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16
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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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
Negation In Relevant Logics (How I stopped worrying and learned to love the Routley Star)
 BULLETIN OF THE SECTION OF LOGIC
, 1999
"... Negation raises three thorny problems for anyone seeking to interpret relevant logics. The frame semantics for negation in relevant logics involves a `point shift' operator . Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A ..."
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Cited by 22 (11 self)
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Negation raises three thorny problems for anyone seeking to interpret relevant logics. The frame semantics for negation in relevant logics involves a `point shift' operator . Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ¸A B to B to be invalid, because the corresponding relevant conditional A (¸ A B) ! B is not a theorem. Yet we often make the inference from A and ¸A B to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x j= \GammaA if and only if x 6j= A. Problem number three involves deciding which is the `real' negation. How can we decide between orthodox negation and the new, `Boolean' negation. In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow u...
Classical BI (A Logic for Reasoning about Dualising Resources)
"... We show how to extend O’Hearn and Pym’s logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a nonconservative extension of (propositional) Boolean BI that includes multiplicative versions of ..."
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Cited by 9 (6 self)
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We show how to extend O’Hearn and Pym’s logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a nonconservative extension of (propositional) Boolean BI that includes multiplicative versions of falsity, negation and disjunction. We give an algebraic semantics for CBI that leads us naturally to consider resource models of CBI in which every resource has a unique dual. We then give a cuteliminating proof system for CBI, based on Belnap’s display logic, and demonstrate soundness and completeness of this proof system with respect to our semantics.
Display Logic And Gaggle Theory
 Reports on Mathematical Logic
, 1995
"... This paper is a revised version of a talk given at the Logic and Logical Philosophy conference in Poland in September 1995. In it, I sketch the connections between Nuel Belnap's Display Logic and J. Michael Dunn's Gaggle Theory. Display Logic and Gaggle Theory Greg Restall  Greg.Restall@anu. ..."
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This paper is a revised version of a talk given at the Logic and Logical Philosophy conference in Poland in September 1995. In it, I sketch the connections between Nuel Belnap's Display Logic and J. Michael Dunn's Gaggle Theory. Display Logic and Gaggle Theory Greg Restall  Greg.Restall@anu.edu.au Nuel Belnap's Display Logic [1] is a neat, uniform method for providing a cutfree consecution calculus for a wide range of formal systems. Mike Dunn's Gaggle Theory [3] is a neat, uniform presentation of the semantics for a wide range of formal systems. In this paper I will show that the two live together happily  many gaggletheoretically presented logics can be given a display proof theory, and that many logics with a display proof theory can be algebraically presented in gaggle theory. 1 Gaggle Theory Dunn [3] introduced the notion of a gaggle as a way to unify many di#erent logics  modal, intuitionistic, manyvalued, and substructural logics are examples of those which f...
Classical logic of bunched implications
 In the informal proceedings of CL&C 2008, an ICALP
, 2008
"... Abstract. We consider a classical (propositional) version, CBI, of O’Hearn and Pym’s logic of bunched implications (BI) from a model and prooftheoretic perspective. We present a class of classical models of BI which extend the usual BImodels, based on partial commutative monoids, with an algebraic ..."
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Abstract. We consider a classical (propositional) version, CBI, of O’Hearn and Pym’s logic of bunched implications (BI) from a model and prooftheoretic perspective. We present a class of classical models of BI which extend the usual BImodels, based on partial commutative monoids, with an algebraic notion of “resource negation”. This class of models gives rise to natural definitions of multiplicative falsity, negation and disjunction. We demonstrate that a sequent calculus proof system for CBI is sound with respect to our classical models by translating CBI sequent proofs into proofs in BI +, a sound extension of sequent calculus for BI. 1
CLASSICAL BI: ITS SEMANTICS AND PROOF THEORY
"... Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including ..."
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Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBIformulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the prooftheoretic level, a very natural formalism for CBI is provided by a display calculus à la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics. 1.
Combining Possibilities and Negations
 Studia Logica
, 1994
"... Combining nonclassical (or `subclassical') logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility (in the presence of conjunction and disjunction), and then we combine the resulting systems with intuitionistic logic. We will fi ..."
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Combining nonclassical (or `subclassical') logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility (in the presence of conjunction and disjunction), and then we combine the resulting systems with intuitionistic logic. We will find that Kracht's results on the undecidability of classical modal logics generalise to a nonclassical setting. We will also see conditions under which intuitionistic logic can be combined with a nonintuitionistic negation without corrupting the intuitionistic fragment of the logic. Many people are interested in logics of modal operators like `necessarily' and `possibly,' and their cousins taken from temporal, epistemic, doxastic and many other concerns. Quite a few people are also interested in negative modal operators, like classical boolean negation, but with some kind of `modal' force. The idea with these sorts of operators is that to evaluate `not p' at a point (world, information ...
Defining Double Negation Elimination
, 1999
"... Dunn mentions a claim of mine to the effect that there is no condition on ‘perp frames ’ equivalent to the holding of double negation elimination ∼∼A ⊢ A. That claim is wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of di ..."
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Dunn mentions a claim of mine to the effect that there is no condition on ‘perp frames ’ equivalent to the holding of double negation elimination ∼∼A ⊢ A. That claim is wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses. 1