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
Alonzo Church (1956). Introduction to Mathematical Logic. Princeton, Princeton University Press.
M. J. Cresswell (1970). Classical intensional logics. Theoria 36 (3):347-372.
Robert Feys (1965). Modal Logics. Louvain, E. Nauwelaerts.
G. E. Hughes (1968/1972). An Introduction to Modal Logic. London,Methuen.
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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