Existence and identity in quantified modal logics.
Richard Routley · Notre Dame Journal of Formal Logic · 1969
The aim of this paper is to present a way in which philosophical objections to the development of a combined quantification and modal logic based on S5 can be overcome.In more detail, the objectives are to show that S5 is immune to criticisms directed at those theorems which distinguish it from SA and T; that problematic theorems 1 of modalised predicate logic like the Bar can formulae [O(3*)/M => (ix) Of(x)] and [O(lt)g(x, ί) i> (it) (>g(x, t)] can be appropriately qualified once existence is explicitly treated; that puzzles over identity can be escaped by a more elaborate treatment of identity than the standard treatment; and that difficulties associated with quantification into modal sentence contexts can be cleared away given these treatments of existence and identity.A combination of these moves suffice, so it will be argued, to meet standard objections, most forcefully presented by Quine 2 , to quantified modal logics.Admittedly a full elaboration of these moves calls for some sentence/statement distinction, some analytic/synthetic (or necessary/contingent) distinction, and some sense/designation (or connotation/denotation) distinction: but although, consequently, it is not to be expected that a combination of these moves will satisfy Quine, they may satisfy some who have been disturbed by the objections Quine raises.§2.Existence in a first-order modalised predicate logic.A semantical system S5R* is obtained by adjoining modal postulates for S5 (with primitive symbol Ή 9 ) to a system R* of first-order predicate logic (with primitive quantifier ( W). R* differs from usual quantification theory in having the predicate constant Έ', read 'exist(s)', added to its primitive symbols, and in interpretation: in place of the frequent interpretation -[(Tlx)f(x)] is true if / is true of all existent items of the domain selected -the following interpretation -[(Tiχ)f(x)] is true if / is true of all possible (consistently describable or designable) items of the domain selected -is preferred 3 .The postulate set of R* is as follows:RO.If A is truth-functionally valid, then A is a theorem.