# Handling Vagueness in Logic, via Algebras and Games

### Introductory Course

Handling Vagueness in Logic, via Algebras and Games, Serafina Lapenta (University of Salerno, Italy) and Diego Valota (The University of Milan, Italy)

Week 1, 9:00 – 10:30, Room 243, Floor 2

In order to capture enough features of the real word in the language of mathematics one needs to represent vague concepts. Several approaches to vagueness can be found in literature, some of them are based on the idea of sharpening vague predicates. A different approach allows to treat vagueness as it is and puts mathematical fuzzy logic at its core, interpreting vague predicates in truth-degrees ranging over [0; 1]. Such logics can be better understood via their semantics and this course aims at giving a clear picture of the algebraic and game-based semantics of some predominant fuzzy logic, after analyzing the notion of vagueness in itself and discussing arguments in favor and against the use of mathematical fuzzy logic to handle vagueness.

The course requires some familiarity with classical logic and boolean algebras. In particular with the algebraic view of classical logic given, for example, in the book “Logic as algebra” by Halmos and Givant.

Course outline:
[If you find any typo/mistake in the slides, please send a message to slapenta@unisa.it and we will update the version]
Lecture 1. From precise to vague predicates. Definition and theories of vagueness. Degrees of truth. Slides.
Lecture 2. Introduction to mathematical fuzzy logic. The logics BL and MTL and related algebraic structures. Slides.
Lecture 3. Lukasiewicz logic: motivations behind it and definitions. Vague predicates in Lukasiewicz logic, MV-algebras and Łukasiewicz-Moisil algebras. Slides
Lecture 4. Hintikka’s Games for classical logics. Giles’s Games for many-valued logics. Slides.
Lecture 5. Ulam-Renyi’s Games for Łukasiewicz logic. Akinator for Łukasiewicz-Moisil logic. Slides.
[we gratefully acknowledge D. Diaconescu and I. Leustean for the preparation of all material involving Lukasiewicz-Moisil algebras and Akinator]

Main references:
The main references are selected chapters of the Handbook of Mathematical fuzzy logic, vol. 1-3.

1.  V. Boicescu, A. Filipoiu, G. Georgescu, and S. Rudeanu. Łukasiewicz-Moisil Algebras. Annals of discrete mathematics. North-Holland, 1991.
2. F. Cicalese and F. Montagna. Ulam-Renyi game based semantics for fuzzy logics. In P. Cintula, C. G. Ferm¨uller, and C. Noguera, editors, Handbook of Mathematical Fuzzy Logic. Volume 3, volume 58 of Studies in Logic. Mathematical Logic and Foundation, pages 1029–1062. College Publications, 2016.
3. D. Diaconescu and I. Leus¸tean. Towards game semantics for nuanced logics. In FUZZ-IEEE 2017, IEEE International Conference on Fuzzy Systems, Proceedings, page To Appear, 2017.
4. A. Di Nola and I. Leus¸tean. Łukasiewicz Logic and MV-Algebras. In P. Cintula, P. Hajek, and C. Noguera, editors, Handbook of Mathematical Fuzzy Logic. Volume 2, volume 38 of Studies in Logic. Mathematical Logic and Foundation, pages 469–584. College Publications, 2011.
5. C. G. Ferm¨uller. Dialogue semantic games for fuzzy logics. In P. Cintula, C. G. Ferm¨uller, and C. Noguera, editors, Handbook of Mathematical Fuzzy Logic. Volume 3, volume 58 of Studies in Logic. Mathematical Logic and Foundation, pages 969–1028. College Publications, 2016.
6. L. Behounek, P. Cintula and C. Noguera. Introduction to mathematical fuzzy logic.
In In P. Cintula, P. Hajek, and C. Noguera, editors, Handbook of Mathematical Fuzzy Logic,Volume 1, volume 37 of Studies in Logic. Mathematical Logic and Foundation, pages 1-101. College Publications, 2011.
7. N.J.J. Smith. Fuzzy logics in theories of vagueness. In P. Cintula, C.G. Ferm¨uller, and C. Noguera, editors, Handbook of Mathematical Fuzzy Logic. Volume 3, volume 58 of Studies in Logic. Mathematical Logic and Foundation, pages 1237–1281. College Publications, 2016.