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Lattices of Modal Logics Lattices Of Modal Logics Frank Wolter Dissertation am Fachbereich Mathematik der Freien Universität Berlin Eingereicht im August 1993 Betreuer der Dissertation Prof. Dr. W. Rautenberg 1 Contents 1 Basic Concepts 10 1.1 Syntax............................................................................................................ 10 1.2 Modal Algebras.............................................................................................. 11 1.3 Generalized Frames........................................................................................ 12 1.4 Completeness and Persistence....................................................................... 13 1.5 Polymodal Logics........................................................................................... 15 1.5.1 Fusions .............................................................................................. 15 1.5.2 What is the upper part of A / j ? ......................................................... 16 2 Sublattices of Mn 20 2.1 Describable Operations...................................................................................... 20 2.2 Subframe Logics ............................................................................................... 23 2.2.1 E xam ples............................................................................................... 25 2.3 Confinal Subframe Logics and other exam ples................................................. 26 2.4 Splittings in Modal L o g i c .................................................................................27 2.4.1 S p littin gs............................................................................................... 27 2.4.2 A General Splitting-Theorem ................................................................ 29 2.4.3 A Counterexample................................................................................. 33 2.4.4 The Use of Splittings.............................................................................. 34 2.4.5 Some Results of W .J.B lok...................................................................... 39 3 Subframe Logics 40 3.1 General P rop erties............................................................................................. 40 3.1.1 S/-splitting-formulas.............................................................................. 42 3.1.2 Subframe Logics and Confinal Subframe Logics above K 4 .................. 44 2 CONTENTS 3.2 Basic Splittings of SNfn ...............................................................................41) 4 The Lattice of Monomodal Subframe Logics 54 4.1 Basic Monomodal Splittings.......................................................................... 54 4.2 Subframe Logics above K4 ( I I ) .................................................................... 56 4.3 A Chain of incomplete Subframe L o g ic s ...................................................... 58 4.4 Subframe Logics above K4 ( I I I ) .................................................................... 61 4.5 Simple incomplete and not strictly 5/-complete L o g ic s ...................................66 4.6 A note on Neighbourhood Sem antics............................................................... 69 4.7 Subframe spectra...............................................................................................69 5 5/-completeness in Polymodal Logic 71 5.1 Connected Logics............................................................................................... 72 5.2 5/-splittings in Lattices of Connected Logics ...................................................73 5.3 The upper part of SAfn ....................................................................................78 5.4 Tense Logics .....................................................................................................84 5.4.1 Descendants and Variants......................................................................87 5.4.2 Subframe logics above K4 ( I V ) ..........................................................95 5.4.3 Proof of Theorem 5.4.2 ..................................................................... 96 5.4.4 Some Remarks on the F M P ............................................................... 106 6 Splittings and ^/-splittings in some sublattices of A/j 109 7 Ä-persistent Subframe Logics without the FMP 113 8 Index 116 9 List of Symbols 118 CONTENTS 3 10 List of Logics 119 11 References 120 12 German Summary 123 13 Curriculum Vitae 124 4 INTRODUCTION Introduction In this paper 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 in­ vestigation 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), which is denoted by Afn. The lattice of normal monomodal logics is also denoted by Af. It is now more than one decade ago that the lattice theoretic point of view on modal logics filled as much papers as for instance completeness theory or correspondence the­ ory. Perhaps the first non-trivial, explicite lattice theoretic theorem was the result of M akinson [71] that the lattice of normal monomodal logics has exactly two co-atoms. It followed a discussion of the lattice of extensions of K4. The first results were obtained by M aksimova [75b], M aksimova & Rybakow [74] and B lok [76]. M aksimova [75a] and Esakia & M eskhi [77] independently proved that there are exactly 5 pretabular exten­ sions of 54. Rauten berg [79],[80] used the technique of splittings to give lattice-theoretic descriptions of most of the standard systems above K 4. B lok [80a] showed that a logic above K 4 is tabular if and only if it has finite codimension. In this paper he also proved that there exist 2N° pretabular logics above /if4, contrasting to the situation above 54. At about the same time systematic research on the lattice Af started. Rauten BERG [77] observed that the lattice Af is distributive and used JÖNSSONS Lemma for proving that tabular logics are finitely axiomatizable and that extensions of a tabular logic are tabular again. It was discovered that the upper part of Af is more complicated than the upper part of the lattice of extensions of K 4. B lok [78] showed that both of the co­ atoms have 2h° lower covers, among which are 2H° incomplete logics. His research on the lattice of normal modal logics culminated in the characterization of the strictly complete logics and the result that a logic is either strictly complete or has degree of incompleteness 2h°. If a logic is not strictly complete, then it has 2K° lower covers. These deep results had one disadvantage: They are negative because they show that all standard systems with the exception of Zf(O T) are not strictly complete and have no interesting positive lattice theoretic property within Af. Thus, for Af lattice theoretic methods did not provide positive results, contrasting to the situation above K4. In the following years and up to now research on lattice theoretic questions is restricted to the lattice of extensions SA of some strong logic A. The main example is of course the lattice of extensions of K 4. In Kracht [93] splittings of £K4 are used to prove the fmp for standard systems above K4. The concept of a canonical formula, introduced in Zakharyaschev [87],[92], coincides in the standard cases with splitting-formulas for sublattices of £K4, as will be shown in this thesis. Nevertheless, I think that it is correct to say that nowadays most of the interesting questions above K4 lie outside the scope of lattice-theoretic notions. Outside £K4 there are investigations from Nagle & T homason INTRODUCTION 5 [85] about the lattice of extensions of Kb and from Segerberg [86] about the extensions of K.Alh. It is obvious that most of the negative results in monomodal logic transfer to lattices of polymodal logics. The situation is even worse. For instance the mentioned result of M akinson does not hold for polymodal logics. It is well known that even the lattice of tense logics has infinitely many incomplete co-atoms. Also the tool of splittings, which is basic for studying extensions of K 4, is not directly applicable to polymodal logics. For instance, K racht [92] shows that the lattice of extensions of the minimal tense extension of A'4 has only the trivial splitting. This situation seems to be one reason for the fact that the interest in polymodal logic lies mainly in definability theory or the investigation of rather specific systems. The aim of this dissertation is to show that W . J. B lo k ’s results and the negative results about lattices of polymodal logic do not force us to restrict lattice theoretic inves­ tigations to lattices of extensions of strong logics. It follows from these results that one cannot formulate interesting and solvable lattice theoretic problems by referring only to the lattice of normal (poly-) modal logics as a whole. But this difficulty is manageable by looking at some proper complete sublattices. Since we do not want to leave the whole lattice out of sight we restrict attention to sublattices D for which for a finitely axiomati- zable logic A its upward projection A t ° :
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