THEAETETUS: Well, do you see what we're looking for?
VISITOR: I think I see a large, difficult type of ignorance marked off from the others and overshadowing all of them.
THEAETETUS: What's it like?
VISITOR: Not knowing, but thinking that you know. That's what probably causes all the mistakes we make when we think.
THEAETETUS: That's true.
VISITOR: And furthermore it's the only kind of ignorance that's called lack of learning.
THEAETETUS: Certainly.
VISITOR: Well then, what should we call the part of teaching that gets rid of it?
THEAETETUS: The other part consists in the teaching of crafts, I think, but here in Athens we call this one education.
VISITOR: And just about all other Greeks do too, Theaetetus. But we still have to think about whether education is indivisible or has divisions that are worth mentioning.
THEAETETUS: We do have to think about that.
VISITOR: I think it can be cut somehow.
THEAETETUS: How?
VISITOR: One part of the kind of teaching that's done in words is a rough road, and the other part is smoother.
THEATETUS: What do you mean by these two parts?
VISITOR: One of them is our forefathers' time-honored method of scolding or gently encouraging. They used to employ it especially on their sons, and many still use it on them nowadays when they do something wrong. Admonition would be the right thing to call all of this.
THEAETETUS: Yes.
VISITOR: As for the other part, some people seem to have an argument to give to themselves that lack of learning is always involuntary, and that if someone thinks he's wise, he'll never be willing to learn anything about what he thinks he's clever at. These people think that though admonition is a lot of work, it doesn't do much good.
THEAETETUS: They're right about that.
VISITOR: So they set out to get rid of the belief in one's own wisdom in another way.
THEAETETUS: How?
VISITOR: They cross-examine someone when he thinks he's saying something though he's saying nothing. Then, since his opinions will vary inconsistently, these people will easily scrutinize them. They collect his opinions together during the discussion, put them side by side, and show that they conflict with each other at the same time on the same subjects in relation to the same thing and in the same respects. The people who are being examined see this, get angry at themselves, and become calmer toward others. They lose their inflated and rigid beliefs about themselves that way, and no loss is pleasanter to hear or has a more lasting effect on them. Doctors who work on the body think it can't benefit from any food that's offered to it until what's interfering with it from inside is removed. The people who cleanse the soul, my young friend, likewise think the soul, too, won't get any advantage from any learning that's offered to it until someone shames it by refuting it, removes the opinions that interfere with learning, and exhibits it cleansed, believing that it knows only those things that it does know, and nothing more.
- Plato, The Sophist 229b - 230d (tr. Nicholas P. White)
There is always a more fundamental question. Kenny Easwaran at Antimetea has written a response to my post What Does Bayesian Epistemology Have to Do With Probabilities? In his post, he raises the question, just what is a probability? I want to take a look at my own assumptions about what a probability is, and what he has to say, and see if this has any relevance for our discussion of Bayesian epistemology.
I will not attempt here to develop a philosophy of probability, like Bayesianism, or frequentism, or anything of that sort. These are accounts of what probabilities mean, but not of what probabilities are. Easwaran and I agree that probabilities, in the sense in which we are using the term, are certain formal constructions. I was assuming a particular set-theoretic construction, because that's what I was taught (although what I'm about to present is slightly different than what I was assuming before, because I didn't remember things quite right).
I had assumed that a probability was defined over what I called a "state space" (which is actually a computer science term, but is not totally inapplicable here) which is a set of equally likely outcomes.
The correct term is, in fact, "sample space," and, according to my textbook (Mathematics: A Discrete Introduction by Edward R. Scheinerman), an ordered pair (S, P) where S is a set of outcomes and P is a function from S (in some formulations, the power set of S is used, but that makes everything else more complicated, and I think all it buys you is a simpler notation) to the real numbers between 0 and 1 (inclusive) such that the sum of P(s) over every s ∈ S = 1.
Once we've got this, give the interpretation of 1 as certain truth and 0 as certain falsity, and so we can map things back to a Boolean algebra. Easwaran constructs this in reverse:
My understanding of the word is that “probability” refers to any function from a Boolean algebra to the real numbers satisfying the following three properties: (1) it is never negative; (2) the tautology is assigned value 1; (3) finite additivity (that is, given two elements whose conjunction is the contradiction, the probability of their disjunction is the sum of their probabilities).
Now, my discussion before was built on this sample space construction, and I was discussing what the members of the set were. Easwaran's construction has the benefit of allowing us to deal directly with propositions, without introducing the possible worlds semantics. This, I think, is why he seems to describe his view as in between (P) and (KPW): he can hold that there is a real sample space, and construct it out of propositions. With his construction, he doesn't need to go much further than what Kripke says explicitly. Ignoring the facts we're not interested in isn't a simplification for practical purposes: it's actually what we want to do.
Now, a benefit of (KPW) proper (that is, the view I originally dubbed (KPW)) over Easwaran's view is that it explains where these probabilities come from, at least in the case of an abstract ideal reasoner: we assign the same possibility to every epistemically possible world, and take a look at how many worlds the proposition in question comes out true in. As Easwaran points out, this may run into trouble, because these may not be defined. Things get tricky with infinite sample spaces: if they are similar enough to the real line (or plane, etc.), then things work out, but otherwise they may not. So my (KPW) may be in trouble. I wonder, though, on Easwaran's view or on (P), where the probabilities are supposed to come from.
The answer to the question in the title of this post may seem obvious (after all, isn't Bayesianism all about probabilities?), but I think that the long discussion that followed Lauren's post on van Fraassen's objection to Bayesianism from quantum mechanics shows that it isn't clear at all - or at least, that it wasn't clear to either of us as we were discussing the issue. I think that I now understand why. In this post, I'm going to give three answers to this question, which I will call The Primitivist Account (P), The Kripkean Possible Worlds Account (KPW), and the Lewisian Possible Worlds Account (LPW). This post will discuss what each view means, and where vagueness enters each account. I will also be identifying three crucial problems with (P) and showing how each of the other views answers these difficulties.
Here are brief definitions of each view, and how each one relates subjective degrees of rational confidence to probabilities (I will explain in more depth later).
