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Math Help - Quantifier question proof

  1. #1
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    Quantifier question proof

    For each of the following, give and explain an example (formulas, structures, and/or valuations) that justifies the claim.
    (Note:
    ⊭ ϕ→∀x.ϕ.
    { ϕ→ψ} ⊭ (∀ x.ϕ)→(∀x.ψ).

    I have no idea how to actually tackle this question and would appreciate any help. thanks
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  2. #2
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    Re: Quantifier question proof

    Quote Originally Posted by badwolf23 View Post
    ⊭ ϕ→∀x.ϕ.
    Let ϕ(x) means "x is married." Then you can find a group of people that includes both marrieds and singles and call one married person x.

    Quote Originally Posted by badwolf23 View Post
    { ϕ→ψ} ⊭ (∀ x.ϕ)→(∀x.ψ).
    I assume that the definition of logical consequence requires evaluating the two formulas (on the left and right of ⊨) in the same structure and the same valuation. Then you can make ∀x. ϕ(x) true and ∀x. ψ(x) false in some structure and choose the valuation that makes ψ(x) false.
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  3. #3
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    Re: Quantifier question proof

    Thank you!!
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