20.3 The Identity Predicate =
There's a special binary predicate, identity, which gets its own symbol: =.
This is the exact same symbol you are familiar with from math: a=b says that a and b are the same thing. In English the verb "is" can also be used this way, like when we say one plus two is three.
When identity is used in this way, it is sometimes called numerical identity. When we say Raquel = Mary, it means there is numerically one person or thing with two names (Raquel and Mary).
Contrast the notion of qualitative sameness, which is not as strong as numerical identity. Quinn and Raquel might own the same car, because they both have Honda Civics. Maybe they even both got the same color, and the same trim packages and everything. But nonetheless they each have their own car; they don't share one.
That idea is qualitative sameness, which is not what we mean by identity. If we said:
Quinn's car is Raquel's car
Quinn's car = Raquel's car
then that means that they literally have a single car: there's only one car that they share.
We write = in FOL this way:
Now you try.
It is perfectly possible to write identity the way we write all the other predicates in FOL. For example, we could write
But since you are already use the the symbol = from math, it is easiest to use it in FOL too.
Notice that the symbol = goes between the terms: clark=superman. Predicates like that are called infix, because they go "in" the middle of the terms, whereas all of our other predicates are prefix, because they go "before" the terms.
Identity is a special predicate for several reasons. It has three properties that are very important:
Reflexive: a=a (for any a)
Symmetric: If a=b, then b=a
Transitive: If a=b and b=c, then a=c
These properties allow it to play important roles in arguments and proofs.
For example, see if you can figure these out.