Part of the reason for the previous confusion is that I was more or less assuming (P), and I think that Lauren had noticed some serious problems with it. First, a word on the reason for my assumption, and then I will try to state Lauren's objections.
(P) may be the dominant interpretation of Bayesianism. I don't really know. But there is good reason why someone reading the literature might think it's the dominant interpretation: it maps especially well to how Bayesian philosophers actually apply Bayesianism. Most philosophers who apply Bayesian reasoning (myself included) do it by simply making up numbers that are supposed to represent their degrees of confidence. Where do these numbers come from? We simply observe that we have varying degrees of confidence about different beliefs, and map these degrees of confidence to the real numbers between 0 and 1. Vagueness comes in from the fact that we don't have mathematically precise degrees of confidence, and our numbers are simply made up from our imprecise degrees of confidence, rather than computed somehow.
Now, as I have said, I believe that three important questions came out in our previous discussion which this theory leaves unanswered:
(P) does not answer these questions. This is, of course, to be expected from a theory called "primitivism," but I think the third question is particularly problematic. In the previous discussion, it was van Fraassen's assumption that we should do this very thing that brought up the issue. However, Bayesianism really needs these principles. Is it possible to provide an analysis of degrees of rational confidence that adequately answers these questions? (KPW) and (LPW) will attempt this very thing.
(KPW) is inspired by Kripke's treatment of possible worlds in terms of state spaces in the 1980 preface to Naming and Necessity, pp. 15-20. Kripke here argues that possible worlds are the same sorts of things as the "states" used in school probability theory, the difference being that possible worlds are maximally specific. Now, consider a view according to which the state space of Bayesian reasoning is the space of all epistemically possible worlds - that is, all the world-states (which are abstractions just like dice-states) which might for all we know be actual. Note that not all of these may be really possible. For instance, the Anselmian God either exists or does not exist, with logical necessity, but his existence and non-existence may both be epistemically possible for a particular person. So, when we say that we have subjective degree of confidence .5 for a given proposition, we are saying that that proposition holds in half of all epistemically possible worlds.
This view will be helped by Lewis's observation about the relationship between propositions and possible worlds: namely, that every proposition picks out a set of possible worlds, the worlds in which it obtains. (Lewis wants this to be a reductive analysis of propositions, but we need not do that.) So, consider any given proposition you believe. There is a set of possible worlds in which that proposition obtains. The set of epistemically possible worlds (for you) is the intersection of the sets for all the propositions you believe.
The (KPW) answer to question (1) has already been given - Bayesian degrees of confidence are probabilities. Let's proceed to give an interpretation of the math on (KPW).
Bayes' theorem is a relation between an initial probability - a probability over some state space S - and a conditional probability - a probability over some subset S' of S. Usually, we consider some proposition p and some evidence e. We already have assigned a particular degree of confidence to p and we want to adjust our confidence in light of learning the new evidence e. We use Bayes's theorem to calculate P(p|e). What has happened here? The new evidence e has eliminated certain formerly epistemically possible worlds - namely, all the worlds according to which ~e. In order to computer P(p|e) we have to know something about the relationship between p and e. In particular, we have to know P(e|p), P(p) and P(e) (all of them over the initial state space S). This involves knowing how many of our epistemically possible worlds certain conditions obtain in.
The (KPW) answer to question (3) should now be quite clear. Probabilities for events like dice rolls are, on this view, actually just special cases of degrees of subjective confidence. Why is there a 1/6 probability of a single die rolling 1? Because in 1/6 of all epistemically possible worlds it will land 1. (We should think of these world-states as covering the whole history of the world, so future events can be handled the same as past or present events.) In our school probability exercises, we simplify the case by supposing there are only 6 worlds. In fact, there are 6 sets of worlds. We know that the worlds will divide more or less evenly (we assume we know with certainty that the die will be rolled), because most of the propositions we are uncertain about vary independently of the result of the die roll. The ones that don't vary independently (e.g. propositions stating that the die is unfair in some particular way) are, for all we know, as likely to favor one side of the die as another.
Vagueness enters (KPW) by virtue of the fact that the worlds are created by us. They don't exist objectively. As such, there is vagueness as to how many worlds there are, and there is vagueness as to whether certain propositions are true in certain worlds. These things are simply not fully defined. We should nevertheless be able to fix upper and lower bounds by considering all of the possible resolutions of the vagueness (actually, we can probably do better by figuring out in advance which resolutions will lead to high values, and which to low values). In practice, however, we do something more like the die role case: we eliminate all the propositions that vary (more or less) independently (as far as we know) of the propositions under consideration, to divide the epistemically possible worlds into sets, and then consider each set as a single underspecified world.
(LPW) is very similar to (KPW) in its answers to the three questions. (LPW) holds that we are talking about real possible worlds, which are epistemically accessible - that is, which might, for all we know, be the world we're in. Vagueness on (LPW) is different. Because the worlds are fully defined and there is an objective truth about how many there are, there is only one source of vagueness properly so-called: vagueness about whether a given world is epistemically accessible. However, there is also second-order uncertainty - uncertainty about whether a certain world is genuinely possible, or whether a given proposition obtains in a certain world.
These two theories improve on (P) by providing explanations for why we use Bayesian reasoning the way we do, and why it works like probability theory at all. They also allow us to define our degrees of confidence much more clearly.
Hello. As a brief introductory reminder, I'm Lauren, Kenny's fiance, and a guest blogger here when I have time (which isn't very often.) However, I am going to take some time to discuss a paper by Bas C. van Fraassen, Conditionalizing on Violated Bell's Inequalities, in which he claims that quantum mechanics creates problems for Bayesian epistemology. I have two main points to make in response, the first is that he doesn't actually need quantum mechanics for his argument, and the second is where he has failed to account for the effect of choosing which events to talk about, which changes the conclusions of his paper. I will treat these in reverse order, though.
A brief summary of van Fraassen's argument is this:
In an experimental set up involving measuring the spin of entangled photons, there are two detectors, each of which has a 50% probability of detecting something (or registering something) for each run of the experiment. (Here I'm going to be slightly sloppy and use "register something" and "detect something" interchangeably to mean "made a positive spin reading".)
However, the detectors are not uncorrelated- the probability of one detecting something is related to the cosine of the angle between the detectors squared. This is well established in quantum mechanics.
