Displaying Modal Logic
Heinr.ich Wansing
I
Kll '" Ill •\l hl> '-"" 1•1 'hi " ' " " ' - --
A C.I.P. Catalogue record for this book is available from the Library of Congress.
CONTENTS
CONTENTS
V
ISBN 0-7923-5205-X
PREFACE CHAPTER ONE Published by Kluwer Academic Publishers, P.O. Box, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
1.1 1.2 1.3 1.4 1.5 1.6 1. 7
I
INTRODUCTION
The problem of Gentzenizing modal logic Standard sequent systems for normal modal logics Rules as meaning assignments Uniqueness Modularity and the Dosen Principle Subformula property, cut, and analytic cut Generality
CHAPTER TWO
2.1 2.2 2.3 2.4 2.5 2.6
IX
I
SEQUENTS GENERALIZED
Modal signs Higher-level sequent systems Higher-dimensional sequent systems Higher-arity sequent systems Hypersequents Natural deduction systems
Printed on acid-free paper
CHAPTER THREE
3.1 3.2 3.3 3.4 3.5 All Rights Reserved © 1998 K1uwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Printed in the Netherlands.
I
DISPLAY LOGIC
Gentzen terms Residuation The Display Theorem Introduction rules Completeness
1
2 4 7 10 10 11 13
15 15 16 17 18 22 24 27 27
30 34 37
43
I
CHAPTER FOUR PROPERLY DISPLAYABLE LOGICS, DISPLAYABLE LOGICS AND STRONG CUT-ELIMINATION
4.1 4.2 4.3 4.4
Properly displayable logics A case distinction and primitive reductions Strong normalization Displayable logics V
47 47 48
51 55
CONTENTS
VI
4.5 4.6
CONTENTS
Characterization of the properly displayable logics Scope of the method
57 62
CHAPTER FIVE I A PROOF-THEORETIC PROOF OF FUNCTIONAL COMPLETENESS FOR MANY MODAL AND TENSE LOGICS 5.1 The problem of functional completeness 5.2 Proof-theoretic semantics 5.3 Functional completeness for Kt and K
65 65 67 71
CHAPTER SIX I MODAL TABLEAUX BASED ON RESIDUATION 6.1 The modal display system DKf 6.2 A display calculus for PDL6.3 Reduction to a set of clauses 6.4 Decidability and completeness 6.5 In the absence of purity
75 75 77 79 82 85
CHAPTER SEVEN I STRONG CUT-ELIMINATION AND LABELLED MODAL TABLEAUX 7.1 The tableau calculus TQS5 7.2 Primitive reductions 7.3 Strong normalization 7.4 Extensions of quantified K 7.5 Equivalence with Fitting's tableaux
87 88 89 94 98 100
CHAPTER EIGHT I TARSKIAN STRUCTURED CONSEQUENCE RELATIONS AND FUNCTIONAL COMPLETENESS 8.1 Preliminaries 8.2 Positive logics 8.3 The higher-level sequent calculus 8.4 Positive proof-theoretic semantics 8.5 Functional completeness for + 8.6 Negation as refutation 8. 7 Constructive logics 8.8 The higher-level sequent calculus a[ 8.9 Constructive proof-theoretic semantics 8.10 Functional completeness for
103 103 105 107 108 112 115 115 119 120 122
at
rv
rvc
CHAPTER NINE I CONSTRUCTIVE NEGATION AND THE MODAL LOGIC OF CONSISTENCY 127 9.1 Disproofs and contrariety 128 9.2 Negation as falsity 136
9.3 9.4 9.5 9.6 9. 7
Negation as inconsistency The relation between negation as falsity and negation as inconsistency Semantics-based nonmonotonic reasoning Choice of parameters Modal logic of consistency over N 4
vii 139 142 144 150 154
CHAPTER TEN I DISPLAYING AS TEMPORALIZING 10.1 Subintuitionistic logics 10.2 Sequent systems for subintuitionistic logics 10.3 Soundness and completeness of DK(o-) 10.4 The connection with temporalization 10.5 Extensions of K(o-) and Kt(K(o-))' 10.6 Proof of strong completeness
155 155 160 161 164 165 167
CHAPTER ELEVEN I TRANSLATION OF HYPERSEQUENTS INTO DISPLAY SEQUENTS 11.1 Hypersequential calculi 11.2 Display calculi 11.3 Mapping hypersequents into display sequents 11.4 Discussion 11.5 Other translations into DL
171 171 174 184 186 187
CHAPTER TWELVE I PREDICATE LOGICS ON DISPLAY 12.1 A sequent calculus for KFOL 12.2 Display of predicate logics 12.3 A route from KFOL to FOL 12.4 Another route to FOL 12.5 Strong cut-elimination 12.6 The Barcan formula 12.7 Remaining proofs
189 191 193 195 201 204 205 207
CHAPTER THIRTEEN I APPENDIX 13.1 DL and Fitch-style natural deduction 13.2 A new axiomatization of Kt 13.3 N4 redisplayed 13.4 Future work
211 211 226 229 234
BIBLIOGRAPHY
235
INDEX
247
PREFACE
The present monograph is a slightly revised version of my Habilitationsschrift Proof-theoretic Aspects of Intensional and Non-Classical Logics, successfully defended at Leipzig University, November 1997. It collects work on proof systems for modal and constructive logics I have done over the last few years. The main concern is display logic, a certain refinement of Gentzen's sequent calculus developed by Nuel D. Belnap. This book is far from offering a comprehensive presentation of generalized sequent systems for modal logics broadly conceived. The proof-theory of non-classical logics is a rapidly developing field, and even the generalizations of the ordinary notion of sequent listed in Chapter 1 can hardly be presented in great detail within a single volume. In addition to further investigating the various approaches toward generalized Gentzen systems, it is important to compare them and to discuss their relative advantages and disadvantages. An initial attempt at bringing together work on different kinds of proof systems for modal logics has been made in [188]. Another step in the same direction is [196]. Since Chapter 1 contains introductory considerations and, moreover, every remaining chapter begins with some surveying or summarizing remarks, in this preface I shall only emphasize a relation to philosophy that is important to me, register the sources of papers that have entered this book in some form or another, and acknowledge advice and support. It may be difficult to immediately perceive a relation to a major philosophical problem in the present work, which is formal, although the proof-theoretic detail sometimes overemphasizes the technical aspects. Nevertheless, this book is not only meant as a contribution to logic, but also to philosophy. Due to the diversity of the philosophical subdisciplines, every general characterization of philosophy is bound to be quite abstract. According to some philosophers, the very aim of philosophy is to distinguish meaningful discourse from meaningless discourse about all kinds of problems and phenomena. More concretely, the aim of logic is often said to be separating the good arguments from the bad ones. From this perspective, logic plays a fundamental role both in philosophy and in scientific inquiry in general, for it is the meaning of the logical operations that is central to determining lX
PREFACE
PREFACE
the sound arguments and the meaningfulness of large parts of discourse. Logical semantics has many faces. In addition to the prevailing paradigm, namely 'realistic', model-theoretic semantics, there are also, for instance, algebraic semantics, game-theoretic semantics, dynamic semantics, and proof-theoretic semantics. In proof-theoretic semantics the meaning of the logical operations is specified in terms of general introduction schemata. They provide a defining framework in the sense that every permissible introduction rule is (or is interchangeable with) an instantiation of such a schema. Different generalizations of the standard notion of sequent suggest different proof-theoretic semantics, and in this book I give, among other things, a proof-theoretic characterization of the tense logical operations and prove other functional completeness theorems. I hope the philosophically inclined reader will regard these discussions and results as they are intended, namely as a contribution to the philosophy of meaning. In particular, it is shown that the idea of meaning-as-use is formally feasible also for modal and tense logics. A general discussion of the proof-theoretic approach to meaning, however, is beyond the scope of this book. Various papers dissolved into parts of this work or became chapters of it in a slightly or more substantially revised form. In particular, the notation and terminology of these papers has also been harmonized, the internal coherence and interdependencies have been highlighted, and new material has been included. To be precise, this ealier work has been distributed as follows:
tured consequence relations, semantics-ba.c;ed nonmonotonic reasoning, temporalization, and negation presented here takes as its starting point concepts developed by him. Moreover, my thinking about logic has been greatly influenced by Johan van Benthem's work. The lecture on first-order logic as modal logic he gave at Leipzig University in 1994 triggered the formulation of first-order display logic presented in Chapter 12. I would like to thank Johan van Benthem and Grigori Mints for inviting me to Stanford in 1994, Dov Gabbay for inviting me to London in 1996, Valentin Shehtman for inviting me to Moscow in 1996, and Makoto Kanazawa for inviting me to Chiba, also in 1996. These visits helped me a lot in thinking about display logic. As to the investigation of display logic, I am also greatly indebted to Rajeev Gore, Marcus Kracht and Greg Restall. I have learnt a lot from their beautiful work, and in particular Marcus Kracht commented upon preliminary drafts of various chapters. I wish to thank Siegfried Gottwald for providing a stimulating working environment at the lnstitue of Logic and Philosophy of Science at Leipzig University and for his supportive attitude toward my proof-theoretic interests. Moreover, I gratefully acknowledge useful comments on various parts of the present book by Seiki Akama, Tijn Borghuis, Kosta Dosen, Jan Jaspars, Grigori Mints, David Pearce, Maarten de Rijke, Valent in Shehtman, and Y de Venema. Of course, I take full responsibility for all remaining defects and idiosyncrasy. Finally, I wish to thank Petra, Kasimir, Friederike, Agnes Dakota, and Carlotta for the most enjoyable family life I can imagine. I gratefully dedicate this book to the memory of my mother, Agnes Wansing.
X
[181] [187] [182] [184], [192] [193] [189]
Chapters 1, 2, and 3 Chapter 5 Chapter 7 Chapter 9 Chapter 11 Section 13.1
[185] [191] [183] [190] [194] [186]
Chapter 4 Chapter 6 Chapter 8 Chapter 10 Chapter 12 Section 13.2
I would like to acknowledge inspiration from and discussions with various colleagues and friends. First of all, I wish to thank Nuel Belnap for having developed display logic and for his advice and encouragement. In particular, I am grateful to him for inviting me to Pittsburgh in 1994, for giving me the opportunity to present the material of Chapter 5 at the Philosophy Department of Pittsburgh University, for discussing strong cut-elimination with me in detail, and for various conversations on first-order display logic when he was a visiting Leibniz Professor at the Center for Advanced Studies of Leipzig University in 1996. Another source of inspiration has been Dov Gabbay. The work on struc-
Xl
Leipzig, March 1998 Heinrich Wansing
CHAPTER 1
INTRODUCTION
Hilbert-style systems are easy to define . . . but they are difficult to use. Gentzen systems reverse this situation by emphasizing the importance of inference rules .... J. Barwise [14, p. 37]
This monograph deals with various selected aspects of proof systems for intensional logics, that is, modal logics in the broad sense, including, for instance, also subintuitionistic and constructive propositional logics with strong negation. But also certain variants of classical firstorder logic will be considered from a modal perspective. The principal formalism to be developed and investigated is a certain refinement of Gentzen's sequent calculus, namely Nuel Belnap's display logic, DL, [16]. However, Dov Gabbay's [67] related general framework of structured consequence relations, higher-level sequent systems for structured consequence relations, higher-arity display sequents, and a labelled modal tableau calculus for (constant domain) first-order S5 will also be examined. The common idea behind these proof-theoretic approaches is that many logical connectives can be introduced by natural operations on structures not as meagre as finite sets. Wheras Belnap considers terms in a natural extension of the structural language of Gentzen's sequent calculus, Gabbay deals with complex datastructures, which essentially are assignments of formulas to individuals of first-order structures. These datastructures exemplify the general idea of Gabbay's theory of labelled deductive systems, LDSs [68]. More specific and familiar examples of LDSs are provided by modal tableaux systems for labelled formulas and by typed lambda calculi. In addition to 'Gentzenizing' modal and non-classical logics as a thematic thread, there will be a pattern of recurring methods, techniques, and ideas. Notably, functional completeness and cut-elimination will be dealt with more than once. We ~hall prove cut-elimination for certain constructive propositional logics f-vc associated with a Tarskitype structured consequence relation (Chapter 8), extend Belnap's general cut-elimination theorem to obtain strong cut-elimination for DL (Chapter 4), and apply this method to a tableaux calculus for the modal predicate logic QS5 (Chapter 7). Functional completeness results will be obtained for f-vc (Chapter 8) and every displayable propositional normal modal and tense logic (Chapter 5). The latter result
r-
1
3
CHAPTER 1
GENTZENIZING MODAL LOGIC
for the first time extends the proof-theoretic characterization of logical constants to modal and tense logics. The meaning of the logical operations thus emerges as a major concern of the present investigation. The utility of the ordinary notion of sequent in classical logic to a large extent rests on the invertibility of introduction rules for the Boolean operations. In Chapter 6 it is shown that invertible display introduction rules for the modal operators are available for logics of functional accessibility relations. Moreover, it will turn out that the theme of negation as refutation, which is used to motivate is relevant to the modal logic of consistency (Chapter 9). Modal logics of consistency arc used as monotonic base systems in semantics-based nonmonotonic reasoning. It will be shown that modal consistency logics extending certain constructive and subintuitionistic logics avoid counterintuitive features of semantics-based nonmonotonic reasoning based on intuitionistic logic. Subintuitionistic logics will then be presented as cut-free display calculi in Chapter 10. Display calculi for modal logics of consistency based on constructive logics with strong negation are studied in [195]. Another important kind of generalized sequent systems are hypersequential calculi. In Chapter 11 it is shown that hypersequents can be simulated by display sequents. As a by-product this embedding gives a cut-free display sequent calculus for Lukasiewicz's three-valued logic. Chapter 12 is devoted to quantification in DL. The analogy between the quantifier prefixes :lx and Vx and the modal operators 0 and D is taken seriously, and the standard introduction rules for 3x and Vx are replaced by modal counterparts leading to a Gentzenization of several variants of classical first-order logic. Fitch-style natural deduction and higher-arity display sequents are considered in Chapter 13. The metalogic of this monograph is classical logic, not because classical logic is meant to enjoy a designated status in the universe of logical systems, but just because it is a famliar background for dealing with and comparing non-classical and intensional logics. Moreover, if confusion is unlikely, we shall not pay much attention to the distinction between using and mentioning a symbol.
Early this century modal logic started as an entirely syntactic enterprise. Its proof theory, however, failed to experience a development comparable to the tremendous rise of modal model theory after the emergence of possible worlds semantics around 1960. In the 1980s various distinguished modal logicians explicitly noticed this undesirable discrepancy between the semantic and the proof-theoretic development of modal logic. Krister Segerberg [29] observed that "Gentzen methods have never really flourished in modal logic, but some work has been done, mostly on sequent formulations". Though proof theory cannot be identified with proof theory in Gentzen-style, axiomatic proof systems, despite their obvious advantages, namely modularity and succinctness, are neither particularly well-suited for actually carrying out and automating deductions, nor for theoretical investigations into the nature of proofs or the proof-theoretic meaning of logical constants. The above quote from Serebriannikov suggests that Gentzen's ideas can be used to develop decent proof systems for modal logic, and in fact, from the 1980s onwards, various generalizations and modifications of standard sequent-style (and natural deduction) proof systems have been devised or taken up again and applied to modal logic. Among these approaches are:
2
rvc,
labelled deductive systems [68] proof systems incorporating combinators [22], [58], [60] relational proof systems [105], [124], [125], [126], Kanger-style calculi [91], [132], [133] higher-level (alias higher-order) proof systems [39], [96] higher-arity proof systems [24], [150] higher-dimensional proof systems [108] hypersequential calculi [12], [134], multiple sequent proof systems [88], [89], and calculi that introduce new structural operations [16], [31], [148].
1.1. THE PROBLEM OF GENTZENIZING MODAL LOGIC
Gentzen's proof-theoretical methods have not yet been properly applied to modal logic. 0. Serebriannikov [154, p. 79]
The latter type of proof systems comprises the display logical treatment of modality, which is the central subject of the present investigation. Within DL we shall present a perspicious and modular sequent-style proof theory for the most important systems of propositional normal modal logic, the basic systems of propositional normal tense logic, a number of subintuitionistic and constructive logics, and variants of classical first-order logic.
4
CHAPTER 1
STANDARD SEQUENT SYSTEMS
1.2. STANDARD SEQUENT SYSTEMS FOR NORMAL MODAL LOGICS
Preparatory to later methodological considerations, in this section various standard Gentzen systems for normal propositional modal logics are reviewed. 1 This survey is, however, by no means meant to be complete. The purpose of this section rather is to give a rough idea of what has been and what can be done in order to present normal modal logics as standard sequent calculi. Such calculi are collections of rule schemata for manipulating Gentzen sequents. A Gentzen sequent is a derivability statement of the form ~ -+ r, where ~ and r are finite sets of formulas. ~ and r are called the antecedent and succedent of ~ -+ r' respectively. A sequent {A1, ... , An} -+ {B1, ... , Bj} is often written as A1, ... , An -+ B1, ... , Bj. This notation suggests conceiving of',' (the comma) as a structural connective in the language of sequents. The antecedent and the succedent of a sequent are then Gentzen terms denoting data and goal structures. Usually, the comma as a structural connective is interpreted as concatenation of premises and conclusions. Antecedent and succedent then consist of sequences of formula occurrences. If one assumes structural rules of permutation, contraction, and expansion of formula occurrences on both sides of the sequent arrow, antecedent and succedent can still be thought of as finite sets. 2 We shall use 'f-' (with and without subscripts) to denote consequence relations between finite sets of sequents and single sequents that satisfy identity, cut, and monotonicity. 3 In other words, if ~ and r are finite sets of sequents and s, s' are sequents, then { s} f- s, ~ ~ U { s'}
f- s
and
f- s ~
r u { s} f- s'
ur
f- s'
5
Matsumoto [122]: (-+ O)o (D-+ )o
o~-+
(-+ O)o (0 -+ )o
~-+
or, A f-
o~-+
or, oA
~'A -+ r f- ~' OA -+ r
r,A f-
~-+
r,OA
or, A-+<>~ f-or, <>A-+<>~-
Here 0~ = {OA I A E ~} [0~ = {OA I A E ~}].If either the rules (-+ 0) 0 and (0 -+ ) 0 or the rules (-+ 0 )o and (0 -+ )o are adjoined to (an appropriate version of) the standard sequent system LCPL for classical propositionallogic, CPL, then the result is a sequent calculus LS5 for S5. Various other modal propositionallogics can be obtained by imposing certain constraints on the structures exhibited in (-+ D)o and (0 -+ )0 , respectively. If r is empty in (-+ O)o [( 0 -+ )o], this yields a sequent calculus LS4 for S4. Ohnishi and Matsumoto also show that if (-+ O)o [(0 -+ ) 0 ] is replaced by (-+ Dh
~
[( <> -+ h
A -+ r f- <>A-+ or],
-+ A f-
0~
-+ DA
one obtains a Gentzen-system LKT for KT (= T). 4 If one just adds (-+ Oh to LCPL, this results in a sequent calculus LK for the minimal normal modal propositionallogic K (see, for instance, [100], [113], [147]). A sequent calculus LK4 for K4 can, for example, be obtained by adding to LCPL the rule
(-+ 0)2
~,0~-+
A f-
0~-+
DA
(see [147]). As is shown in [71], the pair of modal sequent rules(-+ Dh and (0 -+h ~,A-+ 0 f- D~,DA-+ 0 yields a sequent system for KD, and if (-+ D)I is modified into the rule
Standard sequent systems for the axiomatic calculi S4 (= KT4) and S5 (= KT5 = KT4B) have long been known. The following schematic sequent rules for the modal operators 0 (necessity) and 0 (possibility) go back to Curry and Feys, and have been studied by Ohnishi and 1
The notation of various authors will be adjusted to a single convention for naming sequent systems. 2 In the case of Gentzen terms in general, one also has to postulate associativity of the comma. 3 Sometimes, for instance in the context of axiomatic presentations of logical systems, we shall also use 'f-' to denote consequence relations between finite sets of formulas and individual formulas.
(-+ 0)3
~'-+A
f- D~-+ DA,
where ~' results from ~ by prefixing zero or more formulas in ~ by 0, one obtains a sequent calculus for KD4. Shvarts [156] gives a sequent calculus formulation of KD45 by supplementing LCPL with the following rule for 0: [DJ
0~1, ~2-+
or1, r2 f- o~1, 0~2-+ or1, or2,
4 It has been observed by Kripke [93] that the equivalences between DA and --.O--.A as well as OA and --.0--.A cannot be proved by means of Ohnishi's and Matsumoto's rules. In the case of 84, Kripke suggests remedying this by using sequent rules which exhibit both D and 0.
6
7
CHAPTER 1
RULES AS MEANING ASSIGNMENTS
where r2 contains at most one formula. If in addition rl and r2 are required to be non-empty, this results in a sequent system for K45. Avron [6] (see also [155]) presents a sequent calculus LS4Grz for S4Grz (= KGrz). He replaces the rule (--+ 0) 0 in Ohnishi's and Matumoto's sequent calculus for S4 by the rule
In sum, we may say that many normal propositional modal logics can be presented as standard sequent calculi, and that in some cases suitable constraints on the structures exhibited in the statement of the sequent rules for 0 and 0 allow for a number of variations. Obviously, this approach fails to be modular: in general it is not the case that a single axiom schema is captured by a single sequent rule (or a finite set of such rules). In the following section we shall point out several other disadvantages of standard Gentzen systems when applied to modal logic. These disadvantages will become evident when instead of considering the existing standard format we shall concentrate on a number of desirable properties.
(--+ 0)4
D(A
:J
DA), D~ --+ A f-
D~
--+ OA.
In [162], Takano defines sequent calculi LKB, LKTB, LKDB, and LK4B for KB, KTB (=B), KDB, and K4B. The systems LKB and LK4B are obtained from LCPL by including the rules and
(--+ O)n (--+ 0)4BE
r--+ 08, A r- or--+ 8, OA r, Dr--+ 08, 0~, A f-Or--+ 08, ~' DA
respectively. LKTB and LKDB result from LKB by adjoining (D --+ )o and
(o--+ )D
r --+
o~
r- or--+ ~
respectively. Standard sequent systems for several other modal logics can be found in [7 4] and [200]. The sequent calculus for S4.3 ( = S4 + D(DA :J DB) V O(DB :J DA)) in [200] results from LS4 by the addition of the axiomatic sequent D(A V DB), D(DA V B) --+ DA, DB. Shimura [155] obtains a cut-free sequent system LS4.3 by adding to LCPL the rules (D --+ )o and
(--+ D)5 or--+
(D~)
\{DAd ... or--+
(D~)
\ {DAn} t-or--+
o~,
where~=
{A1, ... ,An}· Various normal tense logics have also been presented as ordinary sequent calculi. Nishimura [121], for example, defines sequent calculi LKt and LK4t for the minimal normal tense logic Kt and the tense-logical counterpart K4t of K4. The system LKt contains the following introduction rules for forward-looking necessity [F] ('always in the future', alias D) and backward-looking necessity [P] ('always in the past'):·5
(--+[F]) (--+[P])
r--+ A, [P]~ f- [F]r--+ [F]A, ~ r--+ A, [F]~ f- [P]r--+ [P]A, ~'
where [F]~ = {[F]A I A E ~} ([P]~ = {[P]A I A E ~}).In K4t, these rules are replaced by the following pair of rules:
(--+ [F])4 (--+ [P])4 5
[F]r, r--+ A, [P]r, r--+ A,
[P]~,
[P]E f- (F]r--+ [F]A, ~' [P]E [F]~, [F]E f- [P]r--+ [P]A, ~' [F]E
Nishimura allows infinite sets in antecedent and succedent position. It is proved, however, that if a sequent r --+ ~ is provable, then there are finite SetS f 1 ~ f and ~I ~ ~ SUCh that the Sequent f 1 --+ ~I is provable.
1.3. RULES AS MEANING ASSIGNMENTS
Gentzen-style proof theory is usually associated with a certain 'antirealistic' philosophy of meaning. The idea is that the schematic introduction rules for an n-ary connective f, together with a set of structural assumptions, specify the meaning of f. This idea is usually traced back to a passage on natural deduction from Gentzen's Investigations into Logical Deduction [70, p. 80]. [I]ntroductions represent, as it were, the 'definitions' of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. Moreover, the idea of a proof-theoretic semantics 6 is normally thought of as exemplifying the Wittgensteinian slogan 'meaning is use'. In the Philosophical Investigations [197], Wittgenstein writes in §43 that [f]or a large class of cases~though not for all~in which we employ the word "meaning" it can be defined thus: the meaning of a word is its use in the language. In order to figure as the conceptual basis of a semantic theory, the slogan has to be understood as 'meaning is correct use', for otherwise meaning would depend on the factual linguistic behaviour of certain language users, and hence meaning would be a pragmatic rather than 6 Although 'proof-theoretic semantics' is a rather natural term for referring to introduction rules (in sequent calculi) as meaning assignments, it seems that this term was only coined in the early 1980s by Peter Schroeder-Heister, see his abstract [152]. In 1968, von Kutschera [96] used the term 'Gentzen semantics' and carefully explained the idea of a proof-theoretic characterization oflogical operations.
8
CHAPTER 1
RULES AS MEANING ASSIGNMENTS
a genuinely semantic and speaker-independent notion. Quite evidently, Wittgenstein had this normative conception in mind as becomes clear ' ' for instance, from §558 of the Philosophical Investigations:
Table I. Properties of standard sequent systems.
I sep. I weak sym. I sym. I weak expl. I expl. I (inter) LK LKT LKB LKDB LKTB LK4 LK4B LK45 LKD LKD4 LKD45 LS4 LS4.3 LS4Grz LS5 LKt LK4t
What does it mean to say that the "is" in "The rose is red" has a different meaning from the "is" in "twice two is four"? ... [T]he rule which shews that the word "is" has different meanings in these sentences is the one allowing us to replace the word "is" in the second sentence by the sign of equality, and forbidding this substitution in the first sentence. (My emphasis.) 7 The idea of meaning as correct use has certain consequences for the format of introduction rules in sequent calculi. First of all, if one wants to avoid a (partially) holistic account of the meaning of the logical operations, the meaning assignment should not make the meaning of an operation f dependent on the meaning of other connectives. That is to say, the sequent rules for f should give a purely structural account off's meaning in the sense that they should not exhibit any connective other than f. This property may be called separation (cf. [201]). Moreover, the rules for f should be weakly symmetrical; every rule should either belong to a set of rules (f -+) which introduce f into premises (that is, on the left side of-+ in the conclusion sequent) or to a set of rules (-+ f) which introduce f into conclusions (that is, on the right side of-+ in the conclusion sequent). The sequent rules for f may then be called symmetrical, if they are weakly symmetrical and both (-+ f) and (f -+) are non-empty. The sequent rules for f will be called weakly explicit, if the rules (-+ f) and (f -+) exhibit f in their conclusion sequents only, and they will be called explicit, if in addition to being weakly explicit, the rules in (-+ f) resp. (f -+) exhibit only one occurrence off on the right resp. the left side of -+. Separation, symmetry, and explicitness of the rules imply that in a sequent calculus for a given logic A, every connective that is explicitly definable in A also has separate, symmetrical, and explicit introduction rules. 8 Therefore one would like 7
Note that reference to the fact that Wittgenstein is writing about the use of words in language-games (e.g. in §41 of [197]) fails to substantiate the claim that he had a normative understanding of meaning as use. According to Wittgenstein, the notion of a game is a family-resemblance concept, and therefore, by definition, there is no single defining property that all objects falling under this concept share in common. There are thus Wittgensteinian games without any rules. 8 These rules can be found by decomposition of the defined connective. One has to assume that A has a compositional semantics, or, in syntactic terms, that the deductive role of f(A 1 , ... , An) only depends on the deductive relationships between A 1 , ... , An.
9
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./ ./ ./ ./ ./ ./ ./ ./ ./ ./
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./ ./ ./ ./ --
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-
-
to have rules for both 0 and 0 as primitives. With CPL as our base logic, these rules should allow one to prove OA -+ -.O-.A, -.O-.A -+ OA, OA -+ -.0-.A, and -.0-.A -+ OA. The latter property may be called (inter). It can easily be verified that each of the standard rule systems presented in the previous section fails to satisfy some of the more philosophical requirements introduced so far, see Table 1. 9 But there are further properties that may be required of decent proof systems for modal logics. One of these properties is the unique characterization of the logical operations; another important property is the eliminability of cut. As we shall see in Chapters 5 and 8, versions of cut-elimination play an important role in the proof-theoretic semantics of the logical operations.
9
The rule [DJ of LK45 and LKD45, the rule (-+ D)n, and the rules for (F) and (P) in LKt may be considered either as left or right introducton rules.
10
11
CHAPTER 1
SUBFORMULA PROPERTY, CUT, AND ANALYTIC CUT
1.4. UNIQUENESS
[T]he rules for the logical operations are never changed: all changes are made in the structural rules. [41, p. 352jl 0
Suppose that A is a logical system with a syntactic presentation S in which ~he connec*tive f occurs. Let S* be the result of rewriting f everywhere m S as f , and let AA* be the system presented by the union SS* of S a~d S*. in the language with both f and j*. Let At denote a formula (m this language) that contains a certain occurrence of j, and let A t• denote the result of replacing this occurrence of f in A by f*. The connectives f and f* are said to be uniquely characterized in AA* ~ff for every formula A 1 in the language of AA*, A 1 is provable in SS* I~ Ar is provable in SS*. Unique characterization of the logical operati~ns of a sys~em A can be regarded as a desirable property of a synt~ctic. pr~sentatwn of A. Dosen [39] has proved that unique charactenzatwn IS a non-trivial property and that the connectives in his higher: level systems S4p/ D and S5p/ D for S4 and S5, respectively, are umquely characterized.
We shall take as our basic modal system the minimal normal modal propositionallogic K, that is to say the introduction rules for D and 0 should be such that, together with appropriate structural assumptions, their addition to a suitable sequent calculus for CPL results in a sequent system for K. 11 Sequent calculus presentations of various axiomatic extensions of K should then be obtainable by adding suitable structural inference rules to that basic sequent calculus. The result would be a modular proof-theoretic framework. It would, of course, be nice, if quite a few important systems of normal modal logic turn out to be Gentzenizable in this way. Of interest then is a characterization of the systems which can be Gentzenized and those which cannot.
1.6. SUBFORMULA PROPERTY, CUT, AND ANALYTIC CUT 1.5. MODULARITY AND THE DOSEN PRINCIPLE
In contrast to the axiomatic approach, the standard sequent-style proof th~ory for _normal modal logic fails to be modular, and the very mechamsm ~ehmd ~he ~ange of known possible variations is not very clear. One mi~ht b~ mclmed to agree with Segerberg's [29, p. 30] remark (in connec_twn with natural deduction systems for modal logic) that "only exceptiOnal systems ... seem to be characterizable in terms of reasonably simple rules". Applying Segerberg's methodology, on the basis of the above sequent system LK for K "different logics would have to be characterized by special axioms. This means giving up the idea of finding characteristic rules for those systems." In addition to the absence of symmetrical and explicit introduction rules for D and 0 in ordinary sequent systems, the problem is that it is not quite clear which parameters could be systematically modifed so as to obtain characteristic sequent rules. Structural constraints like those mentioned in Section 1.2 simply do not seem to give enough systematic flexibility. What one would need, it seems, is an extension of the standard Gentzen format ~ha~ (i). conforms to the usual philosophy of meaning present in studIes mspired by Gentzen, and (ii) offers sufficient degrees of structural freedom. The proof theory one would like to have should exemplify a principle that can be traced back to Gentzen and has most emphatically been advocated by Dosen [39, 41, 43]. We therefore refer to it as Dosen's Principle:
Certainly, "Hilbert systems are not suited for the purpose of actual deductions" [29, p. 28]. In order to be computationally attractive, a sequent calculus presentation of a logical system should, however, enjoy certain well-known properties. According to Sambin and Valentini [147, p. 316], it "is usually not difficult to choose suitable [sequent] rules for each modal logic if one is content with completeness of rules. The real problem however is to find a set of rules also satisfying the subformula-property". In sequent systems with separate, symmetrical, and weakly explicit rules the redundancy of cut implies the subformula property, if these systems do not comprise completely weird structural rules like .6.., f --+ 8 f-- .6.. --+ e. In the literature (for instance, [67]) one can find various versions of the cut rule, which may or may not be interreplaceable, depending on the structural sequent rules being assumed. The essence of cut is captured by what sometimes is called 'substitutional cut':
IO This principle is motivated in [43] by the idea of logical constants as punctuation marks in the object language for certain structural features of deductions. 11 In Chapter 6, we shall consider KD as the base logic, in order to obtain an invertible left introduction rule for D. This rule will be used to define a tableau proof procedure for the modal logic of functional accessibility relations.
12
CHAPTER 1
Here A is said to be the cut-formula. In principle, it is desirable to have cut as an admissible rule, that is, as a rule which can be added or deleted without affecting the set of provable sequents. Ohnishi's and Matsumoto's LS5 does not allow cut-elimination. 12 Moreover, cutelimination fails to hold for Takano's sequent calculi for KB, KTB, KDB, and K4B and Nishimura's sequent systems for Kt and K4t. Apart from these systems, the sequent systems presented in the previous section allow cut-elimination. The sequent calculi for S5 in [112], [149], and [150], although admitting cut-elimination, do not have the subformula property. Useful as the subformula property may be, when it comes to proof search in a Gentzen-type formalism with an enriched structural language, the subformula property for the logical vocabulary need neither imply nor be of direct use for syntactic decidability proofs, cf. Section 4.6. Cut-elimination itself does not guarantee efficient proof-search, see [1], [26], so that it may be attractive to work with an 'analytic', that is, subformula property preserving cut rule, if possible. An application of cut ..6.1 ---+A, r1
..6.2, A---+ r2 1- ..6.1, ..6.2---+ r1, r2
is analytic (see [159]), if the cut-formula A is a subformula of some formula in the conclusion sequent ..6. 1, ..6. 2 ---+ r 1, r 2. Let Sub(..6.) denote the set of all subformulas of formulas in ..6.. Applications of the sequent rules
and
(---+ O)s
r---+ oe, A 1- or ---+ e, oA
(---+ 0)4BE (0 ---+ )v
r, or ---+ oe, 0..6., A 1- or---+ oe, ..6., OA r ---+ 0..6. 1- or ---+ ..6.
from Section 1.2 may be said to be analytic if 08 ~ Sub(r u {A}), 0..6. ~ Sub(Or U 08 U {A}), and 0..6. ~ Sub(r), respectively. Takano [162] shows that the cut rule in LS5, LKB, LKTB, LKDB, LK4B, LKt and LK4t can be replaced by the analytic cut rule: every proof in these sequent calculi can be transformed into a proof of the same sequent such that every application of cut (and, moreover, every application of the rules(---+ O)s, (---+ 0)4BE, and(---+ O)v) in this proof is analytic. Although admissibility of analytic cut is a welcome property, in general, unrestricted cut-elimination is to be preferred over elimination of analytic cut. Cut-elimination has great conceptual significance 12
See [123] for an example of an 55-theorem without any cut-free proof in LS5.
GENERALITY
13
and is not just a bizarre or uninteresting preoccupation of some prooftheorists. It justifies certain substitutions of data; in particular it justifies the use of previously proved formulas. In the context of a prooftheoretic interpretation of the logical operations cut-elimination also plays an important role. The elimination of principal applications of cut, i.e. applications in which the premise sequents of the cut rule both have been obtained by introducing the main connective of the cutformula, guarantee conservativity of the meaning assignment given by the introduction rules. Namely, if a formula A does not contain a certain logical operation, the introduction rules for this operation need not be used in a proof of 0 ---+ A.
1.7. GENERALITY
Another methodological aspect is generality. Does there exist a Gentzen-type proof-theoretic framework that allows one in a clear way and in accordance with the idea of a proof-theoretic semantics to present not only normal modal logics, but also relevance, intuitionistic, subintuitionistic and other substructurallogics, combinations of such systems, and subsystems of classical and intuitionistic predicate logic, obtained by generalizing the truth definition for the universal and existential quantifiers? Moreover, is there such a framework suggesting interesting and important logics that have not been investigated so far? While Gabbay's theory of labelled deductive systems is a very rich unifying theory for representing all kinds of inference mechanisms, one may wonder how much generality can be achieved in a framework that does not so heavily (or at least not so obviously) import model-theoretic semantics into proof theory by explicitly taking into account semantic parameters like sets of possible worlds or individual domains as labels of formulas or sequents. As is well-known, intuitionistic logic and David Nelson's systems of constructive logic with strong negation have decent standard Gentzenstyle presentations. If one thinks of these logics in terms of their Kripkestyle possible worlds semantics, there are certain conditions on Kripke models which one might want to give up, viz. persistence of atomic information and reflexivity as well as transitivity of the binary accessibility relation. In fact, under the well-established informational interpretation of Kripke models, giving up or relaxing all or part of these constraints appears to be rather natural (see Chapters 9 and 10). Thus the problem of Gentzenizing the resulting subintuitionistic and subconstructive logics arises, and it is not at all clear whether Gentzenization
14
CHAPTER 1
is feasible within the bounds of the standard format. Fortunately, Rajeev Gore's [75] redisplay of intuitionistic logic immediately suggests display calculi for the subintuitionistic logics. Their adequacy will be proved in Chapter 10, where, moreover, a connection is established with Finger's and Gabbay's [61] notion of 'temporalizing' an arbitrary logic. A study of display calculi for Nelson's constructive logics and extensions of these systems by a consistency operator can be found in [195]. In Chapter 11 it is shown that display sequents can simulate hypersequents, an observation that helps clarifying the relation between two major approaches in the proof theory of non-classical logics. Next, in Chapter 12 it is shown that the modal display calculus naturally induces an extension of DL to first-order logic. The generalization reflects the modal perspective on predicate logic that can be traced back to, for instance, Montague [117], is central to the theory of cylindric modal logic [5], [173], and has recently been forcefully endorsed by Johan van Benthem [20], [21]. This extension to predicate logic considerably enlarges DL's representational power and points to various hitherto unexplored predicate logics that very naturally arise within display logic. There are not many methodological considerations on proof systems for non-classical logics in the literature. Modularity and the idea expressed by the Dosen Principle are mentioned as desirable properties, for example, in [90] and [126]. Methodology has also been addressed in connection with higher-arity proof systems for modal logics [24] and the method of hypersequents [12]. We shall come back to these methodological remarks in Chapters 2 and 11.
CHAPTER 2
SEQUENTS GENERALIZED
Before display logic is introduced in Chapter 3, in th~ present chapter several other ideas of generalizing the standard notiOn of sequent are first briefly reviewed. At the moment, modal proof th:~ry is developing in many directions, see, for example, [188]. In additiOn to DL various other generalizations of the standard Gentzen format~s) have been suggested and investigated. The list of approaches dealt ~Ith below is not intended to be complete in any sense. These formahsms are either partly related, in spirit at least, to DL or otherwise just give an impression of some ways in which Gentzen's notion of sequent may be generalized. Kanger-style calculi using 'spotted for~ulas' [~1), [132] and relational proof systems for modal and non-classical logics [124], [125], [126], for example, are not considered, and Gabb~y's [68] theory of labelled deductive systems is not dealt with here either. Labelled tableaux calculi and structured consequence relations are considered in Chapters 7 and 8.
2.1. MODAL SIGNS
An extended Gentzen-type proof theory for normal modal propositional logics has been suggested by Cerrato [31], [32]. In thi~ se~ue~t calculus framework a formula A may be signed as (A) or [A] mdicatmg that A occurs "in a modal (possible or necessary) way" [31, P·. 231]. Then.~ ~re four types of sequent rules: (i) structural rules includi.ng a ~~~exivity rule and cut, (ii) 'logical rules' for the classical conne:ti~es, (I.n) mod~~ rules for the axiom schemata K, D, T, 4, 5, B, and (Iv) duahty rules· X --+ Y, [A]I- X --+ Y, OA
[A], X--+ Y 1- OA, X--+ Y (A),X--+ Y 1- OA,X--+ Y X--+ Y,[A]I- (·A),X--+ Y X --+ Y, (A) 1- [•A], X--+ Y
X --+ Y, (A) 1- X --+ Y, OA
[A], X --+ Y 1- X --+ Y, (·A) (A), X --+ Y 1- X--+ Y, [•A].
These duality rules make sure that (inter) holds and that one can prove [A] --+ OA, OA --+ [A], (A) --+ OA, and OA --+ (A). Although copyi~g OA and OA as structural elements [A] resp. (A) introduces a certam amount of flexibility, Dosen's Principle fails to be satisfied: some of 15
17
CHAPTER 2
HIGHER-DIMENSIONAL SEQUENT SYSTEMS
Cerrato's modal rules exhibit 0 or 0, like for instance the rules corresponding to 5: [A], X-+ Y f- (OA),X-+ Y
monotonicity of level 1 in Cp/ D, then this gives a higher-level sequent system for intuitionistic propositionallogic IPL. Note that (inter) holds for S4p/ D and S5p/ D. The double-line rules for 0 and O, however, do not satisfy weak symmetry and weak explicitness. As Dosen (private communication) pointed out, the upward directions of these rules can be replaced by:
16
X-+ Y, (A) f- X-+ Y, [OA].
Moreover, Cerrato proves cut-elimination only for his sequent calculus for the basic system K.
0 -+ 1 {A} f- 0 -+ 1 {OA} 2.2. HIGHER-LEVEL SEQUENT SYSTEMS
{A} -+ 1 0 f- { OA} -+ 1 0.
Dosen [39] has developed certain non-standard sequent systems for S4 and S5. In these Gentzen-style systems one is dealing with sequents of arbitrary finite level. Sequents of level 1 are like ordinary sequents, whereas sequents of level n + 1 (n > 0) have finite sets of sequents of level non both sides of the sequent arrow. Moreover, the main sequent arrow in a sequent of level n carries the superscript n, and 0 is regarded as a set of any finite level. The rules for logical operations are presented as double-line rules. A double-line rule
Whereas cut can be eliminated at levels 1 and 2, cut of all levels fails to be eliminable [39, Lemma 1]. Moreover, in Dosen's higher-level framework it is not clear how restrictions similar to the one used to obtain S4p/ D from S5p/ D would allow to capture further axiomatic systems of normal modal propositionallogic. Higher-level sequent systems have also been used to obtain functional completeness results for (substructural subsystems of) intuitionistic and constructive propositionallogics, see [8], [96], [97], [178], [180]. Higher-level sequent systems for structured consequence relations are investigated in Chapter 8.
so involving sequents so, ... , sn, denotes the rules
so
Sl
' ... ' Sn
Dosen gives the following double-line sequent rules for 0 and 0:
X+ {0 -+ 1 {A}} -+ 2 X2 + {X3 -+ 1 X4} X1 -+ 2 X2 + {X3 + {OA} -+ 1 X4}
X1
+ {{A}
-+ 1 0} -+ 2 X2
+ {X3 -+ 1 X4} -+ 1 x4 + {OA}}
x1 -+ 2 x2 + {X3 where + refers to the union of disjoint sets. If these rules are added to Dosen's higher-level sequent calculus Cp/ D for CPL, this results in the sequent system S5p/ D for S5. The sequent calculus S4p/ D for S4 is then obtained by imposing a structural restriction on the monotonicity (or thinning) rule of level 2:
x
-+ 2 Y
r-
x u z1 -+ 2 Y u z2.
The restriction is this: if Y = 0, then Z 2 must be a singleton or empty; if Y i- 0, then Z2 must be empty. If the same restriction is applied to
2.3. HIGHER-DIMENSIONAL SEQUENT SYSTEMS
A 'higher-dimensional' proof theory for modal logics has been developed by Masini [108], [106]. This approach is based on the notion of a 2-sequent. In order to define this notion, various preparatory definitions are useful. Any finite sequence of modal formulas is called a 1-sequence. The empty 1-sequence is denoted by E. A 2-sequence is an infinite 'vertical' succession of 1 sequences, r = { ai}o
i : ak = E}. A 2-sequent is an expression r -+ 6., where r and 6. are 2-sequences. The depth of r-+ 6. (q(r-+ 6.)) is defined as max(qr, q6.). If r-+ 6. is a 2-sequent and A an occurrence of a modal formula in r -+ 6., then A is said to be maximal in r -+ 6., if A is at level k in r or in 6. and k = q(r -+ 6.). A is the maximum in r -+ 6., if A is the unique maximal formula in r -+ 6.. The sequent rules for 0 are based on the idea of "internalizing
18
CHAPTER 2
HIGHER-ARITY SEQUENT SYSTEMS
the level structure of 2-sequents" [108, p. 231]:
a formula: a(.6.---+
r a /),A
(0 ---+)
.6.
---+.6. f---+
r' r
(---+ D)
a,OA ;9
f---+
---+.6.
1-l
A .6. J.L,OA
r' .6.
r (0---+)
a A
f---+ ---+.6. (---+ 0)
1-l
::J
1\ .6. implies Vr does not coincide with
a,OA
where a, ;9, n, and J.L denote arbitrary !-sequences, and A must be the maximum of the premise 2-sequent in (---+ 0) and ( 0 ---+). According to Masini, these introduction rules give rise to a "general basic proof theory of modalities" [108, p. 232]. If added to a 2-sequent calculus for CPL, the above rules result, however, in a sequent calculus for KD instead of the basic system K. This sequent system for KD admits cutelimination, satisfies (inter), and its rules are separate, symmetrical, and explicit, but no indication is given of how to Gentzenize axiomatic extensions of KD. Moreover, it is not clear how Masini's framework may be modified in order to obtain a 2-sequent calculus for K.
n = 1\ .6. Vr,
The sequent s thus is true under a given interpretation if either some formula in .6. is false, or some formula in r is true, and the two places of the sequent arrow correspond to the two truth-values of classical logic. In general, in n-valued logic (with 2 :::; n) one obtains n-place sequents s = .6. 1 I .6. 2 I ... I .6.n, with the understanding that s is true under an interpretation if for every i :::; n, some formula in .6.i has truth-value i; for a comprehensive treatment of sequent calculi for truth-functional many-valued logics see [199]. In [195] display sequent calculi are presented for certain important many-valued logics which are neither truth-functional nor satisfy the Deduction Theorem. That is to say, in these logics the validity of
n,A .6.'
r
19
1\ .6. allows to derive Vr. We shall here briefly review some relevant parts of the work of Blarney and Humberstone [24], who investigate an application of threeplace and, ultimately, four-place consequence relations to normal modal logic. Their approach is congenial to DL with respect to a realization of the Dosen-principle insofar as they emphasize that distinctions between various well-known normal modal logics can "be reflected at the purely structural level, if an appropriate notion of sequenf' is adopted [24, p. 763]. Let r, .6., 8, and 2: range over finite sets of formulas in the modal propositionallanguage with 0 as primitive. The four-place sequent r ---+~ .6. has the following heuristic reading:
2.4. HIGHER-ARITY SEQUENT SYSTEMS
In search of generalizations of the standard Gentzen-style sequent format, it is a natural move to consider consequence relations with an arity greater than 2. It seems that the first higher-arity sequent calculus was formulated by Schroter in the 1950s, see [82], [153]. This formalism is a natural generalization of Gentzen's sequent calculus for CPL to truth-functional n-valued logic. The intended truth-functional reading of a Gentzen sequent s = .6. ---+ r is given by a translation a of s into
if all formulas in r are true and all formulas in 'E are necessary, then some formula in ~ is true or some formula in 8 is necessary. 13 This kind of sequent had independently been used by Sato [149], where a cut-free sequent calculus for 85 is presented containing a left 13
In Chapter 9, we shall encounter another reading, namely: if all formulas in r are true and all formulas in 'E are false, then some formula in ~ is true or some formula in e is false.
20
CHAPTER 2
HIGHER-ARITY SEQUENT SYSTEMS
introduction rule for D that fails to be weakly explicit in the sense of Section 1.3. 14 Blarney and Humbestone's introduction rules forD are:
Since Blarney and Humberstone are primarily interested in semantical aspects of their sequent systems, they do not consider cut-elimination. Although Blarney's and Humberstone's calculi do satisfy Dosen's Principle, it remains unclear whether their approach is fully modular for the most important systems of normal modal propositionallogic. They do not present a structural equivalent of the 5-axiom schema, but rather treat S5 as KTB4. In [121 ], Nishimura uses six-place sequents
f- 0 -+~ DA
(D ..1-)o
(D t)o
f- DA -+: 0.
In order to obtain a sequent calculus for K the following structural rules are assumed:
(R)
f-A-+~ A
(vertical R)
f- 0 -+10
(M)
8 r -+:E
(undercut)
~-+~A
(T)
r, A -+~ ~
(vertical T)
r -+~A ~
8 8' f- r, f' -+E,E'
A
L.\
A A/ Ll., L.\
'
r -+~',A~ f- r -+~,E' ~
'
r -+~ A, ~ f- r -+~ ~ r -+~,A ~ f- r -+~ ~.
Against the background of these rules, the introduction rules (D ..1-)o and (D t)o are interreplaceable with the following rules, respectively: (D ..1-)
r, DA -+~
~ f- f -+~,A~
(D t)
f -+~A~ f- f,DA -+~ ~.
These higher-arity sequents can intuitively be read as follows: if every formula in el is true always in the past, and every formula in r is true, and every formula in e2 is true always in the future, then some formula in :E1 is true always in the past, or some formula in 6. is true, or some formula in :E 2 is true always in the future. Nishimura defines introduction rules for the tense logical operations [FJ and [PJ, which are explicit in the sense of Section 1.3: (-+ [F])
'
The introduction rules for the Boolean operations are straightforward adaptations to the higher-arity case. Here is a simple example of a derivation in this formalism (using some obvious notational simplifications):
([FJ -+)
DA, DB -+ DA 1\ DB (D-!-) A 1\ B-+ A DA -+s DA 1\ DB (D-l-) A 1\ B -+ A, B 0 -+ A,B DA 1\ DB (undercut) 0 -+ A,B DA 1\ DB (Dt) D(A 1\ B) -+ DA 1\ DB
([PJ -+)
The axiom schemata D, T, 4, and Bare captured by purely structural rules, i.e., rules not exhibiting any logical operations: D T 4 B
~ -+~ 0 f- 0 -+~ 0 f- 0 -+~A ~-+~A r -+~',A~ f- r -+~,E' ~ ~ -+~ A
21
r -+~,A ~ f- f -+~ ~.
14 Recently, four-place sequents also have been used to obtain sequent calculi for propositional default and autoepistemic logic, see [25].
(-+ [P])
81; f; 82-+ ~1; ~;A, ~2 81; f; 82-+ ~1; ~' [F]A; ~2 81; f; A, 82-+ ~1; ~; ~2 81; r, [FJA; 82-+ ~1; ~; ~2 81;r;82-+ ~1,A;~;~2 81; f; 82-+ ~1; [P]A, ~~2 81, A; r; 82 -+ ~1; ~; ~2 81; [PJA, f; 82-+ ~1; ~; ~2
In accordance with the Dosen Principle, these rules are held constant in sequent systems for Kt and K4t. The difference between these logics is accounted for by different structural rules, namely (r-trans)
0; f; 0-+ ~;A; 0 0;0;r-+ 0;~;A
(1-trans)
0; r; 0 -+ 0; A; ~ r;0;0-+ A;~;0
in the case of Kt and (r-trans) 4
0; f; r -+ ~'~;A; 0 0;0;r-+ ~;~;A
(1-trans) 4
r; f; 0 -+ 0; A;~'~ f;0;0-+ A;~;~
in the case of K4t. Nishimura observes that although in the introduction rules for (F) and (P) subformulas are preserved from premise
23
CHAPTER 2
HYPERSEQUENTS
sequent to conclusion sequent, cut-elimination fails to hold in the sixplace sequent systems for Kt and K4t. There is, for instance, no cutfree proof of ;p; -+; [F]-.[P]-.p;.l 5 In Section 13.3, we shall introduce four-place display sequents.
of contraction and monotonicity, the above cut rule, and a structural rule of a new kind, namely the modalized splitting rule:
22
2.5. HYPERSEQUENTS
Hypersequents were introduced into the literature, although not under this name, by Pottinger [134], and have later systematically been studied by Arnon Avron [9], [10], [12]. A hypersequent is a sequence r1 -+ ~1
I r2
-+ ~2
I ... I r n-+ ~n
of ordinary sequents (or, more generally, sequents in which ~i and r i are sequences of formula occurrences) as their components. The symbol I in the statement of a hypersequent enriches the language of sequent rules and is intuitively to be read as disjunction. This expressive enhancement "makes it possible to introduce new types of structural rules, and ... to allow greater versatility in developing interesting logical systems" [12, p. 5]. In particular, a distinction may be drawn between internal and external versions of structural rules. The internal rules deal with formulas within a certain component, whereas the external rules deal with components within a hypersequent. Let G, H, H1, H 2 etc. be schematic letters for possibly empty hypersequents. Internal monotonicity, for instance, can be contrasted with external monotonicity:
Cut only has an internal version: G1
1
r1-+ ~1, A 1 H1 G2 1 A, r2-+ ~2 1 H2 G1 1 G2 1 r1. r2 -+ ~1, ~2 1 H1 1 H2
The method of hypersequents allows a cut-free presentation HS5 of 85 satisfying the subformula property. HS5 consists of hypersequential versions of the rules of LS4, in particular, external and internal versions 15
Note that Nishimura allows infinite sets in antecedent and succedent position. It is, however' shown that if a sequent 61; r; 62 -+ :El;~; :E2 is provable, then there are finite SetS 6~ ~ 6;, :E~ ~ :E;, (i = 1, 2), f 1 ~ r, and ~I ~ ~ SUCh that the Sequent 6~; f 1 ; 62 -+ :E~; ~I; :E2 iS provable.
In the context of the methodological discussion in Chapter 1, A vron 's paper [12] is of particular interest because of its remarks on methodology. Avron lists six properties a general framework for "good" proof systems should have: 1. Generality. The framewok should be able to deal with a great diver-
2.
3.
4.
5. 6.
sity of interesting logical systems of different types, and, moreover, should suggest new important logics. Semantics independence. The framework should not be so closely tied to a particular semantics that one can more or less recognize the semantic structures in question immediately. Simplicity of structures. The expressions the framework manipulates should be reasonably simple, so that one obtains a 'real' subformula property. Simplicity of rules. The rules should have a small, fixed number of premises and their applicability should depend only on how the premises have been arrived at in the course of a derivation. The Dosen Principle. The proof systems instantiating the framework should lead to a better understanding of the respective logics and the differences between them.
Avron observes that the framework of ordinary sequents is not capable of handling all interesting logics. There are logics with nice, simple semantics and obvious interest for which no decent, cut-free formulation seems to exist .... Larger, but still satisfactory frameworks should, therefore, be sought. [12, p. 3] According to Avron, higher-level proof systems and display logic are both weak with respect to simplicity of structures. In particular, Avron regards the unrestricted iteration of structural connectives in DL (see Chapter 3) as a weak point, because it violates the 'real' subformula property: structural operations may disappear in the transition from premise sequents to conclusion sequent. As mentioned in Chapter 1, in an enriched structural language of sequents, the subformula property for the logical vocabulary in the object language need not be of immediate use in proof search. This is not to say that the subformula property becomes completely useless. In later chapters we shall frequently use
24
25
CHAPTER 2
NATURAL DEDUCTION SYSTEMS
the subformula property of cut-free display sequent calculi to obtain highly welcome conservative extension results. Avron concedes that except for simplicity of structures, DL "otherwise has all the other properties". However, Avron does not consider properties suggested by the idea of a proof-theoretic characterization of the logical operations with respect to general introduction schemata, see Chapters 5 and 8. This idea of a proof-theoretic semantics is, of course, not in conflict with Avron's requirement of semantics independence, since this requirement is meant to avoid the intrusion of model-theoretic parameters into proof theory. It is not clear whether the application of hypersequents to modal logic can be reconciled with the idea of a proof-theoretic semantics. Since the hypersequential system HS5 is based on LS4, the introduction rules for D in HS5 fail to be even weakly explicit in the sense of Section 1.3.
Benevides and Maibaum [19, p. 45] state a theorem to the effect that, if no further rules are assumed, ~o = K, where ~o may be any (initial) set of modal formulas. Proof systems for certain extensions of K are obtained by adding further proof rules operating within the sets ~i, e.g. the rule associated with the T-schema is
2.6. NATURAL DEDUCTION SYSTEMS
In the liteature one can also find plenty of natural deduction calculi for particular modal logics. Here we shall consider one recent proposal and briefly mention another, older approach, namely Fitch-style natural deduction. Fitch-style natural deduction is then taken up again in Chapter 13. The constructive approach of Benevides and Maibaum. Benevides and Maibaum [19] have suggested presenting D as a 'constructive' connective in a generalized natural deduction system. In this system a modal propositionallogic is presented as a denumerable series of sets of modal formulas ~o, ~1, ~ 2 .... In every ~i, the formulas are indexed by f-r:;, and ~i is supposed to be closed under the usual natural deduction rules for CPL. In addition, for every ~i, there are introduction rules for conjunctions and disjunctions prefixed by D and elimination rules for conjunctions and implications prefixed by D. This latter set of rules consists of the modal distribution principles valid in K and is called R 2 . Moreover, there is a set R 3 of introduction and elimination rules for completely unspecified formulas A in the scope of one occurrence of D, which are based on an interaction between sets ~i, ~i+ 1 : D-1 D-E
f-i+1A f-i DA f-i DA f-i+l A
where the proof of f- i+ 1 A does not depend on an undischarged assumption in ~i+ 1
f-i A f-i DA.
Obviously, these additional rules violate Dosen's Principle. Moreover, no rules are given for () as a primitive operation. Generalized Fitch-style natural deduction. Modular Fitch-style natural deduction proof systems for the most important normal modal propositionallogics have been formulated by Borghuis [27], [28]. At the price of violating the Dosen Principle, this approach is a straightforward and transparent extension of ordinary Fitch-style natural deduction by adding both 'modal import' and 'export rules'. We shall consider Pitchstyle natural deduction in Chapter 13.
CHAPTER 3
DISPLAY LOGIC
The present chapter introduces display logic, DL, a general prooftheoretic schema developed by Nuel Belnap, see [16], [17], [18]. A very general strong cut-elimination theorem for DL and M. Kracht's syntactic and semantic characterization of the properly displayable modal and tense logics will be treated separately in Chapter 4. As will become evident, DL satisfies all the methodological requirements examined in Chapter 1. The following features may be seen as characteristic ofDL: 1. DL extends the structural language of ordinary sequents in a nat-
ural way by introducing abstract operations on structures. 2. Antecedent (succedent) parts of a sequent scan be displayed as the entire antecedent (succedent) of a sequent s' structurally equivalent to s. 3. The introduction rules for the logical operations exploit a certain duality between the antecedent and the succedent position of the sequent arrow. In particular, the introduction rules for the normal modal and tense logical operators rely on the ideas of residuation and Galois connection. 4. There may be more than just one family of structural operations. In the following we shall deal mainly with only the first three ingredients of DL, and hence with only a small part of DL's machinery. Working with more than one family of structure connectives proves extremely useful when it comes to mixed systems, combining logical operations from different kinds of logics, like, for instance, Boolean and relevance logical connectives, see [16], [139], and Chapter 11. 16
3.1. GENTZEN TERMS
In a display sequent, the sequent arrow does not relate finite sets as in ordinary Gentzen sequents, but rather complex 'Gentzen terms'. 17 The structural language of display sequents contains four structural 16
The idea of using premise structuring operations of different sorts has independently been developed by M. Dunn [51] and G. Mints [111]. 17 This notion is, for example, also used in [41]. 27
28
CHAPTER 3
GENTZEN TERMS
connectives, namely the nullary I, the unary operations * and •, and the binary operation o. Every formula of the logical object language under consideration is regarded as a structure, and the structural connectives are used to build up more complex structures in the obvious way, i.e. according to their arity. We shall use X, Y, Z, X 1 , X 2 etc. to denote arbitrary (finite) structures and A, B, C etc. to denote arbitrary formulas. DL's structural language can be motivated by general considerations of what would be desirable and natural if one deals with a binary deducibility relation --+ between possibly complex (finite) data (cf. [67] and Chapter 8). Certainly, it should be possible to combine the data and to transfer the data, to move it around. Moreover, there should be a way of representing the empty data and a way of marking the data as being special in a certain sense or as forming a certain type of structure. Also, the data may occur in more than one context, and this may result in a context-dependency of the data structuring operations being employed. These considerations are natural, general, and do not enforce any particular logical interpretation of the structure operations. Conversely, however, the logical operations may be seen as arising from the available operations on the data. It is the structural language of sequents that together with a format of introduction rules delimits the shape the introduction rules for the logical operations may have, see Chapters 4 and 8. 18 The structure connectives of DL can be understood as follows:
29
the structural connectives is laid down by the basic structural rules of DL. Basic structural rules (1)
(2) (3)
(4)
X o Y--+ Z -H- X--+ Z o *y -H- Y--+ *X o Z X --+ Y o Z -H- X o *z --+ Y -H- *y oX --+ Z X --+ Y -If- *y --+ *X -If- X --+ * * Y X--+ •Y -If- •X--+ Y,
Here X1 --+ Y1 -If- Xz --+ Yz abbreviates X1 --+ Y1 f- X 2 --+ Y2 and Xz --+ Yz f- X1 --+ Y1. The rules (1) and (2) show how data addition, 0 , and data transfer, *, interact in antecedent and succedent position, the rules (3) reveal the context shifting effect of *, and the rules (4) exhibit a certain interplay between antecedent and succedent position of the intensionality sign •. If two sequents are interderivable by means of ( 1) - (4), then these sequents are said to be structurally or display equivalent. The following pairs of sequents are display equivalent on the strength of (1) - (3):
X-+YoZ X-+Y X-+Y X--+ *y
I is the empty structure. o is structure addition. * shifts structures from one side of--+ to the other. • marks the structure in its scope as intensional (or as quantificational, see Chapter 12).
