Knowledge Problems

Monday, August 13, 2007

The Fundamental Operation: Projection

To illustrate a further important point, below is an actual object-property table. It shows 10 objects and 5 attributes. You can think of it as representing 10 objects that just came off an assembly line. Each of those objects has 5 properties, for example `\{P_1,P_2,P_3,P_4,P_5\}=\{text{color},text{shape},text{texture},text{size},text{orientation}\}`. Thus, for example, these objects can have three possible colors, `P_1 in \{0,1,2\}`, which might be `\{text{blue},text{red},text{green}\}`. It can be noted that not all of the objects need be distinct, and in fact there are several identical objects in this set. (Can you find them? Just kidding.)

Object Property
  `P_1` `P_2 `P_3` `P_4` `P_5`
`O_1` 1 2 0 1 1
`O_2` 1 2 0 1 1
`O_3` 2 0 0 1 0
`O_4` 0 0 1 2 1
`O_5` 2 1 0 2 1
`O_6` 0 0 1 2 2
`O_7` 2 0 0 1 0
`O_8` 0 1 2 2 1
`O_9` 2 1 0 2 2
`O_{10}` 2 0 0 1 0

The crucial fact that must be observed is that the distinguishability of objects depends on which attributes are considered. For example, in the table above, we can already note that objects `\{O_1,O_2\}` are indistinguishable, as are `\{O_3,O_7,O_{10}\}`. The objects in these sets correspond to the same point in feature space, even if they are physically distinct objects (which we assume they are). However, if we were to further assume that attributes `\{P_1,P_2,P_3\}` are inaccessible, and were therefore able to consider only attributes `P_4` and `P_5`, we would observe (table below) that several other objects now also become indistinguishable, collapsing to the same point in feature space.

Object Property
  `P_4` `P_5`
`O_1` 1 1
`O_2` 1 1
`O_3` 1 0
`O_4` 2 1
`O_5` 2 1
`O_6` 2 2
`O_7` 1 0
`O_8` 2 1
`O_9` 2 2
`O_{10}` 1 0

In particular, we now find that we have the following sets of indistinguishable objects:


So there are only four discriminable kinds of objects represented in this knowledge system after attributes `\{P_1,P_2,P_3\}` have been dropped and consideration has been restricted to attributes `\{P_4,P_5\}`. These four kinds are called "equivalence classes" because the entities contained in each class are indistinguishable (equivalent) based on the attributes under consideration. The process of "dropping attributes" is called projection and corresponds to an orthogonal (parallel to axes) geometric projection in feature space. In the previous case, we would say that the data is projected onto dimensions `\{P_4,P_5\}`. In general, the projection onto a given set of attributes will produce a collection of equivalence classes containing objects which cannot be disambiguated based on those attributes. This will be extremely important in what follows.

Wednesday, August 01, 2007

A Basic Knowledge Representation Framework

Reminder: There is some mathematical notation in this post. If you use Internet Explorer, you may need to download the free MathPlayer plug-in from Design Science. If you can see the equation below (rather than a bunch of ASCII text), you should already be good to go:

`sum_(k=1)^n k = 1+2+ cdots +n=(n(n+1))/2`

OK. On with the show...

Perhaps the most basic knowledge representation framework is the rectangular object-property table or array, in which rows represent individual objects or entities, and columns represent properties or variables. An example is shown below:

Object Property
  `P_1` `P_2 `P_3` `\ldots` `P_n`
`O_1` 2 1 100 `\ldots` -0.01
`O_2` 3 0 430 `\ldots` 0.23
`O_3` 2 1 `\ldots` 0.30
`\vdots` `\vdots` `\vdots` `\vdots` `\vdots` `\vdots`
`O_m` 3 0 430 `\ldots` -0.43

Sometimes this format is simply called "raw data" or "flat data," because this is the basic representation for data acquired from multiple sensors, channels, etc., and the format used in familiar spreadsheet applications. In the Rough Set literature (pawlak_91), this framework is called an "information system" (pawlak_81), "knowledge representation system" (wong_ziarko_86), "attribute-value system" (ziarko_shan_96) or "information table" (yao_yao_02). In the case where the property values are binary (i.e., present or absent), the format is what Watanabe (1985) calls an "object-predicate table," or "Aristotelian table". It can be given a rigorous mathematical definition, but we'll skip that here, since it's pretty clear what's going on. A great many knowledge discovery problems can be represented in this model (ziarko_shan_96).

