Aristotle’s Critique of Dichotomy

Chapter 3 of the extraordinarily rich book 1 of Aristotle’s Parts of Animals contains a strong critique of the notion of classification by dichotomy, with implications reaching far beyond its original context. The idea that he criticizes is Platonic division into As and not-As, which is intended to result in a binary tree structure (i.e., a tree-shape in which all the branches are binary).

Platonic division was perhaps inspired by the two-sided character of Platonic dialectic, which was concerned with impartially examining the implications of both sides of some disputable question, particularly in the form of arguments for and against some thesis or other. Aristotle’s own dialectic has a more general form that is not bound to arguments for and against, but rather is simply concerned with an impartial examination of the consequences of hypotheses.

But in any case, classification in a world is a different problem from that of impartially examining a single hypothesis.

Ignoring Aristotle’s lesson, and strongly influenced by the more general impoverished notion of logical judgment as grammatical predication, early modern writers on natural history attempted to follow an a priori theory of univocal classification. But for Aristotle, there is no a priori theory of classification. Instead, the starting point is what Kant would call the implicitly schematized manifold of a concrete world.

Aristotle points out that if classification were reducible to the assignment of predicates, then to consistently classify a world or any given collection, there would have to be some one order in which we divide things by one predicate, then another, and so on. By examples he illustrates the fact that by this method, it is impossible to arrive at the division of animal species that we recognize in nature.

He also makes the more general argument that half of the classifying terms in any classification by sequential predication will be negatives, and that negative terms cannot be properly subdivided.

“Again, privative [negative] terms inevitably form one branch of dichotomous division, as we see in the proposed dichotomies. But privative terms in their character of privatives admit of no subdivision. For there can be no specific forms of a negation, of Featherless for instance or of Footless, as there are of Feathered and of Footed. Yet a generic differentia must be subdivisible; for otherwise what is there that makes it generic rather than specific? There are to be found generic, that is to say specifically subdivisible, differentiae; Feathered for instance and Footed. For feathers are divisible into Barbed and Unbarbed, and feet into Manycleft, and Twocleft, like those of animals with bifid hoofs, and Uncleft or Undivided, like those of animals with solid hoofs. Now even with the differentiae capable of this specific subdivision it is difficult enough so to make the classification that each animal shall be comprehended in some one subdivision and not in more than one (e.g. winged and wingless; for some are both — e.g. ants, glowworms, and some others); but far more difficult, impossible, is it to do this, if we start with a dichotomy into two contradictories. For each differentia must be presented by some species. There must be some species, therefore, under the privative heading. Now specifically distinct animals cannot present in their substance a common undifferentiated element, but any apparently common element must really be differentiated. (Bird and Man for instance are both Two-footed, but their two-footedness is diverse and differentiated. And if they are sanguineous they must have some difference in their blood, if blood is part of their substance.) From this it follows that one differentia will belong to two species; and if that is so, it is plain that a privative cannot be a differentia.” (Complete Works, Barnes ed., vol. 1, p. 1000).

Aristotle’s positive conclusion is as as follows:

“We must attempt to recognize the natural groups, following the indications afforded by the instincts of mankind, which led them for instance to form the class of Birds and the class of Fishes, each of which groups combines a multitude of differentiae, and is not defined by a single one as in dichotomy. The method of dichotomy is either impossible (for it would put a single group under different divisions or contrary groups under the same division), or it only furnishes a single differentia for each species…. As we said then, we must define at the outset by a multiplicity of differentiae. If we do so, privative terms will be available, which are unavailable to the dichotomist” (pp. 1001-1002, emphasis added).

This is consistent with Plato’s more general advice that classifiers, like butchers, should “cut at the joints”, i.e., look for natural distinctions rather than imposing artificial ones. Dipping back again to the negative argument, Aristotle adds:

“Now if man was nothing more than a Cleft-footed animal, this single differentia would duly represent his essence. But seeing that this is not the case, more differentiae than this one will necessarily be required to define him; and these cannot come under one division; for each single branch of a dichotomy end in a single differentia, and cannot possibly include several differentiae belonging to one and the same animal.”

“It is impossible then to reach any of the ultimate animal forms by dichotomous division” (p. 1002; see also Classification; Hermeneutic Biology?.)

