On Nonsense and Referents

The Internet hosts many critiques of Objectivism, the philosophy of Ayn Rand, most of which are useless or worse. Michael Huemer’s Why I Am Not an Objectivist (WIANO hereafter) is an exception. It has the great virtue that any Objectivist who engages its arguments can either realize a better understanding of philosophy by overcoming them, or else realize the inadequacy of his understanding by failing to do so.

What follows is a rejoinder to WIANO’s first section, “MEANING.” This is the first revision of a version published earlier here.

Huemer Against Rand on Meaning

Objectivism rejects the analytic-synthetic dichotomy. Michael Huemer accepts it. He correctly recognizes that the basis for Objectivism’s rejection of this dichotomy lies in its identification of the meaning of concepts. Leonard Peikoff writes, in “The Analytic-Synthetic Dichotomy,” “[A] concept means the existents which it integrates. … [It] subsumes and includes all the characteristics of its referents, known and not yet known.” (Introduction to Objectivist Epistemology, Second Edition, p. 99. Emphasis in original. Hereafter cited as ITOE.) Huemer also notes that Objectivists consider concepts to be open-ended, or as he would have them put it, “[T]he meaning of a concept is all of the concretes it subsumes, past, present, and future, including ones that we will never know about.” (WIANO, §1)

Huemer prepares his attack on the Objectivist rejection of the analytic-synthetic dichotomy by reference to the story of Oedipus, who famously — and unwittingly — married his own mother, Jocaste. Huemer wants to show that the Objectivist theory entails that Oedipus could not have married his own mother unwittingly, since he knew that he was marrying Jocaste, and since “Jocaste” and “Oedipus’ mother” have the same referent. It is absurd to think that Oedipus knew he was marrying “Oedipus’ mother” just because he knew he was marrying “Jocaste,” so if Objectivism does entail this, Objectivism is absurd.

After posting the “Jocaste reductio” in WIANO itself, Huemer posted several more-compact variants to Usenet. Here is one:

Jocaste Reductio

1. If a term, A, means the same as a term, B, then in any sentence where
   you see A appear, it is ok to substitute B. (premise)
2. Assume that meaning is reference. (Objectivist theory)
3. “Oedipus’ mother” has the same reference as “Jocaste”. (given by the
   story)
4. Therefore, “Oedipus’ mother” means the same as “Jocaste”. (from 2,3)
5. Oedipus believed he was going to marry Jocaste. (given by the story)
6. It is ok to substitute “Oedipus’ mother” for “Jocaste” in (5). (from
   1,4)
7. Oedipus believed he was going to marry Oedipus’ mother. (from 5,6)

There are several intractable problems with the Jocaste reductio. A reductio ad absurdum must follow the following form: First, assume the original position; second, show that accepting the original position leads to a contradiction or other absurdity. Huemer does a good job of showing the absurdity of an original position, but it is a position original to him, not to Objectivism. In other words, the Jocaste reductio makes use of a straw man.

Huemer’s principal error is at step 2, where he asserts that the Objectivist theory is that “meaning is reference.” This is false. Nowhere in Introduction to Objectivist Epistemology, or anywhere else in the body of work that constitutes Objectivism, does Ayn Rand or anyone else state that “meaning is reference,” or make any claim that reduces to or implies this view.

Introduction to Objectivist Epistemology was an introduction to Ayn Rand’s epistemology, not a comprehensive presentation of it. If it did not present the complete Objectivist epistemology, it certainly did not present a comprehensive theory of meaning, which would be part of a general theory of language. While an Objectivist general theory of meaning would depend upon Ayn Rand’s theory of concepts, the broader theory cannot simply be deduced from what is extant.

The actually existing Objectivist theory is that the meaning of a concept is the existents it integrates. Since Ayn Rand defines a concept as “a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted,” (ITOE, p. 12) the proposition that Huemer should be challenging is this: The meaning of a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted, is the existents that integration integrates. (From here on, whenever I refer to the Objectivist “theory of meaning,” I mean the Objectivist theory of the meaning of concepts.)

Huemer’s principal error is at step 2, but he makes another error at step 6 that invalidates his reductio all on its own. Step 1 of his argument states that “If a term, A, means the same as a term, B, then in any sentence where you see A appear, it is ok to substitute B,” (my emphases). Step 6 of his argument consequently asserts that it is permitted to substitute “Oedipus’ mother” for “Jocaste.” “Oedipus’ mother,” however, is not “a term”; it is two terms (possibly more, depending on how you account for the apostrophe). In this context, assuming that “Oedipus’ mother” is the same as one term is begging the question.

Here is the conclusion Huemer draws from his Jocaste reductio:

The thing that the ideas refer to — the person, existing in physical space — I call the “reference” of the ideas. The reference of a word is the same as the reference of the idea that the word expresses. The sense of a word, however, I identify with the idea that the word expresses. Thus, “Jocaste” and “Oedipus’ mother” have the same reference, but different sense. That’s what we’ve just shown.

I believe it is consistent with Objectivism to posit that neither concepts nor proper names “express ideas” at all. Rather, sentences express ideas. Sentence fragments, such as “his mother” or “Oedipus’ mother,” express incomplete ideas. A fragment like “Oedipus’ mother” is able to express part of an idea precisely because it juxtaposes and interrelates by the rules of English grammar and punctuation the concepts of belonging and motherhood with the proper name “Oedipus,” and it is because of this complex of relations that a meaning-qua-expressed-idea arises.

