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Cutfree Sequent and Tableau Systems for Propositional Diodorean Modal Logics
"... We present sound, (weakly) complete and cutfree tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of po ..."
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Cited by 20 (3 self)
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We present sound, (weakly) complete and cutfree tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points. Although cutfree, the last two systems do not possess the subformula property. But for any given finite set of formulae X the "superformulae" involved are always bounded by a finite set of formulae X L depending only on X and the logic L. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. Each tableau system has a cutfree sequent analogue proving that Gentzen's cutelimination theorem holds for these latter systems. The techniques are due to Hi...
Semianalytic Tableaux For Propositional Normal Modal Logics with Applications to Nonmonotonicity
, 1991
"... The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in ..."
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Cited by 5 (4 self)
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The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in modal logic where \Gamma is required to be empty, resulting in a weaker notion of deducibility. Recently, Marek, Schwarz and Truszczi'nski have given algorithms to compute the stable expansions of a finite theory \Gamma in various such nonmonotonic formalisms. Their algorithms assume the existence of procedures for deciding (strong) deducibility in these monotonic modal logics and consequently such decision procedures are important for automating nonmonotonic deduction. We first give a sound, (weakly) complete and cutfree, semianalytic tableau calculus for monotonic S4R, thus extending the cut elimination results of Schwarz for monotonic K45 and K45D. We then give sound and complete semi...
Semianalytic Tableaux For Propositional Modal Logics of Nonmonotonicity
, 1993
"... The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden ..."
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Cited by 1 (0 self)
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The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in modal logic where \Gamma is required to be empty, resulting in a weaker notion of deducibility. Recently, Marek, Schwarz and Truszczi'nski have given algorithms to compute the stable expansions of a finite theory \Gamma in various such nonmonotonic formalisms. Their algorithms assume the existence of procedures for deciding (strong) deducibility in these monotonic modal logics and consequently such decision procedures are important for automating nonmonotonic deduction. We first give a sound, (weakly) complete and cutfree, semianalytic tableau calculus for monotonic S4R, thus extending the cut elimination results of Schwarz for monotonic K45 and K45D. We then give sound and co...