Modal Logic

Course Information

Course Description

This course introduces modal logic as a general framework for reasoning about necessity, information, agency, norms, preference, social relations, and causation. The first half develops the formal foundations of normal modal logic: syntax, Kripke semantics, Hilbert-style proof systems, soundness, axiom-frame correspondence, canonical models and completeness. The second half consists of guided student seminars on major applications, including epistemic logic, belief revision, dynamic epistemic logic, deontic logic, preference logic, social-network logic, and causal logic.

Assessment

• Homework – 40%. Regular exercises.
• Student presentation – 20%. A presentation on a selected branch of modal logic.
• Final examination – 40%. A comprehensive examination covering core concepts, semantics, proof systems, and key results developed throughout the course.

Weekly Schedule

The course runs for seven weeks, with one four-hour meeting each week. Each meeting consists of a three-hour instructor lecture followed by a one-hour student presentation and discussion.

Week	Instructor Lecture (3 hours)	Student Presentation and Discussion (1 hour)
1	Preliminary. Relations and functions	–
2	Modal Languages and Relational Models. Formation rules, substitution, scope, frames and models, accessibility relations, satisfaction at a world, and basic model-evaluation exercises.	–
3	Semantic Reasoning. Validity, satisfiability, local and global consequence, model versus frame validity, generated submodels, and countermodel construction.	Doxastic Logic and Epistemic Logic
4	Normal Modal Proof Systems. Hilbert system K, modus ponens, necessitation, normality, derived rules, formal derivations, and theoremhood versus consequence.	Deontic/Preference Logic
5	Axiom–Frame Correspondence. Axioms D, T, B, 4, and 5; seriality, reflexivity, symmetry, transitivity, and Euclideanness; correspondence proofs and an introduction to standard translation.	Social Network Logic
6	Canonical Models and Completeness and Further Methods. Strong completeness of K, selected extensions and canonicity; an overview of filtration, the finite model property, and decidability; and a comparative review of the course.	Causal Models and Conditional Logic
7	Final exam.	–

Student Seminar Reading Map

Each seminar should center two foundational primary sources and use at least one companion work for context, criticism, or extension.

Seminar	Foundational Core	Companion Reading
Doxastic/Epistemic Logic	Hintikka 1962; Aumann 1976; Halpern and Moses 1990; Alchourrón, Gärdenfors, and Makinson 1985; Darwiche and Pearl 1997; Plaza (1989) 2007; Baltag, Moss, and Solecki (1998) 2016; van Benthem 2006	Fagin et al. 1995; Kraus, Lehmann, and Magidor 1990; van Ditmarsch, van der Hoek, and Kooi 2007
Deontic Logic	von Wright 1951; Chisholm 1963	Jørgensen 1937; van Fraassen 1972; Makinson and van der Torre 2000; Yan and He 2025
Preference Logic	von Wright 1963; van Benthem and Liu 2007; Liu 2011; Liu & van der Torre 2025	Halldén 1957; Boutilier 1994; Liu 2008
Social Network Logic	Seligman, Liu, and Girard 2011; Christoff and Hansen 2015; Baltag et al. 2019; Christoff et. al. 2026	Liu 2023; Li and Yan 2026
Causal Models and Conditional Logic	Lewis 1973; Galles and Pearl 1998; Halpern 2000	Pearl 1995, 2009; Halpern and Pearl 2005; Stalnaker 1968; Xie and Yan 2024

Student Seminar Requirements

The standard format is a 45-minute presentation followed by 15 minutes of questions and discussion.

• Use at least two foundational primary sources and one reliable survey or later critical source.
• Meet with the instructor at least 5 days before presenting to verify the formal content.
• Provide slides or a handout, and a worked exercise with a solution no later than 24 hours before class.

Required Seminar Components (suggested)

1. The historical problem and the work’s foundational significance.
2. The formal language and intended interpretation of its principal operators.
3. The relevant model class and truth conditions.
4. A proof system, axiom scheme, representation result, or other central metatheorem.
5. A comparison with basic normal modal logic or another system studied in the course.
6. A critical assessment of limitations, disputed assumptions, or an open problem.

