I first became interested in formal epistemology in a roundabout way near the beginning of my graduate education. My first love in philosophy was mainstream epistemology and particularly the justification debate. A professor of mine noticed my love for epistemology and my comparatively rich background in mathematics and took it upon himself to show me just how powerful a tool the probability calculus could be for pursuing epistemological questions. Thus, my interest in formal, Bayesian epistemology was (and mostly continues to be) derivative upon my interest in mainstream epistemology. Accordingly, I have believed (and continue to believe) that there are deep-seated connections between formal and mainstream epistemology; i.e., points of contact that make it possible for each to have something to do with the other. I realize now that this claim is, of course, contestable.
For example, one brief skeptical argument runs like this: mainstream epistemology has to do with knowledge whereas Bayesianism has to do with degrees of belief. Knowledge is an all or nothing affair – you either have it or you don’t – according to mainstream epistemologists whereas degrees of belief, well, come in degrees corresponding to probabilities for the Bayesian. As such, the subjects of both disciplines are too divergent for there to be any straightforward story to tell about how they line up.
In spite of such arguments, it still seems to me that the connection between Bayesian and mainstream epistemology is fairly straightforward. The problem with the short argument above is that it is looking in the wrong place for a connection between the two disciplines. While knowledge is admittedly a qualitative affair, justification is, according to most mainstream accounts, something that comes in degrees: one can be more or less justified for believing one proposition than another; one can be just slightly justified or very highly justified in believing a proposition; etc. And so, it seems that justification is the appropriate concept of mainstream epistemology to align with degrees of belief or probabilities. The degree to which a person is justified in believing 
then is represented by the Bayesian as that person’s rational degree of belief in , or 
.
Or, at least, this is where I have always located the primary point of connection between formal and mainstream epistemology. As far as I can tell, the biggest challenge for anyone who accepts this account is contained in the fact that this account constrains one’s interpretation of probability in a way that most thinkers (in my experience at least) would not find plausible. In order for this truly to be a point of contact between formal and mainstream epistemology, probabilities end up being degrees of justification, or degrees of justified belief; i.e., this account commits one to a justification interpretation of probabilities. So, to shed light upon the question of what is a probability is to shed light upon the definition of justification and vice versa. The problem with this is that both of these concepts are infamously hard to define and attempts to do so are the venue for much debate. Then again, maybe that is further evidence that this is truly a point of contact between the disciplines.
This post is not meant to argue for the identification of justification and posterior probabilities so much as it is meant to raise questions. What problems or challenges arise for those who connect mainstream to formal epistemology in this way? Are there any other plausible places to look for a point of contact that might directly connect the two disciplines? I have recently re-read Tomoji Shogenji’s fascinating paper, “The Degree of Epistemic Justification and the Conjunction Fallacy” (forthcoming in Synthese), wherein Tomoji argues that there is more to justification than simply posterior probabilities … but I remain unconvinced. Thoughts?
Posts
I definitely don’t think rational degree of belief in p should be identified with degree of justification in believing p. Believing p requires having a high degree of belief in p. So one is justified in believing p only insofar as one is justified in having a high degree of belief in p. But one’s rational degree of belief can be much lower than the extent to which one is justified in having a high degree of belief. For instance, I’m about to flip a fair coin. What is my rational degree of belief in it’s landing heads? 0.5; i.e., half-way between the maximum and the minimum. But to what extent am I justified in believing that it will land heads, and thus in having a high degree of confidence in its landing heads? If we identified rational degree of belief with degree of justification in believing, the answer would be that the amount of justification this belief has is half-way between the maximum amount and the minimum amount. But that’s false. It would be deeply irrational for me to believe it will land heads (and thus have a high degree of belief in that proposition). The amount of justification there is for believing it will land heads is either the minimum, or close to the minimum. Therefore, rational degree of belief is not degree of justification in believing.
The mainstream argument works only if belief is an all-or-nothing affair in traditional epistemology, and I don’t think this is true. If a traditional epistemological position is consistent with degrees of belief, then Bayesian epistemology might give one specific explication of this position: The degrees of belief are taken to obey the probability calculus. Whether the position is indeed compatible with Bayesian epistemology then rests on whether each method of justification that is assumed in the traditional position can be interpreted as a justification in Bayesian epistemology.
The upshot of this is that there is no obvious need to interpret `probability’ with the help of `justification’, because Bayesian epistemology can just as well work with anything else that obeys the probability calculus.
Hi Dylan and Sebastian. Thanks for the thoughts! First off, to Dylan, this is a nice clear counterexample. However, it seems a bit quick to me. If I commit to the position that one’s posterior probability is one’s degree of justification, then that doesn’t automatically commit me to saying that
is the point interpreted as complete lack of justification for 
, does it? In fact, it seems more intuitive to me that the minimal value that 
can take be seen as the point at which we are most justified in believing that 
is false, or the maximal value for 
. If this is how one interprets posterior probabilities, then a complete lack of justification in 
is presumably represented by that value of 
that lies halfway between its minimal and maximal values: 
.
There is a parallel to the measures of coherence literature here worth spelling out. Several of these measures (e.g., Fitelson’s) are interpreted in the following way: they take a maximal value whenever there is a maximal degree of coherence that is present in an information set, a neutral value (0 in Fitelson’s case) whenever there is a complete lack of coherence, and a minimal value whenever there is a maximal degree of negative or dis-coherence that is present. The important point is that for these measures, the halfway point is not interpreted as having half the maximal amount of coherence but as having zero coherence.
Sebastian, your right that this point of contact between the two disciplines would seem quite straightforward: probabilities are, after all, degrees of belief for the Bayesian. Nonetheless, I didn’t raise this as an option because I take it as true that belief is an all or nothing affair in mainstream epistemology. This is most definitely true in all of the mainstream epistemology that I have read recently. But I’d be very curious to know whether there is work in mainstream epistemology that doesn’t hold to this. It seems to me that belief only became a quantitative concept when probabilists started talking about it. Am I wrong??
Hi Jonah,
I did not mean that traditional epistemology has sometimes worked with quantitative beliefs, but only that some positions in epistemology admit of degrees of belief, even if not further explicated (I figure much of this is discussed in connection with certainty and doubt). Bayesian epistemology could then be seen as 1) making precise what is meant by degrees of belief, 2) adding conditionalizing as an account of justification in light of new evidence, and 3) drawing a lot of conclusions from these more precise accounts of the two concepts. Any epistemological position that is consistent with points 1 and 2 can then profit from the results in point 3 if the precisifications are accepted.
I have never tried to compare what is said about degrees of certainty and doubt with the probability calculus, though, so I might just be completely mistaken.
