There is a problem with substitutional quantification

Abstract

Whereas arithmetical quantification is substitutional in the sense that a particular quantification is true only if some instance of it is true, it does not follow (and, in fact, is not true) that an account of the truth-conditions of the sentences of a language of arithmetic can be given by a substitutional semantics. A substitutional semantics fails in a most fundamental fashion: It fails to articulate the truth-conditions of the quantifications with which it is concerned. This is what is defended in the paper. In particular, it is defended against remarks to the contrary by Saul Kripke in a well known paper on the topic.