Unknowability of God In Jewish Rationalism IV: Why Unknowable?

In this section I will try to review some of the rational arguments for the unknowability of God that arise in particular from subscription in whole or in part to the Aristotelian system of the predicables described previously. Granted, the system that was inherited by Jewish thinkers had already been pawed over for hundreds of years by Neoplatonists and Islamic thinkers, but a certain degree of core commonality with Aristotle's original system remained. I will certainly not attempt to revisit every medieval rational theological argument, but only those that relate to God's unknowability.

So, now, what are the arguments the medieval Jewish rationalists give as to why it is necessary that God be unknowable? I will sketch out a few that I have seen, but it will imminently become clear that these are not independent arguments, rather variations on a single argument. I somehow feel, though, that the core epistemological issue here is eluding me. I want to find the cornerstone of these arguments — the singular fact from Aristotelean epistemology that positively requires God's unknowability — and I can't quite get it. I had thought that writing the previous post on the predicables would make it immediately obvious to me what the key issue is, but I was mistaken; it is still not clear to me. Perhaps a reader more familiar with Aristotelian thought will be able to provide the missing link for me. If so, I will hope to revise this section at some later time.

At least one common theme that I was able to see is that in the eyes of the medievals, having "knowledge of God" means knowing God's attributes. Thus, for God to be unknowable is for God's attributes to be unknowable, and most of the medieval arguments therefore focus on demonstrating the unknowability of God's attributes. As it turns out, this usually becomes a problem of demonstrating the nonexistence of God's attributes.

In a sense, though, I'm presenting the problem backwards: Unknowability is generally a consequence which falls out from more primary theological concerns, or at least that is how matters are presented. However, I am interested here in Unknowability in a primary sense, and I suspect that this may have been the interest of several medieval authors as well, although they do not say as much. (A completely cynical view would suggest that there is a very keen social imperative to establishing God's unknowability, since people who think they have intimate knowledge of God are often found to act in the most offensive ways toward their fellow humans. This sociological factor was I'm sure not lost on the Jewish thinkers of the middle ages, and thus even if we dismiss the cynical view as too cynical, I doubt that Unknowability can be regarded entirely as a by-product of other theological considerations. There must have been some desire on the part of theologians to establish God's unknowability as a bulwark against the nation of prophets actually behaving like a nation of prophets.)

On the assumption (i.e., my tentative assumption) that it is Unknowability which is a primary fact the medievals sought to demonstrate, it might be noted that there is a certain excessiveness in stripping God of all attributes. Isn't it enough to claim for the purpose of Unknowability just that we humans don't or can't know God's attributes, while yet allowing God to keep His (unknowable) attributes? Is it really necessary that God have no attributes? This seems to have been the position of Philo and his school, and it seems to have been the position of Aquinas as well. He writes in Summa (kreeft_90, p.115):

It is impossible for any created intellect to see the essence of God by its own natural power. For knowledge is regulated according as the thing known is in the knower. But the thing known is in the knower according to the mode of the knower. Hence the knowledge of every knower is ruled according to its own nature. If therefore the mode of anything's being exceeds the mode of the knower, it must result that the knowledge of that object is above the nature of the knower....

The implication here is that "unknowability" is a status that applies only to a set of two entities: (1) a knower and (2) a thing to be known. For a given knower and a given thing to be known, if the capacities of the knower are inadequate to know that particular thing, then we have a state of "unknowability". On this view, it would seem possible to allow unknowability of God without making Him completely naked of all attributes. We could say that God possesses attributes, but that our intellectual capacities fall short of being able to know these attributes. This, however, was not the tact of the Jewish medieval rationalists (with perhaps a couple exceptions). Rather, their approach was to equate unknowability with the complete absence of divine attributes, denying God even the attribute of "intellect" (cf. Maharal, mallin_carmell_75). As Rambam writes (GP I:50), "you must know that He, may He be exalted, has in no way and in no mode any essential attribute, and that just as it is impossible that He should be a body, it is also impossible that He should possess an essential attribute." The reasons behind this extreme position on unknowability seem to have been laid out already by Bachya ibn Pakuda, who gives us our first argument.

The problem of attribute priority

Bachya ibn Pakuda's argument for our essential inability to attribute properties to God is that, being the creator of all attributes, God cannot possess any of these same attributes Himself: "...whatever we could say about Him would refer either to His Essence or His properties, and the Creator of essences and properties cannot be described the way they are described" (pakuda_96, ch.9). It seems to me that what Bachya is saying is that if we were to suppose that God did possess an attribute of some kind, then we would be forced to conclude that God could not have created that attribute. To say otherwise, on Bachya's view, would be absurd. Thus, God cannot have any attributes.

While simple, Bachya's presentation of the problem seems to me to anticipate many of the later discussions. Maharal (mallin_carmell_75) seems to echo this idea when he writes that "Once we realize that all entity originates from His being, we understand that it has nothing in common with Him." We cannot attribute properties to God because doing so gives those properties an existence which is outside the creative control of — and in some sense prior to — God Himself.

