for MV-algebras there were inserted axioms concerning the logical

∗Přírodovědecká fakulta Univerzity Palackého v Olomouci, KAG, 17. listopadu 12, 771 46 Olomouc,[email protected]

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connectives implication and disjunction (for MV-algebras, the disjunction ⊕ is distinctfrom the lattice join ∨ in the induced lattice) as follows:

G(x→ y) ≤ G(x)→ G(y), H(x→ y) ≤ H(x)→ H(y)

G(x)⊕G(y) ≤ G(x⊕ y), H(x)⊕H(y) ≤ H(x⊕ y).

The aim of the paper is to get an algebraic axiomatization of tense operators G, H, Pand F in intuitionistic logic. As mentioned above, an algebraic axiomatization of intuition-istic logic is usually done via relatively pseudocomplemented lattices. Let us recall thata lattice L = (L;∨,∧) is relatively pseudocomplemented if for every a, b ∈ L thereexists a relative pseudocomplement a→ b, i.e. the greatest element with the property(the so-called residuation property):

x ≤ a→ b if and only if x ∧ a ≤ b.

Let us mention two well-known results on relatively pseudocomplemented lattices:

• Every relatively pseudocomplemented lattice is distributive.

• A complete lattice is relatively pseudocomplemented if and only if it is infinitelyjoin-distributive, i.e. if it satisfies a ∧

∨bi; i ∈ I =

∨a ∧ bi; i ∈ I.

We are going to introduce tense operators on a complete relatively pseudocomplementedlattice. Since the logical connective implication in intuitionistic logic is algebraically ax-iomatized by means of relative pseudocomplement and disjunction by the lattice operationjoin, we can apply the axiomatization for MV-algebras and modify it as follows.

Definition. Let L = (L;∨,∧,→, 0, 1) be a bounded relatively pseudocomplemented lattice.Denote by x∗ = x → 0 (the so-called pseudocomplement of x). Unary operators G,H onL are called tense operators if the following conditions hold:

(1) G(1) = 1 and H(1) = 1

(2) G(x→ y) ≤ G(x)→ G(y) and H(x→ y) ≤ H(x)→ H(y)

(3) G(x) ∨G(y) ≤ G(x ∨ y) and H(x) ∨H(y) ≤ H(x ∨ y)

(4) G(x ∧ y) = G(x) ∧G(y) and H(x ∧ y) = H(x) ∧H(y)

(5) x ≤ GP (x) and x ≤ HF (x), where P = H(x∗)∗ and F (x) = G(x∗)∗.

Remark. Let L = (L;∨,∧,→) be an arbitrary relatively pseudocomplemented lattice.One can easily check that there exists two sorts of “extremal” tense operators:

(a) G(1) = 1 = H(1) and G(x) = 0 = H(x) for each x = 1

(b) G(x) = x and H(x) = x for each x ∈ L.

43


It is evident that this is an uninteresting case. A bit more interesting case can be given asfollows:

Example 1. Consider a lattice L visualized in Fig. 1

@@@

@@@@@

@@@

@@@@@

@@@

@@@@@

ss

s

s

s

s

s

ss

0

1

zx y

b

d

a

c

Fig. 1

Then L is a relatively pseudocomplemented lattice which can be equipped with a couple oftense operators G,H which are not “extremal” as follows:

x 0 a b c d x y z 1

G(x) 0 a 0 c y x y 0 1

H(x) 0 0 b y d 0 y z 1

Lemma 2. 1 Let G,H be tense operators on a relative pseudocomplemented bounded latticeL = (L;∨,∧,→, 0, 1). Let x∗ = x→ 0, P (x) = H(x∗)∗ and F (x) = G(x∗)∗. Then

(i) x ≤ y implies G(x) ≤ G(y) and H(x) ≤ H(y)

(ii) x ≤ y implies P (x) ≤ P (y) and F (x) ≤ F (y)

(iii) x∗∗ ≥ FH(x∗∗) and x∗∗ ≥ PG(x∗∗).

It is a natural question if the tense operators G,H satisfy also the equality G(0) = 0 =H(0) which is dual to (1). The answer is as follows.

Lemma 3. 2 Let G,H be tense operators on a bounded relative pseudo-complementedlattice L = (L;∨,∧,→, 0, 1). Then G(0) = 0 if and only if G(x∗) ≤ G(x)∗ and H(0) = 0if and only if H(x∗) ≤ H(x)∗.

Remark. The assumption G(0) = 0 = H(0) is very natural. Due to Lemma 2, it yieldsimmediately F (x) = G(x∗)∗ ≥ G(x)∗∗ ≥ G(x) and, analogously, P (x) ≥ H(x). Since Gand H are considered as general quantifiers and F and P as the corresponding existentialquantifiers, these relations are generally accepted in every logic.

44


In what follows, we will derive a procedure for finding tense operators in every completerelatively pseudocomplemented lattices. For this we need two lemmas which are well-known.

Lemma 4. 3 Let L = (L;∨,∧,→) be a complete relatively pseudocomplemented lattice anda, bi ∈ L for i ∈ I. Then

a→∧bi; i ∈ I =

∧a→ bi; i ∈ I.

Lemma 5. 4 Let L = (L;∨,∧,→) be a complete relatively pseudocomplemented lattice andai, bi ∈ L for i ∈ I. Then∧

ai; i ∈ I →∧bi; i ∈ I ≥

∧ai → bi; i ∈ I.

By a frame (see e.g. [9] or [3], [6]) is meant a couple (T,≤) where T is a non-empty setand ≤ is a quasiorder on T , i.e. a reflexive and transitive binary relation on T . A frame(T,≤) can be considered as a time scale, i.e. if r, s ∈ T and r ≤ s then we say that r is“before” s and s is “after” r. Given an element t ∈ T , all s ≤ t form past tense and allr ≥ t form future in the time scale. Hence, having a relatively pseudocomplemented latticeL = (L;∨,∧,→) which is considered as an algebraic counterpart of intuitionistic logic, wecan form a power structure LT . Then the time scaling by means of (T,≤) enables us todefine past or future of L and hence also of the true-values of formulas or propositionsfrom L. Thus we have in hand an algebraic tool evaluating what said in introduction: fora formula f(x) of L and t ∈ T , G(f(t)) is valid if for each s ≥ t the formula f(s) is validin L. In other words, validity of G(f(t)) cannot exceed validity of f(s) for each s ≥ t.Similarly for H(f(t)) we take s ≤ t and formulas f(s) to express the validity in past. Thefollowing theorem gets a general construction of such tense operators G,H which satisfythe needs given above.

Theorem. Let L = (L;∨,∧,→) be a complete relatively pseudocomplemented lattice and(T,≤) be a frame. For p ∈ LT we define coordinatewise

G(p)(x) =∧p(y);x ≤ y and H(p)(x) =

∧p(y); y ≤ x.

Then G,H are tense operators on LT such that G(0) = 0 = H(0).

References

[1] Birkhoff G.: Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, 3-rd ed., Providence,RI, 1973.

[2] Botur M., Chajda I., Halaš R., Kolařík M.: Tense operators on basic algebras, Intern.J. Theor. Phys., to appear.

[3] Brouwer L. E. J.: Intuitionism and formalism, Bull. Amer. Math. Soc., 1913.

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[4] Burges J.: Basic tense logic, in D.M. Gabbay, F. Günther (Eds), Handbook of Philo-sophical Logic, vol II, D. Reidel Publ. Comp., 1984, 89–139.

[5] Chajda I., Halaš R., Kühr J.: Semilattice Structures, Heldermann Verlag, Lemgo(Germany), 2007.

[6] Chajda I., Kolařík M.: Dynamic effect algebras, Math. Slovaca, to appear.

[7] Chiriţă C.: Tense θ-valued Moisil propositional logic, Inter. J. Computers, Comm. andControl 5(5), (2010), 642–653.

[8] Chiriţă C.: Tense θ-valued Łukasiewicz-Moisil algebras, J. Multiple-valued Logic andSoft Comput., to appear in 2011.

[9] Diaconescu D., Georgescu G.: Tense operators on MV-algebras and Łukasiewicz-Moisilalgebras, Fundamenta Informaticae 81, (2007), 1–30.

[10] Ewald W. B.: Tense and Modal Logic, The Journal of Symbolic Logic 51, (1986),166–179.

[11] Heyting A.: Intuitionism. An Introduction, North-Holland, Amsterdam, 1956.

[12] Rasiowa H., Sikonski R.: The Mathematics of Metamathematics, PWN, Warszawa,1963.

[13] Turunen E.: Mathematics Behind Fuzzy Logic, Springer-Verlag, Heidelberg, 1999.

[14] Wijesekera D.: Constructive modal logic I, Annales Pure Appl. Logic 50, (1990),271–301.

46

A Bilattice-based GQT Framework forInterrogatives and Interrogative Inferences

Ka-fat Chow∗

The studies on interrogatives in logic and formal semantics have been a difficult task be-cause there is not an intuitive and uncontroversial notion of truth values for interrogatives.Thus we see different frameworks for interrogatives with different merits and demerits. Inthis paper, I will formulate a theoretical framework that combines Gutierrez-Rexach ’sGQT-based framework (in [4, 5]) and Nelken and Francez’s bilattice-based framework (in[7, 8]) for interrogatives and derive certain valid inferential patterns involving interrogativesbased on this framework.

Gutierrez-Rexach’s framework is based on Generalized Quantifier Theory (GQT) andtreats a WH-word as an interrogative quantifier (IQ) that requires, in addition to theordinary arguments, an “answer argument” to make a complete proposition. For instance,the truth condition of “who” is represented by “who(Y )(X) = 1 iff PERSON ∩ Y = X”,where X is the answer argument. Thus, under this approach the question “Who sang” issemantically equivalent to the noun phrase “person(s) who sang”.

Nelken and Francez’s framework assumes that interrogatives are of the same semantictype as that of propositions. The denotation of declaratives and interrogatives are thusboth truth values. However, to distinguish the two types of sentences, they adopt 5 truthvalues which are arranged in 2 lattices (hence a “bilattice”). For declaratives, there are 3truth values: t (“known to be true”), f (“known to be false”) and uk (“unknown whether trueor false”). For interrogatives, they borrow the concept of “resolvedness” from [1] and assume2 truth values: r (“resolved”) and ur (“unresolved”). The two groups of truth values arerelated by the resolvedness conditions of interrogatives. For illustration, consider the polarquestion “Did Mary kiss John” whose formal representation and resolvedness condition is“∥?(KISS(m, j))∥ = r iff ∥KISS(m, j)∥ ∈ t, f” (where ∥p∥ denotes the truth value ofp), meaning that “Did Mary kiss John” is resolved iff it is known whether Mary kissedJohn.

In this paper, I will develop a formal framework for interrogatives that is based onNelken and Francez’s framework but with substantial modification. The reason for choos-ing Nelken and Francez’s framework as the basis is that their framework is extensional andis thus easier to manipulate than an intensional framework such as [2]. Moreover, sincetheir framework has a clear definition for truth values for both declaratives and interroga-tives, it is straightforward to define inferential relations between interrogatives and is thusconvenient to study the issue of interrogative inferences under this framework.

Nevertheless, Gutierrez-Rexach’s GQT-based framework also has its merits becauseWH-words do share certain characteristics with ordinary quantifiers. Under Gutierrez-

∗The Hong Kong Polytechnic University, [email protected]

47


Rexach’s framework, certain phenomena related to interrogatives can be studied from theperspective of GQT. Moreover, it is found that IQs also possess certain properties that arethoroughly studied in GQT, such as conservativity, monotonicity, intersectivity, etc. Forthis reason, the framework proposed in this paper will also incorporate certain elements ofGutierrez-Rexach’s framework.

Following the traditional GQT approach (such as in [6]), I will formulate the resolved-ness conditions of IQs as set relations. But since there are now 3 truth values for declara-tives, we first have to define new notions of sets as follows (in what follows, U representsthe universe):

Xt = x ∈ U : ∥x ∈ X∥ = t (1)

Xf = x ∈ U : ∥x ∈ X∥ = f (2)

Xuk = x ∈ U : ∥x ∈ X∥ = uk (3)

Thus, with respect to every concept X, we have 3 sets Xt, Xf and Xuk containingelements that are known to belong to X, known not to belong to X and unknown whetherto belong to X, respectively.

We can now write down the resolvedness conditions of IQs using these notions. Forexample, the resolvedness conditions for “who”, “(everybody except who)” and “which” areas follows:

∥who(−)(B)∥ = r ⇔ (PERSON ∩B)uk = ∅1 (4)

∥(everybody except who)(−)(B)∥ = r ⇔ (PERSON −B)uk = ∅ (5)

∥which(A)(B)∥ = r ⇔ (A ∩B)uk = ∅ (6)

Note that the above conditions treat the IQs “who”, etc. as “strongly exhaustive IQs”.Under this interpretation, the question “who(−)(B)” is resolved iff for every element x,it is known whether x is a person belonging to B. In other words, there is no element xsuch that it is not known whether x is a person belonging to B. This is represented by theset relation (PERSON ∩B)uk = ∅. The resolvedness conditions of other IQs can also beformulated as Suk = ∅ for an appropriate set S.

We can also determine the constituent answer (CA) and sentential answer (SA) to astrongly exhaustive IQ as follows: let Q be a strongly exhaustive IQ whose resolvednesscondition has the form Suk = ∅, then on condition that ∥Q(A)(B)∥ = r, the CA toQ(A)(B)is St, and the SA to Q(A)(B) is the proposition “S = Y ”, where Y is the specific value ofSt in a certain model. This proposition can often be re-expressed as a tripartite structureusing the truth conditions of ordinary quantifiers. For instance, if ∥who(−)(B)∥ = r, thenthe CA to “who(−)(B)” is (PERSON ∩ B)t, i.e. those who are known to be personsbelonging to B. Furthermore, suppose the value of (PERSON ∩ B)t in a certain modelis the singleton set x, i.e. x is the only person known to belong to B, then the SA to

1In this paper, I adopt the notation in [6] that represents a quantified statement in the form of a tripartitestructure Q(A)(B) where Q, A and B represent the determiner, subject (excluding the determiner) andpredicate of the sentence, respectively. When Q is a noun phrase (such as “who”) instead of a determiner,the A argument is empty and is represented by “−”. Thus, “Who sang” is represented as “who(−)(SING)”.

48


“who(−)(B)” is the proposition “PERSON ∩B = x”, which can be re-expressed as thefollowing tripartite structure “(nobody except x )(−)(B)”.

Apart from “strongly exhaustive” questions requesting complete information concerninga subject matter, there are also “non-exhaustive” questions which request only partialinformation, as exemplified by the question “Who for example did John see”. In this paper,WH-phrase “who for example” will be expressed as a non-exhaustive IQ “(at least who)”.The resolvedness condition of this IQ can be written as

∥(at least who)(−)(B)∥ = r ⇔ (PERSON ∩B)t = ∅ ∨ (PERSON ∩B)f = U (7)

The condition above reflects the fact that the question “(at least who)(−)(B)” is resolvediff either one of the following situations holds: (1) at least one element is known to belongto PERSON ∩ B; (2) all elements are known not to belong to PERSON ∩ B. Theresolvedness conditions of other non-exhaustive IQs can also be formulated as St = ∅∨Sf =U for an appropriate set S.

The CA to a non-exhaustive IQ is not unique and so I will provide the set of all possibleCAs which can be determined as follows: let Q be a non-exhaustive IQ whose resolvednesscondition has the form St = ∅ ∨ Sf = U , then on condition that ∥Q(A)(B)∥ = r, the CAset of Q(A)(B) is

CA set =X : ∅ = X ⊆ St, if St = ∅∅, if St = ∅

(8)

For instance, if ∥(at least who)(−)(B)∥ = r, then the CA set of “(at least who)(−)(B)” is

CA set =X : ∅ = X ⊆ (PERSON ∩B)t, if (PERSON ∩B)t = ∅∅, if (PERSON ∩B)t = ∅

(9)

The above piecewise-defined function provides the CA set under two situations. If(PERSON ∩B)t = ∅, by (7) we must have (PERSON ∩B)f = U , i.e. no person belongsto B and so the unique CA should be “nobody”, represented by a singleton consisting of ∅.If (PERSON ∩B)t = ∅, then every non-empty subset of (PERSON ∩B)t, i.e. any set Xsatisfying ∅ = X ⊆ (PERSON ∩ B)t, is an acceptable CA. So all these Xs are collectedinto a set, and the CA can be any member of this set.

To study inferences involving interrogatives, we need to define entailment and equiva-lence relations involving interrogatives. Under the present framework, it is straightforwardto define these notions:

Let S = s1, . . . sn be a set of questions / propositions and q a question, thenS entails q (denoted S ⇒ q), iff in every model, (∥s1∥ ∈ t, r ∧ . . . ∧ ∥sn∥ ∈t, r)⇒ ∥q∥ = r. (10)

Let q1 and q2 be questions, then q1 is equivalent to q2 (denoted q1 ⇔ q2), iff inevery model, ∥q1∥ = r ⇔ ∥q2∥ = r. (11)

49


Based on the resolvedness conditions of IQs and the above definitions, we can derivevalid inferential patterns of IQs. For example, it can be shown that the following equiva-lence, entailment and “interrogative syllogism” are all valid:

who(−)(¬B)⇔ (everybody except who)(−)(B) (12)

who(−)(B)⇒ (at least who)(−)(B) (13)

which(M)(P ), which(M)(S), S ⊆M ⇒ which(S)(P ) (14)

Note that (14) above is a generalization of a result in [3]. An instance of this inferenceschema is that the two questions “Whom does Mary love” and “Who are the men” collec-tively entail the question “Which men does Mary love” (on the understanding that menare persons).

Moreover, we can also discuss the monotonicities of IQs, whose definitions are analogousto those of ordinary quantifiers (such as in [9]). It can be shown that the strongly exhaustiveIQs considered in this paper are non-monotonic in all of their arguments, whereas the non-exhaustive IQs are increasing in all of their arguments within certain restricted domains.For example, it can be proved that within the domain B : (PERSON ∩ B)f = U, “(atleast who)(−)(B)” is increasing in the argument B.

Finally, as pointed out by Nelken and Francez, the relation between a question andits SA can also be seen as an entailment relation. In this framework, we can prove thefollowing:

If p is an SA to q, then p⇒ q. (15)

The above result shows that “p ⇒ q” is a necessary condition for “p is an SA to q”.In other words, we can show that “p is not an SA to q” by showing that “p ⇒ q”, thusproviding us with a method to show that a certain proposition is not a resolved answer toa certain question. For instance, we can show that “(At least) John sang” is not a resolvedSA to “Who sang”, according to the strongly exhaustive interpretation of “who”.

References

[1] Ginzburg, J., Resolving Questions I and II. Linguistics and Philosophy, 18, (1995),459–527, 567–609.

[2] Groenendijk, J. and Stokhof, M., Studies on the Semantics of Questions and the Prag-matics of Answers. PhD. thesis, Universiteit van Amsterdam, (1984).

[3] Groenendijk, J. and Stokhof, M., Type-shifting rules and the semantics of interrogatives.Portner, P. and Partee, B.H. (eds.) Formal Semantics: the Essential Readings, Oxford:Blackwell, (1989), 421–456.

[4] Gutierrez-Rexach, J., Questions and Generalized Quantifiers. Szabolcsi, A. (ed.) Waysof Scope Taking, Dordrecht: Kluwer Academic Publishers, (1997), 409–452.

50


[5] Gutierrez-Rexach, J., Interrogatives and Polyadic Quantification. Scott, N. (ed.) Pro-ceedings of the International Conference on Questions, University of Liverpool, (1999),1–14.

[6] Keenan, E.L., Some Properties of Natural Language Quantifiers: Generalized QuantifierTheory. Linguistics and Philosophy, 25, (2002), 627–654.

[7] Nelken, R. and Francez, N., The Algebraic Semantics of Interrogative NPs. Grammars,3, (2000), 259–273.

[8] Nelken, R. and Francez, N., Bilattices and the Semantics of Natural Language Ques-tions. Linguistics and Philosophy, 25, (2002), 37–64.

[9] Peters, S. and Westerståhl, D., Quantifiers in Language and Logic. Oxford: ClarendonPress, (2006).

51

Fuzzy Functions and Their Representationsin Fuzzy Class Theory

Martina Daňková∗

1 Introduction

In this work, we will put our attention to fuzzy relations that preserve special properties:extensionality and functionality. Such relations will be called fuzzy functions. It willbe shown that analogously to the classical case they can be represented using the crispfunctions and similarity relation. Our framework is Fuzzy Class Theory1 [1] that allowsdeveloping non-classical mathematics over a fuzzy logic.

This contribution is an excerpt of a larger manuscript intended for a journal publica-tion therefore the results will be presented without proofs. In the following section, thebackground logic will be briefly described and the definitions of basic notions will be given.Next, we will explain the notion of fuzzy function and finally, the results relating to theirrepresentation will be shown.

2 Logical framework

Let us work in the framework of Fuzzy Class Theory (FCT) [1], which is Henkin-stylefuzzy logic of higher order. More precisely, it is a schematic extension of a backgroundlogic (that contain crisp equality = and Baaz-delta ∆) by the axiom of comprehensionand the extensionality axiom. The background logic may be various due to our actualrequirements. For the simplicity, we work over infinite-valued first order Łukasiewicz logic.Notice that all the following results pass also for much general first order monoidal t-normbased logic [4].

The language J consists of the following set of basic connectives (&,→,∧,∨) andquantifiers ∀, ∃. Moreover, we define

φn ≡df φ& . . .&φ︸︷︷︸n-times

.

Variables for atomic individuals are represented by lower-case characters x, y, z, . . .; vari-ables for 1’st order classes are upper-case letters A,B,C . . . and variables for 2’nd orderclasses are calligraphic letters A,B, C . . .. Symbols for higher orders will not be neededwithin this contribution.

∗Institute for Research and Applications of Fuzzy Modeling, Faculty of Science, University of Ostravamartina.dankova@ osu.cz

1Another framework for a developing non-classical mathematics over the fuzzy logic is Fuzzy TypeTheory introduced in [5].

52


Interpretation of the connectives is given by the operations ∗,→∗,∧L,∨L, whichtogether form a linearly ordered Łukasiewicz algebra denoted by L with the support L.Baaz-delta ∆ is interpreted as

∆x =

1, if x = 1;0, otherwise.

And the crisp equality = is the standard identity of elements.The language of FCT includes the membership predicate ∈. Moreover, it includes the

class terms of the form x | φ (x – a variable or a vector of variables, and φ – a formula),where y ∈ x | φ stands for φ(y).

Convention 1. For the better orientation, we will use the same terminology on the syn-tax as well as on the semantical level. Moreover, we will omit the specification of sorts,whenever it will be clear from the context.

Convention 2. Throughout the contribution we use the adjective “crisp” to emphasizethat interpretation of the object in concern assigns only values 0 and 1. While in thecase of “fuzzy” objects, we can have the whole spectrum of truth values. These notions arecommonly used in fuzzy sets community.

Due to the space limit, we recommend [1] for more details.

2.1 Basic notions

Let F,≈1,≈2 be fuzzy relations of the type (s1, s2), (s1, s1), (s2, s2), respectively. In thecase of ≈1, we write x ≈1 y for ≈1 (x, y) and analogously for ≈2. Moreover, let R,S arefuzzy relations or fuzzy sets of the same type. Then we define the following properties:

Reflexivity: ReflR ≡df (∀x)⟨x, x⟩ ∈ RSymmetry: SymR ≡df (∀x, y)(⟨x, y⟩ ∈ R→ ⟨y, x⟩ ∈ R)Transitivity: TransR ≡df (∀x, y, z)[(⟨x, y⟩ ∈ R&⟨y, z⟩ ∈ R)→ ⟨x, z⟩ ∈ R]Similarity: SimR ≡df ReflR&SymR&TransRSeparatedness: SepR ≡df (∀x, y)((x = y)→ (⟨x, y⟩ ∈ R))Subsethood: R ⊆ S ≡df (∀x, y)(⟨x, y⟩ ∈ R→ ⟨x, y⟩ ∈ S)Set-similarity: R ≈ S ≡df (R ⊆ S)&(S ⊆ R)Totality: TotR ≡df (∀x∃y)⟨x, y⟩ ∈ R

Moreover, the following set operations can be introduced:

A ∪B =df x | (x ∈ A)⊕ (x ∈ B) strong unionA ∩B =df x | (x ∈ A)&(x ∈ B) strong intersec.A ⊔B =df x | (x ∈ A) ∨ (x ∈ B) unionA ⊓B =df x | (x ∈ A) ∧ (x ∈ B) intersection

The union and intersection of a class of classes are the functions defined as follows:∪A =df x | (∃A ∈ A)(x ∈ A)∩A =df x | (∀A ∈ A)(x ∈ A)∪φ τ =df

∪τ | φ for any term τ , and similarly for

∩53


We will additionally deal with relational compositions defined using a class notation.A systematic study can be find in [2]. We will use three basic relational compositions.

• sup-T composition: R S =df ⟨x, y⟩ | (∃z)(Rxz&Szy)

• BK-subproduct: R S =df ⟨x, y⟩ | (∀z)(Rxz → Szy)

• BK-superproduct: R S =df ⟨x, y⟩ | (∀z)(Rxz ← Szy)

3 Fuzzy functions and their properties

Fuzzy functions are fuzzy relations that fulfill the extensionality and functionality definedby the following formulae:

Extensional: Ext≈1,2 F ≡df (∀x, x′, y, y′)[(x ≈1 x′)&(y ≈2 y

′)&F (x, y)→ F (x′, y′)],Functional: Func≈1,2 F ≡df (∀x, x′, y, y′)[(x ≈1 x

′)&F (x, y)&F (x′, y′)→ (y ≈2 y′)]

Function: Function≈1,2 F ≡df Ext≈1,2 F ∧ Func≈1,2 F

Example 3. Let L = ⟨[0, 1],⊗,⇒,∨,∧, 0, 1⟩ be the standard Łukasiewicz algebra,

x ∼ y = (x⇔ y)3 = (x⇔ y)⊗ (x⇔ y)⊗ (x⇔ y).

The following is the example of a relation that is functional to the degree 1 w.r.t. =,=:

F1(x, y) =

0.5, y = sin(x);0, otherwise.

Moreover, we find out that1. F2(x, y) = y ∼ sin(x) is functional to the degree 1 w.r.t. ∼,∼.

2. F3(x, y) = y ∼ f(x), where f(x) =

1.7x2, x ∈ [0, 0.5];cos(0.9x)− 0.5, otherwise. is functional to

the degree 1 w.r.t. =,∼.3. F4(x, y) = F2 ∨ F3 is functional to the degree 0 w.r.t. ∼,∼. Take x = x′ = 1 and

y = 0.1, y′ = 0.8 then F (x, y)⊗ F (x′, y′) = 1, but y ∼ y′ = 0.

3.1 Set operations and relational compositions of functions

Proposition 4. FCT proves

Function≈1,2 F&Function≈2,3 S → Function≈1,3(F S), (1)Function≈1,2 F&Function≈1,2 S → Function≈2

1,2(F ∩ S), (2)

Function≈1,2 F ∧ Function≈1,2 S → Function≈1,2(F ⊓ S), (3)Function≈1,2(F ⊔ S)→ Function≈1,2 F ∧ Function≈1,2 S. (4)

Proposition 5. FCT proves

(F ⊆ E)2 → [Function≈1,2 F → Function≈1,2 E], (5)

(F ≈ E)2 → [Function≈1,2 F ↔ Function≈1,2 E]. (6)

54


(a) Functional relation F2

w.r.t. ∼,∼(b) Functional relation F3

w.r.t. ∼,∼(c) Non-functional relation F4

w.r.t. ∼,∼

The above formulae together with properties of the relational compositions produces along list of consequences:

Corollary 6. Let

C1 ≡df (E1 ⊆ E2)2 C2 ≡df (F1 ⊆ F2)

2

C3 ≡df (E1 ≈ E2)2 C4 ≡df (F1 ≈ F2)

2

Then FCT proves

C1 → [Function≈1,2(F E2)→ Function≈1,2(F E1)]C1 → [Function≈1,2(F E1)→ Function≈1,2(F E2)]C1 → [Function≈1,2(F E2)→ Function≈1,2(F E1)]C2 → [Function≈1,2(F2 E)→ Function≈1,2(F1 E)]C2 → [Function≈1,2(F1 E)→ Function≈1,2(F2 E)]C3 → [Function≈1,2(F E2)↔ Function≈1,2(F E1)]C3 → [Function≈1,2(F E2)↔ Function≈1,2(F E1)]C3 → [Function≈1,2(F E1)↔ Function≈1,2(F E2)]C4 → [Function≈1,2(F2 E)↔ Function≈1,2(F1 E)]C4 → [Function≈1,2(F1 E)↔ Function≈1,2(F2 E)]

Intersection and union:

Function≈1,2

(( ∩F∈A

F) E

)→ Function≈1,2

( ∩F∈A

(F E))

Function≈1,2

(( ∪F∈A

F) E

)↔ Function≈1,2

( ∪F∈A

(F E))

Function≈1,2

( ∪F∈A

(F E))→ Function≈1,2

(( ∩F∈A

F) E

)Function≈1,2

( ∩F∈A

(F E))↔ Function≈1,2

(( ∪F∈A

F) E

)Function≈1,2

( ∪E∈A

(F E))→ Function≈1,2

(F

∪E∈A

E)

Function≈1,2

( ∩E∈A

(F E))↔ Function≈1,2

(F

∩E∈A

E)

55


4 Fuzzy functions and their representation

Below, results related to representation of a fuzzy function by the crisp function can befound in the form of graded theorems. These results stem from the works [3] and [2]. Theso called crisp function is a relation of the form ⟨x, y⟩ | y = f(x). We can exchange =by ≈2, i.e. we assume that each x is mapped to a neighbourhood of f(x), which can berepresented using ≈2 as y ≈2 f(x). It can be shown that this relation is a function w.r.t.=,≈2 provided that ≈2 is reflexive. The question remains whether this remains true alsofor a function w.r.t. ≈1,≈2.

