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In the preceding six chapters, we constructed a system of ontological categories, which are the objects of any logic that claims to be more than mere formalism, and has at least the potential to address reality. We have considered these categories as a priori possibilities, reserving judgment on the real existential status of each of these types of entities. Having given an extensive intuitive discussion of these categories and having considered a variety of possible alternatives, we are now in a position to subject these entities to logical analysis, followed by the eventual goal of testing whether any of these concepts will help us understand the world as it actually is. For now, we will summarize our results, to serve as a point of reference for further inquiries. These future studies include an examination of modern formal logic and its implicit ontology.
1.0.1 If logic is the way we understand the world, then it must involve assumptions about the world that is, or ontological assumptions.
1.0.2 Logic that lacks ontological assumptions, being purely formal or symbolic linguistics or mathematics, can prove nothing about the world that is. To do so, its terms would have to be predicated of things that are.
1.0.3 A true philosopher's logic, which includes ontology, must be concerned with what type of real-world object his variables represent.
1.0.4 Everyone tacitly holds a theory of ontological categories, expressed in our choice of expressions to describe reality. Ontological theory is a practical necessity of linguistic expression.
1.0.5 The philosopher's logic is a language by which we correspond our ordinary linguistic expressions to things that really are, so it is constrained by ontological reality, not merely some arbitrary rules of grammar or symbolic manipulation.
1.0.6 Ordinary language is a starting point of ontological logic, from which we approach a concept of things that are by modifying our language as needed. To make philosophical concepts intelligible, it is practically necessary to render these in ordinary language.
1.0.7 The exposition of an ontology involves conforming language to categories and distinctions apprehended in part by direct intuition. This use of intuitive knowledge absolves the ontologist of any circular reasoning.
1.0.8 The intuitions underlying ontology should be critically examined. For intuitions that are uncertain, we should construct our logic in a way that it may endure should the intuition fail. Still, we must make some ontological assumptions, or we can never have a theory of reality.
1.0.9 The construction of a categorical system involves the creation of a formal language that we may relate to ordinary speech. Aristotle's Categoriae defines such a formal language; as such, it does not depend on proofs. Our goal, like that of classical ontologists, is simply to describe the relationship between our way of speaking and reality, without constructing or presuming any theory of metaphysics.
1.1.1 Although categoria in Greek means "predication," a grammatical concept, Aristotle is concerned with the real entities represented by predications.
1.1.2 A real entity may be assigned an arbitrary linguistic label or name, which is not the same as the definition of being or meaning behind the name, stating what the thing is (its "essence").
1.1.3 The "essence" of a thing is simply that which makes a thing what it is. We say a thing "has" an essence only out of grammatical convenience; we do not thereby assert that the essence is distinct from the entity. If two distinctly named things are said to have the same essence, this means their names have the same meaning.
1.1.4 We may call entities (not merely words) homonymous if they share a common name, yet have different definitions, and thereby different essences. Entities with a common name and a common definition are called synonymous. Paronymous entities are assigned the same root name, which may be constructed differently for grammatical reasons. Paronymous entities may or may not be similar in essence.
1.2.1 'X is said of Y' or 'X is predicated of Y,' where X and Y are real entities, means that the definition of X may be properly predicated of Y. The definition of a thing is merely a statement of its essence, and an essence is simply that which makes a thing what it is.
1.2.2 If X (or rather, its definition) is predicable of some Y not identical to X, then X is a universal. If X is not predicated of any Y not identical to X, then X is an individual.
Nominalists regard universals (X) as existing in name or concept only, while realists regard them as actually existing. In the strong nominalist view, 'X is said of Y' means only that the name X may be applied to Y, by virtue of the definition of X formally describing Y. In the strong realist view, 'X is said of Y' means that a real entity X somehow is directly predicable of another entity Y, by virtue of its real essence being possessed by Y. In the nominalist view, ontological predication is quasi-linguistic, as a linguistic entity is related to a real entity, while in the realist view, predication reflects a purely ontological relationship, linking entities through a shared real essence.
1.2.3 If a name any meaning, there must be some putative reality that the name signifies or labels. Even if definitions are purely formal, the "essences" they describe are at least real enough for us to meaningfully ask whether they have some relationship to the individuals of which their names are predicated. It is not clear what it would mean for essences to be predicated of individuals.
1.2.4 'X is said of Y' means, at a minimum, that the name of some entity X may be predicated of the entity Y, or at most, that the entity X is itself somehow predicable of Y. In the first (nominalist) view, predication is quasi-linguistic, relating a name to an entity, but in the second (realist) view, predication is purely ontological, relating entity to entity.
1.2.5 'X is in Y,' where X and Y are real entities, means that (1) X is in Y in the ordinary sense of "in"; (2) X is not a part of Y; (3) X is not able to exist separately from Y. We may refer to Y as the subject of X.
1.2.6 If X is in some subject Y, then X is an accident. If X is not in any subject Y, then X is a substance.
1.2.7 The distinctions of 1.22 and 1.26 may be coupled, yielding four a priori possible classes of entities: universal substances, universal accidents, individual substances, and individual accidents.
1.2.8 The existence of a universal substance does not depend on the existence of any determinate individual.
1.2.9 An individual substance is a concrete object that exists independently of subjective perceptions.
1.2.10 A universal accident requires at least a generic subject, though not necessarily an actual individual subject.
1.2.11 The name of a universal accident may be predicated of an individual accident.
1.2.12 The subject of an individual accident may be a substance or another accident.
1.2.13 When the subject of an individual accident is another accident, the first accident modifies its subject, constituting a modified accident.
1.2.14 When the subject of an individual accident is a substance, the accident is intrinsic to the substance and not extensively separable. Further, the being of the accident is proper to the substance.
Entities in the four classes of the ontological square may or may not exist, but it is at least conceivable for them to exist. Even granted their existence, there is room for dispute regarding their existential priority. Plato, for example, made universals primary, while Aristotle regarded individuals as primary.
1.3.1 An individual is said to "instantiate" a universal, defining their existential relationship. The confinement of a universal's essence to a particular object is what constitutes instantiation.
1.3.2 An individual accident is distinct from its universal only in being confined to a particular object, thus an individual accident is nothing more than the instantiation of a universal accident.
1.3.3 An individual substance is more than an instantiation of a species or universal substance, since it may possess other accidental characteristics. We cannot know everything about an individual substance from its universal and its instantiation.
1.3.4 The essence of a universal substance is directly predicable of the object that instantiates it, and the essence of a universal accident is predicable only of the individuated accident, not of the object that is the locus of instantiation.
1.3.5 An individual accident is said to "inhere" in an individual substance, thereby modifying the substance's existence.
1.3.6 Some accidents do not inhere in a discrete individual. Quantity inheres in substance either as a continuum of instantiation or as a collection of individuals treated as a set. Relational accidents may also have multiple subjects.
1.3.7 An individual substance is said to "exemplify" a universal accident when an individuation of that accident inheres in the substance.
1.3.8 Exemplification is necessary if we are to identify the same property in different objects. Exemplification, in turn, requires both distinctions of the ontological square and at least three of the four classes.
1.3.9 An individual substance instantiates a species or universal substance by virtue of exemplifying the properties that define the species. An individual substance is more than an instantiation of the universal, as it possesses other accidental qualities.
1.3.10 The properties that define a species are called its differentiae, as they differentiate the species from other universal substances. The differentiae may also be said to "characterize" the universal substance.
1.3.11 The reality of a species depends on the reality of exemplification of common ontological properties in disparate potential objects. Set theory is an inadequate model for universal substances, since it deals only with collections of individuals, including arbitrary sets whose members share no ontological properties.
1.4.1 There are two kinds of universal substance - species and genera. These are not distinguished in essence, as the same entity could be both a species and a genus, but in predicative relationships. A species is only predicable of individuals, but a genus is also predicable of other universal substances (species or genera).
1.4.2 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.
1.4.3 Suppose an individual A exemplifies all the differentiae of species B, and that 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.
1.4.4 There is no strong analogy between the relationship of individual to species and the relationship of species to genus. An individual instantiates a species, but a species does not instantiate a genus or anything else.
1.4.5 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.
1.4.6 Although a species is more restricted in scope than its genus, it is not a subset or element of the genus. 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.