On the other hand, Bachya's position is not completely convincing. If we were to say, for example, that God has "hands," does that logically imply that "hands" are somehow prior to God? It seems to me it does not. It is just a description of the way God is. To say that a bird possesses feathers is not to say that feathers exist independently and prior to the bird in question. The property "having feathers" is just a description of the way the bird is. In the case of God, it does not seem logically necessary that a property's existence requires the property to have existed prior to or outside of God (in some Platonic realm, for example), and if that's correct, then Bachya's argument loses some force on this account.

However, while the existence of the property in question (e.g., "hands") independent of God may not be logically implied, by naming a property (or by merely stipulating the existence of such a property in God) we do immediately raise the question of why God has this property instead of some other property, i.e., the question of why God is the way He is. This is a broader understanding of Bachya's question, I think. If God indeed has hands, then why does God have hands, instead of wheels or flippers? The same question would apply to psychological attributes as well. If God is indeed angry, then why is God angry, rather than silly or dopey? Since the medieval inclination was to see agency behind everything, in order to answer such question it would evidently be necessary to stipulate the presence of an agent, independent of God, who is responsible for God's having the attributes He has. Even without appeals to agency, unless one were to resort to brute facts (God is the way He is just because), the existence of particular attributes in God would still require some process to explain their presence there.

And if the attributes we attribute to God are shared by other entities as well, matters are actually even worse. Aristotle (Posterior Analytics, mure_01, p.120) informs us that "to know a thing's nature is to know the reason why it is," and that "we possess scientific knowledge of a thing only when we know its cause" (mure_01, p.112). In general, the knowledge that we seek about entities (why they have certain properties) is provided by identifying the superordinate category that endows those entities with the properties in question: "The proper function of science is to provide explanations, the canonical form of which is something like 'Xs are F because they are G.'" (hankinson_95). If an attribute which we apply to God is also applied to other entities as well, and we have scientific knowledge of those other entities in Aristotle's sense, then we will have genuinely identified a superordinate entity (i.e., class description) which is the "generator" of the attribute in question. In this case, since we do indeed "explain" the attribute in question by invoking an ancestral generator to which God as well as the other entities sharing that attribute would owe their properties, we would most likely be realizing Bachya's worst fears.

In summary, as I read it, attributes require explanation, and any explanation of God's attributes will have unacceptable theological consequences. Hence God cannot have attributes. We cannot escape by claiming that God possess attributes that are unknown to us, because whether they are known to us or not, their very existence implies an entity or entities which exist in some sense prior to God. This seems to me to be the liberal reading of Bachya's position.

The problem of compositionality

A second problem with the existence of attributes (and hence with knowing them) is that they introduce a kind of multiplicity into God. Many writers go on at great length about this issue. Rambam, for example, writes (GP I:50) that "no composition whatever is to be found in Him and no possibility of division in any way whatsoever." About God-as-He-is-in-Himself, Ramchal in Derech Hashem similarly tells us that "It is likewise necessary to know that God's essence is absolutely simple, without any structure or additional qualities whatsoever. Every possible perfection exists in Him, but in an absolutely simple manner." Albo (husik_29, p.128) also writes that God's attributes are unified in Him, while in us they are distinct (although his meaning is not clear to me).

It is likewise not entirely clear to me whether the insistence on divine unity is a philosophical imperative or a theological imperative. Aquinas (kreeft_90, p.83) writes that God cannot be composite because composite things must have a cause which causes them to unite, thus again setting up a problem of priority. If there are multiple attributes, we would be forced to seek both the reasons for the particular attributes instantiated in God, as well as the cause of their present combination in that subject. Aryeh Kaplan puts it this way:

The logic behind this is that any additional quality that we would ascribe to God would add an element of plurality to His being. Thus, for example, let us assume that we wish to speak of God's intellect as an entity in itself. We would then have to speak of two concepts, namely God's essence and His intellect. Since this would imply an element of plurality within God, it must be rejected. This is true of any other attribute which we may wish to ascribe to God, and therefore we must say that no such independent attributes exist. But if we cannot ascribe any attribute at all to God, then we must conceive Him as being absolutely simple. This indeed is the consensus of opinion among our great thinkers. Nevertheless, His very simple essence implies every attribute with which God rules the universe (kaplan_90).

It cannot even be said that God is therefore "one" in the numerical sense, because "quantity" too cannot be meaningfully predicated of God (cf. Cordovero, Rambam, etc.). As ibn Gabirol phrases it in Keter Malchut, "Thou art One, and at the mystery of thy Oneness the wise of heart are struck dumb, For they know not what it is... Thou art One, but not like a unit to be grasped or counted, For number and change cannot reach Thee" (zangwill_23, p.83, Ch.2).

Again, note that both here and with respect to Bachya's argument, the conclusion is not just that we cannot know God's attributes. The conclusion is that God cannot have attributes at all. Independent of what humans can or cannot know, if God were objectively to possess attributes, this would set up the objective theological dilemmas of attribute priority or multiplicity described above. On this account, the impossibility of knowing God is entirely self-evident: There is simply nothing to know. Since our knowledge of things is exclusively by way of predication — by attributing qualities to entities, if an entity is assumed objectively to have no qualities, then there is genuinely no possibility of acquiring knowledge about it.