Proposition 7. Let us define

Comp≈1,≈2f ≡df (∀x, y)(x ≈1 y)→ (f(x) ≈2 f(y)),

Ff =df ⟨x, y⟩ | y ≈2 f(x).

Then FCT proves

Sep ≈2 &Tot f → TotFf , (7)

Sym≈2&(Trans≈2)2&Comp≈1,≈2

f → Function≈1,≈2 Ff . (8)

When we have defined the notion of fuzzy function (property Function of a relation Fw.r.t. ≈1,≈2), we assumed the extensionality and functionality together. Extensionalitysays that we can substitute the original inputs (x, y) by the indistinguishable one (x′, y′).The formula representing the functionality is the exact analogy with the classical defini-tion, where we assume that the images of indistinguishable elements are indistinguishable.Relating to this interpretation, we must still keep on mind that ≈1,2 represent the gran-ularity of the input (output) space, which means that they are coarsest relations in oursystem (∀R)(≈1(2)⊆ R) enabling us to distinguish elements of the universe.

Theorem 8. Let f, Ff be as in the previous proposition, moreover, let g be some unaryfunctional symbol. Define

Definition (Ff , g) ≡df (∀x)[⟨x, g(x)⟩ ∈ Ff ↔ (∃y)(⟨x, y⟩ ∈ Ff )]. (9)

Then FCT proves

Sep ≈2 &Tot f&Definition (Ff , g)→ (g ≈ f). (10)

Now, let us address the reverse problem: consider a fuzzy relation F and let us finda crisp function fF such that it is compatible with (≈1,≈2) and its extension to fuzzyrelation FfF is similar to F .

Theorem 9. FCT proves

TotF&Definition (F, fF )&(TotF )&Func≈1,2 F → CompfF , (11)

TotF&Definition (F, fF )&Function≈1,2 F&Refl≈1 → (FfF ≈ F ). (12)

56


An intended class of applications of the introduced theory of fuzzy functions is con-nected with implicative fuzzy rules originally introduced as an approximation of a fuzzyrelation that is functional (to the degree 1). In this case, we obtain an upper approximationof the original relation and moreover, a precision of such approximation can be estimated.But if we have at the disposal only partially functional relation (to the degree 0 < α < 1)then we cannot expect that implicative rules will provide an upper approximation. Thegraded approach gives at least an estimation of a residua between the implicative rules andthe original relation.

5 Conclusions

In this contribution, we have generalized the well known representation theorem for fuzzyfunction into the graded form. The main advantage of our approach is that it incorporatesthe whole scale for degrees of truth, while in the original approach the results were appliedto properties valid in the degree 1. Moreover, evaluation of the degrees of the antecedentsin our graded theorems provides an additional information about a “precision” of the con-sequent. E.g., if ≈ is similarity relation then an evaluation of D in (10) estimates closenessof fFf

≈ f , or in other words, a distance (dual to ≈) between fFfand f .

Hence, we have shown that graded theorems bring a new light into the already wellestablished theory of fuzzy functions. And additionally, the logical framework providesa unified approach to mathematics of fuzzy logic that corresponds with the classical one(also in the notational standards).

Acknowledgements

We gratefully acknowledge support of the grant MSM6198898701 of the MŠMT ČR.

References

[1] Běhounek, L. and Cintula, P. (2005). Fuzzy class theory. Fuzzy Sets and Systems,154(1):34–55.

[2] Bělohlávek, R. (2002). Fuzzy Relational Systems: Foundations and Principles, vol-ume 20 of IFSR International Series on Systems Science and Engineering. KluwerAcademic/Plenum Press, New York.

[3] Demirci, M. (1999). Fuzzy functions and their fundamental properties. Fuzzy Setsand Systems, 106:239–246.

[4] Esteva, F. and Godo, L. (2001). Monoidal t-norm based logic: Towards a logic forleft-continuous t-norms. Fuzzy Sets and Systems, 124(3):271–288.

[5] Novák, V. (2004). On fuzzy type theory. Fuzzy Sets and Systems, 149(2):235–273.

57

A Game Based Approach toQuantification in Łukasiewicz Logic

Christian G. Fermüller∗

1 Introduction and motivation

Already in the 1970s Robin Giles [8, 9] developed a semantic concept that combines dialoguerules for the systematic reduction of arguments involving complex statements to argumentsabout sub-statements with a scheme for betting on the results of dispersive experimentsassociated with atomic statements. The resulting game is sound and complete for (infinite-valued) Łukasiewicz logic Ł. In more recent years, Giles’s game has been generalized indifferent ways to cover a range of important many-valued logics, in particular the twoother fundamental t-norm logics, Gödel- and Product logic (see [10]), Cancellative Hooplogic CHL [2], and Meyer and Slaney’s Abelian Logic A [12], see [1, 3, 5, 4].

All of the just mentioned papers restrict their attention to propositional logics. Thisshould be contrasted with the fact that Giles, in the final sections of [8] and [9], offers a gamebased interpretation of quantifiers. His suggestions and remarks certainly deserve moreattention, although they are impaired by the lack of knowledge about later developmentsin mathematical fuzzy logics [10]. Giles seemingly also did not know about Scarpellini’s [14]proof (from 1962) that the set of valid formulas in first-order Łukasiewicz logic Ł∀—herealways understood in its standard interpretation over [0, 1])—is not recursively enumerable.With hindsight it is clear that Giles aimed at a constructive analysis that leads to a variantof Ł∀, based on approximative evaluation in countable witnessed models (see [11] for thelatter notion).

The main purpose of this work is to provide a thorough analysis of Giles’s game andsome of its variants at the first-order level. This will give rise to various alternative se-mantics. Most importantly it will also provide a basis for studying fuzzy quantifiers, likemany, few, and some.

2 Giles’s Game for propositional logic

To base our discussion on firm ground we briefly have to review Giles’s original game aswell as some of its more recent variations.

Following Giles, we will refer to the players as me and you, respectively. For thebetting phase of the game, each atomic proposition p is associated with an experiment(test) Ep, which has a fixed probability π(Ep) of yielding a positive result. Giles identifies

∗Theory and Logic Group, Technische Universität Wien, [email protected]

58


this probability with a player’s expectation that a trial of Ep will end positively and cashesout this interpretation by the following betting scheme: I promise to pay to you a fixedamount of money, say 1¤, for each of my assertions of p, where a corresponding trial ofEp yields a negative result. Likewise, you have to pay 1¤ to me for each of your assertionsthat does not pass the associated test. Note that it matters whether we assert the sameproposition just once or more often. A final game state at which [p1, . . . , pn] is the multisetof atomic assertions made by you and [q1, . . . , qm] is the multiset of atomic assertions madeby me is denoted by

[p1, . . . , pn | q1, . . . , qm] .

Let us define the risk value of p by ⟨p⟩r = 1 − π(Ep). We can then specify the expectedtotal amount of money (in ¤) that I have to pay to you at the exhibited state by

⟨p1, . . . , pn | q1, . . . , qm⟩r =∑

1≤i≤m⟨qi⟩r −

∑1≤j≤n

⟨pj⟩r .

Note that this value, (simply called my risk) can be negative, i.e., the risk values of therelevant propositions may be such that I expect a net payment by you to me.

To evaluate logically complex assertions, the following dialogue rules are used:

• Asserting A ∨B obliges one to assert either A or B at one’s own choice.

• Asserting A ∧B obliges one to assert either A or B at the opponent’s choice.

• Asserting A ⊃ B obliges one to assert B if the opponent asserts A.

• Asserting ¬A obliges one to pay 1¤ if the opponent asserts A.

The last rule can be eliminated by defining ¬A = A ⊃ ⊥, where ⊥ refers to an experimentthat always fails and thus results in certain payment of 1¤. Giles’s main result can beformulated as follows. (For details we refer to [9] and to [5].)

Theorem 1. For every assignment of risk values to propositional variables I have a strategyfor avoiding positive expected loss of money in the game starting with my assertion of aformula F if and only if F is valid in Łukasiewicz logic Ł.

In the context of investigating certain generalized quantifiers (see below) we will alsorefer to a dialogue rule for so-called strong conjunction, modeled by the Łukasiewicz t-normλx, ymax(0, x+ y − 1), that has been introduced in [6]:

• Asserting A&B obliges one to assert A and B or to assert ⊥ at one’s own choice.

The latter option in replies to attacks on A&B, namely to assert ⊥ instead of A and B,amounts to a version of the principle of limited liability that is discussed in [5].

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3 Giles’s dialogue rules for quantifiers

In lifting his game from the propositional to the first-order level Giles assumes that allelements of the domain of a model in which we evaluate a given statement can be namedby a variable-free term of the language. Since languages are explicitly required to be finite,the set of terms and hence also of domain elements is assumed to be countable.

Giles’s dialogue rules for (standard) universal and existential are as follows:

• Asserting ∀xP (x) obliges one to assert P (t) for any term t chosen by the opponent.

• Asserting ∃xP (x) obliges one to assert P (t) for some term t chosen by oneself.

Terms are always understood to be variable-free, here. Every formula occurring in a gamestate is closed.

Note that, like in a Hintikka-style evaluation game for classical logic, each assertionby the opponent can be attacked at most once. Therefore, just like in the propositionalcase, every game starting with my assertion of an arbitrary formula is finite, ending in astate where a multiset of my atomic assertions have to evaluated against a multiset of youratomic assertions. Giles claims that Theorem 1 can only be established in the following“slightly weakened form” [9]:

Theorem 2. For every assignment ⟨·⟩r of risk values to propositional variables and for ev-ery ϵ the following holds for the game starting with my assertion of a first-order formula F ,where vM (F ) = x in the Ł∀-interpretation M that translates every risk value ⟨p⟩r into thetruth value vM (p) = 1− ⟨p⟩r:(1) There is a strategy for me that will ensure that my expected loss does not exceed 1−x+ϵ.(2) There is a strategy for you that will ensure that my expected loss exceeds 1− x− ϵ.

Constructivist considerations—Giles explicitly refers to Errett Bishop—lead him tothe interesting claim that “with any realistic interpretation Theorem [2] fails” because“we have no constructive means for determining” the supremum of risk values over allelements of the domain and therefore no way to construct corresponding strategies. Withmuch hindsight—remember that Giles wrote in the 1970s—one can assert (in more recentterminology, see [11]) that Giles restricted his attention to computable strategies withrespect to recursive as well as witnessed models for standard (i.e., [0,1]-valued) Łukasiewiczlogic. In the full version of this paper we will provide a proof of Theorem 2 (missing in [8]and [9]) and derive some consequences for variants of Ł∀.

4 Fuzzy quantifiers

We argue that Giles’s characterization of reasoning in first-order Łukasiewicz logic opensthe door to an alternative semantic approach to fuzzy quantifiers, like many, few, andsome, that ties in nicely with recent considerations on this topic by linguists, as we willnow very briefly indicate.

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Let manyx[A(x), B(x)] be the formal counterpart of sentences of the form “Many Asare Bs.” A standard linguist account [13] models the proportional reading of this two-placequantifier by

v(manyx[A(x), B(x)]) = 1 iff |A ∩B| > c · |A|, (1)

where |A| and |B| is the number of domain objects1 satisfying the respective predicate. Theconstant c is supplied by the context of the modeled utterance. The context dependencymight appear unfortunate to mathematical logicians. But it is certainly an adequate, ifnot outright necessary feature in a linguistic context. However, as convincingly arguedby Fernando and Kamp in [7], an adequate model of many should additionally involvesubjective probabilities in evaluating sentences like “Many politicians are criminals.” Fuzzylogicians, on the other hand, will be tempted to complain that virtually all refinements andalternatives to (1) that have been discussed by linguists amount to ‘crisp’ (i.e., bivalent)truth conditions and should be generalized to allow for degrees of truth.

The framework of Giles’s game enables us to come up with straightforward dialoguerules that combines all three above mentioned desiderata for linguistic models of vaguequantifiers: (1) context dependence, (2) involvement of subjective probability, as well as(3) graded truth conditions. The key ingredient is the reference to a mechanism thatpicks witness terms randomly, instead of letting the players choose them, as in the aboverules for ∀ and ∃. (Note that there is some degree of freedom here as to the underlyingprobability distribution. We will argue that this can be exploited in the intended contextof application. In particular, we will indicate a nice connection to the so-called universaldistribution that arisis in algorithmic information theory.) Context dependent thresholdsare modeled by introducing unary connectives Jc with the following semantics:

v(Jc) =

1 if v(A) ≥ c

1− c+ v(A) else.

We assume that Ł is enriched by propositional constants c such that v(c) = c for c ∈[0, 1]∪Q. From the point of view of Giles’s betting scheme this means that c represents astatement that is tested by an dispersive experiment Ec for which the success probabilityis known to be the rational number c. (Note that 0 = ⊥.) This entails that Jc can bedefined by Jc(A) = A ⊃ c.

Given these preparations, we may formulate the following dialogue rule:

• Asserting manyx[A(x), B(x)] obliges one to assert JcB(t) if the opponent assertsA(t) for a randomly chosen term t.

Asserting JcA amounts to investing c¤ in a bet about A. It is up to the opponent toaccept the bet or to call it off. The following rule is easily seen to be adequate:

• Asserting JcA obliges one to assert c (i.e., to pay c¤) if the opponent asserts A.

We will also consider alternative rules for universal quantification as well as for manythat involve strong conjunction. Moreover several variants of dialogue rules for few, some,and most (read as vague quantifier) will be discussed.

1Note that linguists always assume relevant domain sizes to be finite. If desired, the definition can begeneralized in well-known ways.

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5 Conclusion and outlook

The suggested game based approach to (standard as well as fuzzy) quantifiers in Łukasiewiczlogic triggers a lot of questions for further investigation: How does the suggested gamebased semantics relate to traditional truth functional semantics for fuzzy quantifiers? Whatcan be said about the complexity of the satisfiability and validity problems for correspond-ing extensions of Ł? Can one at least approximate these logics by proof systems? Whatother types of quantifiers can be modeled easily in Giles-style games? Can one switch theunderlying logic from Ł to other important fuzzy logics, like Gödel or Product logic? Isthere a connection to supervaluationist accounts of vague quantifiers?

In the full version of the paper we will comment in more detail on some of thosequestions, leaving others to future work.

References

[1] A. Ciabattoni, C. G. Fermüller, and G. Metcalfe. Uniform rules and dialogue gamesfor fuzzy logics. In Logic for Programming, Artificial Intelligence, and Reasoning,volume 3452, pages 496–510. Springer, 2005.

[2] F. Esteva, L. Godo, P. Hájek, and F. Montagna. Hoops and fuzzy logic. Journal ofLogic and Computation, 13(4):532–555, 2003.

[3] C. G. Fermüller. Revisiting Giles’s Game. In O. Majer, A.-V. Pietarinen, and T. Tu-lenheimo, editors, Games: Unifying Logic, Language, and Philosophy, Logic, Episte-mology, and the Unity of Science, pages 209–227. Springer, 2009.

[4] C. G. Fermüller. On Giles style dialogue games and hypersequent systems. In H. Hosniand F. Montagna, editors, Probability, Uncertainty and Rationality, pages 169–195.Springer, Frankfurt, 2010.

[5] C. G. Fermüller and G. Metcalfe. Giles’s game and the proof theory of Łukasiewiczlogic. Studia Logica, 92(1):27–61, 2009.

[6] C.G. Fermüller and R. Kosik. Combining supervaluation and degree based reasoningunder vagueness. In Logic for Programming, Artificial Intelligence, and Reasoning,pages 212–226. Springer, 2006.

[7] T. Fernando and H. Kamp. Expecting many. In Proceedings of SALT, volume 6, pages53–68, 2011.

[8] R. Giles. A non-classical logic for physics. Studia Logica, 33(4):397–415, 1974.

[9] R. Giles. A non-classical logic for physics. In R. Wojcicki and G. Malinkowski, editors,Selected Papers on Łukasiewicz Sentential Calculi, pages 13–51. Polish Academy ofSciences, 1977.

[10] P. Hájek. Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, 2001.

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[11] P. Hájek. On witnessed models in fuzzy logic. Mathematical Logic Quarterly, 53(1):66–77, 2007.

[12] R. K. Meyer and J. K. Slaney. Abelian logic from A to Z. In G. Priest et al., editor,Paraconsistent Logic: Essays on the Inconsistent, pages 245–288. Philosophia Verlag,1989.

[13] B. Partee. Many quantifiers. In Proceedings of ESCOL, volume 5, pages 383–402,1988.

[14] B. Scarpellini. Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalkülsvon Łukasiewicz. Journal of Symbolic Logic, 27(2):159–170, 1962.

63

Multivalued Dependence Logic and IndependenceLogic

Pietro Galliani∗

1 Introduction

One recent direction of research in the field of dependence logic1 consists in the study ofnon-functional dependencies and their properties in the framework of team semantics [4].

In [3], for example, the language of independence logic was defined as the one obtainedby substituting the dependence atoms of standard dependence logic with independenceatoms of the form t2 ⊥t1 t3, where t1 . . . t3 are tuples of terms, not necessarily of the samelength, and where the satisfaction conditions are given by

TS-indep: M |=X t2 ⊥t1 t3 if and only if for all s, s′ ∈ X with t1⟨s⟩ = t1⟨s⟩ there existsa s′′ ∈ X such that t1⟨s′′⟩t2⟨s′′⟩ = t1⟨s⟩t2⟨s⟩ and t1⟨s′′⟩t3⟨s′′⟩ = t1⟨s′⟩t3⟨s′⟩.

In brief, according to this rule a team X satisfies t2 ⊥t1 t3 if and only if the possible valuesof t2 and t3 in X are independent whenever the value of t1 is fixed, or, in other words, ifand only if, given the team X, learning the values of t1 and t2 would allow us to infer onlythe information about t3 that could be inferred from the value of t1 alone.

As was shown in [3], the resulting formalism is expressively equivalent to dependencelogic with respect to sentences2, and it is strictly more expressive than it with respect toopen formulas3

In [1], the properties of generalized quantifiers in the framework of dependence logicwere studied. As a part of this, Engström

1. Pointed out that the rule given for independence atoms by [3] corresponds precisely tothe definition of the embedded multivalued dependencies studied in Database Theory;

2. Defined the multivalued dependence atoms x y, where x and y are tuples ofvariables (not necessarily of the same lengths) and the satisfaction conditions aregiven by

∗ILLC, University of Amsterdam, The Netherlands, pgalliangmail.com1For a comprehensive introduction to this logical formalism, see [6]; furthermore, see the appendix of

this work for a very brief definition of its syntax and semantics.2That is, with respect to definability of classes of models.3That is, with respect to definability of classes of teams.

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TS-multidep : M |=X x y if and only if, for z listing all variables in the domainof X but not in xy and for all s, s′ ∈ X with s(x) = s′(x), there exists a s′′ ∈ Xwith s′′(xy) = s(xy) and s′′(xz) = s′(xz);

3. Proved that multivalued dependence atoms can be used, in this context, to decom-pose branching generalized quantifiers,4 whereas the standard dependence atoms areinadequate to this purpose.

In [2] the expressive power of independence logic with respect to open formulas was charac-terized exactly, thus answering an open problem mentioned in [3]. In brief, this was provedby defining inclusion/exclusion logic, that is, the logic obtained by adding to the languageof first order logic inclusion atoms t1 ⊆ t2 and exclusion atoms t1 | t2, with satisfactionconditions

TS-inc: M |=X t1 ⊆ t2 if and only if for every s ∈ X there exists a s′ ∈ X such thatt1⟨s⟩ = t2⟨s′⟩;

TS-exc: M |=X t1 | t2 if and only if for all s, s′ ∈ X, t1⟨s⟩ = t2⟨s′⟩;

respectively5 and proving that

1. Any independence logic formula is equivalent to some inclusion/exclusion logic for-mula, and vice versa;

2. A class of teams over a fixed domain is definable by a inclusion/exclusion logic formulaif and only if it is expressible in existential second order logic.

However, the corresponding question for multivalued dependence logic was left open, aswas the exact relation between independence logic and multivalued dependence logic. Thiswork will answer both questions at once, by proving that multivalued dependence logic isequivalent to independence logic (and, therefore, to inclusion/exclusion logic too) and, asa consequence, that a class of teams is definable in multivalued dependence logic if andonly if it is expressible in Σ1

1.6

2 Multivalued Dependence Logic is Independence Logic

In this section, we will prove that any multivalued dependence logic formula is equivalent tosome independence logic formula, and vice versa, as long as the domain of the team is fixed.

One direction is easy to show: indeed, the truth condition for the multivalued de-pendence logic is expressible in Σ1

1, and hence any class of teams (w.r.t. a fixed domain)

4That is, to represent a branching quantifier sentence of the form(Q1xQ2y

)ϕ as (Q1x)(Q2y)(∅ y∧ϕ),

and so on, where Q1 and Q2 are generalized quantifiers.5These conditions correspond, in a very exact sense, to the definitions of inclusion dependencies and

exclusion dependencies studied in Database Theory: see [2] for the details.6This is the case as long as the domain is presumed to the same for all teams of the class.

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which is definable through one multivalued dependence logic formulas is also definablethrough some independence logic formula. We can even give an explicit translation: ifz = Dom(X)\x, y then it is not difficult to see that

M |=X x y if and only if M |=X y ⊥x z.

The other direction is slightly more delicate, and in order to prove it we will first needa definition and a couple of lemmas:

Definition 1 (Normal Independence Atoms). An independence logic atom t2 ⊥t1 t3 is saidto be normal if and only if

1. t1, t2 and t3 are tuples of variables, and not just tuples of terms;

2. t1, t2 and t3 are pairwise disjoint.

Lemma 2. Any independence atom is expressible in terms of normal independence atoms.

Proof. Let t2 ⊥t1 t3 be any independence atom, and let x1, x2 and x3 be three tuples ofnew variables, of the same lengths of t1, t2 and t3 respectively. Then

t2 ⊥t1 t3 ≡ ∃x1x2x3(x1 = t1 ∧ x2 = t2 ∧ x3 = t3 ∧ x2 ⊥x1 x3).

Indeed, suppose that M |=X t2 ⊥t1 t3: then, choose the functions Fi so that

Fi(s) = ti⟨s⟩

and let Y = X[F1F2F3/x1x2x3]. Then M |=Y x1 = t1 ∧ x2 = t2 ∧ x3 = t3, trivially, andfurthermore M |=Y x2 ⊥x1 x3, since M |=X t2 ⊥t1 t3.

Conversely, suppose that M |=X[F1F2F3/x1x2x3] (x1 = t1∧ x2 = t2∧ x3 = t3∧ x2 ⊥x1 x3).Then, again for Y = X[F1F2F3/x1x2x3] and all i = 1 . . . 3, it must hold that Y (xi) =ti⟨s⟩. But then, since M |=Y x2 ⊥x1 x3, we have that M |=Y t2 ⊥t1 t3 too. But allvariables occurring in t1t2t3 are already in Dom(X), and therefore

M |=X t2 ⊥t1 t3

Lemma 3. Let y ⊥x z be a normal independence atom, let X be a team whose domainincludes x, y and z, and let w = Dom(X)\x, y, z. Then

M |=X y ⊥x z ⇔M |=X ∀w(x y).

Proof. Suppose that M |=X y ⊥x z: then, by definition, for all s, s′ ∈ X with s(x) = s′(x)there exists a s′′ ∈ X with s′′(xy) = s(xy) and s′′(xz) = s′(xz).

Now consider any two assignments h, h′ ∈ X[M/w] with h(x) = h′(x): by definition,there exist s, s′ ∈ X and m1, m2 ∈ Dom(M)|w| such that h = s[m1/w] and h′ = s′[m2/w].

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But s(x) = s′(x), so by hypothesis there exists a s′′ with s′′(xy) = s(xy) and s′′(xz) =s′(xz). Then consider h′′ = s′′[m2/w]: we have that h′′ ∈ X[M/w], since s′′ ∈ X, andfurthermore

h′′(xy) = s′′(xy) = s(xy) = h(xy);

h′′(xzw) = s′′(xz)m2 = s′(xz)h′(w) = h′(xzw).

Therefore M |=X[M/w] x y and M |=X ∀w(x y), as required.

Conversely, suppose that M |=X[M/w] x y, and let s, s′ ∈ X be such that s(x) =

s′(x). Then take any tuple m ∈ Dom(M)|w|, and consider

h = s[m/w];

h′ = s′[m/w].

Now, Dom(X)\xy is precisely zw: therefore, by the definition of the multivalued depen-dence atom there exists a h′′ ∈ X[M/w] with h′′(xy) = h(xy) and h′′(xzw) = h′(xzw).Since h′′ ∈ X[M/w], we must have that h′′ = s′′[m/w] for some s′′ ∈ X; and for this s′′,we have that

s′′(xy) = h′′(xy) = h(xy) = s(xy)

and thats′′(xz) = h′′(xz) = h′(xz) = s′(xz).

Theorem 4. Multivalued dependence logic is precisely as expressive as independence logic,over sentences and over open formulas considered in teams with finite domain.

Proof. Obvious from the previous results.

Corollary 5. A class of teams over a fixed domain is definable in multivalued dependencelogic if and only if it is expressible in Σ1

1.

3 Conclusions

Multivalued dependence logic, as we just proved, is equivalent7 to independence logic (and,therefore, to inclusion/exclusion logic too). As we saw, the difficulty of the proof is notgreat; however, in the opinion of the author at least, this result carries some nontrivial sig-nificance. Independence logic, inclusion/exclusion logic and multivalued dependence logic,indeed, are at first sight rather different variants of dependence logic, and were developedfor distinct purposes8. However, all of these logics are now known to be equivalent, and

7Even with respect to open formulas!8Namely, the analysis of the notion of informational independence, the study of the interactions between

non-functional database dependencies and team semantics and the decomposition of branching generalizedquantifiers respectively

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their expressive power is Σ11 both with respect to sentences and with respect to open for-

mulas.

In other words, these three logics (and, quite possibly, a number of hitherto unknownones defined in similar ways) characterize a natural cut-off point in the study of extensionsof dependence logic; and, from the point of view of descriptive complexity theory, thissuggests that the study of these formalisms and of their relations with other logics ofimperfect information may be of some use for the study of this important complexity class.

Acknowledgments

I wish to thank Fredrik Engström for his interesting presentation during the Oxford LINTworkshop, which was the inspiration for this work. Furthermore, I thank the EUROCORESLogICCC LINT project for having organized that meeting, as well as for having supportedthis research.

Appendices

A Team Semantics of Dependence Logic

In this appendix, we will give a very short introduction to the language and the teamsemantics of dependence logic. We will not discuss here its properties, its motivations, orits relationship with the corresponding game theoretic semantics ; for a far more thoroughintroduction to this family of logics, we refer to [6] (and to [5] for one to the closely relatedformalism of independence-friendly logic).

Definition 6 (Team). Let M be a first order model, and let V be a finite set of variables.Then a team X over M with domain Dom(X) = V is a set of first order assignments overM with domain V .

Definition 7 (Team Semantics for First Order Logic). Let M be a first order model, letX be a team over it, and let ϕ be a first order formula, which we assume for the sake ofsimplicity to be in negation normal form, with free variables in Dom(X). Then we say thatX satisfies ϕ in M , and we write M |=X ϕ, if and only if one of the following cases holds:

1. ϕ is a first order literal and M |=s ϕ in the usual Tarski semantics for all s ∈ X;

2. ϕ is of the form ψ1∨ψ2 for some ψ1 and ψ2 and there exist two teams Y and Z suchthat X = Y ∪ Z, M |=Y ψ1 and M |=Z ψ2;

3. ϕ is of the form ψ1 ∧ ψ2 for some ψ1 and ψ2, M |=X ψ1 and M |=X ψ2;

4. ϕ is of the form ∃xψ for some variable x and some formula ψ and there exists afunction F : X → Dom(M) such that M |=X[F/x] ψ, where

X[F/x] = s[F (s)/x] : s ∈ X;

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5. ϕ is of the form ∀xψ for some variable x and some formula ψ and M |=X[M/x] ψ,where

X[M/x] = s[m/x] : s ∈ X,m ∈ Dom(M).

The language of dependence logic is obtained by adding dependence atoms of the form=(t1 . . . tn) to the language of first order logic, where n ranges over N and t1 . . . tn areterms. The corresponding semantic rule is

TS-dep: M |=X=(t1 . . . tn) if and only if, for all s, s′ ∈ X such that ti⟨s⟩ = ti⟨s′⟩ for i =1 . . . n− 1, tn⟨s⟩ = tn⟨s′⟩

which is easily seen to correspond to the notion of functional dependence employed bydatabase theorists.

Negated dependence atoms, if allowed, are satisfied by a team X if and only if X = ∅.

References

[1] Fredrik Engström, Generalized Quantifiers in Dependence Logic, arXiv:1103.0396. 2010.

[2] Pietro Galliani, Inclusion and exclusion dependencies in team semantics - On somelogics of imperfect information, Annals of Pure and Applied Logic, 163(1):68–84. 2012.Online first.

[3] Erich Grädel and Jouko Väänänen, Dependence and Independence, Studia Logica (toappear). 2010.

[4] Wilfrid Hodges, Compositional Semantics for a Language of Imperfect Information,Journal of the Interest Group in Pure and Applied Logics, 5(4):539–563. 1997.

[5] Allen L. Mann, Gabriel Sandu, and Merlijn Sevenster, Independence-Friendly Logic: AGame-Theoretic Approach, Cambridge University Press, 2011.

