DS-2006-02:
Bezhanishvili, Nick
(2006)
*Lattices of intermediate and cylindric modal logics.*
Doctoral thesis, University of Amsterdam.

Text (Full Text)
DS-2006-02.text.pdf Download (1MB) |

## Abstract

Lattices of intermediate and cylindric modal logics

Nick Bezhanishvili

Abtract:

In this thesis we study classes of intermediate and cylindric modal

logics. Intermediate logics are the logics that contain the

intuitionistic propositional calculus IPC and are contained in the

classical propositional calculus CPC. Cylindric modal logics are

finite variable fragments of the classical first-order logic FOL. They

are also closely related to n-dimensional products of the well-known

modal logic S5. In this thesis we investigate:

1. The lattice of extensions of the intermediate logic RN of the

Rieger-Nishimura ladder.

2. Lattices of two-dimensional cylindric modal logics. In particular,

we study:

(a) The lattice of normal extensions of the two-dimensional

cylindric modal logic S5^2 (without the diagonal).

(b) The lattice of normal extensions of the two-dimensional

cylindric modal logic CML_2 (with the diagonal).

Our methods are a mixture of algebraic, frame-theoretic and

order-topological techniques. In Part I of the thesis we give an

overview of Kripke, algebraic and general-frame semantics for

intuitionistic logic and we study in detail the structure of finitely

generated Heyting algebras and their dual descriptive frames. We also

discuss what we call frame-based formulas. In particular, we look at

the Jankov-de Jongh formulas, subframe formulas and cofinal subframe

formulas and we construct a unified framework for these formulas.

After that we investigate the logic RN of the Rieger-Nishimura

ladder. The Rieger-Nishimura ladder is the dual frame of the

one-generated free Heyting algebra described by Rieger and

Nishimura. Its logic is the greatest 1-conservative extension of

IPC. It was studied earlier by Kuznetsov, Gerciu and Kracht. We

describe the finitely generated and finite descriptive frames of RN

and provide a systematic analysis of its extensions. We also study a

slightly weaker intermediate logic KG, introduced by Kuznetsov and

Gerciu. KG is closely related to RN and plays an important role in our

investigations. While studying extensions of KG and RN we introduce

some general techniques. For example, we give a systematic method for

constructing intermediate logics without the finite model property, we

give a method for constructing infinite antichains of finite Kripke

frames that implies the existence of a continuum of logics with and

without the finite model property. We also introduce a gluing

technique for proving the finite model property for large classes of

logics. In particular, we show that every extension of RN has the

finite model property. Finally, we give a criterion of local

tabularity in extensions of RN and KG.

In Part II of the thesis we investigate in detail lattices of

two-dimensional cylindric modal logics. The lattice of extensions of

one-dimensional cylindric modal logic, is very simple: it is an

(\omega + 1)-chain. In contrast to this, the lattice of extensions of

the three-dimensional cylindric modal logic is too complicated to

describe. In this thesis we concentrate on two-dimensional cylindric

modal logics. We consider two similarity types: two-dimensional

cylindric modal logics with and without diagonal. Cylindric modal

logic with the diagonal corresponds to the full two-variable fragment

of FOL and the cylindric modal logic without the diagonal corresponds

to the two-variable substitution-free fragment of FOL.

Cylindric modal logic without the diagonal is the two-dimensional

product of S5, which we denote by S5^2. It had been shown that the

logic S5^2 is finitely axiomatizable, has the finite model property,

is decidable, and has a NEXPTIME-complete satisfiability problem. We

show that every proper normal extension of S5^2 is also finitely

axiomatizable, has the finite model property, and is

decidable. Moreover, we prove that in contrast to S5^2 itself, each of

its proper normal extensions has an NP-complete satisfiability

problem. We also show that the situation for cylindric modal logics

with the diagonal is different. There are continuum many nonfinitely

axiomatizable extensions of the cylindric modal logic CML_2. We leave

it as an open problem whether all of them have the finite model

property. Finally, we give a criterion of local tabularity for

two-dimensional cylindric modal logics with and without diagonal and

characterize the pre-tabular cylindric modal logics.

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-2006-02 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 2006 |

Subjects: | Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/2049 |

## Actions (login required)

View Item |