It is not necessary, however, for me to posit any hypothesis about the meaning of sentence fragments to show that the Jocaste reductio is unsound, as we have already seen.

So far, we have learned very little from WIANO beyond how not to construct a counterexample to Ayn Rand’s theory of meaning. Fortunately, Huemer has said that what he thought could be done with “Jocaste” and “Oedipus’ mother” can be done just as well with common terms, like “water” and “H₂O.”

Moving to “water” and “H₂O” will help us build, on the ashes of the Jocaste reductio, a much better counterexample to Objectivism. Most importantly, this move allows us to construct a counterexample without resorting to sentence fragments and the meaning smuggling that they entail. From this point on, I will insist that all the concepts the meanings of which we compare be expressed in single terms, i.e. in one word. This not only helps avoid the confusion and question begging of sentence fragments, it is also consistent with Ayn Rand’s own formulations in Introduction to Objectivist Epistemology, wherein she makes it very clear that a concept must be represented by a single perceptual symbol (ITOE, pp. 10–11 and 177).

Another benefit is that, as we have already seen, under Objectivism a concept always represents a kind of existent, i.e. existents in the plural. “Jocaste” is not a concept under Objectivism, since it refers exclusively to one entity (ITOE, pp. 10–11). Concepts go with universals; proper names go with particulars. Neither “Jocaste” nor “Oedipus’ mother” are valid concepts under Objectivism; both “water” and “H₂O” are.

Finally, both “water” and “H₂O” are bona fide abstractions. This means: in order to form each of these concepts, one looks at the world, at particular concretes, and then “pulls away” — abstracts — something from them. (What is “pulled away,” according to Ayn Rand, is the particular measurements of the concretes that are used as the basis for the concept.) This is important because for something to be an abstraction, it has to have been abstracted from something — from entities. Water and H₂O both actually exist to abstract from. (I call attention to this because another reason that “Oedipus’ mother” and other sentence fragments are invalid in counterexamples to Rand’s theory is that they are not abstractions from concretes, but constructions built out of prior abstractions. This distinction illustrates the all-important inductive nature of Ayn Rand’s theory, so please keep it in mind.)

Non Question-Begging Counterexamples

1. If a term, A, means the same as a term, B, then in any sentence where
   you see A appear, it is okay to substitute B. (Premise)
2. Assume that the meaning of a word (concept) is all the existents which it integrates. (Objectivist theory)
3. “Water” integrates the same existents as “H₂O”. (Premise)
4. Therefore, “water” means the same as “H₂O”. (From 2,3)
5. Oedipus knows he is enjoying a glass of water. (Stipulated)
6. It is okay to substitute “H₂O” for “water” in (5). (From 1,4)
7. Oedipus knows he is enjoying a glass of H₂O. (From 5,6)

Assuming the premises are all true, the obvious conclusion of this argument is that the meaning of a concept is not all the existents which it integrates, or at least it is not just that. Oedipus, just because he knows he is drinking water, does not necessarily know that he is drinking H₂O.

(The failure of this counterexample to Objectivism is that Premise 3 is false. Before I demonstrate this, however, I want to point out how not to respond to this type of challenge to the Objectivist theory of meaning.

On several occasions, I have seen Objectivists or students of Objectivism attempt to rebut this kind of challenge by reference to Oedipus’ “context of knowledge.” This is worse than a dead end. Whoever thinks that context of knowledge holds the key to rebutting this kind of counterexample has certainly completely misunderstood the challenge, and has probably completely misunderstood Ayn Rand’s theory of meaning. The point of contention is not whether ‘water’ and ‘H₂O’ are respectively more primitive and more advanced concepts of the same thing, rather it is whether or not ‘water’ and ‘H₂O’ are concepts of the same thing at all.

Ayn Rand makes it very clear in Introduction to Objectivist Epistemology that different individuals will have different levels of knowledge about entities. A child’s knowledge of water will be less intensive than a chemist’s. Huemer agrees with her. The difference between Rand’s view and Huemer’s is not that one allows for varying contexts of knowledge and the other does not, but in where this knowledge is accounted for in the conceptual scheme. Under Objectivism, concepts are like file folders, containers for knowledge about a concept’s units. Here is the critical passage:

Since concepts represent a system of cognitive classification, a given concept serves (speaking metaphorically) as a file folder in which man’s mind files his knowledge of the existents it subsumes. The content of such folders varies from individual to individual, according to the degree of his knowledge—it ranges from the primitive, generalized information in the mind of a child or an illiterate to the enormously detailed sum in the mind of a scientist—but it pertains to the same referents, to the same kind of existents, and is subsumed under the same concept. [ITOE, pp. 66–67. See also “Meaning and Referent,” ibid., pp. 236–237.]

For Ayn Rand, a child and a chemist have the same concept of water, but the chemist has more knowledge about the units of the concept ‘water.’ Metaphorically, a greater variety of data is filed in the chemist’s copy of the ‘water’ file folder, but the child’s file folder and the chemist’s file folder are — qua container — the same.

