Topics Logic

Question What is a variable and what function does it play in such quantified propositions as "There is at least a thing, x, such that x is F", or "Every x is such that it is F"? Does the variable refer to something in the world? Or does it refer only to things assigned to constants? In other words: does the variable stand for things or words? And if it stands for things, does it stand for named things or even for unnamed things?

Accepted:
March 23, 2011

Comments

We get confused when we assimilate variables to ordinary referring expressions like "Obama". Because, as you realize, there's no good answer to the question "What does 'x' refer to in 'Every x is such that x is F'?", or - to put the question in colloquial English - "What does 'it' refer to in 'Everything is such that it has mass'?" The variable 'x', or the pronoun 'it' as used above, does not stand for anything. It is used in conjunction with the quantificational expression "everything is such that" to make a claim of generality, to say that all things have a certain property. You could view it as shorthand for the claim: o1 is F and o2 is F and ..., where the oi's are all the objects in the universe. Notice that in this infinite expansion of the claim "Everything is such that it is F", the "it" has disappeared. This makes it clear that "it" was never really being used to stand for something in particular. It was used, together with a quantifier (like "everything is such that" or "at least one thing is such that") to make a quantificational claim.

All this is tricky because "it" has another use according to which it does stand for something specific, as when I say "The wine you wanted was hard to find, but you can find it at Trader Joe's." The same is true for other pronominal devices in English. Consider, for instance, the difference between "John loves his mother" and "Every boy loves his mother". In the first, there's a natural reading according to which "his" refers to John. But in the second, "his" does not refer to any particular entity. Rather, it combines with the quantificational noun phrase "every boy" to allow us to state a claim about all boys.