The `m \times n` attribute-value table above represents a set of `m` objects (also called entities or situations), wherein each object or situation (i.e., each row in the table) is described by a set of `n` properties (also called variables, attributes, features, or dimensions). We use these terms interchangeably as do others (goldstone_98,bruner_goodnow_86), although many authors also find it convenient to draw distinctions. In the example above, all properties have been coded numerically, although it makes no difference for our discussion. For example, the `P_1` property might represent "day of week," and the actual names of the days could be substituted for the numerical indices shown. The example also illustrates that different properties may have values drawn from different domains. For example, whereas property `P_1` appears to adopt single-digit integer values, property `P_2` appears to be a (binary) attribute adopting values in the domain `[0,1]`, property `P_3` appears to adopt larger integer values, and property `P_{n}` apparently adopts real (floating point) values, perhaps in the domain `(-1,1)`. The issue of missing values, such as that for property `P_3` on object `O_3`, will not be relevant to this discussion.


We will move to a more concrete example a bit later, but let us first pause to note a number of points about this kind of representation.

  1. An object or entity in this representation is just a list of property values, called a "tuple" in database jargon. Thus, object `O_1` can be described as the property vector or ordered list `[2,1,100,\ldots,-.01]`, which is just convenient shorthand for `[P_1=2,P_2=1,P_3=100,\ldots,P_{n}=-.01]`. Geometrically, then, each object represents a point in an `n`-dimensional "feature space."

  2. An object in this framework is just a list of property values. Two objects having the same set of property values are entirely indistinguishable. Depending on the application, it may still be the case that these objects are physically distinct from one another, but they are nevertheless indiscriminable based on their properties, and therefore completely interchangeable. Identical objects occupy the same point in feature space. There is in some respect a commitment here to something like Russell's "bundle theory" in which an object is nothing more or less than its descriptions (in all possible worlds), there being no "substrata" or essence to it; that is, there is no substrate in which the properties inhere. However, since all we can ever deal with rationally are descriptions of one sort or another, this doesn't seem like a shortcoming unique to this particular representational system. More on this later.

  3. This kind of system can represent many kinds of knowledge. The objects in question may be individual entities, such as donuts on a shelf, where the properties may indicate donut attributes such as flavor, topping, calories, etc. Alternatively, the objects may represent the state of some system at discrete times. For example, an "object" or "situation" might be the weather in Central Park at a given moment, so that `O_1` is the weather at 9am, `O_2` is the weather at 10am, `O_3` is the weather at 11am, and so on. The attributes in this case could be descriptors of weather such as temperature, pressure, precipitation, cloud cover, etc. Thus, the knowledge base in this instance represents a multidimensional time-series or multidimensional signal. This idea is incredibly powerful, because each object/row then represents "the state of the universe" at a given moment in time. By "universe," I mean "universe of discourse," that is, the set of all properties that we are concerned about in a given application, which in the limit we can imagine to include all the properties in the actual universe. (We can imagine!)

  4. A real-valued (i.e., "continuous") or high-cardinality variable such as `P_{n}` in the example above would typically need to be discretized (quantized) for purposes of analysis. Patterns in data often only become (statistically) visible when we step back and view the data at a distance, i.e., at a coarser level of resolution (ziarko_89). To do this, we represent an entire range of variable values with a single symbol, no different than when we round-off decimal numbers to the nearest integer. This process of quantization is actually a matter of categorization, which we will discuss in much greater detail later. In the examples that follow, we will play only with integer values.

  5. In almost all cases of any interest, a given object-property table will be considerably smaller than the maximum size that such a data table might have if all combinations of attribute values were to occur. That is to say, in practice, not every object that can logically occur does actually occur (Mervis_Rosch81). For example, although one might logically conceive of a flying animal that weighs more than 300 pounds, in fact there are no such animals. This means that not every value of the attribute "able to fly" defined on domain `{yes,no}` co-occurs with every value of attribute "weight" defined on the real numbers. The observation that not all objects that can occur do occur, or — more generally — that their probabilities of co-occurrence are not uniform — this is an observation about the very essence of structure (pomerantz_lockhead_91). Structure in data manifests through the unequal co-occurrence of certain sets of attribute values, i.e., the tendency of certain attribute combinations to occur with greater or lesser frequency than other attribute combinations. The "empty locations" in feature space — the events that could logically have happened but did not — are the hallmark of structure. We will return to this important idea a little later.

  6. There is an object-property duality that becomes apparent by rotating the object-property table 90° counterclockwise. Just as it is possible to describe an object as a vector or tuple of property values, it is possible to describe a property as a vector of "object values." A given property then corresponds to a particular point in "object space." This duality is very interesting and, as I understand it, forms the basis for the field of Formal Concept Analysis (FCA), but we will not be concerned with it here.

