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• Pt. 3 of 7

11. Categories and Truth
12. Substance
13. Differentiae and Substances
14. Contraries in Substances
15. Accidents in General
16. Quantity

In our discussion of the four ontological classes, we saw that the distinction between universals and particulars does not entail a distinction in essence. Thus all conceivable essences may be considered as substance or accident, without regard for whether they are instantiated or universal. Aristotle famously identified ten categories or predications of substance and accident. It has long been a matter of dispute whether these were intended to be linguistic or ontological categories. This apparent ambiguity of intention can be seen in Aristotle’s claim, Expressions which are in no way composite signify substance, quantity, quality, relation, place, time, position, state, action, or affection. The Stygian philosopher links the condition of grammatical simplicity to an ontological entity signified. Since non-composite expressions merely signify substance, quantity, quality, etcetera, we ought to regard this list as categories of entities, not linguistic expressions. However, since our scope of discussion is limited to those things that can be signified by grammatically simple terms, there remains the possibility that there are other unknown categories for which we have no words. Conversely, our concern with ontology restricts the semantic scope of the categories, as they only account for terms that signify things that are, thereby excluding prepositions, conjunctions and interjections, for example. The categories are indeed ontological, but their scope is limited by what we are able to make intelligible via language.

The Greek term Aristotle uses for category actually means predication, which is ordinarily a grammatical concept, but as we saw in Part I, the linguistic relation of predication can sometimes correspond to an ontological relationship. Recall that a universal (substance or accident), by definition, can be said of another entity. More precisely, a universal’s mode of being is such that its name may be said of or predicated of another entity. In our oft-used example, man is a universal, and the name ‘man’ may be predicated of Socrates, because the essence that ‘man’ signifies, which is the essence of universal man, is truly instantiated in Socrates. In other words, our grammatical rule allowing ‘man’ to be predicated of Socrates is grounded in the conviction that Socrates instantiates the essence of man; ontology precedes grammar. Further note that ‘man’ is said of Socrates himself, not the name &lquo;Socrates,’ for the name ‘Socrates’ is not a man. Recognizing that grammatical predication can reflect an ontological relationship, we may, for convenience, use the expression predicated of or said of to refer directly to the ontological relationship, so we can simply say, man the species is predicated of Socrates, rather than the name ‘man.’ Predication in this ontological sense refers to the relationship between individual and universal, and between specific and generic universals.

Each higher genus of substance is a predication or category of its subgenera or species. Accidents might be similarly classified in a taxonomic structure, with each more generic accident being predicated of the more specific. We have already seen in Part I that it is not possible to construct a unique hierarchy of genera to classify all substances, nor can we expect any better for the classification of accidents. Nevertheless, this does not preclude the possibility that all things might be classified under a handful of generic categories. Aristotle claims that all things indeed can be classified under ten categories. We have noted that there is no way to prove that this or any other categorization scheme is comprehensive, since we are necessarily confined to intelligible reality for which we can construct words. Nonetheless, we may show the viability of a categorization scheme by showing that the categories do not contain each other, and that they account for all phenomena with which we are familiar.

First, we should explain why all of substance can be regarded as a single category, while accidents are subdivided into several irreducible categories. Substance may be considered as a single super-genus because all genera are certainly kinds of substances, and all substances have the same ineffably simple, declarative mode of being. Among accidents, however, there are apparently widely divergent modes of being: it hardly seems apt to say that a quality exists in the same way as a relation, or as a quantity, or as time. Much less does it seem apt that these different types of accidents should be handled identically in our logic. Here we note a serious departure from modern, formal logic, which concerns itself only with the form of a statement and is indifferent to the content of its predicates. A more robust logic, one that is truly attuned to ontological possibility, rather than what is only semantically coherent, must recognize that different kinds of predicates cannot be used interchangeably in certain contexts that produce nonetheless valid deductions. For example, consider the valid deduction:

(1a) Socrates has red apples.
(1b) Callias has red apples.
(1c) Socrates and Callias have red apples.

A deduction with identical form would be invalid if we substituted quantitative predicates in place of the quality red.

(2a) Socrates has two apples.
(2b) Callias has two apples.
(2c) Socrates and Callias have two apples.

The conclusion does not hold, because quantitative predicates combine differently when their subjects are combined, as they are summed extensively. Socrates and Callias have four apples. Thus accidents of different categories have distinct logical relationships with substances, a fact that presumably has ontological implications, if our logic is not purely semantic.

The classical list of nine categories of accident—quantity, quality, relation, place, time, position, state, action, and affection—follows Aristotle’s intuitions about which attributes of substances pertain to fundamentally different aspects of being. It is not clear how seriously Aristotle took this enumeration, since it is mentioned only in the Categories and the Topics, and only several of these categories (mainly quality, quantity and substance), are explicitly invoked in his later works. Nevertheless, we will scrutinize all of the classical categories, examining whether any of these might be reducible to another. We will also see if modern insights in philosophy and other sciences have revealed the possibility of additional categories.

11. Categories and Truth

If the fundamental categories of substance and accident truly represent irreducibly distinct aspects of being, we should expect these categories to have profound implications in the construction of logical statements. Aristotle in fact makes the bold claim that all affirmations, which he regards as linguistic objects, are composite expressions of terms representing entities in two or more categories, whereas a single category term cannot form an affirmation. Since, in ancient philosophy, the linguistic objects called sentences (whether affirmations or negations) were the locus of truth and falsity, this means that logical truth is confined to cross-category combinations, making the definition of the categories extremely pertinent to a priori analytics.

A modern understanding of linguistics would have us soften this conclusion slightly, since the linguistic objects called sentences are not strictly identical to affirmations or negations. In modern philosophy of language, statements (affirmations or negations) are concepts distinct from the linguistic objects or sentences used to convey them. This subtle distinction between a sentence and its corresponding statement (affirmation or negation) is made necessary by the fact that the same linguistic object can affirm different things in different contexts, even when it contains no equivocal terms. For example, the sentence I am at home, can affirm different things, depending on who is speaking or when the statement is spoken. In a sense, the modern statement-sentence distinction simply broadens the concept of equivocal to apply to composite terms. This modern distinction between the linguistic and the conceptual should not be confused with Aristotle’s notion of statement, which is merely a type of sentence that affirms or denies something. Nonetheless, Aristotle is not concerned with statements as linguistic objects, but with the assertions behind them, so he is really linking the locus of logical truth and falsity to the ontological relationships of entities across categories.

Since truth and falsehood are proper to statements (or rather, truth is a relation between a statement and reality), and statements are necessarily composed of entities from more than one category, it follows that there is no logical truth to be found in entities of single categories. There is a noteworthy exception to this rule, as being in the existential sense does not belong to any of the ten ontological categories. Thus primitive sentences such as X is, or X exists, are ontological affirmations that can be true or false, notwithstanding that they have only a single category term. Being is not a genus, hence not a category, since it cannot be predicated of any entity within the categories. On the contrary, being is the very subject of categorization. We similarly do not regard the ens copulae—the is in X is Y—as having categorical significance, since it is only a copula showing ontological linkage between the entities X and Y, and not a third ontological entity, though it is a third predication in a formalistic sense.

