Modal Logic

General information

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 First-Order 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.


The course will be assessed by means of a final take-home exam.


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 first-order 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 of your plan to attend.

[Linnebo & Shapiro 2019].

Thursday 30 May No class (Ascension).
Monday 3 June Soundness of tableau proof systems.

[Fitting & Mendelsohn 1998] section 2.5.

Thursday 6 June

Exercise set 5, Solution set 5.

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 (mid-term 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. Constant-domain semantics.

[Fitting & Mendelsohn 1998] chapter 4.

Thursday 27 June

Exercise set 7, Solution set 7.

Monday 1 July Variable-domain semantics. The Barcan and Converse Barcan formulas.

[Fitting & Mendelsohn 1998] chapter 4.

Thursday 4 July

Exercise set 8, Solution set 8.

Monday 8 July

Tableau proof systems for first-order 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

Exercise set 10, Solution set 10.

Monday 22 July

Terms and predicate abstraction.

[Fitting & Mendelsohn 1998] chapters 9 and 10.

Thursday 25 July

Exercise set 11, Solution set 11.

Monday 23 September

Take-home exam submission deadline.



  • 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). First-Order 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]