Totality

The last post suggests another nuance, having to do with how “total” and “totality” are said in many ways. This is particularly sensitive, because these terms have both genuinely innocent senses and other apparently innocent senses that turn out to implicitly induce evil in the form of a metaphorically “totalitarian” attitude.

Aiming for completeness as a goal is often a good thing.

There is a spectrum of relatively benign errors of over-optimism with respect to where we are in achieving such goals, which at one end begins to shade into a less innocent over-reach, and eventually into claims that are obviously arrogant, or even “totalitarian”.

Actual achievements of completeness are always limited in scope. They are also often somewhat fragile.

I’ll mention the following case mainly for its metaphorical value. Mathematical concepts of completeness are always in some sense domain-specific, and precisely defined. In particular, it is possible to design systems of domain-specific classification that are complete with respect to current “knowledge” or some definite body of such “knowledge”, where knowledge is taken not in a strong philosophical sense, but in some practical sense adequate for certain “real world” operations. The key to using this kind of mathematically complete classification in the real world is including a fallback case for anything that does not fit within the current scheme. Then optionally, the scheme itself can be updated. In less formal contexts, similar strategies can be applied.

There are also limited-scope, somewhat fragile practical achievements of completeness that are neither mathematical nor particularly ethical.

When it comes to ethics, completeness or totality is only something for which we should strive in certain contexts. About this we should be modest and careful.

Different yet again is the arguably trivial “totality” of preconceived wholes like individuals and societies. This is in a way opposite to the mathematical case, which worked by precise definition; here, any definition is implicitly suspended in favor of an assumed reference.

Another kind of implicit whole is a judgment resulting from deliberation. At some point, response to the world dictates that we cut short our in principle indefinitely extensible deliberations, and make a practical judgment call.

Abstract and Concrete

In contrast to later traditional “metaphysics”, Aristotle recommended we start with the concrete, but then aim to dialectically rise to higher understanding, which is still of the concrete. In any inquiry, we should begin with the things closer to us, but as Wittgenstein said in a different context, we should ultimately aim to kick away the ladder upon which we climbed.

What Aristotle would have us eventually kick away is by no means the concrete itself, but only our preliminary understanding of it as a subject of immediate, simple reference. Beginnings are tentative, not certain. We reach more solid, richer understanding through development.

Aristotle’s discussion of “primary” substance in Categories has often been turned into a claim that individuals are ontologically more primary than form. This is to misunderstand what Categories is talking about. Aristotle explicitly says Categories will be about “things said without combination” [emphasis added], i.e., about what is expressed by kinds of apparently atomic sayings that are used in larger sayings.

The initial definition of substance in the strict or “primary” sense — which he will eventually kick away in the Metaphysics — is of a thing (said) “which is neither said of something underlying nor in something underlying”. (Aristotle often deliberately leaves it open whether he is talking about a referencing word or a referenced thing — or says one and implies the other — because in both cases, the primary concern is the inferential meaning of the reference.)

This initial definition is a negative one that suffices to distinguish substance from the other categories. By implication, it refers to something that is said simply of something, in the way that a proper name is. As examples, he gives (namings of) an individual human, or an individual horse.

“Socrates” would be said simply of Socrates, and would thus “be” — or refer to — a primary substance in this sense. The naming of Socrates is an apparently simple reference to what we might call an object. As Brandom has noted, this picks out a distinctive semantic and inferential role that applies only to references to singular things.

Aristotle then says that more universal namings or named things like “human” and “horse” are also “substances” — i.e., can also refer to singular objects — in a secondary sense, as in “that horse”. Then substance in general is further distinguished, by saying it is something A such that when something else B is said of it, both the naming and the “what-it-is” of B are said of the primary or secondary substance A. (See also Form; Things in Themselves; Definition.)

If a horse as such “is” a mammal of a certain description, then that horse must be a mammal of that description. If a mammal as such “is” warm-blooded, then that horse “is” warm-blooded.

These are neither factual nor ontological claims, but consequences of a rule of interpretation telling us what it means to say these kinds of things. Whether or not something is a substance in this sense is surely a key distinction, for it determines the validity or invalidity of a large class of inferences.

Based on the classification of A as an object reference and B as something said of A, we can make valid inferences about A from B.

When something else C is said of the non-substance B, by contrast, we still have a “naming” of B, but the “what-it-is” or substantive meaning of C does not apply to B itself, but only modifies it, because B is not an object reference. Applying the substantive meaning of C to B — i.e., making inferences about B from the meaning of C — would be invalid in this case.

