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  1. Arnon Avron, Automatic Diagnoses for Properly Stratified Knowledge-Bases.
    we also provide an efficient algorithm for recovering this data. We then illustrate the ideas in a diagnostic system for checking faulty circuits. The underlying formalism is..
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  2. Arnon Avron, A Formula-Preferential Base for Paraconsistent and Plausible Reasoning Systems.
    in models. We show that these natural preferential In the research on paraconsistency, preferential systems systems that were originally designed for paraconwere used for constructing logics which are paraconsistent sistent reasoning fulfill a key condition (stopperedbut stronger than substructural paraconsistent logics. The ness or smoothness) from the theoretical research preferences in these systems were defined in different ways. of nonmonotonic reasoning. Consequently, the Some were based on checking which abnormal formulas nonmonotonic consequence relations that they in-.
     
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  3. Arnon Avron, A Framework for Formalizing Set Theories Based on the Use of Static Set Terms.
    We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF . It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus” for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add ∈-induction and optionally the axiom of choice). Comprehension is formulated as: x ∈ {x | ϕ} ↔ ϕ, where (...)
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  4. Arnon Avron, A Model-Theoretic Approach for Recovering Consistent Data From Inconsistent Knowledge-Bases.
    One 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 knowledge-base that \cause" the inconsistency, and isolate the parts that are \recoverable". We do this by temporarily switching into (...)
     
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  5. Arnon Avron, A New Approach to Predicative Set Theory.
    We suggest a new framework for the Weyl-Feferman 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 type-free, and it reflects real mathematical practice in making an extensive use of statically (...)
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  6. Arnon Avron, A Note on the Structure of Bilattices.
    The notion of a bilattice was rst introduced by Ginsburg (see Gin]) as a general framework for a diversity of applications (such as truth maintenance systems, default inferences and others). The notion was further investigated and applied for various purposes by Fitting (see Fi1]- Fi6]). The main idea behind bilattices is to use structures in which there are two (partial) order relations, having di erent interpretations. The two relations should, of course, be connected somehow in order for the mathematical structure (...)
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  7. Arnon Avron, A Simple Proof of Completeness and Cut-Elimination for Propositional G¨ Odel Logic.
    We provide a constructive, direct, and simple proof of the completeness of the cut-free part of the hypersequential calculus for G¨odel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). We then extend the results and proofs to derivations from assumptions, showing that such derivations can be confined to those in which cuts are made only on formulas which occur in the assumptions.
     
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  8. Arnon Avron, Constructibility and Decidability Versus Domain Independence and Absoluteness.
    We 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 domain-independence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability.
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  9. Arnon Avron, Canonical Constructive Systems ⋆.
    We define the notions of a canonical inference rule and a canonical system in the framework of single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and prove that such a canonical system is non-trivial iff it is coherent (where coherence is a constructive condition). Next we develop a general non-deterministic Kripke-style semantics for such systems, and show that every constructive canonical system (i.e. coherent canonical single-conclusion system) induces a class of non-deterministic Kripke-style frames for which it is strongly sound and (...)
     
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  10. Arnon Avron, Canonical Calculi with (N,K)-Ary Quantifiers.
    Propositional canonical Gentzen-type systems, introduced in [2], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. [2] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [23] extends these results (...)
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  11. Arnon Avron, Four-Valued Diagnoses for Strati Ed Knowledge-Bases.
    We present a four-valued approach for recovering consistent data from inconsistent set of assertions. For a common family of knowledge-bases we also provide an e cient algorithm for doing so automaticly. This method is particularly useful for making model-based diagnoses.
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  12. Arnon Avron, Formulas for Which Contraction is Admissible.
    A formula A is said to have the contraction property in a logic L i whenever A;A;? `L B (when ? is a multiset) also A;? `L B. In MLL and in MALL without the additive constants a formula has the contractionproperty i it is a theorem. Adding the mix rule does not change this fact. In MALL (with or without mix) and in a ne logic A has the contraction property i either A is provable or A is equivalent (...)
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  13. Arnon Avron, General Patterns for Nonmonotonic Reasoning: From Basic Entailments to Plausible Relations.
    This paper has two goals. First, we develop frameworks for logical systems which are able to re ect not only nonmonotonic patterns of reasoning, but also paraconsistent reasoning. Our second goal is to have a better understanding of the conditions that a useful relation for nonmonotonic reasoning should satisfy. For this we consider a sequence of generalizations of the pioneering works of Gabbay, Kraus, Lehmann, Magidor and Makinson. These generalizations allow the use of monotonic nonclassical logics as the underlying logic (...)
     