Jonah,
Ok, that’s interesting. But take the proposition that it won’t land heads on the next 2 flips (~2H). Your rational degree of belief in ~2H is 0.75. If I understand your proposal correctly, that corresponds to halfway between you having no justification to believe it, and having the maximal amount of justification. But I think the degree of confidence required for belief will be significantly higher than 0.75 — I think it’s got to be higher than 0.9. So, insofar as you’re justified in believing ~2H, you’re justified in having a degree of confidence in ~2H that’s higher than 0.9. If your proposal is right, your degree of justification in having that degree of confidence is halfway between having none and having the maximum. But that seems wrong to me. Were you to be that confident in ~2H you’d be very irrational. Your degree of justification in having that degree of confidence is very low — once again, you have almost no justification in being that confident. (I would say you have literally no justification, but that’s not required for the argument.) And therefore you have almost no justification in believing ~2H, it’s lower than halfway between none and the maximum. Therefore, rational degree of belief isn’t degree of justification. What do you think?
Hi there,
Thanks for firing the opening salvos guys. Some first comments:
Jonah: To claim that the degree to which a person is justified in believing p corresponds to the degree of belief in p that the person’s epistemic situation mandates, would not in itself commit one to any particular view of probability, whether or not rational credence functions happen to be probability functions, formally-speaking. The two issues are orthogonal, as far as I can see: a frequentist could buy into the principle just as easily as a personalist. So I entirely agree with Sebastian’s remark that “there is no obvious need to interpret `probability’ with the help of `justification’, because Bayesian epistemology can just as well work with anything else that obeys the probability calculus.”
Dylan: It is at least contentious that “Believing p requires having a high degree of belief in p. So one is justified in believing p only insofar as one is justified in having a high degree of belief in p.” Indeed, one could take the moral of the Preface Paradox to be precisely the denial of the first sentence.
In defense of the identification proposed by Jonah, it strikes me that ‘the degree to which I am justified in believing that p’ ought to be parsed in a similar way to ‘the degree to which I ought to be sorry’: the latter clearly refers to degrees of regret and not to degrees of obligation. Similarly, the former seems (to me at least) to refer to degrees of belief, not to degrees of justification.
Interesting post, Jonah. There are quite a few ideas to pick up on.
I agree with Jonah that the options for interpreting probability are constrained. The interpretation issue isn’t what story to attach to the Bayesian machinery, but rather whether the Bayesian machinery will work under a given interpretation. This raises (perhaps) a broader question, which is whether you think of a given problem like Ramsey or think of it like Keynes. Let me explain.
Ramsey famously contrasted his view on rational belief to Keynes’s view by contrasting the logic of consistency to the logic of truth. Is your formal system, mathematical probability in this case, to be used directly as a procedure for maintaining consistency, or is to be used as a representation language to codify some particular notion of what follows from what? Ramsey argued for the former view, and clobbered Keynes on the messiness of incomparable, interval valued probabilities.
Keynes’s “in the long run, you’re dead” is, although in slogan form, the reply that pretty mathematics is often predicated on empirical assumptions that are false. Some falsities are harmless, others less so. If you are drawn to real data, real psychology, etc., etc., then you’ll likely be drawn to fumbling around with formal prototypes built from the mathematical equivalent of chicken wire and scrap wood rather than insisting on, say, applying a Bayesian lego set, tested and proven safe for children of all ages, with blocks the size of apricots.
(This is not a knock on Bayesians; it is a knock on fundamentalism.)
How then can mainstream and formal epistemology fruitfully interact? I see three options. One is to attempt to apply off-the shelf technology to an informal discussion of a problem. Another is to rummage through the informal descriptions of ideas in traditional epistemology for ideas about how to codify them by the invention of new machinery, or application of unusual techniques. A third is to bring a formal eye to traditional discussions and look for the structure under the jargon-laden talk. Sometimes people’s intuitive discussion of an issue are inconsistent. (And very often they are under-constrained.) Sometimes they are only inconsistent with the machinery that they adopt or are assumed to have adopted. And sometimes there isn’t an inconsistency per se, but rather there are enormous costs associated with doing something one way as opposed to another.
I have a slightly more complicated counterexample along Dylan’s lines.
Suppose we’ve got a coin factory that produces coins with bias anywhere between
and 
. Let the distribution over possible biases be uniform.
Suppose you’re about to flip a coin from the factory. You know that the coin has been tossed 1000 times, and landed heads on 500 of those tosses. I do not know this. It seems like your credence in the coin’s landing heads should be .5, and my credence in the coin’s landing heads should be .5, but your credence of .5 is in some sense better justified than mine. Your partial belief is informed by more evidence than mine, and should be more stable in the face of new evidence than mine.
The broader point is roughly the same as Dylan’s: it looks like there are two different places to fit degrees into the concept of justified belief: I can be justified to degree
in believing 
to degree 
. If that’s right, then you could use higher-order probabilities to represent degrees of justification.
Hi Rachel,
Welcome onboard!
These kinds of cases do indeed provide well-known–and imo legitimate–motivations for providing a more general representation of doxastic states than a simple first-order probability function (e.g. D-S belief functions, sets of probability functions, …).
But I think that Dylan’s point may have been orthogonal to this issue. I think that he was suggesting that: (i) justification/rational acceptability comes in degrees, (ii) there are cases in which a proposition intuitively has a low degree of justification/rational acceptability but does not receive a low probability (I guess he would probably hold the view that ‘lottery propositions’ would provide a good case in point) , so (iii) degree of justification/rational acceptability cannot be identified with probability. As far as I can see, this point does not really hinge on whether the probabilities involved are first- or higher-order.
My view on this, for what it is worth: there are indeed cases in which high-probability propositions are not rationally acceptable, but justification/rational acceptability does not come in degrees in the first place (i.e. in degrees intermediate between 0–unacceptable–and 1–acceptable)). Rather, when we say ‘the degree to which S is justified in believing that P’, we are simply referring to the degree of confidence in P that S is justified in having.
Does this make sense to you?
Hi Jake, you and I seem to have the opposite intuitions on this issue. You say that “justification does not come in degrees (i.e. in degrees intermediate between 0–unacceptable–and 1–acceptable)” but that instead, as I understand you, beliefs are the sorts of things that come in degrees.
I say that, at least according to the mainstream, “beliefs don’t come in degrees” (it seems to me that a belief is the sort of thing that you either fully have or fully don’t according to mainstream epistemology). I also say that justification does (justification is the sort of thing that can be “partial,” “strong,” or “weak” according to the mainstream).
Remembering that the issue is which notions – if any – within mainstream epistemology to attach to probabilities, do you agree that this represents our differences; i.e., do you really think that beliefs come in degrees and justification is an all or nothing thing in the mainstream? I find it odd if you do, but that certainly wouldn’t mean that you’re wrong. In the end, we just need to get the opinion of someone working more deeply in mainstream epistemology these days to see how these concepts really get applied there.