1.4.7 To build a hierarchy of secondary substances, it is necessary to restrict the definition of genus to having fewer differentiae than its species, disallowing the scenario where a species is its own genus.
1.4.8 There is no logical basis for preferring one taxonomic hierarchy to another. If there is an objective order of species and genera, it must be based on metaphysics or natural philosophy.
1.5.1 Genera that are mutually non-subordinate remain so regardless of taxonomic strategy.
A differentia constitutiva is a differentia that defines a genus. This is the sense of "differentia" we have used so far. A differentia divisiva is a differentia that subdivides a genus. With these definitions, we may proceed to the next thesis.
1.5.2 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.
In modern ontology, the term "category" is used for the classes defined by the ontological square, or for any basic ontological concept, rather than the ten predications of substance and accidents. Also, ontology is assumed to be part of metaphysics, so it is usually discussed only as an a posteriori science. In our theses for Part II, we include only what can be determined a priori from our ontological concepts.
We will use the modern terms "object," "property," and "trope" interchangeably with their classical equivalents: individual substance, universal accident, and individual accident. We also use "particular" and "individual" interchangeably, though in later studies we may distinguish these terms to refer to pluralities and singularities.
2.1.1 Set theory cannot serve as a complete theory of categories of being. A set has no existence apart from the elements of which it is constituted, whereas a category or universal is conceptually independent of its individual instances. A universal is not exhaustively specified by its dependence relations with particulars.
2.1.2 A physical law or other principle that applies to counterfactual conditions would be independent of individuated realities and thus necessitate universals.
2.1.3 The attempt to reduce all entities to individuated accidents or "tropes" can succeed only by ascribing to tropes the ontological independence that is characteristic of substance, so such "tropes" are effectively substances. Trope theory is plausible to the extent that we confuse individuated accidents with the microscopic parts of an object, forgetting that a part of a substance is a substance.
2.1.4 Trope theory's supposed advantage of parsimony (since it postulates but one category) is lost when we contemplate phenomena with multiple tropes. We then must suppose that the tropes independently sustain themselves in a cluster, rather than admit a single substance sustaining all these individuated accidents.
2.1.5 To deny substance because an entity can be completely described by its properties or tropes is to regard properties or tropes as constituents of an entity, rather than ways or modes of being. Since accidents, by definition, are existentially dependent on their subject, they cannot be constituents of that subject. On the contrary, substances bear accidents and sustain them in their individuated existence.
2.1.6 Individuation of accidents is ontologically meaningful, since the observation of a property in a particular object does not entail the apprehension of the same property in another object.
2.1.7 Laws that apply to counterfactual conditionals are necessarily relations between universals, whether the terms of the laws are substances or accidents.
2.1.8 A distinction between dispositional (potential) and occurrent (actual or "categorical") states of objects would necessitate universal accidents.
2.1.9 The distinction between potential and actual is of a different order from the distinction of essences into universal and particular, substance and accident. Individuals may be treated both as actuality and potentiality, and a potentially existent individual is not the same thing as a universal, which may be actualized in many individuals.
Property-object ontologists provide some valid criticisms of modern four-category ontology, but fall into some characteristic errors of their own. Here we will summarize these criticisms and errors, as well as some positive theses.
2.2.1 An individual object indeed bears properties and possesses determinate identity conditions, but if we further require it to be countable as a unity, we eliminate entire classes of particular objects. We could not regard parts of continuous substances as countable, though we can treat each extensive part as an individual object in potentia.
2.2.2 Instead of relying on countability, we may instead regard space as a principle of individuation that distinguishes parts so they are not twice counted. Space indeed is the basis for both discrete counting and continuous individuation.
2.2.3 Modern quantum mechanics is not properly interpreted as denying the existence of individual substances; in fact, it explicitly assumes individuality with the particle-exchange operator. Further, without individuation, there would be no basis for counting or quantifying particles without double-counting.
2.2.4 There may be cases where the individuality (in the sense of "thisness") of particles has no bearing on the physics of a phenomenon, but this does not deny their individuality as such. See 2.2.3.
2.2.5 To limit objecthood to maximally collected parts of substance would disqualify pieces of continuous, homogeneous substances as objects. This reduces objecthood to physical contingency, since no piece of substance is necessarily maximal, and no piece of substance could be unambiguously an object.
2.2.6 Ontology should not depend on physical contingency, but on a priori conceptual distinctions such as we discussed in Part I. It is logically invalid to regard intelligible ontological concepts as meaningless simply because they are not accounted for by (poorly understood) theories of physics.
2.2.7 "Many" or "few" cannot be properties of kinds (universal substances), since a universal is abstracted from any individual instantiations. Being numerous is a property of a set or collection of individuals, not of a universal. The universal does not change whether it is instantiated many or few times. A universal is not the set of individuals instantiating it. (See 1.3.11, 1.4.6, 2.1.1)
2.2.8 Universal substances do not bear properties in the same way that individual substances do. Individual substances exemplify properties, having the potential to actually manifest their instantiation. A universal substance bears properties only in the sense that the essence of a property is contained in the essence of the substance.
2.2.9 Embedding an entity X in the phrase "being X" cannot change its category, since the meaning of "being" is specified by the category of X. It is invalid to regard a universal substance X as a universal property, "being X," because "being X" does not depend on being in a subject, as properties do. Similarly, we could not regard "being this particular piece of X" as a property.
2.2.10 If universal substances were merely complex properties, they must have a subject, and the only possible subject would be individual substances. Yet the "dependence" of universal substances on individual objects is not that of an accident-subject relationship. A property P cannot be instantiated without the presence of a subject in which instantiated P inheres, but a kind K can be instantiated without recourse to some other subject.
2.2.11 The relationships among the four ontological classes or categories cannot themselves be regarded as constituting an ontological category of "relations." In ontology, the subjects of a relation are necessarily ontological entities. The four classes or categories are not themselves ontological entities, so the relationships among them are only relations in a formal or linguistic sense. If the classes were (invalidly) considered as entities, they would not be predicable of each other.
2.2.12 "Substance," "accident," "universal" and "particular" applies only to entities, not to the four classes of the ontological square. Since being an entity in one of the four classes is immediately predicable of "being," there can be no intermediate ontological level between entities of the four classes and "being" itself, much less can the classes themselves serve as such an intermediary.
2.2.13 Confining the term "relation" to that which is predicable of multiple ontological entities, we find that only some of these relations are a type of accident (predicable of other accidents). Other relations may exist between entities of different categories, such as instantiation, exemplification, or differentiation. These cross-category relations do not have the same logical constraints as accidental relations, as only the latter have the special sort of predicability formally defined as "in a subject" in Part I.
2.2.14 The existence of a kind might depend on the actual or potential existence of at least one individual instantiating that kind. Even if this dependence is assumed, the kind would not depend on any determinate individual. An indeterminate individual is no individual at all, so the kind is not dependent on any individual.
2.2.15 Given that an object may change its species, it would follow that objects are not ontologically dependent on species. This would refute Platonism, but not other forms of realism.
2.2.16 If an individual object O transmutes from species A to species B, this does not prove that an individual of species A can exist apart from the species, but only that the object O, considered as materia prima, is ontologically independent of universal substances.
At some point, an ontology will have to give an account of physical laws and states. Modern ontologists take physics as a given and try to explain its results a posteriori. We make no such assumption, and instead describe possible models of physical laws and states. Two such models are those of Armstrong and Lowe discussed in Part II. For Armstrong, laws are relations between universal properties having the form, "All Fs are Gs," where the properties F-ness and G-ness are linked by necessity. For Lowe, a physical law is of the form "K is F" or "Ks are F" where K is a kind and F is a property. In Lowe's conception, it suffices for the predication to be true; there is no need for an additional condition of necessity. These models are not mutually exclusive.
2.3.1 (F-ness necessitates G-ness) → (x is F → x is G) is a valid inference when F and G are universal properties, and x is an individual potentially exemplifying these properties. The inference also holds when F and G are each of the form "being K," where K is a kind, and x is an individual that can potentially instantiate that kind.
