Mathematical Logic with Special Reference to the Natural Numbers

Mathematical Logic with special reference to the natural numbers Mathematical Logic with special reference to the nat...

Author: S. W. P. Steen

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Mathematical Logic with special reference to the natural numbers

Mathematical Logic with special reference to the natural numbers

S. W. P. STEEN Sometime Gayley Lecturer in pure mathematics in the University of Cambridge

Cambridge at the University Press 1972

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521080538 © Cambridge University Press 1972 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1972 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 77-152636 ISBN 978-0-521-08053-8 hardback ISBN 978-0-521-09058-2 paperback

To my Wife

Contents

Preface

p. xv

Introduction Chapter 1. Formal systems

1 10

1.1 Nature of a formal system p. 10 1.2 The signs and symbols p . 10 1.3 The formulae p . 12 1.4 Occurrences p . 13 1.5 Rules of formation p . 13 1.6 Parentheses p . 16 1.7 Abstracts p . 18 1.8 The rules of consequence p . 18 1.9 Corresponding and related occurrences p . 21 1.10 The X-rules p . 22 1.11 Definitions and abbreviations p . 23 1.ia Omission of parentheses p . 24 1.13 Formal systems p . 27 1.14 Extensions of formal systems p . 28 1.15 Truth definitions p . 29 1.16 Negation p. 29 HISTORICAL REMABKS TO CHAPTER 1 p. 30 EXAMPLES 1 p. 32

Chapter 2. Propositional calculi 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Definition of a propositional calculus p . 34 Equivalence of propositional calculi p . 35 Dependence and independence p . 36 Models of propositional calculi p . 36 Deductions p . 39 The classical propositional calculus p . 42 #ome properties of the remodelling and building schemes p . 43 Deduction theorem p. 48 Modus Ponens p . 49 [vii]

34

Contents 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17

Regularity p . 51 Duality p . 52 Independence of symbols, axioms and rules p . 53 Consistency and completeness of &c p . 55 Decidability p . 57 Truth-tables p . 58 Boolean Algebra p . 61 Normal forms p . 64

HISTORICAL REMARKS TO CHAPTER 2 p. 65 EXAMPLES 2 p. 68

Chapter 3. Predicate calculi 3.1 Definition of a predicate calculus p . 72 3.2 Models p . 76 3.3 Predicative and impredicative predicate calculi p . 77 3.4 The classical predicate calculus of the first order p . 78 3.5 Properties of the system ^ c p . 79 3.6 Modus Ponens p . 84 3.7 Regularity p . 88 3.8 TOe system
72

Contents 3.27 3.28 3.29 3.30 3.31 3.32 3-33 3.34

ix

Ordinals p. 175 Trans finite induction p. 178 Cardinals p. 180 Elimination of the e-symbol p. 184 Complete Boolean Algebras p. 192 Truth-definitions for set theory p. 193 Predicative and impredicative properties p. 198 Topology p. 199

H I S T O R I C A L REMARKS TO CHAPTER 3 p. 201 E X A M P L E S 3 p. 205

Chapter 4. A complete, decidable arithmetic. The system A oo

213

4.1 The system Aoo p. 213 4.2 The A00-rules of formation p. 213 4.3 The A00-rules of consequence p. 215 4.4 Definition of A00-truth p. 218 4.5 Definition of A00-falsity p. 219 4.6 Exclusiveness of A00-truth and A00-falsity p. 219 4.7 Consistency of Aoo with respect to A00-truth p. 224 4.8 Completeness and decidability of A oo with respect to Aoo-truth p. 225 4.9 Negation in the system Aoo p. 227 4.10 The system B o o (the anti-A00-system) p. 228 H I S T O R I C A L REMARKS TO CHAPTER 4 p. 229 E X A M P L E S 4 p. 230

Chapter 5. Aoo-Definable functions

232

5.1 Calculable functions p. 232 5.2 Primitive recursive functions p. 233 5.3 Definitions of particular primitive recursive functions p. 235 5.4 Characteristic functions p. 243 5.5 Other schemes for generating calculable functions p. 246 5.6 Course of value recursion p. 247 5.7 Simultaneous recursion p. 247 5.8 Recursion with substitution in parameter p. 248 5.9 Double recursion p. 250 5.10 Simple nested recursion p. 252 5.11 Alternative definitions of primitive recursive functions p. 254 5.12 Existence of a calculable function which fails to be primitive recursive p. 258

Contents 5.13 Enumeration of primitive recursive functions p. 260 5.14 Definition of the proof-predicate for A oo p. 265 5.15 The function Val p. 270 H I S T O R I C A L REMARKS TO CHAPTER 5 p. 273 E X A M P L E S 5 p. 275

Chapter 6. A complete, undecidable arithmetic. The system Ao 6.1 6.2 6.3 6.4

278

The system Ao p. 278 A0-truth p. 279 Undefinability of A0-falsity in Ao p. 283 Enumeration of Ao-theorems p. 284

H I S T O R I C A L REMARKS TO CHAPTER 6 p. 286 E X A M P L E S 6 p. 286

Chapter 7. Ao-Definable functions. Recursive function theory

288

7.1 Turing machines and Church's Thesis p. 288 7.2 Some simple tables p. 300 7.3 Equivalence of partially calculable and partially recursive functions p. 304 7.4 The S-d-0' proposition p. 313 7.5 The undecidability of the classical predicate calculus ^c p. 314 7.6 Various undecidability results p. 316 7.7 Lattice points p. 318 7.8 Complete sets p. 323 7.9 Simple sets p. 323 7.10 Hypersimple sets p. 324 7.11 Creative sets p. 327 7.12 Productive sets p. 330 7.13 Isomorphism of creative sets p. 332 7.14 Fixed point proposition p. 334 7.15 Completely productive sets p. 335 7.16 Oracles p. 336 7.17 Degrees of unsolvability p. 343 7.18 Structure of the upper semi-lattice of degrees of unsolvability p. 346 7.19 Example of the priority method. Solution of Post's problem p. 352 7.20 Complete degrees p. 356 7.21 Sequences of degrees p. 361

Contents 7.22 7.23 7.24 7.25 7.26 7.27 7.28

Non-recursively separable recursively enumerable sets p. 364 Cohesive sets p. 365 Maximal sets p. 366 Minimal degrees p. 368 Degrees of theories p. 372 Chains of degrees p. 374 Recursive real numbers p. 375

HISTORICAL REMARKS TO CHAPTER 7 p. 378 EXAMPLES 7 p. 382

Chapter 8. An incomplete undecidable arithmetic. The system A 387 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

The system A p. 387 Definition of A-truth p. 388 Incompleteness and undecidability of the system A p. 390 Various properties of the system A p. 391 Modus Ponens p. 396 Consistency p. 398 Truth-definitions p. 401 Axiomatizable sets of statements p. 403

HISTORICAL REMARKS TO CHAPTER 8 p. 407 EXAMPLES 8 p. 408

Chapter 9. A-Definable sets of lattice points 9.1 9.2 9.3 9.4 9.5

409

The hierarchy of A-definable sets of lattice points p. 409 A*-sets p. 413 Sets undefinable in A p. 416 f-definable sets of lattice points p. 417 Computing degrees of unsolvability p. 419

HISTORICAL REMARKS TO CHAPTER 9 p. 421 EXAMPLES 9 p. 421

Chapter 10. Induction 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Limitations of the system A p. 423 Possible ways of extending the system Ao p. 425 The system E p. 430 The system Aj p. 438 Definition of an Aj-proof p 440 Theorem induction p. 445 The Aj-proof-predicate p. 448

423

xii

Contents 10.8 An example of an Arproof p. 451 10.9 Relations between Ao-theorems and E-correctness p. 455 10.10 Properties of the system Aj p. 459 10.11 Reversibility of rules p. 463 10.12 Deduction theorem p. 469 10.13 Cuts with an A oo cut formula p. 470 10.14 Cut removal with a weaker form of Rd p . 484 10.15 Cut removal in general p. 487 10.16 Further properties of the system Aj p. 497 10.17 The consistency of Aj p. 500 HlSTOBICAL REMARKS TO CHAPTER 10 p . 507

E X A M P L E S 10 p. 509

Chapter 11. Extensions of the system A7

511

11.i The system A' p. 511 11.2 Remarks p. 516 11.3 The hierarchy of systems A(v) p . 518 iz.4 Properties of the systems A{v) p. 518 11.5 The systems A(v)* p. 519 11.6 The definition of A-truth in A'* p. 521 11.7 Consistency of Aj p. 530 11.8 Definition of A(K)-truth p. 535 11.9 Scheme for an A{K)-truth-definition p. 538 11.10 Truth-definitions in impredicative systems p . 540 1 z. 11 Further extensions of the systems A(/c) p. 541 11.12 Incompleteness of extended systems p. 543 11.13 Real numbers p. 544 11.14 The analytical hierarchy p. 553 11.15 On the length of proofs p. 558 H I S T O R I C A L REMARKS TO CHAPTER 11 p. 559 E X A M P L E S 11 p. 560

Chapter 12. Models 12.1 12.2 12.3 12.4 12.5 12.6

Models and truth-definitions p. 563 Models for Aoo p. 565 Models for A o p. 566 Models for Aj9 A<"> p. 567 General models p. 567 Satisfaction p. 568

563

Contents 12.7 Examples p . 570 12.8 Non-standard models p . 571 12.9 A non-standard model for A/ p . 575 12.10 Induction p . 577 12.11 S-models p . 580 12.12 Ultraproducts p . 581 12.13 H-models p . 585 12.14 Satisfaction by A2-predicates p . 590 12.15 He-models p . 593 12.16 Completeness of higher order Predicate Calculi p . 594 12.17 Independence proofs p . 599 HISTORICAL REMARKS TO CHAPTER 12 p. 606 EXAMPLES 12 p. 607

Epilogue

609

Glossary of special symbols

611

Note on references

619

References

621

Index

631

Preface

About ten years ago I conceived the idea of writing a book on the natural numbers because I thought that what had appeared up till then seemed to have reached a point where there was a certain amount of completeness - of course there never will be absolute completeness-and this is one of the attractions of the subject. Anyway it was not until I had retired that I had the time to get down to the task properly. The result is a book which begins with an account of formal languages including the two most basic, namely, the propositional calculus and the predicate calculus, and then goes on to arithmetic; beginning with a very simple arithmetic; finding this inadequate; extending it to overcome this inadequacy; finding the resulting system, though richer in modes of expression, still, but for a different reason, inadequate; extending this in turn to remedy this inadequacy; finding the resulting system has lost some of the 'nice' qualities of its predecessor, but is again, for a new reason, inadequate; extending this and so on. Before I come to develop arithmetic formally, it is convenient to have a primitive notation for the natural numbers (mainly to avoid lengthy circumlocutions) from which the concept of order and the operations of addition and multiplication can easily be obtained. I use sequences of tally marks, this is sufficient for our purposes. The real difficulty with arithmetic, as with other things, enters with the universal quantifier, when we want to make statements about all natural numbers. This use of tally marks is mentioned in the text but in the main is left to the reader to fill in. There are several topics absent from the book which might have been included, these are partly off the main line of development, partly applications of the general theory developed, partly sidelines, etc. Among these topics are: recursive analysis, constructive ordinals, recursive equivalence types, recursive probability theory, the word problem, algorithms, finite automata, A-conversion, combinations, productions, intuitionism, various forms of propositional calculus, many-valued logics, and so on. Of these the constructive ordinals are mentioned several [XV]

xvi

Preface

times because now and again we come across a process which can be continued into the constructible transfinite, but we do not go into it further. The matter developed in this book was developed over the years in a course of lectures delivered at Cambridge, except that very little was said about the contents of Chs. 10, 11, 12, so these three chapters have not come under the fire of criticism of young scholars, and I feel that in consequence that they are not of the same quality as the earlier chapters, particularly the account of cut elimination in Ch. 10. The remaining chapters have been fairly well thrashed out in lecture and I am very appreciative of the comments of my classes and of the elegant demonstrations they gave me from time to time. I hope that I have acknowledged them all. With regard to the language in which the book is written, this is meant to consist of instructions and descriptions and occasionally of pointing out that such and such a procedure would lead to an impossible situation. Later in the book, when treating with ultra products I have transgressed and used Zorn's lemma, but a purist can tear that piece out of the book. Each chapter is followed by a short historical account of the matter treated in that chapter, it is this way that I make acknowledgement to those who first invented the matter, if I have made omissions then I apologize. After the historical account there follow a few examples. Many more examples can be found in books by Rogers (1967), Shoenfeld (1967) and Church (1956). I must thank Professor R. Harrop and Dr N. Routledge for comments on a former, now completely discarded draft which developed a much more complicated system. The present system owes its simplicity to the iterator symbol. I must also thank Dr G.T.Kneebone for reading the draft of Chs. 1-7 inclusive and providing valuable comments, and Drs T. J. Smiley and L. Drake for reading the draft of the remaining chapters and again providing valuable comments; also to the University Press for courtesy and consideration during the production of the book, and finally to my wife for help with the tedious business of making an index. Christ's College Cambridge June 1971

S.W.P.S.

Introduction

Mathematics is the art of making vague intuitive ideas precise and then studying the result. Many examples can be given of the wealth of interesting matter that has arisen when a vague intuitive idea has been made precise. Half the solution to a problem is to state it precisely. Among these vague intuitive ideas is that of natural number and that of preciseness itself, there is also the vague intuitive idea of correctness. In this book we are mainly concerned with making these three vague intuitive ideas precise and with inventing a method whereby our thoughts can be either communicated to others or stored for our own memory. It may be that our concept of natural number arises from our perception of our own heart beats; this gives us a linear (and as far as we can perceive) unending progression, without conscious beginning or ending. The concept of an unending progression of distinct things with a definite starting entity and never returning to any entity previously encountered is the essence of the concept of natural number; this concept is given to us by our perception of our own heart beats. May-be our perception of time arises from our sensing of the circulation of the blood in the brain. This gives a linear background to our thoughts and sense data. It is amusing to imagine a creature with a two-dimensional flow of fluid through its body, such a creature might have a two-dimensional conception of time and be quite unable to conceive of natural numbers as we do. Apart from making the intuitive idea of natural number precise we want to construct a method for communicating our thoughts on natural numbers to other persons and a method of storing them so that they don't get lost and forgotten. This we do by constructing a written language; this consists of a linear series of signs or shapes and will be the main medium in which our thoughts will be made precise. The language will be constructed with the utmost precision so that there can be no dispute as to what we wish to convey. We then translate our thoughts into this language, write them down as we shall say; the reverse process of converting something written in such a language into a thought is called understanding what has been written. Whether another human being I

[ 1 ]

SML

Introduction

gets a similar thought to the writer is a philosophical question which we do not discuss. Our first task then will be to construct a language suitable for our purposes; but we also want to teach it to other persons. There are two methods of doing this. The first method is to take advantage of the fact that we all already know a language of sorts, namely the imprecise language of daily use. This language has been invented over the ages for the purposes of descriptions, commands, instructions, explanations, excuses, deceits, lies, warnings, songs, etc., it is imprecise in that it is impossible to give a precise definition of 'sentence', 'word', 'noun' and many other syntactic terms. For instance, try and give a definition of 'word in the English language' and then test this definition against the works of Shakespeare and see if it is satisfactory. Such a definition to be of any use must be applicable to all writers at all times and not to a particular writer. For instance a definition such as 'any consecutive set of letters (i.e. without a blank between them) written by Shakespeare and printed in the first folio edition is a word and these are all the words' though a precise definition would be quite unacceptable. Since we know the English language, or at any rate some part of it, then we can use it to describe what we are constructing and this will be quite satisfactory; we are quite accustomed to doing this sort of thing. It is unsatisfactory in that it makes our development of arithmetic depend on the English language or at any rate on part of the English language, and we wish our development to be independent of imprecise concepts. This method is the usual method used for teaching a language. But there is another method which is quite satisfactory because if we use it then our development does not depend on any thing outside itself. This method could be used in a modified form for inter-planetary communication, and is a modification of the method by which we learnt English in the first place. It consists of teaching the language by repeatedly writing down the signs of the language (its alphabet) until the pupil has understood which are the correct signs and which are foreign, for instance foreign shapes could be put down from time to time and immediately obliterated. Then repeatedly writing down correctly formed sequences of correct signs and others which are incorrectly formed and immediately erasing these latter; continue them until the pupil has understood which sequences of signs are correctly formed

Introduction

(meaningful sentences). This process would be possible, but lengthy and boring in the extreme. We shall employ the first method in this book. It would be quite possible to use part of the English language instead of inventing a new language to deal with the matter treated in this book. But the result would be impossibly lengthy and much of the matter treated here would hardly have been thought of in the first place if we restricted ourselves entirely to the English language. For example in the English language there is (except for idiomatic variations such as gender, plurals, etc.) only one pronoun namely 'it'. In our language variables correspond to pronouns and we require an unending list of them. In the English language with its one pronoun we have to introduce lengthy circumlocutions such as the 'first', 'the second', etc. to obviate the lack of different pronouns. This has the obvious disadvantage of introducing natural numbers before one has defined them. But the main advantage in a symbolic language is that it makes the metamathematical investigations far simpler and the whole subject far easier to handle. Without it very little headway would ever have been made. The language we shall construct will only be suitable for expressing our thoughts on natural numbers. If we wanted to construct a language for some other purpose, say chemistry, then we should want many more new signs standing for further undefined concepts. The main difficulty in constructing a language for extramathematical purposes is that we are unable to give precise definitions of the concepts used. For instance the solution of the problem 'which came first the egg or the hen ?' is that we are as yet unable to give a precise definition of egg or of hen which would be applicable to all times and places; thus the ancestors of a given present day hen if pursued far enough back into geological times would ultimately contain creatures that no one would now call hen, in between such a creature and the given live hen of today would be a series of creatures changing by imperceptible degrees. Maybe at some future date we shall be able to define ' hen' as a creature with such and such a protein molecule in its genetic code. In other words all extramathematical concepts have furry edges. Thus to count the grains of sand on a beach (i) it is not clear where the boundaries of the beach are (ii) it is not clear what objects are grains of sand, they range from minute pebbles to impalpable dust. We do not want any vagueness of this sort which is inherent in all colloquial languages. Again, words in a colloquial language are used differently as

Introduction

time goes on; in fact it has been said that a word changes its meaning each time it is used. The sentence ' it is artful, aweful and painful' would be used now-a-days in quite different contexts to its use 300 years ago, then it was praise now it would be ridicule. We want to avoid anything like this. Also a phrase like ' a rose red city half as old as time' is utter nonsense but extremely pleasing and conjures up all sorts of images, but we want to avoid this sort of thing too, we want our language to be permanent, definite and absolutely precise. We shall find it very useful to have a primitive form of notation for natural numbers so that when describing our language we can use expressions like 'formula A has more signs than formula B ' . Since our language is designed to talk about the natural numbers then the properties of natural numbers should be avoided when talking about it. But expressions such as the above can easily be eliminated as follows: replace each sign in the formulae A and B by a tally mark, then the expression is an abbreviation for 'the sequence of tally marks obtained from formula B is a proper initial segment of the sequence of tally marks arising from formula A'. Here we have a primitive form of notation for the natural numbers namely: a tally mark standing alone is a natural number, if JT is a natural number then so is Jf* I, namely, the result of adding a tally at the end of the sequence of tallies JV\ these are all the natural numbers. These sequences of tallies form an unending progression of distinct things with a definite starting place and so have the fundamental property of natural numbers. With them we can do addition by juxtaposition, multiplication by replacing each tally of one set of tallies by replicas of the sequence of tallies of another set of tallies. In a similar manner we can deal with other simple situations, thus avoiding the use of the term 'natural number'. The language that we are going to construct is called the object language and the language that we use to describe it is called the metalanguage. The metalanguage was learnt by a method somewhat like the second of the two methods of teaching languages mentioned above. We have just shown how to avoid the use of natural numbers in the metalanguage. Thoughts primarily are languageless and are only put into language for purposes of communication with others or with oneself (memory). We are so accustomed to put thoughts straight into words that one fails to realize that they are languageless and that a language though useful

Introduction

is unnecessary in order to be able to think. If one required a language to think in then a child would be without thoughts at all until it had learnt a language; as it began to learn a language at first it would have only one or two words to think with. Everyone is familiar with the situation when one is momentarily without the right word to express a thought; again when asked to describe some situation an image of the situation arises in the mind and one reads off from it, one puts what one perceives in the mind image into words. When one is working, thoughts not sentences, are coming into being and are perceived first as languageless thoughts, some of them are discarded before being put into words. To construct our language we must first make our tools; these will consist of a tape divided into squares and such that we can always add more tape at the end when desired, so that we are always with spare tape at the end. Also we shall want a pen or other instrument to make marks in the squares: the marks are called signs. At this point we have to rely on the goodwill of the reader to read our writing; by this we mean that he must realize that such and such a mark is intended to represent such and such a sign. If anyone takes the trouble to examine all the letters printed in this, or any other, book he will find that no two letters are exactly alike even if they are supposed to represent the same letter. This fact is sometimes useful in crime detection, in comparing letters typed on different typewriters even if of the same make. One would like this point about goodwill not to occur and to say something like this 'the signs are: a simple closed curve, a simple segment', etc.; but the definition of these things is away beyond the end of the matter treated in this book. The list of signs that we are going to use will first be displayed and we rely on the good sense of the reader to identify any other marks we make with one of them or to realize that it is intended to be a new mark, i.e. one which has as yet not appeared. If a reader is unable to do this then he will be unable to read what has been written, he may as well do something else. The signs will be written in consecutive squares of the tape; the object of the tape is so that the signs are of reasonable size (a sign occupying a square mile would be impracticable; we shall get on with squares in which the letters in this book could be printed), and so that we know when we have come to the end of a sequence of signs by coming to a blank square. Without some such tacit or expressed restriction reading would be impossible, the next symb o 1 might be miles away. If we had an unending sequence of distinct signs

Introduction

then we should be unable to distinguish between every pair and reading would be impossible, in this case there would be two signs which differed so little from each other that even an electron microscope would be unable to detect any difference. For instance if the symbols were circles ^oth inch in diameter lying in a one inch square and with centres at the rational points. Certain sequences of signs will be called well-formed, the rest are called ill-formed. Our language must be constructive, that is to say it must be possible to decide by a terminating process whether a given formula is well-formed or is ill-formed. It is plain that if this was otherwise then our language would be unreadable. The well-formed formulae in our language correspond to the words and sentences of a conversational language. If we were teaching a conversational language then we should stop at this place. Having listed and explained a certain set of words and told how they can be formed into sentences we would then use the language for descriptions, instructions, etc. But in our case our motive is different, we want to obtain the true statements about natural numbers. Truth is a property of statements. The statement ' it snows' is true if and only if it snows. The inverted commas round a statement give us a name of that statement. We wish to give a truth definition for our language which will make precise our vague intuitive conception of arithmetic truth. This we do by noting that our statements are built up from atomic statements by connectives (often called logical connectives, because they can be used in any language). Our atomic statements are equations and inequations. For these we can give a simple and intuitively satisfying truth-definition. The truth of compound statements is then determined in a perfectly precise manner from the truth-values of the statements from which it is compounded. Unfortunately we are sometimes unable to find the truth-value of a statement because we are referred to an unbounded set of more elementary statements. But perhaps this is fortunate because if we had a method which would tell us the truth-value of any statement then all the interest would have gone from mathematics, we could turn the whole thing over to a computer which would tell us the answer. The theory of truth for arithmetic is a very difficult and complicated business. We shudder to think what difficulties we should encounter if we attempted to deal with truth in such disciplines as theology, law or politics, even if we had a suitable language for them,

Introduction

these disciplines make considerable use of the concept of truth; judging by history both ancient and modern it appears that the only way the human race has found of settling the question of truth in any of these disciplines is by the use of force. Even a truth-definition for physics would be extremely difficult to handle. Since our motive is to find true arithmetic statements and since we are without a terminating method of testing a given arithmetic statement for truth then we proceed by what is called the deductive method. We first display a class of statements which we accept as true, these we call axioms, they are statements that anyone studying arithmetic would be bound to accept as true. Then we list certain methods whereby from true statements we can obtain other true statements. These methods are usually expressed as figures, the given true statements are called the premisses and the one that is obtained from them is called the conclusion. They and the axioms virtually play the part of implicit definitions of our primitive signs and logical connectives. Thus starting with the axioms and applying the figures repeatedly we continue to produce more and more true statements. The statements produced in this way are called theorems. But we shall see that in all except the simplest languages some true statement will always elude us, i.e. theorems will be a proper subset of true statements. Our intuition makes us believe that a given meaningful statement is either true or false, that we are without a third possibility. This belief comes from the examination of testable cases-usually reducible to bounded sets. But when we come to deal with unbounded sets then a third possibility arises namely when we are unable to decide between truth and falsity. It seems rather useless to believe that each sentence is either true or false when we are unable to decide which is the case. Anyway we desire our language to be free from all beliefs. The theory of truth belongs to semantics, this is the theory of meaning. The structure of a language belongs to syntax, this includes the signs and the rules which tell us which sequences of signs are well-formed and which figures are deductions. From syntax alone we obtain a language without meaning. Meaning will be given to the languages which we construct, in that, for the arithmetic languages (without going into details) each statement will give rise to a structure consisting of pairs of sequences of tallies and according as these pairs of sequences of tallies are the same or are different

8

Introduction

then the statement will be true or false. A true statement will then correspond to a possible situation as regards these sets of pairs of tallies and a false statement will correspond to an impossible situation as regards these sets of pairs of sequences of tallies. In this way our language will have meaning and the true statements will correspond to possible constructions involving sequences of tallies. The full details of this sketch will become apparent as the book proceeds, but we omit them here. We have just said that our motive is to obtain true arithmetic statements. But we shall only obtain a few of the most elementary such statements. Our interest is in what our languages can do and what their limitations are. That is to say we are interested in the metaiheory of our languages. If we were interested in obtaining true arithmetic statements then we would be writing a book on arithmetic, it would differ from an ordinary book on arithmetic because we should begin with a very full syntactic introduction. Frequently we want to show that each formula which has one property also has another property. We do this by showing outright that the shortest formulae with the first property also have the second property, then we show that if each formula having the first property and shorter than a given formula having the first property also has the second property then the given formula also has the second property. In this way we give sufficient instructions to obtain for a given formula having the first property a detailed demonstration that it also has the second property. This method we call formula induction, it is distinct from mathematical induction. It has often been said that a formal system is just a meaningless game with symbols. In our case this is not so, we have something definite to say and have invented a language in which to say it. It is of course easy to invent a formal system void of meaning, sometimes such systems are useful to investigate, some like Post's productions are obtained by formalizing the essential properties of formal systems such as ours. H I S T O R I C A L REMARKS

Brouwer (1947) suggested that our concept of number derives from our perception of our own heartbeats. Progressions are discussed in Principia Mathematica, Whitehead & Russell. For remarks on formal and informal languages see Carnap (1957). The method suggested for planetary inter-

Introduction

9

course is due to Freudental (1960). The terms 'object language' and 'metalanguage5 are due to Carnap (1957). The idea of using a tape divided into squares with only one symbol on a square is due to Turing (1936, 1967), who used it in his machines. Russell once said 'no two x's are alike'. The term 'new' in the sense in which we have used it is due to Quine (1951). Turing makes remarks (1936, 1967) about the unfeasibility of using an unbounded set of distinct symbols by making a definition of 'neighbourhood' for a symbol. Tarski (1933-56) exhaustively discusses the concept of truth in formalized languages. Brouwer in his intuitionism -ably expounded by Heyting (1956)-discards the hypothesis that a statement is either true or false. This is commonly called T.N.D. (Tertium non datur). Lorenzen (1955) has a useful discussion about formula induction and various other kinds of induction, which are used in the metalanguage.

Chapter 1 Formal systems

i. i Nature of a formal system A formal system is constructed by choosing a set of signs and laying down rules for their manipulation. We have a tape marked into consecutive squares which can always be lengthened, so that we always have vacant squares at the right. The signs are placed on consecutive squares of the tape from left to right, at most one sign to a square. We use capital ell in script type (J?) with or without superscripts or subscripts to denote an undetermined formal system. 1.2 The signs and symbols The signs of a formal system jSf, called J?-signs, must be displayed by representative figures and it must be possible to distinguish between them and to recognise them on different occasions and to decide of an object whether it is intended to represent an J§?-sign or whether it is irrelevant. These conditions are required in order that reading the system be possible. We lack means of displaying an unending list of distinct signs and if we make a list of distinct signs we shall have to stop at some place. Moreover at any stage in our work we shall only require a set of signs that can be displayed, so that the restriction to an initial displayed list of signs is without restriction on our work. The judgement whether a given mark or figure is intended to represent an J5f-sign must be left to the reader. The J5f-signs in the initial displayed list are called primitive ^C-signs. In some formal systems we require a method whereby we can always obtain a new sign of a certain kind, that is a sign which is distinct from any sign of that kind, used up to that place. In order to be able to do this, starting with an initial displayed list, we use what we call compound 3?-signs, that is certain sequences of JSP-signs which can be generated according to some fixed plan. If the scheme of generation can be carried [10]

1.2 The signs and symbols

11

as far as we wish we can generate a new compound J5f-sign according to the scheme at any place we desire. The method we shall adopt is to select a primitive ^f-sign and obtain compound J§?-signs by continually adding superscript primes to the right. For example, in this book we shall only need one scheme of generation, we shall use the sequence which begins as follows: x

x'

x"

x"r

xn"

to obtain an unlimited supply of compound J^-signs which we shall call variables.

We can describe this scheme of generation thus: (i) ' # ' is a variable, (ii) if 2 is a variable then so is 2 n ', (iii) a sequence of primitive JSf-signs is a variable if and only if it is obtained from (i) and (ii). Here S stands for an undetermined primitive or compound JSf-sign and S n/ stands for the result of superscripting a prime at the end (in the next square on the tape) of that undetermined primitive or compound jSf-sign. The clause (ii) could have been worded 'the result of attaching a superscript prime at the end of a variable is a variable'. The sign between capital sigma and the prime is called the concatenation sign. It is to be distinct from the jSf-signs, it is used to augment the English language when we talk about a formal system. A primitive JSf-sign used like the prime above is called a generating sign. The symbols of a formal system, called J§?-symbols, are the primitive ££-signs other than any generating signs together with the compound JS?-signs which are obtained by a scheme of generation. Thus the set of primitive J§?-signs is limited and can be displayed, but if there are any schemes of generation then the jSf -symbols can only be generated as far as desired. We shall use capital sigma with or without superscripted primes or subscripts to denote an undetermined J^f-symbol. Thus £' will stand for an undetermined jSP-symbol, S n ' will stand for the result of superscripting a prime at the right of an undetermined JS?-symbol. We use the notation (v) to stand for an undetermined succession of generating signs and (Sv) for the result of adding another generating sign at the end. Similarly for (#c), (0), (A), and (/i). Thus £',2",2'", ...,£<">, will stand for a sequence of undetermined oSP-symbols. We shall frequently use Greek

12

Ch. 1 Formal

letters and the concatenation sign in this way, it will make our descriptions and instructions easier to follow. There is one further requirement: any linear sequence of J§?-symbols must be uniquely constructed from J§?-symbols. For instance if 1 00 10 01 were JSf-symbols obtained from the primitive JSf-signs 0 and 1 then the linear sequence 1001 could be constructed from J§?-symbols in two different ways. Similarly a linear sequence of generating signs must be uniquely constructed. For instance if' and " were different generating signs then x'" would be ambiguous. Again if we allowed the same generating sign to be used fore and aft so that '"x, "x, and 'x were J§?-symbols as well as x', x" and x1" then x"'x could be constructed in several ways. 1.3 The formulae An JSf-formula is a terminating sequence of J§?-symbols. Thus an J&?-symbol standing alone is an jSf-formula. The null ^-formula is the empty terminating sequence of JSf-symbols. A non-null terminating sequence of generating signs (if jSf contains any such signs) fails to be an jSf-formula. Given a terminating sequence of signs (whether J§?-signs or other signs) it is possible to decide whether it is an J§?-formula or contains signs foreign to the system ££ or contains generating signs incorrectly placed. It is possible to decide of an J5f-symbol whether it is the end symbol of a given ^-formula or is the initial J§? -symbol of that jSf-formula and if neither is the case then it is possible to find the J§?-symbol which precedes it and the j£?-symbol which follows it, our device of the tape divided into consecutive squares does this. We shall use capital phi, psi, chi and omega with or without superscripts or subscripts to denote undetermined jSf-formulae. If
1.3 The formulae

13

sequence of separate formulae O and T in that order, there is then at least one blank square between them on the tape. The absence of concatenation sign denotes that we are dealing with a sequence of separate formulae rather than with a single formula. Usually written O Y o r O , T . 1.4

Occurrences

An occurrence of an ££'-symbol 2 in an J§?-formula O is an initial segment of O which ends with the JSf-symbol 2. Note that by definition of initial segment the part of O which is left when the initial segment is removed is an JSf-formula, the end segment may be null. Thus if O' is an J5f-formula and 2 is an J§?-symbol belonging to a scheme of generation with the prime as generating sign then ®/n2 fails to be an occurrence of the oSf -symbol 2 in the JSf-formula ®'n2n/ because by definition of initial segment ®'n2 fails to be an initial segment of the JSf-formula ®'n2n/, the part left when $' n 2 is removed is only the prime which fails to be an jSf-formula. A given J§?-symbol may have several distinct occurrences in a given S£ -formula. If the last JSP -symbol of the j£?-formula $ is 2 then O itself is an occurrence of 2 in O. The null formula fails to be an occurrence of any jSf-symbol in any jSf-formula. Similarly an occurrence of a non-null ^-formula T in an j£?-formula <J> is an initial segment O' of O which has the end segment T. Note that by definition the remainder of O when
Rules of formation

Any formal system that has been constructed till the present time can be modified so that it uses the rules of formation that we are now going to give. Thus we say that the rules of formation are the same for all formal systems. The symbols of a formal system are of two kinds: proper symbols and improper symbols. The improper symbols are: X abstraction symbol, ( left parenthesis, ) right parenthesis. The abstraction symbol may be absent. There must be some proper symbols. The proper symbols are of two species at most; variables and constants. Each proper symbol has a type associated with it, the improper

14

Ch. 1 Formal systems

symbols are type-less. If 2 is a variable and ' is a generating symbol then 2 n / has the same type as 2. We use small alpha, beta with or without superscript primes for undetermined type symbols. The type symbols are generated according to the following scheme: (i) omicron is a type symbol, (ii) iota with or without superscript primes is a type symbol, (iii) if a, /? are type symbols then so is (nan/?n), (iv) these are the only type symbols. Some of these type symbols may be absent from a given formal system. Thus a formal system might have type symbols only of the types o, (oo), ((oo) o). In writing type symbols we frequently omit the outer pair of parenthesis and omit other parentheses by association to the left. The omitted parentheses can then be replaced uniquely so that the result is a correctly formed type symbol. Thus we sometimes write: ooo

for

((oo)o),

o(oo) for

(o(oo)),

and so on. The rules offormation uniquely associate a type to certain J§? -formulae and enable us to find it or to decide that the J§?-formula is without type. The J§?-formulae which have a type according to the rules of formation are called well-formed formulae (w.f.f.). The rules of formation also define a status (free or bound) for each occurrence of each variable in a well-formed formula and enable us to find it. In this way each occurrence of each variable in a well-formed formula is classified as a free occurrence or as a bound occurrence. When a type symbol is a suffix to a symbol then that symbol stands for an object of that type. The rules of formation are: (i) A symbol 2 a of type a standing alone is a formula of type a. (ii) If 2 is a variable of type a then its occurrence in the formula 2 is free. (iii) If O(nan/?n) is a formula of type (nan/?n) and XF^ is a formula of type J3 then (nO(nan^n)nT^) is a formula of type a. If ®' is a I

II occur-

rence of a variable 2 in the formula ctWc^m then (nO' is a I, ,I 1 p ybound/

1.5 Rules of formation n

15

n

occurrence of the variable 2 in the formula ( O(nan^n) T^), if T ' is / free \ a . p then n I occurrence of the variable 2 in the formula ^¥0 \bound/ (nO(nary»)n*F' is a

I occurrence of the variable 2 in the

formula ("O^n^n/V). (iv) If 2^ is a variable of type /? and Oa is a formula of type a then (nXn2£
1 occur-

rence of a variable 2, distinct from the variable 2^, in the formula Oa then v(nXn2*<J>' is a 11 ,1 occurrence of the variable 2 in the p \bound/ n n formula ( X 2^0^). (v) A formula has a type if and only if a type is given to it by (i), (iii) / free \ and (iv). An occurrence of a variable is I, , I if and only if it ybound/ is I

A according to (i), (ii), (iii) and (iv).

An <j5f-formula which fails to be well-formed is said to be ill-formed. Given an S£-formula we can discover whether it is well-formed or illformed and if it is well-formed then we can find its associated type, the details will be given shortly. According to these rules the null formula is without type. The formation of the formula (nO(nan^n)nT^) from the formulae O^n^n) and T^ is called application. A formula O(nan/?n) of the type (nan/T) is called a junctor of type (nany5n). In the application process a functor of type (nan/?n) is given an argument of type /? and the resulting formula of type a is called the value of the functor for that argument. The formula (nXn2£C) formed from the formula Oa and the variable 2^ is called the abstract of Oa with respect to the variable 2^. The abstract is a functor, by application it can be given an argument of type ft and the resulting formula is of type a, thus (n(nXn2^O«)niF^) is of type a. If the formula O' is a free occurrence of the variable 2^ in the wellformed formula Oa then the occurrence (nXn2^ O' of the variable 2^ in the well-formed formula (nXn2£Oa), which by (iv) is a bound occurrence, is said to be bound by the occurrence of (nXn2^ in (nXn2^O^). The well-

16

Ch. 1 Formal systems

formed part Oa of the well-formed formula (nXn2^ O«) is called the scope of the abstraction of the variable 2^. If O/n(nXnSnHn) is an occurrence of (nXn2n3n) in Oa then each free occurrence of the variable 2 in S is bound by the occurrence O/n(nXnS of (nXn2 in O a . A bound occurrence $ ' of a variable 2 in a well-formed formula O is bound by an occurrence ®" of (nXn2 in ®', and <&" is such that the scope of the occurrence O" of (nXn2 in $ ' is the shortest well-formed formula Y such that O' is an occurrence of 2 in ®"niF. We also say that any well-formed part of S that contains a free occurrence of the variable 2 is bound by the occurrence ®'n(nXn2 of (nXn2 in /n(n' has its mate in O' but some occurrences of left parenthesis in ®' lack mates in ®\ Similarly a proper end segment ®" of O contains an excess of occurrences of right parentheses, each occurrence of a left parenthesis in ®" has

1.6 Parentheses

17

its mate in ®" but some occurrences of right parentheses in O" lack mates in$>". The mates of a consecutive well-formed part of O are mates in O. If O is a single JSP-symbol then O is well-formed and is without parentheses and the lemma follows trivially, otherwise O is of one of the forms: n

3£) or

Suppose the lemma has been demonstrated for well-formed formulae of shorter length than
18

Ch. 1 Formal systems

end segment of O is a non-null proper initial segment of T, similarly with O and W interchanged. L E M M A (ii). / / O and T are well-formed J£?-formulae then either O is a consecutive part of^Vor^Y is a consecutive part of O or O ami! *F fail to overlap. The first two alternatives arise in application and in abstraction. Suppose O is O'nE and Y is E ' T ' , where S, E', ®' and T ' are non-null. Since S is a proper end segment of O it will contain occurrences of right parentheses which lack their mates, but every occurrence of a left parenthesis in H will have its mate in S, since E' is a proper initial segment of T it will contain occurrences of left parentheses which lack their mates, but every occurrence of a right parenthesis in E' will have its mate in E'. Hence E is distinct from E' and so O and T fail to overlap. COROLLARY. / / O is a well-formed ^-formula then the scopes of the various occurrences of X in O, if any, fail to overlap, but the scope of an occurrence may be contained in the scope of another occurrence.

