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QUANTIFICATION IN NONCLASSICAL LOGIC

STUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 153
Honorary Editor:
P. SUPPES
Editors:
S. ABRAMSKY, London
S. ARTEMOV, Moscow
D.M. GABBAY, London
A. KECHRIS, Pasadena
A. PILLAY, Urbana
R.A. SHORE, Ithaca
AMSTERDAM BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
•

QUANTIFICATION
IN NONCLASSICAL LOGIC
VOLUME 1
D. M. Gabbay
King’s College London, UK
and
Bar-Ilan University, Ramat-Gan, Israel
V. B. Shehtman
Institute for Information Transmission Problems
Russian Academy of Sciences
and
Moscow State University
D. P. Skvortsov
All-Russian Institute of Scientific and Technical Information
Russian Academy of Sciences
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier
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09 10 10 9 8 7 6 5 4 3 2 1

Preface
If some 30 years ago we had been told that we would write a large book on
quantification in nonclassical logic, none of us would have taken it seriously: first
-because at that time there was no hope of our effective collaboration; second
- because in nonclassical logic too much had to be done in the propositional
area, and few people could find the energy for active research in predicate logic.
In the new century the situation is completely different. Connections be-
tween Moscow and London became easy. The title of the book is not surprising,
and we are now late with the first big monograph in this field. Indeed, at first
we did not expect we had enough material for two (or more) volumes. But we
hope readers will be able to learn the subject from our book and find it quite
fascinating.
Let us now give a very brief overview of the existing systematic expositions
of nonclassical first-order logic. None of them aims at covering the whole of
this large field. The first book on the subject was [Rasiowaand Sikorski, 19631,
where the approach used by the authors was purely algebraic. Many important
aspects of superintuitionistic first-order logics can be found in the books written
in the 1970-80s: [Dragalin, 19881(proof theory; algebraic, topological, and rela-
tional models; realisability semantics); [Gabbay, 19811 (model theory; decision
problem); [van Dalen, 19731, [Troelstra and van Dalen, 19881 (realisability and
model theory). The book of Novikov [Novikov,19771(the major part of which is
a lecture course from the 1950s)addresses semantics of superintuitionistic logics
and also includes some material on modal logic.
Still, predicate modal logic was partly neglected until the late 1980s. The
book [Harel, 19791and its later extended version [Hareland Tiuryn, 20001 study
particular dynamic modal logics. [Goldblatt, 19841 is devoted to topos seman-
tics; its main emphasis is on intuitionistic logic, although modal logic is also
considered. The book [Hughes and Cresswell, 19961 makes a thorough study
of Kripke semantics for first-order modal logics, but it does not consider other
semantics orgintermediate logics. Finally, there is a monograph [Gabbay et al.,
20021, which, among other topics, investigates first-order modal and intermedi-
ate logics from the Lmany-dimensional'viewpoint. It contains recent profound
results on decidable fragments of predicate logics.
The lack of unifying monographs became crucial in the 1990s, to the extent
that in the recent book [Fitting and Mendelsohn, 19981 the area of first-order
modal logic was unfairly called 'under-developed'. That original book contains

vi PREFACE
interesting material on the history and philosophy of modal logic, but due to its
obvious philosophical flavour, it leaves many fundamental mathematical prob-
lems and results unaddressed. Still there the reader can find various approaches
to quantification, tableaux systems and corresponding completeness theorems.
So there remains the need for a foundational monograph not only addressing
areas untouched by all current publications, but also presenting a unifying point
of view.
A detailed description of this Volume can be found in the Introduction below.
It is worth mentioning that the major part of the material has never been
presented in monographs. One of its sources is the paper on completeness and
incompleteness, a brief version of which is [Shehtman and Skvortsov, 19901; the
full version (written in 1983) has not been published for technical reasons. The
second basic paper incorporated in our book is [Skvortsov and Shehtman, 19931,
where so-called metaframe (or simplicial) semantics was introduced and studied.
We also include some of the results obtained after 1980by G. Corsi, S. Ghilardi,
H. Ono, T. Shimura, D. Skvortsov, N.-Y. Suzuki, and others.
However, because of lack of space, we had to exclude some interesting ma-
terial, such as a big chapter on simplicial semantics, completeness theorems
for topological semantics, hyperdoctrines, and many other important matters.
Other important omissions are the historical and the bibliographical overviews
and the discussion of application fields and many open problems. Moreover, the
cooperation between the authors was not easy, because of the different view-
points on the presentation.1 There may be also other shortcomings, like gaps in
proofs, wrong notation, wrong or missing references, misprints etc., that remain
uncorrected -but this is all our responsibility.
We would be glad to receive comments and remarks on all the defects from
the readers. As we are planning to continue our work in Volume 2, we still hope
to make all necessary corrections and additions in the real future.
At present the reader can find the list of corrections on our webpages
http://www.dcs.kcl.ac.uk/staff/dg/
http://lpcs.math.msu.su/~shehtman
Acknowledgements
The second and the third author are grateful to their late teacher and friend
Albert Dragalin, one of the pioneers in the field, who stimulated and encouraged
their research.
We thank all our colleagues, with whom we discussed the contents of this
book at different stages - Sergey Artemov, Lev Beklemishev, Johan van Ben-
them, Giovanna Corsi, Leo Esakia, Silvio Ghilardi, Dick De Johng, Rosalie
Iemhoff, Marcus Kracht, Vladimir Krupski, Grigori Mints, Hiroakira Ono, Nobu-
Yuki Suzuki, Albert Visser, Michael Zakharyaschev. Nobu-Yuki and Marcus
also kindly sent us Latex-files of their papers; we used them in the process of
typing the text.
'One of the authors points out that he disagrees with some of the notation and the style
of some proofs in the final version.

PREFACE vii
We would like to thank different institutions for help and support -King's
College of London; Institute for Information Transmission Problems, VINITI,
Department of Mathematical Logic and Theory of Algorithms at Moscow State
University, Poncelet Mathematical Laboratory, Steklov Mathematical Institute
in Moscow; IRIT in Toulouse; EPSRC, RFBR, CNRS, and NWO.
We add personal thanks to our teacher Professor Vladimir A. Uspensky, to
Academician Sergey I. Adian, and to our friend and colleague Michael Tsfasman
who encouraged and supported our work.
We are very grateful to Jane Spurr for her enormous work and patience
in preparation of the manuscript - typing the whole text in Latex (several
times!), correcting our mistakes, reading multiple pages (with hardly under-
standable handwriting), making pictures, arranging styles etc. etc. We thank
all those people who also essentially helped us in this difficult process - Ilya
Shapirovsky, Alexey Romanov, Stanislav Kikot for correcting mistakes and typ-
ing, Ilya Vorontsov and Daniel Vorontsov for scanning many hundreds of pages.
We thank all other peopie for their help and encouragement - our wives
Lydia, Marina, Elena, families, friends and colleagues.

This page intentionally left blank

Introduction
Quantification and modalities have always been topics of great interest for lo-
gicians. These two themes emerged from philosophy and language in ancient
times; they were studied by traditional informal methods until the 20th century.
Then the tools became highly mathematical, as in the other areas of logic, and
modal logic as well as quantification (mainly on the basis of classical first-order
logic) found numerous applications in Computer Science.
At the same time many other kinds of nonclassical logics were investigated.
In particular, intuitionistic logic was created by L. Brouwer at the beginning of
the century as a new basis for mathematical reasoning. This logic, as well as its
extensions (superintuitionistic logics), is also very useful for Computer Science
and turns out to be closely related to modal logics.
(A) The introduction of quantifier axioms to classical logic is fairly straight-
forward. We simply add the following obvious postulates to the propositional
logic:
where t is 'properly' substituted for x
where x is not free in A
where x is not free in A.
The passage from the propositional case of a logic L to its quantifier case
works for many logicsby adding the above axioms to the respective propositional
axioms -for example, the intuitionistic logic, standard modal logics S4,S5,K
etc. We may need in some cases to make some adjustment to account for
constant domains, Vx(A V B(x)) 3 ( AV VxB(x)) in case of intuitionistic logic
and the Barcan formula, VxOA(x) > UVxA(x) in the case of modal logics. On
the whole the correspondence seems to be working.

x INTRODUCTION
The recipe goes on as follows:
take the propositional semantics and put a domain D, in each world u or take
the axiomatic formulation and add the above axioms and you maintain corre-
spondence and completeness.
There were some surprises however. Unexpectedly, this method fails for very
simple and well-known modal and intermediate logics: the 'Euclidean logic'
K5 = K +OOp > U p (see Chapter 6 of this volume), the 'confluence logic'
S4.2 = S4 fOOp > D0p and for the intermediate logic KC = H +~p V 1-p
with constant domains, nonclassical intermediate logics of finite depth [Ono,
19831, etc. All these logics are incomplete in the standard Kripke semantics.
In some other cases, completeness theorems hold, but their proofs require
nontrivial extra work -for example, this happens for the logic of linear Kripke
frames S4.3 [Corsi, 19891.
This situation puts at least two difficult questions to us: (1) how should
we change semantics in order to restore completeness of 'popular' logics? (2)
how should we extend these logics by new axioms to make them complete in
the standard Kripke semantics? These questions will be studied in our book,
especially in Volume 2, but we are still very far from final answers.
Apparently when we systematically introduce natural axioms and ask for
the corresponding semantics, we may not be able to see what are the natural
semantical conditions (which may not be expressible in first-order logic) and
converselysome natural conditions on the semantics require complex and some-
times non-axiomatisable logics.
The community did not realize all these difficulties. A serious surprise was
the case of relevance logic, where the additional axioms were complex and
seemed purely technical. See [Mares and Goldblatt, 20061, [Fine, 19881, [Fine,
19891. For some well-known logics there were no attempts of going first-order,
especially for resource logics such as Lambek Calculus.
(B) There are other reasons why we may have difficulties with quantifiers,
for example, in the case of superintuitionistic logics. Conditions on the possible
worlds such as discrete ordering or finiteness may give the connectives them-
selves quantificational power of their own (note that the truth condition for
A > B has a hidden world quantifier), which combined with the power of the
explicit quantifiers may yield some pretty complex systems [Skvortsov,20061.
(C) In fact, a new approach is required to deal with quantifiers in possible
world systems. The standard approach associates domains with each possible
world and what is in the domain depends only on the nature of the world, i.e.
if u is a world, P a predicate, 6 a valuation, then B,(P) is not dependent on
other 0,~(P), except for some very simple conditions as in intuitionistic logic.
There are no interactive conditions between existence of elements in the
domain and satisfaction in other domains. If we look at some axioms like the
Markov principle
l--dxA(x) > 3 ~ 1 7 A ( x ) ,
we see that we need to pay attention on how the domain is constructed. This
is reminiscent of the Herbrand universe in classical logic.

INTRODUCTION xi
(D) There are other questions which we can ask. Given a classical theory I
'
(e.g. a theory of rings or Peano arithmetic), we can investigate what happens
if we change the underlying logic to intuitionistic or modal or relevant. Then
what kind of theory do we get and what kind of semantics? Note we are not
dealing now with a variety of logics (modal or superintuitionistic), but with a
fixed nonclassical logic (say intuitionistic logic itself) and a variety of theories.
If intuitionistic predicate semantics is built up from classical models, would
the intuitionistic predicate theory of rings have semantics built up from classical
rings? How does it depend on the formulation (rmay be classically equivalent
to I",but not intuitionistically) and what can happen to different formulations?
See [Gabbay, 19811.
(E) One can have questions with quantifiers arising from a completely dif-
ferent angle. E.g. in resource logics we pay attention to which assumptions are
used to proving a formula A.
For example in linear or Lambek logic we have that
(3) A -+ (A -+ B).
can prove B but (2) and (3) alone cannot prove B; because of resource consider-
ations, we need two copies of A. Such logics are very applicable to the analysis
and modelling of natural language [van Benthem, 19911. So what shall we do
with 'dxA(x)? Do we divide our resource between all instances A(tl),A(t2),...
of A? These are design questions which translate into technical axiomatic and
semantical questions.
How do we treat systems which contain more than one type of nonclassical
connective? Any special problems with regard to adding quantifiers? See, for
example, the theory of bunched implications [O'Hearn and Pym, 19991.
(F)The most complex systems with regards to quantifiers are LDS, Labelled
Deductive Systems (this is a methodology for logic, cf. [Gabbay,1996;Gabbay,
19981). In LDS formulas have labels, so we write t : A, where t is a label and A is
a formula. Think o f t as a world or a context. (This label can be integrated and
in itself be a formula, etc.) Elements now have visa rules for migrating between
labels and need to be annotated, for example as a:, the element a exists at
world s, but was first created (or instantiated) in world t. Surprisingly, this
actually helps with the proof theory and semantics for quantifiers, since part of
the semantics is brought into the syntax. See [Viganb, 20001. So it is easier to
develop, say, theories of Hilbert &-symbolusing labels. &-symbolsaxioms cannot
be added simple mindedly to intuitionistic logic, it will collapse [Bell,20011.
(G) Similarly, we must be careful with modal logic. We have not even
begun thinking about &-symbolsin resource logics (consider ~x.A(x),
if there
is sensitivity for the number of copies of A, then are we to be sensitive also to
copies of elements?).
(H) In classical logic there is another direction to go with quantifiers, namely
the so-called generalised quantifiers, for example (manyx)A(x) ('there are many

xii INTRODUCTION
x such that A(x)'), or (uncountably many x)A(x) or many others. Some of
these can be translated as modalities as van Lambalgen has shown [Alechina,
van Lambalgen, 19941, [vanLambalgen, 19961. Such quantifiers (at least for the
finite case) exist in natural language. They are very important and they have
not been exported yet to nonclassical logics (only through the modalities e.g.
0,A ('Ais true in n possible worlds'), see [Gabbay,Reynolds and Finger, 20001,
[Peters and Westerstahl, 20061).
Volume 1 of these books concentrates on the landscape described in (A)
above, i.e., correspondence between axioms for modal or intuitionistic logic and
semantical conditions and vice versa.
Even for such seemingly simple questions we have our hands full. The table
of contents for future volumes shows what to be addressed in connection with
(B)-(H). It is time for nonclassical logic to pay full attention to quantification.
Up to now the focus was mainly propositional. Now the era of the quantifier
has begun!
This Volume includes results in nonclassical first-order logic obtained during
the past 40 years. The main emphasis is model-theoretic, and we confine our-
selves with only two kinds of logics: modal and superintuitionistic. Thus many
interesting and important topics are not included, and there remains enough
material for future volumes and future authors.
Figure 1. Chapters dependency structure
Let us now briefly describe the contents of Volume 1. It consists of three
parts. Part I includes basic material on propositional logic and first-order syn-
tax.
Chapter 1 contains definitions and results on syntax and semantics of non-
classical propositional logics. All the material can be found elsewhere, so the
proofs are either sketched or skipped.
Chapter 2 contains the necessary syntactic background for the remaining
parts of the book. Our main concern is the precise notion of substitution based