Finally, before we get to the major limitation of this framework, let us point out a minor limitation: The information system representation described above has been accused of failing to adequately represent relational information, (e.g., arnone_71). As Hahn & Chater (1997) point out, a "bird" is not just a collection of features {wings, beak, feathers, ...}, but is rather a collection of these features having a particular relationship to one another: "A creature with all the right features in the wrong arrangement would not be a bird!" However, while the information system may not be the optimal representation for relational information, it is still possible to represent such information within the system. If, for example, the distance from wing tip to wing tip is an important relational aspect of birds, we can introduce a property that represents this distance, and we can do the same for any other relational qualities. If there it is a relation among many features that is of importance, then we can introduce a new variable to code that as well. This is not to say that the information system gives us the reasoning apparatus by which to deduce new relational information, such as would automatically allow us to know, for example, that "A larger than B and B larger than C entails A larger than C." No, we are simply stipulating that it is possible to represent relational information within the object-property table, and this should be enough.

The Major Limitation

Now, having gotten the more neutral observations out of the way, it is crucial to point out a number of limitations on the sort of knowledge representation system we have been describing above. Principally, the very idea of there being "objects" and "attributes" is a philosophically troublesome one. Firstly, it is just not obvious that we can represent the totality of the external world in terms of objects (Watanabe_85). Who instructs us on what are the "correct" properties by which to distinguish legitimate objects? In the opinion of Bruner, Goodnow, & Austin (1986), all that is required of an attribute is that it be a distinguishable element: "An attribute, in brief, is any discriminable feature of an event that is susceptible of some discriminable variation from event to event. Indeed, if it did not vary it would very likely not be discriminable in any case — the fish will be the last to discover water." So essentially anything discriminable constitutes an attribute, which therefore places almost no constraint on the set of possible attributes for a given system. It goes without saying (maybe) that if we have a discriminable property `P_1` and a discriminable property `P_2`, we can always imagine an additional discriminable property `Q` that is equal to `P_1^2`, or `P_1 \times P_2`, or `\pi P_1 sqrt(P_2) + 42`. So what makes one feature set more legitimate than another? And if what distinguishes two objects is just their features, as our object-property representation scheme assumes, then this ambiguity on what constitutes legitimate features propagates to an ambiguity on what constitutes legitimate objects (watanabe_85).

More provocatively, are there indeed really such things as features or objects at all? This question did not appear to bother the medieval or ancient thinkers, who believed that objects and features were indeed objective aspects of the external world. "Tails" and "hooves" are indeed genuine features of horses, and "horses" are in turn genuine objects. The ancients could debate how significant the possession of tails or hooves is to a horse's horseness, but they could not debate the fact that these objects and attributes genuinely exist. However, in the wake of Berkeley and Kant, the confidence about what constitutes objects and features has evaporated, and it is now clear that the things we subjectively regard as objects and features are themselves the product of a complex interaction between what exists in the external world (distal stimulus) and a sequence of processing performed by our perceptual and cognitive apparatus. Thus, the features to which we have access are symbols employed by a knowledge representation system (our mind) that correspond in some consistent way to aspects of the external world (markman_99), but these features are still notably the product of a classification already imposed by the mind on the world. We cannot of course assume that features are strictly internal symbols, since then we would face a much larger problem of explaining in what way (if any) these features are grounded in reality (harnad_90).

This bifurcation between external and internal features has been given the catchy name of Occam's Cleaver (panaccio_05), which, stated poetically, instructs us that "we should be cautious not take as features of the things signified the features of the signs that signify them. In other words, we should not conflate representational features with ontological ones." This cautionary attitude toward features and objects certainly also characterizes the position of modern cognitive science, where the topic of how objects and features are "made" by minds remains a vigorous area of empirical study, often yielding surprising results. In machine learning and statistics, the question of "What are the features?" plays out practically in the areas of "feature selection" and "feature creation," which include such techniques as Independent Components Analysis and Projection Pursuit. Although there is no final answer to this question, it always being a matter of needs and expectations for a particular application, in some sense a learning theorist might say that the "right" set of features are those which capture the majority of intrinsic structure in a given system. Any set of features having this property is then the "right" set of features, and there may obviously be many such sets.

