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Simple Consequence Relations
 Information and Computation
, 1991
"... 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 (incl ..."
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Cited by 98 (18 self)
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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 nonmonotonic logics) and for a general, semanticsindependent classification of standard connectives via equations on consequence relations (these include Girard's "multiplicatives" and "additives"). We next investigate the standard methods for uniformly representing consequence relations: Hilbert type, Natural Deduction and Gentzen type. The advantages and disadvantages of using each system and what should be taken as good representations in each case (especially from the implementation point of view) are explained. We end by briefly outlining (with examples) some methods for developing nonuniform, but still efficient, representations of consequence relations.
Term Assignment for Intuitionistic Linear Logic
, 1992
"... In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. Th ..."
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Cited by 53 (9 self)
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In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is welltyped). We define a simple (but more general than previous proposals) categorical model for Intuitionistic Linear Logic and show how this can be used to derive the term assignment system. We also consider term reduction arising from cutelimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these, as well as with the equations which follow from our categorical model.
On Negation, Completeness and Consistency
 Handbook of Philosophical Logic
, 2002
"... this paper we try to understand negation from two different points of view: a syntactical one and a semantic one. Accordingly, we identify two different types of negation. The same connective of a given logic might be of both types, but this might not always be the case. The syntactical point of vie ..."
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Cited by 6 (0 self)
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this paper we try to understand negation from two different points of view: a syntactical one and a semantic one. Accordingly, we identify two different types of negation. The same connective of a given logic might be of both types, but this might not always be the case. The syntactical point of view is an abstract one. It characterizes connectives according to the internal role they have inside a logic, regardless of any meaning they are intended to have (if any). With regard to negation our main thesis is that the availability of what we call below an internal negation is what makes a logic essentially multipleconclusion.
Multiplicative Conjunction as an Extensional Conjunction
 Journal of the IGPL
, 1997
"... We show that the rule that allows the inference of A from A\Omega B is admissible in many of the basic multiplicative (intensional) systems. By adding this rule to these systems we get, therefore, conservative extensions in which the tensor behaves as classical conjunction. Among the systems obtain ..."
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Cited by 4 (4 self)
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We show that the rule that allows the inference of A from A\Omega B is admissible in many of the basic multiplicative (intensional) systems. By adding this rule to these systems we get, therefore, conservative extensions in which the tensor behaves as classical conjunction. Among the systems obtained in this way the one derived from RMIm (= multiplicative linear logic together with contraction and its converse) has a particular interest. We show that this system has a simple infinitevalued semantics, relative to which it is strongly complete, and a nice cutfree Gentzentype formulation which employs hypersequents (= finite sequences of ordinary sequents). Moreover: classical logic has a simple, strong translation into this logic. This translation uses definable connectives and preserves the consequence relation of classical logic (not just the set of theorems). Similar results, but with a 3valued semantics, obtain if instead of RMIm we use RMm (the purely multiplicative fragment ...
Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening
 Weakening, forthcoming in the Journal of Symbolic Logic
"... We show that the elimination rule for the multiplicative (or intensional) conjunction is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its ..."
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Cited by 2 (2 self)
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We show that the elimination rule for the multiplicative (or intensional) conjunction is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm , but a proper subset of the set of theorems of RMIm . Hence it still has the variablesharing property. SRm has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm relative to th...