Since truth and falsehood can only be defined with respect to reality, they are not mere semantic concepts. We cannot arrive at truth, and therefore cannot link our language to reality, unless we combine two or more categories in an affirmation (with the exception of simple ontological assertions). This necessity of cross-category combination gives the fundamental categories of substance and accidents a greater significance than merely being the highest genera of entities, but they are essential to the construction of reality, as two entities of the same category cannot constitute an affirmation that is true or false.

The realization that entities of an individual category constitute realities through combination with entities of other categories can motivate the factualist claim that states of affairs are ontologically fundamental. While truthmakers of (non-ontological) affirmations can be no more primitive than the combination of entities from two or more categories, such truthmakers are not necessarily more fundamental than categorical entities. We might construct a truthmaker or state of affairs out of entities from categories A and B, B and C, or A and C. If we were to regard the states AB, BC, and AC as more fundamental than A, B, and C, we would have to define A as nothing more than the intersection of AB and AC, and so on. Nonetheless, the entities A, B, and C would retain a virtual ontological independence, since A can compose a real entity in combination with B or with C, so it is not dependent on any determinate entity for existence, though it cannot constitute an affirmation on its own.

Socrates is not an affirmation, unless we add is or exists to make an ontological affirmation. Socrates walks, by contrast, is an affirmation, linking Socrates to the act of walking, the latter having no claim to reality except when coupled with an entity of the category of substance. The necessity of coupling, however, does not seem to be a good reason to deny reality to Socrates or walking when the composite entity is real. We note that both Socrates and Socrates walks may be considered as (respectively, ontological and logical) affirmations, their truth or falsity being determined by the presence or absence of actual being. In the case of Socrates, grammar requires that we explicitly refer to its being (with is or exists) in order to make an affirmation, but in the case of composites like Socrates walks, being is implicit, for when we say Socrates walks, we mean Socrates really is walking, whereas when we say Socrates, we make no assertion as to whether he really exists. This grammatical distinction should not impede us from recognizing that even single category entities may have claims to truth or falsity, since we could have a grammar where the declaration of an entity entails assertion of its existence. Conversely, composite entities could be constructed without affirmation (e.g., Socrates walking can be considered independently of reality).

In sum, single category entities can have a relation with truth only when their existential being is asserted or denied, while all other kinds of assertions require the coupling of entities from two or more categories. This logical coupling does not abolish the virtual ontological independence of entities in each category. In both ontological and logical affirmations, an assertion of being is essential. Ontological assertions use the existential is or exists, while composite logical assertions use the ens copulae (e.g., fire is hot) to couple two entities with common being. While the is in the latter case is a third logical predication, it is not an ontological entity, since being is not a thing in any category, but rather that which is the object of category theory.

12. Substance

Recognizing that single category entities have claims to truth when reference is made to their being, we now consider the category that is apparently most simply connected to its being, that of substance. The term substance has a broad range of usage, but it generally signifies an underlying, solid, real entity beneath or behind phenomena. Substances are the stuff or things that are which ground all subjective experience into something solid or real. As such, they are independent of other types of entities, which we call accidents and can exist only in a subject. Substance is that which is not in a subject in the formal sense discussed in Part I.

Following the basic intuition of substance as something that exists directly and simply without dependence on another entity, Aristotle asserts that substance in the strictest sense is that which is neither said of a subject nor in a subject. Individual substances are more truly substances because, unlike universal substances, they can never be said of another entity, so their being is radically primitive and non-transferable. The essence or definition of lion can be predicated of entities other than lion, but the essence of Socrates cannot be predicated of anyone but Socrates. Many different entities can be a lion, but only Socrates can be Socrates.

As appealing as this intuition may be, the choice of individual substances as primary substances, leaving species and genera as secondary substances, is based not on logical necessity, but on the empiricist preferences of Aristotle. All we have discussed previously does not lead inevitably to the conclusion that individual substances are more fundamental than universals. Most modern philosophers, being empiricists, would accept Aristotle’s claim without challenge, but there has been a long tradition dating back to Plato that asserts the contrary, namely that concretized individuals derive their existence from participation in the immutable Ideas (universal substances) that constitute their essence. Our highly mathematical theories of physics might easily be interpreted to imply that individual physical entities are but concrete manifestations of mathematical objects. It is by no means demonstrated a priori that individual substances should be considered more primitive than universal substances.

Without judging which type of substance is primary, we can analyze the relationship between universal and individual substances in terms of the predicability of a universal’s name and definition. In Part I, we showed the following facts about the said of relation:

If X is said of a subject Y, then the name of X and the definition of X are predicated of Y.

For example, man is said of Callias, so the name ‘man’ and definition of man (rational animal) are predicated of Callias. In other words, you can properly refer to Callias as ‘man,’ and Callias, like man, is a rational animal. (Again, single quotes denote names, while double quotes denote entities or essences.) Individuals can act only as subjects in this context, as it would be impossible for an individual substance to fulfill the role of X, but species and genera can act either as subjects (Y) or as predications (X), which is one reason for not regarding them as pure or primary subjects like individuals. Irrespective of that subjective assessment, we can affirm that the definitions of individuals may not be predicated of universals. The unity of an individual substance prevents its essence from being predicated of anything other than itself.

The said of relationship does not hold between substance and accident, since being said of a subject is not the same as being in a subject. Accidents, which are in a subject, do not have their name or definition necessarily predicated of the subject. For example, The dog is white, cannot be taken to mean that one can call the dog simply ‘white,’ nor is the definition of white predicable of the dog, since the dog is not a color resulting from the reflection of all visible light frequencies.

By definition, a substance is that which is not in any subject in the formal sense, whereas an accident is in a subject on which it is dependent for its being. A substance has greater ontological autonomy than an accident, since its existence does not depend on being in a subject. Similarly, an individual seems to have greater ontological autonomy than a universal, since the essences of universal substances may be predicated of individual substances, but an individual substance is exclusively qualified to be what it is. Combining these intuitions, we might conclude that individual substances have maximal ontological autonomy, buttressing the Aristotelian position that these should be regarded as primary substances. Nonetheless, universals have as much claim to be considered substances as do individuals.

