Actuality and quantification.

Abstract

A natural deduction system of quantified modal logic (S5) with an actuality operator and quantifiers (ranging, at every world, over domain of actual world) is described and proved to be complete. Its motivation and relation to other systems are discussed. / The language Predicates. 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 free variables). Individual (free variables): u,υ9... . Individual bound variables: x9y,... (I follow 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). 2 Semantics A model is a quadruple M = (W,@9D9I} 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 \Jv(ΞWD(υ), with condition that [/(E!)] (w) = D(w). Note that, corresponding to various intuitive readings of predicates of formal language, and to various metaphysical positions, we might want to impose further conditions on interpretation function; these will often validate extensions of logic described below. An assignment for m is a partial function from individual parameters into Received April 4, 1989; revised June 26, 1989 ACTUALITY AND QUANTIFICATION 499 \Jw(EWD(w) (partial function in order to avoid validating OE!u, which is not derivable in system described. If you want motivation, think of some parameters as formalizing names from fiction.) We define in first instance: Truth in a model, at a world, on an assignment. Truth in a model at a world is Truth at that world in that model on every assignment. Truth in a model is Truth at actual world in model. Validity is Truth in every model; validity of an argument is validity of its associated conditional. Base clause of recursion: An atomic formula, F(uu... ,un), where F is an Az-adic predicate, is True(M, w,α) if and only if (i) a(uχ),... ,a(un) are all defined, and (ii) £ [I(F)](w). Recursion clauses: For truth functional compounds: Standard. For modal and actuality operators: ΏA is True(M, w,a) iff A is True(M, w\a) for every w' G W9 <)A is True(M, w,α) iff 4 is True(M, w',α) for at least one w' E W, OA is True(M,w,α) iffA is True(M,@,α). For ordinary quantifiers: VxA (x/u) is True(M, w,a) iff A is True(M, w,β) for every assignment β such that (i) β(v) = a(v) for every parameter v Φ u, (ii) β(u) is defined, and (iii) β(u)ED(w). 3xA (x/u) is True(M, w,α) iff A is True(M, w,β) for at least one assignment β such that (i) β(v) = a(v) for every parameter v Φ u, (ii) β(u) is defined, and (iii) β(u)EΌ(w). For actuality quantifiers: V°xA(x/u) is True(M, w,α) iff A is True(M, w9β) for every assignment β such that (i) β(v) = a(υ) for every parameter v Φ u, (ii) β(u) is defined, and (iii) jS(iι)eD(@). l°xA (x/u) is True(M, w,α) iff A is True(M,w,β) for at least one assignment β such that