[6] Jouko Väänänen, Dependence Logic, Cambridge University Press, 2007.

69

On Very True Operators on PocrimsRadomír Halaš Michal Botur∗

1 Introduction

Currently, residuated lattices play a crucial role in a fuzzy logic. However, the commonlogical calculi, such as Hájek’s BL-logic [10], are often too weak from the point of viewof applications. In addition to so-called schematic extensions of BL-logic allowing us toextend BL-logic by additional axiom schemes, Hájek [9] proposed the extension of BL-logic which results by adding a new unary logical connective “very true”. This extensionturned out to be one ot the most important ones.

Inspired by the considerations of Zadeh [23], Hájek formalized the fuzzy truth value“very true”. He enriched the language by adding a new unary operation ∗, so called vt-operator, and introduced the propositional logic BL∗. The completenes of BL∗ was provedin [9] by using so-called BL∗-algebras, an algebraic counterpart of BL∗.

Similar considerations were done by J. Rachůnek and D. Šalounová [18] for a moregeneral class of BL-algebras, the so-called bounded commutative residuated lattice orderedmonoids (Rℓ-monoids). Remark that Rℓ-monoids can be taken as an algebraic semanticsof a more general logic then Hájek fuzzy logic. Rℓ-monoids which are not BL-algebrascan be obtanined from BL-algebras by means of their ordinal sums. Note that the pastingof linearly ordered Wajsberg algebras is a linearly ordered BL-algebra, but the pasting ofWajsberg algebras which are not linear, gives bounded commutative Rℓ-monoids which arenot BL-algebras.

Roughly speaking, Rℓ-monoids are just divisible commutative residuated lattices [22],[19], which are categorically isomorphic to a divisible BCK(P) lattices [13], or divisible inte-gral residuated commutative ℓ-monoids [12] or bounded commutative integral generalizedBL-algebras [15].

In our talk we study pocrims with a unary subdiagonal and monotone self-mapping(called vt-operator) which generalize both BLvt-algebras and Rℓvt-algebras. Let us stressthat in general we do not assume neither prelinearity (as for BL-algebras) nor divisibility(as for Rℓ-monoids) as well as lattice ordering of the respective structures. We derivefrom vt-operators, in a purely algebraic and uniform way, superdiagonal and monotoneself-mappings which are in the case of MV-algebras vf -operators, formalizing fuzzy truthvalue “very false”. Remark, nevertheless, that these superdiagonal operators are the dualsto vt-operators only in the case of MV-algebras.

∗Department of Algebra and Geometry, Palacký University Olomouc, Tomkova 40, 779 00 Olomouc,Czech Republic, e-mail: [email protected], [email protected]

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2 Pocrims

Bounded pocrims form a large class of algebras containing as proper subclasses, amongothers, the classes of algebras of some logics, e.g. the class of BL-algebras, i.e. algebrasof the basic fuzzy logic (and consequently the class of MV-algebras, i.e. algebras of theŁukasiewicz infinite valued logic), as well as the class of Heyting algebras, i.e. algebras ofintuitionistic logic.

By a pocrim is meant a partially ordered commutative integral residuated monoid,i.e. a structure A = (A,⊙,→, 1) of type (2, 2, 0) where

(i) (A,⊙,≤, 1) is a partially ordered commutative monoid with a greatest element 1where

x ≤ y iff x→ y = 1 (1)

(ii) ⊙ and → are residuated, i.e. the following adjointness condition holds on A:

z ≤ x→ y iff z ⊙ x ≤ y.

If (A,≤) has a least element 0, a pocrim is called bounded. If a pocrim is a semilatticeor a lattice, we speak of a residuated semilattice or a residuated lattice. Given abounded pocrim A = (A,⊙,→, 1), one can define on A the unary operation ¬ (negation)by

¬x := x→ 0.

It is worth noticing that pocrims are closely related to BCK-algebras introduced by Iséki[14] as an algebraic semantics of BCK-implicational calculus. Namely, pocrims are justBCK-algebras satisfying the condition (P), i.e. BCK-algebras expanded by a binary opa-ration ⊙ which satisfies the identity

(x⊙ y)→ z = x→ (y → z).

On the other hand there are BCK-algebras which do not admit such a multiplication.It is also well known that BCK-algebras are just →, 1-subreducts of residuated lattices

(in the above sense), see e.g. [16].Also notice that algebras of classical propositional fuzzy logics can be characterized in

the class of all Rℓ-monoids as follows:An Rℓ-monoid A is

(a) a BL-algebra [17] iff A satisfies the identity of prelinearity(x→ y) ∨ (y → x) = 1;

(b) an MV-algebra [12] iff A fulfils the law of double negation ¬¬x = x;

(c) a Heyting algebra [20] iff operations ⊙ and ∧ coincide on A.

Let us also mention that bounded commutative residuated lattices which satisfy theidentity of prelinearity are also called MTL-algebras being an algebraic counterpart of theso-called Monoidal t-norm logic (MTL, for short), see [7], [6].

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3 Pocrims with vt-operators

In this section we shall deal with pocrims equipped by vt-operators. As we have mentionedin the introduction, vt-operators were studied on BL-algebras by Hájek and on Rℓ-monoidsby Rachůnek and Šalounová. Another branch of mathematics where special vt-operatorswere successfully used is formal concept analysis (FCA). Among the sereral methods foranalysis of object-attribute data, FCA is becoming increasingly popular. The main aim inFCA is to extract interesting clusters (called formal concepts) from tabular data. Formalconcepts correspond to maximal rectangles in a data table. The number of formal conceptsin data can be extremely large. In order to reduce the number of concepts, R.Bělohlávekand V.Vychodil used so-called hedges, being special cases of vt-operators [2]. Remark thatFCA was founded by R.Wille, for basic notions and results we advice the book [8].

Definition 1. Let A = (A,⊙,→, 1) be a pocrim. A mapping v : A −→ A is called a weakvt-operator (wvt-operator in brief) on A if for any x, y ∈ A:

(1) v(1) = 1

(2) v(x) ≤ x, i.e. v is subdiagonal

(3) v(x→ y) ≤ v(x)→ v(y).

If a wvt-operator v satisfies for any x, y ∈ A

(4) if x ∨ y exists then so does v(x) ∨ v(y), and v(x ∨ y) ≤ v(x) ∨ v(y),

then v is called a vt-operator on A.

Notice that if A is a BL-algebra or an Rℓ-monoid then v : A −→ A is a vt-operatoriff it is a vt-operator in a usual sense [18]. Also any pocrim admits vt-operators, e.g. theidentity operator.

Lemma 2. Let v be a wvt-operator on a pocrim (A,⊙,→, 1) and x, y, z ∈ A. Then

(a) v(0) = 0

(b) x ≤ y =⇒ v(x) ≤ v(y)

(c) v(¬x) ≤ ¬(v(x))

(d) x⊙ y ≤ z =⇒ v(x)⊙ v(y) ≤ v(z)

(e) v(x)⊙ v(y) ≤ v(x⊙ y)

(f) v(x)⊙ v(x→ y) ≤ v(x), v(y).

If v is a vt-operator then

(g) v(x ∨ y) = v(x) ∨ v(y) provided x ∨ y exists.

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If A is a pocrim and v is a vt-operator (wvt-operator) on A then the pair (A, v) iscalled a vt-pocrim (a wvt-pocrim, respectively).

Given a wvt-pocrim (A, v) and a filter F of A then F is called a v-filter of (A, v) ifv(x) ∈ F for every x ∈ F .

For any v-filter F of (A, v) denote by vF : A/F −→ A/F the mapping defined by

vF (x/F ) := v(x)/F.

Proposition 3. If F is a v-filter on a wvt-pocrim (A, v) then vF is a wvt-operator onA/F .

If (A, v) is a wvt-pocrim and θ a congruence or a relative congruence on A, respectively,then θ is called a v-congruence or a v-relative congruence on A if ⟨x, y⟩ ∈ θ implies⟨v(x), v(y)⟩ ∈ θ for each x, y ∈ A.

Lemma 4. If (A, v) is a wvt-pocrim then there is a one-to-one correspondence between itsv-filters and its v-relative congruences.

It is well known [10] that the class of BL-algebras is representable as a subdirectproduct of linearly ordered BL-algebras. Remarkably, it has been proved [18] that alsoevery BLvt algebra is a subdirect product not only of BL-chains (i.e., as a BL-algebra inthe corresponding signature), but, moreover, it is a subdirect product of BLvt-chains (inthe extended signature).

We have seen in Corollary 2 that any pocrim satisfying the prelinearity identity isa subdirect product of linearly ordered pocrims. Hence a natural question arises if thisstatement can be extended to the class of vt-pocrims. For general vt-pocrims we do notknow the answer, but for artinian case the answer is positive:

Theorem 5. Any vt-pocrim (A, v) for which FilA is artinian and satisfying the prelin-earity identity is a subdirect product of linearly ordered vt-pocrims.

4 Operators on pocrims derived from wvt-operators

Every wvt-operator on a pocrim A is, by the definition and Lemma 8, a subdiagonal andmonotone self-mapping of A. Now we use wvt-operators to introduce derived self-mappingsof A that are, among others, superdiagonal and monotone, and in the case of MV-algebrasthey have properties of unary connectives “very false”. For a pocrim A and f : A −→ Adenote by f− : A −→ A a mapping defined by

f−(x) := ¬(f(¬x)).

Example 6. Consider the standard MV-algebra [0, 1]. It is known that the mapping v :[0, 1] −→ [0, 1] defined by v(x) = x2 is a vt-operator on [0, 1]. Then v− : [0, 1] −→ [0, 1] isthe mapping such that v−(x) = 2x− x2 for each x ∈ [0, 1].

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A pocrim A = (A,⊙,→, 1) is said to be normal if it satisfies the identity

¬¬(x⊙ y) = ¬¬x⊙ ¬¬y.

Remark that every BL-algebra and every Heyting algebra is normal [18], hence theclass of normal pocrims is considerably large.

Lemma 7. In any pocrim A we have

(a) ¬¬x ≥ x

(b) ¬(x⊙ y) = y → ¬x = ¬¬y → ¬x = x→ ¬y = ¬¬x→ ¬y

(c) ¬¬¬x = ¬x

(d) ¬¬x→ ¬¬y ≥ ¬¬(x→ y).

Given a pocrim A, one can consider the derived operation ⊕ defined in a usual way asx⊕ y := ¬(¬x⊙ ¬y). If A satisfies the law of double negation ¬¬x = x, then there existnot only the residuum → of ⊙ but one can associate the residuum ⊖ also with ⊕. Indeed,one can easily verify that

x⊕ y ≥ z iff x ≥ z ⊖ y,

where x ⊖ y := x ⊙ ¬y. For MV-algebras this fact is well known and can be found e.g.in [10].

Consequently, on any MV-algebra A one can define not only vt-operators but also dualoperators, vf -operators (very false). We will show that also in a more general case ofpocrims, every wvt-operator on A determines a vf -operator on A under certain additionalconditions. A formal definition of a vf -operator is dual to that of vt-operator:

Definition 8. Let A be a pocrim. A mapping v : A −→ A is called a weak vf-operator(wvf -operator in brief) on A if for any x, y ∈ A:

(1) v(0) = 0

(2) x ≤ v(x), i.e. v is superdiagonal

(3) v(x⊖ y) ≥ v(x)⊖ v(y).

If a wvt-operator v satisfies for any x, y ∈ A

(4) if x ∧ y exists then so does v(x) ∧ v(y), and v(x ∧ y) ≥ v(x) ∧ v(y),

then v is called a vf-operator on A.

One can easily show that any wvf -operator v satisfies also v(1) = 1 by (2) and it ismonotone by (3) and (1) whenever A fulfils the double negation law. Indeed, in this casex ≤ y is equivalent to x⊖ y = 0.Moreover, for any vf -operator v we have v(a ∧ b) = v(a) ∧ v(b) whenever a ∧ b exists.

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Proposition 9. Let (A, v) be a wvt-pocrim. Then we have for any x, y, z ∈ A:

(1) v−(0) = 0, v−(1) = 1

(2) x ≤ v−(x)

(3) v−(x→ ¬¬y) ≤ v(x)→ v−(y)

(4) x ≤ y =⇒ v−(x) ≤ v−(y)

(5) x⊙ y ≤ z =⇒ v−(x)⊙ v−(y) ≤ v−(z)

(6) v−(x)⊙ v−(y) ≤ v−(x⊙ y)

(7) v−(x)⊙ v−(x→ y) ≤ v−(x), v−(y)

(8) v−(x→ y) ≤ v−(x)→ v−(y)

(9) If A is normal then v−(x→ ¬y) ≤ v(¬¬x)→ v−(¬y) ≤ v(x)→ v−(¬y).

(1) and (2) of Proposition 3 show that to any wvt-operator v on a pocrim A one canassign an operator v− satisfying first two conditions of a wvf -operator. Let us show thatif A is normal (especially, if A fulfils the double negation law), then v− verifies also (3) ofDefinition 2, i.e. it is a wvf -operator: applying the normality of A, by Lemma 10(c) and(6) of Proposition 8 we obtain

v−(x)⊖ v−(y) = ¬v(¬x)⊙ ¬¬v(¬y) = ¬¬(¬v(¬x)⊙ v(¬y)) =

¬¬(v−(x)⊙ v(¬y)) ≤ ¬¬v−(x⊙ ¬y) = v−(x⊙ ¬y) = v−(x⊖ y).

There is a natural question under what conditions also the converse is possible, i.e.when v− is a wvt-operator for a wvf -operator v. A close inspection shows that this is notalways the case, for we introduce the following notion:

Definition 10. Let A be a pocrim. A mapping w : A −→ A is called a near wvt-operatoron A if it satisfies (1) and (3) from the definition of a wvt-operator on A and if for anyx ∈ A,

(2’) w(x) ≤ ¬¬x.

Remark that by Lemma 10(a), every wvt-operator on A is a near wvt-operator on A.

Theorem 11. Let A be a normal pocrim and let v be a wvf -operator on A. Then

(i) v− is a near wvt-operator on A

(ii) if, moreover, A satisfies the double negation law, then v− is a wvt-operator.

Thus, for pocrims satisfying the double negation law we can state

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Corollary 12. Let A be a pocrim satisfying the double negation law. Then there is a 1-1correspondence between wvt-operators and wvf -operators on A.

Now we focus on the problem when given a vt(vf)-operator v on A, under what con-ditions the assigned operator v− is vf(vt).

Theorem 13. Let A be pocrim satisfying the condition (C)

“if a ∧ b exists then so does ¬a ∨ ¬b and ¬(a ∧ b) = ¬a ∨ ¬b.”

Then

(i) If v is a vf -operator on A and A satisfies the double negation law then v− is avt-operator on A.

(ii) If v is a vt-operator on A then v− is a vf -operator on A.

From the last statement we directly obtain

Corollary 14. For any pocrim A satisfying the double negation law and the condition (C)there is a 1-1 correspondence between between vt-operators and wf -operators on A.

Remark. It is easy to see that any linearly ordered pocrim fulfils the condition (C).Thus the pocrim A from Example 1 satisfies both assumptions of the above corollary.

Given a pocrim A = (A,⊙,→, 1) and f : A −→ A a mapping, consider the self mapf : A −→ A on A by f := (f−)−, i.e.

f(x) := ¬¬(f(¬¬x)).

As a direct corollary of Proposition 3 and Theorem 2 we obtain

Corollary 15. If v is a wvt-operator on a pocrim A, then v is a near wvt-operator on A.

In the sequel, we shall investigate the setN (A) of all near wvt-operators on on a pocrimA. If w1, w2 ∈ N (A), we put w1 ≤ w2 iff w1(x) ≤ w2(x) for all x ∈ A.

Clearly (N (A),≤) is a (pointwise) ordered set. Let us consider the mapping ˜ :N (A) −→ N (A) given by w 7→ w for every w ∈ N (A).

Lemma 16. The mapping ˜ is closure operator on the poset (N (A),≤).

Proof. Let w,w1, w2 ∈ N (A) be given. Then for a ∈ A(1) ¬¬x ≥ x, hence w(¬¬x) ≥ w(x) and w(x) = ¬¬(w(¬¬(x)) ≥ ¬¬(w(x)) ≥ w(x),

i.e. w ≥ w.

(2) If w1 ≤ w2, then w1(¬¬x) ≤ w2(¬¬y), thus w1(x) = ¬¬(w1(¬¬x)) ≤ ¬¬(w2(¬¬x)) =w2(x), i.e. w1 ≤ w2;

(3) We have ˜w (x) = ¬¬¬¬(w(¬¬¬¬x)) = ¬¬(w(¬¬x)) = w(x), hence ˜w = w.

We say that a near wvt-operator is essential if w is a closed element in (N (A),≤)with respect to the closure operator .

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Proposition 17. If w ∈ N (A) then w and w induce the same operators w− and (w)−.

The following corollary is an immediate consequence:

Corollary 18. If A is a pocrim satisfying the double negation law, then every wvt-operatoron A is essential.

Example 19. Consider the pocrim from Example 2. Then definingv3 : A −→ A byv3(1) = 1, v3(0) = v3(a) = 0, v3(b) = b, v3(c) = v3(d) = 1,then v3 is a mapping satisfying conditions (1) and (3) from the definition of a wvt-operator,not satisfying (2) as 1 = v3(c) ≤ c, but satisfying (2’).Further we have v−1 = v−2 = v−3 , in the pointwise order on N (A) we have v1 < v2 < v3 andv1 = v2 = v3 = v3.

References

[1] Blok W.J., Raftery J.G.: Varieties of Commutative Residuated Integral Pomonoidsand Their Residuation Subreducts, Journal of Algebra, 190(1997), 280-328.

[2] Bělohlávek R., Vychodil V.: Reducing the size of fuzzy concepts by hedges, The 2005IEEE International Conference on Fuzzy Systems, 663-668.

[3] Chajda I., Eigenthaler G., Länger H.: Congruence Classes in Universal Algebra, Hel-dermann Verlag, Lemgo 2003.

[4] Chajda I., Halaš R.: Functional completeness of weak logics with a strict negation,Multiple Valued Logic and Soft Comp., to appear.

[5] Chajda I., Vychodil V.: A note on residuated lattices with globalization, Internat.Journal of Pure and Appl. Mathematics, 27(3) (2006), 299-303.

[6] Di Nola A., Sessa S., Esteva F., Godo L., Garcia P.: The variety generated by perfectBL-algebras: an algebraic approach in a fuzzy logic setting, Ann. Math. ArtificialIntelligence, 35(2002), 197-214.

[7] Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for left continuoust-norms, Fuzzy Sets and Systems, 124(2001), 271-288.

[8] Ganter B., Wille R.: Formal concept analysis. Mathematical Foundations, Springer-Verlag, Berlin, 1999.

[9] Hájek P.: On very true, Fuzzy sets and systems, 124 (2001), 329-333.

[10] Hájek P.: Metamathemtics of Fuzzy Logic, Kluwer, Dordrecht, 1998.

[11] Higgs D.: Dually residuated commutative monoids with identity element do not forman equational class, Math. Jap., 29 (1984), 69-75.

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[12] Höhle U.: Commutative, residuated l-monoids, in: U. Höhle, E.P. Klement (eds.),Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Acad. Publ.,Dordrecht, 1995, pp. 53-106.

[13] Iorgulescu A.: Classes of BCK Algebras-Part I, Preprint Series of the Institute ofMathematics of the Romanian Academy, preprint nr. 1/2004.

[14] Iséki K.: An algebra related to a propositional calculus, Proc. Japan. Acad., 42 (1966),26-29.

[15] Jipsen P., Tsinakis C.: A survey of residuated lattices, in: J.Martinez (ed.), OrderedAlgebraic Structures, Kluwer Acad. Publ., Dordrecht, 2002, pp. 19-56.

[16] Pałasiński M.: An embedding theorem for BCK-algebras, Math. Sem. Notes KobeUniv., 10 (1982), 749-751.

[17] Rachůnek J.: A duality between algebras of basic logic and bounded representable DRl-monoids, Math. Bohemica, 126 (2001), 561-569.

[18] Rachůnek J., Šalounová D.: Truth values on generalizations of some commutativefuzzy structures, Fuzzy sets and systems, 157 (2006), 3159-3168.

[19] Kowalski T., Ono H.: Residuated lattices: an algebraic glimpse at logics without con-traction, Monograph (2001).

[20] Swamy K.L.N.: Dually residuated lattice ordered semigroups, Math. Ann., 159(1965),105-114.

[21] Vychodil V.: Truth-depressing hedges and BL-logic, Fuzzy Sets and Systems,157(2006), 2074-2090.

[22] Ward M., Dilworth R.P.: Residuated lattices, Trans. Amer. Math. Soc., 45(1939),335-354.

[23] Zadeh L.: Fuzzy logic and approximate reasoning, Synthese, 30(1975), 407-428.

78

Lateral Completions of BCK- andPseudo-BCK-Algebras

Jan Kühr∗

A lattice-ordered group (ℓ-group) G is laterally complete if every non-empty set ofpairwise disjoint elements in G has an infimum, i.e., infGX exists for every ∅ = X ⊆G− = x ∈ G : x ≤ e such that x1 ∨ x2 = e for all x1, x2 ∈ X with x1 = x2. Sincethe ℓ-group of automoprhisms of a chain is laterally complete, it follows from Holland’stheorem that every ℓ-group can be embedded into a laterally complete one. Conrad [4]defined a lateral completion of an ℓ-group G as a laterally complete ℓ-group L such that:(i) G is dense in L, in the sense that for every x ∈ L with x < e there exists y ∈ G withx ≤ y < e, and (ii) there is no laterally complete ℓ-subgroup H of L with G ⊂ H ⊂ L.Bernau [2] proved that every ℓ-group has a unique lateral completion.

The negative cone G− of any ℓ-group G equipped with the operations x/y = xy−1 ∧ eand y\x = y−1x ∧ e is an example of a cone algebra [3] or, more generally, of a pseudo-BCK-algebra [5]. A pseudo-BCK-algebra is a structure ⟨A,≤, /, \, 1⟩ where ⟨A,≤, 1⟩ is aposet with greatest element and /, \ are binary operations satisfying the axioms:

x\y ≤ (x\z)/(y\z), y/x ≤ (z/y)\(z/x),1\x = x, x/1 = x,

x ≤ y ⇔ x\y = 1 ⇔ y/x = 1.

For the original definition, see [5]. Clearly, if x\y = y/x for all x, y ∈ A, then ⟨A,≤, \, 1⟩ is aBCK-algebra. Pseudo-BCK-algebras are just the /, \, 1-subreducts of integral residuatedlattices, hence they are equivalent to biresiduation algebras [6]. By a commutative pseudo-BCK-algebra we mean an algebra satisfying the identity

x/(y\x) = (y/x)\y.

Finally, a cone algebra [3] is a commutative pseudo-BCK-algebra which satisfies the iden-tities

(x\y)\(x\z) = (y\x)\(y\z) and (z/x)/(y/x) = (z/y)/(x/y).

MV-algebras are equivalent to bounded commutative BCK-algebras and pseudo-MV-algebras to bounded commutative pseudo-BCK-algebras.

The underlying poset of a pseudo-BCK-algebra need not be a join-semilattice, but theconcept of being laterally complete can be defined as for ℓ-groups; the only difference isthat the condition x1 ∨ x2 = 1 has to be read as ‘supAx1, x2 exists and equals 1’. Alatteral completion of a pseudo-BCK-algebra is then defined as above.

∗Department of Algebra and Geometry, Faculty of Science, Palacký University in Olomouc, 17.listopadu 12, CZ-77146 Olomouc, Czech Republic, [email protected]

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Unfortunately, lateral completions of pseudo-BCK-algebras are not unique in general.Hence we restrict our attention to the class K of commutative pseudo-BCK-algebras forwhich we can prove that if a lateral completion in K exists, then it is unique, and for twoparticular subclasses of K we prove that every algebra has a unique lateral completion.

The first subclass is the class of cone algebras. Bosbach [3] proved that every conealgebra A can be represented as a subalgebra of the negative cone of a suitable ℓ-group.This ℓ-group, say GA, can be chosen in such a way that A is convex in G−

A and generatesG−A as a semigroup. Under these conditions, GA is unique and, among other things, it is

possible to show that A is a conditionally complete lattice iff so is GA. Though the linkbetween lateral completeness of A and GA is not straightforward (it can happen that A islaterally complete and GA is not, or vice versa), we can use the lateral completion of GAin constructing the lateral completion of A. In particular, we obtain lateral completions ofMV- and pseudo-MV-algebras (see [1]).

We also defined the concept of Archimedean cone algebras and show that the repre-sentation of A via GA can be used to construct the Dedekind-MacNeille completion of Awhen A is Archimedean.

The other class under consideration consists of those pseudo-BCK-algebras which havenormal pseudocomplements of deductive systems. By a deductive system in A we meana subset X ⊆ A such that 1 ∈ X and if a, a\b ∈ X, then also b ∈ X. If, moreover,for all a, b ∈ A, a\b ∈ X iff b/a ∈ X, then X is a normal deductive system. For anynormal deductive system X, the relation θX which identifies a, b ∈ A iff a\b, b\a ∈ X isa congruence of A such that the factor algebra A/X = A/θX is a pseudo-BCK-algebra.The deductive systems of A form an algebraic distributive lattice DS(A) where, for everyX ∈ DS(A), the pseudocomplement of X in DS(A) is the deductive system X ′ = a ∈ A :a ∨ x = 1 for all x ∈ X. Then P(A) = X ′ : X ∈ DS(A) forms a complete Booleanalgebra. We equip the set of all partitions1 in P(A) with the following partial order:E ⊑ F iff every F ∈ F is contained in some E ∈ E . This makes the set of partitions inP(A) an upwards directed set.

Now, let A be a commutative pseudo-BCK-algebra such that X ′ is normal for everyX ∈ DS(A). (Of course, all commutative BCK-algebras have this property.) For anypartition E in P(A), let AE =

∏E∈E A/E

′. If E ,F are two partitions in P(A) suchthat E ⊑ F , then the definition of ⊑ ensures that AE can naturally be embedded intoAF . Hence we can form the direct limit L = lim−→AE of the algebras AE where E runsthrough the partitions in P(A). We prove that L is a laterally complete commutativepseudo-BCK-algebra which contains (an isomorphic copy of) A as a dense subalgebra, andconsequently, the lateral completion of A can be constructed within L. In particular, ifA is a finite commutative BCK-algebra and if E1, . . . , Em are the atoms of the Booleanalgebra P(A), then A/E′

1 × · · · ×A/E′m is an MV-algebra which is the lateral completion

of A.1E is a partition in P(A) if E1 ∩ E2 = 1 for all E1, E2 ∈ E with E1 = E2, and supP(A) E = A.

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References

[1] R. Ball, G. Georgescu, and I. Leuştean: Cauchy completions of MV-algebras, Alge-bra Univers. 47 (2002), 367–407.

[2] S.J. Bernau: The lateral completion of an arbitrary lattice group, J. Austral. Math.Soc. 19 (1975), 263–289.

[3] B. Bosbach: Concerning cone algebras, Algebra Univers. 15 (1982), 58–66.

[4] P. Conrad: The lateral completion of a lattice-ordered group, Proc. London Math.Soc. 19 (1969), 444–480.

[5] G. Georgescu and A. Iorgulescu: Pseudo-BCK algebras: An extension of BCKalgebras, in: Proceedings of DMTCS’01: Combinatorics, Computability and Logic,Springer, London, 2001, pp. 97–114.

[6] C.J. van Alten: On varieties of biresiduation algebras, Studia Logica 83 (2006),425–445.

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Substructural Epistemic ModalitiesOndrej Majer∗,

(joint work with Marta Bílková, Michal Peliš and Greg Restall)

A traditional way of representing knowledge in the standard modal framework has beenwidely criticized. The main reason is that the epistemic agents represented in this way aretoo ideal. They are logically omniscient—the do know all the logical truths (it follows fromthe Necessitation rule ϕ/Kϕ) as well as all the logical consequences of the facts they know(the K-axiom K(ϕ → ψ) → (Kϕ → Kψ) entails the modal version of Modus Ponens—from Kϕ and K(ϕ → ψ) entail Kψ). On the top of these two properties, which hold inevery normal modal frame, agents are sometimes assumed to be explicitly aware of theirboth positive and negative knowledge (positive and negative introspection) etc. For thesereasons are S5 representations sometimes called epistemic logics of potential knowledge.

One solution overcoming such criticism was proposed by Majer and Peliš [2] and de-veloped by Bílková et al. [1]. The main idea was to employ relational semantics fordistributive relevant logics—the framework which can be considered as modal and which isweak enough to avoid the unwanted properties of the standard representation. In this pa-per we observe further generalization to weaker (but still distributive) substructural logicsan define some other modalities, which naturally accompain the basic epistemic modality.

The main ingredients of the semantics of distributive substructural logic are a setof information states X, partially ordered by a relation of involvement, plus additionalrelations on X: a ternary relation of relevance (used to interpret implication and fusion),binary relation of compatibility (to interpret negation), and a subset of X called logicalstates. Such a frame accompanied with a persistent valuation of atomic propositions formsa model. The properties of the relations then determine the properties of the correspondingconnectives (e.g. the symmetry of C reflects the fact, that there is only in one negation inthe system rather than two etc.) For detailed overview of these correspondences see [3].

