# Names and quantifiers

‘Socrates is a philosopher’ gets rendered into an appropriate formal language of predicate logic by the likes of $$\mathsf{Fn}$$; ‘Someone is a philosopher’ gets rendered by $$\mathsf{\exists xFx}$$. The syntactic difference between a formal name and a quantifier-tied-to-a-variable vividly marks a semantic difference between the rules for interpreting the two resulting formal wffs. And there isn’t the same sort of immediately striking syntactic difference between the vernacular name ‘Socrates’ and the quantifier ‘someone’.

That’s agreed on all sides, I guess. But many have wanted to say more — namely that, in English, there is no syntactic difference between the name and the quantifier. Thus,
Quine writes that ‘one of the misleading things about ordinary language is that the word “something” masquerades as a proper name’. Well, presumably there is no masquerade if you can easily tell them apart by their surface look. So, more carefully, Quine’s idea is presumably that the English quantifier is just like a proper name as far as surface sentence structure is concerned. Or as Michael Dummett explicitly puts it: ‘As far as the sentence-structure of natural language is concerned, signs of generality such as “someone” and “anyone” behave exactly like proper names — they occupy the same positions in sentences and are governed by the same grammatical rules.’

But this is just not true (as Alex Oliver, for one, has had fun pointing out). Thus, contrast ‘Something wicked this way comes’ with the ungrammatical ‘Jack wicked this way comes’, or ‘Someone brave rescued the dog’ with ‘Jill brave rescued the dog’. Or contrast ‘Foolish Donald tweeted’ with the ungrammatical ‘Foolish someone tweeted’. Compare too ‘Senator, you’re no Jack Kennedy’ with ‘Senator, you’re no someone’. Or what about ‘Hey, Siri!’ compared with ‘Hey, someone!’?

Other quantifiers too aren’t interchangeable with names. Consider ‘Nobody’. We get similar failures of substitution: we can’t replace ‘Nobody wise …’ with ‘Jill wise …’, or replace ‘Foolish Donald …’ with ‘Foolish nobody …’, and preserve grammaticality, etc. And ‘Nobody ever finishes War and Peace’ constrasts with the ungrammatical ‘Jack ever finishes War and Peace’, while ‘Jill never finished War and Peace’ contrasts with ungrammatical ‘Nobody never finished War and Peace’.

And so it goes. Quine’s and Dummett’s claims simply overshoot. English grammar doesn’t treat names and quantifiers exactly on a par. But even if it did, ordinary language would only be “misleading” (in Quine’s word) if there was some tendency for us ordinary speakers to get misled. Now, Mark Sainsbury indeed talks of ‘our tendency to regard quantifiers … as names’. But what is the evidence is that we have such a tendency? This is shown, says Sainsbury, ‘by the fact that Lewis Carroll’s jokes are funny’. But that really is hopeless! After all, Carroll’s wordplay (you know the kind of thing: ‘I see nobody on the road,’ said Alice. ‘I only wish I had such eyes … To be able to see Nobody!’ etc. etc.), apart from being wearyingly unfunny, has nothing specifically to do with confusing quantifiers with names (as Alex Oliver notes, it’s pretty much on a par with the likes of ‘What’s your name?’ ‘Watt.’ ‘I said, what’s your name?’ ‘Watt’s my name’ …)

OK: Quine, Dummett are wrong that English names and quantifiers “are governed by the same grammatical rules”, and even if they weren’t, that would give us no reason to suppose that we are misled (Quine, Sainsbury) by the grammatical similarity or that we tend to regard quantifiers as names.

And yet, and yet … Even if the claim that “signs of generality behave exactly like proper names” is false au pied de la lettre, we are left with the feeling that the sort of exceptions we’ve noted are somehow quirks of idiom rather than deeply significant. We are left with the sense that there is something important and true which ought to be  rescuable from the very familiar kind of remarks from Quine Dummett and Sainsbury. But what is it? We say arm-waving things in our intro logic lectures — but what will we happy to put in black and white?

It is tempting to say this:  the syntax of our formal first order language more perspicuously tracks the semantics of the formal language than the syntax of English names vs quantifiers tracks their semantics. But in a way, this is just too easy to say. Of course  the syntax tracks the semantics in our formal language in a way that is  perspicuous even to beginners — we purpose-designed the language to be exactly that way! And who knows how things are in English when it comes to syntactic or semantic theory — of course that’s not perspicuous at all, as half a century of modern linguistic theory has shown?!

