First-order logic

First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as "there exists an x such that..." (exists x) or "for any x, it is the case that..." (orall x), where x is a member of the domain of discourse. A first-order (recursively-)axiomatisable theory is a theory that can be axiomatised as an extension of first-order logic by adding a recursively enumerable set of first-order sentences as axioms.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

First-order logic is Mathematical logic that is distinguished from higher-order logic in that it does not allow quantification over properties; i.e. it cannot express statements such as "for every property P, it is the case that..." (orall P) or "there exists a property P such that..." (exists P).

Related Topics:
Mathematical logic - Higher-order logic

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Nevertheless, first-order logic is strong enough to formalize all of set theory and thereby virtually all of mathematics. Its restriction to quantification over individuals makes it difficult to use for the purposes of topology, but it is the classical logical theory underlying mathematics. It is a stronger theory than sentential logic, but a weaker theory than second-order logic.

Related Topics:
Set theory - Mathematics - Topology - Sentential logic - Second-order logic

~ ~ ~ ~ ~ ~ ~ ~ ~ ~