Wolfgang Schwarz

Another Break

I'm off to Switzerland for a week or two.

Closure and Justification

John Hawthorne has some nice arguments for the view that knowledge is closed under known implication. I don't know much about knowledge, but it seems to me that there is a good reason to believe that at least justification -- and hence presumably also justified true believe -- is not so closed. The reason is this:

E is some evidence, H and S are alternative and incompatible hypotheses. (Obvious examples are skeptical scenarios, like E = visual evidence of a zebra, H = there is a zebra, S = there is a mule disguised as a zebra.) E strongly supports H: It raises its probability of truth from about 0.3 to about 0.9. And H implies Not-S. Yet E does not raise the probability of Not-S. On the contrary, it raises the probability of S.

Let "S(p)" abbreviate "p is strongly supported by the availble evidence". The picture shows that

S(p) and S(p -> q) does not imply S(q);

S(p & q) does not imply S(p); (let p=-S, q=H)

S(p) does not imply S(p v q); (let p=H, q=-S).

Why Did Frege Give Up?

When I prepared for my exam, I noticed something curious.

Richard Heck, in "The Julius Caesar Objection", claims that

In a letter to Russell, Frege explicitly considers adopting Hume's Principle as an axiom, remarking only that the 'difficulties here' are not the same as those plaguing Axiom V [p.274 in Language, Thought and Logic].

The claim is repeated by Crispin Wright and Bob Hale in the introduction to The Reason's Proper Study (p.11f., fn.21). The letter Heck, Wright and Hale refer to is xxxvi/7 from July 1902.

Tree Prover in Croatian

Berislav Zarnic from the University of Split has translated my tableau prover into Croatian.

More Plans

The exam was okay. It now looks like I will continue to work on my Lewis thesis so that it may be published as a book. My wrists are better, but not yet fully recovered. I'm thinking about spending another week without computers and pens in Switzerland.

Plans

I'll be in Bielefeld tomorrow for my final exams, and then in Poland for a couple of days, so don't be surprised if I don't answer your emails.

The state of my wrists is slowly improving. Maybe I will get back to blogging next week.

TPG Fix

Since it doesn't look I will be able to finish the new version of my Tree Proof Generator anytime soon, I've now added a rough fix for the problem with unrecognized old terms.

Break

Sorry for the recent lack of postings. I'm taking a break from typing to rest my wrists.

Backups

Not only my hands, but also my computers are now threatening to fall apart. While I unfortunately forgot to make backups of my hands, I've just copied all important data from my computers to the server. Most of it is not worth letting the googlebot know, a possible exception being two German scripts (1, 2) I wrote last year about recursiveness, representability, and Gödel's first incompleteness theorem, largely based on chapters 14 and 15 of Computability and Logic, 3rd ed. I've also uploaded some of the songs I made during the past 10 years to this directory, though as with all bad music, it's much more fun creating it than listening to it.

Fictionalism's Ontological Commitment

Fictionalism about a certain discourse is the view that statements belonging to this discourse are to be interpreted like statements in fictional discourse.

Now as Brian has observed, on the common account of fictional discourse, "Fictional(Fa)" implies "(Ex)Fictional(Fx)" (even though it normally doesn't imply "(Ex)Fx"). So one might think that on the common account, fictionalism can't do with fewer entities than realism, even though it can do with different entities. However, the common account is not committed to "Fictional(a != b)" implying "(Ex)(Ey)(x != y)". After all, it usually allows for "(Ex)(Fictional(Fx) and Fictional (not-Fx))", so why not allow for "(Ex)(Fictional(x != b) and Fictional(x = b))"? So maybe one could endorse fictionalism about mathematics and the common account of fictional discourse without being committed to an infinity of entities by claiming that all the "numbers" talked about in mathematics are in fact identical.