Understanding First-order Logic 3.2.2 111 Interpretations of first-order languages An interpretation of a first-order language L is a matching first-order structure S, that is, a structure with a family of distinguished functions, predicates, and constants that correspond to (and match the respective numbers of arguments of) the non-logical symbols in L Some first-order languages, like all those that we have considered so far, are designed for specific structures which are their intended or standard interpretations Other first-order languages are designed for classes of structures For instance, the first-order language containing one binary relational symbol R (plus equality) can be regarded as the language of directed graphs, where the intended interpretation of R in any directed graph is the edge relation in that graph Likewise, the first-order language of (algebraic) groups contains the following non-logical symbols: one binary functional symbol ◦, with intended interpretation being the group operation; one unary functional symbol , with intended interpretation being the inverse operation; and one constant symbol e, with intended interpretation the identity element Note, however, that every first-order language may have many unintended interpretations For instance, the language for directed graphs can be interpreted in the domain of integers, with R interpreted as “divisible by”, or in the domain of humans, with R interpreted as “is a friend of.” Indeed, most of the unintended interpretations are practically meaningless For instance, the language LH can be interpreted in the domain of integers where, for example, the functional symbol m is interpreted as “m(n) = 2n”, f is interpreted as “f(n) = n5 − 1”, the unary predicate M is interpreted as “is prime”, and the unary predicate W is interpreted as “is greater than 2012”, the binary predicate symbols P, C, and L are interpreted respectively as “is greater than”, “is divisible by”, and “has the same remainder modulo 11”, and the constant symbols John, Mary, Adam, Eve are interpreted respectively as the numbers −17, 99, 0, and 10 Of course, such unintended interpretations are not interesting, but they must be taken in consideration when judging whether a given first-order formula is logically valid, that is, true in every possible interpretation That will be discussed later in Section 3.4 however, and we now come back to the meaning of first-order terms and formulae under a given interpretation Once a given first-order language L is interpreted, that is, a matching first-order structure S is fixed, the value in S of every term t from T M (L) can be “computed” as soon as all individual variables occurring in t are assigned values in S, that is, elements of S The meaning of every formula A in F OR(L) can also then be “computed”, just as for propositional logic, from the values of the terms and the interpretation of the predicate symbols occurring in A and the standard meaning of the logical connectives The rules for computing this meaning determine the semantics of first-order logic I will spell out these rules without going into more technical detail than is really necessary