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.