
4It is well known that every propositional logic which satisﬁes certain very natural conditions can be characterized semantically using a multivalued matrix ([Los and Suszko, 1958; W´ ojcicki, 1988; Urquhart, 2001]). However, there are many important decidable logics whose characteristic matrices necessarily consist of an inﬁnite number of truth values. In such a case it might be quite diﬃcult to ﬁnd any of these matrices, or to use one when it is found. Even in case a logic does have a ﬁnite ch…Read more

82The Semantics and Proof Theory of Linear LogicTheoretical Computer Science 57 (2): 161184. 1988.Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding str ong completeness theorems. Finally, we shall i…Read more

60Reasoning with logical bilatticesJournal of Logic, Language and Information 5 (1): 2563. 1996.The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our work. The …Read more

1FourValued Diagnoses for Stratified KnowledgeBasesIn Dirk van Dalen & Marc Bezem (eds.), Computer Science Logic, Springer. pp. 117. 1997.We present a fourvalued approach for recovering consistent data from inconsistent set of assertions. For a common family of knowledgebases we also provide an e cient algorithm for doing so automaticly. This method is particularly useful for making modelbased diagnoses

2An (n, k)ary quantiﬁer is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)ary quantiﬁers form a natural class of Gentzentype systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantiﬁer is introduced. The semantics for these systems is provided using twovalued nondeterministic matrices, a generalization of the classical matrix. In this paper we use a …Read more

8We develop a uniﬁed framework for dealing with constructibility and absoluteness in set theory, decidability of relations in eﬀective structures (like the natural numbers), and domain independence of queries in database theory. Our framework and results suggest that domainindependence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability

87Natural 3valued logics—characterization and proof theoryJournal of Symbolic Logic 56 (1): 276294. 1991.

9One of the most signi cant drawbacks of classical logic is its being useless in the presence of an inconsistency. Nevertheless, the classical calculus is a very convenient framework to work with. In this work we propose means for drawing conclusions from systems that are based on classical logic, although the informationmightbe inconsistent. The idea is to detect those parts of the knowledgebase that \cause" the inconsistency, and isolate the parts that are \recoverable". We do this by temporar…Read more

11Multiplicative Conjunction and an Algebraic Meaning of Contraction and WeakeningJournal of Symbolic Logic 63 (3): 831859. 1998.We show that the elimination rule for the multiplicative conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL$_m$ and RMI$_m$ An exception is R$_m$. Let SLL$_m$ and SR$_m$ be, respectively, the systems which are obtained from LL$_m$ and R$_m$ by adding this rule as a new rule of inference. The set of theorems of SR$_m$ is a proper extension of that of R$_m$, but a proper subset of the set of theorems of RMI$_m$. Hence it still has the variabl…Read more

97What is relevance logic?Annals of Pure and Applied Logic 165 (1): 2648. 2014.We suggest two precise abstract definitions of the notion of ‘relevance logic’ which are both independent of any proof system or semantics. We show that according to the simpler one, R → source is the minimal relevance logic, but R itself is not. In contrast, R and many other logics are relevance logics according to the second definition, while all fragments of linear logic are not

41Relevance and paraconsistencya new approach. II. The formal systemsNotre Dame Journal of Formal Logic 31 (2): 169202. 1990.

59Implicational fstructures and implicational relevance logicsJournal of Symbolic Logic 65 (2): 788802. 2000.We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and prooftheoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Prooftheoretically they correspond to cutfree hypersequential Gentzenty…Read more

25There is a long tradition (See e.g. [9, 10]) starting from [12], according to which the meaning of a connective is determined by the introduction and elimination rules which are associated with it. The supporters of this thesis usually have in mind natural deduction systems of a certain ideal type (explained in Section 3 below). Unfortunately, already the handling of classical negation requires rules which are not of that type. This problem can be solved in the framework of multipleconclusion G…Read more

37FourValued Paradefinite LogicsStudia Logica 105 (6): 10871122. 2017.Paradefinite logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for introducing paradefinite logics. It consists of the fourvalued matrices that expand the minimal matrix which is characteristic for first degree entailments: Dunn–Belnap matrix. We survey and study the expressive power and proof theory of the most important logi…Read more

68Relevant entailmentsemantics and formal systemsJournal of Symbolic Logic 49 (2): 334342. 1984.

