Quantifier Order

With relations (Relational Predicates) a sentence can have two quantifiers, and then their order matters. "Everyone likes someone" and "someone is liked by everyone" sound alike but say very different things. Use what you learned in Truth in a Small World to check them in small worlds.

Key idea

Read quantifiers from left to right. $\forall x\exists y Lxy$: for each person $x$ there is some $y$ that $x$ likes (each may like a different person). $\exists y\forall x Lxy$: there is one person $y$ whom every $x$ likes. The second implies the first, but not the other way round.

Why it works

In $\forall x\exists y Lxy$, the choice of $y$ comes after $x$ is fixed, so $y$ may depend on $x$: Ava likes Ben, Ben likes Cleo, Cleo likes Ava, and everyone likes someone. In $\exists y\forall x Lxy$, $y$ is chosen first, once, and must work for every $x$. That is a much stronger claim: one person liked by all. If such a person exists, then of course everyone likes someone (that person), which is why one direction is valid.

A circle of likes: everyone likes someone, but no one is liked by everyone. Add arrows to make the second sentence true.

Worked examples

Worked example: Translate both

Let $Lxy$: $x$ likes $y$. Translate "Everyone likes someone" and "Someone is liked by everyone."

"Everyone likes someone": for each $x$, some $y$ with $Lxy$.
"Someone is liked by everyone": some $y$ such that every $x$ has $Lxy$.

Answer: $\forall x\exists y Lxy$ and $\exists y\forall x Lxy$

Worked example: In the world above

Is $\exists y\forall x Lxy$ true in the circle of likes?

Try each candidate $y$. Ben is liked by Ava only; Cleo by Ben only; Ava by Cleo only.
No one is liked by all three.

Answer: False, although $\forall x\exists y Lxy$ is true.

Worked example: Someone likes everyone

Translate "Someone likes everyone."

There is an $x$ who likes every $y$.

Answer: $\exists x\forall y Lxy$

Common mistake

Treating $\forall x Ey$ and $\exists y Ax$ as the same. The first lets each $x$ pick its own $y$; the second needs one $y$ for all. A quick test: a circle of three people, each liking the next, makes the first true and the second false. When translating, keep the order of "everyone" and "someone" exactly as the English has it, and read the formula back to check.

Same quantifiers can swap: $\forall x\forall y Lxy$ means the same as $\forall y\forall x Lxy$, and $\exists x Ey$ as $\exists y Ex$.
Mixed quantifiers cannot swap.
$\exists y\forall x Lxy$ implies $\forall x\exists y Lxy$; the converse fails.

How do you evaluate a sentence with two quantifiers in a world? Peel them one at a time. For $\forall x\exists y Lxy$, go through each person $x$ and ask whether $\exists y Lxy$ holds for that $x$: is there an arrow leaving $x$? For $\exists y\forall x Lxy$, go through each candidate $y$ and ask whether every $x$ has an arrow into it. The outer quantifier tells you which question to repeat.

Quantifier order is one of the places where symbols beat English. "Every student read a book" can mean each read some book or all read the same book; the formulas force you to say which.

Quantifier Order

Worked examples

Practice

Builds on

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Sources and further reading