Identity is a special relation: $a = b$ says that $a$ and $b$ are the same object. With it we can say things that were out of reach before, like "only", "someone else" and "the same". It builds on the relations of Relational Predicates.
Key idea
Why it works
Worked examples
Worked example: Two names
- One person, two names.
Answer: $a = t$
Worked example: Only
- Two claims: Ava sings, and anyone who sings is Ava.
Answer: $Sa \land \forall x(Sx \to x = a)$
Worked example: Everyone except
- Ava does not sing.
- Everyone who is not Ava sings.
Answer: $\neg Sa \land \forall x(x \ne a \to Sx)$
Common mistake
Identity also gives "the" a precise meaning. "The team captain sings" says there is exactly one team captain and that person sings; with identity you can write that out (see At Least, At Most and Exactly). Names already pick out one object each, so with names a plain $a = t$ or $St$ is enough.
Notice the shape of "only Ava sings": it is a conjunction, not a single quantifier. Leaving out $Sa$ would give $\forall x(Sx \to x = a)$, which says nobody but Ava sings but allows that Ava does not sing either. Most uses of "only" with a name need both parts.
| English | Symbols |
|---|---|
| Ava is not Ben. | $a \ne b$ |
| Someone other than Ava sings. | $\exists x(Sx \land x \ne a)$ |
| Ava is the only singer. | $Sa \land \forall x(Sx \to x = a)$ |
| Ava sings louder than anyone else. | $\forall x(x \ne a \to Lax)$ with $Lxy$: $x$ sings louder than $y$ |