100Cutfree ordinary sequent calculi for logics having generalized finitevalued semanticsLogica Universalis 1 (1): 4170. 2007.. The paper presents a method for transforming a given sound and complete nsequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finitevalued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truthvalue of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have des…Read more

54A Nondeterministic View on Nonclassical NegationsStudia Logica 80 (23): 159194. 2005.We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzentype rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics whic…Read more

42Multivalued Semantics: Why and HowStudia Logica 92 (2): 163182. 2009.According to Suszko's Thesis,any multivalued semantics for a logical system can be replaced by an equivalent bivalent one. Moreover: bivalent semantics for families of logics can frequently be developed in a modular way. On the other hand bivalent semantics usually lacks the crucial property of analycity, a property which is guaranteed for the semantics of multivalued matrices. We show that one can get both modularity and analycity by using the semantic framework of multivalued nondeterminis…Read more

26CutElimination and Quantification in Canonical SystemsStudia Logica 82 (1): 157176. 2006.Canonical Propositional Gentzentype systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the subformula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to firstorder languages and beyond. We extend the Propositional coherence criterion for the n…Read more

1A FormulaPreferential Base for Paraconsistent and Plausible Reasoning SystemsIn Arnon Avron & Iddo Lev (eds.), Proceedings of the Workshop on Inconsistency in Data and Knowledge, . pp. 6070. 2001.We provide a general framework for constructing natural consequence relations for paraconsistent and plausible nonmonotonic reasoning. The framework is based on preferential systems whose preferences are based on the satisfaction of formulas in models. We show that these natural preferential In the research on paraconsistency, preferential systems systems that were originally designed for for paraconsistent reasoning fulfill a key condition (stopperedness or smoothness) from the theoretical res…Read more

7We construct a modular semantic frameworks for LFIs (logics of formal (in)consistency) which extends the framework developed in [1; 3], but includes Marco’s schema too (and so practically all the axioms considered in [11] plus a few more). In addition, the paper provides another demonstration of the power of the idea of nondeterministic semantics, especially when it is combined with the idea of using truthvalues to encode relevant data concerning propositions

33Two types of multipleconclusion systemsLogic Journal of the IGPL 6 (5): 695718. 1998.Hypersequents are finite sets of ordinary sequents. We show that multipleconclusion sequents and singleconclusion hypersequents represent two different natural methods of switching from a singleconclusion calculus to a multipleconclusion one. The use of multipleconclusion sequents corresponds to using a multiplicative disjunction, while the use of singleconclusion hypersequents corresponds to using an additive one. Moreover: each of the two methods is usually based on a different natural s…Read more

44Gentzenizing SchroederHeister's natural extension of natural deductionNotre Dame Journal of Formal Logic 31 (1): 127135. 1989.

20Formulas for which contraction is admissibleLogic Journal of the IGPL 6 (1): 4348. 1998.A formula A is said to have the contraction property in a logic L if whenever A, A, Γ ⊨ L B also A, Γ & ; L B. In MLL and in MALL without the additive constants a formula has the contraction property if it is a theorem. Adding the mix rule does not change this fact. In MALL and in affine logic A has the contraction property if either A is provable of A is equivalent to the additive constant 0. We present some general prooftheoretical principles from which all these results easily follow

15We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and nonmonotonic logics) and for a general, semanticsindependent classi cation of standard connectives via equations on consequence relations (these include Girard's \multiplicati…Read more

17Combining classical logic, paraconsistency and relevanceJournal of Applied Logic 3 (1): 133160. 2005.

62On modal systems having arithmetical interpretationsJournal of Symbolic Logic 49 (3): 935942. 1984.

Tel Aviv UniversityResearcher

Tel Aviv UniversityRegular Faculty
Tel Aviv, Israel
Areas of Interest
Logic and Philosophy of Logic 
Philosophy of Mathematics 