# Modal Logic 2022/2023 (official course code 569070)

## Organisation

Here an academic year calendar of the UB can be found.

For us it is important to know:

Teaching period (including exam): February 13 — June 30, 2023.
Re-sit period: July 3 — 7, 2023.

The lectures will take place

Mondays: 12:30 -- 13:30;

Wednesdays: 11:30 -- 13:30.

See also here.

The classes are in Calle Montalegre 6 in the Maria Zambrano Seminar which is Aula xxx on the fourth floor.
The start date is Monday, February 13.
TO BE UPDATED
Official schedule information can be found here

The page below will be updated as we proceed.

The final grade is determined by

(A) Homework questions (this may include a mid-term exam); (20 %)

(B) Presentation in class (10 %);

(C) Midterm + Final Exam; (35 + 35 %).

All materials and assignments will also be placed on this page.

Joost J. Joosten is the lecturer of this course.
The literature will consist of among others a reader to be distributed among the participants.

The modal logic course constitutes for 5 European credits and as such comprises 42 contact hours, so that makes 14 weeks, 3 hours each.
(Feb 13 -- 19)
On Monday, we spoke on the role of modal logic in modern mathematical and philosophical logic.
We recalled the syntax of basic modal logic with a single modality. On Wednesdays, Joost could not teach do to
a nasty cough so there was no class.
(Feb 20 -- Feb 26)
We have given the formal definition of the logics K, K4, T, S4 and GL. Using translations into propositional
logic we show consistency of various logics. Moreover, we saw that GLT is inconsistent. We saw that GL contains K4.
We spent quite some time also on the provability reading of the Box and how modal logics can be employed as a
calculus to describe (meta)mathematical 'reality'.
(Feb 27 -- March 5)
We dwelled on the inductive definition of the collection of Natural Deduction Proofs. Then, we showed that we
can actually use natural deduction for modal logics as long as we do not have any open assumptions
when applying the necessitation rule. We proved Loebs formula is provable in PA (under any realisation) using
Goedel's fixpoint theorem and the soundness of K4 of the provability interpretation. We also proved that Santaclause exists.
Finally I managed to update the notes a bit and hereby I upload a preliminary version.
The Homework consists of

(+) of one exercise of your choice from 1.6.18 or 1.6.10;

(+) 2.4.3

(+) two exercises of your choice from 2.4.11, 2.4.12, 2.4.13, and 2.4.14.
and is due Friday March 10. The pdf is password protected. You will be sent the password by mail or you can ask me to
send it to you.
(Mar 6 -- 12)
More on provability logic, also start completeness.
(Mar 13 -- 19)
Modal completeness.
(Mar 20 -- 26)

Frame conditions. Updated version of the reader.
(Mar 27 -- Apr 2)
Holy week!
(Apr 10 -- 16)
No class on Monday! Previous week was Easter holiday.
First and second order translations of modal logic. Start on filtrations.
(Apr 17 -- 23)
Filrations finished. Finite model property and decidability for a myriad of logics.
There is a new version of the reader.

Updated version of the reader.
(April 24 -- 30)
Finite canonical models done and examples given for K, K4 and GL.
Incompleteness LH almost done. Of the new version of the reader we have done all but the last two sections.
(May 1 -- 7) No class on Monday! Wednesday's class was given by Konstantinos Papafilippou for which many thanks
Konstantinos!!! He finished the last observations on Cresswell's model.
Then, he dealt with bisimulations, unravelling and some applications (closure of K under Loeb's Rule).
Van Benthem's characterization theorem.
(May 8 -- 14) Monday was a class in joint session with the Seminari Cuc and it was presented by Konstantinos.
The abstract is given on the Seminari Cuc page. On Wednesday,
Joost took over class again. We saw a proof of the arithmetical fixpoint theorem
and an applications of it to obtain Goedel's variant of the liar sentence (I am not provable).
We saw how indeed this leads to the first Incompleteness theorem. Then we discussed how modal fixpoints lead us
from the second to the first Incompleteness theorem. Next we did some technical preparations for the proofs
of the modal fixpoint theorem.
(May 15 -- 21)
We finished the proof of the fixpoint theorem for GL and saw various applications of it.
(May 22 -- 28)
We proved the arithmetical completeness of GL. Details of the proof can be found in Boolos's The Logic of Provability,
Chapter 8. An updated version of the reader can be found here.

Miriam Kurtzhals pointed out quite some typos and mistakes. They are incorporated in the
new version of the reader.

**FINAL EXAM: Wednesday June 21, From 10:00 -- 12:00 in Aula 410 in the Philosophy building.**

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