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http://intrologic.stanford.edu/exercises/exercise_08_01.html In the link above in question 4, it says p(x) ⇒ ∀x.p(x) http://intrologic.stanford.edu/exercises/exercise_08_02.html However, in the next exercise it also says φ ⊨ ∀x.φ is true.

I don't really get it. Aren't those two the same thing? Why is one valid when another is contingent? Is it because of the difference between implication and entailment?

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The difference is that "$⇒$" is part of the logical language used to form sentences, whereas "$⊨$" is a symbol in the language we use to talk about sentences, so in the first question you have a single sentence "$p(x)⇒∀x.p(x)$", whereas in the second question you have separate sentences "$φ$" and "$∀x.φ$". This matters because free variables are implicitly $∀$-quantified at the sentence level, so the first one is like $∀x.(p(x)⇒∀y.p(y))$ and the second one is like $(∀x.p(x))⊨(∀x.p(x))$.

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