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Introduction to Ontological Categories

Daniel J. Castellano (2007-2008)

The great medieval philosopher John Duns Scotus (c. 1266-1308) distinguished logic from metaphysics by regarding logic as the way the intellect understands the world, while metaphysics is the study of the way the world is. Both logic and metaphysics are defined with reference to some reality ("the world"), considered either conceptually (via logic) or in actuality (via metaphysics). Logic is an a priori science that considers the structure of concepts used to understand reality, while metaphysics is an a posteriori science that attempts to determine how these concepts may describe the world that actually exists. Although logic alone might not tell us what reality actually is, it is nonetheless linked to reality through concepts that define a priori possible realities.

If logic can teach us anything about how the intellect understands reality, it must include more than mere formal logic, the familiar discipline which treats the validity of deductions based on the form or syntax of arguments. A philosopher's logic, one that analyzes how we understand the real world, must concern itself with the content of its concepts, not just how variables and predicates are formally arranged. The laws of deduction would just be arbitrary formalisms if they did not pertain to the structure of possible realities. How could we say "X" and "not X" are mutually exclusive, unless logical concepts tell us something about the structure of possible realities? Logic depends for its philosophical validity on the ontological intuitions contained in its fundamental concepts; it can teach us how we understand reality ("the world" or "things that are") only if the content of its concepts describes potential realities.

Logical concepts that describe potential realities or "things that are" have traditionally been called "categories," an unfortunate term that may be applied to merely formal, syntactic or grammatical distinctions. Here we shall use the term "ontological categories," to indicate we are concerned with categories of entities, "things that are," or "things that have being." We are not analyzing "being" itself, but rather "things" that may be predicated of "being"; that is, things that may be said "to be". Ontological categories are the class of logical concepts that describe possible things that are. Since logic is concerned with understanding reality, if it is more than idle exercise, its concepts must be grounded in sound intuitions about the possibilities of reality.

Seeing that a philosopher's logic inevitably must be grounded in ontology, we may recognize how misguided is the criticism that Aristotle's logic makes ontological asssumptions. Aristotle, as a philosopher, was always concerned with reality, so his logic included ontological assumptions as a matter of necessity. His logic was not intended to be a matter of mere form or syntax, which would render it a purely linguistic exercise. Indeed, when logic is stripped of all ontological assumptions, it effectively becomes an arbitrary formal language that can prove nothing but internally consistent truisms or tautologies. Mathematical logic is fruitful only insofar as it is grounded in sound intuitions about mathematical objects, and it can describe physical reality only to the extent that certain mathematical objects accurately model the properties of physical objects. Take away fundamental intuitions such as number, space, sets and elements, and there is no basis for regarding mathematical proofs as proving anything. Since we are concerned with all of reality, not just that which is described by mathematics, we must resort to a broader array of fundamental concepts, the ontological categories.

The ontological approach to logic, exemplified by Aristotle, Scotus, and most of the classical and medieval Western tradition, diverges widely from the formalistic approach that most modern authors follow. Modern philosophers tend to restrict logic to the form of statements, propositions, and arguments, using either a linguistic or mathematical approach. Treating logic as a formal language, one can analyze its syntax and judge "validity" with complete indifference as to whether linguistic objects correspond to any potential reality. Similarly, the mathematical logician can perform symbolic operations without any concern for what type of real-world object his variables might represent. While these technical disciplines have merit in their own domains, they can never constitute an effective tool for understanding "the world that is," unless they are complemented by a sound set of ontological concepts. Those who would make logic nothing more than symbolic logic, or logistic, undermine any basis for linking logic to "the world that is," as one may define absolutely any symbolic logic with any rules whatsoever. There are many possible symbolic logics, which are really calculi, yet these can be discussed intelligibly only by using classical logic as a meta-logic. True philosophical logic must be grounded in sound ontological concepts.

Since most modern philosophers in the English-speaking world regard logistic (symbolic logic) as logic, without recourse to ontological intuitions, they are incapable of proving anything of substance, and leave all the content of human knowledge to other sciences. Philosophy either becomes a psychological discipline that tries to explain how or why we think the way we do, or it surrenders to scientism, meekly accepting that mathematical logic yields valid results in physical science, which covers all reality. Philosophy in either case consists of syntactic analysis, not a body of knowledge, so there is no set of theses that philosophers of such inclination can agree upon. This discipline of form without content passes for philosophy so ably, that many academic departments merge linguistics and philosophy into a single discipline, or hire philosophy faculty with a background purely in mathematical logic.

A theory of categories is essential to any attempt to ground the validity of logic in the real world. Mathematical logic works in its domain partly because it is based on true categorical intuitions. Everyone, in the practice of conversing or reasoning, implicitly assumes a system of categories, revealing this system in one's choice of expressions to describe reality. If validity means more than meeting an arbitrary set of criteria, there must be some connection between logical concepts and ontology. Physical science and mathematics cannot establish this connection, as they must assume the validity of classical logic. A logic that is concerned with the content of concepts rather than the forms of judgments is not dependent on any science, but rather all other sciences depend on it.

The necessity of categorical theory and a true logic transcending mere symbolism is better appreciated in the German-speaking world, especially among followers of Husserl and Heidegger. Among Spanish-speaking and Italian philosophers, there are various Aristotelian influences to guard against the facile mathematicism that prevails in much of the English-speaking world. Even among Anglophones, there are notable realists who recognize the need for an ontological theory, some of whom we will discuss in Part II. The bulk of our discussion is directed toward those who recognize the need for an ontological foundation, but may disagree on what are the correct categorical concepts.

A system of categories is abused if we make the unjustified leap from a priori logical concepts to the assumption that there must actually exist entities corresponding to these concepts. Against this uncritical idealism, we must remember Wittgenstein's admonition that we should "look, not think," when examining the world a posteriori. Nonetheless, thinking is essential to understanding, so there must be a place for abstract thought in our experience of the world. While we should not assume that a priori categorical distinctions necessarily correspond to real distinctions among existing beings, we cannot deny the practical necessity of having a theory of categories. If logic is to be a language corresponding to things that might really be, we are not free to construct logic any way we please, but must ground it in ontologically valid concepts. Without such a constraint, logic would be just an arbitrary set of operations on symbols.

Constructing an ontological foundation for logic, though practically necessary, might involve circular reasoning if our categories contain linguistic assumptions about reality that logic instead ought to help prove in a theory of metaphysics. Aristotle and his followers avoided this conundrum by using ordinary speech as a mere starting point from which a logical language could be constructed, departing from common usage as necessary. In this way, the development of logic is not utterly dependent on existing grammar and vocabulary, and the metaphysical assumptions they contain, but uses ordinary language as its repertory of symbols, adding new symbols (words) and syntax (grammar) as needed. If a logician is to make his theory intelligible, he must be able to translate his logic into ordinary language.

