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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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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
Grafting Modalities onto substructural implication systems
 Studia Logica
, 1996
"... We investigate the semantics of the logical systems obtained by introducing the modalities 2 and 3 into the family of substructural implication logics (including relevant, linear and intuitionistic implication) . Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" th ..."
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We investigate the semantics of the logical systems obtained by introducing the modalities 2 and 3 into the family of substructural implication logics (including relevant, linear and intuitionistic implication) . Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in a previous paper. Keywords: Kripke semantics, Labelled Deductive Systems, KE system. 1 Introduction The notion of modality is central in pure and applied logic. Many systems presented to formalise some application area require the addition of modality to the language for a variety of reasons: to cater for changes of the system in time, or perhaps for the dependency of the system on the context, or even to bring metalevel notions into the object l...
Theoremhood Preserving Maps As A Characterisation Of Cut Elimination For Provability Logics.
, 1999
"... We define cutfree display calculi for provability (modal) logics that are not properly displayable according to Kracht's analysis. We also show that a weak form of the cutelimination theorem (for these modal display calculi) is equivalent to the theoremhoodpreserving property of certain maps from ..."
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We define cutfree display calculi for provability (modal) logics that are not properly displayable according to Kracht's analysis. We also show that a weak form of the cutelimination theorem (for these modal display calculi) is equivalent to the theoremhoodpreserving property of certain maps from the provability logics into properly displayable modal logics. 1 Australian Research Council International Research Fellow from Laboratoire LEIBNIZCNRS, Grenoble, France. 2 Supported by an Australian Research Council Queen Elizabeth II Fellowship. 1 Introduction Background. Display Logic (DL) is a prooftheoretical framework introduced by Belnap [Bel82] that generalises the structural language of Gentzen's sequents in a rather abstract way by using multiple complex structural connectives instead of Gentzen's comma. The term "display" comes from the nice property that any occurrence of a structure in a sequent can be displayed either as the entire antecedent or as the entire succedent...
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...
Comparing Modal Sequent Systems
"... abstract. This is an exploratory and expository paper, comparing display logic formulations of normal modal logics with labelled sequent systems. We provide a translation from display sequents into labelled sequents. The comparison between different systems gives us a different way to understand the ..."
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abstract. This is an exploratory and expository paper, comparing display logic formulations of normal modal logics with labelled sequent systems. We provide a translation from display sequents into labelled sequents. The comparison between different systems gives us a different way to understand the difference between display systems and other sequent calculi as a difference between local and global views of consequence. The mapping between display and labelled systems also gives us a way to understand labelled systems as properly structural and not just as systems encoding modal logic into firstorder logic. 1
A Theorem Prover for Boolean BI
"... While separation logic is acknowledged as an enabling technology for largescale program verification, most of the existing verification tools use only a fragment of separation logic that excludes separating implication. As the first step towards a verification tool using full separation logic, we d ..."
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While separation logic is acknowledged as an enabling technology for largescale program verification, most of the existing verification tools use only a fragment of separation logic that excludes separating implication. As the first step towards a verification tool using full separation logic, we develop a nested sequent calculus for Boolean BI (Bunched Implications), the underlying theory of separation logic, as well as a theorem prover based on it. A salient feature of our nested sequent calculus is that its sequent may have not only smaller child sequents but also multiple parent sequents, thus producing a graph structure of sequents instead of a tree structure. Our theorem prover is based on backward search in a refinement of the nested sequent calculus in which weakening and contraction are built into all the inference rules. We explain the details of designing our theorem prover and provide empirical evidence of its practicality.
Proofs and Expressiveness in Alethic Modal Logic
, 2001
"... Introduction Alethic modalities are the necessity, contingency, possibility or impossibility of something being true. Alethic means `concerned with truth'. [28, p. 132] The above dictionary characterization of alethic modalities states the central notions of alethic modal logic: necessity, and othe ..."
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Introduction Alethic modalities are the necessity, contingency, possibility or impossibility of something being true. Alethic means `concerned with truth'. [28, p. 132] The above dictionary characterization of alethic modalities states the central notions of alethic modal logic: necessity, and other notions that are usually thought of as being definable in terms of necessity and Boolean negation: impossibility, contingency, and possibility. The syntax of modal propositional logic is inductively defined over a denumerable set of sentence letters p 0 , p 1 , p 2 , . . . as follows: A ::= p  A  (A # B)  #A The other Boolean operations (#, #, #, # and #) are defined as usual. A formula<F10.9