1.7 Abstracts Suppose that ${£] is of type /?, where £ is of type a, then X£. ${£} is of type (fiot). Given an argument 8 of the type a, the result, by the X-rules becomes <j){8), which depends on the formula 8 of type a. Thus the abstract X£. {!;} depends on all formulae of type a. Now consider the following situation: Let ^{9/, £} be of type /?, where T\ is of type (/?a), and £ is of type a, then XT/ . ijr{r], £}, call this K, is of type J3(j3oc) and depends on all formulae of type {/lot). Let A be of type /?(/?(/?a)), then AXT/ . ^{rj, £} is of type ft. Let ^{£, £} be of type /?, where £is of type /?, then X£. {&k7j. i/rty, £}, £}, call this H, is of type (fiot), and since it contains K then it depends on all formulae of type (/?a) including itself! This form of circularity is known as predicativity, and H is called a predicative formula. This circularity seems unsatisfactory, to avoid it we have to make complicated changes in type theory. The situation will be reopened later. 1.8 The rules of consequence Well-formed formulae of a formal system 3? of type o (for some systems further effectively testable structural conditions are required) will be

1.8 The rules of consequence

19

called <£-statements, if there are any further conditions then it must be possible to decide if they are fulfilled or if they are violated. An j£?-statement will be called closed if every occurrence of each variable in the jSP-statement is a bound occurrence, otherwise the jSf-statement will be called open. In some formal systems it is required that an JSf-statement be closed. Certain J?-statements may be called ^-axioms, and if there are J§?-axioms then there must be a method whereby we can decide of an oSf-statement whether it is an J§f-axiom or is distinct from everyjSf-axiom. Thus we could display the JSf-axioms or we could give a description of them by laying down that any J§?-statement satisfying such and such structural conditions is an j£?-axiom (provided we have a method whereby we can decide of an J^-statement whether it satisfies the conditions or fails to do so). Such a description is called an 3?-axiom scheme. There may also be given some rules of procedure called 3?-rules, these are relations between jSf-statements whereby given a set of J§f-statements satisfying certain structural conditions we may by applying one of the jSf-rules produce another J?-statement whose structure depends in some definite manner on the structure of the members of the given set. The given set of J§?-statements is called the premisses (or premiss if there is only one in the given set) and the ££-statement produced by application of the rule is called the conclusion of that rule. It must be possible to decide whether a given JSf-statement is the result of an application of an jSf-rule to given premisses or whether this fails to be the case. The jSf-rules are depicted as follows: O S'

OY S

and so on. There may be conditions on
20

Ch. 1 Formal systems

J§?-statements in the set by application of an j£?-rule is called an <£-proof of the last J&f-statement in the set. In particular cases the partially ordered set might be linearly ordered. A connected tree-like figure consisting of columns of J§?-statements with an JSf -axiom at the head of each column and an application of an J§?-rule between each consecutive vertical pair of members of a column or at a place where two or more columns terminate and are replaced by a single column and whichfinallyends in a single J§?-statement O is called an J?-proof of O in tree-form or an £?-proof-tree of O. The JSf-statement O is called the base of the tree. The portion of the tree which can be reached by proceeding upwards from a given jSfstatement T in the tree is an JSf-proof of T in tree-form and is called the branch of the tree ending in T. An J?}-proof thread is a linear column of JSf-statements which forms a consecutive part of an «Sf-proof-tree. An jSf-proof can be checked since it is possible to decide if a sequence of signs is an J§f-formula and to decide the type of a well-formed JSf-formula and so decide whether it is an J&P-statement and to decide whether an j£?-statement is an jSf-axiom and to decide applications of JS?-rules. The set of J§?-axioms and J5f-rules are collectively known as the J£-rules of consequence.

An 3?-theorem-scheme is a description of a set of JSf-theorems together with instructions for obtaining the JSf-proof of any member of the set. jSf-theorem-schemes are frequently stated when a set of J§?-theorems have JS?-proofs on the same general pattern, we then describe the pattern. In a formal system the J5f-rules are normally J5f}-rule-schemes, that is to say they are descriptions applicable to a variety of cases. O $$' We use the notation ^ *, -^— *, etc. to denote that from an JSP-proof of O (of O and O', etc.) we can find an J5f-proof of T. We use the notation =,

, etc. to denote that T can be obtained from ',

etc.) by the jSf-rules. In this case if the upper formula (formulae) are <J) <J) (J)' $ (J> (J)' jSf-theorems then so is the lower formula. ^- *, 1T. *, = = , etc. are called derived J?-rules. The demonstration of ^ * consists in giving instrucO tions to find an J§?-proof of T from an J§f-proof of O. For = we must give the J§?-rules used to transform O to Y. We also use the notation

1.8 The rules of consequence

21

O'... 0 ~ ,'" • to denote that each of T , . . . , ^ may be obtained from O'... O(0> ( ®', ...,O ^ by the J5f-rules. Similarly we use the notation Xf/t/"' ^(/c) * to denote that from the JS?-proofs of ®',..., O(6I) we can effectively find J§?-proofs for each of Y',..., Y(/c). If # = 0 the above notations reduce to ¥ H T P > and Y'...Y<*>* respectively. These mean that Y', ..., Y<*> are O J§?-theorems. Note that -^ whenever H is an J§f-theorem, in this case 0> ®Y O is unused. Also if -^ then = here again Y is unused.

1.9

Corresponding and related occurrences

Let capital gamma with or without superscripts or subscripts be signs foreign to a formal system J§?. Then ®{F} shall denote a terminating linear succession of J§?-symbols and F's, ®{F, F'} shall denote a terminating linear succession of 3?-symbols F's and F"s, here <&{F'} is to be without occurrences of F, ®{F", P"} without occurrences of F, P , and so on. The terminating linear successions ^{r}, 0{F, F'}, and so on, are called J£-formula-forms. If T, Y' are non-null j£?-formulae then O{T}, {*¥, Y'} shall denote the results of everywhere replacing F, F' by Y, Y' respectively in the JSf-formula-form {F}, and so on. This is done by inserting a piece of tape which exactly contains the J§?-formula Y in place of the square which contains F, this is to be done for each occurrence of F, in O{F}. Similarly each square containing F' is replaced by a piece of tape which exactly contains Y \ Thus if O{F, F'} is Q* ^ Q*. ^ jgf _f o r m u l a e , hence O ,n r n o ,,n r ,n o/// n r n o ,,, w h e r e Q,? ^ without occurrences of F, P , then
22

Ch. 1 Formal systems

because there might be occurrences of S or of T in ®{F}. Suppose 0"{F}nH is an occurrence of S in '{Y{28'}, ^ T ^ ' ^ T 5 ! 2 8 ' } i s c a l l e d a n occurrence of E{2B'} in O I T ^ ' } , SB'} related to the occurrence Y'{2B}nS{3B} of S{3B} in Y{3B} by substitution of Y{©'} for F in d>{F, ©'} and the substitution of SB' for SB in {P^ is a closed JSf-statement-form and Syy is a variable new to 0{F^} then ^{S^} has 2^ as sole free variable provided that some occurrence of F^ in ^{T^} is outside the scope of any occurrence of XF^ in

I.IO The \-rules If lambda is an JSf-symbol then J§? may contain the \-rules applicable to JSf-variables and S£-formulae of certain types. The \-rules for JSf-variables of type /? and JSf -formulae of type a are: (i)

If 2^ is an JSf-variable of type /? and T^ is an JSf-formula of type ft

1.10 TheX-rules

23

and O ^ T J is an JS?-formula of type oc whose jSf-formula-form Oa{F/?} lacks occurrences of the J27-variable 2^ then

is a rule of procedure provided that each occurrence of an J5fvariable 2 in O^Y^} which corresponds to a free occurrence of that variable in T^ is a free occurrence of 2 in O^T^}. (ii) Conversely with the same proviso and the same notation

is a rule of procedure. (iii) Let 0{F} be an J§?-formula-form, 2 an J^-variable. Let O{2} be the scope of an occurrence of (nXn2 in an Jg?-formula Y{(nXn2nO{2}n)}. Let 2 ' be an JSf-variable distinct from and of the same type as the <5f-variable 2. Let ®{F} lack free occurrences of the JS?-variables 2, 2'. Let each occurrence of F in ®{F} be outside the scope of any occurrence of (nXn2' or of (nXn2 that there may be in 0{F} then Y{(nXn2nO{2}n)} Y{(nXn2'nO{2'}n)} is a rule of procedure. It is called change of bound variable. If each occurrence of F in ®{F} lies outside any scope of Xn2 then we say that F is free for 2 in O{F}. i. 11 Definitions and abbreviations We shall frequently introduce new symbols to abbreviate closed formulae of a given formal system J£f. The new symbol is to be considered as everywhere replaced by the jS?-formula it abbreviates. This device is adopted merely to prevent jSf-formulae becoming of unmanageable length; it also facilitates reading, and enables one more clearly to see certain aspects of the structure of an j£?-formula. An jSf-formula occupying a few pages of print would be difficult to assess; such a formula might begin with several lines consisting entirely of left parentheses; but if broken up into parts and new symbols used as abbreviations for the different parts we might arrive at a formula which could be printed on a line and certain features of its construction would be apparent at

24

Ch. 1 Formal systems

a glance. If the new symbol abbreviates a well-formed jSf-formula then it can be given a type. Sometimes we shall introduce formulae containing new symbols to stand for certain other structurally related jSf-formulae. When this is done the new symbol itself fails to stand for an jSf-formula but certain formulae containing the new symbol stand for certain <J§?-formulae. This device is usually adopted if J§? fails to contain the abstraction symbol and its rules. Thus 'N9 could be introduced by (nNnOn)

(n(n£nOn)n
for

where O is of type o and ' $ ' is of type ooo, so that (N9 is of type oo. But if abstraction is present together with the X-rules then we could define N

for

(nXnSn(n(n/SnSn)nSn)n),

where 2 is a variable of type o. Another case of this type of definition which occurs in Ch. 4 is: i\rn[n"ncDn"n], for the formula which results when we everywhere replace 4= by =

=

by 4=

V by

&

& by

v

Here JV, [, ], " and " are new symbols. Sometimes we replace an J5f-formula by another one containing the same proper symbols but in a different order. This rearrangement of the order of J§?-symbols in a well-formed JS?-formula will sometimes be used when the order required by the JS?-rules of formation is different from that to which we have been accustomed. This again makes for easier reading. Thus (x = y) will often be written instead of ((= x)y), — is of type ou. i. 12 Omission of parentheses Lastly we shall frequently omit parentheses according to the following: (i) The outer pair of parentheses may be omitted, (ii) Parentheses may be omitted by association to the left. Thus

ocfiyS

will stand for

(((ayff)y)*).

oc/3(y8) will stand for

((a/?) (yS)).

1.12 Omission of parentheses

25

This device is adopted because it soon becomes difficult to see the structure of a formula on account of a multitude of parentheses. Parentheses round applications could be entirely omitted without affecting the use of a formal system. (iii). / / all parentheses round applications in a well-formed ^-formula are struck out then there is a unique method of restoring them so that the result is a well-formed ^-formula. Let O be a well-formed J§?-formula and let ®' be the result of removing parentheses round those applications which are outside abstracts. The only parentheses left in ®' will then be round or inside abstracts; each abstract is a well-formed J5f-formula, replace it by a new symbol of the same type. Let this transform ®' into ®". Then ®" is a formula void of parentheses, it is a linear sequence of symbols each having a type. We know that it is possible to replace the parentheses so that $" is converted into a well-formed J§?-formula O'" (
(Sd)- times

26

Ch. 1 Formal systems

The symbol next to S y must be of type 5(S<9) or ( . . . ( # W T / ' ) . . . 7/(Sf7]r), where r/f, ...,7}(Sn) are type symbols, otherwise it is impossible to replace parentheses in ®" so that the result is well-formed. If it is S ^ ) of type then parentheses go back as ( n S ( a /( n ... T2 n r £ > > (case (a)). (Sd) -times

Here (Z£ SJosfl) is a well-formed formula of type (... (fid')... <&<*>). Replace it by a new symbol of this type. This converts O" into a shorter formula
(Sn) -times

So we continue, but the process will terminate with case (a) occurring, we can then put in a right parenthesis and replace a part ("S"^/)!^) by a new symbol of type a'. This converts ®" into a shorter formula $ v , we then proceed with Ov. The whole process terminates in at most as many steps as there are proper symbols in O. Thus given a linear sequence of proper symbols we can either insert parentheses uniquely so that the result is a well-formed formula or we can discover that it is impossible to do so. In a similar manner we can deal with parentheses inside abstracts. But if we omit parentheses round abstracts in a wellformed formula then there may be several ways of replacing them so that the result is a well-formed formula. Consider the formula where 2^ and S^ are distinct variables of type J3. This arises from either of the well-formed formulae, both of type (ocfi):

2> 32

1

a{^ *

^ j 12

3

n 3 2

by omission of parentheses round abstractions, and omission of the outer pair of parentheses. Mates are shown by subscript signs. Using the X-rules these two formulae may be replaced respectively by: and

(YS;O^,

provided that the proviso of the X-rules is satisfied.

1.12 Omission of parentheses

27

After this chapter we shall usually omit the concatenation sign. We shall usually write (XS^.OJ for (XS^OJ, the dot before the scope of XS^ makes for easier reading. Also we shall usually write ^ . O J

for

a formula of type ((a/ 1.13 Formal systems To sum up, a formal system is an ordered quartet {Sf ^ , stf, 0>), £f is a display of signs, some of which may be designated as generating signs, ^ is a description of rules of formation, si is a display of axioms or a description of axiom schemes, 2P is a description of rules of procedure. It must be possible to decide of an object whether it comes under one of these cases or is foreign to them, only then is it possible to read the formal system and to check proofs. Thus we say that a formal system is constructive.

A formal system 3? may be without rules of procedure, in this case the jSf-theorems are just the J5f-axioms. A formal system J§? may lack axioms, in this case the formal system J§? is without theorems and we are then only interested in transforming jSf-statements into other ££?-statements by the JSf-rules. Usually we then speak of transforming^7-formulae of a certain type into other ^-formulae, and the formal system ££ is then often used as a system of calculations, say of the value of functors. If we know a procedure which will decide whether an jSf-statement is an jSf-theorem, we can omit the ££-rules and take as jSf-axioms the ^-theorems, because the requirement that it be possible to decide whether an J5f-statement is an JSf-axiom remains satisfied. Thus jSf-rules are only required when we lack a procedure to decide whether an JSf-statement is an JSf-theorem. A formal system J§? is called decidable if we have a procedure to decide if an JSf-statement is an J§? -theorem. But we can write down the jSf-theorems one after the other, so that if we continue long enough any JSf-theorem will appear in the list. To do this we select a new symbol, say • • We then denote a sequence of JSf-formulae O' O" ... T by the formula O'n n n O" n Q n . . . n • " Y of a system J§?' obtained from the system JSf by adding the typeless symbol • • We then give an order of preference, called the alphabetical order, to the symbols, we then order the JSf'-formulae first by length and lexico-

28

Ch. 1 Formal systems

graphically for those of equal length. In this manner we can generate the JSP'-formulae one after the other. When an ££'-formula has been generated we test it whether it is of the form O / n •"<&""•"..."•"^ where the sequence O' ®"... Y is an jSf-proof of Y. If the test is affirmative we write Y down in a list. In this manner we generate the J5f-theorems one after the other without omissions but with repetitions. 1.14 Extensions of formal systems A formal system J2?' is called a primary extension of a formal system oSf if the jSf-symbols are jSP'-symbols of the same type and if the J5?-axioms and JSP-rules are JSf' -axioms and JSP' -rules respectively. Thus a primary extension of a formal system J5f is obtained by doing some of the following operations: adding new symbols to JS?, adding new axioms, adding new rules of procedure. If JSP' is a primary extension of J§? then J§? is called a subsystem of oSf'. JSP is an improper primary extension and an improper subsystem of itself. Two formal systems jSf and J?' are equivalent when their variables are of the same type (by a trivial adjustment we can then use the same symbols for variables in both systems) and when the constants of the one system can respectively be replaced by suitable formulae of the other system of the same respective types in such a way that the theorems of the one system translate via the replacements into theorems of the other system. If the formal system JSf is equivalent to the formal system J2?' and if ££" is a primary extension of J§?' then J§?" is a secondary extension of JSP. Two formal systems, JSP, JSf', can be equivalent merely because of different shapes in the choice of primitive symbols, or because though they both have exactly the same primitive symbols and each symbol has the same type in both systems yet the j£?-axioms are J§?'-theorems and the JSP-rules are derived JSf'-rules and vice versa. jSf, J§?' can be equivalent when the variables are the same and of the same types in the two systems yet the proper constants are different and perhaps of different types. For instance some symbols introduced into 3? by definitional abbreviation might be primitive ££' -symbols of the same type as the jSf-defined symbol. This, in general, would require that the J§?-axioms and J^-rules be distinct from the jSf '-axioms and jS respectively.

1.15 Truth definitions

29

1.15 Truth definitions Let JSf be a formal system and let an jSf-statement be of type o. A truthdefinition for JS? is a set of conditions ZFg applicable to closed JSf-statements. We say that a closed J§?-statement n\n\n\pn\

(c.f. 'The hunting of the shark', 'what I say three times is true'). In a similar manner we can define a falsity-definition (denoted by tFg) for a formal system j£\ If we define ZFg and 3Fy for a formal system J§? then we require that they be exclusive. 1.16

Negation

Let J§? be a formal system and let JSf-statements be of type o. Let N be an JSf-symbol of type 00 and let O be a closed JSf-statement then (nJVnOn) is an J5f-statement. Let ^ be a truth-definition for 3? and let J ^ be a falsity-definition for j£? and let ( T O n ) satisfy J ^ when O satisfies^ and ( T $ n ) satisfy &*# when O satisfies J*>. Then we say that N is a two-valued ^^-negation symbol. If a formal system j£? contains a two-

30

Ch. 1 Formal systems

valued negation symbol then we say that J? is consistent with respect to negation if one of is a closed jSf-statement. H I S T O R I C A L REMARKS TO C H A P T E R 1

The invention of language, written or otherwise, is lost in the mists of antiquity, but the invention of formal systems is of recent date, though the use of special symbols to augment a conversational language is of much older origin. Aristotle used capital letters, in the way we now use variables, to stand for undetermined propositions. Mathematicians used various symbols to denote mathematical terms, operators and concepts. But the first idea of a formal system goes back to Leibniz, Bibliography of Symbolic Logic in J.8.L. i. He wished to invent a formal system (characteristica universalis) which would suffice for all science. As yet we have only got as far as inventing formal systems for logic and mathematics, though one for some sciences such as Newtonian Dynamics or Thermodynamics would be possible. Leibniz also wished to invent a method (calculus ratiocinator) for manipulating statements in his projected formal system. But no school resulted and only fragments, though significant ones, were left. The next investigators who are of interest to us are G. Boole (1847-54), A.de Morgan (1847), E.Schroder (1890-5), G.Frege (1879-1903), G.Peano (1889-1908), and C.S. Pierce (1933). Much of our modern logical and mathematical notation derives from Peano and Schroder. Modern symbolic logic really starts with Boole, his work was greatly enhanced by Schroder. Frege's 'concept writing' is very complicated and has been avoided by all subsequent writers. Boole noticed the similarity in use of the logical constants of conjunction and of disjunction with the mathematical operations of multiplication and addition. Thereby he could bring to bear mathematical methods into logic, this proved very fruitful, and his treatment gave great impetus to the development of formal systems. The Principia of Whitehead andRussell (1910-13) built on the work of Boole, Schroder, Frege and Peano, was the first attempt at a completely formalized language and was worked out in great detail. Even so the

Historial remarks to Chapter 1

31

authors said very little about the construction of such systems, so that they had little to correspond to our Chapter 1. In fact in the first edition they failed to mention a rule of proof which they used on practically every page, namely the rule of substitution. This omission was corrected in the second edition. This monumental work had tremendous influence in the development of formal systems, symbolic logic, etc. The X-symbol occurs there for the first time. The theory of types given is very complicated, it is called the ramified theory of types, this was invented by Russell (1908) to obviate the paradoxes which were cropping up in set theory and in the theory of infinite cardinals. These paradoxes are of a kind known as syntactic because they can be eliminated by change in the syntactic rules, i.e. by change in the rules of formation in the formal system. Variables really stem from Newton and the definition of a variable which we have given is a product of modern symbolic logic and recursive function theory. The concatenation sign is due to Tarski (1933). The idea of using a tape divided into squares is due to Turing (1936). We use it right at the start so as to be in no doubt as to when a formula begins or ends or whether we are dealing with a formula or with a sequence of formulae. The idea of using Greek (or Gothic) letters to augment the syntax language (the language in which we talk about the object language (formal system) we are inventing) is a product of modern symbolic logic, see for instance Carnap (1937). The definition of 'occurrence' we have given is due to Quine (1951). Carnap had great influence in defining a formal system as an ordered quartet. He and Church (1932) established the use of the abstraction symbol. Hilbert and Bernays (1934) introduced the use of proof-threads and formulae-forms in their dissection of proofs. The X-rules were first correctly stated by Church (1941). Definitions as abbreviations are due to Russell and emphasized by Quine (1951), (P.M. vol. 1, p. 11), see also Markov (1954). Lemmas of the type of lemmas (i) and (ii) are due to Church (1941) and lemma (iii) is due to Lukasiewicz who wrote most of his work on the Propositional Calculus without using parentheses. The concept of a decidable system is due to Hilbert, he wanted to find a decision procedure for each formal system, this is now known to be impossible. The concept of truth goes back to the ancient Greeks at least. Epimenides was the first to find something really interesting about it, viz. the antinomy of the liar, this we will come across later in the book. Truth definitions for formal systems stem from Tarski (1933). The

32

Ch. 1 Formal systems

concept of a consistent system is due to Hilbert (1904, 1922, 1930); he wanted to show that formal systems of arithmetic and analysis are consistent. We shall have much to say about this as the book proceeds. Earlier writers like Frege, Peano, and Russell were mainly concerned with proving theorems in their systems. Hilbert was mainly concerned with finding out whether a given formal system was consistent, complete, decidable, etc. In other words he was mainly interested in metamathematics, theorems about theorems, and in the methods used in metamathematics (this term comes from Hilbert), in the methods of proof allowed in metamathematics, here he touched on effectiveness, finiteness, etc. We shall be mainly concerned with what we can do with a given formal system. What concepts we can express in it, whether it is consistent, complete, decidable, etc. Whether it has a truth definition, and so on. We shall usually find that our systems have limitations and that in extending the system to remove such a limitation we get another system which fails to have some 'nice' property which the un-extended system had. Our main interest is to construct a language in which we can talk about the natural numbers, to formalize our intuitive concept of natural number. But we are only interested in the properties of these systems, we are not much interested in proving theorems in the system itself. EXAMPLES 1

1. Give a method for enumerating the well-formed formulae (w.f.f.) of a formal system. 2. A formal system is given as follows: signs: x ' () X,' is a generating sign, x is a variable, if £ is a variable then so is £n/, these are all the variables. Rules of formation: a variable standing alone is w.f.f., if ^ and \[r are w.f.f. then so is (
Examples 1

33

where G and 8 are of type ooo and the other letters are symbols of type o. 4. Apply the X-rule (i) repeatedly to: (Xa(kb((Xa)((irab)))), where

r

x f° Xa6.(a6), i/r for lkab.a(ab)

and obtain

Xa6. a(a(a(a(ab)))).

(baixW0)))

Similarly for and obtain 5. Define:

Xa6 .a(a(a(a(a(ab))))). J

for

'Xabcd.ab(adc)

B

for

Xa&c. a(bc)

C for

Xa&c. acb

W

Xa6. abb

for

show that B(BC(BC))(BW(BBB))C) reduces to J by applying the X-rule (i). And that if T for Xa6. ba then B(B(T(BD(B(TT) (B(BBB) T)))) (BBT)) (B(T(B(TI) (TI))) B) reduces to W by X-rule (i) where D for Xa. aa and / for Xa. a. 6. Apply X-rule (i) to (T^x.xxx) (kx.xxx). 7. If O can be obtained from T by the X-rules then it is possible to do so when all applications of X-rule (ii) come after applications of X-rule (i). 8. A formula is said to be in normal form if it is impossible to apply X-rule (i) to it. Show that if a formula can be reduced to one in normal form by the X-rules then this normal form is unique to within change of bound variable. 9. A system has signs S C ab x ' the variables are x x' x" .... The axioms are Sa, 8b, Sab, Sba, CSxaSxab, CSxbSxba. The rules are -q-jik, — q o

, where £ is a variable and a, p are

strings in a and b and variables, £ fails to occur in a{F}. Put this system into the type notation of a formal system as defined in this chapter.

Chapter 2 Propositional calculi

2.1 Definition of a propositional calculus A propositional calculus is a formal system in which the types of the symbols are among those given by the scheme: (i) o is a type symbol, (ii) if a is a type symbol then so is (ao), (iii) these are the only type symbols. A propositional calculus must have at least one symbol of type o. Symbols of type (ceo) are called connectives. The only variables, if any, are of type o. These are called propositional variables. The abstraction symbol is absent. A pure propositional calculus is a propositional calculus without constants of type o. An applied propositional calculus is without variables. A mixed propositional calculus contains both constants and variables of type o. If 3P' and &" are propositional calculi and if both contain propositional variables then by a trivial change of notation we may use the same symbols for variables in the two systems. Thus we shall take the propositional variables in a propositional calculus to be: p, p', p",... where the prime is a generating symbol. In this chapter we use n with or without superscripts or subscripts to denote an undetermined propositional variable. Suppose we have a propositional calculus 8P which contains constants N and D of types oo and ooo respectively. In terms of these constants by adjoining the abstraction symbol to 3P we can define a constant K of tvpe ooo thus: ^ K for \pp'.N(D(Np)(Np')). This is the normal form for definition of a new symbol. The new symbol stands for a formula. We could have avoided the use of the abstraction symbol as follows: * JTOO' for N{D(NQ)(NQ')), [34]

2.1 Definition of a propositional calculus

35

where 0, O' are formula of type o. In this kind of definition the new symbol K is without definition by itself but is only defined in contexts in which it will be used. By adjoining the abstraction symbol and variables of suitable types the second kind of definition can always be reduced to the first kind. If a propositional calculus is without propositional variables then we can adjoin them and the abstraction symbol solely to use the first kind of definition. Thus using the first kind of definition we see that K is of type ooo and that iiTOO' is an abbreviation for (kpp'.N(D(Np) (Np'))) 00', which by X-rule (i) becomes N(D(N<3>) (iVO')).

2.2 Equivalence of propositional calculi A propositional calculus 3P* is weaker than a propositional calculus 0* under the following circumstances: First add a suffix 1 to each symbol of SP and a suffix 2 to each symbol of 0i\ except that if there are variables then they are to be the same in both systems. This makes the symbols of SP and 8P1 distinct, except possibly variables, if any, which are to be the same symbols in both systems. (i)

SP' is without variables if SP is without variables.

(ii) Definitions are given of the constants of SP' in terms of the constants of SP\ for this purpose the abstraction symbol and variables may be adjoined. (iii) A ^'-theorem O' becomes a ^-theorem O when the constants in O' are replaced by their definitions in terms of the constants of SP and the X-rules are used to eliminate the X-symbol. Thus if 0*' is weaker than SP then SP can express anything that 0*' expresses and ^'-theorems translate into ^-theorems. If a propositional calculus SP' is weaker than a propositional calculus 0* and if 0* is weaker than SP' then SP and 8P' are equivalent. If a propositional calculus 0*' is weaker than a propositional SP and if they fail to be equivalent then 0*' is strictly weaker than SP. It can happen that SP and SP1 are equivalent according to the above definition but some ^-theorem fails to be the translation of any ^'-theorem and vice versa. If some constant of 0* is the same symbol as some constant of 0*1 before we put on the suffixes 1 and 2 and if all such constants correspond in the translation then we say that SP and SP' are equivalent by natural translation.

36

Ch. 2 Propositional calculi

2.3 Dependence and independence Suppose that 3P' has the same constants as 3P except that 8P has a further constant Y and that 3P is equivalent to 0P' by natural translation, then the constant Y is dependent. In this case the constant Y can be defined in terms of the other constants of SP (possibly using the abstraction symbol). We can express this as follows: if a propositional calculus SP contains a constant Y and if there is a definition of Y in terms of the other symbols of 3P such that the system which results when Y is everywhere replaced by its definition is equivalent to 3P then Y is dependent. Similarly if several symbols of 3P can be defined in terms of the remaining symbols of 8P and if the resulting system is equivalent to the original system then these symbols are dependent. If the propositional calculus 3P' is the same as the propositional calculus 3P except that 8P' lacks the axiom have exactly the same constants of compound type. (ii) *Jt has constants of type 0, called elements, some of these are designated the remainder are undesignated, at least one element is designated and at least one is undesignated. (iii) Jt is without axioms. (iv) The rules of Jt enable one to replace any compound ^-statement <J> by a unique element, called the value of the ^-statement O. (v) If we replace ^-constants of type 0 (if any) by suitably chosen fixed ^-elements and propositional variables (if any) by arbitrary ^-elements then a ^-statement O translates into an ^-statement, whose Jl-value is called an Jl-value of
2.4 Models of propositional calculi

37

A ^-statement is called Jt'-valid if each of its ^-values is designated. Since a model Jt is without axioms then it is without theorems. We can only find the Jt-value of ^-statements. We could have taken the designated ^-elements as axioms and reversed the rules, the Jttheorems would then be the ^-statements with designated ^-values. But we wish to use the model to find ^-values of ^-statements so the formulation given above is more natural for our purposes. A model Jt of a propositional calculus & will be called trivial if the ^-value of every compound ^-statement is designated. Every propositional calculus possesses a trivial model. Let F be a ^-constant of type o... o, to obtain a model Jt we have to (SO) -times

fix the ^-elements and give the ^-values of FT' ... T(^, where r', . . . , T ( ^ are ^-elements, the e^-value is to be an ^-element. Thus to obtain a model Jt of a propositional calculus 3P we require to give Jt-tables for each ^-connective. Having obtained a set of tables for these connectives we then have to test if condition (v) is satisfied, that is that ^-theorems always take designated Jt-values. It suffices to show that the ^-axioms are Jt-valid and that the ^-rules preserve ^-validity. Let Jt be a model of a propositional calculus 8P and let F, F' be ^-connectives of the same type. If the tables of F, F' are the same then F and F' are Jt-identifiable. If r', 7" are both designated ^f-elements or both undesignated ^-elements and if the model Jt becomes a model Jt' when T' and r" are both replaced by a new element r of the same designation as r' and r" then r' and r" are indistinguishable. A model Jt of a propositional calculus 2P is basic if each pair of designated elements is distinguishable and similarly for each pair of undesignated elements. A propositional calculus is v-valued if it has a basic model with exactly v elements. We must have v ^ 2. (I an initial segment of (v)). A model Jt of a propositional calculus SP has been defined as a formal system of a special kind, it must then be constructive. This will be the case if the designated and undesignated elements are displayed. Two ^-statements o) and o)f are equivalent if for each O{F}, ${&>} is a ^-theorem if and only if <&{a)'} is a ^-theorem. Let £2W be the class of ^-statements equivalent to the ^-statement 0). Clearly equivalence is a reflexive, symmetric and transitive relation. Hence Q^ is the same as Ow if o)' is equivalent to co. An equivalence class Q will be called designated if and only if a member of Q, is a ^-theorem, then each member of ii is a ^-theorem. This fixes the designated and the undesignated elements

38

Ch. 2 Propositional calculi

and the association of classes to ^-constants of type o. Let F be a ^-connective of type o ...o, let TQ'... £i(/9) have value Q. if and only if F V . . . (d6) is in O where &/A) is in |Q(A) 1 ^ A ^ 6. This gives us tables. These classes and tables form a model of 8P provided the result is constructive and that there is an undesignated class. In this model we take as elements the equivalence classes, and as connectives we take the ^-connectives. The tables we have found provide the rules of the model. It remains to show that the ^-theorems are valid in the model. Suppose that (p{n\ ...,7r(^} containing exactly the variables n', ...,n^ is a 0)theorem, then so is ${(o',...,G/^}, where n' is equivalent to G/, ...,n^ is equivalent to oP\ Thus 0{TT', ...,n^d)} is valid in the model. A propositional calculus 8P is model-consistent if it has a non-trivial model, it is model-complete if it has a model Jt such that every Jt-valid ^-statement is a ^-theorem. Sometimes we say consistent with respect to the model Jt', complete with respect to the model Jt. A propositional calculus SP is functionally complete with respect to a model Jt under the following circumstances: (i) Jt is a model of 3P, (ii) Jt is displayed, (iii) Jt' is an extension of Jt which contains connectives of all types (o ... o) and every possible table for each type, Jt1 has the same (SS6) -times

elements as Jt, (iv) Jt' is a model for a propositional calculus SP', (v) &P' is equivalent to 3P by natural translation. Let Jt be an applied propositional calculus which satisfies conditions (ii), (iii), (iv) of the conditions for being a model. We seek a pure propositional calculus 8P for which Jt is a model. 3P then contains variables p,p\... and the same connectives as Jt. We could take as ^-axioms all Jt'-valid ^-statements provided that this is constructive. Rules would then be unnecessary. If the set of JtWalid ^-statements fails to be constructive then we have to try to discover axioms and rules by trial an error. The problem of finding 3P is the problem of formalizing Jt. Given a formalization of Jt then we might try to find another formalization with the least possible set of axioms or a set of axioms containing the least possible set of symbols. We might try to find a formalization which is complete or which is functionally complete or an equivalent propositional calculus with a single connective. A formalization 8P of Jt is model-consistent because it has Jt for model.

2.5 Deductions

39

2.5 Deductions Deduction is represented in a formal system by application of the rules. The rules are of the form ~,

, ..... where the lower formula can be

constructively obtained from the upper formulae. An example is ——j-*-, known as Modus Ponens, here C is a connective of type 000. A proof in a formal system is a tree -likefigurewith axioms at the tops of the branches and the statement proved at the base and we proceed from the axioms at the tops of the branches downwards to the base by applications of the rules. Thus in a proof the upper formulae of each application of a rule are theorems of the system, the part of the tree ending at an upper formula is a proof of that formula. Thus each statement in the proof is a theorem. A deduction in a formal system is a tree-like figure such that we proceed from the tops of the branches to the base by applications of the rules. Thus a deduction differs from a proof only in that we can have statements other than axioms at the tops of the branches. Thus a deduction in a propositional calculus 3P amounts to a proof in another propositional calculus 3P' obtained from 3P by adding certain ^-statements as additional axioms, they are called hypotheses. We have already used the figure:

4' to denote that i/r may be obtained from (f)'', ...,(Sd) by use of the rules. This amounts to there being a deduction of ^r from $',..., ftSff) as hypotheses. This figure and the figure the figure

?

?

*'''' — must be distinguished from

* *'J —* which means that from the proofs of ^',..., <jtsd)

we can constructively obtain a proof of ^. In this case it may happen that - is impossible. Thus we have three different kinds of figures, namely: rules,

40

Ch. 2 Propositional calculi

(b)

= deductions, 6'

(c)

?

Ssd) * *'/ — * derivations.

In a formal system it usually happens that we are without means to express by a statement of the system that the statements ^',..., ftse\ ft are related in either of these ways. The fact that the statements ',...,
then a theorem C<j)i]f can be taken to express the fact that from a proof of (j) we can obtain one of ft. Similarly if in a formal system we have

then Ccfiijr can be taken to express a figure. Note the following relations between (a), (b), (c): if (a) then (b). If (6) and if

...,&Sd)

(e) if = then C^tp'... ftse)i/r is a theorem. Use of this metatheorem frequently enables us to shorten proofs, the full proof can be

2.5 Deductions

41

found from the meta-theorem. The condition will hold in a propositional calculus if the following condition: (f) if

' "*' = t h e n = ^ " '

for some primitive or defined symbol C

6' 6 of type ooo. For suppose (/) t h e n : if Y'"'*Y

t h e n Cfi(C...

(CftS0ty)...)

is a theorem. We can then define C^ by for C^'( Thus we obtain (e). In a propositional calculus in which (iv) below holds a necessary and sufficient condition that (/) hold is that the conditions (i), (ii), (iii) below should hold, (i) Ctfxfi be a theorem. (ii) Ccfrx be a theorem whenever x is an axiom. (iii) = . (iv) If —^—^

is a rule then

=

0

For suppose (i), (ii), (iii), (iv) and —J In the tree obtained by writing out

— in full replace each

premiss and conclusion x by C^se)x- The tops of the branches become C^^sd) o r CfjFVfi, ...tC^fi*, or Cft^x, where X is an axiom. In the first and last cases we add the proofs of the theorems given in (i) and (ii), in the other cases we add the deductions given in (iii). In the rest of the tree we replace use of a rule by the full proof of the derived rule 6',...,6<® 6 given in (iv). This gives us . Conversely suppose (/): we have = / whence Ccjxj) by (/), that is Ccjxj) is a theorem hence (i). Again we have = where x is a n axiom hence Ctfix by (/) and so (iii). Hence the result.

42

Ch. 2 Propositional calculi

2.6 The classical propositional calculus A pure propositional calculus is called classical if it is equivalent to the following propositional calculus, £PC. Symbols: Noo Dooo p0 ' ( )

negation symbol, disjunction symbol, variable, generating sign, left parenthesis, right parenthesis.

The parentheses can be omitted because we only use application, so that if a formula is well-formed the parentheses can be omitted because there is a unique method of replacing them so that the result is well-formed. This was shown in Ch. 1, lemma (iii). Axiom scheme. DnNn, where TT is a variable. Rules: Remodelling la DDDa)(f)}Jr(o' permutation. Building

n.

Jr.. dilution

m

M*??*" composition

ne

"*»

double negation

o), o)' are called subsidiary formulae and may be omitted, X is a secondary formula and must be present. 0, i/r are called the main formulae. If we omit a subsidiary formula then one occurrence of D is struck out. The premiss la written in full without omission of parentheses is ((D((D((DOJ) $)) ft)) 0)'), which is the only way of replacing the parentheses so that the result is well-formed. Notice that any symbol introduced into a ^ a -proof remains in that 0*c-proof from that place onwards. Hence any ^-theorem must contain the variables which occur in the axioms used in that proof. A formal system S£ is called direct if any symbol introduced into an o5f-proof remains in the J^-proof from that place onwards. Thus the system 2PC is direct.

2.7 Some properties of the remodelling and building schemes

43

2.7 Some properties of the remodelling and building schemes 1 Disjunction is communtative and associative. Commutativity of disjunction:

PROP.

jyTi

I& with both subsidiary formulae absent.

Associativity of disjunction:

Again

D<j)D\jrx

la with left subsidiary formula absent, la with right subsidiary formula absent, la with both subsidiary formulae absent.

—-—— la with both subsidiary formulae absent, ^-^- la with right subsidiary formula absent, LJLA \a with left subsidiary formula absent. Hence disjunction is associative. 2. The schemes la, 116, c are reversible. That is to say, if we have a ^ c -proof of the lower formula of one of these schemes then from that ^ c -proof we can obtain a ^ c -proof of the upper formula (upper formulae) by carrying out the procedure that we are about to describe. Thus in our notation PROP.