In this framework an agent is supposed to be in a state, interpreted as a collection ofdata available to her, which might be incomplete (containing neither ϕ nor ¬ϕ) or/andinconsistent (containing both ϕ and ¬ϕ). A piece of available data becomes knowledgeif and only if it is confirmed by a resource. Which states, in a given state, can serve asresources is specified by a separate binary relation S. The resulting epistemic modality Kis then interpreted as:

x Kϕ iff ∃s(sSx & s ϕ)

In the presented approach the relation S is built on the relations of compatibility andinvolvement already presented in the relevant frames—a source s for a current state xshould be compatible with the current state and should precede it in the involvement

∗Institute of Philosophy of the Academy of Sciences of the Czech Republic, v.v.i., Jilská 1, 110 00Prague 1, Czech Republic, [email protected]

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relation. Depending on the properties of the relation S we can define various kinds offrames, but the problem is if to find an axiomatization for them. For example for thefollowing two classes of frames

sSx iff sCx & s < x and sSx iff sCx & s ≤ x

no axiomatization has been found and it is conjectured in [1] that they are not axioma-tizable. be axiomatized. One solution is to relax the conditions on the source relation.Both of the frames we mentioned require that everything which can serve as a source isin fact considered as a source. The first one on the top of that requires some sort of“independence”—a state cannot serve as a source for itself. If we relax these conditionsand require allow that not all states, which are potentially be sources are in fact consideredas sources (we replace ‘iff’ by ‘only if’ in the definition of the source relation)

sSx only if sCx & s ≤ x

we obtain a class Fg which can be axiomatized using the following modal axioms and rules:

factivity: Kϕ→ ϕ

strong factivity: ¬ϕ ∧Kϕ→ ⊥

normality: K(ϕ ∨ ψ)→ Kϕ ∨Kψ

monotonicity:ϕ→ ψ

Kϕ→ Kψ

In [1] axiomatization over the relevant logic R was provided, a class of frames charac-terized by the axioms have been isolated and the completeness of the axiomatics has beenproved. This result can be generalized to weaker distributive substructural logics. Theonly requirement is that the relations R and C of the system in question are monotonouswith respect to the involvement relation (in [3] these are called “plump” relations).

The second goal of the presented paper is to introduce two modalities related to theK modality and discuss their properties. First we want to introduce the box-like modal-ity related to the diamond-like modality K, with the intended meaning “all the sourcesconfirm ϕ”:

x ϕ iff for any s if sSx then s ϕand discuss its relation to the ‘standard’ dual modality ¬K¬ϕ, which in this context meanssomething different.

From the point of view of the source relation S these modalities are ‘backward looking’.It is quite natural to consider also ‘forward looking’ modalities—modalities for which thecurrent state is a (potential) source. The corresponding modality I adjoint to K, can beconsidered as implicit knowledge: if ϕ holds in all states, for which the current state s isa source, then ϕ is implicit knowledge in s.

x Iψ iff y ψ for all y such that xSy

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This modality has several natural properties. First, everything what is true in thecurrent state is implicitly known:

ϕ→ Iϕ

This condition immediately entails positive introspection (if something is implicitlyknown, it is known, that it is known):

Iϕ→ IIϕ

Finally all that holds in a state is at least implicitly known there and nothing else canbe known to be implicit then facts true in the state:

ϕ→ IKϕ,KIϕ→ ϕ

Formally we can define also a dual modality to I—forward looking diamond like epis-temic operator, but there seems to be no natural interpretation for it.

References

[1] M. Bílková, O. Majer, M. Peliš, and G. Restall. Relevant agents. In L. Beklemishev,V. Goranko, and V. Shehtman, editors, Advances in Modal Logic, pages 22–38. CollegePublications, 2010.

[2] O. Majer and M. Peliš. Epistemic Logic with Relevant Agents. In M. Peliš, editor, TheLogica Yearbook 2008, pages 123–135. College Publications, 2009.

[3] G. Restall. An Introduction to Substructural Logics. Routledge, 2000.

84

Partial First–Order Logical Semantics Basedon Approximations of Sets

Tamás Mihálydeák∗

1 Introduction

In recent years a number of theoretical attempts appeared in order to approximate sets. Forexample, rough set theory was originally proposed by Pawlak, its different generalizationsand granular computing play a crucial role in computer sciences. Rough set theory can beapplied among others in the areas of artificial intelligence, cognitive sciences, medicine andeconomics. It provides a powerful foundation to reveal and discover important structuresand patterns in data and to classify complex objects.

In most cases we have a family of base sets — as subsets of a universe of discourse.In philosophy these sets represent our available knowledge. We consider them as theextensions of our available concepts/properties, and their members are the instances ofthese concepts/properties. In rough set theory (at least in its original version) we emphasizethat there is no way to distinguish two objects belonging to the same base set, whereas ingranular computing we pay more attention to the fact that our knowledge can make a realdifference between two object when they belong to different granules. The primary goal ofdifferent systems of set approximation is to approximate an unknown concept (representedby an arbitrary subset of the universe).

In my presentation, a very general framework of set approximation is proposed. Inthis theoretical system, Pawlak’s theory, its generalizations, the set theoretical systemof granular computing and many other generalizations can be described. Interestinglyenough, our set theoretical framework can be used as a set theoretical semantic base for aspecific first–order logic described as partial first–order logic (or first–order logic permittingsemantic value gaps).

The main idea is the following: Our starting point is a given language of first–orderlogic and we have a finite distinguished subset of its predicate parameters. Its membersexpress available concepts/properties and relations, which we call tools. If we take a usualinterpretation of the given first–order language, we get the interpretation of all predicateparameters. The following question appears: What do we have to change if we use theapproximations of sets generated by the semantic value of our tools as semantic values ofpredicate parameters instead of the sets given by their total interpretation? Our solutionregarding this question is that we may introduce three different partial interpretations of

∗Department of Computer Science, Faculty of Informatics, University of Debrecen, Hungary,[email protected]

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the given first–order language with the help of its total interpretation. As a result of do-ing this, we can compare the “real world” that appears in the total interpretation and itsapproximations that appear in generated partial interpretations. If we take into consider-ation all interpretations that fulfill some requirements, we have the logical possibility toinvestigate what happens to logical laws when we use the approximations of sets insteadof the sets themselves. The common set theoretical framework proves to be a useful toolto compare the results and consequences of different approximations from the logical pointof view.

2 General systems of tool–based approximation of sets

In the following definition the most fundamental sets are given which appear in the settheoretical background of semantics of partial first–order logic based on approximationspace.

Definition 1. Let U be an arbitrary nonempty set.

1. Let PR(U) be the set of representations of all concepts/properties and relations onU , i. e. PR(U) =

∪∞n=1 2

U(n), where U (1) = U and U (n) = U ×U · · · ×U (n-times).

2. Let A(n) ⊆ 2U(n)\∅ be a subset of representations of nonmepty n-argument relations

on U (n = 1, 2, . . . ), A =∪∞n=1A(n) and A = ∅.

3. Let DA be an extension of A such that

(a) A ⊆ DA;

(b) ∅ ∈ DA;

(c) if S1, S2 ⊆ U (n) for some n and S1, S2 ∈ DA, then

i. S1 ∪ S2 ∈ DA;ii. S1 ∩ S2 ∈ DA.

Informally, A is a nonempty set of representations of primitive or available concepts/properties and finite-argument relations on U , and DA contains not only the members ofset A, but the representations of those concepts/properties and finite-argument relationswhich can be defined by their finite conjunctions and finite disjunctions. Usually themembers of DA are considered as well–definable sets by base sets, the members of A. Wewant to use the members of DA to approximate any concept/property or finite-argumentrelation on U .

In order to define approximation space we need the notion of approximation pair whichhas three different versions, weak, semi-strong and strong. Here, only the definition ofweak approximation pair is given.

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Definition 2. The pair of maps ⟨f, g⟩ is a weak approximation pair on concepts/propertiesand finite-argument relations of U , if

1. f, g : PR(U)→ DA;

2. if S ∈ DA, then f(S) = S;

3. if S ∈ PR(U), then f(S) ⊆ g(S).

Informally, the functions f , g determine the lower and upper approximation of anyset with the help of representations of our primitive or available concepts/properties andrelations. The second condition in the definition is not typical: if we look at the setsbelonging to DA as our tools to approximate any set, then it can be a requirement thata tool should be approximated by itself from the lower side. If we give it up, we decreasethe roles of our base sets (members of DA). The nature of an approximation pair dependson how to relate the lower and upper approximations of a set to the set itself.

Using our introduced notations general approximation space can be defined:

Definition 3. Let U,A be sets as in Definition 1, and ⟨f, g⟩ be a weak approximation pair.Then an ordered quadruple ⟨U,A, f, g⟩ is a general weak approximation space (with respectto the weak approximation pair ⟨f, g⟩).

3 Partial semantics of first–order logic basedon approximation spaces

Let L(1) = ⟨LC, V ar, Con, Term,Form⟩ be a given language of first–order logic, andT ⊆

∪∞n=1 P(n) (⊆ Con) be a finite nonempty set of predicate parameters. Members of

set T are called tools.Let Ip = ⟨U, ϱ⟩ be an interpretation of L(1) such that if T ∈ T , then ϱ(T ) = ∅. Tools

determine a general approximation space with respect to the given interpretation.

Definition 4. The quadruple ⟨U,AT , f, g⟩ is a general approximation space generated byset T of tools with respect to the interpretation ⟨U, ϱ⟩ if AT = ϱ(P ) : P ∈ T .

Remark 1. The semantic values of tools with respect to the given interpretation play a dis-tinguished role in the generated partial interpretation: the semantic value of any predicateparameter is approximated by their help.

Tools generate three different partial interpretations of the given language L(1) with re-spect to the given (total) interpretation ⟨U, ϱ⟩, in notation : ⟨U, ϱl⟩T , ⟨U, ϱu⟩T and ⟨U, ϱm⟩Tfor lower, upper and mixed partial interpretation respectively. The differences appear onlyin the definition of interpretation function. Generated interpretation functions will be par-tial ones. For example if P is a one–argument predicate parameter which is not a tool andu ∈ U , then

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• in lower partial interpretation:

1. P is true at u if u belongs to the lower approximation of semantic value of P(i.e. our tools evaluate P as certainly true at u);

2. P is false at u if u belongs to the lower approximation of the complementof upper approximation of semantic value of P (i.e. our tools evaluate P ascertainly false at u);

3. otherwise P is undefined (i.e. our tools are not enough to decide whether P iscertainly true or certainly false at u);

• in upper partial interpretation:

1. P is true at u if u belongs to the upper approximation of semantic value of P(i.e. our tools evaluate P as maybe true at u);

2. P is false at u if u belongs to the lower approximation of the complementof upper approximation of semantic value of P (i.e. our tools evaluate P ascertainly false at u);

3. otherwise P is undefined (i.e. our tools are not enough to decide whether P ismaybe true or certainly false at u);

• in mixed partial interpretation:

1. P is true at u if u belongs to the lower approximation of semantic value of P(i.e. our tools evaluate P as certainly true at u);

2. P is undefined at u if u belongs to the upper but does not belong to lowerapproximation of semantic value of P (i.e. according to our tools u belongs tothe border of P );

3. otherwise P is false (i.e. according to our tools u is outside of the border of P ).

Relying on an interpretation, the different types of generated interpretations and anassignment one can give the semantic rules based on the lower and upper approximationsof semantic values of predicate parameters. We can use the semantic rules of first–orderlogic permitting semantic value gaps.

From the logical point of view, flexibility is the main advantage of our logical framework.It can be recognized on different levels:

1. The generated partial interpretations rely on two theoretical points:

(a) the set of semantic values of tools given by the total interpretation ⟨U, ϱ⟩. Thesesemantic values represent available knowledge, i.e. the total interpretation givesus the representations of available concepts/properties and relations by whichwe approximate any concept/property or relation (with respect to the giveninterpretation);

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(b) the general approximation space generated by tools with respect to the giveninterpretation. By specifying an approximation pair ⟨f, g⟩ different theoreticalsystems of approximation can be gained: Pawlak-type, generated Pawlak-types,approximations used in different systems of granular computing or in very gen-eral versions of the approximation of sets.

2. In the definition of one consequence relation the different notions of models can beused. For example, we can require that all lower models of the set of premises shouldbe a lower/upper/mixed model of the conclusion: it makes possible to investigatedifferent approximations comparably.

Definition 5. Let Γ ⊆ Form be a set of formulae of first–order logic,A ∈ Form be a formula and Ip is a set of total interpretations of the language of first–orderlogic. Then

Γ ⋆• Awith respect to Ip if every ⋆-model ⟨U, ϱ⋆, v⟩ of set Γ is a •-model ⟨U, ϱ•, v⟩ of the formulaA, where ⟨U, ϱ⟩ ∈ Ip and ⋆, • ∈ t, l, u,m, (t stands for total model, i.e. ϱt = ϱ).

Let us consider a simple consequence relation which is valid in first–order logic.

∀x(P1(x) ⊃ P2(x)), P1(a)tt P2(a),

What happens to its validity if we use the different types of approximations of the exten-sions of predicate parameters P1, P2? If our interpretations rely on standard approximationspaces (where f(S) =

∪S′ : S′ ∈ DA, S

′ ⊆ S and g(S) =∪S′ : S′ ∈ DA, S

′ ∩ S = ∅),then we gain the results in the following table, where + means that the given consequencerelation is valid, and − means that it is not.

Interpertation ofpremises conclusion

total lower upper mixedtotal + − − −lower + + + +upper − − + −mixed + + + +

References

[1] Pawlak, Z.: Rough sets. International Journal of Information and Computer Science11(5) (1982) 341–356.

[2] Yao, Y.Y.: On generalizing rough set theory. In: Proceedings of the 9th Interna-tional Conference Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing(RSFDGrC 2003). LNAI 2639, Springer-Verlag (2003) 44–51.

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[3] Skowron, A., Świniarski, R., Synak, P.: Approximation spaces and information gran-ulation. LNCS Transactions on Rough Sets 3400(3) (2005) 175–189.

[4] Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets: An approach to vagueness. InRivero, L.C., Doorn, J., Ferraggine, V., eds.: Encyclopedia of Database Technologiesand Applications. Idea Group Inc., Hershey, PA (2005) 575–580.

[5] Düntsch, I., Gediga, G.: Approximation operators in qualitative data analysis. Inde Swart, H.C.M., Orlowska, E., Schmidt, G., Roubens, M., eds.: Theory and Appli-cations of Relational Structures as Knowledge Instruments. Volume 2929 of LectureNotes in Computer Science., Springer (2003) 214–230.

[6] Csajbók, Z.: Partial approximative set theory: A generalization of the rough settheory. In Martin, T., Muda, A.K., Abraham, A., Prade, H., Laurent, A., Laurent,D., Sans, V., eds.: Proceedings of the International Conference of Soft Computing andPattern Recognition (SoCPaR 2010), December 7-10, 2010., Cergy Pontoise / Paris,France, IEEE (2010) 51–56.

[7] Blamey, S.: Partial logic In Gabbay, D. M., Guenther, F. eds.: Handbook of Philo-sophical Logic, 2nd Edition, Volume 5. Kluwer Academic Publishers, (2002) 261–353.

90

Fuzzy Type Theories: What They Are ForVilém Novák∗

Mathematical fuzzy logic is a well established formal tool that can be effectively appliedin modeling of human reasoning. Since the latter relies on natural language, it is affectedby the vagueness phenomenon. Fuzzy logic provides a well working model of vagueness viadegree theoretical approach. Besides various kinds of propositional and first-order calculi,also higher-order fuzzy logic calculi were developed. In analogy with classical logic, theyare called fuzzy type theories (FTT).

FTT is a generalization of classical type theory introduced by A. Church [2] andL. Henkin [9]; for detailed description of classical type theory, see [1]. Our generalizationconsists especially in replacement of the axiom stating “there are just two truth values”by a sequence of axioms characterizing structure of the algebra of truth values. The truthvalues should form an ordered structure belonging to a class of lattices having several morespecific properties.

The most general class of algebras specially developed to become algebras of truthvalues for FTT are EQ-algebras E = ⟨E,∧,⊗,∼,1⟩ where ∧ is the meet, ⊗ is a multi-plication (a possibly a non-commutative monoidal operation), and ∼ is a fuzzy equality(equivalence). Implication in EQ-algebras is a derived operation defined by

a→ b = (a ∧ b) ∼ a

and so, it is not tied with ⊗. Precise definitions and properties of EQ-algebras are presentedin [23, 7, 8]. If we add the adjunction condition

a⊗ b ≤ c ⇐⇒ a ≤ b→ c,

the bottom element 0 and lattice structure then EQ-algebra reduces to a residuated latticeE = ⟨L,∨,∧,⊗,→,0,1⟩. It can be demonstrated that each residuated lattice is an EQ-algebra but not vice-versa.

Recall that essential role in fuzzy logic is played by MTL-algebras which are prelinearresiduated lattices, i.e. residuated lattices fulfilling the condition (a→ b)∨ (b→ a) = 1. Aspecial operation indispensable for FTT is ∆-operation which, in case of linearly orderedE , can be defined by

∆(a) =

1 if a = 1,

0 otherwise.

Syntax of FTT is a generalization of the lambda-calculus constructed in a classicalway, but differing from it by definition of additional special connectives, and by logical

∗Centre of Excellence IT4Innovations, division University of Ostrava, Institute for Research and Appli-cations of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic, [email protected]

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axioms. The fundamental connective in FTT is that of a fuzzy equality ≡ consideredbetween objects of all types. For truth values, it is interpreted by the operation ∼ (if E isa residuated lattice then by the biresiduation a ↔ b = (a → b) ∧ (b → a)) of the algebraof truth values. For elements of the other types it is interpreted by a reflexive, symmetricand ⊗-transitive binary fuzzy relation.

The first version of FTT was developed for an IMTL∆-algebra of truth values, i.e. theMTL-algebra extended by ∆ and keeping the law of double negation (¬¬a = a) (see [15]).The other distinguished algebra of truth values for FTT is the standard Łukasiewicz∆MV-algebra L = ⟨[0, 1],∨,∧,⊗,→, 0, 1,∆⟩. FTT based on EQ∆-algebra was introducedin [20]. Some other extensions including Łukasiewicz∆-algebra are presented, e.g. in [14].

FTT has a lot of interesting properties. Besides various special formal properties, thegeneralized completeness theorem holds for all kinds of FTT. In [21], extension of FTTenabling to deal with partial functions was introduced. The partiality is realized by meansof special additional value # (undefined) introduced for all types (in case of truth values,#o is identified with ⊥ — the falsity).

Model theory for FTT is extension of that for first-order fuzzy logic. However, it hasinteresting specificities which open various till now unsolved questions (cf. [22]).

One of the main motivations for introducing FTT is to provide a well established formalsystem on the basis of which, the so called FLb-logic (fuzzy logic in broader sense) couldbe developed. The FLb-logic was announced in 1995 in the paper [13]. Its objective was todevelop a formal theory of human reasoning which would include mathematical models ofthe meaning of special expressions of natural language and of generalized quantifiers withregard to presence of vagueness. It should be noted that the paradigm of FLb-logic overlapswith paradigms of the theory of commonsense reasoning (for an overview of its recentdevelopment see [3]) and of much later introduced concept of precisiated natural languageannounced in [28]. The concept of FLb is a sort of glue between the two paradigms thattakes the best of each. It has been slowly developed over the years and at present, FLbconsists of the following theories1:

(a) Formal theory of evaluative linguistic expressions, see [17, 18]).

(b) Formal theory of fuzzy/linguistic IF-THEN rules and human-like approximate rea-soning, see [4, 6, 16, 24, 25, 26].

(c) Formal theory of intermediate and generalized quantifiers, and their syllogisms, see[5, 10, 12, 19].

In this contribution, we will overview state of the art of FTT and present some of thenew results mentioned above. We will also demonstrate that FTT is a reasonable tool fordeveloping models of human reasoning.

1There are some other papers whose topics is related to the topic of FLb (cf. [11, 27]). None of them,however, can be considered as a contribution to the consistent development of FLb as a logical theory.

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References

[1] P. Andrews, An Introduction to Mathematical Logic and Type Theory: To TruthThrough Proof, Kluwer, Dordrecht, 2002.

[2] A. Church, A formulation of the simple theory of types, J. Symb. Logic 5 (1940) 56–68.

[3] E. Davis, L. Morgenstern, Introduction: Progress in formal commonsense reasoning,Artifical Intelligence 153 (2004) 1–12.

[4] A. Dvořák, V. Novák, Fuzzy logic deduction with crisp observations, Soft Computing8 (2004) 256–263.

[5] A. Dvořák, M. Holčapek, L-fuzzy quantifiers of the type ⟨1⟩ determined by measures,Fuzzy Sets and Systems 160 (2009) 3425–3452.

[6] A. Dvořák, V. Novák, Formal theories and linguistic descriptions, Fuzzy Sets andSystems 143 (2004) 169–188.

[7] M. El-Zekey, Representable good EQ-algebras, Soft Computing 14 (2009) 1011–1023.

[8] M. El-Zekey, V. Novák, R. Mesiar, On good EQ-algebras, Fuzzy Sets and Systems 178(2011) 1–23.

[9] L. Henkin, Completeness in the theory of types, J. Symb. Logic 15 (1950) 81–91.

[10] M. Holčapek, Monadic L-fuzzy quantifiers of the type ⟨1n, 1⟩, Fuzzy Sets and Systems159 (2008) 1811–1835.

[11] E. Kerre, M. De Cock, Linguistic modifiers: an overview, in: J. Martinez (ed.), FuzzyLogic and Soft Computing, Kluwer Academic, Boston, 1999, pp. 69–86.

[12] P. Murinová, V. Novák, A formal theory of generalized intermediate syllogisms, FuzzySets and Systems.

[13] V. Novák, Towards formalized integrated theory of fuzzy logic, in: Z. Bien, K. Min(eds.), Fuzzy Logic and Its Applications to Engineering, Information Sciences, andIntelligent Systems, Kluwer, Dordrecht, 1995, pp. 353–363.

[14] V. Novák, Fuzzy type theory as higher order fuzzy logic, in: Proc. 6th Int. Confer-ence on Intelligent Technologies (InTech’05), Dec. 14-16, 2005, Fac. of Science andTechnology, Assumption University, Bangkok, Thailand, 2005.

[15] V. Novák, On fuzzy type theory, Fuzzy Sets and Systems 149 (2005) 235–273.

[16] V. Novák, Perception-based logical deduction, in: B. Reusch (ed.), ComputationalIntelligence, Theory and Applications, Springer, Berlin, 2005, pp. 237–250.

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[17] V. Novák, Mathematical fuzzy logic in modeling of natural language semantics, in:P. Wang, D. Ruan, E. Kerre (eds.), Fuzzy Logic – A Spectrum of Theoretical &Practical Issues, Elsevier, Berlin, 2007, pp. 145–182.

[18] V. Novák, A comprehensive theory of trichotomous evaluative linguistic expressions,Fuzzy Sets and Systems 159 (22) (2008) 2939–2969.

[19] V. Novák, A formal theory of intermediate quantifiers, Fuzzy Sets and Systems159 (10) (2008) 1229–1246.

[20] V. Novák, EQ-algebra-based fuzzy type theory and its extensions, Logic Journal ofthe IGPL 19 (2011) 512–542.

[21] V. Novák, Fuzzy type theory, descriptions, and partial functions, in: S. Galichet,J. Montero, G. Mauris (eds.), Proc. 7th conf. of the European Society for Fuzzy Logicand Technology (EUSFLAT-2011) and LFA-2011, Atlantis Press, Amsterdam, 2011.

[22] V. Novák, Elements of model theory in higher order fuzzy logic, Fuzzy Sets andSystems.

[23] V. Novák, B. de Baets, EQ-algebras, Fuzzy Sets and Systems 160 (2009) 2956–2978.

[24] V. Novák, A. Dvořák, Formalization of commonsense reasoning in fuzzy logic inbroader sense, Journal of Applied and Computational Mathematics 10 (2011) 106–121.

[25] V. Novák, S. Lehmke, Logical structure of fuzzy IF-THEN rules, Fuzzy Sets andSystems 157 (2006) 2003–2029.

[26] V. Novák, I. Perfilieva, On the semantics of perception-based fuzzy logic deduction,International Journal of Intelligent Systems 19 (2004) 1007–1031.

[27] M. Ying, B. Bouchon-Meunier, Quantifiers, modifiers and qualifiers in fuzzy logic,Journal of Applied Non-Classical Logics 7 (1997) 335–342.

[28] L. A. Zadeh, Precisiated natural language, AI Magazine 25 (2004) 74–91.

94

Modal Twist-Structures over Residuated LatticesHiroakira Ono Umberto Rivieccio∗

One of the latest and most challenging trends of research in non-classical logics is theattempt to combine different non-classical approaches together, for instance many-valuedand modal logic [9, 10]. Such interaction offers the advantage of dealing with modal notionslike belief, knowledge, obligations, in connection with other aspects of reasoning that canbe best handled using many-valued logics, for instance vagueness [12, 5] and inconsistency.If the aim is to model human reasoning, it is obvious that all these aspects have to be dealtwith at the same time, therefore such study is also especially interesting from the point ofview of theoretical computer science and AI.

One of the best-known logical systems proposed for handling inconsistent and alsopartial information is the Belnap-Dunn logic [8, 2, 3]. This logic is based on four truthvalues, which can be thought of as the two classical ones plus two additional values meant torepresent, respectively, lack of information and inconsistency (see the famous interpretationproposed by Belnap [2]). Such a simple approach, later on generalized by Ginsberg [11]with the notion of bilattice, proved to be very flexible and has been widely applied indifferent areas of computer science. In [15] Odintsov and Wansing proposed a modalversion of the Belnap-Dunn logic that aims at extending Belnap’s treatment of partialityand inconsistency to the modal setting. In [16] this approach is taken a step further,introducing a modal version of paraconsistent Nelson logic [1] that can also be regarded asa generalization of Odintsov and Wansing’s.

In the present paper we adopt an even more general approach, introducing a class ofalgebras that can be used as a semantics for paraconsistent and many-valued modal logics,encompassing as special cases both the work done in [15] and in [16]. In our attempt to beas general as possible, it is our hope to lay a theoretical framework that can be used forany future study of paraconsistent modal logics.

We first introduce through a concrete construction the algebraic structures we areinterested in, then we show that such class of algebras can be abstractly presented as avariety. Finally, we discuss the logical counterpart of such algebraic semantics.

Definition 1. A modal residuated lattice is an algebra L = ⟨L,⊓,⊔, ·, \, /, f, g, 1⟩ of type⟨2, 2, 2, 2, 2, 1, 1, 0⟩ such that:

(i) ⟨L, ·, 1⟩ is a monoid

(ii) ⟨L,⊓,⊔⟩ is a lattice with associated order ⊑

(iii) for all a, b, c ∈ L: a · b ⊑ c iff b ⊑ a\c iff a ⊑ c/b

(iv) f, g are monotone, i.e., a ⊑ b implies f(a) ⊑ f(b) and g(a) ⊑ g(b) for all a, b ∈ L.∗Research Center for Integrated Science, Japan Advanced Institute of Science and Technology, Nomi,

Ishikawa, Japan, [email protected], [email protected]

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The following construction was inspired by the ones introduced in [15] and [13] (seealso [17, Theorem 3.3] and [6, Theorem 2.3]).

Definition 2. Let L = ⟨L,⊓,⊔, ·, \, /, f, g, 1⟩ be a modal residuated lattice. The full modaltwist-structure over L is the algebra

L = ⟨L× L,∧,∨,⊃,⊂,,¬, ⟨1, 1⟩⟩

of type ⟨2, 2, 2, 2, 1, 1, 0⟩ with operations defined, for all ⟨a1, a2⟩ , ⟨b1, b2⟩ ∈ L×L, as follows:

⟨a1, a2⟩ ∧ ⟨b1, b2⟩ := ⟨a1 ⊓ b1, a2 ⊔ b2⟩⟨a1, a2⟩ ∨ ⟨b1, b2⟩ := ⟨a1 ⊔ b1, a2 ⊓ b2⟩⟨a1, a2⟩ ⊃ ⟨b1, b2⟩ := ⟨a1\b1, b2 · a1⟩⟨a1, a2⟩ ⊂ ⟨b1, b2⟩ := ⟨a1/b1, b1 · a2⟩

⟨a1, a2⟩ := ⟨f(a1), g(a2)⟩¬ ⟨a1, a2⟩ := ⟨a2, a1⟩ .

A modal twist-structure over L is an arbitrary subalgebra A (w.r.t. to the language ∧,∨,⊃,⊂,,¬, ⟨1, 1⟩ such that π1(A) = L, where π1(A) = a1 ∈ L : ⟨a1, a2⟩ ∈ A. We writeA ⊆ L to mean that A is a twist-structure over L.

From the definition it follows immediately that the reduct ⟨A,∧,∨⟩ of any twist struc-ture A ⊆ L is a lattice whose order ≤ is given by

⟨a1, a2⟩ ≤ ⟨b1, b2⟩ iff a1 ⊑ b1 and b2 ⊑ a2

for all ⟨a1, a2⟩, ⟨b1, b2⟩ ∈ L× L. Moreover, the negation ¬ is an order-reversing involutionon ⟨A,∧,∨⟩.

We list some remarkable properties of modal twist-structures:

• The class of modal twist-structures can be finitely presented as a variety of algebrasin the language ⟨∧,∨,⊃,⊂,,¬, e⟩ of type ⟨2, 2, 2, 2, 1, 1, 0⟩.

• For any modal twist-structure A, defining, for all a, b ∈ A,

a→ b := (a ⊃ b) ∧ (¬a ⊂ ¬b)a← b := ¬a→ ¬ba ∗ b := ¬(b→ ¬a)

we have that ⟨A, ∗⟩ is a semigroup and → and ← are, respectively, the right and leftresidual of ∗ w.r.t. the lattice order ≤.

• If L satisfies x · y ≈ x ⊓ y then A ⊆ L is an MN4-lattice as defined in [16]. Infact, MN4-lattices coincide precisely with the ⟨1, 1⟩-free subreducts of modal twist-structures over modal residuated lattices satisfying the equation x · y ≈ x ⊓ y.

