Necessity of identity

In modal logic, the necessity of identity is the thesis that for every object x and object y, if x and y are the same object, it is necessary that x and y are the same object. It is best known for its association with Saul Kripke, who published it in 1971, although it was first derived by the logician Ruth Barcan Marcus in 1947, and later, in simplified form, by W.V.O. Quine in 1953

The derivation in Kripke's 'Identity and Necessity' is in three steps:

The first premiss is simply postulated: every object is identical to itself. The second is an application of the principle of substitutivity: if a = b, then a has all the properties b has, thus from Fa, infer Fb, where F is 'nec(a = --)'. The third follows by elementary predicate logic.

In the later Naming and Necessity, Kripke suggested that the principle could be derived directly, assuming what he called rigid designation. A term is a rigid designator when it designates the same object in every possible world in which that object exists. When a name’s referent is fixed by the original act of naming it becomes a rigid designator. Some examples of rigid designators include proper names (i.e. ‘Richard Nixon’), natural kind terms ( i.e. ‘gold’ or ‘H2O’) and some descriptions.

Proper names are typically rigid designators, but definite descriptions are typically not. So we can speak of "Richard Nixon" referring to the same person in all possible worlds, but the description “the man who won the 1968 election” could refer to many different people. According to Kripke, the proper name “Richard Nixon” can only be used rigidly, but the description “the man who won the 1968 election” can be used non-rigidly. Kripke argues, that if names are rigid designators, then identity must be necessary, because the names ‘a’ and ‘b’ will be rigid designators of an object x if a is identical to b, and so in every possible world, ‘a’ and ‘b’ will both refer to this same object x, and no other, and there could be no situation in which a might not have been b, otherwise x would not have been identical with itself.

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