DNNcjxi)

The scheme l a is reversible. This is clear. The scheme 116 is reversible. Suppose we have a ^ c -proof of then we have a tree whose base is DNDfii/ro). Follow corresponding occurrences of ND(j)^jr up the tree. If NDi/r in a main formula of the lower part of l a then there is a corresponding occurrence in the upper part of la. If there is an occurrence of NDi/r in the main formula of the lower part of II a

44

Ch. 2 Propositional calculi

there fails to be a corresponding occurrence of NDiJr up the tree we may have branches at subsidiary formulae of II6 and the only places where we shall stop will be at main formulae of II a or 116. In any case we shall stop before we reach the top of any branch. In the portion of the tree containing these occurrences of ND and if an occurrence of ND, and Ni/r. The scheme l i e is reversible. We proceed in a similar manner. Given a ^ o -proof of DNN<po) we trace corresponding occurrences of NNcj) up the ^ c -proof-tree and note that NN<j) can only be introduced at applications of II a, c. Replace corresponding occurrences oiNN
3. We have the derived rule 16 77 , . * cancellation.

Suppose we have a ^ c -proof of Dtfxf). Using la repeatedly we obtain D... DO®, where O is a formula of the form i/r'n... ni/ASd\ parentheses are omitted as in la, each ^(A), 1 ^ A ^ S6 is either a variable or the negation of a variable or is NNx^x) where ^(A) is a ^-statement

2.7 Some properties of the remodelling and building schemes {X)

{X

(A)

45

A)

or is NDx to \ where ^ and o/ are ^-statements. By repeatedly using I a and the reversibility of II c, then I a repeatedly we may replace NNx(X) by #(A). (la is used to bring NNx{X) to the left.) By repeatedly using the reversibility of 116 then la repeatedly we may replace NDx^co^ by Nx(A) or NOJ^\ Continuing these three operations as long as possible (this amounts to moving occurrences of N as far to the right as possible) we obtain ^ o -proofs of formulae DD... DD... DHWW, 1 < /i ^ v, (1) where m^\ 1 < /u, < v is a formula formed from a sequence of variables and negated variables. The process must cease because at each of the last two steps we decrease the length of the formula dealt with, and applications of l a are limited. The statements (1) are ^-theorems and their ^-proofs proceed from ^ c -axioms using ^ c -rules la, I I a only, because (1) is without occurrences of ND or NN. Thus each T ( ^ must contain a variable and the same variable negated. Thus D.^DY^ is a £PCtheorem. By repeated use of II6, c, and of course I a, following the inverse order of the applications we made of their reversibilities, we obtain a ^ o -proof of
DC/HO

Di/rco

—*—• TT He —L— DNNcfxo DNNJ/TO) —rkV,r,

by definition of K.

4. (i) Rule 116' is reversible. (ii) Conjunction is commutative and associative. (iii) Disjunction is distributive with conjunction. (iv) Conjunction is distributive with disjunction. (i) From the reversibility of 116 we can obtain ^ o -proofs of DNN
PROP.

46

Ch. 2 Propositional calculi

(ii) We have

* by the reversibility of 116', hence by 116' we get

) , so conjunction is commutative. We have: to show

rz

, JL , * and

TZT.,

Y

,

*.

* by reversibility of 116' * ditto h

l l v

by 116' Hence conjunction is associative. We have to show

reversibility of 116

D
KDHX , DKx Kfx

and

DKfaKfx ' KDjfx

'

We have reversibility of 116' Io>n6,

DKfx^DKfxx,

reversibility I J

ofII6

X Hence conjunction is distributive with disjunction. We use the notation (j) = ^ to denote that ^j-A * and ^ji^ *, for any #, we then say that
2.7 Some properties of the remodelling and building schemes

47

PROP. 5

1. D

idem1*. Kcjxj) 'potency commu- 2*. tativity

2. D 0 0 ' = D$' 3. 2 ) D # ' 0 " = D<j}D<j>f
associ-

3*.

4. BK^>4>'4>" = KD" distri4*. butivity r r

r

r

_ 5*. NK66 = DN6N6 ganlaws 6. iV'iV^ = $5 double negation.

We have to show: rv. * and }/-> *> for any %, and any pairs (j) and ^ Air /

Air/

listed. x{$} is §S: 1 is 16 and II a, 1* is 116' and its reversibility, 2 and 2*, 3 and 3*, 4 and 4* have already been demonstrated. 5*, 6 come from l i e and its reversibility. And 5, if KN' is a ^-theorem then so are N(j> and N$', whence by 116 so is NDcfxj)'. If NDr. Otherwise we note the place where (j) is introduced into the ^ o -proof of x{fi} and then introduce i/r instead. If is introduced by different rules according to which of 1-6 we are considering. Take, for instance 4* with KD'(f)" for $ and BK^" Kficj)" for ^ . KD(f>$'" can be introduced at 116' Replace this by

•

la

48

Ch. 2 Propositional calculi

between this and the base of the ^-proof-tree replace KDcjxj)'^)" by DK"K'" and the ^ c -proof-tree of #{iLD^'0"} is converted into a ^ o -proof-tree of x{DK
where

then

t

D 2. G for \pp'. DNpp'. Then G is of type ooo, it is called the conditional symbol; in our parenthesesless notation we can define G for DN. We have to show (i), (ii), (iii) and (iv). C expresses material implication. (i) DNCJHJ) is a ^ c -theorem, this is known as tertium non datur. We proceed by induction on the construction of , this is known Sbsforumula induction. If ^ is atomic, that is if is a variable, say p, then the result follows at once from the axiom DpNp by la. If it is of the form Nijr and the result holds for ^ then we have

as desired. If (j) is a disjunction Di/r'i/r" and the result holds for xjr' and for \jr" then we have

DNft'DiJr'iJr"

TLa,

la

DNjr" Df'f"

thus DNcjxj) is a ^-theorem for any ^-statement N<j>x is a ^-theorem whenever ^f is a ^ o -axiom. This follows at once from II a. (iii) =X9 this again follows at once from II a.

C(px

2.8 Deduction theorem

49

(iv) If ^ and %-%- are ^ o -rules then DN(j>x these follow at once from II a, l a and the rule in question by putting into the subsidiary formulae. Thus

la

using the rule y with N<j> in the subsidiary formula. The other cases follow similarly. Thus the deduction theorem holds in 3PC. 2.9 Modus Ponens The rule

X=

™,

DojX

where co is subsidiary and can be absent, x is secondary and must be present, is known as Modus Ponens, the rule of detachment or the cut. The formula $ is known as the cut formula. PROP.

7 Modus Ponens is a derived rule in 0>c.

We have to show: Do) may be absent but x must be present. We have to show how we can obtain a ^ o -proof of DOJX when we are given ^ c -proofs of Doxfi and DNfix- The demonstration is by formula induction on the cut formula x, that is we suppose the result holds for the formula or formulae in the ^ o -proof-tree immediately above DNx is a remodelling of a ^ o -axiom then x must be is in the subsidiary formula or secondary formula of the building rule immediately above DN(j>x in the ^ o -proof of DN(j>x so that we have for a one premiss rule _ __ , , DNcpx

m Ia'IIaorc

50

Ch. 2 Propositional calculi

thus we have by our induction hypothesis: la, I I a or c, exactly as before.

Dcox

Similarly for a two premiss rule 116. If N<j> is in the main formula of the building rule immediately above DNcfrx in the ^ o -proof-tree of DN<j>x then this rule can only be II a since
DNx is where x *s Dx'x"

, or

^s J us ^ X'- Hence we obtain

as desired. (6) (j> is D$'$" and the result holds for ^' and $". We have ^ o -proofs of DcoDfi<})" and of DNDfrfi'x* From the reversibility of 116 we can obtain ^ o -proofs oiDN^'x and o£DN<j>"x- Hence by formula induction: cj)" DNfi'x * ,by tormula i induction • ,1 *•

~z—T,——^

^^

Y

A % by formula induction Prop. 3.

(c) <j) is N(j)r and the result holds for '. We have ^-proofs of Da)N Thus if a) is present we have by formula induction V DX(

If w is absent we have

DNfto)

° Dcox

la*

Ha, l a Prop. 3.

2.10 Regularity

2.10 D3

51

Regularity B

for

\pp'

.KCpp'Cp'p.

Thus Bi/r for KC(j)\lrG^f(j>. B is called the biconditional symbol. If is a ^ - t h e o r e m then ^ and ^ are called equivalent. A propositional calculus 3P is called regular if from the ^-proofs of B
8. £PC is regular.

We have to show

W

i.e. from the ^ o -proofs of Bcfifr and ^{^} we can find a ^ o -proof of xifr}We proceed by formula induction on x{n}- X{n} ^s n> w e have: * reversibility of IIb' * Modus Ponens. X{TT} is NTT we have * reversibility of II b' DNN
N

^

,

_

T Ponens. —^-^r^—- * Modus Nijr

X{TT}

is DTTX we have * reversibility of II b'

—— ^^r^

1

*

Modus Ponens.

Thus SPC- is regular. COR.

(i) BSilr

We demonstrate the result by formula induction on x* If x{4>} i s then the result is trivial. If x{} m ^X'i^}X'i^} a n ( i ^ n e result holds for x'ify a n ( i /t"{^}» we have:

52

Ch. 2 Propositional calculi

whence by dilution and permutation

similarly with <j> and ft interchanged, and the result follows. If x{} then we have Cf(j>

•}

whence by lie, l a

CxWxW

} DNxWXNx'W CNx'{tf}Nx'{4>]\ CNx'{fi}Nx'{iJr}

which is COR.

as required

(ii)

?* r *

This follows from Prop. 8 since

These are easily established, consider the second one,

DfiNjr

_ TT Ila, la Di/rNft la

TT _ Ila, la DDfrNjrN
D^lrNf r

r

—^-^ l i e and definition of K.

Similarly for the other cases. 2.11

Duality

PROP. 9

1. B'D commutativity 2*. 3. BDD<})<j>'D'" associativity 3 * . BKK''"KD"D'" distributivity 4 * . BKD'"DK<j>"K<j)'" 5. BNDffl'KNipNfi de Morgan 5*. laws 6. B(j>NN^) double negation.

2.11 Duality

53

Using the Deduction Theorem we have 116' = =

Prop. 6, twice

la, l i e Def. of C, K. (i) Again we have

t Ho, la £

Ub, Prop6 . T

*- Ha

w w

II6,

KDWW

Prop.6.

TT

of

^

'

(ii)

4 now follows from (i) and (ii) by 116' and definition of K. The rest are dealt with similarly and are left as exercises to the reader. The dual of a ^-statement is the result of replacing D by K and K by D throughout
10. If ', where $' is the dual of (p.

In the ^-theorem replace each variable n by Nn, the result is a ^-theorem $", say. The original ^ c -proof-tree becomes a 0>cdeduction from ^-statements of the form DNnNNrr which are ^-theorems hence $" is a ^.-theorem. Now use the de Morgan laws Prop. 5 (5) repeatedly and the result follows. 2.12 Independence of symbols, axioms and rules 11. The symbols, axioms and rules of SPQ are independent Suppose that N can be defined in terms of D N for 7^p.(D,p), of type oo,

PROP.

54

Ch. 2 Propositional calculi

where {D,p) is some ^.-statement built up from D and p alone. Then Ncj) may be replaced by (Z),
\ppr,(N,p,p'),

of type ooo,

f

where (N,p,p ) is some ^-statement built up from N, p,p' alone. The only such ^ c -statements are:

p,Np,NNp,...

and

p',Np',NNp',....

Thus D(j)\]r would be independent of one of its arguments. The axiom DpNp would become one of p,Np,NNp,..., one of these would be a ^-theorem. By the reversibility of l i e repeatedly one of p, Np would be a ^-theorem. This is absurd because a ^-theorem must contain at least four symbols. The ^ - a x i o m DpNp is independent because it is impossible to obtain it from the other axioms. Once a variable is introduced into a ^ o -proof it remains in that ^ c -proof from that place onwards. Hence if we are denied use of the axiom DpNp then the resulting ^-theorems will all contain some variable distinct from the variable p. Lastly the <^c-rules are independent. Suppose we omit the rule la then we are unable to obtain the .^-theorem DNpp because the other «^o-rules increase the length of a .^-statement. If we omit rule II a then we are unable to obtain the ^-theorem DpDpNp because the lower formula of l a with a subsidiary formula, IIb, c begin in a different way. In the case of l a without subsidiary formulae we could only obtain DpDpNp from DDpNpp and this is distinct from the lower formula of l a with DpDpNp or DDNppp or DDppNp as upper formula and these fail to be lower formulae of 116, c. If we omit 116 we are unable to obtain DNDppp, this fails to be the lower formula of II a in a «^o-proof, because the upper formula would then be p which is impossible. It also fails to be the lower formula of II c, if it were the lower formula of I a then the upper formula would be DpNDpp this fails to be the lower formula of lie, if it is the lower formula of II a then the upper formula would be NDpp from

2.12 Independence of symbols, axioms and rules

55

which by idempotency we could obtain a «^o-proof of Np, which is impossible. If we omit l i e then we are unable to obtain the ^-theorem DNNpNp by similar considerations. We have just shown that the symbols of 8PC are independent, yet it is possible to define N and D in terms of a single symbol, S or 8'. This can be done as follows: D4.

S

for \pp' .DNpNp'.

D5.

8'

for

Xpp'.KNpNp'.

then 8 and 8' are both of type ooo. We can then define N for "kp. 8pp. D for And

N

for

D for

\pp'. SSppSp'p'. kp.S'pp. -kpp'.S'S'pp'S'pp'.

The axiom scheme would become SSTTTTSSTTTTSTTTT,

S'S'TTS'nnS'nS'Tm.

Thus there can be syzygies between independent symbols. Independence of axioms can also be shown by finding a model J( for the formal system less one axiom and showing that this axiom fails to be 2.13 Consistency and completeness of 12. SPC is model-consistent Consider the applied propositional calculus J(c with the elements t, f of type o called truth-values, t is designated and / is undesignated, and constants Noo, Dooo of the types shown in the subscripts. The rules are (type symbols are omitted) PROP.

and conversely. It is easily verified that *Jifc is a model for 8PC. Thus £PC is a two-valued propositional calculus. An ~#c-valid ^-statement is called a tautology.

56

Ch. 2 Propositional calculi

13. 8PC is complete with respect to J(c. Let ^ be a ^-statement, by repeated use of the de Morgan laws and double negation move negations to the right as far as possible until they act on variables only, this introduces the connective K. Then using the distributive and commutative laws reduce the resulting ^-statement to a conjunction of disjunctions {conjunctive normal form). PROP.

where i/r\ ..., ijr^\ ...,#',..., x{6) &re variables or negated variables. A disjunctive normal form is obtained by interchanging the roles of K and D. A conjunctive normal form is ~#c-valid if and only if each conjunctand contains as disjunctands a variable n and the same variable negated, otherwise by suitable choice of t and / to replace the variables we can make that conjunctand take the value/. Starting from the axiom DTTNTT and using IIa, l a we can easily ^ c -prove any disjunctand of an J(cvalid conjunctive normal form, and hence the conjunction of these disjunctands. If \[r is the conjunctive normal form of then from a ^ c -proof of i/r we can find one of by using the distributive laws and the law of double negation and the definition of K. Thus an JKC-valid ^-statement is a ^-theorem. Thus £PC is complete with respect to J(c. 14. £^a is functionally complete with respect to ^tc. Let J('c be an extension of Jtc which contains every possible table for every connective of types oo, ooo,.... Let F be one such connective and suppose it has v arguments. If H', ...9H^ are all either t or f then let the ^#o-value of F with these arguments be denoted by T[H',...,H^]. We wish to show that there is a ^-statement containing exactly the distinct variables pf,...,p^ whose *J(C-value when the variables are replaced by H\ ...,H^ is F^',...,#<">]. Consider K...Kqr... q^, where q' isp' if Hf is t otherwise qr is Np',..., q{v) ispM if H^ is t otherwise q(p) is Np then K...Kq'...

has t for */#c-value while K ...Kr' ...&\ where r', ...,r(r) are respectively p' or Np'', ...,p(v) or Np{v) but the set / , ...,rW is different from the set qf, ...,#(j;), h a s / for *JKC-value, on the same replacement. Now form the disjunction of all K ...Kq' ...q(v) for just those sets q', ...,q(v) for which PROP.

2.13 Consistency and completeness of 0>c

57

T[H', ..., JS^] is t. This is a ^-statement which has J(c-value t just in case F reduces to t. Thus ^ c is functionally complete with respect to J(c, 2.14 Decidability 15. &c is decidable It is an effective process to decide whether a ^-statement is Jtc-valid, hence it is an effective process to decide whether a ^-statement is an ^#c-theorem. Furthermore if a ^-statement is found to be a 0>ctheorem by the test of .^-validity then it is an effective process to supply the ^a-proof. For all we need do is to put the ^-statement into conjunctive normal form, this is easily tested for .^-validity and if the test is affirmative it is a routine matter to supply a ^c-proof. We can obtain this result in another way. We note that apart from l a the length of a ^-statement increases as we proceed down a ^ o proof-tree and la leaves the length unaltered, repeated use of l a will reproduce a previous formula so use of l a is limited. Thus given a ^^-statement <J) we can construct all possible deduction-trees with base 0. We can then decide if any of these trees are ^o-proof-trees of (j) or whether each fails to be a ^ c -proof-tree of (j). Thus we can decide if ^ is a ^-theorem and if so we can find a ^ c -proof for it. For instance DNDDNpp'Npp', call it #, fails to be a ^-theorem. It is of the form DNDcfrtyo) with DNpp' for (j) and Np for i/r and p' for 0). PROP.

This can arise (i) . , or (iii) from v from I a ^ , ^ , , or v(ii) ; from I l a ^^^ DNDcfiiJfG) DND(f>i/ra) v ' —J;

,.

only. Case (i) DcoNDcfriJr can arise from v by l a

which brings us back to where we started or from ^ .-T^ , . I l a only. can arise from

^

- 116 only. Now iV^ is NDNpp' which

can only arise from NNp Npr ^rt^^r—7- Ho. l h e second upper formula fails to arise from any other ^-statement by the ^ o -rules and fails to be a ^ c -axiom. Thus case (i) fails to provide a ^ o -proof of %. Case (ii) o) isp' and so fails to provide a ^ o -proof of x- Case (iii) DNcjxx) can only arise from ^ T .

DN(j)(o

l a or -^AT, I l a this we can reject as before since o) is rp'. J DNcpoj

58

Ch. 2 Propositional calculi

Do)N(j) is Dp'NDNpp' and this can only arise from DNtpoj by la, which we started vwith, or from

NDNpp'

^ — , II a, NDNpp' can only arise from

NNp Np'

, 116 and this we can reject as before. Lastly DNcfra) is

DNDNpp'p' and, apart from cases already considered, can only arise from

DNNpp' DNp'p' DNDNpp'p' DNp'p' can only arise from an axiom by la, but DNNpp' can only arise from Dpp' or Dp'p each of which fails to be a ^-axiom. Hence case (iii) fails to produce a ^ o -proof of x- Case (iii) is more easily dealt with if we consider DNi/roj which is DNNpp'. A system with 16 as an independent rule would fail to be decidable in this way. We can add any ^-statement as an extra axiom without causing every ^-statement to be a ^-theorem, provided the statement contains a connective. A variable would then fail to be a ^-theorem. The resulting system would however fail to have many of the properties of ^ c . 2.15 Truth-tables The simplest way of testing a ^-statement for being a tautology is by the use of truth-tables. Suppose that a .^-statement (j) is written in terms of D, N, K, C and B; we replace D, K, C and B by two place functions, d, k, c and 6 respectively, and replace N by a one-place function n. We then replace the propositional variables by elements t a n d / in any manner, always replacing different occurrences of the same propositional variable by the same element. We then evaluate the resulting composition of functions by using the following truth-tables: p t t

f f

st

dpp' t t t

f

f

p' t

V t t

P' kpp' t t

f

f f

f

t

f f f

V t t

f f

P cpp' t t

f

t

f

f

t t

p t t

f f

P' bpp> t t

P t

f

f

t

f

f f

np

st

t

These tables give the values of the five functions for values of the arguments on the same line.

2.15 Truth-tables

59

For example let us test CCKpp'p"CpCp'p" for tautology. This Restatement is a conditional, let us see if it is possible to make it take the value/ by suitable values, t or/, given to p, p' and p". The only way of making a conditional take the value/ is to make the first component t and the second component/. Thus we want to make GKpp'p" take the value t and CpGp'p" take the value/. Both of these are again conditionals so we want p to have the value t and Cp'p" have the value/, this requires p' to have the value t and p" to have the value/. This then fixes the values of p, p' and p" in order that our original statement take the value/. Putting these values for p, p' a>ndpff in the original statement we easily calculate that it is t, thus it must always be t, and so is a tautology. We put the working down as follows: CCKpp'p'VpCp'p"

f t

f

tf ttf t

f The first line with / under the first occurrence of C indicates that we want to make the whole statement take the value/. To do this we must make the first component take the value t and the second component take the value/, this is indicated by placing in the second line t under the second occurrence of C and/ under the third occurrence of 0. Since the third occurrence of C is to take the value/ then its first component must take the value t and its second component must take the value/; this is indicated by placing t in the third line under the second occurrence of p, and/ in the third line under the fourth occurrence of C. We have now found values that p, p' and p" must have if our statement is to take the value/. Because p' must have the value t and p" must have the value/ since Cp'p" is to have the value /, this is centred in the fourth line. In thefifthline we enter the values of p, p' and jp" under the first occurrence of these symbols. In the sixth line we evaluate Kpp1 andfindthat it is t, in the seventh line we evaluate the first component of our original condi-

60

Ch. 2 Propositional calculi

tional and find that it i s / , but we had found in the second line that it should be t, hence it is impossible to make the original conditional take the value/, hence it always takes the value t and so is a tautology. The working can be put down on one line, since except for the last entry we have always used different columns. Thus: CCKpp'p"CpCp'p"

fttttfftft

f

f 176 555 2334 4 The numerals indicate the order in which the letters t a n d / are put in. The final line puts both t a n d / under K indicating an impossibility. Since SPC is decidable we could have another formulation for it, namely: the axiom-scheme 'a tautology is an axiom5 and dispense with rules. The conditions for being a formal system are satisfied because we have a test for being an axiom. Sometimes it is simpler to evaluate directly for all possible argument values, we would then put down the work as follows: p t t t t f f f f

p' t t f f t t f f

p" t f t f t f t f

BBpp'BBpp'fBp'ptf t t t t t t t t

Thus we have a tautology. Independence of axioms and rules may also be shown by means of models. We find a model J(' for the formal system <£' obtained from the formal system jSf by omitting one axiom, axiom-scheme or rule and such that the omitted axiom or axiom-scheme fails to be valid in the model J(' or the omitted rule fails to preserve JS?'-validity. For instance the following three axiom-schemes and Modus Ponens give a Propositional Calculus 0*x equivalent to ^ c , / is a constant.

2.15 Truth-tables

61

(i) dt

(ii) a

3C6S'C(i

(iii) G( Consider the following truth-tables for the connective M.P. Cpp'

v'

V t t t

g

t

t t

f

f

t

g g g

g

t t

f

f

g

g

f

t

t

f f f

t

(i)

Cpp'

(ii) Cpp'

c. (iii) Cpp'

t

t

t

f f f

g

g

f

f

g

f

t t

g t t t t

t t t

t t

t t t

The constant/ is undesignated. The heading of the various columns denotes that the corresponding rule or axiom-scheme fails for that truth-table, but all the other rules and axiom-schemes hold for that truth-table. In this model we have three elements, t is designated,/and g are undesignated. We leave the checking of this table as an exercise for the reader.

2.16 Boolean Algebra A Boolean Algebra is a formal system with the symbols: nooi a

type I Oii Oii

0 1

i

u n

iii

i

iii

ii

( )

name

variable for an element equality inequality null element unit element union intersection complement generating symbol left parenthesis right parenthesis

62

Ch. 2 Propositional calculi

The axioms are given by the following schemes: We have written (a U /?) for ((U a)/?), etc. a = a 0+ 1 0=1

1=0 au 1 = 1 au 0 = a

a no = o a ni = a

ft cz ftl II >-

na = o a = a

ft cz ft II ft

a na = a (a f) /?) = a II/?

(a u B) = a n B

a n (/? n 7) = (a n /?) n 7

a u (/? U 7) = (a u /?) U y

a n (/? U 7) = (a n /?) U (a n 7)

a U (/? n 7) = (oc U /?) n (a U 7)

Note the duality. We have neglected independence. We have written equalities in the customary manner, a, /?, 7 stand for arbitrary elements. We have omitted parentheses wholesale. The rules are:

We could add the symbols and rules of the Propositional Calculus and so get compound statements, but this is unnecessary. Note that if we replace n by K u by D

—

0 1 = *

by by by by by

N

p&Np py Np B NB

and the variables for elements by propositional variables, then the axioms become tautologies of 0*c. Again if we replace the elements by variables for subsets of a given set X, then the axioms become axioms for elementary set theory, as regards union, intersection and complementation, 0 becomes the null set and 1 the given set X. The simplest Boolean

2.16 Boolean Algebra

63

Algebra consists of the two elements {0,1}. This is called the two-valued Boolean Algebra and corresponds to the pair {/, t}. PROP.

16. We have

(i) a U (a n ft) = a

a n (a U /?) = a

(ii) (any3)U/? = aU/? (iii) aU^ = a if and only if We have

(a U ^) H/tf = a n/? a n /? = /?.

a (J (a n /?) = (a n 1) U (a n A) = an(lU/?) = an 1 = a.

Again

(oc(]fi)[)j3=(oc[)/i)(](^[j^) = (a U /?) H 1

Lastly if a U /? = a then

an/?=(au/?)n/?

= /? by (i). D6.

a - > / ? for aUyff a<->y5 for (a->/?) n ( y ) a ^ J3 for a n /? = a.

Notice the correspondence with ^ c on replacing -> by O and <-> by S . PROP.

17. We have

If

a ^ /? a^cZ ft ^ a

//

a ^ /? a^cZ J3 ^ y 0^ a

//

then

a = /?.

then a ^ y. a ^ 1.

a ^ /? tf^w a n y ^ y? n y.

64

Ch. 2 Propositional calculi

//

a ^ /? then afl/J^a

a u y ^ /? U y. a < all/?

a < /? i / cmd owfo/ ^/ fi ^&>

a < /? if and only if a ->/? = 1 a = /? if and only if a <-> /? = 1. These are very easy and are left as exercises for the reader. 2.17 Normal forms Using the commutative, associative, distributive, de Morgan laws and double negation we can express any J^-statement in an equivalent form, either as a conjunction of disjunctions of propositional variables and negated propositional variables or as a disjunction of conjunctions of propositional variables and negated propositional variables. The first case is called the conjunctive normal form (c.n.f.), and the second case is called the disjunctive normal form (d.n.f.). We push the negations to the right as far as they will go so that they act only on propositional variables or negated propositional variables then use double negation as long as possible and lastly the distributive laws. It is like multiplying out an algebraic formula. We can further ensure that in the first case each propositional variable which occurs in the original J^-statement occurs negated or unnegated in each disjunctand in the c.n.f. and dually in the second case in each conjunctand in the d.n.f. This is achieved by disjuncting KpNp in the first case and conjuncting DpNp in the second case, if p is a relevant variable. Clearly we are left with an equivalent J^-statement in each case. The N. and S.C. that an J^-statement be a tautology is that each disjunctand of its c.n.f. contain a variable and the same variable negated. Dually the N. and S.C. that an J^-statement be refutable is that each conjunction of its d.n.f. contain a variable and the same variable negated. We can further ensure that in the c.n.f. each disjunction contains each variable exactly once either negated or unnegated, unless the c.n.f. is a tautology. For we can omit any disjunction which contains a variable and the same variable negated and obtain an equivalent J^-statement. If the original ^r-statement was a tautology by this removal we would finally remove all the disjunctands. Dually for the d.n.f.

Historical remarks to Chapter 2

65

H I S T O R I C A L R E M A R K S TO C H A P T E R 2

Propositional calculi have a long history. Aristotle's syllogistic, when expressed in modern symbology, amounts to a form of singularly predicate calculus or class theory. This is discussed by Lukasiewicz (1951) in detail. The early history of the classical propositional calculus is given in great detail by Bochenski (1951, 1961), his account covers all the middle ages up to Frege and then on to the present time. Church (1956) gives the history of propositional calculi since Boole. Lewis (1918) gives the history from Leibniz to Schroder in great detail. Other works on the early history of logic are Bochenski (1951) which deals with preAristotlean times, the old Peripatetics, the Stoic-Megara school and the last period after Chrysippus. Another work is Moody (1953) which deals with the Medieval period, and lastly Diirr (1951) which deals with Boethius. The definition we have given for a propositional calculus seems to cover all systems which are usually called propositional calculi. The problem of the independence of symbols, rules, axioms, etc. is of quite modern origin. Huntingdon (1904) and Bernays (1926) were the earliest to discuss these matters. Models for propositional calculi came in with truth-tables. Lukasiewicz (1920, 1941) was the first to formalize a three-valued propositional calculus. Post (1921) also considered many-valued propositional calculi. Independence of axioms, etc. was demonstrated by Bernays (1926) using many-valued models, this is called the matrix method. Extensions of formal systems were studied by Lukasiewicz and Tarski (1930). Modus Ponens first appears in Scholastic Logic. In treating the classical propositional calculus we use the notation of Lukasiewicz (1920). This is easily read in conversational English if one reads as follows: read D(j)f as either (j) or j/r, read N(j) as not <j), read Kft as both i/ra) is read: either not either
66

Ch. 2 Propositional calculi

The form we have taken for the classical propositional calculus is due to Gentzen (1934, 1955) (see also Anderson & Johnson (1962) and J. Dorp (1962)), who used the terms 'remodelling scheme' and 'building scheme'. The type of proof used in Prop. 2 is due to Gentzen (1934). Prop. 3 was pointed out to me by A. H. Lachlan. The direct method has been further developed by Schutte (1950, 1960). The de Morgan laws were stated by de Morgan (1867), but were known long before that, and Prop. 5 probably first appeared in connection with Boolean Algebras. The deduction theorem first appears in Herbrand (1930) and has been much used since, notably by Hilbert-Bernays (1934-6) and Church (1956). Tertium non datur or the law of the excluded middle goes back a long way and was first called in question by Brouwer (1908) in the case of infinite classes. He maintained that rules that apply to finite classes might fail for infinite classes. Thus he invented Intuitionism, a form of mathematics which does not accept T.N.D. It gives rise to a propositional calculus. But it is far more complicated than the classical propositional calculus, and so is his form of analysis, much more complicated than classical analysis (see Heyting (1934, 1955, 1956) and Kleene-Vesley (1965)). But his methods go some way to clarifying ones ideas about effectiveness, finiteness, constructiveness, etc. which are conditions that Hilbert (1904) insisted should apply to metamathematical demonstrations. Prop. 7, the elimination of Modus Ponens, is Gentzen'sHauptsatz (1955). We thought that since it can be eliminated then why have it at all. The word' syzygy' is due to the algebraist Sylvester who used it in connection with non-linear relations between algebraic invariants, it means a yoking together. Many proofs of the decision problem for the classical propositional calculus have been given, notably by Church (1956), Kalmar (1935), etc. A correct truth-table for implication was given by Philo of Megara about 300 B.C. Truth-tables were informally used by Frege in special cases, six years later Peirce stated them as a general decision method for the classical propositional calculus. Much of the recent development is due to Lukasiewicz and Post. The term 'tautology' is due to Wittgenstein (1922). D4, 5 are due to Sheffer (1913). The modern treatment of propositional calculi stems from Boole and

Historical remarks to Chapter 2

67

de Morgan in 1847. This was the algebra of logic. MacColl (1877) was probably the first to deal with a true propositional calculus. Frege gave the first formulation of the classical propositional calculus as a formal system in its own right. But his work was for long neglected and so the propositional calculus developed in the older form as in the work of Peirce, Schroder and Peano. Whitehead and Russell appreciated the work of Frege and gave the classical propositional calculus a formulation with negation and implication as primitives, Modus Ponens and substitution as rules. But substitution was not explicitly mentioned, though this omission was noted later. One of their axioms was found to be redundant by Bernays (1926). Nicod (1916) found a formulation of the classical propositional calculus with only one connective and only one axiom and only one rule apart from substitution. Since then a variety of formulations have been discovered. One by Hilbert (Hilbert-Bernays (1934-6) vol. 1) has 12 axioms in 4 groups of 3 axioms each, this was designed to separate out the roles of the connectives N, C, K, D. If in this formulation we omit one axiom then we get a formulation of the intuitional propositional calculus. Between the two world wars the Poles were very active in reasearch on the propositional calculus, see Jordan (1945), Storrs McCall (1969) and H.Skolimowski (1969). Another study is to formalize a partial system of the classical propositional calculus which has only the implication sign, and the object is to find axioms so that exactly all tautologies which only contain the implication sign and propositional variables are theorems of the system. The chief interest of such studies is to find a formulation of the classical propositional calculus which is an extension of this partial system. For instance CCCpp'p"CCp"pCpfffp, with Modus Ponens and substitution is an elegant formulation of the implicational propositional calculus. If to this we add the axiom Cfp, where / is a constant, we get a formulation of the classical propositional calculus. Implication in the classical propositional calculus allows as true an implication with a false antecedent, this seems in some sense unnatural. This has given rise at the hands of Lewis (1918, 1920, 1932) of various other propositional calculi designed to rectify this. He also considered other connectives such as 'possible' and 'necessary'. These give rise to what are called modal logics. We do not discuss them in this book. Much 3-2

68

Ch. 2 Propositional calculi

work has been done on these systems, finding decision procedures, formulations, etc. Gentzen (1955) used Sequenzen, that is figures of the form

where we have used# and q as variables. This behaves in the same way as CK ...Kp'p" ...jP>D ...Dq'q" -.tf*We have written the sequenzen as p' p"...p{v) qfq"-..q^ but have only used it with one lower formula. Gentzen allows the cases when either upper or lower formula may be void. The idea of using axiom schemes is due to v. Neumann (1925), a substitution rule is then unnecessary. Prop. 8, the substitutivity of equivalent statements, is due to Post (1921). Conjunctive and disjunctive normal forms derive from Boolean Algebra. A large number of examples on the propositional calculus are found in Church (1956). Boolean Algebra was invented by George Boole (1847, 1854, 1916). He noticed the resemblance between the behaviour of K and D and + and x in arithmetic. The history of modern symbolic logic can be traced back to Boole. More will be said about Boolean Algebra at the end of Chapters 3 and 12, there we shall consider Boolean-valued settheory as an extension of the classical two-valued set-theory. Sheffer (1913) gave a set of independent axioms for Boolean Algebra.

EXAMPLES 2

1. Complete the demonstration of Prop. 5 for the cases other than 4*. 2. Complete the demonstration of Prop. 8, Cor. (ii) for the first and third cases. 3. Obtain ^-proofs for BBpp'Bp'p, CCpp'CNp'Np. 4. Give a ^-proof for Cpp. 5. State and prove the Deduction Theorem in £PV

Examples 2

69

6. Obtain a ^ - s t a t e m e n t ^ which has the following table: P

p'

Ptf

t t t t

t t

f

Jt

t

t

f

t

st

f

f

f

f

J f f f

f f

t t

f J

t

t

t

7. Define D, B, K in terms of C, N. Define C, D, B in terms of K, N. Define C, D, B, K, N in terms of S and in terms of S'. 8. Show that SSpSqrSpSSrpSSsqSSpsSps is a tautology, where p, q, r, and s are of type o. 9. A Propositional calculus ^ 2 n a s ^ n e &xiom schemes: (i) (ii) (iii) And the rule Modus Ponens. State and prove the deduction Theorem for ^ 2 . 10. Prove the following theorems of ^ 2 : CNpCpp',

CNNpp,

CpNNp,

CCpp'CNpNp'.

11. Check the table giving the independences of the axioms of 3PX. 12. Give definitions of C and / of SPX in terms of D and N of &c, and give definitions of D and JV of £PC in terms of G and fo{^v 13. Show that theorems of 0*x translate into theorems of ^ c by the definitions found in Ex. 12 but that there are theorems of £PC which are unobtainable in this way, show also that there are theorems of ^ which fail to be translations of theorems of ^ c . 14. Show that the axioms of SPC are theorems of ^ and that the rules of 8PC are derived rules of SPX. 15. Prove BNBNpp'Bpp' in 0>c.

16. Show that if- and t then 4 ^ and f X{f} 17. Show that GKGK^xGKN^fxGKf^r'x

is a ^-theorem.

70

Ch. 2 Propositional calculi

18. Define t for Cff in the system 8PV Define K of &c in terms of G and / of ^ . Write out DKptNp without definitional abbreviation in terms of C and/of ^ . 19. Write $ = ijr if and only if B^xjr is a ^-theorem. Show that the result is a Boolean Algebra when U 0 are suitably defined for Restatements. 20. Suppose that a Boolean Algebra 88 has an additional functor * of type u with the axiom schemes: a ^ a*, a** = a*,

0* = 0.

Show that the elements of 88 which satisfy a** = a form a Boolean subalgebra «^* of ^ . When the functions in 88* are defined as follows: write a®

for

a**,

then

a&fi

for

a n /?,

a^jS

for

(all/?)®,

a

for

a®.

(If the elements of 88 are sets of points in a topological space then the specified sets are the regular open sets-open sets which have no pin holes or cracks.) 21. Define for ( a n ^ a x /? for a n /?. Show that with these definitions every Boolean Algebra becomes a Boolean Ring, i.e. a ring in which = 0, if

a + /? = 0 then

a = /?,

Examples 2

71

Conversely show that with the definitions all/? for

a + /3+(axjS),

an/?

for

a x /?,

a

for

1 + a,

every Boolean Ring becomes a Boolean Algebra. In both cases the algebraic zero and unit coincide with the Boolean zero and unit respectively.

