| < 83 older entries | Home | 707 newer entries > |
In §27 of Meaning and Necessity, Carnap announces that all mathematical concepts can be defined without the use of any class expressions. The basic idea is to use Frege's system, but to replace all occurrences of class variables with higher order variables. In particular, the cardinal number of a property F is defined as the second order property of being equinumerous to F (definition 27-4). "Thus, for example, '2' is a predicator of second level" (p.117).
It appears that I'm allowed to make my thesis on Lewis available online. I've put it on a separate page where I might at some time add a couple of other papers I've written.
If anyone knows how to create PDF files that are readable on computer screens from PS files created by OzTeX, please let me know. I've already tried a) ps2pdf on Linux, 2) export as PDF from MacGSView, 3) Acrobat Distiller on Windows, each with all kinds of different settings. I always get files that look nice when printed, but crumbled on screen.
Apple was very quick shipping the (free) replacement adapter.
I've decided to bring order into my thoughts about Fregean thoughts by writing a little paper. If all goes well, I'll hand it in as the termpaper required for my MA. Since my last entry on this topic, I've found out that there is a lively discussion among Frege scholars about the structure of thoughts. Some, in particular Dummett, argue that Frege is, or should be, committed to this view:
My notebook's AC adapter is broken, and I have to wait for Apple to ship a replacement. In the meantime don't expect much blogging or answering of emails.
I hope I won't be arrested by the time the adapter arrives: Today I received my tax assessment for 2001, and somewhat suprisingly it turns out I have to pay 40.000 Euros for that year in which I've earned about 3.000 Euros.
Another nice problem from Brian Weatherson's weblog: Farrington is 50% confident that it's after 4:30, and 50% confident that a certain coin landed tails. Now he comes to know that iff the coin landed tails, some researchers create a brain-in-a-vat duplicate of himself at exactly 4:30 today. What are the probabilities he should assign to the 5 open possibilities:
This appears to be a problem for Lewis' theories of causation:
Let A,B,C,D be any events such that B depends counterfactually on A, and D on C. Now consider the conjunction (fusion) B+C of B and C. If A had not occurred, B+C would not have occurred. For then B would not have occurred, and presumably B+C can't happen without B. And if B+C had not occurred, C would not have occured either, so (unless the absence of B has some surprising effects on D), D would not have occurred. Hence there is a chain of counterfactual dependence between A and D. But since A,B,C,D were arbitrary, this means that every cause causes every effect.
Today I found Montague's paper, and it turns out that I was wrong. Well, Field's presentation was not entirely correct: We shouldn't take Robinson arithmetic itself as R, but some extension of it that contains an additional primitive predicate "True" (T, for short). The extension need not say anything about this predicate. This is why T needn't represent truth in R. (If R says nothing about T, T either represents nothing at all or the inconsistent property, depending on how precisely we define representation.) Montague then shows, very much like Field, that any theory that contains R -- no matter if it's axiomatizable or not --, as well as every instance of
So I've started to actually read Field's papers. Unfortunately I already got stuck on page 4 of "The Semantic Paradoxes and the Paradoxes of Truth". Field there discusses the following restriction of the naive truth schema:
T**) If True(p) then p.
He notes that this is rather weak, since it doesn't even imply that there are any truths at all. Hence, he says, one would presumably add principles like
I've been busy working on the logic book, playing with music software, meeting friends, lazing around, looking after Magdalena (who was ill again), protesting against the war, and thinking that it was a good idea to have voted for Livingstone (as I did when I lived in London). I hope to get back to philosophy soon.
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)
| < 83 older entries | Home | 707 newer entries > |