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~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
< *
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 &'.
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.
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, 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, 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 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).
<
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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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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122
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.
>
CHAPTER 2. BASIC PREDICATE LOGIC
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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130
(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.
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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 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