We may reasonably ponder whether the predicability of the name and definition of a subject is merely an artifact or idiosyncrasy of human language, or if it points to an ontological relationship among substances. The name of X might not be predicable of its subject Y, since the rules of nomenclature can be arbitrary and admit homonyms, so we must implicitly assume that name is used univocally. Further, there are cases where the same word might be used as the name of an accident or a substantial species. Elder is one such word, which can mean either advanced in age (accident) or one who is advanced in age (substantial species). In the first sense, the name ‘elder’ cannot be applied to Methuselah except equivocally, and the definition advanced in age is not directly, univocally predicable of Methuselah, though the name and definition of elder in the second sense are properly predicable of Methuselah.

The definition of an accident is never properly (univocally) attributable to the subject in which an accident inheres. Since a definition is a statement of what it is to be the thing defined, the definition of an accident states what it is to be that type of accident, and all accidents, by definition, cannot exist except in a subject. Thus if the definition of an accident directly pertained to a substance, then that substance could not exist except in a subject. Yet this is impossible, as it is contrary to the definition of substance. Note that we have not proven that the classically enumerated accidental categories are indeed truly accidents and not substances in some or all cases.

Individual substances, or objects, stand in relation to all other entities in the manner described by the following Aristotelian claim:

Claim: All non-objects are either said of individual substances as subjects or in them as subjects.

Aristotle says non-substances in place of non-objects, regarding individual substances or objects as the only true substances, but we avoid this terminology in order not to deny that universal substances are substances. At first glance, the above claim may seem unremarkable, as we have already established that universals are said of other entities and that accidents necessarily exist in a subject. However, this claim includes two stronger assertions: (1) all secondary substances, including higher genera, are directly predicable of individual substances, and (2) universal accidents are in individual substances as subjects.

To see how the first assertion is true for all genera, consider how the genus animal is predicated of the species man and of all individual men. If no individual man (potential or actual) was an animal, then neither could man be a species of animal. In fact, if any individual man was not an animal, then man could not be a proper species of animal, though it might be regarded as such in an imprecise sense, the way biped is a subgenus of reptile, though there are non-reptilian bipeds. We would really mean reptilian biped when we speak of biped as a subgenus of reptile. Similarly, if some men were not animals, man might be considered a species of animal only if it were understood from the context that we are only referring to animal men. In that case, the genus animal would still be predicable of individuals.

The linguistic ambiguities discussed above should remind us that ontology should remain our primary object of study, with language bent to the needs of our task. When we declare a name, we need to clarify whether we are referring to an accident or a universal substance. When we refer to a universal substance, we should clarify whether we are speaking of it in the most generic sense or in a more specific sense (biped or reptilian biped). Language should always conform to our ontological concepts, not the other way around. When we speak of predicability, we do not allow grammar to dictate our ontology, but rather we are describing a situation where grammar happens to conform to ontology. When language is equivocal or makes use of copulae, it produces apparent counterexamples to our ontological predication claims. This does not disprove our ontology, but merely reflects how the artifacts of common language need not conform to philosophical coherence. We find that subject-predicate grammar does reflect a sound intuition regarding the relationship between universals and individuals, and between substance and accidents, though grammar does not always clearly distinguish these two distinctions.

The second assertion we shall consider is that a universal accident is in an individual substance as a subject. Formally, a universal accident is in a universal substance, which, we have noted, is nothing but its essence. Therefore the universal accident may be said to be in the essence of the universal substance. The essence of the universal belongs also to individuals of the species or genus, so the universal accident is also in the essence of the individual. Still, the universal accident is not in in the individual qua individual, but only by virtue of its universal essence. We may also universal accidents (properties) to individual substances (objects) via instantiated accidents (tropes). An instantiated accident or trope is truly in an individual substance as a subject, and a trope is nothing more than a universal accident’s essence, limited by the determinate contingencies of the object in which it inheres. Once again, the universal accident in some sense is in the individual substance, but not as a universal, only as an individuated accident. In either path of analysis (through universal substances or through tropes) we find that universal accidents are in individual substances only in an imperfect way, modified by the contingencies of individuation. For this reason, we give a distinct term for this relationship: exemplification. Regardless of the path on the ontological square we choose, the relationship of exemplification is the same, as universal accidents are "in" individual substances as subjects only as modified by the determinate contingencies of individuation.

Individual substance appears to be logically necessary for all other entities to exist, even if we do not necessarily admit Aristotle’s belief that they are primary substance in the sense of being ontologically more fundamental. It could well be that other entities are equally necessary to individual substances, so there would be a mutual dependence. In any event, we can hardly conceive of a species without at least a potential individual in that species, nor can we conceive of an accident existing without a subject in which it inheres. We shall now consider further how these non-objects relate to individual substance, and it what sense they may be said to have being.

Species and genera of substance are proper substances since they can exist independently of any determinate subject, yet their ideal or abstract character removes them from our intuition of substance, which is grounded in the solidity and ontological unity that is best manifested in individual substances. If we use individual or primary substance as our reference point, we would have to regard a species as more substantial than a genus, as the individual more closely resembles the species than the genus. For example, Callias’s rationality and corporeality pertain to the species man, as do all of his distinctively human traits. Yet his rationality and other distinctively human attributes do not pertain to the genus animal. Thus man tells us much more about who Callias is than does animal.

Another ground for regarding the species as closer to primary substances or objects than a genus is the fact that a genus can be predicated of a species without the species being predicable of the genus, much as universal or secondary substances in general are predicated of primary substances, though primary substances are not predicated of anything else. In other words, with regard to predicability, we can make the analogy:

primary substance : secondary substance :: species : genus

Recall from Part I that this is not a strict analogy, since the predicability relationship between individual (primary) and universal (secondary) substance is not identical to that between species and genus. When we say Callias is a man, we do not mean that Callias is universal man, but only that the essence of the species man pertains to Callias. When we say that, Man is an animal, not only do we assert that the essence of animal pertains to man, which is the extent of the above analogy, but also that man truly is animal in the universal sense. However, as with the individual-species relationship, there is not total identity between the species and its genus, for it would be an obvious error to say that man is the genus animal. Rather, it is animal in a universal (non-particular), though specified sense.

Although a species or sub-genus may be considered closer to primary substance than its genus, there is no basis for regarding one non-subordinate species or sub-genus closer to primary substance than another. We observed in the first part that non-subordinate species or genera had a special claim to objective reality that other secondary substances might lack. Here again we find that non-subordinate secondary substances have a special status, since they are equally substantial in the sense of resemblance to primary substance.

Aristotle says that secondary substances (universals) reveal or give knowledge of primary substances (individuals), since they help specify the essence of the latter. To say what an individual substance is, it is necessary (though not sufficient) to give its species or genus. In answer to What is Callias? we may say in partial reply, Callias is a man, and an animal. In contrast, accidental attributes, such as white or walking, do not necessarily tell us anything about what Callias is. Callias would continue to be himself if he stopped walking, but he would cease to be Callias if he ceased to be a man.

