Elements of Combinatory Loqic REOERIC B. FITCH
::Omb.natory logic has long been thought o hOld great philosophical prom...
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Elements of Combinatory Loqic REOERIC B. FITCH
::Omb.natory logic has long been thought o hOld great philosophical promise. Not mill now. however, has the fulfillment >I that promise been so seriously underaken. and woth wch Impressive results. Professor Fitch here develops the >ystem Q in order to provide one or the nost complete and detailed accounts to Jete ol the construction of a combinatory ogle. The system is provably consistent, •nd It contains all the usual connec· lves and quantifiers of logic, as well as combinators or combinatory logic. Mr. Fotch os especially concerned to develop a logoc capable ot relevant application to philosophical issues ranging from the toundahons of mathematics to the proper lormulalion and analysis of epistemological, psychological. and sociological concepts. A discussion of the method ol subordinate proofs introduces the reader to the rules most basic to working In system Q. Successive chapters cover tuncllon as dealt with in this manner, rules proper too, and the details of tho application ot 0 to the foundations of arithmetic and the logic or circuits. The method or combinatory logic pre· sented here is one both or great power and tar-reaching significance. Despot& the extent of its Implications. however, Professor Fitch's presentation has a clatoty and consistency which will appeal to students In the field as well as to professional logicians. Frederic B. Fitch Is Sterflnll Professor of Philosophy at Yale Universoty.
ISBN -30Q.o1523-2
:Ray
Nakabayashi
Elements of Combinatory Logic
FREDERIC B. FITCH
NEW HAVEN AND LONDON, YALE UNIVERSITY PRESS, 1974
TABLE OF CONTENTS Page Copyright @ 1974- b,y Yale University. This book may not be reproduced, in whole or in part, in an,y form (except by reviewers for the public press), without written permission from the publishers. Library of Cnngress catalog card number: 73-86892 International standard book number: 0-300-01523-2 F.1·..inL..eU ln
Co.,
For~e
Lhe UniLeU fiLaLet::> oi' .iunerlca Uy ihe 1~iu1Ti::i,Y FTiat..i!l!:'; Villa~e,
Mass.
Published in Great Britain, Europe, and Africa by Yale University Press, Ltd., London. Distributed in Latin America by Kaiman & Polan, Inc., New York City; in Australasia and Southeast Asia by John Wiley & Sons Australasia Pty. Ltd., Sydney; in India by UBS Publishers' Distributors Pvt., Ltd., Delhi; in Japan by John Weatherhill, Inc., Tokyo.
PREFACE .... · · · · • · · · · · · · · · · · · · · · · · · · · · · · · · • · · · • · · · · · · · · · · ·vii INTRODUCTION. THE METHOD OF SUBORDINATE PROOFS ..••......••• l CHAPTER l. FUNCTIONS 1. Remarks on Functions ......•••..•...•......... · • · • .29 2. Functional Notation in Combinatory Logic •..•.•.... 3l 3· The Class of Q-Functions ................••....••.. 33 CHAPTER 2. BOOLEAN CONCEPTS AND IDENTITY 4. Rules for Negation, Conjunction, and Disjunction .• 37 5· Rules for Identity •.••................••....•.•.•. 41 6. Boole~n Id~!lLlLl~b·••••••••••••• .....•.•.......... 51 CHAPTER 3· COMBINATORS 7. The Combinators I and K..•..•.................•... 53 8. The Combinators C and T ••..•.•......• · • · · · · • • · • ··.57 g. The Combinators Band w........................... 60 10. Abstraction •...........•....••..•.•..... • · · • · · · • · • 68 CHAPTER 4. FOUNDATIONS OF ARITHMETIC 11. Multiplication, Addition, and Exponentiation ••.•.. 78 12. The Natural Numbers .•.•••...••..•....•.•.•. • •. • · • .83 13. Universality and Existence ..............•..•.• • • • .96 14. Less-or-Equal and Less ..•......•••....•..••.••••. l04 CHAPTER 5. RECURSION AND CIRCUITS 15. Some Special Functions ••....•.•••...••.•..• • • • • · .114 16. Recursive Functions •••.•...•..•••...••..••.•••.•• ll9 17. Self-Referentially Defined Functions •.•.••....•.• l24 18. Sequential Circuits ••.•.•...•....•....••..•.•.•.. 128 CHAPTER 6. A THEORY OF PROPOSITIONS 19. The System QD ....•.••.•.••.......•.••....••....•. 140 20. Definite Classes and Relations ...•...••...•....•. 146 BIBLIOGRAPHY ....•.•.....................•................ 155 INDEX TO ABBREVIATIONS FOR RULES ................•........ 159 v
TABLE OF CONTENTS Page Copyright @ 1974- b,y Yale University. This book may not be reproduced, in whole or in part, in an,y form (except by reviewers for the public press), without written permission from the publishers. Library of Cnngress catalog card number: 73-86892 International standard book number: 0-300-01523-2 F.1·..inL..eU ln
Co.,
For~e
Lhe UniLeU fiLaLet::> oi' .iunerlca Uy ihe 1~iu1Ti::i,Y FTiat..i!l!:'; Villa~e,
Mass.
Published in Great Britain, Europe, and Africa by Yale University Press, Ltd., London. Distributed in Latin America by Kaiman & Polan, Inc., New York City; in Australasia and Southeast Asia by John Wiley & Sons Australasia Pty. Ltd., Sydney; in India by UBS Publishers' Distributors Pvt., Ltd., Delhi; in Japan by John Weatherhill, Inc., Tokyo.
PREFACE .... · · · · • · · · · · · · · · · · · · · · · · · · · · · · · · • · · · • · · · · · · · · · · ·vii INTRODUCTION. THE METHOD OF SUBORDINATE PROOFS ..••......••• l CHAPTER l. FUNCTIONS 1. Remarks on Functions ......•••..•...•......... · • · • .29 2. Functional Notation in Combinatory Logic •..•.•.... 3l 3· The Class of Q-Functions ................••....••.. 33 CHAPTER 2. BOOLEAN CONCEPTS AND IDENTITY 4. Rules for Negation, Conjunction, and Disjunction .• 37 5· Rules for Identity •.••................••....•.•.•. 41 6. Boole~n Id~!lLlLl~b·••••••••••••• .....•.•.......... 51 CHAPTER 3· COMBINATORS 7. The Combinators I and K..•..•.................•... 53 8. The Combinators C and T ••..•.•......• · • · · · · • • · • ··.57 g. The Combinators Band w........................... 60 10. Abstraction •...........•....••..•.•..... • · · • · · · • · • 68 CHAPTER 4. FOUNDATIONS OF ARITHMETIC 11. Multiplication, Addition, and Exponentiation ••.•.. 78 12. The Natural Numbers .•.•••...••..•....•.•.•. • •. • · • .83 13. Universality and Existence ..............•..•.• • • • .96 14. Less-or-Equal and Less ..•......•••....•..••.••••. l04 CHAPTER 5. RECURSION AND CIRCUITS 15. Some Special Functions ••....•.•••...••.•..• • • • • · .114 16. Recursive Functions •••.•...•..•••...••..••.•••.•• ll9 17. Self-Referentially Defined Functions •.•.••....•.• l24 18. Sequential Circuits ••.•.•...•....•....••..•.•.•.. 128 CHAPTER 6. A THEORY OF PROPOSITIONS 19. The System QD ....•.••.•.••.......•.••....••....•. 140 20. Definite Classes and Relations ...•...••...•....•. 146 BIBLIOGRAPHY ....•.•.....................•................ 155 INDEX TO ABBREVIATIONS FOR RULES ................•........ 159 v
PREFACE The system of combinatory logic presented in this book is called the system Q• It contains not only the usual combinatory operators and sentential connectives, but also completely unrestricted quantifiers. This system is applicable to a wide variety of subject-matters. In the first place, Q is a sufficiently strong system to serve as a foundation for elementary arithmetic and to do so in a way that avoids a Gtldel incompleteness of elementary arit.hmet.lc. I11 uL'ue1.· wuJ.:U.::,,
~
c.uuto..i.u.b ctll u.f
~lemt:u"La..cy
a-
rithmetic and not merely a truncated part of it. In the second place, Q is also sufficiently strong to serve as a foundation for a substantial part of mathematical analysis. The details of this are not presented in the present book but they would be similar to the details in the writer's paper, "A Demonstrably Consistent Mathematics." (SeeBibliography.) Quantifiers over the real numbers can be provided for by use of the concept D of Chapter 6. In the third place, the system Q can be employed for representation of sequential circuits and computers. This is shown for sequential circuits in Section 18, and previously in the writer's paper, "Representation of Sequential Circuits in Combinatory Logic," and for computers in two papers by R. J, Orgass and the writer, "A Theory of Computing Machines" and "A Theory of Programming Languages." In these three papers the systems of logic used differ only slightly from Q· Finally, it appears that the system Q can be used in formulating and analyzing fundamental philosophical concepts,
vii
PREFACE The system of combinatory logic presented in this book is called the system Q• It contains not only the usual combinatory operators and sentential connectives, but also completely unrestricted quantifiers. This system is applicable to a wide variety of subject-matters. In the first place, Q is a sufficiently strong system to serve as a foundation for elementary arithmetic and to do so in a way that avoids a Gtldel incompleteness of elementary arit.hmet.lc. I11 uL'ue1.· wuJ.:U.::,,
~
c.uuto..i.u.b ctll u.f
~lemt:u"La..cy
a-
rithmetic and not merely a truncated part of it. In the second place, Q is also sufficiently strong to serve as a foundation for a substantial part of mathematical analysis. The details of this are not presented in the present book but they would be similar to the details in the writer's paper, "A Demonstrably Consistent Mathematics." (SeeBibliography.) Quantifiers over the real numbers can be provided for by use of the concept D of Chapter 6. In the third place, the system Q can be employed for representation of sequential circuits and computers. This is shown for sequential circuits in Section 18, and previously in the writer's paper, "Representation of Sequential Circuits in Combinatory Logic," and for computers in two papers by R. J, Orgass and the writer, "A Theory of Computing Machines" and "A Theory of Programming Languages." In these three papers the systems of logic used differ only slightly from Q· Finally, it appears that the system Q can be used in formulating and analyzing fundamental philosophical concepts,
vii
PREFACE
viii
and perhaps also fundamental psychological and sociological concepts, by methods like those of the writer's paper, "Combinatory Logic and Wbitehead's Theory of Preheksions." The concept of truth, as an example, can be defined for the system Q in a way that avoids the difficulties pointed out by Tar ski. Although this is not done in the prec;ent book, the
INTRODUCTION
method is the same as that described in the writer's paper, "Universal Metalan12:uages Jar Philosophy," except that the hierarchy oi'
ne~ations
THE METHOD OF SUBORDINATE PROOFS
can be dispensed with if the concept
D oi' Chapter 6 is used. Another philosophically important aspect of the system Q is the presence in it oi' unrestricted quantifiers. This importance arises from the fact that such
0.1. This introduction gives a self-contained account
quantifiers cannot even be discussed philosophically except
of a relatively simple method for constructing proofs in
in a lan.,uave that already,
symbolic logic. Familiarity with this method is presupposed
in effect, employs them.
It
should also be mentioned that the system Q, and the extension -+'
V-4-
~-~-
..LV
-!-+----1-----l
..l.llU_l_UU.UI...CU.
~--
.L.U
_..,, __
!
VLld.l-JWC.l
_...
u,
d._!J_fJed.L'
'
l-U
,
Ue
~~
We.l.L
.
'
'
bU_l_LeU
'
101'
representing salient features of the deep structure of English. The writer has reseqrch intereEts in this direction at present. An earlier version of this book took the form of a technical report in a research
pro~ram
on Problem Solving and
in the main body of this book. The method is also described in the auLhor' c Uuuk, SywUul.L--.: Luo.i.c (New
lu.L·k~
Rul..i.ctlU.
F.lt::b;:;,
1°52). 0.2. The concept of implication discussed in this introduction is not used in the system Q of the present book, but it is included here in order to make the presentation of the method of subordinate proofs more complete. Although the
Social Interaction, sponsored by the Office of Naval Re-
concept of implication is not needed in the system Q, a mod-
search, Group Psychology Branch, Contract No. SAR/Nonr-60CJ
ified form of implication could be employed in Q by defining
(16). New Haven, Conn., November lCJ60. The writer has used
'[p ~ q]'
(the sentence that asserts that p implies q) as
this earilier version successfully in lo"ic courses of in-
'[-p V q]'
(the sentence that asserts that 'p' is false or
troductory or intermediate level in Yale University.
'q' is true) .
