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.
Inga got a postdoc in Hamburg, so it looks like we'll be moving back to Germany at the end of the year. It's sad to leave the ANU, but we'll probably return here for at least a few months in 2014. (If only because I don't have another job yet.)
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.
Such norms are easily accounted for in the traditional Bayesian
picture where each perceptual experience is associated with an
evidence proposition E on which any rational agent should condition
when they have the experience. But what if perceptual experiences
don't confer absolute certainty on anything? Jeffrey pointed out that
if there is a partition of propositions { E_i } = E_1,...,E_n such
that (1) an experience changes their probabilities to some values {
p_i } = p_1,...,p_n, and (2) the experience does not affect the
probabilities conditional on any member of the partition, then the new
probability assigned to any proposition A is the weighted average of
the old probability conditional on the members of the partition,
weighted by the new probability of that partition. This rule is often
called "Jeffrey conditioning" and sometimes "generalised
conditioning", but unlike standard conditioning it isn't a dynamical
rule at all: it is a simple consequence of the probability
calculus. To get genuine epistemic norms on the dynamics of belief
through perceptual experience, Jeffrey's rule must be supplemented
with a story about how a given experience, perhaps together with an
agent's previous belief state, may fix the partition { E_i } and
values { p_i } that determine a Jeffrey update. This is the "input
problem" for Jeffrey conditioning.
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?
A common answer, going back to Alan Turing and I.J.Good, is to use
Bayes factors. The Bayes factor B(H) for H is the ratio
(P'(H)/P'(not-H))/(P(H)/P(not-H)) of new odds on H to old odds. Thus
the new odds on H are the old odds multiplied by the Bayes factor. For
example, if the prior credence in H was 0.25 and the posterior is 0.5,
then the odds on H changed from 1:3 to 1:1, and so the Bayes factor of
the update is 3. The same Bayes factor would characterise an update
from probability 0.01 to about 0.03 (odds 1:99 to 1:33) or from 0.9 to
about 0.96 (odds 9:1 to 27:1).
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.
Since the problem concerns the analysis of attitudes de re,
we first have to briefly review what the centred-worlds account might
say about this. Consider a de re belief report "x believes that y is
F". Whether this is true depends on what x believes about y, but if
belief contents are qualitative, we cannot simply check whether y is F
in x's belief worlds. We first have to locate y in these
qualitative scenarios. A standard idea, going back to Quine, Kaplan
and Lewis, is that the belief report is true iff there is some
"acquaintance relation" Q such that (i) x is Q-related uniquely to y
and (ii) in x's belief worlds, the individual at the centre is
Q-related to an individual that is F. For example, if Ralph sees
Ortcutt sneaking around the waterfront, and believes that the guy
sneaking around the waterfront is a spy, then Ralph believes de re of
Ortcutt that he is a spy.
If we want to model rational degrees of belief as probabilities,
the objects of belief should form a Boolean algebra. Let's call the
elements of this algebra propositions and its atoms (or
ultrafilters) worlds. Every proposition can be represented as a
set of worlds. But what are these worlds? For many applications, they
can't be qualitative possibilities about the universe as a whole, since
this would not allow us to model de se beliefs. A popular
response is to identify the worlds with triples of a possible universe,
a time and an individual. I prefer to say that they are maximally
specific properties, or ways a thing might be. David Chalmers (in
discussion, and in various papers, e.g. here and there) objects that
these accounts are not fine-grained enough, as revealed by David
Austin's "two tubes" scenario. Let's see.
Luc Bovens and Wlodek Rabinowicz (2010
and 2011)
present the following puzzle:
Three people are each given a hat to put on in the
dark. The hats' colours, either black or white, has been decided by
three independent tosses of a fair coin. Then the light goes on and
everyone can see the hats of the two others, but not their own. All of
this is common knowledge in the group.
Let's call the three players X, Y and Z. There are eight possible
distributions of hat colours, each with probability 1/8:
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