Natural language statements have quantifiers such as, “most”, “many”, “few”, and “only”. How could ordinary first-order predicate logic with identity (hereafter, FOPL) treat statements containing these vague quantifiers? It seems that FOPL, with only the existential and universal quantifiers at its disposal, is insufficient. I read somewhere that ‘restricted quantification’ notation can ameliorate such problems. Is this true, or are there difficulties with the restricted quantification treatment of vague quantifiers? What are some of the inference rules for restricted quantification notation? For example, in FOPL you have the existential instantiation and universal instantiation inference rules. Are there analogue inference rules for the quantifiers, "many", “most” and “few”? Can you recommend any books or articles that outline, critique or defend restricted quantification? I also read that there are issues with FOPL regarding symbolizing adverbs and events from natural language. Is this true or just a superficial problem? Another complaint about FOPL, (especially Russell’s treatment of statements in the form of “The so and so...”), is that, often there are no obvious correspondences between the grammatical structure of the natural language and its logical notation counterpart. For example, in the English statement, “All men are mortal” to the logical notation, (x)(Mx->Rx), there seems to be no obvious correspondence to the connective ‘->’ from anything in its natural language grammatical structure. In other words, the logical notation seems too contrived. What is the common response to this complaint if any? These seem to be grave problems for the applicability and effectiveness of FOPL to natural language arguments. (I am not referring to the “limits” of FOPL where extensions such as modal, tense, or second-order logic might accommodate the richer parts of natural language, but rather to the apparent inability of any logic(s) dealing with these problems.) Note: Much of these concerns I have come from an article I read by Kent Bach in “A Companion to Philosophical Logic” by Blackwell Publishing. Thanks Kindly for your reply, J Jones

Comments

There are a lot of different questions here, and we need to disentangle some of them.

First, some of the questions you are raising about "most", "few", and the like have nothing to do with their vagueness. Consider, for example, a quantifier I'll write "(Most x)(Fx;Gx)". This is what is called a binary quantifier (similar to your "restricted" quantifiers): Unlike the usual way of representing "all" and "some", it forms a formula from two open sentences. Now, define the quantifier, semantically, so that "(Most x)(Fx; Gx)" is true if, and only if, there are more Fs that are G than there are Fs that are non-G. (More generally, we'd have to talk about satisfaction, but waive this complication.) It can be proven that this quantifier cannot be expressed by any formula of FOPL.

It can also be shown that there is no sound and complete axiomatization of the logic of this quantifier. That isn't to say you can't write down some sound rules. But you can't write down a complete set of rules: No matter how many rules you write down, there will always be valid inferences involving "Most" that are not captured by those rules. It's no surprise, then, that the study of such quantifiers tends to proceed more semantically than syntactically.

There are a couple quantifiers of this kind that have been particularly important in my own work. One could be written: (Eq x)(Fx; Gx), and is defined semantically as: There is a 1-1 correspondence between the Fs and the Gs. It too cannot be defined by any first-order formula and has no sound and complete proof-theory. One can, however, formulate some rules for it. A natural introduction rule, for example, would say that we can infer (Eq x)(Fx;Gx) from any formula that "says" that B(x,y) is a one-one correspondence between the Fs and the Gs. It is harder to formulate elimination rules. (Exercise: Show that Eq and Most are interdefinable.)

It is also fairly easy to see that there isn't going to be any at all natural way of representing "(Most x)(Fx; Gx)" using unary quantifiers. (Obviously, it doesn't mean: Most things are such that, if they are F, then they are G. That's true if most things aren't F. Nothing else works either.) Indeed, when one starts looking generally at natural language quantifiers, it quickly becomes apparent that this is the norm: Most natural language quantifiers are binary. And so it ends up seeming like a kind of happy accident that "all" and "some" can be treated, for some purposes, as unary quantifiers. Hence, most natural language semanticists, nowadays, treat "all" and "some" too as binary. (Actually, that's not quite right, but it's close enough for present purposes. The important point is that, however "all" and "some" get treated, syntactically, it's the same as how "most" and "few" do. There are options here.) It's an option, as you pretty much note, to include "the" in this group, but "the" is always problematic, so there isn't a lot of agreement here. If we do that, then we have a quantiifer: (The x)(Fx;Gx) which is defined semantically as: There is exactly one F, and it is G. In this case, as Russell showed, the quantifier can be defined in first-order terms. But, from the present point of view, that is a mere curiosity.

The three classic papers on these issues one usually sees cited are: Barwise and Cooper, "Generalized quantifiers and natural language", Linguistics and Philosophy (1981); Higginbotham and May, "Questions, quantifiers, and crossing", The Linguistic Review (1981); and Keenan and Stavi, "A semantic characterization of natural language determineers", Linguistics and Philosophy (1986). (This work grew out of earlier work in mathematical logic on so-called generalized quantifiers, begun by Mostowski in the 1950s and then picked up by Montague.) You'll find a good discussion of this issue, as well, in any decent text on semantics: Heim & Kratzer, Chierchia & McConnell-Ginet, and Larson & Segal would all cover it.

Now, as you noted, there are also some other questions about such quantifiers, which concern their vagueness. Vagueness is another very large issue. These quantifiers---certainly this is true of "many" and "few"---also appear to be context-sensitive: What proportion of the Fs need to be G for "Many Fs are G" to be true seems to vary with the occasion. But these are large issues in their own right that don't have much specific to do with quantifiers, so I'll not try to broach them here.

One further point. Toward the end, you write:

These seem to be grave problems for theapplicability and effectiveness of FOPL to natural language arguments.(I am not referring to the “limits” of FOPL where extensions such asmodal, tense, or second-order logic might accommodate the richer partsof natural language, but rather to the apparent inability of anylogic(s) dealing with these problems.)

Waiving the issue about vagueness, there isn't any problem dealing with such quantifiers in a second-order context. Both of the quantifiers I mentioned, "Most" and "Eq", can be defined in second-order logic, so the caveat at the end kind of gives the game away. That said, what perhaps is puzzling about these quantifiers is that, as is the case with second-order quantifiers, there is, as I said, no sound and complete set of rules for them, with respect to the intended semantics. In that sense, there is no "formal" logic for these quantifiers. But, again, that is not to say that one cannot write down some rules for them, and these rules may even be adequate for most or even all of the intuitively valid arguments one might care to give. The really interesting question, to my mind, would be what kinds of resources one actually needs to be able to do that. And that is why it is interesting, to me, how hard it is to write down sensible "elimination" rules for "Eq". For more on that, see my paper "The Logic of Frege's Theorem", which is on my web site. (The example discussed there is more the ancestral, but similar points apply to "Eq".)

Of course, if we don't waive the issue about vagueness, then there are, as I also said, lots of other problems around. But these problems are not special to "few", "many", and the like.

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