# Is it correct that If $\mathcal {A }$ is a model of $\Gamma$, and if $\Gamma \models {\psi}$ then $\mathcal {A } \models \psi$?

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 $\mathcal A \models \psi$?

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 $\mathfrak A \vDash ψ$.