Informally, an interpretation of a logical system assigns meaning or semantic value to the formulae and their elements. More formally, if we consider a language whose non-logical terms include names, function symbols, predicate letters, and sentence letters, then an interpretation of a language specifies the following:

1. (i) a domain, or universe of discourse. This is a non-empty set, and forms the range of any variables that occur in any of the sentences of the language.

2. (ii) For each name in the language, an object from the domain as its reference or denotation.

3. (iii) For each function symbol a function which assigns a value in the domain to any sequence of arguments in the domain.

4. (iv) For each predicate letter a property or relation, specifying which sequences of objects in the domain satisfy the property or stand in the relation to each other.

5. (v) For each sentence letter, a truth-value. The logical constants such as expressions for truth functions and quantifiers will be assigned their standard meanings, via rules such as truth tables specifying how formulae containing them are to be evaluated. See also model theory. For interpretation in general, See meaning, semantics.