2.3.2 Let A be an individual object, B be an instantiation of some kind B', and C be an instantiation of some property C'. Then the following inference holds: (A is B) & (B' is C') → (A is C). This is because an instantiation (B, C) of a universal has the essence of the universal (B', C') that is instantiated.
2.3.3 The dispositional properties of individual objects do not necessarily tell us anything about their species, or vice versa. In fact, objects of a kind only necessarily share the differentiae constitutivae.
2.3.4 Universal substances may be characterized by dispositional or occurrent properties. Nonetheless, universals cannot have determinate states or activities, nor the potential or disposition for such determinate states or acts. Dispositional and occurrent properties of a species can be considered only generically for individuals of that species.
2.3.5 When a property is exemplified by an individual substance or object, it is necessarily local and determinate in its effects and perceptibility. Such local, determinate effects are incompatible with universality, so the accidents perceived in an object are not universal. Nonetheless, individuated accidents are nothing more than instantiations of universal essences, so we truly perceive the essence of the universal in its instantiations. Since a trope modifies the existence of a particular object, it must itself be particular.
2.3.6 The "truthmaker principle" cannot be satisfied in general by ontologically simple entities. There will be many complex statements whose truth can only correspond to states of affairs which are composed of relations among ontologically simple entities. Even the simplest states of affairs are really unions of objects and tropes. States of affairs, in the most general sense, are not ontological entities, but reified propositions.
We find nothing in modern ontological discussions that would add to the four classical ontological classes, nor are there any strong a priori grounds for eliminating any of them.
2.4.1 Abstraction, in the most general sense, is a mental operation whereby we consider an entity independently of some condition of being. We may abstract form from matter, universals from particulars, substance from accidents, or the incorporeal from the corporeal. "Abstract" and "concrete," therefore, cannot be well-defined ontological categories, as the operation of abstraction does not uniquely specify the ontological status of the thing abstracted.
Notwithstanding the above, most modern philosophers regard "abstract" and "concrete" as subclasses of particular objects. While we know that abstract entities can be universals, we should examine whether that is the case for mathematical entities.
2.4.2 Geometrical entities, when abstracted from real physical space, are universals. They are not individuated by specification of size and location in an arbitrary abstract space, since such a space has no determinate objects a reference points for individuation. There is no uniqueness for abstract geometric objects.
2.4.3 When a geometrical entity is iterated in the same abstract space, it is enumerable, but lacks the "thisness" or haeccity essential to individuality. The formal geometrical situation is unchanged by swapping iterations. There is no material distinction since the entities are immaterial. Each geometric object is its own kind.
2.4.4 We may further see that mathematical objects are universals because they can be "said of" individual entities. In other words, the essence or definition of a mathematical object can be predicated of different conceivable individuals.
2.4.5 Numbers can be universal accidents. When a number qualifies a universal accident, it cannot be a particular object, since these cannot qualify universals. In other contexts, a number can be a particular accident.
There is nothing in modern ontology so far that would obstruct us from proceeding with an analysis of the classical ten categories. Mathematical objects are actually better handled with classical universals than with the equivocal abstract-concrete distinction. We further note that it is a mistake to limit the scope of ontology, an a priori science, to that of empirical science.
3.0.1 For convenience, we may use the grammatical term "predicated of" to refer to the ontological relationship it represents, namely the relationship between individual and universal, and between specific and generic universals, as discussed in Part I. If A is "predicated of" B, we may call A a "predication" of B.
3.0.2 Each higher genus of substance is a predication of its subgenera or species. Accidents might be similarly classified in a taxonomy, with each more generic accident being predicated of the more specific. There need not be a unique hierarchy of classification for substances or accidents.
3.0.3 "Substance" may be considered as a predication of all genera of substances, since genera are kinds of substances, and all substances share a common mode of being. Thus substance may be regarded as a single predication or category.
3.0.4 "Accident" is subdivided into several irreducible categories, reflecting divergent modes of being. Each of these categories has different logical relationships with substance. The categories of accident are quantity, quality, space, time, and relation.
3.0.5 All entities for which we have a concept falls under one of the six categories of substance, quantity, quality, space, time, and relation, or is a composite of these. We cannot prove anything about entities for which we have no concept.
3.1.1 Ontological affirmations are simple statements such as "X is" or "X exists," where X may be an ontologically simple entity. The assertion is true only if X is real. Such assertions need consist of only one category term, as "being" is not a genus or category, but the very subject of categorization.
3.1.2 All logical affirmations - that is, all affirmations which are not simple ontological affirmations - are necessarily composed of terms representing entities in two or more categories. A single category term (without an assertion of existence) does not constitute an assertion.
3.1.3 The ens copulae, or "is" in "X is Y," has no categorical significance, though it is a third predication in a formalistic sense. It merely shows the ontological linkage between X and Y.
3.1.4 Truth and falsehood make reference to reality, so they cannot be mere semantic concepts. The fact that logical assertions, or statements that may be true or false, require entities from more than one category means that the categories are essential to the structure of reality.
3.1.5 "Truthmakers" of logical affirmations can be no more primitive than a combination of entries from two or more categories, yet single-category entities remain more primitive than such truthmakers or "states of affairs."
3.1.6 The logical affirmation "X is Y" asserts the reality of composite entity XY, but this necessarily entails the reality of X and Y, even if they cannot exist in isolation.
3.1.7 Both simple and composite entities may be declared with or without an affirmation. Affirmation entails an assertion regarding the being of an entity, either with an explicit term such as "is" or "exists", or through an understanding that such an assertion is implicit in the declaration of a term.
Substance is defined as an entity that is not "in a subject." (See 1.2.6) As such, it has a greater degree of ontological independence than accidents, and may be considered the underlying entity beneath or behind accidental phenomena. The intuitive notion of substance is something that exists directly and simply without dependence on another entity. When we speak of predication below and elsewhere, we are concerned with ontological necessity, not mere rules of grammar.
3.2.1 The essence of an individual substance cannot be predicated of anything other than that individual, though the essence or definition of a universal substance can be predicated of other substances.
3.2.2 The essence or definition of an accident may not be "said of" or predicated of a substance.
3.2.3 All secondary substances, including higher genera, are directly predicable of individual substances.
3.2.4 Universal accidents are "in" individual substances as subjects, but only in an imperfect way, modified by the determinate contingencies of individuation.
3.2.5 An individual more closely resembles its species than its genus, since more of the individual's traits are found in the definition of the species than in that of the genus.
3.2.6 The essence of a genus may be predicated of a species, yet the essence of the species would not be predicable of the genus. A species thus more closely resembles an individual substance in its essential impredicability. (See 3.2.1)
3.2.7 When two universal substances are not subordinate to one another, there is no basis for regarding either one as closer in resemblance to individual substance, in the sense of 3.2.5 or 3.2.6.
3.2.8 The essence of a universal substance always helps specify the essence of the individuals that instantiate it.
3.2.9 Accidents do not necessarily help specify the essence of an individual substance, since they are not always essential to the identity of that individual.
3.2.10 Only accidents essential to a species or genus (differentiae constitutivae) are "in" that species or genus as a subject. Non-essential accidents, even if they are exemplified by an individual of the species or genus, are not "in" that species or genus as a subject.
3.2.11 A species or genus is not "in" an individual, in either the formal or ordinary sense of "in," as no individual can contain the species. A species or genus does not depend on any determinate individual for its existence, nor is there a part-to-whole relationship between universal and individual substances.
3.2.12 Per 1.3.4, the definition of a universal substance is predicable of an individual, but the definition of an accident is never predicable of an individual substance.
3.2.13 A substance may have other substances as its parts, but its accidents are not parts. A part is conceivably separable from the whole, but it is not conceivable for an accident to exist apart from its subject.
3.2.14 Notwithstanding grammatical convention, species and genera are not determinate substances. A universal substance is not a certain "this," but a substance of certain qualification.
3.3.1 The definition of a differentia is predicable of the species or genus it characterizes, but not in the same way that of a universal is predicable of an individual. The essence of the differentia belongs to the essence of the characterized substance, but it does not constitute the full substantial essence.
3.3.2 The essence of a differentia is instantiated in an individual substance via its species, but the differentia's essence still does not constitute a full substantial essence.
3.3.3 A species may be defined per genus et differentia. Since a genus and differentia are both universals, the essence of a species does not change with the fate of individuals that instantiate it. Thus a species is not like a set, which changes as members are added or removed from existence.
3.3.4 A differentia modifies a genus when it characterizes a species, and this modified genus retains all that is essential to the genus.
3.3.5 As we remove differentiae, we progress to higher genera, and as we add differentiae, we define progressively narrower subgenera or species. This is not the same as subdividing a set into subsets, though it is certainly not possible for a subgenus to have more members than its genus, nor can an individual of the subgenus fail to be an individual of the genus.
3.3.6 An individual may admit of degrees of participation in a species, if the differentia defining the species can vary in degree, and the species is characterized by the differentia without regard to degree beyond some threshold.
Contraries are accidents such that one is necessarily present in a subject to the extent that the other is absent, and vice versa. We may improperly refer to substances as "contrary" if they each possess contrary accidents.
3.4.1 Individual substances can receive contrary accidents at different times. If the accident changed into a contrary is a differentia, the individual may change species.
3.4.2 Universal substances cannot change accidents, as the only accidents they have are differentiae, which are essential to them. Thus they cannot receive contraries, nor can they be a locus of action.
3.4.3 An individuated accident or trope cannot be the subject of contraries, even if we allow that a trope may persist in time. If a trope is replaced by an instantiation of a contrary property, the original trope ceases to exist. A trope cannot instantiate different properties over time, as it is nothing other than an instantiation of a given property.
3.4.4 Statements, evaluated in context, do not admit contraries. In other words, a logical statement cannot assert contrary realities.
3.5.1 The definition of an accident is not predicable of its subject, unless the subject is a subordinate accident, in which case the definition is predicable as that of a genus is predicable of a sub-genus.
3.5.2 The name of an accident is univocally predicable of its subject only when the definition is predicable, as in 3.5.1.
3.6.1 Quantity is "how much" of something there is, in number or amount or measure. As it requires a subject, it is an accident.
3.6.2 As an accident, quantity modifies its subject, but cannot abolish or replace it, for it depends on the subject for its existence.
3.6.3 Purely quantitative or mathematical relations cannot yield facts about the quantified properties themselves. In physical equations, we assume the quantities represent amounts of various qualities, but cannot mathematically prove the reduction of one quality to another, as the relations only operate on quantities.
3.6.4 Quantity can modify substance, quality, space, or time. The same kind of quantity may be applied to subjects of different ontological categories.
3.6.5 Continuous quantity, or measure, can admit place, length, area, volume, and time as its subjects. In Euclid's geometry, length, area, and volume are both quantities and the subjects of quantity. Continuous quantity is geometrical when it measures spatial relations, as is the case with all the subjects listed above save time.
3.6.6 In contrast to Euclid, we may use the terms "length," "area," and "volume" to refer to geometric quantities, distinct from the loci (e.g., lines, surfaces, solids) that are the subjects of these quantities. Higher dimensional geometric quantities are at least logically possible.
3.6.7 Number, in the sense of discrete quantity, is needed to quantify distinct iterations of a substance. It is independent of the spatial relations among objects, so it is not geometrical.
3.6.8 Number does not inhere in any particular substance, but in any arbitrarily chosen group of substances. Though the group is a mere formal construct, the relation between the group and its number is objectively real.
3.6.9 Number requires at least a virtual space to serve as a medium of pluralization, to distinguish multiple iterations, even if there are no determinate individuals in abstract space.
3.6.10 A distinct kind of quantity is needed to quantify linguistic meaning, as a sentence with an added phrase contains more meaning than the original sentence, but beyond that we cannot make any comparisons of equality or inequality.
3.6.11 Time is not reducible to spatial extension, but it may be measured by the same kind of continuous quantity as spatial extension. Thus the distinction between geometric and non-geometric continuous quantity is a distinction in subjects, not in kinds of quantity.
3.6.12 In contrast with number and measure, which are extensive magnitudes, there may also be intensive magnitudes that describe qualitative intensity, for which there can be no unit of measurement. Intensive magnitudes lack multiplicative identity, addition, and counting. They only yield knowledge of comparative inequalities and rough equality.
3.6.13 Intensive magnitude might be regarded as embedded in a continuum of extensive magnitude, where the unit of measure is unknown. Even so, intensive magnitude could still be regarded as true quantity, though in imperfect form.
3.6.14 Extensive magnitudes must be conceived as existing in a space of non-overlapping quantities. Counting requires objects to be regarded as distinct from each other, and addition requires a length to act as a standard of measure.
3.6.15 Discrete numbers may be regarded as embedded in a continuum.
3.6.16 As shown in another work, 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 may identify the natural numbers with their analogous measures on the real number line.
3.6.17 Despite the correspondence of number with abstract measures, a number relates to its subject differently than does measure. Number inheres in an arbitrarily defined group (3.6.8), while measure has a well-defined subject. The magnitude of a measure is based on an arbitrary unit, but number has an objective value once its subject is specified.
3.6.18 Both discrete and continuous quantity have only relational claims to objective reality, so they cannot be substances. The reality of number is in its relation to an arbitrarily defined group of objects, while the reality of a measured magnitude is in its relation or proportion to other magnitudes.
3.6.19 Complex numbers can be regarded as extensive quantity when their parameters are regarded as ordered pairs of real numbers in a Cartesian plane. Thus complex numbers need not present a new kind of extensive quantity.
3.6.20 That which is infinite is quantitatively indefinite, so we should not regard "orders of infinity" as definite quantities. The infinite is related to quantity much as a universal is related to an individual, as a background spanning the breadth of possible manifestations.
3.6.21 "Large," "small," "many" and few are not quantities, but relatives that exist only with reference to some measure or number.
3.6.22 Though quantitative relatives (3.6.21) may be contrary to each others, quantity itself need not admit of any contrary.
3.6.23 "Positive" and "negative" can be abstracted from quantity, leaving only magnitudes without contraries as quantities.
3.6.24 Quantity gives the degree to which an accident modifies a substance, or the degree to the which a substance is instantiated. Degree, in turn, is necessarily subject to relations of equality and inequality, which depend on quantity for intelligibility. The concepts of quantity, degree, and equality are thus inextricably linked, though the last two may be applied to non-quantitative categories.
3.6.25 Clearly defined quantitative predicates admit of no degree. There are no degrees of degrees, not even when we construct nested quantitative predicates.
Space is a medium of pluralization, either for universal, abstract entities as in mathematics, or for determinate individuations of substance as in physics. These two concepts of space, abstract and real, are defined in the first two theses below.
4.1.1 Indeterminate Space for Indeterminate Entities: Quantity in the abstract requires at least an abstract space or medium of pluralization for iterations of abstract entities.
4.1.2 Determinate Space for Determinate Entities: A world with real individuations of substance would need a real space for those individuations to co-exist in a way that enables them to be a plurality or merge as parts of a whole substance. Space in this sense is the principle of pluralization for individuated substance.
4.1.3 Relativity may prove (a posteriori) that the magnitudes of spatial displacement and temporal duration are interdependent, but this does not give an ontologically complete account of space and time. Space and time, though quantifiable, are not mere quantities. There is more to time than extension, as indicated by its asymmetry with respect to forwards and backwards.
4.1.4 The "times" of relativity and quantum mechanics are distinct from each other and from metaphysical time. Relativity measures time by external observations of events from a distance, and its characterization of time as "speeding up" or "slowing down" may be interpreted as changes in the rate of intrinsic motion of an object. Quantum mechanical time is a parameter of the evolution of a wavefunction.
4.1.5 If space (in the sense of 4.1.2) is absolute, then "place" is an inherent accident of substance; otherwise, it is a relative accident.
4.1.6 Relative accidents, such as relative position, velocity, and acceleration, only have meaning if we specify a common time for the objects bearing these accidents.
4.1.7 Time has a unique logic, specifying a principle of non-contradiction for other accidents. A substance may endure through time, manifesting contrary accidents at different times, but not at the same time. Each passing instant requires the manifestation of one or another accident.
4.1.8 Quantum mechanics does not prove the simultaneous individuation of contrary properties in a so-called "superposition of states," which is never physically observed. On the contrary, the probabilistic formalism of quantum mechanics presumes the mutual exclusivity of orthogonal states.
4.1.9 The simultaneous possession of contrary accidents by a substance is not psychologically conceivable, since the mind operates in time. Time distinguishes successive states of substances, forbidding contradictions within a state.
4.1.10 It is no contradiction to regard contrary accidents as possible futures, as the future is not definite from the perspective of the present. In order to avoid the conclusion that all things are strongly determined, there must be some reality to past, present, and future. Time is then the medium of action, whereby one of many possible futures is realized in the present, to be fixed forever in the past.
4.1.11 An action necessarily occurs at a definite moment in time. If we add the physical postulate that there is no action at a distance, we may also confine acts to a locality, and define an "event" as the spatiotemporal coordinates of an act.
4.1.12 Relativity does not abolish the reality of past, present, and future, but makes these merely local experiences, as there is no objective simultaneity at a distance. A future event can never precede a past event in a temporal loop.
4.1.13 The physicist's "event" is really a geometric object (4.1.11), not an action. We should not mistake events for actions, nor should we reduce reality to perturbations of a space-time continuum, as though space-time were substantial.
4.1.14 The warpable "spacetime" of general relativity is really a warpable gravitational field, embedded in true, potentially infinite space. This true space is not empirically observable, due to the lack of an absolute reference point. As a mathematical convenience, we may regard motion along the contours of gravitational fields as following the contours of "space" itself, but this is not ontologically probative.
4.1.15 True space, since it is not a substance but a medium of pluralization for substance, is uniform and potentially infinite. Euclidean space best exemplifies these characteristics, but it is only a mathematical model.
4.1.16 Space and time are distinct categories of accident. Space is a medium that makes possible the individuation of substance, while time makes possible the individuation of successive actions and qualities in the same substance. Space and time are each directly linked to substance, so there is no a priori requirement of simultaneity at a distance. We can only measure space and time relatively, but should not confuse measurement models with space and time themselves.
4.2.1 We define "relations" in the sense of "relational accidents" to mean those accidents that are predicable only in relation to another accident of some substance.
4.2.2 "Relatives" are those relations that inhere directly in a specific subject, but only in relation to the property of another subject.
4.2.3 Relations that are not relatives, and thus not inhering in a specific subject, may be conceived as higher order accidents relating two or more accidents.
4.2.4 The subjects of relations are always accidents, and not substances, except indirectly by virtue of their accidents.
4.2.5 A relation is truly an accident, since it is "in a subject" in the formal sense, but it is necessarily an accident of composite subjects.
4.2.6 A relation may exist either as a universal or as a trope. Relations between concrete objects are instantiations of generalized relations.
4.2.7 Relations can be physically real, as there can be real properties that are dependent on relative accidents.
4.2.8 One type of relative may be expressed in the form Rxy, or "x is R of y," where x is an individual accident, R is a relative, and y is a universal accident.
4.2.9 A second type of relative may be expressed in the form Rxy, or "x is R of y," where x is an individual subject (substance or accident), R is the relative, and y is an indeterminate subject.
4.2.10 Not all two-predicate relations are logically similar, as their variables may represent different ontological classes.
4.2.13 Relations may admit of contrariety and reciprocation only when they are two-predicate relations of the form "x is R of y" or higher-predicate relations of the form "(x1, x2, x3...) is R of (y1, y2, y3...)."
4.2.14 For a relation to admit of degree, at least two of its predicates must admit of degree. By this standard, no relatives of the second type (4.2.9) can admit of degree, and only some relatives of the first type (4.2.8) may do so.
4.2.15 If a relation A is capable of reciprocation, then there is some reciprocal relation B for which the following statements hold: (1) (x is A of y) → (y is B of x), and (2) (y is B of x) → (x is A of y).
4.2.16 A relation R is symmetric if Rxy = Ryx; otherwise, it is asymmetric. Symmetric relations are trivially self-reciprocating.
4.2.17 Relatives of the first two types (4.2.8 and 4.2.9), if given properly, are predicated in relation to correlatives (x and y in 4.2.15) that reciprocate. By "given properly," we mean that the correlatives are defined by their essential accidents.
4.2.18 Any reciprocal relation that is predicated of two real subjects is also real, being bound to essential properties.
4.2.19 There is a third type of relatives that do not admit of reciprocation, namely relations defining potential knowledge or perception. These are of the form "x is R of y" where y is an indefinite potential.
4.2.20 The part-to-whole relationship is not a relational accident, because the being of a part is distinguishable from its union with the whole, while for relational accidents, being means being related to something else. Parts of wholes are divisions of substances, unlike relational accidents, which take accidents as their subjects. We may nonetheless relate properties of parts and wholes.
4.3.1 A quality is something in virtue of which a subject possesses some characteristic. It answers the question quale, "what sort/kind is it?"
4.3.2 Quality has greater ontological independence than quantity, as quality is the source of an attribute while quantity is a modification of an existing attribute.
4.3.3 Qualities differentiate essences into kinds. We may call a quality "essential" with respect to a species if it is a differentia of that species. Otherwise, we may call it "accidental" or "incidental."
4.3.4 Those qualities that are reducible to other qualities are called "secondary," while those that are not reducible to other qualities are called "primary."
4.3.5 Qualities may admit of intensive or extensive magnitude. In modern speech, it is common to restrict the term "qualitative" to qualities of intensive magnitude, while extensively measurable qualities are "quantitative." This usage should not obscure the fact that even mechanistic metaphysics requires qualities as subjects of measurement.
4.3.6 In Thomistic metaphysics, a quality inheres in a subject through its form, while a quantity inheres in a subject through its matter. This is congruent with 4.3.1, since a form defines what kind of essence a thing is. Quantity, by contrast, regards substance with respect to iteration and extension.
4.3.7 There are at least four types of qualities: (1) states and conditions; (2) natural capacity and incapacity; (3) affective qualities or affections; (4) shape or external form. 4.3.8 The four types in 4.3.7 may be metaphysically distinguished according to the "mode of determination of the subject to accidental being," which may be taken in regard to: (1) the very nature of the subject; (2) action and (3) passion resulting from its natural principles (matter and form); (4) quantity.
4.3.9 The shape of a substance is its spatial distribution, or how the quantity of substance is distributed in space, without regard for intrinsic characteristics of the substance itself. Thus it is the mode by which substance participates in quantity.
4.3.10 Shape presumes the existence of space, but not of time, so it is, of itself, without movement, as is quantity.
4.3.11 The properties of a substance are distributed in accordance with its spatial distribution, so shape is a true quality, being a means by which substance participates in accidental being.
4.3.12 Shape distinguishes things into kinds, and is indeed the most primitive notion of form.
4.3.13 As shape can only account for static, homogeneous matter, any sophisticated physics would require natural capacities to act.
4.3.14 Natural capacities are true qualities, being dispositional and thus a means by which a substance can possess other accidents. They distinguish substances into kinds.
4.3.15 Natural capacities fully realize their being in the act of occurrence, yet an act of occurrence is not a quality, but a relation (Part V).
4.3.16 The outcomes described by modal logic express the result of dispositions, and thus modal logic cannot serve as the basis of disposition.
4.3.17 Affective quality deals with how a substance is acted upon.
4.3.18 Affective qualities admit only of intensive magnitude, and are not reducible to quantity.
4.3.19 Perceptibles are a kind of affective quality. They may be explained (1) classically, with the perceived quale inherent in the external object, or (2) in a Lockean sense, where the quale is in the mind, with the sensitive faculty as its subject, and the external sensation acting as a stimulus upon the mind.
4.3.20 Non-perceptibles may be affective qualities if they determine the mode in which an object is acted upon. Some properties may be regarded both as natural capacities and as affective qualities.
4.3.21 Affective qualities are dispositional, requiring an action to be manifested occurrently.
4.3.22 Affective qualities differ from natural capacities in relation to the act of occurrence, for affections do not make possible the act they passively receive, but depend on an extrinsic factor.
To speak intelligibly about states and conditions, it is necessary to make some definitions and hypotheses about nature. These will need to be justified in a treatise on natural philosophy. We adopt Aristotle's definition of nature in the Physics as an inherent principle of motion or change. Motive principles are necessarily teleological, being directed toward an end. Thus we may define a state or condition as "good" to the extent that it is in accordance with the subject's nature.
4.3.23 A condition is a quality that determines how a subject is disposed with regard to its nature. A state is a condition that is held for an extended period of time.
Aristotle thought that natures could be possessed by simple bodies, living beings, and intellectual souls. We consider these as a priori possibilities.
4.3.24 In Aristotelian physics, the "end" of a simple body would be its natural place. In modern mechanics, the telos of a simple body is not a fixed end-point, but a direction along a spacetime contour. In both systems, a body blindly follows its natural motion unless acted upon by an extrinsic force. Molecules may also have principles of motion, since their constituent particles may be bound in a way that creates a new nature.
4.3.25 Fundamental particles may be more than simple bodies, having other dynamical principles besides local motion. Such particles may have several distinct principles of motion or change, reflecting different aspects of their nature.
4.3.26 Apparent "natures" of fundamental particles could be natural capacities of more basic entities.
4.3.27 If living things truly have a characteristic principle of motion or change, this may be seen in growth and reproduction, common to all organisms.
4.3.28 The telos of ontogenic growth is the adult form. The telos of reproduction is producing viable offspring. If some form of biological essentialism, even as mild as saltationism, is admitted, then the telos of reproduction would include producing the essential species traits in the offspring.
4.3.29 The teleology of organisms differs from that of simple bodies. Simple bodies have "tendencies" to move or rest in accordance with intrinsic properties such as inertia, whereas organisms each grow to a definite form, and then propagate either that form or at least enough traits for its offspring to grow to some other form and generate further offspring.
4.3.30 We may define an organism's state or condition as good or bad with respect to its telos, the viably functioning adult form. We may speak of such states or conditions in terms of how they affect the organism's ability to grow, survive and propagate.
4.3.31 Animals with behavioral intentionality have mental states that may be considered good or bad with respect to their fitness for survival in a given environment.
4.3.32 Natural principles in an animal may be in conflict with each other, so that what is ontogenically favorable is reproductively unfavorable, or what is advantageous in one environment is unfavorable in another, or instinctive behaviors may not optimize reproductive outcomes.
4.3.34 An intellectual being, which is capable of understanding ideas rather than merely processing representations, has its own proper nature or principle of motion. In addition to intellect or understanding, this nature has volition or will that can act in accordance with the intellect.
4.3.35 The telos of intellect is Truth, for to understand something is to regard it as true, while the telos of the will is called the Good, or that course of action that is in accordance with Truth.
4.3.36 If we acknowledge that Truth and the Good transcend material existence, it follows that the intellectual soul is not confined to material contingency.
4.3.37 In the absence of an ideal teleological Good, there is no basis for any morality other than utilitarianism. "Self-evident" virtues and rights become arbitrary claims.
4.3.38 The telos of the human will may be man's happiness or felicity. The pursuit of happiness may be equivalent or subordinate to the pursuit of the Good, or the pursuit of the Good might be subordinate to the pursuit of happiness.
4.3.39 If the tendencies toward truth and goodness indeed lead toward an Absolute end, that end would be God. In that case, as all intellectual and moral activity tends toward God, the natural end of an intellectual soul can only be fully realized in the Absolute Good which is God.
4.3.40 Habits and dispositions (states and conditions) characteristic of an intellectual soul are defined as "good" and "bad" with regard to the soul's ends of Truth and the Good. Moral habits, such as virtues and vices, and intellectual habits (types of knowledge) are included here.
Substance, quantity, space, time, relation and quality span the entire ontology of intelligible reality. Other known categories are either reducible to these or not ontological at all.
5.1.1 An action is a time-dependent relation.
5.1.2 Actions denoted by transitive verbs inhere in neither the subject nor the object. The active subject is the source of action, while the passive object is the recipient of the action. Transitive actions are predicable of a subject only in relation to some other object.
5.1.3 Actions denoted by intransitive verbs can inhere in a single subject, yet they are truly relations since they have two predicates, namely the same subject considered at different times.
5.1.4 There are six possible kinds of time-dependent relations of substance and accidents: generation, destruction, increase, diminution, alteration, and local motion.
5.1.5 Generation and destruction are changes where substance comes into or out of existence (absolutely or by transformation) with respect to time.
5.1.6 Increase and diminution are changes in quantity (of substance or accident) with respect to time.
5.1.7 Alteration is a change from one quality to another with respect to time.
5.1.8 Local motion is a change in position with respect to time.
5.1.9 Local motion is a relation between the positions of a substance at different points in time. A vector difference provides a complete account of motion in this sense.
5.1.10 Since motion, and therefore velocity, requires some interval of time, velocity is ontologically a relation, being the ratio of motion to the time interval. The expression of instantaneous velocity as a quality of varying intensity is just a mathematical convenience. The same holds for other derivatives of motion.
5.2.1 Affection or passion is the inverse relation of a transitive action.
5.2.2 Aristotelian "position" denotes the relative disposition of substances or their parts. Thus it is a relation among the spatial locations of various substantial parts.
5.2.3 The classical category "state" (distinct from the type of quality in 4.3) describes the qualitative disposition of a subject, or its relation to certain qualities. Such a "state" is a kind of relation, except when it is a simple qualitative predication, in which case it is a quality.
The distinction between potency and act is metaphysical, as are essence and existence, matter and form. These distinctions do not create new ontological categories, but consider the same entity from different aspects of its being.
5.3.1 Act is the realization of being and potency is the capacity for that realization. This distinction does not create two kinds of entities, but considers the same entity as having the fullness of being in some respect or lacking it.
5.3.2 Essence and existence are potency and act in the order of existence. Essence makes an entity what it is, abstracted from its actual manifestation as an existent being. An essence has at least the potential to exist. Existence is not another entity superadded to essence, but the full realization of essence.
5.3.3 Matter and form are potency and act in the order of essence. As neither matter nor form alone constitute a corporeal essence, they are not an ontological division. They are principles of the same essence, namely its capacity for extensive actualization and the determinate realization of this actualization.
5.3.4 Potentiality and actuality transcend the categories, being predicable of entities in every category. They are different ways of viewing the being of an entity, but do not establish a multiplicity of entities.
Mechanists postulated mind, matter, motion and rest as ontological categories. We have already shown that motion is a type of relation. Rest, being the privation of motion, is no entity at all. In relativistic mechanics, motion and rest is definable only in relation to other kinetic states, making both concepts relational. This leaves the ontological status of mind and matter to be determined.
5.4.1 Matter in the sense of corporeal objects is substance.
5.4.2 The dichotomy of quantity into intension and extension does not imply an ontological dichotomy of mind and matter, since there is more to reality than quantity. Further, not all intension need be mental, nor is extension foreign to the mind.
5.4.3 Mind could be a substance or a relation or a quality, according to possible theories of psychology.
Many philosophers have advanced conceptual, semantic, and symbolic categorical systems. While these systems are not necessarily invalid, they are not adequate substitutes for ontological categories.
5.5.1 Kant's categories are conceptual, not ontological. Space and time for him are pure intuitions giving form to the mind's outer and inner senses, not real accidents specifying the determinate being of external objects. His categories of quantity, quality, relation, modality are considered as concepts, not as ontological entities.
5.5.2 Kant regards quantity, quality, and relation as providing content to judgment, while modality denotes the value of the existential copula.
5.5.3 The Kantian categories of conceptual content (quantity, quality, and relation) and conceptual form (space and time) correspond to the five categories of accident we have expounded.
5.5.4 We need not assert that all the classical categories are real entities in order to contend that they are at least a priori possibilities. Not all concepts correspond to reality, but if none of the classical categories described reality, all of reality would be unintelligible to man.
5.5.5 Husserl recognized that some conceptual (formal) categories may correspond to (material) categories of objects considered a priori. This descriptivist stance is congruent with the Scotist notions of logic and ontology we have espoused from the outset.
The concepts of modern physics are reducible to the classical categories or not ontological at all.
5.6.1 Inertia has two senses: (1) a kinematic state of uniform linear motion in some reference frame; (2) a body's resistance to motion. Inertia in the first sense is a relation, and in the second sense is a quality.
5.6.2 A frame of reference is a relation whose predicates are variables defining the kinematic state of a reference object.