Chapter 3 Predicate calculi

3.1 Definition of a predicate calculus A predicate or functional calculus of the first order is a formal system J5" which is a primary extension of a propositional calculus SP, it is obtained from a propositional calculus by adding additional symbols for variables or constants of type 1, calledindividual variables or constants respectively, and adding variables or constants of some or all the types 01, oil, out,..., called predicate variables or constant predicates respectively; there may also be variables or constants of types u, u,..., called variables for functions or constant functions respectively, and there may also be constants of some of the types 0(01), 0(011), ..., (ot) 0(01) (on), ... called quantifiers There must be individual variables given by a scheme of generation, and there must be some predicates. If quantifiers are present then the abstraction symbol is required in order to provide arguments of types ot, oil,... for the quantifiers. If propositional variables and constants are excluded the resulting system (which fails to be a primary extension of a propositional calculus) is also called a predictate calculus. If quantifiers are absent the resulting system is called a free variable predicate calculus of the first order with or without functions as the case may be. If individual variables are discarded so that all well-formed formulae of type 1 are constants then quantifiers are useless and the resulting system reduces to a propositional calculus when propositional variables and constants are written in the form , ^{a}, %{a, /?},..., where a, JS are constant individuals and ^ is a propositional variable or constant, xjr, x,... are predicates of types 01,011,.... If individual variables are discarded and function variables present then in the resulting system there are variable well-formed formulae of type 1 and the system becomes more manageable if these are replaced by individual variables. For these reasons we require individual variables to be present. If the only predicates present are of type 01 then we have a monadic predicate calculus of thefirstorder, if the only predicates present are of types 01 or on then we [ 72]

3.1 Definition of a predicate calculus

73

have a diadic predicate calculus of the first order, and so on. Quantifiers of type O(OL) are called simple quantifiers, quantifiers of other types are called compound quantifiers. A predicate or functional calculus of the second order, J^2, is a primary extension of a predicate calculus of the first order obtained by adding quantifiers (simple or compound) for predicates. Simple predicate quantifiers are of types: read' for at least one',' for an unbounded set', 'for all' etc. 0(00),

O(O(OL)),

0(0(011)),...

generally o(oa), where a is a type of a predicate or of a propositional variable, and compound quantifiers are of types: read 'for all pairs', 'for all except a bounded set of pairs', etc. 0(000),

0(00(01)),

0(0(01)0),

0(0(01) (01)),

0(00(011)),

...

generally o(ootfi),o(ooc/3y),..., where a,fl,y,... are types of predicates or of propositional variables. A predicate or functional calculus of the third order, ^ 3 , is a primary extension of a predicate calculus of the second order obtained by adding variables or constants of types 00,

o(ot),

O(OU),

...

generally ooc where a is the type of a predicate or of a propositional variable, and oa/3 where a, J3 are types of predicates or of propositional variables, ..., these are called predicates of predicates, we could also add some mixed predicates requiring for some arguments predicates and for others requiring individuals, these would have types: 001,

010, 0(01) 1,

01(01),

...

and generally oou,otot, ofiyt,..., where a,/?,7 are types of predicates or propositional variables. A predicate or functional calculus of the fourth order, ^"4, is a primary extension of a predicate calculus of the third order obtained by adding quantifiers of various kinds over predicates of predicates or over mixed predicates. And so on. A predicate calculus of the first order without constant individuals or constant functions or constant predicates or constant propositions is called a pure predicate or functional calculus of the first order with or without functions as the case may be. A pure predicate calculus of the

74

Ch. 3 Predicate calculi

second order is defined similarly. A predicate calculus of the third or higher order is called pure if it is an extension of a pure predicate calculus of one lower order and if only variables (for calculi of odd order) or quantifiers (for calculi of even order) are adjoined. For instance a pure calculus of the third order may have variables and constants of type oo, O(OL), because there may be constants of these types in the pure predicate calculus of the second order of which it is a primary extension, but in the extension only variables of these types are adjoined. A predicate calculus of the first order which is without propositional variables and predicate variables is an applied predicate or functional calculus of the first order, with or without functions as the case may be. Similarly a predicate calculus of odd order is applied if it is obtained from a predicate calculus of one order less by adding only constants and is mixed if both constants and variables are added. A predicate calculus of even order with some constant predicates is mixed, it must have variable predicates of appropriate types. A formal system £P is based on a pure predicate calculus of the first order

J^ if the constants of ZF are constants ofSP and if ^ has symbols of each type for which there are variables in SF and has individual variables (we may use the same symbols for individual variables in SP and in 3F) and if whenever
3.1 Definition of a predicate calculus

75

types from among, i\ i",... and possibly individual constants of some of these types, and possibly functions with values and arguments from these types. The order of a many-sorted predicate calculus is defined as for a one-sorted predicate calculus. A many-sorted predicate calculus IF' is based on a pure one-sorted predicate calculus IF when the constants of J^ are constants of IF' and when the axioms and rules of J^ apply to each sort of individual variable in IF'. We shall show later that it is possible to reduce a many-sorted predicate calculus IF' of the first order to a one-sorted predicate calculus IF of the first order by adjoining to IF some constant one-argument predicates. Each such predicate plays the part of saying that its argument is of a certain sort, distinct predicates referring to distinct sorts. Similarly we could consider many sorts of predicates in predicate calculi of higher orders. A predicate calculus is a formal system hence it must be constructive in accordance with the definition of a formal system which was given in Ch. 1. If two predicate calculi have variables of the same types then by a trivial adjustment of notation we may use the same symbols for these variables in both systems. A predicate calculus IF' is weaker than a predicate calculus IF under the following circumstances: (i) IF' is without variables of type oc if IF is without variables of type a. (ii) The constants of IF' can be defined in terms of the constants of IF. (iii) An J^'-theorem ' are replaced by their definitions in terms of J^-constants and the abstraction rule is used if necessary to eliminate the abstraction symbol, and the same symbols are used for individual variables in both systems. Note that (i) follows from (ii) and (iii). If IF' is weaker than J^ and J^ is weaker than IF' then IF and IF' are equivalent. If IF' is weaker than IF and fails to be equivalent to IF> then IF' is strictly weaker than
76 3.2

Ch. 3 Predicate calculi Models

A model Jfofa, predicate calculus of the first order &> with only simple quantifiers is a formal system which satisfies the following conditions: (i) Jl has exactly the same constants as !F of types other than i, ii, iii, . . . , O, Oi, OLi,

(ii) Jl is without variables but has constants of types 1 and o and of any of the types a, m, ...,oi,ou,... which occur in 3F. The ^-constants of type 0 are designated or undesignated, at least one is designated and at least one is undesignated, the ^-constants of type o are called elements, the constants of type 1 are called individuals, the constants of types u, in,... are called functions and the constants of types oi, on,... are called predicates. (iii) Jl is without axioms. (iv) The rules of t e n a b l e us to replace any ^-statement by a unique element, called the Jf-value of the ^-statement. (v) If ®{
(
3.2 Models

77

For instance if O ^ X T / . Y ! £,?/}} is an ^'-statement whose sole free variable is £, then J( is to contain a constant of type 01, 3 such that Sa has the same J(-value as 0{oc,A7j.x¥{a,/)]}} for each ^-individual a, where again there is to be an ^-constant H' of type oi such that Y{a, /?} has the same J(-value as S'yff for each ^-individual /?. S' will in general vary as a varies. If J^ contains compound quantifiers then clauses (vi) c, (v) require amendment. Suppose J^ contains a compound quantifier of type o(ott) then if X££'.
78

Ch. 3 Predicate calculi

3.4 The classical predicate calculus of thefirstorder A pure predicate calculus of the first order is classical if it is equivalent to the following predicate calculus of the first order, SFC. Symbols: those of £PC together with: type X

name

I

Voc >Pou>~-

01,011,

E A

o(pi)

individual variable predicate variables existential quantifier abstraction symbol

further variables are obtained by superscripting primes, as for propositional variables. Axioms DnNn, where n is a propositional variable, DTT^NTT^, where n is a one-place predicate and £ is an individual variable, etc., for two or more place predicates. Rules Those for £PC together with: Remodelling scheme \b DDtfxfxi) cancellation, D0G) Building schemes II d D(f>{y} o) existential dilution, DE(kt;. ${£>}) o) £ fails to occur free in ^{I\}, TJ free in lie

DN(p{7j}o) generalization, DNE(k£. {£)) (o, where TJ fails to occur free in OJ, 0{I\} and fails to occur free in Here £, TJ denote individual variables, (f>{r)}, 0) denote ^"^-statements, o) is a subsidiary formula and may be absent, (f>, (j>{7)), N, JE7(X£.^{£}), iVi/(X£. <j){£>}) are the main formulae of the lower formulae. We write

D7.

(mm ') for

for

E{u-m)-

#(X£.#(X£\$H£, £'}))>

etc.

3.4 The classical predicate calculus of the first order

79

Note that an #~c-proof is direct, the only formulae that can be omitted are duplicates as in 1b. This rule causes the undecidability of 3FC, as we shall see later. 3-5 Properties of the system ^c 1. The schemes la, b, 116, c, e are reversible. In Prop. 2, Ch. 2 we showed the reversibility of la, 116, c for the system 8?c. Similar demonstrations hold for J r c . 16 is reversible II a, l a . PROP.

l i e is reversible. Suppose we have an J^-proof of In this J^-proof corresponding occurrences of N(EE)) ^{£} can only be introduced at II a, e. If a corresponding occurrence of N(E£)(j){£) is introduced at I I a then introduce N
would become Dn^NnTj which fails to be an J^-theorem. This completes the demonstration of the proposition.

80

Ch. 3 Predicate calculi

2. D$N

* *—ii-LI

n e, conditions on variables are satisfied,

Thus the result holds for (E£) i/r{g} if it holds for i/r{£,}. This completes the demonstration of the proposition. Thus Tertium non datur holds in the system !FC. P E O P . 3 . The Deduction Theorem holds in 3^G with the following restriction. If <j>', ...,(f)(s®\-jr ijr, where l i e fails to be applied to any variable which occurs in ftSd\ then f , . . . , ^f^^JDN^f. When lie fails to be applied to any variable in ^ ^ we say that the variables in
f

for

9

""^9

in J V

The demonstration is similar to that of Prop. 6, Ch. 2. We have just shown that (i) (tertium non datur) holds in 3FC, (ii), (iii) follows exactly as in Prop. 6, Ch. 2 and similarly for (iv) except that if the rule used is I I e>

y DNx 4- H e then ^^ , , IIe,Ia ty DN(j)\jr will only be valid if the variable generalized fails to occur free in (j>, hence the restriction. This completes the demonstration of the proposition. If we apply generalization to each free variable in an jF c -statement the result is called the closure of
..., D provided that the variables in D^SK^CJ are held constant.

3.5 Properties of the system

81

We have DND^SK)o)Di/ro) from Prop. 3 whence the result by permutation and cancellation.

UXi6) for x'>

0=1

D8.

SK

n

3= 1

6= 1

d)

tfthenDN] 6

COR. (ii). If<j>\ ...,

X{6)X{SK)-

fr, where (/> is the closure of (j).

1

We proceed as in Prop. 3 except that if lie is used, say — lie, where a X, variable £ free in an hypothesis $ is generalized then we replace it by

lie, condition on variables is satisfied lie. In this way from $\ ...,
lid

Now repeat for (K\.. , (j)' and the result follows. I

D9.

for

SK

x', K

for 6 =I

X^6)X(SK).

COR. (iii). / / <$> is quantifier-free and is in conjunctive normal form, so that K

(f> is of the form J\ ft6) where ftd) is a disjunction of atomic statements or 6= 1

negations of atomic statements and if C(jx]r is an &c-theorem then We may suppose that in the J^-proof of C^

the free variables in (j>

82

Ch. 3 Predicate calculi

are held constant. Because if one of them, say £, was generalized then this generalization must occur in the J^-proof of C
K

(T{d)

Let ^ be n S 0(*'*°, this distributed becomes £ n fte>r{6)\ call it 0*, 6 = ld' = l

0= 1

r

where 1 ^ r{6) ^ o-{6}, and the summation is over all such r. From 'if N6 DNdilr =?= then , , ' and DN6\jr we obtain DN £ II (j>{e>rmf, whence by ivp* DN(j)*yf T e=i K

the reversibility of 116 repeatedly we obtain DN ]J ^e>T^i/r, for each r. All these J^-proofs are obtained from one tree by replacing part disK

K

junctions of S II {6'r{6)) by II {0' T{6}) and omitting certain branches. Thus T 0= 1

we have })

K

D 2 ^ ^ ' 0=1

6=1 T {

^

for each r. Now consider the places where

enters the J^-proof of D 2 Nftd'T{6})i/r. It will do so either at an 0=1

J^-axiom (T.N.D.) or at IIa or at He only. If it enters by an J^-axiom T{0})^(0,T{0}) replace t h i s b y j]AT?(d 7{e\)M6 TW}\^La' T h i s c o n v e r t s t h e

J^-proof of D S Nfte>TW>ft into an ^-deduction of D S Nftd>T^ft from 0=1

0=1

certain hypotheses <j^e*r^. Once Nfte>r{d)) has entered this deduction it thereafter remains in the subsidiary formulae of building rules because it fails to be governed by a quantifier or by ND or by N in the theorem. We are only considering occurrences of Nft0>T{6}) which correspond to the occurrence of Nfte>T{6}) in ^Nft0'7^. 1

If Nftd>r{d)) enters by l i e then

0=1

is NN(f>(d>T{d}\ where TW is atomic, and we have He,

consider the places where corresponding occurrences of <j>(e>rW> enter the deduction, 00»T0» is an atomic statement so it enters by II a or by (T.N.D.) only. If it enters by (T.N.D.) replace this by

Ila.

3.5 Properties of the system !FC

83

These two cases altogether leave us with an ^-deduction of D 6=1

from hypotheses (jfl>7W\ 1 < 6 < K. In this deduction omit all occurrences of N(jP'r{6)) which correspond to the occurrences of Nft6'7^ we have been considering, so that Nftd>7{6}) fails to be introduced by IIa. We are left with an J^-deduction of ^ from hypotheses 0<^T0», 1 < 6 ^ K, because we have only altered the subsidiary formulae of building rules by omitting from the upper and lower formulae of a rule the same disjunctands and this converts an application of a rule into another application of the same rule, also the parts of upper and lower formulae of remodelling rules have been altered in the same way. These cancellations fail to effect i/r because an occurrence of Nfte>T{d}) in ^N^d'T{6}) fails 6=1

to correspond to any occurrence of Nfte>r{e)) in ^ . The final result when repetitions have been removed, is an ^'-deduction of xjr from hypotheses fto,T{0})i i ^ Q ^ K, this holds for each r we require. an

w

LEMMA. If o),x\ • • • > X^&cfr ^ '> X' • • • > A ^ h ^ ^ where the variables (o, ojr are held constant, then DGJG)', X, • • • > ^ ( l c ) hjr 0 ^. We have Do)G)',x', ...,X{K)\~^CD^O)', by replacing all occurrences of co which correspond to the occurrence of OJ in the first hypothesis by DGJG)'

in

and then putting o)' into the subsidiary formulae, by remodelling, of any rules used: the conditions on variables are satisfied because the variables in co' are held constant. Again we have Di/rco', x'> • • • > y^&JDxlriJr, by replacing o)r by Di/raj' as before in the second hypothesis and then putting ^ into the subsidiary formulae, the condition on variables is satisfied because the free variables in i/r fail to be quantified. Thus we have: (K) D u s i n Ib w e DUG)',/, ...,X ^c ft«>'-> Di/r(o',x'> —itf^rflWr* § §et t h e required result. Returning to Cor. (iii) we have ^ (1>T(1)) ,..., ^(ACjTW)hjr ^ for each r. K

(7(6)

Now the hypotheses arise from distributing n S ^e>e)

so

that we shall

0 = 16' = l

have sets of hypotheses which differ only in their first members; we have already noticed that we may assume that the variables in the hypotheses are kept constant so that we may apply the lemma repeatedly (T(l)

and obtain 2 ), cJ(2)T(2)), 6^T{-K))V^ ty. I n a similar way we have this 6' = 1

C

result when the hypotheses ft2>TW for all possible r are the same,

84

Ch. 3 Predicate calculi o-(i)

whencebythelemmarepeatedlyweget £ 0 (MO , S Continuing in this manner we finally arrive at 0', ...j^H-jr^, as desired. C O R . (iv). / / ^ is £Ae closure of
a

^ d where

6= 1

fte) is a disjunction of atomic statements or negations of atomic statements, and if C^xjr is an &c-theorem then $',..., <^K>>V&C ft-

Consider the proof-tree ofC^i/r, if lid is applied to a variable in <j) and if this variable is held constant in \[r or is absent from i/r then omit that application of lid, the result is an J^-proof-tree of C$*^r where $* differs from (j) by omission of generalizations. Every other variable in (f> is first restricted in ^ and later on generalized in i/r. Variables in (j) fail to get generalized. Now consider the portion of the J^-proof-tree of C is restricted. We have

—-—"K b/ —>—>-— I I c , e t) for i]',...,r D(Et)) (Eg) N<j>DN{Eri) Nf'o) replace the first piece by <J>',..., M\-^c ijr' using Cor. (iii) ,then continue thus: =

as before but without I I d on N,

fr"

ft

as before but without lid on

3.6 Modus Ponens 4. Modus Ponens is a derived We have to show

PROP.

Given the J^>-proofs of the upper formulae we have to show how to obtain an J^-proof of the lower formula. The demonstration is by formula induction on the cut formula $. With one of a), x non-null.

3.6 Modus Ponens

85

(a) x is a remodelling of an axiom then x must be <j) and the result follows at once. Otherwise we demonstrate the more general result SK

D 2 N(j>x (this is to account for all the cancellations of N by 16 between DN(f>x and the next building scheme above it). We use theorem induction on the right upper formula, that is we suppose that the result holds for the SK

formula immediately above D 2 Nx in the J^-proof-tree of this for0=1

mula. We have already shown that the result holds if the right upper forSK

mula is an axiom. If 2 N<j> is in the subsidiary formula of a rule and if the 6=1

result holds for the upper formula or formulae of that rule then it follows at once that it holds for the lower formula of that rule. Thus if we have SK D S N<j>X' 6=1 SK

then we have SK

Daxf> D 2 =r—j^^1 £

* by induction hypothesis

deduction as before,

and similarly for a two-premiss rule. SK

If the whole or part of 2 N<j) is in the main formula of a building rule 6=1 SK

then this building rule can only be I I a since <j> is atomic. By la D 2 Ncfrx 6=1

could become DNifriJr, which since ^ is atomic is different from any of the forms of the lower formulae of building rules other than IIa. Or, by l a SK

D 2 NX could become Dx'ty, where x is Dx'x" 6=1

an(

SK

*S ^

occurs in i/r,

6=1

this could be of the form of the lower formula of the building rules SK

IIa, 6, c,d,e, but then 2 N
contrary to the case considered. Thus the rule can only be II a. In this

86

Ch. 3 Predicate calculi SK

case the formula immediately above D 2 Ntfix is either x i*1 which case 6= 1

the result is trivial or is of the form we are considering. Thus the only SK

non-trivial case is when 2 N(f>x is in the subsidiary formula. By our 0= 1

supposition 16 with N<j> as main formula fails to be the rule immediately SK

SK

above D 2 N<j>X* If the rule is II a with part or all of 2 N6 in the main 6= 1

formula then we have

6=1

f

^

N,

,

6= 1 6=1

where there may be more Nfts in the lower formula so that we have V

diluted with a formula of the form D 2 N<j>x" (x" m a y be null) or a per6=1

mutation of this, and if the result holds for the upper formula then: Dco

^

Dux' then by dilution we easily get DGJX* This completes this case. It is impossible for x! to be null, otherwise 2iV0 would arise from an axiom, which is absurd. (b) <j) is D(f)f<})" and the result holds for $' and $". We have ^-proofs of DOJDC})'(j)" and DND(j)'(j)"x,fromthe reversibility of II b we then have ^Qproofs ofDN'x and DNfi'x- Hence by formula induction: ,. u hypothesis, ^ • A -~— DNfi'x ^—- * ,by •induction

1 DDGJ6'6"

* by induction hypothesis, la, 6. (c) (f> is N
3.6 Modus Ponens

If a) is absent we have

87

N(j)' 16

X (d) is of the form (E£) 0'{£} and the result holds for (/>'{t}}; we have

Do)f{7j}

K

D N < f >w ' { 7 ) } X , . , . . , . . . - * by induction hypothesis;

we wish to demonstrate

It suffices to demonstrate

because from the reversibility of l i e we can obtain an ^>-proof of DN(j)'{rj}x for any variable r\ which fails to occur free in DN'{rt}x9 also immediately above Da)(EE) } there may have been several cancellations of (Eg) $'{£}. K

If 2 (Eg) $'{£} fails to occur in the left upper formula then the result 0=1

follows at once by dilution. Otherwise we use theorem induction on the left upper formula. If the left upper formula is an axiom then (Eg) '{£} is absent. If the left upper formula is the lower formula of any building rule other than the introduction of (Eg) '{g} by II d and if the result holds for the upper formula then we easily obtain DOJX on application of the K

same rule because D 2 (Eg) $'{£} must be in the subsidiary formula 6=1

of that rule. We had an almost similar situation under (a). But if the rule is an introduction of (Eg)
bymductxonhypothesis, by formula induction,

From the demonstration of Prop. 1 we see that the variable 7} in DN'{rj} x obtained from the reversibility of l i e can be any new variable, the

88

Ch. 3 Predicate calculi

variable £ which is restricted by lid! in the upper left formula might be any variable. If we change all free occurrences of £ in the J^-proof of the upper left formula to a new variable then we obtain an J^-proof of a formula which differs from the upper left formula in that now £ is a new variable, but o) and {EE,) <j)'{£] may have suffered a change of variable to this new one. In this case we should end up with DGJ'X, where co' differs from o) in that one free variable in o) has been changed to a new variable (which fails to occur in o)). By change of variable in the J^-proof of Dcox we can get back to Dcox- This completes the demonstration of the proposition. Note that we have given an effective method for eliminating the cut. This consists in taking a highest cut in the proof-tree and either eliminating it outright or replacing it by a cut higher up the proof-tree or by a cut or cuts with simpler cut formulae. Thus applying the process a highest cut ultimately gets replaced by cuts with atomic cut formulae and these can be made to disappear altogether. 3.7 Regularity PROP.

5. !FC is regular.

We have to show

— , /f , ,, *.

We show from this the result and

B6xlr y{6\ r ;C *

follow, the first by the reversibility of 116' and the last follows easily by Modus Ponens which can be eliminated. We proceed by formula induction on x{n}' The cases when ^{77-} is n or is Nx'{rr} or is DX'{TT}x'i71} are dealt with as in Cor. (i), Prop. 8, Ch. 2. If x{n} is (E£) x{n> £} a n ( i ^ n e result holds for ^{TT, £} then we have

whence by IId, la we get

, £} (Eg) xtt,

3.7 Regularity

89

and by lie DN (Eg) X{, £} (Eg) X{f, £} DN(Eg) X{f, Z) (Eg) X& as desired. The variable £ can occur in (j) or in i/r or in both. CoR.(i).

DB^o) Dx{}«>

We proceed by formula induction on xi17}- The details are left to the reader. D10 (^ §)#{£} for N(Eg)N${Q. A is called the universal quantifier. PROP.

6. 1-5 and l*-5* and 6 of Prop. 5, Ch. 2 hold in ^c, also

7. B(Eg)Df{QW!£)t{&fc

7

*-

8. B(^g)D^{g}^2)(ilg)^{g}^; 8.* distributivity of quantifiers. In 7-8* incl. the variable E, fails to occur free in i/r or 0{I\}. 9. BN{E£){Q(A£)N{§\ 9*. 10. B&(El;). lid'

\r\lfn

9 where £ fails to occur free in co or free in
' is a derived rule in ^c. The rule l i d ' is reversible. For 7-9* consider 7. We have from Prop. 2 T TT, la, Ha, DND{£,} \lrD(EE) {£} ty II e, £ fails to occur free in ^ or free in X again

DftQtDftQ^

on d a t u r r e v e r s i b m t y

1 a, 11 a, in

DNf{E£) Dm f the result now follows by life and definition of B.

90

Ch. 3 Predicate calculi

7*, 8, 8* follow similarly and are left as exercises to the reader. 9. We have DN(Eg) N(j>{^} (Eg) N{£} Tertiumnondatur, bydef.of O,,4, DNNNjEg) NJ{g} (Eg) NJ{Q again

DN(E£)N{£}(E£)N{£} DN(Eg) N{£} NN(E£) N<j>{Q

O>

Tertiumnondatur, ° b

the result now follows by 116' and the definition of B. 9* follows similarly and is left to the reader. 10, 10* are trivial. 3.8 The system ^"c The system 8F"Q is like the system !FC except that we add the symbol A of type 0(01) instead of E and replace the building rules II d, e by I I d' D(fi{y} (o rj is absent from 0) and <j) {I\}, D(A £) (j){£) o) £ is absent from
DN(J){rj] OJ

g is absent from {Tt}9 or is everywhere bound in

we then replace D 10 by:

D10'

(mm

for

The systems !FC and tF"G are equivalent via the definitions D 10 and D10'. A convenient system which is equivalent to SFC and to 3f"c is the system IF'Q where we use the building rules lid, l i d ' and one of the definitions D 10, D10'. All our results so far hold for &"c. 7. / / §5 is an ^c-iheorem then so is N<j>, where (j> is obtained from by interchanging D with K9 E with A, n with Nn where n is atomic and conversely. $ is called the dual of
o b t a i n BN.

COR. (i). If= without use of l i e then ——.

3.8 The system &"c

We verify that if % is a rule other than He then -~

91

is a derived rule.

This fails for He. P R O P . 8.

Consider the first, we have Tertium non datur

Ila, la

IIdIa

Again DN{£} ${£}

Tertium non datur

Ila,la

TT,T 11 a, la

DN(E£) Pm

TTp

the result follows from these by 116'. The remainder follow in a similar manner and are left to the reader. 3.9 Prenex normal forms An e^-statement is said to be in prenex normal form if it is of the form (Qic)x{%}y where (Qj) is a sequence of quantifiers, existential or universal or both, and x{%} is an J^-statement void of quantifiers. (Q%) is called the prefix and x{%} is called the matrix. By repeated application of 7-9* incl. of Prop. 6 and Cor. (i) of Prop. 5 we see that each ^,-statement

92

Ch. 3 Predicate calculi

closed J^-statement in prenex normal form with prefix (Qj) and matrix X{ic}. Let y. be the sequence of distinct variables £',..., Qe) in t h a t order and {Q%) a sequence of quantifiers on these variables in the same order. A variable QK\ 1 ^ K ^ 0 is called general if (AE,M) occurs in (Qj) and is called restricted if (#£<*>) occurs in (Qj). I f l ^ y < / c < # then £<"> is called superior to QK\ and £(/c) inferior to £(y).

An ^-statement ^ without bound variables is said to be tautologous under the following circumstances: ^ is built up from other ^ - s t a t e ments joined together by N and D; we replace each of these part statements b y / or by t, but make the same replacement at each occurrence of a variant, we then have a formula built up from/and t (regarded as of type o) by N and D, we then calculate the value of the statement as follows: replace Nf by t, Nt by/; Dffbyf, Dtf, Dft, Dtt by t. The final result is either t or/. If the final result is always t however we make the initial replacements of the part statements then the ^-statement is said to be tautologous. It is easily seen from Ch. 2 that a ^-statement is a ^-theorem if and only if it is tautologous. An J^-statement
K

We shall find that the J^-statement 0 is a disjunction 2 fr{d\ where 01

3.9 Prenex normal forms

93

the disjunctands are of the same logical structure but merely differ by choice of individual variables, they are variants of a common form. l a is used to bring one of the disjunctands \Jrf, ..., i/r^ to the left when we apply lid or lid' to it. 16 is used to discard duplicates as they occur. The final ^ - t h e o r e m is of the form (#'£')... (Qf*>gM) jjr{£',..., £<*>} where each QM is either E or A and ^{£',..., £(7r)} differs from any of the disjunctands ft',..., i/rM merely by change of individual variables. Suppose we have an ^"^-proof of x m prenex normal form. We first modify applications of II a, if necessary, so that they fail to introduce quantifiers. Suppose that Dx'x" i s introduced by I I a then introduce X' x" o n e after the other, suppose NDx'x" *s introduced at II a then introduce Nx',Nx" separately and apply 116, (this forms two branches), if NNx is introduced then introduce x a n ( i a PPty Hc> suppose that (2?£) x! {£} is introduced at II a then introduce #'{£} instead and apply II d, suppose that {A£) #'{£} is introduced at I I a then introduce x'{v} instead, where 7/ is new, and apply II d'. Repeat this process as long as possible and we shall have used II a only with atomic statements or negations of atomic statements as main formulae. Thus we suppose that the ^>-proof of (Qi) ^{j} only uses II a with main formulae which are atomic formulae or negations of atomic formulae. Note that we do this without using 16. Instead of using the system tF'c we shall use an equivalent system which has the rules IId*, d'* in place of rules lid, d\ where D % {&*>}
lid'* in lid'* a) is free for £',..., Qd) and so is 0{I\}, and £',..., £(^ are distinct. An ^t-proof will use rules lid*, d'* whenever possible, so it will be without a sequence of applications of II d followed by 16, etc. We now want to show how to modify an J^-proof of a prenex formula so that applications of 16, IId*, d'* come after applications of IIa, 6, c. Clearly the systems tF'c and ^"c are equivalent. We define the rank of an J^-proof of an J^-formula (Qj) ^{j} in prenex normal form, where % stands for £', ...,£(77), as the ordered Sntuplet {v, v',..., v(n)}9 where v is the number of occurrences of applications

94

Ch. 3 Predicate calculi

of rules II a, 6, c beneath applications of rule 16, v* is the number of applications of rules IIa, 6, c beneath rules IId*, d'* which bind a variable standing in the first argument place in ^{r.},..., v^n) is the number of applications of rules Ila, 6, c beneath applications of rules lid*, d'* which bind variables standing in the 77th argument place in ${%}. These are calculated as follows: take for instance rule 16. Mark applications of 16 in the J^-proof o f (Qj) 0{j}, let these be denoted by K\ ...,#<*>, let /i', ...,/^M be respectively the number of applications of rules Ila, 6, c beneath K', ...,K(n\ then /i' +/i"+... +JLC{K) = v, similarly for the other cases. Ranks are ordered lexicographically. We now show how to modify the J^-proof of the prenex ^-formula {Ql){l} by steps so that all applications of rules 16, IId*,d'* occur below applications of rules II a, 6, c in such a manner that the rank strictly decreases at each step. Thus the process will terminate when the rank is {0,..., 0}, because we shall see that if the rank is greater we can always make a step. We can then find an J^-proof in which all applications of rules 16, lid, dr occur below all applications of rules Ila, 6, c. Consider first rule 16. In what follows we omit mention of l a in many places. Suppose we have, apart from l a

where ^r is atomic or the negation of an atomic formula. Replace this by

j. a.

DDDjfrDfG) We are left with an ^^-proof of (Q%) N(/)0j . Suppose we have DN(j)0)

DNi/ra)

DND
3.9 Prenex normal forms

Replace this by DDNNG) DND^DN^o)

95

116, II a, to introduce N(j) in right upper formula, DNi/rco TTl^ TT . , _T_ . , . . , ^ 1 A l ' ' " +.o introduce ND
Call this case (i). Case (ii) is

Replace by Lj DDNi/rx«> ^— 116, II a to introduce another x i n right

upper formula,

Again we are left with an ^e-proof of (#£) ^6{j} of lesser rank provided we are using a highest case of 16 above 116. The use of rule II a, as already observed, is without any use of 16, so that if we have a highest case of 16 above 116 then the rank has fallen. Suppose we have DD^f 10,

^

He,

DNN<j>(o. Replace this by

DD4>0) DDNNNNa) DNN<j>o).

Again we are left with an J^-proof of (Q%) <j>{%\ of lower rank if we are dealing with a highest case of 16 above lie. Call this case (i). Case (ii) is DD<j><j> Dijra)

DDNNi/ru).

96

Ch. 3 Predicate calculi

Replace this by

DDcjxjyD^o) lie, DDDNNi/r(o

Again we are left with an J^-proof of (Q%) ^{j} which is of lower rank. Now consider rule lid*. Suppose we have

Replace this by

2)2^{9/} o)

TT

Again we are left with an ^"^-proof of (Q%) ^{j} of lower rank. Rule l i d ' is dealt with similarly, but may require a change of variable. Suppose we have

where o)' differs from co by having NN placed over a disjunctand. The other case where NN is placed over (E£) <£{£} is impossible because the theorem is in prenex normal form. Replace this by

Again we are left with an J^o-proof of (Qj) {$.} of lower rank. Rule lid'* is dealt with similarly. Suppose we have

) ${§ Nfco

3.9 Prenex normal forms

97

Replace this by

where (E£)${£} is introduced into the branch above the right upper formula in (a) at applications

Nowwehave

JTO^^-DTO^Q^ DD(E£)f{£\N/

where (i?g) 0{£} is in the subsidiary formula. Hence we shall have the same figure with these occurrences of (EE) <£{£} everywhere replaced by

Now add applications of I I a as follows:

Now from (6), (c) and (d) we obtain _

IIa,etc....

r c

J

IIa, etc.

Use this as the right upper part of (6), then finish up as in (b) and we have placed the application of II & above the application of IIeZ*. In doing this we have had to introduce various variants of <£{£}, this is done without using 16 or any applications of II eZ*, d'* that bind variables earlier in the list than the variables we are binding in (a). The effect of this is that in the rank {v,v',..., v^} the first component is unaltered because we have made our alteration without use of 16, and if the variables we are binding is £f®9 then v',..., j/^-1) are unaltered while if® is decreased by one, the other components may be increased. The total result is a reduction in rank.

98

Ch. 3 Predicate calculi

Suppose we have

a*

Replace this by DDX<j>{V}NXG>

DDXttQNfu '

'

'

by the reversibilityof lid'*,

In the reversibility of lid'* we may take the variables £ to be new and distinct from the variables r\. This allows us to apply lie?'*. As in the case of 16 below II d* we have decreased the rank. The reversibility of II cZ'* is performed without use of 16. For completeness we add: Rule lid'* is reversible. By this we mean that if we have an J^'-proof of D(Ag) ^{£} o) then we can find an J^-proof of JD2^{?/} OJ for some 2^{?/}. In the J^-proof-tree of D(A£,)(J){E]G) note the places where corresponding occurrences of (A£,)^{^} are introduced by lid'*. These will be of the form j>s^}ft/ DXtfrf»}<,fi» LEMMA.

J

In the JF^-proof from these places to D(AE>)^>{£)}(0 the part will remain in the subsidiary formulae everywhere. Hence we may replace all these occurrences of (A£) <j){£] by the disjunction of These can enter by IIa, etc., applied to the upper formulae of (/). In this way we obtain an ^"^-proof of instead of one of D(AE,)(f>{^}o). IIa, etc., as before observed, has been done without use of 16, and the only use of lid*, d'* has been on variables later in the list £',..., g(7r) than the variable £, so that the rank of the J^^-proof of (e) has decreased.

3.9 Prenex normal forms

99

To conplete the demonstration of Prop. 9 we make the alterations discussed above starting from the highest available places. Each time the rank is reduced, and as long as the rank is greater than the lowest rank we can always reduce it. This completes the demonstration of the proposition. 3.10

Let

H-disjunctions

(QV)..>W"Vn))n?>-''>^

(!)

be a closed ^^-statement in prenex normal form where the matrix i/r{£', ...,£(7r)} is quantifier-free. Here each Q^ is either A or E. If Q^d) is E then Qe) is called a restricted variable, if Q^e) is A then £(^ is called a general variable. Let there be n' restricted variables in (1) and let there be n" general variables in (1), then n' + n" = n. Form a list of all ordered nrtuplets of natural numbers {*/,..., *>(7r)} ordered by the sum v' + ...+ v^r) and lexicographically for those of equal sum. Take the initial segment consisting of the first K members. Now write down the list

£'

(2)

£(ff) )

b / o • • • ? Z>K

•

J

where the restricted variables in the yth line are x^v'\..., x^n}), {v*',..., v^} being the vih Tr'-tuplet in our list of Tr'-tuplets, and where the general variables in the first line are in order from left to right if £' is general in (1) or

SiT f)

x",..., x^ '

if £' is restricted in (1).

Suppose that exactly the first A restricted variables in line 6\ 6' < 6 are from left to right the same as in line 6, then the general variables are the same from left to right in these two lines up to and including the general variable immediately following the Ath restricted variable. The remaining general variables in line 6 are in order from left to right the next new variables in the alphabetical list x,x',x",... of variables. An example will make this clear. Let (1) be: (Ex') (Ax") (Ex'") (Ex*) (A&) DNpx^x"x^px'x'"x^,

(3)

where p is a three-place predicate. Let K be 12. For greater clarity we write xv instead of x'...' with v superscript primes. 4-2

100

Ch. 3 Predicate calculi

H-scheme of order 12 variables r

g

r

r

x2 ~xx x3 JX±

xx x2

x2

XQ

_x3 x13

x2 _x3

xx x2 _x3

g x3 x±

x8 x15 x5 x9 Xu

x10 ~xx x7 V'X1 _x2 xl± \_x2 xx x12 X,

x

l

xr

xu

line triplet sum line r 1 a? 1 [1, 1, 1] 3 2 [1, 1, 2]i 2 x 5 [1, 2, 1] 4 3 # 11 3 6 12

[2, 1, 1]. [1, 1, 3]" [1, 2, 2] 7 [1, 3, 1] 4 [2, 1, 2] 5 8 [2, 2, 1] 9 [3, 1, 1] [1, 1, 4]" R 10 [1, 2, 3L u

4

x2

g x2 x2 x2 XQ

x2 6 #! x2 7 # x2 5

iCj

8

tf2

9 10 11 12

XQ

x2 XQ #3 x13 #! x2 x. x2

r xY x± x2 xx xx x2 x3 xx x2 xx xx x2

r

xx x2 xx xx x3 x2 x1 x2 x1 x

l

g x3 x4 x5 x7 x8 x9 x10 xxl x12 xu X

15

x3

x1Q

The first twelve triplets have been written down in the prescribed order. In the column headed ' variables' the restricted variables occur in the first, third and fourth places and the general variables in the second and fifth places. The suffices of the restricted variables agree in order from left to right with the members of the ordered triplet in the same row. The general variables are then put in, x% and xz in the first line, x2 and #4 in the second line, since the first line begins with xx and is followed with x2 and the second line begins with xx then the general variable in the second place in the second line is also x2. Generally the second variable, which is a general variable, is x2 whenever the first variable, which is a restricted variable, is xv In the fourth line the second variable (the first general variable in that line) is x6 because this is the first available new variable in the alphabetical list of variables and this is the first time that x2 has occurred in the first place. Generally whenever the first variable is x2 then the second variable is x6. The H-scheme is obtained by writing down line 1 followed by those lines whose initial segment is the same as in line 1 for as long as possible. Thus lines 1, 2, 5 and 11 agree in having initial segments xxx2xx\ These are followed by lines 3, 6 and 12 which agree in having initial segments x1x2x2. This in turn is followed by line 7 which agrees with the above in having initial segments x±x2. The agreement of initial segments is denoted by bracketing. The 17-seheme of order 12 for (1) is the list (2) arranged by bracketing together lines with equal initial segments and ordering lexicographically within the brackets.

3.10 H-disjunctions Write

<&£2&«5

for

qe

for

101

q&°>&°> &»&>&*>

where ($», Qfi, QpQp and #»> are variables in line 6 of (2). We note that the H-disjunction of order 12 namely: 12

(4)

P

n is a tautology, in fact Dq±q12 is a tautology. 2 #0 fails to be a tautology, be0=1

cause we can only have a tautology when fflffiffl is the same as £f)£f )£f) a n d for 1 ^ e,df < 12 this only occurs when (9=1 and #' = 12. Thus the i?-scheme of order 12 for the statement (3) makes the ^-disjunction of order 12 a tautology. Now consider (Ex±) (Ax2) (Ex3, a?4) (Ax5) DNpx5xAx1px1x2x3. (5) Any £T-scheme for the statement (5) fails to make an iZ-disjunction a tautology because Qd) i s alphabetically later than £i0) while ^2e) is alphabetically later than $*\ hence &e)QP£e) fails to agree with g f ) ^ ' ) ^ ' ) for any 6, dr. Consider again the statement (5), an i7-scheme of order K for (5) gives rise to an if-disjunction which fails to be a tautology for any numeral K. A disjunctand of the if-disjunction of (5) is ,3> V4] xVl xVipxVi xa[vi] xVa,

(6)

where (T[v^\ > vx and p\yx, vs, v^\ > vly v3, v±. Consider the 2-valued model e/T in which the individuals are the natural numbers. We can make (6)

take the Jf-value/ by taking: pv1v2v3=f

for

v1
and pvxv2v3 = t

for

v1 > v2,

the values for v1 = v2 are immaterial. Consider the negation of (5) (Axx) (Ex2) (AxSi xA) (Ex5) Kpxhx±xxls[pxxx2xz.

