I’m reading this book [1] and in Section 1.2 (Semantics) came across a new definition of entailment. In my previous studies on logic, entailment meant that if there existed a set of sentences on the left of the entailment symbol, then all models that satisfy those sentences also satisfy the sentence on the right.
$$ \Gamma \models \varphi $$
But here the book defines entailment as:
$$ I \models F $$
to mean “the formula $F$ evaluates to true under the interpretation $I$.”
Could someone clarify the disparity between these two uses of the symbol $\models$ ? Are we overloading the symbol, or does $\models$ genuinely mean different things in different contexts ? Any insights would be appreciated.
[1]. Bradley, Aaron R., and Zohar Manna. The calculus of computation: decision procedures with applications to verification. Springer Science & Business Media, 2007.