Selected Bibliography

General Modal Logic

• Blackburn, Patrick, Maarten de Rijke, and Yde Venema. 2001. Modal Logic. Cambridge Tracts in Theoretical Computer Science 53. Cambridge: Cambridge University Press. 

Epistemic Logic, Belief, and Dynamic Epistemic Logic

• Alchourrón, Carlos E., Peter Gärdenfors, and David Makinson. 1985. “On the Logic of Theory Change: Partial Meet Contraction and Revision Functions.” The Journal of Symbolic Logic 50 (2): 510-30. 
• Aumann, Robert J. 1976. “Agreeing to Disagree.” The Annals of Statistics 4 (6): 1236-39. 
• Baltag, Alexandru, Lawrence S. Moss, and Sławomir Solecki. (1998) 2016. “The Logic of Public Announcements, Common Knowledge, and Private Suspicions.” In Readings in Formal Epistemology: Sourcebook, edited by Horacio Arló-Costa, Vincent F. Hendricks, and Johan van Benthem, 773-812. Cham: Springer. Expanded from the TARK VII paper (1998), 43-56. 
• Darwiche, Adnan, and Judea Pearl. 1997. “On the Logic of Iterated Belief Revision.” Artificial Intelligence 89 (1-2): 1-29. 
• Fagin, Ronald, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi. 1995. Reasoning about Knowledge. Cambridge, MA: MIT Press. 
• Halpern, Joseph Y., and Yoram Moses. 1990. “Knowledge and Common Knowledge in a Distributed Environment.” Journal of the ACM 37 (3): 549-87. 
• Hintikka, Jaakko. 1962. Knowledge and Belief: An Introduction to the Logic of the Two Notions. Ithaca, NY: Cornell University Press.
• Plaza, Jan A. (1989) 2007. “Logics of Public Communications.” Synthese 158 (2): 165-79. Reprint of the paper originally published in Proceedings of ISMIS 1989, 201-16. 
• van Ditmarsch, Hans, Wiebe van der Hoek, and Barteld Kooi. 2007. Dynamic Epistemic Logic. Synthese Library 337. Dordrecht: Springer. 
• van Benthem, J. ‘one is a lonely number’: on the logic of communication. In Z. Chatzidakis, P. Koepke, and W. Pohlers, editors, Logic Colloquium ’02. ASL, Poughkeepsie, 2006. 20
• van Benthem, J. Dynamic logic of belief revision. Journal of Applied Non-Classical Logics, 17(2):129–155, 2007.
• van Benthem. J. Logical Dynamics of Information and Interaction. Cambridge University Press, 2011.

Deontic Logic

• Chisholm, Roderick M. 1963. “Contrary-to-Duty Imperatives and Deontic Logic.” Analysis 24 (2): 33-36. 
• Jørgensen, Jørgen. 1937. “Imperatives and Logic.” Erkenntnis 7 (1): 288-96. 
• Makinson, David, and Leendert van der Torre. 2000. “Input/Output Logics.” Journal of Philosophical Logic 29 (4): 383-408. 
• van Fraassen, Bas C. 1972. “The Logic of Conditional Obligation.” Journal of Philosophical Logic 1 (3-4): 417-38.
• von Wright, Georg Henrik. 1951. “Deontic Logic.” Mind 60 (237): 1-15. 