Then, van Fraasen imagines a situation where someone named Hilary is asked to predict whether or not one of the detectors registers something. She initially answers that the probability is 50%. She is then told that the other, hereafter referred to as the second, detector did register something, but she is not told what the angle is between the detectors, although she does know of the cosine squared relation. She is then asked the same question, but she now no longer knows what the probability is, because she knows it could be anything between 0% and 100% depending on the angle between the detectors.
Van Fraassen then asks, if Hilary were forced to bet, what the best thing would be for her to do. He concludes that she ought to ignore this new piece of information, even though it is relevant to the probability of the first detector registering something, and to bet the first detector registers something 50% of the time, because, he claims, she would break even doing this. Then, van Fraassen questions why Hilary is justified in ignoring the information about the second detector, since it would change her opinion. This is especially a problem for Bayesian inference, which claims that we should include all relevant evidence in our probability calculus, and as we include more relevant evidence, our probabilities become "better".
I will argue that her initial answer of 50% is actually incorrect, because of the effect of only asking her about situations where the second detector registers something, which is not a sufficiently random subset, with respect to the first detector, of all the events. Thus, her second answer is, in fact, the better answer, and Bayesian inference still stands.
Consider, for a moment, this example. (I'll explain in a moment how it relates.) I have a perfectly fair coin, which you know is fair. I then flip the coin, and ask you to guess whether or not it's heads. You win if, when I ask you, your guess matches the coin. As is well known, you should guess heads 50% of the time, to maximize your likelihood of winning. If you answer yes 49% or 51% of the time, odds are that you'll win less often than if you answered yes 50% of the time. Now, however, imagine that I flip the same fair coin, but that I look at the coin before I ask you to guess whether or not it is heads. If it's tails, then I ask you to guess whether or not it's heads. (If it's heads, I just ignore it and flip the coin again, although you don't know this.) In this case, your likelihood of winning is greatest if you never guess heads. Similarly, if I only asked you when the coin landed heads, you likelihood of winning is greatest if you always answer heads. So, when we ask someone only about a specific subset of events, the properties of that subset are relevant to rate someone should guess at. So then, if you are playing this game with someone and tell them that you're only going to ask them about a certain subset of events, but don't tell them what the subset is, they will be at a loss as to what rate they should guess, and also if they continue to guess yes 50% of the time, they will not necessary break even (depending on your subset), even though the coin lands heads 50% of the time.
Now let's look at van Fraassen's argument again, and ask whether we are, at any point, asking Hilary to guess on only a certain subset of events, and if so, whether the features of that subset was chosen would influence the probability. Recall that we do inform Hilary that the second detector did register something. Now, since the second detector will not always register something, and since we presumably are not lying to her, we are thus picking out a subset of the events, namely, the ones where the second detector goes off. Next we need to question whether this is effectually a random subset with respect to the first detector (the one we are asking Hilary about). If it is, then she will still break even guessing it detected something 50% of the time, but if it's not, then just like in the coin game above, she will no longer break when guessing yes 50% of the time. However, we know that there is a correlation between the first detector registering something and the second detector registering something (namely, that this correlation is related to the cosine of the angle squared), and so this is a NOT an effectually random subset with respect to the first detector. Hence, Hilary will not break even by guessing 50%.
But wait, you say- doesn't the first detector have to register "yes" 50% of the time? Then why doesn't she break even? Yes, the detector does register yes 50% of the time- but only when we're talking about averaging over ALL the events. Similarly, the coin lands heads up 50% of the time, over ALL the flips I make- but not over all the flips I ask you about. Similarly, we're not asking Hilary about all the events- only some of them. If you were asking Hilary about all of them, independent of what the second detector did, then she would break even guessing yes 50% of the time. But this isn't the case, since we're only asking her about ones where the second detector registered something. Thus, the error in van Fraassen's argument is when he says that "For at the right [first} side the clicks come at the 50% rate, and changes in Hilary's personal information or opinions do not affect that at all. Thus Evelyn [a hypothetical person standing at the first detector] at least would be right to advise Hilary to just ignore the ...information [about the second detector]." Evelyn would NOT be right to advise that since Hilary is being asked about a specific subset of the events Evelyn is seeing, and those events DON'T come in at the 50% rate. Evelyn should inform her of the rate for that subset of events.
Van Fraassen then goes on to investigate "Yet by what epistemic principle can one license ignoring evidence that would clearly change one's opinion if heeded?". However, this isn't necessary, because as we've seen, Hilary shouldn't ignore the evidence she has gained, because if she continues guessing 50%, she won't break even, as van Fraassen claimed. Thus, she does have a "better" probability after the evidence from the second detector than she did beforehand.
Now, you may be wondering how "I don't know anything" is "better" than 50%. The reason is that when we initially asked Hilary what the probability was of the first detector registering something and she answered 50%, she was implicitly assuming that there was no correlation between whether or not we asked here about the first detector and what the first detector registered. To be correct, she should have said "Depends- was this a randomly selected run?". As we've seen, it was not. So her answer of 50% is actually wrong- not because the second detector doesn't register something 50% of the time, but because we're asking her about a subset that she knows nothing about instead of the entire set. The extra information tells her that she was wrong in that assumption, and thus, the probability "something between 0% and 100%" is in fact better than "50%".
Now on to my second point, that this doesn't actually require quantum mechanics. Hopefully by stripping away the quantum mechanics, it will become clearer where the flaw is van Fraassen's argument is. So here is an argument isomorphic to van Fraassen's, but without the quantum mechanics.
Consider this case:
Assume that it rains in Timbuktu is 50% of the time.
Also assume that due to the global air flow, ocean currents, and everything else, there is a correlation between whether or not it snows in Philadelphia and whether it rains in Timbuktu. Examples of such relations would be:
1) Whenever it snows in Philadelphia, it always rains in Timbuktu, and never rains any other time. (In this case, it'd snow 50% of the time in Philadelphia).
2) Whenever it snows in Philadelphia, it never rains in Timbuktu, and always rains when it's not snowing in Philadelphia. (In this case again, it'd snow 50% of the time in Philadelphia).
Assume that I know this exact mathematical relation, but that Kenny doesn't. He can know the form of it, but not it's exact mathematical value.
