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 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
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}, 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 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 •". 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 " = KDr. Otherwise we note the place where (j) is introduced into the ^ o -proof of x{fi} and then introduce i/r instead. If 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 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 is D$'$" and the result holds for ^' and $". We have ^ o -proofs of DcoDfi
52

Ch. 2 Propositional calculi

whence by dilution and permutation

similarly with <j> and ft interchanged, and the result follows. If x{

•}

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 DDfrNjrNNN^) 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 ]. 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