5.6.3 Space-time is a mathematical construct relating the extensive aspects of space and time via a pseudometric. Since there is more to time than extension, we cannot infer a categorical merging of space and time.
5.6.4 The warping of space-time does not imply space or time are substances, for it is only the pseudometric relating spatial and temporal extension that is warped.
5.6.5 Momentum, being the quantity of motion, is necessarily relative, as velocity is relative.
5.6.6 Newtonian force is a physical agent, but it need not be substantial, as substance may act through its accidents. The association of force with substantial objects in varying degrees suggests it is a non-essential quality.
5.6.7 The magnitude of force is relative to one's reference frame, suggesting force is a relative accident.
5.6.8 Gravitational force might be replaced with the geometry of space-time, but this could be a mere formal substitution.
5.6.9 Energy in physics is the capacity to do work. In Newtonian mechanics, this is measured as force over distance. It may subsist in potentia or be actualized kinetically.
5.6.10 Physical energy is at most a subclass of Aristotelian energeia, or being-at-work, as it refers only to the being of certain physical changes. As such, energy is not a distinct category of entity, but a mode of being for various qualities or relations.
5.6.11 Classical gravity, or heaviness, is a quality of substance, proportionate to and dependent upon mass.
5.6.12 In relativity, mass is not the quantity of substance, but a relative measure of gravitational inertia. Gravity is defined as the ability of mass to conserve space-time. This is circular, as gravity and mass are defined in terms of each other. This geometric description of gravity does not give ontology.
5.6.13 Electricity and magnetism must be of the same category, under the interrelationship shown by special relativity.
5.6.14 Electromagnetic fields may be substantial entities, generating photons and other manifestations of electromagnetic force.
5.6.15 Charge is a quality of subatomic particles that is necessarily related to electrical force fields.
5.6.16 Treating fields as a property of space erroneously regards space as a substance. More properly, the "space" of which fields are a property is a Cartesian plenum, akin to the modern "vacuum field."
5.6.17 In the Standard Model, substantial fields can be replaced by particle exchange to account for fundamental forces, save gravity. However, the particles exchanged may be merely virtual, or even the product of the field. In any case, no new category is needed.
5.6.18 The particle states of quantum mechanics may be qualities, but this does not imply the non-existence of substance.
5.6.19 The quantum wavefunction measures objective propensities or tendencies of potential being. It is the quantification of the potential to instantiate certain properties.
5.6.20 The exotic properties of nuclear physics and quantum mechanics are all qualities, since they define the form of a substance.
5.6.21 Thermodynamic properties are secondary qualities arising from the behavior of an ensemble of microstates.
5.6.22 If physical laws are nothing more than statistically certain correlations of events, they are merely accidental relations.
5.6.23 If physical laws are causal, they could be descriptions of the dependence of an effect on its cause, or they might themselves be causal agents. In the latter case, they have ontological status.
5.6.24 If physical laws are agents, they may be substantial. Such substantiality must come from an incorporeal metaphysical agent. Since the laws of physics are not logical necessities, the need for metaphysical agency is unavoidable if causal laws are to be enforced.
5.6.25 If spontaneity or randomness did not depend on anything, they would be existential relations, not ontological entities, defined by the privation of a causal relation or agent.
5.6.26 Absolute randomness would imply that a thing does not need to happen for any reason, not having any cause. If this is conceded in one instance, there is nothing preventing a universe suffused with absurdities, contrary to observation.
We have considered physical entities only conceptually, without presuming to decide which are real. Physics can identify sub-categories of entities, but ontology provides the highest a priori categories.
5.7.1 An event in the sense of what is happening involves action, and possibly the circumstances of the action as well. In the latter case, an event is a composite entity; otherwise it is a relation.
5.8.1 States of affairs are generally a synthesis of objects, properties and relations. The only ontologically simple states of affairs are tropes of place, time, quantity or quality.
5.8.2 Facts are judgments regarding states of affairs; thus they are semantic, not ontological objects.
5.9.1 Abstract/concrete is a mental distinction indicating whether an entity is considered independently of some aspect of its being, such as spatiotemporality, corporeality or particularity. Abstract and concrete are not categories of entities, but ways in which entities can be regarded as real.
5.10.1 Intentionality in the Scholastic sense denotes the relation between subject and object in the act of perception. Even if intentional objects are real, they would not require new categories, since intentionality would be a modality of being.
5.10.2 Intentionality in the psychological sense of volition might be a distinct category, but as compared with determinism and randomness it is but a modality.
5.10.3 Treating intention or volition as a reified mental activity would make it a substance, but this is not coherent since activity is relational. At any rate, reified mental activity adds no ontological content, nor does it define a modality.
5.11.1 A class in set theory is a collection of objects characterized by a common property. This is not a species, for a species is not a collection of individuals, but a single essence abstracted from individuation and accidents.
5.11.2 In mathematical definitions, the members of a class may be coextensive with all possible members of a species. This is not the case with classes of physical objects.
5.11.3 In computer science, we define a class by certain properties, and then assign these properties to objects defined to be members of the class. A class is here just a collection of properties. Its "definition" is just a list of differentiae. Objects assigned to the class must be of a genus that can receive the differentiae.
5.12.1 Structure refers to the order or arrangement of a complex object. Structure may be predicated as (1) a quality of the thing constituted, (2) relations among constituents, or (3) substance, namely the thing constituted.
5.12.2 The three senses of structure can be equivalent, when (A) the quality of the thing constituted that is its structure is nothing more than the sum of relations among its constituents, and (B) the structure may be considered to be the composite substance itself when this is nothing more than the sum of its parts and their arrangement.
5.12.3 Condition (B) of 5.12.2 does not hold for composite thoughts or sentences, so the third sense of structure is not applicable to thoughts or sentences.
5.12.4 Conditions (A) and (B) of 5.12.2 hold only for (secondary)qualities that are a synthesis of relations among constituents, and for substances that are a synthesis of parts and their relations. The most fundamental sense of structure is as a relation among constituents.
5.13.1 Privation refers to the absence of an accident in a substance. An absence, as such, is not an entity, so privation is not an ontological category, but only a logical one.
5.13.2 Logical negation is not a category of being, but is opposed to being.
5.13.3 Negation in the sense of a statement opposing an affirmation is not an ontological category, but a type of judgment or proposition.
6.1.1 Degree means that an entity can be "more" or "less" of what it is. An accident varies by degree by being more or less present in a subject.
6.1.2 Degree is proper to quantity and quality, which may vary by extensive or intensive magnitude. Relations admit of degree indirectly by being predicated of such magnitudes. Space and time do not themselves admit of degree, but spatial and temporal relations do.
6.1.3 Substance does not admit of degree, except equivocally when we speak of varying degrees of a differentia.
6.2.1 Opposites are two entities "against" each other, so that one is present to the extent that the other is absent.
6.2.2 The most perfect opposition is between the total presence and absence of a thing. Less radically, two opposites may both exist yet be opposed in aspects of their being; these are contraries. Lastly, some opposites may only be defined with respect to each other, depending on each other for existence; these are correlatives.
6.2.3 A fourth type of opposition, affirmation and negation, pertains not to real entities but to propositions. Still, this opposition is grounded in the conviction that the realities denoted by such propositions are mutually exclusive.
6.2.4 Correlative opposites denote inverse or reciprocal relations, discussed previously. (4.2.15-18)
6.2.5 Contrary opposites are not defined in relation to each other, yet they must have a common property or genus. Contraries may be qualities or relatives. Substances are contraries only incidentally, as they manifest contrary accidents.
6.2.6 Presence and absence are predicable of any simple entity, including substance. Every aspect of the present entity is negated in its absence, including its very being.
6.2.7 A complex entity is present if and only if all the essential simple entities constituting it are present.
6.2.8 Possession and privation refer to the presence or absence of an accident in a subject. An absent accident qualifies as a privation only if its presence in that subject is physically possible.
6.2.9 To be intermediate, that is, partway between two contraries, both contraries must be considered with respect to a common measure that varies by degree. Thus only contraries that vary by degree may have intermediates.
6.2.10 For two contraries A and B, suppose that anything of which A or B is conceivably predicable must necessarily be the subject of either A or B. If this supposition is true, no intermediate is possible. If the supposition is false, an intermediate is possible.