There is always a danger that the limitations of ordinary speech may blind us to certain ontological distinctions, but philosophers in the classical tradition, appealing to direct ontological intuition and subjecting popular concepts to logical scrutiny, have often transcended the limitations of language, and forcing language to conform to their ideas, rather than conforming ideas to language. This can be seen especially in the use of Latin by medieval philosophers, who introduced philosophical terms from Greek, assigned technical definitions to common terms, and often departed from classical syntax in order to give a more precise meaning. The "Subtle Doctor," Duns Scotus, had a most pronounced tendency to deform language to his thought, to the point that his Latin is utterly barbaric by classical standards. Similarly, modern philosophers such as Husserl and Heidegger created their own German vocabularies to articulate concepts that are not captured by ordinary language. As any student of Scotus or Heidegger knows all too well, there is nothing ordinary about the language of non-symbolic logic. The difficulty of classical logic is not merely in learning syntax, but in understanding what concepts are intended by the author. Mathematical logic can be "taught" to a computer, but philosophical logic requires real understanding of conceptual content.

Since ontological categories and their distinctions are in some sense prior to logic, we must appeal at least in part to direct intuition. This intuition pertains to a priori possible realities, not what we think the world actually is, so we are not guilty of circular reasoning when we later use this logical system as a basis for metaphysics. Given the profound importance of building our philosophy on a solid foundation, it is imperative that we examine our ontological intuitions critically. When in doubt, we should err on the side of including concepts, so that our logic is robust enough to endure even if some of our a priori concepts should prove to have no applicability to the real world. Ideally, we would like our categorical system to span all possible realities, but in fact we can only discuss all intelligible realities, being confined by human intellect.

Having considered our general motivation, we may proceed to construct a categorical system, stating our assumptions explicitly, and building our formal language out of ordinary speech, adding distinctions of vocabulary and special syntax as needed. Many others have gone before us, of course, and most of the Western classical tradition has followed the system expounded by Aristotle in his Categoriae, which shall be our starting point. We will follow the general order of Aristotle's work, inserting later commentaries and modifications as needed. Despite the modern criticism that injecting ontology into logic is circular reasoning, Aristotle's theory of categories rarely came into play in the development of his metaphysics or that of his successors. This may account for why the Categoriae is remarkably light on proof, since it is not an analytical or metaphysical work, but the definition of a formal language, one that will better describe potential reality than an ordinary language. The Categoriae, being a logical work, explores the connection between verbal expressions and possible real objects. Thus the term "category," meaning "predication" in Greek, has both semantic and ontological significance.

Part I: Fundamentals of Ontology

Names and Entities

The term "category" bridges the divide between linguistics and ontology. Categoria in Greek means "predication", a primarily grammatical concept, yet Aristotle is principally concerned not with linguistic objects, but with "things that are", as the late J.L. Ackrill (1921-2007) ably argued. At any rate, our present discussion is concerned with ontological categories, and only incidentally with their linguistic counterparts. "Category" in the ontological sense refers to predications of being, meaning modes of being or different ways a thing might exist. In common English and in mathematical logic, a category is just a class or group of objects, but it would be a mistake to apply this notion to our ontological categories. Ontological categories are a priori concepts, not collections of objects; they refer to possible ways a thing may be. If our categorical system is complete, every real entity may be classified under one of the categories, but the categories serve as a classification system for objects only incidentally; their core concept is the different ways a thing may be said to be. Thus categories are truly ontological, as they analyze being, not directly, but through hypothetical "things that are."

Since we are concerned with "things that are," or entities, we should, following Aristotle, take care to distinguish an entity from its name. A name is a linguistic object or label, but a definition of being is the meaning behind the name, or a statement of what a thing is (i.e., its essence). This concern for distinguishing ontology from linguistics motivates the definitions of terms such as homonymous, synonymous, paronymous, which distinguish linguistic objects from the essences they represent. A word as a linguistic object is just a name or label, and we may often use the same name to mean different things. In grammar, we may speak of words being homonyms or synonyms, but in philosophy our primary object of study is not words, but possible entities. Thus we speak of homonymous or synonymous entities rather than words.

Homonymous entities are those which share a common name (word), yet have different definitions, and therefore different essences, since the definition is merely a statement of a thing's essence. Synonymous entities have a common name and a common definition. The Latin equivalents of homonymous and synonymous are "equivocal" and "univocal," respectively. Paronymous entities are those which have the same root name, but may be constructed differently for grammatical reasons, such as the difference between "brave" and "bravery". Paronymous entities may or may not have the same essence. We can create other distinctions among the relationships between entities and their linguistic representations, but these should suffice for our present discussion.

In this discussion I have assumed that entities, "things that are," have an essence. By this assumption I do not necessarily imply that the essence of a thing is distinct from the thing itself, notwithstanding the grammatical convention of saying things "have" an essence. The essence of a thing is merely that which makes a thing what it is, so when I say two things have the same essence, I mean that their names have the same signification. This will be clarified when we elucidate the distinction between individuals and universals.

The Ontological Square

The two most fundamental ontological distinctions in Aristotelianism are those between individuals and universals, and between subjects and accidents. These two distinctions create four types of being in what is known as the ontological square. Neither of these distinctions is without controversy, so we will attempt to provide some justification for them.

First, it is important to emphasize that we are not concerned with mere linguistic distinctions, such as "subject" and "predicate," or "noun" and "adjective". Such linguistic constructs may be useful in expressing ontological distinctions, but we are concerned with language only as a practical necessity for making ontology intelligible. We should not presume to use grammatical convention as a basis for ontology, but instead may appeal to ontological intuition and argument in order to validate certain linguistic choices.

The ontological square was first constructed by Aristotle by considering two concepts. The first is the capacity of an entity to be "said of" or predicated of another entity. An entity X which can be predicated of another entity Y is called a species or genus of Y. The second concept is being "in a subject," meaning that an entity is somehow contained by another entity (which we call the subject), yet not as a part of the subject, nor can the entity exist apart from the subject.

On its face, the concept of being "said of" something appears to be purely linguistic, yet the "something" in question is an ontological entity, not a linguistic object. When we say that "man" or "animal" can be "said of Socrates," we do not mean "Socrates" as a linguistic object, but the real person that the name 'Socrates' signifies. (For clarity's sake, I will use single quotes to indicate the name of a thing, while double quotes will refer to the entity the name signifies, following the convention used by Ackrill in his commentary on the Categoriae.) We can interpret the statement, "Man can be said of Socrates," in two ways. It could mean merely that the name 'man' is predicable of the real individual Socrates, or it could be mean that the thing which 'man' signifies is directly predicable of the individual Socrates. This is the famous metaphysical problem of universals: whether terms of species and genus ought to be regarded as names only (nominalism), or the realities they signify are somehow directly predicable of individual substances (realism). The latter, realist position admits of various gradations, depending on what sense or degree of reality we are willing to ascribe to species and genera such as "man" and "animal."

Regardless of whether we take a realist or nominalist position on the problem of universals, the distinction between that which can be "said of" some entity and that which cannot be "said of" any entity is not purely linguistic, since we are referring to some entity. In the statement "X is said of Y," both X and Y are real entities in the realist view, but the nominalist regards only Y as a real entity, while X is merely a name. In both cases, Y is a real entity, so even for the nominalist the distinction between individuals and universals is only quasi-linguistic.

If a name is more than an empty utterance, then there must be a reality that the name signifies; even the notion of a mere label suggests some thing that is being labelled. To be sure, there is no reason to assume that something labelled by a universal name like 'man' or 'animal' can exist in the same sense that individuals may exist. Nonetheless, a universal term like 'man' signifies some essence that might be regarded as real at least as a concept. Indeed, this conceptualism may more accurately characterize the position of William of Ockham (c. 1287-1347), who is commonly considered a nominalist. Since we are dealing with entities only from an a priori perspective, a conceptualist view of universals is indistinguishable from the realist view, as we are concerned only with conceptually possible realities, not with making assertions about actual reality.

