Correspondence and Canonicity in Non-Classical Logic [PhD Thesis]

257 Pages·2015·2.119 MB·English

by  Sumit Sourabh

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INSTITUTE FOR LOGIC LANGUAGE AND COMPUTATION
Correspondence and Canonicity
in Non-Classical Logic
Sumit Sourabh

Correspondence and Canonicity
in Non-Classical Logic
ILLC Dissertation Series DS-2015-04
For further information about ILLC-publications, please contact
Institute for Logic, Language and Computation
Universiteit van Amsterdam
Science Park 107
1098 XG Amsterdam
phone: [phone]
e-mail: [email protected]
homepage: http://www.illc.uva.nl/
The investigations were partially supported by an Erasmus Mundus scholarship
from the European Commission.
Copyright (cid:13)c 2015 by Sumit Sourabh
Cover art: White Bridge, Leonid Afremov. Source: http://www.afremov.com.
Printed and bound by GVO | Ponsen & Looijen, Ede.
ISBN: 978-90-6464-892-2
Correspondence and Canonicity
in Non-Classical Logic
Academisch Proefschrift
ter verkrijging van de graad van doctor aan de
Universiteit van Amsterdam
op gezag van de Rector Magnificus
prof.dr. D.C. van den Boom
ten overstaan van een door het college voor
promoties ingestelde commissie, in het openbaar
te verdedigen in de Aula der Universiteit
op woensdag 9 september 2015, te 13.00 uur
door
Sumit Sourabh
geboren te Begusarai, India.
Promotor: Prof. dr. Y. Venema
Co-promotors: Dr. A. Palmigiano
Dr. N. Bezhanishvili
Overige leden: Dr. A. Baltag
Prof. dr. J.F.A.K. van Benthem
Dr. W.E. Conradie
Prof. dr. S. Ghilardi
Prof. dr. V. Goranko
Prof. dr. D.H.J. de Jongh
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
To my Parents
v

Contents
Acknowledgments xi
1 Introduction 1
1.1 Outline of chapters . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Sahlqvist correspondence and canonicity 17
2.1 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Syntactic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Algorithmic strategies . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Duality and Canonicity . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Basic algebraic modal correspondence 33
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Meaning Function . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.2 Definite implications . . . . . . . . . . . . . . . . . . . . . 35
3.2 Algebraic correspondence . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 The general reduction strategy . . . . . . . . . . . . . . . . 37
3.2.2 Uniform and Closed formulas . . . . . . . . . . . . . . . . 38
3.2.3 Very simple Sahlqvist implications . . . . . . . . . . . . . 40
3.2.4 Sahlqvist implications . . . . . . . . . . . . . . . . . . . . 44
3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Algorithmic correspondence and canonicity for regular modal
logic 51
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.1 Regular modal logics . . . . . . . . . . . . . . . . . . . . . 53
4.1.2 Kripke frames with impossible worlds and their complex
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
vii