This last observation gives us a way to recover somewhat from criticisms regarding a severe over-commitment to objects and features. Let it be the case that an actual system (in the distal world) can indeed be described in a myriad of ways; that is, using a large or infinite variety of different feature sets. We can still assume that there is at least one such description that will adequately capture all the interesting structure in that system. In other words, there is some property-based description — not necessarily available to a given observer — but there is some description, in some property language, that adequately represents the phenomenon in question. This assumption can be rephrased to state that there is no system in the distal world that is completely impervious to description in some property language — i.e., by some set of attributes or variables. This is not an assumption that every distal system is describable by us — just that it is, in principle, susceptible to description.

The Major Objection

It will probably immediately be objected that in fact there is a perfect and devastating counterexample to the assumption that "there is no system in the distal world that is completely impervious to description in some property language," and that this counterexample is the Deity himself. After all, have not Rambam and a thousand lesser souls admonished us that God is beyond and above all description? Wasn't this their whole point: That there is at least one distal system, God, that is not amenable to any description whatsoever?

My answer is that, yes, in some instances this may have been exactly their point. But, I believe that many thinkers (e.g., Philo, Aquinas, see earlier) probably intended something much less sweeping, and meant by their remarks merely that God is not describable by or to human minds. Not that he is resistant to description altogether. (Can even God not describe God?) This more moderate unknowability proposal poses no problem for us, since all we are assuming of distal systems is susceptibility to some description, not necessarily description by the human mind. The more extreme claim, however — that God is resistant to property-based description of any kind whatsoever, whether the description be accessible by human minds or not — this would in fact pose a problem for us, if it were a coherent claim. But it is not.

It is certainly possible to say that "God is resistant to any description whatsoever," but this is writing a check that cannot be cashed. I do not believe we can claim to offer a coherent concept of "that which cannot be described in any possible description language." That is, we can neither offer an example of such an indescribable entity, nor can we provide an explanation of how or why it is that an entity would be resistant to description in every possible language. In light of this, we have to consider the idea of the absolutely indescribable entity to be incoherent. Now, I do realize that there are arguments made by Rambam and many others as to why it is that God cannot have qualities of any kind, which as I understand the arguments (see earlier posts) all come down to the issue of the unity or non-compositionality of God. And, as I said earlier as part of my "unavoidable heresy", the idea of non-compositionality is also incoherent, and so proofs from that direction cannot ameliorate the incoherence in the notion of a system that has absolute resistance to description.

And so my position here is as follows: The notion of an entity that is completely impervious to description under any possible description language is a notion that is incoherent, and a concept which can neither be explained nor exemplified. Hamilton (1864) writes that "What in reality has no qualities, has no existence in thought, — it is a logical nonentity," and by this I understand him to mean that it is incoherent to talk about a system or phenomenon which has no attributes whatsoever. Such a notion represents not profundity, but stupidity.

That being said, the idea that a system or phenomenon might be indescribable to some observer or by some property language is not stupidity. It may not be especially profound either, as it turns out, since we are well familiar with many observers having limited descriptive abilities; infants, mosquitoes, chess programs, etc. It is entirely coherent to state that a given phenomenon in the distal world is describable by certain observers and not by others. This is the approach we will take in what follows.

Sunday, July 22, 2007

How Is God Unknowable?

As I suggested earlier, in order for us to consider "unknowability of God" to be a coherent notion and to subject it to further examination, we need to be able to offer at least a tentative formal characterization both of the concept "knowable" and the concept "God". The failure or unwillingness to do this is, in my opinion, a tacit admission that such claims really have no meaning at all. As a first step in this direction, we have to be clear that when we use the term "unknowable" we are actually talking about a state of knowledge — a relationship that obtains between two entities, a knower and that which is to be known. This would appear so obvious as to not even need saying; after all, to what could a term such as "unknowable" refer if not to a state of knowledge? However, it seems to me that religious thinkers often feel at liberty to use terms without specifying the context within which those terms have meaning, and in particular I have often seen the term "unknowable" used independently of any knowledge representation framework that would grant it a concrete and fixed meaning. And so I stress that "unknowability" is a state of knowledge, and that therefore the term has a meaning only within the context of a knowledge representation system. This all seems straightforward to me.

Thus, as I say, we need a formal characterization of (1) knowledge, (2) knower, and (3) that which is to be known, before we can claim to understand what it means for "that which is to be known" to be "unknowable." Now, jumping ahead, since this essay is about the "unknowability of God," in the final analysis "that which is to be known" will be not just any phenomenon, but the phenomenon we call "God." However, the issue of unknowability extends far beyond the particular issue of knowing God, and by approaching the matter of knowledge representation formally, we can examine in detail issues concerning the general notion of "unknowability" without specifically making claims about God. In a sense, therefore, the question of whether and how we can know God is ancillary to the larger issue of whether and how we can know anything. So essentially we will be examining the consequences of allowing God to be a member of the class "unknowable," once this class is suitably and formally defined.