If the name 'man' is predicable of Socrates, there is a corresponding relationship between the conceptual essence "man" and the individual Socrates. The essence "man," which is signified by the name 'man,' stands in an indirect relationship with the individual Socrates. Since the name 'man' is predicable of Socrates, the essence it signifies somehow pertains to Socrates. The relationship between universal essences and individuals need not be one of predicability, however, as it is by no means clear how an essence can be "said of" anything. The second distinction of the ontological square may provide some insight into the relationship between universal essences and individuals.

The second fundamental concept in Aristotelian ontology is "being in a subject," which is formally defined by three criteria. An entity X is formally "in" some entity Y, which we call the subject, if and only if:

  1. X is in Y in the ordinary sense of "in," being contained by Y;
  2. X is not part of Y; and
  3. X cannot conceivably exist separately from Y.

These three criteria are motived by the intuitive distinction between substance and accident, where an accident is a property or feature of some thing that is, while a substance is something that can be conceived as existing independently of any other entity. We will explore this intuition, and examine whether the definition of "in a subject" provides an adequate basis for distinguishing substance and accident.

The distinction between substance and accident is readily conceived in our understanding of the macroscopic physical world, where we distinguish between a corporeal object and its properties, such as its color, shape, or texture. These properties are real in some sense, since a grey rock is really distinct from a white rock, but their reality is constrained by their subject, since there is no "greyness" except as manifested in a substantial object such as a rock. These properties, by their very mode of being, presume the existence of a subject. The existential dependence of an accident on its subject is not simply a matter of its existence being contingent upon another being, the way a sculpture requires a sculptor, but it is a more intimate dependence. The existence of the color grey can be manifested only in some subject, so its existence somehow participates in the existence of the subject. Such is not the case with simple contingency, as the sculpture can continue to exist long after the sculptor is dead. The Aristotelian definition of "in a subject" is an attempt to express this intimate union between entities and their properties, that is, substance and accidents.

An accident is that which can exist "in a subject," while a substance is that which is not in any subject, in the formal sense defined. The subject of an accident may be another accident, which in turn may ultimately have some substance as its subject. Since existential dependence is part of the definition of being "in a subject," an accident not only can be "in a subject," but it must be in a subject to the extent it is to exist at all.

Our use of concrete examples serves only to illustrate concepts, not to commit us to particular metaphysical applications. Indeed, Plato would not agree that grey exists in a rock, or that roundness exists in a ball, but he might even invert the order of dependence. Plato held that individual corporeal entities participated in ideal forms. A round ball participates in the idea of "roundness," so that roundness is what has primary existence, while the round ball exists only insofar as it participates in this and other ideas. In Aristotelian language, we might be inclined to call "roundness" an accident or property of the ball, which has no existence except as manifested in existent objects such as the ball.

Plato and Aristotle's contrary views of "what exists in what" (concrete objects in ideas, or properties in concrete objects) derive from differing notions of existence. For Plato, the primitive notion of existence is to be found in the eternal, immutable Ideas or Forms apprehended by the intellect, while for Aristotle, primary being belongs to the concrete reality "out there" that the mind encounters through the senses. Aristotle is the empiricist, so he is generally favored by modern thinkers on this point, but a complete ontology must also take into account the reality of Ideas that Plato considered. This may be accomplished by allowing the possibility of a strong realist interpretation of species and genera. Not to prejudge the question of how real these universal ideas are, we simply note that universals have a different sort of being than individuals, which is the essence of an ontological distinction. The two distinctions of the ontological square, taken together, can allow for both Aristotelian empiricism and Platonist idealism. If universals are real in the Platonic sense, then we may regard individual substances as dependent upon universal substances that are characterized by universal accidents or properties. At the other extreme, if universals are denied reality except as concepts, we may regard individual accidents as dependent on individual substances, with no appeal to universals.

Many modern philosophers presume to render the distinction between substance and accident meaningless by denying the concept of substance. They contend that a thing is fully determined by its physical properties (e.g., momentum, charge, mass, energy) manifested in varying degrees, so it is superfluous to hypothesize the existence of a thing that "possesses" these properties. This position, sometimes called "trope theory," appears to find some justification in quantum mechanics, though it may also be espoused on purely abstract considerations, and applied to disparate areas such as the philosophy of mind, where the "thinker" is said to consist of nothing more than a stream of thoughts. Our distinction between substance and accident, or subject and object more generally, might be a linguistic or anthropological artifact without basis in objective reality.

Trope theory seems to be a logical consequence of a philosophy that considers empiricism to be the only valid epistemology, since empiricism can only give us direct knowledge of properties, not of substances. All scientific data consists of the quantification or characterization of properties, and scientific theory consists of quantitative and qualitative correlations among these properties. The idea that there are real substances "behind" or "beneath" these properties appears to be a metaphysical assumption that cannot be justified empirically.

Modern empiricism is often invoked against the reality of essences of species and genera, so that "essentialism" has become a byword for ascribing reality to arbitrary constructs. Unfortunately, the accusation of essentialism has sometimes obscured the varying degreess of realism expressed by Aristotle and his medieval successors. Just as the crude term "dualism" in the philosophy of mind is sometimes used to make all non-materialists out to be Cartesians, so does "essentialism" tend to lump the Aristotelians with the Platonists.

In fact, essentialism and empiricism need not be mutually exclusive. As Aristotle argued in the Physics, all science must have universals rather than particulars as its object, for if this were not the case, there would be no basis for assuming the universal applicability of its results. For example, if we determine that a pot of water boils at 100 degrees centigrade, there would be no reason to expect another pot of water to boil at the same temperature, unless we were convinced that both pots contained a like substance. Not only is empiricism not at odds with "essentialism", but the entire body of scientific knowledge of physical principles or laws is "essentialist." Without some degree of essentialism, empirical science would just be a vast collection of data.

We do not need to decide metaphysical questions at this stage of our philosophical endeavor, though I have outlined some paths of argument. It suffices to note that the ontological square is sufficiently robust to embrace all the possibilities considered, though if the trope theorists or nominalists are correct, we would end up with at least one empty class (substances or universals, respectively).

Applying both ontological distinctions, we arrive at a maximum of four classes of things defined by the ontological square:

Said of a subjectNot said of a subject
Not in a subjectUniversal Substances
(species, genera)
Individual Substances
In a subjectUniversal Accidents
Individual Accidents

Having considered each of the two distinctions independently, let us now consider their coupled application to each of these four classes of entities. First, we should examine the validity of coupling these distinctions, especially since one of them ("said of" / "not said of") might be quasi-linguistic (on the nominalist assumption), while the other ("in a subject" / "not in a subject") is purely ontological. For this coupling to be valid and all of the four classes to be ontological, we must interpret the "said of" distinction as applying to the names of entities in the classes, not entities in the classes themselves. Thus, when we say, "Species X is said of subject Y," we really mean, "The name of species X may be said of subject Y." Naturally, we assume the name is being used to signify the species, and not as a homonym.

