Leo Esakia on Duality in Modal and Intuitionistic Logics

340 Pages·2014·4.111 MB·English

by  Guram Bezhanishvili (eds.)

Outstanding Contributions to Logic 4
Guram Bezhanishvili Editor
Leo Esakia
on Duality
in Modal and
Intuitionistic
Logics
Outstanding Contributions to Logic
Volume 4
Editor-in-Chief
Sven Ove Hansson, Royal Institute of Technology, Stockholm, Sweden
Editorial Board
Marcus Kracht, Universität Bielefeld
Lawrence Moss, Indiana University
Sonja Smets, Universiteit van Amsterdam
Heinrich Wansing, Ruhr-Universität Bochum
For furthervolumes:
http://www.springer.com/series/10033
Guram Bezhanishvili
Editor
Leo Esakia on Duality
in Modal and Intuitionistic
Logics
123
Editor
GuramBezhanishvili
New Mexico StateUniversity
Las Cruces, NM
USA
ISSN 2211-2758 ISSN 2211-2766 (electronic)
ISBN 978-94-017-8859-5 ISBN 978-94-017-8860-1 (eBook)
DOI 10.1007/978-94-017-8860-1
Springer Dordrecht Heidelberg New YorkLondon
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Preface
ThisvolumeisdedicatedtoLeoEsakia’scontributionstothetheoryofmodaland
intuitionistic systems. Leo Esakia was one of the pioneers in developing duality
theory for modal and intuitionistic logics, and masterfully utilizing it to obtain
some major results in the area. The volume consists of 10 chapters, written by
leading experts, that discuss Leo’s original contributions and consequent devel-
opments that have shaped the current state of the field.
Iwouldliketoexpresssinceregratitudetotheauthorsaswellastothereferees
withoutwhoseoutstandingjobthevolumewouldnothavebeenpossible.Itismy
beliefthatthevolumewillserveasanexcellenttributetoLeoEsakia’spioneering
achievements in developing algebraic and topological semantics of modal and
intuitionistic logics, which have paved the way for the next generations of
researchers interested in this area.
Guram Bezhanishvili
v
Contents
1 Esakia’s Biography and Bibliography . . . . . . . . . . . . . . . . . . . . . 1
2 Canonical Extensions, Esakia Spaces, and Universal Models . . . . 9
Mai Gehrke
3 Free Modal Algebras Revisited: The Step-by-Step Method. . . . . . 43
Nick Bezhanishvili, Silvio Ghilardi and Mamuka Jibladze
4 Easkia Duality and Its Extensions. . . . . . . . . . . . . . . . . . . . . . . . 63
Sergio A. Celani and Ramon Jansana
5 On the Blok-Esakia Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Frank Wolter and Michael Zakharyaschev
6 Modal Logic and the Vietoris Functor. . . . . . . . . . . . . . . . . . . . . 119
Yde Venema and Jacob Vosmaer
7 Logic KM: A Biography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Alexei Muravitsky
8 Constructive Modalities with Provability Smack . . . . . . . . . . . . . 187
Tadeusz Litak
9 Cantor-Bendixson Properties of the Assembly of a Frame . . . . . . 217
Harold Simmons
10 Topological Interpretations of Provability Logic . . . . . . . . . . . . . 257
Lev Beklemishev and David Gabelaia
11 Derivational Modal Logics with the Difference Modality . . . . . . . 291
Andrey Kudinov and Valentin Shehtman
vii
Introduction
LeoEsakia’slifelongpassionformodalandintuitionisticlogicsstartedtodevelop
in the 1960s. Soon after it became apparent that Kripke semantics [28], although
veryattractiveandintuitive,wasnotadequateforhandlinglargeclassesofmodal
logics(thephenomenonofKripkeincompleteness).Itwasalreadyunderstoodthat
Kripke frames provide a nice representation for modal algebras, but a modal
algebracaningeneralberealizedonlyasasubalgebraofthemodalalgebraarising
from a Kripke frame.