New Haven, Connecticut December 28. 1973
0.3. Hereafter the letters 'p',
Frederic B. Fitch
q',
'r',
's', and 't'
will represent arbitrary sentences. It is assumed that if 'p' and 'q' are sentences, so are '[p q"),
'[pV q]'
"or"),
& q]'
(read "p and
(read "p or q" and using the non-exclusive
'[p ~ q]'
(read "if p then q," or less accurately but
often more conveniently as "p imnlies q"), and '-p'
(read "p
is false," or equivalently as "it is not the case that p"). 0. 1+. 'q',
'[p & q]'
'[pV q]'
is called the coniuncti_Q12 of 'p' with
is called the dis,iuns:t_i_lll.l of 'p' with 'q', l
PREFACE
viii
and perhaps also fundamental psychological and sociological concepts, by methods like those of the writer's paper, "Combinatory Logic and Wbitehead's Theory of Preheksions." The concept of truth, as an example, can be defined for the system Q in a way that avoids the difficulties pointed out by Tar ski. Although this is not done in the prec;ent book, the
INTRODUCTION
method is the same as that described in the writer's paper, "Universal Metalan12:uages Jar Philosophy," except that the hierarchy oi'
ne~ations
THE METHOD OF SUBORDINATE PROOFS
can be dispensed with if the concept
D oi' Chapter 6 is used. Another philosophically important aspect of the system Q is the presence in it oi' unrestricted quantifiers. This importance arises from the fact that such
0.1. This introduction gives a self-contained account
quantifiers cannot even be discussed philosophically except
of a relatively simple method for constructing proofs in
in a lan.,uave that already,
symbolic logic. Familiarity with this method is presupposed
in effect, employs them.
It
should also be mentioned that the system Q, and the extension -+'
V-4-
~-~-
..LV
-!-+----1-----l
..l.llU_l_UU.UI...CU.
~--
.L.U
_..,, __
!
VLld.l-JWC.l
_...
u,
d._!J_fJed.L'
'
l-U
,
Ue
~~
We.l.L
.
'
'
bU_l_LeU
'
101'
representing salient features of the deep structure of English. The writer has reseqrch intereEts in this direction at present. An earlier version of this book took the form of a technical report in a research
pro~ram
on Problem Solving and
in the main body of this book. The method is also described in the auLhor' c Uuuk, SywUul.L--.: Luo.i.c (New
lu.L·k~
Rul..i.ctlU.
F.lt::b;:;,
1°52). 0.2. The concept of implication discussed in this introduction is not used in the system Q of the present book, but it is included here in order to make the presentation of the method of subordinate proofs more complete. Although the
Social Interaction, sponsored by the Office of Naval Re-
concept of implication is not needed in the system Q, a mod-
search, Group Psychology Branch, Contract No. SAR/Nonr-60CJ
ified form of implication could be employed in Q by defining
(16). New Haven, Conn., November lCJ60. The writer has used
'[p ~ q]'
(the sentence that asserts that p implies q) as
this earilier version successfully in lo"ic courses of in-
'[-p V q]'
(the sentence that asserts that 'p' is false or
troductory or intermediate level in Yale University.
'q' is true) .
New Haven, Connecticut December 28. 1973
0.3. Hereafter the letters 'p',
Frederic B. Fitch
q',
'r',
's', and 't'
will represent arbitrary sentences. It is assumed that if 'p' and 'q' are sentences, so are '[p q"),
'[pV q]'
"or"),
& q]'
(read "p and
(read "p or q" and using the non-exclusive
'[p ~ q]'
(read "if p then q," or less accurately but
often more conveniently as "p imnlies q"), and '-p'
(read "p
is false," or equivalently as "it is not the case that p"). 0. 1+. 'q',
'[p & q]'
'[pV q]'
is called the coniuncti_Q12 of 'p' with
is called the dis,iuns:t_i_lll.l of 'p' with 'q', l
2
ELEMENTS OF COMBINATORY LOGIC
0
[p ~ q] 0 is called the conditional of 0 p 0 with 0 q 0 , or the implication of 0 q 0 by 0 p 0 , and 0 -p 0 is talled the negation of 0 p 0 or the~ of 0 p 0 • (The single quotation marks are not part of the expressions being discussed. The reference is rather to the expressions between such single quotation marks, or to expressions that these expressions represent.) 0.5. The sentence opo is said to be a direct ~ 0 ~uence of the sentence [p a q] 0 by the rule of conjunction elimination (conj elim). The sentence oqo is likewise said to be a direct consequence of the sentence 0 [p a q] 0 by the rule of conjunction elimination. 0.6. The sentence 0 [p a q] 0 is said to be a direct consequence of the pair of sentences 0 p' and 0 q 0 by the rule of conjunction introduction (conj int). 0.7. The sentence '[pVq] 0 is said to be a direct consequence of the .sentence 'p' by the Lu.le uf
~.i~j Ull(.;
t.lon
1£-
troduction (dis int). The sentence 0 [p V q] 0 is likewise said to be a direct consequence of of the sentence 'q 0 by the rule of disjunction introduction. 0.8. The sentence 0 q 0 is said to be a direct consequence of the pair of sentences 0 p 0 and 0 [p ~ q] 0 by the rule of implication elimination (imp elim). This rule is also called~ Eonens (m p). 0.9. A ~-~ Eroof is defined to be a finite sequence of sentences, written as a vertical column, and such that each sentence of the sequence either is explicitly designated as an ~pothesis Qf !£t sequence or else is in fact a direct consequence (in one of the senses referred to above) of one or a pair of preceding sentences of that sequence. There may be one or more hypotheses of the sequence, and they are designated by writing 0 hyp 0 after each of them. Also, it will be customary to group hypotheses together at the beginning of the sequence and to draw a short horizontal line separating them from the other sentences of the column. A vertical line is drawn to the left of the column or sequence of sentences, indicating that it is a proof.
3
THE METHOD OF SUBORDINATE PROOFS
o.lo. The following columns are proofs. After each sentence there is given an indication either that the sentence is an hypothesis or that it is a direct consequence of specified preceding sentences by one of the rules of direct consequence. The sentences of each column are numbered by numbers on the left so that reference to them is possible. Outermost square brackets are omitted for economy of notation.
Pl
r 2
pV q
3 P
P2
hyp
a
1~ P a 2 p
[p v qJ
hyp l, conj elim 1, conj elim
q
3 q P3
1 ~P 2 p 3 q 4 q
a q_
2
aq
r·q] 3
4
5 P5
1 2
3 4
5 6
7 P6
1 2
3 4
5 6
P r p r
1, dis int 1, 2, conj int
hyp 1, conj elim 1, conj elim 2, 3. conj int
aP
ar
ap a Cq a r]
hyp 1, conj elim 1, conj elim 2, conj elim 2, 3. conj int
p p q q r q [q
hyp 1, conj elim 1, conj elim 3. conj elim 3. conj elim 2, 4, conj int 5. 6, conj int
s r r p q q
hyp hyp
ar aP a pJ a r ar Cp a qJ ~
aq ar
1' conj elim 1, 3. m p 4, conj elim 3. 5. conj int
2
ELEMENTS OF COMBINATORY LOGIC
0
[p ~ q] 0 is called the conditional of 0 p 0 with 0 q 0 , or the implication of 0 q 0 by 0 p 0 , and 0 -p 0 is talled the negation of 0 p 0 or the~ of 0 p 0 • (The single quotation marks are not part of the expressions being discussed. The reference is rather to the expressions between such single quotation marks, or to expressions that these expressions represent.) 0.5. The sentence opo is said to be a direct ~ 0 ~uence of the sentence [p a q] 0 by the rule of conjunction elimination (conj elim). The sentence oqo is likewise said to be a direct consequence of the sentence 0 [p a q] 0 by the rule of conjunction elimination. 0.6. The sentence 0 [p a q] 0 is said to be a direct consequence of the pair of sentences 0 p' and 0 q 0 by the rule of conjunction introduction (conj int). 0.7. The sentence '[pVq] 0 is said to be a direct consequence of the .sentence 'p' by the Lu.le uf
~.i~j Ull(.;
t.lon
1£-
troduction (dis int). The sentence 0 [p V q] 0 is likewise said to be a direct consequence of of the sentence 'q 0 by the rule of disjunction introduction. 0.8. The sentence 0 q 0 is said to be a direct consequence of the pair of sentences 0 p 0 and 0 [p ~ q] 0 by the rule of implication elimination (imp elim). This rule is also called~ Eonens (m p). 0.9. A ~-~ Eroof is defined to be a finite sequence of sentences, written as a vertical column, and such that each sentence of the sequence either is explicitly designated as an ~pothesis Qf !£t sequence or else is in fact a direct consequence (in one of the senses referred to above) of one or a pair of preceding sentences of that sequence. There may be one or more hypotheses of the sequence, and they are designated by writing 0 hyp 0 after each of them. Also, it will be customary to group hypotheses together at the beginning of the sequence and to draw a short horizontal line separating them from the other sentences of the column. A vertical line is drawn to the left of the column or sequence of sentences, indicating that it is a proof.
3
THE METHOD OF SUBORDINATE PROOFS
o.lo. The following columns are proofs. After each sentence there is given an indication either that the sentence is an hypothesis or that it is a direct consequence of specified preceding sentences by one of the rules of direct consequence. The sentences of each column are numbered by numbers on the left so that reference to them is possible. Outermost square brackets are omitted for economy of notation.
Pl
r 2
pV q
3 P
P2
hyp
a
1~ P a 2 p
[p v qJ
hyp l, conj elim 1, conj elim
q
3 q P3
1 ~P 2 p 3 q 4 q
a q_
2
aq
r·q] 3
4
5 P5
1 2
3 4
5 6
7 P6
1 2
3 4
5 6
P r p r
1, dis int 1, 2, conj int
hyp 1, conj elim 1, conj elim 2, 3. conj int
aP
ar
ap a Cq a r]
hyp 1, conj elim 1, conj elim 2, conj elim 2, 3. conj int
p p q q r q [q
hyp 1, conj elim 1, conj elim 3. conj elim 3. conj elim 2, 4, conj int 5. 6, conj int
s r r p q q
hyp hyp
ar aP a pJ a r ar Cp a qJ ~
aq ar
1' conj elim 1, 3. m p 4, conj elim 3. 5. conj int
ELEMENTS OF COMBINATORY LOGIC
P7
P8
l
[p v q] =o [r
2
q
3 4-
r
5
s
6
q
7
pV [q
l 2
p
3 4-
[sVq]=os pvq
5
q
6
q
7 8
s s
a
s]
hyp hyp
pV q s
a
s
a
l' 3. m p 4-, conj elim 2, 5. conj int 6, dis int
s]
proof:
P9
r
vq
4-
5 6
7
0.11. The proof P7 can be said to be a proof of 'pV [q a s]' from (the hypotheses) '[pV q] =o [r a s]' and 'q', and the proof P8 can be said to be a proof of 's' from 'p', ' [ p v q] :::> [ q a r] ' ' and ' [ s v q] :::> s ' • 0.12. EXERCISES. Construct proofs of the following: (1). '[r as] a [r :::> s]' from 'p', 1 p =o r 1 and lp =o S I •
(2).
I
(3).
I
lp v ql. ( 4-). 'p', and 'r' • (5).
I
1
[p v q] a rl from
I [
[
p v q] v
S]
=o rl and
[p v q] a [r V s] 1 from 1 [[p a r] a q] a sl • [ S V q] a rl from lp =o [[pVq]:::> [rVs)] 1 , a [q a q]
S
[r =o [r =o [sa q]]]
1
I
from
1
p1
'
1
r
1
'
1 2
3
hyp hyp hyp l, dis int 2, 4-, m P 5. conj elim 6, dis int 3. 7. m p
a r]
[p v q] =o [ q
a
sequences, when properly constructed, can also be regarded as proofs in a more general sense, and they may be called second-~ Eroofs. Here is an example of a second-order
2, dis int
a
5
THE METHOD OF SUBORDINATE PROOFS
and
I
[pV s] =o
•
0.13. All the proofs considered so far are first-order proofs in the sense of 0.9, since each of them is a sequence of sentences such that every sentence of the sequence is an hypothesis of the sequence or is a direct consequence of preceding sentences of the sequence. But sequences can also be constructed out of sentences and first-order proofs. Such
hyp
r
rV s P a q
r
P
l,
dis int
hyp
3, conj elim
pVq 4-,disint [p a q] =o [pV q] 3-5, implication introduction r a [ [p a q] =o [p v q]] l' 6, conj int
o.l4-. Notice that the above proof is not just a sequence of sentences. It is rather a sequence that consists of two sentences (sentences 1 and 2) as its first two steps, a whole first-order proof (consisting of sentences 3-5) as its third step, and two more sentences (6 and 7) as its last two steps. The whole first-order proof (3-5) is itself merely step 3 of the five-step second-order proof. In other words, the whole first-order proof (3-5) is to be viewed as a single unit (just as a sentence is to be viewed as a single unit) in the construction of the second-order proof. Sentence number 6, which is actually the fourth step of the second-order proof, is a direct consequence of the whole first-order proof considered as a unit. This kind of direct consequence has not previously been considered in this discussion. It is called implication introduction (imp int). In general, any first-order proof that has exactly ~ hypothesis 1 p 1 and that has 1 q 1 as one of its steps can have 1 p =o q 1 as a direct consequence (by implication introduction) in a second-order proof. Observe that if 'p =>· q 1 is a direct consequence of a first-order proof by implication introduction, then 1 p 1 must be the only hypothesis of that first-order proof, and 1 q 1 must be one of its steps (usually in practice, its last step). This requirement is satisfied by '[p a q] :::> [pV q] I in P9 above, because 'p a q 1 is the only hypothesis of the first-order proof 3-5, and 'pV q 1 is the
ELEMENTS OF COMBINATORY LOGIC
P7
P8
l
[p v q] =o [r
2
q
3 4-
r
5
s
6
q
7
pV [q
l 2
p
3 4-
[sVq]=os pvq
5
q
6
q
7 8
s s
a
s]
hyp hyp
pV q s
a
s
a
l' 3. m p 4-, conj elim 2, 5. conj int 6, dis int
s]
proof:
P9
r
vq
4-
5 6
7
0.11. The proof P7 can be said to be a proof of 'pV [q a s]' from (the hypotheses) '[pV q] =o [r a s]' and 'q', and the proof P8 can be said to be a proof of 's' from 'p', ' [ p v q] :::> [ q a r] ' ' and ' [ s v q] :::> s ' • 0.12. EXERCISES. Construct proofs of the following: (1). '[r as] a [r :::> s]' from 'p', 1 p =o r 1 and lp =o S I •
(2).
