Section 05: Rules for identity
The identity predicate is not part of QL, but we add it when we need to symbolize certain sentences. For proofs involving identity, we add two rules of proof.
Suppose you know that many things that are true of \(a\) are also true of \(b\). For example: \(Aa\)&\(Ab\), \(Ba\)&\(Bb\), ¬\(Ca\)&¬\(Cb\), \(Da\)&\(Db\), ¬\(Ea\)&¬\(Eb\), and so on. This would not be enough to justify the conclusion \(a\) = \(b\). (See p. 89.) In general, there are no sentences that do not already contain the identity predicate that could justify the conclusion \(a\) = \(b\). This means that the identity introduction rule will not justify \(a\) = \(b\) or any other identity claim containing two different constants.
However, it is always true that \(a\) = \(a\). In general, no premises are required in order to conclude that something is identical to itself. So this will be the identity introduction rule, abbreviated =I:
Notice that the =I rule does not require referring to any prior lines of the proof. For any constant \(\mathcal{c}\), you can write \(\mathcal{c}\) = \(\mathcal{c}\) on any point with only the =I rule as justification.
If you have shown that \(a\) = \(b\), then anything that is true of \(a\) must also be true of \(b\). For any sentence with \(a\) in it, you can replace some or all of the occurrences of \(a\) with \(b\) and produce an equivalent sentence. For example, if you already know \(Raa\), then you are justified in concluding \(Rab\), \(Rba\), \(Rbb\).
The identity elimination rule (=E) allows us to do this. It justifies replacing terms with other terms that are identical to it.
For writing the rule, we will introduce a new bit of symbolism. For a sentence \(\mathcal{A}\) and constants \(\mathcal{c}\) and \(\mathcal{d}\), \(\mathcal{Ac}\)↺\(\mathcal{d}\) is a sentence produced by replacing some or all instances of \(\mathcal{c}\) in \(\mathcal{A}\) with \(\mathcal{d}\) or replacing instances of \(\mathcal{d}\) with \(\mathcal{c}\). This is not the same as a substitution instance, because one constant need not replace every occurrence of the other (although it may).
We can now concisely write =E in this way:
To see the rules in action, consider this proof: