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.
+
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.
+
+
+
+
CHAPTER 3. KRIPKE SEMANTICS
260
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.
+
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.
+
+
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).
CHAPTER 3. KRIPKE SEMANTICS
262
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.
3.8. TRANSFER O F COMPLETENESS
263
(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 EN, and so by 3.2.11, M I= &$, since every u E M is covered by some R, (uo) (Lemma 1.3.19). 19We use an equivalent definition of a KFE from Lemma 3.5.2.
CHAPTER 3. KRIPKE SEMANTICS
264
Now consider @'
= (F,D+, El),
where
E1(a,b) := {u I M , u k Q(a, b)), and M1 = (@',J1) as in Lemma 3.8.5. Then @' is a KFE and M',uo k I' by Lemma 3.8.5. Since all equality axioms are valid in a', it remains to show that @' k L. It is again sufficient to check that @I k A for any A E I; without Q, since otherwise Q can be renamed. So for an arbitrary KFE-model M2 = (Qr,O), let us show that M2 k A. Since Q does not occur in A the truth values of A do not depend on 0+(Q). So we may further assume that for any u, M1,u != Q(a, b) e u E Er(a,b). M2 remains a KFE-model under this assumption, because
Now put M1 := (a,0). Then M I is also a KFE-model. In fact, the latter means n
(1) ( ~ 1 ,..., an) E Qu(p;) & u E
0 E(ai, bi) * (bl,. . . ,bn) E o,(p[).
(2) ( a , . . . , a n ) E Ou(pt) & u E
n E1(ai,bi) 3 (bl,. . . ,b,) i=
i= 1 Since Mi is a KFEmodel, we have n
E 0,(P[).
1
Now (1) follows from (2), and the observation that (3) E(a, b) E1(a,b) holds for any a , b E D+. To check (3), note that it is equivalent to (4) M , u k a = b > Q(a,b) provided a, b E D,.
But this holds, since
as soon as M is a KFE-model, and M, u k Q(a, a) as we already know. By definition, it follows that M2 = Mi as in Lemma 3.8.5. Now cP l= L implies Ml k A. A = AQ, since A is without equality; hence M2 k A by Lemma 3.8.5. (11) The intuitionistic case is considered in a very similar way. If (I?, A) is an L'-consistent intuitionistic theory, then (I?&U AQ) is also L-consistent. In fact, otherwise by 2.7.13,
12,
3.8. TRANSFER O F COMPLETENESS
265
U E ~ I-& V A? for some finite rl r for some finite A1 A, which implies by 2.8.1. Hence L= I- Arl > V A l by 2.14.5, so (I',A) is L'-inconsistent. By strong completeness, there exists a KFE @ = (F,D f 7E ) such that @ I=L and for some intuitionistic model M = (a,[) and some world uo,M, uo IF ( r Q ~ £ 2AQ). , As in the modal case, we may assume that uo is the root of M, and thus M IlSo we can define a KFE @' := (F,D+, El) such that
£o&.
E1(a,b) := {u1 M, u ll- Q ( a ,b)} and M' = ( @ I , t') as in 3.8.5. Then by 3.8.5, M' is a KFE-model such that M , u IF
iff MI, u Il- A
for any u E M and D,-sentence A. Hence M', uo I t (r,A). @' Il- L is proved as in the modal case - every intuitionistic model over @' corresponds to an intuitionistic model over Q (for the basic language Lo). W Theorem 3.8.8 Let L be a KE-complete s.p.1. or a conically expressive m.p.1. Then L= is also KI-complete. Essentially the same as for 3.8.7, but now we should take care of Proof finiteness. (I) Let us begin with the modal case. Consider an arbitrary formula A in the language of L such that L= Ijf A. Then by 2.14.4 O'E? YL A&,
and thus by completeness, there exists a KFE @ = (F,D+, E ) such that k L and for some world uo in some KFE-model M = (a,J) we have M, uo != O * E U~ {-AQ). As in 3.8.7, we may assume that uo is the root of M and define
where
E1(a,b) := {u I M, u k Q(a, b)).
and M' = (Q', J') according to Lemma 3.8.5. Then M' is a KFE-model. By 3.8.5 we obtain M,U k B& iff M1,u k B for any u E M , for any D,-sentence B using predicate letters only from A, and so M', uo F A . The proof of @' k L is exactly the same as in 3.8.7. (11) In the intuitionistic case suppose L= Y A. Then by 2.14.5, £ 2 YL AQ, i.e. L y A&: > A&, and thus the latter formula is refuted in an intuitionistic
266
CHAPTER 3. KRIPKE SEMANTICS
KFE-model M = (@,<) over a KFE @ validating L. Again we may assume that M is rooted and M, uo {AQ>)
(£2,
for the root uo of M . Now we define @ I , M' as in 3.8.5. Then M' is intuitionistic and M , U IF B~ iff M r , u IF B for any u, whenever B is a D,-sentence in predicate letters from A. Eventually we obtain @' IF L and MI, uo Iy A. i.e. a' separates A from L=. w R e m a r k 3.8.9 The analogue of 3.8.8 does not hold for K-completeness. In fact, Q H and QS4 are IC-complete, as we shall see in Chapter 7, but the equalityexpansions QH', QS4' are K-incomplete, by 3.10.6 below. Corollary 3.8.10 If L is an s.p.1. or a conically expressive m.p.l., then L is KE-complete iff L= is KE-complete. Proof
'If7 follows from 2.16.15, 'only if' - from 3.8.8.
H
Now let us prove an analogue of 2.16.14 for strong completeness. Proposition 3.8.11 Let L be an m.p.1. or an s.p.l., S a semantics of Kripke frames or Kripke sheaves for the corresponding logics with equality. If L= is conservative over L and strongly S-complete, then L is also strongly S-complete. P r o o f Consider the modal case. Suppose I' is an L-consistent theory. Then I'is L'-consistent, due to the conservativity. So I'is satisfiable in an L'-frame (in the corresponding semantics), which is obviously an L-frame. W Corollary 3.8.12 If L is an s.p.1. or a conically expressive m.p.l., then L is strongly KE-complete iff L= is strongly ICE-complete. Proof
'If' follows from 2.14.8 and 3.8.11, 'only if'
3.9
Simulation of varying domains
-
from 3.8.7.
¤
Consider a formula A in a language with or without equality, and let U be a new unary predicate letter. Let Au be the formula obtained from A by relativising all quantifiers under U; formally: Au := A for A atomic; (3xB)u = ~ x ( U ( XA) Bu); ( ' d x B ) ~= Vx(U(x) > Bu); ( B * C ) u = BU * Cu for * E
( 3 ,V, A).
Next, put
A t := 3xU(x) > ALI
3.9. SIMULATION OF VARYING DOMAINS
267
Let M = ( F ,8) be a Kripke model over a rooted frame F = (W,R , D ) with root vo, such that M , vo Il- 3xU(x). Consider the frame FU := (W,R , D u ) with
and a Kripke model M~ := ( F U ,Bu) such that
for P
# U,
( e U ) + ( u ( a ):= ) {V 1 a E D,).
DK
F~ is well defined. In fact, # 0 since M,vo IF 3xU(x); by truth preservation D: C_ D: whenever wRv. T h e model M U is intuitionistic, since M is intuitionistic. W e can obviously extend the relativisation t o formulas with constants. Then the following holds: L e m m a 3.9.1 For any world v in M , for any D,-sentence B that does not contain U M ~ , IF V B. M,V ~ kB~ Proof
Easy by in induction. By definition,
for any atomic D,-sentence B . L e m m a 3.9.2 For every Kripke model M' = ( F ,D1,O1)over a poset F with root vo there exists a Kripke model M = ( F @ V ,8 ) over F with a constant domain such that M,vo I t 3 x U ( x ) and MU= MI. Proof In fact, put
for P E P L n , a = ( a l l . .. ,an). Then obviously M , uo 1l- 3 x U ( x ) and
and for any v E M , P E PLn a E (D,)n i f fM , v I t P ( a ) iff MI, v IF P ( a ) . Therefore MI
= M~
268
CHAPTER 3. KRIPKE SEMANTICS
Theorem 3.9.3 (1) Let F be a rooted poset, A a sentence without equality and urithout occur-
rences of U . Then
(2) Similarly, if A E IF=, then
Proof We prove only (1); the proof of (2) is similar. First, there exist a model Mo over F @ V and a world vo E F such that Mo,vo It 3xU(x) and Mo, vo Iy Av Put M := Mo T vo. Then by the generation lemma and Lemma 3.9.1, M U , v o Iy A. Thus A @ I L ( K ( F f v o ) ) and , so A @ I L ( K F ) . The other way round, suppose A @ I L ( K F ) , i.e. there exists a frame F' = ( F ,Dl) with a model M' such that MI, vo Iy A; now vo is root of F. Consider a model M according to Lemma 3.9.2. Then MI = M u , SO M , vg Iy Au by 3.9.1. Hence A u E I L (C K F ) . Corollary 3.9.4
3.10
Examples of Kripke semantics
According to the general definition from Chapter 2, the Kripke semantics generated by a class C of Kripke sheaves (or KFEs) is
S(C) = { M L ( @ )) @ E C). Let us consider three examples of Kripke semantics.
Definition 3.10.1 A Kripke frame with equality F = ( F ,D , x) is called monic if (*) V u , v E F V a , b E ( D , n D , ) ( a x , b ~ a x b); , F is a KFE with closed equality (or CE-KFE) i f for any i for any a, b E D+ for any u , v E F
(**) Vi Qu,v E F (u&v
+ Va,b E D,
(a xu b H a x, b));
or equivalently, the premise uRiv can be replaced with uR*v. An intuitionistic KFE with closed equality is also called a KFE with decidable equality, or a DE-KFE.
3.1 0. EXAMPLES OF KRTPKE SEMANTICS
269
I f F is presented in the form ( F ,D+, E ) as in Lemma 3.3.2, then the above conditions are rewritten as follows:
(*'I
Qa,b E D+ ( E ( a lb) = 0 V E ( a , b) = E ( a ) n E ( b ) ) ;
(**I)
, Qa,b E D+ QiE ( a ) n E(b) n R ; ' E ( ~ b)
c E ( a ,b).
Definition 3.10.2 A disjoint Kripke sheaf Q, = ( F ,D , p) is called rnonic if for
anya,b~ D+, u , v E F
Q,
is a CE-sheaf (or a DE-sheaf i n intuitionistic case) if (2) V u , v E F Va,bE Du Qi (uRiv & (alv) = (blv) * a = b).
T h e necessary modification for the non-disjoint case is quite obvious. These three types o f KFEs and Kripke sheaves fully correspond t o each other and generate equal semantics: Lemma 3.10.3 (1) A disjoint Kripke sheaf Q, is rnonic iff the KFE G(Q,) is monic. The other way round, a KFE F is rnonic iff the Kripke sheaf O ( F ) is monic. (2) F is a CE-KFE iff O ( F ) is a CE-sheaf.
Thus a Kripke sheaf isomorphic t o Q,.
Q,
is CE iff the KFE G(Q,) is CE. Recall that O ( G ( @ ) is )
Proof
Consider the rnonic case (the CE-case is quite obvious). Let F = ( F ,D l x) be a rnonic KFE, O ( F ) = ( F ,D', p), DL
=
(Du/ x u ) ,pu,(au) = a, i f v E Ful a E Du
C D,.
Assume wR*y, wR*v, wlR*u, wll?v, a, E DL, b,, E Du n D, and a xu b. = p,,,(b,,). Then a x, b since F is rnonic, i.e. p,,(a,) Now, let Q, = ( F ,D l p ) be a rnonic disjoint Kripke sheaf, G ( @ = ) ( F ,Dl, xl). Let u , v E F, a,b E Dh n D l , a X ; b, i.e. a E D,,b E D,, for some w,wl E (R*)-'(u) n ( R * ) - ' ( v ) and pwu(a)= pWtu(b).Then p,,(a) = p,t,(b) since Q, is monic, i.e. a x: b. Example 3.10.4 (Warning) In general i f O ( F ) is rnonic, then F is not necessarily rnonic (even in the intuitionistic case). In fact, let us consider the KFE F = ( F ,D , x ) based on the poset F = { u ,v ) , in which u and v are incomparable, D, = D, = { a ,b), xu= ( D u ) 2 , xv= idD,. Then F is not rnonic, since a xu b, but not a x, b. On the other hand,
@ ( F )= ( F , D 1p),D; ,
=
{au),Dh = {a,, b,),puu = i d ~ ; , p , , = ido;
CHAPTER 3. KRIPKE SEMANTICS
270 This Kripke sheaf is rnonic, since
aR*w & aR*wl + w = w' in O(F). Speaking informally, the individual a, = b, in 8 ( F ) does not know that it corresponds to a, # b,. One can construct a similar example based on a rooted %frame.
Lemma 3.10.5 (1) For any N-modal predicate Kripke frame F, i
F
VXVY(X# y
5N
> U ~ ( X# y)).20
(2) For any intuitionistic Kripke frame F, F
+ DE.
Proof (1) I f u i= a # bfor a , b E D,, t h e n a v E Ri(u); hence u k Oi(a # b).
# b, and t h u s v
k a
# bfor any
(2) We have to show that u IF a = b V a # b for any a , b E D,. In fact, if a = b, then u IF a = b, by definition. If a # b, then v Iy a = b for any v E R(u); thus u IF a # b, by definition.
w Proposition 3.10.6 The logics QK', Q H = are K-incomplete. Moreover, let A be a modal propositional logic such that K A S 4 or S 4 C A c S5, or an intermediate propositional logic # CL. Then QA= is K-incomplete.
c c
Proof In this case a two-element reflexive chain 2 2 validates A. Thus A (and &A=) is valid in the Kripke sheaf @ = ( 2 2 , D , p), in which D = {a, b), Dl = {c}, all = bll = c. Obviously, @ v C E and @ Iy D E . Therefore we obtain QA= y C E in the modal case and QA= Y D E in the intuitionistic case. But by Lemma 3.10.5, QK' k K C E , QH' k K D E , which yields incompleteness.
w The semantics generated by rnonic KFEs (or Kripke sheaves) is denoted by M I C ~ )MIC~,=), , and the semantics generated by CE-KFEs (or Kripke sheaves) is denoted by KCEL=),KC£&). Obviously every rnonic KFE is CE (and similarly for Kripke sheaves). Lemma 3.10.5 shows that every simple KFE (or a simple Kripke sheaf - with inclusions p,,) is also rnonic. So we have:
and similarly for superintuitionistic logics. All these inclusions are actually strict, as we will see later on. 'O~ecall that this formula is denoted by CEi (Section 2.6).
3.10. EXAMPLES OF KRJPKE SEMANTICS
271
Lemma 3.10.7 (1) A K F E (or a Kripke sheaf) is C E iff it validates CE : VxVy (O(x = y) (x = Y)).
>
(2) A n intuitionistic K F E (or a Kripke sheaf) is D E iff it validates D E : VX'J'Y((x = Y)v (0 # Y)). Proof
(1) immediate (2) (for KFEs). Let a , b E D,. b) & 3 v (uRv & a x, b).
Clearly u I# ( a = b) V (a
# b) iff -(a
xu
Example 3.10.4 shows that monic KFEs do not have an adequate logical characterisation. In fact, KFEs F and G(Q(F))have the same modal logic, but only one of them is monic. A similar example can be constructed for monic Kripke sheaves. In some special cases the three semantics introduced are equivalent. Let us give two examples. Recall that simple Kripke sheaves correspond to Kripke frames with trivial equality. Lemma 3.10.8 Every CE-Kripke sheaf over an S4-tree F is isomorphic to a simple Kripke sheaf over F. Proof Let @ = (F,D , p) be the original Kripke sheaf (which we suppose disjoint) and let F = (W, R). Consider the following relation between individuals:
-
a
-
b := 3d (dpa & dpb).
- -
Since F is a tree, is an equivalence relation. In fact, if a b, b c and dpa, dpb, epc, epb, d E D,, e E D,, then both u, v see the world of b, and thus they are comparable. We may assume that u R v . So dpd' for some d' E D,, and also d'pb. Hence dpd', so by (CE) it follows that d' = e, and thus dpc, which implies a c. Then we factorise the domains: D, := D,/ -. It follows that whenever u R v , hence we obtain a PKF (F,5).One can easily show that @ is isomorphic t o O(F). rn
-
-
Definition 3.10.9 A propositional Kripke frame (W, R) is called
directed iff Vx, y E W R(x) n R(y) #
0;
weakly directed iff Vx, y E W (R(x) n R(y) is directed (if it is non-empty) as the frame with the relation R or R-' restricted to it).
272
CHAPTER 3. KRIPKE SEMANTICS
Lemma 3.10.10 Every CE-sheaf over a weakly directed frame i s monic.
Proof Assume that individuals a, b in a given CE-sheaf @ are both related to worlds u, v. Also assume that (alu) = (blu), URW,VRW.Then
and thus (alv) = (blv) by (2). Similarly we obtain (alv) = (blv) in the case, W when a , b are related to some world in R-'(u) n R-'(v). Lemma 3.10.11 Let @ = (W, R, D l p) be a monic Kripke sheaf over a directed frame. Then ML(@) = M L ( F ) for some PKF F over (W, R).
Proof Since (W, R) is directed and the inheritance relation p is transitive, we can construct the direct limit
Namely, we take D: = D + / a
N
-, where
-
b iff a , b have a common inheritor.
There exist natural embeddings i, : D, (I)),such that the diagram
D$ (the injectivity follows from
commutes. Consider the PKF F = (W, R, Do) with (Do), = i,(D,). It can be easily proved that @ and O(F) (see Section 3.3) are isomorphic. Thus, ML(@)= ML(F) by Lemma 3.3.4.
Corollary 3.10.12 to directed frames.
K d i , = M K d i , = KCEdi,,
where dir means the restriction
Definition 3.10.13 A K F E is called a KFE with a constant domain i f all its individual domains are equal. We use the notation F = (F @ V, x) for KFEs with a constant domain (where V = D, for all u E F).
3.10. EXAMPLES OF KRTPKE SEMANTICS
273
Definition 3.10.14 A K F E with a set of possible worlds W is called flabby i f the measure of existence of every its individual is W . So, a KFE is flabby iff it has a constant domain. Now let us describe flabby Kripke sheaves corresponding to flabby KFEs.
Definition 3.10.15 A Kripke sheaf = ( F ,D , p) is called flabby i f there exists a domain Do and a family of surjections po, : Do -+ D, for u E F such that PO, = PO, o puV, whenever u R * v i n F , Lemma 3.10.16 (1) If F = ( F ,D , x) is a flabby KFE, then Q ( F ) is a flabby Kripke sheaf.
(2) For a flabby Kripke sheaf @ = ( F ,D , p ) there exists a flabby KFE F such that Q ( F ) is isomorphic to a.
Proof (1) Let Do be the domain of any world in F and po(a) = a,(= a/ xu) for
u E F.
( 2 ) Let F
= ( F @ Do, x), where
a xu b iff po, ( a ) = po,(b).
Definition 3.10.17 A Kripke sheaf ( F ,D , p) is called meek i f all the maps p,, (for u R * v ) are surjective. A KFE ( F ,D , x ) is called meek i f 'di'du'dv E Ri(u)'db E D, 3a E D, a xu b, i.e. i f any individual i n any accessible world is locally equal to some individual from the present world. Obviously, every flabby Kripke sheaf (or KFE) is meek - if PO, o puv is surjective, then p,, is surjective. Every flabby KFE is meek, since it has a constant domain.
Lemma 3.10.18 Meek Kripke sheaves, meek KFEs, flabby Kripke sheaves, and flabby KFEs generate the same semantics. Proof
Note that every meek Kripke sheaf over a rooted frame is flabby.
Lemma 3.10.19 C D T E ( Q S 4
H
+ Ba).
Proof This follows from Kripke completeness of Q S 4 + B a , see Chapter 7. A syntactic proof is an exercise for the reader.
CHAPTER 3. KRIPKE SEMANTICS
274
R e m a r k 3.10.20 In Chapter 7 we will show that IL(S4-PKFs with constant domains) = Q H ML(N-modal PKFs with constant domains)
= QS4
+CD,
+ {Bal, . . . , B U N ) .
R e m a r k 3.10.21 In a meek Kripke sheaf the following holds:
u bVxB(x,al ,..., a,) iffVc 'c ED, u I=B(c,al ,..., a,) L e m m a 3.10.22 (i) Let @ be an S4-based Kripke sheaf. Then: C D E IL(@) iff B a E ML(@) iff @ is meek. (ii) For an S4-based KFE F ,
F It C D iff F is meek. (iii) For an S4-based P K F F ,
F Il- C D iff every its cone FTu has a constant domain. (lm) For a N-modal Kripke sheaf @, N
@k
A Bai iff
is meek.
i=l (2m) For an N-modal KFE F N
Fk
A Bai if F is meek. i= 1
(3m) For an N -modal P K F F , N
Fb
Bai iff every F r u has a constant domain. i= 1
Proof (1) Let us consider only the 1-modal case; the intuitionistic case is similar. Take (Only if.) Let v ,w E F, vRw and suppose there is bo E D, - p,,(D,). the valuation [ in @ such that E, u k P ( a ) iff vRu and a E p,,(D,). Then v b VxOP(x), since u b P(p,,(a)) for any u such that vRu and for any a E D,. On the other hand, w rj P(bo). It follows that w I# VxP(x) and v rj OVxP(x). (If.) Suppose that 6, u i= VxOP(x) and E, u l j OVP(x) for some valuation J in @ and u E F. Then J, v v P ( b ) for some v such that uRv and some u E D,. Since is meek, we can take a E D, such that b = p,,(a). Then u #I OP(a) and u l#VxOP(x). This is a contradiction.
3.1 0. EXAMPLES OF K R P K E SEMANTICS
275
(2) (Only if.) Let v, w E F,vRw, and suppose there is bo E D, such that Vb E D, b $, bo Take the valuation [ in F such that:
[,u IF Q iff vRu&u
<,u 1 1P ( a ) iff vRu & 3a'
# v,
E D, a xu a'.
Then v It- Vx(Q V P(x)), since v It- P(a) for any a E D, and u IF Q for any u E R(v) with u # V. On the other hand, v Iy Q and v Iy VxP(x) since w IY P(bo). (If.) Suppose that <,u It- Vx(Q V P(x)), [,u Iy Q , u Iy VxP(x) for some valuation in F and u E F. Then v ly P(b) for some v E R(u), b E D,. Since F is meek, we can take a E D, such that b x, a. Then u Iy P ( a ) and u Iy QVP(a). This is a contradiction. (3) This directly follows from (2) and from the obvious fact that a PKF F has a constant domain iff its corresponding KFE with trivial equality is meek.
<
w
As we noticed, every flabby Kripke sheaf is meek. On the other hand, a rooted meek Kripke sheaf is flabby. Therefore, flabby Kripke sheaves generate the semantics equivalent to meek Kripke sheaves. Moreover, this semantics equals the semantics of flabby KFEs. Let CK: be the semantics generated by all PKFs with constant domains. Definition 3.10.23 A Kripke sheaf is called bijective if all its transition maps are bijections.
Let BiC be the semantics generated by all bijective Kripke sheaves. Proposition 3.10.24 CK
c BK.
Proof CK: BK since every PKF with a constant domain F corresponds to the bijective Kripke sheaf Q(F). To prove that the converse inclusion is not true, consider the bijective Kripke sheaf Qo := (W, R, D, p), where
W = { u , ~w), , D = {a, b, c, d), Du = {a), D,
=
{b), D, = {c, d),
R = idw U {(u, v)), p = idg U {(a, b)), see Fig. 3.4. Let
c := 'dxVy(P(x) 3 P(y)), K'
:= q 3 Oq.
Then obviously, Qo k C'
v K',
but Qo I$ C'.
CHAPTER 3. KRTPKE SEMANTICS
Figure 3.4. However, for every PKF F = F @ V, we have
and
IVI > 1 & F V K i
+ FVC'vK';
SO
F k C' V K' only if (F k C' or F != K'). Thus M L ( a o ) 6 CK.
¤
The above example also shows: L e m m a 3.10.25 Proof
CK lacks the collection property (CP) from Section 2.16.
In fact, ML(@o)= ML(@l)n ML(Q2),
where cP1 are other hand,
Q2
the restrictions of iPo respectively to { u , v ) and { w ) ; on the ML(@l),ML(@2)E CK.
Nevertheless, CK and
BK are equivalent, as we shall see below.
= (F, D , p) be a bijective Kripke sheaf over a rooted L e m m a 3.10.26 Let frame F. Then ML(@)E CK.
Proof Let uo be the root of F , Do = D,,, and consider the PKF F = Fa DO. Since p gives rise to a family of bijections between Do and each D,, the Kripke sheaves O(F) and @ are isomorphic. So, ML(@) = ML(F) by Lemma 3.3.4.
w Corollary 3.10.27
CK, = BIC, (r means the restriction t o rooted frames).
3.11. ON LOGICS WTTH CLOSED OR DECIDABLE EQUALITY
277
Lemma 3.10.28 Every logic from BK is CK-complete.
Proof
Let @ be a bijective Kripke sheaf over F. Then
by Lemma 3.7.17. Thus ML(@)is CIC-complete by Lemmas 3.10.26 and 2.16.3. Lemmas 3.10.24, 3.10.28 and Corollary 2.12.4 yield
Corollary 3.10.29 CK 2. BK. Here is the sequence of all Kripke semantics considered above:
3.11
On logics with closed or decidable equality
In this section we show that unlike equality-expansions, extensions with closed or decidable equality may be nonconservative.
3.11.1 Modal case Consider the following modal formula: ~ a := ' 0 3 x P ( x ) > 3xO3y(P(y) A O(x = y)). Note that QS4'
+~
a lQS4'
+ Ba.
We will see that these inclusions are actually strict (Lemma 3.11.4). Nevertheless: L e m m a 3.11.1 QS4"
+ B a = QS4=' + Bal. +
Proof We shall use 'naive reasoning' in QS4=' Bal. Assume that 03xP(x). Then (from Bal) 3xOdy(P(y) A O(x = y)). Using O(x = y)
> x = y, we obtain
Corollary 3.11.2 QS4'
+ Bal krc Ba.
Lemma 3.11.3 Let @ be an S4-based Kriplce sheaf. Then @ I=Bal zff @ satisfies:
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CHAPTER 3. KRIPKE SEMANTICS
Proof (Only if.) Let u , v E F, uRv and suppose there is b E D, such that Va E D, Vw (uRw p,,(a) # p,,(b)). Take the valuation J in cP such that
Then clearly J , u k 0 3 x P ( x ) . On the other hand, suppose J , u k 3xO3y(P(y)A 0 ( x = y)). Then there are a E D,,w E F and c E D, such that [ , w k P ( c ) O(p,,(a) = c). Since [,w k P ( a ) , we have w = v and c = b. We have J , v b O(puw(a)= b), which leads to a contradiction. ( I f . ) Suppose there is a valuation [ in cP and u E F such that J , u != 0 3 x P ( x ) and [, u Ij 3 x O ( 3 y ( P ( y )A O ( x = y)). Then there are v E F and b E D, such that u R v and J , v != P(b). Since cP satisfies (I), there are a E D, and w E F such that vRw and p,,(a) = p,,(b). It follows that J , v k O(p,,(a) = b) and thus J , v k 3 y ( P ( y )A O(pu,(a) = y ) ) . We have a contradiction.
L e m m a 3.11.4 QS4=
+ Bal vKEBa.
Proof Consider the Kripke sheaf cP depicted at Figure 3.5. It is easily seen that cP satisfies (2), but it is not meek since bl does not have a predecessor in uo;thus cP != B a l , @ Ba. W
Figure 3.5. Corollary 3.11.5
3.11. ON LOGICS WITH CLOSED O R DECIDABLE EQUALITY
279
Remark 3.11.6 One can repeat the argument for logics without equality: Bal should be replaced with
3.11.2 Intuitionistic case We shall use the following intuitionistic formulas:
The corresponding logics were considered in [Umezawa, 19561, where many inclusions and equalities between them were proved, in particular:
One can easily see that
QH=
+ Uo C QH' + 1 i U c QH' + U = QH= + VxVy(x = y).
We also have: Lemma 3.11.7 ( I ) QH='
(2) QH'S
+ U = QH=' + Uo; + Uo V A = QH'" U V A for
every sentence A without occurrences
of P. Proof (1) From R(x) > Vyi-R(y) by substitution we obtain
and
x =x Now, x = x and (2) Similar.
1 1
> Vy-7 x = y.
x = y > x = y (in QH")
yield VxVy(x = y).
CHAPTER 3. KRIPKE SEMANTICS
280 L e m m a 3.11.8 (U V J) E (QH=' Proof
+E).
('Naive argument'). Due to 3.11.7 it is sufficient to show that
(uoV J ) E (QH" + E). El and E2 provide
XI, 52
such that
and
Vy(+(x2) 3 +(Y)) If x1 = 2 2 we have Uo,so suppose X I
# 2 2 . Let
It is easily seen that 113xR(x),so due to E, there exists xo such that l l R ( x 0 ) . ¤ Since (so= x l ) V (xo# x l ) , we obtain J .
+ E !=K: U V J . L e m m a 3.11.10 QH + E vKeU V J . Corollary 3.11.9 QH
P r o o f The Kripke sheaf cP depicted at Figure 3.6, does not validate UV J, as it is easily seen. If u k - - G l x P ( x )in some model on cP then vl P(b),V2k P ( q ) for some i and so, u != l l P ( a i ) .
+
Figure 3.6.
Corollary 3.11.11 (1) QH
+ E is K-incomplete.
3.12. TRANSLATIONS INTO CLASSICAL LOGIC
(2) K 4 K& (intuitionistic case). (3) QH"
+ E is not conservative w.r.t.
QH
+E. +
Remark 3.11.12 Kripke-incompleteness (i.e. K-incompleteness) of Q H E was first observed by H.Ono [1973]. Namely, he proved that QH E kK C D but Q H + E C D (the semantics C'T will be defined in Chapter 4). Actually Q H E yKE CD also holds, and Figure 3.7 shows a Kripke sheaf needed for the proof.
+
+
vcT
Figure 3.7
3.12
Translations into classical logic
In Section 1.8 we showed that in many cases Kripke-complete modal or superintuitionistic propositional logics are recursively axiomatisable and complete w.r.t. countable frames. To prove these results, we used the standard translation of propositional modal formulas into classical first-order formulas. In this section we extend this translation to modal first-order formulas, cf. [van Benthem, 19831. Let us consider the two-sorted classical predicate language LZN. The variables of the first sort (ranging over individuals) are taken from the same set V a r as in our basic language L N . The second sort has a countable set of variables (ranging over worlds) W u a r := {w, I n E w}. L2N also contains equality2' 21Strictly speaking, there are two kinds of equality - for worlds and for individuals.
CHAPTER 3. KRIPKE SEMANTICS
282
and binary predicate letters R1,. . . ,RN, U; Ri are of type (worlds x worlds) and U is of type (worlds x individuals); thus atomic formulas are of the form Ri (wm,wn) or U(wm,vn) or W, = W, or vm = v,. ~ 2 ;is the expansion of L2N with (n 1)-ary predicate symbols of type (worlds x individualsn) corresponding t o n-ary predicate symbols PT(x) of our basic language CN. Consider the following classical ~2;-theory:
PT*
+
ro:= {Exd) U {A7 I n, j
2 O),
where Exd := Vwo3voU(wo,vo) A Vvo3woU(wo, vo)A N
A ~ w I ~ w ~ \ J ? I o ( R ~ (AwUI ,(wW~~), V2O U(w2.7 ) VO)), i= 1
The intended interpretation of the formula U(w0,x) is x E D,,. So the conj u n c t ~of Exd respectively mean that every individual domain is non-empty, every individual belongs to some domain, and the domains are expanding with respect to Ri. The formula pj"*(wo, x) asserts that wo k P,l"(x); thus A? means that an atomic formula can be true at world wo only for individuals from D,,. Given a Kripke frame F = (F, D) based on a propositional frame F = (W, pl, .. . ,p ~ )we , can construct the following (associated) L~N-structureF* expanding F* and satisfying Exd, in which the universes of sorts 1 and 2 are the sets W and D+ respectively, and for any u l , u2, u, a
F* != Ri(ul, u2) iff ulpiu2, F* i= U(u,a) iff a E D,. Next, a Kripke model M = (F, D, c) gives rise to an associated ~2;-structure M* satisfying Po and expanding F* such that for any u, a
M* k ~j"*(u,a) iff a € D z & M , u i = Pjn(a). Then we have L e m m a 3.12.1 Every classical model of Exd is associated with some predicate Kripke frame. Every classical model of ro is associated with some predicate Kriplce model. Proof In fact, let p be a model of To. To define M , let D+ be the domain of the first sort, W the domain of the second sort. The accessibility relations pi in A4 are taken from p. Then put for every u E W,
and
i P;*(U, < + ( P t ( a ) ) := {U E W I p =
for a E D:. From the above remarks it follows that p a model of Exd can be presented as F* for some F .
a)), = M*.
In the same way
3.12. TRANSLATIONS INTO CLASSICAL LOGIC
283
Now by induction let us construct an ~ 2 5 - f o r m u l A*(wo, a x ) , the standard translation o f a first-order modal formula A ( x ) :
P?(X)* := PY*(wO, x ) , ( x l = x2)* := ( 2 1 = x2), I* := I, ( B 3 C)* := B* 3 C*, (3yB)" := 3y(U(wo7y)A B*), (QB)*(wo, x ) := Vwl(R(w0,wl)3 B * ( w ~ x, ) ) . (1) M , u k A ( a ) (modally) iff M* != A* ( u ,a ) (classically) L e m m a 3.12.2 for any N-modal Kripke model M , N-modal formula A, u E M , a E D&.
(2) F k A iff V[ (F,[)* k VwoA*(wo) for an CN-sentence A and an N modal frame F. Proof ( 1 ) B y induction. Let us consider the case A ( x ) = O k B ( x ) .
M , u k A ( a ) ~ V u El ek(u) M , u l k B(a) H Vul E e k ( u ) M* k B*(ul, a ) (by the induction hypothesis) H M* t; ~ w ( R k ( u , w>) B*(w,a)) w M* != ( ~ k ~ ) * ( a ) . (2)F ~ = A @ V [ V U E F ( F , [ ) , U ~ ; A HV~VU E F ( F ,J)* t; A*(u) (by ( 1 ) )H ~6 (F,[)*
+VW~A*(W~).
W
L e m m a 3.12.3 (1) M , u IF A ( a ) ifS M* b ( A T ) * ( u , a )for any intuitionistic Kripke model M , an intuitionistic formula A, u E M , a E DZ. (2) F It- A iff for any intuitionistic J , ( F ,J)* k Vwo(AT)*(wo) for an intuitionistic sentence A and an S4-based frame F. Proof ( 1 ) M , u It A ( a ) i f fM , u AT ( a ) (by Lemma 3.2.15) i f fM* k (AT)*(u,a ) (by Lemma 3.12.2). ( 2 ) Easily follows from ( 1 ) .
+
T h e next two definitions are predicate analogues o f Definition 1.8.2. Definition 3.12.4 Let C be a class of N-modal predicate Kripke frames. We say that C is A-elementary (respectively, R-elementary) if the class of associated t 2 ~ - s t r u c t u r e sC* := {F* I F E C ) is A-elementary (respectively, R elementary). Definition 3.12.5 A modal or superintuitionistic predicate logic L is called A-elementary (respectively, R-elementary) if the class V ( L ) of all L-frames is A-elementary (respectively, R-elementary).
CHAPTER 3. KRIPKE SEMANTICS
284
Definition 3.12.6 A predicate Kripke frame F = ( F ,D) i s called countable if the set of its worlds and the set of its individuals ( D f ) are both countable. W e say that F has a countable base if the set of worlds i s countable and that F has a countable domain if D + i s countable. Definition 3.12.7 Let C be a class of Kripke frames, L = ML(')(c) logic. W e say that L has
its modal
the countable frame property (c.f.p.) in C if
L = ML(=)({FI F E C, F is countable )), the countable domain property (c.d.p.) in C if
L = ML(=)({F I F E C, F has a countable d o m a i n ) ) , the countable base property (c.b.p.) in C: L = ML(=)({F I F E C, F has a countable base)). and similarly for the intuitionistic case. Obviously, the c.f.p. implies the c.d.p. and the c.b.p. Proposition 3.12.8 (1) If a class of Kripke frames C i s R-elementary, then its modal predicate logic ML(')(C) i s recursively axiomatisable (i.e. RE). Similarly, the intermediate logic IL(')(c) i s recursively axiomatisable for a n y R-elementary class C of intuitionistic Kripke frames.
(2) If C i s a A-elementary class of Kripke frames, then ML(') (C) has the c.f.p. in C, and similarly for the intuitionistic case. Proof Similar to 1.8.5. Let C be the class of models of an L2N-theory C, C1 the class of all countable models of C. Then by Lemma 3.12.1, the ~2:-models of C U roare exactly the structures of the form (F,E)*, where F k C. Now by Lemma 1.8.4 and Godel's completeness theorem we obtain for any sentence A: A E ML(C) H C U rob \JwoA*(wo) (in the classical sense)
* roU C k Vwo A* (wo) (in classical predicate logic).
Hence the set of all sentences in ML(C) is RE. Since for any formula A, A E ML(C) iff VA E ML(C), the whole logic is reducible to this set, and thus (1) follows. For the intuitionistic case, it suffices to note that IL(')(c) = T ~ ~ ( ' ) ( ~ ) , by 3.2.29, thus IL(=)(c) is reducible to ML(') (c).
3.12. TRANSLATIONS INTO CLASSICAL LOGIC
285
Similarly to the above, we obtain that for any sentence A: A E ML(C1) iff for any countable ~2;-structure p, ( p != C U ro + p I= VwoA* ( ~ 0 ) ) . By Lemma 3.12.1 and the Lowenheim-Skolem theorem, the latter is equivalent to C U roi=VwoA*(wo), and thus (as we have shown) to A E ML(C). Therefore ML(C) = ML(Cl), which proves (2). In the intuitionistic case note that
The following refinement of (2) is also useful. Proposition 3.12.9 Let L be a A-elementary modal or superintuitionistic predicate logic, F an L-frame, M a Kripke model over F , S a countable set of worlds i n M . Then there exists a countable reliable submodel of M containing S , whose frame also validates L . Proof Let V(L)* be the class of models of an t2jptheory C. Then M* I= C. By the Lowenheim-Skolem-Tarski theorem, M* has a countable elementary substructure p containing S . Then p k roU C, so by Lemma 3.12.1 it follows that p = M? for some Kripke model MI. If F1 is the frame of M I , we obtain that k C, i.e. F1 E V(L). : 4 M*, we have for any u E M , A(a) E L(u): Since M
F?
M* I= A(u, a) iff
M?
I= A(u, a).
Hence by Lemma 3.12.3 M, u I=A(a) iff MI, u I= A(a), which means that M1 is reliable.
1
Remark 3.12.10 In the intuitionistic case we can define a slightly different translation. Namely, we can include the axioms of quasi-ordering (or partial ordering) for R in Exd and add the following truth-preservation clause to A?:
For an intuitionistic predicate formula A(x), the ~2:-formula A(zo,x) is constructed as follows:
Pj (x) = P; (z0,x ) ; ( X I = 22) = ( ~ = 1 52); (AloA2) = AloA2 for o = A, V;
286
CHAPTER 3. KRIPKE SEMANTICS
Recall (Section 3.2) that for a class of propositional Kripke frames 2, K(Z) denotes the class of all Kripke frames based on frames from 2. By CK(Z) we denote the class of all frames with constant domains from K(Z). Corollary 3.12.11 Let L be a A-elementary predicate logic, M a Kripke model over a n L-frame F, uo E M . T h e n there exists a countable reliable submodel Mo M over a n L-frame containing uo such that M T ( M ) = MT(Mo). P r o o f For any sentence A @ M T ( M ) (in the language of L) there exists a world UA E M such that M IUA I+ A. Put
and apply the previous proposition. The resulting submodel Mo is reliable, so M T ( M ) 2 MT(Mo). On the other hand, if A @ M T ( M ) , then M,UA I+ A, SO MO,UAI+ A by reliability of Mo. Therefore MT(Mo) C M T ( M ) . Corollary 3.12.12 Let 2 be a class of propositional Kripke frames. (1) If 2 i s R-elementary, then the logics determined over Z ML(=)(K(Z))
and ML(=)(CK(Z)), are recursively axiomatisable. (2) If 2 i s A-elementary, then ML(')(K(z)) ML(')(cK(z)) has the c.f.p. in CK(Z).
has the c.f.p. i n K(Z), and
Analogous properties hold for the intuitionistic case. Proof
For the case of constant domains we have to add the axiom
or, if one prefers,
vzvx U(z, x). Equivalently, we can drop the conjuncts containing U in the definition of
A.
Examples Consider the following R-elementary classes:
W,, the class of all posets of width 5 n;
W,,the class of all posets of height
< n;
Bn,the class of all posets of branching 5 n. So the superintuitionistic predicate logics of the following classes of posets are recursively axiomatisable (and satisfy the countable frame property):
3.12. TRANSLATIONS INTO CLASSICAL LOGIC (b) IL(=)(K(P~n W,)),
287
etc.
If m 2 2, explicit axiom systems for these intermediate logics are unknown. On the other hand, explicit recursive axiomatisations are known for the logics IL(=)(K(P,)), IL(=)(K(P~n am)), IL(=)(K(B,)), see Chapter 7. In particular, we have:
IL(=)(K(B,))
= QH(=)
for m
> 2.
In this connection note that the logic
of frames of finite heights and of finite branching bounded by m in all finite heights is not RE if m > 2, see Chapter 11; recall that the corresponding propositional logic is H B, (Chapter 1). The same holds for the logics IL(K(Wm n U Pn)), IL(K(Pm n U Wn)), etc. n<w n<w Also note that explicit finite axiomatisations of the classes of frames with constant domains in the cases (a), (b) are known, see Chapter 8.
+
Corollary 3.12.13 Every predicate logic detemined over a finite propositional Kripke frame F is recursively axiomatisable, and satisfies the c.d.p. (in K(F)). Proof (cf. [Skvortsov, 19951, [Skvortsov, 19911). It is well known that every finite classical structure is finitely axiomatisable. So the class {F) is Relementary for a finite propositional Kripke frame F ; if F = ( { a l , . . . , a,), R), then up to isomorphism, F is defined by the following C1-sentence:
However in many cases the predicate logics of R-elementary classes of frames are not recursively axiomatisable. For example, for a class 2 of rooted posets, I L ( 2 ) is RE iff 2 is finite, see Chapter 11.
Remark 3.12.14 We have defined a translation from a modal logic language into a classical language based on Kripke frame semantics. Similar translations can be defined for Kripke sheaves and for other Kripke-style semantics considered in further chapters. Recursive axiomatisability and countable frame (or countable domain) property also can be established for logics complete in these semantics. The constructions are rather straightforward, and we do not discuss them in detail.
CHAPTER 3. KRIPKE SEMANTICS
288
We have established the countable domain property for any finite propositional Kripke frame. It turns out that the c.d.p. can be extended t o any countable propositional Kripke frame as well. H. Ono proved this result for the intuitionistic case by a straightforward argument (see Theorem 1.1 in [Ono, 19721731). Here we give a model-theoretic proof for the modal case which is based again on the translation into a classical language.
Proposition 3.12.15 Let F be a propositional Kripke frame of infinite cardinality K. Then (1) ML(')(K(F))
=
the logic IL(')(F) cardinality K.
Corollary 3.12.16
ML(=){(F, D) I Vu E F ID,\ 5 K ) , and similarly for of any intuitionistic propositional Kripke frame F of
Let 2 be a class of countable propositional frames. Then
and similarly for CIC(2).
Proof We prove Proposition 3.12.15 for the modal case. For the intuitionistic case one can repeat the proof or just apply Godel-Tarski translation. (i) We add a set of individual constants of sort 1 of cardinality K
t o C2f: and fix a bijection v:T-tF. We denote the resulting language by ~ 2 ; ~ . Given a Kripke model ((F, D), k), we can construct a model M: by taking an ~2$-modelM+ and putting
of C2$,
M: b u = c, iff u = ~ ( c , ) . Due to the downward Lowenheim-Skolem-Tarski theorem, it has an elementary Obviously, its universe substructure of cardinality K , which we denote by M:'. of sort 1 must coincide with F , and so its ~2f:-reductcorresponds to some Kripke model ((F, D), kt) based on F . Since Dl+ is the universe of sort 2 in M:' , we have ID1+I 5 rc, and thus Vu I DL1 5 K . For any modal predicate formula A(x) we have (F, D),u 1 A(x) iff M$ 1 A(u, X) iff M$' k A(u, X) iff (F, D'), u kt A(x). Therefore, if A(x) @ ML(')(K(F)), it is refuted in a frame (F,Dl) such that lDhl 5 K for any U E F . (ii) Note that if (F, D) has a constant domain, then M: b VzVxU(z, x) and thus M:' k VzVzU(z, x) and (F,Dl) has a constant domain.
3.12. TRANSLATIONS INTO CLASSICAL LOGIC
289
On the other hand, let F be a propositional frame which is well-ordered of the type K, K being an infinite cardinal, cf. [Skvortsov, 19891. Consider the formula K F = -lVx(P(x) V -P(x)), see Section 2.3. Then K F @ IL(')(IC(F)), since K F @ I L ( F @ V) if IVI 2 n. In fact, let (app < K) be a sequence of distinct elements in V, and consider the valuation such that for every a < K
<
Then (,0 lk 1 K F .
Also we have K F E IL(F, D ) if ID+I
< K.
In fact, take an arbitrary valuation J in (F, D); for every a E D+ take an ordinal Pa I K such that < , a IF P ( a ) iff Pa 5 a . Also take a,-, < K such that Va (Pa
< K * Pa I QO).
Then
6, a 0 IF Vx(P(x) v -P(x)). Also note that
ID+[
5 N1 if V a < K ID,I 2 No.
In fact,
/ { a< K I Da $Z
U Dp)l < NI.
Therefore, we cannot replace IDuI 5 K with IDuI 5 No in Proposition 3.12.15 say for F = N2. Let us also note that for F = K KF
IL(F, D) if D, = {ap I
P 5 a),
and 'da
< K ID,/ < K ;
thus K F @IL{(F,D) I V a
< K ID,[
< No)
for F = N1. Let us consider another similar example. Let F be a denumerable tree (with the root OF).,in which Fu is nonlinear (e.g. the tree w* of all finite sequences of natural numbers, or the binary tree {O,l)* of all finite (0,l)-sequences, etc.). Let be the coatomic tree obtained by adding maximal points w, above every branch (maximal chain) T of F. Let us consider the following formula:
Then
CHAPTER 3. KRIPKE SEMANTICS
290 L e m m a 3.12.17
(i) C* E IL(F @ V) for a denumerable V;
(ii) C* # IL(F @ V) i f IVI 2
IF - FI
(e.9. i f IVI 2 2N0).
Proof (i) Suppose that
uo
p q, uo k Vx[(iP(x) > q) > q],ug k 773xP(x), V = {a,
:n
> 0).
Then q 'dn. > 0 3v (uRv&v lP(a,) &v p q), and there exists a chain uORulRu2R. . . such that V n > 0 [u, /= 7 P ( a n ) & ,u, q]. Then w, k 13xP(x) for a branch T containing all u,. This is a contradiction. Vu
(ii) Take different elements a, from V for w, E valuation in F @ V: u
+ P ( a ) iff 37 (U u
Then OF
= W,
(F - F ) , and
&a
the following
= a,);
/= q iff ~ T ( = U w,).
q, OF k --3xP(x).
We also have 7P(a) > q
Vu E F Va E Do u
since for a = a,, there exists v E Fu such that v $! T, i.e. v Thus OF k V X ( ( - ~ ( X3 ) 9) 3 9).
1l P ( a ) .
R e m a r k 3.12.18 This example is also related to the formula K F . It is known [Gabbay, 19811 that the predicate logic Q H K F is Kripke-complete (w.r.t. denumerable atomic trees). Moreover, Q H K F = IL(w*) = IL({(w*, D) : Vu ID,I 5 NO])) (and similarly for the binary tree (0, I)*). The corresponding predicate logic with constant domains is also Kripke-complete:
+ +
QH
+ K F + C D = IL({F' @ V I (0, I)* c F' c w*, F'
is coatomic))
for a denumerable V. On the other hand, the above mentioned example shows that QH+K+CDcIL(w*@V) for a denumerable V. R e m a r k 3.12.19 For the case of constant domains (without equality), Proposition 3.12.15 also shows that
for any countable F and any constant infinite domain V. This follows from 3.4.11: M L ( F @ V) C M L ( F @ V'), provided IVI 2 IV'I.
3.12. TRANSLATIONS INTO CLASSICAL LOGIC
291
The translation into classical logic and the well-known generalised form of the Lowenheim-Skolem theorem shows that M L = ( F @ V) = M L = ( F @ V') for any finite F and infinite V, V', cf. Corollary 3.12.13. The intuitionistic case is quite similar. On the other hand, if a poset F contains an infinite cone then
C* E (IL'(F
@ V) - I L = ( F @
V')),
for IV'I = No, IVI 2 2IFI, cf. Corollary 3.12.13. Thus we obtain the following claim (cf. Theorem 1 in [Skvortsov, 19951): Theorem 3.12.20 IL'(F @ V) = I L z ( F @ V') for any infinite constant domains V, V' liff all the cones i n F are finite.
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Chapter 4
Algebraic semantics 4.1
Modal and Heyting valued structures
As we know from Chapter 1, every propositional Kripke frame F corresponds to a modal algebra M A ( F ) ;so Kripke semantics can be treated as a particular case of algebraic semantics. Likewise, in the predicate case Kripke frames with equality admit a straightforward algebraic generalisation. Recall that by Lemma 3.5.2, every KFE can be represented as a propositional Kripke frame F with individuals such that for every two individuals a, b their measure of equality E ( a , b) is defined. E(a, b) is a set of possible worlds, and thus an element of the corresponding modal algebra M A ( F ) . This suggests for the following generalization of KFEs: replace MA(F) by an arbitrary modal algebra (or a Heyting algebra for the intuitionistic case) and E with a function taking values in this algebra. To make the corresponding semantics sound, E should satisfy the properties cited in Lemma 3.5.2. Thus we come to the following two definitions.
-
Definition 4.1.1' Let 52 be a complete Heyting algebra (a locale'). An 52valued set is a triple (52, D , E), where D is a set, E : D x D R is a map such that for any a , b, c E D
( E l ) E ( a , b) = E(b, a); (E2) E ( a , b) A E(b, c) If also
I E ( a , c).
the triple (52, D, E) is called an 52-valued structure or a Heyting valued structure (H.v.s) over a. D is called its individual domain, the elements of D are called individuals. The elements of 52 are called truth values. ICf. [Borceaux, 1994; Fourman and Scott, 1979; Dragalin, 19881.
294
CHAPTER 4. ALGEBRAIC SEMANTICS
Obviously, (E3) implies that D is non-empty.
Definition 4.1.2 Let R = (0, U, n, -, 0, 1 , 01,. . . ,ON)be a complete modal algebra. An a-valued structure or a modal valued structure (m.v.s) over Cl is a triple ($2, D l E ) , where D is a set, E : D x D ---+ R is a map such that for a n y a , b , c ~D, i E (1,...,N )
The condition (E3) again implies the non-emptiness of D. Since every Boolean algebra is a Heyting algebra (where V is U, A is n and a -+ b = a 3 b = -aUb), an m.v.s ( a , D , E ) over a modal algebra S1 corresponds to the H.v.s ( a b ,D , E), where fib is the Boolean part of a . The other way round, if is an S4-algebra, then an m.v.s. ( 0 , D l E ) , corresponds to the H.v.s. ( a 0 ,D l E ) , where a0is the pattern of 52. Definitions and results for H.v.s. and m.v.s. are often quite similar. In these cases we talk about 'structures' and denote all operations in a Heyting-algebraic style. As in Chapter 3, we call E(a, b) the measure of equality of individuals a , b; they are 'fully equal7if E(a, b) = 1and 'fully different' if E ( a , b) = 0. The truth value E ( a , a ) (which in general may be not equal to 1)is called the measure of existence of a. We also introduce the measure of equality for tuples a = ( a l , . . . ,a,), b = (bl, ... ,bn) E Dn:
and the following abbreviationx2
Eab := E(a,b), Ea := E ( a )
:= Eaa.
We can also include the degenerate case n = 0. A 0-tuple is just sequence), and we put E A A := 1.
(the void
As we said above, Definitions 4.1.1, 4.1.2 generalise the situation in Kripke frames with equality. In that case D corresponds to the set of all individuals D+,the conditions (El)-(E3) in Definition 4.1.2 are exactly the same as in Lemma 3.5.2 (I), and the condition (E4) in 4.1.2 is obviously equivalent to (E4) in 3.5.2(1). Therefore we have he reader will notice that our notation is ambiguous. For example, for a , b E D, Eab abbreviates both E ( a ,b) and E(ab) = Ea A Eb. Nevertheless we use it, when there is no confusion.
4.1. MODAL AND HEYTING VALUED STRUCTURES
295
L e m m a 4.1.3 (1) Let F = ( F ,D , X ) be a KFE, F = (W,R1,. . . ,R p ) , and for every two
individuals a, b E D+ put E ( a ,b) = { u ( M A ( F ) D+, , E ) is an m.v.s.
I
a xu b). Then M V ( F ) :=
, f , E ) is (2) Similarly, i f F is an S4-frame, the triple H V ( F ) := ( H A ( F ) D an H.v.s. Proof In fact, (El)-(E4) in 4.1.2 follow readily from 3.5.2(1). In particular, (E3) means that every individual domain D, is non-empty.
T h e conditions ( E l ) , (E2) in the above Definitions 4.1.1, 4.1.2 correspond t o the symmetry and the transitivity o f equality; more exactly, they are intended t o make the corresponding first-order formulas true (see below). T o prove the reflexivity, the condition (E3) is necessary. The condition (E4) corresponds t o a theorem in 2.6.18(3): X = Y 3 Cli(~=y); it is necessary for verifying the substitution instances o f the equality axiom (Ax17). D , E ) , individuals a, b E D and tuples L e m m a 4.1.4 For any structure (0, a , b,c E Dn: (1) E(a1 b) l m a , a);
< Eaa; (3) E a b A E b c < Eac; (4) E ( a ) < O i E ( a ) (in the modal case); (2) E a b
V
(5)
E ( a ) = 1.
aEDn
Proof
This easily follows from Definition 4.1.2. In fact, E ( a ,b) = E ( a ,b) A E(b,a ) < E ( a ,a ) ;
hence n
Eab =
A
Eaibi 5
i=l Eab A Ebc =
V
aEDn
n
A
Eaiai = Eaa; i=l n n A (Eaibi A Ebici) < Eaici = Eac; i=1 i=1
E ( a ) = V { E a l a l A . . . A Eanan I a l l . . . ,an E D) 2
In the modal case we also have
n
V I"\
aEDi=l
Eaa = 1.
CHAPTER 4. ALGEBRAIC SEMANTICS
296
Definition 4.1.5 A structure F = (52, D , E ) has a constant domain (or briefly, F is a CD-m.v.s. or a CD-H.v.s. or a CD-structure) if the equality is trivial, i.e. for any a,b E D 1 iff a = b, E ( a , b) = 0 iff a # b. Such a structure is denoted simply by (52, D ) . Definition 4.1.6 A structure (52,D , E ) is called flabby if E ( a , a ) = 1 for every a E D and keen if E ( a ,b) = 0 whenever a # b. So a CD-structure is both flabby and keen. Every structure F = (52,D , E ) gives rise t o a flabby structure ( o f the same type) Ffl := (52,D , E f l ) , where E f l ( a ,b) := t o a keen structure Fk,
i f a = b, E ( a , b) otherwise,
:= (52, D , Ek,),
Eke ( a ,b) :=
where
{ E ( a 7 a ) otherwise, ifa=b,
and t o a CD-structure Fcd := (52, D ) . Definition 4.1.7 For a structure F = (52, D , E ) , an n-ary F-predicate (or an 52-valued predicate on D ) is a map A : Dn ---,R. Such a predicate is called strict (or Scott) if A ( a ) < E ( a ,a ) for any a E Dn, and congruential if for any a , b E Dn, for any i E In
For a, b E Dn we will use the notation
L e m m a 4.1.8 For any congruential predicate A : Dn (1) E a b A A ( a )
(2) E a b
< A(a)
--+ 52,
for any a , b E Dn
< A(b), tt
A(b).
P r o o f W e prove ( 1 ) by induction on n. If n = 1, the condition a = I b holds trivially, so the claim is obvious. Suppose ( 1 ) holds for n for any congruential A and consider a , b E Dn+'. Let a = alal, b = blbl. Then (*) A ( a )
Eab = A(a)
Ealbl A Ea'b' 5 A(b1a') A Ea'b'
4.1. MODAL AND HEYTlNG VALUED STRUCTURES
297
since A is congruential. The n-ary predicate B : c H A(b1c) is also congruential. In fact, for c, d E Dn the condition c =i d implies blc =i+l bid, thus
since A is congruential. By the induction hypothesis, A(blal) A Ea'b' = B(a') A Ea'b' 5 B(bl) = A@), therefore by (*), we obtain (1): A(a) A Eab 5 A(b). Now (1) implies
Eab 5 A(a) -t A(b) hence by symmetry Eba 5 A(b)
which eventually implies (2)
A 0-ary predicate A : {A)
-
A(a),
fl can be treated just as an element A(A) of
a. This predicate is always congruential and strict, since E ( A , A ) = 1. L e m m a 4.1.9 (1) A structure F is flabby i f fall F-predicates are strict. (2) A structure F is keen iff all F-predicates are congruential.
Proof (1) (Only if.) E(ai, ai) = 1 implies A(a) 5 E(a, a). (If). Suppose Eao # 1 for some a0 E D. Then the predicate A sending every a to 1is not strict, since A(ao) $ Eao. (2) (Only if.) If a # b, then E a b = 0, and obviously Eaibi A A(a) If a = b, then Eab A A(a) 5 A(a) = A(b).
< A(b).
(If). Suppose E(a0, bo) = ct. # 0 for some a0 # bo and consider the unary F-predicate A sending a0 to (I. and every a # a0 to 0. Then
thus A is not congruential.
Definition 4.1.10 For n-ary F-predicates we introduce the ordering
CHAPTER 4. ALGEBRAIC SEMANTICS
298
Lemma 4.1.11 For an n-ary F-predicate A put As(a):= A(a) A Ea.
Then (1) As 5 A; (2) AS is strict; (3) for any n-ary strict F-predicate
B
(4) i f A is congruential, then As is congruential. Thus ASis the greatest strict predicate "below" A; we call it the strict version of A.
Proof
(I), (2), (3) are obvious.
( 4 ) Suppose a,b E Dn and a =i b. Then
since A is congruential. Also
Eaibi 5 Ebi by 4.1.4(4), hence
(##) E a A Eaibi L Eb, and thus AS(a)A Eaibi
5 A(b) A Eb = AS(b).
by (#), (##).Therefore ASis congruential. Note that trivially AS= A for 0-ary A. Let us now prove a dual to Lemma 4.1.11. Lemma 4.1.12 For an n-ary F-predicate A put
Then
(2) Ac is congruential;
4.1. MODAL AND HEYTING VALUED STRUCTURES
(3) for any n-ary congmential F-predicate
(4) A is strict
B
ifl Ac is strict.
So Ac is the least congruential predicate 'above' A; we call it the congruential version of A.
Proof
( I ) , (3) are obvious.
(2) Suppose a =k b. Then
by well-distributivity (Section 1.2). Now if i = k , then
by 4.1.1(1), (2). If i # k , then ai = bi, so
Thus
Also note that A ( a ) A Eakbk = A(a) A Ebidi
for d = a, i = k. Hence
(4) Since A 5 AC,it follows that AC(a)5 Ea implies A(a)
< Ea.
The other way round, assume that A is strict. Then for any i and d A ( d ) A Eaidi
=i
a,
< Ed A Eaidi = A Eai A Eaidi < Ea, j#i
since Eaidi 5 Eai by 4.1.4(1). By assumption A(a) 5 Ea, so all the disjuncts in AC(a)are bounded by Ea, and thus Ac(a) 5 Ea.
CHAPTER 4. ALGEBRAIC SEMANTICS
300
Remark 4.1.13 Note that the disjuncts A(a) Eaiai corresponding to d = a are obviously redundant in the definition of AC.On the other hand, for a strict A we have Ac(a)= ( A ( d )A Eaidi),
VV
i d={a
since A ( a ) = A ( a ) A Eaiai.
Remark 4.1.14 For an arbitrary predicate A, we can construct a strict congruential version
(*)
( A S ) " ( a= )
( a )= V
V
( A ( d )A Eaidi).
i d={a
In fact, AS is strict, so by Remark 4.1.13
The latter equality holds since Eai A Eaidi = Edi A Eaidi (which easily follows from 4.1.4 (I)),and thus for d =i a
On the other hand,
( A C ) " ( a= ) A C ( a )A Ea = A ( a ) A Ea V
V V ( A ( d )A Ea A Eaidi) i d=ia
by well-distributivity. The first disjunct is redundant as it equals A ( a ) A Ea A Eaiai, hence (*) follows. Note that
AS 5 (As)c5 AC, but in general A and (AS)' are 5-incomparable. E.g. consider a unary A such that A(a) $ Eaa for some a. Then A(a) -$ ( A S ) c ( a )since , ( A S ) " ( a )5 Eaa. And if A(d) A Ebd -$ A(b) for some b,d, then (AS)"(b) A(b), since
A ( d ) A Ebd 5 (AS)"(b). Exercise 4.1.15 Show that (AS)'= A iff A is strict and congruential. Remark 4.1.16 As we are interested in algebraic models satisfying the axioms of equality, atomic D+-sentences should be evaluated by congruential predicates. This requirement also makes sense for logics without equality. A similar situation is in Kripke semantics - as we know from Chapter 3, including equality in semantics is crucial for completeness.
4.2. ALGEBRAIC MODELS
301
Remark 4.1.17 The strictness property is quite natural for evaluation of formulas - the underlying idea is that a D+-sentence may be true only within the "life-zone" of all occurring individuals. But as we shall see, this does not matter for semantics, because replacing every A with AS does not change the notion of validity. However evaluating formulas with arbitrary predicates simplifies the inductive truth definition, see Definition 4.2.4. This happens because strict predicates are not closed under all Boolean operations (for example, under negation, since A(a) 5 Ea does not imply -A(a) 5 Ea if Ea # 1).
4.2
Algebraic models
Definition 4.2.1 A valuation is an m.v.s. (H.v.s.) F = ( a , D , E ) is a map cp : AFD ---, sending every D-sentence to fl such that
whenever P E PLn, a , b E Dn, a = i b. Then the n-ary F-predicate pp : a H cp(P(a)) is called associated to cp and P . The pair ( F ,cp) is called an algebraic model over F . F'rom the definitions it follows that all the predicates cpp are congruential. Definition 4.2.2 A valuation cp is called strict i f every predicate associated to p is strict, i.e. i f cp(P(a))I Ea
for any P E PLn, a E Dn. Definition 4.2.3 For an arbitrary valuation p we define its strict version cpS such that pS(P(a)) := p(P(a)) A Ea.
foranya€Dn, P € p L n . Thus (cpS)p= ( ( ~ p )and ~ , so the definition is sound, since (cpp)Sis congruential by Lemma 4.1.11. Certainly cpS = p if p is strict. Definition 4.2.4 For a valuation cp i n an m.v.s. (respectively, H.v.s.) F = (a,D , E ) we define its (unique) 'large' extension to all modal (respectively, intuitionistic) D-sentences i n the natural way: (1) p ( l ) := 0;
(2) p ( a = b) := E ( a , b);
(3) cp(A V B ) := cp(A) V cp(B);
CHAPTER 4. ALGEBRAIC SEMANTICS
Then for any D-formula A and a distinct list of variables x 2 F V ( A ) of length n, we may define the associated F-predicate
such that ( P A , x ( ~:= )
~ ( [ ~ / ~ l ~ )
for any a E Dn. In particular, =(PP. To simplify notation, we sometimes write c p rather ~ than c p ~ , ~ . Let us check that every predicate c p ~ , , is congruential. We begin with a simple observation.
L e m m a 4.2.5 ( 1 ) cpp remains congruential after fixing some parameters. I n precise terms: let P E PLn, c E Dn, let x , y be disjoint distinct list of variables of
length n and m respectively, let a : I,,, ---+ I , be a n injection, and let B = [ c / x ][ y / x- a ] P ( x ) . T h e n 913,). i s congruential. (2) Let A be a n atomic D-formula without equality with F V ( A ) { x ) . T h e n for any algebraic model (F,cp) with the individual domain D, for any a, b E D cp([alxlA)A Eab i cp([blxlA). Proof
(1) First note that
where
a,-I
(i)
if i G r ( a ) , otherwise.
This is clear, since
B [ y H a] = P ( x ) [ x a . H a ] [ xI-+ c ] . Similarly we have
c p ( [ b l ~ l B=) cp(P(bl)),
4.2. ALGEBRAIC MODELS
303
where b' is constructed from b in the same way as a'. Now a a' =,(i) b', and thus, since cpp is congruential,
=i
b implies
cp([a/y]B) A E a b = cp(P(a')) A E a b I cp(P(b')) = cp([b/y]B), as required. (2) If A is a D-sentence, the claim is trivial. Otherwise we have A = [xrnly1B, for some B as in (I), if A contains P and x occurs m times in A. By (I), c p ~ , , is congruential, hence by Lemma 4.1.8, E(am,bm) A cp(B(am)) I cp(B(bm)). Now since E(am, bm) = Eab, B(am) = A(a) and B(bm) = A(b), we obtain (2). L e m m a 4.2.6 The predicate c p ~ , , is congrmential for any D-formula A with FV(A) C {x).
Proof
By induction on the complexity of A we prove cp([a/xlA) A Eab 5 cp([blxlA).
(1) If A is I,there is nothing to prove. (2) If A is x = c or c = x, then by 4.1.1 we have: cp(A(a)) A Eab = Eac A Eab I Ebc = cp(A(b)). (3) If A is x = x, then cp(A(a)) A Eab = E a a A Eab
I E b b = cp(A(b)).
(4) If A is atomic without equality, the claim follows from Lemma 4.2.5.
(5) In the modal case, if A is OiB, then by Definition 4.1.2(4) and the induction hypothesis cp(A(a))nEab 5 Oicp(B(a))nOiEab = Oi(cp(B(a))nEab) I Oicp(B(b)) = cp(A(b)).
(6) If A is B > C and (PB, (PC are congruential, then cp(A(a)) A Eab = (cp(B(a)) cp(C(a))) A Eab; thus cp(A(a)) A Eab A = (cp(B(a)) cp(C(a))) A cp(B(b)) A Eab I (cp(B(a)) -+ cp(C(a))) A cp(B(a)) A Eab (since c p is ~ congruential) 5 cp(C(a)) A Eab (by properties of Heyting algebras) I cp(C(b)) (since cpc is congruential). -+
+
Therefore
CHAPTER 4. ALGEBRAIC SEMANTICS
304
(7) If A ( x ) is 3y B(y,x ) , then we may assume that y sentence), and thus
cp(A(a))=
# x (otherwise A is a
V cp(B(d1a ) ) ) . dED
So by well-distributivity, and the induction hypothesis applied to B(d,x ) , we obtain
cp(A(a))A Eab =
V
(cp(B(d,a ) )A Eab) i
d€D
(8) If A = B V C , and the claim is proved for B , C, we have
(9) The simple case A = B A C is left to the reader. (10) Let A = VyB(y,x), and assume that y E F V ( B ) ,and that the claim holds for B. Then cp(A(a))A Eab I cp(A(b)) is equivalent to
Eab A
/\ (Ec
-+
cp(B(c,a ) ) )I
cED
/\
(Ec
+
cp(B(c,6 ) ) ) .
c E D
It suffices to show that for any c E D,
Eab A (Ec + cp(B(c,a ) ) )I Ec -* cp(B(c,b ) ) , i.e.
Eab A (Ec -. cp(B(c,a ) ) )A Ec I cp(B(c,b)). The latter follows by properties of Heyting algebras and the induction hypothesis: Eab A (Ec -+ cp(B(c,a ) ) )A Ec 5 Eab A cp(B(c,a ) ) 5 cp(B(c,b ) ) .
Lemma 4.2.7 The predicate cp~,, is congruential for any D-formula A and r ( x )> F V ( A ) . Proof Assume a =i b for a, b of the same length as x. Fix all the parameters of A but xi, so put
B(xi) := A(al,.. . ,ai-1, xi, ai+l,.. . ,an). Then
by Lemma 4.2.6.
4.2. ALGEBRAIC MODELS
305
However, as we mentioned in Remark 4.1.17, the predicate c p is~ not necessarily strict, even for a strict cp. So we introduce another extension of cp to D-sentences. D , E ) be a structure, A a D-sentence of the corresponding type. Let F = Then we put E(A) := / \ { ~ a I a E Dl a occurs in A).
(a,
In particular, E(A) = 1 if A is a usual sentence (without constants from D).
Definition 4.2.8 For a valuation cp in a structure F = ( 0 , D, E ) we define an a-valued function cpb on corresponding D-sentences as follows. (1) cpb(l) := 0;
(2) @ ( a = b) := E ( a , b);
(3) @(A) := cp(A) A E(A) for all other atomic A;
(4) $(A
V
B)
:= E ( A V
B) A ($(A) V cpb(~));
(5) $(A A B ) := @(A) A cpb(~);
> B ) := E(A > B ) A (cpb(A)-+ c p b ( ~ ) ) ; (7) cpb(OiA) := E(OiA) n O,cpb(A) (in the modal case);
(6) $(A
(9) cpb(VxA) := E(VxA) A
A
(Ed 4 cpb([d/x]A)),
dED
W e also define another associated predicate
such that for any a E Dn
Again we often abbreviate
to cp;.
Lemma 4.2.9 Let ( F ,cp) be an algebraic model, with the domain D, A a corresponding D-sentence. Then
Proof
By induction.
(1) If A is I,the claim is trivial. (2) If A is a = b, then by 4.1.4 (1) and symmetry,
cp(A) = Eab < E a A E b = E(A).
CHAPTER 4. ALGEBRAIC SEMANTICS
306
(3) If A is a = a, then p(A) = E a a = E(A).
(4) If A is atomic without equality, the claim holds by definition. (5) If A is a modal formula OiB, then by the induction hypothesis, c p n ( ~=) Q c p f l ( ~n) E(A) = Q ( B ) n OiE(B) n E(A). Next, OiE(B) n E(A) = OiE(A) n E(A) = E(A) by 4.1.4 (4), and thus
(6) If A is B V C , then by the induction hypothesis,
Since E(A) 5 E ( B ) , E ( C ) , we obtain
(7) If A is B A C and the claim holds for B , C , we have
(8) If A is B
> C , then by the induction hypothesis
Let us show that this equals
In fact,
A E(A) 5
The other way round
A E(B),
4.2. ALGEBRAIC MODELS
307
hence,
thus
E ( A ) A (cp(B)-, cp(C))I cp(B)A E ( B ) -,cp(C)A E(C). (9) If A = 3xB(x) and x is a parameter of B ( x ) , then by the induction hypothesis and Definition 4.2.4 we have
If A = 3xB(x) and B ( x ) is a D-sentence, then B ( x ) d E D, E ( B ( d ) )= E ( A ) . So
=
B(d) for any
(10) Finally let A = V x B ( x )and first assume that x E F V ( B ( x ) ) .Then
E ( A )A
A
(Ed -+ E ( A ) A Ed A cp(B(d)))
dED
by the induction hypothesis; thus
@ ( A )5 E ( A ) A
/\
(Ed
+
w(B(d)))= E ( A ) A cp(A).
dED
The other way round,
E ( A )A (Ed -t cp(B(d)))A Ed 5 E ( A ) Ed A cp(B(d)), E ( A ) A (Ed -,cp(B(d)))A Ed I E ( A ) A Ed A cp(B(d)), hence
(*) E ( A ) A (Ed + cp(B(d)))I Ed which implies
+
E ( A ) A Ed A cp(B(d)),
cpw.
E(A) I If B ( x ) is closed, the argument is slightly different, because now B(d)= B ( x ) and E ( B ( d ) )= E ( A ) . By the induction hypothesis we have
CHAPTER 4. ALGEBRAIC SEMANTICS
308
The other way round, (*) implies
L e m m a 4.2.10 Let (F,cp) be an algebraic model with a domain D . Then for any D-formula A with FV(A) c r(x) the predicate cp\,x is congruential; it is also strict i f FV(A) = r(x). Proof
Strictness readily follows from the previous lemma, since
To prove the congruentiality, assume that 1x1 = n and consider a =i b in Dn. If xi @ FV(A), then [a/x]A = [b/x]A, so the inequality
is trivial. If xi E FV(A), then by the previous lemma and Lemma 4.2.7,
cpu([a/x]A) A Eaibi
= cp([a/x]A) A E([a/x]A) A Eaibi < cp([b/x]A) A E([a/x]A) A Eaibi. -
Since a =i b and Eaibi 5 Ebi, we have
thus
cpu([alx]A)A Eaibi I ~~([b/x]~) b y 4.2.7.
L e m m a 4.2.11 Let (F, cp) be an algebraic model with a domain D, cpS the corresponding strict valuation such that for any P E P L n , a E Dn,
i.e. (Lemma 4.1.11) cp;
=(cp~)~.
Then
(cpYh(A) = cpb (A) for any D-sentence A.
4.2. ALGEBRAIC MODELS
Proof
By the choice of cps, for any A E AFD
Since the maps (cpS)b, cph coincide on AFD and they are uniquely prolonged according to 4.2.8, they also coincide on all D-sentences.
Lemma 4.2.12 Let (F,cp) be an algebraic model, A a sentence of the corresponding type. Then
Proof
By 4.2.11,
v h ( 4= ( P ~ ) ~ ( A ) . Since E(A) = 1, by Lemma 4.2.9 we also have
Lemma 4.2.13 Let F = (a,D , E) be a st7*ucture, (F,cp) an algebraic model, A(x) a D-formula of the corresponding type with r(x) = FV(A(x)), 1x1 = n, x distinct. Then
(2) cp(VxA(x)) = 1 iff Ya E Dn E a 5 cp(A(a)). Thus cp(VxA(x)) does not depend on the ordering of x, so we may use the notation cp('?A).
Proof (1) By induction on n. The base is trivial. Consider the step from n to n 1. If x = yz, lyl = n, then VxA(x) = VyVzA(y, z), so by the induction hypothesis and Lemma 1.2.3,
+
as required. (2) Readily follows from (1).
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CHAPTER 4. ALGEBRAIC SEMANTICS
Definition 4.2.14 A closed formula A is called true i n an algebraic model (F, (P) (notation: (F,(P) k A or (F,(P) Il- A i n the intuitionistic case) i f p(A) = 1; an arbitrary formula A is called true i n (F, cp) (with the same notation) if VA is true. Definition 4.2.15 A predicate A : Dn ---,Ct is called true i n a structure F = (a,D, E ) (notation: F t= A or F Il- A ) i f Ea 5 A(a) for any a E D n . Thus FbAiffVa€Dn Ea=A(a). for a strict A,
F I= A iff A(A) = 1. for a 0-ary A, and FbAiffFkAs. L e m m a 4.2.16 The following conditions are equivalent (where b stands for Iti n the intuitionistic case) (1) (F7 cp) b A,
(2) (F7(PS)b A, (3) F t=
( P A , i.e.
V a E Dn Ea 5 cp(A(a)),
(4) F b cpi, i.e. V a E Dn (5) F k P(,;
cpS(A(a))= E a ,
i.e. V a E Dn Ea 5 cph(A(a)),
(6) F b (cps)\, i.e. V a E Dn (cpS)h(A(a))= E a . Proof
By 4.2.12 and 4.2.13.
rn
Definition 4.2.17 A formula A is valid i n a structure F (of the corresponding kind) i f A is true i n every model over F (or equivalently, i n every strict model over F). Validity is denoted again by k in the modal case, It in the intuitionistic case. R e m a r k 4.2.18 If in the definition of validity we also allow for 'valuations7 cp in F = (0,D, E), for which cpp is not congruential (they are just valuations in the corresponding D-structure (Ct, D)), then some QKZ- (or QH=-) theorems become nonvalid. For instance, if P E PL1 and cpp is not congruential, then the formula A := VxVy(x = y A P(x) > P(y)) is not true in ( 0 , cp) (in the sense of Definition 4.2.14).
4.3. SOUNDNESS
31 1
In fact, suppose
Eab A cp(P(a))$ cp(P(b)) for some a, b E D. Then by Lemma 4.2.13 (the proof of which does not use congruentiality),
cp(A) I E a A E b -, (Eab A cp(P(a))-,cp(P(b))) = E a A E b A Eab A cp(P(a))-,cp(P(b)) = Eab A cp(P(a))--, cp(P(b))# 1 , by our assumption.
Lemma 4.2.19 For any valuation cp i n an m.v.s.,
Proof The equality ( 1 ) is checked easily, so let us check (2). In fact, by ( I ) , we have:
Oi E ( A ) . since E ( A ) I
W
4.3 Soundness We begin with an analogue of Lemma 3.2.23.
Lemma 4.3.1 Let F be a modal or Heyting-valued structure with a set of individuals D , and let A ( x ) , B ( x ) be congruent formulas of the corresponding type, 1x1 = n. Then for any a E Dn, for any valuation q5 i n F ;
(11) for A , B without constants, F k (It-)A
F k (It)B.
Proof
(I) Along the same lines as 3.2.22. Consider the equivalence relation on modal (or intuitionistic) formulas: A B iff F V ( A ) = F V ( B ) and for any distinct list x such that r ( x ) = F V ( A ) , for any a E ~ 1 ~ 1
It is sufficient to show that
has the properties 2.3.17(1)-(4).
We only consider the case & = b'. If FV(b'yA) = r ( x ) for a distinct x, we have two options:
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CHAPTER 4. ALGEBRAIC SEMANTICS
(i) If y gZ FV(A), then y
9 V(A),
A[y H z] = A,
By Definition 4.2.4,
/\
cp(v~[alxlA)=
(Ed
cp([alxlA)),
dED
since y does not occur in [a/x]A. Hence by Lemma 1.2.4 and 4.1.1(E3),
By the same reason
so (1)holds in this case. (ii) If y E FV(A), then
and so
cp( [ a l x l ~A) y =
A
ED
cp([alxIQz(A[y
(Ed
2))
=
+
A
cp([dlyl [alxlA)),
d€D
(Ed
+
cp([dlzl[alxl(A[~ zl))). ++
It remains to note that
[dlyl [alxlA = [dl21[alxI(A[y since y
zl),
# BV(A), z $2 V(A).
(2) Supposing A
B , let us show that
We have F V ( A ) = F V ( B ) , FV(VyA) = FV(VyB); let r(x) = FV(VyA). Then
cp([alxlV~A)=
A (Ed A (Ed
+
cp([adlxylA)),
dED
cp([alxlv~B)=
dED
Now A
N
B implies
and the claim follows. The argument for 3 is quite similar.
+
cp([adlxylB)).
4.3. SOUNDNESS
313
-
Consider the case * = 3.We argue similarly t o 3.2.22. If r ( x ) = F V ( A > B ) , then F V ( A ) = x . a, F V ( B ) = x . T for injections a, T . Now A A' & B B' implies F V ( A 3 B ) = F V ( A 1 > B') and
-
cp([a/x](A3 B ) ) = cp([a.a / x / .a ] A> [ a .T / X = cp([a.a / x . a ] A )-+ cp([a.T / X . r ] B ) = cp([a a / x . a ] A r )-+ cp([a.T / X . r ] B 1 ) = cp([a/x] (A' > B')).
. r ] B )=
-
The proof of (4) is trivial. Since E ( A ( a ) ) = E ( B ( a ) ) ,by Lemma 4.2.9 it follows that cp$(A(a))=
cp'(B(a)). (11) is an obvious consequence of (I).