I
(3).
I
lp v ql. ( 4-). 'p', and 'r' • (5).
I
1
[p v q] a rl from
I [
[
p v q] v
S]
=o rl and
[p v q] a [r V s] 1 from 1 [[p a r] a q] a sl • [ S V q] a rl from lp =o [[pVq]:::> [rVs)] 1 , a [q a q]
S
[r =o [r =o [sa q]]]
1
I
from
1
p1
'
1
r
1
'
1 2
3
hyp hyp hyp l, dis int 2, 4-, m P 5. conj elim 6, dis int 3. 7. m p
a r]
[p v q] =o [ q
a
sequences, when properly constructed, can also be regarded as proofs in a more general sense, and they may be called second-~ Eroofs. Here is an example of a second-order
2, dis int
a
5
THE METHOD OF SUBORDINATE PROOFS
and
I
[pV s] =o
•
0.13. All the proofs considered so far are first-order proofs in the sense of 0.9, since each of them is a sequence of sentences such that every sentence of the sequence is an hypothesis of the sequence or is a direct consequence of preceding sentences of the sequence. But sequences can also be constructed out of sentences and first-order proofs. Such
hyp
r
rV s P a q
r
P
l,
dis int
hyp
3, conj elim
pVq 4-,disint [p a q] =o [pV q] 3-5, implication introduction r a [ [p a q] =o [p v q]] l' 6, conj int
o.l4-. Notice that the above proof is not just a sequence of sentences. It is rather a sequence that consists of two sentences (sentences 1 and 2) as its first two steps, a whole first-order proof (consisting of sentences 3-5) as its third step, and two more sentences (6 and 7) as its last two steps. The whole first-order proof (3-5) is itself merely step 3 of the five-step second-order proof. In other words, the whole first-order proof (3-5) is to be viewed as a single unit (just as a sentence is to be viewed as a single unit) in the construction of the second-order proof. Sentence number 6, which is actually the fourth step of the second-order proof, is a direct consequence of the whole first-order proof considered as a unit. This kind of direct consequence has not previously been considered in this discussion. It is called implication introduction (imp int). In general, any first-order proof that has exactly ~ hypothesis 1 p 1 and that has 1 q 1 as one of its steps can have 1 p =o q 1 as a direct consequence (by implication introduction) in a second-order proof. Observe that if 'p =>· q 1 is a direct consequence of a first-order proof by implication introduction, then 1 p 1 must be the only hypothesis of that first-order proof, and 1 q 1 must be one of its steps (usually in practice, its last step). This requirement is satisfied by '[p a q] :::> [pV q] I in P9 above, because 'p a q 1 is the only hypothesis of the first-order proof 3-5, and 'pV q 1 is the
ELEMENTS OF COMBINATORY LOGIC
6
last step of that same first-order proof. 0.15. Some second-order proofs begin with a first-order proof instead of an hypothesis or hypotheses, although the first-order proof itself of course must begin with an hypothesis. Such a second-order proof is said to be categor~· while proofs that begin with one or more hypotheses are said to be hzEothetical. Consider, for example, the following categorical second-order proof which has just two steps, the first of which is a four-step first-order proof: FlO
1 2
3 4
5
1!: q] 1!: r
[p
I p
a:
q
q
s
[ [p
vq 1!: q] 1!: r]
:::>
[
s V q]
hyp 1, conj elim 2, conj elim 3, dis int l- 1+, imp int
Notice that FlO does not have an hypothesis as its first step but rather has a whole first-order proof as its first step, and that this first-order proof ~ have an hypothesis as ~ first step. In other words, FlO, which is a second-order proof, is a sequence of just two thin~s, neither of which is an hypothesis. The first of these two things is a four-step first-order proof (which does have an hypothesis of~~ and not belonging to the second-order proof), while the second of these two things is the sentence '[[p 1!: q] 1!: r] :::> [s V q]', which is a direct consequence of the first-order proof by the rule of implication introduction. Thus FlO is a categorical second-order proof having a hypothetical first-order proof as its first step. 0.16. The concept of a ~-~ Eroof can now be defined as follows: A second-order proof is a sequence of things (written as a column) each of which either is a sentence or is itself a first-order proof (as defined in 0.9), and if it is a sentence it is either an hypothesis (of the second-order proof) or is a direct consequence of preceding things (sentences or first-order proofs) of the sequence. 0.17. There is a special dispensation allowed to those first-order proofs that sPrve as steps of second-order
7
THE METHOD OF SUBORDINATE PROOFS
proofs, and is allowed to them only in their role as steps of second-order proofs. This dispensation permits each such first-order proof to add, as steps of its own, any preceding sentences which are steps of the same second-order proof of which it (the first-order proof) is a step. Such steps of a first-order proof will be said to have been reiterated into the first-order proof from the second-order proof, and this fact will be indicated by writing 'reit' after each such step, followed by a number indicating where the sentence originally appeared in the second-order proof. (For example, see step 6 in Fll below, and steps 5 and 6 in Fl2.) Thereiterated sentence can be used like any other sentence of the first-order proof. Here are two examples of second-order proofs that contain first-order proofs into which sentences have been reiterated:
Fll
1
p
:::>
q
2
q
:::>
r
8
P
:::>
r
1
[p
hyp hyp hyp 1, reit 3, 4, m p 2, reit 5, 6, m p 3-7, imp int
3 4
5 6
7 Fl2
q]
:::>
2
p
:::>
q
3
P
:::>
r
1!: [p
:::>
r]
hyp 1, conj elim 1, conj elim hyp 2, reit 3, reit
4
p
5
p
:::>
q
6
p
:::>
r
7
q
4,
r
4, 6, m p 7, 8, conj int 4-9, imp int
8 9 10
p
q
1!:
:::>
[q
r
1!: r]
5,
m p
0.18. In Fll 'p :::> r' has been proved from the hypotheses 'p :::> q' and 'q :::> r'. This principle, that 'p :::> r' fol-
ELEMENTS OF COMBINATORY LOGIC
6
last step of that same first-order proof. 0.15. Some second-order proofs begin with a first-order proof instead of an hypothesis or hypotheses, although the first-order proof itself of course must begin with an hypothesis. Such a second-order proof is said to be categor~· while proofs that begin with one or more hypotheses are said to be hzEothetical. Consider, for example, the following categorical second-order proof which has just two steps, the first of which is a four-step first-order proof: FlO
1 2
3 4
5
1!: q] 1!: r
[p
I p
a:
q
q
s
[ [p
vq 1!: q] 1!: r]
:::>
[
s V q]
hyp 1, conj elim 2, conj elim 3, dis int l- 1+, imp int
Notice that FlO does not have an hypothesis as its first step but rather has a whole first-order proof as its first step, and that this first-order proof ~ have an hypothesis as ~ first step. In other words, FlO, which is a second-order proof, is a sequence of just two thin~s, neither of which is an hypothesis. The first of these two things is a four-step first-order proof (which does have an hypothesis of~~ and not belonging to the second-order proof), while the second of these two things is the sentence '[[p 1!: q] 1!: r] :::> [s V q]', which is a direct consequence of the first-order proof by the rule of implication introduction. Thus FlO is a categorical second-order proof having a hypothetical first-order proof as its first step. 0.16. The concept of a ~-~ Eroof can now be defined as follows: A second-order proof is a sequence of things (written as a column) each of which either is a sentence or is itself a first-order proof (as defined in 0.9), and if it is a sentence it is either an hypothesis (of the second-order proof) or is a direct consequence of preceding things (sentences or first-order proofs) of the sequence. 0.17. There is a special dispensation allowed to those first-order proofs that sPrve as steps of second-order
7
THE METHOD OF SUBORDINATE PROOFS
proofs, and is allowed to them only in their role as steps of second-order proofs. This dispensation permits each such first-order proof to add, as steps of its own, any preceding sentences which are steps of the same second-order proof of which it (the first-order proof) is a step. Such steps of a first-order proof will be said to have been reiterated into the first-order proof from the second-order proof, and this fact will be indicated by writing 'reit' after each such step, followed by a number indicating where the sentence originally appeared in the second-order proof. (For example, see step 6 in Fll below, and steps 5 and 6 in Fl2.) Thereiterated sentence can be used like any other sentence of the first-order proof. Here are two examples of second-order proofs that contain first-order proofs into which sentences have been reiterated:
Fll
1
p
:::>
q
2
q
:::>
r
8
P
:::>
r
1
[p
hyp hyp hyp 1, reit 3, 4, m p 2, reit 5, 6, m p 3-7, imp int
3 4
5 6
7 Fl2
q]
:::>
2
p
:::>
q
3
P
:::>
r
1!: [p
:::>
r]
hyp 1, conj elim 1, conj elim hyp 2, reit 3, reit
4
p
5
p
:::>
q
6
p
:::>
r
7
q
4,
r
4, 6, m p 7, 8, conj int 4-9, imp int
8 9 10
p
q
1!:
:::>
[q
r
1!: r]
5,
m p
0.18. In Fll 'p :::> r' has been proved from the hypotheses 'p :::> q' and 'q :::> r'. This principle, that 'p :::> r' fol-
ELEMENTS OF COMBINATORY LOGIC
8
lows from 'p ~ q' and 'q ~ r', will be referred to as~ sitivit~ £[implication (trans imp). In Pl2 'p ~ [q r]' has been proved from the hypothesis '[p ~ q] a [p ~ r]'. The converse of this latter result can also be established and is asked for in exercise (l) below. Notice, in connection with reiteration, that although a first-order proof can reiterate steps from a second-order proof to which it belongs, a second-order proof cannot reiterate steps from a first-order proof that belongs to it. Reiteration is therefore a one-directional process. Furthermore, no first-order proof can reiterate from another first-order proof. 0.19. EXERCISES. Construct proofs of the following: (l). '[p ::0 q] a [p ~ r]' from 'p ~ [q a r]'. (2). 'p ~ q' from 'q'. (3). '[p li [p ~ q]] ~ q' categorically (that is,
( 5). ( 6).
'[p a q] ~ r' from 'p ~ [q ~ r]'. ' [ p ~ r] li [ q ::o r] ' from ' [ p V q] ~ r ' • '[((pVs] ~ q] lip]~ [[pVr] a [sVq]]'
categorically. 0.20. A further important rule of direct consequence is disjunction elimination (dis elim). This rule can be used in a second-order proof in such a way that a sentence 'r' is treated as being a direct consequence of the following three things conjointly: (l) a sentence of the form 'p V q', (2) a first-order proof having 'p' as its only hypothesis and having 'r' as one of its steps (usually its last step), and (3) a first-order proof having 'q' as its only hypothesis and having 'r' as one of its steps (usually its last step). The hypotheses 'p' and 'q' must be the same as the 'p' and 'q' in the sentence 'p V q' referred to in ( l). Here are some second-order proofs that use the rule of disjunction elimination:
Pl3
1~ pV q
~~ ~~vp
hyp hyp 2, dis int
hyp 4, dis int l, 2-3, 4-5, dis elim
4
a
( 4).
9
THE METHOD OF SUBORDINATE PROOFS
5 6
Notice that 'q V p' is here playing the role played by 'r' in the general statement of the rule of disjunction elimination. Pl4
l 2 3 4 5 6
7
hyp hyp 2, dis int hyp 4, conj elim 5, dis int l, 2-3. 4-6, dis elim
pV [q li r]
~:v
q
~~a
r
pV q pVq
Notice that ~ numerical references ordinarily precede the reference to dis elim. At step 7 ln Fl4 &Love Ll1etie three references are 'l', '2-3', and '4-6'. When a whole first-order proof is referred to, the method is to write the numbers of its first and last steps and to place a dash between them. First-order proofs are said to be subordinate to the second-order proofs of which they are parts, and they can also be said to be sub£roofs of such second-order proofs; hence the general method of logic being presented here may be called the ~ of subordinate £roofs or the method of subproof~. Some further examples using the rule of disjunetion elimination now follow. Pl5
1
2 3 4 5 6
7 8 9 10 ll
s pV q
,. I' s
P [p
as a s] v [q
li s]
s q a s [p a s] v [q a s] [p a s] v [q a s]
hyp hyp hyp 1, reit 3. 4, conj int 5, dis int hyp 1, reit 7. 8, conj int 9. dis int 2, 3-6, 7-lO, dis elim
ELEMENTS OF COMBINATORY LOGIC
8
lows from 'p ~ q' and 'q ~ r', will be referred to as~ sitivit~ £[implication (trans imp). In Pl2 'p ~ [q r]' has been proved from the hypothesis '[p ~ q] a [p ~ r]'. The converse of this latter result can also be established and is asked for in exercise (l) below. Notice, in connection with reiteration, that although a first-order proof can reiterate steps from a second-order proof to which it belongs, a second-order proof cannot reiterate steps from a first-order proof that belongs to it. Reiteration is therefore a one-directional process. Furthermore, no first-order proof can reiterate from another first-order proof. 0.19. EXERCISES. Construct proofs of the following: (l). '[p ::0 q] a [p ~ r]' from 'p ~ [q a r]'. (2). 'p ~ q' from 'q'. (3). '[p li [p ~ q]] ~ q' categorically (that is,
( 5). ( 6).