*z o *Y--+ *x **X-+Y *y -+X y--+ *X.
The intended declarative meaning of the structural connectives is in part context-sensitive. Context-sensitivity of structure connectives is familiar from ordinary Gentzen systems, where the empty set is to be understood as truth (t) when in antecedent position and as falsity (f) when in succedent position, while the comma can be thought of as conjunction (/\) and disjunction (V) on the right and the left side of --+, respectively. The context sensitive reading of DL's structural operations can be made explicit by a translation T of sequents into tense logical formulas. This translation also reflects the context shift accomplished by *:
This inventory of structure connectives directly refines the structural language of ordinary Gentzen sequents. The constant I replaces the empty set, and the binary o replaces the polyvalent comma. The shift operation * is new. Its inferential behaviour is such that Boolean negation --, can be introduced as a logical operation 'declaratively identical' to *· The unary • is also new, and its inferential behaviour turns out to be suitable for formulating introduction rules for the normal modal and tense logical operations [F] (alias D) 'always in the future', (F) (alias 0) 'sometimes in the future', [P] 'always in the past', and (P) 'sometimes in the past'. The clear and simple inferential behaviour of 18
A case-study development of classical linear logic along these lines can be found in [67].
T(X --+ Y)
hnz
:= T1 (X) :J T2 (Y),
CHAPTER 3
30
RESIDUATION
where Ti (i = 1, 2) is defined as follows:
Moreover, if
A
Ti(A) TI(I) 72(1) TI(*X) T2(*X) TI(X 0 Y) T2(X 0 Y) Tl (•X) T2(•X)
•(F)A •(P)A •[P]A •[F]A
t f
•T2(X) •TI (X) Tl (X) 1\ TI (Y) T2(X) V T2(Y) (P)TI (X) (F]T2(X).
The following definition is taken from Dunn [52, p. 32]: Definition 3.1. Consider two partially ordered sets A = (A,~) and B = (B, ~')with functions f: A-+ Band g: B-+ A. The pair(!, g) is called (fa ~' b (b ~'fa (fa~' b (b ~'fa
iff iff iff iff
a ~ gb); a ~ gb); gb ~a); gb ~a).
Let R be a binary relation on a non-empty set W. Dunn gives various examples of pairs of functions defined on the powerset of W (notation adjusted): if [F]A (P)A [P]A (F) A
·-
{a I Vb (aRb implies bE A)}
·-
{a I 3b (bRa and b E A)} {a I Vb (bRa implies bE A)}
·-
{a I 3b (aRb and bE A)}
-
then ( (P), [F]) and ((F), [P]) are both residuated pairs with respect to the subset relation ~ (and ([F], (P) ), ([P], (F)) are dual residuated pairs). That is, we have: (P)A (F)A
~
B
~
B
iff iff
A~
{a I Vb (bRa implies b rf_ A)} {a I 3b (bRa and b rf_ A)} {a I 3b (aRb and b rf_ A)}
B ~ •(F)A
•[P]B ~A
iff iff
A~
•(P)B, •[F]A ~ B.
If we take these subsets of W to be truthsets of formulas, the examples
3.2. RESIDUATION
iff iff iff iff
{a I Vb (aRb implies b rf_ A)}
then the pair (•(F),•(P)) (as well as (•(P),•(F))) is a Galois connection, and the pair (•[P],•[F]) (as well as (•[F],•[P])) is a dual Galois connection. That is to say:
The rules (4) obviously are correct under the T-translation, and the pair ( (P), [F]) exemplifies the idea of a residuated pair of unary operations.
residuated a Galois connection a dual Galois connection a dual residuated pair
31
[F]B, A~ (P]B.
show that in every normal tense logic the following sequent rules are admissible: 19 (P)A -r B --lf- A -r [F]B (F)A -r B --lf- A -r [P]B B -r •(F)A --lf- A -r •(P)B ·[P]B -r A --lf- •[F]A -r B In general, if Dx is a necessity operator with respect to a binary relation Rx, and Oxv is the possibility operator with respect to the converse relation Rxv (= {(a,b) I (b,a) E Rx}), then (Oxv,Dx) forms a residuated pair. In category-theoretic terms, a residuated pair provides an example of adjointness between functors. The universal and the existential quantifiers can also be understood as a pair of adjoint functors (see [72, Chapter 15] and Chapter 12). Note that the residual of 1\ is implication, i.e.
A 1\ B f- C iff A f- B
:::::>
C.
In logics without the permutation rule and with a non-commutative conjunction n, like non-commutative linear logic, there is a distinction 19
This simple fact has been (re)discoverd several times. It is, for instance, the content of de Rijke's "Switching Lemma" in (143, Lemmata 3.3.6 and 5.5.22], which, though not by that name, can also be found in (171, Corollary 2.75] and (145, Section 2.3 and Theorem 2.4.1]. Recently, it has been interpreted as expressing the Ramsey rule for counterfactual conditionals (146].
32
CHAPTER 3
between left and right residuals: 20 AnBf--C AnBf--C
iff iff
For instance, in the minimal normal tense logic Kt,
B f-A \B A f- C/B
(left residual), (right residual).
1\
::J
Classical, material implication ..., ___ V ... can in succedent position be expressed by means of * and o. The constructive implication of (substructural subsystems of) intuitionistic logic cannot be defined in terms of disjunction and negation. Therefore in general it is natural to relate o in antecedent position with two binary structure connectives or and ol via the following display equivalences: XoY-tZ XoY-tZ
·+ ·+
X--+ Z or Y Y--+ X ol Z.
If o is assumed to be commutative in antecedent position, or and ol collapse, and A oB in succedent position may be interpreted as the implication in question, say, relevant implication A ::Jr B or intuitionistic implication A ::Jh B, cf. [76], [140], and Section 11.2.3. Dunn [52], [53] has defined an abstract law of residuation for n-place connectives f and g. The formulation of this principle refers to traces of operations; see also [141]. 21 Definition 3.2. (Dunn)
An n-place connective
f
has a
trace
(Pl, ... , Pn) f---+ + (in symbols T(f) = (Pl, ... , Pn) t--t +) iff f(A 1, . .. , t, ... , An) -1f- t, if Pi = + (the indicated t is in position i); j(A 1, ... , f, ... , An) + t, if Pi = - (the indicated f is in position i);
if A f- Band Pi=+, then j(A1, ... , A, ... , An) f- j(A1, ... , B, ... , An); if A f- Band Pi=-, then j(A1, ... , B, ... ,An) f- j(A1, ... ,A, ... , An). The operation -) iff
f
has a trace (Pl, ... , Pn) t--t - (T(f) = (Pl, · · · , Pn) t--t
f(A 1 , ... , f, ... , An) + f, f(A 1 , ... , t, ... , An) -1f- f, if A f- Band Pi=+, then if A f- Band Pi=-, then 20
33
RESIDUATION
if Pi = + (the indicated f is in position i); if Pi = - (the indicated t is in position i); j(A1, ... , B, ... ,An) f- j(A1, ... , A, ... , An); j(A1, ... ,A, ... ,An) f- j(A1, ... ,B, ... ,An)·
See also [23, p.325 f.]. As Dunn [52, p. 42] warns, "there is no consistency in the literature about which is the "left" and which is the "right"" residual. Birkhoff, for instance, refers to / as left and to \ as right residuation. 21 Dunn has defined the notions of trace and abstract residuation in algebraic terms; we shall employ logical notation.
..., [F]
(P)
has has has has has
traces trace traces trace trace
(-,-)f---+-, (-,+)f---++ -f---++, +f---++ -f---+-
(+,-)f---+-,
(-,+)f---+-
+f---+-
We define ifT(J) = ( ... ) t--t + ifT(J) = ( ... ) t--tTwo operations f and g are said to be contrapositives in place j iff T(f) = (p1, ... ,pj, ... ,pn) f---+ p implies T(g) = (p1, ... , -p, ... ,pn) f---+ -pj, where-+=- and--=+. Definition 3.3. (Dunn) A pair of n-place connectives f and g satisfies the abstract law of residuation just in case for some j (1 ::; j ::; n), f and g are contrapositives in place j, and
S(f, A1, ... , Aj, ... , An, B) iff S(J, A1, ... , B, ... , An, Aj)· It is an easy exercise to verify that the pairs(/\, ::J ), ( ...,, •), and ( (P), [F]) satisfy the abstract law of residuation. For instance,
S( (P)A, B) iff (P)A f- B
iff A f- [F]B iff S([F]B, A).
Context-sensitivity of structural operations is not a defining property of DL. As Restall [141] emphasizes, there is no loss of generality in disentangling, say, o into two operations oa and ob, oa being in antecedent position and ob being in succedent position. Restall uses two unary structure connectives ~ and 1>, which occur only in succedent position, to define a display proof system for pairs of split negations [54], [56]. Split negations have trace - t--t +, and a pair of such negations satisfies the abstract law of residuation. The structure operations ~ and I> are related to each other by the display equivalence: X--+ ~y iff Y-+ I>X.
3.3. THE DISPLAY THEOREM
DL derives its name from the fact that any substructure of a given display sequent s may be displayed as the entire antecedent or succedent, respectively, of a structurally equivalent sequent s'. In order to state this fact precisely, we need some definitions. An occurrence of a substructure in a given structure is said to be positive (negative) if it is in the scope of an even (uneven) number of *'s. An antecedent (succedent) part of a sequent X -7 Y is a positive occurrence of a substructure of X or a negative occurrence of a substructure of Y (a positive occurrence of a substructure of Y or a negative occurrence of a substructure of X). Theorem 3.4. (Display Theorem, Belnap [16]) For every sequent s and every antecedent (succedent) part X of s there exists a sequent s' structurally equivalent to s such that X is the entire antecedent (succedent) of s'. Proof. We shall follow the elegant proof Restall [140] gives for a structural language with two binary structure operations (and another set of structural rules). Restall defines some auxiliary notions. A context results from a structure through replacing one occurrence of a substructure by the 'Void' (in symbols '-'). If f is a context and X is a structure, then f(X) is the result of substituting X for the Void in f. A context f is called antecedent positive (negative) if the indicated X is an antecedent part (a consequent part) of f(X) -7 Y; f is said to be succedent positive (negative) if the indicated X is a succedent part (an antecedent part) of Y -7 f(X). A contextual sequent has the shape f -7 Z or Z -7 f, and a pair of contextual sequents is said to be structurally equivalent if the sequents are interderivable by means of rules (1) - (4). The Display Theorem then follows from the following lemma. Lemma 3.5. (i) Suppose f is a context in antecedent position. If f is antecedent positive, then f(X) -7 Y is structurally equivalent to X -7 r(Y), where is a context obtained by unraveling the Void in f. If f is antecedent negative, then f(X) -7 Y is structurally equivalent to r(Y) -+ X. (ii) Suppose f is a context in succedent position. If f is succedent positive, then Y -7 f(X) is structurally equivalent to fC(Y) -+X, where fC is a context obtained by unraveling the Void in f. If f is succedent negative, then Y -7 J(X) is structurally equivalent to X -7 fc(Y).
r
35
THE DISPLAY THEOREM
CHAPTER 3
34
The proof is by induction on the complexity of contexts. Case 1: f = -.Then f is antecedent and succedent positive, and r(Y) = JC(Y) = Y. Case 2: f = •g. Then f(X) -7 Y is structurally equivalent to g(X) -7 •Y, and Y -7 f(X) is equivalent to •Y -7 g(X). By the induction hypothesis, these sequents are equivalent to X -7 r(•Y), r(•Y) -7 X, fC(•Y)-+ X, or X -7 fc(•Y). Hence = ga(•-) and fC = gc(•-). Case 3: f = *g. Then f(X) -7 Y is equivalent to *y -7 g(X). Depending on whether g is succedent positive or negative, f(X) -7 Y is structurally equivalent to gc(*Y) -7 X or with X -7 gc(*Y). Therefore, = gc(*-)· Similarly, jC = ga(*-)· by the induction hypothesis, Case 4: f = Z og. Then f(X) -7 Y is equivalent to g(X) -7 *Z o Y. By the induction hypothesis, this sequent is equivalent to X -7 ga (*z o Y) or ga(*Z o Y) -+ X, and hence = ga(*Z o -). Similarly, jC =
r
r
r
ga(-
o
Case 5:
*Z).
f
=go Z. Similar to Case 4. Q.E.D.
Define structures X, Y to be similar to each other (symbolically, X Y) if for every structure Z, we have X -7 Z -H- Y -7 Z
~
and
Z-+X-H-Z-+Y. Corollary 3. 6. (Kracht [92]) The equivalence relation ence relation on the algebra of structures.
~
is a congru-
That is, if X1 ~ Y1, ... , Xn ~ Yn, then for every primitive or defined n-place structure operation f, f(Xl, ... ,Xn) ~ f(Yl, ... , Yn)· Kracht [92] also shows that structures can be brought effectively into a certain normal form. Definition 3. 7. A structure is in normal form of rank 0 if it has the shape A 1 o A 2 o ... oAk o *B 1 o *B2 o ... o *B1;
it is in reduced normal form of rank 0, if all Ai are distinct and all Bj are distinct. A structure is in normal form of rank n + 1 if it has the shape
where Z and all Xi, Yj are in normal form ofrank ~ n. It is in reduced normal form of rank n + 1 if Z is in reduced normal form of rank~ n, if all Xi are distinct, and all Yj are distinct.
36
CHAPTER 3
Theorem 3.8. (Normal Forms, Kracht [92]) Every structure X can be transformed effectively into a term Y in normal form and also in reduced normal form, such that X ;::::; Y.
Kracht uses this Normal Form Theorem to define a calculus of compressed proofs. If a proof system satisfies the Display Theorem, it is said to enjoy the display property. While the rules (1) - (4) suffice to prove the Display Theorem, this particular combination of rules is not the only choice of basic structural rules that guarantees the display property. In particular, in Belnap's seminal paper [16], another set of basic structural rules for * and o is chosen, namely
(1)' (2)' (3)'
X o Y---+ Z --11-- X---+ *y o Z X---+ Y o Z --11-- X o *y---+ Z X---+ Y --ll- *y---+ *X --ll- **X---+ Y
Note that (2)' postulates commutativity of o in succedent position. In [18], Belnap observes among other things that (1) - (3) (the 'A scheme') and (1)'- (3)' (the 'P scheme') are incomparable with respect to partitioning 'contraposition pairs' like
X o Y ---+ *z
INTRODUCTION RULES
in part by the rule Ll-+A B-+f A=:>B,Ll-+f
Then we are not explaining A =:> B alone. We are simultaneously involving the comma not just in our explicans (that would surely be all right), but in our explicandum. We are explaining two things at once. There is no way around this. You do not have to take it as a defect, but is is a fact .... If you are a 'holist', probably you will not care; but then there is not much about which holists much care. [18, p. 81 f.] (notation adjusted) Belnap emphasizes that the display property may be used to keep certain proof-theoretic components as separate as possible. In a proof system enjoying the display property, the behaviour of the structural elements can be described by the structural rules, and the right (left) introductions rules for f can be stated with f(A 1, ... , An) standing alone as the entire succedent (antecedent) of the conclusion sequent. Since j(A1, ... , An) plays no inferential roles beyond being derived and allowing to derive, these left and right introduction rules provide a complete explanation of the meaning of f.
Z ---+ *(X o Y)
into equivalence classes, and that the partitioning of the A scheme may be obtained from a proper refinement ('the GA scheme') by defining X o Y in succedent position as *(*Y o *X). The Display Theorem has technical significance. It allows an " 'essentials-only' proof of cut elimination relying on easily established and maximally general properties of structural and connective rules" [18]; see Chapter 4. The conceptual significance of the Display Theorem resides in the fact that it allows one to improve on the separation requirement of Chapter 1. According to the separation condition the introduction rules for an n-place connective f must not exhibit' any ~ogical operations other than f. Otherwise the account of f's meaning IS - at least partially - holistic. This idea can be strengthened by postulating that the succedent (antecedent) of the conclusion sequent in a right (left) introduction rule must not exhibit any structure operation. Let us refer to this property as segregation. As Belnap explains: ' The nub is this. If a rule for =:> only shows how A =:> B behaves in context, then that rule is not merely explaining the meaning of=:>. It is also and inextricably explaining the meaning of the context. Suppose we give sufficient conditions
for
3.4. INTRODUCTION RULES
The basic structural rules (1) - (4) specify a 'geometry of structures', which can be used to formulate introduction rules for many logical operations known from the field of substructural logic see for instance ' ' ' [48], [168], [180]. Substructurallogics are logical systems that can be obtained from a sequent system for a given logic by dispensing with all or part of the system's structural inference rules. In the present context our main concern is not with substructural logics but with modal and tense logical extensions of classical logic. 22 Therefore we shall not exploit the expressive power even of DL based on a single family of structure connectives, and instead in Table II simply present introduction rules for the Boolean and normal tense logical operations. Note that these introduction rules satisfy the segregation requirement; every right (left) introduction rule is such that the succedent (antecedent) of its conclusion sequent consists of a single formula schema and thus does not exhibit any structural operation. This set of logical operations does not reflect the well-known distinction in substructural logic between internal and combining (alias multiplicative and additive, 22
A=:> B,Ll-+ f
37
Detailed discussions of presenting substructural logics as display sequent systems can be found in [80] and [140].
38
CHAPTER 3 Table II. Introduction rules.
(-+f) (f-+) (-+ t) (t -+)
(-+ /\) (I\ -+) (-+V) (V-+) (-+:::>) (=>-+) (-+=) (=-+) (-+ [F]) ((F]-+) (-+(F)) ((F) -+) (-+ [P]) ((P]-+) (-+ (P)) ( (P) -+)
Boolean rules X-+lf-X-+f f-f-+1 f-1-+t 1-+Xf-t-+X X -+ *A f- X -+ ·A *A -+ X f- ·A -+ X X-+A Y-+Bf-XoY-+A!\B AoB-+Xf-A!\B-+X X-+AoBf-X-+AVB A-+X B-+Yf-AVB-+XoY XoA-+Bf-X-+A:::>B X-+A B -+ Y f- A :::> B -+ *X o Y XoA-+B XoB-+Af-X-+A=B X-+A B-+Y X-+B A -+ Y f- A =: B -+ *X o Y tense logical rules •X -+ A f- X -+ [F]A A -+ X f- [F]A -+ •X X -+ A f- * • *X -+ (F)A * • *A -+ Y f- (F)A -+ Y X A X A
-+ * • *A f- X -+ [P]A -+ X f- [P]A -+ * • *X -+ A f- •X -+ (P)A -+ •X f- (P)A -+ X
alias intensional and extensional) connectives. In particular, the rules for !\ and V are structure dependent, making !\ and V internal. In the presence of (analogues of) the standard structural rules, the distinction becomes redundant, however. The collection of further structural sequent rules in Table Ill contains plenty of redundancies when assumed as a package. In combination with the basic rules (1) - (4) it clearly suffices to justify the labels 'Boolean' and 'tense logical' of Table II.
The (I+) and (I-) rules allow one to add and to remove the empty data in antecedent and succedent position. The rule (I ex) ((ex I)) al-
39
INTRODUCTION RULES Table Ill. Additional structural rules.
(I+ )u (I+ )Lr (I+ )rr (I+ )rl (I- )u (1- )Lr (I- )rr (I- )rt (I ex) (ex I) (I*)t (I*)r (P)t (P)r (C) I (C)r (M)tr (M)u (M)rr (M)rl (A)t (A)r (MN)rt (MN)tt (MN)rp (MN)tp
X-+Zf-IoX-+Z X-+Zf-Xoi-+Z X-+Zf-X-+Zol X-+Zf-X-+IoZ IoX-+Zf-X-+Z Xoi-+Zf-X-+Z X-+Zolf-X-+Z X-+loZf-X-+Z 1-+Xf-Z-+X X-+lf-X-+Z I -+ X -lf- *I -+ X X -+ I -lf- X -+ *I x1 o x2 -+ z r- x2 o X1 -+ z z-+ xi o x2 r- z-+ x2 o xi XoX-+Zf-X-+Z Z-+XoXf-Z-+X
xi -+ z r- xi o x2 -+ z X1 -+ z r- X2 o xi -+ z z -+ xi r- z -+ x1 o x2 z -+ x1 r- z -+ x2 o x1 X 1 o (X 2 o X 3) -+ z -1f- (X1 o X2) o x3 -+ z Z -+ XI o (X2 o X3) -1f- (XI o X2) o X3 -+ Z I -+ X f- I -+ •X X -+ I f- X -+ •I I -+ X f- I -+ * • *X X -+ I f- X -+ * • *I
1
lows t to be derived from arbitrary data (to derive arbitrary data from f). The rules (I*) guarantee the provability of •t --+ f, t --+ ...,f, ...,f--+ t, and f--+ •t. The (P), (C), (M), and (A) rules are counterparts of the familiar rules of permutation, contraction, monotonicity, and associativity, and the (MN) rules capture the necessitation rules of normal modal and tense logic. This diversity of structural inference rules is desirable for a synoptical treatment of (one-family) substructurallogics. In other contexts it may be useful to reduce redundancy and to work with more parsimoniouis collections of structural rules extending (1) (4). Eventually, there are two logical structural rules, namely identity and cut.
40
CHAPTER 3
(id) (cut)
Logical rules 1-p---+p X-tA A-tYI-X-tY
Note that uniqueness is preserved under addition of purely structural rules. For the language with ::J as a defined connective, Kracht [92] observes a nice duality between proofs in DKt. The dual (X ---+ Y)t. of a sequent X---+ Y is defined as yt>---+ xt., where
The identitiy rule (id) can be generalized to arbitrary tense logical formulas.
pl> (A 1\ B)t. ([F]A)t. ( (P)A)t. It. (X o Y)t.
Observation 3. 9. For every tense logical formula A, 1- A ---+ A. Proof. The proof is by a straightforward induction on the complexity of A. For example,
A-t A [P]A---+ * • *A [P]A---+ [P]A
A-t A •A---+ (P)A A---+ •(P)A (P)A---+ (P)A
Definition 3.10. The display sequent system DKt is given by the entire set of the above sequent rules. The calculus DK results from DKt by removing the introduction rules for the backward-looking operators [P] and (P). The calculus DCPL results from DK by removing the introduction rules for the operators [F] and (F). 23
In Section 3.5, DKt (DK, DCPL) is shown to be a sound and complete proof system for Kt (K, CPL). A strong cut-elimination theorem for these sequent systems is established in Chapter 4. It follows that DKt is a conservative extension of DK, and DK is a conservative extension of DCPL. The logical operations are uniquely characterized (cf. Section 1.4) in these sequent systems. Observation 3.11. The logical operations of tense (modal) logic are uniquely charcaterized in DKt (DK). Proof. This is an easy exercise in writing DL derivations. For instance,
23
p At. V Bt. (P)(At.) [F](At.) I xt. o yt.
(•p)l> (A V B)t. ([P]A)t. ( (F)A)t. (*x)t. (•X)t.
•(pl>) At. I\ Bt. (F)(At.) [P](At.) *(X)t. •(Xt.)
The dual rrt. of a proof II is obtained by dualization of all sequents in Q.E.D.
According to Belnap [16, p. 383] this observation constitutes "half of what is required to show that the "meaning" of formulas in Display logic is not context sensitive, but that instead formulas "mean the same" in both antecedent and consequent position. (The Elimination Theorem ... is the other half of what is required for this purpose.)".
A-t A [F]A---+ •A •[F]A---+ A [F]A ---+ [F]* A
41
INTRODUCTION RULES
A-t A [F]* A---+ •A •[F]*A---+ A [F]* A ---+ [F]A
In DCPL all rules exhibiting • may be abandoned.
Q.E.D.
II.
Observation 3.12. (Kracht) If II is a proof of X---+ Y in DKt in the language without material implication, then rrt. is a proof ofYt. ---+ xt. in DKt. Example 3.13. The proof
B-tB AoB-tA AoB-tB AI\B---+A AI\B-tB D(A 1\ B) ---+ •A D(A 1\ B) ---+ •B •D(A 1\ B) ---+ A •D(A 1\ B) ---+ B D(A 1\ B) ---+ DA D(A 1\ B) ---+ DB D(A 1\ B) o D(A 1\ B) ---+ DA 1\ DB D(A 1\ B) ---+ DA 1\ DB is dualized into
At. ---+ At. Bt. ---+Bt. At. ---+ At. o Bt. Bt. ---+ At. o Bt. At.---+ At. V Bt. Bt.---+ At. V Bt. •At.---+ (P)(At. V Bt.) •Bt.---+ (P)(At. V Bt.) Ali---+ •(P)(Ali V Bli) Bli---+ •(P)(Ali V Bli) (P)(Ali)---+ (P)(Ali V Bt.) (P)(Bli)---+ (P)(Ali V Bt.) (P)(Al>) V (P)(Bf>)---+ (P)(Al> V Bl>) o (P)(Al> V Bl>) (P)(Ali) V (P)(Bli) ---+ (P)(Ali V Bli)
42
The interdefinability of [F] and (F) ([P] and (P)) can easily be verified, e.g.: A--+A A--+A *A--+ *A *A--+ *A *A--+ -,A -,A--+ *A * • **A--+ (F)-,A A--+ *'A [F]A --+ • * -,A *(F)-,A --+ • * *A * • *'A--+ *[F]A • * (F)-,A --+ * *A (F)-,A--+ *[F]A • * (F)-,A --+ A [F]A--+ *(F)-,A *(F)-,A--+ [F]A [F]A--+ -,(F)-,A -,(F)-,A--+ [F]A Note also that the introduction rules for-,, 1\, and V, the left introduction rules fort, (P), and (F), and the right introduction rules for f, :J, =, [P], and [F] are invertible in DKt, that is, every premise sequent may be derived from the conclusion sequent. For example, A--+A * • *A--+ (F)A * • *A--+ y
A--+A B--+B AoB--+AI\B AoB--+Y
(F)A--+ Y
(cut)
AI\B--+Y
(cut)
43
COMPLETENESS
CHAPTER 3
X--+ [F]A
X--+AI\B
A--+A [F] --+ •A X--+ •A •X--+ A
A--+ A (M) AoB--+A AI\B--+A X--+A
(--+ D)B X Om Im --+ A f- X --+ DA (0 --+ )B A --+ X f- OA --+ *mlm Om X (--+O)B X--+ A f- *m(*mX Om Im)--+ ()A (0--+)B *mX Om A--+ *mlm f- ()A--+ X. Now the earlier T- translation can be extended according to the intended reading of the modal structural operations as follows: Ti(*mX) Ti(Im) TI(X Om Y)
T2(X Om Y)
Ti(*X) Ti(l) (P)TI (X) 1\
(Y) [F]T2(X) V [F]T2(Y) Tl
Under the extended T-translation, the rules (-+ D)B, (D--+ )B, (--+ 0 )B, and (0 --+)B boil down to the the rules(--+ [F]), ([F]--+), (-+(F)), and ((F) --+ ), except that (0 -+) B uses the fact that ((F), [P]) forms a residuated pair. It should be clear by now that DL is not at all based on merely copying logical operations as structural operations into the language of sequents. As we have seen, residuation principles may justify a context sensitive interpretation of structure connectives that spoils a one-to-one correspondence between the structural and the logical vocabulary. The structure operations of DL register principled ways of manipulating data, and these operations on the data then give rise to introduction rules for the logical operations.
3.5. COMPLETENESS
The structure operation • was introduced in [181]. It allows one to consider (the logical language of) modal and tense logics as being based on one family of structure operations, namely {1, *, •, o}. Modal and tense logics extending CPL may be seen as combining Boolean and intensionallogical operations, and Belnap [16] originally has presented modal logics as systems being based on two different families of structure connectives, a Boolean one and a modal one:
We first consider weak completeness of DKt and DK, i.e. coincidence of Kt (K) and DKt (DK) with respect to provable formulas. Extensions of K and Kt are dealt with in Chapter 4 in the context of M. Kracht's characterization of the properly displayable tense and modal logics.
In [16], Belnap assumes structural rules to the effect that Im = Ib and *m = *b· The binary operations ob and om satisfy the basic structural rules (1)' ~ (3)', and om is used to formulate introduction rules for the modal operators D and 0:
Proof. (i) We may take any axiomatization of Kt and show that the axiom schemata are provable in DKt, and the proof rules preserve provability in DKt. As to the classical base of Kt, we consider Lukasiewicz's three axiom schemata for CPL and modus ponens:
Theorem 3.14. (i) If f- A in Kt, then f-- I --+ A in DKt. (ii) If f-- X --+ Y in DKt, then f-- T(X --+ Y) in Kt.
44
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COMPLETENESS
45
The following is a cut-free proof of the K axiom schema for [F]; the proof for [P] is analogous:
B-tB C-tC B :J C-+ *Bo C B o (B :J C)-+ C A -+ A B -+ C o *(B :J C) A :J B-+ *A o (C o *(B :J C)) A o (A :J B) -+ C o *(B :J C) (A) (A o (A :J B)) o (B :J C) -+ C (PJ A o ((A :J B) o (B :J C))-+ C ((A :J B) o (B :J C)) o A -+ C (A :J B) o (B :J C) -+ A :J C (A :J B) -+ (B :J C) :J (A :J C) I o (A :J B) -+ (B :J C) :J (A :J C) I-+ (A :J B) :J ((B :J C) :J (A :J C))
A-tA B-tB I-tA I-+A:JB A :J B-+ *A oB I o I -+ A 1\ (A :J B) A o (A :J B)-+ B I-+ A 1\ (A :J B) A 1\ (A :J B) -+ B I-+ B
A-t A *A-+ *A *A -+ --.A A -+ A --.A :J A -+ * * A o A *Ao (·A :J A)-+ A *A -+ A o *(--.A :J A) *A o *A-+ *(•A :J A) *A-+ *(•A :J A) --.A :J A-+ A Io(--.A:JA)-+A I-+ (•A :J A) :J A
A-t A *A-+ *A (M) *A o *B-+ *A *A-+ *Ao B --.A-+ *A oB A o --.A-+ B A-+ --.A :J B I o A-+ --.A :J B I-+ A :J (•A :J B).
(C)
A---+A [F]A ---+ •xA [F](A ::::> B) o [F]A ---+ •A •([F](A ::::>B) o [F]A) ---+A B---+ B A ::::> B---+ * • ([F](A ::::>B) o [F]A) oB [F](A ::::>B)---+ •(* • ([F](A ::::>B) o [F]A) oB) [F](A ::::> B) o [F]A ---+ •( * • ([F](A ::::> B) o [F]A) oB) •([F](A ::::>B) o [F]A) ---+ * • ([F](A ::::>B) o [F]A) oB •([F](A ::::> B) o [F]A) o •([F](A ::::>B) o [F]A) ---+ B •([F](A ::::> B) o [F]A) ---+ B [F](A ::::>B) o [F]A---+ [F]B [F](A ::::>B) ---+ [F]A ::::> [F]B I o [F](A ::::> B) ---+ [F]A ::::> [F]B I ---+ [F](A ::::> B) ::::> [F]A ::::> [F]B Necessitation for [F] and [P] is taken care of by the (MN) rules. It remains to derive the familiar tense logical interaction schemata A ::::> [F](P)A and A ::J [P](F)A. For instance:
A---+A •A---+ (P)A A---+ [F](P)A (ii) By induction on the complexity of proofs in DKt. Q.E.D.
Corollary 3.15. (i) In Kt, 1- A iff 1- I ---+ A in DKt. (ii) In K, 1- A iff 1- I ---+ A in DK. Proof. (i) By the previous theorem. (ii) By the fact that every frame complete normal propositional tense logic is a conservative extension of its modal fragment. Q.E.D. The following useful observation has independently been made by several authors (for example [77], [92], [181]) and in combination with the previous corollary allows one to prove strong completeness.
Lemma 3.16. In DKt, 1- X---+ TI(X) and 1- T2(X) ---+X. Proof. By induction on the complexity of X. Q.E.D.
46
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CHAPTER 4
Theorem 3.17. In DKt, I- X-+ Y iff Tl(X) I- T2(Y) in Kt. Proof. (=?): This is Theorem 3.14, (ii). ({::::): Suppose that in Kt, T1(X) I- T2(Y). Hence 1-Kt T1(X) :::::> T2(Y). By Corollary 3.15, 1-nKt I-+ T1(X) :::::> T2(Y) and thus 1-nKt T1(X) -+ T2(Y). Since by Lemma 3.16, I- X -+ T1(X) and I- T2(Y) -+ Y in DKt, an application of cut gives I- X-+ Y. Q.E.D. Corollary 3.18. DK is strongly sound and complete with respect to K. Corollary 3.19. DCPL is strongly sound and complete with respect to CPL.
PROPERLY DISPLAYABLE LOGICS, DISPLAYABLE LOGICS AND STRONG CUT-ELIMINATION
In the present chapter it is shown that every displayable logic enjoys strong cut-elimination. This result sharpens Belnap's very general cutelimination theorem for Display Logic [16]. Moreover, Kracht's characterization of the properly displayable modal and tense logics is presented, together with complete and cut-free display sequent systems for various axiomatic extensions of the basic intensional systems K and Kt.
4.1. PROPERLY DISPLAYABLE LOGICS
We shall first introduce some useful terminolgy. An inference inf is a pair (~, s), where b. is a set of sequents (the premises of in!) and s is a single sequent (the conclusion of in!). A rule of inference R is a set of inferences. If inf E R, then inf is said to be an instantiation of R. R is an axiomatic rule, if~ = 0 for every (~, s) E R. We assume that inference rules are stated by using variables for structures and formulas. Every structure occurrence in an inference inf (a sequent s) is called a constituent of inf (s). The parameters of inf E R are those constituents which occur as substructures of structures assigned to structure variables in the statement of R. Non-parametric formulas in the conclusion of inf are called principal formulas of inf. Constituents of inf are defined as congruent in inf iff they are occupying similar positions in occurrences of structures assigned to the same structure variable. A proper display calculus is a calculus of sequents whose rules of inference satisfy the following eight conditions: Cl Preservation of formulas. Each formula which is a constituent of some premise of inf is a subformula of some formula in the conclusion of inf. C2 Shape-alikeness of parameters. Congruent parameters are occurrences of the same structure. C3 Non-proliferation of parameters. Each parameter of infis congruent to at most one constituent in the conclusion of inf.
47
4S
CHAPTER 4
C4 Position-alikeness of parameters. Congruent parameters are either all antecedent or all succedent parts of their respective sequents. C5 Display of principal constituents. A principal formula of inf is either the entire antecedent or the entire succedent of the conclusion of inf. C6 Closure under substitution for consequent parts. Each rule is closed under simultaneous substitution of arbitrary structures for congruent formulas which are consequent parts. C7 Closure under substitution for antecedent parts. Each rule is closed under simultaneous substitution of arbitrary structures for congruent formulas which are antecedent parts. CS Eliminability of matching principal formulas. If there are inferences infi and inh with respective conclusions (1) X -t A and (2) A-t Y with A principal in both inferences, and if cut is applied to obtain (3) X -t Y, then either (3) is identical to one of (1) or (2), or there is a proof of (3) from the premises of infi and inh in which every cut-formula of any application of cut is a proper subformula of A. Obviously, every display calculus satisfying Cl enjoys the subformula property, that is, every cut-free proof of any sequent s contains no formulas which are not subformulas of constituents of s. If a logical system A can be presented as a proper display calculus, A is said to be properly displayable.
4.2. A CASE DISTINCTION AND PRIMITIVE REDUCTIONS
While Belnap [16] proves that in every properly displayable logic, a proof of a sequent s can be converted into a proof of s not containing any application of cut
(1) X -t A (2) A-t Y (3) X -t Y the proof of the strong cut-elimination theorem shows that every (sufficiently long) sequence of steps in the process of cut-elimination terminates. The elimination process consists of two kinds of actions, principal moves and parametric moves. If the cut-formula A is principal in the final inference in the proofs of both ( 1) and (2), a principal move is performed. Otherwise, if there is no previous application of cut, a parametric move is executed. According to this distinction we define primitive reductions of proofs ending in an application of cut. Let Ili be the proof of (i) we are dealing with, (i = 1, 2).
A CASE DISTINCTION AND PRIMITIVE REDUCTIONS
49
Principal moves. By CS, there are two subcases: Case 1. (3) is the same as (i): II·t
Case 2. There is a proof II of (3) from the premises s 1 , ... , sn of ( 1) and s~, ... , s~ of (2) in which every cut-formula of any application of cut is a proper subformula of A: Ill s1, ... , Sn (1)
II2 I sl, ... ,sm (2) (3)
Ill
I
"-"+
II2
II (3)
Parametric moves. The parametric moves modify proofs on a larger scale than the principal moves. Suppose that A is parametric in the inference ending in (1). The case for (2) is completely symmetrical. What the parametric moves show is that applications of structural rules need never immediately precede applications of cut. In order to define the parametric moves, we inductively define a set Q of occurrences of A, called the set of 'parametric ancestors' of A (in Ill), cf. [16, p. 394]. We start with putting the displayed occurrence of A in (1) into Q. Then, by working up II 1 , we add for every inference inf in I1 1 each constituent of a premise of in/which is congruent (with respect to in!) to a constituent of the conclusion of inf already in Q. What we obtain is a finite tree of parametic ancestors of A rooted in the displayed occurrence of A in (1). This tree either contains an application of cut or not. If so, we do not perform a reduction, but instead consider one of these applications of cut above (1) for reduction. If not, that is, if there is no application of cut in the tree, then for each path of parametric ancestors of A in II 1 , we distinguish two sub cases. Let Au be the uppermost element of the path and let inf be the inference ending in the sequent s which contains Au. Case 1. Au is not parametric in inf. By C4 and C5, it is the entire consequent of s. We cut with II 2 and replace every parametric ancestor of A below Au in the path by Y. Case 2. Au is parametric in inf. Then, with respect to inf, Au is congruent only to itself, and we just replace every parametric ancestor of A below Au in the path by Y. Moreover, we delete II2, which is now superfluous.
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STRONG NORMALIZATION
The result of simultaneously carrying out these operations for every path of parametric ancestors of A in II 1 and removing the initial occurrence of (3) (since now (2) = (3)) constitutes a primitive reduction. Typically we have:
~
~
Z--+A
~ (1)
._....;
II2
(3)
Z--+A Z--+Y
II2
0
Example
51
4.2.
B-+B A-+A AoB-+B A o (A oB) -+ A 1\ B A oB-+ A* o (A 1\ B) A-+ (A* o (A 1\ B)) oB* A-+ A* o ((A 1\ B) oB*) II A o A -+ (A 1\ B) o B* X -+ A A-+ (A 1\ B) oB* X -+ (A 1\ B) oB*
"""
II X-+A A-+A B-+B X-+A XoB-+B X o (X oB) -+ A 1\ B X oB -+ X* o (A 1\ B) X-+ (X* o (A 1\ B)) oB* X-+ X* o ((A I\ B) oB*) X oX -+ (A 1\ B) oB* X -+ (A 1\ B) o B*
Example 4. 3. This very simple example shows that parametric moves may also decrease the number of cuts.
(3)
By C3 and the bottom-up definition of Q, for every inference inf in II1, Q must contain the whole congruence class of inf, if Q is inhabited at all. By C4, Q only consists of consequent parts. Hence, by C2 and C6, the result of such a reduction is in fact a proof of (3), since on the path from (1) to Z--+ A we have the same sequence of inference rules being applied as on the path from (3) to Z --+ Y. If the cut-formula A is parametric in the inference ending in (2), we rely on C7 instead of C6. We shall illustrate the reduction steps with some examples from
A--+A B* o A--+ A A--+BoA A o A* --+ B A o A* --+ Y
II B--+ Y
A--+A Y* o A--+ A A--+YoA AoA*--+Y
4.3. STRONG NORMALIZATION
DCPL.
Example
4.1.
A-+A B-+B AoB-+AI\B A-+ (A 1\ B) oB* (A 1\ B)* o A-+ ·B (A 1\ B)* -+ A :J •B (A :J ·B)* -+ A 1\ B •(A :J ·B)-+ AI\B
II AI\B-+Y
A-+A B-+B AoB-+AI\B AoB-+Y A-+ YoB* Y* o A-+ B* Y* o A-+ ·B Y* -+ (A :J •B) (A :J ·B)* -+ Y
·(A
:J
·B) -+ Y
II AI\B-+Y
The proof of the strong normalization theorem to be given takes its pattern from the proof of strong normalization for typed .\-calculus, cf., for instance, [86, Appendix 2]. We shall follow the argument in [144, Chapter 2, reprinted in [168]] establishing strong cut-elimination for classical linear propositionallogic, CLL. This proof has been extracted from the proof of strong cut-elimination for classical predicate logic in [49, Appendix B] . In Chapter 7, the argument is applied to obtain strong cut-elimination for a labelled tableau presentation of first-order S5. Suppose that II is a proof containing an application of cut. A (onestep) reduction of II is the proof :E resulting by applying a primitive reduction to a subproof of II. If II reduces to :E, this is denoted by II > :E (or :E < II). II is said to be reducible iff there is a :E such that II >:E. Lemma
4.4. If a proof cannot be reduced, then it is cut-free.
Proof. Since the above case distinction is exhaustive, every proof that contains an application of cut is reducible. Q.E.D.
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CHAPTER 4
Definition 4. 5. We inductively define the set of inductive proofs. a Every instantiation of an axiomatic rule is an inductive proof.
b If IT ends in an inference inf different from cut, and every premise si of inf has an inductive proof ITi in IT, then IT is inductive. _
c IT -
IT1
(3)
IT2
cut
is inductive, if every 1: such that IT > 1: is
STRONG NORMALIZATION
53
Proof. By induction on ind(II). If ind(IT) = 1, then IT cannot be reduced. Whence IT is inductive by b or c. If IT is inductive by c, then by definition, ind(IT) > ind(IT'). If IT is inductive by b, then ITj is inductive by definition. If ITj is inductive by a, it cannot be reduced. If ITj is inductive by b, then the reduction of ITj to ITj takes place in the proof of some premise sequent of the final inference of ITj. By the induction hypothesis, ind(ITj) > ind(ITj). Hence ind(IT) > ind(IT'). If ITj is inductive by c, then by definition, ind(ITj) > ind(IIj) and thus ind(IT) > ind(IT'). Q.E.D.
inductive.
Lemma 4. 6. If IT is inductive, and IT > 1:, then 1: is inductive. Proof. By induction on the construction of IT. If IT is inductive by a, then no reduction can be performed. If IT is inductive by b, then every reduction on IT takes place in the ITi's, which are inductive. Hence, by the induction hypothesis, 1: is inductive due to b. If IT is inductive by c, then 1: is inductive by definition. Q.E.D. Definition 4. 7. Let IT be an inductive proof. The size ind(IT) of IT is inductively defined as follows (the clauses correspond to those in the previous definition): a ind(IT)
=
b ind(IT) =
Lemma 4.1 0. Suppose IT ends in an application inf of cut, and IT1 and IT2 are the proofs of the premises of inf. If ITl and IT2 are inductive, then so is IT.
Proof. We must show that every 1: < IT is inductive. For this purpose, we define two complexity measures for IT: r(IT), the rank of IT, and h(IT), the height of IT. r(IT) is the number of symbols in the cut-formula. h(IT) is defined by:
1;
Li ind(ITi) + 1;
c ind(IT) = I:~
h(IT)
= ind(ITI) + ind(IT2).
+ 1.
A proof IT is said to be strongly normalizable iff every sequence of reductions starting at IT terminates.
Lemma 4.8. Every inductive proof is strongly normalizable. Proof. By induction on ind(II). If ind(IT) = 1, no reduction is feasible. If IT is inductive by b, then every reduction is in the premises ITi, and we can apply the induction hypothesis. If IT is inductive by c, then every proof to which IT reduces is inductive and therefore every such proof is strongly normalizable, by the induction hypothesis. But then IT is also strongly normalizable. Q.E.D. Lemma 4.9. Let IT be an inductive proof and let inf be the final inference of IT. If IT > IT' by reducing a proof ITj of a premise sequent of inf, then ind(IT) > ind(IT').
We use induction on r(IT) and, for fixed rank, induction on h(IT). Case 1. 1: is obtained by reduction in IT 1 or IT 2, say IT 1 >IT~. It follows from Lemma 4.9 that ind(ITD < ind(IT 1). Then h(I:) < h(IT). Since IT1 and IT 2 are inductive, by Lemma 4.6, 1: has inductive premises, and by the induction hypothesis for h(IT), 1: is inductive. Case 2. 1: is obtained by reducing inf. Then this reduction was either a principal or a parametric move. Principal move. Case 1. Since 1: proves one of (1) or (2), 1: is inductive by assumption. Case 2. Since for every new proof IT' ending in an application of cut, r(IT) > r{IT'), 1: is inductive by the induction hypothesis for r(IT). Parametric move. We have
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DISPLAYABLE LOGICS
CHAPTER4
55
satisfied. Verification of C8 is also a simple exercise. For instance:
IT=
Ill
Ill
Z-tA
Z-+A Z-+Y
~ (1) (3)
........
I12
E=
I12
0
(3)
and we may assume that there is no application of cut on the path from (1) to Z-+ A.
•X-+ A A-+ Y •X-+ y X-+ •Y
X -+ A * • *A -+ Y * • *X -+ OA OA -+ Y * • *X-+ y
* • *A-+ y *y-+ • *A • * y-+ *A X -+ A A -+ * • *y X-+* • *y • * y-+ *X *y-+ • *X * • *X-+ Y.
Theorem 4.14. Strong cut-elimination holds for DK and DKt.
Ill
Ill
Let IT'= Z-+ A l12 and IT"= z-+ A Z-+Y Consider IT and IT'. Clearly, r(IT) = r(IT'), hence we use induction on the height. Since both l11 and IT" are inductive by b, ind(IT") < ind(ITI). Hence h(IT') < h(IT). By the induction hypothesis for h(IT), IT' is inductive, and thus E is inductive by definition. If the primitive reduction of IT to I: requires cutting with I1 2 more than once, analogously every new IT' and hence I: can be shown to be inductive. Q.E.D.
Corollary 4.11. Every proof is inductive.
Theorem 4.12. Every proof is strongly normalizable, that is, every properly displayable logic enjoys strong cut-elimination.
Proof. By Lemma 4.8 and Corollary 4.11.
Q.E.D.
Corollary 4.13. Cut is an admissible rule. Proof. By Lemma 4.4.
•X-+ A A-+ Y X -+ DA DA -+ •Y X-+ •Y
Q.E.D.
Theorem 4.12 can strightforwardly be applied to DK and DKt. It can easily be checked that in these systems conditions Cl - C7 are
Corollary 4.15. DKt is a conservative extension of DK.
4.4. DISPLAYABLE LOGICS
By conditions C6 and C7, the inference rules of a proper display calculus are closed under simultaneous substitution of arbitrary structures for congruent formulas. The proof of strong normalization can be generalized to logics which for formulas of a certain shape satisfy closure under substitution either only for congruent formulas (of this shape) which are consequent parts or only for congruent formulas (of this shape) which are antecedent parts. Classical linear propositional logic CLL in its standard sequent calculus presentation is an example of such a system; it allows contractions of formulas !A in the antecedent and contractions of ? A in the succedent. (This formulation, of course, presupposes a suitable definition of 'congruent formula' for CLL.) In order to extend the proof of strong cut-elimiation to systems like CLL, first of all C6 and C7 have to be replaced by the more general condition of regularity, see [17]. A formula A is defined as cons-regular if the following holds: (i) if A occurs as a consequent parameter of an inference inf in a certain ruleR, then R contains also the inference resulting by replacing every member of the congruence class of A in inf with an arbitrary structure X, and (ii) if A occurs as an antecedent parameter of an inference inf in a certain rule R, then R contains also the inference resulting by replacing every member of the congruence class of A in inf
CHAPTER 4
56
with any structure X such that X -t A is the conclusion of an inference in which A is not parametric. The notion of ant-regularity is defined in exactly the dual way. In CLL, for instance, !A is cons-regular, and ?A is ant-regular. A formula is said to be regular, if it is either cons-regular or ant-regular. The new condition on rules then is C6jC7 Regularity.
Every formula is regular.
A display calculus simpliciter is a calculus of sequents satisfying Cl C5, C6j7, and C8. If a logic A can be presented as a display calculus, then A is said to be displayable. Obviously, every properly displayable logic is displayable.
Parametric moves. We also have to redefine the parametric moves. Suppose in what follows that the cut-formula A is parametric in both the final inference of Il1 and the final inference of Il2. Moreover, suppose that the trees of parametric ancestors of A in 11 1 and in Ih do not contain any application of cut. If Au is the tip of a path of parametric ancestors of A in Ili, let inf be the inference ending in the sequent which contains Au. Let us call Au significant, if it is not parametric in inf Then, in a proper display calculus we may choose whether we cut every significant tip Au in the tree of parametric ancestors of A in Il1 with Il2 or whether we cut every significant tip Au in the tree of parametric ancestors of A in Il2 with 11 1. Both operations form the essential parts of different primitive reductions. In a display calculus simpliciter this indeterministic choice has to be abandoned. Consider for instance the cons-regular !A in the display calculus for CLL of [17]. In a situation like the following: ~
~
!A!A-tY !A-t Y X-tY
X -t!A
where !A is parametric in the inference ending in X -t!A, we clearly cannot just proceed to the significant tips in the path of parametric ancestors of !A in Il2 and cut with 111, since in general there is no way of proving X -t Y from X X -t Y. But we can safely proceed to the significant tips in the path of parametric ancestors of !A in 11 1 and cut with Il2. Thus, if the cut-formula is cons-regular, we cut with Il2, and if the cut-formula is ant-regular, we cut with 11 1. Obviously, this further restriction on parametric moves does not affect the proof of strong cut-elimination.
CHARACTERIZING THE PROPERLY DISPLAYABLE LOGICS
Theorem
57
4.16. Every displayable logic enjoys strong cut-elimination.
In defining the set of primitive reductions, we have postulated certain restrictions on the order of reduction applications. As Roorda [144, p. 26] emphasizes, such restrictions relativize the strength of a strong cutelimination theorem. The restrictions imposed, however, appear to be rather natural. What they require is that (i) 'stacks' of configurations in which applications of structural rules are followed by a cut are to be reduced 'top-down', and (ii) parametric reductions take into account whether cut-formulas are ant- or cons-regular.
4.5. CHARACTERIZATION OF THE PROPERLY DISPLAYABLE LOGICS
The classes of all properly displayable normal propositional modal and tense logics have been characterized by Marcus Kracht [92]. The idea is to obtain a canonical way of capturing axiomatic extensions of Kt by purely structural inference rules over DKt. 24 Definition 4.17. Let Kt+ a (K +a) be an extension of Kt (K) by a tense logical (modal) axiom schema a, and let DKt + a' (DK + a') be an extension of DKt (DK) by a set a' of purely structural inference rules. Kt + a (K + a) is said to be properly displayed by DKt + a' (DK + a') if (i) DKt + a' (DK + a') is a proper display calculus and (ii) every derived rule of Kt + a (K + a) is the T-translation of a sequent rule derivable in DKt + a' (DK + a').
Now, every axiom schema is equivalent to a schema of the form A:::) B, where A and Bare implication-free. The schema A:::) B has the same deductive strength as the rule
B -t X 1- A -t X. Moreover, if A and B are only built up from propositional variables, t, A, V, (F), and (P), then by classical logic and distribution of (F) and (P) over disjunction, we have
24
Note that the additional rules of Kracht's auxiliary sequent system DLME are not only admissible but also derivable in the display sequent calculus for Kt.
58
CHAPTER 4
CHARACTERIZING THE PROPERLY DISPLAYABLE LOGICS
where every Ci and Dj is only built up from t, 1\, (F), and (P). Therefore A :J B may as well be replaced by the rule schemata
Theorem 4. 20. (Kracht) An axiomatic extension of K is properly displayable iff it can be axiomatized by a set of modal primitive axiom schemata.
D1 --+ Y . . . Dn --+ Y ci--+ Y.
These rule schemata can now be translated into purely structural display sequent rules, using the following translation "1 from formulas of the fragment under consideration into structures:
ry(p) ry((F)A)
ry(A 1\ B)
ry(t)
p
* • *'fl(A) ry(A)
1\
ry( (P)A)
I
•ry(A)
ry(B)
The resulting structural rules
The question whether an axiomatically presented normal modal or tense logic A is properly displayable thus boils down to the question whether A can be axiomatized by primitive axioms over K or Kt. The implicit use of tense logic in the structural language of sequents may help to find simple structural sequent rules expressing less simple modal axiom schemata. The following example is taken from [92].
Example 4.21. The .3 axiom schema D(DA :J DB) V O(OB :J DA) has the primitive modal equivalent (OA 1\ OB) :J ((O(A 1\ OB)
ry(D1) --+ Y ... ry(Dn) --+ Y ry(Ci)-+Y
may still violate condition C3. In order to avoid this obstruction of proper display, it must be required that in the inducing schema A :J B, the schematic formula A contains each formula variable only once. A tense logical formula schema is then said to be primitive if it has the form A :J B, A contains each formula variable only once, and A, B are built up from t, 1\, V, (F), and (P).
Lemma 4.18. Every extension of Kt by primitive axiom schemata can be properly displayed.
+
a' properly displays Kt + a, by condition (ii) of Next, if DKt Definition 4.17, the structural rules in a' may all have the form
X1 --+ Y . . . Xn --+ Y
Z-+Y. This rule has the same deductive strength as the axiom schema
T1(Z) :J viTI(Xi), which is a primitive formula schema.
Theorem 4.19. (Kracht) An axiomatic extension of Kt can be properly displayed in precisely the case that it is axiomatizable by a set of primitive axiom schemata. Kracht [92, p. 113 f.] uses a model-theoretic argument to characterize the properly displayable extensions of K.
59
v O(B 1\ OA)) v O(A 1\ B)),
which in tense logic is equivalent to the simpler primitive schema
(P)(F)A :J (((F) A V A) V (P)A). Application of Kracht's algorithm results in the following structural rule: X-+Y •X-+Y *•*X-+YI-•*•*X-+Y. Kracht also proves a semantic characterization of the properly displayable modal and tense logics. Let :F be a class of (modal or tense logical) Kripke frames. A first-order sentence over two binary relation symbols Rand RV is said to be primitive if has the form ('v') (:J)A, where every quantifier is restricted with respect to R or RV, and A is built up from A, V, and atomic formulas x = y, xRy, xRvy, where at least one of x, y is not in the scope of an existential quantifier.
Theorem 4.22. (Kracht) A class :F of Kripke frames is describable by a set of primitive first-order sentences iff the modal and the tense logic of :F can be properly displayed. The characteristic axiom schemata of many fundamental systems of normal modal and tense logic are equivalent to primitive schemata, and therefore these systems can be presented as proper display calculi, cf. Table IV. A set of structural sequent rules a' is said to correspond to a property of an accessibility relation R (with a modal or tense logical axiom schema a) iff under the T-translation the rules in a' are admissible just in the event that R enjoys the property (the rules in a' have the same deductive strength as a). Every axiom schema a in Table IV corresponds to a purely structural sequent rule a' which can directly be determined from a, see Table V.
CHAPTER4
CHARACTERIZING THE PROPERLY DIS PLAYABLE LOGICS
Table IV. Axioms and primitive axioms.
Table V. Structural rules corresponding t o axiom schemata.
W
D r
4 5 B Altl re 4e
.3 linf linp
V Dv rv
4, 5, B, Altl, rg 4~
v,
* • *I -t Y r I -t Y * • *X -t Y r X -t Y 4' * • *X -t Y r * • • * X -t Y 5' * • *X -t Y r • * • * X -t Y B' * • *(X o * • *Y) -t Z f- Y o * • *X-tZ Altl' * • *(X o Y) -t Z r * • *X o * • *Y-tZ re' X -t y r * • *X -t y 4c' * • • *X -t Y r * • *X -t Y .3' X -t Y • X -t Y * • * X -t Yf--•*•*X-tY linf' = .3' linp' X -t Y • X -t Y * • * X -t Yf--*•*•X-tY V' X -t Y f- •I -t Y D,' .r -t Y r I -t Y r,' •X -t Y r X -t Y 4,' •X -t Y r • • X -t Y 5,' .x -+ Y r- * • * • x -+ Y B,' •(X o •Y) -t Z f- Y o •X -t Z Altl,' •(X o Y) -t Z f- •X o •Y -t Z Te' X -t Y r •X -t Y 4e' • • X -t Y f- •X -t Y V,' X -t Y r * • *I -t Y
schema
primitve equivalent
D'
[F]A ::::> (F)A [F]A :::>A [F]A ::::> [F][F]A (F)A ::::> [F](F)A A::::> (F](F)A (F)A ::::> [F]A A::::>[F]A [F][F]A ::::> [F]A [F]([F]A ::::> [F]B)V V[F]((F]B ::::> [F]A) (F)A ::::> [F](((F)A V A) V (P)A) (P)A ::::> (P](((P)A V A) V (F)A) [F]A [P]A ::::> (P)A [P]A::::>A [P]A ::::> [P][P]A (P)A ::::> [P](P)A A::::> (P](P)A (P)A ::::> [P]A A::::>[P]A [P][P]A ::::> [P]A [P]A
t ::::> (F)t A::::> (F)A (F)(F)A ::::> (F)A (P)(F)A ::::> (F)A (A 1\ (F)B) ::::> (F)(B 1\ (F)A) ((F) A 1\ (F) B) ::::> (F)(A 1\ B) (F)A ::::>A (F)A ::::> (F)(F)A (P)(F)A ::::> ((F)A V A V (P)A)
r'
(P)(F)A ::::> ((F)A V A V (P)A) (F)(P)A ::::> ((P)A V A V (F)A) (P)t ::::>A t ::::> (P)t A::::> (P)A (P)(P)A ::::> (P)A (F)(P)A ::::> (P)A (A 1\ (P)B) ::::> (P)(B 1\ (P)A) ((P)A 1\ (P)B) ::::> (P)(A 1\ B) (P)A ::::>A (P)A ::::> (P)(P)A (F)t ::::>A
61
Corollary 4.25. DKt U r' is a conservative extension of DK U r'. Note that Kracht's algorithm can be dualiz ed. Every schema A :J B is interreplaceable with the rule X-+ A f-- X-+ B. Let .6. (8) be the set of all (all purely modal) axiom schemata from Table IV, .6. ~ .6., e ~ e, .6.' = {a' I a E .6.}, and 8' ={a' I a E 8}. Theorem 4. 23. In DKt U r', 1- I-+ A itf 1- A in Kt U r. In DK U 8', f-- I -+ A iff f-- A in K U 8.
If A and B are only built up from propositio nal variables, f, /\, V, [F], and [P], then by classical logic and distribu tion of [F] and [P] over conjunction, we have j'S:n
Proof. This follows from axiomatizability by primitive schemata. Theorem
Q.E.D.
4.24. Strong cut-elimination holds for DKt U r' and DK U
8'. Proof. The rules in
r' and 8' satisfy conditions C2 ~ C7. Q.E.D.
D·
J'
where every Ci and Dj is only built up from f, V, [F], and [P]. Therefore A ::::> B may be replaced by the rule schemata X-+ C1 ... X-+ C m X-+ Dj·
62
63
CHAPTER 4
SCOPE OF THE METHOD
schemata are translatable into purely structural sequent rules usmg the following translation 77' from formulas of the fragment under consideration into structures:
The relation between gaggle theory and DL has also been investigated and worked out by Gore [81]. A gaggle is an algebra g = (G,~,OP), where ~ is a distributive lattice ordering on G, and OP is a founded family of operations. The latter means that there is an f E OP such that for every g E OP, f and g satisfy the abstract law of residuation (cf. Chapter 3). If one only requires that ~is a partial order, and every ' f E OP has a trace, then g is said to be a tonoid. Restall defines the notion of mimicing structure. An n-place logical operation f mimics antecedent structure if there is a possibly complex n-place structure connective " such that the following rules are admissible:
T~ese
r/(p) r/([F]A) r/(A V B)
p
•77' (A) 17 1 (A) V 11' (B)
1
(f) 77'([P]A) 1]
I
* • *1J'(A)
The resulting structural rules
X -t 17'(CI) ... X -t 1J 1 (Cm) X -t 17'(D1)
s
again may still violate condition C3. In order to avoid the obstruction of proper display, it must be required that in the inducing schema A ::::> B the sche~atic formula B contains each formula variable only once. tense logical formula schema is then said to be dually primitive if it has the form A ::::> B, B contains each formula variable only once and A Bare built up from f, /\,V, [F], and [P]. ' '
A
Theo~em 4.26. An axiomatic extension of Kt can be properly displayed m exactly the case that it is axiomatizable by a set of dually primitive axiom schemata.
For instance, rule T' is equivalent to X -t •Y I- X -t Y and 4' with X -t •Y I- X -t • • Y. Moreover, D' is equivalent to •X o •Y -t *I IX -t *Y, Altt' with X -t Y I- X -t * • * • Y, and V' with I- •I -t X, cf. [181].
4.6. SCOPE OF THE METHOD
-r:he properly displayable modal and tense logics satisfy Dosen's Principle. They are all based on the same set of introduction rules so that ~he logical operations indeed have the same proof-theoretic ~eaning m each of these systems. Kracht 's characterization results show that q~ite a few interesting and important intensionallogics admit a cut-free display sequent calculus presentation. In later chapters the expressive power of DL will be further demonstrated by displaying several nonclassical propositional and predicate logics. Display calculi for various non-normal modal logics may be found in [16]. The generality of DL has been highlighted by Restall [141], who obvserves a close relation between display logic and J. Michael Dunn's Gaggle Theory [52], [53], [55].
= "(A 1, ... , An)
-t X I-
f (A 1, ... , An)
-t X
C(X1, A1) ... C(Xn, An) I- "(A1, ... , An) -t j(A1, ... , An) where "(A 1 , ... , An) is an antecedent part of s, C(Xi, Ai) = Xi -t Ai, if Ai is an antecedent part of "(A1, ... , An), and C(Xi, Ai) = Ai -t Xi, if Ai is a succedent part of ~(A 1 , ... , An)· Dually, f mimics succedent structure if there is a possibly complex n-place structure connective " such that the following rules are admissible: s = X -t " (A 1 , ... , An) I- X -t f (A 1, ... , An) C(X1, AI) ... C(Xn, An) I- j(A1, ... , An) -t "(A1, ... , An)
where "(A1, ... , An) is a succedent part of s, C(Xi, Ai) = Xi -t Ai, if Ai is an antecedent part of "(A 1 , ... , An), and C(Xi, Ai) = Ai -t Xi, if Ai is a succedent part of ~(A1, ... , An)· Theorem 4.27. (Restall [141]) If a logical operation f in a display calculus presentation DA of a logic A mimics structure, then f is a tonoid operator on the Lindenbaum algebra of A. If every logical operation of DA mimics structure, mutual provability is a congruence relation and A has an algebraic semantics. Moreover, Dunn's representation theorem for tonoids supplies also a Kripke-style relational semantics. Another result of Kracht concerns undecidability of display calculi. Consider the fusion or 'independent sum' of Kf and Kf, i.e. the bimodal logic Kf 0 Kf of two functional accessibility relations R1, R2. In this system there are two pairs of modal operators, say, [1], (1) and [2], ( 2) each satisfying the D and the Altl axiom schemata. The structural language of sequents for this logic comes with two unary operations • 1 and •2 satisfying the display equivalence •iX -t Y -If- X -t
•i Y.
64
CHAPTER 4
Clearly, Kf ® Kf has many properly displayable extensions. Using an encoding of Thue-processes into frames of Kf ® Kf, C. Grefe and M. Kracht [92] prove a theorem about the undecidability of decidability.
Theorem 4.28. (Grefe and Kracht) It is undecidable whether or not a display calculus is decidable. According to Kracht, Theorem 4.28 indicates a serious weakness of display logic. However, I find this view unconvincing. The theorem provides insight into the expressive power of DL; it shows that the sub-formula property and the strong cut-elimination theorem for displayable logics fail to guarantee decidability. Undecidability of the decidability of properly displayable extensions of Kf ® Kf is a remarkable property of this particular family of bimodal logics, but is not a defect of DL. The proof of the theorem also shows that it is undecidable whether or not a finite axiomatic calculus is decidable. This clearly fails to be a weakness in any sense of Hilbert-style systems. Nevertheless Kracht's interpretation of Theorem 4.28 as a weakness of DL raises the question whether it is desirable to have a proof-theoretic framework which is bound to deliver decidable systems. On the side of decidability, Restall [140] uses a display presentation to prove, among other things, decidability of certain relevance logics which are not known to have the finite model property. In Chapter 6, we shall present a decidability proof for Kf based on DL.
CHAPTER 5
A PROOF-THEORETIC PROOF OF FUNCTIONAL COMPLETENESS FOR MANY MODAL AND TENSE LOGICS
In what follows we shall use display logic to define a proof-theoretic semantics in terms of general introduction schemata. It will be shown that with respect to this semantics the set of connectives {[F], [P], A, •} is functionally complete for every displayable normal propositional tense logic and the set of connectives {[F], A, •} is functionally complete for every displayable normal propositional modal logic. It seems that there exists no other proof-theoretic characterization of modal operators (apart from intuitionistic implication ::)h) in the literature.
5.1. THE PROBLEM OF FUNCTIONAL COMPLETENESS
Suppose A is a propositional logic with a finite set 2: of primitive finitary operations. Moreover, assume that S is a (strongly) sound and complete semantics for A, and 0 is the class of all finitary operations explicitly definable inS. It may be that 0 is defined by imposing some natural constraints on semantical models, ifS is a model-theoretic semantics. Of course, these constraints must leave the connectives in 2: explicitly definable from finite combinations of connectives in 0. The problem of functional completeness for A consists in finding a proper (possibly a finite) subclass r of 0 such that every connective in 0 is explicitly definable by a finite number of compositions from the elements of r. In a proof-theoretic semantics for A the class 0 of permissible connectives is given by general schemata for introducing finitary operations into premises and conclusions; see, for instance, [96], [97], [151], and [178], [180] for proof-theoretic proofs offunctional completeness for minimal propositionallogic, intuitionistic propositionallogic, Nelson's constructive propositionallogics and various substructural subsystems of these logics. It seems, however, that this approach cannot directly be applied to the standard Gentzen-style proof systems for tense or modal logic. The reason is that the operational rules of these proof systems fail to be explicit or even weakly explicit in the sense of Chapter 1. Ohnishis's and Matsumoto's [122] right rule for 0 in their sequent system for S5, for instance, not only exhibits 0 on both sides of the 65
66
sequent arrow in the conclusion sequent, but even exhibits D in the premise sequent:
D~-+
or, A f-
D~-+
67
PROOF-THEORETIC SEMANTICS
CHAPTER 5
or, DA
(here ~' r range over finite sets of formulas and D~ = {DA I A E ~}). It is unclear how such rules could be captured by a proof-theoretic semantics in terms of introduction schemata. In the present chapter it is shown that the proof-theoretic approach towards functional completeness can be applied to normal propositional modal and tense logics, if these systems are presentable as systems of display logic. Due to the enriched structural language of DL, the tense logical operations [F] (alias D) and [P] can be given explicit introduction rules, see Table II in Chapter 3. As we have seen, many important axiom schemata, like D, T, 4, 5, and B, can be expressed as purely structural rules, which results in a modular Gentzen-style proof theory for the most important systems of normal modal and tense logic. We shall use DL to define a natural proof-theoretic semantics in terms of general introduction schemata. The set of connectives ell = {[F], [P], •, 1\} will turn out to be functionally complete for the minimal tense logical system Kt, and W = {D, •, /\} will be shown to be functionally complete for the minimal normal modal logic K, relative to straightforwardly restricted introduction schemata. The demonstration of functional completeness is such that it immediately reveals ell and W to be functionally complete for any displayable extension of Kt and K, respectively. This notion of functional completeness must not be confused with the notion of temporal completenes over a class of Kripke frames, cf. [30, p. 116 f.]. However, there is a connection. Whereas certain natural temporal operations like 'until', which are first-order definable on Kripke frames, cannot explicitly be defined from [F], [P], and the Boolean connectives, one may ask for a semantics with respect to which [F], [P], and the Boolean operations turn out to be functionally complete. Such a semantics is presented in the present Chapter. Since the systems to be dealt with are based on classical propositional logic, there is no need to consider higher-level sequents as, for example, in the case of intuitionistic logic, cf. [96], [151], Chapter 8. Moreover, in contrast to the proof-theoretic semantics for substructurallogics in [178], [180] and Chapter 8, we need only one right and one left introduction schema.
5.2. PROOF-THEORETIC SEMANTICS
We shall assume the structural language, the logical rules, and the (basic and additional) structural rules of the display sequent systems DKt as defined in Chapter 3. This is our 'structural framework', and we shall refer to it as B (for 'Belnap'). The idea of the proof-theoretic semantics is to delimit the class of admissible connectives by specifying general schemata for introducing (finitary) connectiv~s of a prop_ositionallanguage £ into premises and conclusions, that 1s, on both stdes of the sequent arrow. Of course, these schemata must not be arbitrary, cf. [96]. The constraints on introduction rules developed in Chapters 1 and 3 are relevant again. In particular, a rule schema for an n-ary connective f should exhibit no other connective than f, and the deductive role of formulas f(A 1 , ... , An) should depend only on the deductive relationships between the parameters A1, ... , An· Moreover, the schemata should be conservative (or non-creative), that is, each proof of an J-free formula A should be convertible into a proof of A with no applications of rules characterizing f. This condition boi~s d~wn ~o the requirement that principal applications of cut, i.e. apphcatwns m which both premise sequents are derived by introducing the cut-formula j(A1, ... , An), can be eliminated.
Definition 5.1. The general right introduction schema is this:
where every Xi (i = 1, ... , k) is an unspecified structure, every Yi contains only formulas from A1, ... , An, and every Aj (j = 0, · · ·, n) occurs in some Yi. We have to state a left introduction schema such that the eliminability constraint is satisfied. In order to be able to succinctly formulate such a schema, we define certain operations on sequents.
Definition 5.2. The operations (· )1, (-)r on sequents are simultaneously defined as follows: if Y ifY ifY if Y ifY
=I
=A = •W = =
*w
w1 o w2
68
CHAPTER 5
(Z-+ Y)"
where
z, zl,
~I
and
z2
Z-d Z--tA (Z --t Wt (W --t Z) 1 (Z1 --t WI)r (Z2 --t
w2r
PROOF-THEORETIC SEMANTICS if Y ifY if Y if Y if Y
=I =A = •W = *w = W1 o
w2,
are unspecified structures.
Clearly (Y ---7 Z) 1 is a sequence of sequents, each of which has either the shape A ---7 X, X ---7 A, I ---7 X, or X ---7 I. Moreover, for every occurrence of a formula A in Y, there is exactly one sequent A ---7 X or X ---7 A in (Y ---7 Z) 1• We shall call X the structure corresponding to this occurrence of A in {Y ---7 Z) 1•
Definition 5.3. The general left introduction schema is:
(f
---7)
(Y1
---7
69
in which the cut-formula j(A1, ... , An) is introduced in the inferences ending in ( 1) and (2) and in which (3) is not identical to one of ( 1) or {2), there is a proof of {3) in B + (---7 f) + (f ---7) from the premises of {1) and {2) in which every cut-fromula of any application of cut is a proper subformula of j(A1, ... , An)·
Proof. Suppose (Yi ---7 W) 1 = Si 1 ... Sim. Consider the sequent Si 1. If Si 1 = A ---7 V or V ---7 A, then it is clear from the definition of {·) 1 and that a certain occurrence of A in Yi is a succedent part of Xi -+ Yi or an antecedent part of Xi -+ Yi, respectively. We transform Xi -+ Yi into a structurally equivalent sequent U ---7 A or A ---7 U, respectively, and then apply cut to obtain U ---7 V or V ---7 U, respectively. Then we iterate this process for Si 2 , ••• , Sim {if possible) and finally obtain Xi ---7 (Yi)U. Applying {M) gives X1 o ... o Xk ---7 (Yi)U. Q.E.D.
(·r
W) 1 f- j(A1, ... , An) -+ {YI)U
Example 5.4. (continued) We show in some detail how the procedure is applied. Our starting point is the following application of cut: where W is an unspecified structure and (Yi)U is the result of replacing every occurrence of a formula in Yi by its corresponding structure in (Yi-+ W) 1•
Example 5.4. Consider the following instantiation of(-+ f): X-+ (*Ao•B) o•*l f- X-+ f(A,B).
We have ((*AoeB)o•*I-+W) 1 (*A o •B -+ Wd (• * I -+ W 2 ) 1 (*A-+ W11) 1 (•B-+ Wh) 1 {*I-+ W2) 1 (W1 1 ---7 A)r (B ---7 W1 2)1 (W2 ---7 w11 -+ A B -+ W1 2 w2 -+ 1
1r
and obtain the following instantiation of (f ---7): W11-+ AB-+ Wh W2-+ I f- f(A,B)-+ (*W1 1 o •W1 2) o • *I. We must verify that the eliminability constraint is indeed satisfied.
Observation 5.5. For every proof X1 -+ Y1 ... xk-+ Yk (Yi -+ W) 1 (1) X1 o ... o Xk-+ j(A1, ... , An) {2) j(A1, ... , An) -+ (Yi)U (3) x 1 o ... o xk -+ (Yi)u
X -+ (*A o •B) o • * I W1 1 -+ A B -+ W 12 W2 -+ I X ---7 f(A, B) f(A, B)-+ {*W1 1 o •W1 2 ) o • *I X ---7 (*Wh oeW1 2 ) o•*l
We first display A in X
---7
(*A o •B) o • * I and use it as cut-formula:
•B) o • * I X o * • *I -+ *A o •B (X o * • *I) o * • B-+ *A W1 1 -+A A-+ *((X o * • *I) o * • B) W1 1 -+*((X o * • *I) o * • B) X -+ (*A
Then we display B in W11 cut-formula:
---7
o
*((X o * • *I) o * • B) and use it as
W11 -+*((X o * • *I) o * • B) (X o * • *I) o * • B-+ *W11 (X o * • *I) ---7 *W1 1 o •B W1 1 o (X o * • *I) -+ •B •(Wit o (X o * • *I)) -+ B B-+ W12 •(W1 1 o (X o * • *I)) -+ W1 2 • 0ne might be inclined to object that the schema ( ---7 f) is not general enough, since it permits only connectives with exactly one right introduction rule. This is, however, no real restriction, since multiple right
CHAPTER 5
PROOF-THEORETIC SEMANTICS
introduction rules can be interpreted disjunctively. We shall illustrate this by means of a simple example. 25
Similarly we derive Xh oX1 2 --+ AoD and X1 1 oX1 2 --+ BoD. Applying (-+ <1)' gives X1 1 oX1 2 --+
70
Example 5. 6. Consider the four-place connective
(-+
is interchangeable with the single rule
(-+
X -+ A o C X -+ A o D X -+ B o C X --+ B o D ff- X --+
(--+ (--+ <1) 1 : Consider the following derivation:
X-tAoC X--+BoC X o *C --+ A X o *C -t B (X o *C) o (X o *C) -t
*
(A, B, C, D) oX) o (*
*
(A, B, C, D) oX)--+
*
*
=> (--+
xl)
-+A xll o x12 --+A (X1 1 o Xtz) o *C--+ A Xh oXtz--+ AoC
25
Note that (-+
Xh -+A Xtz -+ B f- Xlt o X1 2 --+
X1 2 -+ B X1 1 o X1 2 --+ B (X1 1 o X1 2 ) o *c--+ B X1 1 oX 12 --+ BoG.
If there is more than one right introduction rule, it seems to be natural to require that every Aj (j = 0, ... , n) occurs in a premise sequent of some right rule.
71
X1 --+ A o C X2 --+A o D X3 -+ B o C X4 --+ B o D ff- x1 o x2 o x3 o x4--+
For the proof of functional completeness we shall need a replacement theorem. Let CA denote an £-formula which contains a certain occurrence of A as a subformula, and let C 8 denote the result of replacing this occurrence of A in C by B. We use [B/A]X to denote the result of replacing every occurrence of A in the structure X by B. The degree of A (d(A)) is defined as the number of occurrences of propositional connectives in A. Let f- X +-+ Y stand for f- X --+ Y and f- Y -+ X.
Theorem 5. 7. If f-A +-7 B in B + (-+ F) + (F --+ ), then f- CA +-+ Cs in B +(-+F) + (F --+). Proof. If d( CA)- d(A) = 0, there is nothing to prove. Suppose therefore that d(CA)- d(A) > 0. Then CA has the form f(Al, ... ,An)A· For every i = 1, ... , k, we consider a certain instantiation of (Yi -+ W) 1 = Si 1 ... Sim. If Siq has the shape A --+ V or V --+ A (q = 1, ... , rn), we instantiate V by B. If Siq has the shape I --+ V or V --+ I, we instantiate V by p o *P, for some propositional variable p. If Siq has the shapeD-+ V or V--+ D and A 1- D, we instantiate V by D. Obviously, every instantiation of any Siq is provable. Therefore by (F --+) we obtain f- f(Al, ... ,An)A--+ [B/A]Yi, and by(--+ F) together with (C) it follows that f- CA --+ CB. Analogously we can show f- CB -+ CA. Q.E.D.
5.3. FUNCTIONAL COMPLETENESS FOR Kt AND K We shall first show that cl>'= {f, t, •, [F], (P), /\,V} is functionally complete for Kt and then prove functional completeness of w' = {f, t, •, [F], 1\, V} forK. Functional completeness of ci> for Kt and \]! forK follows from standard definitions. We first expand on Lemma 3.16.
Lemma 5.8. In DKt: (i) f- X --+ Tl (X). (ii) f- T2(X) --+X. (iii) f- X-+ Y implies f- r1(X)-+ Y. (iv) f- Y-+ X implies f- Y-+ r2(X).
72
CHAPTER 5
Proof. By induction on the complexity of X.
PROOF-THEORETIC SEMANTICS Q.E.D.
In the logical language of Kt we thus have f- X -+ A in DKt iff f- X-+ A in B + (-+f) + (f-+ ). Q.E.D.
Theorem 5.9. ' is functionally complete for Kt. Proof. Suppose that the rules for f are instantiations of (-+ f) and (f-+ ). Then ::::} ::::}
1- 72 (Yi) -+ Yi 1- T2(Y1) 0 ••• o T2(Yk)-+ J(A!, ... , An) f- TI( T2(Y1) 0 ••• 0 T2(Yk)) -+ J(Al, ... , An)
by Lemma 5.8 (ii) by(-+ f) by Lemma 5.8 (iii)
::::} ::::} ::::}
1- f(AI, ... , An)-+ T1h(YI) o ... o T2(Yk))
0 ... 0
T2(Yk)
Proof. The proof is completely analogous to the proof for Kt, except that now the schemata (-+ f) and (f -+) have to be suitably restricted: in every Yi the structural connective • may only occur within the scope of an even number of *'s. Then the definiens Tl(T2(Y1) o ... o T2(Yk)) remains in the language of modal logic. Q.E.D.
by (id) and (! -+) by Lemma 5.8 (iv) by (M) and
display equivalence by Lemma 5.8 (i)
By Theorem 5.7, it follows that j(A1, ... , An) and Tl(T2(YI)
o ...
o
T2 (Yk)) are provably interchangeable.
It still has to be shown that for every primitive operation ~ of Kt, either the rules for ~ are (interchangeable with) instantiations of (-+ f) and (f -+) or ~ is explicitly definable by a finite composition of operations from '. This can be verified by eye and the following straightforward observations: (i) (f -+) is interchangeable with (f-+ )' I -+ X f- f-+ I. (ii) (..., -+) is interchangeable with (..., -+ )' X -+ A f- ·A -+ *X. (iii) The 'multiplicative' rules for 1\ are interchangeable with their 'additive' formulations:
(-+ /\)' (/\ -+)'
X-+A X-+Bf-X-+AI\B A -+ X f- (A 1\ B) -+ X B -+ X f- A 1\ B -+ X.
(iv) The rules for ::J are interchangeable with:
(-+:::) )' (::J-+)' (v) The rules for
(-+=)' (=-+ )'
X -+ B o *A f- X -+ A ::J B *A -+ X B -+ Y f- A ::J B -+ X o Y.
=are interchangeable with:
X-+ B o *A X-+ A o *B f- X-+ A= B *A -+ X B -+ Y * B -+ X A -+ Y f- A = B -+ X o Y.
(vi) t is definable as (p V •p) for some propositional variable p. (vii) (F}A is definable as •[F]·A; [P]A is definable as •(P}•A.
Example 5.4. (continued) We get (•A V [F]B) V [F]•t as definiens for j(A, B). Theorem 5.1 0. '11 1 is functionally complete for K.
Moreover, 1- f(AI, ... ,An)-+ Yi 1- J(AI, ... , An) -+ T2(Yi) 1- J(AI, ··.,An) -+ T2(YI)
73
Corollary 5.11. (i) is functionally complete for every displayable normal propositional tense logic. (ii) '11 is functionally complete for every displayable normal propositional modal logic. Proof. Adding purely structural rules does not affect the above proofs of functional completeness. Q.E.D. The introduction schemata (-+ f) and (J -+) may be regarded as both general and natural, given the underlying structural apparatus. With respect to the structural framework assumed, these schemata may be seen as having priority over the sets of connectives that emerge as functionally complete. Our results highlight the problem of displaying more expressive modal formalisms like systems containing the difference operator D or 'since' (5) and 'until' (U) with semantic clauses:
f= DA
M,t M,t
F S(A,B)
iff iff
M,t
F U(A,B)
iff
where M
=
(:3u E W) t i= u & M, u f= A (:3u E W) tRvu, M, u f= A & for all u' in between u and t, M, u' (::!u E W) tRu, M,u f= A & for all u' in between u and t, M, u'
(W, R, v} is a Kripke model.
f= B f= B,
CHAPTER 6
MODAL TABLEAUX BASED ON RESIDUATION
We consider the extent to which an analogue of the familiar notion of clause can be used for modal tableaux, and show that in DL a straightforward anlogue of the ordinary notion of clause rather than a notion of modal clause can be used in complete tableau calculi for the modal logic Kf ( = KDAltl ), the modal logic of functional accessibility relations, and PDL-, deterministic propositional dynamic logic without Kleene-star. As a corollary, we obtain a decision procedure for Kf and PDL-.
6.1. THE MODAL DISPLAY SYSTEM DKf
A sequent rule s1 ... Sn f- s is invertible in a system if in the system every premise sequent Si can be derived from s. While the right introduction rule for 0 in DL is invertible in every normal modal logic, a left introduction rule suitable for the reduction to clausal form becomes available in extensions of the modal logic of functions, Kf ( = KDAltl). Since in the present context our aim is a calculus of invertible rules, we also have to make an appropriate choice of introduction rules for the Boolean operations. We shall consider the connectives ---., 1\, V, and ~ and assume the familiar structure-free, 'additive' right introduction rule for 1\ and additive left introduction rules for V and ~.
(~ •) (· ~)
*A f- X~ ·A *A ~ X f- ·A ~ X
(~V)
X~AoBf-X~AVB
X~
~)'
A~X
B~Xf-AVB~X
(~ /\)' (/\ ~)
X~A
X~Bf-X~AI\B
AoB~Xf-AI\B~X
(~~)
XoA~Bf-X~A~B
(~~)'
*A
(v
~
X
B
~
X f- A
~
B
~
X
The introduction rules for 0 (= [F]) are such that - under suitable structural assumptions - they lead to Kf as the minimal modal logic:
( ~ D) (0 ~ )'
•X
~
A f- X
* • *A ~ X 75
~
DA
f- DA ~ X
76
A DISPLAY CALCULUS FOR PDL-
CHAPTER 6
In addition to the basic structural rules (1) - (4), (id), (cut) and the above introduction rules for the logical operations, we shall assume a parsimonious collection of further structural rules making sure that the structure connectives indeed have their intended classical meaning and, moreover, that the necessitation rule for 0 and the Alt1 schema -.0--.A ::) OA turn out to be provable. This particular set of additional structural rules allows the remaining rules from Table Ill in Chapter 3 to be derived, rules which in general one would like to have in unfolding the landscape of substructural logics.
(I 1) (Ir)
X1 o (X2 o X3) ---+ Z -lf- (X1 o X2) X1 o X2---+ z f- x2 o x1 ---+ z X oX---+ Z f- X---+ Z x1---+ z f- X1 o x2---+ z
(MN) (rAltl)
I ---+ X f- el ---+ X *X---+ Y f- • * •X
-t
3
1 2
X ---+ Z -ll- I oX ---+ Z X ---+ Z -H- X ---+ I o Z
(A) (P) (C) (M)
1 2
3 o X3
---+ Z
Y
Definition 6.1. The display system DKf is defined as the collection of all these sequent rules. Note that in Kf, 0 = 0, and indeed, in DK, * • *A---+ X f- OA---+ X is the invertible left introduction rule for 0 (= (F)). The rule (0 ---+) captures the D schema (OA ::) -.0-.A), while Alt1 is captured by (rAltl).
Theorem 6.2. The sequent I---+ A is provable in DKfiff A is provable in Kf.
4
T(*. *A---+ X) ·[F]·A::) T2(X) (F)A::) T2(X) [F]A::) T2(X) r(OA---+ X) T(*X---+ Y) •T2(X) ::) T2(Y) (P)[F]•r2(X) ::) r2(Y) (P)-.(F)r2(X) ::) r2(Y) (P)-.[F]r2(X) ::) r2(Y) r(• * •X---+ Y)
77
by functionality
by tense logic by functionality
Finally, we observe that Kft is a conservative extension of Kf. Q.E.D.
6.2. A DISPLAY CALCULUS FOR PDL-
In the language of PDL-, deterministic propositional dynamic logic without Kleene-star, a denumerable supply of basic program variables eo, e1, e2 ... gives rise to basic modalities [eo], [ei], [e2] etc. In the structural language of sequents, each such modal operator is matched by a structure connective •i, i E w. The display rules (4) are replaced by: (4') •i X---+ Y -lf- X---+ •iY, and the introduction rules for [ei] are:
(---+ [ei]) ([ei]---+)
•iX ---+ A f- X ---+ [ei]A * •i *A ---+ X f- [ei] ---+ X.
Proof. We first show that the provability of A in Kf implies f- I ---+ A. In view of Corollary 3.18, it is enough to (i) prove the D and Alt1 axiom schemata and (ii) derive the rule A---+ X f- OA ---+ •X. Here we only show (ii): A-t X *X---+ *A (rAltl) • * •X---+ *A *•X-+•*A * • *A---+ •X (0-+)' OA---+ •X
Whereas ([ei] ---+) captures seriality of the accessibility relation associated with ei, necessitation for [ei] and analogues of Alt1 are accounted for by additional structural rules. That is to say, (MN) and (rAltl) are replaced by:
To prove the converse, we show that f- s implies that r(s) is derivable in Kft, the tense logic of functional accessibility relations. Here we just consider (0---+ )'and (rAltl):
If A is a formula, then A? (test of A) is a program. Other program operations are ; (composition) and U (non-deterministic choice). If a, bare programs, then [(a; b)], [(a U b)] (or simply [a; b], [a U b]), and [A?]
I ---+ X f- •ii ---+ X *X ---+ Y f- •i * •iX ---+ Y
CHAPTER 6
REDUCTION TO A SET OF CLAUSES
are further modal operators. 26 In the intended models of PDL-, the accessibility relations Ra;b, Raub, and RA? associated with [a; b], [a U b], and [A?] are defined as follows:
6.3. REDUCTION TO A SET OF CLAUSES
78
Ra; Rb = {(a, ,8) I 3')' (a,')') ERa & (')', ,8) E Rb} Ra U Rb = {(a, ,8) I (a, ,8) ERa or (a, ,8) E Rb} {(a,a) I A is true at a}.
The meaning of [a; b] and [a U b] in the intended models thus depends on the meaning of [a] and [b], and this dependence is reflected in the following rewrite-style introduction rules for the compound modalities [a; b] and [a U b]:
(--+ [a; b]) ([a;b]--+) (--+[a U b]) ([a U b] --+)
[a]A
o
X --+ [b]A f- X --+ [a U b]A [b]A --+ X f- [a u b]A --+ X
The introduction rules for [A?]B are:
(--+ [A?]) ([A?] --+)
Clauses often are defined as finite disjunctions of literals: 'PO V ... V 'Pn V qo V ... V qm.
If the arrow --+ of a multiple-conclusion Gentzen sequent is understood as material implication, then a clause may be defined as a sequent in which --+ relates finite sets of propositional variables. The Kowalski form of a clause (cf. for instance [11]) is its presentation as a sequent:
In display logic clauses have the form:
X --+ [a][b]A f- X --+ [a; b]A [a][b]A --+ X f- [a; b]A --+ X X --+ [a]A
X--+ *A oB f- X--+ [A?]B *A--+ X B--+ X f- [A?]B--+ X
X--+ Y, where X and Y contain no compound formulas. This is also our notion of a clause in the context of DKf and DPDL-, i.e. no particular notion of 'modal clause' is introduced. 27 Suppose that the introduction rules for a logical connective f satisfy the following conditions:
(t)
Definition 6.3. The sequent system DPDL- is defined as the collection of the above sequent rules together with the structural rules from the previous section that have not been replaced.
(i)
(ii) (iii)
Recall that a structural sequent rule is said to correspond to a property of an accessibility relation R iff under the r-translation the rule is admissible in precisely the case that R enjoys the property. Theorem 6.4. The introduction rules for [a; b] correspond to the definition in intended models of Ra;b, those for [a U b] correspond to the definition of Raub, and those for [A?] correspond to the definition of RA?·
Corollary 6. 5. The sequent I --+ A is provable in DPDL- iff A is provable in PDL-.
Strong completeness of DPDL- (DKf) with respect to PDL- (Kf) follows in close analogy with the proof of Theorem 3.17. 26
A display calculus for relation algebra can be found in [78].
79
(explicitness) the right (left) introduction rules exhibit exactly one occurrence off in succedent (antecedent) position of their conclusion sequent, (separation) the rules exhibit no other connective than j, and every right (left) introduction rule is such that the antecedent (succedent) of the conlusion sequent is completely schematic.
If the display introduction rules for the logical operations satisfy (t), the invertibility of these rules in combination with the Display Theorem immediately provides an efficient method for transforming a given sequent s into an equivalent set of clauses .6. 8 such that .6. 8 f- s and from s every clause in .6. 8 is derivable. We may just successively display the occurrences of compound formulas in s and apply the inverted rules. It can be verified at a glance that the operational rules of DKf satisfy (t). The rules for [a;b]A and [aub]A in DPDL- violate condition (ii).
In contrast to this definition, in [116], for example, modal clauses are defined as formulas of the form (la V (h V ... ) ... )) or D(lo V (h V.·.)··.)), where the li are modal literals. A modal literal is an expression of the form p, --,p, Op, 0--,p, -,Dp or •D•p, where p is a propositional variable. 27
81
CHAPTER 6
REDUCTION TO A SET OF CLAUSES
Since these rules decompose [a; b] and [aUb] into the simpler modalities [a] and [b], however, invertibility still implies efficient transformability into an equivalent set of clauses. The problem with the Kleene-star is the absence of invertible rules for it.
Example 6. 7. Consider D(p ::::> q) ::::> (Dp ::::> Dq), an instantiation of the K axiom schema.
80
Observation 6. 6. The introduction rules of DKf and DPDL- are invertible. Proof. Consider, for instance, (D -+)'and(---+ D):
A---+A **A--+ A •*•*A---+A * • *A ---+ DA * • *A---+ X
X ---+ DA DA ---+ X
A---+A DA ---+ •A X---+ •A •X---+ A
The introduction rules for basic modalities [ei] can be dealt with analogously. As to the invertibility of the introduction rules for the non-basic modalities of PDL-, we have for instance:
X---+ [a; b]A
[a][b]A---+ [a][b]A [a; b]A---+ [a][b]A X ---+ [a][b]A
[a][b]A ---+ [a][b]A [a][b]A ---+ [a; b]A [a][b]A ---+ X
[a; b] ---+ X
[a]A ---+ [a]A o [b]A ---+ X ---+ [a U b]A [a u b]A ---+ X---+ [a]A [a]A [a]A [a]A
[a]A [a]A [a]A [a]A
---+ [a]A [b]A ---+ [b]A o [b]A ---+ [a]A [a]A o [b]A ---+ [b]A o [b]A ---+ [a u b]A [a U b]A ---+ X o [b]A ---+ X
The remaining rules follow similar patterns. Q.E.D.
I---+ D(p ::::> q) ::::> (Dp ::::> Dq) I o D(p ::::> q) ---+ Dp ::::> Dq D(p ::::> q) ---+ Dp ::::> Oq D(p ::::> q) o Op---+ Dq •(D(p ::::> q) o Op)---+ q D(p ::::> q) o Dp---+ •q D(p ::::> q) ---+ •q o *DP * • *(P ::::> q) ---+ •q o *DP * • * (p ::::> q) o Op ---+ •q Dp ---+ • * (p ::::> q) o •q * • *P---+ • * (p ::::> q) o •q * • *P o * • q ---+ • * (p ::::> q) •(* • *P 0 * • q)---+ *(P ::::> q) (p ::::> q) ---+ * • ( * • *P o * • q) *p ---+ * • ( * • *P o * • q) q ---+ * • ( * • *P o * • q)
Example 6. 8. The reduction applies to every modal formula, in particular also to non-theorems of Kf such as instances of the T schema: I---+ Dp ::::> p I o Op---+ p Dp---+ p
* • *P---+ p Example 6.9. Consider [ei; ej]P ::::> [ei][ej]p. (We omit some obvious steps.) [ei; ej]P---+ [ei][ej]P [ei][ej]P---+ [ei][ej]P * •i * * •j * p---+ [ei][ej]P •i * •i •j *P ---+ [ej ]p •j •i * •i •j * p---+ p Example 6.10. Consider [ei U ej]P
::::>
[ei]P A [ej]p.
[ei U ej]P---+ [ei]P A [ej]P * •i *P o * •j *P---+ [ei]P A [ej]P * •i *P o * •j *P---+ [ei]P * •i * P 0 * •j *P---+ [ej]P •i(* •i *P 0 * •j *P)---+ P •j (* •i *P 0 * •j *P)---+ P
82
CHAPTER 6
83
DECIDABILITY AND COMPLETENESS
6.4. DECIDABILITY AND COMPLETENESS
Also note that the rules
Suppose we want to know whether a given sequent s is provable in Kf or PDL-. The tableau method for these systems consists in first reducing s to an equivalent set of clauses ,6. 8 and then testing whether every clause in .6. 8 is provable. In tableaux for, say, cla..:;sical propositionallogic, a clause c is provable iff there is an atom occurring both as antecedent and as succedent part in c. In Kf and PDL-, this necessary condition fails to be sufficient, see Example 6.8. However, there is a natural and efficient proof search strategy. As a first step toward this strategy, note that a provable clause has a proof consisting of finitely many applications of single-premise structural sequent rules to an identity. This follows from cut-elimination.
are admissible, since they are preserved under the T-translation. Here we just consider two modalities:
•X-+ A X-+ DA
* • *A-+
y
DA-+ Y X-+Y
.,.,...
* • *A-+ y *y-+ • *A • * y-+ *A •X-+ A A-+* • *y •X-+ * • *y X-+•*•*Y X-+**y X-+Y
(rD)
Theorem 6.11. (Cut-elimination) If a sequent s is provable in DKf (DPDL-), then there is a cut-free proof of sin DKf (DPDL-). [a][b]A-+ Y X-+ [a][b]A [a;b]A-+ Y X-+ [a;b]A X-+Y
Proof. By a slight modification of Belnap's cut-elimination theorem for display logic [16] or its strengthening to strong cut-elimination, see Chapter 4. The logics in question obviously satisfy Belnap's conditions (C2) - (C7). To ensure cut-elimination, it remains to verify condition (C8), eliminability of principal applications of (cut). For this purpose, we appropriately modify condition C8 and define the rank of a formula.
Like the rules (D --t )' and ([ei] --t ), the rules (rD) and (rDi) correspond to the seriality of the respective accessibility relations. Hence we also have a semantic elimination theorem for these rules in the result of dropping (cut) from DKf and DPDL-. Moreover, applications of contraction, (C), can be eliminated from proofs of clauses.
Observation 6.12. (Contraction-elimination) If a clause c is provable in the display calculus DKf (DPDL -),then there is a contraction-free proof of c in DKf (DPDL -). Proof. We consider only cut-free proofs and successively eliminate the uppermost application of (C). Since all structural rules preserve polarity in the sense of not turning an antecedent part into a succedent part, or vice versa, the duplication of X in the premise of X oX --t Y 1 f-- X --t Y must result from an earlier application of (M), (JT), or (1 ). But then we may take the premise of this rule application and derive X --t Y without using (C).
In the present context, the rank r(A) of a formula A is inductively defined as follows:
r((A
e B))
r(DA) r([a; b]A) r([a U b]A) r([A?]B)
[a][b]A -+ Y
Q.E.D.
C8 Eliminability of matching principal formulas. If there are inferences inf1 and inh with respective conclusions (1) X --t A and (2) A--t Y with A principal in both inferences, and if cut is applied to obtain (3) X --t Y, then either (3) is identical to one of (1) or (2), or there is a proof of (3) from the premises of in!l and inh in which every cut-formula of any application of cut has a rank smaller than the rank of A.
r(p) r(-.A)
X -+ [a][b]A X-+Y
1
r(A) + 1 max(r(A), r(B)) + 1, 8 E {:::),A, V} r([ei]A) = r(A) + 1 r([a][b]A) + 1 max(r([a]A), r([b]A)) + 1 max(r(A), r(B)) + 1.
Corollary 6.13. (Decidability) Provability of sequents is decidable in DKf and in DPDL-. 1,
l
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CHAPTER 6
Proof. The reduction to clausal form reduces the provability problem for sequents to the provability problem for clauses. Cut-elimination licenses considering only proof search trees with the 'no fork property'. By contraction-elimination, the only complexity decreasing rules are (1) - (3) and the right-to-left directions of (F) and (I 1). The latter two rules are completely redundant for proof search. The rules (1) (3) merely register antecedent (succedent) parts on the right (left) of the sequent arrow. None of the non-basic primitive structural rules introduces a starred structure in its conclusion sequent. Hence, for the purpose of bottom-up proof search, we need not take into account the rule X -+ * * Y 1- X -+ Y. Since every sequent is finite, we are justified in finishing the proof search on finite branches. Q.E.D. Corollary 6.14. (Completeness of the strategy) If A is provable in Kf (PDL-), then every clause in 6.1-+A is provable in DKf (DPDL-) without (cut) and (C). We may continue the previous examples with the following (non)derivations:
Example 6. 7. (continued) p-+p **P-+P •*•*p-+p * • *P-+ •p * • *P o * • q -+ •p •( * • *P o * • q) -+ p *P -+ * • (* • *P o * • q)
q-+q *Q-+ *Q • * •q-+ *Q *•q-+•*q * • *P o * • q -+ • * q •( * • *P o * • q) -+ *Q q -+ * • (* • *P o * • q)
Example 6.8. (continued) ? * • *P-+ p Example 6.9. (continued) p-+p **P-+P •j * •j * p-+ p * •j *P-+ •jP •i * •i •j *P -+ •jP •j •i * •i •j * p -+ p
IN THE ABSENCE OF PURITY
85
Example 6.10. (continued) p-+p **P-+P •i * •i * p-+ p * •i *P-+ •iP * •i *P 0 * •j *P-+ •iP •i(* •i *P 0 * •j *P)-+ P
p-+p **P-+P •j * •j * p-+ p * •j *P-+ •jP * •i *P 0 * •j *P -+ •jP •j(* •i *P 0 * •j *P)-+ P
6.5. IN THE ABSENCE OF PURITY
In [11], A. Avron provides "precise criteria for when the existence of good resolution-like proof techniques might be expected" for a sequentstyle presentation of a given logic. Instead of inquiring into the provability of a sequent I -+ A, in the resolution method one tests whether the assumption that A is false results in inconsistency:
A -+ I 1- I -+ I? A vron shows that this procedure is justified if the sequent calculus under consideration is monotonic and 'pure', where the latter means that rule applications are preserved under the addition of structure in both antecedent and succedent position. Note that under the Ttranslation the residuation rules (4) (and (4')) fail to satisfy purity:
eX -+ Y X-+ •Y
does not imply
•X -+ Y o Z X-+ •Y 0 z
The example of a logic not satisfying purity pointed out in [11] is intuitionistic logic. Although normal modal logics have a classical basis, satisfying double negation elimination, the presence of the necessitation rule 1- A/I- DA blocks the test for inconsistency. Clearly, p :J Dp is a non-theorem of Kf; nevertheless the test applies.
Example 6.15. We reduce p :J Dp -+ I to clausal form and test whether the resulting clauses lead to inconsistency. Note that we use the rule (rD). p :J Dp-+ I
*P-+ I *I-+ P
Dp-+ I * • *P-+ I *I-+ • * P • *I-+ *P P-+ * • *I
*I -+ p p -+ * • *I *I-+* • *1 I o *I-+* • *I I -+ * • *I (MN) I -+ • * • * I (rD) I-+**I I-+I
CHAPTER 7
STRONG CUT-ELIMINATION AND LABELLED MODAL TABLEAUX
The method used in Chapter 4 to show that every displayable logic enjoys strong cut-elimination was derived from the proof of strong normalizability of typed .A-terms. It does not only apply to display calculi. The present chapter is devoted to a proof of strong cut-elimination in a labelled tableau calculus for the (constant domain) modal predicate logic QS5. Modal tableau calculi which build in the accessibility relation of possible worlds models were first introduced by Kripke [93] and were later 'linearized' by various authors, notably Fitting [62], [63], [64] and Mints [114]. As in Gabbay's [68] theory of labelled deductive systems, the basic declarative unit of these tableau calculi is not just a formula A, but rather a formula plus label (a, A). In the case of the modal logic S5, the label a may just be a single positive integer, whereas in general it is a non-empty finite sequence of positive integers. Moreover, for S5 the accessibility relation between labels may be universal and hence neglected. In contrast to labelled tableaux, the modal tableau systems of, for example, Rautenberg [137] and Gore [74] do not use labelled formulas. For a general survey on tableau methods for modal and tense logics, see [79]. The use of labels allows to formulate tableau calculi for certain extensions of the minimal normal modal logic K by imposing constraints on accessibility and on occurrences and the shape of labels on tableau branches. These constraints may be regarded as structural in the sense of not referring to any connectives. In order to emphasize the relation to sequent calculi, we shall work with a tableau calculus TQS5 based on the ordinary notion of a sequent. By defining suitable mappings on cut-free closed tableaux it can easily be shown that the result of dropping cut from TQS5 is equivalent to Fitting's tableau calculus for first-order S5 with respect to provable formulas (see Section 7.5). Usually modal tableau calculi are formulated without a cut rule. The admissibility of cut is, however, of interest for constructive proofs of equivalence with Hilbert-type systems; compare [93, p. 82]. Moreover, non-constructive proofs of cut-elimination are of little appeal when it comes to extending the notion of formuals-as-types to modal logic (see, for instance, [28], [106]). It is this respect in which the present chapter may be seen to have significance. Recently, G. Mints [115] presented a cut-elimination proof for certain
87
L
88
CHAPTER 7
PRIMITIVE REDUCTIONS
labelled tableau calculi covering all combinations of the axiom schemata T, B, and 4 above propositional K. Moreover, cut-elimination has been proved not only for displayed S5 (see Chapter 4) but also for a number of other more or less standard sequent calculus presentations of S5; see, for instance, [89], [91]2 8 , [112], [150].
Table VI. The rules of TQS5.
7.1.
THE TABLEAU CALCULUS
LV RV L-. R-. LOS5 RDS5
TQS5 LV
We assume a first-order language C including 0 but without function symbols and equality. The set H supplies denumerably many individual constants not already in C. Bold letters X, Y, Z (possibly with primes or subscripts) denote arbitrary finite sets of labelled first-order formulas. A sequent is an expression of the form X ~ Y, where X is called the antecedent and Y is called the succedent of this sequent. We uses, s1, s2, ... to denote sequents and the 'turnstile' f- to denote derivability between single sequents and finite sets of sequents. Tableau calculi are given by (finite) sets of derivation rules of the form s f- s1, ... , Sn· A tableau for a given sequent s is a tree of sequents rooted ins, such that every node of the tree is an instantiation of one of the derivation rules of the tableau calculus under consideration. A tableau for s is closed if every leaf of any branch of the tableau has the form (a, B) ~ (a, B). The system TQS5 to be considered is a tableau calculus for constant domain first-order S5. As already mentioned, for S5 we may assume that the labels a are positive integers. Let i, j, k, ... range over positive integers. TQS5 is given by the derivation rules in Table VI.
RV
mon id cut
89
X, (a, A V B) -+ Y f- X, (a, A) -+ Y X, (a, B) -+ Y X -+ (a, A V B), Y f- X -+ (a, A), (a, B), Y X, (a, -.A) -+ Y f- X-+ (a, A), Y X -+ (a, -.A), Y f- X, (a, A) -+ Y X, (i, DA)-+ Y f- X, (j, A)-+ Y X-+ (i,DA),Y f- X-+ (k,A),Y, provided k is new on the tableau branch X, (a, VxP(x)) -+ Y f- X, (a, P(t)) -+ Y for any closed term t X-+ (a, VxP(x)), Y f- X-+ (a, P(c)), Y, provided c E H is new on the tableau branch X, Z -+ Z', Y f- X -+ Y f- (a, A) -+ (a, A) X, X'-+ Y, Y' f- X-+ (a, A), Y X', (a, A) -+ Y'
iff (a, A) = (a', A'). A non-parametric formula in the premise of inf is called a principal formula of inf
7.2. PRIMITIVE REDUCTIONS
Our objective is to show that every (sufficiently long) sequence of reduction steps in a process of eliminating applications of cut
(1)
X~
(3) x,x'~Y,Y' (a, A), Y (2) X', (a,
A)~
Y'
. • from a given closed tableau for a sequent Z ~ ZI termmates m a cut-free closed tableau for Z ~ Z'. For that purpose we have to define primitive reduction steps. We shall distinguish between two kinds of reductions, principal moves and parametric moves. If the labelled cut-formula (a, A) is principal in the initial inferences of the closed tableaux for both (1) and (2), the application of cut is called a principal cut. Otherwise, if the cut-formula is parametric in at least one of the initial inferences below (3), the application of cut is said to be a parametric cut. We perform principal moves on principal cuts and parametric moves on certain parametric cuts.
The tableau rules of Table VI are presented by means of rule schemata exhibiting, among other things, variables for formulas, labels, terms, and finite sets of labelled formulas. Every instantiation inf of such a rule R is an inference falling under R. If inf falls under R, we write inf E R. Every labelled formula in inf is called a constituent of inf If inf falls under an operational rule, that is, a rule decomposing one of the primitive logical operations, every constituent in the instantiation of X or Y is said to be a parameter of inf If inf E cut or inf E mon, then every constituent of inf is said to be a parameter of inf A constituent (a, A) in the premise sequent of an inference inf is defined to be congruent to a constituent (a', A') in a conclusion sequent of inf
Principal moves. In order to present the principal moves, we need . . t ant'Isome terminology and some lemmata. If mf E RD 5 5 , t hen t he ms ation of k is called the eigenlabel of inf. If inf E RV, then the displayed constant c is called the eigenconstant of inf.
28
Note that in Kanger's 'spotted formulas', only atomic formulaoccurrences are labelled.
l
90
CHAPTER 7
Lemma 7.1. Consider any inference inf falling under a rule R from TQS5. (i) If every occurrence of a label which is not the eigenlabel of inf is replaced by a label which again is not the eigenlabel of inf, then the resulting inference falls under R. (ii) If every occurrence of a closed term which is not the eigenconstant of inf is replaced by a constant which again is not the eigenconstant of inf, then the resulting inference falls under R. = RD 55 ,
Proof. (i) The only non-trivial case is R but since we may neither replace by the eigenlabel nor replace the eigenlabel, the result of the replacement falls under RD 55 . (ii) Analogous. Q.E.D. Lemma 7.2. In TQS5 every closed tableau IT for a sequent X --? Y can be converted into a closed tableau IT* for X --? Y such that in IT*,
(i) for every inf E RD 55 we have: the eigenlabel i of inf occurs only in sequents below the premise of inj, and i does not occur as eigenlabel of any inf' E RD 55 such that inf -1inf'.
Case 2.1. (a, A) =(a, B V C):
X, X'--? Y, Y' X--?(a,BVC),Y X',(a,BVC)--?Y' X--? (a, B), (a, C), Y X', (a, B) --? Y' X', (a, C) --? Y' IT'2 IT"2 IT '1 X, X'--? Y, Y' X, X'--? (a, C), Y, Y' X--? (a, B), (a, C), Y X', (a, B) --? Y' IT'1 IT'2
the eigenconstant c of inf occurs only in sequents below the premise of inf, and c does not occur as eigenconstant of any inf' E RV such that inf -1- inf'. Proof. (i) By renaming eigenlabels. Consider an inference inf E RD 55 in IT such that there is either no inference falling under RD 55 below the premise of inf, or every such inference has already been subjected to renaming. We replace every occurrence of the eigenlabel of inf below the premise of inf by a new label not already occurring in the tableau. This process is then iterated in order to obtain IT*. By the first part of the previous lemma, correctness of rule applications is preserved. (ii) Analogous. Q.E.D.
There are two cases to be distinguished. Case 1. Ili falls under id: if i = 1 ifi = 2
Case 2. The initial steps in Il1 and Il2 are instantiations of corresponding right and left rules decomposing the cut-formula (a, A).
X', (a, C) --? Y' IT"2
Case 2.2. (a, A) = (a, -.B):
X, X'--? Y, Y' X--? (a, -.B), Y X', (a, -.B) --? Y' X, (a, B) --? Y X'--? (a, B), Y' IT'2 IT'1 X, X'--? Y, Y' X'--? (a, B), Y' X, (a, B) --? Y IT'1 IT'2
(ii) for every inf E RV we have:
(3)
91
PRIMITIVE REDUCTIONS
Case 2.3. (i, A)
= (i, DB):
X, X'--? Y, Y' X--? (i, DB), Y X', (i, DB) --? Y' X--? (k, B), Y X', (j, B) --? Y' IT'2 IT'1
X--? (j, B), Y IT't(j I k]
X, X'--? Y, Y' X', (j, B)--? Y' IT~
Thus for this primitive reduction step we first rename eigenlabels in IT~ a~cording to the proof of Lemma 7.2 (i) and obtain IT~* (such that j does not occur as eigenlabel in IT~*). Then we replace every occurrence of k in IT~* by j. Lemma 7.1 (i) ensures correctness of rule applications, i.e., X--? (j, B), Y Il~*[j /k]
is in fact a closed tableau for X---+ (j, B), Y. Case 2.4. (a, A) = (a, VxP(x)): X, X'---+ Y, Y' X---+ (a, VxP(x)), Y X', (a, VxP(x)) ---+ Y' X---+ (a, P(c)), Y X', (a, P(t)) ---+ Y'
II'1
II'2 X, X'---+ Y, Y' X---+ (a, P(t)), Y X', (a, P(t)) ---+ Y' II~*[t/c]
93
PRIMITIVE REDUCTIONS
CHAPTER 7
92
II~
That is, for this primitive reduction step we take any result II~* of renaming eigenconstants in II~ according to the proof of Lemma 7.2 (ii) {such that t does not occur as eigenlabel in II~*) and replace every occurrence of c in II~* by t. Lemma 7.1 (ii) ensures correctness of rule applications.
Parametric moves. In order to define the parametric moves we have to observe that the tableau rules are sufficiently pure. A tableau rule R is said to be sufficiently pure if whenever Xi ---+ Yi (i = 1, ... , n) can be derived from X---+ Y by means of R, then Xi, X'---+ Yi, Y' can also be derived by means of R from X, X' ---+ Y, Y', provided no side conditions on rules are violated. 29
(2). If there is an applicati~n of cut in the tr~e, ~e do not perform a reduction, but instead consider one such apphcat10n of cut for reduction. If there is no application of cut in the tree, then for every path of parametric ancestors of (a, A) in Ih with respect to in/', we consider its bottommost element (a, A)b· Let s be the sequent containing (a, A )b. There are two possibilities. Case 1. (a, A)b is not parametric in the inference starting at s. We then (i) cut with (1), (ii) replace every parametric an?estor of (a, A) above (a, A)b by X, and (iii) add Y to every succedent m the tableau branch leading from s to (2). Case 2. (a, A)b is parametric in the inference starting at s. We (i) replace every parametric ancestor of (a, A) in the path leading to (a, A)b by X, (ii) add Y to every succedent in the tableau branch leading from s to (2), and (iii) delete IIt, which is now superfluous. Simultaneously carrying out these transformations for every (a, A)b and then removing the inital occurrence of (3) (since now (3) = (2)) constitutes a parametric reduction step. The previous observations ensure that we in fact obtain a closed tableau for (3), since the substitution by X and the addition of Y does not interfere with any side conditions. In Case 1, schematically we have:
Observation 7.3. Every rule is sufficiently pure. Observation 1.4. All rules are closed under uniform substitution of finite sets of labelled formulas for congruent parameters, provided no side conditions are violated. What the parametric moves make clear is that parametric cuts need never immediately precede applications of operational tableau rules. Suppose the cut-formula (a, A) is parametric in the initial inference inf' of the closed tableau II2 for (2). (The case for {1) is similar.) We define a set Q of occurrences of (a, A), called the set of parametric ancestors of (a, A) in II2 with respect to inf' (cf. Chapter 4). We begin with putting the exhibited occurrence of (a, A) into Q, and, working downwards, we include for every inf E II2 each antecedent occurrence of (a, A) of any conclusion of inf. What we obtain is a tree of parametric ancestors of (a, A) rooted in the exhibited occurrence of (a, A) in 29
Purity simpliciter is defined without the proviso concerning side conditions (see, for instance, [11] and Chapter 6).
(3)
(3)
X-+ (a, A), Y
X', (a, A) -t Y'
II~
X", (a, A)b -t Y" II~
11\ X, X" -t Y, Y" X ", (a, A)b -+ Y" X -+ (a, A) ' Y II'!
Example 7.5. Let so = (r, -.E V A), (r, E) ---+ (a, -.(B V -.D)), (r, A), and s 1 = (r, -.E V A), (r, E), (a, C) ---+ (r, A), (a, C).
94
CHAPTER 7
95
STRONG NORMALIZATION
Lemma 7.6. If a closed tableau is not reducible, then it is cut-free. (r, ·E V A), (r, E), (O", B V C) -+ (r, A), (O", C), (ry, H) (r, ·E V A), (r, E) -+ (O", ·(B V ·D)), (r, A) IT (r, ·E V A), (r, E) -+ (r, A) (r, •E), (r, E) -+ (r, A) (r, A) -+ (r, A) (r, E) -+ (r, E), (r, A) (r, E) -+ (r, E)
Proof. Since the above case distinction is exhaustive, every closed tableau that contains an application of cut is reducible. Q.E.D.
Definition 7. 7. We inductively define the set of inductive closed tableaux.
a Every identity {O", A) -+ (O", A) is an inductive closed tableau. b If IT is closed and begins with an inference inf different from cut, and
every conclusion sequent
(r, ·E V A), (r, E), (O", B V C) -+ (r, A), (O", C), (ry, H) (r, ·E V A), (r, E), (O", B V C) -+ (r, A), (O", C) (r, ·E V A), (r, E), (O", B) -+ (r, A), (O", C) so (r, • (B V ·D)) , (O", B) -+ (0", C)
where
in
s1
'(O"=--,C-:::;;-)-+--:(-0",--=C,. ,. )
c
n,
n=
then
of inf has an inductive closed tableau lli
n is inductive.
(3)
llz inductive. IT1
Si
cut
is inductive, if every L: such that IT
>
L: is
Lemma 7.8. If IT is closed and inductive, and IT > L:, then L: is inductive.
n=
(a, •(B V ·D)), (a, B V C) -+ (CJ, C), (TJ, H) (CJ, •(B V •D)), (a, B V C) -+ (CJ, C) (u, •(B V •D)), (a, B)-+ (CJ, C) (u, •(B V •D)), (CJ, C)-+ (a, C) (a, B) -+ (CJ, B V D), (a, C) (CJ, C) -+ (CJ, C) (a,B)-+ (CJ,B),(a,D),(a,C) (CJ, B) -+ (CJ, B)
7.3. STRONG NORMALIZATION
The proof of the strong normalization theorem to be given is derived from the proof ~f strong norma~ization for displayable logics in Chapter 4. I.n order to give a self-contamed presentation, we shall rehearse the entire argument from Chapter 4. Suppose that IT is a closed tableau containing an application of cut. A (one-step) reduction of IT is the closed tableau L: resulting by ap?l!ing a primitive reduction to a subtableau of n. If n reduces to L:, ~his Is denot~d by IT > L: (or L: < IT). IT is said to be reducible iff there IS a L: to which n reduces.
Proof. By induction on the construction of IT. If IT is inductive by a, it cannot be reduced. If IT is inductive by b, then every reduction of IT takes place in the lli's, which are inductive. Hence, by the induction hypothesis, L: is inductive due to b. If n is inductive by c, then L: is inductive by definition. Q.E.D.
Definition 7.9. Let IT be an inductive closed tableau. The size ind(IT) of IT is inductively defined as follows (the clauses correspond to those in the previous definition): a ind(IT) = 1;
b ind(IT) =
L:i ind(ITi) + 1;
c ind(IT) = 2::I:
+ 1.
A closed tableau IT is said to be strongly normalizable iff every sequence of reductions starting at n terminates.
Lemma 7.1 0. Every inductive closed tableau is strongly normalizable.
96
Proof. By induction on ind(IT). If ind(II) = 1, no reduction is possible. If IT is inductive by b, then every reduction is in the conclusions ITi, and we can apply the induction hypothesis. If IT is inductive by c, then every proofto which IT reduces is inductive, and hence, by the induction hypothesis, every such proof is strongly normalizable. Thus, IT is also strongly normalizable. Q.E.D. Lemma 7.11. Let IT be an inductive closed tableau and let inf be the uppermost inference of IT. If IT > IT' by reducing a closed tableau lli of a conclusion sequent of inj, then ind(IT) > ind(IT'). Proof. By induction on ind(IT). If ind(II) = 1, then IT cannot be reduced. Whence IT is inductive by b or c. If IT is inductive by c, then by definition, ind(IT) > ind(IT'). If IT is inductive by b, then ITj is inductive by definition. If llj is inductive by a, it cannot be reduced. If llj is inductive by b, then the reduction of 11j to ITj takes place in the closed tableau of some conclusion sequent of the uppermost inference of 11j. By the induction hypothesis, ind(ITj) > ind(ITj). Hence ind(IT) > ind(IT'). If 11j is inductive by c, then by definition, ind(ITj) > ind(IIj) and thus ind(IT) > ind(II'). Q.E.D. Lemma 7.12. Suppose closed IT begins with an application inf of cut, and ll1 and II2 are the two closed tableaux of the conclusions of inf. If ll1 and II2 are inductive, then so is IT. Proof. We must show that every E
+ ind(II2).
We use induction on r(II) and, for fixed rank, induction on h(IT). Case 1. E is obtained by reduction in ll 1 or II2, say II 1 > IT~. Then r(IT) = r(E), and we use induction on h(IT). It follows from Lemma 7.11 that ind(II~) < ind(III). Then h(E) < h(IT). Since II1 and II2 are inductive, by Lemma 7.8, E has inductive conclusions, and by the induction hypothesis for h(IT), E is inductive. Case 2. E is obtained by reducing inf. Then this reduction was either a principal or a parametric move.
Principal move. Consider the two cases distinguished in the presentation of the principal moves.
97
EXTENSIONS OF QUANTIFIED K
CHAPTER 7
Case 1. Since E is a closed tableau for (1) or (2), 'E is inductive by assumption. Case 2. Since for every new tableau IT' starting with an ap~lication of cut, r(ll) > r(ll'), E is inductive by the induction hypothesis for r(ll).
Parametric move. We have (3)
(3) X--+ (a,A),Y
IT~
11\
X',(a,A)--+ Y'
11\
X", (a, A)b --+ Y"
X, X"--+ Y, Y" X", (a, A)b --+ Y" X --+ (a, A) ' Y
IT~
may assume that there is no application of cut on the path from an d w e , X', (a, A) ---+ Y' to X", (a, A)b --+ Y'. xr--+Y~ X"(A)--+~ , ' , a, b 11 X--+ (a, A), Y X", (a, A)b --+ Y" and IT = IT~ IT' IT' Consider rr and IT'~ Obviously, r(ll) = r(IT'), so we ~se in~ uctio.n on the height. Since both rr 2 and IT" are i.nductive by~' md(IT ) <, ~n~(II2)~ Hence h(IT') < h(IT). By the inductiOn hypothesis for ~(I·I)·, IT IS md~c
Let IT'
=
1
tive and thus 'E is inductive by definition. If the pnmitive reductiOn fIT to 'E requires cutting with ill more than once, analogously every ~ew TI' and hence 'E can be shown to be inductive. Q.E.D.
Corollary 7.13. In TQS5 every closed tableau is inductive. Theorem 1.14. In TQS5 every closed tableau is strongly normalizable. Proof. By Lemma 7.10 and Corollary 7.13.
Q.E.D.
Corollary 7.15. Cut is an admissible rule of TQS5. Proof. By Lemma 7.6.
Q.E.D.
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EXTENSIONS OF QUANTIFIED K
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CHAPTER 7
Table VII. Conditions on accessibility. name of condition general reverse 30 reflexivity transitivity
I
I definition
universal
I
for every n, an is accessible from a for every n, a is accessible from an a is accessible from a if a is a proper initial segment of 7 then T is accessible from a ' any label is accessible from any label
[64, p. 402]), namely:
Table VIII. Accessibility conditions for various modal logics. logic K,KD KT KB, KDB KTB K4, KD4 S4 ( KT4) S5 ( KTB4)
J
right with a single positive integer and (ii) T is not an initial segment of any label occurring on the branch. The label T is available on a branch if it occurs on that branch. The right and left rules for 0 can be stated in such a way that variations among the systems listed in Table VIII can be accounted for by 'structural' side conditions on the left rule (cf.
conditions on access"bTt 1 11 y general general, reflexivity general, reverse general, reflexivity, reverse general, transitivity general, reflexivity, transitivity universal
LD
X, (a,DA)-+ Y 1- X, (T,A)-+ Y for any T accessible from a provided (i) forK, KB, and K4, T must be available on the branch; (ii) for KD, KT, KDB, KTB, KD4, S4, and S5, T must either be available on the branch or be a simple, unrestricted extension of a
RD
X-+ (a, OA), Y f- X-+ (T, A), Y provided T is a simple, unrestricted extension of a
The tableau rules for V, ....,, V and the structural rules mon and cut remain unchanged. If A is any system listed in Table VIII, let TQA be the tableau presentation of its constant domain first-order extension. If we try to reuse the proof of strong cut-elimination for TQS5 in order to establish strong cut-elimination for TQA, we have to be careful since both RD and LD come with complex side conditions. Reconsider the earlier reduction strategy. Suppose we try to reduce a principal cut with cut-formula (a, DB):
7.4.
EXTENSIONS OF QUANTIFIED
X,X' -+Y,Y' X-+(a,DB),Y X',(a,DB)-+Y' X-+(T,B),Y X',(T',B)-+Y' IT'1 IT'2
K
. a non-empty finite Let us recall from [64] · A lab e1 now IS f some . . material . ~:~u~nce o positive mt~gers. We assume a binary relation of 'accessibil1 y etween labels. Thts relation may satisfy certain conditions and a number of such . ' exten· fK h conditions is defined in Table VII · K an d vanous siOns _o t at _can be dealt with by means of labelled tableaux . cert~m p~operties of accessibility between labels. These co d"f reqmre n I tons are specified m Table VIII.
Since a may occur as a label in IT~ and IT~, it is not at all obvious whether in the attempted reduction X,X'-+ Y,Y' X-+ (T', B), Y X', (T', B)-+ Y'
IT't [71/T]
in fact A l_abel T occurring on a tableau branch is said to be a sim l strzcted extension of a label a iff (i) 7 is the result of extendin:
;·o~n;:~
tr;'. This is Fitting's terminology. The traditional term is, of course, 'symme-
X, (T 1 , B)-+ Y
IT't[T' /T]
IT~
and
X', (T1, B)-+ Y' IT'2 1
are closed tableaux for X, (T 1, B) -+ Y and X', (T , B) -+ Y', respectively. We therefore leave the following as an open
101
CHAPTER 7
EQUIVALENCE WITH FITTING'S TABLEAUX
Problem. Can the proof of strong cut-elimination for TQS5 be recycled in order to establish strong cut-elimination for TQA, if A =I S5?
We write CTp(Il,((1),A)) ifii is a closed F-tableau for ((1),A), and we write CTc(II, X--+ Y) if II is a closed G-tableau for X--+ Y. Define J\{(a,A 1), ... ,(a,Ak)} as (A1!\ ... !\Ak) and •V{(a,AI), ... ,(a,Ak)} as ( --.A 1/\ .. .1\•Ak)· If X= { ( (1), A1 ), ... , ( (1), An)} ~nd: = {( (1), B1), ... , ( (1), Bm) }, then from the definition off and g It will become evi-
100
7.5. EQUIVALENCE WITH FITTING's TABLEAUX
dent that We show that without cut the above tableau calculi are equivalent to those of Fitting [64] in the sense that for every quantified modal logic under consideration, there is a closed tableau a la Fitting (F-tableau) for ((1),--.A) iff there is a closed (cut-free) tableau for--+ ((1),A) in the above Gentzen-style (G-tableau). An F-tableau for ((1),--.A) is a tree with root ( (1), --.A). The nodes of the tree are labelled formulas and each ramification is an application of one of the following decompostion or branch extension rules to an earlier node on the branch:
CTc(II,X-+Y) implies CTF(rrf,((1),f\XA·VY)) and
CTp(Il,((1),--.A)) implies CTc(II 9 ,-+ ((1),A)).
The function
f
is inductively defined as follows:
id: II = ( (1), A) -t ( (1), A). ( (1), A A -.A)
(u, ...,...,A) (u, A)
rr' = (u,...,(A V B)) (u,-.A) (u, •B)
(u, A V B) (u, A) I (u, B) (u, DA)
Then (
unrestricted extension of u
(u, VxP(x))
(u, -,\fxP(x)) (u, •P(c)) for any new c E H
(u,P(t)) for any closed term t
Given the definition of A!\ B as •( --.A V --.B), we assume the following derivable decomposition rules for J\{A 1 , ... , An}: (u,A{AI, ... ,An}) (u, AI)
X,((1),AI V A2)-+ Y X, ((1), A1)-+ Y X, ((1), A2)-+ Y rr1 rr2
LV: II =
(u, -.DA) ('y, -,A) provided 1 is a simple,
('y, A) provided 1 is accessible from u and satisfies the appropriate conditions, viz. those stated in LD
X,((1),Ai)-tY )' = rri and rrt =
((1),AXA1iA•VY) IIi
((1), AX!\ (Al V A2) A-.., VY) ((1), (AI V A2))
I
II~
RV:
II
=
Then (X-+
((1),A~;((l),A 2 ),Y and IT =
A{A 1 , ... , An}) I ... I (u,...,An)
(u,-,AI)
L•: II =
II~
X -t ((1), AI V A2), Y X -t ((1),AI),((1),A2),Y rr1
!
( 0'' -..,
( (1), A) ( (1), -.A)
)' =
((1),AXA•Ah~'A2/\•VY)
((1), A X A ·(AI V A2) A..., V Y) II'I
X,((1),...,A) -+Y X -t ((1), A), Y
IT 1 =
( X-t ((1),A),Y )'
rr 1
Ill
An F-tableau for ((1),--.A) is closed if every branch of the tableau contains (a, B) and (a, --.B) for some formula B and label a. We shall define a function ( · )f from closed G-tableaux for sequents {( (1), AI), ... , ( (1), An)} --+ {( (1), BI), ... , ( (1), Em)} to closed F-tableaux and a function (·)9 from closed F-tableaux to closed G-tableaux.
The remaining cases are left to the reader. We may conclude that CTc(II,--+ ( (1), A) implies CTp(IIf, ((1), --.A)). . Now we proceed to the definition of g. Suppose II IS a closed Ftableau for ((1),A). Starting at ((1),A) and working downwards, we build up a closed G-tableau II9. For this purpose we define a map m of
102
CHAPTER 7 CHAPTER 8
labelled formulas and certain pairs of labelled formulas into sequents. The map m applies to ( (1), A), the root of II, as follows:
( (1), A)m = { --+ ( (1), B) ((1),A)--+
if A has the shape ·B otherwise
TARSKIAN STRUCTURED CONSEQUENCE RELATIONS AND FUNCTIONAL COMPLETENESS
When applying (a, .-.A) (a, A)
we may assume that (a, -.-.A)m is a sequent X --+ (a, -.A), Y. Then (a, A)m =X, (a, A) --+ Y. When applying (a, (AI V A2)) (a, AI) I (a, A2)
r.-
we may assu~e that (a, (A1 VA2))m is a sequent X, (a, (AI v A2)) --+ Y. Then (a, Ai) =X, (a, A) --+ Y. When applying (a, -.(AI V A2))
(a, ·AI) (a, •A2) (a, -.(AI V A2))m will be a sequent X--+ (a, (AI V A2)), Y. Then
( ~~: =~~j )m= X--+ (a, AI), (aA
Many proofs of functional completeness for intuitionistic propositional logic and related systems make use of a higher-level natural deduction or sequent-style proof-theoretic semantics, see e.g. [8], [96], [97], [151], [178], and [180]. The idea now is to apply this kind of approach to Gabbay's [67] notion of a Tarski-type structured consequence relation between structured databases ~ and single formulas A. This concept generalizes the ordinary notion of single-conclusion consequence relations and is intended to capture a broad range of non-monotonic reasoning systems, like Defeasible Logic and Probabilistic Nets (for concrete examples and motivation see [67]). Intuitively speaking, a structured database is a structure T together with an assignment of formulas to the elements of the domain ofT. Working within a sequent calculus framework we assume that a structured database ~ is always finite. With every Tarski-type structured consequence relation we by specifying shall associate a certain positive propositionallogic introduction rules for various propositional connectives and constants which naturally arise in the context of structured databases. The main novelty here is Gabbay's idea to conceive of implication as depending on a contraction mapping operating on the underlying structures. It is then shown that this collection of connectives and constants is expressively complete with respect to a semantics in terms of rather natural introduction schemata. In a second step this approach is extended to constructive propositional logics associated with Tarskian structured consequence relations. The constructive negation ,...., of these systems is motivated by the idea of taking refutations seriously and therefore considering the notion of disproof as primitive and of equal importance as the positive notion of proof; see also Chapter 9. Cut-elimination and functional completeness of the extended set of operations will be established.
2 ),
Y.
T~e. decomposition rules for D and V are dealt with in a similar way. II Is. ~he result ~f applying m to the labelled formulas of II. Every ~ransttt.on_ from smgle sequents to sequents obtained in this way is an mstant1at10n of some sequent rule. Hence, if II is a closed F-tableau then IIY is a closed G-tableau. ' Thus, CTF (II, ( (1), -.A)) implies CTa (ITY, --+ ( (1), A)). Since moreover CTa(II,--+ ((1),A) implies CTp(III, ((1),•A)), with respect to provable form~las the method of cut-free G-tableaux is equivalent to that ofF-tableaux.
r.-+
r.-
8.1. PRELIMINARIES
Let us introduce the definitions we (more or less) adopt from [67].
103
POSITIVE LOGICS
CHAPTER 8
104
Definition 8.1. Let T be a theory in first or higher-order logic and let M be the class of all classical, finite T-models 7 with non-empty domain I 7 I· We assume that M contains one-point structures and is closed under a binary operation + {'data addition'), which is defined for pairs of disjoint structures in M. We suppose that 71 and 72 are substructures of 71 + 72. Moreover, for every 7 E M, 0 + 7 and 7 + 0 are defined and equal to 7. If 71, 72 are two disjoint structures in M and x E I 71 I, then an associative substitution operation Sub~2 7!{ x) is defined (yielding the result of substituting 72 for x in 7I}. Also, Sub07(x) is defined and is regarded as deletion of x in 7, and for every one-point structure 7, Sub~71(x) is isomorphic to 71. Let now£ be any propositionallanguage. {1) A {structured) database is any pair (7, <5) such that 7 E M and 8 maps each x E I 7 I to a formula 8(x) of£. We write ~[A] for~= (7(x),8), if 8(x) =A. {2) A Tarskian structured consequence relation for the pair (M,£) is any relation r-- between databases ~ = (7, 8) and single formulas A of £ {denoted by~ r-- A) satisfying the following rules: Identity and Cut for M
(7(x),8) r--A, if 7(x) is a one-point structure and 8(x) =A, If 71, 72 are two disjoint structures in M, ~ = (71, 81), r = (72, 82), ~ r-- A and r[A] r-- B, then r[~] = (7, 8) r-- B, where 7 = Sub~1 72(x) and 8(y) = 8i(y), if y E I 7i I (i = 1, 2).
105
substitutions: (*)
for every 7, 71, 72,73 E M and every II'P on M, Subflcp(r) (Sub~ub~ 1 73 (y,z) 72{x)) = Sub~ub~ 1 7 72(x ), if 1731={y,z} and
If~= (7,8), then II'P(~) = (II'P(7),8
r (I 7 1- {
For every contraction mapping II'P we can define an implication ~'P by means of the Deduction Theorem: II'P((7,8)) r-- A ~'P B
iff (7,8(
Note that Gabbay [67] considers arbitrary contraction mappings II'P' in particular he does not require (*). Suppose that 73 (y, z) is a twopoint structure such that
We write A r-- A for {7{x), 8) r-- A, if 7 is a one-point structure and 8(x) =A. We write ~[A+B] for b.= (Sub~1 +r2 7(x),8), if71(y) and 72(z) are one-point structures, 81 (y) =A, and 82(z) =B. If~= (71, 81) and r = (72, 82), then~+ r = (71 + 72,81 u 82).
With each Tarskian structured consequence relation r-- we associate a propositional system r--+ called r--'s positive propositionallogic.
Definition 8.2. Let M be a class of structures as in the previous definition and let
Definition 8.4. Let M be a class of structures as in Definition 8.1 and let£+ be the language in the propositional constants @and @and the binary connectives E9, @,@)and ~'P' for every contraction mapping II'P on M (that satisfies (*)). Given a structured consequence relation r--, the sequent calculus r--+ is defined by the introduction rules in Table IX.
8.2. POSITIVE LOGICS
106
THE HIGHER-LEVEL SEQUENT CALCULUS
CHAPTER 8
G~
107
Table IX. Introduction rules of f-v+
(f-v (!)) (f-v @) C@H
1-~f-v®
(f-v EB) (EB H
~f-vA ff"vBI-~+ff"vAEBB
(f-v @) C®H (f-v @)) C®H (f-v:J.,) (:J.,H*
l-0f-v@ ~[A) f-v B 1- ~[A+~ f-v B ~[A) f-v B 1- ~[@+A] f-v B
~ = ({cp(7)},6
~[A
+ B] f-v C 1- ~[A EBB) f-v C (i.e. ~[A+ B)= (Sub~,(y)+T2 (z)7(x),6) f"v C 1- ~[A EBB)= (7(x),6') f-v C, where 6' = {(x, A EBB)} U (6- {(y, A), (z, B)})) ~ f-v A ~ f-v B 1- ~ f-v A@B ~[A) f-v C 1- ~[A@B] f-v C ~[B) f-v C 1- ~[A@B] f-v C ~ f-v A 1- ~ f-v A@>B ~ f-v B 1- ~ f-v A@>B ~[A) f-v C ~[B) f-v C 1- ~[A@>B] f-v C (i.e. (7(x),6(x/A)) f-v C (7(x),6(x/B)) f-v C 11- (7(x), 6(xjA@>B)) f"v C) (7, 6(cp(7)/A)) f"v B 1- 11.,((7, 6)) f"v A :J., B 1Lo((7,6)) f-v A :J"' B 1- (7,6(cp(7)/A)) f-v B
11 ((7,6)) f-v A :J"' B
f {cp(7)}) f-v A
B f-v B
~3[~) f-v B
(r, 6(cp(7)/A)) f"v B.
The distinction between two truth constants @ and @ as well as between two conjunctions EB ('intensional' or 'multiplicative') and® ('extensional' or 'additive') is well-known from (intuitionistic) linear logic and relevance logic. Whereas @ reflects 0 at the level of the logical object language, EB reflects data addition+. On the other hand, @and ® are independent of any particular structural elements.
8.3. THE HIGHER-LEVEL SEQUENT CALCULUS
G~
Our next step is to associate with every Tarskian structured consequence relation f"v a higher-level sequent calculus G~ as a basic structural framework for developing a proof-theoretic semantics for f-v+·
Let ~1 = (71, 81), ~2 = (72(x), 82) and 82(x) = B. Let moreover ~ 3 (73,83), I 73 I= {y,z}, 83(y) =A, 83(z) =(A ~'P B), and cp(r3) = Y· Eventually, let ~4 = (r4, 84), where 74 = (SubxSub~ T 3 (y,z) 72 (X ) ' 84 )
Definition 8.5. Let f-v be a Tarskian structured consequence relation for (M, C). The set of all 8~-formulas is the smallest set X such that
=
~nd
84(w) = 8i(w), if wE I Ti
I,
(i = 1, 2, 3). Then the
mterreplaceable with
1
f-v)*
rule(~ 'P
is
- every formula of .C is in X - ifT EX, T EM, and 8:17 I-+ X, then (r,8) for every contraction mapping II'P on M. We use T, U, V, T1, T2, etc. to denote
since we have 1Le(73(Y, z), 83) ~3[A]
and, using(*),
f"v B
f-v
A ~'P B
f-v'P T
EX,
8~-formulas.
Definition 8. 6. Every £-formula is an 8~ -formula of 8-degree 0. If n is the maximum of the 8-degrees of T and the 8~ -formulas assigned by 8, then (7, 8) f-v'P T is an 8~ -formula of 8-degree n + 1. If n is the 8-degree ofT, we write 8d(T) = n. If 8d(T) = 1, then T is said to be a sequent; if 8d(T) > 1, then T is called a higher-level sequent. Definition 8. 7. Every 8~ -formula is an 8-subformula of itself. Each 8-subformula ofT and each 8-subformula of every 8~ -formula assigned
109
CHAPTERS
POSITIVE PROOF-THEORETIC SEMANTICS
Table X. The higher-level calculus G~.
conjunction, disjunction, and falsum. The formu~ation of_ the introduction schemata is subject to the following constramts, which go back to von Kutschera (96]. (See also Chapter 5. Other conditions as discussed in Chapters 1 and 3 are satisfied as well.)
108
(ref) (tra)
Tlr.-T r-'~' T r(T] r-\1' U 1- r(6.] r-'~' U (T,J(
6.
u
by 8 is an S-subformula of (T, 8) r-ep T. Every S-subformula with Sdegree 0 ofT is called a formula component ofT.
(T, 8) lr-T abbreviates (T, 8) r-ep T, for every llep on M. Definition 8.8. Given a structured consequence relation r-, the sequent calculus G~ is defined by the introduction rules in Table X, for every 11 ep, 11 cp' on M.
{i) Rule-schemata characterizing f exhibit apart from on~ occurrence of J no other occurrence of a propositional connective; the role of formulas f(A 1 , ... , An) in deductive contexts depends on the deductive relationships between A1, ... , An only. (ii) The rule-schemata for fare non-creative, that is to say every proof . r- b . . of an j-free formula A in the result of extendmg G+ y mst~ntmtions of these schemata can be converted into a proof of A with no applications of rules characterizing f. Constraint (i) suggests the following right introduction schemata, for every II'P on M: (I) (a)
st
Let Uv be an -formula that contains a certain occurrence of V as an S-subformula, and let Ur be the result of replacing this occurrence of V by T. Let 1- V ~lr.- T denote 1- Vlr.-T and 1- Tlr.-V. Theorem 8.9.
If 1- V ~lr.- Tin G~, then 1- Uv ~lr.- Ur in G~.
Proof. By induction on n
= Sd(Uv)
- Sd(V), using (tra). Q.E.D.
8.4. POSITIVE PROOF-THEORETIC SEMANTICS
Talking about functional completeness makes sense only against the background of a semantics which is either given or being sought. We shall now present as a certain proof-theoretic semantics general schemata for introducing n-ary propositional connectives f(A 1 , .•. , An) into databases and conclusions. These schemata define a space of permissible operations, and it will turn out that every such permissible operation can be defined from the primitive logical vocabulary of r.-+· We tend to consider the introduction schemata as prior to the set of connectives that will be shown to be functionally complete. Note that this is in contrast to Schroeder-Heister's point of view, who regards his functional completeness result for intuitionistic propositional logic as "demarcating the strength" [151, p. 1298] of intuitionistic implication,
(I) (b)
where the st-formulas assigned in lleplik/-. -II'Prik/6-ikJ · · .) (i = 1, · · · t· k·- 1 s·) and b. are unspecified, there are
... ). Moreover, (a) the Tik; and the st-formulas assigned to th~ values ofcp) is an example of an mstantmt10n of (J) (a) and Cr.- @) exemplifies an instantiation of (I) (b). In the latter case, j 2 and b-1 b-2 b..
If\;=
=
st
=
=
110
POSITIVE PROOF-THEORETIC SEMANTICS
CHAPTER 8
Constraint (ii) amounts to requiring (tra)-eliminability. Therefore the rule schemata for introductions into databases for every II'P on M become:
111
f- (U1 11 ~'Pill · · · (Uru ~'Prll Tn) ··.)A "11~ "11~ (Ulu ~'P111 · · · (Uru ~'Prll Tn) · · .)s f- (Ultst ~'Pltst · · · (Urtst ~'Prtst Ttst) ··.)A "11~
(H) (a)
"11~ (Ultst ~'Pltst · · · (Urtst ~'Prtst Ttst) · · .)s,
r[(U1 11 ~'Plll · · · (Ur 11 ~'Prll Tn) · · .) + · · · · · · + (Ullst ~'Pllsl · · · (Urlst ~'Prlsl Tlst) · · .)] ~'P U
where in each case the replacement of A by B is with respect to Ak. Suppose now the rules for CA are instantiations of (I) (a) and (H) (a). By (rei) and the schemata (J) (a) we obtain
r[(U1t1 ~'Pttt · · · (Urtt f-v'Prtt Ttl) · · .) + · · · · · · + (Ultst ~'Pltst · · · (Urtst ~'Prtst Ttst) · · .)] ~'P U ff- f[f(A1, ... , An)] f-vcp U;
(H) (b)
f- (Ulit ~'Plil · · · (Urit ~'Pril Tii) · · ·)A
+ ···
f- (Ulit ~'Plil · · · (Urit ~'Pril Til) · · .)B
+ · ··
· · · + (Ulis;
r[(Ull ~'Pll · · · (Uzu ~'Piu 11) · · .)] ~'P U ff- f[j(A1, ... , An)] ~'P U,
~'Plisl · · · (Urisi ~'Prisi TisJ · · .)AI~CB
and
where uliki is assigned to the element IPliki (TikJ, ... , Uriki is assigned to IPriki ( . . . l.fJl iki (TikJ ... ) ' ull is assigned to IPll (Tl)' ... ' and Uzu is assigned to 'Plu ( ... l.fJll (Tz) ... ) . If n = 0, then for every 11 'P on M, (H) (a) is ~[T] ~"' U f- ~[T + f] ~'P U, ~[T] ~"' U f- ~[! + T] ~"' U, and (H) (b) is not instantiated. The schemata (I) (a), (J) (b) thus completely determine the schemata (H) (a) and (JJ) (b), and the rules of G~ plus these schemata determine how formulas j(A1, ... , An) may be introduced into arbitrary st -formula contexts.
· · · + (Ulis; ~'Plisl
· · · (Uris; ~'Prisi TisJ · · .)si~CA·
The schema (H) (a) and (tra) give for each II'P on M:
and
Let CA denote an £-formula which contains a certain occurrence of A as a subformula, and let C8 denote the result of replacing this occurrence of A in C by B. The degree of A (d(A)) is the number of occurrences of propositional connectives in A.
(UI; 1 ~'Plit · · · (Urit ~'Prit Til) · · .)s + · · · · · · (Uris; ~'Prisi TisJ · · .)B~cpCA f- Cs ~'P CA·
... + (Ulisi ~'Plisl
Hence f- CA "1 I~ CB. If the rules for CA are instantiations of (I) (b) and (H) (b), then by the induction hypothesis and the previous theorem, the schemata (H) (b) give f- CAI~(Un r--'Pn · · · (Utu ~'Piu Tz) ... )s and f- Csi~(Ull ~cp 11 • • • (Utu ~'Piu 11) ··.)A· By (I) (b), we get f- CA "11~ Cs. Q.E.D.
Theorem 8.10. Iff- A "11~ Bin G~ +(J)+(H), then f- CA "11~ Cs
in G~ +(I)+ (H).
Let TA denote an sf-v-formula which contains a certain occurrence of A as a subformula 6f a formula component ofT. Combining the previous two theorems we obtain the desired replacement result.
Proof. By induction on l = d(CA)- d(A). If l = 0, the proof is trivial. Suppose that the claim holds for every l ::::; m, and l = m + 1. Then CA has the form j(A1, ... ,An), where one of the AkA contains the occurrence of A in question and d(AkA) ::::; l. Suppose that f- A "11~ B. By the induction hypothesis, f- AkA "11~ Aks, and by the previous theorem,
Theorem 8.11. If f-A "11~ Bin G~ +(I)+ (H), then f- TA "11~ Ts
in G~ +(I)+ (H).
1
112
FUNCTIONAL COMPLETENESS FOR
CHAPTER 8
113
f--- +
8.5. FUNCTIONAL COMPLETENESS FOR f-v +
We want to show that for every llep on M, S1 = {:::)ep, EB, @, @), @, @} is functionally complete for f-v +. We first show functional completeness with respect to the positive proof-theoretic semantics. In order to be able to do this, we introduce higher-level sequent rules for the constants and connectives in S1. These rules conform to the schemata (I) (a) and (II) (a) resp. (I) (b) and (II) (b):
II'P,(T1,81) r-'P (T r-'P' U)
(r1 , 8I(tp'h)/T))[T f---'~'' U] f---'~' U
f- ~ f-vep @
(f-v @)
f- 0 f-vep @
(@f-v)
~[T] f-vep
U f- ~[T +@ f-vep U
~[T] f-vep
U f- ~[@+ T] f-vep U
(f-v EB)
~ f-vep
r
(EB f-v)
~[T
(f-v @)
~ f-vep
(@f-v)
~[T]
T
+ U] f-vep V f-
r
(r1, 01 (
,(TI,
f-vep U f- ~[T@V] f-vep U
~ f-vep
U f- ~ f-vep T@)U
st
~[T] f-vep V
~[U] f-vep V f- ~[T@)U] f-vep V
(f-v:::)ep' )'
~ = llep' (r,
( :::)ep' f-v )'
~[T f-vep' U] f-vep V f- ~[T :::)cp' U] f-vep V.
r- ~
:::)ep' U] f-vcp V.
Now we translate each -formula T into a formula T with !-degree 0. IfT is an £-formula, then T = T, and (r,o) f-vep V= (r,o) :::)ep T, where if o: x 1--t U, then 8: x 1--t U.
(f!Jf-v)
o) f-vep (T f-vep' u)
<51) f-vep (T f-vep' U) ~[T
U f- ~[T@U] f-vep U
T f- ~ f-vep T@)U
8( tp 1 ( r) /T)) f"--'P U.
llcp' (TI' <51) f-vep (T :::)ep' U)
f-vep T EB U
~[T EB U] f-vep V
~ f-vep
(•)
To see that (:::)ep' f-v)* implies (:::)ep' f-v)', suppose that ~ = (r(y), o(yjT f-vep' U)), I TI I = {x,y},
T ~ f-vep U f- ~ f-vep T@U
~[V] f-vep
(f-v @))
f-vep U f- ~ +
(::)..,+)'
(r1, 81(!f'(rl)/T))[T J
11 ,(r,8) f---'~' (T -::J'~'' U) ( T,
(f-v @)
(f-v..,,f-v..,)
f-vep (T :::)ep' u)
A part from ( f-v:::) ep')' and (:::) ep' f-v )', the above rules are higher-level versions of f-v+'s operational rules. However, (r--:::)ep')' and (:::)ep'f-v)' are interreplaceable with higher-level versions of (f-v:::)ep') and (:::)ep' r-- )*, namely:
Theorem 8.12. In G~ +(I)+ (II), f- T "-11f-v T. Proof. By induction on Sd(T). If Sd(T) = 0, the claim is trivial. Suppose that the claim is true for every l ~ m, and l = m+ 1. Then T is of the form (r, o) r--ep' V. By the induction hypothesis, for ever~ U in the range of o, f- U "-11 f-v U; moreover f- V "-11 f-v V. Hence f- ~ r, 8) f-vep' V) lf-v ((r, o) f-vep' V) and conversely ((r, o) f-vcp' V) lf-v f- ((r, 8) f-vcp' V), by (ref) and (tra). Applying (f-v:::)ep' )'and (:::)ep'f-v)', we obtain f- T "-11f-v T. Q.E.D.
(f-v:::)ep')
(r,O(
(:Jep' r-- )*
llep' (r, o) r--ep (T :::)ep'
u)
f- (r, o(
u.
This is immediately clear for (f-v:::)ep' )'and (f-v:::)ep' ). To see that (:Jep' r-- )' implies (:Jep'r--)*, assume that I TI I= {x,y},
Theorem 8.13. S 1 is functionally complete with respect to the positive proof-theoretic semantics. Proof. Case 1: J(A 1 , ... , An) is defined by instantiations of the schemata (I) (a) and (II) (a). If n = 0, then f = @. Otherwise, from (ref) and (I) (a) we get
114
CHAPTER 8 CONSTRUCTIVE LOGICS 8.6. NEGATION AS REFUTATION
f- Vu = (U111 ~'P111 · · · (Uru ~'Prii Tu)··.)+.·· ... + v1SJ = (Uusl r-'Pllsl ... (Ur1SJ ~'Prlsl T1SJ) ... )1~ f(A1, ... , An)
The idea behind negation as refutation (alias negation as falsity) is very simple and natural: proof theory is not only concerned with proofs but also with refutations (or disproofs), and provability and refutability should be considered as prima facie independent and equally important primitive notions. This point of view has for example been developed by von Kutschera [97], who suggests incorporating into sequent calculi what has later come to be known as Slupecki's notion of inverse consequence [157], [158], that is to say we assume calculi A which are defined by antiaxioms postulated to be refutable in A and by inference rules which specify how formulas refutable in A can be obtained from formulas which have already been refuted in A. In this axiomatic setting, using a structural unary symbol '-' to denote negation in the sense of refutation, we may (i) translate every antiaxiom A of A into -A, (ii) recast inverse inference rules {A1, ... , An} f-A which proceed from the refutability of the formulas in {A 1 , ... , An} to the refutability of A as { -A1, ... , -An} f- -A, (iii) consider -A to be provable in A, if A is refutable in A and (iv) replace - -A by A. In this way, the ordinary notion of an axiomatic calculus may be used. Now, with an axiomatic calculus A we can associate a sequent calculus SA by translating each A-axiom A into the SA-rule f- 0 ~A and translating each A-rule~ f-A into the SA-rule f- ~ ~ A. We may then introduce a unary negation operation "" to represent - in the object language by requiring that ~ -A f- ~ ""A and ~[-A] ~ B f- ~[""A] ~B. Thus, - and"" are supposed to be 'declaratively identical'. Such considerations led von Kutschera [97] to independently developing the propositional part of Nelson's constructive logics N3 and N4 (also known as Nand N-, respectively, cf. [4], [119], [180]), using the very appropriate names 'direct propositionallogic' and 'extended direct propositionallogic'. Nelson's logics will be considered in greater detail in Chapter 9; see also [195]. In particular, the semantics of N3 is presented in Chapter 9 together with an elaborate discussion of the idea of negation as definite falsity.
f- Vn = (U1tl ~'Pltl · · · (Urtl ~'Prtl Tn) · · .) + · · ·
''' + Vfst =
(Ultst ~'Pitst · · · (Urtst ~'Prtst Ttst) · · .)1~ j(A1, ... , An)·
By the previous theorem and (tra) l f- V:2 1+ . • • +V:2S,. 11..... f(A 1' · · · 1 A n ) . f(z = 1, ... , t). Now, using (~ EB), (EB ~), and (tra), we obtain f- ~--
--
---- ----
z
Vi1EB .. · EBVisi I~ j(A1, ... ,An), and applying(@)~) we obtain f- T1® · · · ® Tt I~ f(A1, ... , An)· By (re!), the previous theorem, (tra), and ( ~ EB), we have f- Vi1 + . . . +Vi si I~ Ti, and applications of ( ~ ®) give f- Vi1 + · · · +Vis; I~ T1@) . . . ®Tt. The schemata (II) (a) give f- J(A1, ... , An) T1® ... ®Tt. From Theorem 8.11 we know that f(A1, ... , An) and T1® ... ®Tt are intersubstitutable with resl_l~ct to provable interderivability ~~~- Thus, J(A 1, ... , An) can exphc1tly be defined by a formula in {~'P' EB, @), @}. Case 2: The rules for f (A 1, ... , An) are instantiations of (I) (b) and ( II) (b). If n = 0, then f = @. Otherwise, by (re!), the previous theorem, and (tra), f- (Ull ~'P.ll .... (Ulu r-.-'Plu 1/) ... ) ~~ J:j = (Ull ~'PLI ... (Ulu~'P, @, @}. Q.E.D. =
lr--
r.-)
That S1 is functionally complete for
115
r.-
r--+follows from
Theorem 8.14. The positive proof-theoretic semantics characterizes
~+'
et,
Proof. Restrict the rules of the rules (~ @), (rv~'P' ), (~
r.-
8.7. CONSTRUCTIVE LOGICS
r--+·
Note that the addition of structural inference rules like ~[A+BJ ~ C f- ~[B +A] C to ~ + does not affect the functional completeness of S1. We just have to add higher-level versions of the structural rules in question to e.g. ~[T + UJ r-.-'P V f- ~[U + T] r-.-'P V.
There are various ways of incorporating a notion of negation into Instead of introducing a logical operation -. by means of a structural 'shift operation' as, for instance, in [67], we shall introduce a unary operation "" based on the idea of negation as refutation; cf. also Chapter 9 and [69]. With each Tarskian structured consequence relation ~
r.-
et,
l
116
CHAPTER 8 Table XI. Introduction rules of
(f-v"" EB) H (f-v"-' @)
("-' EB
(""®H (f-v"-' (2)) ("" ® H (f---~::::>'1')
("'=>'PH (f-v"-'~) (~~H
f'vc·
the rules of f-v + ~ f-v~ A r f-v"" B 1- ~ + r f-v"" (A EBB) ~[~A+ ""B] f-v C 1- ~["" (A \fJ B)] f-v C ~ f-v"" A 1- ~ f-v"" (A@B) ~ f-v"" B 1- ~ f-v"" (A@B) ~["" A] f-v C ~["" B] f-v C 1- ~["" (A@B)] f-v C ~ f-v"" A ~ f-v"" B 1- ~ f-v"" (A(S?)B) ~[~A] f-v C 1- ~["" (A(S?)B)] f-v C ~["" B] f-v C 1- ~[~ (A@B)] f-v C f f'v A 11'1'(~ = (T2,1h)) f-v"" B 1- ~(f] f-v"" (A::::>'~' B) (i.e. f = {Tt, J1), ~[f)= (Sub~,(T2 )T2, J), where J(x) = J;(x), ifx E I Ti l(i = 1,2)) (Sub~Tl(x),J) c 1- (Tl,J rl Tl l(x/ ~(A ::::>'1' B))) C, if IT I= {y,z},J(y) = A,J(z) =~ B, and y =
r-
117
CONSTRUCTIVE LOGICS
r-
we shall associate a prepositional system f--'c called f---'s constructive propositionallogic. We obtain f--'c from f---+ essentially by adding rules that specify refutability conditions for f---+ 's binary connectives. Definition 8.15. Let Le be the language £+ extended by the unary connective ""'· Given a structured consequence relation f--', the sequent calculus f--'c is defined by the introduction rules in Table XI.
Note that in some cases, in which II'P interacts with+, one can provide a formula C interderivable with,...., (A ~'P B) such that ® or EEl is the main connective of C.
of a proof rr is inductively defined as follows: the height of an identity A f---A, (f--- @), or an instantiation of (f--- @) is zero. If IT has the form Il1 ... IIm ~f---A,
then h(II) = max{ h(Ili) I 1 ::; i ::; m} + 1. As a complexity measure, we use the Cut-degree d(II) of proofs
Next, we want to prove a Cut-elimination theorem. We say that two structured databases ~1 = (TI, 01), ~2 = (T2, 02) are isomorphic to each other if there is an isomorphism f from T1 to T2 and for every x E I T1 I, 81(x) = 82 (/(x)). The degree d(A) of a formula A is the number of separate occurrences of connectives or constants in A. The height h(II)
Ill
rr2 rr3,
Cut for M
which is defined by d(II) = (d(A), h(III) + h(II2)). We assume that the Cut-degrees are lexicographically ordered (cf. [172]). Theorem 8.11. (Cut-elimination) If 8 f--'c C is provable, then there is a structured database isomorphic to such that f--'c has a Cut-free proof, provided (~'Pf---) is used instead of (~'Pf---)*.
e'
e
e'
c
e
Proof. We show that if f--'c C has a proof with a single application of Cut for M, then there is a structured database 8' isomorphic to such that 8' f--'c C has a Cut-free proof. We can then successively eliminate applications of Cut for M from any given proof. The idea is to replace the subproof IT* that ends in a Cut
e
rr1 IT*
~ f---
rr2
A r[A] f--- B
r[~] f--- B
by a proof IT** of r[~]' f--- B, where r[~]' is isomorphic to r[~] and the Cut-degree of any subproof of IT** ending in an application of Cut is smaller than d(II*). Consider the following case distinction between possible proofs of the premises of Cut. Case 1. The left (right) premise is an identity A f---A: rr*-
Example 8.16. Consider Nelson's propositionallogic N 4. Here
=
rr
r[Ar~ B
~[AF~ A )
(rr* -
Case 2. The last step in the proof of the left premise of Cut is an application of a right introduction rule, whereas the last step in ~he proof of the right premise is not an application of the correspondmg left introduction rule. Schematically, we proceed as follows: IT'2 f'[A] f--- B r[A] f--- B r[~J r-- B
Il1 ~ f--- A
rr1
rr~
~ f---A
f'[~] f--- B
r--
r'[~l B r[~] f--- B
118
THE HIGHER-LEVEL SEQUENT CALCULUS
CHAPTER 8
Case 3. Analogous to the previous case, now considering the right premise of Cut. Case 4. The final steps of the proofs of the left and the right premise of Cut are applications of corresponding right and left rules. This is the principal case, and we consider a number of representative examples.
GS
119
(ii) (Disjunction property) In ~c, 1- 0 ~ A@)B implies 1- 0 ~ A or
1-0 ~B.
(iii) (Constructible falsity) In ~c, 1or 1- 0 ~......,B.
0 ~......, (A@B)
implies 1-
0 ~......, A
Note that ......, fails to be a contrapositive negation in ~c·
IT~
b.= (r(z),b(z/A) f--- B 0 f--- @ b.[A +@ f--- B ((Sub~(Sub~,(x)+r2 (y))T, <5') f--- B (Sub~, (x)+0T, <5') f--- B = (Sub~,(x)T, <5') f--- B II~
8.8. THE HIGHER-LEVEL SEQUENT CALCULUS
IT'2
b.[A] f--- B
IT~
b. f---"' A b. f---"' B b. f---"' (A('0B)
(r, o(
=
b.2[B] f--- C
(Subsub~ rg(y,z)T2(x),b4)
f--- C
b.4[II
~1
r-
r-
A (T, b) B b. [B] l.. . c (r,b)[b.df-vB 2 rb.2[(r,b)[b.r]] f--- C
r-
fl r1 l(x/"' (A-::;"' r- c,
b.1- (r1,b b.l[~[r]]
11"'(~) r-"" B
r
r: iurJ t
b.i[f][ll"'(b.)J
1et' c
t: c,
where, by(*), ~~[f][llcp(~)] = ~d~[f]]. Q.E.D. Corollary 8.18. (i) The system ~c is a conservative extension of~+·
-
every formla of £ is in X if T E X, then - T E X if T, U E X, then (T 0 U) E X ifT EX, rE M, and 8:1 T 1-+ X, then (T,8) ~cp T EX, for every contraction mapping 11 cp on M.
We now use T, U, V, T1, T2, etc. to denote
s['- formulas.
sf'
where, by(*), ~2[(T,8)[~1]] = ~4[llcp((T,8))]. Suppose IT I= {y, z}, 8(y) =A, 8(z) =......, B, ~ = (T2, 82), and y = cp(T) = cp(T2)· Then
r A ll"'(b.) t:"' B b.[f] f-v"' (A -::;"' B)
In order to develop a proof-theoretic semantics for ~c, we need a somewhat richer basic structural framework. In addition to sequent arrows ~cp and a structural negation -, we shall also use a structural connective 0 corresponding to +. Definition 8.19. Let ~be a Tarskian structured consequence relation for (M,£). The set of all s[' -formulas is the smallest set X such that
f["' (A V B)] f--- C
b. 1 f---A ~4
G[
B))) f--- C
Definition 8.20. Every £-formula is an -formula of S-degree 0. If T has S-degree n, then -T has S-degree n. If n is the maximum of the S-degrees ofT and U, then (T 0 U) has S-degree n + 1. If n is the maximum of the S-degrees ofT and the -formulas assigned by 8, then (T,8) ~cp T is an st'-formula of S-degree n + 1. If n is the S-degree ofT, we again write Sd(T) = n. If Sd(T) = 1, then again T is said to be a sequent; if Sd(T) > 1, then T is called a higher-level sequent.
sf'
The notion of an S-subformula of an s[' -formula is defined analogously as for S~ -formulas. Every S-subformula with S-degree 0 ofT is again called a formula component ofT, and iffor every llcp on M, (T, 8) ~cp T, this is again abbreviated by (T, 8) I~T. Definition 8.21. Given a structured consequence relation ~' the sequent calculus G[ is defined by the introduction rules in Table XII, for every llcp' llcp' on M.
120
CHAPTER 8 Table XII. The higher-level calculus Gf.
(f---cp
r-- cp')
(- r--
the rules of G!;:' r f-vcp T 11'~'' (A) f-.-'~'~ U f- A(f) f-.-'~' -(T f--'~'' U) (Sub~T1(x),tS) f---'1' V f- (T1,J 11 T1 l(xj- (T f-.-'~' U))) f---'1' V, if IT I= {y,z},J(y) = T,J(z) =~ U and y = cp(T) A f-.-'~' T r f-.-'~' U f- A + r f-.-'~' T 8 U A[T + U] f-.-'~' V f- A[T 8 U] f-.-'~' V A f-.-'~' -T r f-.-'~' -U f-A+ r f-.-'~' -(T 8 U) A[-T + -U] f-.-'~' V f- A(-(T 8 U)] f-.-'~' V
Let Uv be an Sf -formula that contains a certain occurrence of V as an S-subformula, and let Ur be the result of replacing this occurrence of V by T. Let V "-llr.- T denote Vlr.-T and Tlr.-V, and let V "-llr.-s T ('V and T are strongly interderivable') denote V "11 r.- T and -V r-11 r.- - T. Theorem 8.22. If 1- V "-llr.-s Tin Gf, then 1- Uv "-llr.-s Ur in Gf. Proof. By induction on n = Sd(Uv) - Sd(V), using (tra). Q.E.D.
8.9. CONSTRUCTIVE PROOF-THEORETIC SEMANTICS
The idea now is to characterize an n-ary propositional connective f by both the inference rules used to introduce f(A 1, ... , An) and those used to introduce - j(A1, ... , An) into databases and conclusions. The system Gf serves as the proof-theoretic framework for specifying such inference rules. The heuristic constraints (i) and (ii) from Section 8.4 have to be appropriately modified so that (i) now also refers to the role offormulas - j(A1, ... , An), and 'Gt' in (ii) is replaced by 'Gf'. The schemata for introductions of-f(A 1, ... , An) into conclusions can be motivated as follows. Since we would like to avoid that for some II'P on M both 0 r.-'P - j(A1, ... , An) and 0 r.-'P f(A 1, ... , An) are provable, if not for at least one Aj {1 ~ j ~ n), 0 r.-'P -Aj and 0 r.-'P Aj are provable, the schemata for - f(A 1, ... , An) should not be independent from those for j(A1, ... , An). In fact, since provability and refutability are regarded as equally relevant, the schemata for introducing - j(A1, ... , An) and /(AI, ... , An) into conclusions should mutually depend on each other. We have already defined the schemata (J) (a)
l
CONSTRUCTIVE PROOF-THEORETIC SEMANTICS
121
and (I) (b). Now suppose that by (J) (a), j(A1, ... , An) is provable provided that for some i (1 ~ i ~ t) all Sf -formulas (U1;1 r.-'Plil · · · (Uril f-''Pril Tii) · · .), · · ., (Ul;,; r.-'Plisi · · · (Ur;,; f-''Pri•i TisJ · · .) are provabl_e. Then the refutability of j(A1, ... , An) should follow from the refutability of all of these Sf -formulas. If, by the schema (J) (b), j(A1, ···,An) is provable provided that every Sf -formula (Un r.-'Pn . · · (Utu r.-'Ptu Tt) ... ) is provable, then every refutation of some (Un r.-'Pn . · · (Utu f.-'Ptu 'Il) .. .) should be a refutation of j(A1, ... , An)· . These considerations are transformed into the followmg schemata (III)(a) and (Ill) (b) for introducing- j(A1, ... , An) into conclusions: (Ill) (a)
r
rv'P -Vu 0 ... 0 -V1s 1 ... r rv'P -vt1 0 · · · 0 -Vist II- r rv'P - j(A1, ... , An),
(I II) (b)
r I-
r-'P - V1 1- r r-'P - J(A1, ... , An) ... r rv'P - Vj Ir rv'P - j(A1, ... , An)·
Again, 1t'i1 = (U1; 1 rv'Plil · · · (Uril f-''Pril Til) · · .), · · ., "is; = (Ul;si r.-'Pli•i · · · (Ur;,; rv'Prisi TisJ · · .) (1 ~ i ~ t), Vi = ~Un rv'Pn · · · ~Utu rv'Ptu Tt) .. .) (1 ~ l ~ j), and r is unspecified. If f 1s 0-ary, we stipulate that (Ill) (a) and (Ill) (b) are not instantiated. In the same way as the non-creativity constraint led us from (I) (a) and (J) (b) to (II) (a) and (IJ) (b), it now also leads us from (Ill) (a) and (Ill) (b) to the following schemata (IV) (a) and (IV) (b) for introducing - j(A1, ... , An) into databases:
(IV) (b)
r[-Vii0 ... 0-Vis 1 ] rv'PT 1- r[-J(Al, ... ,An)] rv'PT, r[-V1] rv'PT ... r[-Vj] rv'PT 1- r[-f(AI,···,An)] rv'PT.
For 0-ary J, (IV) (a) and (IV) (b) are not instantiated. Let CA again denote an £-formula that contains a certain occurrence of A as a subformula. Theorem 8.23. If A "-llrvs B in Gf +(I)- (IV), then CA "-llrvs CB
in Gf +(I)- (IV). Proof. By induction on l = d(CA)- d(A). If l = 0, the proof is trivial. Assume that the claim holds for every l ~ m, and l = m + 1. Suppose that CA has the form F(A 1, ... , An), where one of the AkA contains
122
FUNCTIONAL COMPLETENESS FOR
CHAPTER 8
f--'c
123
,,m·--
5 the occurrence of A in question, d(AkA) :::; l, and I- A B. By the 5 induction hypothesis, I- AkA --11 f-...- AkB, and by the previous theorem, I- Vik;A --1lf-...- 5 Vik;A (ki = 1, ... , si), and I- ViA --11f-...-s ViE, where in each case the replacement of A by B is with respect to Ak. We obtain I- CA --11f-...- CB as in the proof of Theorem 8.10. Assume that the rules for -CA are instantiations of (Ill) (a) and (IV) (a). By (re!) we have for every i = 1, ... , t:
f- - Vi1A 0 · · · 0 - Vis;A f- -VilB 0 · · · 0 -Vis;B
r-- - Vi1B 0 r-- -VilA 0
· · · 0 - Vis;B 1 · · · 0 -Vis;A'
The schemata (IV) (a) give 1- -CB 1- -CA
f'-- - Vi1A f'-- -"\lilB
sr-
Let TA denote an formula which contains a certain occurrence of A as a subformula of a formula component ofT. The previous two theorems establish the required replacement property.
Gf
--1lf-...- 8
Bin
Gf +(I)- (IV), then TA
--1lf'-- 8
TB
(-®f-...-) (f-...--@)) (-@)f-...-)
FUNCTIONAL COMPLETENESS FOR
~[-T] f-...-ep U I- ~["' T] f-...-ep U ~ f-...-ep __
T 1-
~ f-...-ep - "'
T
~[-- T] f-...-ep U I- ~[- "'T] f-...-ep U
~ f-...-ep -(T f-...-ep' U) 1- ~ f-...-ep -(T ~ep' U) ~[-(T f-...-ep' U)] f-...-ep V 1- ~[-(T ~ep' U)] r--ep V ~ f-...-ep ( -T 0 -U) I- ~ f'--ep -(T EB U) ~[-T 0 -U] f-...-ep V 1- ~[-(T EB U)] f-...-ep V ~ f-...-ep -T 1- ~ f-...-ep -(T@U)
f'--c
We introduce higher-level sequent rules for the constants and connectives in S 2 = { "', ~ep, EB, @, @), (!), @}, such that these rules conform to one of the schemata (I) - (IV). In addition to the rules from Section 8.5 we also assume:
~[-T] r--ep V ~ r--ep -T
~[-U] f-...-ep V I- ~[-(T@U)] r--ep V ~ f-...-ep -U 1- ~ f-...-ep -(T@)U)
~[-T] r--ep V I- ~[-(T@)U)] f-...-'P V ~[-U] r--ep V I- ~[-(T@)U)] r--ep V.
From these rules it is clear that we have 1- (T 0 U) --11 f-.. -s (T EB U)' 1- _ T --11 f-...-s "' T, and I- (T r--ep U) --11 f'-- 8 (T ~ep U). Note that the rules (f-...-_ ~ep')', (- ~ep'f-...-)', (f-...- -EB)', and (-EB f-...-)' fail to be higherlevel versions of rules from r..-c· However, one can easily verify that (f-...-- ::Jep')' resp. (- ~ep'f-...-) 1 is interchangeable with
(f--'- =>,•) r f--', T 11,,(~) f--', -U 1- ~[f] f--', -(T =>,• U) resp. (- =>,·f--') (Sub~r1 (x),8) f--', V 1- (r1,6 il r1 l(x/- (T =>, U))) f--', V, if 1r 1= {y,z},J(y) = T,8(z) ="' U, and y = t.p(r), and that
8.10.
~f-...- -T 1- ~ f'--ep"' T
~ f-...-ep -U 1- ~ f-...-ep -(T@U)
0 · · · 0 - Vis;A, 0 · · · 0 -Vis;B·
Applying (Ill) (a) we obtain 1- -CB --11f-...- -CA. If the rules for -CA are instantiations of (Ill) (b) and (IV) (b), then by the induction hypothesis and the previous theorem, the schemata (I II) (b) give I-V} A f'--ep -CB and -VjB f'--ep -CA. By (IV) (b), we obtain -CA --1if'--CB· Q.E.D.
Theorem 8.24. If A +(I)- (IV). in
(f'--rv) (rvf'--) (f'---rv) (-rvf'--) (f-...- - ~ep' )' (- ::Jep' r--)' (f-...- -EB )' (-EB f-...-)' (f-...--®)
(f-...- -EB )'
resp. (-EB
(f-...- -EB) ~ f-...-ep -T r ( -EB f-...-) ~[-T + -U]
f-...-)'
is interchangeable with
f-...-ep -U 1- ~
+r
f-...-ep -(T EB U)
resp.
f-...-ep V 1- ~[-(T 0 U)] f-...-ep V.
Consider e.g. the direction from (f-...- - ::Jep') to (f-...- - ~ep' )' • Let I TI I = {x}, 1r 2 1= {x,y}, 1r 3 1 = {z}, and
124
T
CHAPTER 8
FUNCTIONAL COMPLETENESS FOR ~c
125
I
( T1,
81) f-vcp T
ILe' ((T2 , 82 ))
f-vcp -U
By (re!), the rules for 0, (tra), the previous theorem, and the fact that ,..., T ~If-vs -T, we have 1- -Vi1 0 ... 0 -Vis; lf-v -Ti. The schemata (IV) (a) give f- - j(A1, ... , An) lf-v -Ti. Applying (f-v @), we obtain
(T2, 8) f-vcp -(T "Jcp' U) ~ f-vcp -(T f-vcp' U)
(T3, 8') f-vcp -(T "Jcp' U) ~ f---cp
-(T "Jcp' U).
We now translate each Sf -formula T into a formula T with S-de ree 0 as follows: g -
if T is an £-formula, then T = T if T = - U, then T = ,..., T ' if T = U 0 V, then T = U' EB V and ifT = (T,8) f-vcp V, then T = (;,J) "J V .f :\ cp , where I u : x f--7 U, then 8 : x f--7 U.
From the latter we readily obtain
Assume now that the rules for - f(A 1, ... , An) are instantiations of the schemata (HI) (b) and (1V) (b). By (re!), the previous theorem, the fact that ,..., T "-11 f-vs - T, and (tra), f- -Vi If-v -Vi. The schema (HI) (b) gives f- -Vi lf-v - j(A1, ... , An)· Applying (@ f-v), we may conclude that
By induction on Sd(T) we can show that
Theorem 8.25. In
at + (I)- (IV), f- T ~If-vs T.
and from this we can easily deduce
1- -(V1® ... @vt)lf-v- j(A1, ... , An)·
The~rem
8.26. S2 is functionally complete with respect to the constructive proof-theoretic semantics.
Proof. As in the proof of Theorem 8.13, we can show that, if n > 0, ···,An) ~lf-v B, where B = T1 @) ... @) Tt, if the rules for f(Al, ···,An~ are instantiations of (I) (a) and (IJ) (a) resp. B = v1 ® · · · ® Vj, If the rules for f(Al, ... , An) are instantiations of (I) (b) and U!) ~b). n = 0, then f = @or f = @. We have achieved our obJeCtive If we can show that 1- -B -f(A 1, · .. , A n ) , £or n > 0 S . ·-liL... "1 r · uppose that - j(A1, ... , An) IS defined by instantiations of the ~chemata (III) (a) and (IV) (a). By (re!), the rules for 0, the ~revwus theorem, and the _fact that ,..., T ~ 1f-vs - T, we have 1- - Ti 1f-v Vi18 · · · 0 - Vis;· Applymg (@f---), we obtain f- f(Al,
!f
1- -T1@ · · ·@- Ttlf-v- Vi1 0 ... 0 -Vis;· The schema (III) (a) gives
1- -T1@ ... @- Ttlf-v- j(A1, ... , An), and from this we easily obtain
Moreover' clearly f- -Vi If-v -Vi. Hence, by ( f-v @))' 1- -Vi If-v - vl @) ... @) -Vj. By the schema (IV) (b) we can prove -f(Al,···,An) lf-v - V @) ... @) -Vj, from which- f(A 1 , •.. , An) lf-v -(V1® ... @Vj) is 1 derivable. Q.E.D.
It can readily be shown that the constructive proof-theoretic semantics characterizes f-vc· Therefore,
Theorem 8.27. S2 is functionally complete for f-vc·
at,
If (i) the rules of the rules (f-v @), (@ f-v), (f-v @), (f-v "Jcp'), ("Jcp' f-v )*, (f-v - "Jcp' ), (- "Jcp' f-v), (f-v f)~- f), (f f-v) and (- f f-v) (f E { EB, ®, @)}) are restricted to sequents only, (ii) f-v cp is replaced by f-v, and (iii) -T is replaced by ,..., T, then the resulting system is f-vc·
PROOF.
Q.E.D.
Corollary 8.28. S3 = { "',"Jcp,EB,@,@,@} and S4 = { "', "Jcp, EB, @, @, @} are functionally complete for f-vc·
126
CHAPTERS
Gf
Proof. In + {I) - {IV), f- (T@U) (T@)U) ~If-vs -(-T@- U). Q.E.D.
CHAPTER 9
~If-vs -( -T@)
_ U) and f-
Note that again the addition of purely structural inference rules can be accounted for at the level of the underlying basic structural framework in this case
Gf.
CONSTRUCTIVE NEGATION AND THE MODAL LOGIC OF CONSISTENCY
'
In_ conclusion we can say that every propositional connective with cer_tam natural proof and refutation rules can be explicitly defined by a fimte _number of comp?sitions from the elements of Sn (n = 2, 3, 4). If ?~e, hke Schroeder-He1ster does, considers certain connectives as prim~tiVe, the~ what we have done is to describe a semantics in terms of mtroductwn schemata such that Sn (n = 2, 3, 4) emerges as a functionally complete set of connectives.
The aim of this chapter is twofold. First of all, we shall take a closer look at the strong, constructive negation "' introduced in Chapter 8. We shall consider "' from the point of view of a proof-theoretic characterization of negation and argue that negation may be seen as a connecting link between provability and disprovability (refutability). This notion of negation as falsity will be developed against the background of N. Tennant 's [165] considerations of negation in intuitionistic relevant logic, where Tennant also attends to disproofs in addition to proofs. It is shown that negation in intuitionistic relevant logic is a negation as syntactical inconsistency in the sense of Gabbay [66], and that every such negation as inconsistency is a negation as falsity, while the converse is not true. Secondly, we shall consider semantics-based nonmonotonic reasoning as introduced by, again, Gabbay [65]. In this approach, nonmonotonic inference is defined using a modal consistency operator that is interpreted as possibility with respect to the information order in semantical models of a monotonic base logic. It will be shown that certain anomalies of Gabbay's approach can very naturally be avoided using David Nelson's constructive three-valued system N3 [119] as the monotonic base system instead of intuitionistic logic, IPL, or Kleene's three-valued logic, 3. The counterintuitive features of semantics-based nonmonotonic reasoning based on IPL or on 3 also disappear if certain properties of the information order in Kripke models for IPL and model structures for 3 are given up. In the case of IPL this leads to subintuitionistic logics. In this way, the present chapter prepares the ground for Chapter 10. Whereas Chapter 10 is devoted to display sequent calculi for subintuitionistic logics, [195] solves the problem of providing a sound and complete proof-system for the modal logic of consistency over Nelson's four-valued logic N4. In [165], Neil Tennant presents a "rule-based, anti-realist or constructivist account of negation". This account assumes basic contrarieties and uses a simultaneous inductive definition of proofs and disproofs. No appeal is made to the falsity constant f. These considerations lead Tennant to the system IR of intuitionistic relevant logic, a formal system also investigated in, for example, [163], [164]. Like in Nelson's systems, the notion of negation in IR is also based on the concept of
127
sd
128
disproof. However, whereas for Tennant a disproof of a set of formulas ~ is a deduction that reveals ~ as inconsistent, we shall treat the primitive notion of disproof on a par with the notion of proof. Instead of giving rise to IR, this egalitarian approach very naturally leads to Nelson's four-valued constructive logic N4, a system which has recently received a great deal of attention in knowledge representation and logic programming. This part of the presentation takes up and collects ideas already put forward in [97], [102], [130], [127], and [180]. An abstract syntactic definition of negation as inconsistency has been presented by D. Gabbay [66]. The basic idea of this definition is that the negation of B is derivable from A iff an unwanted formula C from a fixed set of unwanted formulas is derivable from A and B together. It can be shown that Nelson's strong negation fails to be a negation as defined by Gabbay (see [180], [69]). We shall suggest a syntactic definition of negation as falsity, of which negation in N4 is a paradigmatic example, observe that negation in IR is a negation as inconsistency (if the requirement of transitivity of deduction is abandoned), and show that every negation as inconsistency is a negation as falsity.
9.1. DISPROOFS AND CONTRARIETY
Tennant [165] argues against the definition of negation in terms of implication and absurdity, familiar from intuitionistic logic, i.e., 'hA := A =>h f, where =>h is intuitionistic implication, and 'h is intuitionistic negation. For Tennant it is a "strategic mistake" to treat the absurdity sign f as a propositional constant. Instead f is to be conceived of as "a kind of structural punctuation mark. It tells us where a story being spun out gets tied up in a particular kind of knot-the knot of patent absurdity, or of self-contradiction" [165, p. 203]. As a punctuation mark, however, f is superfluous, because in using f "[r]ather than writing nothing, we indicate that it's nothing that we intend, by writing something in particular, which is to stand for the nothing that we intend" [165, p. 204]. According to Tennant, a disproof of a set is a deduction showing that the set is inconsistent. Therefore, a disproof of a set ~ of undischarged assumptions is a deduction from ~ that terminates in an occurrence of f as a punctuation mark. If one uses the constant f as a symbol of the object language, this "allows us to assimilate disproofs to the general class of proofs" [165, p. 221]. While Tennant thus dispenses with f as a propositional constant in order to justify treating negation as primitive, his notion of disproof is that of disproof as reductio ad contradictionem.
129
DISPROOFS AND CONTRARIETY
CHAPTER 9
Tennant suggests a simultaneous inductive definition of the following two notions: il is a proof of A from the set ~ of undischarged assumptions, and il is a disproof of the set ~ of undischarged assumptions.
This definition may be regarded as problematic for at, least .two reason~£ (i) There is an obvious asymmetry between Tennant s notions of pro and disproof. The notion of single-conclus~on p:oof would be on a par with the following notion of single-conclusiOn disproof: il is a disproof of A from the set ~ of undischarged assumptions.
Using a notion of disproof as reductio, Tennant does n~t de~l with proofs and disproofs on an equal footing. He does not consider hte:ally concluding that a formula is disprovable fro~ a set of assu~pt10ns. According to Tennant the validity of disproofs iS to be defined m terms of a tomic bases containing conclusionless rules of the form
A1, ... ,An where the atomic sentences Al' ... 'An are mutu~lly inco~siste.nt. How. ddition to reductio ad inconsistency, direct falsificatwn ~lays ever, m a . . 1 I d t a central role in scientific inquiry and reasonmg m gene~a · . n a irec disproof of A we literally draw the conclusion that A iS .disprova~l~ If, for instance, il 1 is a proof of A from ~1 and il2 a dispr~of 0 from ~2, then if we use dotted lines to separate the conclusiOn of a refutation from its premises, ill
il2
··········· A =>h B
· d' of of A =>h B from ~ 1 U ~ 2 . This disproof is a direct arguiS a ispro . · bl (") I Tennant's ment to the conclusion that A =>h B iS dis?r~va . e. u n . system there are merely introduction and ehmmat10n rules; there i~ no systematic distinction between introducing compound formu~as mto proofs and disproofs and eliminating them from proofs and dzs~roofs. Therefore it is not clear how one could obtain an interpretatwn of the constructive connectives from Tennant 's ~ule~ in term~ of. ~ro~fs and disproofs, similar to the 'BHK interpretat10~' of the mt~itioms tic connectives in terms of proofs (or constructions); see for mstance
1
[166], [167]. 31
'BHK' stands for 'Brouwer-Heyting-Kolmogorov'.
130
DISPROOFS AND CONTRARIETY
CHAPTER 9
If we aim at a pr~of-theor~tic, constructive account of the meaning of strong,_ co?str~cti~e negatiOn "", conjunction A, disjunction v, and c~nstructive Imphcatwn -:J h in terms of direct (or canonical) proofs and disp~oofs, we must not only define what it means for a formula A to be directly provable fro~ a finite set of formulas ~ in this language (~--+ A), but also what It means for A to be directly disprovable (or refutable) from.~ (~ +-A). This definition must proceed by induction on the c~mplexity of A. The case where A is atomic does not concern the prov~nce of logic. What is regarded as a direct proof or disproof of an_ atomic sentence depends on the context of argumentation, and in thi~ respe_ct _th~ standards and criteria may differ considerably across vanous disciplines and communities. For compound A the following clauses seem to be very natural: (a)
(b)
0 1
~ --7
2 3
~--+AVB
1 2 3 4
~+-""A
rvA
~--+At\B
~--+A -:Jh B
~+-AI\B ~+-AVB
~+-A
-:Jh B
if if if if
~+-A
if if if if
~--+A
~--+A&~--+B ~--+A
or
~--+
B
~u{A}--+B
~+-A
or
~
+- B
~+-A&~+-B ~--+A
& ~+-B.