T h e o r e m 4.3.2 (Soundness theorem)
(1) For an N-m.v.s. F , the set
(2) For an H.v.s. F , the set
Proof First let us show that the set of valid formulas is substitution closed. The argument resembles the proof of Lemma 3.2.16, but the detail is slightly different. Assume that F = D , E ) k A. Let S = [ C ( x y, ) / P ( x ) ]be a simple formula substitution, where x , y are distinct lists such that r ( y ) C F V ( C ( x ,y ) ) C r ( x y ) . We may also assume that P occurs in A (otherwise S A = A ) and y n F V ( A ) = 0 , due to Lemma 4.3.2 (cf. the proof of Lemma 3.2.16). Since S A is defined up to congruence and validity respects congruence (Lemma 4.3.1), we may further assume that A is clean and B V ( A ) n y = 0 ; then S A is obtained by replacing every subformula of the form P ( x l ) with C ( x l , y ) . By Lemma 2.5.23 we also have F V ( S A ) = y U F V e ( S , A ) , where F V e ( S ,A ) F V ( A ) is the set of essential parameters, see Definition 2.5.22. Let
(a,
c
F V ( A ) = r ( z ) , F V e ( S ,A ) = r ( z l ) , z
= z'z".
For an algebraic model ( F ,cp) let us show that ( F ,cp) i= (IF) S A , i.e. (Lemma 4.2.16) for any b E Dm, c E Dj (where m = lyl, j = Iz'I)
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CHAPTER 4. ALGEBRAIC SEMANTICS
Given tuples b , c , we construct a valuation 77 in F such that for any a E Dn (where n = 1x1) v(P(a)) := cp(C(a1b)); v(B) := cp(B) for any other B E AFD. Let us check that 7 is a valuation. In fact, let a =i d ; then by Lemma 4.2.7
Then we claim that for any formula B(u), with a distinct u such that
and for any d E D' (where 1 = lul)
or in a simpler notation,
Note that SB(d, b) is a D-sentence, since FV(SB) uy, by Lemma 2.5.23. The claim (1) is proved by induction. The case when B is atomic and does not contain P, is trivial. Assume that B is atomic, B = P(x'), x' = u . a, where a : In --,Il. Then we have: B(d) = [d/u][u.a/x]P(x) = [d . a/x]P(x) = P ( a ) , where a = d . a. On the other hand, S B = C ( u . a, y ) , and so [dlb / u , y]SB = [dlb l u , y ] C ( u . a, y) = C ( d . a, b ) = C(a, b).
Thus by definition of 7,
Let B = B1 * B2, S B = SB1 * SB2, where * is V, A or 4.2.4 and the induction hypothesis
>. Then by Definition
where * is the corresponding operation in a. If B = 'dxB1, then by our assumption x @ uy, so S B = VxSB1, and thus by Definition 4.2.4 and the induction hypothesis r](B(d)) =
r\ aED
=
( E a --, r](Bl(a, 4 ) ) =
/\
(Ea
aED
cp(VxSBl(b, d)) = cp(SB(b,d)).
cp(SBi(blaid)))
4.3. SOUNDNESS
315
The case B = 3xB1 is similar and is left to the reader. Now let us verify (#). Take an arbitrary e E Dk-j and the corresponding d = ce E Dk. Then by ( 1 ) we have:
But (3) Id, b / z , YISA = [c,b l z f ,y]SA, [ d / z ] A= [c/zl]A,
since F V ( A ) = r ( z ) , F V ( S A ) = r ( y z f ) . By our assumption, F I= A, and so by (2),( 3 ) ,and Lemma 4.2.16, we obtain cp([c, b / z f ,y ] S A )= v ( [ d / z ] A= ) v ( [ c / z f ] A2) Ec 2 E b A Ec,
i.e. (#) holds. The remaining properties (m0)-(m3), (m5)=, ( s l ) , (s3) from Definitions 2.6.1, 2.6.2, 2.6.3 are verified in a standard way. We check only some of them. Note that the value cp(A) of a propositional formula A in a structure F = (52, D, E ) is exactly the same as in the algebra 52. So, since all propositional axioms are valid in Cl (Lemma 1.2.6), they are also valid in F. Let us check the validity of (Ax12): V x P ( x ) > P ( y ) . In fact, consider an algebraic model (F,cp) . We have
hence
i.e. (F,cp) i= V x P ( x )> P ( y ) ,by Lemma 4.2.16. Let us also show the validity of (Ax14): V x ( P ( x )> q) > ( 3 x P ( x )> q). In fact, cp(Vx(P(x)> q) > ( 3 x P ( x )> q)) = 1 iff cp(Vx(P(x)3 9 ) ) 5 cp(3xP(x)2 q) = cp(3xP(x))-+ iff cp((Vx(P(x)3 9 ) )A cp(3xP(x))I 4 9 ) Transforming the left part of the latter inequality by well-distributivity, we obtain A (cp(P(d)) ~ ( 9 )A ) V cp(P(a)) +
dED
aED
as required. And finally let us consider modus ponens. Suppose F i= A ( x , y ) , A ( x ,y ) > B ( Y ,z ) , where F V ( A ( x ,Y ) ) = r ( x ,Y ) , F V ( B ( Y ,z ) ) = ~ ( Y Z )r (, x )f- r ( z ) = 0 and the lists x y , yz are distinct (and some of them may be empty). Let 1, m, n be
CHAPTER 4. ALGEBRAIC SEMANTICS
316
the lengths of x, y , z respectively. By 4.2.16, for any a E D ~ b, E Dm, c E Dn and valuation cp
Hence E ( a b c ) = E ( a b ) A E(abc) I cp(A(a,b ) ) A cp(A(a, b) 3 B(b, c))
I cp(B(b, c)).
As this happens for any a E DL,we obtain
But
by 4.1.4(5). Thus E(bc) 5 cp(B(b, c)), which implies F I=B(y,z), by 4.2.16. Let us now consider a particular kind of H.v.s. - those arising from S4m.v.s.
Definition 4.3.3 The pattern of an S4-m.v.s. F = (fk,D,E) is the H.v.s. FO= ( 0 ° , D , E ) . An H.v.s. of this form is called 'basic'. More generally:
Definition 4.3.4 A locale is said to be basic if zt is isomorphic to the pattern of a complete S4-algebra. A n H.v.s. over a basic locale is also called basic. For basic H.v.s. the intuitionistic truth definition matches with the modal definition. In precise terms, this means the following.
Lemma 4.3.5 Let cp be a valuation in an S4-m.v.s F = ( a , D , E), and put $(A) := Ocp(A) for every A E AFD. Then $ is a valuation i n FOand $(A) = cp(AT) for every A E IF5
Proof By Lemma 4.2.7, $ is congruential. The required equality is easily proved by induction. Let us consider the induction step for A = 'dxB(x), where B(x) is a D-formula, FV(B(x)) {x). Then according to Definitions 4.2.4, 2.11.1 and Proposition 1.2.7, we have:
4.3. SOUNDNESS
On the other hand,
,
hence
AT) =
( E d . .(BT(d))) 5 O ( E d . v ( ~ ~ ( d ) ) ) ,
dED
and thus
n .(Ed
cp(AT)5
3 v(BT(d))),
d€D
which implies
and eventually
4 A T )= $(A).
1
The case A = 3xB(x) is left t o the reader.
+
L e m m a 4.3.6 Let F = (a,D, E ) be an S4-m.v.s., a valuation i n F O ,$" the same valuation i n F . Then $ ( A ) = $ " ( A ~ for ) every A E IF;. Proof
Apply Lemma 4.3.5 to the case when p = $".
1
Proposition 4.3.7 For any S4-m.v.s. F and intuitionistic formula A ( x ) , F O k A ( x ) ifl F F AT ( x ) ,i.e. IL(')(FO) = s ( M L ( ' ) ( F ) ) .
We may assume that x = F V ( A ) . (+) Suppose F k AT and consider a valuation in FO.Let $" be the same valuation in F ; then + " ( v x A T ( x ) ) = 1, and thus +"(OVxAT(x)) = 1. By Lemma 2.11.7, Q S 4 t- O V X A(~x ) = ( v x A ( x ) ) ~ ;
Proof
+
hence by Lemmas 4.3.2, 4.3.6
+
Since is arbitrary, it follows that FO k A. (+) Suppose F0 k A. For an arbitrary valuation cp in F , let us show that p(VxAT ( x ) )= 1. Let be a valuation in FO described in Lemma 4.3.5. Then by Lemmas 2.11.7, 4.3.2 and since FO k A, we obtain $J
and therefore cp(VxAT( x ) )= 1.
CHAPTER 4. ALGEBRAIC SEMANTICS
318
Definition 4.3.8 W e introduce general algebraic semantics for our four types of logics ( M , M = , S,S=):
A n algebraic semantics is a semantics generated by a class of m.v.s (or H.v.s.). Let us also introduce some particular cases of algebraic semantics.
Definition 4.3.9 For superintuitionistic logics the semantics A&;=) := {IL(=)(F) I F is a basic H.v.s.).
is called basic general algebraic. The question, whether every locale is basic, seems open, and so we do not know if A&;=) and A&,(,) are equivalent. Neighbourhood frames generate a special kind of modal algebras, so we can define the associated algebraic semantics.
Definition 4.3.10 A neighbourhood frame with equality is a triple @ = ( F ,D , E ) such that F is a neighbourhood frame and (MA(F), D , E ) is an m.v.s. The latter m.v.s. is denoted by MV(@). If F is a topological space, we call @ a topological frame with equality and define
The corresponding logics are defined in an obvious way:
Definition 4.3.11 W e define general neighbourhood/topological semantics as follows: IE,(=)
I
:= {ML(=)(@) @
is a neighbourhood frame with equality);
I&,(,) := {IL(=)(@)I @ is a topological frame with equality ). From the definitions and since Kripke frames correspond to a special kind of neighbourhood frames, we have:
Lemma 4.3.12 ICErn(=)5 I&,(=) 5 A&,(=);
4.4. MORPHISMS OF ALGEBRAIC STRUCTURES
4.4
319
Morphisms of algebraic structures
In this section we consider maps between algebraic structures preserving validity; they are analogues o f pmorphisms used i n propositional logic. Let us begin with the intuitionistic case.
-
Definition 4.4.1 Let Fl = (0, D l , E l ) and F2 = (0,D2, E 2 ) be H.v.s. A pR (an '0-valued graph') morphism y : Fl -4 F2 is a map a : Dl x D2 satisfying the following conditions for any a E Dl,b E D2: ( E l ) a ( a ,b) A El ( a ) L E2(b);
(Q1) E l ( a ) 5
V
a ( a , b) ('totality ');
bEDz
(QZ) E2 (b) 5
V
a ( a ,b) ('surjectivity ').
aEDl
If instead of ( E l ) , (EZ), a satisfies the conditions
( I ) a(a1,b l ) A cr(a2,bz) A Ez(b1, b2) L El ( a l ,az) ('injectivit~'); i t is called a p'-morphism. A p(')-morphism a is called a p(')-embedding i f for any a, a' E D l , b, b' E D2 : (E)
E2(b1,b)Aa(a,b)Ia(a,b1).
-
I t is obvious that (11) implies ( E l ) ,and (12) implies ( E 2 ) ,so every p=-morphism F2 is a p-morphism. I t is also clear that every p(')-morphism a : Fl F l , where a-'(a, b) = gives rise t o the converse p(')-morphism a-' : F2 -4 a(b, a ) . This is because t h e conditions ( E l ) , (E2), ( Q l ) , ( Q 2 ) , ( I l ) , (12) are 'symmetrical'. Definition 4.4.2 A p(=)-morphism a is called up(')-equivalence i f both a, a-' are p(')-embeddings, i.e. a satisfies ( E ) and
T h e diagram below shows the correlation between different kinds o f morphisms: p= - morphism
R p= - embedding
a
p - morphism
R
p - embedding
fr
IT
'p - equivalence
p - equivalence
For a = ( a l , . . . ,an) E D y , b = (bl, .. . ,b,) E DF, let a ( a ,b) :=
A
i= 1
a ( a i ,bi).
CHAPTER 4. ALGEBRAIC SEMANTICS
320
L e m m a 4.4.3 If a : Fl ---t F2 is a p-morphism of H.v.s., then for any a, c E D?; b E D?, ~ ( ab), A a ( c ,b) A E I ( c )I &(a, c ) . Proof In fact, the condition (El) yields: a(ci bi) A El (ci) F E2 (bi). On the other hand, from (12) we have:
These two inequalities imply
a(ai, bi) A a ( ~ bi) i , A El (ci) I E I (ai,ci), whence the statement follows easily.
L e m m a 4.4.4 Let a : Fl -+ F2 be a p(=)-morphism, x = ( X I , . .. ,x,), A ( x ) E IF(')), F V ( A ) C x , a E D?, b E DF. Then a ( a ,b) A E I (44)I J32(A(b)). Proof Easy by (El).
W
We will use the following abbreviations:
(PA := (P(A), $"A := $"(A).
Definition 4.4.5 Let a : Fl -+ F2 be a p-morphism of H.v.s. Valuations cpl in Fl and cp2 in F2 are said to be matching if
4% b) 5 (P1P(a)
++
(P2P(b),
for any P E p L n , a E D ; , b € D ? . The above condition is equivalent to
and can be replaced with the following two:
L e m m a 4.4.6 Let a : Fl --t F2 be a p(=)-morphism,x = ( X I , . . . , x,), A ( x ) E IF(=), F V ( A ) C_ x , a E Dy, b E DT. Then 4%b) F (P;uA(a) (P,"A(b), 2 F2 are matching. whenever valuations cpl in Fl and ( ~ in ++
4.4. MORPHISMS OF ALGEBRAIC STRUCTURES
Proof
(*)
By induction on A we check the following two properties:
4% b) A cpTA(a) < cpTA(b);
(**) a(a,b) A cp,"A(b)i cpTA(a). The atomic case is obvious; if A is (12).
( X I = x2), apply
the conditions (Il),
The case A = B A C is trivial. Let A = B V C. Then
by the inductive hypothesis and Lemma 4.4.4; similarly we obtain (**). Let A = B > C. Then we have:
by Lemma 4.4.4. Next,
4% b) A (cpi-B(a)--, cpTC(a))A cpTB(b) 5 ct.(a,b) A (cpYB(a)--, cpTC(a))A cpTB(a) by the induction hypothesis for B
5 ~ ( ab), A cpyC(a) by the property of Heyting algebras 5 cpYC(b) by the inductive hypothesis for C, so we obtain:
Thus
Similarly one can check (**). Let A(x) = 3yB(y,x), y E FV(B). Then by Definition 4.2.1, we have:
Now, since by (Ql),
EI(c)I
V dE D2
a(c,d )
CHAPTER 4. ALGEBRAIC SEMANTICS
322
and by the inductive hypothesis,
we obtain:
So A satisfies (*). To check (**), note: 4 % b ) A cpYA(b) = a ( a , b) A =
V
V
cp,"B(d, b)
dEDz
V
(a(a, b ) A Ez(d) A cp,"B(d, b)) L
cpTB(c,a ) = cp:A(a),
CEDl
dEDz
V
by the inductive hypothesis and since E2(d) I
a(c,d) by (Q2).
cE Dl
Let A(x) = 3yB(x), y @ FV(B); then
Let A(x) = VyB(y, x). For any d E D2 we have: a(a,b) A
A
(E1(c)
-+
cpTB(c, a)) A E2(d)
CEDI
= a(a,b) A
5
V
A
(El(c)
+
cp;'B(c,a)) A E2(d) A
CEDI
( 4 % b ) A E2(d) A a(c, d) A (E1(c)
V
a(cld) (by (Q2))
CEDI +
cpTB(c, a)))
CEDI
(by distributivity) I V ( 4 % b ) A a(c, d) A E l ( 4 A (El( 4
+
cpTB(c1a)))
biE$2)
5
V
((.(a, b) A a(c, d) A cpTB(c,a))
CEDI
(since in Heyting algebras x A (x + y) 5 y) V (a(ca,db) A cprB(c,a)) 5 cp,"B(d,b)
=
CEDI
(by IH). Hence we obtain (*): a ( a , b) A cpTA(a) = a(a,b ) A Ei(A(a)) A
(Ei(c) -+ cpTB(c, a)) CEDI
Similarly, for any c E Dl we have:
5
4.4. MORPHISMS OF ALGEBRAIC STRUCTURES
323
and thus we obtain (**):
a(a,b ) A
= a ( a , b )A E2(A(b))A
A
(Ez(d)-i cp,"B(d,b ) )5
dED2
I E l ( A ( a ) )A A ( E l ( c ) cp;'B(c, a ) )= cp;'A(a). +
&Dl
Lemma 4.4.7 If a : Fl
-
F2 is a p(')-embedding, then
(1) every valuation cpl in Fl matches with some valuation 92 in Fl;
Proof (1) Given pl, we define
cpzP(b) :=
V
( a ( c ,b) A c p l P ( ~ ) )
cED;
for every b E DF. Then
a(a,b) A cp2P(b)= =
V
V
cE D;
(cu(a,b) A a ( c ,b) A cplP(c))
b) A a ( c ,b ) A El(c) A c p l P ( ~ ) )
cED;
(since cplP(c) I El(c) for the valuation c p l ) V (El (a,c ) A c p l P ( ~ )5) cplP(4,
< CED;
by Lemma 4.4.3 and Definition 4.2.1. This
cp2
is a valuation in F2. Indeed,
by ( E ) . Also cpzP(b) I E2(b),since by ( E l ) ,a ( c ,b) A Ei(c) I E2(b) for any c E D;. (2) Let us show that A $! I L ( = ) ( Fimplies ~) A $ 1L(')(F2). We may assume that A is a sentence. Let cpl be a valuation in Fl such that PYA # 1,
and let 972 be a valuation in F2 matching with n=O), cp,A~cp,"A= 1,and thus cp,"A# 1.
Corollary 4.4.8 If a IL(=)(F~).
:
cpl.
By Lemma 4.4.6 (for
Fl - - - F2 is a p(=)-eguivalence,then I L ( ' ) ( F ~ )=
CHAPTER 4. ALGEBRAIC SEMANTICS
324
-
Definition 4.4.9 Let Fl = (52, D l , E l ) and F2 = (52, D2,E2) be two H.v.s. A D2 is called a strong pmorphism if for any a E D l , b E D2 map g : Dl
A strong p'-morphism
is a map g satisfying (ii) and
for any a l , a2 E D l . An isomorphism is a bijection g : Dl (iii).
-+ D2
satisfying
Note that (ii) obviously holds i f g is surjective and that (iii) implies (i). So a surjective g is a strong p=-morphism i f f (iii) holds. I t is also obvious that every isomorphism is a strong p'-morphism and that every strong p'-morphism is a strong pmorphism.
-
Lemma4.4.10 For H.v.s. Fl = (52,D1,E1),F2 = ( n , D 2 , E 2 )and a map g : Dl -+ D2, consider a : Dl x D2 f12 such that
for a E Dl,b E D2. Then g is a strong p-morphism iff a is a p-morphism; g is a strong p=-morphism z f f a is a p'-morphism p-equivalence, p'-embedding, p'-equivalence). Proof
(or a p-embedding,
First note that a satisfies ( E l ) a ( a ,b) A E l ( a ) 5 E2(b).
In fact, this means Ez(b,g(a)) El ( a ) 5 E2(b)7 which obviously holds, by Lemma 4.1.4(1). Let us show that ( E 2 ) for a and ( i ) for g are equivalent. The condition
is equivalent t o E2(b7 g(a)) 5 E l ( a ) ; this follows from ( i ) and implies E2(g(a))5 El ( a ) i f we take b = g(a). The condition (91) E l ( a ) 5 a ( a ,b)
V
bE D2
4.4. MORPHISMS O F ALGEBRAIC STRUCTURES is equivalent to &(a)
I
V
g(a));
bE Dz
this follows from (i), since E2(g(a)) = E2(g(a), g(a))
I V E Z ( ~ , ~ ( ~On )). bE D2
the other ha.nd, this condition implies El (a) E2(g(a)) for any b E D2. Next, the condition
I E2(g(a)), since E2(b, g(a)) I
is equivalent to aEDl
which is the same as the condition (ii). Therefore g is a strong p-morphism iff a is a p-morphism. Next, let us show that g satisfies (iii) if a satisfies (11) and (12). In fact, the condition 1 7A a(a27 b2) A El(al, ~ 2 5 ) E2(b1, b2) (I1) ~ ( ~ bl) means E2(b17g(al))A E2(bz,g(aa)) A E ~ ( a l , a a l ) E2(bi9b2), and (12)
a(a1, bl) A a(a2, b2) A Ez(b1, b2)
I El(a1, a2)
means El(a1, a2). E2(bl,g(al)) A E2(b27g(a2)) A Ez(b1, b2) I These two conditions follow from (iii) and taken together, imply (iii) if we put b~ = b2 = g(a2). The condition
means ~ ~ ( bb)' A, E2(b7g(a))A &(a)
I E2(b1,g(a))
and holds by Definition 4.1.2. The condition
means El(al, a) A E2(b7g(a))A Ez(b) I E2(b, g(a')) and thus follows from (iii) This proves the second part of the Lemma.
CHAPTER 4. ALGEBRAIC SEMANTICS
326
R e m a r k 4.4.11 One can consider a more natural, but slightly more restrictive definition of p(')-morphism. Viz., the conditions ( E l ) , ( E 2 ) can be replaced with a stronger one:
(E') a ( a ,b) I: E2(b) A El ( a ) . In this case the conjunct E l ( a ) in ( E ) , etc. should be omitted. Note that a strong p(=)-morphism g gives rise t o the p(')-morphism a in this stronger sense. P r o b l e m 4.4.12 Does the existence of a p(')-morphism (p(=)-embedding i n our form) imply the existence of a p(=)-embedding satisfying (E')? Now let us consider the modal case. I t is clear that i f F = (0,D l E ) is an D l E ) is an H.v.s. So we can give m.v.s., then its Boolean part F b = (ab, Definition 4.4.13 A p(=)-morphism(respectively, p(')-embedding p(=)-equivalence) a : Fl --,F2 from a p-m.v.s. Fl = (a,D l , E l ) to an m.v.s. Fz = (a,D2,E2) is a p(=)-morphism (respectively, p(=)-embedding, p(=)-equivalence) F$ satisfying the condition of the Boolean parts F!
-
(EO) &(a,b) I: 'Jio(a1b) for all i E I,. Thus a p-morphism is a map a : Dl x D2 conditions (for any a E D l , b E D2):
--, S1
satisfying (EO) and the
( E l ) a ( a ,b) n & ( a ) F and so on. T h e following is rather trivial. L e m m a 4.4.14 If Fl, F2 are m.v.s. over the same S4-algebra, F,", F; are the corresponding H.v.s., a: : Fl - - - F2 is a p(')-morphism (respectively, p(=)embedding, p(=)-equivalence), then the same map a is a p(=)-morphism (respectively, p(=)-embedding, p(')-equivalence) F," F i of the corresponding H.v.s.
-
T h e definition o f matching valuations in the modal case is the same as Definition 4.4.5: Definition 4.4.15 Let a: : Fl ---+ F2 be a p-morphism of m.v.s. Valuations cpl i n Fl and cpz i n Fz are matching i j b, I: (cpiP(a)
for any P E P L n , a
E
cpaP(b)),
Dy, b E Dz.
L e m m a 4.4.16 Let a : Fl ---, F2 be a p(')-morphism, F V ( A ) C x = ( 2 1 , . . . , x,), a E D y , b E D z . Then
whenever valuations cpl i n Fl and
cp2
in Fz are matching.
A ( x ) E MF(=),
4.4. MORPHISMS OF ALGEBRAIC STRUCTURES
327
Proof By induction on A. The atomic case and the cases of A, V, >, 3, V were considered in the proof of Lemma 4.4.6. Consider the case A = OiB. By induction hypothesis, we have:
and thus
Hence, by (EO), (1) a ( a , b ) n OiqlB(a) l n i ~ 2 B ( b ) . By Lemma 4.4.4 (which obviously holds in the modal case),
Let us also define some auxiliary kinds of morphism.
-
-
Definition 4.4.17 Let Fi = ( 0 , Di, Ei), i = 1,2, be two H.v.s (respectively, m.v.s) over the same algebra 52. A regular morphism y : Fl F2is a mapping y : Dl D2 such that for any a, b E Dl (1) El (a, b)
5 E2(y(a), ~ ( b ) ) ,
(2) E l ( % a) = E2(y(a),y(a)). 0-Hvs (respectively, 0-rnvs) denotes the category of H.v.s. (respectively, m.v.s.) over 52 and regular morphisms. In particular, every strong p'-morphism regular morphism.
in the sense of Definition 4.4.9 is a
Remark 4.4.18 For H.v.s. there also exists another kind of morphisms, which we call weak morphisms [Goldblatt, 1984; Borceaux, 19941. A weak morphism is defined as a mapping a : D l x D2 -+ C? with the following properties: E2(b,b')
a ( a ,b')
< a ( a , b);
a ( a , b) A E l ( a , a') 5 a(al, b);
CHAPTER 4. ALGEBRAIC SEMANTICS
328 a ( a , b) A a ( a , b')
5 Ez(b, b');
It is clear that every regular morphism y corresponds to a weak morphism a , such that b) = E(b,?(a))-
4%
Moreover, the category of 0-sets and weak morphisms is equivalent to its full subcategory of complete 0-sets, and in the latter subcategory all weak morphisms correspond to regular morphisms [Borceaux, 19941. However, weak morphisms are not logically faithful (see the discussion below). Thus the above equivalence is not sufficient for showing the equivalence between the semantics of H.v.s. and the semantics of complete H.v.s.
4.5
Presheaves and 0-sets
In this section we consider an equivalent representation of algebraic semantics using presheaves. The connection between presheaves and 0-sets is well known in topos theory. Every presheaf corresponds to an 0-set in a standard way [Fourman and Scott, 19791; this construction can be used to define logics of presheaves. On the other hand, for the case of sheaves there exist a connection in the opposite direction; Higgs's theorem [Higgs, 19841, also cf. [Borceaux, 1994; Goldblatt, 1984; Makkai and Reyes, 1977; Fourman and Scott, 19791, states the equivalence between the category of sheaves and sheaf morphisms and some full subcategory of 0-sets (so called 'complete' 0-sets). It also turns out that the latter subcategory is 'representative', i.e. equivalent to the whole category of 0 sets [Goldblatt, 19841. Unfortunately, this result does not help for our purpose, because the 0-set isomorphisms used in this theorem are not logically faithful. So in this section we describe another construction showing that the logic of any m.v.s. can be presented as the logic of some presheaf. First, let us recall (Chapter 3) that every poset ( F ,R) corresponds to a category Cat(F, R), in which (u, v) is the unique morphism from u to v whenever uRv. For every locale 0 let Cat(0) := Cat(R, 2).
Definition 4.5.1 A presheaf (of sets) over a locale 0 is defined as a functor .F : Cat(0) -,SET.
This means that we have a family of sets (.F(u) : u E S1) and a family of functions (restriction maps)
where u, v E R, u
2 v,
and the following conditions hold:
(1) F ( u , u)= idF(,) (the identity function);
4.5. PRESHEAVES A N D S2-SETS (2) F ( v , w)0 3 ( u ,v ) = 3 ( u ,w ) , provided u
> v 2 w.
A presheaf F is called inhabited i f V { u I F(u)# 0 ) = 1. The set 3 ( u ) is called the (domain) of F at u ; its elements are called individuals (or sections) over u . The set of all the individuals
is called the total domain of 3.
-
Definition 4.5.2 A morphism f : 3 1 F2 of presheaves over fl is a functor morphism (natural transformation) i.e. a family of mappings
such that the following diagram commutes (provided u 1 v ) .
A n isomorphism is an invertible morphism, as usual; this is equivalent to bijectivity of every f,. Presheaves over fl and morphisms constitute a category Psh(S2). Definition 4.5.3 A presheaf 3 over S2 is disjoint if
Lemma 4.5.4 Every presheaf is isomorphic to some disjoint presheaf.
CHAPTER 4. ALGEBRAIC SEMANTICS
330 Proof
Almost obvious. Given a presheaf 3,let F 1 ( u ) := F ( u ) x {u) , 3 ' ( u , v) (a, u) := ( F ( u , v) (a), v) .
Then F' is a disjoint presheaf, and there exists an isomorphism f : 3 4 3' such that fu(a) = (a, u). In fact, f is obviously bijective, and the corresponding diagram commutes, since F1(u,V ) 0 fu : a
H
(a, u)
H
( F ( U ,v)(a),v)
and also
fv 0 3 ( u , v) : a
H
( F ( u , v)(a), v).
Thanks to the above lemma, we can assume that all presheaves are disjoint. In a disjoint presheaf 3 every individual belongs t o a unique 3 ( u ) ; the corresponding u is called the extent of a and denoted by lal.
Definition 4.5.5 If v 5 lal, the restriction of a t o v is
Definition 4.5.1 for disjoint presheaves can be reformulated as follows.
Lemma 4.5.6 Let F be a disjoint presheaf, a
a E , w 5 v 5 la[.
Then
(1) a I (la11 = a;
(2) (alv)jw = alw. The following simple lemma yields an equivalent definition of a morphism of disjoint presheaves as a map of domains preserving extents and commuting with restrictions:
-
Lemma 4.5.7 Let f : F1---+ 3 2 be a morphism of disjoint presheaves. Consider a map g : F: 3; such that
Then g has the following properties. (1) lg(a)l = 14;
(2) g(alv) = g(a) I v. The other way round, given a map g satisfying (I), (2), we can define a morphism f : F1---,3 2 by putting
4.5. PRESHEAVES A N D %SETS Proof
331 H
Straightforward.
Proposition 4.5.8 Let 3 be an inhabited disjoint presheaf over a locale 52, and for a , b E F* let
E ( a , b) =
V X (b).~ ,
Then the triple
E) H V ( 3 ) = (s2,F,
Proof The equality E ( a , b) = E(b, a ) is trivial, since X ( a , b) = X ( b , a). V { E ( a ,a ) I a E F*} = 1holds, because E ( a , a ) = la1 and 3 is inhabited. Since by Lemma 4.5.7,
X ( a , b) is >-stable, that is
So for any u E X ( a , b), v E X ( b , c ) we have (U
A
V)
E
X ( a , b) n X ( b , c ) C X ( a , c).
Hence it follows that
E ( a , b) A E ( b , c ) = V { u A v I u E X ( a , b),v E X ( b , c ) } 5
V X ( ~=,E ( a ,c). C)
Obviously, if the locale fl is basic, isomorphic to EO,for some S4-algebra E, and E is the same as in Proposition 4.5.8, then
Lemma 4.5.9 If F 1 , F2 are isomorphic disjoint presheaves over 52, then the corresponding H.v.s. H V ( 3 1 ) , H V ( F 2 ) are isomorphic; similarly for m.v.s. (and basic 52).
-
Proof Let X 1 , X 2 be the functions corresponding to F 1 , 3 2 described in Proposition 4.5.8. If g : 3; 3; is a bijective map satisfying Lemma 4.5.7 (i), (ii), then one can easily check that X l ( a , b) = X2(g(a),g(b)). Thus E l ( a , b ) = E 2 ( g ( a ) g(b)), , which means that g is an isomorphism of H.v.s. W
CHAPTER 4. ALGEBRAIC SEMANTICS
332
Definition 4.5.10 If 3 is an inhabited presheaf over a locale fl, we define its superintuitionistic logic (with or without equality) via the corresponding H.v.s.:
where 3; is a disjoint presheaf isomorphic to 3. If i-2 is basic, we also define the modal logic
These logics are well-defined. In fact, by Lemma 4.5.9, an isomorphism of disjoint presheaves T I , 3 2 gives rise to an isomorphism of the associated H.v.s. (m.v.s.), and thus the corresponding logics are equal, by Corollary 4.4.8.
Proposition 4.5.11 (1) For any H.v.s. G over a Heyting algebra i-2 there exists a disjoint presheaf G over i-2 such that IL'(G) = IL'(G).
(2) For any m.v.s. G over an S4-algebra f l there exists a disjoint presheaf G over f1° such that ML'(G) = MLZ(G). Proof We prove only (I), leaving (2) for the reader. For every u E i-2 let Go(u) = {(u, a ) 1 a E D, u I Eaa), and let z, be the following equivalence relation in Go(u): (u, a) zU(u, b)
($
u
< Eab
We denote the equivalence class (u, a)/ z, by [u,a]. Now we define the presheaf G as follows:
(**) G(u,v)([u,a]):= [v,a] (provided v
< u).
The map G (u, v) is well-defined, since [u,a] = [u, b] means that u < Eab, which implies v 5 Eab, i.e. [v, a] = [v,b]. Now let us show that IL=(G) = IL'(G). It is clear that (*), (**), really define a presheaf, and we may assume that G is disjoint. G is inhabited, since
Recall that IL=(G) = IL'(HV(G)), where HV(G) = (a,G*, E*) is the H.v.s., in which G* = U G(u) = {[u, a] I a E D, u 5 Eaa), uES2
4.6. MORPHISMS O F PRESHEAVES In particular, we have:
-
E * ( [ u ,a ] )= u A E ( a ) = u. Consider the map a : G* x D
R such that
a ( [ u a, ] ,b) := u A Eab. This map is well-defined, since if [ u ,a] = [ u ,a'], i.e. if u 5 Eaa', then u A Eab = u A Ea'b. Let us show that
(Definitions 4.4.1, 4.4.2).
is a p'-equivalence
(Ql) E(b)5
V
a ( c ,b), since E ( b ) = a ( [ E ( b )b], , b).
c€B*
( Q 2 ) E E * ( [ u a, ] ) 5
V
a([u,a ] ,b), since E * ( [ u a, ] )= u = u A E a a = a([u,a ] ,a).
bED
).(
E*([u',a'l,[u,al)~a([u,al,b)= u' A u A Ea'a A u A Eab 5 u' A Ea'b = a ( [ u l , a ' ]6). ,
(E')
EblbAa([u,a],b)= Eb'b A u A Eab 5 u A Eab' = a ( [ u ,a ] ,b'). Therefore IL'(G) = I L G ( H V ( G ) ) ,by Corollary 4.4.8.
W
C o r o l l a r y 4.5.12 Every algebraic semantics can be obtained from a class of presheaves. I n particular, the general algebraic semantics A& is generated by the class of all presheaves.
4.6
Morphisms of presheaves
L e m m a 4.6.1 Let 3 1 , F2 be two inhabited disjoint presheaves over a locale a, g : Fl -+ F 2 a morphism (in the sense of Lemma 4.5.7). Then g is a strong morphism frbm H V ( F 1 ) to H V ( & ) (in the sense of Definition 4.4.17). Proof Let H V ( F i ) = (0,F,*,E i ) . Then
CHAPTER 4. ALGEBRAIC SEMANTICS
334
(see Proposition 4.5.8). Since g is a morphism, we have:
So u E Xl(a, b) implies
and thus u E Xz(g(a), g(b)). Hence Xl(a, b)
C Xz(g(a), g(b)), and so
On the other hand, Xl(a, a) = {u l u
l lal),
and thus -&(a, a) = lal; similarly, E2(g(a),g(a)) = Ig(a)l; hence
El (a, a) = E2(g(a),g(a)) holds, and the proof is completed. So we obtain
Proposition 4.6.2 There exist functors HV: Psh(i2) --, a-Hvs for any locale a; MV: Psh(i-2) 4 a-mvs for any basic locale a , such that H V ( 3 ) , M V ( 3 ) are defined as above, and for a strong morphism f , HV(f) = MV(f) = f . The converse to Proposition 4.6.2 is in general false.
Example 4.6.3 Let F1and F2 be inhabited disjoint presheaves shown a t Fig. 4.1. Let g : FT -+ F; be the map such that
for x = v or w. Then g is not a morphism of presheaves. On the other hand, g is a strong morphism of the corresponding H.v.s. (Definition 4.4.17), since Ez(a, b) = u = El (a, alu).
4.6. MORPHISMS OF PRESHEAVES
Figure 4.1.
-
Proposition 4.6.4 Let F1 and F2 be two inhabited disjoint presheaves over a locale Every strong morphism f : H V ( 3 1 ) H V ( F 2 ) (Definition 4.4.17)
a.
is a morphism of presheaves if
satisfies the following condition:
-
-
Moreover, every strong H.v.s.-morphism f : H V ( F 1 ) + H V ( F 2 ) is a morphism of presheaves 3 1 3 2 iff for any strong H.v.s.-morphismg : H V ( 3 1 ) H V ( F 2 ) the following holds: (*) Qu > v Qa ~ g ( F l ( u ) ) QEb3 2 ( v ) (Ez(a,b) = v
* alv= b).
Proof (If.) Let f be a morphism of H.v.s. Let us show that f ( x l v ) = f ( x ) l v for x E F l ( u ) , v < u. Take a = f ( x ) , b = f(x1v). Then Ez(a,b) 2 E l ( x , x l v ) = v , and thus E 2 ( a ,b) = v and alv = b, by the condition (*) applied to f . (Only if.) Assume that every strong H.v.s.-morphism H V ( F 1 ) ---t H V ( F 2 ) is a morphism of presheaves F 1 32. Suppose there exists an H.v.s.morphism g : H V ( F 1 ) ---+ H V ( F 2 ) that does not satisfy (*), i.e. for some u > v , 5 E F 1 ( u ) , b E F ~ ( vwe ) have Ez(a, b) = v , alv # bfor a = f ( x ) E 3 2 ( u ) . By our assumption, g is a morphism of presheaves, and thus g(xlv) = alv. Consider the m'ap f : 3; -+ 3; defined as follows
-
( Y ) :=
{ gb ( y )
if y = xlv, otherwise.
Then f is not a morphism of presheaves, since f ( x l v ) = b, but f ( x ) l v = alv f b. On the other hand, f is a strong morphism of H.v.s. In fact, let us show that E l ( y , z ) E z ( f ( y ) ,f ( 2 ) ) for y, z E F;. It is sufficient to consider the case y f z = X ~ V .Then
<
CHAPTER 4. ALGEBRAIC SEMANTICS
336 since
Definition 4.6.5 Let 3 1 , F2be (inhabited) disjoint presheaves over a locale a. A pre-morphism from Fl to F2is a map g : F; --+ 3; preserving extents, i.e. such that (1) Ig(a)l = lal for any a . A pre-morphism g is called a pmorphism if it satisfies (3)
for a n y u E R , b E F 2 ( u )
A p'-morphism
(5)
is a p-morphism satisfying
i f a ~ F l ( u ) v, I u , then v = V{w 5 v I g(a)lw = g(alv)lw}.
A p-embedding is a p-morphism satisfying
(4).
Note that (2)
g(alu) = g(a>lu
implies (5); (3) holds for every surjective pre-morphism; (4) holds for every injective morphism. The definition of a p(')-morphism of presheaves is rather complicated, but it exactly corresponds to the definition of a p-morphism of H.v.s. (Definition 4.4.9). More precisely, the following holds. L e m m a 4.6.6
-
(1) A map g : 3;+ 3; is a p(')-morphism of presheaves zff g is a strong HV(32). p(')-morphism of corresponding H.v.s. g : HV(31)
-
(2) If there exists a p-embedding g : .Fl F2 (respectively, p'-morphism) of disjoint presheaves, then I L ( F I ) C IL(F2) (respectively, IL=(F1) C IL= (F2)).
(Cf. Lemma 4.4.7 stating the converse inclusion: IL(')(F~) Proof (i)
Let us check items (i)-(iii) from Definition 4.4.9. E l ( a ) = Ez(g(a)) @ lal = lg(a)l
@
(1).
G IL(=)(F~).)
4.6. MORPHISMS OF PRESHEAVES (ii)
E2(b) 5
V
Ez(b,g(a))
This condition (#) implies (5). In fact, let a E Fl(u), v 5 u. Take a1 = a , a2 = alv); then allv = a2lv Thus
(recall that Ig(a)l = u 2 v = lg(alv)1, by (1)). On the other hand, (5) implies (#). Indeed, assume that a1 E 31(ul),a2 E .F2(u2), u 5 u1 A u2, a l ( u = azlu. Take the sets
for i = 1,2. Then by (5),
(recall that then
is a complete Heyting algebra). And if v = wl A w2, wi E Xi,
Note that a p(')-morphism is not necessarily a morphism (because the condition (2) does not always hold). On the other hand, (1) and (2) implies (5); in fact, g(a)lv = g(a1v) = g(a1v)lv. Also note that every surjective pre-morphism is a p-morphism: the condition (3) holds if b = g(a) for some a E F:. Every isomorphism (i.e. a bijective morphism) is a p'-morphism; in fact, (4) follows from (2) and the bijectivity.
rn
CHAPTER 4. ALGEBRAIC SEMANTICS
338
4.7 Sheaves In this and the next section we look more closely at topological semantics. First let us show that in this case presheaves can be replaced with sheaves. Recall (cf. [Godement, 19581) that a sheaf (over a locale 0)is a presheaf 3 satisfying two conditions: (Fl)
V ui,
if u =
a,b E F ( u ) and (alui) = (blui) for all i E I , then a
= b;
iEI
(F2): if u =
V
ui, ai E 3 ( u i ) and (ailui A uj) = (ajlui Auj) for all i, j E I,
iEI
then 3a E F ( u ) Vi E I (alui) = a i . Every disjoint presheaf 3 over a topological space (W, 0)gives rise t o the canonical sheaf defined as follows. For x E W consider the set
with the equivalence relation (a =, b) := x E E(a, b ) . The equivalence class a, := (a/ =,) is called the germ of a at x. The set 3 ( x ) := (FO(x)/ r,) of all the germs at x is called the stalk (or the fibre) of 3 at x. We define F ( u ) as the set of maps from u to .F sending every x E u to some germ at x. More precisely, F ( u ) := {f E
3 ( x ) ( Vx E u 3v
C u 3a E F ( v ) (z E v & Vy E v f (y) = a,)),
XEU
and for u C U , let
r
F ( u ,v ) ( f ) := (f v), the restriction of the map f from u to v. The following fact is well-known [Godement, 19581: Proposition 4.7.1 is a sheaf; the family of mappings a, : F ( u ) ,,(a)
-
=f H
is a morphism of presheaves; a is isomorphic iff 3 is a sheaf.
F ( u ) such that
Vx E u f (x) = a,
4.8. FIBREWISE MODELS
For a E 3*let us denote (a). 6 := aIaI L e m m a 4.7.2 Let 3 be a presheaf over a topological space, sheaf. Let H V ( 3 ) = ( f l , ~E), , HV(F) = ( 5 2 , p , E). Then for any a , b E F E(a, b) = ~ ( 6b). ,
F its
canonical
P r o o f Let f l be the Heyting algebra of the given topological space. Then by Definition 4.3.4, x E E ( a , b)
3u E f l (x E u & u 5 la/ n lbl & alu = blu) 3u E 52 (x E u & u 5 la1 n 161 & Vz E u-a, = b,) 3u E fl (x E u & u 5 la1 n lbl & 6lu = blu) u 2€ ~(6~6). & &
-
L e m m a 4.7.3 Consider the same F, F as in the previous lemma. Let g : P" be the map such that g(a) = TL for a E F*. Then g is a strong p'-morphism H V ( 3 ) ---+ HV(F).
F*
P r o o f By Lemma 4.7.2 and because g is surjective (see remarks after Definition w 4.4.9). Proposition 4.7.4 IL'(3)
= IL=(?),
M L = ( 3 ) = ML=(F)
Proof Follows from the previous lemma, by Lemma 4.4.10 and Corollary 4.4.8; recall that in the modal S4-case p(')-morphisms are just the same as in the intuitionistic case. Corollary 4.7.5 7&is generated by the class of all sheaves over topological spaces. Question 4.7.6 Is the semantics A& generated by the class of all sheaves over locales (or complete modal algebras)?
4.8 Fibrewise models In this section we show that validity in a presheaf 3over a topological space can be defined via a forcing relation between points and formulas, similarly to Kripke semantics. Instead of Kripke models, we can use 'fibrewise models', which are collections of classical models on fibres of 3 . As in Definition 3.2.2, a system of domains over a non-empty set W (whose elements are called possible worlds, or points) is a pair (W, D ) , where D is a family of non-empty sets D = (Du)uEW. A classical valuation in a non-empty set V is a map [ from P L to relations on V such that J ( P ) Vn, whenever P E P L n (VOis treated as a certain singleton). A (modal) valuation in a system of domains (W, D) is a family (vu)uEW,where
CHAPTER 4. ALGEBRAIC SEMANTICS
340
7 7 2 ~ is a classical valuation in D, such that vu(p) E { u ) for every proposition letter p. A (modal) model over (W,D ) is a triple ( W ,D , v),where 7 is a valuation
in ( W ,D ) . Definition 4.8.1 Let 3 be a presheaf over a topological space (W,0).We associate with 3 the system of domains 3+:= (W,D ) with D, = F ( u ) (the fibre at u). A fibrewise model over F is a model over F f . Definition 4.8.2 For a fibrewise model M = (W,D , v ) we define the (modal) forcing relation M , u k A (or briefly: u != A ) between u E W and a Du-sentence A: M , u k P j iff u E % ( P i ) ; M , u k P p ( a ) i f l a € % ( P p ) (form>O); M , u b a = b iff a equals b;
M , u k 3 x A iff 3a E D, M , u != [ a / x ] A ; M , u k V x A iff Va E D, M , u k [ a / x ] A ; M , u k UA((al),, . . . , (a,),) iff there exist an open U such that u E U , U G l a l l n ... n\a,l a n d V v ~U M , v k A ( ( a l ),,..., (a,),). Let us check that forcing is well-defined. This has t o be done only for the last item. More precisely, we have t o show that i f L e m m a 4.8.3 Let M = (3, D,q)be a fibrewise model. Consider the algebra R as follows: 52 = M A ( W , 0)and define the map fj : AFF*
where F V ( A ) x, A ( a ) = [a/x]A. Then f j is a valuation on 3 and (*) is true for every A(a) E MFF*. P r o o f 6 is a valuation since
(here a, = b, for u E E ( a ,b), by the definition o f germs). T h e second claim is proved by induction on the length o f A. Let us consider two cases.
4.9. EXAMPLES OF ALGEBRAIC SEMANTICS
(i) w Efj(a= b ) @ w E E ( a , b ) @ 3 U 3 w ( a l U = b l U ) w a w = b , @ w a = b; (ii) w E .rj(UA(a))i f f 3U 3 w (U G fj(A(a))) iff3U3wVv~U (vkA(a)) w b U A ( a )i f f , & Vv E U ( v k A ( b ) ) ) 3U 3 w 3 b E 3 ( ~ ) ~ (=a bw
341
k
(*I (**>
I t is clear that (*)+(**). Conversely, assume (**). Since (a, = b,), there exists an open V U containing w such that (Vi ( a i l V ) = (bilV));thus (Vv E V a , = b,), and so v A(b,), b y (**). P r o p o s i t i o n 4.8.4 A modal formula is valid i n 3 iff it is true i n every fibrewise model over 3. P r o o f I t is sufficient t o show that every valuation cp i n 3 equals f j for some fibrewise valuation Q. In fact, define 77 b y
is well-defined, since a, = b, implies x E E ( a , b), and therefore u E cp(P(a)) i f f u E cp(P(b)),b y Definition 4.2.1.
Q
4.9
Examples of algebraic semantics
D e f i n i t i o n 4.9.1 A presheaf F over a locale 52 is called constant i f all its domains are the same (usually, non-empty) and all its restriction maps are identity functions. A constant presheaf over 52 with the domain D is denoted by C(52,D ) . A presheaf F has a constant domain (or briefly, is a CD-presheaf) i f every 3 ( u ,v ) is bijective (for u 2 v ) .
I t is clear that every constant presheaf is a sheaf. I t is also clear that every CD-presheaf 3 over C2 is isomorphic t o the constant sheaf C(52,3 ( 1 ) ) . One can easily see that for a CD-m.v.s. (52,D ) the corresponding presheaf constructed i n Proposition 4.5.11 is isomorphic t o C(52,D ) . Using these observations, we obtain: P r o p o s i t i o n 4.9.2 The following classes correspond to equal semantics: (1) the class of constant presheaves;
(2) the class of CD-presheaues; (3) the class of CD-m.v.s. (or H.v.s.).
R e m a r k 4.9.3 Note that HV(C(52,D ) ) # (Cl, D). In fact, the domain o f HV(C(52,D ) contains not only the individuals a E D with E ( a ) = 1, but also the restrictions alu such that E(a1u) = E ( a , a J u ) .
CHAPTER 4. ALGEBRAIC SEMANTICS
342
Thus we obtain the algebraic semantics with constant domains denoted by A. Historically, it was introduced earlier than the general semantics d& [Ra,'.Iowa and Sikorski, 1963, Ch. X, $15, Ch XI], [Takano, 19871. Similarly to Lemma 3.10.25 we have: L e m m a 4.9.4 The semantics 7 and A do not satisfy (CP). P r o o f Consider the Kripke sheaf Oo from the proof of Proposition 3.10.24. From the proof of Lemma 3.10.25 we know that ML(Q0) is an intersection of CK-complete logics. So it suffices to show that ML(Oo) @ A; in fact, it is easily seen that F != (C' V K t ) implies F I= C' or F k K' for every CD-m.v.s. F , similarly t o Proposition 3.10.24. Definition 4.9.5 A presheaf 3 is called rnonic if all 3 ( u , v ) (for u 2 v) are injective. An m.v.s. F = (0,D, E ) is called rnonic if E(a, b) E (0, (E(a) n E(b))) for all a, b E D;
(4.1)
similarly for H.v.s. L e m m a 4.9.6 If an m.v.s. F is rnonic, then the associated presheaf 3 (Proposition 4.5.11) is rnonic. P r o o f It is sufficient to show that [v,a] = [v, b] implies [u,a] = [u,b], whenever E ( a ) n E(b). But [v,a] = [v,b] (i.e. v 5 E(a, b)) only if E ( a , b) # 0, v u and so, E(a, b) = E ( a ) n E(b), and thus u E(a, b) by our assumption.
< <
<
L e m m a 4.9.7 Let 3 be an inhabited disjoint monic presheaf. Then the m.v.s. M V ( 3 ) is rnonic. P r o o f Assume that a, b E F*,E(a, b) # 0, u = la1 n Ibl. Then alv = blv for some v 5 u, and alu = blu (since 3 is monic). Thus E ( a ) n E(b) = u = U{v 5 u 1 (a/v)= (blv)) = E(a, b).
From Lemmas 4.9.6 and 4.9.7 we obtain: Proposition 4.9.8 The classes of rnonic m.v.s and rnonic presheaves generate the same semantics. This semantics is denoted by MA. Note that the semantics M K introduced in Section 3.4 is generated by rnonic presheaves over Kripke spaces. Let us state some simple logical properties of rnonic m.v.s, which will be used later on. Recall that C E denotes the formula (x # y) > U(x # y), DE denotes the formula (x = y) V (x # y).
4.9. EXAMPLES OF ALGEBRAIC SEMANTICS L e m m a 4.9.9 CE is valid in every monic m.v.s. Proof
For a valuation cp we have: cp(a # b) = ( E ( a ) n E(b)) - E(a, b),
so cp(a f: b) is open, being either E ( a ) n E(b) or 0. C o r o l l a r y 4.9.10 D E is valid in every monic H.v.s. 2.2.2).
H
(see 1.4 and Lemma
The converse is not true: an H.v.s. validating D E may be not rnonic. Corollary 4.9.11 A=' C C M ~ ( A if) A is an m.p.l., ASd C CMA(A) if A is an s.p.1. P r o p o s i t i o n 4.9.12 Let A be a propositional logic, Q A its quantified version (see. Section 2.4). Then CMA(QA) = Cd(QA). P r o o f Assume that an m.v.s. F = (a,D, E ) separates some sentence B E M F from Q A ; let us find a monic m.v.s. with the same property. Consider F' = (Q, Dl E'), where
It is clear that F and F' validate the same propositional formulas, that is Q A M L ( F r ) . There exists a valuation cp in F such that cp(B) # 1, and cp is also a valuation in F, since a # b implies cp(P(b)) n E'(a, b) = 0 5 cp(P(a)), and cp(P(a)) 5 E(a) = E'(a). Corollary 4.9.13 If the quantified version Q A of a modal propositional logic A is A-complete, then QA=' is a conservative extension of QA, i.e. Q A = (QAEc)" (cf. Section 2.9). We can also define corresponding restrictions of neighbourhood and Kripke semantics. Definition 4.9.14 I := A n I & (neighbourhood semantics with constant domains), M I := M A n I E (mono-neighbourhood semantics), MK: := M A n ICE (mono-Kripke semantics). Note that M I corresponds t o monic presheaves over neighbourhood frames.
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Chapter 5
Metaframe semantics 5.1
Preliminary discussion
In this chapter we study some further variations of Kripke semantics. In all these semantics we keep to the main idea that the truth value of a formula depends on a possible world and a formula O A is true at some world iff A is true at every accessible world.
Kripke bundles Let us first consider S4-frames. Our starting point is Kripke sheaf semantics. Recall (Section 3.5) that a Kripke sheaf over a propositional S4-frame F is a system of domains D = (D, I u E F ) , together with transition maps p, : D, D, parametrised by pairs (u,v) E R; p,,(a) is an 'inheritor' of an individual a E D, in the world v. In Kripke sheaves inheritors of different individuals can collapse in some accessible world. But can we give more freedom to individuals allowing them to have several inheritors in the same accessible world (perhaps, in their own)? Following this idea we come to Kripke bundles [Shehtman and Skvortsov, 19901, where transition maps p,, are replaced with 'inheritance' (or 'counterpart') relations p,, D, x D,; ap,,b is read as 'b is an inheritor (or a counterpart) of a in v' (or a v-inheritor of a). If the domains D, are disjoint, we can equivalently use the global inheritance relation p C D f x Df on the set of all individuals D+ and define p,, as p n (D, x D,).
-
Here we assume that the relation p is reflexive and transitive - which is analogous to the properties of transition maps from Definition 3.6.1. Models in Kripke bundles are defined in the same way as in Kripke sheaves. Forcing is defined by induction, but with a special care about the 0-case. The first idea is t o extend the definition of forcing from Kripke sheaves to Kripke bundles as follows. For a Du-sentence A put
(V)
M , u k CIA iff for any v E R(u) every v-version of A is true at M, v.
CHAPTER 5. METAFRAME SEMANTICS
Figure 5.1. An example of a Kripke bundle: p is the smallest partial ordering containing the indicated arrows; R = {(u,v)).
In Kripke sheaves v-versions of Du-sentences are obtained by replacing every occurrence of every a E Du with aJv, the unique inheritor of a in v. In Kripke bundles inheritors may be not unique, so we can define a v-version of A as a result of replacing every occurrence of every individual with some of its vinheritor. But this definition gives too much freedom, because then we can refute some theorems of QK= (and QK). To see this, consider the formula B := VxO(x = x) and the following Kripke bundle.
Figure 5.2.
5.1. PRELIMINARY DISCUSSION Then (for any model M )
because a = a has a false v-version bl = bz. Hence M, u !# B. The same argument applies t o the formula without equality
It is refuted at M, u for any model M such that M, v I=P(bl) and M, v +i P(bz). In fact, then P ( a ) > P ( a ) has a false v-version P(bl) > P(b2). In these examples undesirable v-versions of D,-sentences are obtained by replacing different occurrences of the same individual a with its different inheritors b l , b2. There is the simplest 'way out' - to forbid such a splitting of individuals within the same D,-sentence. So we adopt the following definition: (vv)
a v-version of a D,-sentence A s a D,-sentence obtained by replacing all occurrences of every a E D, in A with some its v-inheritor a'.
In other words, v-versions of A are of the form f . A for maps f : Const(A) ---+ D,, where Const(A) is the set of all constants from A, such that apf (a) for any a E Const(A). These maps are 'local analogues' of the maps p,, in Kripke sheaves; the latter are defined on the whole D,. It seems natural to write the condition (V) (which is analogue of (. . .)) in a form similar to Definition 3.6.4.
But such a 'definition' is ambiguous. In fact, as we know (Section 2.4) the same D,-sentence can be presented as A(a1, . . . ,a,) (= [al, . .. ,a,/xl, . . . ,a,] A) for different formulas A(x1, . . . , x,). For example, (a = a) = [alx] (x = x) = [a,a l x , y] (x = y). Now consider the same Kripke bundle model M as above. By applying (V') we obtain M , u i+ [a,a l x , Yl 0 ( x = Y), but
M, u b [alx]U(x = x). However the ambiguity of (V') disappears after fixing one of the presentations of a D,-sentence. For example, if we use only maximal generators,1 O(a = a ) is regarded as [a/x]O(x = x), so M , u l= (a = a) from this viewpoint. This 'Recall that a maximal generator is obtained by replacing all occurrences of a constant with the same variable, cf. Section 2.4.
CHAPTER 5. METAFRAME SEMANTICS
348
restricted version of (V') exactly corresponds to the combination of (V) and (VV). Although the use of maximal generators and the condition (VV) seem ad hoc, later on we shall see that this is sufficient for (and closely related to) soundness. There is another natural way to avoid ambiguity in definition (V') - instead of evaluating Du-sentences, we can evaluate formulas under variable assignments. This approach is well-known in classical logic. In full detail and in a more general context it will be considered in Section 5.9. Here we only sketch the main idea. The truth value of a formula A(x) with a list of parameters x a t a world u is defined under a 'finite assignment' (or a Du-transformation) [x H a], where a is a tuple in D, (cf. 2.4). Then (V') changes to
v''
M, u k OA(x) [x H a] iff Vv E R(u) Vbl .. . b, (Vi aip,,bi
=+ M, v iA(x) [x H b]).
Now the formulas 0 ( x = x), O(P(x) > P(x)) become true - in fact, e.g.
is equivalent to Vv E R(u) Vb(ap,,b+
M,vkx=x
[XH
b]).
But soundness still remains a problem. Viz., consider the same Kripke bundle on Fig. 5.2. Then according to (V"), the QK'-theorem
A : = (x = y > U ( x = y)) is refuted. In fact (for any model 111)
but M, u
lf' O(x = Y) [x,Y
a , a],
since
M , v lf' 0 ( x = Y) [x,Y
H
bl, b2]
A similar example exists in modal logics without equality. Consider the following QK-theorem:
which is a substitution instance of axiom (Ax13) from 2.6.10. This formula is refuted in the same Kripke bundle in the model M such that M , u k P ( x , x ) [x H a], M, v k P ( x , y) [x, y H bi, bj] iff i = j.
5.1. PRELIMINARY DISCUSSION In fact, M, u I= OP(x, x) [x H a], since M, u k P(x, x) [x H a], M , v != P ( x , x ) [x H b i ] . But M, u
F DP(x, Y) 15,Y
++
a , a],
since
M , v If P ( x , Y) 1x7 Y
b l , b21.
These counterexamples, together with the soundness theorem (Proposition 5.2.12), justify the truth definition stated in (V), (VV) or its equivalent (V') using maximal generators.
Functor semantics C-sets are another generalisation of Kripke sheaves, where we allow for many transition maps between the same possible worlds. This happens, because a quasi-ordered set is replaced with an arbitrary category C. More precisely, let C be a category with the set of objects W. Consider the frame F = (W, R), where uRv iff there exists a morphism from u to v, i.e. iff the set C(u, v) of morphisms from u t o v is non-empty. F is called the frame representation of C. A C-set is defined as a functor from C to SET; it can be regarded as a system of domains (F,D) together with a family of functions
respecting composition and identity, cf. Definition 3.6.1. Obviously, Kripke sheaves over F are nothing but CatF-sets. In a C-set, the v-inheritors of an individual d E D, are pad for all a E C(u, v). Given a model M in a C-set (F, D, p), we define forcing, so that
M, u UB(dl, . . . , d,) iff Vv E R(u) Vcr E C(u,v) M , v
t=
B(p,dl,.
. . ,padn).
Similarly to Kripke sheaves, in C-sets we may not worry about different presentation of a D,-sentence as B(d1,. . . ,d,), because p, are functions. A vinheritor of a D,-sentence is obtained by replacing all individuals a E Du with their images p,a under the same transition map pa, cr E C(u, v).
Validity Validity in a Kripke bundle or in a C-set is defined again as the truth in all models. But there is another problem: the set of valid formulas may be not substitution closed! To show this, consider the category C with a single object2 21.e. its frame representation F is a reflexive singleton.
CHAPTER 5. METAFRAME SEMANTICS
350
u and two arrows: id and y such that y o y = y (and certainly, id o y = y 0 id = y; C is just a two-element monoid). Let F be a C-set in Fig. 5.3 with Du = {a,b), p,(a) = P,@) = b.
Figure 5.3. Then the formula p > Up is valid in F (since F k p 3 Up) OP(x) is not. In fact, consider a model in which u P(a),u u UP(a), because u P(p,a).
, but
P(x) 3 P(b); then
Exercise 5.1.1 Let us call a formula A intuitionistically valid in a C-set (or in a Kripke bundle) if AT is valid. Show that the formula p V ~ is p intuitionistically valid in the same C-set as above, whereas P(x) V l P ( x ) is not intuitionistically valid. Exercise 5.1.2 Consider the Kripke bundle (6 with the single domain D, = { a , b ) such that p,, = {(a, a), (b, b), (a, b)), see Fig. 5.4. Show that the formula P ( x ) 3 UP(x) is not valid and P ( x ) V ~ P ( x is) not intuitionistically valid in G. So it turns out that the set of all formulas (modal or intuitionistic) that are valid in a given Kripke bundle or a C-set, is not necessarily a predicate logic. To obtain a sound semantics, we replace validity with the notion of strong valzdzty - a formula is said to be strongly valid if all its substitution instances are valid. At first glance, two new semantics seem independent. But actually the semantics of C-sets is stronger. Furthermore, their natural combination (the 'C-bundle semantics') is strongly equivalent to C-sets. C-bundles are defined analogously to C-sets, with the following difference - we replace functions p, : D, D, with relations p, C D, x D, parametrised by morphisms a E C(u, v); the clause (V') is modified as follows:
-
('+)
M, u OA(a1,. . . ,a,) iff VU E R(u) Vy E C(U,U)Vbl,. . . , b, (Vi aip,bi
i
M,u
A(b1,. . .,b,)) ,
5.2. KRJPKE BUNDLES
Figure 5.4. with the same requirement about A. It turns out that for any C-bundle there exists a C-set with the same valid formulas; therefore every Kripke bundle also corresponds to some C-set. This is not very surprising, because 'functionality' of C-bundles is hidden in the fact that v-versions of A are of the form f . A for 'local functions' f , cf. (VV).
5.2
Kripke bundles
Now let us turn to precise definitions for the polymodal case.
Definition 5.2.1 A n (N-modal) Kripke bundle over an N-modal propositional frame F = (W,RI, . . . ,R N )is a triple IF = (F, D, p), in which D = (D, I u E F) is a system of domains over F and p = (pi,, I uRiv, 1 5 i 5 N ) is a family of relations pi,, C_ D, x D, such that for any i , u , v
i.e. dom(pi,,) = D,; this means that every individual has inheritors in all accessible worlds. The frame F is called the base of F,and the domain D, the fibre at u. For the 1-modal case we use the notation p,,
rather than pl,,.
Definition 5.2.2 A n intuitionistic Kripke bundle is a 1-modal Kripke bundle (F, D , p) over an S4-frame F = (W,R) satisfying the following two conditions: (#2)
every p,,
is reflexive;
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Definition 5.2.3 A n intuitionistic Kripke bundle (F, D, p) is called a Kripke quasi-sheaf i f every p,, is the identity function o n D,, i.e. i f V a , b E D, (ap,,b
a = b).
Obviously, every Kripke sheaf is a l-modal Kripke bundle, in which all p,, are functions. It is also clear that in the intuitionistic (and in the S4-) case every Kripke sheaf is a Kripke quasi-sheaf. The other way round, Lemma 3.6.3 shows that every Kripke bundle, in which pi,, are functions satisfying 'coherence conditions', gives rise to a Kripke sheaf. In a sense, Kripke sheaves are exactly Kripke bundles of this kind. As mentioned in the previous section, a Kripke bundle can be presented in another equivalent form.
Lemma 5.2.4
IF = (F,D,p) be a Kripke bundle, in which the domains D, are disjoint.3 Consider the total domain
(1) Let
with relations Pi := U{pauv I U&V)
for 1 5 i 5 N . Then we obtain the propositional frame of individuals (also called the total frame of IF): Fl = (Dl, ply.. . PN).
-
Let .rr : D1 W be the map such that ~ ( a=) u for a E D,. Then .rr is a p-morphism Fl + F . Moreover, for N = 1, IF is an intuitionistic Kripke bundle, iff Fl is an S4-frame. (2) Conversely, a p-morphism of modal frames T
: Fl =
( W rpi,. , . . ,p ~ -H) F = (W, R1, . .. ,R N )
gives rise to a system of disjoint domains D, = .rr-l(u) for u E W and a family of relations pi,, = pi n (D, x D,) for u R i v , 1 5 i 5 N , which forms a modal Kripke bundle. For the case N = 1, the corresponding Kripke bundle is intuitionistic iff both Fl,F are S4-frames. Sometimes it is convenient to denote the base F = (W, R1,. . . , RN) of a Kripke bundle IF by F o = (DO, R:, . . .,R;) and the frame of individuals Fl by (Dl, Ri, . . . , Rh). Thus we come to the following equivalent definition of Kripke bundles: 3As in the case of Kripke sheaves, cf. Section 3.5.
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Definition 5.2.5 A Kripke bundle (modal or intuitionistic) is a p-morphism between two propositional frames (resp., modal or intuitionistic).
The choice between two equivalent definitions o f Kripke bundles is a matter of convenience. I t is clear that quasi-sheaves correspond t o pmorphisms o f S4-frames n : Fl Fo such that Fl 1 n-'(u) is a discrete frame, i.e. a quasi-sheaf is a Kripke bundle with discrete fibres.
-
Exercise 5.2.6 Show that Kripke sheaves correspond t o coverings, i.e. t o Kripke bundles with the unique lift property:
n ( a ) R v + 3!b(apb & n(b) = v). Definition 5.2.7 A valuation E i n a (modal) Kripke bundle F = ( F ,D,p) is just a (modal) valuation i n its system of domains D = ( D , I u E W ) , cf. Definition 3.2.4. The pair M = (IF,c) is called a Kripke bundle model.
The inductive definition o f forcing is more delicate than for Kripke frames or Kripke sheaves, because (as explained in Section 5.1) in the clause for we should explicitly indicate all the individuals occurring in A and use the presentation of D,-sentences described in Lemma 2.4.4. Forcing in Kripke bundle models is quire similar t o Kripke frames or Kripke sheaves; the only differenceis in the 0-clause: Definition 5.2.8 Let IF = ( F ,D , p) be a Kripke bundle, F = (W,R 1 , . . . ,R N ) , M = (IF,[) a Kripke bundle model. We define the forcing relation M , u k A between worlds u E F and Du-sentences (with equality) by induction:
M , u k Pf iff u E Eu(P,"); M , u t= P p ( a ) i f fa E J u ( P p ) (for m > 0); M , u != a
=b
iff a equals b;
M , u k B V C iff ( M , u k B or M , u k C);
M , u k B > C iff ( M , u f B o r M , u k C ) ; M , u != 3 x A iff 3a E D, M , uk [ a / x ] A ;
M , Z Lk Cii [ a l , .. . ,a,/xl,. . . , x,] Bo i f f VV E Ri ( u )Vbl, . . . , b, E D, ( V j a j piuubaj M , v k [bl,...,b,/xl,. ..,x,]Bo) where F V ( B o )= { X I , . . ,x,) and a l , .. . ,a, E D, are distinct.
*
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So in the latter case we present a D,-sentence O i B as Oi[a/x]Bofor a formula
Bo and an injective [ x H a ] , recall that Bo is called a maximal generator of B and it is unique up to renaming of x. This clause can be equivalently presented in the form discussed in 5.1:
(V)
M , u b O A iff for any v E R ( u ) every v-version of A is true at M , v.
In particular, for a sentence B we have:
M , u k O i B iff V V E R i ( u ) M , v k B , exactly as in standard Kripke semantics. Definition 5.2.9 A (modal) predicate formula is called true i n a Kripke bundle model if its universal closure V A is true at every world of this model. A formula is called valid i n a Kripke bundle IF if it is true i n every model over IF. As above, the sign k denotes the truth in a model and the validity in a Kripke bundle. The following is trivial (cf. Lemma 3.2.20(1)): Lemma 5.2.10 For a Kripke bundle model M and a modal formula A ( x 1 , . . . , x,) M A ( x l ,...,x,) ifl V u E M V a l ,..., a, E D , M , u k A ( a l , ..., a,).
+
Recall that A(a1, . . . ,a,) denotes [a1, .. . ,a,/xl, . . . ,X , ] A ( X I ,. . . ,xn), cf. 2.4. Consider the set of valid formulas
This set is closed under MP, Gen and 0-introduction, cf. 3.2.27. But it may be not substitution closed (cf. Exercise 5.1.2), so we introduce the following notion. Definition 5.2.11 A formula A E M F ~ = ) is called strongly valid i n a Kripke bundle F (respectively, strongly valid with equality) i n F if all its MFN (respectively, MFG) -substitution instances are valid i n IF; notation: IF k+ A (resp., IF k+= A). Let Then we have Proposition 5.2.12
(2) The strong validity of a formula A follows from the validity of all its mshifts Am, i.e.
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Speaking informally, Proposition 5.2.12(2) means that A is strongly valid iff it is valid with arbitrarily many extra parameters. The proof is postponed until Section 5.13. More exactly, this is a particular case of a general soundness theorem for metaframe semantics, cf. Section 5.12. However, let us sketch a general plan of the proof. At the first stage we prove (2). Recall that substitution instances of a formula A are the strict substitution instances of its m-shifts Am. So it is sufficient t o check the following
Claim. ML:=)(IF) is closed under strict substitutions. The proof is very similar to the substitution closedness of Kripke sheaves (Proposition 3.6.17) in the particular case of strict substitutions. In the proof of (I), the strong validity of axioms, due to (2), follows from the validity of their shifts. The latter is proved in a straightforward way. For propositional axioms it is easier to use proposition 5.3.7 (see below), which in its turn, follows from (2). The substitution closedness of ML(')(F) is obvious. The closedness under M, Gen, 0-introduction follows from the same property of MLI=)(P) (again,due to (2)). For example, for MP note that (A > B)n = An > Bn, so Bn E ML?)(F) whenever An, (A > B)n E ML!=)(F). Thanks to 5.2.12(1), we may call ML(")(P) the modal logic of F. Also note that ML(')(P) is the largest modal logic contained in ML(=)(F) (so to say, the substitution interior of ML(=)(F)). Let us point out again that for Kripke sheaves the notions of validity and strong validity are equivalent, because the set of valid formulas is already substitution closed. Exercise 5.1.2 shows that this is not always the case in Kripke bundles.
Exercise 5.2.13 Give an example of a 1-modal Kripke bundle, in which p,, are functions, but the set of valid formulas is not substitution-closed. We also remark that the notion of strong validity a priori depends on the language. In fact, it might happen that IF l= SA for any MFN-substitution S , but not for any MFG-substitution. But 5.2.12(2) shows that this is impossible, so ML'(F) is conservative over ML(F).
Definition 5.2.14 For a class C of Kripke bundles we define
Definition 5.2.15 Kripke bundle semantics K B N is generated by the class of all N-modal Kripke bundles. KBN-complete logics are called Kripke bundle complete.
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Definition 5.2.16 (Cf. Definition 3.2.32) If F is a propositional Kripke frame, KB(F) denotes the class of all Kripke bundles based on F . The logic NIL(') KB(F)) is called the Kripke bundle modal logic (with equality) determined over F. Similarly we define KB(C) for a class of propositional frames C and the Kripke bundle modal logic determined over C.
5.3
More on forcing in Kripke bundles
Now let us present an arbitrary Kripke bundle as a collection of propositional Kripke frames. This presentation is more convenient from the technical viewpoint, and it will lead us to a more general notion of a metaframe.
Definition 5.3.1 Let D = (D, I u E W) be a system of disjoint domains over a propositional modal frame F = (W,R1,. ..,RN). For n 2 0 we define the nth level of D as the set Dn := U{DE I u E W). We also identify DA and D, (strictly speaking, they are not equal), so that our new definition of D1 corresponds to that in Lemma 5.2.4. Also note that Do = W, since D: = { u ) , see Definition 3.2.4. Note that every valuation E = ([u)uEW in a system of disjoint domains is associated with the following function [+ sending n-ary predicate letters to subsets of Dn: CU(P2). E+(PC) :=
u
uEW
In particular, we have
c
<+(pi) W. Recall that for any function 8 such that 8(P;) C_ Dn, there exists a valuation [ such that 8 = [+: tU(p;) := e(p;) n D;. To define accessibility relations on the sets Dn, we first recall the notation from Lemma 5.2.4: Pi := U { ~ i u IvuRiv). Recall the subordination relation between n-tuples (see Introduction):
a sub b := V j , Ic ( a j = ak
* bj = bk).
Definition 5.3.2 Let IF = (F,D, p) be a modal Kripke bundle over F = (W, R1, . . . ,RN). For n > 0 , 1 5 i 5 N , we define the relation on Dn: aRFb iff V j ajpibj & a sub b.
So i n particular, Rf = pi. For n = 0 we put RP := Ri . The frame Fn := (Dn,R;, . . . , R;) is called the nth level of IF
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Exercise 5.3.3 Show that if pi is transitive, then R7 is also transitive. Again in the 1-modal case we write Rn rather than RT. Now the inductive clause for UiB from the definition of forcing can be rewritten in the following form. L e m m a 5.3.4 Under the conditions of Definition 5.2.8, let B be an N-modal formula with FV(B) r(x), 1x1 = n. Then for any u E F and a E DZ M, u k= OiB(a)
iff
(*) Vv E Ri(u) V b E D,"(aR7b + M, u k= B(b)) where as usual, B(a) denotes [a/x]B . variables Proof ('Only if7.) Suppose M, u k OiB(a). As explained after Definition 5.2.8, this means that every v-version of B(a) is true a t M, v (for any v E &(u)), and u-versions are obtained according to (VV). To check (*), let us show that for a R l b , B(b) is a v-version of B(a). In fact, every occurrence of ak in B(a) either replaces an occurrence of xk in B or replaces an occurrence of x j in B for j # k - if a j = ak. In the first case an occurrence of ak in B(a) corresponds to an occurrence of bk in B(b). Similarly, in the second case an occurrence of a j = ak in B(a) corresponds to an occurrence of bj in B ( b ) , and bj = bk, since aRqb. ('If'.) Suppose (*), and consider a presentation of B(a) as [c/x]Bo with a distinct c. Then Bo is a maximal generator of B(a) = [a/y]B(y), so by 2.4.5(2), Bo is obtained by a variable substitution from B , i.e. Bo = [x - a / y ] B for some a E Cmn, where 1x1 = n, lyl = m. To show that M , u k Oi[c/x]Bo, consider v E Ri(u) and d E D: such that Vk ckpidk. Then
= Also (c . a ) sub ( d . a ) , since c,(~)= c,(j) implies a(i) = a ( j ) and thus d,(j) (remember that c is distinct). Therefore (c . g )R y ( d . a); hence M, v I= [ d . a / y ] B by (*), i.e. M , v != [d/x]Bo by (**). Eventually M , u != Oi[c/x]Bo by Definition 5.2.8. W
Since the world u is fully determined by a tuple a E Dn, we can drop u from the notation of forcing and just write M k A(a) (or M k [a/x]A, to be more precise). Now (*) becomes almost the same as in the propositional case: M t= OiB(a) iff V b (aR7b +-M i= B(b)). Since by definition of R7, a is subordinate to b, the truth value of UiB(a) is well-defined for any tuple of individuals a = (al,. . . , a n ) in the same world; some of ai's may coincide.
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So IF corresponds to the sequence of propositional frames Fn, in which Fo is the basic propositional frame. A Kripke bundle model M = (IF,[) is then associated with the family of propositional Kripke models (Mn)nEw, where Mn = (Fn,En) and for any k, tn(P;) := E+(Pt). Fn (respectively, Mn) is called the n-level of IF (respectively, M ) . This presentation of Kripke bundles allows us to describe the propositional fragment ML~=)(F)of the predicate logic ML(')(F). L e m m a 5.3.5 Let M = (IF, E) be an N-modal Kripke bundle model, and let Mn = (F,, En) be the n-level of M . Then for any N-modal propositional formula A and for any a E Dn M n , a l= A iff M M An(a). Proof Easy, by induction on the length of A. The case A = UiBreadily follows from Lemma 5.3.4. W
L e m m a 5.3.6 Let IF be an N-modal Kripke bundle, A a propositional N-modal formula. Then for any n 2 0
Proof
Lemma 5.3.5 implies that for any model M over IF,
In fact, note that by 5.2.10,
iff Mn M A by 5.3.5. Hence IF k An iff b'E (IF, J) b An iff
(Fn, En) k A.
Finally note that every valuation 8 in Fn is En for some valuation 6 in IF. In fact, one can put tf (PF) := O(P:) for any k and define arbitrary values of t+ Fn b A. on other predicate letters. Hence V( (Fn, En) k A Proposition 5.3.7 For an N-modal Kripke bundle IF and an N-modal propositional formula A, IF l=+ A ifl b'n Fn l= A, i.e.
Proof
Follows from Lemma 5.3.6 and Proposition 5.2.12.
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5.4
359
Morphisms of Kripke bundles
D e f i n i t i o n 5.4.1 Let IF = (F,Dl p), G = (G,D', p') be Kripke bundles. A pair f = ( f o ,f l ) is called an equality-morphism (briefly, =-morphism) from F to G (notation: f : F -= 6) if (1) fo : F
(2) f l : Fl
--
G is a morphism of propositional frames, G1 is a morphism of propositional frames,
(3) f l is a fibrewise bijection, i.e. every flu := fl 1 D, is a bi3ection between DtL and D;o (.).
fo is called the world component, f l the individual component of ( f o ,f i ) . A n =-morphism ( f o ,f l ) is called a p=-morphism if fo is surjective (and thus, a p-morphism); an isomorphism i f fo, fl are isomorphisms.
I t is clear that Definition 3.3.1 for Kripke frames is a particular case o f 5.4.1. From the definition one can easily see that the following diagram commutes (where .rr, .rr' are the pmorphisms corresponding respectively t o F, 6). fl
Exercise 5.4.2 Show that in Definition 5.4.1 the condition ( i ) follows from (ii) and (iii). Exercise 5.4.3 Show that i f ( f o ,f l ) is an =-morphism and fo is an isomorphism, then fl is also an isomorphism.
--
L e m m a 5.4.4 I f f : F -+= F' is an =-morphism of Kripke bundles, then the map fn sending a to f - a is a morphism Fn FA. Proof B y definition, the statement holds for n = 0 , l . Let us show that fn respects subordination. In fact, suppose a sub b; then f l ( a i ) = f l ( a j ) implies ai = aj (since f is a fibrewise bijection), and hence bi = bj (due t o a s u b b), f l ( b i ) = f l ( b j ) . Thus ( f . a)sub(f . b). T h e other way round, ( f . a ) sub ( f . b) implies a s u b b; the reader can show this i n a similar way. Now the monotonicity o f fn ( n > 1 ) follows easily. In fact, i f aREb, then for every i , ai R k bil and thus f 1 ( a %R)L f 1 ( b i ) , which implies ( f . a )Rin ( f . b ) ,since f respects subordination.
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To show the lift property for fn ( n > l), assume ( f . a)Rknc. Then for every i, fl(ai)Rici, and thus by the lift property of f i , there exists bi such that aiRkbi and ci = fi(bi). Put b := (bl,. . . ,b,); then asubb. In fact, since f is a fibrewise bijection, we have
ai
= aj
iff f l ( a i ) = f i ( a j )
and
bi = bj i f f c i= c j . Since ( f . a ) subc, we also have
Thus ai = aj implies bi = bj. So we obtain aRFb and fn(b) = c.
Definition 5.4.5 Let f = ( f o , f l ) : F -= (6. For Kripke bundle models M = ( I F , [ ) , MI = (IF',[') we say that f is an =-morphism from M to M' (notation: f : M -= MI) if
M , u i= P ( a ) for any P E PLn, a
E
MI, fo(u) t= P ( f 1 . a).
D:, n > 0; or in another n ~ t a t i o n : ~
a E <+(P)iff ( f i.a) E < I f ( p ) , and also M , u I= P
++
M', fo(u) k P
for any P E PLO Then we obtain an analogue of Lemma 3.3.11
L e m m a 5.4.6 If ( f o , f l ) : M -+= M 1 then for any u E M and for any D,-sentence A M , 21 + A iff M r ,fo(u) != f i . A, (*) where f l - A is obtained from A by replacing every c E D, with f l (c). D,Proof By induction. Let us consider only the 0-case, i.e. A = O i B ( a ) ,where a E DZ. We also assume that n > 0; the easy case n = 0 is left for the reader. So suppose M , u !#OiB(a), a E DZ, and let us show that
We have v E Ri(u), b E D,R such that M , v !#B ( b ) and a R l b . By the induction hypothesis, we obtain M', f o ( v )y B ( f l . b). We also have ( f l . a ) R ; ( f l . b) by Lemma 5.4.4, and f o ( ~ ) Rfo(v), i since fo is a morphism. Hence MI, fo(u) !#Di B ( f 1- a). 4Cf. Section 5.3.
