Question

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

Answer 1

Correct Answer - Option 2 : No one can fool everyone all the time
F(x,y,t) = Person 'x' can fool person 'y' at time 't'.

For better understanding propagate negation sign outward by applying Demorgan's law.

\(\forall_x \exists_y(\neg F(x,y,t))= \neg \exists_x \forall_y \forall_t(F(x,y,t))\)

Now converting \(\neg \exists_x\forall_y\forall_t(F(x,y,t))\)to English is simple.

\(\neg \exists_x\forall_y\forall_t(F(x,y,t)) =\)There does not exist a person who can fool everyone all the time.

Which means "No one can fool everyone all the time".

Hence the correct answer is No one can fool everyone all the time.