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  14. Arnon Avron, Gentzen-Type Systems, Resolution and Tableaux.
    In advanced books and courses on logic (e.g. Sm], BM]) Gentzen-type systems or their dual, tableaux, are described as techniques for showing validity of formulae which are more practical than the usual Hilbert-type formalisms. People who have learnt these methods often wonder why the Automated Reasoning community seems to ignore them and prefers instead the resolution method. Some of the classical books on AD (such as CL], Lo]) do not mention these methods at all. Others (such as Ro]) do, but (...)
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  15. Arnon Avron, Logical Non-Determinism as a Tool for Logical Modularity: An Introduction.
    It is well known that every propositional logic which satisfies certain very natural conditions can be characterized semantically using a multi-valued 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 (...)
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  16. Arnon Avron, LFIs with Marco's Schema.
    We 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 truth-values to encode relevant data concerning propositions.
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  17. Arnon Avron, Multi-Valued Calculi for Logics Based on Non-Determinism.
    Non-deterministic matrices (Nmatrices) are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically 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 (the latter is new here). We use the Rasiowa-Sikorski (R-S) decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such (...)
     
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  18. Arnon Avron, Many-Valued Non-Deterministic Semantics for First-Order Logics of Formal (in)Consistency.
    A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family (...)
     
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  19. Arnon Avron, Many-Valued Non-Deterministic Semantics for First-Order Logics of Formal (In)Consistency.
    A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family (...)
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  20. Arnon Avron, Non-Deterministic Matrices and Modular Semantics of Rules.
    We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multi-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set (...)
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  21. Arnon Avron, Non-Deterministic Semantics for Logics with a Consistency Operator.
    In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. Da Costa’s approach has led to the family of logics (...)
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  22. Arnon Avron, On Negation, Completeness and Consistency.
    We have avoided here the term \false", since we do not want to commit ourselves to the view that A is false precisely when it is not true. Our formulation of the intuition is therefore obviously circular, but this is unavoidable in intuitive informal characterizations of basic connectives and quanti ers.
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  23. Arnon Avron, Processing Information.
    We introduce a general framework for solving the problem of a computer collecting and combining information from various sources. Unlike previous approaches to this problem, in our framework the sources are allowed to provide information about complex formulae too. This is enabled by the use of a new tool — non-deterministic logical matrices. We also consider several alternative plausible assumptions concerning the framework. These assumptions lead to various logics. We provide strongly sound and complete proof systems for all the basic (...)
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  24. Arnon Avron, Strong Cut-Elimination, Coherence, and Non-Deterministic Semantics.
    An (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 Gentzen-type 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 two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence (...)
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  25. Arnon Avron, Simple Consequence Relations.
    We 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 non-monotonic logics) and for a general, semantics-independent classi cation of standard connectives via equations on consequence relations (these include Girard's \multiplicatives" and \additives"). We next investigate the (...)
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  26. Arnon Avron, Stability, Sequentiality and Demand Driven Evaluation in Data Ow.
    We show that a given data ow language l has the property that for any program P and any demand for outputs D (which can be satis ed) there exists a least partial computation of P which satis es D, i all the operators of l are stable. This minimal computation is the demand-driven evaluation of P. We also argue that in order to actually implement this mode of evaluation, the operators of l should be further restricted to be e (...)
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  27. Arnon Avron, Safety Signatures for First-Order Languages and Their Applications.
    In several areas of Mathematical Logic and Computer Science one would ideally like to use the set F orm(L) of all formulas of some first-order language L for some goal, but this cannot be done safely. In such a case it is necessary to select a subset of F orm(L) that can safely be used. Three main examples of this phenomenon are: • The main principle of naive set theory is the comprehension schema: ∃Z(∀x.x ∈ Z ⇔ A).
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  28. Arnon Avron, Tonk- A Full Mathematical Solution.
    There 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 multiple-conclusion Gentzen-type (...)
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  29. Arnon Avron, The Method of Hypersequents in the Proof Theory of Propositional Non-Classical Logics.
    Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by many researchers (...)
     
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  30. Arnon Avron, The Semantics and Proof Theory of Linear Logic.
    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 investigate (...)
     
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  31. Arnon Avron, Two Types of Multiple-Conclusion Systems.
    Hypersequents are nite sets of ordinary sequents. We show that multiple-conclusion sequents and single-conclusion hypersequents represent two di erent natural methods of switching from a singleconclusioncalculusto a multiple-conclusionone. The use of multiple-conclusionsequentscorresponds to using a multiplicative disjunction, while the use of single-conclusionhypersequents corresponds to using an additive one. Moreover: each of the two methods is usually based on a di erent natural semantic idea and accordingly leads to a di erent class of algebraic structures. In the cases we consider here (...)
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  32. Arnon Avron, 5-Valued Non-Deterministic Semantics for The Basic Paraconsistent Logic mCi.
    One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5-valued characteristic nondeterministic matrix for mCi. This provides a quite non-trivial example for the utility and effectiveness of the use of non-deterministic many-valued semantics.
     
