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One of the fundamental units in logical analysis is that function/operatori lovingly known as “the quantifier.” Most logic texts content themselves with just two: “all” and “some,” formally symbolized as “” and “” respectively. Thus, to say that, “All X is p,” one is asserting that every (or any) instance of X is also an instance of p, or is characterized by p, etc. Similarly, when someone says only that, “Some X is p,” the claim is made that, if one looks hard enough, one will find at least one instance where X is p. There are ways of precisifying (one of those \$5.00 words philosophers love to use) the above statements, but there is hardly any need to do so here. It suffices to have a general idea. Two points I’ll mention in passing. First, in most formal contexts (substructural logics are an example of an exception), “all” and “some” are defined as being interchangeable using “not”: thus, “not-All X is not-p” is taken to mean “Some X is p,” and conversely. Secondly, these are not the only quantifiers possible: “many” and “most” are also examples. But these last two are difficult to formalize (to say the least) and by a polite convention among logicians they are generally ignored wholesale.

As the title of this post states, I wish to talk about what I am calling the implicit “all”; uses of the “all” quantifier in which that quantifier is functioning but not explicitly stated. This happens quite often, in point of fact, and is not problematical in itself. Where problems do arise is when that usage is not merely implicit, but actively denied as a means of evading the consequences of what someone has actively stated or written. When this happens, we are faced not merely with a logical error, but an overt act of dishonesty. The dishonesty becomes not merely overt but blatant when, even after the implicit “all” is pointed out, the individual continues to deny it.

Let us begin with a simple example, to get some sense of where we’re headed with this. Consider the statement, “Dogs have four legs.” An important clue to the implicit “all” is when the subject is a group, class, or collection of some type. For example, “dogs” in the preceding does not refer to this or that dog, nor to this or that individual’s small or large pack of friendly canines. Rather, it refers to ALL dogs, to the class or collection of dogs.ii The emphasis on “all” in the preceding is neither accidental nor casual; it is, rather, the point to be made.

Now, the real world is quite a bit sloppier than the far country of mathematical idealization and pure logical formalities. So of course there are such things as three legged dogs, even a few with only two, and many of us have seen the charming videos of them variously scooting about, sometimes with the assistance of a wheeled mechanism to hold up their front or rear quarters. Such supposed counter-examples to the implicit “all” are, of course, nothing of the kind. The ancient phrase, “the exception that proves the rule,” is hardly some arcane formula unknown to the common mind. And invoking recondite formal absolutes in the name of evading the reality of what was claimed scarcely raises our intellectual credibility, nor does it erase our moral responsibility. When we speak of “dogs” without qualification, we mean all dogs, even as (realistically) a few exceptions fall through the cracks. Because we still mean the three-legged dog would or should have four legs regardless of those twists of fate that deprived the creature of its natural quadrupedal status. Exceptional examples do not invalidate the rule; they illustrate it.

Let me present a few more examples. Sentences 1 – 3 below are chosen deliberately because they are outrageous, even though there will be some persons who would actually accept &/or assert them as factually correct. Let me be absolutely clear up front that I reject every one of these statements without qualification, and I reject them precisely because they involve an implicit “all” (even though each of them also has a exceptional cases for which the relevant statement – with sufficient sophistry and misdirection – could pretend to make it true.) The statements are:

1. Black men don’t take responsibility for their own children.
2. Trans-sexual persons are confused and dishonest about their own sexuality.
3. Undocumented immigrants are a danger to our society.

In one form or another, all three of these claims have been asserted in recent years, as anyone who pays attention to the news and society can readily attest. So I am not appealing to recondite abstractions here.

Now, any persons not actively outraged by these claims stand convicted of being themselves actively outrageous. None of 1 – 3 is even remotely defensible by any rational standard. (Clearly, though, many hypocrites and ideologues will still attempt to defend them.) And who among us could possibly be so willfully obtuse as to fail to recognize that 1 – 3 are all making their childish assertions about an entire class of people? About, that is, ALL of them? And how would we respond to any such claimant who went on to declare, “But I didn’t actually say all! No, no! I clearly only meant some of them!” Who in the world could possibly be so galactically credulous as to be even momentarily gulled by such a fatuous, monstrous, and absolutely bald-faced lie? Persons asserting any or all of 1 – 3 clearly mean to denounce an entire class of persons with their claims; they mean to denounce all of them.

And yet, it must certainly be true that, for each of 1 – 3, someone or several such ones can be found for whom the appropriate statement is nominally true. There are also three-legged dogs. Again, this changes nothing. The implicit “all” is there, and it is the thing that is being asserted. Exceptions that prove the rule only prove that the rule was asserted as a rule, as an all.

But here we encounter a significant difference between 1 – 3 and “dogs have four legs”: statements 1 – 3 are false. How might this alter the problem?

It seems a fair point. Suppose one wishes to make a true statement about some collection or other that is highly negative, and the exceptional cases (that prove the rule) are the ones that don’t evince that strongly negative quality? Is that sufficient reason to abandon qualification and go ahead with the promiscuously sloppy “all”? You can probably guess what my answer will be from the immediately preceding rhetoric.

To put it simply, it is not an attack on a three-legged dog to day that “dogs have four legs,” even with the implicit “all.” For one thing, the dog doesn’t care what words we say (unless one of those words is either “walk” or “dinner”), as long as we say it with a pleasant &/or cheerful tone of voice. If, on the other hand, one were to say something like “Republicans have no respect for the Constitution,” one has crossed a line. Perhaps it is true that most Republicans are Trump supporters (and it is certainly true that Trump supporters have no respect for the Constitution), yet there are many Republicans who are adamantly opposed to Trump and his viciously neo-fascist disregard for the rule of law. Other examples can be found without much difficulty. (Consider THIS discussion from a while back.)

It is generally pretty lazy to lean on the implicit “all.” How much harder would it be to say, “So many Republicans today have no respect for the Constitution,” as opposed to the example given above? But it is frankly indefensible to deny using that implicit “all” when one gets called out on the matter.

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i For purposes of rhetorical balance and readability I’ll use the terms interchangeably, though it should be understood that my doing so is, formally, quite sloppy of me.

ii For reasons I’ll not trouble anyone with here, I’m suspicious of “set theoretic” talk of sets, classes, and elements. This is the reason I keep hedging my language in the above and the following.