Not to belabor the point, but again, the approach here is not to declare God knowable or unknowable, but rather to examine what it would mean for God to be knowable or unknowable in a certain capacity, and what consequences would follow from such a status. Indeed, if we were to foolishly declare God to be unknowable, we would be invoking the very knowledge that we claim it is impossible for us to possess! Heschel (heschel_51) in fact has it right that "... he who insists that God is in every way unknowable claims to know that what he says cannot be known." This alone would seem to make many of the classical statements on God's unknowability and incomprehensibility highly suspect. Our goal in science and philosophy, as Nagel writes, is "... to reach a position as independent as possible of who we are and where we started, but a position that can also explain how we got there." And, as Heschel suggests, the matter of "how we got there" is the very first challenge that must be raised against the position that "God is unknowable". Simply, how do Rambam and Ramchal and Steinsaltz and Kaplan come to a position where they know that God is unknowable? In adopting this position, they are in fact professing a knowledge of God that is far more sweeping than what any of their opponents had ever claimed. To be able to assert that "it is inconceivable that man's understanding should be able to penetrate to the essence of the Infinite" (berkovits_04) would seem to itself demand a transcendent theory of both "man's understanding" and the "essence of the Infinite" that is far beyond anything that any thinker has ever offered. Likewise, to argue that "we have neither the words nor the mental processes that would enable us to actually describe God or understand Him" (Kaplan) would seem to require one to posses God-like insight into the structure of the Mind of Man and Mind of God.

Our purpose in this essay, therefore, is not to make such allegations, and our little project will hopefully not fall victim to such presumptions. (It will fall victim to other presumptions, however.) The current project is not to declare a particular entity (i.e., God) to be knowable or unknowable to human beings, but rather to provide a mathematical framework for discussing such questions intelligently, and one that might provide some general insights into the issue of knowability that will allow us to use the concept in a meaningful manner.

Still, one matter that must trouble the reader as we proceed is how we can possibly propose to obtain a formal characterization of the concept of God, or rather, how we can allow God to be subsumed in a class (i.e., the class "unknowable") that admits a formal characterization. Doesn't the very possession of any such formal characterization immediately make God knowable, so that whatever we might then prove about God's knowability would have been trivially assumed from the start? In other words, by the very act of formally specifying those three elements, (1) knowledge, (2) knower, and (3) that which is to be known, and then assuming that a process called "God" can be represented within the specified system, haven't we already committed ourselves to the idea that God is knowable?

Yes and no. And maybe. It is difficult to answer this question in the abstract, so as we go along I will try to be very clear on exactly what assumptions I am making about the entity or phenomenon known as God. And while almost all of the assumptions will be heretical according to someone, they are also so weak that it would be difficult to claim that these assumptions alone constitute knowledge of God.

So then, what does it mean, formally, to say that something can be know or cannot be known? There are a thousand legitimate answers to this question, for a thousand conceivable knowledge representation systems. In the next section I will review a very elementary knowledge representation scheme, and review a straightforward definition of unknowability that can be used within that system to express the issues we are talking about.

Thursday, May 17, 2007

Kabbalism and Rationalism: A Point of Agreement

So, in the end, what have the Jewish theologians told us about the knowability of God? On the kabbalistic account, God is represented by an unknowable component (En Sof) and a knowable component (Sefirot). On the rationalistic account, God is represented by an unknowable component (Essence and Attributes) and a knowable component (Actions). We therefore seem to have a point of agreement between the kabbalistic and rationalistic theological systems; in particular, both concede that God must be partially knowable and partially unknowable.

This agreement is, in itself, not very surprising. In the first place, both the kabbalistic and rationalistic systems are ultimately outgrowths of the Jewish religion, and therefore both require in the end that God remain somehow knowable to human beings. Were God to be completely and utterly cut off from human experience, nothing in the Jewish religion would make any sense, and, for the most part, Jewish thinkers are not willing or able to go this far. "Far from being able to serve as the basis for religion, the Absolute One renders religion meaningless" (berkovits_04).