The first class of entities are those whose names (used synonymously) can be said of a subject, but are not in any subject, in the formal sense. These are the universal substances, or species and genera. 'Man' or 'animal' can be said of Socrates, but neither the species "man" nor the genus "animal" can be said to be in Socrates as accidents, since these universals can conceivably exist apart from Socrates, for example in Callias or in a dog. We do not consider here whether a species or genus could exist independently of any individual, but only affirm there is no determinate individual upon which the existence of a universal depends.

Since species and genera are not in a subject, they may be regarded as a kind of substance. Nonetheless, Aristotle and most of his successors had misgivings about the reality of universal substance, and considered that its existence might be meaningless or void unless it is potentially or actually manifested in an individual subject, even though a universal substance depends on no determinate individual for its existence. For this reason, species and genera were called "secondary substances," as though their existence were somehow derivative of individual or "primary" substances.

We should not let the nomenclature of "primary" and "secondary" substance prejudge the question of the ontological dependence of universals on individual substances. The ontological square does not decide this question of ontological dependence, but only affirms a distinction between the two classes, defined only by the fact that the names of universals may be predicated of individuals, but not vice versa. This distinction could just as easily accommodate the view of Plato, who made individuals ontologically dependent on universals through his doctrine of "participation."

The second class in the ontological square is of entities that are not in a subject, nor can their names be said of a subject. These are individual substances or objects, which Aristotle and most empiricists hold to be the primary entities of our world, except for trope theorists who deny that it is necessary for these to exist. Abstracting from the question of whether individual substances actually exist, we can still describe this conceivable mode of existence, which is nothing other than our intuitive notion of "stuff," namely a concrete thing that exists independently of our subjective experience. For example, though we may perceive the moon as white or yellow, bright or dark, depending on our subjective circumstances, we hold as certain that the moon is a real object that exists independently of our changeable perceptions. As Einstein remarked, the moon exists even when we have our backs turned to it.

Nonetheless, trope theorists perceive a difficulty with the notion of substance, since we can empirically interact with a thing only through its properties, by measuring its brightness, mass, or position, so it is not clear what would be left if we were to strip away all these properties. The notion that there is a concrete existent thing somehow "underlying" or "standing beneath" (sub stare) appearances, properties, or phenomena seems problematic if we try to describe this "substance" abstracted from its properties.

Substance is practically necessary if anything concrete exists. On the trope theorists' assumption, that only properties exist, there is no solid basis for the reality of anything. Perceptibles such as color and brightness, and even seemingly concrete properties such as mass and energy, vary according to the circumstances of the observer. If these are all that exist, we could hardly account for the ability of multiple observers to have perceptions of a common object that vary according to intelligible laws, as for example, when viewing different phases of the moon or star constellations from different latitudes. The fact that the variability of perception is constrained by certain rules suggests that there is an underlying reality which different observers perceive differently. This underlying reality might be the very "laws of physics" themselves rather than individual substances. However, these "laws of physics" are relational, and unintelligible except on the assumption of things being related.

As an example, a person may observe a train moving at 100 km/hr, while another person aboard the train perceives no such motion. This difference in perception may be explained by inertial mechanics as expressed by a Galilean transformation, but this computation presumes that both observers are observing the same train. Without this assumption, there is no basis for applying the Galilean transformation. Relational laws cannot substitute for the grounding of reality that substance provides. The justification of individual substance is best left for a detailed metaphysical analysis or a philosophy of physics, but hopefully this argument outline will persuade the reader that we are not wasting our time by defining a class of entities called individual substances.

The third class of entities consists of those things that are "in a subject" and have names that can be "said of" some other entity. These are universal accidents or properties, abstracted from their individual manifestations in particular substances.

Plato regarded all universals as primary realities, including these accidental universals. For example, the property "round," which may be abstracted from its manifestation in various objects, may be considered as the Idea of "roundness" in which all round objects participate in order to be. Aristotle, in contrast, saw universal accidents as dependent upon individual substances for their existence, so that in the rest of the Categoriae he seldom bothers to repeat the distinction between universal and individuated accidents, treating both of these as equally dependent on individual substances.

This disparity of opinion arises from different approaches to conceiving of universal accidents. Plato simply abstracts universals from individuated accidents, and deduces that, since individuated accidents cannot exist without the universal, their existence is derived from the universal. Aristotle, on the other hand, derives universal accidents directly from the double distinction of the ontological square, which explicitly recognizes that universal accidents, though their names may be "said of" individuals, they are nonetheless "in a subject," namely secondary substances, which in turn are dependent on primary substances. The dependence of secondary substances on primary substances, we have noted, is not a necessary consequence of the ontological square, so we cannot decide the question of ontological dependence between individual substances and universal accidents.

Universal accidents are inconceivable without at least a generic subject, though there need not be an actual individual subject. Although the individual accident "Callias' generosity" cannot exist apart from the individual substance Callias, "generosity" in the universal sense is not dependent on Callias, since it can be manifested in other individuals. Nonetheless, the very notion of "generosity" implies some generic subject, though it need not imply an actually existing individual subject who is generous.

The entity of which a universal accident's name may be predicated is an individuated accident. Thus 'white' may be predicated of an individual manifestation of color in a particular object, as the definition of the universal "white" certainly applies to a particular instance of "white." The definition of "white" would not apply to a white substance, though we may say, "X is white," meaning white inheres in X.

Individuated accidents, the fourth class of entities, are "in a subject," but not "said of" any entity. Only the most determined solipsist can deny the existence of individuated accidents, since they are the stuff of all experience. Less obvious, however, is the way in which individual accidents may be said to inhere in individual substances, so we should examine more closely the formal definition of "in a subject."

An individual accident is "in a subject," which could be a substance or another accident. When the subject is another accident, as in "dark blue," where the subject of "dark" is "blue," the first accident modifies or qualifies its subject. It is contained by the subject, not as a part, and cannot exist separately from the subject. Similarly, when the subject is a substance, the accident is contained the substance, not as part, and cannot exist separately from the substance. It is one thing for an accident to be bound up intimately with another accident, effectively constituting a modified accident, but it is less clear how such a close union might exist between an accident and a substance. The accident is contained by the substance in a way that the accident may be considered proper to the substance, so we can say, "X is white," as if there were some sort of existential union.

This union is not denied when we say that the accident is not a part of the substance, but we merely exclude an extensive part-to-whole relationship. If we split a substance along some extensive dimension, such as space, each of its parts would be a substance, though reduced in a particular dimension. Division of a substance along an extensive dimension cannot separate a substance from its accident, since if that were the case, the accident would be extensively outside the substance, not contained by the substance. The seemingly arbitrary requirement in the definition of "in a subject" that an accident is in a subject not as a part merely clarifies that an accident must be intrinsic to the substance, and not something extensively separable. The formal notion of "in a subject" contains a distinction between intensive and extensive participation in a substance. This is important, because at a very basic stage of the development of our thought, there appears to be something special about extensive properties such as spatial volume. This will be fleshed out only much later, when we distinguish intensive and extensive magnitudes.

The last part of the definition of "in a subject" is that an accident cannot exist separately from its subject. This third requirement does not inevitably follow from the first two parts of the definition. An entity might be contained by a subject, not as a part, yet it might still be able to exist separately from the subject. An example of this scenario would be two intermingled coextensive substances, such as dyed water or ignited iron. Although we could say in ordinary speech that the dye is in the water or that fire is in the iron, without either substance being part of the other, it is nonetheless certain that both substances in each union fully retain their essences, and can exist independently of the other.

