Multiple choice

Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. Which one of the statements below expresses best the meaning of the formula $∀x∃y∃t(¬F(x,y,t))$?

  1. Everyone can fool some person at some time.

  2. No one can fool everyone all the time.

  3. Everyone cannot fool some person all the time.

  4. No one can fool some person at some time.

Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The formula ∀x∃y∃t(¬F(x,y,t)) reads: For all x (everyone), there exists some y (some person) and some time t such that x cannot fool y at time t. This means No one can fool some person at some time. More colloquially: Every person has at least one other person they cannot fool at at least one moment. Option A says everyone CAN fool someone (wrong - we have ¬F). Option B says no one can fool everyone all the time (different quantifier structure). Option C says everyone cannot fool some person all the time (wrong - says ∀t not ∃t). Option D correctly captures: there's at least one person each of us fails to fool at some point.