'[p a q] ~ r' from 'p ~ [q ~ r]'. ' [ p ~ r] li [ q ::o r] ' from ' [ p V q] ~ r ' • '[((pVs] ~ q] lip]~ [[pVr] a [sVq]]'
categorically. 0.20. A further important rule of direct consequence is disjunction elimination (dis elim). This rule can be used in a second-order proof in such a way that a sentence 'r' is treated as being a direct consequence of the following three things conjointly: (l) a sentence of the form 'p V q', (2) a first-order proof having 'p' as its only hypothesis and having 'r' as one of its steps (usually its last step), and (3) a first-order proof having 'q' as its only hypothesis and having 'r' as one of its steps (usually its last step). The hypotheses 'p' and 'q' must be the same as the 'p' and 'q' in the sentence 'p V q' referred to in ( l). Here are some second-order proofs that use the rule of disjunction elimination:
Pl3
1~ pV q
~~ ~~vp
hyp hyp 2, dis int
hyp 4, dis int l, 2-3, 4-5, dis elim
4
a
( 4).
9
THE METHOD OF SUBORDINATE PROOFS
5 6
Notice that 'q V p' is here playing the role played by 'r' in the general statement of the rule of disjunction elimination. Pl4
l 2 3 4 5 6
7
hyp hyp 2, dis int hyp 4, conj elim 5, dis int l, 2-3. 4-6, dis elim
pV [q li r]
~:v
q
~~a
r
pV q pVq
Notice that ~ numerical references ordinarily precede the reference to dis elim. At step 7 ln Fl4 &Love Ll1etie three references are 'l', '2-3', and '4-6'. When a whole first-order proof is referred to, the method is to write the numbers of its first and last steps and to place a dash between them. First-order proofs are said to be subordinate to the second-order proofs of which they are parts, and they can also be said to be sub£roofs of such second-order proofs; hence the general method of logic being presented here may be called the ~ of subordinate £roofs or the method of subproof~. Some further examples using the rule of disjunetion elimination now follow. Pl5
1
2 3 4 5 6
7 8 9 10 ll
s pV q
,. I' s
P [p
as a s] v [q
li s]
s q a s [p a s] v [q a s] [p a s] v [q a s]
hyp hyp hyp 1, reit 3. 4, conj int 5, dis int hyp 1, reit 7. 8, conj int 9. dis int 2, 3-6, 7-lO, dis elim
ELEMENTS OF COMBINATORY LOGIC
10 Pl6
1
[qVr]=>s
2
[p 1i q] V [p li
3
~~
4
qVr
s s li [qVr]
1, 9, m p 9, 10, conj int
I qV r
6
~~
11
q
1i r
I qVr
1~ [p li q]V [p li r]
~ ~~liq 4
qV r
Ip
li [q V 1i r
p
7
p
8
r
9
qV r
10 11
12 Pl8
I
5
6
1 2
3 4
5
hyp hyp hyp
3, conj elim 4, dis int hyp 6, conj elim 7, dis int 2, 3-5, 6-8, dis elim
5 7 8 9 10 Pl7
li
r]
r]
r~
1i [qVr] p li [qVr] [r => s] => ([r => s] => r] [p li q] V [r => s]
~~
I
q
li q
vs
6 7 8
r => s [r ~ s] => [[r => s] => r] [r => s] => r
9
r
10 11
s
12
qvs
q
hyp hyp 2, conj elim 3, dis int 2, conj elim 4, 5, conj int hyp 7, conj elim 8, dis int 7, conj elim 9, 10, conj int 1, 2-6, 7-ll, dis elim hyp hyp hyp 3, conj elim 4, dis int hyp l, reit 6, 7, m p 6, 8, m p 6, 9, m p
vs
10, dis int 2, 3-5, 6-11, dis elim
THE METHOD OF SUBORDINATE PROOFS
11
0.21. The converse of Pl7 can also be established, that is, '[p li q]V [p li r]' can be proved from the hypothesis 'p 11 [ q V r]'. The method is to start by using conj elim to separate 'p li [qVr]' into its two main parts 'p' and 'qVr', and then to proceed from the steps 'p' and 'qVr' somewhat as Pl5 proceeds from steps 's' and 'p V q'. Incidentally, it is not possible to use conj elim to separate '[p li q] V [p li r]' into parts. This is because conj elim can be applied only to a conjunction, while '[p li q]V [p li r]' is obviously a disjunction rather than a conjunction (though its two main parts are conjunctions). 0.22. In constructing proofs, the following guiding principles are useful: (l) It is generally best to start by using elimination rules and reiteration as far as possible, the elimination rules so far available being conj elim, m p (imp elim), and dis elim. (.2) In u.cder tu ebLabllBh a con-
junction, first use elimination rules and reiteration as far as possible, then use any introduction rules that appear to be needed, and finally use conj int to obtain the required conjunction. (3) In order to establish an implication, first use elimination rules and reiteration as far as possible, and then use implication introduction to obtain the required implication. Be sure to reiterate any needed steps into the first-order proof required for implication introduction. Within this first-order proof start by using elimination rules and end by using introduction rules. (4) In order to establish a disjunction, first use elimination rules and reiteration as far as possible, then use whatever introduction rules that seem to be needed, and finally use disjunction introduction to obtain the required disjunction. This method may fail, however, when one disjunction is to be proved from another disjunction, unless care is taken to proceed in the following way: Try to use dis elim as the overall strategy, as was done, for example, in obtaining step ll of Pl5 from step 2. In doing this, attempt to establish the required disjunction by first obtaining it by dis int within each of
ELEMENTS OF COMBINATORY LOGIC
10 Pl6
1
[qVr]=>s
2
[p 1i q] V [p li
3
~~
4
qVr
s s li [qVr]
1, 9, m p 9, 10, conj int
I qV r
6
~~
11
q
1i r
I qVr
1~ [p li q]V [p li r]
~ ~~liq 4
qV r
Ip
li [q V 1i r
p
7
p
8
r
9
qV r
10 11
12 Pl8
I
5
6
1 2
3 4
5
hyp hyp hyp
3, conj elim 4, dis int hyp 6, conj elim 7, dis int 2, 3-5, 6-8, dis elim
5 7 8 9 10 Pl7
li
r]
r]
r~
1i [qVr] p li [qVr] [r => s] => ([r => s] => r] [p li q] V [r => s]
~~
I
q
li q
vs
6 7 8
r => s [r ~ s] => [[r => s] => r] [r => s] => r
9
r
10 11
s
12
qvs
q
hyp hyp 2, conj elim 3, dis int 2, conj elim 4, 5, conj int hyp 7, conj elim 8, dis int 7, conj elim 9, 10, conj int 1, 2-6, 7-ll, dis elim hyp hyp hyp 3, conj elim 4, dis int hyp l, reit 6, 7, m p 6, 8, m p 6, 9, m p
vs
10, dis int 2, 3-5, 6-11, dis elim
THE METHOD OF SUBORDINATE PROOFS
11
0.21. The converse of Pl7 can also be established, that is, '[p li q]V [p li r]' can be proved from the hypothesis 'p 11 [ q V r]'. The method is to start by using conj elim to separate 'p li [qVr]' into its two main parts 'p' and 'qVr', and then to proceed from the steps 'p' and 'qVr' somewhat as Pl5 proceeds from steps 's' and 'p V q'. Incidentally, it is not possible to use conj elim to separate '[p li q] V [p li r]' into parts. This is because conj elim can be applied only to a conjunction, while '[p li q]V [p li r]' is obviously a disjunction rather than a conjunction (though its two main parts are conjunctions). 0.22. In constructing proofs, the following guiding principles are useful: (l) It is generally best to start by using elimination rules and reiteration as far as possible, the elimination rules so far available being conj elim, m p (imp elim), and dis elim. (.2) In u.cder tu ebLabllBh a con-
junction, first use elimination rules and reiteration as far as possible, then use any introduction rules that appear to be needed, and finally use conj int to obtain the required conjunction. (3) In order to establish an implication, first use elimination rules and reiteration as far as possible, and then use implication introduction to obtain the required implication. Be sure to reiterate any needed steps into the first-order proof required for implication introduction. Within this first-order proof start by using elimination rules and end by using introduction rules. (4) In order to establish a disjunction, first use elimination rules and reiteration as far as possible, then use whatever introduction rules that seem to be needed, and finally use disjunction introduction to obtain the required disjunction. This method may fail, however, when one disjunction is to be proved from another disjunction, unless care is taken to proceed in the following way: Try to use dis elim as the overall strategy, as was done, for example, in obtaining step ll of Pl5 from step 2. In doing this, attempt to establish the required disjunction by first obtaining it by dis int within each of
ELEMENTS OF COMBINATORY LOGIC
12
the first-order proofs needed for dis elim, as was done at steps 6 and 10 in Pl5. Be sure to reiterate sufficiently into each of these two first-order proofs. Then by use of dis elim obtain the required disjunction as a step of the secondorder proof. 0.23. EXERCISES. Construct proofs of the following: 'r' from 'p =:J r', 'q ::J r'' and 'pV q' • ( 1) . '[p q] v [p r]' from 'p &. [qVr]'. (2). '[q &. r] &. [s &. p]' from I pI 1 'p :::> [qVr]', ( 3). 'q :::> [r &. [q :::> s]]'' and 'r :::> [q &. [r :::> s]]'. 's' from 'q V r', 'q ::>[[qVr] :::> s]' ' and (4). 'r :::> [[rVq] :::> s]'. 'rV s' from 'p &. q' and '[p :::> [q :::> r]] V ( 5). [q :::> [p :::> s]]'. o.24. Third-order proofs can be constructed which make
a
a
cf fir.::;t-
way that second-order proofs make use of first-order proofs. Here are two examples of third-order proofs. Pl9
1 2 3 4 5 6 7 8 9 10 11
P20
1 2 3 4 5 6
p
:::>
[q
p p
:::> :::>
q [q
p p :::> q p :::> q :::> r p ::J r [p ::J q]
:::>
~~
:::>
r]
q [q r
:::>
r]
::J [p ::J r]
p ::J r q :::> r
I'"
hyp hyp
r]
:::>
1' reit hyp 2, reit 4, 5, m p 3. reit '+' 7, m p 6, 8, m p 4-9, imp int 2-10 imp int hyp hyp hyp hyp
r
reit 2, 8, m p l'
THE METHOD OF SUBORDINATE PROOFS 7 8 9 10
~~
:::>
13 hyp 2, reit 7. 8, m p 3. 4-6, 7-9. dis elim
r
r
Notice that the first-order (innermost) subproofs in P20 receive reiterations directly from steps 1 and 2 of the main third-order proof, instead of receiving them indirectly by way of second-order proofs as is the case in Pl9. Direct reiteration of the kind used in P20 will be allowed if the reiterated step comes from a preceding step of a higher-order proof that encloses the proof receiving the reiteration. 0.25. The system of logic that has so far been described can be further extended in a similar way by allowing fourth-order proofs, fifth-order proofs, and, in general, nth-order proofs for every finite positive integer n. It will now be assumed that this extension has been made. Here are two examples of categorical fourth-order proofs: P21
1 2
p ::J q pV r
3 4
5 6
7 8
9 10 11
P22
l 2
3
IL",' ~ ~v
r
qV r
[pVr] ::J [qV r] [p::Jq]=>[[pVr]::J[qVr]J q ::J r P ::J r q ::J r
4
5
::J q
6
7
hyp hyp hyp l, rei t 3, 4, m p 5, dis int hyp 7, dis int 2, 3-6, 7-8, dis elim 2-9, imp int 1-lO,impint hyp hyp l , reit hyp 2, reit
4,
::J r
5,
m p
3, reit
ELEMENTS OF COMBINATORY LOGIC
12
the first-order proofs needed for dis elim, as was done at steps 6 and 10 in Pl5. Be sure to reiterate sufficiently into each of these two first-order proofs. Then by use of dis elim obtain the required disjunction as a step of the secondorder proof. 0.23. EXERCISES. Construct proofs of the following: 'r' from 'p =:J r', 'q ::J r'' and 'pV q' • ( 1) . '[p q] v [p r]' from 'p &. [qVr]'. (2). '[q &. r] &. [s &. p]' from I pI 1 'p :::> [qVr]', ( 3). 'q :::> [r &. [q :::> s]]'' and 'r :::> [q &. [r :::> s]]'. 's' from 'q V r', 'q ::>[[qVr] :::> s]' ' and (4). 'r :::> [[rVq] :::> s]'. 'rV s' from 'p &. q' and '[p :::> [q :::> r]] V ( 5). [q :::> [p :::> s]]'. o.24. Third-order proofs can be constructed which make
