I trying to come up with a valid formalisation of the quote "You can fool all the people some of the time, and some of the people all the time, but you cannot fool all the people all the time".

I have the following expression which I believe to be a correct answer:

(∀p.∃t.𝖥𝗈𝗈𝗅(p,t)) ∧ (∃p.∀t.𝖥𝗈𝗈𝗅(p,t)) ∧ ¬∀p.∀t.𝖥𝗈𝗈𝗅(p,t)

The above expression states that all people will be fooled for some time AND some people will be fooled for all of the time AND not all people will be fooled all of the time. I think the expression is correct because it covers all the scenarios in the quote.

However, I am not sure if my reasoning is correct. Any insights are appreciated.


Your interpretation of the first clause has the time qualifier following the people qualifier, so it means that for every person, there is some time -- dependent on the person -- at which you can fool that person. Since the time is dependent on the person, this doesn't mean there is any single time at which you can fool all the people. It seems to me Lincoln might have meant there is a single time at which you can fool all the people, in which case the time qualifier has to come first. Of course, the quote is logically ambiguous between these two meanings, and we can't ask Lincoln which he intended.

  • $\begingroup$ Would you say that (∃t.∀p.𝖥𝗈𝗈𝗅(p,t))∧(∀t.∃p.𝖥𝗈𝗈𝗅(p,t))∧¬∀t.∀p.𝖥𝗈𝗈𝗅(p,t) is a better answer then? $\endgroup$ – ceno980 Aug 9 at 12:16
  • $\begingroup$ @ceno980 But you've changed the order of the quantifiers in the second clause too, so now you've changed its meaning. There are four possible interpretations of the statement which differ by quantifier order in the first two clauses. You and I can't say which is better because we don't know which Lincoln intended. But you can try to precisely describe (in words) what you think Lincoln intended and then ask whether your expression matches that meaning. $\endgroup$ – BallBoy Aug 9 at 12:40
  • $\begingroup$ I also believe that in the quote, Lincoln meant that at a specific time you can't fool all people. $\endgroup$ – ceno980 Aug 9 at 12:43

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