Additionally assume that Kenny and I are both aware that it's not snowing here in Philadelphia.
Finally, assume Kenny has a friend in Timbuktu.
Now, assume I tell Kenny that if it rains in Timbuktu, I will make him hot chocolate. Kenny would like to know what the odds are that I'm going to make him hot chocolate. So Kenny calls his friend in Timbuktu. We expect the conversation to goes something like this:
Kenny: "Hi. What are the odds that it is going to rain there?"
Kenny's friend: "50%."
However, the conversation really should go like this:
Kenny: "Hi. What are the odds that is going to rain there?"
Kenny's friend: "I know that it depends on whether it's snowing over there, but I don't know how."
It's wrong for Kenny's friend in Timbuktu to say 50%, because the probability of it raining in Timbuktu is actually conditional on the probability of snow in Philadelphia, and I am forcing the case where it's not snowing in Philadelphia. Essentially, I'm making a cut and ignoring the days when it snows in Philadelphia. So, the relevant probability isn't the probability that it rains in Timbuktu on any day, but the probability that, on days it isn't snowing in Philadelphia, it rains in Timbuktu. Now, if Kenny bet that in 50% of these cases he'd get hot chocolate, as van Fraassen recommends, he's not necessarily going to average out even- in the first case, he's never going to get any hot chocolate. In the second case, he'll always get hot chocolate. Thus, the "extra information" that it depends on whether it's snowing in Philadelphia is not at all irrelevant, nor should he ignore it. I hope this case is somewhat clearer than the quantum mechanical case in his paper.
A collection of writings have come down to us under the name "Dionysius the Aereopagite" (after Acts 17:34) which effectively form the foundation of the tradition of Christian mysticism. Most scholars today believe the writer lived in Syria, c. 500 AD. The general consensus is that he couldn't have written earlier than this because he seems to have been influenced by 5th century Neo-Platonists. All this by way of background; I don't have any particular opinion as to when the writer lived or by whom he was influenced.
The principle work of "Dionysius" is only a few pages long and is called "On Mystical Theology." His surviving book-length works are The Ecclesiastical Hierarchy, The Celestial Hierarchy, and The Divine Names. I read "On Mystical Theology" recently, first in the original Greek (available online), and then in the English translation included in Bernard McGinn's The Essential Writings of Christian Mysticism. The work begins with a discussion of the divine darkness beyond understanding, i.e. of mystical, non-discursive knowledge of God. Predictably, this section is virtually incomprehensible (the English translation doesn't help that much, and if it followed the text more closely it would help even less). What is interesting to me, however, is the discussion of the different types of theologies - that is, of the types of "God-talk" that are possible while preserving God's status as beyond knowledge and intellect. This passage is interesting in and of itself, and when I read the Greek it was the part I felt I understood, so I was even more puzzled when I read the translation and found that the things I thought I understood weren't there! I'm going to first give the very beginning of the treatise in my translation and McGinn's for flavor (I should note that McGinn's translation is an adaptation of an anonymous one published in 1923), and then translations of a large chunk of chapter 3 to see if we can figure out what is going on.
| McGinn | Pearce |
|---|---|
| 1.1 Trinity beyond all essence, all divinity, all goodness! Guide of Christians to divine wisdom, direct our path to the ultimate summit of your mystical lore, most incomprehensible, most luminous, and most exalted, where the pure, absolute, and immutable mysteries of theology are veiled in the dazzling obscurity of the secret silence, outshining all brilliance with the intensity of their darkness, and surcharging our blinded intellects with the utterly impalpable and invisible fairness of glories surpassing all beauty. | 1.1 Trinity beyond existence and beyond divinity and beyond goodness, guide of Christians to godly wisdom, direct us on [the way] of mystical discourses beyond ignorance and beyond assertions and at the highest peak; inside [of it] the pure, the uncorrupted, the unturning mysteries of theology according to that which is beyond light have been unveiled, a darkness of silent mystical secrets, in the darkest [place], that which is beyond appearance, beyond luminescence, and in the entirely impalpable and the unseen thing of the splendor beyond name, beyond filling the sightless mind. |
The first thing you will notice is, of course, that the McGinn translation is rather more polished. Mine is more literal. (Well, actually, the first thing you might notice, is that I was serious about this being incomprehensible.) I had originally wanted to follow "Dionysius" literally and user the prefix "hyper-" where I have used the word "beyond," since he has the prefix huper in the Greek, but I couldn't get that to make sense in English. At any rate, keep in mind that this whole section is about hyper-this and hyper-that; later I will try to explain why that is and draw an interesting conclusion from it.
"Dionysius" goes on, as I have said, for a page or two in this fashion, before getting to chapter 3 which is, as I have said, what I'm interested in:
| McGinn | Pearce |
|---|---|
| 3 In the Theological Outlines [a lost work] we have set forth the principle affirmative expressions concerning God, and have shown in what sense God's holy nature is one, and in what sense three; what is within it which is called Paternity, what Filiation, and what is signified by the name Spirit; how from the uncreated and indivisible good, the blessed and perfect rays of its goodness proceed, and yet abide immutably, one both within their origin and within themselves and each other, co-eternal with the act by which they spring from it; how the super-essential Jesus enters an essential state in which the truth of human nature meets it; and other matters made known by the oracles [i.e. scripture] were expounded in the same place.