Leo’smaininterestatthetimewasGödel’stranslation[23]oftheintuitionistic
propositional logic Int into Lewis’ modal system S4, and the corresponding
classesofHeytingalgebrasandS4-algebras.InfluencedbytheworkofStone[38,
39], Tarski (and his collaborators McKinsey and Jónsson) [26, 27, 30–32], and
Halmos [25], Leo realized that the missing link between the algebraic and
relationalsemantics of these systems is topology. This yielded the notion of what
we now call (quasi-ordered) Esakia spaces (namely quasi-ordered Stone spaces
withadditionalproperties)andtherepresentationofS4-algebrasasthealgebrasof
clopen subsets of Esakia spaces. This representation extends to full duality
between the categories of S4-algebras and Esakia spaces. In his discussions with
Sikorski, Leo also realized an apparent need for duality for Heyting algebras. He
wasabletoobtainsuchadualityasaparticularcaseofhisdualityforS4-algebras,
thus obtaining a powerful machinery to study modal logics over S4 and
superintuitionistic logics (extensions of Int). These ground-breaking results were
publishedinEsakia’s1974paper[10],whichremainsoneofthemostcitedpapers
by Leo.
Aroundthesametime(mid1970s),GoldblattandThomasoncametothesame
realization, and developed what later became known as the descriptive frame
semantics for modal logic. These findings were published in Goldblatt [21, 22].
Note that although Esakia worked with quasi-ordered Stone Spaces, replacing a
quasi-order with an arbitrary binary relation in an Esakia space yields the
descriptive frame semantics of Goldblatt and Thomason.
The machinery Leo developed was powerful in many respects. In particular,
whatwenowcalltheEsakialemmawasaconsequenceofhisduality(infact,Leo
developed the lemma to obtain the morphism correspondence of his duality). As
ix
x Introduction
was shown by Sambin and Vaccaro [35] it plays a crucial role in developing the
Sahlqvist completeness and correspondence in modal logic. Subsequently, many
generalizations of Sahlqvist’s theorem have been obtained that utilize Esakia’s
lemma.
The volume opens with the chapter by Mai Gehrke which discusses Esakia
duality for S4-algebras, and how to derive Esakia duality for Heyting algebras
from it. Gehrke provides a more general setting for this approach, which also
yieldsthecelebratedPriestleydualityforboundeddistributivelattices[33,34].All
this is done utilizing the theory of canonical extensions, a very active field of
research of today. Gehrke also discusses Esakia’s lemma and gives a modern
account of how to construct free finitely generated Heyting algebras and their
Esakia duals.
ThedualdescriptionoffreefinitelygeneratedHeytingalgebrasandS4-algebras
wasinitiatedbyEsakiaandhisstudentGrigoliainthemid1970s.Theydeveloped
the so-called coloring technique [19, 20] which became very useful in describing
‘‘upper-parts’’of the dual spaces ofthe freefinitely generatedHeytingand modal
algebras. This important topic was further developed in the 1980s by Shehtman,
Rybakov, Grigolia, and Belissima. In the 1990s, Ghilardi published a series of
papers which gave a novel perspective on the topic. This paved the way for the
follow-on papers by N. Bezhanishvili, A. Kurz, M. Gehrke, D. Coumans, S. van
Gool,andothers.Anup-to-datesurveyofthistopicisgiveninthechapterbyNick
Bezhanishvili, Silvio Ghilardi, and Mamuka Jibladze.
Over the years, several generalizations of Esakia duality have been developed.
To name a few, Leo himself generalized his duality to the setting of bi-Heyting
algebrasandtemporalalgebras[11,13](seealsoF.Wolter[40]),G.Bezhanishvili
generalized Esakia duality to monadic Heyting algebras [1], S. Celani and
R. Jansana generalized it to weak Heyting algebras [9], and G. Bezhanishvili
and R. Jansana to implicative semilattices [3]. The chapter by Sergio Celani and
Ramon Jansana discusses Esakia duality for Heyting algebras and its generaliza-
tionstoweakHeytingalgebrasandimplicativesemilattices.Italsodiscusseshow
toobtainthedualsofmapsbetweenHeytingalgebrasthatonlypreservepartofthe
Heytingalgebrastructure.TheseturnouttobepartialEsakiamorphismsthatplay
a crucial role in developing Zakharyaschev’s canonical formulas, which provide
anaxiomatizationofsuperintuitionisticlogics(aswellastransitivemodallogics).
EsakiaspacesarecloselyrelatedtothecelebratedVietorisconstruction.Infact,
originally Esakia defined his spaces by means of the Vietoris space of a Stone
space [10]. This was the precursor of the coalgebraic semantics for modal logic.
ThistopicandtherelatedrecentdevelopmentsarediscussedinthechapterbyYde
Venema and Jacob Vosmaer.
