Probably Possible

A dilemma, or apparent dilemma, looms for anyone who subscribes to the following three theses:

(1) There are certain closure principles that hold for rational beliefs; for example, rational beliefs are closed under conjunction and modus ponens:

(a) If we’re rational to believe that p and we’re rational to believe that q, then we’re rational to believe that p&q.

(b) If we’re rational to believe that p and we’re rational to believe that p ⊃ q, then we’re rational to believe that q.

(You might worry that the above principles are false as they stand, because of cases in which we believe that p and believe that q (for example), but have not put one and one together. If you like, add the relevant bells and whistles to get your preferred versions of the principles.)

(2) To be rational to believe a proposition, we need to be rational in assigning it a sufficiently high credence.

(3) There are many propositions we are rational to believe.

Let’s not question (2). Now suppose that to be rational to believe a proposition, we need to assign it a credence of 1. Then (3) seems false, for it seems that there are not many propositions to which we would be rational in assigning a credence of 1. But suppose that to be rational to believe a proposition, we just need to be rational in assigning it a sufficiently high credence that may well fall short of 1. Then it seems that (1a) and (1b) are false. Let’s say that we are rational to believe that p and also that q, and that our credences in p and in q are each 0.95. Let’s say that 0.95 is the minimum credence we need to assign to a proposition to be rational to believe it. Then, for all the probability axioms say, our rational credence in p&q may well be less than 0.95, in which case we are not rational to believe it. Similarly, let’s say that we are rational to believe that p and also that p ⊃ q, and that our credences in p and in p ⊃ q are both 0.95. Then, for all the probability axioms say, our rational credence in q may well assume a value in the interval [0.9, 1]. Suppose our credence in q is 0.91. Then, if 0.95 is the minimum credence we need to assign to a proposition to be rational to believe it, we are not rational to believe that q.

How might we resolve this dilemma? On Tuesday, at my midterm review, I argued that there is a way of thinking about the relation between belief and credence so that we can preserve closure without having to hold that rational belief requires certainty. (The talk drew heavily from one of wo’s blog posts, but I’m solely responsible for anything stupid I said!) Think of rational belief in terms of the ruling out of possible worlds. To be rational to believe that p is to be rational to rule out all worlds in which not-p. And to be rational to rule out worlds in which not-p is to be rational to assign those worlds a credence of 0. Now this would mean that rational belief requires certainty, unless we restrict the space of worlds within which ruling out takes place. Let’s hold that one is rational to believe that p iff one is rational to rule out all not-p worlds within the relevant set of worlds. If m is the proposition that is the relevant set of worlds, then one is rational to believe that p iff one is rational to assign a conditional credence of 1 to p given m, and a sufficiently high credence to m. Let’s hold that what is relevant and what counts as sufficiently high vary with context.

For the time being, let’s set aside the following questions:

(i) What exactly is m?

(ii) What determines what counts as sufficiently high?

(iii) Whose context restricts the relevant space of possible worlds?

Assuming that something like the above account of rational belief is correct, we can see how we can preserve closure, and yet not hold that rational belief requires certainty. Suppose that we are in a context in which m is relevant, and that the minimum credence we need to assign to m to be rational to believe that a proposition is true is 0.95. Suppose that we are rational to believe that p and rational to believe that q. Then on the above account, Cr(p|m) = Cr(p ⊃ q|m) = 1. Then Cr(q|m) = 1. Then Cr(q&m) = Cr(m). But Cr(q) ≥ Cr(q&m). So Cr(q) ≥ Cr(m) = 0.95. If we’re rational to believe that p and also that p ⊃ q, the credence we are rational to assign to m sets the lower bound for the credence we are rational to assign to q

One might worry that on this account of rational belief, we would still have very few rational beliefs—if Cr(p|m)=1, then we are rational in being disposed to place limb, life, and soul on the conditional bet that p is true given m. But there are hardly any such bets, or so one might think. To mitigate the worry, however, note that we may well be rational to make such bets when either m entails p, or m, together with anything else that we are certain of, entails p. Suppose I’m certain that there appears to be a table in front of me. Suppose also that m is something like `Appearance reflects reality on this particular occasion’. Then it seems rational for me to assign a conditional credence of 1 to there being a table in front of me, given that appearance reflects reality on this particular occasion.