Preference Logic

• Boutilier, Craig. 1994. “Toward a Logic for Qualitative Decision Theory.” In Principles of Knowledge Representation and Reasoning: Proceedings of the Fourth International Conference (KR’94), edited by Jon Doyle, Erik Sandewall, and Pietro Torasso, 75-86. San Francisco: Morgan Kaufmann. 
• Halldén, Sören. 1957. On the Logic of “Better”. Library of Theoria 2. Lund: C. W. K. Gleerup; Copenhagen: Ejnar Munksgaard.
• Kraus, Sarit, Daniel Lehmann, and Menachem Magidor. 1990. “Nonmonotonic Reasoning, Preferential Models and Cumulative Logics.” Artificial Intelligence 44 (1-2): 167-207. 
• van Benthem, Johan, and Fenrong Liu. 2007. “Dynamic Logic of Preference Upgrade.” Journal of Applied Non-Classical Logics 17 (2): 157-82. 
• van Benthem, Johan, Patrick Girard, and Olivier Roy. 2009. “Everything Else Being Equal: A Modal Logic for Ceteris Paribus Preferences.” Journal of Philosophical Logic 38 (1): 83-125. 
• von Wright, Georg Henrik. 1963. The Logic of Preference: An Essay. Edinburgh: Edinburgh University Press.
• Liu, F. 2008, Changing for the Better: Preference Dynamics and Agent Diversity, Ph.D. thesis, ILLC, University of Amsterdam.
• –––, 2010, “Von Wright’s ‘The Logic of Preference’ Revisited”, Synthese, 175(1): 69–88.
• –––, 2011a, “A Two-Level Perspective on Preference”, Journal of Philosophical Logic, 40(3): 421–439.
• –––, 2011b, Reasoning about Preference Dynamics (Synthese Library 354), Berlin: Springerd.

Social Network Logic and Bridge Readings

• Acemoglu, Daron, Munther A. Dahleh, Ilan Lobel, and Asuman Ozdaglar. 2011. “Bayesian Learning in Social Networks.” The Review of Economic Studies 78 (4): 1201-36. 
• Baltag, Alexandru, Zoé Christoff, Rasmus K. Rendsvig, and Sonja Smets. 2019. “Dynamic Epistemic Logics of Diffusion and Prediction in Social Networks.” Studia Logica 107 (3): 489-531. 
• Christoff, Zoé, and Jens Ulrik Hansen. 2015. “A Logic for Diffusion in Social Networks.” Journal of Applied Logic 13 (1): 48-77. 
• DeGroot, Morris H. 1974. “Reaching a Consensus.” Journal of the American Statistical Association 69 (345): 118-21. 
• Granovetter, Mark. 1978. “Threshold Models of Collective Behavior.” American Journal of Sociology 83 (6): 1420-43. 
• Jackson, Matthew O., and Asher Wolinsky. 1996. “A Strategic Model of Social and Economic Networks.” Journal of Economic Theory 71 (1): 44-74. 
• Seligman, Jeremy, Fenrong Liu, and Patrick Girard. 2011. “Logic in the Community.” In Logic and Its Applications: 4th Indian Conference, ICLA 2011, edited by Mohua Banerjee and Anil Seth, 178-88. Lecture Notes in Computer Science 6521. Berlin: Springer. 
• Seligman, J., Liu, F., & Girard, P. 2013. Facebook and the epistemic logic of friendship. In B. C. Schipper (Ed.), Proceedings of the 14th conference on theoretical aspects of rationality and knowledge (pp. 229–238).
• 刘奋荣，社会认知逻辑，北京：清华大学出版社，2023 年.

Causal Models, Counterfactuals, and Conditional Logic

• Galles, David, and Judea Pearl. 1998. “An Axiomatic Characterization of Causal Counterfactuals.” Foundations of Science 3 (1): 151-82. 
• Halpern, Joseph Y. 2000. “Axiomatizing Causal Reasoning.” Journal of Artificial Intelligence Research 12: 317-37. 
• Halpern, Joseph Y., and Judea Pearl. 2005. “Causes and Explanations: A Structural-Model Approach. Part I: Causes.” The British Journal for the Philosophy of Science 56 (4): 843-87. 
• Lewis, David. 1973. “Causation.” The Journal of Philosophy 70 (17): 556-67. 
• Pearl, Judea. 1995. “Causal Diagrams for Empirical Research.” Biometrika 82 (4): 669-88. 
• Pearl, Judea. 2009. Causality: Models, Reasoning, and Inference. 2nd ed. Cambridge: Cambridge University Press. 
• Stalnaker, Robert C. 1968. “A Theory of Conditionals.” In Studies in Logical Theory, edited by Nicholas Rescher, 98-112. American Philosophical Quarterly Monograph Series 2. Oxford: Basil Blackwell.
• Xie, K., Yan, J. 2024. “A Logic of Desire based on Causal Inferences”. Journal of Logic and Computation, 2024.