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Math Help - First Order Logic: Why is this sentence valid?

  1. #1
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    First Order Logic: Why is this sentence valid?

    Let φ(x) and ψ be first order formulas so that x is not free in ψ.
    Show why this sentence is valid:
    ( (∀x
    φ(x) ) ψ) (
    ∃x( φ(x) → ψ) )

    I hope someone can give me a simple semantical explanation
    (no need to show a natural deduction)
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  2. #2
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    Re: First Order Logic: Why is this sentence valid?

    The semantics of first-order logic is designed to reflect our meta-level concepts of "implies," "for all," etc. Do you see why this formula is true when the connectives are interpreted as these natural-language concepts? Hint: for the left-to-right direction, consider the cases when ∀xφ(x) is true and when it is false.
    Thanks from Zhai
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