And here I will pause to issue my first heresy alert: This notion of divine unity or structurelessness is one which I will most likely be forced to reject in another few posts. It is simply not possible to speak meaningfully of a dynamic entity which has no structure or no qualities. Ramchal already warns us that "admittedly, this is something far beyond the grasp of our understanding and imagination, and there hardly exists a way to express it and put it into words," but the fact remains that if we wish to explore the nature of God and his involvement in the universe, we simply cannot accept the notion of His structurelessness. The idea of objective structurelessness makes God into a nothing. Not a nothing in the sense that He has no describable attributes, but a nothing in the absolute sense that there is no possibility of His interaction or relationship with the universe, physical or otherwise. On Ramchal's account, God would be far less that a piece of chalk or lump of clay. Indeed, a wisp of tissue paper would be infinitely more powerful than God were we to insist upon his complete structurelessness.

The much-overtaxed excuse that God's structurelessness is "far beyond the grasp of our understanding and imagination" provides no remedy whatsoever. If we are willing to here admit into our theology an idea that is positively absurd on its face, then we might just as well do so whenever we please, and there is no longer any sense in which the process can be considered a rational theology. We could just say that it is "far beyond the grasp of our understanding and imagination" that a universe can create itself, and be done with it right there. If we are to pursue any kind of rational theology, then God must have structure, which means that God must have composition and/or attributes in some sense. This is simply an unavoidable heresy. As to whether such a view makes God contingent on his components or attributes, and thus gives those components or attributes priority in some sense, it may do. On the other hand, it may also be the case that the components could not exist in isolation, and that (like our own bodies) the constituent parts are in most respects inferior rather than superior to the whole. We will have more to say about this when the time comes.

The problem of subsumption of God within a class

Another problem with attributes as applied to God is the resulting subsumption of God within a class. I have not seen this problem directly expressed in this form by a medieval writer, but it has certainly been attributed to them often (and I assume correctly) by modern writers. Here is Aryeh Kaplan's version which I quoted in an earlier post:

Since everything conceivable — including any category of thought that the mind can imagine — was created by God, there is nothing conceivable that can be associated with Him. Let us say that I want to think about God. There is, however, no category in my mind in which I can place Him. Therefore, trying to depict God is like trying to see without eyes. When I try to see where there are no eyes, all I see is nothing. Similarly, when I try to think about God, all that my mind can depict is nothing (kaplan_85, p.89,90).

Kaplan's first statement simply echoes Bachya's position. The second line seems at first to suggest something different, though — that we cannot understand God because we have no category for God. However, as we mentioned earlier, classification is predication and predication is classification. Therefore, all that Kaplan is saying here is again that God cannot have attributes, the reason being that which Bachya has already laid out.

A closely related assertion that is often made is that God cannot be known because God cannot be defined. This would appear to be a weaker assertion than the previous, because definitions are a subset of the predicables. If we have already ruled out the applicability of any attributes to God, then we have certainly also precluded definitions. However, a weaker position might leave open the possibility that some attributes of God could be known, provided they are not essential (i.e., definitional) attributes. In any case, Louis Jacobs (jacobs_57) explains the problem this way:

God cannot be defined for definition is genus plus differentia. If, for example, man is defined as a rational creature, there is first the statement of the genus — the group to which he belongs — and then the statement of how he differs from other members of that group. We say that man is a member of the group 'creatures' and that by possessing reasoning faculties he differs from all other members of that group. But on any advanced view of Theism, God cannot belong to a group, for this would imply that the group to which He belongs is greater, i.e. more embracing than He.

Because in Aristotelian philosophy a definition requires specification of a genus, and genera are (as Porphyry points out) in some sense "generative" of their constituent species and therefor prior to those species, it is not possible that God has a genus. It may be noted that in Aristotle's system not everything actually needs a definition, since obviously such a demand would invoke an infinite regress. But nevertheless, for something to possess a definition requires that particular something to possess a genus, and therefore God cannot be said to possess a definition. Moreover, on the view of many, it cannot even be said that God belongs to the genus of "existing," since long-standing tradition dating to Aristotle asserts that "existence" is not a legitimate genus. As Albo writes (in typically muddled fashion),

In reality, however, it [existence] is neither a definition nor a description, God having no definition. For a definition is composed of genus and difference, but the word existent is not a genus which is predicated of all its subjects synonymously, as a genus is... For there is no genus in the world which includes God and another. The word existent is not applied synonymously to God and to other things. God's existence is real (absolute), whereas the existence of other existing things is acquired from His existence. But if existent is not a genus which includes God and other things, He has no difference (husik_29, p.36).

If the above explanation of God's resistance to definition is not adequate, Aquinas provides another. Aquinas (kreeft_90, p.81) points out that every member of a genus must have a difference, but yet possess the same quiddity (which is contributed by the genus). Hence, existence and quiddity must differ. But this is not so in God, and therefore he cannot have a genus. Whether this reasoning is entirely circular, which I believe it is, we will leave for another blog. The main point is that God cannot be known because God cannot be classified. The reason why God cannot be classified is again, at root, because any attribute when predicated of God (which immediately establishes a class in which God is a member) would in some Aristotelean sense assume precedence over God, which is theologically unacceptable.

The idea that God has no genus also has significant fallout for the notion of similarity between God and other entities. Aquinas (kreeft_90, p.89) puts it very simply by stating that entities in different genera cannot be compared, and since God has no genus, and hence no entity is in the same genus as God, there is not any possibility of comparison between God and other entities. Simultaneously, the lack of genus has consequences for relation between God and other entities, as Rambam points out (GP I:56):

Know that likeness is a certain relation between two things and that in cases where no relation can be supposed to exist between two things, no likeness between them can be represented to oneself. Similarly in all cases in which there is no likeness between two things, there is no relation between them. An example of this is that one does not say this heat is like this color, or that this voice is like this sweetness. This is a matter that is clear in itself. Accordingly, in view of the fact that the relation between us and Him, may He be exalted, is considered as nonexistent — I mean the relation between Him and that which is other than He — it follows necessarily that likeness between Him and us should also be considered nonexistent....

We may pause here to ask how convincing such arguments should be to a contemporary theologian. We no longer give metaphysical significance to predicates or categories, and thus for God to be subsumed in some arbitrary category would seem to hold little theological danger. If God were to be a member of the class of "things that have hands," would we consider this category to be somehow greater than God? Many categories may be constructed based on arbitrary feature conjunctions (Barsalou_83), but which have little or no ontological significance. The mere fact that "sitting on a yellow swivel chair in New Jersey within reach of a black coffee mug, purple pen, and unpaid credit card bill" defines a definite category which includes myself, some other people, and perhaps some animals, there is no deep ontological significance to this category. It is just an arbitrary grouping of entities that does not correspond in any way to the deep structure of the world. Most such categorizations have no deeper meaning, so merely belonging to a category does not by itself carry much epistemological weight.

Even so, it may not be possible to completely rid ourselves of the problem of class subsumption. While we can think of many categories with no ontological significance, many of the attributes on the basis of which such categories are formed would have been considered "accidents" — attributes which might have been and might yet be otherwise. The medievals were forced to reject such attributes of God for another reason (next post). Because of this, the kind of ad-hoc category contrived above is not the kind of category God could ever belong to. Therefore we are left asking whether we should still be troubled by the predication of essential attributes to God and what this entails for class subsumption. As it turns out, though, we can't really answer this question. In modern thought, there is not really any notion of "essential attributes," and so we are just returned to the question of whether God can have attributes at all.

So how persuasive is this argument of "unknowability due to unclassifiability"? It doesn't seem to me that this argument can any longer carry the weight it once did. By placing God in a category (or, more likely, a set of categories), we do not automatically promote those categories to an ontological status superior to God. The very notion of placing God in a category cannot therefore be offered by itself as providing a definitive demonstration of God's unknowability. However, if we do allow God to be classified in some fashion, we can still not escape Bachya's dilemma regarding the sources of God's attributes. There is no solution to that problem, but there may not need to be.

Unknowability of God In Jewish Rationalism III: The problem with predicates

What follows is a modest digression into the topic of why the medieval thinkers were so insistent that God cannot be defined or categorized. It is their philosophical intuition on this point which, independent of any Scriptural attestations, leads thinkers such as Rambam into denying the application of any predicates to God. I will freely admit that I have not seen all the sources on the issue (which probably number in the thousands, across a half dozen languages), or even the best of the sources, so I will make do with what I have. Hopefully someone out there will be kind enough to illuminate matters further for me.

Let us begin by rehearsing the meaning of predication. A predicate is something that is affirmed or denied of a subject, yes, but we should strive to gain some more clarity on the matter than this. (It is precisely such clarity which has been the Holy Grail of every logician since Aristotle.) We could do worse than beginning with William Hamilton's concise summary of the nature of thought, which includes a few words on the nature of predication; if nothing else, it may help to clarify a few terminological issues. (W. Hamilton was a Scottish philosopher whose chief notoriety comes from his frequent impeachment in the writings of J.S. Mill. However, I think we can assume that there are good reasons Mill chose Hamilton as his prey, not simply the prospect of an easy kill.)

When we think a thing, this is done by conceiving it as possessed of certain modes of being, or qualities, and the sum of these qualities constitutes its concept or notion... As these qualities or modes are only identified with the thing by a mental attribution, they are called attributes... as it is only in or through them that we say or announce aught of a thing, they are called predicates, predicables, and predicaments, or categories, these words being here used in their more extensive signification... as it is only in and through them that we recognize a thing for what it is, they are called notes, signs, marks, characters... finally as it is only in and through them that we become aware that a thing is possessed of a peculiar and determinate existence, they are called properties, differences, determinations... (hamilton_64 p.55)

On Hamilton's view, as I read it, a predicate is a verbalizable quality of an object; however, the very same quality may also be called a "note", "sign", "determination", etc., when it is considered in certain contexts appropriate to those designations. Thus, really, a predicate is just some distinguishable quality of a perceived entity. What cannot be distinguished cannot be predicated of an entity, whereas conversely, anything than can be distinguished can be predicated. Thus, the necessary and sufficient condition for predication to take place is the existence of at least one distinguishable property or quality.