INTRODUCTION xiii
on re-naming of variables. This classical topic is well known to all students in
logic. However none of the existing definitions fits well for our further purposes,
because in some semantics soundness proofs may be quite intricate. Our a p
proach is based on the idea that re-naming of bound variables creates different
synonymous (or 'congruent') versions of the same predicate formula. These ver-
sions are generated by a 'scheme' showing the reference structure of quantifiers.
(Schemes are quite similar to formulas in the sense of [Bourbaki, 19681.) Now
variable substitutions (acting on schemes or congruence classes) can be easily
arranged in an appropriate congruent version. After this preparation we intro-
duce two main types of first-order logics to be studied in the book - modal
and superintuitionistic, and prove syntactic results that do not require involved
proof theory, such as deduction theorems, Glivenko theorem etc.
In Part I1 (Chapters 3 - 5) we describe different semantics for our logics and
prove soundness results.
Chapter 3 considers the simplest kinds of relational semantics. We begin
with the standard Kripke semantics and then introduce two its generalisations,
which are equivalent: Kripke frames with equality and Kripke sheaves. The first
one (for the intuitionistic case) is due to [Dragalin,19731,and the second version
was first introduced in [Shehtman and Skvortsov, 1990]. Soundness proofs in
that chapter are not obvious, but rather easy. We mention simple incomplete-
ness results showing that Kripke semantics is weaker than these generalisations.
Further incompleteness theorems are postponed until Volume 2. We also prove
results on Lowenheim - Skolem property and recursive axiomatisability using
translations to classical logic from [Ono, 19721731and [van Benthem, 19831.
Chapter 4 studies algebraic semantics. Here the main objects are Heyting-
valued (or modal-valued) sets. In the intuitionistic case this semantics was stud-
ied by many authors, see [Dragalin, 19881, [Fourman and Scott, 19791, [Gold-
blatt, 19841. Nevertheless, our soundness proof seems to be new. Then we show
that algebraic semantics can be also obtained from presheaves over Heyting (or
modal) algebras. We also show that for the case of topological spaces the same
semantics is given by sheaves and can be defined via so-called 'fibrewise mod-
els'. These results were first stated in [Shehtman and Skvortsov, 1990],but the
proofs have never been published so far.2 They resemble the well-known results
in topos theory, but do not directly follow from them.
In Chapter 5 we study Kripke metaframes, which are a many-dimensional
generalisation of Kripke frames from [Skvortsov and Shehtman, 19931 (where
they were called 'Cartesian metaframes'). The crucial difference between frames
and metaframes is in treatment of individuals. We begin with two particular
cases of Kripke metaframes: Kripke bundles [Shehtman and Skvortsov, 19901
and C-sets (sheaves of sets over (pre)categories) [Ghilardi, 19891. Their prede-
cessor in philosophical logic is 'counterpart theory' [Lewis, 19681. In a Kripke
bundle individuals may have several 'inheritors7 in the same possible world,
while in a C-set instead of an inheritance relation there is a family of maps. In
2 ~ h e
first author is happy to fulfill his promise given in the preface of [Gabbay, 19811: "It
would require further research to be able to present a general theory [of topological models,
second order Beth and Kripke models] possibly using sheaves".

xiv INTRODUCTION
Kripke metaframes there are additional inheritance relations between tuples of
individuals.
The proof of soundness for metaframes is rather laborious (especially for
the intuitionistic case) and is essentially based on the approach to substitutions
from Chapter 2. This proof has never been published in full detail. Then we
apply soundness theorem to Kripke bundle and functor semantics. The last
section of Chapter 5 gives a brief introduction to an important generalisation of
metaframe semantics - so called 'simplicia1 semantics'. The detailed exposition
of this semantics is postponed until Volume 2.
Part I11 (Chapters 6-7) is devoted to completeness results in Kripke seman-
tics. In Kripke semantics many logics are incomplete, and there is no general
powerful method for completeness proofs, but still we describe some approaches.
In Chapter 6 we study Kripke frames with varying domains. First, we in-
troduce different types of canonical models. The simplest kind is rather well-
known, cf. [Hughes and Cresswell, 19961, but the others are original (due to
D. Skvorstov). We prove completeness for intermediate logics of finite depth
[Yokota, 19891, directed frames [Corsiand Ghilardi, 19891,linear frames [Corsi,
19921. Then we elucidate the methods from [Skvortsov, 19951 for axiomatising
some 'tabular' logics (i.e., those with a fixed frame of possible worlds).
Chapter 7 considers logics with constant domains. We again present dif-
ferent canonical models constructions and prove completeness theorems from
[Hughes and Cresswell, 19961. Then we prove general completeness results for
subframe and cofinal subframe logics from [Tanaka and Ono, 19991, [Shimura,
19931, [Shimura, 20011, Takano's theorem on logics of linearly ordered frames
[Takano, 19871and other related results.
Here are chapter headings in preparation for later volumes:
Chapter 8. Simplicia1semantics
Chapter 9. Hyperdoctrines
Chapter 10. Completeness in algebraic and topological semantics
Chapter 11. Translations
Chapter 12. Definability
Chapter 13. Incompleteness
Chapter 14. Simulation of classical models
Chapter 15. Applications of semantical methods
Chapter 16. Axiomatisable logics
Chapter 17. Further results on Kripke-completeness
Chapter 18. Fragments of first-order logics
Chapter 19. Propositional quantification

INTRODUCTION
Chapter 20. Free logics
Chapter 21. Skolemisation
Chapter 22. Conceptual quantification
Chapter 23. Categorical logic and toposes
Chapter 24. Quantification in resource logic
Chapter 25. Quantification in labelled logics.
Chapter 26. E-symbolsand variable dependency
Chapter 27. Proof theory
Some guidelines for the readers. Reading of this book may be not so easy.
Parts 11, I11 are the most important, but they cannot be understood without
Part I.
For the readers who only start learning the field,we recommend to begin with
sections 1.1-1.5, then move to sections 2.1, 2.2, the beginning parts of sections
2.3, 2.6, and next to 2.16. After that they can read Part I1 and sometimes go
back to Chapters 1,2 if necessary. We do not recommend them to go to Chapter
5 before they learn about Kripke sheaves. Those who are only interested in
Kripke semantics can move directly from Chapter 3 to Part 111.
An experienced reader can look through Chapter 1and go to sections 2.1-2.5
and the basic definitions in 2.6, 2.7. Then he will be able to read later Chapters
starting from Chapter 3.

xvi INTRODUCTION
Notation convention
We use logical symbols both in our formal languages and in the meta-language.
The notation slightly differs, so the formal symbols A, 3, = correspond to
the metasymbols &, =+,H; and the formal symbols V, 3, V are also used as
metasymbols.
In our terminology we distinguish functions and maps. A function from A
to B is a binary relation F C A x B with domain A satisfying the functionality
condition (xFy & x F t =+ x = z), and the triple f = (F,A, B) is then called a
map from A to B. In this case we use the notation f : A ---+ B.
Here is some other set-theoretic notation and terminology.
2X denotes the power set of a set X ;
we use for inclusion, C for proper inclusion;
R o S denotes the composition of binary relations R and S:
R o S := {(x,y) 1 3 t (xRz & zSy));
R - ~
is the converse of a relation R;
Idw is the equality relation in a set W;
idw is the identity map on a set W (i.e. the triple (Idw,W, W));
for a relation R W x W, R(V), or just RV, denotes the image of a set
V 5 W under R, i.e. {y I 3x E V xRy); R(x) or Rx abbreviates R({x));
dom(R),or prl(R), denotes the domain of a relation R, i.e., {x I 3y xRy);
rng(R), or prz(R),denotes the range of a relation R, i.e., {Y I 3x XRY};
for a subset X C Y there is the inclusion map jxy : X -Y (which is
usually denoted just by j ) sending every x E X to itself;
R 1V denotes the restriction of a relation R to a subset V, i.e.
R 1V = R n(V x V), and f V denotes the restriction of a map f to V;
for a relation R on a set X R- :=R - Idx is the 'irreflixivisation' of R;
1
x
1denotes the cardinality of a set X;
I
, denotes the set (1,...,n);I. := 0 ;
X M denotes the set of all finite sequences with elements in X ;
(Xi Ii E I ) (or (Xi)iEr) denotes the family of sets Xi with indices in the
set I;
U Xi denotes the disjoint union of the family (Xi)iET,
i.e. I
J Xix {i};
icI i E I

INTRODUCTION xvii
w is the set of natural numbers, and T, denotes wm;
Cmn= (In)Imdenotes the set of all maps a : I
, -In(for m, n E w ) ;
Tmn denotes the set of all injective maps in Em,;
T, is the abbreviation for T,,, the set of all permutations of I,.
Note that we use two different notations for composition of maps: the compo-
sition of f : A -
-
i
B and g : B -C is denoted by either g .f or f o g. So
(fog)(x) = (9 .f)(x) = g(f (x)).
Obviously,
C m n # O i f f n > O o r m = O ,
T m , # O i f f n > m .
A map f : I
, -In(for fixed n) is presented by the table
We use a special notation for some particular maps.
Trans~ositions
02 E Tn for n 2 2, 15 i <j 5 n.
In particular, simple transpositions are a; :=a
; for 1< i 5 n;
Standard embeddings (inclusion maps).
a?" E T,,, for 0 5 m <n is defined by the table
In particular, there are simple embeddings al;L := +
for m 2 0;
0, :=a? is the empty map I
. ---+ In(and obviously, Con = (0,)).
Facet embeddings S
; E Tn-l,n for n > 0.
In particular, 6
; =a:-'.
Standard projections a?" E Cmn for m >n > 0.
In particular, simple projections are a
: := a
:
'
'
'
" for n > 0.

xviii INTRODUCTION
It is well-known that (for n > 1) every permutation a E T, is a composition
of (simple) transpositions. One also can easily show that every map from C
,
,
is a composition of simple transpositions, simple embeddings, and simple pro-
jections. In particular, every injection (from T,,) is a composition of simple
transpositions and simple embeddings, and every surjection is a composition of
simple transpositions and simple projections, cf. [Gabriel and Zisman, 19671.
The identity map in C,, is id, := id^, = a;4" = a", and it is obvious that
id, =a; oayi whenever n 2 2, j < i.
Let also AT E El, be the map sending 1to i; let A
; E Can be the map with
the table
( :)
For every a E C
,
, we define its simple extension a+ E Em+l,n+l such that
a(i) for i E I,,
a+(i) :=
n + l i f i = m + l .
In particular, for any n we have (a;)+ = 6
;
:
; E Cn+l,n+a:
for i E I,,
i f i = n + l .
We do not make any difference between words of length n in an alphabet D
and n-tuples from Dn. So we write down a tuple (al, ...,a,) also as a1 ...a,.
denotes the void sequence;
Z
(
I
a
1
)(or lal) denotes the length of a sequence a;
ap denotes the join (the concatenation) of sequences a, p; we often write
x1 ...xn rather than (xl, ...,x,) (especially if n = I), and also a x or
(a,x) rather than the dubious notation a(%);
For a letter c put
ck : = c ...C .
w
k
For an arbitrary set S, every tuple a = (al,...,a,) E Sncan be regarded as
a function In -S. We usually denote the range of this function, i.e. the set
{al,...,a,) as r(a). Sometimes we write b E a instead of b E r(a). Every map
a : I
, -Inacts on Snvia composition:
Thus every map a E C
,
, gives rise to the map .rr, : Sn -Sm sending a to
a . a. In the particular case, when a = 61 is a facet embedding and a E Sn,we
also use the notation ~1
:=ng; and
A
nla := a - ai :=ai :=a . bn = (al,...tai-l,ai+l,...,an).
Hence we obtain

INTRODUCTION xix
Lemma 0.0.1 (1)
.rrT .To =
whenever a E Sn,a E Em,, 7 E Ck,.
(2) I
f a is a permutation (a E T,), then T, is a permutation of Sn and
To-1 = (r,)-l.
Proof (1)Since composition of maps is associative, we have
a . (a . T ) = ( a .a ) .r.
We use the following relations on n-tuples:
Lemma 0.0.2 Let S # 0, a E Em,. Then
r,[Sn]= {aE SmI asuba),
where asuba denotes the property Vi,j (a(i) = a(j) +ai = aj), cf. (a).
Proof In fact, if a = b .a , then obviously a ( j ) = a(k) implies a j = ak. On
the other hand, if a suba, then a = b .a for some b; just put b,(%) := ai and
add arbitrary bk for k @ r(a).
Lemma 0.0.3 For IS/ > 1, a E Em,, a is injective iff .rr, : Sn + Smis
surje~tive.~
Proof If a is injective, then for any a E Sn,a(i) = a(j) +i = j +ai = aj,
i.e. a suba. Hence by Lemma 0.0.2, n, is surjective.
The other way round, if a(i) = a(j) for some i # j, take a E Smsuch that
ai # aj. Then a suba is not true, i.e. a 9.rr, [Sn]
by 0.0.2.
Lemma 0.0.4 For I
S
1 > 1, a E Em,, a is surjective iff r, is injective.
Proof Suppose a : I
, -Inis surjective and a,b E Sn,.rr,a # nub. If
.rr,a and .rr,b differ at the j th component, then ai # bi for i Insuch that
a($)=j. On the other hand, let a E Em, be non-surjective, j EI
, - rng(a). Let
c,d E S, c # d. Takea= cn; b =cf-'den-j. Then a#band-ir,a=n,b=cm.
w
Hence we obtain
Lemma 0.0.5 For I
S
1 > 1, a E C,,, a is bijective ijf -ir, : Sn + Smis
bijective.
3Clearly, if I
S
1 = 1, then T , is bijective for every a E C,,.

xx INTRODUCTION
We further simplify notation in some particular cases. Let T
: := .rrs;, so
facet embedding 61 eliminates the ith component from an n-tuple a E Sn. Let
also
I
T
: :=ran,
- 71; :=T ~ ; ,
where a? E is a simple projection, a; E is a simple embedding.
Thus
T? (al,...,an) = (al, ...,an,a,) for n > 0,
~ ; ( a )= a - an+l = ( a ~ ,
...,an) for a = (al,...,an,an+l) E Dntl, n >0.
We say that a sequence a E Dn is distinct, if all its components at are
different.
Lemma 0.0.6 If a, T : I
, ---,I,, a # T and I
S
1 2 n, then a . a # a . 7 for any
distinct a E Sn.
Proof If for some i, ~ ( i )
# a(i), then a,(i) # a,(i).
Lemma 0.0.7 (1) For T E Em,, a E Ekm,
( r . u ) + = r +. u + .
(2) For a E Em,,
a+-a+m=a;.a
Proof Straightforward. ¤
Lemma 0.0.8 (1) Let a E Sn, b E Sm,r(b) C_ r(a). Then b = a . o for
some a E Em,.
(2) Moreover, z
f b is di~tinct,~
then u is an injection.
Proof Put a(i) = j for some j such that bi = aj.
41n other words, b is obtained by renumbering a subsequence of a.