In contrast, for Huemer, concepts are not containers for our knowledge about existents; they themselves, to some imprecise extent, are the knowledge we have about existents. The file folder and (at least some of) its contents are now fused into a single mental entity — an “idea” in Huemer’s terms. This fusion is what causes, what makes necessary, the sense-reference distinction. If a concept-cum-idea is constituted by an individual’s particular and limited knowledge of the referents of that idea, i.e. if a chemist’s concept of water is his knowledge of water, then the exact meaning of his concept is personal to him, and it cannot include all the characteristics of the concept’s referents — only those known to him.

So while Ayn Rand would say that a chemist has more sophisticated knowledge about the units of the concept ‘water,’ Huemer might say that the chemist has a more sophisticated concept. This is where the battle lines are drawn: these counterexamples, if they stand, will show that is incoherent to conceive of a concept as a container for knowledge, and will force us to make a new container, called “sense,” that is somehow (Huemer has not said how) related to “reference,” i.e., to entities. How these counterexamples will force us to move our knowledge out of concepts and into “sense” is by means of showing that two distinct and discrete concepts with clearly different meanings yet have the same referents. If the meaning of a concept, in other words, is something more, above and beyond, what it refers to, this “something more,” this remainder, is the precipitate material from which Huemer will scrape together “sense.”)*

The failure of the H₂O counterexample to Objectivism is that Premise 3 is false. “Water” and “H₂O” do not subsume the same concretes, and believing otherwise is just sloppy thinking. Water is composed (largely) of H₂O, but that does not mean water and H₂O are identical.

The quickest and easiest way to see this is just to remember that water — what is in a lake, for example — is a mixture. It is a mixture of several elements: the molecular compound H₂O, dissolved gasses, microbes, dissolved solids, considerable empty space, and a few odd atoms of ²H₂O, (so-called “heavy water”). I am sure there are many, many things mixed up in water of which I am perfectly ignorant, and probably many things of which even our best scientists are unaware. Obviously, water is not just H₂O. Therefore, the existents subsumed by “water” and by “H₂O” overlap, but these concepts do not describe concentric and congruent areas of our world.

It does no good to object that it is merely “contingent” that water is a mixture, as philosophers coming from Huemer’s perspective might want to do. First, it does no good because this is a return to begging the question. The notion of “contingent” facts depends on the meaning of a concept not being all of the concretes it subsumes, which is exactly what is in dispute here. (For the argument that shows that “contingency” depends on a mistaken view of concepts, precisely the mistaken view of concepts that Huemer advances, see Leonard Peikoff’s article “The Analytic-Synthetic Dichotomy” in ITOE, pp. 88 ff, especially pp. 106–111.) Second, it does no good because, even were all water in existence chemically pure, a single molecule of H₂O would still have vastly different properties than a macroscopic quantity composed of articulated molecules. In other words, a single molecule of H₂O would be H₂O, but it would not be water. (A single molecule cannot flow, change state from solid to liquid to gas, or do just about anything else that water does.) Furthermore, if one could gather however many molecules it would take to make a droplet of recognizable water, but were to affect them in such a way that they could not coalesce, it would no longer be a matter of numbers: even lots of H₂O molecules still would not be water.

Those who uncritically thought Huemer’s criticism more persuasive than Ayn Rand’s theory are, I hope, now be beginning to grasp the subtlety and power of her system. In any case, it should be clear now that neither Huemer’s own Jocaste reductio, nor the refined water-H₂O version he himself has endorsed do any good.

(The real object of Huemer’s counterexamples, we should remember now, is not to establish the sense-reference distinction, but to prevent the collapse of the analytic-synthetic dichotomy. To say that this latter distinction is dearly held by many philosophers is a very great understatement. If the analytic-synthetic distinction turns out to be spurious, then critical foundations of a centuries-old and academically dominant philosophic enterprise will be washed away. Asking the intellectual progeny of Hume, Kant, and Frege to abandon the analytic-synthetic distinction is like asking the Pope to deny the Trinity. Because so much is at stake, it is worthwhile to go far beyond the requirements of rebutting WIANO, which have already been met and more than met, and demonstrate further the unassailable strength of Ayn Rand’s theory of meaning.)

Hearts and Kidneys

The terms “cordate” (meaning a creature having a heart) and “renate” (meaning a creature having a kidney) are nonce words that are useful in setting up another counterexample to Objectivism. It is commonly thought (among those who use the cordate-renate example, but not among entomologists) that all creatures with hearts are creatures with kidneys, and all creatures with kidneys are creatures with hearts. So:

1. If a term, A, means the same as a term, B, then in any sentence where
   you see A appear, it is okay to substitute B. (Premise)
2. Assume that the meaning of a word (concept) is all the existents which it integrates. (Objectivist theory)
3. “Cordate” integrates the same existents as “renate”. (Premise)
4. Therefore, “cordate” means the same as “renate”. (From 2,3)
5. Oedipus knows he is enjoying fillet of cordate. (Stipulated)
6. It is okay to substitute “renate” for “cordate” in (5). (From 1,4)
7. Oedipus knows he is enjoying fillet of renate. (From 5,6)

As in the case of water-H₂O, the argument fails at Premise 3. In fact, insects such as grasshoppers have simple hearts, but their open circulatory systems do not filter wastes through kidneys. We have already dealt with the question-begging “contingency” objection, but, for the sake of a richer exposition, let us imagine that no grasshoppers or similar creatures existed. Were that the case, it would still be possible that, at some point, somewhere in the universe, such a creature might evolve. Since Ayn Rand’s theory shows very clearly how concepts are open-ended, there is no question that newly evolved or newly discovered non-renate cordates are integrated under the concept ‘cordate’ from the inception of the concept. (Introduction to Objectivist Epistemology explains the open-ended nature of concepts in detail. Deep familiarity with ITOE’s argument on this point is absolutely essential to understanding the failure of the cordate-renate counterexample.)