An e~rl~. suggestion of treating the notion of disproof (or refutation) as pnmitive can be found in von Kutschera's [97) motivation of 'direct' and 'extended direct' propositional logic. These systems are precisely D. Nelsons's (see [4], [119]) constructive propositional logics N3 and N 4 respectiv_ely. In f~ct, the above clauses (a) 1 - 3 are nothing but the rules for mtroducmg /\, V, and -:Jh on the right of --7 in a standard sequent calc~lus presentation of N4 (and positive logic). Moreover, if cl~~se (a) 0 IS strengthened into an equivalence and viewed as a defi~ztwn_ of+- by_ means of--+ and "", then the clauses (b) 1 - 4 are the r~ght mtro~ucti?n ~ules _for negated negations, conjunctions, disjunctions? an~ Imphcatwns m N4 respectively. Negation introduction on th~ ~Ight IS thus only defined for negations of compound formulas. But this JUSt reflects what has already been said about refuting atomic sentences. In other words, negative atomic information has to be treated on a par with positive atomic information. The left introduction rules are such that they guarantee the elimi-
131
nation of principal cuts: (a')
1 2 3
AU{AAB}--tC A U {A VB}--+ C A u f U {A Jh B}--+ C
if if if
A u {A,B}--+ C A u {A}--+ C & A U {B}--+ C A--+ A & f u {B}--+ C
(b')
1 2 3
AU{,..,~A}--tC
if if if if
AU{A}--tC ~ u {""'A} --+ C & A U { ~ B} --+ C ~ U {,..,A,~B}--+ C ~ U {A, "'B}--+ C.
4
A U {"'(A 1\ B)} --+ C A U {"'(A V B)}--+ C A u {"'(A "Jh B)}--+ C
As L6pez-Escobar [102] has observed, this treatment of negation in terms of disproofs avoids an unpleasant problem caused by the nonconstructive nature of intuitionistic negation. The problem arises in the context of the BHK interpretation of the intuitionistic connectives in terms of canonical proofs. Since -.hA is defined as A -:Jh f, according to this interpretation, a proof of -.hA is a construction that converts every proof of A into a proof of f. Since there is no (possible) proof off, a proof of -.hA would convert any proof of A into a non-existent entity. If we assume that the existence of a proof of -.hA precludes the existence of a proof of A, then a proof of -.hA would convert a non-existent object into a non-existent object. This is at the very least obscure. Some advocates of the BHK interpretation seem to be aware of the problem. In addition to the notion of proof, Troelstra, for instance, uses the notions of "hypothetical proof" [167] and reduction of an "alleged proof ... to an absurdity" [166). Moreover, he admits that "the notion of contradiction is to be regarded as a primitive (unexplained) notion" [167, p. 9). Nevertheless, this does not solve the problem of transformations into non-existent objects. As a remedy, L6pez-Escobar [102] has suggested supplementing the BHK interpretation by the notion of (canonical) refutation. He gives the following disproof-interpretation of the intuitionistic connectives /\, V, and -:J h and the constructive negation ,. ._. (notation adjusted):
i.) the construction c refutes A 1\ B iff c is of the form (i, d) with i either 0 or 1 and if i = 0, then d refutes A and if i = 1 then d refutes B, ii.) the construction c refutes A VB iff c is of the form (d, e) and d refutes A and e refutes B, iii.) the construction c refutes A -:Jh B iff c is of the form (d, e) and d proves A and e refutes B, viii.) [t)he construction c refutes ""A iff c proves A.
133
CHAPTER 9
DISPROOFS AND CONTRARIETY
A proof of""' A is thus not interpreted as a proof of A ~h f, but rather as a refutation of A. This seems to be the most natural and intimate way of linking proofs and disproofs by means of negation. Lopez-Escobar uses the following notion of provable sequent with respect to which N 4 emerges as sound: { A1, ... , An} --7 A is valid iff there is a construction 1r such that 1r( c1, ... , cn) proves A, whenever c1, ... , Cn are constructions proving A1, ... , An (if 1 :::=; n). A sequent 0 --7 A is valid iff a construction exists that proves A. Moreover, Lopez-Escobar assumes that no construction both proves and disproves the same A. Note that {A, ""'A} --7 B is valid under the stronger assumption that no formula A is both provable and disprovable. 32 The interaction between proofs, negation and disproofs developed above does not have direct proofs and disproofs as disjoint classes of deductions. Instead, the difference between proofs and disproofs is an intentional one: what may be regarded as a disproof of something may be viewed as a proof of something else. If this something is A, the something else is ,. ._, A. Taking ,. .,., as primitive and using reductio ad contradictionem as the natural deduction introduction rule for ""' would also avoid the problem of transformations into non-existent objects. 33 We have that II is a direct proof of ""'A from the set .6. of undischarged assumptions iff II is of the form
is disprovable. In a private communication, Tennant replied that if an atomic basis contains the rule
132
A B then "a single application of that very rule constitutes . . . a dir~ct disproof of {A, B}". Such a rule registers the joint contrariety of I~S atomic premises. But then the disprovability of singleton sets of a~omic sentences amounts to the selfcontrariety of these atoms. Irrespective of whether there are selfcontrary atoms or not, in the present context nothing is assumed about the refutability of atoms. In IR each rule resulting in a disproof is an indirect rule, exemplifying the idea of disproof as reductio. For compound A, every such elimination rule instantiates the following schema:
A where the deduction V specifies the refutation conditions of A, and where the absence of a formula below the horizontal line presents a "logical dead-end", as Tennant puts it. For example, then the rv-elimination rule of IR states that if II is a proof of A from .6., then
II'
""A and II' is a deduction showing that .6. U {A} is inconsistent. However, according to Tennant,
.6. 11
""A
is a disproof of {"'"'A} U .6.. A proof of '""A, however, e?ds in a~ application of ,..._,-Introduction, and, according to Tennant, ~t thus ~ails to be a disproof, even though the introduction of'"" A reqmres a disproof of A, namely if 11 is a disproof of .6. U {A} then
[d]isproofs have no 'conclusion'. Disproofs arise only through the terminal application of elimination rules. They cannot arise from the application of introduction rules. Terminal application of introduction rules produces only proofs, not disproofs.[165, p. 206]
.6.
'
AD-(i)
11
Therefore, Tennant just cannot provide clauses for a disproof-interpretation, for defining the notion of disproof by saying that a direct disproof II of A is a direct proof of "'A would mean that II terminates in an application of an introduction rule, quod non. Tennant's approach contains direct proofs, but no direct disproofs of compound formulas, i.e. no deductions not merely revealing the inconsistency of some data, but rather leading to the conclusion that a certain compound formula
(i)
rvA is a proof of ,..._,A from .6., where 0 prefixed to the discharg_e stroke indicates that A must have been used in 11. 34 In contrast to this meaning assignment to ""', the meaning of negation as falsity is essentially captured by clauses (a) 0 and (b) 1. Obviously, Tennant's introduction and elimination rules for ""' are not the natural deduction counterparts of the sequent rules for negation in N 4. For instance, in combination with the introduction and
32
A more comprehensive critical discussion of the BHK interpretation can be found in [180], which suggests generalizing the BHK interpretation into a semantical framework for various constructive substructurallogics. 33 Note that this a problem Tennant is not concerned with.
34
,,
The notation ~,A means~ U {A}, where A f/- ~-
135
CHAPTER 9
DISPROOFS AND CONTRARIETY
rules for intuitionistic relevant implication :Jr, Tennant's negatiOn rules allow the principle of contraposition to be proved:
Note, however, that due to the failure of transitivity of deduction in IR, this does not imply that 1- {,..._,A, A} -t B. 35 Therefore, whereas Tennant's notion of disproof as reductio gives rise to IR, the egalitarian approach, which treats the notions of proof and disproof in their own right, leads to Nelson's constructive four-valued system N4. Although in [165] Tennant does appeal to disproofs, it still seems appropriate to quote from Pearce [127, p. 5] (notation adjusted):
134 elimi~ation
(A :Jr B)0-(iv)
AD-(ii)
BD-(i) (i)
=""=B==o-=(=i~=·i)=::;===~B~ (ii) ""A (iii) ""B ::::lr"-' A (iv) (A ::::lr B) :J ("-'B :J"-'A) where
0 prefixed to the discharge stroke indicates that the assumption
t~us marke~ ~eed
not have been used in the deduction, but may be It has been used. Interpreting ,. ._, as falsity in the sense of refutab_Il~ty al~ne; however, does not justify the provability of the ~ontraposit_w~ prm_ciple, and, indeed, this principle fails to be provable m N4. T~Is IS as It should be, since there need not necessarily be a co~structwn 1r such th,at if_1r' is a construction converting any proof of A I~to a proof of B, 1r ( 1r) IS a proof converting any disproof of B into ~ disproof of ~- ~he weaker contraposition rule A -t B 1-,...., B -trv A Is no~ an ad~ISSible r~le of ~4 either and fails to be supported by the dispro~f-mterpretatwn, for If there is a construction converting any proof o~ A m to a ~roof of B, there need not necessarily be a construction con:ertmg any ~Isproof o~ B into a disproof of A. In this respect, the not10~ of negatiOn as falsity differs from, for example, the notions of negatiOn advo_c~ted by Lenzen [101] and Restall [142], who consider the contrapositiOn rule as absolutely indispensable for any negation. Furthermore, e_very negation as inconsistency in Gabbay's [66] sense (see below) satisfies contraposition as a rule. A more specific criticism conc~rns ~enn~nt's first introduction rule for implication, which states that If II IS a disproof of ~ U {A}, then discharge~ ~f
~ AD-(i)
'
II
A
(i)
::::lr
B
is a proof of A :Jr B from ~ . Th"IS ru1e, It · seems, confers a nonconstructive meaning to :Jr. Using the rule we can derive ,...., A :Jr (A ::::lr B) for any formula B: ,..._,A0-(ii)
(A
AD-(i) (i)
::::lr B) (ii)
"'A ::::lr (A
=>r
B).
[N]either Dummett nor subsequent adherents to his anti-realist theory of meaning (... Tennant ... ), have gone beyond the notions of verification or proof as the sole conveyors of meaning. In particular, none of them takes the step of interpreting "'A as a (constructive) disproof of A. Tennant also inquires into the origin of our understanding of the meaning of negation. According to him, this origin "is to be found in our sense of contrariety", and contrariety among at least some atomic sentences of a language is a necessary condition of the language's learnability. This prerequisite of our grasp of the meaning of negation does not depend on the use of an explicit unary negation connective: "it is enough to have a few pairs of antonyms ... , or contraries of a more general kind". In fact, the presence of antonyms and contrary notions appears to be indispensable for concept formation and information acquisition in general. 36 Our grasp of the meaning of negation is thus based on predications which we use to communicate distinctions. One and the same physical object cannot, according to the same scale, be both huge and tiny; most actions fail to be simultaneously both moral and immoral, etc. If information is indeed a difference that makes a difference, contrariety among atomic sentences of a language is central to the notion of linguistic information processing. Interestingly, these considerations of the atomicity of negation have a formal counterpart in N4. In this system every formula has a unique negation normal form ( nnf) with respect to the congruence relation of strong equivalence, 37 i.e. negations can be pushed towards the atoms. 'Positivization', the replacement in nnf's of (strongly) negated atoms by new atoms not already in the language, results in a faithful embedding of N4 into positive (intuitionistic) logic 35 However, in [37] it has been observed that in IR, f- AA"' A-+ (AV B) =>r (A A B), "which does not seem justifiable from the point of view of inferential
relevance" [37, p. 257]. 36 A more detailed analysis of oppositeness of meaning between lexical items may be found in (104, eh. 9]. 37 In N 4 two formulas A, B are said to be strongly equivalent if not only 1- A-+ Band f- B-+ A, but also f- "'A-+ "'Band 1- "'B-+ "'A.
136
NEGATION AS FALSITY
CHAPTER 9
(see [127]). Following Tennant's explanation of our understanding of negation, atomicity of strong negation in N4 accounts for the equal importance of positive and negative atomic information. To put it in a slogan: literals have equal rights. Suppose that for any formula A, +A denotes the removal of negation from A by positivization of A's nnf, and +~ = {+A I A E D.}. According to Pearce [128], a negation in a logical system -t is hard iff
1-
~ -t
A iff 1-
+~ -t
+A.
(a)
(!3) ('y)
9.2. NEGATION AS FALSITY
Consider a sententiallanguage containing a unary operation *. 38 A twoplace relation -t between finite sets of formulas and single formulas in this language is called a single-conclusion consequence relation iff for all formulas A, B and finite sets ~' r of formulas:
(i) (ii)
1- A-t A ~ -t A, r U {A} -t B 1- D. U r -t B
(reflexivity) (cut)
A binary relation +- between finite sets of formulas and single formulas is called a single-conclusion *-refutation relation iff for all formulas A, B and finite sets ~' r of formulas: (i) (ii)
1- *A +- A 1- A+- *A D.+- A, r U {*A}+- B 1- ~ ur +- B
(!3') (1')
the relation -t defined by ~ -t A iff ~ +- *A is a single-conclusion consequence relation; for every formula A, not both 1- 0 +- A and 1- 0 +- *A; there is a formula A such that not 1- A +- A.
The conditions (a) and (a') express the idea ofnegatio~ ~a conn~c~ing link between proofs and refuations, whereas the remammg conditiOns reflect the contrariety between the notions of proof and disproof linked in this way. In particular, it is reasonable to assume that not every formula A is a refutation of A from {A} and that not every formula A is interderivable with its negation *A. If * satisfies both (a) and (a') for a single-conclusion consequence relation -t and a single-conclusion *-refutation relation +-, then negation as falsity is a vehicle for either keeping -t and dispensing with +or keeping+- and dispensing with -t. Then not only_ the ~ouble _negation law A -t * * A but also its converse * * A -t A IS easily denvable and we have the rule ~+-*A
r U {A}+- B
1- ~ U r +-B.
(*-reflexivity)
Analogously, A +- * * *A and * * *A +- A. Clearly,. the_ relatio? +defined by (a) is a single-conclusion *-refutation relatiOn Iff * satisfies
(*-cut)
1- A-t** A. Let us refer to an ordered pair (-t, +-} as a system, if -t is a single conclusion consequence relation and +- is a single-concl~sion *refutation relation. If S = ( -t, +-} consists of a single-concl uswn consequence relation -t and just any binary relation +- between finite sets formulas and single formulas, then if* satisfies (a) and (a'), S is a ~ys tem. (To see this, note that since *A-t *A, by (a), *A+- A, and smce A-t A, by (a'), A+- *A. If~+- A and r U {*A} +-f!, then, by (a), ~ -t *A and r u {*A} -t *B. Applying (cut) we obtam ~ U r -t *B,
We assume that membership in -t and +- is determined by a set of inference rules. If -t is a single conclusion consequence relation, then * is a negation as falsity in -t iff 38
the relation +- defined by ~ +- A iff ~ -t *A is a single-conclusion *-refutation relation; for every formula A, not both 1- 0 -t A and 1- 0 -t *A; there is a formula A s.t. not both 1- A-t *A, 1- *A-t A.
If+- is a single conclusion *-refutation relation, then * is a negation as falsity in +- iff
(a') Pearce observes that hard negation cannot be contrapositive. Akama [2] suggests using hardness as a defining characteristic of the notion of strong negation. In the next section a definition is given of the notion of negation as falsity.
137
We take advantage of context sensitivity and reuse the symbol '*' from the structural language of display logic.
138
NEGATION AS INCONSISTENCY
CHAPTER 9
and (o:) gives ~ U r +---B.) IfS= (~, +---) is a pair consisting of any two-place relation~ between finite sets offormulas and single formulas and a single-conclusion *-refutation relation +---, then S is a system if* satisfies (o:'). What can be said in favour of (*-reflexivity) and (*-cut)? In particular, one might wonder why a refutation relation should not also be a single-conclusion consequence relation, preserving falsity instead of truth. In fact, inverse consequence due to Slupecki et al. [157], [158] and also W6jcicki's [198] notion of dual consequence provide examples of such falsity-preserving relations. Obviously, if one reads ~ +--- A as 'if the formulas in ~ are false, then A is false', then +--- should turn out to be reflexive and transitive. However, this approach is inappropriate if we want to introduce negation as falsity (in the sense of refutability) by means of+---: if~ +--- A is defined as ~ ~ *A, then *A ~ *A implies *A +--- A, whereas *A +--- *A would imply *A ~ * * A. Moreover, the rule ~ ~ *A r u {A} ~ *B f- r u ~ ~ *B hardly supports interpreting * as a negation operator. The reading of ~ ~ A appropriate for our purpose is 'there is a proof of A from ~'. And if this means that there is a refutation of *A from ~' then A ~ A just translates into A +--- *A. Conversely, if ~ +--- A is interpreted as 'there is a disproof of A from ~' and if this means that there is a proof of *A from~' then *A+--- A translates into an instance of (reflexivity), and (*-cut) translates into (cut). The conditions (*-reflexivity) and (*cut) are hence the obvious counterparts of (reflexivity) and (cut), if* is introduced as negation as falsity. Note that a more uniform proof-theoretic definition of negation as falsity is available in terms of four-place sequents
with the following reading: if every formula in r is true and every formula in I: is false, then some formula in ~ is true or some formula in e is false. Blarney and Humberstone's reflexivity and cut rules from Chapter 2 now reappear as notational variants: (odd reflexivity) (even reflexivity)
f- A I 0-+ A I 0
(odd cut) (even cut)
r I I: -+ A, ~ I e A, r I I: -+ ~ I e r I I:-+~ I A, e r I A, I:-+~ I e
f-
0 I A-+ 01 A 11-
r I I: -+ ~ I e r I I:-+~ I e
139
As a counterpart of conditions (a) and (a') one obtains
(o:")
f- *A I 0 ~ 0 I A f- 01 *A~ A 10
10 ~ 01 *A 01 A~ *A 10
f- A f-
There are equally obvious counterparts of conditions (/3), (/3'), (1), and ('y').
(!3")
for every formula A, neither both f- 0 10 ~ 01 A and f- 0 10 ~ 0 I *A nor f-
('y")
0I 0~
A
I 0and
f-
0 I 0 ~ *A I 0
there is a formula A, such that neither both ff-
I 0 ~ *A I 0and 0 I A ~ 0 I *A and A
ff-
I 0 ~ A I 0 nor 0 I *A ~ 0 I A
*A
Definition 9.1. A unary operation * is called a nega_ti?n as f~lsity ,in a four-place consequence relation~ iff *satisfies conditions (a ), (/3 ),
and ('y"). Moreover, in this setting there are separate, symmetrical, and explicit introduction rules for strong negation, namely: ( ~'"" odd) ( ~'"" even) ('""~odd) ('""~even)
r 11: ~ ~ 1A, e r- r 11: ~ "'A, D.. I e r 11: ~ D.., A 1e r- r 11: ~ D.. 1"'A, e r 1A, 1: ~ D.. 1e r- "'A, r 11: ~ D.. I e A, r 11: ~ D.. 1e f- r 1""'A, 1: ~ D.. I e
However, with four-place (single-conclusion) consequenc: relat~ons we lose comparability with Gabbay's notion of ne~ation ~ mco_nsistency. Instead of developing a generalization of negatiOn as mconsi~tency to four-place Gentzen sequents, we shall introduce f~ur-place display sequent arrows in Section 13.3 and use this generalizatiOn of DL to redisplay Nelson's useful paraconsistent logic N4.
9.3. NEGATION AS INCONSISTENCY
Consider again a propositional language containing ~ unary conn~c tive *· Think of a logical system as being given by a ,smgle-~onclusiOn consequence relation ~ over this language. Gabb~y s (66]_ Idea for a syntactic definition of negation (as inconsistency) m a logica~ system is that f- A ~ *B iff A together with B leads to some undesirable C
140
from a set ()* of unwanted formulas. In this context, the object language counterpart of bunching premises together is conjunction /\, governed by its sequent rules in positive logic (i.e. (a) 1 and (a') 1). Hence, let us suppose that 1\ is already in the language or that it can conservatively be added. Gab bay defines * as a negation (as inconsistency) in -+ iff there is a non-empty set ()* of formulas which is not the same as the set of all formulas such that for every finite set ~of formulas and every formula A we have:
Moreover, ()* must not contain any theorems. If such a collection of unwanted formulas exists, it can always be taken as {C 11- 0 -+ *C}, since by (reflexivity) the latter set is non-empty, if * is a negation. There is an equivalent definition, which does not refer to (}*. Namely, * is a negation in -+ iff for every finite set ~ of formulas and every formula A the following holds:
h IT' is a disproof of ~' C, B. By the claim for disproofs, IT' is a wd. ere f f ~ B and hence by rv-lntroduction, IT is a proof of"" B Isproo o ' ' El' . t' then from~. If IT ends in an application of rv- Imma lOll, ~,c
IT'
""A
Lemma 9.2. Suppose C is a theorem of IR. Then in IR, (i) if II is a disproof of~ U {C}, then IT is a disproof of~' and (ii) if II is a proof of A from~ U {C}, then IT is a proof of A from~. Proof. By simultaneous induction on the construction of proofs and
disproofs in IR. For example, if II is a proof of A from {A}, then ~ = 0 and C = A, thus A is a theorem of IR. If II terminates in an application of rv-Introduction, then A = ""B and II has the form: ~
C BD-(i)
' 'II'
rvB 39
See (101 J for a critical discussion of Gab bay's definition in terms of intuitive criteria of adequacy.
A
. . f f ~ C U {"-'A} while IT' is a proof of A from ~'C. IS a disproo o ' ' f f A f ~ and However, by the claim for proofs, IT' is also a pr~o o rom "" A hence the above disproof figure also represents a disproof of ~ U { }. The remaining cases are equally simple. Q.E.D.
Observation 9.3. Negation in IRis a negation as inconsistency. Proof Put(}*= {rvAI\A I A is a formula}.lf ~-+"-'A, then ~U{A}-+ "" AI\.A follows by /\-Introduction. For the converse, note that for every
B,
rvBI\B ""B B rv(rvBI\B)
1- ~-+*A iff ::JC (1- 0-+ *C & 1- ~ u {A}-+ C). 39 In [69] the notion of negation as inconsistency (alias inferential negation) is extended to a novel kind of non-monotonic inference relations between structured databases, called structured consequence relations. In IR, the deducibility relation is not unrestrictedly transitive and hence fails to be a consequence relation. The basic idea of negation as inconsistency, however, does not depend on this assumption. We want to show that negation "" in IR is a negation as inconsistency, and for this purpose we only assume that (reflexivity) holds.
141
NEGATION AS INCONSISTENCY
CHAPTER 9
is a proof in IR of "" ("" B 1\ B) from 0. If now IT is a proof of ""B 1\ B from~
U {A}, then ~u{A}
IT
""(rvB 1\ B)
rvB 1\ B
. d' f of ~ U {A} U {"-' ("-' B 1\ B)} and, by the disproof part IS a Isproo bt · oof of "" A of the previous lemma and ""-Introduction, we o am a pr from ~. Q.E.D. Negation in minimal (intuitionistic, classical) senten~ial lo?ic, MPbL (IPL CPL) can also be shown to be a negation as m consistency . y identifying (); with the set of all explicit contradictions in the respectiVe language. . · t' fies contrapoA tated above every negation as mconsistency sa IS . . s· B B by defimtlon there ss ' sition as a rule. Suppose A -+ B · mce * -+ * ' . · ) is a C E (}* such that {*B' B} -+ C. Performing an applicatiOn of (cut . { B A} -+ C and hence *B -+ *A· Therefore, strong negawe obt am * ' · M reover every tion in Nelson's N4 is not a negation as inconsistency. o ' .. negation as inconsistency validates the law of exclu~ed contr~di~I~~' *(*A 1\ A). Since *A-+ *A, we have ~*A,~}-+ B, son;e(*A 1\ A). -+ B for some B E (} ' which means VJ -+ . (\ A H ence, * A '
o:
NEGATION AS FALSITY VS. NEGATION AS INCONSISTENCY 143 142
CHAPTER 9
bHence ' negation . . in Bel nap ' s [15] "use ful four-valued logic" also fails t o e a negatiOn m Gabbay's sense.
9.4. THE RELATION BETWEEN NEGATION AS FALSITY AND NEGATION AS INCONSISTENCY
Our aim · as mconsistency . . as fals. t is. to show that e.ver~ nega t IOn is a negation I y, I.e. we want to JUStify the following picture: negation as falsity negation as inconsistency
N4
IR MPL IPL CPL
If * is a negation as in;en~sfiency, · t relation-+ sequence
:e
A f- 0.\0.\A.
are give~ a single-conclusion con-
of formulas and sin~le for~~:: bym:~~ ~l~ti~n ~l betwe(en finite sets tio £ 1 ·t I e nmg cause a) for nega } . . n as a SI y. t remains to be shown that (-+ the defined ~ is in fact a single-conclusion * r ' ~ . IS a sy~tem, I.e. moreover, that conditions ({3) and (r) are relatwn, and,
sat~s~~~-atwn
J." l()_btservation 9.4. Every negation as inconsistency is 1a si y. a negation as
;roof. It must be shown that the defined
relation~
satisfies ( *-reflexiviy), (*-cut), ({3), and (r). (*-reflexivity): *A~ A is cl f A and the definition Since 0-+ *(*A 1\ A and ear rom * -+*A there is a C such that 0 -+ *C and {A} U {*j} *A} -+ *A 1\ A, and thus by the defi "t" f A -+ ' hence A -+ * * A ' m Ion o ~ ~ *A ( ) follows from the definition of~ 'd ( t) £ · *-cut : The (*-cut )-rule r U {A} ~ B Th" an cu or -+. Assume b. ~ *A and . IS means b. -+ A and r U {A} B A of (cut) gives b. u r -+ *B , w h"1ch means b. u r ~ -+ B* · n application · d ( Suppose that both 0 -+ A and 0 -+ *A £ A ' as reqmre . {3): E ()* such that A -+ B or some · Then there is a B theorem uod n . However, by (cut), -+ B, that is,()* contains a A A' q on. (1): Suppose that for every formula A A-+ *A d * -+ . Then there is a B E ()* such that A -+ B . 0 ' an ing_ (cut) to the latter and *A-+ A, we obtain 0 B-+t *Ah. Ap?ly()* IS non-empt ·t . · u t en, smce y, I must contam a theorem; a contradiction. Q.E.D.
of~-
Obviously, the construction in the proof does not work for any unary connective. Consider, for instance, 0 or 0• in the smallest normal modal logic K. In K, 0 and 0• do not satisfy (*-reflexivity). We have thus obtained a non-trivial generalization of Gabbay's definition. This definition of negation as falsity covers Nelson's strong negation, a recognized negation living beyond the realm of negation as inconsistency. In view of its atomicity, in [69) we have referred to strong negation in N4 as a rewrite connective. 40 This choice of terminology emphasizes the problem of ensuring that a generalization of Gabbay's definition of negation in order to capture strong negation in N 4 does not result in too general a notion of negation. One may thus wonder whether the notion of negation as falsity also encompasses unary connectives which fail to be 'negations' on intuitive grounds. The conditions (a), ({3), ('"y) and the basic properties of -+ and ~ may appear to be rather weak and perhaps insufficient. But in fact, positive normal modalities do not present counterexamples. Consider prefixes of the shape 0.\. If 0.\ in normal modal logic is to figure as a negation as falsity, then, m an axiomatic setting, the following rule must preserve validity:
~A,
~I.~
Thus, if B is any tautology, then f- 0.\0.\B. For the possible worlds models this implies that every world has at least one successor. If now ({3) were satisfied, then for every formula A, we would have ff A 1\ 0.\A. In particular, ff 0.\B 1\ 0.\0.\B. Hence ff 0.\B, quod non, since 0.\B cannot be falsified in serial Kripke models. For prefixes of the form 0.\, 0.\0.\(p V •P) enforces a property of the accessibility relation also validating 0.\(p V •p). Thus, f- (p V •p) 1\ 0.\(p V •p), in contradiction of ({3). In general, negative modalities do not fail to be negations as falsity. In [69) we have observed that if'"'"' is interpreted as 0•, all equivalences which axiomatize N4 in Hilbert-style hold in Kripke models in which the accessibility relation forms pairs {t, s I t'Rs and s'Rt}. In such models every negative modality is a negation as falsity. According to Lenzen's [101) catalogue of principles indispensable for any genuine negation, the notion of negation as falsity is not only too general, since it does not require the contraposition rule, it is also too restrictive, because it calls for double negation introduction: A f- * * A. As already remarked, in some systems of normal modal logic the latter fails to hold for*:= 0•. Lenzen, however, takes 0• to be a weak form of negation and hence rejects A f- * * A as a necessary property of negations. While
n
40
Note that in (69) N4 is called N.
144
CHAPTER 9
SEMANTICS-BASED NONMONOTONIC REASONING
on ~he one hand this attitude may be welcomed because negation as falsity turns out to impose a significant constraint, on the other hand the same type of objection can be raised against the weaker rule f-A
I
f- **A,
which together with contraposition as a rule and
(6) there is a formula A such that not A f- *A constitutes Lenzen's list, namely, for every tautology A, a theorem of K.
O...,O...,A is not
9.5. SEMANTICS-BASED NONMONOTONIC REASONING
In this section, we discuss Gabbay's idea of basing nonmonotonic inference on semantic consequence in IPL extended by a consistency operator and Turner's suggestion of replacing the intuitionistic base system by Kleene's three-valued logic 3. It is shown that a certain counterintuitive feature of these approaches can be avoided by using Nelson's constructive logic N3 instead of intuitionistic logic or Kleene's system. The _addition of a c~nsistency operator to the more fundamental paraconsistent constructive logic N4 is considered in [195]. In order to avoid fixpoint definitions of nonmonotonic inference Gabbay [65] suggested basing nonmonotonic deduction on semanti~ consequence in a logical system extended by a consistency operator M (see also [34] and [35]). MA is to be read as 'it is consistent to assume at this stage that A'.
Definition 9. 5. A formula A is said to nonmonotonically follow from a set of assumptions Ll = {A 1, ... , An} ( Llr--A) if there are formulas B1, ... , Bm such that A1, ... ,An ~ B1 Al, ... ,An,Bl ~B2
A1, ... ,An,Bl, ... ,Em~ A,
~nd the a~xiliary relation ~ is defined as follows: cl' ... ' ck ~ c If there exist extra assumptions D 1 • • . D · such that (i) {C C ' ' J 1' ... ' kl MDl, ... ' MDj} is consistent, and (ii) {Cl, ... , ck, MDl, ... 'MDj} F C.
145
This notion of nonmonotonic inference requires, of course, a clear semantics for consistency statements MA. Gabbay's idea is to interpret M as possibility with respect to the 'information ordering' !:; in Kripke models for intuitionistic logic, that is, MA is true at an information state t iff there is a state u such that t !:; u and A is true at u. Let us refer to the result of extending IPL by M as CG. Assuming that nonmonotonic inferences are appropriate only in the presence of incomplete information, Turner [169) suggested using G~b bay's definition of nonmonotonic inference based on a system of partzal, in effect Kleene's system 3. Turner considers model structures (I,!:;), where I is the set of all partial interpretations of the atomic sentences and !:; is a 'plausibility' relation on I, that is, a reflexive transitive relation such that t ~ u implies t ~ u, where ~ is the natural 'information ordering' on partial interpretations. Consistency statements MA are evaluated as true at an information state t E I like in CG, MA is defined to be false at t, if A is false at every information state u which is at least as plausible as t, and MA is evaluated as undefined at t otherwise. Let us call Turner's system CT. Gabbay's andTurner's approaches both sucessfully deal with various counterintuitive features of McDermott's and Doyle's [109] nonmonotonic formalism. In McDermott's and Doyle's logic, for instance, {-,Mp} is nonmonotonically inconsistent, since the non-derivability of -,p forces Mp to be assumed. Moreover, {(Mp :J q), ...,q} {Mp, ...,p}
is inconsistent; is satisfiable;
{M(p A q), -,p} M(p A q) fv Mp.
is satisfiable;
However Gabbay's and Turner's nonmonotonic systems suffer from an' . other weakness (see [103]), namely the fact that Mp :J p ...,p, since m CG as well as in CT, {(Mp :J p), M-,p} I= ...,p and {(Mp :J p), M-,p} is consistent. In the literature it has been suggested interpreting Mp :JP as 'p is true by default'. Intuitively Mp :J p ...,p clearly fails to be sound, no matter that also Mp :J p p, because ...,p should simply not be nonmonotonically derivable from the assumption that p is true by default. According to Lukaszewicz [103], this weakness renders it problematic to apply Gabbay's and Turner's definitions of nonmonotonic inference to formalizing common-sense reasoning. We shall see that if Gabbay's definition of nonmonotonic inference is based on semantic consequence in Kripke models for Nelson's system N3, then (Mp :J p) A M "'p F"' p fails to hold. N3 combines t_he ~dvant~ges_ of (i) having an intuitionistic and hence a genuine implicatiOn sahsfy~ng the Deduction Theorem and (ii) semantically being based on partial, three-valued interpretations. As we shall see, this is exactly what is
r.-
r.-
r.-
SEMANTICS-BASED NONMONOTONIC REASONING
CHAPTER 9
146
needed to overcome the problem with the approaches of Gabbay and Turner. Moreover, we shall discuss justifying the choice of a suitable base logic and in the course of this consideration discuss the suggestion of evaluating MA as true at an information state t iff the negation of A fails to be true at any possible extension of t. It will turn out that this rather strong notion of consistency is definable in N3 itself and, moreover, its definition in N3 directly expresses a natural and basic constraint on formalizing M. Before that, however, we shall look at the intuitionistic and the Kleene base logics. lntuitionistic base. Assume we have a non-empty set Atom of propositional variables. CG is the theory of the class of all intuitionistic Kripke models in the language {'h, M, ~h, 1\, V} over Atom. An intuitionistic Kripke frame is a structure (I,~), where I is a non-empty set and ~ is a reflexive transitive relation on I. An intuitionistic Kripke model is a structure (I,~' v), where (I,~) is an intuitionistic Kripke frame, v is a valuation function assigning to each propositional variable a subset of I, and for every p E Atom and every t, u E I: (persistence 0 ) ift ~ u, then t E v(p) implies u E v(p). Kripke [94] suggested the following 'informational' reading of frames (I,~): I is a set of information states and ~ is the relation of possible expansion of information states over time. Let M = (I,~' v) be an intuitionistic Kripke model, t E I and A a formula. M, t f= A (A is verified at t in M) is inductively defined as follows: M,tf=p M,tf=BI\C M,tf=BVC M,t F B -:Jh M,t F 'hB M,t F MB
c
iff iff iff iff iff iff
t E v(p), p E Atom M, t f= Band M, t f= C M, t f= B or M, t f= C (VuE J) if t ~ u, then M,u (Vu E J) if t ~ u, then M, u 3u E I, t ~ u and M, u f= B
~Mpp
9) gested basing Gabbay's defTurner (16 sug 1 d 1 gt·c 3 In K leene 3-valued base. · · r Kl ene's three-va ue o · inition of nonmonotomc m~ere~ce ?n e A B = "' A V B' 3 a non-constructive implication lS defined by ~ def r~sulting in the following truth table for ~:
u
0
1 u 1 u 1 1
0 u 1
1 1 u 0
. is to be read as 'undecided'. Since there where the thtrd truth value u d t h ld in Kleene's are no tautologies, the Deduction Theorem oes no o logic. ' tics for CT makes use of partial interp~e~ations. f>:Turner s seman. . . from the set of proposttional vanpartial interpretatwn lS a maplpmg { } The natural 'definedness 0
s~t trut~. va£luesa~~;~v::d~;i:g~ ~ ~: ~artial inter~
f= ~
B => M,u B
f= C
If .6. is a set of formulas, then M, t f= .6. iff M, t .6.. A formula A is said to be entailed by .6. (.6.
f= A for every A E f= A) iff for every and every t E I: M, t f= .6.
intuitionistic Kripke model M = (I,~' v) implies M, t f= A. Whereas formulas A in {'h, ~h, 1\, V} satisfy (persistence) if t ~ u, then t
Mp
147
f= A implies u f=
A,
(persistence) fails to hold for arbitrary formulas. The Deduction Theorem does not hold either. Although {Mp, Mp ~h q} f= q, in the following model t f= Mp and t ~ (Mp ~h q) ~h q:
.............
______________
'T"· 148
(a) t f=-"'"'P, i.e., t f=+ p. By (persistence+), u f.=+ p for every u such that t ~ u. But since t f=+ M ,.._, p, there is a state u such that t ~ u
as follows:
M,t f.=+ p M, t f=- p
iff iff
M, t f=+ B 1\ C M,tf=- BI\C
iff iff
M,t f.=+ BVC M,t f.=- BVC
iff iff
M, t f=+ B :J C M,t f.=- B :J C M,t f=+,.._,s M,tf=-,.._,B
iff iff iff iff
M,t f.=+ MB M,t f.=- MB
iff iff
t(p)=1, = 0, M, t f= + M, t f=-
pE Atom
t(p)
p E Atom B and M, t f= + C B or M, t f=- C M,t f.=+ B or M,t f.=+ C M, t f=- B and M, t f=- C
M, t f=- B or M, t f= + C M, t f= + B and M, t f=- C M,t M,t :lu E Vu E
f.=- B f.=+ B
I, t I, t
~ u and ~
M, u f= + B u implies M, u f.=- B
It can be shown that every M-free formula A satisfies the following persistence properties:
F + A implies u F + A then t F- A implies u F _ A.
(persistence+) if t ~ u, then t (persistence-)
149
SEMANTICS-BASED NONMONOTONIC REASONING
CHAPTER 9
if t
~
u,
Anomalies .analysed. When applied to the theory {Mp :Jh } the nonmonotomc consequence relations based on CG and CT t p ' t to be · ou . pro blemat.Ic, smce Mp :Jh p f-v 'hP and Mp :J p f-v "'"'P byurn virtue (·~)(l{)(~Mp :Jh)p), M·hP} and {(Mp :J p), M "'"'P} being con~istent and 11 P :Jh P , M·hP} F •P and {(Mp :J p), M "'"'P} f=+,.._, W h ll p. es a now take a closer look at why (ii) holds true.
Intuitionistic base. Suppose t F Mp :J p t ~ M· Th sumption means that -, . "fi d h ' I .hP· e second ashP IS ven e at some poss1ble expansion of t and hence, ~ue to (persistence), p cannot be verified at t. If t ~ -, ' then there IS a state w such that t c w and w I th hdP, t f= M w· h h 1 P· n o er wor s P: It t e fi.rst ~~u~p~ion, t F p, quod non. Hence, using th~ sem~ntlc c~ause for mtmt~omstlc negation, 'hP is verified at t. If, as in partla~ logic, we use ~arti~l valuations and define the negation of p to ~e~enfied ~t a s~ate Iff pIS falsified at that very state, then, of course, . P :Jh PIS venfied at t and "'"'Pis not, this does not impl th t · venfied at t. Y a P Is
~
Kleene 3-~alued base. Suppose that t f=+ (Mp :J p) 1\ M "'"'P and t ~+ "'"'P· Consider the following two cases (as in [l03]):
and u f.=- p, quod non. (b) Neither t f.=- •p nor t f.=+,.._, p. By the verification conditions for implications, t f.=- Mp, since t f.=+ Mp :J p. Hence u f.=- p for every u such that t ~ u. In particular, t f.=- p, quod non. Clearly, in this case the problem arises by defining A :J B as ,.._,A V B.
Constructive base. Nelson's constructive logic N3 (cf. [119], [4]) combines the virtues of intuitionistic logic (viz. its constructive implication) and partial logic (viz. its suitability for representing incomplete information). From a philosophical point of view, one main advantage of Nelson's logic is that it allows formulas to be falsified on the spot in intuitionistic Kripke frames. A Nelson model is a structure (I,~' v), where (J, ~) is an intuitionistic Kripke frame and v a mapping that assigns to each w E I a partial interpretation Vw such that for every propositional variable p and every t, u E J: (persistenceri) ift ~ u, then Vt(P) = 1 implies vu(P) = 1. (persistence())
if t ~ u, then Vt (p)
= 0 implies Vu (p) = 0.
Let M = (J, ~' v) be a Nelson model, t E I and A a formula in the language {"-',M, :Jh, 1\, V} over Atom. The notions M, t f.=+ A (A is verified at t in M) and M, t f.=- A (A is falsified at t in M) are inductively defined as follows: iff M,t \=+ p iff M,t \=- p iff M,t \=+ B 1\C iff M,t \=- B 1\C iff M,t\=+ BVC iff M,t\=-BVC M,t \=+ B =>h C iff M,t \=- B =>h C iff iff M, t \=+,..., B iff M,t\=-....,B iff M,t \=+ MB iff M,t \=- MB
Vt(p) = 1, p E Atom Vt(P) = 0, p E Atom M,t \=+Band M,t \=+ C M,t \=- B or M,t \=- C M,t \=+ B or M,t \=+ C M,t \=-Band M,t \=- C (VuE 1) if t ~ u, then M,u \=+ B M,t \=+Band M,t \=- C
M,t \=M,t \=+ 3u E 1,t VuE 1, t
::::::>
M,u \=+ C
B B
~ u and M,u \=+ B ~ u implies M, u \=- B
Nelson's system N3 is the theory of the class of all Nelson models in the language {,...,, :Jh, 1\, V}. It can easily be shown that every formula A in this language satisfies (persistence+) and (persistence-). Contraposition does not hold in N3. Moreover, provable equivalence fails to be
CHOICE OF PARAMETERS
150
151
CHAPTER 9
logic is, according to Gabbay, given by a certain approximation of the "main formal equation for M" (for consistent formulas A):
a congruence. relation on the set of formulas. If formulas A and B are defin~d as bemg strongly equivalent iff both A and B and their stron '"" A '"" B provably equivalent, then it can be at strong ~qmvalence IS a congruence relation in N3. In [180] it is arg~ed that m th: context of abstract information structures these are esirable properties. Obviously, N3 is related to three-valued logic. If one permits onl Nelson models, one obtains a three-valued logic that diffe/s rom the _well-known systems of Kleene, Lukasiewicz, and Bochvar insofar, as It has the following truth table for ~h:
:~gatwns
~nd
~re
show~
(*1) A If -.B iff A 1-MB or rather its semantic counterpart
d
iff A f= MB. Gabbay points out that in a certain naturally defined intuitionistic
(*2) A
~ne-state
A
~h
B
1 u
0
1
u
0
1 1 1
u
0 1 1
1 1
Kripke model M one can show that
(*3) M, A lF -.B iff M, A f= MB, where A and B are formulas in { -.h, 1\, ~h} and the information states are the formulas in this fragment themselves. The justificatory power of this approximation is not quite clear. Whereas (*2) asserts an equivalence between the existence of certain countermodels and a property of all models, (*3) is a claim about one particular model. Moreover, since the consistency operator M was introduced in order to define a notion of nonmonotonic consequence, and nonmonotonic reasoning is usually triggered by incomplete information about the world, it seems natural to use a system of partial logic as the monotonic base system. Gabbay's evaluation clause for consistency statements MA is derived from the semantic notion of consistency in classical logic, that is, A is consistent, if there is a model that verifies A. Classically, this is equivalent to the requirement that not every model falsifies A. In partial logic, however, the equivalence breaks down, since A may be undefined at every state of a model structure. At first glance the following notions of M, s f= MA appear to be plausible:
This system can be axiomatized by adding the axiom schema ( (A ~ ~h'"" (A ~h rv A)) ~hA to the familiar Hilbert-style axiomatiza~ twn of N3, cf. [84]. Let us refer to the above extension of N3 as C3d. In C3d we have adopted Turne~'s 'dynamic' falsification conditions for consistency statements _MA. This, _howev~r, is a deviation from the remaining f=-_ clauses.' which ~xemphf~ falsification 'on the spot'. If we want to stick to the Idea of direct falsification, then Turner's f=- -clause for formulas MA should be replaced by the less general
":'A)
M,t f=- MB iff M,t F+'"" B.
a
resulting system, C3, like C3d, is not only void of the problematic eatures of McDermott's and Doyle's approach; we also have
b
~he
{(Mp ~hp), M
rv
p} ~~3drv p,
9.6.
lF -.B
c
f=+ A (3t E J) s c t & M, t lf=- A (Vt E J) s C t => M, t lf=- A
(3t E I) s C t & M, t
Obviously, c implies b, and in the context of Nelson models for N3, also a implies b. However, consider again Mp ~hp. If p is falsified at every point in a model, then clause b renders Mp ~h p true at every point, in other words, p is true by default everywhere, which is absurd. But clause c, that has been suggested in (184), may also be viewed as problematic. This clause restores (persistence+) and (persistence-) for the entire language, and persistence of consistency statements may be an unwanted property. Nevertheless this persistent notion of consistency has some nice properties. Consider (*1) again. The intuition behind
{(Mp ~hp), M'"" p} ~ts'"" p.
CHOICE OF PARAMETERS
~part from (i) (implicitly) claiming that the semantic clause for M m C_G captu_res the_ notion of consistency in a plausible way and (ii) c?nsid~rably Impro~m~ on ~cDermott's and Doyle's approach, Gabbay give~ ~Ittle fur_ther JUStificatiOn for using an intuitionistic base system. AdditiOnal evidence for the suitableness of working with intuitionistic
(*1) seems to be this: I
I
L
152
CHAPTER9
CHOICE OF PARAMETERS
153
(*4) A fl•hE iff Af-vB. If on,e w~nts to confine :he meaning of M in the base system by a general . mam for~al equation', then the following is a natural equivalence (agam for consistent formulas A):
(*5) A, 'hE
f= f
w~er~ f may ~e defined
iff A
f=
ME,
asP 1\ 'hP, for some p E Atom. What one ob-
~am~ ~sa_ c~nsistency oper~tor slightly stronger than Gabbay's, namely mtmtiO~Istlc double negation 'h 'h· 41 The verification conditions for
'h'hA Imply those for Gabbay's MA:
t
f= 'h 'hA
iff iff only if
(Vu E I) t ~ u :::::} u ~ 'hA (VuE I) t ~ u :::::} ((3w E I) u ~wand w (3u E I) t ~ u and u f= A.
f= A)
Since in intuitionistic logic 'h 'h 'hA is equivalent to 'hA, one still has {(M~ ::J~ ~), _M:'hP} f= 'hP and therefore {(Mp ::Jh p)}f-v•hP· Now, in N3 mtmt10mst1c negation can be defined by -, h A -def A "' A , an d . -'h"' the consistency operator M given by clause c and falsification on the spot turns out to be definable by MA =def 'h ,..._, A:
t
f=+,..._, A
::J "'""'A
iff iff iff
(Vu E I) if t [;;;; u, then u f=+"' A implies u not (:3u E J) t [;;;; u and u f=+,...., A (VuE I) t [;;;; u implies u ~+,..._,A;
iff iff
t F=-"' A ::J ,....,,..._, A t f=+"' A and t f=- ,. . , . . , A tf=+rvA.
f=+ A
Note that, due to (persistence-) and the reflexivity of _, c for the defi ne d M we have
M, t
f=-
MA iff
M, t
f=+"' A
iff VuE I, t [;;;; u implies M, u
f=-
A,
that is, ~mer's falsification conditions amount to falsification on the spot. Obviously, the defined M satisfies (*5): I A semantic ~reatment of intuitionistic double negation as a modal opera- regar ds 'h'h as a necesstty · operator tor can be found m [40] · Note that Dosen D. However, he. notes that one "can prove DA0 A h. h = ..., ..., , w tc goes some way towards explamng why intuitively 0 ... has some features of possibility" ([40 p. 16), notation adjusted). ' 4
iff iff iff
A,"' E f=+ f A f=+,...., B ::J f A f=+ --, ""' E A f=+ ME.
The counterintuitive results of McDermott and Doyle and the problem with Gabbay's and Turner's systems do not arise in N3 either: the assumption sets {Mp =>h q,"' q}, {,...., Mp} are satisfiable, {Mp,"' p} and {M(p/\q),,....., p} are not satisfiable, M(pl\q)f-v Mp, and {(Mp ::Jh p),
Mrv p} ~!t 3
"-' p.
According to Clarke and Gabbay [35, p. 177 f.] (notation adjusted), "[i)t could be viewed as a justifiable criticism of the intuitionistic system that MC ::Jh C is equivalent to CV 'hC, i.e. neutral with respect to C or 'hC. This", they continue, "is not the usual intention behind defaults." Note that neither in C3d nor in C3, nor in N3 we have that MA ::Jh A is equivalent to AV ""'A. It seems as if the fact that on the basis of CG and CT we have Mp ::Jh p f-v 'hP and Mp ::Jh p f-v ""'p respectively has been considered the main obstacle to working with Gab bay's definition of nonmonotonic inference instead of fixpoint definitions. We have seen that this obstacle can easily be removed by using Nelsons constructive logic N3 as the underlying base system, that is, by working with C3d or C3. The beauty of Gabbay's definition results from the flexibility provided by the choice of the underlying base logic. In principle any logic given by a class of 'information models' will do. What is needed is some kind of information order [;;;; to interpret the consistency operator M. The various persistence conditions and the presence or absence of properties of [;;;; (like reflexivity, seriality, transitivity , etc.) give rise to a semanticsdriven landscape of subsystems of C3d, C3, but also of CG and CT. As the inspection of the anomalies of CG and CT has shown, in the absence of (persistence), the derivation of Mp ::Jh p f-v 'hP is blocked for CG, and in the absence of either (persistence+) or reflexivity of[;;;;, the derivation of Mp ::Jh p f-v ""'Pis blocked for CT. There thus exists a large variety of different notions of consistency and hence notions of nonmonotonic consequence, which may be compared, tested against benchmark problems, and applied in knowledge representation. Subintuitionistic subsystems of IPL obtained by dispensing with properties like transitivity of[;;;; will be considered in detail in Chapter 10. It should also be pointed out that in semantics-based nonmonotonic reasoning there is no need for the underlying base logic to be monotonic. It is well-known that in IPL the persistence property corresponds to the
154
CHAPTER 9
validity of the monotonicity axiom schema A ~h (B ~h A). If thus Mp ~h p f-v 'hP is avoided by abandoning the persistence requirement in CG, one obtains a relevance logical base system, see Chapter 10. There is nothing wrong with basing nonmonotonic inference on a nonmonotonic logic, since nomonotonicity as such is only a symptom, comparable to the absence of contraction or permutation of premise occurrences in substructurallogics. What is important, however, is the naturalness of the nonmonotonic inference mechanism or rather its formal representation. Gabbay's definition describes such a simple and natural mechanism. There is an obvious open problem, namely axiomatizing C3d and C3.
9.7. MODAL LOGIC OF CONSISTENCY OVER N4
In the present chapter, we have discussed the modal logic of consistency over IPL as suggested by Gabbay [65] and the modal logic of consistency over Kleene's three-valued logic 3 as defined by Turner [169]. It was observed that conterintuitive features of the resulting nonmonotonic consequence relations can be avoided in a very natural way by replacing IPL and 3 with N3, or by replacing IPL with a suitable subintuitionistic logic. While N3 is a system of partial logic, Nelson's N 4 is both partial and paraconsistent. Since in applications one has to cope not only with partial but also with inconsistent information, in general the monotonic logic of information on which nonmonotonic consequence is semantically based should be partial and paraconsistent. With strong negation primitive, the latter means that {A, '"'" A} does not entail arbitrary formulas. In [195] it is shown that C4, the modal logic of consistency over Nelson's paraconsitent logic N4, can be faithfully embedded into the modal logic S4. Moreover, it is shown that this embedding can be used to obtain cut-free display sequent calculi for both C4 and N4. From this presentation one can also straightforwardly obtain a complete display sequent calculus for C3.
CHAPTER 10
DISPLAYING AS TEMPORALIZING
This chapter is about display sequent calculi for subintuit~onisti~ l?gics, that is, logics obtained from intuitionistic sen~e.ntiall~gic by_givmg up or relaxing all or part of the following conditlO~s:_ ~I) persi~tence of atomic information, (ii) reflexivity of the accessibihty relat10n ~ (iii) transitivity of ~- The sequent calculi are obt_ain~d. fr~m. sequ~nt systems for certain 'temporalizations' of t~e ~ubmtmtwmstlc logics. However we will not consider full temporahzat10ns, but only one particular c~nstruction which is available because the temporalizing and the temporalized system are complete with respect to the same class of Kripke models. The subintuitionistic logics are m_otiv~~ed _b~ ext~nd ing the well-known informational interpretation of mtmtwmstic Knpke models.
10.1. SUBINTUITIONISTIC LOGICS
One particularly appealing feature of intuitionistic propositionallogic, IPL, is that it may be regarded as the logic of cumulative research, see [94]. It is sound and complete W:ith respect ~o the class of all no~-empty sets I of information states which are quasi-ordered by a relat10n ~ of 'possible expansion' of these states, and in which atomi~ formulas _established at a certain state are also verified at every possible expans10n of that state. There are thus three constraints which are imposed on the basic picture of information states related by ~: ~i)_ the pers~~:ence (alias heredity) of atomic information, (ii_) th~ _re~ex_Ivity, and (m) the transitivity of~- The persistence of every mtmtwmstlc formula emerges as the combined effect of (i) and (iii). Although a Kripke frame, that is a binary relation over a non-empty set, admittedly provides an extr~mely simple model of information dynamics, and, moreover, each of the conditions (i) - (iii), as well as their com~ina~ion~, m~y. be of value for reasoning about certain varieties of scientific mqmry, It IS nevertheless interesting and reasonable to consider giving up all or some of these conditions. Evidently, conceiving of information pro~ress _as a steady expansion of previously acquired insights is extremely Ide~hzed, and the basic model of such a progress should leave room for mcorporating revisions, contractions, and merges of information as well. If
155
T• 156
I
CHAPTER 10
I
persist~nce
is given up, !:;;; can no longer be understood as a relation of possible e~pa_nsion. This reading, however, may be replaced by more gen~rally thmkmg of !:;;; as describing a possible development of infor~atiOn s_tates. Development thus need not imply the persistence of mformat10n. . When talking about the development of information states one mig~t want to di~pense with the assumption that such states always possibly develop mto themselves. There might be information states which ~n practice simply 'must' be changed, say, in the light of overwhelmmg a_nd unde~i~ble evidence. In other words, it may make sense not to reqmre reflexivity of!:;;;. Even more obviously, ~:;;;read as possible development need not be transitive in general. If t C u and u c the intermediate state u may just be indispensable order to at w from t. _In~or~ation obtained at w may, for instance, depend on conceptual distmctiOns or certain findings which are available at w only b:cause w develops from u rather than from t directly. Moreover' one might doubt that information development can ever lead to 'deacl ends'. Such an optimistic attitude would amount to requiring seriality of 1:;;;.42 I~ seell_ls only fair to say that the evaluation of intuitionistic formu~as m Knpk~ models (J, !:;;;, v}, where (J, !:;;;} is a Kripke frame and v IS a ~otal assig~m.ent function to interpret the propositional variables, provides deep msight into the nature of the intuitionistic connectives. It sets apart intuitionistic conjunction and disjunction, which are evaluated 'on the spot', from intuitionistic negation 'h and intuitionistic
i~
a-;ri~~
~ Indee~, IPL is also characterized by a class of Kripke models where c IS J~st senal, but not necessarily reflexive and transitive. In these so-called rudimentary Kripke models (46], however, not only persistence is postulated ~or e:ery i~tuitionistic formula A, but also converse heredity, which says that If A IS venfie~ at e:ery state into which a given state t may develop (in one step), then A Is_venfied_at t already. Upon reflection this is a very strong, yet not completely Implausible constraint. Suppose, for instance, you are playing chess, a~d ev~ry _possibl~. move will put you into a winning position. Then you are m a ;vmnmg positlOn already. Dosen (44] has also shown that in order to ~haractenze IPL, converse heredity can be replaced by ancestrality: if t venfies A, ~he~ there is _a state u such that u ~ t and u verifies A. In Kripke models sati~fymg heredity and ancestrality both ~ and its inverse relation must be ser_Ial. Thus, according to this conception, development not only is always possible, but also never starts from scratch. In the absence of persistence, the requirement of symmetry would also make sense. However, with this condition we would leave the 'subintuitionistic sector'. .
4
SUBINTUITIONISTIC LOGICS
157
implication ~h, which are evaluated 'dynamically' with their evaluation clauses referring to !:;;;. These verification conditions (together with persistence as postulated for propositional variables and derived for arbitrary formulas) illuminate why the various modal translations faithfully embed IPL into 84. The completeness theorem then shows that the class of all Kripke models satisfying (i) - (iii) in fact characterizes IPL. Whereas, however, the interpretation of the intuitionistic connectives in Kripke models is in the first place laid down by the verification clauses, the conditions (i) - (iii) appear as degrees of freedom which may or may not be postulated. Hence, if Kripke frames are ju~t the right kind of structures for the language Lh of IPL, then the basic 'intuitionistic' system seems to be the logic of the class K of all models based on any Kripke frame whatsoever in that language. We shall follow Dosen (47] and refer to this subintuitionistic system as K(a). Although the interpretation of Lh in Kripke models conveys a neat understanding of the intuitionistic connectives by treating conjunction and disjunction as Boolean, while treating negation and implication as intensional, it should be clear that this is not the only possible partition into Boolean and intensional operations of IPL. Sylvan (161], for instance, presents IPL as an extension of classical propositional logic formulated in Boolean negation --, and Boolean conjunction I\. See also the nonhereditary Kripke models for IPL in (44]. In K(a) an implication p ~h q is interpreted as 'strict implication' O(p ~ q) and 'hP is interpreted as 'strict negation' D(p ~ f) in the minimal normal modal propositional logic K. It is thus natural to regard K(a) as the logic embedded into K by the modal translation a which for each A E Lh replaces every implication and negation in A by its corresponding strict implication and strict negation, respectively. This is the perspective under which K(a) is studied by Dosen (47]. More generally, if A is a modal extension of ~lassi~al ~ropositional logic, A(a) is defined as {A E Lh I f-A a (A)}. Ax10matizatiO~S of K (a) and various extensions of it have been presented by Corsi (36]. The axiomatization of K(a) in Table XIII is taken from (47]. Both Corsi and Dosen show that K(a) is also complete with respect to the class K of Kripke models based on frames having a strongly generating sg d 1 · h' state, that is, a 'base state' from which every state can eve op wit m one step. This is the kind of models for subintuitionistic logics that independently has been investigated by G. Restall (138], who defines validity of an intuitionistic formula A in K 89 as verification of A at the base state of every model M E Ksg· Restall presents an axiomatization SJ of the intuitionistic formulas valid in Ksg in this sense. Since K(a) is
158
CHAPTER 10
SUBINTUITIONISTIC LOGICS
Table XIII. An ax:iomatization of K(a). Al A2 A3 A4 A5 A6 A7 A8 A9 AlO mp:Jh weakening adjunction
axiom schemata A :>hA ((A =>h B) 1\ (B =>h C)) ((C :>hA) 1\ (C =>h B)) (A 1\ B) :>hA (A 1\ B) =>h B A =>h (AV B) B :>h (A V B) ((A =>h C) 1\ (B =>h C)) (A 1\ (B V C)) :>h ((A A f =>hA rules A, (A =>h B) f- B A f- (B =>hA) A, B f-A 1\ B
=>h (A =>h C) =>h (C =>h (A
159
r,t,p
1\
B))
p,t =>h ((A V B) :>h C) B) V (A 1\ C))
p
Figure I. A lattice of subintuitionistic logics.
the set of all intuitionistic formulas verified at any state of any M E K clearly K (a) ~ SJ. Conversely, if A is not verified at some state t of ~o~e m~del from K, consider the submodel generated by t and extend Its p~ssible dev~lo_pment' relation to make t strongly generating. The res_ultmg mo_del1s m Ks 9 , and by induction on A it can be shown that A IS not venfied at t. Thus, SJ is just another axiomatization of the system K(a).
O~r aim_ is to fin~ decent sequent calculi for K(a) and some interesting axiOmatic d" 1 d' extensiOns of this system. Recently' R . G ore' [75] h as ' reISp a~e IPL, that is, has presented IPL as a system of display logic that differs from the diplay calculus for IPL I·n [16] . Th"IS proof sys. te~ DI~L, ~Irectly suggests Gentzen systems for the subintuitionistic ~o?Ics, smce It cap~~r~s persistence of atomic information (p), reflexI.VIty (r) and transitivity (t) of ~' respectively, by separate structural znference rules. In the present chapter, it is shown that the sequent sys~ems resulting from giving up all or part of these structural rules are m fa~t sound and complete with respect to the corresponding classes of Knpke models. Moreover, seriality (s) of~ can also be expressed by a P_urely structural rule. We arrive at the lattice of subintuititionistic logics exhibited in Figure I.
It can easily be verified that the systems in this lattice are pairwise distinct. Interestingly, our characterization results reveal a connection with Finger's and Gabbay's [61] notion of temporalizing (or adding a temporal dimension to) a given logic. Temporalization is a method for combining an arbitrary logical system with a propositional tense logic. It turns out that the display calulus DK(a) for K(a)is obtained from a sequent calculus for an extension of the combined system Kt(K(a)), which is the result of temporalizing K(a) by the minimal tense logic
Kt. The above lattice of subintuitionistic logics is shaped from a semantical point of view. This taxonomy is different from the substructural hierarchy of systems weaker than IPL, cf. [48], [180]. The possible worlds models developed by Dosen [42] for these systems may, however, also be viewed as information models, see [179], [180]. Whereas the route from K(a) to IPL is paved by conditions on Kripke models, the substructural subsystems of intuitionistic logic arise from the standard sequent calculus presentation of IPL by a systematic variation of the hitherto standard structural inference rules. It would be interesting, of course, to clarify the relations between both ways of 'going subintuitionistic'. Persistence in combination with transitivity, for example, corresponds to the monotonicity axiom A :Jh (B :Jh A). Indeed, display logic has been designed for the very purpose of Gentzenizing substructural logics (and combinations of such systems). It thus turns out to be general enough a schema for dealing with both routes to weakening IPL.
