Take the usual language of first-order logic from introductory
textbooks, without identity and function symbols. The vast majority of
sentences in this language are satisfied in models with very few
individuals. You even have to make an effort to come up with a sentence
that requires three or four individuals. The task is harder if you
want to come up with a fairly short sentence. So I wonder, for any given number n, what is the shortest
sentences that requires n individuals?
It is widely agreed that conditionalization is not an adequate norm
for the dynamics of self-locating beliefs. There is no agreement on
what the right norms should look like. Many hold that there are no
dynamic norms on self-locating beliefs at all. On that view, an
agent's self-locating beliefs at any time are determined on the basis
of the agent's evidence at that time, irrespective of the earlier
self-locating belief. I want to talk about an alternative approach
that assumes a non-trivial dynamics for self-locating beliefs. The
rough idea is that as time goes by, a belief that it is Sunday should
somehow turn into a belief that it is Monday.
Let's assume that propositional attitudes are not metaphysically
fundamental: if someone has such-and-such beliefs and desires, that is
always due to other, more basic, and ultimately non-intentional
facts. In terms of supervenience: once all non-intentional facts are
settled, all intentional facts are settled as well.
Then how are propositional attitudes grounded in non-intentional
facts? A promising approach is to identify a characteristic
"functional role" of propositional attitudes and then explain facts
about propositional attitudes in terms of facts about the realization
of that role. (We could also identify the attitude with the realizer,
or with the higher-order property of heaving a realizer, but that's
optional.)
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.