Proving Vacuous Quantification in First-Order Logic

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The discussion centers on proving the theorem \exists x (P) \rightarrow P, where x is not a free variable of P, within first-order logic. The initial query seeks a rigorous justification for this identity, expressing a desire for clarity despite an intuitive understanding. Participants clarify that if P is a naked propositional symbol, the proof is straightforward as it becomes a tautology. The conversation reveals that P is actually a predicate without x, leading to the conclusion that the identity holds true. Ultimately, the original poster resolves their confusion and thanks the contributors.
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I'm trying to prove a theorem which makes use of the identity \exists x (P) \rightarrow P (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.
 
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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.
 
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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.
 
Never mind, I've got it. Thanks anyway.
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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