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.
The Dualist 13 (2006) is finally available online, including my paper "The Ontological Status of Dreams in Berkeleian Metaphysics". Unfortunately, the main index site is still badly broken. Hopefully it will soon be fixed. In the meantime, the direct link to my paper works.
There are some typesetting errors in the PDF (most importantly: footnote numbering is messed up, and some logical symbols are boxed out), and I haven't seen the print version to know if it contains these errors as well. I was never shown any proofs and I also found some spelling errors, and at least one place where a sentence is missing a word. Such is life. But the content is, I hope more interesting than the form, so that's what I will focus on and ask readers to focus on.
I wrote this paper over two and a half years ago, and it's now been just over a year since the paper was accepted, so there are definitely things I would do different if I were writing it today. Most of this is simply the sloppiness, unclarity, and general lack of polish that one expects from a Sophomore philosophy student. However, having just re-read the paper, I think that there is only one place in which these flaws touch the core of the argument, and that is in defining just what the argument is supposed to show. I will try to correct this flaw here, by outlining the general flow of argument as it appears in the published paper and explaining how it ought to differ.
The paper deals with the problem of dreams, given a Berkeleian idealist framework. In particular, it is focused on the question of whether "esse is percipi" places dreams on par with waking life, ontologically. Since, for the Berkeleian, perception defines reality, and dream perceptions are bona fide perceptions, it seems that the answer is yes. However, Berkeley claims (Dialogues 235) that "by whatever method you distinguish things from chimeras on your own scheme, the same, it is evident, will hold also upon mine" (emphasis original). The paper argues that this is in fact true: that is, that a particular solution to the epistemological problem of dreams, drawn from Leibniz's "On the Method of Distinguishing Real From Imaginary Phenomena," succeeds in solving Berkeley's ontological problem of dreams.
The descriptions of the nature of this "ontological problem" and its solution are the critical flaw of the paper. On p. 34, I say that the paper will argue that it is possible for a Berkeleian to draw "an ontological distinction" between dreams and waking life (this is the sentence with the missing word - my draft says "between dream worlds and the actual world"). I go on to talk about an ontology with four "levels" and use the word "real" in scare quotes a lot. Later, in sect. 5, I talk about some things being more or less real than others. I'm no longer entirely sure what the things I actually said mean, but, having studied a bit of W.V.O. Quine and David Lewis in the last year, I think I am prepared to say more accurately what it was that I was trying to say before:
In the famous Quinean formula, "to be is to be the value of a variable." That is, the things that are in the domain of quantification for the universal quantifier are the items of our ontology. However, context often causes the domain of quantification to vary, and, although existence is, as a matter of the way the world is, almost certainly an absolute, black and white sort of thing, "relative existence" enters the picture via language and discursive context.
Given this picture of being, I claim that Berkeley's position (that is, the position I believe to be entailed by his system, although he doesn't discuss the issue) with regard to dream worlds is almost exactly analogous to Lewis's position with regard to non-actual possible worlds: Lewis says that while, strictly speaking, all possible worlds exist, most of the time we restrict our quantifiers to the actual world, and the modal realist has as much reason as anyone for doing this.
I divide Berkeley's world into four "ontological levels:" "The level M of minds, the level RP of 'real' perceptions, the level DP of dreamed or hallucinated perceptions, and the level T of thoughts and volitions" (p. 51). These "ontological levels" are in fact domains of quantification, with every level including the level before it, in addition to the entities it specifies (that is, for instance, DP includes M and RP, but not T - it includes minds and all their perceptions). Strictly speaking, the Berkeleian is going to have to either say that ideas are modes of minds and so M is all that, in metaphysical rigor, exists, or that ideas are bona fide entities and so all of it exists. However, what the paper seeks to show, is that the use of these distinct domains of quantification is well justified by Berkeley's system and, especially, there are good reasons which the Berkeleian can admit - reasons having nothing to do with mind-independent entities - for frequently quantifying over only RP, and saying that those things are "real."
With these thoughts in mind, I present for your consideration, "The Ontological Status of Dreams in Berkeleian Metaphysics". You are encouraged to post your comments and criticisms here.
(Note: I tried to write this post last night and lost it when my powerbook overheated. Here goes the second time.)
David Lewis is best known for his modal realism, the view that all possible worlds exist in precisely the same sense that the actual world exists. He holds this view because he believes that it solves all sorts of philosophical problems related to modality, counterfactuals, properties, and so forth. However, there are a number of philosophers who think that the benefits of modal realism can be had without actually supposing that the possible world really exist. These philosophers Lewis calls ersatzers and in the section entitled "Paradise on the Cheap" in his book On the Plurality of Worlds Lewis attempts to reply to the ersatzers.
