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I have to prove some Propositional Logic assertions.

Given this one: $\alpha \models \beta \Leftrightarrow (\alpha \Rightarrow \beta)$ is valid

Where $\models$ is entailment

The answer is: $\alpha \Rightarrow \beta$ holds in those models where $\beta$ holds or where $\neg\alpha$ holds. This is precisely the case if $\alpha \Rightarrow \beta$ is valid.

I don't understand this answer. Could someone clarify it for me. I don't know what holds means in this context. English is not my first language.

I know that a sentence to be valid needs to be True in all models. Also that $\alpha \models \beta \Leftrightarrow$ in every model in which $\alpha$ is true, $\beta$ is also true.

Thanks

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2 Answers

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You can think of the phrase "[insert something here] holds" as "[that same something] is [insert property here]". More concretely, "holds" in this context is a sort of expression that refers to some thing "holding true" or always having the property of being true.

So, when we say $\alpha \Rightarrow \beta$ holds, we mean that $\alpha \Rightarrow \beta$ is in fact true.

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so in this case the assertion is incorrect and it should be: $\neg\alpha \models \beta \Leftrightarrow (\alpha \Rightarrow \beta)$ is valid? –  ivan.freire Jan 19 '12 at 2:41
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Definition :

$\Gamma$ - set of one or more sentences

$S_1$ - conjunction of the elements of $\Gamma$

$S_2$ - sentence

then : $\Gamma \models S_2$ iff $\lnot(S_1 \land \lnot S_2)$

First of all $\alpha$ cannot be notation of the set and the sentence at the same time . But if we define $\Gamma$ as :

$\Gamma=\{\alpha\}$ then we can ask whether : $\Gamma \models \beta \Leftrightarrow (\alpha \Rightarrow \beta)$

Now let us observe following case :

$\Gamma$$= $ $\{ $ S likes ice cream $\}$

$S_1=$ S likes ice cream

$S_2=$ S is a man

then $\Gamma$ doesn't entail $S_2$ because :

"S likes ice cream" and "S is not a man" are not logically inconsistent because they can both be true .

Therefore if $\Gamma=\{\alpha\}$ then assertion : $\Gamma \models \beta \Leftrightarrow (\alpha \Rightarrow \beta)$ isn't valid .

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