How “for every” (∀) works

∀x says something about every piece on the board. One counterexample makes it false.

F∀x (Cube(x)→Small(x))

Every cube must be small. a is a large cube, so a is the counterexample.

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How it works

∀x (Cube(x) → Small(x)) reads “for every piece x, if x is a cube, then x is small.” To check it, walk through the pieces one at a time. Any piece that breaks the rule is a counterexample.

The → matters. Without it, ∀x Cube(x) says every piece is a cube, which is a much stronger claim.

Worth memorizing

One counterexample makes ∀ false.
“For every” is true when nothing fits.
With no dodecahedra, every dodecahedron is large, and small, and anything else.
∀ usually pairs with →.
“Every cube is small” is ∀x (Cube(x) → Small(x)).

Practice

0 of 5 right

Question 1

∀x (Tet(x)→Small(x))

Question 2

∀x (Dodec(x)→Large(x))

Question 3

∀x (Cube(x)→Large(x))

Question 4

∀x (Small(x)→RightOf(x, a))

Question 5

∀x Dodec(x)

Next: There is∃x needs one witness: a single piece that makes the rest true.