I'm thinking about how to introduce the semantics of predicate logic to
beginning philosophy students. In particular, I'm interested in the
interpretation of predicates and quantifiers. Last year in logic class, it
seemed that most students were rather unhappy with the formal recursion on
truth we were teaching them.
So I've just picked 15 random logic textbooks to see how they are doing
it.
Group 1 (functions and sets): Interpretations are introduced as
entities that assign to each n-ary predicate symbol a class of n-tuples
of elements of the domain. (Machover, Beckermann, Bostock, Newton-Smith,
Mendelson, Kutschera, Allen/Hand, Bühler)
Hereby I stipulate that "fb13" is to denote the first human born in the
13th century. Hence it might seems that "fb13 was born in the 13th
century" is analytically true, true by definition. But if analytic truths
are closed under logical implication, "somebody was born in the 13th
century" would also be analytically true. Which it is not.
I don't think tinkering with closure under logical implication will help.
Hereby I stipulate that "fb23" is to denote the first human born in the
23rd century. However, if recent progress in civilization continues, there
might well be no humans in the 23rd century. And if no humans are born in
the 23rd century, "fb23 is a human born in the 23rd century" is false. So
it cannot be true by definition.
In my last posting, I argued that to escape the cardinality problem
for thoughts Frege perhaps has to give up
1) For any things there is at least one concept under which all and only
those things fall.
Now (1) is clearly false if, as I think, all there is are objects --
that is, if it makes sense to quantify over absolutely everything. But if
not, as Frege thinks, denying (1) is not an option. A concept is a
function from things to truth values. Given that functions are not
themselves things, how could there fail to be such functions?
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."