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  33. Arnon Avron, What Reasonable First-Order Queries Are Permitted by Trakhtenbrot's Theorem?
    Around 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 rst-order 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 databases (...)
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  34. Arnon Avron & Anna Zamansky, A Triple Correspondence in Canonical Calculi: Strong Cut-Elimination, Coherence, and Non-Deterministic Semantics.
    An (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 Gentzen-type 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 two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence (...)
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  35. Arnon Avron (2014). Paraconsistency, Paracompleteness, Gentzen Systems, and Trivalent Semantics. Journal of Applied Non-Classical Logics 24 (1-2):12-34.
    A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form , or of the form , and all the active formulas of its premises belong to the set . In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence criterion (...)
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  36. Arnon Avron (2014). The Classical Constraint on Relevance. Logica Universalis 8 (1):1-15.
    We show that as long as the propositional constants t and f are not included in the language, any language-preserving extension of any important fragment of the relevance logics R and RMI can have only classical tautologies as theorems (this includes intuitionistic logic and its extensions). This property is not preserved, though, if either t or f is added to the language, or if the contraction axiom is deleted.
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  37. Arnon Avron (2014). What is Relevance Logic? Annals of Pure and Applied Logic 165 (1):26-48.
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  38. Anna Zamansky & Arnon Avron (2012). Canonical Signed Calculi with Multi-Ary Quantifiers. Annals of Pure and Applied Logic 163 (7):951-960.
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  39. Ofer Arieli, Arnon Avron & Anna Zamansky (2011). Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics. Studia Logica 97 (1):31 - 60.
    Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued (...)
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  40. Arnon Avron, Oskar Becker, Johan van Benthem, Andreas Blass, Robert Brandom, L. E. J. Brouwer, Donald Davidson, Michael Dummett, Walter Felscher & Kit Fine (2009). Jagadeesan, Radha, 306 Japaridze, Giorgi, Xi. In Ondrej Majer, Ahti-Veikko Pietarinen & Tero Tulenheimo (eds.), Games: Unifying Logic, Language, and Philosophy. Springer Verlag. 377.
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  41. Arnon Avron & Beata Konikowska (2009). Proof Systems for Reasoning About Computation Errors. Studia Logica 91 (2):273 - 293.
    In the paper we examine the use of non-classical truth values for dealing with computation errors in program specification and validation. In that context, 3-valued McCarthy logic is suitable for handling lazy sequential computation, while 3-valued Kleene logic can be used for reasoning about parallel computation. If we want to be able to deal with both strategies without distinguishing between them, we combine Kleene and McCarthy logics into a logic based on a non-deterministic, 3-valued matrix, incorporating both options (...)
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  42. Arnon Avron, Jonathan Ben-Naim & Beata Konikowska (2007). Cut-Free Ordinary Sequent Calculi for Logics Having Generalized Finite-Valued Semantics. Logica Universalis 1 (1):41-70.
    . The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general (...)
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  43. Anna Zamansky & Arnon Avron (2006). Cut-Elimination and Quantification in Canonical Systems. Studia Logica 82 (1):157 - 176.
    Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula 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 first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with (...)
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  44. Arnon Avron (2005). A Non-Deterministic View on Non-Classical Negations. Studia Logica 80 (2-3):159 - 194.
    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 Gentzen-type 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 which is (...)
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  45. Arnon Avron & Beata Konikowska (2001). Decomposition Proof Systems for Gödel-Dummett Logics. Studia Logica 69 (2):197-219.
    The main goal of the paper is to suggest some analytic proof systems for LC and its finite-valued counterparts which are suitable for proof-search. This goal is achieved through following the general Rasiowa-Sikorski methodology for constructing analytic proof systems for semantically-defined logics. All the systems presented here are terminating, contraction-free, and based on invertible rules, which have a local character and at most two premises.
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  46. Arnon Avron (1999). Review: John C. Mitchell, Foundations for Programming Languages. [REVIEW] Journal of Symbolic Logic 64 (2):918-922.
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  47. Arnon Avron, Furio Honsell, Marino Miculan & Cristian Paravano (1998). Encoding Modal Logics in Logical Frameworks. Studia Logica 60 (1):161-208.
    We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deduction-style proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO.
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  48. Ofer Arieli & Arnon Avron (1996). Reasoning with Logical Bilattices. Journal of Logic, Language and Information 5 (1):25--63.
    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. (...)
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  49. Arnon Avron (1994). What is a Logical System? In Dov M. Gabbay (ed.), What is a Logical System? Oxford University Press.
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  50. Arnon Avron (1992). Whither Relevance Logic? Journal of Philosophical Logic 21 (3):243 - 281.
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