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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 '17 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 '17 at 15:40
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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 '17 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 $\endgroup$ – man_in_green_shirt Jan 22 '17 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 '17 at 21:24
  • $\begingroup$ @psmears yeah, good catch, thanks! $\endgroup$ – Bram28 Jan 22 '17 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 '17 at 10:47

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