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
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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
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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 between 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 t o 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 [Rasiowa and 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 relational 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, 19791 and its later extended version [Harel and Tiuryn, 20001 study particular dynamic modal logics. [Goldblatt, 19841 is devoted to topos semantics; 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 intermediate 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 problems 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 1980 by 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 material, 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 viewpoints 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 Benthem, Giovanna Corsi, Leo Esakia, Silvio Ghilardi, Dick De Johng, Rosalie Iemhoff, Marcus Kracht, Vladimir Krupski, Grigori Mints, Hiroakira Ono, NobuYuki 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 t o 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 understandable 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 typing, 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.
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Introduction Quantification and modalities have always been topics of great interest for logicians. 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 t o be closely related to modal logics. (A) The introduction of quantifier axioms to classical logic is fairly straightforward. 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 logics by 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 ( A V 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.
INTRODUCTION
x
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 correspondence 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 f OOp > D 0 p 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 conversely some natural conditions on the semantics require complex and sometimes 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 themselves 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 (r may 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 different 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 considerations, 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 (many x)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, [van Lambalgen, 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 ('A is 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 ourselves 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 syntax. Chapter 1 contains definitions and results on syntax and semantics of nonclassical 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 versions are generated by a 'scheme' showing the reference structure of quantifiers. (Schemes are quite similar t o 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 introduce 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 incompleteness 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, 19721731 and [van Benthem, 19831. Chapter 4 studies algebraic semantics. Here the main objects are Heytingvalued (or modal-valued) sets. In the intuitionistic case this semantics was studied by many authors, see [Dragalin, 19881, [Fourman and Scott, 19791, [Goldblatt, 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 models'. 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 predecessor 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 first e author is happy t o fulfill his promise given in the preface of [Gabbay, 19811: "It would require further research t o be able t o 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 semantics. 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 introduce different types of canonical models. The simplest kind is rather wellknown, 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 [Corsi and 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 different 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, 19871 and other related results. Here are chapter headings in preparation for later volumes: Chapter 8. Simplicia1 semantics 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-symbols and 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 t o 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 t o 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 1 and 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 R
o
for inclusion, C for proper inclusion;
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; I d w 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 usually denoted just by j ) sending every x E X to itself; R R
1 V denotes the restriction 1 V = R n (V x V), and f
Y (which is
of a relation R to a subset V, i.e. 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;
1x1 denotes 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 I i E I ) (or (Xi)iEr ) denotes the family of sets Xi with indices in the set I;
U Xi
icI
denotes the disjoint union of the family (Xi)iET,i.e. IJ iEI
Xix {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 compoC is denoted by either g . f or f o g. So sition of f : A --iB and g : B (f og)(x) = (9 . f )(x) = g(f (x)). Obviously, Cmn#Oiffn>Oorm=O, Tm,#Oiffn>m. A map f : I, In (for fixed n) is presented by the table
-
We use a special notation for some particular maps. Trans~ositions02 E T n for n
2 2, 1 5 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
for m 2 0; In particular, there are simple embeddings al;L := + 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.
INTRODUCTION
xviii
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 projections. 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 1 to 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) :=
a(i) n+l
for i E I,, ifi=m+l.
In particular, for any n we have (a;)+ =;:6;
E Cn+l,n+a:
for i E I,, ifi=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(Ia1) (or lal) denotes the length of a sequence a;
a p 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 = ( a l , . . . ,a,) E Sn can 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, In acts on Sn via composition:
-
-
gives rise to the map .rr, : Sn Sm sending a to Thus every map a E C, 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 nla := a - ai := ai := a . bn = (al ,... tai-l,ai+l,...,an). A
Hence we obtain
INTRODUCTION L e m m a 0.0.1
xix
(1) .rrT
whenever a E Sn, a E Em,,
7
.To =
E Ck,.
(2) If a is a permutation (a E T,), To-1 = (r,)-l.
Proof
then
T,
is a permutation of Sn and
(1) Since composition of maps is associative, we have a . ( a . T ) = ( a . a ) . r.
We use the following relations on n-tuples:
L e m m a 0.0.2 Let S # 0 , a E Em,. Then
r,[Sn]= {a E SmI a s u b a ) , where a s u b a 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 sub a , then a = b . a for some b; just put b,(%) := ai and add arbitrary bk for k @ r ( a ) . L e m m a 0.0.3 For IS/ surje~tive.~
> 1, a
E Em,, a is injective iff .rr,
:
Sn + Sm is
Proof If a is injective, then for any a E Sn,a ( i ) = a ( j ) + i = j + ai = a j , i.e. a sub a. 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 sub a is not true, i.e. a 9.rr, [Sn]by 0.0.2. L e m m a 0.0.4 For IS1 > 1, a E Em,, a is surjective iff r, is injective.
Proof .rr,a and a($)= j . c , d E S,
-
Suppose a : I, In is surjective and a, b E Sn,.rr,a # nub. If .rr,b differ a t the j t h component, then ai # bi for i In such that On the other hand, let a E Em, be non-surjective, j E I, - rng(a). Let c # d. T a k e a = cn; b =cf-'den-j. Then a # b a n d - i r , a = n , b = c m .
w
Hence we obtain L e m m a 0.0.5 For IS1 > 1, a E C,, bijective. 3Clearly, if IS1 = 1, then
T,
a is bijective ijf
is bijective for every a E C,,.
-ir,
:
Sn
+
Sm is
INTRODUCTION
xx
We further simplify notation in some particular cases. Let T: := .rrs;, so facet embedding 61 eliminates the i t h component from an n-tuple a E Sn. Let also T I: := ran, - 71; := T ~ ; , where a? E Thus
is a simple projection, a; E
is a simple embedding.
T? (al, . . . ,an) = (al, .. . , a n , a,) for n > 0, ~ ; ( a )= a - an+l = ( a ~. ., ., a n ) for a = (al, . . . , a n ,an+l) E Dntl, n
> 0.
We say that a sequence a E Dn is distinct, if all its components at are different. L e m m a 0.0.6 If a, T : I, distinct a E Sn. Proof
---,I,,
a#
T
and IS1 2 n, then a . a
If for some i, ~ ( i #) a ( i ) , then a,(i)
# a . 7 for any
# a,(i).
L e m m a 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.
¤
L e m m a 0.0.8 (1) Let a E S n , b E Sm, r ( b ) C_ r(a). Then b = a . o for some a E Em,. (2) Moreover, zf b is d i ~ t i n c t ,then ~ u is an injection.
Proof
Put a(i) = j for some j such that bi = a j .
41n other words, b is obtained by renumbering a subsequence of a.
Contents Preface
v
Introduction
ix
I
1
Preliminaries
1 Basic propositional logic 1.1 Propositional syntax . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Algebraic semantics . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Relational semantics (the modal case) . . . . . . . . . . . . . . . 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . -1.3.2 Kripke frames and models . . . . . . . . . . . . . . . . . . 1.3.3 Main constructions . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Conical expressiveness . . . . . . . . . . . . . . . . . . . . 1.4 Relational semantics (the intuitionistic case) . . . . . . . . . . . . 1.5 Modal counterparts . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 General Kripke frames . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Canonical Kripke models . . . . . . . . . . . . . . . . . . . . . . 1.8 First-order translations and definability . . . . . . . . . . . . . . 1.9 Some general completeness theorems . . . . . . . . . . . . . . . . 1.10 Trees and unravelling . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 PTC-logics and Horn closures . . . . . . . . . . . . . . . . . . . . 1.12 Subframe and cofinal subframe logics . . . . . . . . . . . . . . . . 1.13 Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Tabularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Transitive logics of finite depth . . . . . . . . . . . . . . . . . . . 1.16 A-operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Neighbourhood semantics . . . . . . . . . . . . . . . . . . . . . .
xxi
3
3 3 5 11 19 19 19 24 30 32 37 38 40 44 47 48 52 58 65 68 70 72 76
CONTENTS
xxii
2 Basic predicate logic 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Variable substitutions . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Formulas with constants . . . . . . . . . . . . . . . . . . . . . . . 2.5 Formula substitutions . . . . . . . . . . . . . . . . . . . . . . . . 2.6 First-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 First-order theories . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Deduction theorems . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Godel-Tarski translation . . . . . . . . . . . . . . . . . . . . . . . 2.12 The Glivenko theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.13 A-operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Adding equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Propositional parts . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Semantics from an abstract viewpoint . . . . . . . . . . . . . . .
I1
Semantics
79 79 81 85 102 105 119 139 142 146 151 153 157 158 172 180 185
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 M e t a f r a m e 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 Simplicia1 frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
I11
Completeness
481
6 K r i p k e 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 K F . . . . . . . . . . . 515 6.6 Directed frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
+
CONTENTS
xxiv 6.7 6.8 6.9 6.10 6.11
Logics of linear frames . . . . . . . . . . . . . . . . . . . . . . . . Properties of A-operation . . . . . . . . . . . . . . . . . . . . . . A-operation preserves completeness . . . . . . . . . . . . . . . . . Trees of bounded branching and depth . . . . . . . . . . . . . . . Logics of uniform trees . . . . . . . . . . . . . . . . . . . . . . . .
524 528 532 536 539
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
Part I
Preliminaries
This page intentionally left blank
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 P L = {pl,p2,. . . ) of proposition letters, the classical propositional connectives A , V , >, I and the unary modal connectives 01,. . . , ON; the derived connectives are introduced in a standard way as abbreviations: 7 A : = (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,uence of natural numbers a = kl . . . k, from I F , 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 t h e term 'unimodal' t o 'monomodal'.
4
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
As usual, for a finite set of formulas r , A r denotes its conjunction and V I' 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 01 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 preserving 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 P L , 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 substitution 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 ~ this n 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 L P N 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 1 5 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 +r is axzomatised by the set r. A logic is called finitely 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 A4 AD AM A2 A3 AGrz AL A5 AB Ati At:!
:=Op>p, > OOp, : = 0 0 p > Up, := OOp > OOp (McKinsey formula), := OOp > OOp, :=O(pAOp>q)VO(qAUq>p), := O(O(p 3 Op) > p) > p (Grzegorczyk formula), := O(Op > p) > Up (Lob formula), := OOp > Up, := OOp 3 p , := OlOzp 3 p, := 0 2 0 1 > ~ p, := Op
6
CHAPTER 1. BASIC PROPOSITIONAL LOGIC := K + OT, := K A4, := K 4 + A T , := S 4 AM, := S 4 A2, := K 4 A3, := S 4 AB, := K A5,
+ + + + + +
+ AT,
T
:= K
D4 D4.1 S4.3
:= D + A 4 , := D 4 + A M , := S 4 O(Op > q) v O ( 0 q > p), := S 4 AGrz (Grzegorczyk logic), := K AL, (Godel-Lob logic),
Grz GL K.t
+ + + := K 2 + Atl + At2.
The corresponding N-modal versions are denoted by D N , TN etc.; so for example, DN:=KN+{O~TI~<~
(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, A r m denotes its restriction to m-formulas. The sets A [m are 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 Nmodal 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, A Y A r o 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 superintuitionistic logics: EM AJ AJAPl APn K P
:=
p V 7 p (the law of the excluded middle);
..-- -.p V 7 l p (the weak law of the excluded middle); := := := :=
7 7 p v ( 1 1 p 3 p); EM; p n V ( p n > A P n - 1 ) (for n > 1); ( 7 p > q V r ) > ( i p > q) V ( ~ >pr) (Kreisel-Putnam formula);
(
)
)
n
V pi (Gabbay-De V p j > j+i V p j > i=o (i~o j+i := ( P > 4) v ( 9 3 PI; Pi 3
Jongh formulas);
n
i=O
IG, HJ LC CL
V (pi pj); O
+ +
The following inclusions are well-known: HcHJcLCcCL,
A superintuitionistic logic C is called consistent if I6C; C is said to be intermediate if H C CL. It is well-known that every consistent superintuitionistic propositional logic is intermediate.
c
L e m m a 1.1.1
(1) Some theorems of K N :
8
CHAPTER 1. BASIC PROPOSITIONAL LOGlC
(2) The following rules are admissible i n every modal logic: Monotonicity rules Replacement rules
A>B
A-B OiA s O i B
OiA 3 OiB (3) Some theorems of S 4 :
o o p = Op; (Op v Oq) = u p v Oq.
(4) A theorem of S4.2:
-
oo(Api) i
onpi. i
Lemma 1.1.2 Some theorems of H :
A
(5) i=1( ( P i 3 q ) 3 q ) -
(A
i=1pi>*) 3 9 ) .
Lemma 1.1.3 (Propositional replacement rule) The following rule is admissible 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 S Y N T A X
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 ) i f f ( AA 3 A ) E C for
(11) Let A be an N-modal logic, A E ( A + r )ifl
some finite A
r U { A ) a set
5 Sub@).
of N-modal formulas.
Then
1
A 3 A t A for some r 2 0 and some finite A i Sub(I').
(
(111) Let A be a 1-modal logic, T
(1) (
k=O
r U { A ) E CPl.
Then A E ( A+I?) i f f
( AU k A ) 3 A) t A for some r 2 0 and some finite
A
Sub(r)
- i n the general case;
(2)
( AO V A3 A) E A for some r 2 0 and some finite -
(3)
provided T
A 5 Sub(r)
G A;
( AA A UA 3 A) E A for - provided K 4
some finite A
C Sub(I')
2 A;
(4) ( ACIA 3 A) E A for some finite A c S u b ( r ) - provided S 4 2 A . Similarly one can simplify the claim (2) for the case when A is an N-modal logic containing T N , K ~ Nor, 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:
i f formulas from I? and
r'
do not have common proposition letters.
(2) For N-modal logics:
i f formulas from I? and I" do not have common proposition letters. (3) For 1-modal logics:
i f formulas from I? and I" do not have common proposition letters. I n 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 1 1 A E H . Proof In fact, A E C L implies --A theorem.
E CL, so we can apply the Glivenko
1.2. ALGEBRAIC SEMANTICS
1.2
11
Algebraic semantics
For modal and intermediate propositional logics several kinds of semantics are known. Algebraic semantics is the most general and straightforward; it interprets 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. Neighbourhood 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 the greatest element. Note that (*) can be written as
--+
0 and 1 = a
--+
a is
In particular, a--+b=liffaIb. Also recall that an implicative lattice is always distributive: ( a V b) A c = (aA c) V (b Ac), (a A b) V c = (a v c) A ( b v c ) . 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),
j€J
/\ a j := max{b 1 V j E J b 5 aj). j€J
A complete lattice is implicative iff it is well-distributive, i.e., the following holds: aA
(v
j € J
aj) =
v j€J
3Cf. [Rasiowa and Sikorski, 1963; Borceaux, 19941.
(aAaj).
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
12
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 ak
(aj + b j )
A
< bk.
j E J
The latter holds, since by 1.2.1(*), it is equivalent to
Lemma 1.2.3
Proof
hence
and thus
(2) hence
(I)
1.2. ALGEBRAIC SEMANTICS and thus
Lemma 1.2.4 A ( v l -+ u ) = ( V v i + u ) . iEI
iEI
Proof ( 2 ) vi 5 V vi implies iEI
v v i + u SVi
+ u;
iEI
hence
(I Since ) A(Vi
'u ) 5 vi
-)
U,
iEI
it follows that for any i E I
Hence
(V vi) A A ( v i iEI
+U
)
iEI
V (vi A A iEI
( ~-+ i
u ) )5 U .
iEI
Eventually A ( v i -+u) 5 V v i t u . iEI
iEI
w A Boolean algebra is a particular case of a Heyting algebra, where a V i a = 1. In this case V, A, --+, 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) 7
in Heyting algebras and its analogue
in Boolean algebras.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
14
Definition 1.2.5 A n N-modal algebra is a structure
0 = ( 0 , n , u,
-,
0, l , n l , . . . , O N ) ,
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 i f the Boolean algebra fib is complete. We also use the dual operations
For 1-modal algebras we write 0 , 0 rather than 01, O 1 (cf. Section 1.1.1). Definition 1.2.6 A topo-Boolean (or interior, or S 4 - ) algebra is a 1-modal algebra satisfying the inequalities
I n this case CI is called the interior operation and its dual 0 the closure operation. A n element a is said to be open i f Ua = a and closed i f O a = a. Proposition 1.2.7 The open elements of a topo-Boolean algebra 0 constitute a Heyting algebra: a0= (a0, n , u, 4, o),
i n which a
Proof
-,b = O(a 3
b). Moreover, i f fl is complete then f1° also is, and
Cf. [ ~ c ~ i nand s e Tarski, ~ 19441; [Rasiowa and 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 [Rasiowa and Sikorski, 19631 Definition 1.2.8 A valuation i n 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 ) = c p ( 4 3 cp(B); (5) ~ ( n i A = ) ni~(A).
The pair ( a , cp) is then called an (algebraic) model over f l . A n 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 i n the algebra f l (notation: S2 k A) i f it is true i n every model over S2.
L e m m a 1.2.9 Let S2 be an N-modal algebra, S a propositional substitution. Let cp, 7 be valuations i n 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.
L e m m a 1.2.10 (Soundness l e m m a ) 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 S2 k SA, take an arbitrary valuation cp in f l , and consider a new valuation 7 according to (4) from Lemma 1.2.9. So we obtain
i.e. S2 i= SA. The classical tautologies are valid in f l , 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: n i x Oiy, (*) x Y because x 5 y implies
< *
<
nix
= Oi(x n y) = n i x
Now since
( a a b) n a
n Oiy.
< b,
by monotonicity (*), we have
O ( a ZI b) n Oa 5 Ob, which implies
U ( a 3 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.
C H A P T E R 1. BASIC PROPOSITIONAL L O G I C
16
D e f i n i t i o n 1.2.11 ML(52) is called the modal logic of the algebra 52. W e also define the modal logic of a class C of N-modal algebras ML(C) := ~ { M L ( s ~I 52 ) E C}. Note that ML(52) is consistent iff the algebra 52 is nondegenerate, i.e. iff 0 # 1 in 52.
-
D e f i n i t i o n 1.2.12 A valuation in a Heyting algebra 52 is a map cp : P L It has a unique extension cp' : LPo 0 such that
--
0.
A s i n the modal case, the pair (52, cp) is called an (algebraic) model over 52. A n intuitionistic formula A is said to be true i n (52, p ) if pl(A) = 1 (notation: (52, cp) b A); A is called valid in the algebra 52 (notation: 52 k A ) i f 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 i n 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 ( S o u n d n e s s l e m m a ) For a Heyting algebra 52, the set
is a superintuitionistic logic. D e f i n i t i o n 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
D e f i n i t i o n 1.2.16 A valuation cp i n an S4-algebra f2 is called intuitionistic i f it is a valuation i n i.e. i f its values are open.
17
1.2. ALGEBRAIC SEMANTICS
Definition 1.2.17 Godel-Tarski translation is the map tic to 1-modal fomulas defined by the following clauses:
(-)T
from intuitionis-
IT = I ; qT = Oq for every proposition letter q; ( A A B ) =~ A A ~B ~ ; ( A V B )=~ A ~ v B ~ ;
( A3
B)T = O(AT 3
BT).
Lemma 1.2.18 (OAT F 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
w
1.1.1. Lemma 1.2.19 Let (1) Let ip,
be an S4-algebra.
1C, be valuations in
such that for any q E PL
Then for any intuitionistic formula A, i p ' ( ~ )= * ( A T ) .
I n 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)
+(cT)
= ipl(B)-+ ipl(C)= + ( B T )-+ = 0 ( + ( B T )3 = +(O(BT> C T ) )= $ ( ( B 3 C ) T ) .
(2) (Only if.) Assume nok A. Let $ be an arbitrary valuation in be the valuation in a0such that
+(cT))
a, and let ip
for every q E PL. By (1) and our assumption, we have:
Hence f2 k AT. k AT. By ( I ) , for any valuation cp in (If.) Assume ip(AT)= 1. Hence a0FA.
a0we have ipl(A) = ¤
C H A P T E R 1. B A S I C P R O P O S I T I O N A L LOGIC
18
Let us now recall the Lindenbaum algebra construction. For an N-modal or superintuitionistic logic A, the relation between N-modal (respectively, intuitionistic) formulas such that N~
A-AB
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 L i n d ( A ) of a modal logic A is the
set LPN/ -A with the operations
[A] n [B] := [AA B], [A] u [B]:= [A v B], -[A]
:= [iA],
0 := [ l ] ,
T h e o r e m 1.2.21 For a n N-modal logic A
(1) L i n d ( A ) is a n N-modal algebra;
Definition 1.2.22 The Lindenbaum algebra L i n d ( C ) of a superintuitionistic logic C, is the set L%/ -c with the operations
[A] r\ [B]:= [A A B], [A] v [B] := [A v B], [A] 4 [B] := [A > B], 0 := [ I ] .
T h e o r e m 1.2.23 For a superintuitionistic logic C, (1) L i n d ( C ) is a Heyting algebra;
i s valid i n a modal algebra f2 Definition 1.2.24 A set of modal formulas (notation: f2 k r ) i f all these formulas are valid; similarly for intuitionistic formulas and Heyting algebras. I n this case C2 is called a r-algebra. The set oJ all r-algebras is called a n algebraic variety defined by r .
1.3. RELATIONAL SEMANTICS (THE MODAL CASE)
19
Algebraic varieties can be characterised in algebraic terms, due to the wellknown Birkhoff theorem [Birkhoff, 19791 (which holds also in a more general context):
Theorem 1.2.25 A class of modal o r Heyting algebras i s a n algebraic variety iff it i s 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 T h e poset M N of N-modal propositional logics (ordered by inclusion) i s dually isomorphic t o the set of all algebraic varieties of N-modal algebras; similarly for superintuitionistic logics and Heyting algebras.
Relational semantics (the modal case)
1.3
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
T h e possible is that which either i s o r 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 a t 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.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
20
Definition 1.3.1 An N-modal (propositional)Kripke frame is an (N+l)-tuple F = ( W ,R 1 , .. . ,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 often we write u E F rather than u E W . For a Kripke frame F = ( W ,R 1 , . ..,R N ) and a sequence a E IF we define the relation R , on W :
(Recall that .k is a void sequence, I d w is the equality relation, see the Introduction.) Every N-modal Kripke frame F = ( W ,R1, . . . ,R N ) corresponds t o an N modal algebra
where
n,U,
- are the standard set-theoretic operations on subsets o f W , and
OiV := { X I R i ( 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 : P L ---+ 2W. A Kripke model over a frame F is a pair M = (F,0 ) , where 6 is a valuation i n F . 0 is extended to all formulas in the standard way, according to Definition 1.2.8:
(1) 0 ( 1 ) = 0 ; (2) 6 ( A A B ) = 0 ( A )n 0 ( B ) ; (3) e ( A V B ) = @ ( AU) 6 ( B ) ;
(4) 6 ( A > B ) = 6 ( A ) 3 6 ( B ) ; (5) 0 ( 0 i A ) = Ui0(A)= {U I R ~ ( u2) 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 o f the model M (or that w forces A). The above definition corresponds t o the well-known inductive definition o f forcing in a Kripke model given by (1)-(6) in the following lemma. Lemma 1.3.3
(1) M , u k q z f f
(3) M , u ~ B A C iff
u E 0(q) (for q 6 PL);
(M,ukBandM,uI=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
(7) M , u k i B
iff
iff
( M , u b B impliesM,ubC);
M,uI$B;
(8) M , u k O i B
i f f 3v E R i ( 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,.. . , R N ) is a pair ( F ,8 ) , in which 8 : { p l , . . . , 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 ) i f it is true at every world of M ; A is satisfied in M i f 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 ) i f it is true in every model over F . A set of formulas is valid in F (notation: F != I?) i f 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.
C H A P T E R I . BASIC PROPOSITIONAL LOGIC
22
(2) For a class C of N-modal frames, the set
is an N-modal logic. Definition 1.3.9 M L ( F ) (respectively, M L ( 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 ) is not always a modal logic; it is closed under M P and 0-introduction but not necessarily under substitution. The following is 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 )n V ( 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 Vx3y x R y ; 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 ( S 4 . 2 ) consists of all S4-frames with Church-Rosser property (or confluent, or piecewise directed): V x ,y , z ( x R y & x R z
+ 3t ( y R t & z R t ) ) ,
or equivalently,
R - ~ O R GR O R - ' ;
1.3. RELATIONAL SEMANTICS (THE MODAL CASE)
23
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 y R z
V zRy)),
or equivalently
R-~ORLI~URUR-~; 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 ascending chains xlR - x z 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 (respectively, serial, transitive) if it contains TN (respectively, D N , 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.) i f it is determined by some class of finite frames; A has the countable frame property (c.f.p.) i f 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 i n some A-frame (respectively, i n a countable/finite A-frame). (2) M L ( V ( A ) ) is the smallest Kripke-complete extension of A ; so A is Kripkecomplete i f fA = 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 semantics; counterexamples were found by S. Thomason, K. Fine, V. Shehtman, J. Van Benthem, cf. [Chagrov and Zakharyaschev, 19971. But incomplete propositional 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 x R - y iff xRy & x # y, see Introduction. 6'countable' means 'of cardinality &'.
<
C H A P T E R 1. BASIC PROPOSITIONAL LOGIC
24
Definition 1.3.14 A n 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 translation, 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). If M = (F, 8) is a Kripke model, then M 1 V := ( F 1 V, 8 V),
where (6 1 V)(q) := Q(q)n V for every q E PL, is called its submodel (the restriction to V). A set V C_ W is called stable (in F) i f for every i, Ri(V) & V. In this case the subframe F 1 V and the submodel M 1 V are called generated. Definition 1.3.17 F' = (V, R i , . . . , R h ) is called a weak subframe of F = (W, R1, .. . ,RN) if Rb 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.
c
c,
,--
We use the signs 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 i n the previous definition. The smallest stable subset W f u 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
L e m m a 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 i n a frame F = (W, R1,. . . , RN) is a sequence (uo,j o , u l , . . . ,jm-l,um) such that uo = U , u, = v, and uiRj,uifl for i = O , . . . , m - 1.
1.3. RELATIONAL SEMANTICS (THE MODAL CASE)
25
For the particular case N = 1 we have ji = 1 for any i , so we can denote a path just by (u0,u1,.. . ,urn). Now Lemma 1.3.19 can be reformulated as follows: L e m m a 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,.. . ,R N ) is the frame F' := (W, R1,. . . , R N ,R,', .. . ,R;'). A nonoriented 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) i f it is stable i n F*, i.e. both Riand R;'-stable for every i = 1 , . . . , N. F itself is called connected if W is connected i n F . A cone in F* (as a subset) is called a (connected) component of F. L e m m a 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 )
H
is an equivalence relation on W. The following is well-known:
L e m m a 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,
M L ( F 1 V).
(2) M L ( F )
The same holds for bounded models, with obvious changes. We also have: L e m m a 1.3.26
(1) M L ( F ) =
n ML(Ffu).
UGF
(2) MT(M)
=
n MT(M
uEM
Proof
TU)
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
26
( 1 ) By Lemma 1.3.25(2), M L ( F ) C M L ( F f u ) for any u E F , and thus
Now let A # M L ( F ) . Then, by definition, there is a Kripke model M over F and a world u such that M , U Y A . By Lemma 1.3.25(1) MTu, u I+ A, and so F f u !+A.
(2) By 1.3.25(1),M T ( M ) 5 M T ( M MT(M)
u ) , so
n
c
MT(M
uEM
,
u).
On the other hand, if A 6M T ( M ) ,then M , u I+ A for some u E M ; hence M f u , u I+ A by 1.3.25 ( I ) , thus A # M T ( M f u ) .
Definition 1.3.27 The disjoint sum ( o r disjoint union) of a family of frames Fj = (W,, R l j , .. . , h j )for , j E J , is the frame U Fj := ( W ,R 1 , . . . ,R N ) , j€J
where W =
u
W j := U(wjx { j ) ) ,
j€J
j€J
( x ,j ) R i ( y , j ' ) iff j = j' & x h j y . Obviously, in this case F,! := of
U Fj isomorphic to Fi. jcJ Hence we obtain
L e m m a 1.3.28
ML
Proof
Let F :=
)
U F! (j€J
)
U Fj 1 (Wi x ( 2 ) )
(i€J
=
is a generated subframe
n ML(F'). jGJ
U Fj. By Lemma 1.3.26, j€J
ML(F)=
n vEF
ML(F,,) =
n0
ML(FT(u,j ) ) .
j€JuEFJ
But the embedding of Fj in F yields an isomorphism FjTu 2 FT(u,j ) , hence
by Lemma 1.3.26, which implies the main statement.
H
1.3. RELATIONAL SEMANTICS (THE MODAL CASE)
Remark 1.3.29 One can show that MA also show that M L
CGJ ) U Fj
2
27
n MA(F').
One can
jGJ
ML(Clj) for a family of modal algebras
(f2j)jEJ . Together with Lemma 1.3.7, this yields an alternative proof of 1.3.28. Definition 1.3.30 Let F = (W, R1, . . . ,R N ) , F' = (W, Ri, . . . , R h ) be two frames. A map f : W + W' is called a morphism from F to F' (notation: f :F F') i f it satisfies the following conditions:
-
+ f (u)R: f ( v ) ) (monotonicity); E W' Vi ( f (u)Riv' + 3v ( u R i v & f (v) = v'))
(1) Vu, v E W tJi (uR,v
(2) Vu
E
W Vv'
A surjective morphism is called a pmorphism (notation: f f is called an isomorphism from F onto F' (notation: f monotonic bijection and f-' is also monotonic.
(lift property).
: F ++ :F
F'). F') i f it i s a
As usual, F and F' are called isomorphic (notation: F 2 F') if there exists an isomorphism from F onto F'. We write F + F' if there exists a pmorphism from F onto F'. Note that the conjunction (1) & (2) is equivalent to 'cone preservation':
-
Definition 1.3.31 Let M = (F, Q), M' = (F', 0') be two Kripke models. A (p-)morphism (respectively, an isomorphism) of frames f : F F' i s said to be a (p-)morphism (respectively, an isomorphism) from M (on)to M' if for every q E PL, u E W M, u I= q iff M', f (u) I= q. As in the case of frames, morphisms of models are denoted by by +, isomorphisms by 2. L e m m a 1.3.32
-
(1) The composition of frame morphisms F
-
-,
p-morphisms
-
F' and F' F" is a F"; similarly for p-morphisms of frames and for frame morphism F (p-)morphisms of Kriplce models.
F is generated ifl the inclusion map is a morphism (2) A subframe F' F' +, F; similarly for Kriplce models. (3) A restriction of a frame morphism to a generated subframe is a frame morphism and moreover, a restriction to a cone is a p-morphism onto a cone; similarly for Kripke models.
(4)
2 i s an equivalence relation between Kriplce frames; the same for Kripke
models.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
28
Proof
(1) Easily follows from cone preservation.
(2) In fact, the cone preservation for j is equivalent to R1(x) = R ( x ) for
x E F'.
(3) Follows from (1) and (2); note that the restriction of f : F ---t G to F' F is the composition f - j , where j : F' --F-isithe inclusion map. The image of a cone is a cone by cone preservation.
(4) Trivial, since a composition of isomorphisms is an isomorphism.
1.3. RELATIONAL SEMANTICS (THE MODAL CASE)
29
L e m m a 1.3.33 (Morphism lemma)
(1) Every morphism of Kripke models is reliable, i.e. if f : M for any u E M , for a n y modal formula A,
-
MI, then
M , u != A iff MI, f ( u )k A.
(2) I f f : M
++
MI, then for a n y modal formula A,
(3) If F ++F1then M L ( F ) C ML(F1). Proposition 1.3.34 Every variety V(r) is closed under generated subframes, p-morphic images, and disjoint sums. Proof
W
Follows from 1.3.25, 1.3.28 and 1.3.33.
R e m a r k 1.3.35 This proposition resembles the 'only if' part of the Birkhoff theorem 1.2.25. The analogy becomes clear if we note the duality between Kripke frames and their modal algebras - generated subframes correspond to homomorphic images of algebras, pmorphic images to subalgebras, and disjoint sums to direct products. However the converse to 1.3.34 is not true. A precise model-theoretic characterisation of Kripke frame varieties (using ultrafilter extensions) was given by van Benthem, cf. [van Benthem, 19831, [Blackburn, de Rijke and Venema, 20011. We shall return to this topic in Volume 2. Exercise 1.3.36 Show the following analogue of 1.2.26: the poset of Kripkecomplete N-modal propositional logics is dually isomorphic to the poset of all N-modal Kripke frame varieties.
I i E I ) be a family consisting of all different components of a propositional Kripke frame F. T h e n F U Fi.
Proposition 1.3.37 Let (Fi
iEI
P r o o f An isomorphism is given by the map f sending x to (x, i ) whenever x E Fi. This map is well-defined, since different components are disjoint by 1.3.24. If xRjy and x E Fi, then y E Fi as well, SO f (x) = ( x ,i)Rji(y, i ) = f ( y ) . The implication f (x)Rji f ( y ) + xRj y is trivial. Proposition 1.3.38 Let (Fi I i E I ) be a family of all different components of
a propositional Kripke frame F . Then (1) for any morphism f : F -t G, every fi := f 1 Fi is a morphism;
-
(2) for a n y family of morphisms fi : Fi ---+ G, the joined m a p f := morphism F
U fi iEI
G.
is a
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
30 Proof
( 1 ) Since every component is a generated subframe, this follows from Lemma 1.3.32(3). (2) Let Rj be a relation in F, Sj the corresponding relation in G. For x E Fi we have Rj ( x ) C Fi, so
f [ R j(%)I
= fi [Rj( x ) ]=
sj( f i( x ) )= Sj ( f ( x ) ) ,
since f f Fi = fi and fi is a morphism.
In this book we are especially interested in K4- and S4-frames; by Lemma 1.3.11, K4-frames are transitive, and S4-frames are quasi-ordered sets. For a K4-frame (W, R) there is the equivalence relation %R:= ( Rn R-l) U Idw in W ; its equivalence classes are called (R-)clusters, and we can consider the quotient set as a frame. Definition 1.3.39 Let F = (W, R) be a K4-frame. Let W" := W/ = R be the
set of all R-clusters, and let u" be the cluster of u . Then the frame F" := (W", R"), where R" := { ( u " , ~ " )I u R v ) , is called the skeleton of F . A singleton cluster { u ) is called trivial (respectively, degenerate) i f u is reflexive (respectively, irreflexive). L e m m a 1.3.40 (1) R" is transitive and antisymmetric; if R is refiexive, then R" is a partial
order. (2) The map u ++ u" is a p-morphism from F onto F" Proof
Straightforward.
L e m m a 1.3.41 I f f : F
-+
G and F is connected,then G is connected.
urn)is a path from u to v in F , then ( f ( u O )j, ~..,. , If (uo,jo, . . . f (urn))is a path from f ( u )to f (v) in G.
Proof
1.3.4
Conical expressiveness
Definition 1.3.42 A n N-modal propositional logic L is called conically expressive i f there exists a propositional N-modal formula C ( p ) with a single
proposition letter p such that for any propositional Kripke model M with M t= L, for any u E M , M , u t = C ( p )@ M f u t = p . Obviously, an extension of a conically expressive logic (in the same language) is conically expressive. A simple example of a conically expressive logic is K4; the corresponding C ( p ) is p A Up.
1.3. RELATIONAL SEMANTICS (THE MODAL CASE)
Exercise 1.3.43
+ Ci2p> 04pis conically expressive. (b) Recall that K4.t = K 2 + 0 1 0 2 > p + O l O l p > p + O l p 3 0 ? p . that K4.t + O 1 0 2 p > O z O l p is conically expressive. (a) Show that K
Show
Now given C from Definition 1.3.42, for a formula A E LPN put
Let us show that 0*behaves like an S4-modality in logics containing L.
Lemma 1.3.44 For any N-modal Kripke model M such that M != A for any u E M ,for any N-modal formula A
where R* i s the same as in Lemma 1.3.19. Proof Similar to soundness of the substitution rule. Let Mo be a Kripke model over the same frame as M ,such that for any u E M
Then by induction we obtain for any propositional formula X ( p ) for any u E M.
Hence for any u E M and for C from 1.3.42
and so by 1.3.42,
M,u!= O * A
Moluk p. But by Lemma 1.3.19and the choice of Mo
Hence the claim follows.
Lemma 1.3.45 If a modal logic A i s conically expressive and O* i s defined as A above, then the rule - i s admissible i n A O*A Proof Since by 1.13.3, A has an exact model, it suffices to show that for any Kripke model M ,for any A, M k A implies M 'F O * A . But this readily follows from 1.3.44.
Lemma 1.3.46 If a n N-modal propositional logic A is conically expressive, then the following formulas are in A (for k < N ) :
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
32
Proof Again we can consider an exact model and show that these formulas are true at every world. This easily follows from 1.3.44, the reflexivity and the transitivity of R*, and the inclusions R* 2 Rk o R*, Rk C: R*. W
1.4 Relational semantics (the intuitionist ic case) Definition 1.4.1 Assume that F = (W, R) is an S4-frame, i.e. MA(F) is a topo-Boolean algebra. Then the algebra of its open elements (or, equivalently, of stable subsets of F ) HA(F) := MA(F)'
is called the Heyting algebra of F . S4-frames are also called intuitionistic propositional. A n intuitionistic valuation i n F is defined as a valuation in HA(F), i.e. as a valuation in W with the following truth-preservation(or monotonicity) property (for every q E P L ) :
The corresponding Kripke model M = (F, 8) is also called intuitionistic. 6' is extended to the map 6" : I F + HA(F) according t o Definition 1.2.72. Definition 1.4.2 For an intuitionistic Kripke model M = (W, R, 6') we define the intuitionistic forcing relation between worlds and intuitionistic formulas as follows: M , u I k A := U E ~ ' ( A ) . Now we readily obtain an alternative inductive definition of intuitionistic forcing, cf. Lemma 1.3.3.
Lemma 1.4.3 Intuitionistic forcing has the following properties: M, u It q
iff u E 8(q) (for q E P L ) ;
M,ulk B A C iff M,ulFBvC
iff
M,ult- B & M , u l k C ; M,uII-BvM,ulkC;
1.4. RELATIONAL SEMANTICS (THE INTUITIONISTIC CASE)
M,uIF B 3 C
33
i f f VV E R(u) ( M , v IF B J M , v It- C);
iff V V E R(u) M,vIYB;
M,uIt7B M 7 u 1 tB - C
iff V V E R(u) (M,vIF B e M,vIF C).
L e m m a 1.4.4 I n intuitionistic Kripke models the truth-preservation holds for any formula A:
(TP) Proof
Vu,v~F(uRv&M,~ltA+M,vlkA).
Trivial, since @(A) is stable.
¤
Definition 1.4.5 A n intuitionistic formula A is said to be valid in an S4-frame F (notation: F IF A) i f H A ( F ) It- A, i.e. if A is true i n every intuitionistic model over F. Similarly, one can reformulate the definitions of satisfiability, etc. (1.3.5, 1.3.6) for the intuitionistic case. We also have a relational analogue of Lemma 1.2.19, for which we need the following Definition 1.4.6 Let F be an S4-frame, M a Kripke model over F . The pattern of M i s the Kripke model Mo over F such that for any u E F, q E P L M o , u k q i f l M , u I= Oq. Obviously, Mo is an intuitionistic Kripke model; Mo = M if M is itself intuitionistic. L e m m a 1.4.7
Under the conditions of Definition 1.4.6, we have
(1) for any u E F and intuitionistic formula A Mo,u It- A iff M , U I= A ~ ,
Proof
Obviousfrom1.2.19.
H
Together with Lemma 1.2.14 and the trivial observation that F Iy I,this implies L e m m a 1.4.8 Let F be an S4-frame. Then the set of all intuitionistic formulas valid i n F is an intermediate logic. This logic is called the intermediate logic of F and denoted by IL(F). For a set of intuitionistic formulas I?, an intuitionistic I?-frame is an intuitionistic propositional frame, in which I? is intuitionistically valid. The class of all these frames is called the intuitionistic (Kripke frame) variety of I? and denoted by V' (I?). Definition 1.3.9 is obviously transferred to the intuitionistic case. Respectively the notation M L , MT changes to IL, IT. Lemmas 1.3.10 and 1.3.13 and Proposition 1.3.15 also have intuitionistic versions; the reader can easily formulate them.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
34
L e m m a 1.4.9 Let F be an S4-frame, M an intuitionistic model over F . (1) If M I is a generated submodel of M , then M1 also is intuitionistic, and for any A E LPo, u E M I
(2) If Fl is a generated subframe of F then
Proof ( 1 )From the generation lemma 1.3.25 and the definition of intuitionistic forcing (1.4.1). ¤ ( 2 ) From Lemma 1.3.25(2) and Lemma 1.4.7(2). L e m m a 1.4.10 Let F' be a subframe of a Kripke S4-frame F . Then every intuitionistic valuation 6' i n F' can be extended to an intuitionistic valuation i n F.
Proof In fact, take a valuation 6 in F such that B(q) = R(O1(q))for any q E PL. rn The three subsequent lemmas readily follow from Lemmas 1.4.7, 1.3.26, 1.3.28 and 1.3.33. L e m m a 1.4.11 For an S4-frame F
L e m m a 1.4.12 For S4-frames Fi, i E I
L e m m a 1.4.13 ( M o r p h i s m l e m m a ) Let M , M' be intuitionistic Kripke models, and let F, F' be S4-frames. (1) I f f : M + M ' , then f is reliable: for any u E M , for any intuitionistic formula A, M , u IF A iff M ' , f ( u )IF A.
(2) I f f
:
M
-
M ' , then M I!- A iff M' I!- A.
(3) If F
+ F',
then I L ( F ) E I L ( F 1 ) .
1.4. RELATIONAL SEMANTICS (THE INTUITIONISTIC CASE)
35
Lemma 1.4.14 Let F be a n S4-frame. T h e n
Proof If cp is an intuitionistic valuation in F, consider the valuation cp" in F" defined by ~ " ( 9 ):= {u" I u E ( ~ ( 9 ) ) . It is clear that p" is well-defined and intuitionistic. By Lemma 1.3.40, it follows that the map sending u to u" is a p-morphism of Kripke models, and thus
(F, p) k A iff (F", p") k A by Lemma 1.4.13. To complete the proof, note that an arbitrary intuitionistic valuation in F" can be presented as cp", with
+
For a class C of S4-frames let C" be the closure of {F" isomorphism, and let Posets be the class of all posets.
1
F E C) under
Lemma 1.4.15
(2) VJ(I?)" = V1(r) n Posets for any set of intuitionistic formulas I?. (3) Every Kripke-complete intermediate logic i s determined by some class of posets: L = IL(V1(L) n Posets). Proof
(1) Follows readily from 1.4.14. (2) In fact, F IF I? e F" IF I? by 1.4.14, so ~ ' ( r ) " C V1(I?). The other way round, if a poset G E V' (I?), then G 2 G" E V1(r)".
(3) Note that L
= IL(v'(L))
for a complete L and apply (I), (2).
So instead of V1(I') we can use the reduced intuitionistic variety
V " ( r ) := ~ ' ( r )n Posets. Now we obtain an analogue of 1.3.34: Proposition 1.4.16 Intuitionistic Kripke frame varieties and reduced intu-
itionistic Kripke frame varieties are closed under generated subframes, p-morphic images, and disjoint sums.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
36
Proof For intuitionistic varieties this follows from 1.4.9, 1.4.12 and 1.4.13. For reduced intuitionistic varieties we can also apply 1.4.15(2) and note that the class of posets is closed under the same three operations. Let us describe reduced intuitionistic varieties for some intuitionistic formulas.
Proposition 1.4.17
V" ( E M ) = V " ( C L ) consists of all trivial frames, i.e. frames of the form (W,I d w ) ; V " ( A J ) = V " ( H J ) = V ( S 4 . 2 ) " consists of all confluent posets; V" ( A Z ) = V" ( L C ) = V ( S 4 . 3 ) " consists of all nonbranching posets; V W ( A P n )consists of all posets of depth 5 n;? V W ( A I W n )consists of all posets of width 5 n;8 V W ( A G n )consists of all posets F , where for any x E F , IHA(FT $)I I n.
-
Definition 1.4.18 A quasi-morphism between S4-frames F = (W,R ) and F' = (W',R') is a monotonic map h : W W' with the quasi-lift property: V x E W Vy' E W' ( h ( x ) R ' y l + 3y ( x R y & h ( y ) =R. y')). For intuitionistic Kripke models M = ( F ,8 ) , M' = ( F ' , 8') a quasi-morphism from M to M' is a quasi-morphism of their frames such that for any q E PL M , x It q iff MI, h ( x ) It q.
A quasi-p-morphism is a surjective quasi-morphism. The following is clear:
L e m m a 1.4.19 A quasi-(p)-morphism of frames h : F ---t F' gives rise to a quasi-(p)-morphism of their skeletons h" : F" -t, F'" such that h N ( u " ) = h(u)" . Quasi pmorphisms are reliable for intuitionistic formulas:
L e m m a 1.4.20 (1) If h is a quasi-morphism from M to MI, then for any x E M , for any intuitionistic formula A ,
M , x IF A iff MI, h ( x ) It A. (2) If there exists a quasi-p-morphism from F onto F' then I L ( F ) C I L ( F 1 ) . Proof
By Lemmas 1.4.20 and 1.4.14.
7See Section 1.15.
sSee Section 1.9.
1.5.
MODAL COUNTERPARTS
1.5
Modal counterparts
The following lemma can be easily proved by induction.
Lemma 1.5.1 Let S = [ C / p ] be an intuitionistic substitution, ST := [CTIp]. Then S 4 t- (SA)T = s T A T for any intuitionistic formula A. This lemma implies
Proposition 1.5.2 For any I-modal logic A 2 S 4 the set T~ := {A E
13%
1
E
A)
is a superintuitionistic logic. Lemma 1.5.3 For any S4-algebra 52, m L ( 5 2 ) = 1 ~ ( 5 2 ' ) Proof
This is a reformulation of 1.2.19(2).
Definition 1.5.4 The above defined logic TA is called the superintuitionistic fragment of A; the logic A is called a modal counterpart of TA. For a set
of intuitionistic formulas let
rT:= {AT I A E I?).
Theorem 1.5-5' Every propositional superintuitionistic logic L = H the smallest modal counterpart: r ( L ) := S 4 + rT.
+ r has
Theorem 1.5.6" Every propositional superintuitionistic logic has the greatest modal counterpart. I n particular, the greatest modal counterpart of H is Grz. The greatest modal counterpart of L is denoted by a ( L ) .
Theorem 1.5.7 (Blok-Esakia) (1) u(H
+ r) = Grz + r T .
(2) The correspondence between superintuitionistic logics and their greatest modal counterparts is an order isomorphism between superintuitionistic logics and modal logics above Grz. See [Chagrov and Zakharyaschev, 19971 for the proof of 1.5.7 (as well as 1.5.5 and 1.5.6).
Proposition 1.5.8 If a modal logic A > S 4 is Kripke-complete, then TA is also Kripke-complete. More precisely, %L(C) = I L ( C ) . g[Dummett and Lemmon, 19591. and Rybakov, 19741, [ ~ s a k i a 19791. ,
lo[Maksimova
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
38
Proof
In fact, %IL(c) =
equality follows from 1.4.7.
n WL(F)= n IL(F) = IL(C). The second FEC
F EC
W
Lemma 1.5.9 For any intermediate logic L ,
Proof
An exercise.
W
Theorem 1.5.10 (Zakharyaschev) The m a p I- preserves Kripke-completeness so if a n intermediate logic L i s Kripke-complete, then
For the proof see [Chagrov and Zakharyaschev, 19971 Remark 1.5.11 Unlike r , the map a does not preserve Kripke-completeness; a counterexample can be found in [Shehtman, 19801.
1.6
General Kripke frames
'General Kripke frame semantics' from [Thomason, 19721'' (cf. also [Chagrov and Zakharyaschev, 19971) is an extended version of Kripke semantics, which is equivalent to algebraic semantics. Definition 1.6.1 A general modal Kripke frame is a modal Kripke frame together with a subalgebra of its modal algebra, i.e. @ = (F,w ) , where W MA(F) is a modal subalgebra. W i s called the modal algebra of @ and also denoted by MA(@);its elements are called interior sets of @. So for F = (W, R1, . . .,RN), W should be a non-empty set of subsets closed under Boolean operations and Eli : V H OiV (Section 1.3). Instead of Oione can use its dual OiV := RL'V. Definition 1.6.2 A valuation in a general Kripke frame @ = (F,W) i s a valuation in MA(@).A Kripke model over @ is just a Kripke model M = (F,e) over F, i n which 8 is a valuation i n @. A modal formula A i s valid i n @ (notation: @ b A) if it i s true in every Kripke model over @; similarly for a set of formulas. Analogous definitions are given in the intuitionistic case. Definition 1.6.3 A general intuitionistic Kripke frame is @ = (F,W), where F i s a n intuitionistic Kriplce frame, W & HA(F) is a Heyting subalgebra. W is called the Heyting algebra of @ and denoted by H A ( @ ) ;its elements are called interior sets. ''In that paper it was called 'first-order semantics'.
1.6. GENERAL KRIPKE FRAMES
39
Definition 1.6.4 A n intuitionistic valuation in a general intuitionistic Kripke frame @ = (F, W ) is a valuation in HA(@). A n intuitionistic Kripke model over @ is of the form M = (F,O), where 0 is an intuitionistic valuation i n @. A n intuitionistic formula A is valid i n @ (notation: @ It A) if it is true in every intuitionistic Kripke model over @; similarly for a set of formulas. Obviously we have analogues of 1.3.7, 1.3.8, 1.4.7 and 1.4.8:
L e m m a 1.6.5 For any modal formula A and a general Kripke frame @ @ k A iff MA(@)k A;
analogously, for intuitionistic A and @ @ IF A iff HA(@) k A;
L e m m a 1.6.6 (1) For a general modal Kripke frame @ the set
i s a modal logic; if @ is intuitionistic, then IL(@):= {A I @ IF A) is a n intermediate logic, and moreover, IL(@)= ?h.IL(@). (2) For a class C of N-modal general frames the set
is a n N-modal logic; i f the frames are intuitionistic, then
is an intermediate logic, and IL(C) = %fL(C). L e m m a 1.6.7 (1) For a n N-modal Kripke model M = (FIB) the set of all definable sets
WM := {O(A) I A E CPn) is a subalgebra of MA(F).
(2) Similarly, for an intuitionistic Kripke model M = ( F ,O), A E CPo) is a subalgebra of H A ( F ) .
Proof
(Modal case) In fact, by definition
W h = {@(A) I
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
40
So we define the corresponding general frames Definition 1.6.8 For a Kripke model M = (F,O), the general frame G F ( M ) := (F, WM) (or GF'(M) := (F, w&) i n the intuitionistic case) i s called associated. L e m m a 1.6.9 GF(M) b A iff all modal substitution instances of A are true i n M; similarly for the intuitionistic case. Proof (If.) For M = (F, e), let 77 be a valuation in F such that q(pi) = O(Bi) for every i . An easy inductive argument shows that
for any n-formula A, cf. Lemma 1.2.9. (Only if.) The same equality shows that for any modal (or intuitionistic) W substitution S, O(SA) = v(A) for an appropriate valuation 7 in GF(M). Definition 1.6.10 If = (F,W) i s a general Kripke frame, F = (W, R1, . . . , RN), V W, then we define the corresponding general subframe:
where w~v:={XflVIX€W).
If V is stable, r V i s called a generated (general) subframe. The cone generated by u i n a is the subframe @Tu:= a (WTu). The definition of a subframe is obviously sound, because W 1 V is a modal algebra of subsets of V with modal operations OivX := OiX n V. The following is a trivial consequence of 1.3.25 and 1.3.26. L e m m a 1.6.11 For a general frame a , (1) i f V i s a stable subset, then ML(Q) E ML(@ V);
Exercise 1.6.12 Define morphisms of general frames and prove their properties.
1.7
Canonical Kripke models
Definition 1.7.1 The canonical Kripke frame for a n N-modal propositional logic A is FA:= (WA, R l , ~ ,... , RN,n), where WA i s the set of all A-complete theories, x R i , ~ yiff for any formula A, and OiA E x implies A E y. The canonical model for A i s MA = (FA,O h ) , where
1.7. CANONICAL K R I P K E MODELS
41
Analogously, the canonical frame for a bounded modal logic Arm is FArm := (WAF,,R1,Arm,. .. , RN,Arrn), where WAr, is the set of all maximal A-consistent sets of N-modal m-formulas,
i f ffor any m-formula A, U i A E x implies A E x. The canonical model for A lm is M A r m := (FA[,, OArm), where OArm(pi)= OA(pi) for i m.
<
Definition 1.7.2 The canonical frame for an intermediate propositional logic C is F e := (We, Re), where We is the set of all C-complete intuitionistic (double) theories, x R c y i f fx C y. The canonical model for C is M e := ( F c , Ox), where
The corresponding definitions for bounded intermediate logics must be now clear, so we skip them. The following is well-known, cf. [Chagrov and Zakharyaschev, 19971, [Blackburn, de Rijke and Venema, 20011.
Theorem 1.7.3 (Canonical model theorem) For any N-modal or intermediate logic A and m-formula A (of the corresponding kind):
(2) MA[TTL,Y b A i f fA E Y ;
(3) M A r m k A i f f M A k A i f f A ~ A .
Definition 1.7.4 The general canonical frame of an N-modal (respectively, int u i t i o n i s t ~ ~logic ) A is cPA := G F ( M A ) (respectively, @; := G F 1 ( M A ) ) . Theorem 1.7.5 (General canonical model theorem) For a modal (respectively, intermediate) propositional logic A M L ( @ * ) = A (respectively, I L ( Q A ) = A ) . Proof By Lemma 1.6.9, ML(cPh) consists of all formulas A such that M A k S A of any modal substitution S. By 1.7.3, the latter is equivalent to S A E A . Since A is substitution closed, it follows that ML(cPA) = A. The same argument works for the intuitionistic case. An alternative proof of 1.7.5 can be obtained from the algebraic completeness theorem 1.2.21 and the following observation:
Proposition 1.7.6 M A ( @ A ) L i n d ( A ) for a modal logic A ; H A ( @ = ) L i n d ( C ) for an intermediate logic E.
C H A P T E R 1. BASIC PROPOSITIONAL LOGIC
42
Proof (Modal case.) T h e map y : L i n d ( A ) ---+ M A ( @ A )sending [ A ]t o OA(A) is well-defined, since M A k A F B whenever A -A B by Theorem 1.7.1. y is obviously surjective and preserves the modal algebra operations. In fact,
according t o Definitions 1.2.20 and 1.3.2, and similarly for the other operations. Finally, note that y ( [ A ] )= y ( [ B ] )i f fOA(A) = 8 a ( B ) iff M A k A
= B i f f ( A = B ) E A i f f [A]= [B]
by 1.7.1 and the definitions. So y is an isomorphism. Definition 1.7.7 A general frame @ = ((W,R1,. . . , R N ) W , ) (modal or intuitionistic) is called descriptive if it satisfies the following conditions:
is differentiated (distinguishable): for any two different points x , y there exists an interior set U E W separating them, i.e. such that x E U ++ y E u; tightness: V x ,y, i (VU E W ( x E OiU + y E U ) + x R i y ) (in the modal case), V x ,y (VU E W ( x E U + y E U ) + x R l y ) (in the intuitionistic case); compactness: every centered subset X C W (i.e. such that n X l # 0 for any finite XI C X ) has a non-empty intersection (in the modal case); i f a pair ( X , Y ) of subsets of W is centered (i.e. Xl Uyl for any finite XI C X , Y I c Y ) , then X $Z IJY (in the intuitionistic case).
n
n
A differentiated and tight general frame is called refined. Lemma 1.7.8 A generated subframe of a refined frame is refined. Proof Distinguishability is obviously preserved for subframes. Let us check tightness in the modal case. W e use the same notation as in 1.7.7. Suppose V C W is stable, x , y E V and
This is equivalent t o
and thus (since V is stable and x , y E V ) t o (2)
VU E W ( x E OiU =+ y E U ) .
