Quantifier Concepts

Would it be quixotic to suppose that quantifiers hold the secret to human success?[1] Could the student of quantification theory be studying the ultimate differentia that separates humans from the rest of nature? That would be a delightful result for the logically minded; and I think there is actually a good deal to be said for it. For it is plausible to suppose that what other animals lack, cognitively speaking, and we splendidly possess, is the ability to engage in quantifier-driven reasoning. We grasp what Quine called the “apparatus of quantification” but they do not—though they no doubt grasp much else. That apparatus, to put it briefly, involves the existential and universal quantifiers, variable binding, embedding, scope, domain, and a distinctive syntactic form—not to mention the non-standard quantifiers “most”, “a few”, “many”, and others. Suppose all this represented in the human language of thought.  Then we can surmise that other animals, though cognitively gifted in many ways, lack an internalization of the apparatus of quantification—though they may well entertain singular and general concepts, truth functions, psychological concepts, etc. At any rate, there are possible beings that have mastery of a conceptual apparatus just like ours except that quantification is not included. The question is what capacities they would thereby lack that we possess, and which confer signal advantages on us. What do quantifiers do for you? What mental achievements do they make possible?

Quantifiers are obviously deeply embedded in our thinking, so it is not easy to tease out their contribution, but certain areas of human thought clearly depend on them.  First, science: where would science be without the universal quantifier? A law is precisely a generalization about all things of a certain kind (we can include ceteris paribus laws). If you don’t grasp the concept all, you don’t know what a law is. Similarly, you have to grasp that if some things lack a certain property then it is not a law that all things have that property (I omit some obvious qualifications). Quantificational reasoning is essential to scientific thought (animals don’t seem strong with science): science consists of universally quantified propositions. Second, mathematics: this too is shot through with quantificational structure (it was mathematics that caused Frege to invent modern quantification theory). The most basic axiom of arithmetic is universally quantified: for every number, there exists a successor number. Peano’s axioms are quantificational in form. The embedding of quantifiers is rife in mathematics. Geometry is much the same: we have theorems about all triangles, circles, etc. Moreover, according to some views, arithmetic reduces to quantification theory (plus set theory, which is itself formulated by means of quantifiers). Standard first-order predicate logic is clearly quantificational, but so is second-order logic (which greatly increases expressive power). Propositional logic discerns no quantifiers in its formulas, but it is tacitly quantificational itself, since sentence letters are interpreted generally: for any p and q… We understand it to express universal propositions. That is what logical necessity consists in. Modal logic involves quantification over possible worlds (necessity and universality are close cousins). Inductive logic involves moving from singular premises to general conclusions and would be impossible in the absence of the concept everything. Falsification depends on there being some counter-instance to a generalization. All this would be impossible for beings without a mental representation of the apparatus of quantification. When we reason we move from the particular to the general and the general to the particular, and this requires grasping how all and some work; not to grasp these principles would be a severe cognitive deficit (“quantifier derangement syndrome”). If Russell is right, definite descriptions are not possible without quantification (do animals grasp definite descriptions?); they are built from the quantifiers “all” and “some”. Many pronouns function as bound variables. Lastly, cosmology requires the use of the ultimate universal quantifier: for it concerns the nature of everything (ditto metaphysics). Here we ascend from specific domains to the entire domain of the universe. It is remarkable that we have such an all-encompassing concept—we can think about everything there is. Can animals ever think about the whole enchilada? I doubt it: they think specific and particular, local and limited. Maybe their thought is largely demonstrative, or maybe it employs a medium alien to human thought. In any case, our cognitive resources include the extensive and intricate apparatus of quantification, which greatly expands our powers of mental representation and hence our understanding of the world.[2] In turn, this enhanced understanding feeds into our actions and mode of life. We are quantifying creatures (other creatures could be rational beings but not quantifying beings).

Let me note two further features of quantifier concepts that set them apart. We know from the work of logicians that they are not semantically singular terms but a sui generis type of expression; they occupy their own category in mental grammar. It is sometimes said that they are second-order concepts, i.e., concepts of concepts, and this sets them apart from their first-order brethren. To grasp them, you have to be able to ascend a level and predicate them of a concept: this requires a cognitive leap, a new mode of mental representation. Creatures with only first-order concepts are not guaranteed to be capable of achieving this new level, however hard they think. Presumably, it occurred at some point in human cognitive evolution, perhaps triggered by a specific mutation affecting brain circuitry, and not shared by other species. Perhaps we have a gene for quantification! Some piece of brain rewiring caused us to be able to grasp second-order concepts like all and some, where there was no such grasp before. Then we were off to the races, with science, logic, mathematics, and cosmology on the horizon. A new cognitive trick catapulted us to the next intellectual level. Imagine if you lacked these concepts and were stuck at the level of the specific and particular: then a super-scientist rewires your brain to give you a grasp of quantification. Wouldn’t that be a stunning intellectual breakthrough, opening up vast avenues of new understanding and reasoning power? The child picks it all up automatically along with the other remarkable resources of human language, but that doesn’t mean it isn’t a signal achievement—imagine losing it one day! Quantification is a classy mental act, belonging only to the intellectual elite, by no means proletarian.

