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An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
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Cited by 139 (16 self)
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Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
Logical Pluralism
 To appear, Special Logic issue of the Australasian Journal of Philosophy
, 2000
"... Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism th ..."
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Cited by 23 (5 self)
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Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism there is not one true logic; there are many. There is not always a single answer to the question “is this argument valid?” 1 Logic, Logics and Consequence Anyone acquainted with contemporary Logic knows that there are many socalled logics. 1 But are these logics rightly socalled? Are any of the menagerie of nonclassical logics, such as relevant logics, intuitionistic logic, paraconsistent logics or quantum logics, as deserving of the title ‘logic ’ as classical logic? On the other hand, is classical logic really as deserving of the title ‘logic ’ as relevant logic (or any of the other nonclassical logics)? If so, why so? If not, why not? Logic has a chief subject matter: Logical Consequence. The chief aim of
Natural Deduction for NonClassical Logics
, 1996
"... We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke m ..."
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Cited by 11 (3 self)
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We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.
Categorial Grammars with Negative Information
 A notion in focus, de Gruyter
, 1995
"... this paper we discuss some possibilities of introducing negative information in the formalism of categorial grammar. Traditionally, formal grammars, including categorial grammars, admit positive information only. For instance, one postulates the rule S)NP,VP but not \GammaS)NP,\GammaVP; here \GammaA ..."
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Cited by 4 (2 self)
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this paper we discuss some possibilities of introducing negative information in the formalism of categorial grammar. Traditionally, formal grammars, including categorial grammars, admit positive information only. For instance, one postulates the rule S)NP,VP but not \GammaS)NP,\GammaVP; here \GammaA stands for the negation (complement) of category A.
Implementing Modal and Relevance Logics in a Logical Framework
, 1996
"... We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, ..."
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Cited by 2 (2 self)
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We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework. 1 INTRODUCTION The origins of natural deduction (ND) are both philosophical and practical. In philosophy, it arises from an analysis of deductive inference in an attempt to provide a theory of meaning for the logical connectives [24, 33]. Practically, it provides a language for building proofs, which can be seen as providing the deduction theorem directly, rather than as a derived result. Our interest is on this practical side, and a development of our work on ap...
Duality Theorems for Partial Orders, Semilattices, Galois Connections and Lattices (IULG Preprint)
, 1993
"... LatticeOrdered Stone Spaces are shown to be the dual spaces of partial orders or meet semilattices. These results are subsequently extended to obtain a duality between galois connections and ?frames. Galois connections are viewed as negationlike operators. The representation of galois connections ..."
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Cited by 1 (0 self)
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LatticeOrdered Stone Spaces are shown to be the dual spaces of partial orders or meet semilattices. These results are subsequently extended to obtain a duality between galois connections and ?frames. Galois connections are viewed as negationlike operators. The representation of galois connections leads us to a representation theorem for lattices, using the identity homomorphism on a lattice L. Rephrased, the representation theorem for lattices asserts that for any lattice L there is a complete, concrete Boolean algebra B, and a closure operator c : B ! B, such that L can be imbedded in the complete lattice of stable elements of B. B can be taken to be the powerset of a latticeordered Stone Space, in which case L is identified as the lattice of all clopen and stable subsets of the space. Representation is subsequently extended to a duality theorem for lattices and canonical Lframes. . Duality Theorems for Partial Orders, Semilattices, Galois Connections and Lattices Chrysafis H...
An O(n log n)Space Decision Procedure for the Relevance Logic B+
, 2000
"... In previous work we gave a new prooftheoretical method for establishing upperbounds on the space complexity of the provability problem of modal and other propositional nonclassical logics. Here we extend and rene these results to give an O(n log n)space decision procedure for the basic posit ..."
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In previous work we gave a new prooftheoretical method for establishing upperbounds on the space complexity of the provability problem of modal and other propositional nonclassical logics. Here we extend and rene these results to give an O(n log n)space decision procedure for the basic positive relevance logic B + . We compute this upperbound by rst giving a sound and complete, cutfree, labelled sequent system for B + , and then establishing bounds on the application of the rules of this system. Keywords: Relevance Logics, Computational Complexity, Labelled Deduction Systems, Sequent Systems. 1
Carnap's Tolerance, Language Change and Logical Pluralism
, 2000
"... In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap's Principle of Tolerance from a pluralism which maintains that there are different, equally "good" logical consequence relations on the one ..."
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In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap's Principle of Tolerance from a pluralism which maintains that there are different, equally "good" logical consequence relations on the one language. I will argue that this second form of pluralism does more justice to the contemporary state of logical theory and practice than does Carnap's more moderate pluralism.
OrderDuality, Negation and Lattice Representation
 Wansing, De Gruyter, Berlin
, 1996
"... this paper is to extend this framework to the case of a possibly nondistributive calculus, possibly also lacking an orthonegation operator. The solution we present can be extended to the case of logical systems with a variety of additional operators (Hartonas 1994b). 1.1. Negation and Duality ..."
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this paper is to extend this framework to the case of a possibly nondistributive calculus, possibly also lacking an orthonegation operator. The solution we present can be extended to the case of logical systems with a variety of additional operators (Hartonas 1994b). 1.1. Negation and Duality
Modelling Truthmaking
, 1998
"... ... In this paper I give a consistency proof, by providing a model for the theses of truthmaking in my earlier paper. This result does two things. Firstly, it shows that the theses of truthmaking are jointly consistent. Secondly, it provides an independently philosophically motivated formal model fo ..."
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... In this paper I give a consistency proof, by providing a model for the theses of truthmaking in my earlier paper. This result does two things. Firstly, it shows that the theses of truthmaking are jointly consistent. Secondly, it provides an independently philosophically motivated formal model for relevant logics in the `possible worlds' tradition of Routley and Meyer [8, 16, 17].