I f @ is tight, (2) implies xRiy, therefore ( 1 ) also implies x R i y , which means tightness o f @ r V .
1.7. CANONICAL KRIPKE MODELS
43
Remark 1.7.9 Compactness is not always preserved for generated subframes; the reader can try to construct a counterexample.
Descriptive frames can also be characterised as canonical frames of modal algebras. These frames resemble canonical frames of modal logics. Recall that a (proper) filter in a Boolean algebra is a <-stable (proper) subset closed under meets. A maximal filter is maximal among proper filters. Definition 1.7.10 For an N-modal algebra fl, consider the set R+ of its max-
imal filters with the accessibility relations:
and with the interior sets of the form
for all a E f l . The resulting general frame canonical or the dual frame of f l .
a+:= ( a + , R1,... ,RN) is called the
Recall that a (proper) prime filter in a Heyting algebra is a <-stable (proper) subset X closed under A and such that
Definition 1.7.11 For a Heyting algebra fl, consider the set R+ of its prime filters with the accessibility relation xRy iff x y and with the interior sets h ( a ) = {x E R+ I a E x) for a E a. The general frame a+:= (R+,R) is called the canonical or the dual frame of a . Theorem 1.7.12 (Tarski-Jonsson) (1) If 0 is a modal algebra, then MA(O+) S a .
(2) If f l is a Heyting algebra, then HA(fl+) r f l .
A required isomorphism is the Stone map h from Definitions 1.7.10 and 1.7.11. In particular, for the Lindenbaum algebra we again obtain the frame isomorphic to cPA (with an obvious notion of isomorphism): Theorem 1.7.13 For a modal or intermediate logic A
This is because the maximal (prime) filters of Lind(A) exactly correspond to A-complete theories. Theorem 1.7.14 A general Kripke frame is descriptive iff it is isomorphic to some frame f l + .
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
44
This result was proved in [Goldblatt, 19761 and earlier in [Esakia, 19741 for S4-frames. For a more available proof see [Chagrov and Zakharyaschev, 19971. The subsequent lemma follows from the above, but also has an independent proof based on the properties of MA. L e m m a 1.7.15 Every general canonical frame cPA i s descriptive. From the definitions it is evident that for any general Kripke frame ( F ,W), M L ( F ) ML(F, W) (or IL(F) IL(F, W) in the intuitionistic case). Moreover, in many cases these two logics coincide.
c
Definition 1.7.16 A set of N-modal formulas r is called d-persistent (respectively, r-persistent) if for a n y descriptive (respectively, refined) general frame (F, W), (F, W) I= r + F I= r; similarly, for intuitionistic formulas. The following observations are trivial consequences of soundness: L e m m a 1.7.17 (1) A modal or intermediate propositional logic i s d-persistent (respectively, r-
persistent) iff it i s axiomatisable by a d-persistent (respectively, r-persistent) set. (2) A s u m of d-persistent (respectively, r-persistent) logics i s d-persistent (respectively, r-persistent). Proposition 1.7.18 Every d-persistent modal or intermediate logic is canonical, and therefore Kripke complete. Proof (Modal case) an C= A by 1.7.5, and cPA = (FA,WA) is descriptive by 1.7.15. So for a d-persistent A it follows that FA k A. Thus for propositional modal logics d-persistence implies canonicity and canonicity implies Kripke completeness. The converse implications are not true - however, all three properties are equivalent for elementary logics, by the Fine-van Benthem theorem, see 1.8.3 below.
1.8
First-order translations and first-order definability
Let elNbe the classical first-order language with equality and binary predicate letters R1,. . . , RN. So classical tlN-structures are nothing but N-modal frames. The classical truth of an tlN-formula cp in a frame F is denoted by F k cp as usual. Definition 1.8.1 A n t l ~ - f o r m u l cp a corresponds t o a n N-modal propositional formula A if for a n y N-modal Kripke frame F ,
1.8. FIRST-ORDER TRANSLATIONS A N D DEFINABILITY
45
F I=A (modally) H F k cp (classically). Similarly, an L 1 -formula cp corresponds t o an intuitionistic propositional formula A i f for any intuitionistic Kripke frame F, F I t A F k cp. Definition 1.8.2 The class Mod(C) of all classical models of a first-order theory (i.e. a set of first-order sentences) C is called A-elementary; if C is recursive (respectively, finite), this class is called R-elementary (respectively, elementary). A n N-modal or intermediate logic A is called A-elementary (respectively, R-elementary, elementary) if the class V ( A ) (or V 1 ( A ) ) is of the corresponding kind i n the language LIN (or L l l ) . A modal or intermediate logic determined by a A-elementary (respectively, R-elementary, elementary) class of frames is called quasi-A-elementary (respectively, quasi-R-elementary, quasi-elementary). l 2 Theorem 1.8.3 (Fine-van Benthem) Every quasi-A-elementary modal or intermediate logic (in particular, every Kripke complete A-elementary logic) is d-persistent.
For the proof see [van Benthem, 19831, [Chagrovand Zakharyaschev, 19971. Let us recall the well-known 'standard' translation from modal formulas into first-order formulas, cf. [van Benthem, 1983].13 Let t l ; be t h e language obtained by adding countably many monadic predicate letters P I , P2,. . . t o L I N . Every N-modal formula A is translated into an ~ 1 ; - f o r m u l aA*(t) with at most one parameter t , according t o the rules: "P Pi(t),
1*(t) =I, ( A 3 B)*(t) = Ak(t) 3 B*(t), (OkA)* ( t )= V x ( R k(t,x ) > A* ( x ) ) ,where A* ( x ) is obtained from A*(t) b y substituting x for t (together with renaming o f bound variables, i f necessary).14 Every Kripke model M = ( F ,cp) over a frame F = (W,el,. . . , Q,) clearly cor), . . .). responds t o a classical ~1;-structure M* = (W,e l , . . . ,e,, c p ( p ~ cp(pa), Lemma 1.8.4 (1) Let M = ( F ,cp) be a Kripke model over a frame
F = (W,e l , . .., e
~ )Then . for
any a E W, for any N-modal formula A ,
M , a b A iff M* F ~ * ( a ) . (2) For an9 N-modal frame F , for any N-modal formula A
F k A ifl
vcp ( F ,cp)*
k MA* ( t ) .
12According t o another terminology, a quasi-A-elementary logic is called A - e l e m e n t a r i l y
determined, etc. similar translation for intuitionistic formulas was first introduced in [Mints, 19671. 14Variable substitutions for first-order formulas are considered in detail in Chapter 2. 13A
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
46
Proof (1) Easy by induction. E.g. in the case A = O k B we have: M,akA
@
H H
F k A
(2)
H H
b'b E &(a) M , u l k B b'b E ek(a) M* != B*(b) (by the induction hypothesis)
.
M*~=VX(R~(~,X)~B*(X))UM*~(O~B)*(~). VpVa(F,p),akA~Vcpb'a(~,p)*b~*(a)(by(l)) V p (F,p)* k MA* (t).
In the next proposition we use the following notation. For an L~N-theory C, [C[ denotes the set of its LlE-theorems (in classical first-order logic). Proposition 1.8.5 (1) For any L ~ -theory N C, ML(Mod(C))
sI
[C]. In particular, every quasiR-elementary modal o r intermediate logic is recursively axiomatisable.
(2) Every quasi-A-elementary modal or intermediate logic has the c.f.p.
(3) Every finitely axiomatisable (quasi-)A-elementary modal or intermediate logic is (quasi-)elementary. Proof (1) Since all ~1:-models of C are exactly the structures of the form (F,p)*, where F I= C, by Lemma 1.8.4 and Godel's completeness theorem, we obtain
(#>
A E ML(Mod(C)) H H
C b VtA*(t) (in the classical sense) C t- b'tA*(t) (in classical first-order logic).
Since A can be uniquely restored from A*, this proves the 1-reducibility. Next, if C is recursive, then ML(Mod(C)) is RE as it is 1-reducible to the set of theorems of a recursive classical theory. Now (1) follows by the well-known Craig's lemma, cf. [Boolos, Burgess, and Jeffrey, 20021. To prove (1) in the intuitionistic case, note that by 1.5.8, IL(C) = T ~ ~ ( C ) l so IL(C) is 1-reducible to ML(C). (2) Let Co be the class of all countable LlE-models of C. Similarly to the above, we obtain that A E ML(C0) iff for any countable LIE-structure M, (M b C + M l= VtA*(t)). By the Lowenheim-Skolem theorem, the latter is equivalent to C t- b'tA*(t), and thus (by (#)) to A E ML(Mod(C)). Therefore ML(Mod(C)) = ML(Co), which proves (2). The intuitionistic version follows readily. (3) (Modal case) If ML(Mod(C)) = K N A for a formula A, then by (#), C t VtA*(t) in classical logic, which implies El t- WA*(t) for some finite C1 C. Hence p k WAf(t), where p = AX1, so A E ML(Mod(cp)) (by (#)), and thus ML(Mod(C)) = K N + A C ML(Mod(cp)).
+
c
On the other hand, Mod@) Mod(cp), hence ML(Mod(cp)) C ML(Mod(C)). Therefore ML(Mod(C)) = ML(Mod(9)) is quasi-elementary.
1.9. SOME GENERAL COMPLETENESS THEOREMS
47
+
The case when KN A is A-elementary is left to the reader. (Intuitionistic case.) Assume that L is finitely axiomatisable and A-elementary. Then by definition, r ( L ) is also finitely axiomatisable. Since by 1.5.9, v l ( L ) = V(r(L)), it follows that r ( L ) is A-elementary. Then (as we have just proved) T(L) is elementary. Now the equality V1(L) = V(r(L)) implies the elementarity of L. It remains to consider the case when an intermediate L is finitely axiomatisable and quasi-A-elementary. Then we can argue as in the modal case. In fact, if IL(C) = H A for C = Mod@), then AT E ML(C), so by (#), C t- vt(AT)*(t). Then we can replace C with a single cp and obtain that IL(C) = IL(Mod(cp)).
+
w
1.9
Some general completeness theorems
In this section we recall two general results on Kripke-completeness - the Sahlqvist theorem and the Fine theorem. Definition 1.9.1 A Sahlqvist formula is a modal formula of the form O,(A > B ) , where cu E I F , B is a positive modal formula (i.e. built from proposition letters using I , T , V, A, Oi, mi), and A is built from proposition letters and
their negations using the same connectives, so that subformulas of A of the form C V D or OiC containing negated proposition letters are not within the scope of any O j . Theorem 1.9.2 (Sahlqvist theorem) Let A be a modal logic axiomatised by
Sahlqvist formulas. Then A is d-persistent and A-elementary. For the proof see [Chagrov and Zakharyaschev, 19971, [Blackburn, de Rijke and Venema, 20011. Definition 1.9.3 A subset V in a transitive frame F = (W, R) is called a n
antichain if its different worlds are incomparable: Vx, y E V(xRy
+ x = y).
The width of F is the maximal cardinality of finite antichains i n cones of F i f it exists and m otherwise. So for a finite n, F is of width
< n iff
This property corresponds to the modal formula AW, (cf. 1.3.11) and to AIW, in the intuitionistic case (cf. 1.4.17).
5 n is an extension of K4 + AW,; an intermediate logic of width < n is an extension of H + AIW,. All these are called the logics of finite width. Definition 1.9.4 For a finite n, a modal logic of width
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
48
The width of a transitive modal logic A can be defined explicitly as the least n such that Awn E A (and co if Awn 6A for any n);similarly for the intuitionistic case. T h e o r e m 1.9.5 (Fine theorem) Every logic of finite width (modal or intermediate) is Kripke-complete and moreover, has the c.f.p. This result (for the modal case) was first proved in [Fine, 19741; for a shorter proof see [Chagrov and Zakharyaschev, 19971. The intuitionistic case follows from the observation that L I- AIW, implies T ( L )t- Awn together with 1.5.8 and 1.5.5.
1.10
Trees and unravelling
Recall that a path in a frame is a sequence of related worlds together with indices of the accessibility relations (Definition 1.3.20). Definition 1.10.1 The depth of a world u i n F (notation: d ~ ( u or ) d(u) if there is no confusion) is the maximal length of a path from u i n F i f it exists and co otherwise. Let us introduce two other kinds of 'paths' and 'depths' in Kripke frames. Definition 1.10.2 A distinct path in a frame F is a path, i n which all the ) dd(u)) worlds are different. The distinct depth of u in F (notation: d d ~ ( u or is the maximal length of a distinct path from u i n F i f it exists and co otherwise. Definition 1.10.3 A strict path in a transitive frame F = (W,R) is a path (uO, ~ 1 ,. .. ,urn),in which for any i < m (or equivalently, all the worlds are i n different clusters). Respectively, the strict depth of u i n F (notation: s d F ( u ) or s d ( u ) ) is the maximal length of a strict path from u i n F i f it exists and co otherwise. Obviously, sd(u) = s d ( u ) = s d ( v ) if u
GR
v , and s d ~ ( u=) s d ~ (u"). -
Exercise 1.10.4 Show that d ( x ) 5 n iff x k O n l , where
OA := 01A A .. . A
DNA. Definition 1.10.5 A world u in a frame is called a dead end (respectively, maximal; quasi-maximal) i f d ( u ) = 0 (respectively, dd(u) = 0 ; s d ( u ) = 0). Dead ends i n a tree are also called leaves (as well as maximal worlds i n reflexive or transitive trees). A maximal cluster i n a K4-frame F is a maximal point i n F". So quasi-maximal points in a K4frame are exactly the points of the maximal clusters.
1.10. TREES AND UNRAVELLING
49
Definition 1.10.6 A frame F = (W,R1,. . . ,R N )with a root uo is called a tree (or an N-tree) i f
a
for any x
# uo there exists a unique pair (i,y ) such that yRix.
Hence we readily have
Lemma 1.10.7 A frame F is a tree with a root u a unique path from u to x i n F .
zfl for
any x E F there exists
Definition 1.10.8 The height of a world x i n a tree F (notation: h t F ( x ) , or h t ( x ) ) is the length of the (unique) path from the root to x . Definition 1.10.9 A successor of a world u i n a poset15 F = (W,R) is a minimal element i n the set R-(u) of all strictly accessible worlds. ,OF(u)(or P ( u ) ) denotes the set of all successors of u i n F . F is called successive i f for any u , R- ( u ) = R ( P F ( u ) ) .The branching at u in F is I,BF(u)1. Every finite p.0. set is clearly successive. For rooted successive p.0. sets we can also define the height of x as the minimal length of a 'successor-path' from the root to x .
Definition 1.10.10 Let Tw = w m be the set of all finite sequences of natural numbers. The 1-modal universal tree is the frame
where u ~ v : = 3 k v~=wu k . The N-modal universal tree is
where u C ~ vN := 3k E w ( v = u k & k
-
i - 1 (mod N)).
Sometimes we drop N i n FNTw and C ~ N if it is clear from the context. The intuitionistic universal tree is the p.0. set16 IT, := (T,, Q ) , where u Q v := 3w ( v = u w ) . So Q is the reflexive and transitive closure of C, and the set of immediate successors of u in IT, is just P(u) = c ( u ) . From the definitions it follows easily that FNT, is really a tree with the root A. ISMore generally, we may assume that F is transitive antisymmetric. I6This is a p.0. tree according t o Definition 1.11.10.
50
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
Definition 1.10.11 A subframe F C FNT, or F C IT,, that is stable under 3 (or equivalently, under all 7i,1 5 i 5 N ) , is called a standard tree (or a standard subtree of the corresponding frame).
Let us fix the following standard trees: the universal n-ary trees, or the universal trees of branching n, for finite n > 0:
FNT, := FNT, 1 T N n , ITn := ITw T,, where Tn is the set of all (0, 1, . . .,n - 1)-sequences;
universal trees of depth (or height) k :
where
~ , := k {U
E T,
I l(u)5 k ) ,
l ( u ) is the length of sequence u, n 5 w. In particular, T: contains only the root A. Obviously, l ( u ) = ht(u) in FNTw.
Figure 1.1. F2T: Sometimes we also use the notation T," rather than T,, for 0
< n 5 w.
Definition 1.10.12 Let F be a successive poset. A svbfrarne G F is called greedy if for any u E G either PF(u) G or PF(u)n G = 0 . A greedy standard tree is a greedy standard subtree of FNT, or IT,. Definition 1.10.13 Let F = (W,R1,.. . ,R N ) be a frame with a root uo. The unravelling of F is the frame F# = ( W # ,R?,. . . ,R$), where W # is the set of
all paths in F starting at uo and
a R j # iff ~ 3v fi = ( a ,j, v ) .
1.10. TREES AND UNRAVELLING
51
Unravelling was first introduced probably in [Sahlqvist, 19751; later it was used by many authors, cf. [Gabbay, 19761, [van Benthem, 19831, [Gabbay and Shehtman, 19981. The proof of the following lemma is straightforward. Lemma 1.10.14 (1) F# is a tree.
(2) The map .rr : W# from F# onto F.
-
W such that .rr(uo,. . . ,urn) = urn, is a p-morphism
Now let us show how to increase branching in a standard way. Definition 1.10.1517 Let F = (W, R1,. . . ,R N ) be a frame with root uo. Put Wl : =
iw
i f there exists a path of positive length (a 'loop') from u0 to uo, W - { u o ) otherwise
w'
Let
:= (Wl
x w)U { ( u 0 ,-1)).
R:,. . . , R& be the relations on W' such that
The frame F'
:= (W', R:,
. . . , RL)is called the (w-)thickening of F
Informally speaking, we make w copies of every world of Wl and connect all the copies of two worlds if the original worlds are connected in F ; we also add the root ( u o ,-1). From the definition we readily obtain Lemma 1.10.16 (1) For any x E W',
R: (x)is either empty or denumerable.
(2) the map y : ( a ,n) H a is a p-morphism from F' onto F . Lemma 1.10.17 Let F = (W, R1,. . . ,RN)be a tree, i n which every set Ri(x) is either empty or denumerable. Then F is isomorphic to a greedy standard tree.
Proof Let Fk be the restriction of F to the worlds of height 5 k. We define the embeddings f k : Fk++ F N T ~by induction. If uo is the root of F , we put
l7
[Gabbay and Shehtman, 19981.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
52
If f k is already defined, we extend it to fk+' as follows. If htr(y) = k and Ri(y) # 0 , then &(y) is denumerable, by our assumption. So let ey,i : w --t Ri(y) be the corresponding bijective enumeration. Then put
+
Since F is a tree, every world of height (k 1) is e,,i(j) for a unique triple (y,i, j ) , so f k + ' is well-defined. Since ey,i is a bijection, it is clear that f k + l is an embedding. then by Its image rng(fk+l) is greedy. In fact, if u = f k ( y ) and Ri(y) # 0, our construction, f k + ' gives rise to a bijection between &(Y) and
Eventually, the required isomorphism is f :=
U fk. k>O
Proposition 1.10.18 (1) Every countable rooted frame is a p-morphic image of a greedy standard tree.
(2) Every serial countable rooted N-frame is a p-morphic image of FNT,. Proof (1) Let F be the original frame. By 1.10.14 and 1.10.16 we have pmorphisms
By 1.10.17, (FV)flis a tree. Since by 1.10.16, every R I ( x ) is either empty or denumerable, the same property holds for (F')fl, and thus it is isomorphic to a greedy standard tree, by 1.10.17. (2) If F is serial and f : G -N F , then G is also serial, due to the lift property. By definition, every serial greedy standard subtree of FNT, is FNT, itself.
1.11 PTC-logics and Horn closures Definition 1.11.1 A modal propositional formula is called closed (or constant) if it is a 0-formula, i.e. if it does not contain proposition letters.
Lemma 1.11.2 If F,F1are N-modal frames and f : F + F1, then F' l= A for any closed N-modal formula A.
FkA u
1.I 1. PTCLOGICS AND HORN CLOSURES
53
Proof Let M, M' be arbitrary Kripke models respectively over F, F'. By induction it easily follows that M, x k A iff MI, f (x) != A for any x E M and closed A. Hence F, x != A iff F', f (x) k A. This implies F k A +=+ F' k A since f is surjective. W Proposition 1.11.3 Every logic axiomatisable by closed formulas is canonical. P r o o f By Theorem 1.7.3, every axiom of the logic is true in the canonical model. For a closed formula, this is equivalent to validity in the canonical frame.
w Definition 1.11.4 A pseudotransitive N-modal formula has the form O,Okp 3 Opp, where p E PL, a , p E I F . This formula is called one-way if a = A. A PTC-formula is a formula which is either pseudotransitive or closed. A PTClogic is a modal logic axiomatised by a set of PTC-formulas. One-way P T C -
formulas and logics are defined similarly. For example, the following formulas are pseudotransitive:
Thus many well-known logics are PTC, e.g. D , K 4 , D 4 , S4, T, B, K.t, K4.t, S5. Since every pseudotransitive formula is obviously Sahlqvist, from the Sahlqvist theorem and Proposition 1.11.3 we obtain Proposition 1.11.5 (1) A pseudotransitive N-modal formula A = O,Okp 3 Opp corresponds to
the L l N - f o r m ~ l a ~ ~ AO
:= Vx, v(3u(uRax A uRpv)
> xRkv),
(2) Every PTC-logic is canonical and A-elementary. Later to the well-known transitive closure, there exists a 'closure' under pseudotransitivity. This is a rather simple fact from classical logic, but still let us recall its proof briefly. Further on arbitrary lists of individual variables are denoted by x, y. etc. and arbitrary lists of individuals by a, b. etc. As usual, we write cp(x, y,z) to indicate that all parameters of the formula cp are among x, y , z. Definition 1.11.6 A n L I N -sentence of the form
is a called a universal strict Horn clause, if cp(x, y, z) is a (non-empty) conjunction of atomic formulas. 18Strictly speaking, this becomes an LIN-formula after writing uR,x and uRpv in terms of the basic predicates R1, . . . , RN.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
54
Proposition 1.11.7 Let F = (W, e l , . . . ,e N ) be an N-modal frame, I? = {$k k E I) a set of universal (strict) Horn clauses
: Then there exists a frame F
(2) F; i= r; (3) If G k I? and f : F monotonic.
I
such thatlg
-
G is monotonic, then f
:
F :
-
G is also
We can say that F; is the 'smallest' full weak extension of F satisfying I?.
Proof We construct F : as the union of a sequence of frames Fm= (W, @,I,. . . , e , ~ ) , beginning with Fo= F. Namely, let:
ej+= Uemj,
+
~r+=(~,e:,..-,e~).
m
Then F o have:
2 . . .Fm5 F m + l . . . $ F,:
and thus (1) holds. Furthermore, we
(4) if cp(z) is a conjunction of atomic formulas, then the following conditions are equivalent: (i) (ii) (iii)
F : I= cp(a); 3mVk 2 m Fk I= cp(a); 3m FmI= cp(a).
In fact, (ii)+(iii) is trivial. (iii)+(i) holds, since Fm2 F;, and the truth of positive formulas is preserved by monotonic maps. (i)+(ii) easily follows from and P: = UPmj. Pmj P(m+~)j m
~r+
Now let us prove (2). Assume != cpk(a,b,ck), ik = j. Then by (4), Fmb cpk(a, b,ck) for some m, and thus (a, b) E Q ( ~ + by ~ ) definition. ~ Hence Frf I= R j (a, b). Therefore F : t= qjk. To check (3), we assume that G t= I?, f : F ---+ G is monotonic and show by the induction that f : Fm G is monotonic for any m. To make the step, assume ~ e ( , + l ) ~ bThen . there are two cases. If aemjb, then G t= Rj(f (a),f (b)) by induction hypothesis. If Fmt= 3zk(~k(a, b, zk), ik = j, then G i= 3zkpk(f(a), f (b), zk), since monotonic f : Fm G preserves the truth of positive formulas. Now G != qjk implies G I= R j (f (a), f (b)).
-
IgRecall that
2 denotes a full weak subframe, cf. 1.3.17.
1.1 1. PTC-LOGICS AND HORN CLOSURES
55
By Proposition 1.11.5 every pseudotransitive modal formula corresponds to a universal strict Horn formula. So Proposition 1.11.7 yields
Corollary 1.11.8 Let be a set of pseudotransitive N-modal formulas, F an N-modal frame. Then there exists an N-modal frame F ' such that
-
(3) for any N-modal frame G , i f G i= I' and f : F f : F: G is monotonic.
-
G is monotonic, then
Definition 1.11.9 The frame F: described i n Corollary 1.11.8 is called the pseudotransitive closure of F (under I?), or the I?-closure of F . If A = K N r + A is a PTC-logic with the set of pseudotransitive axioms I' and a set of closed axioms A, then F: is also called the A-closure i f it validates A.
+
Definition 1.11.10 If A is a PTC-logic, then the A-closure of a tree is called a A-tree. K4-trees (respectively, S4-trees) are also called transitive trees (respectively, p . ~ .trees). In the particular case when A = D N I?, with a set of pseudotransitive axioms I?, the r-closure of FNT, is called the full standard A-tree.
+
Proposition 1.11.11 (1) Let A be a PTC-logic. Then every countable rooted A-frame is a p-morphic image of a greedy standard A-tree.
+
(2) If A = D N I?, with a set of pseudotransitive axioms r, then every serial countable rooted A-frame is a p-morphic image of the full standard A-tree. (3) Every countable rooted S4-frame is a p-morphic image of IT,.
--
Proof ( 1 ) Let F be such a frame. By Proposition 1.10.18, there exists f : @ F for F is some greedy standard tree a. By Proposition 1.11.7, the map f : monotonic, where I? is the set of pseudotransitive axioms of A. The lift property F. So F is a p-morphic obviously holds for this map, since it holds for f : @ image of @,f. Finally note that all closed axioms of A are valid in a:, by Lemma 1.11.2. (2) By Proposition 1.10.18(2), in this case we can take @ = FNT,. (3) In fact, IT, is the full standard S4-tree.
-
Corollary 1.11.12 (1) Every PTC-logic A is determined by the class of greedy standard A-trees. (2) Every serial PTC-logic A is determined by the full standard A-tree.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
56
Proof ( 1 ) A is complete and A-elementary, so it has the c.f.p by Proposition 1.8.5. Then by Proposition 1.11.11(1) and the morphism lemma, every A $ A is refuted in a greedy standard A-tree. ( 2 ) By the same argument using 1.11.11 (2). ( 3 ) By ( 2 ) , since IT, is the full standard SPtree. The claim about H follows by 1.5.7, since H = T ~ 4 .
-
Proposition 1.11.13 IT, i s a p-morphic image of IT2. Proof Let T : w w be a map such that every set ~ - l ( nis) infinite (such a map obviously exists, since there is a bijection between w x w and w ) . Next, every a E T2 can be uniquely presented in the form O n l l . . .Onk1Onk++',where n1, . . . ,n k + l 2 0 , and On means 0 . .0. Now put
* n
f ( a ):= ~ ( n l ) ~ ( n. .27(nk). ). We claim that f is a required pmorphism from IT, onto IT2. In fact IT, is a Horn closure of F,T,, so by 1.11.7, it suffices to check monotonicity of f w.r.t. C. But this is the case, since f ( a 0 ) = f ( a ) , and f ( a l )= f ( a ) ~ ( n k + l ) . For the lift property, suppose f ( a ) c u ; then u = f (a)mfor some m. By the construction of r , there exists n n k + l such that ~ ( n=) m. So we can take p = onll...OnklOnl,
>
then f ( P ) = 7 ( n l ) . . T ( n k ) 7 ( n )= f (a)m= u, and obviously, a 4 ,B. Since is the transitive closure of the lift property now follows easily. It remains to note that T sends the root to the root, which implies the W surjectivity of f .
<
c,
Corollary 1.11.14 Every countable rooted S4-frame i s a p-morphic image of IT2.
Proof
By Propositions 1.11.13 and 1.11.11.
W
Definition 1.11.15 A standard I-tree i s called strongly standard if together with any sequence an it contains all the sequences am for m < n. Lemma 1.11.16 Every countable tree is isomorphic t o a strongly standard tree. Proof Given a tree F, we construct a required isomorphism f . f ( x ) is defined by induction on h t ( x ) . If x is the root, put f ( x ) := A. If f ( x ) is defined and p ( x ) = { y o , . . . ,Y n ) , put f ( y i ) := f ( x ) i ; similarly if P ( x ) is countable. (Of course this definition depends on the chosen ordering of P ( x ) . ) W
1.11. PTC-LOGICS AND HORN CLOSURES
57
Definition 1.11.17 An S4-tree is called branchy if for any u I,B(u)l 5 1. Every branchy tree is clearly infinite. L e m m a 1.11.18 If F is a strongly standard branchy tree, then F -+ IT2. Proof
We define the p-morphism g as follows:
where E is the remainder of n modulo 2. g is obviously monotonic. The lift property follows, since g(x)O = g(xO), g(x)l = g(xl) and a branchy standard H tree always contains x0, x l together with x. L e m m a 1.11.19 If every cone in a countable S4-tree F is nonlinear, then F -+ IT2. Proof By 1.11.18 and 1.11.16, it is sufficient to pmorphically map F onto a branchy tree. Let F = (W, R). For x E W let h(x) be the least element in {y E R(x) I IP(y)l > 1). h(x) clearly exists, since this set has a minimal element, which must be unique. Now h is a morphism, i.e. a p-morphism onto its image. In fact, the monotonicity is obvious. The lift property follows since
To show the latter, suppose h(x)R- h(y). Since yRh(y) and xRh(x)Rh(y), it follows that x, y are R-comparable (remember that F is an S 4 tree). If yRx, we obtain yRh(x)R-h(y), and h(x) has at least two successors, which contradicts the choice of h(y). Thus xR-y. The image G of h is a branchy tree, since h(x) = h(h(x)), and thus &(h(x)) = h[P~(h(~))l.
Definition 1.11.20 A p.0. tree is called effuse if it has a cone without linear subcones.
Propositiolr 1.11.21 If F is an efSzlse countable tree, then M L ( F ) = S 4 and thus I L ( F ) = H. Proof If a cone FTu does not contain linear subcones, then by Lemma 1.11.19 and Proposition 1.11.13, FTu -+ IT2 + IT,, and thus M L ( F ) C_ ML(IT,) = S 4 by the generation lemma, the morphism lemma and Corollary 1.11.12. The converse inclusion is trivial by soundness.
58
1.12
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
Subframe and cofinal subframe logics
Definition 1.12.1 A modal or intermediate propositional logic is called subframe if its Kripke frame variety is closed under taking subframes. Definition 1.12.2 For a transitive Kripke frame F = (W,R) a subset V and the subframe F' = F 1 V are called cofinal i f R(V) R-'(V).
c
W
Definition 1.12.3 A transitive modal or an intermediate propositional logic A is called cofinal subframe i f its Kripke frame variety V(A) or V"(A) is closed under taking cofinal subframes. Definition 1.12.4 Let F, G be Kripke frames of the same kind. A subreduction from F to G is a p-morphism from a subframe of F onto G. A reduction is a subreduction defined on a cone. A subreduction of transitive frames is called cofinal i f its domain is cofinal. If G is rooted and there exists a reduction (respectively, a subreduction, a cofinal subreduction) from F to G, we say that F is reducible (respectively, subreducible, cofinally subreducible) to G. If G is arbitrary we also say that F is reducible to G i f it is reducible to every cone i n G. By the morphism and generation lemmas every variety V(A) or VW(A)is closed under reductions. Thus for a subframe (respectively, cofinal subframe) A it is closed under subreductions (respectively, cofinal subreductions). Definition 1.12.5 A class of first-order structures is called universal, if it is a class of models of a set of universal first-order sentences. Definition 1.12.6 A modal or intermediate propositional logic A is universal, i f V(A) (or Vw(A)) is universal. The well known Tarski-Los theorem states that a class is universal iff it is closed under substructures. So a A-elementary propositional logic is universal iff it is subframe. The following result yields a criterion of universality for subframe logics; for the proof see [Chagrov and Zakharyaschev, 19971, Theorem 11.31, or [Wolter, 19971. Definition 1.12.7 A modal or intermediate propositional logic A has the finite embedding property if for any Kripke frame F , F validates A whenever every finite subframe of F validates A. T h e o r e m 1.12.8 (Wolter) For any subframe modal logic A the following properties are equivalent (1) A is universal and Kripke-complete;
1.12. SUBFRAME AND COFINAL SUBFRAME LOGICS
(3) A i s d-persistent;
(4)
A i s r-persistent;
(5) A has the finite embedding property and is Kripke-complete. Later on we will need the implication (1)+(4) from this theorem, so let us give some comments on its proof. (1)+(2) is obvious, (2)+-(3) is the Finevan Benthem theorem. For (3)+(4), note that every refined A-frame (F, W) has a descriptive extension (F', W') also validating A (the so-called 'ultrafilter extension'). By d-persistence it follows that F' I= A, and thus F k A, since A is subframe. However an example from [Chagrov and Zakharyaschev, 19971 shows that subframe monomodal logics may be Kripke-incomplete. On the other hand, subframe intermediate and K4-logics enjoy better properties. Their main new feature is axiomatisability by special 'subframe formulas' defined below. In this definition we assume that pa are different proposition letters corresponding to worlds a of a finite Kripke frame F . Definition 1.12.9 For a transitive frame F = (W,R) with root 0 put
SM-(F)
:=PO A
A
aRb
n ( p a > Opb) A
-aRb
O ( p a > l o p b )A
/\
ORa
OpaA
X M - ( F ) := S M - ( F ) A /
\
CSM-(F) := S M - ( F ) A 00 S M ( F ) := 1 S M - ( F ) , C S M ( F ) := 7 C S M - ( F ) , X M ( F ) := 1 X M - ( F ) . S M ( F ) , C S M ( F ) , X M ( F ) are respectively called the (modal) subframe, the cofinal subframe, and the frame formula of F." Obviously, the conjunct
A
7 p a is redundant if the underlying logic is S4 and
a#0
all frames are reflexive. These formulas as well as the next theorem originate from [Fine, 19741, [Fine, 19851, [Zakharyaschev, 19891. They are particular kinds of Zakharyaschev canonical formulas, see [Zakharyaschev, 19891, [Chagrov and Zakharyaschev, 19971. Theorem 1.12.10 Let F be a j n i t e rooted transitive Kripke 1-frame. T h e n for
any transitive Kripke 1-frame G (1) G
v X M ( F ) iff G is reducible t o F ,
(2) G +i S M ( F ) iff G is subreducible t o F , 2 0 ~ M ( Fis) also called the Jankov-Fine, or the chamcteristic formula.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
60
(3) G Ij C S M ( F ) i f f G is cofinally subreducible t o F . Let us recall the idea of the proof. For example, if M = (G, 8 ) YCSM(F), then we obtain a cofinal subreduction from G to F by putting f (x) = a iff M, x t= pa. The other way round, if f is such a subreduction, we construct a countermodel M for C S M ( F ) by putting M, x k p, iff f (x) = a . The next two theorems from [Zakharyaschev, 19891 are also specific for the transitive case:
Theorem 1.12.11 A transitive 1-modal logic is subframe (respectively, cofinal subframe) iff it is axiomatisable by subframe (respectively, cofinal subframe) formulas above K4. Theorem 1.12.12~' Every cofinal subframe modal logic has the f.m.p. Corollary 1.12.13 Let A. be a subframe K4-logic. T h e n for a n y 1-modal logic A 2 A o , A is subframe (respectively, cofinal subframe) iff it i s axiomatisable by subframe (1 -espectively, cofinal subframe) formulas of Ao-frames above A o . Proof 'If' easily follows from 1.12.11. To prove 'only if', suppose A is subframe; the case of cofinal subframe logics is quite similar. By 1.12.11, we have
A
=K4
+ {SM(F) I ( F ) E A ) , A0 = K 4 + {SM(F) 1 S M ( F ) E n o )
hence
A
= A0
+ {SM(F) I SM(F) E A - Ao).
Now S M ( F ) $! A. implies F t= Ao. In fact, since A. is Kripke-complete by 1.12.12, there exists a Ao-frame G such that G Ij S M ( F ) . Then G is subreducible to F by 1.12.10, and since A. is a subframe logic, it follows that F I= Ao. We shall use this corollary especially for the case A. = S4.
Example 1.12.14 S4.1 = S4+OOp > OOp is a cofinal subframe logic (McKinsey property is obviously preserved for cofinal subframes). It can be presented as S 4 + CSM(FC2), where FC2 is a 2-element cluster - one can check that an S4-frame has McKinsey property iff it is not cofinally subreducible to FC2. Similarly the logic K4.1- := K 4 + 0 1 V 0 0 1 is cofinal subframe; it is presented as K 4 + CSM(FC1), where FC1 is a reflexive singleton and characterised by the (first-order) condition
Thanks to completeness stated in 1.12.12, Theorem 1.12.8 has the following transitive version: 'lFor subframe transitive logics this fact was first proved in [Fine, 19851.
1.12. SUBFRAME AND COFINAL SUBFRAME LOGICS
61
Theorem 1.12.15 (Zakharyaschev) For any subframe modal logic A 2 K 4 the following properties are equivalent: (1) A is universal;
(3) A is d-persistent;
(4) A is r-persistent; (5) A has the finite embedding property.
+
By applying 1.12.10 to subframe logics A = A0 {SM(Fi) I i E I}described in 1.12.13, we can reformulate the finite embedding property as follows:
For any Ao-frame G, i f G is subreducible to some Fi (i E I ) , then some finite subframe of G is subreducible to some Fj (jE I). Hence we obtain a sufficient condition for elementarity of subframe logics above K 4 or S4.
Proposition 1.12.16 (1) A subframe K4-logic is A-elementary i f above K 4 it is axiomatisable by subframe formulas of irreflexive transitive frames.
(2) A subframe S4-logic is A-elementary if above S4 it is miomatisable by subframe formulas of posets. Proof We prove only (1);the proof of (2) is similar. By 1.12.15, it is sufficient to check the finite embedding property for K4 {SM(Fi) I i E I), where Fi are irreflexive K4-frames. So for a K4-frame G subreducible to some Fi, we find a finite subframe subreducible to Fi. Given a subreduction f : G' -+ Fi, for G' C G, it is sufficient to construct a finite G" G' such that f 1 G" : G" ++ Fi. This is done by induction on IFi/. If Fi is an irreflexive singleton, everything is trivial. Otherwise, let u be the root of Fi, and let f ( a ) = u. Obviously, a is irreflexive. For every v E P(u) there exists a cone G!,, 2 G'T a such that f G!,, : G!,, ++ Fi f v - this follows from 1.3.32(3). Then by the induction hypothesis, there is a finite G i 5 GL such that f G l : Gc + Fi f v . Finally put G" := { a ) U U G; (as a subframe of GI). This G" is the
+
v€8(u)
required one; in fact, monotonicity is preserved by restricted maps, and the lift property easily follows from the construction. For subframe logics axiomatisable by a single subframe formula the converse also holds:
Proposition 1.12.17
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
62
+ S M ( F ) is elementary iff F is irrejlexive. (2) A subframe logic S4 + S M ( F ) is elementary iff F is a poset. (1) A subframe logic K 4
Proof (1) If F contains reflexive points, we can replace each of them by the chain (w, <). The resulting frame F' is reducible to F , since a reflexive singleton is a pmorphic image of (w, <). However, finite subframes of F' are not reducible to F, since a pmorphic image of a finite irreflexive K4-frame is always irreflexive (e.g. because finite irreflexive KPframes are exactly finite GL-frames). Thus K 4 + S M ( F ) does not have the finite embedding property in this case. (2) Similarly, if F contains nontrivial clusters, we can replace each of them by (w, 5). Then we obtain a frame F' reducible to F . Every finite subframe of F' is a poset, so it is not reducible to F (e.g. because it is a Grz-frame).
R e m a r k 1.12.18 In general, the converse to 1.12.16 is not true. For example, the trivial logic S4+p > U p is obviously subframe and elementary, but it cannot be axiomatised by subframe formulas of posets. In fact, every formula S M ( F ) for a nontrivial F , is valid in every S5-frame G (which is a cluster), since G is not subreducible to F . Moreover, there is a conjecture that elementarity of a logic axiomatisable by a finite set of subframe formulas is ~ndecidable.'~ Theorem 1.12.15 has an analogue for cofinal subframe logics. Definition 1.12.19 A world i n a transitive frame is called inner i f its cluster is not maximal. The restriction of a frame F to inner worlds is denoted by F - . Definition 1.12.20 Let F , G be K4-frames and suppose that G is finite. A cofinal subreduction f from F to G is called a cofinal quasi-embedding i f f - ' ( x )
is a singleton for any inner x . If such a subreduction exists, we say that G is a finite cofinal quasi-subframe of F . Definition 1.12.21 A modal or intermediate propositional logic A has the finite cofinal quasi-embedding property if for any Kripke frame F , F validates A whenever every its finite cofinal quasi-subframe validates A. T h e o r e m 1.12.22 (Zakharyaschev) For any cofinal subframe modal logic
the following properties are equivalent: (1) A is elementary;
(2) A is quasi-A-elementary; 22M. Zakharyaschev, personal communication.
1.12. SUBFRAME AND COFINAL SUBFRAME LOGICS
(3) A is d-persistent;
(4) A
has the finite cofinal quasi-embedding property.
Note that unlike the previous theorem, 1.12.22 does not include r-persistence. Let us now consider the intuitionistic case. Now we assume that qa are different proposition letters indexed by worlds of a finite poset F . Definition 1.12.23 For a poset F = (W, <) with root 0 put
a
/\
gal
a€ W
/\
i
)
/\ qb > qa V a@ /\ qb , b
go, C S I ( F ) := C S I - ( F ) > go, X I ( F ) := X I P ( F )
XI-(F)
:= CSI-(F) A
a
> go.
S I ( F ) , C S I ( F ) , X I ( F ) are respectively called the (intuitionistic) subframe, cofinal subframe, and frame formula of F.23 Note that S I ( F ) is an implicative formula and C S I ( F ) is built from proposition letters and >, 1. Then similarly t o the modal case, we have (cf. [Zakharyaschev, 19891, [Chagrov and Zakharyaschev, 19971): Theorem 1.12.24 Let F be a finite rooted poset. Then for any poset G
(1) G
v X I ( F ) if/ G is reducible to F;
(2) G '$ S I ( F )
ifl G
is subreducible to F;
(3) G I$ C S I ( F ) iff G is cofinally subreducible to F .
The idea of the proof is quite similar to 1.12.10. E.g. if M = (G,B) X I ( F ) , we obtain a reduction f from G to F by putting
Iy
The formulas from 1.12.23 can be simplified. For example, in C S I - ( F ) we can replace the conjunct 7 /\ qa with 1 gal where max(F) is the set a€ W
of maximal points of F; also can be replaced with
/\
A
a€max(F)
qb occurring in the second conjunct of X I P ( F )
b
a€O(b)
Other versions of these formulas are described in [Chagrov and Zakharyaschev, 19971 and [Shimura, 19931. The next result is also due to [Zakharyaschev, 19891, cf. [Chagrov and Zakharyaschev, 19971. 2 3 X ~ ( Fis) also called the Jankov, or the characteristic formula of F .
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
64
Theorem 1.12.25 For a n intermediate propositional logic A the following properties are equivalent:
(1) A is a subframe (respectively, cofinal subframe) logic;
(2) A is axiomatisable by subframe (respectively, cofinal subframe) formulas; (3) A i s axiomatisable by implicative formulas (respectively, (2, I)-formulas). Let us now formulate analogues of Theorems 1.12.12, 1.12.15 and 1.12.22 for intermediate logics. Theorem 1.12.26 Every cofinal subframe intermediate logic has the finite model property.
This result follows from [McKay, 19681 and 1.12.25; also see [Chagrov and Zakharyaschev, 19971. Theorem 1.12.27 (1) A n intermediate logic i s subframe i f it i s universal.
(2) Every subframe intermediate logic is r-persistent. Theorem 1.12.28 Every cofinal subframe intermediate logic is A-elementary and d-persistent.
To formulate a modal analogue of this result, we need Definition 1.12.29 A transitive frame is called almost irreflexive if all its nonmaximal clusters are degenerate. A n S4-frame is called blossom if all its nonmaximal clusters are trivial. Theorem 1.12.30
(1) A cofinal subframe S4-logic i s A-elementary if above S4.1 it is axiomatisable by cofinal subframe formulas of blossom frames. (2) A cofinal subframe K4-logic is A-elementary if above K4.1- it is axiomatisable by cofinal subframe formulas of almost irreflexive transitive frames. Proof ( 1 ) By 1.12.22, it suffices to check the finite cofinal quasi-embedding property for our logic A. So suppose GYA. If G y S 4 . 1 , the further proof is easy, so let G b S4.1. Then there is a finite rooted blossom frame F such that C S M ( F ) E A and a cofinal subreduction f from G to F . Let us show that then there exists a subframe G o G such that f Go is a cofinal subreduction to F and f is injective on f ( F - ) . Moreover, we can choose Go so that f [Go] F - , and thus Go is finite. The proof is by induction on the cardinality of F . Let u be an element of G with f ( u ) = OF, where OF is the root of F . For every a E OF), choose an
r
arbitrary v E G such that u 5~v and f ( v ) = a. Note that if a is maximal in F and G has McKinsey property, we can choose a maximal v , since f is cofinal. Then fa := f I (GTv), is a cofinal subreduction from G T v to FTa. Hence by the induction hypothesis, there exists a finite subframe Ga of G such that fa 1 G, is a cofinal subreduction from G to F T a. Put
Then Go obviously has the required property. (2) is proved similarly.
1.13
Splittings
The following result is a modal version of [Jankov, 19691. T h e o r e m 1.13.1 Let F be a finite rooted Kripke frame. T h e n for any modal algebra 0 the following properties are equivalent:
(1) fl k' X M ( F ) ; (2) M A ( F ) i s a subalgebra of a homomorphic image of $2.
L e m m a 1.13.2 For a n y I-modal logic A frame F XM(F) E A i f l A
> K 4 and
a finite rooted transitive
ML(F);
in particular, for a n y I-modal formula A
If F i s reflexive, then the latter i s also equivalent to S 4 + A I- X M ( F )
+
R e m a r k This lemma shows that the pair of logics ( K 4 X M ( F ) , M L ( F ) ) K 4 is either below M L ( F ) or 'splits' the set of extensions of K4: every A above K 4 X M ( F ) . The original Jankov formula [Jankov, 19691 is defined as follows.
+
>
Definition 1.13.3 Let F = (W,5 ) be a finite p.0. set, S1 = HA(F). T h e n
where {pa I a E a) is a set of distinct propositional variables corresponding t o the elements of S1 and w is the subgreatest element of 52.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
Figure 1.2. F2
Theorem 1.13.4 Let F be a finite rooted Kriplce frame. Then for any Heyting algebra 0' the following properties are equivalent: (1) 0'
v XI(F);
(2) HA(F) is a subalgebra of a homomorphic image of 0'.
Corollary 1.13.5 Let F be a finite rooted p.0. set. Then for any p.0. set G the following properties are equivalent: (1) G l y XI(F);
(2) there exists u E G such that GU+ F.
Lemma 1.13.6 For any superintuitionistic logic A XI(F) E A iff A
g IL(F),
i n particular, for any intuitionistic formula A
XI(F) E (H
+ A) iff A $! IL(F).
Hence for example, we obtain
Lemma 1.13.7 Let A be a propositional fomula. Then:
+ A) i f f A $! IL(F2) (Fig. 1.2); 6 (H + A) iff A $! IL(F3) and A $! IL(F4) (Figs. 1.3, 1.4).
(1) AJ E ( H (2) A J -
Proof
Note that
H + AJ = H + XI(F2), H + A J - = H + XI(F3)A XI(F~). Then apply Lemma 1.13.6
1.13. SPLITTINGS
Figure 1.3. F3
Figure 1.4. Fq
68
C H A P T E R 1. B A S I C P R O P O S I T I O N A L L O G I C
1.14
Tabularity
Definition 1.14.1 A modal or intermediate propositional logic i s called tabular if i t is determined by a finite modal o r Heyting algebra, o r equivalently, by a finite Kripke frame. Lemma 1.14.2 In the notation of 1.10.2, if F has a root uo, dd(u0) i s finite and R(x) i s finite for a n y x E F , t h e n F is finite. Proof By induction on k = dd(uo). The statement is trivial for k = 0. For the step note that
By the induction hypothesis, every W T u for uORu, uo
# u
is finite, since
d d ( u ) < dd(u0). Hence W is finite.
W
Consider the following N-modal k-formulas
where
O A := O I A v . . . v O N A .
These formulas have the following first-order characterisations.
Lemma 1.14.3 For a frame F = (W, R 1 , .. . ,R N ) put R := R1 U . . . U R N . Then (1) F, x k A l t k iffIR(x)I < k;
(2) F , x I= A H k iff d d f ( x ) < k . For intermediate logics we have the same characterisation with different formulas.
Lemma 1.14.4 For a poset F = (W,R)
Hence we obtain a characterisation of tabular modal logics similar to that in [Chagrov and Zakharyaschev, 19971. Proposition 1.14.5
1.14. TABULARITY
69
(1) A n N-modal propositional logic A i s tabular i f fA E Altk A AHk for some k.
(2) A n intermediate propositional logic A is tabular i f fA I- IGk for some k. P r o o f The claim 'only if' follows from 1.14.3 and 1.14.4. To show 'if', use the canonicity of AHk, Altk, I G k and the observation that every rooted frame validating Altk A AHk (or IGk) is finite. L e m m a 1.14.6 If @ = (F,W) i s a f i n i t e refined frame, then ML(@)= M L ( F ) ( o r IL(@)= I L ( F ) in the intituionistic case). P r o o f (Modal case) For x # y let U,, be an interior set such that x E U,,, U,,. Since F is finite, it follows that
y $?
Thus all subsets of F are interior, so the valuations in and F are the same. Therefore valid formulas are the same. (Intuitionistic case.) Suppose F = (W, R) is an intuitionistic frame and for y # R ( x ) let V,, E W be such that z E V,, y # V,, . Then R(x) =
n
vx, E
w.
yeR(x)
So for every R-stable U we have
and again it follows that @, F have the same valuations.
¤
Proposition 1.14.7 Every tabular modal or intermediate propositional logic i s r -persistent. P r o o f (Modal case) Consider a tabular logic A and a refined frame = (F, W) validating A. By 1.14.5, A t- Altk A AHk for some k, hence @ != Altk A AHk. According to the characterisation given in 1.14.3, the classes V(Altk), V(AHk) are universal, so Altk A AHI, is r-persistent by Theorem 1.12.8 and thus valid in F and therefore in every cone F f u. So by 1.14.5, @ T u is finite. By 1.7.8, @ u is refined and by 1.6.11, @ f u != A . Therefore, F 1' u != A by 1.14.6. Since u is arbitrary, by Lemma 1.3.26, it follows that F k A. In the intuitionistic case use the same argument and 1.14.4 instead of 1.14.5.
r
w Proposition 1.14.8 For finite modal frames G, F , ML(G) reducible t o F; similarly for the intuitionistic case.
M L ( F ) iff G is
T h e o r e m 1.14.9 Every tabular modal or intermediate logic is finitely axiomatisable. Moreover, a transitive modal o r intermediate logic i s finitely miomatisable by frame formulas.
70
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
Transitive logics of finite depth
1.15
The results of this section are based on [Hosoi, 19671, [Hosoi, 19691.
L e m m a 1.15.1 For a poset F
Proposition 1.15.2 Let Z, be an n-element chain. Then
for every infinite chain F . In particular, IL(Z1)= H and H AP, c IL(Z,) for n 2 2.
+
Proposition 1.15.3 H depth n .
+ AP,
+ APl
is a classical logic
is determined by the class of finite posets of
+
Proof As we know, APn-frames are S4-frames of depth 5 n , so H AP, is a subframe logic. Thus it is determined by finite posets of depth 5 n. It remains to note that every poset of depth < n is a generated subframe in a poset of depth n . W Hence by unravelling we obtain
Corollary 1.15.4 H+AP, = I L ( I T Z ) , where IT: is the standard tree of depth n and branching w . Hence by induction on n we obtain
Corollary 1.15.5 For every m, n E w there exists k E w such that ( H + AP,) [m
= IL(IT;)
[m,
where I L ( I T t ) is the tree of depth n and branching k . So to say, m letters can distinguish only finitely many successors of any point.
Definition 1.15.6 A logic L is called locally tabular if all its finitely generated Lindenbaum algebras Lind(L [ m ) are finite; in other words, if for every m there exist finitely many non-L-equivalent formulas in m propositional letters. Corollary 1.15.7 Every logic H
+ AP,
is locally tabular.
Proposition 1.15.8 Every locally tabular logic has the f.m.p. L e m m a 1.15.9 H
+
=H
+ AP,
for n
20
1.1 5. TRANSITIVE LOGICS OF FINITE DEPTH Proof
71
A frame F is of depth 5 n iff F is not reducible t o iff F ItThus H + AP, and H + have the same finite frames, and
it remains to show that the latter logic has the f.m.p. But this logic is subframe, since the class of frames of depth 5 n is closed under subframes. So we can apply Fine's theorem. Hence by Lemma 1.13.6 we obtain Proposition 1.15.10 For a superintuitionistic logic L
L
IL(Zn+l) i f f H
+ APn c L.
This proposition shows that the structure of superintuitionistic logics splits into an (w 1)-sequence of slices.
+
Definition 1.15.11 For a finite n, the nth slice is the set of superintuitionistic logics S, := {L E S I H AP, E L IL(Zn));
+
the wth slice is
S, := { L I H
C L c LC).
i.e. So contains only the logic of the empty 'chain' Zo. Let S, be the set of logics of slice n (for 0 5 n 5 w). Proposition 1.15.10 implies Proposition 1.15.12
U
(1) S =
Sn,
n<w
(2) SnnSm= 0 for n
# m,
(3) i f L is a superintuitionistic logic, then L E Sn, where
=
(4)
{
zax{n E w / L i IL(Z,))
the slices S1 i s one-element: S1= { H
if V n L E IL(Z,), otherwise
+ APl) = CL,
(5) the slice Sz i s a decreasing (w + 1)-chain, (6) for n
> 3 all slices S,
are of cardinality 2 N ~
Proposition 1.15.13 For a finite n
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
72
P r o p o s i t i o n 1.15.14 Sn = { I L ( F ) I F i s a poset of depth n ) . Every logic of slice S, i s determined by a disjoint union of finite posets of depth n. P r o o f Local tabularity is clearly inherited by extensions, so it holds for all logics of finite slices. Therefore all logics of finite slices have the f.m.p., and ¤ hence the proposition follows. As we mentioned, L C =
IL(Zn). The set nEw
is a decreasing w-chain; this readily follows from Proposition 1.15.14, since IL(Zn) = L C APn is a simple extension of L C in the n t h slice. Also H = ( H APn), since H has the f.m.p. and every finite poset is of fi-
n +
+
nEw
nite depth. P r o p o s i t i o n 1.15.15 Sn are sublattices of the lattice of superintuitionistic logics. Every finite slice S, i s embeddable in all slices Sn for m n w by maps A,:, L H L n IL(Z,) (and IL(Z,) = LC).
< <
+ +
P r o o f Note that L1 L2 iff Xmn(L1) C Xmn(L2), since L = Xmn(L) APm = Xmn(L) (H AP,), for L E S,; moreover, X,,(L) = X,k(L) APn = Xmn(L) ( H AP,) for k 2 n,by distributivity. Also
+ + + +
and A,
( L + L') = X,,(L)
+ Xmn(Lf),again by distributivity.
In this section we study an embedding of S, in in [Hosoi, 19691.
W
(A-operation) introduced
Definition 1.16.1 For A E I F put 6A := p V ( p > A), where p is a proposition letter that does not occur i n A. For a superintuitionistic logic L put A L := H {bA I A € L).24
+
The original definition [Hosoi, 19691 is A L := {S'A I A E L), where 6'A := > p ) > p) (the 'A-version' of Peirce's law). Let us show that both definitions are equivalent. ((p > A)
L e m m a 1.16.2 H
+ 6A = H + S'A for
any formula A.
