Belief Update across Fission
When an agent undergoes fission, how should the beliefs of the fission results relate to the pre-fission beliefs? This question is important for the Everett interpretation of quantum mechanics, but it is of independent philosophical interest. Among other things, fission scenarios demonstrate that ‘self-locating’ information can affect the probability of uncentred propositions even if an agent has no essentially self-locating uncertainty. I present a general update rule for centred beliefs that gives sensible verdicts in cases of fission, without relying on controversialmetaphysical or linguistic assumptions. The rule is supported by the same considerations that support standard conditioning in the traditional framework of uncentred propositions. 1 The Problem 2 Conditioning and Self-location 3 Shifted Conditioning 4 Transition Probabilities 5 Ockhamism 6 Diachronic Rationality 7 Consequences and Conclusions 1 The Problem
Fred’s home planet, Sunday, is surrounded by two moons, Monday and
Tuesday. Tonight, while Fred is asleep, his body will be scanned and destroyed; then a signal will be sent to both Monday and Tuesday where he will be recreated from local matter. A lot of ink has been used on the question of how to describe scenarios like this. Can people survive teleportation?Which of the persons awakening on the two moons, if any, is identical to the person going to sleep on Sunday? I want to look at a different question: what should
Fred’s successors believe when they awaken on Monday and on Tuesday?
More precisely, how should their beliefs relate to Fred’s beliefs before he went to sleep on Sunday?
Brit. J. Phil. Sci. 66 (2015), 659–682 The Author 2014. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved.
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The two questions are independent. For the present topic, it does not matter whether the two ‘successors’ are identical to Fred, either temporally or absolutely. If you think Fred would not survive the double teleportation so that two new persons come into existence on Monday and Tuesday, it still makes sense to ask how the beliefs of these persons should relate to the beliefs of
Fred. Imagine you are designing intelligent amoebae that regularly undergo fission. What update process would you implement for the amoebae’s beliefs so as to make optimal use of the previously collected information?
The case of Fred gets more interesting if he doesn’t know what is going to happen. Suppose Fred learns that a fair coin will be tossed while he is asleep; if it lands heads, the signal to Tuesday will be cut so that he only gets teleported to Monday. In fact, the coin lands tails and the signal isn’t cut, but Fred doesn’t know this. Now how confident should his successors be that they are on Monday? What should they believe about the outcome of the coin toss? What should Fred’s Monday successor believe once he learns that he is on Monday?
Why care about this far-fetched situation? There are several reasons. I will argue that fission cases illustrate an important fact about the relevance of selflocating evidence and thereby cast doubt on a popular approach to the dynamics of rational belief. They also shed new light on the connection between objective chance and rational credence, and on the possibility of rational disagreement among agents with the same evidence and priors. Fred’s predicament also bares an obvious resemblance to the Sleeping Beauty problem (Elga ; Lewis ), which has caused some concern among philosophers who want to give different answers to the two problems (see, for example,
Lewis [2007a]). The model I will present can alleviate these worries.
Finally, Fred’s situation is not as far-fetched as it may at first appear.
According to the Everett interpretation of quantum mechanics, what are commonly regarded as chance events are really branching events in which all possible outcomes determinately occur, although on different branches of the universe. Thus, if a particle is in a superposition of two states, M and T, and you measure the relevant state, one of your successors will find the detector indicating M, the other T. If you give intermediate credence to the
Everett interpretation, your beliefs are divided between a branching hypothesis and a non-branching hypothesis, just like Fred’s. 2 Conditioning and Self-location
When Fred’s successor on Monday wonders whether he is on Monday or on
Tuesday, what he lacks is not objective information about the universe, but self-locating information about himself. He knows that the universe contains a
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Monday successor and a Tuesday successor. What he doesn’t know is whether he himself is the former or the latter.
I will model this kind of ignorance by assuming that degrees of belief attach to centred propositions whose truth value can vary between different locations within a world. A useful heuristic, due to Lewis (), is to identify centred propositions with properties: Fred’s successors give some degree of belief to ‘being on Monday’ and some to ‘being on Tuesday’. Suitably regimented, the relevant space of properties forms a Boolean algebra, closed under conjunction, disjunction, and negation. To keep distracting technicalities at bay, I will pretend for most of this article that this algebra is finite and hence isomorphic to the full powerset algebra on its atoms. These atoms are known as ‘centred (possible) worlds’. So a centred proposition is a set of centred worlds.
Intuitively, each centred world represents a maximally specific way a thing might be.