Proofs with Identity

Although we employ a special symbol for identity, indicating its importance, from a logical point of view identity is just another relation. Identity is an example of what we call an "equivalence relation", namely a relation which is reflexive, symmetric and transitive.  In proofs with identity, we need to add rules that exploit these three features.  Here are our three rules:

We can infer an identity of the form a = a at at any time.

Any identity statement is symmetrical. We can reverse the order of an identity, even within a wff.

If two things are identical, then any predicate which holds of one member of the identity pair, holds of the other.

Let's see these rules in action. We'll begin with the following proof:

(x)(Fx ⊃ Gx), Fa, a = b /// Gb

This is pretty straightforward. We'll instantiate with UI, and then use MP to get Fa. Then we can get Gb  by the Substitutivity of Identity. Here's the proof.

Let's prove the following valid argument:

Our dictionary is as follows:

Our translation:

Now to the proof. Our conclusion is an existential generalization. Bu so is our premise. So to instantiate  we'll need a flagged constant, that is, we'll need to use a constant new to the derivation. Essentially, we'll choose "a", a new constant, and that individual will be identical to Fred.  So whatever we can say about Fred can be said about that individual. And we'll be able to say that that individual is not happy, since Fred is not happy. And we'll be able to generalize existentially from the fact that that individual is (as identical to Fred, the tallest man in London), to the tallest man in London being unhappy.

Exercises: Properties of Relations and Proofs for QL with Identity