240bviously, the definition of AL does not depend on the choice of p; but we fix p t o make
6 A unique.
1.1 6.
A-OPERATION
73
Proof On the one hand, obviously QH k 6A us prove
> &'A.
On the other hand, let
In fact, obviously
hence
and thus by the deduction theorem 1.1.4
This implies
hence
by the deduction theorem. Now since
is a substitution instance of 6'A, (5) implies (0). The next lemma shows the semantical meaning of 6A.
Lemma 1.16.3 Let F be a rooted poset with root OF. Then FIt 6Aiff Q u # O F F
T u l t A.
Proof (Only if.) Suppose F t u Iy A, u # OF, SO M Iy A for some Kripke model M = (FTu, 0). By truth preservation it follows that M , u Iy A. Then consider M' = (F, J ) such that
Obviously M ' is intuitionistic. Also MI, u Iy A, since p does not occur in A. Now from M', u I t p and MI, OF Iy p it follows that M', OF Iy SA; thus F Iy 6A. (If.) Suppose F Iy &A,i.e. M Iy 6A for some Kripke model M over F. Then by truth preservation M I OF Iy 6A; hence M I OF ly p, and M , u I t p, M , u Iy A for some u # OF. By the generation lemma it follows that M l u , u Iy A; thus FTu Iy A. W
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
74 Hence we obtain
+
Proposition 1.16.4 If F Il- L, then 1 F It. AL, where 1 adding a root below F .
+ F is obtained by
+
Proof In fact, if 1 F Iy SA, then by 1.16.3, F l u Iy A for some u E F ; thus F Iy A by the generation lemma. rn L e m m a 1.16.5 For any superintuitionistic logic L, SA E AL iff A E L
Proof We consider a particular case, when L is Kripke-complete. The claim easily follows from Lemma 1.16.3 (and Proposition 1.16.4). In fact, if A $ L = I L ( F ) for a poset F, then AL C I L ( l F ) and SA $2 I L ( l F ) . w
+
+
A predicate analogue of 1.16.5 will be discussed later on in Section 2.13. L e m m a 1.16.6
(1) H k A
(2) H t- 6(A1 > A2)
Proof
> 6A.
> (6A1 > SAz).
An easy exercise; also see Chapter 2.
Proposition 1.16.7 For propositional superintuitionistic logics L1, La. (1) A L
c L;
(2) L1 C L2 iff AL1
C AL2;
(3) L 1 = L2 iff AL1
= AL2.
Proof (1) AL L follows from H t A > 6A. (2) 'Only if' is obvious. To show 'if', suppose L1 L2 and A E L1 - L2; then 6A E AL1 - AL2 by 1.16.5, and thus AL1 AL2. (3) A trivial consequence of (2). W
L1
Note that (2) means that A is a monotonic embedding S L2 also implies AnLl E AnL2 for any n. Now the deduction theorem implies
+
+
L e m m a 1.16.8 A ( H r ) = H S r , where 6 r := {SA A n ( H I?) = H P I ? , where b n r := {bnA I A E r ) .
+
+
I
---+
S. By (2))
A E I?). Hence
m
Proof In fact, if L = H + I ' k A then HI-
/\ Bi 3 A for some B 1 , . . . , B , i=l
Sub(r). Then SBi E Sub(&?), and so H H +sr.
+ dl? t- bA.
Thus AL = H
E
+ 6L c w
75
1.16. A-OPERATION
Obviously APm+l = bAPm, so bnAPm = APm+,, in particular AP, = b m l . Hence we conclude that A n ( H AP,) = H APm+,, and thus for any n A(H+APn) = H + A P n + l ; and H+APn = A n ( H + I ) = An-'(CL) for n > 0. Also by Proposition 1.16.7,
+
+
A(IL(Zn)) c IL(Zn+,) for n
> 0;
+ AP2 for L C CL, and thus
The inclusion is proper, since AL AZ = ( P 3 4) v (4 3 P ) @ AL. So we obtain
C ACL
Proposition 1.16.9 AL E Sn+1and S, in itself.
for L E Sn, n E w. T h u s A embeds Sn in
=H
Note that A is not a lattice embedding; more precisely, it preserves joins not meets. L e m m a 1.16.10 If A1 E (L1 - La) and A2 E (La ALl n ALa - A(L1 n L2).
-
Lr), then &A1V &A2E
P r o o f For Kripke-complete L1 and L2 (in particular, for all logics of finite slices) this readily follows from Lemma 1.16.3. Namely, if A1 6 L2 = IL(F2) and A2 $! L1 = IL(Fl), then the frame 1 Fl u F2 separates 6A1 V bA2 from nL ~ ) . A(L~ The lemma actually holds for arbitrary logics; its analogue for predicate logics will be discussed in Section 2.13.
+
Proposition 1.16.11 For superintuitionistic logics L1, L2
(2) A(L1 n L2) = ALl n ALz iff Ll and Lz are g-comparable. P r o o f (l)followsfromLemma1.16.5: L1+L2=H+L1UL2,soA(L1+L2) = H 6L1 U 6L2 = ( H 6L1) ( H 6L2) = AL1 AL2. (2) follows from 1.16.10.
+
+
+ +
+
Proposition 1.16.12 AL = L iff L = H . S o AL c L for any L
+
P r o o f If AL = L, then L C H Iimplies L C A n ( H all n E w. Hence L n ( H AP,) = H . n
+
# H.
+ I ) = H + APn for
R e m a r k 1.16.13 A similar argument shows that for any superintuitionistic logic L,
A"L=H
(*I
nEw
in other words, the 'w-iteration' of A is trivial. (*) readily implies Proposition 1.16.12. In Volume 2 we will prove that 1.16.12 and (*) do not transfer to the predicate case.
CHAPTER 1. BASIC PROPOSITIONAL LOGIC
76
1.17
Neighbourhood semantics
Neighbourhood semantics is a generalisation of Kripke semantics. In this case 'possible worlds' are regarded as points of an abstract 'space' or a 'neighbourhood frame'. In such a frame every world has a set of 'neighbourhoods', and OA is true a t w iff A is true in all worlds in some neighbourhood of w, that is, in all the worlds that are 'rather close' to w . So in neighbourhood semantics 'necessary' is interpreted as 'locally true'. Here is a precise definition. Definition 1.17.1 An n-modal (propositional) neighbourhood frame is an (n+ 1)-tuple F = (W, O1,. . . ,UN), such that W # 0 , Oi are unary operations in 2W satisfying the identities:
As in Kripke frames, the elements of W are called possible worlds, or points; u E UiV is read as 'V is an i-neighbourhood of u'. The basic identities mean that the intersection of two i-neighbourhoods of u is also an i-neighbourhood of u, every extension of an i-neighbourhood is again an i-neighbourhood and that W is an i-neighbourhood of any u. However a neighbourood of u may not contain u, or may even be empty. Obviously, an N-modal neighbourhood frame F corresponds to the N-modal algebra M A ( F ) := (2W, U, n, -, 0 , W, 0 1 , . . . , O N ) . The following is a trivial consequence of definitions, cf. Section 1.3. L e m m a 1.17.2 Every Kripke frame F = (W, R l , . . .,R N ) corresponds to a neighbourhood frame N d ( F ) = (W, O1,. . . ,ON), such that M A ( N d ( F ) ) = MA(F).
-
Definition 1.17.3 A neighbourhood model over a neighbourhood frame F is a pair M = (F, 0), in which 0 : P L 2W is a valuation. 19 is extended to all formulas, according to Definition 1.2.8. We use the same terminology and notation as in Kripke semantics. For a formula A, we write: M , w k A (or w k A) instead of w E @(A),and say that A is tme at the world w of M (or that w forces A). So we have:
M, x b OiA iff 0(A) is an i-neighbourhood of x. Definition 1.17.4 A modal formula A is true in a neighbourhood model M (notation: M k A) if it is true at every world of M; A is valid in a neighbourhood frame F (notation: F k A) if it is true in e v e q model over F. A set of formulas l? is valid in F (notation: F k I?) if every A E l? is valid.
1.1 7. NElGHBOURHOOD SEMANTICS
77
Similarly to Lemmas 1.3.7 and 1.3.8 we obtain
L e m m a 1.17.5 For a n y modal formula A and a neighbourhood frame F
F k A iff M A ( F ) k A. Lemma 1.17.6 (1) For a neighbourhood frame F the set
is a modal logic. (2) For a class C of N-modal neighbourhood frames the set
is a n N - m o d a l logic. Definition 1.17.7 T h e logic M L ( F ) (respectively, M L ( C ) ) i s called the modal logic of F (respectively, of C ) , or the modal logic determined by F (by C ) . A modal logic i s called neighbourhood complete if it i s determined by some class of neighbourhood frames. From Lemma 1.17.2 we have:
L e m m a 1.17.8 Every Kriplce-complete propositional logic i s neighbourhood complete. The converse to the previous Lemma is false [Gabbay, 19751, [Gerson, 1975a], [Shehtman, 19801, [Shehtman, 20051. There also exist examples of modal logics that are incomplete in neighbourhood semantics [Gerson, 1975b], [Shehtman, 1980], [Shehtman, 20051. The question of whether all intermediate propositional logics are neighbourhood complete (Kuznetsov's problem [Kuznecov, 19741 ), is still open.
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Chapter 2
Basic predicate logic 2.1
Introduction
The main notion of this chapter is first-order logic. Similarly to the propositional case, we define a logic as a set of formulas that contains some basic axioms and is closed under some basic inference rules. Here the crucial point is the substitution rule, which is important because we would like to distinguish between logics and theories. On the one hand, every axiomatic logical calculus (postulated by a set of axioms and inference rules) generates a 'theory' - the set of all theorems. Usually theories are supposed to collect properties of a certain kind of objects. Many well known theories, such as Peano arithmetic, Tarski's elementary geometry, Zermelo-Fraenkel set theory, were developed for that purpose. On the other hand, we may be interested in theories that do not depend on 'application domains' and express the basic 'logical laws'. For example, the proposition Every human has a f a t h e r and a mother
expressed by a formula
A
= Vx(H(x)
> 3yF(y, x) A 3zM(z, x)),
is a specific property of humans that does not hold for all living creatures. The formula B = Vx(H(x) > 3yM(y, x)) also expresses a true property of humans, which does not hold in other cases. But the implication A>B (allowing us to deduce B from A) is a logical law - its truth does not depend on the meaning of the predicates H, F, M.
80
CHAPTER 2. BASIC PREDICATE LOGIC
So the laws of logic should sustain replacing of predicates by arbitrary formulas. We can regard them as schemata for producing theorems, and define a logic just as a substitution closed theory.' A standard example is classical first-order logic, the set of all theorems of classical predicate calculus. Numerous classical theories contain it as a fixed basic part. In the nonclassical area there is a great variety of logics deserving special attention. Of course study of nonclassical theories (such as Heyting arithmetic or modal set theories) is also interesting and important, but due to the lack of time, we postpone it until Volume 2. Study of nonclassical logics in this volume is closely related to study of different semantics. Unlike the classical case, there are many options here. From our viewpoint, a semantics S for a certain class of logics (say, C) should include the notions of a 'frame' and 'validity'. A semantics S is 'sound' for C if the set of all formulas valid in any S-frame is a logic from C.2 Thus to check soundness, it is necessary to prove that the substitution rule preserves 'validity' in a 'frame'. In this respect there is a big difference between the classical and nonclassical cases. In classical logic we may not care about formula substitutions, and they are usually not discussed in textbooks and monographs.3 Classical predicate calculus is traditionally formulated using axiom schemes rather than substitution rule, and soundness is proved without mentioning substitutions. But in our nonclassical studies we deal with rather exotic types of frames and the proof of soundness may be nontrivial. Therefore let us first take a closer look at the syntactic notion of a logic, and especially at the substitution rule. Its intuitive meaning is clear: given a firstorder formula A we can deduce every formula [C/P(xl,. . . ,x,)]A obtained by substituting a formula C for an atomic formula P(x1,. . . ,x,). More exactly, to obtain [C/P(xl,. . . ,%,)]A, one should replace every occurrence of P(x1,. . . ,x,) with C and every occurrence of P(y1,. . . ,y,) (using other variables yl, . . . ,y,) with the corresponding version of C, [ ~ 1 ,. .. ,yn/ 21,. . . ,xn]C. In its turn [yl,.. . ,y,/xl,. .. ,x,]C is obtained by a 'correct' replacement of parameters XI,.. . , x, with yl, . . . , y,. Thus the definition of a formula substitution [C/P(xl,. . . , x,] relies on the definition of a correct variable substitution [yl, . . . , yn/xl,. . . ,x,]. Our approach to variable substitutions is rather nonstandard and resembles [Bourbaki, 19681. But it is convenient from the technical viewpoint, see Section 2.3 for further details. l0f course this definition is rather conventional, and there exist examples of 'logics' that are not substitution closed. 'Some authors still use 'semantics' that are not sound in this sense [Kracht and Kutz, 20051, [Goldblatt and Maynes, 20061. This is not so convenient, because it may be difficult to describe all 'frames' characterising a given logic. 3 ~ i t few h exceptions, such a s [Church, 19961, [Novikov, 19771.
2.2. FORMULAS
2.2
81
Formulas
The expansion of a propositional language to a first-order language is defined in a standard way. Let Var = {vl, v2,. . .), PLn = I i 2 0) (n 2 0) be fixed disjoint countable sets. The elements of Var and PLn are respectively called (individual) variables, and n-ary predicate letter^.^ An atomic formula without equality is either I, or Pf (a proposition letter), or P?(xl,. . . ,x,) for some n > 0, 21,. . . ,x, E Var. Atomic formulas with equality can also be of the form x = y, where x, y E Var and '=' is an extra binary predicate letter.5 Note that our basic language does not include constants or function letters; we will return to these matters in Volume 2. Also note that the language is countable, but we shall consider its uncountable expansions with constants. Classical (or intuitionistic) predicate formulas (with or without equality) are built from atomic formulas using the propositional connectives A, V, >, and the quantifiers V, 3; in N-modal predicate formulas6 the unary connectives Oi, 1 5 i 5 N , can also be used. The abbreviations -A, T, A = B , OiA have the same meaning as in the propositional case; x # y abbreviates -(x = y). For a formula A, a list of variables x = XI . . .xn and a quantifier & E {V, 31, QxA denotes &xl ... &xnA.
{Pr
Definition 2.2.1 The (modal) degree d(A) of a modal predicate formula A is defined by induction: d(A) = 0 for A atomic; d(A A B ) = d(A v B ) = d(A d(VxA) = d(3xA) = d(A); d(OiA) = d(A) 1.
> B ) = max (d(A),d(B));
+
So d(A) = 0 iff A is a classical formula. AF, I F , M F N denote respectively the sets of atomic, intuitionistic, and Nmodal formulas without equality; the corresponding sets of formulas with equality are denoted by AF', I F = , MFG; we omit the subscript N if N = 1 or if N is clear from the context. Sometimes we write IF(=), MF$). etc. in order to combine the cases with and without equality in a single statement. The and called the (basic) N-modal first-order set MF~=) is also denoted by language; LA=,=) is (basic) classical (or intuitionistic) first-order language. We skip a routine proof of the following standard statement (cf. slightly different versions in [Shoenfield, 19671, [Bourbaki, 19681).
LE)
Lemma 2.2.2 (Parsing lemma) Let A, B , A', B' be formulas, *, *' E {v, A, > ) binary connectives such that A * B = A' *' B'. Then A = A', * = *I, and B = B'. 4We also use the symbols P,Q, R, . . . (sometimes with subscripts) as names of predicate letters; p, q, r, . . . as names of proposition letters and x, y, . . . as names of variables. 5'=' is also used as a metasymbol. 6To avoid confusion, we use the letter N instead of n, because in many cases n denotes the number of variables in a list.
CHAPTER 2. BASIC PREDICATE LOGIC
82
The definition of free and bound occurrences of variables in formulas is wellknown, but we shall now give it in a more formal way.
Definition 2.2.3 A n occurrence of a letter c in a word a is a triple (a,i ,c ) such that c is the i t h letter of a; an occurrence of a word p in a is a triple (a,i , /3) such that P is a subword of a starting from the i t h letter of a . Free and bound occurrences of variables in formulas are now defined by induction.
Definition 2.2.4
All vanable occurrences i n atomic formulas are free. If (A, i, x) is free (bound), then (OjA, i Let C = (A * B), where
+ 1,x) is free
(bound).
* is a binary connective,
IAl = 1. If (A, i ,x) is free (bound), then (C, i + 1,x) is free (bound). If (B, i , x) is free (bound), then (C,i 1 2, x) is free (bound).
++
Let B = VyA or 3yA. If ( A ,i ,x) is free (bound) and x # y, then (B, i + 2,x) is free (bound). All occurrences of y i n B are bound. The occurrence (B, 2, y) is called strongly bound. FV(A) denotes the set of parameters (free variables) of a formula A, i.e. of all variables having free occurrences in A. A closed formula (or a sentence) is a (respectively, I S ( = ) ) denotes the set of all formula without parameters. N-modal (respectively, intuitionistic) sentences. BV(A) denotes the set of bound variables of A (i.e. vsriables having bound occurrences in A). We also use the notation
MSE)
A variable is called new for A if it does not occur in A. Recall that a universal closure of a formula is usually understood as the result of the universal quantification over all its parameters. But such a definition is a priori ambiguous. To fix a unique notation for the universal closure, one can take the parameters in a certain order, for example as follows.
Definition 2.2.5 The standard list of parameters of a formula A is the set of its parameters FV(A) ordered i n accordance with their first occurrences i n A. The standard universal closure VA of a formula A is the sentence VxA, where x is the standard list of parameters of A. For any ordering y = yl . . .y, of FV(A) the sentence VyA is called a universal closure of A.
r.
The set of all universal closures of formulas from a set I? is denoted by Note that in all the logics considered in this book the universal closures of the same formula are always equivalent, so we can deal only with standard universal closures, i.e. with {VA I A E I?) instead of
r.
83
2.2. FORMULAS
For a subformula QxB of a formula A (where Q is a quantifier), every free occurrence of x in B, as well as the first occurrence of x, is called referent to the first occurrence of Q. More precisely:
Definition 2.2.6 Let (A, i , QxB) be an occurrence of QxB i n A. If (B, j, x) is a free occurrence of x i n B , then ( A ,i j 2, x) is an occurrence of x referent to (A, i, Q); (A, i 1,x ) is also referent to (A, i, Q). All variable occurrences i n A referent to the same occurrence of a quantifier are called coreferent (or correlated). W e also say that an occurrence of a quantifier binds all referent occurrences of variables.
+ +
+
Now the reference structure of a formula A can be defined as a function sending every bound occurrence of a variable in A to the occurrence of a binding quantifier. But the following definition is more convenient.
Definition 2.2.7 Let A be a formula of length n. The reference function of A is a function r f A defined on the set {i
I 1 5 i 5 n & (A, i ,x) is a bound variable occurrence for
some x)
such that r f ~ ( i = ) j whenever (A, i , x) is referent to (A, j, Q) (for some variable x and a quantifier Q). So r f sends ~ positions of bound variables to positions of their binding quantifiers.
Definition 2.2.8 Let be a new symbol, regarded as an extra variable ('joker'). The stem of a formula A is a formula A- obtained by replacing every bound occurrence of every variable i n A with a. The scheme of A is a pair .A, := (-4-7 T ~ A ) Thus BV(A-) {a), FV(A-) = FV(A). Reference functions can be represented graphically, by connecting occurrences of quantifiers with referent variable occurrences. For example, for A := Vx(P(x) 3 3yQ(y, x)) the reference function is pictured as follows:
and the scheme as follows:
(Note that the stem itself has a different reference function!)
CHAPTER 2. BASIC PREDICATE LOGIC
84
Remark 2.2.9 Instead of this graphic representation, one can number quantifier occurrences and add corresponding superscripts to referent variable occurrences, cf. [Kleene, 19671. Alternatively, the scheme of a formula can be defined by induction as follows.
Definition 2.2.10
.A,:= A for A atomic;
, ( A* B )
-
:=
O j A := Dj
(A* for * E {v,A, 3 ) ; A;
& (for Q E rence of x with
{V, 3)) is obtained from Qx.A, by replacing every occurand connecting it with the first occurrence of &.
*
Remark 2.2.11 Strictly speaking, this definition shows us how to reconstruct a graphic representation of .A,. In terms of functions, it means that ~ f ( ~ is* ~ r f with ~ some shifts, and so on. Such an explication does the union of r f and not seem useful however. Schemes can also be defined by induction without appealing to formulas:
Definition 2.2.12
Every atomic formula is a scheme. If S1, Sz are schemes, then (S1 * S2) is a scheme. If S is a scheme, then OjS is a scheme. If S is a scheme, x E Var, then there is a scheme obtained from QxS by replacing every occurrence of x with and connecting it with the first occurrence of Q. It is obvious (a strict proof is by induction) that for any scheme S in the sense of this definition, there is a formula A such that S = . So in our syntax we can deal with schemes rather than formulas. In a systematic way this approach is developed in [Bourbaki, 19681.' One can even argue that schemes better correspond to human intuition about first-order logic.8 But in our book, we prefer to keep to the traditional notion of a formula and use schemes only incidentally, for technical purposes. 'with minor differences - instead of quantifiers Bourbaki uses the E-symbol (denoted by and defines 3xA as an abbreviation for [ T % ( A ) / X ] A . 8 ~ a t u r a language l does not use bound variables and hides them in the reference structure, cf. the sentence Every triangle has at least two acute angles. T)
~ )
2.3. VARIABLE SUBSTITUTIONS
2.3
85
Variable substitutions
It is well-known that a 'logically correct' variable substitution is not a simple replacement of variables, due to possible variable collisions. For example, if A is 3y (x # y), and we want the formula VxA > [y/x]A to be (classically) valid; it is incorrect to define [y/x]A as 3y (y # y). Many authors consider such substitutions as 'bad' and simply do not allow them; formally, [Y/x]A is a 'good' substitution if free occurrences of x are not within the scope of any quantifier over y; as they say, y is free for x in A, cf. [Kleene, 19521, [Kleene, 19671, endel el son, 19971. But this restriction does not help for defining formula substitutions, because e.g. [3yQ(x,y)lP(x)lP(y) should be [Y/xI~YQ(x, Y), and the latter substitution [ylx] is 'bad'. A well-known way to solve the problem is renaming of bound variables, cf. [Kleene, 19631. For example, in the formula 3y (x # y) we can rename the bound y by ,z and define [y/x](3y (x # y)) as 32 (y # z). But this variable z can be chosen in many different ways - it can be arbitrary except for the original y. If the result of a substitution should be unique, we have to fix one of these options. For example, we can always take the first variable allowed for renaming (in the list of all variables) [Kolmogorov and Dragalin, 20051. But this definition is technically inconvenient and rather unnatural , because the alphabetical order of variables is not related to logic at all. So we propose another approach. A substitution is considered as a relation not function; the result of a substitution is unique only up to congruence (correct renaming of bound variable^).^ We regard [y/x]A not as a single formula, but as a member of a certain class of formulas. This resembles the well-known mathematical notation f (x)dx of a primitive function, which is unique up to adding a constant. Using schemes instead of formulas simplifies all the details: two formulas are congruent if they have the same scheme. A formula is called clean if all its quantifiers bind different variables and none of its bound variables is free. Every formula A can be transformed into an equivalent clean formula AO, without bound occurrences of x or y. Then we define [y/x]A as [y/x]AO. Since there are no variable collisions, the latter substitution is made by a straightforward replacement. Now we pass to the details. Let x = (XI,. . . , x,) be a list of variables (n > O).1° Later on we use the following notation: gCongruence corresponds t o a-equivalence in X-calculi. 1°1f there is no confusion, we also denote the list ( x l , . . . , x,) by X I . . .x,.
CHAPTER 2. BASIC PREDICATE LOGIC
86
r ( x ) for the set {xl,. . . ,x,}; z E x for ' z occurs in x', i.e. for z E r(x);ll xy for the concatenation of the lists x, y; x n S for 'the sublist of x containing the elements of S'; x - S for 'the sublist of x obtained by removing the elements of S', etc.
A list is said to be distinct if all its members are different. If x = ( x l , .. .,x,), y = (yl,. . . ,y,) are lists of variables and all x l , . . . ,x, are distinct, we define the variable substitution [y/x]as the following function Var Var:
-
Note that the same definition can be given in the case when some of the xi are equal, but x sub y. Of course a variable substitutior, is nothing but a function V a r ---+ V a r that changes only finitely many variables. So a composition of substitutions is a substitution. For a list of variables z = zl . .. z,, put
We also use the dummy substitution
[/I,
which is the identity function on V a r .
Definition 2.3.1 A variable transformation of a (distinct) list x to y is the finite function (the set of pairs) { ( x l ,yl), . . . ,(x,, y,)}; this function is denoted by [xI-+ If a transformation [x H y] is bijective (i.e. y is distinct), it is called a variable renaming. If also V ( A ) C r(x) for a certain formula A, [x H y] is called a variable renaming in A. A bound variable renaming in A is a variable renaming in A fixing all parameters of A.
YI.
Note that [xH y] is a variable renaming iff the corresponding substitution [y/x]is a permutation of V a r . For a set of variables S, [x H yIs denotes the restriction of [x ++ y] to r(x)n S . We also use the abbreviations
A transformation [xH y]is called proper if xi # yi for every i . We say that a transformation [xH y] represents the substitution [y/~].12 Obviously, every substitution is represented by infinitely many transformations, but only one of them is proper. It is easy to describe how a transformation [xI-+ y]acts on (modal) predicate formulas. Let A[x H y] be the result of simultaneous replacement of all (both llThis notation is only occasional. 12Sometimes transformations are also called 'substitutions', but we avoid this terminology.
2.3. VARIABLE SUBSTITUTIONS
87
free and bound) occurrences of xi in A with yi (for i = 1,. . . ,n). We can say that A[x H y] is obtained from A by a 'straightforward' renaming of variables.13
Lemma 2.3.2 Let [x I-+ y'], [y H z] be variable transformations such that y' = y . a for some sujection u . Then
(2) for any predicate formula A such that V ( A ) r(x) (A[x H yl])[y H Z] = A[x H z . a ] . In particular, (A[x H y])[y H X] = A if
[X H
y] is a variable renaming in A.
Proof
(1) Trivial: [x H y'] sends xi to y,(i), and next [y +-+ Z] sends
to
(2) Also obvious, formally, one should argue by induction on the length of A.
w In a more general situation the following holds (the proof is similar).
Lemma 2.3.3 Let [x H y'], [y H z] variable transformations such that r(yl) 2 r ( y ) . Then for any predicate formula A,
The formulas A and A[x ++ y] may be not logically equivalent in classical logic. For example, this is the case for A = 3x1 (P(x1) A -P(yl)) and 4 x 1 H yl] = 3yl(P(yl) A TP(Y~)). Later on we will describe 'admissible' transformations that do not affect the truth values of formulas, cf. Lemma 2.3.19.
Lemma 2.3.4 Let A be a predicate formula, [x H y] a variable renaming in A. Then (1) free (respectively, bound) occurrences of variables in A correspond to free
(bound) occurrences of variables in A[x H y]: (A, i, xj) is free (bound) iff (A[x ++ y], i, yj) is free (bound); (2)
r f A =rf~[x++~].
Proof
By induction. We denote A[x H y] by A'.
If A is atomic, then A' is atomic, so all variable occurrences are free, and the claim is trivial. 13The notation [y/x]A is reserved for a 'correct' variable substitution with renaming of bound variables, see below.
88
CHAPTER 2. BASIC PREDICATE LOGIC If A = ( B * C), then A' = (B' * C'). Variable occurrences in A coming from B are of the form (A, i 1, xj) where (B,i, xj) is an occurrence in B. Then
+
+
(A, i 1, xj) is free (bound) iff (B, i , xj) is free (bound) (by Definition 2.2.4) iff (B', i, yj) is free (bound) (by induction hypothesis) iff (A', i 1, yj) is free (bound) (by 2.2.4).
+
+
Moreover, a bound occurrence (A,i l , x j ) is referent to a quantifier occurrence (A, k 1,Q) iff (B, i, xj) is referent to (B,k, Q ) :
+
+
In fact, according to 2.2.5, this happens iff either i = k 1 or this occurrence of x j in A comes from a free occurrence in a subformula D starting at position k 3 (more precisely, A, k 1, QxjD) is an occurrence in A and (D,i - k - 1, xj) is a free occurrence in D).
+
+
By the same reason,
= r fB' by induction hypothesis, it follows that Since r f ~
A similar argument applies to variable occurrences in C; we leave formal details to the reader. The case A = O k B is also left to the reader If A = QxkB, then A' = QykB1. Now if j # k, then as above, we obtain: (A, i 2, xj) is free (bound) iff (B, i, xj) is free (bound) iff (B', i, yj) is free (bound) iff (A', i 2, yj) is free (bound). Note that yj # yk, since [x ++ y] is a bijection. A bound occurrence (A, i 2, xj) is referent to (A, k 2, Q1) iff (B, i, xj) is referent (B, k, Q1):
+
+
+
+
2.3. VARIABLE SUBSTITUTIONS
and similarly,
r f A , ( i + 2 ) = k + 2 @ r f B , ( i ) =k . Hence
~ f A (+i 2 )
= rfA'(i + 2 ) .
And if ( A ,i ,xk) is an arbitrary occurrence, it is bound, as well as (A',i ,yk). Then r f ~ ( i=) 1 ~ r f ~ = ~ l(. i )
Definition 2.3.5 Two predicate formulas A, B are called congruent (notation: A B ) if they have the same scheme. Definition 2.3.6 Let A be a predicate formula. A bound variable renaming in A is a variable renaming in A fixing all parameters of A. So this is a bijective transformation [ xH y] such that V ( A )C_ r ( x ) and xi = yi for xi E F V ( A ) .
Definition 2.3.7 We say that B is strongly congruent to A (notation: A B ) i f there is a bound variable renaming [xH y] in A such that B = A[x H y]. Lemma 2.3.2 shows that
is an equivalence relation on formulas.
Lemma 2.3.8 If [ x++ y] is a bound variable renaming in A, then A y]. Thus strong congruence implies congruence.
A[x H
Proof By Lemma 2.3.4 the reference functions coincide. As for the stems, A and A- may differ only in bound variables. Since xi = yi for xi E F V ( A ) , A and A' = A [ x H y] also differ only in bound variables. Now consider bound variable occurrences. We have: occurs in A- a t position j iff some xi occurs in A at position j iff some yi occurs in A' at position j (by 2.3.4) iff occurs in (A1)-at position j . Thus A- = (A1)-,and eventually A A A'.
Lemma 2.3.9 (1) If A = ( B * C ) A A', then A' = (B' * C') for some B' g B , C'
(2) If A = QyB A A' and y 6 B V ( B ) , then for some variable z and formula B', A = QzB' and B' B [ y ct z].
C.
6FV(B)
90
CHAPTER 2. BASIC PREDICATE LOGIC
Proof (1) Let A = ( B * C) A A'. Since A' begins with (, it must have the form = (B' *' C'). Then ,A, = ( a * and .A'. = (,B'.*',C'), hence & , * = *', & = .C',by 2.2.2 (or strictly speaking, 2.2.2 implies the coincidence of the corresponding stems; the reference functions coincide, because they are inherited from A and A').
a,
(2) Assume A = QyB A A' and y 6 BV(B). By definition, is obtained from Q y a by replacing y with and adding connections to the first Q. = it follows that A' has the form QzB' (otherwise A' does not If begin with Q) and .B'.=&[y 21, (U)
A,
++
where , also z $? FV(B). Since [y H z] means (B- [y ++ z], r f ~ ) and ~ rfB[,,,]. It is also clear that B-[y H z] = (B[y H y # BV(B), r f = 21)- (a formal proof is routine by induction); therefore
FYom
(t9,(flu) we have B' A B[y H z ] .
Definition 2.3.10 A formula A i s called clean if there are n o variables both free and bound i n A and the number of occurrences of quantifiers equals the number of bound variables, i.e. diflerent occurrences of quantifiers i n A bind different variables i n A. This is equivalent to the following inductive definition: Definition 2.3.11
Every atomic formula is clean. If formulas A, B are clean, * E { A , V >), and BV(A) n V(B) = BV(B) n V(A) = 0 , then (A * B ) is clean. If A i s clean, x E V a r , and x
6BV(A), then QxA is clean.
Lemma 2.3.12 If [x H y] i s variable renaming i n a clean formula A, then A[x H y] i s clean. Proof By induction. Again we denote A[x H y] by A'. If A is atomic, then A' is atomic. In this case, if [x H y] is bound for A, then A' = A. Suppose A = ( B * C), B , C are clean,
2.3. VARIABLE SUBSTITUTIONS
B', C' are clean by induction hypothesis. By Lemma 2.3.4,
Hence BV(B1) n V(C1) = 0 . Similarly we obtain BV(C1)n V(B1) = 0, and thus A' = (B' * C') is clean. Suppose A = QxiB, B is clean, xi # BV(B). Then A' = QyiB', and by induction hypothesis, B' is clean. By Lemma 2.3.4, yi $ BV(B1). Thus A' is clean by 2.3.11.
L e m m a 2.3.13 Let A be a clean formula, [x ++ y] a variable transformation such that BV(A) n r(xy) = 0.Then A[x H y] is clean. Proof By induction, similar to the previous lemma. We leave this as an exercise for the reader.
L e m m a 2.3.14 If A A A' and A,A1 are clean, then A Proof
& A'.
By induction on the complexity of A, we prove the claim for any A'
If A is atomic and A A A', then A' = A, and the claim is trivial If A = ( B * C), then by 2.3.9(1), A' = (B' * C') for B B B' and C & C' by the induction hypothesis, i.e.
B', C A C'. So
for some bound variable renamings [x H y], [z H t]. Obviously, we may assume that r(x) = V(B), r(z) = V(C). Then [x H y] U [z H t] is a function. In fact, the lists x, z do not have common bound variables, since A is clean. Both functions [x H y], [z ++ t] accord on free variables as they fix them. The same argument shows that [y ++ x] U [t H z] is a function, and thus [x H y] U [z H t] is bijective. So we obtain a bound variable renaming that transforms A into A'. If A = QyB is clean, y does not have bound occurrences in B. Then by Lemma 2.3.9(2), A' = QzB', for B' A B[y H z]. Note that B' is clean as a subformula of A'. B is clean as a subformula of A, B[y H z] is clean by 2.3.12 ([y H z] becomes a variable renaming if we prolong it in a trivial way - as the identity function on V(B) - {y)). So by the induction hypothesis B ' g B[y ++ z],
CHAPTER 2. BASIC PREDICATE LOGIC
for some bound variable renaming [ x I--+ u] in B f . Since A' is clean, z @ BV(B1),so [ xH u] fixes z. Thus
(&zB')[xI--+ u] = &z(B[yH z]).
(I)
Hence by applying [ z w y] we obtain
( Q z B 1 ) ( [I--+x U ] 0 [z
y ] ) = QyB.
(2)
Note that [X
I--+
U ] 0 [Z H
y] = [ ( x- Z ) Z
H
(U - Z ) Y ]
is a bound variable renaming in B', so ( 2 ) implies
If A = OiB 4 A', then A' = OiB1,and B A B'. By induction hypothesis, B 2 B', which easily implies A & A'.
Proposition 2.3.15 For any clean formula A,
g ( A )= Proof
A (A)
n
(clean formulas).
Immediate from 2.3.12, 2.3.14, 2.3.8.
Proposition 2.3.16 Every predicate formula A i s congruent t o some clean formula (called a clean version of A). Proof
By induction on the complexity of A.
If A is atomic, it is already clean. If A = OiB and B A Bo for a clean Bo, then obviously A OiBo is clean.
+ OiBo and
If A = ( B * C )and B A Bo, C A Co for clean Bo,Co,then A A (Bo*Co). (Bo* Co)may be not clean, but we can make it clean by an appropriate bound variable renaming. In fact, let BV(Bo) = r ( x ) , and let [ x H y] be a bijection such that r ( y )n V ( C o )= 0 . then B1 := Bo[xw y] is clean by Lemma 2.3.12, and
Similarly there exists C1
Co such that
2.3. V A N A B L E SUBSTITUTIONS Since
B V ( B l )n FV(Cl) = r ( y )n FV(Co)= 0 , by 2.3.11, it follows that (Bl * G I )is clean.
B1 A Bo and C1
Co implies (B1 * C1) A (Bo * Co) A A.
If A = QxB and B A Bo for a clean Bo, there are two cases. If x E F V ( B ) ,then also x E FV(B0) (since free variable occurrences in a formula are at the same positions as in its stem, and B- = Bo). Thus x # B V ( B o ) ,so QxBo is clean. Now
so A A QxBo. Finally, if x @ F V ( B ) , then also x
x
E
e FV(Bo). However, it may be that
BV(B0).
So we rename x into a new variable y @ V ( B o ) . Then [x H y] can be prolonged to a bound variable renaming in QxBo; thus &xBo A &y(Bo[xH y ] ) by 2.3.8, and Bo[x ++ y] is clean by 2.3.12. Hence the formula Qy(Bo[x y ] ) is clean by 2.3.11, and we have proved that it is congruent to A.
The next proposition gives us a convenient inductive definition of congruence that does not appeal t o schemes.
Proposition 2.3.17 Congruence is the smallest equivalence relation N-modal predicate formulas with the following properties: (1) QxA
(2) A
-
-
B
-
between
&Y(A[x Y ] ) for a: @ B V ( A ) , y q! V ( A ) , Q E {V, 3); QxA
-
&xB for & E {V, 3 ) ;
Proof
First note that congruence really has properties (1)-(4). ( 2 ) , (3), ( 4 ) follow from the inductive definition of schemes 2.2.12. (1)follows from 2.3.8. In fact, let z be the list of all other variables occurring in A, then [xz ++ yz] is a bound variable renaming in A, and it transforms QxA into Qy(A[xH y]). So these formulas are congruent. Now consider an arbitrary equivalence relation satisfying (1)-(3)and show that congruent formulas are --related. So we prove that for any B
-
CHAPTER 2. BASIC PREDICATE LOGIC
94
by induction on IAl. If A is atomic, then obviously A A B implies A = B, so the claim is trivial. If A = (A1 * A2), then as we saw in the proof of 2.3.14, for some B1 Al, B1 A A2 we have B = (B1 * B2). Then by induction hypothesis, Ai Bi, and thus A B by (3). We skip a similar simple case when A = OiA1. Finally, suppose A = QxC A B. Since A=&, it follows that B = QyD for some D l y. By Proposition 2.3.16, C is congruent to a clean formula Co. Next, by a bound variable renaming (Lemmas 2.3.8, 2.3.12) we transform Co into a congruent clean formula C1 such that s,y $ BV(C1). Since ICI < IAl, we have C C1 by the induction hypothesis, and hence A = QxC QxCl by (2). We also have QyD A QxC QxC1,
-
-
N
where the latter readily follows from C A C1. Since congruent formulas have the same occurrences of parameters, we obtain that y @ FV(QxCl), and thus y $ V(QxC1) by the choice of C1. Hence
which implies y]. D AC~[XH By induction hypothesis, D - C1[x++yl, So
B := QYD
-
e y c l [ ++ ~ yl
by (2). On the other hand, by ( I ) ,
and we already know that A
-
QxC1. Therefore A
-
B.
The characterization given in Proposition 2.3.17 resembles the definition of a-equivalence in A-calculi, cf. ... It is worth noting that there exists an equivalent description of congruence (or a-equivalence) via variable swaps, cf. [Gabbay and Pitts, 20021: Lemma 2.3.18 Congruence is the smallest equivalence relation modal predicate formulas with the following properties:
-
N
between N -
B i f l B i s obtained from A by replacing some of its subformula QxC with D = Qy(C[xy yx]), where y $! FV(C), Q E {V, 31,
(1') A
N
(2)-(4) from 2.3.17.
2.3. VARTA BLE SUBSTITUTIONS
Proof
Congruence satisfies (l'), since for y
# FV(C),
where z is a list of all other variables from V(C), and thus
by 2.3.12. It also satisfies (2)-(4) as noted in 2.3.17. The other way round, (1') obviously implies (1) from 2.3.17, since A[xy ++ yx] = A[x ++ y] for y # V(A). Thus every equivalence relation satisfying (l'), W (2)-(4) contains by 2.3.17. To complete the whole picture, let us also give a description of a congruence class of a clean formula in terms of transformations. However this description will not be used in further studies.
Lemma 2.3.19 Let A be a clean formula. Then every formula congruent to A can be obtained by a transformation [x H y] such that
for any subformula of A of the form QxiB, where Q is a quantifier, the following holds: (1) if x j E F V ( B ) and j
# i then yj # yi;
(2) if yi E F V ( B ) then yi E r(x).
Proof [Sketch]If A A C , then all parameters of these formulas are at the same positions. Since A is clean, all occurrences of every bound variable xi in A are coreferent and since r f A = r f C , in C they are replaced with the same bound variable yi. So C = A[x H y] for a list x of bound variables and some y. Now suppose QxiB occurs in A. If x j has a free occurrence in B and j # i, this occurrence of xi corresponds to an occurrence of yj in the formula Qyi(B[x H y]) occurring in C. So yj # yi, since otherwise r f c # r f ~ Thus . ( 1 ) holds. If yi has a free occurrence in B and yi # r(x), this occurrence remains free in B [ x ++ y], and thus referent to the first occurrence of Q in Qyi(B[x H y]), so r fc # r fA. Thus (2) holds. On the other hand, transformations described in the statement of the lemma, preserve the scheme of A. In fact, any occurrence of xi free in a subformula B of QxiB occurring in A becomes an occurrence of yi in B[x H y]. This occurrence of yi is also free, since otherwise it is referent t o another occurrence of a quantifier Q' over yi in C: C = . . . Qyi .. . Qfyi.. .yi . . . . . .
B[x++yl
CHAPTER 2. BASIC PREDICATE LOGIC
96
-
But this Q1yi comes from some occurrence of &'xj in A, and j clean. Then A = . . . Qxi ... Q'xj . . .xi . . . . .. . ..
# i , since A is
Now xi is free in a subformula D of Q1xjDoccurring in B , while yi = yj. This contradicts the condition (1). Also every occurrence of yi free in a subformula E [ x H y] of QyiE[xH y] occurring in C comes from a free occurrence of xi in E from a subformula QxiE of A. In fact, otherwise yi E F V ( E ) .
(since A is clean, xi @ B F ( E ) ) ,so yi E r ( x ) by (2). Then yi = xj for j # i (since yi E F V ( E ) ) ,and yi = yj, since [ x H y] fixes yi. This again contradicts (1). Therefore A A[x H y ] . W Now we can define how variable substitutions act on formulas.
Definition 2.3.20 For a variable transformation [ xH y] and a scheme S , we define S [ x H y] as the result of replacing every occurrence of xi in S by yi. Thus
A [ x H Y]= (A-[xHY ] , T ~ A ) . Lemma 2.3.21 Let A be a formula, [ x H y] a variable transformation such that r ( x y )n B V ( A ) = 0 . Then
Proof
Easy by induction on [At.We consider only the case A = QzB.
since a does not occur in x. Now by induction hypothesis
thus
, & z B , [ xH y] =
B [ x H y] =
&Z
But
Q z ( B [ x Yl) = A[x YI, again since z does not occur in x. Hence the claim follows. ++
+ +
2.3. VARIABLE SUBSTITUTIONS
97
Definition 2.3.22 Let A be a predicate formula, [y/x] a variable substitution.
Then we define [y/x]A as an arbitrary formula B such that A [ x H y] =
a.
Let us show soundness of this definition. Lemma 2.3.23 Every predicate formula A has a clean version A" such that
~ ( x Y )n BV(AO) = 0. Then [y/x]A A" [x +-+ y]. Proof To obtain A", take an arbitrary clean version and make an appropriate bound variable renaming. Then
by Lemma 2.3.21. From the definition we readily obtain Lemma 2.3.24 A
A' + [y/x]AA [y/x]A1
Note that [y/x] and [x H y] do not depend on the ordering of x; in precise terms, if 1x1 = n, a E Y,, then [ x .a H y - a] = [X H y]. lemma Therefore congruent formulas have the same substitution instances under every variable substitution. The next lemma contains some simple properties of variable substitutions. Lemma 2.3.25 Forany predicate formula A, substitutions [y/x],[yl/x'],[y'l/x"],
[x/u], [z/u] and quantijier Q
(4) [y"/x"][yl/xl]AA [y/x]A,where [y/x] = [yfl/x"].[yl/x'] (the composition of functions on V a r );
CHAPTER 2. BASIC PREDICATE LOGIC
98
(11) [ y / x ] A9 [ynlzn].. . [ ~ l / z l ] [ z n /-~..n[ z] l / x l ] A for distinct variables z l , . . .,z., 6 F V ( A ) U r ( x y ) ; so every variable substitution in a formula can be presented as a composition of substitutions of the form [ y l x ] (simple substitutions);
(13) Q y [ y / x [ AA Q x A if y is distinct, r ( y ) n F V ( A ) = r ( ~n)r ( x ) = 0 .
Proof ( 1 ) Since [ y / x ] A= A [ x I+ y ] , it follows that F V ( [ y / x ] A )= V(,Alx t-i y ] ) (or, to be more precise, F V ( A V [ xH y ] ) - ( 0 ) ) . So we should take the set V ( & = F V ( A ) and replace every xi occurring in this set with the corresponding yi; this gives us exactly
( 2 ) Trivial. ( 3 ) The case y = x is trivial, so suppose y # F V ( A ) . Let A" be a clean version of A such that B V ( A O )n r ( x y ) = 0.As we know
hence by 2.3.17,
(#I) QY [YIXIAA Qy(AO[X t-i Y ] ) . Since y
# F V ( A ) , by 2.3.17 we also have
Now obviously, A A A0 implies
(by 2.3.17 or just by the definition of a scheme). So from ( # I ) , (#2),(#3) we obtain Q y [ y / x ] A QxA. (4) Let A" be a clean version of A such that B V ( A O )n r(xx'x"yy'y") = 0.
Then
[ y l / x ' ] AA AO[x'H y']. By Lemma 2.3.13, the latter formula is clean, and obviously its bound variables are the same as in A'. So
[y"/x"][ y l / x ] AA [y'l/x'l](AO[x' t - i y']) 2 ( A 0[x' H y'])[x'l t-i Y"].
2.3. VARIABLE SUBSTITUTIONS
99
Since [yl/xl]= [ylt/xlt],we can always add variables to both x1 and y', so we may assume that r(xl) FV(A)(= FV(AO)),
>
We can also write
A0[xl H yl] = A0[u H V] , where [U H V]
:= [XI
H
yIA,
since x1 does not contain bound variables of A. Then r(u) = FV(AO),so r(v) = FV(AO[u ++ v]). Similarly we have
where [W H Z]
:= [XI'
HY"]~O
[U+.+VJ.
SOr(w) = r(v), and thus v = w - a for some surjective a E CI.wl,l,l.Hence
by Lemma 2.3.2. On the other hand, for [y/x] r(x) F V ( A ) . Then we have
>
=
[y'/xl] 0 [yI1/x"] we may assume that
and it remains to show that
In fact, suppose xj = ui E FV(A). Then
by the choice of [u H v],
[W ++
z].
(5) By (4), it suffices to check that [y/x] . [z/u] and [[y/x]z/u]coincide on parameters of A. In fact, the first substitution sends every ui to zi and next t o [y/x]zi and every xj @ r(u) to yj. So if r(x) n FV(A) E r(u), all parameters of A beyond r(u) remain fixed. The second substitution sends ui directly to [y/x]zi and also fixes other parameters; thus the claim holds. (6) This is a particular case of (5) when x = z. Then [y/x]z = y.
(7) Apply (5) to the case y := v, x := u, z := u, u := x. Note that
FV(A) n {u} C r(x) iff {u} G -FV(A)
u r(x).
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(8) Consider a clean version A" of A such that r(xyz) n BV(AO)= 0.Then QzA G &zAO and &zAOis clean by 2.3.11. So
(8.1) [y/x]QzA
(&zAO)[xH Y] = &z(AO[x++ Y]).
On the other hand,
AO[x
YI
4%
[YIxIA,
hence
(8.2) Qz(AO[xH y])
&z[y/x]A
by 2.3.16. Now (8) follows from (8.1) and (8.2).
(9) Let A', B0 be clean versions of A and B such that
BV(AO)n FV(A) = BV(BO)n F V ( B ) = 0. Then (A0 * B O )is a clean version of (A * B ) and by 2.3.21 we have:
[y/x](A * B ) A (A0 * BO)[xH y] = (AO[xH y] * B O [ xH y]) ([ylxIA * [ylxIB). (10) Note that in this case
[y"/x"] . [y1/x'] = [y"y'/x"xl] and apply (4)
(11) Since zi @ FV(A), from (6) we obtain
(#)
[ylzl[zlxlA
[ylx[A
By induction from (10) we also have
(tin)
[zlxlA A [znlxnl . . . [zl/xl]A,
since r ( x ) n r(z) = 0 and z is distinct. In the same way from (10) we obtain
since r(z) n r ( y ) = 0 and z is distinct. Now (11) follows from (#),(##),(###)and 2.3.24.
2.3. VARIABLE SUBSTITUTIONS (12) Follows from (8) by induction on
101 121.
For the step, suppose z = zlzl and
then by 2.3.17(2),
Qzl [y/x]QzlA
QZ[Y/X]A.
On the other hand, by (8)
[y/x] QzA A Qzl [YIxIQz'A; hence (12) follows.
(13) Apply induction on 1x1 = lyl. The base follows from (3). For the step, suppose y = yly', x = xlx' and
Qyl[y'/x']A A QxlA. Then by 2.3.17(2),
On the other hand, by (lo),
hence by 2.3.17(2),
BY (1217
Q Y ~ [ Y ~ / X ~ I [A Y ~[ YI ~ I~XI ~ AI Q Y ~ [ Y ~ I X ~ I A , hence by 2.3.17(2) and (3)
Now (13) follows from (*2), (*3), and (*I).
Exercise 2.3.26 Describe the composition of substitutions (or of corresponding transformations) explicitly.
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Formulas with constants
2.4
Although our basic languages do not contain individual constants, we will need auxiliary languages with constants. So let D be a non-empty set; we assume that D n Var = 0 . Let L N ( D ) be the language CN expanded by individual constants from the set D . Formulas of the language LN(D) (respectively, Co(D)) are called N-modal (respectively, intuitionistic) D-formulas; the set of all these formulas is denoted by M F ~ = ) ( D ) ~ ~ (respectively, IF(')(D)). Obviously, every predicate formula (in the ordinary sense, i.e. without extra constants) is a D-formula. A D-sentence is a D-formula without parameters; MS~=)(D) and IS(=)(D) denote the sets of D-sentences of corresponding types. Definition 2.4.1 Let x = (XI,.. . ,x,) be a list of distinct variables, a = ( a l l . . ., a,) a list of constants (indiwiduals) from D (not necessarily distinct). Then the D-transformation [x H a] is a finite function {(XI,ax), . . . , (x,, a,)} sending every xi to ai, i = 1,.. . , n. The D-instance [a/x]A of a D-formula A under [x H a] is obtained by simultaneous replacement of all free occurrences of X I , . . . ,x, in A respectively with a l , . . . ,a,. Strictly speaking, [a/x]A is defined by induction on [A[: [a/x]P(y) := P([a/xIy), where [a/x]y is a tuple z such that lzl = [yl and for any j,
[a/x]P:= P i f P EPLO, [a/x](B * C) := ([a/x[B c [a/x]C) if
* E {V, A, >},
So we can also denote [a/x]A by A[x H a] if r(x) n BV(A) = 0.Normally we use the notation [a/x]A in the case when both A is a usual formula and [a/x]A is a D-sentence (which is equivalent t o FV(A) 2 r(x)). A formula A is called a generator of every D-sentence [a/x]A. . . For D-formulas we define schemes, clean versions and congruence in the natural way, briefly, by M F ( = ) ( D ) . 15Recall that jci is obtained by eliminating xi from x; similarly for bi. 140r
2.4. FORMULAS WITH CONSTANTS L e m m a 2.4.2 (1) A A B + [a/x]A A [a/x]B for any D-transformation [x H a] and D-formulas A, B.
(2) If x is a distinct list of variables 1x1 = n, a E Dn, then for any predicate formula A, for any a E T, [(a - (.)/XI A = [ a / ( x . a-I)] A.
(3) For any predicate formula A, for any distinct list x y
-
[a/yl [y/xI A A la/xl A.
(4) Let x , z be distinct lists of variables, 1x1 a : I,
Proof
= n , lzl = m 5 n, and let I,. Let A be a formula such that r ( z . a ) n BV(A) = 0 . Then
[a/.] [(z . a ) 1x1A A [(a . (.)/XI A.
(1)It is clear that
[alx]A = d x H a] (a strict proof is by induction). So = .B.implies [a/x]A = [a/x]B. (2) Note that [x ++ a . a] = [x . a-l H a] - each of these maps sends xi to a,(i), and %,-I (j) t o a j . Now the claim follows from 2.4.1. (3) As noted above, constant substitutions respect congruence. So we can prove the claim for a clean version A" of A, where BV(AO)n r(xy) = 0 . In this case it is equivalent t o ~
~
(A0[x H y] ) [y ++ a] = A" [x H a]. This holds, since obviously [X H y] 0
[y H a] = [X H a].
But again, a strict proof is by induction. (4) Similar to (2). Consider a clean version A0 of A, with BV(AO)n r ( x z ) = 0 . Then the claim reduces t o
which follows from
c
We also use a somewhat ambiguous notation A(x) to indicate that FV(A) r(x); in this case [a/x]A is abbreviated to A(a). The abbreviation A(a) is convenient and rather common, but it leads to some confusion: it may happen that a D-sentence B can be presented as [ a l , . . . , a,/xl,. . . , x,]A for different formulas A. For example, P ( a , a ) = [a/x]P(x,x) = [a,a l x , y]P(x, y). Such an ambiguity may be undesirable (cf. Section 5.1), so we will mainly use 'maximal' representations described as follows.
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Definition 2.4.3 A formula A is called a maximal generator of a D-formula B i f B = [a/x]A for some bijective D-transformation [xH a]. Since [a/x]A does not depend on the variables xi that are not parameters of A, in the above definition we may further assume that r(x) FV(A), and thus a is the list of all constants occurring in B .
L e m m a 2.4.4 Every D-formula has a maximal generator. P r o o f Let a = a1 . . .a, be a list of all constants occurring in a D-sentence B, x = x1 . . .x, a list of different new variables for B. The formula A := B[a H x] (obtained by replacing every occurrence of ai with xi) is a maximal generator of
B,since xi @ BV(B), and thus [a/x]A = A[x H a] = (B[a H x])[x H a] = B.
a L e m m a 2.4.5 (1) If B = [a/x]A for a formula A and a bijection [x H a], then A = B[a H
XI. (2) If A1,A2 are maximal generators of B, then A2 [y/x]Al for some variable renaming [x H y] (and of course, A1 is obtained from A2 i n the same way).
More precisely, if B = [a/x]A1 = [a/y]A2 for bijections [x H a],[y H a], then A2 A [y/x]Al. (3) A maximal generator of a D-formula B is a substitution instance of any generator of B under some variable substitution.
Proof
(1) We check that
by induction on \ A ( . This is clear for atomic A = P ( y ) , when [a/x]A = A[x H a] (cf. Lemma 2.3.2). All induction steps are routine; let us consider only the case A = QxiB for a quantifier Q. By definition,
Since B is a usual formula and [x [ailxi]B; thus
H
a] is a bijection, ai does not occur in
by the induction hypothesis. Therefore (1) holds for A. (2) If A1 is a maximal generator of B, then B = [a/x]A1 for some bijection [x H a]. Similarly, B = [a/y]A2 for a bijection (y H a]. Now let A: be a clean version of A1 such that BV(A7) n r(xy) = 0 .
