“To say that things are identical is to say that they are the same.” (Noonan 2008). This is the notion of numerical (or absolute) identity, which the tradition distinguished from, for example, qualitative identity, i.e. when two objects share some properties. For the scope of this work with “identity” we will mean numerical identity. Moreover we will deal with contemporary characterizations of identity.
It is only with Frege and Peano (late XIX century) that we achieved the conceptual framework we use today in logic and in philosophy of language (see their fundamental works (Frege 1879) and (Peano 1889)). For example, Peano distinguished between different forms of predication: the difference between “Cats are feline” (inclusion between classes) and “Mark is human” (membership of an element to a class) was not clearly formulable before the XIX century.
Identity is generally considered a binary relation. However this poses a problem: is identity a relation between objects or between names for objects? The question is not as naïve as it seems: Frege, founder of modern logic, in (Frege 1884), accepted one of Leibniz's characterizations of identity as his definition of equality:
Eadem sunt quorum unum potest substitui alteri salva veritate1
This sentence hides a confusion between use and mention, as observed by (Church 1956, p. 300), that corrects:
(S) “Things are identical if the name of one can be substituted for that of the other without loss of truth.”
We have to add the clause that the substitution must occur in referential contexts, because in opaque (or intensional) contexts names for the same thing could not be substituted salva veritate2.
This characterization of identity is of linguistical flavour: it deals with substitutions of names denoting objects, and is pre-theoretical. In fact, (S) is a formulation of what is generally called the substituting principle.
There are other ways of thinking about the same notion. Identity can also be thought in one of the following alternative ways:
- •
as the relation everything has to itself and to nothing else
- •
as the smallest equivalence relation
- •
as the identity relation: Δ={(x,x)|x ∈ D} over a domain of discourse D.
We will now see how to embed the notion of identity in a formal system.