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For the converse, suppose M', fo(u) l+OiB(fl a). Then there exist w and c E DE such that (fi - a ) R l c , fo(u)Raw, M', w Cj B(c). By Lemma 5.4.4, c = fi - b for some b E R l ( a ) , and (say, from alRibl) it easily follows that uRiv, where v is the world of b. Hence M, v Cj B(b) by the induction hypothesis, and thus M, u CjOiB(a). L e m m a 5.4.7 Let (fo, f l ) : F -= G, and let M' be a Kripke bundle model over 6. Then there exists a model M over F such that (fo, f l ) : M ---+= MI. Proof If M' = ((6, $I)one , can take
for P E PLn, n
> 0, and
for P E PLO. Hence we obtain an analogue of Proposition 3.3.13. Proposition 5.4.8 If there exists a p=-morphism f : F MLT(IFf) and thus, NIL' (F) ML=(F1).
c
*=
IF', then M L I ( P )
Proof Let (fo, f l ) : IF = (F, D, p) -w= P' = (F', Df,p'), and assume that IFf A(x1,. . .,xn). Then for some u' E F', a l , . . . , a n E DL,, for sofne model M' over IF' we have MI, uf If A(atl, . . . ,a;).
By the previous lemma, there exists a model M over IF such that
Since fo and all
flu
U'
are surjective, there exist u, a l , . . . ,a,, such that = fo(u), a', = fl,(al), . . . , a; = fi,(an),
and thus by Lemma 5.4.6, we obtain M, u Cj A(al,. . . , a n ) . Therefore, IF' Cj A implies P t+ A.
A particular case of an =-morphism is an inverse image morphism obtained by the pullback construction as follows. L e m m a 5.4.9 Let IF = (F, D, p) be an N-modal Kripke bundle, f : G morphism of propositional frames, and put
-
Fa
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where n : Fl follows.
-w
Fo corresponds to IF. Consider the relations on X defined as (u', a)p:(vr,b) i f f u'Riv' & apib,
where pi and Ri are the relations respectively in F 1 and G. Let also n1(u',a ) := u', cp(ul,a ) .- a. Then (1) n1 : ( X ,pi,
.. . ,p k ) + F';
(2) ( f , cp) : G -= Proof
IF, where
(6
corresponds to n'.
In all the cases the monotonicity is obvious.
To check the lift property for .TI, suppose u l R ~ v ' Then . f (ul)&f (v') by rnonotonicity, and thus there exists b such that apib and n(b) = f (v'). Hence (v', b) E X , ~ ' ( v b) ' , = b, (u',a)p:(v1,b) as required. It is clear that cp is a fibrewise bijection; in fact,
a E D f ( u ~iff) ~ ( a=) f (u')iff (u', a ) E X iff ( u l , a )E DL,. Checking the lift property for cp is left t o the reader.
W
For the Kripke bundle G constructed in Lemma 5.4.9 we shall use the notation f,IF and say that it is obtained by change of the base along f . A particular case of this construction is a generated subbundle. Explicitly it is defined as follows.
Definition 5.4.10 Let F 1 V be a generated subframe of a propositional frame F, IF = ( F ,D , p) a Kripke bundle over F . Then we define
IF 1 V
:= (F', D',
where F' = F 1 V , D' = D 1 V , p' is an appropriate restriction of p, i.e. piuv = piuv for ( u ,v ) E Ri n ( V x V ) . IF 1 V is called a generated subbundle of IF (more precisely, the restriction of IF to V ) .
5.4. MORPHISMS OF K R P K E BUNDLES
-
Lemma 5.4.11 Let j : F 1 V F be the inclusion morphism, cp : D" the inclusion map. Then (j, p) : IF [ V --t=IF. Proof
363 + D1
An easy exercise.
Exercise 5.4.12 Show that IF f V is isomorphic to j,P. In particular, if F f V is a cone F T u , then IF 1 V is also called the cone of IF and denoted by IF t u . Lemma 5.4.13
Proof Follows readily from Lemmas 5.4.6 and 5.4.7; an exercise for the reader (cf. the proof of Lemma 1.3.26). Lemma 5.4.14 Let IF be a Kripke bundle, IF f V its generated subbundle. Then
Proof Follows for the previous lemma and the observation that every cone in
IF 1 V is a cone in IF. Finally let us show that G = f,IF gives rise to the pullback (or 'coamalgam') diagram:
In precise terms, this means the following Lemma 5.4.15 Let f : G -+ F be a morphism of propositional frames, IF a Kripke bundle over F, T : Fl + F the associated p-morphism, P = f,P and (f, cp) : (6 += F the =-morphism described i n 5.4.9. Also let ( f , $ ) : W += P be an arbitrary =-morphism from a Kripke bundle W over G to IF. Then (1) there exists a unique map g : G1 -+ Fl such that the following diagram (where T is associated with W) commutes.
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(2) g is a n isomorphism between H1 and G I . Proof (i) The diagram commutes iff for any x E H, xl(g(x)) = ~ ( x and ) p(g(x)) = $(x), i.e. iff g(x) = (T(x),$(x)); note that (T(x),$(x)) E G1 since .rr . $ = f . T . SO g is well-defined. (ii) We can consider the category P of Kripke bundle morphisms of the form (f, $) : W -+ IF, where IHI is an arbitrary Kripke bundle over G. A morphism (in P ) from (f, $) : W -+F to (f,$I): W' -+ P is defined as a map h, for which the following diagram (where T' : Hf -+ G is associated with H') commutes.
The assertion (i) shows that our specific (f,p ) is a terminal object of P. But actually the above argument applies to any (f, p) : (6 + IF from P. In fact, we only need to find a unique y E GI such that xl(y) = T(X)& p(y) = $(x). So it suffices to show that
. this holds since p is a fibrewise bijection between whenever f (u) = ~ ( a ) But x'-' [u] and .x-' [f (u)].
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Thus all objects in IJ are terminal, and all morphisms are invertible. So the map g from (i) is bijective. To complete the proof of (ii), it remains to show the monotonicity of g; the monotonicity g-l follows by the same argument. So let Si be the relations in H I . Then for any y, z E H I
and similarly ysiz
+-r1(g(y))R:r'(g(z)).
Thus ysiz =+ ~ ( Y ) P : ~ ( z ) by definition (see Lemma 5.4.9).
5.5
Intuitionistic Kripke bundles
Now let us consider the intuitionistic case. First note the following: L e m m a 5.5.1 If F is an intuitionistic Kripke bundle, then all the F, are S4kames. Proof
An exercise.
H
Definition 5.5.2 A valuation 5 i n an intuitionistic Kripke bundle F is called intuitionistic i f every set J + ( P r ) = lJ{J,(G) I u E W} is stable i n F,, i.e. i f Rn([+(P2)) J+(PF). The model (F, J ) is also called intuitionistic i n this case. Intuitionistic forcing in intuitionistic Kripke bundle models is defined via modal forcing similarly to the case of Kripke frames (Definition 3.2.12): Definition 5.5.3 Let M be an intuitionistic model over an intuitionistic Kripke bundle F = ( F ,D , p ) , F = (W, R), u E M , A an intuitionistic D,-sentence. Then we put M,uIt-A:= M , u ~ A ~ . Now we obtain an analogue of Lemma 3.2.13: L e m m a 5.5.4 Under the conditions of Definition 5.5.3 (1) M, u IF B iff M, u b B (for B atomic);
(3) M , u I t B V C iff (M,uIk B o r M , u I F C ) ;
(4) M , u It- B(a) > C(a) ifl Vu 'v R(u) Vb E D; (aRnb & M, v IF B ( b ) + M, v IF C(b));
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(5) M , u I t 3 x B 283a E D, M , u It- [alx]B ;
(6) M , u I t V x B ( x ,a ) iff Vv E R ( u ) Vd E D, V b E Dz (aRnb =+ M , v I t B(d, b ) ) ;
(8) M , u I t a # b i 8 a does not equal b.
Here we assume that in all cases we evaluate D,-sentences at u. In (4), (6) and (7) we also assume that a E D t is a list of individuals, which replaces a list of variables containing the parameters respectively of B > C or V x B , B . Let us consider only the case (4);the others are an exercise for the reader. In the fixed model M we have
Proof
u I t B ( a ) > C ( a )iff u I= ( B ( a )> C ( a ) ) T= o ( B ( > ~C)( a~) T )by Definitions 5.5.3, 2.11.1 iff Vv E R ( u )V b E D t (aRnb + v != ~ ( b> C ) ( b~) T )by Lemma 5.3.4 iff Vv E R ( u )V b E D; ( a R n b & v It B ( b ) v It- C ( b ) )by Definitions 5.5.3, 5.2.8.
As in the modal case, we can drop u from the notation; so e.g. (4), (6) are written as follows: (4)
M I t ( B 3 C ) ( a )iff V b ( a R n b & M It B ( b )
(6)
M It- V x B ( x , a ) iff VdVb ( ( d b )E Dn+' & aRnb + M It- B ( d ,b)).
M It- C ( b ) ) ;
Now we obtain some facts similar to those proved in Section 3.2.
L e m m a 5.5.5 Let M be the same as in Lemma 5.5.4. sentence A ( a ) M IF A(a) & a R n b + M I t A ( b ) . Proof
Then for any D1-
By Definition 5.5.3 and Lemma 5.3.4.
Definition 5.5.6 Let F be an intuitionistic Kripke bundle; M a Kripke bundle model over F. The pattern of M is the Kripke bundle model Mo over F such that for any u E IF and any atomic D,-sentence A
It is clear that the model Mo is intuitionistic.
L e m m a 5.5.7 Under the conditions of Definition 5.5.6, we have for any u for any D,-sentence A Mo, u It- A iff M , u k A ~ .
E IF,
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P r o o f By induction, similar to Lemma 3.2.15. Let us consider only the case A = B(a) > C(a). We have Mo,uIl-Aiff Vv E R(u) Vb E D," ( a R n b & Mo, v Il- B(b) =+ Mo, v Il- C(b)) (by Lemma5.5.4) iff Vv E R(u) Vb E D," (aRnb & M , v I= B(b)T =+ M, v t= C(b)T) (by the induction hypothesis) iff M, u k O(B(a)T > C(a)T) (by Lemma 5.3.4 and Definition 5.2.8). The latter formula is AT by Definition 2.11.1. L e m m a 5.5.8 Let M be an intuitionistic Kripke bundle model with the accessibility relation R, A(x) an intuitionistic formula with all its parameters i n the list x, 1x1 = n. Then for any u E M
Proof The claim is similar to Lemma 3.2.18, but the proof is different; now it is based on soundness (Proposition 5.2.12). By Definition 3.2.12, in the given model M we have: u It- VxA(x) iff u k ( v x A ( x ) ) ~ and By Lemma 2.11.7, in QS4 the latter formula is equivalent to OVXA(X)~, thus u k ( V ~ ( X )iff) u~ b O V X A ( X )(by ~ Proposition 5.2.12) iff Vv E R(u)V a E Dc v b A(a)T (by Definition 5.2.8 and Lemma 5.2.10) iff Vv E R(u)Va E D," v It- A(a) (by Definition 5.5.3).
Exercise 5.5.9 Let M be an intuitionistic Kripke bundle model with the accessibility relation R, u E M , A(x, y ) an intuitionistic formula with FV(A) xy, 1x1 = n, lyl = m. Then for any u E M , c E D r M, u IF VxA(x, c) iff Vv E R(u)V a E D t Vd E D r (cRmd +-M, u IF A(a, d ) ) . Definition 5.5.10 A n intuitionistic predicate formula A is called true i n an intuitionistic Kripke bundle model M (notation: M IF A) i f VA is true at every world of M ; valid i n an intuitionistic Kripke bundle F (notation: P IF A) i f it is true in all intuitionistic Kripke models over P; strongly valid in an intuitionistic Kripke bundle all its substitution instances are valid i n IF.
F (notation: IF IF+ A) i f
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L e m m a 5.5.11 M IF A(x1,. . . , x n ) ifl Vu E M V a l , . . . , a n E D, A(al,-.-,an).
Proof
M,u I t
Similar t o Lemma 3.2.20.
Proposition 5.5.12 Let IF be an S4-based Kripke bundle, A E I F = . Then
(2) the following three assertions are equivalent: (a) IF I t + A, (b) tlm F IF Am,
(c) F k+ AT.
Proof (1) Let A = A(x), 1x1 = n. (Only if.) Assume IF It- A. Let M be a Kripke bundle model over F. By Lemma5.5.11, we havetoshow M , u k AT(a)(= A(a)T) for a n y u E F, a E DZ. By Lemma 5.5.7 we have: M o , u Il- A(a) iff M , u t= ~ ( a ) ~ . By Lemma 5.5.11, IF I t A implies M o , u I t A(a), and thus we obtain F k AT. (If.) Assume F k AT. Then for any intuitionistic model M , any u E M, a E D," we have M , u k A(a)T(= AT(a)) by Lemma 5.5.11, i.e. M , u It- A(a). Hence IF I t A, by Lemma 5.5.11. (2) (a)+(b) is obvious. (b)+(c). Assume F I t Am, i.e. IF k (Am)T. Since (Am)T = (AT)m, we obtain F k+ AT by Proposition 5.2.12. (c)+(a). Assume IF kf AT, and for an arbitrary IF'-substitution S let us show F I t SA, i.e. IF k (SA)T. By Lemma 2.11.5, (SA)T = sT(AT) is a QS4'-theorem, and thus it is valid in IF, by Proposition 5.2.12. By assumption, F I= ST(AT); thus we obtain IF I= ( s A ) ~ . Therefore IF It A+. rn
Proposition 5.5.13 For an intuitionistic Kripke bundle F the set of strongly valid formulas IL(=)(F) := {A E IF(=) I F IF+ A) is a superintuitionistic predicate logic (with or without equality), moreover,
Proof Similar t o Proposition 3.2.29. By Proposition 5.5.12, F I t + A iff IF k+ AT, which means A E IL(=)(IF) iff AT E ML(=)(P). Thus IL(')(IF) = s ( ~ ~ ((F)). = )
rn
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369
Definition 5.5.14 The set IL(')(F) is called the superintuitionistic logic of F (respectively, with or without equality). Now we easily obtain intuitionistic analogues of some assertions from the previous section.
L e m m a 5.5.15 Let ( f o , f l ) : M -= M' for intuitionistic Kripke bundle models M , M'. Then for any u E M and for any intuitionistic D,-sentence B M , u ll- B i f f M', f ~ ( u Il-) f l . B ,
. .
P r o o f From Lemma 5.4.6, Definition 5.5.3 and the observation that f l B~ = (fl
.W T .
Proposition 5.5.16 If F and (6 are intuitionistic Kriplce bundles and IF +' 6 , then I L I ( F ) ILT ((6) and IL' (P) IL'(6). Proof The first statement is an exercise for the reader. For the second, note that IF += 6 implies ML'(P) C_ ML'((6) by Proposition 5.4.8; hence s ( M L Z ( F ) ) s(ML'(G)), i.e. IL'(F) C IL'(G), by Proposition 5.5.13.
L e m m a 5.5.17 For an intuitionistic Kripke bundle IF IL(=)(F)= ~ { I L ( = ) (1-FU ) 1
F).
P r o o f Follows readily from Lemma 5.4.13, Proposition 5.5.13, and the equality
L e m m a 5.5.18 Let P be an intuitionistic Kripke bundle, F 1 V its generated subbundle. Then I L ( = ) ( FG ) IL(=)(FI. v). Proof
Similar to Lemma 5.4.14.
M
For the intuitionistic case the notion of a Kripke bundle can be slightly extended.
-
Definition 5.5.19 An intuitionistic Kripke quasi-bundle is a quasi-p-morphism T : Fl Fo between S4-frames, cf. Definition 1.4.18.~
A Kripke quasi-bundle can be presented equivalently as a triple F = (Fo,D , p ) with a system of domains D = ( D , I u E W ) and a family of relations p = (p,, I u R v ) satisfying ( # 2 ) , (83) from Definition 5.2.2 and
5Recall that quasi-p-morphisms are intuitionistic analogues of p-morphisms, cf. Chapter 1.
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However one should be careful about using quasi-bundles. Since an intuitionistic quasi-bundle is not necessarily a modal bundle, it is not a surprise that the semantics of quasi-bundles is not sound for modal logics. But it is not sound for intuitionistic logic either! To obtain soundness, we have to reduce the class of quasi-bundles, see Section 5.17 for details.
Exercise 5.5.20 Try to give an inductive definition of intuitionistic forcing in a Kripke quasi-bundle (intuitionistic valuations are defined similarly to Kripke bundles). Note that the definition of forcing is not so straightforward.
A familiar truth definition for a formula 3xA is irrelevant in this case, as the following counterexample shows. Example 5.5.21 Consider a quasi-bundle F pictured at Fig. 5.5. bl E D,, does not have inheritors in ug, but bl is its own inheritor in ul = R ug. SOIF is still a quasi-bundle.
Figure 5.5. R is universal; p is universal on {al, ag). Consider an intuitionistic model M
=
(IF, 5) such that [+(P)
=
{b). Putting
we obtain M, u1 IF 3xP(x), but
M, ug IpL 3xP(x). So since u1Rug, there is no truth-preservation for 3xP(x). But as we shall soon see (cf. Lemma 5.14.15), truth-preservation (or 'monotonicity') is necessary for soundness.
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371
So here is an alternative definition:
M , u It- 3 x B ( x ,a ) iff 3v x u 3d E D, 3 b E D,"(b xna & M , v It B(d, b)). (*) Due to the quasi-lift property (1.4.18) of quasi-bundles, (*) can be rewritten as
M , u It- 3 x B ( x ,a ) iff 3vRu 3d E D, 3 b E D:(bRna & M , v I t B ( d ,b ) ) . (**) Note that this definition for the 3-case is dual t o the clause for the V-case in Lemma 5.5.4(6). The modified definition provides the desired truth-preservation property, as we shall see later on.
L e m m a 5.5.22 Let M = ( F , [ ) be an intuitionistic Kripke bundle model, and let tn be a propositional valuation i n its n t h level Fn, n 1 0 such that tn(pk) = <+(PC)for any k 2 0. Let Mn = (Fn,En) be the corresponding propositional Kripke model. Then every Mn is intuitionistic, and for any intuitionistic propositional formula A and for any a E Mn
Proof
Mn is intuitionistic, according to Definition 5.5.2. Mn, a I t A iff Mn, a k AT (by Definition 1.4.1) iff M k ( A T ) n ( a )(= ( A n ( a ) ) T ) (by Lemma 5.3.5) iff M IF An(a) (by Definition 5.5.3).
L e m m a 5.5.23 Let IF be an intuitionistic Kripke bundle, A a propositional intuitionistic formula. Then for any n > 0
P It- An iff Fn It- A. F I t An iff IF k ( A n ) T ( =( A T ) n )(by Proposition 5.5.12) iff Fn k AT (by Lemma 5.3.6) iff Fn It- A (by Lemma 1.4.7).
Proof
Proposition 5.5.24 For an intuitionistic Kripke bundle
F,
P r o o f V n F, It A iff V n Fn It- AT (by Lemma 1.4.7) iff IF k+ AT (by Proposition 5.3.7) iff P IFf A (by Proposition 5.5.12). Let us also make some remarks on Kripke quasi-sheaves.
L e m m a 5.5.25 Let P be a quasi-sheaf over an (intuitionistic) frame F . Assume that u MR v i n F , i.e. u , v are i n the same cluster. Then p,, is a bijection between D, and D, with the converse p,,.
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Proof First, note that ap,,b implies bp,,a. In fact, by Definition 5.2.1, there exists a' such that bpVua1.Then by transitivity, apUua1,and thus a = a', since F is a quasi-sheaf. Obviously, the same argument shows that bp,,a implies ap,,b, and thus p,, = It remains to show that both p,,, p,, are functions, and again we can check this only for p,,. So assume aop,,al, aopuVa2;then alp,,ao, and thus by transitivity, alpv,a2, which implies a1 = a2, by Definition 5.2.3. W So, similarly to the case of Kripke sheaves (Section 3.5), we can factorise a quasi-sheaf F and obtain a quasi-sheaf over the partially ordered frame F": Lemma 5.5.26 Let F = (F,D, p) be the same as in Lemma 5.5.25, with F = (W, R) and consider the corresponding frame of individuals F1 = ( D l , R1); let n : F1 --++ F be the associated p-morphism. Let F" = (D", R"), (F1)" = ((Dl)", (R1)") be their skeletons (Definition 1.3.39). Then (1) The map n" : a" bundle F").
(2)
H
n(a)" is a p-morphism (and thus defines a Kripke
F" is a Kripke quasi-sheaf.
(3) There exists a p=-morphism F
-r,
F".
Proof (i) n is well-defined. In fact, a Z R I b only if n(a) Z R n(b), due to the monotonicity of T. Next, a"(~')"b" iff aR1b (by Definition 1.3.39), only if n(a)Rn(b) (since n is p-morphism), iff n(a)"RWr(b)" (by Definition 1.3.39). To check the lift property for n", suppose x" R-v", i.e. T(a)Rv, by Definition 1.3.39. Since T is a p-morphism, there exists b such that aR1b and n(a) = v, and thus a"(R1)"b", v" = n(b)" = ~"(b"). (ii) Suppose n"(a") = ~"(b"), i.e. ~ ( a Z)R I n(b) and also (a")(R1)"(b"), i.e. aR1b. To obtain a" = b", we have t o show that bR1a. By the lift property, there exists a' such that bR1a' and n(a1) = n(a). But then aR1a' by transitivity, and since IF is a Kripke quasi-sheaf, it follows that a' = a. Hence bR1a. (iii) Let us show that (fo, fl) : P -= F", where fo(u) := u", f l(a) := a". In fact, these maps are propositional pmorphisms, by Lemma 1.3.40, so it remains to show that (fo, fl) is a fibrewise bijection, i.e. that f l restricted to D, is a bijection. To show the injectivity, suppose n(a) = n(b) = u, but a # b. Then a" # b", since in Kripke quasi-sheaves different individuals in the same fibre are not related. For the surjectivity, note that an arbitrary element of (7r")-l(u") is of the form b", with u" = n"(bm) = ~ ( b ) " . Let us find a E D, such that a" = b". Let v := n(b); then u W R v, and by Lemma p,, is a bijection between D, and D,, whose converse is p,,. Now take a := p,,; it follows that a Z R I b, and a E D,. W Definition 5.5.27 P" is called the skeleton of F.
5.5. INTUITIONISTIC KRIPKE BUNDLES So we obtain an analogue of Lemma 3.7.18: L e m m a 5.5.28 IL(')(P)
= IL(=)(P") for a Kripke quasi-sheaf F .
Proof The inclusion IL(=)(P) C IL(=)(P") follows from Lemma 5.5.26 and Proposition 5.5.6. To prove the converse, suppose M = (IF,cp) is an intuitionistic model, A is an intuitionistic sentence and M Iy A, and let us show IF" Iy A. Let f : IF + IFN (Lemma 5.5.26), and let us define a valuation $ in F" such that $+(P) = { f - a 1 a E cp+(P)). for any predicate letter P. Then
+ is an intuitionistic valuation. In fact, suppose
c = f . a E $+(P), c ( R " ) ~ ~and , let us show that d E $+(P). By Lemma 5.4.4, there exists b E Rn(a) such that d = f . b (Fig. 5.6). Then from a R n b we obtain b E cpf (P). Thus by definition, d E $+(P).
a
c= f-a
Figure 5.6. So $ is intuitionistic, and we also have f : (F, cp) += (IF", $). In fact, by definition, a E cp+(P) only if f . a E $+( P ) , and it remains to show the converse. Suppose f . a E ++(P), i.e. f . a = f . b for some b E cp+(P). Then a == b (by the definition of f ) , and thus a E cp+(P), since cp is intuitionistic. By Lemma 5.4.6 we obtain (for a given sentence A and for any world u): M , u It Aiff (F",+),uW It A. Since M
Iy A by our assumption, we conclude that IF" IF A.
Thus in the intuitionistic case it is sufficient to consider Kripke quasi-sheaves only over posets. On the other hand, for Kripke bundles analogous reductions are not always possible, because the domains of equivalent worlds may be of different cardinality. So we have to consider Kripke bundles over arbitrary quasi-ordered sets. Lemma 5.5.16 shows that we can use only cones, but this does not simplify anything, because a quasi-ordered cone may still have several equivalent roots with non-equivalent domains.
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5.6
Functor semantics
In Chapter 4 we defined validity in a presheaf over a locale and showed that the presheaf semantics is equivalent t o algebraic semantics. Functor semantics is based on an alternative definition of validity in a presheaf (over an arbitrary category), which is mainly due to Ghilardi, Makkai and Reyes. In this section we define this semantics in a more general polymodal setting. Let us begin with a notion of a precategory. This is actually a coloured multigraph, or a labelled transition system. Similarly to a category (cf. 3.4.3), it consists of objects (or points) and morphisms (arrows), but a priori without composition, and without special identity morphisms. Every morphism is coloured.
-
Definition 5.6.1 A n N-precategory is a tuple C = ( X , Y , o, t , cr), where X, Y are non-empty classes; o, t : Y ---+ Y and c r : Y ( 1 , . . . ,N ) are functions. X is called the class of objects (notation: ObC), and Y is the class of morphisms (notation: Mor C). o (f ), t ( f ), cr(f) are called respectively the origin, the target, and the colour o f f . A n i-morphism is a morphism of colour i . W e also use the following notation: Mori(C) := c r P 1 ( i )(the class of all i-morphisms), C(U,V):= { f I ~ ( f = ) U, t ( f ) = v ) (the class of all morphisms from u to v), Ci(u,v) := Mori(C) nC(u,v) (the class of all i-morphisms from u to v).
W e shall usually consider small precategories, where X, Y are sets. I n this case the Kripke frame (W,R1, . . . ,RN) is called the frame representation of C (notation: FR(C)) if W = ObC and Ri = {(u,v) 1 Ci(u,v) # 0 ) . Note that a 1-precategory is nothing but a multigraph. It is also clear that a category is a 1-precategory expanded by identity morphisms and composition of consecutive morphisms (cf. Section 3.6); its frame representation is an S 4 frame. Every 1-precategory C can be extended to a category C*; this construction is similar to reflexive transitive closure of a binary relation. Namely, morphisms in C* are paths (sequences of consecutive morphisms) in C: aI a2 UrJ --+ U1 --+ U2 ---'
. . . ---+
u,-1
a,
u,
(where uo, . . . ,u, may be not all different), and we also add all identity morphisms 1, (as new). The composition of morphisms in C' is defined as the join of sequences:
and of course we define l,ocp:=cpol,:=cp for any cp E C*(u,v). If a (small) category C has a single object, then MorC* is the free monoid generated by the set MOTC.
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375
D e f i n i t i o n 5.6.2 Let C be a (small) N-precategory with the set of objects W . A C-preset is a triple IF = (C,D , p), i n which D = ( D , I u E W ) is a system of domains on W and p = (pa I a E M o r C ) is a family of functions indexed by morphisms of C , such that pa : D, + D, for cu E C ( u , v ) . The objects of C are called possible worlds of IF, and the elements of D, are called individuals at u . If d E D,, a E C ( u ,v ) , the image p,(d) is called the a-inheritor of the individual d i n the world v . Note that an individual d E D, can have different a-inheritors i n the same world v for different morphisms a. D e f i n i t i o n 5.6.3 Let C be a category based on a precategory Co, i.e. C = (Co,o , 1 ) . A Co-preset IF is called a C-set (more precisely, an intuitionistic C-set) if it preserves composition and identity morphisms: (iv) Pa00 = pa
0
pp;
for any a,@E M o r C , u E ObC. Let also give an alternative definition o f C-presets. D e f i n i t i o n 5.6.4 A prefunctor F : C --, C' from an N-precategory C to an N precategory C' is a map sending objects of C to objects of C' and morphisms of C to morphisms of Cf%nd preserving colours, origins and targets: i f f E Ci(u,v ) , then F ( f ) E Ci(F(u),F ( v ) ) . W e also extend this definition to the case when C is an N-precategory and C' is a I-precategory: i f f E Ci(u,v ) , then F ( f ) E C f ( F ( u )F, ( v ) ) . So we see that a C-preset (C,D , p) is nothing but a prefunctor F : C C V ~S E T such that F ( u ) = D,, F ( a ) = pa. Conversely, every prefunctor F : C --+ S E T such that every F ( u ) is non-empty, can be considered as a preset. O n the other hand, a preset F = (C,D , p) is also associated with the precategory of individuals C 1 ( F ) such that ObC1(F):= D1 :=
U
D,,
uEObC
C1(F)i(alb) := { ( a ,b , a ) I b = p,(a), a E MoriC). W e also obtain an etale prefunctor E whenever a E D,, and E ( a , b, a ) = a.
:
C 1 ( F ) C V ~ C such that E ( a )
=
u
P r o p o s i t i o n 5.6.5 (1) Let F : C --t S E T be a C-preset, and let E : C 1 ( F )--, C be the corresponding
etale prefinctor. Then E has the following properties 'We may assume that ObC i l M o r e = 0; otherwise, C can b e replaced with an isomorphic category having this property.
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Figure 5.7. Etale prefunctor. (i) tlu E 0 b C 3a E ObC1(F) E(a) = u (surjectivity on objects); (ii) tla E ObC1(F) tla E M a r c (E(a) = o ( a )
+ 3!f
(F(f) = a & o(f) = a ) )
(the unique lift property). (2) Let G : C' ^.t C be a prefunctor which is surjective on objects and has the unique lift property. Then C' is isomorphic to C1(F) for some C-preset F . P r o o f (1) (i) F is surjective on objects since every set F(u) is non-empty. (ii) Suppose a E C(U,V)and a E D,. Then E ( f ) = a & o ( f ) = a holds iff f = ( a ,b, a ) , where b = pa (a). (2) For an object u put D, := F(u) := G-' [u]. For a morphism a E C(u, v), put F ( a ) := p, := {(a, b) I3f ( o ( f ) = a & t ( f ) = b & G ( f ) = a ) } . The precategories C' and C1(F) are isomorphic. In fact, they have the same objects, and there is a bijection between C1(F)(a, b) and Cf(a,b) sending (a, b,a) to the the unique lift of a beginning at a , i.e. to f such that G ( f ) = a , o ( f ) = a.
If C is category, F : C -A SET is a C-set, then we can also make C1(F) a category by putting (a,b,a) 0 (b,c,p) := ( a , c , a o p ) ; 1, := (a,a,l,) for a E D,. Lemma 5.6.6 If F is a C-set, then E : C1(F)
-A
Proof In fact, E((a, b, a ) o (b, c, P)) = a o E(L) = l~(a).
F is a functor. =
E(a, b, a) o E(b, c, P), and
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Definition 5.6.7 For a propositional N-frame F = (W, R 1 , .. .,R N ) we define the associated N-precategory C = CatoF as follows:
ObC := W,
Ci(u,v) :=
otherwise.
From definitions it is clear that every Kripke sheaf based on a frame F is a CatoF-preset; if F is an S4-frame, we obtain a CatF-preset (where C a t F is the category defined in Section 3.5.3).
Remark 5.6.8 Note that if F is not an S4-frame, it may happen that a CatoFpreset is not a Kripke sheaf, because it may not satisfy the coherence conditions (I)*, (2)* from Lemma 3.6.3. Definition 5.6.9 (cf. Definitions 5.2.7, 5.2.8). A (modal) valuation in a Cpreset F i s a valuation in the system of domains D = (F(u) I u E ObC). A model over F is a pair (F, 6)) where [ is a valuation in F . The forcing relation M , u t= A between a world u E W and a D,-sentence (with equality) A is defined by induction:
M , u k Pi iff u E [,(Pi); M, u k P r ( a ) i f fa E [,(Pr) (for m > 0); M , u k a = b iff a equals b;
M,u
+ OiB(a) iffVv E R;(u) VO E C ~ ( U , VM) , v 1 B(pa . a ) ;
M , u k 3xA iff 3a E D , M , u k [a/x]A; M, u
+ VxA iff V a E D,
M, u 1 [alx]A.
Here as usual i n the 0-case B(a) means [a/x] B, and we assume that FV(B) = X. Note that the truth condition for Oi[a/x] B obviously extends to the case when FV(B) x. The following definition is similar to the case of Kripke bundles. Definition 5.6.10 A (modal) predicate formula A is called
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true in a C-preset model M if its universal closure VA i s true at every world of M ; valid in a C-preset F if A i s true in every model over F ; strongly valid i n a C-preset F if all substitution instances of A are valid in F. We use the same signs as above: != for truth and validity, l=+ for strong validity. The following is a trivial consequence of the definitions: L e m m a 5.6.11 For a C-preset model M and a modal formula A(x)
M I=A(x) i f f Vu E M Q a E DZ M , u k A(a). For a C-preset F let
The proof of the following soundness result is postponed until Section 5.13 (Proposition 5.13.2). Proposition 5.6.12 For a C-preset F: (1) ML(')(F) is a modal predicate logic;
Definition 5.6.13 For N-modal predicate logics we introduce the functor semantics FSN generated by presets over N-precategories. Similarly to Kripke bundles, we introduce levels for C-presets. Definition 5.6.14 Let F = (C, D,p) be a C-preset over a n N-precategory C. Let u s define relations Rr o n Dn for n > 0 , l 5 i 5 N : a R a b iff 37 E MoriC p,
. a = b.
Also let R: := Ri, Fn := (Dn,Ry, . . . , R k ) . The nth level of F is the frame Fn := (Dn, R;, . . .,R;). W e again abbreviate R? t o Rn i n the 1-modal case. From the definition of forcing 5.6.9 we readily obtain L e m m a 5.6.15 Let C be a n N-precategory, M = ( F ,[) a model over a C-preset F = (C,D,p). Let B be a n N-modal formula with FV(B) x, 1x1 = n. Then for any u E F and a E DZ
M , u k OiB(a) iff Vu E Ri(u) V b E Dz (aRab + M,u l= B(b)).
5.6. FUNCTOR SEMANTICS
379
L e m m a 5.6.16 If C is a category and F is a C-set, then every Fn is an intuitionistic propositional frame. P r o o f An exercise.
W
<
Definition 5.6.17 A valuation in a C-set F is called intuitionistic i f every set <+(PC)= U{<,(P;) I u E W ) is stable in F,, i.e. i f Rn(J+(P;)) C_ <+(Pp). The model (IF,<) is also called intuitionistic. Definition 5.6.18 Let M be an intuitionistic model over a C-set F = ( C , D,p), u E M , A an intuitionistic D,-sentence. Then we put
L e m m a 5.6.19 Under the conditions of Definition 5.6.18 (1) M , u It- B iff M , u k B (for B atomic);
(3) M , u I t B V C i f f (M,uIF B o r M , u I t C ) ;
(4) M , u It B ( a ) > C ( a ) i f f
Vv E R ( u )V p E C(u,u ) ( M ,u It B(p, . a ) + M , u It C ( p , . a);
(5) M , u It- V x B ( x , a ) i f f Vu E R ( u )V p E C(u,u)Vc E D, M , u != B(c, p, . a ) ; (6) M , u It- 3 x B iff 3a E D, M , u It [alx]B ; (7) M , u It. l B ( a ) iff Vu 6 R ( u ) V p E C(u,u)M , u W B(p,. a ) ;
(8) M , u It a
Proof
# b iff a does not equal b.
Similar to 5.5.4, with obvious changes.
L e m m a 5.6.20 Let M be the same as i n Lemma 5.6.19. Then for any D1sentence A ( a ) M It A ( a ) & aRnb + M It- A ( b ) . Definition 5.6.21 Let F be a C-set, 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 A
The model Mo is intuitionistic, and similarly to Lemma 5.5.7, we obtain
L e m m a 5.6.22 For any u E F, for any D,-sentence A M0,u IF A iff M,U k
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L e m m a 5.6.23 Let M be an intuitionistic model over a C-set with the accessibility relation R, A(x) an intuitionistic formula, 1x1 = n. Then for any u E M
M , u It- V x A ( x ) i 8 V v E R ( u )V a E Dc M , u IF A ( a ) . Proof
Similar to Lemma 5.5.8 using soundness (5.6.12).
Definition 5.6.24 A formula A E IF(=) is called true i n an intuitionistic C-set model M (notation: M
It- A ) i f V A is true
at every world of M ; valid i n a C-set F (notation: F
It- A ) i f it is true i n all intuitionistic models
over F ; strongly valid in a C-set F (notation: F
It+A ) i f all its IF(=)-substitution
instances are valid in F . L e m m a 5.6.25 For an intuitionistic C-set model M and an intuitionistic formula A M It- A ( x ) iff V u E M V a E DE M , u IF A ( a ) . Proposition 5.6.26 Let F be a C-set, A E I F T . Then (1) F I F A i f f F k A T
(2) The following three assertions are equivalent:
(a) F It-+ A ; (b) V m F IF Am;
(c) F k+ A T . Proof
Cf. Proposition 5.5.12.
Proposition 5.6.27 For a C-set F the set
I L ( = ) ( F ):= { A E IF(=) I P ~t-+ A ) is a superintuitionistic predicate logic (with or without equality), and IL(=)( F ) =T NIL(=)( F ) . Definition 5.6.28 The set I L ( = ) ( F )is called the superintuitionistic logic of F .
5.7. MORPHISMS O F PRESETS
5.7
Morphisms of presets
Now let us define truth-preserving morphisms of presets. Definition 5.7.1 (Cf. Proposition 5.6.5.) A prefinctor @ : D --, E has the lift property if
V U E ObVVp E MorE(@(a)= o ( p )
+ 3 f E MorV
@ ( f )= p ) .
The reader can see that this is a generalisation of the lift property for propositional frame morphisms. Definition 5.7.2 Let F be a C-preset, F' a C'-preset. A pair y = ( Q 7 9 ) is called an =-morphism from F to F' (notation: y : F += F') i f (1) @ : C 2, C' is a prefunctor with the lift property;
(2) @ = (9, I u E ObC) i s a family of bijections
(3) @ respects morphisms, i.e. for any p E C ( u ,v ) the following diagram commutes:
Q v
y is called a p=-morphism i f it also has the property
(4)
@ is surjective on objects.
The above conditions (2), (3) mean that 9 is a functor morphism ('natural transformation7) from F to F' . @. Definition 5.7.3 For y = yn : Fn -+ FA as follows:
(a7@) : F
whenever a E F ( u ) ~n, > 0 . L e m m a 5.7.4 yn : Fn
-=
FA.
-+=
F', n
2 0 , we define the map
382
CHAPTER 5. METAFRAME SEMANTICS
Figure 5.8.
Proof We consider only the case n > 0, and assume N = 1 to simplify notation. To prove the monotonicity, suppose aRnb, i.e. 3p E C(u,v)Vi bi = F(p)(ai). Then by 5.7.2(3),
Thus Q, . b = F1(@(p)). (Qu . a ) , and so by definition, ~ , ( a ) ( R ' ) ~ y , ( b ) . To check the lift property, suppose a E Fn, a' = 9, F'(ul), b1 E F'(v1). Then
. a, a'Rrnb',
a' E
3X E C'(ul, v') b' = F1(X) . a'. Since cP has the lift property, there exist v E ObC and p E C(u,v) such that X = cP(p). Then put
By definition, aRnb, and we also claim that b' = yn(b) = Q, . b (Fig. 5.8).
5.7. MORPHISMS O F P R E S E T S In fact, let b"
= 9, . b.
B y 5.7.1, the following diagram commutes: Q v
ai Thus b"
= F1(X). a' = b'.
W
D e f i n i t i o n 5.7.5? Let y : F -= G be an =-morphism of presets. y is called an =-morphism from a model M = ( F , J ) to M' = ( F 1 , J ' ) (notation: M ' ) if f : M -=
, > 0; and also for any P E P L n , a E F ( u ) ~ n
for any P E P L O M' for preset models M L e m m a 5.7.6 If y : M -= then for any u E M and for any F(u)-sentence A
= ( F ,J ) ,
M'
= (F',J1),
M , u b A iff M I , ~ ( uk )7 1 . A. If the models are intuitionistic, then for any intuitionistic F(u)-sentence A M , u It- A i f f M 1 , y o ( u )It- yl . A. P r o o f By an obvious modification of the proofs of 5.4.6, 5.5.15 based on Lemma 5.7.4; an exercise for the reader. W L e m m a 5.7.7' Let y : F -= G , and let M' be a model over G . Then there exists a model M over F such that y : M -= Mr. Proof
The same as for Lemma 5.4.7.
w
P r o p o s i t i o n 5.7.8 If there exists a p=-morphism F += F', then MLf ( F ) C M L T ( F ' ) and thus, ML' ( F ) M L = ( F 1 ) . Similarly, for the intuitionistic case: I L I ( F ) c I L S ( F ' ) and IL'(F) G IL'(F1). Proof
Cf. Propositions 5.4.8, 5.5.16.
7Cf. Definition 5.4.5. *Cf. 5.4.7.
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Let us now construct inverse image morphisms. Lemma 5.7.9' Let F be a C-preset, E : C' --, C the corresponding etale prefunctor, G : V -+ C an arbitrary prefunctor with the lift property. Consider a precategory 2)' with the class of objects
{ ( u , a ) 1 u E ObD, a E ObC', G ( u ) = E ( a ) ) and with D1((.la), (a1b ) ) := {(Pl f
I P E D ( u ,211, f
E C1(a,b),
G ( P ) = E ( f )).
Then (1) There exists an etale prefunctor E' : 2)' E1(P1f ) = P.
-+
V such that E ' ( u , a ) = u ,
(2) There exists a prefunctor with the lift property. G' : D'
G 1 ( u , a )= a , G 1 ( p f, )
-+
C' such that
= f.
(3) The following diagram commutes:
(4) If
F' is a D-preset corresponding to E', then there exists (G, Q ) : F' --+= F such that Q u ( u ,a ) = a.
(5) If C, D are categories, F is a C-set and G is a functor, then D' becomes a category with the composition
and F' becomes a D-set. Proposition 5.7.10 Every =-morphism of presets ( @ , Q ) : F' -= F i s the composition of the inverse image morphism O : Q?,F += F and a prefunctor isomorphism (over the same precategory) E : F' + @,F. Definition 5.7.11 Let C, D be N-precategories. V is called a full subprecategory of C (notation: D Lo C ) if 9Cf. Lemma 5.4.9.
5.7. MORPHISMS OF PRESETS (1) ObV
c ObC,
(2) V u , v E ObVVi 5 N Ci(u,v) = V i ( u , v ) . If C, D are categories and D Go C, then D is called a full subcategory of C (notation: V c C ) i f (3) for any consecutive morphisms f , g i n V
f o g (in V )= f
0
g (in C).
D e f i n i t i o n 5.7.12 A full sub(pre)category D of a (pre)category C is called a conic sub(pre)category (notation: 2) E(0) C) i f Here is a typical example. D e f i n i t i o n 5.7.13 For a (pre)category C and u E ObC the cone generated b y u (notation: C f u ) is defined as the full sub(pre)category with the class of objects { v I C*(U,v ) # 0 ) . T h u s every object o f C f u is accessible from u via a path o f morphisms. D e f i n i t i o n 5.7.14 Let F be a C-preset, V Go C. The restriction of F to V (notation: F 1 V ) is the V-preset taking the same values as F on objects and morphisms of V . For u E ObC we define the cone o f F generated by u : F t u := F (CTu).
r
Note that i f F is a C-set and V
C C, then obviously, F 1 D is a V-set.
L e m m a 5.7.15 Let F be a C-preset, V soC. Let J : V C be the inclusion prefunctor (sending every object and every morphism of V to itself). Also let = (idF(u)1 u E ObV). Then
*
( J , Q ) : F rV +=F. Proof
Obviously, J has t h e lift property and 9 respects morphisms.
L e m m a 5.7.16
(1) For a C-preset F
(2) For a C-set F
Proof
Along the same lines as Lemmas 5.4.13, 5.5.17.
L e m m a 5.7.17
(1) If F is a C-preset, V LoC, then ML(=)(F)
(2) If F is a C-set, V
5 C, then IL(=)( F )
Proof
c ML(=)(F I. v ) .
Easy, from 5.7.16.
c IL(=)(F 1 D).
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Bundles over precategories
5.8
Let us now combine two generalisations of Kripke sheaves: Kripke bundles and presets. Definition 5.8.1 A modal C-bundle over an N-precategory C is a triple F = (C,D, p), i n which
I u E W)
D = (D, p = (p,
of c; Pa
C D,
is a system of domains;
I a E MorC) x D,,
is a family of relations parametrised by morphisms
dom p, = D, for a
E C(u,v).
A n intuitionistic C-bundle over a category C is a modal C-bundle satisfying two extra conditions: (1) pl, is reflexive (on D,) for u E W,
cf. the corresponding conditions for intuitionistic Kripke bundles i n Definition 5.2.2. The notion of a C-bundle generalises both Kripke bundles and C-sets. In fact, C-sets are just C-bundles with functions p,, and Kripke bundles based on a frame F are exactly CatoF-bundles (where CatoF is the precategory from Definition 5.6.7). But it turns out that the semantics of C-bundles is strongly equivalent (in the terminology of Section 2.16) to the semantics of C-sets, as we show below. Hence it follows that the semantics of Kripke bundles ICB is included in the functor semantics FS. But actually, FS is stronger than KB, as we shall see later on (Section 5.10). To define forcing in C-bundles, we first extend Definition 5.3.2 to this case: Definition 5.8.2 For a C-bundle F = (C,D, p) put a R a b iff
a s u b b & 3f EMariC V j a a j p f b j
for a , b E Dn, n > 0. Also put Ry := Ri, the corresponding relation i n FR(C). Definition 5.8.3 A (modal) valuation i n a C-bundle F = (C, D, p) is a valuation in the system of domains D. A model over F is a pair (F, t), where [ is a
valuation i n P. Now let us give a preliminary definition of forcing.
5.8. BUNDLES OVER PRECATEGORIES
387
Definition 5.8.4 (cf. Definitions 5.2.7, 5.2.8, 5.6.9). The forcing relation M , u t= A between a world u E W and a Du-sentence (urith equality) A is defined by induction:
M , u I= P: i f f u E & ( P i ) ; M , u k P," ( a ) iff a E EU (P,") (for m > 0); M , u b a = b iff a equals b; M,uIfl.;
M , u I= Eli B ( a ) iff Vv E Ri(u) Vb E DE ( a R l b + M , v b B ( b ) ) (for lal = n); M , u b 3 x A iff 3a E Du M , u b [ a / x ] A ;
Again in the U-case B ( a ) means [a/x]B , and we assume that F V ( B ) = x. But this definition should be justified. In fact, as we know, the presentation of the same formula in a form B ( a ) is not unique. So we should prove that forcing does not really depend on this presentation. To show this, we find a precategory corresponding to the same 'metaframe' relations Rr.
Proposition 5.8.5 Let IF = ( C , Dl p) be a C-bundle over an N-precategory C. Then there exists an N-precategory C' and a Cf-preset F' such that
FR(C) = FR(C1);
D,
= F f ( u )for
any u E ObC;
E w , 1 5 i 5 N (where Ry and Rin are the nth level accessibility relations in IF and F' respectively).
Ry = Rin for all n
Moreover, if IF is an intuitionistic C-bundle over a category C (in particular a Kriplce bundle over an S4-frame, then C' is also a category and Ff is a C'-set. Proof
Put
Cf(u,v ) := { f is a function D,
-
D,
I 37 E Ci(u,v) f C p,),
and let p; := f in IF'. So we replace every relation p, C D, x D, with the set of all functions from D, to D, included in this relation.
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Let us show that Rin = R> In fact, suppose aRinb; then asub b, and there exists f E C:(u, v) such that b = pP; - a = f . a , i.e. for all j , bj = f ( a j ) Since f E C:(u,v), there exists y E Ci(u,v) such that f p,; thus ajp,bj, which implies aRYb. The other way round, suppose a R l b . Then the set g = {(al, bl), . . . , (a,, b,)) is a function (since a sub b), and g C p, for some y E Ci(u, v). This g can be prolonged to a total function f : D, D, such that f p,, since domp, = D, by Definition 5.8.1. Hence b = f - a = pP; . a , which implies aRinb. If C is category, then C' also becomes a category after we define 0 as the composition of functions and lDU a s the identity function id^". In fact, iff C p, and g pp, then f o g pa o pp C pa,@ (by Definition 5.8.1); idDUC plU,since pl, is reflexive by 5.8.1.
-
R e m a r k 5.8.6 The crucial point of this construction is 'local functionality' in the definition of forcing for a Kripke bundle. This corresponds to the choice of distinct individuals a l , . . . , a, in the inductive clause for 0, or to the conjunct asub b in the definition of R r . R e m a r k 5.8.7 Note that the argument fails for intuitionistic Kripke quasibundles, because the totality of p,, is not guaranteed in this case. Therefore a model M = (P, 6 ) corresponds to the model M' = (F', t),and we can define M, u != A as MI, u 1A. Then we obtain the properties described in Definition 5.8.4, and we can further define validity in IF and the logic
in the modal case and IL(=)(F) := IL(=)(F') in the intuitionistic case. Hence we obtain the semantics of C-bundles, which is obviously strongly equivalent to the functor semantics, and the following Corollary 5.8.8 KBN 5 F S N , lCBint 5 FSint. Proof In fact, for a Kripke bundle IF with the base F there exists a CatoFpreset F' such that ML(') (IF) = ML(=) (F') .
5.9
Metaframes
The definition of forcing in Kripke bundles and C-sets via forcing in propositional frames F, motivates a further generalisation of Kripke semantics. So far the relations R r were derived from other relations or functions. Now let us consider arbitrary metaframes; they are defined as sequences of propositional frames of tuples, without special requirements for accessibility relations.
5.9. METAFRAMES
389
Definition 5.9.1 Let Fo = (W, R1,. . . ,RN) be a propositional Kripke frame, D a system of disjoint (nonempty) domains over W . Let Fn = (Dn, R;, .. . ,Rg), n > 0, be propositional Kripke frames such that Dn = U Dc. Then the pair IF = ((F,,),€,, D ) is called an N-metaframe uEW
based on Fo.As usual, W is called the set of possible worlds and D1 the set of individuals of F . Fn is called the n-level of F . For a E DZ we denote the domain D, by D(a). We shall often identify a metaframe F with the sequence (Fn)nEw,i.e. consider it as a graded propositional frame. Note that the worlds of Fn are n-tuples of individuals, and the original worlds from W are '0-tuples'. Thus we may define D: := { u ) , DO := W1 ROz := Ri. We say that an n-tuple b is an i-inheritor (counterpart) of an n-tuple a if aRYb. As usual, i(=I) is not indicated in the notation if N = 1. By default we assume that an arbitrary metaframe and all its components are denoted as in 5.9.1.
Definition 5.9.2 A (modal) valuation i n an N - m e t a h m e IF is a valuation i n its system of domains. A (modal) metaframe model over F is a pair M = (IF, [), where is a (modal) valuation i n IF.
<
So a valuation in IF is a graded propositional valuation; n-ary predicate letters are evaluated in IF as propositional letters in Fn. Our goal is now to define forcing in metaframe models. This definition is not a straightforward generalisation of the corresponding definitions in functor and Kripke bundle semantics. The reason is that in arbitrary metaframes it is impossible to define forcing for D,-sentences in the natural way - this will be shown later on. Remember that every D,-sentence is obtained by replacing parameters with D,-individuals in a formula A(x), or by applying a D,-transformation [x H a] to A(x), cf. Definition 2.4.1. This transformation itself (not only its result A(a) 'forgetting' the original A(x) and [x H a]) is essential in our definition. Moreover, we define forcing for a formula A(x) with respect to an 'ordered assignment' (x, a ) (arranging a transformation in a certain order). In the classical case (and in all our earlier Kripke-type semantics) the two versions of the truth definition (with D-sentences or variable assignments) are equivalent. The original Tarski's definition is given in terms of assignments. So we begin with
Definition 5.9.3 A n ordered assignment (at a world u ) i n a metaframe IF = ((Fn)nEw, D) is a pair (x, a), where x is a distinct list of variables, a E Dz. If the list x is empty, then by definition, a is just the world u .
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We define forcing in a model M as a relation M , ( x ,a ) I= A between ordered assignments and formulas. It can also be regarded as a relation between tuples a E Dn and pairs ( A ,x ) , so we use the alternative notation M , a != A [x].This is read as ' a forces a pair ( A ,x ) ' , or 'a formal expression A[x]'.1° It is convenient to give such a definition for the case when F V ( A ) C r ( x ) . Let us make some remarks about notation. Remember that according to the Introduction, for a tuple a = ( a l , . . .,a,) and a : I, --+I,, we denote
and
a - a a = a , = a . S y = ( a l ,...,ai-l,ai+l ,..., a,). A
The same notation applies to lists of variables; thus in Definition 5.9.4, x . a means ( x , ( ~ ).,. . , xu(,)) and x - xi means x . SF for a list of variables x = ( x l , .. . , xn), a : Im In. However the case m = 0 is special; then a = 0, is the empty map, and we define x .On as the empty list, a . 0 , as the world of a (i.e. a . 0 , = u a E Dt).
-
Note that every atomic formula P p ( y ) with r ( y ) C r ( x ) , 1x1 = n , has the form P p ( x r ( l ) ,. . . ,xu(,)), or P ~ ( - xa ) , for some F E C,. Definition 5.9.4 For a metaframe model M and a modal formula (with equality) A we define forcing M , a I= A [x]under an ordered assignment ( x ,a ) (such that F V ( A ) C r ( x ) ) l l by induction:
(2) M , a b P j " ( x . a ) [ x ]iff ( a . a ) E [+(Pjm)(for m > 0);
(3) M , a i=
P~O
[x]iff u E eJ+(PjO)(for a E D c ) ;
(4) M , a I= ( x i = x j ) [ x ]iff ai = a j ; (5) M , a k B
3
C [ x ]iff ( M , a Y B [ x ]or M,a!= C [ x ] ) ;
(6) M , a b B V C [ x ]i f f ( M ,a != B [ x ]or M , a b C [ x ] ) ; (7) M , a I= B /\ C [ x ]iff ( M ,a
B [ x ]& M , a I=C [ x ] ) ;
(8) M , a I= U i B [x]iff 'db (aRab + M , b I=B [ x ] ) ; (9) if y @' x , a E DZ, then M , a i= 3yB [x]iff 3c E D, M , (ac) k B [ x y ] , M , a b 'dyB [x]iff 'dc E D, M , (ac) I= B [ x y ] ; ' O ~ h enotation A[x] should not be mixed up with A(x) denoting a formula with parameters in x . "If x is empty, a is a possible world u , this is written as M , u F A [ I , or just M , u F A.
5.9. METAFRAMES (10) M, a b 3xiB [XI iff M, a - ai I= 3xiB [X - xi]; M,al=VxiB[x] iffM,a-ait=VxiB[x-xi]. So by Definition 5.9.4, we obtain: M , a b l B [ x ] iff M , a v B [ x ] ,
-
Also note that if D is a system of domains over W, m , n > 0, then every map a E C, gives rise to the map T , : Dn Dm sending a to a . a , and it is clear that T, maps every D z to D F exactly as in the Introduction. Do = W For the empty map 0, E Con we can define the map TD, : Dn sending every tuple to its world; sometimes we use a simpler notation and write TB rather than n ~ . , In a metaframe the maps T, are called jections. They have the properties mentioned in Lemma 0.0.1:
if a E T n , then
T,
is a permutation of Dn and
T,-I
-
= (T,)-l.
L e m m a 5.9.5 Let IF = ((Fn)nc,, D) be an N-metaframe, a E Em,. r,,[Dn] = {a E Dm I asuba).12 Proof
Then
rn
Follows from Lemma 0.0.2 applied to every D,.
L e m m a 5.9.6 Let IF be a metaframe, in which not all domains are one-element. Then for a E Em,, a is injective iff T, : Dn -+ Dm is surjective. Proof
Follows from Lemma 0.0.3.
H
We also use notation from the Introduction:
recall that
. . , a n ) = ( a l , . . . , a,, a,) for n > 0, $!(al,. n?(a) = a - ai for a E Dn, n > 0, TTT;(a) = a - an+l for a E Dn+', n
2 0.
Remembering that a$ = Dl E Col, we also have n$(al) = u for a1 E D,. Weshalluse thesame notationr, for listsof variables; inparticular,r~(x1 ...x,) is the empty list. Note that in all the clauses except (a), (lo), Definition 5.9.4 is essentially the same as in earlier Kripke-type semantics. For example, consider Kripke bundles. Our new definition (5.9.4) corresponds to the old one (5.2.8) as follows. I2Here similarly to 5.3.2, usuba denotes the property Vi,j (a(i)= u(j)=$- ai = a j ) .
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M, a I= A[x] (in the new sense) is equivalent to M, u k [a/x]A (in the old sense), where u is the world of a. So 5.9.4(2) M , a k P r ( x - a ) [x]- a - c € [ + ( P r ) becomes
exactly as in 5.2.8. For evaluating a formula A we use assignments (x, a ) with r(x) 1 FV(A). The requirement r(x) = FV(A) is insufficient, e.g. in clause ( 5 ) , because it may be that F V ( B > C) # FV(B), FV(C). We have t o be careful about the quantifier clauses. In fact, if r(x) > FV(3yB), then either y 6 r(x)13 or y E r(x). In the first case we obtain the clause (9), which is clear. But if y E ~ ( x ) the , clause (9) is inapplicable, because (xy, ac) is not a correct assignment - y is repeated twice. So before adding y to x, we should eliminate it from x and its value from a. This leads us to clause (10). Sometimes it is convenient to join (9) and (10) in a single clause. To do this, let us introduce some more notation. For a (distinct) list of variables x and y E Var we put x-y:=
x-xi
ify=xi, if y 6 x, .
and xlly := ( x - Y)Y. These lists can be obtained by corresponding transformations, viz. put
provided 1x1 = n;14 then
The associated jections are denoted as follows:
1 3 ~ h i is s always the case if ~ ( x= ) FV(3yB). 14+ means the simple extension, see the Introduction.
5.9. METAFRAMES
393
So we can combine (9), (10) from Definition 5.9.4 in a single clause: M, a I= 3yB [x] iff 3c E Du M, 7rxll,(ac) i= B [xlly], and similarly for V. In fact, if y y = xi, then
(9
+ 10)
6 x, then 7rxll,(ac) = a c and xlly = xy. And if
The clause (8) generalises a version of the truth definition for OB(a) in Kripke bundles given in 5.3.4. A similar version for functor semantics was considered in 5.6.15. In fact, in the case of Kripke bundles (8)
M, a I=O B [x] e Vb(aRnb + M, b k B [x])
corresponds to
i.e. t o (*) in 5.3.4. Let us also recall another reading of the 0-clause in Kripke-type semantics considered so far: (8.2) M , u k C7C iff for any v E R(u), for any v-version C' of C in v, M, v k C', where C is a Du-sentence. (8.1) transforms to (8.2) if we define v-versions of a Du-sentence [a/x]B as D,-sentences [b/x]B for b E Rn(a) n D," (i.e. for b that are v-inheritors of a ) . In a Kripke sheaf a E Dc has a unique v-inheritor pu,.a. In functor semantics inheritors also result from map actions: b = p, . a, for p E C(u, v). In Kripke bundles we define the relation Rn in a special way. In metaframes we can also call Rn the 'inheritance relation' between n-tuples and read (8) as (8')
M , a k O B [x] iff for any inheritor b of a , M, b k B [x]
But now we cannot always define inheritance relations between D,-sentences and transform (8) to (8.2) or (8.1). This happens, because a D,-sentence can be presented as [a/x]A in different ways, as mentioned in 2.4. Different presentations may lead to different sets of inheritors for the same Du-sentence. The following example shows that this is crucial. Example 5.9.7 Consider a Du-sentence C := P ( a , a). Then
where B1 := P ( x , y), B2 := P ( x , x).
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Put
X l := R2(aa) = { b E D2 1 ( a a ) R 2 b ) , X2 := { ( d d ) E D2 1 aR1d), B1 := { [ b l x y I B i 1 b E X i ) = { P ( b ) I b E X i ) , B2 := { [ d / x ] B 1I a R 1 d ) = { P ( b ) 1 b E X 2 ) .
Since X 1 consists of all inheritors of aa, we may call the members of I31 'inheritors of [ a a / x y ] B l ' .Similarly, the members of BZ are 'inheritors of [a/x]Bz'. But it may happen that X I # X2,15 SO B1 # Bz, and thus we cannot properly define inheritors of C . In this case ( 8 ) cannot be rewritten as (8.1). In fact, in a metaframe model M = (F,
e)
M, aa b O P ( x lY ) [ X Y ] e X I C J + ( P ) , M , a k CIP(x,x) [XI H X 2 J f ( P ) . So there exists a model, in which
-just
put
So we see that in arbitrary metaframes there is no reasonable way to define the relation M, u i= A for D,-sentences A. That is why we define metaframe forcing in the form M, a b A [ x ] . Now let us define the truth in a metaframe model. Definition 5.9.8 A modal formula A is called true i n a metaframe model M = ( F ,t') (notation: M k A ) i f M, a k A [x]for any ordered assignment (x,a ) i n P such that F V ( A ) r ( x ) . Definition 5.9.9 A modal formula A is called valid i n a metaframe F (notation: IF k A ) if A is true i n all models over P. Let
MLI=)(P) := { A E MF$)
IF
+ A).
Definition 5.9.10 A formula A E MF&=) is called strongly valid (respectively, strongly valid with equality) i n a metaframe P if all its M F N - (respectively, MFG-) substitution instances are valid i n P (notation: IF k f A or respectively, IF b+= A). Let M L ( F ) := { A G MFN I F b+ A ) .
ML=(IF) := { A E M F ' I F b f = A ) . Definition 5.9.11 Let F = (C, D , p ) be a preset over an N-precategory C. Then its associated metaframe is Mf (F) := ((F,), D ) , where F, is the n-level of F (Definition 5.6.14). I n the same way we define Mf ( F ) for a Kripke bundle F . I 5 ~ h reader e can easily construct such an example.
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395
So a Kripke bundle or a C-preset model M = (F,<) corresponds to a metaframe model Mf (M) := (Mf (F),<), and the two definitions of forcing 5.9.4, 5.6.9 are obviously the same: L e m m a 5.9.12 Let F be a preset over an N-precategory (or a Kripke bundle), M a model over- F. Then for any formula A E MF~,) for any ordered assignment (x, a ) in Mf (F) with FV(A) C r(x) (1) M t= [a/x]A iff Mf (M), a I= A [x]
(2) M k A iff Mf (M) i= A. P r o o f (1) By straightforward induction on the length of A, according to 5.9.4, 5.6.9. (2) follows from (I), 5.6.9, 5.2.10. Proposition 5.9.13 For a C-preset or a Kripke bundle F:
Proof
Follows readily from 5.9.12.
R e m a r k 5.9.14 One can also define a metaframe Mf (@) for any Kripke quasibundle @ (cf. 5.5.19) such that
for any modal formula A. The counterexample 5.5.19(a) shows that the set ML(Mf (@))is not always a modal predicate logic. Definition 5.9.15 A metaframe IF is called modally sound (respectively modally sound with equality), in brief, m(')-sound, if ML(')(F) is an m.p.1. (=). Our goal in the next sections (5.10-5.12) will be to describe the class of msound metaframes; as we shall see, m=-soundness is equivalent to m-soundness (Theorem 5.12.13). But first, let us point out some peculiarities of forcing in arbitrary metaframes. This notion is still too broad, because M, a != A [x]may depend on the ordering of x and a and also on the variables xi that do not occur in A. E x a m p l e 5.9.16 Let F = (W, R) be a reflexive singleton {u). Consider a metaframe IF over F such that D, = {cl, cz), RI
I 5 j),
:= {(ci, cj) i
R~ := {(a, a ) I a E 02)U ( ( ~ 1 ~C2~, C Z( )~, 2 ~~12,~ 2 ) ) .
So R2 is the same as the corresponding relation in the Kripke bundle in Fig. 5.9, but without the pair (clc2,czc2).
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Figure 5.9. We do not specify the frames Fn for n > 2; for example, Rn can be universal on DE. Consider a metaframe model M = (IF, J ) such that
J + ( P ):= {c2c2},J f ( Q ) := {cz} for certain P 6 P L 2 , Q E PL'. Let A : = O P ( x l , x 2 ) ,then in M we have
but k ~A
(since ( ~ 1 ~ 4e2(c2c2)). 2 ) So M , a k A [x]depends on the ordering of x and a. Now put B := O Q ( x ) ,then (in M ) ~
1
[2 X I X I ]
cl k B [x](since alR1cz), clcl k B [ x y ](since ( C I C I ) R2(c2c1)), but
clc2
2 ) clca B [ x y ](since R2(c1c2)= ( ~ 1 ~ and
b' Q ( x ) [ x Y ] ) .
So M , a k B [ x ]depends on the variables in x that do not occur in B .
R e m a r k 5.9.17 The above counterexample may cause doubts about the clause (10) in Definition 5.9.4. In fact, as M , a k A [ x ]may depend on the values of xi that do not occur in A, it seems improper t o make M , a k 3xiB [x]equivalent to M , a - ai I= 3xiB [x- xi]. Instead we can define an alternative forcing relation k* by the same clauses (1)-(9) and the following modification of (10). (lo*) M , a k * 3xiB [ x ]iff 3 b E Dn (b .6: = a . 61 & M , b k* B [ x ] ) iff 3c E D ( a ) M , [c/ai]aI=* B [x], where [c/ai]ais obtained by replacing ai with c at position i;16 and similarly
M , a k * V x i B [ x ]iff V b E Dn (b - : 6
=a
. 6;
+ M , b I=*B
[x])
iff b'c E D(a) M , [c/ai]aI=* B [ x ] . In arbitrary metaframes the relations k and k* may not be equivalent: 16So b .6? = a . 67 iff b = [ c / a i ] afor some c E D(a).
5.10. PERMUTABILITY AND W E A K FUNCTORIALITY
397
Exercise 5.9.18 Consider a metaframe model M from Example 5.9.16 and formula C = 7Q(xl) A A, where A = OP(xl, x2). Show that
If* 3xlC [x~xz]. Hint: recall that M, clcl t= A [xzxl] and M, clcz If A [x~xz]. 3xlC [xlxz] and M,clcz M , c ~ kc ~
On the other hand, later on we will show (see the end of Section 5.12) that
t= and t=* are equivalent in logically sound metaframes.
5.10
Permutability and weak functoriality
Now let us describe a class of metaframes without pecularities mentioned a t the end of Section 5.9. This class will be used in the further description of m-sound metaframes. First we give the following
Definition 5.10.1 An N-metaframe IF is called permutable if s, is monotonic for every permutation a:
V a E T n Vi
E
IN b'a, b (aR2b + s,(a) Rrs, (b)),
Note that for n = 0 , l this condition holds trivially, since in these cases Tfl = {idn}. Also note that to show permutability, it is sufficient to check monotonicity of jections corresponding to (simple) transpositions a;, where 2 5 k 5 n, which are generators of the symmetric group. Since for any permutation a , s,-I = (s,)-l, we obtain the following equivalent condition: L e m m a 5.10.2 A metaframe IF is permutable iff for any permutation a E T n , s, is an automorphism of the n-level F,.
Proof (Only if.) s, is a bijection for any a E T n by Lemma 0.0.1. (.ir,a)Rn (rub)+ a R n b follows from 5.10.1 by applying so-I= (so)-'. Exercise 5.10.3 Show that in a 1-modal permutable metaframe, where not all domains are one-element, the relation R2 cannot be a linear ordering. Definition 5.10.4 A metafiame IF is called weakly functorial (w-functorial) if for any injection a Tmn (where m 5 n), n, is a p-morphism Fn * Fm. From 5.9.6 we already know that s, is surjective for a E T,,. To show w-functoriality, it suffices to check that n, is a (p-)morphism, whenever a is a simple transposition a; € T, or a simple embedding a; € C,,,+l, because every injection is a composition of functions of this kind.17 Remembering that n;a = a - a,+l for n # 0, and s:a is the world of a (for a E D l ) , we obtain: "See Introduction.
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398
L e m m a 5.10.5 A metaframe F = ((Fn)nG,, D ) is w-functorial ifl IF is permutable and for any n > 0 , i E I N , the following conditions hold: (1) V n > 1 Va,b E Dn (aRYb + (a-an)R:-l(b-
b,)) (the monotonicity for
7ry-l); (2) V n > Ova, b E Dn Vc E D1 ( a R l b & (ac) E Dn (the lift property for T;);
+ 3d
(ac)~~+l(bd))
(3) Qa E D,Qb E D, (aR!b + uRiv) (the monotonicity for n:);
(4) VU,u (uRju + Va E D J b
E
D, a ~ t b (the ) lift property for T:).
L e m m a 5.10.6 Let IF be a metaframe. (1) F is permutable iff for any model M over F , for any ordered assignment ( x ,a ) , for any formula A with F V ( A ) C x and for any permutation a E T n , where n = 1x1,
(2) F is w-functorial i f f (*) holds for any model M over IF, for any ordered assignment ( x , a ) , for any injection a E T,,, where 1x1 = n 2 m 2 0, and for any formula A with F V ( A ) C r ( x . a ) . Note that i f a is injective and (x,a ) is an ordered assignment, then the list x . a is distinct, so ( x . a, a . a ) is also an ordered assignment. Proof The case n = 0 (when a, x are empty and A is a sentence) is trivial. ( I ) 'Only if' is proved by induction on A , and the proofs o f ( 1 ) and ( 2 ) can be made in parallel. So we assume that F is respectively permutable or transformable and check the equivalence (*). The model M is fixed, so we drop it from the notation. Let y := x - a ; then r ( y ) C r ( x ) .
Let A be atomic; then since F V ( A ) C r ( y ) , it follows that
for some j, k 2 0,
7
E
Ckm
(not necessarily injective). Hence by definition,
a b A [ x ]i f f i f fa -
(7
. CT) E <+(P:)
O n the other hand, ( a . a ) I=A [y]i f f ( a . a ) . r E <+(P;), and since ( a .a ) . r = a . ( r . u ) , (*) follows. The cases when A = I , B A C , B V C , B
> C , are trivial.
5.10. PERMUTABILITY AND W E A K FUNCTORIALITY Let A = OiB. Then a b A [x]iff
( 3 ) V b E R y ( a ) b b B [x] which by induction hypothesis, is equivalent to (4) V b E R r ( a ) 7rU(b)k B [rU (x)].
On the other hand, .rr,(a) k A [n,(x)] iff (5) Vb' E R:(T, ( a ) )b' I= B [T,(x)]. But Ra ( T , ( a ) )= T , [Rr( a ) ] ,since (5). Let A
T,
is a pmorphism. So (4) is equivalent to
= 3yB.
Put T := E ~ - ~p ,:= E ~ . , - ~ . Then ~ ~ = Tl + , ~ l ( ~~ . = ~ p+. ) l l ~ So according t o our definitions and notation (see condition (9+10) in Section 5.9), for any appropriate assignment ( x ,a ) we obtain1'
( 6 ) a k 3 y B [x]i f f3c E D, (ac) . T + b B [ ( x y ). T + ] , and (7) a . a ! = 3 y B [ x - a ] i f f 3 c E D (u( a . ~ ) c ) . ~ + k B [ ( ( x . a ) y ) . ~ + ] . In view of ( 6 ) , ( 7 ) , to apply the induction hypothesis, it suffices t o find an injective A' such that
(10) ( x y ) . (T+ . A') = ( x y ), (a+ - p+); (11) (ac) . (T+ . X I ) = (ac) - (a+ . P+). Note that x . a - y is a distinct list of variables from x - y. So by Lemma 0.0.8, x . a - y = ( x - y) . X for some injection A.
C l a i m T . X = a . p. For the proof note that X . T = X - y , ( x - a ). p = x - a - y, and thus
IsRemember that ( x y ) . T + = xlly, ( ( x. u)y) . p+ = ( x . o)lly.
400
CHAPTER 5. METAFRAME SEMANTICS
Since x is distinct, the claim now follows by Lemma 0.0.5. Hence by Lemma 0.0.7, T+ . A+ = a+ . p+, so (lo), (11) hold for A' := A+. (11) Now let us prove the 'if' part in both assertions, (1) and (2). We assume (*) and check the p-morphism properties of T, (respectively, for a E , T or a E T,). To check the monotonicity, we also assume a R l b , and show that n,(a)Ry x,(b). Let A = OiP(x,(x)), where x = (xl, . . . ,x,) is distinct. Consider the model M = (F,J) such that ( + ( P ) = {r,(b)). Then by Definition 5.9.4
and so M , a k A[x]. Hence by (*), M, ra(a) b A [ru(x)l, which means by Definition 5.9.4, 7ru(a)RYbffor some b' E E+(P). By the choice of J, b' = r,(b). Therefore xu (a)R y r , (b). Note that the same argument is valid in the particular case m = 0; then we have a = 0,, A = O i P for P E PLO,Ef ( P ) = {7rB(b)). To check the lift property, we assume x,(a) R r b ' and find b E Rr(a) such that r,(b) = b'. Consider the same formula A = OiP(7r,(x)), and the model M = (IF, E) such that J + ( P ) = {b'). Then
and thus M, na(a) l= A [ra(x)l. Hence by (*), M,a
+ A [XI,
for some b E RT(a). But then r, (b) E c + ( P ) , according to 5.9.4, thus x,(b) = W b', by the choice of J. Lemma 5.10.6(1) shows that in permutable metaframes forcing M, a b A[x] is invariant under simultaneous permutations of x and a. So we can also define forcing under unordered assignments. Such an assignment (at a world u) is nothing but a D,-substitution (Section 2.2), i.e. a function [a/x] sending every xi to ai. An ordered assignment (x, a ) corresponds to the unordered assignment [a/x] = {(XI,a l ) ,. . . ,(x,, a,)), where n = 1x1. So in a model M over a permutable metaframe we can define
5.1 0. PERMUTABILITY AND WEAK FUNCTORIALITY
401
This definition is sound; in fact, [a/x]= [b/y] iff (y, b) is obtained from (x, a) by some permutation, i.e. iffy = x - a , b = a . a for some a E T,. So by Lemma 5.10.6(1), M , a k A [x]iff M , b I= A [y]. Furthermore, in w-functorial metaframes the truth value of a formula A depends only on individuals assigned to parameters of A (or only on a possible world if A is a sentence). Viz., suppose r(y) FV(A) = r(x). Then x = y . a for some injection a. So by Lemma 5.10.6(2),
>
M , a k A [y] iff M , a . a i = A[x]. Thus forcing M, a k A [y] reduces to M, b k A [x] for an appropriate b. In particular: L e m m a 5.10.7 If r(x) = FV(A), z is a distinct list of other variables and c is a tuple of individuals (from the world of a) of the same length, then M , a i = A [x] i f f M , a c i = A [xz]. Since in w-functorial metaframes we may use only the forcing relation M, a k A [x]for r(x) = FV(A), in Definition 5.9.4 we need only the clause (9) for A = VyB or 3yB, because clearly y @ FV(A) = r(x). Exercise 5.10.8 Show that in a permutable metaframe for any model M M , a I= 3xiB [x]iff 3c E D(a) M, [c/ai]a I=B [x], where the tuple [c/ai]a is obtained from a by replacing ai with c; cf. Remark 5.9.17. Hint: (x - X C ) X =~x . a and ( a - ai)c = ([claila) . a for a E T,. So by induction on IAl, one can easily show that
for a model M over a permutable metaframe for any A E MF', denotes the modified forcing from Remark 5.9.17.
where k*
To show that forcing in w-functorial metaframes is congruence independent, we begin with an auxiliary lemma. L e m m a 5.10.9 Let M be model over an arbitrary metaframe, A a modal forr(x). Also let y' @ mula, (x, a ) an assignment in M such that FV(A) V(A), y @ BV(A), x' = [y'ly] x, and A' = A[y H y']. Then M, a I= A [x] iff M, a I=A' [x'].
CHAPTER 5. METAFRAME SEMANTICS
402 Proof
This statement is quite obvious - it means that the truth value of a formula does not depend on the name of a certain free variable. Formally we argue by induction on the complexity of A. Let us consider three cases. (1) If A = P ( x . a),then A' = P ( x l . a). So
M , a k A [x]iff x,a E e
+ ( ~ iff) M , a l=A'
[x'].
( 2 ) Let A = 3 z B . Then z # Y , ~ 'by the assumption of our lemma, so 3zB1, (xllz)' = x'llz, n,l, =T ,I, Then by the modified clause (9+10) of Definition 5.9.4 and the induction hypothesis, A'
=
M , a != A [ x ]& 3c E D(a) M,7rXtl~,(ac) I= B [xllz] @ 3c E D ( a ) M , nXt 1, (ac) I= B' [x'llz] @ M , a I= 3zB1(=A') [x']. ( 3 ) Let A = OiB, a E Dn. By Definition 5.9.4 and the induction hypothesis, M , a I = A [ x ]H V ~ ERa(a) M , b k B [x] # Vb E Rl ( a ) M , b I= B' [x']e M , a I= EliB' (= A') [x']. All other cases are rather trivial.
L e m m a 5.10.10 Let M be model over a w-functorial metaframe, and let A, A' be congruent modal formulas. Then for any assignment [ a l x ] with F V ( A ) c
.(XI and thus
MFA-M!=A~. Proof We apply Proposition 2.3.17. Consider the equivalence relation between formulas:
A
N
B
:= for any assignment [ a l x ]such that H M , ] a / x ]I=B.
F V ( A ) ,F V ( B )
r(x),
M , [ a l x ]k A It suffices to check that
satisfies the conditions (1)-(4) from 2.3.17.
( 1 ) M , [ a / x ]k Q y A @ M , [ a / x ]F Qz(A[yH z ] ) provided y V ( A ) , Q E {V,3>and F V ( Q y A ) C r ( x ) .
# BV(A), z
!$
(Obviously, F V ( Q z ( A [ yH z ] ) )= FV(QyA).) Consider e.g. the case Q = 3. We may assume r ( x ) = F V ( Q y A ) , since the metaframe is w-functorial. Then y, z @ r ( x ) ,so we have
M , [ a l x ]k 3yA e 3c E D ( a ) M , [aclxy]! = A 3c E D(a) M , [aclxz]k A[y ++ Z ] (by Lemma 5.10.9 ) e M , [ a l x ]F 3 z ( A [ yH z ] ) .
5.10. PERMUTABILITY AND WEAK FUNCTORIALITY
-
403
-
(2) Assuming A B let us prove QyA QyB (for Q = 3). We again assume r(x) = FV(A), so y r(x), and thus we obtain
M, [a/x]t= 3yA H 3c E D(a) M, [ac/xy] t= A H (since A 3c E D(a) M, [aclxy] I= B H M, [a/x]I= 3yB.
-
B)
-
(3) This property holds trivially, by the definition of forcing. (4) Assuming A B let us show OiA UiB: M, [a/x] t= OiA H V b E Rl(a) M, [b/x] k A R l ( a ) M, [b/x]k B H M, [a/x]I= OiB.
H 'v'b E
Hence we readily obtain L e m m a 5.10.11 Let F be a w-functorial metaframe. Then for any congruent modal formulas A, A' IF~AHFFA~. R e m a r k 5.10.12 Wecannot extend Lemma5.10.10 toarbitrary(notw-functorial) metaframes. For example, suppose r(xz) = FV(B), z $ BV(B) and y, y' are distinct variables that do not occur in B . Then the formulas 3y(B[z I-+ y]) and 3yt(B[z H y']) are congruent. However by 5.9.4 and 5.10.10 M,ab k 3y(B[z H y]) [xy] H 3c E D(a) M , a c I= B[z H y] [xy] @ 3c E D(a) M, ac I= B [xz],
(*I) while
M, a b k 3y1(B[zH y']) [xy] @ 3c E D(a) M, abc I= B[z H y'] [xyy'] @ (*2)
3c E D(a) M, abc t= B [xyz].
Now, e.g. if B = OP(x, z ) , then (*I) means
and (*2) means 3c 3d 3e 3e' ((abc)~"+l(de'e) & de E ['(P)). If a metaframe is permutable, but not w-functorial, these conditions may be not equivalent, cf. Example 5.9.16. On the other hand, we do not know if w-functoriality is necessary for Lemma 5.10.10. One can try to construct counterexamples explicitly, as in the proof of Lemma 5.10.6.
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404
5.11
Modal metaframes
Definition 5.11.1 An (N)-modal metaframe is a w-functorial metaframe satisfying (mmz) Qi E IN Qa,bl, b2 ((a,a)R; (bl, bz) + bl = b2). An equivalent form of (mm2) is Ri(A)
A
1
:= {(a, a) a E
A, where
D').
