linguistic:voice_change, structural:summarization, information:classification, information:highlighting
Evaluation of truth values

A formula evaluates to true or false given an interpretation and a variable assignment μ that associates an element of the domain of discourse with each variable. The reason that a variable assignment is required is to give meanings to formulas with free variables, such as y = x {\displaystyle y=x}. The truth value of this formula changes depending on the values that x and y denote.

First, the variable assignment μ can be extended to all terms of the language, with the result that each term maps to a single element of the domain of discourse. The following rules are used to make this assignment:

Next, each formula is assigned a truth value. The inductive definition used to make this assignment is called the T-schema.

If a formula does not contain free variables, and so is a sentence, then the initial variable assignment does not affect its truth value. In other words, a sentence is true according to M and μ {\displaystyle \mu } if and only if it is true according to M and every other variable assignment μ ′ {\displaystyle \mu '}.

There is a second common approach to defining truth values that does not rely on variable assignment functions. Instead, given an interpretation M, one first adds to the signature a collection of constant symbols, one for each element of the domain of discourse in M; say that for each d in the domain the constant symbol c is fixed. The interpretation is extended so that each new constant symbol is assigned to its corresponding element of the domain. One now defines truth for quantified formulas syntactically, as follows:

This alternate approach gives exactly the same truth values to all sentences as the approach via variable assignments.