Again, in the treatise on Divine Names, we have considered the meaning, as concerning God, of the titles of Good, of Being, of Life, of Wisdom, of Power, and of such other names as are applied to him [Divine Names, chpaters 4-8]. Further, in the Symbolic Theology [another lost work] we have considered what are the metaphorical titles drawn from the world of sense and applied to the nature of God; what is meant by the material and intellectual images we form of him, or the functions and instruments of activity attributed to him; what are the places where he dwells and the raiment in which he is adorned; what is meant by God's anger, grief, and indignation, or the divine inebriation; what is meant by God's oaths and threats, by his slumber and waking; and all sacred and symbolical representations. And it will be observed how far more copious and diffused are the last terms than the first, for the Theological Outlines and the discussion of the divine names are necessarily more brief than the Symbolic Theology. (Brackets McGinn's) | 3 Therefore, in the Theological Hypotyposes we praised the most dominant things of the cataphatic theology: how the divine and good nature is called 'simple' [i.e. 'one']; how [it is called] 'triune;' what [nature] is called 'paternal' and what 'filial;' what the theology of the Spirit wants to clarify; how the lights in the heart of the [nature] of goodness grow out of the immaterial and indivisible good, and [yet], subsisting in it and in themselves and in one another, remain alone without growth moving about; how Jesus, [though] beyond existence, took on existence with the truly human growths; and however many other things concerning the discourses have been made clear in the Theological Hypotyposes, were praised. But in On The Divine Names, how [the divine nature] is called 'good,' how [it is called] 'being,' how [it is called] 'life,' and 'wisdom,' and 'power,' and however many other things of the understanding are divine names, [were praised]. ['divine names' = Gr. 'theonyms' – what a nifty word!] And in the Symbolic Theology, certain metaphorical names [Gr. 'metonyms' – another nifty word] for the divine nature from sensible things, certain divine shapes, certain divine outlines and parts and tools, certain divine places and worlds, certain desires, certain sufferings and wraths, certain drunkennesses and carousings, certain oaths and imprecations, certain sleeps and certain wakings and however many other forms are holy falsehoods of symbolic God-patterns.
And I think you have seen how very many more words the last things take up than the first things: for also it was necessary that the Theological Hypotyposes and the exposition [lit. 'unfolding'] of the divine names should be a shorter discourse than the Symbolic Theology. The general view given by thought [must] account for [lit. 'set up'] as much as is denied of the opposite: just as even now when we were entering the darkness beyond thought we did not find a short discourse but, [rather, it was] entirely non-discursive and without understanding. |
Now, I think that what I am about to say is compatible with the McGinn translation, so I'm not too worried about my interpretation being out to lunch, but I sure wouldn't have thought of this if I hadn't also read the Greek, and I didn't understand it very well, so I'm waiting to be corrected in terms of my interpretation of "Dionysius," but I'm nevertheless going to tell you what I think, if for no other reason than that it is independently interesting, regardless of historical accuracy.
Traditionally, especially in Eastern Christianity, theology is divided in apophatic and cataphatic forms. Apophatic theology says what God is not (he is infinite, atemporal, unlimited, immaterial, etc.), and cataphatic theology says what God is (he is good, loving, powerful, three, one, etc.). Now, it seems to me that Dionysius uses these terms rather differently (well, he actually doesn't, in this work, use the word apophatikos, but he uses some cognates). That is, traditionally the words are interpreted with the etymologies "affirming away from" and "affirming toward," whence saying what something is not vs. saying what it is. But there is reason in this passage to suppose that what Dionysius really means is not "affirming away from" but "away from affirming," and, similarly, "toward affirming" - that is, he means discursive and non-discursive knowledge of God. In my translation I consistently used "discourse" and its cognates for logos and its cognates, and transliterated the word "cataphatic" (and also "hypotyposis," but I'm coming to that).
This, then, is the first distinction in theology: the mystical theology in which we know the unknowable and speak the ineffable in the cloud of brilliant darkness beyond light, existence, understanding, and language (note that if this made too much sense, it would mean that I had actually succeeded in giving a discursive account of what can, according to "Dionysius," be known only non-discursively, and so his theory would be false, so you shouldn't get too upset if you didn't understand it), and the discursive theology of the understanding. In God's true nature he is understood to be beyond the understanding, and so only accessible to this non-discursive knowledge-beyond-knowing, insofar as he is accessible at all. We cannot speak literally of God. (The debate on whether we can speak literally of God continues to this day, with the majority of traditional monotheists on the side of "Dionysius" to date.)
This is where things get really interesting. Although we cannot speak literally of God, nevertheless not all ways of speaking of God are equivalent. This seems intuitively true: it is very different to say that God is good or loving than to say that God is our Father, or that Christ is the Bridegroom of the Church, or - in an even more extreme example - that "a mighty fortress is our God." But, if we deny that we can speak literally of God, then what is the distinction here? "Dionysius" actually gives an analysis of this, having written books on each of the three divisions.
The first division is that of the "Hypotyposes." This word usually means outlines or some such, and so McGinn has translated it as such, but I'm not sure that's what it means in "Dionysius," exactly, because I think there's reason to suppose that "Dionysius" is playing on the etymology. A tupos is a pattern, imprint, outline, or some such, and hupo is the opposite of "Dionysius'" favorite prefix, hyper. God, he says, is hyper-existent, hyper-good, and so forth. What then are existence and goodness? They are hypo-types of God! That is, God is, strictly speaking, beyond existence and goodness, but existence and goodness don't misdescribe God as the "holy falsehoods of symbolic God-patterns" strictly speaking do; rather, they fall short of describing him, which is, after all, what hupo means.
The next category is that of the divine names. I'm not totally certain what the difference between the divine names and the hypotyposes is supposed to be, so let's simply assume that they are items that fall on the boundary of the hypotyposes and the symbolic theology. Goodness and existence are actually among the examples of this category, so "Dionysius" must not think that they are strictly hypotyposes, the way threeness, oneness, Fatherhood, Sonhood, Spirithood, the "lights of goodness" (whatever that means), and Christ's humanity are, but I chose them above because they are among the examples of hyper-attributes in chapter 1.
The final category is the symbolic theology, which contains the truly metaphorical. These are cases where we describe God in terms of sensible things (but apparently Fatherhood and Sonhood are not sensible?), and succeed in saying something true about him by this means.
This, at any rate, is what I got out of it. I encourage all you Bible translation bloggers to try your hand at interpreting/translating "Dionysius" to stretch your Greek muscles a little more and to tell me if you come to the same conclusions (and critique my translation!). I'd also appreciate any comments on this account as either (1) an interpretation of Dionysius, or (2) an actual assertion about our knowledge of God from anyone who has anything to say about such things.
My apologies for the lack of posting. I've been very busy this semester. I do, however, have a moment tonight, and I have been thinking about Moore's argument for the existence of the physical world. I've been thinking about it in part because I actually read him for the first time a last month (although I was already somewhat familiar with his argument from reading Wittgenstein's On Certainty), and in part because a friend of mine recently (on my suggestion) made a remark about Berkeley in a contemporary philosophy class and was summarily dismissed with an argument almost identical to Moore's.