For the next lemma recall that a s u b b @ V j , k ( a j =ak n?a = aa, for a E Dn. L e m m a 5.11.2 (m%) (mm:)
+ bj = bk),
For an N-modal metaframe for any n > 0, i E IN we have
VaQb (aRpb + asub b); VaQb (aRp b + (n? a )R;"
(nl4b)) .
Proof In fact, (mm,) follows from (mm2), since (aj, ak) = a . ATk (see the Introduction). So for a w-functorial metaframe aRpb implies (aj, ak)R?(bj, bk), and we can apply (mmz) to (aj, ak) and (bj, bk). To check (mm:), note that n; : Fn+l+ Fn in a w-functorial metaframe. Thus by the lift property, a R l b implies 3c (aa,) R;+ '(bc), and by (mmn+l), it follows that c = b,. Hence
The next lemma will be mainly used later on, in the intuitionistic case (section 5.14). L e m m a 5.11.3 If an N-metaframe satisfies ( m m ) , i E IN and a E Em,, then n, [Dn] is Rr-stable. Proof If n,(a)Rrb, then .rr,(a) sub b, and the latter yields asub b, i.e. b E n,[Dn], by 5.9.5. rn Exercise 5.11.4 Show that for a certain i, (mm,) is equivalent to Ry-stability of all sets n,[Dn] for c E C,, n > 0. Hint: a E Dm can be presented as n,c for some tuple c with different n = Icl. components, where a E C,,
5.11. MODAL METAFRAMES
405
-
Definition 5.11.5 A metaframe F is called functorial if every its jection .rr, : Dn + Dm is a morphism F, Fm. L e m m a 5.11.6
A
metaframe is modal iff it is functorial.
P r o o f (Only if.) Assume that a metaframe IF is modal. Recall that every function a E Em, is a composition of simple injections and simple projections.lg Since F is w-functorial, it is sufficient to show that every jection 7rT (= xu?) is a p-morphism. Its monotonicity is already stated by (mm,). To check the lift property, assume T?! (a) = (aa,)~y+l(bc). Then (aa,) sub (bc) by (mmn+l), and hence b, = c. Thus bc = bb, = .rrn(b). We also have aRPb by Lemma 5.10.5(1), since F is w-functorial. (If.) We assume that F is functorial and check (mma). In fact, suppose (a, a) = .rr! ( a )R:(bl, b2). Since 7rk is a pmorphism, (bl, b2) = .rrl (b) for some b, i.e. bl = bz = b. rn The next lemma shows that forcing in functorial metaframes 'respects variable substitutions'. L e m m a 5.11.7 Let M = (F, 5) be a model over a functorial metaframe. Then, for any modal formula A
(#)for any a E Em,, for any distinct lists of variables x = (xl, .. . , x,), y = ( ~ 1 ,... ,ym) such that FV(A) r(y), and for any a E Dn: M, [a/x]I= [x . a / y ] A @ M, [ a . a / y ] k A. Recall that [ x . a / y ] A is defined up to congruence. But this does not matter in functorial metaframes, by 5.10.10. Proof By Lemma 5.10.10, the claim (#) does not change if we replace A (in A. So we may replace A with its clean version A0 both sides) with any B such that BV(AO)n r(xy) = 0 (and FV(AO)= FV(A) r(y)); then we have [x.a/y]A~A0[y~x.a]. To simplify notation, we now assume A0 = A and proceed by induction on A.
.
Let A = P;(~.T),
T
E
Ek,.
Then [x.a/y]A = P;((x.a).~) = P;(x.(~.T)).
So by Definition 5.9.4, we have:
Let A = (yj = yk), then [x. o l y ] A = (xu(j) =
lgSee Introduction.
So we have
406
CHAPTER 5. METAFRAME SEMANTICS Let A = 3zB, then z $! r(xy) by the choice of A, and so [X
- u/y] A = 3z [x . u/y] B = 3.2 [(xz) . a+/yz] B.
Hence M, [a/x] t= [x. u/y]A w 3c M, [ac/xz] I= [(xz) . u+/yz] B @ 3c M, [(ac) . C T + / ~ Z ]b B (by the induction hypothesis) @ 3c M, [(a . U)C/YZ] I= B e M, [a - a / y ] b A. Let A = OiB.By Definition 5.9.4 M, [a/xl b [nu(x)/yI A @ V b E R l ( a ) M7 [b/xl@[nu( 4 1 ~B1 w V b E R l ( a ) M, [x,(b)/y] b B (by the induction hypothesis)
On the other hand, M, [n,(a)/y] I= A is equivalent to Vb' E Rr(.ir, (a)) M, [blly]b B .
Now, since n, is a p-morphism, we have Ry(n,(a)) (1) @ (2).
(2) = T, [Rl(a)].
Thus
All other cases are obvious.
Hence it follows that variable substitutions preserve validity. L e m m a 5.11.8 Let IF be a functorial metaframe, A a modal formula such that IF t= A. Then for any variable substitution [y/x], IF I= [y/x]A.
Proof Since [y/x]A does not depend on the variables beyond FV(A), we may assume that r(x) = FV(A); then r(y) = FV([y/x]A). Let z be a distinct list such that r(z) = r(y); then y = z - a for some transformation u. By Definitions 5.9.8, 5.9.9 and Lemma 5.10.7, IF I= [y/x]A iff for any model M over F, for any assignment [a/z] in IF M, [a/zl [ylxlA. By 5.11.7, the latter is equivalent to M, [a.a/x] I=A, which follows from IF k A. Therefore, [y/x]A is valid in
IF.
The next lemma can be considered as a converse to 5.11.7 in a stronger form. L e m m a 5.11.9 Let IF be a metaframe such that for any model M over IF, for any quantifier-free formula A
5.1 1. MODAL METAFRAMES
a) for any a E C,, such that F V ( A )
407
for any distinct lists x of length n and y of length m r ( y ) ,for any a E Fn
M , a I= ( [ x. a / y ] A ) [x]H M , a . a I= A [ y ] . T h e n F i s functorial.
Proof Given an arbitrary a for .rr,.
E C,
let us check the p-morphism properties
( 1 ) aRab + (r,a) Ry(r,b).
In fact, assume aRab. Consider the formula A := O i P ( y ) , P E PLm and the model M = (IF, () such that ( + ( P )= { r u b ) . Then [r,x/y] A = O,P(r,x), M , b I= P(r,x) [ X I , and so
M,a
[ r , x l ~ lA 1x1.
Hence by assumption (#)we have M , r,a b A [ y ] and , thus
for some b' E Ry(r,a). Hence b' E ( + ( P ) ,i.e. b' = rub,by the choice of (, so we obtain (r,a)Ry(r,b). (2) ( r U a ) R y b + f 3 b E Ra(a) rub= b'.
Assume (r,a)Rybf. Consider the same formula A = O i P ( y ) and the model M = (F, 8) such that 8+(P)= {b'). Then
and so
M , rua I= A [Y]. Thus by assumption (#),
i.e. a R l b for some b such that
Hence rub E 8+(P),i.e. rub= b' Obviously, if m = 0, a = O n , then we can use the formula A p E PLO)both in ( 1 ) and (2).
=
Oip (with
Remark 5.11.10 Note that in the above proof we use the condition (#) only for formulas of the form O i P ( y ) or Oip.
CHAPTER 5. METAFRAME SEMANTICS
408 Hence we obtain
Proposition 5.11.11 For a metaframe
P the following properties are equiva-
lent: (1)
P is functorial;
(2) 5.11.9
0)holds for
any formula A;
(3) 5.11.9 (#)holds for any quantijier-free A. Actually (#) in (2) should be formulated as the equivalence
M, a It- ([x. u/y]A)
[XI
H M, a . a IF A [y]
for any congruent version of [x . a/y]A. Note that variable substitutions are defined up to congruence, while in non-functorial metaframes forcing may be sensitive to congruence. But for a quantifier-free A, the formula [x . a / y ] A is unique, so there is no ambiguity in (3). Proposition 5.11.12 Let M be a model over a functorial metaframe, u E M , and let A, A* be modal formulas, [a/x], [a*/x*]assignments giving rise to equal A*. Then D,-sentences: [a/x] A = [a*/x*]
Proof Recall that (Lemma 2.4.2) a Du-sentence [a/x]A can be presented in the form [b/y]B, where b is a list of distinct individuals (from D,) and r(y) = FV(B); this presentation is unique up to the choice of a distinct list y. So let us show that (1) M, [alx]I= A iff M, [bly] I= B and (I*) M, [a*lx*] != A iff M, [bly] != B. Since both these assertions are similar, it is sufficient to check (1). Let x' be a sublist of x such that FV(A) = r(xt), a' the corresponding sublist of a (i.e. if x' = x . T for an injection T, then a' = a . T). Thus (2)
M, [ajx] I=A iff M, [a'lx'] I= A
by Lemma 5.10.6, since F is w-functorial. Here b is a list of distinct individuals from a', and B is obtained from A' by identifying those variables x j in x', which correspond t o equal individuals aj in a' (cf. the proof of Lemma 2.4.2). Thus if m = lx'l = la'l and n = lyl = Ibl is the number of distinct individuals in a', then for some function a E , C we have a' = b . a, B = [y . a/xt]A, Now by 5.11.7(#),
Since b - a = a', (1) follows from (2) and (3).
5.12. MODAL SOUNDNESS
409
Again we have the converse to the above proposition in the following stronger form. Corollary 5.11.13 Let IF be a metaframe such that for any model M over F, for any u E M , quantifier-free formulas A, A* and assignments [a/x], [a*/x*] in F such that [a/x]A = [a*/x*]A*.
(##) M , a
A [x]e M, a* t= A* [x*].
Then F is functorial. Proof Note that (##)implies 5.11.9 (8) for a quantifier-free A, since [a/x]([xa/y]A) = [ a . a/y]A by 2.4.2(4). So we can apply Lemma 5.11.9.~~ Therefore modal (i.e. functorial) metaframes are exactly those in which forcing can be defined for D,-sentences, cf. Section 5.9.
5.12
Modal soundness
In this section we show that modal metaframes are exactly m(')-sound metaframes Recall that by Definition 5.9.8, for a metaframe model M = (F, M t= A if M, a k A [x]for any ordered assignment (x, a) in F with FV(A) C r(x). For a w-functorial metaframe we can fix the list x as the following simple lemma shows.
c),
L e m m a 5.12.1 Let M = (IF,() be a model over a w-functorial N-metaframe F, A an N-modal formula. Then the following conditions are equivalent.
(2) M, a k A [x]for any ordered assignment (x, a ) i n IF such that FV(A) = r(x);
(3) there exists a list of distinct variables y containing FV(A) such that M, b k A [y] for any ordered assignment (y, b ) i n F. P r o o f Obviously, (1) implies (2), and (2) implies (3). The other way round, assuming (2), let us show (I), i.e. M, [bly] k A for any assignment [b/y] with r(y) FV(A). By 0.0.7, we have x = y . a for some injection a , hence by Lemma 5.10.6, M, [b/y] k A is equivalent to M , [b. a / x ] k A, which holds by our assumption. Finally, assuming (3), let us check (2). In fact, again we have x = n,(y) for an injection a , and a = r,(b) for some b, due to the surjectivity of nu (see ¤ Lemma 5.9.6); thus 5.10.6 can be applied.
>
Recall that MLI=)(F) denotes the set of all formulas valid in F (Definition 5.9.9). "This equality holds for a quantifier-free A; for an arbitrary A it should be replaced by congruence.
410
CHAPTER 5. METAFRAME SEMANTICS
L e m m a 5.12.2 For any metaframe F: (1) ML(=)(F)is closed under necessitation and generalisation. (2) If F is zu-functorial, then MLI=)(F)is closed under modus ponens. (3) If P is w-functorial, then for any B E MFG
(4) If P is functorial, then MLI=)(F)is closed under strict substitutions. Proof ( 1 ) Rather trivial. Note that by definition (cf. (8) and ( 9 f 1 0 ) in Section 5.9)
M , a b O i B [ x ]e V b E Rl(a) M , b I= B [ x ] ; and
M , a k V y B [ x ]e V c E D, M,7rXll,(ac)I= B [xlly]. (if a E DE). (2) If M, [ a / x ]b B > C and M , [ a / x ]b B , for any assignment ( x ,a ) with a fixed x containing F V ( B > C ) F V ( C ) , then we have M , [ a / x ]!= C . Hence M k C by Lemma 5.12.1. (3) Obviously, it suffices to check the first equivalence. (+) follows readily from (1). (+) Suppose F b V y B and consider a model M over F. Let r ( x ) = FV(VyB), then for any assignment ( x ,a ) we have M , a k V y B [x].Since y @ x , this means that M , ac != B [xy] for any assignment ( x y ,ac). Hence M b B by Lemma
>
5.12.1. (4) Suppose F k A. Consider a strict substitution S = [ C / P ( y ) ]where , P is an m-ary predicate letter, y = ( y l , .. . ,y,) is a distinct list containing F V ( C ) , and let us show that F k S A . Recall that a substitution instance [ C / P ( y ) ] A is defined up to congruence and can be obtained from a clean version A' of A by replacing every subformula of the form P ( y f )with [ y f / y ] C(y' may be not distinct). By Lemma 5.10.10, for any model M over F we have
and thus
FbAiffFbAO. So we may assume that A is clean (i.e. A = A"). For a model M = (IF,<)and a list of distinct variables x such that r ( x ) = F V ( A ) 2 F V ( [ C / P ( y ) ] A we ) , have to show that M F S A , i.e. that
(4.1) M , [ a / x ]!= [ C / P ( y ) ] Afor any assignment [ a / x ] .
5.12. MODAL SOUNDNESS
Since A is clean, B V ( A ) n r ( x ) = 0. Consider the model N = ( F , 7 ) such that
7 + ( P )= { b I M , [b/yI i= C ) , $ ( Q ) = E+(Q) for any other predicate letter Q. We claim that
By our assumption, F i= A, thus N , [ a / x ]b A, and so (4.2) implies (4.1). To show (4.2), let us prove
(4.3) N , [ a / z ]t= Biff M , [ a / z ]i= S B for any subformula B of A and for any assignment [a/z]with r ( z ) 2 F V ( B ) , T ( Z ) nB V ( B )= 0. The proof is by induction on the complexity of B . If B = P ( z . a ) for some map a , then by Definition 5.9.4 and the choice of N, N , [a/z]i= B iff ( a . a) E T,-+(P)iff M , [ a .a / y ] I= C. By Lemma 5.11.7 in a functorial metaframe we have
M , [ a .u / y ] l= C iff M , [ a / z ]b [z . a / y ] C(= S B ) , and thus B satisfies (4.3). If B = B1 A B2 and (4.3) holds for B 1 , B 2 , we obtain it for B :
N , [ a / z ]I= B iff N , [ a / z ]k B1 & N , [ a / z ]!= B2 iff M , [ a / z ]i= S B 1 & M , [ a / z ]i= SB2 iff M , [ a / z ]b SB1 A SB2 (= S B ) . The cases B = B1 > B 2 , B = B1 V B2 are similar to the above, and the cases B = I,B = Q ( z . a ) (Q # P ) are trivial. If B = OiB1, and (4.3) holds for B1, then for an n-tuple a we have:
N , [ a / z ]i= B iff V b E R r ( a ) N , [ b / z ]i= B I iff V b E Rr ( a ) M , [ b / z ]k SB1 iff M , [ a / z ]i= Oi(SB1)(= S B ) , i.e. (4.3) holds for B . If B = 3vB1, then by our assumption, v @ r ( z ) . Also r ( z v )n B V ( B 1 ) = 0 , since B is clean. So by Definition 5.9.4 and the induction hypothesis we have
N , [ a / z ]l= B iff 3c (ac E Dn+l & N , [aclzv]t= B1) iff 3c(ac E D ~ + '& M , [aclzv]i= SB1) iff M , [ a / z ]b 3v(SB1)(= S B ) . ~ ' So (4.3) holds for B . The case B = VvB1 is left to the reader. 21Notethat in our case S3vB1 = 3vSBl
CHAPTER 5. METAFRAME SEMANTICS
412
Lemma 5.12.2 (3), (4) yields the following convenient description of the set of strongly valued formulas ML(=)(F) for a functorial metaframe IF. Proposition 5.12.3 Let IF be a functorial N-metaframe. Then ML(=)(F)
=
=
{A E MF$) {A E MFA=)
I IF l= An for I P l= A" for
any n E w ) any n E w ) .
Proof Recall that 3 = V A ~so , the second equality follows from 5.12.2(3). Since An is a substitution instance of A, we have
The other way round, suppose V n IF k An. By Lemma 2.5.30, every substitution instance SA of A is congruent to a formula of the form [y/x]SIAn, where S1 is a strict substitution. By 5.12.2(4) and 5.11.8, the latter formula is valid in IF. Thus SA is valid, by 5.10.11. Corollary 5.12.4 If IF is a functorial metaframe, then MLZ(IF) is conservative over ML(IF). R e m a r k 5.12.5 For an arbitrary N-metaframe IF, we can only state that of a formula A E MFN ML'(IF)" C ML(F). In fact, if all MF:-instances are valid in IF, then all its MFN-instances are also valid, but not the other way round. So we cannot claim that ML(F) C ML=(IF). The next two lemmas are full analogues of 5.3.5 and 5.3.6, so we leave their proofs to the reader. L e m m a 5.12.6 Let IF be an arbitraq N-metaframe, M = (IF,() a model over 0 such IF, and let (" be a propositional valuation i n its n-th level Fn, n that En(pk) = [+(PC)for any k 2. 0. Let Mn = (Fn,en) be the corresponding propositional Kripke model. Then for any N-modal propositional formula A, for any a E Fn and for any assignment (x, a )
>
L e m m a 5.12.7 Let F be a w-functorial N-metaframe, A an N-modal propositional formula. Then P k A n zffFnkA. R e m a r k 5.12.8 The w-functoriality of IF is essential in Lemma 5.12.2, which is used in the proof of 5.12.7. By 5.12.3 and 5.12.7 we obtain: Proposition 5.12.9 For a modal metaframe IF
5.12. MODAL SOUNDNESS
413
Corollary 5.12.10 A11 theorems of K N are strongly valid i n every N-modal metaframe P. Proof
Obvious, since they are valid in all propositional frames Fn.
H
Exercise 5.12.11 Show that Proposition 5.12.9 (and Corollary 5.12.10) holds for any N-metaframe F (not necessarily functorial). Hint: for a substitution instance A' of a propositional formula A and for an assignment (s,a) (where FV(Af) r ( x ) and a E D n ) the truth of M, a k A' [x]can be checked in the propositional frame F,. However the claim of the above exercise is not so important, because as we will show in Theorem 5.12.13, only modal (i.e. functorial) N-metaframes validate Q K ~ ) Therefore . only modal metaframes are interesting from the logical point of view. L e m m a 5.12.12 If a metaframe IF is functorial, then the set ML(')(F) is closed under necessitation, generalisation, modus ponens, and arbitrary substitutions. P r o o f For a functorial IF, ML(=)(P) is substitution closed, by Lemma 2.5.27. (Note that ML(=)(F) is the largest substitution closed subset, the 'substitution interior7of ML(=) (IF).) By Lemma 5.12.2 and Proposition 5.12.3, it follows that ML(=)(F) is closed under necessitation and generalisation (for a functorial F) as well as under modus ponens (for an arbitrary metaframe IF). For example, consider generalisation. Suppose A E ML(')(IF). Then An E ML(=)(P) for any n, and hence (VxA)" = VxAn 6 ML(=)(F) by 5.12.2. Hence VzA E ML(')(P) by 5.12.3. H T h e o r e m 5.12.13 (Soundness t h e o r e m ) Let the following properties are equivalent:
F be an N-metaframe. Then
(1) F is modal;
(2) ML(F) is a predicate N-modal logic without equality;
(3) ML=(F) is a predicate N-modal logic with equality;
P r o o f (1) + (3), (1) + (2). For a modal metaframe F, let us check the strong validity of the predicate axioms and the axioms of equality, i.e. the validity of their n-shifts (for n 2 0). We fix a model M = (IF, and do not indicate it in the notation of forcing.
c)
CHAPTER 5. METAFRAME SEMANTICS
414
A1 := V y P ( x ,y) > P ( x ,z ) , for y, z 4 r ( x ) ,y # z. Suppose [ablxz]!= V y P ( x ,y), ab E D;+l; then by 5.9.4, for any c E D,, [abclxzy]!= P ( x ,y), which is equivalent to Vc 'c D,, ac 'c E+(P). Hence ab E ( + ( P ) ,and thus [ablxz]!= P ( x ,z). Therefore [ablxz]!= Al. A2 := P ( x ,z ) > 3yP(x,y), where y, z # r ( x ) ,y # z. Similarly to the previous case, assume [ablxz]I= P ( x ,z ) , i.e. ab E J + ( P ) . Then for some c E D, (viz. for c = b ) , [abclxzy]!= P ( x ,y), and thus [ablxz]!= 3 y P ( x ,y). So [ablxz]!= A2.
A3 := V Y ( Q ( X2 ) P ( x ,Y ) ) 3 ( Q ( x )3 VyP(x,Y ) ) , where Y # r ( x ) . Assume [./XI != V Y ( Q ( X3) P ( x , Y ) ) , a E D;. Then for any b E D,, we have
If also [ a / x ]k Q ( x ) ,i.e. a E c+(Q),then for any b E Du, [ablxy]!= Q ( x ) . Hence [ablxy]I= P ( x ,y) for any b, and thus we obtain
Therefore M != A3
A4 := V y ( P ( x , y )3 Q ( x ) )3 ( ~ Y P ( ~3, YQ )( x ) ) ,Y e x Assume [./XI != QY ( P ( x ,Y ) 3 Q ( x ) ) and [alxl != ~ Y P ( Y X ),. Then [ablxyl P ( x ,Y ) for some b E D,, and also
Thus [ablxy]!= Q ( x ) ,i.e. a E E+(Q),which implies [ a / x ]!= Q ( x ) . Therefore M k A4.
A5 := ( x = x). [alx]!= x = x holds trivially, since a = a. A6 := ( y = z > ( P ( x ,y) > P ( x ,z ) ) , y, z 6r ( x ) . We may assume that y, z are distinct, since otherwise M != A6 is trivial. If [abclxyz]I= y = z , and [abclxyz]k P ( x ,y), then by Definition 5.9.4, b = c and ab E J + ( P ) ,which implies [abclxyz] P ( x ,z ) by the same definition. Therefore M I= A6.
5.12. MODAL SOUNDNESS
415
The implications (2) + (4), (3) =+ (5) are obvious. ( 5 ) + (4) follows by Remark 5.12.5. ( 4 ) + (1). We assume Q K N C ML(IF) and show that IF is functorial. ( I ) F is permutable, i.e.
a R r b + T , ( a )Ryn,(b) for any a E T,, n > 0. Suppose a R l b , a E DE. Consider the QKN-theorem
, = { ~ ~ ( b Then )). and a model M = ( I F , J ) such that J + ( P )= { ~ , ( a ) ) J+(Q) M , b Q(nu( x ) )[XI and M , a I=(P(na( x ) ) )A OiQ(na(x))[XI. Hence M , u 3 x ( P ( ~ , ( x )A) o i Q ( ~ , ( x ) ) ) .
+
Since M , u b B 1 , we have
i.e.
M , c b ( P ( x )A O i Q ( x ) )[XI for some c E D:. Thus c E ( P ) ,i.e. c = .rr,(a) and cRyd for some d E E+(Q), i.e. for d = ~ , ( b )Therefore . 7r,(a)R~~,(b). (11) IF is w-functorial. We apply Lemma 5.10.6.
cf
whenever , a E D,, b E D,. (IIa) aRf b U R ~ V In fact, assume aRfb. Consider the QKN-theorem
where p E PLO,and a model M = ( I F , < ) such that J+(p)= { v ) . Then (by Definition 5.9.4) M, b b p [y],(since T D ( b ) = v ) , and so
since aRf b. Thus
M , u b 3y Oip. Since M, u k B2, we have M , u I=Oip, which implies uRiv.
(IIb) ( a c ) ~ : + l ( b d+ ) a R l b for n > 0. Suppose ( a c ) ~ T + l ( b dConsider ). the QKN-theorem
where 1x1 = n, y $! r ( x ) , and a model M = (IF,[)such that <+(P)= { b ) . Then M , bd k P ( x ) [xy].
CHAPTER 5. METAFRAME SEMANTICS Since (ac)R:+' (bd), we have
and so M, a != 3yOiPn(x) [XI. Now from M , a k Bg the choice of M.
[XI,
we obtain M, a k OiP(x) [x],and thus aRTb by
(IIc) ac E Dnfl & aRTb + 3d ( a c ) ~ y + ' ( b d )for n
> 0.
In fact, assume ac E D;+l, a R l b , and consider the QKN-theorem
where 1x1 = n, y @ x, and a model M = (P,J) such that J + ( P ) = {b}. Then M , b k P ( x ) [XI,and so
Since M, a k B4 [x],we also have
i.e. M, ae != OiP(x) [xy]for any e E D,. In particular, M, ac != OiP(x) [xy]. So there exists g = (gl, . . . ,gn+1) E Dn+' such that
Hence (gl, . . . ,g,) = b , i.e. (ac)~:+'(b~,+~),and so we can take d = gn+l. (IId) uRiv
+ Va E D,
3b E D, aRjb.
Consider the QKN-theorem
for p E P L O and a model M = (IF,<)such that < + ( p ) = {v). Then we obviously have M, v k p, and M, u t= Oip. Since M, u != B5, it follows that M, u != Vy Oip. Thus
which yields M, b != p [y] for some b E ~ : ( a ) . By definition, the latter means that 7ra(b) E <+(p), i.e. 7ra(b) = v, or b E D,. (111) (a,a )RS(b1, b2) =+ bl = bar for a E D,, bl,
b2
E D,.
This property (mm2)easily follows from the validity of the QKZ-theorem x = y > Oi(x = y). In fact, obviously, a a 1 x = y [xy]. Thus aa k x =
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y > Oi(x = y) [xy] implies a a I= Oi(x = y) [xy], which is equivalent t o (111). So we have proved (5) + (1). To check (111) for the case without equality (i.e. for the proof of (4)+(1)) we need a longer argument. Let (a, a)R:(bl, ba), a E D,. Consider the QKN-theorem
Take a model M = (F,5) such that
Then M, a k OiQ(x, x) [XI,since M , b I= Q(x, x) [x] for any b. We also have M, a I= P ( x ) 1x1; hence M, u I= 3x(P(x)AQQ(x, x)). Thus from M , u b B6 we obtain:
for some a l , a2 E D,. Then M, alas k P(x1) A P(x2) [xlx2], which implies a1 = a2 = a, since [+(P)= {a). Now we have
and so M, bib2
Q(xi, x2) [xixz],
since (a, a)Rq(bl, ba). Thus (bl, bz) E tS(Q), i.e. bl = b2. Therefore IF is modal.
W
Theorem 5.12.13 means that modal metaframes are exactly m-sound (and also m'-sound) metaframes. This allows us to introduce the metaframe semantics M E and M N for modal predicate logics (with or without equality) generated by the class of N-modal (or functorial) metaframes. This is the largest sound modal Kripke-type semantics generated by metaframes. Therefore we obtain a precise criterion of logical soundness in metaframes. In the next section this criterion will be applied to Kripke bundles and C-sets. Now let us make some remarks on terminology. According t o general definitions, a metaframe validating a modal logic L (predicate or propositional), should be called an 'L-metaframe'. If L is a propositional logic, 'L-metaframes7 are those, for which every F, is a propositional L-frame (Proposition 5.12.9); such metaframes are not necessarily modal. But if L is a predicate logic, then IF is an L-metaframe iff L C ML(=)(IF); so every L-metaframe is modal (Theorem 5.12.13).
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In particular, for a propositional logic A, QA-metaframes are just modal Ametaframes. So e.g. F is an S4-metaframe iff every F, is reflexive and transitive. An S4-metaframe is also called propositionally intuitionistic. On the other hand, a QS4-metaframe, is a 1-modal S4-metaframe. Finally let us explain why the modified forcing i=*described in 5.9.17 does not really change the notion of logical soundness. To see this we consider an arbitrary modification !=I of forcing satisfying the clauses (1)-(9) from Definition 5.9.4. Then we define the notions of truth, validity and strong validity for k1 in the natural way. Let ML'(')(F) be the corresponding set of strongly valid formulas in a metaframe IF. If this set is an m.p.1. (=), we say that F is modally sound for k1 (respectively, without or with equality). This notion is described as follows. Proposition 5.12.14 Consider a modified forcing relation +I satisfying 5.9.4(1)(9). (1) If Q K N
ML1(F), then IF is a modal metaframe;
(2) If b1 is equivalent to k i n modal metaframes, 2.e.
M, a k1 A [x] H M, a I=A [x]
for any model M over a modal metaframe, for any assignment (x,a ) i n M and formula A with FV(A) C r(x), then for any metaframe P the following conditions are equivalent: (i) F is modally sound for b1 (with or without equality); (ii) P is modally sound; (iii) IF is modal. Moreover, ML'(=)(F) = ML(')(F)
for modal (i.e. modally sound) IF
Proof (1) We can repeat the part (4) =+ (1) from the proof of 5.12.13. The argument does not use the clause (lo), because it does not involve forcing M, a -JyB[x] with y E r ( ~ ) . ~ ~
+
(2) Obvious.
Thus a 'reasonable' modification of 5.9.4(10) does not affect logical soundness. This applies to the forcing k* from 5.9.17, because it is equivalent to b in modal metaframes. In fact, by 5.9.4 we have
"One can easily rewrite the proof using only clean formulas and forcing M, a k A [x] with = FV(A).
T(X)
5.13. REPR..ESENTATION THEOREM FOR MODAL METAFRAMES 419
Now let 1x1 = n, and let a E T n be a permutation
then ((x - xi)xi) . a = X. By 5.10.6 we can further write M , a != 3xiB [x] w 3c E D(a) M, ((a- ai)c) . a I=B [x]. But ( ( a - ai)c) .a = (al, . . . , ai-1, c, ai+l,. ..,an), so the existence of c is equivalent to the existence of b E Dn such that b - bi = a - ai and
M, b k B [x]. Thus k satisfies the inductive clause (lo*) from 5.9.17 for 3 (and similarly for V). Hence the equivalence of M, a != A [x]and M, a b* A [x]easily follows by induction on \A].
5.13
Representation theorem for modal metaframes
Recall that a preset F = (C, D, p) over an N-precategory C gives rise to the metaframe Mf (F) = (Fn)nEwwith the relations
cf. Definition 5.6.14. By 5.9.13, ML(')(M~ (F)) = ML(')(F). We also know that every Kripke bundle F = ( F ,D, p) is associated with a metaframe Mf ( F ) = (Fn)nEw(cf. Definition 5.3.2) as well as with a preset F' (over some precategory), cf. 5.8.5. By Proposition 5.8.5 these constructions are coherent, i.e. Mf (F) = Mf (F'). Recall (Definition 5.3.2) that the relations in Mf (F) are a R l b iff V j aajpibj & a sub b. By 5.9.13, ML(')(F)
= ML(=)(M~ (F)); hence ML(')(F)
= ML(')(F').
Lemma 5.13.1 (1) If F is a preset over an N-precategory C (or an N-modal Kripke bundle),
then Mf (F) is an N-modal metaframe.
(2) If F is a C-set over a category C (or an intuitionistic Kripke bundle), then Mf (F) is a QS4-metaframe. Proof (1) By the above remark, it is sufficient to consider only presets. To show the and show that .nu : Fn 4 F,. To check the w-functoriality, consider a E T,,,
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monotonicity, suppose aR:b, 7 E Ci(u1v)Now note that (3) p,
u = na(a), v = ~ g . ( b )Then . p, - a = b for some
. ( a . a ) = (p, . a ) . a.
In fact, (PY. ( a . f f ) ) j = P r ( ( a ' f f ) j ) = Pr(au(j)) = (Py .a)u(j) = ((Py . a ) . a ) j . So by (3) we have: p,
. ( a . a) = (p, . a) . a = b . a.
Thus (a - a )R l ( b . a ) . Next, let us check the lift property for nu. If ( a . a)RYb1, u = ng.(a), v = nO(bl), then p, . (a . a ) = b' for some y E Ci(u, v). By (1) it follows that (py ' a ) . a = b', i.e. b' = n u b , where b = p, . a , and thus aR/b. Therefore nu : F,, + F,. To check the property 5.11.1 (mmz), suppose (a, a)Ra(bl, bz), a E D,, bl , bz E Dv. Then for some y E Ci(u, v)
i.e.
bl = p,(a) = b2. Thus (mm2) holds. (2) If F is a C-set, then every F, is an S4-frame by 5.6.16. The same holds for intuitionistic Kripke bundles by 5.5.1 or by the observation that if IF is an intuitionistic Kripke bundle, then the corresponding C-preset F' is a C-set (Proposition 5.8.5), and Mf (F) = Mf (F').
Therefore by Proposition 5.9.13, Theorem 5.12.13, Proposition 5.12.3, and Corollary 5.12.4 we obtain Proposition 5.13.2 Let F be a preset over an N-precategory C (or an N modal Kripke bundle). Then (1) ML(') (F) = ML(') (Mf (F)) is an N-m.p.1.(=);
(3) ML'(F)
is conservative over ML(F).
(4) if F is C-set or an intuitionistic Kripke bundle, then ML(=)(F) > QS~('). Let us now show that all countable modal metaframes can be represented by C-sets. Definition 5.13.3 We say that D = (D, : u E W) is a system of countable domains if every D, is countable.23 In this case we also say that a metaframe with a system of domains D has countable domains. 2 3 ~ e c a lthat l according t o our terminology, a countable set may be finite.
5.13. REPRESENTATION THEOREM FOR MODAL METAFRAMES 421 T h e o r e m 5.13.4 (Representation theorem) (1) Let F be an N-modal metaframe with countable domains. Then F = Mf (F') for some preset F' over an N-precategory C. Moreover, if IF is a QS4-metaframe, then C is a category and F' is a C-set.
(2) There exists a modal (QS4-) metaframe F,for which (1) does not hold, and all but one domains are countable. Proof (1) Let F = ((Fn)nEw, D). Consider the N-precategory C with ObC = W and
Note that if F is a QS4-metaframe ( N = I), then C is a concrete category, i.e. the composition of morphisms is the composition of functions and identity morphisms are identity functions. Recall that Rn are reflexive and transitive in this case; thus f o g E C(u, W) whenever f E C(u, v), g E C(v, w), URVRW,and also idD, E C(u, u) for u E W. Consider the C-set F' = (F, D , p), in which pf = f for f E Ci(u, v) and F = FR(C). Let us ensure that P = Mf (F') according to Definition 5.6.14, i.e.
for a E DZ, b E D t , (recall that p j = f). In fact, i f f . a = b , f E Ci(u,v), then aR;b by (1.1). The other way round, suppose aRyb. Let a l , . . . , a n , a n + l . . . be an enumeration of D, starting a t our a = (al, . . . ,an). Using Lemma 5.10.5 (2), by induction we construct a sequence b l , . .. ,b,, bn+l . . . in D, such that b = (bl, . . . ,b,) and (al, . . . ,a k ) ~ f ( b l ,. .. ,bk) for all k > n. Then the function f sending each ak t o bk belongs to Ci(u,v). In fact, ( a l , . . . ,ak)Rf(bl,.. . ,bk) for k 2 n, by our construction, and this is also true for k < n, by monotonicity of T,, where (T : I k Inis the inclusion map. Thus (1.2) holds. Finally note that (1.2) implies F ( = FR(C)) = Fo:
-
In fact, suppose uRiv, a E D,. By 5.10.5(4), a ~ , ! bfor some b E D,, thus by ) so Ci(u,v) # 0. (1.2), f ( a ) = b for some f E C ~ ( U , Vand The other way round, if f E Ci(u, v), then a ~ ,f! (a) and f (a) E D,. Hence uRiv by 5.10.5(3). (2) Consider a QS4-metaframe = ((Fn)nEw,D ) , such that
Fo = (W, R) is a 2-element chain: W = {UO,ui), R = W x W - ((211,UO));
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a R n b @ a s u b b & b s u b a ++ Vid(ai = a j D,",, j 5 k .
* bi = bj) for a E D&, b E
To check that IF is a modal metaframe, we apply 5.10.2, 5.10.5. First, for a permutation a E T n , n, is an automorphism of Fn. In fact, b,(i) = iff a R n b iff Vi, j (ai = a j ++ bi = bj) iff Vi, j (a,(i) = a,(j) (a.a)Rn(b.a). A similar argument shows that 7r0 is monotonic for any a E T,,. Second, w; has the lift property. In fact, suppose aRn-lb, i.e. asub b and b sub a. Let u, v be the worlds of a, b, respectively. Then for any c E D, there is d E D, such that (ac)Rn(bd): if c = a* for some i, take d = bi; otherwise take d # bi for any i (since D, is infinite, such an individual d always exists). The case n = 1 described in 5.10.5(4) is trivial. By definition, the property (mmz) also holds: (a, a)R2(bl,b2) + bl = b2. All the F, are S4-frames, since the relation sub is reflexive and transitive. Now suppose F corresponds to a C-set F' = (F, D, p). Let p E C(uO,ul); then p, is a function from Duo to D,, . But ID,, I > ID,, 1, so there exist a l , a z E Duo, b E D,, such that a1 # a2 and p,(al) = p,(a2) = b. Then (al, a2)R2(b,b) in Mf (F'), but not in IF. One can easily construct similar examples for any N > 1. H
*
Therefore modal metaframes with countable domains are nothing but C-sets. However we do not know if the semantics of modal metaframes is stronger than functor semantics (either in the case of C-presets or C-sets). For the intuitionistic case (discussed below) this question is also open.
5.14
Intuitionistic forcing and monotonicity
5.14.1
Intuiutionistic forcing
Now let us consider the intuitionistic case. We shall begin with a simple observation that QS4-metaframes generate a sound semantics for superintuitionistic logics via Godel-Tarski translation (Proposition 5.14.7). But this semantics is probably not maximal in the intuitionistic case, so our goal will be to identify a larger class of 'intuitionistic sound metaframes'. Let us first give a definition of intuitionistic forcing in S4-metaframes. Definition 5.14.1 A valuation J in an S4-metafr-ame F is called intuitionistic if it is intuitionistic in every Fn7i.e. for any predicate letter Pjn, n 2 0
The pair (F,E ) is called an intuitionistic metaframe model. Henceforth until Section 5.19, we shall consider mainly S4-metaframes.
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423
Definition 5.14.2 An intuitionistic model M = (F,c) gives rise to the forcing relation M , a It. A [x] (where a E Dn, 1x1 = n , A E IF', FV(A) C r(x)) defined by induction:
(11) M , a It P;(x. a)[x] iff (a.a) E J+(P2);
(111) M , a It- (xj= xk) [x] iff
aaj= a k ;
(IV) M , a It (B A C) [x] iff M , a It B [x] and M , a It C [x], (V) M , a IF (B v C) [x]iff M , a It B [x] or M, a It C [x]; (VI) M, a It. (B 2 C) [x] iff V b E Rn(a) (M, b It B [x]+ M, b It C [x]); (VII) M , a It- VyB [x] iff V b E Rn(a) Vc E D(b) M , ~ , ~ ~ , ( b cIt-) B [xlIy]; M, a It- 3yB [x] iff 3b E (Rn)-'(a) 3c E D(b) M, ~ , ~ ~ , ( bIct )B [xlly]. This definition of forcing is motivated by Godel-Tarski translation. E.g. the intuitionistic forcing for VyB resembles the modal forcing for UVyB. In more detail the connection between modal and intuitionistic forcing will be discussed later on. However note that for the ]-case the intuitionistic and the modal definition differ, unlike Kripke frame or Kripke sheaf semantics. This situation was already discussed for quasi-bundles in section 5.5. Definition 5.14.3 Let F be an S4-metaframe. An intuitionistic formula A is called true in an intuitionistic model M overF (notation: M It- A) if M, a It A[x] for any assignment (x, a) with r(x) 2 FV(A); valid in F (notation: IF It A) if it is true in all intuitionistic models over IF; strongly valid (respectively, strongly valid with equality) in F if all its IF(respectively, IF')-substitution instances are valid in F. Strong validity is denoted by It.+ (respectively, It.+=). Similarly to the modal case, we use the notation
Note that ILE(IF) is the largest substitution closed subset of IL~=)(F). As we have pointed out for Kripke bundles, it may be the case that F It SA for any IF-substitution S, but not for any IF=-substitution; so ILZ(F) may be not conservative over IL(F), cf. Remark 5.12.5 for the modal case. Similarly to the modal case we have IL=(F) n IF E I L (F).
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Definition 5.14.4 A metaframe IF is called intuitionistic(') sound (i(=)-sound, for short) if IL(')(F) is an s.p.1. (=). As we shall see in Section 5.16, i'-soundness implies i-soundness. Now let us show that all QSCmetaframes are i=-sound. This happens because intuitionistic forcing for QSCmetaframes corresponds to modal forcing via Godel-Tarski translation. Definition 5.14.5 Let F be a QS4-metaframe;M = (IF, J) a metaframe model. The pattern of M is the model Mo = (IF,to) such that
for any j, n 2 0. It is clear that Mo is an intuitionistic model and Mo = M if M itself is intuitionistic. L e m m a 5.14.6 Let M be the same as in Definition 5.14.5. A E IF' and any assignment (x,a) with FV(A) C x
Then for any
Proof By induction, cf. Lemmas 3.2.15, 5.5.7 If A = P(n,x), o E C,, 1x1 = n, then Mo, a IF A [x]u r,a E J$(P) H Rm(w,a) C J + ( P ) , M, a I t OA [x]u Qb E Rn(a) n,(b) E Jf ( P ) e w,(%(a)) C E+(P).
-
But &(.rr,a) = w,(Rn(a)) for a morphism w, : F, F, (see the remark after Definition 1.3.30), so the statement is true in this case. For the induction step let us only check the quantifier cases. Suppose A = QyB and the statement holds for B. Then
The argument for the 3-case is slightly longer as the intuitionistic and the modal definitions differ in this case. Suppose A = 3yB and the statement holds for B. We have
-
-
The latter holds if M , a k AT - just take b = a. The other way round, suppose M, b b AT 1x1. As we know, QS~(') t- AT OAT (Lemma 2.11.2), so M b AT OAT by soundness (5.12.13), and thus M, b I= OAT [x], which implies M, a k AT.
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425
The case A = 3xiB is reduced to the above:
All other cases are left to the reader. Proposition 5.14.7
Let IF be an QS4-metaframe, A E I F = . Then
(1) IF It- A iffIF != A ~ .
(2) The following three assertions are equivalent: (a) IF It-+ A; (b) Vm IF It- Am;
(c) IF b+ AT. (3) IL(')(F)
=T
~ ~ ( ' ) ( Fand ) therefore IL(=)(F) is an s.p.l.(=).
P r o o f Similar to 5.5.12, 5.5.13. (1) (Only if.) Assume IF It A. For a metaframe model M over P let us show M b AT, i.e. M, a t= AT [x] for any appropriate assignment (x, a). By Lemma 5.14.6, the latter is equivalent to Mo, a It A [x], which follows from our assumption. (If.) Assume IF k AT. Let M be an intuitionistic metaframe model over IF, and let us show M It- A, i.e. M , a It A [x]for any appropriate assignment (x, a). Since Mo = M , by Lemma 5.14.6, M , a It A [x] iff M , a != AT[x], and the latter follows from IF k AT. (2) The proof is completely analogous to 5.5.12. Use 5.12.3 for (b)'+(c) and soundness (5.12.13) for (c)+(a). H (3) follows readily from the equivalence (a)w(c) in (2). Proposition 5.14.7 shows that QS4-metaframes are i(')-sound. As we shall see later on, the class of i-sound metaframes is larger, but the question, whether i-sound metaframes generate a stronger semantics, is open. The same happens to i'-soundness. Corollary 5.14.8 If F is a C-set (or an intuitionistic Kripke bundle), then IL(=) (F) is an s.p. 1.(=); moreover, IL(') (F) =T ML(=)(F) . Exercise 5.14.9 Using 5.14.6, check the following properties of intuitionistic forcing in QS4-metaframes analogous to the properties of modal forcing (cf. Lemma 5.10.6(2), Lemma 5.10.10, Lemma 5.11.7, Proposition 5.11.12). Let IF be a QS4-metaframe, M an intuitionistic model over IF. Then (1) for any ordered assignment (x,a), for any a E T,,, where 1x1 = n 1 m and for any formula A E IF' with FV(A) 2 r ( x . a )
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( 2 ) for any congruent intuitionistic formulas A, B and any ordered assignment ( x ,a ) with F V ( A ) G r ( x )
(3) for any a E C,, for any distinct lists of variables x = ( X I , . . .,x,), y = ( y l , . .. ,ym), for any intuitionistic formula A with F V ( A ) E r ( y ) , for any a E Dn
M , a It- ( [ x. n / y ]A ) [ x ]
M , a . a It- A [ y ] ;
( 4 ) if A, A* are intuitionistic formulas, [ a / x ] ,[a*/x*]assignments giving rise to equal D,-sentences: [ a / x ]A = [a*/x*]A*, then M , a It- A [ x ]u M,a* It- A [x*]. As we shall see later on, all the claims in 5.14.9 extend to arbitrary i'-sound metaframes. Now let us consider arbitrary SCmetaframes. Let us first note that the quantifier clauses in Definition 5.14.2 are analogues of the combined clause (9+10) for modal forcing from Section 5.9. Now if we consider two options in the intuitionistic case, we obtain
(VII.l) if y $ x, then M , a It- V y B [x]iff V b E Rn(a)Vc E D ( b ) M , bc It B [ x y ] , M , a It I y B [ x ]iff 3 b E ( R n ) - l ( a )3c E D ( b ) M , b c It- B [ x y ] ;
(VII.2) for I 5 i 5 n M , a It- V x i B [x]iff Vb E Rn(a) Vc E D(b) M , (b-bi)c It- B [ ( x - x i ) x i ] iff Vd E T: ( R n( a ) )Vc E D ( d ) M , dc It B [ ( x- xi)xi]. M , a It 3xiB [x]iff 3 b E (Rn)-'(a) 3c E D ( b ) M ,dc It- B [ ( x- xi)xi]. Note that (VII.l) corresponds to the clause ( 9 ) from Definition 5.9.4. Let us show that in QS4-frames (VII.2) corresponds to the clause 5.9.4(10), i.e. M , a It- V x i B [x]iff M , ^ai It- VxiB[5ii], M , a It 3 x i B [ x ]iff M , G It- 3xiB [5ii]. In fact, by (VII.l)
M , a - ai It V x i B [ x- xi iff Vd E Rn-'(a-ai) Vc E D ( d )M,dc It- B [ ( x - x i ) z i ] iff Vd E Rn(7r:a) Vc E D ( d ) M , d c It B [ ( x- xi)xi]. Now in QS4-metaframes T: is a morphism, so the latter is equivalent to (VII.2). But in arbitrary SCmetaframes (VII.2) may not be true, because it may happen that nY(Rn(a))# Rn(r:(a)), cf. Section 5.9.24 The reader can construct a counterexample as an exercise, cf. 5.9.16, 5.9.18. 240r note t h a t
*
M , a I t - V x i B [x] M , a - a i It-VxiB [x-xi] may not imply
M , a It- W x i B [x]a M , a - ai It- W x i B [x- xi]
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427
Later on we shall see that (VII.2) and its analogue for the ]-case hold in all i(=)-sound metaframes.
Remark 5.14.10 These comments motivate an alternative definition of forcing (It-") for the quantifier case instead of (VII): This can be rewritten as follows: There also exists an altrnative version (IF*) resembling 5.9.17: All these definitions are equivalent in i(')-sound metaframes. Now we obtain an analogue to Lemma 5.12.2 (I), (2) and Corollary 5.12.12. Proposition 5.14.11
Let IF be an S4-metaframe. Then
(1) ILI=)(F) is closed under generalisation and modus ponens;
(2) IL(=) (IF) is closed under formula substitutions;
(3) the following conditions are equivalent: (a) IL(')(F)
is an s.p.1. (=);
(b) QH(=) g IL(=)(F); (c) QH(=)
c ILI_=)(P).
Proof (1) (I) Let us first consider generalisation. Let M be an intuitionistic model over IF, B E IF(=). Assuming M It- B , let us prove M It VyB, i.e. M , a It- VyB [x] for any assignment ( x , a ) with r(x) 2 FV(VyB). Suppose 1x1 = n. If y @ x, then
M , a It VyB [x]iff Vb E Rn(a)Vc E D(b) M,rXlly(bc) It B [xlly]. But M , ~ ~ ~ ~ , It( bB c[xlly] ) holds, since M It. B . (11) Now let us consider modus ponens. Let M be an intuitionistic model M over P. Assuming M It B, B > C , let us check that M IF C. Let x FV(C), y = FV(B) - x. By our assumption, for any assignment (XY7 a b ) 7 M, ab It- B [xy]; M , ab IF ( B 3 C)[xy].
c
Since every relation Rn is reflexive, we readily obtain (for any assignment (XY a b ) ) (3.1) M , ab IF C [xy]. Now let us show that
CHAPTER 5. METAFRAME SEMANTICS In fact, (3.1) implies
as one can easily check. By (c), we also have
since VyC 3 C is an intuitionistic theorem. Now (3.2) follows from (3.3), (3.4), and the reflexivity of Rn. (2) Note that the composition of substitutions is a substitution, cf. the argument in the modal case in 5.12.12. (3) The implications (a) + (b), (b) + (c) are obvious. (c) + (b) is also obvious, IL(')(P) is substitution closed. So assuming (c), let us check (a). Due to (2), it remains to consider modus ponens and generalisation. First note that IL!,==)(F)is closed under congruence. In fact, suppose A E IL4(IF) and A B. Then (A B ) E QH(=) by 2.6.9, and hence (A > B ) E QH(') (by AX^), MP and substitution). So by the assumption (c), (A 2 B) E IL(')(IF). Since by (1) IL!,==)(IF) is closed under MP, it follows that B E IL!=)(IF). Now the closedness of IL(=)(F) under modus ponens follows easily. In fact, suppose A, A 3 B E IL(')(P).
"
Then S, SA, S(A > B) E IL(=)(F) for any formula substitution. By Lemma 2.5.13, S(A > B ) A (SA > SB). Since IL~=)(P)is congruence closed and MP-closed (by (I)), this implies S B E IL(=)(P). Eventually B E IL'(IF). Finally, let us consider generalisation. We suppose A E IL(')(IF) and show that VxA E IL(')(P). This does not follow directly from (I), because a substitution instance of VxA is not always of the form VxA' for a substutition instance A' of A (cf. Lemma 2.5.13), and so we need a little detour. So for a substitution S , let us prove that IF It- SdxA. Let y be a new variable, y @ V(VxA) U FV(S); then by 2.3.25(13)
Hence by 2.5.12 and 2.5.13(3), SdxA A Sdy[y/x]A A VyS[y/x]A. As we already know, the validity in IF is closed under congruence and generalisation. So it is sufficient to show that IF Ik S[y/x]A. By Lemma 2.5.14 (for complex substitutions) we can rename the bound variables of S; so
5.14. INTUITIONISTIC FORCING AND MONOTONICITY
with x $! BV(S1). Now we introduce yet another new variable obviously have
429 XI.
Then we
By 2.5.17, there exists a formula substitution Sosuch that x @ FV(So) and
But then by 2.5.15 we obtain
Eventually from (#I)-(d4) and 2.3.24 it follows that
It remains t o notice that validity in IF respects variable substitutions. In fact, if IF It B, then IF It VxB by (1). By 2.6.15(xxv), VxB > [ylx]B is an intutionistic theorem, so it is valid in IF by our assumption (c). By (I), we can apply modus ponens, thus IF It [y/x]B. Therefore, since A E IL(')(P), we obtain F It [x/xl][y/z]SoA; thus IF It S[y1x1A by (#5),since (IF) is closed under congruence.
ILL')
Remark 5.14.12 The above proof cannot be simplified by concluding IF I/So[y/x]A directly from A E IL(')(IF), before we know that [y/x]A E IL(')(F). Since Sodoes not always commute with [ylx] for x E FV(So), we had to replace x with x' and Sowith S1. Corollary 5.14.13
2-'
-soundness implies i-soundness.
Proof QH' C ILE(IF) implies Q H = (QH')" implies i-soundness by 5.14.11(3).
5.14.2
C (IL=(F))" C IL(P), which rn
Monotonic metaframes
Now let us consider an important property of intuitionistic sound metaframes. Definition 5.14.14 An S4-metaframe IF is called monotonic(') the condition
if it satisfies
(im) M , a l t A[x] & aRnb + M , b It A[x] for any A E IF('), for any intuitionistic model M = (IF, J), a distinct list x of length n containing FV(A) and any a, b E Dn. We already mentioned this property in Section 5.5 when quasi-bundles were discussed. The next lemma proves that monotonicity is really necessary.
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L e m m a 5.14.15 If the formula B := p > (T 3 p) is strongly valid i n an S 4 metaframe IF (in the language with or without equality, respectively), then IF is monotonic('). P r o o f Let us check (im) for an arbitrary formula A. Consider the following substitution instance B' := A > (T > A ) of B. Assume a R n b and M, a It A [x].Then M, a It T > A [x] since M , a I I- B' [x] and Rn is reflexive. Obviously, M, b It T ( = I> I) [x], and thus M, b It A [x]. The idea of the previous proof is quite clear; B' expresses the property (im) for A - it states that if A is true now, then T > A is also true, and thus A will always be true in the future, by the definition of forcing for implication. Hence we obtain
P r o p o s i t i o n 5.14.16 Let F be an S4-metaframe such that H Then F is monotonic(=).
c
IL(=)(F).
Let us now describe monotonic metaframes in terms of accessibility relations.
D e f i n i t i o n 5.14.17 A n S4-metaframe F is called semi-functorial (or sm, n > 0 functorial, for short) i f for any a E C;, (00) V a ,b E Dn ( a R nb + ( r u a )Rm(rub)), i.e. if every T , is monotonic. L e m m a 5.14.18 A n S4-metaframe F is semi-functorial iff the following holds. (11)
F is permutable (see Definition 5.1 0.1), i.e. (Oo) holds for all (simple) permutations a.25
(12.2) V n > 0 V u ,v E W V a E DZ V b E D: Vc E D, Vd E D, ((ac)Rn+l( b d ) + aRnb). (I3)
V n > 0 V a ,b E Dn ( a R n b + (aa,) Rn+' (bb,))
(Also note that (12.1) can be replaced by its particular case with n = 1.) P r o o f Every a E , C is a composition of permutations, simple embeddings and simple projections. So it suffices to check the monotonicity of r, for a = ( I ; ( n 2 0 ) , a? ( n > 0 ) and for all permutations. Now recall that a: = W 41,T D , ( a ) = u for a E D z , TO: ( a ) = aa,, ru=(ac) = a. D e f i n i t i o n 5.14.19 A semi-functorial metaframe is called semi-functorial with equality (briefly, s=-functorial) if it satisfies the condition (cf. 5.11.1) 25Cf.Definition 5.10.1, Lemma 5.10.5.
5.14. INTUITIONISTIC FORCING AND MONOTONICITY (mm2) Qa, bl, b2 ((a, a)R2(bl,bz)
+ bl = b2).
L e m m a 5.14.20 In an s-functorial metaframe all properties (mm,)
Qa, b (aRnb + a sub b)
are equivalent for n
> 2. So (mm,)
holds in every s'-functorial
metaframe.
Proof (mm2) implies (mm,), since a R n b implies (ai, aj)R2(bi, bj) by (Il), (12.2); cf. Lemma 5.11.2. The other way round, suppose (mm,). Let a = (a,a)R2(bl,b2) = b , and consider the projection a E C,z such that
Then r,a = an, r u b = blb;-'. By s-functoriality (r,a) Rn ( r u b ) , hence by (mm,), (7r,a)sub(r, b), and thus bl = b2. W An inductive argument shows that s'-functoriality implies monotonicity. The steps of the proof are almost trivial, due to the definition of forcing 5.14.2, so the only problem is the atomic case. Recalling that
and <+(P)is Rn-stable, we see that monotonicity exactly corresponds to (00). Now let us give more details. L e m m a 5.14.21 A metaframe is s(')-functorial iff it is monotonic(').
Proof (Only if.) By induction on the complexity of A. j (7r,x) [x] iff r,a E <+(Pjm). If A = Pjm(r,x), then (im) holds, since a It Pm By monotonicity of P, a R n b implies (7r,a)Rm(7r,b), and since M is intuitionistic, we obtain r,b E <+(Pjm), i.e. b It A [x]. If A = (xi = xj), aRnb, we have all- xi = x j [x] iff ai = a j , blt xi = x j [x] iff bi = bj.
By (I4), a sub b, so ai = a j implies bi = bj. The induction step for A, V is obvious. Let A = B > C . Suppose aRnb and M I b Iy A[x], i.e. M, c IF B [x], M I c Iy C [XI for some c E Rn(b). By transitivity, aRnc and thus M, a Iy A [x]. Let A = VyB. Suppose y @ x , a R n b and M, b Iy A [x].Then by Definition 5.14.2, M I cd Iy B [xy] for some c E Rn (b) and some d E D(c). By transitivity, aRnc, and thus M, a Iy A [x].
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Now suppose A = VxiB, aRnb, M, b Ijc A [x]. Then by Definition 5.14.2, M, bi, Ijc A [Zi]. Recall that Si = =,a for a =.:6 So since n, is monotonic, a R n b implies Si~"6i,and thus M, Si Ijc A [Zi], as we have already proved. Therefore M, a ljc A [x],by Definition 5.14.2. Let A = 3yB, y 6 x, and suppose a R n b and M, a IF A [x]. Then M, cd IF B [xy] for some c E (Rn)-l(a) and some d E D(c). By transitivity, we have cRnb, and hence we obtain M, b IF A [XI. The case A = 3xiB is similar t o A = VxiB and is left to the reader. (If.) Assuming (im), let us show that P is monotonic('). Note that ( m m ) ~ readily follows from (im) for A = (xl = x2). If aRnb, then M, a I t Pm(rr,x) [x] in the intuitionistic model M = (IF,[) such that [+(Pm)= Rm(s,a). Thus by (im), M, a IF Pm(n,x) [x], i.e. n u b E ¤ [+ ( P m) , and so (n,a) Rm(rr, b ) . A
Now we obtain Proposition 5.14.22 Every i(')-sound S4-metaframe is s(=)-bnctional. R e m a r k 5.14.23 Monotonicity is an intrinsic property of intuitionistic forcing, thus Lemma 5.14.8 explains why we confine ourselves to s-functorial metaframes in our further considerations.
5.15
Intuitionistic soundness
In this section we prove intuitionistic soundness for a certain class of S4metaframes. Its description is more complicated than in the modal case. As explained in the previous section, we certainly need s-functoriality corresponding to monotonicity. We also need the 'quasi-lift property', an intuitionistic analogue of the lift property for jections rr,. But unlike the modal case, this is yet insufficient, and we need two extra conditions related to interpretation of quantifiers. In SCmetaframes we use the notation znfor the corresponding equivalence relation z ~ onn Dn; it is called n-equivalence. Definition 5.15.1 A n S4-metaframe F is called quasi-functorial (i-functorial, for short) i f all its jections s, are quasi-morphisms. So we have L e m m a 5.15.2 A metaframe IF = ((Fn)nEw, D) is i-functorial iff it is s-functorial (i.e. all rr, are monotonic) and all rr, have the following quasi-lift property: (la)
V a E Dn Vb' E Dm((n,a)Rmb'
+ 3 b E Dn(aRnb & s u b zmb')).
5.15. INTUITIONISTIC SOUNDNESS
Recall that in the modal case we defined functorial metaframes, where all r, are morphisms, and also weakly functorial metaframes, where T, are morphisms for injective a . The intuitionistic analogue of functoriality is quasi-functoriality, but as we shall see now, there is no need in the intuitionistic analogue of weak functoriality. Lemma 5.15.3 Let M be an intuitionistic model over an i-bnctorial wimetaframe P. Then we can replace Rn with enin the truth definition for 3:
M , a It 3yB [x] iff 3 b zna 3c E D(b) M,.rr,lly(bc)It B [xliy]. Proof In fact, if there exists bc E Dn+' such that b R n a and M , r x l l y ( b c I)t B [xlly], then, by the quasi-lift property (la:), there exists b'c' E Dnf l such that (be)R ~ +(b'c'), ' b' zna , (recall that n,; (b'c') = b'). Now by ( o b i ) ,
and thus by monotonicity 5.14.14 (im),
The modified definition of forcing in the 3-case resembles the familiar definition for intuitionistic Kripke models, but with xn replacing '='. So in our semantics, a 'witness' for 3yB does not always exist in the present world, but should exist in an equivalent world (cf. section 5.5). This reflects the basic idea that intuitionistic models26 'distinguish' worlds up to z O ,individuals up to z l , n-tuples of individuals up to e n .
Remark 5.15.4 One can see that in the above proof monotonicity and quasi~ . we also lift property are used explicitly only for injections a? and ~ ~ 1 1But use the monotonicity of forcing relying on the monotonicity of n, for arbitrary (perhaps, non-injective) a, cf. Lemma 5.14.21. Therefore in further definitions we suppose monotonicity of all nu. The next lemma shows that in functorial metaframes the intuitionistic truth definition for the 3-case is quite analogous to the modal one. This simplifies the proof of Lemma 5.14.6. 2 6 ~ u to e t h e monotonicity of forcing, cf. 5.14.14.
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L e m m a 5.15.5 Let M be a n intuitionistic model over a functorial metaframe. Then for any B E IF', y 6 x
Proof The same as in 5.15.3, but now T,; is a morphism, so we can take b' = a . Note that forcing is monotonic, because in a functorial metaframe every ruis monotonic. H R e m a r k 5.15.6 In a similar way one can show that the equivalence M, a It 3xiB [x]H 3c E D(a) M, aic It- B [xixi]. holds in functorial metafkames. Then we readily have M , a It 3xiB [x]
* M,ai I t 3xiB [xi].
The same equivalence for V was already proved in 5.14, cf. the condition (VII.2).
Intutionistic metaframes As we already know, functoriality is equivalent to modal soundness (Lemma 5.11.6 and Theorem 5.12.13). But i-functoriality does not yet imply intuitionistic soundness as we shall see in section 5.16. Now let us consider two other soundness conditions specific for the intuitionistic case. Definition 5.15.7 A n intuitionistic metaframe 0-metaframe, for short) is an i-functorial metaframe satisfying (20) (forward 2-lift property)
V a E Dn Vb E Dm+' ( ( r u a )Rm( r ~ b+ ) 3c E Dn+l (aRn(r;c) & (ru+c)Rm+' b)) . (3a) (backward 2-lift property)
V a E Dn V b E D m + ' ( ( ~ ~ b ) R m ( ~+u 3c a ) E Dn+' ( ( ~ 3 c ) R " a& bRm+' (r,+ c))). Recall that al;. E Tm,m+l, a; E Tn,n+l are simple embeddings and a+ E Cm+l,,+l is the simple extension of a E C,,, see the Introduction. So a metaframe is intuitionistic iff it satisfies (Oo), ( l a ) , (2a), (30) for any a . Definition 5.15.8 A weak intuitionistic (wi-) metaframe is an s-functorial metaframe satisfying ( l a ) , (2a), (3a), for all injections a. Note that in this case (Oa) holds for arbitrary o. Definition 5.15.9 A n intuitionistic metaframe with equality (an i=-metaframe, for short) and respectively, a weak intuitionistic metaframe with equality (a wi'-metaframe) is an s'-functorial i-metaframe (respectively, an s'-functorial wi-metaframe).
5.15. INTUITIONISTIC SOUNDNESS
Figure 5.10. Forward 2-lift property
Figure 5.11. Backward 2-lift property.
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L e m m a 5.15.10 z=-metaframes (respectively wi'-metaframes) are exactly imetaframes (respectively, wi-metaframes) satisfying (mmz) from 5.11.1. Our next goal is to show that i'-metaframes are exactly i=-sound metaframes (Theorem 5.16.13) and at the same time - wi=-metaframes (Theorem 5.16.10(3)). R e m a r k 5.15.11 Let us give a motivation for 2-lift properties. The starting point is the following simple observation on tuples (called the 'lift property' in [Skvortsov and Shehtman, 19931). Let
then there exists c E Dn+' such that a = n;c and i.e. b = a,(l) . .. a,(,)b,+l; ) the case when Rk n,+c = b; in fact, put c = ab,+l. This shows (20), ( 3 ~ for are the equality relations. See Fig. 5.12 below.
Figure 5.12. n,a is a 'common part' of b and a; c is their 'join'. We can also check these properties for a metaframe Mf (IF) associated with a C-set F = (C, D, p) over a category C. In fact, if for a E DE and b E D,"+' we have nus = (a,(l), . . . ,a,(,))Rm(bl,. . ., b,) = n F b , then there exists a morphism p V i bi = p,(~,(~)).Then consider
E
C(u, v) such that n+mb = pp . (n,a), i.e.
and extend it to C
:= ~ ' b , + ~ E D,n+l.
It follows that
and thus obviously (nu+c ) ~ ~ + l b .
5.15. INTUITIONISTIC SOUNDNESS
437
So (2a) holds; (3a) is checked in a dual way. We see that in the first example the condition(2a) expresses the'co-amalgamation property' of tuples (Fig. 5.12).
Figure 5.13. The second example corresponds to Fig. 5.14. Here p; sends a to p, - a; similarly for p:. Again c is the 'join' of c' and b; it exists, because they have a 'common part' n y b . L e m m a 5.15.12 In i-functorial metaframes (20) is equivalent to
V a E DnVb E ~ ~ + ' ( . r r , axmn F b =+ 3c E ~ ~ + ' ( a ~ ~ ( n&;7r,+c c ) xmf' b1)(2-a) Similarly (3a) is equivalent to
As we can see, these conditions are obtained from (20) and (30) by changing Rm, Rm+l t o z m , z m + l .
Proof (20) + (2-0). We assume (20) and prove a stronger claim than (2-0). Namely, for a E Dn+l, b E Dm+' such that (7r,a)Rm(nyb) we find c' E Dn+l such that aRnn;4c' & nU+c1xmb . First we apply (20) to a , b and find c shown in Fig. 5.13. Then using the quasi-lift property ( l a f ) we find c'. Finally we note that aRn (rTc)Rn(.rr;c') by monotonicity (On;). (2"a) + (20). Assume (2-a). Given a , b such that (n,a) Rm(n? b), by the quasi-lift property ( l a ) , we can find a' E Rn(a) with n,al xmn y b , see Fig. 5.15. Then by applying (2-a) to a' and b , we obtain c such that a' xn n;c, b xm+l.rr,+c. Hence by transitivity aRnnn+c. n,+cRmf l b is obvious. The proof of (3a) @ (3-0) is dual, so we skip it.
C H A P T E R 5. METAFRAME SEMANTICS
Figure 5.14.
Figure 5.15.
5.1 5. INTUITIONISTIC SOUNDNESS
439
Lemma 5.15.13 In i-functorial metaframes (20) is equivalent to
and dually for (3a). Proof
Similar t o Lemma 5.15.12 (an exercise).
Now we are going to prove intuitionistic analogues of the results from Sections 5.11 and 5.12. The property ( l a ) together with s-functoriality means that T, is a quasip-morphism. In modal logic we have a stronger requirement that T, is a p morphism. In fact, in intuitionistic models, due to monotonicity (im), z n equivalent worlds are indistinguishable. Thus equality transforms into zn in the intuitionistic case. The next assertion is an intuitionistic analogue of Lemma 5.10.6.
Lemma 5.15.14 Let IF be an s(=)-functorial metaframe. Then P is weakly functorial iff for any A E IF('), for any assignment (x,a) of length n, for any injection a E Tmn such that F V ( A ) c x - a , for any intuitionistic model M = (P, E ) M , a I t A[x] e M , a - a It A [ x . a]. (*i) So (*i) means that M , a It A[x] does not depend on the choice of a list
x
> FV(A) and possible renumbering of variables.
Proof (+) is proved by induction on the complexity of A (cf. Lemma 5.10.6). The atomic cases are obvious. The cases when the main connective of A is LJ or A, are also easy, and we leave them to the reader. Let A = B > C. Suppose M, a ly A [x], i.e. M , b It- B [x]; M , b Iy C [ x ] for some b E Rn(a). Then ( a . a )Rm(b . a ) by s-functoriality, and so
by the induction hypothesis, thus M, a . a Iy A [x . o]. The other way round, suppose M, a . a Iy A[x . a], i.e. M, b' It B [ x . a ] , M, b' Iy C[x - a]. for some b' E Rm(a . a). Then by ( l a ) , b . a zmb' for some b E Rn (a). Now by monotonicity,
and thus by the induction hypothesis,
Hence M, a Iy A[x].
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Now let us consider the case A = VyB. Supposing M, a . a ly A [x . a], and let us prove M, a Iy A [x]. By definition, we have c E Rm(a . a), d E D(c) such that Ml ."(x.u)lly(cd) IY B [ ( x . a>llvl. Since (r,a)Rmc = r?(cd), by applying (2a), we obtain b E Dn, e E D1 such that aRnr;(be) = b & ( b . a)e = xu+(be)~"+'(cd). Hence by s-functoriality,
where 1 is the length of (x . a ) 11y. Now note that
for some 7, as we saw in the proof of 5.10.6. Hence by a standard argument we have the same for arbitrary tuples:
Then by monotonicity we obtain
and since ( x . a)lly = (xlly) - r , by the induction hypothesis, it follows that
as required. But aRnb, thus a Iy VyB [x], The other way round, suppose a Iy A [x]. Then
for some b E Rn(a), e E D(b). Hence by the induction hypothesis,
where
T
is the same as above, i.e. we obtain
But (a.a)Rn(b - a ) , by s-functoriality; thus eventually,
by Definition 5.14.2.
5.15. INTUITIONISTIC SOUNDNESS
441
The case A = 3yB is dual to A = VyB. The proof for this case is obtained by inverting the accessibility relations Rn, changing Iy to It. and applying (3a) instead of (20). In more detail, suppose M , a . a IF A [x - a]. By definition, there exists c E (Rm)-'(a. a), d E D(c) such that
From cRm(n,a), by (3a), we obtain b E Dn, e E D1 such that b n; (be) = bRna & (cd)~ ~ + ' - (a)e. Hence by s-functoriality,
where 1 is the length of (x a)ll y. Then by monotonicity and since ( x . a)ll y = (XI1 y) .T by the induction hypothesis, it follows that nxllv(be) IF B [xllyl. Since aRnb, we obtain a Iy VyB [x]. To show the converse, suppose a IF A [x]. Then nxlly(be) IF B[xllyl for some b E (Rn)-'(a), e E D(b). Hence by the induction hypothesis,
This is the same as "(x.u)lly((b. ale) IF B [(x.a)llyl. Since by s-functoriality, ( b a)Rn(a - a ) , it follows that
(+) Assuming the equivalence (*i) for suitable formulas A, let us prove the properties (la), (2a), (3a) for an injection a E T,,. (la)
Let a E Dn, b' E Dm, and assume (n,a) Rmb'. Consider the formula
and the intuitionistic model M = (IF,J) such that
(recall that IF is an S4-frame). Then b' E Ef (P)-J+(Q), thus M, n,a Iy A [nux]. Therefore by (*i), M, a Iy A [x], i.e. there exists b E Rm(a) such that M, b It P(x,x) [x] and M, b Iy Q(n,x) [x]. Thus r u b E ((+(P) - J+(Q)) = zm(b'), i.e. b' zmnub.
CHAPTER 5 . METAFRAME SEMANTICS
442 (20)
Let a E Dn, c = (.rr1;2c)d E D ~ + ' and , assume that (.rr,a)Rm(n1;2c). Consider the formula A := VyP(n,x, y) with x = ( X I , . .. ,x,), y 6x, and the intuitionistic model M = (F, <) such that
<+(P)= {c' E D ~ + '1 T ( c ' R ~ + ' c ) ) . Since c
6<+(P),we have
and so M , r,a Iy A[n,x], since (n,a)Rm(.iryc). Thus by (*i), M , a Iy A [ x ] ,i.e. there exists b = (n7b)d E Dn+' such that
But (n,x) y = nu+( x y ) , thus nu+(b)$! ( + ( P ) ,so by definition of M , r,+ (b)Rm+'c, as required.
(3u)
The proof is quite similar to the previous case. Now let a E Dn, c (n1;2c)dE Dm+', (.rryc)Rm (.rr,a) .
=
Consider the formula A := 3yP(n,x, y), where x = ( X I , . . . ,x,), y @ x , and the intituionistic model M = (IF,<) such that <+(P)= R ~ + ' ( c ) . Since c E <+(P),we have M , c Il- P(.rr,x, y) [n,x, y], and so M,.rr,a ItA [.rr,x], since ( n y c ) R m(.rr,a). Thus by (*i) M , a It- A [ x ] i.e. , there exists b = (.rr;b)d E Dnf' such that (.rr3b)Rnaand M , b Ik P(K,x, Y ) [xY], which is equivalent to
i.e. nu+b E <+(P),and hence cRm+'(n,+ b ) .
As in the modal case, Lemma 5.15.14 allows us to arbitrarily change a list x 2 F V ( A ) in M , a It- A [ x ] ;e.g. we can take x = F V ( A ) . So in the truth conditions for quantifiers we may always assume y $! x , and thus we have
a It- V y B [ x ]iff V b E Rn(a) Vc E D ( b ) b c It- B [ x y ] ; a Il- 3yB [x]iff 3bRna 3c E D ( b ) bc It- B [xy]. (or equivalently, bRna may be replaced with b zn a in the clause for 3, cf. Lemma 5.15.3). Now let us prove an intuitionistic analogue of Lemma 5.10.9.
L e m m a 5.15.15 Let M = (IF,<) be an intuitionistic model over an S4metaframe, A E IF('), F V ( A ) C r ( x ) . Also let y 6 B V ( A ) , y' $! V ( A ) , X' := [y'/y]X , y' $! r ( x ) , and A' := A[y H y']. Then M , a It- A [x]iff M , a It- A1[x'].
5.15. INTUITIONISTIC SOUNDNESS
443
Proof Since F is a wi(')-metaframe, due to the previous remarks, we prove the equivalence only for x = FV(A). We argue by induction and consider only the atomic and the quantifier cases; other cases can be easily checked by the reader. Let A = P ( x a). Then A' = P ( ( x . a)') = P(xl - a ) , so
-
alt- A[x] iff ( a . a ) E <+(P) iff alt A1[x']. Let A = VzB and suppose the assertion holds for B. Then z # y, y' by the assumption of the lemma, and z @ x, by our additional assumption; so z $ x'. Thus a It- A [x] iff Vb E Rn(a)Vc E D(b) b c It- B [xz];
a It- A' [x'] iff Vb E Rn(a)Vc E D(b) bc It- B' [x'z]. Since (xz)' = x'z, the right parts of these equivalences, are also equivalent by the induction hypothesis. So the assertion holds for A. If A = 3zB, the argument is similar: a It- A [x] iff 3 b zna 3 c E D(b) bc It B [xz];
a It- A' [x'] iff 3 b xn a 3c E D(b) b c It- B' [x'z] , and again the right parts of these equivalences are equivalent. Let us prove an intuitionistic analogue of 5.10.10. L e m m a 5.15.16 Let F be a wi(')-metaframe, M an intuitionistic model over IF Then: (c)
M , a It- A[x] e M , a I t A1[x]
for any congruent formulas A, A* and any appropriate assignment (x, a).
Proof (Cf. Lemma 5.10.10.) We assume x = FV(A), 1x1 = n. It is sufficient t o consider the case when A' is obtained from A by replacing its subformula QyB with Qy'B', where Q is a quantifier, y' 9 V(A), y @ BV(B), B' = B[y y']. Now the proof is by induction. (1) If A = VyB, then A1 = Vy'B'. By our assumption, x = FV(A), s o y @ x, and thus a k VyB [x] @ Vb E Rn (a) Vd E D(b) ad It- B [xy] H Vb E Rn(a)Vd E D(b) ad It- B' [xy'] (by Lemma 5.15.15 ) e a It Vy'B'
[XI.
(2) If A = 3yB, the proof is almost the same. We have A' = 3y1B'. Suppose y', y 9x, and thus a k 3yB [x] 3 b xn a l d E D(b) ad It- B [xy] H 3 b xn a 3 d E D(b) ad Ik B' [xy'] (by Lemma 5.15.15 ) e a It 3y1B' [x].
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444
(I), (2) prove the base of the induction. The step is rather routine, so we only consider the case A = VzC, A1 = VzC1. We have x = FV(A), z @ x, and thus a It A [x]e 'v'b E Rn(a)Vd E D(b) ad It- C [xz] % 'v'b E Rn(a)Vd E D(b) ad It B1 [xz] (by induction hypothesis ) e a II- VzB1 (= A1) [x].
The remaining cases are left to the reader.
W
Let us prove the intuitionistic analogue of 5.11.7. Lemma 5.15.17 Let F be an s(=)-functorialrnetaframe. Then P is an i-rnetaframe iff
(#)
for any a E C,, distinct lists of variables x = (XI,.. . ,x,), y = (PI,. . . ,ym), and formula A E IF(') with FV(A) C r(y), for any model M over P,a E Dn: M , a I t [ ( x - a ) / y ] A [x]w M , a . a It A[y].
Similarly to 5.11.7, this equivalence is stated for all congruent versions of [X
- ~IYIA.
Proof ('Only if'.) Recall that [(x . a)/y]A is defined up to congruence and can be obtained from a clean version AO of A such that BV(AO)n r(xy) = 0 (and FV(AO)= FV(A) y ) by replacing [y c-, x . a]. By Lemma 5.15.16 we may assume that A" = A. Now we argue similarly to 5.11.7 by induction. (1) If A is yi = yj or P ! ( ~ T), . T E Ctm or A is, the proof is the same as in 5.11.7. (2) Suppose A = B 3 C. If a Iy [(x . a)/y]A 1x1, i.e. for some b E Rn(a)
and 1x1, b lpL [(x.~ ) I Y I C then by the induction hypothesis b . a II- B [y] and b . a Iy C [y].
Since a R n b implies ( a . a )R m ( b . a) by (Oa), we have a - a Iy (B > C) [y]. The other way round, if
then for some c E Rm(a . a ) c
IF B [y] and c IY C [y].
5.15. INTUITIONISTIC SOUNDNESS
445
Now by ( l a ) ,there exists b E Rn(a) such that b . a z m c . Then by monotonicity b-a
It- B [ y ]and b
a Iy C [ y ] .
So by the induction hypothesis, b IF [(x.a ) / y ] B[x]and b ly [ ( x .a ) / y ] C[XI.
Since aRnb, this implies
a 1Y [ ( x ~ . )/YIA [XI. (3) Let A = VzB, then z $ r ( x y ) by the choice of A so
If
a IPL [ ( x.~ ) I Y I A [XI, then there exists b E Rn(a),d E D ( b ) such that bd
Iy [ ( x z )- a+lyz]B [xz].
So by the induction hypothesis (b. a)d = ( b d ) . a+ Iy B [ y z ] .
Since ( a . a )Rm(b- a ) , it follows that
The other way round, suppose a.0 Iy A[y].Then for some c' E Rm(a-a),d E D(cl) c'd I#B [ y z ] . SOfor c := c'd we have (nua)Rmc'= TTC.Hence by (2a)there exists b E Dn+' such that aRn(nn+b)and (nu+b ) ~ ~ + ' c . Put b' := n y b , then
Since
(no+b )R ~ " C
Iy B [ y z ] ,
by monotonicity we have
nu+b Then by the induction hypothesis,
Iy B [ y z ] .
446
CHAPTER 5. METAFRAME SEMANTICS
But b = b'bn+l, while aRnb'. Hence by the definition of forcing
as required. (4) The case A = 3zB is dual t o the previous one. The proof makes use of (3a),cf. the 3-case in the proof of 5.15.14. ( 5 ) The cases of A, V are almost trivial. ('If'.) (Cf. the proof of 5.15.14.) For a E Em, let us check the properties ( l a ) , ( 2 ~ 1(, 3 ~ ) .
(la)
Let a E Dn, b' E Dm, (.rrua)Rmb'. Consider A := P ( y ) > Q ( y ) , lyl = m and the intuitionistic model M = (IF, c) such that
E f ( P ) = Rm(b'), (F+(Q)= Rm(b')- em(b'). Then b' E [+ ( P ) - <+(Q),thus M , r u a IY A [ y ] .Therefore by (#),
i.e. there exists b E Rm(a) such that M , b IF P(r,x) [ x ]and M , b Iy Q ( r U x )[XI. Thus rub E ((F+(P)- (F+(Q))= em(b'),i.e. b' zmrub.