Another worry with the account is whether it merely shifts the bump in the rug. Someone who subscribes to the Lockean Thesis, and holds that one is rational to believe that p iff one is rational to assign a credence to p that meets a sufficiently high threshold, is at risk of being impaled on the horns of the dilemma. On the current account of rational belief, there is also a threshold to be met: one needs to assign a certain minimum credence to m for one to be rational to believe that p. Does the account face similar problems to those faced by someone who subscribes to the Lockean Thesis? The worry that it does cropped up a number of times during my talk’s Q&A, and I’m afraid I botched up my reply to it. Here’s another attempt to assuage the worry.

Subscribing to the Lockean Thesis seems to be bad for someone who likes (1a) and (1b). But holding that one needs to assign a sufficiently high credence to m to be rational to believe that p need not lead to a similar problem, depending on what m is supposed to be exactly. Let’s say that when one is rational to believe that p, m is what one is rational to presuppose, or rational to take for granted (whatever it may be to presuppose something or to take it for granted). Then we might think that if what is rational for one to presuppose is closed under conjunction or modus ponens, we are led to similar worries that someone who subscribes to the Lockean Thesis faces, but at the level of what is rational to presuppose, rather than at the level of what is rational to believe.

But for the current account of rational belief to work, we don’t need to hold that what is rational for one to presuppose is closed under conjunction and modus ponens. Suppose that you’re in a context in which you’re rational to believe that p, and Cr(p|r&s) = Cr(p|r&t) = Cr(p|r&s&t) = 1. Suppose that you are rational to presuppose that r&s, and you are also rational to presuppose that r&t, but you might not be rational to presuppose that r&s&t, because your credence in r&s&t might be below the minimum required for you to be rational to believe that p. Nonetheless, so long as you presuppose that r&s or you presuppose that r&t (but not both at the same time), you are rational to believe that p.

One might think that on an intuitive understanding of what it is to presuppose something, if one is rational to presuppose that r, and one is rational to presuppose that s, and one is rational to presuppose that t, then one is rational to presuppose that r&s&t. But it also seems intuitive that one shouldn’t presuppose too much! While one may be rational to presuppose any one of a number of things, one shouldn’t presuppose all of them.

In any case, I don’t want to put too much weight on intuitions about how the term `presupposition’ is used in daily life, since it’s really meant to be a technical notion. To be rational to believe that p, there is some special proposition, m, to which one has to assign a sufficiently high credence, and which carves out the not-p possibilities to which one needs to assign a credence of 0. While I’ve been speaking of whether one is rational to presuppose that m, or to take m for granted, it remains an open question whether to be rational to believe that p, one needs to assume any propositional attitude (e.g. that of presupposition, taking for granted) towards m, over and above assigning it a sufficiently high credence. If the answer is `No’, then the above worry dissipates. Of course, more will have to be said about this m that I’ve been talking about, but I’ll try to answer questions (i), (ii), and (iii) in more detail when I’ve got the energy to do so.

Suppose that being amazed that p, being relieved that p, and being happy that p are factive attitudes. Does having such attitudes require one to know that p? Williamson, in the first chapter of Knowledge and Its Limits, argues that the answer is “Yes”. The oddity of the following sentences provides support for his view:

• Sue doesn’t know that you’re a great singer, but she is amazed that you’re one.
• Xuxa is relieved that her son has arrived safely, but she doesn’t know that he has; for all she knows, he died in the war, and people are just trying to keep her away from the awful news.
• They were happy to have found a barn for shelter, but they didn’t know, although they truly believed, that it was a barn; after all, they couldn’t rule out the possibility that it was a barn facade.

One upshot of Williamson’s view is that if some sort of global scepticism is right, then not only is there little that we know about—there’s little that we can be said to be happy about, or amazed about, or relieved about. (The oddity of the above sentences remains if we replace “Sue is amazed that you’re a great singer” with “Sue is amazed about your being a great singer”, “Xuxa is relieved that her son has arrived safely” with “Xuxa is relieved about her son’s safe arrival”, etc.)

Another upshot is that if “knows that p” is context-sensitive, then (perhaps derivatively) so are terms like “happy that p”, “amazed that p”, and “relieved that p”. (Since happiness, amazement, and relief come in degrees, it might be a matter of context how much happiness, for example, is needed for one to be said to be happy. But this is not the kind of context-sensitivity that I’m concerned with here.) Suppose that we start off in a context in which it is correct to say both that Sue knows that you sing well, and also that she is amazed that you sing well. But suppose that standards go up, and we find ourselves in a new context in which it is no longer correct to say the former. Then it is no longer correct to say the latter. If contextualism about knowledge attributions is right, then it seems that whether you can be said to be happy, or amazed, or relieved about something, depends on the epistemic standards operating in the contexts in which such things are said. (And a similar story could be told if subject-sensitive invariantism is correct. If whether one knows that p is subjective-sensitive, then so is whether one is happy that p, or amazed that p, or relieved that p.)