160
SOUNDNESS AND COMPLETENESS OF DK(a)
CHAPTER 10
161
10.2. SEQUENT SYSTEMS FOR SUBINTUITIONISTIC LOGICS
In [75], Rajeev Gore has defined a display calculus for IPL using the Godel-McKinsey-Tarski translation of IPL into S4. This sequent system, DIPL, contains separate structural rules for the persistence of propositional variables and the reflexivity and transitivity of the 'possible expansion' relation (;;;. It therefore proves suitable for defining sequent calculi for the subintuitionistic logics we are interested in. To prove the adequacy of these calculi, it proves useful to translate sequents into formulas of the language Ct(£), which is the language of tense logic defined over the set of formulas of the logical object language £as the set of atoms. In our case£= £h. To be precise:
7i(A) 71(I) 71(*X) 71(X 0 Y) 71 (•X)
= = =
A t •7z(X) 71 (X) A 71 (Y) (P)71(X)
7z(I) 7z(*X) 7z(X o Y) 7z(•X)
f
•71 (X) 72 (X) V 7z(Y) [F]7z(X).
Consider the partition of the introduction rules for the logical vocabulary in Table XIV. We can conveniently define various sequent ~ystems by combing these modules of operational rules on top of the logical and structural rules of Chapter 3, see Table XV.
Definition 10.1. Let Atom be a denumerable set of propositional variables. Lh is the smallest set b. such that -Atom~~'
Theorem 10.2.
- f, t E b., - if A, B E b., then (A A B), (A V B), (A
~h
B) E b..
Lt(Lh) is the smallest set b. such that - £h
~b.,
- if A, B E ~' then ·A, (A A B), (A V B), (A ~ B), [F]A, (P)A E b.. In Lh the unary operation 'h (intuitionistic negation) may be defined by 'hA :=A ~h f. In Lt(Ch) as in Lt the unary operations [P] and (F) can be defined by [P]A := •(P)•A, and (F)A := •[F]•A, respectively. When it comes to defining functions over the set of formulas of Lt(£), like the depth of parsing trees, one has to be careful, if£ and Lt share some logical vocabulary. The resulting 'double parsing problem' can be circumvented either by renaming in £, or by restricting the base clause in the inductive definition of Lt(£) to monolithic formulas, that is, formulas not built up from Boolean connectives, see [61]. The declarative meaning of the structural connectives I, *, •, and o in DL is given by the translation r from Chapter 3, which now sends sequents into formulas of Lt(Lh):
where
Ti
(i = 1, 2) is defined as follows:
f-Kt
Proof. See Chapter 3.
A iff f-nKt I~ A. Q.E.D.
Theorem 10.3. The systems DKt, DKt(K(a))' and DK(a) enjoy strong cut-elimination. Proof. These systems are proper display calculi, that is, their rules satisfy the conditions which in (16] have been shown to guarantee cutelimination and which in Chapter 4 have been proved to guarantee strong cut-elimination (for a certain set of primitive reductions). Q.E.D. Corollary 10.4. DKt, DKt(K(a))' and DK(a) enjoy the subformula property.
10.3. SOUNDNESS AND COMPLETENESS OF DK(a)
We want to prove completeness with respect to models (I,(;;;, v) b~ed on ordinary possible worlds (or Kripke) frames (I,~), wher~ I .Is a non-empty set, ~ ~ I xI and v: Atomxi --7 {0, 1}. ?ur aim IS ~o prove that for every intuitionistic formula A, I ~ A IS provable. m DK(a) iff A is valid in every Kripke model. We shall de~ne.a n~ti~n of valid sequent such that K f= X ~ Y ('X ~ Y is vahd m K ) Iff in DKt(K(a))' f- X ~ Y, in order to derive that for every formula
SOUNDNESS AND COMPLETENESS OF DK(o-)
162
163
CHAPTER 10
A E eh, K F I---+ A iff in DK(a) this notion of valid sequent is
Table XIV. Modules of introduction rules.
K
Boolean rules I
F X---+ y
I (-+ f)
We thus have to define K
I (t-+) <-+ t)
Definition 10.5. We state t') by induction on A: X-+A Y-+Bf-XoY-+A!\B AoB-+Xf-AAB-+X
(-+ /\) (/\ -+) (-+V) (V-+)
X -+ *A f- X -+ -.A *A -+ X f- -.A -+ X XoA-+Bf-X-+A:::>B X -+ A B -+ Y f- A :::> B -+ *X o y tense logical rules
(-+ [F)) ([F]-+) (-+ (F)) I ((F) -+) (-+ [P]) ([P]-+) (-+ (P)) ( (P) -+)
•X -+ A f- X -+ [F]A A -+ X f- [F]A -+ •X
F T(X---+ Y).
iff K
X -+ * • *A f- X -+ [P]A A -+ X f- [P]A -+ * • *X X -+ A f- •X -+ (P)A A -+ •X f- (P)A -+ X .. .. mtmtiOmstJc rules •X o A-+ B f- X-+ A :::>h B X -+ A B -+ Y 1- A :::>h B -+ •( *X o Y)
I
o-(p) o-(t) o-(A !\B) o-(A :::>h B)
Boolean (I), (11), tense Boolean (I), (11), intuitionistic, tense Boolean (I), intuitionistic
v(p, t) = 1, p E Atom 0=1 1= 1
M, t f= B and M, t f= C M, t f= B or M, t f= C M, t ~ B or M, t f= C M,t~B
(VuE I) (VuE I) (VuE I) (3u E I)
t [:;;; u implies M,u f= B ::J f t [:;;; u implies M,u f= B ::J C t [:;;; u implies M,u f= B u [:;;; t and M, u f= B
=
=
p t
=
= =
o-(A) !\ o-(B) [F](o-(A) :::> o-(B))
=
anguage
Ct Ct(Ch) eh
=
f o-(A) V o-(B) [F](o-(A) :::> f)
Given the above evaluation clauses for intuitionistic implications and 1 negations, we may also work with the translation T from sequents into et, which is defined like T except that TI(A) = a(A). Evidently, K f= T(X ---+ Y) iff K f= T'(X -+ Y). However, as we shall see, T more clearly brings to the fore the connections with temporalization.
Table XV. Sequent calculi.
I operational rules
in model M at
We say that A is valid in model M = (I,[:;;;, v) (M f= A) iff for every t E I: M, t f= A. If C is a class of models, A is said to be valid in C iff A is valid in every M E C. If it is clear which model M is being considered, we shall write 't f= A' instead of 'M, t f= A'. The modal translation a from eh into et is defined by:
X-+ A f- * • *X -+ (F)A * • *A -+ Y f- (F)A -+ Y
•X -+ *A f- X -+ -,hA *A-+ Y f- -.hA-+ •Y
iff iff iff iff iff iff iff iff iff iff iff
M,tf=p M,tf=f M,tf=t M,t F B !\ c M,t F BVC M,t F B ::J c M,t F -.B M,t F -.hB M,t F B ::Jh c M,t F [F]B M,t F (P)B
X-+AoBf-X-+AVB A-+X B-+Yf-AVB-+XoY Boolean rules 11
(-+ :::>) (:::>-+)
DKt DKt(K(a))' DK(o-)
I---+ A. The obvious candidate for
f= A, for every A E et(eh)· define M, t f= A ('A is verified
(f -+)
I system
f-
Theorem 10.6. K
f= T(X---+ Y)
iff f-nKt(K(a))' X---+ Y.
Proof. <:=:By induction on proofs in DKt(K(a))'. =?:By a slight adjustment of the standard construction of a canonical model in normal tense logic, see Section 10.6. Q.E.D. Corollary 10. 7. K
f= A
iff f-nKt(K(a))' I ---+ A.
164
EXTENSIONS OF K(a) AND KT(K(a))'
CHAPTER 10
Corollary 10.8. For every A E .Ch: K
f=
A iff f-oK(a) I--+ A.
Proof. By the fact that DKt(K(a))' is a conservative extension of DK(a), which follows from the subformula property for DKt(K(a))'. Q.E.D.
10.4. THE CONNECTION WITH TEMPORALIZATION
Temporalization is a method of combining an arbitrary logical system C by a n?rmal propositional tense logic Tin the language with [F] and (P) (or m the language with 'since' and 'until', in which [F] and (P) are definable), see [61]. In this approach a logic A is conceived of as a triple (.CA, f-A, FA), where .CA is a language, f-A is an (axiomatically presented) inference system, and FA is a semantics (given by a notion of validity with respect to a class CA of models that characterizes f-A)· The temporalized system T(A) is obtained as a component-wise combination of T and A; it can be regarded as the result of adding a temporal dimension (or point of view) to A. In general, the language LT(.CA) is defined as exemplified by the definition of .Ct(.Ch)·
Definition 10.9. .CT(.CA) is the smallest set
~such
that
-£A~~'
-if A,
BE~'
then •A, (A 1\ B), (A V B), (A ::J B), [F]A, (P)A E
~-
Definition 10.10. Axiomatization of T(.C). - The axioms of T. - The inference rules of T. -For every A E .CA, if l-A A, then 1-T(£) A. In defining validity with respect to the class of all combined models MT(.C) one has to ensure that for every formula A from .CA and every MA E CA: MA FA or MA~ A. If (.CA, l-A, FA) is a subintuitionistic or a normal modal logic, this can be achieved by considering designated, 'current' states and defining validity in a model as verification at its designated state. This is correct, since generated submodels preserve validity for such logics. Suppose (J, !:;, v) E CT and g: I ---+ CA· A model of T(.C) is a triple (I, !:;,g).
165
Definition 10.11. Let MT(.C) = (I,!:;, g) be a model of T(.C) and let M .CA be the set of monolithic formulas of .CA· MT(.C)' t F A ('MT(.C) verifies A at t E I') is defined as follows:
t MT(.C),t FA MT(.C),t F MT(.C)' t F MT(.c)l t F MT(C),t F MT(£), t F MT(£), t F
B 1\ c BVc B ::) c -B [F)B (P)B
iff iff iff iff iff iff iff
g(t) =MA and MA f= A, A EM .CA MT(.C), t f= B and MT(.C), t f= C MT(.c)' t f= B or MT(.C)' t f= C MT(.C)' t ~ B or MT(£), t f= C MT(.C)' t ~ B (Vu E I) t !:; u implies MT(.C), u F B (:Ju E I) u!:; t and MT(£), u f= B
A formula A is valid in MT(L) = (J, !:;, g) (MT(£) F A) ~f MT(£) verifies A at every t E I. If CA = CT, it is natural to consider only combined models (I,!:;, g), where g is a mapping from I into models based on (I,!:;) and t is replaced by the more specific
t (I,!:;,g),tFA iff g(t),tf=A, AEMLA Let us refer to this smaller model class as K' and to the set of formulas valid inK' as Kt(K(a))'. Obviously, K' f= A iff K FA holds ,true ~or every A in .Ct(Ch)· Hence K' F A iff 1- I--+ A in DKt(K(a)). Whtle Kt(K(a)) ~ Kt(K(a)Y, the converse is no~ t~ue. T_he formula [F](A ::J B) ::J (A ::Jh B), for instance, though vahd m K IS not a th~orem of Kt(K(a)). The tender relation between K(a) and Kt(K(a)) may be viewed as the key to Gentzenizing K(a) in DL.
10.5. EXTENSIONS OF K(a) AND Kt(K(a))'
Our aim now is to characterize certain subintuitionistic logics between K(a) and IPL. We say that a structural rule corresponds to a cond~tion on Kripke models, if the following holds: a Kripke model M satisfies the condition iff the rule preserves validity in M.
Theorem 10.12. Let A be a logic in .Ch faithfully embedded by a into a propositional normal tense logic T, and let ~ be a fi_nite set of structural sequent rules. Suppose T is sound and co_mplet~ w1th re_s~ect to a class C of Kripke models obtained from K by 1mposm~ cond_1t10ns corresponding to the rules in ~- If these conditions are satisfied m the canonical model for Kt(K(a)Y U ~ as defined in Section 10.6, then DK(a) U ~is a display calculus for A.
166
p s r t
PROOF OF STRONG COMPLETENESS
Table XVI. Some correspondences.
10.6. PROOF OF STRONG COMPLETENESS
condition persistence seriality reflexivity transi ti vi ty
rule p' s' r' t'
For the convenience of readers not familiar with the Henkin-style proof of strong completeness for Kt (see, for instance, (1361) we shall prove the completeness of Kt(K(a))' with respect to K in some detail. In what follows f- stands for derivability of sequents in DKt(K(a))'. Let X 0 , Y 0 , Z 0 etc. stand for structures which may be denumerable 0 o-nestings of ordinary, finite structures. Define the set SUB(X ) by
P-+ X 1- •p-+ X, p E Atom •X o •Y -+ *I 1- X -+ *y X -+ •Y 1- X -+ y X -+ •Y 1- X -+ • • y
if X 0 = xp otherwise
Proof. We may reason as follows: {::}
f-A A f-T a(A) T
{::} {::}
1
X 0 ::5 Y 0 iff X 0 is a o-nesting of elements from SUB(Y
f-nKt(K(a))'u~ f-nK(a)u~
I --+ A I --+ A Q.E.D.
;abdl~t_)CVI
listfs correpondences between structural sequent rules and on I wns on rames and valuations.
Obs~rvation
10.13. (i) For every Kripke frame :F = (I C)· C. . iff every model based on :F -{r!s ;;)nal · ' · . n M or every Knpke mo d e1 M : M satisfies p iff p' preserves validity m .
((~'e)flFexive, transit~ve)
Let ..6. ~ {p,s,r, t} and ..6.'
o
xg
The relation ::5 between possibly infinite structures is defined by
{::} c F {1--+ A) {::} c F T{l --+ A)
valida~~ ~'
Proof. The prooffor s, r, and t is standard. For p observe that
0 ).
Thus, (•A2o*A1) ::5 *A1 o(•A2oA3), whereas (A2o*A1) ~0 *Al oe(A2 o A ). Note that ::5 is reflexive and transitive. We write X f- Y if there 0 0 3 0 is a finite X such that X ::5 X and f- X --+ Y. If not X l- Y (X If Y),0 X 0 is called ¥-consistent. X 0 is said to be maximal ¥-consistent,0 if X 0 is ¥-consistent and for every Z sucht that Z ~ X we have Z o X f- Y. As a first step towards completeness we show that every ¥-consistent structure can be extended to a maximal ¥-consistent structure.
Lemma 10.16. If X 0 is ¥-consistent, then there is a maximal ¥-con0
sistent Z 0 such that X ::5 Z
0
.
Proof. Let X1, X2, ... be an enumeration of all finite structures. Let
= {r IrE ..6.}.
. Obse:vation 10.14. The canonical model for Kt{K(a))' U ..6. d fi d . m Section 10.6 satisfies the cond'Itwns corresponding to the rules ein ne ..6.'. Q.E.D.
167
CHAPTER 10
f- •p--+ p ·
~or~llary 10.15. In DKt{K{a))' U ..6.', f- X --+ y 'ff {X . validd.m the class of all. Kri Pke mo d els satisfymg . . the Iconditions T --+ correY) Is spon mg to the rules m ..6.'.
1
Xo :=
XO
;
X' { n+l :=
X~ o Xn X~
if X~ o Xn lfY otherwise
Define Z 0 as OiEwXI. We want to show that Z 0 is maximal ¥-consistent.0 (i) Y-consistentcy. Observe first that every X~ is ¥-consistent. If Z were not ¥-consistent then there would be an n such that X~ l- Y, quod non. (ii) Maximality. Suppose Y' ~ Z 0 . Y' = Xj for some positive integer j. 0 Hence Y' = Xj ~ Xj+l, and hence Xj o Y' f- Y. But then Y' o Z f- Y. Thus, Z 0 is maximal ¥-consistent.
Q.E.D.
Lemma 10.11. Suppose X 0 is maximal ¥-consistent. Then 0
(i) (A1/\ A2) ::5 X 0 iffboth A1 ::5 X and A 2 ::5 X (ii) --.B ::5 X 0 iff B ~ X
0
.
0 .
168
(iii) X 0 1- A implies A :::5 X 0 • Proof. (i) :::::}: Suppose (A11'\A2) :::S X 0 but, say, A 1 ~ X 0 . Then A 1oX 0 1Y, that is, (:JX :::5 X 0 ) and 1- A 1 oX --+ Y. Hence (3X :::5 X 0 ) 1- A1
o
A2 oX -+ Y
{:::> {:::>
(i) ~: Suppose A1 :::5 X 0 and A2 :::S X (3X :::5 X 0 ) 1- A1 1\ A2 oX -+ Y
{:::> {:::>
but (A 1 1\ A 2) ~ X 0 . Then
(3X :::5 X 0 ) 1- A1 o A 2 oX -+ Y (3Z :::S X 0 ) 1- Z -+ Y quod non
(ii) :::::}: Suppose ·B :::S X 0 and B :::S X 0 • Since 1- -,BoB--+ Y, it follows that X 0 1- Y, quod non. (ii) ~: Suppose B ~ X 0 and •B ~ X 0 • Then 0
{::} {::} {::} {::}
(:JX :::5 X ) (:JZ :::5 X 0 ) (:JZ :::5 X 0 ) (:JZ :::5 X 0 ) (:JZ :::5 X 0 )
1- BoX --+ Y and (:JX' :::5 X 1- Z o *y --+ B 1\ ·B 1- Z o •(B 1\ ·B) --+ Y 1- Z o I --+ Y 1- Z--+ Y contradiction
0)
Definition 10.19. The canonical model system Kt(K(a))' is defined by
= the set of all maximal £-consistent structures
- I
If ~ is a finite set of further structural rules, the canonical model for Kt(K(a))'U~ is defined as for Kt(K(a))', except that £-consistency is defined with respect to 1-Kt(K(CT))'ut.· Consider MK~(K(CT))'· ~ote that if t c u then A ~h B :::5 t implies A~ B :::S u. The aim now 1s to show thata~ intuitionistic formula A is verified at a state t E I iff A :::5 t. For this purpose we first prove the following Lemma 10.20. For every t, w E I and every BE .Ch:
0
Proof. (i) Suppose (P)B :::S t and consider Z = 0( { -,C I •(P)C :::5 t}U{B} ). is £-consistent. Otherwise, (3•Al o .. . o•An) •(P)•Ai :::S t
zo
and
=> {:::>
0
Proof. Suppose ·[F]B :::5 X and the set Z 0 {·B}) were not £-consistent. Then
{::} {::} {::} :::::} {::} {::} {::} {::} :::::}
(:JA1 o ... (:JA1 o ... (:JA1 o ... (::IA1 o ... (::JA1 o ... (::JA1 o ... (:JA1 o ... (:JA1 o ... (:JA1 o ... X 0 1- f
o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1o An :::5 Z 0 ) 1contradiction
0,
then O({C
1
= 0( { c I [F]C :::5 xo} u
A 1 o ... o An o ·B --+ f A 1 o ... o An --+ f o *'B A 1 o ... o An --+ ··B A 1 1\ ... 1\ An --+ B [F](A 1 1\ ... 1\ An) --+ •B [F]A 1 1\ ... 1\ [F]An --+ [F]B (F]A 1 1\ ... 1\ [F]An --+ I o [F]B (F]A 1 o ... o [F]An o •[F]B --+ I [F]A1 o ... o [F]An o •[F]B --+ f Q.E.D.
We now define the canonical model for Kt(K(a))'.
t
(ii) w ~tiff (•(P)B :::5 t:::::} ·B :::5 w).
(iii): Suppose X 0 1- A but A~ X 0 • Then there are X 1,X2 :::S X 0 such that 1- X1 --+A and 1- A--+ Y o *X2. Hence 1- X 1 o X 2 --+ Y and hence X 0 1- Y, quod non. Q.E.D. Lemma 10.18. Let X be £-consistent. If •[F]B :::5 X [F]C :::5 X 0 } U { ·B}) is £-consistent.
=(I,~' v) for the
(i) (P)B :::5 t implies (3u E I) u ~ t and B :::5 u.
1- ·BoX' --+ Y
0
MKt(K(CT))'
- t ~ u iff ([F]A :::S t implies A :::5 u) - (Vp E .Ch) (Vt E I): v(p,t) = 1 iffp :::5
(3X :::5 X 0 ) 1- A1 1\ A2 oX -+ Y (3Z :::5 X 0 ) 1- Z -+ Y quod non 0,
169
PROOF OF STRONG COMPLETENESS
CHAPTER 10
=>
(3-.Al o ... o •An) 1- •A1 1\ ... 1\ •An -+ ·B (3-.A 1 o ... o •An) 1- -.(P)••A1 1\ ... 1\ •(P)·•An -+ -,(P)...,...,B (3-,A 1 o ... o •An) 1- -,(P)Al o ... o •(P)An o (P)B -+ f t 1- f contradiction
zo can be extended to a maximal £-consistent p E I. By definition and Lemma 10.17 (iii), p ~ t. .. (ii) :::::}: Let w ~ t and •(P)B :::5 t. Then (P)B ~ t and, by defimt10n ~f ~' [F](P)B ~ w. Hence •[F](P)B :::5 w. Since 1- •[F](P)B --+ •B, 1t follows by Lemma 10.17 (iii) that ·B :::S w. ~: Suppose not w ~ t. Then there is aB such that [F]B :::S wand ·B :::5 t. Since 1- ·B--+ •(P)[F]B, -,(P)[F]B :::5 t. Thus, it is not the case that •(P)B :::5 t :::::} -,B :::5 w. Q.E.D.
Lemma 10.21. For every t E I and every A E .Ch:
MKt(K(u))''
t FA
iff A :::St. Proof. If A E Atom, the claim holds by definition; if the main connective of A is a Boolean operation, use Lemma 10.17. A = [F]B. ~: If [F]B :::5 t, then, by the definitio? of ~' (Vu E I) t ~ u implies B :::5 u. By the induction hypothesiS, t f= [F]B. :::::}:
170
CHAPTER 10
CHAPTER 11
Suppose [F]B ~ t. ~hen •[F]B ::5 t and, by Lemma 10.18 X= [F]C ::5 t} U {•B}) IS £-consistent. Thus X has a · '1 f ~( { C I extension, say w. By the definition of C 't C Mmaxima -consistent th . d t. h _, - w. oreover --,B -< w B e m uc Ion ypothesis and the cas fi B I '. - . y hence t ~ [F]B. e or oo ean negatwn, w ~ B, A = (B -.:J h C). (Vu E J) t c
If (B -.:J C) ::5 t, then, by the definition of c . (B h -.:J C) -< u By th . d t" h _, and the clause for Boolean implication. (Vu E e/)nt ~ IO~ ytothesis (B ::> C), and hence t f= (B -.:J C) ' - u Imp Ies u f= [F](B -.:J C) ~ t d h h . :=;.: Suppose (B :>h C) ~ t. Then t ~ (B -.:Jh C). an ence t ~ [F](B ::> C), which is equivalent to -
A = (P)B.
(ii),
. {::} {::} {::}
TRANSLATION OF HYPERSEQUENTS INTO DISPLAY SEQUENTS
{=;
·
u Imp 1Ies
If (P)B -< t then b L · t and Y emma. 10.20. (1), there is a (P) - u. Hence, by the mductlon hypothesis . uppose B ~ t. Then --,(P)B ::5 t. By Lemma 10.20
{=:
u E I such that u c t (P)B ::::;.· S -
F
!:"
L .
B-.::
'
(VuE/) (u ~ t::::;. (--,(P)B ::5 t::::;. --,B-< u)) (VuE I) (u ~ t::::;. --,B-< u) {VuE/) (u ~ t::::;. u ~ t ~ (P)B
JJ)
A = 'hB. Treat 'hB as B :>h f. Q.E.D. Lemma 10. 22. Every max · 1 y . . £-consistent. Ima -consistent structure IS also maximal Proo{ Suppose xo is maximal Y-consistent. Since f- To (Y) 10 17 (ii) '72(Y) -< xo ---:l- Y, T2(Y) Z ~ xo. Then there are X . X -< 'xo . ow assume that L 1' 2 such that ~ z o X 1 y and 'X2 ---:l- 'T2(Y). Since X---+ y f- X---+ . ---:lT2(Y) 1\ 'T2(Y). Thus f- z oX o T2(Y), we obtam f- .zox1 oX2--;)1 X 2 ---+ f, and thus X 0 Is maximal fconsistent. Q.E.D.
~ X . Hence, by Lemma
J
Theorem 10 23 DKt(K( ))' · all Kripke model~· if K L Xa Yis complete with respect to the class of · r- ---+ , then ~ X ---+ Y. Proof. Suppose If X ---:l- y Th X . y . to a maximal Y-consist . en IS -consistent and can be extended xo S ent and hence also £-consistent xo that is X -< · uppose now for reductio that xo F X---+ Y. Then ' -
xo F 71(X) ::> 72(Y)
{::} 71(X) ::> 72(Y) :S
xo
by Lemma 10.21. Obviously,~ T1(X)o(T1(X) -.:J T2(Y))---+ To (Y) s· moreover ~X---+ 7 (X) d ~ 2 . mce T2(Y)) ---:l- Y. Thus1 (3Z ~ o 72(Y) ---+ Y, we obtain~ X o (71 (X) ::> - X ) ~ Z ---+ Y. In other words xo ~ y q uo d non. Q.E.D. ' ,
As we have seen in Chapter 2, the study of nonclassical and modal logics has led to the development of many generalizations of the ordinary notion of sequent. In view of the diversity of these types of proof systems it becomes increasingly important to investigate their interrelations and their advantages and disadvantages. In this chapter, we shall consider the relation between display logic and the method of hypersequents independently introduced by G. Pottinger (134] and A. Avron [7]. It will be shown that hypersequents can be simulated by display sequents. Whereas hypersequents are finite sequences of ordinary sequents, display sequents are obtained by enriching the structural language of sequents, thus turning the antecedent and the succedent of a sequent into 'Gentzen terms', see also [41]. As Avron [12, p. 3] points out, "although a hypersequent is certainly a more complex data structure than an ordinary sequent, it is not much more complicated, and goes in fact just one step further". The translation of hypersequents into display sequents illuminates how display sequents generalize ordinary sequents still further than hypersequents. In contradistinction to the application ofhypersequents to modal logics, the modal display calculus comprises introduction rules for the modal and tense logical operators which can arguably be interpreted as meaning assignments; see Chapter 1. Therefore the translation of hypersequents into display sequents has something to recommend preferring display logic over the hypersequential formalism. We shall consider three case-studies of translating hypersequents into display sequents, corresponding with three different interpretations of hypersequents in [12]. 11.1. HYPERSEQUENTIAL CALCULI
Hypersequents have systematically been studied in a series of papers by A. Avron [7], [9], [10], [12}. As explained in Chapter 2 already, a hypersequent is a sequence r1 ---+ ~1 I r2 ---+ ~2
I ... I r n
-+
L1n
of ordinary sequents (or sequents in which ~i and ri are sequences of formula occurrences) as their components. The symbol 'I' in the state-
171
172
ment of a hypersequent is intuitively to be understood as disjunction. Hypersequents allow one to draw a distinction between internal and external versions of structural rules. The internal rules deal with formulas within a certain component, whereas the external rules deal with components within a hypersequent. If G, H, HI, H 2 etc. are used as schematic letters for possibly empty hypersequents, one obtains, for example, the following external and internal expansion rules:
HI I H2 I H3 f- HI I H2 I H2 I H3 versus
HI 1r,A,6.-+ e 1H2
r- HI 1r,A,A,6.-+ e 1H2.
Cut only has an internal version:
GI 1 ri-+ 6-I,A 1HI G2l A,r2-+ 6.2l H2 GI 1G2l ri,r2-+ 6.I,6.2I HI 1H2 The method of hypersequents allows a cut-free presentation GSS of SS satisfying the subformula property. In this system, which is presented in Section 11.1.2, the introduction rules for 0 take the form
HI I 06. -+ A I H2 f- HI I 06. -+ DA I H2 HI 16., A -+ r 1H2 r- HI 1 6., DA -+ r 1H2, where 06. ={DB I BE 6.}. With these rules, internal monotonicity cannot be replaced by internal monotonicity for atoms only, as has been emphasized by Hacking [85]. In the terminology of Chapter 1, the right introduction rule for 0 fails to be even weakly explicit: it exhibits D in one of the premise sequents and also in the antecedent of the conclusion sequent into which DA is introduced. Therefore this rule cannot be viewed as assigning a meaning to 0 by specifying in a purely schematic way how a formula with 0 as the main connective is introduced into conclusions. In [12], Avron demonstrates the versatility and usefulness ofhypersequential calculi by presenting cut-free hypersequent systems for several logics for which an appropriate cut-free Gentzen calculus presentation turned out to be a problem for a long time. Among these systems are Lukasiewicz 3-valued logic L3, the modal logic SS, and Dummett's LC. (In this chapter, we shall use the names 'L3', 'SS', and 'LC' to denote certain familiar semantic specifications of these systems.) In the completeness proofs for the sequential presentations, A vron considers various interpretations of hypersequents. These interpretations give rise to different translations of hypersequents into display sequents. In Section 11.3, these translations will be dealt with separately.
173
HYPERSEQUENTIAL CALCULI
CHAPTER 11
11.1.1. GL3 The problem with L3 is to find a cut-free sequent calculus presentat~on such that the sequent arrow reflects the 'official' implication connective of this logic, i.e., f- AI, ... , An-+ B iff
(\/)
F AI
:J (A2 :J .. · :J (An :J B) ... ).
The n-contraction sequent systems of Prijatelj [135] for Lukasiewicz's many-valued logics, for example, fail to satisfy cut-eliminati?n·. A ~y persequential calculus GL3 for.L3 satisfying.(\/) an~ cut-eln:n_mati~n has been presented in [10]. This system consist~. of (I) the axwJ?atic rule f- A -+ A and the above internal cut rule, (n) the standard mternal structural rules, apart from internal contraction, (iii) the standard external structural rules, (iv) the rule (MX):
a 1ri r2 r3 -r ~11~2,~31 H ' G' 1r1,ri
G I ri,r~,r~ I H-r ~i,~~,~~ I H -r ~~,~i I r2,r~ -r ~2,~~ I fg,r~ -r ~3,~3'IH
and (v) the following introduction rules for the primitive operations of L3:
c 1r
-+ 6., A 1 H G 1r', B -+ 6.' 1H G 1r', r, (A :J B) -+ 6.', 6. 1 H
Glf,A-tB,6.IH G 1 r -+ (A :J B), 6. 1 H
Glf,A-+6-IH G 1 r -+ 6., ·A 1 H
c 1r
-+ 6., A 1 H G 1 r, ·A -+ 6. 1 H
c
c 1 A, r
-+ 6. 1 H 1 (AAB),r-+ 6.1 H
GIB,r-+6-IH -+ 6. 1 H
c 1 (A A B), r
G 1r -+ 6., A 1H G 1r -+ 6., B I H c 1 r -+ 6., (A A B) 1 H
Glf,A-+6-IH Glf,B-+6-IH c 1r, (A v B)-+ 6.1 H G
c 1r
r
-+ 6., A 1 H -+ 6., (A v B) 1 H
1
c 1r G
1
r
-+ 6., B 1 H -+ 6., (A v B) 1 H
11.1.2. GSS
It seems that no cut-free ordinary sequent system for SS is known. The cut-free sequent calculus presentations of SS due to Kanger [91], Sato [150], Belnap [16], Indrzejczak [89], and the present author [181 ], [182],
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for instance, all make use of a generalized notion of sequent or sequent calculus. Avron [12] presents a cut-free hypersequent calculus GS5 for SS. This system consists of the axiomatic rule f- A -+ A, the internal cut rule, the standard internal and external structural rules, the above hypersequential introduction rules including the introduction rules for D, and the 'modalized splitting rule': G 1 or1, r2-+ 0.:11, .6.2 1 H G 1 or1 -+ 0.:11 1 r2-+ .6.2 1 H 11.1.3. GLC
The superintuitionistic propositional logic LC axiomatized by Dummett [50] is the logic of the class of all linearly ordered intuitionistic Kripke models. As emphasized by Avron, the cut-free sequent system for LC presented in [160] has the disadvantage of comprising infinitely many schematic rules. This drawback is avoided in the hypersequent system GLC for LC in [9]. In GLC, components of hypersequents are constrained to be single-conclusion ordinary sequents. GLC consists of the axiomatic rule f- A -+ A, the internal single-conclusion cut rule, the standard external structural rules, the standard single-conclusion internal structural rules, hypersequential versions of the single-conclusion introduction rules of intuitionistic logic, and two further structural rules, the 'intuitionistic splitting rule' and the 'communication rule': Glf,L1-+AIH
G1 1 r1 -+ A1 1 H1 G2 1 r2-+ A2 1 H2 G1 1 G2 1 r1 -+ A2 1 r2-+ A1 1 H1 1 H2
11.2. DISPLAY CALCULI
As we have discussed at length, in DL complex structures are built up using certain structure connectives. In order to present as sequent systems logics combining logical operations with an essentially distinct inferential behaviour, there may be more than one family of structure operations. In the one-family case, the structural language of sequents comprises the nullary I, the unary operations * and •, and the binary operation o. Lukasiewicz 3-valued logic combines connectives from two different families of operations, namely on the one hand combining (extensional, additive) conjunction and disjunction and on the other
hand internal (intensional, multiplicative) negation and imp~ication. To distinguish the corresponding families of structure connectives, the · structure operatiOns may b e subscri"pted·· I.l ' o.~' *l·' and •i versus le, * and •c· In the present cont ext , h owever, 1.l -- I c, *l· = *c, and 0 ~'- ~ so that we are left with distinguishing between °i and 0 c· T~e ·~- Cl clear and simple inferential behaviour of the structure connecf Ives IS laid down by the following basic structural rules of DL: Basic structural rules (1.1) (1.2)
(2.1) (2.2) (3) (4)
X oi Y -+ Z -H- X -+ Z oi *Y -1f- Y -+ *X 0 i Z X oc Y -+ Z -11- X -+ Z Oc *Y -11- Y -+ *X Oc Z X -+ y Oi z -11- X oi *z-+ y -11- *y Oi X -+ z X -+ Y oc Z -11- X oc *z -+ Y -11- *y 0 c X -+ Z X -+ Y -1f- *y -+ *X -11- X -+
**Y
X-+ •Y -11- •X-+ Y
As before X 1 -+ Y1 -11- X 2 -+ Y2 abbreviates X1 -+ Y1 I- X2-+ Y2 and X -+ y 'I- X 1 -+ y 1_ If two sequents are interderivable by me~ns of 2 2 rules (1) __ (4), then these sequents are said to be structurally or display equivalent. . . . The intended declarative meaning of the structure connectives I.s m part context-sensitive, and the earlier translation 7 of sequents mto tense logical formlas is now defined as: 7(X-+ Y) := 71(X) ::::> 72(Y), where the translations 7i (i = 1, 2) are defined as follows: 7i(A) 71 (I) 71(*X) 71(X oi Y) 71(XocY) 71(•X)
A =
f
72(*X) (X 72 °i Y) 72 (X °c Y)
= =
' 71(X)
=
T2(X) + 72(Y) 72(X) V 72(Y)
72(•X)
=
(F]72(X)
t
=
•72(X)
=
71(X) 0 71 (y) 71(X)/\71 (y) (P)71(X)
=
A+ B is defined as -,A ::::> B, which in classical logic and thus H ere . A v B and A 0 B in normal tense logic is, of course, eqmvalent to . ' 1 . ll . is defined as the dual of A+ B (i.e. •(A ::::> ·B)), which c ass~ca. ~IS equivalent to A 1\ B. Moreover, if t and fare not assumed as pnmitive, t is defined asp::::> p, for some atom p, and f is define~ as •t. The rul~: (1) _ (4) obviously are correct under the 7-translat10n, and the pa
( (P), [F]) exemplifies the idea of a residuated pair of unary operations, see Chapter 3. Recall that DL derives its name from the fact that any substructure of a given display sequent s may be displayed as the entire antecedent or succedent, respectively, of a structurally equivalent sequent s'. In order to state this fact precisely, some definitions are needed. An occurrence of a substructure in a given structure is said to be positive (negative) if it is in the scope of an even (uneven) number of *'s. An antecedent (succedent) part of a sequent X -+ Y is a positive occurrence of a substructure of X or a negative occurrence of a substructure of Y (a positive occurrence of a substructure of Y or a negative occurrence of a substructure of X).
Theorem 11.1. (Display Theorem, Belnap [16]) For every sequent s and every antecedent (succedent) part X of s there exists a sequent s' structurally equivalent to s such that X is the entire antecedent (succedent) of s'.
sequential calculus for L3, contains combining conj.m~c~ion a~d disjun~ tion and (internal) negation and implication as pnm1t1ve. Smce w~ ~1d not make these distinctions in previous chapters, we shall exphc1tly state the introduction rules to be considered, reusing the symbols 'A' and 'V': absurdity (-+ f)
X -+ I f- X -+ f
(f-+)
f-f-+1
Internal negation (-+ •)
X-+ *A f- X-+ ·A
(• -+)
*A-+ X f- ·A-+ X
Internal implication
X As has been emphasized in Chapter 3, the Display Theorem has both technical and conceptual significance. It allows an "'essentials-only' proof of cut elimination relying on easily established and maximally general properties of structural and connective rules" [18]; see Chapter 4. Moreover, the Display Theorem allows the removal of a certain amount of holism in the proof-theoretic semantics of the logical operations. If the meaning of an n-place connective f is specified by it's introduction rules, then these introduction rules must not exhibit any logical operations other than f. Otherwise f's meaning is - at least in part - holistic. This idea may be strengthened by postulating that the succedent (antecedent) of the conclusion sequent in a right (left) introduction rule must not exhibit any structural operation. The display property guarantees this segregation principle. In addition to the basic structural rules, every display sequent system to be considered will contain the logical rules (id) and (cut). Logical rules (id) (cut)
A -+ B f- X -+ A :J B X -+ A B -+ y f- A :J B -+ *X Oi
Oi
y
Combining conjunction (-+ /\) (/\ -+)
X-+A X-+Bf-X-+At\B A-+ X f-A 1\ B-+ X; B-+ X f-A 1\ B-+ X
Combining disjunction (-+V)
X -+ A f- X -+ A V B;
(V-+)
A-+X
X -+ B f- X -+ A V B
B-+Xf-AVB-+X
Forward-looking necessity (-+[F])
•X -+ A f- X -+ [F]A
([F] -+)
A -+ X f- [F]A -+ •X
Intuitionistic implication
f-A-+ A X-+ A
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176
A-+ Y f- X-+ Y
In the absence of unrestricted contraction and monotonicity rules, the distinction between internal (intensional, multiplicative) and combining (extensional, additive) conjunction and disjunction can be drawn. The display calculus to be defined in Subsection 11.2.1, like the hyper-
•X
Oi
A -+ B f- X -+ A :Jh B B -+ y f- A :Jh B -+ •( *X
X -+ A
Oi
Y)
As we have seen, introduction rules for the other normal tense logical modalities are also easily available, for example:
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Note that (Pi) is required in Lemma 11.2, because inst~ad of (--+ :J) and (::=>--+) we did not formulate order-sensitive introductiOn_ rules for the left- und right searching implications known from Categonal Grammar.
Backward-looking possibility
(--+ (P)) ( (P) --+)
X--+ A f- •X--+ (P)A A --+ •X f- (P)A --+ X
In addition to the display sequent rules introduced so far, we shall also consider further purely structural rules, where o is any of oi and oc:
{I)
X --+ Z -lf-- I
{Ai)
X1 oi {X2 oi X3) --+ Z -lf-- {X1 oi X2) X1 oi X2--+ Z f- X2 oi X1 --+ Z X Oi X--+ z f- X--+ z X1 --+ Z f- X1 oi X2--+ Z
{Pi) (Ci)
{Mi)
o
X --+ Z
X --+ Z -lf- X --+ I oi
o Z
X3 --+ Z
xl Oc {X2 Oc X3) --+ z -lf- (Xl Oc X2) Oc x3 --+ z {Pc) X1 Oc X2 -t Z f- X2 Oc X1 -t Z X Oc X --+ Z f- X --+ Z {Cc) {Me) X1 -t Z f- X1 Oc X2 -t Z {Mix) I--+ Z oc {*Xl oi *X2 oi *X3 oi Y1 oi Y2 oi Y3) oc Z', I --+ Z Oc {*X~ oi *X~ oi *X~ oi Y{ oi Y~ oi Y3) oc Z' f-f-- I--+ Z oc (*Xl oi *X~ oi Y1 oi Y{)oc oc( *x2 Oi *X~ oi y2 oi Oc ( *x3 oi *X~ oi y3 oi Y3) Oc Z' {Ac)
yn
{MN) (rP) (rT) (r4) {rB)
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DISPLAY CALCULI
I --+ X f- •I --+ X p --+ X f- •p --+ X * • *X --+ Y f- X --+ Y * • *X --+ Y f- * • • * X --+ Y * • *(X o * • *Y) --+ Z f- Y o * • *X --+ Z
The rule {Mix) is a special rule to be used in the display presentation ofL3.
Lemma 11.2. In every display calculus extending the logical rules, the basic structural rules, the (I) rules, (Me), (Cc), (Pi) and the above introduction rules by purely structural rules we have: (i) f- X --+ T1(X) and f- T2(X) --+ X; (ii) f- X--+ Y implies f- T1{X)--+ Y, and f- Y--+ X implies f- y --+ T2(X). Proof. By induction on the complexity of X. Q.E.D.
11.2.1. DL3 We shall define a display sequent calculus for L3 satisfying condition (<::!) and cut-elimination. For the puq~ose of displaying L3 we may completely neglect the structure operatiOn •·
Definition 11.3. The display sequent calculus DL3 consists of the rules (1) - (3), (id), (cut), the introduction rules for -,, :J, 1\, ~' and the rules (I), (Ai), {Pi), (Mi), (Ac), (Pc), (Cc), (Me), and (Mlx). The truth-functional semantics of L3 is given by the following matrices:
T
T T
I
I
I I I
_L
_L
j_
(\
_L
_L
V
T
I
_L
: =>
T
I I T
_L
T T
T
T
T
_L
T
T
T
T
_L
I
T T
I I
I
T I
_L
_L
_L
_L
I
In L3 T is the only designated value, and a formula A is valid in L3 if under' every valuation of atomic formulas, A ~ece~ves ~he value T · The following axiomatic presentation HL3 of L3 1s g1ven m [10]: Axiom schemata:
A::=> (A::=> B) (A::=> B) : => ((B ::=>C) : => (A=> B)) (A::=> (B ::=>C)) : => (B : => (A=> C)) ((A::=> B) :J B) :J ((B :J A) :J A) (({(A::=> B)::=> A) ::=>A) : => (B =>C)) => (B =>C) (A 1\ B) ::=>A (A 1\ B) : => B (A :J B) : => ((A::=> C) : => (A=> (B 1\ C))) A :J (A V B) B :J (A V B) (A::=> C) :J ((B :J C) :J ((A V B) :J C)) {N1) (-,B ::=>-,A)::=> (A :J B)
{Il) {12) {I3) (I4) (15) {Cl) (C2) (C3) (Dl) {D2) (D3)
Inference rule: A, (A :J B)/ B
Theorem 11.4. (i) If f- A in HL3, then f- I --+ A in DL3. . (ii) If f- X --+ Y in DL3, then p T(X --+ Y) m L3.
180
Proof. (i) It suffices to show that the axiom schemata of HL3 can be proved in DL3, and that modus ponens preserves provability in DL3. The only non-trivial cases are given by the implicational schemata (I4) and (I5). Proofs in DL3 can be obtained by following Avron's [10] derivations of (I4) and (I5) in GL3 using the translation of hypersequents into display sequents presented in Section 11.3. This will give rise to display proofs with two final applications of (Cc). The sequents *A -+ (A ::::) B) and B -+ *I oc A oc (*A oi B) are easily derivable; for the latter use monotonicity and (Mix). In the case of (I4), leaving out some simple steps, we obtain: *A -+ (A ::::) B) B -+ *I Oc A oc (*A oi B) (A::::) B)::::) B-+ **A oi (*I Oc A Oc (*A oi B)) ((A::::) B)::::) B) oi (B::::) A)-+** A oi (*I Oc A Oc (*A oi B)) *(*I Oc A Oc (*A oi B)) oi ((A::::) B)::::) B) oi (B::::) A)-+ A *(*I Oc A oc (*A oi B))-+ (I4) *(I4) -+ *I oc A Oc (*A oi B) I Oc *(I4) Oc *(*A Oi B)-+ A I Oc *(I4) oc *(*A oi B) oi ((A::::) B)::::) B) oi (B::::) A)-+ A I Oc *(I4) Oc *(I4) -+ *A oi B I Oc *(I4) Oc *(I4) -+ A ::::) B B -+ B (I Oc (*(I4) oc *(I4))) oi ((A::::) B)::::) B)-+ B A-+ A (I Oc (*(I4) Oc *(I4))) oi ((A::::) B)::::) B) oi (B::::) A)-+ A
11.2.2. DS5 In normal modal logic enough structural assumptions are made to make the distinction between combining and internal disjunction and conjunction disappear. We are thus dealing with one-familiy logics and may just use o instead of oi or oc· Definition 11. 'l. The sequent system DS5 consists of the logical rules,
the basic structural rules, the rules (I), (A), (P), (C), (M), the introduction rules for the logical operations of S5, and the rules (MN), (rT), (r4), and (rB). Theorem 11.8. 1-nss X-+ Y iff F=ss r(X-+ Y).
In modal logics without the symmetry schema B, the completeness proof with respect to a display formulation uses the fact that for every frame complete normal modal logic, the extension by (P) is conservative. As we have seen in Chapter 4, the modal display calculus offers a modular sequent-style treatment of many normal modal logics. The rules (rT), (r4), and (rB), for instance, correspond with the familar axiom schemata T, 4, and B, respectively. Instead of (r4) and (rB) one could also use a structural rule corresponding with the modal axiom schema 5, namely
I Oc *(I4) Oc *(I4) -+ (I4) I -+ (I4) Oc {I4) Oc (I4) I-+ (I4) Also a proof of (I5) may be obtained by starting from B -+ *I oc A oc (*A oi B). (ii) By induction on proofs in DL3. Q.E.D. Corollary 11. 5. I-HL3 A iff 1-nL3 I -+ A.
Using Lemma 11.2, this result can be strengthened to strong completeness, cf. [77]. Theorem 11.6. 1-nL3 X-+ Y iff FL3 r(X-+ Y).
F TI(X)::::) in L3. By the previous corollary and completeness of HL3, we have 1-nL3 I-+ r1(X)::::) r2(Y) and thus 1-nL3 r 1(X)-+ r2(Y). Since by Lemma 11.2, 1- X-+ r1(X) and 1- r2(Y)-+ Y in DL3, two applications of the cut rule give 1- X-+ Y. Q.E.D. Proof.(=>): This is Theorem 11.4, (ii). c~=): Assume that
T2 (Y)
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* • *X -+ Y 1- • * • * X -+ Y.
11.2.3. DLC There is more than one way to display logics containing a constructive implication but lacking an involutive negation. One may, for example, retain the involutive structural operation * and use a modal translation of the logic under consideration, if such a translation is available and the modal logic in question is displayable, or one may work with a version of display logic due toR. Gore [75], [76] in which the structure operations * and o are replaced by two binary operations o1\ and ov with the following Ti-translations: T1(Xo"Y) TI(XovY)
= =
Tl (X) 1\ Tl (Y) Tl(X)...;.. Tl(Y)
r2(X o" Y) T2(X Oy Y)
=
T2(X) -:Jh T2(Y) T2(X) V T2(Y),
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where -;- is the residual of disjunction, namely the subtraction operation from dual intuitionistic logic. The basic structural rules for these operations are as follows:
Definition 11. 9. The sequent system D54.3 results from D55 by giving up (rB) and adding the structural rule
182
X -+ Y X
Oy
o/\
Z
--lf-
X
y-+
z
-If-
X -+
o/\
Y-+ Z
-If-
Y
o/\
X-+ Z
-If-
Y-tXo/\Z
y
-if-
X -+
z Oy y
+
X
Oy
z
and we would have the following introduction rules for
~h
Oy
(r.3)
z-+ y
and V.
Proof. See Chapter 4.
m(A 1\ B) m(A V B) m(A ~h B)
I--+ Z ov (X1 of\ At) ov Z' I--+ Y ov (X2 o/\ A2) ov Y' ff- I--+ Z ov Y ov (X1 o/\ A2) ov (X2 of\ At) ov Z' ov Y'
Dp, for every atom p
f m( A) 1\ m( B) m( A) V m( B) D(m(A)
~
m(B)).
Theorem 11.11. For every formula A in the language of LC, I- A in LC iff f- m(A) in 54.3.
B) V (B ~ A) can then be
I--+Aoi\A I--+Boi\B I --+ (A of\ B) ov (B of\ A) I ov (A of\ B)--+ (B of\ A) I ov (A of\ B) --+ (B ~A)
Definition 11.12. The sequent system DLC consists of the logical rules, the basic structural rules, the rules (I), (A), (P), (C), (M), the introduction rules for f, ~h, 1\, and V, and the structural rules (MN), (rT), (r4), (rP), and (.3). Let D54.3t be the result of adding the introduction rules for (P) and the following structural rule to D54.3:
I --+ (A ~ B) ov (B ~ A) I --+ (A ~ B) V (B ~ A)
I --+ X f- I --+
In the present context, however, working with the modal translation approach has the advantage of translating hypersequents into the same version of display logic in each of the examples considered. Axiomatically, the modal logic 54.3 results from the familiar axiomatic presentation of 54 (= KT4) by adding the .3 schema D(DA ~ DB) V D(DB :J DA). This schema corresponds with the weak connectedness of the accessibility relation R in modal Kripke models: ~
* • *X --+ Y.
Q.E.D.
m(p) m(f)
In order to prove the characteristic axiom schema of LC, the display version of Avron's communication rule may be used:
VrVsVt((Rrs 1\ Rrt)
--+ Y I- •
The Godel-McKinsey-Tarski translation of intuitionistic propositionallogic into 54 amounts to a faithful embedding of LC into 54.3. This translation m is defined as follows:
X --+ A B--+ Y f- (A ~h B) --+ X of\ Y X --+A of\ B f- X --+ (A ~h B)
~
* • *X
Theorem 11.10. f-ns4.3 X--+ Y iff FS4.3 T(X--+ Y).
A --+ X B --+ X f- (A V B) --+ X X --+ A ov B f- X --+ (A V B)
The characteristic axiom schema (A proved as follows:
X --+ Y, •X --+ Y,
* • *X.
In the next section, we shall consider the system DLC U D54.3t. Like the other display calculi defined in this paper, DLC U D54.3t enjoys cut-elimination and the subformula property with respect to the logical operations. This follows from the very general strong cut-elimination theorem for display calculi satisfying certain conditions on the shape of sequent rules in addition to eliminability of principal cuts; see Chapter 4. In particular, it follows that
((s = t) V Rst V Rts)).
Lemma 11.13. DLC U D54.3t is a conservative extension of both DLC and D54.3.
Since any generated subframe of a transitive weakly connected frame is connected, (i.e., Vs\lt((s = t) V Rst V Rts)), and generated subframes preserve validity, 54.3 is characterized by the class of all linearly ordered modal Kripke models, see for instance [73, p. 29f.].
Theorem 11.14. In DLC, f- X--+ Y iff
1
f= T(X--+ Y)
in LC.
184
Proof. Weak soundness and completeness is proved in [75], using Lemma 11.13. Strong completeness follows using Lemmata 11.2 and 11.13. Q.E.D.
Proof.
<=> <=> <=>
11.3. MAPPING HYPERSEQUENTS INTO DISPLAY SEQUENTS
11.3.1.
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MAPPINGS INTO DISPLAY SEQUENTS
CHAPTER 11
{::}
f-oL3
p(H)
f-oLa f-oLa
I -+ Tz (po (H)) I-+
f-HL3 f-GL3
Lemma 11.2 Observation 11.17 Corollary 11.5 [10, §5.2] Q.E.D.
Note that the rules of GL3 are indeed derivable under the p-translation.
Lukasiewicz 3-valued logic
If b.= {AI, ... , An}, let *b.= {*AI, ... , *An} and (oib.) = A1 oi ... oi An; if b. = 0, let *b. = oib. = I.
11.3.2. The modal logic 85 Definition 11.19. The translation ry0 of ordinary sequents into display structures is defined by
Definition 11.15. The translation p0 of ordinary sequents into display structures is defined by
TJo(b. --t f)= e((o *b.)
0
(or)),
and the translation ry of non-empty hypersequents into display sequents is defined by The translation p of non-empty hypersequents into display sequents is defined by
p(s1
I ... I sn)
=I-+ Po(si)
oc · .. oc
Po(sn)·
Definition 11.16. (Cf. Avron [10]) For every non-empty hypersequent G, its translation
- if G = A1, ... , An -+ B, then
I ... I sn)
ry(s1 Theorem 11.20.
f-os5
=I-+ TJo(si)
o •.• o
ryo(sn)·
ry(H) iff f-cs5 H.
Proof. Let ry 0 (si I ... I sn) = ryo(sl) o ... o ryo(sn)· Then Tz(TJo(H)) is exactly the interpretation
<=> <=> <::::>
ry(H) I-+ Tz(ryo(H))
f-os5 f-os5
FHS5
Lemma 11.2 Theorem 11.8 [12, Section V]
Q.E.D.
Note that Tz(ry0(H)) is in the language of 85 and that also T1(X) remains in the language of 85, in the sense that in 85, (P)A is equivalent to -.0-.A. 11.3.3. Dummett's LC Definition 11.21. Let s be a single-conclusion sequent A1, · .. ,An -+ B. The mapping ( 0 of s into a display structure is defined by
Observation 11.17. For every non-empty hypersequent H in the language of L3,
f-oLa
p(H)
( 0 (s)
=
•(*Al o •(*Az o ... • (*An oB) ... )).
The translation ( of non-empty hypersequents with single-conclusion components into display sequents is defined by
((s1
I ... I sn) =I-+ (o(sl) o ... o (o(sn)·
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This translation reflects the interpretation cp8 of s used in Avron's completeness proof for GLC, namely A1 => (A2 => ... => (An => B) ... ). The interpretation c/JH of a hypersequent H = s 1 I ... I sn with singleconclusion components Si is the formula cp81 V ... c/Jsn.
The structures used in the framework should be built from the formulae of the logic and should not be too complicated (for human understanding and for computer implementation). Most important- the subformula property they allow should be a real one.
Observation 11. 22. Let the translation r* of display sequents into formulas be defined like the earlier translatioin r, except that rt(A) = m( A), for formulas A in the language of LC. For every non-empty hypersequent H with single-conclusion components, r2((o(H)) = m(cfJH). Lemma 11.23. In DLC U DS4.3t: (i) f-- X -t ri(X) and f-- r2(X) -t X; (ii) f-- X -t Y implies f-- ri(X) -t Y, and f-- Y -t X implies f-- Y -t r2(X). Proof. By induction on the complexity of X. Q.E.D. Theorem 11.24. In DLC, f-- ((H) iff f--GLC H. Proof. ~ ~
~
~ ~
f--oLC ((H) f--oLCuDS4.3t I-t r2((o(H)) f--os4.3 I-t r2((o(H)) f--os4.3 I-t m(cfJH) f--Lc c/JH f--GLC H
Lemmata 11.23 (i), 11.13 Lemma 11.13 Observation 11.22 Theorems 11.11 and 11.14 (12, Section III.l.4] Q.E.D.
187
In a footnote to this paragraph, he remarks that "[a] use of "structural connectives" that can arbitrarily be nested usually violates this principle. It seems to me that this is the weak point of Belnap's framework of Display Logic". The point is that the subformula property for the logical operations implied by (strong) cut-elimination in display logic (see Chapter 4) fails to be of immediate use in proof-search and other applications of cut-elimination. As Avron [12] observes, also in hypersequent systems cut-elimination normally does not imply Craig interpolation. The subformula property of display calculi may, however, still be used to prove conservative extension results. As often, there is a price to be paid for greater generality. A major advantage of DL is the simple and natural treatment of modal operators it allows. The display introduction rules for these operators can be viewed as meaning assignments and in combination with purely structural necessitation rules give rise to cut-free sequent systems for the minimal normal modal and tense logics K and Kt. Modularity in the sequent-style presentation is achieved by a correspondence between many important modal axiom schemata and purely structural display sequent rules, see Chapter 4. Irrespective of the aspect of greater generality, the modal display calculus provides a reason for preferring the method of display sequents over the method of hypersequents at least in one important application area, namely modal and tense logic.
11.4. DISCUSSION 11.5. OTHER TRANSLATIONS INTO DL
We have seen that there are straightforward and perspicious translations of hypersequents into display sequents. The components of a hypersequent can be translated using the structure connectives of display logic which are employed in the introduction rules for the internal logical operations and the modalities, while the structure connective 'I' in a hypersequent is translated by 'oc', the structural analogue of combining disjunction (conjunction) in succedent (antecedent) position. Whereas hypersequents as defined by A vron may be expected to work neatly for one-family systems and logics comprising connectives from two different families of logical operations, DL has been developed as a framework for systematically combining connectives from n > 1 families of logical operations. In [12, p. 2], Avron requires of a "good" proof system to satisfy among other things the following condition:
Recently, translations of other kinds of proof systems into DL have also been considered in the literature. In [115], G. Mints has presented systems of indexed sequents for the normal modal propositional logics obtained by combining the axiom schemata B, T, and 4 over K. Mints gives a detailed proof of cut-elimination for these systems of indexed sequents and shows how they can be translated into theoremwise equivalent display sequent calculi. R. Gore [80] raises the questions whether higher-level sequents can be translated into display sequents and whether 'Basic Logic' [148] can be embedded into DL and vice versa. Display logic might thus serve as a background theory used to compare with each other various kinds of generalized sequent systems.
CHAPTER 12
PREDICATE LOGICS ON DISPLAY
... predicate logic can be regarded as a special kind of propositional modal logic
S. Kuhn [95, p. 145] This chapter provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Benthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic, DL, by introduction rules for the existential and the universal quantifier. These rules for Vx and :Jx are analogous to the display introduction rules for the modal operators 0 and 0 and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal 'modal' predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules. The chapter has two main aims, namely 1. presenting a uniform proof-theoretic schema for both substructural subsystems of classical first-order logic, FOL, and various subsystems of FOL obtained by relaxing Tarski's truth definition for the existential and universal quantifiers, and 2. introducing these quantifiers into the framework of DL.
Recently, DL has received a great deal of attention. The original approach has been refined, extended, and re-applied to well-known logics and applied to new ones, see also [80], [118], [140]. However, so far all these developments have remained at the level of propositional logic. The addition of quantifiers to DL is briefly discussed in [16]: Quantifiers may be added with the obvious rules: (UQ)
Aa f- X
X f- Aa
(x)Ax f- X
X f- (x)Ax
provided, for the right rule, that a does not occur free in the conclusion . . . . The rule for the existential quantifier would be dual. ... [A]s yet this addition provides no extra illumination. I think that is because these rules for quantifiers are "structure free" (no structure connectives are involved; ... ) . One upshot is that adding these quantifiers to modal logic brings along Barcan and its converse ... willy-nilly, which is an indication of an unrefined account; alternatives therefore need investigating. [16, p. 408 f.] 43 189
190
The crucial observation here is that the standard quantifier rules arc structure-free. In what follows a structural account is developed of existential and universal quantification in DL, which is based on ideas put forward by, for instance, R. Montague [117] and, more recently, J. van Benthem [20] (see also the references listed in [95]). The key to this structural and more refined account of quantification is to look at the quantifier prefixes :Jx and Vx as modal operators. It has often been observed that the modal operators D ('it is necessary that') and 0 ('it is possible that') can be regarded as universal and existential quantifiers respectively. This is actually already Leibniz' idea that a proposition is necessarily true, if it is true in every possible world, whereas a proposition is possibly true, if it is true in some possible world. The Kripke-style possible worlds semantics generalizes this idea by introducing a binary relation of accessibility between worlds. There is thus a widely shared conviction that the truth conditions for D and 0 depend on the meaning of the universal and the existential quantifier in the meta-language of classical predicate logic. But then, conversely, the meaning of Vx and :lx may be explicated in terms of necessitiy and possibility, i.e. in terms of accessibility between 'worlds'. Moreover, from this modal perspective one may expect an illumination of universal and existential quantification. In particular, one may expect that the lattice of normal modal logics leads to an interesting landscape of predicate logics, in which classical first-order logic is just one among other 'first-class' citizens. This is the perspective on predicate logic advocated by van Benthem [20], [21]. According to van Benthem, the modal perspective reveals the abstract semantical core of quantification. Let M be any first-order model and let a, (3, ... range over variable assignments in that model. The standard Tarskian truth definition for the existential quantifier is:
M
f= 3xA[a]
iff for some assignment f3 on IM
I: a
=x
f3 and M
f= A[f3]
In the more general semantics the concrete relations =x between variable assignments are replaced by abstract binary relations Rx of 'variable update' between 'states' a, (3, /, ... from a set of states S. Assuming an interpretation of atoms containing free variables, the truth 43
191
A SEQUENT CALCULUS FOR KFOL
CHAPTER12
Belnap concludes this paragraph with the remark: "Introducing a family for each constant helps". While I shall take up the suggestion for further investigation, I shall neither try to elaborate this remark nor attempt to relate it to the treatment of quantifiers presented in the present paper.
definition for the existential quantifier becomes: M
f= 3xA[a]
iff for some (3 E S : aRxf3 and M
F A[J)]
or in the standard notation of modal logic,
'
M, a
f= 3xA
iff for some (3 E S : aRxf3 and M, f3
FA
Thus to every individual variable x there is associated a transition relati~n Rx on states. The resulting minimal predicate logic, KFO~, is nothing but thew-modal version of the minimal normal mo_dallogic K. In order to obtain an axiomatization of KFOL, one may JUSt take any axiomatic presentation of K and replace every occurrence of 0 and D by one of 3x and Vx, respectively.
12.1. A SEQUENT CALCULUS FOR KFOL
The idea now is not only to explicate the truth conditions of the quantifiers from the modal perspective but also to obtain from it structuredependent sequent rules for :Jx and Vx. DL offers a structural accou~t of 0 and 0 and therefore suggests a structural account of the quantifiers. Recall from Chapter 3 M. Dunn's definition of a residuated pair.
