David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Studia Logica 38 (3):277 - 296 (1979)
The main purpose of this paper is to define and study a particular variety of Montague-Scott neighborhood semantics for modal propositional logic. We call this variety the first-order neighborhood semantics because it consists of the neighborhood frames whose neighborhood operations are, in a certain sense, first-order definable. The paper consists of two parts. In Part I we begin by presenting a family of modal systems. We recall the Montague-Scott semantics and apply it to some of our systems that have hitherto be uncharacterized. Then, we define the notion of a first-order indefinite semantics, along with the more specific notion of a first-order uniform semantics, the latter containing as special cases the possible world semantics of Kripke. In Part II we prove consistency and completeness for a broad range of the systems considered, with respect to the first-order indefinite semantics, and for a selected list of systems, with respect to the first-order uniform semantics. The completeness proofs are algebraic in character and make essential use of the finite model property. A by-product of our investigations is a result relating provability in S-systems and provability in T-systems, which generalizes a known theorem relating provability in the systems S 2° and C 2.
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References found in this work BETA
G. E. Hughes (1968/1972). An Introduction to Modal Logic. London,Methuen.
Alonzo Church (1944). Introduction to Mathematical Logic. London, H. Milford, Oxford University Press.
Alonzo Church (1956). Introduction to Mathematical Logic. Princeton, Princeton University Press.
S. K. Thomason (1972). Semantic Analysis of Tense Logics. Journal of Symbolic Logic 37 (1):150-158.
Saul A. Kripke (1965). Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi. In J. W. Addison, A. Tarski & L. Henkin (eds.), Journal of Symbolic Logic. North Holland 135-135.
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