Someone who subscribes to Williamson’s view, but who, unlike Williamson, is either a global sceptic or a contextualist, might well embrace such consequences of the view. The global sceptic isn’t committed to saying, on Williamson’s view, that there’s a dearth of happiness, amazement, or relief in the world, even though she has to say that there’s little that one can be happy, amazed, or relieved about. The contextualist can maintain that in ordinary contexts, one can be said to be happy, amazed, or relieved about lots of things; it’s only in extraordinary contexts that there’s little that one can be said to be happy, amazed, or relieved about.

Now I suppose that if the global sceptic doesn’t find her view that we know very little discomfitting, she would have no qualms holding that there’s little we can be happy, amazed, or relieved about. But what about the contextualist? I find it strange that whether Xuxa can be said to be happy that her son has arrived safely varies with context, such that in one context, we can say that she is happy that her son has arrived safely, but in some other context, we can’t, even though in both contexts, her subjective feeling of happiness, her degree of confidence in her son’s safe arrival, and her justification for thinking that her son has arrived safely, are the same. Perhaps, this won’t seem so strange to me if in both contexts, there is something common that we can say that Xuxa is happy about. But what is it? Of course, in both contexts, we can say that Xuxa is happy in her belief that her son has arrived safely, but it’s not her belief that she’s happy about.

Perhaps, the lesson to draw is as follows. Stalnaker (Inquiry, Ch. 4) and Lewis think that questions about the content of belief and questions about our practice of belief attributions should be separated. Perhaps, questions about the content of happiness and questions about our practice of happiness attributions should likewise be separated. Xuxa is happy, and the content of her happiness seems to be just this: her son’s safe arrival. Whether we can say that she’s happy that he has arrived safely might depend on other things. If Xuxa’s son actually died in the war, but she mistakenly thinks that he has arrived safely, and therefore feels happy, it won’t be correct to say that she’s happy that he has arrived safely. But this doesn’t change that fact that she feels happy, and that the content of her happiness is about her son’s safe arrival.

Suppose I know that coin A is fair, and have no information whatsoever about whether coin B is fair. To what degree should I believe that coin A will land heads if tossed, and that coin B will land heads if tossed?

In both cases, my intuitions say “1/2″. But it is sometimes pointed out that there’s an important difference between the two cases that one’s degrees of belief should reflect. In the first case, one has evidence about whether coin A will land heads—one knows that the chance of the coin landing heads is 1/2. In the second case, however, one has no evidence whatsoever about whether coin B will land heads—one has no idea what the chance of the coin landing heads is.

I’m not sure, however, why the difference between the two cases should be reflected in one’s degrees of belief. Set aside degrees of belief for the moment, and let’s just focus on good (?) old all-or-nothing beliefs. In the first case, I suspend belief on whether the coin will land heads. In the second case, I suspend belief too. But suppose someone points out—correctly—that the two cases are different in the way described above. It doesn’t seem reasonable to conclude that this difference should somehow be reflected in your suspension of belief. But then, why might it be reasonable to make such a demand when it comes to degrees of belief?

Beliefs represent the world—or so goes received wisdom. I believe that Canberra is the capital of Australia iff I represent the world as being such. I don’t believe that Singapore lies north of the equator iff I don’t represent the world as being such.

If belief is all-or-nothing, then either we represent the world as being such-and-such, or we don’t. But Frank Ramsey, according to Richard Jeffrey, has “sucked the marrow” out of such a notion of belief. The correct way to think about belief, according to Jeffrey, is to think of it as coming in degrees. Assuming that this is right, how should we (re-)read the claim that beliefs represent the world?
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In which direction do you see the dancer in the picture spin? I saw her spin clockwise initially, but with some practice, managed to get her to spin anti-clockwise as well.

