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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 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.
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.)
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.
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?
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.
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.)
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.
First, the puzzle:
In a certain country there are two Gods, called A and B. One of them (A or B, you don't know which) only tells the truth, the other one only falsehoods. One day you meet a God in this country and want to find out whether it's A or B. You're only allowed to ask a single yes/no question. Unfortunately, you don't understand the language of the Gods (even though they understand yours). All you know is that their words for "yes" and "no" are "qwer" and "poiu", but you don't know which of these means "yes", and which "no". With what question will you be able to find out whether it's A or B you're talking to?
I can't really say that I have made up this puzzle. Well, I have made it up, but I took all the main ingredients from puzzles by George Boolos, who himself owes them mainly to Raymond Smullyan and a computer scientist whose name I forgot.
My logfiles show that an alarming number of people (namely more than 10 per day) look at my blogger and diffbot scripts. Don't do that. Don't use them. I have almost finished programming much better versions of both that might be worth trying. But since they have been in the "almost finished" state for quite some time now, I thought I'd better add this note here.
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