Identity

Logic

In logic, the identity relation is normally, (by definition), the transitive, symmetric, and reflexive relation that holds only between a thing and itself. That is, identity is the two-place predicate, "=", such that for all x, y, "x = y" is true iff x is y.

Related Topics:
Logic - Transitive - Symmetric - Reflexive - Relation - Predicate - True - Iff

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More usefully, it can be expressed formally in second-order logic or in set theory: For all objects x, y, if for all properties F, Fx iff Fy, then x = y.

Related Topics:
Second-order logic - Set theory

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It is an axiom of most normal modal logics that for all x, if x = x then necessarily x = x.

Related Topics:
Axiom - Modal logic

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(These definitions are of course inapplicable in some area of quantified logic, such as fuzzy logic and fuzzy set theory, and with respect to vague objects.)

Related Topics:
Fuzzy logic - Fuzzy set theory - Vague object

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