This point of agreement between kabbalistic theology and rational theology is also unsurprising for the following simple reason: Rambam casts a large shadow. While the concept of En Sof may predate Rambam (although Kaufmann Kohler and Isaac Broydé, writing in the Jewish Encyclopedia, attribute the term to Azriel of Girona, who was Rambam's junior by a generation) and in any case owe more to Neoplatonic than Aristotelian ideologies, it seems unlikely that kabbalists thinking and writing about En Sof in any succeeding generation could have overlooked the 800-pound gorilla in the room, which is Rambam's denuded conception of God-as-He-is-in-Himself. What was left for the kabbalists after Rambam had completed his work was merely to pile poetical grandiosities upon the concept. From the kabbalistic standpoint, En Sof could certainly not be permitted to be less unknowable than Rambam's God, and since it is logically impossible for anything to be more unknowable than Rambam's God, the kabbalistic conception of the Unknowable was essentially forced into rough equality with that of Rambam. At least it seems so to me.

Still, it is beyond certain that both schools would reject a conclusion such as that "God is composed of two parts, one knowable and one unknowable." It is almost as certain that both schools would reject a more cleverly-worded conclusion such as that "God is composed of two aspects, one knowable and one unknowable." Cordovero writes, for example,

In the beginning Ein Sof emanated ten sefirot, which are of its essence, united with it. It and they are entirely one. There is no change or division in the emanator that would justify saying it is divided into parts in these various sefirot. Division and change do not apply to it, only to the external sefirot (matt_90).

There are hundreds of other testimonials to the absolute unity of God and to His imperviousness to division into "aspects," much less "parts". That being said, if it is to be admitted that God is somehow knowable and somehow unknowable (which is the only reasonable conclusion that can be drawn from both the rational and kabbalistic treatises), and if we are not to play games of equivocation with our own mundane terminology, such as by permitting "knowable" and "unknowable" both to inhere in the same subject without contradiction, then we are forced to accept one of the previous statements or a close cognate, however distasteful. Thus, kabbalists and rationalists alike must accept something like the following: "God is composed of two aspects or parts, one knowable and one unknowable."

One can choose to reject such a statement outright, but only at the risk of completely collapsing the distinction between "knowable" and "unknowable". If one rejects the above statement and all its cognates, it is no longer meaningful to talk about "En Sof vs. Sefirot" or about "Essence vs. Actions". There can be no meaningful distinctions to be had, and it is all holy nonsense. Certainly, one can debate what precise meaning we should attach to the notion of "part" or "aspect" in this context, and we will try to be more formal about this below, but if we desire that there be a distinction between knowable and unknowable, and we still wish to respect the law of contradiction, then it must be the case that there are in some sense multiple parts or aspects of God. So it is.

In any case, despite their point of agreement, neither the kabbalistic nor rationalistic system provides us with an explicit formal definition of "knowability," nor does either provide an explicit "knowledge-theoretic" explanation of the relation between the knowable and unknowable components/aspects of God. It is these two omissions that I will make a primitive attempt to address in the following section.

Sunday, May 13, 2007

Unknowability of God In Jewish Rationalism V: So What's Left?

Having concluded our little survey of Jewish rationalism, let us summarize. The position on God's unknowability found among many in the Jewish rationalist school (including, as I see it, Bachya, Rambam, and Albo) is that God's attributes are completely unknowable in the strongest possible sense. That is, there is no known predicate (i.e., one for which we can specify the classes induced by the predicate) which can be applied to God. Not even the predicate "one" or the predicate "exists" (if the latter even qualifies as a predicate) can be applied to God. The reason that we cannot know any of God's attributes is simply that God does not possess attributes, period. Naturally, objections both ancient and modern have been raised against this strong position on unknowability. I earlier mentioned the objections of Gersonides, Aquinas, and Berkovits, and I will return to consider their positions at the end of this essay (should I actually make it that far). Here, I just want to consider where the "strong unknowability" position leaves us in our relationship to God.

It could not have escaped a thinker of Rambam's caliber that by making God utterly unknowable and outside of any relation or likeness to mundane entities he was flirting with the possibility of God's complete irrelevance to the world of men, and thus of opening the door to the utter vanquishment of religion. He knew that while there might be some intellectuals who could achieve satisfaction and meaning through sublime meditations on the transcendental nothingness of the Unknowable Absolute, this task could hardly satisfy the majority of religious people, for whom "ponderation on ultimate unknowing" would scarcely constitute a religion of any sort at all, let alone the religion of the Bible that they had been taught from childhood. For the Jewish religion to make any kind of sense, God must have some tangible influence in the world of men — despite His unknowability. The solution that the Rambam proposes (and in which he is followed by many others) is that God is indeed yet knowable — not in His attributes, but through his actions:

Every attribute that is found in the books of the deity, may He be exalted, is therefore an attribute of His action and not an attribute of His essence, or it is indicative of absolute perfection (GP I:53)... Thus, God's emanation or connection with the world is his actions, and this is the intent of God's message to Moses in Exodus 33 (GP I:54)...