Only accidents essential to a species or genus are properly in that species or genus. Non-essential accidents, even if they are exemplified by an individual of a certain species or genus, are not in that species or genus as a subject. For example, suppose generosity is exemplified by Callias. We cannot conclude from Callias is generous, that Man (the species) is generous, so generosity is not in the species man. The non-essential accident exists only in the individual.

Species and genera, like all substances, are not in a subject in the formal sense. Recall that being in a subject in the formal sense means: (1) in the subject in the ordinary sense of in, (2) not as a part of a whole, and (3) not capable of existing without a subject. A species or genus fails the first and third conditions, considering individuals as potential subjects. A universal cannot be contained by any individual, so it is not in any individual in the ordinary sense of the term. It is obviously wrong, and not just in a grammatical sense, to say that man is in Socrates, as though manhood itself could be contained by one individual. A universal substance does meet the second condition, as it is not in a part to whole relationship with its individuations or any other entity, which is why set theory is an inadequate model of species and genera. There is a difference between the set of all numbers that happen to be integers and the essential concept of integer—a number which is divisible by one—considered generically. Regarding the third condition, a universal might to be at least capable of instantiation in order to exist, but it is not dependent on any determinate individual for its existence, and an indeterminate individual is no individual at all. Thus universal substances are not to be regarded as accidents inhering in individuals; indeed, they fail the defining criteria of accidents, as they are true substances.

As discussed in Part I, the definition of a universal substance is predicable of its individual subject; e.g., the definition of man is predicated of Callias. The definition of an accident, by contrast, is never predicable of an individual substance, even though an accident may be in a substance as a subject. This contrast reinforces our sense that species and genera are more substantial than accidents, since their definition is predicable of concrete objects in a way that is impossible for accidents.

Our intuition of substance as a thing that exists, simply and absolutely, is congruent with the condition that being in a subject cannot be a part to whole relationship. A part abstracted from the whole may be conceived as existing, but it would be impossible to conceive of an accident as existing apart from its subject, though they may be formally separated. It is inconceivable for an accident to exist apart from its subject, but a part may conceivably exist apart from the whole, even if it is actually impossible. Physical chemistry provides an example where the part seems inextricably mixed with the whole. An atom in a molecule may share electrons with other atoms, sometimes even creating mixed orbitals, so it is not really separable from the other atoms. Even in this extreme case, we may consider the molecule as potentially subdivided into atoms, so we can at least conceive of each atom’s existence independently. This is not possible for accidents, as it is contrary to the essence of an accident for it to exist apart from a subject. The part to whole relationship is therefore exclusive to substances.

Lastly, a species or genus ought not to be regarded as a determinate substance. Remembering that ontology should not depend on grammar, we must ignore the grammatical necessity of referring to universals as though they were determinate objects. As Aristotle has it, a species or genus does not signify a certain this, but rather a certain qualification. This does not mean that a universal substance is a quality, like white, but rather a secondary substance is a substance of certain qualification. This qualification, we shall see, is defined by differentiae.

13. Differentiae and Substances

We have seen how universals are said of or predicated of individuals, and how genera are predicated of species, yet there is a further kind of predication relating differentiae to substances. A differentia, recall, is a universal accident or property that characterizes the essence of a species or genus. Since the essence of the differentia belongs to the essence of the characterized species or genus, it follows that the differentia’s definition or statement of essence is predicable of the species or genus.

This mode of predication is slightly different from that between individuals and universals. When we say, Snow is white, this does not mean that snow is a color resulting from the reflection of all visible light frequencies, so the definition of the differentia white is not predicable of the substance snow the same way that a universal’s definition is predicable of an individual. A differentia characterizes an essential aspect of a substance, but, since it is an accident, it cannot by itself constitute a full substantial essence. Still, the differentia’s relation to its characterized substance is stronger than merely being in a subject, for the entire essence of the differentia belongs to the essence of the substance, so that the differentia’s definition is truly predicable of the substance. We may not say that the essence of white is snow, but it is an essential aspect of snow.

When an individual exemplifies a property, it (1) instantiates a species characterized by the property and (2) its being is modified by an instantiation of the property. A property can be a differentia only by virtue of its role in characterizing or defining a species, so exemplification along the first path, via species, is more directly pertinent to differentiae. We have noted how the definition and essence of a differentia are predicable of a species. Since the essence of a species, in turn, is fully predicable of an individual (though still falling short of identity), it follows that the essence of the differentia is also instantiated in the individual. For example, when we say, Man is rational, we mean that the essence of rational belongs to the essence of man. Since the entire essence of "man" is instantiated in each individual man, then the essence of "rational" is necessarily included in this instantiation. Thus each individual man is rational, though there is more to the individual than rationality. Exemplification may be regarded as the predication of the essence of a universal property in an individual object, in a manner that does not constitute a full substantial essence, nor does it admit of identity between property and object.

The importance of differentiae in characterizing essences has naturally led to the classical and medieval philosophical tradition of defining species per genus et differentia. A common example of this form is, "Man is a rational animal," where the species "man" is defined by the differentia "rational" applied to the genus "animal." We should emphasize that this construction is distinct from the set theoretic definition: "Men are the set of all animals that are rational." A set has no existence apart from the elements of which it is constituted, so the set of men would change as individual men come into or depart from existence. The universal "man," however, does not change with the fate of each individual man, but is perfectly knowable and unchanging in its essence, defined as it is by the universals "rational" and "animal." Set theory deals with individuals and groups of individuals (sets), but the terms of a definition per genus et differentia consists entirely of universals.

The role of a differentia in this type of definition is to modify the genus, much as a trope modifies an individual substance. The essence of the modified universal substance retains all that is essential to the genus, so the modified essence is different, yet not contrary to that of the genus. In order to preserve the essence of the genus, the differentia must not modify or negate any of the properties that are essential to the genus. These essential properties characterize the genus much as the differentia will characterize the species or subgenus. We could regress in this fashion to higher genera until we reach those that are distinguished from undifferentiated substance only by a few properties. Differentiae distinguish universal substances within each genus, and ultimately from undifferentiated substance.

Descending from the highest genera, we can visualize the addition of differentiae as subdividing substance into progressively narrower classifications, but we should not confuse this process with the set theoretic analogue of subdividing a set into subsets or elements. There is, nonetheless, a real connection between these two processes. As we subdivide a genus by adding differentia, a set of individuals of that genus will necessarily have subsets (possibly empty sets in some instances) of individuals in each newly defined subgenus or species. Clearly, since the adding of differentiae imposes an essential constraint on a universal substance, it is not possible for the subgenus to have more members than the genus, nor, since the differentia cannot be contrary to the essential properties of the genus, can a member of the subgenus fail to be a member of the genus. Here we speak of set theoretic membership equivocally, since an individual is not properly a member of a genus or subgenus, but rather is a member of the set of objects that individuate that genus or subgenus.