Jonah: (1) Regarding talk of ‘degrees of belief’ in the ‘mainstream’ literature, note that one very commonly finds reference to ‘degrees of confidence’, which, I take it is a possibly more attractive way of rephrasing the talk of degrees of belief in the Bayesian literature; Ayer’s The Problem of Knowledge (1956) springs to mind, in which he speaks of knowledge as the ‘right to be certain’, but if you carry out a quick google, you will find many more mainstream references. (2) Regarding talk of’ degrees of justification’, I agree with you that this kind of linguistic practise is–imo unfortunately–commonplace in recent mainstream epistemology. So I do not think that ‘justification is an all or nothing thing in the mainstream’; I think that the mainstream is guilty of an unhelpful abuse of language. It would be interesting to track down the first use of the phrase.
Dylan: I’m not actually sure that I fully understand your point, but let me give it a go. First thing; why I should agree with you that the degree of confidence required for belief will be significantly higher than 0.75, or specifically why it’s got to be greater than 0.9. I take it that any threshold for belief will be a complicated matter that is partly a matter of pragmatics. My threshold may be higher or lower depending on the context, for example. And I think it’s quite plausible that a degree of justification
could be quite sufficient in many contexts to warrant belief. For the second point, see my reply to Rachael below…
Rachael: Thanks for the example! However, I’m not convinced that there is a problem here for the position I’m proposing. With regards to the proposition
, that the coin will land heads, it seems right that either person in this example would say that they are not justified at all in believing it, and this is captured accurately with the probabilities both being equal to .5 (the point at which we are said to lack all justification for 
). However, regarding the meta-epistemological question of how justified we are respectively with regard to our degrees of justification, we are on unequal ground. But such meta-justification has as its object a different proposition than 
, right? And pertaining to that proposition, our probabilities are also unequal – I have much less justification than you for saying that the probability of the next toss being a heads is .5 and that statement is also much less probable for me given that I have no evidence for believing it and you have a lot. So this just seems to strengthen my point, or am I missing something??
Oh yeah…and thanks for your helpful and clarifying thoughts Greg! I’m curious: where (if anywhere specifically) do you see the position that I’ve laid out in this post fitting into the categories that you lay out in your discussion above?
From a Bayeasian perspective, I don’t think adding a notion of ‘justification’ solves any epistemological problem, for you will have to introduce some distinction between the degree of justification a proposition gives to another, and the degree of justification YOU BELIEVE it gives. The virtue of Bayesianism is that it gives you a reason to DISPENSE of the notion of knowledge: ALL there is out there, in the minds of people, are beliefs, and people have higher or lower degrees of confidence in those beliefs. If people are rational, these degrees of confidence must respect probability calculus, and so they must be ‘Bayesian’ (what has THE SAME epistemological relevance as saying that there operations with numbers must respect arithmetic: being ‘Bayesian’ is as important to rationality as being ‘Al-Khwarizmian’). Once you have recognised that everything epistemically-relevant that exists is individuals’ beliefs, then you will call ‘knowledge’ to those beliefs of others that you accept (you can also add that the beliefs have been formed by a reliable method), and so, the CONCEPT of ‘knowledge’ transforms into just a POLITE way of referring to some beliefs you accept, and has NOTHING philosophically mysterious.
The interesting philosophical question, once you have recognised this, is: what are the ways to master our beliefs in the most EFFICIENT way?, i.e., what are the procedures that have the virtue of producing in us high degrees of belief in true propositions, and low degrees of belief in false propositions? But this is mainly an EMPIRICAL question. This does not mean that it is ‘philosophically unimportant’: rather on the contrary, it means that philosophy must turn out more empirical (and more formal, as well, for the discussion about those procedures is necessarily mathematical).
I had in mind the Levi-Kyburg exchanges over the years. See in particular Isaac’s paper in the volume I did with Bill Harper.
Your post proposes that the relation ‘X is justification for Y’ be understood (roughly) in terms of ‘Y is confirmed by X’. The project then is one of how to sort out various issues that arise in applying the standard Bayesian machinery to GoFE (Good ol’ Fashion Epistemology). Several issues arise in doing this. One is how to reconcile a numerical representation of support/justification (or represention an agent’s beliefs about support/justification, depending on the interpretation of the calculus, and whether you are interested in representing propositional justification or doxastic justification) with full belief / full acceptance.
There are a couple of methodological options.
First, if you are starting with the Bayesian machinery, then it will appear obvious that the lottery paradox is no paradox at all since closure under intersection and consistency must be maintained, otherwise your machinery breaks down. So the threshold-acceptance principle is what gets tossed out. That’s the Jeffrey-Carnap view. Every decade a new generation comes of age and publishes this point, sometimes as an ‘impossibility result’, others as an historical finger-wagging about how Kyburg must have been napping.
But here is another way to go. It is the 1950s. You have a laundry-list of objections to the standard Bayesian machinery, from which you argue that, despite its (emerging) popularity, it does not and cannot make direct contact with representations of rational belief. Suppose you write a book, call it The Logic of Rational Belief, sketch an alternative machinery and mention a toy example near the end about lottery tickets to illustrate why closure under intersection is problematic. The lottery example isn’t a paradox for you, either. After all, the solution is baked into the approach you’ve sketched in the book. The example is just an illustration. Indeed, you may not even bother to list the example in the index of your book.
A third option is to take away Kyburg’s point that his theory of evidential probability deeply rejects closure under intersections, but you might think that there are ways to keep some of the machinery of classical probability while weakening additivity. In other words, would some kind of set-based Bayesianism work? Here you have philosophical issues in mind –full belief, closure conditions for justified/rational/supported belief– but are exploring a variety of systems to see whether weakening closure under intersections is good enough, or whether you run into other limits as you start looking at these weaker (more general?) forms of Bayesianism.
This is the terrain that Kyburg and Levi worked out in their friendly rivalry through the 80s and 90s. And you can quickly see that neither philosopher was napping at all.
Jake: Here’s why I think my example is a variant of Dylan’s.
Suppose we accept that (i) justification/rational acceptability comes in degrees. I say (ii*) there are cases in a proposition intuitively has different degrees of justification/rational acceptability for different persons but the same probability for both of them, so (iii) degree of justification/rational acceptability cannot be identified with probability.
The higher-order probabilities were meant to be a way out of the problem for somebody who insists that justification comes in degrees, but feels the pull of the idea that probabilities are degrees of belief, not degrees of justification. Your solution is just to deny that justification comes in degrees, which sounds both simple and plausible to me. (Jonah has yet another way out; see below.)
