In predicate logic, is the "environment" only needed when free variables are present?

Suppose there exist no free variables in a given predicate logic formula. Is then a model alone sufficient to fully interpret the formula and make inferences? Don't we need an environment or variable lookup table (as often mentioned)?

It is not so that the model alone allows you to make inferences and that you don't need an environment at all; it is rather that, if no free variables occur, then you may as well start with an empty environment. Quantifiers modify the current environment, so you do need one. However, if there are no free variables around, then any of the assignments in the initial environment will not be used, and all variable values will come from the quantifiers anyway. Therefore, by trimming unnecessary assignments in the initial environment, you get an empty one.

StackExchange Computer Science Q#109795, answer score: 3