Applications of modal logic in mathematics and cognitive science 3501-S95-10-OG

Modal logics form a rich and important class of logics. Modal sentential calculi are obtained by extending the classical sentential calculus with modal operators like “it is possible that”, “it is necessary that”, “It has been the case that”, “It will be the case that”, “It is known that”, etc. In recent decades modal logics have become very popular and useful in many areas, in particular in philosophy, mathematics, linguistics, and computer science.

The aim of the seminar is a discussion and a detailed analysis of several applications of modal logics in mathematics and cognitive science. In particular, the following issues will be discussed:

• Applications of modal logics in Peano arithmetic, set theory, and topology
• Applications of modal logics to linguistics in the development of natural language semantics (Montague semantics; formal concept analysis)
• Applications of modal logics in the representation and specification of knowledge (both static and dynamic), belief revision, knowledge discovery and mining

Type of course

Master's seminars
elective seminars

Mode

Classroom

Prerequisites (description)

Completion of the course „Logic I” (for the students of IF) or some equivalent logic course (for the students from other faculties). Knowledge of the classical propositional calculus, first order logic and the basics of set Theory. The completion of the following courses will be an advantage: Logic II (lecture and tutorial) in IF UW; Modal Logic (monograph lecture) in IF UW.

Learning outcomes

After taking the seminar, the student:

1. Indicates typical applications of modal logics in cognitive science
2. Characterizes modal logics, their language and semantics
3. Classifies modal logics with respect to their applications in a given domain
4. Explains the significance of modal logics in the context of their application
5. Identifies the dependencies between modal logics and the interpreted domain
6. Knows the proofs of basic theorems
7. Estimates the philosophical adequacy of the presented theories

Assessment criteria

Preparation and activity – 30%
A presentation with a written draft, at least one time in a semester – 70 %

Bibliography

M. Aiello, J. van Benthem, G. Bezhanishvili, Reasoning about space: the modal way, Journal of Logic and Computation 13 (2003), 889-920

S. Artemov, Logic of Proofs, Annals of Pure and Applied Logic 67 (1994), 29-59

A. Avron, On modal systems having arithmetical interpretations, Journal of Symbolic Logic 49 (1984), 935-942

J. van Benthem,, A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, 1997

P. Blackburn, J. van Benthem, F. Wolter (eds.), Handbook of Modal Logic, Elsevier 2007

G. Boolos, The Logic of Provability, Cambridge University Press 1993

B. F. Chellas, Modal Logic, Cambridge University Press 1980

G. Chierchia, S. McConnell-Ginet, Meaning and Grammar: An Introduction to Semantics, The MIT Press, Cambridge 2000

D. R. Dowty, R. E. Wall, S. Peters, Introduction to Montague Semantics, Reidel, Dordtrecht 1989

R. Fagin, J. Y. Halpern, Y. Moses, M. Y. Vardi, Reasoning about Knowledge, The MIT Press, Cambridge MA, 1995

M. Fitting, The Logic of Proofs, semantically, Annals of Pure and Applied Logic 132 (2005), 1-25

D. Gabbay, F. Guenthner (eds.), Handbook of Philosophical Logic, Kluwer, 2003

E. Goris, Logic of proofs for bounded arithmetic, Lecture Notes in Computer Science 3967 (2006), 191-201

P. Hajek, P. Pudlak, Metamathematics of First-Order Arithmetic, Springer, 1993.

M. Kracht, The Mathematics of Language, Berlin 2003

P. Kremer, G. Mints, Dynamic topological logic, Annals of Pure and Applied Logic 131 (2005), 133-158

P. Lindstrom, Provability Logic – a short introduction, Theoria 62 (1996), 19-61

J. D. McCawley, Everything that Linguists Have Always Wanted to Know about Logic (But Were Ashamed to Ask), University of Chicago Press, 1993

J.-J.Ch. Meyer, W. van der Hoek, Epistemic Logic for AI and Computer Science, Cambridge University Press 1995

B. Partee, A. ter Meulen, R. E. Wall, Mathematical Methods in Linguistics, Kluwer 1990

M. de Rijke, A system of dynamic modal logic, Journal of Philosophical Logic 27 (1998), 109-142

S. Shapiro (ed.), Intensional Mathematics, North-Holland 1985

R. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics 25 (1976), 287-304

H. de Swart, Introduction to Natural Language Semantics, CSLI Publications, Stanford, CA 1998

H. Wansing (ed.), Knowledge and Belief in Philosophy and Artificial Intelligence, Akademie Verlag, 1995

Additional information

Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 3501-S95-10-OG in USOSweb