(For some, it is perhaps easier to understand the open-ended nature of concepts if they imagine Ayn Rand’s theory this way: “A concept means all possible existents it integrates.” From this perspective, even in our imagined grasshopperless world, a non-renate cordate is possible, and is therefore integrated, from the outset, under the concept ‘cordate.’

It is, however, extremely misleading to formulate Ayn Rand’s theory in such a way that it depends upon the notion of “possibility.” If we understand Ayn Rand’s theory of meaning to be something like “A concept means all possible existents it integrates,” then whatever is meant by “possible” becomes a limit on the meaning of each individual concept. A pressing need to identify the exact meaning of “possible” obviously follows. The essential (and intractable) problem becomes: we do not know what is and what is not possible. To know what is and is not possible means: to be omniscient. It was “impossible” for a mammal to lay eggs until the platypus was found doing just that.

Under Objectivism, “possibility” is a concept that pertains only to the realm of human action, and so an authentically Objectivist treatment of possibility would be incompatible with the loose use of “possible” necessary for describing the open-ended range of a concept. Even so, we can avoid the use of “possibility” and still perhaps clarify the Objectivist theory for those who are unmoved by “possible” cordates.)

It might help bring into focus what, exactly, Ayn Rand designates to be the meaning of a concept (what Huemer calls the “reference” of a concept) if we use the subjunctive mood. Sentences in the subjunctive mood express these kinds of ideas:

• If I were a rich man … all day long I’d biddy biddy bum.
• Were I the King of England, I would still prefer French food.
• I would like to go with you to the Gustav Mahler concert, but I have to become a lumberjack instead.
• Were I to draw a round square, I would afterwards dance a jig.

Consider the last subjunctive sentence: It is not a statement about geometry or dance; it is a statement about my disposition. The (im)possibility of drawing a round square has nothing to do with the truth (or falseness) of the statement. In other words, I, who would dance a jig had I drawn a round square, by saying so, feign no hypotheses about squares whatsoever.

Back in our imagined world where there are no grasshoppers, two zoologists are studying a sketch of a fanciful creature, a grasshopper, their world’s unicorn. As a lark, they have imagined its internal structure down to the minutest detail, and have even postulated an open circulatory system, and therefore, they are considering a creature with a heart but without kidneys. The first zoologist says to the second, “Dear fellow, one thing is certain: if there were such a creature, it would be a cordate, but not a renate.” In saying so, this zoologist feigns no hypotheses about cordates: he is not saying that it is possible for there to actually be a cordate that is not a renate, and obviously he is not saying that it is impossible for there to be a cordate that is not a renate. He is not saying anything about the possibility of non-renate cordates at all.

This is what Ayn Rand means when she talks about the open-ended nature of concepts: if something would satisfy the measurements required of a cordate, it would be a cordate, and the meaning of cordate is: everything that does, anything that has, and anything that would satisfy the measurements required of a cordate. If something would satisfy the measurements required of an X, it would be an X, and the meaning of X is: everything that does, anything that has, and anything that would satisfy the measurements required of an X.

Rand herself, it should be noted, does not use the subjunctive mood to help explain her theory of meaning. Rather, her chosen explanatory metaphor is algebraic:

A concept is like an arithmetical sequence of specifically defined units, going off in both directions, open at both ends and including all units of that particular kind. For instance, the concept “man” includes all men who live at present, who have ever lived or will ever live. An arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means only that whatever number of units does exist, it is to be included in the same sequence. …

The relationship of concepts to their constituent particulars is the same as the relationship of algebraic symbols to numbers. In the equation 2a = a + a, any number may be substituted for the symbol “a” without affecting the truth of the equation. … In the same manner, by the same psycho-epistemological method, a concept is used as an algebraic symbol that stands for any of the arithmetical sequence of units it subsumes. [ITOE, pp. 17–18.]

Of the three ways of conceptualizing Rand’s theory of meaning that I have presented — with possibility, with the subjunctive perspective, and with algebra — each is better than the last, and Rand’s own is by far the best. It should now be clear exactly how ‘cordate’ subsumes and includes non-renate cordates, irrespective of whether or not there are any non-renate cordates, and how any concept subsumes and includes an infinite series of concretes, irrespective of whether those concretes are extant, and irrespective of whether those concretes are even “possible.” Just in case the relationship of concepts to existents is still unclear, Ayn Rand’s algebraic metaphor enables some rather interesting and instructive new examples.

It is my own coinage to say that the meaning of a concept is “specific yet indeterminate,” and that is exactly what we are seeing with respect to the meaning of ‘cordate.’ It is specific: a cordate must be a creature and have a heart, yet indeterminate: we do not know all of the traits of cordates, nor do we know just what kinds of cordate or what all individual cordates exist or may exist. In the same way, the value of x in |x| = 5 is specific yet indeterminate. The value of x can be 5 or it can be -5, but neither 5 nor -5 is (alone) the value of x. The value of x is specific: it must be 5 or -5, yet indeterminate: it may be 5 or -5.

