Modal Logic 2024/2025 (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 10 — June 27, 2025.
Re-sit period: June 23 — July 4, 2025.
The lectures will take place
Mondays: 15:30 -- 17:00;
Wednesdays: 15:30 -- 17:00.
See also here.
The classes are in Calle Montalegre 6 in Aula 412 on the fourth floor.
The start date is Monday, February 10.
Official schedule information can be found here
The page below will be updated as we proceed.
The final grade is (as per teaching plan) determined by
(A) Homework questions (this may include a mid-term exam); (40 %)
(B) Presentation in class (10 %);
(C) Midterm + Final Exam; (25 + 25 %).
All materials and assignments will also be placed on this page.
See also here and here.
Joost J. Joosten is the lecturer of this course.
The course counts with a teaching assistant: Vicent Navarro Arroyo.
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.
TO BE UPDATED
(Feb 10 -- 14)
We started the course, gave an introduction to Modal Logic, started by defining the basic modal logics, K, K4, etc.
and did some examples of concrete Hilbert style derivations. We have proven some inclusions and non-inclusions of so-called normal modal logics.
Basically, we have covered the first two chapters of the Book I am writing.
(Feb 17 -- 21)
We have made some exercises in class and then finished Chapter 2 of the reader. Next we started an inductive definition of set
of Natural Deduction proofs in modal logic. As homework you are asked to choose two exercises from
2.4.11; 2.4.12; 2.4.13 and 2.4.14.
(Feb 24 -- Feb 28)
We have given Natural deduction systems for CPC, IPC and normal Hilbert logics. For the modal case we have shown equivalence of the
two formalisms. Upon request there is a new version of the reader.
(Mar 3 -- March 7) We have proven completeness for modal logics through the canonical model.
The new reader is now available.
(Mar 10 -- 14)
We have proven some frame correspondences and frame completeness for various modal logics.
The new reader is now available.
The next batch of homework consists of 4.5.4; 4.5.21, 5.4.3 and 5.4.12.
(Mar 17 -- 21)
We have spoken more on frame conditions and seen the frame incompleteness of the logic KH.
(Mar 24 -- 28)
We have seen three constructions on frames that yield means to show modally undefinable properties.
Next we have spoken on bisimulations and on finite approximations of bisimulations.
The new reader is now available.
(Mar 31 -- Apr 4)
We saw how the first order translation gives us access to all kinds of first-order techniques:
compactness and countable saturated models. Using unraveling we have seen that Löbs rule is admissible for K.
(Apr 7 -- Apr 11)
We had a question hour on Monday and then on Wednesday the Midterm exam.
(Apr 21 -- 25)
No class on Monday nor Wednesday (Sant Jordi)! Previous week was Easter holiday.
(Apr 28 -- May 2)
On Monday there was the great "apagón": no electricity at the entire Iberian peninsula.
Classes in the afternoon were institutionally canceled.
On Wednesday we discussed some matters about the exam and then revisited omega-saturated models of first order logic.
We have seen how omega-saturated models are m-saturated too.
(May 5 -- 9) We have seen the relation between omega-saturation and m-saturation.
We have proven the van Benthem Characterisation Theorem. We have also revisited filters and ultrafilters and have given the definition of
the ultrafilter extension of a modal frame.
The new reader is now available.
(May 12 -- 16)
We studied ultrafilter extensions of frames and models.
(May 19 -- 23)
We finished our chapter on ultrafilter extensions and then moved on to filtration, finite filtrations and decidability.
The new reader is now available.
(May 26 -- 30)
We addressed some questions on the homework set. Then we moved on to Admissible rules.
We have covered Sections 12.1 and 12.2 of the updated reader.
Furthermore, we gave a high-level overview of the paper Best Solving Modal Equations by Silvio Ghilardi.
(June 5 -- 9)
Guest lectures by Mojtaba Mojtahedi on the admissible rules of GL. Mojtaba made notes.
