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Premise: ∃𝑥𝐹(𝑥)→∃𝑥𝐺(𝑥)

Desired Conclusion: ∃𝑥(𝐹(𝑥)→𝐺(𝑥))

  • You have to be more specific. In case you need a syntactic proof, what system of logic do you work with? I.e. what axioms and rules of inference do you have? Without such information, it is virtually impossible to answer your question. – Mad Hatter Dec 12 '19 at 8:45
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Hint 1: The claim is clearly only valid in nonempty domains (you cannot prove something exists where nothing exists), so there is an unwritten assertion that something does exist.

Hint 2: You may syntactically prove by Excluding the Middle. Either there does exists something satisfying F (ie ∃x F(x)), or there does not.

Here's a skeleton using Fitch notation for Natural Deduction.

 |  ∃x x=x              existential import
 |_ ∃x F(x) → ∃x G(x)   premise
 |  ∃x F(x) v ~∃x F(x)  law of excluded middle
 |   |_ ∃x F(x)         assumption
 |   |  :
 |   |  ∃x (F(x)→G(x))
 |   +  
 |   |_ ~∃x F(x)        assumption
 |   |  :
 |   |  ∃x (F(x)→G(x))
 |  ∃x (F(x)→G(x))      disjunction elimination
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You won't be able to prove it, since these are not equivalent.

The premise says if there exists an x such that F(x) is true then there exists an x (not necessarily the same x) such that G(x) is true.

In the desired conclusion it is saying that there exists an x such that if F(x) is true then G(x) is true with that same x.

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  • 2
    Hi Cedric Martens. They're not equivalent, but the argument form is a valid one. The conclusion ∃x(Fx → Gx) is a very "mild" claim. It's logically equivalent to ∃x(~Fx v Gx), which does indeed follow from the premise. – Adam Sharpe Dec 11 '19 at 19:43

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