A penultimate algebraic example should put to rest any lingering doubts about the relationship between a concept and the existents it integrates. The purpose of this example is to show that integration or subsumption is an epistemological relation, not a metaphysical relation, and as such, does not depend in any way on the ontological status of marginal cases. (By “marginal cases,” I do not mean borderline cases, such as something that we cannot classify easily in an existing conceptual scheme (e.g. slime molds). Rather, I mean non-extant units of a concept, e.g. past cordates, future cordates, or “possible” cordates.)

Suppose you are trying to figure out the dimensions of a square-shaped something, perhaps a patio, and you know its area is 64 square feet, but do not know the length of any side. Let x be the length of a side; x² then equals 64. The value of x can be 8 or -8, but since length cannot be negative, -8 is discarded: the square is 8 feet on each side.

If you had not been investigating the relationship of a length to an area, then -8 might have been an applicable solution to the equation. Because of the identity of the subject of the equation, because what is being measured is a patio, only the positive solution is valid. If the subject of the equation were changes in temperature, where values could be negative or positive, then both 8 and -8 might have been valid solutions. The equation x² = 64 itself does not specify what kind of thing is being measured. It is an abstraction, a “pulling away” from reality. The equation x² = 64 can apply to anything whatsoever that is analyzable in the specified relations. It just says: some units of something squared will equal 64 units of something. Specifying that x times itself must equal 64 feigns no hypotheses about the nature of x.

In just the same way, ‘cordate’ feigns no hypotheses about the nature of cordates. It is impossible for x to be negative when x is a length, and we can pretend it might have been impossible for a creature with a heart to live without kidneys, but whether that is possible or impossible does not alter the meaning of ‘cordate’ any more than patios alter the meaning of x² = 64.

Finally, to concretize the conceptual-subsumption-as-algebraic metaphor: I find it useful to think of concepts as open-ended functions. A function is a species of relation. A relation is a set of ordered pairs wherein the first components of the ordered pairs are the input values and the second components are the output values. Functions are relations that, for each input, there is exactly one output. The kind of function I have in mind looks like this:

• {…(a,0),(b,0),(c,0),(d,1),(e,1),(f,1),(g,0)…}

In a function or relation, the set of input values is called the “domain.” The set of output values is called the “range.” In a concept-function, there are only two possible output values: 0 and 1, or No and Yes, so the range of any concept-function is:

• {0,1}

For a concept-function, the domain of all possible inputs is: any entity.

• {… a,b,c,d,e,f,g …}

In the domain of a concept-function, each variable stands for an entity and all of its characteristics. (It is vital to understand that the addendum, “and all of its characteristics” is strictly tautological. Any thing is nothing apart from its characteristics.)

The meaning of a concept is a sub-set of the domain of a concept-function, consisting of those entity-inputs that are paired with a Yes value. In the current example, that sub-set is:

• {… d,e,f …}

Applying the concept ‘cordate’ to the present concept-function requires considering some entity-inputs:

• a = a brick
• b = the concept ‘cordate’
• c = an apple pie
• d = a grasshopper
• e = a lemur
• f = myself
• g = justice

It is important to note that both the domain of a concept-function and the sub-set of the domain which constitutes the meaning of a concept are infinite and open-ended. There are infinite values on the number line between 1 and 1.5 and between 1.5 and 2, but nonetheless the values between 1 and 2 are a specified and limited segment of the number line. Likewise, the meaning of a concept is an infinite but specified segment of a larger infinity.

Triangles, Trigons, and Polygons-180

In an earlier version of this rebuttal, I concluded with my explosion of the cordate-renate counterexample. A commenter posting as “Lewis” was kind enough to provide another counterexample that I think is worth examining:

I don’t think your defense for cordates and renates really hold [sic] up. There could be a causal or logical connection between two concepts that wouldn’t allow you to rescue the Objetctivist [sic] theory the way you did. For example:

Concept A: All planar figures with three sides.
Concept B: All planar figures with angle sum 180.

Those two concepts are logically connected and have the same referents, namley [sic] all triangles, but you could know the meaning of either one of them with out [sic] knowing this very fact.

Lewis’s ad hoc definitions of concepts A and B are not quite rigorous enough to hold up under examination, so consider these improvements, which give each concept a name that is a single perceptual symbol, and define each in terms of genus and differentia:

• Trigon: a three-sided polygon.
• Polygon-180: a polygon the measurements of the interior angles of which add up to 180 degrees.

In Huemer’s counterexample form, trigon and polygon-180 compare like this:

1. If a term, A, means the same as a term, B, then in any sentence where
   you see A appear, it is okay to substitute B. (Premise)
2. Assume that the meaning of a word (concept) is all the existents which it integrates. (Objectivist theory)
3. “Polygon-180” integrates the same existents as “trigon”. (Premise)
4. Therefore, “polygon-180” means the same as “trigon”. (From 2,3)
5. Oedipus knows he is drawing a trigon. (Stipulated)
6. It is okay to substitute “polygon-180” for “trigon” in (5). (From 1,4)
7. Oedipus knows he is drawing a polygon-180. (From 5,6)

This counterexample is exceptional. A defender of the Objectivist theory has 3 promising options.