6.2.11 An intermediate could be a compound of varying degrees of two contraries, or at least construed as sharing a common measure with the contraries. However, to be an intermediate in more than name, it is necessary to be capable of inhering in the same subjects as the contraries, and not to merely share an arbitrarily defined common measure.
6.2.12 Contraries that have intermediates are not negations of each other; rather they are opposed by maximum and minimum degrees of some property.
6.2.13 An accident is more truly an intermediate when it is a pure compound of the contraries, without admixture of other features.
6.2.14 Substance can admit of any category of accident; quality admits of quantity, relation or time; quantity admits of place or relation; relation admits of time.
6.2.15 From the predicative relationships in 6.2.14 and the lack of contraries in quantity, place and time, only these predications remain for contraries: A substance can have contrary qualities and relations; a quality or quantity can have contrary relations.
6.2.16 Contraries have the same genus, so contrary qualities could not inhere in the same substance unless they differed in time, respect (quantity and degree), or relation. Contrary relations cannot be predicated of the same subjects unless they differ in time.
6.3.1 Affirmation and negation are logically complementary statements, so that one is true to the extent that the other is false.
6.3.2 Either an affirmation is true and its negation is false, or the affirmation is false and the negation is true; no intermediate condition is possible. This is because the negation is defined to encompass any putative intermediate statements.
6.3.3 How we regard the truth values of statements with non-existent referents depends on whether we interpret an affirmation to necessarily affirm the existence of all referents. If so, both the affirmation with non-existent referent and its negation are false. If not, we cannot ascribe any truth value to the statements, since we do not know what is affirmed.
6.3.4 An affirmation and negation are opposed only when the realities they assert or deny are considered at the same in the same respect in the same subject.
6.4.1 When the essence of entity A pertain to the being of its subject B, then the name of A may be predicated of B, and we may say simply that A is predicated of B.
6.4.2 Inverting the statements in 6.2.14, we find: quality is predicable of substance (or other qualities); quantity is predicable of quality or substance (or other quantities); place is predicable of quantity or substance; relation is predicable of quantity, quality, or substance (or other relations, improperly); time is predicable of quality, relation or substance.
6.4.3 Qualities and quantities are predicable of entities in the same category. Relations are predicable of other relations only in the improper sense that their predicates or arguments may be the arguments of a higher order relation. This is formal composition, not ontological predication.
6.4.4 Place and time are not predicable of entities in the same category, nor are they the subjects of any accident. Spatial relations between quantities are not accidents of place, but of the quantities related. Time is treated as an extensive quantity only by considering quantities of spatial displacement.
6.4.5 Quantity, relation and time may be predicated of quality in distinct ways. Quantity gives the degree of a quality. Specifying time of a quality does not modify its essence, but restricts its relationship to its subject. A relation can compare a quality to other qualities, degrees or times.
6.4.6 Quantity may admit of place when it modifies substance, and may admit of relation under any circumstances. Specifying place does not modify quantity itself, nor necessarily the substance quantified, but enables relative comparisons to other quantities.
6.4.7 Relations need not inhere in any particular subject, yet may be predicated of multiple subjects.
6.4.8 The arguments or variables of a relation are not predicated of the relation itself, but of the subject of the relation.
6.4.9 Time is the only other category predicable of relation. Time may modify relation extrinsically, or by incorporating time-dependent variation into the definition of the relation. In the latter case, the relation takes a time variable, an extensive quantity related to spatial displacement, not time itself.
6.5.1 Entities may vary according to value, being "good" or "bad" to various degrees. Good is what an entity ought to become or ought to do. The notion of Good assumes a metaphysical teleology of which it is the object.
6.5.2 Metaphysical good is the realization of the metaphysical end or telos of an entity. An entity is good simply and absolutely if this end is fully realized; otherwise it admits of degrees of good or bad. When an extrinsic influence directs a thing away from its natural end, it is contra phusis.
6.5.3 Being an entity's degree of actualization, metaphysical good transcends the categories, though it is predicable of any entity of any category. There is no category of "good essences."
6.5.4 Moral goodness is applied to intelligent actions or dispositions to act, requiring an intellect and a will. The moral good is what one ought to will, or the metaphysical good of intelligent volition.
6.5.5 Physical good is applicable to entities lacking volition. Animals capable of emotional pleasure may find physical good in such pleasure. Other lifeforms have physical good in health, while inanimate objects have the good of structural integrity. These are all types of metaphysical good: pleasure perfects appetition; health perfects vital processes; and structural integrity most perfectly realizes and object's essence.
6.5.6 The moral good is not reducible to physical good, since intellect and will are not reducible to appetition. Further, physiology does not dictate behavior, especially moral behavior.
6.5.7 As described in 4.3.23-40, good and evil may be predicated of habits and dispositions, which may belong to intellectual substances, animals and simple bodies. Metaphysical and physical good may apply to all of these while moral good applies only to the first.
6.5.8 Good and bad qualities or relations are contraries, though good and bad are not themselves entities. Substances are improperly good and bad insofar as they manifest such accidents. Substance, of itself, being predicated simply of being, is good to various degrees, but only its privation is bad.
6.5.9 The contrary of a good thing is necessarily bad, yet the contrary of a bad thing might also be bad.
6.6.1 Priority is a sequential, direction-dependent order of entities. Types of priority include temporal, existential, scientific, semantic, and by value (goodness). That which is neither prior nor posterior in each order is defined as "simultaneous."
6.6.2 In the temporal order, prior and posterior are "before" and "after," the order in which events are experienced and effects follow causes.
6.6.3 In the existential order of priority, an object A that cannot exist without object B existing at some point is existentially dependent upon or posterior to B. Object A may not be temporally prior to the original existence of B. Existential dependence may be a matter of logical, metaphysical or physical necessity.
6.6.4 Causality is a particular type of existential order, where every effect must have a cause as its source. The cause must be temporally prior to or simultaneous with its effect. Causal priority is distinguished from the temporal and the logical by the fact that the being of the effect is derivative of the being of the cause.
6.6.5 Priority in the order of science or knowledge refers to the order in which concepts of entities are intelligible. This order is an artifact of discursive thinking, yet has an objective structure and is independent of culture.
6.6.6 The semantic order, or order of speech, is culturally contingent, yet on a large scale, it has an objective logical structure.
6.6.7 The order of value, "better" or "worse," is least proper. It refers not only to metaphysical or moral good, but also to the utility of things, which is relative. In the order of science, good is prior to evil.
6.7.1 Change is the effect of action, and has the same classification as action: generation, destruction, increase, diminution, alteration, and local motion.
6.7.2 Generative change is the presence of an entity once absent, and destruction is the absence of an entity once present. Only contingent beings are subject to generation and destruction. Substances are most properly the subjects of generation and destruction.
6.7.3 Increase and diminution produce an entity of greater or lesser magnitude. Quantity, quality and relation are subject to this change. Only spatial and temporal relations are subject to this change.
6.7.4 Alteration is the substitution of one quality for another.
6.7.5 A quality is always replaced by another quality in the same genus unless there is an accompanying change in the kind or quantity of substance.
6.7.6 Local motion effects the new position of an object. Some local motions may effect substantial change or transmutation.
6.8.1 Modality describes the existential status of an entity, not a variation in its essence. Existential modalities include possible/impossible, existent/non-existent, necessary/contingent. Existence is the actualization of an essence, and modality describes the existential actuality of an entity.
6.8.2 Existence is a primitive concept; its negation, non-existence, can be understood only formally.
6.8.3 Entities that are incapable of existence, for logical, metaphysical or physical reasons, are impossible, while those that may exist are possible.
6.8.4 Possibility may be expressed as fractional probability, yet probability is contingent upon choice of condition.
6.8.5 A necessary being not only exists with probability 1, but it is logically (or metaphysically, or physically) inconceivable for it not to exist.
6.8.6 A contingent being can exist only if some other thing actually exists. A contingent being may exist with probability less than or even equal to 1, as long as it is hypothetically possible for it not to exist.
The relationship between ontology and metaphysics can be further developed only when we have expounded a system of deductive logic and epistemology, reserved for another work.
© 2008 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org
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