Secondly, quantifier concepts are unique as to content: no other concept is such a bad candidate for empiricist treatment.[3] How could the concepts all and some be derived by a copying operation from sensory stimuli? They are not concepts of a sensory quality, or of any mental operation. I am tempted to call them abstract, but that is just a vague way to register their distinctness from other types of concept. I would guess they are innate—for how could they be picked up from observation of the environment? They enter our thought at an early age and shape it pervasively, but their origins are obscure. They are part of the universal human lexicon, but they name nothing and describe nothing. Form the thought “Everything changes” and ask yourself what is going on in your mind: you will find no discernible constituent corresponding to the quantifier—no image or feeling or disposition. There is nothing…concrete here. Yet you mentally took in all of reality! How is that possible? What do you have that your cat doesn’t have? I mean: how is the concept of universality mentally represented? Can we have a description theory of it? How about a causal theory? Neither seems remotely feasible. We are used to the words (and their corresponding logical symbols) but what is the content exactly? Where is the cognitive science of quantification? What we have here is a complex and intricate biological adaptation of enormous utility but quite opaque in its mode of operation. It took logicians thousands of years to identify it and describe its logical character, but its psychology is not even in its infancy (or its neuroscience). The point I am urging is that it has some claim to distinguish us from other thinking beings on our planet. Let us grant that bees, whales, and dolphins have communication systems, along with associated cognitive structures—but it is a further claim to maintain that they understand quantification as humans do.[4] All humans do understand it (short of pathology), but there is no evidence that other animals can engage in quantificational reasoning (just consider the difficulties of embedded quantifiers).

It is not implausible to suppose that humans go through an ontogenesis in which “all” begins locally and then gradually widens to take in more and more of reality. Thus the child initially applies “all” to all the marbles on the table or all the apples he can see, later expanding the domain to include all the marbles or apples on earth. But that isn’t enough to yield the adult concept: the child must include all past and future marbles and apples, as well as any found elsewhere in space. Then there are all the possible marbles and apples. Finally, we reach everything there is. The original concept (innately present, we can suppose) already contained this potential, but it undergoes a process of maturation that ends with the cosmic all. This would be in conformity with standard views of linguistic and conceptual development. But the process has a special interest because the concept is so all-encompassing in its nature: its enormous reach signifies a kind of supremacy among concepts—it is the king of all concepts, as it were. Every other concept is subordinate to it, literally. Doubtless, it is a concept that has fueled the acts of many a despot or madman, or metaphysician or cosmologist (a “theory of everything”). God is described as all-powerful, all-knowing, and all-virtuous: the recipient of every estimable universal quantification. So much majesty revolves around this concept—its place in human thought is unrivaled.[5] Once the child has fully absorbed this concept (or it has fully matured within her) she becomes a being of a different cognitive order from the run of terrestrial animals, including her former self. Morality is stamped with it too: duty is what everyone ought always to do in any circumstances (remember Kant’s categorical imperative, in which universalization is paramount). We would not be the cognitive (and emotional) beings we are without this capacious and ubiquitous concept. When Aristotle enunciated his famous syllogism beginning “All men are mortal” he was drawing attention to the mighty power of that little word “all”: once you know that all F’s are G you know something of high significance from which many interesting things follow. In it may reside our capacity for the type of thought that defines human nature.[6]


Colin McGinn

[1] I am slightly misusing the word “quixotic” here, but the alliteration was irresistible.

[2] George Eliot reminds us of a downside to this mental advantage over other animals: “But this power of generalizing which gives men so much the superiority in mistake over the dumb animals…” (Middlemarch, 592) Our ability to generalize lays us open to errors of thought unknown to animals lacking this capacity; and it must be said that quantifiers can cause us no end of trouble—especially the standing temptation to abuse “all” in the presence of “some” (quantificational malfeasance).

[3] I know this is saying a lot given empiricism’s poor track record, but a bit of overstatement may be forgiven in the light of the fact that one never hears much about quantifier concepts from empiricists (I don’t recall Hume discussing them at all). They are expected to take care of themselves.

[4] Given other differences between human and animal thought, it might be more apt to compare humans to other hominids now extinct. What if Neanderthals matched humans cognitively except where quantification is concerned? That could be the reason for our relative success.

[5] What is the connection between death and the universal quantifier? Simply this: when you die it is all over. Everything about you has gone. You are now nothing. The quantifiers say it all. We understand what death is because we can use quantifiers this way.

[6] People often discuss this question as if it is all-or-nothing matter—either we share thought with animals or we don’t. But a more nuanced discussion can focus on whether there are any areas of human thought inaccessible to other thinking beings. Thought may not be homogeneous in its nature and origins (similarly for language). Quantification may have been added quite late in the game.

4 replies
  1. jgkess@cfl.rr.com
    jgkess@cfl.rr.com says:

    “Perhaps we have a gene for quantification!”. How Dawkinsian of you. I approve the tendency, even if not the application in this particular case. Why shouldn’t quantification reduce to a creature’s capacity to note and use frequencies of event types—all, some, none—in the exercise of its intelligence? Language enables in our case the externalization of latent ” thoughts” and “concepts”, upon which idly (or not so idly) to meditate. Cumulative cultural epistemic inheritance gives us the rest—philosophy, science, morality.

    • Colin McGinn
      Colin McGinn says:

      No idea how this is supposed to work. Is this the old idea of “general intelligence” as the basis of all knowledge? Is the child supposed to “abstract” all and some from observations of groups of things? Is this some sort of “copying”?

  2. jgkess@cfl.rr.com
    jgkess@cfl.rr.com says:

    No, ” general intelligence” is not the basis of all knowledge, but it is the basis of some. The reflexive output of modular information-processing carried the day in ancestral environments. Reflection on that output, given language, in more modern environments, hurries us ever forward. But from where the circumspection of reflection itself? Easy: the trial-and-error wisdom of cumulative cultural epistemic inheritance.


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