## MVL

# Course on Many-Valued Logics (Autumn 2014)

## Contents of the page

- News
- Contents of the classes
- Course material
- Practicalities
- Grading and homework assignments
- Course Description and Prerequisites

## Contents

The course covers the following topics:

- Basic Logic and Monoidal t-norm Logic.
- Substructural logics and residuated lattices.
- Cut elimination and completions.
- Lukasiewicz logic.

More specifically, this is the content of each single class:

**September, 1:**Introduction, motivations, t-norms and their residua. Section 2.1 (up to Lemma 2.1.13) of the Course Material 1.**September, 5:**Basic Logic, Residuated lattices, BL-algebras, linearly ordered BL-algebras. Section 2.2 and 2.3 (up to Lemma 2.3.16) of the Course Material 1.**September, 8:**Lindenbaum-Tarski algebra of BL, algebraic completeness. Monodical t-norm logic, MTL-algebras, standard completeness. The rest of Course Material 1 (excluding Section 2.4) and Course Material 2.**September, 12:**Ordinal decomposition of BL-algebras. Mostert and Shield Theorem. Course Material 3.**September, 15:**Ordinal decomposition of BL-algebras (continued). Algebrizable logics and equivalent algebraic semantics. Course Material 4.**September, 19:**Algebrizable logics and equivalent algebraic semantics (continued). Course Material 4.**September, 22:**Algebrizable logics and equivalent algebraic semantics (continued): Leibniz operator and implicit characterisations of algebraizability. Course Material 4.**September, 26:**Leibniz operator and implicit characterisations of algebraizability (continued). Course Material 4. Gentzen calculus and the substructural hierarchy. Course Material 5 (to be continued).**September, 29:**Structural quasi-equations and $N_2$ equations. Residuated frames. Course Material 5 (Continued).**October, 3:**Analytic quasi-equations, dual frames, and MacNeille completions. Course Material 5 (Continued).**October, 9:**Atomic conservativity, closing the circle of equivalencies. Course Material 5 (Continued).**October, 10:**Lukasiewicz logic and MV-algebras. Mundici’s equivalence. Course Material 6.**October, 17:**The duality between semisimple MV-algebras and Tychonoff spaces. Course Material 7.

## Course material

The material needed during the course can be found below.

- Course material 1
- Course material 2
- Course material 3
- Course material 4
- Course material 5
- Course material 6
- Course material 7
- An example of a possible final exam can be downloaded here.

The homework due during the course can be found below.

- Homework 1 (Deadline 12th September)
- Homework 2 (Deadline 19th September)
- Homework 3 (Deadline 26th September)
- Homework 4 (Deadline 3d October)
- Homework 5 (Deadline 10th October)
- Homework 6 (Deadline 17th October)

## Practicalities

### Staff

- Lecturer: Luca Spada

### Dates/location:

- Classes run from the 1st of September until the 17th of October; there will be 14 classes in total.
- There are two classes weekly.
- Due to the high number of participants classrooms will change weekly, datanose.nl will always be updated with the right classrooms.

## Grading and homeworks

- The grading is on the basis of weekly homework assignments, and a written exam at the end of the course.
- The homework assignments will be made available weekly through this page.
- The final grade will be determined for 2/3 by homeworks, and for 1/3 by the final exam.
- In order to pass the course, a score at least 50/100 on the final exam is needed.

### More specific information about homework and grading:

- You are allowed to collaborate on the homework exercises, but you need to acknowledge explicitly with whom you have been collaborating, and write the solutions independently.
- Deadlines for submission are strict.
- Homework handed in after the deadline may not be taken into consideration; at the very least, points will be subtracted for late submission.
- In case you think there is a problem with one of the exercises, contact the lecturer immediately.

## Course Description

Many-valued logics are logical systems in which the truth values may be more than just “absolutely true” and “absolutely false”. This simple loosening opens the door to a large number of possible formalisms. The main methods of investigation are algebraic, although in the recent years the proof theory of many-valued logics has had a remarkable development.

This course will address a number of questions regarding classification, expressivity, and algebraic aspects of many-valued logics. Algebraic structures as Monoidal t-norm based algebras, MV-algebras, and residuated lattices will be introduced and studied during the course.

The course will cover seclected chapters of the following books.

- P. Hájek, ‘
*Metamathematics of Fuzzy Logic*‘, Trends in Logic, Vol. 4 Springer, 1998. - P. Cintula, P. Hájek, C. Noguera (Editors). ‘
*Handbook of Mathematical Fuzzy Logic*‘ – Volume 1 and 2. Volumes 37 and 38 of Studies in Logic, Mathematical Logic and Foundations. College Publications, London, 2011 - R. L. O. Cignoli, I. M. L. D’Ottaviano e D. Mundici, ‘
*Algebraic Foundations of Many-Valued Reasoning*‘, Trends in Logic, Vol. 7 Springer, 2000 - D. Mundici. ‘
*Advanced Lukasiewicz calculus and MV-algebras*‘, Trends in Logic, Vol. 35 Springer, 2011.

### Prerequisites

It is assumed that students entering this class possess

- Some mathematical maturity.
- Familiarity with the basic theory of propositional and first order (classical) logic.

Basic knowledge of general algebra, topology and category theory will be handy but not necessary.

**Comments, complaints, questions**: mail Luca Spada

## Geometrical dualities for Łukasiewicz logic

This is the transcript of a *featured talk* given on the 15th of September 2011 at the XIX Congeresso dell’Unione Matematica Italiana held in Bologna, Italy. It is based on a joint work with Vincenzo Marra of the University of Milan that was published in Vincenzo Marra and Luca Spada. **The dual adjunction between MV-algebras and Tychonoff spaces**, *Studia Logica ***100**(1-2):253-278, 2012. S*pecial issue of Studia Logica in memoriam Leo Esakia (L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema Editors). *

The article develops a general dual adjunction between MV-algebras (the algebraic equivalents of Łukasiewicz logic) and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. Such a dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. Further the duality theorem for finitely presented objects is obtained from the general adjunction by a further specialisation. The treatment is aimed at emphasising the generality of the framework considered here in the prototypical case of MV-algebras.

Geometrical dualities for Łukasiewicz logic

## The dual adjunction between MV-algebras and Tychonoff spaces

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to aduality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.

The dual adjunction between MV-algebras and Tychonoff spaces