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.

Is Logic Gibberish?

We are familiar with the standard notation of predicate logic in which we have what is called variable binding. Thus we have a symbol for (say) existence followed by an “x” and then a formula in which a predicate and bound variable occur (“Ex(Fx)”). How should we read this? It is common to hear it read as “There exists an x such that Fx”. What does this mean? The letter “x” is supposed to be replaceable by a singular term such as a proper name, a pronoun, or a demonstrative. So, a substitution instance might be “There exists a London such that London is crowded” or “There exists a him such that he is tall”. But these sentences are nonsense; it is not grammatical to place a singular noun in this position. What is wanted is clearly a predicate like “city” or “man”: then we can say “There exists a city such that that city is crowded” or “There exists a man such that that man is tall”. The correct logical form would then be something like “EF(that F is G)”, read as “There exists an F such that that F is G”, where “F” and “G” are predicate expressions. But this is not what logical notation says under the usual reading: this makes sense but that does not. Alternatively, we might try saying that the usual formula is to be read “There exists an object x such that x is F”, so that we have “There exists an object x such that x is a city and x is crowded”. That looks closer to the usual notation, but it contains a funny construction, viz. “an object x”. Put aside whether “object” is an adequate count noun and ask yourself what could be meant by following this word with an “x”. Presumably a substitution instance would be “an object London” or “an object it”: but these are also nonsense. That occurrence of “x” after “E” cannot be replaced by such an expression on pain of producing gibberish. It really doesn’t belong there at all; what belongs there is a count noun or predicate variable replaceable by a count noun, as in “There exists a city such that”. So, predicate logic should not be written in the standard way but rather along the lines of “EF(that F is G)”; otherwise the notation is gibberish. Even that formula has its drawbacks in the shape of the expression “that F”, since it is obviously not deictic and it has no anaphoric antecedent. One wants to say instead “There exists an F, x, such that Gx”, as in “There exists a city, x, such that x is crowded”, so as to have an antecedent to work with. But this also is gibberish, as can be seen by substituting a proper name or pronoun for “x” throughout. There is nothing wrong with saying “London is crowded”, but what is meant by “There exists a city, London”? Is this just a funny way of saying “London exists”? In fact, no one ever translates the standard formula this way, instead saying simply “There exists an x such that x is a city and x is crowded”—which takes us back to our first point. There is simply no intelligible way to interpret this initial occurrence of “x”. Hence predicate logic is gibberish and needs to be overhauled into something that actually makes sense. The whole quantifier-plus-variable symbolism is a mistake.[1]

[1] An Aristotelean might pounce and suggest that Aristotle’s subject-predicate logic is the way to go. Thus, we will have “Some men are philosophers” and “All men are mortal” where the first two words in these sentences are construed as logical subjects. Or else we will need to devise a whole new grammar and notation for expressing logical relations involving “some” and “all”.

I’m reading Goethe’s “Italian Journey”. He remarks during his visit to Rome: “The past year has been the most important one in my life; it does not matter whether I die now or last a while longer, in either case I am content.” People used to ask me whether there was anything about England I missed when I moved to the United States; I would reply, “Yes, Italy”.

Why Did You Die?

 

You were my dear old friend

That word is too short for you

Now you’ve gone and left me

And I don’t know what to do

 

I’d walk with you by my side

On summer days and winter nights

I thought you’d always be around

Like the clear blue sky

 

So why did you have to die?

And leave me here to cry

What harm did I do to you?

That you could ever justify

 

I asked what’s on your mind

I hung on every word

And you hung on every word of mine

We never went unheard

 

There was no doubt in our bond

I could see it in your eyes

But now it’s over and beyond

I’m left with only memories

 

So why did you have to die?

And leave me here to cry?

What harm did I do to you?

That you could ever justify

 

Why did you die?

Oh, why did you die?