Jonah: If I’ve got you right, your gloss on my example is:
J1) Both of us are justified to degree .5 in fully believing that the coin will land heads (that is, neither justified in fully believing it nor justified in fully believing its negation).
J2) But you are more justified in your degree of justification than I am (since you have more evidence).
My gloss on the example is:
R1) Both of us are justified in believing to degree .5 that the coin will land heads (that is, justified in a having disposition that leads us to accept bets on heads at better-than-even odds, among other things).
R2) But you are more justified in believing it than I am (since you have more evidence).
I see that introducing meta-justification is a way of getting around the worry that there appear to be two different degreed notions here (belief and justification). I’m a little unsure about how meta-justification is supposed to work. Is my meta-justification for
the degree to which I’m justified in believing I’m justified in believing 
? Or the degree to which I’m justified in believing to degree .5 that I’m justified in believing that 
? Or what?
Rachael,
I’m not sure I understand your R2. In particular, does ‘believing it’ mean having an all-or-nothing belief that the coin will land heads? If so, it doesn’t seem plausible to me, since the greater evidence doesn’t justify believing it will land heads. But maybe that’s not what that phrase meant.
Jonah,
In response to Jonah, I must confess that I don’t have a knock down argument that belief requires a credence that’s greater than 0.75. Here are a couple things I can offer on behalf of the premise. Believing p seems to be incompatible with regarding not-p as a serious possibility. After all, thinking the Moore-paradoxical ‘p but possible not-p’ seems to be in some sense incoherent. Once you think that p might be false, in some sense you’re revising your position on p from when you just thought to yourself ‘p’. And I think believing p is something like thinking to yourself ‘p’ (without qualification). But if your degree of belief in p is only 0.75, then it seems you are regarding not-p as a serious possibility. Belief, it seems to me, must involve more confidence than that.
As far as the Preface Paradox goes, I didn’t really understand your point at 4/18-8:59am. In fact, I think the Preface considerations bear out my assumption. You can say in a preface “I’m sure I made a mistake in this book somewhere” while making all the claims in the book at the same time only because there are so many claims in the book. But you can’t say ‘P, Q, R, S, T, and I probably just said something false’. When you say the last thing, it will seem like you’re not really asserting the first five things. But, if you could believe with only a 0.75 credence, it’s hard to see why this would be. If your credence in each of these propositions was just 0.75, we can imagine that believing all these things doesn’t entail any probabilistic incoherence. (I’m assuming here that assertion is a way of expressing belief, and that we can draw conclusions about the nature of belief based on what we can[not] assert. This isn’t uncontroversial, but many think it’s right, and I like it.)
I should actually have said “justified in believing it to degree .5″. (A very annoying slip of the keyboard.) I was thinking along something like the robustness lines that Jonah points to below.
Or rather (looking back at it and realizing I said higher-order probabilities were the way to capture degrees of justification), I think what Jonah says about robustness is a good way of cashing out the vague thought I had, and probably better than the higher-order probabilities I had in mind.
Hi Jesús; so you don’t believe that the notion of justification can serve a purpose for the Bayesian, but I’m still wondering: is there anything keeping you from saying that nonetheless posterior probabilities represent formally what is basically meant in the mainstream literature by ‘justification’? That is, even if you don’t think that justification is a useful concept for the Bayesian, do you think that justification is a good place conceptually to show that formal and mainstream epistemology contact each other?
Rachael: I’m basically making this up as I go now, but I’ll try to answer your question. It seems that meta-justification corresponds to the strength of the evidence upon which I have my justification (upon which I base my probability assignment), and I’m not sure that I would want to try to represent this sort of justification using more probabilities; this type of justification seems to me to be of a different – though certainly related – type. So, I surmise that the measure for this type of justification would not be the probability per se, but rather a measure of the robustness of my probability assignment – i.e., how difficult it is to move in the face of new evidence. Take your coin example with
being the proposition that the coin is fair. The person who has degree of justification 
because he has no evidence whatever will easily be moved from that assignment (give this person 10 heads in a row and the J(p) will have diminished greatly) whereas the other person who has the same degree of justification based upon 1000s of trials will not easily be moved (give this person 10 heads in a row and 
will still sit around 
). This difference then will be captured in a measure of robustness and this notion would seem to capture the mainstream concept of meta-justification. What do you think??
Jonah, what you say about robustness makes a lot of sense. I’m not completely sure I endorse the line that probabilities are degrees of justification rather than degrees of belief (I have difficulty remaining convinced of anything), but I can see how what you said makes sense of the worry I was gesturing at.
Jonah,
I have an obvious comment regarding your proposal. Forgive me if it has already been discussed, but I seem to be missing some crucial aspect of the proposal. Here is my concern: Probability assumes possibility, and if we are to avoid putting the cart before the horse, then I don’t see how your account works when it comes to judgments of possibility (e.g. belief in ‘x is possible’ where x is in the set of possibilities on which Pr is defined).
Hi Jeff,
If one has a non-zero degree of belief in some impossibility, then I would refrain from calling this a probability, so long as probabilities correspond to justification. This is because there simply is no justification for believing in square circles and so, according to a justification interpretation of probability, there cannot be a non-zero probability assigned to an impossibility. In other words, your comment is only a worry if the probability of an impossibility can be non-zero (because any non-zero probability assignment implies possibility), but upon this justification account of probability, this cannot happen.
This could become a problem for me if one could come up with a case where we have justification for believing in an impossibility, but I don’t think there are such cases. What do you think?
Hi Jonah,
Thanks for the reply. What are you taking as the domain of Pr?
Cheers,
Jeff
A Boolean algebra of propositions
Which one?
Well, I want to be quite general here, so how about the Boolean algebra containing all of those propositions that are either true or false, but not both.
Hi Jonah,
I think it is a little odd to say that you want to be quite general and then refer to /the/ Boolean algebra such that …. Putting that aside, I’m not sure what you mean, since propositions do not have truth values in and of themselves but rather with respect to a given valuation. You say that you want to restrict attention to Boolean algebras that contain all of those propositions that are true or false, but not both. So, it seems that you have some independent conception of propositions, i.e. something that goes beyond simply being an element of an abstract Boolean algebra. Whatever that conception is, let P denote the set of these “propositions that are either true or false, but not both” — again, I’m not sure what the assignment of truth values is doing. Now, you want to restrict attention to those Boolean algebras that contain P, but this is nothing more than a restriction on the cardinality of the abstract algebras in question.
