Degrees of Self-locating Belief, Essential Indexicals, and Sleeping Beauty

Transcription

1 Degrees of Self-locating Belief, Essential Indexicals, and Sleeping Beauty Namjoong Kim Sunkyunkwan University

2 Introduction In this talk, I will explain the concept of degree of belief (or subjective probability), explain why we need centered propositions as the contents of our belief, present my theory about how to update our degrees of belief, and talk about the recent debate on the relation between the Sleeping Beauty problem and the Many World Interpretation of QM.

3 Degrees of Belief Belief is not really an all or nothing aairs; it admits of degrees. You might be reasonably sure that the president was guilty without being absolutely certain. This raises a problem: How do we measure one's degree of belief in a proposition p? Frank Plumpton Ramsey thought that we can use a gamble in order to measure one's degree of belief in a statement p. Think about this betting: (Bp) If p, then you receive $1; otherwise, you get nothing. According to your opinion, the expected value of Bp is $1 Pr (p) + $0 Pr ( p) (which is justpr (p)) where Pr ( ) is your degree-of-belief function. Now, if you are rational, the expected value of Bp will be the same as the price at which you will buy or sell Bp. Thus, your rational degree of belief in p is the price at which you will buy or sell Bp.

4 Strict Conditionalization If so, how are you supposed to update your degrees of belief? The most popular answer is by strict conditionalization: For any proposition p Pr n+1 (p) = Pr n (p/e) where Pr n+1 ( ) and Pr n ( ) are your degree-of-belief functions at t n+1 and t n and e is the totality of what you learn with full certainty. (Here is the denition of conditional probability: for any probability function Pr( ), for any propositions p and q, Pr (p/q) = Df. Pr (pq) /Pr (q).) For example, suppose that you believe to the degree of 1 that (h) a certain coin lands heads 6 conditional on the assumption that (t) it is tossed. In this case, Pr new (h) = Pr old (h/t) = 1 6.

5 Jerey Conditionalization Richard Jerey pointed out that we do not always learn something with full certainty. Suppose that you see a bird that looks like a magpie rather than a raven, but you are not completely sure. Indeed, you are eighty percent sure that (m) the bird in front of you is a magpie but believe to the degree of twenty percent that (r) it is a raven. Consider the possibility that (t) the wedding ring, which disappeared last night, was actually stolen by an animal. At the previous moment, you assigned the subjective probability of.75 to t conditional on m but assigned only that of.25 to t conditional on r. In this situation, if you are rational, to what degree will you believe t? According to Jerey, Pr new (t) = Pr old (t/m) Pr new (m) + Pr old (t/r) Pr new (r) = =.65.

6 Jerey Conditionalization (continued) In general, your degree of belief in p is updated in accordance with this rule: Pr n+1 (p) = Pr n (p/e i ) Pr n+1 (e i ), i I where {e i } i I are the propositions whose probabilities are directly determined by your experience. (Here, {e i } i I is supposed to be a partition.)

7 Propositions So far, we have discussed propositions as bearers of probabilities. But what kind of entities are they? In philosophy, they play several important roles: 1. Propositions are bearers of truth-value. For example, the proposition expressed by Namjoong is a philosopher is true, but that expressed by Namjoong is a girl is false. 2. Propositions are objects of belief. For instance, I believe that Einstein won his Nobel Prize thanks to his brilliant work of photoelectronic eect. Thus, although we do not know the innards of propositions, we know what theoretical roles they play. Also, many philosophers, including Gottlob Frege, thought that propositions have xed truth-values; in other words, they neither have dierent truth-values for dierent objects nor change their truth-values as time passes.

8 I and Today However, some sentences have dierent truth-values regarding dierent objects in dierent contexts. For instance, I am a professional philosopher is true of Namjoong Kim, but the same sentence is false of President Park. Likewise, Today is a rainy day might be true on Monday but false on Tuesday. If so, shouldn't the propositions expressed by those sentences also have variable truth-values?

9 I and Today (continued) Not necessarily. Perhaps, I am a professional philosopher, when uttered by me, expresses the proposition that Namjoong is a philosopher, while, when uttered by the president, it expresses the proposition that President Park is a philosopher. Of course, the former is true but the latter false. Thus, a sentence can have dierent truth-values in dierent contexts simply by expressing dierent propositions.

