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Higher order quantification

Higher order quantification, which means quantification not over the individuals belonging to a given universe of discourse, as in first-order logic, but also over sets of individuals and sets of n-tuples of individuals. (Alternatively, the properties and relations that specify these sets may be quantified over.) This gives rise to second-order logic. The process can be repeated. Quantification over sets of such sets (or of n-tuples of such sets or over properties and relations of such sets) as are considered in second-order logic gives rise to third-order logic; and all logics of finite order form together the (simple) theory of (finite) types. The membership relation, expressed by ∊, can be grafted on to first-order logic; it gives rise to set theory. The concepts of (logical) necessity and (logical) possibility can be added.

This narrower sense of logic is related to the influential idea of logical form. In any given sentence, all of the nonlogical terms may be replaced by variables of the appropriate type, keeping only the logical constants intact. The result is a formula exhibiting the logical form of the sentence. If the formula results in a true sentence for any substitution of interpreted terms (of the appropriate logical type) for the variables, the formula and the sentence are said to be logically true (in the narrower sense of the expression).