Understanding First-order Logic 101 If t1 , , tn are terms in L and f is an n-ary functional symbol in L, then f (t1 , , tn ) is a term in L We denote the set of terms of L by TM(L) The set of variables occurring in a term t is denoted by VAR(t) Terms that not contain variables are called constant terms or ground terms Example 80 Some examples of terms in the first-order languages for some of the structures we have seen include the following In the language LN : First, 0, s(0), s(s(0)), etc are constant terms The term s( s(0) ), where s occurs n times, is denoted n That term is called the numeral for n6 We are less formal from this point onward, and allow ourselves to use the common, infix notation, as well as omitting outermost parentheses in terms whenever that does not affect the correct reading With that in mind, other examples of terms include: • +(2, 2), which in a more familiar notation is written as + 2; ã ì(3, y ), written in the usual notation as × y ; ã (x2 + x) ì 5, where x2 is an abbreviation of x ì x; ã x1 + s((y2 + 3)×s(z )), etc In the ‘human’ language LH : • Mary; • x; • m(John) (meant to denote “the mother of John”); • f(m(y )) (meant to denote “the father of the mother of y ”) The inductive definition of terms generates the respective principle of induction following the general scheme presented in Section 1.4 Proposition 81 (Induction on terms) Let L be any first-order language and P a property of terms in L, such that: every constant symbol in L has the property P; every individual variable in L has the property P; if t1 , , tn are terms in L that have the property P and f is an n-ary functional symbol in L, then the term f (t1 , , tn ) has the property P Then every term in L has the property P Let us now illustrate definitions by recursion (recall Section 1.4) on the inductive definition of terms by formally defining for every term t the set of variables VAR(t) occurring in that term The term n, which is a syntactic object, that is, just a string of symbols, must be distinguished from the number n, which is a mathematical entity