This is not altogether disjoint from the modern notion of predication, in which a predicate is defined simply as a relation, a function that maps its arguments to TRUE or FALSE (sowa_00; see also Sowa's mathematical review). That is, a predicate accepts a tuple of attributes or variables, and returns for each tuple a value of TRUE or FALSE. In some cases, a predicate can be expressed by a simple rule (i.e., intentionally), as for the unary predicate "is_positive_number(x)." Such an intentional description of a predicate is generally only viable when there is some simple algorithmic representation for the predicate in question (in the previous case, a division-by-2 remainder test), but more generally a predicate is just described by the set of entities for which it evaluates to TRUE, or, alternatively, by the set of entities for which it evaluates to FALSE. In these cases, the predicate is said to be described extensionally.

As an aside, classically, a predicate differs from a proposition in that a proposition is a sentence that has truth or falsity (Aristotle, On Interpretation, edghill_01b p.42). Thus, "a proposition is a portion of discourse in which something is affirmed or denied of something" (mill_36 p.51). In other words, a proposition is usually formed by explicitly evaluating a predicate or combination of predicates on a subject. Thus, "went to the store on Tuesday" is a predicate designating the class of people who went to the store on Tuesday. It is a relation which maps individual people (or their names) to TRUE or FALSE. A proposition which uses this predicate might then be "John went to the store on Tuesday," which evaluates to TRUE or FALSE depending on whether "John" is a member of the class represented by the predicate in question. Propositional logic deals with how truth is preserved when truth-bearing entities such as propositions are combined in various ways.

In the case of binary attributes (e.g., attributes which are either present or absent), the intentional description for a predicate would simply be the set of attributes shared by all the objects in the predicate-defined class, whereas the extensional description would be the set of objects themselves. As Hamilton puts it (hamilton_64 p.105), "The comprehension [intension] of a concept is nothing more than the sum or complement of the distinguishing characters, attributes, of which the concept is made up; and the extension of a concept is nothing more than the sum or complement of the objects themselves, whose resembling characters were abstracted to constitute the concept."

Note: See the note preceding the previous post about viewing mathematics. (You should see some red-colored mathematics below, if things are working on your end.)

To be a little more formal, a predicate (like a relation) is just some subset of the Cartesian product of the set of attributes or features by which the entities are described (i.e., some region of the "feature space"). For example, if we have a set `cc{X}` of three binary variables/attributes `cc{X}={x_1,x_2,x_3}`, each variable adopting value 0 or 1, then the full Cartesian product `ox cc{X}` is the set

`ox cc{X}={[000],[001],[010],[011],[100],[101],[110],[111]}`,

which is the set of every ordered combination of the attributes, i.e., the complete set of possible entities in the universe of discourse. (The notation `{010}` is shorthand for `{x_1=0,x_2=1,x_3=0}`, etc.) Any subset of `ox cc{X}` then defines a predicate. Thus, the subset `{000,010,101,110}` defines a ternary predicate, a relation which returns TRUE for just the previously specified objects (3-tuples), and returns FALSE for all other objects. This predicate defined by the set `{000,010,101,110}` has no compact intentional expression (i.e., rule), whereas, for example, the predicate defined extensionally by the set `{100,110,101,111}` admits the simple intentional description of `x_1=1` which we might capture verbally with a simple predicate label such as has_feature_x1(x1,x2,x3). (Note, however, that invoking an intentional description like this necessarily introduces inductive bias, which rears its head when new objects outside this set are observed, i.e., when the feature space is expanded.) The issue of simplicity and complexity (i.e., compressibility) of relations is a very deep one, and not directly relevant to this discussion. Also deep but not immediately relevant is the nature of the distinction between intentional and extensional description, and whether the two forms are not just points on a continuous spectrum of compressibility that includes many levels of "intentionality" or "extensionality" between the absolute poles of "intentional" and "extensional".

On the other hand, what is deep and immediately relevant is that the establishment of a predicate (intentionally or extensionally) immediately induces a categorization scheme on the universe; in particular, with the introduction of a predicate, two classes of entities are immediately distinguished — those for which the predicate evaluates true, and those for which the predicate evaluates false. Here then we begin to see the connection between predication and categorization: Indeed, "as soon as we employ a name to connote attributes, the things, be they more or fewer, which happen to possess those attributes, are constituted ipso facto a class... It is a fundamental principle in logic, that the power of framing classes is unlimited, as long as there is any (even the smallest) difference to found a distinction upon. Take any attribute whatever, and if some things have it, and others have not, we may ground on the attribute a division of all things into two classes; and we actually do so the moment we create a name which connotes the attribute" (mill_36 p.76,79). Thus, predication is categorization.