Contents
Preface v
Introduction ix
I Preliminaries 1
1 Basic propositional logic 3
. . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Propositional syntax 3
. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Formulas 3
1.1.2 Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
. . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Algebraic semantics 11
. . . . . . . . . . . . . . .
1.3 Relational semantics (the modal case) 19
. . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Introduction -- 19
. . . . . . . . . . . . . . . . . .
1.3.2 Kripke frames and models 19
. . . . . . . . . . . . . . . . . . . . . .
1.3.3 Main constructions 24
. . . . . . . . . . . . . . . . . . . .
1.3.4 Conical expressiveness 30
. . . . . . . . . . . .
1.4 Relational semantics (the intuitionistic case) 32
. . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Modal counterparts 37
1.6 General Kripke frames . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . .
1.7 Canonical Kripke models 40
. . . . . . . . . . . . . .
1.8 First-order translations and definability 44
. . . . . . . . . . . . . . . .
1.9 Some general completeness theorems 47
. . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Trees and unravelling 48
. . . . . . . . . . . . . . . . . . . .
1.11 PTC-logics and Horn closures 52
. . . . . . . . . . . . . . . .
1.12 Subframe and cofinal subframe logics 58
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.13 Splittings 65
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.14 Tabularity 68
. . . . . . . . . . . . . . . . . . .
1.15 Transitive logics of finite depth 70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.16 A-operation 72
. . . . . . . . . . . . . . . . . . . . . .
1.17 Neighbourhood semantics 76
xxi

xxii CONTENTS
2 Basic predicate logic 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Formulas 81
2.3 Variable substitutions . . . . . . . . . . . . . . . . . . . . . . . . 85
. . . . . . . . . . . . . . . . . . . . . . .
2.4 Formulas with constants 102
. . . . . . . . . . . . . . . . . . . . . . . .
2.5 Formula substitutions 105
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 First-order logics 119
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 First-order theories 139
. . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Deduction theorems 142
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Perfection 146
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Intersections 151
. . . . . . . . . . . . . . . . . . . . . . .
2.11 Godel-Tarski translation 153
. . . . . . . . . . . . . . . . . . . . . . . .
2.12 The Glivenko theorem 157
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 A-operation 158
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 Adding equality 172
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.15 Propositional parts 180
2.16 Semantics from an abstract viewpoint . . . . . . . . . . . . . . . 185
I1 Semantics 191
Introduction: What is semantics? . . . . . . . . . . . . . . . . . . . . . 193
3 Kripke semantics 199
. . . . . . . . . . . . . . . . . . . . . . . .
3.1 Preliminary discussion 199
. . . . . . . . . . . . . . . . . . . . . . .
3.2 Predicate Kripke frames 205
. . . . . . . . . . . . . . . . . . . . .
3.3 Morphisms of Kripke frames 219
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Constant domains 230
. . . . . . . . . . . . . . . . . . . . .
3.5 Kripke frames with equality 234
. . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Introduction 234
. . . . . . . . . . . . . . . . .
3.5.2 Kripke frames with equality 235
. . . . . . . . . . . . . . . . . . . . . .
3.5.3 Strong morphisms 239
. . . . . . . . . . . . . . . . . . . . . .
3.5.4 Main constructions 241
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Kripke sheaves 243
. . . . . . . . . . . . . . . . . . . .
3.7 Morphisms of Kripke sheaves 253
. . . . . . . . . . . . . . . . . . . . . . .
3.8 Transfer of completeness 259
. . . . . . . . . . . . . . . . . . .
3.9 Simulation of varying domains 266
. . . . . . . . . . . . . . . . . . . .
3.10 Examples of Kripke semantics 268
. . . . . . . . . . . . .
3.11 On logics with closed or decidable equality 277
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.11.1 Modal case 277
. . . . . . . . . . . . . . . . . . . . . . .
3.11.2 Intuitionistic case 279
. . . . . . . . . . . . . . . . . . .
3.12 Translations into classical logic 281

CONTENTS xxiii
4 Algebraic semantics 293
4.1 Modal and Heyting valued structures . . . . . . . . . . . . . . . . 293
4.2 Algebraic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
4.3 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
4.4 Morphisms of algebraic structures . . . . . . . . . . . . . . . . . 319
4.5 Presheaves and Sl-sets . . . . . . . . . . . . . . . . . . . . . . . . 328
4.6 Morphisms of presheaves . . . . . . . . . . . . . . . . . . . . . . . 333
4.7 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
4.8 Fibrewise models . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
4.9 Examples of algebraic semantics . . . . . . . . . . . . . . . . . . 341
5 Metaframe semantics 345
5.1 Preliminary discussion . . . . . . . . . . . . . . . . . . . . . . . . 345
5.2 Kripke bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
5.3 More on forcing in Kripke bundles . . . . . . . . . . . . . . . . . 356
5.4 Morphisms of Kripke bundles . . . . . . . . . . . . . . . . . . . . 359
5.5 Intuitionistic Kripke bundles . . . . . . . . . . . . . . . . . . . . 365
5.6 Functor semantics . . . . . . . . . . . . . . . . . . . . . . . . . . 374
5.7 Morphisms of presets . . . . . . . . . . . . . . . . . . . . . . . . . 381
5.8 Bundles over precategories . . . . . . . . . . . . . . . . . . . . . . 386
5.9 Metaframes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
5.10 Permutability and weak functoriality . . . . . . . . . . . . . . . . 397
5.11 Modal metaframes . . . . . . . . . . . . . . . . . . . . . . . . . . 404
5.12 Modal soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
5.13 Representation theorem for modal metaframes . . . . . . . . . . 419
5.14 Intuitionistic forcing and monotonicity . . . . . . . . . . . . . . . 422
5.14.1 Intuiutionistic forcing . . . . . . . . . . . . . . . . . . . . 422
5.14.2 Monotonic metaframes . . . . . . . . . . . . . . . . . . . . 429
5.15 Intuitionistic soundness . . . . . . . . . . . . . . . . . . . . . . . 432
5.16 Maximality theorem . . . . . . . . . . . . . . . . . . . . . . . . . 452
5.17 Kripke quasi-bundles . . . . . . . . . . . . . . . . . . . . . . . . . 465
5.18 Some constructions on metaframes . . . . . . . . . . . . . . . . . 467
5.19 On semantics of intuitionistic sound metaframes . . . . . . . . . 469
5.20 Simplicia1frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
I11 Completeness 481
6 Kripke completeness for varying domains 483
. . . . . . . . . . . . . . . . .
6.1 Canonical models for modal logics 483
6.2 Canonical models for superintuitionistic
logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
6.3 Intermediate logics of finite depth . . . . . . . . . . . . . . . . . . 501
6.4 Natural models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
6.5 Refined completeness theorem for QH +KF . . . . . . . . . . . 515
6.6 Directed frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

xxiv CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . .
6.7 Logics of linear frames 524
. . . . . . . . . . . . . . . . . . . . . .
6.8 Properties of A-operation 528
. . . . . . . . . . . . . . . . .
6.9 A-operation preserves completeness 532
. . . . . . . . . . . . . . .
6.10 Trees of bounded branching and depth 536
. . . . . . . . . . . . . . . . . . . . . . . .
6.11 Logics of uniform trees 539
7 Kripke completeness 553
. . . . . . . . . .
7.1 Modal canonical models with constant domains 553
. . . . . .
7.2 Intuitionistic canonical models with constant domains 555
. . . . . . . . . . . . . . . .
7.3 Some examples of C-canonical logics 558
. . . . . . . . .
7.4 Predicate versions of subframe and tabular logics 563
. . . . . . . . . . . .
7.5 Predicate versions of cofinal subframe logics 565
. . . . . . . . . . . . . . .
7.6 Natural models with constant domains 573
. . . . . . . .
7.7 Remarks on Kripke bundles with constant domains 577
. . . . . . . . . . .
7.8 Kripke frames over the reals and the rationals 579
Bibliography 593
Index 603

Chapter 1
Basic propositional logic
This chapter contains necessary information about propositional logics. We
give all the definitions and formulate results, but many proofs are sketched or
skipped. For more details we address the reader to textbooks and monographs
in propositional logic: [Goldblatt, 19871, [Chagrov and Zakharyaschev, 1997],
[Blackburn, de Rijke and Venema, 2001], also see [Gabbay, 1981], [Dragalin,
19881, [van Benthem, 19831.
1.1 Propositional syntax
1.1.1 Formulas
We consider N-modal (propositional) formulas1 built from the denumerable set
PL = {pl,p2,...) of proposition letters, the classical propositional connectives
A, V, >, Iand the unary modal connectives 01,
...,ON; the derived connec-
tives are introduced in a standard way as abbreviations:
7A:= (A 3 I),
T := (I>
I),
(A =B) := ((A > B) A (B > A)),
OiA :=l U i l A for i = 1,...,N.
To simplify notation, we write p, q, r instead of pl, pz, p3. We also use standard
agreements about bracketing: the principal brackets are omitted; A is stronger
than V, which is stronger than > and -. Sometimes we use dots instead of
brackets; so, e.g. A 3. B > C stands for (A > (B > C)).
For a seq,uenceof natural numbers a = kl ...k, from IF, 0, abbreviates
Okl ...Ok,. UA denotes the identity operator, i.e. OAA= A for every formula
A. If a = k k, 0, is also denoted by 0;(for r 2 0 ).
w
7
'
Similarly, we use the notations O,, 0;.
'1-modal formulas are also called monomodal, 2-modal formulas are called bimodal. Some
authors prefer the term 'unimodal' to 'monomodal'.

4 CHAPTER 1. BASIC PROPOSITIONAL LOGIC
As usual, for a finite set of formulas r , A r denotes its conjunction and VI'
its disjunction; the empty conjunction is T and the empty disjunction is I.We
also use the notation (for arbitrary r )
If n = 1, we write q instead of 0
1 and 0 instead of O1.
The degree (or the depth) of a modal formula A (denoted by d(A)) is defined
by induction:
d(pk) = d(l) = 0,
d(A A B) = d(A V B) = d(A 1B) = max (d(A),d(B)),
d(OiA) = d(A) f l .
LPN denotes the set of all N-modal formulas; LPo denotes the set of all formulas
without modal connectives; they are called classical (or intuitionistic2).
An N-modal (propositional) substitution is a map S : LPN -+ LPN pre-
serving Iand all but finitely many proposition letters and commuting with all
connectives, i.e. such that
{ k I S(pk) # pk} is finite;
Let ql, ...,qk be different proposition letters. A substitution S such that
S(qi) = Ai for i 5 k and S(q) = q for any other q E PL, is denoted by
[Al,...,Ak/q17...,qk]. A substitution of the form [Alq]is called simple. It is
rather clear that every substitution can be presented as a composition of simple
substitutions.
Later on we often write SA instead of S(A);this formula is called the sub-
stitution instance of A under S, or the S-instance of A. For a set of formulas
r , Sub(r) (or S u b ~ ( r ) ,
if we want to specify the language) denotes the set of
all substitution instances of formulas from I?.
An intuitionistic substitution is nothing but a O-modal substitution.
2 ~ n
this book intuitionistic and classical formulas are syntactically the same; the only
difference between them is in semantics.

1.l. PROPOSITIONAL SYNTAX 5
1.1.2 Logics
In this book a logic (in a formal sense) is a set of formulas. We say that a logic
L is closed under the rule
(or that this rule is admissible in L) if B E L, whenever A1,...,A, E L. A
(normal propositional) N-modal logic is a subset of LPN closed under arbitrary
A, A 3 B
N-modal substitutions, modus ponens ( ), necessitation (&)
and containing all classical tautologies and all the formulas
AKi :=Oi(p 3 q) 3 (Dip > Oiq),
where 15 i 5 N
K N denotes the minimal N-modal logic, and K denotes K1. Sometimes we
call N-modal logics (or formulas) just 'modal', if N is clear from the context.
The smallest N-modal logic containing a given N-modal logic A and a set
of N-modal formulas I? is denoted by A +I?; for a formula A, A +A is an
abbreviation for A+{A). We say that the logic K N+ris axzomatised by the set
r. A logicis calledfinitely axiomatisable (respectively, recursively axiomatisable)
if it can be axiomatised by a finite (respectively, recursive) set of formulas.
It is well-known that a logic is recursively axiomatisable iff it is recursively
enumerable. A logic A is consistent if I@ A.
Here is a list of some frequently used modal formulas and modal logics:
AT : = O p > p ,
A4 :=Op > OOp,
AD : = 0 0 p > Up,
AM :=OOp > OOp (McKinsey formula),
A2 :=OOp > OOp,
A3 :=O(pAOp>q)VO(qAUq>p),
AGrz := O(O(p 3 Op) >p) > p (Grzegorczyk formula),
AL := O(Op >p) > Up (Lob formula),
A5 := OOp > Up,
AB :=OOp 3 p ,
Ati :=OlOzp 3 p,
At:! :=0 2 0 1 ~
>p,

:=K + OT, T :=K +AT,
:=K +A4,
:=K4+AT, D 4 := D + A 4 ,
:=S4 +AM, D4.1 :=D 4 +AM,
:= S4 +A2, S4.3 :=S 4 +O(Op > q) v O(0q >p),
:= K 4 +A3, Grz :=S 4 +AGrz (Grzegorczyk logic),
:=S4 +AB, GL :=K +AL, (Godel-Lob logic),
:= K +A5, K . t :=K 2 +Atl +At2.
The corresponding N-modal versions are denoted by DN, TNetc.; so for ex-
ample,
D N : = K N + { O ~ T I ~ < ~ < N ) ,
and so on.
A superintuitionistic logic is a set of intuitionistic formulas closed under
intuitionistic substitutions and modus ponens, and containing the followingwell-
known axioms:
(Ax81 (P 3 r ) ((9 r ) 3 ( P V ~
3 r)),
The smallest superintuitionistic logic is exactly the intuitionistic (or Heyting)
propositional logic; it is denoted by H.
The notation A +l? and the notions of finite axiomatisability, etc. are used
for superintuitionistic logics as well.
An m-formula is a formula without occurrences of letters pi for i > m.
LPN[m denotes the set of all N-modal m-formulas. If A is a modal or a
superintuitionistic logic, Arm denotes its restriction to m-formulas. The sets
A [mare called bounded logics.
An extension of an N-modal logic A is an arbitrary N-modal logic containing
A; extensions of a logic A are also called A-logics.
Members of a logic are also called its theorems; moreover, we use the notation
A l- A as synonymous for A E A. A formula A in the language of a logic A
is A-consistent if 1 A @ A. An N-modal propositional theory is a set of N-
modal propositional formulas. Such a theory is A-consistent if all conjunctions

1.1. PROPOSITIONAL SYNTAX 7
over its finite subsets are A-consistent and A-complete if it is maximal among
A-consistent theories (in the same language).
In the intuitionistic case we also consider double theories that are pairs of
sets of intuitionistic formulas. For a superintuitionistic logic A, a double theory
(l7,A) is called A-consistent if for any finite sets G r,A. C A, AYAro 3
V A,. A A-consistent double theory (
I
'
, A) is called A-complete if r U A = LPo.
Let us fix names for some particular intuitionistic formulas and superintu-
itionistic logics:
E M
A J
AJ-
APl
APn
K P
:= p V 7 p (the law of the excluded middle);
.
.-
- -.p V7 l p (the weak law of the excluded middle);
:= 77p v (11p 3 p);
:= E M ;
:= pnV(pn>APn-1) (for n > 1);
:= (7p > q V r) > ( i p > q) V ( ~ p
> r) (Kreisel-Putnam formula);
n
Pi 3 V pj > V pj > V pi (Gabbay-De Jongh formulas);
( i ~ o( j+i ) j+i ) i=o
:= (P> 4) v ( 9 3 PI;
n
i=O
IG, := V (pi pj);
O<i<j<n
HJ := K C := H+AJ(Jankov7slogic);
L C := H +AZ (Dummett's logic);
C L := H +E M (classical, or Boolean logic).
The following inclusions are well-known:
H c H J c L C c C L ,
A superintuitionistic logic C is called consistent if I6C; C is said to be interme-
diate if H c C CL. It is well-known that every consistent superintuitionistic
propositional logic is intermediate.
Lemma 1.1.1 (1) Some theorems of KN:

