In "Gandalf's solution to the Newcomb problem" (2013), Ralph Wedgwood proposes a new form of decision theory, Benchmark Theory, that is supposed to combine the good parts of Causal and Evidential Decision Theory.
There's an exciting new theory in cognitive science. The theory began as an account of message-passing in the visual cortex, but it quickly expanded into a unified explanation of perception, action, attention, learning, homeostasis, and the very possibility of life. In its most general and ambitious form, the theory was mainly developed by Karl Friston -- see e.g. Friston 2006, Friston and Stephan 2007, Friston 2009, Friston 2010, or the Wikipedia page on the free-energy principle.
Suppose I say (*), with respect to a particular gambling occasion.
(*) A gambler lost some of her savings. Another lost all of hers.
There is an implicature here that the first gambler, unlike the second, didn't lose all her savings. How does this implicature arise?
Imagine the universe has a centre that regularly produces new stars which then drift away at a constant speed. This has been going on forever, so there are infinitely many stars. We can label them by age, or equivalently by their distance from the centre: star 1 is the youngest, then comes star 2, then star 3, and so on, without end. The stars in turn produce planets at regular intervals. So the older a star, the more planets surround it. Today, something happened to one (and only one) of the planets. Let's say it exploded. Given all this, what is your credence that the unfortunate planet belonged to the first 100 stars? What about the second 100? It would be odd to think that the event is more likely to have happened at one of the first 100 stars than at one of the next 100, since the latter have far more planets. Similarly if we compare the first 1000 stars with the next 1000, or the first million with the next million, and so on. But there is no countably additive (real-valued) probability measure that satisfies this constraint.
Two initially plausible claims:
- Sometimes, a possible chance function conditionalized on a proposition A yields another possible chance function.
- Any rational prior credence function Cr conditional on the hypothesis Ch=f that f is the (actual, present) chance function should coincide with f; i.e., Cr(A / Ch=f) = f(A) for all A (provided that Cr(Ch=f)>0).
Claim 1 is a supported by the popular idea that chances evolve by conditionalizing on history, so that the chance at time t2 equals the chance at t1 conditional on the history of events between t1 and t2. Claim 2 is a weak form of the Principal Principle and often taken to be a defining feature of chance.
How much can you say about the world in purely logical terms? In first-order logic with identity, one can construct formulas like '(Ex)(Ey)~(x=y)'. But arguably, this doesn't yet mean anything. As we learned in intro logic, formulas of first-order logic have no fixed interpretation; they mean something only once we provide a domain of quantification and an assignment of values to predicate and function symbols. As it happens, '(Ex)(Ey)~(x=y)' doesn't contain any non-logical predicate and function symbols, so to make it mean anything we just need to specify a domain of quantification. For example, if the domain is the class of Western black rhinos, then the formula says that there are at least two Western black rhinos.
You can't predict the stock market by looking at tea leaves. If an episode of looking at tea leaves makes you believe that the stock market will soon collapse, then -- assuming your previous beliefs did not support the collapse hypothesis, nor the hypothesis that tea leaves predict the stock market -- your new belief is unjustified and irrational. So there are epistemic norms for how one's opinions may change through perceptual experience.
Suppose a rational agent makes an observation, which changes the subjective probability she assigns to a hypothesis H. In this case, the new probability of H is usually sensitive to both the observation and the prior probability. Can we factor our the prior probability to get a measure of how the experience bears on the probability of H, independently of the prior probability?
Dilip Ninan has also argued on a number of occasions that attitude contents cannot in general be modelled by sets of qualitative centred worlds; see especially his "Counterfactual attitudes and multi-centered worlds" (2012). The argument is based on an alleged problem for the centred-worlds account applied to what he calls "counterfactual attitudes", the prime example being imagination.
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