There has been a lively debate in recent years about the
relationship between graded belief and ungraded belief. The debate
presupposes something we should regard with suspicion: that there is
such a thing as ungraded belief.
Compare earthquakes. I'm not an expert on earthquakes, but I know
that they vary in strength. How exactly to measure an earthquake's
strength is to some extent a matter of convention: we could have used
a non-logarithmic scale; we could have counted duration as an aspect
of strength, and so on. So when we say that an earthquake has
magnitude 6.4, we characterize a central aspect of an earthquake's
strength by locating it on a conventional scale.
Philosophers (and linguists) often appeal to judgments about the
validity of general principles or arguments. For example, they judge
that if C entails D, then 'if A then C' entails 'if A then D'; that
'it is not the case that it will be that P' is equivalent to 'it will
be the case that not P'; that the principles of S5 are valid for
metaphysical modality; that 'there could have been some person x such
that actually x sits and actually x doesn't sit' is an unsatisfiable contradiction; and so on. In my view, such judgments
are almost worthless: they carry very little evidential weight.
The following principles have something in common.
Conditional Coordination Principle.
A rational person's credence in a conditional A->B should equal the
ratio of her credence in the corresponding propositions B and A&B;
that is, Cr(A->B) = Cr(B/A) = Cr(B)/Cr(A&B).
Normative Coordination Principle.
On the supposition that A is what should be done, a rational agent
should be motivated to do A; that is, very roughly, Des(A/Ought(A))
> 0.5.
Probability Coordination Principle.
On the supposition that the chance of A is x, a rational agent
should assign credence x to A; that is, roughly, Cr(A/Ch(A)=x) = x.
Nomic Coordination Principle.
On the supposition that it is a law of nature that A, a rational agent
should assign credence 1 to A; that is, Cr(A/L(A)) = 1.
All these principles claim that an agent's attitudes towards a certain
kind of proposition rationally constrain their attitudes towards other
propositions.
Humeans about laws of nature hold that the laws are nothing over
and above the history of occurrent events in the world. Many
anti-Humeans, by contrast, hold that the laws somehow "produce" or
"govern" the occurrent events and thus must be metaphysically prior to
those events. On this picture, the regularities we find in the world
are explained by underlying facts about laws. A common argument
against Humeanism is that Humeans can't account for the explanatory
role of laws: if laws are just regularities, then then laws can't
really explain the regularities — so the charge —
since nothing can explain itself.
In discussions of the raven paradox,
it is generally assumed that the (relevant) information gathered from an
observation of a black raven can be regimented into a statement of the
form Ra & Ba ('a is a raven and a is
black'). This is in line with what a lot of "anti-individualist" or
"externalist" philosophers say about the information we acquire
through experience: when we see a black raven, they claim, what we
learn is not a descriptive or general proposition to the effect that
whatever object satisfies such-and-such conditions is a black raven,
but rather a "singular" proposition about a particular object --
we learn that this very object is black and a raven. It seems
to me that this singularist doctrine makes it hard to account for many
aspects of confirmation.
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.