Interpretations in zeroeth order logic were relatively simple. For each variable we got to assign it one of two values, true or false. Technically there were infinitely many variables and hence infinitely many interpretations but for any particular statement, or finite set of statements, only finitely many variables occur and so we could enumerate all the interpretations. This was in fact the basis of the method of truth tables.
First order logic has many more interpretations. We can, for example, construct an interpretation as follows. We begin by choosing a pair of sets, called the “inner domain” and “outer domain”, such that the inner domain is a subset of the outer domain. Then we assign an element of the domain to each variable and an element of the outer domain to each parameter. To each predicate we assign a relation, which you can safely think of as a Boolean-valued function, on the outer domain. To each unary relation we assign a unary relation, which you should think of as a function which takes a single argument, belonging to the domain, and gives you a Boolean value, i.e. true or false. To each binary predicate we assign a binary relation, i.e a Boolean function of two arguments. To each ternary predicate we assign a ternary relation, and so forth. An atomic expression is considered true if the function associated to its predicate, when evaluated at its arguments, has the value true. Then from the atomic expressions we can assign truth values to larger and larger expressions, much as we did in zeroeth order logic. We handle Boolean operator just as we did there. Equality is handled in the obvious way, with \({ ( A = B ) }\) evaluating to true if the two variables or parameters have been assigned the same value. The one complication is that when we assign truth values to a quantified expression we check only those values of the variable in the inner domain, and we substitute them only for free occurrences of the variable in the inner expression.
A statement in first order logic is said to be valid if it is true for every interpretation of the type described above. Valid statements play much the same role for first order logic as tautologies did for zeroeth order logic.
There is no analogue of truth tables for first order logic because we have no hope of listing the possible interpretations of a statement.
Some textbooks imply, or even directly state, that all interpretations are of the type described above. They are wrong, for reasons we will discuss in the set theory chapter.