8 CHAPTER 1. BASIC PROPOSITIONAL LOGlC
(2) The following rules are admissible in every modal logic:
Monotonicity rules Replacement rules
A > B A - B
OiA 3 OiB OiA s OiB
(3) Some theorems of S4:
o o p =Op;
(Opv Oq)=u p v Oq.
(4) A theorem of S4.2:
oo(Api)
- onpi.
i i
Lemma 1.1.2 Some theorems of H:
(5) i=
A1((Pi3q)3q)- (
(
Ai=1pi>*) 3 9 ) .
Lemma 1.1.3 (Propositional replacement rule) The following rule is ad-
missible in every modal or superintuitionistic logic:
We can write this rule more loosely as
i.e. in any formula C we can replace some occurrences of a subformula A with
its equivalent A'.
To formulate the next theorem, we introduce some notation. For an N-modal
formula B, r > 0, let
for a finite set of N-modal formulas A, let

1.1. PROPOSITIONAL SYNTAX 9
Theorem 1.1.4 (Deduction theorem)
(I) Let C be a superintuitionistic logic, ~ u { A )
a set of intuitionistic formulas.
Then:
A E (C +r) iff ( AA 3 A ) E C for some finite A 5 Sub@).
(11) Let A be an N-modal logic, r U {A) a set of N-modal formulas. Then
A E (A+r)ifl
( A 3 A t A for some r 2 0 and some finite A i Sub(I').
1
(111) Let A be a 1-modal logic, r U { A ) E CPl. Then A E ( A+I?) iff
T
(1) ( (AU k A )3 A) t A for some r 2 0 and some finite A Sub(r)
k=O
- in the general case;
(2) (AO V A3 A) E A for some r 2 0 and some finite A 5 Sub(r)
- provided T G A ;
(3) (AA A UA 3 A) E A for some finite A C Sub(I')
- provided K 4 2 A ;
(4) (ACIA 3 A) E A for some finite A c Sub(r)
- provided S4 2 A .
Similarly one can simplify the claim (2) for the case when A is an N-modal logic
containing T N , K ~ N ,
or S4N; we leave this as an exercise for the reader. But
let us point out that for the case when S4N A, n > 1, 0, is not necessarily an
SPmodality, and it may happen that for any A, A E (A+r)is not equivalent
Corollary 1.1.5
(1) For superintuitionistic logics:
if formulas from I? and r' do not have common proposition letters.
(2) For N-modal logics:
if formulas from I? and I" do not have common proposition letters.
(3) For 1-modal logics:
if formulas from I? and I" do not have common proposition letters.
In some particular cases this presentation can be further simplified:

CHAPTER 1. BASIC PROPOSITIONAL LOGIC
(a) for logics above T :
( A+I?) n ( A+r')= A +{OrA v OrA' I A E r , A' E I"; r 2 0);
(b) for logics above K4:
(c) for logics above S4:
Therefore we have:
Proposition 1.1.6
(1) The set of superintuitionistic logics S is a complete well-distributive lattice:
Here the sum of logics C A, is the smallest logic containing their union.
,€I
The set of finitely axiomatisable and the set of recursively axiomatisable
superintuitionistic logics are sublattices of S.
(2) The set of N-modal logics M N is a complete well-distributive lattice; the
set of recursively axiomatisable N-modal logics is a sublattice of M N .
Proof In fact, for example, in the intuitionistic case, both parts of the equality
are axiomatised by the same set of formulas
Remark 1.1.7 Although the set of all finitely axiomatisable 1-modal logics is
not closed under finite intersections [van Benthem, 19831, this is still the case
for finitely axiomatisable extensions of K4, cf. [Chagrov and Zakharyaschev,
19971.
Theorem 1.1.8 (Glivenko theorem) For any intermediate logic C
1 A E H iff -A E E iff 1 A E CL.
For a syntactic proof see [Kleene,19521. For another proof using Kripke models
see [Chagrov and Zakharyaschev, 19971,Theorem 2.47.
Corollary 1.1.9 If A E CL, then 11A E H .
Proof In fact, A E CL implies --A E CL, so we can apply the Glivenko
theorem.

1.2. ALGEBRAIC SEMANTICS 11
1.2 Algebraic semantics
For modal and intermediate propositional logics several kinds of semantics are
known. Algebraic semantics is the most general and straightforward; it inter-
prets formulas as operations in an abstract algebra of truth-values. Actually
this semantics fits for every propositional logic with the replacement property;
completeness follows by the well-known Lindenbaum theorem.
Relational (Kripke) semantics is nowadays widely known; here formulas are
interpreted in relational systems, or Kriplce frames. Kripke frames correspond to
a special type of algebras, so Kripke semantics is reducible to algebraic. Neigh-
bourhood semantics (see Section 1.17) is in between relational and algebraic.
Let us begin with algebraic semantics.
Definition l.2.13 A Heyting algebra is an implicative lattice with the least
element:
fi = (0, A, V, +, 0).
More precisely, (St, A, V) is a lattice with the least element 0, and + is the
implication in this lattice, i.e. for any a,b, c
(Here I is the standard ordering in the lattice, i.e. a 5 b iff a A b = a.)
Recall that negation in Heyting algebras is l a := a --+ 0 and 1= a --+ a is
the greatest element.
Note that (*) can be written as
In particular,
a - - + b = l i f f a I b .
Also recall that an implicative lattice is always distributive:
(aV b) Ac = (aA c) V (bAc),
(a A b) Vc = (a v c) A (bvc).
A lattice is called complete if joins and meets exist for every family of its
elements:
V a j :=min{b I b'j J a j 5 b), / a j :=max{b 1 V j E J b 5 aj).
j€J j€J
A complete lattice is implicative iff it is well-distributive, i.e., the following
holds:
a A (vaj) = v (aAaj).
j € J j € J
3Cf. [Rasiowa and Sikorski, 1963; Borceaux, 19941.

So every complete well-distributive lattice can be turned into a Heyting algebra.
Let us prove two useful properties of Heyting algebras.
Lemma 1.2.2
Proof We have to prove
which is equivalent (by 1.2.1(*)) to
But this follows from
a k A (aj + b j ) < bk.
j E J
The latter holds, since by 1.2.1(*),it is equivalent to
Lemma 1.2.3
Proof (I)
hence
and thus
(2)
hence

1.2. ALGEBRAIC SEMANTICS
and thus
Lemma 1.2.4
A ( v l -
+ u) = ( V v i + u).
i E I i E I
Proof (2)
vi 5 V vi implies
i E I
v v i + u SVi + u;
i E I
hence
(
I
)
Since
A ( V i '
u) 5 vi -
) U ,
i E I
it follows that for any i E I
Hence
(Vvi)A A ( v i + U ) V(viA A( ~ i
-+ u))5 U .
i E I i E I iEI i E I
Eventually
A ( v i -+u) 5 V v i t u .
i E I i E I
w
A Boolean algebra is a particular caseof a Heyting algebra,where aVia = 1.
In this case V, A, --+,
7 are usually denoted by U, n,3,-. Then we can consider
U, n, -, 0 , l (and even U, -, 0)as basic and define a 3 b := -a U b.
We also use the derived operation (equivalence)
in Heyting algebras and its analogue
in Boolean algebras.

Definition 1.2.5 An N-modal algebra is a structure
0= (0, n, u, -, 0, l , n l , . ..,ON),
such that its nonmodal part
is a Boolean algebra, and Diare unary operations in I;t satisfying the identities:
Oil = 1.
S2 is called complete if the Boolean algebra fib is complete.
We also use the dual operations
For 1-modal algebras we write 0 , 0 rather than 01,
O1 (cf. Section 1.1.1).
Definition 1.2.6 A topo-Boolean (or interior, or S4-) algebra is a 1-modal
algebra satisfying the inequalities
In this case CI is called the interior operation and its dual 0 the closure opera-
tion. A n element a is said to be open if Ua = a and closed if Oa = a.
Proposition 1.2.7 The open elements of a topo-Boolean algebra 0 constitute
a Heyting algebra:
a0= (a0, n, u, 4, o),
in which a -
,b = O(a 3 b). Moreover, if fl is complete then f1° also is, and
Proof Cf. [ ~ c ~ i n s e ~
and Tarski, 19441; [Rasiowaand Sikorski, 19631. W
Following [Esakia, 19791,we call S2O the pattern of a.It is known that every
Heyting algebra is isomorphic to some algebra no[Rasiowaand Sikorski, 19631
Definition 1.2.8 A valuation in an N-modal algebra is a map cp : PL -
+ 0.
The valuation cp has a unique extension to a,ll N-modal formulas such that

1.2. ALGEBRAIC SEMANTICS
(4)cp(A 3 B) = cp(4 3 cp(B);
(5) ~ ( n i A )
= n i ~ ( A ) .
The pair ( a , cp) is then called an (algebraic)model over fl. An N-modal formula
A is said to be true in the model ( a ,cp) if cp(A) = 1 (notation: ( a ,cp) k A); A
is called valid in the algebra f l (notation: S
2 k A) if it is true in every model
over S2.
Lemma 1.2.9 Let S2 be an N-modal algebra, S a propositional substitution.
Let cp, 7 be valuations in S1 such that for any B E PLrk
(4) 7(B) = cp(SB).
Then (4) holds for any N-modal k-formula B.
Proof Easy, by induction on the length of B.
Lemma 1.2.10 (Soundness lemma) The set
ML(S2) := {A E LPN 1 f l k A)
is a modal logic.
Proof First note that ML(fl) is substitution closed. In fact, assume f l !
= A,
and let S be a propositional substitution. To show that S
2 k SA, take an
arbitrary valuation cp in fl, and consider a new valuation 7 according to (4)
from Lemma 1.2.9. So we obtain
i.e. S
2 i= SA.
The classical tautologies are valid in fl, because they hold in any Boolean
algebra. The validity of AKi follows by a standard argument. In fact, note that
in a modal algebra Oi is monotonic:
(*) x <Y * n i x <Oiy,
because x 5 y implies
nix = Oi(x ny) = nix nOiy.
Now since
(a a b) na <b,
by monotonicity (*), we have
O(a Z
I b) nOa 5 Ob,
which implies
U ( a3 b) 5 (Oa 3 Ob),
This yields the validity of AKi.
Finally, modus ponens and necessitation preserve validity, since in a modal
algebra 1<a implies a = 1,and O i l = 1.

Definition 1.2.11 ML(52) is called the modal logic of the algebra 52.
We also define the modal logic of a class C of N-modal algebras
ML(C) :=~ { M L ( s ~ )
I52 E C}.
Note that ML(52) is consistent iff the algebra 52 is nondegenerate, i.e. iff
0 # 1 in 52.
Definition 1.2.12 A valuation in a Heyting algebra 52 is a map cp : P L --0.
It has a unique extension cp' :LPo -0 such that
As in the modal case, the pair (52,cp) is called an (algebraic) model over 52.
A n intuitionistic formula A is said to be true in (52,p ) if pl(A) = 1 (notation:
(52, cp) b A); A is called valid in the algebra 52 (notation: 52 k A ) if it is true
in evenJ model over 52.
We easily obtain an intuitionistic analogue of Lemma 1.2.9:
L e m m a 1.2.13 Let 52 be a Heyting algebra, S a propositional substitution. Let
cp, 77 be valuations in f2 such that for any B E P L
( 4 ) rl'(B) = cpl(SB).
Then ( 4 ) holds for any intuitionistic formula B.
Similarly we have
L e m m a 1.2.14 (Soundness l e m m a ) For a Heyting algebra 52, the set
is a superintuitionistic logic.
Definition 1.2.15 IL(52) is called the superintuitionistic logic of the algebra
$2. Similarly to the modal case, we define the superintuitionistic logic of a class
C of Heyting algebras
Definition 1.2.16 A valuation cp in an S4-algebra f2 is called intuitionistic i f
it is a valuation in i.e. if its values are open.

1.2. ALGEBRAIC SEMANTICS 17
Definition 1.2.17 Godel-Tarski translation is the map (-)T from intuitionis-
tic to 1-modal fomulas defined by the following clauses:
I T = I ;
qT = Oq for every proposition letter q;
( A A B ) ~
= A ~
A B ~ ;
( A V B ) ~
= A ~ v B ~ ;
( A3 B)T = O(AT3 BT).
Lemma 1.2.18 (OATF AT)E S 4 for any intuitionistic formula A.
Proof Easy by induction; for the cases A = B V C, A = B A C use Lemma
1.1.1. w
Lemma 1.2.19 Let be an S4-algebra.
(1) Let ip, 1C, be valuations in such that for any q E PL
Then for any intuitionistic formula A,
i p ' ( ~ )= *(AT).
In particular,
cp'(A) = p(AT)7
if ip is intuitionistic.
(2) For any intuitionistic formula A,
Proof
(1)By induction. Consider only the case A = B > C. Suppose
Then we have
i p ' ( ~> C ) = ipl(B)-+ ipl(C)= +(BT)
-+ +(cT)
= 0(+(BT)
3 +(cT))
= +(O(BT> CT))= $((B 3 C)T).
(2) (Only if.) Assume nok A. Let $ be an arbitrary valuation in a,and let ip
be the valuation in a0such that
for every q E PL. By (1)and our assumption, we have:
Hence f
2k AT.
(If.) Assume k AT. By ( I ) , for any valuation cp in a0we have ipl(A)=
ip(AT)= 1. Hence a0FA. ¤

18 C H A P T E R 1. BASIC PROPOSITIONAL LOGIC
Let us now recall the Lindenbaum algebra construction. For an N-modal
or superintuitionistic logic A, the relation N~ between N-modal (respectively,
intuitionistic) formulas such that
A - A B iff ( A = B ) E A
is an equivalence.
Let [A] be the equivalence class of a formula A modulo -A.
Definition 1.2.20 The Lindenbaum algebra Lind(A) of a modal logic A is the
set LPN/ -A with the operations
[A]n[B] := [AAB],
[A]u [B]:= [Av B],
-[A] := [iA],
0 := [ l ] ,
Theorem 1.2.21 For an N-modal logic A
(1) Lind(A) is an N-modal algebra;
Definition 1.2.22 The Lindenbaum algebra Lind(C) of a superintuitionistic
logic C, is the set L%/ -c with the operations
[A]r [B]:= [AA B],
[A]v [B] := [A v B],
[A] 4 [B] := [A > B],
0 := [I].
Theorem 1.2.23 For a superintuitionistic logic C,
(1) Lind(C) is a Heyting algebra;
Definition 1.2.24 A set of modal formulas is valid i n a modal algebra f2
(notation: f2 k r ) if all these formulas are valid; similarly for intuitionistic
formulas and Heyting algebras. In this case C2 is called a r-algebra. The set oJ
all r-algebras is called an algebraic variety defined by r .

