Let's look at the third type of case in which credences can come apart from known chances. Consider the following variation of the Sleeping Beauty problem (a.k.a. "The Absentminded
Driver"):
Before Sleeping Beauty awakens on Monday, a coin is
tossed. If the coin lands tails, Beauty's memories of Monday will be
erased the following night, and the coin will be tossed again on
Tuesday. If the Monday toss lands heads, no memory erasure or further
tosses take place. Beauty is aware of all these facts.
When Beauty awakens on Monday morning and learns that today's toss
has landed tails (alternatively: that the Monday toss has landed
tails), how should that affect her credence in the hypothesis that the
coin is fair?
Next, undermining. Suppose we are testing a model H according to
which the probability that a certain type of coin toss results in
heads is 1/2. On some accounts of physical probability, including
frequency accounts and "best system" accounts, the truth of H is
incompatible with the hypothesis that all tosses of the relevant type
in fact result in heads. So we get a counterexample to simple
formulations of the Principal Principle: on the assumption that H is
true, we know that the outcomes can't be all-heads, even though H
assigns positive probability to all-heads. In such a case, we say that
all-heads is undermining for H.
Suppose we are testing statistical models of some physical process
-- a certain type of coin toss, say. One of the models in question
holds that the probability of heads on each toss is 1/2; another holds
that the probability is 1/4. We set up a long run of trials and
observe about 50 percent heads. One would hope that this confirms the
model according to which the probability of heads is 1/2 over the
alternative.
(Subjective) Bayesian confirmation theory says that some evidence E
supports some hypothesis H for some agent to the extent that the
agent's rational credence C in the hypothesis is increased by the
evidence, so that C(H/E) > C(H). We can now verify that observation of
500 heads strongly confirms that the coin is fair, as follows.
Most programming languages have conditional operators that combine a
(boolean) condition and two singular terms into a singular term. For
example, in Python the expression
'hi' if 2 < 7 else 'hello'
is a singular term whose value is the string 'hi' (because 2 < 7). In
general, the expression
x if p else y
denotes x in case p is true and otherwise y. So, for example,
Time-slice epistemology is the idea that epistemic norms are
history-independent: whether an agent at a time satisfies an epistemic
norm is always determined by the agent's state at that time,
irrespective of the agent's earlier states.
One motivation for time-slice epistemology is a kind of
internalism, the intuition that agents should not be epistemically
constrained by things that are not "accessible" at the relevant
time. Plausibly, an agent's earlier beliefs are not always accessible
in the relevant sense. If yesterday you learned that yew berries are
poisonous but since then forgot that piece of information, it seems
odd to demand that your current beliefs and actions should
nevertheless be constrained by the lost information.
Fred has bought a duplication machine at a discount from a series
in which 50 percent of all machines are broken. If Fred's machine
works, it will turn Fred into two identical copies of himself, one
emerging on the left, the other on the right. If Fred's machine is
broken, he will emerge unchanged and unduplicated either on the left
or on the right, but he can't predict where. Fred enters his machine,
briefly loses consciousness and then finds himself emerge on the
left. In fact, his machine is broken and no duplication event has
occurred, but Fred's experiences do not reveal this to him.
An evil scientist might have built a brain in vat that has all the
experiences you currently have. On the basis of your experiences, you
cannot rule out being that brain in a vat. But you can rule out
being that scientist. In fact, being that scientist is
not a skeptical scenario at all. For example, if the scientist in question
suspects that she is a scientist building a brain in a vat, then that
would not constitute a skeptical attitude.
Decision theoretic representation theorems show that one can read
off an agent's probability and utility functions from their
preferences, provided the latter satisfy certain minimal rationality
constraints. More substantive rationality constraints should therefore
translate into further constraints on preference. What do these
constraints look like?
Here are a few steps towards an answer for one particular
constraint: a simple form of the Principal Principle. The Principle
states that if cr is a rational credence function and ch=p is the
hypothesis that p is the chance function, then for any E in the domain
of p,
In On the Plurality or Worlds, Lewis argues that any account
of what possible worlds are should explain why possible worlds
represent what they represent. I am never quite sure what to make of
this point. On the one hand, I have sympathy for the response that
possible worlds are ways things might be; they are not things
that somehow need to encode or represent how things might be. On the
other hand, I can (dimly) see Lewis's point: if we have in our
ontology an entity called 'the possibility that there are talking
donkeys', surely the entity must have certain features that make it
deserve that name. In other words, there should be an answer to the
question why this particular entity X, rather than that other entity
Y, is the possibility that there are talking donkeys.
Noam Chomsky's New Horizons in the Study of Language and Mind
contains a famous passage about London.
Referring to London, we can be talking about a location or
area, people who sometimes live there, the air above it (but not too
high), buildings, institutions, etc., in various combinations (as in
'London is so unhappy, ugly, and polluted that it should be destroyed
and rebuilt 100 miles away', still being the same city). Such terms as
'London' are used to talk about the actual world, but there neither
are nor are believed to be things-in-the-world with the properties of
the intricate modes of reference that a city name
encapsulates. (p.37)
I don't know what Chomsky is trying to say here, but there is
something in the vicinity of his remark that strikes me as true and
important. The point is that the reference of 'London' is a complex
and subtle matter that is completely obscured when we say that
'London' refers to London.