The first type of ersatzism to be dealt with is linguistic ersatzism. According to this view, possible worlds are linguistic constructs and therefore have only the ontological status of abstract objects like mathematical sets (whatever that might be) and not the status of concrete objects like the actual world. The linguistic ersatzer sets up a "world-making language" and asserts that possible worlds are maximal consistent sets of sentences in this language.
Lewis's first objection to this view (pp. 150-157) is that the ersatzer is required to assume certain facts about modality, which he is supposed to be explaining. In particular, Lewis wants to know whether it is possible for a particle to be both positively and negatively charged. Since these are (presumably) distinct predicates ("negatively charged" means more than just "not positively charged"), an axiom will be needed to enforce this rule (it's not a basic rule of the language). Lewis thinks that the ersatzer's axioms must come from some facts of modality distinct from his theory, so that his theory doesn't actually explain the facts of modality. In shot, Lewis claims that the ersatzer must be a primitivist about modality. To drive the point home, he wonders whether, according to the ersatzer, it is possible that there is a talking donkey. Again, Lewis says an axiom will be needed, and the axiom will depend on primitive facts of modality.
I suspect that Lewis is mistaken in his argument and that there is a reply open to the ersatzer based on distinguishing between different types of modality. By a "type of modality" I here mean a set of meanings that modal terms (e.g. "possible," "necessary," "impossible," etc.) can take in a certain context. Let's first distinguish between these types of modality and then consider the reply. The terms of modality can all be defined in terms of one another, so in my descriptions I will use whichever term is easiest to define.
In his discussion of possible worlds (and possible talking donkeys and positively and negatively charged particles) Lewis is talking about metaphysical or real modality. I've never heard a generic explanation of this type of modality that was more illuminating than its name, so suffice it to say that something is really possible just in case it really might have been actual, whatever really means. Lewis thinks this means that it really is actual for someone (the word "actual", according to Lewis, is an indexical like "here" or "now" - the actual world is just whatever world the speaker is in).
The next type of modality is narrowly logical or formal modality. A sentence is formally impossible relative to a language just in case the deductive calculus of that language can be used to derive an explicit contradiction (e.g. "A & ~A" in propositional logic) from it. Note that formal modal statements can be predicated of sentences, not of propositions and not of anything else. Also note that formal modal statements are always relative to a language.
The next concept is semantic or conceptual modality . A proposition is conceptually necessary just in case its truth is implicit in the definitions of the terms involved. For instance, it is conceptually necessary that a bachelor be an unmarried man (unless the context indicates that "bachelor" means "the recipient of a bachelor's degree" or something else, in which case it is not conceptually necessary).
Finally, there is broad logical modality. A proposition is broadly logically possible just in case all sentences that express it in some idealized language are formally possible relative to that language and it is conceptually possible. Now, of course, to give a full account of broad logical modality, you would have to give an account of the idealized language involved, but that's another story. Let's just suppose that there is some best formal logical language and we know what it is.
Now, Lewis seems to think, as I do, that real possibility and broad logical possibility are coextensive. Suppose the ersatzer also takes this view. Then what is the ersatzer doing? Well, he already has formal possibility by way of his language. The axioms Lewis wants him to add are the conceptually necessary truths. This isn't actually a problem for the ersatzer, because these are truths about language so they carry no additional commitments.
Considered from this direction, Lewis's objection seems a bit silly, especially when you consider that at the end he criticizes the ersatzer's introduction of axioms about talking-donkeyhood, saying "The job was to analyse modality ... It was not also part of the job to analyse 'talking donkey' (p. 156). The argument actually goes roughly like this (most of this is paraphrase of Lewis; the parts in italics I've added on behalf of the ersatzer):
LEWIS: Is it really possible that a single particle should be both positively and negatively charged?ERSATZER: I don't know. What do you mean by positive and negative charge?
LEWIS: It's your theory, so you tell me what positive and negative charge are.
ERSATZER: Well I don't know what positive and negative charge are, but if the definition of positive charge is such as to exclude its coexistence with negative charge in a single particle, then the answer to your question is yes. If the definition doesn't exclude this, then the answer is no.
LEWIS: Aren't you assuming facts about modality independent of your theory?
ERSATZER: No, I'm just assuming that terms like "positive charge" mean something, and that they mean the same thing when we're talking about modality as they do in the real world.
LEWIS: Well, then tell me this: is it possible that there should be a talking donkey?
ERSATZER: What do you mean by "talking donkey?"
LEWIS: You know, a donkey that talks!
ERSATZER: Well, it so happens that I know more about donkeys and talking than about positive and negative charge. A donkey is a certain arrangement of matter, and talking is a certain event having to do with vocal chords and sound waves, and these arrangements of matter are possible, so, yes, it is possible that there should be a talking donkey.