1.3. RELATIONAL SEMANTICS (THE MODAL CASE) 19
Algebraic varieties can be characterised in algebraic terms, due to the well-
known Birkhoff theorem [Birkhoff, 19791 (which holds also in a more general
context):
Theorem 1.2.25 A class of modal or Heyting algebras is a n algebraic variety
iff it is closed under subalgebras, homomorphic images and direct products.
Since every logic is complete in algebraic semantics, there is the following
duality theorem.
Theorem 1.2.26 The poset M N of N-modal propositional logics (ordered by
inclusion) is dually isomorphic to the set of all algebraic varieties of N-modal
algebras; similarly for superintuitionistic logics and Heyting algebras.
1.3 Relational semantics (the modal case)
1.3.1 Introduction
First let us briefly recall the underlying philosophical motivation. For more
details, we address the reader to [Fitting and Mendelsohn, 20001. In relational
(or Kripke) semantics formulas are evaluated in 'possible worlds' representing
different situations. Depending on the application area of the logic, worlds
can also be called 'states', 'moments of time', 'pieces of information', etc. Every
world w is related to some other worlds called 'accessible from w', and a formula
O A is true at w iff A is true at all worlds accessible from w; dually, OA is true
at w iff A is true at some world accessible from w.
This corresponds to the ancient principle of Diodorus Cronus saying that
The possible is that which either is or will be true
So from the Diodorean viewpoint, possible worlds are moments of time, with
the accessibility relation 5 'before' (nonstrict).
For polymodal formulas we need several accessibility relations corresponding
to different necessity operators.
For the intuitionistic case, Kripke semantics formalises the 'historical a p
proach' to intuitionistic truth by Brouwer. Here worlds represent stages of our
knowledge in time. According to Brouwer's truth-preservation principle, the
truth of every formula is inherited in all later stages. 1 A is true at w iff the
truth of A can never be established afterwards, i.e. iff A is not true at w and
always later. Similarly, A > B is true at w iff the truth of A implies the truth
of B at w and always later. See [Dragalin, 19881, [van Dalen, 19731 for further
discussion.
1.3.2 Kripke frames and models
Now let us recall the main definitions in detail.

Definition 1.3.1 An N-modal (propositional)Kripke frame is an (N+l)-tuple
F = (W,R1,...,R N ) ,such that W # 0 , Ri 5 W x W. The elements of W are
called possible worlds (or points), Ri are the accessibility relations.
Quite oftenwe write u E F rather than u E W .
For a Kripke frame F = (W,R1,...,R N )and a sequence a E IF we define
the relation R, on W :
(Recall that .
k is a void sequence, Idw is the equality relation, see the Intro-
duction.)
Every N-modal Kripke frame F = (W,R1, ...,R N ) corresponds to an N-
modal algebra
where n,U, - are the standard set-theoretic operations on subsets of W , and
OiV := { XI Ri(x) V ) .
M A ( F ) is called the modal algebra of the frame F.
Definition 1.3.2 A valuation in a set W (or in a frame with the set of worlds
W ) is a valuation in MA(F), i.e. a map 0 : PL ---+ 2W. A Kripke model over
a frame F is a pair M = (F,0), where 6 is a valuation in F. 0 is extended to
all formulas in the standard way, according to Definition 1.2.8:
(1) 0 ( 1 )= 0;
(2) 6(AA B ) = 0(A)n0(B);
(3) e(AV B ) = @(A)
U 6(B);
(4) 6(A> B ) = 6(A)3 6(B);
(5) 0(0iA)= Ui0(A)= {UI R ~ ( u )
2 B(A)).
For a formula A, we also write: M ,w k A (or just w k A) instead of w E 6(A),
and say that A is true at the world w of the model M (or that w forces A).
The above definition corresponds to the well-known inductive definition of
forcing in a Kripke model given by (1)-(6) in the following lemma.
Lemma 1.3.3 (1) M,u k q zff u E 0(q) (for q 6 PL);
(3) M , u ~ B A C iff ( M , u k B a n d M , u I = C ) ;
(4) M,u+ B V C iff ( M , u +B or M , u k C ) ;

1.3. RELATIONAL SEMANTICS (THE MODAL CASE)
(5) M , u k B > C iff ( M , u b B impliesM,ubC);
(7) M , u k i B iff M,uI$B;
(8) M , u k OiB iff 3v E Ri(u)M , v k B.
(9) M , u k U,B iff Qv E R,(u) M,v k B ;
(10) M , u b O , B iff 3v E R,(u) M , v k B.
Definition 1.3.4 An m-bounded Kripke model over a Kripke frame F =
(W,R1,...,RN)is a pair (F,8), in which 8 : {pl,...,p,) -+ 2W; 8 is called
an m-valuation. In this case 0 is extended only to m-formulas, according to
Definition 1.3.2.
Definition 1.3.5 A modal formula A is true in a model M (notation: M k A )
if it is true at every world of M ; A is satisfied in M if it is true at some world
of M. A formula is called refutable in a model if it is not true.
Definition 1.3.6 A modal formula A is valid in a frame F (notation: F k A )
if it is true in every model over F. A set of formulas is valid in F (notation:
F != I?)if every A E is valid. In the latter case we also say that F is a r-frame.
The (Kripke frame) variety of I? (notation: V ( r ) )is the class of all r-frames.
A formula A is valid at a world x in a frame F (notation: F,x k A) if it is
true at x in every model over F ; similarly for a set of formulas.
A nonvalid formula is called refutable (in a frame or at a world).
A formula A is satisfiable at a world w of a frame F (or briefly, at F,w ) if
there exists a model M over F such that M , w k A.
Since by Definitions 1.2.8 and 1.3.2, 8(A)is the same in F and in M A ( F ) ,
we have
Lemma 1.3.7 For any modal formula A and a Kripke frame F
F k A iff M A ( F )k A.
Thus 1.2.10 implies:
Lemma 1.3.8 (Soundness lemma)
(1) For a Kripke frame F , the set
is a modal logic.

22 CHAPTER I . BASIC PROPOSITIONAL LOGIC
(2) For a class C of N-modal frames, the set
is an N-modal logic.
Definition 1.3.9 M L ( F ) (respectively, ML(C)) is called the modal logic of F
(respectively, of C), or the modal logic determined by F (by C), or complete
w.r.t. F (C).
For a Kripke model M , the set
is called the modal theory of M .
M T ( M )isnot always a modal logic;it is closedunder M P and 0-introduction
but not necessarily under substitution.
The followingis a trivial consequence of definitions and the soundness lemma.
Lemma 1.3.10 For an N-modal logic A and a set of N-modal formulas I?,
V ( A+I
'
) = V ( A )
nV(I'). In particular, V(KN+I
'
)= V(r).
Let us describe varieties of some particular modal logics:
Proposition 1.3.11
V ( D ) consists of all serial frames, i.e. of the frames (W,R) such that
Vx3yxRy;
V ( T )consists of all reflexive frames;
V ( K 4 ) consists of all transitive frames,
V ( S 4 ) consists of all quasi-ordered (or pre-ordered) sets, i.e. reflexive
transitive frames;
V ( S 4 . 1 ) consists of all S4-frames with McKinsey property:
V(S4.2) consists of all S4-frames with Church-Rosser property (or con-
fluent, or piecewise directed):
Vx,y , z ( x R y & x R z +3t (yRt & zRt)),
or equivalently,
R - ~ O R GROR-';

V ( K 4 . 3 ) consists of all piecewise linear (or nonbranching) K4-frames,
i.e. such that
Vx,y, z ( x R y& x R z +( y = z V yRz V zRy)),
or equivalently
R - ~ O R L I ~ U R U R - ~ ;
V ( K 4+AW,) consists of all transitive frames of width I
:n;4
V ( G r z ) consists of all Notherian posets, i.e. of those without infinite as-
cending chains xlR-xz R- x3 ...;5
V ( S 5 )consists of all frames, where accessibility is an equivalence relation.
Due to these characterisations, an N-modal logic is called reflexive (respec-
tively, serial, transitive) if it contains TN(respectively, DN, K ~ N ) .
Definition 1.3.12 Let A be a modal logic.
A is called Kripke-complete if it is determined by some class of frames;
A has the finite model property (f.m.p.) if it is determined by some class
of finite frames;
A has the countable frame property (c.f.p.) if it is determined by some
class of countable frame^.^
The following simple observation readily follows from the definitions.
L e m m a 1.3.13
(1) A logic A is Kripke-complete (respectively, has the c.f.p./
f.m.p.) iff each of its nontheorems is refutable in some A-frame (respec-
tively, in a countable/finite A-frame).
(2) M L ( V ( A ) )is the smallest Kripke-complete extension of A ; so A is Kripke-
complete iffA = M L ( V ( A ) ) .
All particular propositional logics mentioned above (and many others) are
known to be Kripke-complete. Kripke-completeness was proved for large families
of propositional logics; Section 1.9 gives a brief outline of these results. However
not all modal or intermediate propositional logics are complete in Kripke seman-
tics; counterexamples were found by S. Thomason, K. Fine, V. Shehtman, J.
Van Benthem, cf. [Chagrov and Zakharyaschev, 19971. But incomplete propo-
sitional logic's look rather artificial; in general one can expect that a 'randomly
chosen' logic is compete.
Nevertheless every logic is 'complete w.r.t. Kripke models' in the following
sense.
4See Section 1.9.
5Recall that xR- y iff xRy & x # y, see Introduction.
6'countable' means 'of cardinality < &'.

Definition 1.3.14 An N-modal Kripke model M is exact for an N-modal logic
A if A = MT(M).
Proposition 1.3.15 Every propositional modal logic has a countable exact
model.
This follows from the canonical model theorem by applying the standard trans-
lation, see below.
1.3.3 Main constructions
Definition 1.3.16 If F = (W,R1,. ..,RN) is a frame, V 2 W, then the frame
is called a subframe of F (the restriction of F to V).
I
f M = (F,8) is a Kripke model, then
M 1V := ( F 1 V, 8 V),
where (6 1 V)(q) := Q(q)nV for every q E PL, is called its submodel (the
restriction to V).
A set V C_ W is called stable (in F) if for every i, Ri(V) & V. In this case
the subframe F 1V and the submodel M 1V are called generated.
Definition 1.3.17 F' = (V,Ri, ...,Rh) is called a weak subframe of F =
(W,R1, ...,RN) if R
b c Ri for every i and V c W. Then for a Kripke model
M = (F,Q), M' = (F',8 /' V) is called a weak submodel. If also W = V, F' is
called a full weak subframe of F.
,
-
-
We use the signs c, a, C, 2 to denote subframes, generated.subframes,
weak subframes, and full weak subframes, respectively; the same for submodels.
Definition 1.3.18 Let F, M be the same as in the previous definition. The
smallest stable subset Wfu containing a given point u E W is called the cone
generated by u; the corresponding subframe F f u := F 1 (WTu) is also called
the cone (in F ) generated by u, or the subframe generated by u; similarly for
the submodel M f u :=M 1 (Wfu). A frame F (respectively, a Kripke model M)
is called rooted (with the root u) if F = F f u (respectively, M = Mfu).
We skip the simple proof of the following
Lemma 1.3.19 WTu = R*(u), where R* is the reflexive transitive closure of
(R1U ... URN), i.e. R * = U R,.
,€IF
Definition 1.3.20 A path of length m from u to v in a frame F = (W,R1,...,
RN) is a sequence (uo,jo,ul,...,jm-l,um) such that uo = U, u, = v, and
uiRj,uifl for i = O,.. .,m - 1.

For the particular case N = 1we have ji = 1for any i, so we can denote a path
just by (u0,u1,...,urn).
Now Lemma 1.3.19 can be reformulated as follows:
Lemma 1.3.21 x E FTu iff there exists a path from u to x in F .
Definition 1.3.22 The temporalisation of a propositional Kripke frame F =
(W,R1,...,RN) is the frame F' := (W,R1,. ..,RN,R,', ...,R;'). A non-
oriented path in F is a path in F'.
Definition 1.3.23 Let F = (W,R1,. ..,RN) be a propositional Kripke frame.
A subset V E W is called connected (in F) if it is stable in F*, i.e. both Ri-
and R;'-stable for every i = 1,...,N. F itself is called connected if W is
connected in F . A cone in F* (as a subset) is called a (connected) component
of F.
Lemma 1.3.24
(1) The component containing x E F (i.e. the cone F' T x ) consists of all
y E F such that there exists a non-oriented path from x to y.
(2) The components of F make a partition.
Proof
(1) Readily follows from Lemma 1.3.21.
(2) Follows from (1) and the observation that
{(x,y) I there exists a nonoriented path from x to y)
is an equivalence relation on W. H
The following is well-known:
Lemma 1.3.25 (Generation lemma) Let V be a stable subset in F,
M = (F,6) a Kripke model. Then
(1) For any u E V ,for any modal formula A,
(2) ML(F) M L ( F 1V).
The same holds for bounded models, with obvious changes.
We also have:
Lemma 1.3.26 (1) ML(F) = n ML(Ffu).
UGF
(2) MT(M) = n MT(MT U )
uEM
Proof

