I've just started working through C. Chang and H. Keisler's Model Theory independently right now, and I'm reading through the first chapter, but I'm a little confused by the nature of a valid sentence. So suppose we have language, $\mathscr{S}$, and a sentence built from these, $\varphi$. According to the text, we say that $\models\varphi$ if and only if $$\forall A\subseteq \mathscr{S},A\models \varphi$$However, earlier the text also says,
Two distinguished models are the empty set $\emptyset$ and the set $\mathscr{S}$ itself.
It seems to me that this implies that no $\varphi$ may be valid then, as $$\nexists \varphi,\emptyset\models\varphi$$due to the fact that there are no sentence symbols from which to construct any $\varphi$. Is it the case then that no sentence ever valid in our model theory of sentential logic? In which case why do we even care about validity? Or is it to be inferred that in defining validity we must ignore the empty model?
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$\begingroup$Let $S$ be a sentence symbol, and let $\varphi$ be $\neg S$; then $\varnothing\not\vDash S$, since $S\notin\varnothing$, so $\varnothing\vDash\varphi$.
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