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Vacuous quantification

  1. Jul 14, 2007 #1
    I'm trying to prove a theorem which makes use of the identity [itex]\exists x (P) \rightarrow P[/itex] (where x is not a free variable of P). Intuitively, I want to believe it, but since I'm trying to do things rigorously, I'd like to be able to justify it to myself. Can anyone offer a suggestion as to how I'd derive the identifier from the usual axioms of first-order logic? (I'm sure that I'm missing something totally obvious). Thanks.
     
  2. jcsd
  3. Jul 14, 2007 #2
    I don't understand. P is a naked propositional symbol in FOL? If so, then your done
    (regardless of whether it has any quantifiers attached to it, or not). It's a tautology.
     
    Last edited: Jul 14, 2007
  4. Jul 14, 2007 #3
    P is a predicate in which x doesn't appear. Anyway, I know that it is a tautology, but I'm trying to prove it rigorously from FOL's axioms.
     
  5. Jul 16, 2007 #4
    Never mind, I've got it. Thanks anyway.
     
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