Since an accident is existentially dependent on its subject, the relationship between an accident and its subject must be more intimate than that of two intermingled substances. It is not enough to be coextensive, united in extension and in physical union, but there must be a union at some existential level. This existential unity is reflected in our intuition of "property," which entails that the substance manifesting the property may claim it as part of its own being, as "proper" to itself. When we say "the white horse," we attribute "white" to the horse itself, a psychological fact expressed in the grammatical expression, "The horse is white." Of course, we cannot prove ontology from grammar or psychology, but we can see that our common sense notion of a "property" is consistent with the formal definition of an entity "in a subject," and that all three parts of the definition are necessary for this congruence.

The four classes of entities in the ontological square are conceptual a priori possibilities that may or may not correspond to metaphysical reality. The experience of phenomena practically compels us to accept the reality of accidents, but substance, being much more elusive, has been denied by many modern thinkers. Still, it does us no harm to include all four classes, for if it turns out, for example, that universal substances do not exist, we will simply have an empty class, and our ontology is more robust than reality requires. On the other hand, those who dismiss parts of the ontological square (such as substances or universals) are left with an impoverished conceptual vocabulary, ill-equipped to account for the existential relationships among all four classes of entities, which are necessary to make reality intelligible.

Relationships among Ontological Classes

The four ontological classes, being defined by the "said of" and "in a subject" relational distinctions, necessarily stand in some existential relationship to one another. The traditional nomenclature for these relationships is indicated in the diagram below. In particular, we note that the basic logical concept of "exemplification" requires both ontological distinctions, and implicitly requires all four classes of the ontological square at least as a priori concepts. An exploration of these existential relationships, grounded in the two fundamental ontological distinctions, should make the conceptual necessity of the ontological square more evident.

Relationships within Ontological Square

The relationship between individuals and universals is generally called "instantiation," whereby we say that an individual "instantiates" or "is an instance of" some universal. For example, the whiteness of a particular sheet of paper may be said to "instantiate" or "be an instance of" the universal "white." This does not imply that the whiteness of the sheet of paper is actually doing anything, but we are defining an existential relationship between the individuated whiteness and "white" considered in abstraction from any particular manifestation. The whiteness of a particular piece of paper is distinct from universal "whiteness" only in that it is confined to a particular object, whereas the universal has no such limitation. It is this confinement or projection of the universal's essence in a particular object that constitutes instantiation.

For substances, universals and individuals are less intimately related, since the individual substance may be more than an instance of a species, whereas an individual accident is nothing else but an instance of its universal. The reason for this is that an individual substance may have accidental characteristics beyond those which define its species. For example, an individual horse instantiates the species "horse," yet there is more to this particular horse than having the features common to all horses. It may have its own peculiar appearance, gait, or habits that are not contained in the essence defined by the species "horse." Nonetheless, instantiation means the same thing for substances as it does for accidents, being the confinement or projection of a universal essence in a particular object. The difference is that for substances, the instantiation relation is not the only thing distinguishing universals and particulars, so we cannot determine everything about a particular object simply by knowing its universal and its instantiation.

Also, for substances, the universal essence is directly predicable of the object that is the locus of instantiation, whereas for accidents this is not the case. The essence of "horse" is directly predicable of a particular horse, so that anything that may be said of the universal may also be said of the individual. By contrast, the essence of "white" is not directly predicable of a white sheet of paper, for the paper is not a color. Only the whiteness of the paper has the essence of "white;" the paper only provides a locus for the instantiation of white, made necessary by the fact that accidents depend on substances for existence.

The relationship between individual accidents and substances is usually called "inherence," so an individual accident is said to "inhere in" an individual substance. Inherence, being applicable only to individuals, is a specific type of being "in a subject." Being "in a subject" in the formal sense, we have noted, means being intrinsic to the subject, not as an extensive part, and being incapable of existing independently of the subject. When the subject is an individual substance, any of its intrinsic accidents (whether essential or non-essential to its species), may be said to inhere in the substance. The accidents that inhere in the substance modify the substance's existence. Despite this way of speaking, the accidents are not necessarily doing anything by "inhering" in the substance, but we are simply denoting an existential relationship between an individual substance and its individuated properties or accidents. The intrinsic accidents of an individual substance or object are inherent to the substance, and are extensively inseparable from it.

Not all individual accidents inhere in an individual substance. Extrinsic accidents, such as quantity and number, regard substance as composed of continuous or discrete parts or instantiations. When speaking of a determinate three liters of water, the accident "three liters" tells us how much water there is, or the extent to which the universal "water" is instantiated. "Three liters" inheres in individuated water only insofar as that water is considered as being composed of a continuous quantity of instantiation. With numbers of discrete objects, the lack of individual inherence is more pronounced: the "four" in "four horses" does not inhere in any individual horse, but only in the collection of individuals treated as a set. Relational accidents (to be discussed in Part IV) certainly need not inhere in one substance to the exclusion of others, as they can explicitly take multiple substances as their subject. Although all individuated accidents ultimately depend on individuated substance as their subject, extrinsic and relational accidents need not inhere in a single individual substance.

The intrinsic accidents that may inhere in an individual substance depend on that substance for existence, yet they also modify the existential state of the substance. The closeness of this symbiotic relationship between objects (individual substances) and their tropes (individual accidents) has led some philosophers to draw a metaphysical identity between the two classes, or to dispense with substance altogether as superfluous. However, the collapse of this distinction would eliminate the important concept of "exemplification," and render any ensuing logic nearly impotent.

The conceptual necessity of "exemplification," the relationship between universal accidents (properties) and individual substances (objects) can be appreciated by considering some examples. When an individuated accident, such as the whiteness of a horse, inheres in a substance (the horse), that substance may be said to "exemplify" the universal accident. The concept of exemplification allows us to say "This horse is (exemplifies) white," and "This sheet of paper is (exemplifies) white," where "white" in both statements has the exact same meaning. Without exemplification, there would be no basis for attributing the same property to different objects. Yet the identification of common properties properties instantiated in disparate substances is the very basis of any scientific or intellectual enterprise. Unless all metaphysical and scientific theory is doomed to be mere linguistic exercise, the concept of exemplification must correspond to potential reality. The intelligibility of reality depends on exemplification.

Exemplification, in turn, requires the ontological distinctions between substance and accident and between universals and individuals. If there were no such thing as substance, we could only say, "This instance (or manifestation) of white is (universal) white," not, "This object is white." If there were no universals, we could not even make the first statement. More importantly, without universals, we could never say that "white" in both statements, "This horse is white" and "This sheet of paper is white," means the same thing, making virtually all intellectual or scientific analysis impossible. Exemplification is a powerful concept that allows us to analyze existent objects that differ in some ways, yet are the same in other respects. If we acknowledge that such things may exist, we are practically required to admit that there can be such a thing as exemplification. This, in turn requires us to admit the necessity of both distinctions in the ontological square.

Having established the existential relations among three of the ontological classes, we are left only with universal substances. The other three classes, as we have just shown, are practically indispensible to any theory of reality, but this fourth class, consisting of genera and species, has been challenged by many eminent and capable philosophers throughout history on a priori and a posteriori grounds. We should examine the relationships between universal substances and other entities with special care, to see whether universal substances add anything to our ontology.

