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".
Let [] and <> express alethic necessity and alethic possibility, let @ stand for
'actually', and L for 'it is unalterable that'. We are going to prove that
if something happens, then it is unalterable that it happens.
We need the following principles:
- A <-> <>@A.
Something is the case iff it is possibly actually the case.
- <>A -> L<>A.
If something is alethically possible, one cannot make it
alethically impossible.
- L(A -> B) -> (LA -> LB).
If A -> B and A are both unalterable, then so is B.
- If A is provable then LA.
Logical truths are unalterable.
Here is the proof, with a sea battle for illustration.
Alvin Goldman has just been giving this year's summer school here
in Cologne. When he put forward his view that what distinguishes good
ways of belief formation from other ways is their truth-conduciveness, I
found myself disagreeing and claiming that there is no general principle that
distinguishes the good ways from others. This is somewhat surprising
given that I've often claimed in recent times that the only epistemic
criterion for evaluating belief-formation is truth-conduciveness. Here
is how I think the two claims can go together.
In the old days, it was common to exclude individual constants from
quantified modal logic in favour of Russellian descriptions. I can see
how this works if we have either fixed domains (the same individuals
populating all worlds) or possibilist quantifiers. But in such systems
individual constants don't cause much trouble anyway. Can one also make
the description move in more liberal systems? I don't see how, but I guess
I'm just missing something obvious.
Consider a formula "possibly, a is F". We want to replace the name "a" by a description "the A".
Does the description get narrow scope ("possibly, the A is F") or wide scope ("the A is
possibly F")? Either way, we seem to get the wrong result.
There is a mistake on page 49 of Lewis's "Counterfactual dependence
and time's arrow" (1979). Since the mistake seems to be repeated all the
time, it might be worth pointing it out.
Page 49 is where Lewis lists similarity standards for his analysis
of counterfactuals. The analysis, recall, says that "if A were the
case, then C" is true iff the closest A-worlds are C-worlds (or, more
precisely, iff either there are no A-worlds or some A&C-worlds are
closer to the actual world than any A&~C world). Closeness is a matter
of similarity, and Lewis indicates what the relevant respects of
similarity might be for certain ordinary counterfactuals in section
3.3 of his 1973 book, and again in the 1979 article on counterfactual
dependence. Roughly, the closest A-worlds are those that perfectly
match the actual world across as much of spacetime as possible without
diverse and widespread violations of the actual laws. This won't do
for indeterministic worlds, where generally no laws need to be
violated at all in order to ensure perfect match of futures even after
earlier divergence. So Lewis restricts his standards to deterministic
worlds, returning to the indeterministic case in the 1986 postscript
to the 1979 paper.