(7)

Thus we can obtain a satisfaction of (7) over the model *A^ We shall demonstrate later the general proposition that if the Hdisjunctions of (1) all fail to be tautologies then there is a satisfaction of the negation of (1) over the 2-valued model in which the individuals are the natural numbers. Note that we lack a method for deciding whether

102

Ch. 3 Predicate calculi

there is a numeral K such that the if-disjunction of order K is a tautology. If we had such a method then the system &c would be decidable, we show later on that the system ^c is undecidable. From the tautology (4) we may obtain an ^^-proof in normal form of the statement (3). Apply universal quantification to the variables x5, x7, x8, xQ, x10, xll9 x12, xu, x15, x16 successively, these variables occur at one place only in (4) so the condition on variables in the rule for universal quantification is satisfied. Delete these variables from the H-scheme of order 12 this leaves: line 1 2 5 11 3 6 12 7 4 8 9 10

x6

The tautology (4) has become 12

(4.1) where q'd is qe if d = 1,2 otherwise q$ is (Ax5) q£{d)&>^Q0)x5, Now apply existential quantification to the fourth variable in every disjunction of (4.1) except q1 and q2 and delete those variables from the J?-scheme of order 12. The disjunction (4.1) becomes 12

(4.2) 0= 1

where qf{ = ql9 & = q2 otherwise q" is {Ex^){Axb)q^^^x^xb.

In

(4.2) ql is the same as q[x so cancel qu, also ql, qfQ, q[2 are the same so cancel ql and ql%, also q% is the same as ql so cancel q%. Thus (4.2) becomes (4.3) D...Dq1q2qlqlq';qlqlql0. 7-times

3.10 if-disjunctions

103

The deleted H -scheme of order 12 has become: line 1

X2

xz

x±z

x2 xz "x x2 x^

3 7 4 9 10

In the disjunction (4.3) the variable x4 occurs only in q2 hence we may apply universal quantification to it, we can then apply existential quantification to the fourth variable in q2 this makes q2 the same as ql so cancel ql and (4.3) becomes: (4.4) 6-times Delete these variables from the ZT-scheme. This is indicated above by a stroke through them and through 5. Now apply existential quantifiers to the third variable in each disjunction of (4.4) except qx and cross these variables out of the ^-scheme. The disjunction (4.4) becomes Z>...Z>&>xsxtx5,

(4.5)

A = 2, 3, 4, 7, 9, 10.

f

In the disjunction (4.5) q 2", q%, q? are the same, so are q± and qfg. Cancel duplicates and we obtain the disjunction (4.6)

Uio and the deleted i/-scheme: r

g

xx

x2

xz

xlz

r

r

g

JUX

JUX

JUZ

L

line i

2 10

In the disjunction (4.6) the variable xls occurs only in q±Q and the variable XQ occurs only in q% hence we may apply universal quantification to them, we can then apply existential quantification to the first variables in q'l

104

Ch. 3 Predicate calculi

and q^Q. Cross out these variables from the deleted //-scheme. q*l and qx0 have now become the same, so omit q'i0. The disjunction (4.6) has become: Vxj) {Ax2) (Exs, x4) (Ax6)

qxxx2xzx4xh,

and the deleted //-scheme is: r

g

r

r

g

X-^

X%

I X^

X^

Xg

L

line JL

2 4

The variable xz occurs free only in qx so we may apply universal quantification to it, we can then apply existential quantification to the third and fourth variables in qv This makes q± the same as q2, so cancel q2. We then obtain the disjunction D(Ex3, x4) (Ax5) qQV^x3x4xb{Exx)

{Ax2) (Exs, x4) (Ax5) qxxx2xzx4xb. (4.7)

In the disjunction (4.7) the variable x2 occurs free only in the first disjunctand, so we may apply universal quantification to it, we can then apply existential quantification to the first variable. This makes the disjunctands the same, cancel one of them, and we are left with (3). The //-scheme for (1) of order K can be written down on a fixed plan for any numeral K, hence the place number of a general variable is uniquely determined by the place number of the superior restricted variables. Thus for the statement (3) the place number of the second variable is uniquely determined by the place number of the first variable, and the place number of the fifth variable is uniquely determined by the triplet of the place numbers of the three superior restricted variables. P R O P . 10. If for some numeral K the H-disjunction of order K of a closed ^Q-statement in prenex normal form is a tautology then is an ^'ctheorem.

The method of demonstration is the same as that given in the worked example. We apply quantifications to the various disjunctands of the //-disjunction of order K and delete the corresponding variables from the //-scheme, and cancel duplicates as they occur. At any stage in the proceedings a variable in the //-scheme is available if it is at the end of its line in a deleted //-scheme and is a restricted variable or is a similarly

3.10 H-disjunctions

105

situated general variable which fails to occur elsewhere in the deleted H-scheme. If there is always an available variable until all the variables are deleted from the H-scheme then we obtain an J^-proof of (j). Suppose that at some stage there fails to be an available variable, then in the deleted if-scheme at that stage the variable at the end of each line is a general variable and each such variable occurs elsewhere in the deleted H-scheme. The lines in the deleted if-scheme are always distinct because identical lines get deleted as soon as they arise by cancelling duplicates. A variable can only occur once as a general variable in a deleted if-scheme but it can occur again as a restricted variable. For instance the variable x2 occurs once as a general variable and six times as a restricted variable in the complete if-scheme of order 12 for the statement (3). The restricted variables which precede a general variable in a line of an if-scheme are alphabetically earlier variables. Thus if there fails to be an available variable then each general variable £ at the end of a line occurs again as a restricted variable in another line which ends in an alphabetically later general variable TJ. In turn there is another general variable alphabetically later than TJ and so on without end. This is absurd because the if-scheme of order K is displayed. Thus there is always an available variable and we may continue to quantify and remove duplicates until we obtain an J^-proof of 0. This demonstrates the proposition. P R O P . 11. If cj) is an ^'c-theorem in prenex normal form then there is a numeral K such that the H-disjunction of order K is a tautology. According to Prop. 9 the J^-proof of (j) can be modified to one in normal form. We then have a tautology (8) 9= 1

where each i/r^ differs from ${£,',..., £(77)} by change of individual variables. From (8) we can obtain 0 by l a , 6, IId, d'. Let 3F'Q be the same as ^'Q except that the individual variables are x*, x*', #*",..., and let ^'c (J 3F'* be the same as the system tF'c except that the variables are those of !F'C and those of 3F'Q . We now change the individual variables in (8) to those of ^'Q by superscripting an asterisk to each variable. In this way let ^(A) become ^(A)* and (8) become (8*). We will give a method of changing the individual variables in (8*) to J ^ variables in such a way that (8*)

106

Ch. 3 Predicate calculi

is changed into part of an //-disjunction. Let (8*) become (9) by this change. Clearly if we change an individual variable at all its occurrences to another one then a tautology remains a tautology. Thus (9) will be a tautology and part of an //-disjunction. Thus there will be a numeral K such that the H-disjunction of order of K is a tautology. Let (j> be

where if ft is zero the initial set of universal quantifiers is absent. Let the variables in ijr'*,..., ^JrOO* be (10) We now replace the ^f-variables by J^-variables in (10) from left to right. We first replace through (8*), (10) the first [i variables in each line of (10) by x',..., x^ respectively, that is ££*, ...,££* are replaced by x' at all their occurrences,..., ^ * , . . . , ffl* a r e aU replaced by x^ at all their -statement, occurrences. Let (8*) then become (8'), it is an ^'Q^^'Q clearly it is a tautology and we can obtain (8) from it. This is because all the other general variables in (8*) are distinct from x\ ...,x^\ By this change (10) becomes (10'). Secondly if the lines V and v" of (10) agree in having the same initial segment and if the next variable is different and is a general variable then we may alter this general variable in one of the lines so that they are both the same. Suppose these general variables are $*>* and $?A>* so that the segments £*, ...,£* and £*, ...,£<*>* are the same. In passing from (8') to we shall at some stage generalize g(#A)* a n ( j a ^ a n o ther stage we shall generalize ^ A ) * . Suppose that we generalize £^A)* before we generalize £^A)*- We can modify the order of quantifying the individual variables in (8') so that we generalize £££A)* immediately after (except for permutations) generalizing ^ A) *» This follows because when we are about to generalize ££?A)* every as yet unquantified variable will be distinct from ^ A ) * so that any variable in lines other than line v' which is quantified between the generalizations of and £££A)* could have been quantified before the generalization of . ^ n y quantifications on variables in line v' which occur between the generalizations of ^ A ) * and £^A)* (which must be restrictions, because the

3.10 ^-disjunctions

107

initial segments of the two lines are the same) can take place immediately after the generalization of £££A)*, any such variable is distinct from £*£A)*, again because the initial segments are the same. Having made these modifications so that we generalize £^A)* immediately after (except for permutations) generalizing £*?A)* we now everywhere replace £$?A) * by £
108

Ch. 3 Predicate calculi

variable £p[0] is the same as a general variable in line p[p[0]], say p2[d], and so on. Since the scheme (10) is displayed we must have for some fi' < ju,"'.

Thus we have a general variable in line p^Xff] is equal to a restricted variable (which is available) in line p^'^ld] and so it is impossible to generalize the former until the latter has been restricted, but this restricted variable is inferior to a general variable which must be generalized first, this in turn is the same as a restricted (available) variable in line pfl"'1'2[6] which must be restricted first, and so on until, a general variable in line ps^'[d] is the same as a restricted (available) variable in line p^'id], but this variable is £a itself. Thus finally, we are to restrict a restricted variable £a in line a before we generalize an inferior general variable. This is absurd. Thus there is always an available restricted variable distinct from any general variable. Replace the alphabetically earliest such restricted variable (which is an 3F'* -variable) at all its occurrences by the first as yet unused 3F'Qvariable. This leaves the general variables unaffected. Thus an available variable (which is an J^^f-variable) can always be replaced by an SF'G- variable without upsetting our build-up of an //-scheme by renaming of general variables. Finally each 3F'£-variable is replaced by an tF'cvariable in such a way that the resulting disjunction is a part disjunction of an if-disjunction, because we have chosen the general variables so that this should be so. This completes the demonstration of Prop. 11. 3.11 Validity and satisfaction An J^-statement is called generally valid over Jf when (vii) below has been demonstrated: (i) We replace a part (E%) i/r{£} of 0 by N(AE)N{Q. (ii) (a) If n is a propositional variable which occurs in ^ then we replace it by t or by/. (6) If n is a one-place predicate variable which occurs in <j> then we replace each of nv v = 0,1,2,... either by t or b y / . (c) If n is a two-place predicate variable which occurs in then we replace each ofnvK v, K = 0,1,2,..., either by t or b y / .

3.11 Validity and satisfaction

109

(d) similarly for many-place predicate variables, (iii) We replace the free individual variables by numerals. (iv) We replace a part (A!;) i]r{£) of $ by t if and only if i/r{v} is replaced by t for v = 0,1,2,... otherwise we replace {AE) ^{£} b y / . (v) We replace Dff b y / and Dft, Dtf, Dtt by t. (vi) We replace Nt b y / and Nf by t. (vii) $ reduces to t however the replacements (ii) (iii) are carried out. We lack a test for general validity. This will be demonstrated in Ch. 7. A closed J^-statement is said to be satisfiable over J^ if it can be shown to reduce to t for at least one replacement under (ii), (iii). For example consider: DN(A^DN^}x{QDN{A^^{Q(A^x{^ ( n ) We show that it is impossible for (11) to reduce to/when the above process is carried out. If (11) reduces t o / then N(AE)DNi/r{!;}x{£} and DN(AE) i/r{Q (Ag) x{£} must both reduce t o / . In order that this happen DN\lr{v}x{v} a n ( i ^ M must both reduce to t for each numeral v, and x{v] must reduce to / for at least one numeral, say A:. Then DNJ/T{K} X{K} reduces to t hence ^{K} must reduce t o / , this is absurd. Thus (11) always reduces to t no matter how the replacements are carried out. Consider (3), give t, / to jpX/jiV in any manner, if pv'v"v'" is t for some set v', v'\ v'" of numerals then (3) is t, iipv'v"v'" is always/then (3) is t. 12. A closed ^c-statement is an ^c-theorem if and only if it is generally valid over JV. The J^-axioms are generally valid over JV*. The J^-rules preserve general validity over JV*. Thus J^-theorems are generally valid over Jf. Now suppose that the J^-statement
110

Ch. 3 Predicate calculi

the value / for some assignment of values t, f to each TTV' ... y(A) for each predicate variable which occurs in <j) and for each set of arguments which occurs in an H-disjunction. If a set of arguments fails to occur in any if-disjunction then we give nv'..MX) the value t. Let i^K be an assignment of values t, f to each TTV' ... y(A) which occurs in HK, the JBT-disjunction of order K. irK will only give values to TTV'' ...v^ for those argument sets v' ...v^ which occur in HK. Let J(K be the set of assignments i^K. Now HK is a part disjunction of He for K < 6, hence one i^e will contain all the values given by at least one i^K. We can express this by saying that at least one irK can be extended to become a ^ or that a *Ve with domain of definition restricted to that of i^K becomes a i^K. Each *JlK contains at least one ^ Hence there is a valuation i^ which defines TTV'.. Mx) for each argument set which occurs in some HK and which gives the value / to (j). Now this valuation says that for any values given to the restricted variables there are values that can be given to the general variables, which values depend on the values given to the superior restricted variables, in such a manner that ^ takes the value/. Then N<j) takes the value t and for any values given to the general variables in N, which values depend on the values given to the superior general variables, in such a way that N<j> takes the value t. But this is to say that N<j) is satisfiable over JV9 and we have finished. This is called the denumerable model. COR (i). An ^c-theorem is valid over^V*K (the set of natural numbers < K). The J^-axioms are valid over JVK for any K and it is easily verified that the J^-rules preserve validity over J/%K so the result follows. ValiK

dity over J^K is decidable, we need only replace (J57£) ${£} by 2 0{#} 0=0 K

(and (A £)${!;} by n {@}) replace {6} by a propositional variable p^d) 6= 0

and evaluate by truth-tables. COR. (ii). An ^-statement which is valid over JVK for each natural number K but which fails to be valid over JV can be found. Consider the conjunction P of the following J^-statements: {Ax) Npxx (Ax, x', x") CK
3.11 Validity and satisfaction

111

It is clear that P fails to be satisfiable over any JVK. Hence NP is valid over every JVK. But P is satisfiable over Jf (let pxx' be x < x'), hence NP fails to be valid over J^. Another example is the negation of the conjunction Q of the following ^-statements: (Ex)(Ax')Npx'x {Ax, x', x", x'") GKKpxx"px'x"pxulxpx'"x' (Ax) (Ex1) pxx'. Again it is clear that Q fails to be satisfiable over any ^ fiable over JV (let pxx' be Sx = x').

but is satis-

3.12 Independence 13. The symbols, axioms and rules of J ^ are independent. We have to show that N, D, E, p^x\ #(A) are independent. Clearly p is independent otherwise ^-theorems would be without occurrences of p, similarly for x and the other variables. But note that if we omit p we get an equivalent system, similarly for x and the other variables. The only closed J^-formulae of type 00 formed from D, E, X and variables are: PROP.

\p.p, *kp.E(\x.p), \p.E(kx.DpE(kx'.p)),

Xp.Dpp, Xp.DpE(Xx.p), etc.

but these all using A-rule (i) give J3A00, where A stands for any one of the above formula. Hence if we took A as a definition of N then BNtfxfi Whence

DpNp

* 16 P and this is absurd because an ^"o-theorem must contain D. Thus the symbol N is independent. The only closed ^-formulae of type 000 without occurrences of D are: Xpp'.
112

Ch. 3 Predicate calculi

E is independent because the only J^-formula of type 0(01) that we can construct from N, D and variables is Xp0L. (j>, where (j) is of type o and fails to contain E, but this fails to be closed and so violates the conditions for a definition. The demonstration that the ^-axioms are independent is the same as for &>c. The ^,-rules are independent. First, rule l a is independent, because if we omit rule la then we are unable to obtain the J^-theorem DNpp. Any J^>-proof of DNpp fails to use IId,e, because once E enters an .^-proof then it remains in that #^-proof from that place till the base. Thus any J^-proof of DNpp will be a ^ c -proof possibly using 16. Any J^-proof of DNpp will fail to use 116, c because it is without occurrence of NN or of ND, hence an J^-proof of DNpp which omits la proceeds from the axiom DpNp using 16, II a only. We are unable to apply 16 to DpNp so we can only apply II a obtaining DtfiDpNp where (j) is an ^cstatement built up from D, N and p only, other variables and E must be absent because if they were introduced into the ,^c-proof by II a then they would remain in the ^ o -proof from that place to the base. We can only use 16 on Dcj)DpNp if c. Note that the demonstration that 16 is a derived rule in £PC requires use of all ^ c -rules whence if we omit a ^ c -rule we are denied use of the dependence of 16 on the other ^-rules. The ^ - r u l e s II d, e are independent because they are the only means of obtaining J^-theorems starting with DE, DNE respectively, or E, NE respectively (when the rule is used without subsidiary formula). Clearly there are such J^-theorems, for example:

D(Ex)pxNpx and

DN(Ex)px(Ex)px.

For the rule 16 we note that if we are denied the use of rule 16 then we are unable to obtain: (Ex') (Ax") (Ex" Any J^-proof of (3) which fails to use 16 also fails to use IIa, 6, c (these

3.12 Independence

113

lengthen the formula) and must start with the axiom px'x"xmNxfx"xr" it must then use la, lie, lid twice l i e and lastly lid. But it is impossible to generalize only the first occurrence from the left of x'". 3.13 Consistency 14. The system &c is model consistent. We show that fFc has a model with a sole individual a and two elements t,f of which t is designated a n d / is undesignated. It has the constants N, D, E. N and D obey the same rules as in the model J(c for 8PC. The PROP.

rulefor2?is

It is easily verified that these rules give a model for ^c. 15. The system !FC is consistent with respect to negation. We have to show that if ^ is a closed ^-statement then at least one of and Ncj) are J ^ theorems, then so are (j) and DN
3.14 ^Q with functors The system J ^ with functors or constant individuals or both can be dealt with as the system ZFC. We have the additional rule: v free in

^'

a b s e n t

from

114

Ch. 3 Predicate calculi

This is called the rule of substitution. Here a is a term of type i and is free in ^{a}, i.e. if g is a variable which is free in a then corresponding occurrences of £ are free in ,.,«*,,. for ^x'x" .pfxgxx'hfx"x, where / is of type u9 g is of type in and h is of type tu so that Pf0g01hf20 is of type out. Similarly p^^ y,Xtpafxgxa for where/ is of type u and g is of type in and a is of type t, so that pafogoa is of type oi. The rule 11/would then amount to a rule for changing predicates, but this could always be done in the axioms before we began. 3.15 Theories A theory &~ based on ^c is an applied predicate calculus in which certain statements are specified as axioms, it may have some extra rules. Suppose that the ^"-axioms can be displayed, let <j) be their conjunction, and let $ be the closure of (j>. If ty is a ^"-theorem by the deduction theorem Prop. 3, Cor. (ii) we obtain the ^-theorem D^i/r. Suppose that Dcoi/r and DNtyx are ^""-theorems then Co)^r and DNtyDN^x, whence by Modus Ponens , but Modus Ponens can be eliminated from J^, thus an ^ c " ^ e o r e m > hence by 16 DN^Dux, i.e. C^DCJX is an J^-theorem. K

Now suppose that is n 4^ where (j^e\ 1 ^ 6 < K, are disjunctions 0=1

of atomic statements or negations of atomic statements, from the J^>theorem C(J)DOJX we obtain by Prop. 3, Cor. (iv) <j)r, ...,ftK)\r#r>Da)x, thus is a ^"-theorem. 3, COR. (V). Modus Ponens can be eliminated from a theory whose axioms are disjunctions of atomic statements or negations of atomic statements. Suppose that we have 6',..., ^Y^' Doj^Jr and ^',...,ftK))r&'DNftx then we

PROP.

c

c

have C$DG)X as above whence we have 0', ...,^K)\-^DO)X,

by Prop. 3,

3.15 Theories

115

Cor. (iv). A theory whose axioms contain free variables would normally be based on &"'c rather than on ^c. The rule for substitution of variables merely converts an axiom scheme (as used in ^c) into a set of particular axioms. A theory whose axioms are disjunctions of atomic statements or negations of atomic statements is called a theory in free disjunctive form, free variables are allowed. COR. (vi). If jJ.

is a rule in a theory in free disjunctive form then

^— *,

where co is subsidiary.

We have ^Sffi taking N<j) for to. Thus Cffl, now is a ^-theorem, whence by Modus Ponens twice we get DNtfxo if we have DNi/ro). But Modus Ponens can be eliminated. COR. (vii). The deduction theorem holds in a theory whose special rules are without restrictions on variables. The demonstration is the same as before, the extra rules of the theory behave just like the e^c-rules other than l i e . COR. (viii). Modus Ponens can be eliminated from a theory without axioms and whose special rules are without restrictions on variables or iutroductions of E. We proceed as in Prop. 4. The case when $ is atomic and DN is introduced by a special rule then this acts just like a case of introduction by II a.

3.16 Many-sorted predicate calculi A K-sorted classical predicate calculus of the first order is formed from the symbols: type

xc • XL{K)

C : L(K)

pOL>,...,pol(K)

OL', ..., OL(K)

name

individual variable of the first sort individual variable of the /cth sort one-place predicate variable

116

Ch. 3 Predicate calculi

type name es as Poi'i > Voi'i"> - • • 9 PoiWiW tyP shown two-place predicate variables generally Poe\..eW> where e^, types as shown A-place predicate variables for 1 ^ 6 ^ A, is one of *',..., *M E',..., EM O(OL'), ..., O(OL{K)) existential quantifiers of types shown X abstraction symbol ' generating symbol N oo negation symbol D ooo disjunction symbol ( ) parentheses The axioms are T.N.D. for all atomic statements. The rules are those of fFc with restriction and generalization for each type of individual variable, denoted by l i d ' , ...,IId{K\ He', ...,IIe (lc) . We denote the /csorted classical predicate calculus by ^CK. The situation is just as if in $FC we labelled the variables as x^-K+d\ 1 ^ Q ^ K, and stated lid, e separately for each 6,1 < 6 ^ A:. But the main difference is in the argument places of the predicates. Let J r ( J ) be !FC plus constant one-place predicates 8\ ...,$ (AC) and additional axioms DNS'^S'Z,..., DNSMgSMg. We give a method for translating J^-statements into J ^ - s t a t e m e n t s in such a way that 3^CK' theorems translate into jF^-theorems. The translation of an ^CKstatement is obtained as follows: f

(a) <j) is atomic, say P01{B')^L^)X%I). (A)

K n S^x^'^+^p^e')

. .a^&o), the translation is l(0 (A))^-/+^...^^

(A)

+«.

v=l

If two of the individual variables are the same then we omit an occurrence of S followed by that variable, (6) ^ is Ni/r, its translation is Ni/r\ where ^r' is the translation of ^ , (c) ^ is J D ^ X its translation is Di/r'x', where i/r', %' are the translations oii/r\X respectively. (d) $ is {E0g0)f{!;e)9 its translation is (Eg') f'{£'}, translation of

where ^'{g'} is the

3.16 Many-sorted predicate calculi

117

(e) (j)r is the translation of
if and only if its transla-

tion into 3~ is a 3~ -theorem, (ii) If 3~Kis consistent with respect to negation then so is 3T. (iii) If ^ is consistent with respect to negation then so is 2TK. (iv) There is an effective method whereby given a 3~K-proof of a &~Kstatement (j) we can find a & -proof of the translation of

we can find a ^K-proof of (p.

(ii) follows from (i), so does (iii). Ad. (ii) if y is inconsistent with respect to

118

Ch. 3 Predicate calculi

negation then we can ^"-prove (j> and N(j) for any ^"-statement 0, for the case when ' is the translation of

^/J; f —, £ fails to occur free in
£ is a variable of sort 6, and where 0'{F} is the translation of 0{F} and o)f is the translation of a). But this is still a case of lid. Similarly for lie. An axiom DnNn translates into Dn'Nn', where n' is the translation of zr, this is a case of T.N.D. and so is a ^""-theorem, add its ^"-proof. The other ^-axioms translate into ^"-axioms by definition of 3~. Thus half of (iv) is demonstrated. For the second half of (iv) first suppose that 2TK is ^CK\ omit each occurrence of S^iv))af-K'^v)+dlv)) and replace the remaining occurrences of &'/*»•&»> by dp. Omit K and II in (A), also replace (Eg) by (EV>£0) whenever S<®£ occurs in the scope of (E£). All this converts the ^"-statement x[r into a ^-statement
Thus

(Exi)pxi by (Ex)(KSixpx). (Axi)pxi becomes N(Exi)Npxi N(Ex) (KStxNpx) (Ax)NKpSxNpx (Ax) CSxpx.

3.16 Many-sorted predicate calculi

119

Let O be the conjunction of those ^-axioms used in the ^-proof of <j>. Let Y be the translation of O into 3T. By the Deduction Theorem we have CO
I(i)

Iaa

and the rule

o) is subsidiary and may be omitted. 0{/?} is called a variant of
120

Ch. 3 Predicate calculi

Prop. 9 goes through because we can push all applications of I(ii) back into the axioms and so above rules la, b, lid, d'. In the definition of tautology a = a must be given t. Prop. 9becomes: P R O P . / 9 . An I^c-proof can be modified so that all applications of rules IIa, b, c, I(ii) occur above all applications of rules 16, lie?, df. An / ^ - p r o o f then divides into two parts; the first part is a free variable J^-proof of a disjunction. The second part consists of applying quantifiers. In Prop. 10, 11 we need to change 'tautology' to ' / ^ - t h e o r e m ' . In the definition of 'generally over e/T' we add: v = v is given t and v = fi, where v is different from pi, is given/. Prop. 12 and 14 carry over without difficulty.

P R O P. / 1 3 . Tine rule I(ii) is redundant otherwise the symbols, axioms and rules of IZFC are independent. We need only show that the symbol / is independent of the other symbols and that I(i) is independent of the other axioms. The symbol / is independent of the other symbols of I!FC because if there was definition of / in terms of the other symbols o£IcFc, i.e. in terms of the symbols of £FC then this will be without use of any free variables, because a free variable in the definiendum must be present as a free variable in the definiens, this leaves only N, D and E and bound variables and with these we are unable to construct a formula of type OIL Similarly the axiom I(i) is unobtainable from the axioms of ^c. P R O P. /14.1!FC is consistent with respect to negation. If we could /J^-prove <j) and N for some /^ o -statement
obtain an ^ - p r o o f of D 2 Nla^oc^KNrfxf) where Io^0)oc{e) were the cases of I(i) used in the original proofs In this replace Io&e)aSe) by Dn^Nn^ and K

we are left with an^ o -proof of D021 NDn^Nn^KN^, but this is absurd e 0 1 because D 2 NDn{d)Nnid)KN
Dll.

V!£)#£} for

read 'there is exactly one thing with the property

3.17 Equality

121

Given a theory 2T there is another theory 2T1 effectively obtainable from 3~ such that ^""-theorems are ^"'-theorems and the ^"'-axioms are disjunctions of atomic statements of their negations. To obtain &' we replace any ^""-axiom of the form K
Dpoca) Dlotfio) n , ,. , ——jT—n——, for a one-place predicate,

Dpocyo) Dlafiaj Dpfiyco '

Dpyocco Dlafia) DpyfSo) '

for a two-place predicate, including / itself,

and so on, and similar rules for Npoc, Npocy, Npyoc, etc. We now show that rule I(ii) is obtainable from these special cases. We proceed by formula induction. (a) 0{a} is atomic; the result holds by hypothesis. (6) "{ot} and the result holds for ^'{<x} and for §J"{a}, we have

DDIa/3aj<j)'{/3}

DD'{p}<j)"{$}(*) as desired.

122

Ch. 3 Predicate calculi

(c) (j){a) is N(j>'{a) and the result holds for '{a}, we have by formula induction: (c') '{&} is atomic, the result holds by hypothesis; (c") <j){(x} is N"{a} o) whence by the reversibility of l i e we have D"{<x} a), thus D"{<x}(o

DNN$"{(}} o) as described. (c'")
\

*s a n i^roduction of E, replace this by Di/r{y,ot}(o

D(E£) f{£, /?} o) and the result follows. This completes the demonstration of the proposition. #«}

PKOP.18

(Aj)CI&4{Q

ft*) {

Ad. (i)

^a} DNIoc£6(a} r<> }

Ila r / .. x

I(ii)

. ,, . T MTr „ usmg the axiom nDIocgNIotg,

3.17 Equality

Ad. (i')

123

(AMI&M* reversibilityofllc*', ^-L- * substitution, M.P. using axiom I(i),

Ad. (ii)

—i-i-2—

lift

and axiom/(i),

Ad. (no 0=1 (

one /? ^, 1 < d ^ /c, must be a otherwise lower formula is/, this is absurd if upper formula is t, hence KIaa
DIoc/So) DI/3yoj

— and The first comes from Iaoc DIaocco DI/SOCG).

The axiom expresses the reflexiveness of equality, the first of the derived rules expresses the symmetry of equality and the second derived rule expresses the transitivity of equality. A relation (a formula of type ou) is called an equivalence relation if it is reflexive, symmetric and transitive. 3.18 The predicate calculus with equality and functors We showed before that !FC with functors is virtually the same as ^c without functors in that we can easily translate from the one into the other. We now demonstrate a similar result for / J ^ in a different way. Let I^a be I^c with functors and possibly with constant individuals (functors without argument places). We translate I^Cf into a theory y without functors or constant individuals in such a way that I^cf *ne~ orems translate into ^theorems and ^"-theorems which are translations of /J^-statements are translations of /J^-theorems. If we had started

124

Ch. 3 Predicate calculi

with a theory based on I^ct then a similar translation is obtained. The translation is as follows. Replace lag Iga Igrj Iafi ^{a}

by lag, where a is a constant and g is a variable, by lag, where a is a constant and g is a variable, by Igyj, where g and rj are variables, by (Eg) Klaglfig, where a and /? are constants, by (Eg) K
{fa',..., a«} by (Eg) K${g} (EV',..., ^>) i L P y . . . rf* g n /<*> 0-1

where a', ...,a(/c) are constants, but if any of the a's are variables then omit the corresponding Ia^Tj^ and the corresponding quantifier, use different predicates F for different functors / . by Ncj), where , Dcfii/r by D^ty, where <j) and ijr are the transforms of (j> and ft respectively, (Eg) (j){g) by (jEg) ~${g}, where ^{g} is the transform of Repeat until all the functors have been eliminated and all occurrences of constants a occur as lag, lastly, replace lag by a{g} using different oneplace predicates a for different constants a. We shall require some new axioms in the translated system, namely CKFV'...VMgFV'...VMg'Igg>^ (EV)a{V}, CKa{V}a{V'}IVV',

*

(1)

for all F and a that we have introduced. If we had started with a theory 3" based on l^Ci then we should also require the translations of the «^"axioms as axioms in the translated system, similarly for ^~-rules. 19. An !Ffstatement is a ^-theorem if and only if its translation is a ^-theorem. Let

" be the conjunction of the following three /^-statements: {A%",..., gW) (A^') (EC, C) W,

..., ZCn'%%'",

(AC, C, £"') KOK7r%'CTr'CC'K%"'OKn"CC7r"CC'ICC',

(12) (13)

3.21 The reduction problem

141

where £', £" are new. Let the predicate variables which occur in $' be nm9 ...,7T^K+2\ then these are binary. We show that if is satisfiable then so is (j)". Let p"f,..., p(K+® be a set of logical functions over Jf which satisfy £2] = * if and only if £ is the place number of the ordered pair <£1? £2) m o u r ordering of all ordered pairs of natural numbers, then p',p", ...,p^K+2) clearly satisfy <j>". We now show that if " is satisfiable then so is (p. If p',p", ...,#(*+2) is a set of logical functions which satisfy <j>" over ^V, then p'"9 ...,^(/c+2) is a set of logical functions over JV which satisfy >>'"] and ^'{77",7rV",...,*/A>,y,...,yW} take the value ^ for some natural numbers 7', ...,y ( ^, where 0' is obtained from (j> by replacing 77^> by p(°\ From (13) we have TT" = J/, 77/;/ = v" where takes the value t. Thus all together (p is satisfied if and only if $" is satisfied. The prenex normal form of (j)" contains (A— 1) initially placed universal quantifiers followed by existential quantifiers as long as A — 1 ^ 3. Proceed until we get exactly three initially placed universal quantifiers. Call the result
F

and (Ag)m, (A£,V)(CU7)Cn£n7i, (A£,V,QCI&iCnKmi£ and (A£tV,QCI&iChrgn&i,

for all singularity and binary predicates which occur in $'" including the equality relation itself. The prenex normal form of
25.f^4 binary ^-statement

{A?, £",£'")(W, ...,*«)#*', -Mx\ g', ...,y^} in Skolem 8-normal form is satisfiable if and only if a binary <^c-statement (A £', £", £") (Ey',..., rf») ir{n; £', £", £", y'9..., ^ \ also in Skolem 8-normal t Prop. 25 follows closely the proof given by Kalmar and Suranyi (1947), though the symbolism has been changed to that of the present author. It is reproduced by permission of the publisher, the Association for Symbolic Logic.

142

Ch. 3 Predicate calculi

form and containing only a single binary predicate, is satisfiable. Also there is an effective method for finding xjr given (j).

Let

(A?, £", nW, ...,^ ) )^K....,»«; £', i", Z",y', ..

be the given binary ^ - s t a t e m e n t in Skolem ^-normal form, where } = t,

(15)

where (j) is what ^ becomes when n', ...,TT^ are replaced by p', ...,p (A) respectively and the propositional connectives are replaced by their respective truth-functions. of c/T and ordered pairs (v^r), j / ^ ) of Jo so that for 1 ^ &, d" , . . . , <^+3>, ^+3>>} = ^,

(17)

where T denotes the logical function which arises from
which consists of the natural numbers J¥', triads of the forms (y{6'\ v«n9 0> and <^'>, v^, 1> and2/, ...,^ A ) . The triads (v^, v^"\ 0> will play the role of the pairs (v^'\ it0"*} while the triads (y^\ v^e"\ 1> serve merely to express the coincidence of the first and second components of pairs. Thusfor

T

= <„>',()>

3.21 The reduction problem

143

we shall define Qrcr = t if and only if v' = v", QCTT = t if and only if K' = AC". v is the first component of r is expressed by QVT, K is the second component of r is expressed by QTK. p' is characterized as the only element of J for which Qp'p' = t, p" is characterized as the only element of J for which NQp"p' = t, pW is characterized as the only element of J for which Qp^pW-V = t, 3 ^ 6 < A. The triads r = (y\ K', 0) are characterized by QrpW = t, we distinguish between the natural numbers and the triads (y", K", 1} = cr by Q^V = * and Qp'cr = / , then the triads or can be characterized by' o* is different from p', ...,#(A) and o-is different from a triad (y'9 K',0) and Q^'cr = / ' . Now if (15) holds then, for numerical functions />(iv), ...,/o(/*+3),

Let

^" denote the set of triads (J/, /c', 0>, ^ denote the set of triads (vf, K',1), & denote the set of {p\ ..., #(A)}, ,/K* denote the set of natural numbers,

We define a binary logical function Q over f by the following table, in this Q has the same truth-value at an entry in the table as the statement standing at that entry.

Qxx'

x' = K xf = X' = {K\ K", 1 >

x' = pW 1 ^ 6f < A [K(d =t= 2)(6>r = 1)

x = pi®

x =x

X*X

D

\ [K(0 = d'+l)(d'* 1 <6> D(0' = 1) (6>' = A)

1) x 3F x v = K' v" = K x 4= a

v — K' */ = /c'

144

Ch. 3 Predicate calculi

Consulting the table we see that (a2) (a6) (6) (c)

Qxx = t if and only if (iff) x = p\ NQxp' = t iff x = p", QxpW-» = t iff x = pW (3 < 6 < A), QxpM = t iff xe^, KKNQxxNQxpWQp'x = t iff

(d) KQxp'Y[NQxpWNQp'x

=t

iff

0= 2

Further supposing that r = (yr, v", 0), a = then (e) (/) (g) (h) (i)

Qvr = t QW = ^ QTV = t Qcrv = t QTO- = t

iff iff iff iff iff

(j)

Q(TT = t

iff

v = v\ V = K\

v" = v, /c" = p, / = K', ACr/ = ^ .

Also s u p p o s i n g t h a t rx = , r 2 = (v'2, v%, 0> (fe)

if

(Z) for

KKKQT1orQT2orQ(rT1Qo'r2

1 ^ 6 < A,

/[^;2/',2/(A)]

Write

y[>;*/',...,*/]

for for

= tf t h e n

rx = r2,

KKNQxxNQxy^Qy'x' KQxy' \{

KNQxy^NQy'x.

e=2

Then for arbitrary elements a/, x", x'" of ^ / and if y' = p',...,y(x) we have (a)

by

KKKQy'y'NQy"y'

U

(ae),

= p{X)

1 < 6> < A,

Qy{e)yV-1)CDDKQx'x'Qx"x"KNQx'y'NQx"y'

6= 3

2 KQxfy^-^Qxffy^-^KBQxfxf"Qx"xmBQx'"xfQx'"xfr.

(18)

This says that if #' = a" = #<*>, 1 < 6 ^ A, then BQx'x'"Qx"x"' and BQxmx'Qx'"x\ and that ^ = ^ , 1 < 6 ^ A. (6) By(c),(d),(/)and(A), CKI[x';y'9

y™] I[x"; y\ y™] (Eu) KK7[u;

y',...,

y™] Qx'uQux".

(19)

3.21 The reduction problem

145

This says that if x', X"EJV then there is a triad (xr, x", 1 >. (c) By (c), (b) (d), (e), (/), (»), (fir), (A) and (j)

CKCKQx'x"Qx'x'"Qx"x'"CKQx"x'Qx'"x'Qx'"x". (20) This says that iix'ejV, x"e$~, x'" e <%, y' = p',..., y™ = p™ then if x' is the first component of x" and also is the first component of x'" then x" and x'" have the samefirstcomponents, and similarly for the second components. (d) By (b), (d) and (k), CK... KQx'^Qx"^y{x'";y',

...,yw]

6 times

Qx'x'"Qx"xr"Qx"lx'Qx"lx" n BQy^x'Qy^x".

(21)

0=1

This says that if x\ xfre^ and o^^e^/" and if x', x" have the same first and the same second components then Bp^x^p^xlxl, 1 < 6 ^ A, where Finally by (c), (6), (e), (g), (I) and the definition of T the fact that for x',x\xmejr and x* = p^x'xW, ...,^+ 3 ) = j W / / (15) holds, 3 C

0 = 4:

n

7 [

^)

y 9

^

( i 7 % i s B m myU

)

0 1

fi+3

n 0,0=1

(22)

This says that for any #', a;", X'^JV there are a;iv,..., x(^+%/T such that if ^ ^ r = ,xP\ 0> then (15) holds. Now form the conjunction of (18),..., (22) inclusive and place (Ax', x", xr") (Ey'9..., i/(A)) in front and replace Q by the predicate variable q, we obtain an .^-statement of the form (14), viz.: x',x",x"') (Ey'9 ...,yW) (En) ',..•,y(A),u,x'9...9x^^;uhl,...,^+3^J.