(20)
Let a E Dn, c = (nl;Lc)d E Dm+', and assume that (r,a)Rm(rI;.c). Consider the formula A := V z P ( y ,z ) with lyl = m , z $ r ( y x ) ,and the intuitionistic model M = (IF,[) such that
[ + ( P ) = {c' E D ~ + '1 ~ ( c ' R ~ + ' c ) ) . Since c $ <+ ( P ) ,we have
and so M , n,a Iy A [ y ] ,since ( r u a ) R m ( r T c ) . Thus by (#), M , a QzP(.rr,x, z ) [ x ] i.e. , ~there ~ exists b = (rl;b)d E Dn+l such that
Iy
aRn (.rr; b) & M , b Iy P(.rr,x, z ) [xz]. But ( r u x ) z = .rr,+(xz), thus r,+(b)$Z E+(P), so by definition of M , ru+( b ) R m fl c , as required.
(3a)
The proof is similar to (2a). We assume a E Dn, c = ( r T c ) d E Dm+', (.rrTc)Rm(.rr,a)and consider the formula A := 3 z P ( y ,z ) , where lyl = m, z @ r ( y x ) , and the intituionistic model M = ( P , t ) such that { + ( P )= Rm+'(c). Checking the details is left to the reader.
27~ is tessential that here we use VzP(n,x, z ) , a congruent version of [(x. a ) / y ] A with a new variable z (not xi E r(x) - r(n,x)).
5.15. INTUITIONISTICSOUNDNESS
447
Note that unlike the modal case (Lemma 5.11.9, Proposition 5.11.11), we cannot consider only quantifier free formulas A in the proof of 'if7 in the above lemma, because (20)~(3a) are related to quantifiers. Now we obtain an intuitionistic analogue of 5.11.12:
Proposition 5.15.18 Let M be a model over a wi(')-metaframe, u E M , A, A* E IF('), and let [ a / x ] , [a*/x*]be assignments i n M giving rise to equal D,-sentences: [ a / x ]A = [a*/x*]A*. Then
(##)
* M, a* It- A [x*]
M, a Il- A [x]
Proof Similar t o 5.11.12, but now we apply 5.15.14, 5.15.17 instead of 5.10.6, 5.11.7. Proposition 5.15.18 allows us to define forcing for D,-sentences [ a / x ] Ain i-metaframes in the same manner as we did in Kripke sheaves, Kripke bundles and modal metaframes. In the modal case this definition is successful, due to the property (mm,) VaVb (aRnb + a sub b), cf. Lemma 5.1 1.2, Proposition 5.11.12. This property also holds in i'-metaframes (5.14.20), but not always in i-metaframes (cf. Example 2* in 5.16). But imetaframes still enjoy the property (##), perhaps thanks to the following weaker version of (mm,).
Exercise 5.15.19 Show that (mm,")
VaVb (aRnb + 3b' znb a sub b')
holds in i-functorial metaframes. Hint: choose a such that a s u b a & a suba and apply ( l a ) t o c such that a = r,c. Now we can prove intuitionistic analogues of results from Section 5.12. The following lemma is an analogue of 5.12.1.
L e m m a 5.15.20 Let M = (IF,<) be a model over an w(=)-metaframe IF, and let A E IF('). Then the following conditions are equivalent. (1) M It- A ;
(2) M, a It- A [x]for any ordered assignment ( x ,a ) i n P such that FV(A) = r(x);
>
(3) there exists a list of distinct variables r(y) FV(A) such that M, b IF A[y] for any ordered assignment (y, b) i n P. &)
Thus r(l) does not depend on the choice of y. Proof 5.10.6.
Along the same lines as 5.12.1, but now we apply 5.15.14 instead of
CHAPTER 5. METAFRAME SEMANTICS
448
The next lemma is an analogue of 5.12.2. L e m m a 5.15.21 Let IF be an i(')-metaframe.
Then
(1) ILI=)(F) is closed under modus ponens;
(2) for any B E
IF(')
(3) IL~=)(P)is closed under generalisation;
(4) IL(=)(P) is closed under strict substitutions. Note that (1) for B E IF holds in every uri-metaframe IF.
Proof (1) Similar to the proof of 5.12.2(2), but now we use Lemma 5.15.14 (and the reflexivity of Rn 5.15.20). (2) Cf. the proof of Lemma 5.12.2(3); again we use 5.15.14 and the reflexivity of Rn. (3) Follows from (2). (4) The argument is essentially the same as in the proof of 5.12.2(4). Assuming that S = [C/P(y)] is a strict substitution ( F V ( C ) y ) and P It- A, we show that P It- SA, i.e. M , a IF S A [x]for any intuitionistic model M and any assignment (x,a ) with r ( x ) = FV(A) FV(SA).
>
By Lemma 5.15.16, we may assume that A is clean, BV(A) n r ( x ) = 0. Consider the model N = (P, 77) such that v + ( P ) = {b 1 M , b C [Y]), v+(Q) = (+ (Q) for any other predicate letter Q. This model is intuitionistic, by monotonicity of P. Now it suffices t o prove by induction that
(#) N , a I t B [ z ] iff M , a I t S B [ z ] for any subformula B of A and for any assignment (z,a) with r(z) F V ( B ) , r(z) n B V ( B ) = 0.
>
If B = P ( z . a ) for a map a , then SB = [z . a / y ] C . By Definition 5.14.2, the choice of N, and Lemma 5.15.17 we have ) ~ , a . a I t -C [ y ] iff ~ , a l[ tz . o / ~ ] [z], C N , a I F B [ z ] iff ( a . a ) E r l + ( ~iff and thus
(fl) holds.
5.15. INTUITIONISTIC SOUNDNESS
449
If B = B1 > B2 and (g) holds for B1, Bz, we obtain it for B: N, a I t B [z] iff Vb E Rn(a) (N, b IF B1[z] 3 N, b I t Bz [z]) iff V b E Rn(a) (M, b I t SB1 [z] + M, b I t SB2 [z]) iff M, a I t SBI 3 SB2 (= S B ) [z] (where n = la/). Suppose B = VvB1 and (#) holds for B1. By our assumption, v $! r(z), so we have N, a I t B [z] iff V b E Rn(a) Vc E D ( b ) N, bc It- B1 [zv] iff V b E Rn (a) Vc E D(b) M, bc It SB1 [zv] iff M, a IF Vv(SB1) (= S B ) [z]. Thus (#) holds for B . The cases B = B1 A Bz, B1 V Bz, 3vB1 are left to the reader, and the case B = Iis trivial.
Now similarly to 5.12.3 we obtain the following Proposition 5.15.22 Let IF be an i(')-metaframe. IL(=)(F)
= =
{A E IF(=) {A E IF(=)
Then
I F It- An for any n E w ) ,
I IF IF An for any n E w ) .
Therefore IL=(F) is conservative over IL(IF) We also obtain the following analogues of 5.12.9, 5.12.10. Proposition 5.15.23 For an i(=)-metaframe IF
Corollary 5.15.24 All theorems of H are strongly valid in every 2-metaframe. Proof
H C IL(Fn) if Fn is an SCframe.
4
R e m a r k 5.15.25 Note that we cannot extend Lemma 5.15.21 and Proposition 5.15.22 to wi-metaframes, but Proposition 5.15.23 still holds (cf. Exercise 5.12.11 for the modal case). However in the next section we will show that only i(=)metaframes can validate predicate all QH-axioms (and theorems). T h e o r e m 5.15.26 (Soundness theorem) For any i(')-metaframe is an s.p.1. (=).
F,IL(=) (F)
CHAPTER 5. METAFRAME SEMANTICS
450
Proof Soundness of the inference rules is proved by the same argument as in 5.12.12. Now similarly to 5.12.13, we prove the strong validity of the predicate axioms (and the axioms of equality - for i'-metaframes); all these formulas can be supposed clean. So consider an intuitionistic model M = (IF, [).
A1 := b'yP(x, y ) 1P ( x , z ) , for y, z $ x, y # z , 1x1 = n. It is sufficient t o show that ab It- b'yP(x, y ) [ x z ]+ ab IF P ( x , y ) [ x z ] .
So assume ab It V y P ( x ,y ) [ x z ] ;then by 5.15.14 a IF b'yP(x, y ) [x].Hence (by the reflexivity of Rn) for any c E D ( a ) , ac It P ( x , y ) [ x y ] ,which is equivalent to a c E [ + ( P ) , and thus to ac It P ( x , z ) [ x z ] . Therefore ab It- P ( x , z ) [ x z ] .
A2 := P ( x , z ) > 3 y P ( x , y ) , where y , z $ x, y # z , 1x1 = n. Assume ab It- P ( x , z ) [ x z ] ,i.e. ab E [+ ( P ) , which is equivalent to ab It P ( x , y ) [ x y ] . Since a zn a, by the definition of forcing it follows that a It 3 y P ( x ,y ) [x]; hence ab It 3 y P ( x ,y ) [ x z ]by 5.15.14.
A3 := V y ( Q ( x )3 P ( x , y ) ) 3 ( Q ( x )3 V y P ( x 7Y ) ) , where Y $ x, 1x1 = n. Assuming
let us show
In fact, (3.1) means (3.3) b'b E R n ( a ) V C ED ( b ) b e l t ( Q ( x ) > P ( x , y ) ) [ x y ] ,
and we have to show for any b E R n ( a ) (3.4) blt Q ( x ) [ x ]=+b l t V y P ( x , y ) [ x ] .
So we assume
and prove that
which means (3.7) b'd E Rn(b) b'c E D ( d ) d c l t P ( x , y ) [ x y ] .
5.1 5 . INTUITIONISTIC SOUNDNESS Let b R n d ; by monotonicity, (3.5) implies (3.8)
d
Q(x) [XI;
then d E E+(Q), and thus (3.9)
VCE D(d) d c l t Q(x) [xy].
On the other hand, Rn is transitive, so d E Rn(a), and (3.3) implies (3.10)
VcE D ( d ) d c l t (Q(x) 3 P(x,y))[xy].
Eventually from (3.9) and (3.10) we obtain 'dc E D ( d ) d c It P ( x , y) [xy],
which yields (3.6). Therefore (3.4) holds. A4 := V Y ( P ( ~ , Y2) Q(x)) 3 ( ~ Y P ( x , Y3) Q(x)), for Y Assuming (4.1)
a
vy(P(x, Y) 3 Q(x))
# x, 1x1 = n.
[XI
let us show (4.2)
a
( 3 y P ( x , ~2 ) Q(x))
[XI.
Consider an arbitrary b E Rn(a) such that (4.3)
b It 3yP(x, y) [x].
We have to show that (4.4)
bit-Q(x)[x].
By (4.3) we have for some b' znb, c E D(b1), (4.5)
b'c It P ( x , y) [xy].
Now aRnb', so by (4.1) (4.6)
b'c -!I ( P ( x , Y) 3 Q(x))
[XY].
From (4.5), (4.6) we obtain (4.7)
b'c 11- Q(x) [xyl,
which implies (4.8)
b' It Q(x) 1x1,
and the latter implies (4.4) by monotonicity, since b' znb.
CHAPTER 5. METAFRAME SEMANTICS
452
A5
:= (x = x). The proof is trivial.
A6 := (y = z
>.
P(x, y)
> P ( x , z)), y, z $ x, y, z
are distinct, 1x1 = n.
In fact, assume (6.1) abc I t y = z [xyz] and
From (6.1) we readily obtain b = c, and (6.2) implies (abc) sub d by 5.14.20 (14). Hence dn+l = dn+2. Now we have to check that (6.3) d I t P ( x , y) [xyz] + d IF P ( x , z) [xyz]. So suppose (6.3) d lk P ( x , y) [xyz]. By 5.15.18, this implies
Since dn+l = dn+2, (6.4) is the same as (6.5) (dl, . . .,dn, dn+2)
P ( x , 2) [xz].
which (again by 5.15.18) implies d IF P ( x ,z) [xyz].
Thus (6.3) holds.
Recall that QSCmetaframe is i'-functorial. So we can conclude that IL(=)(F) is a superintuitionistic predicate logic for any QSCmetaframe IF. Thus soundness theorem 5.15.26 implies Proposition 5.14.7.
5.16
Maximality theorem
In this section we prove necessary conditions for intuitionistic soundness. To this end, we present Definitions 5.15.7, 5.15.8, 5.15.9 in a more convenient form. But first let us recall basic properties of i-metaframes and introduce some their versions. (Oa) Va, b E Dn (aRnb =+ ( ~ , a Rm ) (rub)),
5.16. M A X I M A L I T Y THEOREM
453
(la)
V a E Dn Vb' E Dm((7rua)Rmb'+ 3 b E Dn(aRnb & r u b zmb')).
(20)
(forward 2-lift property)
(30)
(backward 2-lift property)
V a E Dn V b E Dm+' ((7rl;b)Rm(n,a) bRmfl (xu+c)));
+ 3c E Dn+l((.rr;c)
Rna &
(mm2) Va, bl, 62 ((a,a)R2(b1,b2) =+ bl = bz). Recall that i-metaframes are the S4-metaframes satisfying (Oa), ( l a ) , (2a), (3a) for all a; wi-metaframes satisfy the same conditions for all injective a ; (w)i'metaframes also satisfy (mmz). The correlation between different kinds of metaframes is shown in the diagram below. i-functorial metaframes
wi-metaframes ---+ s-functorial metaframes
i-metaframes
i'-metaframes
-
wi'-metaframes
+s=-functorial metaframes
By Lemma 5.14.14 the condition V a (00) (i.e. s-functorality) is equivalent to the conjunction of the following properties: (11)
(Oa) holds for all permutations a;
(12.1) Vn
> 0 Vu,v E W V a E DZ Vb E D," ( a R n b + uRv);
(12.2) Vn > 0 Vu, v E W V a E DZ V b aRnb), (13)
E
D: Vc E ED, Vd E D, ((ac)Rn+l(bd) =+
Vn > 0 Va, b E Dn (aRnb + (aan)Rn+'(bbn)).
We also need the following conditions:
(12.4) Vn, a, b (x;(a)Rnb =+ 3c E Dnfl (aRnf 'c & (7r;c) xn b)); (14.1) for any a , b E D1
CHAPTER 5. METAFRAME SEMANTICS
454
(14.2) for any n > 0, a , b E Dn+l (x;a)Rn (ryb)+ 3c E Dn 3d, e E D ( c ) (aRn+' ( c d ) & ( c e )Rn+'b); (15.1) (aa,) Rn+'b & bn
# bn+l
3 c E R n ( a ) ( ( c % ) zn+'b);
(15.2) for any a E Dn, b E Dn+2, n > 0
(15.3) for any a E Dn, b E D ~ + n~>, 0
*
(r3"
b)Rn+' (aa,) & bn # bn+ 1 3 c E Dn 3d E D ( c ) ( c R n a & bRn+2( c ~ d ) ) ;
and also the following:28
(15')
for any a E D l , b E Dn+l, n > 0 3c E Dn b zn+lcc,. ( a ,a)R2(bn,b,+~) & b, # bn+l
Later on we will show (Lemma 5.16.9) that each of (I5), (15') is equivalent t o (15.1) & (15.2) & (15.3).
L e m m a 5.16.1 I n i-functorial metaframes (&.I), (14.2) are respectively equivalent to (14.1") for any a , b E D1
( I 4 . r ) for any n > 0 , a , b E Dn+l
Proof Note that (14.1) is ( 2 0 1 ) . In fact world of a E D l ) , so ( 2 0 1 ) is
0 1 = a:
(and r0, ( a ) = ?r:(a) is the
'da,b E D' (nO(a)RO?rO(b) + 3c E ~ ~ ( a R l n :&c ?rO:(c)R1b)).
ri
.
Now if c = ~ 1 ~then 2 , ( c ) = c l , n + ( c ) = c2. By the same reason, (14.1") 0 1 is ( 2 - 0 1 ) from 5.15.12. So by 5.15.12, (14.1) @ (14.1"). Analogously, (14.2) is (20;) and (14.2") is (2"0;), so (14.1) % (14.2"). 2 8 ~ e c a lthat l
6is obtained
by eliminating cl from c.
5.1 6. M A X I M A L I T Y THEOREM
455
The next lemma is similar to 5.11.3 and 5.11.6. L e m m a 5.16.2 Let IF be an s-functorial metaframe, and let a E (Em, - Tmn), m 2 2. Then IF is s'-functorial (i.e. satisfies (mm2)) iff .rr,[Dn] is Rm-stable. Recall that .rr, is surjective iff a is injective. Proof (Only if.) Assume that n,[Dn] is Rm-stable, but F does not satisfy (mm2). Suppose (aa)Rz(blba), b l # b2, and i, j E Im, i # j, a ( i ) = p(j). Now letz9 a := am; then a = an . a E .rr,[Dn]. Next, let c := br1b2by-i. Then ci # c j , so ~ ( a s u b c )and , thus c 9 n,[Dn], by Lemma 5.9.5. On the other hand, consider T E Cn2 such that ~ ( i =) 2, ~ ( j =) 1 for j # i. Then a = .rr,(aa), c = 7r,(blb2), and thus aRmc by s-functoriality. Therefore c E (Rm(.rr, [Dn]) - .rr, [Dn]),which contradicts our assumption. (If.) If IF is s'-functorial, it satisfies (mm,) (Lemma 5.11.2). Then by Lemma 5.11.3, n, [Dn] is Rm-stable. Let us now reformulate the condition (la). Note that for an s-functorial metaframe IF, ( l a ) means that .rr, is a quasi-pmorphism from (Dn,Rn) onto (Dm,Rm). L e m m a 5.16.3 Every s-functorial metaframe satisfies ( l a ) for any permutation a . Proof s-functoriality implies permutability (cf. Definition 5.10.1), so by 5.10.2, for any a E T n , n, is an automorphism of Fn,and thus a (p)morphism. L e m m a 5.16.4 Let IF be an s-functorial metaframe (1) If IF satisfies (la?) for any simple embedding a;,
then it satisfies ( l a ) for
any injection a . (2) If IF satisfies (la;) a.
and (la?) for every n, then it satisfies ( l a ) for any
Proof Follows from 5.16.3 and the observation that the composition of quasimorphisms is a quasi-morphism. L e m m a 5.16.5 Let F be an s-functorial metaframe. Then (1) IF satisfies ( l o ) for any injection a i;Br it satisfies (12.3) and (12.4);
(2)
H satisfies ( l a ) for any a iff it satisfies (I2.3), (12.4), and (15.1);
(3) If IF is 3'-functorial,
then
(a) F satisfies (15.1);
(b) ( l a ) holds for any a iff it holds for any in3ection a (iff IF satisfies (12.3) and (I2.4), by (1)). 2gFor the notation a m , see the Introduction.
CHAPTER 5. METAFRAME SEMANTICS (12.3) can also be replaced with its particular case corresponding to n = 1, cf. Lemma 5.14.18. Note that in (12.3) the worlds of a and b are zO-equivalent, but not necessarily equal.
Proof ( 1 ) (12.3) is the quasi-lift property (10,) and (12.4) is the quasi-lift property ( l a ; ) for n > 0. Since 0 1 is a:, we can apply Lemma 5.16.4(1). ( 2 ) Similarly, we can apply 5.16.4(2). Now (15.1) is (2a:), up to the conjunct b, # b,+l in the premise. But b, = b,+l obviously implies the conclusion of (15.1) -just put c = b; thus (15.1) @ (la:).
(3) (a) By (mm,+l), (aa,+l)Rn+lb implies b, = b,+l, so the premise of (15.1) is false. (b) Obviously follows from ( 2 ) and (3)(a).
Now let us turn t o properties ( 2 a ) , (3a). It is clear that ( 3 0 ) is dual to ( 2 a ) , so it can be analysed in a similar way. L e m m a 5.16.6 Let IF be an s-functorial S4-metaframe. Then
and
( 3 a ) & (37) + ( 3 ( a . 7 ) ) .
Proof We prove only the first implication; the second one is an easy exercise for the reader. So assume ( 2 0 ) & (27). Let a E Em,, 7 E Ckm; then u . T E Ckn Let a E Dn, c = c ' c ~ +E ~Dk+l, and assume that
Then by (27), there exists d
= dfdm+l E
Dm+' such that
( n , a ) R m ( n y d ) = d', ( d . T + ) R ~ + ' c .
Now by ( 2 u ) , there exists b = bfb,+l E D n f l such that aRn(n;b) = b' and (nu+ b) = (b' . a)b,+lRm+ld. Thus ~ ( , . , ) + b = b - ( a -7)' = ((b'. a ) . r)bn+1 = ((nu+b) . T+)R~"( d . T + )
by s-functoriality. So by the transitivity of R ~ + 'it, follows that (n(,,,)+ b ) R k + l c .
w
5.16. MAXIMALITY THEOREM
457
L e m m a 5.16.7 Let IF be an s-functorial S4-metaframe. Then (2a), (3a) hold for any permutation a E T,.
Proof Let us check (30). Let
Put b = (c' ~ - l ) % + ~Then . by s-functoriality, nZ;b = (c' . a - ' ) ~ ~ ( ( aa .) . a-l) = a. We also have:
ng+b = ((c' - a-l)
.
=c ' ~ = + c,~
and thus cRn+l(nU+ b) by the reflexivity of Rn+' So to prove (20), (30) for all injective a (respectively, for all a), it suffices to check them only for simple embeddings a; (respectively for all a;, a"_).This is the same as with ( l a ) in Lemma 5.16.7. L e m m a 5.16.8 Let F be an s-functorial S4-metaframe. Then
(1) (20) & (30) holds for any a iff (14.1) & (14.2). (2) (20) & (3a) holds for any a i f f (14.1) & (14.2) & (15.2) & (15.3). (3) If F is s'-functorial,
then
(a) F satisfies (15.2) & (15.3). (b) (20) & (3a) holds for any a iff it holds for any injection a (2.e. iff satisfies (14.2) & (I4.2), by (2)).
IF
Note that a R n b iff (aan)Rn+'(bb,) by s-functoriality; so we can respectively change (15.2); similarly with (15.3).
Proof (1) As we noticed in the proof of 5.16.1, (201) is equivalent to (I4.1), and ( 2 ~ ; ) is equivalent to (14.2). (30) is obtained from (2a) by replacing every Rk with its converse and permuting a with c. But this replacement does not change (I4.1), (14.2); so (30;) is equivalent to (2a;). (2) ( 2 a 9 is
If g = cd, c E Dn, d E D(c), then
CHAPTER 5. METAFRAME SEMANTICS
458 so (20:)
is equivalent t o
Note that this implication holds if b,
= b,+l.
In fact, put
then (aan)Rn+l(.rrl;+'b) implies a R n c by s-functoriality, and c ~ =dcb,+ld = bRnf' b by reflexivity. Therefore ( 2 a 9 is equivalent to (15.2). Thus an equivalent of ( 3 0 2 ) is obtained from (15.2) by replacing Rn+', Rn+2 with their converses, which is an equivalent of (15.3). (3)(b) By 5.14.20 s-functorial metaframes satisfy (mm,+l). Thus (aan)Rn+l (.rr;+'b),implies (aa,) sub (.rrY+lb), and hence b, = b,+l. So (15.2) holds trivially. Similarly we obtain (15.3).
L e m m a 5.16.9 Let IF be an s-functorial metaframe satisfying. Then (1) (15.1) & (15.2) is equivalent to each of (I5), (15'). (2) If IF also satisfies (I2.4), then each of these conditions implies (15.3) and thus (15) @ (15') @ (15.1) & (15.2) & (15.5'). Proof
Let us first show that (15) is equivalent to (15').
(15) for n = 1
+ (15') for n = 1.
Assume (15) for n = 1:
b'a E D1 b'd E D3 ( ( a ,a )R~(d2,ds) & d2 # d3 + 3 e E D 2 d x3 ee2).
(*)
To check (I5'), suppose ( a ,a ) R 2 ( b l ,b2), bl # b2. Let us find c E D1 such that (bl,bz) x2 (c,c). Put d := blb2bl. By (*) there exists e E D2 such that d x3 ee2 = elezez. If a E T 2 3 is such that a ( 1 ) = 2, a ( 2 ) = 3, then blb2bl . a = blb2, e e 2 - a = eze2; so by s-functoriality it follows that blb2 z2e2e2, and we can put c = e2.
+
Suppose (I5), (15') hold for n, and let us prove (15') for n 1. SOwe assume b E Dn+2, a E D l , ( a ,a )R2(bn+l,bn+2), b,+l # b,+z, and find c E D,+' such that b xn+2 c k + l . By (15') for n, for bl = b 2 . . . bn+2 there exists c' E D n f l such that bl x n f l c'ck. Then we apply (15) to a := c', b and obtain c E Dn+l such that b zn+2 ccn+1. A
+
(15') for n 1 =+- (15) for n. In fact, assume (15') for n 1. Suppose
+
Then (a,, an)Rn+l (b,+l, bn+2) by s-functoriality, hence by (I5'), 3c E Dn+l b xn+2C&+l.
5.16. M A X I M A L I T Y THEOREM
+
(15.1) & (15.2) (15). In fact, assume (I5.1), (15.2). Suppose
(aan)Rn+l6l. Now we can apply (15.2) t o a and 6lbl = b2.. . bn+'bn+2bl (this is possible, since b,+l # bn+2). So there exist e E Dn, d E D(e) such that (ee,d)~"+~(&bl). Hence by s-functoriality (deen)Rn+2b,and thus by (15.1) there exists c E Dn+' such that b x ~ c k++ l .~ (15') + (15.1). In fact, assume (15'). Suppose a E Dn, (aan)Rn+'b, b, # b,+l. Then by s-functoriality, (a,, an)R2(bn,b,+l). Hence by (I5'), there exists c E Dn such that b =,+I c ~ Hence . ( a a n ) R n + ' ( c ~ )by transitivity, and eventually aRnc, by s-functoriality. Therefore (15.1) holds. (15') + (15.2). In fact, assume (15'). Suppose
a E Dn, b E D,+~, (aa,) R"+' (bl . . .bn+l) = xyf 'b; then aRn (bl . . . bn), (an, an)R2(bn,&+I) by s-functoriality. NOWby (15') applied to d := bn+2b1 there exists e = cocl . . .cn E Dn+' such that
. . .bn+l
E
Dn+2,
Then by s-functoriality, and so aRn(cl . . . k ) . Therefore for c := e l . . . G , we have aRnc and ( c ~ QRn+2b ) as required. Finally let us show that together with (I2.4), (15') implies (15.3). Assume (15'). Let a E Dn, b E Dnf 2, and (T;+' b) Rn+' (aa,). Then by (12.4) there exists c E Dn+2 such that b ~ ~ and + aa, ~ czn+lcl . . .%+I. Now (a,, an)R2(cn,%+I) by s-functoriality, so by (15') (applied to a,, cl .. . G + ~ we ) obtain d E Dn+l , such that cl . . . c,+l ="+' dd,+l. Thus
again by s-functorality. By s-functoriality we also have
so (d2. ..dn+l)Rna, which proves the conclusion of (15.3) (where d2 . . . d,+, stands for c, dl stands for d).
CHAPTER 5. METAFRAME SEMANTICS
460
Note that again (I5), (15') become obvious if bn # bn+l is replaced with b, = bn+l - take c = b. Therefore we obtain the following equivalent form of Definitions 5.15.7, 5.15.8, 5.15.9. Proposition 5.16.10 Let IF be an S4-metaframe. Then (1)
P is a wi-metaframe iff F is an s-functorial metaframe satisfying (12.3) & (12.4) & (14.1) & (14.2);
(2) F is an i-metaframe iff F is a wi-metaframe satisfying (15) (or equivalently, (15')); (3)
F is a wz=-metaframe iff IF is an z5-metaframe iff F is a wi-metaframe satisfying (mmz).
Proof (1) If a metaframe is s-functorial, then by 5.16.5, the quasi-lift property for injections is equivalent to (12.3) & (I2.4), and by 5.16.8, the 2-lift properties for injections are (together) equivalent to (14.1) & (14.2). (2) By 5.16.5, in s-functorial metaframes the quasi-lift property for all maps is equivalent to (12.3) & (12.4) & (15.1). By 5.16.8, the 2-lift properties for all maps are equivalent to (14.1) & (14.2) & (15.2) & (15.3). So if P is quasi-functorial, it is weakly functorial and satisfies (15.1) & (15.2). By 5.16.9, the latter implies (15). The other way round, if P is weakly functorial and satisfies (I5), then by 5.16.9, it satisfies (I5.1), (I5.2), (15.3). Also by (I), it satisfies (12.3) & (12.4) & (14.1) & (I4.2), which yields the quasi-lift and the 2-lift properties. (3) By 5.16.5, for monotonic= metaframes the quasi-lift property for all injections implies it for all maps. By 5.16.8, the same happens to the 2-lift properties. Proposition 5.16.11 Let F be an S4-metaframe such that Q H Then P is an i(=)-metaframe.
IL(')(F).
Proof By Proposition 5.14.16, F is monotonic(=). So by Proposition 5.16.10 it is sufficient to check the properties (12.3), (12.4), (I4.1), (I4.2), (15): (12.4) a R n b =. V d E D(a) 3c E Dn+l ((ad)Rn+lc & (T;c) znb);
The proofs in both cases are quite similar. Suppose aRnb, n > 0 d E D(a) (or uRv, d E D,). Consider the QH-theorem
for p E PLOand its substitution instance.
5.1 6. M A X I M A L I T Y THEOREM
461
where 1x1 = n , y @ x . Let M := (IF,'$)be an intuitionistic model such that
Thus '$+( P )- '$+ ( Q ) = xn (b),
in particular b E c+(P)- '$+(Q),hence M , a assumption M IF A we obtain
Iy
P ( x ) > Q ( x )[ x ] ,so from the
and thus
M , ad
IPL
( P ( x )3
[xyl-
So there exists c such that
( a d ) ~ , + l c& c IF P ( x ) [xy]& c
Iy
Q ( x )[ x y ] .
Since x = ( x y ) . (T;, by definition of forcing it follows that
which proves (12.4). The changes for (12.3) are now obvious: n = 0 and '$+ ( P ) = R ( u ) , '$+(Q)= W - R-'(u).
For these properties the proofs are also similar, so we check only (14.2). Suppose (rl;a)Rn(r;c) and consider the intuitionistic model M := ( I F , J ) such that '$+(P)= 0"'' - (Rn+l)-'(c). By the assumption, the QH-theorem (the substitution instance of the same A l )
B
:= 3yVzP(x,z )
> V z P ( x ,z ) ,
where 1x1 = n , y, z $'! x , y # z , is true in M . But M , c Iy P ( x ,z ) [ x z ] ,and so since (r;a) Rn(r;c), we have M , .rrl;a Iy V z P ( x z )[ x ]. Now M IF B implies M , r;a Iy 3yVzP(x,z ) [ x ] ;hence M , a Iy V z P ( x ,z ) [xy].So there exists bde E Dn+2 such that aRn+' ( b d ) and M , bde Iy P ( x ,z ) [ x y z ] .The latter is equivalent to be $ c + ( P ) ,i.e. to ( b e ) R n f' c , by the definition of J. Thus (14.2) holds.
Recall that by Proposition 5.16.10, this property is essential only for the case without equality.
CHAPTER 5. METAFRAME SEMANTICS
462
Suppose (aan)Rn+'Cl. Take a distinct list x of length n, y, z @ x, y and consider the following formulas:
#z
Obviously, A is a QH-theorem (a substitution instance of the axiom A2 := P ( x n ) > 3 z P ( z ) ) . Consider the intuitionistic model M := (F,<) such that
Then obviously
Now we claim that (0.1) M , a
Iy 3 z B [x].
In fact, suppose the contrary. Then there exists de E Dnf l such that
( 1 ) dRna and d e It- B [xz]. Hence
By the choice of M, dRna implies d
# [+(Q2) and thus
Now from (2), (3) we obtain
which according t o the choice of M, is equivalent to
By ( 1 ) we also have
E ,(5). Now by ( 6 ) By our initial assumption, (aa,)Rn+'El, hence ( d e ) ~ ~ + 'by we obtain
5.16. M A X I M A L I T Y THEOREM On the other hand, by (0)
and 21~1 IPL
P2(y, x, z) [xzy].
This contradicts (7) and the reflexivity of Rn+2. So we have proved that a ly 3zB [x].But M It A, therefore
However note that
) then by definition, dRna, which implies In fact, if (in M) d Iy Q ~ ( x [XI, (ddn)Rn(aan) by s-functoriality. But the latter means (dd,) $ E+(Q1), i.e. d ly Q1 (x, x,) [XI. Eventually from (8) and (9) it follows that
So there exists e E Dn+' such that
which is equivalent to
by (0). Then b := en+le fits for the conclusion of (15).
H
R e m a r k 5.16.12 So we see that IF is an i(=)-metaframe if IL(')(F) contains A0 := (p > .T> p) (cf. the proof of 5.14.12) and Al, A2 from the proof above. T h e o r e m 5.16.13 (Maximality theorem) Let P be an S4-metafiame. Then the following conditions are equivalent:
(2) QH
c IL(=) (IF);
P r o o f In fact, (2) + (3) by Proposition 5.16.11; (3) (1) + (2) is trivial.
* (I) by Theorem 5.15.26.
CHAPTER 5. METAFRAME SEMANTICS
464
Therefore we can introduce the intuitionistic semantics of metaframes MF~,=) generated by i(')-metaframes. Theorem 5.16.13 shows that (using the terminology from Section 2.12) this is the greatest sound semantics of S4-metaframes for superintuitionistic logics, because it is generated by all S4-metaframes strongly validating QH(')-theorems. Therefore the list of properties of i(')-metaframes, yields a precise criterion of intuitionistic soundness for metaframes. In the next section we will apply this criterion to describe intuitionistic soundness in Kripke-quasi-bundles. Note that the classes of weak i-metaframes, i-metaframes and i=-metaframes are actually different. Let us consider two simple examples.
Example 5.16.14 Let IF1 = ((Fn)nEw, D), be an SCmetaframe such that DO
= {uo,ul), (Do,RO)is a two-element chain;
(Dn, Rn) is a tree of height 2 with the root d:, n-tuples from D,, (which are incomparable).
i.e. the tuple @ sees all
On one hand, by Proposition 5.16.10, P1is a wi-metaframe: the s-functoriality, (I2.3), (12.4), (I6.1), (16.2) hold obviously. For example, for (16.2): if n > 0, (al, . . . ,an, an+l) E (D,)n+l, (cl, . . . ,&, &+I) E (Dv)n+l,( a i , . . . ,an) Rn(cl,.. .,cn), then we can put (bl,. . . ,b,, d, e) := (do,. . . ,do, do, cn+l) if u = UO,and (bl,. . . ,bn,d,e) := (al,. . . , a n ,an+l,&+l) if ( a l , . . . , a n ) = ( ~ 1 ,... ,cn). On the other hand, IF1 is not an i-metaframe, since (15) fails for n = 1. In fact, (do,d O ) ~ ~ (d2), d ~but , there does not exist b' = (bo, bl, bl) z3(dl, dl, d2).
Example 5.16.15 Let IF2 be the metaframe with the same D as F1 and with the universal R'O, RIn = Rn U (Rn)-l, see the following figure: Again, IF2 is a wi-metaframe, and even an i-metaframe: (15) holds since (do,do,do) x3 (dl, d2,d2), etc. On the other hand, IF2 is not an 2'-metaframe - (mm2) fails, since (do,do)Rl2(dl,d2), but dl # d2.
5.1 7. KRIPKE QUASI-BUNDLES
Therefore, the class of i-metaframes (which are sound for the intuitionistic logic without equality) is larger than the class of i=-metaframes. Recall that in the modal case there is no difference between logics with or without equality in this respect; sound metaframes are same (see Theorem 5.12.13). Recall also that IL(IF) = ILZ(F) n IF is the fragment without equality for any i=-metaframe F (by Proposition 5.15.22). Let us also show that the class of i'-metaframes is larger than the class of QS4-metaframes (which are sound for the modal case). Example 5.16.16 Consider an Shmetaframe F such that W = {UO,UI),uo = u1, Duo = {do), Dul = {dl, d2), aRnb iff a = (bl, . . . ,b,) V 3i 5 l ( a = (di,. . . , di) & b = (dl-i, .. .,dl-i)) Obviously, Rn = x n . On the one hand, IF is not a modal metaframe, because (12.3) fails; in fact, (al, a2) E D?, , a1R1ao, and 73b E Duo (ala2)R2(ao,b). On the other hand, F is an i=-metaframe. E.g. the condition (12.3) holds: one can take (bi, . .. , b;, bb) = (al, . . . , a n ) since R" = zn. And (16) also :: has an ="+I- COPY holds, because the (n 1)-tuple (ao, . . . ,ao, ao) E 'D (al, . . . ,a l , al) E DE,fl, and we can stick together ( n 1)-tuples in Dul .
+
+
Note that we can consider M 3 , as a semantics for superintuitionistic logics (with or without equality) and MFG, also as a semantics for superintuitionistic logics without equality. We still do not know if all these semantics are equal to
M3g.
5.17
Kripke quasi-bundles
Recall that according to Definition 5.5.19, a Kripke quasi-bundle is a quasi-pmorphism between S4-frames, i.e. a monotonic surjective map with the quasi-lift property. Every Kripke quasi-bundle .rr : Fl Fo, where Fi = (Wi, Ri), is associated with a system of domains Du := {T-'(u) I u E Fo)and a family of inheritance relations pu, := Rl n (D, x D,). It also corresponds to a metaframe constructed
-
CHAPTER 5. METAFRAME SEMANTICS
466
as in the case of Kripke bundles, cf. Definition 5.3.2; the n-levels are F, = (Dn, Rn), with a R n b iff V j ajRlbj & a sub b. So Ri = Ri for i = 0 , l . Proposition 5.17.1 Let IF = ( F ,(D,)) be a metaframe corresponding to an intuitionistic Kripke quasi-bundle. Then IF is an i-metaframe iff IF is an i=metaframe zff the following conditions hold:
(2)
if n > 0, and (ad) E Dn+', all ai and d are distinct; b E Dn (bi are not necessarily distinct), and Vs asR1bs, then there exists c E Dn+' such that dR1&+l & Vs I n (b, z1c,) & Vs, t I n (b, = bt @ c, = ct);
(3)
if n > 0, a, c E Dn+', all ai are distinct, all ci are distinct and Vs a,R1c,, then there exists (bde) E Dn+2 such that d, bl, . ..,b, are distinct (but perhaps e = d or e = bi) such that
These conditions allow us to 'move' individuals from one world to another. Proof First, P is an S4-metaframe: all Rn are reflexive and transitive, since the relations R1, RO,sub are reflexive and transitive. Now let us check the properties of i'-metaframes stated in Proposition 5.16.10. The monotonicity is almost obvious. In fact, Vi aiR1bi implies Vi a,(i) R1ba(~); and a sub b implies (n,a) sub (nab). (12.4) is the quasi-lift property, which follows from the definition of a Kripke quasi-bundle. (16.1) is the same as (1). (12.3) follows from (2). In fact, b = (bl,.. . ,b,) zn c = (cl,. . . ,%) iff V s 5 n (b, z1 c,) & b subc & csub b; we also have (ad)Rn+'c, since (al, . . . ,an)Rn(cl,. . . ,%) and d R 1 ~ + lThe . requirement in (2) that d, a l , . . . ,a, are distinct, is overcome due to the 'local functionality'. It remains to check (16.2) or its equivalent version 5.16.8 (16.2"). So let a, c E Dn+l, n y a zn nTc. Then a s u b c and csuba. The cases when a,+' E { a l , . . . ,a,), %+l E {cl , .. . ,c,) are obvious. Thus we may assume that all a j and all bj are distinct. Now (3) yields us a tuple (bde) E Dn+2 such that aRn+l(bd), (be)Rn+lc. Also note that the conditions (I), (2), (3) are necessary: they definitely hold if IF is an i-metaframe. In fact, they respectively follow from (I6.1), (I2.3), (16.2). Therefore, the logic IL(=)(F) of a Kripke quasi-bundle IF is superintuitionistic iff the conditions ( I ) , (2), (3) hold. We call such quasi-bundles intuitionistic (or,
5.18. SOME CONSTRUCTIONS O N METAFRAMES
467
briefly, i-quasi-bundles); they generate a sound semantics for superintuitionistic logics. Every intuitionistic Kripke bundle is clearly an i-quasi-bundle. Note that a metaframe associated with a Kripke quasi-bundle is not necessarily modal. For instance, the i'-metaframe F from Example 5.16.16, which is not modal, corresponds t o the Kripke quasi-bundle (W,D , P ) , W = { u o ,ul),u0 x 211, Duo = {do), = { d l ,d 2 ) ~ u ~ = , , ~iddui ( 2 = 0, I ) , ~ u ~ =~ ( (u 4 ,~dl-i)). - ~ T h i s explains why the construction o f the C-set corresponding t o a Kripke bundle described in the proof o f Proposition 7.8.11, fails for Kripke quasi-bundles.
5.18
Some constructions on metaframes
D e f i n i t i o n 5.18.1 Let IF = ( F , (D,)) be an N-metaframe based on F = (W,R1, . . . , R N ) and let V C W . Then the submetaframe IFIV is the restriction of F to V , i.e. it has a system of domains is DIV := ( D , I u E V ) with relations RYIV := Rr 1 (DIV),; i n particular, ROIV= R0 [ V . A submetaframe FIV is called generated i f V C W is Ri-stable for every i = 1,..., N . L e m m a 5.18.2 (1) If FIV is a generated submetaframe of F and all the maps TD, i n IF are monotonic (in particular, i f F is a modal or a wi-metaframe), then each of its n-level ( F I V ) , is a generated subframe of F,. (2) Every submetaframe of an N-modal metaframe is an N-modal metaframe (3) Every generated submetaframe of an i- (resp., z=-, wi-) metaframe is also an i- (resp., i=-, wi-) metaframe.
D e f i n i t i o n 5.18.3 A cone F l u of a metaframe IF (for u E F O )is its restriction to the subset (cone) R*(u)i n the base F O . N
Recall that R* is the reflexive transitive closure o f
U Ri. i= 1
D e f i n i t i o n 5.18.4 A metaframe model M = ( F , J ) over IF (N-modal or intuitionistic) gives rise to the model MIV := (IFIV,tIV) (the restriction o f M t o V ) such that (tIV)+(Pjn)= t+(Pjn) n (DIV)". L e m m a 5.18.5 Let F be an N-modal metaframe satisfying (12.I), or a wimetaframe, let M = ( I F , t ) be a model (N-modal or intuitionistic, respectively), IFlV a generated submetaframe, and let M l V be the corresponding submodel of M . Then (1) M , a IF ( k ) A[x]iff MIV, a It ( k ) A[x] for any appropriate assignment ( a ,x) (modal or intuitionistic, respectively).
CHAPTER 5. METAFRAME SEMANTICS
468
Proof By induction on the length of A. Note that in the intuitionistic case it is essential that in a wi-metaframe an inductive clause for 3 can be rewritten with = instead of R , see Section 5.11. Thus we cannot state (1) for an arbitrary rn metaframe. Note that every valuation (N-modal or intuitionistic) in FlV is also a valuation in F and it coincides with its restriction to FIV. Hence we obtain
Proposition 5.18.6 (1) ML')(F) M L ~ =( )P I V ) and ML(')(F) C ML(')(FIV) for a generated submetaframe of an N-modal metaframe P. (2) IL?)(P) & IL!=)(FIV)and I L ( = ) ( F ) IL(')(FIV)for a generated submetaframe IF. metaframe of an i- (&-, 2-')
Proposition 5.18.7
n
n
M L ~ = ) ( F ~and u ) ML(=)(IF) = ML(')(FTU)for an ucw uEW N-modal metaframe F .
(1) ML?)(F) =
(2) IL:=)(F)
=
n I L ~ ) ( F T U )and I L ( = ) ( F )n I L ( = ) ( F T Ufor) an i- (wi-,
uEW
u F W
F - ) metaframe IF.
P r o o f If M = ( I F , 6)I+ A then M , a I+ A [x]for some ( a ,x ) , and thus MTu, a A [XI, for u = rD(a). rn Definition 5.18.8 Let be a family of metaframes of the same type ( N , j ) , Fj, = (Djn,R;, . . .,RjnN). Their modal or intuitionistic), IFj = ( ( F j n )D disjoint sum (union) U Pj is defined as the metaframe F = ((F,), D ) such that j€J
Fn = ( D n ,R;, . . . ,R;), where Dn = (J D z , and for a , b E Djn w€Fo
L e m m a 5.18.9 Let F =
U IFj.
Then
j€J
( I ) Fn 2
u Fjn;
j€J
(2) F is an N-modal metaframe, or respectively an i- (wi-,i=-) metaframe iff all Fj are.
5.19. ON SEMANTICS OF INTUITIONISTIC SOUND METAFRAMES 469 Proof
U Fjn
(1) The map
jEJ
-
F, sending ( a , j ) (for a E Dj") to ( ( a l , j ) ,.. . ,( a n , j ) ) ,
is an isomorphism. (2) An exercise. L e m m a 5.18.10 (1) ML?)(uIF,.)
= nMLL=)(IFj) for N-modal metaframes.
j
(2) IL?)
(U~
j
j =)
j
n IL(=) (Fj) for i- (wi-, i=)-metaframes. j
(3) Similarly for IL, M L Proof On one hand, each IFj is isomorphic to a generated submetaframe of is isomorphic to a cone in the UIFj. On the other hand, every cone in j
j
corresponding IFj. Therefore the semantics M F ~ ) , MF~,=)satisfy the collection property (CP) (see Section 2.16).
5.19
On semantics of intuitionistic sound metaframes
Note that all definitions related to i(')-soundness can be readily extended to arbitrary (not necessarily S4-) 1-metaframes. Namely, a valuation [ in a 1metaframe is called intuitionistic if Rn(tf ( P y ) ) J+(Pjn)for any predicate letter Py, n 0,cf. 5.14.1. Now the definitions of forcing, validity, and strong validity are rewritten in a straightforward way (cf. Definitions 5.14.2, 5.14.3). The notations ILL=), IL(=) are also used in this case. Finally a metaframe IF is called i(=)-sound if IL(=)(F) is an s.p.l., cf. Definition 5.14.4. To conclude our description of intuitionistic sound metaframes, we now prove the following statement.
>
Proposition 5.19.1 For every i(')-sound 1-metaframe IF there exists an S4metaframe IF' such that IL(=)(IF') = IL(=)(F). Hence IF' is also i(=)-sound, so it is an i(') -metaframe, by 5.16.13. Therefore we can conclude that the semantics MF:,=,) of arbitrary i(=)metaframes actually equals the semantics of all i(')-sound metaframes. To prove Proposition 5.19.1, we need two auxiliary notions.
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Definition 5.19.2 A propositional frame F = (W, R) is called weakly reflexive (or w-reflexive, for short) i f the transitive closure of R is reflexive (cf. [Dogen, 19931). A 1-metaframe IF is called w-reflexive if all its levels Fn are w-reflexive. Definition 5.19.3 A propositional frame F = (W, R) is called co-serial if Vu3v vRu. A 1-metaframe IF is co-serial i f all F, are co-serial. Obviously, every reflexive propositional frame (and 1-metaframe) is w-reflexive and co-serial. L e m m a 5.19.4 Let F be a 1-metaframe. Consider the formulas B1 := 3zT and B2 := (T > p) > p.
(1) If B1 E IL(F), then IF is co-serial. (2) If IF is co-serial and B2 E IL(F), then F is w-reflexive. Proof (1) Suppose B1 E IL(F), a E Dn. Let x = (xl,. . . ,xn), z $ x. Consider an arbitrary intuitionistic model M = (IF,[). Then M, a Il- Bl[x], so there exists b E (Rn)-'(a) satisfying the condition: 3c E D(b) M , bc IF T[xz]. This implies co-seriality.
(2) Suppose IF is co-serial, Bz E IL(P). Given a E Dn,choose b E by co-seriality. Consider the substitution instance
(Rn)-'(a),
= ((Rn)*o of B2, and an intuitionistic model M = (F,[) such that [+(Pn) Rn)(a). Then M , a IF (T > Pn(x))[x].
In fact, if c E Rn(a) and M , c It T , then c E [+(Pn), and so M , c IF Pn(x) [x]. Thus M, a I t P n ( x ) , since M, b I t Bi [x] and bRna. Thus a E <(Pn)[x], so a((Rn)* o Rn)a. Hence the w-reflexivity of IF follows.
Definition 5.19.5 A 1-metaframe F is monotonic(') i f it satisfies
(im) M , a I t A [x] & a R n b + M, b It A [x] for all for A E IF('), x 2 F V ( A ) and a, b E Dn, cf. 5.14.14. Proposition 5.19.6 Let F be a co-serial-metaframe, B := p B E IL(') (F), then IF is monotonic(=).
>
(T
> p). If
5.19. ON SEMANTICS OF INTUITIONISTIC SOUND METAFRAMES 471
Proof 30 Let us check (im) for a formula A E IF(=). Consider a substitution instance B' := A > ( T > A) of B. Assume a R n b and M , a It A [XI. Due t o co-seriality, there exists cRna. Since M, c It B' [x],it follows that M, a It ( T > A) [XI. Now M, b It T [x] implies M, b I t A [XI. H Corollary 5.19.7 If for a I-metaframe IF, QH G IL(=)(IF) and even, if {p = ( T > p), 3 t T ) C IL(')(F), then IF is w-reflexive, co-serial, and monotonic('). P r o o f Note that (p = (T> p ) ) E IL(')(P) iff p 3 ( T 3 p), (T 3 p) 3 p E IL(=)(F). Then apply 5.19.4, 5.19.6. w Definition 5.19.8 For a w-reflexive metaframe IF = ((Dn,Rn),€,, fine its transitive closure, the S4-metaframe IF* = ((Dn,Rn*),€,, Rn* = Rn+ is the transitive closure of Rn.
D) we deD), where
L e m m a 5.19.9 Let IF be a w-reflexive monotonic(=)-metaframe. Then
P r o o f It is clear that J is an intuitionistic valuation in IF iff J is an intuitionistic valuation in IF*, since <+(Pn) is Rn-stable iff it is Rn*-stable. So it is sufficient to show that for every model M = (F,J) and the correfor every ordered assignment (x, a) such that sponding model M* = (IF*, x FV(A), the following holds:
>
e),
M, a It A [x] u M*, a It- A
[XI.
We proceed by induction and consider only three non-trivial cases. (1) A = B 2 C. If M , a Iy A [x],then there exists b E Rn(a) such that M, b It- B [x] and M, b Iy C [x]. Then M*, b It B [x] and M*, b Iy C [x] by the induction hypothesis, and obviously b E Rn*(a). So M*, a Iy A [XI.
The other way round, let M*, a Iy A [x]. Then there exists b E Rn*(a) such that M*, b It- B [x] and M*, b Iy C [x]. So M , b It B [x] and M, b Iy C [x]. Take d E Dn such that aRn*dRnb. Then M, d Iy A [x] and by monotonicity M, a Iy A [x] as well. (2) A = 3yB [x]. If M , a It- A [x], then there exist b E (Rn)-'(a) and
c E D(b) such that M , .ir,lly(bc)It- B [xlly]. So M*, 7rxlly(bc)IF B [xlly] and M*, a It A [x],since aRn*b. The other way round, let M*,a IF A [XI. Then there exist b, d E Dn and c E D(b) such that3' bRndRn*a and M*, 7rXlly(bc)It- B [ X I [ y]. Hence M,"ir,lly(bc)It B [xlly], so M , d It A [x], and thus M , a It A [x] by monotonicity. 30Cf. Lemma 5.14.15. 3 1 N ~ t ethat R* o R = R o R* is the transitive closure of R.
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472
(3) The case A = QyB [x]is similar.
Lemma 5.19.9 and Corollary 5.19.7obviously imply Proposition5.19.1; namely we put F' = F*. The previous consideration in this sectionallows us toextendLemma5.14.1 l(3) to arbitrary 1-metaframes: Proposition 5.19.10 Let F be a 1-metaframe. Then the following conditions are equivalent:
(3) F is w-reflexive and monotonic(=), and F* is an i(')-metaframe. Moreover, these conditions are equivalent to
(4) the following formz~lasare in IL(=)(P): p
(T 3 p), 3zT, Al := 3yp 3 p, A2 := P(x) 3 3zP(z)
Here A1 and A2 are the formulas used in the proof of Proposition 5.16.11. Note that all formulas in (4) are QH-theorems, so (2) readily implies (4). P r o o f (3)+(1). By Lemma 5.19.9 and Theorem 5.15.26 (soundness). (2)+(3). By Corollary 5.19.7, Lemma 5.19.9, and 5.16.11. (4)+(3). Assume (4). Then F is w-reflexive and monotonic(=) by 5.19.7, so by 5.19.9 IL(')(F) = IL(')(F*). So formulas from (4) are in IL(=)(F*), and therefore IF* is an i(')-metaframe, cf. the proof of Proposition 5.16.11 and rn Lemma 5.14.15, or cf. Remark 5.16.12. By the way, basing on Proposition 5.19.10 (1) % (3), we can give an explicit description of the class of all i(')-sound metaframes. Viz., to check that IF* is an i(=)-metaframe, we can rewrite all conditions from the definition of an i(=)metaframe (cf. Section 5.15 and 5.16) with Rn* replacing Rn. In particular, the condition (00) becomes
or equivalently
a R n b + (rUa)Rm*(nub).
Here is a description of monotonicity(') for arbitrary metaframes, which we state without a proof. L e m m a 5.19.11
5.20. SIMPLICIAL FRAMES (I) A I-metaframe F is monotonic iff it satisfies (i) a R n b + (nua)Rm*(nub)for any a E Em,;
(ii) aRnbRnc
3c'(aRnc' & cRn*c & c'Rn*c);
(iii) aRnbRnc + Ve E D(c) 3d3g E D(c) (aRnd & (dg)IP("+')(ce)); (iv) cRnbRna =+ Ve E D(c) 3d3g E D(d) (dRna & ( c e ) ~ * ( " + ' ) ( d ~ ) ) . (2)
IF is monotonic' iff it is monotonic and satisfies (mm2).
Here the condition (i) corresponds to (00) in F* and expresses the monotonicity for atomic formulas P(nux), cf. the proof of 5.14.11. The condition (ii) means that all levels Fn are weakly t r a n ~ i t i v e . ~ ~ Actually the conditions (b), (c), (d) express the monotonicity for the implication and the quantifiers Q, 3 respectively, i.e. for formulas P ( x ) > Q(x), Q Y ~ ( x , Y )3yP(x, , Y). The interested reader can try to restore the missing details. We point out that the description of i(')-soundness in 5.19.11 is more complicated than the notion of an i(')-metaframe, but the semantics is the same.
5.20
Simplicia1 frames
Introduction In the last section of this chapter we briefly describe the next step in generalising Kripke-type semantics. Recall that in the usual Kripke semantics for predicate logics we have a system of nested domains D = (D, I u E F), and every individual from D, is considered as its own inheritor in the domains D, of all worlds v accessible from u. In Kripke sheaves, Kripke bundles and functor semantics accessibility relations (or functions) between individuals are introduced; they describe inheritors of individuals in accessible worlds. Accessibility relations between n-tuples of individuals (for n > 1) can be derived from these accessibility relations on individuals. At the next step, in metaframes, accessibility relations Rn (or R r , in the polymodal case) between n-tuples of individuals for different n 2 1 may be arbitrary, and only the requirement of soundness puts some constraints on these relations. But n-tuples a = (al, . . ., a,) are obtained from individuals existing in the same world. So all n-tuples are taken from the set Dn = U{DZ I u E F). Now let us consider more general kind of frames, in which n-tuples are 'abstract'; so the sets Dn for n 2 1 are a priori independent, and unlike the case of metaframes, Dn is not constructed from the set of 'actual' individuals D 1= 32A propositional frame F = (W, R ) is called weakly transitive if
Vu,u, w E W ( u R v R w + 3t ( u R t & wR*t & t R ' w ) ) .
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474
-
U{D, I u E F). But then for any a E C, = (I,)'", we should introduce a map w, : Dn Dm transforming 'abstract' tuples. Their metaframe analogues are 'jections' T, transforming 'actual' tuples: .rr, (al, . . . ,a,) = (a,(l), .. . ,a,(,)). But in the 'abstract' case w, are chosen arbitrarily. The definitions of valuations, forcing, validity, and strong validity can be given quite similarly to metaframe semantics.33 And again the predicate logic ML(')(F) or IL(=)(F) for such a frame F is the set of formulas strongly valid in IF. Next, we can find constraints corresponding to logical soundness or to other natural properties of forcing, as we did in Chapter 5 for metaframes. An 'abstract' n-tuple a E Dn corresponds to the 'real' n-tuple (wx; (a), .. . , wxR(a)),where A? E El,, Xr(1) = i , see the Introduction. wA;(a) can be regarded as the ith 'component' of a. But this correspondence in general is not bijective - different tuples may have the same 'components'; moreover, for a permutation a E T,, 'n-tuples' a and w,(a) may have different (and even disjoint!) sets of 'components'. Nevertheless, this correspondence is useful; as we shall see in Volume 2, this helps associate logically sound simplicial frames with metaframes. As in metaframes, we can identify Do with the 'underlying propositional frame' F = (W, R1,. . .,RN). And we can introduce the individual domains of a world u E W as D, := {a E D1 I w ~ , ( a )= u). More generally, put DE := n s t ( u ) for n 2 1, where 0, E Con is the empty function. Then Dn = U{DZ I u E F ) is a partition. But in general, an 'abstract' n-tuple a and its 'components' wni, (a) for 1 5 i 5 n may be in domains of different worlds from F.
Forcing in simplicial frames Now let us turn to precise definitions. We begin with the case without equality.
Definition 5.20.1 A (formal) simplicial N-frame based on a propositional 4 ---f + Kripke frame F = (W, R1,. . .,RN) is a tuple F = (F, D l R , T ) , where D = + (Dn I n E w) is a sequence of (non-empty) sets, R = (RT I n E w, 1 5 i 5 N ) is a family of relations R? C_ Dn x Dn; so we have propositional N-modal frames: F, := (Dn, RA,. . . ,R:). As usual, we assume that Fo := (W, R1,. . . ,RN) is the original frame F. And finally, w = (w, I a E U Em,) is a family of m,n
mappings ('abstract jections ') nu : Dn ---t Dm for a E C , .
So Dn (for n 2 1) are sets of 'abstract' n-tuples (in particular, D1 is the set of 'individuals'), R r are accessibility relations between n-tuples, and T, are mappings ('jections') transforming 'abstract' tuples. Put D; := T G ~ ( Ufor ) n > 0 , u E W, where 0, is the empty function from Con;it may be called the n-tuple domain of the world u. In particular, for n = 1 we sometimes write D, rather than DA and call this set the individual domain of the world u. Obviously {Dg I u E W) is a partition of Dn. 33Up t o some details about the equality, which are briefly discussed later on.
5.20. SIMPLICIAL FRAMES
475
The i t h component of an abstract n-tuple a E Dn, n > 0 is [a]i := xri, (a) E D l , where sin E Tin, ain(l) = i (i.e. ( a l , . . . ,a,) . sin = ai). In general, we do not assume that a and its components 'live' in the same world, i.e. that ns,(a) = x ~ , ( [ a ] i ) .We do not even assume that [all = a for a E D l - an 'individual' may be non-equal to its own 'component'; they can even 'live' in different worlds. But as we shall see later on, in natural semantics (for modal or superintuitionistic logics) we may consider only frames, in which rid, (a) = a for any n; such frames are called T-identical. In these frames we also have [all = a for a E D l , and x ~ ( u = ) u for u E W, i.e. D: = x,l(u) = {u). As usual, for the monomodal (and the intuitionistic) case we denote accessibility relations on n-tuples by Rn, without the subscript i = 1. The notations Rn+ (respectively, Rn* ) for the transitive (respectively, reflexive transitive) closure of U R r are usual, cf. Chapter 1. We also consider i
=,
the corresponding equivalence relations :=% ~ * n on Dn. Obviously, every N-metaframe gives rise to a simplicial N-frame,with Dn,R r , xu described in Section 5.9. Definition 5.20.2 A valuation i n a simplicial N-frame IF is a function J sendDn; i n particular, ing every n-ary predicate letter Py to a subset J+(Py) J+(P;) W (cf. Definition 5.9.2). A valuation J i n a simplicial 1-frame (or more briefly, a simplicial frame) is intuitionistic i f every J+(Pj")is an Rn-stable subset of Dn:
cf. Definition 5.14.1. A simplicial model is a pair M
=
(IF, J).
Definition 5.20.3 A n assignment of length n i n a simplicial frame IF is a pair (x, a), where a E Dn, and x is a distinct list of variables of length n.
Now we can define forcing for modal formulas without equality, cf. Definition 5.9.4. Definition 5.20.4 A simplicial N-model M = (IF, J) gives rise to the forcing relation M, a A [x], where A E MFN, (x, a) is an assignment i n IF, F V ( A ) 5 r(x). The definition is by induction, modifying 5.9.4 i n a natural way. W e assume that (x, a) is of length n.
+
(Atl)
M, a Y 1 [XI;
(At2)
M, a 1 Pjm(x - a )
[XI
ifl ~ , Ea Jf (Pjm) (for a E C,,)
and similarly for the other Boolean connectives;
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CHAPTER 5. METAFRAME SEMANTICS
(Ql)
M, a != 3yB [x]iff3c E Dnfl(.rr,;c = a & M, c I= B [xy]), M , a != VyB[x] iffVc E Dn+'(.rru;c = a + M , c != B[xy]) (for Y 94;
(Q2)
M, a I= 3xiB [x]iff M, TJ: a != 3xiB [x- xi], M , a != VxiB[x] iff M, .rrb;a b 3xiB[x - xi], wherex-xi = x . b , " .
Hence we obtain the truth conditions for 7 and Oi similar to those in Section 5.9. One can see that in a simplicial frame corresponding to a metaframe the clauses (At2), (U), (Ql), (Q2) become equivalent to (2), (8), (9), (10) from Definition 5.9.4. In fact, for (Ql) note that in a metaframe T,; c = a iff c = ad for some d E D(a). Let us also note that in 'T-identical' simplicial frames (where Tid,a = a for any a E Dn) both cases of the inductive clause for the quantifier can be presented in the following uniform way (cf. (9+10) in Section 5.9): M , a I= 3yB [x]iff 3c E D ~ + ~ ( T = , ~~C
~& M, ~c t= B- [ ~ (~l y ] ) a
(where m = Ix - yl). In fact, if y = xi, this clause is equivalent to (Q2), since = 6," and xll y = (x - xi)xi; and if y @ x, then it is equivalent to (Ql), since .rr,x-ua = .rrid,a = a and xll y = xy.
EX-,
R e m a r k 5.20.5 Again we can propose a reasonable alternative definition of forcing I=* differing in the clause (Q2): M , a k * 3xiB [x] iff 3c E Dn(r6;c= r6;a & M , c k B[x]), and similarly for Vxi B. But this definition actually leads to the same natural semantics of modal simplicial frames as Definition 5.20.4; thus both versions are equivalent. However we do not know if these definitions give equal (or equivalent) maximal logically sound semantics. Similarly we can define forcing for the intuitionistic case. Definition 5.20.6 An (intuitionistic) simplicial model M = (IF, <) gives rise to forcing for intuitionistic formulas A and assignments (x, a ) with r(x) FV(A) defined by the following inductive clauses (cf. Definition 5.14.2):
>
M , a It Pjm(.rr,x) [x] iff .rr,a E [+(Pjm); M , a Ik ( B A C) [x] ifl M, a It. B [x] and M, a IF C [x]; M , a I F ( B v C ) [x]i f l M , a I F B o r M , a I F C [x];
5.20. SIMPLICIAL FRAMES
477
M , a I k ( B > C ) [x]i f f V b ~ D ~ ( a R ~M,bII-B b& [ x ] + M , b I l - C [x]); M , a IF VyB [x] iff Vc E ~ ~ + l ( a ~ (c) ~ r+, M ; , c It B [xy]) (for y @ x);
M , a It 3 y B [x] i83c E Dn+l(r,;(c)Rna
& M , c It B [xy]);
M , a I t QxiB [x] iff M,.rra;al= QxiB [x- xi] for a quantifier Q. Again we have uniform presentations of quantifier clauses in T-identical simplicial frames.
Sound and natural simplicial frames Definition 5.20.7 A n N-modal formula A is true i n a simplicial model (notation: M l= A) i f M , a k A [x]for any assignment (x, a) such that FV(A) C r(x). The definition for the intuitionistic case is similar. Definition 5.20.8 A n N-modal formula A is valid i n a simplicia1 frame F (notation F k A) i f it is true i n all models over IF. The definition for the intuitionistic case is similar. ML-(F) denotes the set of all N-modal formulas (without equality) valid in F; the notation IL- ( I F ) is similar.
Definition 5.20.9 A formula is strongly valid i n P if all its substitution instances (without equality) are valid. The set of all formulas strongly valid i n F is denoted by ML(F) i n the modal case and by IL(F) i n the intuitionistic case. A simplicial frame F is logically sound if ML(F) or IL(F) is a modal or a superintuitionistic logic respectively; we call such a frame m-sound or i-sound respectively. Logically sound simplicial frames generate 'maximal semantics'. But unlike the case of metaframes, we do not know an explicit description of these semantics, and we may conjecture that they are rather complicated (cf. Section 5.16, 5.19 for intuitionistic sound metaframes). Therefore in Volume 2 we shall describe more convenient classes of simplicial frames, where forcing satisfies natural properties of 'logical invariance' similar to those we had for metaframes in Sections 5.10, 5.11,5.15. These classes generate 'almost maximal' semantics (and maybe even maximal, but this is yet unknown).
Definition 5.20.10 A simplicial frame F is called modally transformable (or m-transformable) if the following condition3* M , a I= A [x]H M , T , ~I=A [r,x]
>
holds for any injection a E T,,(n m), for any model M = any assignment (x, a) such that FV(A) r(r,x). 34Cf. 5.10.6(*).
( t r fm)
(IF,(),
and for
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CHAPTER 5. METAFRAME SEMANTICS
The definition of i(')-transformable frames for the intuitionistic case is similar - they satisfy the condition35
M, a II- A [x]H M, r,a I=A [r,x].
( t r f 2)
for intuitionistic models M and formulas A. Recall that the case n = 0 (when A is a sentence and a = ido = Ao) is trivial. Loosely speaking, in transformable frames forcing M, a != (It-)A[x] does not depend on the choice of a list of variables x containing FV(A). More precisely, M, a k A [x] with r(x) > FV(A) can be reduced to M, a k A[x] with r(xl) = FV(A), where x' = r,x, a is an injection. Moreover, the choice of x' is inessential, since for any enumerations x, x' of FV(A), there exists a permutation a such that x' = T,X. This observation allows us to use only the first case in the inductive clause for quantifiers, cf. Section 5.6. L e m m a 5.20.11 Let P be an m-transformable simplicial frame, M = (P, E ) a simplicial model. Then for any congruent formulas A, A' and for any x, a such that FV(A) = FV(A1) C r(x): M , a I= A [x]H M , a != A [x],
(*)
and similarly for the intuitionistic case. Definition 5.20.12 A simplicial frame P is called modally s-transformable (or for any distinct lists x of length n, m ms-transformable) if for any a E C,, respectively, for any modal formula A such that FV(A) 5 r(y), for any model M over F,and for any a E Dn :
Similarly for the intuitionistic case, we define is-transformable simplicial frames with the following condition:
M , a II- ([r,x/y] A) 1x1
* M,
T
U
[I~ A [Y].
(trf is)
This condition expresses the invariance of forcing under variable substitutions (cf. Lemma 5.11.7 for metaframes). Recall that variable substitutions are defined up to congruence, and the congruent versions of A are all the results of applying the identity substitution to A (cf. Section 2.3). So in ms-transformable simplicial frames forcing is congruence invariant:
[/I
M, a b A [x]e M , a II- A' [x]
(*)
for any congruent formulas A, A' (apply (trfsm) with y = x, a = id,), and similarly in the intuitionistic case. On the other hand, we can show that forcing is congruence invariant in every transformable simplicial frame as well (cf.
5.20. SIMPLICIAL FRAMES
479
Lemmas 5.10.11 and 5.15.16 for metaframes in the modal and the intuitionistic cases respectively). Anyway, if the condition (*) holds, then the choice of any congruent version of [.rr,x/y]A in (trf sm) does not matter.
Definition 5.20.13 A simplicial frame F is called m-natural (respectively, inatural) if it is logically sound and transformable (respectively, s-transformable). The classes of all m-natural and i-natural simplicial frames generate the 'natural' semantics of simplicial frames SFmand SFint. In Volume 2 we will also describe other classes of simplicial frames (called 'modal' and 'intuitionistic') generating the same semantics SFmand SFint; this description generalises the work done in Sections 5.11 and 5.15 above. Theorems 5.12.13, 5.16.13, along with 5.10.6, 5.11.7, 5.15.14, 5.15.17, show that for every logically sound metaframe the corresponding simplicial frame is natural; thus MFm 5 S F m and MFint 5 SFint. On the other hand, we do not know if every logically sound simplicial frame is natural and even if the semantics of natural simplicial frames equals the 'maximal' semantics of all logically sound simplicial frames. It is also unknown if the semantics of simplicial frames (natural or 'maximal;') are stronger than the semantics of metaframes.
Equality in simplicial frames To conclude this preliminary exposition, we briefly discuss problems with interpretation of equality in simplicial frames. Recall that in metaframes the clause for equality has the following form:36
a k x j = xk [x]iff a j
= ak,
(=I
where a E Dz,1x1 = n. This definition admits several reasonable generalisations for simplicial frames; in metaframes all these versions are equivalent. First, for the formula X I = 2 2 , x = (XI,xz), and a E D2, we can put
where ci2(l) = j (so (al, a2) . a j 2 Second, we can define
=aj
in metaframes).
a b xl = x2 [x] iff a = n,l (b) for some b E D',
where a' E C21 is the 'diagonal' map; a l ( j ) = 1 for j E I2(so T,I (b) = (b, b) in metaframes). These two definitions are clearly equivalent for 'real' tuples, but they differ for 'abstract' tuples. For example, in general we cannot assert that (a)) = a if x,,, (a) = ~ , , , ( a ) . T,~,(T,I (b)) = b for b E D1 or that T,I (.r, 36We do not mention a model M as equality does not depend on the valuation.
CHAPTER 5. METAFRAME SEMANTICS
480
There exist other more 'exotic' versions of forcing, e.g.
a I= X I = 2 2 [x]iff rUl2(a)= r,,,(a)
& a = r,l(r,,,(a)).
All these interpretations are equivalent to (=) (with n = 2, i = 1, j = 2) in metaframes, although they are non-equivalent in simplicial frames. To express these definitions (i.e. generalisations of (=) to simplicial frames) for arbitrary n and i, j E I n we use projections rain'extracting' components of an n-tuple (ruin (al,. . . ,a,) = ai) and 'pairing jections' r x ? where X; E 3k ' Czn, Xyk(l) = j, Xjnk(2) = k, i.e. r ~(al, ? . . . ,a,) = ajak. 3k Thus we obtain the following interpretations of equality in simplicial frames
a !=+ x j
= xk
[x]iff rUj, (a) = rukn (a),
(=+)
the 'componentwise' or the upper interpretation, a !=- x j = xk
[XI
iff 3 b E D' r ~ ? & ( a=) n,l(b)
(=-)
the 'diagonal' or the lower interpretation, and also the 'combined' interpretation
a
xi = x j [x]iff rujn (a)= rokn (a) & r x (a) ~ =~r , ~ (ruin(a)).
(=*)
We suppose that the upper interpretation is the most straightforward generalisation of the condition (=) in metaframes. However, some logical properties of the lower interpretation k- seem better. In Volume 2 we will consider a general approach to interpretation of equality in simplicial frames covering all these versions - various interpretations of equality a b xi = x j [x]depend on triples of the form (ruin(a), n,, (a), rx,;, (a)). But such a general approach seems too complicated, so we simplify it by using 'pairs' r ~ % ( aE) D2. Thus, for example, the upper interpretation has a simplified version: a I=+'x j = xk [x]iff TA? E I + , 3k
where I+= {b E D2 1 r,,, (b) = r,,, (b)). Although this condition is not equivalent to (=+) in arbitrary simplicial frames, for 'natural' simplicial frames this reduction works quite well.
Part 111
Completeness
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Chapter 6
Kripke completeness for varying domains Completeness proofs in this chapter are based on various canonical model constructions. They originate from canonical models in modal propositional logic and Henkin's completeness proof in classical predicate logic. The main idea is that worlds in canonical models are consistent (or even syntactically complete) theories, and individuals are identified with individual constants of these theories.
Canonical models for modal logics
6.1
Recall (Definition 2.7.7) that for a first-order modal theory I?, Dr denotes the set of individual constants occurring in I?, c(=)(F)is the set of all Dr-sentences in the language of I?. Definition 6.1.1 A (first-order) modal theory I? i s called L-consistent zf k
Y L r\ 7
Ai for any A1,. . .,Ak E E, o r equivalently, if I?
YLI .
A modal theory
i= 1
i s called L-complete if it i s maximal (by inclusion) among L-consistent theories in the same language. Lemma 6.1.2 Let E be a modal theory. (1) If I? is L-consistent and I? k L l A , then I'
Y LA.
(2) If I' i s L-complete, A E L(')(I'), then
in particular,
E (where
denotes the set of all sentences i n L ) .
484 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
(3) If I? is L-complete, then for any A E C(=)(r)
(4)
Every L-consistent I' satisfying the equivalence (3), is L-complete.
Proof (1) If l7 kL 1 A and r t - A, ~ then I? kL I , by MP. (2) 'If' is obvious. To show 'only if', suppose A 6f I?. Then I? c I? U {A), hence r U {A) kL I,and thus I? kL A > I(= -A) by 2.8.1. Hence rYLA by (1). (3) 'Only if' readily follows from (1) and (2). To show 'if', suppose the contrary - that both A, 7 A are not in r. Then as in the proof of (2), we obtain I? kL -A and similarly, l? kL l l A , which implies the L-inconsistency of
r.
(4) Suppose the equivalence in (3) holds for any A E ,C(=)(r), but r is Lincomplete. Then r C r" for some I?' in the same language, so there exists A E ( r ' - I?). By (3), we have 1 A E I'G r', thus r' is L-inconsistent. ¤
Similarly t o the propositional case, we have the following
Lemma 6.1.3 Let (2)
(A r\ B ) E
r iff
r
be an L-complete theory. Then for any A, B E C(=)(r):
(A E I' and B E I ');
(ii) (A v B ) E I? iff (A E r or B E r); (iii) (A > B) E l7 iff (A 6f r or B E I ').
Proof (i) (If.) A A B > A is an instance of a tautology, so if (A A B ) E I?, we have r kL A by MP. Hence A E I' by 6.1.2 (2). Similarly B E r. (Only if.) If A, B E r, we apply the instance of a tautology A > ( B > AAB), MP and 6.1.2 (2). (ii) (If.) Similarly to (i), use A > A V B or B > A V B. (Only if.) Suppose A, B # r. Then -A, -B E I? by 6.1.2(3). Since 1 A > ( 1 B > -(A V B)) is an instance of a tautology, we obtain -A, 1 B EL l ( A V B ) , hence 7 ( A V B) E I?, and thus (A V B ) 6f I' by 6.1.2 (3). (iii) (Only if.) Supposing (A > B ) , A E r, we obtain B E I' by M P and 6.1.2 (2). (If.) If A 6f I?, then 7 A E r by 6.1.2 (3). Since -IA > (A > B ) is an instance df a tautology, by MP and 6.1.2 (2), it follows that (A > B ) E r. ¤ If B E I?, we use B > (A > B ) in the same way. Recall that denotes the set of all sentences in a logic L, 0*r:= {O,A I A E I',a E I F ) (if N is fixed).
Lemma 6.1.4 Let L, L1 be N-m.p.l.(=),
r
an L-complete theory. Then
6.1. CANONICAL MODELS FOR MODAL LOGICS (1) I' is L1-complete iff
(2) Suppose L1 = L o*Sub(@)G r.
485
zCr .
+ @ for a set of sentences O.
Then r an L1-complete ifl
Proof (1) 'Only if' follows from Lemma 6.1.2 (2). To show 'if', assume r. By Lemma 6.1.2 (3), (4), it suffices to check that r is LI-consistent. Suppose the contrary; then for a finite X r, t-L, A = 7 ( r \ X ) . Put A := l ( A X ) . Then by Definition 2.7.1, A has a generator r. But then I' k L FAl, whence E L A1, and Al E L1. Hence vA1 E therefore I? k L A implying the L-inconsistency of I?. (2) By ( I ) , l? is L1-complete iff I?. Since n*Sub(O) C z , the 'only if' part of (2) follows readily. For the converse, we assume 0*Sub(O) r and show that I'. In L 8 is L-provable in fact, by the deduction theorem 2.8.4, every A E ~*S.ub(@), hence r k L A, therefore A E I?, by 6.1.2.
c
c
c
zc
+
c
Lemma 6.1.5 (Lindenbaum lemma) Every L-consistent theory can be extended to an L-complete theory in the same language. Proof Similar to the propositional case. If L(=)(F) is countable, we consider its enumeration Ao, A1,. . . and construct a sequence of L-consistent theories I' = Fo rl,. . . such that rn+l= rnU {A,) or r,+l = I?, U {lAn}. This is possible, since one of r, U {A,}, r, U {lA,} is L-consistent (otherwise I?, t-L 7An, l l A n by Lemma 2.8.1, and thus Fn is L-inconsistent). Eventually the union r, := Urnis L-consistent and L-complete by 6.1.2(4). n
If the language is uncountable, we apply transfinite induction, but actually we do not need this case in the sequel. Definition 6.1.6 An L-complete theory r is called L-Henkin if for any Drsentence 3xA(x) there exists a constant c E Dr such that
We say that
r is (L, S)-Henkin if r is L-Henkin with Dr
=S
Due to Lemma 6.1.3 (iii), every L-Henkin theory I' has the following existence property: (EP) if 3xA(x) E
r, then
for some c E Dr, A(c) E I'.
Moreover, we have Lemma 6.1.7 Every L-Henkin theory r satisfies the condition (EP') 3xA(x) E
r ifl
for some c E Dr , A(c) E
r
486 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Proof Since L F A(y) > 3xA(x) for a new variable y (Lemma 2.6.15(ii)), we have F L A(c) > 3xA(x), and so (A(c) > 3xA(x)) E I?. Thus Lemma 6.1.3 implies the converse of (EP), therefore (EP') holds. Exercise 6.1.8 Show that every L-complete theory with the existence property is L-Henkin.
Lemma 6.1.9 Let I' be an L-consistent theory, D r Then there exists an (L, S)-Henkin theory I?' r .
>
C
S, IS1
=
IS - Drl = No.
Proof The set of all S-sentences is countable, so let us enumerate all Ssentences of the form 3xA(x) : 3xlAl(xl), ~ x ~ A ~ ( x. .z. )Choose , distinct constants ck from (S- D r ) such that ck does not occur in 3xiAi(xi) for i k, and Put r, := r u {dXkAk(xk) > Ak(ck) I k E w).
<
Let us show that
r,
is L-consistent. Suppose the contrary. Then for some k
Take the minimal k with this property. Let
Then
ri B , 3yAk(y) 3 Ak(c) F L 1 (where y = xk, c = ck). Hence by Lemma 2.8.1,
Then from the propositional tautology ~ ( >pq) (1) (2)
r,B F L ~ Y A ~ ( Y ) ; r,B E L l A k ( ~ ) .
Since c does not occur in (3)
> p A l q we obtain
U {B), by Lemma 2.7.11, it follows that
r,B t - ~V Y ~ A ~ ( Y ) ,
and thus by 2.6.15(xiii),
R o m (1) and (4) it follows that r k - l~B , contrary t o the choice of k. Therefore r, is L-consistent, and by Lemma 6.1.5, we can extend it to I".
rn
6.1. CANONICAL MODELS FOR MODAL LOGICS
Lemma 6.1.10 Let A be a modal sentence, IS1 = No. Then L F A iff (A E I? for any (L, S)-Henkin theory r ) . Proof 'Only if' follows from Lemma 6.1.3. To prove 'if', suppose L Y A. Then { l A ) is an L-consistent subset of MFL=)(B), and so by Lemma 6.1.9, there exists an (L, S)-Henkin theory r {-A). W
>
For S C_ S' and I?' C MF~=)(s'), let r'lS := ~ ~ M F L = ' ( s ) (the restriction of
r' to the domain S). Obviously, r'lS is consistent (complete) if I?' is consistent (respectively, complete), but for a Henkin (L, S')-theory I?', r'lS may be not a Henkin (L, S)-theory. Henceforth we fix a denumerable set S*,the 'universal set of constants'. We call a set S C S* small if (S*- S ) is infinite.