• If L is a modal algebra [7], then A ⊆ L is a BK-lattice as defined in [15]. In fact,BK-lattices coincide precisely with the ⟨1, 1⟩-free subreducts of modal twist-structuresover modal algebras.

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• In general, it is possible to associate to any sub-quasi-variety of residuated latticesa quasi-variety of twist-structures in a canonical way. Along the same lines of thework done in [13] for bilattices, we introduce an algorithmic procedure that allowsto obtain an axiomatization of any quasi-variety of twist-structure starting from anaxiomatization of the corresponding sub-quasi-variety of residuated lattices.

We introduce a modal logic T = ⟨Fm, |=T ⟩ as the syntactic counterpart of modaltwist-structures in the following way. For any modal twist-structure A ⊆ L, we considerthe set D ⊆ A of designated elements defined by

DA := ⟨a1, a2⟩ ∈ A : 1 ⊑ a1.

We then define T as the consequence relation in the language ⟨∧,∨,⊃,⊂,,¬, e⟩ associatedto the class of logical matrices of the form ⟨A, DA⟩ where A is an arbitrary modal twiststructure. That is, for all formulas Γ ∪ φ ⊆ Fm, we set Γ |=T φ if and only if, for anytwist-structure A and any homomorphism h : Fm→ A from the formula algebra into A,it holds that h[Γ] ⊆ DA implies h(φ) ∈ DA.

Notice that T is paraconsistent in that it can happen that h(φ) = h(¬φ) ∈ DA. It isalso a global consequence relation, in the sense that it validates the following rule:

p ⊃ qp ⊃ q

We show how to axiomatize T and prove that it is algebraizable (in the sense of [4]) withrespect to the class of modal twist structures.

We also show that the constructions and results mentioned above can easily be adaptedto the setting of non-modal twist structures, both on a logical and on an algebraic level,thus generalizing previous work done on twist-structures over Heyting algebras (see forinstance [14, 17, 6]).

Finally, let us mention some further problems and possible developments:

• Study modal twist-structures from a universal algebraic point of view. For instance,characterize the congruences of any modal twist-structure in terms of those of itslattice factor, along the same line of the work done on bilattices in [13].

• Investigate the local consequence relation associated with the logic of modal twist-structures T .

• Extend the modal twist-structure construction described above in order to add modaloperators to the residuated bilattices of [13].

• Introduce and study extensions of T that have a special interest for particular ap-plications, whose semantics can be provided by sub-quasi-varieties of modal twist-structures. For this we can use the procedure mentioned above for obtaining anaxiomatization of quasi-varieties of twist-structures from a presentation of the corre-sponding class of residuated lattices.

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References

[1] A. Almukdad and D. Nelson. Constructible falsity and inexact predicates. The Journalof Symbolic Logic, 49(1):231–233, 1984.

[2] N. D. Belnap. How a computer should think. In G. Ryle, editor, Contemporary Aspectsof Philosophy, pages 30–56. Oriel Press, Boston, 1976.

[3] N. D. Belnap, Jr. A useful four-valued logic. In J. M. Dunn and G. Epstein, ed-itors, Modern uses of multiple-valued logic (Fifth Internat. Sympos., Indiana Univ.,Bloomington, Ind., 1975), pages 5–37. Episteme, Vol. 2. Reidel, Dordrecht, 1977.

[4] W. J. Blok and D. Pigozzi. Algebraizable logics, volume 396 of Mem. Amer. Math.Soc. A.M.S., Providence, January 1989.

[5] F. Bou, F. Esteva, L. Godo, and R. Rodríguez. On the minimum many-valued modallogic over a finite residuated lattice. Journal of Logic and Computation, 2011. Toappear.

[6] M. Busaniche and R. Cignoli. Residuated lattices as an algebraic semantics for para-consistent Nelson’s logic. Journal of Logic and Computation, 19(6):1019–1029, 2009.

[7] A. Chagrov and M. Zakharyaschev. Modal Logic, volume 35 of Oxford Logic Guides.Oxford University Press, 1997.

[8] J. M. Dunn. The algebra of intensional logics. Ph. D. Thesis, University of Pittsburgh,1966.

[9] M. Fitting. Many-valued modal logics. Fundamenta Informaticae, 15:235–254, 1992.

[10] M. Fitting. Many-valued modal logics, II. Fundamenta Informaticae, 17:55–73, 1992.

[11] M. L. Ginsberg. Multivalued logics: A uniform approach to inference in artificialintelligence. Computational Intelligence, 4:265–316, 1988.

[12] P. Hájek and D. Harmancová. A many-valued modal logic. In Proceedings IPMU’96.Information Processing and Management of Uncertainty in Knowledge-Based Systems,pages 1021–1024, Granada, 1996. Universidad de Granada.

[13] R. Jansana and U. Rivieccio, Residuated bilattices. Soft Computing, 2011, DOI10.1007/s00500-011-0752-x.

[14] S. P. Odintsov. On the representation of N4-lattices. Studia Logica, 76(3):385–405,2004.

[15] S. P. Odintsov and H. Wansing. Modal logic witn Belnapian truth values. Journal ofApplied Non-Classical Logics, 20:279–301, 2010.

[16] U. Rivieccio. Paraconsistent modal logics. Submitted to 7th Methods for Modalitiesworkshop, Osuna, Spain, 10-12 November 2011.

[17] C. Tsinakis and A. M. Wille. Minimal varieties of involutive residuated lattices. StudiaLogica, 83(1-3):407–423, 2006.

98

Elementary Submodels of Kripke ModelsTomasz Połacik∗

Informally, a Kripke model can be viewed as a structure comprising a collection of first-order classical structures, called the worlds of the model, and (weak) homomorphismsdetermining the accessibility relation between the worlds. The attractive possible-worldinterpretation on one hand, and strong connections with classical model theory on theother, make Kripke models an interesting and powerful tool in semantical investigationsof constructive first-order theories. It turns out that many important problems, such asindependence or non-provability, can be successfully tackled with use of Kripke semantics.In the existing literature there are many results of this kind concerning particular con-structive first-order theories such as first-order intuitionistic arithmetic and intuitionisticset theory, to mention only these two prominent examples. However, the general theoryof Kripke models is still not well developed, and general methods of construction Kripkemodels of constructive theories are not well understood. This contrasts with status of theresearch in classical model theory which offers many deep general results and interestingtools for construction various models of classical first-order theories.

Obviously, the notion of Kripke model, as described above, may be viewed as a gen-eralization of that of classical first-order structure, since any first-order structure (withthe identity homomorphism) can be viewed as the simplest example of a Kripke model.This fact motivates research in the theory of Kripke models viewed as a counterpart ofclassical model theory. In the development of this theory it seems to be natural to acceptthe following basic postulates: first, all the notions introduced for Kripke models shouldcoincide in the simplest case with the standard model-theoretic notions; moreover, theresults obtained for Kripke models applied to the simplest model should coincide with theresults of classical model theory. We will present a contribution to Kripke model theoryunderstood in this way.

One of the most basic notions that we encounter in model theory is that of substructure.As was observed by A. Visser in [6], this notion can defined for Kripke models in severaldifferent ways. We can either consider models on the same frame, where the worlds of sub-models are substructures of the worlds of the original model, or we can define a submodelto be the result of restricting the frame of the given model, or we can combine both of theseoperations. All of these possibilities were considered in the literature, see [1], [6] and [2],respectively. But if we agree to accept the postulates presented above, it seems that weshould accept the third notion as the correct one. There are at least to reasons for thischoice. First, this notion of submodel coincides with the classical notion of substructurein the case of the simplest Kripke model. Moreover, the well-known classical Tarski-Łośpreservation theorem becomes a particular case of the result proven in [2], stating that

∗University of Silesia, Katowice, Poland, [email protected]

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the class of the formulas that are preserved under Kripke submodels is exactly the class ofuniversal formulas.

After accepting the definition of submodel of a Kripke model, in a natural way we canask about elementary submodels. However, one can easily notice that the operation ofrestricting the frame changes, in general, the theory of the given Kripke model since, forexample, any one-node submodel of a given Kripke model always validates the Principleof Excluded Middle. Furthermore, as was noticed in [1], even when we consider submodelsover the same frame as the given Kripke model K, whose worlds are elementary substruc-tures (in the classical sense) of the corresponding worlds of K, we do not necessarily getan elementary Kripke submodel of K. As a simple example consider two classical struc-tures M and N that are models of true arithmetic, and a homomorphism f : M→ N whichis not elementary. Now, let K be the Kripke model whose worlds are M and N and whoseonly morphism is f . Since the standard model of arithmetic N is a prime model of truearithmetic, it is an elementary substructure of the models M and N. So, the model N ,consisting of two copies of the standard model and identity as a morphism, is a submodelof K. But, of course, N validates the classical theory of true arithmetic, while K does not.So, although its worlds are elementary substructures of the corresponding worlds of K, themodel N cannot be elementary submodel of K. These examples show clearly that if wewant to construct an elementary submodel of a given Kripke model, we must take intoaccount also the properties of the morphisms of the model in question.

From the characterization of partially elementary models given in [4], it follows thatthe root any Γ-elementary extension Kripke model K, viewed as the submodel of K, is itsΓ-elementary submodel. Of course, the models considered force strong intuitionisticallynot valid principles including instances of the Principle of Excluded Middle. In the sequel,we will present examples of elementary submodels that need not force any instances of thePrinciple of Excluded Middle and any other non-intuitionistic principles.

In order to present the details, we recall basic definitions and introduce some auxiliarynotions that are needed to present our result. We fix a first-order language L and assumethat all the models considered are models for L.

Following [6], we define a Kripke model as a functor K from a small category A to thecategory M of classical first-order structures with (weak) homomorphisms as its arrows.We write α→f α′ if f is an arrow between α and α′. To keep the notation simple, we willconfuse an arrow f with its corresponding homomorphism K(f). For each object α of A,K(α) is a world of the model K. The forcing relation K on K is defined in the usual way.

Now we can give definitions of a submodel and elementary submodel of the given Kripkemodel. Let K : A → M and M : B → M be Kripke models. We say that K is a submodelofM if A is a subcategory of B and for every α in A, the world K(α) is a substructure ofthe worldM(α). For the sake of simplicity, we state the definition of elementary submodelfor rooted models only. Given two Kripke models K and M with the roots κ and µ,respectively. We say that K is an elementary submodel of M if

• K is a submodel ofM, and

• κ K A[a] iff µ M A[a], for every sequence a from K(κ), and for every formulaA(x).

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To state our result, we have to introduce an auxiliary notion. For simplicity, we refer itto models over the same frame. Let us consider Kripke models K : A→M andM : A→M.A pair (a ; b) of sequences of elements of K(α) and and of M(α) is an elementary map iffor every formula A(x) we have K(α) |= A [a] if and only if M(α) |= A [b]. We saythat an elementary map (a ; b) between the classical structures K(α) andM(α) is upwardspreserved iff for every α→f α′, (fa ; gb) is an elementary map between K(α′) andM(α′).

Now we can state the promised theorem.

Theorem 1. Let K : A → M and M : A → M be rooted Kripke models over the sameframe. Additionally, assume that

(1) for every α, the structure K(α) is an elementary substructure of M(α),

(2) for every α, all elementary maps between K(α) and M(α) are upwards preserved,

(3) all the worlds of K and M are ω-saturated.

Then the Kripke model K is an elementary submodel of the Kripke model M.

Note that the assumption that the Kripke models are built over the same frame isnot necessary and can be weakened. Let us comment on other assumptions of the theo-rem. The assumption (1) is obvious. The condition (1) describes an anticipated property ofmorphisms. As we already know, a condition of this form is necessary. Finally, the assump-tion (1) is crucial for proving the existence of required bisimulations between the model Kand its submodel, and in consequence, that the submodel in question is elementary. Wedo not know whether this assumption can be weakened or waved.

Note also that, by a suitable relativization of the assumptions, our theorem can begeneralized to a theorem concerning Γ-elementary submodels. Other generalizations andvariants are also possible.

References

[1] S. M. Bagheri and M. Moniri. Some results on Kripke models over an arbitrary fixedframe. Mathematical Logic Quarterly, 49:479–484, 2003.

[2] B. Ellison, J. Fleischmann, D. McGinn, and W. Ruitenburg. Kripke submodels anduniversal sentences. Mathematical Logic Quarterly, 53(3):311–320, 2007.

[3] W. Hodges. A Shorter Model Theory. Cambridge University Press, Cambridge, 1997.

[4] T. Połacik. Partially-elementary extension Kripke models: A characterization andapplications. The Logic Journal of the IGPL, 14(1):73–86, 2006.

[5] T. Połacik. Back and forth between first-order Kripke models. The Logic Journal ofthe IGPL, 16(4):335–355, 2008.

[6] A. Visser. Submodels of Kripke models. Archive for Mathematical Logic, 40(277–295),2001.

101

A Partial Characterization of Axiomatizability ofFirst-Order Logics of Linear Kripke Frames over

Constant DomainsNorbert Preining∗

1 Introduction

Kripke frames as possible semantics for modal logics were introduced by S. A. Kripke in thelate fifties and early sixties. While the origin of this notion is disputable, the influence ofthe possible world interpretation has been enormous. This new type of semantics providedan attractive model theory that seemed more manageable than the previous algebra-basedsemantics. One of the reasons for its early success was that well known logical systems,like S4, S5 and Intuitionistic Predicate Logic, were shown to be characterised by naturalfirst-order properties of their frames. Kripke himself in [6] used these frames to prove thecompleteness of Intuitionistic Predicate Logic. For a more detailed presentation of theseand related topics see [3, 4].

Detailed studies of intermediate predicate logics based on linear Kripke frames havebeen carried out by several researchers. For example, the general structure of linear Kripkeframes and their logics is discussed in [8], the logics defined by Kripke frames determinedby ordinals on constant domains are analysed in [7], and the logics based on Kripke framesR and Q with constant domains are determined in [11].

An open question in this area is which Kripke frames over constant domains give rise toaxiomatizable logics. Partial results have been obtained in the past (as mentioned above),but a full characterization is still missing.

In this extended abstract we add a few chapters to this book by showing that certainclasses of Kripke frames give rise to non-recursively enumerable logics, narrowing down theclass of possibly axiomatizable logics.

2 Logics of Kripke frames with constant domains

Let L be a countable first-order language which includes the propositional constant ⊥. Fora set U , let LU denote the extended language with constants for all elements of U . Theset of all closed atomic formulas of LU is denoted by A(LU ), the set of all sentences of LUis denoted by S(LU ).

∗Universita degli Studi di Siena, Dipartimento di Scienze Matematiche ed Informatiche “Roberto Mag-ari”, E-mail: [email protected]

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A Kripke frame is a partial order K = (W,R). In the following we will only considerlinear Kripke frames, characterized by the linearity axiom (A → B) ∨ (B → A), i.e. totalorders, and we will denote the order by ≼. Furthermore, we will only consider logicsdefined by linear Kripke frames on constant domains, where frames with constant domainsare characterized by the quantifier shift axiom ∀x(A ∨ B(x)) → (A ∨ ∀xB(x)) where xmust not occur in A. Elements of Kripke frames will also be called worlds.

With the following definitions we adapt the standard definitions (which can be found in[9] for example) to the special case of Kripke models based on linear frames with constantdomains.

Definition 1. Let K = (W,≼) be a linear Kripke frame. For any non-empty set U , thetuple (K,U) is called a Kripke model. A relation val(K,U) ⊆ W × A(LU ) is said to be avaluation of the Kripke model (K,U) if val(K,U) satisfies the following conditions: For allw, w1 and w2 in W it holds that w1 ≼ w2 and val(K,U)(w1, A) implies val(K,U)(w2, A), andthat val(K,U)(w,⊥) does not hold.

The valuation val(K,U) can be extended to a relation on W × S(LU ) inductively:

val(K,U)(w,A ∧B) iff val(K,U)(w,A) and val(K,U)(w,B)

val(K,U)(w,A ∨B) iff val(K,U)(w,A) or val(K,U)(w,B)

val(K,U)(w,A→ B) iff val(K,U)(v,A) implies val(K,U)(v,B)

for any v with w ≼ vval(K,U)(w, ∀xA(x)) iff val(K,U)(w,A(u)) for any u ∈ Uval(K,U)(w, ∃xA(x)) iff val(K,U)(w,A(u)) for some u ∈ U .

Definition 2. The logic defined by a linear Kripke frameK = (W,≼) on constant domains,denoted by L(K), is the set of all L-formulas A such that for all Kripke models (K,U), allvaluations val(K,U) of (K,U), and all worlds w ∈ W , val(K,U)(w,A

′) holds, where A′ is aclosure of A.

Logics of linearly ordered Kripke frames in general have been studied, e.g., by Ono [8].Some special cases of logics defined by linearly ordered Kripke frames on constant domainshave also been considered in the literature: for example, L(Q) and L(R) in [11], and L(α)for ordinals α in [7].

We continue by defining logics based on complete linear rings of sets, which will beused as a turning point between logics of Kripke frames and Gödel logics. They form aspecial case of logics defined by complete pseudo-Boolean algebras (also called completeHeyting algebras) (cf. [9]).

Definition 3. Let S be a complete linear ring of sets. Recall that 1 =∪

S and 0 =∩

S .A tuple (S , U) is called a model based on S and U if U is a non-empty set. An assignmentval(S ,U) of (S , U) is a mapping from A(LU ) to S with val(S ,U)(⊥) = 0.

The assignment val(S ,U) can be extended to a function from S(LU ) to S inductively inthe usual way using union and intersection.

Following [9] we define the logic based on a complete linear ring of sets.

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Definition 4. The logic defined by a complete linear ring of sets S , denoted by L(S ), isthe set of L-formulas A such that for all models based on S and some non-empty set Uand all assignments val(S ,U), val(S ,U)(A

′) = 1 holds, where A′ is a closure of A.

The equivalence of Kripke models based on valuations and Kripke models based oncorresponding upsets (or ‘propositions’) is well known (cf. [3, Chapter IX] and [5]). Usingthis equivalence we obtain the following Lemma.

Lemma 5. Let K = (W,≼) be a linear Kripke frame and Up(K) the induced completelinear ring of sets. Then L(K) = L(Up(K)).

In the following we will use only mappings into complete sets of rings, i.e., into upsetsof a Kripke frame to defined valuations.

3 Scattered linear Kripke frames with constant domain

A Kripke frame is called scattered if Q cannot be embedded into the Kripke frame (aslinear order).

We will show that the logic of any scattered linear Kripke frame with constant domainis non recursively enumerable by reducing classical validity of a formula in all finite modelsto the validity of a formula in the given logic. This is an extension of the proof given in [2],transfered for Kripke frames and extended to arbitrary linear Kripke frames.

Definition 6. A formula is called crisp if all occurrences of atomic formulas are eithernegated or double-negated.

Lemma 7. If A and B are crisp and classically equivalent, then also L(K) |= A ↔ B.Specifically, if A(x) and B are crisp, then

|= ∀xA(x)→ B ↔ ∃x(A(x)→ B) and|= B → ∃xA(x)↔ ∃x(B → A(x)).

Proof. Given an interpretation val, define val′(C) = K if val(C) = ∅ and = ∅ if val(C) = ∅for atomic C. It is easily seen that if A, B are crisp, then val(A) = val′(A) and val(B) =val′(B). But val′ is a classical interpretation, so by assumption val′(A) = val′(B).

Theorem 8. If K is scattered, then L(K) is not recursively enumerable.

Proof. We show that for every sentence A there is a sentence AK s.t. AK is valid in L(K)iff A is true in every finite (classical) first-order structure.

We define AK as follows: Let P be a unary and L be a binary predicate symbolnot occurring in A and let Q1, . . . , Qn be all the predicate symbols in A. We use theabbreviations x ∈ y ≡ ¬¬L(x, y) and x ≺ y ≡ (P (y)→ P (x))→ P (y). Note that for anyinterpretation val, val(x ∈ y) is either ∅ or K, and as long as val(P (x)) = K for all x (inparticular, if val(∃z P (z)) = K), we have val(x ≺ y) = K iff val(P (x)) ⊂ val(P (y)). LetAK ≡

S ∧ c1 ∈ 0 ∧ c2 ∈ 0 ∧ c2 ≺ c1 ∧∀i[∀x, y∀j∀k∃z D ∨ ∀x¬(x ∈ s(i))

] → (A′ ∨ ∃uP (u)) (1)

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where S is the conjunction of the standard axioms for 0, successor and ≤, with doublenegations in front of atomic formulas,

D ≡ (j ≤ i ∧ x ∈ j ∧ k ≤ i ∧ y ∈ k ∧ x ≺ y)→→ (z ∈ s(i) ∧ x ≺ z ∧ z ≺ y)

and A′ is A where every atomic formula is replaced by its double negation, and all quan-tifiers are relativized to the predicate R(i) ≡ ∃x(x ∈ i).

Intuitively, L is a predicate that divides a subset of the domain into levels, and x ∈ imeans that x is an element of level i. If the antecendent is true, then the true standardaxioms S force the domain to be a model of PA, which could be either a standard model(isomorphic to N) or a non-standard model (N followed by copies of Z). P orders theelements of the domain which fall into one of the levels in a subordering of the truthvalues.

The idea is that for any two elements in a level ≤ i there is an element in a not-emptylevel j ≥ i which lies strictly between those two elements in the ordering given by ≺. Ifthis condition cannot be satisfied, the levels above i are empty. Clearly, this condition canbe satisfied in an interpretation val only for finitely many levels if Up(K) does not containa dense subset, i.e., K is scattered, since if more than finitely many levels are non-empty,then

∪ival(P (d)) : val |= d ∈ i gives a dense subset. By relativizing the quantifiers in A

to the indices of non-empty levels, we in effect relativize to a finite subset of the domain.We make this more precise:

Suppose A is classically false in some finite structure val. W.l.o.g. we may assume thatthe domain of this structure is the naturals 0, . . . , n. We extend val to a K-interpretationvalK with domain N as follows: Since Up(K) contains infinitely many subsets, we canchoose c1, c2, L and P so that ∃x(x ∈ i) is true for i = 0, . . . , n and false otherwise,and so that valK(∃xP (x)) = K. The number-theoretic symbols receive their naturalinterpretation. The antecedent of AK clearly receives the valuation K, and the consequentreceives Ig(∃xP (x)) = K, so valK 2 AK .

Now suppose that val 2 AK . Then val(∃xP (x)) = K. In this case, val(x ≺ y) = K iffval(P (x)) ⊆ val(P (y)), so ≺ defines a strict order on the domain of val. It is easily seenthat in order for the value of the antecedent of AK under val to be greater than that ofthe consequent, it must be = K (the values of all subformulas are either ⊆ val(∃xP (x))or = K). For this to happen, of course, what the antecedent is intended to express mustactually be true in val, i.e., that x ∈ i defines a series of levels and any level i > 0 is eitherempty, or for all x, and y occuring in some smaller level there is a z with x ≺ z ≺ y andz ∈ i.

To see this, consider the relevant part of the antecedent, B = ∀i[∀x, y∀j∀k∃z D ∨

∀x¬(x ∈ i)]. If val(B) = K, then for all i, either val(∀x, y∀j∀k∃z D) = K or val(∀x¬(x ∈

i)) = K. In the first case, we have val(∃z D) = K for all x, y, j, and k. Now supposethat for all z, val(D) = K, yet val(∃z D) = K. Then for at least some z the value of thatformula would have to be ⊃ val(∃z P (z)), which is impossible. Thus, for every x, y, j, k,there is a z such that val(D) = K. But this means that for all x, y s.t. x ∈ j, y ∈ k withj, k ≤ i and x ≺ y there is a z with x ≺ z ≺ y and z ∈ i+ 1.

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In the second case, where val(∀x¬(x ∈ i)) = K, we have that val(¬(x ∈ i)) = K forall x, hence val(x ∈ i) = ∅ and level i is empty.

Note that the non empty levels can be distributed over the whole range of the non-standard model, but since Up(K) and K contains no dense subset, the total number of nonempty levels is finite. Thus, A is false in the classical interpretation valc obtained from valby restricting val to the domain i : ∃x(x ∈ i) and valc(Q) = val(¬¬Q) for atomic Q.

4 Discussion

Although the above results provides us with knowledge about a much larger class thenknown before, there are several directions we are planning to extended this result. Onthe one hand, the full characterization of axiomatizability is still missing. For countableKripke frames we can use the characterization of axiomatizability of Gödel logics givenin [2] together with the equivalence result given in [1] to obtain a full characterization ofaxiomatizability of countable linear Kripke frames on constant domains.

Combined with the above results what remains are non-scattered linear Kripke framesthat are not countable, thus containing a copy of ω1 (see [10]). Extending other resultsfrom [2] should clarify the image, but will not give a full characterization.

Another direction that seems to be tractable is to use methods similar to the above toattach general (non-linear) Kripke frames over constant domains.

References

[1] Arnold Beckmann and Norbert Preining. Linear Kripke frames and Gödel logics.Journal of Symbolic Logic, 71(1):26–44, 2007.

[2] Matthias Baaz, Norbert Preining, and Richard Zach. First-order Gödel logics. Annalsof Pure and Applied Logic, 147:23–47, 2007.

[3] Dov M. Gabbay. Semantical Investigations in Heyting’s Intuitionistic Logic, volume148 of Synthese Library. D. Reidel Publishing Company, 1981.

[4] Robert Goldblatt. Mathematical modal logic: A view of its evolution. J. Appl. Log.,1(5-6):309–392, 2003.

[5] Philip Kremer. On the complexity of propositional quantification in intuitionisticlogic. J. of Symbolic Logic, 62(2):529–544, June 1997.

[6] Saul A. Kripke. Semantical analysis of intuitionistic logic. I. In Formal Systemsand Recursive Functions (Proc. Eighth Logic Colloq., Oxford, 1963), pages 92–130.North-Holland, Amsterdam, 1965.

[7] P. Minari, M. Takano, and H. Ono. Intermediate predicate logics determined byordinals. Journal of Symbolic Logic, 55(3):1099–1124, 1990.

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[8] H. Ono. On finite linear intermediate predicate logics. Studia Logica, 47(4):391–399,1988.

[9] Hiroakira Ono. A study of intermediate predicate logics. Publ. Res. Inst. Math. Sci.,8:619–649, 1972/73.

[10] Joseph G. Rosenstein. Linear orderings, volume 98 of Pure and Applied Mathematics.Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.

[11] Mitio Takano. Ordered sets R and Q as bases of Kripke models. Studia Logica,46:137–148, 1987.

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The Role of Modal Logic in Formalizing AnalogicalReasoning∗

Dagmar Provijn†

In [5], D’Hanis proposed an adaptive logic approach for analyzing metaphors. In thispaper, a first attempt to formalize the analysis of analogical reasoning in the developmentand defense of scientific theories is presented. This formalization will be based on theinteractionist view on metaphors described in [5] and the prioritized adaptive logics aselaborated in [2]. From a general logico-philosophical point of view, ‘analogical reasoningwill refer to processes in which inferences are made based on certain similarities betweentwo domains which can be two objects, two classes of objects, . . .’ [7, pp. 24–25], or as ‘akind of reasoning that applies between specific exemplars or cases, in which what is knownabout one exemplar is used to infer new information about another exemplar’ [6, p. 106].In terms of D’Hanis’ interactionist view on metaphors, which is itself based on Barsalou’sdefinition of concepts from [1], we may consider analogies to contain a ‘primary subject’and a ‘secondary subject’ in such a way that information is transferred from the secondarysubject to the primary subject. The difference between metaphors and analogies is thedegree of accessibility of the information that constitutes a concept. In the analysis ofmetaphors, the secondary subject is used in a non-literal sense and the analysis consistsof transferring as much as possible of the easy accessible information that constitutesthe concept of the secondary subject to the primary subject. I agree with D’Hanis thatanalogies rather deal with the transfer of less accessible information, which requires actualreasoning processes based on context-bound information.

More specifically, what will be presented is a formalization of the analysis of analogiesof the form ‘all X’s are/behave like Y’s’. So, the paper will focus on the reasoning thatis involved in the analysis of analogies that interpret/expect specific classes of objectsto be like or to behave like some other classes of objects. Rather than applying non-literal predicates that are interpreted as literally as possible (a technique applied in [5]),analogies will be presented by means of a sequence of sets that contain a set of premisesand sets of expressions concerning the expected behavior of the primary subject basedon the properties of the second subject. Hence, the premise set will contain context-bound information on being X and being Y. In the first set of expected behavior, it willbe presupposed that being X shows the same properties as being Y. So, if the premiseset contains the formula (∀x)(Y x ⊃ Ax), then the second set (or another set that has alower preference or reliability) may contain (∀x)(Xx ⊃ Ax). The prioritized adaptive logicfrom [2] that will be applied to formalize this idea simply applies an adaptive strategy to

∗The author is Research Fellow of the Fund for Scientific Research – Flanders (FWO-Vlaanderen).†Centre for Logic and Philosophy of Science, Universiteit Gent, Belgium, [email protected]

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a sequence of sets, first to the set of premises, next to the set of most preferred (mostreliable) expectancies, etc.

Adaptive logics are formal logics showing the specific feature of ‘adapting themselvesto the premises’. They are developed to characterize defeasible forms of reasoning as thereare induction, abduction, compatibility, etc. An adaptive logic in standard format – asdefined in [4] – is characterized by a triple:

1. a lower limit logic LLL, roughly speaking a deductive Tarski-logic, being compactand for which there is a positive test,

2. a set of abnormalities Ω, characterized by a specific logical form,

3. and an adaptive strategy.