a
a
cf fir.::;t-
way that second-order proofs make use of first-order proofs. Here are two examples of third-order proofs. Pl9
1 2 3 4 5 6 7 8 9 10 11
P20
1 2 3 4 5 6
p
:::>
[q
p p
:::> :::>
q [q
p p :::> q p :::> q :::> r p ::J r [p ::J q]
:::>
~~
:::>
r]
q [q r
:::>
r]
::J [p ::J r]
p ::J r q :::> r
I'"
hyp hyp
r]
:::>
1' reit hyp 2, reit 4, 5, m p 3. reit '+' 7, m p 6, 8, m p 4-9, imp int 2-10 imp int hyp hyp hyp hyp
r
reit 2, 8, m p l'
THE METHOD OF SUBORDINATE PROOFS 7 8 9 10
~~
:::>
13 hyp 2, reit 7. 8, m p 3. 4-6, 7-9. dis elim
r
r
Notice that the first-order (innermost) subproofs in P20 receive reiterations directly from steps 1 and 2 of the main third-order proof, instead of receiving them indirectly by way of second-order proofs as is the case in Pl9. Direct reiteration of the kind used in P20 will be allowed if the reiterated step comes from a preceding step of a higher-order proof that encloses the proof receiving the reiteration. 0.25. The system of logic that has so far been described can be further extended in a similar way by allowing fourth-order proofs, fifth-order proofs, and, in general, nth-order proofs for every finite positive integer n. It will now be assumed that this extension has been made. Here are two examples of categorical fourth-order proofs: P21
1 2
p ::J q pV r
3 4
5 6
7 8
9 10 11
P22
l 2
3
IL",' ~ ~v
r
qV r
[pVr] ::J [qV r] [p::Jq]=>[[pVr]::J[qVr]J q ::J r P ::J r q ::J r
4
5
::J q
6
7
hyp hyp hyp l, rei t 3, 4, m p 5, dis int hyp 7, dis int 2, 3-6, 7-8, dis elim 2-9, imp int 1-lO,impint hyp hyp l , reit hyp 2, reit
4,
::J r
5,
m p
3, reit
ELEMENTS OF COMBINATORY LOGIC
14 8
11 r
P :::> r
9 [p
lO ll
[q
:::>
:::>
r]
r] :::>
[q :::> r] [ [p :::> q] :::> [p
:::>
:::>
r])
6, 7, m p 4-8, imp int 2-9. imp int l-10, imp int
0.26. Notice that in P22 the reiteration of 'q :::> r' in the innermost subordinate proof could have been made directly from step l of the main proof instead of indirectly by way of step 3. Another way to shorten P22 would be to write it in the following way: P23
l 2 3 4
rq"
lp :::> q q :::> r P :::> r [p :::> q]
5 6,
[q
~
T']
~
:::>
[p
[[:r
:::> -:J
r] 'l]
-:J
[:r :::> r]J
hyp hyp l, reit 2, 3. trans imp 2-4, imp int l-5. imp int
In P23 we have shortened P22 by omitting the details of the inference from 'p :::> q' and 'q :::> r' to 'p :::> r'. This inference is said to be by transitivit~ of imElication (trans imp). The details can be found in Pll, which establishes the principle of the transitivity of implication. P23 can be regarded simply as an abbreviation or shorthand for P22, and the reference 'trans imp' in P23 can be viewed as a reference to Pll where details can be found for rewriting P23 out in full as P22. We may regard transitivity of implication as a derived rule of inference, the derivation of it being given in Pll. The other rules that we have so far been using are underived, or £rimit~y~. rules of inference (such as conj int, m p, and so on). In fact, all rules of direct consequence are underived or primitive, while all derived rules (such as trans imp) may be said to be rules of indirect ~ seguence. All rules for which abbreviated names have been given are rules which can be freely used in constructing proofs. Thus transitivity of implication can be used in proofs just as freely as the underived rules (the rules of direct consequence). Here is another example of the use of
THE METHOD OF SUBORDINATE PROOFS
15
trans imp: p24
l~p
2 3
p :::> [q ::>·[rV s)J [rVs]::>r
4
q :::> [r V s]
5
q :::> r
hyp hyp hyp 1, 2, m p 3, 4, trans imp
0.27. Another derived rule of inference is the rule of added condition (add cond) according to which 'p :::> q' may be inferred from 'q'. This rule is derived as follows: P25
r 2 3 4
hyp hyp l, rei t 2-3, imp int
rp q p :::> q
Here is an example of a proof that uses the rule add cond: P26
l 2 3 4 5 6 7 8 9 10
[p
:::>
q]
:::>
[p
r]
:::>
hyp hyp hyp
p q p :::> q [p :::> q] p :::> r p r q :::> r p :::> [q :::> r]
:::>
[p
:::>
r]
3. add cond 1, reit 4, 5, m p 2, reit 6, 7, m p 3-8, imp int 2-9. imp int
Notice that P26 is the converse of Pl9. 0.28. EXERCISES. Construct proofs of the following: (1). '[pV[qVr]] :::> [[pVq]Vr]' categorically. ( 2) • 0 [ s p] :::> [ q v r] 0 from 0 [ p :::> q] v [ s :::> r] 0 • (3). 'q' from '[pV [[p :o q] :::> q]] V [[p -o q] -o [[p -o q] :o q]]', and 'p :::> q'. ( 4) • 0 [ q v [ r a s]] ::J p 0 from 0 [ [ q v t] v r] ::J p 0 • 0.29. Any sentence which is a step of a proof may be repeated as a later step of the same proof. This is because the same reason can be given for the later step as for the
a
ELEMENTS OF COMBINATORY LOGIC
14 8
11 r
P :::> r
9 [p
lO ll
[q
:::>
:::>
r]
r] :::>
[q :::> r] [ [p :::> q] :::> [p
:::>
:::>
r])
6, 7, m p 4-8, imp int 2-9. imp int l-10, imp int
0.26. Notice that in P22 the reiteration of 'q :::> r' in the innermost subordinate proof could have been made directly from step l of the main proof instead of indirectly by way of step 3. Another way to shorten P22 would be to write it in the following way: P23
l 2 3 4
rq"
lp :::> q q :::> r P :::> r [p :::> q]
5 6,
[q
~
T']
~
:::>
[p
[[:r
:::> -:J
r] 'l]
-:J
[:r :::> r]J
hyp hyp l, reit 2, 3. trans imp 2-4, imp int l-5. imp int
In P23 we have shortened P22 by omitting the details of the inference from 'p :::> q' and 'q :::> r' to 'p :::> r'. This inference is said to be by transitivit~ of imElication (trans imp). The details can be found in Pll, which establishes the principle of the transitivity of implication. P23 can be regarded simply as an abbreviation or shorthand for P22, and the reference 'trans imp' in P23 can be viewed as a reference to Pll where details can be found for rewriting P23 out in full as P22. We may regard transitivity of implication as a derived rule of inference, the derivation of it being given in Pll. The other rules that we have so far been using are underived, or £rimit~y~. rules of inference (such as conj int, m p, and so on). In fact, all rules of direct consequence are underived or primitive, while all derived rules (such as trans imp) may be said to be rules of indirect ~ seguence. All rules for which abbreviated names have been given are rules which can be freely used in constructing proofs. Thus transitivity of implication can be used in proofs just as freely as the underived rules (the rules of direct consequence). Here is another example of the use of
THE METHOD OF SUBORDINATE PROOFS
15
trans imp: p24
l~p
2 3
p :::> [q ::>·[rV s)J [rVs]::>r
4
q :::> [r V s]
5
q :::> r
hyp hyp hyp 1, 2, m p 3, 4, trans imp
0.27. Another derived rule of inference is the rule of added condition (add cond) according to which 'p :::> q' may be inferred from 'q'. This rule is derived as follows: P25
r 2 3 4
hyp hyp l, rei t 2-3, imp int
rp q p :::> q
Here is an example of a proof that uses the rule add cond: P26
l 2 3 4 5 6 7 8 9 10
[p
:::>
q]
:::>
[p
r]
:::>
hyp hyp hyp
p q p :::> q [p :::> q] p :::> r p r q :::> r p :::> [q :::> r]
:::>
[p
:::>
r]
3. add cond 1, reit 4, 5, m p 2, reit 6, 7, m p 3-8, imp int 2-9. imp int
Notice that P26 is the converse of Pl9. 0.28. EXERCISES. Construct proofs of the following: (1). '[pV[qVr]] :::> [[pVq]Vr]' categorically. ( 2) • 0 [ s p] :::> [ q v r] 0 from 0 [ p :::> q] v [ s :::> r] 0 • (3). 'q' from '[pV [[p :o q] :::> q]] V [[p -o q] -o [[p -o q] :o q]]', and 'p :::> q'. ( 4) • 0 [ q v [ r a s]] ::J p 0 from 0 [ [ q v t] v r] ::J p 0 • 0.29. Any sentence which is a step of a proof may be repeated as a later step of the same proof. This is because the same reason can be given for the later step as for the
a
ELEMENTS OF COMBINATORY LOGIC
16
earlier one, since the same sentence constitutes both steps. Instead of actually giving the same reason, however, we will simply refer back by number to the previous occurrence of the sentence and write 'rep', meaning repetition. Here is an example of this process:
P27
l 2
3
p :::> p
p :::> p
same proof as the original step. Here are three proofs that use repetition:
p28
hyp l, rep 1-2, imp int
In P27 we have established that 'p :::> p' is provable. We may use 'p :::> p' as a step in any proof (with 'p' replaced by any sentence we wish), and give as a reason, reflexivit~ £[ im£lication (refl imp). This reason may be thought of simply as a reference to P27, since the method of P27 can be used to establish 'p :::> p' as a step in any proof. Thus if in some proof we have.
P29
P30
Of course reflexivity of implication may be regarded as being a derived rule of inference, but only in the sense of enabling us to insert 'p :::> p' anywhere in a proof, and not in the sense of enabling us to infer a sentence from one or more other sentences in a proof. 0.30. Repetition is to be sharply distinguished from reiteration. In reiteration there is a transference of a sentence from one proof to another proof that is subordinate (interior) to the latter. In repetition there is no transference to another proof. The repeated step is a step of the
pV p
r~ r~
6
p
hyp hyp
2, rep hyp
4, rep l' 2-3, 4-5, dis elim hyp
2lipp 3 p B. p
rVq
2
q
~
~p
5
I;
6
7 8
this is to be understood as a shorthand for,
p :::> p
l
2 3 4 5
'+
refl imp
hyp ••• , rep ••. , imp int
17
THE METHOD OF SUBORDINATE PROOFS
I
l, rep 1, 2, conj int hyp hyp hyp :;, rep hyp 2, reit 5. 6, m p
p
:::>
p
:::>
p
1, 3-4, 5-7, dis elim
p
0.31. Hereafter if one (or each) of the subproofs that are used in connection with dis elim is of the form,
r~
then it (or they) may be omitted. Thus P28 and P30 could be rewritten more simply thus:
P31
lrpVp p
hyp
1 2 3 4 5
pV q q :::> p
hyp hyp hyp
6
p
2 P32
I;
:::>
1, dis elim
p
2, reit 3, 4, m p 1, 3-5, dis elim
ELEMENTS OF COMBINATORY LOGIC
16
earlier one, since the same sentence constitutes both steps. Instead of actually giving the same reason, however, we will simply refer back by number to the previous occurrence of the sentence and write 'rep', meaning repetition. Here is an example of this process:
P27
l 2
3
p :::> p
p :::> p
same proof as the original step. Here are three proofs that use repetition:
p28
hyp l, rep 1-2, imp int
In P27 we have established that 'p :::> p' is provable. We may use 'p :::> p' as a step in any proof (with 'p' replaced by any sentence we wish), and give as a reason, reflexivit~ £[ im£lication (refl imp). This reason may be thought of simply as a reference to P27, since the method of P27 can be used to establish 'p :::> p' as a step in any proof. Thus if in some proof we have.
