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Is the following statement correct?

if $ \alpha \models (\beta \vee \gamma) $ then $ \alpha \models \beta \vee \alpha \models \gamma $ or both.

I guess it is, but how would you prove it?

Piotr
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  • In mathematics, "or" means "one, or the other, or both," especially with the notation you're using (at least the $\vee$ is standard for inclusive "or"). – Matt Samuel Nov 21 '15 at 00:06

1 Answers1

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It is correct that :

if $\alpha \vDash (\beta ∨ \gamma)$, then $α⊨β$ or $α⊨γ$ ?

No, it is not.

Consider as $α$ the formula $p∨¬p$ and $p$ in place of $β$ and $¬p$ as $γ$.

Community
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Mauro ALLEGRANZA
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  • Can you expand a bit on this answer? Surely if we accept the law of excluded middle, then we must have either $p \lor \neg p \models p$ or $p \lor \neg p \models \neg p$? – mrp Nov 22 '15 at 11:00
  • See Logical consequence or entailment : "A formula $A$ is a semantic consequenceof a set of statements $\Gamma : \Gamma \models A$, if and only if there is no model $\mathcal{I}$ in which all members of $\Gamma$ are true and $A$ is false. We have to apply it to propositional logic, using truth-valuations. – Mauro ALLEGRANZA Nov 22 '15 at 11:05
  • Ah, I see, because if we assume $p \lor \neg p \not\models p$ and $p \lor \neg p \not\models \neg p$, then the two models $\mathcal{I}$ and $\mathcal{I}'$ that are witnesses may be different. Thank you. – mrp Nov 22 '15 at 11:55