Posts on: Teaching
I taught two courses this year that I haven't taught before. One of
them was our 4th-year undergraduate course on mathematical logic,
"Logic, Computability, and Incompleteness". As usual, I ended up writing
my own textbook. Here it
is as PDF and here as
HTML.
Why yet another textbook? Two reasons mainly. One is that many
existing textbooks are addressed at maths students. This shows up not
only in the examples and illustrations, but also in the fact that
comparatively little time is spent motivating, explaining, and
discussing definitions, proof ideas, or results. I wanted more of
that.
I'm teaching an intermediate/advanced logic course this semester. So
I had to ask myself how to introduce the semantics of quantifiers, with
an eye on proving soundness and completeness. The standard approach,
going back to Tarski, defines a satisfaction relation between a formula,
a model, and an assignment function, and then defines truth by
supervaluating over all assignments. The main alternative, often found
in intro logic textbooks, is Mates' approach, where ∀xA(x) is defined as
true in a model M iff A(c) is true in every c-variant of M, where c is a
constant not occurring in A.
I've been teaching a course called Logic 2: Modal Logics for the past few years. It's an intermediate logic course for third-year Philosophy students, all of whom have taken intro logic. I'm not entirely convinced that a second logic course should focus on modal logic, but it works OK.
One nice aspect of modal propositional logic is that models, proofs, soundness, completeness, etc. are not as trivial as in classical propositional logic, but easier than in classical predicate logic. I also like the many philosophical applications. I spend a week on epistemic logic, another on deontic logic, one on temporal logic, and one on conditionals.
Anyway, I've just uploaded my lecture notes to github, in case anyone is interested. The LaTeX source is there as well.
The semester has now ended and I've returned to working on some long overdue stuff. (More on that soon.)
One thing that has kept me busy during the semester was the philosophy of language course (German) that I've taught. Obviously, this got way out of control. (I wrote 100 pages of handouts because I was so dissatisfied with the available textbooks. I missed two things in particular: applications of results from semantics and philosophy of language to other areas of philosophy (e.g., how the discovery of rigid designation and a posteriori necessity provided the basis for things like type-B materialism and Cornell realism), and an intelligible sketch of how all the different parts of the subject fit together: Grice's analysis of meaning, Kripke's observations about names, Lewis's theory of convention, Montague's model-theoretic semantics, etc. I'm not sure if in the end I did that better, but I've definitely learned a lot in that seminar.)
I'm preparing an introductory course on Wissenschaftstheorie that I'm supposed to teach next semester in the institute of library science. Unfortunately, the textbooks currently available in German are not nearly as good as many English ones.
Another (related) problem is that I'm not sure what Wissenschaftstheorie actually is. Well, I believe it is roughly the same as philosophy of science. But looking through German textbooks and the course guide of my predecessor, apparently some people think it also includes some or all of history and sociology of science, general epistemology, methodology, logic, philosophy of language, and stuff like hermeneutics and dialectics (whatever that is). I guess I'll stick to philosophy of science, even if that means using old textbooks by Carnap and Hempel.
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.
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)