P29
P30
Of course reflexivity of implication may be regarded as being a derived rule of inference, but only in the sense of enabling us to insert 'p :::> p' anywhere in a proof, and not in the sense of enabling us to infer a sentence from one or more other sentences in a proof. 0.30. Repetition is to be sharply distinguished from reiteration. In reiteration there is a transference of a sentence from one proof to another proof that is subordinate (interior) to the latter. In repetition there is no transference to another proof. The repeated step is a step of the
pV p
r~ r~
6
p
hyp hyp
2, rep hyp
4, rep l' 2-3, 4-5, dis elim hyp
2lipp 3 p B. p
rVq
2
q
~
~p
5
I;
6
7 8
this is to be understood as a shorthand for,
p :::> p
l
2 3 4 5
'+
refl imp
hyp ••• , rep ••. , imp int
17
THE METHOD OF SUBORDINATE PROOFS
I
l, rep 1, 2, conj int hyp hyp hyp :;, rep hyp 2, reit 5. 6, m p
p
:::>
p
:::>
p
1, 3-4, 5-7, dis elim
p
0.31. Hereafter if one (or each) of the subproofs that are used in connection with dis elim is of the form,
r~
then it (or they) may be omitted. Thus P28 and P30 could be rewritten more simply thus:
P31
lrpVp p
hyp
1 2 3 4 5
pV q q :::> p
hyp hyp hyp
6
p
2 P32
I;
:::>
1, dis elim
p
2, reit 3, 4, m p 1, 3-5, dis elim
ELEMENTS OF COMBINATORY LOGIC
18
0.32. The rule of constructive dilemma (cnst dil) is now presented. It treats 0 r V s 0 as a consequence of three things: (l) a sentence 0 pVqo (or 0 qVp 0 ) , (2) a proof with single hypothesis 0 p 0 and with 0 r 0 as one of its steps, and (3) a proof with single hypothesis 0 q 0 and with 0 S 0 as one of its steps. This rule is clearly a derived rule, since the effect of it can be gotten by uses of underived rules, specifically by uses of dis elim and dis int. For example, P33 uses cnst dil, but P3~ gives the same effect by using dis elim and dis int. Also, since it is a derived rule, we speak of a "consequence" obtained by use of the rule, rather than a "direct consequence." P33
l~pV [o 2 3 ~
5
I!. , s ::> p
::> s]
7 8 l
p
2
3
~ ~ ::> p
~
l[s::>p]Vs
5 6
7 8
9 lO
hyp hyp 2, add cond hyp ~. conj elim ~. conj elim 5. 6, m p l, 2-3. 4--7, cnst dil
[O o e]]
q ::> s s [s::>p]Vs
6
P34-
tp
a
r
v
[q
a
[q ::> s]]
hyp hyp 2, add cond 3, dis int hyp 5, conj elim 5, conj elim 6, 7, m p 8, dis int l, 2-4, 5-9, dis elim
~: :· ,,
[s::>p]Vs [s ::> p]Vs
0.33. EXERCISES. Construct proofs of the following: 0 (l). [[p a r]V] ::> [s :::> r] 0 categorically. 0 ( 2) • [ p v ql a [p v q] 0 from 0 [ p a q] v [ p a q] 0 • (3). oq :::> [p :::> r] from [p :::> q] :::> ro. (~). o[p :::> r] :::> [[pV q] :::> [[q :::> s] :::> [rV s]]] o 0
0
THE METHOD OF SUBORDINATE PROOFS
19
categorically. 0 o.~. The sentence [p" q]o (or 0 p "q 0 if outer brackets are omitted for brevity) is the coimElication or the ~ conditional of 0 p 0 with 0 q 0 , and it is read, "p if and only if q" or "p coimplies q." The sentence 0 p " q o will be regarded as an abbreviation for 0 [q ::> p] a [p => q] 0 • The rule of ~ ponens for coimplication (m p c) asserts (first form) that oqo is a consequence of 0 p 0 and 0 p "q 0 , and also (second form) that 0 p 0 is a consequence of'- 0 q 0 and 0 p = q 0 • This is a derived rule in both its forms (hence we speak of "consequence" instead of "direct consequence"). The derivation of the first form of it is established by P35 thus: P35 2l p p "' q 3 [q ::> p] p :::> q q
;I
a
[p => q 1
hyp hyp 2, rep, def 3. conj elim l, ~. m p
The derivation of the second form is similar. Notice that the reason for step 3 of P35 is repetition and definition. This means merely that step 2 is being repeated, but in a form in which 0 p " qo is replaced by the expression for which 0 p " q 0 is, by definition, an abbreviation. 0.35. Another important derived rule is the rule of ££imElication introduction (coimp int). This rule asserts that op " q 0 is a consequence of two proofs, one of which has 0 p 0 as its only hypothesis and has oqo as a step, and the other of which has 0 q 0 as its only hypothesis and has 0 p 0 as a step. This is a derived rule in the sense that any case of using it, for example, hyp
II II
p - q
hyp
... ' ... ,
co imp int
ELEMENTS OF COMBINATORY LOGIC
18
0.32. The rule of constructive dilemma (cnst dil) is now presented. It treats 0 r V s 0 as a consequence of three things: (l) a sentence 0 pVqo (or 0 qVp 0 ) , (2) a proof with single hypothesis 0 p 0 and with 0 r 0 as one of its steps, and (3) a proof with single hypothesis 0 q 0 and with 0 S 0 as one of its steps. This rule is clearly a derived rule, since the effect of it can be gotten by uses of underived rules, specifically by uses of dis elim and dis int. For example, P33 uses cnst dil, but P3~ gives the same effect by using dis elim and dis int. Also, since it is a derived rule, we speak of a "consequence" obtained by use of the rule, rather than a "direct consequence." P33
l~pV [o 2 3 ~
5
I!. , s ::> p
::> s]
7 8 l
p
2
3
~ ~ ::> p
~
l[s::>p]Vs
5 6
7 8
9 lO
hyp hyp 2, add cond hyp ~. conj elim ~. conj elim 5. 6, m p l, 2-3. 4--7, cnst dil
[O o e]]
q ::> s s [s::>p]Vs
6
P34-
tp
a
r
v
[q
a
[q ::> s]]
hyp hyp 2, add cond 3, dis int hyp 5, conj elim 5, conj elim 6, 7, m p 8, dis int l, 2-4, 5-9, dis elim
~: :· ,,
[s::>p]Vs [s ::> p]Vs
0.33. EXERCISES. Construct proofs of the following: 0 (l). [[p a r]V] ::> [s :::> r] 0 categorically. 0 ( 2) • [ p v ql a [p v q] 0 from 0 [ p a q] v [ p a q] 0 • (3). oq :::> [p :::> r] from [p :::> q] :::> ro. (~). o[p :::> r] :::> [[pV q] :::> [[q :::> s] :::> [rV s]]] o 0
0
THE METHOD OF SUBORDINATE PROOFS
19
categorically. 0 o.~. The sentence [p" q]o (or 0 p "q 0 if outer brackets are omitted for brevity) is the coimElication or the ~ conditional of 0 p 0 with 0 q 0 , and it is read, "p if and only if q" or "p coimplies q." The sentence 0 p " q o will be regarded as an abbreviation for 0 [q ::> p] a [p => q] 0 • The rule of ~ ponens for coimplication (m p c) asserts (first form) that oqo is a consequence of 0 p 0 and 0 p "q 0 , and also (second form) that 0 p 0 is a consequence of'- 0 q 0 and 0 p = q 0 • This is a derived rule in both its forms (hence we speak of "consequence" instead of "direct consequence"). The derivation of the first form of it is established by P35 thus: P35 2l p p "' q 3 [q ::> p] p :::> q q
;I
a
[p => q 1
hyp hyp 2, rep, def 3. conj elim l, ~. m p
The derivation of the second form is similar. Notice that the reason for step 3 of P35 is repetition and definition. This means merely that step 2 is being repeated, but in a form in which 0 p " qo is replaced by the expression for which 0 p " q 0 is, by definition, an abbreviation. 0.35. Another important derived rule is the rule of ££imElication introduction (coimp int). This rule asserts that op " q 0 is a consequence of two proofs, one of which has 0 p 0 as its only hypothesis and has oqo as a step, and the other of which has 0 q 0 as its only hypothesis and has 0 p 0 as a step. This is a derived rule in the sense that any case of using it, for example, hyp
II II
p - q
hyp
... ' ... ,
co imp int
ELEMENTS OF COMBINATORY LOGIC
20
can be treated as a shorthand for a series of steps that use imp int and conj int instead: hyp
r:
... '
p ::> q
imp int
hyp
... ...
' imp int [q ::> pJ Cp ::> qJ ' ••• , conj int p .. q ' rep, def 0.36. Notice that 'q ::> p' can be read as "p if q» and that 'p ::> q' can be read as "p only if q,» so that '[q ::> p] [p ::> q]' (that is, 'p " q') can be read as "p if q, and p only l.i q, t;hat; l.S, "p l.i ana. only l.i q." Here are some proofs that use coimp int and m p c. (The rule m p c, modus ponens for coimplication, could also be called coimplication elimination, since it is the elimination rule for coimplication.) q ::> p
a
...
a
P36
l 2 3 4 5 6 7 8
9 lO ll
12
l3 14
r
" s p r p r r " s s p s p s p s r - s r p r [p li r]
a
a a
a
-
[p li s]
hyp hyp 2, conj elim 2, conj elim l, reit 4, 5. m p c 3. 6, conj int hyp 8, conj elim 8, conj elim l' reit 10, ll, m p c 9. 12, conj int 2-7. 8-13. co imp int
THE METHOD OF SUBORDINATE PROOFS P37
l
hyp hyp hyp l, rei t 3, 4, m p c 2, 3-5, cnst dil hyp / hyp l, rei t 8, 9, m p c 7, 8-10, cnst dil 2-6, 7-ll, coimp int
r "' s
2
3 4 5 6 7 8 9 10 ll
12
rr:" rE'
"
21
s
pV s
.. s
pV r [p V r] "' [p V s]
Notice that in connection with the use of cnst dil in P37 trivial proofs of the form ~ ~ have been omitted. 0.37. Still another derived rule concerned with cairnplication is the rule of transitivit~ £[ coimplication (trans coimp). This rule asserts that 'p "'r' is a consequence of 'p ~ q' and 'q "r'. The derivation of this rule is left as an exercise. There is also a derived rule of ~ ~ 2f coimplication (sym coimp) which asserts that 'q = p' is a consequence of 'p "q', and a derived rule of reflexiv~ 2f coimplication (refl coimp) which asserts that 'p " p' is provable. 0.38. EXERCISES. Construct proofs of the following: (l). 'p " r' from 'p " q' and 'q " r' • (In other words, derive the rule of transitivity of coimplication.) (2). '[[p ::> r] li [q ::> r)] "[[pVq] ::> r]' categorically. ( 3) . (4).
( 5). ( 6).
( 7).
a
' [ p li r] "' [ q s] ' from 'p l! q' and 'r .. s' • '[p V r] " [q V s]' from 'p - q' and 'r - s' . , [p ::> r] [q ::> s]' from 'p • q' and 'r " s'. ,. '[p !! r] [q " s)' from 'p q' and 'r s'. 'p [ [p = p] " p)' categorically.
.
. ..
..
ELEMENTS OF COMBINATORY LOGIC
20
can be treated as a shorthand for a series of steps that use imp int and conj int instead: hyp
r:
... '
p ::> q
imp int
hyp
... ...
' imp int [q ::> pJ Cp ::> qJ ' ••• , conj int p .. q ' rep, def 0.36. Notice that 'q ::> p' can be read as "p if q» and that 'p ::> q' can be read as "p only if q,» so that '[q ::> p] [p ::> q]' (that is, 'p " q') can be read as "p if q, and p only l.i q, t;hat; l.S, "p l.i ana. only l.i q." Here are some proofs that use coimp int and m p c. (The rule m p c, modus ponens for coimplication, could also be called coimplication elimination, since it is the elimination rule for coimplication.) q ::> p
a
...
a
P36
l 2 3 4 5 6 7 8
9 lO ll
12
l3 14
r
" s p r p r r " s s p s p s p s r - s r p r [p li r]
a
a a
a
-
[p li s]
hyp hyp 2, conj elim 2, conj elim l, reit 4, 5. m p c 3. 6, conj int hyp 8, conj elim 8, conj elim l' reit 10, ll, m p c 9. 12, conj int 2-7. 8-13. co imp int
THE METHOD OF SUBORDINATE PROOFS P37
l
hyp hyp hyp l, rei t 3, 4, m p c 2, 3-5, cnst dil hyp / hyp l, rei t 8, 9, m p c 7, 8-10, cnst dil 2-6, 7-ll, coimp int
r "' s
2
3 4 5 6 7 8 9 10 ll
12
rr:" rE'
"
21
s
pV s
.. s
pV r [p V r] "' [p V s]
Notice that in connection with the use of cnst dil in P37 trivial proofs of the form ~ ~ have been omitted. 0.37. Still another derived rule concerned with cairnplication is the rule of transitivit~ £[ coimplication (trans coimp). This rule asserts that 'p "'r' is a consequence of 'p ~ q' and 'q "r'. The derivation of this rule is left as an exercise. There is also a derived rule of ~ ~ 2f coimplication (sym coimp) which asserts that 'q = p' is a consequence of 'p "q', and a derived rule of reflexiv~ 2f coimplication (refl coimp) which asserts that 'p " p' is provable. 0.38. EXERCISES. Construct proofs of the following: (l). 'p " r' from 'p " q' and 'q " r' • (In other words, derive the rule of transitivity of coimplication.) (2). '[[p ::> r] li [q ::> r)] "[[pVq] ::> r]' categorically. ( 3) . (4).
( 5). ( 6).
( 7).
a
' [ p li r] "' [ q s] ' from 'p l! q' and 'r .. s' • '[p V r] " [q V s]' from 'p - q' and 'r - s' . , [p ::> r] [q ::> s]' from 'p • q' and 'r " s'. ,. '[p !! r] [q " s)' from 'p q' and 'r s'. 'p [ [p = p] " p)' categorically.
.
. ..
..
