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I'm not certain that I have understood the definiton of $\models $ correctly. I this statement correct?

If $\mathcal {A } $ is a model of $\Gamma $, and if $\Gamma \models {\psi}$ then $\mathcal {A } \models \psi $

Here $\Gamma $ is a set of formulas, and with $\mathcal {A } $ is a model of $\Gamma $ it is ment that "$\mathcal A $ is a model for every formula in $\Gamma $".


Also to understand the difference between $\mathcal {A } \models \psi $ and $\phi \models \psi $ can I say that,

if $\phi \models \psi $, then $\mathcal A \models \psi $ implies $\mathcal A \models \psi $?

Thanks in advance!

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Yes and yes.

If $\mathfrak A$ is a model of $\Gamma$, then every formula $\gamma \in \Gamma$ is true in $\mathfrak A$, i.e. for every $\gamma \in \Gamma :\mathfrak A \vDash \gamma$.

But $Γ \vDash ψ$ means that $\psi$ is true in every model of $\Gamma$, thus also in $\mathfrak A$, i.e. $\mathfrak A \vDash ψ$.

Summing up :

$Γ \vDash ψ$ iff for every $\mathfrak A$, if $\mathfrak A \vDash \gamma$, for every $\gamma \in \Gamma$, then $\mathfrak A \vDash \psi$.

About $ϕ \vDash ψ$, this is nothing else that $\Gamma \vDash \psi$, with $\Gamma = \{ ϕ \}$.

Thus

$ϕ \vDash ψ$ iff for any $\mathfrak A$, if $\mathfrak \vDash ψ$, then $\mathfrak A \vDash ψ$.

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