Modal Logic

Organisation

The lectures will take place
Thursdays: 17:30 -- 19:00;
Fridays: 12:30 -- 14:00.
In Aula 412 of Calle Montalegre 6. The Start date is Thursday September 22.

The final grade is determined by
(A) Weekly homework questions; (30 %)
(B) Midterm take-home exam; (30 %)
(C) Final Exam; (40 %).

All materials and assignments will also be placed on this page.
The lecturers are Felix Bou and Joost J. Joosten. The best way to contact one of us is by sending us a mail. You can also come around to see if we're in. Joost is in the Montalegre building in Room 1095 with phone number +34 934031939.


Week 1	Week 2
Week 3	Week 4
Week 5	Week 6
Week 7	Week 8

Week 1

(Sept 19 -- 23)
On Thursday we had an introduction with motivation of our study and then started on Jansana's reader and went all the way up to Definition 5.1. Most important lesson for today: there are various notions of validity around in modal logics. Please convince yourself of the fact that frame-validity is closed under substitutions.

On Friday we finished Section 5.3 of Chapter 1 (that is sent to the students) of the reader. Maybe the most important lesson for today was the Soundness Theorem for K. Also we have seen the difference between a formal Hilbert-Style proof (which did not make us particularly happy) and a semi-formal proof.

The homework for this week consists of
(1) Give a fully formal Hilbert-Style proof (so, each step in the proof should be included) of Proposition 5.8 Item 6.
(2) Give semi-formal proofs of Proposition 5.8 Items 2 and 4.
(3) Prove that the implications in Proposition 5.8 Items 3--6 cannot be reversed in general. (What theorem do you need to use here?)
(4) Prove that indeed KT4B and KT4E are equivalent logics.

Students that wish some additional exercises to practice can find some here.

Week 2

(Sept 26 -- 30) We finished Chapter 1 and started on Chapter 2 proving very carefully Proposition 1.1. The homework of this week consists of Exercises 4,9, and 10 on Page 29 of ReaderExcerpt2 that has been sent to you by mail.

Week 3

(Oct 3 -- 7) We finished Chapter 2 about frame definability, and we explained the main issues of the canonical model construction involved in most completeness proofs. We finished at Corollary 2.2 in Chapter 3. The homework of this week consists of Exercise 11 on Page 30 of ReaderExcerpt3 (that has been sent to you by mail), and the two Exercises on Pages 40-41.

Week 4

(Oct 10 -- 14) We proceeded till Page 37 of ReaderExcerpt3 that was sent to you by mail. We conclude with the remark "There are logics that are not canonical but are frame complete. One example is provability logic, called also GL. We will prove frame completness of GL later, once we will have at our disposal the filtrations technique." at the bottom of Page 37. Here is the homework of Week 4 to be handed in on Thursday.

Week 5

(Oct 17 -- 21) We finished Chapter 3 showing that GL is not canonical (the notes in ReaderExcerpt5 are much closer to what has been explained in the classroom than previous versions), and we started the chapter on finite frames. In this chapter, we proved that the finite frame property provides an alternative technique for proving frame completeness. During the lecture there was some trouble with the proof of Lemma 1.5, the trouble was caused by not paying attention to the definition of R_{\sim}. You can check ReaderExcerpt5 for the details about proving this lemma (we have also added to the notes another construction, different than the one explained in the classroom, to get a differentiated model equivalent to one Kripke model). The two exercises for this week can be found at the last page of ReaderExcerpt5.

Week 6

(Oct 24 -- 28) In Week 6 we have finished Chapters 4 and 5 (skipping van Benthem's Logic). The homework of this week consist of three items (we refer to ReaderExcerpt6):

(1) Consider Proposition 3.2 on Page 48. All is fine, as [ ] A is an abbreviation for \neg < > \neg A. Now consider a modal language where [ ] is a primitive symbol rather than an abbreviation. What should the logic state about the primitive symbol [ ]? What should we change in the Filtration so that we can still prove Proposition 3.2?

(2) Fill out the details of Lemma 1.5 on Page 54. Probably you will have to use Lemma 1.4 on Page 53. Note that we took in class a different approach by changing the closure properties of \Sigma.

(3) On Page 57, write down the proof of Claim 2 in your own words providing extra details where you think they are needed. Prepare this so that you can give a short PRESENTATION of about 5 minutes of the outline of the proof of Proposition 2.2.

If you have any questions about the homework, please do contact Felix or Joost.

Week 7

(Oct 31 -- Nov 25) During these three weeks, the reader has been fully finished.

Week 8

(Nov 28 -- Dec 2)
We started on our 6 lectures session on Provability logics. Here are the slides of the first two lectures. HOMEWORK: Loeb's rule allows us to conclude A if we have proven []A --> A. Show that over K4 Loeb's rule is equivalent to Loeb's axiom.

More weeks will be added later.