Probably Possible

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)

Have been reading an article by Timothy Williamson that I find quite interesting:
http://www.philosophy.ox.ac.uk/faculty/members/docs/infiniteheads.pdf

Suppose that a fair coin is tossed infinitely many times at one second intervals starting from t1, and the tosses are all independent. Let (H1…) be the event that every toss of the coin lands heads. Let (H1) be the event that the first toss at t1 lands heads, and (H2…) be the event that every toss from t2 (separated from t1 by one second) onwards lands heads.

There’s an intuition that (H1…), although extremely unlikely, is possible, and hence, we should not be certain that ~(H1…) - we should be more confident that (H1…) than we are that A & ~A. Williamson shares the intuition, but argues that with a few other plausible assumptions, it leads to contradiction.

Let “>” be read as “is more probable than” and “≥” be read as “is as least as probable as”. The following, Williamson thinks, is a “promising principle” for comparative probability:

(!) If X and Y are each incompatible with Z, then
(a) X > Y iff X v Z > Y v Z
(b) X ≥ Y iff X v Z ≥ Y v Z

But now, let X be (H1…), Y be A&~A, and Z be ~(H1) & (H2…). Since (H1…) and A&~A are both incompatible with ~(H1) & (H2…), we get:

(H1…) > A&~A iff (H1…) v (~H(1) & (H2…)) > A&~A v (~H(1) & (H2…))

(H1…) v (~H(1) & (H2…)) is equivalent to (H2…), and A&~A v (~H(1) & (H2…)) is equivalent to (~H(1) & (H2…)). Hence we get:

(H1…) > A&~A iff (H2…) > (~H(1) & (H2…))

But ~H(1) & (H2…) and (H1…) differ only on the outcome of the first toss; hence,

(H1…) > A&~A iff (H2…) > (H1…)

Now Williamson thinks that (H1…) ≥ (H2…), which means, given the equivalence above, that it’s not the case that (H1…) > A&~A. But this contradicts what intuition purportedly tells us. What is Williamson’s reason for thinking that (H1…) ≥ (H2…)? He writes:

“H(1…) and H(2…) are isomorphic events. More precisely, we can map the constituent single-toss events of (H1…) one-one onto the constituent single-toss events of H(2…) in a natural way that preserves the physical structure of the set-up just by mapping each toss to its successor. H(1…) and H(2…) are events of exactly the same qualitative type; they differ only in the inconsequential respect that H(2…) starts one second after H(1…). That H(2…) is preceded by another toss is irrelevant, given the independence of the tosses. Thus H(1…) and H(2…) should have the same probability.”

I have to admit that my mathematically naive intuition is that (H2…) > (H1…). Suppose that this is not correct, that (H1…) ≥ (H2…). Since (H1) & (H2…) ≥ (H1…), transitivity yields (H1) & (H2…) ≥ (H2…). But it seems to me that (H1) & (H2…) < (H2...). Imagine there's an evil demon who can choose to exercise his ability to tell, before we start to toss a coin infinitely, whether it'll land heads on every toss or not. You've previously been condemned to be tortured for eternity, but the evil demon offers you a way out. He gives you two tickets, one which says (H2...) and the other which says (H1). The coin has yet to be tossed, and the demon, who has thus far refrained from exercising his ability, tells you that you'll be spared from eternal torture if every ticket you hold says something true. Before the demon exercises his unique ability, he gives you the choice to throw away the ticket which says (H1) if you so desire. Should you throw it away?

It seems to me that you should. If (H1) is true, throwing the ticket away does not make it less likely that you'll be spared. If (H1) is false, then throwing it away is better than keeping it, since keeping it means that there's an additional possibility, and hence, additional danger, that you won't be spared. But this means that keeping the ticket decreases the probability that you'll be spared, i.e., (H1) & (H2...) < (H2...).

For all I know, defending the intuition that (H2...) > (H1…) might be fraught with insuperable mathematical problems. In any case, I find the intuition about as strong as the intuition that (H1…) > A&~A. If, by keeping the ticket which says (H1), the additional possibility that I won’t be spared doesn’t mean a lower probability that I’ll be spared, then it’s hard to see why the mere possibility that (H1…) means that I shouldn’t be certain that ~(H1). If it does not make any difference whether I keep or throw away the ticket saying (H1), then I’m inclined to think that a ticket saying (H1…) is worth exactly the same as one saying A&~A.

[In Part I, I raised a problem for the Lockean Thesis. In Part II, I considered, as well as rejected, a purported resolution of the problem. Here’s Part III, in which I explain why the problem arises.]

Consider, once again, the following:

(1) I’m slightly more confident that B than that A, and I believe that A is equivalent to B.

(4) I’m slightly more confident that B than that A, and I’m 95% confident that A is equivalent to B.

Contrary to what one would expect were the Lockean Thesis true, (1) seems infelicitous, whereas (4) seems fine. Where lies the source of this intuition?

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[I planned to write only two blog posts on the Lockean Thesis, but now, it seems that there’ll be a Part III. No more after that, or I might as well write a paper. In this post, I consider, and then reject an attempt to resolve the problem I raised for the Lockean Thesis in my previous post. In my next post, instead of trying to resolve the problem, as I said I would, I’ll try to locate the source of our intuitions for thinking that (4) to (6) sound better than (1) to (3).]

It seems that I can be rational in believing that Mark Twain is Samuel Clemens with less than complete confidence. But suppose I’m offered a bet that will win me a dollar if Mark Twain is Samuel Clemens, and condemn me to eternal torture if not. I’ll decline the bet. Question: in doing so, do I still believe that Mark Twain is Samuel Clemens?

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