There is now also an updated version of the reader since there were some errors in some exercises in the previous version.
After addressing some issues --mostly concerning Exercise 8.5.9-- I have now uploaded a (hopefully) final version of the reader for
this edition of the course.
FINAL EXAM: Tuesday, June 10, 11--13. Seminari de Filosofia (l'aula seminari Núria Folch que està situat entre l'aula 403 i una de les entrades a l'aula Magna).
Question and answer
Question
What is turing degree?
Answer
You can either find a general definition online but for the sake of the exam, it is also fine to consider any arithmetical Turing degree.
The latter you have seen with Computability.
Question
What is "locally consistent"?
Answer
Falsum is not derivable using the local derivability relation.
Question
Q
Answer
Thanks, it should have been "For example, for the first two equivalence classes in Exercise 8.5.8."
But this is just a suggestion and you can choose the examples as you wish.
Question
Moreover, I think section 4 of exercise 8.5.9 is also wrongly formulated.
Answer
Oh well, the trace on M is defined as the trace on M' :-)
Question
In Exercise 8.5.9. (Hedgehog), point 4, the definition of tr_M(A) does not depend on A. Should the condition "M,x||-A" be added?
Answer
That is entirely correct. This constitutes a correction to the reader and as such is noted down for bonus.
Question
In Exercise 7.3.4, are there any specific reasons to refer to the "Creswell’s model as exhibited in Boolos’s book",
or can we interchangeably use the exposition from your notes?
Answer
Indeed, please follow the reader which in the new version will say: "Consider Creswell’s model from Figure 7.2. "
Question
Regarding Exercise 4.5.14 (Exponential size models),
I believe you said in class that functions like 2^(x^0.001) count as exponential, could you please confirm this?
Answer
Confirmed. In the new version I will simply ask for superpolynomial.
Question
2') In Exercise 8.5.9, point 4, what is the definition of tr_M'(A)? We cannot define in the way tr_M(A) is defined, because M' is not converse well-founded, whereas the definition of ord_M(x) depends on M being converse well-founded.
2'') In Exercise 8.5.9, point 4, tr_M(A) is defined to be a subset of \omega, and the next line contains the formula "tr_M'(A) \cap \omega", thus either "\cap \omega" is redundant or a different definition of trace is meant.
Answer
You are entirely right and it was silly to oversee this. I have changed the exercise accordingly.
Question
In point (4), what is the definition of ord_M([[A]]_M)? As [[A]] is a set rather than a single node,
the definition in 5.2.5 does not apply.
Note that the exercise suggests that this "ord_M([[A]]_M)" never contains \infty.
Answer
You make two points here. The first I consider minor: it is quite common to extend a function
on a domain to a function on subsets over that domain. For mere clarity this standard construction is now mentioned
in the hopefully final version of the reader (Version 10). The other point is more serious.
In your career you will find many situation where a paper contains an error.
Often there are various ways to repair. Also here. One can include either omega as allowed value or one can exclude the point (0,0).
In the the new version of the reader I have chosen to do the first.
Question
Apparently, the statement of point (5) is false.
E.g., take $A := \Diamond \Box \bot \wedge \Diamond \Diamond \top$, then [[A]]={(0,0)},
whence [[A]] is neither disjoint from nor contains the set ${(0, 0)} \cup \{(n,\infty) | 1\leq n\in \omega\}$.
Answer
We stand corrected. See the new version of the reader.
Question
In point (2), it is unclear what are the "first two equivalence classes" in the context of Exercise 8.5.8.
Did you mean something like "all equivalence classes of modal depth 0 and 1?" (or 1 and 2?)
Or based on which ordering are those "first two" equivalence classes chosen?
Answer
Of course I meant, the first two in the trans-galactic modal order and you are free the formulas that you
think are most informative. The point is that it is good to play with a definition and
apply it to some basic and simple cases. That was
the kind of skill/competence that I tried to transmit here.
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