Politicians can fool FOL confusion

Question

In the above problem why in the last part of the sentence is the negation also enclosing Person(Y)? Shouldn't it just enclose Fools(x,y,t) only because the question implies "can't fool" not "can't people"?

Answer 1

"$\forall t\forall y [Person(y)\implies\neg Fools(x,y,t)]$" - which I think is your alternate candidate for the final clause of the expression above (with parentheses added for clarity) - means "For every person and at every time, $x$ cannot fool that person at that time." In better English, this is "$x$ can't fool anyone ever."

The point is that "$x$ can't fool all of the people all of the time" should be understood as

not($x$ can fool all of the people all of the time),

which is expressed as $$\neg(\forall t\forall y(Person(y)\implies Fools(x,y,t)))$$ as in the OP.

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Incidentally, as a slight quibble I'd observe that we also need to specify that $t$ ranges over times, so e.g. $$\neg(\forall t\forall y(Person(y)\wedge Time(t)\implies Fools(x,y,t)))$$ is better. But that's not the main point here.