One of the most fascinating features of adaptive logics is their dynamic proof theory(rendering annotated proofs), allowing to explicate a person’s reasoning and especiallythose forms of reasoning that are undecidable and for which there is no positive test.Therefore the proof theory is necessarily dynamic. The LLL and the set of abnormalitiesΩ determine the inference rules of the proof system. Whenever an axiom is added to theLLL that connects the abnormalities to triviality, a logic is obtained that is called theupper limit logic ULL. Each line of an annotated dynamic proof has a possibly emptycondition, being a finite set of abnormalities, written at the end of the line. The rules ofLLL apply unconditionally, those of the ULL apply conditionally. The underlying idea isthat the formula derived at the line, is derived provided all members of the condition ofthe line behave normally.

The abnormalities Ω and the strategy settle the Marking definition. This Markingdefinition takes into consideration the lines that are derived at the stage of the proof.Whenever a formula of a condition of a line is derived, this line is considered as out;i.e. the formula of the line is not derived at that stage. Formulas of unmarked lines areconsidered as derived at the stage.

The adaptive logic from [2] that will be applied as a basic logic to deal with analogicalreasoning treats the premises and the expectancies (that incorporate a specific preferenceor reliability ranking) in a specific order. Γ0 will be the set of premises, Γi (i > 0) will besets of expectancies. In the latter, we will find the expressions on the expected behaviorof certain objects (the primary subject of the analogy), analogously to the behavior of theobjects referred to by the secondary subject. Consequences of members of the Γi (i > 0)will or will not be joined to Γ0. By ‘consequences’ we shall mean classical logic (CL)-consequences (if some member of Γi (i > 0) is inconsistent, the consequence set will betrivial). The clue to the whole enterprise is to interpret expectancies within the modallogic T of Feys (which is von Wright’s M). Σ ⊢T A will denote that iA | A ∈ Γi ⊢T A.Remember that each Γi is a set of closed formulas of L (the standard predicative language).So, the lower limit logic is T. The upper limit logic, say T+ is obtained by adding to Tthe axiom “A ⊃ A”.

The abnormalities are of the form ∃(iA ∧ ¬A); with all A ∈ Fa (Fa being the setof atoms, i.e. primitive formulas and their negations). In fact, as most often in adaptive

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logics, we will need to consider disjunctions of these abnormalities in which i is alwaysthe same number. By Dabi(∆) we denote the disjunction

∨∃(iA ∧ ¬A) | A ∈ ∆. A

Dabi-consequence Dabi(∆) of Σ will be called minimal iff there is no ∆′ ⊂ ∆ such thatDabi(∆′) is a Dabi-consequence of Σ.

The applied strategy will be the Reliability strategy. This strategy relies on the ideathat any n-tuple Σ defines a set of unreliable formulas. This set of unreliable formulas isdefined as the set of factors of the minimal Dab-consequences of Σ:

U i(Σ) =∪∆ | Dabi(∆) is a minimal Dabi-consequence of Σ

A formula A is derivable on the condition ∆i in a proof from Σ iff A ∨ Dabi(∆) isT-derivable from Σ.

It will be shown that the presented formalization of analogical reasoning can deal withthe analysis of different analogies at the same time. It does not matter whether they are ata same level of preference or not. The results of the formalization will also be confrontedwith some results from the research on analogical reasoning in the works of William Harveyin the context of the discovery of the blood circulation – [8].

References

[1] Lawrence W. Barsalou. Intra-concept similarity and its implications for inter-conceptsimilarity. In S. Vosniadou and A. Ortony , (eds.), Similarity and Analogocial Reason-ing. Cambridge University Press, Cambridge, pp. 76–121, 1989.

[2] Diderik Batens, Joke Meheus, Dagmar Provijn and Liza Verhoeven. Some AdaptiveLogics for Diagnosis. Logic and Logical Philosophy, 11/12, pp. 39-65, 2003.

[3] Diderik Batens. Content Guidance in Formal Problem Solving Processes. In O. Pombo,A. Gerner (eds.), Abduction and the Process of Scientific Discovery. Centro de Filosofiadas Ciências da U. de Lisboa, Lisboa, pp. 121–156, 2007.

[4] Diderik Batens. Adaptive Logics and Dynamic Proofs. Mastering the Dynamics ofReasoning. 201x. Forthcoming.

[5] Isabel D’Hanis. A Logical Approach to the Analysis of Metaphors. In L. Magnani,N.J. Nersessian & C. Pizzi, (eds.), Logical and Computational Aspects of Model-BasedReasoning. Kluwer Academic, Dordrecht, pp. 21–37, 2002.

[6] Dedre Gentner. Analogical Reasoning, Psychology of. In L. Nadel (ed.), Encyclopediaof Cognitive Science. Nature Publishing Group, London, pp. 106-112, 2003.

[7] Joke Meheus. Analogical Reasoning in Creative Problem Solving Processes: Logico-Philosophical Perspectives. In F. Hallyn (ed.), Metaphor and Analogy in the Sciences.Kluwer, Dordrecht, pp. 17-34, 2000.

[8] Dagmar Provijn. Bloody Analogical Reasoning. 201x.

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An Expansion of First-order Belnap-Dunn LogicKatsuhiko Sano∗ Hitoshi Omori†

1 Introduction and Motivation

One of the well-known systems of non-classical logics is the four-valued logic of Belnap andDunn, which was originally developed by Belnap [1, §81] and later examined in detail byDunn. The motivation of Belnap’s system is known to be lying in the question “How acomputer should think” which is the title of one of his articles [3]. As is known, two valuesin the semantics of classical logic are often read as true and false. But in the semantics ofthe system of Belnap and Dunn, there are four values, namely true only, false only, bothtrue and false, and neither true nor false.

There are several works which extend or generalize the system of Belnap and Dunn. Forexample, there are some works by Wansing and Shramko in which they consider a systemwith sixteen values in its semantics. This is a result of generalizing what Wansing callsthe “powerset formulation” [13, p. 369], the technique that can recapture the constructionof the four-valued logic of Belnap and Dunn from the two-valued classical logic. Anotherdirection of research gave birth to the theory of bilattice and even trilattice, which is basedon the algebraic aspect of the semantics of Belnap-Dunn logic.

Based on this, the purpose of the present paper is twofold. First, we try to expand theBelnap-Dunn logic, in the scope of predicate calculus, by enriching it with a new connectivewhose intended reading is ‘- is designated’ or ‘- has designated values’. We shall write theconnective to be added as and refer to the system to be developed as BD in thefollowing. The connective is not new since we can find it, for example, in works ofD’Ottaviano, Kachi and Baaz. Indeed, in the first two works [5] and [6], the connectiveis added to systems of three-valued logic and in the work [2] of Baaz the connective isconsidered in the context of infinite-valued Gödel logic which is a system of fuzzy logic.However, there seem to be no studies for the four-valued case of Belnap-Dunn logic, atleast to the best of authors’ knowledge. Therefore, this paper investigates both inthe semantics and the natural deduction axiomatization, and provides some basic resultsincluding the completeness result. The main theorem is actually a general completenessresult with respect to P( 0, 1 )-valued semantics, not restricted to the syntax of BD(see Theorem 7).

Second, we show that the addition of to Belnap-Dunn logic leads us to a unifiedperspective on the systems of three-valued logic with and their related four valuedsystems. More concretely, we obtain a simple natural deduction system for the system J3

∗School of Information Science, JAIST, [email protected]†JSPS and Graduate School of System Informatics, Kobe University, [email protected]

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of D’Ottaviano, and also a better formulation of the system SPL of Kachi. We alsorecapture the three-valued Łukasiewicz’s implication by natural deduction rules with thehelp of (see Corollary 10). Moreover, we know that (i) the system J3 is known tobe “inter-definable” with the system LFI1 of Carnielli, Marcos and de Amo [4, Remark3.2] having the different syntax from J3, i.e., having the implication symbol ⊃ and theconsistency operator instead of ; (ii) the system BD can be seen as a generalizedsystem of J3; (iii) there is a four-valued system BS4 of Omori and Waragai [9], whichkeeps the same syntax as LFI1 and generalizes the system LFI1. With these three facts inmind, we consider the relation between the two systems BD and BS4 which is a naturalgeneralization of the relation between the two systems J3 and LFI1. Furthermore, based onthese unified perspective provided by, we obtain, as corollaries of our main theorem, somenew completeness results for first-order versions of systems SPL and BS4 (see Corollary 9and Corollary 12, respectively), and some alternative proofs for the completeness resultsfor first-order versions of J3 and LFI1 (see Corollary 11 and Corollary 12, respectively).

2 Syntax and Semantics

Our syntax consists of the following vocabulary: a fixed finite set P of propositional connec-tives, the universal and existential quantifiers ∀ and ∃, a countable set x0, x1, . . . of vari-ables, a countable set c0, c1, . . . of constant symbols, and a countable set P0, P1, . . . ofpredicate symbols, where we associate each predicate Pk with a fixed finite arity. We regard0-ary predicate symbols as propositional letters. Let us denote this syntax by LP (note thatwe only write the information of propositional connectives in P). This paper is concernedwith expansions of Belnap-Dunn logic, and so, we always assume that ∼,∧,∨ ⊆ P. Ourproposal is to add to Belnap-Dunn logic a new unary propositional connective , whosemeaning is ‘- is designated’ or ‘- has a designated value’. We define the set of formulas inL∼,∧,∨, as follows:

α ::= P (t1, . . . , tn) | ∼ α |α ∧ β |α ∨ β |α | ∀x. α | ∃x. α,

where ti is a term, i.e., a variable or a constant symbol. We say that a formula is propo-sitional if it constructed from propositional letters by using the propositional connectives.We define the notions of free and bound variable, and sentence as usual. We write α[x/t]to mean the result of substituting all the occurrences of free variable x in α by the term t,renaming the bound variables, if necessary, to avoid variable-crashes. We denote a set offormulas by Γ, Σ, etc. Given any set Γ of formulas, we define ∼ Γ := ∼ γ | γ ∈ Γ andΓ := γ | γ ∈ Γ .

An interpretation I is a pair ⟨D, v ⟩ where D is a non-empty set D and we assignv(c) ∈ D to each constant c and assign both the extension v+(P ) ⊆ Dn and the anti-extension v−(P ) ⊆ Dn to each n-ary predicate symbol P . Given any interpretation ⟨D, v ⟩,we can define P( 0, 1 )-valuation v to all the sentences of LP expanded by d | d ∈ D inductively as follows: as for the atomic sentences P (t1, ..., tn),

1 ∈ v(P (t1, ..., tn)) iff ⟨ v(t1), . . . , v(tn) ⟩ ∈ v+(P ),0 ∈ v(P (t1, ..., tn)) iff ⟨ v(t1), . . . , v(tn) ⟩ ∈ v−(P ).

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As for the formulas of the form α,

1 ∈ v(α) iff 1 ∈ v(α),0 ∈ v(α) iff 1 /∈ v(α).

The rest of the clauses are exactly the same as in the case of P( 0, 1 )-valued semanticsof Belnap-Dunn logic (the reader can refer to [10]). Given any set Γ ∪ α of sentencesand a class M of interpretations, we write Γ |=M α to mean that, for any interpretation I= ⟨D, v ⟩ from M, if we have 1 ∈ v(γ) for all γ ∈ Γ, then we also have 1 ∈ v(α). When Γ= α , we usually write α |=M β instead of α |=M β. We use α =∥=M β to mean thatα |=M β and β |=M α. Let us denote the class of all the interpretations by M4. We alsodefine M3b (or, M3n) to be the class of all interpretations that satisfies v+(P ) ∪ v−(P ) =Dn (v+(P )∩v−(P ) = ∅, respectively) for all predicate symbols P , where n is the associatedarity with P .

If we regard the values 1 , 1, 0 , ∅, and 0 of sentences as t (true only), b (both),n (neither), and f (false only), respectively, then we can reformulate our semantics interms of four-valued semantics. For example, we can give the truth table of α as inTable 1 below. Our semantic definition for justifies our intuitive reading of as ‘- hasa designated value’, because our set of designated values amounts to t,b .

α αt tb tn ff f

α Tα

t tn ff f

α→L β t n f

t t n fn t t nf t t t

α αt tb tf f

α αt tb fn ff t

α ⊃ β t b n f

t t b n fb t b n fn t t t tf t t t t

Table 1: Truth Tables for Propositional Connectives of Our Main Interest

Let f be an n-ary function on P( 0, 1 ) and f the n-ary connective such thatv(f (β1, . . . , βn)) = f(v(β1), . . . , v(βn)) for any interpretation I = ⟨D, v ⟩ and any sen-tences βi. A propositional sentence α(p1, . . . , pn) of LP positively defines (or, negatively de-fines) the connective f within M if α(p1, . . . , pn) =∥=M f (p1, . . . , pn) (or, α(p1, . . . , pn) =∥=M∼ f (p1, . . . , pn), respectively). f is weakly definable within M in LP if there existspropositional sentences α(p1, . . . , pn) and β(p1, . . . , pn) of LP such that α positively definesf within M and β negatively defines f within M. A propositional sentence α(p1, . . . , pn)of LP defines f within M if α positively defines f within M and ∼ α negatively definesf within M. f is definable within M in LP if there exists a propositional sentenceα(p1, . . . , pn) of LP such that α defines f within M. For a difference between definabilityand weak definability in L∼,∧,∨,, the reader can refer to Proposition 4.1.

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3 Natural Deduction for BD-logic with The natural deduction rules RBD for Belnap-Dunn logic is all the introduction and elimi-nation rules and of ∧, ∨, ∀, and ∃, and the De Morgan’s laws of them. Let F be a (possiblyempty) finite set of functions on P( 0, 1 ). Let us list all the relevant rules needed in ourmain theorem:

αα 1 α∨ ∼ α 2

α∧ ∼ αβ

3α∧ ∼ α

βEXP

α∨ ∼ α LEM,

f (α1, ..., αn)

φf (α1, ..., αn)f1

∼ f (α1, ..., αn)

ψf (α1, ..., αn)f2

,

where f ∈ F is an n-ary functions on P( 0, 1 ). Remark that if a propositional sentenceφf (p1, ..., pn) of L∼,∧,∨, positively defines f within M4 and a propositional sentenceψf (p1, ..., pn) of L∼,∧,∨, negatively defines f within M4, we can automatically producethe rules for .

Define RBD := RBD ∪ 1,2,3 , RK3 := RBD ∪ EXP and RLP :=RBD∪LEM. We useRF to mean the set f1,f2| f ∈ F of all rules for f | f ∈ F.

Given any set R of rules and any set Γ ∪ α of formulas, Γ ⊢R α means that we canchoose some finite Γ′ from Γ such that there is a α’s derivation from Γ′ in the calculus whoserule set is R. Given any sets Γ and Π of formulas, we define Γ ⊢R Π if Γ ⊢R α1 ∨ · · · ∨ αnfor some αi ∈ Π.

4 A General Strong Completeness Result

Let us introduce some terminology. We say that a set Γ of formulas is a R-theory if it isclosed under ⊢R, i.e., if Γ ⊢R α then α ∈ Γ for any formula α. A R-theory Γ is primeif α ∨ β ∈ Γ implies that α ∈ Γ or β ∈ Γ for any α and β. A R-theory Γ is saturatedif the following hold: i) ∀x. α ∈ Γ iff α[x/c] ∈ Γ for any constant c, and ii) ∃x. α ∈ Γiff α[x/c] ∈ Γ for some constant c. Finally, we say that a R-theory Γ is -consistent ifα ∈ Γ or ∼ α ∈ Γ, and α∧ ∼ α /∈ Γ, for any α.

Lemma 1. Let 2,3 ⊆ R and Γ a prime R-theory such that Γ 0 β for some β.Then, Γ is -consistent.

Lemma 2. Let RBD ⊆ R and Γ 0R α. By adding countably new constant symbols, wecan extend ⟨Γ, α ⟩ to ⟨Γ+,Π+ ⟩ such that Γ ⊆ Γ+, α ∈ Π+, Γ+ 0R Π+, (β ∈ Γ+ orβ ∈ Π+) for all β, and Γ+ is a prime and saturated theory.

Definition 3. Let Γ be a R-theory. We define the Henkin interpretation IΓ = ⟨D, v ⟩derived from Γ as follows: D is the set of all constant symbols in the syntax and, for anyn-ary predicate symbol P :

v+(P ) := ⟨ t1, . . . , tn ⟩ |P (t1, . . . , tn) ∈ Γ ,v−(P ) := ⟨ t1, . . . , tn ⟩ | ∼ P (t1, . . . , tn) ∈ Γ ,

and, for any constant symbol c, v(c) = c.

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Lemma 4 (Truth Lemma). Fix a finite set F of functions on P( 0, 1 ) and assume thatf | f ∈ F is both positively and negatively definable within M4 in L∼,∧,∨,. Let RFis the set of rules for f | f ∈ F and RBD∪RF ⊆ R. Given any prime, -consistent,and saturated R-theory Γ, the following holds on the Henkin interpretation IΓ = ⟨D, v ⟩:for any sentence α,

1 ∈ v(α) iff α ∈ Γ,0 ∈ v(α) iff ∼ α ∈ Γ.

Definition 5. Let f be any n-ary function f on P( 0, 1 ). We say that f is positivelyBooelan if, for any interpretation I = ⟨D, v ⟩ and any sentences α1, . . . , αn,

1 ∈ v(f (α1, . . . , αn)) iff BOOL( 1 ∈ v(αi), 0 ∈ v(αi) | 1 ≤ i ≤ n ),

where BOOL(X) means a Boolean combination constructed from a set X of atoms. Simi-larly, we also define the notion of negative Booleanness.

Lemma 6. Let f be any n-ary function on P( 0, 1 ). Suppose that f is both positivelyand negatively Boolean. Then, f is weakly definable within M4 in L∼,∧,∨,.

Proof. We only check the positive definability of f . Put X := 1 ∈ v(αi), 0 ∈ v(αi) | 1 ≤i ≤ n. First, let us transform BOOL(X) into a conjunctive normal form. Then, all thenegations are (if any) in front of some atoms having the form as 1 ∈ v(αi) or 0 ∈ v(αi).Our literals have one of the following forms:

1 ∈ v(αi) ( pi), 1 /∈ v(αi) ( ∼ pi),

0 ∈ v(αi) ( ∼ pi), 0 /∈ v(αi) ( ∼ ∼ pi).As suggested in the above, we can find the characterizing sentence of each literal by thehelp of . We replace each literal with the corresponding sentence in our conjunctivenormal form, and then, replace all meta-level conjunctions and disjunctions with ∧ and ∨in the syntax. Then, it is clear that the resulting sentence positively defines f .

The following witnesses a difference between weak definability and definability inL∼,∧,∨,.

Proposition 4.1. Let f be a constant function sending all values of t,b,n, f to b.Then, f is weakly definable within M4 in L∼,∧,∨,. However, it is undefinable withinM4 in L∼,∧,∨,.

If f is both positively and negatively Boolean, let us fix φf and ψf of L∼,∧,∨, suchthat φf positively defines f and ψf positively defines f . By Lemmas 1, 2, 4, and 6, weobtain:

Theorem 7 (Main Theorem). Let ⟨M,R⟩ be one of ⟨M4,RBD ⟩, ⟨M3n,RK3 ⟩ and⟨M3b,RLP ⟩. Let F be a finite set of functions on P( 0, 1 ). Suppose that f is bothpositively and negatively Boolean (f ∈ F ). Then, given any set Γ ∪ σ of sentences ofL∼,∧,∨,∪f | f∈F , Γ |=M σ iff Γ ⊢R∪RF

σ.

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5 Some Applications of Main Theorem

5.1 Kachi’s SPL

Kachi [6] proposed to add to Kleene’s strong three-valued logic the truth operator T , whosetruth condition in t,n, f -valued semantics is given as in Table 1 above. He called theresulting logic Simple Partial Logic (we denote it by SPL). Since the designated valueis only t in Kleene’s strong three-valued logic, Tα means ‘α is designated’ and so wecan regard α as a generalization of Tα to our semantics. Kachi, however, did notemploy |=M3n as his consequence relation. Instead, he used the following [6, p.144] in ourterminology.

Definition 8. Let Γ ∪ α be a set of sentences. We say that ⟨Γ, α ⟩ is weakly valid(written: Γ |=w α) if there is no interpretation I ∈ M3n such that 1 ∈ v(γ) (γ ∈ Γ) and1 ∈ v(∼ α). We also say that ⟨Γ, α ⟩ is strongly valid (written: Γ |=s α) if there is nointerpretation I ∈ M3n such that 0 /∈ v(γ) (γ ∈ Γ) and 0 /∈ v(∼ α).

Kachi [7] gave a sound and complete tableau calculus for |=w and |=s, though the set Γof assumptions is always finite and the syntax is in the propositional level. An underlyingidea of his tableau calculus is to introduce a new notion of lined formula to handle thestrong validity, where non-lined formulas and the tableau rule for them are for the weakvalidity. In order to capture |=w and |=s proof-theoretically, however, there is no need tointroduce such new syntactic notion. A key consists in using (or T ) to ‘internalize’ |=w

and |=s into |=M3n .

Corollary 9. Let Γ ∪ α be a set of sentences. Γ |=w α iff Γ |=M3n∼ ∼ α andΓ |=s α iff ∼ ∼ Γ |=M3n α. Therefore, Γ |=w α iff Γ ⊢RK3∼ ∼ α and Γ |=s αiff ∼ ∼ Γ ⊢RK3 α.

5.2 A P( 0, 1 )-Valued Generalization of Łukasiewicz’s Implication

Łukasiewicz defined the truth table of→L in t,n, f -valued semantics as in the followingtable (cf. [8, p.18]. We need to identify 1, 1/2, and 0 of [8, p.18] with t, n, and f ,respectively) as in Table 1. As suggested in [11] and [12, p.323], we can reformulate thistruth table as:

1 ∈ v(α→L β) iff (1 ∈ v(α) implies 1 ∈ v(β)) and (0 ∈ v(β) implies 0 ∈ v(α)),0 ∈ v(α→L β) iff 1 ∈ v(α) and 0 ∈ v(β).

Then, the proof of Lemma 6 automatically generates the following proof rules of →L inthe syntax L∼,∧,∨,,→L :

α→L β

(∼ α ∨ β) ∧ (∼ ∼ β∨ ∼ α) L1,

∼ (α→L β)

α∧ ∼ β L2.

Let us define RBD→L:= RBD ∪ L1,L2 and RK3→L

:= RK3 ∪ L1,L2 . Our maintheorem gives us the following:

Corollary 10. Let Γ ∪ α be a set of sentences. Γ |=M3n α iff Γ ⊢RK3→Lα and

Γ |=M4 α iff Γ ⊢RBD→Lα.

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5.3 D’Ottaviano and da Costa’s J3

The system J3 was developed with the motivation to present a system which meets thethree criterions of Jaśkowski for systems of paraconsistent logic. Besides ∨, ∧, and ∼, J3

also contains a connective which can be read as ‘- is designated’ or ‘- has a designatedvalue’ within three-valued semantics, and D’Ottaviano regards as a connective for a kindof modality with the influence of Łukasiewicz. Unlike Łukasiewicz, however, D’Ottavianochose 1, 1/2 as the designated values from 1, 1/2, 0 [5, p.79]. Therefore, it is naturalto see the truth values for J3 as t,b, f in our setting. Then, the semantics of is givenas in Table 1 above. So, we can state that our is a generalization of to t,b,n, f -valued semantics. D’Ottaviano gave Hilbert-style axiomatization of J3 and showed itsstrong completeness result. By Theorem 7, we can give a considerably simpler proof of J3

by providing a natural deduction style axiomatization RLP to J3.

Corollary 11. Let Γ ∪ α be a set of sentences. Γ |=M3bα iff Γ ⊢RLP α.

5.4 Omori and Waragai’s BS4 & Carnielli, Marcos and de Amo’s LFI1

Carnielli, Marcos and de Amo [4] discovered that their LFI1 on the syntax L∼,∧,∨,,⊃is inter-definable with J3 with respect to the three-valued semantics above. Recently,Omori and Waragai [9] took the same syntax as LFI1 and proposed the system BS4 asa generalization of LFI1 into four-valued setting. It has been unknown that a first-orderextension of BS4 enjoys a strong completeness result. We can also derive this unknownresult as a corollary of Theorem 7. First, let us transform the following four-valued truthtables of and ⊃ as in Table 1 into P( 0, 1 )-valued semantics:

1 ∈ v(α) iff (1 ∈ v(α) and 0 ∈ v(α)) or (0 ∈ v(α) and 1 ∈ v(α)),0 ∈ v(α) iff (1 ∈ v(α)⇐⇒ 0 ∈ v(α)).

We can provide α ⊃ β with the following P( 0, 1 )-valued semantic clauses:

1 ∈ v(α ⊃ β) iff 1 ∈ v(α) implies 1 ∈ v(β),0 ∈ v(α ⊃ β) iff 1 ∈ v(α) and 0 ∈ v(β).

We could apply the argument of the proof of Lemma 6, but here we can obtain the strongerdefinability result:

Proposition 5.1. Within M4, ∼ p∨q defines p ⊃ q and (p∨ ∼ p)∧(∼ p∨ ∼ ∼ p)defines p.

Conversely, is definable in L∼,∧,∨,,⊃. Define ⊥ := q ∧ q∧ ∼ q for some fixed qand ¬α := α ⊃ ⊥. Then,

Proposition 5.2. Within M4, ¬¬p defines p.Let us give the natural deduction rules to and ⊃:

α ∧ α∧ ∼ αβ

1 αα∨ ∼ α 2 α 3

∼ α ∧ α∼ α 4 ∼ α∧ ∼ α

α 5

α ∨ (α ⊃ β)⊃1

∼ (α ⊃ β)α∧ ∼ β ⊃2

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and the usual introduction and elimination rules for ⊃. Define RBS4 (or, RLFI1) as theresult of adding the above rules of and ⊃ to RBD (or, RLP, resp.). Then, we obtain thefirst-order completeness results of LFI1 and BS4:

Corollary 12. Let Γ∪α be a set of L∼,∧,∨,,⊃-sentences. Γ |=M3bα iff Γ ⊢RLFI1

αand Γ |=M4 α iff Γ ⊢RBS4

α.

Proof. Fix the syntactic translation τ from L∼,∧,∨,,⊃ to L∼,∧,∨, by Propositon 5.1.Similarly, we define the translation σ from L∼,∧,∨, to L∼,∧,∨,,⊃ by Propositon 5.2.We establish the left-to-right direction of BS4 alone. Assume Γ |=M4 α. Then, we obtainτ [Γ] |=M4 τ(α). By Theorem 7, τ [Γ] ⊢RBD τ(α). It follows that σ[τ [Γ]] ⊢RBS4

σ(τ(α)).Therefore, Γ ⊢RBS4

α by σ(τ(β)) ⊣⊢RBS4β for any β of L∼,∧,∨,,⊃.

One of the key ideas of this proof is the connective can be seen as “decomposed” intotwo connectives ⊃ and in BS4. It is, then, natural to question the expressive power of⊃ in the system BS4. Let us consider the syntax L∼,∧,∨, .

Lemma 13. Let φ(p) be any propositional sentence of L∼,∧,∨, whose propositional letteris p alone. Then, for any interpretation I = ⟨D, v ⟩, we have one of the following: (i)v(φ)(b) = v(φ)(n) = f , (ii) v(φ)(b) = v(φ)(n) = t, (iii) v(φ)(b) = b and v(φ)(n) = n,where we regard v(φ) as the derived unary function on t,b,n, f .

Proposition 5.3. is undefinable within M4 in L∼,∧,∨, . Therefore, ⊃ is also unde-finable within M4 in L∼,∧,∨, .

Proof. Suppose for contradition that p is definable by φ(p) within M4 in L∼,∧,∨, . Fixsome interpretation I = ⟨D, v ⟩. Then, v(φ)(b) = t and v(φ)(n) = f (recall Table 1). Acontradiction with Lemma 13.

References

[1] A. R. Anderson, N. D. Belnap, and J. M. Dunn. Entailment, volume 2. PrincetonUniversity Press, 1992.

[2] M. Baaz. Infinite-valued Gödel logic with 0-1-projections and relativisations. InP. Hájek, editor, Gödel’96: Logical Foundations of Mathematics, Computer Science,and Physics, volume 6 of Lecture Notes in Logic, pages 23–33. Springer, 1996.

[3] N. Belnap. How a computer should think. In G. Ryle, editor, Contemporary aspectsof philosophy, pages 30–55. Oriel Press, 1976.

[4] W. Carnielli, J. Marcos, and S. de Amo. Formal inconsistency and evolutionarydatabases. Logic and Logical Philosophy, 8:115–152, 2000.

[5] I. M. L. D’Ottaviano. The completeness and compactness of a three-valued first-orderlogic. Revista Colombiana de Matemáticas, 19:77–94, 1985.

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[6] D. Kachi. Validity in simple partial logic. Annals of the Japan Association for Phi-losophy of Science, 10:139–153, 2002.

[7] D. Kachi. Partial logic as a logic of extensional alethic modality. Journal of the JapanAssociation for Philsophy of Science, 34(2):13–22, 2007. (In Japanese).

[8] G. Malinowski. Many-Valued Logics, volume 25 of Oxford Logic Guides. ClarenonPress, Oxford, 1993.

[9] H. Omori and T. Waragai. Some observations on the systems LFI1 and LFI1∗. InProceedings of the First International Workshop Data, Logic and Inconsistency, 2011.

[10] G. Priest. Paraconsistent logic. In D. Gabbay and F. Guenthner, editors, Handbook ofPhilosphical Logic, volume 6, pages 287–393. Kluwer Academic Publishers, 2 edition,2002.

[11] G. Restall. Łukasiewicz, supervaluations and the future. Logic and Philosophy ofScience, 3:1–10, 2005.

[12] K. Sato. Proper semantics for substructual logics, from a stalker theoretic point ofview. Studia Logica, 88(2):295–324, 2008.

