
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

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

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

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

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

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

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.

21A New Approach to Predicative Set TheoryIn Ralf Schindler (ed.), Ways of Proof Theory, De Gruyter. pp. 3164. 2010.We suggest a new framework for the WeylFeferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF , and is suitable for mechanization. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they deﬁne in an absolute way, independent of the extension of the “surrounding universe”. The language of P ZF is typefree, and it reﬂects real mathematical practice in making an extensi…Read more

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

27Multivalued Calculi for Logics Based on NondeterminismLogic Journal of the IGPL 13 (4): 365387. 2005.Nondeterministic matrices are multiplevalued structures in which the value assigned by a valuation to a complex formula can be chosen nondeterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one . We use the RasiowaSikorski decomposition methodology to get sound and complete proof systems employing finite sets of mvsigned formulas for all propositional logics based on such structu…Read more

5Around 1950, B.A. Trakhtenbrot proved an important undecidability result (known, by a pure accident, as \Trakhtenbrot's theorem"): there is no algorithm to decide, given a rstorder sentence, whether the sentence is satis able in some nite model. The result is in fact true even if we restrict ourselves to languages that has only one binary relation Tra63]. It is hardly conceivable that at that time Prof. Trakhtenbrot expected his result to in uence the development of the theory of relational dat…Read more

7Implicational 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

22The Classical Constraint on RelevanceLogica Universalis 8 (1): 115. 2014.We show that as long as the propositional constants t and f are not included in the language, any languagepreserving extension of any important fragment of the relevance logics R and RMI can have only classical tautologies as theorems . This property is not preserved, though, if either t or f is added to the language, or if the contraction axiom is deleted

119Ideal Paraconsistent LogicsStudia Logica 99 (13): 3160. 2011.We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every threevalued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n valued logics, each one of …Read more

34Decomposition proof systems for gödelDummett logicsStudia Logica 69 (2): 197219. 2001.The main goal of the paper is to suggest some analytic proof systems for LC and its finitevalued counterparts which are suitable for proofsearch. This goal is achieved through following the general RasiowaSikorski methodology for constructing analytic proof systems for semanticallydefined logics. All the systems presented here are terminating, contractionfree, and based on invertible rules, which have a local character and at most two premises

8Review: John C. Mitchell, Foundations for Programming Languages (review)Journal of Symbolic Logic 64 (2): 918922. 1999.

Tel Aviv UniversityResearcher

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