Mill makes a point that we do not predicate a class of an individual — we predicate membership in a class of the individual, or a name representing an attribute (mill_36 p.78). It is further notable (and Hamilton, for one, does not miss the opportunity to note it at great length) that there exists an inverse relationship between the sizes of the intention and extension of a concept or predicate. As the size of the intension increases through expansion of the set of attributes shared by the objects in the class, the extension is simultaneously decreased by the elimination of objects not sharing the specified features. Intuitively, the more rigorous the intentional description, the fewer objects can satisfy it. Conversely, as the extension of a concept or predicate increases through addition of non-redundant objects to the class, the intention is decreased by elimination of attributes which those objects fail to share. We will say more about this later.

It seems to me that the classical authors were generally concerned with unary predicates — those which accept only a single argument. Often these are "is-a" predicates, such as is_a_dog(x) or is_a_human(x), which return TRUE when x is a dog or human, respectively, although binary predicates such as has_a(x,y) or is_made_from(x,y) are also entertained. Such binary predicates would return TRUE when it is true (for instance) that object x possess a property y or that object x is made from y, respectively. (Lest anyone think that such primitive predicates as "is-a" are hopelessly antiquated, these sorts of relations are still very much current in modern ontologies, description logics, semantic networks, etc. The reason for their continued utilization is the same reason that found Aristotle pondering them 2000 years ago: There is a small set of common predicates which we humans use to describe our world, and any mechanical system that ultimately hopes to interact intelligently with humans must therefore cope with common predicates designating possession, composition, subsumption, etc.)

Aristotle, later amplified by Porphyry, distinguishes several different semantic classes of predication later to become known as "the predicables" (mill_36 p.77). It can probably go without saying that Aristotle himself is less than entirely clear on this issue (smith_95) — else he would not have so easily entertained great minds for two millennia — but in his Topics (pickard-cambridge_01 p.191) he at least lays out the following four types of predication: definition, property, genus, accident. In later treatments, definition seems to be replaced by the predicables species and differentia thus yielding the most common version of the hierarchy of predicables: genus, species, differentia, proprium, accidens, as is given in Porphyry's Introduction. Below, I review the two schemes as one, even though there may in fact be "radical differences" between the two, as suggested by the Wikipedia article (actually a 1911 Britannica article). In particular, the Britannica author suggests that Aristotle's system is the more secure because all of the predicates deal with universals (i.e., abstractions), whereas Porphyry's scheme by involving "species" intermixes predication of universals and individuals. I don't know whether I agree with that assessment or not, so I will just leave it alone.

Let us very speedily review these types of predicates, while simultaneously trying not to be sucked into Aristotle's interlocking theories of causation and "the categories". First, to paint the larger picture, it is best to think of the universe (of discourse) in terms of an inclusion hierarchy. The figure below shows a set of 10 individuals (bottom row). These individuals may be objects or events or any other entities which are susceptible to predication (i.e., which have distinguishable qualities), but it is only these individuals, i.e., the nodes in the bottom row, that are actually observable. The distinguishable qualities (i.e., feature set) in this universe are represented by capital English letters. (The letters themselves are, of course, just meaningless symbols; i.e., the label "ACH", for example, simply means "object having property A, and property C, and property H, and no other properties".) I think it is correct to say that this is the model of the structure of the world that most of the ancients were working with. They observed (as do we) that entities tend to have many common properties, and they took these common property clusters to be a guide to the deep structure of the universe; a structure in which entities share common features not by accident, but because in some sense these entities share a common genesis, a common connection to a particular generative node in an underlying (unobservable) reality.



The inclusion hierarchy model which results from this line of thinking is not wrong, per se. In fact, it can be an appropriate model both for understanding certain kinds of accretion-based artifacts (e.g., multiple-author documents) and certain natural systems (e.g., genetic inheritance, as per cladistics). Many other systems can be reasonably and profitably simplified into inclusion hierarchies (e.g., medical knowledge). The strict inclusion hierarchy is, however, certainly an incomplete model for any system of more than rudimentary complexity.

In any event, the method by which the inclusion hierarchy is inferred from observations is by recursive abstraction of common elements. For example, we note that in the figure three of the observed entities {ACH, ACI, ACJ} all share the feature complex AC. We therefore abstract the AC complex away from the entities, thus signaling our belief that there is an underlying entity (hidden node) which contributes this AC complex to all the entities which posses it. The same process allows us to abstract the feature complex AF from entities {AFK,AFL}. Recursively, we then note that hidden nodes AC and AF share the feature A in common, which can then be abstracted in the same way. After we construct the inclusion hierarchy in this manner, we may choose to apply special titles such as "genus" and "species" to some of the hidden nodes thus inferred. In the figure, AC and AF could be considered species of A, if certain conditions hold (discussed below). With this image in mind, we can now return to the definitions of "the predicables".

Genus and Species: Porphyry (barnes_03) explains that genus relates to "genesis" in the sense that genus is the "origin" for the things collected under it: Moreover, "genus is what is predicated, in answer to 'What is it?', of several items which differ in species; for example, animal." Species are then the subclasses within a genus, which are in turn comprised of individuals.