2.5. FORMULA SUBSTITUTIONS
By 2.4.2(1), A1 g A; implies B = [a/x]Al
[a/x]Ai = A; [x H a].
Hence B [ a H X]
(Ay[x H a])[a H X] = A;.
Similarly, there exists a clean A; 2 A2 such that
Therefore
and the latter formula is [y/x]A1 by 2.3.22. (3) Let C be a generator of B, thus B = [b/z]C for some [z H b]; and let A be a maximal generator of B, with B = [a/x]A, for a bijection [x H a]. We may assume that r(x) FV(A), r ( z ) F V ( C ) and r(b) = r ( a ) is the set of all constants occurring in B. Since a is distinct, every hi equals to some a j , so b = a - Tfor some surjective map r : I, +I,. Thus B = [ a .r / z ] C = [a/x]A,
c
and we have by 2.4.2(4) [a.T/z]CA [a/x][x.r/z]C.
Hence by (2), A
2.5
[x . r/z]C.
Formula substitutions
Definition 2.5.1 A (simple) formula substitution is a pair (C, P ( x ) ) , where C is a predicate formula, P ( x ) is an atomic equality-free formula. The substitution ( C , P ( x ) ) is usually denoted by [C/P(x)]. More exactly, [C/P(x)] is called an MFN-, (MF,'-, I F - , IF=-) substitution zf the formula C is of the corresponding type. Definition 2.5.2 For a substitution [C/P(x)],
is called the set of parameters,
the set of bound variables. A substitution [C/P(x)]iscalled strict i f F V [ C / P ( x ) ]= i.e. if F V ( C ) r(x).
0,
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106
Definition 2.5.3 Let A be a clean predicate formula, S = [ C / P ( x ) ]a formula substitution such that B V ( A ) n F V ( S ) = 0 . Let B be a result of replacing all subformulas of A of the form P ( y ) with [ y / x ] C . Every formula congruent to B is denoted by S A and is called a substitution instance of A under S . More precisely, S A is defined by induction:
S P ( Y ) [YIxIC, S A g A if A is atomic and does not contain P, SOiA A Q S A , S ( A * B ) ( S A* S B ) for * E {v, A, 21, S Q z A QzSA for Q E {V, 3).
A formula S A is called a substitution instance of A, or more exactly, an MF$)- (IF(=)-)substitution instance i f S is an MF,$=)-(IF(=)-)substitution. Due to the assumption B V ( A )n F V ( S ) = 0 , in S A the parameters of S do not collide with the existing bound variables from A. Note that applying S to A does not affect occurrences of equality in A, but may introduce new occurrences if C contains equality.
Lemma 2.5.4 Let A be a clean formula, S = [ C / P ( x ) ]a formula substitution such that F V ( S )n B V ( A ) = 0 . Then
for any variables u , v such that v $! V ( A ) ,u E F V ( A ) , and u ,v $.! F V ( S ) . Proof Since v $.! V ( A ) ,we can prolong [u H v] to a variable renaming in A by fixing all variables from V ( A )- { u ) . So A[u H v] is clean by 2.3.11 with the same bound variables as A, and Definition 2.5.3 is applicable to this formula. Now we argue by induction on IAl. If A
= P ( y ) ,then
A[u ++ v] = P ( [ v / u ] y )S, A
[ y / x ] Cand ,
S(A[uH v ] ) [[v/u]y/x]C. By assumption, u $! F V ( S ) = F V ( C ) - r ( x ) ,so we obtain
S ( A [ uH v ] ) [v/u]SA by applying 2.3.25 (7). Let A = QzB, then S A A QzSB. Hence (1)
[ v / u ] S AA [v/u]QzSB A Q z [ v / u ] S B
by 2.3.25 (8); note that z
# u , since u E F V ( A )
2.5. FORMULA SUBSTITUTIONS
By the induction hypothesis, [v/u]SB A S(B[u H v]), hence &z[v/u]SB A &zS(B[u I+ v])
(2) by 2.3.6. Now by 2.5.3
&zS(B[u I+ v]) A S&z(B[u H v]),
(3)
so from (I), (2), (3) we have [v/u]SA n SQz(B[u H v]). It remains to note that Qz(B[u H v]) = A[u H v], since z
# u. Therefore the claim holds for A.
If A = ( B * C), we can use 2.3.25 (9) and the distribution of S and [u H v] over *. Note that if u does not occur in B (or in C), the main statement trivially holds for B (or C), and the argument does not change. The details are left to the reader. The case A = OiB is trivial.
Lemma 2.5.5 Let A, B be congruent clean formulas, S a formula substitution such that BV(A) n F V ( S ) = BV(B) n F V ( S ) = 0 . Then SA S B .
Proof
By induction on IAl = IBI.
If A is atomic, then A = B , and there is nothing to prove. If A = (Al * A2), then by Lemma 2.3.9(1), B = (B1 * B2) for A1 Az B2. Hence
and SAi A SBi by the induction hypothesis. Eventually SA 2.3.17. We skip the easy case when A = OiAl
Bl,
S B by
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Suppose A = QxAl for a quantifier Q. Since A is clean, x @ BV(A1), so by Lemma 2.3.9(2), for some y @ FV(A1), B1
We may also assume that y $ BV(A1). (Otherwise consider A2 such that y $ BV(A2), then
Al
so A2 can be used instead of Al.) Thus
and SB1 A S(A1[x H y]) A [y/x]SA1 by the induction hypothesis and Lemma 2.5.4 (which is applicable, since y $ V(A1) and x, y @ F V ( S ) by the assumption of the lemma). Hence by 2.3.24(3)
SB = Q y S B 1 2 Qy[y/x]SAl
QxSA1 = SA.
Now we can define substitution instances of arbitrary formulas.
Definition 2.5.6 A substitution instance S A of a predicate formula A under a simple substitution S is an arbitrary formula congruent to SAO,where A" is a clean version of A such that F V ( S ) n BV(AO) = 0. A strict substitution instance is a substitution instance under a strict substitution. Lemma 2.5.5 shows soundness of this definition, i.e. that the congruence class of SAOdoes not depend on the choice of A". Note that according t o the definition, for a trivial formula substitution S = [P(x)/P(x)] and a formula A, SA denotes an arbitrary formula congruent t o A. Lemma 2.5.7 Let [C1/P(x)], [C2/P(x)] be formula substitutions such that C1 A C2. Then for any predicate formula A, [Cl/P(x)]A A [C2/P(x)]A. Proof We denote [Ci/P(x)] by Si. Let A0 be a clean version of A such that FV(Si) n BV(AO) = 0 for i = 1,2. Obviously we can construct such A" by an appropriate bound variable renaming from an arbitrary clean version. Now SiA SiAO,so we show SIAOA S2A0 by induction on IAOI. To simplify notation, put B := A". If B = P ( y ) , then S i B = [y/x]Ci, so S I B A S 2 B follows from 2.3.24. If B is atomic and does not contain P, the claim is trivial.
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109
The induction step easily follows from the distribution of Si over all connectives and quantifiers. E.g. suppose B = QyB1; then y @ FV(Si), so
by 2.3.17. By induction hypothesis,
hence QY SiBi I QY S2B1
by 2.3.17, and therefore SIB
S2B.
All the remaining cases are left to the reader. Now let us consider complex substitutions. Definition 2.5.8 For atomic equality-free formulas Pl(xl), . . . ,Pk(xk) (with different predicate letters PI, . . . ,P k and distinct lists x l , . . . ,xk) and formulas C1, . . . ,Ck we define the complex formula substitution
as the tuple (Cl, . . . ,Ck, Pi(xl), .. . ,Pk(xk)). The set of its parameters and bound variables are respectively
and B V [ c l , . . . ,Ck/pl(xi), . . - Pk(xk)] := ~ ( x .1..xk).
A substitution without parameters is called strict. Now we have an analogue of Definition 2.5.3. Definition 2.5.9 For a substitution S = (C1,. . . ,Ck/l(xl), . . . , Pk(xk)] and a clean formula A such that FV(S) n BV(A) = 0 , a substitution instance SA is defined up to congruence by induction:
SP, (Y) A [y/xi]Ci, S A A if A is atomic and does not contain P I , . . . , Pk, SOiA A QSA, S(A * B ) (SA * S B ) for * E {v, A, I), SQzA A QzSA for Q E {V, 3 ) . We also have an analogue of Lemma 2.5.5.
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Lemma 2.5.10 If A, B are clean formulas, A A B , S is a complex formula substitution and
BV(A) n F V ( S )
= BV(B)
nFV(S) = 0 ,
then S A G S B . Proof The same as in 2.5.5 (including an analogue of 2.5.4).
H
So the following definition is sound.
Definition 2.5.11 For an arbitrary formula A and a formula substitution S , we define S A as SAO,for a clean version A" o f A such that F V ( S ) n BV(AO)= 0 . Hence we readily obtain
Lemma 2.5.12 For any predicate formulas A, B and a formula substitution S ,
AGB=.SAASB. The inductive definition 2.5.9 now extends to arbitrary formulas:
Lemma 2.5.13 Let S be a formula substitution. Then for any formulas A, B ( I ) SOiA A OiSA, (2) S ( A * B ) A ( S A * S B ) for
* E {V,A,>),
(3) S&zA A QzSA for & E {V, 31,z @ F V ( S ) .
Proof
(1) Let A" be a clean version of A such that F V ( S ) n BV(AO) = OiAO is a clean version of OiA, so
0.
Then
By definition, SAOA SA, hence
therefore (1) holds.
(2) An exercise for the reader. (3) Let A" be a clean version of A such that F V ( S ) n BV(AO) = 0 , z @ BV(AO). Then &zAO is a clean version of QzA and
By definition,
SQzA A S&zAOA QzSAO, SAOA SA; hence
&zSAOA QzSA.
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111
This implies (3). The next lemma shows that the result of applying a substitution does not really depend on the names of its bound variables. We prove this only for simple substitutions, leaving the general case to the reader.
Lemma 2.5.14 Let [C/P(x)] be a formula substitution, [x H y] a variable renaming such that r(y) n FV(C) = 0 . Then for any formula A,
Proof If A = P(z), we have
while [z/xlC A [zlyl [y/xIC by 2.3.25(6). If A is atomic and does not contain P , the claim is trivial. Now we can argue by induction on IAl. Put
If A = QuB, we may assume that u @ FV(Sl)(= FV(S2)) -otherwise consider A' A A of the form Qu'Br, where u' $! FV(S1). S2B, then SiA A QuSiB by 2.5.3; hence S1A A S2A by Suppose S I B 2.3.17. Other cases are also based on 2.5.3 and 2.3.17; we leave them to the reader. Lemma 2.5.15 Let S = [C/P(u)] be a simple formula substitution, [y/x] a variable substitution such that FV(C) n r(x) = 0 . Then for any predicate formula A, S[y/x]A A [y/x]SA. Proof Since S respects congruence by Lemma 2.5.12, [y/x] respects congruence by 2.3.24 and both S and [y/x] distribute over all connectives and quantifiers in an appropriate clean version of A (Definition 2.5.3, Lemma 2.3.25), it suffices to consider only the case when A is atomic. The nontrivial option is A = P(z). Then
Now the claim follows by 2.3.25(5).
rn
Lemma 2.5.14 shows that in some cases variable substitutions commute with formula substitutions. The next lemma considers situations where formula substitutions 'absorb' variable substitutions.
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Lemma 2.5.16 Let [ C / P ( x ) ]be a simple formula substitution, A a predicate formula, [ y / z ]a variable substitution such that r ( z ) n F V ( A ) = r ( z )n r ( x ) = r ( y )n r ( x ) = 0 . Then
Note that [ y / z ] Cis defined up to congruence, but the congruence class of [ [ y / z ] C / P ( x ) ]does A not depend on the choice of a congruent version of [ y / z ] C , thanks to Lemma 2.5.7.
Proof The same idea as in 2.5.14 shows that it is sufficient to consider only the case when A = P ( u ) is atomic (and by the assumption, r ( z )n r ( u ) = 0 ) . In this case the claim becomes
The latter congruence follows from 2.3.24. In fact, by 2.3.24,
from r ( z ) n r ( u x ) = 0 , by 2.3.25(10) we have
and similarly from r ( x )n r ( y z ) = 0 ,
Since [ y u / z x ]= [ u y / x z ]this , implies (*). The previous lemma easily transfers to complex sibstitutions:
Lemma 2.5.17 Let
be a formula substitution, A a predicate formula, [ y / x ]a variable substitution such that r ( z )n F V ( A ) = 0 and r ( y z )n r ( x l ,. . . ,x k ) = 0 . Then where So = [[y/z]C1,. . . , [ y / z ] C k / P l ( x l .) ., . , Pk(xk)].Note that r(z)nFV(So)= 0.
Proof Again everything reduces to the case of atomic A. But in this case S W acts as a simple substitution, so we can apply 2.5.16. Lemma 2.5.18 [ [ c / xB] / q ]A A [ c / x ] [ B / qA] for a propositional formula A, a list of proposition letters q, a list of constants c , a distinct list of variables x , a list of predicate formulas B , r ( x )n F V ( A ) = 0 .
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113
Proof The same argument as above reduces everything to the case when A is atomic, i.e. a proposition letter. Then the claim is trivial. Lemma 2.5.19 Every complex substitution acts on formulas as a composition of simple substitutions. More precisely, if S = [Cl, . . . ,Ck/Pl(xl), . . . ,Pk(xk)] is a complex substitution, P,! is of the same arity of Pi and P,! does not occur i n C1,. . . ,Ck for i = 1 , . . .,k , then for any formula A
Proof Since substitutions respect congruence and distribute over all connectives and quantifiers over non-parametric variables (by Lemma 2.5.13), we may prove the claim for a congruent version of A, in which the parameters of S are not bound. In this case it suffices to check the claim for an atomic A. If PI,. . . ,Pkdo not occur in A, there is nothing to prove. So let A = Pi(y). Then by definition SA A [ylxi]Ci , while
So the claim holds. The composition of substitutions reduces to a single (complex) substitution as the following lemma shows.
Lemma 2.5.20 Let So = [Co/Pi(xo)],Sl = [C1,. . . ,Ck/Pl(xl), . . . , Pk(xk)] be formula substitutions. Then for any formula A
where
S2 =
[SoCl, . . . , S o C k / P ~ ( x l ).,. . ,Pk(xk)].
Proof Similarly to the previous lemma, it suffices t o check this only for A = Pi(y). In this case we have
Lemma 2.5.14 shows that a formula substitution acts in the same way after renaming bound variables. So we may assume that r(xi) n FV(C0) = 0 . Then by Lemma 2.5.15
This completes the proof.
Lemma 2.5.21 For any formula substitutions So, S1, there exists a formula substitution S such that for any formula A SoSlA A SA.
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Proof
By Lemma 2.5.15
for some simple formula substitutions S2,. . . ,S,. Then we can use induction W on n and Lemma 2.5.16. Now let us consider parameters of substitution instances. We begin with a simple remark that a strict substitution instance of a formula A may be not a sentence if A is not a sentence. Intuitively it is clear that free variable occurrences in a substitution instance [C/P(x)]A may be of three kinds: (1) those derived from original free occurrences in A if they occur in atoms not containing P (and thus not affected by the substitution); (2) members of y in subformulas of the form [y/x]C replacing occurrences of P ( y ) in A;
(3) those produced by parameters of the substitution wherever P ( y ) is replaced with [y/x]C. Parameters of the first two types are called essential. Here is a precise definition for an arbitrary substitution.
Definition 2.5.22 A parameter z E FV(A) is called essential for a formula substitution [C/P(x)] i f one of the following conditions holds: (1) there exists a free occurrence of z i n A within an atomic subformula that
does not contain P; (2) there exists a free occurrence of z i n A as some y j within an occurrence of P ( y ) , where y = yl . . . y, and xj E FV(C). The set of all essential parameters of A for S is denoted by FVe(S, A). Now let us prove the above observation on parameters of SA in more detail.
Lemma 2.5.23 Let A be a formula, S = [C/P(x)] a simple formula substitution. Then FV(SA) = FV(S) U FVe(S, A) i f P occurs i n A FV(SA) = FVe(S, A) otherwise.
Proof The second claim obviously follows from 2.5.22(1). To prove the first, we argue by induction. We may assume that A is clean, BV(A)n F V ( S ) = 0 . For atomic A there are two cases. (1) A = P(y). Then SA A [y/x]C and by Lemma 2.3.25(1), FV(SA) = F V ( S ) U rng[x H yIc. By definition, rng[x H y ] =~FVe(S,A) in this case, cf. 2.5.22(2).
2.5. FORMULA SUBSTITUTIONS
115
( 2 ) A does not contain P. Then F V ( A ) = F V e ( S ,A). For A = OiB we have S A = O i S B and thus F V ( S A ) = F V ( S B ) . Since P occurs in A iff it occurs in B and F V e ( S , A ) = FVe(S, B ) , the claim follow readily. For A = B * D , where * is a binary connective, the proof is similar to the previous case; note that F V e ( S , B * D ) = F V e ( S ,B ) U F V e ( S , D). For A = QuB we have
F V ( S A ) = F V ( S B ) - { u ) . By induction hypothesis, F V ( S B ) = F V ( S ) U F V e ( S ,B ) , since P occurs in B. So it remains to show that
F V e ( S , A ) = F V e ( S ,B ) - { u ) . In fact, (1) z has a free occurrence in QuB within an atom that does not contain
P iff z # u and z has the same kind of occurrence in B ; ( 2 ) z has a free occurrence in QuB within P ( y ) as described in 2.5.22 (2) iff z # u and z has the same kind of occurrence in B .
From the previous lemma we obtain
Proposition 2.5.24 Let A be a formula, S stitution such that P occurs in A. Then
=
[ C / P ( x ) ]a simple formula sub-
(1) F V ( S ) C F V ( S A ) C F V ( S ) U F V ( A ) , (2) F V ( S A ) = F V ( A ) if S is strict, (3) for any subformula B of A, F V ( S B ) 5 F V ( S A ) U B V ( A ) .
Proof
( 1 ) Note that F V e ( S ,A ) 2 F V ( A ) . ( 2 ) Follows from ( 1 ) .
F V ( S B ) = F V ( S )U F V e ( S ,B ) G F V ( S ) U F V e ( S ,A ) U ( F V e ( S ,B ) - F V e ( S ,A ) ) = F V ( S A ) U ( F V e ( S ,B ) - F V e ( S , A)).
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Now note that according t o Definition 2.5.22, the set FVe(S, B ) - FVe(S, A) contians only variables that are free in B , but not free in A, so this set is contained in BV(A). Hence (3) follows. ¤
Remark 2.5.25 The reader can try to prove this proposition directly without using Lemma 2.5.23. This does not seem easier. Definition 2.5.26 For a set of formulas I? MF$) (respectively, IF(=)), its substitution closure is the set of all their substitution instances of the corresponding kind: Sub(r) := {SA I A E I?, S is an MF$)- (IF(=)-) formula substitution). The universal substitution closure of sures16 of formulas from Sub@').
r
is the set Sub(r) of all universal clo-
Since every N-modal formula is also N1-modal for N1 > N , there is some ambiguity in this definition. But usually it is clear from the context, what kind of formulas we consider.
Lemma 2.5.27 (1) Sub(Sub(r)) = Sub(r) --
(2) Sub(Sub(r)) A =(I?) for a set of sentences r (where A means that these sets are the same up to congruence).
Proof (1) Every B E Sub(r) has the form S A for some A E r and formula substitution S. Then for any formula substitution S1, S l S A E Sub(r) by Lemma 2.5.21. (2) Let us show that for any B E Sub(!?) and for any substitution S1,Vs1~ is congruent to a formula from Sub(l?). We have B = VzSA for some A E r, substitution S and r(z) = FV(SA). We may also assume that F V ( S ) n BV(A) = 0 (otherwise we replace A with a congruent formula). Since A is a sentence, we have FV(SA) = F V ( S ) by 2.5.23. Now let y be a distinct list of new variables such that lyl = lzl and r ( y ) n FV(S1) = 0.Then VzSA A Vy[y/z]SA by 2.3.25(13), and so by 2.5.12
16Cf. Definition 2.2.5
2.5. FORMULA SUBSTITUTIONS Now by 2.5.16 [y/z]SA A S2A for some formula substitution Sz (note that the condition r(z) n BV(S) = 0 holds, since r(z) = FV(S)). Hence by 2.3.17(2) and 2.5.12
Since r(y) n FV(S1) = 0, from 2.5.13(3) it follows that
Eventually, by (*), (**), (* * *) we obtain
and the latter formula is in
Sub(I'), by
2.5.21.
Now let us define 'minimal' non-strict substitution instances of predicate formulas. Let 4 , . . . ,Pk be all predicate letters (besides equality) occurring in a formula A, Pi E PLni, and put
>
where every xi is a distinct list of variables of length ni. Next, let m 0, and let P,! be different (m ni)-ary predicate letters (i = 1 , . . . , k), z = 21. . . zm a distinct list of new variables for A. Then we call P,' the m-shift of Pi; an m-shift of the formula A is Am A [C/P]A, where
+
We also put A0 := A. Obviously FV(Am) = FV(A) U r(z) if A is not purely equational, i.e. it contains some predicate letters other than '='; for purely equational A, Am = A. Note that Am is a substitution instance of any An; this substitution instance is strict iff m 5 n. Lemma 2.5.28 Let S = [C/P(x)] be a formula substitution. Then for any formula A and m 2 0
where we assume that the list of extra parameters z of Am, Cy is disjoint with X.
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Proof
By definition, for a certain substitution S1,
and FV(S1) = r(z). So, as before, we have to check the claim only for atomic
A (without equality). If A = Pi(y), then (for some z)
By our assumption, FV(Sl) = r(z) is disjoint with x. So the claim follows by W Lemma 2.5.15. Exercise 2.5.29 Deduce 2.5.28 from 2.5.20. L e m m a 2.5.30 Every substitution instance S A of a formula A is obtained by a variable renaming from a st&t substitution instance of Am for some m 2 0, e.g. for m = IFV(S)I.
Proof Let us first show this for a simple substitution S = [C(x,y)/P(x)] and a formula A containing P. Let P I , . . . ,Pk be a list of all other predicate letters A be their m-shifts, where m = ly 1. Next, let z occurring in A, and let Pi, . . . ,P be distinct list of new variables. Then
where every Pi(xi) is an atomic formula with distinct xi. In fact, by Lemma 2.5.20
hence by the same lemma
and the latter formula is (congruent to) [C(x,z) / P(x)]A. So [C(x,z) / P(x)]A is a strict substitution instance of Am. Since by 2.5.9(2)
this proves our claim. Now we can apply induction. As we know, every complex substitution is a composition of simple substitutions. So it is sufficient to show that applying a
2.6. FIRST-ORDER LOGICS
119
simple substitution S t o a formula [y/z]SoAm,where S o is strict, can also be presented in this form. Note that we may assume that r(z) n FV(S) = 0 - otherwise change the list of extra parameters of Am. So by 2.5.15 we have
As we have already proved, for B
SoAm
for some k , strict substitution S1 and variable renaming [t ++ u]. By 2.5.28,
where S2 is a strict substitution. Thus
and S1S2 is strict as required. Let A" be a universal closure of Am (for m 1 0); thus A" = Vzl . . . VzmAm for a sentence A. For a set of formulas I?, let Sub(r) be the set of all their substitution instances, Sub(r)the set of all universal closures of formulas from Sub(r). Both these sets are closed under congruence if I' is a set of sentences.
2.6
First-order logics
Definition 2.6.1 An (N-)modal predicate logic (m.p.1.) is a set L such that
MFN
(mO) L contains classical propositional tautologies; (ml) L contains the propositional axioms
(m2) L contains the predicate axioms (for some fixed P, q and arbitrary x, y): (Ax12) (Ax13) (Ax14) (Ax15)
VxP(x) > P(y); P(y) 3 3xP(x); Vx(q > P(x)) > (q > VxP(x)); Vx(P(x) > q) > (3xP(x) > q);
(m3) L is closed under the rules A, (A 3 B)
B
(Modus Ponens, or MP);
CHAPTER 2. BASIC PREDICATE LOGIC
A
- (Necessitation, or 0-introduction); OiA
A
- (Generalisation, or V-introduction)
VxA (for any x E Var).
(m4) L is closed under MFN -substitutions.
Definition 2.6.2 An (N-)modal predicate logic with equality (m.p.l.=) is a set L MFG satisfying (m0)-(m3) from 2.6.1 and also (m4') L is closed under MFG -substitutions; (m5') L contains the axioms of equality (for arbitrary x, y and fixed P):
Definition 2.6.3 A superintuitionistic predicate logic (s.p.l.) is a set L IF such that ( s l ) L contains the axioms of Heyting's propositional calculus H (cf. Section 1.1.2); (s2) = (m2) L contains the predicate axioms; (s3) L is closed under the rules (MP), V-introduction, see (m3); (s4) L is closed under IF-substitutions. Definition 2.6.4 A superintuitionistic predicate logic with equality (s.p.l.=) is a set L IF' satisfying (s1)-(s3) from 2.6.3 and (s4') L is closed under IF'-substitutions; (s5') = (m5') L contains the axioms of equality.
c
Further, by a 'first-order logic' we mean an arbitrary logic, modal or superintuitionistic, with or without equality. Elements of a logic are called theorems, and we often write L I- A instead of A E L.
Definition 2.6.5 A logic L (modal or superintuitionistic) is called consistent ifI$L.
MN (respectively MG, S, S = ) denotes the set of all N-m.p.1. (respectively, N-m.p.l.=; s.p.1.; s.p.l.=). The smallest N-m.p.1. (respectively, N-m.p.l.=, s.p.l., s.p.l.=) is denoted by Q K N (respectively, by Q K E , Q H , QH'). L +I? denotes the smallest m.p.1. containing an m.p.1. L and a set r C M F . This notation is obviously extended to other cases (m.p.l.=, s.p.l., s.p.l.=). It is well-known that every theorem of QH(') can be obtained by a formal proof, which is a sequence of formulas that are either substitution instances of axioms or are obtained from earlier formulas by applying inference rules cited in (s3). The same is true for Q K ~ )but , with the rules from (m3). The notion of
2.6. FIRST-ORDER LOGICS
121
+
a formal proof extends to logics of the form QH(') +I?, Q K ~ )I',with the only difference that formulas from I' can also be used as axioms. By applying deduction theorems, we can reduce the provability in L +I? to provability in L in a more explicit way, see Section 2.8 below. Definition 2.6.6 The quantified version of a modal (respectively, superintuitionistic) propositional logic A is
QA := QKN
+A
(respectively, QA := QH + A ) .
Definition 2.6.7 The propositional part of a predicate logic L is the set of its propositional formulas: L, := L n LN (for an N-modal L); L, := L n Lo (for a superintuitionistic L).
The following is obvious. Lemma 2.6.8 (1) If L is an N-m.p.1. or an s.p.l., then L, is a propositional logic of the corresponding kind.
(2) If L is a predicate logic with equality, then L, = (Lo),.
A well-known example of an s.p.1. is the classical predicate logic
where EM = p V l p (see Section 1.1). An s.p.l.(=) L is called intermediate iff L QCL(=). Note that QCL(') is included in (and thus, in any m,p.l.(=)). The rule (m4) means that together with a formula A, L contains all its MFN-substitution instances (and similarly for ( s 4 ) ) . In particular, L contains every formula congruent to A, because it is a substitution instance under the dummy substitution. Hence we easily obtain
QKE)
Lemma 2.6.9 If A
- -
B then (A = B ) E L (for any m.p.l.(=) or s.p.l.(=) L ) .
A A B implies (A = B ) (A = A), and (A = A) = [ A / p ] ( p p), thus (A = A) E L by (mO), (m4) (or ( s o ) , (s4)). Hence (A B) E L.
Proof
Lemma 2.6.10 Let A, B be formulas in the language of a predicate logic L. Then for a variable x @ FV(B): (1) L t- Qx(B > A)
>.
B > QxA,
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e
Proof (1) Consider the substitution S = [A,B/P(x), q ] ; note that x FV(S). By Lemma 2.5.13, up to congruence, S distributes over > and Vx (since x $ FV(S)). Congruence also distributes over > and Vx, by 2.3.17. Thus S(Axl4)
Vx(B
> A) 3.
B 2 VxA,
and so the latter formula is in L. The proof of (2) is similar.
Definition 2.6.11 Let be a set of formulas i n the language of a predicate logic L. A n L-inference of a formula B from r a sequence A1, . . . ,A,, i n which A, = B and every Ai is either a theorem of L, or Ai E or Ai is obtained from earlier formulas by applying MP, or Ai is obtained from an earlier formula by V-introduction over a variable that is not a parameter of any formula from I?. If such an inference exists, we say that B is L-derivable from r , notation: r FL B .
r,
Note that we distinguish inferences from proofs; the latter may also use substitution and 0-introduction. From definitions we easily obtain
Lemma 2.6.12 k L A iff L F A. Proof 'If'. If L k A, then A is an L-inference (from 0 ) . 'Only if'. By induction on the length of an L-inference of A from 0 . Recall the simplest first-order analogue of the propositional deduction theorem:
Lemma 2.6.13 If
U {A) k L B , then r F L A
>B
Proof Standard, by induction on the length of an inference of B from F U {A). (i) If B E L U I?, then A > B follows by M P from B and B > (A > B), which is a substitution instance of (Axl). (ii) If B is obtained by MP from C and C > B and by the induction hypothesis r FL A > C, A 3,C > B , note that
from a tautology (or an intuitionistic axiom (Ax2)); hence I? k L A > B by MP. (iii) Suppose B = VxC, k L A > C by induction hypothesis and x is not a parameter in r U {A), then r k L Vx(A > C). By Lemma 2.6.10, L t- Vx(A > C) > (A > B ) , therefore F L A > B by MP. (iv) If B = A, then (A > B ) = (A > A), which is L-derivable by a standard argument; see any textbook in mathematical logic.
r
Hence we obtain an equivalent characterisation of L-derivability.
2.6. FIRST-ORDER LOGICS
123
Lemma 2.6.14 Let r be a set of N-modal (or intuitionistic) predicate formulas, L an N-modal (or superintuitionistic) predicate logic (with or without equality). Then for any N-modal (or intuitionistic) fomula B, r FL B zff there exists a finite X C I? such that
As usual, we also include the case X = 0 , with T as the empty conjunction. Of course the notation A X makes sense, due to the commutativity and the associativity of conjunction in intuitionistic logic.
Proof Since every inference from I' contains a finite number of formulas from I?, it is clear that I? F L B iff there exists a finite X I? such that X FL B . So we have to show that
1x1.
The proof is by induction on B iff L t- B by 2.6.12. If X = 0, then But L t- B >. T > B (this is an instance of (Axl)), so by MP, L F B implies Lt-T>B. The other way round, L I- T > B implies L t- B , since L I- T. Therefore LkBiffLt-T>B. Suppose (1) holds for X (and any B). Then it also holds for X U { A ) . In fact, by 2.6.13 and our assumption
The latter is equivalent to
due t o
(2) follows in a standard way by the deduction theorem from
and
The next lemmas collects some useful theorems and admissible rules for different types of logics.
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124
Lemma 2.6.15 The following theorems (admissible rules) are i n every firstorder logic L:
(9 Bernays
rules:
B>A
A 3 B
B ~ V X A'
3xA 1B
if x @ FV(B);
(iii) variable substitution rule:
-.
A
[ylxlA '
> B ) 3 (QxA > QxB),
(iv) Vx(A
(v) monotonicity rules for quantifiers
A>B QxA 3 Q x B ' (vi) replacement rules for quantifiers
(vii) Vx(A A B )
-
(viii) 3x(A V B ) (ix) VxA (x) 3xA
--
3xA V 3xB;
--
-
> C)
( C > VxA) if x @ FV(C); (3xA > C ) if x @ FV(C);
73xA;
(xiv) 3 s ( C > A) (xu) 35 (A
VxA A VxB;
i f x $! FV(A);
(xi) Vx ( C > A)
(xiii) Vx TA
QxA = Q x B '
A i f x $ FV(A);
EA
(xii) Vx (A
A=B
3
( C 13xA) if x $ FV(C);
> C ) > (VxA > C )
i f x @ FV(C);
(xvi) 3x 1 A > 1VxA; (xvii) 3x(A V C)
= 3xA V C
i f x @ FV(C);
2.6. FIRST-ORDER LOGICS
(xviii) Q x ( A A C ) = Q x A A C, if x $ F V ( C ) , Q E {V,3); (xix) 3 x ( A A B ) > 3 x A A 3 x B ; (xx) V x A V V x B 3 V x ( AV B ) ; (xxi) V x A V C (xxii) Q x Q y A
-
> V x ( AV C ) i f x $ F V ( C ) ; Qy Q x A for Q E {V, 3 ) ;
(xxiii) Q x A r Q ( x . a ) A for a quantifier Q, a distinct list x and a permutation a of In, where n = 1x1; (xxiv) 3xVyA > Vy3xA; (xxv) V x A > [ y / x ] Afor a variable substitution [ y / x ] ;
-
-
(xxvi) V x ( A = B ) > ( Q x A = Q x B ) ; (xxvii) a ( A
A')
> ~ ( [ A / P ( xB) ] [ A 1 / P ( x B ) ]) ,
-
if B is non-modal (moreover, i f P ( x ) is not within the scope of modal operators in B ) ; (xxviii)
A [ A / P ( x )B]
5
A' [A1/P(x)]B
(replacement rule)
with the same restriction as i n (xxvii). So (xxviii) shows that up to equivalence, the universal closure V A does not depend on the order of quantifiers. Similarly t o the propositional case (Section 1.1), the replacement rule (xxviii) can be written as follows:
B ( . . . A . . .) = B ( . . . A 1 ...) Proof (i) Readily follows from 2.6.10. (ii) By Lemma 2.5.13 we obtain
[ A / P ( x ) ] ( V x P ( x3) P ( y ) ) A V x A > [ y / x ] A (note that x (ii).
F V [ A / P ( x ) ] ) So . since L contains (Ax12),it also contains
The particular case of this is V x A > A. Hence V x A > A easily follows by induction on 1x1 and the trnasitivity of 3. The dual claims for 3 are proved in a similar way,
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126
(iii) If L t- A, then L t- VxA. Since L t- VxA > [y/x] A by (ii), we obtain L t [ylx]A by MP. Therefore L is closed under variable substitution, since every variable substitution is a composition of simple substitutions. (iv) By the deduction theorem, it is sufficient to show Vx(A
B ) FL QxA
QxB.
First consider the case Q = V. We have the following 'abridged' Linference from Vx(A > B): 1. Vx(A 3 B) 3. A
B by (ii)
2. Vx(A > B ) by assumption
3. A 3 B by 1,2, MP 4. VxA
> A by (ii)
5. VxA 3 B by 3, 4, transitivity
6. VxA > VxB by 5, (i). Here we apply the transitivity rule and the Bernays rule to L-derivability from l?; the reader can easily see that they are really admissible in this situation. For the case Q = 3 the argument slightly changes in items 4-6.
> 3xB by (ii) 5. A > 3xB by 3, 4, transitivity 6. 3xA > 3xB by 5, (i).
4. B
(v) If L I- A > B, then L t- Vx(A 3 B) by generalisation. Since L t- Vx(A B ) >. QxA > QxB by (iv), we obtain L F QxA > QxB by MP.
>
-
(vi) If L t- A r B, then L t- A > B , B > A by (Ax3), (Ax4)17 and MP. Hence L t- QxA > QxB, QxB > QxA by (v), and thus L t- QxA QxB by C, D A-introduction -, which is admissible in L. CAD (vii) Since L F A A B > A by (Ax3) and substitution, it follows that L IVx(A A B ) > VxA, by (v), and thus Vx(A A B ) t - VxA. ~ Similarly from (Ax4) we obtain
hence Vx(A A B) I-L VxA
VxB,
171n the modal case we may use AX^), (Ax4) as classical tautologies.
2.6. FIRST-ORDER LOGICS by A-introduction, and therefore
L I- Vx(A A B) 1VxA AVxB, by the deduction theorem. To show the converse we may also use the deduction theorem. In fact, we have the following abridged inference from VxA A VxB: 1. VxA A VxB by assumption
2. VxA A VxB 1VxA by AX^), substitution 3. VxA by 1,2, M P 4. VxA
> A by
(ii)
5. A by 3,4, MP. A similar argument shows VxA A VxB E L B . Hence VXAAVXBE L AAB, by A-introduction and therefore VxAAVxB F L Vx(A A B). (viii) I t is sufficient to show
and
L I- 3x(A V B )
3zA V 3xB.
For the first, we can use the V-introduction rule:
which is admissible in L, due to (Ax5). So it remains to show
L k 3xA 13x(A v B), 3xB 2 3x(A v B).
But these follow by (v) from A > AVB, B instances of AX^), (Ax7).
> AVB, which are substitution
The converse L t 3x(A V B) 2 3xA V 3xB follows by the Bernays rule from Lt- A v B > 3xAv3xB. For the latter we can also use V-introduction after we show
But A > 3xA V 3xB follows by transitivity from A > 3xA (ii) and 3xA 3xA v 3xB (Ax6). The argument for B > 3xA V 3xB is similar.
>
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128
(ix) L I- VxA > A by (ii). L t- A LtA>A.
> VxA
follows by the Bernays rule from
(x) The proof is similar to (ix). (xi) We have Vx(C > A) FL C > VxA by Bernays' rule, and thus L t- Vx(C > A) >, C > VxA by Deduction theorem. For the converse, first note that
by the abridged inference C, C
> VxA,
VxA, VxA
> A, A,
hence C>VXAFLC>A by Deduction theorem, and thus C
> VxA t-L Vx(C > A),
since x @ FV(C). Therefore
L t C 3VxA. >Vx(C 3 A). (xii) Along the same lines as in (xi), using the second Bernays' rule and the theorem A > 3xA. We leave the details to the reader. (xiii) Readily follows from (xii), with C = 1. (xiv) By Deduction theorem, this reduces to 3xC follows by the abridged inference
> A, C tL3xA. The latter
(xv) By the Bernays rule and deduction theorem from A
> C I-L VxA > C.
By Deduction theorem, the latter reduces to A > C,VXAt-L C, which we leave as an easy exercise for the reader. (xvi) = (xiv) for C = 1.
2.6. FIRST-ORDER LOGICS (xvii) By (x), L k 3xC = C, so the admissible replacement rule B1 = B2 yields
L F 3xAV3xC = 3xAVC.
Since also L k 3x(A v C)
= 3xA V 3xC
by (viii), and we obtain (xvi) by transitivity for r. (xviii) If & = 'v', the argument is similar to (xvi), using (ix), (vii), and the replacement rule B1 = B2 AAB1-AAB2 Let & = 3. Then L k ~ ~ ( A A>C3 x) A A C follows from (xvii), (x), and the replacement rule
for A1 = 3x(A A C), A2 = 3xA, B1 = 3xC, B2 = C. Finally, to show Lt3xAAC> ~ ~ ( A A C ) we argue as follows. First we obtain
by the deduction theorem and A-introduction. Hence
by (v); this rule is still admissible in L-inferences from C , since b'xintroduction is admissible. So by the deduction theorem,
tL C >. 3xA > 3x(A A C). The latter formula is equivalent to 3xA A C
3
3x(A A C).
In fact, 3 x A A C tL3x(AAC), since 3xA A C tLC and 3xA A C EL 3xA, and we may use C >.3xA 3x(A A C) and M P t o obtain 3x(A A C). Therefore
t-L 3xA A C > 3x(A A C).
>
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(xix) The proof is similar to (vii). From AX^), (Ax4) by monotonicity we obtain L t- 3x(A A B) > 3xA, 3x(A A B) > 3xB. Hence L t 3x(A A B ) theorem.
> 3xA A 3xB by A-introduction
and the deduction
(xx) The proof is similar to (viii). First we note that Lt-VxA>AvB by transitivity from VxA
> A,
A
> A V B.
Similarly Lt-VxB>AvB. Hence by V-introduction,
and (xviii) follows by the Bernays rule. (xxi) Almost the same as (xx). Apply the Bernays rule to VxA V C (xxii) By (ii), L F VyA
> A V C.
> A; hence L t- VxVyA > VxA
by monotonicity and L t- VxVyA
> VyVxA,
by the Bernays rule. The converse is obtained in the same way. The case of 3 is similar. (xxiii) Since a is a composition of elementary transpositions, it is sufficient to consider a = So let x = yxixi+lz, then x . a = yxi+lxiz. We have (in L) tQZA = Q X ~ + ~QZA
OZ~+~.
ex,
by (xxii), hence
by (vi), i.e. we obtain QxA
-
Q(x . a)A.
(xxiv) Since L t- VyA > A, we obtain L t- 3xVyA > 3xA by monotonicity; hence L t- 3xVyA > Vy3xA by the Bernays rule.
2.6. FIRST-ORDER LOGICS
131
(xxv) First consider the case when x n y = 0 . We argue by induction on n = 1x1. The base n = 1 was proved in (ii). Next, if x = xlxl, y = ylyl, and we know that L t- Vx'A 3 [y'/x']A, then by (v), L t Vx'A > Vxl [y'/xl]A. By (v) again,
L
Vxi [y1lx'1A > [yllxl][y1lx']A,
hence
L EVxA 3 [~llxl][y'/x']A, by transitivity. Since x1 @ y', the conclusion is [y/x]A as we need. Now in the general case, let z be a distinct list of new variables, lzl Then as we have proved, L k VxA > [z/x]A, and thus
= n.
L t VxA > Vz[z/x]A by the Bernays rule. We also have
from the above, so by transitivity
Since [y/z][z/x]A
[y/x]A, this completes the argument.
(xxvi) We have the following theorems in L: 1. A = B. 3, A > B (Ax3) 2. Vx(A = B ) >, Vx(A > B )
3. Vx(A 3 B) >. QxA > QxB 4. Vx(A
= B ) 3.
QxA 3 QxB
1, monotonicity (v)
(iv) 2, 3, transitivity.
Hence Vx(A r B ) tL QxA > QxB. In the same way (using Ax4) we obtain Vx(A = B ) FL QxB > QxA. Hence by propositional logic Vx(A = B) t LQxA
-
QxB,
which implies (xxvi) by the deduction theorem. (xxvii) To simplify the notation, we write B(A) instead of [A/P(x)]B. So we show V(A = A') FL ~ ( B ( A = ) B(A1)) by induction on the length of B and then apply the deduction theorem.
132
CHAPTER 2. BASIC PREDICATE LOGIC
If B = B1 * B2 for a propositional connective hypothesis we have
-
*, and
by the induction
V ( A= A') E L ~ ( B( AI ) B1(A1)),V ( B ~ ( Ar) B2(A1)), hence we deduce (by (xxv)) B1( A ) = B1 (A'), B2(A) E B2 (A1). Now we can apply the admissible propositional rule
and obtain B ( A ) E B(A1).Since V ( A also applicable.
-
A') is closed, V-introduction is
I f B = QyB1,we change it to a congruent formula so that y @ F V [ A / P ( x ) ] . Then B ( A ) = QyBl(A),B(A1)= QyB1(A1).If by the induction hypothesis B(A r A') B1(A) B1(A1),
-
then
V ( Az A') EL Vy(Bl(A) B I ( A ' ) ) . Hence we deduce QyBl(A)= QyB1(A1) by (xxvi) and M P and then apply generalisation. In the case when B = OiBl the propositional replacement rule (1.1.1)can be used, the details are left to the reader.
-
(xxviii) If L t- A A', then by generalisation L t- V ( A E A'). Hence by (xxvii) and MP, L k ~ ( B ( A=) B ( A i ) ) . Now we can eliminate ti b y (xxv) and MP.
Lemma 2.6.16 Theorems in logics with equality:
(4) xi Proof
= y1 A
. . . A xn = yn
3. P ( x l , .. . ,xn)
= P ( y l , .. . ,y,).
2.6. FIRST-ORDER LOGICS
(1) From (Ax17)by substitution [x= z/P(x)]we obtain
Hence by substitution [x/z](2.6.15(iii))
This implies x = y k L y = x (due to (Ax16)),whence L t x = y > y = x by the deduction theorem.
(2) From (Ax17) y = z 3. P(Y) 3 P(z) by substitution [x= y/P(y)]we have
This is equivalent t o (2)by H.
and
y
=x
Hence by (I), x = y k L P(y) P(x) = P(y),and thus
FL P(y) 3 P(x).
> P(x), so
by A-introduction x = y k-L
by the deduction theorem. Now we apply the substitution
where
B := [xxl/xz]A and X I 51 V(A).Then
Note that
[ylxl. [xx1/zxI= [xlylxzl, since this substitution sends x to variables. So by 2.3.25(4),
X I ,z
to y and does not change other
CHAPTER 2. BASIC PREDICATE LOGIC and thus (#I) implies
Hence
L t- [X/XI](X= y 3. [xlx/xz]A = [xly/xz]A)
by 2.6.15(iii). The latter formula is congruent to
Now we can again apply 2.3.25(4):
It remains t o note that [xx/zxl]A % [x/z]A, [ Y X / ~ X ~ [I x ~ Y I x ~ I A[ Y I ~ I A , since x1 does not occur in A. (4) By the deduction theorem, i t suffices to show
For this we show by induction that
for a list of new variables z = (21, . . . , z n ) . The case m = 0 is trivial. Suppose (#3,)
holds; to check (#3,+1),
assume
Then by (b'zm+l)-introduction (since zm+l is new) V ~ m +(Am l
= Bm) ,
where
Hence by 2.6.15 (ii) and M P
--
[~m+l/~m+lIAm [~rn+l/~m+lIBm.
(#5)
The assumption (#4) implies xm+l = yrn+l, so by (iii) we have From (#5), (#6), by transitivity we obtain [xm+l/zm+l]Am = [ym+l/ %+I. Now since (#2) is (#3,), the claim is zm+l]Bm, i.e. Am+1 proved.
2.6. FIRST-ORDER LOGICS
-
Lemma 2.6.17 Theorems in QCL (and thus, in any m.p.1.): (1) 3 x ( A 2 C)
(VxA 1 C ) if x @ FV(C);
Proof 1. We have in QCL: 1. 73x(A
> C) = tlxi(A > C )
(Lemma 2.6.15(xiii))
> C) = A A -47 (by a propositional tautology) 3. Vxy(A > C) = Vx(A A -C) (2, replacement)
2. -(A
= VxA A 7 C (2.6.15(xix)) > C) (by a propositional tautology). VxA A 4' = ~ ( V X A
4. Vx(A A 4') 5.
Hence we obtain (a) d x ( A
> C)
-
~(VXA > C ) (by transitivity from 1, 3, 4, 5).
This implies (I), due to the admissible rule
(ii) Take C = Iin (1) (iii) We have in QCL: (a) ( C > A)
= (4V A)
-
(from a propositional tautology)
(b) 3x(C 3 A) r 3 x ( l C V A) (1, replacement) (c) 3 x ( 4 V A) = 4'V 3xA (2.6.15(xvii))
> 3xA (from a propositional tautology) (e) 3x(C > A) = C > 3xA (by transitivity from 2, 3, 4).
(d) 1 C V 3xA
C
(iv) We have in QCL: (a) A V C
=.
4' > A (from a propositional tautology)
= V x ( 4 ' > A) (1, replacement) tlx(7C > A) =, 4' > VxA (2.6.15(xi))
(b) Vx(A V C) (c)
(d) -C
> VxA.
-
VxA V C (from a propositional tautology)
CHAPTER 2. BASIC PREDICATE LOGIC
136
(e) Vx(A V C) r VxA V C (by transitivity from 2, 3, 4).
Lemma 2.6.18 Theorems in modal logics (where variables): (1) OVxA
> Vx 0A;
(2) 3 x O A
03xA;
(3) x = y
> O,(x
0 E {mi, Oi),
= y) for N-modal logics with equality,
x is a list of
a E IF.
Proof (1)L k VxA > A (2.6.15 (ii)), hence L I- OVxA > OA by monotonicity (1.1.1), which is also admissible in the predicate case. Therefore L t- OVxA > V x O A , by the Bernays rule. (2) Similar to (I), using A > 3xA. (3) Let us first prove x = y > Oi(x = y). So assuming x = y, we prove o i ( x = y). 1. x = y
>.
Oi(z = x)
> Oi(z = y) Ax17, substitution
2. O i ( z = x ) > O i ( z = y) 3. Vz(Oi(z = x)
1, x = y , MP.
> Oi(z = y))
4. Ui(x = X) > Oi(x = y)
[Oi(z = x)/P(x)].
2, Vz-introduction (if z is new).
3, Ax12, MP.
6. Oi(x = y) 4, 5, MP. Hence L F x = y > Oi(x = y). For arbitrary a apply induction and monotonicity rules, cf. Lemma 1.1.1.
w
We use special notation for some formulas. Intuitionistic formulas: C D := Vx(P(x) V q) > VxP(x) V q (the constant domain principle); Is CD- := Vx(1P(x) v q) > Vx1P(x) v q; M a .- 1 1 3 x P ( x ) > 3 x l l P ( x ) (strong Markov principle); .- 13xP(x) v 3x11P(x); M a f .U P := ( l p > 3xQ(x)) > 3 x ( l p > Q(x)); .- 17Vx (P(x) v 1 P ( x ) ) ; K F .:= Vxl(Ql(x1) V lQl(x1)); := V~n(Qn(xn) v (Qn(xn) 2 AP,f_l)) (n > 1); := VxVy (x = y V 7 x = y) (the decidable equality principle); := VxVy ( l l x = y > x = y) (the stable equality principle); := 3xP(x) 3 VxP(x);
2.6. FIRST-ORDER LOGICS
137
Modal formulas: Bai := VxOiP(x) > OiVxP(x) (Barcan formula for Oi); CEi := VxVy(x # y > Oi(x # y)) (the closed equality principle for mi). In particular, AUl
> P(y)),
VxVy(P(x)
AUF
VxVy(x = y).
All the above intuitionistic formulas except AU;, AU,, and AUF are classical theorems. Classically both formulas AU, and AUE state that the individual domain contains a t most n elements, so they are logically equivalent. This also holds in intuitionistic logic: Lemma 2.6.19
(1) QH' I- AU,'
> AU,
(and so QK; t AU;
+ AU, = QH= + AU; QH + AU; = QH + AUl.
(2) QH'
(3)
(and QKG
3 AU,).
+ AU,
= QKG
+ AU;).
Proof (1) Since Pi(xi) A (xi = xj) implies Pi(xj). (2) Consider the formula
(the 'quantifier-free matrix' of AU,) and the substitution
Then xi
S(AUl) =
= xi
i
implies
V
xi = x j ; therefore QH'
i<j
+ AU,
> V xi = x j i<j
t AU;.
(3) [TIPl(x)]AUl is equivalent t o AU;. So QH is trivial.
+ AUl I-
AU; . The converse
¤
Note that the argument is not valid for S(AU,), because S renames bound variables xi in AU,. Lemma 2.6.20
(1) For any modal formula A and a list of variables x
QS4F O V x O A ~ O V x A 0, 3 x O A = O 3 x A . 18First introduced by A. Grzegorczyk.
CHAPTER 2. BASIC PREDICATE LOGIC
138
(2) For any superintuitionistic formula A
QH + K F I- VX7 - A
> 1-VX A;
moreover, QH + K F = QH
+ VX- l P ( x ) > 7 - y ~~ ( ~ 1 .
Proof ( 1 ) OVxOA > VxOA follows from Op > p. The other way round, we obtain OVxA > VxOA by 2.6.16(1),hence U2VxA> UVxUA by monotonicity, and thus OVxA > OVxOA (since S 4 k Up > C12p). For the second formula the proof is similar, using A > 3xA. (2) For L := QH V x - l P ( x ) > l l V x P ( x ) let us show that L t- K F . In fact,
+
by substitution. On the other hand,
H t- 7 - ( p V -p) by the Glivenko theorem, hence
QH k 1 - ( P ( x ) V - P ( x ) ) , and thus
QH k V x 7 i ( P ( x )v i P ( x ) ) . So by MP it follows that L I- 7-Vx(P(x) V ~ P ( x ) ) (K=F ) . The other way round, let us show that
QHK
:= QH
+ K F t- V x - l P ( x ) > 1-VxP(x).
By the deduction theorem, this reduces to
and next t o
V x l l P ( x ) ,~ V X P ( FQHK X) I. I t suffices t o show that
VX--P(x), - P ( x ) tQH ~ V X ( P (VX~)P ( x ) ) , which follows from
(1)
VX--P(x),V X ( P ( XV) - P ( x ) ) kQH P(x).
To show (I), note that
(11) and
VX~~P(X),VXv ( P- P( (Xx)) ) k Q H --P(X), P ( x ) v ~ P ( x )
--p(x), p ( x ) v +(x) I-QH P ( x ) , (111) since P ( x ) t- P ( x ) and {--P(x), - P ( x ) ) is inconsistent.
2.7. FIRST-ORDER THEORIES
2.7
139
First-order theories
In this section we again consider formulas with extra individual constants. Definition 2.7.1 Let L be an N-m.p.1. (=) (respectively, an s.p.1. (=)). A n N modal (respectively, an intuitionistic) formula with extra constants A is called L-provable, or an L-theorem (notation: tL A) if it has a maximal generator in L, i.e. if it can be presented i n a form A = [c/x] B, with B E L, a distinct list of constants c, and a distinct list of variables x (cf. Lemma 2.4.4). For A without constants, we also assume that A iff A E L. Lemma 2.7.2 tL A i f f A has a generator in L (for A containing constants). Proof 'Only if' follows from 2.7.1. The other way round, if A has a generator B E L, then its maximal generator A1 can be presented as [y/x]B (Lemma 2.4.5); so Al E L by 2.6.15 (iii). L-provability does not depend on the presentation of a formula: Lemma 2.7.3 If B , B' are maximal generators of A and B E L, then B' E L. Proof By Lemma 2.4.5, B' A [z/y] B for some variable substitution [zjy]. So B' E L, by Lemma 2.6.15. L-provability respects MP: Lemma 2.7.4 If kL A and k L A > B, then kL B . Proof Suppose (A > B ) = [c/x](Al > B l ) , for distinct c, x, and (A1 > B1) E L. Then A1 is a maximal generator of A, so kL A implies A1 E L, by Lemma 2.7.3. Hence B1 E L, by MP. L-provability respects 0-introduction: Lemma 2.7.5 If tL A, then t-L OiA. Proof Suppose A = [c/x]Al for distinct c, x , and A1 E L. Then OiAl E L and DiA = [c/x]OiA1. H L-provability also respects substitution into propositional formulas: Lemma 2.7.6 If L is a predicate logic (of any kind), A is a propositional formula, A E L, S = [B1,. . . ,B,/ql,. . . , q,] is a substitution of formulas with constants for propositional letters, then kL SA. Proof Let Gi be a maximal generator of Bi, so Bi = [ci/xi] Gi for an injective [ci/xi]. Take distinct lists y l , . . . , y, of variables non-occurring in any Gi and put G: := [yi/xi]Gi, GI := Gi ,..., GL, q := q l , . . . , q n , c := c1 ...cn, y := y l . . .y,. Then by Lemma 2.4.2(3), Bi A [ci/yi] G:, and thus
(Lemma 2.5.18). Since [G1/q]A E L, it follows that kL SA.
140
C H A P T E R 2. BASIC PREDICATE LOGIC
Definition 2.7.7 A n N-modal theory (respectively, an intuitionistic simple theory, with or without equality) i s a set of N-modal (respectively, intuition-
istic) sentences with individual constants. Dr denotes the set of individual constants occurring i n a theory I?. So according to the terminology from Section 2.3, r is a set of N-modal or intuitionistic Df-sentences: F C or I? I S ( ' ) ( D ~ ) .The set of all Dr-sentences (of the corresponding kind) L(=) (I'):= MS$) ( ~ r (or ) IS(=)( ~ r )is) called the language of I?.
