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I'm trying to determine whether there exists a model for the following logical formula: $(p_1 \to (p_2 \lor p_3)) \land(p_2 \to \neg p_3) \land ((p_1 \lor p_3) \to \neg p_2)$. Here's my understanding of the formula: If p1 is true, then either p2 or p3 must be true, and if p2 is true, then p3 must be false. However, this seems to create a conflict because if p1 is true, p2 cannot be false. Can anyone confirm whether a model exists for this formula or explain why it might not be possible? Thank you!

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2 Answers 2

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Set all variables to false, that way each implication has a false left-hand-side and thus it's satisfied.

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  • $\begingroup$ Thank you for your response. I thought that (p2 → ¬p3) implied that if p3 is false, then p2 must be true. However, if setting all variables to false satisfies the formula, it seems I may have misunderstood the logic. $\endgroup$
    – cozen
    Nov 7, 2023 at 21:36
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Be lazy: encode your problem in the input language of your favourite SAT solver and feed the resultung term to it. If it answers "SAT" it'll also provide a model. If not, then there's no model.

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