Definition 6.1.11 An L-place is a Henkin L-theory with a small set of constants. The set of all L-places is denoted by VPL.
Lemma 6.1.12 Let u be a world in a Kripke model M over an N-modal predicate Kripke frame F = (F,D ) and assume that M k L. Then the set of D,-sentences that are true at u
Proof
This is an easy consequence from the definitions. Let us only give an k
argument for L-consistency. If
t L A Ail then B 1
i= 1
k
:=
A
Ai can be presented
i= 1
as [c/x]Bo for Bo E L, c E D,M, and then VxBo E L. Hence by assumption k
M, u ZL VxBO,which implies M, u b [c/x] Bo = 7 one of the Ai must be false at M, u.
A
i=l
Ai by Lemma 3.2.18. Thus
¤
Definition 6.1.13 Let M be a Kripke model for a modal logic L, in which every individual domain is a small subset of S'. The map from (the worlds of) M to VPL sending u to r, is called canonical and denoted by U M , L (or by V M , or by v if there is no confusion). Lemma 6.1.12 extends to Kripke sheaves, Kripke bundles, or other kinds of models for L. Moreover, VPL is the set of all 'places' r, of all possible Kripke models (and of metaframe models) for L with small domains. To show this, we shall now construct a certain predicate Kripke frame (the canonical frame), whose worlds are L-places and in which every 'world' r has the individual domain D r , while I' is the set of all Dr-sentences true at this world. Since all the domains are small, there are infinitely many spare constants at every world.
488 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS For an N-modal theory I' and i = 1,. . .,N, put
Definition 6.1.14 For L-places tions: r RLi rt := o;r G rl. Lemma 6.1.15
r,rl we
r RLi I?' i f f Dr G Dr.
define canonical accessibility rela-
& Oil?' n C ( r ) G r
Proof (Only if.) Assume I'RLir'. For any c E Dr we have k L O ( P ( c )3 P ( c ) ) ,so O ( P ( c )3 P ( c ) ) E I? by 6.1.2(2), and thus ( P ( c )3 P ( c ) )E 0;I' g I". Therefore c E D r ) , which shows Dr c Dr.. (If.) Assume Dr C D p , Oil?' n C ( r ) C I?. Next, assume O i B E r. Then B E I?', since otherwise T B E I", O i l B E Oir' n L ( r ) 2 I', and thus r is inconsistent. Therefore Oil? r'.
c
The crucial property of canonical relations is the following
Lemma 6.1.16 For any L-place I? and for any Dr-sentence A such that O i A E I?, there exzsts an L-place I" such that rRLiI" and A E I". Proof Let us show that the theory ro := O ~ I ' U { Ais} L-consistent if O i A E I?. Suppose t Ll\ B j 3 1 A for some formulas B j E Oil?. Then t~Oi(l\ B j ) 3 j
O i l A , hence L t A O i B j
j
> 1 0 i A , I' F L T O ~ Aand , thus by Lemma 6.1.2 ( 2 ) ,
j
1 0 i A E r. This is a contradiction. Now by Lemma 6.1.9, we can construct an L-place I" such that ro r' (remember that Dr is small, so there exists a small S such that IS- DrI = No). By Definition 6.1.14, ~ R L ~ I " . Lemma 6.1.17 For any I' E V P L and A E L ( r ) we have
r z f f ( A E I" for (2) O i A E I' i f f ( A E r' for
(1) O i A E
r' E R L i ( r ) ) ; each r' E R L i ( r ) ) .
some
Proof ( 1 ) 'Only if' follows from 6.1.16. The other way round, if O i A = 7 O i l A $ I?, then O i l A E I?, hence 1 A E I" (i.e. A q! I?') for any I" E RLi(r). ( 2 ) 'Only if' follows from Definition 6.1.14. To show 'if', assume O i A $ I?, then 1 O i A E I',and since k L 7 0 i A > O i l A , this implies O i l A E r . Hence by W ( I ) , 1 A E r1for some I" E RLi(I'). Definition 6.1.18 The canonical Kripke frame with varying domains of an N-modal logic L without equality is V F L := ( V P L ,R L ~.., . , R L N ,D L ) , where ( D L ):= ~ Dr.
6.1. CANONICAL MODELS FOR MODAL LOGICS
489
Definition 6.1.19 For an N-modal logic L with equality we define the canonical Kripke frame with equality as
where c ( X L ) r d : = ( c = d) for C,d E D r . Due to the standard properties of equality 2.6.16(i), (ii), ( x L ) ~ is an equivalence relation on Dr (sometimes we denote it just by ~ r ) We . also have
since L -l (x = y) > Oi(x = y); thus from Definition 3.5.1.
XL
satisfies the RLi-stability condition
Definition 6.1.20 The canonical model of an m.p.1. (=) L is
where (J~)r(Pkm):= {CE (Dr),
I P ~ ( CE r)) .
Note that V M ~ = )is well-defined, since P r ( c 1 , . . . ,cm),(CI = dl), . .. , (c, dm) E r implies P r ( d l , . . . , d m ) E I?, thanks to 2.6.16(v). The main property of canonical models is the following
=
Theorem 6.1.21 (Canonical model theorem) VML=),~ I=A iffA E I' for any L-place
r E VPL and A E ~ ( = ) ( r ) .
Proof By induction on the length of A. The base readily follows from the definitions. The inductive step for classical connectives and quantifiers follows from 6.1.3 (i)-(iii) and 6.1.7 (EP'), properties (i)-(iv), (vi) of Henkin theories. The case A = OiB follows from Lemma 6.1.17 (1). Corollary 6.1.22 For any modal formula A, V M ~ = I= ) A iff L k A. Proof If A,is a modal sentence, then by Corollary 6.1.lo, A E L implies A E for any L-place r; hence V M ~ = ) , I= A by Theorem 6.1.21. The other way round, if A @ L, then by 6.1.10, A @ for some Henkin (L, S)-theory F, where S is infinite and small. So r is an L-place, and thus V M ~ = )r, !+A by Theorem 6.1.21. The claim for an arbitrary formula A is reduced to the claim for VA, since W V M ~ = 'k A iff V M ~ = k ) VA, and A E L iff BA E L.
490 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS Definition 6.1.23 A n m.p.l.(=) L i s called V-canonical if V F ~ = )b L. This property is sufficient for completeness: Corollary 6.1.24 Every V-canonical m.p.1. i s strongly Kripke complete. Every V-canonical m.p.l.= i s strongly K F E (or Kripke sheaf) complete. P r o o f In fact, let L be a V-canonical m.p.1. (=), roan L-consistent theory; by Lemma 6.1.9, there exists an L-place r ro.By 6.1.21, V M ~ = ) , Tk r, so rois satisfied the L-frame V F ~ = ) .
>
Now let us consider canonical models for two N-modal logics L1 C L2. Every Lz-consistent theory is clearly L1-consistent, thus VPL, 2 VPL,. Moreover, we obtain L e m m a 6.1.25 Let L1
L2 be N-m.p.l.(=).
Then
(2) VML, i s a generated submodel of VML,. (3) I f L1 i s canonical, then VFL2k L1.
Proof (1) Let r be an L1-place. Then VML1,I' k iff 1;;C I? (by the canonical model theorem) iff I' is L2-complete (by 6.1.4(1)). This is equivalent to I' E VPL2,since r E VPL, already enjoys the Henkin property and has a small set of constants. (2) In fact, r k implies k OiA for any A E thus A I= for any A E RLli(r). SO due to (I), VPL, is stable in VFLl. VML2 is a submodel of VMLl, since V M L l , r I= P ( a ) iff P(a) E I' iff V M L 2 , rk P(a). (3) Follows from (2) and the generation lemma 3.3.18.
G
G
z;
z
Proposition 6.1.26 If L i s a V-canonical N-m.p.1. (=), r is a set of N-modal pure equality formulas, then L1 = L + r i s also V-canonical. Proof By Lemma 6.1.25, VFL, is a generated subframe of VFL, so L C_ ML(=)(VFL) ML(=)(vFL,). On the other hand, VMLl I= I? by the canonical model theorem, thus since all formulas in I' are constant, VFLl k I?. Therefore VFL, k L1. Corollary 6.1.22 shows that V M ~ = )is an exact model for L. The next proposition is a further refinement of this fact. Proposition 6.1.27 (1) Every m.p.1. (=) L has a countable exact Kripke model (respectively, K F E -
model). (2) Moreover, if L 2 Lo for a A-elementary canonical m.p.l.(=) Lo, such a model exists over a n Lo-frame.
6.1. CANONICAL MODELS FOR MODAL LOGICS
491
(3) Moreover, every L-consistent theory is satisfiable i n some model described i n (2).
P r o o f We can apply Corollary 3.12.11 to L = Lo, M = V M ~ = ) ,F = VF~=) and uo such that M,UO!= I? (for a given L-consistent I?). Note that F != Lo by 6.1.33. So by 3.12.11 there exists a countable reliable Mo 5 M over an Lo-hame such that MT(Mo) = M T ( M ) = L and uo E Mo. H Then Mo, uo b I' by reliability. Obviously, the logics Q K N and Q K ; are V-canonical. Let us also show V-canonicity for some other simple modal predicate logics. For an N-modal theory I?, a E IF let
oar := {O,A I A E r). Also let RLil ...i k := R ~ i 0l . . . 0 R L ~ ~ and let RLA be the equality relation Idvp,, cf. Proposition 1.11.5. L e m m a 6.1.28 Let a E I F , I'RL,A.
Then
P r o o f (1) By induction on the length of a. If a = A , everything is obvious. If (i) holds for P, a = k.0 and rRL,A, then there exists I" such that I'RL~I?', rlRLaA. By Definition 6.1.14 and the induction hypothesis we obtain
(2) Ad absurdum. Suppose A E A, 0,A E (-r). Then -0,A E I?, and thus H O,yA E r . By (I), this implies TA E A, which makes A inconsistent. T h e o r e m 6.1.29 Let A be a propositional one-way PTC-logic. Then the logics Q A , Q A = are V-canonical. Proof Every constant axiom is valid in the canonical frame, since it is true in the canonical model. If A = Okp > O p p E L = &A'=', then RLp 5 RLk. In fact, suppose I ' R L ~ A ,C]kB E I?. Since I-L OkB > OpB, we have U p B E I?. By Lemma 6.1.28, it follows that B E A. By Proposition 1.11.5 and Lemma 3.2.34, we obtain VFL != A.
492 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Example 6.1.30 The following counterexample shows that Theorem 6.1.29 does not extend to arbitrary PTC-logics, because they may be ICE-incomplete. Consider the logic K5 := K OOp > Op; its Kripke frames are characterized by the 'Euclidian' condition
+
VuVvVw (uRv A URW> vRw).
(€1
Then QK5' is ICE-incomplete. To see this, first note that
QK5' k K o OAUl
3
OAUl.
In fact, consider a Kripke sheaf model M over a Euclidian frame, and suppose M, u C= OAUl, i.e. for some v E R(u) M , v C= AUl, which means that IDv/ = 1. Then ID,] = 1 for any w E R(u). In fact, by ( E ) we have vRw & wRv, so there are transition functions p,,, p,,; and by the properties of Kripke sheaves it follows that they are bijections. So ID,! = 1. However QK5' If OAUl > OAUl. To see this, consider a Kripke bundle F over the base F = (W, R), where W = { u , ~ i , v 2 ) ,R = W x ( W - {u)), Du = {a), D,, = { b ) , D,,
=
{el, cz), p = D+ x (D+ - {a)).
It is clear that F is Euclidean. Every level Fn = (Dn,Rn) is also Euclidean, because on n-tuples from D,, or D,, the relation Rn is equivalent to d sub c & c sub d (which is an equivalence relation) and anRnd for any d # an. Therefore F k QK5= by 5.3.7 and soundness of Kripke bundle semantics. On the other hand, F OAUl > OAUl, since vl b AUl and v2 AU1. Therefore QK5' ft OAUl > OAUl by soundness.
v
Actually many simple and natural predicate logics are not V-canonical. So they are either Kripke-incomplete, or the completeness proof requires more work. One of the options may be to take an appropriate submodel of VML, for which an analogue of Lemma 6.1.21 still holds. This leads us to the following definition.
C RLi on VPL is called (i-)selective if for any L-place I? and for any A E C ( r ) such that OiA E I', there exists an L-place I" such that r R i r l and A E I". Definition 6.1.31 A relation Ri
So we obtain an analogue of Lemma 6.1.17:
Lemma 6.1.32 For any selective Ri, for any r E VPL and A E C ( r ) we have (1) OiA E I' iff (A E I" for some I" E Ri(I'));
(2) OiA E I' ifl (A E Proof
r' for each I" E R i ( r ) ) .
The same as in 6.1.17, but 'only if' in (1) now follows from 6.1.31. W
6.2. CANONICAL MODELS FOR SUPERINTUITIONISTIC LOGICS 493 Lemma 6.1.16 means that the relation RLi is i-selective, and thus it is the greatest i-selective relation on VPL. Now similarly to the canonical frame, we can introduce quasi-canonical Kripke frames (VPL,R1,. . . ,R N ,DL [, XL])with arbitrary i-selective relations Ri and the corresponding quasi-canonical models (with the valuation &). Then we obtain the following analogues of Lemma 6.1.21 and Corollary 6.1.22:
Lemma 6.1.33 Let M be a quasi-canonical Kripke model for an m.p.l.(=) L. Then (1) for any L-place
and A E L ( r )
(2) for any formula A MkAiffLl-A. Another option is t o use subsets of VPL still satisfying Corollary 6.1.10 (and thus, Corollary 6.1.22). Various modifications will be considered in this chapter later on.
6.2
Canonical models for superintuitionist ic logics
Now let us turn t o the intuitionistic case. We would again like to represent possible worlds in Kripke (or other) models by sets of formulas. But now we should take false formulas into account as well, because in intuitionistic logic the falsity of A is not equivalent to the truth of TA. For this purpose we need double theories introduced in 2.7.12.
Definition 6.2.1 If L is a superintuitionistic logic (with or without equality), a theory (I?, A) is called L-consistent if ( r , A) yl,i.e. YL
for any finite I'l C I?, Al
/\r1 3 Val
G A.
It is obvious that I'n A = 0 whenever
(I',A) is L-consistent.
Definition 6.2.2 An L-consistent theory (I',A) is called L-complete if r U A = L(I', A) So in this case A = L ( r , A) - I',which is abbreviated as (-I'). Also note that for any c E L ( r , -I?) we have (P(c) > P(c)) E I', since otherwise (P(c) > P(c)) E (-I'), while the theory ( 0 ,{P(c) > P(c))) is inconsistent. So it follows that D-r Dr. A similar argument using P(c) A -P(c) shows that Dr g D-r. So we can write Dr rather than D(r,-q. Moreover, we shall often say that I' (rather than (I?,-F)) is an L-complete intuitionistic theory.
c
494 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Definition 6.2.3 An L-complete intuitionistic theorg F is called L3-complete if it satisfies the existence property:
(EP) if 3xA(x) E I?, then A(c) E I' for some c E Dr. More specifically, an L3-complete F is called (L3, S)-complete if Dr = S .
Definition 6.2.4 An L3-complete (respectively, an (L3, S)-complete) theory I? is called L3V-complete (respectively, an (L3V, 5)-complete) if it satisfies the following coexistence property: (Av) if VxA(x) E (-I') and A(d) E F for all d E Dr, then Vx(A(x) V C) E F for some formula C E (--I?). The intended meaning of this property (after we prove the canonical model theorem) is the following. If at a world F of the canonical model VxA(x) is false, but A(d) is true for every individual d, then obviously VxA(x) must be false at some strictly accessible world. (Av) allows us to find a formula C such that I'IF Vx(A(x) V C); so C should be true at all strongly accessible worlds, where A(d) is refuted for some individual d. For example, this property holds if C exactly defines the set of all strictly accessible worlds (and the canonical model theorem holds). For a logic L containing the formula C D the condition (Av) may be simplified: (Ac) if VxA(x) E
(--I?) , then A(d) E (-F) for some d E Dr.
In fact, let C D E L. If (Av) holds and VxA(x) E (-F) , but A(d) E F for all d E Dr, then Vx(A(x) V C) E F for some C E (-F). Since C D E F, we obtain (VxA(x) V C) E F, which leads to a contradiction. Thus (Ac) holds. Conversely, if (Ac) holds then the premise of (Av) is false, so (Av) holds. Note that (Ac) is an analogue to the property (V) of forcing in Kripke models with constant domains. Since the Gijdel-Tarski translation for 3 is just 3, the definition of intuitionistic forcing for the 3-case is the same as classical. This explains, why the existence property is defined in the same way. On the other hand, the intuitionistic V translates as modal 0V, and thus the intuitionistic definition for the V-case consists of the 'classical' part dealing with individuals from a certain world and of the 'modal' part dealing with accessibility relations between worlds. So property (Av) is responsible for the classical part of intuitionistic universal quantification. In the case of constant domains ( V X B ) is ~ equivalent to VxBT, and thus (Ac) is just the dual of (E). 3V-complete theories seem to be a better intuitionistic analogue of Henkin theories than 3-complete theories. But the latter can be used for completeness proofs in the simplest cases (for example, for QH(')). Lemma 6.2.5 Every L-complete theory r has properties 6.1.3(i), (iii), 6.1.2(2):
6.2. CANONICAL MODELS FOR SUPERINTUITIONISTIC LOGICS 495 ( A v B ) E i~ f f ( A ~ or r B ~ r ) ;
r E L A iff A E r
(for A E L(=)(r)).
Every L3-complete I' also satisfies 6.1.7(EP). 3xA(x) E Proof
iff (A(c) E r for some c E Dr).
An exercise.
W
Now let us prove the crucial property of L3-complete theories analogous to Lemma 6.1.9 for Henkin theories. For intuitionistic theories (I?,A), (I", A') put ( r , A) 3 (r', A') := r E
I" & A r A'.
For a complete theory O also let
which means (I',A) 5 ( 0 , - 0 ) . Lemma 6.2.6 Let (I',A) be an L-consistent theory, D[r,&) = S C S', IS'I = IS' - SI = No. Then there exists an (L3, St)-complete (and even an (L3V, S f ) complete) theory 0 such that (I',A) 5 0. Proof The idea of the proof is quite standard. Namely, we can successively extend the pair (I', A) by adding S1-sentences either to I' or to A. Moreover, together with adding 3xA(x) to r, we always add A(c) for some new constant c from (S' - S). And together with adding VxA(x) to A, we add A(c) for a new c - but only if this is consistent. This construction provides the existence and the coexistence properties. If we need only an L3-complete theory 0 (I?,A), we can use the same procedure, but without mentioning V-formulas. Let IF(=)(s') = {Bk I k E w ) . Choose distinct constants ck from (S' S ) such that ck does not occur in B o y . . ,Bk. Next, define a sequence of Lconsistent theories
as follows. If Bk = 3xA(x), then A(ck)),A,) if ( r k { (( rr kk lAk{Bk, u {Bk}) otherwise U
rk+lAk+l
=
U {Bk}, Ak) is L-consistent,
If Bk is not of the form 3xA(x), then (rk+l, Ak+l) :=
( r k u {Bk), Ak) if ( r k u {Bk}, Ak) is L-consistent, (rk, Ak U {Bk)) otherwise.
496 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS Let us show that (rk+l, is L-consistent if (rk, Ak) is. Ak U {Bk)) are both inconsistent, then First, if (rkU {Bk), A,) and (rk, ( r k ,Ak) is also inconsistent. In fact, in this case
for some finite A', A"
5 Ak, and thus
where A = A' U A". Now since p > q, p V q kH q, we obtain r k k L V A, i.e. ( r k ,Ak) is inconsistent. Second, if Bk = 3xA(x) and ( r kU {3xA(2),A(ck)),Ak) is inconsistent, then
for some finite A
C Ak; therefore by 2.7.11
since ck does not occur in
rkU Ak, which implies
by 2.6.10 and MP, and thus (rkU {Bk),Ak) is inconsistent. Next, put 0 := U r k . By construction, the theory ( 0 , - 0 ) is L-consistent kEw
and has the existence property. To obtain an ~3b'-'-com~lete theory, we should slightly refine the above construction. Viz., if Bk = VxA(x), we put
(rkfl,
Ak+l) :=
(rk U {Bk), A,) ( r k ,Ak U {Bk, A(ck))) (rk,
Ak U {Bk))
if this theory is L-consistent, if this theory is L-consistent and (rku {Bk),Ak) is L-inconsistent, otherwise.
In other words, we add Bk either to rkor to Ak keeping consistency, and also add A(ck) to Ak whenever this is possible. The consistency of ( r k ,Ak) again implies the consistency of ( r k + l , &+I), Ak U {Bk)) is consistent, as we have seen since either ( r k u {Bk),Ak) or (rk, above. Finally, let us check (Av) for 0 . Suppose Bk = b'xA(x) @ 0 and A(c) E 0 for all c E Do. Then Bk E &+I, and A(ck) @ Ak+l, SO by construction, (Fk,Ak U {Bk,A(ck)) is L-inconsistent, and thus,
6.2. CANONICAL MODELS FOR SUPERINTUITIONISTIC LOGICS 497 for some finite A
C Ak.
Hence by 2.7.11
Since 0 is L3-complete, it satisfies 6.1.3(v), so it follows that Vx(CVA(x)) E 8, where C = V A E (-0). Lemma 6.2.7 Let A be an intuitionistic sentence, IS1 = No. Then: L F A iff (A E r for any (L, S)-place l?) iff (A E I? for any (LV, S)-place I?). Proof
Cf. Lemma 6.1.10 for the modal case.
Let I'lS := I? n IF^=) for S C S t and r G IF$=). We also fix a denumerable universal set of constants S* and consider small subsets of S* as in Section 7.1. Definition 6.2.8 A n L-place (respectively, an LV-place) is an L3-complete (respectively, an L3V-complete) theory with a small set of constants. V P L and UPL denote the sets of all L-places and LV-places respectively. Obviously, UPL C V P L . We also have an analogue of Lemma 6.1.12: Lemma 6.2.9 Let u be a world in an intuitionistic Kripke model M over a frame F = ( F ,D ) and assume that M k L. Then
is an L3-complete theory. Proof
Almost obvious. As for L-consistency, note that if Ai E I?, for i =
1 , . . ., k and L k
k
m
i= 1
j=1
A Ai > V
Bj, then the latter implication is true in M, and
so M , u IF B j for some j. Thus (I?,, -I?,)
is L-consistent.
W
Note that I?, is not necessarily L3V-complete. Similarly to the modal case we define canonical maps: Definition 6.2.10 Let M be a Kripke model for a superintuitionistic logic L, in which every individual domain is a small subset of S * . The map from (the worlds o f ) M to V P L sending u t o I?, is called canonical and denoted by V M , L (or by v ~ or, by v). Lemma 6.2.9 also extends to Kripke sheaves, bundles and to metaframe models in modal metaframes.' The intuitionistic canonical model theorem (see below) shows that V P L is the set of all L-places I?, in Kripke models for L with small domains. Now let us define some accessibility relations on V P L and UPL. lFor intuitionistic metaframes the situation may be different.
498 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Definition 6.2.11 For r , r' E VPL, we say that
r' is conservative over I? (notation: I? 5 I") i f r = r'l Dr; I?' is proper over I' (notation: l? < I?') if r c r l l D r . Let 5 be the reflexive closure of <, i.e.
The inclusion relation denoted by RL.
It is clear that
G on VPL (i.e. the set { ( r , I")
E VPj
I r C I?'))
is also
r C I?' iff r < r' V r 5 I"
Definition 6.2.12 Let W C VPL. A partial ordering R RL is called selective o n W i f for any I? E VPL the following two conditions hold: (3) i f (Al 3 A2) E (-I?), then A1 E
r' and A2 @ l?'
for some
r' E R ( r ) n W;
(V) if VxA(x) E (-I?), then A(c') # I" for some I?' E R ( r ) n W and some C' E Dr.. These conditions can be rewritten in an equivalent form: (3')
(Q')
r iff for any I" E R ( r ) n W (A1 E I" + A2 E I?'); VxA(x) E r iff A(cl) E r1for any r' E R ( r ) n W, c' E Dr, . (A1 3 A2) E
Lemma 6.2.13 (1) The relation RL is selective on VPL and UPL.
(2) The relation 5 is selective on UPL.
Proof (1) ( 3 ) If (A1 > A2) @ I?, then r YL (A1 > A2) (Lemma 6.2.5), i.e. the theory (I? U {Al), {A2)) is L-consistent. So by Lemma 6.2.6, there exists an LV-place I" t (I? u {Al), {A2)). Thus r I?', Al E I', A2 # rl. (Q) Suppose VxA(x) @ I', then I? YL VxA(x). Take c E (S* - Dr); then the theory (r,{A(c))) is L-consistent (otherwise r kL A(c), which implies I' kL QxA(x)). Hence by Lemma 6.2.6, there exists r' E UPL such that r' k (I?,{ A ( c ) ) ) ,i.e. I" 2 r and A(c) @ r ' . (2) ( 3 )Suppose (A1 3 A2) E (-I?) and consider two cases. (a) If A1 $! r , then by (1) we obtain I" r such that A1 E F1, A2 @ I". So Al E ((rllDr) - r ) , and thus I? < r'. (b) If A1 E I?, then to show (IV), we can take r' = r , since A2 I-L (A1 > A2), and thus (A1 > Az) @ F implies A2 @ l7. (V) Suppose QxA(x) E (-r) and consider two cases. (a) If A(c) @ I' some c E Dr, there is nothing to prove: one can take I?' = I?. (b) If A(c) E I? for all c E Dr, we can apply 6.2.4 (Av). So there exists a formula C E (--I?) such that Vx(A(x) V C ) E r. Now as in the proof of (I), we
c
>
6.2. CANONICAL MODELS FOR SUPERINTUITIONISTIC LOGICS 499
>
obtain r' I? and e E ( D p - D r ) such that A(e) 6 I?'. Since Vx(A(x) V C) E C I?' and 'dx(A(x) V C) FL A(e) V C, we also have (A(e) V C) E r', and thus C E I?' by Lemma 6.2.5 (iii). Hence C E (rflDr) - r , and eventually I' < I".
rn
Note that the previous argument fails for VPL. Definition 6.2.14 The canonical frame of a superintuitionistic predicate logic L is defined as VFL := (VPL,RL, DL), where r R L r l := I?
I?', (DL)r := Dr
similarly to the modal case (cf. Definition 6.1.18). The canonical frame with equality of an s.p.l.(=) L is
where, as i n Definition 7.1.10,
The stability condition for
XL
holds trivially, due to the definition of RL.
Definition 6.2.15 The canonical model of an s.p.l.(=) ( v F ~ = ) ,&), where
L is V M ~ = ) :=
This valuation is obviously intuitionistic. Definition 6.2.16 W e also introduce two kinds of quasi-canonical frames and models: UF~=):= V F ~ = ) I U P U ~ ,M ~ = ):= V M L = ) I U P ~ ,
FL=)
US (respectively, UI Mi=)) is the same as UF~=) (respectively, U M ~ = ) ) , but with the relation 5 instead of RL. T h e o r e m 6.2.17 (Canonical model theorem) Let L be an s.p.1. (=). Then for any L-place r and A E L ( r ) , VM~=),I?IF A iff A E
r;
similarly for both quasi-canonical models UML, U 5 ML and for all their selective submodels.
P r o o f By induction on the length of A (cf. Lemma 6.1.21). The inductive step for A, V, 3 uses Lemma 6.2.5. In the cases of > and 'd use 6.2.12 (I), (b').
500 CHAPTER 6. KRTPKE COMPLETENESS FOR VARYING DOMAINS Corollary 6.2.18 If M = V M ~ = ) ,U M ~ = )or u < M ~ = ) ,then
for any A E IF('). Now we obtain an analogue of 6.1.25: L e m m a 6.2.19 Let L1 C LZ be s.p.l.(=).
Then
(1) VPL, = {rE VPL, I V M L , , IF L2), (2) V M L , is a generated submodel of V M L , . P r o o f Almost the same as in the modal case. The proof of (1) uses a slightly different argument about consistency. Viz., let us show the L2-consistency for an L1-place O L2. Suppose the contrary: Lz k / \ A > V V for finite A C O, V -0; then E kQH / \ A > V V for a finite E L2 & 0, and hence QH t- /\(A U E) > V V , thus ( 0 , -0) is L1-inconsistent, which contradicts the assumption that O is an L1-place. For the proof of (2) note that I' IF L2 implies A It- Lz for any A 2 I? since W the model is intuitionistic.
c
>
Definition 6.2.20 A n s.p.1. (=) L i s called V-canonical if L U-canonicity and Us-canonicity are defined analogously.
IL(')(vF~=)).
Similarly t o 6.1.24, we obtain Corollary 6.2.21 Every (quasi-) canonical superintuitionistic logic is strongly Kripke complete (Kripke sheaf complete i n the case of a logic with equality). By definition we readily have Proposition 6.2.22 The logics QH, QH= are V-canonical. We also have an analogue of Proposition 6.1.26: Proposition 6.2.23 If L is a V-canonical s.p.l.=, I' i s a set of pure equality intuitionistic sentences, then L I' i s also V-canonical.
+
Proof
The same as in the modal case, but using Lemma 6.2.19.
¤
Corollary 6.2.24 The logics with decidable and with stable equality Q H = = ~ QHT DE and QH=' = QH= SE are V-canonical.
+
+
Therefore, we obtain the following completeness result. T h e o r e m 6.2.25
(1) Intuitionistic predicate logic QH i s strongly Kripke frame complete.
6.3. INTERMEDIATE LOGICS O F FINITE DEPTH (2) The logics with equality QH', QH'~, QH" complete.
501
are strongly Kripke sheaf
Note that (ii) can also be deduced from (i) by applying Theorems 3.8.8, 3.8.4.
Remark 6.2.26 In Section 6.4 we shall show that Q H = ~is actually Kripkecomplete. However V-canonicity is rare. In general one may expect that superintuitionistic predicate logics with simple axioms are not V-canonical. For example, classical logic Q C L is obviously not V-canonical; in fact, every QCL-place (as well as every L-place for any L) is not maximal in the corresponding canonical frame, since it may be extended after adding new constants to the language. Another trivial observation is that Q C L I- CD, but C D is not valid in VFQcL, since as we have just noted, every QCL-place sees another place with a larger domain. Moreover, the same argument shows that VFL is of infinite depth (for any logic L). Thus every superintuitionistic logic containing a propositional finite depth formula Pk , k 1 (see Section 1.1.2) is not V-canonical. Nevertheless for logics with additional axioms listed in Chapter 2 we can sometimes prove Us-canonicity. Let us now consider such examples.
>
6.3
Intermediate logics of finite depth
Let us begin with some simple, but useful remarks.
Lemma 6.3.1 Let L, L' be superintuitionistic logics, (I?, A) an intuitionistic theory. (1) If ( r , A) is QH-consistent, ZC I', then
(I',A) is L-consistent.
(2) If (I?,A) is L3-complete, then it is L13-complete iff
(3) If ( r , A) is L3'd-complete, then it is L13'd-complete iff L'
z
r. r
Proof (i) Suppose ( r , A) is L-inconsistent, E r, and let us show that (I?,A) is QH-inconsistent. We have Fr, /\ rl > V Al for some finite rl G r , Al G A. So a generator A of this formula is in L; hence WA E I?. Thus I? tQH A, and so I? t QAI'l ~ > V A l , which implies kQH V Al, i.e. the QH-inconsistency of ( r , A). (ii) (If.) Assume that p C r and (I?, A) is L3-complete. Then it is QHconsistent, and thus L'-consistent, by (i). Now L'-completeness and the existence property readily follow from L3-completeness. (Only if.) Suppose r, and take A E T - I'. Since Q H -I T > p, it follows that tLT > A, and thus (I?, A) is L1-inconsistent. (iii) The proof is similar to (ii).
z
502 CHAPTER 6. KMPKE COMPLETENESS FOR VARYING DOMAINS R e m a r k 6.3.2 Every QCL-place F has all the properties from Lemma 6.1.3 (with L = QCL), so it obviously satisfies the conditions 6.2.12 (>), (V). In both cases we can take r' = r. In fact, (A1 > A2) @ r; implies -(Al > A2) E I?, and thus A1 E r , A2 6r. Next, if VxA(x) 6I?, then lVxA(x) E r , and thus 3 x ~ A ( x E) I', by classical logic (cf. 2.6.17(ii)). Hence by the existence property, for some c E S we have lA(c) E r, i.e. A(c) @ I?. Now let us consider Kuroda's axiom. Recall that K F is valid in Kripke frames F = (W, R, D) with the McKinsey property: Vu E W3v E W(uRv & v is maximal), and similarly for Kripke sheaves. As noted a t the end of the previous section, there are no maximal worlds in the canonical frame VFL, so VFL does not have the McKinsey property (for any logic L). However let us show that the McKinsey property holds in u < F ~ = )whenever L contains K F . L e m m a 6.3.3 Let L be an s.p.1. (=) containing K F , ( r , 0 ) an L-consistent theory. Then ( r , 0 ) is QCL-consistent. Proof In fact, suppose tqcL-(A r l ) . Then ( A r l ) has a generator A such that -A E QCL. By the Glivenko Theorem 2.12.1, then 1 A E Q H K F L, and thus kL l ( A r l ) . SO the QCL-inconsistency of ( r , 0 ) implies its Linconsistency.
+
c
L e m m a 6.3.4 Let L be an s.p.1. (=). (1)
r is a QCL-place iff I'
is maximal in U I F L .
(2) If K F E L, then for any L-place r there exists an QCL-place I" 2 r Proof (1) Every QCL-complete theory is maximal among consistent theories in the same language, so every extension of I? is conservative. By 6.3.2, r is a QCLV-place, i.e. r E UPL. -
(2) If already Q C L C I?, then according to 6.3.2, r is a QCLV-place, so we we can take I" = r , The other way round, if an LV-place r is not maximal in U5FL, then r < A for some A E UPL, so there exists A E ((AlDr) - r ) . Then A = B(a) for some B(x) E IF(') and a tuple a from Dr. By 6.2.17 it follows that U S ML((=),F Iy A V -A, and thus
Hence Vx(B(x) V l B ( x ) ) $!
r,and thus QCL c I? (again by 6.2.17).
Otherwise there exists B E (QCL - F). Obviously, the theory ( r , 0 ) is L-consistent, and thus QCL-consistent by 6.3.3. By Lemma 6.2.6, there exists a QCLV-place r' I?. We also have r < r' since B E (I?' - r ) n I F .
>
6.3. INTERMEDIATE LOGICS O F FINITE DEPTH
503 H
The argument for logics with equality is similar. So we readily obtain completeness:
+
+
T h e o r e m 6.3.5 The logic Q H K F (respectively, Q H = K F ) is strongly determined by Kripke frames (respectively, Kripke sheaves) satisfying the McKinsey property. Now let us prove completeness for intermediate logics QHP; introduced where the logic AL is in Section 2.4. Recall that QHP; = A"'(QcL), axiomatised by formulas bA = p V (p > A) for A E L, p not occurring in A; QHPZ-frames are the intuitionistic frames of depth 5 k. L e m m a 6.3.6 If I?, I?' are L-places,
r < I?'
and AL
G I?, then L E r'
P r o o f Let B E L ( r ) n ( r ' - l?) and suppose L g I", A E L - I?'. Then B V ( B > A) is a substitution instance of 6A, so B V ( B > A) E AL. On the other hand, (B > A) 6 I?; in fact, otherwise (B > A) E I' G I", which together with B E I?', implies A E I" contradicting the choice of A. Now since also B 6r , we obtain B V ( B > A) 6 by Lemma 6.2.5. H Hence it follows that AL I?, in a contradiction to the assumption.
r,
L e m m a 6.3.7 If QHP;
L, then the frame U I F L is of depth 5 k.
P r o o f Suppose rl < . . .rr, is a chain in U I F L . We have L C QHP; = rl, which yields Q C L = Q H P ~ r k , by applying Lemma Ak-'(QcL) 6.3.6 (k - 1) times. Thus rkis maximal in u ~ Fby~Lemma , 6.3.4(i). T h e o r e m 6.3.8 The logic QHP;(=) for k (respectively, Kripke sheaves) of depth k.
2 1 is determined by Kripke frames
P r o o f We already know that L = L P is ~ sound for frames of depth k, so it suffices to show that U ~ F L is of depth k. By Lemma 6.3.7, U
+
504 C H A P T E R 6. KRIPKE COMPLETENESS F O R VARYING DOMAINS Definition 6.3.10 Let L be a superintuitionistic logic, I? an L-place. W e say that E is a characteristic formula for I? if E E (-r) and A V ( A > E ) E I? for any A E L(=)(I?). Or equivalently, this means that ( A 2 E ) E I? for any A E (-I?), i.e. E is the weakest Dr-sentence outside r. If a characteristic formula exists, it is clearly unique up to equivalence in I? (the relation ( A r B) E I?). The next lemma uses the operation A introduced in Section 2.13 L e m m a 6.3.11 If I? is a AL-place, which is not an L-place, then there exists a sentence E E IF(') (without extra constants), which is characteristic for I?. Proof Let E E (L - I?). Then p V ( p > E ) E A L for p not occurring in E l and thus A V ( A > E ) E I? for any A E C(I?). W Corollary 6.3.12 If QHP; L for some n exists a characteristic sentence.
> 0, then for any L-place there
+
Proof Recall that Q H P ~= A ( Q H P ~ - ~for ) k > 0 and LP: = Q H I is inconsistent. For an L-place I?, take the least k (> 0) such that I? is an w ~ H ~ t - t h e oThen r ~ . apply Lemma 6.3.11.
6.4
Natural models for modal and superintuit ionistic logics
As we can see, canonical models work well only for some particular predicate logics. In subtler cases it is convenient to use so-called 'natural models' obtained from canonical models by a sort of 'selective filtration', together with canonical maps (cf. Definition 6.1.13). Let us begin with some simple remarks on canonical maps. The following is a trivial reformulation of the canonical model theorem. L e m m a 6.4.1 Let M be a Kriplce model for a modal logic L, i n which every individual domain is a small subset of S* (cf. Definition 6.1.13). Then for any world u E M , for any A E L ( u )
i.e. canonical maps are reliable. The same holds i n the intuitionistic case, with obvious changes. L e m m a 6.4.2 Let M be the same as i n the previous lemma, relations i n M . Then for any u , v E M
i.e. canonical maps are monotonic.
Rithe accessibility
6.4. NATURAL MODELS
505
P r o o f Almost obvious: if O i A E v M ( u ) ,i.e. M , u k U i A , and u a v , then M , u k A , i.e. A E v M ( v ) . In the intuitionistic case, if u R v , then v ~ ( uE) v ~ ( v )by, truth-preservation. H
However note that in general v~ is not a morphism of Kripke frames. The lift property does not always hold, because it may happen that r u R L i r v for incomparable u , v E F , but there does not exist w E R i ( u ) such that I?, = I?,. Definition 6.4.3 Let L be a predicate logic (N-modal or superintuitionistic, without or with equality), F = (W,R 1 , . . . ,R N ) a propositional frame of the corresponding kind. An L-map based on F is a monotonic map from F to (the propositional base of) V F ~ = )i.e. , a map h such that for any u , v E F
In the intuitionistic case we say that h is an LQ-map i f all h ( u ) are LV-places, i.e. h is actually a map to UFL. Definition 6.4.4 Let h be an L-map based on a propositional frame F . The predicate Kripke frame associated with h is F ( h ) := ( F ,D ) , where D, = (DL)h(U)for u E F . If L is a logic with equality, the Kripke frame with equality associated with h is F' ( h ) := ( F ,D , x), where
c x u d iff ( c = d ) E h ( u ) for u E F and c,d E D,. The L-model associated with h is ~ ( ' ) ( h ):= ( ~ ( = ) ( h~)(, h ) )where , < ( h ) , ( p r ) := { c E D p I p F ( c ) E h ( u ) } foru E F. All these frames and models are well-defined, since uRiv implies h ( u ) R ~ i h ( v ) (which means h ( u ) C h ( v ) in the intuitionistic case).
-
-
V F L and the associated L-model M ( h ) Definition 6.4.5 An L-map h : F are called natural i f h = U M ( ~ ) , L i.e. , (h,id) : M ( h ) V M L is reliable: for any u E F , A E C ( u )
This definition is an analogue t o the canonical model theorem for M ( h ) . In this case we readily obtain M ( h ) != L. R e m a r k 6.4.6 Lemma 6.4.1 shows that every predicate Kripke model M with small domains can be presented as a natural Kripke model M ( h ) in a unique way; namely, put
and similarly in the intuitionistic case.
506 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
-
VFL based on F = (W, R1,. .. ,RN) Definition 6.4.7 A modal L-map h : F is called selective if it satisfies the condition
-
(0) if u E F and OiA E h(u), then there exists v E Ri(u) such that A E h(v).
L e m m a 6.4.8 An L-map h : F
VFL is natural igP it is selective.
Proof The 'only if' direction is trivial, since in a natural model M(h) we have M(h), u b A iff A E h(u) (or VML,h(u) I= A, by the canonical model theorem) for any A E C(u). To prove 'if', we check the above equivalence by induction on the length A (for any u). Let us consider the case A = OiB; other cases easily follow from Lemma 6.1.3. If OiB @ h(u), then h(u)ljOiB, and thus h(u) k O p B . By ( 0 ) , there exists v E Ri(u) such that 1 B E h(v), which is equivalent to B @ h(v). Hence M(h), v !+B by the induction hypothesis. Since v E Ri(u), we obtain M(h),u!#OiB. The other way round, if h(u) b OiB and uRjv, then h(u)RLih(v),by monotonicity. So h(v) k B, and thus v b B , by the induction hypothesis. Since v is arbitrary, this implies u k OiB. Note that the condition ( 0 ) is a generalisation of selectivity from Definition 6.1.31 (where F is a subframe of VFL). Moreover, by (0)and monotonicity it follows that the image of h is a selective subframe of VFL. The intuitionistic analogue of 6.4.7 is the following
-
VFL based on F = (W, R) Definition 6.4.9 An intuitionistic L-map h : F is called selective if it satisfies the following two conditions (cf. 6.2.12 (>), (d)). (3) if (A1 3 A2) E (-h(u)), then A1 E h(v), A2 @ h(v) for some v E R(u);
(d) if 'v'xA(x) E (-h(u)), then A(c)
-
# h(v) for some v E R(u) and c E D,.
L e m m a 6.4.10 An intuitionistic L-map h : F selective.
VFL is natural iff it is
Proof Similar to Lemma 6.4.8. The 'only if' part is trivial. To prove 'if', we show by induction on the length of A E L(u) (for any u) that M ( h ) , u Il- A iff A E h(u) (or VML, h(u) Il- A). Let us consider only three cases. Let A = B > C . If A # h(u), then by (>), there exists v E R(u) such that B E h(v), C @ h(v). By the induction hypothesis, we obtain v Il- B, v Iy C. Hence u ly B > C. The other way round, if u Iy B > C , then there exists v E R(u) such that v It B , v Iy C. Thus h(v) Il- B , h(v) Iy C by the induction hypothesis. Since h(u)RLh(v), it follows that h(u) Iy B > C.
6.4. NATURAL MODELS
Let A = 3xB. Then u II- A iff 3c E Du u It- B(c) iff 3c E Du h(u) IF B(c) by the induction hypothesis. Since h ( u ) R ~ h ( v )the , latter is equivalent to h(u) ltA. Let A = VxB. If A @ h(u), then by (V), there exist v E R(u), c E D, such that B(c) @ h(v). By the induction hypothesis, v Iy B(c). Hence u Iy A. The other way round, if u Iy VxB, then there exist v E R(u), c E D, such that v Iy B(c). Thus h(v) Iy B(c) by the induction hypothesis, and therefore H h(u) Iy A, since h(u)RLh(v). Sometimes we specify the terminology and say, e.g. that h is an (L, R1)-map if R1 is a selective relation on VPL such that uRv implies h(u)R1h(v). In the intuitionistic case we also use the term LV-map if h is a map t o UFL; M(h) is called a natural LV-model. If an L-map h is injective and also UR~V iff h(u) R ~ i h ( v ) , then M(h) is clearly a selective submodel of the canonical model. Also note that the naturalness condition ( 0 )is a weak analogue of lift p r o p erty. In the intuitionistic case an L-map h over a p.0. set is called proper if
in particular, if h is L, I-natural and proper, then
Obviously this condition does not imply the injectivity for h. Similarly in the intuitionistic case one can consider proper natural models on a quasi-ordered F; in the modal case for extensions of QT proper natural models on reflexive frames can be used. Informally speaking, in these cases in natural models we do not have to 'repeat' the same L-place in strictly accessible worlds, thanks to reflexivity. VML a Lemma 6.4.11 Let L be a modal or superintuitionistic logic, MI selective submodel, Fl the propositional frame of MI, h : F -H Fl. Then h is a natural L-map.
Proof By Lemmas 6.4.8, 6.4.10, it is sufficient to show that h is selective. The monotonicity of h follows from the definitions. In the modal case, to check ( 0 ) , suppose OiA E h(u). By selectivity of MI, there exists I? E M1 such that h(u)RiLr and A E r. Then by the lift property for h, there exists v E Ri(u) such that h(v) = I?, so A E h(v). In the intuitionistic case (I),(V) are checked by a similar argument, which H we leave to the reader.
508 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Lemma 6.4.12
(1) Let L be an N-modal logic, r E VPL. Then there exists a Kripke model M based on a standard greedy tree F such that YM(A)= r and M != L. (2) If L contains OiT for 1 5 i 5 N , then one can take F = FNTw. (3) If L is 1-modal and L 2 QS4, then the claim holds for F = IT,.
Proof By Proposition 3.12.9, there exists a countable reliable M1 VML verifying r. Let Fl be the propositional frame of M I . We may assume that is the root of M1 (otherwise, replace M1 with M1 T I?). By Proposition 1.10.18, there exists a pmorphism h from some greedy standard tree F onto Fl sending h to I?. By Lemma 6.4.11, h is a natural L-map, = h ( ~= )r. so we can take M = M(h). Thus VI~.I(A) In the case (2) all the formulas OiT are true in VML, hence they are true in its selective submodel MI. This means that Fl is serial, thus we can take F = FNT,, by Proposition 1.10.18. In the case (3) we use Proposition 1.11.11 to construct a pmorphism h : IT, Fl sending A to r.
-
Hence we obtain
Proposition 6.4.13 Let FNT, be the set of all greedy standard subtrees of FNTw . Then
Now let us modify this proposition for the logics described in Theorem 6.1.29.
Proposition 6.4.14 Let A be a propositional one-way PTC-logic. Let GT(A) be the set of all greedy standard A-trees. Then
Proof (i) By Theorem 6.1.29, the logic L = QA is Kripke complete. By Proposition 1.11.5, it is also A-elementary. So by Proposition 3.12.8, it has the c.f.p., i.e. L = ML(KVo(A)), where Vo(A) is the class of all countable A-frames. Hence by Lemma 3.3.21, L = ML(KVl(A)), where Vl(A) is the class of all countable rooted A-frames. Next, by Proposition 1.11.11, every frame from V l (A) is a p-morphic image of a frame from GT(A). Therefore ML(KGT(A)) C L, by Corollary 3.3.14. The converse inclusion is a trivial consequence of the definitions. Of course, instead of applying 3.3.14, we could apply Lemma 6.4.12. In the case (ii) the argument is the same, based on the corresponding facts H about frames with eaualitv.
6.4. NATURAL MODELS For logics with closed equality there is a similar result: Proposition 6.4.15
(2) Let A be a propositional one-way PTC-logic. Then QA=
+ C E = ML'(ICGT(A)).
Proof (ii) We modify the argument from the previous proof. Kripke sheaf completeness follows from Theorem 6.1.29 and Proposition 6.1.26. The logics are A-elementary, since CE corresponds t o a first-order property of KFEs. As in the proof of Lemma 6.4.12, we obtain that every L-consistent theory r is satisfied a t the root of a Kripke sheaf model M = M(h) based on a standard greedy A-tree. Let Q, be the Kripke sheaf of M. Since M b C E , it follows that Q 'F CE. Then by Lemma 3.10.8, Q, is isomorphic to a simple Kripke sheaf a' (in fact, to apply this lemma, one should take the corresponding Horn closure). So F is satisfied in a Kripke frame from QGT(A).
Recall (Definition 1.10.10) that the intuitionistic universal tree is IT, := (Tw, 4), where a < P iff 3y ( 0 = a y ) . So in the intuitionistic case we have the following analogue of 6.4.12.
Lemma 6.4.16 Let L be a superintuitionistic logic, R' a selective relation on VPL. Then for any L-place r there exzsts a proper natural (L, R1)-model on a standard (complete) subtree F of IT, such that VM(A)= I?. We shall often denote v ~ ( uby ) r, if there is no confusion. It is clear that for v E P(u) one can choose an L-place F, non-equal to F,, due to the reflexivity. Therefore there exists a natural (L, R1)-model on IT,: in fact, one can copy L-places I?, from maximal points u of F at all points from IT,Tu. This natural model is in general improper. But if for example, R' = RL, then one can directly construct a proper natural model on IT,; recall that every L-place I? in the canonical model can be properly extended by extending its 'domain' Dr. Hence we obtain completeness: Theorem 6.4.17
Now let us prove a somewhat stronger form of completeness (cf. Dragalin 119881).
510 CHAPTER 6. KRJPKE COMPLETENESS FOR VARYING DOMAINS T h e o r e m 6.4.18 Let F be an effuse countable tree. Then
(1) IL(KF) = QH; ML(KF) = QS4; (2) IL=(K&F) = QH=; ML=(KF) = QS4';
(3) ILz(KF) = QH=d, ML(KF) = QS4'". Proof (1)In the modal case the argument is similar to Proposition 1.11.21. We have F r u -B IT2 + ITwfor some u. Hence by Proposition 3.3.14, ML(KF1u) G QS4. Thus by the generation lemma, ML(KF) C ML(KFTu) G QS4. The remaining statements are left to the reader. Corollary 6.4.19 QH(=)(~) is determined by the universal n-branching tree IT, for any n 2 2. If a relation R' is selective on U P L (for an intermediate logic L), then similarly to 6.4.12, for any I? E U P L one can construct a proper natural (LV,R1)model on a standard greedy subtree F of IT, and a natural (LV, R1)-model on IT, (not necessarily ~ r o ~ e rsuch ) , that FA = F. In particular, one can apply this construction t o the selective relation on UPL. However unlike the case of VFL, we cannot state the existence of a proper (LV, <)-model on the whole IT,. In particular, we obtain
<
L e m m a 6.4.20 If QHP; is a frame of depth k .
<
CL
and u is a proper (LV, <)-model on F, then F
L e m m a 6.4.21 If QHPE L then for any I? E U P L there exist a proper (LV, <)-model u on a standard (greedy) subtree F of IT: and an (LV, <)-model on ITkw such that U ( A ) = r. Thus we obtain the following completeness result for Q H P ~ , k
> 0.
Proposition 6.4.22
( I ) QHP; = IL(K(IT:)), (2) (QHP;=
= IL= (KE(IT;)).
In Section 6.10 we will axiomatise the superintuitonistic predicate logic of the frames IT: of finite depth k and finite branching n; it is just the logic of all posets of depth k and branching n (obviously, all these posets are finite). But first we shall give some simple remarks on natural models based on posets of finite heights. The intuitionistic conditions of naturalness for this case can be slightly modified.
<
<
L e m m a 6.4.23 Let M be an intuitionistic (L, Rf)-model on a poset F of finite depth. Then
(1)
M is natural iff the following conditions hold (for any u E F):
6.4. NATURAL MODELS
511
(3')
If (A1 > A2) E (-I?,) v E Re (u);
(V)
if VxA(x) E (-r,) and A(c) E for some v E Re(u).
(2)
If M is an (LV, Rf)-model, then the condition (3')implies (Av').
and A1 @ r,, then (A1
r,
> A2)
@ I?, for some
for all c E Dru, then VxA(x) $!
rv
Proof
+ ( 2 ) If A1 E I?,, A2 @ rvfor some v E Re(u) then (A1 > A2) @ r,. (V') + (V) The proof is similar. (3)+ (1') Obviously, if (A1 > A2) @ r,, then A1 E I?, and A2 @ I?, for a (2')
maximal v E R(u) such that (A1 > A2) @ I?,.
(3') &
(V)
* (V) If VxA(x) @ r, and A(c) E r, for all c E S, then by (Av), for
an L3V-complete r,, we have Vx(A(x) V C) E r, for some C @ r,. Then (C > VxA(x)) @ r, (otherwise VxA(x) E r,) and by (I)', (C > VxA(x)) @ rvfor some v E Re(u); hence VxA(x) @ I?,.
Let us now prove two general lemmas on natural models.
Definition 6.4.24 A subtree F = (W,R1,. . . , RN) FNT, is said to be small if Vu E F V i E IN ( c i (u) - Ri(u)) is infinite. I n the transitive (or intuitionistic) case we should take PF(u) instead of Ri(u) i n the above condition. Thus every world of a small subtree F has infinitely many successors in FNT, unused in F . Note that every denumerable tree, in which every point is of finite height, is isomorphic to a small subtree of FNT, (or IT, in intuitionistic case).
512 CHAPTER 6. KRIPKE COMPLETENESSFOR VARYING DOMAINS
Lemma 6.4.25
Then every L-map based on a small subtree F of FNT, can be extended to a natural L-map based on a (small) subtree F' 5 FNT, such that F' > F .
(1) Let L be an N-m.p.1.
(2) The same holds for an s.p.2. L and a small subtree F of IT,.
Proof (1) By induction we construct an increasing sequence of small trees
and a sequence h = ho
F' := UFn is small, n
c hl
2 . . . of L-maps hn
Fn+1- Fn
: Fn ---,VF~=)such
that
N
c U
~i
(F,) and hn+l 'cures the defects' of
i= 1
h, :
(*) for any u E F,, if OiA E h,(u), then 3v E Ri,,+l(u) A E h,+l(v), cf. Lemma 6.1.16. Actually we may assume that Fntl - F, consists only of is the v constructed by (*). Thus the joined map h' := Uh, : F' ---t VF~,=) n
selective, i.e. natural, by Lemma 6.4.8. More explicitly this is done as follows. For any N-modal S*-sentence OiA choose the corresponding 'Skolem function'
such that rRLiFOiA(r)for any r E $(OiA). This function exists by the axiom of choice. Now if hn is constructed and (*) holds for Fn-1, then we extend h, to hn+1 by putting
whenever - is the list of all N-modal S*-sentences, u E F , - Fn-l and hn(u) E dorn(fo,~,,,) (i.e. OiA, E h,(u)). So it follows that (*) holds for Fn - in fact, if OiAm E hn(u), the corresponding v is u(2mN i - 1). It also is small (if Fn is small). In fact, infinitely many ~i-successors follows that they are of the form u(kN i - 1) of u E F, - Fn-1 do not appear in with odd k. (2) In the intuitionistic and in the modal transitive case the definition is slightly modified. In the transitive case one should take P(u) instead of ci (u). In the intuitionistic case the condition (*) is replaced with two (where IF denotes forcing in the canonical model):
+
+
(**) if hn(u) Iy A 3 B, then 3v E P(u) (hn+l(v) IF A & hn+l(v) IY B ) , (* * *)
if hn(u) Iy VxA(x), then 3v E P(u) 3c E Dh,+l(v) hn+l (v) Iy A(c).
6.4. NATURAL MODELS
513
Similarly t o the modal case, we use corresponding Skolem functions fA3*, ~VXA(X). f A 3 ~ is defined on the complement of [ ; ( A 3 B), and r R LfA3B(I'), -t t ( B ) f ~ > ~ (E r[:(A) ) W Analogously, fvxA(x) is defined on -6: (VxA(x)). Note that in the intuitionistic case we can construct an LV-map or LVL-map, or even LV<-map h' if the original map h is of the corresponding kind. For LV< we should not repeat ruin future worlds and similarly for any selective relation (in the intuitionistic or in the reflexive modal case). Lemma 6.4.25 directly implies Lemma 6.4.12, which is its particular case for a singleton frame on {A), with h ( ~ =) I'. A small tree F' from 6.4.25 is clearly isomorphic to a greedy one (or to the whole FNT,, or to IT, in the intuitionistic case). Actually the proof of 6.4.25 is a more explicit version of the proof of 6.4.12. In fact, the intermediate L-maps h, correspond to steps in calculating Skolem functions used in a well-known proof of the downward Lowenheim-Skolem theorem. The construction in 6.4.25 extends L-maps by adding new points above the existing ones. Now we shall describe a construction allowing us to add new points (theories) below the existing theories. The main idea is to extend theories in a conservative way, by adding only sentences with extra constants. Although we will further apply this construction to the linear case, we formulate it for arbitrary trees, see Lemma 6.4.30 below. Definition 6.4.26 The domain of an L-map h (and of the corresponding L:= U Dh(u). model) is
DL
-
uEF
-
Definition 6.4.27 Let h : F VF~=)and h' : F' VF~=)be two L-maps. We say that h' is conservative over h if for every u E F n F':
(2) h(u) 5 hf(u) (cf. 6.2.11). We say that h' is a conservative extension of h if it is conservative and F
F'.
The condition (1) means that additional constants in h'-domains are 'new' for h.
-
L e m m a 6.4.28 If h : F VFL=) and h' : F' -+ VF~=) are two L-maps on subtrees F, F' of FNTw (IT, in the intuitionistic case) and h' is conservative over h, then h' can be prolonged to a conservative extension h" : F U F' ---, V F ~ = ) of h. Usually this lemma is applied to the case F'
cF
514 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Proof Consider the case when F' c F. We construct hl'(v) for v E F - F' by induction on its length l(v). To simplify notation, we write I?,, rl, r: for h(u), hl(u), hl'(u) respectively; similarly D, abbreviates Dh(,), etc. Assume that I?: satisfying (i) and (ii) is already constructed (with I?: = I'; for u E F n F') and consider v E &(u) (or v E P(u) in the intuitionistic case). Let us begin with the intuitionistic case. Put (I?, A ) := (r:ur,, -I?,). Then D(r,*) = D: U D,. Let us show that ( r , A) is L-consistent. In fact, suppose I-L
A1(c,c") A A(c, d )
> B(c, d ) ,
where r (c)
C Dv n D t = D,
(by (i) for r:),
A ' = A A ~ , A = A A ~ ,B = VB,, A; k
i
T
(d)
C D, - D,,
r(cl') C D t - D,, A~ E r,, B? E -r,,
E r:,
j
so A' E I?:, A E I?,, B E -r,. P u t A1'(c) := 3xA1(c,x), where x is a distinct list corresponding t o c". Then k L A"(c) A A(c, d ) > B(c, d ) , and
It follows that r, is L-inconsistent, which is a contradiction. Now, by Lemma 6.2.6, we extend ( r , A ) t o an L3-complete I?: k (I',A); moreover, we choose D: such that D: n c D, U D t ; then D t n ~ h + D,, since D t n Dh+ = D, C D,. Obviously, C r: and ??:ID, =I?,, i.e. rv5 r:. Therefore by repeating this construction for all v E ( F - F'), we obtain a conservative extension of F. Now let us consider the modal case. We put
~ h +
again Dr = Dt U D,. Let us check the L-consistency of I?. Suppose
I-L 1(A1(c,c") A A(c, d)), where A' = AAL, A = A A j , CliAi E ,:?I k
A, E
r,, so CliA1 E r:,
A E
r,
j
(c, d, c" are just the same as in the intuitionistic case). Similarly to that case, D, = I?,. Since put A1'(c) := 3xA1(c,x). Then k L 1(A1'A A) and UiA" E I?, RLir, (in h), i.e. OJ?, r,, we obtain A" E I?, . Hence ,?J is L-inconsistent, which is a contradiction. with a set of Now by Lemma 6.1.9 we extend I' to an L-Henkin theory C D,. Then r:RLir: and [:?I D, = r,; in constants D: such that D t n fact, A E -rVimplies 1 A E rVC r:, so A $ r:.
c
~ h +
Lemma 6.4.29 Under the conditions of Lemma 6.4.28, let F U F' be a small subtree of FNT, (or IT,). Then h' can be prolonged to a natural conservative extension ha of h over a (small) Subtree Fa 2 F U F'.
Proof
Apply Lemma 6.4.25 t o h" constructed in Lemma 6.4.28.
6.5. REFINED COMPLETENESS THEOREM F O R QH + K F L e m m a 6.4.30 Let F be a subtree of FNT, (or IT,).
F - := F
-
515
Let h be an L-map over
{ A ) . Let I' be an L-place such that
n D t 2 D, for all u E F (or equivalently, for all u E R i ( ~ )i ,E I N , or for all u E P ( A ) resp.)
(1) Dr
(2) in the intuitionistic case: r l ( D r n D?) C I?, for all u E P(A) n F (hence for all u E F ) ; in the modal case: 0: ( r l ( D r n D:)) C ru for all u E R i ( ~ ) . Then there exists a conservative extension hl' of h to F such that h " ( ~ = ) r. P r o o f Repeat the proof of Lemma 6.4.28 word for word. For constructing hl'(v) for v E R ~ ( A(i.e. ) for u = A), we use (ii) in the intuitionistic case and (i) in the modal case. R e m a r k 6.4.31 Strictly speaking, Lemma 6.4.28 cannot be applied directly here. We cannot just prolong h t o F by putting = r l ( D r n D t ) , since the latter does not always have the existence property. But this formal obstacle is not really essential.
6.5
Refined completeness theorem for QH
+K F
Definition 6.5.1 A p.0. set F = (W,R) is called coatomic i f it satisfies the McKinsey property:
V u 3v ( u R v & R ( v ) = { v ) ) , 2.e. every world sees a maximal world. Recall that formula K F is valid in all frames over coatomic p.0. sets. Definition 6.5.2 A p.0. set F R-' is non-branching.
=
(W,R) is called tree-like if it is rooted and
Definition 6.5.3 Let IT! be the set of all w-paths (i.e maximal chains) i n the
universal tree IT,. Let us extend IT, to the tree-like p.0. set IT, = IT, i n which IT; is the set of maximal points and
for u E IT,,
z E IT;. A set G E IT! gives rise to the subframe IGT , :=
of
IT, with the set
IT, 1 ( I T , U G )
of maximal elements G .
u IT!,
516 CHAPTER 6. K R P K E COMPLETENESS FOR VARYING DOMAINS --G
Obviously, IT, is coatomic and tree-like. IT, is coatomic iff the paths from G cover IT,, i.e.
V u E IT, 32 E G u E Z .
(a)
---G . It is also clear that if IT, is coatomic, then there exists a denumerable G such that IT, is also coatomic - in fact, for each u E IT, choose a G' path z, E G containing u and put GI = { z , I u E IT,). - - G I
L e m m a 6.5.4 Let L be a superintuitionistic logic containing K F . Then for any G IT! and for any L-place I? -G
(1) there exists an L-natural model M on IT, such that vM(h) = I?; -G
(2) similarly, there exists an LSV-natural model M on IT, with UM(A)= r .
Proof First, by Lemma 6.4.16 we find an L-natural model M over IT,. Obviously, we may assume that the set UD, is small, i.e. it is a subset of S* (cf. u
Section 7.1). Now for every path z consider the L-consistent theory
I k E w) E G we put DL := D,, and k (I?, 0 ) ,where I' := UI?,, . By Lemma 6.3.3, = {uk
k
this theory is QCL-consistent. So by Lemma 6.2.6, there exists a QCLV-place ,'I 2 I'with Dr, DL. -G The resulting map h : IT, VFL sending v to r,, is L-natural, since the conditions (>), (V) for a QCL-place I?, hold trivially. It is also clear that I', C r, whenever u E z (and I', 5 r, in the proof of (2)). W
>
-
Note that this construction can be applied to an arbitrary selective relation
R on VPL (or UPL); to obtain an (L, R)-natural model, one should only check that r,RI?, holds for u E z E IT! in the resulting model. Proposition 6.5.5 Q H 4 poset IT,.
+ K F = IL(Km,)
-G = IL(KIT,)
for any coatomic
One can easily obtain the corresponding completeness results for logics with K F , and in K - for QH=d + K F . equality, in the semantics ICE - for QH'
+
6.6
Directed frames
In this section we prove completeness results from [Corsi and Ghilardi, 19891. We consider the following formulas: M L := VxOOP(x) > OOVxP(x), ML* := OVxOUP(x) > OOVxP(x), F := 0 3 x P ( x ) > 03xOP(x).
6.6. DIRECTED FRAMES
517
Definition 6.6.1 A propositional Kripke frame F = (W,R) is called directed if the relation R is reflexive, transitive and such that
A predicate Kripke frame F = (W,R, D ) is called directed if its base (W,R) is directed. In this section we use specific propositional frames: Definition 6.6.2
IT, is called the subordination frame; IT, + 1 (IT, with the added top element oo) is called the subordination frame with the greatest element; I T , + C , (IT, with the added greatest countable cluster C,) is called the subordination frame with the greatest cluster; the elements of C, are denoted by coo, ml,. . . ; the ordinal product IT, . w is called the multiple subordination frame; recall that this is the set w" x w ordered by the relation
(a,n)R(P,m)iff n < m V ( n = m & a
40).
-
Definition 6.6.3 Let L be a l-m.p.1. or s.p.1. A subordination L-map is a V F L , where F is one of the subordination frames from natural L-map h : F Definition 6.6.2, with the following properties
(1) the set
~ h := + U Dh(u) is small; uEF
(3) i f F = I T ,
. w , then h is a subordination map on each copy of IT,.
I n this case M ( h ) is called a subordination L-model.
>
-
Lemma 6.6.4 Let L QS4 and let I? be an L-place. Then there exists a subordination L-map g : IT, V F L such that g ( h ) = I?.
-
Proof By 6.4.12, we can construct a natural L-map h : IT, V F L such that h ( h ) = I?, so now we are going to transform it into a subordination L-map 9To satisfy the conditions 6.6.3 ( I ) , ( 2 ) , we use renaming of constants. For this purpose we should first trace all constants appearing in D:. Viz., for cE we say that h introduces c at stage a if a is minimal in the set { p E
~ h +
518 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
IT, I c E D h ( ~ ) )The . same constants can be introduced a t different stages, but they are all incomparable. Now let S* = {en I n E w) be an enumeration of our universal set of constants. We choose its small subset S and present it as a two-dimensional array: S = { ~ n , , 1 72. E W , a E w ~ ) .
-
Next, we define g : IT,
VFL as follows.
where A, is obtained from A by replacing all occurrences of each constant en with cn,p such that 0 a and h introduces en a t stage p; this is uniquely determined by n and a. Of course we have to ensure that g(a) is really an L-place. First note that A can be also obtained from A, by renaming constants. This operation respects provability, so FL A, +kL A for any Dh(,)-formula A; formally, one should ar-
<
gue by induction on the inference of A. Thus Y L1
k
k
(Ai), implies YL 1 A Ai, i= 1 i= 1 i.e. the L-consistency of h(a) implies the L-consistency of g(a). The Henkin property is almost obvious. In fact, if 3xA,(x) = (3xA(x)), E L(g(a)), then 3xA(x) E C(h(a)). By the Henkin property for h(a), there is n such that (3xA(x) > A(en)) E h(a). Therefore (3xA,(x) > A,(%,,)) E g(a). Now g satisfies the condition 6.6.3 (I), since the new set of constants S is small. For (2), note that if a 10 , then Dh(p)- Dh(,) consists exactly of the constants introduced at stage P, so all constants in D g ( ~-) D,(,) are of the W form cn,p. Thus if P y,they cannot appear in g(y).
>
Lemma 6.6.5 Let L QS4.2. If h is a subordination map such that h ( h ) then U Cl-h(a) is L-consistent.
> L,
aEwm
Proof Suppose
U 0-h(a) is L-inconsistent.
Then for some n,
is L-inconsistent. Take the smallest such n; obviously n Note that a /< 0 implies 0 - h ( a ) C 0-h(P). In fact, by monotonicity of h,
U
U- h(a)
lalSn
C ~ E U ~
> 0.
Now A E 0 - h ( a ) implies OA E h(a), hence UUA E h(a) by S4, so OA E O-h(a) C h(P), i.e. A E U-h(P). Hence it follows that U 0 - h ( a ) is L-inconsistent. lal=n Now since every 0 - h ( a ) is /\-closed, by joining conjunctions, we obtain different sequences a ~ l c. ~. .,,a&, of length n and formulas Ai E O-h(aiki) such that {Ai I 1 5 i 5 m) is L-inconsistent. (Note that ai are not necessarily distinct.) Every Ai can be presented as [ci/xi]Bi, where r(ci) Dh(ai),xi is a
c
6.6. DIRECTED FRAMES
519
distinct list of variables and Bi is a Dh(ai)-~enten~e. Since h is a subordination map, the sets r(ci) are disjoint, so we may assume that all xi are disjoint. By Lemma 2.7.10 tAi implies k Vx-. Bi ,
-A i
i
where x = xl . ..G . By 2.6.15(xvi), we obtain
kL -.3x
A Bi7 and thus
kL
A 3xiBi
1
i
i
Ai3xiBi (0-introduction
by 2.6.15(xxvii), since xi are disjoint. Hence k ~ 001 is admissible, and (Up > OOp) E S4), which implies
and thus
by Lemmas 1.1.1, 1.1.3. But h(aiki) k OAi, so h(aiki) k 3xiOBi, which implies h(aiki) b 03xiBi by 2.6.18 and soundness; thus
and so h(ai) != 0 0 0 3 x i B i 7 by 1.1.1and soundness. This shows that 0 0 3 x i B i E 0-h(ai), so eventually (*) implies the L-inconsistency of U 0 - h ( a ) contrary to the choice of n. lal=n-1 Lemma 6.6.6 Let L 2 QS4.2. Then for every L-place I? there is an wsubordination model M with UM ( A ) = r .
Proof Let us define h as follows: By Lemma 6.6.4, there is a subordination model Mo with Put h(0, a) := UM, ( a ) for all a E wm.
UM,
(A) =
r.
By Lemmas 6.6.5 and 6.1.9, for every n there is an L-place rn+1 such that U UPh(n, a). By Lemma 6.6.4, DF,+~ 2 U Dh(n,a) and rn+l aEwm
> aEwm
there is a subordination model Mn+1 with UM,+~(A) = r,+l. h(n 1, a ) := UM,+~ ( a ) for all a E wm.
+
Then put
520 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Erom Lemmas 6.6.4, 6.6.6 and 6.1.9 we obtain Theorem 6.6.7 The logic QS4.2 is determined by the w-subordination fmme.
c
Definition 6.6.8 An L-consistent theory I? MFs is called a nearly ( L , S ) Henkin theory if I' E L Qx A ( x ) , whenever r F L A(c) for all c E S . Lemma 6.6.9 (1) Let L be an I-m.p.l., S a small set of constants. If r is a nearly ( L ,S)-Henlcin theory and A is an S-sentence such that r U { A ) is L-consistent, then I' U { A ) can be extended to an ( L ,S)-Henkin theory.
+
(2) Let L > QK Ba. If I' is a nearly ( L ,S)-Henkin theory then 0-I' is a nearly ( L ,S)-Henkin theory. Proof
Standard.
Lemma 6.6.10 Let L > QS4 + ML* and let I' C MFs be L-consistent and L-0-closed (i.e. A E r only if r E L OA). Let also S 5 S f , IS'I = IS' - SI = No. Then there exists an L-consistent theory I" C MFs# such that I? C r' and the following conditions hold: (1) O O l B E
r'
or O B E r' for any B E M F s t ;
(2) if Q x B ( x ) E MFsl and O O i Q x B ( x ) E r', then OO7B(c) E I" for some C E S'. Proof Let { A i I i E w ) be an enumeration o f MFsl. W e define the following sequence o f theories:
I", := r;
(1)
r, u { O O ~ A , ,O O ~ B ( C ) ) ,
where c does not occur in r, U {A,) i f I', U {OO-IA,) is L-consistent and A, = V x B ( x ) ; (2) rn U {UO-A,) i f r, U {no-A,) is L-consistent and A, is not o f the form Qx B ( x ) ; (3) I?, U {OA,) otherwise. Each r, is clearly L-0-closed. W e only need to show by induction that r, is L-consistent. In fact, rois L-consistent by assumption. For the induction step, suppose is not obtained by ( 2 ) . that some r, is L-consistent, but rn+1is not. Then Suppose rn+1is obtained by ( 1 ) . Then there are sentences B1,. . . ,B , E r, such that L t B1 A .. . A B , A 0 0 1 ' ~ 'B~( x ) > 7 O O l B ( c ) . Since c does not occur in I?, or in B ( x ) , it follows that
6.6. DIRECTED FRAMES
Since L 2 QS4
+ ML*, we have
hence L I- OBI A
.. . A OB, > OOQx B(x).
Since r, is L-0-closed, we obtain
This contradicts the L-consistency of rnU { 0 0 7 A n ) . If rn+lis obtained by (3), then I?, U {OOlA,) is L-inconsistent, and so I?, k L --tOOIAn.If r,+I is L-inconsistent, then rnk L C I A n . By Necessitation, rnk L CIOTA,, and so I?, is L-inconsistent, contrary t o hypothesis. Thus each I?, is L-consistent. Therefore r' := U l?, is L-consistent and L-0-closed by a standard argunEw
ment. The conditions (1) and (2) hold by construction, so I?' is an (L, S)-Henkin theory. rn
Lemma 6.6.11 Let L > QS4. If r C M F s is an L-consistent theory satisfying conditions (1) and (2) of Lemma 6.6.10, then the following holds: (1)
0-r is a nearly (L, S)-Henkin theory;
(2) If I" 2 0-I? is L-consistent then 0-I?'
C 0-l?.
(3) If I' t LA implies A E r , then (1) and (2) above imply 6.6.10(1) and 6.6.1 O(2).
Proof (1) Suppose 0-I? FL Qx A(x). Then Qx A(x) @ Owl?, and so OQx A(x) @ I?. By 6.6.10(1), then 00-Qx A(x) E r , and by 6.6.10(2), OOlA(c) E r for some c. Since I' is L-consistent, I' F L OA(c), and so 0-I? bLL A(c). (2) Suppose A $2 0 - r . By 6.6.10(1) it follows that OOTA E I?, so OTA E 0-r and O l A E r ' . Since r' is L-consistent, OA @ I" and so A @ 0-l?'.
(3) We first check 6.6.10(1). If O O l A @ I?, then OTA @ 0-I?, and so 0-I' bLL 0-A. Hence 0-I'U{OA) is consistent. From (2) it follows that 0 - ( 0 - r ~ {CIA)) C 0-I? and so A E 0-r and OA E I?. Now suppose I? k~ 0 0 i Q x A(x), but for all c E S , I? bLL OOlA(c). Then OO7A(c) $2 I?, and by 6.6.10(1), OA(c) E for all c. Then A(c) E 0-I' and 0-r t-L Qx A(x). It follows that I? tLOVx A(x). This contradicts the L-consistency of I?.
Lemma 6.6.12 Let L 2 QS4.2 + ML*. Then for every L-place r there is a subordination model M with the greatest cluster such that vM(h)= r .
522 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Proof We construct the corresponding h as follows. Let Mo be a subordination model such that v ~( A, ) = r . Such a model exists a I" := (J 0 - h ( a ) by Lemma 6.6.4. Let h = VM,, on wm. By ~ e m k 6.6.5, aEwm
is L-consistent. One can easily t o see that I?' is L-0-closed. Let S" > S' = U Dh(a), IS'' - S'I = No. By Lemma 6.6.10 there is an L-consistent set -
aEwm
I"' > I" of St'-sentences with properties 6.6.10(1) and 6.6.10(2). Let A1, . . . be an enumeration of all St'-sentences B such that 0-I?" U {B) is L-consistent. For any n we define r,, as a Henkin (L, 57")-theory extending 0-I?'' U {A,). C ., In fact, for We have 0 - F a C r p whenever crRP for all cr,P E IT, a E ww and n E w we have 0-I?, = O-O-I?, 2 0-r" C r m n . For n,m E w we have, by Lemma 6.6.11(2), I?,, 0-I?" rm, heref fore, crRP implies
+
s
0-ra c rD.
If OA E I?,,, then O-J?" U {OA) is L-consistent. By Lemma 6.1.16, there is an L-place I?* such that I?,,Rhr* and A E I?*. Since we also have 0-r,, I?", 0-I?,, U {A) is L-consistent and A = A, for some m. By applying Lemmas 6.6.4, 6.6.12, 6.1.9 we obtain
Theorem 6.6.13 The logic QS4.2 frame with the greatest cluster.
+ ML* is determined by the subordination
>
Lemma 6.6.14 Let L QS4.2.1+ ML*. Then for every L-place I? there is a subordination model M with the greatest element such that with ~ M ( A=) I?. Proof Let h Iww, r" be the same as in the proof of 6.6.12. Let us show that 0-I? is"an (L, St')-Henkin theory. Suppose that both A @ 0-I"' and 1 A @ 0-I?". Then OA @ I?" and O l A @ I?". By Lemma 6.6.10(1), O O l A E I?" and 0 0 l ~ E A I"',in contradiction with the L-consistency of I?". By Lemma 6.6.11(1), 0-I'" is a nearly (L, St')-Henkin theory, and so it is an (L, S1')-Henkin theory as well. For any a E ww we have 0-I?, = O-O-ra 0-r" and Dra S". Let OA E 0-I"'. If A q! 0-r", then 1 A E 0-r" and so 0-A E I?" and O l A E 0-r" contradicting the L-consistency of 0-I?".
c
c
By applying Lemmas 6.6.4, 6.6.14, 6.1.9 we obtain
Theorem 6.6.15 The logic QS4.2.1+ML* is determined by the subordination frame with the greatest element. Now let us consider the intuitionistic case. Analogously to Lemma 6.6.4 we have
Lemma 6.6.16 Let L subordination map g : F
> QH and --+
let r be an L-place. Then there exists a V F L such that g ( h ) = r.
Proof Same as Lemma 6.6.4, using Lemma 6.4.16 instead of Lemma 6.4.12. We need to check the existence property instead of the Henkin property, but this is trivial.
6.6. DIRECTED FRAMES
523
We also obtain an analogue t o Lemma 6.6.6:
>
+
Lemma 6.6.17 Let L QH J and let h be a subordination map such that L 2 h ( h ) . Then U h ( a ) is L-consistent. aEITW
Proof The argument is basically the same as in the proof of 6.6.17. By monotonicity, a /I h ( a ) C h(/I). So the L-inconsistency of U h(a)
< +
implies the L-inconsistency of
U
aEITW
h ( a ) for some n. Consider the least such n.
lalsn
As in 6.6.5, we obtain Ai E h ( a i k i ) with different aiki of length n such that { A i j I 1 5 i , j 5 m ) is consistent. Next, we choose maximal generators Bi of Ai, SO for distinct ci and xi Ai = [ c i / x i ] B i . Hence F L V x i l Bi, which implies F L 1 A 3xiBi by 6.6.10 (xvi, xxvii), and i
next
i
since QH F p > l ~ p .On the other hand, h ( a i k i ) k 3 x i B i , thus h ( a i ) 1 3 x i B i , which implies
since h ( a i ) k -3xi Bi V-3xi of U h ( a ) .
Bi by A J . Now (*), (**) imply the L-inconsistency
lal=n-1
+
Corollary 6.6.18 Let L > QH J and let h be a subordination map such that L > h ( ~ ) Then . U h ( a ) is L-consistent for any standard subtree T of I T W . aET
Proof
Obvious.
+
Lemma 6.6.19 Let L > QH J and let r be an L-place. Then there exists an w-subordination model M such that U M ( A ) = r. Proof
Same as Lemma 6.6.6, replacing modal lemmas with their analogues.
w
From Lemmas 6.6.16, 6.6.19 and 6.2.6 we obtain
Theorem 6.6.20 The logic QH+J i s determined by the w-subordination frame.
>
+
+
Lemma 6.6.21 Let L QH K F J and let be an L-place. Then there exists an subordination map with the greatest element h : I T , + V F L such that h ( h )= r.
524 CHAPTER 6. KRTPKE COMPLETENESS FOR VARYING DOMAINS Proof Let g be the subordination map given by Lemma 6.6.16. By Lemmas 6.6.17, 6.2.6 and 6.3.4 there exists a QCL3-complete LV-place I?' U g(a).