As mentioned previously, individual substances are not merely instantiations of universals, as is the case with accidents. An individuated accident is nothing more than an instance of an essentially identical universal. There is no difference between the whiteness of a particular sheet of paper and generic "whiteness," save that the first is an individual instance of the universal. With substance, however, individuals and universals are distinguished by more than instantiation. In the expression, "Callias is a man," the individual Callias may be considered an instance of the species "man," yet it would be rash to assume that the only difference between "Callias" and "man" is that one is an instance of the other. Callias may have qualities other than those that define "man," such as curly hair or knowledge of arithmetic. Callias and other men, as substances, are not mere drops of some amorphous substance "manhood."

Individual men belong to the species "man" by virtue of having the same properties that define "man." In other words, they exemplify the accidents that differentiate the species "man" from other species. These accidents that define a species or genus are called the differentiae of the species or genus. A universal substance is defined by a set of universal accidents (differentiae), and any individual substance exemplifying these accidents instantiates the universal substance (species or genus).

Differentiae might not be the only means of defining a universal substance, as we could conceivably take an arbitrary collection of individuals and define them to be a species or genus. For example, Socrates, Callias, Callias' horse, and a particular rock might collectively be considered to define an arbitrary species X. If, by definition, these are the only individuals of species X, how should we address the question, "What is X?" X is not simply the set of Socrates, Callias, Callias' horse, and the rock, as this would just be a collection of individual substances. X, by contrast, is a certain kind (species) of being somehow defined by this arbitrary collection of individuals. To answer "What is X?" as a definition of the universal, we would need to find what all the individuals characterized by X have in common, and what distinguishes them from individuals that are excluded from X. In this case, the individuals have been selected so that they have no unique set of common properties, save perhaps "being a member of the set {Socrates, Callias, Callias' horse, this rock}." This purely formalistic defining pseudo-property does not really do anything to help describe any of the individuals in the set. For instance, knowing Socrates is a member of this set does not tell me anything about what Socrates is like, since being a member of the set does not entail the possession of any determinate ontological property.

When several individual substances exemplify some shared ontological properties that are not merely formalistic, we can use these shared properties or accidents to define a species and differentiate it from other species. The existence of species as a kind of being can occur if and only if exemplification is real. We do not thereby conclude that universal substances would have the same sort of claim to reality as universal accidents. Still, these universal substances do tell us something about the individuals that instantiate them. If I know that "Socrates is a man," then I know that Socrates is a rational being. This tells me something about what Socrates is like, as opposed to a mere formalistic knowledge that he is the member of the set {Socrates, Callias, Napoleon, Shakespeare, etc.}. The species "man" is a kind of being, and Socrates instantiates that kind, which tells us something of what it is like to be Socrates.

Set theory is an inadequate model for universals, since membership in a set alone does not show instantiation of something universal, but gives a mere list of individuals. A universal is something that transcends the individuals, each of which is an instance of the universal. Since we are concerned with instantiating something ontological, the arbitrary formalism of set theory will not suffice, as that permits pseudo-species such as X discussed above.

We should emphasize that an individual substance instantiating a universal substance is not the same as an element being a member of a set. "Being a member of set Y" is a purely formal property with no ontological content. The set itself is defined by the elements it contains. A universal substance, by contrast, is defined by universal accidents, not by individual objects instantiating it. In fact, the individuals of a species may even possess accidental qualities that do not pertain to the species. Further, individual substances do not stand in a part-to-whole relationship with the universal; each individual fully instantiates the universal, and nothing is taken away from the universal should an individual be removed. The element A in the set Y = {A, B, C} is only a part of the set Y, as B and C are needed to complete the set. However, when we say Callias is a man, it is not necessary for Robin Hood also to exist in order for the species "man" to exist and be fully defined. The species has a degree of existential independence from any of its particular individuals, which a set thoroughly lacks in relation to its elements.

It is sometimes convenient to use set theoretic notation to describe the instantiation of a species, as we can define "the set of all individuals exemplifying property Z." This does not properly define the species as a univeral, though it may define the set of all possible members of a species. The universal is hinted at by reference to a common property and by the reference to "all," yet this "all" confines us to some determinate individuals, whereas ontological universals have no such constraint. For the reasons outlined above, it would be a serious error to regard set theory as describing the relationship between universals and particulars.

Many will argue that most, if not all, species and genera are arbitrary formalisms that are vulnerable to the same critique I have applied to membership in an arbitrary set. Without pretending to solve the problem of universals so early in our inquiries, we can observe that universal substances defined by accidents exemplified by individual substances are on firmer ground than the formal set discussed above. If we acknowledge that universal accidents are real, being real properties of individual substances, then it follows that the universal substances defined by such accidents describe real aspects of the being of individual substances. This is more than can be said of the pseudo-species X discussed earlier.

Although universal substances might describe real existential aspects of individual substances, they would not seem to merit a claim to the same level of objective reality that individuated objects can possess. Even after restricting ourselves to defining species and genera according to ontological properties, we are still left with a broad degree of arbitrary discretion in how we classify objects. For example, we could define a species of animal that includes any individual between three and four feet in length, a construction that surely tells us more about our mental artifice than it does about the reality of things. The statement that a particular snake is a member of this species does tell us something about the snake, namely its length, but we should not think that the species of "three-to-four-foot-long animals" is itself something that exists in anything but formal reality, as it is hardly a less arbitrary classification than our pseudo-species X.

Universal substances make us acutely aware of the problem of distinguishing ontology from linguistics. How we label or name things involves assumptions about which things ought to be distinguished from each other. What range of hues should we include as "green"? If we are not concerned with the objective nature of things at this point, we do not yet need to answer questions about which accidents are "true" accidents and which species are "true" species. Yet, without this study of nature, how can we know that our linguistically constructed accidents and species might correspond in any way to extra-mental reality?

If anything is certain, we can at least know that there exist perceptible phenomena, whether sensibly perceptible or intellectually apprehensible. Without worrying about whether there are primary substances behind these phenomena, we can distinguish these perceptibles as finely or as crudely as we like, and in any case we will be describing a real difference, namely a difference in the manifestation of certain properties. Even if the green and blue balloons I see are hallucinations, I can nonetheless know that green and blue are different from each other. Universal accidents are knowable realities even if there are no individuals that exemplify them in extra-mental reality.

Universal substances that are defined by knowable universal accidents also have some claim to extra-mental reality, if we admit there are such things as individual substances. The existence of the species "green leaves" certainly follows if we admit that individual leaves actually exist which exemplify the universal accident "green." We reserve judgment, for the moment, as to whether the species would have any claim to existence if there were no individual leaves that are actually green. Assuming such individuals may exist, we are not defining something unreal when we define the species, as there certainly are individuals that instantiate the species. We have the same range of liberty in defining species that we do in defining universal accidents, limited only by our range of perception and apprehension.

The four classes of being defined by the ontological square span a range of a priori possible entities. It remains for a theory of metaphysics to determine which of these classes of entities really exist. Meanwhile, we can continue to analyze these ontological fundamentals on an a priori level and attempt to determine which, if any, are ontologically prior or more fundamental. By "ontologically prior," I mean capable of existing independently (at least conceptually) of entities in other classes.

