Results 1  10
of
80
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 ..."
Abstract

Cited by 182 (17 self)
 Add to MetaCart
(Show Context)
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
Weakly Distributive Categories
 Journal of Pure and Applied Algebra
, 1991
"... There are many situations in logic, theoretical computer science, and category theory where two binary operationsone thought of as a (tensor) "product", the other a "sum"play a key role. In distributive and autonomous categories these operations can be regarded as, respect ..."
Abstract

Cited by 140 (20 self)
 Add to MetaCart
(Show Context)
There are many situations in logic, theoretical computer science, and category theory where two binary operationsone thought of as a (tensor) "product", the other a "sum"play a key role. In distributive and autonomous categories these operations can be regarded as, respectively, the and/or of traditional logic and the times/par of (multiplicative) linear logic. In the latter logic, however, the distributivity of product over sum is conspicuously absent: this paper studies a "linearization" of that distributivity which is present in both case. Furthermore, we show that this weak distributivity is precisely what is needed to model Gentzen's cut rule (in the absence of other structural rules) and can be strengthened in a natural way to generate  autonomous categories. We also point out that this "linear" notion of distributivity is virtually orthogonal to the usual notion as formalized by distributive categories. 0 Introduction There are many situations in logic, theoretical co...
Proofs nets for unitfree multiplicativeadditive linear logic
 18th IEEE Intl. Symp. Logic in Computer Science (LICS’03
, 2003
"... A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfr ..."
Abstract

Cited by 53 (5 self)
 Add to MetaCart
A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfree monomial proof nets can correspond to the same cutfree proof. Thus the problem of finding a satisfactory notion of proof net for unitfree multiplicativeadditive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory. 1
Substructural Logics on Display
, 1998
"... Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek ca ..."
Abstract

Cited by 50 (16 self)
 Add to MetaCart
Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponentialfree linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic BiLambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bilinear, Birelevant, BiBCK and Biintuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...
Linearly Distributive Functors
 J. Pure Appl. Algebra
, 1997
"... This paper introduces a notion of \linear functor" between linearly distributive categories that is general enough to account for common structure in linear logic, such as the exponentials ( ! , ? ), and the additives (product, coproduct), and yet when interpreted in the doctrine of autonomous ..."
Abstract

Cited by 28 (7 self)
 Add to MetaCart
This paper introduces a notion of \linear functor" between linearly distributive categories that is general enough to account for common structure in linear logic, such as the exponentials ( ! , ? ), and the additives (product, coproduct), and yet when interpreted in the doctrine of autonomous categories, gives the familiar notion of monoidal functor. We show that there is a biadjunction between the 2{categories of linearly distributive categories and linear functors, and of  autonomous categories and monoidal functors, given by the construction of the \nucleus" of a linearly distributive category. We develop a calculus of proof nets for linear functors, and show how linearity accounts for the essential coherence structure of the exponentials and the additives. Introduction What is the \appropriate" notion of a functor between linearly (formerly \weakly") distributive categories? In [CS92] we were content to think of the functors between linearly distributive categories as being ...
From proof nets to the free * autonomous category
 Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05
, 2005
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
(Show Context)
Orderenriched categorical models of the classical sequent calculus
 LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS
, 2003
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contra ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
(Show Context)
It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cutreduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish nondeterministic choices of cutelimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.
On proof nets for multiplicative linear logic with units
 In Jerzy Marcinkowski and Andrzej Tarlecki, editors, Computer Science Logic, CSL 2004, volume 3210 of LNCS
, 2004
"... Abstract. In this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the wellknown theory of unitfree multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. Th ..."
Abstract

Cited by 21 (5 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the wellknown theory of unitfree multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. Furthermore, the identifications enforced on proofs are such that the proof nets, as they are presented here, form the arrows of the free (symmetric) *autonomous category. 1