22
ELEMENTS OF COMBINATORY LOGIC
0.~9. It is to be recalled that '-p' is the negation of 'p' and is read, "pis false" or "it is not the case that p." Different systems of logic can be constructed, depending upon what rules for negation are added to the rules so far presented. The system of classical ~-~ 19gic is obtained by adding a rule called indirect Eroof (ind pr) to the present rules. This rule asserts that 'p' is a direct consequence of any proof that has '-p' as its only hypothesis and has a pair of contradictory steps, 'q' and '-q' (one of which could be the hypothesis '-p' itself), All the other negation rules that are described below are derivable from the rule of indirect proof and so are derived rules in the classical system. 0.40. If instead of adding indirect proof to previous rules, we add the rules negation introduction (neg int) and negation elimination (neg elim), we obtaln tne Heyting ~ ~ of intuitionistic logic. These two rules are as follows: The rule of negation introduction asserts that '-p' is a direct consequence of any proof that has 'p' as its only hypothesis and has a pair of mutually contradictory steps, 'q' and '-q' (one of which could be the hypothesis 'p' itself), Notice that here 'p' is the hypothesis, while in indirect proof it is '-p' that is the hypothesis. The rule of negation elimination asserts that 'p' is a direct consequence of any pair of mutually contradictory steps, 'q' and '-q' 0.41. The ~ system is obtained by using as negation rules the rule neg elim together with the following further rules for negation which are described below: double nega~ elimination (neg 2 elim); double negation introduction (neg2 int); negative conjunction elimination (neg conj elim); negative conjunction introduction (neg conj int); negative disjunction elimination (neg dis elim); and negative disjunction introduction (neg dis int). These six rules are as follows: The rule of ~ negation elimination asserts that 'p' is a d.c. (direct consequence) of '--p', while the rule of double negation introduction asserts that the latter
THE METHOD OF SUBORDINATE PROOFS
23
is a d.c. of the former. The rule of !l:egative conjunction elimination asserts that '-p V -q' is a d.c. of '[p a q]', while the rule of negative con,junction introduction asserts that the latter is a d.c. of the former. The rule of nega~ disjunction elimination asserts that '-p -q' is a d. c. of '-[pV q] ', while the rule of negative dis,jujlction introduction asserts that the latter is a d.c. of the former. 0.42. It will now be shown that from the rule of indirect proof all the other negation rules are derivable. First, the following proof shows that the rule of double negation elimination is derivable by use of the rule of indirect proof (and so is derivable in classical two-valued logic):
a
P38
1~--p
hyp hyp
21
~-P
~~~ --P
1, reit
4
p
2-3, ind pr
Notice that in the above use of indirect proof one of the two mutually contradictory steps of the subordinate proof is itself the hypothesis of the subordinate proof. 0.43. The rule of negation introduction is derivable from the rule of indirect proof in the sense that any use of negation introduction can be replaced by uses of indirect proof. More specifically, any use of negation introduction such as the following, hyp
-P .•• , neg int can be replaced by the following uses of ind pr and neg elim: 2 hyp •.• , neg2 elim
If'
-P
••• , ind pr
22
ELEMENTS OF COMBINATORY LOGIC
0.~9. It is to be recalled that '-p' is the negation of 'p' and is read, "pis false" or "it is not the case that p." Different systems of logic can be constructed, depending upon what rules for negation are added to the rules so far presented. The system of classical ~-~ 19gic is obtained by adding a rule called indirect Eroof (ind pr) to the present rules. This rule asserts that 'p' is a direct consequence of any proof that has '-p' as its only hypothesis and has a pair of contradictory steps, 'q' and '-q' (one of which could be the hypothesis '-p' itself), All the other negation rules that are described below are derivable from the rule of indirect proof and so are derived rules in the classical system. 0.40. If instead of adding indirect proof to previous rules, we add the rules negation introduction (neg int) and negation elimination (neg elim), we obtaln tne Heyting ~ ~ of intuitionistic logic. These two rules are as follows: The rule of negation introduction asserts that '-p' is a direct consequence of any proof that has 'p' as its only hypothesis and has a pair of mutually contradictory steps, 'q' and '-q' (one of which could be the hypothesis 'p' itself), Notice that here 'p' is the hypothesis, while in indirect proof it is '-p' that is the hypothesis. The rule of negation elimination asserts that 'p' is a direct consequence of any pair of mutually contradictory steps, 'q' and '-q' 0.41. The ~ system is obtained by using as negation rules the rule neg elim together with the following further rules for negation which are described below: double nega~ elimination (neg 2 elim); double negation introduction (neg2 int); negative conjunction elimination (neg conj elim); negative conjunction introduction (neg conj int); negative disjunction elimination (neg dis elim); and negative disjunction introduction (neg dis int). These six rules are as follows: The rule of ~ negation elimination asserts that 'p' is a d.c. (direct consequence) of '--p', while the rule of double negation introduction asserts that the latter
THE METHOD OF SUBORDINATE PROOFS
23
is a d.c. of the former. The rule of !l:egative conjunction elimination asserts that '-p V -q' is a d.c. of '[p a q]', while the rule of negative con,junction introduction asserts that the latter is a d.c. of the former. The rule of nega~ disjunction elimination asserts that '-p -q' is a d. c. of '-[pV q] ', while the rule of negative dis,jujlction introduction asserts that the latter is a d.c. of the former. 0.42. It will now be shown that from the rule of indirect proof all the other negation rules are derivable. First, the following proof shows that the rule of double negation elimination is derivable by use of the rule of indirect proof (and so is derivable in classical two-valued logic):
a
P38
1~--p
hyp hyp
21
~-P
~~~ --P
1, reit
4
p
2-3, ind pr
Notice that in the above use of indirect proof one of the two mutually contradictory steps of the subordinate proof is itself the hypothesis of the subordinate proof. 0.43. The rule of negation introduction is derivable from the rule of indirect proof in the sense that any use of negation introduction can be replaced by uses of indirect proof. More specifically, any use of negation introduction such as the following, hyp
-P .•• , neg int can be replaced by the following uses of ind pr and neg elim: 2 hyp •.• , neg2 elim
If'
-P
••• , ind pr
ELEMENTS OF COMBINATORY LOGIC
24
THE METHOD OF SUBORDINATE PROOFS
Uses of neg 2 elim themselves can be reduced to uses of ind pr, as has already been shown in P38. The rule of negation elimination will next be derived by use of ind pr: P39
1
p
2
-P
3 4
hyp hyp hyp
l;q
1, rei t
-P
2, rei t
5 6
q
3-5. ind pr
Double negation introduction is derivable as follows by use of negation introduction, which, as we have seen, is itself derivable from ind pr: P4-0
lrp :
/I4 --P .~=p ¥
hyp hyp l, relL 2-3, neg int
The remaining negation rules so far mentioned can be fairly easily derived from ind pr, neg int, neg elim, together with the introduction and elimination rules for double negation, conjunction and disjunction, and so these derivations are left as exercises: 0.44. EXERCISES. Using any of the rules ind pr, neg int, neg elim, neg 2 int, neg 2 elim, conj int, conj elim, dis int, and dis elim, derive: (l). neg conj int. (2). neg conj elim. (3). neg dis int. (4). neg dis elim. 0.45. Of the four rules to be derived as exercises above, all but neg conj elim are derivable in the Heyting system (the system which uses neg elim and neg int as the sole basis of its treatment of negation), while all four of them are taken as primitive (underived) rules in the Fitch system. (It should also be mentioned that although the two double negation rules are taken as primitive in the Fitch
I
25
system, only one of them, neg2 int, is even derivable in the Heyting system.) All four of the negative conjunction and negative disjunction rules, furthermore, are special cases ((1) and (5) below) of the~ of De Morgan (d m). This rule is a derived rule in the classical system of two-valued logic and in the Fitch system. I t has eight cases listed below and asserts that each sentence in the left-hand column below is a consequence of, and has as a consequence, the correspending sentence to the right of it. (l). -lp a qJ -P V -q (2). -l-P a qJ p v -q (3). -lP a -qJ -P V q ( 4). -l-P a -qJ p v q ( 5). -[p v q] -P a -q (6). -[-p v q] P a -q ( 7). -[p v -qJ -P a q (8). -[-p v -q] P a q In the Heyting system the above sentences on the left can be shown to be consequences of the corresponding sentences on the right, but only in case (5) is a right-hand sentence a consequence of the corresponding left-hand sentence in that system. (Notice that in the Heyting system sentences that begin with a ne~ation symbol are more likely to be provable than those that do not. This is because the rule neg int, available in the Heyting system, is so powerful with respect to proving sentences that begin with a negation symbol.) A derivation will now be given in the classical two-valued system (and in the Fitch system) of case (2) of De Morgan, going from left to right. '[-H]' means not valid in Heyting. PH
r-P
2
3 4 5
hyp
--P Vh -ql
l,
r--P p p v -q
hyp
neg conj elim [-HJ
3. neg 2 elim [-H] 2, 3-4, cnst dil
It would not be correct to go directly from step 2 to step 5
ELEMENTS OF COMBINATORY LOGIC
24
THE METHOD OF SUBORDINATE PROOFS
Uses of neg 2 elim themselves can be reduced to uses of ind pr, as has already been shown in P38. The rule of negation elimination will next be derived by use of ind pr: P39
1
p
2
-P
3 4
hyp hyp hyp
l;q
1, rei t
-P
2, rei t
5 6
q
3-5. ind pr
Double negation introduction is derivable as follows by use of negation introduction, which, as we have seen, is itself derivable from ind pr: P4-0
lrp :
/I4 --P .~=p ¥
hyp hyp l, relL 2-3, neg int
The remaining negation rules so far mentioned can be fairly easily derived from ind pr, neg int, neg elim, together with the introduction and elimination rules for double negation, conjunction and disjunction, and so these derivations are left as exercises: 0.44. EXERCISES. Using any of the rules ind pr, neg int, neg elim, neg 2 int, neg 2 elim, conj int, conj elim, dis int, and dis elim, derive: (l). neg conj int. (2). neg conj elim. (3). neg dis int. (4). neg dis elim. 0.45. Of the four rules to be derived as exercises above, all but neg conj elim are derivable in the Heyting system (the system which uses neg elim and neg int as the sole basis of its treatment of negation), while all four of them are taken as primitive (underived) rules in the Fitch system. (It should also be mentioned that although the two double negation rules are taken as primitive in the Fitch
I
25
system, only one of them, neg2 int, is even derivable in the Heyting system.) All four of the negative conjunction and negative disjunction rules, furthermore, are special cases ((1) and (5) below) of the~ of De Morgan (d m). This rule is a derived rule in the classical system of two-valued logic and in the Fitch system. I t has eight cases listed below and asserts that each sentence in the left-hand column below is a consequence of, and has as a consequence, the correspending sentence to the right of it. (l). -lp a qJ -P V -q (2). -l-P a qJ p v -q (3). -lP a -qJ -P V q ( 4). -l-P a -qJ p v q ( 5). -[p v q] -P a -q (6). -[-p v q] P a -q ( 7). -[p v -qJ -P a q (8). -[-p v -q] P a q In the Heyting system the above sentences on the left can be shown to be consequences of the corresponding sentences on the right, but only in case (5) is a right-hand sentence a consequence of the corresponding left-hand sentence in that system. (Notice that in the Heyting system sentences that begin with a ne~ation symbol are more likely to be provable than those that do not. This is because the rule neg int, available in the Heyting system, is so powerful with respect to proving sentences that begin with a negation symbol.) A derivation will now be given in the classical two-valued system (and in the Fitch system) of case (2) of De Morgan, going from left to right. '[-H]' means not valid in Heyting. PH
r-P
2
3 4 5
hyp
--P Vh -ql
l,
r--P p p v -q
hyp
neg conj elim [-HJ
3. neg 2 elim [-H] 2, 3-4, cnst dil
It would not be correct to go directly from step 2 to step 5
ELEMENTS OF COMBINATORY LOGIC
26
by neg 2 elim. This is because the rules do not apply to £arts of sentences that are steps in proofs, but only to ~ sentences that are steps in proofs. o.~6. EXERCISES. Proofs of the remaining cases of d m in the classical and Fitch systems are left as exercises, also proofs in the Heyting system of those cases of d m that are derivable in the Heyting system. 0.~7. The following proof establishes the rule of ~ eluded middle (ex mid) in the classical system. This rule asserts that 'p V -p' is provable (for every sentence 'p'). It is not derivable in either the Heyting or Fitch systems. '[-H)' means, as before, not valid_ in Heyting, while '[-F)' means not valid in Fitch. P42
~I 3
~
5 6 7 8
r-[p j_p
v
-p]
I p v -P -[p
v
-P p V -P --[p v -p] p V -P
-p]
hyp hyp 2, dis int 1, reit 2-~. neg int [-F) 5. dis int 1-6, neg int [-F) neg 2 elim [-H)
Steps 1-7 of ~+2 can be taken to show that the double negation of excluded middle is derivable in the Heyting system, even though excluded middle itself is not derivable. 0.~8. The main negation rules that do not hold in the Heyting system are ind pr, neg 2 elim, cases (1)-(~). (6)-(8) of d m going from left to right in the table in 0.~5. and ex mid. The main negation rules that do not hold in the Fitch system are ind pr, neg int, and ex mid. The rules neg 2 int and neg elim hold in all three systems. 0.~9. Another rule that holds in all three systems is the rule of~ tollendo ponens (m t p). This rule asserts that 'q' is a consequence of any of the following pairs of sentences: (1). 'p' and '-p V q'
THE METHOD OF SUBORDINATE PROOFS
27
(2).
'p' and 'q V -p' '-p' and 'p V q' (~). '-p' and 'q V p-' The derivation of case (1) of m t p is as follows, the other cases being similarly derivable:
(3).