An individual’s participation in a species may admit of degrees. Although no individual substance is more substantial than another, each may exemplify the differentia of the species to varying degrees, if the differentia is a property that admits of degree, and the species is characterized by the differentia without regard to degree. For example, if the definition of athlete is strong man, without regard to the degree of strength beyond some minimum threshold, then individuals may be said to be athletes to varying degrees if they have varying degrees of strength. If athlete were defined in a way that specified the degree, such as a man strong enough to lift at least 100 pounds, then one individual could not be more of an athlete than another; one either meets the criterion or does not.

14. Contraries in Substances

Contraries are mutually exclusive accidents such that one is necessarily absent in a subject to the extent that the other is present in that subject, and vice versa. A substance, by extension, may be improperly called contrary to another substance if they each possess contrary accidents. A white horse and a black horse, for example, may be improperly regarded as contraries, though it is really black and white that are the contraries. Since a substance does not exist in a subject, but exists simply and absolutely, the only contrary of a substance qua substance would be the privation of that substance, which is not an entity.

Individual substances can receive contrary accidents at different times, thereby changing the qualifications of their being. When the accident changed is a differentia, the individual may even change its species. Only an individual substance, not a universal, can be a locus of action, since a change in accident is essential to action. In Greek natural philosophy, this would have important implications, since for the Greeks, nature (physis) is a principle of change or motion. Though the nature of a thing would often be identified with a universal essence, the thing which actually moves or changes is invariably an individual. Ironically, the universal essences that are principles of motion do not themselves move or change, creating a conundrum for the problem of motion that Aristotle would address in his works on natural philosophy.

Aristotle contends that individuated accidents, unlike individual substances, cannot admit contraries. This may be because tropes are confined to a specific moment in time, while substance is a substratum that persists in time, having various qualities over time, including those that are utterly contrary to each other. Even if we were to conceive of some trope as persisting in time, it could nonetheless not admit any property contrary to itself without ceasing to be. For example, if some instance of whiteness should be replaced by an instance of blackness, we would say that the white trope ceased to exist, not that it now instantiates blackness. This is because the very notion of a trope entails that it instantiate a certain property; an instantiation of a different property is a different trope. If we were to grant tropes the ability to instantiate different properties, we would effectively be giving them the status of substantial objects, as trope theorists do. An individual substance, by contrast, is more than the instantiation of a species, as it may possess additional accidental features. Thus its existence is not irrevocably tied to instantiation of a determinate species. The same cannot be said for a trope, which is nothing else but the instantiation of a property.

Statements do not admit contraries, for even though facts may change, a statement must be evaluated in the unique context in which it is asserted. The sentence He is alive, may be both true and false, depending on when it is spoken and who he is, but the assertion or statement represented by a particular instance of the sentence, He is alive, is context-specific, and therefore cannot be both true and false. This principle of non-contradiction implies that a statement cannot admit contrary realities, so there can be no statement, He is alive and not alive. We can construct such a sentence grammatically, but there can be no intelligible assertion behind that sentence, as logic is independent of grammar.

15. Accidents in General

The independence of logic from grammar must be considered when evaluating the familiar subject-predicate form of sentences representing statements. This form is not a culturally arbitrary constraint upon thought, so much as an expression of our intuitions about reality. Things that we regard as substantial are rendered grammatically as nouns, while those that are non-substantial yet inherent in substance are given a predicative form. We have outlined the basis for these intuitions in Part I, without pretending to exhaustively determine a posteriori which things are truly substances and which are truly accidents. Our use of examples helps articulate the concepts, without necessarily pronouncing on the ontological status of particular entities.

Whether the subject of an accident is a substance or another accident, the definition of an accident is not applicable to its subject. When the subject is a substance, the definition of the accident is clearly impredicable of it, as a substance cannot be in a subject. When the subject is another accident, as blue can be the subject of dark in dark blue, the definition is necessarily impredicable if one accident is not subordinate to the other. In cases where one accident is subordinate to each other, as in colorful blue, where colorful is essential to blue, the definition is predicable in the way that of a genus is predicable of a sub-genus. Lastly, the definition of an accident is trivially predicable of its subject where the accident is formalistically regarded as its own subject, as in blue blue. Such predication, however, corresponds to no ontological relationship save identity, as there is no ontological distinction between blue blue and simply blue.

The name of an accident is univocally predicable of its subject only when the definition is predicable, under the constraints described above. Naturally, the name of an accident may always be equivocally predicated of its subject, due to the arbitrary nature of names. We should take care that names are univocally predicated, referring to the ontological constraints defined above, rather than relying on common grammatical usage.

In his discussion of the nine categories of accident, Aristotle does not explicitly distinguish individual accidents (tropes) from universal accidents (properties). As we analyze each category, we will compensate for this omission, clearly indicating the distinction between individual and universal accidents.

16. Quantity

A dominant intellectual paradigm of the last three centuries has been the emphasis on quantitative analysis for understanding reality. Advances in measurement techniques and mathematical theory have made possible two scientific revolutions in the seventeenth and late nineteenth centuries, the latter being accompanied by a technological and industrial revolution that has firmly established physical science as an authoritative intellectual force in the popular mind. Quantitative science may be a victim of its own success, however, as the technological fruits of quantitative analysis may have blinded scientists and philosophers to the metaphysical limitations of this approach. There is much more to reality than quantity, and it would be a careless mistake to regard all other categories of being, including substance, space, time, and various qualities, as reducible to quantity. It is possible, though unproven, that all things in the natural world are quantifiable, as St. Augustine and St. Thomas Aquinas interpreted the Biblical dictum, Deus omnia fecit in numero, pondere et mensura (Wisdom 11:21) Yet even assuming that everything is quantifiable, it does not follow that everything is quantity. Nonetheless, since the seventeenth century, most men of science have assumed that the quantitative analysis of entities such as qualities effectively reduces these entities to quantity. In line with this view, the mechanists Newton and Pascal re-interpreted the above-cited Biblical verse as endorsing the notion that everything is quantity, signifying a serious shift in European thought, in both the Catholic and Protestant worlds. To understand the metaphysical limitations of quantitative analysis, we must appreciate a clear distinction between quantity and that which it quantifies.

Quantity is how much of something there is, in number or amount or measure. As it is necessarily predicated of something, quantity is an accident, whose subject may be a substance or another accident. As an accident, quantity modifies the being of its subject, but does not thereby abolish the subject, upon which it depends for existence. Quantity can modify the quality brightness, as there can be more or less brightness, but this fact alone does not abolish brightness or reduce it to quantity. Indeed, the existence of the quantity modifying brightness depends on the existence of its subject, the quality brightness. A thorough treatment of the irreducibility of specific qualities to quantity is best reserved for a discussion of natural philosophy. Here, we will consider the relation of quantity to its subject only in general a priori ontological terms.

