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I have a proof problem which I am stuck on. 

{ } ⊢ ( ∃x Fx ∨ ∃x ~Fx)

I already figured it out in order to get a∨b, I will have to do ~a→b or ~b→a. However, I have absolutely no idea how to get to there ~∃x Fx → ∃x ~Fx or ~∃x ~Fx → ∃x Fx.

These rule of inference are allowed: http://pastebin.com/4CFaYVzT and http://pastebin.com/V3GsrvUG

Can someone guide me through this problem? Thanks!

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    I'm confused. This is false if the domain is empty.
    – user4702
    Nov 4, 2013 at 4:32
  • You need to add your quantifier rules to your Pastebin file!
    – Paul Ross
    Nov 4, 2013 at 20:41
  • @PaulRoss - Added.
    – Derek 朕會功夫
    Nov 4, 2013 at 21:29

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You can assume the negation of this sentence and easily derive a contradiction from it. By DeMorgan's laws it becomes the conjunction of two negations. Each of these negations is a negated existential sentence which is the equivalent of a universally quantified negation. That is, you have the conjunction of "everything is not F" and "everything is not not F". Separate them via conjunction-elimination and instantiate them and you have a contradiction. It might look like ~Fa and ~~Fa.

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