>
aEITW
Let h ( a ) = g ( a ) for all a E I T W ,h ( m ) = I?'. Since I?' is <-maximal and 5 is selective on UPL, we observe that A E I?' and B 6I?' whenever ( A > B) 6I?' and VxA(x) E I?' whenever A ( c ) E I?' for all c E Dp. Therefore h is natural. Hence we obtain Theorem 6.6.22 The logic Q H + K F frame with the greatest element.
6.7
+J
is determined by the subordination
Logics of linear frames
In this section we prove strong Kripke-completeness for the logic Q L C w.r.t. denumerable linear Kripke frames, or moreover, w.r.t. Kripke frames over the set of nonnegative rationals Q+. The first proof of this result found by Corsi [Corsi, 19921 used the 'method of finite diagrams', which is a modification of the original S. Kripke's method of semantic tableaux for the linear case, and it was rather laborious and complicated. Another proof using linear natural models was given in [Skvortsov, 20051. Now we present a new, essentially simplified version of the latter proof basing on Lemma 6.4.28. Let Clin be the class of all linearly ordered sets, CKn the class of all denumerable sets from Clin. Recall that every set from Crn is embeddable in Q. Theorem 6.7.1
Moreover, strong completeness holds i n all these cases. Lemma 6.7.2 Theorem 6.7.3
If F is a countable linearly ordered set, then Q -+ F .
6.7. LOGICS O F LINEAR FRAMES
525
Proof From the previous lemma it follows that Q+ + F for any rooted F E CFn. Then apply Theorem 6.7.1 and Proposition 3.3.14. Soundness Q L C 5 IL(IC(Can)) follows from 3.2.34. Let us prove strong completeness. Let L Q L C be a superintuitionistic predicate logic.
>
Definition 6.7.4 A finite L-chain is an L-map over a finite linearly ordered set W = {0,1,. . .,k), 0 < 1 < . .. < k, i.e. a sequence of L-places M = ( r o , r l , . . , r k ) such that ri E for z < n. Recall that D& := UDr, = i
Dr, C o m m e n t We shall construct a natural L-model step-by-step; L-chains will be finite fragments of this model at different stages of the construction. As usual, if say, (A 3 B ) @ r j ,then we have t o insert a new point above j , i.ea new theory r' extending rj such that A E ~ ' ,B #I?'. But in general we cannot put this I?' above the whole n-chain, because it may happen that ( A > B ) E rj+1.Then we have to insert a new point between j and ( j + l ) ,and after extending the theory (rjU{A), {B)) to r' by Lemma 6.2.6, we should extend rj+lto a new theory in a conservative way, etc. and thus obtain a conservative extension (I?:+] I i > j) of (riI i > j), which together with the 'old' ( r i I i 5 j ) and the 'new' r' makes a (k+2)-element L-chain extending the original (k+l)-element L-chain. To make this, we apply Lemma 6.4.28 (its condition is satisfied due to the linearity axiom). L e m m a 6.7.5 Let M for a certain j < k
=
(ro, rl,. . . ,r k ) be a finite L-chain and assume that
Then there exists a finite L-chain M' = ( r o , . . . ,l?j,I';+l, r5+2,.. . ,r;c+l) such that Dr;+l n D L = D r j , MIJ({O,.. .,k 1) - { j 1)) is a conservative and respectively: extension of
+
h
(1') AI E
A2 @
+
i n case (I),
(2) A(c) @ ri+lfor some c E Dr;+l - Dr, i n case (2). C o m m e n t Between rj and rj+1we insert a new theory r'. satisfying the ?tl standard falsity condition for (A > B) or VyA(y). This positlon is exact, for it is impossible to put after and it is not necessary to put it before r j . Along with putting ri+lbefore rj+1, we extend the members of the former subchain r j + l , . . . ,r k to r ; + 2 , . . . , (by applying Lemma 6.4.28). Of course, we may not preserve the conservativlty, if we use (say, in - rj or in r j + l , etc.) the constants that are already present in r j + ] , rj+2, etc. Thus we should add only new constants.
526 CHAPTER 6. KRlPKE COMPLETENESS FOR VARYING DOMAINS Proof Consider the following L-consistent theory (I?,A) in the case (1) or (2) respectively:
(2) ( r , A ) := (rj,{A(c))), with c # D L and By Lemma 6.2.6, there exists an L-place l?;+l such that ( r , A) 5 and the = D r j (i.e. we add only new constants). Then DF;+~n 5 condition (1') or (2') (respectively) holds. If j = k, the construction terminates. So consider the case j < k. Put Bo := (A1 > A2) or Bo := VxA(x) in case (1) or (2) respectively; then Bo E ( r j + l lDrj) - r;+l. Let us show that l?;+llDrj C r j + l . In fact, let B E rl+,lDr,. By LC, ( B 3 Bo) V (Bo 3 B) E r j . Also ( B > Bo) 9 rj, since otherwise B E implies Bo E I';+,. Thus (Bo > B ) E rj C rj+1,and so B E rj+1,since BOE rj+l. Now Lemma 6.4.28 yields a conservative extension (I?;+,, . . . , of ( r j + l ,. . . ,r k ) such that G r;+2.This completes the construction; obviously M'I W is a conservative extension of M .
DL
Proposition 6.7.6 Let r be an L-place. Then there exists a natural L-model M over a subframe (not necessarily generated) F Q+ with the root 0 such that v M (0) = r . Proof Rather straightforward. Consider an enumeration BO,B1, B 2 , ... of the set I F ( S * ) of S*-sentences (where S* is the universal set of constants as usual), in which every sentence occurs infinitely many times. We construct an exhausting sequence of finite subsets of Q+: (0) = Wo C Wl C . . . and a sequence of finite L-chains hn : Mn V F L over Wn (for n E w ) such that hn(0) = r (for all n) and every restriction Mn+1I Wn is a conservative extensions of Mn. The inductive step is as follows. Assume that Mn is already constructed, with Wn = {uO,.. . ,uk), hn(ui) = r u i , and Bn has the form (A1 2 A2) or VxA(x), so that Bn E (-rui) for some i 5 k. Then we choose the greatk such that B, @ rujand construct an L-chain Mn+l over Wn+1 = est j {uO,. . . ,u j ,v, uj+l, . . . ,u,), where u j < v < uj+l, according to Lemma 6.7.5; so Mn+11 Wn is a conservative extension of Mn. Now put W := Wn; M := IJ Mn, with h(u) = hn(u) whenever u E Wn.
-
<
Un
n
Thus h(0) = F and every MIWn is a conservative extension of Mn. By the construction it is clear that the L-map h is natural. In fact, if (A1 > A2) E -h(u) for some u E W, then (A1 3 Az) E -hn(u) for some n such that u E Wn and Bn = (A1 > A2) (recall that (Al > A2) occurs infinitely many times in our enumeration). Let Wn = {uo,. . ., u k ) , u = ui. Then Wn+l = {UO, - .. , U j , v , ~ j + l . , .. ,uk) for some j 2 i, and A1 E hn+l(v), A2 @ hn+l(v). So A1 E h(v), A2 @ h(v) (by conservativity) and v > u j 2 ui = u in W. The case VxA(x) E -h(u) is quite similar. rn
6.7. LOGICS O F LINEAR FRAMES
527
Corollary 6.7.7 QLC is strongly Kripke complete.
Remark 6.7.8 The completeness proof given in [Skvortsov, 20051 follows the same scheme, but instead of Lemma 6.4.28, it uses the notion of a 'prechain' under a finite L-chain M = (r,,, . . . , r U , ) over W = {ul,. . .,uk). Such a where ( r , A) is an L-consistent prechain has the form ((I?,A), rul,. . . ,),?I theory. It should satisfy a special condition, when it can be extended it to a finite L-chain M' = (I?;,, I'll,. . .,r;,) such that (I?,A) 5 I?;, and M'I W is a conservative extension of M. But this notion makes sense only for L-chains M , where r,, < . . . < I?,, (i.e. all extensions rui< rui+, are proper). So for constructing a natural LC-model (cf. the proof of Proposition 6.7.6) one has to take care of properness of all inclusions between L-places. But Lemma 6.4.28 allows us t o avoid these complications in the above proof. Corollary 6.7.9 Let FQ := (Q+, DQ) be a Kripke frame over Q+, in which D,&:= w x {v E Q+ I v 5 u). Then IL(FQ)= QLC. Note that here ID, -
U D,I
V
= No
for every u E Q+.
Proof Let F' = (Q+, Dl) be a Kripke frame with denumerable domains. Then we can construct a pmorphism f = (fo, fl) : FQ + F'. Namely, we put fo := idQ , and for any u E Q+ fix a surjective map f: : w x {u) DL. Then
-
-
we define fu : D,& DL such that fulw x {v) = f: for any v 5 u. Obviously, (fo, f l ) , where f l = (f, 1 u E Q+), is a p-morphism.
From Chapter 3 it follows that QLC= is characterised by all KFEs over FQ = (Q+, DQ). On the other hand, Q L C = ~# IL=(F&). Moreover, there does not exist a single KFE F over Q+ such that IL=(F) = QLC=, or a Kripke frame F over Q+ such that IL'(F) = Q L C = ~ .In fact, one cannot refute both formulas lVxy(x = y) and l l V x y ( z = y), which are obviously not in Q L C = ~ , in the same KFE over Q+ (e.g. since Q+ IF J). Now let us consider the modal case. Let us prove the result on strong Kripke completeness of QS4.3 from [Corsi, 19891. Our proof follows the same lines as 6.7.1. Finite L-chains are defined as in 6.7.4, i.e. 0 < 1 < . - .< k corresponds t o r O R L r l R L . .R L r k (with perhaps rj+1RLrj for some j). Lemma 6.7.5 is replaced with the following L e m m a 6.7.10 Let M = (ro, r l , . . . ,r k ) be a finite L-chain such that for a certain j 5 k OA E rj - r j + l . Then there exists a finite L-chain
such that D r . + , n D& = Drj, MII({O,. . .,k extension of and A E r;+l.
hf
+ 1) - {j+ 1)) is a conservative (*)
528 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS YOA. Note that in this case -(rj+lRLI'j), since rj k OA, while rj+1
Proof There exists an L-place I?;+, such that 0-rj u {A) E I?;+, and Dr;+l n = D r j (we add only new constants); so rjRLI';+, and A E I?$+, . . ,rk)to as required. Now we can apply Lemma 6.4.28 for extending . .,r i + l ) in a conservative way so that r;+lRLI'$+2; it is sufficient to check that (**I n-(r;+l IDrj) r j + ~ ;
DL
c
recall that Dr;+l n D& = Drj. In fact, let O B E r$+,, B E L(Dr,). Note that D(OB 3 TA) V O(O-A since QS4.3 C r j . Also O(OB > 1A) $2 rj, since otherwise (DB 1 B ) E rUj -A) E r5+1(remember that r j R ~ r $ + , ) and , so O B E r$+, would imply TA E r;+,. Hence O(O7A 3 B) E rj,so (0-A > B ) E r j + l , since r j R L r j + l . Now B E rj+lfollows from OTA E r j + l . W
Proposition 6.7.11 QS4.3 i s strongly Kripke complete. Proof Repeat the proof of 6.7.7 with the inductive step for Bn = OA. Note that i L: j & OA E rj + OA E r i . So we choose the largest j such that OA E rj and then apply 6.7.10. Now we obtain full analogues of the intuitionistic completeness results:
Theorem 6.7.12
Again strong completeness holds in all these cases.
Corollary 6.7.13
6.8
Properties of A-operation
In this section we use completeness to prove syntactic properties of the Aoperation introduced in Section 2.13. First let us introduce some notation. For a poset F , let F- be its restriction to the set of all nonminimal elements (of course, it may happen that F- = F . ) Then by induction we define:
6.8. PROPERTIES OF A-OPERATION
529
If F is an intuitionistic Kripke frame or a KFE (or a Kripke sheaf) over F, let F- be the restriction F IF-; F-, := F 1 F-,. Proposition 6.8.1 For any intuitionistic formula A and k
20
P r o o f We write A as A(x) for r(x) = FV(A). (If.) Let us consider Kripke frames or KFEs; the case of sheaves is similar. Suppose F Iy ~ ~ , P A ( = x )VY(P(Y)V (P(Y) 3 A(x))) for disjoint x, y. Then there exist a Kripke model M over F, a world w, and tuples a , b in D, such that M, w Iy P ( a ) , P ( a ) > A(b). So there is v E R(w) (where R is the accessibility relation in F ) such that M , v It P ( a ) and M, v Iy A(b). Then w # v, so v is not minimal, i.e. v E F-. Let M- be the restriction M F-; since it is a generated submodel, we have M-,v Iy A(b) by 3.3.18. Hence F- Iy A. (Only if.) Suppose F- Iy A(x). Then by 3.3.18 there is a model M over F such that M, v Iy A(a) for some v E F- and a tuple a in D,. Then consider a model M' over F differing from M only on P , i.e. such that for any w, Q # P and a tuple of individuals b of the corresponding length
r
M', w Ik Q(b) iff M, w IE Q(b), and also such that M',w It- P(c) iff w # uo for any c E D;, where uo is the root of F . Now by induction it easily follows that M', w Ik B iff M, w It- B for any world w and D,-sentence B that does not contain P . Thus
Iy A(a); MI, v lk P(c), and on the other hand, M', uo Iy P(c). Hence M', v
and therefore M', uo Iy 76kA implying F
Iy
6kA.
rn
So the validity of 6A at some world means that 'A will be true tomorrow'. All dkA are KE-indistinguishable from 6.4; this also follows from their deductive equivalence: QH SkA = QH 6A,
+
+
cf. Proposition 2.13.22. Corollary 6.8.2 Let F be a rooted predicate Kripke frame (or a KFE, or a Kripke sheaf).
530 CHAPTER 6. KlUPKE COMPLETENESS FOR VARYING DOMAINS (1) F Il- SkA i f fVv E F- FTv Il- A. (2) AL
c IL(')(F)
(3) An L
Proof
iff L
IL(=)(F-).
C IL(=)(F) i f fL C IL(=)(F-,), for
n E W.
(1) By 3.3.21
F- Il- A iff Vv E F- F-Tv Il- A. It remains to note that F - f v = Ffv. (2) Readily follows from 6.8.1, the definition of AL and soundness. (3) Follows from (2) by induction on n, since F-(n+l)= (F-,)-. Lemma 6.8.3 Let F = ( F ,D) be an intuitionistic Kripke frame, i n which the intersection of all individual domains is non-empty, and let a0 E D,. Next,
n
uEF
let M be an intuitionistic Kripke model over F , L an s.p.1. such that M Il- 1. Consider an intuitionistic Kripke model M' = 6M obtained by adding a new root uo with the domain {ao) below M , such that M' F = M and M',v Ilp iff v = uo. Then M' IF Sub(SL). Proof If A = p V ( p > B ) E SZ (where B E Z, and p does not occur in B ) , then every strict substitution instance of A has the form C V ( C > B') for a sentence C and B' E SZLb(z) SOlet us show
z.
Mf,uo It C V (C > B'). In fact, otherwise M', uo Iy C, C > B', and then M', v Iy B' for some v E M . Since M is a generated submodel, it follows that M, v ly B' contradicting M It-
L.
Lemma 6.8.4 For any intuitionistic formula A and an s.p.1. L
Proof Although this lemma is syntactic, we know only a model-theoretic proof. (If.) Obvious, by the definition of AL. - (Only if.) Again we write A as A(x). Suppose L Y A(x), or equivalently, L y A x (x). Then by the canonical model theorem and the generation lemma 3.3.12, the there exists a Kripke model M over F with a root vo such that M, vo Il- and M, vo Iy A(a) for a tuple a in Duo. Consider the model M' = SM defined in Lemma 6.8.3. By that lemma we have
z
M', uo IF =(BE). Since M and M' coincide on F, an inductive argument as in the proof of 6.8.1 shows that M',wItB* M,wIt B
6.8. PROPERTIES OF A-OPERATION
531
for any w E M and a D,-sentence B that does not contain p. Hence M', uo Iy A ( a ) , and thus MI, vo Iy-bA(a). Now put I' := Sub(bL). By truth-preservation M ' , vo I t I' implies MI It- I'; hence by Lemma 3.2.31 applied t o I? and L = QH, we obtain
-
Sub(bz) YQH b A ( x ) ,
i.e.
AL
YQH S A ( x )
by the deduction theorem. Proposition 6.8.5 For any s.p.1. L1, L2 (1) L1 C L2 iff A L 1
AL2;
(2) L1 = L2 i f f A L 1 = AL2. Proof (i) 'Only if' is trivial. For 'if' note that A E L1 by 6.8.4. (ii) Follows from (i).
-
L2 implies b A E A L 1 - AL2
rn
So as in the propositional case, A is monotonic. Let us now prove an analogue to Lemma 1.16.10.
L e m m a 6.8.6 For s.p.1.s L1 and L2 and sentences A1, A2, if Al E ( L 1 - L a ) and A2 E (L2 - L l ) , then (6A1 V bA2) E (AL1 n ALz - A ( L 1 n L 2 ) ) . Proof
Since AL1 F bA1 and AL2
t 6A2, it follows that
Next, L2 y A1 implies L2 y p > A1 (cf. the proof of Lemma 2.9.4). Similarly L1 y p > A2. So the theories ( 1 2 U { p ) , { A l ) ) and (TIU { p ) , { A 2 ) ) are QHconsistent, and by strong completeness there exist Kripke models M I , M2 with roots u1, u2 such that M1,ul I t ( T 2 ~ { p{)A, l ) )and M2,u2 IF ( Z l u { p ) , { A 2 ) ) . Obviously we may assume that there is a common individual a0 in the domains of the worlds u l , 212. Then consider the model M' := b(M1UM2) described in Lemma 6.8.3. Since Mi IF it follows that M1 U M2 ItHence
z,
m.
MI, uo It- S u b ( b ( m ) ) by Lemma 6.8.3. We also have Mi, ui Iy p > Ai, hence MI, uo Iy p > Ai. Since also MI, uo ly p, we obtain MI, uo Iy bAl V 6A2. Therefore A(L1 n L2) Y 6A1 V bA2 by Lemma 3.2.31. rn Let us mention the following analogue of Proposition 1.6.11. Proposition 6.8.7 Let L1 and L2 be superintuitionistic predicate logics.
(2) A ( L 1 n L2) = AL1 n AL2 iff L1 and Lg are comparable by inclusion.
532 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
6.9
A-operation preserves completeness
Definition 6.9.1 Let F = (W,R) be a non-empty poset, m a x F the set of its ---t maximal points G := (G, I w E maxF) a family of rooted posets. Then F ?? := F U G, is a poset obtained by putting disjoint isomorphic
+ wEmaxF
+
copies of G, above the corresponding points w E rnax F . Speaking precisely, i f
Figure 6.1. F G,
=
(V,, R,), then
where
+ e.
F + Z = (W,X),
-
U
W := W - U
V,,
W - := W - m a x F
wEmax F
(cf. Section 1...),
Obviously
v; := vwx {w)
+ -+ for w E max F, are disjoint R-stable subsets in F+ G , and (F+ G ) 1 Vh r G,. If v, is the root of G,, then
W'
:= W-
u {(v,,
w)I w E max F)
6.9. A-OPERATION PRESERVES COMPLETENESS is an
z--stable subset of F + G , ( F + 2)1W '
and W'
-+
1
n Vd
Z
533
F . It is also clear that
= {(v,, w ) ) for w E rnax F.
Definition 6.9.2 Let F = (F, D ) be a predicate Kripke frame over a poset F and for w E max F let G , = (G,, DL) be Kripke frames over rooted posets G , (with roots v,) such that D, = DL(v,). Then we define a Kripke frame +:= (F+ G , D ) , in which D ( w ) = D ( w ) for w E W - andD(v,w ) = DL(v) for v E G,.
F+Z
Definition 6.9.3 Similarly for Kripke sheaves F = (F, p, D ) and G , = (G,, p L , DL), with roots v,, where w E rnax F , and D, = DL(v,), we define a -+ Kripke sheaf F G := (F z,p,B),in which
+
+
for u E F - , w E R ( u ) n max F, v E G,; in all other cases p is inherited from F or G , in the natural way.
+
Definition 6.9.4 We can also define a KFE F' = F 2 for KFEs F = ( F ,D , x ) and G , = (G,, DL,x,), w E m a x F such that D(w) = DL(v,) and = -, (x$),, for w E maxF. The details are left to the reader. Definition 6.9.5 Let Co and C be classes of rooted predicate Kripke frames. where Then Co C denotes the class of all Kripke frames of the form F -+ F E Co and G = ( G , I w E maxF) is a family of frames from C. Similarly we define a class Co C of KFEs or Kripke sheaves for classes Co and C of rooted KFEs or Kripke sheaves respectively.
+
+Z,
+
Definition 6.9.6 A tree of finite height is called uniform if all its maximal points are of the same height.
Recall that IT: (for k > 0, n 5 w ) is the tree of n-sequences of length < k. Obviously all these trees are uniform. -+ Lemma 6.9.7 Let F be a uniform tree of height k , = F G . Then F-k e + U G . Similarly, if F is a predicate frame over F and G , are predicate frames over G,, then (F 2)-t
+
+
= UG.
Obvious. The levels I < k in F and F are the same, and the level k in W F consists of the roots of posets G , for w E max F.
Proof -
3 ~ denotes h the domain function in G,, so we denote the domain at a world u in G , by Dh(u); D(u) (respectively, D(u))is the domain a t u in F (respectively, in F ) .
534 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
Proposition 6.9.8 (1) Let L be an s.p.l., C a class of rooted predicate Kripke frames, and let F o be a uniform tree of height k , Co = IC(Fo). Then
a kC IL(')(c~ ~ + C) iff L C IL(')(c). (2) Similarly for KFEs or Kripke sheaves, with Co = ICE(Fo).
Note that if k = 0; then F is a singleton and Co
+ C is equivalent to C.
-+ Proof (If.) Let G = (G, I w E maxFo) be a family of frames from C. Then by 3.5.24 I L ( 2~) = I L ( G ~ )2 IL(C).
n
wEmax F
By Lemma 6.9.7, for a frame Fo over Fo
so L
C IL(C) implies L
IL((Fo
+ -+ G)-k), which is equivalent to
+ by 6.8.2. Since G is arbitrary, this implies
(Only if.) Assuming 5 IL(Co + C), let us show that L E IL(F) for any F E C. Given F with a domain V at the root, consider the constant family
Then there exists a frame Fo over Fo such that copies of F can be stuck to its maximal points. For example, we can put Fo = Foa V, the frame with the constant domain V. Then (Fo 2 ) - k g F1
+
U
vEmax Fo
By our assumption
hence by 6.8.2, L
C IL((F0 + G)-k) -+
= IL(F).
6.9. A-OPERATION PRESERVES COMPLETENESS
535
Proposition 6.9.9 Let L be an intermediate predicate logic (with or without equality) strongly complete w.r.t. a class C of Kripke frames or K F E s . Then (or KE(IT;+') is strongly complete w.r.t. Co C, where Co = K(IT:+') respectively).
+
+
Hence AhL is Kripke-complete or Kripke sheaf complete w.r.t. Co C. Note that we cannot state Kripke completeness of ah^ for a (strongly) Kripke complete logic L with equality, because in this case Co is still KE(IT:+'). Proof Soundness follows from 6.9.8. Now for a Ak(L)V-place ro,let us construct a proper (Ah~,VS)-map(I?, u E IT:"), beginning with FA = r o .
(**) if hj(u) IY A > B, then 3v
E P(u) nFj+l(hj+l(u) It-
I
A & hj+'(u) I? B),
+
After we reach a point w of level k in IT:+ l , we obtain a (k 1)-element sequence rA< . . . < r,, so ah^ C r A implies L C I?, by Proposition 6.9.8 Now we can use the strong completeness of L and obtain a natural model with our I?, at the root of a frame from C. So we obtain a natural model over a frame from Co C. Clearly, this construction gives us a natural model over a frame from Gf.Naturally, for a logic with equality, we construct places r, for in the language with equality and hence we obtain a natural KFE. u E IT:+' There exists one special case. Namely, if for some point u E IT: there do not exist places > ,?I i.e. I?, is maximal in U FAk(L), then by Lemma 7.3.2(i) r, is QCLV-place. In this case the naturalness conditions hold with I?, = I?, and we can put I?, = r, for all v E (IT;+' u). And again for points w of level k we can apply the strong completeness of L, because I?, = r, is an LV-place since L QCL. W
+
<
c
Note that the condition L Namely:
C
QCL(')
is necessary in proposition 6.8.7.
Proposition 6.9.10 Let L be a superintuitionistic predicate logic, L QCL('), and h > 0. Then ah^ # IL(')(c~ C) for any class C C KE and any
+
c02 I C & ( I T , ~ + ~ ) .
+
Proof Consider a sentence A E L - QCL(') and suppose ah^ = IL(=)(c~ C). Then L IL(')(c), by Corollary 6.8.6, so A E IL(')(c) and hence obviously 1 1 A E IL(=)(CO C). On the other hand, TTA 6AhL since AhL AL QCL(=). w
c
+
Now let us reformulate the previous completeness result (Proposition 6.8.7) k the uniform finite branching n > 0 instead of T.: for the tree ~ , with
536 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS Proposition 6.9.11 Let L be an intermediate predicate logic strongly complete w.r.t. a class C of Kripke frames. Let k 5 0, n > 0, let L = A ~ L @ for a set @ of sentences V-perfect in ah^. Let Co = IC(IT,k+'). Let L' = AkL Bn[O]), then L' is strongly complete w.r.t. C Co.
+
+
Recall that
+
ah^ E L.
Proof Soundness follows from Lemma 7.8.5. Now, let I' be a L'V place. Again we construct a LfV<-mapover ITi+'(I?, IE IT:+') in such a way that FA = I?o and L F, for all v E max(IT,k+') and the naturalness condition (2') for all u E IT: with v E P(u) holds. Then for any v E max(IT,k+') we apply the strong completeness for L and find a natural L-model on a frame F, E C with I?, in its root; these models give us a natural where F E Co. model over F (F, I v E max IT:+'), Let us describe the inductive step. Let I?, for u E IT: be already constructed. If L = AkL @ I?,, then we put C , = r, for all v E (IT$+' T u). And if A k L + @ I?,, then we apply Lemma 7.8.8 to obtain L'V-places C , > I', for n points v E P(u). Note that a characteristic formula E for I' exists by 2 C and L r, so there exists 1 < k such Lemma 7.4.21. In fact, here I? and AzL I?. Finally, if n' = 0 (in applying of Lemma 7.8.8), that A'+'L T u); then I', is a QCL-place, and again we put ,'I = C , for all (v E (IT:+' here we use that L C QCL, cf. the concluding argument in the proof of 7.10.8.
+
+ s
A similar statement holds for KFE. Note that the completeness for L P ~ + +~ Br, = IL(ITtf ')) and for LP;+,++,+ Br, = IL(IT;+') are the particular cases of these statements for L = QLC.
6.10
Trees of bounded branching and depth
As we shall see in Volume 2, the s.p.1. IL(IC(Fin)) determined by all finite posets is not recursively axiomatisable. Still we can explicitly describe 'approximations' of this logic of bounded height and width or branching. Recall (from Chapter 1) that the propositional axiom F n :=
i j (Pi 3 v
i=O
~j
j#i
characterises posets of width 5 n. Similarly, the axiom
or its equivalent version: n
B1.k := A ( q 3 pi
i=O
.
3
q) A A ( q > p i V P ~>) q
i<j
6.10. T R E E S OF BOUNDED BRANCHING AND DEPTH
537
identifies posets of branching 5 n among posets of finite depth. We also know that H Br, is the propositional logic of finite n-ary trees and Q H Brl = Q H WI. In this section we study the propositional axiom Brn in predicate logics. Now it is not sufficient for axiomatising Kripke frames over finite n-ary trees, as we shall see in Volume 2. But still it works properly above Q H P ~ ,i.e. for posets of bounded height. So consider classes B: := Bn n P+ of posets of depth j h and branching jn (for n, k > 0) and also classes W ! := Wn fl Pz of posets of depth k and width n. Actually the predicate logics are determined by rooted posets, so we and w,kT. Up to isomorphism, can deal only with the corresponding classesT:B each of these classes contains finitely many finite posets. Therefore by 3.2.20, the logics IL(K(B:)) and ~ ( K ( w , k )are ) uniformly recursively axiomatisable. Note that W,kT C_ B T: 2 w ~ L - ~ Also . note that every finite poset belongs to w,k (hence to Bk) for some n, k; thus pinT = U w,kT = U B:t. It follows that n,k>O n,k>O the logic of finite predicate Kripke frames IL(IC(Fin)) is in 11;. Now we shall find finite axiomatisations for the logics IL(K(B:)) and thus obtain a simpler and more natural resentati at ion for IL(K(Fin)). First we need some lemmas.
+
+
+
<
<
L e m m a 6.10.1 Let r be an L-place, Br, E L, and assume that there exists a characteristic sentence E for I' such that the theories (I'U {E),Ai) are Lconsistent for i = O, 1,.. . ,n. Then (I'U {E),Ai U Aj) is L-consistent for some i f j.
Proof Suppose (I' U {E), Ai U Aj) is L-inconsistent for any i Then for any pair (i, j ) with i # j , we have
# j.
c
for some finite Atj G Ai, A;i Aj. Hence by the Deduction theorem,
where
A!
:= U ( A & u
ci := V A;.
j#i On the other hand, I-L E A (E > Ci) > Ci and (I?U {E),A,) is L-consistent, so (E 3 Ci) 6I?. Since ( ( E 3 Ci) V ( E > Ci. > E ) ) E I' (by the property of characteristic formula, see 6.3.10), it follows that ( E 3 Ci. 3 E ) E r. But
538 CHAPTER 6. KRTPKE COMPLETENESS FOR VARYING DOMAINS since Br; E L. Since I? is an L-place, we obtain E E I', which contradicts Definition 6.3.10. L e m m a 6.10.2 (Main l e m m a o n n-branching) Let I' be an L-place, Brn E L, and let E be a characteristic formula for I?. Assume that Dr = S c S', ISf- SI = IS1 = No. Then there exist L-places r l , . . . ,I?,, m 5 n such that (a)
r u {E) C ri for i = 1 , . . .,m
(b) if ( B
> C ) E (--I?) and B
(so ~ R L F ~ ) ;
@ r , then ( B > C) @ ri for some i.
Proof Let {(B 3 C) E (-I?) I B @ r) = {Ak I k E w ) . We define a family of theories A; L ( r ) for k E w , i > 0 such that
c
and for any k there exists nk
(2)
(3)
< n with the following properties
(I'u {E), A;) is L-consistent for 1 5 i 5 nk,
(rU {E), A:
U A;) is L-inconsistent for any different i, j
5 nk.
Hence by Lemma 6.10.1, nk 5 n. To begin the construction, put Ah := 0 for all i > 0 (i.e. no = 0). A;+, is constructed as follows. Consider Ak = B > C. There are two cases. (I) If ( r u {E),A; U { B > C)) is L-consistent for some i 5 nk, then take the minimal i with this property and put A:+, := A; u {B 3 C), A;+, := A; for all j # i (thus nk+l = nk). (11) If for any i 5 nk, (I? U {E), A; U {B > C)) is L-inconsistent, then put ank+l k+l .- { B > C) and A;+l := A; for all i i nk (thus nk+l = nk 1).
+
The properties (1),(3) hold trivially by definition, the same with (2) in case (I). To check (2) in case (11), we have to show that ( r U {E), {B > C)) is L-consistent, i.e. ( E 3. B 3 C ) # I'. In fact, B @ r and B V ( B > E ) E r , since E is a characteristic formula, thus ( B > E) E I?. So (E 3. B > C) E I? implies ( B >, B > C) E r, which contradicts ( B > C) @ I?. Therefore ( E 3. B > C) $ r . Next, put m := max{nk I k E w ) (thus m 5 n) and
for 1 i i i m. Since the sequence (A;)kE, is increasing, (2) implies the Lconsistency of every (I? U {E), Ai). So there exist LV-places ri + (I'U {E),Ai). Thus condition (a) holds. (b) holds by our construction, since B C = Ak for some k, whenever B @ r ; so C E Ai. W
6.1 1. LOGICS O F UNIFORM TREES
539
+
L e m m a 6.10.3 Assume that Q H P ~ Brn C L. Then for any L-place I' there exists a proper natural (LV, I)-model N = (I', I u E F ) based on a greedy standard subtree of IT: such that FA = r and a natural (LV, <) -model on IT: (not necessarily proper). P r o o f We construct I', using Lemma 6.10.2, which gives us the condition of naturalness from 6.4.9. If 0 < m < n, we repeat some theories; if m = 0 then the construction terminates at this point. Recall that characteristic formulas used in Lemma 6.10.2 always exist by 6.3.12. The construction terminates at some height 5 h , due to Lemma 6.3.7. Finally we repeat the theories from the rn maximal points of the resulting subtree at all leaves of IT:. = Q H P +Brn ~ for any n, k
Proposition 6.10.4 IL(K(B~))= IL(K(IT:)) 0, where IT: is the n-ary tree of height k . Proof
>
rn
Readily follows from 6.10.3.
In particular, for n = 1 we have:
+
Corollary 6.10.5 Q H P ~ L C is determined by an k-element chain. Corollary 6.10.6 IL(K(1Fin)) = ~ ( Q H P :
+ Br,)
=~(QHP:
+
P r o o f In fact, H W, t- B r n and H due to Kripke completeness). Thus
+ W,).
n,k
n,k
+ APk A B r ,
I- W, for some m (e.g.
+
On the other hand, the logics Q H P ~ Wn for k 2 3, n 2 2 are Kripke incomplete [Skvortsov, 20061; the proof will be given in Volume 2. Our conjecture (based on some results from Volume 2) is that all the corresponding logics IL(K(w,~))are not finitely axiomatisable.
6.11
Logics of uniform trees
As we shall see in Volume 2, generalised versions of formulas B r , allow us to axiomatise the intermediate predicate logic of all finite trees explicitly. Recall that the logic determined by an arbitrary finite poset is recursively axiomatisable (Chapter 3), but the proof of this fact provides only an implicit axiomatisation. In this section we make a step towards the logic of all finite trees clarifying main ideas of the whole construction; here we describe an explicit axiomatisation of an s.p.1. determined by an arbitrary 'uniform' tree of finite depth.
540 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS Definition 6.11.1 A tree F is called (levelwise) uniform if all its cones F T u with roots u of the same height are isomorphic.
A levelwise uniform tree F of depth h + 1 gives rise to an h-sequence t = (ti I i < h) of non-null cardinals ti; ti is the branching i,f3(u)l for all points u E F of height i < h (obviously, the branching at a maximal point is 0). The other way round, every h-sequence t of cardinals corresponds to a unique tree Tt of depth (h 1) and of branching ti at points of height i; this tree is clearly levelwise uniform. Obviously Tt is finite (respectively, countable) iff every ti is finite (respectively, countable). The corresponding h-sequences are called hf- (respectively, ht-)sequences . Every denumerable uniform tree can be represented as a subtree of IT,:
+
Tt
= { a E w*
1 la1 5 h, Vi ai < ti}.
Every h-sequence t of cardinals gives rise to the truncated hsw-sequence t t , in which every infinite ti is replaced with w . We shall construct explicit axiomatisations for all denumerable levelwise uniform trees and show that
for any uncountable Tt. Note that for h = 0 there is the empty sequence corresponding t o a one-element tree Tk of depth 1. A constant sequence t = (n, .. . ,n ) of length h 2 0 corresponds to a uniform tree Tt = IT:+' considered if n = w . in Section 6.10 or to Definition 6.11.2 Let A be a predicate formula and let q,po,pl,. . . ,p, be different proposition letters non-occurring in A, n > 0. We put
and define the formulas
We will also use the notation Brn[A], Br; [A] for B r t , BrLA.
pi > Br, and (A > ( i=o 1 q) > Brk respectively. Thus B r i and BrkL are equivalent to Br, and Brk. n
Obviously, B T and ~ BrkA are equivalent to
A
>V
Now let us show that both axioms are actually deductively equivalent (cf. Lemma 2 for Brn and Br;):
Lemma 6.11.3 QH
+ BT!
= QH
+ BrkA (for n > 0).
6.1 1. LOGICS OF UNIFORM TREES Proof
( 2 ) Let p
:= pl
541
...pn, C := C1. .. Cn, where C i := /\ pj
and let us
j#i
show that
V Cj,
~ ,:= Let B be the premise of B T ~CzT
j#i
the premise of [C/p] BT;. It suffices to check that
hence we can obtain n
n
FQHVC~V , Ci>q, i=O i=O
B , [C/p]Br:=
i=O and thus [ C l p l ~ r , AB
~ Q 9, H
which implies ( 1 ) . For the proof of (2) first note that
To show this, we argue in H. Assume q and
A ( q > pi
i<j
V p j ) . Then
l\ (pi V
i<j
V C i by distributivity - in fact, this should be a i=O disjunction of conjunctions, in which every conjunction contains either pi or pj from each distinct pair (i,j); so there is at most one pi missing from that conjunction. Thus
p j ) , which is equivalent to
We also have
since pi is present in every Cj for j
# i.
Hence
542 CHAPTER 6. KRIPKE COMPLETENESS FOR VARMNG DOMAINS
> pi) 3 q is a conjunct in B,
by (B), since (q
by (9), (7). Thus
Next, (12) by (3) and since A > q is also a conjunct in B. Next, we obviously have ci FH pj, for j
# i, hence
ci FH q > P j ,
and consequently
since B kQH (q
> pj) > q.
Thus
by (13). Eventually (2) follows from (ll),(12), and (14). (G)Similarly,
n
where
D :=
V pi.
i=O In fact, FH D = pi V p i , hence
FH ( D 3 p i )
-- (pi > p i ) ,
6.11. LOGICS O F UNlFORM TREES
and thus
pi>pi.>piI-~D>pi.>pi. Since I-H p i
> D , this implies
It also obvious that
for i < j . So by ( 1 6 ) , (17) the premise of B r f :
implies the premise of [ D ,p l , . . . ,p~ / q , p i , . . . ,pn]BrLA :
Since the conclusions of [ D ,p r claim ( 1 5 ) follows.
, . . . ,p; / q , p l , . . . ,P,] BrAA and B r f coincide, the
Lemma 6.11.4 The following formulas are QH-theorems: (1) A > ~ r f ,A > B T ; ~ ,
Proof ( I ) A trivial exercise. (2) Note that A1 > A 2 , A z (3) B y Lemma 1.1.2(5),
> D I- A1
3
D.
Since BrAA is equivalent to B > ( A 3 q. > q), where B does not depend on A and q, the claim follows. For B r f replace q with V p i . (4) Follows from (2), since
I> A.
i
W
544 CHAPTER 6. KRIPKE COMPLETENESSFOR VARYING DOMAINS For a set of formulas 0 we define the sets
Then
Q H + B= ~ Q~ H +
B~L@
QH+@
by 6.11.3 and 6.11.4 (1). We also put
B r , := Brk := T , BT; := Br'
:= 0 .
L e m m a 6.11.5 Let Q 1 , 0 2 be sets of sentences and let L be an s.p.l., n (1) If L +
(a) L
(b) L
(2) If L
+
0 1
C L +0 2 ,
> 0.
then
+ B r , [el]C L + B r , [e2] provided 0 2 is V-perfect i n L ; + B r n [@:I C L + B r , [@:I for any 0 1 , 0 2 . =L+e2,
then
(a) L + B r ,
[el] = L + B r , [02] provided el,0 2 are V-perfect i n L ; (b) L + B r , [0:]= L + B r , [Q:].
Proof Cf. Lemma 2.8.7 (on &operation). Recall that 0' is V-perfect in QH for any 0.
L e m m a 6.11.6 Let F be an intuitionistic Kripke frame of finite depth, 0 n < w . Then
<
~ r E; IL(F) ifS Vu E F(IP(u)l > n + F t u It. A). So BT; states that 'the branching is
< n until A becomes true'.
Proof Let 5 be the accessibility relation in F . (If.) Suppose BrkA ,A I L ( F ) , so due to the finiteness of height, there exists a Kripke model over F and u E F of maximal height refuting BrkA, i.e. UlF (9 3 pi) 3 q, u IF (A 3 q), u IF q > p i V p j for i
# j, u Iy q, V v > u v
IF q.
Then u Iy q > pi, so there exist a minimal vi > u such that vi It- q, vi Iy pi, and obviously vi E P ( u ) Since u It- q > pi v pj, it follows that vi I t pj for all j # i. So all vi differ, and thus IP(u)l > n. Since u lpL q, u It- A > q, it follows that u Iy A, therefore F t u Iy A. (Only if.) Suppose F u Iy A and there exist different vo, . . . ,v , E P(u). Consider an intuitionistic model over F such that u Iy A and
6.11. LOGICS O F UNIFORM TREES
545
(Such a model exists, since pi, q do not occur in A.) Then u since u Iy A. Also u IF q > p i Vpj for i # j , since
Finally note that v F IgL BrAA
Iy q > pi; thus u Iy q > pa and
Remark 6.11.7 Note that QH
Iy q and u I t A > q,
u I t (q
> pi) > q.
Hence
¤
+ BrkEM = QH + Br;,
and so
(recall that E M = p V i p ) . In fact, BrLEMt- ( E M > q) > Br;, whence we deduce (q V yq > q) > Br;. The latter is equivalent to t- ( ~ >q q) > Br;, which readily implies Br;, since H k ((q > pi) > q) > (-9 > q). From the semantical point of view, the statement we have proved means that if the branching is 5 n in all nonmaximal points then it is 5 n everywhere. However the equality of logics does not directly follow from this semantical argument, once we have not yet proved the completeness of QH BrFM.
+
Now let us prove a natural analogue of the main lemma on n-branching, 6.10.2, for Br?. We begin with an analogue of 6.10.1.
Lemma 6.11.8 Let L be an s.p.l., O a set of sentences that are V-perfect in L. Assume that Br: L, I' E V P L and L O I'. Also let E be a characteristic formula for I?, and assume that (I' U {E), Ai) are L-consistent theories in the language L(I') for i = 0,1,. . ., n. Then (I'U {E),Ai U Aj) is L-consistent for some i # j.
+
+
Proof Since L O I',there exists A E Sub(O) such that A E (-I?). In fact, if B E L O - I',then by V-perfection, there exist A1,. . . ,Ak E Sub(@) such that k L A1 A.. .AAk > B ; by Lemma 2.7.11 we may assume that Al, . . . , Ak E C(I'), since Dr is denumerable and the parameters of Ai can be replaced with constants from Dr. Hence Ai E (-J?) for some i. Now we proceed as in the proof of Lemma 6.10.1 and find Ci (for 0 I iI n) such that ((E> Ci) > E) E I' and ( E > Ci V Cj) E I' whenever i < j . Finally, since B~L@ ,O L,
+
So by L-consistency, this formula is in I?. Hence (A > E) $ I?, which contradicts the property of characteristic formula E. W
Lemma 6.11.9 Let L be an s.p.l., O be a set of sentences that are V-perfect in L, I? E V P L . Also assume that $ I? and Br: L for a finite n. Let E be a characteristic formula for I?. Then there exist rl,. . . ,I?, E V P L (where m 5 n) such that
546 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS
(2)
r u { E ) g rifor
i = 1,. . .,m (and thus, F < r i ) ;
(3) if (A1 > Aa) E (-I?) and A1 $2 F, then ( A 1 > A2) $2ri for some i .
Proof
Along the same lines as 6.10.2, using Lemma 6.11.8 instead o f 6.10.1.
Definition 6.11.10 Let t = (ti I i < h ) be an ht-sequence, h > 0. Put
+
C + ( t ) := { h 1 ) U { i < h I i # 0, ti-1 < t i ) , C ( t ) := ( 0 , h 1 ) U { i < h ( i # 0 , ti-1 # t i ) .
+
The levels from C f ( t )and C ( t ) are respectively called +-critical and critical fort. For i E C ( t ) , i < h , put i* := min{j i+ :=
{
>i Ij
E C(t)),
i f V j > i tt 5 ti, > i 1 t j > t i ) otherwise.
Obviously, i < i* 5 i+ and i+ E C + ( t ) ,although i+ may be non-equal t o min{j > i I j E C + ( t ) ) (the next element of C + ( t ) ) . I t is also clear that for any u E Tt o f level i < h, the tree Tt t u is of depth hi := h + 1 - i , and Tt t u r T i := Ttti, where t i := (ti,ti+1,.. . ,th-l). L e m m a 6.11.11 Let t = (ti I i < h ) be an ht-sequence, and let i E C ( t ) , 0 i < h. Assume that the logics
<
are strongly complete w.r.t. the classes of frames K(T;*) and K(T;"+)respec, assume tively. Let Oi+ be a set of sentences that are V-perfect i n Q H P $ + ~ and ~ = Li+ . that Q H P ~ + Qi+ Then the logic L z. :~ i-i * ( L i * ) Brti [Oi+]
+
+
is strongly complete w.r.t. K ( T i ) , and therefore Li
= IL(K(Ti)).
W e may also put Li. : = Q H + I i f f i * = k + 1 and similarly, Oi+ := { I )i f fi+ = k
+ 1.
So ~,k+' is an 'empty tree' and its logic is inconsistent. Also note that Q H P $ + ~ -5~ Li, since LP;+,-,, c Li*, SO Q H P ~ + = ~ -Ai*-' ~ (LPl+l-i*) G ~ i-i *( L i * )c Li.
6.1 1. LOGICS OF UNIFORM TREES
547
Proof Fix a point uo E Tt of height i, so T," = Tt T uo. First let us show that Tt is an Li-frame. In fact, A ~ * - ~ ( L C ~ *IL(Tt), ) since Lie is valid at all points of height > i* - i in Tt, i.e. at levels > i* in Tt. Also B r t E IL(Tl) for any A E Oi+ by Lemma 6.11.6, since A E IL(T~)=IL(T,~ Tv) for any v of level j > if and the branching of Tt is 5 ti at any point of level i j < i+ (to minimise calculations, we consider levels in the tree Tt not in T;) . Now we prove the strong completeness. Let R be the accessibility relation in Tt. Consider an LiV-place r . We put r,, := r for root uo of Tt and construct (inductively) LiV-places I', for points u E T; of levels from i till i+ or i*. During our construction we take care of the naturalness property (1)' from Lemma 6.4.23 and the property of Vs-models: r, 5 I?, if uRv in T;. Let us begin with the 'trivial' case: Li Oi+ C r,,. Then we put r, := I?,, for all v E R(uo) of levels < i+. And for any w of level i+ we obtain a natural 5 I?, by the strong completeness of Li+; in model over Tt f w = T:* with ,?I ', is an Li+-place, since Li+ = QHPZfl Oi+ Li Oi+ r,,. Note fact, I that the naturalness condition ( 1 ) ' holds for all points v of levels j < i+; in fact, if (A1 > A2) E (-r,), then (A1 > A2) @ I?, for any w E R(v) of level i f . Now consider the 'nontrivial' case Li Oi+ r,,. Then if ti < w, we apply Lemma 6.11.9 (for n = ti, O = Oi+) and find LiV-places r, for v E P(uo) such that I?,, < r, for all v and the naturalness condition (3)' for r,, holds, cf. the proof of 6.10.3 (recall that a characteristic formula E exists by Proposition 6.3.12, since LP:+l C Li). Then we repeat the same construction for points v E P(uo) and so on, till level i*; note that t j = ti for all levels j such that i 5 j < i*. If for a certain v, Li Oi+ r, and Lemma 6.11.9 is inapplicable, then we proceed as in the 'trivial' case, and repeat this LV-place r, at all points above v till level i+. If we successfully arrive at a point w of level i*, then we obtain a sequence of LiV-places I?, < . . . < I?,. By Lemma 6.3.1 we can conclude that L,. r, (i.e. r, is an Lie-place), since I?, is a ~ ~ * - ~ ( ~ i - ) - ~Now l a c we e . apply the strong completeness of Lie and obtain a natural model over Tt f w = T,"'. There also exists a degenerate case, when at point uo (or at some intermediate point) Lemma 6.11.9 should be applied to m = 0, so it does not produce any LV-place I', > r,,. But in this case the naturalness condition (1) trivially holds for I?,,, so we can repeat the place ,?I at every v E Tt uo and thus obtain a required natural model. and Finally let us mention the case ti = w. Now if (A1 > A2) E (-I?,,) A1 @ I?,,, we construct an LV-place I?, such that I?,, U {A1) C_ r,, A2 @ r, (thus (A1 > A2) @ rwand r,, < r,), cf. the proof of Lemma 6.2.13(2). So we obtain infinitely many LQ-places I?, for v E P(u0) - if there are only finitely many, then we can repeat one of them at all other points v E P(uo). We apply this construction to all points from ,B(uo) and further on, till level i*. Eventually we obtain LpV-places I?, for all points w of level i* and use the strong completeness of Li- as in the previous case.
<
+
+
+
+
+
It is clear that Lemma 6.11.11 yields a V-perfect axiomatisation of IL(K(T2))
548 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS for any ht-sequence t and i E C ( t ) ;in particular, for IL(K(Tt))= IL(K(Tt)).
Proposition 6.11.12 Let t = (ti I i < h ) be an ht-sequence. Consider the sequence Qi for i E C ( t ) constructed by induction on h 1 - i:
+
@i := S,P,-i(Oi*) U
(Brt,[@a])v for i < h.
Then all Oi are V-perfect in L P ~ + , ~ ,
and these logics are strongly Kripke-complete w.r.t. K(Ti). In particular,
is strongly Kripke-complete w.r.t. K(Tt). Proof Every Qi is Q-perfect in LP;+l and
c
c
by Proposition 2.13.33. Recall that L P ~ + IL(K(Tt)) ~ IL(K(Tt) and also that every set of the form OV is V-perfect in QH and V-perfection is U-additive. So we can apply Lemma 6.11 . l l at the induction step i.
ei
Note that APhfl-i = b~+,-,I E for any i, since I E Oh+l and S$-i 0 (6h+l-i* 1)= 6h+l-il. Recall that L P ~ ~ + ~ + A P ~ + ~ - ~by~roposition2.13.31,so~~~+l~i~ =LP~+~-~ LP;+, Oi for all i E C ( t ) . Also note that for h = 0 and the empty h-sequence t we have C ( t ) = ( 0 ,I ) , 0 1 = { I ) , Oo = ( 6 1 ) = { A P l ) , and this axiom generates classical logic QCL = IL(K(Tk)). For a constant sequence t = ( n ,. . . , n ) of length k > 0 (where 0 < n L: w) we have C ( t ) = (0,k 11, Ok+1 = { I ) . If n is finite, then Oo = { 6 i + l l )U { B r k ) = {APk+l,Br,), so we obtain the axiomatisation L P ~ + Br, ~ for the uniform tree T k f l described in Section 6.10. If n = w, then Oo = {APk+l), so we obtain a standard axiomatisation L P ~ + of IT:+ ~ l ; recall that B r z = 0 . In some cases the axiomatisation described in Proposition 6.11.12 can be simplified. Note that by Lemmas 2.9.2 and 6.11.5, 6; and Br,[(. . .)'I preserve deductive equivalence, so every Oi can be replaced with its simpler deductive equivalent. E.g. if A E Oi. and A E Oi for some i < h, then we can eliminate 6$-iA from b$-,(Of), as this formula follows from A. Proposition 6.11.12 can be strengthened as follows. 0
+
+
+
6.11. LOGICS OF UNIFORM TREES
549
P r o p o s i t i o n 6.11.13 Let t = (ti I i < h ) be an ht-sequence, Lo an s.p.1. such that LP;+~ C Lo C IL(IC(Tt)). For each i E C [ t ]let @i be a set of sentences, which i s V-perfect in Lo whenever i E C + ( t ) . Assume that LO+Oh+1 = Q H + I , and for any i < h in C ( t )
Then Lo
+ @i = I L ( K ( T l ) )for all i E C ( t ) ; in particular, +
P r o o f Again we proceed by induction on h 1 - i , and the argument at the induction step fully repeats the proof of Proposition 6.11.12, cf. the proof of Lemma 6.11.11, which is actually a particular case of this construction for Lo = LPh++l. H So Proposition 6.11.12 yields an infinite axiomatisation for any nonconstant
t , since any set of the form ( B r E ) v is infinite, while Proposition 6.11.13 allows us t o find a finite axiom system for the case of increasing t. Viz., consider an increasing ht-sequence t = (ti I i < h ) , i.e. to < tl <
.. .
< th-1,
ik+l
:= h
< il < . . . < i k < h and put except for 0. Let t' be the corresponding subsequence: t i := tij for 0 < j 5 k + 1, so with the critical levels 0 = io
+ 1, SO all these levels are +-critical
Obviously,
i+ = Z* = min { j '
> i I t j > t i ) = ij+l
< <
for i = i j ,0 j k. Then we define the finitely axiomatisable logics
L ( t ) := LPh++l
+ Brt; + { B r t ; [APh+l-ij+l]I j
< k).
Note that if tk = w , the axiom Brt; = T is redundant.
T h e o r e m 6.11.14 IL(IC(Tt)) = L ( t ) for any increasing ht-sequence t . P r o o f We apply Proposition 6.11.13 to Lo = L ( t ) and O i= {APh+l-i) for each i E C ( t ) = {io, . . . ,irc,ik+l) (where io = 0 , ik+1 = h 1). Obviously L ( t ) G I L ( K ( T t ) ) by Lemma 6.11.6, since the branching is t i at all points of levels < i j + l , and APh+l-iji1 E I L ( K ( T j ) ) for i i j + 1 . The sets Oi are Q-perfect in Lo by 2.8.13(2). For i = i j E C ( t ) , j < k we have
>
+ <
550 CHAPTER 6. KRIPKE COMPLETENESS FOR VARYING DOMAINS and Brti [Oi+]= Brt; (AP/t+l-ij+l) E LO. Thus
+
Finally by 6.11.13, IL(K(Tt)) = LO Oo = Lo +APh+1 = L(t), since APh+1 E L(t)On the other hand, for all non-increasing ht-sequences t the corresponding logics IL(K(Tt)) are not finitely axiomatisable [. . .I; a proof will be given in Volume 2. Let us also present a simplified infinite axiomatisation for IL(K(Tt)) for a decreasing t . Proposition 6.11.15 Let t = (ti tl 2 . . . > th-l; then
I i < h)
be a decreasing ht-sequence: to 2
P r o o f Readily follows from Proposition 6.11.13. Note that here C + ( t ) = { h + l ) , i+ = h + l for any i < h, Brti[Oh+l]= {Brt) and Br,'; is H-equivalent to Brt,. Finally we obtain axiomatisations for uncountable levelwise uniform trees. T h e o r e m 6.11.16 Lett be an h-sequence with some ti able). Then IL(K(Tt)) = IL(K(T,+)).
> w (i.e. Tt is uncount-
P r o o f Obviously Tt + TtSw, so by Proposition 3.3.14, IL(K(Tt)) 2 IL(K (Ttt)). On the other hand, IL(K(Tt)) IL(K(Ttsw)),since all the axioms used in the axiomatisation Oo for TtSw (Proposition 6.11.12), are also valid in Tt. More precisely, we can show (by induction on h 1- i) that Oi C IL(K(Tt+f u))for points u of level i, for any i E ~ ( t t ) The . argument is almost obvious, cf. soundness proof for Lemma 6.11.11. In fact, recall that BT? = 0 , so the levels i with ti = w do not bring anything new to the axiomatisation Oi, and thus to Oj for j 5 i.
>
+
Therefore we really have an explicit recursive axiomatisation for an arbitrary levelwise uniform tree of any cardinality. For an h-sequence of cardinals t = (ti I i < h) let us also consider a poset Tt 1 obtained by adding the top element. Then the following holds.
+
+
Proposition 6.11.17 L e t t = (ti I i < h) be an ht-sequence. Then IL(K(Tt 1))= Q H P ~ ~ J + ~60, where the sets bifor i E C[t] are constructed just as in Proposition 6.11.12, but with a difference at the beginning
+ +
6.11. LOGICS OF UNIFORM TREES
All of them are V-perfect i n QHP:+~. This logic is strongly complete (w.r.t. K(Tt
+ 1)).
We can also obtain a finite axiomatisation of IL(K(Tt
+ 1)) for an increasing
t (cf. Theorem 6.11.14) and a simplified infinite axiomatisation for a decreasing t (cf. Proposition 6.11.15). We also have
for an uncountable t (cf. Theorem 6.11.16). One can also consider logics with equality and obtain the following consequence from the results of Section 3.9:
Proposition 6.11.18 Let t be an h-sequence ( h 2 0 ) . Then
+
and similarly for Tt 1. The corresponding logics are strongly complete w.r.t. these classes of Kripke frames or sheaves.
4So t o say, the depths increase by 1.
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Chapter 7
Kripke completeness for constant domains In this chapter we prove completeness results in Kripke semantics with constant domains for modal logics containing Barcan axioms and superintuitionistic logics containing CD.
7.1
Modal canonical models with constant domains
Let us first describe canonical models with constant domains for modal logics. Since the Barcan formula is not valid in frames with varying domains, it is clear that every m.p.1. containing B a is not V-canonical. Thus canonical models should be restricted to L-places having a fixed denumerable domain So. Here is a precise definition. Definition 7.1.1 Let L be an N-modal logic containing Bai for 1 5 i 5 N . Let CPL be the set of all L-places (from VPL) with the set of constants So. The canonical frame and the canonical model with a constant domain for L are defined as CFL := VFLICPL, CML := VMLICPL. For the sake of simplicity, the notation of relations and the domain function in CFL is the same as in VFL. So in particular, DL is constant: DL(I') = So. It is clear that mentioning the superset S* is not necessary. Thus we can consider CPb as consisting of (L, So)-Henkin theories rather than L-places. Now we have to show that the restricted relations RLi are selective on CPL. Unfortunately, a direct analogue of Lemma 6.1.9 does not hold for the case of constant domains, because one cannot extend an L-consistent theory r C MF~;) to an (L, So)-Henkin theory I? without adding new constants.' But the following lemma suggests the way out. l ~ h corresponding e counterexample for the intuitionistic case was constructed by Ghilardi.
553
CHAPTER 7. KRIPKE COMPLETENESS
554
Lemma 7.1.2 Let L be an N-modal predicate logic, L F Bai for every i 5 N . Let r E CPL, A E MF~=)(SO)and OiA E for some i. Then there exists r' E CPL such that Oil? I?' and A E I?'.
c
This lemma means that the relations RLi on CPL are selective, cf. Definition 6.1.31 (recall that r R L i r l ++ Oil? C r' ++ Oil?' C_ r , cf. Definition 6.1.14).
Proof Let us enumerate the set of modal So-sentences of the form 3xA(x), viz. {3xAk(x) I k > 0). Then we construct a sequence of finite subsets (rklk E w ) of such that for any k E w
MFL~)
We put ro:= {A) and rk:= u {3xAk(x) > Ak(c)) for some c E So (say, for the first one in an enumeration of So), such that ( 0 ) holds. Let us show the existence of this c. In fact, by the inductive hypothesis, Oi(/\rk-1) E r . We also have
(say, for a new variable y). Thus
Now by the Barcan axiom,
and thus by property (vi) of Henkin theories
oi (A r k - 1 Let I?,
:=
U rk.
r
A ( 3 d k ( x ) 3 ~ ~ ( c ) E) ) for some c E So.
Then
0-r U r,
is L-consistent.
In fact, suppose
kEw
n i r FL ark) for Some k. Then r kL O i 1 ( A r k ) , i.e. 1 0 i ( l \ r k ) E r . This contradicts ( 0 ) . Finally by Lemma 6.1.5, Oil? U I?, can be extended to an (L, So)-complete theory r'. Clearly I" is an (L, So)-Henkin theory, and 0;r U {A) I?'. Remark 7.1.3 The above lemma implies that every L-consistent theory of the form 0-r U r o with finite To, can be extended to an (L, So)-Henkin theory. In fact, 7.1.2 is applicable, since in this case Oi(A r o ) E l? (otherwise 7 0 i ( A r o ) E r, which implies Oil(l\ r o ) E r , i.e. -(A I'o) E Oil?). This argument fails if rois infinite. In fact, if Oi(A rb) E r for any finite subset rb of roand 3yA(y) has Henkin property (w.r.t. r ) for any rb, then there exist constants c' E So satisfying Arb A A(cl) for all finite I'b G ro,but we cannot state the existence of a single constant c for all these rb, and thus for the whole infinite set r o .
7.2. INTUITIONISTIC CANONICAL MODELS
555
Now since the relations R L are ~ selective on CPL, we readily obtain the main property of the canonical model with a constant domain (cf. Lemma 6.1.21):
Theorem 7.1.4 For any I? E CPL and A
€
MF~=)(SO),
Corollary 7.1.5 For a logic L with Barcan axioms, for any formula A,
Proof (Cf. 6.1.22). We can consider only the case when A is a sentence. 'Only if' follows from 6.1.3 and Theorem 7.1.4. To check 'if', suppose A $! L; then by 6.1.10, we have A $ I? for some (L, So)-Henkin theory I?; hence CML,r lf A, by W Theorem 7.1.4. Definition 7.1.6 A m.p.l.(=) is called C-canonical if CFL k L. Corollary 7.1.7 Every C-canonical m.p.1. is strongly CK-complete. Every Ccanonical m.p.l.= is strongly CKE-complete. Proof
¤
Similar to Corollary 6.1.24. N
For an N-m.p.l.(=) A put AC := A
+ /\ Bai.
i=1 The next result is an analogue of 6.1.29 for constant domains; the proof is very similar.
Theorem 7.1.8 Let A be a propositional PTC-logic. Then the logics QAC, QAC= are C-canonical. Proof The argument for closed axioms is trivial, cf. 6.1.29. If A = OpOkp > 0,p E L = $A(=) + Ba, then R; o RL, C RLk. In fact, suppose r R ~ p A i ,I ' R L ~ A ~U,k B E r . Then OsOkB E I? (note that B E L(A1) = L(I'), since the domain is constant), and thus from OpOkB 3 O,B E L C r it follows that U,B E r . Therefore B E A2, since rRL,A2. So W we obtain AlRkA2, and eventually CFL b A. This theorem is actually a particular case of a more general completeness result by Tanaka-Ono, which will be proved in Section 7.4.
7.2
Intuitionistic canonical models with constant domains
Let us now consider intuitionistic models. Let L be a superintuitionistic predicate logic (with or without equality) containing the formula CD. Then L is definitely not V-canonical, and we have to extract a subframe of VFL with a fixed denumerable domain So.
CHAPTER 7. KRIPKE COMPLETENESS
556
Definition 7.2.1 Let CPL be the set of all (L3V,So)-complete intuitionistic theories; they are called CDL-places. Recall that (Section 7.2) for L containing C D , L3V-completeness is equivalent t o the property VxA(x) E
(Ac)
iff for all c E SoA(c) E
r.
This reflects the property of forcing in intuitionistic Kripke models with constant domains: u IF VxA(x) iff for all c E D(u) u IF A(c). As in the modal case, we omit So from our notation. Definition 7.2.2 The canonical frame and the canonical model with a constant domain for a superintuitionistic logic L containing C D are (respectively) CFL := V F L ~ C P Land CML := VMLICFL. Again we use the same notation of relations and domains as in VFL; so (DL(I?) = So) and r R L r r iff r C rr. Note that R L coincides with 5 on CPL, since every proper extension of contains additional SO-formulas (see Definition 6.2.11). Now let us show that the relation RL is selective on CPL. The property 6.2.12(V) follows readily from (Ac); so it remains t o check 6.2.12(>). Lemma 7.2.3 Let L be a superintuitionistic predicate logic containing C D , r E CPL, and let ( r U r o , Ao) be an L-consistent theory with finite ro,A,. Then there exists an L3V-complete theory rr t ( r U r o , Ao) (i.e. r U ro C rr,A. n I? = 0).
Proof Similar t o Lemma 6.2.6. We enumerate IF(")(so) as {Bk ( k E w) and construct an increasing sequence of finite theories ( r o ,Ao) i . . . 5 (rk, Ak) 3 .. . such that (I' U rk,Ah) is consistent for any 5 E w. The construction is as follows. ( I ) If Bk is not of the form 3xA(x) or VxA(x), we define (rk+17Ak+1):=
{
( r k U {Bk), A k ) if (I? u rkU {Bk), Ak) is L-consistent, ( r k ,A k u {Bk}) otherwise.
Then the consistency of ( F u r k ,Ak) implies the consistency of ( r U r k + l , &+I); this is proved as in Lemma 6.2.6. (2) If Bk = 3xA(x) and (rU rkU {Bk), Ak) is L-consistent, then
and thus V X ( / \r
k A
A(x) 3
V Ak) @ I?
7.2. INTUITIONISTIC CANONICAL MODELS Hence by (Ac), ( A rkA A(c)
>
V Ak) g' r for some c E So,
and so r ~ A r k A ~ (3 c V)
~
k
,
by Lemma 6.2.5(v). Thus
i.e. the theory
(I' U rk+l, Ak+l), where
is L-consistent. Otherwise, if
(rU r k U {Bk), Ak) is L-inconsistent, we put
then as in (I), we obtain that ( r U r k + l , Ak+1) is consistent. (3) If Bk = VxA(x) and
then by axiom CD,
Hence ( A r k > V ~ k ~ ~ ( ~ ) ) @ r for some c E So. Take this c and define
then (I? U rkfl, Ak+l) is L-consistent. Otherwise, if r k~ r k >
V Ak V VXA(X),
we define (rk+l,Ak+l) := (rk U VxA(x), Ak); then (I? U r k + l ,Ak+l) is again L-consistent. Eventually, the theory l?' := r U U rk (or and (I? U rot Ao) 5 r' as required.
kEw
U rkitself) k€w
Note that only property (Ac) of the original theory proof, and the existential property is not used.
is L3V-complete,
¤ is essential in the above
CHAPTER 7. KRIPKE COMPLETENESS
558
L e m m a 7.2.4 Suppose L t- CD, E CPL, (A1 > A2) @ r . Then there exists I?' E CPL such that I'g I?' (i.e. I'RLrl), A1 E F', A2 @ I?'. In other words, the relation RL on CPL satisfies 6.2.12 (3)and is selective. Proof If (A1 > A2) @ I?, then the theory (I'U {Al), {A2)) is L-consistent, ¤ and so we can apply Lemma 7.2.3.
Remark 7.2.5 Note that in Lemma 7.2.3 the sets Fo, A. are finite, and thus it is actually equivalent to Corollary 7.2.4. If roU A. is infinite, one cannot always find an L3V-complete extension for an L-consistent theory (r U r o , Ao) (as we have mentioned, this was shown by Ghilardi). From Corollary 7.2.4 and the conditions 6.2.12(>), (V) we obtain the main property of the canonical model: T h e o r e m 7.2.6 For any I' E CPL and A E IF(')(&)
Corollary 7.2.7 For a logic L containing CD, for any formula A
Proof By Lemma 6.2.6, for an L-unprovable So-sentence A there exists an L3V-complete theory F E CPL such that A @ F. Thus CML,r Iy A, by Theorem 7.2.6. rn Now we can repeat Definition 7.1.6 for the intuitionistic case. Definition 7.2.8 A n s.p.1 (=) L is called C-canonical if CFL k L. From Corollary 7.2.7 we obtain Proposition 7.2.9 Every C-canonical s.p.1. containing C D is Kripke-complete.
7.3
Some examples of C-canonical logics
Unlike the case of varying domains, there are quite a few natural examples of C-canonicity; some of them were first found by H. Ono. Let Q H C := Q H CD, Q H C K := Q H C K F . Sometimes we also use the notation QAC, QACK for an arbitrary intermediate propositional logic A.
+
+
T h e o r e m 7.3.1 The logics Q H C = , Q H C , QHC'~, QHC=' are C-canonical. Proof
Similar to Proposition 7.2.15.
Proposition 7.3.2 Classical logic Q C L is C-canonical.
¤
7.3. SOME EXAMPLES O F C-CANONICAL LOGICS
Proof The frame C F Q c ~is discrete, i.e. r R Q c L r l iff Lemma 6.3.4(i) (recall that RL = in CFL).
559
r = rl,cf. the proof of W
C-canonicity also holds for many logics of finite depth. Lemma 7.3.3 Let k
> 0 , and let L be a superintuitionistic logic.
(1) If C D A APk E L, then CFL is of depth
(2) If L = QHC
5 Ic.
+ APk, then CFL is of depth k.
Proof (1) Suppose rkc rk-1c . . . c rl c roin CFL, and let Ai E (ri-1 - r i ) for 1 i k. Then by induction on i it follows that
< <
< <
<
(for 1 i k). In fact, Bi+1 = Ai+1 V (Ai+1 > Bi) for 0 I i k-1 (where Bo = I). Then Bo @ ro,by consistency. Suppose Bi @ r i , but Bi+l E ri+i. Then Ai+l E ri+l or (Ai+l > Bi) E ri+1. The first option contradicts the choice of Ai+1. In the second case we have (Ai+l 3 Bi) E ri+l ri. Since Ai+i E r i , we obtain Bi E r i , which contradicts the inductive hypothesis. Therefore Bk @ r k , while Bk is a substitution instance of APk. This contradicts Corollary 7.2.7. (2) Recall that L t+ APk-1 = qk-I V (qt-1 3 - .. 3 (92 V (q2 3 q1 V 141))) (since APk-1 is refuted in a Ic-element chain with a constant domain). By Corollary 7.2.7 there exists I?k-1 E CPL such that rk-1Iy A P k - - ~Then . r k - 1 Iy qk-I, r k - I Iy qk-1 > APk-I, and so there exists rk-z rk-1 such that r k - 2 Ik qk-1, r k - 2 Iy APk-2. By induction we obtain a chain r k - I C r k - 2 C . - .C rl C l?o such that ri Iy APi, ri Iy qi, ri-1 IF qi for i > 0.
>
Remark 7.3.4 Note that the argument used in the above proof of (1) fails for the case of varying domains, because the formula Ai can contain additional constants not occurring in ri,thus we may not find an appropriate substitution instance of APk refuted in r k . This observation somewhat explains why propositional axioms cannot describe the finite depth in this case and we need predicate axioms taking new individuals into account. Lemma 7.3.5 Let L be a superintuitionistic logic containing 2 . Then every cone in CFL is linearly ordered, i.e. for any r , I",rl' E CPL, I? I?' and r C rl' imply that r' and I?" are C-comparable.
Proof Suppose A1 E (I"-I?") and A2 E (I?" -I"). Then I? Iy A1 3 A2, A2 3 A1, and thus I? Iy (A1 > A z ) V (A2 > A1). This contradicts CML IF L.
CHAPTER 7. KRIPKE COMPLETENESS
560
Proposition 7.3.6 The following superintuitionistic predicate logics are C canonical:
(1) QLC
(3) Q H
+ C D = Q H + C D A AZ;
+ C D A AZ A APk for k > 0.
They are respectively determined by the following classes of Kripke frames with constant domains: (1) linear Kripke frames;
(2) Kripke frames of depth k ;
(3) k-element chains. These logics are also determined by countable frames of the corresponding kind. (1) was proved in Ono [Ono, 19831, [Minari, 19831.
+
Remark 7.3.7 The logic Q L C CD was also characterised algebraically in [Horn, 19691. Note that it is also equivalent to the logic IF introduced in [Takeuti and Titani, 19841. Remark 7.3.8 Note that canonical and quasi-canonical frames with varying domains (for any logic) obviously do not validate A Z . In fact, two extensions of an L-place with incomparable sets of new constants are always RLincomparable. Thus the logic Q L C and all its extensions are not V-, U-, or Us-canonical. Still QLC is Kripke complete, as we showed in Section 6.7. Now consider the logics Q H C K := Q H C
+ KF, QS~CK' := Q S 4 C + MK',
where
MK+ := OVXU(P(X)3 UP(X)). Note that Q S 4 C K + contains S4.1.
Lemma 7.3.9 (1) Let L be a superintuitionistic logic containing Kuroda formula K F . Then its C-canonical frame C F L is coatomic. Moreover, for any CDL-place I' E CPL there exists a CDQCL-place I" >
r.
(2) Let L be an m.p.1. containing QS~CK'.
Then C F L is coatomic.
Note that QCL-complete theories are maximal in C F L (cf. Lemma 6.3.4(i)).
7.3. SOME EXAMPLES O F C-CANONICAL LOGICS
561
Proof (1) For r E CPL, we construct a sequence r = roC rl C . . . in CPL as follows Let Ao, A1 . . . be an enumeration of I F . For a CDL-place 0 a formula A is said t o be critical if CML, 8 Iy B(A V -A). Since L t KF, we have kL llV(Av-.A), which follows from kQcL V ( A V ~ A ) by the Glivenko theorem. Hence 0 Iy -a(A V -A) for any 0 and A. Now assume that I?, is constructed and A, is critical for r,. Since r, Iy lV(A, V TA,), there exists > I?, such that rn+1It V(A, v TA,). If A, is not critical for I?,, we put r,+l := r,. This procedure makes all formulas noncritical - A, is always noncritical for l'n+l. Eventually we obtain a theory r' := U r n . It is clear that (r', -I?') is Ln
consistent, otherwise r, kL V A for some n and finite A 5 -I" 5 -r, - but then (r,, -I?,) is L-inconsistent. Since every -I?, has the (EP), so does r ' . So r' is an L-place, and moreover, Q C L C r'. In fact, by the deduction theorem, B E QCL implies Sub(p V i p ) kqH B; hence there exist formulas Ao, . . . ,A, such that V(Aov-Ao), . . . ,V(A,V-A,) J-QH B; therefore r' kQH B, whence B E I" By 6.3.3, r' is a QCL-place, and therefore an LV-place (by 6.3.2), and it is maximal in CFL. (2) Similar to (1). Again starting from r E CPL, we construct a sequence = r o R L r l . . . in CFL. Let M F l be the set of all So-formulas with a t most one parameter, and let Ao, Al . . . be an enumeration of MFl. Now we call a formula A critical for 0 E CPL if C M L , O IyvO(A> CIA). Suppose I?, is constructed and A, is critical for I?,. Since L Q S ~ C K + , we have r, k O ~ O ( A , > CIA,), so there exists E RL(I',) such that rn+lb VO(A, > oA,). If A, is not critical for r,, we put r,+l := r,. Then we claim that the theory
>
k
is L-consistent. In fact, otherwise
U 0-r,
n=l
is L-inconsistent for some finite k. k
But this is impossible, since r n R L r k + i for n 5 k
+ 1, and thus n=l U 0-r, ~
C
-
rk+l. Now by the Lindenbaum lemma, there exists an L-complete A I". It follows that A is a Henkin theory. In fact, consider an arbitrary formula A,(y) of the form 3xB(x) > B(y), where FV(B(x)) {x) as usual. By our construction and reflexivity, is a Henkin theory, != V y ( A n ( ~> ) OA,(y)). Since there exists c E So such that k A,(c). We also have r,+l != A,(c) > UA,(c), so I',+l != OA,(c). Hence A,(c) € O-Fn+l G A. Therefore A E CPL. It remains to note that A is maximal - assuming ARLA', let us show that A'). Suppose A = A, E A (i.e. A != A,) for an So-sentence A' = A (or A
>
c
CHAPTER 7. KRIPKE COMPLETENESS
562
A,. By construction r,RLA, so A k V(A, > OA,), which is the same as W A k A, > CIA,, and thus A k OA,, which implies A, E A'. R e m a r k 7.3.10 Note that the proof in case (1) almost repeats the proof of Lemma 7.2.3, when we construct a QCLIV-complete extension I?' of r . But in our case we cannot directly apply Lemma 7.2.3 to the logic QCL(') and empty r o , A o , because the basic theory r is not in general ~ ~ ~ ( ' ) 3 - c o m ~ l eNor te. we can apply Lemma 7.2.3 to an L3-complete I? and A. = 0 , To = QCL(') (or T'o = {v(A V 1 A ) I A E IF('))), since I? is infinite. Proposition 7.3.11 The logics Q H C K and Q L C C K are C-canonical. The former is determined by Kripke frames with constant domains and the McKinsey property, and the latter by chains with the greatest element and constant domains and also by countable chains with the greatest element and constant domains.
+
Recall that Q H C D A APk t- K F for any k, since a poset of a finite depth is clearly coatomic. R e m a r k 7.3.12 Note that the logic of the weak excluded middle with constant domains QH+CDAAJ is not C-canonical. Recall that a Kripke frame validates A J iff it is directed, i.e. V V ~ ~ ' V ~ ~ W&( V v ~~RRwW ) This . property fails for the canonical frame, and our 'extension lemma' 8.2.3 does not allow us to extend the L-consistent theory I?' U rr' (where I? c I?' and I? c I?") to an L3V-complete theory with the same set of constants So. The logic Q H C D A A J is actually Kripke-incomplete [Shehtman and Skvortsov, 1990; Ghilardi, 19891, see also Volume 2. But there is no difficulties with the logic Q H C D A A J A K F , as we shall see now.
+ +
L e m m a 7.3.13 Let L be a superintuitionistic logic containing C D A A J A K F . Then for any I? E CPL the set {A E CPL I I? 2 A) has the greatest element.
>
Proof By 7.3.9, there exists a CDQCL-place A r . Suppose A is not the greatest in C F L ~ I ?then , there exists another I?, and thus A A l , since A is maximal. So there CDQCL-place A1 exists A E I F ( & ) such that A IF A, Al Iy A; hence Al Il- 1 A by maximality of Al. Thus r Iy 1 A V --A, which contradicts kL TA V 1-A.
>
+
+
Proposition 7.3.14 The logics Q H C A J A K F and Q H C C D A A J A A P ~ for k > 0 are C-canonical. The former is determined by Kripke frames with constant domains and top elements, and the latter, by the same kind of frames of depth k. Definition 7.3.15 A subordination L-map with constant domains is a natural L-map from a subordination frame to CFL.
>
+
Q H C D and let I? be an LV-place. Then there L e m m a 7.3.16 Let L exists a subordination L-map with constant domains g : IT, ---+ CFL such that g ( ~ =) r .
7.4. PREDICATE VERSIONS Proof
563
From Lemma 7.2.4 and an analogue to Lemma 6.4.16.
>
+
+
+
-
L e m m a 7.3.17 Let L QH CD K F J and let I? be an LV-place. Then there exists a subordination L-map with constant domains h : IT, 1 CFL such that h ( h )= r. Proof
+
Let g be the subordination map given by Lemma 7.3.16. Let I" =
{ A I T-A E
r).
Suppose A E g(a) for some a E ITW.Then kL 1 A V 7 1 A by A J , r bLL l A , therefore r 1-A and A E I?'. Moreover, I" is a QCLV-place. In fact, it is clearly L-consistent and closed under deduction in L. Since for every Dpsentence A, E L 1 1 A V T T T A , I? contains either T T A or A -, so J? contains either A or 1 A . To check the existence property, suppose A(c) @ I" for all c. Then by AJ we deduce that 1 A ( c ) E F for all c and so V x l A ( x ) E r r'. Since FQH V x l A ( x )3 13xA(x),we have 13xA(x) E r' and so 3xA(x) # .'?I To show the coexistence property, suppose VxA(x)@ I?'. Then l V x A ( x ) E I?'. But Q H + K F t- l l ( l V x A ( x )> 3 x ~ A ( x )hence ), ( l V x A ( x )2 3x-.A(x)) E I?' and 3 x l A ( x ) E r'. By existence property, -A(c) E I" for some c and so
c
A(C)
g rl.
Since I?' is maximal, we see that A E I" and B @ I?' whenever ( A > B ) # I". Therefore h extending g by h ( m ) := I" is natural. From Lemmas 7.2.3 and 7.3.17, we obtain
+
+
+
T h e o r e m 7.3.18 The logic QH C D K F AJ is determined by the subordination frame with the greatest element and constant domains.
7.4 Predicate versions of subframe and tabular logics The results on canonicity of some families of predicate logics presented in this and the next section are due to H. Ono, T. Shimura and Y. Tanaka. But the original proofs are essentially simplified, thanks to canonical models and some results on propositional logics. Definition 7.4.1 Let L be a modal or superintuitionistic predicate logic with constant domains. The canonical general frame of L is
where W L is { e L ( A ) I A E M F . A J ( S ~in) ) the modal case or { o ~ ( AI )A E I F ( & ) ) in the intuitionistic case. L e m m a 7.4.2 C@Lis well-defined.
CHAPTER 7. KRIF'KE COMPLETENESS
564
Proof Similar to 1.6.7. Note that WL is a subalgebra of MA((CFL),), since by definition
OL (7A) = -eL(A), 6~( A A B ) = 0~(A) n ~ L ( B )0~ , (OiA) = O ~ ~ L ( A ) .