For those who may not be familiar, Moore's argument looks something like this:
This simple argument seems to be part of the reason why many contemporary analytic philosophers do not consider idealism a live issue (something that I intend to make it my business to change). However, it seems to me to have two enormous and equally simple defects:
Concerning the first problem: Berkeley's famous maxim is esse est percipii; "to be is to be perceived." Moore thinks he proves the existence of an external world by showing us his two hands. In fact, this proves that the external world exists in precisely the way Berkeley says it does: it's esse is percipii. To speak more clearly, if the fact that I can see two hands is conclusive evidence that the two hands exist, then Berkeley's view is correct, and perception defines reality (at least for hands). If the existence of hands was mind-independent, then even though I could see Moore's hands, they might not exist.
However, there is a bit more to Moore's argument than this, and he may be able to level an objection at Berkeley after all. The real meat of what Moore is getting at is in the third premise: Moore and Berkeley do not agree on what it means for something to be a physical object. Moore thinks that physical objects have a mind-independent reality. His real argument here, though, is that he is more certain that his hands exist as mind-independent physical objects than he is of the premises of any skeptical or idealistic argument about the physical world. Therefore, he thinks, it is more reasonable to hold to the existence of hands, and therefore physical reality, than to bow to the skeptical arguments. This, however, is where he gets into trouble with Descartes.
Descartes' argument for the existence of his soul is in fact quite similar to Moore's argument for the existence of physical substance. Moore says "I am perceiving physical objects, therefore physical objects exist;" Descartes says "I am having a subjective experience of the world, therefore my soul exists." Both Descartes and Moore are making the same mistake: they are taking a phenomenological point and drawing from them conclusions about metaphysical substances which are not objects of experience. Moore has proven that physical substance exists and Descartes has proven that mind exists, but neither have proven it in the way he seems to think he has. Moore has observed that there are perceptions, and Descartes has observed that there are subjective experiences, and these do, in my view, show that body and mind, respectively, exist in the only way possible and the only way that is meaningful for human beings. However, neither show anything about whether mind or body are metaphysical entities, and whether they are on par ontological or whether one depends on the other. Descartes simply assumes that thinking things are immaterial substances, and Moore simply assumes that perceived things are mind-independent substances, and it seems to me that these statements have approximately equal degrees of justification: namely, none whatsoever.
Thus we can see that Moore's argument cannot possibly serve as a physicalist response to Berkeley. In his defense, Moore himself did not intend his argument to respond to idealists, but to skeptics, and as an argument against skepticism it fares somewhat better. Also in Moore's defense, he wrote a separate paper, "The Refutation of Idealism," which, as the name suggests, is targeted specifically at the idealist position. This paper is sitting on my bookshelf waiting to be read. In the meantime, I hope that this post will motivate some of you to consider Berkeley's positions if not seriously enough to accept them, at least seriously enough to attempt to provide a real argument against them.
In my recent post on the inerrancy of the autographs, I made the following passing comment:
There is uncertainty everywhere in this world. If we are (rationally, justifiably) certain about anything at all, it is only about very elementary logically necessary propositions like '2+2=4', and even here there is some question (uncertainty!) about whether we are, or ought to be, truly certain.
"no one can mistakenly believe that there are beliefs"
- Richard Feldman, Epistemology, p. 125
Middle knowledge is a problem that has been bothering me for quite some time now. It goes like this: middle knowledge is knowledge of the truth or falsity of counterfactuals of freedom, where a counterfactual of freedom (sometimes called a counterfactual of creaturely freedom) is a statement about what some agent having libertarian free will would do in a purely hypothetical situation, i.e. one that never has and never will occur. Libertarian free will means that one is free because one could do otherwise than one actually does. So, for instance, if human beings (including me) have libertarian free will (as I believe they do), then the statement "if Kenny was offered $1 million to kill someone yesterday, he wouldn't have done it" is a counterfactual of freedom: yesterday has come and gone, and I can assure you that no such situation arose yesterday. It seems that middle knowledge ought to exist, because it seems that God ought to have it. First, there are a few places in Scripture that seemingly make claims about counterfactuals of freedom, but I find this to be unimportant to the discussion, as the Bible is presumably speaking in a 'loose and popular' sense and not in 'metaphysical rigor.' What I find more problematic is this: assuming that human beings have libertarian free will and that God has foreknowledge of their actions, but there is no such thing as middle knowledge, it would be the case that God wouldn't have known that Adam would sin if he hadn't created this world, the one in which Adam did sin. That is, because God created this world, and it is a world in which Adam sins, God knew from eternity that Adam would sin. However, if God had created a different world - say, for instance, one that didn't have any beings with libertarian free will (other than, I suppose, God himself) - God would never have known what would have happened if he had created Adam and Eve and the garden. Furthermore, if God has foreknowledge but not middle knowledge, then God does not now know with absolute certainty whether Adam would still have sinned if he had placed the Tree of the Knowledge of Good and Evil three inches east of where he in fact placed it. This seems more than a little problematic!
So suppose that there is middle knowledge (and God, being omniscient, has it). In addition to allowing God to make more informed decisions, it is useful for theodicy, in that it allows us to take the step that William Lane Craig does in explaining the justice of the condemnation of those who are under non-ideal conditions with regard to Christian salvation in terms of what Craig calls "transworld damnation" (William Lane Craig, "Is 'Craig's Contentious Suggestion' Really So Implausible?," Faith and Philosophy 22 (2005): 358-361). Craig's interlocutor, Raymond Van Arragon, gave one definition of "transworld damnation" as follows, and Craig seems to accept his formulation: "The property of being such that in every feasible world in which one exists, one does not accept Christ" (ibid. 359, emphasis original). What this means is that we might say that for everyone who rejects Christ and is condemned to hell, there was absolutely nothing that could have been done, no evidence that could have been offered, such that he would accept Christ. Leibniz offers a similar defense from a more fatalistic perspective: he thinks that all properties are essential, and therefore person A's spending eternity in hell is an essential characteristic of person A, and if A's eternal fate were changed, A would no longer be A. In other words, according to Leibniz, it is logically necessary that (A exists)->(A will spend eternity in hell). Not a very comforting thought, but the idea, again, is that if A went to heaven, A would simply not be A.