1. Deny Premise 3 on the basis that ‘polygon-180’ is a spurious concept.
2. Deny Premise 3 on the basis that “trigon” and “polygon-180” are used equivocally.
3. Deny Premise 3 on the basis that ‘trigon’ and ‘polygon-180’ in fact integrate (marginally) different existents.

(I must note at this point that I am not sure which of these three options Ayn Rand herself would have taken, if any of them. My own views on how Objectivists can defend against the trigon counterexample are just that.)

i. Polygon-180 is a spurious concept. In order to form a concept by abstraction, there are a number of prerequisites:

• You must have some entities from which to abstract.
• You must have two or more entities that will become units of the concept.
• You must have additional entities that will be contrasted against the entities that will become units of the concept.
• You must have what Rand calls a Conceptual Common Denominator, defined as: “The characteristic(s) reducible to a unit of measurement, by means of which man differentiates two or more existents from other existents possessing it.” (ITOE, p. 15)

In the case of ‘polygon-180,’ then:

• You must have some polygons. (These could be inscribed on a sheet of paper, or stamped into tiles, for example.)
• You must have at least 2 polygons that are polygons-180 (though you do not know that they are polygons-180 until you have formed the concept and fixed it with the word “polygon-180.”)
• You must have at least one more polygon that is not a polygon-180, such as an octagon, square, or dodecahedron.
• You must have consciously isolated the aspect of polygons that they all have interior angles, and considered that these interior angles can be added together to form a sum. The Conceptual Common Denominator prerequisite for ‘polygon-180’ is: polygons the interior angles of which add up to n-degrees.

Given these materials, the next step would be to measure the angles and calculate the angle sums of enough of the given polygons to discover that two of them, those of the polygons-180, added up to exactly 180 degrees, while those of the others did not. At this point, the two (or more) polygons-180 could be integrated into the concept ‘polygon-180,’ while the concept was being concretized into the word “polygon-180.”

Now there is sufficient reason to dismiss ‘polygon-180’ as a spurious concept (and, in my view, more than sufficient reason): no one would ever go through the process just described. No one has ever gone through the process just described. No one will ever go through the process just described.

The blindingly obvious reason why no one would ever actually go through the process of forming ‘polygon-180’ is that when anyone looks at a “polygon-180,” what he sees is a three-sided figure, a triangle or trigon. The notion that anyone ever would or could “miss” the sides and integrate based on angle sums is, flatly, absurd (especially considering that the prior-requisite concept of ‘polygon’ explicitly integrates sides!). Put more technically, angles depend on sides, they are a consequence of sides; sides are prior to angles both “metaphysically” and epistemologically.

Furthermore, the conditions and even the actions required to form the concept ‘trigon’ and the concept ‘polygon-180’ are essentially the same. The only differentiating factor is the choice of Conceptual Common Denominator (which in the former case would be “n-sided polygon.”) This means, since the only thing that could cause ‘polygon-180’ to be formed would be a bizarre choice of Conceptual Common Denominator, anyone forming the concept ‘polygon-180’ would simultaneously and implicitly be forming the concept ‘trigon.’ In other words, if Oedipus knows he is drawing a polygon-180, he knows (at least implicitly) that he is drawing a trigon.

It does not matter if the reverse is not true: if Oedipus knows he is drawing a trigon, he may not therefore know, even implicitly, that he is drawing a polygon-180. (It seems one would not implicitly form the concept ‘polygon-180’ as part of the process of forming the concept ‘trigon,’ though I believe an argument could be made to the contrary.) In fact, if someone were to try to form the concept ‘polygon-180,’ ‘trigon’ would not simply get formed alongside as an implicit concept. Rather, the attempt to conceptualize ‘polygon-180’ would implode, the obvious fact that the proto-polygons-180 were all three-sided would be unmistakable, and the result would be forming the concept ‘trigon’ after all, immediately containing the knowledge that three-sided polygons have interior angles the sum of the measurements of which are 180 degrees. Thus, we can reject Premise 3 of the trigon counterexample on the basis that “polygon-180” can have no independent meaning or etiology, and therefore is just another word for trigon. In other words, Premise 3 is no good because “polygon-180” does not integrate any existents at all; it is merely a crystallized sophism.

I myself find option (i) convincing. As an argument, however, it lacks rhetorical force. Those still sharing Huemer’s point of view will probably object that, even if the concept ‘polygon-180’ is empirically impossible due to its etiology, it is nevertheless logically possible. I think this objection is an anemic sophism, and in the context of all that has been seen so far, represents a desperate rear-guard action on the part of the sense-reference partisans. Rather than quitting the field and declaring victory, however, I shall consider the Objectivists’ second option. Perhaps it will turn out to be more conclusive.

ii. “Trigon” and “polygon-180” are used equivocally. What do the concepts in column A have in common that differentiates them from the concepts in column B?

In my treatment of the Objectivists’ first option, I avoided the more common name for a three-sided polygon, “triangle,” in favor of the more obscure “trigon” (from which we get “trigonometry”). This was to differentiate in advance two very closely related concepts that we tend to treat as one concept in practice. I use “triangle” for the broader concept that includes both planar triangles and the little plastic tiles that children learn shapes from (which are, of course, 3-dimensional). I have used “trigon” for the narrower concept which subsumes and includes only “three-sided polygons.”

