Quantifiers Deconstructed

Quantifiers Deconstructed

How should we interpret the quantifiers of the predicate calculus? Here is one suggestion: “Ex(Fx)” should be read “There exists an individual, call it x, such that Fx”.[1] There is an obvious problem with this: it commits a use-mention fallacy. The first occurrence of “x” should be in quotation marks so that the whole reads “There exists an individual, call it “x” such that Fx”. Then the first “x” is mentioned and the second used: the reference of the first “x” is the letter “x”, but the second refers to an object x. This is like saying “There exists an individual, call it “Herbert”, such that Herbert is F”. This is not what the original formula attempts to say, since it uses “x” throughout and does not mention it, thus securing co-reference. Further, who is calling this existing object “x”? It is likely not itself already called “x” by anyone, so we are being invited so to christen it; it is nowcalled “x”. That is its name. But the original formula says nothing about naming an object “x” thereby creating a new name. What we have here is an unholy mishmash of use and mention not a case of anaphoric co-reference. And what about the universal quantifier—does it say “For all individuals, call them “x”, x is F”? Why call them all “x”—what purpose does this serve? And how can the first “x” co-refer with the second? This is clearly a hopeless way to gloss the original formula.

            But we might take a hint from this failure and go metalinguistic throughout. We might paraphrase the original formula as follows: “There exists a term such that substituting this term into the open formula “F” gives a truth”.[2] Here we don’t incoherently combine a mentioned expression with a used expression: we speak ofexpressions throughout, never of objects. We quantify over expressions, affirming the existence of at least one that produces truth when joined with “F”. Thus the “x’s” of predicate calculus never actually range over objects; the only reference that is going on is to symbols. Is this the correct way to interpret the usual formulas? There is the problem that not all the relevant objects might have terms denoting them: not every object has a name. We might get over this problem by exploiting the descriptive and demonstrative resources of language, but a more fundamental problem remains, namely that the formulas we are aiming to gloss are plainly not intended as metalinguistic statements. They say nothing about language, terms, substitution, etc. They purport to speak only of objects in the extralinguistic world. We don’t want the formulas themselves to commit us to an act of semantic ascent, i.e., reinterpreting them as really about language. That is not what the inventors of the standard notation intended to convey. So this way of trying to make sense of “Ex(Fx)” is not going to work. We are left with no satisfactory way of reading the formulas of the predicate calculus. The only reason students manage to read meaning into them is by tacitly appealing to the underlying proposition, whose form they do not reveal. This is a highly unsatisfactory state of affairs. We really have no logic of “all” and “some”.

[1] I came across this formulation somewhere on the Internet but can no longer trace where. It did occur in an otherwise expert piece of writing. At least the author realized that he or she had to say something to explain what the standard formulas mean.

[2] This is the way Russell tended to think about quantification: statements of existence were supposed to be about “propositional functions” and to involve inserting terms into their argument places. He was never very careful about use and mention. The notation we now have reflects this sloppiness.