Nested quantifiers: why order matters

∀x ∃y and ∃y ∀x use the same symbols and say different things.

F∀x ∃y Adjoins(x, y)

Every piece must touch some piece. b touches nothing, so b is the counterexample.

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How it works

Read quantifiers left to right. In ∀x ∃y Adjoins(x, y), you pick x first, then find a y for that x. Each piece can have its own neighbor.

In ∃y ∀x, you pick y first and it has to work for every x. One piece has to serve them all.

Read quantifiers left to right.
In ∀x ∃y, y can change with x.
Every person has a mother: each person may have a different one.
In ∃y ∀x, one y must work for every x.
One person is everyone’s mother. A much stronger claim.

0 of 4 right

Question 1

∀x ∃y Adjoins(x, y)

Question 2

∃x ∀y (Cube(y)→Adjoins(x, y))

Question 3

∀x ∃y (x≠y∧SameRow(x, y))

Question 4

∃x ∀y (y≠x→LeftOf(x, y))