Wednesday Cognitive Science Blogging: What Are the Odds Princeton's David Lewis Understands Probability Properly?
: Sleeping Beauty: Reply to Elga: "Researchers at the Experimental Philosophy Laboratory... Sleeping Beauty...
...On Sunday evening they will put her to sleep [P-]. On Monday they will awaken her briefly. At first they will not tell her what day it is [P], but later they will tell her that it is Monday [P+]. Then they will subject her to memory erasure. Perhaps they will again awaken her briefly on Tuesday... depend[ing] on the toss of a fair coin: if heads they will awaken her only on Monday, if tails they will awaken her on Tuesday as well.... We shall need to consider her credence functions at three different times.
Let P be her credence function just after she is awakened on Monday. Let P+ be her credence function just after she’s told that it’s Monday. Let P- be her credence function just before she’s put to sleep on Sunday.... Elga (2000) argues that P(HEADS) = 1/3.... I disagree, and argue that P(HEADS) =... 1/2.... Elga’s argument applies in the first instance to the case that it is tossed after; but he thinks, and I agree, that the answer to our question should be the same in both cases....
When Beauty awakens during the experiment, three centred epistemic possibilities are compatible with her total evidence: H1: HEADS and it’s Monday, T1: TAILS and it’s Monday, T2: TAILS and it’s Tuesday.... Beauty gains no new uncentred evidence, relevant to HEADS versus TAILS, between the time when she has credence function P- and the time when she has credence function P. The only evidence she gains is the centred evidence2 that she is presently undergoing either the Monday awakening or the Tuesday awakening: that is, (H1 or T1 or T2)....
My argument.... (L1) Only new relevant evidence, centred or uncentred, produces a change in credence; and the evidence (H1 or H2 or H3) is not relevant to HEADS versus TAILS (Premiss). (L2) P(HEADS) = 1/2 = P(TAILS). Quod erat demonstrandum....
Further consequences.... (L6) P+(HEADS) = 2/3....
At the time of P+.... Beauty knows that there will be a future toss of a fair coin. There is a well-known principle which says that credences about future chance events should equal the known chances.... [It] would seem to say also that P+(HEADS) = chance(HEADS) = 1/2.... [But] imagine that there is a prophet whose extraordinary record of success forces us to take seriously the hypothesis that he is getting news from the future by means of some sort of backward causation.... What should we do? If the prophet’s success record is good enough, I say we should take the prophet’s advice and disregard the known chances.... When Beauty is told during her Monday awakening that it’s Monday... she is getting evidence... about the future.... That’s new evidence: before she was told that it was Monday, she did not yet have it. To be sure, she is not getting this new evidence from a prophet or by way of backward causation, but neither is she getting it just by setting her credences equal to the known chances. The news is relevant to HEADS, since it raises her credence in it by 1/6.... Therefore... we cannot rely on... P+(HEADS) = chance(HEADS).... I admit that this is a novel and surprising application of the proviso, and I am most grateful to Elga for bringing it to my attention. Nevertheless I find it fairly convincing, independently of wishing to follow where my argument leads...
- Sleeping Beauty has gone to sleep on Sunday.
- Sleeping Beauty has been awakened on Monday.
- When Sleeping Beauty woke, she thought that the experiment might be further along that that it might be Tuesday
- Sleeping Beauty has just been told that it is, in fact, Monday, and that the memory-disruption part of the experiment has not yet been conducted.
- The experimenters are about to flip their fair coin.
- If the coin comes up tails, they will awaken her on Tuesday; if it comes up heads, they will let her sleep until Wednesday.
- The experimenters are about to disrupt her short-term memory so that, if the coin comes up tails and she will be awakened on Tuesday, she will not then remember waking up on Monday.
Now Lewis asks: what are Sleeping Beauty's odds--her "credence" P+--that the coin flip will come up heads?
What Sleeping Beauty knows is that:
- They are about to flip a fair coin.
- They are about to disrupt her short-term memory, so that if they do wake her up on Tuesday she will not remember waking up on Monday.
And Lewis says that Sleeping Beauty's odds that the coin flip will be heads are not 1/2 but 2/3--that somehow the fact that if the coin falls tails and she wakes up tomorrow she will be confused exercises a mysterious force on the coin that raises the odds that the coin will fall heads by 1/6 to 2/3.
What if Lewis's argument? I read it, and all I see is incoherent word-salad:
There is a well-known principle which says that credences about future chance events should equal the known chances.... I reply that the principle requires a proviso.... Imagine that there is a prophet whose extraordinary record of success forces us to take seriously the hypothesis that he is getting news from the future by means of some sort of backward causation. Seldom does the prophet tell us outright what will happen, but often he advises us what our credences about the outcome should be, and sometimes his advice disagrees with what we would get by setting our credences equal to the known chances. What should we do? If the prophet’s success record is good enough, I say we should take the prophet’s advice and disregard the known chances.
Now when Beauty is told during her Monday awakening that it’s Monday, or equivalently not-T2, she is getting evidence--centred evidence--about the future: namely that she is not now in it. That’s new evidence: before she was told that it was Monday, she did not yet have it. To be sure, she is not getting this new evidence from a prophet or by way of backward causation, but neither is she getting it just by setting her credences equal to the known chances. The news is relevant to HEADS, since it raises her credence in it by 1/6; see my (L7). Elga agrees; see his (E6). Therefore the proviso applies, and we cannot rely on it that P+(HEADS) = chance(HEADS) and P+(TAILS) = chance(TAILS).
Can anybody present me with a possible world in which Lewis's argument that P+(HEADS)=2/3 is not incoherent, and wrong?
Suppose that after the experimenters ask Sleeping Beauty her P+(HEADS), the experimenters discover that the memory-disruption does not work. So they cancel the experiment. But, they say, we will flip the same coin, and ask her what her P++(HEADS) is. Does Lewis still say her P++(HEADS)=2/3? If so, why? If not, why is P++(HEADS)≠P+(HEADS)?
