Extensional contexts are usually defined as positions in a
sentence at which co-refering terms can be substituted without
affecting the truth-value of the sentence. So 'Cicero' occupies an
extensional position in 'Cicero denounced Catiline', but not in
'Philip said that Cicero denounced Catiline'. One might think that a
term t occupies an extensional position in A(t) if and only if all
instances of the following schema are true:
(LL) x=y -> A(x) <-> A(y).
'x=y' is true iff 'x' and 'y' co-refer, and 'A(x) <-> A(y)' is true
iff 'A(x)' and 'A(y)' have the same truth-value. So to say that all
instances of (LL) are true is to say that
Two rather different things sometimes seem to go under the name
"norms of assertion", and it might be useful to keep them
apart. Often, e.g. by Williamson, norms of assertion are characterised
as constitutive norms of a particular speech act. Roughly, a
constitutive norm for an activity X is a norm you must obey, or try to
obey, in order to partake in activity X. The rules of chess are a
paradigm example: to play chess, you have to move the pieces in a
particular way across the board. The other kind of "norm of assertion"
would be a genuine social norm that is normally in force when
people make an assertion.
Suppose tonight you will fission into two persons. One of your
successors will wake up Mars and one on Venus. There are then two
possibilities for how things might be for you tomorrow: you
might wake up on Mars, and you might wake up on Venus. These are
distinct centered possibilities that do not correspond to distinct
uncentered possibilties. There is just one possibility for the
world, but two possibilities for you. Indeed, the two possibilities
are two actualities: you will wake up on Mars, and you will
wake up on Venus. It is tempting to go further and say that there are also two
possibilities for you now. I want to discuss three quite
different reasons for making this move.
In today's installment we take a look at the "imaging analysis" of subjunctive conditional probability. We will find that the analysis is fairly empty, and therefore fairly safe. In particular, it seems invulnerable to a worry that Robbie Williams recently raised in a comment on his blog. Let's begin with an example.
What if the government hadn't bailed out the banks? Some
of them would almost certainly have gone bankrupt, and other
companies would probably have followed.
Here we have some sort of conditional probabilities: "if A, then probably/almost certainly C". But they aren't ordinary conditional
probabilities of the kind that go in the ratio formula, P(A/B) =
P(AB)/P(B). I do not believe that if the government actually
didn't bail out the banks (but only made everyone believe it did),
then some of the banks went bankrupt. That is, my ordinary
conditional probability in the bankruptcies given that there was no
bailout is fairly low. Nevertheless, I believe that if the government
hadn't bailed out the banks, some of them would probably have
gone bankrupt. My subjunctive conditional probability in the
bankruptcies given no-bailout is high.
One of the grave threats to the development of mankind in general,
and philosophy in particular, is the assumption that the objects of
propositional attitudes can be expressed by that-clauses. The
assumption is often smuggled in via a definition, e.g. when propositions
are defined as things that are 1) objects of attitudes and 2)
expressed by that-clauses. No effort is made to show that anything
satisfies both (1) and (2) -- let alone that the things that satisfy (1)
coincide with the things that satisfy (2).
When reading technical material outside philosophy, I am often
struck by the widespread use of non-rigid names and variables. A
typical example goes like this. You introduce 'X' to stand for, say,
the velocity of some object under investigation. When you want to say
that at time t1, the velocity is 10 units, you put exactly this into
symbols: 'at t1, X = 10'. If the velocity changes, we get a violation
of the necessity of identity:
At t1, X = 10.
At t2, X = 20.
Or suppose you have a population of n objects with various
velocities. Your statistics textbook will tell you that the variance
of the velocity in the population is defined as
Compare the following two ways of responding to the weather report's
"probability of rain" announcement.
Good: Upon hearing that the probability of rain is x,
you come to believe to degree x that it will rain.
Bad: Upon hearing that the probability of rain is x, you
become certain that it will rain if x > 0.5, otherwise certain that
it won't rain.
The Bad process seems bad, not just because it may lead to bad
decisions. It seems epistemically bad to respond to a "70%
probability of rain" announcement by becoming absolutely certain that
it will rain. The resulting attitude would be unjustified and irrational.
Apropos Williamson. The following question came up last year when
we discussed The Philosophy of Philosophy in Canberra. I
thought it had a sensible answer that we just couldn't figure out, but
then Dorothy Edgington raised the same question at the recent
phloxshop workshop in Berlin, and even though there were quite a few
Williamsonians present, there was no agreement on what the answer is,
and the proposals didn't sound very convincing.
The question is simply how, on Williamson's account, we can have
knowledge of substantial metaphysical necessities, e.g. of the fact
that gold necessarily has atomic number 79. Williamson explains that
when we counterfactually imagine gold having atomic number 78 (knowing
that it has number 79), we will "generate a contradiction", because we
hold "such constitutive facts [as atomic number] fixed" (p.164). But
the distinction between constitutive and not-constitutive facts can
hardly be analysed as the distinction between whatever we happen to
hold fixed and the rest, given Williamson's commitment to strong
mind-independence of metaphysical modality. So what justifies our
holding fixed the atomic number?
Suppose we want to follow Frege and distinguish an expression's
denotation from its sense. Suppose also we take the
denotation of a predicate to be its extension: the set of its instances. The following argument
appears to show that this leads to trouble.
- All humans are featherless bipeds, and all featherless bipeds are
human, but there could have been featherless bipeds that are not
human. In short, (Ax)(Hx <-> FBx) & <> (Ex)(~Hx & FBx)).
- By existential generalisation over the predicate positions, it
follows that (EX)(EY)((Ax)(Xx <-> Yx) & <> (Ex)(~Xx &
Yx)).
- If things in predicate position denote sets of individuals, this
can be read as: there is a set X and a set Y such that X and Y have
the same members and it is possible for something to be a member of Y
and not of X.
- But if X and Y have the same members, then they are identical; and then
nothing could belong to "one of them" without also belonging to "the
other".
- Hence things in predicate position do not denote sets of
individuals.
The argument is modeled on a brief passage (p.13) in Tim
Williamson's latest
paper on the Barcan Formula. Williamson there argues against the
plural interpretation of second-order quantifiers. On this
interpretation, the sentence in (2) can be read as "there are things
xx and things yy such that all xx's are yy's and all yy's are xx's and
it is possible for something to be one of the yy's but not of the
xx's". Williamson objects that if the xx's just are the yy's,
then it is not possible for something to belong to "the ones" without
also belonging to "the others".
Here is an attempt at an argument against formulating causal decision theory in
terms of counterfactuals (loosely following up on the discussion in the previous
post). The point seems rather obvious, so it is probably old. Does anyone know?
Suppose you would like to go for a walk, but only if it's not
raining. Unfortunately, it is raining heavily, so you have
almost decided to stay inside. Then you remember Gibbard and
Harper's paper "Counterfactuals and two kinds of expected
utility".