To recollect, Exodus 33 is perhaps the Bible's theological manifesto sui generis, if one reads it that way, of course: Moses had asked God, "Now, if I have truly gained Your favor, pray let me know Your ways, that I may know You and continue in Your favor," and "Oh, let me behold Your Presence." God had responded:

I will make all My goodness pass before you, and I will proclaim before you the name Lord, and I will grant the grace that I will grant and show the compassion that I will show. But," He said, "you cannot see My face, for man cannot see Me and live." And the Lord said, "See, there is a place near Me. Station yourself on the rock and, as My Presence passes by, I will put you in a cleft of the rock and shield you with My hand until I have passed by. Then I will take My hand away and you will see My back; but My face must not be seen.

Rambam sees in this passage evidence that God is providing an affirmative answer to Moses' first question concerning God's ways (actions). God shows "goodness" and "compassion" in the world of men, and it is through these actions that He can be known. The second entreaty of Moses, to behold God's Presence (God's attributes), is however denied. "Man cannot see Me and live." And the final statement that "you will see My back" can again be taken, as by Philo, as a reinforcement of the idea that it is solely by the effects of God's actions in the world that He can be known.

Albo makes a similar reading of Psalm 104, which begins "Bless the Lord, O my soul; O Lord, my God, you are very great; You are clothed in glory and majesty."

Alluding to the first aspect [God's essense], he says, "O Lord my God, Thou art very great," i.e. from the side of Thy quiddity Thou art very great, so that man can not speak about Thee, and with all this, "Thou art clothed with glory and majesty," i.e. from the side concerning which it is possible to speak about Thee, namely from the visible activities which come from Thee. They show Thy glory and Thy majesty. Therefore he describes in the sequel the creations which come from God, and which point to God's excellence and perfection by the perfection which is visible in them. (husik_29, p.6)

But what does it mean to say that God is knowable in His actions? I have not seen a coherent account of how one could know something about the actions of an agent, and simultaneously not know anything about the attributes of that agent. Is not to say that "agent X does action Y" just to say that "agent X has the attribute of doing action Y"? Is not an animal which digs holes in the earth just an animal that has the attribute of "burrowing"? How does one ascribe an action to an agent without simultaneously ascribing attributes? If God's attributes are utterly unknowable (or even nonexistent), then surely his actions are unknowable (or nonexistent) as well. I am hoping that I might find somewhere in Rambam a more coherent explanation of how this can work, but until then I cannot say much more.

A different problem related to God's actions which weighed on the minds of many medieval thinkers (and which indeed seemed considerably more pressing than the one I just mentioned), is the question of what the notion of "action" entails for alteration in the state of the Deity. From our own mundane experience, we recognize that (intelligent) action is generally precipitated by some sequence of changes in the cognitive state of the acting agent. We (as agents) speculate, we discover, we hypothesize, we deliberate and revise, we decide, we will, and then we finally act. However, the medieval thinkers absolutely rejected the idea that any such sequence of cognitive state-changes could obtain in the case of actions by the Deity. God's state cannot change, both because this would imply plurality in the Deity (multiple states), and because it would imply that the Deity's present state is not one of ultimate Perfection. If God is Perfect, why would he ever need to undergo change? The typical medieval opinion is that change is a defect, and so cannot be predicated of God (cf. Albo, husik_29, p.129).

The problem is even more keen on the view (cf. Aquinas, kreeft_90, p.136) that God's intellect, knowledge, and essence are all one and the same. If such is the case, a change in God's intellect would imply a change in God's essence, and, as Maharal suggests, "it is better to remain silent than say such things" (mallin_carmell_75).

The solutions to this problem are as a whole quite unsatisfying. Albo, for example, cannot deny of God a notion such as "will," because if God could be said to lack will, this would make him by any account an inferior kind of agent (if an agent at all). And yet to allow the notion of divine "will" leads to a lengthy sequence of logical dilemmas. Here is Albo capturing the conundrum:

