Quantifiers, substitutional and objectual

Understood substitutionally, ‘Something is F’ is true provided one of its substitution instances (a sentence of the form ‘a is F’) is true. This contrasts with the objectual understanding, on which it is true provided ‘is F’ is true of some object in the domain of the quantifier. Substitutional quantifications have quite different truth-conditions from objectual ones. For instance, ‘Something is a mythological animal’ is true if understood substitutionally, since the substitution instance ‘Pegasus is a mythological animal’ is true. But understood objectually, the sentence is not true, since there are no mythological creatures to make up a domain for the quantifier.

Since substitutional quantifiers do not need domains over which they range, it is easy to introduce substitutional quantifiers which bind predicate or sentential variables, even variables within quotation marks. One reason for interest in substitutional quantification is the hope that it may provide a way to understand discourse which appears to be about numbers, properties, propositions and other ‘troublesome’ sorts of entities as being free of exceptional ontological commitments. Whether natural language quantification is sometimes plausibly construed as substitutional is not, however, clear.