Definition 3.1 Consider two partially ordered sets A = (A,~) and B = (B, ~') with functions f: A---+ Band g: B---+ A. The pair (!,g) is called
residuated a Galois connection a dual Galois connection a dual residuated pair
iff iff
(fa~' b iff a~ gb);
iff iff
(fa ~' b iff gb ~a); (b ~' fa iff gb ~ a).
(b ~'fa iff a~ gb);
In category-theoretic terms, a residuated pair provides an ex~mple _of adjointness between functors, and also the universal and the existential quantifiers can be understood as a pair of adjoint functors (see [72, Chapter 15]). For every binary relation Rx on a non-empty set S of states, we may define the following functions on the powerset of S:
VxA 3xvA VxvA :JxA
....-
{a I Vb (aRxb implies bE A)} {a I 3b (bRxa and bE A)} {a I Vb(bRxa impliesb EA)} {a I 3b (aRxb and bE A)}
192
DISPLAY OF PREDICATE LOGIC
CHAPTER 12
193
Table XVII. Introduction rules for Vx and :Jx. (--+Vx) (Vx --+)
•xX --+ A f- X --+ VxA A --+ X f- VxA --+ •xX
(--+ :Jx) (:Jx --+)
X --+ A f- * •x *X --+ :JxA * •x *A --+ X f- :JxA --+ X
That is, we have:
To exploit this Gentzen duality between Vx and 3x, for every individual variable x, we introduce a structure connective •x· The basic structural rules for •x are:
12.2.
DISPLAY OF PREDICATE LOGICS
Since our main interest is in the behaviour of the quantifiers Vx and :lx as modal operators, we shall concentrate on a first-order language £ without individual constants and equality. We assume that £ contains ::J, --,, and V as primitive, whereas the remaining Boolean connectives and 3 are defined as usual. Note that the above introduction rules for 3x can be derived under this definition.
X --+ •x Y -H- •xX --+ Y.
One driving force behind van Benthem's investigation into generalizations of Tarski's truth definition is the interest in decidable predicate logics. When it comes to extensions of KFOL, van Benthem
We obtain the structure-dependent introduction rules for Vx and 3x in DL stated in Table XVII. In addition to these introduction rules we need further structural assumption in order to take care of the necessitiation rules in axiomatic presentations of normal tense logics:
would like to find logics (1) that are reasonably expressive, (2) that share the important meta-properties of predicate logic (such as interpolation: effective axiomatizability, perhaps even 'Gentzenizability') and (3) that m1ght even improve on this, by being decidable. (20, p. 11]
Let us refer to the result of adding these sequent rules to the display calculus DCPL for classical propositionallogic defined in Chapter 3 as DKFOL. Then f-nKFOL I --+ A iff f-KFOL A. The proof is completely analogous to the display of K in Chapter 3. Derivations in DKFOL are also just re-written derivations of DK; consider, for instance, the following cut-free proof of the Vx-version of the K axiom schema: Px--+ Px VxPx --+ •xPx Vx(Px ::::> Qx) o VxPx--+ •xPx •x(Vx(Px ::::> Qx) o VxPx) --+ Px Qx--+ Qx Px ::::> Qx--+ * •x (Vx(Px ::::> Qx) o VxPx) o Qx Vx(Px ::::> Qx)--+ •x(* •x (Vx(Px ::::> Qx) o VxPx) o Qx) Vx(Px ::::> Qx) o VxPx--+ •x(* •x (Vx(Px ::::> Qx) o VxPx) o Qx) •x(Vx(Px ::::> Qx) o VxPx) --+ * •x (Vx(Px ::::> Qx) o VxPx) o Qx •x(Vx(Px ::J Qx) o VxPx) o •x(Vx(Px ::J Qx) o VxPx) --+ Qx
(C)
•x(Vx(Px ::J Qx) o VxPx) --+ Qx Vx(Px ::J Qx) o VxPx--+ VxQx Vx(Px ::J Qx) --+ VxPx ::J VxQx I o Vx(Px ::J Qx) --+ VxPx ::::> VxQx I--+ Vx(Px ::J Qx) ::::> VxPx ::::> VxQx
KFOL satisfies these requirements. In comparison to deciable fragments of FOL obtained by restricting the language of first-order logic, say, to admit only special quantifier prefixes, KFOL .is a deci~ab~e system in the full first-order language. Decidable predicate logics m the full language of first-order logic have also been known from the field of substructurallogics for a long time. The system which is nowadays referred to as BCK predicate logic, for example, also enjoys decidability; cf. [83], [177]. The question arises whether it is possible to pr~sent both the family of substructural subsystems of FOL and the family ?f extensions of KFOL within a single proof-theoretic schema. As we will see, the predicate logical extension of DL suggested in the present paper provides such a uniform proof-theoretic framework. Moreover, we not only obtain sequent systems (i) for the familiar substruct~ral subsystems of FOL - extending the non-associative Lambek logic, N~L, as the minimal substructural predicate logic - and (ii) for extensiOns of KFOL, but also (iii) for substructur3.1 subsystems of extensions of KFOL. The above cut-free proof of the K schema, for instance, makes use of the contraction rule (C), which is lacking in BCK predicate logic. The minimal displayable classical predicate logic is then the resu~t of depriving DKFOL of all its structural rules (apart f~om the ba:'Ic structural rules for the structure connectives and the logical rules (Id) and (cut), see Chapter 3). Let us refer to :his syst~m as D~FOLmin· We thus obtain three 'landscapes' of classical predicate logics:
194
CHAPTER12 FOL
A ROUTE FROM KFOL TO FOL
£
FOL
NLL
KFOL
KFOL
DKFOLmin
195
are equivalent to the axiom schemata 3y3xB :::::> 3x3yB and 3y\fxB :::::> \fx3yB respectively. The latter are Sahlqvist formulas, i.e. their corresponding first-order frame conditions can be straightforwardly computed. Instead of searching for correspondences between axiom schemata and frame conditions, in the present context we are interested in correspondences between axiom schemata and structural sequent rules. We say that a first-order axiom schema corresponds to a set of structural rules iff (i) the rules allow proving the schema, and (ii) in the presence of the schema, the structural rules are axiomatically derivable under a suitable translation T from sequents into formulas. Since in antecedent position a structure connective is interpreted as backward-looking possibility and, in general, we do not assume symmetry of the binary relations Rx, this translation sends sequents to formulas of a 'tense logical' counterpart of the first-order logic £ under consideration. If the tense logical counterpart Lt is a conservative extension of £, then for every first-order formula A, we obtain that A is provable in the display system iff A is provable axiomatically. The persistence of atomic information, for instance, can be expressed by the purely structural sequent rule: X -t Py f- •xX -t Py.
•x
Figure II. Three 'kites' of predicate logics.
Van Benthem [20] investigates capturing predicate logics intermediate
~etween ~FOL and FOL by frame conditions corresponding to additi~na~ axwm schemata. There are, however, familiar predicate logical prmc1ples extending KFOL, which are not reflected by conditions on frames alone. As van Benthem observes, simple instances of the axiom schema:
(*) A
:::::>
VxA,
if x does not occur free in A
taken fr?m the axiomatization of FOL in [59] fail to be naturally translatab~e mto frame conditions. The schema Py :::::> \fxPy, where Py is ~tom1c, ~or example, corresponds to persistence (or heredity) of atomic mformatwn not depending on x: M, a
f= Py and a.Rx/3,
then M, {3
f= Py.
In o~der to detect frame correspondences, van Bent hem treats ( *) inductively._ In one case, however, frame correspondence is only achieved by assummg ~x an? Vx to be S4 modalities. The case where A = \fyB or A= 3yB gives nse to two subcases: (*1) where y = x and (*2) where Y i= x. In the first case one obtains analogues of the well-known axiom schemata 4 and 5, namely [x]B :::::> [x][x]B and (x)B ::::> [x](x)B. These s~h_e~ata co~respond to the transitivity and Euclidicity of the accessibility relatiOns Rx- Case (*2) is less obvious. Van Benthem observes that in S4 the proof rules B ::::> \fxB f- \fyB\fx\fyB B ::::> VxB f- 3yB\fx3yB
Our aim now is displaying various predicate logics between KFOL and FOL by adding purely structural rules to certain fixed collections of logical, structural and operational sequent rules of display logic. 44 The introduction rules for Vx and 3x thus remain unaltered, and all variation is achieved at the level of structural rules. In the following we shall consider two particular routes from from KFOL to FOL.
12.3. A ROUTE FROM KFOL TO FOL There is a simple obstacle to presenting FOL as a purely axiomatic extension of KFOL. The problem arises with atomic £-formulas like Rxy and Ryx, since \fx\fyRxy :::::> \fx\fyRyx, though a theorem of FOL, fails to be a theorem of KFOL. Identifying Rxy and Ryx does not help, because then \fx3yRxy 1\ --.\fx3yRyx, for example, would not be satisfiable. The familiar operation [yjx]A of substituting y for every 44
As we saw in Chapter 1, there also exist sequent calculi for Kin the ordinary Gentzen-style. However, there is nothing that deserves to be called a correspondence theory with respect to standard modal Gentzen calculi, neither in terms of frame conditions nor in terms of axiom schemata.
197
CHAPTER 12
A ROUTE FROM KFOL TO FOL
free occurrence of x in A can be regarded as a syntactic device exactly for overcoming this obstacle. Consider the axiomatization of FOL in [59]:
for every x, y. An axiomatic presentation of KFOL* results from the axiomatization of KFO L by adding
196
1.1 all universal closures of tautology schemata and of the following quantifier schemata 1.2 - 1.4, 1.2 \t'x(A :::> B) :::> (\fxA :::> \t'xB), 1.3 (*), 1.4 \t'xA :::> [yjx]A, if y is free for x in A, 1.5 A, A :::> B 1- B. Schema 1.4, but not its instance \t'xA :::> A allows \fx\fyRxy :::> \t'x\t'yRyx to be axiomatically derived: 1 2 3 4 5 6 7 8 9 10 11
\t'x\fyRxy :::> Rz1z2 Vz1 \t'zz(\t'x\t'yRxy :::> Rz1zz) \t'z1 \t'zz\t'x\t'yRxy :::> Vz1 \t'zzRz1z2 \t'x\t'yRxy :::> Vz1 \t'zz\t'x\t'yRxy \t'x\t'yRxy :::> Vz1 \t'zzRz1zz \t'z1 \t'zzRz1zz :::> Ryx \t'x\fy(\fz1 \t'zzRz1z2 :::> Ryx) \t'x\t'y\fzl\t'zzRzlzz :::> \t'x\t'yRyx \t'zl\t'zzRzlzz :::> \t'x\t'y\t'zl\t'zzRz1z2 \t'z1 \t'zzRz1z2 :::> \t'x\t'yRyx \t'x\t'yRxy :::> \t'x\t'yRyx
1.4, propositionallogic 1.1, 1 1.2, 2 1.3, propositional logic propositionallogic, 4, 3 1.4, propositionallogic 1.1, 6 1.2, 7 1.3, propositionallogic propositionallogic, 8, 9 propositional logic, 5, 10
Note that clause 1.1 amounts to postulating a necessitation rule for prefixes \fx. Van Benthem (20] suggests treating the substitution operator as a further necessity-type normal modal operator. Thus, substitution is not regarded as a meta-language device of the syntactic presentation, but is treated as part of the object language of first-order (or poly-modal) logic. The new modality is denoted by [Sxy], and formulas [Sxy]A are interpreted by means of a doubly indexed 'accessibility relation' Ax,y:
M, a F [Sxy]A iff for all {3
E S:
if aAx,y/3 then M, f3
FA
If [Sxy] is indeed to be conceived of as substitution of variables, then in addition to the K schema and a necessitation rule for [Sxy], further principles have to be postulated. In particular, substitutions commute with negation, which means that the relations Ax,y are functions and [Sxy]A is equivalent to (Sxy)A (:= ---,[Sxy]---,A). The system KFOL* is formulated in the language £*, which extends C with operators [Sxy],
[Sxy](A :::>B) :::> ([Sxy]A :::> [Sxy]B); A 1- [Sxy]A; and
Py;
sub1
[Sxy]Px
sub2
[Sxy]Pz
sub3
[Sxy]---,A
sub4
[Sxy](A :::> B)
sub5
[Sxy]VxA
sub6
[Sxy]VzA
\t'z[Sxy]A, if z =1- x,y;
sub7
[Sxy]VyA
\fyA, if x does not occur free in A;
sub8
[Sxy]A
:::>
---,\fx---,A;
sub9
[Sxy][Sxy]A
=
[Sxy]A;
sub10
[Sxy][Syx]A
-
[Sxy]A.
-
Pz, if z =1- x; ---,[Sxy]A; ([Sxy]A :::> [Sxy]B);
-
\fxA;
The extension of the logical object language is mirrored by a corresponding extension of the structural language of sequents: every operator [Sxy] comes with a structural connective •x,y· The inferential meaning of •x,y is given by its basic structural rules analogous to the basic structural rules for •x:
X-+ •x,yY 1- •x,yX-+ Y; •x,yX -t Y 1- X -+ •x,yY In succedent position •x,y is to be read as the forward-looking necessity operator with respect to Ax,y, in antecedent position as the backwardlooking possibility operator with respect to Ax,y· Since [Sxy] is a normal modality, we also postulate:
(MN•x,y)
I-+ X 1- I-+ •x,yX X -t I 1- X -+ •x,yi·
Proposition 12.1. The schemata sub1- sub10 correspond to the following structural rules:
198
CHAPTER 12 rsub1.1 rsub1.2
X ----+ •x,yPx f- X ---+ Py; X ---+ Py f- X ---+ •x,yPx;
rsub2.1 rsub2.2
X ----+ •x,yPz f- X ---+ Pz, if z =/= x; X ----+ Pz f- X ---+ •x,yPz, if z =/= x;
rsub3.1 rsub3.2
X----+ •x,y * •x,yY f- X---+ *Y; X ----+ Y f- •x,y * •x,y * X ---+ Y;
rsub4
= rsub3.2;
rsub5.1 rsub5.2
X----+ •x,y •x Y f- X---+ •xY; X---+ •xY f- X---+ •x,y •x Y;
rsub6.1 rsub6.2
X ----+ •x,y •z Y f- X ---+ •z •x,y Y, if Z =/= x, y; X ---+ •z •x,y Y f- X ----+ •x,y •z Y, if Z =/= X, y;
rsub7.1
X----+ •x,y •y Y f- X----+ •yY,
rsub7.2
if x does not occur free in any formula in Y; X ----+ •y Y f- X ---+ •x,y •y Y, if x does not occur free in any formula in Y;
rsub8
X----+ •x,y
rsub9.1 rsub9.2
X-+ •x,y •x,y Y f- X----+ •x,yY; X ---+ •x,yY f- X ---+ •x,y •x,y Y;
rsub10.1 rsubl0.2
X ---+ •x,y •y,x Y f- X ----+ •x,yY; X---+ •x,yY f- X---+ •x,y •y,x Y.
* •xY f-
X----+ *Y;
Proof. Relegated to Section 12.7. Q.E.D.
We shall refer to the result of augmenting DKFOL by rsub1 - rsub10 the basic structural rules for the structure connectives • x,y, (M N • x,y ) ,' an d (---+ [Sxy]) •x,yX ---+ A f- X ---+ [Sxy]A ([Sxy] ---+) A---+ X f- [Sxy]A----+ •x,yX
A ROUTE FROM KFOL TO FOL
from DKFOL and DKFOL* by adjoining the following sequent rules:
f-KFOL*
A iff f-DKFOL* I-+ A.
X ---+ * •x *A f- X ----+ VxvA A ---+ X f- VxvA ----+ * •x *X I ---+ X f- I ---+ * •x *X X ---+ I f- X ---+ * •x *I
(---+ Vx) (Vxv----+)
(MN•x)t
An obvious question is whether DKUKtFOL* can be faithfully embedded into DKtFOL under a suitable translation
p(X---+ Y)
(yfxfVyRyy:::) Rxx (yfxfVyRyy:::) Rxy (yfxfVyRyy:::) Ryx
PI(X)---+ p2(Y)
(yfxfVzRzz:::) Rxx (yfxfVzRzz:::) Rxy (yfxfVzRzz:::) Ryx
(y/xfRyy:::) Rxx (yfxfRyy:::) Rxy (yfxfRyy:::) Ryx
It is unclear to me how this could be achieved by a systematic definition of (yfxt as a function on£, and I therefore prefer to deal with [yjx], which in contrast to [Sxy] is also defined for terms, as a meta-linguistic device. In the above axiomatization, it is 1.3 and 1.4 which extend KFOL. To obtain a characterization of 1.4, we extend [yjx] to a function on structures by defining:
[yjx]I [yjx](X o Y) [yjx] *X [yjx]•z X
I
[yjx]X o [yjx]Y *[yjx]X •z[yjx]X
Proposition 12.3. Schemata 1.3 and 1.4 correspond to
Proof. See Section 12.7. Q.E.D.
Consider now the languages Lt and£* U£t, which are obtained from £and£* by the addition of backward-looking universal quantifiers Vxv for every variable x. The systems DKtFOL and DKUKtFOL* resul~
:=
of sequents into sequents such that p2([Sxy]A) = [yjx]p2(A). We would then need a Gentzen dual of [y/x] to supply Pl([Sxy]A). As it seems, however, there is no natural such Gentzen dual (yfxt Consider, for example, VyRyy ::J Ryy, VzRzz ::J Ryy, and Ryy ::J Ryy, which are provable in FOL. Since Ryy = [yjx]Rxx, Ryy = [yjx]Rxy, and Ryy = [yjx]Ryx, the following formulas would have to be provable:
as DKFOL*. Corollary 12.2.
199
r 1.3
X ---+ Y f- X ---+
r1.4
X---+ •xY
f-
•x Y,
X---+ [yjx]Y,
if x does not occur free in any formula in Y; ify is free for x in every formula in Y.
201
CHAPTER 12
ANOTHER ROUTE TO FOL
Proof. Straightforward, using a translation 7(X -+ Y) := 71 (X) ::)
This work is closely related to the algebraic study of restricted firstorder logic, cf. [120). The addition of substitution operators to cylindric modal logic is dealt with in [107] and [174). Alechina and van Lambalgen [3], [98] investigate the fine structure of quantification resulting ~rom explicitly considering dependency on parameters, while Meyer-Vwl [110) considers substructural quantification in terms of choice processes. Kuhn [95) prefers to overcome the problem of representing atomic £-formulas by using the idea of a variable-free formulation of predicate logic. This idea, too, has a distinctive poly-modal flavour, and we turn to it in the next section.
200
72(Y) of sequents into .Ct-formulas analogous to the 7-translation defined in Chapter 3. The tense logical extension of KFOL under consideration is the system KtFOL, which results from KFOL by adding the following axiom schemata and rule: 45
A ::) \fx--,\f£•A; A ::) \fxv•\fx--,A; \fxv(A::) B) ::) (\fxvA::) \fxvB}; A / \fxvA. Q.E.D. Let b. ~ { 1. 3, 1.4}, and let b.' be the set of rules corresponding to the schemata in Ll.
Corollary 12.4. (i) f-KtFOLUll. A only if f-oKtFOLull.' I -+ A. (ii) f-oKtFOLull.' X -+ Y only if f-KtFOLull. 7(X -+ Y).
Corollary 12.5. For every £-formula A, cisely the case that f-KFOLull. A.
f-oKFOLull.'
I -+ A in pre-
Proof. By the previous corollary, for every .Crformula A, f-oKtFOLull.' I-+ A iff f-KtFOLull. A. The claim follows from the fact that KtFOLU b. is a conservative extension of KFO L U b.. (One can verify that both systems are characterized by the same class of models.) Q.E.D. Note that KtFOL U b. and KFOL U b. are propositional theories not closed under uniform substituion. Andreka, van Benthem, and Nemeti emphasize that the generalized semantics invites the introduction of new vocabulary, reflecting distinctions not usually found in first-order logic. Examples are irreducibly polyadic quantifiers :Jy binding tuples of variables y, or modal calculi of substitutions (5, p. 712). Indeed, extended modal formalisms open up many routes from KFOL to FOL, and there is an extensive literature on modal approaches toward first-order logic. The central reference is Y. Venema's work on cylindric modal logic [171 ), [173), in which restricted first-order logic in the language with equality is extended by a set of 'irreflexivity rules'. 45 A new axiomatization of the minimal normal tense logic Kt with slightly stronger interaction (between future and past tense) axiom schemata can be found in Chapter 13. This axiomatization was inspired by the display presentation of Kt.
12.4.
ANOTHER ROUTE TO
FOL
The generalized truth definition does not allow atomic £-formulas to be treated as atoms of the full system FOL. The idea behind the variablefree presentation of FOL is to consider only n-place atomic £-formulas pn (n > 0), in which n distinct variables occur exactly once and in a fixed order. From these atoms, which can be taken as n-sorted sentence letters, representations of the remaining atoms of .C are generated by applying certain necessity-type normal modal operators. These operators are: 46 [r] [s]
[i)
rotation switch identification
The vocabulary of the language £** comprises these modalities, the atoms pn, ::), ., and the quantifier prefix \fx. Every atom pn is a sortn formula. If A is a sort-n formula, then also \fxA, [r]A, [s]A, [i]A, and -,A are sort-n formulas. If m ::; n, A is a sort-n formula, and B a sort-m formula, then (A ::) B) and (B ::) A) are sort-n formulas. Note that the usual definitions give rise to sort-n truth and falsity constants tn and rn. In the structural language of sequents this is reflected by constants In. We use An, Bn, en etc. as schematic letters for sort-n formulas and sometimes omit superscripts if no confusion can arise. [zp A := [l]A, and [l]n+l A := [l][l]n A, for l E {[r), [s); [i]} and ~ 2:: :· Formulas [r]An, [s]An, and [i]An receive a natural mterpretatwn .m models M = ((D), V), where (D) is the set of all sequences of fimte or denumerably infinite length of elements from a non-empty domain 46
For the sake of greater uniformity, our notation diverges from that in (95).
,..,., "'
202
ANOTHER ROUTE TO FOL
CHAPTER 12
D, and V maps atoms pn to n-tuples of elements from D, cf. [95]. If d = (d1, ... , dn, .. . ) E (D), then a sort-n formula An is true at d in
(M, d
f= An)
M
according to the following inductive definition:
M,d F pn M,df=·B M,d f= (B :J C) M,d f=VxB M,d f= [r]B M,d f= [s]B 1 M,d f= [s]Bm, (2::; m) M,d F [i]B
iff iff iff iff iff iff iff iff
(d1, ... ,dn) E V(Pn);
not M,d f= B; M,d f= B implies M,d f= C; for every dE D, M, (d, d2, ... , dn)
f=
B;
M, (dn, d1, · · ·, dn-1) F B; M,d F B; M, (d2, d1, d3, ... , dn) F B; M,(dl,dl,d3, ... ,dn) F B.
Kuhn [95] defines further modal operators and uses them to give an axiomatization of FOL in the language £**. These definitions are rather tedious, although the defined connectives again receive a natural interpretation in models ((D), V). Our aim is verifying that in Kuhn's axiomatization, every axiom schema which is not minimally derivable corresponds to a purely structural sequent rule. As far as schemata exhibiting defined modalities are concerned, this is a completely straightforward matter, once the definitions have been restated for the structural companions •r, •8 , and •i of [r], [s], and [i] respectively. We shall therefore consider only one simple paradigmatic case to explain the general method of rewriting these axiom schemata as structural sequent rules. Suppose in the following that m = max( n, k).
and also In is a sort-n structure. If X is a sort-n structure, then so are *X, •rX, •sX, and •iX. If m ::; n, X is a sort-n structure, and y a sort-m structure, then (X o Y) and (Y o X) are sort-n structures. We use xn, yn, zn etc. as schematic letters for sort-n structures and sometimes omit superscripts if no confusion can arise. The structure constants 1n are governed by rules analogous to those for I, and the system DKFOL** results from DKFOL by adding (i) introduction rules for [r], [s], and [i] analogous to those for Vx, and (ii) necessitation and basic structural rules for •n •s, and •i analogous to those for •x· We now restate the earlier definitions for the structure connectives •r,
•s, and
•i·
Definition 12. 7.
The axiom schema we consider is [d]ij [db A
Proposition 12.8. The schema [dJij[d]jiA
Definition 12.6.
[rJ(m+l)-k([r][s])k-l(An 1\ tk) 2. [rJ;;lAn := [rJZ-lAn [r]k[r]j[i][r]j 1 [r];; 1 A if j > k 1 3. [dJjkA := { A[rh[r]j+l[i][r]j~drJ;; A if j < k
X X
1. [r]kAn :=
if
j = k
Formulas [r]nAn and [r];;:- 1 An are sort-n formulas, and [d]jkAn is a sortmax(j, k, n) formula. Let dt be the result of deleting the first k components of (d1, ... , dk, .. . ) =d. If the length of d is at least m, then
f= [r]kAn M, d f= [r];; 1 An M,d F [d]jkA M,d
iff iff iff
M,(dk,dl, ... ,dk-l,dt) f=An M, (dz, ... , dk, d1, dt) f= An M, (d1, ... , dj-1 1 di, dj} FAn.
In order to display poly-modal logics in the given vocabulary, we inductively define sort-n structures. For each n 2: 1, every sort-n formula
203
--t --t
= [d]ij A.
= [d]ijA corresponds to
•ij •ji Y I- •ijY; •ij Y I- •ji •ij Y.
Proof. Straightforward, if in succedent position •n •s, and ~i. are trans; lated as [r], [s], and [i] respectively, and in antecedent position as (r), (sr, and (ir respectively. Q.E.D.
In the primitive vocabulary, Kuhn's axiomatization contains the following minimally non-derivable schemata:
=
2.1 -.[l]A [l]-.A, for lE {[r], [s], [i]}; 2.2 [r]A 1 =A\ 2.3 [s]A 1 =: A1 . Proposition 12.9. The schemata
structural sequent rules:
2.1-2.3 correspond to the following
THE BARCAN FORMULA
CHAPTER 12
204
r2.1.1 r2.1.2 r2.2.1 r2.2.2 r2.3.1 r2.3.2
X--+ •t * •tY I- X--+ *Y; X --+ Y I- •t * •t *X --+ Y;
205
for an application of cut like
xi --+ • T yi 1- xi --+ y1.l
X --+ •x,y •y A
xi --+ yi I- x--+. T yl.l
rsub7.1
x1 --+. yi I- xi --+ yi. s
'
XI --+ yi I- X --+ •sYl.
•yX--+ y would require applying cut to the uppermost 'parametric ancestors' of A in the derivation of •yX --+ A and replacing every such ancestor by an occurrence of Y. If x occurs free in Y, the preservation of rule applications is, however, not given:
Proof. The cases of 2.2 and 2.3 are obvious. The case of the functionality schema 2.1 is analogous to that of sub3. Q.E.D.
Also on this route to FOL, the inferential meaning of the modal operators involved- in particular, the inferential meaning of the universal quantifier - is laid down 'once and for all' by their introduction rules. These rules exhibit only one occurrence of the operations in question either in antecedent or in succedent position in the conclusion sequent (and exhibit no other logical operation). Therefore these introduction rules may qualify as inferential meaning assignments; see Chapter 2.
X--+ •x,y •y Y(x) X--+ •yY(x) •yX --+ Y(x)
not rsub7.I
12.6. THE BARCAN FORMULA 12.5. STRONG CUT-ELIMINATION It is well-known that the Barcan formula
BF 'v'xDA ::J DVxA
In Chapter 4 it was proved that every displayable logic enjoys strong cut-elimination. This result shows that no matter in which order the elimination steps in Belnap's [16] general cut-elimination proof for DL are applied, every sufficiently long sequence of one-step reductions terminates in a cut-free proof. 47 Recall that a displayable logic is a system admitting a display presentation such that its rules satisfy certain conditions regarding their shape, and the introduction rules for the logical operations are such that principal cuts can be eliminated. In particular, the rules are supposed to be closed under certain substitutions of structures for formulas, a constraint rather sensitive to side-conditions on rules. The side-conditions on rsub7.1, rsub7.2, r1.3, and rl.4 violate this closure condition and therefore the 'wholesale' elimination theorem does not apply. Consider for instance rsub7.1. The reduction strategy
and its converse
BFc D'v'xA ::J VxDA
correspond to the assumptions of constant domains and persistence of individuals along the accessibility relation respectively; cf. for example [64]. As has been observed by Belnap [16], adding the obvious str~cture free rules, i.e. (UQ), for the universal quantifier to DL results m the provability of BF and BFc. Consider:
A--+A DA-+ •A (UQ) 'v'xDA-+ •A •'v'xDA --+ A (UQ) •'v'xDA-+ 'v'xA VxDA --+ D'v'xA I o 'v'xDA --+ D'v'xA I --+ 'v'xDA ::J D'v'xA
47 Provided stacks of configurations in which applications of structural rules are followed by a cut are reduced top-down.
I
l
A--+ A (UQ) 'v'xA--+ A D'v'xA--+ •A eD'v'xA--+ A D'v'xA--+ DA (UQ) D'v'xA --+ 'v'xDA I o D'v'xA --+ 'v'xDA I --+ D'v'xA ::J VxDA
207
CHAPTER 12
REMAINING PROOFS
The structural account of the quantifiers as modal operators blocks these proofs of BF and BFc. 48 In the generalized semantics neither BF nor BFc are valid. The latter principles beome provable in the presence of further structural sequent rules, however.
Note also that the display introduction rules for 3x and Vx may remain unchanged in display sequent calculi for intuitionistic predicate logic. As in the Kripke semantics for intuitionistic predicate logic, the non-classical behaviour of the existential and universal quantifer results from the non-classical interpretation of implication and negation. In Chapter 10 intuitionistic logic was displayed using a modal translation into S4. Also if one displays intuitionistic logic more directly by dispensing with the structure operation *, interpreting the structure operation o in succedent position as intuitionistic implication, =>h, and defining intuitionistic negation 'h by 'hA := A ~h f (see [76], [140]), the classical interdefinability of 3x and Vx fails. In the direct approach one obtains the following introduction rules for intuitionistic negation and implication:
206
Proposition 12.1 0. BF and BFc correspond to rBF rBFc
X--+ •x • Y I- X--+ • •x Y; X --+ .. x I- X --+ •x • Y.
Proof. Straightforward, if in succedent position • is translated as D, and in antecedent position as D's Gentzen dual Ov. Q.E.D. What have we achieved? In the preceding we have formulated structure-dependent introduction rules for the universal and the existential quantifer by treating them as modal operators. In this extension of Belnap's display formalism to quantifier logic, the rules for Vx and 3x proved to be general enough to avoid the derivability of the Barcan formula BF and its converse BFc. Moreover, we have considered two routes from the resulting minimal predicate logic DKFOL to full classical first-order logic. In the languages with V primitive, every additional axiom schema and also BF and BFc turned out to be expressible by purely structural sequent rules. It is also worth emphasizing that this extension of DL to predicate logic naturally 'Gentzenizes' various families of interesting substructural systems. Let DKFOL* . and ** mzn DK F OL min refer to the result of removing all structural rules apart from the basic structural rules of their structure connectives, (id) and (cut) from DKFOL* and DKFOL** respectively. In addition to the 'kites' of systems in Figure II, we obtain two further logical 'landscapes' depicted in Figure Ill. 48 Fitting [64] points out that what he takes to be the "most obvious" axiomatization of a minimal normal modal extension of FOL, namely one using the rule of universal generalization
(--+ 'h) (·h --+)
X o A --+ I I- X --+ 'hA X --+ A I- 'hA --+ X o I
(--+~h) (~h--+)
X
o A --+ B I- X --+ A ~h B X --+ A B --+ Y I- A ~h B --+ X
£*
£**
DKFOL*
DKFOL**
o
Y
DKFOL;',in
Figure Ill. More 'kites' of predicate logics.
A => B I- A => VxB, if x does not occur free in B,
allows proving BFc but not BF. The provability of both BFc and BF can be avoided by using an axiomatization like the one in [59], which refers to universal closures instead of postulating universal generalization, cf. [87, p. 179ff.]. It should be clear, however, that such axiomatizations are rather remote from reflecting in an axiomatic setting the idea of capturing the meaning of Vx and 3x by means of introduction and elimination rules as meaning assignments. In DL, however, the introduction rules for Vx and 3x may be argued to allow such an interpretation; cf. [181].
12.7. REMAINING PROOFS
In order to verify the correspondences to be established, we extend £* by the backward-looking quantifiers Vxu and [SxyJ and refer to the resuiting language as £!. The system K tFO L *, the tense logical version
1
208
of the minimal predicate logic with a modal substitution operator, results from KtFOL U KFOL* by adding the following axiom schemata and rule:
A-+A *A-+ *A -,A-+ *A \lx•A -+ •x * A •x\lx•A -+ *A A -+ * •x \lx•A [Sxy]A-+ •x,y * •x\fx--,A rsub8 [Sxy]A -+ *\lx•A [Sxy]A-+ --,\fx·A
A :J [Sxy]•[SxyJ•A; A :J [SxyJ•[Sxy]•A; [SxyJ(A :J B) :J ([SxyJA :J [Sxy]"B); A/ [SxyJ"A. We extend the tran~l~tion T from sequents into Lt-formulas (used in the proof of PropositiOn 12.3) to a translation T' from sequents into .c;- formulas by defining:
T!(•x,yX) T2(•x,yX)
•[SxyJ'Tl(X) [Sxy]T2(X).
Proof of Proposition 12.1. Consider first the direction from structural rules to axiom schemata. The left-to-right direction of sub4 is just the K schema for [Sxy], which is minimally derivable. The derivations of sub1, sub2, sub5- sub7, sub9, and sub10 are simple. In the case of the left-to-right direction of sublO, for example, we have:
For the right-to-left direction of sub4 it is enough to observe that 49 this schema is equivalent to the right-to-left direction of sub3. Like the cut-free proof of the K schema in Section 12.1, the derivation of ([Sxy]A :J [Sxy]B) :J [Sxy](A :J B) involves applications of the mantonicity and contraction rules. We now turn to the direction from axiom schemata to structural 1 rules. In most cases, the axiomatic derivation under the T -translation is accomplished merely by the transitivity of :J. The derivation of rsub3.1, rsub3.2, and rsub8, however, involves tense logical principles: rsub3.1
A-+A [Sxy]A -+ •x,yA rsublO [Sxy]A -+ •x,y •x,y A •x,y[Sxy]A-+ •x,yA •x,y •x,y [Sxy] -+A •x,y[Sxy]A-+ [Sxy]A [Sxy]A-+ [Sxy][Sxy]A
1 2 3 4
rsub3.2
Less obvious are the derivations of both directions of sub3:
A-+A [Sxy]A-+ •x,yA •x,y[Sxy]A-+ A *A -+ * •x,y [Sxy]A ·A-+* •x,y [Sxy]A [Sxy]•A -+ •x,y * •x,y[Sxy]A rsub3.1 [Sxy]•A-+ *[Sxy]A [Sxy]•A-+ •[Sxy]A
209
REMAINING PROOFS
CHAPTER 12
A -+ A rsub3.2 •x,y * •x,y * A -+ A * •x,y *A-+ [Sxy]A *[Sxy]x]A-+ •x,y *A •[Sxy]A-+ •x,y *A •x,y•[Sxy]A-+ *A •x,y•[Sxy]A-+ ·A •[Sxy]A-+ [Sxy]•A
The derivation of sub8 follows the pattern of the derivation of the rightto-left direction of sub3:
rsub8
r!(X) ::::> [Sxy]••[SxyJ•Tt (Y) [Sxy]••[SxyJ•Tt (Y) ::::> ::::> •[Sxy]•[SxyJ•Tt (Y) Tt (X) ::::> •[Sxy]•[SxyJ•TI(Y) T1(x) ::::> •T1(Y)
T1 (X) ::::> r2(Y) •T2 (Y) ::::> •T1 (X) •T2(Y) ::::> [SxyJ•[Sxy]••TI(X) -.[Sxy]•-.T2 (Y) ::::> •[Sxy]Tt (X) •[Sxy]••T2 (Y) ::::> [Sxy]•Tt (X) 6 -.r2(Y) ::::> [SxyJ1Sxy]•ri(X) 7 -.[SxyJ1Sxy]-.rl(X) ::::> T2(Y) 8 -.[SxyJ••[Sxy]•Tl (X) ::::> T2(Y)
1 2 3 4 5
1 2 3 4
T1 (X) ::::> [Sxy]••'v'Xv•Tl (Y) [Sxy]-.-.\I£•TI (Y) ::::> ::::> -,\fx-.\lxv•TI (Y) Tt (X) ::::> -,\fx-.\lxv•Tt (Y) T1 (x) ::::> •T1 (Y)
sub3 modus ponens, 1,2 KtFOL*, 3
CPL, 1 KtFOL *, CPL, 2 KtFOL*, 3
sub3, 4 KtFOL*, 5 CPL, 6 CPL, 7
sub8 modus ponens, 1,2 KtFOL*, 3 Q.E.D.
In other words, schema sub4 corresponds to quasi-functionality ('each point is related to at most one point'). 49
210
CHAPTER 12
CHAPTER13
Proof of Corollary 12.2. Refer to DKUKtFOL*
[Sxy]} ([SxyJ -7) (MN•x,y)t (-7
X
-7
* •x,y *A f-
X
+
[SxyJA * •x,y *X
-7
A -7 X f- [SxyJA -7 I -7 X f- I -7 * •x,y *X X
-7
I f- X
-7
* •x,y *I
as DKtFOL*. Since f-KtFOL* A just in case f-oKtFOL* I -7 A the claim follows by KtFOL* (DKtFOL*) being a conservative exte~sion of KFOL* (DKFOL*). Q.E.D.
APPENDIX
Many important logical systems have proof-theoretic presentations of more than one type, say, an axiomatization, a natural deduction proof system, and a sequent calculus presentation. Usually, this is a rather fortunate situation. It may happen that certain axiom schemata are characterizable by algebraic or relational properties expressible in an interesting fragment of first-order logic, and that Gentzen-style proof systems lend themselves to automated deduction. As we have seen, display logic is an elegant and powerful refinement of Gentzen 's sequent calculus and meets quite a few methodological requirements of a more philosophical and a more technical nature. It would be nice to relate the modal display calculus to natural deduction proof systems for intensional logics, and in the present chapter we shall relate DL to generalized Fitch-style natural deduction systems for modal logics. Moreover, we shall show that DL may have repercussions on the axiomatic presentation of logical systems and define an apparently new axiomatization of Kt. Eventually, we shall consider a generalization of DL, namely four-place display sequents. We shall redisplay Nelson's system N4 and obtain a display sequent calculus for N3 by the addition of a purely structural sequent rule.
13.1. DL AND FITCH-STYLE NATURAL DEDUCTION
A full-circle theorem 50 for a given logic A says that certain proof systems S1 , ... , S4 for A of the four maybe most important types of inference systems (axiomatic, natural deduction, tableaux, sequent calculi) are all equivalent in the following sense: - Every proof of a formula A from formulas A 1 , ... , Ak in 8 1 can be transformed into a proof of A from A 1 , ... , Ak in S2; - every proof of A from A 1 , ... , An in S4 can be transformed into a proof of A from A1, ... , Ak in S1. 50 This term was pointed out to me by Tijn Borghuis, who found it in (13, Chapter XI].
211
212
CHAPTER 13
0
FITCH-STYLE NATURAL DEDUCTION Table XVIII. Additional axiom schemata and frame conditions.
Hilbert-style
<equent cakulu'
natuml d'ductioa
tableaux
Figure IV. A full circle.
To establish such a full circle, one has to make sure that in each case the data A 1 , ... , Ak are structured in the same way. If one, for instance, considers proofs from sets of assumptions, then these proofs have to be transformed into proofs from (representations of) the same data structures. Moreover, in the case of normal tense logic there is a whole lattice of logics rather than one designated logical system. The standard syntactic presentation of these systems is in Hilbert-style and is thus modular. If one wants to establish a full circle for the most important systems of normal tense logic, the problem is that in the literature there seem to be no modular natural deduction presentations of these systems, which are not obtained by simply adding axiom schemata to presentations of the minimal normal tense logical system Kt. We shall therefore first of all extend T. Borghuis' generalization of Fitch-style natural deduction [27], [28] to a modular proof theoretic framework for normal tense logic. We shall then relate these natural deduction systems to display calculi. As is clear from Chapter 6, a proper application of the tableau method within display logic, including a reduction to Kowalski clausal form, is available for logics of functional accessibility relations. More generally, display tableaux will therefore be obtained simply by inverting the rules of the display sequent systems for the logics under consideration.
0
(AI} (A2} (A3} (A4} (A5}
axiom schema [F]A :::> [F][F]A [F][F]A :::> [F]A A:::> (F}(P}A A:::> (P}(F}A (F) A :::> [F]( (F}A V A V (P}A)
(A6}
(P}A :::> [F]( (P}A V A V (F) A)
frame condition
VxVyVz((x < y 1\ y < z) :::> x < z) VxVy(x < y :::> 3z(z < y 1\ x < z)) Vx3y(x < y) Vx3y(y < x) VxVyVz((x < y 1\ x < z) :::> :::> (z < y V y < z V z = y)) VxVyVz((y < x 1\ z < x) :::> :::> (z < y V y < z V z = y))
contribution to the emergence of a general proof theory of intensional logic.
Hilbert-style. Consider once more the language of normal_ tense _logic in the vocabulary of classical propositional logic CPL (mcludmg t 'truth' and f 'falsity') together with the unary connectives [F] 'always in the future', (F) 'sometimes in the future', [P] 'always in the.~ast', and (P) 'sometimes in the past'. The minimal normal propos1t10nal tense logic Kt can be axiomatized as follows: AO Al A3 A5 Rl
all tautologies; (F}A := -.[F]-.A; A:::> [F](P}A; [F](A :::> B) :::> ([F]A :::> [F]B); A 1- [F]A;
RO A2 A4 A6 R2
A, A:::> B 1- B; (P}A := -.[P]-.A; A:::> [P](F}A;
[P](A :::> B) :::> ([P]A :::> [P]B); A 1- [P]A.
We also consider a number of further axiom schemata, which are presented in Table XVIII along with their defining first-order properties on Kripke frames (I, <). 51 A (Hilbert-style) axiomatic proof of A from a finite set b. = {A 1 , ... , Ak} is a finite sequence of formulas B1, · · ., Bn, where Bn = A and for each Bj (1 ~ j ~ n), either
Hilbert-style
a;,play '""""'calculus
213
F;tch-,yl'
1. Bj E b., or 2. B · is an instantiation of an axiom schema, or 3. B~ is derivable from earlier items in the sequence by means of RO, R1, or R2.
display tableaux
Figure V. Another full circle.
We write PH(TI, A, b.), if n is an axiomatic proof of A from b..
Although the proof of the full-circle theorem to be given is neither complicated nor particularly deep, it may be of some interest as a
51 Instead of using familar names for these schemata, for reasons of uniformity we here simply number them as (Al} - (A6}.
l
214
Fitch-style. Tijn Borghuis [27], [28] obtains a modular natural deduction proof-theoretic framework for the most important normal modal propositional logics by adding both modal import and export rules to the Fitch-style natural deduction system for CPL. This method can be extended to the above systems of normal tense logic. 52 A characteristic feature of Fitch-style natural deduction is that the extent of proofs and parts of proofs which are called 'subordinate proofs' is indicated by vertical lines. If~ = {A1, ... , An} and n > 0, then [F]..6. = {[F]A1, ... , [F]An}i if n = 0, then [F]..6. = 0. We adopt analogous conventions for [P], (F), and (P). For Kt, the notion PN(IT, A, ..6.) ('IT is a proof in natural deduction of A from the finite set of assumptions ..6.') is inductively defined as follows:
FB
F2
PN(I A,A,{A}) PN(I t, t, 0)
F3
PN(I
F4
~~
, A1 1\
1\
PN(I IT, A 1\ B, .6.) => PN
F7
(I ~ (I ~
PN(I IT,A,.6.) => PN(I
~VB
PN(IIT,B,.6.) => PN(I
~VB
PN(
[~:
F!O
PN (I
Fll
PN(I IT1,A,.6.1), PN(I IT2,--.A,.6.2) =>
n, A, A,),
PN (
n,, ~A, A,)
~:
~:
,B, (A,
u A,) I
=> PN (
{~B))
~~
Fl4
PN(I IT, A, .6.) => PN
Fl5
PN(I IT, A, .6.) => PN
Fl6
PN(I n1,B,.6.I), PN(I n2,--.B,.6.2) => PN (
~~
(I [)Q (I [)Q
,A, A U
{A,, ... ,A.))
I IT '[F]A, [F].6.) I IT '[P]A, [P].6.)
, (F)A, (.6.1 u .6.2) \ {[F]--.A})
(F) A
,AVB,.6.)
Fl7
PN(I n1, B, .6-I), PN(I n2, --.B, .6.2) => PN (
~~
, (P)A, (.6.1 U .6.2) \ {[P]--.A})
(P)A
Fl8
, B, A, U A,)
~1 .~B, (A, u A,) I {B))
PN(\ ll, A, A)
,A V B,.6.)
PN(I n1,B,.6.1), PN(I IT2,--.B,.6.2) =>
~:
--.[F]-.A
Note that the rule (Fl) below does not exactly reproduce Borghuis' 4-import rule.
=> PN (
PN(I n1, A, .6.1), PN(I IT2, --.A, .6.2) =>
PN (
52
PN(\
F13
'B, .6.)
,G,A,u(A,I{A))u(A,I{B)))
,A :J B,.6. \{A})
PN(I IT,B,.6.) => PN(I
'A, .6.)
PN(I IT1,C,.6.1), PN(I IT2,C,.6.2), PN(I IT3,A v B,.6.3) =>
~~ B
F9
A2, .6.1 U .6.2)
A2
PN(I IT, A 1\ B, .6.) => PN F6
,B,A,uA,)
~
A1 F5
~:
PN (
,A,{f}) PN(I IT1,A1,.6.I),PN(I IT2,A2,.6.2) => PN (
PN(I n1,A,.6.1), PN(I n2,A :J B,.6.2) => PN(
F12 Fl
215
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
'--.[F]-.A, (.6.1 u .6.2) \ {--.(F) A})
216
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
F19
PN(\ Ill, B, ~I), PN(\ II2, ·B, ~2) PN (
~~
'·[P]•A,
•[P]•A
F20
F21
PN(\ IT,
PN(\ IT,
A,~) A,~)
=> =>
PN (
PN (
we shall associate the following tableau rules with the axiom schemata (A1) - (A6):
=>
(~1 u ~2) \ {·(P)A })
II [P)\ (F) A , [P)(F)A, [P](F)A
(T1) (T2) (T3) (T4) (T5) (T6)
~)
II [F)\ (P)A , (F)(P)A, [F)(P)A
=>
PN
(F2)
PN(\ IT,
A,~)
=>
PN
(F3)
PN(\ II,
A,~)
=>
PN
(F4)
PN(\ IT,
A,~)
=>
PN
(F5)
PN(\ IT, (F) A,~) PN
(F6)
(I ~)((F) A
PN(\ IT, (P)A, ~) PN
(I ~)((P)A
(I [F) (I [;rr~\~ (I ~F)(P)A
, [F)[F]A,
, (F)(P)A,
~)
(I ~P)(F)A
, (P)(F)A,
~)
[F)\ II , [F]A, A
~)
[F)[F)~) [F)~)
, [F)((F)A V A V (P)A),
~)
=> V A V (F)A)
, [F)((P)A V
A
V (F)
• *• *X *• *• X
the tree are sequents, and every branching is an instantiation of one of the reduction rules. A closed tableau is a finite tableau such that every leaf is of the form A ---+ A, I ---+ t, or f ---+ I. We write C/(11, X ---+ Y) if 11 is a closed tableau for X ---+ Y.
Display sequent calculi. Display sequent calculi are the proof systems of their corresponding tableau refutation systems. The display sequent calculi are obtained by reading the tableau rules 'bottom up' and adding axiom schemata (i.e., axiomatic sequent rules), namely 1- A ---+ A, 1- I ---+ t, and 1- f ---+ I. Thus, every proof in a display sequent calculus amounts to a closed display tableau. Recall from Chapter 4 that the sequent systems under consideration enjoy strong cut-elimination. We shall now define the transformations needed to form the circle.
=> V A V (P)A)
X---+ ••Y 1- X---+ •Y •X ---+ Y 1- • • X ---+ Y X---tYI-*•*•X---+Y X---tYI-•*•*X---tY
---+ Y 1- * • *X ---+ Y X---tY •X---+Y ---+ Y 1- * • *X ---+ Y X---tY •X---+Y A tableau for a sequent X ---+ Y is a tree with root X ---+ Y. The nodes of
The additional axiom schemata are taken into account by the following rules:
(Fl)
217
From axioms to natural deduction. We define the mapping ( · )H of axiomatic proofs into natural deduction proofs. Suppose PH(II, A,~), where~= {A1, ... , An}· Case 1: A = Aj E ~- Take any enumeration B1 ... Bk of~\ {Aj} and
A),~)
Display tableaux. Tableau calculi are refutation methods consisting of certain rules that manipulate sequents. The intuition behind tableaux is semantic in nature: if the premise sequent of a tableau rule has a countermodel, then so has at least one of its conclusion sequents. Display tableaux are built up from display sequents; see Chapter 3. The inverted versions of the basic structural rules, the introduction rules in Table II (Chapter 3) and the additional structural rules in Table V (Chapter 4) preserve countermodels in this sense. In particular,
B1 set IIN =
Bk
Aj· Case 2: A ~ ~' and A is an instantiation of one of the axiom schemata.
A1 Then IIN =
:
An 11''
where 11' is a certain natural deduction proof of A from 0. We shall present 11' here only for (a characteristic sample of) the tense logical
The effect of (·) H can be summarized as
axiom schemata:
(F) A •(F)A ·[F]·A (F)A :J •[F]·A
·[F]·A [F]·A (F) A •[F]·A :J (F)A
[F]
A
[F] I (P)A [F](P)A A :J [F](P)A
219
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
218
A:JB A B
[F]B [F]A :J [F]B [F](A :J B) :J ([F]A :J [F]B)
[F]IA [F][F]A I [F]A :J [F][F]A
[F]IA [F]A [F][F]A :J [F]A
I
I1F)(P)A A :J (F)(P)A
I [F]( (F) A V A V (P)A) (F) A :J [F]( (F) A V A V (P)A)
From natural deduction to closed tableaux. We inductively define the mapping (· f of natural deduction proofs into closed tableaux. If ~={AI, ... , An}, then o.6. =(AI o ... o An)i if n = 1, then o~ =AI; and if~ = 0, then o~ =I. A proof of A from~= {A1, ... , An} will be mapped to a closed tableau foro~ -t A. 53 Fl:
ITT =A-+ A
F2:
rrr
=I-+ t
F3:
(F)A
Case 3: A (/. ~' A is not an instantiation of one of the axiom schemata, and A is obtained from earlier items in IT by means of (i) RO or (ii) R1 or R2. (i): In this case there is a formula C and there are proofs III, II2, such that PN(III, C, .6.I ~ ~) and PN(II 2 , C :J A, ~ 2 ~ ~). Take any enumeration BI ... Bk of~\ (~I U ~ 2 ) and set BI
(ii): In this case A = [F]B or A= [P]B, and there is a proof III such that PN(III, B, 0). Then rrN =
or
An [P] 1 rr1
A
o~ 1
Ill
o o~ 2 -+ A1 A A2
lh
F5:
We deal with only one rule. Suppose CT(II1, o~-+ A o~-+ A ITT = Il1 A 1\ B -+ A AoB-+A A-+A
F6:
We deal with only one rule. Suppose CT(II 1 , o~ -+ A).
1\
B).
o~-+
rrr =
A VB o~ -+ A o B o~ o *B-+ A Ill
53 This is justified since, due to the structural tableau (and sequent) rules, A 1 o ... o An can legitimately be viewed as the set {A 1 , ... , An}, and I can be viewed as 0.
220 F7:
221
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13 Suppose CT(II1, o~1 ---+C), CT(II 2, o~2 ---+C), and CT(II3, o~3 ---+A V B). IJT =
o o~ 2 ---+ B ---+ B o * o ~ 2 A -+ B o * o ~ 2 A o ~ 2 ---+ B
o~ 1
o~ 1
o~3 U (~1
\{A})
U (~2
II 1
\ {B})---+ C
o~ 2 ---+
*A oB II 2 ·A ---+ *A o B *A-+ *A oB A o *A---+ B *A---+ *A oB *A o *B---+ *A *A-+ *A A-t A
o~3oo(~1 \{A})oo(~2\{B})oi-tC
o~3 o o(~l \{A}) o o(~2 \ {B})---+ C o *I *C o o~ 3 o o(~ 1 \{A}) o o(~ 2 \ {B})---+ *I
o~3---+
(C o *(o~l \{A})) o (C o *(o~2 \ {B} )) Il3 AVB-t(Co*(o~l \{A}))o(Co*(o~2\{B})) A---+ C o *(o~l \{A}) B---+ C o *(o~2 \ {B})
Fll: Suppose CT(II 1, o~ 1 -+ A) and CT(II 2, o~2 ---+ ·A). Then, by the previous construction, there is a tableau II 3 such that CT(II3, o~1 U ~2 -+f). rrr = o(~1
F8:
---+ B o * o ~ 2 II 1 A ---+ B o * o ~ 2 o~ 2 -+
B *A oB II2 A : : > B -+ *A o B A-tA B-tB o~ 2 -+
o~
F9:
Suppose CT(II 1 ,o~---+ B). IJT =
\ {B})-+ ·B
o(~l
B
o~ 1
Ao
U (~2
o(~1 \ {B}) o o(~2 \ {B})---+ ·B \ {B}) o o(~2 \ {B})-+ (B ::::>f) (B ::::>f)-+ ·B o(~l \ {B}) U (~2 \ {B}) oB-+ f (B ::::>f) -+ *B : (B : : > f) o I ---+ *B II 3 (B : : > f) -+ *B o *I B---+ B f---+ *I f-+I
Suppose CT(II 1, o~ 1 ---+A) and CT(II 2 , o~ 2 ---+ A::::> B). IJT =
o~ 1 o o~ 2 -+
\ {B})
\{A}---+ A::::> B
F12:
Suppose CT(II 1 , o~ 1 -+A) and CT(II 2, o~ 2 ---+ •A). Then there is a tableau II 3 such that CT(II 3 , o~ 1 u ~2---+ f). IJT = o(~l
\ {•B})
U (~2
\ {•B})---+ B
o(~1
\ {·B}) o o(~2 \ {·B}) -+ B o(~1 \ {•B}) o o(~2 \ {•B})---+ ·B : : > f ·B : : > f-+ B o(~ 1 \ {·B}) o o(~2 \ {·B}) o ·B ---+ f (--,B : : > f) o I -+ B : (•B::::>f)-tBHI II 3 *B o (·B : : > f) ---+ *I *B o (·B : : > f) -+ I (--,B : : > f) ---+ * * B o I *B-+ ·B f---+ I *B---+ *B B-tB
222
223
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
f -t -.[F]-.A ::::> f -t *[F]-.A ([F]-.A ::::> f) o I-t *[F]-.A [F]-.A ::::> f -t *[F]-.A o *I [F]-.A -t [F]-.A f -t *I
[F]-.A [F]-.A Fl3: Suppose CT(II 1 , o~ -t A). IIT =
o~ o
A 1 -t A rr1
::::>
Fl4: Suppose CT(II 1, o~ -t A),~= {A1, ... , An} =j:. 0. rrr = -t [F]A o(F]~ -t [F] o ~ • o [F]~ -to~
f-ti
o(F]~
[F] o ~ -t [F]A •[F] o ~ -t A [F] o ~ -t •A rr1
• o [F]~ o (*An o ... o *A2) -t A1 • o [F]~ -t A1 o(F]~ -t •A1
and let II5 =
-.[F]-.A -t (F)A *[F]-.A -t (F)A *(F)A -t [F]-.A • * (F)A -t -.A • * (F)A -t *A *(F)A -t • *A * • *A-t (F)A A-t A.
o(~ U ~2)
\ {[F]-.A} -t (F)A {[F]-.A} -t -.[F]-.A U ~2) \ {[F]-.A} -t [F]-.A ::::> f
o(~1 U ~2) \
Suppose CT(II 1,I -t A). Then rrr
=
I-t [F]A ei -t A rr1.
Fl5: Suppose CT(II 1, o~ -t A), ~ = { A1, ... , An} =j:. 0. rrr = o[P]~ -t
[P]A o(P]~ -t [P] o ~ • * o~ -t * o [P]~ o[P]~ -t
[P] o ~ -t [P]A • *A-t *(P] o ~
*• *o ~
A1 o (*An o ... o *A2) -t A1 A1 -t A1 I-t [P]A
= • *A -t *I rr1.
Fl6: Suppose CT(II 1 , o~ 1 -t B) and CT(II 2 , o~ 2 -t -.B). Then there is a tableau II3 such that CT(II 3, o~ 1 U ~ 2 -t f).
II4
F17: Analogous to the previous case. F18: Suppose CT(II 1 , o~ 1 -t B) and CT(II 2 , o~2 -t -.B). Then there is a tableau II 3 such that CT(II3, o~1 U ~2 -t f). Let II4 =
-.(F)A -.(F)A
::::> ::::>
f -t -.[F]-.A f -t *[F]-.A
-.(F)A -t [F]-.A e(-.(F)A) -t -.A •( -.(F)A) -t *A -.(F) A -t • *A *(F)A -t • *A) * • *A-t (F)A A-t A.
[P]A 1 -t * • * o ~ A 1 -to~
Suppose CT(II 1, I-t A). Then rrr
o(~1
II5
f -t *I f -t I
224 Fl9:
CHAPTER 13
FITCH-STYLE NATURAL DEDUCTION
225
Analogous to the previous case.
A-+ ((F)A o A) o (P)A A o *(P)A-+ (F) A o A and let II 3 = A-+ (F) A o A *(F)A o A-+ A A-+ A. •A -+ ((F) A o A) o (F) A *((F) A o A) o •A -+ (P)A Let Il4 = •A-+ (P)A
o~
F20: Suppose CT(II 1, o~ -+A). ITT =
-+ [P](F)A • * (F)A-+ * o 6 *(F)A-+ • * o6 * • * o ~ -+ (F)A Ill
F21: Analogous to the previous case. (Fl): Suppose CT(II1, o6-+ A),~= {A 1, ... ,An} f 0. liT=
A-+A
o[F]~
-+ [F][F]A • o [F]~ -+ [F]A • • o[F]~-+ A • o [F]~-+ •A o[F]~-+ • • A o[F]6-+ •A o[F]~ -+ [F] o ~ [F] o 6 -+ •A
(F)A-+ [F]((F)A V A V (P)A)
* • *A-+ [F]( (F) A V A V (P)A) and let II 5 =
• * • *A-+ (F)A V A V (P)A • * • *A -+ ((F) A V A) o (P)A • * • *A o *(P)A-+ (F) A V A • * • *A o *(P)A-+ (F)A o A • * • *A-+ ((F) A o A) o (P)A II2 Il3 Il4. o~-+
If 6 =
0, then we get rrr
=
[F]I-+ [F][F]A •[F]I -+ [F]A • • [F]I-+ A •[F]I-+ •A [F]I-+ • •A [F]I-+ •A I-+ A.
Then
rrr =
o~-+
[F]((F)A V A V (P)A) (F)A II 5
Ill
(F6): Similar to the previous case.
From this definition it is clear that Lemma 13.2. PN(IT, A, { A1, ... , An})
(F2): Similar to the previous case. o~-+ (F)(P)A * • * • o6-+ (F)(P)A • o ~-+ (P)A
=?
CT(ITT, A1 o ... o An -+ A).
Ill
From closed tableaux to sequent calculus. The mapping ( · )8 from closed tableaux into sequent calculus proofs is as simple as could be: just turn the closed tableaux and their sequents upside down. Obviously,
(F5): Assume CT(II1, o6-+ (F) A). Let II 2 =
Lemma 13.3. CT(IT,A1o ... oAn-+ A)=? Ps(IT 8 ,Alo ... oAn-+ A).
(F3): Suppose CT(II 1 ,o~-+ A). ITT = (F4): Similar to the previous case.
* • *A-+ ((F)A o A) o (P)A * • *A o *(P)A-+ (F) A o A * • *A -+ (F)A o A * • *A o *A-+ (F) A * • *A-+ (F)A A-+A
From sequent calculus to axioms. In order to inductively define the mapping (-)H from sequent calculus proofs into axiomatic proofs, we use the earlier translation T of sequents into tense logical formulas; see Chapter 3. The axiomatic sequents are thus translated into instantiations of the axiom schema A ::J A. Suppose Ps(IT, X -+ Y) and Ps(x,~Y', X1 o ... o Xk -+ Y'). By the induction hypothesis there is
1
226
CHAPTER 13
an axiomatic proof rrH of T(X-+ Y) from
A NEW AXIOMATIZATION OF KT
0.
(x,~Y' )H =
Tl(Xk)
rrH II',
where II' is a certain axiomatic proof of T(X' -+ Y') from T(X -+ Y). Sequent rules with two or three premise sequents are dealt with similarly. We shall not specify II' here; the existence of II' is clear from the possible worlds semantics.
227
While AO and RO are inherited from the underlying classical propositionallogic (CPL), and Al and A2 are just definitions, the remaining axiom schemata and rules exhibit a certain division of labour as far as the tense logical operations are concerned: A5, A6, Rl and R2 indicate that [F] and [P] are normal necessity-type operators, whereas A3 and A4 specify some interaction between the future and the past. The aim of this section is to present an alternate axiomatization of Kt, one which underlines the interaction between the forward and backward oriented modalities, but puts less emphasis on normality as a salient feature of the presentation. To achieve this, the K axiom schemata A5 and A6 are replaced by the following regularity rules: R3
(A=::> B) f- ([F]A
=::>
[F]B)
R4
(A=::> B) f-- ([P]A
=::>
[P]B).