Quantity is often invoked as a basis for reducing some qualities to other more fundamental properties. Though there may be genuine instances where a secondary quality is reducible to some more fundamental or primary quality, such reduction cannot be deduced from quantitative relationships alone. The Newtonian force law, F = ma, considered as a necessary mathematical relation, only establishes that the magnitude or quantity of force is equal to the product of the quantities of mass and acceleration. Quantity is related to quantity; this mathematical relation alone cannot establish that force is constituted of mass and acceleration. To show that one quality is derivative of another, we would have to establish that all of its essence or nature is contained in its constituents. We would need genuinely physical analysis to establish this dependence; mathematical analysis alone cannot prove this, though it may be suggestive of it. In the reduction of temperature (on a thermodynamic scale) to molecular motion, quantitative analysis must be supplemented by physical observations that all the phenomena of temperature may be accounted for by molecular motion. We cannot establish the emergence of qualities solely from the mathematical relations among other properties, as these only relate the quantitative aspects of properties, not the properties as such. Purely mathematical equations yield no qualities, rather we assume that the variables represent amounts of various qualities.

Quantity can modify substance or accidents such as quality, space, and time. Although quantity can modify different subjects, this does not necessarily suffice to establish a distinction among kinds of quantity. Three is three, regardless of whether it modifies space or time. There are, nonetheless, ontologically meaningful distinctions among kinds of quantity, such as discrete and continuous, intensive and extensive, each of which may be applicable only to certain classes of subject.

Ancient and modern philosophers have generally agreed that there is a meaningful distinction between discrete and continuous quantity, but there has been a broad range of opinions concerning which classes of entities are subjects of each type of quantity. Classically, space and time have been regarded as subjects of continuous quantity, as are length, area and volume. Some modern physicists and philosophers, relying on speculative interpretations of quantum mechanics, have suggested that space and time may be grainy, or constituted of discrete units. Notwithstanding quantum theoretical limitations on the spatiotemporal confinement of particles, space and time themselves each must be a continuum, unless we should have particles constantly blink in and out of existence as they passed from one discrete unit to the next, a supposition that disregards Occam’s razor almost to an order of infinity. Although there may be limited resolution by which we can specify the spatiotemporal location of a particular particle, there is nothing preventing some other particle from occupying the gaps between the grains of space and time in the first particle’s reference frame. If the graininess of space and time were absolute, no particle could be between two grains; a particle must utterly vanish from one grain and rematerialize in the one adjacent. This would be absolute nonsense for macroscopic objects, as a simple sweep of one’s arm through the air would effect an inconceivable number of quantum leaps. A less parsimonious physics is scarcely conceivable. At any rate, for all practical, empirical purposes, space and time are continuous, and mathematically, our calculus presumes a continuum, reinforcing our a priori suspicions toward discrete space and time.

Classically, continuous quantity has been subdivided into geometric and non-geometric quantity, with the former including place, length, surface, and volume, while time is the only non-geometric continuum. Place and time are classically regarded as distinct categories of accidents that are modified by quantity, but length, surface and volume are nothing but quantities themselves. Euclid expressed this latter conviction in his assertion that A line is length. By this, he meant that the geometric concept of line consists of nothing else than some quantity of length; the line is not some substance that has length.

The Euclidean concept of a line as a geometric measure, though valid, is not the only possible way to conceive of a line. The line might be considered as a locus of points in some abstract space, and thus the length would be a measure of this locus, or even a measure of space. Similarly, we could adopt Euclidean notions that Area is surface, and Volume is solid, or we could regard area as the measure of some locus or part of space called surface, and think of volume as the measure of some locus or part of space called solid. Alternatively, we could define solid as a substance that has volume, and do similarly for surfaces and lines, if we admit lower dimensional substances.

The concept of lines, surfaces, and solids as loci rather than measures enables us to give a more complete account of geometry. While there is no distinction among lines of the same length, there are many different surface configurations that could have the same area, and the same is true for volume. Further, a line, surface or solid conceived as a locus can have properties other than length, area or volume, since it may have curvature and gradient properties. Generally speaking, we may characterize n-dimensional continuous loci that are locally homeomorphic to Euclidean space as manifolds. In plain English, manifolds are Euclidean lines, surfaces or solids that are deformed or curved in various ways. The conceptual validity of manifolds implies that length, area, and volume are distinguishable from the geometric entities that bear these quantities.

In modern geometry, we can generalize beyond three dimensions analytically, though we cannot conceive of higher dimensional objects in the mind’s eye, nor do we see higher spatial dimensions in nature. Some theoretical physicists have postulated the existence of higher dimensions, but in most cases these are just dynamic degrees of freedom, not truly spatial dimensions. It is not immediately clear why there are only three spatial dimensions observed in nature, nor is it evident why we cannot directly apprehend higher dimensions, even though the human mind is capable of conceiving many things it has not seen. Aristotle, apparently at a loss, proposed a mystical explanation in the Physics, which Galileo famously ridiculed in the beginning of his Dialogo sopra i due Massimi Sistemi del Mondo. Yet Galileo himself could do no better than to show that after an object has moved in three dimensions, it has nowhere else to go. Against this naive empiricism, modern mathematicians and theoretical physicists note that there is no formal mathematical barrier to higher spatial dimensions. Nonetheless, it is invalid to infer from mathematical formalism alone that higher dimensions are legitimate physical or metaphysical possibilities.

Since a priori ontology is concerned with all things that may exist, we can tentatively admit the potential existence of higher dimensions of space, as the coherent mathematical models of such space prove at least a conceptual cogency. However, such analytical models cannot be regarded as true geometries in the classical sense, as we do not know if they are the measure of any truly possible world. For the Greeks, geometry was irrevocably tied to physical space, but modern mathematics has abstracted itself from physical reality, at the expense of not knowing whether an empirically validated mathematical model truly describes geometric reality in the classical sense: a measure of the world. In the modern sense, geometry is not the measure of any world, but measure considered abstractly. Modern abstract geometry, nonetheless, is a true measure by classical standards, as it encompasses the measures of length, area, and volume, as well as possible higher-dimensional continuous measures. It remains to be seen if there are other types of continuous quantity beyond abstract geometry.

The Greeks, from Pythagoras to Aristotle, conceived of number as limited to discrete quantities. Thus the Pythagorean dictum that everything in nature is number must be understood as restricted to natural numbers (positive non-zero integers). It was known to Greek geometers that natural numbers can be considered as embedded in a continuum, yet there are many objects in nature that can only be counted discretely. While the measure, size or volume of a substance can be a continuous quantity, discrete quantity is needed to quantify distinct iterations of a substance, even those widely separated in space. Number is independent of the spatial relations among objects, so it is not geometrical. Numerical quantity is a strange sort of accident, since it does not inhere in any particular substance, but is in any arbitrary group of substances we choose. Despite the role mental operation plays, experience compels us to admit that, however arbitrary our selection of a group may be, the number of that group is physically real. It may be arbitrary for us to define the apples in this basket as a group, but once the group is defined, the number of the group, say three, is non-negotiable. What is objectively real is the relation between three and this arbitrarily defined set. This is an example of how the use of arbitrary mental operations can nonetheless lead to knowledge of physical reality.

