# Dual of identity relation?

Does anyone have any intuitions about what the dual of the identity relation might be? I.e. is there a 'natural' concept expressed by a statement such as 'it is not the case that a is not identical to b', where this latter statement can be taken to express a statement dual to the statement 'a is identical to b'?

• What do you mean by "'natural' concept"? And, did you mean to use two 'not's in the first of the two statements? – Alexis Dec 28 '14 at 13:11
• In classical logic this is not a question. Do you mean an Intuitionist double negative -- "It cannot be reliably known whether a and b are different"? Some nonstandard analysis relies on 'infinitesimals' or 'spatial monads' injected in this way. – user9166 Dec 28 '14 at 14:48

You might be looking for something like the complement of the apartness relation. Apartness focuses on distinguishing things, instead of identifying them.

In practice, we would tend to have a strong logical presumption that `a` is not identical `b` based on one being called "a" and the other called "b" because the difference in names implies a spatial and/or temporal difference in the conditions under which each is encountered. The burden of proof falls on those claiming "Hesperus is Phosphorus."

The premise that classic philosophic logic has some strong principle of identity is a bit suspect in itself. The Ship of Theseus has been an important paradox since the early days of philosophy.

The real problem is that "the same" requires disambiguation if we're talking in a logical context. Thus formal languages in computing often have multiple predicates for sameness [e.g Lisp's `eq`, `eql`, and `equal`].

Pierce's pragmatic maxim perhaps points the way. `a` is not identical `b` if and only if the effects of `a` are not identical to the effects of `b`.

The obvious logical answer would be to negate the expression, identical, (ie. not identical). Another possibility is, "is different" (ie. A is different from B).

• Note that the question was not about the contradictory of the identity relation, but about its dual. In classical logic, as implied by @jobermark in his comment, identity is self-dual. My answer, on the other hand, allows for a constructive stance, according to which such self-duality of identity does not necessarily hold. – J Marcos Dec 30 '14 at 12:29