Actuality and quantification.
1990; Duke University Press; Volume: 31; Issue: 4 Linguagem: Inglês
10.1305/ndjfl/1093635586
ISSN1939-0726
Autores Tópico(s)Logic, programming, and type systems
ResumoA natural deduction system of quantified modal logic (S5) with an actuality operator and "rigid" quantifiers (ranging, at every world, over the domain of the actual world) is described and proved to be complete.Its motivation and relation to other systems are discussed. / The languagePredicates.One logical predicate: E!, "exists".Individual constants if you want, though for simplicity I'll ignore them (constants thought of as formalizing names or other "rigid designators" ought to behave like the free variables).Individual "parameters" (free variables): u,υ 9 .... Individual bound variables: x 9 y,... (I follow the conventions of Thomason [16] here).Truth functional connectives: &, v, D, ~.Modal operators: D (necessity), 0 (possibility), O (actuality).Ordinary quantifiers: V, 3. "Actuality" quantifiers: V°, 3°.The usual formation rules (bound variables never occurring free). SemanticsA model is a quadruple M = (W,@ 9 D 9 I} where Wis a set (of "worlds"), @ E W (@ is "the actual world"), D is a function assigning to each wEWa (not necessarily nonempty) set as its domain, and /is an interpretation function assigning to each Λ-adic predicate a function assigning to each w G Wa set of ^-tuples drawn from \J v(ΞW D(υ), with the condition that [/(E!)] (w) = D(w).Note that, corresponding to various intuitive readings of the predicates of the formal language, and to various metaphysical positions, we might want to impose further conditions on the interpretation function; these will often validate extensions of the logic described below.An assignment for m is a partial function from the individual parameters into
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