Identity (Identity) lets logic count, without any numbers. "At least two", "at most one" and "exactly one" can all be written with quantifiers, $=$ and $\neq$.
| English | Symbols |
|---|---|
| At least one thing is $F$. | $\exists x Fx$ |
| At least two things are $F$. | $\exists x\exists y((Fx \land Fy) \land x \ne y)$ |
| At most one thing is $F$. | $\forall x\forall y((Fx \land Fy) \to x = y)$ |
| Exactly one thing is $F$. | $\exists x(Fx \land \forall y(Fy \to y = x))$ |
Key idea
Why it works
Worked examples
Worked example: At least two
- Two variables, both squares, and different from each other.
Answer: $\exists x\exists y((Sx \land Sy) \land x \ne y)$
Worked example: At most one
- Any two squares are the same square.
Answer: $\forall x\forall y((Sx \land Sy) \to x = y)$
Worked example: Exactly one
- Something is a square, and any square is that thing.
- In the world, $b$ is the only square.
Answer: $\exists x(Sx \land \forall y(Sy \to y = x))$; true.
Common mistake
These formulas can be checked in a world like the one above. For "at least two red shapes", look for two different red objects: $a$ and $b$ will do. For "at most one", look for two different objects with the property; finding them makes the sentence false. For "exactly one", find one and check there is no second.
The pattern extends to any number. "At least three" needs three variables, all pairwise different: $x \neq y$, $x \neq z$ and $y \neq z$. It gets long, but it shows something remarkable: counting, which seems to need numbers, can be done with logic alone.