1.1.8 Free Variables versus Bound Variables

Discussion of the concept of a bound variable.

Although they are both called variables, free and bound variables are in reality two very different things. (In fact, some formulations of first-order logic even use two distinct kinds of symbol for what we have lumped together under the heading `variable'.) As an analogy, try thinking of a free variable as something like the pronoun ``she'' in the sentence ``She even has a stud in her tongue''. Uttered in isolation, this would be somewhat puzzling, as we don't know whom ``she'' refers to. Normally though, such an utterance would be made in an appropriate context, this context being either a non-linguistic one (imagine, for example, the speaker pointing towards a heavily tattooed biker, in which case we would say that ``she'' was being used deictically or demonstratively) or a linguistic one (for example with the preceding sentence being ``Anna is heavily into body piercing'', in which case the name ``Anna'' would supply a suitable anchor for an anaphoric interpretation of ``she'').

The point of this analogy is: Just as the pronoun ``she'' requires something else as a complement (namely, contextual information) in order to supply a suitable referent, formulae containing free variables will require additional information on how to link the free variables to the entities in the model. So just supplying a model isn't sufficient.

Sentences, on the other hand, are relatively self-contained. For example, consider the sentence . This sentence claims that every individual is a moron. Roughly speaking, the bound variable x in acts as a kind of placeholder. In fact, the use of is completely arbitrary; the sentence would mean exactly the same thing. Both sentences are simply a way of stating that no matter what we take the second occurrence of (or ) as standing for, that entity will invariably be a moron. In any model of appropriate vocabulary this sentence (and in fact any sentence over that vocabulary) will either be true or false.

Our discussion of the interpretation of first-order languages in first-order models will further clarify these distinctions (indeed, most of the actual work involved in interpreting first-order logic focuses on the correct handling of free and bound variables).