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for the elastic power of human nature; and they dislike hope and
courage in others, thinking you unfeeling in exact proportion to your
cheerfulness.
Morally this same habit of fear deteriorates, because it weakens and
narrows, the whole nature. So far from following Luther's famous
advice—Sin boldly and leave the rest to God—their sin is their very
fear, their unconquerable distrust. These are the people who regard
our affections as snares and all forms of pleasure as so many
waymarks on the road to perdition—who would narrow the circle of
human life to the smallest point both of feeling and action, because
of the sin in which, according to them, the whole world is steeped.
They see guilt everywhere, but innocence not at all. Their minds are
set to the trick of terror; and fear of the power of the devil and the
anger of God weighs on them like an iron chain from which there is
no release. This is not so much from delicacy of conscience as from
simple moral cowardice; for you seldom find these very timid people
lofty-minded or capable of any great act of heroism. On the contrary,
they are generally peevish and always selfish; self-consideration
being the tap-root of their fears, though the cause is assigned to all
sorts of pretty things, such as acute sensibilities, keen imagination,
bad health, tender conscience, delicate nerves—to anything in fact
but the real cause, a cowardly habit of fear produced by continual
moral selfishness, by incessant thought of and regard for
themselves.
Nothing is so depressing as the society of a timid person, and
nothing is so infectious as fear. Live with any one given up to an
eternal dread of possible dangers and disasters, and you can
scarcely escape the contagion, nor, however brave you may be,
maintain your cheerfulness and faculty of faith. Indeed, as timid
folks crave for sympathy in their terrors—that very craving being
part of their malady of fear—you cannot show them a cheerful
countenance under pain of offence, and seeming to be brutal in your
disregard of what so tortures them. Their fears may be simply
absurd and irrational, yet you must sympathize with them if you

wish even to soothe; argument or common-sense demonstration of
their futility being so much mental ingenuity thrown away.
Fear breeds suspicion too, and timid people are always suspecting ill
of some one. The deepest old diplomatist who has probed the folly
and evil of the world from end to end, and who has sharpened his
wits at the expense of his trust, is not more full of suspicion of his
kind than a timid, superstitious, world-withdrawn man or woman
given up to the tyranny of fear. Every one is suspected more or less,
but chiefly lawyers, servants and all strangers. Any demonstration of
kindness or interest at all different from the ordinary jogtrot of
society fills them with undefined suspicion and dread; and, fear
being in some degree the product of a diseased imagination, the
'probable' causes for anything they do not quite understand would
make the fortune of a novel-writer if given him for plots. If any one
wants to hear thrilling romances in course of actual enactment, let
him go down among remote and quiet-living country people, and
listen to what they have to say of the chance strangers who may
have established themselves in the neighbourhood, and who, having
brought no letters of introduction, are not known by the aborigines.
The Newgate Calendar or Dumas' novels would scarcely match the
stories which fear and ignorance have set afoot.
Fearful folk are always on the brink of ruin. They cannot wait to see
how things will turn before they despair; and they cannot hope for
the best in a bad pass. They are engulfed in abysses which never
open, and they die a thousand deaths before the supreme moment
actually arrives. The smallest difficulties are to them like the straws
placed crosswise over which no witch could pass; the beneficent
action of time, either as a healer of sorrow or a revealer of hidden
mercies, is a word of comfort they cannot accept for themselves,
how true soever it may be for others; the doctrine that chances are
equal for good as well as for bad is what they will not understand;
and they know of no power that can avert the disaster, which
perhaps is simply a possibility not even probable, and which their
own fears only have arranged. If they are professional men, having

to make their way, they are for ever anticipating failure for to-day
and absolute destruction for to-morrow; and they bemoan the fate
of the wife and children sure to be left to poverty by their untimely
decease, when the chances are ten to one in favour of the
apportioned threescore and ten years. Life is a place of suffering
here and a place of torment hereafter; yet they often wish to die,
reversing Hamlet's decision by thinking the mystery of unknown ills
preferable to the reality of those they have on hand.
Over such minds as these the vaticinations of such a prophet as Dr.
Cumming have peculiar power; and they accept his gloomy
interpretations of the Apocalypse with a faith as unquestioning as
that with which they accept the Gospels. They have a predilection
indeed for all terrifying prophecies, and cast the horoscope of the
earth and foretell the destruction of the universe with marvellous
exactitude. Their minds are set to the trick of foreboding, and they
live in the habit of fear, as others live in the habit of hope, of
resignation, or of careless good-humour and indifference. There is
nothing to be done with them. Like drinking, or palsy, or a nervous
headache, or a congenital deformity, the habit is hopeless when
once established; and those who have begun by fear and suspicion
and foreboding will live to the end in the atmosphere they have
created for themselves. The man or woman whose mind is once
haunted by the nightly fear of a secreted burglar will go on looking
for his heels so long as eyesight and the power of locomotion
continue; and no failure in past Apocalyptic interpretations will shake
the believer's faith in those of which the time for fulfilment has not
yet arrived. It is a trick which has rooted, a habit that has
crystallized by use into a formation; and there it must be left, as
something beyond the power of reason to remedy or of experience
to destroy.

OLD LADIES.
The world is notoriously unjust to its veterans, and above all it is
unjust to its ancient females. Everywhere, and from all time, an old
woman has been taken to express the last stage of uselessness and
exhaustion; and while a meeting of bearded dotards goes by the
name of a council of sages, and its deliberations are respected
accordingly, a congregation of grey-haired matrons is nothing but a
congregation of old women, whose thoughts and opinions on any
subject whatsoever have no more value than the chattering of so
many magpies. In fact the poor old ladies have a hard time of it; and
if we look at it in its right light, perhaps nothing proves more
thoroughly the coarse flavour of the world's esteem respecting
women than this disdain which they excite when they are old. And
yet what charming old ladies one has known at times!—women quite
as charming in their own way at seventy as their grand-daughters
are at seventeen, and all the more so because they have no design
now to be charming, because they have given up the attempt to
please for the reaction of praise, and long since have consented to
become old though they have never drifted into unpersonableness
nor neglect. While retaining the intellectual vivacity and active
sympathies of maturity, they have added the softness, the
mellowness, the tempering got only from experience and advancing
age. They are women who have seen and known and read a great
deal; and who have suffered much; but whose sorrows have neither
hardened nor soured them—but rather have made them even more
sympathetic with the sorrows of others, and pitiful for all the young.
They have lived through and lived down all their own trials, and
have come out into peace on the other side; but they remember the

trials of the fiery passage, and they feel for those who have still to
bear the pressure of the pain they have overcome. These are not
women much met with in society; they are of the kind which mostly
stays at home and lets the world come to them. They have done
with the hurry and glitter of life, and they no longer care to carry
their grey hairs abroad. They retain their hold on the affections of
their kind; they take an interest in the history, the science, the
progress of the day; but they rest tranquil and content by their own
fireside, and they sit to receive, and do not go out to gather.
The fashionable old lady who haunts the theatres and drawing-
rooms, bewigged, befrizzled, painted, ghastly in her vain attempts to
appear young, hideous in her frenzied clutch at the pleasures
melting from her grasp, desperate in her wild hold on a life that is
passing away from her so rapidly, knows nothing of the quiet dignity
and happiness of her ancient sister who has been wise enough to
renounce before she lost. In her own house, where gather a small
knot of men of mind and women of character, where the young
bring their perplexities and the mature their deeper thoughts, the
dear old lady of ripe experience, loving sympathies and cultivated
intellect holds a better court than is known to any of those miserable
old creatures who prowl about the gay places of the world, and
wrestle with the young for their crowns and garlands—those
wretched simulacra of womanhood who will not grow old and who
cannot become wise. She is the best kind of old lady extant,
answering to the matron of classic times—to the Mother in Israel
before whom the tribes made obeisance in token of respect; the
woman whose book of life has been well studied and closely read,
and kept clean in all its pages. She has been no prude however, and
no mere idealist. She must have been wife, mother and widow; that
is, she must have known many things of joy and grief and have had
the fountains of life unsealed. However wise and good she may be,
as a spinster she has had only half a life; and it is the best half
which has been denied her. How can she tell others, when they
come to her in their troubles, how time and a healthy will have
wrought with her, if she has never passed through the same

circumstances? Theoretic comfort is all very well, but one word of
experience goes beyond volumes of counsel based on general
principles and a lively imagination.
One type of old lady, growing yearly scarcer, is the old lady whose
religious and political theories are based on the doctrines of Voltaire
and Paine's Rights of Man—the old lady who remembers Hunt and
Thistlewood and the Birmingham riots; who talks of the French
Revolution as if it were yesterday; and who has heard so often of
the Porteus mob from poor papa that one would think she had
assisted at the hanging herself. She is an infinitely old woman, for
the most part birdlike, chirrupy, and wonderfully alive. She has never
gone beyond her early teaching, but is a fossil radical of the old
school; and she thinks the Gods departed when Hunt and his set
died out. She is an irreligious old creature, and scoffs with more
cleverness than grace at everything new or earnest. She would as
lief see Romanism rampant at once as this newfangled mummery
they call Ritualism; and Romanism is her version of the unchaining
of Satan. As for science—well, it is all very wonderful, but more
wonderful she thinks than true; and she cannot quite make up her
mind about the spectroscope or protoplasm. Of the two, protoplasm
commends itself most to her imagination, for private reasons of her
own connected with the Pentateuch; but these things are not so
much in her way as Voltaire and Diderot, Volney and Tom Paine, and
she is content to abide by her ancient cairns and to leave the
leaping-poles of science to younger and stronger hands. This type of
old lady is for the most part an ancient spinster, whose life has worn
itself away in the arid deserts of mental doubt and emotional
negation. If she ever loved it was in secret, some thin-lipped
embodied Idea long years ago. Most likely she did not get even to
this unsatisfactory length, but contented herself with books and
discussions only. If she had ever honestly loved and been loved,
perhaps she would have gone beyond Voltaire, and have learned
something truer than a scoff.

The old lady of strong instinctive affections, who never reflects and
never attempts to restrain her kindly weaknesses, stands at the
other end of the scale. She is the grandmother par excellence, and
spends her life in spoiling the little ones, cramming them with sugar-
plums and rich cake whenever she has the chance, and nullifying
mamma's punishments by surreptitious gifts and goodies. She is the
dearly beloved of our childish recollections; and to the last days of
our life we cherish the remembrance of the kind old lady with her
beaming smile, taking out of her large black reticule, or the more
mysterious recesses of her unfathomable pocket, wonderful little
screws of paper which her withered hands thrust into our chubby
fists; but we can understand now what an awful nuisance she must
have been to the authorities, and how impossible she made it to
preserve anything like discipline and the terrors of domestic law in
the family.
The old lady who remains a mere child to the end; who looks very
much like a faded old wax doll with her scanty hair blown out into
transparent ringlets, and her jaunty cap bedecked with flowers and
gay-coloured bows; who cannot rise into the dignity of true
womanliness; who knows nothing useful; can give no wise advice:
has no sentiment of protection, but on the contrary demands all
sorts of care and protection for herself—she, simpering and giggling
as if she were fifteen, is by no means an old lady of the finest type.
But she is better than the leering old lady who says coarse things,
and who, like Béranger's immortal creation, passes her time in
regretting her plump arms and her well-turned ankle and the lost
time that can never be recalled, and who is altogether a most
unedifying old person and by no means nice company for the young.
Then there is the irascible old lady, who rates her servants and is
free with full-flavoured epithets against sluts in general; who is like a
tigress over her last unmarried daughter, and, when crippled and
disabled, still insists on keeping the keys, which she delivers up
when wanted only with a snarl and a suspicious caution. She has
been one of the race of active housekeepers, and has prided herself

on her exceptional ability that way for so long that she cannot bear
to yield, even when she can no longer do any good; so she sits in
her easy chair, like old Pope and Pagan in Pilgrim's Progress, and
gnaws her fingers at the younger world which passes her by. She is
an infliction to her daughter for all the years of her life, and to the
last keeps her in leading-strings, tied up as tight as the sinewy old
hands can knot them; treating her always as an irresponsible young
thing who needs both guidance and control, though the girl has
passed into the middle-aged woman by now, shuffling through life a
poor spiritless creature who has faded before she has fully
blossomed, and who dies like a fruit that has dropped from the tree
before it has ripened.
Twin sister to this kind is the grim female become ancient; the gaunt
old lady with a stiff backbone, who sits upright and walks with a firm
tread like a man; a leathery old lady, who despises all your weak
slips of girls that have nerves and headaches and cannot walk their
paltry mile without fatigue; a desiccated old lady, large-boned and
lean, without an ounce of superfluous fat about her, with keen eyes
yet, with which she boasts that she can thread a needle and read
small print by candlelight; an indestructible old lady, who looks as if
nothing short of an earthquake would put an end to her. The friend
of her youth is now a stout, soft, helpless old lady, much bedraped
in woollen shawls, given to frequent sippings of brandy and water,
and ensconced in the chimney corner like a huge clay figure set to
dry. For her the indestructible old lady has the supremest contempt,
heightened in intensity by a vivid remembrance of the time when
they were friends and rivals. Ah, poor Laura, she says, straightening
herself; she was always a poor creature, and see what she is now!
To those who wait long enough the wheel always comes round, she
thinks; and the days when Laura bore away the bell from her for
grace and sweetness and loveableness generally are avenged now,
when the one is a mere mollusc and the other has a serviceable
backbone that will last for many a year yet.

Then there is the musical old lady, who is fond of playing small
anonymous pieces of a jiggy character full of queer turns and
shakes, music that seems all written in demi-semi-quavers, and that
she gives in a tripping, catching way, as if the keys of the piano were
hot. Sometimes she will sing, as a great favour, old-world songs
which are almost pathetic for the thin and broken voice that chirrups
out the sentiment with which they abound; and sometimes, as a still
greater favour, she will stand up in the dance, and do the poor
uncertain ghosts of what were once steps, in the days when dancing
was dancing and not the graceless lounge it is now. But her dancing-
days are over, she says, after half-a-dozen turns; though, indeed,
sometimes she takes a frisky fit and goes in for the whole quadrille:
—and pays for it the next day.
The very dress of old ladies is in itself a study and a revelation of
character. There are the beautiful old women who make themselves
like old pictures by a profusion of soft lace and tender greys; and the
stately old ladies who affect rich rustling silks and sombre velvet;
and there are the original and individual old ladies, who dress
themselves after their own kind, like Mrs. Basil Montagu, Miss Jane
Porter, and dear Mrs. Duncan Stewart, and have a cachet of their
own with which fashion has nothing to do. And there are the old
women who wear rusty black stuffs and ugly helmet-like caps; and
those who affect uniformity and going with the stream, when the
fashion has become national—and these have been much exercised
of late with the strait skirts and the new bonnets. But Providence is
liberal and milliners are fertile in resources. In fact, in this as in all
other sections of humanity, there are those who are beautiful and
wise, and those who are foolish and unlovely; those who make the
best of things as they are, and those who make the worst, by
treating them as what they are not; those who extract honey, and
those who find only poison. For in old age, as in youth, are to be
found beauty, use, grace and value, but in different aspects and on
another platform. And the folly is when this difference is not allowed
for, or when the possibility of these graces is denied and their utility
ignored.