[13] H. Wansing. The power of belnap: Sequent systems for SIXTEEN3. Journal ofPhilosophical Logic, 39:369–393, 2010.

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From Theory Revision to Logic: A Formal Linkbetween the Logic of Comparative Similarity CSL

and Distance Based RevisionCamilla Schwind∗

Abstract

We study a formal link between distance based theory revision and the logic ofcomparative similarity CSL. Introduced in [10] to express a form of qualitative sim-ilarity comparison between concepts, CSL is an extension of description logic by abinary modal operator. It allows to formulate assertions of the form “concept A ismore similar to concept B than to concept C”. Semantically, CSL is characterized byKripke structures where the set of possible worlds is equipped with a distance func-tion. Distance based theory revision defines a revision operator by means of a distancefunction between models. Thus both formalisms are based on distance spaces. Weshow a correpondence result between these two formalisms: the CSL operator canbe formulated by means of a distance based revision function and a distance basedrevision function can be expressed by means of the similarity operator.

1 Introduction

In this paper we establish a formal link between a logic, namely CSL and distance basedbelief revision. CSL is an extension of description logic by a binary modal operator thatmakes it possible to represent a notion of comparison between concepts.

The logic of comparative similarity, CSL and theory revision are different formalismsfor commonsense reasoning that address very different issues. CSL is a logic that makesit possible to make assertions about the similarity of concepts. The language of CSL is anextension of description logic by a new binary modality ⇔ that allows to express that aconcept (or an individual) is closer to one concept than to another one. Thus the formulaC ≡ (A⇔ B) says that concept C is closer to concept A than to concept B. CSL semanticsis defined in terms of distance spaces that is a set of objects (or individuals) together with adistance function between these objects. Belief revision is the process of changing beliefs toadapt the epistemic state of an agent to new information. Belief revision such as introducedby Alchurron et al [1] is a formalism that characterizes revision operators in a very generalframework. Distance based belief revision is a semantic approach to belief revision andrelies on a distance function defined between models. Thus the semantics of distance basedbelief revision is also defined by using distance spaces.

∗Laboratoire d’Informatique Fondamentale, LIF-CNRS, Université de la Marseille, 163 Avenue de Lu-miny – F-13288 Marseille Cedex09, case 901, [email protected]

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In this paper, we present the following correspondence results:

1. Every CSL formula A ⇔ B can be defined by a boolean expression over model setsand the revision function.

2. Every revision function can be translated into an expression using CSL formulas.

Links between conditional logics and belief base revision have been studied by severalauthors [5, 6, 8, 4]. Many of these articles concern the Ramsey rule and revision in thegeneral case (and not distance based revision).

Notation and terminology In the rest of this paper, we denote by FP (or simply F)the set of propositional formulas over a set P of propositional variables. A belief set is adeductively closed set of propositional formulas and K denotes the set of all belies sets.K⊥ is the inconsistent belief set. For a formula ϕ ∈ F , Mod(ϕ) is the set of models of ϕ.Given a set of interpretations M , we define FOR(M) = ϕ ∈ F : M ⊂ Mod(ϕ) the setof propositional formulas satisfied by all models in M .

2 Background

We introduce the two formal systems, the logic CSL and distance based belief revision.Both formalisms are characterized semantically by means of distance spaces.

Definition 1. A distance space is a pair (∆, d) where ∆ is a non-empty set and d is adistance function from ∆×∆ to R≥0 satisfying the following condition:

∀x, y ∈ ∆, d(x, y) = 0 iff x = y (id)

Two further properties are usually considered: symmetry and triangle inequality. Wedo not refer to them in this paper, see a discussion in [2, 3].

The distance between two sets A ⊆ ∆ and B ⊆ ∆ is defined by

d(A,B) = infd(x, y) : x ∈ A, y ∈ B (inf)

Definition (inf) is not a distance function since d(A,B) = 0 iff A∩B = ∅. As a special caseof Definition (inf), the distance between an indiviual w ∈ ∆ and a set A ∈ ∆ is d(w, A)and is noted

d(w,A) = infd(w, y) : y ∈ AHence d(w,A) = 0 iff w ∈ A. If A = ∅, then d(w,A) =∞. We call (∆, d) a min-space ifadditionally it satisfies the following condition on the existence of a minimum of a set ofdistances:

If A = ∅ and B = ∅ then ∃x0 ∈ A∃y0 ∈ B s.t. d(x0, y0) = d(A,B). (min)

In the rest of this paper, we will always consider distance spaces that are min-spaces.Intuitively, for CSL semantics, the set ∆ will be a set of objects. For theory revision,

∆ will be a set of possible worlds, namely of propositional models. But as we will seebelow every object or individual can be seen as a propositional model, namely as the setof propositional variables, that hold for this object.

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2.1 The Logic of Comparative Concept Similarity with singletonconcepts CSL

The logic CSL has been introduced in [10, 11, 12] as a purely qualitative formalism forknowledge representation and reasoning about comparative similarity. As an extensionof description logic its alphabet contains a finite set of concept names Cnom and a finiteset of individual names Aind. The language LCSL contains all concepts formed out fromconcept names and singletons by applying boolean connectives and the ⇔ operator. ⊥,⊤are propositional formulas. If C ∈ Cnom, then C is a propositional formula, if a ∈ Aind, ais a propositional formula. If ϕ, ψ are propositional formulas then ϕ ⊓ ψ,¬ϕ and ϕ ⇔ ψare propositional formulas.

As usual, ϕ ⊔ ψ = ¬(¬ϕ ⊓ ¬ψ) and, dealing with enumerative concepts, we writea1, . . . , an in place of a1 ⊔ . . . ⊔ a1.

The semantics of CSL is defined in terms of Kripke models based on distance spaces.

Definition 2 (CSL-distance model). A CSL-distance model is a triple I = (∆, d, .I) where:

• ∆ is a non-empty set of objects (or possible worlds),

• (∆, d) is a distance space,

• .I is the evaluation fuction that assigns to each concept name Ai ∈ Cnom a set AIi ⊆ ∆and to each individual name ai ∈ Aind an object aIi ∈ ∆. AIi can be seen as the setof objects (or possible worlds) “belonging” to concept Ai. We have also ⊥I = ∅. Forcomplex formulas, .I is defined inductively as follows: (¬ϕ)I = ∆ \ ϕI , (ϕ ⊓ ψ)I =ϕI ∩ ψI , (ϕ ⇔ ψ)I =

w ∈ ∆

∣∣d(w, ϕI) < d(w,ψI).

If (∆, d) is a min-space, I is called a CSL-distance min-model. For the rest of thispaper, we consider exclusively min-models.

We say that a formula ϕ holds in w ∈ ∆ (we note I, w |= ϕ) if w ∈ ϕI . We say that aformula ϕ is valid in a model I (and note I |= ϕ) if ϕI = ∆. We say that a formula ϕ isvalid (and note |= ϕ) if ϕ is valid in every CSL-distance model.

The concept names C ∈ Cnom as well as the singletons a where a ∈ Aind can beseen as corresponding to the propositional variables of the language. Furthermore, wecan identify each world (or object) w ∈ ∆ with the set of formulas m(w) that hold in w:m(w) = ϕ ∈ LCSL : w ∈ ϕI.

2.2 Distance based revision

Revision has been introduced as an operation that assigns a belief set to a belief set and aformula. Revision is generally not formulated as a logic, the revision operator ∗ has notbeen introduced as a modal operator. And no axiomatics has been defined for revision.Instead, postulates have been formulated in [1], called AGM postulates after their authors,that characterize properties of revision operators in a very general way. Formally, a revisionoperator is a function that assigns to a belief set Γ and a formula ϕ another belief set,namely the revised belief setΓ∗ϕ. Hence every belief revision operator ∗ is a function∗ : K ×F −→ K .

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Definition 3 (Revision operator [1]). Let Γ be a belief set and ϕ, ψ ∈ F propositional for-mulas. We call AGM revision operator every operator ∗ for which the following postulateshold

K∗1 Γ∗ϕ is a belief set .

K∗2 ϕ ∈ Γ∗ϕ

K∗3 Γ∗ϕ ⊆ Cn(Γ ∪ ϕ)

K∗4 If ¬ϕ /∈ Γ then Cn(Γ ∪ ϕ) ⊆ Γ∗ϕ

K∗5 Γ∗ϕ = ⊥ iff ⊢ ¬ϕ

K∗6 If ϕ ≡ ψ1 then Γ∗ϕ = Γ∗ψ

K∗7 Γ∗(ϕ ∧ ψ) ⊆ Cn((Γ∗ϕ) ∪ ψ)

K∗8 If ¬ψ /∈ (Γ∗ϕ) then Cn((Γ∗ϕ) ∪ ψ) ⊆ Γ∗(ϕ ∧ ψ)

In this paper, we will use a characterization of revision operators in terms of model setsidentifying a formula (or a formula set) with the set of models satisfying it. This approachyields a rather simpler algebraic characterization of revision operators.

Definition 4 (Revision function). Le U be a set of sets (model sets).|: U × U −→ U is a revision function on U if for any M,N,L ∈ U

(S1) M | N ⊆ N

(S2) M ∩N ⊆M | N

(S3) if M ∩N = ∅ then M | N ⊆M ∩N

(S4) if N = ∅ then M | N = ∅

(S5) (M | N) ∩ L ⊆M | (N ∩ L)

(S6) if (M | N) ∩ L = ∅ then M | (N ∩ L) ⊆ (M | N) ∩ L

Intuitively, Γ∗ϕ is a belief set that contains ϕ and is as close as possible to Γ. Accordingto K∗2 (or to (S1)) every model of Γ∗ϕ is a model of ϕ, that means that the models of therevised base Γ∗ϕ are a subset of the models of ϕ. It is natural to assume that the revisedbase is as close as possible to the original base Γ. In order to measure closeness, distancefunctions between models have been introduced. The models of a belief setΓ revised by aformula ϕ are the models of ϕ that are closest to the models of Γ according to the distancefunction. This yields a specific class of revision operators, namely distance based revisionoperators. Distance based revision relies on a distance function between models and isdefined semantically as a binary function on sets of models.

1ϕ ≡ ψ iff ⊢ ϕ↔ ψ

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Definition 5 (Distance based revision function [7, 9]). Given a distance space (∆, d) andA,B ⊆ ∆

A |d B = b ∈ B : ∃a ∈ A ∀a′ ∈ A, b′ ∈ B d(a, b) ≤ d(a′, b′)

If (∆, d) is a distance space, we call |d a distance based revision function.

Remark 2. A |d B = b ∈ B : d(A, b) = d(A,B)

Theory revision is defined in terms of theories and formulas: the revision of a beliefsetΓ by a formula ϕ is again a belief setΓ∗ϕ. In order to formulate a distance basedrevision operator we claim that the revision result Mod(Γ) |d Mod(ϕ) can be representedby a formula. Such revision function is called definability preserving. The revision resultis not necessarily definability preserving as shown by Lehmann and co-authors [7], whoconstruct an example where the revision result Mod(Γ) |d Mod(ϕ) is a set of models thatis not representable by a set of forumlas. Note that in the finite case, i.e. when the set ofpropositional variables is finite, the distance based revision result is always representableby a set of formulas.

Definition 6. Given a belief set Γ, a formula ϕ and a distance based revision func-tion |d, the corresponding distance based revision operator ∗d is defined by Γ ∗d ϕ =FOR(Mod(Γ) |d Mod(ϕ) )

Theorem 7. [9] A distance based revision operator ∗d satisfies the AGM postulates if

1. |d respects identity id,

2. (∆, d) is a minspace

3. |d is definability preserving.

3 Correspondence Result

CSL as well as distance based revision are characterized semantically by means of a dis-tance space (∆, d). Although these two formalisms are different formally as well as wrtthe reasoning issues they address, it seemed promising to study whether they might beinterdefinable. We have found two correspondence results that we present here.

3.1 From distance based revision to CSL

Here we show that the CSL modality ⇔ can be defined in terms of a distance basedrevision function |d . We show that the set of objects for which the CSL formula α ⇔ βis true can be obtained as a boolean algebraic expression built from the revision function|d and set operations (∪,∩, \,=) over the sets Mod(α) and Mod(β) .

Theorem 8 (Correspondence theorem). Let (∆, d) be a distance space and |d a distancebased revision function on 2∆. Let be I = (∆, d, .I) a CSL interpretation, α, β ∈ LCSLand w ∈ ∆. Then

w ∈ (α ⇔ β)I iff (w |d αI ∪ βI) ∩ βI = ∅

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Intuitively the expression of Theorem 8 means the following: given a min-model I =(∆, d, .I), we know that α ⇔ β holds in w if w is closer to set αI than to set βI . Thisis the case if the revision of set w by set (αI ∪ βI) does not contain elements of βI i.e.contains only elements of αI (since αI |d βI is contained in βI). This means that all of βI

is more far away from w than (at least some elements) from αI .

Proof of theorem 8. Case 1: α ≡ β and βI = ∅. Then (α ⇔ β)I = ∅, hence ∀w,w /∈(α ⇔ β)I . On the other hand, (w |d αI ∪ βI) ∩ βI = (w |d βI) ∩ βI = w |d βI = ∅(by postulate (S4)).Case 2: αI = ∅ and βI = ∅. Then (α ⇔ β)I = ∅ and w /∈ (α ⇔ β)I . We have then(w |d αI ∪ βI) ∩ βI = (w |d βI) ∩ βI = w |d βI = ∅, since βI = ∅ (by postulate(S4)).Case 3: αI = ∅ and βI=∅. Then (α ⇔ β)I=∆ and for all w ∈ ∆, (w |d αI∪βI)∩βI = ∅.Case 4: αI = ∅ and βI = ∅. We show both directions.

“⇒”: Since w ∈ (α ⇔ β)I , ∃a0 ∈ αI and ∀b′ ∈ βI , d(w, a0) < d(w, b′). Let be b ∈ βI .Then d(w, a0) < d(w, b). Suppose for the contrary that b ∈ w |d αI ∪ βI . Then ∀b1 ∈αI ∪ βI , d(w, b) ≤ d(w, b1). But then we have also d(w, b) ≤ d(w, a0) contradiction!

“⇐”: Let (w |d αI ∪ βI) ∩ βI = ∅. Then ∀x ∈ ∆, if x ∈ βI then x /∈ w |d αI ∪ βI .This means that ∀x ∈ ∆, if x ∈ βI then ∃y ∈ αI ∪ βI such that d(w, y) < d(w, x). Butthen, since (∆, d) is a minspace, we have that ∀x ∈ ∆, if x ∈ βI then ∃y ∈ αI such thatd(w, y) < d(w, x). We conclude that w ∈ (α ⇔ β)I .

3.2 From CSL to distance based revision

Here we show that the distance based revision function |d can be expressed equivalentlyby means of the logic CSL. Given a CSL language and a min-model I = (∆, d, .I), thefollowing equation holds for the distance based revision function |d over ∆.

Theorem 9. Let be (∆, d, .I) a CSL model and A,B ⊆ ∆ two F-definable subsets of ∆.Then the distance based revision operator |d can be equivalently defined by the followingexpression.w ∈ A |d B iff w ∈ B and either (w ∈ A or ∃a ∈ A such that a ∈ (¬(w ⇔ FOR(B)) ∨(FOR(B) ⇔ w))I and ∀a′ ∈ A, a′ ∈ (¬(FOR(B) ⇔ w))I)

Proof. “⇒”: Let be w ∈ A |d B. This means that ∃a ∈ A and d(a,w) = d(A,B). But thend(a,w) = d(a,B) and ∀a′ ∈ A, d(a′, B) = d(a′, w) which means that d(a′, B) ≮ d(a′w).From this follows easily the left hand side.

“⇐”: If w ∈ A ∩ B then d(w,w) = 0 = d(A,B). Suppose that w /∈ A and∃a ∈ A such that a ∈ (¬(w ⇔ FOR(B)) ∨ (FOR(B) ⇔ w))I and ∀a′ ∈ A, ′ ∈(¬(FOR(B) ⇔ w))I . Then (i) d(a,w) ≮ d(a, FOR(B)) and d(a, FOR(B)) ≮ d(a,w)and (ii) ∀a′ ∈ A, if a′ = a then d(a′, FOR(B)) ≮ d(a′, w). From (i) we conclude that∃a ∈ A such that d(a,w) = d(a,B) and by (ii) ∀a′ ∈ A, a′ = a, d(a′, B) = d(a′, w). Thisentails that ∃a ∈ A, d(a,w) = d(A,B). But then we have w ∈ A |d B.

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4 Conclusion and outlook

The first correspondence result presents one boolean condition on a revision function equiv-alent to a CSL formula. The inverse result expresses a revision function in terms of a CSLformula. But this second result is not a real equivalence with one formula. Instead wehave found a logical expression containing CSL formulas. The first part of this expres-sion, namely ∃a ∈ A such that a ∈ χI , where χ = (¬(w ⇔ B) ∨ (B ⇔ w)) can beexpressed by one formula χ ⇔ ⊥, that is true when χI = ∅. But then, we cannot expressthe rest of the expression that asserts a property for all a′ = a. In order to be able torefer explicitely to objects we need something that allows us to express within the logicthat element a belongs to the extension of formula χ. This might be a new variant of CSL(and of description logic), some sort of “hybrid” desciption logic. This might be an issueof further research.

References

[1] C. E. Alchourron, V. Gärdenfors, and D. Makinson. On the logic of theory change:Partial meet contradictions and revision functions. Journal of Symbolic Logic,50(2):510–530, 1985.

[2] Régis Alenda, Nicola Olivetti, and Camilla Schwind. Comparative concept similarityover similarity, tree automata, and diophantine equations. In Martin Giese and ArildWaaler, editors, Tableaux2009, Lecture Notes in Computer Science, pages 651–665.Springer, 2009.

[3] Régis Alenda, Nicola Olivetti, Camilla Schwind, and Dmitry Tishkovsky. Tableaucalculi for slCSL over minspaces. In Anuj Dawar and Helmut Veith, editors, CSL,volume 6247 of Lecture Notes in Computer Science, pages 52–66. Springer, 2010.

[4] Nir Friedman and Joseph Y. Halpern. Conditional logics of belief change. In AAAI,pages 915–921. AAAI Press, 1994.

[5] Laura Giordano, Valentina Gliozzi, and Nicola Olivetti. A conditional logic for beliefrevision. In Jürgen Dix, Luis Fariñas del Cerro, and Ulrich Furbach, editors, JELIA,volume 1489 of Lecture Notes in Computer Science, pages 294–308. Springer, 1998.

[6] Laura Giordano, Valentina Gliozzi, and Nicola Olivetti. Weak agm postulates andstrong ramsey test: A logical formalization. Artif. Intell., 168(1-2):1–37, 2005.

[7] Daniel J. Lehmann, Menachem Magidor, and Karl Schlechta. Distance semantics forbelief revision. J. Symb. Log., 66(1):295–317, 2001.

[8] Hans Rott. A nonmonotonic conditional logic for belief revision. part 1: Semanticsand logic of simple conditionals. In André Fuhrmann and Michael Morreau, editors,The Logic of Theory Change, volume 465 of Lecture Notes in Computer Science, pages135–181. Springer, 1989.

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[9] Karl Schlechta. Coherent Systems. Number 2 in Studies in Logic and Practical Rea-soning. Elsevier, Amsterdam, 2004.

[10] Mikhail Sheremet, Dmitry Tishkovsky, Frank Wolter, and Michael Zakharyaschev.Comparative similarity, tree automata, and diophantine equations. In Geoff Sutcliffeand Andrei Voronkov, editors, LPAR 2005, volume 3835 of Lecture Notes in ComputerScience, pages 651–665. Springer, 2005.

[11] Mikhail Sheremet, Dmitry Tishkovsky, Frank Wolter, and Michael Zakharyaschev. Alogic for concepts and similarity. J. Log. Comput., 17(3):415–452, 2007.

[12] Mikhail Sheremet, Frank Wolter, and Michael Zakharyaschev. A modal logic frame-work for reasoning about comparative distances and topology. Ann. Pure Appl. Logic,161(4):534–559, 2010.

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Many-Valued Logics for Dynamical Semantics ofthe Atomic Self-Reference Statements

Vladimir Stepanov∗

On the self-reference quantor

Let P (x) be a predicate formula of a fragment of the type-free second-oder languagewithout ∀- and ∃-quantors, in which predicates can take other predicate as arguments. Asit is known, the self-reference might be expressed with the help of the fixed-point axiom.There are several ways of marking such statements, described in literature. For example,we might use the formula l ↔ P (l), and further on to operate just with a single quantity l.As for us, for the same aim we would use the quantor of self-reference Sx combined withthe axiom of self-reference. Something similar to our application of the symbol S wehave for the first time found in the paper by Johnstone [3]. In that paper the symbol Splayed somewhat peculiar part of the “filter” of self-reference and, moreover, was used asan operator. By our application of S symbol those expressions which are obtained withits help appear automatically as well formed formulas, while the symbol Sx itself obtainsall the properties of a quantor, including the function of the binding the x variable in thesubquantor formula.

By quantor Sx we designate the fact, that the formula SxP (x) refers to itself andsatisfies the self-reference axiom (see Feferman [2]):

SxP (x) ↔ P (SxP (x)),

where P (SxP (x)) is obtained from P (y) by the (correct) substitution of SxP (x) for thevariable y. The expression SxP (x) is read as

self-referential with respect to x P of x.

The variable of x for formula SxP (x) is bounded by the quantor Sx.Let P (x) be constructed by ∨∧ →↔ ¬ from atomic predicate Tr(x), which satisfies

Tarsky axiom:

Tr(x) ↔ x.

We are interested, however, with atomic S-formulas which do not possess any S-quantorsin the subquantor formula. For this reason we shall weaken the definition of self-reference,basing ourselves on the works by American logician Charles Pierce.

∗Supported by the RFBR N 09-06-00125.Dorodnicyn Computing Centre of RAS, Vavilov str. 40, Moscow, 119333, Russia, [email protected]

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On the Pierce’s case

The dynamic approach in research of self-reference was foreshadowed in the lectures of1865 by C.Pierce. There he presented the Liar as a sentence of infinite length obtained asa result of substitutions iteration of the Liar into itself [1] :“............S2: What is here written is not true.Similarly concerning S2 Pierce says that we get an infinite number of propositions:What is here writtenThe statement that that is falseThe statement that that is falseThe statement that that is false..............and so on to infinity”Actually, C.Pierce used implicitly the self-reference axiom, which was explicitly formulatedand proved by S.Kleene in 1936 as the recursion theorem.

Assume that in our language the predicate variant of an Equivalence theorem is fulfilled.Then by a recurring of replacement of SxP (x) by P (SxP (x)) in the right member ofthe self-reference axiom we obtain the following infinite sequence of the formulas:

SxP (x) ↔ P (SxP (x)) ↔ P (P (SxP (x)) ↔ P (P (P (SxP (x))))...

One should take into account the sequence of the form:

P (•), P (P (•)), P (P (P (•))), ...

This sequence will be used for the definition of approximation of the self-reference,which look as follows:

SxP (x) =def ⟨P (x), P (P (x)), P (P (P (x))), . . . ⟩

This definition will be named as a Pierce’s approximation and will be designated as Sx,in distinction from the symbol Sx, which designate the “real” self-reference, but not itsapproximation.

The last sequence reminds driving along the trajectory of a dynamic system. It suggestsa possibility of using of dynamic systems to develop self-reference statements models.

On the interpretation of a coherent case

Let I be the interpretation of our language, described in [5]. ThenI(x) = x ∈ 0, 1 - the set of classical truth values.Let P (x) be constructed by ∨∧ →↔ ¬ from atomic predicate Tr(x), free of S-symbol.ThenI(P (x)) = p(x); it’s viewed as a map p : 0, 1 −→ 0, 1; in what follows,I(SxP (x))= < pn(x0) >,n ∈ Z | x0 ∈ 0, 1 is the set of motions of the dynamicalsystem (0, 1, p), associated with SxP (x). In this case pn = p(pn−1).I(¬SxP (x))= < ¬(pn(x0)) >,n ∈ Z | x0 ∈ 0, 1.

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Such an application of the sign ¬ will be named as external and will be designated by∼ sign.

All of the preceding can be illustrated with the following diagram:

Sx(Tr(x)∨ Sx(Tr(x)∧ SxTr(x) Sx¬Tr(x) (1)¬Tr(x)) ¬Tr(x))p(x) = 1 p(x) = 0 p(x) = x p(x) = ¬x (2)

x0 =1 1111... 1000... 1111... 1010... (3)x0 =0 0111... 0000... 0000... 0101... (4)

11/11 00/00 11/00 10/01 (5)

3 ?d? ?

0 1k?d? ?

0 1??d d? ?

0 13

+? ?

0 1(6)

T F V A (7)After application to the initial formulas of external negation ∼ we will receive the followingpattern:

∼Sx(Tr(x)∨ ∼Sx(Tr(x)∧ ∼SxTr(x) ∼Sx¬Tr(x) (1’)¬Tr(x)) ¬Tr(x))p(x) = 1 p(x) = 0 p(x) = x p(x) = ¬x (2’)

x0 =1 0000... 0111... 0000... 0101... (3’)x0 =0 1000... 1111... 1111... 1010... (4’)

00/00 11/11 00/11 01/10 (5’)

k?d? ?

0 13 ?d? ?

0 1??d d? ?

0 13

+? ?

0 1(6’)

∼T ∼F ∼V ∼A (7’)

Expressions of the types 11/00, 01/10, ... appearing in the lines (5) and (5’) coding theregular cut-offs of the two sequences from 0 and 1, which are shown in the lines (3), (4) and(3’), (4’). In lines (6) and (6’) trajectories of the discrete dynamic systems are represented.They are constructed on one-dimensional discrete phase space.

Taking into account the mentioned above limitations for the predicate formulas, let usdefine the binary operations on the atomic S-formulas, where symbol means ∨∧ →↔:I(SxP (x)SyQ(y))= (pn(x0)qn(y0)), n ∈ Z | x0 = y0 ∈ 0, 1.

As soon as the S-formulas at left passess the sign of external negation ∼, then thedefinition is modified in accordance to the formula describing the external negation.

The dynamical interpretation I generates a 16-valued logic, which is the fourth Carte-sian direct power of classical propositional logic C2 with the following matrix Mc

2 =< 1, 0,¬,∨,∧, 1 >, i.e.:

Mc16 = (Mc

2)4 = < 11/11, 10/01, 11/00, ..., 00/11, 01/10, 00/00,¬,∨,∧, 11/11 > =

< T, A, V, ...,∼V, ∼A, ∼T ,∼, ∨, ∧, T >.

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The T(rue),A(ntinomy),V(oid) and its negations are the set of truth-values for atomicself-reference statements. The figure (see below) presents the schematical lattice diagramfor the 16-valued logic in the coherent case of the interpretation I . The estimations of theformulas of the types (A∨V), (A∧V) and some others are presented in the matrix Mc

16

as dots.It’s known, see [4], that the following lemma is valid:

Lemma 1. The sets of tautologies for Mc2 and Mc

16 coincide.

The lattice diagrams for our logical values are shown below:

∼A

∼A∨∼V

A∼V

A∨V

V

∼A∧∼V A∧V

TdTT

*

HHHYt t t t

@@@Id d t t d d

t t t t@@@I

dTT*

HHHY

∼T

On the interpretation of incoherent case

For the coherent case the initial values of letters x0 and y0 were just the same, x0 = y0,and this property stipulated the name for that case.

We shall retain the coherency for the case when the interacting formulas (or theirexternal negations) possess the equal estimations - either A, or V, - and for that reasonare interpreted over the equal dynamical systems:

(A ∼A) = (A ∼A)

For the case when atomic formulas within the combined formula will be different in theirestimations, there are many ways for determination of operations. Most “liberal” of themconsists in the following:I(SxP (x)SyQ(y))= (pn(x0)qn(y0)), n ∈ Z | x0 ∈ 0, 1, y0 ∈ 0, 1.

As it is evident, the initial values of the letters x0 and y0 are independent, and theyare forming four possible combinations. Then the incoherent estimation will consist of thesequence of the four coherent estimations of such an appearance:

A∨V = A∨V, A∨∼V, ∼A∨V, ∼A∨∼V,

A∨∼V = A∨∼V, A∨V, ∼A∨∼V, ∼A∨V,∼A∨V = ∼A∨V,∼A∨∼V, A∨V, A∨∼V,∼A∨∼V = ∼A∨∼V, ∼A∨V, A∨∼V, A∨V,

and so on. Let’s note that the sets of formulas to the right of equality sign for all four∨-incoherent formulas coincide.

This case of incoherent interpretation demands the further analysis.

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References

[1] M. Emily, Pierce’s Paradoxical Solution to the Liar’s Paradox. NDJFL, v. XII, no.3, July 1975.

[2] S. Feferman, Toward useful type-free theories I., The JSL, 1984, vol. 49, no. 1,pp. 75-111.

[3] A. Johnstone, Self-Reference, the Double life and Godel, Logique et Analyse,1981, vol. 24, no. 2, pp. 35–47.

[4] A. Karpenko, The Development of Many-Valued Logics, Moscow: LKI Publishers,2010. 488 p.

[5] V. Stepanov, Semantics of self-reference: a dynamical systems approach, The Bul-letin of Symbolic Logic, 2004, vol. 10, no. 2, pp. 272–272.

[6] V. Stepanov, Propositional logic of reflexive sentences, Automatic Documenta-tion and Mathematical Linguistics, 2007, vol. 41, 3, pp. 80-88.