We can see already that there are going to be major problems. How does one know whether a given abstracted class should be given the special label of "genus" or "species"? Porphyry (barnes_03 p.6) already notes that "between the most general and the most special are other items which are at the same time both genera and species (but taken in relation now to one thing and now to another)." In other words, genus and species appear to be relative designations. Mill points out, for example, that "animal" is evidently a genus with respect to "man", but is a species with respect to "substance". One notion can always be regarded as subordinate or superordinate to another (hamilton_64 p.136).

It is possible to simply embrace the relativity of such terms, which amounts to an admission that the "generality" or "specificity" of classes are in the eye of the beholder, contingent on context, etc., etc. However, while this may work for the pure logician, it is not acceptable to Aristotelians who view the inferred inclusion hierarchy as descriptive of the order in the natural world. In their view, the classes abstracted at some level must have attributed to them Genus status with absoluteness. The Genus "animal" is not like the genus "shoe lace", although both are inferred by the selfsame process of abstraction; rather, the class "animal" corresponds to the structure of the world in a profound way that the class "shoe lace" does not. However, what exactly it is that makes something genuinely a capital-G Genus or capital-S Species (rather than no-capital genus or species) is rarely clear. Mill (mill_36 p.78) indicates that for Aristotelians, Genus and Species must reflect the essence of the subject, where the difference between essential attributes and non-essential attributes is just that the former are involved in the class name. (On Mill's interpretation, the essence of a subject is the essence (i.e. intension) of the class in which it is a member, this being the only notion of "essence" which he allows. This is clearly not how the ancients understood essence, however.) In any case, the appeal to essence just makes the entire endeavor completely circular, and provides no justification why one class is a Genus, a second is a Species, and a third is neither at all.

Mill provides a plausible, though weaker, approach to the designations of Genus and Species (mill_36 p.80). He writes that genera and species are those classes which are set apart from other classes by "vast numbers of features". Thus, plant is set off from animal by thousands of features, for example. This is a reasonable view, both because of naturalistic considerations relating to common descent in biology, and because it provides the rudiments of a procedure for assessing genericity and specificity. (Unfortunately, without constraints on features, the procedure cannot be implemented, but this is a common problem in all inference schemes.) Thus, the genera are classes that are separated from each other by vast numbers of features (classically, by all features, except perhaps "Being"), while species are the subclasses of a genus that are separated by vast numbers of features. Having now suitably confused matters, let us continue with our definitions.

Differentia: Differentia or differences are the distinctions between species within a genus. For example, the genus "animal" contains the species "human", "horse", "crab", etc. Whatever makes these subclasses distinct from one another — i.e., whatever features one of these subclasses possesses over and beyond what is contributed by its genus, that is its difference: As Porphyry writes, "a difference is that by which a species exceeds its genus" (barnes_03 p.10). "This surplus of connotation — this which the species connotes over and above the connotation of the genus — is the Differentia, or specific difference; or, to state the same proposition in other words, the Differentia is that which must be added to the connotation of the genus, to complete the connotation of the species" (mill_36 p.82). On Porphyry's view, "a difference is what is predicated, in answer to 'What sort of so-and-so is it?'" (barnes_03 p.10).

In the illustration, species AC differs from its genus (A) by the property C. Thus, C is its difference. Likewise, species BDC differs from its genus (B) by properties DC, which therefore constitute its difference. Again, this explanation raises many questions; for example, what kind of difference constitutes a capital-D Difference? Porphyry (barnes_03 p.9) indicates that "it is in virtue of those differences which make a thing other [not just otherlike] that divisions of genera into species are made... not just anything that happens to separate under the same genus is a difference but rather something which contributes to their being and which is a part of what it is to be the object" (barnes_03 p.11). Thus, true Differences must in some sense be categorical differences.

Property: The least-often discussed of the predicables, property, is a quality in members of the species which is non-essential, but which is characteristically present in members of that species (and in no others). For example (for argument's sake), possession of opposable thumbs or ability to laugh are special characteristics of human beings. However, opposable thumbs and laughter are not essential qualities of a human such that an individual's humanity would be called into question by their absence (as it would, for example, by the absence of rationality). Thus, properties are characteristic, but non-essential, qualities.

Accident: One can say that accident is any predication that does not fall into the other predicable categories. Loosely, accident is a quality of an individual that may have been other than what it is. The blue color of a house is an accident. The house could have been red, or yellow, or pale green, and at some other time it might indeed be one of those other colors. Accidents, in general, are any predications that are not universal and (like properties) do not go to the essence of the subject. Aristotle in Interpretation (edghill_01b p.53) points out that accidents do not combine in the subject to form a unity. (It is not clear to me what he means by this.) Accidents can also be described as those predicates which fail to counter-predicate with the objects they modify (see below).