Now, Quine, Dummett and Sainsbury aren’t  aiming to contrast the known with the unknown: they make a positive (even if false) claims about English, after all. But then, to repeat the question, just what that is instructive and importantly true can we extract from those incorrect claims about quantifiers “behaving exactly like” (?!) names in English but not in first order languages? I have (or rather had) my lecture patter. But, as I think about the relevant bit of my revised IFL book, I’d be very interested to know what others say to their students!

### 6 thoughts on “Names and quantifiers”

1. These do not look like genuine grammatical variants in English, but I think different idiomatic expressions. ‘Someone brave rescued the dog’ is just ‘Someone who is brave rescued the dog’ and that does not contrast much with ‘Jill brave rescued the dog’, which is just, ‘Jill, who is brave, rescued the dog’.

1. I’m not sure what work you want the notion of “idiom” to do here. But surely, ‘Someone brave rescued the dog’ does contrast with ‘Jill brave rescued the dog’ — the first is grammatical (acceptable, idiomatic, non-deviant), the second isn’t. And that’s a flat counterexample to Dummett’s “As far as the sentence-structure of natural language is concerned, signs of generality such as ‘someone’ and ‘anyone’ behave exactly like proper names — they occupy the same positions in sentences and are governed by the same grammatical rules”. Unless you assign to Dummett a more technical notion of grammatical rule than I recall him using.

1. Dummett should have said “almost exactly,” then;
or as David Makinson says below, “up to a point.”
Counter-examples just don’t work, in philosophy,
like they do in mathematics (is what it looks like).

2. One remark that one could make to one’s students on this matter is that in ordinary English, terms like “something” and “everything” look like they are simple devices, but from the point of view of modern logic they are complex, with two hidden ingredients: a quantifier and a variable that it binds. In English neither component is made explicit, but they correspond roughly to the word “some” attached to “thing”, likewise the word “every” attached to “thing”. In modern logic these two ingredients are clearly separated.

In logic, the variable has a syntax that is in many respects like that of a proper name — they can both be subject to predication and enter into statements of relationship; names can be substituted for (free occurrences of) variables to obtain meaningful expressions. This is echoed by the fact that the English grammar of “something” and “everything” is, up to a point, like that of proper names.

On the other hand, in logic the bare quantifier has a quite different logical syntax, more like that of connectives such as “and” and “or” when taken in isolation from the statements that they connect. This is echoed by the fact that the English grammar of “some” and “every”, as words without “one” or “thing” attached, is quite different from that of proper names.

One of the great advances of modern logic over traditional ways of analysing logical structure as reflected in, say, Aristotelian logic, was to make the composite nature of “something” and “everything” explicit, and to decompose them into their parts. And when we are faced with a negative word like “nothing”, it decomposes into three parts: the variable, the quantifier, and the negation.

1. What Makinson said. At least conceptually.
But pedagogically I took it slow. First I would introduce predicate/name syntax as a plausible notation for capturing, for instance, the commonalities (logically relevant commonalities) among:
Edna slept, Maynard slept, George slept
and
Edna wept, Edna slept, Edna laughed.
not to mention
Edna kicked George, George kicked Edna, George kicked himself.

And, of course, they had prop. calc inferences to hand.
Then I ask what follows from Edna laughed.
Lots of things, but they converge on some variant of
Someone laughed.
Le, symbolizes Edna laughed.
How to symbolize Someone laughed.
Ls? Well, no because that would make it seem that
Edna laughed hence Sadie laughed was valid.
So a special symbol for someone. Nothing roman, ‘cause those are all taken.
So, let’s try L∃.
Discussion with students about what that means as distinct from designators like e and s. Well, they’re for who and what, whereas ∃ is for how many (one or more, in this case).
Then depending on the feel of the room, I either guide them to moving it to the front (∃L ), because it feels like a remark about the predicate rather than the predicate remarking…
Or, I move to looking at the notation’s deficiencies wrt what follows from:
Edna laughed and fell.
Which leads inexorably(?) to the variable as pronoun.

3. I think that if it is necessary to say something, then David Makinson’s suggestion is a good one. But is it necessary, or even desirable? I’m not “left with the sense that there is something important and true which ought to be rescuable from the very familiar kind of remarks from Quine, Dummett and Sainsbury”; and I’m not sure there is anything “instructive and importantly true can we extract from those incorrect claims about quantifiers”. Perhaps if the remarks weren’t familiar, and hadn’t been made by philosophers as eminent as Quine and Dummett, nobody would think it something worth raising with students, except perhaps as an amusing aside with reference to Carroll.

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