MsE)(D~)
The next definition is an analogue of 2.6.11.
Definition 2.7.8 A n L-inference of a formula with constants B from a theory I' is a sequence A1,.. . ,A,, i n which A, = B and every Ai is either Lprovable, or Ai E I', or Ai is obtained from earlier formulas by applying MP or V-introduction. If there exists an L-inference of B from I', we say that B is L-derivable from I' (or an L-theorem of I'); notation: I'k L B. Then we easily obtain an analogue of 2.6.14:
Lemma 2.7.9 Let I' be an N-modal or intuitionistic first-order theory (with or without equality), L a predicate logic of the corresponding kind. Then for any N-modal or intuitionistic formula B (perhaps, with extra constants)
I' k L B ifl there exists a finite X Proof
I' such that t-I, /\ X
> B.
The argument is essentially the same as for 2.6.14; we check that
for a finite theory X and a formula with constants B , by induction on If X = 0 , (*) means
FI, B
@
t L
T
3
1x1.
B,
which follows in the same way as in 2.6.14. For the induction step we need the equivalence
which follows by Lemma 2.7.6 from ( 2 ) in the proof of 2.6.13.
Lemma 2.7.10 Let J? be an N-modal theory, Oil? := { O i A ( A E I?). (1) If I? k L A, then OiI? t-~,OiA.
> B , then OiI? k L O i A O i B . r, then by 2.7.5, Proof ( 1 ) W e apply 2.7.9. If L t- /\X > A for a finite X L t Eli(/\ X ) > O i A , and thus L t- /\ O i X > OiA. Hence Oil? E L OiA. ( 2 ) By (I),r F L A > B implies Oil? t LU i ( A > B). Then we can apply the (2) If I? F L A
axiom A K i and MP.
W
2.7. FIRST-ORDER THEORIES
141
Another useful fact is the following lemma on new constants.
Lemma 2.7.11 Let L be an N-modal or superintuitionistic logic, I? a modal (respectively, intuitionistic) theory, A(x) a formula with constants (resp., modal or intuitionistic), x a variable not bound in A(x), and assume that a constant c does not occur in I? U { A ( x ) ) .Then the following conditions are equivalent:
Proof (1) 3 ( 2 ) . Assume I? t-L A(c). Then by 2.7.9, t-L FI > A(c) for some finite rl I?. Let B := I?,. Then for some injective [yx +-+ dc],
Since by assumption c does not occur in I?, it does not occur in B , so we have
Hence
A ( x ) = A(c)[cH x] = [ d / ~ I A o ( x ) , and thus
B
A(x) = [dlyl(Bo Ao(x)).
So t LB > A(x), therefore r t LA(%). ( 2 ) 3 (3). Assume r t LA ( x ) ,then by 2.7.9, for some finite Fl I? we have FL B > A ( x ) , where B = /\I?l. So for some injective [ y H dl, B > A ( x ) = [dlyl(Bo3 Ao(x)) and (Bo 3 Ao(5))E L. Since B is closed, x @ F V ( B o ) ,thus by the Bernays rule, (Bo > VxAo(x))E L. We also have
and so the latter formula is L-provable. Therefore r EL QxA(x). ] (x), (3)3 (1).Let Ao(x)be a maximal generator of A(x),then A(x)=[ d / y A. and also VxA(x)> A(c) = [dc/yx](VxAo(x) > Ao(x)). But (VxAo(x)> A o ( x ) )E L by 2.6.13 (ii), so
Then I? F L VxA(x)implies I'k L A(c) by MP. In the intuitionistic case it is convenient to use theories of another kind.
CHAPTER 2. BASIC PREDICATE LOGIC
142
Definition 2.7.12 A intuitionistic double theory (with or without equality) is a pair (I?, A), i n which I?, A are intuitionistic sentences (respectively, with or without equality). D(r,A)(=DruA) denotes the set of constants occurring in I'U A; the language of (I', A) is L ( r ,A) := I F ( ' ) ( D ( ~ , ~ ) ) . Definition 2.7.13 Let L be an s.p.l., (I',A) an intuitionistic theory. A n intuitionistic formula A (with constants) is L-provable i n (I?, A) if I' E L A V V Al for some finite A1 E A.
I'F L A implies (I',A) F L A. So, as we assume V 0 := 1, This provability respects M P as well: Lemma 2.7.14 If (1', A) k L C and (I?, A) F L C
B , then (I?,A) k L B.
Proof First note that 'I k L A V V Al implies I' F L AV V A2 for any A2 2 A,, since FQH A V V A1 > A V V A2. The latter follows from the intuitionistic tautology P v q >pV(qVr). So if (I',A) F L C and (I',A)
for some finite A1
C A.
t-LC 2 B, then r t LC v V A l and
But by 2.7.6
since
H t - ( p V r ) A ( ( p > q ) V r )> q V r (the latter follows from p V r , ( p > q ) V r FH q V r , which we leave t o the reader). ¤ Hence I' t-L B V V A l , and thus ( r , A) t-L B .
2.8
Deduction theorems
We begin with an analogue of Lemma 2.6.13. Lemma 2.8.1 For a predicate logic L, a first-order theory I' and formulas with constants A, B of the corresponding kind
Proof By an easy modification of the proof of 2.6.13, using Lemma 2.7.6. The details are left to the reader. rn Lemma 2.8.2 Let A be a predicate (intuitionistic) formula and let S be an IF(')-substitution such that FV(A) n F V ( S ) = 0 . Then
2.8. DEDUCTION THEOREMS
143
P r o o f Let FV(A) = r(x), so VA = VxA. Let A" be a clean version of A, such that BV(AO)n F V ( S ) = 0 . Then SA S SAO,and thus
since r(x) n FV(S) = 0.But by Lemma 2.6.15 (xx),
where FV(SAO)= r(xy). By 2.6.15 (ii),
Hence QH(=) I-
VSAO
3 VXSAO,
and thus by ( I ) , (2) we obtain QH(') I- VSA
> s(~A).
Here is an example showing that the requirement
is necessary. Let A = P ( x ) , S = [Q(x,y)lP(y)l, so we have FV(A) n FV(S) = {x). Then
but
QH k+ VxQ(x,x) 3 VYQ(X,Y). T h e o r e m 2.8.3 (Deduction t h e o r e m for superintuitionistic logics) Let L be an s.p.l.(=), I? an intuitionistic theory. Then for any A E IF(=) L + r I - A iff ; f S ( r ) k L ~ .
+ +
c +
P r o o f (If.) Sub(I') G L I', hence =(I?) L I?. So Szlb(r)t LA implies L I? kL A, and thus L I? I- A, since L I? is closed under L-provability. (Only if.) It is sufficient to show that the set {A I Szlb(I?) k L A) is a superintuitionistic logic. The conditions (s1)-(s3) from Definition 2.6.3 are obconsists vious (e.g. for generalisation we apply the Bernays rule, since =(I?) of sentences). To check (s4), assume that vAl,. . . ,FAk t LA for A1,. . . , Ak E Sub(r). Consider a substitution S = [C(xy)/P(x)], and let z be a distinct list such that
+
k
+
r(z) n (r(y) U IJ FV(Ai) U FV(A)) = 0 . Let us show that Sub(I?)tLSA. i=l We may also assume that BV(C) n r(xyz) = 0.Consider another substitution S' = [C1/P(x)],where C' := [z/y]C.
CHAPTER 2. BASIC PREDICATE LOGIC
144
A VAi > A
Since V A ~.,. . ,FA, F L A, we have
1
E
L, and thus
But
and so
k
(1) L I-
/\
V S ' ( ~ A ~3) SIA. i= 1 On the other hand, since F V ( S f ) n FV(Ai) = 0, by Lemma 2.8.2, we have
and thus
k
Hence
Sub(!?)t LA
i=l
S1(vAi), and by (1) we obtain S u b ( r ) k L S'A,
which implies Sub(r) k L v z s f ~ , by V-introduction. But by 2.5.27(ii), [y/z]SIA A SA, and thus by 2.6.15(ii),
QH I- VzSIA > SA. Therefore Sub(r) F L SA, as stated. Note that the previous proof can be simplified if I' is a set of sentences: then we choose substitution instances A, in such a way that F V ( A , ) n F V [ C / P ] = 0, and we can readily apply Lemma 2.8.2. For an N-modal theory A put
Theorem 2.8.4 (Deduction theorem for modal logics) Let L be N-m.p.l.(=), r an N-modal theory. Then for any N-modal formula A L
+ r I- A
iff
OmSub(I') F L A.
a
2.8. DEDUCTION THEOREMS
145
Proof OmSub(I') kL A clearly implies L +r k A. For the converse, note that the set {A I CImSub(r) kL A) is substitution closed; this is proved as in the previous theorem. It is also closed under 0-introduction. In fact, OalB1,..., O,,Bn FL Aiff t-LO,,BlA
...AIJa,Bn > A.
By monotonicity and the distribution of U iover A in K N , this implies
tLOiOal B1 A .. . A OiO,, B,
3
OiA.
Hence O""Sub(r) EL OiA.
1
Definition 2.8.5 An m.p.1. (=) is called conically expressive if its propositional part is conically expressive. Lemma 2.8.6 For any conically expressive N-m.p.l.(=) L, theory A and formula A, O*A FL A iff A U = ' kL A,
where
lJ*A := {O*B I B E A).
Proof 'If' readily follows from L k O*p > 0,p. For the general proof of the converse we need the strong completeness of Q K N , see below. But in particular cases (like Q S 4 or QK4), O*B is equivalent to a finite conjunction of formulas 1 from Um{B), SO 0 * A EL A obviously implies OmA kL A. Hence we obtain a simplified version of the deduction theorem for conically expressive modal logics:
Theorem 2.8.7 Let L be a conically expressive N-m.p.E.(=), theory. Then for any N-modal formula A
r
an N-modal
L f I ' F A iff n * S u b ( r ) E L ~ ,
Proof
Follows from 2.8.4 and 2.8.6.
Here is a simple application of the deduction theorem. Lemma 2.8.8 QH
+ W* I- KF, where
w* = Vx((P(x) 3 VyP(y)) 3 VyP(y)) 3 VxP(x), K = VxiiP(x) > iiVxP(x). Proof
It is sufficient to show that
or, equivalently, W*,V X ~ ~ P ( ~'v'xP(x) X), FQH VxP(x). But this is obvious, since the premise Vx((P(x) > VyP(y)) > VyP(y)) of W* , the presence of is equivalent to Vx((P(x) > I)> I),i.e. to V x l ~ P ( x ) in 1VxP(x). 1
146
CHAPTER 2. BASIC PREDICATE LOGIC
2.9
Perfection
Let A1.. .Am be a list of formulas (not necessarily distinct); we define their disjoint conjunction
where Si is a formula substitution transforming Ai in such a way that predicate letters in all conjuncts become disjoint. E.g. we can put := P g k + i ( ~ ) ; S~PL(X)
then Py occurs in SiAi only if 1 E i (mod m). The formula
is equivalent to V(A!
A ... A ~
i),
in all our logics, see below. Similarly we define a disjoint disjunction:
For a theory O put OV := {A"\A E O, k 2 0 ) ,
0" = {A1
. . . A A, I m > 0, A l l . . . ,Am E O).
Thus 8" contains conjunctions of variants of formulas from O in disjoint predicate symbols, and OAV= {(Al
A .. . A Ak)k ( k > 0,
All .. . ,Ak E O).
Obviously, H + O = H + O v = H + o A= H + Q A v . For theories e l , . . . ,Om we also define the disjoint disjunction
(the set of disjunctions of variants of formulas from e l , . . . ,Om) and the extended disjoint disjunction: (el
C...$ O m ) * ={(A1
I k>O,
A1 E 01, ... ,Am E Om).
In this section we consider only superintuitionistic logics
Definition 2.9.1 Let L be an s.p.l.(=), of L. W e say that
sets of formulas in the language
2.9. PERFECTION
.
el L-implies 0 2
(notation: 'dB E
el and 0 2
0 2
01
0 2
if
3A E 0 1 L t- A
are L-equivalent (notation: 01
> B.
-L
02)
if
01 < L
Q2 and
01.
01 sub-L-implies
0 2
(notation:
el is sub-L-equivalent to 0 2
01
I y b0 2 )
if Sub(Ol) IL Sub(02);
(notation: 01 -? 0 2 ) if
Sub(&) -r, Sub(&). Here are some simple properties of these relations. Lemma 2.9.2 (1)
01
(3)
01
(4)
=+Q1
0 2
02.
iff Sub(01) I L 0 2 .
0 2 = = ~+
$ 0 C_ 2
~ + 0 1 .
(5) @ ~ - S , " ~ @ ~ * L + Q ~ = L + Q ~ . Proof
(1) Note that L I- A
> B implies L t S A > S B for any substitution S.
(2) Follows from (1).
(3) By ( I ) , Sub(&) IL 0 2 converse is trivial. (4)
01
IY
0 2
==+ Sub(0l) = Sub(Sub(Ol)) I L Sub(&).
clearly implies e2C L
The
+ 01.
(5) Follows from (4).
2.9.2(2), (5) allow us sometimes to identify L-(sub)-equivalent sets of formu-
las. Definition 2.9.3 A set of sentences 0 is called V-perfect i n a logic L i f the following holds
(If-p) for any A E Sub(@) there exists B E Sub(@) such that L I- B
i.e. Sub(@) sL Sub(@). Lemma 2.9.4 ('d-p) follows from its weaker version
> VA,
148
CHAPTER 2. BASIC PREDICATE LOGIC
@-p-) for any A E 0 , k
2 0 there exists B
E Sub(@) such that L k B
> A".
Proof In fact, assume (V-p-). If A' E Sub(@), then VA' = s A ~for some substitution S and A E 0 , k 2 0. By (V-p-), L k B > Ak for some B E Sub(@);
L t S ( B > A") = (SB > VA').
¤
Since SB E Sub(@), (V-p) holds for A'.
Definition 2.9.5 A set of sentences @ is called A-perfect in a logic L if (A-p) for any A1,. . . ,Am E Sub(@) there exists B E Sub(@) such that L t- B Al A...AA,.
>
Note that the condition (A-p) for m = 2 implies (A-p) for arbitrary m , by induction. Lemma 2.9.6 @ is A-perfect in L iff sub(@) SL Or\
Proof Note that if every Ai is a substitution instance of A: E @, then A1 A . . . A A, is a substitution instance of A: A . . . A6. Definition 2.9.7 A set 8 is called AV-perfect (in L) if it is both V-perfect and A-perfect. Obviously, a perfect set (of any kind) in L is also perfect in all every L' 2 L. The next proposition suggests equivalents to 2.9.7.
Proposition 2.9.8 Let @ be a set of sentences. Then the following conditions are equivalent:
(3) for any Al, . . . , Am E Sub(@) there exists B E Sub(@) such that
L t B > VA1 A . . .VA,
(or equivalently, L t B
> V ( A ~A . . . A A,));
Informally speaking, (4) means that a AV-perfect set of axioms allows for the simplest natural form of the deduction theorem.
Proof ( 1 ) + ( 2 ) . I f L k B i > A " f o r i = l ,..., m a n d L k B > B I A . . .AB,, then L k B 3 F A ... A=. (2) (3). If VA: E S U ~ ( A ? ) for , i = 1,.. .,rn,then for k 2 max(m, kl,
*
a substitution instance of (Al A . . . A Ak)k implies VA; A .. . AVA; QH (where e.g. Ai = A, for m < i 5 k). (3)+ (4). If L @ t- A, then by the deduction theorem 2.8.3, L I- VA1 A . . . A VA, > A for some A1,. . . ,A, E Sub(@). The converse implications (and (4) + (1))are obvious.
. . . , k,),
+
in
2.9. PERFECTION
149
Similarly in the case of V- or A-perfection we can somewhat simplify the deduction theorem: Lemma 2.9.9
(1) If @ is V-perfect in L, then L
(2) If 0 is A-perfect in L, then L
+ 0 t- A iff
+ 0 t- A iff Sub(@) FL A.
L F BB
>A
for some B E
Sub(@).
+
Proof (1) By 2.8.3, L @ F A iff Sub(@)FL A, so Sub(@) FL A implies L+@t-A. The other way round, if Sub(@) F L A, then
for some A1,. ..,A, E Sub(@). By (V-p) there exist Bi E Sub(@) such that L t- Bi > BAi; thus Sub(@) EL A. (2) 'If' is again obvious. The other way round, suppose Sub(@)FL A, i.e.
-
VAl A
for A1,. . . ,A,
.. . ABA, kL A
E Sub(@). Since
by the deduction theorem we obtain L ~ B ( A ... ~ AA A , ) I J A . By (A-p) then LFB>AIA
... AA,
for some B E 0 ; hence by monotonicity L t BB
> V(Al A .. . A A,)
and eventually
Lemma 2.9.10 (1) For any theory 0 , the theories GV,0",OAVare respectively V-perfect, A-perfect, and AV-perfect.
(2) For any theories 0 1 , . . .,,@ ,
the theory ( 0 1 \j . . . i/ Om)* is V-perfect
Proof (1) Straightforward. (2) By 2.6.15(xx)
hence
-
QH F (Al \j . . . \j Thus (V-p-) holds.
> ((A1 i/ . . . i/
150
CHAPTER 2. BASIC PREDICATE LOGIC
So every recursively axiomatisable s.p.1. has a recursive AV-perfect (in Q H ) axiomatisation. lg Sometimes one can construct simpler perfect axiomatisations for Q H Q (or for L O, with a superintuitionistic L). E.g. if a set Oo is V-perfect, then Q t (or (Oo U 8')" for any 0 ) is AV-perfect, etc. To construct the sets Ov, QA,QvA from O we usually need infinitely many additional universal quantifiers or conjuncts, so these sets are infinite for any TI)." But sometimes finite perfect extensions also finite set Q (unless 8 {I, exist.
+
+
Lemma 2.9.11 If 01, .. . , O m are AV-perfect, then
Proof V-perfection follows from Lemma 2.9.10. To show A-perfection, consider Cl ,... , C k E (01 i/ ... i/ Om)*, with
Aij E Qj. As noted above, we may assume that different Ci have no common predicate letters. Then every Ci can be presented as
where Aij E Sub(Qj). By VA-perfection (2.9.8(3)), there exist B j E Sub(Qj) such that k
L F B, > V / \ A : ~ . i=l Then
Remark 2.9.12 Another kind of perfection was introduced in [Yokota, 19891, cf. also [Skvortsov, 20041. A theory 8 is called arity-perfect in a logic L if for any A E Sub(@)there exists a closed substitution instance B' of some B E 8 such that L t B' > A (and thus L t B1>VA). ''Of course every s.p.1. itself is At/-perfect, but such a trivial axiomatisation is not recursive. 20Formally speaking, Qv and 0'' are infinite even for Q = {I}, but repetitions and dummy ~ quantifiers can be eliminated, so we can say that { I ) = = {I}.
2.10. INTERSECTIONS
151
Our notion of V-perfection is weaker, because B in 2.9.3(V-p) is not necessarily closed21. Note that the sets OA, 0"' are arity-perfect.
Lemma 2.9.13 L-sub-equivalence preserves all forms of perfection.
2.10
Intersections
Proposition 2.10.1 Let L be a superintuitionistic predicate logic, and let I',rl be sets of sentences such that formulas from I? and r1have no common predicate letters. Then:
(2) (cf. [Ono, 1973, Theorem 5.51) If L I- C D then
(3) If L
+ r I- C D and L + I?' I- C D , then ( L + ~ ) n ( L + r l ) = L + C D + { A V B I A ~BrE, I " ) .
Proof
(1) Let (L +I?) n ( L
+ r l ) k A. Then by Theorem 2.8.3, k
1
i= 1
j=1
L I - A V A ~ ~andA L I - A V B ~ ~ A for some A l l . . .Ak E Sub(r), B1,. . .B1 E Sub(rl). Thus
Assuming that IFV(Ai)l = m, IFV(Bj)I - = n, we obtain that VAi V V B ~ is a substitution instance of Am V Bn. So we obtain the first equality in (1)Now note that by substitution and omitting dummy quantifiers,
if m < m l , n s n l . Hence
where s = max(m, n), and the second equality follows. 'lRecent counterexamples by D. Skvortsov show that V-perfection is strictly weaker than arity-perfection.
CHAPTER 2. BASIC PREDICATE LOGIC
152
+
- -
(2) We use (1). Assume that A E I?, B E r'. Obviously, L A0 v B0 t- AV B , thus A V B E (L +I?) n (L r'). So it remains to show that for any m E w
+
~ e Am t = Vul
Then L
. . .Vum Am, B" = Vvl . . .Vv, Bm, and
+ A V B F C and L F C > A" V B", by CD. Hence (*) follows.
(3) follows from (2): take L
+ C D as L. H
Remark 2.10.2 The assumption L F C D in Proposition 2.10.1(2) is necessary. [Ono, 19721731gives an example of predicate logics for which (**) does not hold Similarly one can describe intersections of modal predicate logics. Proposition 2.10.3 Assume that L is a N-modal predicate logic, and have no common predicate letters. Then
r,I"
(1) (L + r ) n (L + r l ) = L+{o~A~vD~B~IAE B IE'r,' ; m , n E w ; a , P E I P ) . For particular classes of 1-modal logics this presentation can be simplified, cf. Section 1.1: (a) For logics above Q T :
(b) For logics above Q K 4 :
(c) For logics above Q S 4 : ( n + r ) n ( n + r l ) = L + { O A r n v ~ B n lA E ~ B, E r r ; m , n ~ w ) . (2) If L t- Ba, then:
( L + I ' ) ~ ( L + I " ) = L + { U , A V O ~ B I A E I 'B, E ~ ' ; ~ , P € I E ) . Similarly, to (i), there exist simpler versions for the cases L B a , Q K 4 Ba or Q S 4 Ba.
+
Therefore we have
+
> QK4 +
2.11. GODEL-TARSKI TRANSLATION
153
P r o p o s i t i o n 2.10.4 The complete lattices of predicate logics (superintuitionistic or modal) are well-distributive, i.e. they are Heyting algebras. Moreover, the intersection of two recursively axiomatisable logics is recursively axiomatisable. The intersection of finitely axiomatisable logics is finitely axiomatisable for superintuitionistic logics QH C D and for 1-modal logics > QK4 B a .
>
+
+
Later in this book we will show that the intersection of finitely axiomatisable superintuitionistic predicate logics may be not finitely axiomatisable, if one of them does not contain C D .
2.11
Godel-Tarski translation
-
D e f i n i t i o n 2.11.1 Godel-Tarski translation for predicate formulas is the map (-)T : I F = M F = defined by the following clauses: AT = CIA for A atomic; ( A A B )=~ A ~ A B ~ ; ( A V B )=~ A ~ v B ~ ; ( A 3 B)T = O(AT 3 B T ) ; ( ~ x A=) 3~x A T ; ( v x A )= ~ CiVxAT. For a set r I F = put rT := { A T I A E r ) . Godel-Tarski translation is obviously extended to formulas with constants. Since QS4= t- ( x = y)
-
O ( x = y ) , we may also define ( x = y)T just as x = y.
L e m m a 2.11.2 QS~(') t- OAT Proof
= AT
for any A E I F ( = ) .
By induction on the length of A.
L e m m a 2.11.3 (1) Let [ x H y] be a (simple) variable transformation. Then for any A E IF', ( A [ xH Y ] )=~ A ~ [ X H y]. (2) For any formula with constants A E I F E ( D ) , for any D-transformation [XH a17 ( [ ~ I x I A=) [alxl ~ (AT).
Proof If z
( 1 ) Easy, by induction on ! A [ .We consider only the case A = V z B .
# x , we have ( A [ xH y])T = ( V z ( B [ xH Y ] ) ) T = O V z ( B [ xH Y ] ) ~ = O Y ~ ( B ~H [X y ] ) (by the induction hypothesis) = ( O Y ~ B *[)X Y ] = AT [ X Y].
If z = x, we have
(A[x
~ 1 = )( Q y~ ( B [ x ~ 1 ) =) O~ V y ( B [ x ~ 1 ) ~
= 0 V Y ( B T [ xH y ] ) (by = ( O V X B [~X ) y] =
(2) An exercise.
the induction hypothesis) [X Y].
CHAPTER 2. BASIC PREDICATE LOGIC
154
Lemma 2.11.4 If A A B for A , B E IF=, then AT A BT.
Proof We use Proposition 2.3.17. Consider the equivalence relation
on IF'. It is sufficient t o show that 2.3.17. We check only (1) for & = V:
satisfies the conditions (1)-(3) from
( V X A )A ~ (Vy(A[x H ~ 1 ) ) ~ The remaining (routine) part is left t o the reader. In fact, by 2.11.1 and 2.11.3 ~ U Q Y ( A ~ [ X YI), (Vy(A[x Y I H T = UVY(A[X Y I ) = which is congruent t o ( V X A ) = ~ OVxAT by 2.3.17.
Lemma 2.11.5 Let S = [B(x,y)/P(x)] be an IF(=)-substitution, and consider the MF(=)-substitution
Then for any A E IF(')
(*) Proof
Q S ~ ( ' )t ( s A ) ~
-
S
~
A
~
By induction on A.
If A = P(z), then (SA)T = BT(z, y), and
S
~ =As T (~n p ( z ) ) = O B ~ ( Zy), ,
which is equivalent to B ~ ( zy, ) by Lemma 2.11.2 If A is atomic, A # P(z), the claim is trivial.
.
If A = VzC is clean and also BV(A) n F V ( S ) =
But
Q S ~ ( ' )F ( s c ) ~ -
-
then
sTcT
by the induction hypothesis, and hence
- by replacement. So
(*) holds.
220therwise we can consider a congruent formula with this property.
2.11. GODEL-TARSKI TRANSLATION If A = 3zC, then
Now again we have Q S ~ ( =I-) (SA)T and replacement.
-
STAT by the induction hypothesis
If A = C V D , then
(SA)T = ( S C V S D ) ~= (SC)T V ( s D ) ~ , STAT = ST(CT V DT) = STCT v sTDT. r STCT, ( S D ) ~= S
By the induction hypothesis Q S 4 Iwe can apply a replacement.
~ , SO D
The case A = C A D is almost the same, and we skip it.
If A = C II D , then (sAlT = ( S C I) SD)T = O((SC)T II STAT = S T ( o ( C T I)D ~ ) = ) n(sTcT 2 s ~ D ~ ) . In this case we can also use QS4-theorems
(sc)~
= STcT,
( s D ) ~= s T D T
and replacement.
Lemma 2.11.6
g QS~(') + Sub(rT) for any r E IF(=)
Proof By Lemma 2.11.5 we have ( s A ) ~ E QS~(') + STAT for any simple substitution S . Now recall that every substitution is reducible to simple ones. Lemma 2.11.7 For a list of variables x and a n IF(')-formula
A(x),
-
Proof By induction on the length of x . The base: QS4 t- AT OAT (2.11.2). For the induction step, assume O\dxAT = ( V X A ) ~Then . in QS4 we obtain: 0'dy\dxAT = ~ l ' d ~ 0 ' d x(Lemma A~ 2.6.202)- O V ~ ( V X A (by ) ~ assumption and H replacement) = ( b ' y ~ x A ) ~ . Proposition 2.11.8 For any m.p.1. (=) L containing QS4 the set T~ := {A E
IF(=)I
E
L)
~
156
CHAPTER 2. BASIC PREDICATE LOGIC
Proof We check that TL satisfies the conditions (s1)-(s5). By Proposition 1.5.2, AT E QS4 for every propositional intuitionistic axiom A. The same holds for predicate axioms and for the axioms of equality, as one can easily see. It is also clear that TL satisfies (s3) from Section 1.2. Corollary 2.11.6 shows that TL is closed under IF(=)- substitutions. Definition 2.11.9 T h e above defined s.p.l.(=) TL is called the superintuitionistic fragment of the m.p.1 (=)L; a n m.p.l.(=) L i s called a modal counterpart of the s.p.l.(=) TL, cf. Definition 1.5.4. Lemma 2.11.10 For a n y A E IF(=),
Proof By induction on the length of the proof of A. An alternative proof ¤ makes use of completeness, see below. Lemma 2.11.11 Let L = QH(')
+r be a n s.p.1. (=).
T h e n for a n y A E IF(=),
A, i.e. Proof Suppose L t- A. Then by Theorem 2.8.3, Sub(r) tQH(=) there exists formulas Ax,. . . , An E r and substitutions S1,. .. ,Sn such that n
QH(=) t-
A V S ~ A ~> A.
a= 1
Hence
Then by Lemma 2.11.10,
n
QS~(') t I\(~s,A,)~. i=l
3 AT.
-
o~sTAT. ) ~ Therefore By definition and Lemma 2.11.5, QS~(') t ( ~ s ~ A ~ ~ S u b ( rkQS4c=, ) AT, i.e. Q S ~ ( = ) rTk AT by Theorem 2.8.4.
+
As we shall see (cf. Proposition 2.16.17), in many cases the converse of 2.11.11 also holds, i.e. ) ), QH(=) +- r = T ( ~ ~ 4 (+=r T
QH
+r(=) k A
iff QS~(')
In particular, it is well known that
[Schiitte, 19681, [Rasiowa and Sikorski, 19631. Let us make another simple observation.
+ r F AT.
2.12. THE GLIVENKO THEOREM
157
+
Lemma 2.11.12 If an s.p.1. (=) L = QH(') I' has modal counterparts, then Q S ~ ( =+ ) rTis the smallest modal counterpart of L.
+
Proof Suppose L = TA. Then obviously, QS~(') rTC A. By Lemma 2.11.11, L k A only if QS~(') rTk A. On the other hand, if LYA, then A VAT, and thus QS~(') rTVAT. Therefore Q S ~ ( =+) rTis a modal counterpart of L.
+
+
Remark 2.11.13 The paper [Pankratyev, 19891 states that every s.p.1. has modal counterparts. However the proof in this paper requires some verification, which we postpone until Volume 2. We do not know if there exist the greatest modal counterparts in the predicate case, and Theorem 1.5.6 has no direct analogue:
Theorem 2.11.14 QH.
(1) [Pankratyev, 19891 QGrz is a modal counterpart of
(2) [Naumov, 1991] There exists a proper extension of QGrz which is also a modal counterpart of QH. These matters will also be discussed in Volume 2.
2.12
The Glivenko theorem
Proposition 2.12.1 (Predicate version of the Glivenko theorem) any intuitionistic predicate formula A
Proof
Let QCL k 1 A . Then by the deduction theorem 2.8.3
for some formulas A,. Then by Lemma 1.1.2, we have (1) QH k
A 71V(~,
v TA,)
3
-A.
S
By Lemma 2.6.20(11),
By Corollary 1.1.9, QH F %l(A, V l A , ) , so by applying (2), we obtain:
QH
+ K F t-
7 7 B ( ~ , v YA,) S
Hence by ( I ) , QH
+ K F t- 1 A .
For
158
CHAPTER 2. BASIC PREDICATE LOGIC
Corollary 2.12.2 For any
A V-formula A,
1
QCLtA iffQH+KFtA. Proof
By induction on A (exercise).
H
We say that an intermediate predicate logic L satisfies the Glivenko property if for any formula A, LtiA QCLI-iA and that L has the classical
A V- fragment if for any
1
A V-formula A,
1
T h e o r e m 2.12.3 The following conditions are equivalent (for a superintuitionistic predicate logic L): (1) L satisfies the Glivenko property;
(2) L has the classical
7
A V-fragment;
(3) Q H f K F G L C Q C L .
In other words, Q H the property (i) or (ii). Proof
+ K F is the smallest intermediate predicate logic with
By Lemma 2.6.20(11),
-
+
Thus (i), (ii) imply Q H K F C L. (i) also implies L G QCL. In fact, suppose A E (L - QCL). Since Q H I- A > 7-A and Q C L t- A TTA, we then have 7 l A E (L - QCL), which contradicts the Glivenko property. Finally, (ii) implies L Q C L as well. In fact, assume (ii) and suppose A E (L - QCL) and Q C L t- A B, where B is a A V-formula; then Q C L t TT(A = B), whence Q H f K F t- 1 7 ( A r B), by Proposition 2.12.1. As we have proved, L I- K F . So we obtain L t= B), whence L t-A-- = ~ T Bby, Lemma 1.1.2. Since A E L, we also have -7A E L, and thus 7 7 B E (L-QCL), which contradicts (ii). H
-
1
The same result holds for the logic with equality QH'
2.13
+KF.
A-operation
Definition 2.13.1 For a predicate formula A put bA := p V ( p > A), where p is a proposition letter that does not occur in A. For a superintuitionistic predicate logic L put AL := Q H (6A I A E L) and also AOL := L, An+'L := AAnL by induction.
+
2.13. A-OPERATION
L e m m a 2.13.2 The following formulas are theorems of QH: (1) A 2 6A
Proof (1) A FQH p > A and p 3 A FQH bA; hence A FQH 6A. Then apply the deduction theorem. (2) First note that by AX^), P > (A3B), p > A F Q ~ p > B .
Hence by V-introduction
Now (2) follows by the deduction theorem. (3) We have Q H I- VA > A by 2.6.15. Hence Q H t- R A eventually Q H -I ~ V A> V6A by the Bernays rule.
> 6A by (I), and W
So AL can be axiomatised as follows.
L e m m a 2.13.3 AL = Q H + (6A I A E
z).
Proof Let EL:= Q H + {GA I A E 2);then clearly K L 5 AL. The other way round, if A E L, then VA E so @?A E EL. By Lemma 2.13.2, &?A implies V6A; so 6A E h L . Therefore AL 2 h ~ . rn
z,
+
In [Komori, 19831 AL was defined as Q H {6'A I A E L), where 6'A := > p) > p. As in the propositional case, both definitions are equivalent:
((p > A)
L e m m a 2.13.4 Q H Proof
Quite similar to 1.16.2 using the deduction theorem.
L e m m a 2.13.5 (2) LI
+ 6A = Q H + 6'A for any formula A.
(1) AL
W
L.
C L2 ===+ AL1 5 AL2.
(3) A(QH
+ I ) = QCL.
(4) AL c QCL for
every consistent s.p.1. L.
Proof (1) Since Q H t A > 6A for any A. (2) Trivial. (3) A(QH I ) C QCL, since Q C L t 6A for every A. The other way round, QCL C A(QH I ) , since Q C L = Q H EM and E M = p V ~p is just 6 1 = p V ( p > I). (4) follows from (3) by Proposition 2.12.5.
+
+
+
w
CHAPTER 2. BASIC PREDICATE LOGIC
160
In Chapter 6 we will prove the converse to (2), similarly to 1.16.7 for the propositional case. The logics QHP: := An(QH I)= An-'(QCL)
+
were first introduced in [Komori, 19831 (where they were denoted by A k ( w ) , for W := Q H I). Let us now turn to a finite axiomatisation of Q H P ~ presented in [Yokota, 19891. First we generalise 6-operation as follows [Yokota, 19891.
+
Definition 2.13.6 Let
where P E P L ~ ,y i s a distinct list of variables, r(y) n FV(A) = 0 . If P does not occur in A, we use the notation 6kA rather than 6k,pA23 Also put
( i n particular, SEA = A). For a set O of formulas and y = bk, 6; etc. put
W e also introduce
-
b&O :=
U 620,
kELd
U
b:O:=6 k . . . 6 k O = n
6kl...k,@
k l , ...,k,EW
for n E w . ~ ~ Obviously,
szo E 6:O.
Also note that 60A = 6A and Pn = 6z1. Hence
Lemma 2.13.7
(1) Q H
+ 6A k dkA.
23Cf. Definitions 1.2.1 and 1.2.4 in [ ~ o k o t a 19891. , 24The latter notation is somewhat ambiguous as it means P I ,. . . , P, that do not occur in A. 2 5 ~ n[ ~ o k o t a 19891 , the set 6,t{l) was denoted by P;.
. . . 6k,P,A
for dzfferent
2.13. A-OPERATION
161
Proof (1) Sk,pA = Qy [P(y)/p] 6A, so we can apply substitution and generalisation. (2) Follows from (1). (3) Follows from (2) by induction. The next lemma shows that 6-type operators behave as 0-modalities. Lemma 2.13.8 The following are theorems of Q H : (1) A 3 O A ;
(2) O ( A 2 B ) 3. O A 3 O B ,
where
0= 6 k , ~or 6 k >...k , .
Proof (I), (2). For 0 = bk,P this follows from 2.13.2 (I), (2) by substitution [P(y)/p] and generalisation. For 0= d k l . . . k , apply induction. (3) Similar to 2.13.2 (3). Q H F VA > A implies Q H I- OVA > O A by (I), (2); hence QH t- OVA > 0 A by the Bernays rule. (4) We may suppose n = 2 and use induction for the general case. So let us show QHt-O(AAB)=OAAOB. Since Q H t- A A B
> A, by ( l ) , (2) and (MP) we obtain
Similarly
Q H ~ O ( A A B ) ~ O B , and thus
QHt-O(AAB)>OAAOB. For the converse note that
by AX^), hence
Q H ~ O A > O ( B ~ A A B ) by ( I ) , (2) and next
QH~-OA~OOB>O(AAB) again by ( l ) , (2) and the transitivity of 3 . The latter formula is equivalent to
O A A OB 3 O ( A A B ) .
a
162
C H A P T E R 2. BASIC PREDICATE LOGIC
Proposition 2.13.9 Q H I- 6 r A > 6 p A for n 5 m. Proof Readily follows from 2.13.8(1), since 6 r Lemma2.13.10
(2) Q H
¤
= 6F-"b;.
(1) Q H k P , f > 6 Y A , Pn > 6 z A forany formulaA.
+ b y 0 c Q H + P,f
and Q H
+ 6E0 C Q H + P,
for any theory 0 .
Proof
I>A.
(1) By 2.13.8 ( 2 ) , since Q H I( 2 ) Follows from ( 1 ) . Lemma 2.13.11 QH
Proof
+ 6k,... knA I- 6l
1,A whenever ll 5 k l , . . . ,1, I k,.
By substitution and elimination of dummy quantifiers.
W
The same argument actually proves a stronger claim. Since elimination of dummy quantifiers is legal in Q H , we obtain:
Proposition 2.13.12 6 & 0
6:0
rn
Proof Lemma 2.13.13 For an s.p.l(=) L and sets of formulas (1) If L
+
0 1
CL
+
02,
01, 0 2
then
+ 6 2 0 1 C L + 6;@2 provided 8 2 is V-perfect in L; + 6 r (07)G L + 6; (0;). (2) If L + =L + then = L + 6;@2 provided el,0 2 are V-perfect in L ; (a) L + (b) L + 6; (07)= L + 6; ( 0 ; ) . (a) L (b) L
0 1
02,
Proof
+
( 1 ) (a) If A E 0 1 C L L I-- B1 A . . . A B,
c L + 0 2 , then by 2.13.8
>A
S u b ( 0 2 ) -l A, i.e. for some B 1 , . . . ,Bm E S u b ( 0 2 ) . Then by
the monotonicity of 6;,
thus
k
L F (/"\ 6,"Bi) i=l
&,"A.
Since S k ~ Ei Sub(6kOz), it follows that L Therefore S k 0 1 C L 6:02.
+
+ 6;02 I- 6EA.
2.13. A-OPERATION
163
(b) Readily follows from (a), since Ov is V-perfect in Q H and Q H + O = QH 0'.
+
(2) Follows from (1).
Lemma 2.13.14 V-perfect in L.
26
If a set Q is V-perfect in an s.p.1. (=) L, then S&,O is also
For an arbitrary 6kA E 6 k 0 , with A E O, let us check the property 2.9.3 (V-p-). We have
Proof
( b k A ) m ( ~=) 'dzb'x(P(x, z) V (P(x, z)
3
Am(z)))),
where xz is a distinct list of new variables, lzl = m, 1x1 = k, P does not occur in Am(z). This formula is clearly QH-equivalent to 6k+,Am(z). Since 8 is V-perfect, there exists B E Sub(@) such that
L t B 3 VzAm(z), and thus
L k B 3 Am(z), We may also assume that r(xz) n F V ( B ) = 0 . Hence
by the monotonicity of 6k+m and
as we can choose P that does not occur in Am(z) and in B.
therefore (V-p-) holds for 6kA. Proposition 2.13.15 If O is V-perfect in L, then 6 & 0 and 6:0 in L, for all n 0.
>
Proof
For 6:Q
=
sL...6;
are V-perfect
O proceed by induction on n. And for 6&@ use
n
sub-equivalence: b& O N G 620. ~ Alternatively, for 6 & 0 one can apply a direct argument generalising the proof of Lemma 2.13.14. In fact,
26Cf. [Yokota, 1989, Proposition 1.4.61.
CHAPTER 2. BASIC PREDICATE LOGIC
164 implies
L k B 3 Am(z) and next
L I- 6 L m B > S;+,Am(z), with
= (SEA)";
Q H t- 6,",,Am(z) again we can present b;+,B,
(6;A)m
= VzVx(Pn(x, Z) V
( C ~ ; A )as~
Pn(x, Z) > Vx(Pn-l (x, Z) V (Pn-l(x, Z) > . ..))));
+ m)-ary predicate letters in both formulas. Lemma 2.13.16 If L = Q H + Q and O is V-perfect in QH, then
with the same set of (k
Proof
By 2.13.13(2a)
A L = Q H + S L = QH+S@, since Q H
+ L = L = Q H + 8 and both Q and L are V-perfect in Q H . Thus
Since by 2.13.7 b k Q
C AL, it follows that
Proposition 2.13.17 (1) If L = Q H AnL = Q H SF@. for any n 2 0.
+
(2) If L = Q H
AL = Q H + 6&0.
4
+ O and Q is V-perfect in Q H , then
+ 8 , then
Proof (1) By induction on n. If n = 0, then AL = L, 620 = O, so the statement is trivial. For the induction step suppose
Then by 2.13.16 and 2.13.15,
(2) Follows from (I) and the observation that Q H V-perfect in Q H . In particular, for G = {I) we have
+ O = Q H + Qv and Ov is
2.13. A-OPERATION
Corollary 2.13.18
27
165
QHP:
=Q
H
+ 6:{1).
Now let us show that every bgA for k > 1 (and hence every bkl,...,k,), is deducible from byA. Lemma 2.13.19 Q H
Proof
+ 6;tA k (bkA 3 A) > A for any k, n 2 0.
By 2.13.8(2),
B 3 A k Q6 ~ : ~ > &$A, hence
(1) bkA 3 A, B 3 A, bkB FQH A. On the other hand, by the deduction theorem
From (I), (2) we obtain28
Hence (again by the deduction theorem)
Note that bkA is bk,pA for some P that does not occur in A, but (3) still holds if P occurs in B . Now by induction it follows that
for any n. In fact, for n = 0 this is a substitution instance of an axiom, and for the induction step we can apply (3). Lemma 2.13.20 Q H
Proof
+ bY6.jA t 6YSj+lA for j
> 0, m > 0.
Put
L
:= QH
+ bYbj,pA, B := Vz(Q(y, z) V (Q(y, z) 3 A)),
where lyl = j, z @ r(y), r ( y z ) n FV(A) = 0 and P, Q do not occur in A. Then
&,,PA = V Y ( ~ ( YV) (P(Y) 3 A)), bj+l,QA = VYB. Hence by substitution [B/P(y)]
(1)
L t by'dy(B
27Cf. [Yokota, 1989, Theorem 1.4.71. 28Cf. Lemma 1.2.3(4) from [Yokota, 19891.
V
(B
3
A)).
166
CHAPTER 2. BASIC PREDICATE LOGIC
By Lemma 2.13.19 (for k = 1, n = m
(2) QH
+ 6;""~
+ 1) we have t (61,sA > A) > A.
By 2.13.11, L F &?+'A, and thus
Hence by substitution [Q(y, z)/S(z)]
Since z @ FV(A), we also obtain QH F A 2 B ; thus
which implies
(6) L I - B V ( B > A )
> B,
and therefore
(7) L t- Vy (B v ( B 3 A)) 3 VyB, by the monotonicity of Vy. Eventually from (I), (7) and 2.13.8(2) we obtain
as required.
Lemma 2.13.21 QH
+ 6;""~
I- 6YSkA for k
1 1, m 2 0.
Proof By induction on k . The case k = 1 is trivial. If the statement holds for k we obtain it for k 1:
+
by Lemma 2.13.20 and the induction hypothesis. Proposition 2.13.22" For any A E I F ( = ) (1) QH
+ 6T.A I- 6;A
for k, n 2 0;
Proof (1) implies (2), by 2.13.11. The case k = 0 in (1) also follows from 2.13.11, and the case k = 1 is trivial. So it remains to prove (1) for k > 1. By induction on n - m let us show
heor or em 1.3.5 from [Yokota, 1989) is the particular case of this proposition for A = 1. Our proof is similar to that paper.
2.13. A-OPERATION
167
for 0 5 m 5 n. If m = n, this is trivial. For the step, suppose ( 3 ) for m. By 2.13.21 we have
So by the induction hypothesis we obtain
i.e. ( 3 ) holds for m - 1. Finally note that ( 1 ) is ( 3 ) for m = 0. From 2.13.22 it follows that
for any k
> 1. We also have
Lemma 2.13.23 QH Proof
+ SIA = QH + dOA.
In fact QH + ~ o = A QH
+ P v ( p 3 A) I- V y ( P ( y )V ( P ( y ) 3 A ) ) = blA.
by substitution [ P ( y ) / p ]and generalisation.
+
W
+
However it may happen that QH &,"Ac QH 6rA for some A and n > 1. E.g. QH P, Y P,f for n > 1 (cf. [Ono, 19831); recall that Pn = 621, P,f = SYI. Now we can strengthen 2.13.17:
+
Proposition 2.13.24
(2) If L = QH
Proof
+ O and O is V-perfect in QH, then A n L = QH + 6;O.
(1) If L = QH
+ O, then
Follows from 2.13.17 and 2.13.22.
By applying 2.13.24 to O = {I) we obtain
Theorem 2.13.25 QHP:
= QH
+ P,f
for n > 0.
Now let us show perfection for finite axiomatisations of QHP: and obtain an alternative proof of Theorem 2.13.25. This proof is not so straightforward, but does not use the infinite axiomatisation described in 2.13.18. However we still need 2.13.22 for this proof.
CHAPTER 2. BASIC PREDICATE LOGIC
168
Lemma 2.13.26 H t- (C> q) > C , where C =
n
A (pi V (pi > q)),n 2 0. i=l
Proof By induction on n. The case n = 0 is trivial. Consider the inductive step from n to n 1. By the deduction theorem it
+
suffices to prove
C ~ ~ F H C , nfl
for
C = r\ (pi V (pi> q)), which obviously follows from i= 1
for I < i
< n + 1. Put
Then
Ht- C
-
B A (pi V (pi > q)),
and thus
C 3 4, B , Pi FH Hence by the deduction theorem,
Together with the induction hypothesis
(2) implies
C 3 9, Pi F- q. Hence (1) follows by the deduction theorem again.
Lemma 2.13.27 Let L be a predicate superintuitionistic logic, O a set of formulas, k 0. Then
>
(1) If @ is A-perfect i n L, then bk0 is A-perfect i n L.
(2) If O is 'd-perfect i n L and L 2 dTO for some n > 0 , then bkO is V-perfect i n L. (3) If O is AV-perfect i n L and L _> bTQ for some n
perfect i n L.
> 0 , then 6kO is A'd-
2.13. A-OPERATION
Proof ( 1 ) Consider arbitrary formulas
for i = 1,. . . ,m, where y and x are disjoint lists of variables, lyl = k, B i ( x ) E Sub(@),F V ( B i ( x ) )C x , F V ( A i ( Y ,x ) ) C X Y By A-perfection there exists B E Sub(@)such that
we may assume that r ( y ) n F V ( B ) = 0. Put
By Lemma 2.13.26, QH F ( C > B ) > C , hence
and so
by standard properties of quantifiers (Lemma 2.6.15). Since
we obtain the required property for
A Di. i
(2) Suppose A E Sub(@),
for lyl = k , where P does not occur in A,
for lzl = m, r ( y )n r ( z ) = 0 ,r ( y z ) n F V ( A ) = 0 , Q E PLk+m,where Q does not occur in Am.
By V-perfection there is B E Sub(@)such that L I- B > V z A m ( z ) ;we may assume that F V ( B ) n r ( y z ) = 0 and Q does not occur in B. Put
CHAPTER 2. BASIC PREDICATE LOGIC Since L F B
> Am (z), it follows that (3) L F C > ( G ~ A ) ~ .
Since by the assumption (2) of the lemma 6;"B E Sub(b;"Q) L, Proposition 2.13.22 implies L k SZ+,B. Thus by Lemma 2.13.19, L t- ( C > B) 3 B. Also L k B > C by 2.13.8, hence
and so
(5) L t C V ( C > B ) > ( G k A ) m , by (3) and (4); eventually,
where C' := Vy(C V (C
> B)) E [C/Q(x)] SkPQBlQ E P L ~
(note that here V y is a dummy quantifier). Since C1 E Sub(6kQ), this completes the proof.
Similarly to 2.13.27(1) one can show Proposition 2.13.28 (1) If O is A-perfect in L, then 6LO is A-perfect in L (and thus 6&Q1 6:O
are also A-perfect i n L ) . (2) If O is AV-perfect i n L, then 6LQ is AV-perfect i n L (and 6;O1 well).
6:Q
as
Lemma 2.13.27(2) allows us to prove 2.13.25 without applying 2.13.17; cf. the proof of 2.13.16. Since {I) is AV-perfect in Q H , we obtain Lemma 2.13.29 (1) The sets {P,) and {P:)
are A-perfect in Q H for n > 0.
(2) The sets {P,) and {P:) are AV-perfect in Q H + P,f for all n, r > 0 (note that the case n 2 r is trivial). Remark 2.13.30 Moreover, these sets are arity-perfect in the sense of [Yokota, 19891. Proposition 2.13.31 Q H P ~= Q H
+ P,f + Pn for r > n > 0.
2.13. A-OPERATION
171
So Pn and P,f are deductively equivalent in Q H P f for a sufficiently large r. Proof
By induction on n. Consider the induction step. We have
QHP:+~ = A(QHPR) = QH
+ ~(QHP:) c Q H + P,? + Pn+l,
by Lemma 2.13.13 (la) applied to L = QH
+ P,?,
8 2 := {Pn),
01 = QHP:,
+
+
note that L 01 = QHP: = L O2 by the induction hypothesis, and O2 is tl-perfect in L by Lemma 2.13.29 (ii). rn Theorem 2.13.25 is clearly a particular case of this statement for n = r. Therefore we obtain an alternative proof of Theorem 2.13.25 that does not use 2.13.18. So we have Corollary 2.13.32 Q H
+ P,f = Q H + P: + Pn for n < r.
Similarly we obtain
+
Proposition 2.13.33 If L = Lo 0 , O is V-perfect in Lo and ST@ some r > 0, then Lo AnL = Lo 6;O and 6 c 0 is tl-perfect in Lo for all k, n 0.
+
>
c Lo for
+
P r o o f By induction on n. The case n = 0 is trivial. Consider the induction step from n to n 1. First note that by 2.9.9,
+
QH t- 6yA JI 676;A. Hence 6[6;0 L, and so 6Ef '0 = 6k(6;O) is V-perfect in Lo by the induction hypothesis and Lemma 2.13.27. Next, by the induction hypothesis and Lemma 2.13.13 (la) Lo
+ An+'L = Lo + 6(AnL) = Lo + b(Lo + hnO k - ) C Lo+6(6;0) -
= Lo+b;+lO.
The converse inclusion Lo
c - L~ + A ~ + ' L
+ 6,
is trivial.
+
+
So AnL = Lo 6;0 in particular, AnL = Lo SEO, whenever Lo AnL. Note that 6;O G AnL for any L and O L and n 0, by Lemma 2.13.7. Also recall that b[O Q H P ~for any O by 2.13.10.
c
c
Corollary 2.13.34 If L = Lo AnL = Lo 6;(Ov) for all k , n
+
+ O and 40 & Lo for > 0.
>
some r
> 0, then Lo +
Actually we need only the case k = 0 of this statement, because the case k = 1 (and thus, the case k > 0) readily follows from 2.13.24, without any restriction on Lo (recall that S;Lo Lo by 2.13.8 (1)). And 2.13.33 shows that if Lo contains 6:O for some (sufficiently large) r > 0, then one can reduce the case k = 1 to k = 0 in an axiomatisation of AnL above Lo.
c
CHAPTER 2. BASIC PREDICATE LOGIC
172
Adding equality
2.14
In this section we study correlation between a logic without equality L and its minimal extension with equality L=. There exists an obvious translation from formulas with equality to formulas without equality just replacing equality with an ordinary predicate letter. In some cases specific equality axioms can be reduced to finitely many formulas - then we obtain a reduction for the corresponding decision problems.
Definition 2.14.1 For an N-m.p.1. L, let L= := QKZ + L, and for an s.p.1. L, let L= := QH= L. L= is called the equality-expansion of L. The other way round, for an m.p.l.= (s.p.l.=) L, we define the equality-free fragment as Lo := L n M F N (respectively, L n IF).
+
that does not occur For any modal formula with equality A and Q E P L ~ in A, we define AQ as the formula obtained from A by replacing all occurrences of '=' with Q. For a set of N-modal formulas I? put
rQ:= {AQ I A E r ) , nWr:= {&B I B E r, a E IF}. Consider the following sets of formulas (for N
2 0)
EN :={VxVy(x = y > y = x ) , VxVyVz(x = y A y = z N) U { V ~ V y ( x = y 3 > i ( x = y )11 )
n
>x
= z),
Vx z = x )
u{v(A ~ i = y i > ~ G (,..., x i x,)=P,(yl,...,
yn))In,k21,P~#Q), i=l where all xj , yj are different. For a formula A, let M F N , be ~ the set of all N-modal formulas built from predicate letters occurring in A:
We shall omit N if it is clear from the context and use the notation Ow&, o " & ~ , MFA, &A,
~2.
Lemma 2.14.2 If A A Sub(EN), then OWEN
Proof
vA.
The only nontrivial case is when
for P E PLn, x = ( x l , .. . ,x,), y = ( y l , .. . , yn). Put n
X F
y := /\xi = yi. i=l
Then
V A =v ( xG y
3, B
-
[ylxB ] ). We prove n m E N k Q K N VA by induction on the complexity of B.
2.1 4. ADDING EQUALITY
173
If B is atomic and does not contain P , the claim is trivial. If B = P(z) for some list z (maybe not distinct), then [y/x]B = P ( t ) , where t = [y/x]z, i.e. every xi is replaced with yi, whenever it occurs in z. Then obviously,
By Lemma 2.6.10 (xxvii) we also have
Now (1) and (2) imply
and it remains to apply generalisation. If B
= B1 @ Bz
for a propositional connective 8 and
-
by the induction hypothesis, then the claim follows by an argument in classical propositional logic - note that the rule A
A1, B
= B'
A@B=A'@B" is admissible in the QKN-theory {x = y ) (with x , y considered as constants) and apply the deduction theorem. If B
= OiC
and
we obtain by Lemma 2.7.10
Since
K N Oi(p = q) 3, n i p = Oiq (Lemma 1.I .I), it follows that U W E ~~Q K ,O ~ ( X = y) >. OiC = O~[Y/X]C.
Now EN FQK,
Z j = yj
3 Cli(xj = yj).
implies OWENkQKN x E y 3 Oi(x From (4) and (5) we have
and thus the claim holds for B.
y).
(4)
CHAPTER 2. BASIC PREDICATE LOGIC
174
If B = VzC, we may assume z @ x y (otherwise, after renaming z , B and [y/x]Bchange t o equivalent congruent formulas).
