A while ago, I was discussing Adam Rieger's alleged paradox in Frege's
ontology (here, here, and here). I'm still confident that the Russellian
version of the paradox can be blocked. But on second thought, the
cardinality version of the paradox appears to be much more difficult. Here
it is again.
1) For any things there is at least one concept under which all and only
those things fall.
2) For each of these concepts, there exists the thought that Ben Lomond
falls under it.
3) All these thoughts are different.
4) All thoughts are objects.
From (1)-(3) it follows that there are more thoughts than objects (2^k
if k is the number of objects), contradicting (4).
Ulrich Blau is professor of logic in Munich. For the last 30 years or so he's been working on an enormous book in which he solves all known and several unknown problems in logic, foundations of mathematics, and philosophy in general. If you ever come across that book (it's not published yet), I'd strongly recommend you just skip the non-technical introduction (and conclusion). It's really getting much better where the formulas begin. Anyway, in the introductory chapter I found a silly question that I once discovered myself when I still went to school:
Does this question have an answer?
(In Blau's version, it goes "Can you answer the question you are now reading either affirmatively or negatively?")
When I take a break from philosophy I often find myself creating utterly useless
computer programs. Today, for example, I've spent some hours on Quines.
A Quine is a program that outputs its own source code. (Quines are so called
because Quine, in "The Ways of Paradox" if I recall correctly, introduced the
self-denoting expression "'appended to its own quotation' appended to its own quotation".)
Making Quines is a lot of fun, and also a good training to avoid
use/mention mistakes. I've just written several JavaScript Quines. Here is a particularly
neat one (try it!):
for(i=0;c=[",","'",'"',"for(i=0;c=[",
"][('320202120121023202424').charAt(i++)];)document.write(c)"
][('320202120121023202424').charAt(i++)];)document.write(c)
I've finished my thesis on Lewis' metaphysics. I'll make it available online as soon as I've found out that I'm allowed to do so. (Only "unpublished books" are accepted at the contest, and I don't know if online publication counts as publication.) Anyway, it's German, and doesn't contain many new ideas, especially if you've been reading my blog for the last couple of months.
Next, I have to find out how to register the thesis at my university. Then I will officially be given 4 months to finish it. I also have to find out if it's okay to hand in the finished thesis before registering.
I've been thinking about yesterday's problem from Brian Weatherson's
interactive philosophy blog. Instead of a solution I've found a name:
"Forrest's Paradox" (see §2.5 in Lewis, On the Plurality of
Worlds).
Knowing the name, it is now easy to create even stranger problems of the
same kind. First a reformulation of the original problem.
I'm trying to finish my thesis before February 1st. So this David Lewis blog might eventually become a more general philosophy blog again soon. For the remainder of this month, I probably won't be blogging very much.
By the way, I made a fool of myself by asking physicists about whether elementary particles are extended. As expected, the answer is that the question doesn't make sense in quantum mechanics.
Shelby Moore: "The specification (by definition of specification) does not
allow deviations which would violate the specification."
What, in general, does it mean that something A satisfies a predicate 'F'?
Traditionally, there are three candidates:
1) 'A is F' means that A is F. That' all. Simple predications can't be analysed.
2) 'A is F' means that A instantiates the property F. Except in some special cases,
in particular the case where 'F' is 'instantiates'.
3) 'A is F' always means that A instantiates the property F.
It is not entirely obvious how to locate Lewis here. In some places, when
discussing Armstrong's request for analyses (or truthmakers) for
predication, he sounds like he favours (1): "the statement that A has F
is true because A has F. It's so because it's so. It just is." ("A world
of truthmakers", p.219 in Papers)
Brian Weatherson tells me that Lewis does mention Goodman's 'New Riddle' as
a task for natural properties in "Meaning without use: Reply to Hawthorne".
Lewis says here that we should not be scared off by "Kripkenstein's
challenge (formerly Goodman's challenge)" to find a distinction between
natural and unnatural extrapolation (p.150 in Papers in Ethics and
Social Philosophy, similar remarks can be found in the introduction to
Papers in Metaphysics and Epistemology). So the first suggestion
is very probably right.
(Reading Brian's comments it now seems to me when I argued that natural
properties can't solve the New Riddle I've been confusing it with the Old
Riddle. All the New Riddle requires is an objective distinction between
good and bad extrapolations. That induction based on good extrapolations
might nevertheless yield systematically false predictions ("not work") is the
Old Riddle.)
I think these conditions match the dot-matrix test better than the ones I
proposed earlier. They are more complicated, but closer to the matrices
and not too unnatural:
A property F is natural to the extent that the following conditions are
satisfied, where (1), (3) and (5) weigh heavier than (2), (4) and (6).
1) The Fs resemble each other intrinsically.
2) The Fs resemble each other extrinsically.
3) Anything that exactly resembles an F intrinsically is itself F.
4) Anything that exactly resembles an F extrinsically is itself F.
5) There are few intrinsic F-gaps.
6) There are few extrinsic F-gaps.
Something y is an intrinsic (extrinsic) F-gap if it isn't F and there are
Fs x and z such that y intrinsically (extrinsically) resembles both of them
more closely than x intrinsically (extrinsically) resembles z.
RL satisfies all conditions except (3), whereas R only properly satisfies
(1).