VOICES.
Far before the eyes or the mouth or the habitual gesture, as a
revelation of character, is the quality of the voice and the manner of
using it. It is the first thing that strikes us in a new acquaintance,
and it is one of the most unerring tests of breeding and education.
There are voices which have a certain truthful ring about them—a
certain something, unforced and spontaneous, that no training can
give. Training can do much in the way of making a voice, but it can
never compass more than a bad imitation of this quality; for the very
fact of its being an imitation, however accurate, betrays itself like
rouge on a woman's cheeks, or a wig, or dyed hair. On the other
hand, there are voices which have the jar of falsehood in every tone,
and which are as full of warning as the croak of the raven or the hiss
of the serpent. These are in general the naturally hard voices which
make themselves caressing, thinking by that to appear sympathetic;
but the fundamental quality strikes up through the overlay, and a
person must be very dull indeed who cannot detect the pretence in
that slow, drawling, would-be affectionate voice, with its harsh
undertone and sharp accent whenever it forgets itself.
But without being false or hypocritical, there are voices which puzzle
as well as disappoint us, because so entirely inharmonious with the
appearance of the speaker. For instance, there is that thin treble
squeak which we sometimes hear from the mouth of a well-grown
portly man, when we expected the fine rolling utterance which
would have been in unison with his outward seeming. And, on the
other side of the scale, where we looked for a shrill head-voice or a
tender musical cadence, we get that hoarse chest-voice with which
young and pretty girls sometimes startle us. This voice is in fact one

of the characteristics of the modern girl of a certain type; just as the
habitual use of slang is characteristic of her, or that peculiar
rounding of the elbows and turning out of the wrists—which
gestures, like the chest-voice, instinctively belong to men only and
have to be learned before they can be practised by women.
Nothing betrays feeling so much as the voice, save perhaps the
eyes; and these can be lowered, and so far their expression hidden.
In moments of emotion no skill can hide the fact of disturbed feeling
by the voice; though a strong will and the habit of self-control can
steady it when else it would be failing and tremulous. But not the
strongest will, nor the largest amount of self-control, can keep it
natural as well as steady. It is deadened, veiled, compressed, like a
wild creature tightly bound and unnaturally still. One feels that it is
done by an effort, and that if the strain were relaxed for a moment
the wild creature would burst loose in rage or despair—and that the
voice would break into the scream of passion or quiver down into
the falter of pathos. And this very effort is as eloquent as if there
had been no holding down at all, and the voice had been left to its
own impulse unchecked.
Again, in fun and humour, is it not the voice even more than the face
that is expressive? The twinkle of the eye, the hollow in the under
lip, the dimples about the mouth, the play of the eyebrow, are all
aids certainly; but the voice! The mellow tone that comes into the
utterance of one man; the surprised accents of another; the fatuous
simplicity of a third; the philosophical acquiescence of a fourth when
relating the most outrageous impossibilities—a voice and manner
peculiarly Transatlantic, and indeed one of the American forms of fun
—do we not know all these varieties by heart? have we not veteran
actors whose main point lies in one or other of these varieties? and
what would be the drollest anecdote if told in a voice which had
neither play nor significance? Pathos too—who feels it, however
beautifully expressed so far as words may go, if uttered in a dead
and wooden voice without sympathy? But the poorest attempts at
pathos will strike home to the heart if given tenderly and

harmoniously. And just as certain popular airs of mean association
can be made into church music by slow time and stately modulation,
so can dead-level literature be lifted into passion or softened into
sentiment by the voice alone.
We all know the effect, irritating or soothing, which certain voices
have over us; and we have all experienced that strange impulse of
attraction or repulsion which comes from the sound of the voice
alone. And generally, if not absolutely always, the impulse is a true
one, and any modification which increased knowledge may produce
is never quite satisfactory. Certain voices grate on our nerves and set
our teeth on edge; and others are just as calming as these are
irritating, quieting us like a composing draught, and setting vague
images of beauty and pleasantness afloat in our brains.
A good voice, calm in tone and musical in quality, is one of the
essentials for a physician—the 'bedside voice' which is nothing if not
sympathetic by constitution. Not false, not made up, not sickly, but
tender in itself, of a rather low pitch, well modulated and distinctly
harmonious in its notes, it is the very opposite of the orator's voice,
which is artificial in its management and a made voice. Whatever its
original quality may be, the orator's voice bears the unmistakeable
stamp of art and is artificial. It may be admirable; telling in a crowd;
impressive in an address; but it is overwhelming and chilling at
home, partly because it is always conscious and never self-
forgetting.
An orator's voice, with its careful intonation and accurate accent,
would be as much out of place by a sick-bed as Court trains and
brocaded silk for the nurse. There are certain men who do a good
deal by a hearty, jovial, fox-hunting kind of voice—a voice a little
thrown up for all that it is a chest-voice—a voice with a certain
undefined rollick and devil-may-care sound in it, and eloquent of a
large volume of vitality and physical health. That, too, is a good
property for a medical man. It gives the sick a certain fillip, and
reminds them pleasantly of health and vigour. It may have a
mesmeric kind of effect upon them—who knows?—so that it induces

in them something of its own state, provided it be not overpowering.
But a voice of this kind has a tendency to become insolent in its
assertion of vigour, swaggering and boisterous; and then it is too
much for invalided nerves, just as mountain-winds or sea-breezes
would be too much, and the scent of flowers or of a hayfield
oppressive.
The clerical voice again, is a class-voice—that neat, careful, precise
voice, neither wholly made nor yet natural—that voice which never
strikes one as hearty nor as having a really genuine utterance, but
which is not entirely unpleasant if one does not require too much
spontaneity. The clerical voice, with its mixture of familiarity and
oratory as that of one used to talk to old women in private and to
hold forth to a congregation in public, is as distinct in its own way as
the mathematician's handwriting; and any one can pick out blindfold
his man from a knot of talkers, without waiting to see the square-cut
collar and close white tie. The legal voice is different again; but this
is rather a variety of the orator's than a distinct species—a variety
standing midway between that and the clerical, and affording more
scope than either.
The voice is much more indicative of the state of the mind than
many people know of or allow. One of the first symptoms of failing
brain power is in the indistinct or confused utterance; no idiot has a
clear nor melodious voice; the harsh scream of mania is proverbial;
and no person of prompt and decisive thought was ever known to
hesitate nor to stutter. A thick, loose, fluffy voice too, does not
belong to the crisp character of mind which does the best active
work; and when we meet with a keen-witted man who drawls, and
lets his words drip instead of bringing them out in the sharp incisive
way that should be natural to him, we may be sure there is a flaw
somewhere, and that he is not 'clear grit' all through.
We all have our company voices, as we all have our company
manners; and, after a time, we get to know the company voices of
our friends, and to understand them as we understand their best
dresses and state service. The person whose voice absolutely

refuses to put itself into company tone startles us as much as if he
came to a state dinner in a shooting-jacket. This is a different thing
from the insincere and flattering voice, which is never laid aside
while it has its object to gain, and which affects to be one thing
when it means another. The company voice is only a little bit of
finery, quite in its place if not carried into the home, where however,
silly men and women think they can impose on their house-mates by
assumptions which cannot stand the test of domestic ease. The
lover's voice is of course sui generis; but there is another kind of
voice which one sometimes hears that is quite as enchanting—the
rich, full, melodious voice which irresistibly suggests sunshine and
flowers, and heavy bunches of purple grapes, and a wealth of
physical beauty at all four corners. Such a voice is Alboni's; such a
voice we can conceive Anacreon's to have been; with less
lusciousness and more stateliness, such a voice was Walter Savage
Landor's. His was not an English voice; it was too rich and accurate;
yet it was clear and apparently thoroughly unstudied, and was the
very perfection of art. There was no greater treat of its kind than to
hear Landor read Milton or Homer.
Though one of the essentials of a good voice is its clearness, there
are certain lisps and catches which are pretty, though never
dignified; but most of them are painful to the ear. It is the same with
accents. A dash of brogue; the faintest suspicion of the Scotch
twang; even a little American accent—but very little, like red-pepper
to be sparingly used, as indeed we may say with the others—gives a
certain piquancy to the voice. So does a Continental accent
generally; few of us being able to distinguish the French accent from
the German, the Polish from the Italian, or the Russian from the
Spanish, but lumping them all together as 'a foreign accent' broadly.
Of all the European voices the French is perhaps the most
unpleasant in its quality, and the Italian the most delightful. The
Italian voice is a song in itself; not the sing-song voice of an English
parish schoolboy, but an unnoted bit of harmony. The French voice is
thin, apt to become wiry and metallic; a head-voice for the most
part, and eminently unsympathetic; a nervous, irritable voice, that

seems more fit for complaint than for love-making; and yet how
laughing, how bewitching it can make itself!—never with the Italian
roundness, but câlinante in its own half-pettish way, provoking,
enticing, arousing. There are some voices which send you to sleep
and others which stir you up; and the French voice is of the latter
kind when setting itself to do mischief and work its own will.
Of all the differences lying between Calais and Dover, perhaps
nothing strikes the traveller more than the difference in the national
voice and manner of speech. The sharp, high-pitched, stridulous
voice of the French, with its clear accent and neat intonation, is
exchanged for the loose, fluffy utterance of England, where clear
enunciation is considered pedantic; where brave men cultivate a
drawl and pretty women a deep chest-voice; where well-educated
people think it no shame to run all their words into each other, and
to let consonants and vowels drip out like so many drops of water,
with not much more distinction between them; and where no one
knows how to educate his organ artistically, without going into
artificiality and affectation. And yet the cultivation of the voice is an
art, and ought to be made as much a matter of education as a good
carriage or a legible handwriting. We teach our children to sing, but
we never teach them to speak, beyond correcting a glaring piece of
mispronunciation or so. In consequence of which we have all sorts of
odd voices among us—short yelping voices like dogs; purring voices
like cats; croakings and lispings and quackings and chatterings; a
very menagerie in fact, to be heard in a room ten feet square, where
a little rational cultivation would have reduced the whole of that
vocal chaos to order and harmony, and would have made what is
now painful and distasteful beautiful and seductive.

BURNT FINGERS.
An old proverb says that a burnt child dreads the fire. If so, the child
must be uncommonly astute, and with a power of reasoning by
analogy in excess of impulsive desire rarely found either in children
or adults. As a matter of fact, experience goes a very little way
towards directing folks wisely. People often say how much they
would like to live their lives over again with their present experience.
That means, they would avoid certain specific mistakes of the past,
of which they have seen and suffered from the issue. But if they
retained the same nature as now, though they might avoid a few
special blunders, they would fall into the same class of errors quite
as readily as before, the gravitation of character towards
circumstance being always absolute in its direction.
Our blunders in life are not due to ignorance so much as to
temperament; and only the exceptionally wise among us learn to
correct the excesses of temperament by the lessons of experience.
To the mass of mankind these lessons are for the time only, and
prophesy nothing of the future. They hold them to have been
mistakes of method, not of principle, and they think that the same
lines more carefully laid would lead to a better superstructure in the
future, not seeing that the fault was organic and in those very initial
lines themselves. No impulsive nor wildly hopeful person, for
instance, ever learns by experience, so long as his physical condition
remains the same. No one with a large faculty of faith—that is,
credulous and easily imposed on—becomes suspicious or critical by
mere experience. How much soever people of this kind have been
taken in, in times past, they are just as ready to become the prey of
the spoiler in times to come; and it would be sad, if it were not so

silly, to watch how inevitably one half of the world gives itself up as
food whereon the roguery of the other half may wax fat.
The person of facile confidence, whose secrets have been blazed
abroad more than once by trusted friends, makes yet another and
another safe confidant—quite safe this time; one of whose fidelity
there is no doubt—and learns when too late that one panier percé is
very like another panier percé. The speculating man, without
business faculty or knowledge, who has burnt his fingers bare to the
bone with handling scrip and stock, thrusts them into the fire again
so soon as he has the chance. The gambler blows his fingers just
cool enough to shuffle the cards for this once only, sure that this
time hope will tell no flattering tale, that ravelled ends will knit
themselves up into a close and seemly garment, and heaven itself
work a miracle in his favour against the law of mathematical
certainty. In fact we are all gamblers in this way, and play our
hazards for the stakes of faith and hope. We all burn our fingers
again and again at some fire or another; but experience teaches us
nothing; save perhaps a more hopeless, helpless resignation towards
that confounded ill-luck of ours, and a weary feeling of having
known it all before when things fall out amiss and we are blistered in
the old flames.
In great matters this persistency of endeavour is sublime, and gets a
wealth of laurel crowns and blue ribands; but in little things it is
obstinacy, want of ability to profit by experience, denseness of
perception as to what can and what cannot be done; and the
apologue of Bruce's spider gets tiresome if too often repeated. The
most hopelessly inapt people at learning why they burnt their fingers
last time, and how they will burn them again, are those who,
whatever their profession, are blessed or cursed with what is called
the artistic temperament. A man will ruin himself for love of a
particular place; for dislike of a certain kind of necessary work; for
the prosecution of a certain hobby. Is he not artistic? and must he
not have all the conditions of his life exactly square with his desires?
else how can he do good work? So he goes on burning his fingers

through self-indulgence, and persists in his unwisdom to the end of
his life. He will paint his unsaleable pictures or write his unreadable
books; his path is one in which the money-paying public will not
follow; but though his very existence depends on the following of
that paying public, he will not stir an inch to meet it, but keeps
where he is because he likes the particular run of his hedgerows;
and spends his days in thrusting his hand into the fire of what he
chooses to call the ideal, and his nights in abusing the Philistinism of
the world which lets him be burnt.
And what does any amount of experience do for us in the matter of
friendship or love? As the world goes round, and our credulous
morning darkens into a more sceptical twilight, we believe as a
general principle—a mere abstraction—that all new friends are just
so much gilt gingerbread; and that a very little close holding and
hard rubbing brings off the gilt, and leaves nothing but a slimy,
sticky mess of little worth as food and of none as ornament. And
yet, if of the kind to whom friendship is necessary for happiness, we
rush as eagerly into the new affection as if we had never
philosophized on the emptiness of the old, and believe as firmly in
the solid gold of our latest cake as if we had never smeared our
hands with one of the same pattern before. So with love. A man
sees his comrades fluttering like enchanted moths about some
stately man-slayer, some fair and shining light set like a false beacon
on a dangerous cliff to lure men to their destruction. He sees how
they singe and burn in the flame of her beauty, but he is not
warned. If one's own experience teaches one little or nothing, the
experience of others goes for even less, and no man yet was ever
warned off the destructive fire of love because his companions had
burnt their fingers there before him and his own are sure to follow.
It is the same with women; and in a greater degree. They know all
about Don Juan well enough. They are perfectly well aware how he
treated A. and B. and C. and D. But when it comes to their own turn,
they think that this time surely, and to them, things will be different
and he will be in earnest. So they slide down into the alluring flame,

and burn their fingers for life by playing with forbidden fire. But have
we not all the secret belief that we shall escape the snares and
pitfalls into which others have dropped and among which we choose
to walk? that fire will not burn our fingers, at least so very badly,
when we thrust them into it? and that, by some legerdemain of
Providence, we shall be delivered from the consequences of our own
folly, and that two and two may be made to count five in our behalf?
Who is taught by the experience of an unhappy marriage, say? No
sooner has a man got himself free from the pressure of one chain
and bullet, than he hastens to fasten on another, quite sure that this
chain will be no heavier than the daintiest little thread of gold, and
this bullet as light and sweet as a cowslip-ball. Everything that had
gone wrong before will come right this time; and the hot bars of
close association with an uncomfortable temper and
unaccommodating habits will be only like a juggling trick, and will
burn no one's heart or hands.
People too, who burn their fingers in giving good advice unasked,
seldom learn to hold them back. With an honest intention, and a
strong desire to see right done, it is difficult to avoid putting our
hands into fires with which we have no business. While we are
young and ardent, it seems to us as if we have distinct business with
all fraud, injustice, folly, wilfulness, which we believe a few honest
words of ours will control and annul; but nine times out of ten we
only burn our own hands, while we do not in the least strengthen
those of the right nor weaken those of the wrong. We may say the
same of good-natured people. There was never a row of chestnuts
roasting at the fire for which your good-natured oaf will not stretch
out his hand at the bidding and for the advantage of a friend.
Experience teaches the poor oaf nothing; not even that fire burns.
To put his name at the back of a bill, just as a mere form; to lend his
money, just for a few days; or to do any other sort of self-
immolating folly, on the faithful promise that the fire will not burn
nor the knife cut—it all comes as easy to men of the good-natured
sort as their alphabet. Indeed it is their alphabet, out of which they
spell their own ruin; but so long as the impressionable temperament

lasts—so long as the liking to do a good-natured action is greater
than caution, suspicion, or the power of analogical reasoning—so
long will the oaf make himself the catspaw of the knave, till at last
he has left himself no fingers wherewith to pluck out the chestnuts
for himself or another.
The first doubt of young people is always a source of intense
suffering. Hitherto they have believed what they saw and all they
heard; and they have not troubled themselves with motives nor facts
beyond those given to them and lying on the surface. But when they
find out for themselves that seeming is not necessarily being, and
that all people are not as good throughout as they thought them,
then they suffer a moral shock which often leads them into a state
of practical atheism and despair. Many young people give up
altogether when they first open the book of humanity and begin to
read beyond the title-page; and, because they have found specks in
the cleanest parts, they believe that nothing is left pure. They are as
much bewildered as horror-struck, and cannot understand how any
one they have loved and respected should have done this or that
misdeed. Having done it, there is nothing left to love nor respect
further. It is only by degrees that they learn to adjust and apportion,
and to understand that the whole creature is not necessarily corrupt
because there are a few unhealthy places here and there. But in the
beginning this first scorching by the fire of experience is very painful
and bad to bear. Then they begin to think the knowledge of the
world, as got from books, so wonderful, so profound; and they look
on it as a science to be learned by much studying of aphorisms.
They little know that not the most affluent amount of phrase
knowledge can ever regulate that class of action which springs from
a man's inherent disposition; and that it is not facts which teach but
self-control which prevents.
After very early youth we all have enough theoretical knowledge to
keep us straight; but theoretical knowledge does nothing without
self-knowledge, or its corollary, self-control. The world has never yet
got beyond the wisdom of Proverbs and Ecclesiastes; and Solomon's

advice to the Israelitish youth lounging round the gates of the
Temple is quite as applicable to young Hopeful coming up to London
chambers as it was to them. Teaching of any kind, by books or
events, is the mere brute weapon; but self-control is the intelligent
hand to wield it. To burn one's fingers once in a lifetime tells nothing
against a man's common-sense nor dignity; but to go on burning
them is the act of a fool, and we cannot pity the wounds, however
sore they may be. The Arcadian virtues of unlimited trust and hope
and love are very sweet and lovely; but they are the graces of
childhood, not the qualities of manhood. They are charming little
finalities, which do not admit of modification nor of expansion; and
in a naughty world, to go about with one's heart on one's sleeve,
believing every one and accepting everything to be just as it
presents itself, is offering bowls of milk to tigers, and meeting armed
men with a tin sword. Such universal trust can only result in a
perpetual burning of one's fingers; and a life spent in pulling out hot
chestnuts from the fire for another's eating is by no means the most
useful nor the most dignified to which a man can devote himself.

