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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)?

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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.

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  • $\begingroup$ How do "quantifiers modify the current environment"? $\endgroup$ – apen May 26 at 0:17
  • $\begingroup$ Quantifiers bind variables to values. $\endgroup$ – dkaeae May 26 at 6:35
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The wikipedia article First-order logic calls the "environment" a variable assignment. A formula with no free variables is called a sentence. A model alone is sufficient to fully interpret a sentence. However, all deduction systems for first order logic that I know need formulas with free variables as intermediary deductions, and hence need the concept of a variable assignment for an interpretation of their inferences.

In general, this seems to be necessary. Some specific first-order theories (like Presburger arithmetic) allow quantifier elimination, but all theories that I know which allow quantifier elimination are decidable. So my guess would be that any deduction system for an undecidable first-order theory will need to work with free variables (or some equivalent concept) and hence will need the concept of a variable assignment (or some equivalent concept) for an interpretation of their inferences.

To see that the issue is related to the deductions mostly and not to interpretation, one can look at alternative semantics like game semantics.

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