Satisfiability of Sets of Formulas

We next introduce sets of formulas as a way of talking about potentially infinite conjunctions of formulas. We will need this when reducing reasoning in first-order logic to reasoning in propositional logic.

We say that interpretation $I$ is a model for a set of formulas $S$, written $I \models S$, iff for each $F \in S$, $I \models F$. In other words, we view a set of formulas as a (potentially infinite) conjunction; when $S$ is finite then $I \models S$ is the same condition as $I \models \bigwedge_{F \in S} F$.

Example: the set $\{p_1, \lnot p_1 \lor p_2, \lnot p_2 \lor p_3, \lnot p_3 \lor p_4, \ldots \}$ is an infinite satisfiable set. The example of one satisfying interpretation is the interpretation which evaluates all variables to $true$.

Clearly, if $I \models B$ and $A \subseteq B$, then also $I \models A$.

We say that a set of formulas $S$ is satisfiable iff there exists an interpretation $I$ such that $I \models S$.