DÉSŒUVREMENT.
Perhaps we ought to apologize for using a foreign label, but there is
no one English word which gives the full meaning of désœuvrement.
Only paraphrases and accumulations would convey the many subtle
shades contained in it; and paraphrases and accumulations are
inconvenient as headings. But if we have not the word, we have a
great deal of the thing; for désœuvrement is an evil unfortunately
not confined to one country nor to one class; and even we, with all
our boasted Anglo-Saxon energy, have people among us as
unoccupied and purposeless as are to be found elsewhere. Certainly
we have nothing like the Neapolitan lazzaroni who pass their lives in
dozing in the sun; but that is more because of our climate than our
condition, and if our désœuvrés do not doze out of doors, it by no
means follows that they are wide awake within.
No state is more unfortunate than this listless want of purpose which
has nothing to do, which is interested in nothing, and which has no
serious object in life; and the drifting, aimless temperament, which
merely waits and does not even watch, is the most disastrous that a
man or woman can possess. Feverish energy, wearing itself out on
comparative nothings, is better than the indolence which folds its
hands and makes neither work nor pleasure; and the most
microscopic and restless perception is more healthful than the dull
blindness which goes from Dan to Beersheba, and finds all barren.
If even death itself is only a transmutation of forces—an active and
energizing change—what can we say of this worse than mental
death? How can we characterize a state which is simply stagnation?
Not all of us have our work cut out and laid ready for us to do; very

many of us have to seek for objects of interest and to create our
own employment; and were it not for the energy which makes work
by its own force, the world would still be lying in barbarism, content
with the skins of beasts for clothing and with wild fruits and roots for
food. But the désœuvrés know nothing of the pleasures of energy;
consequently none of the luxuries of idleness—only its tedium and
monotony. Life is a dull round to them of alternate vacancy and
mechanical routine; a blank so dead that active pain and positive
sorrow would be better for them than the passionless negation of
their existence. They love nothing; they hope for nothing; they work
for nothing; to-morrow will be as to-day, and to-day is as yesterday
was; it is the mere passing of time which they call living—a moral
and mental hybernation broken up by no springtime waking.
Though by no means confined to women only, this disastrous state
is nevertheless more frequently found with them than with men. It is
comparatively rare that a man—at least an Englishman—is born with
so little of the activity which characterizes manhood as to rest
content without some kind of object for his life, either in work or in
pleasure, in study or in vice. But many women are satisfied to
remain in an unending désœuvrement, a listless supineness that has
not even sufficient active energy to fret at its own dullness.
We see this kind of thing especially in the families of the poorer class
of gentry in the country. If we except the Sunday school and district
visiting, neither of which commends itself as a pleasant occupation
to all minds—both in fact needing a little more active energy than
we find in the purely désœuvré class—what is there for the
unmarried daughters of a family to do? There is no question of a
profession for any of them. Ideas travel slowly in country places, and
root themselves still more slowly, even yet; and the idea of woman's
work for ladies is utterly inadmissible by the English gentleman who
can leave a modest sufficiency to his daughters—just enough to live
on in the old house and in the old way, without a margin for
luxuries, but above anything like positive want. There is no
possibility then of an active career in art or literature; of going out as

a governess, as a hospital nurse, or as a Sister. There is only home,
with the possible and not very probable chance of marriage as the
vision of hope in the distant future. And that chance is very small
and very remote; for the simple reason—there is no one to marry.
There are the young collegians who come down in reading parties;
the group of Bohemian artists, if the place be picturesque and not
too far from London; the curate; and the new doctor, fresh from the
hospitals, who has to make his practice out of the poorer and more
outlying clientèle of the old and established practitioners of the
place. But collegians do not marry, and long engagements are
proverbially hazardous; Bohemian artists are even less likely than
they to trouble the surrogate; and the curate and the doctor can at
the best marry only one apiece of the many who are waiting. The
family keeps neither carriages nor horses, so that the longest tether
to which life can be carried, with the house for the stake, is simply
the three or four miles which the girls can walk out and back. And
the visiting list is necessarily comprised within this circle. There is
then, absolutely nothing to occupy nor to interest. The whole day is
spent in playing over old music, in needlework, in a little desultory
reading, such as is supplied by the local book society; all without
other object than that of passing the time. The girls have had
nothing like a thorough education in anything; they are not specially
gifted, and what brains they have are dormant and uncultivated.
There is not even enough housework to occupy their time, unless
they were to send away the servants. Besides, domestic work of an
active kind is vulgar, and gentlemen and gentlewomen do not allow
their daughters to do it. They may help in the housekeeping; which
means merely giving out the week's supplies on Monday and
ordering the dinner on other days, and which is not an hour's
occupation in the week; and they can do a little amateur spudding
and raking among the flower-beds when the weather is fine, if they
care for the garden; and they can do a great deal of walking if they
are strong; and this is all that they can do. There they are, four or
five well-looking girls perhaps, of marriageable age, fairly healthy
and amiable, and with just so much active power as would carry

them creditably through any work that was given them to do, but
with not enough originative energy to make them create work for
themselves out of nothing.
In their quiet uneventful sphere, with the circumscribed radius and
the short tether, it would be very difficult for any women but those
few who are gifted with unusual energy to create a sufficient human
interest; to ordinary young ladies it is impossible. They can but
make-believe, even if they try—and they don't try. They can but
raise up shadows which they would fain accept as living creatures if
they give themselves the trouble to evoke anything at all, and they
don't give themselves the trouble. They simply live on from day to
day in a state of mental somnolency—hopeless, désœuvrées,
inactive; just drifting down the smooth slow current of time, with not
a ripple nor an eddy by the way.
Quiet families in towns, people who keep no society and live in a
self-made desert apart though in the midst of the very vortex of life,
are alike in the matter of désœuvrement; and we find exactly the
same history with them as we find with their country cousins,
though apparently their circumstances are so different. They cannot
work and they may not play; the utmost dissipation allowed them is
to look at the outside of things—to make one of the fringe of
spectators lining the streets and windows on a show day, and this
but seldom; or to go once or twice a year to the theatre or a
concert. So they too just lounge through their life, and pass from
girlhood to old age in utter désœuvrement and want of object. Year
by year the lines about their eyes deepen, their smile gets sadder,
their cheeks grow paler; while the cherished secret romance which
even the dullest life contains gets a colour of its own by age, and a
firmness of outline by continual dwelling on, which it had not in the
beginning. Perhaps it was a dream built on a tone, a look, a word—
may be it was only a half-evolved fancy without any basis whatever
—but the imagination of the poor désœuvrée has clung to the
dream, and the uninteresting dullness of her life has given it a mock
vitality which real activity would have destroyed.

This want of healthy occupation is the cause of half the hysterical
reveries which it is a pretty flattery to call constancy and an
enduring regret; and we find it as absolutely as that heat follows
from flame, that the mischievous habit of bewailing an irrevocable
past is part of the désœuvrée condition in the present. People who
have real work to do cannot find time for unhealthy regrets, and
désœuvrement is the most fertile source of sentimentality to be
found.
The désœuvrée woman of means and middle age, grown grey in her
want of purpose and suddenly taken out of her accustomed groove,
is perhaps more at sea than any others. She has been so long
accustomed to the daily flow of certain lines that she cannot break
new ground and take up with anything fresh, even if it be only a
fresh way of being idle. Her daughter is married; her husband is
dead; her friend who was her right hand and manager-in-chief has
gone away; she is thrown on her own resources, and her own
resources will not carry her through. She generally falls a prey to her
maid, who tyrannizes over her, and a phlegmatic kind of despair,
which darkens the remainder of her life without destroying it. She
loses even her power of enjoyment, and gets tired before the end of
the rubber which is the sole amusement in which she indulges. For
désœuvrement has that fatal reflex action which everything bad
possesses, and its strength is in exact ratio with its duration.
Women of this class want taking in hand by the stronger and more
energetic. Many even of those who seem to do pretty well as
independent workers, men and women alike, would be all the better
for being farmed out; and désœuvrées women especially want
extraneous guidance, and to be set to such work as they can do, but
cannot make. An establishment which would utilize their faculties,
such as they are, and give them occupation in harmony with their
powers, would be a real salvation to many who would do better if
they only knew how, and would save them from stagnation and
apathy. But society does not recognize the existence of moral
rickets, though the physical are cared for; consequently it has not

begun to provide for them as moral rickets, and no Proudhon has yet
managed to utilize the désœuvrés members of the State. When they
do find a place of retreat and adventitious support, it is under
another name.
The retired man of business, utterly without object in his new
conditions, is another portrait that meets us in country places. He is
not fit for magisterial business; he cannot hunt nor shoot nor fish;
he has no literary tastes; he cannot create objects of interest for
himself foreign to the whole experience of his life. The idleness
which was so delicious when it was a brief season of rest in the
midst of his high-pressure work, and the country which was like
Paradise when seen in the summer only and at holiday time, make
together just so much blank dullness now that he has bound himself
to the one and fixed himself in the other. When he has spelt over
every article in the Times, pottered about his garden and his stables,
and irritated both gardener and groom by interfering in what he
does not understand, the day's work is at an end. He has nothing
more to do but eat his dinner and sip his wine, doze over the fire for
a couple of hours, and go to bed as the clock strikes ten.
This is the reality of that long dream of retirement which has been
the golden vision of hope to many a man during the heat and
burden of the day. The dream is only a dream. Retirement means
désœuvrement; leisure is tedium; rest is want of occupation truly,
but want of interest, want of object, want of purpose as well; and
the prosperous man of business, who has retired with a fortune and
broken energies, is bored to death with his prosperity, and wishes
himself back to his desk or his counter—back to business and
something to do. He wonders, on retrospection, what there was in
his activity that was distasteful to him; and thinks with regret that
perhaps, on the whole, it is better to wear out than to rust out; that
désœuvrement is a worse state than work at high pressure; and that
life with a purpose is a nobler thing than one which has nothing in it
but idleness:—whereof the main object is how best to get rid of
time.

THE SHRIEKING SISTERHOOD.
We by no means put it forward as an original remark when we say
that Nature does her grandest works of construction in silence, and
that all great historical reforms have been brought about either by
long and quiet preparation, or by sudden and authoritative action.
The inference from which is, that no great good has ever been done
by shrieking; that much talking necessarily includes a good deal of
dilution; and that fuss is never an attribute of strength nor
coincident with concentration. Whenever there has been a very deep
and sincere desire on the part of a class or an individual to do a
thing, it has been done not talked about; where the desire is only
halfhearted, where the judgment or the conscience is not quite clear
as to the desirableness of the course proposed, where the chief
incentive is love of notoriety and not the intrinsic worth of the action
itself—personal kudos, and not the good of a cause nor the
advancement of humanity—then there has been talk; much talk;
hysterical excitement; a long and prolonged cackle; and heaven and
earth called to witness that an egg has been laid wherein lies the
germ of a future chick—after proper incubation.
Necessarily there must be much verbal agitation if any measure is to
be carried the fulcrum of which is public opinion. If you have to stir
the dry bones you must prophesy to them in a loud voice, and not
leave off till they have begun to shake. Things which can only be
known by teaching must be spoken of, but things which have to be
done are always better done the less the fuss made about them;
and the more steadfast the action, the less noisy the agent. Purpose
is apt to exhale itself in protestations, and strength is sure to
exhaust itself by a flux of words. But at the present day what Mr.

Carlyle called the Silences are the least honoured of all the minor
gods, and the babble of small beginnings threatens to become
intolerable. We all 'think outside our brains,' and the result is not
conducive to mental vigour. It is as if we were to set a plant to grow
with its heels in the air, and then look for roots, flowers and fruit, by
the process of excitation and disclosure.
One of our quarrels with the Advanced Women of our generation is
the hysterical parade they make about their wants and their
intentions. It never seems to occur to them that the best means of
getting what they want is to take it, when not forbidden by the law—
to act, not to talk; that all this running hither and thither over the
face of the earth, this feverish unrest and loud acclaim are but the
dilution of purpose through much speaking, and not the right way at
all; and that to hold their tongues and do would advance them by as
many leagues as babble puts them back. A small knot of women,
'terribly in earnest,' could move multitudes by the silent force of
example. One woman alone, quietly taking her life in her own hands
and working out the great problem of self-help and independence
practically, not merely stating it theoretically, is worth a score of
shrieking sisters frantically calling on men and gods to see them
make an effort to stand upright without support, with interludes of
reproach to men for the want of help in their attempt. The silent
woman who quietly calculates her chances and measures her
powers with her difficulties so as to avoid the probability of a fiasco,
and who therefore achieves a success according to her endeavour,
does more for the real emancipation of her sex than any amount of
pamphleteering, lecturing, or petitioning by the shrieking sisterhood
can do. Hers is deed not declamation; proof not theory; and it
carries with it the respect always accorded to success.
And really if we think of it dispassionately, and carefully dissect the
great mosaic of hindrances which women say makes up the
pavement of their lives, there is very little which they may not do if
they like—and can. They have already succeeded in reopening for
themselves the practice of medicine, for one thing; and this is an

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Quantification In Nonclassical Logic Dov Gabbay Dimitrij Skvortsov

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