Let me return to my original remark about the agent’s judgments concerning possibility. Pr is supposed to represent the agent’s degrees of belief. We shouldn’t allow a case where the agent assigns a positive degree of belief to something that the agent does not regard as a possibility. When asked for the domain of Pr, you responded that the domain of Pr is a Boolean algebra. What are “possibilities” with respect to a given Boolean algebra, A? The possibilities with respect to A are those elements of A that lie strictly above the bottom element of A. That is, fixing an algebra determines a set of possibilities. Also, in general, distinct algebras give rise to distinct sets of possibilities. Now why am I concerned about the domain of Pr? As noted above, Pr is supposed to represent that agent’s degrees of belief, and I am assuming that we should not allow the agent to assign a positive degree of belief to something that he does not regard as a possibility. Suppose that A is the domain of Pr, but that the set of possibilities determined by A does not coincide with the set of things that the agent judges to be a possibility. In such a case it is possible for Pr(x) to be greater than zero for some x that the agent does not judge to be a possibility. We can avoid this by requiring that the domain of Pr is a Boolean algebra A such that the set of possibilities determined by A coincides (or is at least a subset of) the set of thing that the agent judges to be possible. Let us assume that the domain of Pr is an algebra A that satisfies this requirement. Let x be a possibility with respect to A, i.e. an element of A that lies strictly above the bottom element of A. Given that A is assumed to satisfy the indicated restriction, I assume that the agent has some positive degree of justification for believing that “x is possible” — after all, such an x is assumed to be in the set of things that the agent judges to be possible. On your account this would seem to require that
Pr(“x is possible”) > 0. Even if this makes sense formally in the sense that “x is possible” is representable as some element p in A, it seems odd to say the agent has a positive degree of belief in p, since p would also represent “the agent judges x to be possible” and then Pr(p) seems to say that the agent has a positive degree of belief in “the agent judges x to be possible”. Now perhaps that doesn’t strike you as odd. However, let’s consider the situation in a bit more detail. Could Pr(p) be some value less than one, say Pr(p) = .5? This would mean that the agent has a .5 degree of belief in “the agent judges x to be possible”. So, by assumption, the agent judges x to be possible but assigns degree of belief less than one to “the agent judges x to be possible”. This seems crazy to me. We are left with the possibility that Pr(p) = 1, but then, according to your proposal, this would commit us to saying that the agent has maximal justification for believing that “the agent judges x to be possible”, which might not be so bad. Anyway, there is no need to respond, since I suppose that we should both get back to serious work — most importantly, I don’t want you to feel that you need to waste time responding to what are likely superficial comments on my part.
All the best,
Jeff
Jonah:
‘posterior’ probability, in a Bayesian framework, only refers to temporal order: it is the probability you attach to a proposition X after acquiring new information. If you attached p(X/E) p(Y/E) at t, and now, at t+1, you are sure that Y is true (so, your new ‘evidence’ is E&Y), now instead of attaching p(X/E) to X you take its new probability is p(X/E&Y). This is what ‘coherence’ demands. If now X is much more probable for you, you may say it has been ‘justified’ by Y.
.
But what is essential is that, besides the logical relations between X, Y and E (that can even NOT exist: I mean, they can be logically independent, though you give it SUBJECTIVELY a high CONDITIONAL probability to some of then, conditioned by the others), there is nothing ‘objective’ in the connection between those propositions that can stand for the ‘justification relation’. Or, stated differently, what is important in the Bayesian approach is how well ‘justified’ you BELIEVE a proposition is, not how well justified it IS (this is what the Bayesian approach can not even EXPRESS!).
.
A I said, the only way of introducing something like ‘objective justification’ here is by a touch of ‘reliabilism’: introducing some ‘discoveries’ telling under what conditions we are more likely led to attach high probability if it is true.
Point well-taken Jesús … thanks! As I see it however, there is Bayesianism in its simplest glory as you outline, and then there is Bayesianism beefed up with certain constraints on probability assignments in addition to coherence (I’m thinking of the Principal Principle and other such additional requirements on rational degrees of belief). And I wonder if the mainstream concept of justification fits smoothly in with one of these more objective takes on Bayesianism.
With respect to the notion of justification as usually understood in traditional epistemology the following passage by van Fraassen might provide some insight:
“What I hope for is some reconciliation of the diverse intuitions of Bayesians and traditionalists, within a rather liberal probabilism. The old we might call defensive epistemology, for it concentrates on justification, warrant for, and defense of one’s beliefs.
The whole burden of rationality has shifted from justification of our opinion to the rationality of change of opinion.
This does not mean that we have a general opinion to the effect that what people find themselves believing at the outset is universally likely to be true. It means rather that rationality cannot require the impossible. We believe that our beliefs are true, and our opinions reliable. We would be irrational if we did not normally have this attitude toward our own opinion. As soon as we stop believing that a hitherto held belief is not true, we must renounce it — on pain of inconsistency!”
Laws and Symmetry. Oxford: Oxford University Press, 1989.
van Fraassen as many other philosophers of science (today recognized as formal epistemologists) proposes that the main problem in contemporary epistemology (formal or not) is to justify belief change, rather than the problem of how to justify static bodies of belief. This dismissal of the notion of justification is independent of whether one adopts a Bayesian position in epistemology or not. One can see this problem even if one does not adopt a Bayesian position. Peirce was not a Bayesian but he saw the problem quite clearly.
On the other hand van Fraassen hopes for a reconciliation of traditional epistemology and Bayesian epistemology by adopting a form of probabilism that permits the non-paradoxical reconstruction of central notions of the traditional approach like qualitative belief, conditionals, etc. Here it is another crucial quote from van Fraassen:
“Personal or subjective probability entered epistemology as a cure for certain perceived inadequacies in the traditional notion of belief. But there are severe strains in the relationship between probability and belief. They seem too intimately related to exists as separate but equal; yet if either is taken as the more basic, the other may suffer. […] I would like to propose a single unified account, which takes conditional personal probability as basic.
There is a third aspect of opinion, besides belief and subjective grading, namely supposition. Much of our opinion can be elicited only by asking us to suppose something, which we may or may not believe. Here supposition will be a central ingredient of an unified probabilistic picture.”
Fine-grained opinion, probability and the ,logic of full belief, JPL 24, 349.
Since 2005 or so Rohit Parikh and I tried to develop this idea formally. Currently the argument has the form of a long essay that will be part of a book. But the published results show that the approach does permit the reconstruction of some of the central notions of traditional epistemology within a Bayesian approach. This has a cost though. One has to adopt as a primitive a notion of conditional probability (supposition epistemologically) and this is equivalent to abandon Kolmogorov’s successful foundational program in the theory of probability. But many philosophers have recently proposed to adopt primitive conditional probability for independent reasons, so perhaps the price is not so high. In any case, I think that it is indeed possible to find a deep and interesting connection between traditional and formal epistemology within the bounds of a liberal form of probabilism.