Then by V-introduction we deduce (in the same theory)
and hence
x = y 3. Vz(C = [y/x]C)
by Lemma 2.6.10(xi). Finally
Vz(C G [Y/x]C) 3. v z c = Vz[y/x]C.
by Lemma 2.6.10(xxviii).
follows from (6) and (7) by transitivity and v-introduction.
If B = I z C , we also have (6) by the induction hypothesis. Then, instead of (7) we use Vz(C = [y/x]C) 3. 3 z C = 3z[y/x]C,
(8)
which follows from 2.6.10(xxvii).
Lemma 2.14.2 has an intuitionistic version:
Lemma 2.14.3 If A E Sub(Eo), then Eo kqH A. Proof
Very similar to 2.14.2. Again we prove
by induction. The reader can check that in all cases the argument is based only on intuitionistic logic. w P r o p o s i t i o n 2.14.4 Let L be a n N-m.p.1. T h e n for any N-modal formula A with equality that does not contain Q
If L i s conically expressive, then
2.14. ADDING EQUALITY
175
Proof First note that EN is a set of axioms for L= above L (they are all L=theorems and the usual axioms follow from EN). So by the deduction theorem 2.8.4, L= k A e O m S u b ( E ~k) L A.
All formulas from O m S u b ( & ~are ) of the form o , ~ B , where B E Sub(EN), so they are QKN-provable in Om&Nby 2.14.2, generalisation and O-introduction. Thus L' t A @ Oo0EN t LA. Replacing '=' with Q does not affect the L-inference (more precisely, the equivalence r F L A iff I'QtL AQ is checked by induction), so
and thus
am&?F L AQ =+ L= k A.
The other way round, suppose L= t A. Then Om&$ t LAQ. Let S be a formula substitution replacing every atomic formula P ( x ) with T for any P # Q that does not occur in A. We claim that Om&$ E L B =+ 0 " E j E L SB. In fact,
(10)
S(Q(x,y) 3. P ( x ) = P(y)) = (x = y 3. T r T )
is obviously L-provable, so (10) holds for any B E I$ - £ 2 . Then we can argue by induction, since S distributes over propositional connectives and quantifiers. Eventually
which completes the proof. The observation about conical expressiveness follows from Lemma 2.8.7. W Proposition 2.14.5 Let L be an s.p.1. Then for any A E IF= without occurrences of Q L= t- A @ &$ t LAQ e £2 tL AQ. Proof
Follows the same lines as 2.14.4. By 2.8.3,
Hence by Lemma 2.14.3 and generalisation,
which is equivalent to
&$ t LAQ. The implication
&,& E L AQ =+ £ 2 t ~ AQ, is again proved by replacing redundant predicate letters with T.
CHAPTER 2. BASIC PREDICATE LOGIC
176
In many cases the equality-expansion is conservative. To show this, for logics without equality we define a weak analogue of equality - indiscernibility. Namely, for P E PLn and variables x, y we put
where of course, all zjs are different and x, y # zj. For a predicate formula A, put InA(x,y) := &1np(x, y) 1 P occurs in A).
Lemma 2.14.6 For P E PLn
Proof Almost the same as for 2.6.16(iv). We show by induction that .
.
i=l
for a list of new variables z = (zl, . . . ,zn). For the induction step we again consider A, := [ X I , . . . ,xm/zl,. . . ,z,]P(z), Bm := [yl,. . . 1yrn/z17... 1 zm]P(z).
By the induction hypothesis, b'zm+l-introduction and 2.6.10(ii) we obtain
We also have
In fact, B,
= P(y1,. . . ,y,,
z,+l,. . . ,z,), so
by 2.6.10(xxv), and (2) is a conjunct in the latter formula. From ( I ) , (2) by transitivity it follows that
2.14. ADDING EQUALITY
177
Lemma 2.14.7 ( I ) If L is a conically expressive N-m.p.1, then for any N-modal predicate formula A that does not contain Q
(2) If L is an s.p.l., then for any A E I F
Proof We have to prove the corresponding substitution instances of the 'axioms' from (o*)E? in L.
-
( I ) Reflexivity of Q. We have [ x / z j ] P ( z ) [ x / z j ] P ( z by ) H , hence I n p ( x , x ) by A- and Vzintroduction. Thus V x I n A ( x ,x ) in the intuitionistic case, V x O * l n ~ ( x ), in the modal case - by A-, 0 " - and V-introduction. (2) Symmetry of Q. By H we have
-
-
[ ~ l z j l P ( z ) [ ~ l z j l P ( zt-L ) [ ~ l z j l P ( z ) [xlzjlP(z), hence I n p ( %y) , I-L I n p ( y ,x ) and next I n ~ ( xy), E L I n ~ ( yx ,) by applying V z B > B, A- and V-introduction. Then L t- I n ~ ( xy), > I n ~ ( yx ,) by the deduction theorem. In the modal case this implies L I- O*InA( x ,y) > O*InA(x,y) by the monotonicity of O*. So it remains to generalise over x, Y .
(3) Transitivity of Q. Since G is transitive in H , we have
[x/zj]P(z) [t/zj]P(z), which implies
I n p ( x ,Y ) , I ~ P ( tY) ,F L I n p ( x , t ) again by standard arguments with A and 'd, and hence
Since x , y, t are fixed in this proof, in the modal case we may apply Lemma 2.7.10 for 0*:
O * I ~ A (y), X ,CI*In'4(y,t ) t-L O * I ~ A ( tX) ., Thus
L t- v ( U * ~ n A (y) x , A u * I n ~ ( ty), 3 ~ * I ~ At () )x , by the deduction theorem and v-introduction.
CHAPTER 2. BASIC PREDICATE LOGIC
178
( 4 ) VXVY(Q(X, Y ) 3 'JiQ(x,Y ) ) . By 1.3.46 L I- D*p > OiO*p,hence L tVx'v'y(O*In~(x, y ) > O i O * I n ~ ( xy)) , by substitution and V-introduction. (5)
v ( i=l " ( u * ) I n ~ ( xyi)i , 3. P ( x ) for P E P L n n M F ~ ,X~=,
-
(XI,
P(Y)
..., x,), y = ( y ,..., ~ y,).
hence
~ * I ~ A (yl), x I,., . ., O*InA(xn,y)F L P ( x )
P(Y) by the reflexivity of O*. Now we can apply the deduction theorem and v-introduction.
Theorem 2.14.8 L= is a conservative extension of L, i.e. (L=)" = L for any superintuitionistic predicate logic L and for any conically expressive m.p.1. L. Proof Let A be a formula without equality of the corresponding kind and = A. L= t A implies O * E t~LA assume that Q does not occur in A. Then in the modal case (by 2.14.4) and F L A in the intuitionistic case (by 2.14.5). Hence L F A > A and respectively L t- A E >~ A by the deduction theorem. By applying the substitution [O*InA(x, y ) / Q ( x ,y)] in the modal case, and [ I n A ( xy, ) / Q ( x ,y)] in the intuitionistic case, we obtain
12
0*&2
or respectively,
L
A[I~A(x, y ) / Q ( x ,Y)I&? .
3
A.
Therefore L I- A by Lemma 2.14.7.
Problem 2.14.9 Does Theorem 2.14.8 hold for an arbitrary m.p.1. L? Remark 2.14.10 T. Shimura and N.-Y. Suzuki obtained the following stronger result for the superintuitionistic case. Fix a binary predicate letter P , and let E>(x,y ) := 'v'z(P(x,z )
G
P ( y ,z ) ) .
Then for any L E S, and for any r 5 IF= consisting of pure equality formulas with only positive occurrences of '=',
(cf. [Shimura and Suzuki, 1993, Theorem 31).
2.14. ADDING EQUALITY
179
Now recall a well-known definition from recursion theory [Rogers, 19871. Definition 2.14.11 For sets of words (in a finite alphabet A ) X, Y we say that
X is m-reducible to Y (notation X 5 , Y ) i f there exists a recursive function f : Am -+ A" such that X = f - l [ Y ] ;X is m-equivalent to Y (notation: Xr,Y)ifX<,Y andY5,X. Theorem 2.14.12 If L is an s.p.1. or a conically expressive m.p.E., then L
=,
L=. L 5 , L=, since L= is conservative over L (2.14.8); the corresponding function f sends every formula without equality to itself (and all other words to I,say). L= 5 , L , since for a formula A without Q, Proof
L= t A iff L I- / \ ( o " ) E ~ .IAQ I
.
.
by 2.14.4, 2.14.5. So the reducing function sends every formula A without Q to A ( o * ) £ ~ 3 AQ and every nonformula to I. If a formula A contains Q, first replace Q with another binary predicate letter that does not occur in A. Let us now consider some specific axioms of equality.
Definition 2.14.13 A logic (with equality) L is said to have stable (respec-
tively, decidable, closed) equality i f it contains the corresponding formula ( S E , D E , or C E , see Section 2.4). For a superintuitionistic predicate logic L (without equality), let
Similarly, for an N-m.p.1. L, we define
where CEi denotes C E for Oi. Obviously, L= C L=S L=d. Soon we will show that Q H = ~ is conservative over QH. However, L'~ is not always conservative over L; the corresponding example will be given later on. As we do not substitute formulas for '=' in predicate logics with equality, the following lemma is an easy consequence of the deduction theorem. Lemma 2.14.14
(1) L + D E I - A i f f L I - D E > A
for a superintuitionistic logic with equality L and a formula A; similarly for S E .
CHAPTER 2. BASIC PREDICATE LOGIC
180 N
(2) L +
A
CEi F A iff^ k- CISk
i= l
N-modal logic with equality. For a I-modal logic L
similarly, for L
2.15
N
i=1
CEi > A for some k E w , where L is a n
> Q T this can be simplified:
> QK4, QS4, cf. Section 1.1.
Propositional parts
Let us explicitly describe the construction of the propositional part L, for a predicate logic L without equality, cf. [Ono, 19721731for the case of intermediate logics. Since all the sets PLn are countable, we can consider a bijection
Then for any formula A, let r ( A ) be the result of replacing each atomic subformula of the form P(x) with r o ( P ) and erasing all occurrences of quantifiers. For a set of formulas r let
Obviously r(A) is equivalent (in Q H or in Q K N ) to a substitution instance of A (cf. Lemma 2.6.15(I)(iv)). On the other hand, a propositional formula A is a substitution instance of r(A). Thus we obtain
+
+
Proposition 2.15.1 L, = K N r ( L ) or L, = H r ( L ) for a predicate logic L without equality (respectively, N-modal or superintuitionistic). Proposition 2.15.2
Proof By induction over a proof of A in QKN (or in Q H ) we show that r ( A ) E K N (resp. r ( A ) E H). It is easily checked that r ( A ) E K N if A is a substitution instance of a Q K N - axiom (for example, r(A) = (B > B) for predicate axioms). If A is obtained by (MP) or (necessitation), then r(A) is also obtained by the same rule. If A = VxB, then T(A) = r ( B ) .
Lemma 2.15.3 n(Sub(r)) = S u b , ( ~ ( r ) )for any r C I F (or I? C M F N ) , where Sub, denotes closure under propositional substitutions (of the corresponding type).
2.15. PROPOSITIONAL PARTS
181
Proof Since every substitution is a composition of simple substitutions, we can consider only simple substitutions. It is easily proved that
(by induction on B). On the other hand, every propositional formula can be presented as n(C) for some predicate formula C.
+
+
Proposition 2.15.4 (L I?), = L, n ( r ) for a modal (or superintuitionistic) logic L + r . I n particular, (QKN I?), = K N ~ ( r ) (,Q H r).rr = H ~ ( r ) .
+
P r o o f [Modal case.] Let A E (L
+
+
+
+ r).By the deduction theorem,
for some formulas A, E Sub(r). Then
while r ( A , ) E Sub,(n(r)) (by Lemma 2.15.3). Thus n(A) E (L, The converse inclusion is obvious by Proposition 2.15.1.
+n(r)). H
Definition 2.15.5 The quantified version of a modal (respectively, superintuitionistic) propositional logic A is: QA := Q K N
+A
(respectively, QA := Q H
+ A).
L e m m a 2.15.6 (1) For any propositional logic A and a set of propositional formulas I?,
(2) For a propositional S4-logic A, T(QA)
> QTA.
Proof (1) Consider the intuitionistic case. Q(A +I?) is the smallest s.p.1. containing A r , while Q A l? is the smallest s.p.1. containing QA U r . These two logics coincide, since every s.p.1. containing A also contains QA.
+
+
(2) By 2.11.11, QTA k A implies Q S 4 Q ~ A GT ( ~ ~ ) .
+ ( T ~ ) kT AT, hence QA t- AT. Thus
182
CHAPTER 2. BASIC PREDICATE LOGIC
Remark 2.15.7 We do not know if the equality T ( Q ~ = ) QTA holds for any propositional S4-logic. For example, this is unknown for A = S4.2Grz. Definition 2.15.8 30 A predicate logic L is called a predicate extension of a propositional logic A if L is called a conservative extension of A (i.e. if L, = A). P r o p o s i t i o n 2.15.9 For any modal or superintuitionistic propositional logic A, Q A is a predicate extension of A. Proof
Follows readily from Proposition 2.15.4 since n(A)
C A.
rn
Obviously, Q A is the weakest predicate extension of A. To describe the greatest predicate extensions of propositional logics, we use the formula AUl = Vx'dy(P(x) > P ( y ) ) from Section 2.6; recall that
for any predicate logic L and
for any predicate logic L with equality. For a formula A with equality, let AT be the result of replacing each occurrence of (x = y ) in A with T. Then (A = AT) € ( Q K E AUl) (respectively, QH= AUl).
+
+
Lemma 2.15.10 Let A be a predicate formula without e6uality. Then A E QKN AUi n(A) (in the modal case) or A E Q H AUl n(A) (in the intuitionistic case).
+
+
+
+
P r o o f First, given n(A), we can restore all occurrences of quantifiers from A, because they are dummy in n(A). Next, every occurrence of no(P) in n(A) coming from some P ( x ) in A, can be replaced with VP(X), which is equivalent t o P ( x ) in Q K N AUl.
+
P r o p o s i t i o n 2.15.11 Let L be a predicate logic without equality, modal or superintuitionistic, and let A be a propositional logic of the corresponding kind. Then L , = A iff Q A c L c Q A + A U 1 . Proof
(QA
+ AUi),
= A by Proposition 2.15.4 since
+
On the other hand, if L, = A then L C Q A AUl. In fact, A E L implies n(A) E L, = A and A E Q A AUl by Lemma 2.15.10. rn
+
30Cf. [Ono, 19731.
2.15. PROPOSITIONAL PARTS Therefore Q A logic A.
183
+ AUl is the greatest predicate extension of a propositional
C o r o l l a r y 2.15.12 The greatest intermediate predicate extension of an intermediate propositional logic A is (QA
+ AUl) n QCL = QA + AUl v q v ~
Proof By Proposition 2.10.1 (2); recall that Q C L CD E Q H + A U 1 v q v l q .
=
q .
QH
+ q V ~q
and
Let us also mention the following result on the number of predicate extensions.
T h e o r e m 2.15.13~' (1) Every nonclassical superintuitionistic propositional logic has uncountably
many predicate extensions.
(2) Every modal propositional logic which does not contain S 5 , has uncountably many predicate extensions. Now let us give another description of the greatest predicate extensions.
P r o p o s i t i o n 2.15.14 (I) Let L be a predicate logic (of any kind). Then the following conditions are equivalent:
+ A = L + n(A) for any predicate formula A without equality; (b) QH(=) + AUl L or QKE)+ AUl L
(a) L
(for the superintuitionistic or the modal case, respectively);
+
(c) L = QA(') AUl for a propositional logic A (superintuitionistic or modal, respectively). Moreover, these conditions imply (d) for any predicate formula A (in the language of L ) there exists a propositional formula A' such that L + A = L + A'. (11) If L is a logic without equality, all conditions (a)-(d) are equivalent.
Proof
+
( I ) ( a ) =+-(b). L AUl = L since n(AU1) = (no(P) > no(P)) E H (or KN). (b) + ( a ) . n(A) E L A by Proposition 2.15.1. On the other hand, A E Q H AUl n(A) or A E Q K N AU1 n(A), by Lemma 2.15.10. 31 [Suzuki,
19951.
+
+
+
+
+
184
CHAPTER 2. BASIC PREDICATE LOGIC (a) =+ (d). If L is a logic with equality, we take A' = AT). (b)
3
(c). We can replace any A from L with A' because (b) implies (d).
(11) (d) + (a). Let L
+ A = L + A' for a propositional formula A'. Then A' E (L + A), = L, + 7r(A) C L + 7r(A).
On the other hand, x(A) E (L
+ A).
We say that L is a predicate logic with degenerate predicates if L satisfies the condition (I)(d) from Proposition 2.15.14. Thus, Q H AUl and Q K N AUl are the weakest logics with degenerate predicates. On the other hand, there AUl and satisfying (I)(d), exist logics with equality incomparable with QH' e.g. L = QH' 3 x 3 (x ~ # y AVz(z = x V z = y)).
+
+
+
+
It is clear how to describe the propositional fragments for these logics with equality. L e m m a 2.15.15 Let L and L' be predicate logics containing AUl such that L is without equality and L' is with equality. Then the following conditions are equivalent: (1) L:, = L,;
Proof 32
(1) 3 (2). By Proposition 2.15.14, (b) + (a). (3) + (1). Let A be a propositional formula, A E L=. Then by the deduction theorem, the following formula is a theorem of L:
for some k 2 0 and formulas A1,. . . ,Ak E MFG. Then its substitution instance BT belongs to L and ~ ( B T €) L,. Since r((Vx(x = x ) ~ = ) T and 7r(([x/z]A,)~)= 7r(([y/z]A,)~),we obtain r ( A ) E L, i.e. A E L,. (2) + (3). If (L')' = (L')' = L then L= C L', and L= = L' - because (A = AT) E L' for any formula with equality A (recall that AT is a formula without equality). H
h he implication (3) =+ (2) is proved in Proposition 2.14.8 for all superintuitionistic logics and for 1-modal logics above QK4. Here we give a slightly simpler proof that fits for any modal predicate language.
2.16. SEMANTICS FROM AN ABSTRACT VIEWPOINT
185
Corollary 2.15.16 Let L be a predicate logic with equality, A a propositional logic. If QA= c L QA= AUl
+
c
then L, = A.
Proof
5L
&A= implies (&A=), Next, (&A=
+ AUI),
by Lemma 2.15.15, and (&A
QA=
+ AU1
G L, C (&A= + AUI),. =
(QA
+ AUl),
+ AUl);
= (QA
+ AUl),,
= A, by Proposition 2.15.11.
Corollary 2.15.17 Let L be a predicate logic without equality. Then (1) (L'), = ( L = ~ ) ,= (L=S), = L, for a superintuitionistic L;
(2) (L'), Proof
= (L'"),
= L,
for modal L.
(Intuitionistic case.) Let L, = A, L
C QA + AUl.
Then
and we can apply Corollary 2.15.16. The modal case is analogous. Proposition 2.15.14 and Lemma 2.15.15 show that there exists a natural bijection between intermediate propositional logics and extensions of'the logics QH(') AUl (with or without equality), and similarly for the modal case. The logic &A= AU1 is a maximal predicate extension (with equality) of a propositional logic A. Nevertheless, for any propositional logic A the greatest predicate extension with equality does not exist. Indeed, later on we will show that L, = A for the logic
+
+
which is obviously incompatible with QA=
2.16
+ AUl.
Semantics from an abstract viewpoint
Consider a language with the set of well-formed formulas a , and a set of its subsets A 2' closed under arbitrary intersections. Elements of A are called A-logics. We define a semantics for the set A as a quadruple 9 = (a,A,U, k), in which U is a class (whose elements are called 24-frames), and I=is a binary relation between U and (called the validity relation) such that the set
CHAPTER 2. BASIC PREDICATE LOGIC
186
is a A-logic for any F E U. Ls(F) is called the A-logic of the frame F (in 8 ) . Sometimes we write F E 9 instead of F E U. Since A is intersection closed, we obtain that for any class C C U, the set Ls(C) = n{La(F) I F E C) is also a A-logic; it is called the A-logic of C . Logics of this kind are called Scomplete, and logics of the form Ls(F) are called simply 9-complete. As stated in [Ono, 1973, Theorem 2.21, completeness does not always imply simple completeness. A semantics 9 gives rise to the logical consequence relation (between r C @ and A E a):
I? ks A iff Q F E U (I? C Ls(F) + A
E Ls(F)).
We say that F is a r-frame (in 9) if I? Ls(F). Thus r ks A means that A is valid in every I?-frame. If F is a r-frame such that F !+A, we say that F separates A from a set I?. One can recognise a particular case of Galois correspondence here. This correspondence is derived from the relation != in the standard way; cf. [Chapter 5, Theorem 191[Birkhoff, 19791. So
is a closure operation on 2', complete logics. Hence we have
and the closed sets of formulas are just the S-
Lemma 2.16.1 CS(r) is the smallest 9-complete logic containing I?. C s ( r ) is called the 9-completion of I?. Here is a simple criterion of completeness:
Lemma 2.16.2 A logic L is complete in a semantics S'iff every formula A E (@- L) is separated from L by a frame in 9. Proof
In fact, by 2.16.1, L is complete iff Cs(L) = L.
We can compare different semantics for the same set of logics. A semantics S1 is reducible to S2 (notation: S1 5 82) iff Cs, (L) Csl (L) for any L E A.
Lemma 2.16.3 (1) 91 5 S2 ifl every S1-complete logic is S2-complete.
(2) S1 5 9 2 iff for any 91-frame F, Lsl ( F ) is 92-complete. Proof (1) Suppose 91 5 S2. Then for any logic L,
c
(L)
c c ~(L).l
c ~ 2
If L is S1-complete then L = Csl (L) (by Lemma 2.16.1), and hence L = Cs, (L) which implies 92-completeness of A (again by Lemma 2.16.1). Conversely, assume that 91-completeness implies Sz-completeness. Then Cs, (L) is 92-complete, and thus Cs,(L) Csl(L) by Lemma 2.16.1.
c
2.1 6. SEMANTICS FROM AN ABSTRACT VIEWPOINT
(2) 'Only if' is an immediate consequence of (i) and Lemma 2.16.1. To prove 'if' consider an arbitrary S1-complete logic
If each Lsl (F) is 92-complete, we have that Lsl(F) = L$,(*F) for some class QF. Then
showing that A is 92-complete.
Definition 2.16.4 Semantics S1 and S2 are called equivalent (S1 = 92) if S1 5 S2 and S2 3 S1; we say that S1 is weaker than S2 (notation: S1 4 S 2 ) if Sl 5 S2 but not S2 3 S1. Lemma 2.16.3 readily implies: L e m m a 2.16.5 S1 z: S2 iff S1-completeness is equivalent to S2-completeness. Definition 2.16.6 A semantics S1 is simply reducible to S2 (notation: S1 S2) if simple S1-completeness implies simple Sa-completeness, i.e.
C
Semantics 91 and 9 2 are called simply equivalent (notation: S1 E S 2 ) if S1 G S2 and S2 C S1. We do not usually distinguish between simply equivalent semantics; from this viewpoint a semantics 9 = (a,A,U, I=) can be identified with the corresponding set of simply complete logics {Ls(F) I F E 2.4). L e m m a 2.16.7 S1 C S2 implies S1 5 92. Proof ones.
Obvious, since complete logics are intersections of simply complete H
E x a m p l e 2.16.8 For any set of logics A there exists a 'trivial semantics' T,in which U = A (i.e. 'frames' are just logics) and L b A iff A E L, i.e. LT(L) = L). Definition 2.16.9 We say that a semantics 6 has the collection property (CP) if every S-complete consistent logic is simply 9-complete. By Lemma 2.16.7, for semantics with the (CP), reducibility is equivalent to simple reducibility, so their equivalence implies simple equivalence. But in general equivalent semantics may be not simply equivalent; some counterexamples will be given later on.
CHAPTER 2. BASIC PREDICATE LOGIC
188
Example 2.16.10 In modal propositional logic we have semantics with the (CP): finite
4
Kripke
4
topological
4
algebraic S trivial.
'Finite semantics' consists of all finite Kripke frames, with the usual definition of validity. So in this semantics simple completeness means tabularity and completeness means the f.m.p. Algebraic semantics is simply equivalent to trivial, since every logic is algebraically complete, by the Lindenbaum theorem. For superintuitionistic propositional logics we have the following diagram: finite 4 Kripke 4 topological 5 algebraic S trivial. The question, whether topological semantics is trivial, is Kuznetsov's problem mentioned in Section 1.17. Recall that in this book we consider four types of predicate logics - modal or intuitionistic, with or without equality. So we alter the notation Ls(F) respectively. For example, if F is a frame in a semantics S for N-modal predicate logics with equality, for {A E MF; I F != A} we use the notation M L = ( F ) rather than Ls(F); M L ( F ) refers to semantics without equality. Similarly in the intuitionistic case we use the notations IL'(F), IL(F). Every semantics 9 for logics with equality generates a semantics So for the same kind of logics without equality, with the same frames and validity. So LSo ( F ) = Lg(F)O. Loosely speaking, we call So-complete logics 'Scomplete'.
Remark 2.16.11 However it may happen that there exists a semantics for logics without equality Sf, with the same frames as 9 (and So), but with a slightly different notion of validity. In these cases we define M L ( F ) (or IL(F)) as Ls(F) and then prove that Sf = So. We will encounter such a situation in Chapter 5. Definition 2.16.12 Let S be a semantics for N-m.p.1. We say that S admits equality if there exists a semantics S1 for N-m.p.l.= such that 9 9; , i.e.
c
Proposition 2.16.13 If S admits equality and an N-m.p.1. L is S-complete, then L' is conservative over L. Proof Almost obvious. Suppose L is N-modal, L = ML(C) for a class of Sframes C and M L ( F ) = MLF(Fl)O for any F E C. Then
Put C1 := {Fl I F E C), then for any A E M F N
2.1 6. SEMANTICS FROM A N ABSTRACT VIEWPOINT
189
Thus ML=(C1)O = L, i.e. ML'(C1) is conservative over L = ML(C). Since ML'(C1) is a logic with equality containing L, it follows that L= C ML'(C1) and thus L= is also conservative over L. H Now let us prove a simple result on correlation between completeness of L and L= . Proposition 2.16.14 Let L be an m.p.1. or an s.p.l., 9 a semantics for the corresponding logics with equality. If L= is conservative over L and 9-complete, then L is also 9-complete. P r o o f Consider the modal case only. Suppose L= = ML'(C) for a class of Sframes C. Then by conservativity, L = (L=)O = ML'(C)O, and obviously, ML'(C)O = ML(C). Thus L is Scomplete. Corollary 2.16.15 If L is an s.p.1. or a conically expressive m.p.1. and L= is complete in a semantics 9, then L is also %-complete. Proof
rn
By 2.16.14 and 2.14.8.
Finally let us show the existence of modal counterparts using completeness. Definition 2.16.16 Let M be a semantics for m.p.l.(=) extending QS4(=). Then we define its intuitionistic version, the semantics % for s.p.1. (=), with the same frames as M such that for any M-I-frame F,
Proposition 2.16.17 Every %I-complete s.p.l.(=) has a modal counterpart:
P r o o f The inclusion 2 is proved in Lemma 2.11.11. For the converse suppose Q H + I' YA. By assumption QH r is %I-complete, so there exists an Mframe F such that I? E L%(F), but A Lw(F). By Definition 2.16.16, L w ( F ) = TLM(F),hence rT LM(F), AT !$ LM(F). Since M is a semantics for modal logics above QS4, we have LM(F) _> QS4 r T . Consequently Q S 4 rTVAT, i.e. T(QS4 r T ) YA. W
+
+
+
e
+
As we shall see in Volume 2, the smallest modal counterpart of an %complete logic may be M-incomplete. But completeness always transfers in the other direction: Proposition 2.16.18 If L Proof
> QS~(') is M-complete, then TL is %I-complete.
Suppose L = LM(C)for a class of M-frames C. Then
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Part I1
Semantics
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INTRODUCTION: WHAT IS SEMANTICS
Introduction: What is semantics? In Chapter 2 we proposed a general approach to semantics, which is very formal and does not help understand the meaning of language expressions. For example, this approach allows for a degenerate semantics, where 'frames' are just logics and L I= A iff L k A. In this case the completeness theorem is a triviality, but of course such a semantics does not explain anything.l To describe more plausible semantics we need to define notions of a model and of the truth in a model that could tell us 'how the logic works'. This is not a serious problem for classical first-order logic: in this case model theory can be developed within the standard semantics based on the well-known Tarski truth definition. Due to Godel's completeness theorem (GCT), the standard semantics works properly, and thus alternative types of semantics (such as sheaves, forcing, polyadic algebras) are of less importance in the classical case. The situation in nonclassical first-order logic is quite different. GCT has no direct analogues, and incompleteness phenomena enable us t o consider various semantics, without any obvious preference between them. Nonclassical model theory is still rather miscellaneous, and this book is aimed at organising some part of it. The book does not cover all the semantics in equal proportion. So we begin this part with a brief survey of important results in the area and some references for further reading. GGdel7scompleteness theorem: discussion First let us explain why GCT is not always transferred to nonclassical logics. Actually there exist two forms of Godel's theorem:
(GCTl) A formula A is a theorem of classical predicate calculus iff A is valid in any domain. (GCT2) Every consistent classical theory has a model. These two statements are more or less equivalent: (GCTl) is equivalent to (GCT2) for finitely axiomatisable theories. However in the nonclassical area it is essential to distinguish between logics and theories. A theory is an extension of some basic calculus by additional axioms (and perhaps inference rules) postulating specific properties of objects being studied; a typical example is Heyting arithmetic, which formalises the intuitionistic viewpoint on natural numbers. A logic is a theory whose theorems may be considered as 'logical laws' common for a certain class of theories and not depending on specific 'application domains'. We can treat logical laws as 'Of course in general we may have a new logic and several syntactical translations into well-known logics. Such translations can be viewed a s semantics and can be very illuminating. See [Gabbay, 1996, Chapter 11.
194
INTRODUCTION: W H A T IS SEMANTICS
schemata for producing theorems, and define a logic just as a substitutionclosed theory.2 So we can say that (GCT2) is a property of classical theories, and (GCTl) is a property of classical logic. Some results similar t o (GCT2) can be proved for nonclassical theories. In fact, it is well known that every consistent first-order normal modal theory S is satisfied in some Kripke model, i.e. there exists a Kripke model M and a possible world w E M such that (M, w) k A for any A E S. Moreover, M can be chosen uniformly for all consistent theories [Gabbay, 19761. Analogous claims are true for superintuitionistic theories considered as pairs of sets of formulas, cf. [Dragalin, 19881. However finding nonclassical analogues of (GCTl) is more problematic. These are completeness theorems of the following form: (GCTlt) a formula A is a theorem of a logic L iff A is L-valid. The main problem is in finding 'reasonable' notions of validity, for which (GCTl') may be true. In classical logic this is validity in a domain. In a general case, because we require logics to be substitution-closed, we also need 'substitutioninvariant' notions of validity. So for a nonclassical logic L, analogues of domains are frames (or model structures), from which we can obtain models of theories based on A if we specify interpretations of basic predicates; a formula is valid in a frame iff it is true in every model over this frame. It is usually required that the set of all formulas true in a model is a theory, and thus the set L(F) of all formulas valid in a frame F is also a theory.3 The difference between models and frames is the following substitution property implying that L(F) is a logic: (SP) The set of all formulas valid in a frame F is substitution closed.
Generally speaking, there may be many kinds of 'frames' and 'validity'. We say that a class of frames C generates a semantics S(C) = {L(F) I F E C).
A logic L is called complete in S(C) iff for every formula A @ L there exists a frame IF E C, such that formulas from L are valid in F, whereas A is not. Equivalently, a logic is complete iff it can be presented as an intersection of logics of the form L(F) for some frames F. For example, in the standard classical semantics 'frames' are just sets, and the notion of validity is well known. So (GCT1) means that the classical predicate logic is complete in this semantics. 2 0 f course this definition is rather conventional, and there exist examples of 'logics' which are not substitution-closed. 3 ~ h i set s depends also on the language, where formulas are constructed.
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Examples of incompleteness in first-order logic In Chapter 1 we gave a picture of semantics in nonclassical propositional logic and pointed out some rather strong completeness results. But in first-order logic the situation becomes worse: there are very few completeness theorems known, and incompleteness is very frequent. Incompleteness already appears for logics extending the classical predicate logic QCL. In fact, consider any formula A valid in all finite domains, but refutable in some infinite domain (and thus refutable in any infinite domain, by the Lowenheim-Skolem theorem). The logic QCL+A is incomplete, because the set of all finitely valid formulas is not recursively axiomatisable [Trachtenbrot, 19501. This argument seems to be just a trick with definitions. But in nonclassical logic there exist more natural examples of incompleteness. The first example of this kind was discovered by H.Ono [Ono, 19731, who proved that the intermediate logic of the strong Markov principle QHE =Q H
+~
~ x P ( >x 3)x l l P ( x )
is incomplete in the standard Kripke semantics (K). Recall that X: is generated by predicate Kripke frames; such a frame is a triple IF = (W, R, D ) , in which (W, R) is a propositional Kripke frame; D is a system of 'expanding individual domains', i.e. a family of non-empty sets indexed by possible worlds (Dw)wEW, such that Qu, v (uRv J Du
Dv).
The logics Q S 4 and Q H are known to be K-complete. But their minimal equality extensions (QS4', QH') are incomplete if the symbol '=' is interpreted in every domain just as the identity. This is due to the formulas D E = QxQy (x = y V ~ ( =xy))
(decidable equality principle)
and
C E = QxQy ( O ( x = y)
> x = y)
(closed equality principle)
which are valid in all Kripke frames, but nonprovable respectively in QH' and in QS4'. But we can also interpret equality in a world w in another natural way, as an equivalence relation in D,. Then we come to the semantics KE of Kripke frames with equality (KFEs), in which QS4=, QH= become complete; so KE is stronger than K. The latter observation and the definition of KE for the intuitionistic case first appeared in [Dragalin, 19731 and then in [ ~ r a ~ a l i19881. n, Note that in the classical case these two approaches are equivalent: every 'non-normal' model, in which equality is an arbitrary equivalence relation can be 'normalised' by identifying equal elements, so that its elementary theory does not change. For a Kripke frame with equality a similar construction is possible: one can factorise every individual domain through the corresponding equivalence
INTRODUCTION: WHAT IS SEMANTICS
196
relation. Thus we obtain a Kripke sheaf; it can be defined as a propositional Kripke frame (W, R) together with a system of individual domains (Dw)wEW and transition maps puv : Du ---+ D, for every R-related pair u, v, satisfying the natural transitivity condition:
and such that puu is the identity map for every R-reflexive u. So every KFE corresponds t o a Kripke sheaf with the same modal logic. Note that predicate Kripke frames correspond to those Kripke sheaves in which all transition maps are inclusions. Although the semantics K& was introduced for dealing with equality, it happens to be stronger than K also for logics without equality. This can be seen again by analysing Ono's counterexample Q H E . Algebraic and Kripke-type semantics The semantics K& also is inadequate in many cases, in particular for quantified versions of all intermediate propositional logics of finite depths. K-incompleteness of these logics was proved by H. Ono in [Ono, 19731, and the proof is easily transferred to KE. Another counterexample (found independently by S. Ghilardi and V. Shehtman & D. Skvortsov) is presented by the intermediate logic of 'the weak excluded middle' with constant domains: QHJD =Q H
+ ~p V
lip
+ b'x(P(x) V q) 3 (b'xP(x) V q).
So one can try to generalise K& in a 'reasonable way' . Generalisations can be done at least in two directions. The first direction leads to the algebraic semantics A&. In the intuitionistic case a frame in A& is a Heyting-valued set [Fourman and Scott, 19791. This is a set of 'individuals', in which every individual has a 'measure of existence' (an element of some Heyting algebra) and every pair of individuals has a 'measure of equality'. For the modal case Heyting algebras are replaced by modal algebras. The neighbourhood (or topological) semantics I& is a particular case of A&, involving only algebras of topological spaces (or algebras of propositional neighbourhood frames for the modal case). Both semantics A&, I& are not much investigated and seem to be rather strong. We do not know if there exist A&-incomplete logics. On the other hand, we do not know simple and natural examples of logics that are A&-complete, but KE-incomplete. Another direction leads from K& to the semantics of Kripke quasi-sheaves (KQ), and further to the semantics of Kripke bundles (KB). In Kripke bundles transition maps p, : D, D, are replaced with transition relations puv 2 D, x D, collected in the unified accessibility (or inheritance, or counterpart) relation between individuals. Every individual has an inheritor (not necessarily unique) in any accessible world. The idea of counterparts first appeared in [ ~ e w i s 19681, , and in a formal setting - in [Shehtman and Skvortsov, 19901. A Kripke bundle can be defined
-
INTRODUCTION: WHAT IS SEMANTICS
197
p) onto a as a pmorphism from a (propositional) frame of individuals (D+, frame of possible worlds (W, R). Then Kripke sheaves correspond exactly to etale maps, similarly to the well-known fact in sheaf theory [Godement, 19581. In Kripke bundles an individual may have several inheritors even in its own possible world. Kripke quasi-sheaves are a subclass of Kripke bundles, in which this is not allowed. Further generalisations of K B are the functor semantics (FS), in which 'frames' are set-valued functors (or presheaves over categories); the metaframe semantics, and finally the hyperdoctrine semantics generalising both metaframe and algebraic semantics. In hyperdoctrine semantics every logic is complete, but this semantics is very abstract, and thus not suitable for large-scale applications. There are several other options, which will be considered later in this book (Volume 3), such as Kripke-Joyal-Reyes semantics [Goldblatt, 1984; Makkai and Reyes, 19951 or 'abstract realisability' [Dragalin, 19881. Also we will discuss the approaches where the basic language is modified, e.g. by adding the existence predicate [Fine, 19781, various kinds of quantifiers [Garson, 19781or nonstandard substitutions [Ghilardi and Meloni, 1988). This may help the reader to find his way through the landscape of first-order logic.
Substitution property: discussion General Kripke-type semantics have some peculiarities: it turns out that in Kripke bundles (and even in Kripke quasi-sheaves) the above mentioned substitution property fails. In fact, a very simple counterexample shows that the (intuitionistic) validity of p V l p does not necessarily imply the validity of P(x) V -IP(x) (cf. Chapter 5). To avoid this, the 'straightforward' notion of validity has to be changed. So we introduce strong validity in a frame: a formula is said to be strongly valid iff all its substitution instances are valid (in the original sense). Then the set L(F) of all formulas strongly valid in a frame IF is a logic (called the logic of IF). One may argue that after such a modification the notion of frame becomes almost useless and can be replaced by the notion of model. For, in the same manner we can define a 'logic' L ( M ) of an arbitrary Kripke model M as the set of all the formulas 'strongly verifiable' in M , i.e. L(M) contains a formula A iff all substitution instances of A are true in M . Then we get a first-order version of general propositional frames mentioned above. In this semantics (GCT2) follows from (GCTl), and thus every modal or superintuitionistic first-order logic is complete. None the less, there is some advantage in dealing with Kripke bundles (or metaframes) rather than Kripke models. For, let us recall the two kinds of substitutions considered in Chapter 2: a strict substitution does not add new parameters to atomic formulas; a shift adds a certain fixed list of new parameters to every atomic formula. Recall that a simple substitution instance Ak of a formula A is obtained by replacing every atom PC(x1,. . . ,x,) . . ,~x ,n ,y1,. . . ,yk), where yl,. . .yk are variables not occurring in by P ~ + ~ ( .X A. As we know, every substitution is a composition of some strict and some
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INTRODUCTION: W H A T IS SEMANTICS
simple substitution and a variable renaming. One can prove that the set L-(P) of all formulas valid in a Kripke bundle P is closed under strict substitutions and variable renaming. Therefore the logic of F can be derived from L-(IF) as follows: L(F) = {A V k 2 0 E L-(IF)).
I
So to check strong validity of a formula A there is no need to verify all its substitution instances, but only the Ak are sufficient. In other words, strong validity is nothing but validity with arbitrarily many extra parameters. Unlike that, the set of formulas true in a Kripke model M is usually not closed under strict substitutions. So in general there is no way to describe the logic of M other than checking the truth of all possible substitution instances of formulas. This may be very difficult even in the propositional case; in algebraic terms, this means describing equations in a subalgebra of the modal algebra of a Kripke frame with a given set of generators. So the semantics of Kripke bundles is in some sense 'more constructive' than the semantics of Kripke models (not frames!). Anyway the algebraic semantics A£ is the strongest semantics with (SP), that we know of. The Kripke-type semantics stronger than the Kripke bundle semantics Kt3 (such as the functor semantics 3s or the metaframe semantics M F ) do not have the (SP) either, but here again only simple substitutions are essential for strong validity.
Chapter 3
Kripke semantics 3.1
Preliminary discussion There is no remembrance in former things; neither shall there be any remembrance of things that are to come with those that shall come after. (Ecclesiastes, 1.11.)
In propositional modal logic Kripke semantics is widely used and is very helpful. As pointed out in Chapter 1, natural logics turn out to be Kripke-complete, and moreover, many of them have finite model property. This makes Kripke semantics an efficient model-theoretic instrument in the propositional case. In the first-order case one can try to generalise the propositional Kripke semantics in a straightforward way. Consider the first-order language C1with a single modal operator (Section 2.1). Let (W, R) be a 1-modal propositional Kripke frame (Definition 1.3.1). We can define a 'first-order Kripke model' over (W, R) as a collection of classical models parametrised by possible worlds:
where every Mu is a classical L-structure, i.e. at the world u the basic predicates are interpreted as in Mu. This is not yet enough for a formal definition, because we have to answer the following two questions: What are the individuals in M? How are the quantifiers interpreted? The simplest way is to assume that every individual is an element of some Mu; more exactly, if D, is the domain of Mu, then M has the set of individuals
Df =
U D,. uEW
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200
We can say that D, consists of the individuals existing in the world u. Thus some individuals may exist in one world, but may not exist in another world. A more realistic example is given by a moving lift. Here W is the set of moments of time, R is the earlier-later relation, D, is the set of people inside the lift at the moment u. From this viewpoint, it is natural to quantify only over existing individuals; thus Vxcp(x) must be true at the world u iff cp(a) is true for every a E D,. In our example, if P(x) is interpreted as x i s a child,
k QxP(x) means that at the moment u (it is true that)
then u
only children a r e i n t h e l i f t .
Eventually, we can define the forcing relation u cp between a world u and a D,-sentence (cf. 2.2) cp by induction on the complexity of cp, so that
u k PC(a1,. .. ,a,) VXCP
I=
21
k~
X P
u k o ~
iff iff iff iff
Mu P,"(al,. . . ,a,) (classically) Qa E D, u k [a/x]cp 3a E D, u k [a/x]cp Vv (uRv & cp is a D,-sentence + v
k cp)
and with the standard clauses for the classical propositional connectives. Returning again to our example, for a certain individual a (say, Robert Smith) u k OP(a) means: at the moment u it is true that always i n t h e f u t u r e Robert Smith w i l l use t h e l i f t only while he i s a child.
This is an essential point: we cannot state u k P ( a ) for ax individual a that is not present in D,; and thus to check u 1OP(a), we have to consider only those v in R(u), which have a in their domain. Once the forcing relation is defined, we can say that a formula cp is true in M if u k vcp for any u E W. If FV(cp) = { X I , . . . ,x,), the latter is equivalent to
u
k
[al, . . . ,a,/xl,
. . . ,x,] cp for any a l l . . . ,a,
E D,.
Now what formulas are true in every Kripke model? This set can be axiomatised, but unfortunately, it may not be a modal predicate logic in the sense of Chapter 2. To see what happens, consider the formula
which is a theorem of K (propositional) and is obviously true in all models defined above. However take the substitution instance of a:
This formula is not always true. For, consider the model with two possible worlds u, v and two individuals a, b, such that
3.1. PRELIMINARY DISCUSSION
R = {(u, u), (u, v)), D, u
P(a) A P(b), v
= {a, b), D, = {a), P(a), see Fig. 3.1.
Figure 3.1. Then we have u O(P(a) A P(b)), since P ( a ) A P(b) is not a D,-sentence, but u OP(a). Here is a more natural analogue of this example: if John and Mary always t a l k when they a r e i n t h e l i f t t o g e t h e r ,
it may be not the case that John always t a l k s i n t h e l i f t .
Now there are several options t o choose: A. If we need semantics for the logics described in the previous chapter, we can try to amend the above definition.
B. Alternatively, we can change (actually, extend) the notion of a logic and try to axiomatise the 'logic' which is complete with respect to the above interpretation.
C. We can accept other definitions of semantics, using different kinds of individuals and different interpretations of quantifiers. In this Volume we accept OPTION A and keep the same notion of logic. There is not so much to change in the original definition: it is enough to assume that
202
CHAPTER 3. KRIPKE SEMANTICS
'individuals are immortal', which in precise terms, means the expanding domain condition: (ED) u R v + D u g D,. Of course this requirement contradicts the viewpoint expressed in the epigraph to this section.' But it saves the logic, and as we will see later on, this semantics is sound and complete for a few well-known logics, such as QK, QS4, etc. Furthermore, (ED) is essential in semantics of intuitionistic logic. In this case (W, R) is an S4-frame, and we would like to keep the intuitionistic truthpreservation principle (TP) from Lemma 1.4.4 for atomic D,-sentences:
The inductive definition of forcing from Section 1.3 can be extended t o the first-order case so that u It 3xcp iff 3a E D, u IF [alxly. Now, without (ED), we cannot guarantee the truth-preservation for all D,sentences. For, take a model based on a two-element chain:
w = { ~ 7 ' ) 1 R = {(u,u),( u , ~ )( ,v , ~ ) ) Du , = {a), Dw= {b), u It P(a), v Iy P(b).
Figure 3.2. In this case the truth-preservation holds for P(a) (because a exists only in u), but it fails for 3xP(x), because obviously,
To restore the truth-preservation, we can change the definition, e.g. as follows: u It 3xy iff 'dv E R(u) 3a E D, u Ik [a/x]y, 'Commonsense understanding of individuals is not easy to formalise, and there were many philosophical debates on that subject. 'What is an individual? - This is a good question!' - Dana Scott writes in [Scott, 19701. Note that in labelled deductive systems, individuals are labelled and may be deemed t o have internal structure. Wait for a later volume on this.
3.1. PRELIMINARY DISCUSSION
203
cf. the clause for the implication in the intuitionistic case. Another alternative is: u IF 3 x 9 iff 3a E D, Vv E R(u) u It [a/x]cp, which means that only 'immortal' individuals are considered as existing. But nevertheless without (ED), the semantics is not sound, and some intuitionistic theorems may be false. For example, consider the formula P(x) > 3xP(x). In the above model we have u It P(a), but u Iy 3xP(x), since uRv, and v Iy P(b), b being the only individual in v. Thus u Iy P(a) > 3xP(x). We will return to this subject in Volume 2 of our book. Note that anyway, the principle (ED) is quite natural for intuitionistic logic. It normalises the situation, and the logic QH becomes sound and complete in an appropriate semantics. OPTION B is close to Kripke's treatment in [Kripke, 19631; it will be also considered later on (in Volume 2). OPTION C includes many different approaches; we mention some of them. OPTION Cl (modal case). As before, a Kripke model is a collection of classical structures, but quantifiers now range over all possible individuals i.e. over the whole set D+. This is so-called possibilist quantification. As individuals may not exist in some worlds, we have t o add the unary existence predicate E to our language. Now instead of D,-sentences we evaluate D+-sentences (where arbitrary individuals are used as extra constants). The truth condition for V becomes u
+ Vxcp
iff Va E D+ u
[alzlcp.
[alxlcp may also hold if a # D,.2 Note that u Option B can be realised within this approach as well, if we interpret Vxcp(x) as Vx(E(x) > (p(x)). It turns out that the minimal logic QK is sound, but incomplete in this semantics. This is due to the formula
found by Ruth Barcan and named after her. It is easily checked that the Barcan formula is true in every model, but is not a theorem of QK, as the following countermodel (of the type A ) shows (Fig. 3.3). VxOP(x). On the other hand, Here we have u OP(a), and so u v VxP(x), and thus u '& OVxP(x) since uRv. Although axiomatising the minimal complete logic for the C1-semantics is simple, this approach is controversial, because quantification over all individuals is rather ambiguous, at least in natural language.3 Still option C1 is possible and useful by formal reasons.
+
2 ~ natural n language a non-existing individual can still have a name in the world u. We can say: S i r Isaac Newton discovered the laws of motion, using the name of a person given after the event had happened. 3Quantification in natural language is a complicated subject, not t o be discussed here. We remark only that in most cases quantification refers to the actual world, a s in A l l students
CHAPTER 3. KRIPKE SEMANTICS
Figure 3.3. OPTION C2. This is actualzst quantification (see [Fitting and Mendelsohn, 1998]), a combination of B and Cl. Now u (P is defined when p is a Df-sentence, but u PF(al,. . . , a n ) is put to be false when some of a l , . . . , a n do not exist in the world u: only actual individuals are allowed to have atomic proper tie^.^ The inductive definition of forcing is the same as in the, cases A, B. However, there is a difference in the definition of the truth in a model:
+
+
+
( ~ ( x l ., . ,x,) iff u cp(a1,. . . , a n ) for anyu, for any a l , . . . , a n in D+ (not only in Du).
M
This approach again breaks the logic: the formula VxP(x) > P(y) may be false, while Vy(VxP(x) > P(y)) is always true. FURTHER OPTIONS. One may argue that one and the same individual cannot exist in different worlds. In fact, nobody would identify a newly born baby and an old man or woman. This suggests we consider disjoint individual domains, together with transition maps (or relations) between them, as mentioned in the Introduction to Part 11. Another idea is to treat individuals as changing entities and consider indzvidual concepts, that are partial functions from W to the set D+ (of 'individual images'). have supervisors, but there also exists possibilist quantification, as in Any student has a supervisor. In the latter case it is not quite clear whether the quantifier ranges over the whole D+ or not. Cf. [ ~ r o n ~ a u19981 z , studying these matters in Russian. 4 ~ o t that e in the case C1 this is not required, and basic predicates may be true for nonexisting individuals. This happens in natural languages as well. For example, Mike likes Socrates may be evaluated as true in the actual world.
3.2. PREDICATE KRIPKE FRAMES Now u
P ( a l , . .. ,an) is defined if al(u), . . .,an(u) exist, and u
VX(P(X)iff Va (a(u) exists
+u
cp(a)).
In this case the logic grows much larger than QK and even becomes not recursively axiomatisable. We shall return to this topic in Volume 2.
3.2
Predicate Kripke frames
Now let us turn to precise definitions and statements. Definition 3.2.1 A system of domains over a set W empty sets D = (DU)uEW.
#
0
is a family of non-
Definition 3.2.2 Let F = (W, R1,. . . ,R N ) be a propositional Kripke frame. A n expanding system of domains over F is a system of domains D over W such that Vi E IN Vu,v E W (uRiv +-D, C D,).
A predicate Kripke frame over F is a pair F = (F, D), i n which D is an expanding system of domains over F . The set D, (sometimes denoted by D(u)) is called the individual domain of the world u; the set D+ := Du
u
uEW
is the total domain of F; the frame F (also denoted by F,) is called the propositional base of F. The following observation is an easy consequence of the definition.
Lemma 3.2.3 Let F Du C Du.
=
(F,D) be a predicate Kripke frame, v
E FTu.
Then
Proof By Lemma 1.3.19, v E R,(u) for some (I: E I F . Then we can apply induction on la1 In fact, for cr = h we have v = u, so D, = D,; if cr = pi and uR,v, then uRpwRiv for some w, so D, E D, D, by the induction hypothesis and 3.2.2. W By default, we denote an arbitrary propositional Kripke frame by F =
(W, R1,. . . ,R N ) and an arbitrary predicate Kripke frame by F = (F, D). For an individual a E Df the set
is called the measure of existence (or the extent). Since the system of domain D is expanding, E(a) is a stable subset of F .
CHAPTER 3. KRIPKE SEMANTICS
206
For a tuple a E (D*)n we also introduce the measure of existence
n n
E(a) :=
.(ai)
= {u E
W ( r ( a ) C D,).
i=l
The set is also stable in F. In particular, for an empty a we define E ( A ) as W (since r(A) = 0). Definition 3.2.4 A (modal) valuation J in a predicate Kripke frame F is a function associating with every predicate letter a member of the set 2Dr,
Pr
n
uEW
i.e. a family of m-ary relations on individual domains:
where J u ( P p ) C D r . TO include the case m = 0, we assume that DE = {u); so tu(P;) is either {u) or 0 . The pair M = ( F , J) is called a (predicate) Kripke model over F . We may call the function Ju sending every n-ary predicate letter to an nary relation on D,, a local valuation i n F at u ; this is nothing but a classical valuation in D,. So we can say that J is a family of local valuations (Ju)uEw. Definition 3.2.5 Let M = (F,J) be a Kripke model. For u E M, the classical model structure Mu := (D,,Ju) is called the stalk (or the fibre) of M at u. On the other hand, we may call a function sending every n-ary predicate letter to an n-ary relation on D+, a global valuation in F . Let us also define another kind of valuation in predicate Kripke frames. Definition 3.2.6 A global valuation i n a predicate Kripke frame F is a function y sending every n-ary predicate letter P ( n > 0) to an 'n-ary 2W-valued predicate on D+', i.e. to a function y(P) : (D+)n + 2W such that for any a E ( D f ) n , y(P)(a) C E(a), and every q E P L O to a subset of W . So y 1 P L O is a propositional valuation in the propositional frame F. We can also regard it as an O-ary predicate, i.e. as a function (D+)O --+ 2W, where (D+)O = {A). In this case the condition y(q)(a)
c E(a)
trivially holds for any a E (D+)' (i.e. for A), since E ( A ) = W. L e m m a 3.2.7 Let F = ( F ,D) be a predicate Kripke frame. (1) For any valuation 6 i n F there exists a global valuation 6+ such that for any P E PLm, m > 0, for any a E (D+)m
and for q E P L O Of((?)= {u I QZL(P)= (~11,
3.2. PREDICATE KRIPKE FRAMES (2) Every global valuation y equals <+ for a unique valuation €. Proof
(1) From the definition it is clear that
for any P E PLn, n
> 0, a E (D+)n.
(2) Put
C U ( P , := ) {a I u E y ( P r ) ( a ) ) for m > 0 and
e,(p,O) := {u> if U E y(P,O), 0 otherwise. Then for P E PLm, m > 0
u E y ( P ) ( a )iff a c & ( P ) iff u E t f ( p ) ( a ) , and similarly for m = 0. The uniqueness of [ is also clear; y = <+ means that for P E PLm ( m > O ) , a E (D+)m, u E W
and for q E PLO E ~ ( 9@ ) u E €u(Q).
Definition 3.2.8 A 1-modal predicate Kripke frame F is called %-based, or intuitionistic i f its propositional base F, is an S4-frame. I n this case a modal valuation t i n F (and the corresponding Kripke model) is called intuitionistic i f it has the truth preservation (or monotonicity) property:5
(TP)
u R v = . J u ( p ~ )C t v ( p r )( i f m > O ) ; uRv & u E JU(P;)+-v E € v ( P j ) .
To define forcing, we use the notion of a D-sentence introduced in Section 2.4. Definition 3.2.9 For a Kripke model M = ( F ,J ) , F = ( F ,D ) , F = (W,R1,. . . ,R N ) we define the (modal) forcing relation M, u I= A (in another notation: <,u k A, or briefly: u k A) between worlds u E W 6 and D,-sentences A. The definition is inductive: 5Recall that we omit the subscript '1' in the 1-modal case. 'Sometimes we write u E F or u E M instead of u E W.
