Modal Logic
General information
 Seminars: Mondays 12:00—14:00, Room 021, Ludwigstr. 31.
 Problem classes: Thursdays 14:00—15:00, Room D 102, RichardWagnerStr. 10.
 Language: English.
 Prerequisites: Familiarity with the fundamentals of propositional and predicate calculus is required. Knowledge of standard metalogical results (soundness, completeness) is desirable but not required.
 Lecturer:
 Benedict Eastaugh (benedict.eastaugh@lrz.unimuenchen.de).
 Office: Room 123, Ludwigstr. 31.
 Office hours: Mondays 15:00—17:00, or by appointment.
Course description
It is possible that it will rain tomorrow; it is necessary that you are identical to yourself; I believe that the sky is green; it is obligatory to laugh at your logic teacher’s jokes. These sentences are all examples of modal judgements: assertions involving the use of modals like ‘possible’, ’necessary’, ‘believe’, and ‘obligatory’. Modal judgements have been studied through the lens of logic since antiquity, and modal logic experienced a renaissance in the second half of the 20th century, sparked by the introduction of possible worlds semantics, where modal operators like “it is necessary that” and “it is possible that” are interpreted as quantifying over possible worlds or states of affairs. This course will introduce you to the fundamentals of classical propositional and quantified modal logic. We will study proof systems and possible world semantics for different modal logics, and see how the formal regimentation of modality allows us to make precise (and sometimes helps us resolve) philosophical puzzles concerning existence, identity, and reference.
Content and readings
For details, see the schedule below.
The essential readings for the course will largely be drawn from the textbook FirstOrder Modal Logic [Fitting and Mendelsohn 1998]. The first half of the course will be concerned with classical propositional modal logic, including: possible worlds (Kripke) semantics; tableau and Hilbert proof systems; the main propositional modal logics such as K, T, S4, and S5; and soundness and completeness theorems for these logics. In the second half of the course we will look at quantified modal logic.
Each week has a list of required readings, which will be necessary to follow the course. I have also supplied a list of supplementary readings, which complement or augment the primary material for that week, either by providing historical or philosophical background; by providing an alternative view on the technical material; or by providing additional material which goes beyond that required for the course, so that you can start to learn further topics by yourself.
In addition to the readings, every Monday there will be a list of exercises for you to complete. We will then go through them (and possibly others) in the problem classes on Thursdays.
Assessment
The course will be assessed by means of a final takehome exam.
Schedule
Date  Topic  Readings and exercises 
Monday 29 April  Introduction and course overview. Modal operators and possible worlds. Fundamentals of possible worlds semantics. 
[Fitting & Mendelsohn 1998] sections 1.1–1.7. Supplementary: [Garson 2018] sections 1, 2, and 6. 
Thursday 2 May  Exercise set 1, Solution set 1.  
Monday 6 May  Logics as frame conditions. Some important modal logics. 
[Fitting & Mendelsohn 1998] sections 1.8–1.9. Supplementary: [Garson 2018] sections 7 and 8. 
Thursday 9 May  Exercise set 2, Solution set 2.  
Monday 13 May  Modal and firstorder definability. Löb's axiom. S5, equivalence relations, and universality. 
[Zach et al.] sections 2.1–2.4. Supplementary: [Zach et al.] sections 2.5–2.7, [Blackburn et al. 2001] chapter 3. 
Thursday 16 May  Exercise set 3, Solution set 3.  
Monday 20 May  Tableau proof systems for propositional modal logics. 
[Fitting & Mendelsohn 1998] sections 2.1–2.4. 
Thursday 23 May  Exercise set 4, Solution set 4.  
Monday 27 May 
Excursion: Martin Fischer on modalities in arithmetic at the 1st Pisa–MCMP Meeting (13:00–13:55, room C113, Theresienstr. 41). Attendance is free but please notify events.mcmp@lrz.unimuenchen.de of your plan to attend. 

Thursday 30 May  No class (Ascension).  
Monday 3 June  Soundness of tableau proof systems. 
[Fitting & Mendelsohn 1998] section 2.5. 
Thursday 6 June  
Monday 10 June  No class (Whit Monday).  
Thursday 13 June 
Please submit the assignment before the class, either by email or by leaving your solutions in my pigeonhole in the secretary’s office. 
Exercise set 6 (midterm assignment), Solution set 6. 
Monday 17 June  Completeness of tableau proof systems. 
[Fitting & Mendelsohn 1998] section 2.5. 
Thursday 20 June  No class (Corpus Christi).  
Monday 24 June  Quantified modal logic. Necessity de re and de dicto. Constantdomain semantics. 
[Fitting & Mendelsohn 1998] chapter 4. 
Thursday 27 June  
Monday 1 July  Variabledomain semantics. The Barcan and Converse Barcan formulas. 
[Fitting & Mendelsohn 1998] chapter 4. 
Thursday 4 July  
Monday 8 July 
Tableau proof systems for firstorder modal logics. 
[Fitting & Mendelsohn 1998] chapter 5. 
Thursday 11 July 
Exercise set 9, Solution set 9 [questions 1 and 2 only for now]. 

Monday 15 July 
Identity and existence. 
[Fitting & Mendelsohn 1998] chapters 7 and 8. 
Thursday 18 July  
Monday 22 July 
Terms and predicate abstraction. 
[Fitting & Mendelsohn 1998] chapters 9 and 10. 
Thursday 25 July  
Monday 23 September 
Takehome exam submission deadline. Exam. 
Bibliography
 Blackburn, P., M. de Rijke, and Y. Venema (2001). Modal Logic. Cambridge University Press.
 Blackburn, P., J. van Bentham, and F. Wolter, eds. (2007). Handbook of Modal Logic. Elsevier.
 Fitting, M. and R. L. Mendelsohn (1998). FirstOrder Modal Logic. Springer.
 Garson, J. W. (2013). Modal Logic for Philosophers. Cambridge University Press.
 Garson, J. (2018). Modal Logic. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2018 Edition). [link]
 Linnebo, Ø. and Shapiro, S. (2019). Actual and potential infinity. Noûs 53(1):160–191. [link]
 Zach, R. et al (2018). Boxes and Diamonds: An Open Introduction to Modal Logic. Open Logic Project. [link]