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:
  1. Pat Morandi’s notes on dualities,
  2. Tutorial in Buenos Aires.
• Lecture 5 and 6 (Non classical logic) references:
  1. Y. Venema, Algebras and Coalgebras, in: J. van Benthem, P. Blackburn and F. Wolter (editors), Handbook of Modal Logic, 2006, pp 331-426.
  2. 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.

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.

Course on Many-Valued Logic at ILLC

Starting form the 1st of September 2014, I will teach a course on Many-Valued Logics at the University of Amsterdam.  The webpage with all the details can be found here.

Logica II per Informatica

This year I will teach the course “Logica II” for the M.sc. degree in Computer Science.  I think I will follow Moore’s modified method for the course (see also the wikipedia entry for the original method).

There are some freely available notes, looking promising,  I will use during the course.

Other reference (standard books) that may be used as sources of inspiration during the course are:

• Mendelson, E.  Introduction to Mathematical Logic. Chapman & Hall 2009.
• Mundici, D.  Logica – Metodo breve. Springer Verlag 2011.
• Asperti, A. and Ciabattoni, A. Logica a Informatica. McGraw-Hill 2003

Further information will follow on this website.