CHAPTER 3. KMPKE SEMANTICS
208
M , u b Pf iff u E J U ( P f ) ; M , u k P,"(a) i f f a
a JJ(P,") ( f o r m > 0);
M , u k a = b iff a equals b,
M , u k B > C iff ( M , u ! # B o r M , u k C ) ;
Remark 3.2.10 The above truth definition for propositional letters seems peculiar. It is more natural to use truth values l and 0 rather than { u ) and 0 . But our definition will be convenient later on, in Chapter 5. Now we readily obtain an analogue of Lemma 1.3.3:
L e m m a 3.2.11 For any Kriplce model:
u!=OiB iff 3v E Ri(u) v k B ; u k 7 B iffu!#B; u k O,B iffVv E R,(u)v k B ; u k 0,B iff 3v E R,(u) v != B . Similarly to the propositional case, we give the following inductive definition of intuitionistic forcing.
Definition 3.2.12 Let M = ( F ,J ) be an intuitionistic Kriplce model. We define the intuitionistic forcing relation M , u It A between a world u E F and an intuitionistic Du-sentence A (also denoted by J , u Il- A or just by u It A ) by induction:
M , u Il- A iff M , u k A for A atomic;
M , U I ~ B A ~C ~ ~ M , u I ~ - B & M , u I I - c ; ~ M , u l t B v C iff (M,ulk B o r M , u l k C ) ;
7 ~ o t that e B is a Dm-sentence,since D is expanding. 8 ~ e r& e is an abbreviation of 'and', and further on in this definition Vv E R(u) abbreviates 'for any v in R(u)',etc.
3.2. PREDICATE KlUPKE FRAMES M , u It- 3xB iff 3a E D, M, u It- [a/x]B; M, u Il- VxB iff Vv E R(u) Va E D, M, v Il- [a/x]B. Hence we easily obtain L e m m a 3.2.13 The intuitionistic forcing has the following properties M , u I k l B iffVv 'v R(u) M,vIy B; M , u I t - a f b i f f a , b ~D, are not equal.
Definition 3.2.14 Let F be an S4-based Kripke frame; M a Kripke model over F. The pattern of M is the Kripke model Mo over F such that for any u E F and any atomic Du-sentence without equality A
Obviously, every Kripke model M over F has a unique pattern; Mo is an intuitionistic Kripke model and Mo = M if M is itself intuitionistic. L e m m a 3.2.15 If Mo is a pattern of M , then for any u E M , for any intuitionistic Du-sentence A
Proof Easy, by induction on the length of A. The atomic case follows from Definition 3.2.14 (and is trivial for A of the form a = b). Let us check only the case A = VxB; other cases are left to the reader. Mo, u It- VxB iff Vv E R(u) Va E D, Mo,u Il- [alx]B (by Definition 3.2.12) ) ~ the induction hypothesis). iff Vv E R(u) Va E D, M, u b ( [ ~ / x ] B (by On the other hand, by Definition 3.2.9 M, u It- AT(= O V X B ~iff) Vv E R(u) Va E D, M, u k [ a / x ] ( ~ ~ ) , and it remains to note that ( [ ~ / x ] B=) ~[a/x](BT), by 2.11.3.
W
L e m m a 3.2.16 Let u, v be worlds in an intuitionistic Kripke model M . Then for any intuitionistic Du-sentence A
Proof This easily follows by induction. The case when A = P ( a ) is atomic follows from the monotonicity property (TP). The cases A = B > C, A = VxB follow from the transitivity of R. For A = 3xB note that M , u It A implies M , u It- [a/x]B for some a E D,, and hence M , v Il- [a/x]B by the induction hypothesis, which yields M, v It B. Other cases are trivial.
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CHAPTER 3. KRIPKE SEMANTICS
L e m m a 3.2.17 Let M be an N-modal predicate Kripke model, A ( x ) an N modal formula with all its parameters in the list x , 1x1 = n . Then for any U E M M , u k V x A ( x ) iff Va E DZ M , u t= A(a). Proof By induction on n. The base is trivial and the step is almost trivial:
u k VyVxA(y,x ) i f fVb E D, u k VxA(b,x ) i f fVb E Du VC E Dc u k A(b, c ) (by the induction hypothesis) i f fV a E D c f l u k A ( a ) . In the intuitionistic case we have the following L e m m a 3.2.18 Let M be an intuitionistic Kripke model with the accessibility relation R , u E M , A ( x ) an intuitionistic Du-formula with all its parameters in the list x, 1x1 = n . Then
Proof By induction, similar t o the previous lemma. The base (with n follows from Lemma 3.2.16. For the induction step we have:
=
0)
u It- V y V x A ( y ,x ) i f fVv 'v R ( u )Vb E D, v IF V x A ( b , x ) (by 3.2.13) i f fVv E R ( u )Vb E D, Vw E R ( v )Vc E D: w IF A(b, c ) (by the induction hypothesis) i f fVw E R ( u )Va E D;+l w IF A ( a ) . In the latter equivalence, 'if' follows from the transitivity o f R and the inclusion D, C D, for w € R ( v ) ; t o prove 'only i f 7 ,for given u , w , a, take v = w and b = a l , c = (an,.. . ,an). Definition 3.2.19 A modal (respectively, intuitionistic) predicate formula A is said to be true in a Kripke model (respectively, intuitionistic Kripke model) M i f its universal closure V A is true at every world of M . This is denoted by M t= A i n the modal case and by M It A in the intuitionistic case. The set of all modal (respectively, intuitionistic) sentences that are true i n M is denoted by M T ( = ) ( M ) (respectively, I T ( ' ) ( M ) ) and called the modal (respectively, the intuitionistic) theory of M (with or without equality).
L e m m a 3.2.20 Let M be an N-modal or intuitionistic predicate Kripke model with a system of domains D , A(x1,. . . ,x,) a predicate formula of the corresponding type. Then ~ an). M k (I!-)A(xl, ...,xn) i f f V U EMVa17...,an E Du M , u k ( l k ) A ( a,..., Proof Follows easily from Lemma 3.2.17, 3.2.18. E.g. in the intuitionistic case we have M It- V x A ( x )i f fVuVv € R ( u )Va E D,R M , v It A ( a )
i f fVvVa E D: M , v It A(a). In the latter equivalence, 'if' is trivial; t o show 'only i f ' , take u = v.
W
3.2. PREDICATE K R W K E F R A M E S
21 1
D e f i n i t i o n 3.2.21 A modal (respectively, intuitionistic) predicate formula A is said to be valid in a Kripke frame (respectively, S4-based Kripke frame) F if it i s true i n all Kripke models (respectively, intuitionistic Kripke models) over F . Validity is denoted by F intuitionistic case.
I= A in the modal case and by F
Il- A in the
D e f i n i t i o n 3.2.22 Let C be a set of modal (respectively, intuitionistic) sentences. W e say that C is valid i n a Kripke frame F (of the corresponding type) i f every formula from C is valid in F ; or equivalently, we say that F is a C-frame (or C E M L ( = ) ( F ) , C 5 I L ( = ) ( F ) ) . F t= C (or F Il- C ) denotes that C is valid in F . The class of all C-frames is denoted by V ( C ) and called modally (respectively, intuitionistically) definable (by C). L e m m a 3.2.23 Let F be an N-modal (respectively, intuitionistic) Kripke frame, A an N-modal (respectively, intuitionistic) propositional formula. Then F k A iff F , k A (respectively, F I t A iff F , II- A), i.e. L ( F ) , = L(F,), where L is ML(=) or IL(=). P r o o f Easy from definitions. Valuations in F , are in principle the same as valuations of proposition letters in F ; more precisely, a valuation J in F corresponds to J' in F , such that
for any q E P L O , and both [, 6' are extended to propositional formulas in the same way. So ( F ,[), u 1 A iff (F,, J'), u b A and similarly for the iptuitionistic case. Finally note that every propositional valuation in F , is [' for some valuation E in F . H
L e m m a 3.2.24 Let M be an S4-based predicate Kripke model, Mo its pattern. Then for any A E IF=, Mo I t A iff M b AT, and thus
A E I T ( M 0 ) iff Proof
E MT(M).
From 3.2.13, 3.2.18.
¤
P r o p o s i t i o n 3.2.25 Let F be an S4-based Kripke frame, A an intuitionistic formula with equality. Then
P r o o f Let A = A ( x ) , 1x1 = n. (Only if.) Assume F Il- A. Let M be a Kripke model over F . By Lemma 3.2.20, we have to show M , u k A T ( a ) (= A ( a ) T ) for any u E F , a E Dl. B y Lemma 3.2.15 we have: Mo, u It- A ( a ) iff M , u t= ~ ( a ) ~ .
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CHAPTER 3. K R P K E SEMANTICS
By Lemma 3.2.20, F IF A implies Mo,u It- A(a), and thus we obtain F k A ~ . (If.) Assume F k AT. Then for any intuitionistic model M and u E F , a E DZ we have M , u k AT(a)) by Lemma 3.2.20, i.e. M , u IF A(a) by Definition 3.2.12. Hence F IF A, by Lemma 3.2.20. L e m m a 3.2.26 Let A(x), B ( x ) be congruent N-modal (respectively, intuitionistic) formulas, 1x1 = n , and let M be an N-modal (respectively, intuitionistic) Kripke model. Then for any u E M , a E DZ
M, u k (IF) A(a) iff M , u t= (IF) B(a). Proof We begin with the modal case. Consider the following equivalence relation between N-modal formulas: A B := FV(A) = F V ( B ) and for any distinct list of variables x such that FV(A) = r(x), for any u E M , a E D:'
Our aim is t o show that A A B implies A B. By Proposition 2.3.17 this implication follows from the properties (1)-(4). So let us check these properties for
-
(1) QyA Qz(A[y H z]) for Y $! BV(A), # V(A). We consider only the case & = 3. Suppose FV(3yA) = r ( x ) for a distinct x. Then there are two subcases. (i) y # FV(A) (and thus y # V(A)). In this case A[y I-+ z] = A , so 3z(A[y H z]) = 3zA, [a/x]3yA = 3y[a/x]A, [a/x]3zA = 3z[a/x]A. By 3.2.9 we have M , u k 3y[a/x]A H M, u k [a/x]A, since y does not occur in [a/x]A. Similarly,
so (1) holds. (ii) y E FV(A). Then again (since y
# r(x))
[a/x]3yA = 3y [a/x[A, [a/x]3z(A[yH z]) = 3z[a/x[(A[y H z]). Hence
Similarly M , u != [a/x]3z(A[y++ z]) @ 3d E D, M , u t= [a/x][d/z](A[yH 21).
3.2. PREDICATE KRIPKE FRAMES
213
But since y $! BV(A), z $! V(A), we have [ ~ / Y ] A = A [ ~Y d = (A[Y ] ++z])[z~
d= [d/z](A[y ] H z]).
Thus (1) holds in this case too.
-
(2) Supposing A
B, let us show that for & = ~ Y AQ ~ B N
the case Q = 3 is quite similar. Obviously, FV(A) = F V ( B ) implies FV(VyA) = FV(VyB); let r ( x ) = FV(VyA). Then
and similarly
Now if y E FV(A)= F V ( B ) , then we use the hypothesis (#)for A, B , xy, ad. If y @ FV(A), then FV(A) = F V ( B ) = r(x),
so we can use (#) for A, B , x , a. Anyway
(3) Supposing A A', B B', we prove that (A * B) consider the case * = A. Obviously FV(A A B )
= FV(A) U
FV(B)
= FV(A1) U
N
(A'
* B').
Let us
FV(B1) = FV(A1A B').
Let r(x) = FV(AAB). Then its subset FV(A) is r ( x . a ) for some injection a and F V ( B ) = r ( x . T ) for some injection T. Obviously [a/x]A = [a - a/x . o]A, [a/x]B = [a.T/X . T]B, and similarly for A', B'; hence M , u I=[alx](A A B ) % M, u k [a/x]A & M, u I=[alx]B e M , u k [ a . a / x . a ] A & M , u k [a-T/x.T]B. Similarly M , u t= [a/x](A1A B') Since A
-
A', B
% M, u
t= [ a . a/x . a]Ar & M , u k [ a . r / x . r]B1.
B', this implies (A A B )
N
(A' A B').
CHAPTER 3. KRIPKE SEMANTICS
214 (4) Supposing A
B, let us show OiA
OiB.
Obviously, FV(OiA) = FV(A) = FV(B) = FV(UiB). Next,
and similarly for B. Since A
B, these two conditions are equivalent.
, 2.11.3. So In the intuitionistic case note that A B implies AT A B ~ by by applying the modal case and Lemma 3.2.15 we obtain: M , u IF A(a) iff M , u t= ~ ~ (iffaM ), u I=~ ~ (iffaM), u It B(a).
L e m m a 3.2.27 Let A be a modal (respectively, intuitionistic) formula valid i n a Kripke frame (respectively, an S4-based Kripke frame) F . Then every formula congruent to A is valid in F . In other words, the sets ML(')(F), IL(')(F) are closed under congruence. Proof
By Lemmas 3.2.26, 3.2.17.
L e m m a 3.2.28 For any classical Du-formula A M , u k A iff Mu k A (in the classical sense). P r o o f By induction. The classical and modal truth definitions coincide in this case. T h e o r e m 3.2.29 ( S o u n d n e s s t h e o r e m ) (I) The set ML(=)(F) of all modal predicate formulas (with equality) valid i n a predicate Kripke frame F is a modal predicate logic (with equality). (11) The set IL(')(F) of all intuitionistic predicate formulas (with equality) valid i n an %-based Kripke frame F is a superintuitionistic predicate logic (with equality). Moreover, IL(') (F) = T ~ ~ ( (F) ' ) . Proof (I) Let F be a predicate Kripke frame. The axioms of K, are obviously valid, since they are valid in every propositional Kripke frame and we can apply 3.2.23. Let M be a Kripke model over F. The classical first-order axioms and the axioms of equality are true in every Mu; so by Lemma 3.2.28, they are true in M. It is also easy to check that modus ponens preserves the truth in a Kripke model. In fact, suppose M t= A, A > B , and let x be a list of parameters of (A > B ) , 1x1 = n. Then by 3.2.20(1), for any u E M, a E DZ
3.2. PREDICATE KRIPKE FRAMES
hence M , u I= [a/x]B. Thus M k B by 3.2.20(1). To verify 0-introduction, note that M , u k [a/x]OA iff Vv E R(u) M, v != [a/x]A, and the right hand side of the equivalence follows from M k A by 3.2.20. Next, consider the V-introduction rule. If M i= A(y, X I ,. . . ,x,), then by Lemma 3.2.20, Vu Vb, a l , . . . , a n E D, M , u k A(b, a l , .. . ,an), and thus Vub'al, . .. , a n E D, M , u k b'yA(y,al, . . . ,a,); hence M I= b'yA(y, XI,. . . ,2,) again by Lemma 3.2.20. Thus V-introduction preserves the truth in M. The only nontrivial part of the proof is to show that the substitution rule preserves validity. So suppose F !=A,and consider a simple formula substitution S=[C(x, y)/P(x)], where P occursg in A, P E P L n , n > 0'' and x , y are disjoint lists of different variables such that r(y) C FV(C) C ~ ( x Y ) . Recall that S A is obtained by appropriate replacements from a clean version A0 of A such that BV(AO)n r ( y ) = 0.By Lemma 3.2.27, A and A" are equivalent with respect t o validity, so we may assume that A is clean and BV(A)nr(y) = 0 . As we know from 2.5.24, r(y) = F V ( S )
c FV(SA) c FV(A) U F V ( S ) ,
so BV(A) n FV(SA) = 0. Let us fix a distinct list z such that r(z) = FV(A) U F V ( S ) (for example, put z = y t , where r ( t ) = FV(A)); so r ( y ) & FV(SA) E r(z). Due t o our assumption, r(z) n BV(A) = 0. Now for an arbitrary Kripke model M = (F, let us show that M 1 SA, i.e. for any u E F and c E D r
c),
where m = 121. To verify this (for certain fixed u and c), consider another model M1 = ( F , q) such that for any v E FTu, a E D t MI, v k P ( a ) iff M , v k C(a, c'), where c' is the part of c corresponding to y (i.e. if y = zl . . .zk, then c'=Cl ...I&); g ~ h case e when P does not occur in A is trivial. 1 ° ~ h e r eis a little difference in t h e case when n = 0 (then x is empty), but we leave this t o the reader.
216
CHAPTER 3. KRIPKE SEMANTICS for any other atomic D,-sentence Q M1,v =i Q iff M , v k Q.
Thus % ( P ) = {a E Dt 1 M, v k C(a, c')). Now consider a subformula B of A. Since FV(B) 5. FV(A) U BV(A), we can present B as B(z, q) 11, where q is distinct, r(q) = BV(A). Then by 2.5.24
So we can also present S B as (SB)(z, . . . q). -. Certainly we can use the same presentation in the trivial case when P does not occur in B (and S B = B). Now let us prove the following Claim. For any v E F T u and tuple a in D, such that la1 = Iql M1,v I= B(c,a) iff M , v i= (SB)(c,a).
(2)
This is proved by induction. It is essential that B (as a subformula of A) is clean, otherwise the argument is inapplicable, because S B can be constructed by induction only for clean formulas. The case when B is atomic and does not contain P is trivial, then S B = B and MI, v t= B(c, a ) H M, v I= B(c, a) by the definition of MI. If B is atomic and contains P, it has the form P(t1,. . . ,t,), where tl, . . . ,tn are variables from the list zq, q consists of parameters of B that are not in z (i.e. r(q) = F V ( B ) n BV(A)). So we have B = P((zq) - a ) for some a : In Ik, k = Izql. Then
-
B(c, a ) = [ca/zq]P((zq). a ) = P((ca) . a ) , (SB)(c, a ) = [ca/zq]SP((zq). a ) = [ca/zq]C((zq). a, y) = C((ca) . a , c'). Recall that y transforms to c' when z transforms to c. So in this case MI, v k B(c, a) iff M, v I= (SB)(c,a ) by the definition of M1. If B = B1 VB2, then M1,v t= B ( c , a ) iff (M1,v k Bl(c,a) V M1,v k Bz(c,a)) iff (M,v k (SBl)(c, a ) or M, v k (SB2)(c, a)) iff M , v k (SB)(c,a). ''Of course, this only means that F V ( B ) 2 r(zq).
3.2. PREDICATE KRIPKE FRAMES
217
If B = 3tB1, then M1,ui=B(c,a) iff 3 d D,~ M l , v i = B l ( c , a ' ) i f f 3 d ~D, M , v k (SBl)(C, a') (by the induction hypothesis) iff M, v k (3t SBl)(c, a)(= (SB)(c,a)). Here a' denotes the tuple obtained from a by putting d in the position corresponding to t , i.e. if t = qi (in q), then bi = d and bj = a j for j # i. Let us explain the first equivalence in more detail; the third equivalence is checked in the same way. In fact, we have
where Gi is obtained by eliminating qi = t from q and respectively Bi is obtained by eliminating ai from a , since t is bound in B. So by definition, M , u I= B(c, a) iff 3d E Dv M, u k [d/t][cBi/zGi]Bl, and the latter Dv-sentence is Bl(c, a'). If B = OiB1, then Ml,v I= B(c,a) iff Vw E Ri(v) M1,w I= Bl(c,a) iff Vw E Ri(u) M , w I= (SBl)(c,a) (by the induction hypothesis) iff M, v k Oi(SB1) (c, a)(= (SB)(c, a)). The remaining cases: B = C
> D, C A D, Vx C(x, z) are quite similar.
Now (1) easily follows from (2). In fact, (1) is equivalent to
for any a E DE since r(q) n FV(SA) = 0 . By (2), this is equivalent to
which holds since F i= A. (11) By Proposition 3.2.25, F It A iff F I= AT. This is exactly the same as A E IL(')(F) iff AT E ML(=)(F). So we obtain: IL(=)(F) = T ~ ~ ( = ) and ( ~ ) , W thus IL(=)(F) is an s.p.1. (=), by Proposition 2.11.8. In accordance with Section 2.16, the set ML(')(F)
is called the modal logic
of F (respectively, with or without equality). Similarly, for an intuitionistic F ,
its superintuitionistic logic is IL(')(F). For a class of N-modal frames C we define the modal logic determined by C (or complete w.r.t. C)
and similarly for superintuitionistic logics.
CHAPTER 3. KRIPKE SEMANTICS
218
Definition 3.2.30 (N-modal) Kripke (frame) semantics KN (or KG, for logics with equality) is generated by the class of all N-modal predicate Kripke frames. Similarly intuitionistic Kripke (frame) semantics is generated by the class of all intuitionistic predicate Kripke frames. Kg)-complete or K!,',)-complete logics are called Kripke (frame) complete (or K-complete i f there is no confusion).
xi,=)
Here is another version of soundness. L e m m a 3.2.31 Let M be a modal (respectively, intuitionistic) Kripke model, L an m.p.1. (=) (respectively, s.p.l.(=)), r a modal (respectively, intuitionistic) theory without constants, such that M !=(Ik) U l?. Then for any A E IF(=),
I' FL A +-M k(Ik) VA. P r o o f By induction on the length of an L-inference of A from I'. If A E r the claim is trivial. so the claim is also trivial. If A E L , then VA E It is clear that the truth in M respects MP, cf. the proof of 3.2.29. Finally, if A = VsB and M i= (Ik) VB, then also M k (It-) VA, since QH tH VA = VB and we can apply soundness.
r,
Definition 3.2.32 If F is a propositional Kripke frame, K(F) (or K F ) denotes the class of all predicate Kripke frames based on F. ML(')(IC(F)) is called the modal logic (with equality) determined over F. Similarly we define K(C) (or KC) for a class of N-modal propositional frames C and the modal logic determined over C. W e also define the superintuitionistic logic determined over a propositional S4-frame F as IL(=)(KF) and similarly for a class of S4-frames C . We may regard KC as a semantics for a certain class of logics. In particular, for a propositional modal logic A, the class K V ( A ) generates the semantics of all Kripke-complete logics containing A. Proposition 3.2.33 K!;) is the intuitionistic version of K V ( S 4 ) . IL(=)(c) = T ~ ~ ( = for ) ( any ~ ) class of S4-based predicate frames C .
Thus
Proof According to Definition 2.16.16, the first assertion is equivalent to 3.2.25. The second one is checked in the proof of 2.16.18. H L e m m a 3.2.34 Let F be a predicate Kripke frame over a propositional Kripke frame F . Then ML(F), = M L ( F ) . P r o o f Rather trivial. The definition of validity for propositional formulas in F is the same as in F. H
3.3. MORPHISMS O F KRTPKE F R A M E S Proposition 3.2.35 For a class of propositional Kripke frames C,
and (IL(=)(KC)), = IL(C)
in the intuitionistic case.
Proof
We begin with the modal case. By Lemma 3.2.34, we obtain
( M L ( = ) ( K C ) ) ,= ( ~ { M L ( = ) ( FI F ) E KC)), ~ { M L ( =( )F ) I F E C } = ML(') (c).
I
= ~ { M L ( = ) ( F ) ,F E KC)
=
w
3.3
Morphisms of Kripke frames
In this section we extend the notions of a frame morphism, a generated subframe and a pmorphism to the predicate case.
Definition 3.3.1 Let F = ( F ,D ) , F' = ( F ' , D') be predicate Kripke frames based on F = (W,R1,... ,R N ) ,F' = (W',Ri, . . . ,Rh) respectively. A morphism from F to F' is a pair f = ( f o ,f l ) such that
( I ) fo : F (2)
fl =
+ F'
is a morphism of propositional frames;
(f1u)ucw;
(3) every flu : D,
D>O(u)i s a surjective map;
(4) ~ R i * v flu
I' Du.
= fiv
fo is called the world component, f l the individual component of ( f o ,f l )
Definition 3.3.2 Let f = ( f o , f l ) be a morphism from F to F'. f is called an equality-morphism (briefly, =-morphism) if every flu is a bijection. f is called a pmorphism i f fo is surjective (i.e. fo is a p-morphism of propositional frames). f is called a p'-morphism
i f it is a p-morphism, and an =-morphism.
f is called an isomorphism zf it is an =-morphism, and fo is an isomorphism of propositional frames.
C H A P T E R 3. K R I P K E SEMANTICS
220
Definition 3.3.3 A morphism o f predicate Kripke models M = ( F , E ) and M' = (F1,<') is a morphism ( f o ,f l ) of their frames F and F' preserving the truth values of atomic formulas, i.e. such that for any P E P L m , m 0, u E F ; bl, ...,bm E D ,
>
M ,u
(
1
-
7
m
i f f M ' , fo(u) != P ( f i u ( b l ) ,. . fiu(bm)). - 7
A p-morphism of Kripke models is a morphism of Kripke models, which is a p-morphism of their frames. Similarly for =-, p=-morphisms and isomorphisms.
-
T h e notation ( f o ,f l ) : M M' means that ( f o ,f i ) is a morphism from M t o M ' , and similarly for frames. As in t h e propositional case, pmorphisms o f predicate Kripke frames (models) are denoted b y +. W e also use the notation --= for =-morphisms, -+= for p=-morphisms, 2 for isomorphisms. Lemma 3.3.4
(1) For a predicate Kripke frame F = ( F ,D ) , the identity morphism idF := ( i d w ,f l ) , where flu := id^^ for any u , is an isomorphism.
-
(2) For morphisms of predicate Kripke frames ( f o ,f l ) : F F' F", consider the composition ( f 0 ,f l )
O
(90,gl) := (g0,gl) -
This i s a morphism F
( f 0 ,f l )
:= ( f 0
---+
F' and (go,gl) :
90, ( f l u . O S I U ) U G F ) .
+F".
(3) The composition of =-morphisms is an =-morphism; similarly for p(=)morphisms and isomorphisms.
(4) The statements (1)-(3) also hold for Kripke models, with obvious changes. Proof
An easy exercise; use Lemma 1.3.32.
W
PKFE)
T h u s we obtain t h e categories o f N-modal predicate Kripke frames and (=)-morphisms, with the composition o and the identity morphism i d F . One can easily check the following Lemma 3.3.5 Isomorphisms in P K F ~ are ) exactly isomorphisms i n the sense of Definition 3.3.2.
In a similar way we can define the categories PKF~,') o f intuitionistic Kripke o f N-modal Kripke models, PKM!,=) o f intuitionistic Kripke frames, models; the details are left t o the reader.
PKME)
Definition 3.3.6 Let F = (W,R1,.. . , R N )be a propositional Kripke frame. A predicate Kripke frame morphism over F is a morphism of the form ( i d w ,f l ) : ( F ,D) ( F ,Dl). +
3.3. MORPHTSMS OF KRIPKE FRAMES
221
T h e following is obvious Lemma 3.3.7 The composition of (=)-morphisms over F is an (=)-morphism over F . Thus we can also consider the categories o f frames over F and (=)-morphisms. But the case with equality is not so interesting, because all =-morphisms over F are isomorphisms. Definition 3.3.8 Let F , F' be propositional Kripke frames, h : F' + F , and let F = ( F ,D ) be a predicate Kripke frame. Then we say that a Kripke frame F' = (F', D') is obtained from F by changing the base along h i f DD:, = Dh(u) for all u E F . This frame F' is denoted by h,F. Lemma 3.3.9 Under the conditions of Definition 3.3.8, there exists a 'canonical' =-morphism ( h , g ) : h,F -= F ; i f h is a p-morphism, then ( h , g ) is a p=-morphism.
Proof
Put gu := idDl : DL
Then ( h , g ) : (F', D')
-+
+ Dh(%).
( F ,D ) b y Definition 3.3.1.
Now let us show that every morphism is represented as a specific composition:
-
Proposition 3.3.10 Every (=)-morphism of predicate Kripke frames ( f o , f l ) : F' F can be presented as a composition of an (=)-morphism over the propositional base F' of F' and the canonical morphism:
- (idw,, f l )
F'
(fo)*F
(fo, 9)
F
Moreover, (idwl,f l ) is a unique morphism ( i d w ~h,l ) over F' such that ( f o , f l ) = (idwt,h l ) 0 ( f 0 , g ) . P r o o f In fact, (idw1,h l ) 0 ( f o , 9 )= ( f o , f l ) i f f idw, o fo = fo (which is true) and for any u flu
= hlu
0
gu = hlu
0
ido: = hlu,
i.e. i f f f l = h l . Proposition 3.3.11 If ( f o , f l ) : M ---+(=) M', then for any u E M , B E MS~)(D,) M , u I= B i f f M', fo(u) f l u . B , where flu . B is obtained from B by replacing occurrences of every a E Du with flu(a>. If the models are intuitionistic, the same holds for the intuitionistic forcing and B E IS(=)(D,).
CHAPTER 3. KRIPKE SEMANTICS
222
Proof By induction on the complexity of B we prove the claim for any U. Let us check only two cases (1) B = OjC. Then
by the induction hypothesis. Since fa is a morphism, we have R(i(fo(u))= fo[Rj(u)]. By Definition 3.3.1, f lu(a) = flv (a) for any a E D,, v E R j (u); hence f l u . C = flu . C for any Du-sentence C. So M , u k B iff Vv' E R(,(fo(u)) M',vl b flu - C iff M', fo(u) I= Oj(flu . C) (= f l u .B). (2) B = 3xC(x, a), where 3xC(x, y) is a generator of B, a is a tuple from D,. Then M , u != B iff 3b E D, M , u != C(b,a) iff 3b E Du M', fo(u) != C(flu(b), f l u . a) (by induction hypothesis) iff 3c E Die(,) M', fo(u) b C(c, f l u . a) (since fiu[Du] = D;o(u)) iff M', fo(u) 3zC(x, f l u . a). The intuitionistic case now follows easily by Godel-Tarski translation. Proposition 3.3.12 Let (fa, f l ) : F -+ F', and let M' be a Kriplce model over F'. Then there exists a unique model M over F such that (fo,f i ) : M -+ M', and similarly for +, -=, +=. If M' is intuitionistic, then M is also intuitionistic. Proof In fact, if M' = (F',J1), then M = (F,J) is unjquely determined by the equalities Jw(Pr) := {b E D r I flu. b E E;o(w)(Pr))
for any v E F, m
> 0, where flu.
. . ,bm) := (flv(bl), . . . ,flu(bm)),
and Ju(p')
:=
{ {v) 0
if J>ocu) (PkO) = {fo(v)), otherwise.
Now suppose J' is intutionistic. Then VRUimplies fo(v)R1fo(u)and next J;,(,)(P,") G J>,(,)(Pr) (for m > 0), whence by definition
In the same way, we obtain
Thus J is intutionistic.
3.3. MORPHISMS OF KRIPKE FRAMES
223
Now let M be a Kripke model over F and consider changing the base of F. By applying 3.3.12 to the canonical morphism ( h , g ) : h,F -, F , we obtain a unique Kripke model M' over h,F such that ( h ,g) : M' -, M . We also say that M' is obtained from M by changing the base along h and denote M' by h,M.
Proposition 3.3.1312 If there e ~ i s t s a ~ ( = ) - r n o r p h i s r n ~ + ( =then ) ~ 'M , L(')(F)~ ML(=)(F'), and similarly, IL(=)( F ) C IL(=)(F')for intuitionistic Kripke frames. Proof Consider the modal case only. Let ( f o ,fl) : F + F', and assume that F' !+ A ( x l , . . . , x n ) . So by Lemma 3.2.20, for some u' E F', a l , . . . ,a, E DL,, for some model M' over F' we have M', u'
!+ A ( a i , . . . ,a;).
By Proposition 3.3.12, there exists a model M over F such that
( f o ,fi) : M
+
M'.
Now since fo and flu are surjective, there exist u , a l l . . . ,an, such that
and thus by Lemma 3.3.11, we obtain M , u Therefore, F' A implies F !+ A.
!+ A(a1, . . . ,a,).
Proposition 3.3.14 If there exists a p-morphism F' -+ F , then ML(=)(KF')c ML(=)( K F ) (and IL(=)(KF')E I L ( = ) ( K F )in the intuitionistic case). Proof
Consider the modal case. Let h : F'
++
F . For any F E KF we have
by Lemma 3.3.9 and Proposition 3.3.13. Hence
Definition 3.3.15 Let F = ( W ,R1, . . .,R N ) be a propositional Kripke frame, F = ( F ,D ) a predicate Kripke frame, V C W . A subframe of F obtained by restriction to V is defined as where D 1 V := ( D u ) u E V . If M = ( F , J ) is a Kripke model, we define the submodel M i v := ( F t t V), where (t t V)(P!r) := ( ~ u ( P k m ) ) u € v . If V is stable, the subframe F 1 V and the submodel M 1 V are called generated.
v,e
12Cf. [Ono, 1972/73], Theorem 3.4.
CHAPTER 3. KRIPKE SEMANTICS
224 The notation M I
cM
means that M I is a submodel of M .
Definition 3.3.16 A submodel Ml c M is called reliable i f for any u E M I , for any D,-sentence A M1,ukA i#M,ukA.
In this case obviously, M T ( M ) 2 M T ( M 1 ) . Definition 3.3.17 Similarly to the propositional case (Definition 1.3.14), we define cones, rooted frames and rooted models:
L e m m a 3.3.18 (Generation l e m m a ) Let F be a predicate Kriplce frame, M a Kripke model over F, V a stable set of worlds i n F . Then (1) there is a morphism ( j ,i ) : F 1 V ---t F (and M 1 V ---t M ) , in which j : V ---+ W is the inclusion map and every i , is the identity map on the corresponding domain; (2) M
1 V is a reliable submodel of M ;
(3) ML(')(F)
c ML(')(F
1 V); similarly, for the intuitionistic case.
Proof (1) Obvious. ( 2 ) Apply Lemma 3.3.11 and (1). (3) Every valuation J' in F 1 V equals J 1 V for a valuation J in F such that Ju = [h whenever u E V. Such a f obviously exists, e.g. put Eu(P):=
{ tL(P) 0
if u E V, otherwise.
If [' is intuitionistic, then [ is also intuitionistic, since V is stable. Now the claim follows from (2).
Definition 3.3.19 I f f : F ---+ F' is a morphism of Kripke frames and V is stable in F , then f t V := ( j ,i ) o f , where ( j ,i) is a morphism from 3.3.18(1), is called the restriction of f to V. Restrictions of Kriplce model morphisms are defined in the same way. Lemma 3.3.4 readily implies
L e m m a 3.3.20 A restriction of an (=)-morphism (of Kriplce frames or models) to a generated subframe (or submodel) is an (=)-morphism (respectively, of frames or models).
3.3. MORPHISMS O F KRIPKE F R A M E S If G = F 1 V , we also denote f IV by f IG. L e m m a 3.3.21
n ML(=)(FTU)
(1) ML(')(F) =
uEF
and analogously in the intuitionistic case. (2) Every Kripke complete modal or superintuitionistic logic is determined by a class of rooted predicated Kripke frames:
where L is M L or I L , C f : = { F f u I F E C, u E F ) . Proof ( 1 ) Similar to Lemma 1.3.26 (an exercise). ( 2 ) Follows from ( 1 ) .
D e f i n i t i o n 3.3.22 The notions 'path', 'connectedness', 'component', 'nonoriented path' are obviously extended to the predicate case. Viz., a predicate Kripke frame is called connected i f its propositional base is connected; a nonoriented path in ( F ,D) i s the same as in F ; a component in ( F ,D) is its restriction to a component i n F . Now we have a predicate version of 1.3.38.
P r o p o s i t i o n 3.3.23 Let ( F i ( i E I ) be a family of all different components of a predicate Kripke frame F . Then (1) for any morphism f : F
--+ G ,
every f IFi is a morphism;
(2) for any family of morphisms fi = (gi, hi) : Fi morphism U fi : F -+ G defined as ( g , h ) , with
g :=
U gi,
-
G there is a joined
iEI
h := ( h u ) U E hu~:= , (hi)u for u E Fi;
iEI
(3) every morphism f : F
+
G is presented as
U( f 1Fi). iEI
Proof ( 1 ) Follows from Lemma 3.3.20.
CHAPTER 3. KRIPKE SEMANTICS
226
(2) g is a morphism of propositional bases, by Propositon 1.3.38. Every h, is surjective, since it coincides with some (hi),. Finally, if u E Fi, uRjv, then v E Fi, thus for any a E D,
hu(a) = (hi)u(a) = (hi)v(a) = hv(a), i.e. h,
= h,
D,.
(3) Trivial, by definition.
w We write F1 ++ F2 to denote that there is a pmorphism from F1 onto F2. Similarly to the propositional case, we give Definition 3.3.24 A predicate Kripke frame F1 is called (=)-reducible to a rooted predicate Kripke frame F2 if there exists u E F1 such that F1T U --H(=) F2. Definition 3.3.25 Let C1, C2 be classes of predicate Kripke frames. We say that C1 is (=)-reducible to C2 if for any F2 E C2 for any v E F2 there exists F1 E C1 that is (=)-reducible to F2 T v. In the similar way reducibility is defined for classes of propositional Kripke frames.
Cl red(') C2 denotes that C1 is (=)-reducible to C2. It is clear that reducibility implies reducibility.
=-
Proposition 3.3.26 If Cl red(') C2 for classes C1, Cg of predicate Kripke frames, then L(')(c~) L(=)(C2) (where L denotes ML in the modal case and IL in the intuitionistic case). Proof Suppose Cl red(') C2. Then for any Fa E C2, v E F 2 there is F1 E C1, u E F1 such that F1l u -+(=) F 2 f v. Hence by 3.3.18 and 3.3.13
( F ~ )E L(')(F~) by 3.3.21. Therefore L(')(c~) and thus L(=) any F2 E C2, which implies L(=)(C1) L(=)(c~).
L(')(F~) for
L e m m a 3.3.27 Let F be a propositional Kripke frame, u E F . Then K(FTu) = { F t u I F E KF). We shall denote the latter class by (KF) T u. Proof The inclusion (KF) f u C_ K ( F I' u) is trivial. The other way round, if F' = ( F u, Dl), then F' = F T u for the frame F = ( F ,D), where
D, :=
Dh if v E F l u , DL otherwise.
The system of domains D is expanding, since by 3.2.3, DL & DL for any v E w F T u. Thus F1E (KF) u.
r
3.3. MORPHISMS OF KRIPKE FRAMES
227
Proposition 3.3.28 Let C be a class of N-modal propositional Kripke frames,
Then ( I ) K ( C t ) = (KC)T (= { F l u I u E F , F E KC))
(3) IL(')(KC) = I L ( = ) ( K ( c T ) )i f C is a class of S4-frames. Proof
( 1 ) In fact,
and we can apply 3.3.27. (2), (3) follow from (1) and 3.3.21(2).
L e m m a 3.3.29 Let Fl, F2 be propositional Kripke frames such that Fl red F2. Then ( K F l )red' ( K F 2 ) . Proof By assumption, there exists h : Fl for any F2 E KF2 there is a canonical
tu
-tt
Fz for some u. So by 3.3.9,
and h,F2 E K(F1 T u ) = (ICFl) T u (by 3.3.27). Since h is surjective, for any v E F2, there is w E FIT u such that h ( w ) = v . Then by 1.3.32, the restriction of h is a pmorphism
and it follows that
(h32)Tw
+
F2 t v
by the restriction of y. Since (h,F2) T w E ( K F l ) w , we obtain ( K F l )red= (KF2).
w
Proposition 3.3.30 If C1 red Cz for classes C1, Cg of propositional Kripke frames, then (KC1)red' (KC2). Proof Let F2 E KC2, i.e. F2 E KF2 for some F2 E C2; then F2 T v E (ICF2)t v = K(F2 v ) . Since C1red C2, there exists Fl E C1 such that Fl red (F2 T v ) . By Lemma 3.3.29, (KFl)red' K(F2 T v ) ; thus Flred'(F2 t v ) for some F 1 E KFl C KC1. Therefore (KC1)redZ (KC2). ¤ Corollary 3.3.31 If C1 red C2 for classes C1,C2 of propositional Kripke frames, ) I L ( = ) ( K C ~E) I L ( = ) ( K C ~for ) S4then M L ( = ) ( K C ~ E ) M L ( = ) ( K C ~(and frames).
CHAPTER 3. KRIPKE SEMANTICS
228
Proof
W
By 3.3.30 and 3.3.26.
Definition 3.3.32 We say that a Kripke frame ( F ,D') is obtained by domain restriction from ( F ,D ) if for some V C D+, DL = D, n V . In this case we denote ( F ,Dl) by (F,D ) n V .
Obviously this definition is sound i f f V n D, # 0 for any u E F. Proposition 3.3.33 ( F ,D ) red ( F ,D ) n V . Proof
I f F has a root v, there exists a p-morphism over F
id^, g ) : ( F ,D ) + ( F ,Dl) = ( F ,D ) n V. In fact, we can define g, as an identity map on DL sending every a $ V t o some fixed element of V . Then obviously, g, 1 D, = g, whenever u&v. rn Definition 3.3.34 Let ( F i ) i € ~be a family of predicate Kripke frames, Fi = (Fi,Di). The disjoint sum (or disjoint union) of the family (Fi)iCIis the frame
where D(u,i) := (Di)u x (2) for i E I , u E Fi. If for each i E I , Mi = ( F i ,Qi) is a Kripke model over Fi, the disjoint sum (union) of ( M i ) i Eis~ the Kripke model
where Q(u,i)(P):= {((a192 1 , - - + (an, , 2 ) ) 1 (al,. . . , an)
(Qi)u(P))
for P E PLn, n > 0 and := @(u,i)(p)
{{
( ) ) if (Qi)u(P)= { u } , otherwise
for P E PLO. Obviously, there exists an isomorphism from Fk (respectively, M k ) onto a generated subframe in U Fi (respectively, U Mi), given by the pair ( f o , f l ) iEI
iEI
such that fo(u) := ( u ,k ) ,
flu(.)
:= ( a ,k ) .
3.3. MORPHISMS OF KRIPKE FRAMES
229
In particular, the definition of O(,,i) yields
Now we have an analogue of 1.3.37. Proposition 3.3.35 Let ( F i I i E I ) be a family consisting of all different components of a predicate Kripke frame F . Then F 2 U Fi. iEI
Proof A required isomorphism is the map f from the proof of 1.3.37 together with the family id^,)^^^ (as usual, we assume that F = (F,D ) ) . 1 Proposition 3.3.36
for the modal case. I L ( = ) ( F ~ )IT(=) , G
intuitionistic case.
>
uI M~
= I,
n I T ( = ) ( M { )for the
Proof Similar to Lemma 1.3.28 using Lemma 3.3.21; a simple exercise for the reader. Corollary 3.3.37 Kripke frame semantics erty (cf. Definition 2.16.9).
ICE),@,=) have the Collection Prop-
Proposition 3.3.38 Every modally or intuitionistically definable class of predicate Kripke frames is closed under generated subframes, disjoint sums and p(=)morphic images.13 Proof Validity is preserved for generated subframes by 3.3.18, for p(')-morphic images by 3.3.13, for disjoint sums by 3.3.36. 1 Finally let us prove a predicate analogue of Lemma 1.3.44. L e m m a 3.3.39 Let L be a conically expressive N-m.p.l.(=). Then for any N modal Kripke model M such that M != L for any u E M , for any Du-sentence
A M , u k C l * A ~ b ' v ER * ( u ) M , v ! = A , where R* is the same as in Lemma 1.3.19. 13p'-morphic images
- if
the class is definable by a set of formulas with equality.
CHAPTER 3. KRIPKE SEMANTICS
230
Proof Similar to 1.3.44 and the soundness of the substitution rule in 3.2.29. Let Mo be a propositional model over the same propositional frame as M , such that for any v E M M o , v I = p + +M , v k A . Then by induction we obtain for any propositional formula X ( p ) for any v E M. Mo,v I= X ( p ) @ M , v I= X ( A ) . Hence for any u E M and C from 1.3.42
and so by 1.3.42,
M,ukO*A*
Moru/=p.
But by Lemma 1.3.19 and the choice of Mo
Hence the claim follows.
3.4
Constant domains
Definition 3.4.1 A predicate Kripke frame F has a constant domain (or F is a CD-frame ) iff D, = D, for any u , v E F.
A CD-frame (F,D ) , in which V = D, for all u E F, is denoted by F 8 V. Proposition 3.4.2 (1) A rooted N-modal predicate Kripke frame F has a constant domain zff
Vi E IN F t= Bai. (2) A rooted intuitionistic Kripke frame F has a constant domain iff F
It - C D .
Proof (1) (Only if.) Suppose M is a Kripke model over F and M I u t.= Oi3xP(x). Then M I v P ( a ) for some v E R i ( u ) , a E D, = D,. Hence O i P ( a ) and thus M , u k 3s O i P ( x ) . M,u (If.) Let uo be a root of F. If F does not have a constant domain, then D, # D, for some U , V E F , and thus D, # D,, or D, # D,,. Consider U, ~so the first option. Then there exists a path u o R i , u ~.,. . , U ~ - ~ R ~=, u D,, G D,, G .. . D,. It follows that for some k D,, c D,,,, . Thus for some i ,u , v we have D, c D,, v E Ri(u). Let a0 E ( D , - D,). Consider a model M = ( F ,[) such that (for a certain P E P L 1 )
c
JW(P) :=
{ao) if a0 E D,, 0 otherwise.
3.4. CONSTANT DOMAINS
231
Then we have M , v 3 x P ( x ) , and thus M , u Oi3xP(x); but M , u 3 x O P ( x ) , since a0 6D,. ( 2 ) Let R be the relation in F. (Only if.) I f M is an intuitionistic Kripke model over F and M , u It. V X ( P ( X ) V q), but M , u Iy q and M , u ly V x P ( x ) , then M , v Iy P ( a ) for some v E R ( u ) , a E D, = D,. Hence M , u Iy P ( a ) V q, which contradicts M , u IF V x ( P ( x )V q) and U R U . ( I f . ) Let u be root o f F , and suppose D, # D, for some v E W = R ( u ) . Since u R v , we have D, c D,, so there exists a0 E ( D , - D,). Consider the valuation [ in F such that for any w
i f wRu, { w ) otherwise,
0
for a certain P E P L 1 , q E PLO and t w ( Q )= 0 for all other predicate letters Q. I t is clear that [ is intuitionistic. Under this valuation we have u Iy q, v Iy P ( a o ) ,and thus u Iy V x P ( x )V q; but u IF V x ( P ( x )V q). In fact, suppose uRw. I f w R u , then obviously D, = D,, and thus w Ik P ( a ) for any a E D,; on the other hand, w @u implies w It. q. Therefore u Iy C D . ¤ from Definition 3.4.1 it is clear that the class o f CD-frames is closed under generated subframes.
Definition 3.4.3 Modal and intuitionistic Kripke semantics with constant domains are generated by CD-frames:
CK$)
:=
ex:,=,):=
{ML(')(x)
/
{IL(')(x)
IX
X is a class of N-modal CD-frames); is a class of intuitionistic CD-frames).
T h e class o f CD-frames is not closed under disjoint sums (the reader can easily construct a counterexample), but there is an equivalent semantics generated by a class with this property. Definition 3.4.4 A predicate Kripke frame F = ((W,R1,. ..,R N ) ,D ) is called a local CD-frame if it satisfies
(*) ViVuVv (uRiv =+ D, = D,). Lemma 3.4.5 (1) For an N-modal frame F, F 1
A
Bai iff every cone FTu
~EIN
has a constant domain iff F is a local CD-frame. (2) Similarly, for an intuitionistic F, F is a local CD-frame i# F IF C D iff all cones in F have constant domains. Proof
Easily follows from 3.4.2.
W
C H A P T E R 3. KRIPKE SEMANTICS
232 Proposition 3.4.6
(1) The class of local CD-frames is closed under disjoint sums, generated sub-
frames and p-morphic images. (2) CD-frames and local CD-frames generate equivalent semantics. Proof ( 1 ) B y 3.4.5 and 3.3.38 this class is modally (or intuitionistically) definable, so we can apply 3.3.38. (2) Note that every CD-frame is a local CD-frame; on the other hand, i f X is a class o f (N-modal) local CD-frames, then b y Lemma 3.3.21,
and all the FTu are CD-frames.
W
Exercise 3.4.7 Show that the class o f CD-frames is not closed under p-morphic images, but closed under p'-morphic images.
Let us now describe morphisms o f local CD-frames. Lemma 3.4.8 Every connected local CD-frame is a CD-frame. Proof In a local CD-frame, i f u, v are in the same component, then D, = D,. This easily follows b y induction on the length o f a non-oriented path from u t o v. w Proposition 3.4.9 (1) If ( f o ,f l ) : ( F ,D )
++ ( F ' , Dl) for a connected CD-frame ( F ,D ) , then all the maps flu (and certainly their targets D(fo(u))for u E F coincide and ( F ' , D') is a connected CD-frame.
(2) Conversely, assume that ( F ,D ) , ( F ' , Dl) are connected CD-frames, f F -+F' and g : D, ---+ DL is a surjective map. Then ( f , f i ) ( F ,D ) (F', D') for f i = ( S ) ~ E F .
:
-+
Proof ( 1 ) Again we show flu = flu by induction on the length o f a non-oriented path from u t o v . I t suffices t o consider the case when uRiv. T h e n flu = f l u 1 D,, SO flu = flu as functions, since D, = D,. T h e targets coincide, due t o the surjectivity.
W e know from Lemma 1.3.39 that F' is connected. B y 3.4.6, ( F ' , Dl) is local C D , so it is C D by 3.4.8.
3.4. CONSTANT DOMAINS (2) Trivial by definition.
The pmorphism described in 3.4.9 is briefly denoted by f @ g.
I i E I ) a family of all its G are exactly the maps of diflerent components. Then the p-morphisms F the form U fi, where every fi : Fi -w G (as a p-morphism onto its image) has
Corollary 3.4.10 Let F be a local CD-frame, (Fi
iEI
the form described in 3.4.9. Proof Note that every F i is a CD-frame by 3.4.8 and apply Proposition 3.3.23.
w
Every Kripke frame over a propositional frame F is reducible to some Kripke frame F a V; for example, with a singleton V. More precisely, the following proposition holds (its intuitionstic version was proved in [Ono, 19721731). Proposition 3.4.11 Let F = (F, D) and F' = F' @ V be predicate Kripke frames such that F is reducible to F' and ID,I 2 IVI for any u E F. Then F is reducible to F' and thus14 ML(F) 5 ML(F1) (and IL(F) E IL(F') for the intuitionistic case). Proof
Assuming h : F f v -tt F' T u, let us show
To simplify notation, we assume that F = F f v and F' = F'Tu. Le! a0 be a fixed element of V, g, : D, --+ V a surjective map. For every u E F consider V. the following surjective function g, : D,
-
%'(a)
:=
gv(a) if a E Dv, a0 otherwise.
It is clear that uRiul implies g, = gut 1 D,, so we obtain ( h , g ) : F thus ML(F) 5 ML(F1) by Proposition 3.3.13.
-+ F',
and
Remark 3.4.12 Proposition 3.4.11 is not transferred to the case with equality. In fact, 3.4.11 implies that
IL(F @ V)
c I L ( F @ V') if IVI > IV'I
But a similar assertion does not hold for logics with equality (even for infinite
V'). In fact, consider the following formula [Skvortsov, 19891:
1 4 ~ Proposition y
3.3.26.
CHAPTER 3. KRIPKE SEMANTICS
234 L e m m a 3.4.13
F @ V II- A0
zfi
Vu E F IHA(Ffu)l 2 IV1.15
Proof Let M be an intuitionistic model over F @ V and let u E F . Consider the sets
Ea := {v E F f u I M, v II- P ( a ) )
for a E V. It is clear that Z a E HA(FTu). (If.) If IHA(FTu)l IVI for every u E F , then Ea = zb for some a # b (from V) - otherwise the map a H Za embeds V is H A ( F f u). Then1' M, u II- a # b A (P(a) = P(b)), and thus M Iu II- Ao. (Only if.) Suppose IVI 5 IHA(Ffu)l for some u E F. Let h : V - - H A ( F f u ) be an injection, and consider a model M over F such that for any w
2
M, w IF P(a) iff w E h(a). Then Ea = h(a), and thus Za # Eb, whenever a # b. Hence M, u whenever a , b E V, a # b, and thus M, u Iy Ao.
Iy P ( a )
-
P(b)
Therefore, if a p.0. set F contains an infinite cone F t u and V' is countable, but IVI > 2IFl, we have A. E ( I L Z ( F@ V) - IL'(F @ V')).
3.5 3.5.1
Kripke frames with equality Introduction
Let us first make some informal comments about interpreting equality. In Kripke frame semantics this interpretation is the simplest (Definition 3.2.9): u k a = b iff a equals b. But does this properly correspond to our intuitive understanding of equality? In fact, we should evaluate a = b with respect to a certain world, so it might happen that a and b are the same in a world u, but different in another world. Examples of this kind are quite popular in literature. But as mentioned in the Introduction to Part 11, there are more formal reasons for modifying the notion of predicate Kripke frame. For example, Lemma 3.10.5 (see below) shows that the principle of decidable equality DE is valid in any intuitionistic Kripke frame. But from the intuitionistic point of view, equality may be undecidable. In particular, this happens in intuitionistic analysis - for real numbers or functions. So we should choose the interpretations of equality appropriately. In classical model theory there exist two ways of dealing with equality. The standard way is to interpret equality as coincidence of individuals ('normal 150f course, we can replace 2 with < if we accept Axiom of Choice. 16Recall that M, u It- a # b iff a , b are different.
3.5. KRIPKE FRAMES W I T H EQUALITY
235
models'). The second way is to interpret equality as an equivalence relation preserving values of basic predicates. It is well-known that in classical logic these two approaches are equivalent, because we can always take the quotient domain modulo the equivalence relation corresponding to equality, and obtain a logically equivalent 'normal' interpretation. Kripke semantics for intuitionistic and modal logics corresponds to classical 'normal models', but the second approach also makes sense. So we can interpret equality in a predicate Kripke frame as an equivalence relation on every individual domain D,. Thus we obtain the notion of a predicate Kripke frame with equality ( K F E ) and the corresponding semantics KE. This semantics is stronger than the semantics of Kripke frames K for formulas with equality the crucial formula D E can be refuted in a KFE. Moreover, as we shall see, KE is stronger than K for logics without equality. We will also describe an equivalent semantics of 'Kripke sheaves'. Kripke sheaves are obtained from KFEs by taking quotients of individual domains through the corresponding equivalence relation.
3.5.2
Kripke frames with equality
Definition 3.5.1 A predicate Kripke frame with equality (KFE) is a triple (F, D , x ) , in which (F,D ) is a predicate Kripke frame and x is a valuation for the binary predicate symbol '= ' satisfying the standard equality axioms. In more detail, i f F = (W,R1,. .. ,RN), then x = (=c,),~w is a family of equivalence relations ( x u C D, x D,), which is &-stable for every i E IN:
The following lemma gives an alternative definition of KFEs presenting them as a kind of 0-sets (see Chapter 4).
-
Lemma 3.5.2 (1) Let (F, D, x ) be a KFE, F = (W,R1, . . . ,RN).Consider 2 W , such that for any a, b E D + the function E : D + x D+ E(a,b) = { u I a xu b).
Then the following holds: ( E l ) E ( a , b) = E(b,a ) ;
(E4) Ri(E(a, b ) ) 2 E(a, b) for i E I N . (2) Given a propositional Kripke frame F , a non-empty set D+, and a function E satisfying (E1)-(Ed), we can uniquely restore the corresponding K F E as (F, D , x ) , with
236
C H A P T E R 3. KRIPKE SEMANTICS
Proof
( 1 ) ( E l ) , ( E 2 ) follow respectively from the symmetry and the transitivity of xu. The reflexivity of xu implies ( E 3 ) - it follows that u E E ( a , a ) for a E D,, and such an a exists since D, # 0 . ( E 4 ) follows from the Ri-stability of x. (2) Note that
E ( a , b) = E ( a , b) n E(b, a ) C E ( a , a ) by ( E l ) , (E2) and similarly, E ( a , b) C E(b, b). Thus u E E ( a , b) implies u E E ( a , a ) n E(b,b), i.e. x,G D, x D,. ( E l ) implies the symmetry of xu, and ( E 2 ) implies the transitivity.
( E 3 ) implies the non-emptiness of every D,. obvious.