(23)

We have just shown that if (14) can be satisfied then so can (23), namely by the logical function Q given by the table. Now suppose that (23) can be satisfied over JV*, we wish to show that (14) can also be satisfied over JVI

146

Ch. 3 Predicate calculi

By hypothesis we can find a binary logical function over JV*, say Q, and ternary numerical functions over JV

such that on replacing q by Q, y(0) u

tftox'x"^,

by

by a)xrx"x'\ "x'"

(4

in (23) and #', x'\ x'" by ^', ^ , v1" respectively we obtain the value t. Consequently we have CKI[x';y',x'x"x'";y',...,y™] QX'O)X'X"X'"Q
(20') and (21') are obtained by writing p^x', x"x'" for xf-e\ 1 ^ d < 3 p+3

3

/t+3

e=i

0=1

e,6'=i

K n I[p^x'x"x'";y', t/W]0 n I&»;y',ym]K

n

QpWx'x"x'"Te, e, x'x"x'"QTe> g. x'x"x'"pWx'x"x'"W{Q ru^"

KKQT6 e,x'x"x"Y»

;y',...,

y™, (22")

V3,,«^1

for any x',x",x'"ejV and for y«» = ^x'x"x'", 1 < 6> < A. Choose a fixed member of Jr, say 1. Write a(B) for ^ 1 1 1 . Then by (18) for any x',x",x'"eJr,

Qx'x'x"x'"x'x'x"x'",

(23')

NQx"x'xV'x'x'x"x"',

(23")

'/f-iVa;"/

(23^)

(3 ^ 6 < A)

CKQv'v'Qv"v"KBQv'v"lQv"v'"BQv'"v'Qv"'v",

and

CKNQv'x'v'v"v"lNQv"x'v'v"v"'KBQv'v'"Qv"v"lBQv"'v'Qvl"v", e l)

l

6 1)

l

GKQvY - v'v"v" Qv'Y - v'v"v'"KBQv'v'"Qv"v" BQv'"v'Qv'"v".

(24') (24")

3.21 The reduction problem

147

From (23<*>), 1 ^ 6 ^ A, we obtain for v' = v" = v"' = 1, Qa'a',

(25')

NQa"a\

(25")

QaWaP-*,

3 < d < A.

(25<«)

w

Now we show that for any v\ v", v eJV* and 1 < 6 ^ A,

<9 = 1 in (24') put

^WV"

for

i/,

a'

for

/',

^

for

^,

detach (23') and (25'), and we get (26J) and (262). For 6> = 2 in (26^) put X"v'v"vw for i; and use (23") we get %WW, In (24") put

/WV"

(27") for

v\

a"

for

J/',

j;

for

^w,

and replace I W V " by a' in virtue of (26^), then detach (27") and we are left with (26J) and (262). Supposing that we have shown (262(^~1)) for some 3 ^ 6 ^ A then in (262(^-1)) put F W V " for v use the resulting equivalence in (23(^) and we get

in (24<*>) p u t

a«» v

for

v\

for

v\

for

v"\

replace ^-^v'vnvm by a^"1* in virtue of (262^(-1)) then detach (25<*>) and (27^) and we are left with (26f >) and (26^). Consequently we may replace y^v'v"^" by a(^ for 1 ^ 6 ^ A whenever it occurs as an argument of Q. In particular (18'), (20'), (21'), (22'), (23") hold for any v', v\ v^eJf and y^ = dd\ 1 ^ 6 < A.

148

Ch. 3 Predicate calculi

Let JV' denote the set of elements of N for which I\y\ a', a(A)] holds. In virtue of (22'), Jf' has a member. Choose a fixed element b o{J^\ say b = p l v l l l (see (22')). We define predicates^?', ...,^ (A) overJ^' by pWVK = Qddh1>2VKb

(1 ^ 6 ^ A, v9 KeJT').

We now show that these satisfy (14) over^T'. Let y,/ce^'and^ = r1}2VKb. By (22") we have ^(A) ^ = ft)> (28)

LEMMA.

Qvp

(v = v',d=l),

(29)

Q^

(/c = v",0' = 2).

(30)

For any element qe^V' for which Qqa,

(28')

Qvq,

(29') (30')

, we have BQa^qp^vKfor v, mjV', 1 s? 6 < A, i.e. Indeed, for r = U>VKI we have by (19') y[r;a',... ,a«],

BQa^qQa^p. (28")

Q^r,

(29")

Qr/c.

(30")

In (20') put v for )/, ^ for v", r for i/" detach l[v; a', oW], (28) (28"), (29), (29") and we obtain ^ (31) In (20) put v for v', q for v", r for v'" detach i[v; a', a ], (28'), (28"), (29'), (29") and we obtain ^ ,„,,, In (20) put K for v', p for v", r for v'" detach /[i^; a', a ], (28), (28"), (30), (30") and we obtain ^ m"^ In (20) put K for v', q for v", r for v'" detach /[/c; a', a ], (28)', (28"), (30'), (28"), (30'), (30") and we obtain Qrq.

(31'")

3.21 The reduction problem

149

Finally in (21') put p for x\ q for x\ r for x"' detach (28), (28'), (28"), (31), (30'), (31"), (31"') and we obtain which is the lemma. Now let v1', v", v1" be elements of J^\ and let

(this holds also for 1 ^ 6 ^ 3 by definition). By (28') v*, ..., i^+3) also belong to JV'. Let te e, = Td^v'v"v"\ 1 ^ 0, 6' ^ ji + 3. By (28") we obtain,

i

'

^

(32)

>,

(33)

\

(34)

e

{

A,

...,1^^}.

(35)

The lemma now gives from (32), (33), (34) BQa<%,t r: p<W>">,

1 ^ 6 < A, 1 ^ 6', 6" < /i + 3.

Thus by the definition of T we infer from (35)

i.e. the binary logical functions p',..., p^ over ^T' satisfy (14). This completes the demonstration of the proposition. 3.22

Method of semantic tableaux

We shall show in Ch. 7 that ^Q is undecidable, that is to say that any proposed method of deciding whether an J^-statement is an fFctheorem or otherwise will fail to give a result in some cases. We now give a method for deciding whether an J^-statement is generally valid or whether its negation can be satisfied. Of course the method will fail to give a result in some cases. The method consists in trying to find a counter example by the use of semantic tableaux. We try to make a given J^statement
150

Ch. 3 Predicate calculi

Consider an J^-statement (j>. We form two columns and call one an f-column and the other one a t-column, each column will be called opposite to the other and they will be said to correspond. We place take the value/, by the alternatives given by these columns, we have to give both t and / to x> since this is impossible we discard this alternative. If every column is closed then our attempt to make take the value/has failed and so is generally valid over any domain. The process terminates, except for substitutions for variables, when we arrive at atomic formulae. Thus a semantic tableau will (i) terminate by all columns being closed in which case
3.22 Method of semantic tableaux

151

f. The method always comes to a definite conclusion, if the first or the last case holds, but fails if the second case arises. If all the atomic formulae are singulary we can obtain a counter example, if that case arises, over a bounded domain. But if there are binary predicates then we have the general case (see Prop. 25), and unbounded domains may be required and the functions introduced in dealing with a general statement in an /-column may introduce such complications that we are unable to tell whether certain atomic formulae occur in both of two opposite corresponding columns. Consider GKN(Ex) Kpxp'x(Ax) Cp"xpxN(Ex) Kp'x
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

KN(Ex) Kpxp'x(Ax) Gp"xpx N(Ex)Kpxp'x (Ax) Gp"xpx (Ex)Kp'xp"x Cp"apa we have substituted for a variable

Kp'ap"af p'a p"a Np"a 1

:

(x) pa

2

:

(0) (xi) N(Ex) Kprxp"x (xii) (Ex) Kpxp'x (xiii) Kpap'a we have substituted for a

variable

(xiv) (xvi)

pa

1 : (xv) p'a

2

p"a

Below entry (xiii) the/-column splits into two columns indicating the two ways in which (xiii) can be made/. The second of these two columns can be closed because (xv) agrees with (vii). Now there is only one/-column and the f-column splits indicating the two ways in which (v) can be made t. This gives us the pair of opposite columns labelled 1, 1 and the pair of opposite columns labelled 2, 1. Both these columns can be closed because the last member of t-2,1 (x) agrees with (xiv) and the last member of/-1,1 (xvi) agrees with (viii). Thus our attempt to make (0)/has failed. Hence (0) is generally valid and so is an ^^-theorem. From the tableau we can find an ^,-proof of (0), as follows: We start with the hypotheses: (ii), (iii), (iv) and (vi)x from these we deduce pa as shown in the tableau but we had already produced p 'a so that we get Kpap'a and hence (xii), thus we have: N(xii), (iii), (iv),

152

Ch. 3 Predicate calculi

where (vi)^ is (vi) with x instead of a. By the Deduction Theorem we have (iii),(iv)

\-C(vi)xCN(xii)(xii),

(iii), (iv) f-C(iv) (xii), Prop. 6 (iii) (iii) (iii) \-CN(xii)N(iv), hC(iii) CiV(xii) N(iv), which is equivalent to (0). Now consider CK(Ex) KpxNp'x(Ex) Kp'xNp"x{Ex) KpxNp"x

(0)

Our tableau is: t (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (xv)

(Ex)KpxNp'x (Ex) Kp'xNp"x KpaNp'a pa Np'a Kp'bNp"b p'b Np"b p"a

f (ix) (x) (xi) (xii)

(0) (Ex)KpxNp"x KpxNp"x KpaNp"a pa 1 : (xiii) Np"a 2 : (xiv) KpbNp"b

(xvi) pb 21 : (xvii) Np"b 22 (xviii) p'a (xix) p"b

In this case the tableau terminates without the tableau being closed, and we are able to read off a counter example, namely the domain consists of two elements a and b, pa, p"a and p'b are t while pb, p'a and p"b are/. We take pa to be t because as far as making the entry (ix)/we need only take (xiii)/independently of the value of (xii), but to make (i) and (ii) t we must have (xii), i.e. (iv) t. In the ^-column (i) gives rise to (iii), (iv) and (v) and (ii) gives rise to (vi), (vii) and (viii). In the/-column (xii) and (xiii) are the two alternative ways in which (xi) can be made/ of these (xii) closes with (iv). We still have (xiv) and this gives rise to two columns 21 and 22, of these 22 closes with (viii). This column consists of (ix), (x), (xiii), (xiv), (xvii).

3.22 Method of semantic tableaux

153

But column 1 consisting of (ix), (x), (xiii), (xiv), (xvi) can be continued with (xviii) and (xix) from (v) and (viii) respectively, and this column is still open, we could go on with KpcNp"c but it is pointless to do so. The final ^-column consists of (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (xv) and thefinal/-column consists of (0), (ix), (x), (xi), (xiii), (xiv), (xvi), (xviii), (xix). We read off the counter example by the values given for the predicates p,p',p" at a, 6, in these two final columns. Now consider: C(Ax)(Ex')KDpxx'p'xx'Dpx'xp'xx(Ex) {Ax^BKpxxp'x'x'Kpxx'p'xx' (0) t Dpxap'xa Dpaxp'xx

(0)

1 pxb 11 pxx paa pab pbb

f Kpxxp'bb Kpxbp'xb 2p'xb 12 p'bb 21 pxx pab pbb

paa pbb

22 p'bb p'ab

p'ab p'bb

This gives 4/-columns each of which correspond to the single ^-column at present consisting of only two entries. Now the ^-column similarly splits up into 4 columns, thus 1 pxa LI pax paa pba pab

I2p'xx p'aa

2 p xa 21 pax paa

paa pba

pab

p'bb

p'aa

p'ba

22 p'xx p'bb p'ba p'aa

Each of these four ^-columns corresponds to each of the four/-columns. To make (0) take the value / we require that a pair of corresponding columns be open. We have entered in the columns all the results of substitution over a two-element domain. Thus we require to test 16 cases, viz. any ^-column with any/-column, we tabulate the results:

154

Ch. 3 Predicate calculi /-ll /-12 /-21 /-22 /-ll /-12 /-21 /-22 /-ll /-12 /-21 /-22 /-ll /-12 /-21 /-22

t-ll t-ll t-ll t-ll t-12 t-12 t-12 t-12 t-2l t-2l t-2l t-2l t-22 t-22 t-22 t-22

closed by closed by closed by open closed by closed by closed by closed by closed by closed by closed by open open closed by closed by closed by

paa pab paa paa p'bb paa p'bb paa pab paa p'bb pfbb p'bb p'bb

Thus we have found three open pairs Each of these gives a way of making (0)/over a two-element domain. Namely /-22 and *-ll t paa pab pba

/ p'bb p'ab

S-22 and t-2l

/-ll t

p'bb p'ab

p'bb p'ab p'ba

t paa

p'ba

/

pab

and *-22 / paa pbb pab

p'aa

the values omitted can be chosen arbitrarily.

3.23 An application of the method of semantic tableaux As an application of the use of semantic tableaux we demonstrate the following proposition: 26. Let Cifiijr be a closed ^-theorem where neither N and i/r are in prenex normal form with matrixes ^ 0 and i/r0 respectively, where 0 is in conjunctive normal form while XJTQ is in disjunctive normal form and that PROP.

3.23 An application of the method of semantic tableaux

155

both are constructed from atomic statements as described in the K

n 0=1'

enunciation of the proposition. Suppose also that (f>0 is of the form

while i/rQ is of the form 2 ^o6) a n d that $f} is of the form 2 $0% while 0 = 1

7T = 1

$ f * is of the form n ^ofl where 0O^ and i/rff], are atomic statements or 77=1

negations of atomic statements. Now form the semantic tableau for Ci/r; it will be closed. In forming the tableau we first treat the quantifiers introducing variables and functions and constants as required by the rules of forming a semantic tableau. When the quantifiers have been treated we have entries 0* m ^ n e *-column and ty* in the /-column, where these differ from ^)f0, ...,^0K) and fr'o,. >.,froX)respectively by the substitutions we made above. Thereafter the single columns t and / both split; we will put it down as follows: the /-column splits into 1 ( ^ columns corresponding to the 1 ( ^ ways in which ^Q* can be made/, then each of these columns splits up into 2(^ columns indicating the 2 ( ^ ways in which ^{j* can be made / , and so on until we have the Ath split into A(^ columns indicating the A(^ ways in which ^0A)* can be made/. Altogether we have ]>> x 2(^ x ... x A(^ columns on the /-side. These columns contain one conjunctand from each of the A rows ifr'o*,..., ^0A)*> so each column indicates a way in which ijr* can be made/by making one conjunctand in each of the conjunctions ^ 0 *,..., ^oA)*/- Each of these columns is to have a corresponding opposite column on the tf-side; we further transfer an atomic statement which is negated in the /-side into the corresponding column on the £-side. Now we go over to the f-side. Each of the columns on the £-side will split up into 1^ columns indicating the various ways in which 0O* can be made t. Then each of these columns splits up into 2("> columns indicating the various ways in which ^o* can be made t, and so on until the /cth split into K(V) columns indicating the various ways in which
156

Ch. 3 Predicate calculi

Thus each side of the final tableau has l ( ^x ... x A<^x 1<">X ... x i6v) columns which correspond in pairs, we continue these columns with substitutions on the free variables, then if we make all the atoms in one / - column/and all the atoms in the corresponding ^-column t then we shall make C, or case (iv) the common atom 8 can arise in both sides from tjf. With each pair of corresponding columns we associate the ^-statement S, if 8 is the common atom and we have case (i), N8 if we have case (ii),/ if case (iii), and lastly t if we have case (iv). Now suppose that we have associated an ^-statement with each subtableau up to a certain point and that a set of these subtableaux join (on going up) or split (on going down), then to the subtableau formed from their union we associate the disjunction of the statements associated with each of them if the split is due to treatment of 0, but their conjunction if the split is due to treatment of ijf. Thisgivesusan J^-statement x* > composed of y's, / and t, by N, D and K, associated with the whole tableau which we can effectively find when the closed tableau is given. Now the structure of both sides of the tableau is the same, because each column has its corresponding column and when one splits so does the opposite one. Now in the f-side omit every entry except atoms which close columns and those which arise from i/r, and replace these atoms by 8, N8, f or t according as which of the four cases occurred. We are left with a tableau for Cx* f*j because if a pair of corresponding columns closed by case (i) then the situation is exactly the same as before, and similarly in case (ii), in case (iii) we have/, but in order to get a counter example we ought to have t, and in the last case we have t, but to get a counter example we ought to have/. Now the entry/ can be replaced by Ky'Ny'', this in a ^-column becomes y' with Ny' below it; this gives y' in that column and in the corresponding column, hence the column is closed as before.

3.23 An application of the method of semantic tableaux

157

Similarly the entry t can be replaced by Dy'Ny', this in an/-column becomes y' with Ny' below it; this gives y' in that column and in the corresponding column, hence the column is closed as before. Thus columns which were originally closed by a's or /?'s are now closed entirely by y's. We have the closed tableau for Cx* ^o which is an ^.-statement free of quantifiers. We call Xo the sentential power of (p. Similarly if we interchange the treatment of the two sides we get the tableau for C(j)*x*X* is the same in both cases, the structure is the same in both cases, if one divides so does the other and for the same reason. We want to put the quantifiers back so as to get the quantificational power of X'

Note that if x* i s / then N<j) is an ^-theorem because the tableau for Ncp is closed on its own, and if Xo is t then \jr is an J^-theorem because then the tableau for i/r is closed on its own. Note t h a t / and t can be omitted from x because in the formation of x we can replace KSf and KfS by / , DSt and DtS by t, DSf and DfS by 8, lastly KSt and KtS by S. S o / and t will disappear unless the final result is / or t, but we have discarded this case. Note also that only those y's are used in x which occur positively in both (j) and ^r or occur negatively in both (j> and i/r. Where an atom is said to occur positively in co if it is on the same side of the tableau as a), otherwise it is said to occur negatively in o). Thus only some of the y's are used in x* We restrict a variable which replaces a term which originated from treatment of (f> because terms which at their first occurrence arise from 0 do so by treatment of existential quantifiers. We generalize a variable which originated from treatment of T/T because terms which at their first occurrence arise from i/r do so by treatment of universal quantifiers. Let us return to case (ii) where we were unable to decide whether the tableau closed or was open. A tableau might show that certain compositions of functions were the same, i.e. that f...gx = l...mx for all x, we

158

Ch. 3 Predicate calculi

only need on the t side an .^-statement which has this as an interpretation. By substitutions for variables we can arrive at great complications as regards composition of functions. A composition of singulary functions can be written as a word, i.e. a string of letters or primed letters from a given set called the alphabet, then if we have/. ..gx = Z.. .mx for all x a consecutive part/...
CKKDNy'yDay'DKyNay'DDKPyKyy'KNyp. The tableau is given on facing page. There are 64 pairs of corresponding columns, viz. v, v' where v, v' — 1,... ,8. In the next table we show that the tableau is closed by giving the value of the associated ^-statement.

In many cases corresponding columns can be closed in several different ways. Thus 55' can be closed in three different ways, viz. by oc and NOL being in column 5 which gives/, by y and Ny being in column 5' which gives us t, and lastly by y being in both columns. We have now associated an ^-statement with each column. As we go up the tableau these columns unite by treatment of the *-side. Accord-

3.23 An application of the method of semantic tableaux

159

KKDNy'yDay'DKyNay' DNy'y Day' DKyNay' Ny' a

7

a

KyNa

/

7

KyNa

7'

7 Na

Not 1

2

7

KyNa

r'

4

7'

7 Na

7 Na

3

KyNa

5

8

6

7

/?

Ny

n

V

8'

DDKfiyKyy'KNyfi Kfiy Kyy' KNyfi

7 Ny

r

7

y i

/?

12'

Ny

Ny 4'

5'

Y

ing to our rules for finding x we take the disjunction of the ^-statements associated with lines: 8. lvf and 2v\ 3 / and 4J/, 5v' and 6 / , 7^' and 8J/, The results we show in the following table: ll'/,',7 21/, * 31^,y 41'*, y

12'/,y 22/ 32'y 42'y

13'/,/ 23/, y' 33y 43y

14'/,yf 24/, y' 34'y' 44y

I5y,tfy 25/, t 35/,t,y 45't,y

16'/,y 26/ 36'y 46'y

lVf,t,y,y',Dyy' 18/,y,/ 2f7J,t,y/ 28/, y' 3Vf,t,y,y',Dyy' 3&'y,y'9Dyy' 47% y, y', Dyy' 48'y, y', Z>yy'

Here ^ v' indicates the disjunction of (2v— 1) v' and (2p) p'. In some cases

there are several different ways in which the disjunction can be taken. Thus 47' can give either y, y', Dyy' or t according as to which entry in 77' and 87' we use. The next stage is the disjunction of the pairs (2v— 1) v' and (2y) v'. The result is put down in the next table: 12/,yl3/,y' 14/,y' 15/,t,y 16/,y 17/,t,y,y',Dyy' 26'y 2Tt,y,y\Dyy' 28'y,y',Dyy' 22'y 23'y' 24'y' 25'r,y

160

Ch. 3 Predicate calculi

Again we take the disjunction of the two members in each column. l't,y

2'y 3'y',

4'y' 5%y9 6'y Tt,y,y',Dyy'

%'y,y

We now form the conjunctions of consecutive pairs corresponding to the junction of columns on the/-side. This gives

y; y'; KyDyy',y;

KyDyy',Ky'Dyy',Kyy',y,y',Dyy'.

We have to do this a second time, getting

Kyy;

y, KyDyy', KKyy'Byy', Ky'Dyy', Kyy1

and lastly we require the conjunction of these, however we do this we get something equivalent to Kyy'; this then is the sentential power. We can easily verify that

CKyy'D*K/3yKyy'KNyj3 and CKZDNy'yDay'DKyNay'Kyy' are tautologies. 3.24

Resolved ^Q

The resolved !FC or the resolved I^c is obtained as follows: 7} is a new symbol of type t(ot), the axioms and rules are the same as for ?FC or I^c which ever is the case, except that the symbol E is omitted and the rules lid, e are discarded and the following rule replaces them: H where we have written D21 nM&

for

Here f\^(j>{^) plays the part of a thing which has the property ( It is a term of type 1, it can only be used after {^\ can be used. The theorems of H^c are the ^-theorems in resolved form. By this we mean that if
3.24 Resolved ^c

161

tion for these new functions. The system IHFC is without quantifiers but it is sometimes convenient to have a system using both lid, e and rule H, we then have a system like SFO with functors and constant individuals. A related system is the following system based on I^c\ it is together with the symbol i of type t(ot), we write as usual D22

h

m

for

We have the rule

is read 'the thing with the property (X£^{£})'. The formula is of type i and can be used only after the premisses of J have been proved. The resulting system, which again fails to be a formal system, is denoted by J^Q. From J we obtain r,

From the hypotheses of J' we obtain by !FC (Ag, n) CKK<j>{g)

whence from J and (Eg) K
whence
162

Ch. 3 Predicate calculi

this is called an e-term. £ is called the bound variable of the e-term }. The symbol E and the rules lid, e are omitted and replaced by E where co is subsidiary and can be omitted, this is called the e-rule or the E-rule, we say that this application of the e-rule belongs to the e-term

§ fails to occur free in o) and in If we allow the rule of substitution then rule E' becomes a case of the rule of substitution and so can be dispensed with. If we write D24 then rules E and E' translate respectively into rules lid, e. If we write D25 then we have a definition for the universal quantifier. Thus in the systems E3FC and EISFQ we have definitions for both quantifiers, if we have the rule of substitution then we need only rule E. These two systems are formal because the e-term e^{£} can be used without restriction. The e-term, €

£0{£}> i s r e a d 'the most ^-like thing'; then
un-0-like thing is §J-like', if this is the case then everything is ^-like. The term e^{£} can be used when {A^)N(j){^ is an J^-theorem. For instance if r is a variable for a rational number then er(r2 = 2) is read' the rational number whose square is most nearly equal to 2' but we have (er(r2 = 2))2 =f= 2, so that e(r2 = 2) might be any rational number (see Ch. 12). It is readily seen that if ^ is an i?J^-theorem without occurrences of any e-term then the jE/«^-proof of
3.24 Resolved &c

163

particular term for which 0 holds. If however {oc} fails for each term a then €^{£} could be any term. This expresses the reversibility of E'. LEMMA. The rule of substitution can be eliminated from First push all substitutions back into the axioms.

Suppose that we have j^\) {

U(p\oCj 0)

in an iJ/J^-proof-tree, where £ fails to

occur free in D0{Tt} o). In this tree replace all corresponding occurrences of £ by a. Axioms remain axioms, applications of rules remain applications of the same rules or repetitions and we are left with an EI^FCproof-tree of D${(X}G). In other words, substitutions may be pushed back to the axioms. Note in particular that rule E is preserved. 27. Rules 116, c, e are reversible in 116 is reversible. We have to show PROP.

Consider the iJ/J^-proof-tree of DNDfrfi'a) and note the places at which a related occurrence oiNDficj)" enters this tree. This will occur at IIa, 6 only. If a related occurrence NDtfi^cfil enters the tree at I I a then replace it by Nfa and if at II6 then strike out the upper right formula and the branch above it and replace ND^>[ (j>[ byN[^>l was part of all of the main formula. Axioms remain axioms. IfND^fil occurs in the main formula of rule E, for instance if we have

If in this we strike out related occurrences o£D—<j>" then it becomes

164

Ch. 3 Predicate calculi

this fails to be a case of any rule. We shall have to call the occurrence of ND—(j>" in the c-term related to the occurrence of ND—$" in the upper formula. By this device we preserve the proof. This completes this case. l i e is reversible. We have to show —^ . *. Dj In the U/J^-proof-tree oiDNN^o) consider the corresponding or related occurrences of NN<j>. These will be introduced at II a, c only, omit NN at each of these occurrences throughout the proof-tree and we are left with a tree with Drfxo at its base. Applications of rules remain applications of the same rules or repetitions except that rule E can become upset if a related occurrence of NN occurs in the main formula of that rule axioms remain axioms. If NNcj) occurs in the main formula of rule E, use the device as before. We are left with an EI&o-proof-tree of He is reversible. We have to show

DN{g}a)

±

Again in the U/J^-proof-tree oiDN(Eg) ${g\ o) consider the corresponding or related occurrences of N(Eg) <£{£} and in these omit (Eg) or the part related to it. These occurrences can only be introduced into the tree at II a, e, when we omit (Eg) applications of rules remain applications of the same rules except that II a, e can become repetitions, axioms remain axioms, but rule E can be upset if (Eg) occurs in the main formula of that rule. We use the same device as before, and the result follows. 28. Modus Ponens is a derived rule o We have to show

PROP.

D(j)(j) DN<j>x

*' Prom the 2?/J^-proofs of the upper formulae we require to find proof of the lower formula. We proceed as in Prop. 4, by formula induction on the cut formula. There are only three cases, namely when the cut formula is atomic, a disjunction or a negation. The last two cases are dealt with exactly as before. The first case is also dealt with as before but the theorem induction

3.24 Resolved ^c

165

requires the additional consideration of rule E. Thus suppose that in our theorem induction we have a case of rule E:

by the reversibility of l i e we obtain D{^}(0, thus we require:

Here {e^N${£}} is 7ra'*...a(/c)s|:, where a(A)* is the result of replacing all free occurrences of £ in a(A) by a, where a is egNnot'... a(A). Thus 0{£} contains fewer occurrences of e than
i

I{U)

'

where a contains e-terms but J3 is without e-terms. If o) and 0{I\} are without c-terms then Dcj){^}(i) is without e-terms. Consider the highest such case of rule I(ii), follow the related occurrences oflocfi up the tree until we come to the highest case of/a'/?' where a! contains e-terms but /?' is without them. Then in the tree above this each IyS must be such that y, S both contain e or neither do. Equations of the type of IyS just mentioned are incapable of producing equations of the type of Iccfl above. Hence the equation IyS must arise from an axiom T.N.D., but this introduces NlyS as well. The descend-

166

Ch. 3 Predicate calculi

ants of NlyS will be in the subsidiary formula of the application of I (ii) which eliminated the e-term This leaves an e-term in the lower formula of that application of I(ii) which will have to be eliminated lower down the tree. This, in turn, in the same manner, will introduce another eterm, which in turn will have to be eliminated still further down the tree and so on without end. This is absurd. Hence the deduction must be without e-terms altogether. Prop. 29 enables us to show the consistency with respect to negation of theories based on EI^C whose axioms are without the e-symbol and which are verifiable. That is to say: a closed statement of the theory without €-terms can be decided using a suitable definition of truth for the theory. If the axioms ^',..., <j>^ of the theory are valid then so is every theorem (j) of the theory which lacks e-terms because a proof of G with a single binary predicate p. In SS^C we define equality

(£ = ?) for

K(AQBpttp£V(AQBp&p£V.

= is a symbol of type on, we use the more familiar way of writing equalities. More generally equality can always be defined in a theory 2T

3.25 The systems 3&!FC

167

which contains a terminating sequence of predicates, but is without functions. We define (£ = v) for the conjunction of

taken over all the predicates contained in the theory ^ . It is more usual to define D26

(£ = ?,) for

(AQBpZZp&i,

(g * V)

for

and take C(AQ Bp^pv^E, = v) as an axiom, or to define D26'

(£ = ?) for

{AQBp&pr,^

(£ * V)

for

N(£ = V),

and take O(AQBp^p^v (£ = v) as an axiom. In either case if (£ = TJ) we have {AQ Bp^ptyj and (AQBpg^pw^ so that by 3FC pat; may be replaced by par/ and#£a by pyoc in any ^ - s t a t e ment, in other words we have -a) (a (a-a)

aand nd

*>(

Dcj>{fi}oj

by regularity <j) being built up from the sole predicate p, this is the same axiom and rule as for the equality symbol / . D27

(asyff)

for

(A£)Cp£ap£fi9

(ac/?)

for

K(oc c /?) (a # fi).

Read (a c y?) as 'a is contained in /?'. The things which stand in the relation p to oc are less extensive than the things which stand in the relation pto/3. <= is called the inclusion symbol. D28

Smoi for

{A£)Np£a.

'oc is without ^-predecessors', or oc is empty. We can regard the predicate p as being an ordering relation. If pocfi we shall call oc an immediate p-predecessor to /? and /? an immediate psuccessor to oc. We can then identify fi with the class of its immediate ppredecessors, and so identify the relation# with the membership relation. This amounts to reading pocfl as 'oc stands in the relation p to /?' or as c a is a member of the class of things which stand in the relation ptofi'. Now /? is of type i and (Agpg/3) is of type (01). Let A be a symbol of type t(ot) then A(A£p£/?) is of type £ and this term is uniquely fixed by fi. Thus

] 68

Ch. 3 Predicate calculi

if we identity /? with A(Xgp£/?) then we have formalized the above informal exposition. To make this identification we require (/? = A(X|p£/?)) more fully:

BptfpgVkZpgfi)

and J3p/?gp*(Xgp&ff) £

In conformity with previous uses of (X£^{£}) we define D29 If we have a set ^ of things oc, /?,..., and are given a truth-value for each oipococ, pa/3, pfiot, pfifi,..., then for fixed /? we can form the class of things oc for which pafi istf.This gives us another set, say Sf*, whose members are classes of the members of £?, and there is a (1-1)-correspondence between £f and 5^*. Every member of ^ * is a class of members of SP and there is a (1-1)-correspondence between £f and ^ * whereby oc of S? corresponds to the member a* of £?*, where a* is the class of members of £f which stand in the relation^ to oc, call this class (£pz). The relation poc/3 translates into p*oc*fi* or (oc*e/3*), read 'a* is a member of the class /?*'. With this interpretation ^ * is a set of classes whose members are also classes whose members in turn are again classes, and so on, stopping, if at all, only at the null or empty class which is without members. The things a*, /?*, as defined, are subclasses of £f*, and between these subclasses we define the relation p* which we interpret as the membership relation. The members of ^ * are classes of members of «$^*. Classes can be combined by certain operations to yield new classes. Thus two classes can combine to form their union or their intersection. From a single class we form its complement, and so on. Thus we can extend ^ * if necessary so as to contain these and other combinations. We define D30

(#0/?) for

gKpgocplzfl intersection,

D31

(<*U/?) for

^Dpfop^p

union,

D32

oc

^Np^oc

complement.

for

We shall require as axioms or theorems

Bpg(a<\fi)Kp&p£P, Bp£(a{jp)Dpfrpgp,

Bp&Npga.

This suggests that we have in general but this leads to an absurdity. Take NpijT} for ${>)}, then the suggested

3.25 The systems 0^^c

169

axiom becomes: Bpr/^Np^NpT/i], now substitute the term £Np£t; of type i for y and we get: Bp(£Npgi;)(£Npgg)Np(£Npgg)(£Npgli) which is absurd. If instead we take: C(EQpy^Bpy^{^}<j){rj}, and proceed as before we merely arrive at (AQ Np(£Npg£) £, i.e. the term £Np£l; fails to stand in the relation^ to anything. It will be a maximal element in the ordering given by the relation p. D33 @£ for D 34 *g£ for iV(3£,

£ is a proper class.

Thus if we adjoin the symbolA we obtain constant terms of type t of two kinds, sets and proper classes. Let us then change the notation for variables of type t and take them to be: X, X', X",.... Then we define another sort of variable, called set variables, by relativization, thus: D35

(Ex){x) for

D36

${x}

These give

(Ax){x} for

(EX) K(SX{X},

for

XK(SX(f>{X}. (AX)C&X
We shall want some information as to whether a given class is a set or is a proper class. For convenience we use the letters x, y, z, u, v, w with subscripts or superscripts as variables for sets and X, Y, Z, U, V, W with subscripts or superscripts as variables for classes. We define: D 37

{X, Y}

for

uD(u = X)(u=

Y), pair class,

T> 38 SX

for

il(Ev) KpuvpvX,

D 39 PX

for

H(u c X),

power class,

D40

{X}

for

{X,X},

unit class,

D41

(X,Yy

for

{{X}, {X, F}},

ordered pair,

D42

(X, Y,Z) for (X, (Y,Z)), ordered triplet, etc., in vtuplets, (xr, ...,x^v)}, pointed brackets are put back by association to the right,

D43

V

for &(u — u),

universal class,

D 44

A

for H(u 4= u),

null class,

union class,

170

Ch. 3 Predicate calculi

D45 D46

for p(UV)X, U stands in the relation X to F. 2 u)(Eu, v) K (w = (u, v}) puXpv Y, direct product. 2 X (XxX), 3 X (X2xX),etc, / ®x H(Ev)p(v, u) X, domain, %(Eu)p(y, u)X, range, mx u X or or CnvxX u)(Eu, v) K(w = (u, v}) p(v u) X, converse, X BelX (X c F2), relation, X"Y H(Ev) Kpv Yp(u, Vs) X, transform of Y by X, (Au, v, w) CKp(u, v} Xp(w, v) X(u = w), oneUnX valued relation, $ ib(Eu,v)K(w = (u,v})puv, membership relation, / fi)(Eu, v) K(w = (u, v}) (u = v), identity relation, %(Eu, v, w) K\z =
D47 D48 D49 D 50 D50 D51 D52 D53 D54 D 55 D56 D57 D58 D59 D60 D61

D62 D63 D64 D65

(UXV) (XxY)

3.26 Set theory

171

3.26 Set theory The system £8!FC consists of the binary !FC with exactly one predicate p and the class-forming symbolA of type t{ot), so far we have only given definitions, if we wish to use the system as a set theory we shall have to add some axioms and rules. For instance we want some information as to which classes are sets, and conditions when a member of a class satisfies the defining statement of that class, i.e. for which ^ ^ - s t a t e m e n t s <j) do we have: Bp7/£{t;}cj){rj} as a ^J^-theorem? We have already seen that we are debarred from having this without qualification. Some classes must fail to be members of the universal class, the class of all sets. If all classes were sets we should have pVV and the ordering relation p would fail to be irreflexive. So now we add some axioms and rules and obtain a theory £P based on ^G which we call set theory. In future we write (ote/3) instead of pocft. We take as our definition of equality: D66

(X=Y)

for

(Au)B{ueX) (ueY), and (aeyff)

for N(aej3).

This is the extensional approach, common in mathematics. Two classes are called equal if they have exactly the same members. We add the rule: D(AU)K(XeU)(YeU)o) D(X= T)co " As axioms to tell us which classes are sets we take: Ax. 1. @2#, Ax. 2. <S{x,y}, XX.X.. O.

\£)± JU)

Ax. 4.

CUnX(SX"y,

these say that the union class of a set is a set, the pair class of two sets is a set, the power class of a set is a set and lastly the transform of a set by a one-valued relation (which may be a class) is a set. The free set variables in these axioms mean that we may only quantify with set quantifiers. We add the following rules to tell us when a class satisfies the defining statement of that class: D(xey)(o Jrv ^-1 . •——~~

~——~ .

DD(ueX)(ueY)(o D(ueXuY)oj '

DN(xey)o) xv & .

•

DND(ueX)(ueY)(o DN(ueX{jY)
172

Ch. 3 Predicate calculi

DN(ueX) o) D{ueX)a) '

D(ueX) co ' DN(ueX)co'

D(Ev)((v,u)eX)G>

DN(Ev)((v,u)eX)o)

D(ueX)u> KiO . -^rr,

r

T^z

DN(ueX)a>

i^r—.

±4

D((v,u)e(VxX))(oD«u,v)eX)(o

0.

D((v,u)eX)o) D((u,v,w}eCnv2X)o)'

R8".'

D((u,w,v)eX)
'

DN«v,u)e{VxX))o)DN((u,v)eX)(o DN«y,u)eX)a) DN«v,w,u)eX)a) DN((u,v,w}eCnv2X)o)m DN((u,w,v)eX)(o DN({u9v,wyeCnv3X)co'

30. The rules R2', ...,R9" are reversible. Take for instance rule R 4'. We have an ^-proof of D(ueX) o), we wish to find an^-proof of DN(ueX) co. In the y-proof of D(ueX)o) note the places where related occurrences of (ueX) enter the e9^-proof. These will be at R4', I l a or T.N.D., viz. DN(ueX) (ueX), in the first two cases replace (ueX) by N(ueX) and similarly for all related occurrences, in the third case add above T.N.D. DN(ueX)N(ueX). This is an ^-theorem, from R4" with N(ueX) for o). Now replace related occurrences of (ueX) by N(ueX) and we have an ^-proof of DN(ueX). Similarly for R4" and the other rules. PROP.

D67.

U{i}

for

p(Ex',...,xM)K(y=(1c)){x',...,x%

where £ stands for x'9..., a^^ and <j> for (x',..., x^v)}. 31. / / all the bound variables in $ {t)} are set variables and ift) contains the complete list of free variables in
PROP.

and

DN{t)}d)

and conversely.

Prop. 31 says that we can replace the suggested general rule (*) applied to ^-statements without bound class variables by 8 particular cases

3.26 Set theory

173

R2', ...,R9". If all the bound variables in
(xex) by

(Ey){x = y)(xey).

Suppose t) is <#',...,x^} and 0{t)} is x^X)ex^\ We have: R 5', 5" (a) (z#i/> cX ~ (xy) e@X with 2; for v and <#?/> for w, R 9', 9"

<#z2/> cX ~ <#2/z> eCnv3X,

R8', 8" R 5', 5"

~ eCnv2CnvsX. (6)

R 8', 8" R5', 5"

- <#2/> e@ Cnv2CnvsX, (xyz) eX - (zxy) eCnv2X.

(c)

- <#2/> e@Cnv2X.

From these we get: R 5', 5"

(d) <2/z'... BM> eX - <»'... ^ > e^X,

from (a) with ?/ for z, x' for a; and (xr ...x(p)) for 1/, hence by repetition:

(( /<-times

(/) (x'j/Y... *W> eX ~ <«/'a;V... «w> df 2 ?,.Z, from (6) with ?/' for a;, a;' for z and (x" ... x w ) for «/. (/') is equivalent to <x' ...xfir>)e@'gi&sX, (writing ^ 2 for Cnv2 and ^3 for Cnv3), hence by repetition: (g) <»y ... 2 3

(

/t-times

< z V y ... 2/^> 6X ~ {x'x") € ^ ^ 2 X , from (c) with x' fora;, x" for«/ and ($'... t/^) for 2.

