Why the ordering of the quantifiers matters here?

Question

$\newcommand{\fool}{\operatorname{fool}}$I want to transfer the statement

You can fool all the people some of the time

into predicate logic.

Now I am using fool(X,T) to mean that I fool person X at time T.

Now I know that the correct form is $\exists T\ \forall X\ \fool(X,T)$.

But I want to understand why is it not correct if we change the ordering of the quantifiers to $\forall X\ \exists T\ \fool(X,T)$.

The first one is saying there exists a time that I can fool all people right ?

Ans the second one is saying for all the people out there, There exists a time that I can fool you , right ?

What is the difference ??

What makes the first once correct but not the second.

Answer 1

$(\exists t)(\forall x) \operatorname {ICanFool}(x,t)$ says: "I can fool everybody at the same time". Switching the order of quantifiers, $(\forall x)(\exists t)\operatorname {ICanFool}(x,t)$ says: "for each person $x$, I can fool $x$ at some time" — that is, at some time $t_x$, depending on $x$. It's a weaker statement. The times $t_{x_0}$, $t_{x_1}$ may be different: you duped John at 2pm Oct 5 2013, but you will trick Sally tomorrow at 3:15pm.

The second one can be rendered into more idiomatic English as "I can fool anybody sometime [at some times]". The first one I think is more likely the way to symbolize "I can fool everybody [all of the people] some of the time", i.e. "Sometimes, I can fool everybody" — all at once, that is. It's that stronger statement that's meant in the first part of the famous line attributed to Lincoln, "You may fool all the people some of the time, you can even fool some of the people all of the time, but you cannot fool all of the people all the time." Anyway, the stronger statement (the first) is bolder rhetoric, so I infer that's what is intended.

Granted, if the statement were "Some of the time, you can fool all of the people", then the first, stronger reading would clearly be the correct one; but in the interest of rhetorical parallelism, it isn't — so it's ambiguous. However, spoken intonation, dynamics and timing (suprasegmental features, as linguists would say) make it possible to say "you can fool all of the people... some of the time" and clearly mean the stronger statement. Regretably we don't have a recording of the original performance.

Interestingly, though the quote is attributed to Lincoln, there's no evidence of him having said or written it. It was other (Prohibitionist!) politicians who used it, two decades after Lincoln's assassination, and attributed it to him. See http://quoteinvestigator.com/2013/12/11/cannot-fool/. In any case, these words were used in the political arena, where cautious understatement is not the norm, and not in front of a logic class.

Answer 2

To expand on AJ stas's comment, the first formulation, $\exists T \forall X fool(X,T)$, first you assert the existence of (at least one) single Time, T. Then you say that for EVERY person X, $fool(X,T)$, i.e, every single person can be fooled at that exact time.

When you reverse the quantifiers, $\forall X \exists T fool(X,T)$, you are asserting that for every single person, there exists (at least one) time (that depends on that person!) for which that person can be fooled. So, I might be fooled today and you might be fooled tomorrow.

In the first example, there's a time, like 3:17PM, where every person in the world can be fooled at once.

I'm pretty sure that the second version more accurately states the colloquial meaning of the phrase.

Answer 3

The difference is that in the second case, different people are fooled at different times. Suppose I can fool all Christians on even-numbered days and all Muslims on odd-numbered days, and everyone is either a Christian or a Muslim. That doesn't mean there is any day on which I can fool all of the people.