A Note on Substitutional Quantification
Martin Davies · Noûs · 1980
Suppose thatL is a language with the first-order substitutional quantifiers HI and instead of the referential quantifiers VI and 3. Then it is natural to offer the following definitions of validity and consequence, for a (closed) sentence o- of L: o- is valid if for each (substitutional) valuation V of L, V(o-)= T; o- is a logical consequence of a set X of sentences of L if for each valuation V of L such that V(r) = T for all r E X, V (o-) = T. In an influential paper ([1]), Dunn and Belnap show that upon these definitions strong completeness does not hold (a sentence may be a logical consequence of a set X but not provable from X), but claim that (weak) completeness does hold provided only that L has infinitely many (closed) terms. In this note I show that the claim about completeness is not quite correct: there may be valid sentences of L which are not provable, if the infinite set of terms inL is generated by finitely many constants and function letters. First, let us review the facts about strong completeness which were set out in [1]. (To give full generality, I drop the assumption thatL has infinitely many terms.) Suppose that (F is a one-place context for terms, thatX is {I t: t is a term,} and that o is the sentence (H)'Ax. Then o is a logical consequence of X but is not provable from X. By noticing that strong completeness fails even when L has only finitely many terms we can avoid any temptation to think, in cases where L has infinitely many terms, that the failure of strong completeness