Suppose the predicate \( F(x, y, t)\) is used to represent the statement that…

Formal reading: For every person x there exists a person y and a time t such that x does not fool y at time t (i.e. ∀x∃y∃t ¬F(x,y,t)).

Equivalence: This formula is equivalent to ¬∃x∀y∀t F(x,y,t). In words: it is not the case that there exists a person who can fool every person at every time.

Natural-language paraphrase: "No one can fool everyone all the time."

Why other phrasings are incorrect:

• The statement "Everyone can fool some person at some time" affirms that for each person there exists someone and a time they can fool; the given formula instead asserts that for each person there exists someone and a time they do not fool, so these are opposites.

• The statement "Everyone cannot fool some person all the time" means each person has someone they never fool (failure for every time), whereas the formula requires only the existence of at least one time when they do not fool that person (weaker condition).

• The statement "No one can fool some person at some time" denies that any fooling ever occurs (no person fools any person at any time), which is a much stronger claim than the given formula that only requires each person has at least one person and one time they fail to fool.