> el Lastly

(j) (i7x') {(x'x"... z(">> eX) ~ >

174

Ch. 3 Predicate calculi

(i) (j) is atomic. Thus (j> is either #(A) ex^ or #(A) €x
If A ^ JLL we have If /£ < A we have By (A)
Hence altogether:

(x'... ^ > cX is equivalent to ( ( ( (

2

/i—A — 1- times

=X

Nowwrite

3

2

A —1-times

and

= X.

7 x ("^3 1 ••• (^3^11 x ( F x . . . ( F x « f ) ...))•••)) for X and we obtain: fi—A — 1-times

>

A —1-times

l x( (

3

| (

/t—A —1-times

Since
V x (Fx...(Vx«f)...))•••))• A —1-times

it follows from the definitions of F, x , <^2,

Thus any two ^-statements may be put into equivalent forms with exactly the same free variables. (ii) "{t)}. By induction hypothesis we have

?? as desired, (iii) ^{x} is

By induction hypothesis we have and

3.26 Set theory

175

as desired. (iv) ^{r.} is (Ey) (j)'{y, £}. By induction hypothesis we have:

We have-

Wlfr.S}

yP>

This completes the demonstration of the proposition. 3.27 Ordinals The system £? is very useful for model making. For instance we can define the natural numbers thus: 0 for A,

1 for {0}, 2 for {0, {0}}, 3 for {0, {0}, {0{0}}},...

so that Sv is defined as the set of lesser natural numbers. The first transfinite number co is then defined as the class of all natural numbers. To carry this through we have to^-prove that the natural numbers defined as above are sets, otherwise they are debarred from membership of other sets and the process breaks down. @A is easily ^-proved so are: (&(xn y), &(x u y), ©(# x y), (S&x, (&&x, (&x, etc.

and these suffice to show that the natural numbers as defined above are sets. But to show that 0) is a set so that the process can continue into the transfinite we require another axiom. We could take So itself as an extra axiom, but it is usual to take: Ax. 5.

(Ex) KN^mx(Ay) C(yex) (Ez) K(zex) (y c z).

This ensures the existence of a set containing an unending strictly increasing sequence of sets. It is called the axiom of infinity. An ordinal is defined as the class of lesser ordinals and is well-ordered

176

Ch. 3 Predicate calculi

by the membership relation #. The successor of an ordinal x is then x U {#}, and the limit of a class of ordinals X is 2X. D68 Xi^eY

for K(X* c 7 u Y U I)(AU)CKN
i.e. X is well-ordered by Y; for any two members #, x1 of X we have DD(xYx') (x' Yx) (x = x'), and every non-empty subclass of X has a least member in the ordering Y. The official definition of an ordinal is D69

OrdX

for KX1Te£{X c PX),

i.e. X is well-ordered by the membership relation and members of members of X are members of X. It is easy to show Ord 0, i.e. OrdA. On for ftOrdx. On is the class of all set ordinals. D70

X< Y

for

XeY,

X^Y

for D(X < Y) (X = Y).

We shall use a, 6, c, d, a',..., as variables for set ordinals. We have _ ,Tr// l w , x , aea, NK(aeb) (pea), etc.

Any member of an ordinal is an ordinal. In fact an ordinal is the class of all lesser ordinals. The tricotomy holds DD{a < b) (a = b) (b < a). a c: On any ordinal is a subset of and a member of any larger ordinal. On itself is an ordinal but is a proper class SPr On. Thus we are unable to form the successor of On and so the antinomy of the greatest ordinal is avoided. An ordinal is either a member of On or is On itself. D71

LimX

D72

X+l

D73

1 for 0 + 1 , 2 for

or MaxX for

Xu{X}.

1 + 1, etc.

for SX.

3.27 Ordinals

177

We have G(X c: On) OrdHX. The limit of a class of ordinals is an ordinal. We have

C(X <= &n) KOrd I>X(Aa) C(aeX) (a ^ SZ), C(X c &n)KOrd?:X(Aa)C(X

c a)(2Z ^ a),

the properties of the limit ordinal of a class of ordinals. W e haVe

(Ax) B(x + leffn) (xe&n), N(a < b < a+1).

D 74 Kz

for

&(Ea) D(x = a + 1) (x = 0), ordinals of the first kind,

D 75 Kn

for

0w — iTj,

We have

ordinals of the second kind.

C(aeKn) K(a = Sa) (a 4= 0), C(aeKj)D(a = S a + l)(a = 0).

D76 w for the members of w and the members of the members of co are all of the first kind. We have Ord OJ, SOJ and a) e Ku. o) is a set ordinal of the second kind. Members of a) are called intergers. We use i,j,i',... as variables for integers. The principle of Mathematical Induction can be obtained in the form

provided that (j) is without bound class variables. D77

(X ~ Y)

for

(Ez)KKKUn2zRelz(@z

= X)(0tz = Y).

X is similar to Y and both are sets. D78

Sq

for

&$(x ~ y).

The similarity relation for sets. Then

D 79

Fin

for

^(JJ/i) (i ~ x)

InFin

for

S(^4i) j^(i d^ x)

the class of finite sets. D 80

the class of infinite sets.

178

Ch. 3 Predicate calculi

Having defined the integers we can then define rational numbers as triplets of integers, then real numbers as Dedekind sections of rational numbers and lastly complex numbers as ordered pairs of real numbers. This is further discussed in Ch. 7, § 2 8. We are then ready to develop analysis and as explained in § 32 of this chapter we can introduce all topological concepts. An ordinal is either of the first kind or of the second kind or the ordinal is On. If X is a non-void class of ordinals then IIX is the least member of X. Thus for I c f e w e have TlXeX and £m(X{\ IIZ). 3.28 Transfinite induction The principle of transfinite induction is (Aa) C{a} {a 4-1} (^46) C(Aa) O(a
(Aa) provided that is without bound class variables. This comes from G(X c: On) XWeS'. This allows us to prove properties of ordinals by transfinite induction, since the class of ordinals without the property, if non-void, will have a first member. By a proof by transfinite induction we mean the reductio ad absurdam of the existence of a least ordinal without the property in question.

NimX X^On, (Ev)K{veX)£m{X(\v)9 X^On

N(Ev) K(veOn - X) Sm((0n -X)(]v) _____ ^ X^On

(Aa) C(a c X) (aeX) X = On

We also want to define functions by transfinite induction. It makes for easier reading if we use F, G, H, F',..., as variables for functions, corresponding small letters if they are sets, and B,S,T,R',... as variables for relations, corresponding small letters if they are sets. We want then to define F'a by means of the behaviour of F for arguments less than a. Now F [ a is the function F with arguments restricted to a. Hence the induction should have the form

3.28 Transfinite induction

179

where G is a previously defined function. Thus we shall have (AG) (E\F) (KF^n6n(Aa) (F'a = G\F [a)). The method of demonstrating this is to take the union of all partial solutions. Thus //

for f(Eb) (Kf^nb(Aa) C(a < b) (f'a = G'(f [ a)),

then show that 2 i / has the required property. Something like this is done in detail in Ch. 11. The addition, multiplication and exponentiation of ordinals are defined by transfinite induction thus: D81

+6

for

(

(

)( (Ea) K(v = a = 2a) (v = 2 +1'a),

D82

xb

for tiv(DDK(u = O)(v = 0)(Ea)K(u = a+l)(v

=+bx'ba)

(Ea) K(u = a = 2a) (v = 2 x b'a), D 83

expb for

uv(DDK(u = 0) (v = 1) (Ea) K(u = a + 1) (v = x b exp'ba) (Ea) K(u = a = 2a) (v = 2 e ^ ^a).

The first of these defines the function +bi the second the function x b and the last the function expb. They are all of the form F(a = Gl(F [a). Weusuallywrite (6xa) a

b

for

x^a

for exp'b a.

(a + b),(axb) and ab are all set ordinals. They satisfy some of the usual rules of addition, multiplication and exponentiation. But some rules fail, e.g. l+(o = a). It can be verified that the ordinal (a + b) is isomorphic as regards order to the order type obtained when we stick the order type b at the end of the order type a, and that the ordinal (a x b) is isomorphic as regards order to the order type (c,d),cea, deb ordered by last differences, i.e. (c,d) < (c',df) ifd < d'ovd = d'andc < c\ The ordinal ab is isomorphic to as regards order to the ordering by last differences of functions over b with values in a, but with only a bounded number of non-zero values, i.e. if/and g are two such functions then/ < g if f'd < g'd, where d is the greatest ordinal for which

180

Ch. 3 Predicate calculi

f'd =f= g'd. In fact we could have taken these properties as definitions of addition, multiplication and exponentiation of ordinals provided we have shown yTT m xWeT where

D84

RIsom(*'^\

for

KKKKUn2RRelRSJR = X01R = Y(Au,v)C(u,veX)B(uSv)(RcuTR'v), i.e. R is a (1-1 ^correspondence between X and Y such that two members of X stand in the relation S if and only if their images in Y by R stand in the relation T. Any class of ordinals is well-ordered by £, hence a decreasing sequence of ordinals terminates, otherwise the sequence would be without first member and so would violate the condition of being well-ordered. 3.29 Cardinals A cardinal number is frequently defined as the class of classes similar to a given class. We shall define a cardinal as an ordinal which is dissimilar to any lesser ordinal. This is less general than the usual definition because there may be classes dissimilar to any ordinal. D85

f

for

t(Aa) (C{a ~ x) (bea),

then f is the least ordinal which is similar to the set x. f is called an initial ordinal or the cardinal integer of the set x in case x is finite. The class of ordinals similar to a given ordinal is non-void, because a is similar to itself. Hence the class of ordinals similar to a given ordinal exists and being a class of ordinals has a least member U. But if x is any set then set theory as we are developing it may turn out to be so poor in modes of expression that the (1-1)-correspondence required to show that x is similar to an ordinal may be missing. Thus the concept of cardinal is relative, that is relative to the set theory used. We have U = ti. We divide ordinals into classes, the members of one class being similar to each other, except that the first class is to consist of all the integers. The ordinals of the second class are similar to co, they are the denumerable

3.29 Cardinals

181

ordinals. The least member of class III is denoted by ti, it is the least nondenumerable ordinal. Q for 26(6 ~ o)). D 86 Note that this use of the word' class' is distinct from ' class' as opposed to 'set 5 . D87 JT for &(Eb)(a = $), JV* is the class of integers and initial ordinals. D88

JT'

for

JV-G),

J/*' is the class of initial ordinals. Jf' being a class of ordinals is wellordered, hence there is an isomorphism between the initial ordinals and the ordinals. Let X be this isomorphism. D89 G)a or Ka for K'a. Then GJ0 = G) = No, fi = K1? etc. These cardinals are called alephs. We can define addition, multiplication and exponentiation for cardinals. If otj9 je J is a class of cardinals then 2 % is the cardinal of the union of classes of cardinals otj for jeJ. In forming the union we require that the representative classes be distinct. This is achieved by using ordered pairs {a,j}, aeAjSOCj. Then the ordered pairs are distinct for different jeJ. If the cardinals are initial ordinals & then 5 itself is a representative class of that cardinal. The cardinal of the product of the class of cardinals ocj,jeJ is the cardinal of the class of functions / over J such that/'Jea^-. This amounts to picking out one member from each otj and doing this in all possible ways. This raises the question as to whether there is any such function at all. The statement of the existence of such a function is known as the Multiplicative Axiom, Axiom of Choice (A.C.) or Zermelo's Axiom, If it failed then the product of an unbounded set of cardinals would be zero. The axiom is: Ax. 6.

(EF) (Ax) KF^n VBSmx(Fixex\

This is a very strong form of the axiom of choice because it allows for the simultaneous choice from each set of an element of that set. The axiom of choice occurs frequently in mathematics, sometimes it is possible to avoid it by a more elaborate proof. At the end of Ch. 12 we sketch a demonstration of the independence of the axiom of choice from the other axioms of set theory.

182

Ch. 3 Predicate calculi

Using the axiom of choice we can show that every set can be wellordered, conversely if every set can be well-ordered then A.C. (Ax) (EaJ)KK(f^na) Un2f{x = /"a), so that f'b for b < a well-orders x. By transfinite induction we define a function G so that G^nOn and (Aa) (G'a = F'(x-@(G [a))), where F is the function postulated in Ax. 6. Then G'O = F(x, the member chosen from x by F, G'l = F\x — {G'O}), the member chosen from x-{GiO) by F, etc. Then G'b for b < a well-orders x. If we use Ax. 6 then every set can be well-ordered, hence every set is similar to an ordinal and so all cardinals are alephs, and hence the tricotamy will hold for cardinals. But without the axiom of choice there may be cardinals without any order relationship with any aleph. The exponentiation of cardinals is defined by: a^ is the cardinal of the class of functions over /? with values in a. Thus 2so is the cardinal of the real numbers. The equation 2**o = Nx is known as the Continuum Hypothesis (C.H.). It is now known to be independent of the other axioms of set theory and a brief sketch of this is given at the end of Ch. 12. It can be shown that the sum, product and exponent of alephs is an aleph. The equation 2Na = XSoc is known as the Generalized Continuum Hypothesis (G.C.H.). Again it is now known to be independent of the other axioms of set theory and a brief sketch of this is given at the end of Ch. 12. Many statements about cardinals are now known to be independent of the axioms of set theory. But there are some important theorems about cardinals. D90 <x
We shall show £ < Px. Each member y of x gives rise to a subset {y} otx, hence x can be put into (1-1 ^correspondence with a proper subset of Px, thus f ^ Px. Note that x and Px are sets. Suppose that % = Px, then there is a (1-1)-correspondence between x and Px. Let o-{y} be the correlate of y by this correspondence. Let a be the class of members of y such that y1cr{y\. Let a be the correlate of z so thata = cr{z}. If 2;ea then 2:ecr{2:} by definition of a and Prop. 31, but cr{2;} = a

3.29 Cardinals

183

andsozea. Again if zea then by definition of a ze(x{z} and Prop. 31,i.e. zeoc. We have an absurdity in either case, hence X #= Px. 33. If a ~ /?' and ft c ft and J3 ~ a' and a,' c a, then a ~ /?. Let / map a (1-1) onto /?' <= y? and gr map /? (1-1) onto a' c: a. We can clearly assume that oc(]j3 = A. Now (oc\Jj3) is the disjoint union of sequences PROP.

cr':(b,g%rg'b,...) (be/S),
LEMMA

(i). For any set a there is a well-ordered set w, such that w ^ P4a

and w ^ U.

Consider the class w of well-orderings of a and subsets of a. A well-ordering of a is a class of ordered pairs hence is in P 3 a. Thus w c= P^cc.w is isomorphic to a class of ordinals and so is well-ordered and is isomorphic to an ordinal. If w ^fiethen w would be order isomorphic to a well-ordered subset of a and so w would be order isomorphic to a proper subset of itself, which is impossible. Hence lemma (i). For the moment we assume that 2 P a = Pa. LEMMA

(ii). If y and 8 are disjoint sets such that y u S = P(2y) then

$>Py. 2y denotes the union of two disjoint copies of y, say yx and y2 If/maps y U S onto P{yx U y2) ^ Pji x Py2, then the image of y projected into

184

Ch. 3 Predicate calculi

Pyx is only a proper part of Pyv since y1 < Pyv and hence if £ is outside the projection, / must map some subset of S onto £ x Py2, which means that 8 ^ Py. Whence lemma (ii). Now we have P 3 a ^ w + Psa ^ P 4 a + P 3 a ^ P 4 a by the assumption. Thus by G.C.H. either: w + Pzoc = P 4 a

or

Consider the first case. w + Pzoc = P 4 a = P(2P 3 a), by the assumption. We have, by lemma (ii), w ^ P 4 a, but w ^ P^oc, hence w = P±oc. Thus there is a (1-1 ^correspondence between P^oc and w, thus jF^a is wellordered, but a can be embedded in P^a, hence a can be well-ordered, and we are done. In the other case we have w + P>a = P,a then we have w ^ P 3 a, hence w = P 3 a, and we are done, as before, or w < P3oc, and whence by G.C.H. w < P 2 a, but then ^ = P 2 a, and we are done as before, or W where af]o) = A, then easily 2iy? = Pip, for 0 < i < 4. Let y be new. 2/? = P(au^u{y}) = P(a U w) = /?. Also yff^ y^uy ^ 2/?, so /?Uy= Now 2.2/* = 2/*uW = 20, and similarly 2P^/? = P ^ , for all i. Hence our argument can be applied to /?, and so ft can be well-ordered. But a can be embedded in /? in a natural manner, hence a can be well-ordered. Thus Prop. 33 is demonstrated. 3.30 Elimination of the e-symbol The choice effected by A.C. can also be effected by the e-symbol, thus (Ax)CN£>mx(ey(yex)ex), then tivCNdfrnviu = ey(yev)), is the required function that picks out a member from each set. We could also have the rule. C

provided that
3.30 Elimination of the e-symbol

185

The system Sf with the e-symbol and rule C is called the system CSP. We could also, as in D 24, define the existential quantifier in terms of the e-symbol. The system SP with rule II d replaced by rule E and rule II e' replaced by rule E' is called the system ESf. We could then dispense with rule E' in favour of a rule of substitution. The system CE£f is the system ESf plus rule C. 35. If an £f-statement is a CSf-theroem then it is an £f-theorem. We shall show that if the e-symbol is used in a C^-proof of an Sfstatement ^ then it can be eliminated leaving an e-free proof of ^ , which is thus an ^-proof. Thus (J) is an ^-theorem. This proposition says that if CSP is inconsistent with respect to negation then so is Sf. For if we can C^-prove the ^-statements then these C^-proofs can be transformed into ^-proofs and so S? would be inconsistent as well. This means that the axiom of choice is consistent with the other axioms of set theory. First we replace the rule C by the rule E, this converts a C^-proof into an 2^-proof, where ESP is the system Sf with the rule E added. To do PROP.

this, consider rule C, say, _ ,,

,}J^

, and consider the places where

related occurrences of (EE) <j){£) entered the C^-proof, these will be of the form

n/grrx^/m /> w n e r e 0'{£} i s

a

variant of ^{g} by I(ii), replace

the lower formula by D^'{e^'{^}} GJ', and similarly for all descendents. Entrance by II a can be replaced by entrance by II d, the special ^-rules fail to introduce D(E£) $${£} OJ. The original Cc^-proof-tree becomes an E£fproof-tree of the original statement, because these related occurrences of (E£)(j){Q all occur in the subsidiary formulae, except I(ii), from their introduction to the application of the C-rule under discussion. Thus we require to eliminate the e-symbol from an E^-proof of an ^-statement, free from the e-symbol. We next replace the special ^rules by axioms, thus: if _ ,

is an

Dyrco

^-rule replace it by CD^o)Dj/rco, we recover the rule by Modus Ponens, which can be eliminated from a theory in free disjunctive form. To make use of this result we replace rule R1 by its free variable form viz. DK(XeU)(YeU)a>. ,,, .,.r, ,TT,., —Y^i— because by the reversibility of lie if we can ^-prove n v

186

Ch. 3 Predicate calculi

the upper formula of R1 then we can ^-prove the upper formula of its free variable form. We similarly replace R 5 ' and R5" by D((oc,u)eX)oj —^7—'v.

and

D((v9u)eX)o) " \'v,

,. . respectively.

This assures the reversibility of l i e ' because A can then only be introduced at l i e ' and so the demonstration of reversibility of l i e ' goes through as before, The axioms can be put into resolved form, viz.: Ax. 1. (LxeiXx}), Ax. 2. ({x,y}e{{x,y}}), Ax. 3. (Pxe{Px}) Ax. 4. CCK{(u, v)eX) ((w, v)eX) B{yeu) {yew) (X"ye{X"y}). From these the original axioms may be recovered. The system SP can now be put into free disjunctive form, so that Modus Ponens can be eliminated. If we retain the axiom of infinity then we replace it by: (£2e{Q}), (Aefi), C{ueQ)({u}e£l), then Q, contains the unending set: {A}, {{A}}, {{{A}}},.... Call the resulting system Sf\ then Modus Ponens can be eliminated from Sf'. Let ^ be an E^-theorem then 0 is an ESP'theorem, let x be the closure of the ^'-axioms used in the ES^'-"proof of <j), then by the Deduction Theorem G^r^> is an iJ^-theorem. From its .EJ^-proof we wish to eliminate the e-symbol. By hypothesis the e-symbol is absent from <j> also the ^'-axioms are without the e-symbol, hence Cx is without the €-symbol. The method we shall adopt is to replace e-terms by other terms which lack the €-symbol in such a manner that the JE^'-proof remains correct. Consider first the simple case when all the E-rules belong to the same e-term. Suppose _ . r

,*

is an E-rule used in the l£^'-proof of 6,

where \jr is e-free and closed; in this replace the e-term e^{Q by a, we are left with a tree with (f> at its base because this is €-free. The above E-rule becomes a repetition, all applications of other rules are preserved except other applications of E-rules, which by supposition belong& to the same e-term. For instance -^ , , ,1^—, E becomes Z)^{e^{£}}6/ n

,\{

, which fails to be a case of any rule. But if we add ^{a} as an extra

axiom we can obtain Di/r{x} o)f by II a, each case of rule E in the j&^'-proof of $ can be replaced by a case of II a; hence we obtain an ^-deduction of (j) from the hypothesis ^{a}. Similarly we get an ^-deduction of
3.30 Elimination of the e-symbol

187

for all d for which i/r{a{6)} occurs as the main upper formula of an E-rule. Since we are supposing that all E-rules belong to the same e-term then ^r is the same in each case. Thus we get an ^-deduction of (j) from the hypotheses ^{a'}, ifr{oc"}, ...,i/r{oc(p)}, where this is the complete list of main formulae in the upper formulae of E-rules used in the V

proof of (j). Hence we obtain an J^-deduction of $ from 2 ^{#(7!r n= l

On the other hand we have Dft{aP>}at*> -i.V_L. _L •

)

TT

But Modus Ponens can be eliminated, and so we get an .^-deduction of (f> from the hypotheses Ni/r{ocf}, Ni/r{ot"},..., Ni/rfaM}, where we have the same set of a',..., a^ as before. Thus by the Deduction Theorem we obtain the ^-theorems CM<x'}$,...,CMoiM}$, and ON 2 {<*<«>}0. Whence 0= 1 v

d)

(e)

C 2 ^{o^ }(j> and CNHi/r{x }^) are J^-theorems and so by Modus Ponens 0=1

is . Thus this simple case we have eliminated the e-terms, and obtained an ,^-proof of {%}, where j stands for £,',..., QK\ We may also suppose that the .BJ^-proof is without free variables of any type, because we can replace any that there may be by constants of the same type, and for this purpose we need only use one constant of each of the types required. This will fail to affect (j) because it is closed, also the ii/J^-proof will remain an JS/J^-proof. Now replace each occurrence of rule II d by a corresponding occurrence of the E-rule, this will replace the end formula by ${&} where a stands for a',a", ...,a<*>, Before going any further we examine the structure of e-terms. The order of an e-term is the greatest numeral A such that we can find a sequence er, e",..., e(A) of e-terms such that e ( ^ fails to occur bound in e(^ but occurs in eld) i.e. occurs free in e(^. The rank of an e-term is defined as the greatest numeral fi such that we can find a sequence of e-terms e', e",..., e(^ such that e ( ^ occurs bound in e(^, i.e. e ( ^ contains a free variable in the scope of the binding variable of e^e\ Clearly both sequences of e-terms are terminating. Also if a is an e-term occurring free

188

Ch. 3 Predicate calculi

in the €-term j3 then the replacement of oc by another e-term lacking the binding variable of fi has no effect on the rank of/?. And if a is an e-term occurring in the e-term /? and containing the binding variable of /? then a is of lower rank than /?. Also the rank of an e-term is unaltered when we change a free variable to a new variable. We propose to eliminate the e-terms by replacing e-terms by other terms as we did in the simple case when all E-rules belonged to the same e-term, thus in the E-rule

., *. j ^

we replace the e-term e^t/r{Q

wherever it occurs by a, then the above application of the E-rule becomes a repetition; the application of the E-rule ^ , c , i>>^—; becomes Dj/r{e^{Q}ojf -r^. \ ^I0) , which fails to be an application of any rule if (a 4= fi), but we can obtain the lower formula from the hypothesis i/r{(x}. The subsitution of a for €^{£} may alter the end formula {a[} from ^{a'},...,
, f* —^,

1 < 0 ^ K. By the

Di/r{ei/r{^}}(i)^)3

J

deduction theorem and &c this would give us: C 2 ft{ote)} 2 ^=1

6=1

without using the E-rule belonging to the e-term e^{^}. On the other hand we have by Modus Ponens

but Modus Ponens can be eliminated. Thus if we introduce Ni/r{oc'}, Ni/r{a"},..., Ni/r{a^K)} as hypotheses we obtain an iJ^-deduction of
the e-term e^{^}.

of C Yl Ni/r{oL(e)}
This gives us an E^c--proof

6= 1

C 2 ifr{ot^} 2 ^{cii^} and 6=1

CNYii]f{a!&)} ^{ci}

6=1 K

we obtain by Modus Ponens 2 {&(i}}, where ai0) is a. Modus Ponens 6= 0 K

can be eliminated so we obtain an JS/J^-proof of 2 ^{cti^}. The reason for

3.30 Elimination of the e-symbol

189

replacing rule II d by rule E is to avoid the existential quantifier binding an e-term, thus:

will fail to occur instead, we shall have in the lower formula

which has an e-term of higher rank. But if there are other applications of the E-rule in the jE/J^-proof of 0{ct} then the substitution of af® for €^{^} may destroy them. The following cases can arise for the E-rule

(i) eMB occurs at most only in fi, the E-rule is then 1 this becomes _ } *• / {. which is the E-rule.

Dfai}} y (ii) €j^{^} occurs in x{v} a n d possibly in fi as well the E-rule is then DM*},*}* this becomes w h i c h is t h e E . r u l D a a x{*x{v h ) ^ here y stands for (iii) One or both of/?, # where S = €£#{£} is contained in y = (so that in the first case y is of the form y'{/?}), and x{v} * the form X'{y'{V}}; the E-rule is then comes •=- /f

/r

^./ / r ^ , ^

D

^

{

^ ^ ^

this } } } a >

be

"

which fails to be an E-rule, f is new to

Dx{?M{v{Q)}}<* X'{y'{^i}}' Similarly if y is of the form y'{8}, and x(v)^s of the form ^'{y'{9/}}, the E-rule is XxrxPss

^ j c h f a ii s to be an E-rule. Similarly if y is of the form

y'{(3,8} and x is of one of the forms x'{Y{y, *}}5 X'{7'{P> y}}> X'fy'fy* V}}>etc-

In all these cases the e-term ^x{j'{Q} ^ °f higher rank than the term y'{/?} for which a substitution is being made.

190

Ch. 3 Predicate calculi

(iv) The only remaining case is when one or both of /?, S are contained in y so that y = y'{/?}, etc., but the variable 7/ in x{v} fau"s to occur in a part y'{ij} of x{v}- I*1 this case the E-rule remains correct as can be seen from case (iii) by omitting y' in xij'iv)} m the lower formulae. In case (ii) the e-term belonging to the E-rule is of higher order than y but is of the same rank as y, but in case (iii) it is of higher rank than y. Thus if we make the substitution for an €-term of highest rank and among those of highest rank we choose one of highest order then cases (ii), (iii) will fail to arise; we can proceed as in the first case where all the E-rules belonged to the same €-term. The result of the elimination of one K

€-term is an UJ^-proof of a disjunction 2 ^ {ct(^}. A second application 0=0

will produce a similar disjunction of these disjunctions which is merely a longer disjunction of the same kind. Finally we can eliminate all the V

E-rules and are left with an ^-proof of a disjunction 2 <j>{oSe)}. If there e=i

are any €-terms left in this disjunction then replace them all with the same new free variable, the result is an J^-proof of a free variable disjunction which is €-free. This is possible because all substitutions had been pushed back into the axioms before we started so that an e-term V

in the J^-proof of the disjunction 2
into the J^-proof at an axiom and if it is replaced by a new variable axioms remain axioms and rules are preserved. From the free variable disjunction we easily obtain (E%) ^{j}. This completes the case when $ is without universal quantifiers. We may suppose that (f> is in Skolem F-normal form, i.e. is of the form: (E$) (At)) fr{£,t)} where the matrix ifr{%,t)} is quantifier-free. Since . Now an ^-proof is the same as an J^-deduction from hypotheses, say X. Thus if we can get an !FCdeduction of ' from hypotheses $, then we can do the same for (j). Also if we have an .257^^-deduction of from hypotheses 3£, then we also have an JS^^-deduction of (j)f from hypotheses 36. Lastly, if we can convert this U^rdeduction of 0' from hypotheses 36 into an J^-deduction of $' from hypotheses $ then we can convert an i£J^-deduction of <j) from

3.30 Elimination of the e-symbol

191

hypotheses dc into an J^-deduction of (j) from hypotheses X. Thus a C^-proof of (j) can be converted into an ^-proof of . Now suppose that there are universal quantifiers in $. Introduce sufficient new functors so that we can replace each general variable by a function of its superior restricted variables. Then omit all the universal quantifiers, we are left with : (^7J)^{J, fj}. From the E1FCproof of we can obtain one of (Z£j) ${£, fj} as follows: We have the ^,-theorem C(A*)) ${l, t)} <j>{l,fe}whence

C{E$ (At)) <%, t)} (Ed flj, fj}.

Thus from the iJJ^-proof of (E%)(Ati)){%,\)} we can obtain one of (E%) {%, f j}. Thus from the i?J^-proof of ^ we obtain one of (E%) 4>{h f?}> where f denotes the sequence : /',/", ...,/(7r) of functors a n d / ^ j signifies that the argument places off{6) are filled with the superior restricted variables only and so may fail to embrace all the variables in £. From the case when the end formula was without universal quantifiers obtain an ^,-proof of a disjunction where

a'9a",...,a«>

are sequences of terms of type t, €-free but whichmay contain/',/", ...,/(7r). Moreover the ^ c -proof of this disjunction is a free variable ^ c -proof free from the substitution rule, because all substitutions had been pushed K

back into the axioms before we began. If in 2 {&{6\ \(X(6)} and its £?ce=o

proof we replace each atomic formula by a propositional variable, distinct formulae by distinct propositional variables, identical formulae by the same propositional variable, we are left with a ^-proof. Thus K

2 {&{d), f cte)} arises from a ^ - t h e o r e m by substitution of atomic for0=1

mulae of a theory for propositional variables. The same will hold if we replace the terms f^a(fl) by new free individual variables using distinct variables for distinct terms, the same variable for different occurrences of the same term. Consider the term / ^ a ( ^ and the number of distinct occurrences of /',/", ...,/(7r) which are contained in it. Call this number the complexity of f(d)a{fl). Then the complexity of/'a ( ^,... ,/(7r)a(^ are all the same. We associate this complexity number with K

Now arrange the disjunctive terms of 2 (f>{a(e\ \a{6)} in order of increasing 0=0

192

Ch. 3 Predicate calculi

complexity from left to right. We suppose that duplicates have been omitted from the disjunction. Then f(6)a(/l) is distinct from f^aW if and only if 6 4= d' or JJL =)= /JL'. Also f^a(fl) can only occur in a(/O if [i < /if. Now replace /'a', ...,/ (7 %',/a", ...,f(n)a(lc) by new individual variables £',..., £<™>, so that /% is replaced by £«/-«*+«>, then £ 0{a<*>, fa<*>} 0=0

becomes: 2 ^{b^, ^'

7r+1)

, ...,^'

7r+7r)

}. In this disjunction the variables

0=0

, ...,Qv'n) are absent from the first (v—1) disjunctions because can only be a member of the sequence a(A) or be contained in a member of the sequence a(A) when JLC < A, also f^ai/l) fails to occur in or be any member of the sequence a^\ Now we can generalize the variables QK~1)rr+1, ...,£,K'n then apply lid repeatedly to the terms in b(/c) this converts the last disjunctand to (E%) (At))^^,^}. We can proceed similarly with each disjunctand obtaining a disjunction of K disjunctions all being (EAjc)(At))
3.31 Complete Boolean Algebras In some Boolean Algebras any subset has an l.u.b. that is, if 9£ is a subset of a Boolean Algebra 88 then there is an element a of 88 such that /? ^ a for each element /? of 9C, and if /? ^ y for each element ft of 9£ then a < 7. We denote a by l.u.b. 9£. Similarly a g.l.b. can exist; in any case if the l.u.b. exists then we call the Boolean Algebra complete. PROP.

36. If 88 is a complete Boolean Algebra then l.u.b. & = g.l.b.3?,

where 2£ is the set of complements of members of SC. We sometimes write U at for l.u.b. {aj, and similarly f|
PROP.

iel

35. l.u.b. l.u.b. a^ = l.u.b. aij9 iel

jej

iel,jej

«nu A = UMA), iel iel

iel

A

iel

iel

iel

3.31 Complete Boolean Algebras

Uai

=

0 iff

cx>i = 0 for

193

iel,

iel

fl oci = 1 iff

a* = 1 for

ie/.

The V is the membership symbol. A Boolean Algebra is said to satisfy the countable chain condition if every disjoint set of non-zero elements is countable. Two elements of a Boolean Algebra are said to be disjoint if their intersection is zero. The l.u.b. acts like the union of an unbounded set and the g.l.b. acts like their intersection. These correspond to the Existential and Universal Quantifiers respectively. Distributive Laws In some complete Boolean Algebras there are extensions to the distributive laws corresponding to unbounded sets. Thus: PI U ay = U n aiT(i) and ieljej

re J1 iel

(J D % = fl U ocT(j)j. jej

iel

lfjj

Here J1 denotes the set of functions with domain / and range in J. But these laws can fail in some complete Boolean Algebras. 3.32 Truth-definitions for set theory A truth-definition for a formal system can be given by formula induction. First a truth-definition is given for closed atomic statements, then the truth-definition for closed compound statements are obtained in the usual manner by truth tables, if we are seeking a standard two-valued truth-definition. If closed atomic formulae are lacking as in ^Q then we usually give a definition of validity. In set theory the closed atomic statements are of the form a closed statement of this form will be true if and only if this in turn will be true if and only if t{x<j){x}} & ^ { a ? } } & (Ey) (x{x} ey),

for some x> but this will be true if and only if which is what we had before, so we must abandon this method.

194

Ch. 3 Predicate calculi

Another way of giving a truth-definition is to construct a model. By this we mean a class of elements V and truth-values for all atomic statements of the form (aeb), where a and b are two elements of F. But we can easily be more general because we can take the truth-values to be members of a Boolean Algebra ^ . Let then ||ae6|| and \\a = b\\ be the members of the Boolean Algebra associated with the statements (aeb) and (a = b) respectively. From the Boolean values of these statements we can find the Boolean values of compound statements thus:

= ¥\l aeV

from these we obtain

\\Df\\ = H\\ U

U

||##

aeV

We write \=$ for ||^j| = 1, and h^ for §1 is an ^-theorem. We want to show that, if h^ then |= <j>, i.e. all ^-theorems take the Boolean value 1. First of all we easily show that f= DNcjxj), and, if \= i/r then = |
It is impracticable to introduce all the members of V at once. We proceed by a transfinite process. We start with Vo = A. The elements to be added to Va to produce Va+1 will be functions over Va with values in 38. These will correspond to new ' sets'. Thus/{#} defined over Va with values in £8 will be the 'characteristic function' of a new set. Clearly Vx = {A}. If a is a limit ordinal Va= \J ¥#. p<*

The step from Va to Va+1 is defined as follows: we assume that the

3.32 Truth-definitions for set theory

195

members of Va have been defined and that Vp ^ Vy for fi < y ^ a and that for a, b e Va we have

\\aeb\\=JJJ\K(xeb)(a = x)\\,

(1)

||a = 6||= n \\C(xea)(xeb)\\n fl \\C(xeb)(xea)\\, II

II

' ' I I

V

/ V

xeBa

' II

' '

II

*

' *

(2)

' II '

^ '

xe&b

and that for a, b, c, eVa we have^ a = a,

(3)

(a = b)(b = c)(a = c),

(5)

(a = 6) (bee) (aec),

(6)

eb) (b = c) (aec),

(7)

and we also assume that every member of Va is a function whose values are in the Boolean Algebra 3$. aeVfi+1, aeVp, fi < ot then

if if

aeT^,a;e^a then x,ye@a then

2a = Vfi9

(8)

\\xea\\ = a{o;},

a{#}n |a? = y\\ ^ «{i/},

(9) (10)

a function which satisfies (10) is called extensional. We first put into Va+1 every member of Va. Next we generate each function / from Va to 0$. As @f — Va for each new /, the value of \\x = y\\ has been determined for each x,ye3>f. Hence for each x,ye@f the value of/{#}n ||» = y|| is determined, we discard all functions/for which this value is ^ f{y}. Thus we restrict our choice to extensional functions. For xeVa we define \xea\ to be a{x] for each new a. We now define ||a = 6|| by (2). If a, beVa this duplicates a known result. If aeVa, beVa+1, beVa, this is an acceptable definition, since Q)a c Va 3)b = Va, hence ||#ea|| is determined by (1) and ||#e&|| by b{x}, so ||G(a;ea)(aje6)|| and ||C(a;e6)(icea)|| are both determined. Similarly in the other cases. So (2) holds for Va+V Now if ae@b, beVa+1, beVa we define: ||ae6|| for U \\K(xeb)(a = x)\\ xe9b

this is an acceptable definition since for each xe3)b \K(xeb) {a = x)\\ is determined by (9) and (2). 7-2

196

Ch. 3 Predicate calculi

We have to show that (1) holds in F a + 1 . The only case to consider is beVa+1, beVa, aeSb. In this case S)b = Va, so that aeVa, then we have \\aeb\\ = \\K(aeb)(a = a)\\ ^ U \\K{xeb){a = x)\\.

(11)

xe9b

Now take xeQ)b = Va. Then by (9) \\K(xeb)(a = x)\\ < b{a] = \\aeb\\, thus

U \\K(xeb)(a = x)\\ ^ \\aeb\\, xe9b

then by (11) we infer (1) for F a + 1 . It remains to check that (3)-(7) inclusive hold in Va+1. By (2) we conclude (3), since \\C(xea) (xea)\\ = 1. Also (2) gives (4). LEMMA

We have thus whence

(i). IfxeSb then \\xeb\\ n ||6 = c\\ ^ \\xec\\. ^ ^ ^ ^C(pceh) ( ^ c ) | | = ^ n n ^xec^ ? ||ae&|| n \\C(xeb) (xec)\\ ^ \\xec\\, ||a;e6||n 0 ||O(#e&)(#ec)|| ^ ||a;ec|| if xeQ)b. xe2b

By (2) the lemma follows. (ii). If a, beVa and ceVa+1 then (7) holds. If ceVa then by assumption (7) holds. Thus suppose that ceVa. Then LEMMA

2c = Va. By (8) and Va c Va+1 we have Sib c Sc. Then by lemma (i) \\K{xeb) {a = x)\\ n \\b = c\\ < \\K{xec) {a = x)\\9

whence using (1) ||ae6|| n ||6 = c\\ ^ U K\\(xec) (a = x)\\ xe@c

= \\aec\\ by (1), which is (7). We now verify (5) for oc + 1. Let xeQ)a. Then, by lemma (i) we have ||a?ea|| n ||a = 6|| ^ \xeb\. So \\K(a = 6) (6 = o)H n ||ajea|| ^ ||aje6|| n ||6 = c\\. Case (i) beVa, by lemma (i) ||ge&||n||& = c|| ^ ||^ec||.

(12) (13)

3.32 Truth-definitions for set theory

197

Case (ii) beVa. Then 3)b = Va9 so that xe3)b. Then by lemma (i) we again have (13). Therefore (13) holds in either case. By (12), (13) we obtain = c)\\n\\xea\\ < \\xec\\. So

\\K(a = 6) (6 = c)\\ U \\xea\\ ^ ||a;ea|| -> \\xec\\

and then

\\K(a = 6) (6 = c)\\ ^ \\G(xea) (xec)\\

whence

\\K(a = b)(b = c)\\ < fl ||C(sea) (n;ec)||.

