# Thread: Quantifier question proof

1. ## 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

2. ## Re: Quantifier question proof

Originally Posted by badwolf23
⊭ 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.

Originally Posted by badwolf23
{ 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.

Thank you!!