Problemes Filosofiques 3 (Problemes Filosofiques 3; Official code: 363783, Year 2023--2024)

Organisation

This is a 6 course worth for credits (EC).
We will be reading the following texts:

• Number and Numbers; Alain Badiou.
• Menon; Plato.
• Thinking about mathematics; Stewart Shapiro.

S'ha d'actualitar aquesta pàgina.
Per al text de Meno, m'han recomanat la traducció següent.
De moment només penjo el pla docent de l'any passat.
Els docents son el Joost J. Joosten per 2 crèdits i en Vicent Navarro Arroyo per 4 crèdits.
El Vicent Navarro Arroyo és professor associat dins del ICREA Acadèmia del Joosten.

Here is an academic year calendar of the UB.

The second semester for the graduate courses runs from February 6, 2023 -- May 26, 2023. Lecturer and course coordinator: Joost J. Joosten
The text below shall be updated as we go along in our course.

This is a course of 6 credits which corresponds to 45 contact hours and the students are supposed to dedicate at least 75 more hours of individual work. Since we do three hours a week, this will correspond to 15 weeks.
Official information regarding the course is published at the course pages of the UB. From there, one can redirect to the courses. The lectures will take place in Aula 410 in the Montalegre Building.

The lecture schedule is as follows:
Mondays 11:00 -- 12:00
Tuesdays 11:00 -- 13:00

We strongly advice students to follow the course in the so-called avaluació úinca mode. Here we will have take-home exercises, a mid-term exam and a final exam.

Date and location midterm exam: TBA;

Date and location final exam: TBA.

The distribution of points in the final grade is as follows:

Take-home exercises: 40 %

Midterm exam: 30 %

Final exam: 30 %

Students may also decide --even though we stronly would like to discourage this-- to participate in the so-called avaluació única.

Date and location avaluació única exam: June 9, 2023, 12:00 -- 14:00, Aula 410.

Date and location resit exam: July 6, 2023, 12:00 -- 14:00, Aula 410.

As mentioned, the lecturer is Joost J. Joosten and the best way to contact is by sending an email. You can also come around to see if I'm in: the Montalegre building in Room 4045 with phone number +34 934031939.

There will be no lectures on
April 2 -- April 10;
May 1
June 5

Objectives

Logic can be described as the art of valid reasoning. In the first part we will see how one can reason in a logical system that allows for modalities of propositions. These modalities can range from `necessary’ to `known’ . We shall have a special interest in the provability interpretation as well giving rise to a frame-work in which, for example, Gödel’s second incompleteness theorem can be formulated.

Further, we shall see how giving a constructive reading to the connectives gives rise to a different logic: constructive or intuitionistic logic. Naturally this requires a ontological stance very different from platonism/realism underlying classical logic. If time allows we shall see how contstructive logic can be related both to classical and to modal logic.

• To get an understanding how modalities add a subtle and complex dimension to reasoning;
• See a couple of standard modal logics and reason in them;
• Understand the ontological presupposition that underly constructive reasoning;
• Understand the ontological presupposition that underly classical reasoning;
• Understand the fundamental difference and tension between the two.
• Learn formal reasoning systems in Natural Deduction style;
• Learn formal reasoning systems in Gentzen Deduction style;
• Learn and understand proofs by induction
• Study modal semantics;
• Learning how to apply the soundness theorem to obtain non-derivability results;
• Learning how to apply the completenss theorem to obtain provability without actually exhibiting a concrete proof

THE REMAINDER OF THIS PAGE NEEDS UPDATING


Week 1	Week 2
Week 3	Week 4
Week 5	Week 6
Week 7	Week 8
Week 9	Week 10
Week 11	Week 12
Week 13	Week 14

Week 1

(Feb 6 -- 12)
We dwelled on what logic is about, what ontological assumptions are subsumed by classical logic and how this can be challenged and how this is challenged by constructive logic. In the tradition of Brouwer, this is often also known as intuitionistic logic refering the intuition involved in Kant's A priori synthetic knowledge of mathematics. We also discussed how various anomalies plagued foundations of mathematics and logic at the turn of the 19th century.

Week 2

(Feb 13 -- 19) We dealt with syntax of propositional logic and discussed proofs by induction for natural numbers and for formulas. Then we started to do derivations for propositional logic considering just the rules for conjunction and implication. I have written some notes on the theory we dealt with in class. The homework is posted on Overleaf and you got the link through the Campus Virtual. Let me know if you have any complications with that.

Week 3

(Feb 20 -- Feb 26) We continued doing Natural Deduction proofs. We have seen that all rules except RAA are both acceptable both from the BHK perspective and from the classical perspective. Finally, we have seen some proofs with RAA. There is new homework on the Overleaf.

Week 4

(Feb 27 -- March 5) We have done more exercises with Natural Deduction and seen examples of the Curry-Howard Isomorphism. You can consult a first draft of the reader . The password needed to open the pdf will be sent to you by mail through the Campus Virtual. There is some new homework on the Overleaf concerning the Curry-Howard Isomorphism.

Week 5

(Mar 6 -- 12) We spoke about semantics and introduced the notion of Kripke semantics. After mentioning the soundness theorem we could establish some non-derivability results in IPC.

Week 6

(Mar 13 -- 19) We made some further exercises that involved Kripke models. We gave a full proof of monotonicity of the forcing relation. Inductive proofs are (hopefully) getting clearer now.

Week 7

(Mar 20 -- 26)

Examples of Kripke semantics at work.

Week 8

(Mar 27 -- Apr 2) Double negation translation.

Week 9

(Apr 10 -- 16) No class on Monday! Previous week was Easter holiday. On Tuesday, April 11 we will have at the regular time, at the regular place no class but rather the MIDTERM exam. The best way to prepare for the midterm is by making many exercises. There is a new reader of which you will have to study part of Section 3 and Section 6. Updated Reader

Week 10

(Apr 17 -- 23) We started on the modal logic part of this course. In particular, we gave the formal definition of a Kripke model and of truth of a modal formula at a world in a model.

Week 11

(April 24 -- 30) We have seen the definition of logic K and of Logic K4 and have done some basic proofs in the corresponding Hilbert proof systems.

Week 12

(May 1 -- 7) No class on Monday!

Week 13

(May 8 -- 14)

Week 14

(May 15 -- 21)

Week 15

(May 22 -- 28)

FINAL EXAM: ???

Question and answer

Question

Answer

A.:

Question

Answer

Question

Answer

Question

Answer

Question

Q.: The meaning of the second item is to list all the normal logics that fulfils the statement? Is this also correct?

Answer

A. No, this is not correct. You should either prove that indeed it holds for any normal modal logic or otherwise find a particular normal modal logic for which the implication fails. Even though the answer is easy, the question is sligthly tricky...