Now, I don't know that I would want to accept Craig's claim, even if the problem of middle knowledge were solved, but there are clearly many reasons why a Christian would want to say that God has middle knowledge besides just this. So what is the problem of middle knowledge? It is the question of "in virtue of what are conterfactuals of freedom true?" That is, in general, when we say that a proposition is true, we mean something like the classic correspondence theory of truth (no, I haven't read the long article I just linked to) - we claim that it corresponds to some "way things are" out in the world. But what is the "way things are" out in the world such that counterfactuals of freedom are true? If it is facts about my character, I don't have libertarian free will (I act always according to my character, as in compatibilist free will). If it is divine will, I don't have libertarian free will (God compels my hypothetical actions). It seems that if I can actually do otherwise in actual situations, I should be able to hypothetically do otherwise in hypothetical situations! But since the hypothetical situation never happens (the proposition is counterfactual), I never have the chance to act one way or another.
This is known as the 'grounding objection,' and is discussed in another paper by William Lane Craig, which I just finished reading online, entitled "Middle Knowledge, Truth-Makers, and the 'Grounding Objection.'" I was hoping for a fantastic solution from Craig that would solve all of my problems but, alas, I was quite disappointed. Craig's paper actually only argues that oponents of middle knowledge have not demonstrated that counterfactuals of freedom need truth-makers, and that the whole theory of truth-makers is convoluted, controversial, and quite possibly false. Furthermore, even those who embrace the theory often hold that not all propositions need truth-makers. This is all well and good, but what, I ask, about truth-conditions?! Surely propositions must have truth-conditions! If a sentence is meaningful (i.e., actually expresses a proposition), there must be some condition in virtue of which it is true (or false). I'm not talking about verificationism here - that is, I'm not claiming that in order for a sentence to be meaningful there must be some process by which we can determine its truth. Rather, I am claiming that in order for a sentence to be meaningful there must be some way that existing things in the universe are such that the sentence is true or false. Craig doesn't seem to address this in either of papers I read, but perhaps he could do so with a kind of limited modal realism:
I'm not sure what is meant by a "feasible world" in the transworld damnation discussion, but suppose it means a world that would have occurred had God acted differently than he actually did. That is, God acts differently, and then leaves the other beings with libertarian free will to act as they wish. Now suppose that all feasible worlds exist - perhaps on a lower ontological plane than the actual world, or perhaps we should say like David Lewis that 'actual' is simply an indexical term, picking out one of the feasible worlds and they are all on the same ontological level. Now we will need to claim that you and I are in fact 'transworld entities' - that is, we are made up not just of our selves in the actual world but of our counterparts across all feasible worlds. If this is the case then we actually do choose whether counterfactuals of freedom about us will be true or false, and middle knowledge is saved. However, the whole transworld damnation thing only works if God picks out certain feasible worlds to create. So perhaps we should say that only the people who exist in the actual world are 'real' and God has created all of the feasible worlds which contain one or more 'real' people, and that he does so specifically so that counterfactuals of freedom will have meaning.
At this point, I think we are worse off than we started. (1) It's still not clear that God really has middle knowledge, because he only knows about what free beings do in worlds he creates (he just created a lot of worlds). (2) What happens to the merely 'feasible' people (those who do not exist in the actual world)? Are there 'feasible' heavens and hells? Or will those people simply cease to exist at the end of the actual world? (3) Are merely feasible people actually people with moral value and so forth? (4) If the feasible worlds are real, how do I know I'm in the actual world and not one of those? Furthermore, how do I know I'm a 'real' person (transworld entity) and not merely feasible? If I'm merely feasible, doesn't that mean that God still doesn't have (complete) middle knowledge about me?
In short, problem not solved, by me or by the Craig papers I read. Does anyone have better suggestions? Craig seems to have written a lot on the subject. Perhaps one of his other papers solves this problem?
Since my post on Berkeley's metaphysics generated so much interesting discussion, I thought I would write a post on Berkeley's taxonomy of ideas. A particularly interesting (to me) aspect of this discussion is the way it plays into his critique of John Locke's theory of abstraction. Also of interest is the way this view (may have) influenced Immanuel Kant's epistemology of metaphysics. I'll skip lightly over that last one, because I don't understand Kant very well (who does?), but I re-read the Prolegomena recently and was thinking about this, so I'll float a few ideas by toward the end of this post and perhaps people who understand Kant better than I do will pick up on them.
The first point that must be made here is just what an 'idea' is. Berkeley is a sort of revisionist Lockean (one might say that Berkeley is a Lockean the way Arminius is a Calvinist) and, as such, essentially every one of his views is set up in either agreement or opposition with Locke's views. Berkeley would have had Locke's Essay Concerning Human Understanding as a logic textbook when he worked on his B.A. degree at Trinity College, Dublin.[1] The point of this digression is that Berkeley uses words such as 'matter,' 'idea,' and 'abstraction' according to Locke's definitions. This is, in my oppinion, sometimes a problem for him: for instance, he shocks people by claiming that he doesn't believe in matter or the capacity of abstraction, but he actually means that he doesn't believe in John Locke's theories of matter and abstraction. According to modern, everyday usage, Berkeley believes in both.
Returning to the point, Locke defines the word 'idea' at Essay 2.1.1 as "the object of thinking" and goes on to divide ideas into 'simple' and 'complex.' Simple ideas are the basic, fundamental constructs or sensations - for instance, a particular shade of the color red - which cannot be broken down into constituent ideas, and out of which all of our complex ideas are composed (see Essay 2.2). Thus complex ideas are collections of simple ideas which are united with one another in a single "object of thinking," as in the case of the red water bottle on my desk. Locke's principle point here is empiricism: it is his contention that all of our complex ideas are built up from simple ideas, and we get simple ideas from sense perception.
All of this Berkeley accepts. There is much more that Locke says about ideas, and it is certain that all of it heavily influenced Berkeley, and certainly some of what I will now say about Berkeley is only repeat of distinctions made by Locke. Nevertheless, there are important differences, especially, as has been said, with regard to abstraction.
