Section 3: Semantics for identity
Identity is a special predicate of QL. We write it a bit differently than other two-place predicates: \(x\) = \(y\) instead of \(Ixy\). We also do not need to include it in a symbolization key. The sentence \(x\) = \(y\) always means ‘\(x\) is identical to \(y\),’ and it cannot be interpreted to mean anything else. In the same way, when you construct a model, you do not get to pick and choose which ordered pairs go into the extension of the identity predicate. It always contains just the ordered pair of each object in the UD with itself.
The sentence ∀\(xIxx\), which contains an ordinary two-place predicate, is contingent. Whether it is true for an interpretation depends on how you interpret \(I\), and whether it is true in a model depends on the extension of \(I\).
The sentence ∀\(x\) \(x\) = \(x\) is a tautology. The extension of identity will always make it true.
Notice that although identity always has the same interpretation, it does not always have the same extension. The extension of identity depends on the UD. If the UD in a model is the set {Doug}, then extension(=) in that model is {<Doug, Doug>}. If the UD is the set {Doug, Omar}, then extension(=) in that model is {<Doug, Doug>, <Omar, Omar>}. And so on.
If the referent of two constants is the same, then anything which is true of one is true of the other. For example, if referent(\(a\)) = referent(\(b\)), then \(Aa\) ↔ \(Ab\), \(Ba\) ↔ \(Bb\), \(Ca\) ↔ \(Cb\), \(Rca\) ↔ \(Rcb\), ∀\(xRxa\) ↔ ∀\(xRxb\), and so on for any two sentences containing \(a\) and \(b\). However, the reverse is not true.
It is possible that anything which is true of \(a\) is also true of \(b\), yet for \(a\) and \(b\) still to have different referents. This may seem puzzling, but it is easy to construct a model that shows this. Consider this model:
UD = {Rosencrantz, Guildenstern}
referent(\(a\)) = Rosencrantz
referent(\(b\)) = Guildenstern
for all predicates \(\mathcal{P}\), extension(\(\mathcal{P}\)) = ∅
extension(=) = {<Rosencrantz, Rosencrantz>,
<Guildenstern, Guildenstern>}
This specifies an extension for every predicate of QL: All the infinitely-many predicates are empty. This means that both \(Aa\) and \(Ab\) are false, and they are equivalent; both \(Ba\) and \(Bb\) are false; and so on for any two sentences that contain \(a\) and \(b\). Yet \(a\) and \(b\) refer to different things. We have written out the extension of identity to make this clear: The ordered pairhreferent(\(a\)),referent(\(b\))i is not in it. In this model, \(a\) = \(b\) is false and \(a\) ≠ \(b\) is true.