5.1.4 Interpretations and Variant Assignments

Let's suppose we've fixed our vocabulary. (Note: Whenever we talk of a model from now on, we mean a model of this vocabulary, and whenever we talk of formulae, we mean the formulae built from the symbols in that vocabulary.) We now give two further technical definitions which will enable us to state the satisfaction definition in a concise manner.

Interpretations

First, let be a model, let be an assignment of values to variables in , and let be a term. The interpretation of with respect to and is if is a constant, and if is a variable. We denote the interpretation of by .

Variant Assignments

Another concept we need is that of a variant of an assignment of values to variables. So, let be an assignment of values to variables in some model, and let be a variable. If is an assignment of values to variables in the same model, and for all variables other than , then we say that is an x-variant of . Variant assignments are the technical tool that allows us to try out new values for a given variable (say ) while keeping the values assigned to all other variables the same.