LEWIS: I asked you to analyze modality - why are you analyzing talking donkeys?
ERSATZER: How on earth am I supposed to tell you whether a talking donkey is possible without establishing what is meant by the words "talking donkey?"
Of course, this discussion is sympathetic to the ersatzer; from Lewis's perspective the objection is not so silly since I don't suppose he thinks these facts are just linguistic. Nevertheless, it seems to me that this reply on the part of the ersatzer is a simple and effective one.
D. M. Armstrong is the best known proponent of a currently quite popular understanding of natural laws. Laws so understood are, as a result, called Armstrong-Laws, or A-Laws for short. These are distinguished from L-Laws, named for David Lewis. L-laws are identical to regularities in events (but not all regularities are laws). Unlike L-Laws, A-Laws are actual metaphysical entities, which exist independently of their instances. That is, according to this theory, the Law of Universal Gravitation is a thing out there in the universe (not in the mind) which actually makes massive objects move toward one another. The attraction (no pun intended) of A-Laws is that they seem to explain why there should be regularity in the world at all, whereas L-Laws simply state the regularities. Armstrong-type theories posit that there is actually something out there which makes the regularities occur. Now, despite Armstrong's naturalist/physicalist claims, this thing must be transcendent and non-physical (not any more so than Armstrong's "states of affairs," but that's another story).
Philosophers usually talk for simplicity about laws of the form "all Fs are Gs" or "all Fs are followed by Gs," but, of course, the real laws that physicists talk about are not like this at all. The real laws are things like F=ma or K=(1/2)mv^2. (Note that I say the real laws are like this - we don't actually live in a Newtonian universe, so these are not examples of actual natural laws, or at least not fundamental ones - macrophysics is usually considered by philosophers to be one of the "special sciences" like geology or psychology, and these are supposed to follow from the true theory of microphysics, whatever that might be.) It is not clear to me (perhaps because I haven't read the positive part of Armstrong's book What is a Law of Nature? - I've only read the critique of "naive regularity theory" so far) how Armstrong's specific claim (not held by all Armstrong-type theories) that laws are relations between universals is supposed to deal with these sorts of laws, which aren't actually about Fs being Gs. As a result, there doesn't seem to be any reason why we should posit multiple laws of nature: why not just conjoin them?
If we do this, we've got a transcendent, non-physical entity responsible for the orderliness and regularity of the world, "and this all men call God." Hmm...
Of course, if you are concerned about confusing this entity which, for all we know, is impersonal with personal conceptions of God or with some religious theory, you might not want to give it that name, but at the very least you've got the Heraclitean logos (not to be confused with the Johannine logos), a fundamental ordering principle of the universe, and this certainly seems to be a god-like thing. Of course, if we were actually positing God in a more traditional sense, he is supposed to be a necessary being and to create freely, so this would explain why the laws are as they are, but, whatever the case, we seem to have here at the very least something that might be reasonably described as an impersonal, disinterested (small-g) god, and maybe we've got a good deal more than that.
(For the record, I believe in a sort of regularity theory instead, despite believing that God wills at every moment that the laws hold; this is because I believe that laws are strictly identical with true law statements, where these statements are purely descriptive in nature, or something like that.)
Compatibilism is belief in actions that are both free and determined. Usually, one hears such phrases as "what I will to do, I must do" (I think Hume phrases it something like this) or "I am free to act according to my nature." The idea is that human beings have determinate natures and they act as their natures determine. They are free because nothing outside determines their actions.
Theories that posit a more robust freedom of the will are called "libertarian" (no relation to the political theory referred to in my tagline). Usually one hears phrases like "I am free because I might have done otherwise." (Of course, if actions were completely random, that wouldn't be freedom either, so I believe that libertarians must posit a type of action that is neither free nor determined.)
Foreknowledge is often considered to be a problem for the latter type of free will. In order for it to be possible for there to be knowledge of something, there must be a fact of the matter about it, and even the existence of such a fact (a fact about what I will choose which is already the case before I choose it) has often been thought problematic. This is often solved by simply saying something about how it is my future choice that is the truth-maker for this fact. In an eternalist framework (one that views all times as existing equally, and does not give a priveleged position to the present), it seems unproblematic that my future choice should make something true now. In fact, if we take relativity seriously enough, then it isn't really that different from, for instance, a choice I made while in Greece making some fact true here in the U.S. Certainly that is unproblematic!
A further objection arises in terms of someone actually knowing the truth-value of the fact (especially me knowing what choice I will make before I will make it, through some other means than deciding). It is thought that this will interfere with free will since someone knowing the fact "pins it down" as it were. However, if eternalism is true, the fact is already "pinned down" - that future time exists, and in that future time, I make the choice. Still, it seems that my knowledge could interfere with my freedom in this sort of case.
