Semantic Consequence is Transitive

Theorem

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF, \GG$ and $\HH$ be sets of $\LL$-formulas.

Suppose that:

\(\ds \FF\)

\(\models_{\mathscr M}\)

\(\ds \GG\)

\(\ds \GG\)

\(\models_{\mathscr M}\)

\(\ds \HH\)

Then $\FF \models_{\mathscr M} \HH$.

Proof

Let $\MM$ be an $\mathscr M$-structure.

By assumption, if $\MM$ is a model of $\FF$, it is one of $\GG$ as well.

But any model of $\GG$ is also a model of $\HH$.

In conclusion, any model of $\FF$ is also a model of $\HH$.

Hence the result, by definition of semantic consequence.

$\blacksquare$