I'm trying to semantically prove the following argument (sorry about the formatting - I'm new to stackexchange):

if P ⊨c Q then ⊨k 򪪪(P → Q)

However, I don't know how to translate or relate the classical logic on the left to the Kripke logic on the right. Can anyone help?


There's really no translation to be done. Here's an outline of a proof.

Assume P ⊨c Q. To show ⊨k 򪪪(P → Q), assume for contradiction that 򪪪(P → Q) is not valid. This means that there's a world which falsifies P → Q, that is, in which P is true but Q is false. But that contradicts the assumption that Q follows classically from P.

  • 1
    Is what you want to assume to obtain the contradiction ~⊨{sub}k{/sub} 򪪪(P → Q) or ~ 򪪪(P → Q) ? (I'm not familiar w/ Kripke logic so I'm not sure whether "assume X is not valid" is supposed to refer to the entailment or the implication) – guest1806 Oct 18 '18 at 16:23
  • 2
    @guest1806 I mean the former: that it's not the case that ⊨k 򪪪(P → Q). – Eliran Oct 18 '18 at 16:29

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