Logic 2: Modal Logic (Spring 2022)

This is a follow-on course to Logic 1, focusing on modal extensions of classical propositional and predicate logic.

Modal logic is used to reason about possibility and necessity, knowledge and belief, permission and obligation, past and future, and a variety of other topics. The first part of the course will introduce standard models and proofs for propositional modal logic, with a brief look at the meta-logical properties of soundness and completeness. We will then go through a range of philosophical applications, studying the logic of knowledge, the logic of obligation, the logic of time, and logical properties of "if-then" constructions. Finally, we will turn to quantified modal logic. We will look at the choices between constant and variable domains, rigid and non-rigid names, and discuss whether standard predicate logic should be weakened to a "free" logic.

Course organiser

Dr Wolfgang Schwarz (wolfgang.schwarz@ed.ac.uk)

Office hour: Friday 11:00-12:00 and by appointment

My office is room 8.06, 40 George Square.

Course administrator

Ann-Marie Cowe (philinfo@ed.ac.uk)


Lecture notes with exercises will be made available each week, and are the only required reading. (See the syllabus below.)

If you want to look ahead, here are the notes from last year (PDF). The content will change for this year, however, so pay attention to what I upload to the syllabus below!

If you want to get a wider perspective, you may find one or more of the following books useful (listed in increasing difficulty):

  • Rod Girle, Modal Logics and Philosophy, 2nd edition, 2009
  • Graham Priest, An Introduction to Non-Classical Logic, 2nd edition, 2008
  • G.E. Hughes and Max Cresswell, A New Introduction to Modal Logic, 1996


  • Lecture 1: Wednesday 12:10-13:00, Appleton Tower Lecture Theatre 2
  • Lecture 2: Friday 13:10-14:00, 40 George Square Lecture Theatre B
  • Tutorial Group 1: Monday 11:10-13:00, Appleton Tower Room 2.12
  • Tutorial Group 2: Tuesday 11:10-13:00, Appleton Tower Room 2.12

Tutorials start in week 2. Only the first hour of tutorials is compulsory.

If you'd like to change your tutorial group, please use the "Group Change Request form" on the timetabling website.


In addition to the final exam, which accounts for 50% of the grade, there will be two take-home tests, the first counting 20%, the second 30%.

The first take-home test was released on Monday 28th February, to be completed by Thursday 3rd March.

The second take-home test will be released on Monday 28th March, to be completed by Thursday 31st March.

The final exam will take place on Thursday 28th April. The test will be released online at 10am and has to be completed by 1pm.

Provisional Syllabus

Week 1 (17/01): Modal Operators

Lecture notes for week 1 (PDF)

Slides for lecture 1 (PDF)

Slides for lecture 2 (PDF)

Answers to the exercises (PDF)

The language of modal propositional logic. Reasoning about necessity and possibility. Flavours of modality. Axiomatic systems.

Week 2 (24/01): Possible Worlds

Lecture notes for week 2 (PDF)

Slides for lecture 3 (PDF)

Slides for lecture 4 (PDF)

Optional: More tree exercises (PDF)

Answers to the exercises (PDF)

Basic possible-worlds semantics for modal propositional logic. The tree method for establishing validity and finding counterexamples.

Week 3 (31/01): Accessibility

Lecture notes for week 3 (PDF)

Optional bonus text: Natural deduction proofs (PDF)

Slides for lecture 5 (PDF)

Slides for lecture 6 (PDF)

Answers to the exercises (PDF)

Adding an accessibility relation to possible-worlds models. Properties of the accessibility relation and corresponding logical systems.

Week 4 (07/02): Models and Proofs

Lecture notes for week 4 (PDF)

Slides for lecture 7 (PDF)

Slides for lecture 8 (PDF)

Answers to the exercises (PDF)

Soundness and completeness for trees and the axiomatic method. A brief look at the logic of provability.

Week 5 (14/02): Epistemic Logic

Lecture notes for week 5 (PDF)

Slides for lecture 9 (PDF)

Slides for lecture 10 (PDF)

Answers to the exercises (PDF)

The logics of knowledge and belief. Gaining information as excluding possibilities. Modal logics with multiple modalities. Interaction principles.

Week 6 (28/02): Deontic Logic

Lecture notes for week 6 (PDF)

Slides for lecture 11 (PDF)

Slides for lecture 12 (PDF)

Answers to the exercises (PDF)

The logic of obligation and permission. Ideal-worlds models. Some puzzles and paradoxes. Neighbourhood models. The concept of conditional obligation.

Week 7 (07/03): Temporal Logic

Lecture notes for week 7 (PDF)

Slides for lecture 13 (PDF)

Slides for lecture 14 (PDF)

Answers to the exercises (PDF)

The logic of past, present, and future. Worlds and times. Branching time. `Now'.

Week 8 (14/03): Conditionals

Lecture notes for week 8 (PDF)

Slides for lecture 15 (PDF)

Slides for lecture 16 (PDF)

Answers to the exercises (PDF)

Material conditionals. Strict conditionals. Lewis-Stalnaker conditionals. If-clauses as restrictors.

Week 9 (21/03): Towards Modal Predicate Logic

Lecture notes for week 9 (PDF)

Slides for lecture 17 (PDF)

Slides for lecture 18 (PDF)

Answers to the exercises (PDF)

Optional: More translation exercises (PDF)

Predicate logic recap. Modal fragments of predicate logic. Modality de dicto and de re. Identity and descriptions.

Week 10 (28/03): Semantics for Modal Predicate Logic

Lecture notes for week 10 (PDF)

Slides for lecture 19 (PDF)

Slides for lecture 20 (PDF)

Answers to the exercises (PDF)

Constant domain semantics and variable domain semantics. Quantification and existence. Trans-world identity.

Week 11 (04/04): Review Week

Slides for lecture 21 (PDF)

Slides for lecture 22 (PDF)

All lecture notes in one file (PDF)