## Corso di “Algebra della Logica” alla scuola AILA 2017

This year I teach a course (12 hours) a the AILA summer school of logic. Below one can find the slides of my first three lectures and some references.

**Lecture 1**(Classical propositional logic and Boolean algebras)**Lecture 2**(Algebraic completeness of propositional calculus)**Lecture 3**(Abstract Algebraic Logic)**Lecture 4**(Dualities) lecture material:**Lecture 5 and 6**(Non classical logic) references:- Y. Venema, Algebras and Coalgebras, in: J. van Benthem, P. Blackburn and F. Wolter (editors),
*Handbook of Modal Logic*, 2006, pp 331-426. - 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.

- Y. Venema, Algebras and Coalgebras, in: J. van Benthem, P. Blackburn and F. Wolter (editors),

Lecture notes by Guido Gherardi (Computability Theory).

## 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

## A discrete representation of free MV-algebras

We prove that the $n$-generated free MV-algebra is isomorphic to a quotient of the disjoint union of all the $n$-generated free MV$^{(n)}$-algebras. Such a quotient can be seen as the direct limit of a system consisting of all free MV${}^{(n)}$-algebras and special maps between them as morphisms.

A discrete representation of free MV-algebras

## Advances in the theory of muLPi algebras

Recently an expansion of \rm L {\rm\Pi\frac{1}{2}} logic with fixed points has been considered. In the present work we study the algebraic semantics of this logic, namely \mu L\Pi algebras, from algebraic, model theoretic and computational standpoints.

We provide a characterisation of free \mu L\Pi algebras as a family of particular functions from [0,1]^{n} to [0,1]. We show that the first-order theory of linearly ordered \mu L\Pi-algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered \rm L {\rm\Pi\frac{1}{2}} algebras. Furthermore, we give a functional representation of any \rm L {\rm\Pi\frac{1}{2}} algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of \mu L\Pi algebras is in PSPACE.

Advances in the theory of muLPi algebras