Neither points, nor lines, nor planes actually exist. All of these are abstractions. Since a polygon is “a closed planar figure with line-segment sides,” obviously no polygons exist. Since a trigon is a three-sided polygon, obviously no trigons exist. No one has ever seen a trigon, or any other polygon. What has been seen on paper are various necessarily imperfect representations of polygons.

I point this out not to illustrate something about abstractions in general. I am not saying that the referents of what Objectivism calls “first-level” abstractions (abstractions from perceptual concretes, like water) are the only referents that exist. I am not saying that if you cannot perceive it, it does not exist.

Water does exist. Length does exist. Justice does exist. Justice is like water and length, in this respect: “Water,” “length,” and “justice” are all referring terms. “Water” refers directly to a perceptual concrete. You can perceive water directly, see it, touch it, hear it, taste it, smell it. “Length” refers to an attribute of perceptual concretes. You cannot see length itself, but you can see various long entities. “Justice” refers to certain kinds of actions performed by perceptual concretes: men.

“Unicorn” and “trigon” do not refer to perceptual concretes, but they are often used as if they did. The concepts in column A have in common that they refer to perceptible existents, which differentiates them from the concepts in column B, which do not.

“Trigon,” when it is used correctly, refers to relationships between perceptual concretes. (Any three things can have three points assigned to them, and then these three points-on-things, by their interrelation(-in-selective-conscious-focus), become a trigon, a kind of pseudo-concretization of the spatial relations among these three things.) When we treat a trigon as if it itself were a perceptual concrete, “trigon” becomes like “unicorn” — a non-referring term. This is committing an error which I will term “the reification of the trigon.”

When ‘trigon’ is reified (and I suspect this usually happens because of our habit of conflating the concepts ‘trigon’ and ‘triangle’), it seems as though trigons were things with properties, like trees, or rocks, or anything else. From there it is a short journey to believing that ‘trigon’ and ‘polygon-180’ refer to the same things, when in fact they refer to different implications of one relation that can be ascribed to any three things.

This may seem like a distinction without a difference, but it is not. If trigons and polygons-180 do not exist (are not reified), then they do not have sides or angles. Instead, the infinite variety of things that we might consider as being related trigonometrically are varying distances apart. Viewed this way, we can reject Premise 3 of the trigon counterexample on the grounds that neither “trigon” nor “polygon-180” refer to anything at all.

Perhaps some partisan of the sense-reference distinction will want to try and come up with a reworked and expanded version of the trigon counterexample that does not depend on the reification of the trigon. Right now, I cannot imagine how that would be done, and so, like option (i), I find option (ii) convincing. Just in case there are readers who do not, let us consider option (iii).

iii. ‘Trigon’ and ‘polygon-180’ in fact integrate (marginally) different existents. Every time a side is added to a polygon, 180 degrees are added to the sum of its interior angles. All squares, rectangles, trapezoids, and rhombuses, for example, have interior-angle sums of 360 degrees. The proof that a trigon’s interior angles must add up to 180 degrees is rather simple, and once it is grasped, it is easy to see how squares and other quadrilaterals must have interior-angle sums that add up to 360 degrees. (All polygons can be dissected into tessellations of triangles.)

The only way to know that a polygon-180 cannot have 4, 5, 6, or more sides is to learn, one way or another, implicitly or explicitly, what I have just presented. Any initial acceptance of the trigon counterexample depends on our knowing that “all trigons are polygons-180” and “all polygons-180 are trigons.”

As we saw in the post mortem of the cordate-renate counterexample, Ayn Rand’s algebraic conception of meaning feigns no hypotheses concerning the nature of a concept’s referents. Therefore, since the concept ‘polygon-180’ is genetically dependent on the abstracting away of all consideration of sides, under Objectivism, the meaning and the “reference” of ‘polygon-180’ necessarily include 4-sided polygons-180.

Of course, we know that there are not and cannot be any 4-sided polygons-180. So much the worse for the concept ‘polygon-180.’ If one were, by some bizarre chance, to ever form this spurious pseudo-concept, twice its half-life would be exactly as long as it would take to discover that any 4-sided polygon must have more than 180 degrees worth of interior angles.

Once the concept ‘polygon-180′ becomes incoherent, where does the knowledge that all trigons’ interior angles add up to 180 degrees go? Into the container marked “trigon.” This may be the paradigmatic example of knowledge about the units of a concept.

Astute readers will have noticed that not only is (iii) the most elegant and forceful reply to the trigon counterexample, it is also the one that follows the form established for “water” and “H₂O” and “cordate” and “renate.” Hopefully it is now clear that this is the correct general method for rebutting counterexamples in the mold of the Jocaste reductio.

Conclusion and Summary

Objectivism, as we have seen, offers a far more subtle theory of meaning than at least one of its critics has comprehended. Yet Michael Huemer’s Why I Am Not an Objectivist stands to this day, unretracted by its author, in part because Objectivists have been insufficiently rigorous in defending their theory of meaning. Is it that too many Objectivists have been satisfied to know in their guts that Huemer and like critics are wrong, without actually knowing it in their heads, where it counts? Ayn Rand did not agree with this lackadaisical approach to philosophy:

If you feel nothing but boredom when reading the virtually unintelligible theories of some philosophers, you have my deepest sympathy. But if you brush them aside, saying: “Why should I study that stuff when I know it’s nonsense?—you are mistaken. It is nonsense, but you don’t know it … not so long as you are unable to refute them. [The Ayn Rand Letter, Vol. III, No. 8, January 14, 1974. “Philosophy: Who Needs It (Part II)”]

It is my hope that WIANO’s days are numbered, even if too few of Ayn Rand’s admirers have seen fit to engage it. Let us review the reasons why I am hopeful.

