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How would you formalise: "There are no geniuses but Newton was a genius"?

I thought it could be: $$\neg\forall x(Px) \land Pa$$ and also $$\neg\exists x (Px) \land Pa$$These both seem to make sense and formalise the sentence, however they are not identical statements. The way it is worded is really confusing!

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    $\begingroup$ Are you using are and was synonymously? I could interpret it as, there are no geniuses now, but there were geniuses at some point. $\endgroup$
    – GFauxPas
    Jan 22, 2017 at 15:38
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    $\begingroup$ Yes I think it is meant to be synonymous... that was all the information I was give. Sorry, didn't make it clear that Px: x is/was a genius; a: a is/was Newton $\endgroup$
    – Andy
    Jan 22, 2017 at 15:40

2 Answers 2

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The second one looks right to me. If by $Px$ you mean "$x$ is a genius" and $a$ is Newton, then your statements translate to:

1) it's not true that every person is a genius, and it's true that Newton is a genius;

2) there's doesn't exist anyone who is a genius, and it's true that Newton is a genius

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    $\begingroup$ Thanks, so that means $\forall x (\neg Px)\land Pa$ is the same as (2)... intuitively it feels strange! Thanks for the help. $\endgroup$
    – Andy
    Jan 22, 2017 at 15:48
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    $\begingroup$ "No one is a genius" doesn't intuitively feel the same as "everybody is not a genius" to you? Don't worry, you'll get used to it :) Remember that you can accept an answer if your doubts have been cleared $\endgroup$ Jan 22, 2017 at 15:56
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One way to formalize this would be $\bot$, since that statement is a contradiction ... Unless of course the point of the sentence is that there are no geniuses now, though we have had geniuses in the past, like Newton.

One way to formalize that would be to use a predicate $G(x,t)$ which says that $x$ is a genius at time $t$. So then:

$\neg \exists x G(x,t_{now}) \land \exists t (t<t_{now} \land G(newton,t))$

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    $\begingroup$ Might want to add something like $... \land t < t_{now}$ (for "was" rather than "will be") :-) $\endgroup$
    – psmears
    Jan 22, 2017 at 21:24
  • $\begingroup$ @psmears yeah, good catch, thanks! $\endgroup$
    – Bram28
    Jan 22, 2017 at 21:25
  • $\begingroup$ Thank you very much for your help! The question was in the worksheet that did not want us to use dyadic predicates, but it's definitely much more accurate! $\endgroup$
    – Andy
    Jan 23, 2017 at 10:47

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