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Abduction of Generalizations in a Modal AdaptiveFramework

Frederik Van De Putte Tjerk Gauderis∗

Abstract

Abduction can be described as the process of forming explanatory hypotheses. Inthis talk, we will be mostly concerned with one specific type of abductive reasoning, i.e.one where we abduce a general hypothesis in order to explain an observed correlation,called law abduction. We call this pattern “Abduction of Generalizations” (AG). Aftera general informal discussion of AG, we will argue that in order to formally model it,we have to be able to distinguish between on the one hand, the explanatory frameworkof an abduction, and on the other hand mere observational data and generalizations,in the object language of our logic. In this talk, we will do so with the aid of themodal necessity operator. We will show how AG can be modelled by three differentadaptive logics, each formulated in a first order modal framework.

Extended Abstract

“Abduction” was defined by Peirce as “the process of forming an explanatory hypothesis”[6, p. 216]. This concept has known a wide variety of more specific characterizations,not all of them compatible. Consequently, a lot of different reasoning patterns such asinference to the best explanation, singular fact abduction, analogical reasoning, etc. havebeen identified as abductive. However, in this talk we are concerned with one specific“pattern of abduction” (to use the phrase from [7]). Consider the following example:

Pineapples taste sweet.Everything that contains sugar, tastes sweet.Pineapples contain sugar

In the literature on abduction, inferences of this kind have been called “law abduction”[7], and “rule abduction” [8]. We will use the term Abduction of Generalizations (henceforthAG) for this specific pattern. An AG is typically of the following form:

(AG) We observe that everything that is A is also B.Also, we consider being C as an explanation for being B.Therefore, we raise the hypothesis that all A are C, as an explanation for the factthat all A are B.

∗Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, 9000 Gent, Belgium,frvdeput.VanDePutte,[email protected]

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Put in other words, by an AG we generate hypotheses that explain observed correlationsbetween two properties. We will explain how this definition can be operationalized in afirst-order modal language. As we will argue, law abduction is ubiquitous in both everydayand scientific reasoning, and it is commonly recognized as a valid – be it fallible – way toextend one’s knowledge beyond its merely deductive consequences.

Notwithstanding the importance of AG, little effort has been made so far to study thecharacteristics of this inference pattern, and to capture it by means of formal logic tools.Most research in AI and non-monotonic logic has focused on singular fact abduction. Thisapproach neglects the importance of other patterns of abduction. These distinct inferencepatterns are associated with distinct contexts, justifications and methodological issues, andeach of them seems just as good a candidate for a formal study as singular fact abduction.

The present talk consists of two parts: (i) an informal analysis of the pattern we callAG, and (ii) a survey of three formal logics that model this inference pattern.

In the first part, we will provide a brief analysis of the phenomenon of AG. We willdescribe this pattern informally, showing that it is a widespread inference pattern bothin everyday and scientific reasoning. Next, we will explain why this pattern has beenneglected in formal (philosophical) logic, artificial intelligence and philosophy of science.Finally, it will be argued that AG is typically a more scientific kind of abduction, and relateit to two other important reasoning methods in scientific practice: analogical reasoning andcommon cause abduction.

Next, we will argue that one has to distinguish between what we call “mere generaliza-tions”, and “the explanatory framework” in the object language of any logic that modelsthis reasoning pattern. This observation provides the bridge between our informal analysisand the formal systems which we shall present afterwards. That is, each of these systemsis formulated in the language of the well-known modal system S4, such that the operator can be used to express that a certain piece of information is part of the explanatoryframework.

In the second part of the talk, we will propose three different logics by which one maymodel AG. The logics are formulated in the standard format of adaptive logics, which hasshown its great strength in the formal analysis of various forms of defeasible reasoning, andhas already been successfully applied to inductive generalization [1], singular fact abduction[5] and theoretical abduction [4].

After a brief introduction to this framework, we will present the three distinct adaptivelogics LAr

∀, ILAr and LAr

∃ and discuss their most salient characteristics. Notably, theseare the first logics (with a proof theory and semantics, characterizing a consequence relationthat maps sets of formulas onto other sets of formulas) that have been developed for AG.

Each of the logics is based on a specific conception of how best to think of AG. Thefirst one takes law abduction as a primitive inference pattern, and can be seen as themost straightforward of the three. The second logic provides a reconstruction of AG bya combination of inductive generalization and singular fact abduction, elaborating on anidea from [8] and [3]. Finally, the third logic is based on the idea that abductions shouldbe performed in a uniform way, and delivers instances of AG as a side-effect.

As we will argue, each of these logics produces abductive hypothesis that are in a verygeneral sense rational: they are the weakest possible explanations for the explanandum;

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they are mutually compatible with the set of classical consequences of the premises; theyare not circular; etc.

However, we will also discuss some prototypical cases in which the three systems differ.This will lead to our final conclusion, i.e. that whether or not any of these logics is a suitablecandidate to model AG, depends on whether one conceives of AG as self-motivating, or ifone rather thinks that the motivation for an AG follows from other, more basic, ampliativeprinciples.

The aim of our research on this topic is foundational: it provides the basis upon whichfurther refinements can be built. We will not discuss enhancements of the proposed systemsin order to deal with prioritized background knowledge, possibly inconsistent theories, orinsufficient conditionals. Also, we will not consider abductions on the basis of probabilisticrelations between certain predicates. Various solutions to deal with these problems havealready been developed in the context of singular fact abduction and inductive generaliza-tion – see e.g. [2] –, and each of them can be easily carried over to the systems we willpresent here.

References

[1] Diderik Batens. Logics for qualitative inductive generalization. Unpublished paper,2010.

[2] Diderik Batens and Lieven Haesaert. On classical adaptive logics of induction. Logiqueet Analyse, 173–175:255–290, 2001. Appeared 2003.

[3] Peter A. Flach and Antonis C. Kakas. Abductive and Inductive Reasoning: Backgroundand Issues. In Peter A. Flach and Antonis C. Kakas, editors, Abduction and Induction.Essays on their Relation and their Integration, volume 18 of Applied Logic Series, pages1–27. Kluwer Academic Publishers, Dordrecht, 2000.

[4] Tjerk Gauderis. Modelling abduction in science by means of a modal adaptive logic.Foundations of Science, 2011. Forthcoming.

[5] Joke Meheus. A formal logic for the abduction of singluar hypothesis. UnpublishedPaper, available at http://logica.ugent.be/centrum/preprints/fma3.pdf.

[6] Charles Peirce. The Essential Peirce, volume 2. Indiana University Press, Bloomingtonand Indianapolis, 1988.

[7] G. Schurz. Patterns of Abduction. Synthese, 164:201–234, 2008.

[8] Paul Thagard. Computational Philosophy of Science. MIT Press/Bradford Books,Cambridge, MA, 1988.

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Handling Unnamed Abnormal Objects in(Mathematical) Adaptive Logic Theories∗

Peter Verdée†

Summary In this talk I shall elaborate a method to handle unnamed abnormal objectsin mathematical adaptive logic theories. In (some models of) most first order mathemati-cal theories, unnamed objects occur either because the theory has existentially quantifiedaxioms or because of the Löwenheim Skolem theorem. For some purposes, among whichthe formalization of non-trivial inconsistent mathematics, it is useful to select only thoseinterpretations of mathematical theories that are minimal with respect to some abnor-mality criterion. In general, adaptive logics are excellent tools to provide such selectionsemantics with a syntactic counterpart, immediately warranting indispensable propertieslike Classical Recapture and (Strong) Reassurance. So it is natural to use adaptive logicsas the underlying logics of mathematical theories. However, in their usual form, adaptivelogics turn out not to be suited for handling unnamed abnormal objects. In this talk Iexplain why this is the case and I propose a method to solve this problem, without losingthe positive aspects of the standard adaptive logic approach.

In this talk, I shall discuss a method to handle unnamed abnormal objects in (math-ematical) adaptive logic theories in a syntactic way. An adaptive logic theory is a theorywith first order axioms and an adaptive logic as its underlying logic. Adaptive logics arenon-monotonic logics (cf. [3] and [4]) with a selection semantics and a remarkable dynamicproof theory. Adaptive logics in Standard Format are defined by means of a regular LowerLimit Logic (LLL), a set of abnormalities and a strategies. Adaptive logic consequences ofa set of premises are those formulas that are true in all most normal LLL-interpretationsof the premises.

Inconsistency adaptive logics, for example, are able to prove a useful amount of in-teresting consequences of inconsistent premises, without leading to triviality, by assumingconsistency wherever possible. Consequently, adaptive logic theories are good candidatesto formalize paraconsistent/inconsistent mathematics, in the vein of Priest’s work (cf. [6]and [5]). Generally speaking, adaptive logics are ideal instruments to syntactically formal-ize any theory that is the result of selecting models of a usual theory (with a monotonicunderlying logic) by means of a formal criterion. Mathematical adaptive logic theories are

∗The results presented in this paper are (partly) developed in collaboration with Diderik Batens.†This research is financed by the FWO (Fund for Scientific Research – Flanders)

Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, B-9000 Gent, Belgium,E-mail: [email protected]

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usually not semi-recursive, and precisely because of this property they are not limited byGödel’s incompleteness theorems, which can be a useful strength.

In order to be able to devise mathematical adaptive logic theories, one needs a wayto handle unnamed abnormal objects. Unnamed objects are as good as unavoidable inmathematics. In axiomatic theories the non-logical axioms often do not name the objectswhose existence is postulated by the axioms. It is, for example, very common for axiomaticset theories not to name the sets that exist according to the axioms. A prominent exampleis the axiom of (restricted1) comprehension in Zermelo Fraenkel set theory or in naive settheory. It postulates, for an arbitrary property, the existence of a (sub)set that containsall elements that have that specific property.

Also in case of theories with constants that refer to objects, there still might be unnamedobjects. A typical example is Peano Arithmetic (PA). Although the intended objects ofthe theory (the natural numbers) are named by the numerals 0, S0, SS0, and so on, theexistence of non-standard models suggests that not all objects PA quantifies over can benamed by those numerals. Non-standard models exist for most first order mathematicaltheories (in view of the Löwenheim-Skolem theorem), whence the issue of unnamed objectsis ubiquitous in mathematical theories.

Usual adaptive logics are unable to handle these issues properly. Consider a usual incon-sistency adaptive logic such as CLuNsm. At the propositional level, the abnormalities areinconsistencies of primitive formulas, i.e. the set of abnormalities is Ω = A∧¬A | A ∈ P,where P is the set of primitive formulas. This works perfectly and allows concluding q frompremises p, r ∧ ¬r and ¬p ∨ q, because the premises do not require p to be inconsistent(although they do require r to be inconsistent).

At the predicative level, however, the standard approach can be considered quite bluntwhen confronted with of unnamed objets. The set of abnormalities becomes Ω = ∃(A ∧¬A) | A ∈ P, where ∃B denotes the existential closure of the free variables in B and P alsocontains formulas in which free variables occur. For named objects this works as expected:Pa,¬Pa ∨ Qa,Pb,¬Pb ⊢CLuNsm Qa, because the premises do not require a to behaveinconsistently with respect to P . For unnamed objects, however, there is no way to makea distinction between objects with respect to abnormal behavior. Consider the premise set∃x(Px∧¬Px∧¬Rx),∃x(Rx∧Px∧ (¬Px∨Qx)). One would expect that the existentialP -inconsistency would not require all unnamed P to be inconsistent. Nevertheless ∃xQxis not a CLuNsm-consequence of the premises. This is due to the fact that there is onlyone way to recognize unnamed inconsistent objects as abnormal. ∃x(Px∧¬Px) is the onlyderivable abnormality. The interpretation where all (unnamed) objects are P -inconsistentis just as abnormal as the interpretation where only one P -object is abnormal. Evenalthough there is a clear difference between an unnamed object that exists according tothe first premise (it has property ¬R), and the object that exists according to the secondpremise (it has property R), CLuNsm does not allow the assumption that the secondobject is consistent with respect to P , because of the P -inconsistency of the first object.

Adaptive logics are syntactic approaches to selecting the most normal interpretation.Syntactic approach typically suffer from the described bluntness when it comes to unnamed

1The axiom of restricted comprehension is also called the axiom of specification.

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objects. Ontological solutions, on the other hand, can treat unnamed object individually.Intuitively speaking, a syntactic selection of models defines (ab)normalities as formulas ofthe object language of the logic, whereas an ontological approach expresses (ab)normalitiesas metatheoretic properties of the model and the (named and unnamed) objects in itsdomain.

So adaptive logics in their usual form are too blunt to properly handle unnamed objects.But the ontological approaches such as Priest’s LPm, Minimally Inconsistent Logic ofParadox, suffer from problems that are more drastic. They do not have a proof theory andthey do not warrant Strong Reassurance and Classical Recapture. These negative resultsare proven and extensively discussed in [2] and [1].

Let me quickly explain Strong Reassurance. The problem of Strong Reassurance comesto the inability to select a most normal model from a set of models. It occurs with setsof formulas Γ when there are models that have infinitely many abnormal objects and nomodels with only finitely many abnormal objects. I illustrate the problem in its simplestform. Suppose there is a model M0, such that the set of abnormal objects is the set ofnatural numbers N. Then there is also a model M1 such that the set of abnormal objectsis N−0 (just assign all constants to the next object in line and change the assignment ofpredicates accordingly), a model M2, with abnormal objects N− 0, 1, and so on. Usualsensible ontological approaches (based on the set inclusion order relation), will requirethat N is less normal than N − 0 (one more abnormality), N − 0 is less normal thanN−0, 1, and so on. Because this series is infinite, there is no most normal model, whenceno most normal model can be selected.

Adaptive logics in Standard Format, in contrast, always warrant Strong Reassuranceand Classical Recapture. But then again, as mentioned, the usual adaptive logics haveturned out to be too blunt for the unnamed objects that occur in most mathematicaltheories.

Fortunately there is an adaptive logic solution to the bluntness in case of unnamedobjects, without giving up this warrant. Basically it comes to making the abnormalitieslonger. Where an abnormality in an inconsistency adaptive logic is of the ∃(A ∧ ¬A) and¬ denotes the classical negation, the new kind of abnormalities have the form ∃(A∧¬A∧B)∧∃(A∧¬A∧ ¬B). The subformula B enables putting identifying details about the un-named object in the abnormalities. In this way, objects without a name but with differentproperties can be differentiated. Although this is the basic idea, the actual adaptive logicsthat implement this idea require more technicalities due to some smaller additional prob-lems, resulting in a combination of standard adaptive logics. Nevertheless, this technicalsolution is still reasonably elegant and does exactly what one would expect.

In the talk, after having introduced the main problem and having presented the tech-nical solution, I shall argue how this adaptation solves the problem and how it can beunderstood philosophically. I shall show how it works by means of a few examples: twovery clear toy examples, the case of (restricted) comprehension and finally finitistic incon-sistent arithmetic as an extension of PA with an undetermined largest object.

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References

[1] Diderik Batens. Linguistic and ontological measures for comparing the inconsistentparts of models. Logique et Analyse, 165–166:5–33, 1999. Appeared 2002.

[2] Diderik Batens. Minimally abnormal models in some adaptive logics. Synthese, 125:5–18, 2000.

[3] Diderik Batens. A universal logic approach to adaptive logics. Logica Universalis,1:221–242, 2007.

[4] Diderik Batens, Kristof De Clercq, Peter Verdée, and Joke Meheus. Yes fellows, mosthuman reasoning is complex. Synthese, 166:113–131, 2009.

[5] Graham Priest. In Contradiction. A Study of the Transconsistent. Nijhoff, Dordrecht,1987.

[6] Graham Priest. Minimally inconsistent LP. Studia Logica, 50:321–331, 1991.

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On the Crispness of ω and an Arithmetic with aBi-Simulation in a Constructive Naive Set Theory

Shunsuke Yatabe∗

1 Introduction

It is well-known that the comprehension principle does not imply a contradiction on sub-structural logics without the contraction rule. A significance of these naive set theories isto prove a fixed point theorem, a general form of the recursive definition: For any formulaφ(x, · · · , y), the comprehension principle implies there is a term θ such that

(∀x)[x ∈ θ ≡ φ(x, · · · , θ)]

within many substructural logics [1] [5]. This allows us to define, for example, the set ofnatural numbers ω as follows (this is seemingly an inductive definition):

(∀x)[x ∈ ω ≡ [x = 0 ∨ (∃y)[y ∈ ω ∧ x = suc(y)]]]

Similarly, any partial recursive function on ω is numeralwise representable in such theories.To know how much we can develop an arithmetic in these naive set theories is to know thelimit of the power of the general form of the recursive definition.

However, it is not known whether ω is a crisp set, i.e. tertium non datur, (∀x)[(x ∈ω) ∨ (x ∈ ω)], holds1 for ω, or not. It is an important problem because it examinesthe nature of sets defined by a general form of the recursive definition. In this paperwe give a partial answer to this. The framework of this paper is Constructive Naive SetTheory CONS, a naive set theory within Full Lambek predicate calculus with exchangeand weakening rule FLew∀ (which is a intuitionistic predicate logic minus the contractionrule), therefore CONS is very constructive. We prove:

Theorem 1. CONS does not prove tertium non datur holds for ω.

This gives a negative answer to the question: the crispness of ω is not a theorem ofCONS.

The structure of this paper is as follows: we introduce FLew∀ and CONS in section 2.Next we introduce a hidden motivation in section 3: it is to stress how Leibniz equality is

∗Collaborate Research Team for Verification, National Institute of Advanced Industrial Science andTechnology, Japan, [email protected]

1There are some variations of the definition of the crispness: ω is cirsp if t ∈ ω → (t ∈ ω)⊗ (t ∈ ω) forany t (we can apply a kind of the contraction rule to ω) in [3]: it is easy to see that tertium non daturimplies this in Łukasiewicz logic.

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difficult to handle. The difficulty in proving the crispness seems to be due to this. Thereforewe define a bi-simulation relation, which is more easy to use, and its equivalent classes ω.In 4.1, we prove that the analogy of theorem 1 for ω: we construct a non-terminate circularautomaton fix which is a counter-example of tertium non datur for ω. We modify it toshow theorem 1 in section 4.2: we unfold fix to be the counterexample in theorem 1.

2 Preliminaries

Our framework in this paper is Full Lambek predicate calculus with exchange (e) andweakening (w) FLew∀ (which is a intuitionistic predicate logic minus the contraction rule).

α ⊢ α ⊥ ⊢Γ1 ⊢ α Γ2, α,Θ ⊢ β

Γ1,Γ2,Θ ⊢ βcut

Γ ⊢ α β,Π ⊢ γα→ β,Γ,Π ⊢ γ

Γ, α ⊢ βΓ ⊢ α→ β

Multiplicative connectives:

Γ, α, β,Σ ⊢ δΓ, α⊗ β,Σ ⊢ δ

Γ ⊢ α Σ ⊢ βΓ,Σ ⊢ α⊗ β

Additive connectives:

Γ, αi,Σ ⊢ δΓ, α1 ∧ α2,Σ ⊢ δ

Γ,⊢ α Γ ⊢ βΓ,⊢ α ∧ β

Γ, α,Σ ⊢ δ Γ, β,Σ ⊢ δΓ, α ∨ β,Σ ⊢ δ

Γ ⊢ αiΓ ⊢ α1 ∨ α2

Quantifiers (a is not a free variable in Γ ⊢ ∀xα and Γ, ∃xα ⊢ β respectively):

Γ, α[x := s] ⊢ βΓ, ∀xα ⊢ β

Γ ⊢ α[x := a]

Γ ⊢ ∀xαΓ, α[x := a] ⊢ β

Γ, ∃xα ⊢ βΓ ⊢ α[x := s]

Γ ⊢ ∃xα

Structural rules:Γ, β, α,Σ ⊢ δΓ, α, β,Σ ⊢ δ e

Γ ⊢ δΓ, α,Σ ⊢ δ w

We note that BCK-logic is its fragment, and it is crucial for the proof of the main theoremthat FLew∀ satisfies the disjunction property and the existence property.

Definition 2. Let CONS be a set theory within FLew∀, which has a binary predicate ∈and terms of the form x : φ(x), and the following two ∈-rules:

α[x := s],Γ ⊢ βs ∈ x : α,Γ ⊢ β

Γ ⊢ α[x := a]

Γ ⊢ a ∈ x : α

It is easy to see that CONS enjoyes cut elimination and proves the fixed point theorem.We note that CONS is equal to Uwe Petersen’s LiDλ except his notation [4]. As LDi

λ,any λ-term can be defined in CONS, therefore we have the full form of Curry-Howardisomorphism by itself: any proof of CONS corresponds some λ-term defined in CONS.As for an arithmetic, we note that the definition of 0 and suc(y) has many choices: 0 isdefied as ∅ = x : ⊥ and suc(y) is defined as y in Zermelo style, for example, in [3].

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3 Non-extensionality and bi-simulation ∼Identity is a key to develop mathematics. Leibniz equality, a = b ≡ (∀z)[a ∈ z ≡ b ∈ z],has been widely used in the study of naive set theories, and it is a contractive relation:

Lemma 3. For any s, t, CONS proves

s = t→ (s = t)⊗ (s = t)

= is too strict to use because of the contractiveness. For example, it is well-known thatit is different from the extensional equality, a =ext b ≡ (∀z)[z ∈ a ≡ x ∈ b]. For the axiomof extensionality implies a contradiction in many naive set theories [2]. So Leibniz equalityis more like syntactical identity (Grisin logic is classical logic minus contraction rule [1]):

Lemma 4 (The literal identity property). A naive set theory within Grisin logic provesthat if ⊢ t = u is provable then t and u are syntactically identical.

We note that Terui proved the similar theorem for LAST [5]. The proof is just aneasy application of cut-elimination theorem (an analogy of this lemma holds for any logicwhich can enjoy cut-elimination).

Such syntactical character seems to prevent developing arithmetic nicely. For example,the fixed point theorem only proves the existence of θ, and it does not guarantee theuniqueness of such terms. Therefore we cannot prove the uniqueness of sets, like ω, definedby the fixed point theorem. Thus, if we use =, we cannot take the full advantage of thepowerful forces of a general form of the recursive definition. However, when consideringthe mathematical proposition, = is not necessary, so that a weaker identity relation shouldbe reasonable.

How about to use the extensional equality =ext? There are a few problems2. Here weconcentrate on this problem: =ext is not enough to be a base of developing an arithmetic,i.e. it is still too strict. For example, even if a =ext 0, a =ext 1. This means that, twoseries,

• 0, suc(0), suc(suc(0)), suc(suc(suc(0))), . . .

• a, suc(a), suc(suc(a)), suc(suc(suc(a))), . . .

are completely different with respect to =ext. Therefore =ext is not enough base to developan arithmetic on. In this paper, we define an identity relation in the naive set theory sothat we can take the full advantages of the ability of the fixed point theorem, and it hassome advantages of developing an arithmetic under it.

To solve the last problem, this paper introduces a bi-simulation relation ∼ as an alter-native identity relation for developing an arithmetic. The motivation for introducing ∼ isto express the hereditary extensional equality.

2First, unfortunately, =ext is not crisp, which would mean arithmetic with a generalized recursivedefinition is essentially non-crisp. Second, it does not satisfy a =ext b→ [φ(a) ≡ φ(b)] for any φ. However,it might not be a problem because this might hold for any arithmetical formulae.

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Definition 5. ∼ is a binary relation which is defined as follows:

• (∀x, y)[x =ext y → x ∼ y],

• 0 ∼ a ≡ [(∀y)(∀x)x ∈ a ≡ x ∈ suc(y)], i.e. a =ext suc(b) for any b,

• suc(a) ∼ suc(b) ≡ a ∼ b.

∼ is a bi-simulation relation over natural numbers: a ∼ b represents that a’s behaviorwith respect to suc is similar to that of b’s.

Next, for any natural number m ∈ ω, we define an equivalence class [m] by ∼ of m.We also define ⟨ω,∼⟩ as follows:

Definition 6. • For any a, [a] = x : x ∼ a,

• ω is a set of ∼-equivalence classes whose representative element is a natural number:

ω = [n] : n ∈ ω

We can develop an arithmetic over ⟨ω,∼⟩ by using the general form of the recursivedefinition. Hereafter we write a ∈ ω in case (∃x ∈ ω)a ∈ [x].

4 The non-crispness of ω

By using the fixed point lemma and the bi-simulation, we can implement automata as (non-crisp) members of ω. In this section, we introduce that we can prove the unprovability ofthe crispness of both ω and ω by using a non-terminate automaton.

4.1 The non-crispness of ω

First, let us introduce a simple automaton fix such that fixsuc−→ fix.

Definition 7. fix is a fixed point of the successor function suc with respect to ∼:

suc(fix) ∼ fix

More precisely, fix is defined by the fixed point theorem : (∀x)[(x ∈ fix) ≡ (x ∈ suc(fix))].

We note that fix =ext x : x = fix if we use Zermelo ordinals, and it is crisp becauseof the crispness of =. From a viewpoint of behavior, all standard natural numbers (crispmembers of ω) are automata which eventually terminates. However, in the case of fix, thisautomaton never terminates. Therefore we can see that fix is a non-standard element ofω in the following sense:

Lemma 8. (1) CONS does not prove fix ∈ ω,

(2) CONS does not prove fix ∈ ω

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Proof. Let us prove (1). Assume otherwise: ⊢ “fix ∈ ω”. Then, there should be a finiteproof such that its consequence is ⊢ “(∃x ∈ ω)x ∼ fix”. Its predecessor should be ⊢“suc(m) ∼ fix” for some m ∈ ω3. Therefore m ∼ fix by the definition of fix. In this way,the proof is an infinite regress, and the proof never achieves the bottom case, w ∼ 0 forsome w, in finite steps. Therefore there is no finite proof of ⊢ “fix ∈ ω”.

As for (2), assume otherwise: this means that “x ∼ fix” ⊢ ⊥. Let us remember how suchproof looks like. The proof of “x ∼ t” ⊢ ⊥ is as follows: “x ∼ t” ⊢ “x− 1 ∼ t”4

and “(x− 1) ∼ t” ⊢ “(x− 2) ∼ t”, etc. Finally, it achieves the sequent “∗ ⊢ (x−m) ∼ b”for some b which is one of the bottom cases, e.g. b =ext ∅ (in Zermelo style), etc. It implies⊢ ⊥ when x −m = 0. However, in the case of fix, again, the proof should be an infiniteregress.

This proof shows that neither the proposition which construct the fixed point fix on suc(unique up to ∼) nor its negation can be proved in the finite length in CONS.

Therefore, we have

Lemma 9. CONS does not prove (fix ∈ ω) ∨ (fix ∈ ω).

Proof: Otherwise, since CONS satisfies the disjunction property, fix ∈ ω or fix ∈ ωis a theorem of CONS.

4.2 The non-crispness of ω: unfolding fix

In this section, we prove theorem 1. The problem to prove theorem 1 is as follows: fix isclearly not a member of ω. For it is not of the form x : x = a since it is constructed byusing the fixed point theorem. Therefore let us define a (possibly partial) function whichunfolds fix.

Definition 10 (rank). rk is a relation over set and (possibly) natural numbers:

⟨x, y⟩ ∈ rk ≡ [(x ∼ 0 ∧ y = 0) ∨ (∃z0, z1)[⟨z0, z1⟩ ∈ rk ∧ x =ext z0 ∧ y = suc(z1)]]

Roughly speaking, rk unfolds the nested box, and counts how many singletons arenested. We note that the formula “s =ext t”, i.e. (∀z)[z ∈ s ≡ z = t], is crisp because ofthe crispness of =, and any natural numbers are in the range of rk.

We do not know whether rk can unfold fix, i.e. fix ∈ dom(rk), or not. However, atleast we can say that CONS does not reject fix ∈ dom(rk), i.e there is no s such that⟨fix, s⟩ ∈ rk. For, if so, it proves ¬(∃x ∈ ω)x ∼ fix, which contradicts lemma 8. So let usextend the CONS by adding the following axiom:

Definition 11. CONS is an extension of CONS by adding

• the new axiom “fix ∈ dom(rk)”,

• the new constant c which satisfies ⟨fix, c⟩ ∈ rk.3Its m-steps predecessor should be “0 ∼ t” where fix ∼ suc(suc(· · · (t) · · · )).4Let us assume x−m could be a negative number. I omit the detail.

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The intuitive image of c is suc(suc(suc(· · · ))), an infinite stream of suc (i.e. · · · in Zermelo style). Therefore, we can prove the analogy of lemma 8 as follows:

Lemma 12. (1) CONS does not prove c ∈ ω,

(2) CONS does not prove c ∈ ω,

(3) CONS does not prove c ∈ ω ∨ c ∈ ω.

Proof. Otherwise, we can prove the negation of lemma 8.

This gives a proof of theorem 1: the theorem is proved since CONS, an extension ofCONS, cannot prove (∃x)x ∈ ω ∨ x ∈ ω.

5 Conclusion

We proved that constructive naive set theory CONS does not prove tertium non datur forω. Since it implies the crispness, this gives a negative answer to the question.

We remark that we introduced ⟨ω,∼⟩ to show the above theorem, and this system itselfseems to have many interesting aspects. For, the theorem shows that we can never provethe negation of that ω, which is seemingly defined inductively, contains an infinite objectΦ. This highlights the co-inductive character of CONS.

References

[1] Cantini, Andrea. 2003. The undecidability of Grisın’s set theory. Studia logica 74:345-368

[2] Grisin, V. N. 1982. Predicate and set-theoretic caliculi based on logic without con-tractions. Math. USSR Izvestija 18: 41-59.

[3] Hajek, Petr. 2005. On arithmetic in the Cantor-Łukasiewicz fuzzy set theory.Archive for Mathematical Logic 44(6): 763 - 82.

[4] Petersen, Uwe. 2000. Logic Without Contraction as Based on Inclusion and Unre-stricted Abstraction. Studia Logica 64: 365-403.

[5] Terui, Kazushige 2004 . Light Affine Set Theory: A Naive Set Theory of PolynomialTime. Studia Logica 77: 9-40.

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