Thanks for your intriguing and helpful thoughts Horacio … I look forward to seeing them spelled out further in the forthcoming book!
Gosh, looks like I’m late in coming to the party …
Regarding Rachael’s example, this is actually a variant on something Henry Kyburg was talking about three decades ago. Here are two possible solutions.
(1) The information derived from the extensive testing of the coin does get encoded in a new probability distribution, call it P*, but not in a new probability for heads. So P(H) = P*(H) = 0.5. The way it gets encoded is that probabilities for runs are changed. So our distribution when we are lacking information about the bias or lack of bias in the coin has to give a higher probability for P(H1 & H2) than P(H1) P(H2). But the sort of information we obtain that carries us from P to P* will reduce the probability of runs toward the minimum, so P*(H1 & H2) < P(H1 & H2). This is, if I recall, the solution favored by the Lund school folks like Gardenfors and Sahlin.
(2) We can take seriously Kyburg’s notion of interval-valued probabilities. Then P(H) may be, say, [.4, .6] whereas P* (H) = [.48, .52]. We speak casually when we give point values to the probabilities. This approach does not foreclose the encoding of some information in other parts of the distribution, but it does provide a way for it to show up right in the probability given to H.
Quick question for Greg: why say that with the Bayesian machinery, “closure under intersection and consistency must be maintained”? Maybe this is just a misreading of what you mean by “machinery,” but surely you’ll agree that we can throw out closure of knowledge (or of justification) under conjunction and still maintain a threshold-acceptance criterion, perhaps one that is context-sensitive, for justification.
Thanks for the thoughts Tim and good to have you commenting on C&I! I’m curious as to what you think of the option that I present above of encoding the new information (the degree of meta-justification) in Kyburg’s example with some measure of the robustness of one’s probability assignment (??).
Hi Tim, Jonah-
My remarks were a bit condensed. A few years ago I looked at 1-monotone capacities as a study for an extremely weak logic of rational belief (JoLLI 2006). (I called the little logic there ‘system Y’, as in ‘Why on earth would you adopt such a logic?’. I digress.) The point of the exercise, for me, was to see that I could not get under Kyburg’s theory of acceptance with even this extremely weak system; but Y is what’s under Bovens and Hawthorne’s bit on Lockean acceptance in MIND, as it it is under all Bayesian / set-based Bayesian / convex Bayesian probabilistic logics.
So, I don’t deny that one can come up with various ways for a Bayesian to limit closure –and the literature is full of examples– although I do think it is a weird move, since presumably you are doing this with respect to a specified joint distribution. (And then the criticisms that you are throwing away perfectly good information in order to ‘accept’ seems right.) Even so, at bottom such proposals are doing something different than what Kyburg was after.
Forgive the blatherings of an old man; but I took a look at Shogenji’s essay to be published in Synthese and this led me to think that perhaps I should mention the following items from my publications dating back to 1963. I do not claim that Shogenji or any other the others participating in discussions of measures of support or degree of justification and the like adopt views precisely like mine. Far from it; but there is sufficient similarity to warrant reference to these publications.
In recent years I took to classifying alleged indices of evidential support, warrant or confirmation into three types: probability, maximizing and satisficing. See my 2002 paper. I agree with all of those who think that probability is not a useful measure of evidential or inductive support. It cannot be a maximizing measure because the best supported hypotheses would be those already settled and established as carrying probability of standard value 1. It cannot be a satisficing measure because this leads to violations of deductive closure. Pace my good friend Henry Kyburg, allowing violations of deductive closure undermines the function of an inquirer’s set of full beliefs as a standard for serious possibility.
Many authors think that some function of p(x/y) and p(x) is a good index of evidential support. I myself once thought so (Levi, 1963) and argued, as Shogenji does from a vision of the inquirer seeking error free information to this conclusion. I soon realized this to be a mistake and this idea is abandoned in Gambling with Truth and “Information and Inference”. As explained in “Information and Inference” and in many places since, this proposal fails to address the point that when inquirer X is in state of full belief K and wishes to determine whether to add x to K and form the deductive closure, K may entail that x and x’ carry the same truth value even though x and x’ are not equivalent in the minimal state of belief whatever that might be (the veil of ignorance). In that case p(x/y) = p(x’/y) while p(x) does not equal p(x’) . So the difference (or ratio or log ratio) of the posterior and the prior cannot be a good index of evidential support. Yet, Shogenji and many others are enamored of such functions as measures of evidential support, degrees of justified belief or whatever.
Why?
Isaac Levi
1963
(a) “Corroboration and Rules of Acceptance”, British Journal for the Philosophy of Science v.13, pp.307-13.
1965
(a)”Deductive Cogency in Inductive Inference,” Journal of Philosophy v.62, pp. 68-77.
1967
(a) Gambling with Truth, New York: A.Knopf. Reissued in paperback without revision in 1973 by MIT Press.
1967
(c) “Information and Inference,” Synthese 17, pp.369-91.
1971
(b) “Truth, Content and Ties,” Journal of Philosophy, v.68, pp.865-76..
1976
(a)”Acceptance Revisited,” in Local Induction, ed. by R.Bogdan, Dordrecht: Reidel, pp.1-71.
(b) “A Paradox for the Birds,” in Essays in Memory of Imre Lakatos, ed. by R.S. Cohen et al. Dordrecht: Reidel, pp.371-8.
1979
b) “Inductive Appraisal:, Current Research in Philosophy of Science, ed. by P.D.Asquith and H.E.Kyburg, East Lansing, Mich.: PSA., pp.339-51.
(d) “Abduction and Demands for Information,” The Logic and Epistemology of Scientific Change, ed. by I. Niiniluoto and R.Tuomela, Amsterdam: North Holland for Societas Philosophica Fennica, pp.405-29.
1980
(a) The Enterprise of Knowledge: An Essay on Knowledge, Credal Probability and Chance, Cambridge, Mass.: MIT Press.
1982
(c) “Self Profile” (pp.181-216) and “Replies” (pp.293-305) in Profiles of Henry E.Kyburg, Jr. and Isaac Levi ed. by R. Bogdan, Dordrecht: Reidel.
1983
(b) “Truth, Fallibility and the Growth of Knowledge,” Language, Logic and Method ed. by R.S.Cohen and M.Wartofsky, Reidel, pp.153-4 followed by comments by I.Scheffler and A.Margalit and replies.
(h) “Conjunctive Bliss,” Behavioral and Brain Sciences, v.6, pp.254-255.
1984
(a) Letter on “The impossibility of inductive probability,” in Nature, v.310, p.433.
(b) Decisions and Revisions, Cambridge: Cambridge University Press.
