Equivalent logical expression for the Well Formed Formula (WFF), ~(∀x) F[x] is

Key rule: Negation of a universal quantifier: ~(∀x)F[x] is equivalent to ∃x (~F[x]).

• Apply quantifier negation: replace the universal quantifier with an existential quantifier and move the negation inside: ~(∀x)F[x] ≡ ∃x (~F[x]).

• Intuition: If it is not true that every x satisfies F, then there must be at least one x for which F is false.

• Related equivalence to avoid confusing: ~(∃x)F[x] ≡ ∀x (~F[x]) — negating an existential gives a universal negation, which is a different (usually stronger) statement.

Check the given expressions briefly:

• ∀x (~F[x]) — asserts every x fails F, stronger than 'not every x satisfies F'; not equivalent.

• ~(∃x) F[x] — equivalent to ∀x (~F[x]), again stronger than the original negation; not equivalent.

• ∃x (~F[x]) — correct: there exists an x for which F is false, exactly what ~(∀x)F[x] expresses.

• ∀x F[x] — asserts every x satisfies F, which contradicts the negation and is not equivalent.

Conclusion: The equivalent logical expression is ∃x (~F[x]).