Famously, Lewis used to believe Nassau Street ran roughly east-west; that the railroad nearby ran roughly north-south; and that the two were roughly parallel. Lewis’ three beliefs were mutually inconsistent. In classical logic, everything follows from an inconsistency. But not everything was true according to Lewis’ belief state, because he did not explicitly believe the conjunction of the three sentences. In one fragment of his mind, Lewis believed that Nassau Street ran roughly east-west and that Nassau Street and the railroad nearby were roughly parallel. In another fragment, Lewis believed that the railroad ran roughly north-south and that the railroad and Nassau Street were roughly parallel. But no fragment was such that the conjunction of each sentence in the inconsistent triple was true according to that fragment. Rather, the inconsistency occurs across fragments and not within fragments. When Lewis detected the inconsistency in his belief system, “the fragmentation was healed” and “straightaway [his] beliefs changed”. Consistency was restored, and he now thought that Nassau Street and the railroad both ran roughly northeast-southwest.

Assume that we have an agent who believes p in fragment F1, but ~p in fragment F2, where p is a contingent proposition.# For Lewis, fragments are represented by sets of possible worlds. So for each world in F1, p is true, and for each world in F2, ~p is true. As a first pass, healing a fragmented mind consists in conjoining F1 and F2 into one fragment F. But of course, since F is just another set of worlds, we need worlds at which both p and ~p can be true. Worlds at which both p and ~p are true are impossible worlds, where an impossible world, following Restall, can be thought of as the superimposition of two possible worlds. We can superimpose two worlds, w* and w**, to get w as follows (following Reschner and Brandom): for all worlds w* and w**, and for any proposition A, w is the world such that A is true at w if and only if A is true at either w* or w**. As easily seen, if p is true at w* and ~p is true at w**, then there is a w such that, by superimposition, p and ~p is true at w. Simplifying, let us assume that F1 consists of just one world w*, at which p is true, and that F2 consists of just one world w**, at which ~p is true. Then F consists of the impossible world w, at which p and ~p are true, and which results from superimposing w* and w**. So, by conjoining F1 and F2, the agent ends up believing an explicit contradiction p and ~p in F. Assuming our agent is minimally rational, but not a paraconsistent logician, contradictions are recognized as bad. So, to restore consistency in F, the agent has to give up one of the beliefs in F1 or F2.

Lewis, however, was a firm denier of impossible worlds. On his view, the above agent cannot heal his fragmented mind by: (i) conjoining the two fragments F1 and F2; (ii) noticing the contradiction in F; and (iii) giving up one of the beliefs in F1 or F2. So, rather than prescribing:

(A) conjoin F1 and F2 into F –> aware of contradiction in F –> restore consistency in F,

it seems that Lewis has to prescribe something along the lines of:

(B) aware of contradiction across F1 and F2 –> restore consistency between F1 and F2 –> conjoin F1 and F2 into F.

Unfortunately, Lewis does not explain what it means to heal a fragmented mind. But if it is right that he has to prescribe something like (B), then how exactly would such an explanation of successful healing look like? In particular, if an agent has a fragmented mind, how can he be aware of a contradiction across fragments without actually believing it? Do we have to postulate a super agent, who does not believe either p or ~p, but just monitors the belief system of the agent? Or can an agent, as it were, gaze at his fragments from above, without any of them being the dominating one, at least for the moment?

# Since each fragment represents a set of possible worlds, and since each possible world verifies all logical and mathematical truths, fragmentalization will not help explain why we can fail to believe such truths.

JCB and WHT

Let’s say I ask you to suppose that p. Naturally, in supposing that p, you won’t at the same time suppose that not-p, even if you’re inclined to believe the latter. But is there anything in the concept of “suppose” such that supposing p means not supposing not-p? I’m inclined to answer `No’. The request that you suppose p and also that not-p might well make sense. Say, I make the request because I want to show you that from a contradiction, anything follows. And if I then request that you suppose that p, without also supposing that not-p, I need not be saying anything redundant. Say, I make this request because the next thing I want to show you is that p makes q very likely, and I no longer need you to suppose that not-p. But the first request won’t make sense, and the second request will be redundant, if the answer is `Yes’. How then do we explain why it’s so natural that when asked to suppose that p, one doesn’t also suppose that not-p?

Assuming ex contradictione quodlibet, it’s not surprising that in the majority of cases, when I ask you to suppose that p, there is a strong implicature that you don’t also suppose that not-p—the point of asking you to suppose that p is defeated if you also suppose that not-p, and that which I want to show follows from p, or is made likely by p, now follows trivially from “p and not-p”.

