Wolfgang Schwarz

Postbote Update

I've fixed a couple of (five, to be precise) problems in Postbote.

What does Russell's Paradox Teach in Semantics?

On Friday, I wrote:

Conclusion 2: If we want to avoid Bradley's regress, there is no reasonable way to defend the principle that every meaningful expression of our language has a semantic value. (Russell's paradox is an independent argument for the same conclusion.)

Today, I was trying to prove the statement in brackets. This is more difficult than I had thought.

Semantic paradoxes usually (always?) arise out of an unrestricted application of schemas like

English Sentences

Friends who know English better than I often tell me that when I write English, my sentences get too long and complicated. So I noticed with considerable relief this resolution from the University at Buffalo on open source software.

Frege's Semantics and Bradley's Regress

Frege believes that predicate expressions have semantic values (Sinne and Bedeutungen) which can't be denoted by singular terms. Hence "the Bedeutung of 'is a horse'" does not denote the Bedeutung of 'is a horse'. Before the discovery of Russell's paradox, the only reason he ever gave for this view -- apart from claiming that it is a fundamental logical fact that just has to be accepted -- is that otherwise the semantic values of a sentence's constituents wouldn't "stick together". The more I think about this reason, the less convincing I find it.

Whitespace

That new Whitespace programming language looks fun. It uses only three different whitespace characters. So I've been thinking about a possible language with just a single character. The only information contained in the source code of such a program would be the code's string length. The compiler would have to read all instructions from the properties of this number, e.g. its digits, its prime factors, etc. I couldn't come up with anything that looks even remotely feasible though. (The cheap trick of course is to interpret the string length as the Gödel number of some C code.)

Time to Move

The war and the Spring, that broke out almost simultaneously, both distract me from philosophy. I also have to think about where to go when I move out of my flat in about two weeks time. Should I stay in Berlin and enjoy another cheap and relaxed summer, or should I rather go to Bielefeld and enjoy some reasonable philosophy? Unfortunately, in Germany the quality of philosophy departments is inversely proportional to the attractiveness of the cities where they are located.

Could Frege's Ontology be a Henkin Model?

Frege uses second-order quantification in both his formal and informal works. So far, I have always assumed that his second-order logic is standard second-order logic. But couldn't it also be second-order logic with Henkin semantics, which would in fact be a kind of first-order logic (compact, complete and skolem-löwenheimish)? Unfortunately, I know far too little about second-order logic to answer this question.

Are there any second-order statements that are satisfiable in standard semantics, but not in Henkin semantics? (I guess there must be: Wouldn't second-order logic with standard semantics have to be complete otherwise? Not sure.) If so, do any of Frege's theorems belong to these?

Questions about Imaginative Resistance

I've finished the exercises. I still have to put together some of the solutions, but since Word always crashes when I draw complicated tables and trees, I've decided to take a break in order to save my mental health. (In fact, Word not only crashes frequently in these cirumstances, it also deletes the currently open file while crashing.) So now I'm working on the Frege paper again, which I really want to finish soon.

Brian Weatherson has posted a couple of interesting entries on imaginative resistance.

Exercises and Puzzles II

I've finally managed to introduce the provability predicate and its properties without mentioning representability and recursiveness. The exercise is then to derive Löb's theorem and Gödel's incompleteness theorems. Unfortunately these deductions are not as simple as I thought they were. Probably too difficult for an introductory book.

I've also just made up this puzzle, which is not very difficult I think. ("Not very difficult" even in the ordinary sense of "not very difficult", not only in the David Chalmers sense.)

Exercises and Puzzles

I'm still doing exercises for the logic book. This is rather unpleasant because I have to use Microsoft Word. Getting back to Word after using reasonable document formats (like LaTeX) and editors (like Alpha) for a while is a very frustrating experience.

At the moment, I'm trying to find nice and simple versions of Gödel's Theorems that still leave something formal to prove (like deducing Löb's Theorem from provability properties). This turns out to be difficult because I don't have the space to introduce the concepts of representability and recursiveness.