1985
(c) “Illusions about Uncertainty,” British Journal for the Philosophy of Science, v.36, pp.331-340.
1991
(a) The Fixation of Belief and Its Undoing: Changing Beliefs Through Inquiry, Cambridge: Cambridge University Press.
1996
(a) For the Sake of the Argument: Ramsey Test Conditionals,, Inductive Inference and Nonmonotonic Reasoning, Cambridge: Cambridge University Press.
2001
(a) “Inductive Expansion and Nonmonotonic Reasoning,” Frontiers in Belief Revision edited by Mary-Anne Williams and Hans Rott, Dordrecht: Kluwer, 7-56.
2002
c “Maximizing and Satisficing Measures of Evidential Support,” Reading Natural : Essays in the History and Philosophy of Science and Mathematics, edited by David Malament, Chicago: Open Court, 315-333.
2004
(b) Mild Contraction: Evaluating loss of information due to loss of belief. Oxford: Oxford University Press.
2006
(a) “Replies”, Knowledge and Inquiry: Essays on the Pragmatism of Isaac Levi ed. by E.J.Olsson, Cambridge: Cambridge University Press, 327-380.
2008
(a) ”Degrees of Belief,” Journal of Logic and Computation, 18, 1-21
Hi Isaac,
I have a question for you about non-adjunctive systems. The modal system EMN, which Henry and Choh Man proposed as a qualitative representation of threshold accepted belief, is axiomatizable, sound and complete. Also, I’m looking at encoding Henry’s system within description logic, which are by design sound, complete and tractable. I’m not sure I can get these nested direct inference statements that seemed to want to allow, but the basic machinery looks like it will survive translation.
Given these nice properties of these two systems of logic, why should one worry about limits placed on (unrestricted) closure under intersections?
Best, -Greg
Dear Greg, Unfortunately I do not know the modal system EMN although I suspect I am familiar with it in some guise or other. So I cannot answer you with complete confidence. But if the aim of uncertain inference is, as I would put it, to shift from one standard for serious possibility to another where such standards determine spaces over which probability measures both determinate and indeterminate (including interval valued) are defined, the shift should be from one deductively closed theory to another. Threshold rules for rationalizing such changes can be constructed that respect such closure. If you don’t like my approach, try L.J.Cohen’s or some other such variant on Shackle measures. One could even do it using Henry’s approach to probability. His objection is that he doesn’t like partition sensitivity. I do not think this objection is crushing. But that is where the action is to my way of thinking.
But I may be speaking irrelevantly because I do not know what EMN is.
Best,
Isaac
Thanks Isaac!
Dear Greg, I should have delayed my response until I could cudgel my memory and reread some old papers (I do not have the Kyburg and Teng paper). I do have papers on neighborhood semantics. In particular, I have papers by Arlo Costa and Arlo Costa and Pacuit and these papers provide characterizations of nonadjunctive epistemic logics that satisfy the axioms E, M and N according to a standard code for neighborhood semantics. I do not regard the fact that Kyburg’s approach can be formulated in neighborhood semantical terms (using confections like EMN) as providing any sort of support for fragment of Kyburg’s epistemology that recommends believing, accepting or knowing that h iff h carries high probability No doubt nonadjunctive modal logics can give systematic characterizations of some operator like “is highly probable”. But I think we should let it go at that and not seek to make “highly probable” more significant than it actually. In my own work, I am interested in factors that warrant changing states of full belief. If it is highly probable that h when K is the investigator’s state of full belief, that is neither necessary nor sufficient for inductively expanding K by adding h and closing under logical consequence. No matter how attractive the formal properties of EMN or any other non adjunctive ‘logic’ might be, I have never seen (from Henry K or anyone else) a convincing reason for changing my mind.
This is the same answer as I gave before but now I have a better idea of what EMN is.
Best,
Isaac
Dear Isaac,
It is fairly straight-forward to define an AGM revision operator for the class of modal logics that contains EMN, and I anticipate that there will be available options other than AGM to choose from. So, I don’t see technical barriers to doing revision on non-adjunctive logics, either.
Now, the devil is in the details — of course. But, these ideas seem new and perhaps worth mentioning.
Best, Greg
Dear Greg,
I am not concerned with whether you can define a revision operator. My interest is in justifying changes in states of full belief where agent X is committed to such a state when X is committed to distinguish between what is seriously possible and what is not in terms of consistency with that state and where the set of standard real valued credal probability functions Q(x/y) that are permissible according to X are restricted to those where y is a serious possibility and for fixed y, Q(x/y) is a probability defined over the powerset over the set of serious possibilities.
Best,
Isaac
Dear Isaac and Greg:
I think that Isaac is right in pointing out that the philosophical problem related to the normative adequacy of acceptance rules in terms of high probability is not solved by appealing to a neighborhood representation. But I also think that the use of classical modalities to represent high probability operators is a big improvement with respect to previous attempts to determine the logical commitments of philosophers like Kyburg who are convinced about the philosophical adequacy of these rules of acceptance. Previously to this development Kyburg suggested that the classical rule of Adjunction has to be abandoned if you embrace these rules. The representation via classical modalities shows that what needs to be abandoned is the Kripkean semantics for the modalities (and there are independent reasons that indicate that the Scott-Montague approach offers an interesting and more flexible alternative to Kripke’s semantics). Kyburg does not need to touch the underlying logic which remains classical. What is abandoned is the axiom C:
.
Of course one can use a non-adjunctive logic rather than deploying a modal operator, but the representation in terms of neighborhoods shows that one does not need to do that. One can embrace high probability acceptance rules without giving up classical logic. This does not offer a philosophical argument supporting the conceptual adequacy of high probability acceptance rules. But it shows that the logical commitments of philosophers like Kyburg are less radical than previously thought. Moreover once one sees the connection between high probability operators and classical modalities one can formulate these bridges at the first order level. Then an epistemic interpretation of first order classical logic is apparent. For example, there is an interesting connection between the Barcan schema and the lottery paradox as I indicated in various recent articles. This connection has not been yet explored in detail, I think. All this seems to indicate that the neighborhood representation is the right logical tool for philosophers who appeal to high probability acceptance rules. But still the follower of Kyburg needs to establish that these rules can be applied successfully. Of course Kyburg thought that this is indeed the case, but this has been disputed by other philosophers, like Isaac.
The development of a neat logical account for the acceptance rules eliminates some obstacles for its application and makes possible a better understanding of the way the rules work, but per se it cannot replace a philosophical debate about the normative or descriptive adequacy of the rules.
Best, H.
Dear Greg and Horacio
I agree with Horacio that if NA&NB -> N[A&B] is abandoned where ‘NA’ is interpreted as “it is highly probable that”, there is nothing objectionable philosophically as far as I can see. Indeed, rejection of the principle is the right thing to do.