Just because, say, warm-blooded as such “is” a quality, there is no valid inference that mammals “are” qualities, or that that horse “is” a quality. The concern here is with validity of a certain kind of inference and interpretation, not ontology (or epistemology, either).

In the Metaphysics, the initial referential notion of substance as something underlying is explicitly superseded through a far more elaborate development of “what it was to have been” a thing that emphasizes form, and ultimately actuality and potentiality. The appearance of what might be mistaken for a sort of referential foundationalism is removed. (See also Aristotelian Dialectic.)

I also think he wanted to suggest that practically, a kind of preliminary grasp of some actuality has to come first in understanding. Actuality is always concrete and particular, and said to be more primary. But potentiality too plays an irreducible role, in underwriting the relative persistence of something as the “same” something through change, which motivated the earlier talk about something underlying. The persistence of relatively stable identities of things depends on their counterfactual potentiality, which can only be apprehended in an inferential way. (See also Aristotelian Demonstration.)

It does make sense to say that things like actuality and substance inhere more in the individual than in the species, but that is due to the meanings of actuality and substance, not to an ontological status.

Interpretation

It seems to me that the main thing human reason does in real life is to interpret the significance of things. When we think of something, many implicit judgments about it are brought into scope. In a way, Kant already suggested this with his accounts of synthesis.

In real-world human reasoning, the actually operative identity of the things we reason about is not the trivial formal identity of their names or symbols, but rather a complex one constituted by the implications of all the judgments implicitly associated with the things in question. (See also Identity, Isomorphism; Aristotelian Identity.)

This is why people sometimes seem to talk past one another. The same words commonly imply different judgments for different people, so it is to be expected that this leads to different reasoning. That is why Plato recommended dialogue, and why Aristotle devoted so much attention to sorting out different ways in which things are “said”. (See also Aristotelian Semantics.)

I think human reason uses complex material inference (reasoning based on intermediate meaning content rather than syntax) to evaluate meanings and situations in an implicit way that usually ends up looking like simple summary judgment at a conscious level, but is actually far more involved. A great deal goes on, very rapidly and below the level of our awareness. Every surface-level judgment or assertion implicitly depends on many interpretations.

Ever since Aristotle took the first steps toward formalization of logic, people have tended to think of real-world human reasoning in terms modeled straightforwardly on formal or semi-formal logical operations, with meanings of terms either abstracted away or taken for granted. (Aristotle himself did not make this mistake, as noted above.) This fails to take into account the vast amount of implicit interpretive work that gets encapsulated into ordinary terms, by means of their classification into what are effectively types, capturing everything that implicitly may be relevantly said about the things in question in the context of our current unity of apperception.

A typed term for a thing works as shorthand for many judgments about the thing. Conversely, classification and consequent effective identity of the thing depend on those judgments.

As a result of active deliberation, we often refine our preconscious interpretations of things, and sometimes replace them altogether. Deliberation and dialectic are the testing ground of interpretations.

In general, interpretation is an open-ended task. It seems to me that it also involves something like what Kant called free play. (See also Hermeneutics; Theory and Practice; Philosophy; Ethical Reason; The Autonomy of Reason; Foundations?; Aristotelian Demonstration; Brandom on Truth.)

Classification

Like definition, classification sometimes gets an unjustified bad name. It is not a kind of truth about the world, but rather plays an expressive role, helping to explain the meaning of what is said. Classification makes it possible to substitute simple names for arbitrarily complex conditions or adverbial expressions, allowing the underlying complexity to be abstracted away, or reconstructed as needed.

In Book 1 of Parts of Animals, Aristotle shows great sophistication about this. He explicitly argues against Plato’s recommended procedure of “division” or repeated application of binary distinctions, noting that many significant real-world distinctions are better approached as n-ary or manifold than as binary.

He also explicitly notes how difficult and arbitrary it ends up being to develop real-world classifications in a strictly hierarchical manner, arguing for a more holistic approach, which cannot be reduced to a sequential application of lower-level operations.

Good real-world classifications are arrived at through dialectical trial, error, and iterative self-correction over time. Conversely, behind every ordinary referential use of simple names for things is a complex implicit dialectical/semantic development. (See also Aristotle’s Critique of Dichotomy; Hermeneutic Biology?; Difference; Aristotelian Identity; Substance; Aristotelian Semantics.)