Number, following its Greek root, aggregate, may be considered a principle of aggregation, which is both a mental operation and also a description of physical reality when what the mind aggregates co-exists in the same physical reality. The assumption that what we perceive exists in a single reality permits us to aggregate any plurality of perceived objects and achieve a true result regarding their number. When number is applied to abstract objects, we must at least conceive of individuated objects as existing in the same reality if we are to speak coherently of their number. Number requires at least a virtual space to serve as a medium of pluralization, so that we can distinguish multiple iterations, even if there are no determinate individuals in abstract space. To count real individuals, of course, we would need a real extensive space to serve as the medium of pluralization.

Oddly, Aristotle regards language as admitting non-geometric discrete quantity. Spoken language may be measured in discrete syllables, but it is unclear why this should be regarded as distinct from number. Perhaps Aristotle was not satisfied that the quantity of speech can be measured by counting syllables or by duration of time. The physical process of speech can clearly be reduced to temporal extension and number of utterances, so we could only need a distinct class of quantity if we regard language with respect to its meaning, rather than its physical process. Plato argues in his Cratylus dialogue that parts of meaning can be broken down to the level of words, or even syllables and letters. Regardless of where we draw the line at the fundamental components of language, it is clear that more meanings are conveyed with the utterance of more components, though it is not clear that each component should be counted equally. We can know that an existing sentence with a phrase added is longer than the original sentence, but we have no objective way to determine whether two completely different sentences are of equal length. This is a highly impoverished notion of quantity (even by the standard of intensive magnitudes), since we can only draw inequalities between linguistic entities and their subsets.

Aristotle singles out time as the only entity admitting non-geometric continuous quantity. Time is non-geometric in the classical sense, because it does not measure the extension of physical space. From the continuity of motion, causality, and indeed of reality, we infer the continuity of time, lest we should suppose that everything is repeatedly created and annihilated. Understandably, those familiar with modern theoretical physics have challenged these notions about time. First, it is argued that time is geometric, not merely because modern geometry admits any abstract measure, but even in the stricter sense of spatial extension, for relativity seems to imply that a temporal displacement in one frame of reference is a spatial displacement in another frame. Notwithstanding this interpretation, it should be evident that time is not reducible to mere spatial extension, as we can gather from the fact that one may freely move forward or backward in space, but this symmetry breaks down with time. We will elsewhere examine the distinction between space and time, but here it suffices to observe that even if time is different from space, in principle it can be measured by the same kind of quantity as spatial extension, as we do in actual practice using the real number continuum. (It is arguably a mistake to measure time as extension, as indicated by the paradoxes this generates, but it is unclear what other notion of quantity ought to be used.) We do not make ontological distinctions among kinds of quantities solely on the basis of the subject of quantity; e.g., we use the same kind of number to count apples and bananas. Thus the classical distinction between geometric and non-geometric does not define two kinds of quantity, but the same kind applied to different subjects.

In our mention of the quantity of language, we alluded to an impoverished form of quantity that only admits of inequality comparisons. This is intensive magnitude, to be distinguished from number and measure, which are extensive magnitudes. An intensive magnitude describes qualitative intensity to which we cannot ascribe a number or measure. For example, I can know that one object feels hotter than another, but not whether something feels twice as hot. It may be that qualitative intensity has a true number or measure, but this may not be knowable. We may say that intensive magnitude, far from being an altogether distinct kind from extensive magnitude, is an imperfect form of quantity, lacking certain properties of extensive magnitudes such as multiplicative identity, addition and counting. With intensive magnitude, we can only know comparative inequalities and rough equality.

In the example of language, we noted that the length of two sentences, as measured in meaning, can only be compared if one is a subset of the other. Intensive magnitude in general admits a somewhat broader range of comparison. If I know object 2 is hotter than object 1, and object 3 is hotter than object 2, then I certainly know object 3 is hotter than object 1. I can also contrast differences in intensive measures; for example, I can observe that the difference in heat intensity between objects 1 and 2 is less than the difference between objects 3 and 4.

There is substantial reason to doubt whether there are true intensive magnitudes, or whether these simply reflect our ignorance of the rule of measure for certain phenomena. For example, the intensity or brightness of light may now be extensively quantified or measured, though for most of history it was only experienced as an intensive magnitude. Our sensation of heat and cold does not correspond linearly to temperature, and it varies by individual. There seems to be no strong reason in principle why this intensity of sensation might not someday be quantified extensively. If that occurs, we may then plot our subjective sensation intensities against a defined neurological measure, and regard the formerly intensive magnitudes as embedded in a field of extensive magnitudes. Even if it could be proved that all intensive magnitudes are embedded in extensive magnitudes, there would be nothing preventing us from accepting intensive magnitude as true quantity, albeit in imperfect form.

Just as intensive magnitudes may be embedded in a continuum of extensive magnitude once the measure is learned, so may discrete numbers be embedded in a continuum. As an illustration, consider the square roots of prime numbers, irreducible to one another, as embedded in the real number line. Extensive magnitudes, whether discrete or continuous, share the common trait that they can only be conceived as existing in an extensive space of non-overlapping quantities. In order to count, we need to consider the objects as distinct from one another, to avoid double-counting. In order to add lengths, we need a length to act as a standard of measure; then we conceive of that measure superimposed iteratively on the length measured, and if two lengths are added, we conceive of them as lain end to end, with the measure applied to both. Addition entails an abstract spatial extension, while multiplication entails the acceptance of a common measure or multiplicative identity. Counting, seemingly the simplest of operations, is not possible without considering objects extensively, and indeed, as I have contended elsewhere, counting implicitly assumes addition. Counting discrete quantities may be considered a special case of operations on extensive measures in abstract space, where each object counted has a measure of 1. Number, therefore, may be regarded as a type of measure, and we are justified in identifying the natural numbers with their analogous measures on the real number line.

Although numbers correspond to abstract extensive measures, a number relates to its subject differently than a measure does. A number cannot inhere in an individual subject as such, but in a group of subjects. Since this group of subjects can be defined arbitrarily, it would seem that number is not capable of the same kind of objective reality as a physical property. While we may believe that the properties of mass and charge truly modify an electron’s mode of being, it is dubious that an objective threeness inheres in a group of three apples, since it is only by a mental operation that we did not choose fewer or more apples to count. Nonetheless, it is certainly an objective truth that the set of apples in question is three in number rather than two or four. The reality of numerical quantity consists not in the objective inherence of twoness or threeness, but in the relation between the abstract number and the (arbitrarily) specified group.

