Knowledge Problems

Saturday, April 07, 2007

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!


  • I found this to be a great overview of some of the technical terms that we run into in the Moreh, among other places... thanks! - looking forward to the next post...

    By Anonymous revel-dropout, at Apr 15, 2007, 12:24:00 PM  

  • Thanks a lot! I will try to do the next one before another two months go by!

    By Blogger Big-S Skeptic, at Apr 15, 2007, 1:18:00 PM  

  • Hi,
    I have a suggestion.
    How about a running commentary on the Rambam's Moreh Nevuchim from beginning to end. One small piece a day. When it is completed, it can be sold as a book!

    By Anonymous Anonymous, at Apr 26, 2007, 3:47:00 PM  

  • Hey anonymous. It's a great idea! Unfortunately, I'm now starting Volume II, so I would have to backtrack to do this project. But it would be worthwhile (certainly for me). There are already many books on GP, so I don't know how valuable it would be for anyone else! Anyway, David Guttmann has been essentially doing this for two years, and I could never "out-Rambam" David. But still a nice idea. Why don't you give it a shot yourself?

    By Blogger Big-S Skeptic, at Apr 26, 2007, 6:21:00 PM  

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