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Labelled Resolution for Classical and Non-Classical Logics

D. M. Gabbay and U. Reyle
Studia Logica: An International Journal for Symbolic Logic
Vol. 59, No. 2, Combining Logics II (Sep., 1997), pp. 179-216
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015935
Page Count: 38
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Labelled Resolution for Classical and Non-Classical Logics
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Abstract

Resolution is an effective deduction procedure for classical logic. There is no similar "resolution" system for non-classical logics (though there are various automated deduction systems). The paper presents resolution systems for intuitionistic predicate logic as well as for modal and temporal logics within the framework of labelled deductive systems. Whereas in classical predicate logic resolution is applied to literals, in our system resolution is applied to L(abelled) R(epresentation) S(tructures). Proofs are discovered by a refutation procedure defined on LRSs, that imposes a hierarchy on clause sets of such structures together with an inheritance discipline. This is a form of Theory Resolution. For intuitionistic logic these structures are called I(ntuitionistic) R(epresentation) S(tructures). Their hierarchical structure allows the restriction of unification of individual variables and/or constants without using Skolem functions. This structure must therefore be preserved when we consider other (non-modal) logics. Variations between different logics are captured by fine tuning of the inheritance properties of the hierarchy. For modal and temporal logics IRS's are extended to structures that represent worlds and/or times. This enables us to consider all kinds of combined logics.

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