The reflexivity of xu is
The Ri-stability of x follows from (E4).
E ( a , b) is called the measure of identity of a and b. E ( a , a ) is called the measure of existence (or the extent) of a , and is also denoted by E ( a ) . Definition 3.5.3 A KFE-model over a Kripke frame with equality F = ( F ,D , x) is a pair M = ( F ,0 , where J is a valuation i n ( F ,D ) respecting the relations xu, i.e. such that for every P E P L n , n 2 1 and a l , . . . ,a,, bl, . . . ,bn E D+
( a l , . . . , a n ) E & ( P ) & a1 xu bl & . . . & a, xu bn
* (bl,. . . ,b,)
E [,(P).
Then we define forcing M , u =i A (for u E F and a D,-sentence A ) by the same conditions as i n Definition 3.2.9, with the following difference:
A KFE-model M is called intuitionistic if the corresponding model without equality ( F ,J ) is intuitionistic. In this case we define forcing M , u IF A according to Definition 3.2.12 with the only difference: A modal predicate formula A is called true i n M i f FA is true at every world of M ; similarly for the intuitionistic case. Definition 3.5.4 A modal (respectively, intuitionistic) predicate formula A is called valid i n a KFE (respectively, S4-based KF'E) F iff it is true i n all KFEmodels (respectively, intuitionistic KFE-models) over F . Again we use the notation M 1 A, F 1A for the modal case; M It A , F It A for the intuitionistic case. Lemma 3.2.11 obviously transfers to KFE-models. Lemma 3.2.13 has the following analogue.
3.5. KRIPKE FRAMES WITH EQUALITY
M , u IF 7 B i f f Vv E R ( u ) M , v
L e m m a 3.5.5
ly B;
M,uIta#biff V v ~ R ( u ) a $ , b . We also have an analogue of 3.2.16:
L e m m a 3.5.6 For an intuitionistic model M and D,-sentence A M,uIFA&uRv
+
M,vIkA
Next we obtain analogues of 3.2.17, 3.2.18, 3.2.20 by the same arguments.
L e m m a 3.5.7 (1) Let M be an N-modal KFE-model, A ( x ) an N-modal formula withFV(A)= r ( x ) , 1x1 = n . Then
(i) for any u E M M , u t= V x A ( x ) iff V a E Dun M , u I= A ( a ) , (iz) M t= A ( x ) iff V u E M V a E D; M , u k A ( a ) . (2) Let M be an intuitionistic KFE-model with the accessibility relation R , A ( x ) an intuitionistic formula with F V ( A ) = r ( x ) , 1x1 = n. Then (i) for any u E M M , u It- V x A ( x ) i f f Vv E R ( u ) V a E Dz M , v IF A ( a ) , (ii) M It- A ( x ) iff V u E M V a E DZ M , u Il- A ( a ) .
F be an S4-based KFE, M a KFE-model ove,r F . The pattern of M is the intuitionistic KFE-model Mo over F such that for any u E F and any atomic D,-sentence A without equality Definition 3.5.8 Let
M o , u Ik A iff M , u b OA. Let us check soundness of this definition:
L e m m a 3.5.9 The pattern exists for every S4-based KFE-model. Proof According to 3.5.3, we have to show that for any P E PLn, a , b E D; such that Vi ai xu bi, M , u I= O P ( a ) iff M , u I= O P ( b ) . In fact, by 3.5.1 for any v E R ( u ) , a; v, bi; hence M , v I= P ( a ) iff M , v I= P ( b ) by 3.5.3. Then
Vv E R ( u ) M , v b P ( a ) iff Vv E R ( u ) M , v t= P ( b ) , which implies (*). Thus Mo always exists; it is intuitionistic by definition.
(*)
238
CHAPTER 3. KRTPKE SEMANTICS
Now there is an analogue of 3.2.15. L e m m a 3.5.10 Let Mo be the pattern of M; then for any u E M and for any intuitionistic D,-sentence A
and for any intuitionistic sentence A Mo It A iff M k Proof The same as for 3.2.15, with a difference in the case when A is atomic of the form a = b. Then we have Mo,uIFa=b~ax,b, M, u != AT(= O(a = b)) @ Vu E R(u) a x, b, which is equivalent to a xu b by Definition 3.5.1.
W
Proposition 3.5.11 Let F be an S4-based KFE, A E I F = . Then
Proof
The same as for 3.2.25 based on 3.5.7 and 3.5.10.
Corollary 3.5.12 For a class C of S4-based KFEs
Proof
Followsfrom3.5.11 asintheproofof2.16.18.
H
The set of formulas valid in a KFE F is denoted by ML(')(F) (or IL(=)(F)). This notation is not quite legal before we show soundness; the proof of soundness is postponed until Section 3.6. L e m m a 3.5.13 For any KFE (F, D, x ) , ML(F, D)
C ML(F, D, x ) .
P r o o f For formulas without equality the definitions of forcing in (F,D, x) and (F,D) are the same. Every valuation in (F, D l x ) is a valuation in (F, D), so refutability of a formula in (F,D, x) implies its refutability in (F, D). R e m a r k 3.5.14 Not all valuations in (F, D) are admissible in (F, D l x), so it may happen that ML(F, D, x) # ML(F, D). A trivial counterexample is (F, D, x ) where F is a reflexive singleton, D is two-element and x is universal. Then obviously (F, D) !+ 3xP(x) > QxP(x), while (F, Dl x) k 3xP(x) > VxP(x). On the other hand, predicate Kripke frames can be regarded as a particular kind of KFEs.
3.5. KlUPKE FRAMES WITH EQUALITY
239
L e m m a 3.5.15 ( I ) Every Kripke frame F = (F, D ) is associated with a simple K F E F = = (F, D , x ) , i n which xu= idD, for any u E F . Then ML(=)(F) = ML(=)(F=), and IL(=)(F) = IL(')(F,) in the intutionistic case.
(2) A K F E (F, D , x ) is simple iff V a ,b E D+(a # b + E(a, b ) = 0 ) , where E i s the measure of identity (3.5.2). Proof (1) Valuations and the corresponding forcing relations in F and F= are just
the same. (2) In fact, in a simple KFE, u E E ( a , b) implies a = b. The other way round, if E ( a , b) = 0 whenever a # b, then a xu b holds only for a = b.
Proposition 3.5.16 Let C be a class of N-modal propositional frames. Then ML(KEC) = ML(KC). Similarly, for a class C of intuitionistic propositional frames, I L (ICEC) = IL (KC). Proof Consider the modal case. If A E ML(KEC), then A is valid in every KFE over C. In particular, for any F E KC, A is valid in the associated simple frame F,. So by Lemma 3.5.15, A E n { M L ( F ) I F E KC) = ML(KC). The other way round, suppose A E ML(KC). Then for any (F, D , x ) with F E C, we have A E M L ( F , D ) C M L ( F , D , x ) by Lemma 3.5.13. Hence A E ML(KEC).
3.5.3
Strong morphisms
In this section we consider strong morphisms defined in an obvious way - as morphisms of Kripke frames preserving equality. A larger class of morphisms will be considered in Section 3.7. Definition 3.5.17 Let F = (F, D, x ) , F' = (F', D', x') be Kripke frames with equality. A strong (p-)morphism from F to F' is a (p-)morphism of frames without equality f = ( f o ,f l ) : (F, D) (F', Dl) (Definition 3.3.1) such that for any u E F, a , b E D+
-
A strong isomorphism is a strong morphism which is an isomorphism of frames without equality.17 Strong morphisms of KFE-models are defined as morphisms of their frames satisfying the reliability condition from Definition 3.3.2; similarly for strong p-morphisms and strong isomorphisms.
CHAPTER 3. KRIPKE SEMANTICS
240
Obviously, i f the KFEs F = ( F ,D , x), F' = (F', Dl, x ) are simple, then a strong KFE-morphism from F t o F' is nothing but an =-morphism from (F,D ) t o (F', Dl). strong Strong morphisms o f KFEframes and models are denoted by -=, p-morphisms by -H=, strong isomorphisms by E. Lemma 3.3.4 easily transfers t o KFEs: L e m m a 3.5.18 (1) For a KFE F = ( F ,D , x ) the identity morphism idF :=
is a strong
isomorphism. (2) The composition of strong morphisms (in the sense of 3.3.4) is a strong morphism; similarly for p-morphisms and isomorphisms.
This yields the categories o f N-modal KFEs and strong morphisms, intuitionistic KFEs and strong morphisms, and similarly, o f KFE-models. L e m m a 3.5.19 Strong isomorphisms are exactly isomorphisms in the category of N-modal KFEs and strong morphisms.
Proof
H
An exercise.
Now we have an analogue o f Proposition 3.3.11 L e m m a 3.5.20 If ( f o , f l ) : M --+= M', for KFE-models M , MI, then for any u E F and for any Du-sentence B M , u != B
M', fo(u)
flu
.B ,
where flu . B is obtained from B by replacing occurrences of every c E D, with fl,(c). If the models are intuitionistic, the same holds for any intuitionistic D,sentence (and the intuitionistic forcing). L e m m a 3.5.21 Let ( f o , f l ) : F -= F' be a KFE-morphism, and let M' be a KFE-model over F'. Then there exists a unique model M over F such that ( f o ,f l ) : M += M ' , and similarly for ++=.
Proof
As in the proof o f 3.3.12, we put
for any u E F , P 6 PLm. Then M = ( F ,[) is a KFE-model. In fact, we have t o check that a E tU(P) & Q i ~xu i bi b E (,(P),
*
i.e. M', fo(u)
P ( f l u . a ) & a xu b
* MI, fo(u) b P ( f l , . b ) ,
(1)
3.5. KRIPKE FRAMES W I T H EQUALITY where
a x u b : = ' d i a i xu bi. But (1) holds, since
by the definition of a strong KFE-morphism, and
by the definition of a KFE-model. The claim ( f o ,f i ) : M uniqueness of M' follow easily.
-
M' and the
Hence we have an analogue of 3.3.13:
P r o p o s i t i o n 3.5.22 If F += F' for KFEs F , F', then M L ' ( F ) E M L = ( F 1 ) , and similarly, IL'(F) C I L = ( F 1 ) in the intuitionistic case. Proof
Along the same lines as 3.3.13, now using 3.5.21, 3.5.7, 3.5.20.
3.5.4
Main constructions
Now let us extend the definition of subframes and submodels 3.3.15 to the case with equality.
D e f i n i t i o n 3.5.23 Let F = (W,R1,. . . ,R N ) be a propositional Kriplce frame, F = ( F ,D l x ) a KFE, V W . A subframe of F obtained by restriction to V is
where D V is the same as in 3.3.15, x r V := ( x u ) u c v . If M = ( F ,J ) is a KFE-model, we define the submodel
where J 1 V is the same as i n 3.3.15. If V is stable, F generated.
1V,M
V are called
r
It is obvious that M V is a KFE-model, since J 1 V coincides with J on V . The definitions of reliability, rooted frames (models) and cones are trivially extended to frames and models with equality.
L e m m a 3.5.24 ( G e n e r a t i o n l e m m a ) Let F be a KFE, M a KFE-model over F , V a stable set of worlds in F . Then (1) M
r V is a reliable submodel of M ;
(2) M L ( = ) ( F ) & ML(=)(F
V ) ; similarly, for the intuitionistic case.
CHAPTER 3. KRIPKE SEMANTICS
242
Proof Almost the same as for 3.3.18. We use the same map (j,i ) and apply Lemma 3.5.20. To prove ( 2 ) ,we need a valuation J in F such that Ju = Jh for u E V, viz. JU(P) :=
otherwise.
This is really a valuation; J respects xu for u E V, since it coincides with the valuation 6'; and for u @ V there is nothing to prove. We define cones exactly as in 3.3.17:
Definition 3.5.25
Then we obtain an analogue of 3.3.21:
L e m m a 3.5.26 ( I ) M L ( = ) ( F )=
n ML(=)(F~u)
u€F
and similarly for the intuitionistic case. (2) For a class C of N-modal or intuitionistic KFEs,
where L is M L or IL, CT is the class of all cones of frames from C .
L e m m a 3.5.27 Let F be a propositional Kriplce, u E F , then KE(F T u ) = (KEF) 1u , where (KEF) f u := {F u I F E KEF).
Proof Similar to 3.3.27. Given a KFE G = ( F t u , D , x),we have to show that G is F T u for some K F E F over F. It suffices to extend the domain function D to the whole F ; x is extended in the trivial way (as identity at every world). We do this by putting
Then D' is obviously expanding, since F t u is a generated subframe and D, 5 DL for any w E F . W This implies an analogue of 3.3.28:
Proposition 3.5.28 Let C be a class of N-modal or intuitionistic propositional Kripke frames. Then
3.6. KRSPKE SHEAVES
(2) L(=)(KEc) = L(=)(KE(cT)), where L is M L or IL respectively. Proof
Similar to 3.3.28; apply 3.5.27 and 3.5.26(2).
Let us also define disjoint sums: Definition 3.5.29 Let Fi = (Fi, Di, x i ) be predicate Kripke frames with equality. Then iEI
where D(u, i) := Di(u) x {i), (a, i) f o r i E I, u E Fi, a , b E D,.
(b, i) := a
Since Fi is isomorphic to a generated subframe of
U F i , we obtain an analogue
L e m m a 3.5.30 ML(') IL(=)
(U Fi) iEI
3.6
=
1
U Fi G I
=
n ML(')(F~),
xi
b
and similarly
i€I
n lL(=)(Fi)for the intuitionistic case.
zEI
Kripke sheaves
Now let us consider semantical equivalents of KFEs called 'Kripke sheaves'. To define them, let us begin with the SCcase. In this case a Kripke sheaf is a set-valued functor defined on a certain category. Recall that a category consists of objects ('points') and morphisms ('arrows') between some of these points. In precise terms, it can be defined as a tuple.
where X, Y are non-empty classes, o, t , I are functions:
1:x-Y;
Y x Y --+ Y . X is called the class of objects of C and is also denoted by ObC. Y is called the class of morphisms of C and is also denoted by MorC. For a morphism f , the object o (f ) is called the origin of f , and t ( f ) the target of f . The notation f : a b or a f b means that o ( f ) = a and t ( f ) = b; this is read as 'f is a morphism from a to b' (or between a and b). C(a, b) denotes the class of all morphisms in C between a and b. I(a) is called the identity morphism of a and is also denoted by 1,. There are also the following conditions ('axioms'): o is a partial function
- -
CHAPTER 3. KRIPKE SEMANTICS
244 (1) la: a
-+
a;
(2) a o
is defined i f f o ( 0 ) = t(a)(i.e. i f f arrows a, 0 are consecutive);
(3) i f x
5y
(4) if x
-
(5) if x
a
y
P
---+ z ,
P
-+
then x
z
aoP
z;
u , then ( a o 0)o y = a o (0o y ) ;
y, then 1, o a = a o 1, = a.
A standard example o f a category is S E T , the category o f sets, in which objects are arbitrary sets, morphisms are maps. There is a well-known canonical way o f associating a category C = CatF with an S4-frame F = (W,R) [Goldblatt, 19841. Viz., we put ObC := W, MorC := R, C ( u , v ) := { ( u , ~ ) )1,, := ( u , u ) . So arrows o f C just represent t h e relation R. T h e composition is defined according t o the 'triangle rule': ( u ,v) 0 ( v ,w ) := (u,w). Definition 3.6.1 A n (S4-based, or intuitionistic) Kripke sheaf over an S 4 frame F is a SET-valued co-functor defined on C a t F . This means that a Kripke sheaf is a triple = ( F ,D , p ) where ( F ,D ) is a system of domains, p = (p,, I u R v ) is a family of functions p,, : D, -+ D, (transition maps) satisfying the following functoriality conditions: (1)
for every u E F, p,
= idD, (the identity function on D,);
(2)
u R v R w ( i n F ) implies p,, o pv,
= p,,.
W e call puv(a) the inheritor of the individual a (from D,) i n the world v . The domains D, are also called the fibres or the stalks of a. T o extend this definition t o arbitrary frames, recall that an arbitrary propositional Kripke frame F = (W,R1,.. . , R N ) is associated with an S4-frame F* := (W,R*),where R* is the reflexive transitive closure o f R1 U . . . U RN. Definition 3.6.2 A Kripke sheaf over a propositional Kripke frame F is a triple = ( F ,D , p) such that cP* := ( F * ,D , p) is an S4-based Kripke sheaf over F*. T h e frame F b y @., ( F ,D , p) Note that in R*, i.e. p = (p,,
is called the (propositional) base o f cP = ( F ,D , p) and denoted is called N-modal i f F is N-modal. Definition 3.6.20 p is a system o f functions parameterised b y 1 u R * v ) , satisfying the conditions 3.7.1 ( I ) , ( 2 ) for R*.
Lemma 3.6.3 Let F = (W,R1,.. . ,R N )be a propositional Kripke frame, ( F ,D ) a predicate Kripke frame, p = (pi,, I u R i v , 1 5 i I N ) a family of functions pi,, : D, D, satisfying the following 'coherence' conditions:
-
3.6. KRJPKE SHEAVES
(1) if u & ~ u ~ R ~. .,.u k~R i k u for some k 2 0, then P ~ ~ U U ~ O P ~ ~..ZO LP i~k ~Uk t~L= O ~id^,,;
(2) if u R ~ ~ u ~ R ~. u~k R u izk .v .and U R ~ ~ V ~. .u,Rj,u, R ~ ~ Uthen ~ . Piouul
OPilzlluz O ' ' ' OPikukv
= P i o u v l O P j l v l v z O ~' ' Opjmw,u.
Then there exists a unique Kripke sheaf ( F ,D, p*) such that pi,, uRiv.18
= PC, whenever
Proof For any pair ( u , ~ E) R* we can define a function pt, : D, -4 D, such that p:, := idD, and pz, := p ~ o u , l ~ p ~ , , l , Z ~ ~ - ~ ~ pfor ~ k ,any k V path ( u ,iO,u1, i l ,. ..,uk, i k ,v ) . The conditions ( I ) , ( 2 ) show that pz, is well defined. Then it follows that p* determines a Kripke sheaf over F*. H
Of course for S4-based Kripke sheaves the above conditions ( I ) , ( 2 ) follow from Definition 3.6.1. So every Kripke sheaf can be presented in the form described in 3.6.3. Viz. for = ( F ,D , p) put pi,, := p,, whenever uRiv. Then the conditions 3.6.3 (I), ( 2 ) hold and the corresponding Kripke sheaf ( F ,D , p*) is just a. So 3.6.3 yields an alternative definition of a Kripke sheaf. If (F,D , p) is a Kripke sheaf, a E D,, uR*u, we sometimes use the notation alv := p,,(a) and alu := (allv,.. . ,a,lu) for a = ( a l , . . . ,a,) E DZ. But this notation should be used carefully, because it is ambiguous if the fibres are not disjoint (then it may happen that p,,,(a) # p,,,(a) for a E D,, n D,,). Definition 3.6.4 A valuation i n a Kripke sheaf @ = ( F ,D, p) is just a ualuation i n the frame ( F ,D ) . If is a valuation, M = (a,<) is called a Kripke sheaf model over D. Forcing relation M , u k A between a world u in an N-modal Kripke sheaf model and an N-modal D,-sentence. A is defined by the same clauses as in predicate Kripke frames (Definition 3.2.9), with the only difference:
<
<
Definition 3.6.5 A valuation i n an S4-based Kripke sheaf (and the corresponding Kripke sheaf model) is called intuitionistic if it satisfies the following conditions:
Then the'intuitionistic forcing in M is defined by the following clauses (cf. Definition 3.2.10):
u IF PP(a1, . . . ,a,) iff (a1, . .. , a,) E
<, ( P g ) (for n > 0 ) ;
u It- P$ iff u E <,(P;); lsSo in particular, pi,,
= i d D , if uRiu;pi,, = pjuv if uRiv and uRjv.
CHAPTER 3. KRIPKE SEMANTICS
246
u IF ( B 3 C ) ( a l , .. . ,an) iff
u l t 3 x A ( x ) iff 3a E D, u I I- A(a); u It V x B ( x ,a l , . . . ,an) iff Vv 'v R ( u )Vc E D, v ItB(c, p,,(al), . . . ,pu,(an)). L e m m a 3.6.6 Let u , v be worlds in an intuitionistic Kriplce sheaf model M . Then for any intuitionistic D,-sentence A ( a ) M , u It A(a) & uRv Proof
+
M , v It- A(puv - a ) .
Similar to 3.2.16.
L e m m a 3.6.7 The intuitionistic forcing relation has the following properties: M , u II- -B(a) iff Vv E R ( u ) M , v IpL B(p,, . a ) , M , u II- a # b iff Vv E R ( u ) p,,(a) Proof
# puw(b).
Obvious from 3.6.5.
Definition 3.6.8 An N-modal predicate formula A is called true in an N-modal Kripke sheaf model M if V A is true at every world of M . A is called valid in a Kriplce sheaf @ if it is true under any valuation in @. The definitions for the intuitionistic case are similar. The notation for the truth and validity is the same as in the case of Kripke frames.
L e m m a 3.6.9 Let M be an N-modal Kriplce sheaf model, A ( x ) an N-modal formula with F V ( A ( x ) )= r ( x ) , 1x1 = n . Then for any u E M M , u t= V x A ( x ) iff Va E DE M , u != A(a). Proof
Similar to 3.2.17.
L e m m a 3.6.10 Let M be an intuitionistic Kriplce sheaf model with the accessibility relation R , u E M , A ( x ) an intuitionistic D,-formula with F V ( A ( x ) )= r ( x ) , 1x1 = n . Then M , u IF V x A ( x ) i f fQv E R ( u )Qa E Dz M , v It- A(p,, . a). Proof
Similar to 3.2.18.
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247
Definition 3.6.11 Let @ be an S4-based Kripke sheaf; M a model over F . The pattern of M is the model Mo over F such that for any u E F and any atomic D,-sentence without equality A
As in the case of Kripke frames, the pattern is an intuitionistic model and it always exists.
L e m m a 3.6.12 If Mo is a pattern of a Kripke sheaf model M , then (1) for any u E M , for any intuitionistic D,-sentence A
(2) for any A E IF'
MoIkA i f f ~ k ~ ~ . Proof
Similar to 3.2.15 and 3.2.24.
¤
Definition 3.6.13 For a set C of modal (or intuitionistic) sentences, a C-sheaf is a Kripke sheaf (of the corresponding type) validating every formula from C . @ k C (or @ Ik C ) denotes that C is valid in @. The class of all C-sheaves is denoted by VKs(C) and called modally (respectively, intuitionistically) definable (by C). L e m m a 3.6.14 Let @ be an N-modal (respectively, intuitionistic) Kripke sheaf over a propositional frame F , A an N-modal (respectively, intuitionistic) propositional formula. Then @ k (It)A iff @, k (Ik)A. Proof Similarly to the case of Kripke frames (Lemma 3.3.32), validity for propositional formulas in @ is the same as in F . P r o p o s i t i o n 3.6.15 Let
Proof
Q,
be an S4-based Kripke sheaf, A E I F = . Then
Along the same lines as 3.2.25, now using 3.6.11.
¤
Now we have an analogue of 3.2.26: L e m m a 3.f3.16 Let A(x), B(x) be congruent modal (or intuitionistic) formulas, 1x1 = n, and let M be a modal (respectively, intuitionistic) Kripke sheaf model. Then for any u E M , a E Dz
M , u k (It) A(a) iff M, u I= (IF) B(a). Thus the set of formulas valid i n a Kripke sheaf is closed under congruence.
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P r o o f Along the same lines as 3.2.26. Again we consider the equivalence relation on modal formulas A
-
B iff FV(A) = F V ( B )
and for any distinct x with FV(A) = r(x), for any u E M , a E
DL~'
We have t o check the properties 2.3.17(1)-(4) for this relation. For (1)-(3) the proof is the same as in 3.2.26. For (4) there is a slight difference: now M , u k [a/x]OiA w Vv E Ri(u) M, v I= [p,, and similarly for B. So A
B implies OiA
-
. a/x]A
OiB.
¤
T h e o r e m 3.6.17 (Soundness t h e o r e m )
(I) The set of all modal predicate formulas (with equality) valid in a Kripke sheaf is a modal predicate logic (with equality).
(11) The set of all intuitionistic predicate formulas (with equality) valid in an S4-based Kripke sheaf is a superintuitionistic predicate logic (with equality). P r o o f Along the same lines as 3.2.29. The main thing is t o check that formula substitutions preserve validity. So we assume that cP k A for a formula A and a Kripke sheaf Q, and show that cP k SA for S = [C(x,y)/P(x)], where P E P L n occurs in A, the list x y is distinct, and r ( y ) C_ F V ( C ) C r(xy). By Lemma 3.6.16 we can replace A with a congruent formula, so we assume that A is clean, BV(A) n r(y) = 0.Again we choose a distinct list z such that FV(A) U r ( y ) = r(z); then
Let m = lz 1. Given a model M = (a,J), we show that for any u E @, c E D r
Let c' be the part of c corresponding t o y. We define MI = ( a , q ) similarly t o 3.2.29: for any v E
@Tu,
a E Dz
M I , v I=P ( a ) iff M, v t= C(a, c'lv); for any other atomic D,-sentence Q M l , v t = Q iff M , v k Q .
3.6. KRTPKE SHEAVES
249
Then every subformula of A has the form B(z, q), where q is distinct, r(q) = BV(A); so by 2.5.24 we present S B as (SB)(z, q). Then we prove the claim: Vv E @I.uV a E ~ $ 1(M1,v k B(clv, a) % M , v t= (SB)(clv,a)) by induction. The only difference with 3.2.29 is in the case B = OiB1: M I , v I= B(clv, a) iff Vw E Ri(v) MI, w I= Bl(clw, alw) iff Vw E Ri(v) M, w l= (SBl)(clw, alw) (by the induction hypothesis) iff M, v I= Oi(SBl)(clv, a)(= (SB)(clv,a)). Here we use the equality (a1v)lw = alw, which follows from Definition 3.6.1. H
R e m a r k 3.6.18 One can similarly define forcing and validity for the case when p does not satisfy the coherence conditions (I), (2) from 3.6.3. But then the set of validities is not necessarily substitution-closed. These 'frames' are a special kind of Kripke bundle considered in Chapter 5; note that validity in Kripke bundles is not substitution closed either, cf. Exercise 5.2.13. Every predicate Kripke frame corresponds t o a Kripke sheaf, in which uRiv always implies D, E D, and p, is the inclusion map, i.e. p,,(a) = a for any a E Du . More generally, every KFE F = (W, R1,. . . ,RN, D, v) corresponds to a Kripke sheaf O(F) constructed as follows. Let a, be the class of a E D, modulo xu. We define O(F) as the Kripke sheaf with the fibres DL := {a, I a E D,) and the transition maps p,,(a,) := a, for u R ~ v O(F) ; is well defined, due to Lemma 3.6.3 and since uRiv implies D, C D, and x, C x u . The following is almost obvious: L e m m a 3.6.19 Valuations (both modal and intuitionistic) in a KFE F and in the Kripke sheaf O(F) are associated. Namely, a valuation in F corresponds to the valuation O([) in O(F) such that
c
<,
for P E P L n , n > 0, and O(e),, coincide on PLO. The other way round, every valuation in O(F) has the form O(E) for some valuation ( in F . E is intuitionistic ifl O(J) is intuitionistic. Proof The above definition of O(E) is sound; recall that ( a l , . . . , a n ) E t,(P) depends only on classes of a l , . . . ,a, modulo x u . Now if 77 is a valuation in O(F), we define E by
where a, := ((al),,
. . . , (a,),) for a = ( a l , . .. ,a,), n > 0.
This definition is sound, because a, = b, iff Vi ai xu bi. Then obviously 77 = O ( 0 The argument for the intuitionistic case is left to the reader.
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250
L e m m a 3.6.20 (1) For any N-modal formula A ( x l , . . . ,x,), a valuation E i n F , for any u E
F ; a l l . . . ,a,
E D,:
6, u I= A(a1,. . . , a n ) (in F ) iff Q(E),u k A ( ( a l ) u l . .. ,(an),)
(in @ ( F ) ) ;
and similarly for the intuitionistic case. (2) If F is an N-modal KFE, A is an N-modal formula, then F O ( F )k A.
+ A iff
(3) If F is an intuitionistic KFE, A is an intuitionistic formula, then F Il- A iff Q ( F ) Il- A.
P r o o f (1) By induction on the complexity of A. For example,
O i B ( a ) iffVv 'v R i ( u ) c , u k B ( a ) iff Vv Jv R $ ( u ) O(E),v [,u Q ( 6 ) ,u O i B ( % ) , since a, = p,, . a,. The remaining cases are left to the reader. The claims ( 2 ) , (3) now follow from 3.6.19, 3.6.9, 3.6.10.
B ( a , ) iff H
Due to Theorem 3.6.17 and Lemma 3.6.20, we obtain
T h e o r e m 3.6.21 ( S o u n d n e s s t h e o r e m ) (I) The set of all modal predicate formulas (with equality) valid in a KFE is a modal predicate logic (with equality). (11) The set of all intuitionistic predicate formulas (with equality) valid i n an S4-based KFE is a superintuitionistic predicate logic (with equality). These logics are denoted by M L ( = ) ( F ) ,IL(=)(F) as usual. According to the general definitions from Section 2.16, the modal predicate logic of a class of Kripke sheaves 3 is
The superintuitionistic logic I L ( = ) ( ~is) defined analogously. The other way round, every Kripke sheaf is equivalent to one of the form O ( F ) . To show this, let us introduce a convenient subclass of Kripke sheaves.
D e f i n i t i o n 3.6.22 A Kripke sheaf is said to be disjoint i f all its fibres are disjoint.
-
D e f i n i t i o n 3.6.23 A n isomorphism between Kripke sheaves ( F ,D, p) and DL such that f, . pd, = f , . p,, ( F ,D', p') is a family of bijections f , : D, whenever u R * v .
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251
A more general notion of morphism will be discussed in the next section. It is almost obvious that isomorphic Kripke sheaves have the same modal (or superintuitionistic) logics; for a precise proof, one should check the equivalence M, u t= B(a) iff M', u k B(f, - a) for any D,-sentence B(a) if it holds for any atomic D,-sentence. We leave this as an exercise for the reader. L e m m a 3.6.24 Every Kripke sheaf is isomorphic to a disjoint Kripke sheaf over the same propositional frame.
Proof In fact, we can replace each D, with DL = {(a,u) I a E D,) and change the functions p,, appropriately, viz., put pL,(a, u) := (p,,(a), v). For a disjoint Kripke sheaf (F, D, p) and a E D,, uRiv, we sometimes write aRiv and say that a is Ri-related t o v. Now let @ = (F,D, p) be a disjoint Kripke sheaf. Consider the KFE G(@):= (F, D', XI), where I w E F, u E F f w ) , DL :=
U{D,
I
xh := {(a, b) E (D;)~ (alu) = (blu)).
Here is an equivalent presentation of G(@) in the form described in Lemma 3.5.2: D'+ ..D1 .
u
uEF
f o r a D,, ~ b~ D,; u , v E F . It follows easily that DL Dh and x; x:, whenever uRiv; thus G(@) is really a KFE. Speaking informally, DL absorbs D, and the domains of all R*-predecessors of u; x; makes every individual from D, equivalent to all its predecessors. So This observation is used there is a natural bijection between D, and DL/ XI. in the proof of the following
c
L e m m a 3.6.25 The Kripke sheaves @ and O(G(@)) are isomorphic.
Proof We have @ ( G ( @ = ) ) (F, D",pl'), where D" = (DL,' X~),,=F,and the transition map pewsends every equivalence class a / XL to a / xh (for a E DL). Now, every class a" = (b/ xL) E D t contains a single element a from D,, namely, a = (blu). So there exists a well-defined bijection 8, : D: -+ D, such that Q,(b/ -6) = blu. (For the surjectivity, note that (alu) = a for a E D,.) Finally, uRiv implies
CHAPTER 3. KRIPKE SEMANTICS
252 and
8,(pE,(b/ x:)) = 8,(b/ x:) = blv. Thus p, -6, = 8, . pz,, which means that the family of functions (8, is an isomorphism between the Kripke sheaves Q(G(@))and a .
Iu E F)
rn
So we can introduce semantics generated by predicate Kripke frames with equality. Due to Lemmas 3.6.20 and 3.6.25, the same semantics are generated by Kripke sheaves.
Definition 3.6.26 The N-modal Kripke sheaf semantics ICE$=) is generated by the class of all N-modal predicate Kripke frames with equality (or Kripke sheaves). Similarly the intuitionistic Kripke sheaf semantics KE~;) is generated by the class of all intuitionistic KFEs (or Kripke sheaves). Logics complete i n these semantics are called Kripke sheaf complete, or just KE-complete. As we pointed out at the end of Subsection 3.4.2, every predicate Kripke 5 ICE$) and actually 4 frame is equivalent a simple KFE. Thus K E E ) , as we shall see later on; the same is true for the intuitionistic semantics. Similarly to Corollary 3.3.37 we obtain
ICE)
KE)
Proposition 3.6.27 Kripke sheaf semantics has the collection property. Proof
Obvious, by Lemma 3.5.30.
W
For Kripke and Kripke sheaf semantics there also exists a stronger version of completeness.
Definition 3.6.28 A n N-modal theory I? is called satisfiable i n an N-modal Kripke frame (respectively, K F E ) F if there exists a model (respectively, KFEmodel) M over F and a world u E M such that M , u I= r , i.e. M, u I= A for any A E r. A n intuitionistic theory (I?,A) i s called satisfiable in a n intuitionistic Kripke frame (or K F E ) F if there exists a n intuitionistic model (or KFE-model) M over F and a world u E M such that M, u IF ( r , A), i.e. M, u IF A for any A E r a n d M , u Iy B for any B E A. Lemma 3.6.29 A theory (modal or intuitionistic) i s satisfiable i n a n L-KFE iff it is satisfiable i n a Kripke sheaf validating L.
Proof In fact, by Lemma 3.6.20, F and Q(F) have the same logic (of the corresponding kind); a sentence A is satisfiable at F , u iff it is satisfiable at O(F),u . On the other hand, by Lemma 3.6.25, every Kripke sheaf Q, is isomorphic to H O(G(@)),so satisfiability in cP and G(@)is the same.
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253
Definition 3.6.30 A n N-modal predicate logic L (with or without equality) i s called strongly Kripke complete (respectively, strongly Kripke sheaf complete) if every L-consistent N-modal theory (respectively, with or without equality) i s satisfiable i n some Kripke L-frame (respectively, L-KFE). Similarly, a n intuitionistic predicate logic L (with/without equality) i s called strongly Kripke (Kripke sheaf) complete i f every L-consistent intuitionistic theory (with/ without equality) i s satisfiable i n some Kripke L-frame (L-KFE). Thanks to Lemma 3.6.29, the notions of strong completeness are the same for Kripke sheaves and KFEs.
Lemma 3.6.31 Every strongly complete m.p.l.(=) or s.p.1. (=) is complete ( i n the corresponding semantics of Kripke frames or Kripke sheaves). Proof In fact, in the modal case, if A @ L, then {TA) is L-consistent. So if L is strongly complete, then 1 A is satisfiable in an L-frame F , and thus F separates A from L. Therefore L is complete by Lemma 2.16.2. In the intuitionistic case, if A @ L, then ( 0 ,{A)) is L-consistent; the remaining argument is the same as in the modal case.
3.7
Morphisms of Kripke sheaves
Kripke sheaf morphisms are a natural generalisation of predicate Kripke frame morphisms defined in 3.3.1:
Definition 3.7.1 Let cP = ( F , D, p), cP' = (F', D',pl) be Kripke sheaves over the frames F = (W, R1,. . . ,RN), F' = (W', R i , . . . ,R h ) . A morphism from cP t o @ is a pair f = (fo, f l ) satisfying the conditions (1)-(3) from Definition 3.3.1 and also (5) the following diagram commutes whenever uRiu:
=-
Du
D>o(u)
flu The latter condition can be briefly written as
(for a E Du, U R ~ V but ) , strictly speaking, this makes sense only for disjoint sheaves.
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254
Exercise 3.7.2 Show that under the conditions of 3.7.1 the diagram ( 5 ) commutes whenever uR*v.
-
Definition 3.7.3 A morphism of Kripke sheaf models M = (a,() and M' = (@',<') is a morphism ( f o , f l ) : @ @' such that for any P E PLm, m 2 0, U E @ , ~ E D T M , u !F P ( a ) i f fM', fo(u) P ( f l u . a). The notions '=-morphism', 'p(')-morphism', 'isomorphism' are transferred to Kripke sheaves in an obvious way. Now we easily obtain an analogue to Lemma 3.3.4:
L e m m a 3.7.4 (1) The identity morphism id+ := ( i d w , id^^)^^^) is an isomorphism of
Kripke sheaves.
-
(2) The composition of morphisms ( f o , f l ) : @ --+ @' and (go,gl): @' ---+ @I' defined as ( f o 0 go, f l u 0 glu)uEw)is a morphism @ @ I , similarly for all other kinds of morphism. This yields us different categories of Kripke sheaves as in the case of Kripke frames. Next, we have analogues to 3.3.6, 3.3.8-3.3.10:
Definition 3.7.5 A Kripke sheaf morphism over a propositional Kripke frame F is a morphism of Kripke sheaves over F , in which the world component is the identity map.
-
Definition 3.7.6 Let @ = ( F ,D , p ) , @' = (F', D',pl) be Kripke sheaves, h : F' F a morphism of propositional frames. We say that a' is obtained by changing the base along h i f DL = Dh(?) for any u E F and ph, = for any pair ( u ,v) E Ri.We use the notatzon h,@ for @'. R e m a r k 3.7.7 This definition is sound in the case when h is only monotonic, i.e., for a 'more traditional category of sheaves'.
--
Proposition 3.7.8 Under the conditions of the Definition 3.7.6, there exists a 'canonical' =-morphism ( h , g ) : h,@ --+= @. Every morphism ( h ,f l ) : @" @, where @: = F', can be uniquely presented as a composition @ --+ I' h,@ @ of a morphism over F' and the canonical morphism. The next claim is analogous to 3.3.11:
Proposition 3.7.9 If ( f o , f l ) : M --+= M' for N-modal Kripke sheaf models M , MI, then for any u E M , A E M S ~ ) ( D , ) M,u
+A
iif M Mfo(u) ',
flu
.A,
where flu . A is obtained by replacing every a E D, with fl,(a). The same holds in the intuitionistic case.
3.7. MORPHlSMS OF KRIPKE SHEAVES
255
Proof We check only the 0-case. Let R i , Ri be the accessibility relations in M , M'. If A = U i B ( a ) (for a formula B ( x ) ) ,then M , u I=A iff Vv E R i ( u ) M , v t. B(pu,.a) iff VV E R ~ ( uM', ) fo(v) b B(fl,-(p.a)) by the induction hypothesis. By 3.7.1 (5), the latter is equivalent to vv E Ri(u) M', fo(v) I= B(p>o(u)fo(V) . ( f l u .a ) ) ,
which is the same as
Since f satisfies the conditions 3.3.1 (1)-(3),we have fo[Ri(u)]= R:(fo(u)). Eventually M , u I=A iff M', fO(u)!= O i B ( f l u. a ) W
as required.
L e m m a 3.7.10 Let f : @ -+ @' be a morphism of Kripke sheaves, M' a Kripke sheaf model over a'. Then there exists a unique model M over such M'. The same holds for all kinds of morphism. If M' is that f : M intuitionistic, then M is also intuitionistic.
-
Proof
Similar to 3.3.12; an exercise for the reader.
rn
Proposition 3.7.11 Let and be Kripke sheaves. If there exists a p(=)morphism from onto ( P z , then ML'(Q1) C ML'(Q2) (or IL=(B1) C I L Z ( Q 2 )in the intuitionistic case). Proof
Similar to 3.3.13. Use Lemmas 3.6.9, 3.6.10 and 3.7.10.
W
In the case of disjoint Kripke sheaves there exists en equivalent definition of a morphism. Let us first present disjoint Kripke sheaves in an equivalent form.
Definition 3.7.12 A morphism of propositional Kripke frames
is called etale i j it has the unique lift property
In this case every restriction f I (F T u ) is an isomorphism to F' T f ( u ) . In fact, this is a bijection, due to the monotonicity and the unique lift property. Its converse is also monotonic, due to the lift property.
Proposition 3.7.13
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256
(1) Let @ = ( F ,D , p) be a disjoint Kripke sheaf over a frame F= (W,R 1 , . . . ,R N ). Consider the frame of individuals F f := (D+, . .. ,p$), where
pt,
Let T : D+ ~ ( a=) u
-
W be the map sending every individual to its world (i.e. a E D,). Then T : F+ ++ F is etale.
(2) Conversely, for any etale p-morphism of propositional frames T : F' -h F there exists a disjoint Kripke sheaf = ( F ,D, p), i n which F' is the frame of individuals and D, = r-l ( u ). Proof (1) The monotonicity and the lift property of T hold, since p,, from D, to D, whenever uRiv. The unique lift property
holds, since (p+)*induces a function p,, R* , according to 3.6.1, 3.6.2.
: D,
-
is a function
D, for any pair ( u ,v ) E
( 2 ) In fact, for uR*v we can define the function p,, : D, ---+ D, by the ~ ( b=) v & a(pl)*b. Since T is etale, this b is condition p,,(a) = b always unique. The unique lift property also implies the properties 3.6.1 (I), ( 2 ) for ( F * ,Dl P ) .
Proposition 3.7.14
( I ) Let @ = ( F ,Dl p), a' = (F', D', p') be disjoint Kripke sheaves over N modal frames, and let T : F+ ++ F , T' : F'+ + F' be the corresponding etale morphisms. Also let ( f o , f l ) : @ a' be a morphism (in the sense of 3.7.1). Consider the map f,f : D+ ---+ Dl+ of total domains such that flf(a) = flu(a) for a E D, (i.e., f: = U f l u ) . Then flf : Ft ++ F'+
-
uEF
and the following diagram commutes:
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257
f: is surjective on all fibres; it is a p-morphism whenever fo is a pmorphism.
-
-
(2) Conversely, every morphism g : F+ F'+, for which every g 1 D, is a surjective map onto Dj0(,), equals fif for some morphism ( f o , f l ) :
a'.
P r o o f (1) Let us show that fif is monotonic. Suppose aprb, a E D,, b E D,. Then b y definition, b = alv, fif(a) = fl,(a), fT(b) = fl,(b), u&v. By 3.7.1,
and thus f f (a)p:+f ,f (b). T h e commutativity o f the diagram is almost obvious: for a E D,,
-
D;o(u). But fo(u) = fo(rl(a)). since f l u : D, Now we can check the lift property for fif . In fact, suppose a E D,, f , f ( a ) = flU(a)p:+b1.Then b' = flU(a)lv1for some d E R:(ul), where fi,(a) E D,, , i.e.,
Since fo has the lift property, we obtain v E R i ( u ) such that fo(v) = v'. Then by definition, ap+(alv). O n the other hand, we obtain
b y 3.7.1. Therefore f: is a morphism. I t is surjective, since every f l u is surjective, b y 3.3.1. (2) Given g, we define f l := ( f l u ) u E F , with f l u = g 1 D,. Then for any a E D,, uRiv flv(alv) = g(alv) E Dfo(v). Since ap:(ajv), it follows that fl,(a) = g ( a ) p : f g ( a l v )i.e. , g(alv) = g(a)lfo(v). Thus ( f o , f i ) satisfies 3.7.1 ( 5 ) , and so it is a morphism, g = f:. T h e definitions o f a subframe, etc. from Section 3.3 can also be transferred t o Kripke sheaves. Definition 3.7.15 Let = ( F ,D , p ) be a Kriplce sheaf over a propositional frame F = (W,R 1 , . . .,R N ) ,and let V & W . Then we define the corresponding subsheaf as follows:
where F
r
1 V is the same as in Definition
1.3.13.
For u E F , the subsheaf generated by u or the cone cPfu is defined as (WTu).
CHAPTER 3. KRIPKE SEMANTICS
258 Exercise 3.7.16 sheaves.
(1) Prove an analogue to Generation lemma 3.3.18 for Kripke
(2) Define disjoint sums of Kripke sheaves and prove an analogue t o Proposition 3.3.36. L e m m a 3.7.17 For a Kripke sheaf @ over a frame F ML(=)(@) = ~){ML(=)(@Tu) I u E F),
and similarly for the intuitionistic case. Proof
Cf. Lemma 3.3.21.
Note that if u X R v (i.e. if u, v are in the same cluster) in a Kripke S4-frame F , then in every predicate Kripke frame F = (F, D) we have D, = D,. So we can define the skeleton of F :
F" := (F",D"), where F" is the skeleton of F (Definition 1.3.40) and for a cluster u", D,",
:=
D, Analogously, one can easily see that for u MR v in an S4-based Kripke sheaf is a bijection between D, and D, with the converse p,, . Thus for any model over @, @ = (F, D, p), the function p,,
u k A(a1,. . . ,a,) iff v k A(alIv,. .. ,anlv). So we can properly define the skeleton Q," of an S4-based Kripke sheaf @. Then we obtain an analogue to 1.4.14: L e m m a 3.7.18 For an intuitionistic Kriplce frame F , IL(F) = IL(F"); similarly, for intuitionistic Kriplce sheaves. P r o o f Consider the case of Kripke sheaves. For an intuitionistic valuation in Q, there exists an intuitionistic valuation 6" in @" such that
t
-
for any P, u. Then the map u H u" is a p-morphism ((a,() (@",tW). It remains to note that every intuitionistic valuation in Q," has the form t". Therefore, in the semantics ICEint is generated by Kripke sheaves over posets (and similarly, for the semantics ICint). Finally note that Kripke sheaf morphisms are appropriate also for KFEs. F' of KFEs just as an arbitrary morphism Viz., we can define a morphism F of associated Kripke sheaves O(F) 8(F1). Every strong morphism of KFEs corresponds to a morphism in this sense, but not the other way round. E.g. consider KFEs F = ( F ,D, E) and F' = (F, Dl, E'), where F is a reflexive singleton u, ID, I = 1, IDL1 = 2 and E, E' are universal. Then obviously @(F) 2 O(Ff), but there is no strong isomorphism from F to F'.
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3.8. TRANSFER OF COMPLETENESS
3.8
259
Transfer of completeness
In this section we show that in some simple cases Kripke and Kripke sheaf completeness transfer t o extensions. We begin with a lemma analogous to 1.3.45. Its proof is based on the existence of exact KFEmodels for logics with equality, which will be proved in Chapter 6 (cf. Proposition 6.1.27). A L e m m a 3.8.1 If an m.p.l.(=) L is conically expressive, then the rule - is O*A admissible in L. P r o o f Since L has a characteristic model, it suffices to show that for any KFE-model, for any A, M k A implies M I=O*A. We can argue similarly to 3.2.29, if we consider the Kripke model with the accessibility relation R*. If M k A(xl,. . .,x,), then for any u E M for any a l , . . .,a, E D,, M, u k A(a1,. . . ,a,), by 3.5.7. Hence, M, u k O*A(al,. . . , a n ) by 3.3.39. But obviously, O*A(al,. . . , a n ) = (O*A)(al,.. . ,an). Thus M k O*A(xl,. . . , x,), by 3.5.7. Definition 3.8.2 A pure equality sentence is a predicate sentence with equality that does not contain predicate letters other than '=I. In particular, every closed propositional formula is a pure equality sentence. T h e o r e m 3.8.3 If a conically expressive m.p.l.= or an s.p.l.= L is Kripke frame (respectively, Kripke sheaf) complete and C is a pure equality sentence in the language of L, then L C is also Kripke frame (respectively, Kripke sheaf) complete.
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P r o o f (I) Modal case. If L f C Y A, then L Y O*C > A, since by Lemma 3.8.1, L C t- O*C. By completeness, there exists a Kripke frame (or a KFE) F k L and a Kripke (respectively, KFE) model M over F such that M, u k O*C A -A for some u. Hence MTu, u b O*CAlA by Lemma 3.3.18 ( I ) , and thus MTu b C, by Lemmas 3.3.39 and 1.3.19. But C is a simple equality sentence, so its truth value a t every world does not depend on the valuation. Thus F f u b C. We also have FTu 1 L by Lemma 3.3.18 (2), and therefore A is refuted in an (L C)frame. (11) Intuitionistic case. The proof is very similar. If L C Y A, then L y C > A. By completeness, there exists an intuitionistic Kripke frame F It L and a Kripke model M over F such that M , u Iy C > A for some u. Hence M, v IF C and M, v Iy A for some v accessible from u. Then M f v , v It C and M f v , v Iy A, by Lemma 3.3.18 (I), and thus MTv It- C (by 3.2.16). Since C is a pure equality sentence, it follows that FTv k C . FTv k L by Lemma 3.3.18 W (2), so A is refuted in an (L C)-frame.
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An analogue o f this theorem holds for strong completeness: Theorem 3.8.4 Let L be an s.p.l.(=) or an m.p.l.(=), I' a set of pure equality sentences in the language of L. If L is strongly Kripke (respectively, Kriplce sheaf) complete, then L r is also strongly Kripke (Kripke sheaf) complete.
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Proof ( I ) Modal case. If a theory A is (L+r)-consistent, then it is (L+Umr)consistent, since obviously L+r t- Omr. So the theory OmI'UA is L-consistent. By strong completeness, there exists a Kripke frame (or a K F E ) F k L such that M , u k Ooor U A for some world u in a Kripke model M over F . Then M , u k O,B for any B E I?, a E I F , SO by 3.2.11 it follows that M , v I= I' for any v E R * ( u ) . Then as in the proof o f 3.8.3, it follows that FTu is an ( L r)-frame satisfying A at u. (11) Intuitionistic case. I f a theory ( A ,9)is L+r-consistent, then the theory (I?U A ,E) is L-consistent. By strong completeness, it is satisfiable in an L-frame Q, at some world u. Then similarly t o the modal case, we obtain that ( A ,Z) is satisfiable in the ( L I')-frame CJTu.
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Now let us consider equality expansions and prove some results generalising [Shimura and Suzuki, 19931. Lemma 3.8.5 Let M = (Q,,6 ) be an L-modal (respectively intuitionistic) Kripke model over N-modal (respectively, an S4-)KFE CJ = ( F ,D+, E ) such that M b (It-) E: and Q does not occur in A. Let CJ' := ( F ,D+ , El), where
E1(a,b) := { u I M , u k Q(a,b)), MI := (a',J ' ) , where for any u E F, P
E
PL
J u ( P ) zf P occurs i n A or P = Q , otherwise. Then (1)
a'
is an N-modal (respectively, S4-) K F E ;
(2) MI is a modal (respectively, intuitionistic) KFE-model;
(3) for any u E M , for any modal (respectively, intuitionistic) Du-sentence B in the same predicate letters as A M , u k (It)B& 2 8 M ' , u k (IF) B. The same statements are true under the assumption M k (It)EQ, with the following changes: 6' = J ; (3) holds for any D,-sentence B without occurrences of Q.
3.8. TRANSFER OF COMPLETENESS Proof
(1) In fact,
M != ~ x ~ Y ( Q ( x ,3YQ(Y,x)) ) implies Ef(a,b) E Ef(b,a ) for any a , b E D+;hence obviously Ef(a,b) = Ef(b,a). Similarly the other members of E are responsible for the properties (2)-(4) from Lemma 3.5.2. For example, M, u I= VxQ(x, x) means that M, u for any a E D,, hence uE
Q(a, a )
U Ef(a,a). a€D+
(2) Next, let us show that M' is a KFE-model. In fact, according to 3.5.2 and 3.5.3, we have to check
If P t occurs in A, this means
: I which follows from M I=. If P t = Q, (*) is equivalent to
: I which also follows from. If Pp does not occur in A, P t # Q, (*) holds trivially. (3) Let MF~=)(D,) be the set of all D, sentences constructed from MF~=). By definition of MI, Mf,ukBiffM,ut=B for any u E M , for any atomic B E MFA(D,), and thus the equivalence M , u k B& iff M f , u I =B holds for any atomic B E MFT(D,). By induction we obtain that it holds for any B E MFT(D,), since @, @' are based on the same (F, D).
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Theorem 3.8.6 (1) Let C be a class of N-modal propositional Kripke frames such that ML(C) is conically expressive. Then MLE(KEC) = ML(KEC)'.
(2) If C is a class of propositional Kripke S4-frames, then IL=(KEC) = IL(K&C)=.
Proof (1) Since ML(C) is conically expressive, ML(K&C) is also conically expressive. So let us show that if a modal sentence A does not contain Q, then
(+) Suppose
o*( Then
~ £ 2 )3 AQ @ ML(KEC).
(A&:)
M , c~0*
A
7~~
for a KFE-model M over F E C and for some u E M. By the generation lemma
Hence by a KFE-analogue of Lemma 3.3.39, M T u k E:. Then by Lemma 3.8.5, M T u , u != 7AQ implies ( M T u)', u k 7A. Since ( M T u)' is a KFE-model over F f u, we have A @ ML=(ICE(C T u)), whence A @ ML'(KEC) by Proposition 3.5.28. (+) Suppose for some F E C, A @ ML'(K&F). Then there exists a KFE @ = (F,D, x) and a model N = (@,J) such that N, u !# A at some world u. Consider a model M over @ with the trivial equality such that for any v E M , a , b D, ~ M , v i = Q ( a , b ) i f f a x , b, and M,vl=BiffN,v!=B for any D,-sentence B without Q and equality. Then M I= E,: as one can easily check, and N = M', so we can apply Lemma 3.8.5. Thus N , u !+ A implies M , u I# AQ, and therefore El*(/\E:) > AQ $ ML(KEF) C ML(KEC). The equivalence (#) together with Proposition 2.14.4 show that sentences without Q are the same in MLZ(KEC) and ML(KEF)'. Now if A contains Q, it can be replaced with another letter Q' that does not occur in A. If A' := [Qf(x,y)/Q(x, y)] A, then A = [Q(x,y)/Q1(x,y)] A'; so for any logic L, A E L iff A' E L. Therefore all sentences in these logics are the same. Since for any formula A and a logic L, A E L iff VA E L, the logics are really equal.
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(2) Similarly, in the intuitionistic case due to Proposition 2.14.5, it is sufficient to show that for a sentence A and an extra predicate letter Q
&z
The proof is quite similar to the modal case. (+) Suppose > AQ # IL(KEC). Then for some intuitionistic KFE-model M over F E C and for some u E M M Iu Ik 82 and MI u
Iy A&,
so by the generation lemma MfulkE:
and M ~ U , U I ~ A ~ .
Then by Lemma 3.8.5, ( M T u)', u
IY A,
which implies A # IL'(KE(Cf)), and so A @ ILZ(ICEC) by Proposition 3.5.28. (+) Given @ = (F,Dl x) and an intuitionistic N = (@,[) such that N, u ly A, the same construction as in the modal case yields an intuitionistic M such that N = Mr. We leave the details to the reader. ¤
Theorem 3.8.7 Let L be a strongly ICE-complete m.p.1. or s.p.1. Then L= is also strongly ICE-complete. Proof (I) Consider the modal case first. Let r be an L'-consistent N- modal theory. We may assume that Q does not occur in - to avoid Q, we can appropriately rename all letters in I?. Then the theory Cim&$U I'Qis L-consistent. In fact, otherwise by 2.8.1
for some A1,. . . , Ak E r . Hence L= F 7(Al A .. . A Ak) by 2.14.4, which means that I? is L'-inconsistent. Since L is strongly KE-complete, there exists a KFE cP = (F,D+, E)19 such that cP k L and for some model M = (a,[) and some world uo we have M, uo b
rQu om&$.
We may also assume that uo is the root of M; in fact, otherwise M and @ can be replaced with M f u o , and @ t u g ,respectively, since by Lemma 3.3.18, ~f uo,uo b rQu OOO&$
and @ f u o!= L.
Thus M,uo b O,BQ for any B E