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Normal modal logic
About: Normal modal logic is a research topic. Over the lifetime, 2649 publications have been published within this topic receiving 75735 citations.
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01 Jan 1993
TL;DR: In this paper, the completeness and decidability of GL and other modal logics are discussed, including the fixed-point theorem, the fixed point theorem, letterless sentences, and analysis.
Abstract: 1. GL and other systems of propositional modal logic 2. Peano arithmetic 3. The box as Bew(x) 4. Semantics for GL and other modal logics 5. Completeness and decidability of GL and K, K4, T, B, S4, and S5 6. Canonical models 7. On GL 8. The fixed point theorem 9. The arithmetical completeness theorems for GL and GLS 10. Trees for GL 11. An incomplete system of modal logic 12. An S4 -preserving proof-theoretical treatment of modality 13. Modal logic within set theory 14. Modal logic within analysis 15. The joint provability logic of consistency and w-consistency 16. On GLB: the fixed point theorem, letterless sentences, and analysis 18. Quantified provability logic with one one-place predicate letter Notes Bibliography Index.
447 citations
440 citations
IBM1
TL;DR: In this article, the authors consider interpretations of modal logic in Peano arithmetic determined by an assignment of a sentencev * ofP to each propositional variablev. They show that a modal formula, χ, is valid if ψ* is a theorem ofP in each interpretation.
Abstract: We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev
* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.
438 citations
TL;DR: The operator M (usually read "possible") is extended so that Mp is true whenever p is consistent with the theory, and any theorem of this form may be mvahdated if ~p ~s is added as an axiom.
Abstract: Tradmonal logics suffer from the "monotomclty problem"' new axioms never mvahdate old theorems One way to get nd of this problem ts to extend traditional modal logic in the following way The operator M (usually read "possible") is extended so that Mp is true whenever p is consistent with the theory Then any theorem of this form may be mvahdated if ~p ~s added as an axiom This extension results m nonmonotomc versions of the systems T, $4, and $5 These systems are complete in that a theorem is provable in a theory based on one of them just if it is true m all "noncommittal" models of that theory, where a noncommittal model ts one m which as many thmgs are possible as possible Nonmonotomc $4 is probably the most interesting of the three, since it is stronger than ordinary $4 but has all the usual inferential machinery of $4 There is a straightforward proof procedure for the sententlal subset of nonmonotomc $4. This approach to nonmonotonlc logic may be applied to several problems in knowledge representation for arUficml mtelhgence Its main advantages over competmg approaches are that tt factors out problems of resource hmltattons and allows the symbol M to appear m any context, since M is a meaningful part of the language
434 citations