MVL

Course on Many-Valued Logics (Autumn 2014)

This page concerns the course `Many-Valued Logics’, taught at the University of Amsterdam from September – October 2014. 

Contents of the page

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.

The homework due during the course can be found below.

Practicalities

Staff

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

October 28th, 2014 | Luca Spada | Tags: , , , , , , , , , .
Categories: Teaching | Comments: 15 Comments

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.

August 28th, 2014 | Luca Spada | Tags: , , , , , , .
Categories: News, Teaching | Comments: No Comments

LPi Logic with Fixed Points

We study a system, $\mu$L$\Pi$, obtained by an expansion of  L$\Pi$ logic with fixed points connectives. The first main result of the paper is that $\mu$L$\Pi$ is standard complete, i.e. complete with regard to the unit interval of real numbers endowed with a suitable structure.
We also prove that the class of algebras which forms algebraic semantics for this logic is generated, as a variety, by its linearly ordered members and that they are precisely the interval algebras of real closed fields. This correspondence is extended to a categorical equivalence between the whole category of those algebras and another category naturally arising from real closed fields.
Finally, we show that this logic enjoys implicative interpolation.

LPi Logic with Fixed Points

October 11th, 2011 | Luca Spada | Tags: , , , , , , .
Categories: Preprint | Comments: No Comments