10 I and Today (continued) Similarly, Today is a rainy day, when uttered on Monday, expresses the proposition that Monday is a rainy day, while, when uttered on Tuesday, it expresses the proposition that Tuesday is a rainy day. So, while the sentence may have dierent truth-values by expressing dierent propositions, the propositions perhaps have xed truth-values.

11 I and Today (continued) In both cases, the main idea is that although propositions have xed truth-values (regardless of contexts), sentences may have dierent truth-values by expressing dierent propositions. However, many philosophers think that this idea does not work in all cases.

12 Essential Indexicals Suppose that Superman lost his memory. Luckily, all his supernatural powers are intact, but he does not remember who he is and so he does not know that he has those powers. Living a normal life, he reads newspapers and learns many things about this cool guy, Superman. So, if someone asks him Can Superman y? then he will say Yes! Superman can y. Still, if someone asks him Can you y? then he will say No! I cannot y.

13 Essential Indexicals (continued) A few months later, his girlfriend nds him and tells him the truth: You are Superman. You can y. He believes her and says I can y. In this case, what does Superman learn from his girlfriend? It cannot be that Superman can y, because he already knew it. So something else is expressed by his utterance of I can y.

14 Centered Proposition According to David Lewis, that something else is a centered proposition. To some extent, they are similar to propositions of the traditional type. 1. Centered propositions are bearers of truth-value. For example, the centered proposition expressed by I can y is true of Superman, but the same centered proposition is false of me. 2. Centered propositions are objects of belief. For instance, before his girlfriend tells him the truth, Superman did not believe the centered proposition expressed by I can y but did believe the proposition expressed by Superman can y. Of course, not everything is the same. Especially, The same centered proposition may have dierent truth-values regarding dierent objects at dierent times.

15 Sleeping Beauty Problem Now consider this example: On Sunday night, Beauty, a perfectly rational agent, knows that the following things will happen during the next three days: Next moment, a group of experimenters put her to sleep and toss a fair coin. Case 1: (h) The coin lands heads. Then, they awaken her on Monday, and then she is put to sleep again, and left asleep until the end of the experiment. Case 2: (t) The coin lands tails. Then, they awaken her twice, the rst time on Monday and the second on Tuesday. Between the two awakenings, they inject a drug that erases the memory of Monday. On Wednesday, they wake her up and the experiment ends. Let t s be Sunday night, t m Monday morning, and t t be Tuesday morning. At t m, she learns a centered proposition expressed by (w) Beauty wakes up with the memory up to t s. Here comes a question: what is Beauty's rational degree of belief at t m in h?

16 Sleeping Beauty Problem The Halfer argument: At t m, she experiences nothing unexpected. At t s, she believed h to the degree of 1. So her degree of belief at 2 t m in h should remain to be 1. 2 The Thirder argument: From her point of view at t m, there are three open possibilities. h&mon The coin has landed heads and today is Monday. t&mon The coin has landed tails and today is Monday. t&tue The coin has landed tails and today is Tuesday. On one hand, Pr m (h/mon) = Pr m (t/mon) because, if she were told that it is Monday, she would assign the same degree of belief to h and t. On the other hand, Pr m (mon/t) = Pr m (tue/t) because, if she were immediately told that the coin has landed tails, then she would assign the same degree of belief to mon and tue. As a result, Pr m (h&mon) = Pr m (t&mon) = Pr m (t&tue) = 1 3. Note that h&mon is the only case where the coin lands heads. Hence, Pr m (h) = 1 3.

17 Shifted Jerey Conditionalization In my Sleeping Beauty and Shifted Jerey Conditionalization, I suggested the following rule for updating degrees of belief in centered propositions: For any centered proposition p, Pr n+1 (p) = i,j Pr n (p is true at t j /e i is true at t j ) Pr n+1 (e i & [now = t j ]) The core idea: to make a judgment of how probable p is, you need to take your temporal location into consideration. This is because, depending upon what time it is, your observations will have dierent degrees of relevance to p. For example, consider the centered proposition expressed by (r) It rains now and that expressed by (p) There is a precipitation now. If it is Summer, p will be positively relevant to r but, if it is Winter, p will be negatively relevant to r. (See my paper for a more detailed explanation of SJC.)

