5.1.6 Truth in a Model

We can now define what it means for a sentence to be true in a model :

A sentence is true in a model if and only if for any assignment of values to variables in , we have that:

If is true in we write:

This elegant definition of truth beautifully mirrors the special, self-contained nature of sentences. It's based on the following observation: It doesn't matter at all which variable assignment is used to compute the satisfaction of sentences. Sentences contain no free variables, so the only free variables we will encounter when evaluating one are those produced during the process of evaluating its quantified subformulae (if it has any). But the satisfaction definition tells us what to do with such free variables, namely, to try out variants of the current assignment and see whether they satisfy the matrix or not. In short, you may start with whatever assignment you like; the result will be the same. It is reasonably straightforward to make this informal argument precise, and the reader is encouraged to do so.