Philosopher Istvan Aranyosi sings:

We say things like the following quite often:

(1) Jack is probably home right now.
(2) She probably forgot her keys.
(3) I’m probably going to hell.

Now consider the following example. (More of such examples are discussed by Andy Egan in “Epistemic Modals, Relativism and Assertion”.) Suppose that on the basis of good but non-conclusive evidence that Jack is home right now, you utter (1). You seem entitled to utter (1). But I’ve seen Jack at the office just a minute ago, and know that he can’t be home right now. Overhearing what you’ve just said, I reply, “No, that’s not true”. Is my reply appropriate?

Some have quite a strong intuition that it’s appropriate. But what do I deny in replying, “No, that’s not true”? Am I denying that Jack is probably home right now, or am I denying that he is home right now? Consider the following examples:

A: It seems to me that Jack should be home by now.
B: No, you’re wrong.
C: Yes, it does seem that way to you.

A: It seems to me that philosophy is a waste of time.
B: Well, you’re wrong!
C: Yes, it does seem that way to you.

A: I think that truth is relative.
B: No, it isn’t.
C: Yes, you do think that truth is relative.

A: I believe that work is more important than exercise.
B: No, that’s not true.
C: Yes, you’re right that you have such a belief.

A: I’m quite certain that Jack is at home right now.
B: No, I’m afraid you’re wrong.
C: Yes, you’re indeed quite certain that he’s at home.

B’s replies to A seem to be very natural ones to make. But it also seems that none of them is a denial of what is said at face value. Suppose you say, “It seems to me that philosophy is a waste of time”, and I reply, “Well, you’re wrong”. Is my reply appropriate? You might think that it isn’t, if you take it to be a reply to what is said at face value; after all, it isn’t my business to set you right on how things seem to you to be. But it’s a very natural reply to make, especially in contrast with the kind of reply that C gives. In telling you that you’re wrong, I’m denying that philosophy is a waste of time, and not that it seems that way to you. (Of course, you could try to be difficult and say in response to me, “Well, I didn’t say that philosophy is a waste of time—I just said it seemed that way”.)

Question: might something similar be going on in the “probably” example? When you utter, “Jack is probably home right now”, and I say, “No, that’s not true”, am I denying that Jack is at home, rather than denying what is said at face value, namely, that Jack is probably at home?

Suppose we tweak the example slightly. Suppose I have very good but inconclusive evidence that Jack’s in his office. Now imagine that the following exchange takes place:

You say: Jack is probably at home right now.
I say: No, he’s probably still in his office. (Thanks to JC for suggesting this example.)

Since I’m not sure that Jack isn’t at home, I do not want to deny that he is. But does this show that in saying “No”, I’m denying that he is probably at home? No. For on the same grounds, we might have had the following exchanges instead:

You say: I believe that Jack is at home right now.
I say: No, I believe he’s still in his office.

You say: I’m quite certain that Jack is at home right now.
I say: No, I’m quite certain that he’s still in his office.

Both exchanges above sound rather natural. In the first exchange, it’s implausible that in saying “No”, I’m denying that you believe that Jack is at home. In the second exchange, it might not seem that I’m denying outright that Jack is at home, but it’s also implausible that in saying “No”, I’m denying that you’re quite certain that Jack is at home. Is what’s going on in the last three exchanges relevantly similar? Perhaps, saying “No” in each case is just a way of protesting that you should not believe, should not be quite certain, should not think it probable, that Jack is at home.

One more example:

You say: I have it on good evidence that Jack is at home right now.
I say: No, I have it on good evidence that he’s still in his office.

In saying “No”, am I denying that you’ve good evidence that Jack is at home? I guess the answer depends partly on my conception of evidence. If I think that my having good evidence that p is compatible with your having good evidence that not-p, then I’m not denying that you’ve good evidence that Jack is at home—perhaps, I’m just denying that Jack is at home on the basis of my evidence that he’s still in his office, or perhaps, I’m just denying that your evidence is better than mine. But if I think that my having good evidence that p is incompatible with your having good evidence that not-p, then in saying “No”, I do seem to be denying that you have good evidence that Jack is at home right now. Perhaps, what’s going on in the “probably” example is similar to what’s going here, in which case by saying, “No, that’s not true”, I’m indeed denying that Jack is probably at home.

and so, to spruce it up, here’s a picture of JC and I having coffee somewhere in Sydney:

(Picture taken by Ole Koksvik)