...the term voluntary applies where the agent desires and does a given thing at a given time, which he did not desire before. The term voluntary does not apply to a thing which is always in the same condition. It follows therefore that the one who has will changes from a condition of not willing to a condition of willing. If so he is affected by, and receives change from another, for a thing can not be active and passive at the same time in the same relation. But God can not change, nor be affected by another. Nor can He be affected by Himself, for He would then be composed of two elements, an element by virtue of which He acts, and an element by virtue of which He is acted upon. But there can not be any composition in God at all, as we shall see. Nor can God be active and passive at the same time. It follows therefore that God's activity can not be voluntary, since He can not change. Moreover, an agent possessed of will lacks the thing which he desires. But God does not lack anything which He desires at any time and did not have before. Similarly one who exercises a choice chooses one of two things because that thing is more suitable to him than the other. He therefore lacks the thing which he chooses before he has chosen it. It follows, therefore, that God's activity can not be due to choice or will. But if He does not act with choice and will, nor like a natural agent, as light comes from the sun, as we explained, how can God be called an agent? (husik_29, p.13)... If God does not act with will, He can not act at one time rather than another, a supposition which leads to the doctrine of eternity in the absolute sense, and to the rejection of all the miracles in the Torah. Prayer ceases to be of any avail in time of distress, and right conduct and repentance are useless, and so on..." (husik_29, p.16).

The possible solution that God's actions in history (for example, the miracles retold the Bible) were somehow "planned out" from eternity (or from Creation), and therefore did require any proximal change in God's will, is dismissed by Albo as positively absurd:

Thus they say, God stipulated with the works of creation that the Red Sea should divide, that the fire should not burn Hananiah, Mishael and Azariah. From this it seems that their purpose was not to attribute to God a newly originated will at the time of the occurrence of the miracle. Therefore they say that the miracle occurred when it did through an eternal will, which determined that the miracle should take place when it did. But this opinion is far from intelligible. For if we examine the expression eternal will, we find that it is a spurious conception, and points to necessity rather than will. For the question remains as it was originally. When the world was created or when the miracle occurred through His eternal will, was it possible for God to postpone it to another time, or not? If He could postpone it, then the eternal will was nullified. For the reason He did not postpone it was not because of the eternal will, but because He did not desire at that moment to postpone it. If He had desired to do so, He would have postponed it. And if it was not possible for Him to postpone it because so the eternal will had decreed, then there is no such thing as will any more, and He becomes an agent acting by necessity and not with will, since He could not postpone the act if He would. We should have to say the same thing about all miracles and all acts proceeding from Him at any time, that they happen by necessity, and that the thing could not help being when it is, for so it has been determined by God's eternal will, which can not be changed... (husik_29, p.15)

And so what Albo, following Rambam, is forced to fall back on is simply the notion that we cannot understand what kind of an agent God actually is. That is, we do not know what it means for God to have "will" or "desire", or to make a "decision". These concepts simply cannot mean the same thing for Him as they do for us, and we thus have absolute equivocation in terminology. Albo again:

Because at first sight it might seem that there is a change in God's will, he says, "For My thoughts are not your thoughts, neither are your ways My ways...," i.e. as there is a difference between His knowledge and our knowledge, the two not belonging to the same class at all, so there is a difference between His will and our will, and between His ways and our ways, there being no similarity between them at all (husik_29, p.23)... It is clear therefore that since we see acts emanating from God which are similar to those acts which emanate from a voluntary agent, we speak of God as desiring and willing; though we can not understand how will and desire reside in God without causing change and affection. This is unknown to us, as the nature of His knowledge is unknown to us (husik_29, p.145).

Indeed, on Albo's opinion, the only alternative to absolute equivocation is to attribute ignorance to God (as does Ralbag, to whose views Albo is evidently contrasting his own below):

There is no way out of this difficulty except by saying either that His knowledge is not of the same kind as our knowledge, and that it does not cause change in Him as our knowledge causes change in us, or to commit the grievous heresy of attributing ignorance to God and saying that He does not know any particular thing that originates in the world, but the universal only, that He did not know Moses while he was in existence any more than He knew him before he existed and after he ceased to exist and that God does not know opposites, else He would have different kinds of knowledge. The best solution is to say that His knowledge follows His wisdom, and as His wisdom is of His very essence and not something added to His essence, and is absolutely unknown, so His knowledge and His will and His power, which follow His wisdom, are absolutely unknown (husik_29, p.21).

So if God's will is not like our will, and God's knowledge is not like our knowledge, and therefore God's actions are not like our actions, then how is it meaningful to talk about knowing God through his actions? I hope someone can explain this to me. In any event, on the rational view it seems that this is the very limit of what we can know about God.

Next section: Wrap-up of the Introduction! Hallelujah! (Yes, sadly, everything till now was part of the Introduction. Hopefully things will move more quickly now!)

Sunday, April 29, 2007

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.

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!