<logic> A 2-adic predicate, say Ixy, asserting that its two arguments are identical. Customarily symbolized by "=" and written in infix notation, "x=y". While all systems of polyadic predicate logic can express identity as easily as any other 2-adic relation, a system is said to be "with identity" iff it also contains axioms, axiom schemata, and/or rules of inference determining how "=" is to be used. Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity.
See first-order theory with identity, predicate logic with identity, interpretation, normal
[Glossary of First-Order Logic]
<logic, philosophy of science> the logical relation of numerical sameness, in which each thing stands only to itself. Although everything is what it is and not anything else, philosophers try to formulate more precisely the criteria by means of which we may be sure that one and the same thing is cognized under two different descriptions or at two distinct times. Leibniz held that numerical identity is equivalent to indiscernibility or sameness of all the features each thing has. But Locke maintained that judgments of identity are invariably made by reference to types or sorts of things. The identity of individual persons is an especially troublesome case. Recommended Reading: Colin McGinn, Logical Properties: Identity, Existence, Predication, Necessity, Truth (Clarendon, 2001); Eli Hirsch, The Concept of Identity (Oxford, 1992); Andre Gallois, Occasions of Identity: A Study in the Metaphysics of Persistence, Change, and Sameness (Clarendon, 1998); Individuation and Identity in Early Modern Philosophy, ed. by Kenneth F. Barber and Jorge J.E. Garcia (SUNY, 1994); Self and Identity: Contemporary Philosophical Issues, ed. by Daniel Kolak and Raymond Martin (Prentice Hall, 1991); and Particulars, Actuality, and Identity Over Time, ed. by Michael Tooley (Garland, 1999).
[A Dictionary of Philosophical Terms and Names]
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