Species and Genera

Of our four classes, universal substances have come under the heaviest scrutiny on both the metaphysical and conceptual levels. In the late Middle Ages, William of Ockham accepted universals as concepts for understanding reality, yet denied they had any claim to reality. Others took a more extreme nominalism, arguing that universals exist in name only. Such dim views of universals have become more widespread in the modern era, due to an unwarranted faith in empiricism as the only valid epistemology.

Ironically, the empirical sciences ultimately depend on a mathematical theory of physics that must assume the reality of universal substances in order to have any force. For example, if it is a law of nature that a body in motion tends to stay in motion, this holds for any body, not just an individual that we actually observe. The idea that certain laws of nature are universal - that is, they hold even when abstracted from individual instances - implies that the terms of these relations, expressed as mathematical formulae, are universals, not individuals.

Aristotle long ago realized that the study of nature necessarily deals with universals, since we are trying to find principles that apply universally to substances of a species. If science only dealt with individuals, it would just be a collection of facts, and the fact that one moving body remains in motion would have no bearing on whether another should obey the same principle. Of course, we could argue that similar substances behave similarly only because they have the same properties, so natural laws might only require universal accidents, not universal substances. Regardless of their metaphysical status, universal substances can be useful analytic tools, classifying individual substances that exemplify shared universal properties.

Universal substances, considered conceptually, can be related to one another in hierarchical schemes such as those used in taxonomy. We will explore this structure at some length, beginning with the primitive concepts of species and genera, the two fundamental types of universal substance. The distinction between species and genus is not in substantial essence (for the same thing could be both a species and a genus), but in predicative relationships. Loosely speaking, a species is only predicable of individuals, but a genus is also predicable of other universals, such as species or other genera. By "predicable," we mean the relationship whereby the name of a universal may be "said of" an individual. Strictly speaking, only names, not entities, are predicable of entities, but there is an ontological relationship underlying predicability, since the essence signified by the name is properly attributable to the entity of which the name may be predicated. Thus the essence of the species "man" is proper to all individuals of whom 'man' is predicable. For this reason, we may colloquially speak of species being predicable of individuals, and of genera being predicable of species.

The predicative relationships among individuals, species and genera are described in this Aristotelian thesis:

If one thing (B) is predicated of ("said of") another (A) as of a subject (A), then all things (C) said of what is predicated (B) will be said of the subject (A) also.

In other words, if B is predicated of A and C is predicated of B, then C is predicated of A. If A is in an individual, B is a species, and C is a genus, we could use this thesis to show that the genus of a species is necessarily predicable of individuals in that species. For example, if Socrates is a man, and man is a species of animal, then it follows that Socrates is an animal. As persuasive as this example may seem, the thesis is problematic, as J.L. Ackrill has noted, for it asserts a transitive principle of predication without justification.

We should restate the thesis, being more explicit about the ontological content of predicability, to more accurately gauge its validity.

If the essence of a species B is instantiated in an individual A, and there is a genus C whose essence is attributable to species B, then it follows that the essence of genus C is also instantiated by individual A.

When we say, "Socrates is a man," we mean that Socrates instantiates the essence of the species "man." When we say "Man is an animal," we mean that the essence of the genus "animal" is attributable to the species "man." Since the species "man" is a universal, it consists of nothing but its essence, as there is no concrete individuation to admit of non-essential inherent properties. Individuals such as Socrates may possess non-essential properties such as curly hair, but these cannot pertain to the species "man," or we would really be speaking of some other species. Seeing that a species is nothing but its essence, the fact that the essence of some genus is attributable to the species means that the essence of the genus is in the essence of the species. Thus the essence "man" contains the essence of "animal." It follows then, that any individual instantiating the species also instantiates its genus.

The predicative relationships among individual, species and genus may also be approached by examining differentiating properties. An individual in a species exemplifies all the properties that are the differentiae defining the species. To be a man, an individual must exemplify all the properties that define "man." A species, in turn, must have in its essence all the differentiae that define its genus, or it would fail to be in the genus. In order for "man" to be a species of animal, it must have in its essence all the defining characteristics of animals. Thus the differentiae of the genus are a subset of the differentiae of the species. If follows, therefore, that an individual exemplifying all the differentiae of the species will also exemplify all the differentiae of the genus. We may state this thesis more generally:

Suppose an individual A exemplifies all the differentiae of species B.

Suppose there is a genus C, all of whose differentiae are also differentiae of species B.

Then individual A must exemplify all the properties that differentiate genus C.

The thesis is proven by logical necessity. If an individual exemplifies all the differentiae of a species, it must also exemplify all the differentiae of the genus. If exemplifying the differentiae of a universal substance suffices to instantiate the substance, then an individual instantiating a species necessarily instantiates the genus.

For the scenario in the above thesis to hold, it suffices for a genus to have the same differentiae as the species or some subset of these. Thus we could say a species is its own genus. This is contrary to common usage, and does not help us apprehend a non-trivial structure to secondary substances, so we define a genus more restrictively, requiring that a species should possess at least one differentia in addition to those of its genus. In other words, the differentiae of the genus are a proper subset of the differentiae of the species.

There has been a dangerous tendency, beginning with Aristotle, to regard the relationship between a species and its genus as fully analogous to the relationship between an individual and its species. Ancient Greek did not possess an indefinite article, so it was perhaps less obvious to Aristotle that the term 'man' performs different functions in the sentences, "Socrates is man," and, "Man is animal." In the first sentence, we would now say, "Socrates is a man," making it clear that Socrates is not universal man. In the second sentence, by contrast, we are indeed speaking of universal "man," so there can be no question of "man" instantiating a universal or exemplifying properties. Man, being a universal, cannot instantiate or exemplify anything, yet the relationship between man and animal does seem at least loosely analogous to that between Socrates and man, as Socrates is one of many individual men, and "man" is one of many species of animal.

We can expound the proper relationship between species and genus by considering the relationships among the classes of the ontological square. A species does not exemplify properties, but is differentiated by them. A genus G of species S is a universal that is differentiated by some of the same properties as species S, but by no others. Conversely, species S is differentiated by all the differentiae of G, and possibly others as well. These additional differentiae are what make the species more restricted in scope, thus seeming to be a subset or element of the genus. These set theoretic concepts are dangerously misleading, since a genus does not consist of species in the way a set consists of elements, and a species cannot be a subset of anything since it does not consist of individuals the way a set consists of elements. Still, there appears to be a rough analogy between the concepts.

In order to grasp the hierarchical aspect of the relationship between species and genus, we must define genus in the more restrictive sense of having fewer differentiae than its species, as noted earlier. From this definition, we can develop a hierarchy of secondary substances.

Let us consider the universal substance "vertebrate," which is instantiated by any individual exemplifying the property of having a spinal column. There are many species, such as "mammal," "reptile," "primate," "man" (obviously, we are not restricted to the taxonomic definition of species), that are instantiated by any individual exemplifying the property of having a spinal column, as well as additional differentiae proper to each species. Mammals have mammary glands in females, reptiles have embryos surrounded by amniotic fluid, primates have opposable thumbs, and humans have the capacity to reason. The biological exactitude of these definitions is irrelevant here, for we can define species freely, without reference to phylogeny. It suffices that a defined species is differentiated by ontological properties that can at least potentially be exemplified by real individuals.

