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I am struggling with the way to write a clear and mathematical proof of logical theorems. Take for example the theorem $\Gamma \models A, \Gamma \subseteq \Delta$ implies $\Delta \models A$. I can prove it this way:

If every formula in $\Delta$ is validated, then that validates every formula in $\Gamma$, which validates $A$.

But how do I prove such thing with mathematical language?

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  • $\begingroup$ What exactly dies "validated" mean here? Spell that out a bit, and you will have your desired proof. $\endgroup$ May 15, 2013 at 18:19
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    $\begingroup$ If the word "validated" was used in the definition of $\models$ (and was, therefore, defined beforehand), then you might not need to spell it out, but otherwise I agree with Peter that you should. $\endgroup$ May 15, 2013 at 20:55

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A proof of the above fact needs the definitions :

(i) $Γ \vDash A$ iff for every interpretation $\mathcal M$, if all formulae in $\Gamma$ are true in $\mathcal M$, then also $A$ is

and :

(ii) $\Gamma \subseteq \Delta$ iff for every formula $\alpha$, if $\alpha \in \Gamma$, then $\alpha \in \Delta$.

Consider now an interpretation $\mathcal M$: due to the fact that $\Gamma \subseteq \Delta$, if all formulae in $\Delta$ are true in $\mathcal M$, then $\mathcal M$ will be also a model for all formulae in $\Gamma$ (which are among those in $\Delta$).

But $A$ is true in every model of $\Gamma$; thus, for every interpretation $\mathcal M$, if $\mathcal M$ is a model for all formulae in $\Delta$, then also $A$ is true in $\mathcal M$, i.e.

$\Delta \vDash A$.

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