Names let us talk about particular things (Predicates and Names). Quantifiers let us talk about all things, or some, without naming them. That is what "All dogs are mammals" and "Some birds cannot fly" need.
Key idea
Why it works
| English | Symbols |
|---|---|
| Everything is red. | $\forall x Rx$ |
| Something is red. | $\exists x Rx$ |
| Nothing is red. | $\neg \exists x Rx$, or $\forall x\neg Rx$ |
| Not everything is red. | $\neg \forall x Rx$, or $\exists x\neg Rx$ |
| Something is not red. | $\exists x\neg Rx$ |
Worked examples
Worked example: Evaluate in a world
- Check every object: $a$ red, $b$ red, $c$ blue.
- One counterexample, $c$, is enough to make it false.
Answer: False
Worked example: Some
- One witness is enough: $a$ is red.
Answer: True
Worked example: Nothing
- "Nothing" denies that something is: $\neg \exists x Tx$.
- Equivalently, everything is not a triangle: $\forall x\neg Tx$.
Answer: $\neg \exists x Tx$
Common mistake
Worked example: Not everything
- Deny the universal: $\neg \forall x Lx$.
- In the world, $a$ is not large, so $\forall x Lx$ is false and its negation is true.
Answer: $\neg \forall x Lx$, true here.
Any variable will do: $\forall x Rx$ and $\forall y Ry$ say exactly the same thing, since the variable only marks which places the quantifier governs. We use $x$, $y$ and $z$ for variables and keep $a$, $b$, $c$ for names, so a reader can always tell a placeholder from a particular object.
Watch the scope of a quantifier: it covers only what comes right after it. In $\exists x Fx \land Gx$ the $\exists x$ covers only $Fx$, and the $x$ in $Gx$ is left free, so the formula is not a complete sentence. Use parentheses, $\exists x(Fx \land Gx)$, to make the quantifier cover both.