What I am calling 'Berkeley's Taxonomy of Ideas' seems to have acheived its full development in the second edition of his Treatise on the Principles of Human Knowledge.[2] Berkeley divides 'ideas' - that is, direct objects of the mind - into three classes: perceptions, thoughts, and volitions. These types of ideas are distinguished by their source, as well as by their nature. Perceptions are those ideas that are imposed upon us from without (a crucial point in Berkeley's metaphysical argument; see my previous post): as long as my eyes are open, I perceive by sight, and I cannot (directly) control what ideas enter my mind in this way. Thoughts are ideas that are like perceptions except that they are produced by my own volition - I make them up. Volitions are the objects of the will which, among other things, produce thoughts. Like Locke, Berkeley believes that new simple ideas enter the mind only through perception, and our thoughts are created by putting simple ideas which we previously perceived together into different complex ideas.
Where Berkeley and Locke differ really substantially is in generalization. How is it that we are able to think about 'triangle' in general? Even more problematic for Berkeley is how we can think of spirits (Principles 27, 140) when these are not direct objects of the mind but, rather, are minds themselves and Berkeley has rejected the idea of any kind of resemblance between ideas and non-ideas (Principles 8ff.). Berkeley believes (and I am inclined to agree with him) that Locke's answer to the problem of generalization is virtually incomprehensible. Regarding "the general idea of a triangle," Locke says, "it must be neither oblique, nor rectangle, neither equilateral, equicrural, nor scalenon; but all and none of these at once" (Essay 4.7.9). On this section, Berkeley comments,
This is the idea which [Locke] thinks needful for the enlargement of knowledge, which is the subject of mathematical demonstraction, and without which we could never com to know any general proposition concerning triangles. That author acknowledges it doth 'require some pains and skill to form this general idea of a triangle.' ibid. But had he called to mind what he says in another place, to wit, 'That ideas of mixed modes wherein any inconsistent ideas are put together cannot so much as exist in the mind, i.e. be conceived.' [3.10.33] I say, had this occurred to his thoughts, it is not improbable he would have owned it to the above all the pains and skill he was master of to form the above-mentioned idea of a triangle, which is made up of manifest, staring contradictions. (Essay Toward a New Theory of Vision 125)
Nevertheless, there is the point of the phenomenology of the situation: we do, it seems, think about triangles in general, and not only particular triangles. How is this possible? Berkeley came to refer to his account of this phenomenon as 'general notions.' Things like spirits are not properly ideas, but we nevertheless mean something when we speak of them. As he says, "the words will, soul, spirit do not stand for different ideas, or in truth, for any idea at all but for something which ... cannot be like unto, or represented by, any idea whatever. Though it must be owned, at the same time, that we have some notion of soul, spirit, and the operations of the mind ... in as much as we know or understand the meanings of those words." (Principles, second edition, 27)
As he explains at New Theory of Vision 128 and elsewhere, Berkeley believes that these 'general notions' consist of two parts: a word or symbol, and a decision procedure. That is, when we talk about 'triangle' the symbol is the word 'triangle' either read or heard, or a picture of some particular triangle. Then we have some procedure by which, given any particular idea, we can determine whether it falls under this notion. When we reason 'abstractly' about triangles, we reason only from the 'essential characteristics' - that is, the characteristics specified in the decision procedure, namely, being a three-sided polygon. Ultimately, the symbols and decision procedures can be broken into simple ideas.
Of course, this doesn't exactly work for spirits, because there are no particular ideas that are spirits as there are particular ideas that are triangles, but the process is similar. We have certain 'relational notions' that are indeed general notions of how things might be related to one another and by applying these we can define spirits and so forth. In particular, a mind or spirit (Berkeley uses the two words more or less interchangeably) is that in which ideas inhere, and we have other examples of the inherence relation, and thus we can from an exceedingly vague general notion of a mind.
Now, why have I been thinking about this in connection with Kant? Well, Kant's conclusion in the Prolegomena (which I think I understand, even if I understand only pieces of the arguments for it) is that the proper objects of the understanding are "the possible objects of experience" and most metaphysics which has been done so far is based on attempts to go outside this boundary, which can lead to nothing but wild speculation. Instead, Kant believes, metaphysics must be 'reigned in' to it's proper domain: true metaphysics exists on the boundary of human reason, that is, at the point where the possible objects of experience come to meet the things which can never be direct objects of experience. By the nature of the boundary, we can conclude that at least some such things (e.g., the external world, souls, God) do in fact exist, and by examining the boundary closely we can come to conclusions about them. At sect. 57, Kant argues that "we can ... never cognize ... intelligible beings according to what they may be in themselves, i.e., determinately ... [but] we can still at least think this connection by means of such concepts as express the relation of those beings to the sensible world."[3] He proceeds to take God as an example, and concludes that "we ... do not attribute to the supreme being any of the properties in themselves by whcih we think the objects of experience ... but we attribute those properties nonetheless to the relation of this being to the world." In sect. 58, Kant gives an example of such a relation and how it is understood by analogy: (1) as "the promotion of the happiness of the children ... "is to the love of the parents [so] ... the welfare of mankind ... is to the unknown in God ... which we call love." Thus Kant seems to me to say very much the same thing as Berkeley regarding the development of so-called 'abstract' ideas.
In conclusion: this is yet another example of how Berkeley doesn't get nearly enough credit, considering two very well known philosophers, Hume and Kant, seem to have received most of their best ideas, directly or indirectly, from him. Not to imply that Hume and Kant are unworthy of the credit they receive, but with regard to the ideas I find most impressive, Berkeley got there first. Kant's formulations are, on the whole, more precise and rigorous (one might even say that Kant brought new meaning to those words in the world of metaphysics), but they are also proportionately more opaque to those of us who have not yet had a chance to devote several years to studying them! Berkeley, on the other hand, is, for me, one of the easiest philosophers to read. So, enough of me - go read Berkeley!
[1] According to Kenneth Winkler's introduction to his abridged version of Locke's Essay, the Essay was required reading for BA students at Trinity beginning in 1692. Berkeley entered Trinity in 1700.
[2] See the editors' note on Principles 140 in Michael Ayers, ed., George Berkeley: Philosophical Works Including the Works on Vision.
[3] tr. Gary Hatfield