18 Applying SJC to Sleeping Beauty We apply SJC to the Sleeping Beauty problem: Pr m (h) = Pr s (h is true at t m /w is true at t m ) Pr m (w& [now = t m ]) +Pr s (h is true at t t /w is true at t t ) Pr m (w& [now = t t ]). In this instance of SJC, h is true at t m means that (it is true at t m that) the coin lands heads, w is true at t m means that Beauty wakes up at t m with the memory up to t s, and w& [now = t m ] means that Beauty wakes up now with the memory up to t s and it is t m now. Similarly for h is true at t t, w is true at t t and w& [now = t t ]. (Replace t m with t t.)

19 Applying SJC to Sleeping Beauty (continued) We apply SJC to the Sleeping Beauty problem: Pr m (h) = Pr s (h is true at t m /w is true at t m ) Pr m (w& [now = t m ]) +Pr s (h is true at t t /w is true at t t ) Pr m (w& [now = t t ]). On one hand, Pr s (h is true at t m /w is true at t m ) = Pr s (h) = 1 2. This is because, at t s, Beauty fully expected that she would wake up at t m with the memory up to t s. On the other hand, Pr s (h is true at t t /w is true at t t ) = 0.0. This is because, if she is waking up on Tuesday, then the coin has already landed tails. Since she cannot be sure at t m that it is Monday, Pr m (h) = 1 2 Pr m (w& [now = t m ]) < 1 2 This means that the Halfer view is wrong.

20 Duplicate Beauty Problem Next think about this example: On Sunday night, Beauty, a perfectly rational agent, knows that the following events will happen during the next two days: Next moment, a group of experimenters put her to sleep and toss a fair coin. Case 1: (h) The coin lands heads. Then, they awaken her on Monday, and then she is put to sleep again, and left asleep until the end of the experiment. Case 2: (t) The coin lands tails. Then, they duplicate Beauty, awaken Beauty and Duplicate in separate rooms, and then put them into sleep again. On Tuesday, they wake her (or them) up and the experiment ends. Let t s be Sunday night and t m Monday morning. Let b be Beauty and d be Duplicate. At t m, she learns a centered proposition expressed by (w ) At t m, I wake up with the memory up to t s. Here comes a question: what is Beauty's rational degree of belief at t m in h?

21 Applying SJC to Duplicate Beauty SJC is not designed for the cases in which the agent does not know who she is. However, it is not dicult to modify SJC for such cases, and it will have the following instance: Pr m (h) = Pr s (h is true of b/w is true of b) Pr m (w & [me = b]) +Pr s (h is true of d/w is true of d) Pr m (w & [me = d]). In this instance, h is true of b means that (it is true of b that) the coin lands tails, w is true of b means that Beauty wakes up at t m with the memory up to t s, and w & [me = b] means I wake up now with the memory up to t s and I am Beauty. Analogously for h is true of d, w is true of d, and w & [me = d]. (Replace Beauty with Duplicate.)

22 Applying SJC to Duplicate Beauty (continued) Pr m (h) = Pr s (h is true of b/w is true of b) Pr m (w & [me = b]) +Pr s (h is true of d/w is true of d) Pr m (w & [me = d]). On one hand, Pr s (h is true of b/w is true of b) = 1. This is 2 because, at t s, she fully expected that Beauty herself would wake up at t m. On the other hand, Pr s (h is true of d/w is true of d) = 0.0. This is because, if Duplicate wakes up at t m with Beauty's memory up to t s, then the coin has already landed tails. At t m, Beauty cannot be sure that she is Beauty. Thus, Pr m (h) = 1 2 Pr m (w& [me = d]) < 1 2. On Monday, Beauty assigns a smaller probability to the coin's landing heads than 1 2.

23 Superposition According to quantum mechanics, a physical object, especially a microparticle such as electrons, protons, and so on, may be in a multiple number of seemingly incompatible physical states at the same time. Think about the famous double slit experiment, in which a particle (e.g., a photon) passes through slits A and B. Intuitively, passing through A is incompatible with passing through B. However, you can observe an interference pattern on the screen, which means that the wave having passed through slit A interacted with that having passed through slit B. Hence, we can say that passing through slit A was superposed with passing through slit B.

