Fuzzy Logic
2017/18
A semestral course at the
Faculty of Informatics, Vienna University of
Technology, in the winter term of 2017/18.
Lecturer: PrivDoz. Petr Cintula
PhD
Contact: cintula@cs.cas.cz
Credits (3 EC)
There are 25+ exercises in the
slides of the corresponding lessons. Students are supposed to choose 15 of them
and be able to explain the solutions of the chosen ones at a short exam. For
best marks the student will be required to answer additional question for
material from lessons 2-5.
Aim
The aim of the course is to
introduce contemporary Mathematical Fuzzy Logic as a formal tool for reasoning with graded notions. The course gives
motivations for the area; introduces several formal systems of fuzzy logic with
stress on Lukasiewicz and Gödel–Dummett logics; shows
their position in the logical landscape; covers basic and selected advanced
mathematical results for both propositional and predicate logic; review recent
trends in the development of the area, and showcases their possible
applications.
1. Motivations and historical origins of mathematical
fuzzy logic (2 hours) [slides]
motivation from computer science,
engineering, linguistics, and philosophy; short history of MF; introduction to
Łukasiewicz and Gödel–Dummett
propositional logics: proof systems and standard semantics
2. Completeness of propositional Łukasiewicz and
Gödel–Dummett logics (6 hours) [slides]
completeness with respect
to general and linear semantic; standard completeness
3. Additional properties of propositional Łukasiewicz and
Gödel–Dummett logics (3 hours) [slides]
functional representation;
finite model property; computational complexity
application: fuzzy logic and
probability
4. Predicate Łukasiewicz and Gödel–Dummett logics (3 hours) [slides]
syntax, semantics,
axiomatizability; completeness; skolemization; witness semantics;
application: fuzzy set theory; formal
fuzzy mathematics; counterfactual implications
5. The growing family of fuzzy logics (3 hours) [slides]
(left-)continuous t-norms
and their residua; Mostert--Shield's theorem; Product logic, Esteva and Godo’s
MTL; the quest for basic fuzzy logic; logics in richer propositional languages;
core semilinear logics as a general theory for fuzzy logics,
application:: fuzzy logic as logic of
resources/costs and logical omniscience
paradox
6. Further lines of research in MFL and open problems (1 hour) [slides]
recent development in the
core theory of MFL; game theory; model theory; proof theory
Prerequisites
Basic knowledge about classical
propositional and first-order logic as covered, e.g., in "Theoretische Informatik und
Logik".
Study materials
1. Libor Běhounek, Petr Cintula, and Petr Hájek: Introduction
to Mathematical Fuzzy Logic. In
Petr Cintula, Carles Noguera, and Petr Hájek (eds). Handbook of Mathematical
Fuzzy Logic. Studies in Logic, Mathematical Logic and Foundations, vol. 37 and
38, London, College Publications, 2011.
2. Cintula, Petr, Fermüller, Christian G. and
Noguera, Carles, "Fuzzy Logic", The Stanford Encyclopedia of
Philosophy (Fall 2017 Edition), Edward
N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2017/entries/logic-fuzzy/
>.
3.
Petr Cintula, Petr Hájek, and Carles Noguera
(eds.). Handbook of Mathematical Fuzzy
Logic, College Publications, 2011.
4. Petr
Hájek. Metamathematics of Fuzzy Logic.
Kluwer, 1998.