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In my course notes of my modelling course, there is proof for the following proposition:

If the proof system is sound and complete, then consistency and satisfiability coincide.

In some part of the proof, they claim the following:

If $T$ is not satisfiable, then it holds in a trivial way that every formula is true in every model of $T$, hence, for arbitrary formula $\varphi$, $T \vDash \varphi$ and $T \vDash \neg \varphi$, and by completeness $T \vdash \varphi$ and $T \vdash \neg \varphi$. We conclude that $T$ is inconsistent.

Can someone explain this in a more 'practical' way, e.g. by providing an example. More precisely, I don't really understand the claim "If $T$ is not satisfiable, than it holds in a trivial way that every formula is true in every model of $T$".

Thanks in advance

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  • $\begingroup$ Are you comfortable with the notion of vacuous truth? For example, do you understand that, if there are no unicorns, then all unicorns are green? If so, then you can analogously understand that, if there are no models of $T$, then all models of $T$ satisfy $\varphi$. $\endgroup$ Commented Jan 20, 2018 at 14:10
  • $\begingroup$ Welp, never heard of that and wasn't mentioned in my book, this makes things a lot clearer obviously! Thanks a lot! $\endgroup$ Commented Jan 20, 2018 at 14:19

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"$T$ is not satisfiable" is short for "There are no models such that every statement in $T$ holds."

We want to test the statement that "In all models of $T$, $\varphi$ holds." To do this, we would need to check all models of $T$. If $\varphi$ holds in all of them, the statement is true. If there is any single model where $\varphi$ doesn't hold, the statement is false.

However, there are no models of $T$. Therefore, it is not possible for us to find a model where $\varphi$ doesn't hold, and so $\varphi$ must hold in all models of $T$.

(This is a classic sort of vacuous truth. People often have trouble with this sort of thing when starting mathematical logic, but it's worth familiarising yourself with it so that inferences like this become natural to you.)

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  • $\begingroup$ Welp, never heard of that and wasn't mentioned in my book, this makes things a lot clearer obviously! Thanks a lot! $\endgroup$ Commented Jan 20, 2018 at 14:19

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