Definition: Perhaps the most contentious of all the predicables, definition can be considered a derived predicable, given by the formula "genus + differentia" (Topics, pickard-cambridge_01 p.195; smith_95 p.52). That is, the definition of an entity is given by stating its genus followed by its differentia. The classical example is the definition of "human" as rational animal, or "animal + rational", where "animal" designates the genus to which humans belong, and "rational" designates the Difference which humans possess over and above the genus. Aristotle also describes definition as "an account which signifies what it is to be something" (smith_95 p.51).

Aristotle offers some additional hints to what definition means in his view: Along with genus, definition is one of the "essential predicates" which say of an entity "what it is" (smith_95 p.53-54), and which involve "essential attributes" — those which are essential to the nature of the object (Posterior Analytics, mure_01 p.116). In his Topics (pickard-cambridge_01 p.190) he indicates that "the peculiar" can be divided into definition (indicating essence) on the one hand, and property on the other. Indeed, the criterion of essentiality is really the ultimate test for definitionality: A weaker test, that of counter-predication (see below), does not in fact distinguish definition from property. Therefore, a definition must counter-predicate with the entity it describes as well as explain the essential nature of that entity (smith_95 p.53-54). Predicates which fail to do the latter, but which still counter-predicate, are simply properties. Predicates which fail to do both are accidents.

The notion of counter-predication is (in true Aristotelian fashion) simple but slippery. The idea is that a definition specifies both the necessary and sufficient conditions for the entity to be what it is (swoyer_06 p.141). Thus, if the definition of human is "rational animal," this means that human implies rational animal and rational animal implies human. Thus, the definition expresses the bilateral implication `mbox{human} \leftrightarrow mbox{rational animal}`. In other words, the class of `cc{X}` and the class of `cc{Y}` are exactly and necessarily the same classes when `cc{Y}` is the definition of `cc{X}`. (This notion of definition is what is known as a "concept" in formal concept analysis.) One can see that this counter-predication property holds for both property predicates as well as definition predicates, since a property is a peculiar quality shared by all instances (e.g., opposable thumbs). All humans and only humans have opposable thumbs (for argument's sake). Thus, an animal is a human if and only if it has opposable thumbs. Likewise for definition: an animal is a human if and only if it possess rationality. As indicated above, for a predicate to be a definition, beyond counter-predication it must also be descriptive of essential qualities, rather than nonessential qualities.

Aristotle provides no formal procedure for determining the definition of entities, although he disparages Plato's method of division (smith_95 p.52). However, he does indicate in Posterior Analytics (mure_01 p.179) that if we look for commonalities among entities, we can "persevere until we reach a single formula, since this will be the definition of the thing." This certainly suggests the kind of factoring procedure mentioned above for inferring inclusion hierarchies (figure above), but it is unclear what is meant here by "a single formula". For example, if the inclusion hierarchy is 37 levels deep, how do we know at which level are constituted the "definitions"? Perhaps connected to his reluctance to provide a procedure is Aristotle's insistence that definitions are not susceptible to proof or demonstration (Posterior Analytics, mure_01 p.162): "...there is no identical object of which it is possible to possess both a definition and a demonstration... all demonstrations evidently assume and posit the essential nature..." In other words, our knowledge of a thing's definition, since it is related to its essential nature, cannot come from any deductive argument; it must be assumed. This seems to make sense on a syllogistic view of proof; definition is Genus plus Differentia, but the Differentia cannot be implied by the Genus, since it is the nature of Difference not to be implied by Genus. In other words, if a Difference were implied by a Genus, then the Difference would simply be part of the description of the Genus, and be no difference at all.

It need not be mentioned that the entire system of "the predicables" is horribly question-begging, and leaves unanswered the question of which abstractions have a genuine existence as universals and which do not, as well as the essential nature (I know, I know) of universals such as "genus" or "definition". However, to one extent or another, this is the system that most religious thinkers of the middle ages had to wrestle with, and by acknowledging it we may better understand the particular positions they were forced to stake out. We will try to understand some of these theological positions vis à vis definition, predication, and categorization in the next section, but for the time being we will just conclude by pointing out that the Aristotelian system itself cannot be considered entirely wrong. In the wake of Occam and Mill and Wittgenstein and others, we can no longer seek "definitions" or "genera" in the absolute sense that the Aristotelians sought them; nevertheless, there is structure in the world, and this structure in many instances has a certain hierarchical character. Properties and features, whether they inhere in objects themselves or are merely mental constructions, do tend to cluster into groupings or modes. As Mill puts it, it seems undeniable that there are vast numbers of qualities by which plants differ from animals — far more than the number on which one plant differs from another, or one animal from another. Until this structure is acknowledged and modeled in some manner, science is impossible. Moreover, in some sense, the "universals" underlying this structure turn out to be even more real than the Aristotelians imagined them to be. Barring exercises in antirealism, there really are underlying genetic structures which explain the commonalities observed among organisms, and the manner in which these commonalities accrete and emerge is genuinely something that can be reasonably approximated by an inclusion hierarchy. As in many other areas of science (e.g., psychology), the instincts and ideas were right, but applied to the wrong constituents and at the wrong level of abstraction. Enough for now. חג שמח!