Any individual instantiating one of the named species must also be a vertebrate, since a spinal column is one of the defining characteristics of each species. The possession of additional characteristics does not abolish the presence of a spinal column, which is the only requirement to being a vertebrate. These additional characteristics, however, do determine whether an individual belongs to one or another of the named species. All mammals must be vertebrates, but a vertebrate need not be a mammal, so in some sense, being a vertebrate is more primitive or fundamental than being a mammal. We might express this hierarchy in a set theoretic form, showing that the set of all individuals that instantiate "mammal" is a subset of the set of individuals that are vertebrates. If we are dealing with actually existing individuals only, it is possible that all vertebrates that actually exist happen to be mammals, so the set of mammals would be a subset of the vertebrates only in an improper sense, since they are coextensive. If we are dealing with all potential or conceivable individuals, then the set of potential mammals would be a proper subset of the set of potential vertebrates. In either case, the set belonging to a species must be a subset of the set belonging to the genus, while there is no converse necessity, showing an asymmetry in the relationship between species and genus, which suffices to establish a sort of hierarchical ordering.

It is clear that we could create genera of genera, with progressively fewer differentiae, so that the individuals of each subgenus must be members of the higher genera as well. This would establish a hierarchy of universals, similar to what exists in biological taxonomy. Nonetheless, we should not lose sight of the fact that there are many ways to classify individuals, so we could change the order in which we add differentiae, creating different hierarchies. For example, we could divide animals first into vertebrates and invertebrates, and then subdivide each group into tetrapods and non-tetrapods, or we could first divide animals according to their number of limbs, and then subdivide each group according to the presence or absence of a spinal column. The fact that we choose the vertebra as a more fundamental distinction reflects a belief in an objective order, such as phylogeny, rather than a merely formal order that is valid regardless of which hierarchy we choose.

Just as there remains some doubt as to the ontological status of universal substances, so there must necessarily be some ambiguity regarding the reality of their hierarchical relations. As we are logically free to select differentiae in any sequence, constructing many different hierarchies, the existence of such hierarchy cannot prove that one differentia is truly more fundamental than another, and that therefore the corresponding hierarchy of genera would represent an objective order. Nonetheless, to the extent that any species or genus is real, we may acknowledge that all of the taxonomic hierarchies we can construct are real, though there is yet no objective basis for preferring one over another. Such a basis, if it exists, ought to be found in natural philosophy and metaphysics, which seek hierarchies of causation and being.

Non-Subordinate Genera

Restricting our attention to the merely formal notion of taxonomic hierarchies, we can nonetheless identify genera that are not subordinate to one another in the manner discussed above. Aristotle makes an inference about the differentiae subdividing such genus that may help inform the development of a theory of categories. Differentiae subdividing genera which are (1) different and (2) not subordinate to one another are different in kind. Note that here we are discussing the differentiae divisivae that subdivide genera, whereas all prior discussion referred to the differentiae constitutivae that define genera, distinguishing them from other genera. We can now state Aristotle's assertion more formally:

Suppose there are two genera, A and B, such that A is different from B. In other words, either A or B has a differentiating characteristic (differentia constitutiva) that is not shared by the other genus.

Further suppose that neither genus is subordinate to the other, so A is not a sub-genus of B, nor is B a sub-genus of A.

It follows that the differentiae divisivae of A and B are different in kind. That is, the differentiae that subdivide A into subgenera and species cannot be applied to B, and vice versa.

Before examining this assertion, we should recall that subordination of genera entails a kind of predicability, where the higher genus (or rather, its name) is predicated of the lower. Since every mammal necessarily has the properties of the genus "animal," it follows that 'animal' is predicated of the genus (or species) "mammal." "Mammal" is subordinate to the genus "animal" because it has all the differentiae of "animal." For a genus to be non-subordinate to another, it must lack at least one of the differentiae constitutivae of the other. If neither genus A nor genus B is subordinate to the other, then both must lack at least one of each other's differentiae. This does not mean they may not have some things in common. For example, a lizard is not a type of rock, nor is a rock a type of lizard, though they do have some things in common, such as corporeality and solidity. A lizard lacks the mineral density essential to a rock, and a rock lacks the biological attributes essential to a lizard, so neither may be said to be a subgenus of the other.

Mutual non-subordination of genera is interesting because it admits of an objectivity that otherwise may be lacking in the construction of taxonomic hierarchies. When dealing with subordinate genera that share more than one defining characteristic or differentia, we found that we could structure our hierarchies arbitrarily, according to which differentia we regarded as more fundamental. When there is no relationship of subordination between two genera, on the other hand, there are no choices we can make that would make them subordinate. This suggests a level of reality that is more solid than the arbitrary taxonomic hierarchies we have discussed. No taxonomic choice could make a lizard become a type of rock, nor a rock a type of lizard, since each lacks at least one differentia constitutiva of the other. The non-subordination of these genera is non-negotiable, and will exist independently of our choice of taxonomic strategy.

It remains to be seen if the differentiae subdividing two distinct, non-subordinate genera must be different in kind. Suppose this were not the case, and that a differentia subdividing genus A was the same in kind as a differentia that subdivides genus B. This differentia D is a property that subdivides both genera, so A has the sub-genera or species A1 and A2, defined by the presence or absence of D, and likewise B may be subdivided into B1 and B2. There is no logical contradiction here, as we can see from the simple example that both lizards and rocks, though non-subordinate genera, can be subdivided according to color or size. Aristotle's assertion therefore does not hold in its strongest sense, but, if we make certain qualifications, it does contain a correct intuition.

A genus might have differentiae divisivae that are proper to itself, as, for example, we can divide living things by criteria that would be inapplicable to non-living things. Since non-subordinate genera must each have at least one differentia constitutiva that is not shared by the other genus, any differentia divisiva that subdivides one genus in terms of one of its distinctive differentiae constitutivae must be distinct in kind from any differentia divisiva of the other genus. We can see this concretely by observing that any subdivision of lizards that makes explicit reference to a vital function would be inapplicable to rocks, which lack life, and similarly any subdivision of rocks according to their mineral properties would be inapplicable to lizards, which lack a mineral character in general. Non-subordination of genera implies the possibility of creating taxonomies within each genus that are proper to each genus and not subordinate to the taxonomy of the other genus. These subdivisions appear to be more objectively grounded, as they are based on the unequivocally non-subordinate differentiae constitutivae of each genus. This finding comports with our intuition that it is more objectively meaningful to subdivide rocks according to their mineral characteristics and lizards according to their vital properties than to arbitrarily subdivide both according to size or color. Non-subordinate hierarchies within each non-subordinate genus seem to tell us more about the real nature of things and appear to be less arbitrary than taxonomies that mix members of non-subordinate genera according to common characteristics.

Secondary substances, the least valued of the four ontological classes by modern philosophers, offer perhaps the best hope of providing a system of classification for things that are. We would do well, therefore, not to dismiss these out of hand, and to retain them in our conceptual repertory. As stated on several occasions, we are not asserting the metaphysical reality of any of these ontological entities, but are considering them on a conceptual level, as necessary constituents of any attempt to understand reality. Though we have already taken care to provide some justification for these classical concepts, we would do well to examine how they have held up under modern scrutiny.

Continue to Part II

© 2007 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org

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