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I am trying to prove $$\vdash \exists x (Px \to \forall x Px)$$ with a formal deduction, but I am stumped. Does anyone see what to do?

Hugh Mungus
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  • (It's false in an empty model; true in nonempty ones.) Do you understand how to prove it informally? – Patrick Stevens Mar 13 '17 at 11:23
  • You will need to disclose which formal system you want to work in. There are many possible choices, and we can't just take a guess at which one you're using. – hmakholm left over Monica Mar 13 '17 at 11:26
  • See also https://math.stackexchange.com/questions/412387/why-is-this-true-exists-xpx-rightarrow-forall-y-py (and questions linked from there) -- it is not primarily about formal proofs, but some of the answers do provide them in various systems. – hmakholm left over Monica Mar 13 '17 at 11:27
  • This is a question from my textbook, so I'm not sure how it could be false. The question just asks to show the above is true, though it falls in the section dealing with first-order predicate logic. Informally, we know that $A\to B$ is false only when $A$ is true, and $B$ is false, so we would want that to be the case for the above to be false. – Hugh Mungus Mar 13 '17 at 11:28
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    You can see here and many many similar post under drinker's paradox. – Mauro ALLEGRANZA Mar 13 '17 at 12:22
  • Thanks! Didn't know this was a well known question. – Hugh Mungus Mar 13 '17 at 12:24
  • @HughMungus What specific proof system do you work with? There are many variants! – Bram28 Mar 13 '17 at 13:21

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Here is a formal proof ... but remember that the system you have to use may have its inference rules defined differently!

enter image description here

Bram28
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