Theorem 13. 5. The Full-Circle Theorem: 1. 2. 3. 4. 5.
R3 is obviously derivable from Rl, A5 and RO; R4 is derivable from R2, A6, and RO. The interaction between the future and the past is tightened by dispensing with the axiom schemata A3 and A4 in favour of the schemata:
PH(II,A,{Al, ... ,An}) implies PN(IIN, A, {A1, ... , An}) implies CT((IIN)T, A1 o ... o An -+A) implies Ps(((IIN)T) 8 ,Al o ... o An-+ A) implies PH((((IIN)T)S)H,A,{Al,···,An})
Proof. By the previous lemmata. Q.E.D.
Corollary 13.6. The tableau calculi presented are strongly complete with respect to their corresponding possible worlds semantics: A1, ... , An p A iff there exists a closed display tableau for A1 o ... o An -+A.
13.2. A NEW AXIOMATIZATION OF Kt The presentation of the minimal normal propositional tense logic Kt as a display calculus suggests an interesting alternate axiomatization of Kt. Consider again the standard axiomatization of Kt from the previous section: AO A1 A3 A5 R1
all tautologies; (F)A -,[F]-.A;
=
A:::::> [F](P)A; [F](A :::::> B) :::::> ([F]A A I- [F]A;
:::::>
[F]B);
RO A2 A4 A6 R2
A, A::::>B I- B; (P)A •[P]-.A; A:::::> [P](F)A; [P](A :::::> B) :::::> ([P]A A I- [P]A.
A7
[F]( (P)A =::>B)
=::>
(A=::> [F]B);
A8
[P]( (F) A
=::>
(A
A9
[P](A
=::>
[F]B)
=::> ( (P)A =::>
B);
AlO
[F](A
=::>
[P]B)
=::>
((F) A
B).
=::>
B)
=::>
[P]B); =::>
The latter schemata are obviously valid on Kripke frames and hence derivable in Kt. We have achieved our aim, if from AO, A7- AlO and RO - R5 we can derive A3- A6. The derivation of A3 and A4 is trivial. We may, for example, use Rl to obtain [F]( (P)A =::> (P)A) and then apply RO to the latter and the following instance of A7: [F]( (P)A =::> (P)A) =::> (A =::> [F](P)A). The derivation of A5 and A6 is less trivial. 54 Table XIX exhibits a detailed derivation of A5 which parallels an analogous derivation of A6.
=
:::::>
[P]B);
54
Note that it would be enough to derive the distribution of [F] and [P] over conjunction: [F](A 1\ B) :::::> ([F]A 1\ [F]B), [P](A 1\ B) :::::> ([P]A 1\ [P]B), cf. [33, Exercise 4.5 (b), p. 122].
228
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24
CHAPTER 13
N4 REDISPLAYED
Table XIX. An axiomatic derivation of A5.
The new axiomatization of Kt was extracted from a certain cutfree proof of the K axiom schema in the modal display calculus. The proof in question is instructive from the point of view of proof search in display logic, but there are shorter cut-free display proofs of K, from which a derivation of A5 and A6 can be obtained. The first fifteen lines of the previous derivation of A5, for example, may be replaced by the following deduction:
A :J (B V ·(A :J B)) [F]A :J [F](B V -.(A :J B)) [P]([F]A :J [F](B V •(A :J B))) [P]([F]A :J [F](B V •(A :J B))) :J :J ( (P)[F]A :J (B V ·(A :J B))) (P)[F]A :J (B V •(A :J B)) (A :J B) :J ( -.(P)[F]A V B) [F](A :J B) :J [F](-.(P)[F]A V B) ([F](A :J B) 1\ [F]A) :J [F](•(P)[F]A V B) [P](([F](A :J B) 1\ [F]A) :J [F](-.(P)[F]A V B)) [P](([F](A :J B) 1\ [F]A) :J [F](-.(P)[F]A V B)) :J :J ((P)([F](A :J B) 1\ [F]A) :J (•(P)[F]A V B)) (P)([F](A :J B) 1\ [F]A) :J (•(P)[F]A V B) (P)[F]A :J (B V -.(P)([F](A :J B) 1\ [F]A)) [F]((P)[F]A :J (B V •(P)([F](A :J B) 1\ [F]A))) [F]((P)[F]A :J (B V •(P)([F](A :J B) 1\ [F]A))) :J :J ([F]A :J [F](B V •(P)([F](A :J B) 1\ [F]A))) [F]A :J [F](B V •(P)([F](A :J B) 1\ [F]A)) ([F](A :J B) 1\ [F]A) :J :J (F](B V •(P)([F](A :J B) 1\ [F]A)) [P](([F](A :J B) 1\ [F]A) :J :J [F](B V •(P)([F](A :J B) 1\ [F]A))) [P](([F](A :J B) 1\ [F]A) :J :J [F](B V •(P)([F](A :J B) 1\ [F]A))) :J :J ( (P) ([F) (A :J B) 1\ [F) A) :J :J (B V •(P)([F](A :J B) 1\ [F]A))) (P)([F](A :J B) 1\ [F]A) :J :J (B V •(P)([F](A :J B) 1\ [F]A)) (P)([F](A :J B) 1\ [F]A) :J B [F]((P)([F](A :J B) 1\ [F]A) :J B) [F]((P)([F](A :J B) 1\ [F]A) :J B) :J (([F](A :J B) 1\ [F]A) :J [F]B) ([F](A :J B) 1\ [F]A) :J [F]B [F](A :J B) :J ([F]A :J [F]B)
CPL
R3, 1 R2, 2 A9 RO, 3, 4 CPL, 5 R3, 6 CPL, 7 R2, 8 A9
1 2 3 4
5 6
([F](A :J B) 1\ [F]A) :J [F]A [P](([F](A :J B) 1\ [F]A) :J [F]A) [P](([F](A :J B) 1\ [F]A) :J [F]A) :J :J ((P)(([F](A :J B) 1\ [F]A) :J A) (P)(([F](A :J B) 1\ [F]A) :J A (A :J B) :J (•(P)([F](A :J B) 1\ [F]A) V B) [F](A :J B) :J (F](-.(P)([F](A :J B) 1\ [F]A) V B)
229
CPL
R2, 1 A9 RO, 2, 3 CPL, 4
R3, 5
RO, 9, 10 CPL,ll
R1, 12 A7
13.3. N 4 REDISPLAYED
RO, 13, 14
In this section we shall restrict the structural language of display logic to containing three structure operations: the constant I, the unary *, and the binary o. Moreover, we shall increase the arity of the sequent arrow. A display sequent now has the form
CPL, 15
R2, 16
A9
where every Xi is a structure built up from the formulas, I, *, and o. Since we intend to assume commutativity of o in both antecedent and succedent position, we shall lay down only 'one-sided' basic structural rules, namely:
RO, 17, 18 CPL, 19 R1, 20
Basic structural rules
A7 RO, 21, 22 CPL, 23
(a) (b) (c) (d)
x1 o Y 1 x2-+ X3l x4 X1 1 x2 o Y-+ x3 1 X4 X1 1 X2-+ x3 o Y 1 X4 x1 1 x2-+ X3 1 x4 o Y
-11- x1 1 x2 o *Y-+ X3 1 x4 -H- X1 o *Y 1 X2-+ X3l x4 -H- x1 1 x2 -+ x3 1 X4 o *Y -11- x1 1 x2-+ X3 o *Y 1 X4
Two four-place sequents are said to be structurally equivalent if they are interderivable by means of rules (a)- (d). These rules make sense under the following translation T of sequents into formulas of the language of
1
230
CHAPTER 13
N4 REDISPLAYED
Table XX. Introduction rules for four-place sequents.
N4 and N3. 55
where 7i (i = 1,2,3,4) is defined as follows: 7l(A) 72(A)
73(A) 74(A)
71(1) 73(1)
72(1) 74(1)
7l(*X) 72( *X) 73(*X) 74(*X) 7l(X 0 Y) 72(X 0 Y) 73(X 0 Y) 74(X 0 Y)
A
"'A t f
72(X) 7l(X) 74(X) 73(X) 71 (X) !\ 71 (Y) '"'"' 71 (X)/\ '"'"' 7l{Y) 73(X) V 73(Y) '"'"'73(X)V '"'"'73(Y).
Here t is defined asp ~hp, for some p E Atom, and f is defined as ""'t. Again, an occurrence of a substructure in a given structure is said to be positive (negative) if it is in the scope of an even (uneven) number of *'s. An o-antecedent (e-antecedent) part of a sequent xl I x2 --7 x3 I x4 is a positive occurrence of a substructure of X 1 or a negative occurrence of a substructure of X 2 (a positive occurrence of a substructure of X2 or a negative occurrence of a substructure of XI). An o-succedent (esuccedent) part of x1 1 x2 --+ x3 1 x4 is a positive occurrence of a substructure of x3 or a negative occurrence of a substructure of x4 (a positive occurrence of a substructure of X4 or a negative occurrence of a substructure of X3).
Theorem 13. 7. (Display Theorem) For every sequent s and every aantecedent (e-antecedent) part X of s there exists a sequent s' structurally equivalent to s such that X is the entire o-antecedent (e-antecedent) of s'; and for every sequent sand every o-succedent (e-succedent) part X of s there exists a sequent s' structurally equivalent to s such that X is the entire o-succedent (e-succedent) of s'. Proof. Analogous to the proof of the Display Theorem in Chapter 3. Q.E.D. 55
231
We reuse the symbol 'o' and the letter 'T' from previous chapters with new meanings, because the context precludes any ambiguity.
x1 1 X2 -+ *A 1 x4 r- X1 1 X2 -+ ~A 1 x4 x1 1 x2-+ x3 1 **A r- X1 1 x2-+ x3 1 "'A *A 1 X2 -+ x3 1 x4 r- "'A 1 Xz -+ X3 1 x4 x1 1 **A-+ x3 1 x4 r- x1 1 "'A-+ x3 1 x4 x 1 X2-+ A 1 x4 Y 1 x2 -+ B 1 X4 rr- x o Y 1 x2 -+ A AB 1 x4 x1 1 X2 -+ x3 1 A oB r- X1 1 X2 -+ x3 1 A AB (-+ 1\ even) A oB 1 Xz -+ X3 1 X4 r- A AB I X2 -+ X3 I X4 (/\-+ odd) (/\-+ even) x1 1 A-+ X3l x x1 1 B -+ x3 1 Y rr- x1 1 A AB -+ x3 1 x o Y x 1 X2-+ A oB 1 x4 r- x1 1 X2-+ A v B 1 x4 (-+V odd) x1 1 x-+ x3 1 A x1 1 Y-+ x3 1 B r(-+V even) r- x1 1 x o Y -+ x3 1 A v B A 1 x2-+ x 1 x4 B 1 X2 -+ Y 1 x4 r(V-+ odd) r- A v B 1 x2 -+ x o Y 1 x4 x1 1 A oB -+ x3 1 x4 r- X1 1 A v B -+ x3 1 x4 (V-+ even) X1 o A 1 Xz-+ B 1 X4 r- X1 I X2-+ A ~h B I X4 (-+~h odd) (-+ ~h even) x 1 X2 -+ A 1 *x3 Y 1 X2 -+ x3 1 B rr- x o Y 1 x2 -+ x3 1 A ~h B xl I Xz-+ A I x4 B 1 x2 -+ x3 1 x4 r(~h-+ odd) r- A ~h B 1 x2 o *x1 -+ x3 1 X4 A o *B 1 *x1 -+ x3 1 x4 r- x1 1 A ~h B -+ x3 I x4 (~h-+ even)
( -+~ odd) ( -+""' even) (odd ~-+) (even "'-+ ) (-+ 1\ odd)
As logical rules we assume appropriate versions of identity and cut, namely: logical rules (odd id) (even id)
r- AII-+AII r- IIA-+IIA
(odd cut) (even cut)
X1 1 X2 -+ A 1 x4 A 1 Xz -+ X3 1 x4 r- X1 I X2 -+ X3 I X4 X1 1 X2 -+ X3 1 A X1 1 A-+ x3 1 X4 r- X1 1 Xz -+ X3 I X4
Moreover, we have the separate, symmetrical, explicit, and segregated introduction rules for the logical operation of N 4 and N3 in Table XX. In addition to the basic structural rules, the logical rules, and the introduction rules, we postulate a tight collection of further structural rules, stated in Table XXI. As a package, these rules allow one to derive plenty of other structural rules which one would like to consider sep-
232
Table XXI. Further structural rules for four-place sequents.
(It)
(r) (A) (P) (C)
(M)
Corollary 13.12. In 4DN4, {i) 1- x1 o *x2 1 1-+ TI(X1)/\ ""'T2(X2) 1 1. {ii) 1- T3(X3)v "-'T4(X4) 11-+ x3 o *x4 11.
Xz --+ X3 I X4 -H-I o X1 1 Xz --+ X3 1 X4 I Xz --+ X3 I X4 -H- X1 1 I o Xz --+ X 3 1 X4 I Xz --+ X3 I X4 -11- X1 1 Xz --+ I o X3 1 X4 I Xz --+ X3 I X4 -11- XI 1 Xz --+ X3 1 I o X4 X o (Y o Z) 1 Xz --+ X3 1 X4 -11- (X o Y) o z 1 X2 --t X 3 1 X 4 xi 1 x o (Y o Z) --+ x3 1 x4 -11- xi 1 (X o Y) o z--+ x3 1 X 4 X 0 y I Xz --+ x3 I x4 1- y 0 X I Xz --t x3 I x4 X1 1 x o Y --+ x3 1 x4 1- xi 1 Y ox --+ x3 1 x4 x ox I Xz --+ X3 I X4 1- X I Xz --+ X3 1 X4 xi 1 x ox --+ x3 1 x4 1- xi 1 x --+ x3 1 x4 X I Xz --+ X3 I X4 1- X o Y 1 Xz --+ X3 1 X4 xi 1 x --+ x3 1 x4 1- xi 1 x o Y --+ x3 1 x4
X1 XI xi XI
233
N 4 REDISPLAYED
CHAPTER 13
I
Proof. (i): We shall omit some obvious steps in the derivation: I I X 2 -+ I I 72 (X 2) 1 1 x2 -+ *T2(X2) 1 1 1 1 x2 -+""'T2(X2) 1 1 x1 11-+ T1(XI) 11 * x2 11 -+""'72{X2) 1I x1 o *x2 1 1-+ TI(XI)/\ ""'72(X2) 1 1 x1 1X2 11-+ TI(X1)/\ "'72(X2) 11 (ii): Similar. Q.E.D.
Theorem 13.13. In 4DN4, 1- xl N4, 1- T(Xl 1 x2-+ x3 1 X4).
arately in a synoptical treatment of substructural constructive logics (see also Chapter 6).
I x2-+ x3 I X4)
I x4
if and only if in
The previous corollary and two applications of (odd cut) give the desired result: from X 1 o *X2 I I-+ T1(XI)/\ "'T2(X2) I I and TI(XI)/\ "-' T2(X2) I I -+ T3(X3)V "-' T4(X4) I I we obtain xl 0 *x2 I I -+ T3(X3)V "'T4(X4) I I. This sequent together with T3(X3)V "'T4(X4) I I -+ x3 0 *x4 I I gives xl 0 *x2 I I -+ x3 0 *x4 I I which is display equivalent to xl I x2-+ x3 I x4. Q.E.D.
in
Proof. (i): By induction on axiomatic proofs in N4. (ii): By induction on proofs in 4DN4. Q.E.D.
Definition 13.14. The sequent system 4DN3 results from 4DN4 by the addition of the purely structural sequent rule
Corollary 13.10. 1- A in N4 iff 1- I I I-+ A I I in 4DN4. In combination with this weak completeness property, the next lemma is crucial for the proof of strong completeness.
1-AIA-+XIY. In 4DN3 we may define f as pi\ ""'P for some p E Atom, and t as ,.,_,f. The additional rule corresponds to the ex falso schema (AI\ ""'A) ::J h B, which in an axiomatic context marks the difference between N4 and N3. Hence,
Lemma 13.11. In 4DN4, (i) 1- x1 1 1-+ T1(XI) 1 I; (ii) 1- 1 1 x2-+ 1 1 T2(X2); (iii) 1- T3(X3) I I-+ x3 I I; and (iv) 1- I I T4(X4) -+I I x4. Proof. By induction on Xi.
-+ x3
Proof. (::::}:)This is Theorem 13.9 (ii). (-{::=:) Suppose in N4, 1- T(Xl I x2 -+ x3 I X4)· By Corollary 13.10, in 4DN4, 1- I I I-+ T(Xl I x2-+ x3 I X4) I I. Therefore in 4DN4,
Definition 13.8. The system 4DN4 is defined as the collection of the said display sequent rules. Theorem 13. 9. (i) If 1- A in N4, then 1- I I I-+ A I I in 4DN4. (ii) If 1- xl I x2-+ x3 I x4 in 4DN4, then 1- T(Xl N4.
I x2
Theorem 13.15. 4DN3 is strongly sound and complete with respect to N3.
Q.E.D.
l
234
CHAPTER 13
We shall not consider generalizing the strong cut-elimination theorem for DL from Chapter 4 to four-place sequents. Note, however, that in this higher-arity setting, the subformula property of the sequent systems for N 4 and N3 follows from cut-elimination by merely inspecting the introduction rules.
BIBLIOGRAPHY
1. 13.4. FUTURE WORK 2. The recent rapid development of modal proof theory is still ongoing. In order to approach a more stable situation, future work will have to focus on clarifying the relative advantages and disadvantages of and interrelations between major formalisms such as higher-arity sequent systems, higher-level sequent systems, relational proof systems, and DL. A translation of systems of indexed sequents into DL is presented in [115]; the relation between the method of hypersequents and DL is dealt with in Chapter 11. Research on display logic still has to address part of the agenda determined in Belnap's seminal paper [16]. In particular, the relation between the cut-elimination property of displayable logics and the interpolation property remains to be investigated. Moreover, in spite of the syntactic and semantic characterization of the properly displayable normal modal and tense logics, further case-studies of displayable nonclassical logics may still be of interest, because displayablity can, for instance, also be achieved via suitable modal or tense logical translations; see Chapters 10, 11 and [195]. Other issues that have not been addressed so far (or at least not in this book) are, for example, a detailed presentation of the relations between DL and type-theoretical grammars (see [118]), a systematic investigation of the practical use of DL for decidability problems, the computational complexity of display calculi, and the possibility of a formulas-as-types analysis of proofs in display sequent systems. Finally, there is plenty of work to be done on implementations of display calculi. An implementation of Gore's display calculus for relation algebra as an Isabelle theory can be found in [38].
3. 4. 5.
6. 7. 8.
9.
10. 11. 12.
13. 14.
15.
16.
M. D'Agostino and M. Mondadori. The Taming of the Cut. Classical Refutations with Analytic Cut. Journal of Logic and Computation, 4, 285-319, 1994. S. Akama. What is Strong Negation in a System? Draft, Teikyo University of Technology, 1995. N. Alechina and M. van Lambalgen. Generalized quantification as substructural logic. Journal of Symbolic Logic, 61, 1006-1044, 1996. A. Almukdad and D. Nelson. Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231-233, 1984. H. Andreka, J. van Benthem and I. Nemeti. Back and Forth Between Modal Logic and Classical Logic. Journal of the Interest Group in Pure and Applied Logics, 3, 685-720, 1995. A. Avron. On modal systems having arithmetical interpretations. Journal of Symbolic Logic, 49, 935-942, 1984. A. Avron. A Constructive Analysis of RM. Journal of Symbolic Logic, 52, 939-951, 1987. A. Avron. Gentzenizing Schroeder-Heister's natural extension of natural deduction. Notre Dame Journal of Formal Logic, 31, 127-135, 1990. A. Avron. Using Hypersequents in Proof Systems for Non-classical Logics. Annals of Mathematics and Artificial Intelligence, 4, 225-248, 1991. A. Avron. Natural3-valued Logics- Characterization and Proof Theory. Journal of Symbolic Logic, 56, 276-294, 1991. A. Avron. Gentzen-Type Systems, Resolution and Tableaux. Journal of Automated Reasoning, 10, 265-281, 1993. A. Avron. The Method of Hypersequents in the Proof Theory of Propositional Non-Classical Logics. In: W. Hodges et al. (eds.), Logic: From Foundations to Applications, Oxford Science Publications, Oxford, 1996, 1-32. E. Earth and E. Krabbe. From Axiom to Dialogue. De Gruyter, Berlin, 1982. J. Barwise. An Introduction to First-Order Logic. In: J. Barwise (ed.), Handbook of Mathematical Logic, North Holland, Amsterdam, 5-46, 1977. N.D. Belnap. A Useful Four-Valued Logic. In: J.M. Dunn and G. Epstein (eds.), Modern Uses of Multiple- Valued Logic, Reidel, Dordrecht, 8-37, 1977. N.D. Belnap. Display Logic. Journal of Philosophical Logic, 11, 375-
235
236
BIBLIOGRAPHY
417, 1982. Reprinted with minor changes as §62 of A.R. Anderson, N.D. Belnap, and J.M. Dunn, Entailment: the logic of relevance and necessity. Vol. 2, Princeton University Press, Princeton, 1992. 17. N.D. Belnap. Linear Logic Displayed. Notre Dame Journal of Formal Logic, 31, 14-25, 1990. 18. N.D. Belnap. The Display Problem. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 79-92, 1996. 19. M. Benevides and T. Maibaum. A Constructive Presentation for the Modal Connective of Necessity (D). Journal of Logic and Computation, 2, 31-50, 1992. 20. J. van Benthem. Modal Foundations of Predicate Logic. Logic Journal of the Interest Group in Pure and Applied Logics, 5, 259-286, 1997. 21. J. van Benthem. Exploring Logical Dynamics. CSLI Publications, Stanford, 1996. 22. K. Bimbo and J.M. Dunn. Two Extensions of the Structurally Free Logic LC. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. 23. G. Birkhoff. Lattice Theory. American Mathematical Society, Providence, 19673 . 24. S. Blamey and L. Humberstone. A Perspective on Modal Sequent Logic. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 27, 763-782, 1991. 25. P. Bonatti. Sequent Calculi for Default and Autoepistemic Logics. In: P. Miglioli et al. (eds.), Theorem Proving with Analytic Tableaux and Related Methods, Proceedings TABLEAUX '96, Lecture Notes in AI 1071, Springer Verlag, Berlin, 127-142, 1996. 26. G. Boolos. Don't eliminate cut. Journal of Philosophical Logic, 13, 373-378, 1984. 27. T. Borghuis. Interpreting Modal Natural Deduction in Type Theory. In: M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 67-102, 1993. 28. T. Borghuis. Coming to Terms with Modal Logic. PhD thesis, Department of Computer Science, University of Eindhoven, 1994. 29. R. Bull and K. Segerberg. Basic Modal Logic. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. II, Extensions of Classical Logic, Reidel, Dordrecht, 1-88, 1984. 30. J. Burgess. Basic Tense Logic. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. II, Extensions of Classical Logic, Reidel, Dordrecht, 89-133, 1984. 31. C. Cerrato. Modal sequents for normal modal logics. Mathematical Logic Quarterly, 39, 231-240, 1993. 32. C. Cerrato. Modal sequents. In: H. Wansing (ed.), Proof Theory of Modal Logic, 141-166, Kluwer Academic Publishers, Dordrecht, 1996. 33. B. Chellas. Modal Logic: An Introduction. Cambridge University Press, Cambridge, 1980.
BIBLIOGRAPHY 34.
35.
36. 37.
38.
39. 40. 41. 42. 43. 44.
45. 46. 47.
48. 49. 50. 51. 52.
237
M. Clarke. Intuitionistic Non-Monotonic Reasoning - further results. In: Y. Kodratoff (ed.), ECAI 88. Proc. 8th European Conference on Artificial Intelligence, Pitman Publishing, London, 525-527, 1988. M. Clarke and D. Gabbay. An Intuitionistic Basis for Non-Monotonic Reasoning. In: P. Smets et al. (eds.), Non-Standard Logics for Automated Reasoning, Academic Press, London, 163-178, 1988. G. Corsi. Weak Logics with Strict Implication. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 33, 389-406, 1987. G. Crocco and L. Farinas del Cerro. Structure, Consequence Relation and Logic. In: D. Gabbay (ed.), What is a Logical System?, Oxford University Press, Oxford, 239-259, 1994. J .E. Dawson and R. Gore. A Mechanised Proof System for Relation Algebra using Dispaly Logic. Australian National University, Canberra, 1998. K. Dosen. Sequent-systems for modal logic. Journal of Symbolic Logic, 50, 149-159, 1985. K. Dosen. Intuitionistic Double Negation as a Necessity Operator. Publications de L 'Institute Mathematique (Belgrade), 49, 15-20, 1984. K. Dosen. Sequent systems and groupoid models I. Studia Logica, 47, 353-389, 1988. K. Dosen. Sequent systems and groupoid models II. Studia Logica, 48, 41-65, 1989. K. Dosen. Logical Constants as Punctuation Marks. Notre Dame Journal of Formal Logic, 30, 362-381, 1989. K. Dosen. Ancestral Kripke Models and Nonhereditary Kripke Models for the Heyting Calculus. Notre Dame Journal of Formal Logic, 32, 580-597, 1991. K. Dosen. Modal Logic as Metalogic. Journal of Logic, Language and Information, 1, 173-201, 1992. K. Dosen. Rudimentary Kripke models for the intuitionistic propositional calculus. Annals of Pure and Applied Logic, 62, 21-49, 1993. K. Dosen. Modal Translations in K and D. In: M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 103127, 1993. K. Dosen and P. Schroeder-Heister (eds.), Substructural Logics, Clarendon Press, Oxford, 1993. A. Dragalin. Mathematical Intuitionism. Introduction to Proof Theory. American Mathematical Society, Providence,1988. M. Dummett. A Prepositional Calculus with a Denumerable Matrix. Journal of Symbolic Logic, 24, 96-107, 1959. J. M. Dunn. A 'Gentzen' System for Positive Relevant Implication (Abstract). Journal of Symbolic Logic, 38, 356-357, 1973. J. M. Dunn. Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation and Various Logical Operations. In: J. van Eijk (ed.), Logics in AI, Proc. European Workshop
238
53.
54.
55.
56. 57.
58.
59. 60. 61. 62. 63. 64.
65.
66. 67.
68. 69.
BIBLIOGRAPHY JELIA 1990, Lecture Notes in Computer Science 478, 31-51, SpringerVerlag, Berlin, 1990. J.M. Dunn. Partial-Gaggles Applied to Logics with Restricted Structural Rules. In: K. Dosen and and P. Schroeder-Heister (eds.), Substructural Logics, Clarendon Press, Oxford, 63-108, 1993. J .M. Dunn. Perp and star: two treatments of negation. In: J. Tomberlin (ed.), Philosophical Perspectives (Philosophy of Language and Logic) 7, Ridgeview, Atascadero, 331-357, 1994. J.M. Dunn. Gaggle Theory Applied to Modal, Intuitionistic, and Relevance Logics. In: I. Max and W. Stelzner (eds.), Logik und Mathematik: Frege-Kolloquium 1993, de Gruyter, Berlin, 335-368, 1995. J.M. Dunn. Generalized Ortho Negation. In: H. Wansing (ed.), Negation. A Notion in Focus, de Gruyter, Berlin, 3-26, 1996. J.M. Dunn. A Comparison of Various Model-theoretic Treatments of Negation: A History of Formal Negation. In: D. Gabbay and H. Wansing (eds.), What is Negation, Kluwer Academic Publishers, Dordrecht, 2351, 1998. J.M. Dunn and R. Meyer. Combinators and Structurally Free Logic. Logic Journal of the Interest Group in Pure and Applied Logics, 5, 505-537, 1997. H. Enderton. A Mathematical Introduction to Logic. Academic Press, New York, 1972. M. Finger. Towards Structurally-free Theorem Proving. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. M. Finger and D. Gabbay. Adding a Temporal Dimension to a Logic System. Journal of Logic, Language and Information, 1, 203-233, 1992. M. Fitting. Tableau Methods of Proof for Modal Logics. Notre Dame Journal of Formal Logic, 13, 237-247, 1972. M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht, 1993. M. Fitting. Basic Modal Logic. In: D. Gabbay et al. (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1, Logical Foundations. Oxford University Press, Oxford, 365-448, 1993. D. Gabbay. Intuitionistic Basis for Non-Monotonic Logic. In: Proc. 6th Conference on Automated Deduction., Lecture Notes in Computer Science 138, Springer-Verlag, Berlin, 260-273, 1982. D. Gabbay. What is Negation in a System?. In: F. Drake and J. Truss (eds.), Logic Colloquium '86, Elsevier, Amsterdam, 95-112, 1988. D. Gabbay. A general theory of structured consequence relations. In: K. Dosen and P. Schroeder-Heister (eds.), Substructural Logics, Clarendon Press, Oxford, 109-151, 1993. D. Gabbay. Labelled Deductive Systems: Volume 1. Foundations. Oxford University Press, Oxford, 1996. D. Gabbay and H. Wansing. What is Negation in a System? Negation in
BIBLIOGRAPHY
70.
71. 72. 73. 74.
75. 76.
77.
78.
79.
80. 81.
82. 83.
84. 85.
239
Structured Consequence Relations. In: A. FUhrmann and H. Rott (eds.), Logic, Action and Information, de Gruyter, Berlin, 328-350, 1996. G. Gentzen. Investigations into Logical Deduction. In: M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North Holland, ~m sterdam, 1969, 68-131. English translation of: Untersuchungen uber das logische SchlieBen. Mathematische Zeitschrift, 39, I 176-210, II 405-431, 1934. L. Goble. Gentzen systems for modal logics. Notre Dame Journal of Formal Logic, 15, 455-461, 1974. R. Goldblatt. Topoi. The Categorical Analysis of Logic. North-Holland, Amsterdam, 1979. R. Goldblatt. Logics of Time and Computation. CSLI Publications, Stanford, 2nd edition, 1992. R. Gore. Cut-Free Sequent and Tableau Systems for Propositional Normal Modal Logics. PhD thesis, University of Cambridge Computer Laboratory, Technical Report No. 257, 1992. R. Gore. Intuitionistic Logic Redisplayed. Technical Report TR-ARP1-1995, Australian National University, 1995. . R. Gore. A uniform display system for intuitionistic and dual mtuitionistic logic. Technical Report TR-SRS-2-95, Australian National University, 1995. R. Gore. On the Completeness of Classical Modal Display Logic. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 137-140, 1996. R. Gore. Cut-free Dispaly Calculi for Relation Algebras. In: D. van Dalen and M. Bezem (eds.), CSL96: Selected Papers of the Annual Conference of the Association for Computer Science Logic, Lecture Notes in Computer Science 1258, Springer-Verlag, Berlin, 198-210, 1997. R. Gore. Tableau Methods for Modal and Temporal Logics. To appear in: M. D'Agostino, D. Gabbay, R. Hiihnle, and J. Posegga (eds.), Handbook of Tableau Methods, Kluwer Academic Publishers, Dordrecht, 1998. R. Gore.Substructural Logics on Display. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. R. Gore. Gaggles, Gentzen and Galois: Cut-free Display Calculi and RelationM Semantics for Algebraizable Logics. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. S. Gottwald. Mehrwertige Logik. Akademie-Verlag, Berlin, 1989. V.N. Grishin. A nonstandard logic and its application to set theory (in Russian). In: Studies in Formalized Languages and Nonclassical Logics (in Russian), Moscow, Nauka, 135-171, 1974. . . . Y. Gurevich. Intuitionistic Logic with Strong NegatiOn. Studw Logtca, 36, 49-59, 1977. I. Hacking, What is Logic?, The Journal of Philosophy 76 (1979), 285-
240
86. 87. 88. 89. 90.
91. 92.
93.
94.
9S. 96.
97.
98.
99. 100. 101. 102. 103.
BIBLIOGRAPHY 319. Reprinted in: D. Gabbay (ed.), What is a Logical System?, Oxford University Press, Oxford, 1994, 1-33. J.R. Hindley and J.P. Seldin. Introduction to Combinators and .ACalculus. Cambridge University Press, Cambridge, 1986. G. Hughes and M. Cresswell. An Introduction to Modal Logic. Methuen, London, 1968. A. Indrzejczak. Generalised Sequent Calculus for Propositional Modal Logics. Logica Trianguli, 1, 1S-32, 1997. A. Indrzejczak. Cut-free Double Sequent Calculus for SS. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. P. Jackson and H. Reichgelt. A general proof method for first-order modal logic. In: Proc. 9th International Joint Conference on Artificial Intelligence, 942-944, 1987. S. Kanger. Provability in Logic. Almqvist & Wiksell, Stockholm, 19S7. M. Kracht. Power and Weakness of the Modal Display Calculus. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 93-121, 1996. S. Kripke. Semantical analysis of modal logic I: Normal modal propositional calculi. Zeitschrift fiir mathematische Logik und Grundlagen der Mathematik, 9, 67-96, 1963. S. Kripke. Semantical analysis of intuitionistic logic I. In: J. Crossley and M. Dummett (eds.), Formal Systems and Recursive Functions, North-Holland, Amsterdam, 92-129, 196S. S. Kuhn. Quantifiers as Modal Operators. Studia Logica, 39, 14S-1S8, 1980. F. von Kutschera. Die Vollstandigkeit des Operatorensystems { ....,, 1\, V, :::>} fur die intuitionistische Aussagenlogik im Rahmen der Gentzensemantik. Archiv fiir Mathematische Logik und Grundlagenforschung, 11, 3-16, 1968. F. von Kutschera. Ein verallgemeinerter Widerlegungsbegriff fur Gentzenkalkule. Archiv fiir Mathematische Logik und Grundlagenforschung, 12, 104-118, 1969. M. van Lambalgen. Natural deduction for generalized quantifiers. In: J. van der Does and J. van Eijck (eds.), quantifiers, Logic, and Language, CSLI Publications, Stanford, 22S-236, 1996. J. Lambek. The mathematics of sentence structure. American Mathematical Monthly, 65, 1S4-170, 19S8. D. Leivant. On the proof theory of the modal logic for arithmetic provability. Journal of Symbolic Logic, 46, S31-S38, 1981. W. Lenzen. Necessary Conditions for Negation Operators. In: H. Wansing (ed.), Negation. A Notion in Focus, de Gruyter, Berlin, 37-S8, 1996. E.G.K. L6pez-Escobar. Refutability and Elementary Number Theory, Indagationes Mathematicae, 34, 362-374, 1972. W. Lukaszewicz. Non-Monotonic Reasoning. Formalization of Commonsense Reasoning. Ellis Horwood, Chichester, 1990.
BIBLIOGRAPHY 104. 10S.
106.
107. 108. 109. 110. 111. 112.
113.
114. 11S. 116.
117.
118.
119. 120. 121.
241
J. Lyons. Semantics, Vol. I, Cambridge University Press, Cambridge, 1977. W. MacCaull. Relational Proof System for Linear and Other Substructural Logics. Journal of the Interest Group in Pure and Applied Logics, 5, 673-697, 1997. S. Martini and A. Masini. A computational interpretation of modal proofs. In: H. Wansing (ed.), Proof Theory of Modal Logic, 213-241, Kluwer Academic Publishers, Dordrecht, 1996. M. Marx and Y. Venema. A Modal Logic of Relations. Technical Report IR-396, Vrije Universiteit Amsterdam, 199S. A. Masini. 2-Sequent calculus: a proof theory of modalities. Annals of Pure and Applied Logic, 58, 229-246, 1992. D. McDermott and J. Doyle. Non-Monotonic Logic I. Journal of Artificial Intelligence, 13, 41-72, 1980. W. Meyer-Viol. Instantial Logic. PhD thesis, University of Amsterdam, Institute for Logic, Language and Computation, 199S. G. Mints (Mine). Cut-elimimation theorem in relevant logics. The Journal of Soviet Mathematics, 6, 422-428, 1976. G. Mints. Cut-free calculi of the SS type. Studies in constructive mathematics and mathematical logic. Part II. Seminars in Mathematics, 8, 79-82, 1970. G. Mints. Gentzen-type systems and resolution rules. Part I. Propositional Logic. In: P. Martin-Lofand G. Mints (eds.), COLOG-88, Lecture Notes in Computer Science 417, Springer-Verlag, Berlin, 198-231, 1990. G. Mints. A Short Introduction to Modal Logic. CSLI Lecture Notes 30, CSLI Publications, Stanford, 1992. G. Mints. Indexed systems of sequents and cut-elimination. Journal of Philosophical Logic, 26, 671-696, 1997. G. Mints, V. Orevkov, and T. Tammet. Transfer of Sequent Calculus Strategies to Resolution for S4. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 17-31, 1996. R. Montague. Logical Necessity, Physical Necessity, Ethics and Quantifiers. Inquiry, 2S9-269, 1960. Reprinted in: R. Thomason (ed.), Formal Philosophy. Selected Papers of Richard Montague, Yale University Press, New Haven, 71-83, 1974. M. Moortgat. Categorial Type Logics. In: A. ter Meulen and J. van Benthem (eds.), Handbook of Logic and Language, North-Holland, Amsterdam, 93-177, 1997. D. Nelson. Constructible falsity. Journal of Symbolic Logic, 14, 16-26, 1949. I. Nemeti. Algebraizations of Quantifier Logics, an introductory Overview. Studia Logica, 50, 48S-S69, 1991. H. Nishimura. A Study of Some Tense Logics by Gentzen's Sequential Method. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 16, 343-3S3, 1980.
242
122. 123. 124.
125. 126.
127.
128.
129.
130.
131.
132.
133.
134. 135. 136.
BIBLIOGRAPHY
BIBLIOGRAPHY M. Ohnishi and K. Matsumoto. Gentzen Method in Modal Calculi. Osaka Mathematical Journal, 9, 113-130, 1957. M. Ohnishi and K. Matsumoto. Gentzen Method in Modal Calculi, II. Osaka Mathematical Journal, 11, 115-120, 1959. E. Orlowska. Relational interpretation of modal logics. In: H. Andreka, D. Monk and I. Nemeti (eds.), Algebraic Logic. Colloquia Mathematica Societatis Janos Bolyai 54, North Holland, Amsterdam, 443-471, 1988. E. Orlowska. Relational proof systems for relevant logics. Journal of Symbolic Logic, 57, 1425-1440, 1992. E. Orlowska. Relational Proof Systems for Modal Logics. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers Dordrecht, 55-77, 1996. ' ~· Pear.ce. n Reasons for Choosing N. Technical report 14/91, Gruppe fur Log1k, Wissenstheorie und Information, Free University of Berlin 1991. ' D. Pearce. Reasoning with Negative Information 11: Hard Negation, Strong Negation and Logic Programs. In: D. Pearce and H. Wansing (eds.), Nonclassical Logics and Information Processing Lecture Notes in AI 619, Springer-Verlag, Berlin, 63-79, 1992. ' D. Pearce. Answer Sets and Constructive Logic, 11: Extended Logic Programs and Related Non-monotonic Formalisms. In: L. Pereira and A. Nerode (eds.), Logic Programming and Non-Monotonic Reasoning, MIT Press, Cambridge (Massachusetts), 457-475, 1993. D. Pearce and G. Wagner. Reasoning with Negative Information I: Strong Negation in Logic Programs. In L. Haaparanta et al. (eds.), Language, Knowledge, and Intentionality, (Acta Philosophica Fennica 49), Helsinki, 430-453, 1990. D. Pearce and G. Wagner. Logic Programming with Strong Negation. In: P. Schroeder-Heister (ed.), Proc. Workshop on Extensions of Logic Programming, Lecture Notes in AI 475, Springer-Verlag, Berlin, 311326, 1990. A. Pliuskeviciene. Cut-free Calculus for Modal Logics Containing the Barcan Ax~om. To appear in: M. Kracht, M. de Rijke, H. Wansing, and M. ZakharJaschev (eds.), Advances in Modal Logic '96, CSLI Publications, Stanford, 1998. A. PliuskeviCiene. Extended Disjunction and Existence Properties for Some Predicate Modal Logics. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. G. Pottinger. Uniform, cut-free formulations ofT, S4 and S5 (Abstract). Journal of Symbolic Logic, 48, 900-901, 1983. A. Prijatelj. Bounded Contraction and Gentzen-style Formulations of Lukasiewicz Logics. Studia Logica, 57, 437-456, 1996. W. Rautenberg. Klassische und nichtklassische Aussagenlogik. Vieweg, Braunschweig, 1979.
137. 138. 139. 140.
141. 142. 143. 144. 145.
146.
147. 148. 149.
150. 151. 152. 153.
154.
243
W. Rautenberg. Modal tableau calculi and interpolation. Journal of Philosophical Logic, 12, 403-423, 1983, 14, 229, 1985. G. Restall. Subintuitionistic Logics. Notre Dame Journal of Formal Logic, 35, 116-129, 1994. G. Rest all. A Useful Substructural Logic. Bulletin of the Interest Group in Pure and Applied Logic, 2, 137-148, 1994. G. Restall. Displaying and Deciding Substructural Logics 1: Logics with Contraposition. Technical Report TR-ARP-11-94, Australian National University, Canberra, 1994. To appear in: Journal of Philosophical Logic, 27, 1998. G. Restall. Display Logic and Gaggle Theory. Reports on Mathematical Logic, 29, 133-146, 1995 (published 1996). G. Restall. Combining Possibilities and Negations. Studia Logica, 59, 121-141, 1997. M. de Rijke. Extending Modal Logic. PhD thesis, University of Amsterdam, Institute for Logic, Language and Computation, 1993. D. Roorda. Resource Logics. PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, 1991. D. Roorda. Dyadic Modalities and Lambek Calculus. In: M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 215-253, 1993. M. Ryan, P.-Y. Schobbens, and 0. Rodrigues. Counterfactuals and Updates as Inverse Modalities. In: Y. Shoam (ed.), TARK 'g6. Proceedings Theoretical Aspects of Rationality and Knowldege, Morgan Kaufmann, San Francisco, 163-173, 1996. G. Sambin and S. Valentini. The modal logic of provability. The sequential approach. Journal of Philosophical Logic, 11, 311-342, 1982. G. Sambin, G. Battilotti, and C. Faggian. Basic logic: reflection, symmetry, visibility. To appear in: Journal of Symbolic Logic. M. Sato. A Study of Kripke-type Models for Some Modal Logics by Gentzen's Sequential Method. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 13, 381-468, 1977. M. Sato. A cut-free Gentzen-type system for the modal logic S5. Journal of Symbolic Logic, 45, 67-84, 1980. P. Schroeder-Heister. A Natural Extension of Natural Deduction. Journal of Symbolic Logic 49, 1284-1300, 1984. P. Schroeder-Heister. Uniform proof-theoretic semantics for logical constants (Abstract). Journal of Symbolic Logic 56, 1142, 1991. K. Schroter. Methoden zur Axiomatisierung beliebiger Aussagen- und Priidikatenkalkiile. Zeitschrijt fur mathematische Logik und Grundlagen der Mathematik 1, 214-251, 1955. 0. Serebriannikov. Gentzen's Hauptsatz for Modal Logic with Quantifiers. In: I. Niiniluoto and E. Saarinen (eds.), Intensional Logic: Theory and Applications (= Acta Philosophica Fennica, 35), 79-88, 1982.
244 155. 156.
157. 158. 159. 160. 161. 162.
163. 164. 165.
166. 167. 168. 169. 170.
171. 172. 173. 174.
BIBLIOGRAPHY
BIBLIOGRAPHY T. Shimura. Cut-Free Systems for the Modal Logic S4.3 and S4.3Grz. Reports on Mathematical Logic, 25, 57-73, 1991. G. Shvarts. Gentzen style systems for K45 and K45D. In: A. Meyer and M. Taitslin (eds.), Logic at Batik '89, Lecture Notes in Computer Science 363, Springer-Verlag, Berlin, 245-256, 1989. J. Slupecki, G. Bryll and V. Wybraniec-Skardowska. The theory of rejected propositions I. Studia Logica 29, 75-119, 1971. J. Slupecki, G. Bryll and V. Wybraniec-Skardowska. The theory of rejected propositions 11. Studia Logica 30, 97-142, 1972. R. Smullyan. Analytic cut. Journal of Symolic Logic 33, 560-564, 1968. 0. Sonobo. A Gentzen-type Formulation of Some Intermediate Propositional Logics. Journal of Tsuda College, 7, 7-14, 1975. R. Sylvan. Intuitionistic Logic - Subsystem of, Extension of, or Rival to, Classical Logic. Philosophical Studies, 53, 147-151, 1988. M. Takano. Subformula property as a substitute for cut-elimination in modal propositional logics. Mathematica Japonica, 37, 1129-1145, 1992. N. Tennant. Perfect Validity, Entailment and Paraconsistency. Studia Logica, 43, 179-198, 1984. N. Tennant. Natural Deduction and Sequent Calculus for Intuitionistic Relevant Logic. Journal of Symbolic Logic, 52, 665-680, 1987. N. Tennant. Negation, Absurdity and Contrariety. In: D. Gabbay and H. Wansing (eds.), What is Negation?, Kluwer Academic Publishers, Dordrecht, 199-222, 1998. A. Troelstra. Arend Heyting and his Contribution to Intuitionism. Nieuw Archief voor Wiskunde, 24, 1-23, 1981. A. Troelstra and D. van Dalen. Constructivism in Mathematics, Vol. I, North-Holland, Amsterdam, 1988. A. Troelstra. Lectures on Linear Logic. CSLI Lecture Notes 29, CSLI Publications, Stanford, 1992. R. Thrner. Logics for Artificial Intelligence. Ellis Horwood , Chichester , 1984. A. Urquhart. Many-valued Logic. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. Ill, Reidel, Dordrecht, 71116, 1986. Y. Venema. Many-Dimensional Modal Logic. PhD thesis, University of Amsterdam, Mathematical Institutue, 1991. Y. Venema. Meeting strength in substructurallogics. Studia Logica, 55, 3-32, 1995. Y. Venema. Cylindric Modal Logic. Journal of Symbolic Logic, 60, 591-623, 1995. Y. Venema. A modal logic of quantification and substitution. In: L. Czirmaz, D. Gabbay and M. de Rijke (eds.), Logic Colloquium '92, 293-309, CSLI Publications, Stanford, 1995.
175. 176. 177. 178. 179.
180. 181. 182.
183. 184. 185. 186. 187.
188. 189.
190.
191. 192.
193.
194.
245
G. Wagner. Logic Programming with Strong Negation and Inexact Predicates. Journal of Logic and Computation, 1, 835-859, 1991. G. Wagner. Vivid Logic. Know ledge-Based Reasoning with Two Kinds of Negation. Lecture Notes in AI 764, Springer-Verlag, Berlin, 1994. H. Wang. A Survey of Mathematical Logic. Science Press, Peking, 1962. H. Wansing. Functional Completeness for Subsystems of lntuitionistic Propositional Logic. Journal of Philosophical Logic, 22, 303-321, 1993. H. Wansing. Informational Interpretation of Substructural Propositional Logics. Journal of Logic, Language and Information, 2, 285-308, 1993. H. Wansing. The Logic of Information Structures. Lecture Notes in AI 681, Springer-Verlag, Berlin, 1993. H. Wansing. Sequent Calculi for Normal Modal Propositional Logics. Journal of Logic and Computation, 4, 125-142, 1994. H. Wansing. Strong Cut-Elimination for Constant Domain First-Order S5. Journal of the Interest Group in Pure and Applied Logics, 3, 797810, 1995. H. Wansing. Tarskian Structured Consequence Relations and Functional Completeness. Mathematical Logic Quarterly, 41, 73-92, 1995. H. Wansing. Semantics-based Nonmonotonic Inference. Notre Dame Journal of Formal Logic, 36, 44-54, 1995. H. Wansing. Strong cut-elimination in Display Logic. Reports on Mathematical Logic, 29, 117-131, 1995 (published 1996). H. Wansing. A new axiomatization of Kt. Bulletin of the Section of Logic, 25, 60-62, 1996. H. Wansing. A Proof-theoretic Proof of Functional Completeness for Many Modal and Tense Logics. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 123-136, 1996. H. Wansing (ed.). Proof Theory of Modal Logic, Dordrecht, Kluwer Academic Publishers, 1996. H. Wansing. A Full-Circle Theorem for Simple Tense Logic. In: M. de Rijke (ed.), Advances in Intensional Logic, Kluwer Academic Publishers, Dordrecht, 173-193, 1997. H. Wansing. Displaying as Temporalizing. Sequent Systems for Subintuitionistic Logics. In: S. Akama (ed.), Logic, Language and Computation, Kluwer Academic Publishers, Dordrecht, 159-178, 1997. H. Wansing. Modal tableaux based on residuation. Journal of Logic and Computation, 7, 719-731, 1997. H. Wansing. Negation as Falsity: a Reply to Tennant. In: D. Gabbay and H. Wansing (eds.), What is Negation?, Kluwer Academic Publishers, Dordrecht, 223-238, 1998. H. Wansing. Translation of Hypersequents into Display Sequents. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. H. Wansing. Predicate Logics on Display. To appear in: Studia Logica.
246 195. 196. 197. 198. 199. 200. 201.
BIBLIOGRAPHY H. Wansing. Displaying the Modal Logic of Consistency. To appear in: Journal of Symbolic Logic. H. Wansing {ed.). Two special issues on generalized sequent systems. Logic Journal of the Interest Group in Pure and Applied Logics, 1998. L. Wittgenstein. Philosophical Investigations. Blackwell, Oxford, 1953. R. W6jcicki. Dual Counterparts of Consequence Operations. Bulletin of the Section of Logic, 2, 54-57, 1973. R. Zach. Proof Theory of Finite-valued Logics. Diplomarbeit, Institut fiir Computersprachen, Technische Universitiit Wien, 1993. J.J. Zeman. Modal Logic. The Lewis-Modal Systems. Clarenden Press, Oxford, 1973. J. Zucker and R. Tragesser. The adequacy problem for inferential logic. Journal of Philosophical Logic, 7, 501-516, 1978.
INDEX
2-sequent 17 3 127, 144, 147 4DN3 233 4DN4 232f. (*-cut) 136 (*-reflexivity) 136, 143 A
adjoint functor 31, 191 Akama, S. 136 Alechina, N. 201 ancestrality 156 Andreka, H. 200 antiaxiom 115 anti-realism 7 ant-regular 56 A-scheme Avron, A. 6, 22 ff., 171 f., 174, 182, 184, 186 f.
B B 67 BCK 193 Barwise, J. 1 Basic Logic 187 Belnap, N.D. 1, 27, 34, 36 f., 40, 42, 47 f., 67, 82, 142, 173, 176, 189 f., 204 ff., 234 Benevides, M. 24 van Benthem, J.F.A.K. 14, 189 f., 193 f., 196, 200 BHK interpretation 129 ff. Birkhoff, G. 32 Blarney, S. 19 ff., 138 Bochvar, D.A. 150 Borghuis, T. 25, 211 f., 214
c C3 150, 153 f. C3d 150, 153 f. C4154
CG 145 f., 148, 153 f. CT 145, 147 f., 153 CLL 51, 55 f. CPL 5, 9, 11, 16, 18, 24, 40, 43, 46, 141 f., 213 f., 227 canonical model 166, 169 Cerrato, C. 15 f. Clarke, M. 153 classical logic 2 clause 79 Kowalski form of 79, 212 component 22, 171 compositionality 8 connective combining vs. internal 37f., 174 f. consequence relation dual138 inverse 115, 138 single-conclusion 136 *-refutation 136 structured 103 f. conservativity 13, 23, 40, 45, 55, 61, 67, 109, 118, 164, 183, 195 cons-regular 55 constituent 47, 88 constructible falsity 119 context 34 context-sensitivity 29, 33, 43, 175 contraction-elimination 83 contraction mapping 104 f. contraposition 134, 146, 149 contrapositive 33, 119 contrariety 135 correspondence 59, 78, 83, 165, 194 f., 203 f., 206, 213 corresponding structure 68 Corsi, G. 157 Curry, H.B. 4 cut4, 11,40, 136,138,172,178,204, 231
248
INDEX
INDEX
analytic 12 -elimination 12 f., 16 f., 21, 40, 82, 88, 173, 234 strong 48, 51 ff., 57, 87, 161, 187, 204 -formula 12, 48, 89 principal13, 67, 89, 131 Cut for M 104 -degree 117 elimination 105, 116 f. D DCPL 40, 46, 50, 191 DIPL 158, 160 DK 40, 43 ff., 55, 60 f., 191 DKf 76, 78 ff., 83 ff. DKt 40, 43 ff., 55, 57, 60 f., 67, 71, 73, 161 DKt(K(a)) 161 ff. DKt(K(a)) 161 ff. DK(a) 159, 161 ff. DKFOL 191 ff., 202, 206 DKFOL* 198 f., 210 DKFOL** 202 DKFOLmin 193 f. DKFOL;',.in 206 f. DKFOL;;in 206 f. DKtFOL 198 ff. DKtFOL* 210 DK U KtFOL* 198 f., 210 DL3 179 f., 184 f. DLC 183, 186 DLC U DS4.3t 183, 186 DPDL- 78 ff., 83 ff. DS4.3 183 DS4.3t 183 DS5 181, 183, 185 decidability 12, 64, 83, 193 Deduction Theorem 19, 105, 145 f. definedness ordering 147 direct propositionallogic 115 extended 115 disjunction property 119 displayable 56, 73 properly 48
display equivalence 29, 175, 229 display logic 23, 27 ff., 64 display property 36 f. Display Theorem 34 ff., 79, 176, 230 disproof interpretation 131 f. Dosen, K. 10, 16 f., 152, 156 f., 159 Dosen Principle 10 f., 14 f., 19, 21, 23, 25, 62 f. Doyle, J. 145, 150, 153 Dummett, M. 135, 172, 174, 185 Dunn, J.M. 27, 30, 32 f., 191 E e-antecedent / succedent 229 eigenconstant 90 eigenlabel 90 (even cut) 138 (even reflexivity) 138 explicitness (of a rule) 8, 11, 24, 65, 172 extension (of a label) 98
F FOL 189, 193 ff. Feys, R. 4 Finger, M. 14, 159 Fitting, M. 87, 98, 100, 206 formula Barcan 189, 205 converse 205 congruent 47, 55, 88 monolithic 160 primitive 58 f. dually 62 principal 47, 82, 88 Sahlqvist 195 sort-n 201 unwanted 128, 140 F-tableau 100 full-circle 211 f. Full Circle Theorem 226 functional completeness 65, 71 ff., 112 ff., 124 ff.
G Gt 107 f., 110 f., 113 Gt" 119 ff., 124 ff. GL3 173, 184 f. GLC 174, 186 GS5 172 ff., 185 Gabbay, D.M. 1, 13 ff., 87, 103, 105, 127 f., 134, 139 f., 142 ff., 150 ff., 159 gaggle 63 Galois connection 30, 191 dual 30, 191 GA-scheme 36 generality 13, 23 Gentzen, G. 7, 10 Gentzen sequent 4 Gentzen term 4, 27, 171 Gore, R. 14, 63, 158, 160, 181, 187, 234 Grefe, C. 64 G-tableau 64 H HL3 179 f., 185 Hacking, I. 172 height of a proof 53, 116 f. of a tableau 97 higher-arity proof system 3, 18 ff. higher-dimensional proof system 3, 17 f. higher-level proof system 3, 16 f., 66 Hilbert system 1, 11, 64 holism 8, 36, 176 Humberstone, L.l. 19 ff., 138 hypersequent 22, 171 hypersequent system 3, 22 ff., 171 ff.
IPL 127, 141 f., 144 f., 153 ff., 155 ff. IR 127 f., 133, 135, 140 ff. identification 201 identity (reflexivity) 4, 40, 136, 138, 176, 231
249
Identity 104 implication intuitionistic 128, 206 intuitionistic relevant 134 material 32 strict 157 lndrzejczak, A. 173 inductive proof 52 inductive tableau 95 information ordering 147 interpolation 187, 234 K K 5, 10 f., 16, 18, 24, 40, 43 ff., 59 f., 73, 98, 143, 187, 191, 195, 227 Kf64, 75 Kt 6, 12, 21, 33, 40, 43 ff., 57 f., 60, 62, 71 ff., 187, 200, 213 f., 226 ff. K4t 6, 12, 21 K4 5 f. K4B 6, 12 K45 6 KB 6, 12,98 KT 5, 98 KTB 6, 12, 98 KGrz 6 KD 11, 18 KDB 6, 12,98 KD4 5, 98 KD45 5 K(a) 157 ff., 165 ff. Kt(K(a)) 159 Kt(K(a))' 165 ff. KFOL 191, 193, 199 KFOL * 196 ff., 209 KtFOL 200, 207 KtFOL* 207, 209 KtFOL U KFOL* 207 Kanger, S. 88, 173 Kanger-style calculi 3 Kleene, S.C. 150 Kracht, M. 27, 35 f., 41, 43, 47, 57 ff., 61 ff.
250
INDEX
Kripke, S.A. 5, 87, 146 Kripke model 73 intuitionistic 146, 155 ff. rudimentary 156 Kuhn, S. 189, 201 ff. von Kutschera, F. 7, 109, 115, 130 L L3 172 f., 177 f., 184 LC 172, 174, 182 f., 185 f. labelled deductive system 3, 13, 87 van Lambalgen, M. 201 Lambek Calculus 105 language game 8 Leibniz, G.W. 190 Lenzen, W. 134, 143 Lopez- Escobar, E.G.K. 131 f. Lukasiewicz, J. 43, 150 Lukaszewicz, W. 145 M MPL 141 f. Maibaum, T. 24 many-valued logic 19, 173 Masini, A. 17 f. Matsumoto, K. 5 f., 12, 65 maximal ¥-consistent 167 McDermott, D. 145, 150, 153 Meyer-Viol, W. 201 mimicing structure 63 Mints, G. 27, 87, 187 modal literal 79 model structure 147 modularity 3, 7, 10 f., 14, 181, 187, 212 monotonicity 4 Montague, R. 14, 190 multiple sequent system 3 N N3 115, 127, 130, 144 ff., 149 f., 152 f. 211, 230, 234 N4 115 f., 127£., 130 ff., 134 ff., 139, 142 ff., 154, 211, 229 ff., 234 NLL 193 f.
INDEX
natural deduction 24 f., 211 f., 214 ff. negation as inconsistency 128, 139 ff. atomicity of 135, 143 Boolean 28, 157 hard 136 intuitionistic 128, 206 double 152 normal form 135 split 33 strict 157 strong, as falsity (refutation) 115, 127, 136 f., 142 f. Nelson, D. 13, 65, 115, 127 f., 130, 135, 139, 143 f., 149, 154, 211 Nelson model 149 Nemeti, I. 200 Nishimura, H. 6, 21 f. non-monotonic reasoning 103, 146 ff. normal form 35 reduced 35 Nor mal Form Theorem 36
Pottinger, G. 22, 171 Prijatelj, A. 173 principal move 49, 53, 89, 96 f. proof-theoretic semantics 7, 65, 108 f., 114, 120 f. pure 85, 92 sufficiently 92 P-scheme 36
Q QS5 87 quasi-functionality 209 R Rautenberg, W. 87 rank of a formula 82 of a proof 53 of a tableau 96 of a structure 35 (ref) 108 refutation (disproof) 115, 127 ff. relational proof system 3 replacement 71, 108, 110 f., 121 f. residual 31 f. residuated pair 30, 43, 191 dual 30, 191 residuation abstract law of 32, 63 Restall, G. 32 f., 62 ff., 134, 157 de Rijke, M. 31 Roorda, D. 57 rotation 201 rule branch extension 100 communication 174 double line 16 duality 15 export vs. import 214 invertible 42, 79 splitting 174
0 o-antecedentfsuccedent 229 occurrence negative vs. positive 34 (odd cut) 138 (odd reflexivity) 138 Ohnishi, M. 4 ff., 12, 65 p PDL- 77 f. parameter 47, 88 parametric ancestor 49, 92, 205 parametric move 49, 53 f., 56, 92, 97 partial interpretation 14 7 Pearce, D. 135 f. persistence (heridity) 13, 146, 148 ff., 155 f., 159, 166 converse 156 plausibility relation 147 positivization 135
S4.3 6, 182 f. S4Grz 6 S5 4 f., 10, 12, 16, 19, 21 f., 65, 98, 100, 172 ff., 185 SJ 157 f. Sambin, G. 11 Sato, M. 19, 173 Schroeder-Heister, P. 7, 108 Schroter, K. 18 S-degree 107, 119 Segerberg, K. 3, 10 segregation 36 f., 176 semantics independence 23 separation (of a rule) 8, 11 Serebriannikov, 0. 2 Shimura, T. 6 Shvarts, G. 5 simplicity 23 size of a proof 52 of a tableau 95 Slupecki, J. 115, 138 S-subformula 107, 119 strong completeness 45, 167 ff., 226, 233 strong normalization 51, 94 ff. strongly equivalent 135, 150 structural connective 4, 27 f. families of 27, 42, 174 f. structural rule 11, 19, 21, 61 additional 39, 76, 166, 178, 232 basic 28 f., 175, 229 external vs. internal 22, 172 structured database 104 subformula property 11 f., 22 f., 48, 161, 187 subintuitionistic logic 13, 155 ff. subordinate proof 214 substitution 48, 104 f., 195 ff. switch 201 Sylvan, R. 157 symmetry (of a rule) 8, 11
5 S4 4 f., 10, 16, 98, 154, 157, 160, 182 f., 194, 207
l
251
T TQS5 88, 97, 100
252
INDEX
Takano, M. 6, 12 Tarski, A. 189 temporal completeness 66 temporalization 159, 164 Tennant, N. 127 ff., 132 ff. three-valued logic 144 f., 147, 150, 154, 172 f. (tra) 108 -elimination 110 trace 32 Troelstra, A. 131 Thrner, R. 144 ff., 150, 152 typed .\-calculus 1, 51
u undecidability 63 f. uniqueness 10 V
Valentini, S. 11 Venema, Y. 200 Void 34
w Wittgenstein, L. 7 f. W6jcicki, R. 138