But if the box operator represents ‘it is fully believed that’ or ‘it is certain that’ or ‘it is known that’ where the set of certainties or corpus of knowledge is the standard of serious possibility, there is plenty to object to even if the ‘underlying logic’ is classical.
We would have systems of modal judgment like it is not seriously possible that ~A and it is not seriously possible that ~B yet it is seriously possible that ~Aor~B.
Of course, as Horacio, if I remember rightly points out, one can abandon the duality of the possibility operator and the necessity operator. But that does not help. It only reinforces the point that the full beliefs construed using the box cannot be the standard for serious possibility.
This is not a merely verbal point. P(X/~Aor~B) would be well defined but neither P(X/~A) nor P(X/~B) would be.
Horacio seems to think that Henry’s position is better off if one abandons ‘Kripkean Semantics’ for a version of neighborhood semantics. Not if the necessity operator is to serve as a standard for serious possibility.
There may be other useful applications of the approach Horacio favors. We should always keep an open mind about that.
But Henry’s romance with high probability (and I would include the romance with high probability as infinitesimally less than 1) cannot satisfy our requirements for a standard for serious possibility
Of course, as LJ Cohen rightly noted (see also Gambling with Truth) one might reinterpret probability so that high probability rules for adding new information to the belief state yield a deductively closed set. This is easy to do using Shackle measures.
The point of emphasizing this is that the yearning for high probability rules that so many authors display might be assuaged by the use of Shackle measures.
I want to insist that high probability rules are neither necessary nor sufficient for developing the contributions that make Henry’s work so important. But that is another story.
Isaac
Dear Isaac & Horacio,
Thanks much for this excellent exchange. Horacio, I think you are right about Risky Knowledge not quite capturing acceptance in EP. Here is another data point: Both description logics and EMN are complete, but one of the last conversations that I had with Henry was about a completeness result for EP itself. He waved this away as a category mistake to even ask. (!). (Kyburg & Teng have a soundness result in Uncertain Inference). If either of you recall discussions with Henry on this point, I’d be keen to hear about it.
Dear Horacio & Isaac,
I agree with Horacio’s take on this, but would put the emphasis a bit differently: technically, closure conditions and revision operations are possible to build on a qualitative version of Kyburgian evidential probability. That’s not to say that the mere possibility of doing so is an argument for adopting the Kyburgian view; but, the relative ease of getting these two features does remove these two technical objections from use in this philosophical argument.
As an aside, I have taken to viewing Kyburg’s EP as essentially a description logic. It is a system for reasoning about (reference) classes, basically, where the probability intervals enter in as terms simply for determining which classes to close under the rules of richness and specificity (i.e., those whose intervals ‘conflict’) and which to ignore. The actual numeric values of the interval do not show up until the end, when you apply the strength rule to the surviving classes. What seems promising about this view is that it yields tremendous technical advantages over a (convex) Bayes probability logic. (Good luck computing with credal sets!)
Again, I agree that this doesn’t settle the philosophical dispute between the views. But, in so far as there have been complaints lodged against EP for not being a proper logic or for EP being technically unwieldy, it appears to me that this type of objection in the end may very well be stood on its head: the technical features of the system, stemming from the (apparent) ability to translate the core of the theory into various types of guarded fragments of first-order logic with extremely nice properties, may well be EP’s strongest advantage.
Dear Greg: I basically agree with you. But I have a doubt that is based on recent work done by Makinson and Hawthorne and Paris (independently). Notice that neither Kyburg and Teng or our paper with Eric (Pacuit) prove a representation result. The idea is to propose a logical system that seems adequate given that it satisfies properties that it needs to satisfy and fails to satisfy other problematic properties like C.
Now independently Makinson and Hawthorne and more recently Paris have looked at the non-monotonic notions of logical consequence generated by the following equation (for all measures w, and threshold t in [0,1]:
Paris shows in an impressive paper (published in the Review of Symbolic Logic) that the notion of consequence that thus arises cannot be axiomatized. This continues the interesting work initiated by Makinson and Hawthorne. Now consider:
The notion
can be seen as a notion of belief. Is this notion completely axiomatizable? Astonishingly I do not think that this has been investigated, although this is a natural question to ask. Logically, perhaps it is the most natural question to ask here. The notion of belief should obey the axioms of EMN among others. But do these axioms suffice to characterize the operator of high probability (now a doxastic operator, rather than a conditional)? I posed the question in a way that it can be checked by looking at the recent proofs of non-axiomatizability. Unfortunately Paris’s paper is not easy to read (it presumes a fair amount of high powered mathematics). Makinson and Hawthorne’s paper is more clear and perhaps it suffices. In any case, it seems to me that we still do not understand clearly the logical commitments of a high probability test. In conditional logic the question has been avoided for years by changing theme and embracing tests based on infinitesimal probability (which are well behaved logically). Paris and Makinson and Hawthorne focused their attention on the high probability test that Kyburg and others proposed. Perhaps the results established by Paris can be presented in a more direct way, but prima facie the result indicates that the notion of consequence induced by the test cannot be characterized logically. It would be interesting to see whether the notion of belief induced by a natural modification of the test is characterizable. If the answer is not, this would indicate that Kyburg and Teng’s notion of `risky knowledge’ is not graspable via the use of standard logical methods. As I said above it is incredible that this is still an open question at this point.
Best, H.
This is a belated reply to Isaac’s May 8 post that refers to my paper (I was away when his post appeared). In the paper that Isaac refers to, I proposed the following measure J(h, e) of epistemic justification.
In the process of formulating this measure, I reasoned that P(h|e) determines the risk of h being false, while P(h) determines the potential gain in truth, which I took to be the amount of information h carries. Isaac reminds us in his post that he argued against this approach in his article “Inference and Information” in Synthese (1967) and other places. The central thesis of Isaac’s 1967 Synthese article is that “information does not decrease with an increase in probability in the sense in which probability determines fair betting quotients” (370). I have my reason for holding onto my own view (I do not accept Isaac’s principle A-2 on page 374), but I am not going into this issue here. I want to point out instead that the main part of my paper is not tied to the fate of the particular view about information that I adopted in the paper.
For example, Isaac distinguishes the probability function
that determines the informational value of h, from the credal probability function 
. Since 
determines the degree of risk and 
determines the amount of potential gain in information, their roles are comparable, respectively, to those of P(h|e) and P(h) in my paper. So, those who like this part of Isaac’s view can systemically replace P(h|e) and P(h) in my paper with 
and 
, respectively. The resulting measure of epistemic justification would be J*(h, e) instead of J(h, e) above.
Of course, Isaac uses