The foreknowledge of God is considered to be a special case which is somewhat easier to get out of: since God exists atemporally, he witnesses all moments simultaneously, and so he simply observes me making my future choices. But why should foreknowledge had by an agent within time be any more problematic than this?
Consider the "Grandfather Paradox" as a famous example. Here is David Lewis's formulation of the problem:
Consider Tim. He detests his grandfather, whose success in the munitions trade built the family fortuen that paid for Tim's time machine. Tim would like ntohing so much as to kill Grandfather, but alas he is too late. Grandfather died in his bed in 1957, while Tim was a young boy. But when Tim has built his time machine and traveled to 1920, suddenly he realizes that he is not too late after all. He buys a rifle; he spends long hours in target practice; he shadows Grandfather to learn the route of his daily walk to the munitions works; he rents a room along the route; and there he lurks, one winter day in 1921, rifle loaded, hate in his heart, as Grandfather walks closer, closer,....
- David Lewis, "The Paradoxes of Time Travel," American Philosophical Quarterly, 13
Perhaps the concern is something like this: suppose that Tim goes through the same reasoning we have just gone through, and determines that he didn't and won't kill Grandfather, and therefore doesn't try. Further suppose that we have the correct theory of truth-conditions for counterfactuals (including counterfactuals of freedom - suppose that these have truth-values), and, on this theory, the statement "if Tim had tried to kill Grandfather, he would have succeeded" is true. Then we seem to have at best a case of circular causation, and maybe even worse difficulties. Consider an explanation of why Grandfather didn't die. It might go something like "although Tim could have killed Grandfather had he tried, he did not attempt to kill Grandfather because he knew that he didn't kill Grandfather." Or we could condense it into the even worse sentence "Tim chose not to kill Grandfather because he knew that he didn't kill Grandfather." Let C represent "Tim chose not to kill Grandfather" and D represent "Tim didn't kill Grandfather." It is now the case that (according to libertarians) C is the truth-maker of D, but D is the reason for C. What a headache!
But is this really worse than circular causation? I'm not sure. And, honestly, circular causation doesn't bother me too much anyway. It's not any worse than an infinite chain of causation, both of which are fine if there is a sufficient reason outside the chain or circle for why the chain or circle is. However, in this case, it doesn't seem that God can be invoked as the reason, because then we would no longer have libertarian free will (either God would be the truth-maker for C and not Tim alone or, worse, God would be the truth-maker of D and not Tim's free choice).
Might libertarians escape through their denial of psychological determinism? That is, Tim's knowledge that he didn't and won't kill Grandfather doesn't actually prevent him from trying in any deterministic way, so perhaps he might still have tried, and succeeded, and then it would have been eternally true that Tim killed Grandfather - but then Tim would not have been born, would not have built a time machine, would not have killed Grandfather, etc. Perhaps, however, this is only an argument against (one dimensional) time travel. Perhaps if we remove the causal problem it will work.
However, libertarians don't want to deny that our beliefs, etc., influence our choices, so it seems that we would still have a circularity problem. Perhaps a degree of uncertainty solves the problem. That is, I think it is very unlikely - say, probability .05 - that I will ever audition to be a television actor. I think this based on my plans, decisions, etc., in the present. But suppose I somehow gain additional information that makes it very likely - but not certain - that I will audition as a television actor in the future. It seems to me that whether this help will depend on its source. If someone I believe to be very skilled in such matters tells me, based on extensive psycho-analysis, that I am highly likely to make such a choice, this is not very problematic. But if I get a message from the future, that may be more problematic. We may have the circularity problem again.
But consider Tim once more. Suppose Grandfather is not his grandfather, but my great-grandfather (my grandfather not having been born in 1921). Tim doesn't know that he didn't and won't kill Grandfather, but I do. This doesn't seem to suffer from the same problem of explanatory regress. However, we must then ask the question of whether I am free to tell Tim what I know.
These are serious problems, but I'm not convinced they are unsolvable. We are walking along our epistemic boundaries here, and mind-bending difficulties here and there are to be expected.
The reason I am interested in these problems, is that it seems like detailed knowledge of brain states, etc., might provide the kind of information that runs into these difficulties. Then again, it could simply be that between the Uncertainty Principle of quantum mechanics and the limits of our ability to gather and process information (we can only know so much about a person's present brain states, and to then figure out what stimulus the person will experience in enough detail to make predictions may be impossible for humans) may make it impossible for us to reach this kind of knowledge. If there are no temporal beings capable of gathering and processing enough information to do this sort of thing, then the theoretical possibility of such a thing is probably unproblematic.