(14)

One can start with xe@c and go through a similar argument to obtain \\K(c = 6) (6 = a)|| ^ n ||0(sec) (sea)||.

(15)

xeQic

By (4), (2), (14) and (15) \\K(a = 6) (6 = c)|| ^ ||a = c||, which gives (5). We next verify (6) for a + 1. Let xeQJc, By (5) we have \\K(a = b)(b = x)\\ <\\a = x\\. Therefore \\a = b\\ n \\K(xec) (b = x)\\ < \\K(xec) (a = «)||, summing both sides over xe3)c and using (1) we get (6). Finally we verify (7) for oc + 1. Let xeQsb. By lemma (i) ||a;e6||n ||6 = c\\ ^ \\xec\\. Therefore Then by (6)

\\K(xeb) (a = x)\\ n ||6 = c\\ ^ \\K(a = x) (xec)\\. \\K{xeb) {a = x)\\ n ||6 = c\\ ^ \\aec\\.

Sum on the left over xeS)b and we obtain (7) by (1). We have now obtained a universe V and a Boolean value for each atomic statement (aeb), where a,beV. So we have given a generalized truth-definition for set theory. Now write ^ ^ _ 1; for h (j> for

fi is an 5^-theorem.

It is possible to ^-prove that if h^ then t=^, i.e. all ^-theorems take the value 1. Thus C becomes a model for SP. The proof must take place in some system which can deal with ordinals because they are essential to the construction of V. The system £P is such a system.

198

Ch. 3 Predicate calculi

An interesting application of this method for denning truth for set theory is that by suitable choice of the Boolean Algebra 0&, we can show that A.C., G.C.H., etc. take values different from 1 and hence are non-theorems of SP. Thus it is impossible to ^-prove them. On the other hand we can find another type of model, namely an 'inner model9, constructed as follows: we start with the null set and by a process of transfinite induction we define all the sets which can be obtained from it by repeatedly performing the operations allowed for the construction of new sets from old ones, e.g. union, complementation, etc., In this way it seems clear that all sets are given an ordinal and so V is well-ordered, so that A.C. holds. It can also be shown that G.C.H. holds in this model. All this can be done in the system SP. Thus A.C. and G.C.H. are consistent with set theory if this is itself consistent. Altogether A.C. and G.C.H. are independent of the other axioms of set theory. The full details are lengthy. 3.33 Predicative and impredicative properties In a second order predicate calculus we can have bound predicate variables and hence we can form properties Ax
property variables of the first order,

X2, X'2,...

property variables of the second order, etc.

The order of a property is the greatest of (i) the order of a free property variable in it, (ii) the successor of the order of a bound property variable in it. Then \x
r>/A/A\—

*' ^ e

revers

ibility of

3.33 Predicative and impredicative properties

199

lie', here A is to be a property of order the same or less than that of the variable X. We obtain this result from l i e ' by everywhere replacing related occurrences of X in the part of the tree above the upper formula of l i e ' by the property A which must be of the same or less order as X, A property is called predicative if it fails to be defined in terms of itself, otherwise it is called impredicative. Thus in ^^ we only have predicative properties. If we try to give a definition of validity to ^l?\ which contains impredicative properties, then we get into trouble because we should require (AX) (/>{X} to be valid if and only if ^{A} is valid for all properties A, but one of these is 3£
D91 Topx for (Au,v) (u, vex.-^.un vex) & (Ay) (y <= x-^Hyex); read lx is a topology'. D 92 xTopy for Topx & y — 2#; read 'x is a topology for y\ D93 xOpy for Topx &y ex; read ' y is an open set in the topology x\

D94 xCly for

Topx&Op(x-y);

read 'y is a closed set in the topology x\ D 95 TopSpx for (Ey) (Topy & x = Sy); read 6x is a topological space'. D 96 IndisTopx for (Ey) (x = {y, 0}); read 'a; is an indiscrete topology'. D 97 DisTopx for (Ey) (x = Py)\ read (x is a discrete topology'.

200

Ch. 3 Predicate calculi

D98 uTopxNeighy for (Ev) (v ^ u&yev &xOpv) &u c x\ read 'u is a neighbourhood of?/ in the topology x\ D 99 yTopxLimz for 2 c= a; & Topx & (^4w) (uTopxNeighy -> (i£#) (# e w n z & # 4= ^/)); read '?/ is a limit point of the subset z of the topological space x\ D100 Z*TOPX for $(yTopxLimz) [) z; read 'the closure of the subset z of the topological space x\ D101 IntzTopx for y(zTopxNeighy); read 'the interior of the subset 2 of the topological space x\ D102 BdyzTopx for z* T o ^ n ( # - z ) * T o ^ ; read 'the boundary of the subset z of the topological space x\ D 103 zBaseTopx for z ^ x & (^y, u)(yex& uTopxNeighy ->(Ev)(yev&vez&y c ^)); read '2 is a base for the topology a?'. D 104 Se^a; for Topx & (#z, ti;) (zBaseTopx & w"z = w); read ' x is separable', x is separable if a; has a countable base. D105 zDenseTopx for z* To ^ = X; read'2 is dense in the topological space x\ D106 uGovw for w ^ Su; read '^ covers w\ D 107 Sep [y, z] Topx for y^opx n s = A & z*T°vx n y = A; read '^/ and z are separated in the topology x\ D 108 GonnTopx for .4 (y, z) (a? = y u z &fifep[y,z] T o ^ . -».y = A v z = A); read 'a; is a connected topological space'. D 109 HausSpx for Topo; & (^4^, v) (^, v e 2# & ^ 4= v. -> (.E/^, ^') (^n wr = A & wTopxNeighu & w'TopxNeighv)); read '# is a Hausdorff space'. (,4y) (fe) (?/(7ora & z(7ora &z^y& (Ew) (w"z c
Compx for

and so on. If we are dealing with a single topology for a topological space

3.34 Topology

201

then we can omit Topx in the above definitions. The whole of general topology can now be formalized without undue difficulty. H I S T O R I C A L REMARKS TO C H A P T E R 3

Predicate calculi differ from propositional calculi by the adjunction of quantifiers, whose intended meaning always has something to do with the cardinal number of things which satisfy a certain statement. Quantifiers were first introduced by Frege (1879). Somewhat later and independently quantifiers were used by Pierce who introduced the term 'quantifier'. Thereafter their use becomes general, though the notation for them varies. The various orders of predicate calculi is due to Russells' theory of types and perhaps to Frege's Stufen and Schroder's Mannigfaltigkeiten. Lowenheim and Skolem in effect gave a treatment of the first order predicate calculus with equality. But the first explicit formulation of the classical predicate calculus of the first order as a formal system in its own right is in the first edition of the book by Hilbert and Ackermann (1928). Thereafter it was much studied as a formal system. Many-sorted predicate calculi were discussed by Schmidt (1938) and Wang (1952). Models by Kemeny (1949) among others. Predicative and impredicative predicate calculi arises from the unqualified use of the concept of' all'. Russell's (1906) P.M. vol. 1, Ch. II, vicious-circle principle, designed to avoid paradoxes, was 'no totality can contain members defined in terms of that totality'. The term 'impredicative is due to Poincare (1905) who condemned impredicative definitions, as did Weyl (1918). In developing the classical predicate calculus of the first order we again use the direct formulation due to Gentzen (1934) and further studied by Schiitte, (1950-1,1960). Prop. 4, the elimination of M.P., is Gentzen's Hauptsatz, the demonstration we give is due to Lorenzen (1951). The prenex normal form is due to Skolem who also found other normal forms. Props. 9,10 and 11 are due to Herbrand (1930) as is the discussion on -ff-disjunctions, hence their name. Much has been contributed to the concepts of validity and satisfaction by Tarski (1933). Prop. 12, the completeness theorem of the classical predicate calculus of the first order is due to Herbrand (1930), Godel (1930), Lowenheim (1915) and Skolem (1920). Cor (ii) is due to Lowenheim (1915). Prop. 13, the independence of the axioms and rules of the classical predicate calculus of the first order was first considered by Godel

202

Ch. 3 Predicate calculi

(1930) and the consistency by Hilbert-Ackermann (1928). The discussion of theories is due to Tarski (1935-6). The classical predicate calculus with equality is implicit in the work of Pierce and Schroder, but its first treatment as a system in its own right is in Hilbert and Ackermann (1928) and again in Hilbert and Bernays (1934-6). Prop. 19, the elimination of axiom schemes, is due to Skolem (1959) who used a formulation of set theory due to Godel (1940). Early attempts at finding a decision procedure for the classical predicate calculus of the first order were unsuccessful (because as we shall see in a later chapter there is none), so research turned to finding decision procedures for special classes of statements. One of the earliest of these was by Lowenheim (1915). He gave a decision procedure for the monadic predicate calculus of the first order. This was followed by work by Skolem (1919, 1920) and Behmann (1922) on the monadic predicate calculus of the second order and of the first order with equality. Prop. 22 is due to Bernays and Schonfinkel (1928) and Prop. 23 is due to Godel (1933), Kalmar (1933) and Schiitte (1934) A detailed account of all known decision procedures for special classes of statements has been given by Ackermann (1954), see also Church (1956). A related type of problem is the reduction problem. Here we try to find special classes of statements such that any statement of the predicate calculus of the first order is equivalent as regards validity to one in the special class, there is a corresponding problem for satisfiability. A simple case is the class of statements in prenex normal form. These special classes are called reduction classes. Prop. 24 is due to Lowenheim (1915) and Godel (1933). The Skolem V- and ^-normal forms are of course due to Skolem (1925) and lemma (iv), the restriction to exactly three universal quantifiers is due to Godel (1933). Prop 25, where we have only one predicate and that one binary is due to Suranyi (1943) and Kalmar (1947) whom we follow closely. Many reduction types with a variety of prefixes have been found by Kalmar and Suranyi, and account of them is given by Suranyi (1959), see also Church (1956). The method of semantic tableau is due to Beth (1955, 1959) and Hintikka (1953, 1955) and Prop. 26 is due to Craig (1953) and Kleene (1952, 1967). The idea of a resolved predicate calculus is due to Hilbert [H-B, II], but the 'T/' symbol (written i) and its use is due to Peano (1897), Frege (1893, 1962) and (1905, 1956). The V symbol is due to Hilbert [H-B].

Historical remarks to Chapter 3

203

He demonstrated two theorems about the elimination of the e-symbol, the first is virtually Prop. 35. An historical account of the predicate calculus has been given by Hermes & Scholz (1932). Many examples are to be found in Church (1956). The system 88!FC is suggested by Kalmar & Suranyi's reduction type whose only predicate is a single binary one. Definitions D26 and 27 go back to Leibniz He took two classes to be the same if they contained exactly the same members. This is the extensional approach, and is used in classical mathematics. But one might consider two classes to be distinct, even if they contained exactly the same members on the ground that the rules for membership were different. This is called the intensional approach, we shall have to speak about it again in later chapters. The concept of set and class as defined in D 33 and 34 is due to v. Neumann (1925) who used it to avoid the syntactic paradoxes which had crept into set theory since Frege and Cantor. The definitions D 37-57 are largely from P.M. But the definition of an ordered pair has received simplification at the hands of Wiener (1912) and Kuratowski (1921). The rules R2'-9" are modifications of the axioms used by Godel (1940) in his account of set theory. Prop. 31 on normal classes is due to Godel (1940). When P.M. first appeared it was considered to have a few blemishes. One was the complicated type theory. Several writers have tried various ways of simplifying this, notably Leon Chwistek (1921, 1927) and F.P.Ramsey (1926), with his simple theory of types. Another thing thought by some to be a blemish is the axiom of infinity. Perhaps the reason for this may be compared to the reasons for considering Euclid's axiom of parallels to be a blemish on his work in geometry (first formalized by Hilbert (1922)), namely, one might think that it should follow from the other axioms. But apparently this is not the case. In fact in Godel's (1940) formulation of set theory the axioms previous to the axiom of infinity are consistent, this is seen by taking ae/3 to be always false. But with the axiom of infinity this is no longer the case, because this axiom postulates the existence of a set. There are many ways of formulating an axiom of infinity, all one requires is the existence of some set with an infinity of members. The last thing thought by some to be a blemish in P.M. is the axiom of reduction. Here this is avoided by the distinction between classes and sets. Set theory is very useful for making models of various mathematical conceptions. Thus we easily get a model for ordinal numbers. So we give

204

Ch. 3 Predicate calculi

a short account of them. Fuller accounts are given in Cantor (1895, 7), Godel (1940), Bernays-Fraenkel (1958), Sierspinski (1928, 1958), Bachmann (1955). We have indicated in Ex. 41-4 inclusive how to develop the algebra of ordinals. Ordinals and cardinals were first invented by Cantor, but finite ordinals and cardinals were known to the Greeks. Ex. 43 (xi) is known as Cantors normal form for ordinals. When we introduce ordinals we immediately require a new axiom, namely the axiom of infinity, its object is to ensure that the ordinals we define are sets, so the process of ordinal construction can proceed. We largely follow Godel's (1940) account of ordinals and cardinals. The alephs are defined as certain ordinals, but there may be other cardinals that are incompatible with the alephs, if we define cardinals as classes of similar classes. It is at this point that we come across A.C., first introduced by Zermelo (1904). Prop. 32 is due to Cantor and Prop. 33 to Cantor and Bernstein (1905). Cantor's theorem immediately gives rise to C.H. and G.C.H. For many years the logical position of these and A.C. was unknown. Then Godel (1940), by constructing the constructible universe, was able to find an 'inner model' in which A.C, G.H. and G.C.H. were all satisfied provided that set theory itself is consistent. Thus A.C, C H . and G.C.H. are consistent with set theory provided set theory is consistent. The proof of this takes place in set theory. Godel's method of inner models, as shown by Shepherdson (1951,2,3), is incapable of showing that the negations of A.C, CH. or G.C.H. are consistent with set theory. Many years were to pass before Cohen (1963, 4, 5) developed an entirely new method for dealing with independence proofs, this was based on the denumerable model found by Skolem. This was a remarkable breakthrough, the original paper was couched in such strange terms that only the most resolute of professional logicians could understand it. Now however, monographs are appearing and the method is explained so that it is available to the general mathematician. There is a monograph by Cohen (1965), Another method of doing the same things was discovered by Solovay and Scott (1970). This proceeds by forming Boolean valued models and gives a generalization of twovalued truth. This method arose because it was noticed that the main feature of Cohen's 'forcing' method was the semi-order it gave rise to. Accounts of this method are given by Rosser (1969), Scott (1966a, 6), Jensen (1967). By suitable choice of Boolean Algebra models can be found in which A.C or C.H. or G.C.H. fail. The complete Boolean Alge-

Historical remarks to Chapter 3

205

bras required came in after Boole, they are discussed by Halmos (1963) and Sikorski (1960). We just show how to set up the model. The full proof that it is a model for set theory together with the independence proofs is given by Rosser (1969). Thus with Godel's result that A.C., C.H. and G.C.H. are consistent with set theory the final result is that A.C., C.H. and G.C.H. are independent of the other axioms of set theory. The proof takes place in set theory so that the result just stated only holds if set theory is itself consistent. This is parallelled in mathematical history by Caley's proof in Euclidean geometry that the axiom of parallels is independent of the other axioms of Euclidean geometry provided these themselves are consistent. The elimination of the e-symbol is due to Hilbert and two theorems about it are given in H-B (1934-6), we give one of these because it is tantamount to the consistency of A.C. with the other axioms of set theory. That is we give an effective method of converting a contradiction in set theory plus A.C. into a contradiction in set theory itself. The chapter closes with some topological definitions, many of which occur in the latter parts of P.M., they show again how useful the system S? is for talking about all sorts of mathematical concepts. Other works on set theory are: Suppes (1960), Halmos (1960), Skolem (1962), Sierpinski (1951), Fraenkel and Bar-Hillel (1958), Fraenkel (1953) with a complete bibliography to 1953 and Fraenkel (1946). EXAMPLES 3

1. Complete the demonstration of Prop. 6. 2. Complete the demonstration of Prop. 8. 3. Put into prenex normal form (Ax) CpxC(Ax')p'x'x(Ax")p"x% B(Ex) (Ex')pxx'(Ex')

(Ex)pxxf.

4. Put the «^>-proof of the prenex normal forms of C(Ax)

CpxpfxC(Ex)px(Ex)p'x,

C(Ex)

NNpxNN(Ex)px,

BK(Ax) Cpxpfx(Ax) Cp"xp'x(Ax) CDpxp"xp'x, into normal form.

206

Ch. 3 Predicate calculi

5. Obtain H-disj unctions for (Ex) (Ax') (Ex") DDpxx'xpxx'x"Npx"xx', (Ax) (Ex') (Ex") (Ax")DDpxx'x"Npxxx"px"x'x, (Ex) (Ax') (Ex") (Ex'") DDpx"x'xNpx"'x'x"px"'x'x. 6. Obtain semantic tableau for the statements of Ex. 5. 7. Find the equivalent of (Ax, x', x") (Ey) KDpxx'p'x'yCKp"x"xp'x'yDpyx"p"xx" according to Prop. 25. 8. Find which of the following are ^-theorems: (Ax, x', x") (Ex®\ x®>, x®)

DCpxx^x^px'x"x^Cpxx^x^ px"x'x®\ (Ax, x') (Ex", x'") CCpxx"px'x"'Cpx"xpx'"x'.

9. Find which of the following are J^-theorems: (Ex) (Ax', x") (Ex"1) GKpxx'px'x"Gpx'"xpx"x', (Ex) (Ax', x") (Ex1") CCpxx'px'x"Cpx'"x"px'x. 10. Reduce (Ex) (Ax', x") (Ex"1) (Axir) CCpxx'xCpx"x"'x'Gpx"xiYxf"pxiYx'x to the form given in Prop. 24. 11. Put in resolved form, using the e-terms (Ax)(Ex')(Ax"){x,x',x"}, (Ex) (Ax') (Ex") <j){x, x', x"}. 12. Define Re

Show that:

for

$ft'$"(x = x'),

&e

for

SAr&"(x = x").

9t?X = MX, 2«X= VxX, S {x, x'} = x U x', X{x} = x, £"{x} = x, C(X c X') (Q)X c CK(X c X') (X" c Xm)CX«X* c X'«XXm,

Examples 3

207

13. Show that (X x X')n {X" x Z i v ) = ( I n X") x (X'n X iv ), U X') = S Z U S I ' , X = SPX, CN£mX{X' c SS(Z x X')), X c P2X,

2 2 / = V,

14. Construct an applied predicate calculus of the second order which has exactly one constant predicate R which is binary. State axioms for order and express 'every non-empty class has a least member in the ordering R\ 15. Set up axioms for a group using one ternary predicate. 16. Find the Skolem $-normal forms and the Skolem F-normal forms for the following: C(Ax)px(Ex)px9 C(Ax) Cpxp'xC(Ex)px{Ex)p'x, C(Ex) (Ex')pxx'(Ex')

(Ex)pxx'.

17. Give the demonstration of Prop. 24, lemma (iii). 18. Show that the conditions (A), (B), (C) and (D) below are necessary in order that (Ag, £', £") (Er/) ^{£, £', g", TJ} be satisfiable. There is a nonempty set S of 4 x 4 tables which satisfy <j) such that: (A) If To is a table of S, v, v\ v" = 1, 2, 3, 4 then there is a table T of S such that: [T/1] = [TO/1] and [T/123] = [T0/v', v', v"]. (B) If To is a table of S then there is a table T of S such that: [T/1] = [TO1] and

T = [T/1114].

(C) If T 1; T 2 are tables of 2 then there are tables T, T", T" of S such that:

[T/l] = [T^l],

[T/2] = [T 2 /l],

T = [T/1224],

[T'/2] = p y i ] ,

[T'/l] = p y i ] ,

T' = [T/1214],

[T"/2] = [Tj/1],

[T'/l] = [T 2 /l],

T" = [T"/1134].

208

Ch. 3 Predicate calculi

(D) If Tl9 T2, T 3 are tables of S then there is a table T of such that: [T/l] = p y i ] ,

[T/2] = [T 2 /l],

[T/3] = [T 3 /l].

19. Apply Prop. 20 to decide which of the following are J^-theorems. (i) (Ex) (Ax') GBp'xpBp'x'p, (ii) B(Ex)Gpxp'x(Ex,x')Cpxp'x, (iii) (Ax') G(Ax) CpxCpx'p'xCpC(Ax)pxp'xf. 20. Apply Prop. 22 to decide (Ax, x', x") (Ey,yf)

Cpxx'Cpyx"DKpyx'py'x'Kpxx"py'x".

21. Apply Prop. 22 to decide (Ax, x') (Ex")

KCpx"xKCpxx'pxx"Cpxx"GNpxx'Kpx"xpx'x".

22. Find the binary SFC statement equivalent to (Ex) (Ax') (Ex") Cpxx'x"px'x"x as regards satisfiability according to Prop. 24. 23. Continue Ex. 22 to find a binary J^-statement with exactly one binary predicate which is equivalent to (22) as regards satisfiability according to Prop. 25. 24. Ditto for (Ex) (Ax')(Ex")\(Ax'") Cpxx'x"px'x"x'", (Ex) (Ax')(Ex", x"1) Cpxx'x"x'"x"px"xx'x"xm. 25. Demonstrate Prop. 12 for the system I^Q. 26. Demonstrate Prop. 14 for the system I^c. 27. Show that the singularly I!FC is decidable. The only predicates in the singulary I!FC other than / are singulary. 28. Show that B(X= Y)(Au)B(ueX)(ueY) and B(X = Y) (A U) (XeU) (YeU) are independent. [See Robinsohn, J.S.L. 4, 69.] 29. Show by formula induction X= Y B where F is free for X, Y in

Examples 3

30. Show and

209

B<j>{ Y}(AX) C(X = Y) 4>{X} B{Y}(EX)K(X = Y)${X}.

31. Show

B(XeY) (Eu) (u = X) K(u = X) (weY).

32. Show

X = y(yeX).

33. Show

(A£,',...,^)Bf y=S

where S is like y except for containing ijr at some places where y contains ^ and £', ...,£,(v) exhaust the variables with respect to which those occurrences of 0, i/r are bound in y, S.

where \]r is like 0, except for containing 8 at some places where $ contains y and £',..., £^p) exhaust the variables with respect to which there occurrences of y, S are bound in
jy 7 / V

T7"\

Rel (X, Y)

/ A

\ ID V

Tr

(Au, v) BuXvu Yv X=Y

36. Show

{u, v} = {x, y) DK(u = x)(v = y) K(u = y) (v = x)

37. Show

(u,v) = (x}y) K(u = x)(v = yY

38. Show

(Ay) (y = Fy).

39. Show

(Av) B(v = y) vXx y = X'x

40. Show (Ax)BKUnX{xeSiX){Eu)(Av)B{u = v {Ax)BKVnX{xe9X) (Ay)B(y = X'x)yXx, Y&nX (Aw)B(weY) (Eu) K{ueX) (w = Y&nX

Z&nX

(Y'u,u)y

(Au)C(ueX)(Y'u = Z'u) Y =Z

210

Ch. 3 Predicate calculi

41. Defining the sum a + b of two ordinals a, b as the ordinal isomorphic as regards order to the order type obtained by sticking the order type b at the end of the order type a, show that: (i) If 0 c i < c 2J then6 2 < 6X. (vi) If a = 6 + c, c is called a remainder of a and 6 is called a segment of a. Show that the number of remainders of an ordinal is finite. (vii) An ordinal a is called decomposable if a = b + c, 0 < b, c < a. Otherwise indecomposable. Show that the least positive remainder of a positive ordinal is indecomposable. (viii) If cx < c2 are both remainders of an ordinal a, then cx is a remainder of c2. (ix) The least positive remainder of an ordinal a is a remainder of every other remainder of a. (x) The only positive remainder of an indecomposable ordinal is itself. (xi) The only positive indecomposable remainder of an ordinal is its least positive remainder. (xii) If c is indecomposable and b < c then b + c = c. (xiii) If c > 0 and b + c = c whenever b < c then c is indecomposable. (xiv) Any ordinal a is the sum of a finite decreasing sequence of decreasing indecomposable ordinals. (xv) If

a = c1 + c2+...+cn,c1

> c2 > ... > cn

and

cl9c29...,cn

in-

decomposable then cx is the greatest indecomposable ordinal < a. (xvi) If A is a set of indecomposable ordinals then HA is indecomposable. (xvii) Every ordinal can be uniquely represented as a finite sum of non-decreasing indecomposable ordinals. 42. Defining the product ab of two ordinals a, b as the ordinal order isomorphic to the set of ordered pairs (c9d} c < a,d 0 and axb = a2b. (iv) If a = be, 0 < 6,1 < c, then b < a and c ^ a.

Examples 3 (v) If abx < ab2 then bx < b2. (vi) If axb < a2b then ax < a2, (vii) If ab± = ab2, 0 < a then bx = b2. (viii) lie < ab t h e n u n i q u e l y c = ab1 + d,b1
211

< a.

(ix) If 0 < a then b = ac + d, d < a, uniquely. (x) If c is indecomposable, 1 < c then ac is indecomposable, (xi) If 0 < a then the least indecomposable ordinal greater than a is aco.

(xii) If c is indecomposable and positive then the next greater indecomposable ordinal is ca). (xiii) Every indecomposable ordinal is divisible on the left by every lesser positive ordinal and the quotient is indecomposable. (xiv) An ordinal is prime if it is greater than unity and is different from the product of any two lesser ordinals. Show that every ordinal > 1 is the product of a finite number of primes. (xv) Show that the number of right divisors of an ordinal is finite. 43. Defining ba for ordinals a, b as the ordinal order isomorphic to the order type of functions over a with a finite number of non-zero values in b ordered by last differences, show that: (i) This order type is that of an ordinal. (ii) If 0 < a < 6, 1 < c then ca < cb. (iii) If 0 < a < 6, 0 < c then ac < bc. (iv) ca+b = ca.cb, 0 < a, 6, c. (v) If b is a limit ordinal, 1 < a then ab is the limit of all ordinals 0 a for c < b. (vi) (aa)c = abc, 0 < a, 6, c. (vii) o)a is indecomposable for a > 0. (viii) If 0 < a, 1 < c then a < ca. (ix) If 0 < d, 1 < c then there is exactly one ordinal a such that ca ^ d < ca+1.

(x) Every indecomposable ordinal ^ co is of the form a)a for some a > 0. (xi) Every ordinal a > 0 can be uniquely expressed in the following normal form: „ d = fc)ci. 71^ + (i) 2 . 7l2 + . . . + (if™. 7lm

w h e r e cx> c2> ... > cm a n d nx,n2, ...,nm a r e i n t e g e r s . 44. A n e-ordinal is a n o r d i n a l e w h i c h satisfies e = co6. S h o w t h a t : (i) I f e0 = co + a)*0 + w(6>CU) + ... t h e n e 0 = 6>e«.

212

Ch. 3 Predicate calculi

(ii) e0 is the least e-ordinal. (iii) If 1 < c < e0 then e0 = ce«. (iv) If 0 < c, c0 = c, cn+1 = G)cn then limcn is the least e-ordinal not less than c. 45. Show that there exist ordinals which satisfy a = o)a. [Consider c0 = ^> cn+i = Mcn>

c

= limcn.]

46. Show that: (Ax, x') D(x = x') KxNx' is consistent in a universe of one element but is inconsistent in any other universe. 47. H{z', ...,2(7r)} is a conjunction of identities and differences zv = zp zv 4= z^ which contains exactly one of these for each pair of v,ja,l < v, /i ^ 7T. By using the disjunctive normal form show that a statement F {ft, ft > X, •••> —^\

--->z{n\ x',...,x(0)}

can be expressed in the form

A

2 HK{z',..., z(7r)} FK{cj), ..., = , J, ?}, where 5 stands for 2',..., z(7r) and j for /c = l

x',...,xf®. Hence show that (E%)(A%)F{$,..., = , J , j } is equivalent to one of (Z?s) (^l j) HKFK for some /c, 1 ^ K ^ A. 48. Show that DN(Ax) (Ey) <j>{x, y] (Ex, x', x", x'") (DKKKKK{x, x'} (j>{xf, a;"} 0{xr/, x"'} <£{x, x"} {x, x'"}
x")

is generally valid. Ackermann J.8.L. ai (1956), p. 197. 49. Show that (i) (Ex) DN4>{x, x) {Ay) NK{x, y) N{y, y}, (ii) N(Ax) K<j>{x, x} (Ey) K<j>{x, y) N<j>{y, y), (iii) N(Ax) DK<j>{x, x} (Ey) K{x, y) Ntfy, y} K<j>{x, x) (Ey) K<j>{y, x)

are generally valid. Oglesby (1962).

Chapter 4 A complete, decidable arithmetic. The system Ao

4.1 The system Aoo In this chapter we construct the formal system Aoo. It is a very simple arithmetic with familiar fundamental concepts. These are: the natural number zero, the successor function, the operation of repeatedly applying a function, the operation of forming functions by abstraction, equality and inequality between numerical expressions, and the logical connectives, conjunction and disjunction. The atomic statements are equations and inequations between numerical terms, compound statements are built up from atomic statements by conjunction and disjunction. Negation, material implication and material equivalence are definable, but existential quantification and universal quantification are unrepresentable. We give definitions of AQ0-truth and of A00-falsity for closed AOostatements, and show that they are exclusive properties. We also show that a closed A00-statement is A00-true if and only if it is an Aoo-theorem. Thus the system Aoo is consistent in the sense that Aoo-theorems are Aoo-true; and is complete in the sense that A00-true AOo-statements are Aoo-theorems. We give a procedure which applied to a closed A00-statement will terminate and tell us whether it is A00-true or is A00-false. Thus the system Aoo is decidable. 4.2 The Aoo-rules of formation To construct the system Aoo we first list the A00-signs and attach a type to each proper A00-symbol and give each Aoo-symbol a name which will assist the reader in understanding how the system was first conceived. Parentheses round type symbols are usually omitted by association to the left as explained in Ch. 1. (See table overleaf.) The required unending sequence of variables is obtained by repeatedly attaching primes, thus: x, x', x", .... The table order of the proper and improper symbols is called their lexicographic order. In the system Aoo [ 213 ]

214

Ch. 4 A complete, decidable arithmetic. The system Aoo symbol 0 8 = =1= & V

name

type I

11 Oil

oil 000 000

J

ui(iu)

X

i

zero successor function equality inequality conjunction disjunction iterator operator variable

The improper isymbols are, as usual: X ( )

abstraction operator left parenthesis right parenthesis generating sign

the natural numbers are represented by certain formulae of type i called numerals. These formulae are defined by the following rules: (i) 0 is a numeral, (ii) if v is a numeral, then so is (Sv), (iii) these are the only numerals. Thus 0, (SO), (8(80)), (8(8(80))), ... are the numerals. Note that a numeral is a formula, distinct numerals are distinct formulae. An Aoostatement is an A00-formula of type o, according to the universal rules as given in Ch. 1. The only abstracts allowed are of types u, ui, uu, etc., i.e. (kx> P> 7> $ £,7j,£,v >tyiX> M p ,
to denote undetermined numerals, to denote undetermined formulae of type i, called numerical terms, to denote undetermined variables of type i, to denote undetermined statements, to denote undetermined functors of type a, ui, etc.

4.2 The Aoo-rules of formation

215

In the last case the type can be introduced as a subscript if desired. From now on we usually omit the concatenation sign. Thus = otfi stands for an undetermined equation between numerical terms, &
(<* = /?) for

D113

(&>ft)for

D114

(?W^) for

= a/?,

(a 4= fi) for

+ a/?,

&$ft,

v#\

We shall frequently use the parenthesis convention, thus: ^ Vftv x stands for ((<}) vft)vx)

an

d hence for ((V ((V (j>) ft))x)>

) ((V ft)x))Note that, by the parenthesis convention, = a/3 stands for ((= a)/?), but the defined expression (a = /?) has its complete set of parentheses. As in Ch. 1 we shall use ^{F}, oc{T}, etc. as statement-forms, and numericalterm-forms respectively, <£{£} stands for an undetermined statement containing an undetermined variable free, this arises from the statementform 0{F} by substitution. We shall also use the notation ^{©}, ^{?}, as explained in Ch. 1, here £ denotes an ordered set of variables. Sometimes we subscript type symbols to F, F',... in formulae-forms to denote that substitutions will only be made for these types. Functors of types u, iu9 and so on will be called functions of natural numbers or simply functions. The iterator symbol J is such that JpoifS is a numerical term whenever a, /? are numerical terms andp is a function of type ut9 Jp is a function of type ui. Thus «/ converts a function of type in into another function of type in. 4.3.

The AoO-rules of consequence

The system Aoo has the following axiom schemes: Ax 00 .1

(oc = a),

Axoo.2.1

Offa#0)f

216

Ch. 4 A complete, decidable arithmetic. The system Aoo

Ax 00 . 2.2

(0 4= Sot),

Axoo.3.1 Axoo. 3.2

where a, ft are closed numerical terms and p is a closed function of type in. i... i is the result of striking out each parenthesis and the zero in the Sn- times

formula (Sn) and then replacing each occurrence of S by a corresponding occurrence of i and replacing the parentheses by association to the left. Ax 00 .4.1 Ax oo .4.2

(X£. p{$) PP'... P* = p{p}pf... /?<*>, (7

where /){£} is of type i... i and both sides are closed and £, £' fail to occur Sn-timea

free in p{TL] and I \ is free for £, £' in p{Tt}. When written in full the parentheses are put back by association to the left, viz.: This axiom allows us to apply an argument to a function. We really only need functions of at most two arguments but it is sometimes useful to have functions of any number of arguments. This makes these two axioms more complicated. In them p{£,} is of type i... i so it must be of the form: where a{£, £',£", ...,£(7r)} is a numerical term, or it could be; (Xg'(Xg*(...(Xg<'-»5)...)) or or

(H'(H"

with appropriate modifications if n = 0,1,2. By repeatedly applying Axoo- 4.1, 4.2 we can change any Ef® to a new variable. The first axiom scheme states a familiar property of equality and the two parts of the second axiom scheme state a familiar property of the successor function. We require both parts because if we worked with only one part then some properties of inequality which we require would fail. The third axiom scheme shows how the iterator operator acts. This may become clear if Ax00. 3.2 is written in ordinary mathematical notation: suppose that / is a function of two arguments, then {{Jf)m{n+l)=f{n,{Jf)mn)).

4.3 The A00-rules of consequence

217

If we write g[n, m] for ^ffmn we obtain: g[n + l,m]

=f[n,g[n,m]].

g[3,m]=f[2,g[2,mj\

Thus

since g[0,m] = m by Axoo. 3.1. In general g[n,m] = / [ » , / [ » - 1 , [ f [ » - 2 , ...,/[2,/[l,/[0,m]]]... ]]]]. The system Ao0 has the following rules of procedure: ^{a} v
p

where a and /? are closed numerical terms and the A00-statement form
where a and J3 are closed numerical terms and co is a closed A00-statement and is subsidiary as in R 1, (a #= /?) and (Sot 4= #/?) are the main formulae. Note that since a and /? are closed in R 1 then a variable whether free or bound is unaffected by applications of R 1. The remaining rules are labelled in a different manner because they are some of the rules of &c. In listing them, except in one case, we omit the condition that the A00-statements in them be closed because they can only be used in A00-proofs when this is so. In the exceptional case the condition must be stated otherwise free variables could be introduced into an Aoo-proof. I.

Remodelling rules 0)' V
w'vfv^vw' II.

permutation

Building rules (a) -X— dilution

(b'\ ^yC°

^V0)

composition

218

Ch. 4 A complete, decidable arithmetic. The system Aoo

In II (a) (j) is closed. In II (&') the order of the premisses is immaterial. The Aoo-statements a>, a)' are subsidiary and may be omitted, the other Aoo-statements are the main formulae. The A00-statement x is secondary and must be present. We have omitted parentheses by association to the left and the outer pair is usually omitted. The rules are known by the names beneath them. 4.4 Definition of A00-truth We say that a closed numerical term 7 determines a numeral v under the following conditions: (i) 7 is v. (ii) v can be obtained from 7 by replacements of the following kinds: (a) replace an occurrence of ^paO by a corresponding occurrence of a; (b) replace an occurrence of Jrpoc(Sj3) by a corresponding occurrence of (c) replace an occurrence of X£.p{£}/?/?'.../?(7r) by a corresponding occurrence of p{/?}/?'... /?(7r); (d) replace an occurrence of X£./?{£}/?/?'... /?(7r) by a corresponding occurrence of X£' ./ where £ and £' fail to occur free in p{TL} and TL is free for £ and £' in p{Tt}. If 7 determines the numeral v then 7 = v is an Aoo-theorem and the above replacements give a special type of Aoo-proof of 7 = v. We say that an A00-statement is A00-true if and only if it satisfies the following conditions ^ 0 . (i) is an equation between closed numerical terms and both terms determine the same numeral, (ii) (j) is an inequation between closed numerical terms and these determine distinct numerals, (iii)

4.9 Negation in the system Aoo The system Aoo lacks a negation sign, nevertheless negation can be Aoo-defined as follows: D 115 N[(' 0/'] for the result of replacing: = * & V throughout 0. Similarly material can be A00-defined: D116

["^->f"]

T> 117

["<-> f " ]

by * by = by v by & implication and material equivalence

for for

We use square brackets and inverted commas in this type of definition to denote that (j) fails to occur in N[""]"] is (j). Thus our definition of negation is classical and different from the intuitionist negation. 7. AOo is consistent and complete with respect to negation. We have to show that if <j) is a closed A00-statement then exactly one of is an AoO-theorem by Prop. 3. If "] is A00-true and so by Prop. 3 is an A00-theorem. Thus
228

Ch. 4 A complete, decidable arithmetic. The system Aoo

We can divide the closed A00-statements into two classes so that one class consists exactly of the negations of the other class. To do this we enumerate the Aoo-formulae by length and order those of equal length lexicographically. We then run through the list testing for being an Aoo-statement, when we come across a closed A00-statement we put it in the first list provided its negation is absent from the segment of that list so far obtained, otherwise we place it in the second list. We could obtain other definitions of AOo~truth and AOo-falsity which preserve the property of exclusiveness and such that N["
rule

o)r &
Destruction rules l l

a

v

X' concentration

,
(<j> V i/r) & a) dispersion

a 4= /? & a)

R2' £^ In these rules OJ and cof are subsidiary and can be absent, x is secondary and must be present. The Boo-axioms are A00-false and the B00-rules preserve A00-falsity. Thus the Boo-theorems are A00-false and so B oo is consistent with respect to A00-falsity. On the other hand a closed Aoo-false Boo-statement is a Boo-theorem. Thus B oo is complete with respect to A00-falsity. To show that B oo is complete with respect to Aoo-falsity we use formula induction.

7

a

(vi) 0 < x, (vii) a

a n d

and

oc < p AJ[x,x']+A2[x'',xlff].

3. Prove the following E-theorems or derived rules: (i) x x B2[x, x'] = x' x B2[x, x'], (ii) pO = o-O p(8x) = cr(Sx) px = ax

510

Ch. 10 Induction (iii) x < 2X,

(iv) px < ex 2 px" < 2
(vi) px < p(Sx) x ^ px (vii) Max[x, x'] -^x" = ikfao;[a;^-x", x' ^ xff], (viii) A2[2xx,2xx' + l] = I. 4. Call an A-statement decidable if either it or its negation is an A r theorem. Show that if ^ is decidable then we have a method for deciding its truth value. 5. Let

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