Another important result in modal logic associated with Esakia’s name is the
so-called Blok-Esakia theorem. It establishes that the lattice of normal extensions
ofGrzegorczyk’smodalsystemS4.Grzandthelatticeofsuperintuitionisticlogics
are isomorphic. The modal system S4.Grz was introduced by Grzegorczyk [24],
who proved a topological completeness of S4.Grz, and showed that the Gödel
embedding of Int into S4 also embeds Int into S4.Grz. In Esakia’s terminology,
Introduction xi
both S4 and S4.Grz are modal companions of Int. Grezegorczyk’s modal system
S4.Grz was one of the favorite modal systems of Leo. He investigated it in great
detail.Inparticular,EsakiashowedthatS4.Grzisthelargestmodalcompanionof
Int. He also showed that each superintuitionistic logic L has the largest modal
companion,obtainedbyaddingtheGrzegorczykaxiomtotheGödeltranslationof
L. This yields the Blok-Esakia theorem, which was obtained independently by
Blok[8]andEsakia[12].SeveralgeneralizationsoftheBlok-Esakiatheoremwere
obtained by A. Kuznetsov and A. Muravitsky [29], F. Wolter and M. Zakharyas-
chev [41–43], F. Wolter [40], and G. Bezhanishvili [2].
The chapter by Frank Wolter and Michael Zakharyaschev is dedicated to the
Blok-Esakiatheorem,whilethechapterbyAlexeiMuravitskyprovidesanoutline
of the intuitionistic modal logic KM which is closely related to the Gödel-Löb
provability logic GL. In particular, it discusses the generalization of the Blok–
Esakia isomorphism to an isomorphism between the lattices of all normal
extensions of KM and GL, respectively. This isomorphism has a further
generalization. Namely, in [18] Esakia introduced the modalized Heyting
calculusmHC and announced that the isomorphism between the lattices of all
normalextensionsofKMandGLextendstoanisomorphismbetweenthelattices
of all normal extensions of mHC and K4.Grz—the modal system obtained by
adding to the well-known modal system K4 a version of Grzegorczyk’s axiom.
The syntax and semantics of the intuitionistic modal logic mHC are discussed in
the chapter by Tadeusz Litak. The chapter also proves the isomorphism between
the lattices of all normal extensions of mHC and K4.Grzannounced in [18], and
discusses the important related issues of well-foundedness, scatteredness, and
constructive fixed point theorems, as well as interpretations of constructive
modalities in scattered topoi.
Leo Esakia was also one of the pioneers in developing the topological
semantics for modal logic. In the 1970s he proved that if we interpret modal
diamondasthederivativeofatopologicalspace,thenGListhemodallogicofall
scattered spaces [14, 15]. This result was obtained independently and slightly
earlier by Simmons [36]. Whether or not a given space is scattered depends on
whetherornottheassembly(i.e.,theframeofnuclei)oftheframeofopensofthe
spaceisBoolean[37].Thisandrelatedissuesabouttheassemblytowerofagiven
framearediscussedinthechapterbyHaroldSimmonsinthesettingofpoint-free
topology. The topological semantics of the provability logic GL and the
polymodal provability logic GLP is reviewed in the chapter by Lev Beklemishev
and David Gabelaia, who also point out interesting connections between the
topological semantics of GLP, large cardinals, and consistency issues in set
theory.
As we pointed out, Esakia and Simmons were the first who developed the
topological semantics for the provability logic GL. In [16], Esakia introduced a
weakening of the modal system K4, which he termed weakK4 and denoted by
wK4. He showed that when interpreting modal diamond as the derivative of a
topologicalspace,wK4isthemodallogicofalltopologicalspaces,andthatK4is
themodallogicofallspacessatisfyingtheso-calledT -separationaxiom(alower
d
xii Introduction
separationaxiomproperlysituated between T andT ,asserting thateach pointis
0 1
locally closed). These results were originally obtained by Leo in the 1970s, but
werepublishedforthefirsttimeonlyin2001.Furtherresultsinthisdirectionwere
obtainedbyLeoandhiscollaboratorsinthefollow-uppapers[17,4–7],aswellas
byShehtmanandhisschool,JoelLucero-Bryan,andothers.Thelastchapterinthe
volume by Andrey Kudinov and Valentin Shehtman is dedicated to the
derivational semantics of modal logic and other related issues.
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