In this thesis we investigate modal logics from a lattice theoretic point of view. There are essentially two well-known methods of research on lattices of modal logics. The investigation of the lattice of modal logics as a whole, and the local investigation of the lattice of extensions of some strong logic. In this thesis we will provide a third one by investigating proper complete sublattices, which are not filters within the whole lattice, but compactness preserving. The framework of our investigation is the lattice of normal n-modal logics, i.e., normal modal logics with n modal operators.
Author(s): Frank Wolter
Publisher: Freie Universität Berlin
Year: 1993
Language: English
Commentary: Scanned by author; DjVu'ed, OCR'ed by Envoy
Pages: 125
City: Berlin
Title page ......Page 1
Table of contents ......Page 3
Introduction ......Page 6
1.1 Syntax ......Page 12
1.2 Modal Algebras ......Page 13
1.3 Generalized Frames ......Page 14
1.4 Completeness and Persistence ......Page 15
1.5.1 Fusions ......Page 17
1.5.2 What is the upper part of N2? ......Page 18
2.1 Describable operations ......Page 22
2.2 Subframe logics ......Page 25
2.2.1 Examples ......Page 27
2.3 Confinal subframe logics and other examples ......Page 28
2.4.1 Splittings ......Page 29
2.4.2 A general splitting-theorem ......Page 31
2.4.3 A counterexample ......Page 35
2.4.4 The use of splittings ......Page 36
2.4.5 Some results of W.J.Blok ......Page 41
3.1 General properties ......Page 42
3.1.1 Sf-splitting-formulas ......Page 44
3.1.2 Subframe logics and confinal subframe logics above K4 ......Page 46
3.2 Basic splittings of SNn ......Page 47
4.1 Basic monomodal splittings ......Page 56
4.2 Subframe logics above K4 (II) ......Page 58
4.3 A chain of incomplete subframe logics ......Page 60
4.4 Subframe logics above K4 (III) ......Page 63
4.5 Simple incomplete and not strictly Sf-complete logics ......Page 68
4.7 Subframe spectra ......Page 71
05 Sf-completeness in polymodal logic ......Page 73
5.1 Connected logics ......Page 74
5.2 Sf-splittings in lattices of connected logics ......Page 75
5.3 The upper part of SNn ......Page 80
5.4 Tense logics ......Page 86
5.4.1 Descendants and variants ......Page 89
5.4.2 Subframe logics above K4 (IV) ......Page 97
5.4.3 Proof of theorem 5.4.2 ......Page 98
5.4.4 Some remarks on the FMP ......Page 108
06 Splittings and Sf-splittings in some sublattices of N2 ......Page 111
07 R-persistent subframe logics without the FMP ......Page 115
08 Index ......Page 118
09 List of symbols ......Page 120
10 List of logics ......Page 121
11 References ......Page 122
12 German summary ......Page 125