24 Decoherence An even stranger phenomenon occurs when we install detectors near the slits. When the locations of the particles are recorded (measured), the interference patter disappears and particles behave like, well, particles rather than waves. This phenomenon is called decoherence. Which consequence results from decoherence is interministic, that is, a truly probabilistic matter.

25 Many World Interpretation of QM So how do we interpret these weird phenomena? One interpretation is that passing through A and passing through B are not really incompatible, but they can occur at the same time. According to this view, we can regard the universe as having a tree-like structure, each branch consists of the states to which the given physical system collapses as a result of decoherence. (The diagram below illustrates the branches resulting from measuring a particle's spin.)

26 Quantum Beauty Problem Consider Beauty's twin Qeauty, who participates in an experiment similar to Beauty's. The experimenters put Qeauty to sleep on Sunday and ip a coin. If the result of the ip is tails, they do an x-spin measurement on an electron in a superposition of x-spin up and x-spin down (with probabilities of one half corresponding to each outcome per the Born rule). If the result of the measurement is x-spin up, they wake Qeauty on Monday then put her back to sleep. If the result of the measurement is x-spin down, they wake Qeauty on Tuesday then put her back to sleep. Finally, if the coin lands on heads, the experimenters wake Qeauty on Monday then put her back to sleep. (continued to the next slide)

27 Quantum Beauty Problem (continued) Assuming that Qeauty is an Everettian, her case mirrors her sister's in many ways; Qeauty in a world where tails is ipped is similar to Beauty in a world where tails is ipped by Lewis's lights: Qeauty has two successors, both of whom are psychologically continuous with pre-branching Qeauty but not psychologically continuous with each other. Likewise, Beauty has two post-sunday successors, both of whom are psychologically continuous with Sunday Beauty but not psychologically continuous with each other because of the drug. Also, as in the Beauty case, Qeauty has self-locating uncertainty: that is, she may be on one of many branches, yet when she wakes up, she does not know which branch she is on. Given the similarities between the Beauty and Qeauty cases, Lewis's claim that the same account of probability should govern both cases seems reasonable. (Peterson Qeauty and the Books 2011, p. 369)

28 Relation between Beauty and Qeauty

29 Lewis's Argument P1. If MWI is correct, then [Pr Beauty m (h) = 1 2 i Pr Qeauty m (h) = 1 2 ]. P2. According to Born equation, Pr Qeauty m (h) = 1 2. C. Therefore, if MWI is correct, then Pr Beauty m (h) = 1 2 ; that is, MWI implies the Halfer view. P1 is justied by the structural similarity between the Beauty and Qeauty cases. P2 is uncontroversial. Hence, if you want to reject the conclusion, then you need to say why the mentioned similarity is not enough to estabilish P1.

30 Applying SJC to Beauty and Qeauty Remember that w was the centered proposition expressed by At t m, I wake up with the memory up to t s. Let w be the centered proposition expressed by At t m or t t, I wake up with the memory up to t s.' Pr b m (h) = Pr b s (h is true of b/w is true of b) Pr b m (w & [me = b]) +Pr b s (h is true of d/w is true of d) Pr b m (w & [me = d]). Pr q m (h) = Pr q s (h is true of q/w is true of q) Pr q m (w & [me = q]). Remember that Prm b (h) < 1. No duplication occurs as a result of 2 tossing the coin or the x-spin measurement. Consequently, SJC has a simpler instance for the Qeauty case. Since Qeauty is sure at t s that she will wake up at t m and that the coin is fair, Pr q s (h is true of q/w is true of q) = Pr q s (h) = 1. Since Qeauty is 2 sure at t m that she is Qeauty and she is waking up, Pr q m (h) = 1. 2 Hence, the Beauty and Qeauty cases are substantially dierent.

31 Conclusion So far, I explained the concepts of degree of belief and centered proposition, presented a general theory about how to change your degrees of belief on receiving evidence with centered contents, criticized the Halfer by making use of the mentioned theory, and, nally, argued that there is an important disimilarity between the original and quantum versions of the Sleeping Beauty problem.