Editing First-order logic (section)

===Higher-order logics===
{{Main|Higher-order logic}}

The characteristic feature of first-order logic is that individuals can be quantified, but not predicates. Thus
:<math>\exists a ( \text{Phil}(a))</math>
is a legal first-order formula, but
:<math>\exists \text{Phil} ( \text{Phil}(a))</math>
is not, in most formalizations of first-order logic. [[Second-order logic]] extends first-order logic by adding the latter type of quantification. Other [[higher-order logic]]s allow quantification over even higher [[type theory|types]] than second-order logic permits. These higher types include relations between relations, functions from relations to relations between relations, and other higher-type objects.  Thus the "first" in first-order logic describes the type of objects that can be quantified.

Unlike first-order logic, for which only one semantics is studied, there are several possible semantics for second-order logic. The most commonly employed semantics for second-order and higher-order logic is known as ''full semantics''. The combination of additional quantifiers and the full semantics for these quantifiers makes higher-order logic stronger than first-order logic. In particular, the (semantic) logical consequence relation for second-order and higher-order logic is not semidecidable; there is no effective deduction system for second-order logic that is sound and complete under full semantics.

Second-order logic with full semantics is more expressive than first-order logic. For example, it is possible to create axiom systems in second-order logic that uniquely characterize the natural numbers and the real line. The cost of this expressiveness is that second-order and higher-order logics have fewer attractive metalogical properties than first-order logic. For example, the Löwenheim–Skolem theorem and compactness theorem of first-order logic become false when generalized to higher-order logics with full semantics.