In question (e), I have to prove: A ≡ (KB ⊭ S) and (KB ⊭ ¬S) is satisfiable, where KB and S are propositional variables. I am not able to follow the solution given in the image above as to why it is replacing S with another propositional variable P. Even if we do take P, one of (KB ⊭ P) and (KB ⊭ ¬P) becomes true and other false making A as false. So anyway here's what I tried, I find some interpretation to make statement A false, if I cannot find such interpretation then A is a tautology. From what I understand, (KB ⊭ S) is true (i.e. KB does not entail S is true) when premise KB is true and the consequence S is false. So if I take KB ≡ TRUE and S ≡ TRUE making (KB ⊭ S) ≡ FALSE making A ≡ FALSE, hence A is not a tautology, whereas the answer given in above image is Yes.

  • $\begingroup$ So I read the question wrong. It was to prove the statement in question was satisfiable by some interpretation and that's what the selected answer proves. Right? @MohammadRostami. So I am editing the question. $\endgroup$
    – Venky
    Commented Aug 18, 2021 at 3:59
  • $\begingroup$ Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics. You can use LaTeX. Don't forget to give proper attribution to your sources! $\endgroup$
    – D.W.
    Commented Aug 18, 2021 at 6:34

1 Answer 1


When you say $\alpha\models KB$, this means that $models(\alpha)\subseteq models(KB)$ that $models$ of $\alpha$ is used for an assignment that evaluates formula $\phi$ to true.

Under above interpretation, suppose $KB\equiv True$, and $S$ be a formula that isn't Tautology and just satisfiable (i.e. $A\vee B$), in propositional logic, hence we can conclude that

$$True\nvDash S\wedge True\nvDash \neg S.$$

Because $$models(KB)\nsubseteq models(S)\wedge models(KB)\nsubseteq models(\neg S).$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.