Modal Logic

LMU München, Summer Semester 2019

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 schedule is both provisional and incomplete, and will be revised as the course progresses.

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.


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 and completeness of tableau proof systems.

[Fitting & Mendelsohn 1998] section 2.5.

Thursday 6 June

Review of the soundness and completeness proofs for tableau systems.

No exercises, but you should read through section 2.5 of [Fitting & Mendelsohn 1998] in preparation.

Monday 10 June No class (Whit Monday).
Thursday 13 June Exercise set 5.
Monday 17 June Quantified modal logic. Necessity de re and de dicto. Constant-domain semantics.

[Fitting & Mendelsohn 1998] chapter 4.

Thursday 20 June No class (Corpus Christi).
Monday 24 June Comparing constant- and variable-domain semantics. The Barcan and Converse Barcan formulas.
Thursday 27 June
Monday 1 July
Thursday 4 July
Monday 8 July
Thursday 11 July
Monday 15 July
Thursday 18 July
Monday 22 July
Thursday 25 July


  • 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]