Often the factors that determine a phenomenon don't determine it
uniquely. Sometimes this changes the phenomenon itself.
Take language. Plausibly, the meanings of our words are somehow determined by
patterns of use, but these patterns aren't specific enough to fix,
say, a unique extension or intension for our language. There is a
range of precise meaning assignments all of which fit our use equally
well. One might leave it at that and say that it is indeterminate
which of these precise languages we speak. But this misses
something. It misses the fact that we don't speak a precise
language. For example, in a precise language, "Mount Everest has sharp boundaries"
would be true, but in English it is false. The logic of a precise
language would (arguably) be classical, but the logic of English is
not.
When we face a decision and work out what we should do, we gain information about what we will do. Taking into account this information can in turn affect what we should do. Here's an example.
(I) In front of you are two opaque boxes, one black one white. You can
open one of them and keep whatever is inside. Yesterday, a perfect (or
almost perfect) predictor tried to predict what you would choose. If
she predicted that you'd take the black box, she put a million dollars
in the white box and two dollars in the black box. If she predicted
that you'd take the white box, she put a thousand dollars in the black
box and one dollar in the white box. Which box do you open?
Let's say that at the beginning of your deliberation, you are
completely undecided, giving 50 percent credence to the hypothesis
that you'll end up opening the black box. Standard formulations of
causal decision theory then say that opening the white box has greater
expected payoff: since there's a 50 percent probability that it
contains a million, the expected payoff is 500000.50, which is a lot
more than what you could possibly find in the black box. However, choosing to open the white box would
provide you with highly relevant information: it would reveal
that the predictor has (almost certainly) put only one dollar in the white box and a thousand in the black box. As
a rational decision-maker you should take that information into
account. Many putative "counterexamples" to causal decision
theory, such as those
in Richter
1985 and Egan
2007, are based on this observation.
Lewis, in "Causal Decision Theory" (1981, p.308):
Suppose we have a partition of propositions that distinguish worlds
where the agent acts differently ... Further, he can act at
will so as to make any one of these propositions hold, but he cannot
act at will so as to make any proposition hold that implies but is
not implied by (is properly included in) a proposition in the
partition. ... Then this is the partition of the agent's
alternative options.
That can't be right. Assume I "can act at will so as to make hold"
the proposition P that I raise my hand. Let Q be an arbitrary fact
over which I have no control, say, that Julius Caesar crossed the
Rubicon. Then I can also act at will so as to make P & Q true. (By
raising my hand, I make it true, by not raising it I make it false.)
So, by Lewis's definition, P is not an option, since I can act at will
so as to make a more specific proposition P & Q true (a
proposition that implies but is not implied by P). By the same
reasoning, all my options must entail Q. So they don't form a
partition: they don't cover regions of logical space where Q is
false.
Consider a long list S1...Sn of sentences such that (a) each Si
is trivially equivalent to its predecessor and successor
(if any), and (b) S1 is not trivially equivalent to Sn.
For example, S1 might be a complicated mathematical or logical
statement, and S1...Sn a process of slowly transforming S1 into a
simpler expression. For another example, S1...Sn might be statements
in different languages, where each Si qualifies as a direct
translation of its neighbor(s) but S1 is not a direct translation
of Sn.
I recently accepted a Chancellor's Fellowship at the University of Edinburgh. So it looks like the next stop, after six years in Australia, will be Scotland. Woop!
Over the weekend I made a website that lets you search through the works of David Lewis. It's not perfect: a lot of the documents contain garbled words from OCR, the character encoding is messed up, and it doesn't show page numbers of matches. Maybe I'll fix that eventually. Also, three papers are currently missing from the index because I don't have them in PDF form: "Nachwort (1978)", "Lingue e Lingua", and "Review of Olson and Paul, Contemporary Philosophy in Scandanavia".
[Update: See the changelog for updates.]
In a large election, an individual vote is almost certain to make
no difference to the outcome. Given that voting is inconvenient and time-consuming,
this raises the question whether rational citizens should bother to
vote.
It obviously depends on the citizen's values. For a completely
selfish person, the answer may well be 'no'. Different election
outcomes typically don't matter too much for an ordinary citizen's
selfish interests; and a miniscule chance of a medium-sized gain does
not offset the cost in time and inconvenience.
In her 2012 paper "Subjunctive
Credences and Semantic Humility" (2012), Sarah Moss presents an
interesting case due to John Hawthorne.
Suppose that it is unlikely that you perform a certain physical
movement M tomorrow, though in the unlikely event that you
contract a rare disease D, the chance of your performing M is
high. Suppose also that the combination of contracting D and
performing M causes death. Then many judge that the objective
chance of 'if you were to perform M tomorrow, you would die' is low,
but the conditional objective chance of this subjunctive given that
you perform M is high.
The intuitive judgments Moss reports are
An amusing passage from a recent
paper by Erik and Martin Demaine on the hypar, a pleated hyperbolic
paraboloid origami structure:
Recently we discovered two surprising facts about the
hypar origami model. First, the first appearance of the model is much
older than we thought, appearing at the Bauhaus in the late
1920s. Second, together with Vi Hart, Greg Price, and Tomohiro Tachi,
we proved that the hypar does not actually exist: it is impossible to
fold a piece of paper using exactly the crease pattern of concentric
squares plus diagonals (without stretching the paper). This discovery
was particularly surprising given our extensive experience actually
folding hypars. We had noticed that the paper tends to wrinkle
slightly, but we assumed that was from imprecise folding, not a
fundamental limitation of mathematical paper. It had also been
unresolved mathematically whether a hypar really approximates a
hyperbolic paraboloid (as its name suggests). Our result shows one
reason why the shape was difficult to analyze for so long: it does not
even exist!
So the hypar joins the ranks of phlogiston,
the planet Vulcan,
the largest
prime, or the quintic
formula: objects of inquiry that turned out not to exist.
Here is a coin. What would have happened if I had just tossed it?
It might have landed heads, and it might have landed tails. If the
coin is biased towards tails, it is more likely that it would have
landed heads. If it's a fair coin, both outcomes are equally
likely. That is, they are equally likely on the supposition that
the coin had been tossed. Let's write this as P(Heads // Toss) =
1/2, where the double slash indicates that the supposition in question
is "subjunctive" rather than "indicative".