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True or False: if $\alpha \vDash (\beta\land\gamma)$ then $\alpha \vDash \beta$ and $\alpha \vDash \gamma$.

Answer: true because for the and statement to be correct both $\beta$ and $\gamma$ would have to be the same.

True or False: if $\alpha \vDash (\beta\lor\gamma)$ then $\alpha \vDash \beta$ or $\alpha \vDash \gamma$ (or both)

For this one I am unsure. any insight?

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  • $\begingroup$ In the first, $\beta$ and $\gamma$ would not have to be the same. Just taking any two distinct things provable from $\alpha$; these won't be the same by our choice, but $\alpha$ certainly proves their conjugation. Do you have the Completeness Theorem at your disposal? $\endgroup$ – Hayden Mar 25 '15 at 0:56
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The first one is indeed true, but not for the reason you give. For example, if we set $\alpha \equiv Q \land P$, $\beta\equiv P$, and $\gamma\equiv Q$ (where $P$ and $Q$ are propositional variables), then the premise becomes $$ Q\land P \vDash P\land Q$$ which certainly is the case, even though $P$ and $Q$ are not the same.

Rather your argument should be that for if we have a valuation/interpretation that satisfies $\beta\land\gamma$, then by definition of $\vDash$ it also has to satisfy $\beta$ and $\gamma$ separately.

In the second question, consider what happens if we set $\alpha \equiv P \lor Q$, $\beta\equiv P$, and $\gamma\equiv Q$? Does the assumption hold? Does the purported conclusion?

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  • $\begingroup$ yes it should hold as well then. if we have an interpetation that satisifes b V y then by definition it has to satisfy b or y, or can satisfy both. is my thinking sound? $\endgroup$ – user2510809 Mar 25 '15 at 1:08
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    $\begingroup$ @user251080: No. The question doesn't ask about a single interpretation. Find your definition of $\alpha\vDash\beta$ -- it speaks about all interpretations that satisfy $\alpha$. So in order for the second claim to be true it must either be the case that all interpretations that satisfy $\alpha$ also satisfy $\beta$, or that all these interpretations also satisfy $\gamma$. It won't work that some of them satisfy $\beta$ and others satisfy $\gamma$. $\endgroup$ – hmakholm left over Monica Mar 25 '15 at 1:20

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