An oddity about philosophers, and especially logicians, is that when they talk about “quantity” they are not talking about numbers, or numerical counts. Rather, they are talking about the ways things can be gathered together (or singled out) using words like “all” or “some.” These ideas are called “quantifiers.” I want to do three things (briefly, as always) here: say a little about the “basic” quantifiers (“all” and “some”), say a little about how they get dropped from common discourse and argument – whether from laziness or deliberate obfuscation – leading to much gratuitous confusion. Finally, I want to say something about quantifiers that typically do not make it onto philosophers’ or logicians’ lists, yet are at least as common in ordinary discourse and argument as the “principal” two are. My purpose here (as always) is not to lead you onto the path of righteous proof making, but simply alert the reader to the importance of these operators so that they might not slip by quite so stealthily in the future.
The second greatest sin in logic is to allow things to pass implicitly; the greatest sin is to block the road of inquiry, which is one of the things that happens when concepts are allowed to pass implicitly. Allowing things to remain implicit means that vague statements are permitted, by innuendo, to become concrete, thus leading us astray (blocking inquiry) from the directly stated vagueness. Sometimes things really are ambiguous, and they must be allowed to stay that way until real data, rather than jumping at conclusions, enables us to clear up the ambiguity. That, or recognize that the ambiguity is not – or, at least, not yet – cleared.
The first thing we need to understand when we start tossing about quantifiers is that these functions can only ever have any meaning within the scope of a “universe of discourse,” the broadly characterized subject matter about which we are speaking. It is often fun to flatter ourselves into imagining that our universe of discourse is, in fact, THE UNIVERSE. But such flattery is a remarkably childish piece of willful delusionalism. Our universe of discourse can never be larger than that around which we can effectively wrap our cognitive machinery, and THE UNIVERSE is always much, much bigger than that. Even those scientists who work on gravitational cosmology, with their grand designs on the whole of observable physical reality, are not even remotely close to dealing with THE UNIVERSE. After all, no amount of their clever mathematics will ever provide even the slightest insight into, say, 16th C. English literature (which is also part of THE UNIVERSE). Admittedly, this latter does include Shakespeare, but if you’re going to presume to talk about THE UNIVERSE, then you also have to be prepared to talk about sweet Will.
So we begin with a universe of discourse – call it “UoD” – some reasonably determinate subject matter of inquiry and discussion. That subject matter need not be entirely definite. But it must be adequately constrained so that finite minds can effectively engage it. (The italicized words all represent technical terms that I’ll not go into here. Just be aware that real differences are being marked.)
So, the first piece of the squirrelly mass to deal with is “all”. When we say that “ALL x is P”, it seems as though we’ve made a sweeping declaration about our entire UoD. And in a sense, we have … but maybe not quite the sweeping declaration we thought. You see, the above does not actually assert that there really ARE any x’s such that they are P’s, only that, if there are such x’s, THEN they are all P’s. The fancy(er) way of saying this is that that “all” has no existential “bite.” It makes no claim about the UoD in as an actual fact, only about what it “must be” should other conditions hold. Conditions like, there is anything that is P, therefore all such things must be P. For example, asserting that “ALL lunar colonists are bald” is vacuous, since there are no lunar colonists who are bald or otherwise.
That little bastard “some,” however, is a whole other can of worms. To say that “Some x is P” means that there IS some “x” out there (whether it is P or not.) In other words, that “some” quantifier carries existential “bite,” it declares that you have definite and determinable contact with ontological reality. To say that “some Martians are green,” is to assert the existence of Martians, regardless of whether any of them are green. Many purely mathematical contexts disregard such issues; often times even in philosophical discourse where formal logic is being used in a naïve, if not reckless fashion. In other philosophical situations people will employ what is known as “free logics”, where any existential content to the “some” operator is explicitly denied. (Free logics seem more frequently studied to see what, if any, are their interesting characteristics, rather than actually applied to other problems.)
A few examples now will help us understand the need for being explicit. Consider the sentence, “Reporters are lazy.” There are many reasons why persons might say this: they don’t think reporters adequately research their topics, or they too often give a free-pass to politicians rather than asking the tough questions. But what is the topic under discussion here? Is it some reporters, or all reporters? By the bye, when I suggested above that “they too often give a free-pass to politicians,” was that some politicians or all politicians? Did you notice the ambiguity the first time you read it? Just the littlest bit of consideration will make it evident that how one chooses to quantify reporters (and politicians), one will end up saying very different things. The UoD also comes into play here. For example, does that universe of discourse include all reporters, everywhere, or only national reporters in mainstream media?
While simple, the above examples will hopefully suffice to establish the need for some care in their handling. This opens our segue to the third part, the “other” quantifiers, the one’s many philosophers do not trouble to talk about, yet which people use in conversation all the time. It is not that “all” and “some” are the most interesting of such operators. Rather, they are the easiest to manage on a formal basis. Other quantifiers include such operators as “few,” “many,” “most,” “almost all,” “almost none,” and some of a peculiarly mathematical flavor – “uncountably many,” for example. “Few,” “many,” and “most” seem to share with “some” that “existential bite,” in that they assert the existence of what they are quantifying. Yet “few” and “most” (at least) seem to rotate around “all” as contraries. Thus “few” can be interpreted as “’all’ minus ‘most’”; similarly, “most” becomes “’all’ minus ‘few’.” “Many” is especially hard to characterize, because how many is “many.” This seems to depend on context. If a general suffers 5% casualties in an action, that is a rather horrifying number – that is not just many, it is too many. On the other hand, in other contexts (other UoD’s), 5% loss (say, in the degradation of a signal from a deep space probe) is so trivial that not only is it not many, it is almost none.
A recent personal experience might help illustrate why such distinctions are important. (As always, personal anecdotes need to be taken with a grain of salt.) All of the following is a deliberate paraphrase to minimize any chance of backtracking the interaction. Anyway, in response to the recent Melania Trump plagiarism kerfluffle at the Republican National Convention (evidently basic copy editing is one of them hard word things that the right wing can’t be troubled with), I mentioned in a public thread that there are not MANY sins that I view with as much disdain as plagiarism. A respondent suggested murder as a possibly more serious sin. Now, while I confess that I did not provide the foregoing emphasis on “many” that I did in the previous sentence (limitations of purely ASCII text), I am still mystified at how a person of even the most moderate level of intelligence or integrity could confuse “many” with “any.” Yet that is what my interlocutor did.
Intelligent conversation depends upon saying all of what we mean, and exactly what we mean. This will not, of course, prevent willfully stupid persons from fabricating indefensible interpretations of what we’ve said. But it will provide us with the basis for responding to such twaddle.