With continuous measure, there is also a degree of arbitrariness, but this is in our choice of measurement unit rather than our choice of subject. Since all units measuring physical properties are arbitrary (including so-called natural units, where ħ = c = 1), the value we ascribe to a property’s magnitude has no objective significance; what is objective is the relation or proportion of the magnitude to other magnitudes, be it our unit of measurement or some other measured value.

In both discrete and continuous cases, we find that quantity has only a relational claim to objective reality (though each is relative in a different way). Aristotle was therefore justified in rejecting the Pythagorean claim that everything is a number. Number is not a substance, but an accident, and with respect to physical objects, it can only be a relative accident. We must similarly reject the claims of any modern Pythagoreans who would reduce everything to quantity, falsely deducing from the supposition that everything is quantifiable that everything is quantity. The same quantities may modify different kinds of properties, and the numerical values of these quantities are arbitrary. Quantity is mere abstraction unless it supposes an entity (substantial or accidental) whose being it modifies. Even assuming its omnipresence among all other entities, this would not abolish the need for other entities. The power of quantitative science lies in the fact that the magnitudes of disparate properties can be quantitatively related, but it is unjustified to presume that such quantitative relations necessarily abolish the distinctions among properties. To take a clear example, acceleration is utterly dependent quantitatively on velocity, but no one thereby argues that velocity and acceleration are the same thing or equivalent. Nonetheless, there may be cases where a quantitative relation does reflect (but does not prove) that the two properties related are the same thing or aspects of the same thing. Such cases must be subjected to metaphysical and physical scrutiny.

Quantity in classical philosophy is restricted to positive real numbers, since it is confined to natural numbers and geometric measures. Both types of quantity may be regarded as extension in an abstract Euclidean space. The development of modern quantitative physics has heightened our awareness of quantity in non-spatial dimensions, including many physical properties that admit negative values. Interestingly, all physical observables can only be measured to have real number values, even in quantum mechanics, where there is an imaginary component to the time-dependent wavefunction. Abstract mathematics admits of more generalized notions of quantity, such as complex numbers, which are used in theoretical physics, but it is by no means clear that they have any ontological status beyond that of computational devices, since all observables are real-valued. Conceptually, complex numbers can be regarded as extensive quantity when we regard the parameters a, b, in a + ib as ordered pairs of real numbers in a Cartesian plane. This operation only reaffirms that real numbers span the entire range of possibility for extension.

Since philosophy does not confine itself to physically observable entities, we may inquire into the status of infinite quantities or orders of infinity expressed in formal mathematics. Cantor’s concept of orders of infinity is problematic if we attempt to regard it as a more generalized notion of quantity. For example, the transcendental numbers (numbers that are not the root of any polynomial) are a proper subset of the irrational numbers, yet they are of the same order of infinity as the irrationals. While order of infinity may be a valid concept, it seems to depart from even the most elementary features of quantity, so that it would be equivocal to regard it as generalized quantity. At any rate, we regard the extension of the physical world and all it contains as finite, and there are strong metaphysical reasons to regard all corporeal entities as finite in extension and all determinate creations as finite in all their properties. Nonetheless, natural theology admits the infinity of God, and some natural philosophers contend that time is infinite. Supposing the reality of infinity, it is not clear if infinity ought to be regarded as quantity or the transcendence of quantity. In formal mathematics, it is necessary to invoke a generalized concept of equality to deal with infinities, making it doubtful that these should be regarded as true quantities. That which is infinite is quantitatively indefinite (e.g., 2/0 = 3/0, etc.), so it is certainly an error to regard infinities as though they were definite quantities. The infinite is related to quantity in much the same way as a universal is related to an individual, as a background spanning the breadth of possible individuations or manifestations.

Greek philosophers were fascinated by the concept of contraries or opposites, applying this idea to virtually every subject of analysis. In the case of quantity, Aristotle concluded that this has no contrary, since he recognized only positive real values as quantities. Apparent contraries such as large, small, many and few are considered relatives, not quantities, since they can only exist with reference to something else. Nonetheless, large, small, many and few resemble quantity more than comparative relatives (larger, smaller, more and fewer) do, as they can inhere directly in the subject they modify. Still, Aristotle is correct to call large and small relative, for we cannot say something is simply large or small without reference to some other extensive measure. Similarly, there can be many or few only relative to some other number. The number or measure serving as a reference may modify some other object, or it may reflect abstract expectations of what the size or number of the considered entity ought to be typically. In any case, it is clear that such relatives are not quantities, though they do relate quantity.

As Plato affirmed, A thing cannot admit contraries at the same time and in the same respect and in relation to the same thing. Further, contrary accidents need some common parameter with respect to which they are in opposition to each other, so they must have some similarity in kind. Aristotle finds it problematic to assert that the quantitative relatives large and small are contraries, because a thing apparently can be considered large and small at the same time. Yet a thing is not both large and small in relation to the same thing, so these relatives are not prevented from being contraries per Plato.

Large, small, many and few are relations between quantities, and they are indeed contrary to each other. This does not mean that numbers and measures themselves admit of contrariety. The most likely example of contrary quantities would be negative numbers or magnitudes, which may be regarded as opposites to their additive inverses. However, we can abstract the plus or minus sign from quantity, regarding positive and negative as qualities or some other sort of accident, while the absolute value or magnitude remains a real-valued quantity. Whether we incorporate positive and negative signs into quantity depends on exactly what is our notion of quantity. From the above discussion, we should be prepared to arrive at a general description.

Quantity tells us the degree to which an accident modifies a substance, or the degree to which a substance is instantiated individually or multiply. Since quantity is about degree, it has the distinction of being subject to the relations of equality and inequality, which have no meaning apart from quantity. It may be objected that by defining quantity in terms such as degree or equality, I am guilty of implicitly circular definition. This may be so, but when we are dealing with the most fundamental categories of being, it will not suffice to formally define words in terms of other words, but rather we must clarify our concepts by seeing how they relate to other intuitive concepts, from which we hope to sharpen our intuition of the concepts being studied. It is not a trivial observation that quantity, degree, and equality are inextricably related, since the last two terms are often equivocally applied to ostensibly non-quantitative categories, yet when this is done, we are really quantifying entities in those categories, without abolishing the need for non-quantitative categories.

Ironically, though quantity pertains to degree, quantitative predicates admit of no degree. For example, there are not varying degrees of participation in four-footedness; one is either four-footed or not. Quantitative predicates express the degree of participation in some property (footedness), but there are no degrees of participation in a clearly defined quantitative predicate. Degree begins and ends with quantity; there are no degrees of degrees. Formally, at least, we can construct quantitative predicates of quantitative predicates: three threes of apples, but this is just nine, which we would not say is more three than three. Similarly, a triad of tripods is not more three-footed than a single tripod. Thus even nested quantitative predicates cannot yield degrees of degrees.

Continue to Part IV

© 2008 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org