Lemma 7.4.3 CQL i s refined. Proof It is again similar to the propositional logic. CQL is differentiated, since r # A implies r 9 A (or vice versa), and thus by Theorem 7.1.4 (or 7.2.6 in the intuitionistic case), I? E eL(A), A @ BL(A) for some A. Tightness in the modal case
follows from the definition of RL,( and 7.1.4. In the intuitionistic case this is written as VA(I'kA+A!=A)+rLA and follows from 7.2.6.
W
Remark 7.4.4 CQL is not descriptive, as S. Ghilardi noticed. Proposition 7.4.5 For any m.p.1. or s.p.1. L, CQL validates L,. Proof Again we consider only the modal case. We can argue as in the proof of 1.7.5. Viz., consider a formula A E L, [ k and a valuation $ in CQL. Choose Bi , let S := [BI,.. . ,Bk/pl,. . . ,pk]. Then S A E L, such that $(pi) = 0 ~ ( B i )and so$(A) =QL(SA) = C P L by 1.2.9andthecanonicalmodeltheorem7.1.4. W Proposition 7.4.6 If a propositional modal or intermediate logic A i s r-persistent, then the predicate logic Q A C i s C-canonical. Proof Let L = QAC. By 7.4.5, CQL != A. By 7.4.3, this frame is refined, so by r-persistence, we obtain CFL k A. Since CFL has constant domains, it also W validates the Barcan formula, and thus the whole L.
Hence we obtain the main result from [Tanaka and Ono, 19991: Theorem 7.4.7 (TanakaOno) For any universal propositional modal logic A, the logic Q A C is C-canonical. [and so, strongly Kriplce complete] Proof By Theorem 1.12.8, A is r-persistent, so Q A C is C-canonical by Proposition 7.4.6.
This allows us to construct many examples of canonical predicate logics. In almost the same way we prove the result from [Shimura, 19931:
7.5. PREDICATE VERSIONS O F COFINAL SUBFRAME LOGICS
565
Theorem 7.4.8 (Shimura) If A is a subframe intermediate logic, then Q A C is C-canonical. Proof 7.4.6.
By Theorem 1.12.27, A is r-persistent. So we can apply Proposition
The next result for the intuitionistic case was proved in [Ono, 19831; the modal version is probably new.
Theorem 7.4.9 If a propositional modal or intermediate logic A is tabular, then Q A C is C-canonical. Proof
By Proposition 1.14.7, A is r-persistent. So we can apply 7.4.6.
7.5
Predicate versions of cofinal subframe logics
W
In this section we prove completeness results from [Shimura, 20011. Now we use an auxiliary 'pre-canonical' model dual to the Lindenbaum algebra of So-sentences.
Definition 7.5.1 Let L be an N-m.p.1. with constant domains, C f PL the set of all L-complete theories with the set of constants So. W e define the following propositional frames and models: the pre-canonical frame of L is C f FL is defined as i n the canonical model: rRiA
:= (C'PL,
VA E MFN(So)(OiA E
R1,. .. ,RN), where Ri
r + A E A);
the pre-canonical model of L is C + M L := ( C f F ~B:), , where O Z ( ~:= ~ {l? ) E C+PL I pi E l?);
the pre-canonical general frame of L is
I
C+@L := (C'FL, {IAl A E MFN(SO))),
where IAI := {I' E C+PL I A E l?). Lemma 7.5.2 C+@L is well-defined. Proof We have to show that the sets IAl constitute a subalgebraof MA(Cf FL). This follows from the equalities:
The first two are just reformulations of 6.1.2(iii) and 6.1.3(i). For the third, note that the inclusion 10iAl OilAl readily follows from the definition of Ri.
CHAPTER 7. KRIPKE COMPLETENESS
566
>
The converse lOiAl OilAl is proved a s in propositional logic and similarly to 6.1.16. In fact, suppose OiA r. Then the theory
e
is L-consistent. For otherwise we have O i r EL A, hence I? FL OiA by Lemma 2.., which contradicts OiA I?. Now by the Lindenbaum lemma 6.1.5 there exists an L-complete A 2 Ao. So A E IlAl = -IAl, and also rRiA. Thus I? @ OilAl.
e
As complete theories may be not Henkin, the canonical model theorem for this model holds only for propositional formulas; this explains the name 'precanonical'. Proposition 7.5.3 Let L be an N-m.p.1. with constant domains. (1) For any N-modal propositional formula A, for any L-complete r
(2) For any N-modal propositional formula A,
(3) ML(C+@L)= L,.
Proof (1) Standard, by induction on the length of A, using properties of complete theories 6.1.3. We also need an analogue of 6.1.16 for complete theories; its proof uses 6.1.5. (2) 'If' follows from (I), since L-complete theories contain For 'only if' use the Lindenbaum lemma 6.1.5. (3) (( If AI 6) L,, then by (2), C+ML YA, and thus C + a L YA. ( 2 ) The claim (1) can be written as 82(A) = IAl. So we can argue as in the proof of 1.7.5. Viz., consider a formula A E L, [ k and a valuation in Cf @ L . Choose Bi such that $(pi) = IBiJ = B ~ ( B ~and ) , consider a substitution S := [&,. . . ,Bk/pl,. . . ,pk]. Then SA E L, so $(A) = 82(sA) = Cf PL by 1.2.9 and (2).
z.
+
We also have an analogue of Theorem 1.7.13: Theorem 7.5.4 Let L be an m.p.1. with constant domains. Consider the modal algebra Lind(L(S0)) of all So-sentences up to the equivalence t-L A = B. Then
Lemma 7.5.5 C+QL is descriptive.
7.5. PREDICATE VERSIONS OF COFINAL SUBFRAME LOGICS Proof It is again similar t o the propositional case. Cf QL is differentiated, since I' # A implies r A, and thus IAl for some A. Tightness VA ( r E CIilAI + A E IAI) + I'KA
567
E IAl, A
6
follows from the definition of Ri and the equality lOiAl G OilAl we have already proved. For compactness, note that if V is a set of interior sets with non-empty finite intersections, then the theory ro:= {A I IAl E V) is L-consistent. In fact, if
rotL1,then t
k
- ~
7
- a contradiction.
A
i= l
r cro,so n v 3 r .
Ai for some IAll,. . . , lAkl E V, SO
k
k
n /Ail = I i=Al Ail = 0 i=l
By the Lindenbaum lemma, there exists an L-complete
Alternatively, one can apply Theorem 7.5.4 and Proposition 1.7.14. Quite similar constructions and proofs can be made in the intuitionistic case.
Definition 7.5.6 For an s.p.1. L with constant domains, let C+PL the set of all L-complete intuitionistic theories with the set of constants So. Then we define the pre-canonical model, the pre-canonical frame and the pre-canonical general frame of L:
where
Lemma 7.5.7 I n the intuitionistic case Cf QL is well-defined.
Proof Let us show that {IAl I A E I F ( S o ) ) is a subalgebra of H A ( C f F L ) . It is sufficient to check the equalities:
The first one is trivial; the second and the third follow from Lemma 6.2.5. The equality IA > BI = IAl -+ IBI is checked as in the propositional case. In fact, ]A > BI IAI -+ IB1, since IA > BI n IAl IBI. The latter follows from the implications
c
To show the converse, suppose (A > B) @ r . Then the theory (I' U {A), {B)) is L-consistent. For otherwise we have r,A FL B, which implies I' k L A > B , by the Deduction theorem, and next (A > B ) E r by 6.2.5. Thus by Lemma 7.2.3 there exists an L-complete A 5 (I'U {A), {B)), hence r A E (\A1- IBI), and therefore r 6(IAl -+ IBI).
c
C H A P T E R 7. KRIPKE COMPLETENESS
568
The next proposition is an analogue of 7.5.3.
Proposition 7.5.8 Let L be an s.p.1. with constant domains. Then (1) for any intuitionistic propositional formula A , for any L-complete
I'
C+ML,I' It- A i f f A E I'; (2) for any intuitionistic propositional formula A, C + M L It- A iff A E L ; (3) IL(C+@L) = L,.
Proof Repeats the proof of 7.5.3, with minor changes; for example, use 1.2.13 instead of 1.2.9. We leave the details to the reader. W Hence we obtain
Theorem 7.5.9 For any s.p. 1. L with constant domains, C+QL Lind(L(So))+, where Lind(L(S0)) is the Lindenbaum algebra of So-sentences i n L . Lemma 7.5.10 For any s.p.1. L with constant domains, C f Q L is descriptive. Proof
Similar to 7.5.5.
W
Lemma 7.5.11 I n each of the following cases the pre-canonical frame C+@L is coatomic: (1) L is an s.p.1. containing Q H C K ;
(2) L is an m.p.1. containing Q S ~ C K ' . Proof ( 1 ) Follows the lines of the proof of 7.3.9. (2) Since L 2 S4.1, we can apply the same argument as in the propositional W case for S4.1 - its canonical frame is coatomic. Now we can prove the main completeness results of this section. We begin with a theorem from [Shimura, 19931.
Definition 7.5.12 A world t i n a transitive frame is called terminal i f it is accessible from every nonmaximal cluster and its cluster t" is maximal and either trivial or degenerate. A finite transitive frame is called weakly directed if it contains a terminal world. Theorem 7.5.13 Let A be an intermediate propositional logic axiomatisable by cofinal subframe formulas of weakly directed finite rooted posets. Then Q A C K is C-canonical.
7.5. PREDICATE VERSIONS O F COFINAL SUBFRAME LOGICS
569
+
Proof If A = H {CSI(F) I F E 3)for a set 3 of weakly directed posets, L = QACK, let us show that CFL It- C S I ( F ) for F E 3. Suppose the contrary, then CFL is cofinally subreducible to some F E 3. By Theorems 1.12.22, 1.12.30, there exists f : G ++ F for a cofinal G c CFL f uo (for some uo) such that f-'(x) consists only of maximal points, whenever x is maximal in F . Then let us prolong f to f ' : G' -w F for a cofinal G' C C'FL T UO. Since K F E L, C+FL is coatomic by Lemma 7.5.11, so it suffices to extend f only to maximal points of C f F L . We can do this as follows. Let y E Cf PL- CPL be a 'new' maximal point (so y is a classical non-Henkin theory). We put ff(y) := t, where t is a (fixed) terminal world of F. Thanks to the McKinsey property, the subframe G' := G U (maximal points of C+FL) is cofinal in C+FL. Since f' sends maximal points t o maximal points, the lift property is preserved. The monotonicity of f ' follows from the definition: if U R ~and ~ u, E G, y E G' - G, then u is nonmaximal, so f (u) is nonmaximal (Here by the choice of f , while fl(y) is terminal; hence f '(u) = f (u)RF f F ~ RF , denote the accessibility relations in G, F respectively.) Therefore, C f F L is subreducible to F, so CfFL Iy C S I ( F ) , and thus Cf FLIy A. On the other hand, Cf @L It. A by 7.5.8 and Cf @L is descriptive (7.5.10), hence C+FL IF A by d-persistence (Theorem 1.12.28). This is a contradiction. The next lemma and theorem are also taken from [Shimura, 19931.
Lemma 7.5.14 Let L be an s.p.1. containing Q H K , and assume that a cone CML f u has finitely many maxzmal worlds. Then C+ML f u has the same mmzmal worlds. Proof Let X I , . .. , xn be all maximal worlds in CML T u, and suppose that y is another maximal world in C'ML f u. By distinguishability, there exist Ai E xi - y; then l A i E y, since y is classical. Hence l A 1 A . .. A l A n E y, and thus ~ ( 1 A Al ... lA,) @ u, which is equivalent to n
(
n
1
V
i=l
Ai
>
$Z u, and to CML,u Iy
n
1
V
i=l
Ai. But then
CML,v I'r 7 V Ai for some v E RL(u),while by the McKinsey property, vRLxi i=l n for some i. Then xi IF Ai contradicts xi IF 1 V Ai. i=l
Theorem 7.5.15 Let A be a cofinal subframe intermediate propositional logic containing a formula CSI(Vn) for some finite n, where Vn = IT;. Then Q A C K is C-canonical.
+
Proof By Theorem 7.5.13, Q H C K CSI(Vn) is C-canonical, so CFL ItCSI(Vn) (where L := QACK). This means that CFL is not cofinally subreducible to Vn, and thus every cone CFL u contains at most (n - 1) maximal
CHAPTER 7. KRIPKE COMPLETENESS
570
points, otherwise, due to the McKinsey property 7.3.9, there exists a cofinal subreduction to Vn. Now we can show that CFL IF C S I ( F ) whenever C S I ( F ) E A. In fact, otherwise CFL is cofinally subreducible to F and by 1.12.27, there exists f : G -+ F for a cofinal G CFL T u such that for every maximal x E F, f-'(x) contains only maximal points of CFL. Then G is cofinal in C+FL T u. In fact, by Lemma 7.5.14, C+FL T u has the same maximal points as CFL 1 u, and they are all in G, as G is cofinal in CFL T u. By 7.5.1 1, the set of maximal points is cofinal in C+FL. Therefore f is a cofinal subreduction from C+FL to F . Now we can use the same argument as in the proof of Theorem 7.5.13 t o show that C+ FLIF A, which leads to a contradiction. Hence we obtain another proof of 7.3.14. Corollary 7.5.16 QACK is C-canonical for A
=H
+ AJ .
+
Proof In fact, A = H CSI(V2) - a cofinal subreduction to Vz does not exist iff a frame is confluent. Let us now prove the main result from [Shimura, 20011 with slight additions. T h e o r e m 7.5.17 Let A be a modal propositional logic miomatisable above S4 by cofinal subframe formulas of weakly directed posets. Then QACK' is C canonical. Proof Similar to the proof of 7.5.13. It is sufficient to show that CFL != C S M ( F ) , provided CSM(F) E A and F is a weakly directed poset. Suppose the contrary, then CFL is cofinally subreducible to F . By Theorems 1.12.22, 1.12.30, there exists f : G -++ F for a cofinal G CFL T uo such that for any maximal x E F , f-'(x) consists only of maximal points. Then we prolong f to f' : G' ++ F for a cofinal G' G C'FL T uo. Since by 7.5.11 C+FL is coatomic, we extend f to maximal points of C+FL by sending every y E C+PL - CPL to a certain terminal world of F . Hence Cf FLWCSM(F). On the other hand, C+@Lk C S M ( F ) by 7.5.3 and C+@Lis descriptive (7.5.5), hence Cf FL i= C S M ( F ) by d-persistence (1.12.30(4), 1.12.22), which leads to a contradiction. W Corollary 7.5.18 ( 1 ) Let A be a cofinal subframe intermediate logic containing the formula
SI(FS3). Then L = QACK is C-canonical.
(2) The same holds for a A-elementary cofinal subframe A SM(FS3) and L = Q A C K + .
> S4.1 containing
7.5. PREDICATE VERSIONS O F COFINAL SUBFRAME LOGICS
571
Proof (1) As we know from 7.5.13, if C S I ( F ) E L and F is a weakly directed finite rooted poset, then CFL It C S I ( F ) . So assuming that CFL Iy C S I ( F ) , and F is not weakly directed we can show that C S I ( F ) @ A. By assumption, CFL is cofinally subreducible t o F. Let F = (W, R) and for any a E W put ii := R(a) n max(F). We claim that {ti I a E W - max(F)) is a chain in (2max(F),c ) . In fact, if ii, b are 2incomparable and ml E ii - &,7b2 E b - ti, then F contains a subframe (where 0 is the root):
So F, and thus CFL, is subreducible to FS3. But then CFL Iy SI(FS3), which contradicts the canonicity of QHCK SI(FS3) L. Therefore the sets ii constitute a chain, so there must be the least of them, say 6. But then every non-maximal a sees every m E iio which means that F is weakly directed. This is a contradiction. (2) If C S M ( F ) E A, then we may assume that F is a blossom frame (by 1.12.30), and hence a poset (since S4.1 2 A). Now we can repeat the argument from the proof of (1) replacing C S I with CSM.
+
Lemma 7.5.19 (1) A rooted S4.1-frame is not cofinally subreducible to FSl iff the set of its inner points is a quasi-chain.
(2) A rooted S4.1-frame is not cofinally subreducible to FS2 ifl every maximal point is accessible from every inner point.
Proof (1) (If.) If f is a subreduction of G to FS1, f(a1) = 1, f(a2) = 2, then both a l , a2 see points in f -'(3), so they are inner in G. a l , a2 are incomparable, since 1, 2 are. Thus inner points are not in a quasi-chain.
(Only if.) If G is coatomic, has a root a0 and two incomparable inner points a l , az, then the partial map f : G -+ FS1 sending ai to i and every maximal point to 3 is a cofinal subreduction.
CHAPTER 7. KRIPKE COMPLETENESS
572
( 2 ) (If.) If f is a cofinal subreduction of G to FS2, f(a1) = 1 and a1 sees a maximal point a2, then f (a2) = 2. Similarly there exists a maximal a3 E f (3). So a1 is an inner point and a1 does not see as. (Only if.) Suppose G is coatomic with root a0 and has an inner point a l , which does not see a certain maximal point as. Then the map defined on ao, a1 and all maximal points of G is a cofinal subreduction if it sends ai to i and all other maximal points to 2.
Proposition 7.5.20 (1) Let A be an intermediate propositional cofinal subframe logic containing
C S I ( F S 1 ) or C S I ( F S 2 ) . Then L
= QACK
is C-canonical.
(2) The same holds for a A-elementary cofinal subframe A C S M ( F S 1 ) or C S M ( F S 2 ) and L = QACK+.
> S4.1 containing
+
+
Proof (1) By Theorem 7.5.15, the logics QHCK C S I ( F S l ) , QHCK C S I ( F S 2 ) are C-canonical, so CFL It C S I ( F S 1 )or CFL It C S I ( F S 2 ) . Now assuming C S I ( F ) E A let us show CFL It- C S I ( F ) . In fact, if CFL ly C S I ( F ) , i.e. CFL is cofinally subreducible t o F , then F is weakly directed. For, the validity of C S I ( F S i ) is preserved by cofinal subreductions, thus F It C S I ( F S 1 )or F It C S I ( F 2 ) .Therefore F is weakly directed by Lemma 7.5.19. Hence CFL It- C S I ( F ) by Theorem 7.5.15. To prove ( 2 ) ,use the same argument applying 7.5.17.
Figure 7.1. FS1.
Figure 7.2. FS2.
Figure 7.3. FS3.
Since incompatible sets are always strongly incompatible in the canonical models of the intermediate propositional logics, we can prove that all cofinal subframe propositional logics are canonical.
Question 7.5.21 Let L be an s.p.1. with constant domains. Is every incompatible subset of CPL strongly incompatible?
7.6. NATURAL MODELS W I T H CONSTANT DOMAINS
+
573
+
Question 7.5.22 Is QA C D K F strongly complete for every cofinal subframe logic A ? What happens i n the case A = H + C S I ( F S 3 ) where FS3 is the frame in Fig. 7.32
7.6
Natural models with constant domains
In this section we transfer the method described in section 6.4 to constant domains.
Lemma 7.6.1 Let M be a Kripke model for a modal logic L with the constant domain So. Then for any world u E M , for any So-sentence A
M , u k A iff C M L ,V M ( U ) k A. The same holds i n the intuitionistic case, with obvious changes. Lemma 7.6.2 Let M be the same as in the previous lemma, R, the accessibility relations in M . Then for any u , v E M
Definition 7.6.3 Let L be an m.p.l.(=) containing Barcan axioms or an s.p.1. (=) containing C D , F = (W,R 1 , . . . ,R N ) a propositional frame of the corresponding kind. A CDL-map based on F is a monotonic map from F to (the propositional base of) C F ~ = ) . Definition 7.6.4 Let h be a CDL-map based on a propositional frame F . The predicate Kripke frame associated with h is F ( h ) := F @ So. If L is a logic with equality, the Kripke frame with equality associated with h is F'(h) := ( F ,D , x), where c xu d iff ( c = d ) E h ( u ) .
, The CDL-model associated with h is ~ ( = ) ( h:=) ( ~ ( = ) ( h~)(, h ) )where J t h ) u t P r ) := { C E Sgm I p r ( c ) E h ( u ) )
-
for u E F .
Definition 7.6.5 A CDL-map h : F C F L and the associated CDL-model M ( h ) are called natural if h = V M ( ~ ) , Li.e. , for any u E F , A E C ( u )
-
M ( h ) k ( I t ) Aiff C M L ,h ( ~b)( I t ) A (eA E h ( ~ ) ) . Lemma 7.6.6 For a modal logic L , a CDL-map h : F it is selective (in the sense of Definition 6.4.7). [Shimura, 19931
CFL is natural iff
CHAPTER 7. KRIPKE COMPLETENESS
574
H
Proof The same as for 6.4.8. Definition 7.6.7 An intuitionistic CDL-map h : F (W,R ) is called selective if it satisfies the condition (3)if (A1 3 Az) E ( - h ( u ) ) , then A1 E h ( v ) , A2
---t
CFL based on F
# h ( v ) for some v
E
=
R(u).
Note that now we do not postulate the analogue of condition (V) from Definition 6.4.9; it obviously holds for v = u , since h ( u ) satisfies (Ac). Lemma 7.6.8 An intuitionistic CDL-map h : F
---,CFL
is natural iff it is
selective. Proof Similar to 6.4.10. The only difference is the case A = V x B in the 'if' part: If A # h ( u ) ,then by (Ac),there exists c E D, such that B ( c ) # h(u). By induction hypothesis, u Iy B(c). Hence u Iy A. The other way round, if u Iy V x B , then since the domain is constant, there exists c E D, such that u Iy B(c). Thus h ( u ) Iy B ( c ) ,which implies h ( u ) Iy A. Lemma 7.6.9 Let L be a modal (resp., superintuitionistic) logic containing
Barcan formulas (resp. C D ) , MI C M L a selective submodel, Fl the propositional frame of M I , h : F + Fl. Then h is a natural CDL-map. Proof
Cf. Lemma 6.4.11.
Lemma 7.6.10
(1) Let L be an N-modal logic containing Barcan formulas for all mi,I' E CPL. Then there exists a CD-Kripke model M based on a standard greedy tree F such that V M ( A )= I' and M k L. (2) If L contains OiT for 1
(3) If L is 1-modal and L Proof
< i < N , then one can take F = FNT,.
> QS4, then the claim holds for
F = IT,
Cf. Lemma 6.4.12.
w
Proposition 7.6.11
Proof
Follows from the previous lemma, cf. 6.4.13.
Proposition 7.6.12 Let A be an N-modal propositional PTC-logic. Then (1) QAC = ML(CKGT(A)).
W
7.6. NATURAL MODELS W I T H CONSTANT DOMAINS
Proof Similar to Proposition 6.4.14. (1) By 7.1.8, Q A C is CK-complete. Then by 1.11.5 and 3.12.8 we obtain L = ML(CKVo(A)), and hence L = ML(CKVl(A)), by 3.3.21. (We use the same notation Vo,V1 as in the proof of 6.4.14.) Now by 1.11.11 and 3.3.14 it follows that ML(CKG'T(A)) L. (2) The proof is similar. In the same way we obtain an analogue of Proposition 6.4.15
Proposition 7.6.13
(2) Let A be a propositional PTC-logic. Then Q A C Z + C E = ML'(CKGl(A)). Natural models with constant domains are a particular case of natural models described in section 6.4; only the notation is slightly simplified in this case.
Definition 7.6.14 Let L be a predicate logic, F a propositional Kripke frame, and let Ri ( 1 I i I N ) or R' i n the intuitionistic case be relations on CPL. A n (L, R;, . . .,Rh)-model on F is a mapping from F to CPL, i.e. N = (I', 1 u E F ) where r, are ( L ,So)-places such that
A n L-model N is called natural if the conditions from Definitions 6.4.7, 6.4.9 (for the modal or the intuitionistic case respectively) holds (with So replacing SU). Recall that in the intuitionistic case the condition (tl) from Definition 6.4.9 holds obviously already with v = u,since theories in CPL are Ltl-complete, i.e they satisfy Ac from Definition 7.2.1. As in Section 6.4, a natural (L, RL)-model is called a natural L-model. (Recall that = R L in the intuitionistic case with constant domains.) All main properties of natural models and crucial lemmas from Section 6.4 (together with their proofs) can be directly transferred to the case of constant 'I instead domains: namely 6.4.8, 6.4.10, 6.4.11, etc. (now one has to use r or , of L-places ( S , r ) or (S,,I',) respectively). Therefore the logics with Bai or with C D (in the N-modal or in the intuitionistic case respectively) satisfy the main completeness results about the universal tree T, (or its subtrees). Let us formulate these completeness theorems for the intuitionistic case.
RE
Proposition 7.6.15
CHAPTER 7. KlUF'KE COMPLETENESS
576
+
Proposition 7.6.16 Q H C D A APh = IL(CK(IT,h)) for h > 0 and similarly for the corresponding logics wtth equality. Note that in the case of constant domains a natural analogue of 7.4.12 holds for any logic L containing APh. Now let us prove completeness results (w.r.t. coatomic trees) for the logic Q H + C D A K F . Here the situation is subtler than for varying domains. Namely, for an arbitrary path w = ( u o , u l , .. .) in IT, we cannot always extend an Lconsistent theory I? = U rUito an LV-place I?, in the same language (i.e. iEw
with constants from So). For example, if So = {c, I i E w), and for each i , P(q) E r u t , while -+xP(x) E I?,, then I?, satisfying (Ac) does not exist. That is why we cannot prove completeness of Q H + C D A K F w.r.t the coatomic tree IT, with a constant domain. Moreover Q H C D A K F is not determined by @ w, cf. Lemma 3.12.17. We do not know if this logic is complete w.r.t. IT, @ V with uncountable V, but anyway natural models with denumerable domains are insufficient. Instead we can show completeness w.r.t. the class of (denumerable) coatomic 4 trees of the form IT, (for G C IT:) (cf. [Gabbay, 19721).
+
m,
L e m m a 7.6.17 Let L be a superintuitionistic predicate logic containing C D A 4 K F . Then for any (L, So)-place r there exists a subset G 2 IT! such that IT, is coatomic and for any G' G there exists an L-natural model N = (I?, I u E -G
'
IT, ) (with the constant domain So) such that
= I?.
Proof First we construct an L-natural model (I?, I u E IT,) on IT,. Recall that a subset X of IT, is called dense if Vu R(u) n X # 0 . We put X B := {u E IT, I B E I?,) for QCL-theorems B E IF(=) (without constants from So); actually it is sufficient to consider only theorems of the form B = v(AV l A ) , cf. Remark 7.3.1). These sets are dense since L I- 11B (by the Glivenko theorem), thus 1 B $ I?, ifor any u. Call a path generic if it intersects every X B ; let Gr be the set of generic -G paths. Obviously, Vu E ITw3w E G(u E w ) , i.e. the tree IT, is coatomic. Now for any generic path w E G, we take the set r, := (J F,. Then I?, E CPL. In
z
c c
UEW
c
c
fact, if kL ATl > V Al for finite Fl F2,Al -F2, then I?l I?,, Al -Fv 3xA(x) E rvand A(c) E I?, I?, for for some v E w; also, if 3xA(x) E I?,, then some v E w, c E So. Also by genericity, Q C L I?,, thus I?, is (QCL, So)-place 4' and condition (Ac) for I?, (in a natural model on IT, for G' C G,) holds. W Recall that although G r may be uncountable, there always exists a denu-G merable subset G G r such that IT, is coatomic. Therefore we obtain the following completeness result.
7.7. REMARKS ON K R I P K E BUNDLES
577
+
Proposition 7.6.18 The logic Q H C K := Q H CD A K F is complete (in semantics CK) w.r.t. the class of all coatomic trees moreover one can take only denumerable coatomic trees.
z;
However we do not know if Q H C K is determined by a single coatomic tree of the form T .I: But still it is determined by a single denumerable coatomic -4; tree, e.g. by Smorynski's sum HIT, of all coatomic trees refuting nontheorems of the logic. Note that the sum over all denumerable atomic trees of the form 4 IT, is not itself denumerable. Another alternative is to add maximal elements not above paths, but above points of IT, as explained below. Definition 7.6.19 A subset X IT, gives rise to a denumerable tree IT: := IT,UX* where X * = {u* I u E X ) is the set of maximal elements u* := (u, -1) such that u* is an immediate successor of u. Obviously the tree IT:
is coatomic iff X is a dense subset of IT,.
L e m m a 7.6.20 Let L be a superintuitionistic predicate logic containing C D A K . Then for any I? E CPL and for any X & IT, there exists an L-natural such that I?, = I?. model N (with constant domain) on IT: P r o o f We simply extend an L-place ,?I Lemma 8.3.6.
(for u E X ) to an L- place I?,.
for any dense X
Proposition 7.6.21 Q H C K = IL(CIC(IT:)) particular, for X = IT,).
C_
by
IT, (in
A similar completeness result for logics with equality (w.r.t. CICE or w.r.t. K) is quite obvious.
7.7
Remarks on Kripke bundles with constant domains
Now let us make some observations on completeness in more general semantics than IC or ICE. Proposition 7.6.16 clearly shows that Q H CD A APh is determined by the tree IT: in the semantics of Kripke sheaves or even quasi-sheaves (recall that APh is strongly valid in any Kripke quasi-sheaf based on a frame of depth 5 h). In particular, Q C L = IL(CIC(IT1)) = IL(CIC&(IT1))= IL(K&(IT1)), where IT1 2 Z1 is a one-element poset. On the other hand, for Kripke bundles the situation changes drastically:
+
Proposition 7.7.1 IL(CKB(IT1)) = Q H
+ CD.
578
CHAPTER 7. KRTPKE COMPLETENESS
Proof Let M be a Kripke model based on IT, with constant domain w refuting a nontheorem A, i.e. M, h Iy A. Take a Kripke bundle F based on {u0) in which Duo = {a, I u E IT,) U {bi, ci I i E w) is quasi-ordered by the following relation R1: a,Rla, iff u Q v E IT,, for U,v E IT,, biRla, for all i E w,v E IT,, and biRlbj for all i, j E w. Obviously, F has a constant domain -every ci is incomparable with all elements of Duo; every a, has infinitely many predecessors bi (i E w). Consider the following map x from Duo onto w: ~ ( c i= ) i and ~ ( b i= ) ~ ( a , )= 0 for any i E w, u E IT,, and consider the model M' over F for the formula A' (with an additional parameter z ) such that:
where the predicate letter P' is substituted for P in A1 One can easily check that M1,uo It ~ l ( a , , d l , ... ,d,) iff M , u k B(x(dl), . . . , ~ ( d , ) ) for any formula B(x1,. . .,xn) and u E IT, (by induction on B). Thus, MI, vo Iy A1(A), and so A 6 IL(F). W In the same way we can simulate every Kripke model with a constant domain by a Kripke bundle model based on one-element poset, so that individuals replacing additional parameters of formula A' correspond to worlds of the original Kripke frame.
Corollary 7.7.2 IL(CKB(F)) = Q H
+ C D for any propositional base F .
Therefore in the semantics of Kripke bundles the propositional base is not so important as the structure of individual domains.
Remark 7.7.3 Note that the above corollary does not transfer to logics with equality - e.g. the formula Vxy(x = y) > p V l p is strongly valid in every Kripke bundle over a one-element base. Remark 7.7.4 One can also try to describe the logic IL(KB(IT1))determined by a one-element base with varying domains. We may conjecture that this logic is QH, but it is still unclear how to simulate arbitrary Kripke frames with varying domains within a single domain in a single world. Remark 7.7.5 Let us also mention the semantics of Kripke quasi-sheaves. One can easily show that IL(CKQ(ITl)) = QH CD and IL(KQ(IT1)) = QH for the w-chain ITl; in fact, every Kripke model based on IT, can be simulated in a Kripke quasi-sheaf over ITl similarly to Proposition 7.7.1. Branching of
+
7.8. KRIPKE FRAMES OVER THE REALS AND THE RATIONALS
579
individuals in a quasi-sheaf replaces the branching of worlds. For the case of varying domains, note that for constructing a natural model over IT, we may assume that individual domains of all worlds at the same level coincide. Similarly, IL(CICQ(F)) = Q H C K and IL(KQ(F)) = Q H K F for an 4-X (w 1)-chain F; atomic trees IT, or IT, can be used here. On the other ~ hand, note that IL(CKQ(F)) = Q H CD A APh and IL(ICQ(F)) = L P for any poset F of finite height h > 0 (in particular, for an h-element chain).
+
+
7.8
+
Kripke frames over the reals and the rationals
In this section we consider specific extensions of the intermediate logic of linear ordered sets with constant domains QLCC, viz. the logic of frames over the rational and the real line. Both logics happen to be finitely axiomatisable. The first result is not a big surprise, but the second is a 'happy chance7 making a contrast with classical logic. These results were first proved in [Takano, 19871 who also observed that strong completeness theorem holds in both cases. We use the following two axioms:
Ta. (VxP(x) > 3xQ(x)) > 3x(P(x) > r ) V 3x(r > Q(x)).
L e m m a 7.8.1 Q H C
+ Ta = Q L C C +Tat.
+
Proof (C). It suffices to show that QLCC Tat t Ta. So we argue in Q L C C Tat. We have ( Q b ) 3 r) V (r 1Q(x))
+
by A Z , hence (Q(x) > r) V 3x(r > Q(x)), Vx((Q(x) > r ) V 3x(r > Q(x))), and thus
by CD. Similarly we have (2)
Vx(r
> P(x)) v 3x(P(x) 3 r).
From (1) and (2) we obtain either the conclusion of Ta:
CHAPTER 7. KWPKE COMPLETENESS
580
Now assume the premise of T a (congruent to the premise of T a t ) :
Hence by Tat we have
Consider the first option and assume
By 2.6.15 we also have
and thus
(31,(5) 1- 3 x ( P ( x )3 r ) . Now assume
Since by 2.6.15
V x ( r 3 P ( x ) ) t- r 2 V x P ( x ) , we have
(3)k r 3 V x P ( x ) , hence
( 3 ) ,( 4 ) t- r 3 3xQ(x), and thus (31, ( 4 ) 1- r 2 ~ Y Q ( Y ) .
Now obviously
r
3
~ Y Q ( Y )~, Y Q ( Y 3 ) Q ( x ) 1- r 3 Q ( x ) ,
hence
(31,(4,~ Y Q ( Y 3 ) Q ( x )t- r 3 Q ( x ) , and thus (3), ( 4 ) 1- ~ Y Q ( Y 3 ) Q(x)*2. r 2 Q ( x ) .
Therefore by monotonicity
and eventually
(317 ( 4 ) ,( 6 ) t- 347- 3 Q ( x ) ) .
7.8. KRIPKE FRAMES OVER THE REALS AND THE RATIONALS So we have proved the conclusion of T a . (2). We have Q H T a t- Ta', since obviously [3xQ(x)/r]T a show Q H T a t- AZ, note that
+
+
is clearly equivalent to (p > r ) V ( r > p).
581
Ta'. To
W
In this section we consider subsets of 8 , in particular, Q, R+ = [O,+co), Q+ = R+ n Q as propositional Kripke frames with the accessibility relation 5.
Theorem 7.8.2 (1) Q L C C = IL(CKQ) = IL(Q
w).
(2) Q L C C is strongly complete w.r.t. CKQ and Q
w.
Proof (1) The inclusion C follows readily from Proposition 7.3.6. The proof of 2 is quite similar to 6.7.3. By Lemma 6.7.2, Q+ + F for any rooted countable chain F. Then we can apply Propositions 7.3.6, 3.3.14. (2) By the Shimura theorem, Q L C C is strongly Kripke complete, hence it is also strongly complete w.r.t. the class of countable chains with domain w. By 6.7.2 we know that Q+ -+ F for any countable chain F. Hence Q+ @ w + F O w by 5.1 and eventually strong completeness w.r.t. Q+ O w follows by 5.1.
Definition 7.8.3 Let Fo c F be linear Kripke frames with a constant domain V , R the accessibility relation i n F . Let M = (F, O), Mo = (Fo,00) be intuitionistic Kripke models. M is called a right extension of Mo if for any V-sentence A O+ (A) = R(O$(A)). (*r) So we have
M,uI=Aiff3v~Fo(vRu&M~,vIt-A) Note that (*r) readily implies @(A) = O+(A) n Fo,i.e. Mo A right extension is obviously unique (if it exists).
c M.
Definition 7.8.4 Let Fo F be linear Kripke frames with a constant domain V , R the accessibility relation in F . Let M = ( F , O), Mo = (Fo,00) be intuitionistic Kripke models. M is called a left extension of Mo i f for any V-sentence A, for any u E M u E @(A) iff R(u) n Fo c O$(A).
CHAPTER 7. K R P K E COMPLETENESS
582
The latter condition can also be written as follows 0+(A) = oRO$(A), where
(*1)
c
ORU := {U I R ( U ) ~ F O u).
(*1) is also equivalent to
Lemma 7.8.5 Let F @ V be a linear Kripke frame over F = (W, R), Wo and let Mo = (FIWo,80) be a Kripke model such that
cW
(2r) for any V-sentence VxB(x)
Then there exists a right extension of Ma over F Note that the converse to (2r) obviously holds, so (2r) is equivalent t o
Proof
We define the valuation 0 by the equality from 7.8.3
for all atomic V-sentences A. 0 is obviously intuitionistic, since R is transitive. Then we check (*r) for any V-sentence A by induction. (I) Consider the case A = B > C. By the induction hypothesis, it suffices to show that for any u ) Vv E R(u) (VE R(@(B)) u E R ( ~ $ ( A ) iff
+ v E R(~$(c))
('Only if'). Assuming u E ~ ( 0 : (A)), URVand v E R(B,~(B)),let us check v E R(B$ (C)). By assumption there exists ulRu such that u1 I t B > C and vlRv such that vl I t B (where I t refers to Mo). Let wl = max(ul,vl); then wlRv. Now ul I t B > C implies wl It- B > C ; vl Ik B implies wl I t B. Thus wl Ik C, and so v E R(O$(C)). ('If'). We suppose u @ R(B$(A)) and show
By assumption (lr), there exists q E R-'(u) n Wo. Since u @ R(B$(A)), we have ul Iy A. Then for some vz E R(vl),vz It- B, but vz W: C.
7.8. KRTPKE FRAMES OVER THE REALS AND THE RATIONALS
583
Now put v = max(u,v2). Then v2 It- B implies v It B , and thus v E R ( ~ $ ( B ) It ) . remains to check that v # R(Q$(C)). If v = va, then v # R(e$(C))follows from v2 W C. If v = u, then u # R(6$(B > C ) ) implies u # R(o;(C)). This is because C > ( B 3 C ) is an intuitionistic tautology, so Mo It- C > ( B > C ) , i.e. 6: ( C ) E 6$ ( B 3 C ) . (11)I f A = B A C, by the induction hypothesis it suffices to show
~ ( 6 (;B A c))= ~ ( 6 (;B ) )n R(O$(c)).
(#)
The inclusion 'C'follows from 6$(B A C ) C 6; ( B ) and 6; ( B A C ) C O$(C). The other way round, suppose u E R ( ~ $ ( B n ) )R(B$(C)). Then vl E Q$(B),v2 E 6; ( C ) for some vl, vz E R-l(u). If ZJ = max(vl, up),then obviously v E @ ( B A C ) , and thus u E R ( ~ $ ( ABC ) ) .This proves '2' in (#). The remaining cases are left t o the reader; note that for the case A = VxB(x) one can use the assumption (2r). W
Remark 7.8.6 ( r l ) ,(r2)are not only sufficient, but necessary for the existence of a right extension. (r2) follows from (*r) for A = VxB(x), ( r l ) from (*r)for A = T. Lemma 7.8.7 Let F O V be a linear Kripke frame over F and let Mo = (FIWo,60) be a Kripke model such that
= (W,R ) , W2
cW
(21) for any V-sentence 3xA(x)
Then there exists a left extension of Mo over F Proof
Similar to Lemma 7.8.5. Now we define
@+(A) :=
OR^$ ( A ) ,
for atomic V-sentences A and prove (*1) by induction for arbitrary A. Suppose A = B V C. By the induction hypothesis, (*l)for A is equivalent to oR6$( B )U 0 ~ 6 (;C )= ORB$ ( B v C ) . The inclusion 'G'is obvious, so let us check '2'. Suppose u CIR6$(C).Then
# OR6$(B)U
3v, w E R(u)(v # 0; ( B ) & w # 0; ( C ) ) . Since R is linear, vRw or wRv. Then respectively, v # 6$(B V C ) or w # 6$(B V C ) ,so anyway u # OR6$( B V C ) .
CHAPTER 7. KRlPKE COMPLETENESS
584 Suppose A = B
> C. By the induction hypothesis we can rewrite (*I)as
(+-) Suppose u # nR8$ (A). Then w $2 8; ( A ) for some w E R(u),and thus v E 8: ( B )- 8; ( C ) for some v E R(w) C R(u). Obviously, v @ O R ~ $ ( C Since ). Mo is intuitionistic, B$(B) C CIR6,f(B). This proves (+). (+) Suppose u R v , v E ( u R ~ $ ( B-)U~8of(C)). Then w # 8$(C) for some w E R(v) R(u) and v E O R ~ $ ( Bimplies ) w E 8 o f ( ~ )Hence .
The case A = B A C is obvious. The case A = 3xB(x) follows from ( 2 9 , since its converse holds trivially. If A = QxB(x), we have
u # O f ( A )H 3a E v u
# 8+ ( B ( a ) )=
(~(a)),
(**)
c
since the domain is constant and by the induction hypothesis. But 6$(A) 8; ( B( a ) ) ,so u # 8+( A ) implies u @ ORB$( A ) . The other way round, if u # O R ~ , ~ ( Athen ) , v # 6of(b'xB(x))for some v E R(u)r l Fo. So v @ Oof(B(a))for some a E V, and thus u # 8+(A)by (**).
Lemma 7.8.8 Let M = (Q+ OV, 6 ) be an intuitionistic Kripke model, i n which every instance of Ta' i s true. Then there exists an extension 8* of 8 such that (R+ V, B*) is an intuitionistic model.
Proof The proof consists of two parts. (I) First we define a model over the set
where C T B ( ~:= ) inf R
f3(3yB(y)).
We would like to define a model Mo with a valuation 80 over S as a right extension of 8:
OO(A):= 5 ( @ ( A ) ) . By Lemma 7.8.5, it suffices to prove that for the relation 5 on S, for every V-sentence VxA(x),
To prove this, we assume that for a certain a E S
7.8. K H P K E FRAMES O V E R THE REALS AND T H E RATIONALS
585
and show that for some q E Q+
Case 1. a E Q + . Then M , a It- VxA(x), so a E Of (VxA(x)) G 5(O+(VxA(x))), so q satisfies (7.3). Case 2. a = ag(y,$! Q+. By definition, M, 0 W 3y(3zB(z) > B(y)). Now let us show
= a
Suppose the contrary. Then M, 0 It- A(a) > 3zB(z) for some a E V. By the assumption (#), M, q, It- A(a) for some q, 5 a. So M, q, It 3zB(z), and hence a 5 q,, by the definition of a ~ ( ~ Therefore ) . a = qa E Q, and we have a contradiction. Hence (b) holds. On the other hand, by the assumption of the lemma,
Hence M,O W VxA(x) > 3yB(y), so there is q E Q+ such that M, q It VxA(x) but M, q W 3yB(y). Hence q 5 a , by the definition of a . Thus q satisfies (7.3). This proves (7.1). (11) Next, we construct M* with a valuation O* over IWf as a left extension of Mo: M*,aIkA#Va~S(cu5a+M~,aItA). We use Lemma 7.8.7 to show the existence of M*. So we have to check that for every V-sentence 3xB (x)
To prove this, suppose cu E D ~ O ~ ( ~ X B ( that X ) ) ,is
Let us show that cr E
U 0 ~ 6 $ ( ~ ( a ) i.e. ) , for some a E
V
aEV
Now we argue in Mo. Case 1. 0 It- 3y(3zB(z) > B(y)). Then there is a E V such that 0 It- 3zB(z) > B(a), and this a is what we need. Case 2. 0 W 3y(3zB(z) > B(y)). So 0 Iy 3y(3z) 3 B(y)), and hence ffqy) E S. Let us show that B B ( ~ )Iy 3yB(y). Suppose the contrary. Then ag(y)It3yB(y), so O B ( ~ Ik ) B(a1) for some a1 E V. Then by the assumption of Case 2,
CHAPTER 7. KRTPKE COMPLETENESS
586
so there exists CT E S such that a I t 3zB(z), and a I y B(a). Thus a since a ~ ( , I)t B(a). Hence 3q E Q+ a < Q < O B ( ~ ) ,
< OB(~),
and then Mo, q 11- 3zB(z), which implies M, q I t 3zB(z). But this contradicts q < C T B ( ~ ) . Now, by (7.5), Mo,a If 3yB(y) implies a ~ (< ~ a. ) But then there is q E Q such that ag(,) < q < a. Since UB(,) < q, it follows that Mo,q Il- 3yB(y). Hence Mo, q Il- B(a) for some a E V, and this a satisfies W (7.6). Thus (7.4) is proved. Theorem 7.8.9 Q H C
+ T a = IL(CKR).
Proof (2). Let us show that R @ V I t Ta. So consider an arbitrary model over R @ V, a E R and show
Suppose the contrary. Then there exists ,f3 5 a such that ,6 Il- VxP(x) 3 3zQ(x), but
PUL 3x(P(x) 3 r ) , P W 3x(r 3 Q(x)).
Thus for any a E V, there is y, 2 P such that y, It P ( a ) , y, Iy r . Similarly, for any b E V there is 6, 2 p such that bb I t r , db W &(a). F'rom y, W r and It r it follows that y, < hb, for every a , b E V. Now put E := supaGvya. Then /3 5 E and E I t VxP(x). Also ,f?I t VxP(x) 3 3xQ(x), hence E Il- 3xQ(x). On the other hand, E 5 db for any b E V, so E W 3xQ(x). This contradiction proves (7.7) ( 2 ) . We assume Q H C TaYA for a sentence A and construct a countermodel for A over R+. Since R+ is a generated subframe, this implies A @ I L (CICR). By Lemma 7.8.1, Q L C C Tar bL A. By Theorem 7.8.2(2), we obtain a model M = (Q+ V, 0) and qo E Q such that every instance of Ta' is true at M,qo and M,qo Yf: A. We may assume that qo = 0. By Lemma 3.3.18, M' := M 1 Q+ is a model, in which every instance of Ta' is true. Hence by Lemma 7.8.8 there is an extension M * of M' over R+. rn
+
+
Proposition 7.8.10
(2) Q H C
(1) Q L C C
+ K F = IL(CIC([O,11n 0 ) ) .
+ Ta + K F = IL(CK[O,11).
We skip the proof, since it is very similar to Theorems 7.8.2 and 7.8.9.
7.8. KRTPKE FRAMES OVER THE REALS AND THE RATIONALS
587
Proposition 7.8.11 For any linearly ordered set F , the following conditions are equivalent: (1) F is Dedekind complete (2.e. every non-empty subset with an upper bound
has a supremum); (2) IL(C1CF) F Ta'. Proof The proof of (1) (2) is the same as for ( G ) in 7.8.9. Let us show (2)*(1). Suppose (Z), but not (1). Then there exists a nonempty X C W with an upper bound, but without a supremum. Let Y be the (non-empty) set of upper bounds of X. We index both X and Y by elements of V of cardinality IWI so that X = {Ja I a E V ) and Y = {va I a E V ) . Consider a model M over W a V, in which
*
(i) w It- P ( a ) HJ, 5 w, (ii) w It- Q(a) H
va < W.
Let us show that for any w E W
The first equivalence is obvious:
by (i) and the definition of Y. Next, by (ii) W It- Q(a) -3w
> 70,
which implies w E Y. Conversely, suppose w E Y. Since w is not a supremum of X , there is a E V such that rl, < w. Hence w IF Q(a), and thus w It- 3xQ(x). Therefore
Now consider any J E X. Let us show that
In fact, for any a
E
6 ~ a ??a 7 IPLQ(a), Va
IF~~Q(Z)
by (3), hence
C IY 3zQ(z) 3 Q(a).
CHAPTER 7. KRIPKE COMPLETENESS In fact, let us show
for any a E V. Consider <'= max(<,Q). Then (3). This implies (7) and (6). But (2) implies
6' Iy 3yQ(y) by
E'
E
X,
E 5 <', E'
I t P(a), and
H
which contradicts (4), (5), (6). Proposition 7.8.12
(2) If a fomula A does not contain V or 3, then IL(CICW) k A iff IL(CICQ) t. A. Proof
(1) IL(CICQ) C IL(CICR) readily follows from Theorems 7.8.2 and 7.8.9, while IL(CKQ) # IL(CICR) by Proposition 7.8.11. (2) Let us consider only formulas without occurrences of V.
Let A(xl,. . . ,x,) be a V-free formula, and suppose IL(CKQ) ly A. Consider a Kripke model Mo = (Q @ V, e), q € Q, a l , . . . ,an E V, such that Mo, q Iy A(al,. . . ,an). We define M over W similarly to the proof of 7.8.4:
for every cu E W and atomic V-sentence B. Now we prove (fl) for any V=free B. but unlike Lemma 7.8.5, we do not need the assumption (7.1). Hence M, q Iy A(a1,. . . ,an). 3-free formulas are considered similarly using Lemma 7.8.7 instead of Lemma 7.8.5.
Takano [ ~ a k a n o19871 , also proved that IL(IC(W - 0 ) ) tually his method proves the following statement.
=
QLC
+ CD. Ac-
Theorem 7.8.13 If W C R and both W and F-W are dense in W, F = (W, <),
then IL(CICF)
= I L ( F @ w) = Q L C
+ CD.
C IL(Q@w). So given a sentence B and a countermodel M = (Q @ w, E ) for B, we construct a model MI over F @ w refuting B. The construction consists of two stages. Proof It is sufficient to show that I L ( F @w)
7.8. KR.IPKE FRAMES OVER THE REALS AND THE RATIONALS
589
(I) At the first stage we take a denumerable subset Wo of W that is dense in B, Fo= (Wo,<), and construct a model Mo = (Fo @ w, Q) refuting B and satisfying the following condition: inf [$(A) @ (W - Wo) for every w-sentence A. R Put
(+)
S := {inf [ + ( A ) I A is an w-sentence) - Q
and enumerate the sets Q U S and Wo, i.e. put Q U S = {uk I k E W ) and Wo = {v, I n E w). Now since WOand R- W are dense in B, there exists an order embedding h : Q U S B such that h[Q] = Wo and h[S] B - W.
-
To construct h, we can apply the back-and-forth method and define in turn h(uk) and h-l(v,) for new elements of Q U S and Wo respectively, preserving the order <. Or we can define h(uk) in the following direct way, by induction on k. Suppose h(ui) for i < k are already defined, let
If uk E Q, we put h(uk) = vn for the least n such that v, E (h(a), h(u,)). And if u E S , we choose h(uk) in (h(ul),h(u,)) - W. Then definitely every v, E Wo is h(uk) for a suitable k, so to say, n becomes 'the least possible' at some stage of the construction. We leave the routine technical details to the reader. Let us show that VX
Q (inf X E Q U S + inf h[X] = h(inf X)).
R
R
R
(*>
In fact, let infBX = uk. Then h(uk) 5 h(u) for all u E X . Suppose infR h(X) > h(uk). By density, there exists v E Wo such that h(uk) < v < infR h(X); let v = h(u), u E Q. Then u is a lower bound of X , so u 5 uk, and hence v = h(u) 5 h(uk). This is a contradiction. Obviously, the restriction ho = hlQ is an order isomorphism between Q and Wo, so h r l : Wo 4 Q is a pmorphism. Now by Lemma 5.1 we obtain a model Mo = (Wo @ w, t o ) such that
for every w-sentence A. This model is a required one, because (*) readily implies (+). In fact, ikf [:(A)
= h(inf [+(A)) E h[Q u S] G Wo U (R - W).
R
(11) Now we apply Lemma 7.8.4 and obtain a right extension M1 of Mo. This model is a required one over W. It is sufficient to check the condition (r2).
CHAPTER 7. KRTPKE COMPLETENESS
590
So we take an arbitrary wo E W , assume that
Va E V 3u E Wo (U 5 wo & M0,u It A ( a ) ) ,
(7a)
and find u E Wo such that
u 5 wo and Mo,u It Q x A ( x ) .
(7b)
Put uo := infR{u E W o I Mo, u It V x A ( x ) ) . Case 1. uo E W Oand Mo, uo I t VxA(x).
Let us show that uo 5 wo. Suppose the contrary: wo < U O . Then by density there exists u1 E W o such that wo < ul < uo. On the other hand, for every a E V there exists u E W o such that u I wo < uz and Mo,u It- A ( a ) , so Mo, ul It A(a) as well. Hence Mo, ul It- V x A ( x ) and ul < U O , which contradicts the choice of uo. Thus we conclude that u = uo satisfies (7b). Case 2. uo E Wo and Mo,uo Ijc VxA(x). Then Mo, wo ljc A(a) for some a E V. By (7a), there exists u E Wo such that u 5 wo and Mo,u It- A ( a ) . Then uo < u , and so MO,u It- V x A ( x ) by the definition of uo. Thus u satisfies (7b).
Case 3. uo W o . Then u0 @ W by (+). Let us show that uo < wo. Suppose the contrary. Then wo < U O , since wo E W and uo W . So by density, there exists w E Wo such that wo < w < uo. Then Mo, w Ijc V x A ( x ) , by the definition of uo. So Mo,w Iy A(a) for some a E V. On the other hand, M o , u It A ( a ) for some u 5 wo, and hence Mo, w It A ( a ) as well. Since u 5 wo < w , this is a contradiction. wo and Mo,u It Now since uo < wo, there exists u E W Osuch that u I V x A ( x ) . This u satisfies (7b).
Corollary 7.8.14 IL(CIC(R - Q ) ) = I L ( ( R - Q) @ w)= Q L C
+ CD.
Remark 7.8.15 The above argument can be readily transferred to the logics with equality. In fact, Lemmas 7.8.5 and 7.8.7 are extended easily. The conditions ( T * ) and (l*) for equality:
and
~ , u I ~ a = b ~ 3 v ~ W ~ (Mo,vIPLa=b) w R v & respectively follow from ( r l ) and (11). Thus we obtain the following analogues of Theorems 7.8.2, 7.8.9 and 7.8.12:
7.8. KRIPKE FRAMES OVER THE REALS AND THE RATIONALS
591
(1) If W and (R - W) are dense in R then
IL=(CKE(W)) = QLC=
+ CD,
IL=(CK(W)) = Q L C = ~+ C D ;
in particular
+
IL=(CKEQ) = ILE(CKE(R - Q ) ) = QLC= C D , IL=(CKQ = IL=(CK(R - Q ) ) = Q L C = ~ CD.
+
(2)
+
IL'(KER) = QLC= C D + T a , IL=(K(R)) = QLC'~ + C D + T a .
These results imply corollaries for the Kripke semantics with nested domains. Recall that IL(K(Q)) = QLC, IL=(KEQ) = QLC=. IL=(KQ) = Q L C = ~ . Hence by the method from section 3.9 we obtain Proposition 7.8.16 If subsets W and R - W are dense i n R, then
IL(KW) = Q L C , ILE(KEW) = QLC=, IL=(KW) = Q L C = ~ Let us finally mention some simple consequences of the above results. T h e o r e m 7.8.17 The logics IL=(KR), ILE(KER) are recursively axiomatisable. Proof
Follows from 7.8.9 and 3.9.4.
H
Corollary 7.8.18 IL(KR) is recursively axzomatisable. P r o o f In fact, we already know that IL=(KR), ILZ(K&R)are its conservative rn extensions (by 3.8.6 and 2.16.13). As we know, the completeness proof for Q L C can be extended to logics with equality, so IL=(CKEQ) = QLC', IL'(CKQ) = Q L C = ~ . As we know from 7.8.2 and 7.8.14
IL(CK(R - Q)) = IL(CKQ) = QLC
+ CD.
Thus 7.8.18 implies Proposition 7.8.19
IL(K(R - Q)) = IL(KQ) = QLC; IL=(K&(R- Q)) = IL=(KEQ) = QLC=; IL=(K(R - Q)) = IL'(KQ) = QLC'~
CHAPTER 7. KRIPKE COMPLETENESS
592
Perhaps Takano's proof for R - Q can also be transferred to the case with nested domains, but the embedding method provides the result rather easily. As for an explicit axiomatisation for IL(ICR), IL=(ICR), IL'(KER), one can easily see that T a is refuted in a frame over R. In fact, informally speaking, T a means the following. Suppose VxP(x) > 3yQ(y) is true, say a t world 0 E R. Let v be the g.1.b. of the set {u I u I t r ) (if this set is empty, then T a holds trivially). Now if v Iy b'xP(x), i.e. u Iy P(a) for some u 2 v, then 0 IF P ( a ) > r. Otherwise v IF VxP(x), and then v I t 3yQ(y), i.e. r I t Q(b) for some b, so 0 IF r > Q(b). If the domain is constant, both individuals a, b are in Do, but in the case of nested domains a, b can exist only in larger domains D, or D,, so the formula T a is refutable. Constructing a counterexample is left as an exercise for the reader. Conjecture.
IL(KR) = Q L C
By Theorem 3.8.6, this conjecture implies the equality
and (as one can show), ILZ(KR) = Q L C = ~ .
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Index 0-valued structure, 294 abstract n-tuple, 474 abstract jection, 474 admissible rule, 5 algebraic model, 15, 16, 301 algebraic semantics, 11, 196, 317 general, 317 with constant domains, 342 algebraic variety, 19 antichain, 47 arity-perfect, 150 associated F-predicate, 301, 302 general frame, 40 metaframe of a preset, 395 precategory, 377 atomic formula, 81 axiomatise, 5
bounded logic, 6 branch S4-tree, 56 branching a t a world, 49
C-set, 349 superintuitionistic logic of, 380 canonical accessibility relation for L-places, 488 frame, 40 Kripke frame for a modal propositional logic, 40 Kripke frame with varying domains with equality, 489 Kripke frame with varying domains without equality, sub 489 model for a modal propositional logic, 40 model for an intermediate propositional logic, 40 base of a Kripke bundle, 351 model of a superintuitonistic predbasic classical or intuitionistic icate logic, 499 first-order language, 81 model of predicate modal logic, 489 basic H.v.s., 316 model theorem, 41, 489, 499 basic locale, 316 sheaf, 338 category, 243 Bernays rule, 124, 128 binds, 83 of sets, 244 Boolean CD-H.v.s., 296 algebra, 13 CD-m.v.s., 296 logic, 7 CD-structure, 296 bound CE-sheaf, 268 occurrence, 82 change of the base, 362 variable, 82 changing the base, 221, 223, 254 renaming, 89 characteristic formula, 59, 63 renaming in a formula, 86 Church-Rosser property, 23 of a complex formula substitu- class tion, 109 R-elementary, 283
INDEX A-elementary, 283 definable, 211 intuitionistically definable, 211 modally definable, 211 of frames complete, 217 classical formulas, 4 logic, 7 predicate logic, 121 propositional connectives, 3 valuation, 339 clean formula, 90 clean version, 92 closed, 13 equality, 179 formula, 82 propositional formula, 52 closure operation, 13 clusters, 30 coatomic poset, 515 coexistence property, 494 cofinal subframe, 57 subframe formula, 59, 63 subreduction, 57 subset, 57 coherence condition, 244 compactness, 42 complete logic, 194 complete w.r.t., 21 complex formula substitution, 109 component, 225 component of an abstract n-tuple, 475 composition of predicate Kripke frame morphisms, 220 cone, 24, 224, 241 in a general Kripke frame, 40 of a Kripke bundle, 363 of a preset, 385 confluent, 23 congruent formula, 89 congruential, 296 conic sub(pre)category, 385 conically expressive propositional logic, 30
conjunction disjoint, 146 A-perfect set, 148 AV-perfect set, 148 connected component, 25 connected frame, 25 connected subset, 25 connectedness, 225 conservative, 498 extension of an L-map, 513 consistent, 5, 7 logic, 120 constant domain, 296 presheaf, 341 propositional formula, 52 coserial propositional frame, 469 countable base, 284 base property, 284 domain property, 284 frame property, 23, 284 counterpart, 345, 389 relation, 196 D-formula, 102 D-instance, 102 D-sentence, 102 D-transformation, 102 A-elementary class, 45 A-elementary logic, 45 d-persistent set, 44 DE-sheaf, 268 dead end, 48 decidable equality, 179 Deduction theorem, 9 for superintuitionistic logic, 143 definable class, 211 degenerate cluster, 30 A-operation, 72 depth of a world, 48 descriptive general frame, 42 differentiated (distinguishable) general frame, 42 directed predicate Kripke frame, 516 directed propositional Kripke frame, 516
INDEX disjoint conjunction, 146 disjunction, 146 presheaf, 329 sum, 26 of predicate Kripke frames, 228 of predicate Kripke models, 228 union, 26 of predicate Kripke frames, 228 of predicate Kripke models, 228 disjunction disjoint, 146 extended disjoint, 146 distinct depth, 48 list, 86 path, 48 domain expanding system of, 205 level system of, 356 of an L-map, 513 restriction, 233 system of, 205, 339 double theories, 7 Dummett's logic, 7 dummy substitution, 86 effuse tree, 57 elementary class, 45 elementary logic, 45 equality =-morphism of Kripke bundles, 359 (=)-reducible, 226 =-morphism of Kripke bundle model, 242 =-morphism of predicate Kripke frames, 219 =-morphism of preset models, 383 =-morphism of presets, 381 equality morphism of Kripke bundles, 359 equality-expansion, 172 equality-free, 172 equivalence, 13 etale morphism, 255 etale prefunctor, 375
605 exact Kripke model, 23 existence property, 494 expanding system of domains, 205 extended disjoint disjunction, 146 extension, 6 ~ - ~ ~ ~ 296d i ~ ~ t ~ , associated, 302 fibre, 338 fibrewise model, 339, 340 forcing in, 340 Fine theorem, 48 Fine-van Benthem theorem, 45 finite L-chain, 525 finite embedding property, 58 finite model property, 23 finitely axiomatisable, 5 V-perfect set, 147 forcing in a C-bundle model, 386 in a fibrewise model, 340 in a metaframe model, 390 in a simplicial N-model, 475 formula characteristic, 59, 63 clean, 90 closed, 82 cofinal subframe, 59, 63 complex substitution, 109 congruent, 89 frame, 59, 63 intuitionistic subframe, 63 Jankov, 63 Jankov-Fine, 59 modal subframe, 59 preset model, 378 simple substitution, 105 strongly valid in a C-set, 380 strongly valid in a simplicia1frame, 477 strongly valid in an intuitionistic Kripke bundle, 367 true, 246 true in an algebraic model, 309, 310
INDEX true in an intuitionistic C-set model, 380 valid, 246 valid in a C-set, 380 valid in a structure, 310 fragment, 172 frame, 26, 194 almost irreflexive, 64 blossom, 64 formula, 59, 63 neighbourhood, 76 of individuals, 255 representation, 349 rooted, 224 free occurrence, 82 full standard A-tree, 55 full subcategory, 385 full subprecategory, 384 fullweak subframe, 24 functor semantics, 197 functorial metaframe, 405
H.v.s. basic, 316 H.v.s. valuation, 301 height of a world, 49 Heyting algebra, 11 algebra of a general Kripke frame, 39 algebra of an S4-frame, 32 Heyting propositional logic, 6 valued structure, 293 valued set, 196
IF-, 105
I F = - , 105 i-neighbourhood, 76 i(')-sound metaframe, 423 identity morphism, 220, 243 implication, 11 individual, 293 component of a morphism, 219 domain in a simplicia1 frame, 475 Gabbay-De Jongh formula, 7 domain of a world, 205 r-frame, 186 in a preset, 375 general of a metaframe, 388 algebraic semantics, 317 variables, 81 canonical frame, 41 inheritance relation, 196 canonical model theorem, 41 between D,-sentences, 393 intuitionistic Kripke frame, 39 inheritor, 244, 345, 389 modal Kripke frame, 38 inner world, 62, 64 neighbourhood, 318 interior generated algebra, 13 general subframe, 40 operation, 13 subbundle, 363 set, 39 subframe, 24 set of a general Kripke frame, 38 of a predicate Kripke frame, 224 intermediate submodel, 24 logic of a Kripke frame, 33 of a predicate Kripke frame, 224 logic of width < n, 47 Generation lemma, 224, 241 predicate logic, 121 generator, 102 intuitionistic germ, 338 forcing, 32 Glivenko theorem, 10 in an intuitionistic metaframe global valuation, 206 model, 423 greedy standard tree, 50 intuitionistic formula, 4 greedy subframe, 50 strongly valid with equality in a Grzegorczyk formula, 5 QS4-metaframe, 423
INDEX formula valid in, 236 valid in a QSCmetaframe, 423 intutionistic, 236 Kripke frame variety, 33 pattern of, 236 Kripke model over a general Kripke frame, 39 reliable submodel, 241 submodel of, 241 Kripke quasi-bundle, 369 logic, 6 strong isomorphisms, 239 strong morphisms, 239 metaframe model, 422 predicate logic Kreisel-Putnam formula, 7 strongly Kripke compelet, 252 Kripke bundle, 196, 345, 352 base of, 351 propositional Kripke frame, 32 complete logic, 23, 355 subframe formula, 63 substitution, 4 cone of, 363 theory, 210 fibre of, 351 forcing relation, 353 (L3, S)-complete, 494 formula true in an intuitionistic (L3V, S)-complete, 494 model, 367 L-complete, 493 intuitionistic, 351, 368 L-consistent, 493 intutionistic pattern, 366 L3-complete, 494 inverse image morphism, 361 valuation, 16, 32 valuation in a general Kripke frame, isomorphism, 359 39 level of, 356 modal, 351 valuation in a simplicia1 1-frame, modal formula strongly valid, 354 475 modal logic of, 355 (')-sound metaframe, 423 metaframe, 434 model, 353 p=-morphism, 359 isomorphic Kripke frames, 27 restriction of, 363 isomorphic of Kripke models, 27 semantics, 355 isomorphism of predicate Kripke frames, strongly valid formula in, 367 219 superintuitionistic logic of, 368 isomorphism of predicate Kripke modtrue modal formula, 354 els, 219 valid in an intuitionistic formula, isomorphism of presheaves, 329 367 valid modal formula, 354 Jankov formula, 63 valuation in, 353 Jankov's logic, 7 =-morphism of, 242 Jankov-Fine formula, 59 jection, 391 Kripke frame, 11 abstract, 474 K-complete, 218 complete, 218 KFE KFE satisfiable, 252 predicate formula valid in, 211 flabby, 272 subframe of, 241 semantics, 218 variety, 21 with a constant domain, 272 with equality, 195 with closed equality, 268 monic, 268 with decidable equality, 268 -model, 236 Kripke model, 19
INDEX fibre of, 206 A-complete, 7 stalk of, 206 A-consistent, 6 over a general Kripke frame, 38 A-logic of a class, 186 predicate formula, 210 A-logics, 186 Kripke quasi-sheaf, 196. 197, 352 A-closure of a propositional Kripke skeleton of, 372 frame, 55 Kripke semantics with constant domains, A-logics, 6 231 language of a theory, 140 Kripke sheaf law of the excluded middle, 7 S4-based, 244 leaves, 48 bijective, 275 lemma on new constants, 141 KE-complete, 252 letter complete, 252 occurrence of, 82 disjoint, 250 level of a metaframe, 388 fibres of, 244 levelwise uniform tree, 539 flabby, 272 lift property, 27 intuitionistically definable class, 247 of a prefunctor, 381 intutionistic, 244 Lindenbaum algebra, 18, 19 isomorphism of, 250 Lindenbaum lemma, 485 meek, 273 list modally definalbe class, 247 distinct, 86 monic, 268 local CD-frame, 231 simple, 270 local valuation, 206 stalks of, 244 locale, 293 strongly complete, 252 basic, 316 valuation in, 245 locally tabular logic, 70 Kripke sheaf model, 245, 246 Lob formula, 5 forcing in, 245 logic, 5, 193 intuitionistic, 245 cofinal subframe, 57 intuitionistic forcing in, 245 complete, 194 Kripke sheaf semantics consistent, 120 N-modal, 252 locally tabular, 70 intuitionistic, 252 subframe, 57 Kuroda's axiom, 502 universal, 58 Kuznetsov's problem, 77 of a frame, 197 of finite width, 47 L-derivable, 122 logical consequence relation, 186 L-equivalent, 147 L-implies, 147 MFG-, 105 L-inference, 122, 140 MFN-, 105 L-map, 505 m-bounded Kripke model, 21 L-place, 487, 497 m-equivalent , 179 L-provable, 119 m-formula, 6 L-theorem, 119 m-reducible, 179 LV-map, 505, 507 m-shift, 117 LV-place, 497 m-valuation, 21
INDEX m.v.s. valuation, 301 matching valuations, 320, 326 maximal cluster, 48 filter in a Boolean algebra, 43 generator of a D-formula, 103 predicate extension, 151 world, 48 Maximality theorem, 463 McKinsey formula, 5 McKinsey property, 23, 502 measure of equality, 294 measure of existence, 294 metaframe, 388, 389 i-functorial, 432 i(')-sound, 423 s-functorial, 430 s'-functorial, 430 QS4, 418 S 4 valuation, 422 cone, 467 coserial, 469 disjoint sum (union), 468 functorial, 405 individual of, 389 intuitionistic, 418, 434 intuitionistic valuation in, 469 intuitionistic with equality, 434 intuitionistic S 4 valuation, 422 intuitionistic(=)-sound, 423 level of, 389 m(')-sound, 395 modal, 404 subsubitem model, 389 modally sound, 395 with equality, 395 model forcing in, 390 restriction of, 467 monotonic(=), 429 of a preset associated, 395 permutable, 397 possible world of, 389 propositionally intuitionistic, 418 quasi-functorial, 432 semantics, 197
semi functorial with equality, 430 semi-functorial, 430 true in an intuitionistic model, 423 valuation in, 389 weak intuitionistic, 434 weak intuitionistic with equality, 434 weakly functorial, 397 modal C-bundle, 386 modal (propositional) formulas, 3 modal algebra of a general Kripke frame, 38 modal algebra of the frame, 19 modal counterpart, 37 modal degree, 81 modal formula pseudotransitive, 53 true in a simplicia1 model, 477 valid in a simplicia1 frame, 477 strongly valid in a metafame, 394 true in a metaframe model, 394 valid in a metaframe, 394 modal logic, 21 V-canonical, 490 complete, 217 determined over a class of propositional frames, 218 determined over a propositional frame, 218 neighbourhood complete, 77 of a neighbourhood frame, 77 determined, 21 of a class, 16 of a class of frames, 217 of a Kripke frame, 217 of a presheaf, 332 of the algebra, 16 of width 5 n, 47 without equality, 489 modal predicate formula, 81 A, 81 modal predicate logic strongly Kripke complete, 252 strongly Kripke sheaf complete, 252 modal subframe formula, 59
modal theory, 210 L-complete, 483 L-consistent, 483 satisfiable, 252 modal valuation, 205, 389 in a C-bundle, 386 modal valued structure, 294 modally sound metaframe, 395 modally transformable simplicial frame, 477 model, 193, 194, 241 modal over a system of domains, 339 rooted, 224 simplicial, 475 over a C-bundle, 386 structures, 194 monic H.v.s., 342 monic m.v.s., 342 monic presheaf, 342 mono-Kripke semantics, 343 mono-neighbourhood semantics, 343 monotonic(=) metaframe, 429 monotonicity, 27 morphism, 243 regular, 327 weak, 328 of Kripke frames, 27 of predicate Kripke frames, 219 of predicate Kripke models, 219 of presheaves, 329 multiple subordination frame, 516
neighbourhood frame, 76 with equality, 318 neighbourhood model, 76 true in, 76 neighbourhood semantics, 11, 196 with constant domains, 343 new variable, 82 Nijtherian poset, 23 nonbranching, 23 normal propositional N-modal logic, 5 objects, 243 occurrence, 83 bound, 82 coreferent , 83 correlated, 83 free, 82 strongly bound, 82 of a letter, 82 of a word, 82 R-predicate, 296 a-valued set, 293 R-valued structure, 293 one-way PTC logics, 53 open, 13 ordered assignment, 389 origin, 243
p'-morphism strong, 324 p'-morphism of H.v.s., 319 p'-morphism of predicate Kripke frames, 219 p'-morphism of presheaves, 336 N-modal predicate logic, 119 p(')-embedding of H.v.s., 319 with equality, 120 N-modal (propositional) Kripke frame, p(')-embedding of m.v.s., 326 p(')-equivalence, 319 19 N-modal (propositional) substitution, p(')-equivalence of m.v.s., 326 4 p(')-morphism of m.v.s., 326 p-embedding of presheaves, 336 N-modal algebra, 13, 16 N-modal first-order language, 81 p-morphism strong, 324 N-modal propositional theory, 6 of H.v.s., 319 n-tuple domain in a simplicial frame, of Kripke frames, 27 475 of Kripke models, 27 natural LV-model, 507 negation, 11 of predicate Kripke frames, 219
INDEX of predicate Kripke models, 219 of presheaves, 336 p.0. tree, 55 parameter essential for formula substitution, 114 parameters standard list, 82 (free variables), 82 of a complex formula substitution, 109 of a substitution, 105 parsing lemma, 81 path, 25, 225 pattern, 15, 209 of S4-m.v.s., 316 pattern of a Kripke sheaf model, 247 permutable metaframe, 397 piecewise directed, 23 piecewise linear, 23 points, 19, 76 positive modal formula, 47 possible points, 339 possible world, 19, 76, 339 of a preset, 375 of a metaframe, 388 pre-morphism of presheaves, 336 precategory, 374 associated, 377 of individual, 375 predicate extension, 182 formula, 211 Kripke frame, 195 L-map, 505 S4-based, 207 countable, 284 directed, 516 intuitionistic, 207 morphism over a propositional frame, 220 with equality, 235 Kripke model, 206 L-map, 505 letter, 81 logic R-elementary, 283
A-elementary, 283 propositional part, 121 superintuitionistic, 12 Scott, 296 strict, 296 true in a structure, 310 prefunctor, 375 etale, 375 preset, 375 level of, 378 modal valuation in, 377 preset formula strongly valid, 378 true, 378 preset model formula, 378 valid, 378 presheaf constant, 341 constant domain with, 341 disjoint, 329 domain of, 328 extent of an individual in, 330 individual of, 328 inhabited, 328 isomorphism of, 329 monic, 342 morphism of, 329 restriction of an individual in, 330 section of, 328 total domain of, 328 over a locale, 328 presheaves p'-morphism of, 336 p-embedding of, 336 p-morphism of, 336 pre-morphism of, 336 proper, 498 filter in a Boolean algebra, 43 prime filter in a Heyting algebra, 43 transformation, 86 proposition letter, 3, 81 propositional formula closed, 52 constant, 52 propositional frame
612 co-serial, 469 weakly reflexive, 469 weakly transitive, 473 propositional Kripke frame directed, 271, 516 weakly directed, 271 propositional logic quantified version of, 121 tabular, 68 propositional replacement rule, 8 propositionally intuitionistic metaframe, 418 pseudotransitive closure of a propositional Kripke frame, 55 pseudotransitive modal formula, 53 one-way, 53 PTC-formula, 53 PTC-logic, 53 one-way, 53 pure equality sentence, 259 quasi-A-elementary logic, 45 quasi-canonical frame, 499 Kripke, 493 quasi-canonical model, 493, 499 quasi-elementar y logic, 45 quasi-lift property, 36 quasi-maximal world, 48 quasi-morphism, 36 quasi-p-morphism, 36 R-elementary class, 45 R-elementary logic, 45 r-persistent set, 44 recursively axiomatisable, 5 reducible, 57 reference structure, 83 referent, 83 refined general frame, 42 refutable, 21 regular morphism, 327 relation counterpart, 196 inheritance, 196 selective, 492
INDEX transition, 345 relational (Kripke) semantics, 11 reliable submodel, 224 replacement rule, 125 representation theorem for modal metaframes, 420 restriction of a morphism, 224 restriction of a preset, 385 rooted frame, 24, 224, 241 rooted model, 224 Kripke, 24 rule Bernays, 124, 128 variable substitution, 124 Scomplete A-logic, 186 Sahlqvist formula, 47 Sahlqvist theorem, 47 satisfiable, 21 intuitionistic theory, 252 modal theory, 252 scheme, 83 Scott predicate, 296 selective intuitionistic L-map, 506 selective partial ordering, 498 selective relation, 492 semantics, 185 basic general algebraic, 317 functor, 197 Kripke frame, 218 metaframe, 197 neighbourhood, 196 topological, 196 semi-functorial metaframe, 430 sentence, 82 set of, 211 separates, 186 set A-perfect, 148 Av-perfect, 148 small, 487 universal, 58 sheaf, 338 C-sheaf, 247 simple formula substitution, 105 simple substitution, 4
INDEX simplicial frame, 474 T-identical, 475 i-natural, 480 ~('1-transformable, 478 is-transformable, 478 lower interpretation of equality, 480 m-natural, 479, 480 modally s-transformable, 478 modally transformable, 477 natural semantics of, 479, 480 upper interpretation of equality, 480 valuation in, 475 simplicial model, 475 skeleton of a Kripke quasi-sheaf, 372 skeleton of a transitive Kripke frame, 30 Skolem function, 512 slice, 71 small set, 487 small subtree, 511 Soundness lemma, 16, 21 stable equality, 179 stable set, 24 stall, 338 standard classical semantics, 194 standard list of parameters, 82 standard subtree, 50 standard tree, 50 standard universal closure, 82 strict depth, 48 strict path, 48 strict predicate, 296 strict substitution, 105, 109 instance, 108 strict valuation, 301 strong (p-)morphism of predicate Kripke frames with equality, 239 strong isomorphism of predicate Kripke frames with equality, 239 strong p'-morphism, 324 strong p-morphism, 324 strong validity, 197
strongly bound occurrence, 82 strongly congruent, 89 strongly standard tree, 56 structure flabby, 296 keen, 296 sub-L-equivalent, 147 sub-L-implies, 147 subframe, 23 logic, 57 of a predicate Kripke frame, 224 submetaframe, 467 generated, 467 submodel reliable, 224 1of a predicate Kripke frame, 224 subordination frame, 516 multiple, 516 with greatest cluster, 516 subordination L-map, 516 subreducible, 57 substitution closure, 116 universal, 116 instance, 4, 108, 109 strict, 108 dummy, 86 interior, 355 property, 194 strict, 105, 109 variable, 86 subtree small, 511 successive frame, 49 successor, 49 superintuitionistic fragment, 37 superintuitionistic logic determined over a class of propositional frames, 218 determined over a propositional frame, 218 superintuitionistic logic of a C-set, 380 superintuitionistic logic of a Kripke frame, 217 superintuitionistic logic of a presheaf, 332
INDEX superintuitionistic predicate logic, 120 universal tree V-canonical, 500 1-modal, 49 with equality, 120 N-modal, 49 superintuitonistic logic of the unordered assignment, 400 algebra, 16 unravelling, 50 system of domains, 205, 339 valid, 15, 16, 19, 21 tabular propositional logic, 68 a t a world, 21 target, 243 in a frame, 21 Tarski-Jonsson theorem, 43 in a general Kripke frame, 38, 39 temporalisation, 25 intuitionitic formula, 33 theorem of a logic, 120 validity theory, 150, 193 relation, 185 (L, S)-Henkin, 485 strong, 197 L-Henkin, 485 valuation, 19 tightness, 42 classical, 339 topo-Boolean algebra, 13 global, 206 topological frame with equality, 318 in a general Kripke frame, 38 topological semantics, 196, 318 in a metaframe, 389 total domain of a presheaf, 328 local, 206 transformation matching, 320, 326 proper, 86 modal, 205 transition map, 244 modal in a system of domains, 339 transition relations, 345 strict, 301 transitive trees, 55 variable tree, 49 new, 82 branch S4, 56 renaming, 86 renaming in a formula, 86 effuse, 57 levelwise uniform, 539 substitution, 86 trivial cluster, 30 substitution rule, 124 trivial frames, 36 transformation, 86 true, 15, 16 weak law of the excluded middle, 7 at the world, 20 weak morphism, 328 in a model, 21 weak subframe, 24 truth, 193 weak submodel, 24 values, 293 weakly functorial metaframe, 397 -preservation, 33 weakly reflexive propostional frame, 469 ZA-frames, 185 weakly transitive propositional frame, unique lift property, 255 473 universal closure, 82 width, 47 standard, 82 width of an intermediate logic, 48 universal logic, 58 Wolter theorem, 58 universal set, 58 word universal strict Horn clause, 53 occurrence of, 82 universal substitution closure, 116 world
INDEX inner, 62, 64 component of a morphism, 219 Zakharyaschev theorem, 60, 62
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