We saw first that the Jocaste reductio is a formal and a substantive failure, based in a gross misunderstanding of Ayn Rand’s theory of meaning, namely the conflation of her theory of meaning with the broader, and untenable, “Meaning is reference.”

Second, we saw that “water” and “H₂O” do not, in reality, subsume the same concretes. We then saw the same for “cordate” and “renate” as well as “trigon” and “polygon-180.” It is vitally important to note what was common to all three rebuttals: a recourse to reality. In the “water” counterexample, we saw that, in reality, “water” refers to a macroscopic mixture, while “H₂O” refers to a molecular structure. In the “cordate” counterexample, we saw that, in reality, grasshoppers are cordates but not renates. In the “trigon” counterexample, we saw that the concept represented by “polygon-180” is, in reality, no concept at all. It is no coincidence that, in each case, an investigation of reality led to the realization that the counterexamples depended on untenable submerged premises.

Third, while considering the cordate-renate counterexample, we began to see the implications of Ayn Rand’s algebraic view of meaning. The crucial prerequisite for understanding why all counterexamples like those considered here must fail is understanding the algebraic nature of concepts. The meaning of a concept is specific yet indeterminate. The meaning of a concept is everything that does, anything that has, and anything that would satisfy the measurements required of a unit of that concept. Requiring an entity to have certain measurements within a specified range, in itself, feigns no hypotheses about the full variety of unknown entities that might yet be found to conform to the specified measurements. Thus, the meaning of ‘cordate’ is augmented by the classification of grasshoppers, but it is not reformed.

Consider now that the proximate goal of all of these counterexamples was to establish a general need for a sense-reference distinction. Professor Huemer was not attempting, with his Jocaste reductio, to bring to light an exceptional case. On the contrary, his thinking was that the need to distinguish between a concept’s meaning and its reference was a general need, one that was illustrated by his examples but by no means limited to them. This means that there should have been, if the sense-reference distinction were truly necessary, myriad examples at Huemer’s disposal, each showing that Ayn Rand’s theory of meaning led to absurdities or paradoxes. Instead, we saw that supposedly ideal counterexamples all disintegrate upon examination. By far the best (that is, “most stealthily pernicious”) counterexample I have ever encountered is the trigon counterexample, but even if it were to stand on its own, it would require the company of the water counterexample and many, many others in order to establish the general need for a general distinction. For those that still believe Huemer-style counterexamples have merit: where are they?

They are nowhere. What Huemer needs to do, in order to make a counterexample work, is to show that a concept and knowledge about its units are one integral mental entity. He needs to show, contra Rand, that there is some “idea expressed” by ‘water’ that is different than the “idea expressed” by ‘H₂O.’ In order to do that, however, he needs to show that the Objectivist theory of concepts is incoherent unless it is secreting knowledge about the units (the referents) of concepts into the concepts themselves. He needs to show, in other words, that each and every concept is bound to an idea about the nature of its referents. He needs to show that Ayn Rand’s ‘cordate’ implies a complex of hypotheses about the nature of cordates.

The opposite of this is what I have shown: Ayn Rand’s open-ended, algebraic theory of concepts feigns no hypotheses about the referents of a concept whatsoever.

The Space Opened Up By Letting Go of Hypotheses

Sir Isaac Newton, in the scholium of the second edition of his Mathematical Principles of Natural Philosophy, made a famous statement of his philosophy of induction:

Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. … I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction.

In the original Latin, Newton’s statement “I frame no hypotheses” was rendered “hypotheses non fingo,” a phrase that is now translated “I feign no hypotheses,” and which, in this form, has made several appearances here.

What Newton could not explain and refused to hypothesize about was the nature of the force that motivated gravitation’s “attraction at a distance.” Einstein would later fill in this blank with his General Theory of Relativity. The beauty and brilliance of Newton’s inductive method was that, using it, he did not need to feign a hypothesis about the cause of gravitation in order to describe, with great mathematical precision, what objects in gravity actually did.

Newton’s theory of gravitation became, in other words, the container that would lie open, waiting for later physics to fill it with an account of the causes of gravitation. This is the very model of induction. To understand the awesome power of Newton’s conceptualization of gravity, one need only consider that, before Newton, the terrestrial phenomena of falling bodies and the celestial phenomena of orbiting bodies were not understood to be at all related. Newton’s theory, literally, brought heaven to earth. His integration of the terrestrial and celestial spheres revealed man’s mind to be something numinous, prompting Alexander Pope to rhapsodize:

Nature, and Nature’s Laws lay hid in Night.
God said, Let Newton be! and All was Light.

Ayn Rand’s theory of concepts, like Newton’s theory of scientific method, is inductive. Both are based on observation followed by the integration of observations into concepts. Rand described how concepts are formed from the observation of entities. Newton described how the concept of gravity arises from the observation of falling bodies. Because both methods are properly and self-consciously inductive, they lead to open-ended integrations that allow for the expansion of knowledge. Both methods feign no hypotheses about their objects. And so, as Newton integrated the celestial and terrestrial spheres, Ayn Rand united meaning and reference.