P43
1 2
3 ~
5 6
p
-P V
~~p q
o.50.
q
hyp hyp hyp 1, reit 3, ~. neg elim 2, 3-5, dis elim
EXERCISES. (l). Give a proof of excluded middle that uses only ind pr, dis int, and reit. (2)# The rule of m t p could have been chosen as a primitive (underived) rule, and then neg elim would be derivable using m t p and dis int. Give such a derivation of m t p. (Notice that this result shows that neg elim cannot be regarded as objectionable without also regarding either m t p or dis int as objectionable.) 0.51. The system Q of the main body of the present book does not use all the negation rules of classical two-valued logic. It uses only those negation rules that the Fitch system uses or could consistently use. In particular, Q does not use ind pr, neg int, or ex mid, although limited forms of these rules are permitted in which the sentences involved are of the form 'a z b' or 'Na' or negates of either of these. (For example, see 5.8 and 12.12.) 0.52. The various implication rules described above are not included among the rules of the system Q. If 'p ~ q' is treated as an abbreviation for '-p V q', however, the rule of modus ponens is then seen to be merely a case of m t p and therefore to hold in system Q. On the other hand, if 'p ~ q' were to be defined in the way just indicated, the rule of implication introduction would fail to hold in Q.
ELEMENTS OF COMBINATORY LOGIC
26
by neg 2 elim. This is because the rules do not apply to £arts of sentences that are steps in proofs, but only to ~ sentences that are steps in proofs. o.~6. EXERCISES. Proofs of the remaining cases of d m in the classical and Fitch systems are left as exercises, also proofs in the Heyting system of those cases of d m that are derivable in the Heyting system. 0.~7. The following proof establishes the rule of ~ eluded middle (ex mid) in the classical system. This rule asserts that 'p V -p' is provable (for every sentence 'p'). It is not derivable in either the Heyting or Fitch systems. '[-H)' means, as before, not valid_ in Heyting, while '[-F)' means not valid in Fitch. P42
~I 3
~
5 6 7 8
r-[p j_p
v
-p]
I p v -P -[p
v
-P p V -P --[p v -p] p V -P
-p]
hyp hyp 2, dis int 1, reit 2-~. neg int [-F) 5. dis int 1-6, neg int [-F) neg 2 elim [-H)
Steps 1-7 of ~+2 can be taken to show that the double negation of excluded middle is derivable in the Heyting system, even though excluded middle itself is not derivable. 0.~8. The main negation rules that do not hold in the Heyting system are ind pr, neg 2 elim, cases (1)-(~). (6)-(8) of d m going from left to right in the table in 0.~5. and ex mid. The main negation rules that do not hold in the Fitch system are ind pr, neg int, and ex mid. The rules neg 2 int and neg elim hold in all three systems. 0.~9. Another rule that holds in all three systems is the rule of~ tollendo ponens (m t p). This rule asserts that 'q' is a consequence of any of the following pairs of sentences: (1). 'p' and '-p V q'
THE METHOD OF SUBORDINATE PROOFS
27
(2).
'p' and 'q V -p' '-p' and 'p V q' (~). '-p' and 'q V p-' The derivation of case (1) of m t p is as follows, the other cases being similarly derivable:
(3).
P43
1 2
3 ~
5 6
p
-P V
~~p q
o.50.
q
hyp hyp hyp 1, reit 3, ~. neg elim 2, 3-5, dis elim
EXERCISES. (l). Give a proof of excluded middle that uses only ind pr, dis int, and reit. (2)# The rule of m t p could have been chosen as a primitive (underived) rule, and then neg elim would be derivable using m t p and dis int. Give such a derivation of m t p. (Notice that this result shows that neg elim cannot be regarded as objectionable without also regarding either m t p or dis int as objectionable.) 0.51. The system Q of the main body of the present book does not use all the negation rules of classical two-valued logic. It uses only those negation rules that the Fitch system uses or could consistently use. In particular, Q does not use ind pr, neg int, or ex mid, although limited forms of these rules are permitted in which the sentences involved are of the form 'a z b' or 'Na' or negates of either of these. (For example, see 5.8 and 12.12.) 0.52. The various implication rules described above are not included among the rules of the system Q. If 'p ~ q' is treated as an abbreviation for '-p V q', however, the rule of modus ponens is then seen to be merely a case of m t p and therefore to hold in system Q. On the other hand, if 'p ~ q' were to be defined in the way just indicated, the rule of implication introduction would fail to hold in Q.
28
ELEMENTS OF COMBINATORY LOGIC
Its place has to be taken by the somewhat weaker rule of restricted implication introduction (res imp int). This latter rule, which is derivable in system Q, states that the implication 'p ~ q' is a consequence of two things: (l) the disjunction 'p V -p', and (2) a proof having 'p' as its only hypothesis and having 'q' as one of its steps. The derivain the Fitch system is as follows: P44
1
p V -P
2 n n+l n+2
rr
-P V q p ~ q
CHAPTER l FUNCTIONS
hyp hyp
1. l, 2-n, cnst dil n+l, rep, def
Remarks
£E
Functions
1.1. In the usual notation of mathematics we might define a function f(x) in such a way that f(x) = x 2 + 6x + 3, and another function g(x) in such a way that g(x) = 2x + 5. These are merely examples of functions to illustrate the ordinary notation. These functions should actually be referred to as f and g, rather than as f(x) and g(x). 1.2. A function may be aEplied to a number. For example, if the function f described above is applied to the number 6, the result is f(6), and the defining equation for f 2 2 shows that f(6) = 6 + 6 + 3 = 75. Similarly, g(5) = 15, so that the result of applying g to 5 is 15. Generally for each function there is specified a so-called ~ £f ~ nition. This is the class of things (such as integers or fractions or real numbers, etc.) to which the fuction can be meaningfully applied. Thus in applying f to 6 we assumed that 6 was in the domain of definition of f, By an ~ ~ of a function will be meant anything to which the function may be applied. The domain of definition of a function can then be said to be the class of applicands of that function. The result of applying a function to one of its applicands is called the ~ of the function for that applicand. Thus 75 is the value of f for the applicand 6, while 94 is its value for the applicand 7. The class of all values of a
29
28
ELEMENTS OF COMBINATORY LOGIC
Its place has to be taken by the somewhat weaker rule of restricted implication introduction (res imp int). This latter rule, which is derivable in system Q, states that the implication 'p ~ q' is a consequence of two things: (l) the disjunction 'p V -p', and (2) a proof having 'p' as its only hypothesis and having 'q' as one of its steps. The derivain the Fitch system is as follows: P44
1
p V -P
2 n n+l n+2
rr
-P V q p ~ q
CHAPTER l FUNCTIONS
hyp hyp
1. l, 2-n, cnst dil n+l, rep, def
Remarks
£E
Functions
1.1. In the usual notation of mathematics we might define a function f(x) in such a way that f(x) = x 2 + 6x + 3, and another function g(x) in such a way that g(x) = 2x + 5. These are merely examples of functions to illustrate the ordinary notation. These functions should actually be referred to as f and g, rather than as f(x) and g(x). 1.2. A function may be aEplied to a number. For example, if the function f described above is applied to the number 6, the result is f(6), and the defining equation for f 2 2 shows that f(6) = 6 + 6 + 3 = 75. Similarly, g(5) = 15, so that the result of applying g to 5 is 15. Generally for each function there is specified a so-called ~ £f ~ nition. This is the class of things (such as integers or fractions or real numbers, etc.) to which the fuction can be meaningfully applied. Thus in applying f to 6 we assumed that 6 was in the domain of definition of f, By an ~ ~ of a function will be meant anything to which the function may be applied. The domain of definition of a function can then be said to be the class of applicands of that function. The result of applying a function to one of its applicands is called the ~ of the function for that applicand. Thus 75 is the value of f for the applicand 6, while 94 is its value for the applicand 7. The class of all values of a
29
30
ELEMENTS OF COMBINATORY LOGIC
function for all its applicands is called the domain £f ~ of the function. 1.3. If we wish to refer to a whole series or class of applicands of a function, we use a letter from the end of the alphabet called a variable. Thus in the equation, f(x). x 2 + 6x + 3, the letter 'x' serves as a variable. Used in this way, a variable is called an argument of a function. More specifically, 'x' is said to serve as the argument of f in the above equation. (In practice, applicands are often also called arguments. Thus we might say that 6 is the argument of f in the equation f( 6) = 62 + 6 2 + 3, but this is a less accurate way of speaking. It is more accurate to regard the argument of a function as a ~ which is being used as a variable.) Thus by way of a variable we may refer not to just one applicand but rather to a whole class of applicands. 11' the numbers 1, 2, and ; are among the appl1cands referred to by the variable 'x', then in asserting the equation, f(x) = x 2 + 6x + 3, we are also implicitly asserting the equations f(l) = l + 6 + 3, f(2) = 4 + 12 + 3, and f( 3) = 9 + 18 + 3. The class of applicands to which a variable refers is called the range of the variable, and each such applicand is called a ~ of the variable. One variable, for example, might have the class of rational numbers for its range, while another might have the class of real numbers. When a variable is being used as an argument for a function, the range of the variable would not be expected to be more extensive than the domain of definition of the function, but might be less extensive. In many cases they would coincide. 1.4. We may also use a variable to refer ambiguously to all the values of a function for all applicands in the range of some argument of the function, say 'x'. For example, 'y' is used as just such a variable when we write y = f(x). In this case we distinguish the variable 'y' from the argument 'x' by calling 'y' the deEendent variable and 'x' the ~ Eendent variable. ~
FUNCTIONAL NOTATION IN COMBINATORY'LOGIC
31
1.5. Some functions are ~-place functio~s (often called functions £f ~ variables) in contrast to the ~-Elace functions we have so far been considering. An example of such a two-place function is the function h defined by the equation, h(x,y) = x 2 + 2xy + y 2 Such a function must be applied to two applicands, rather than merely to one applicand. For instance, the value of h for the pair of applicands (1,0) is l, since the defining equation for h gives h(l.O) = l + 0 + 0 = l. Similar considerations apply to threeplace functions, four-place functions, and so on. 2.
Functional Notation Qf
Combinate~
Logic
2.1. We now introduce a notation that has become established in the branch of logic called combinate~ logic. Instead of writing 'f(6)' for the application of f to 6, we write '(f6)', or. if no ambiguity is caused by doing so, we write simply 'f6'. (The quotation marks are not themselves part of the notation. See 0.4.) Also, in the case of a twoplace function, we would write '((hl)O)' instead of 'h(l,O)', and the notation '((hl)O)' can be further simplified to '(hlO)' and even to 'hlO' if no ambiguity results. (It would be understood that the number ten is not referred to in such an expression, because two-digit numbers would be written in some special form, such as '!0'.) The fact that the expression '((hl)O)' can be abbreviated as '(hlO)' is a special instance of the general convention in combinatory logic that omitted parentheses are always understood as inserted as far to the left as possible. Thus, in general, '(abc)' is an abbreviation for '((ab)c)', while '(abed)' is an abbreviation for '(((ab)c)d)', and so on. For example, '(ab(cde)fg)' would be an abbreviation for '((((ab)((cd)e))f)g)'. Also, outermost parentheses may be omitted when no ambiguity results from doing so. 2.2. Since the value of the two-place function h for the pair of applicands (1,0) is written as 'hlO' in combinatory logic, it is seen that there is a double use of application. First h is applied to l to obtain hl, and then
30
ELEMENTS OF COMBINATORY LOGIC
function for all its applicands is called the domain £f ~ of the function. 1.3. If we wish to refer to a whole series or class of applicands of a function, we use a letter from the end of the alphabet called a variable. Thus in the equation, f(x). x 2 + 6x + 3, the letter 'x' serves as a variable. Used in this way, a variable is called an argument of a function. More specifically, 'x' is said to serve as the argument of f in the above equation. (In practice, applicands are often also called arguments. Thus we might say that 6 is the argument of f in the equation f( 6) = 62 + 6 2 + 3, but this is a less accurate way of speaking. It is more accurate to regard the argument of a function as a ~ which is being used as a variable.) Thus by way of a variable we may refer not to just one applicand but rather to a whole class of applicands. 11' the numbers 1, 2, and ; are among the appl1cands referred to by the variable 'x', then in asserting the equation, f(x) = x 2 + 6x + 3, we are also implicitly asserting the equations f(l) = l + 6 + 3, f(2) = 4 + 12 + 3, and f( 3) = 9 + 18 + 3. The class of applicands to which a variable refers is called the range of the variable, and each such applicand is called a ~ of the variable. One variable, for example, might have the class of rational numbers for its range, while another might have the class of real numbers. When a variable is being used as an argument for a function, the range of the variable would not be expected to be more extensive than the domain of definition of the function, but might be less extensive. In many cases they would coincide. 1.4. We may also use a variable to refer ambiguously to all the values of a function for all applicands in the range of some argument of the function, say 'x'. For example, 'y' is used as just such a variable when we write y = f(x). In this case we distinguish the variable 'y' from the argument 'x' by calling 'y' the deEendent variable and 'x' the ~ Eendent variable. ~
FUNCTIONAL NOTATION IN COMBINATORY'LOGIC
31
1.5. Some functions are ~-place functio~s (often called functions £f ~ variables) in contrast to the ~-Elace functions we have so far been considering. An example of such a two-place function is the function h defined by the equation, h(x,y) = x 2 + 2xy + y 2 Such a function must be applied to two applicands, rather than merely to one applicand. For instance, the value of h for the pair of applicands (1,0) is l, since the defining equation for h gives h(l.O) = l + 0 + 0 = l. Similar considerations apply to threeplace functions, four-place functions, and so on. 2.
Functional Notation Qf
Combinate~
Logic
2.1. We now introduce a notation that has become established in the branch of logic called combinate~ logic. Instead of writing 'f(6)' for the application of f to 6, we write '(f6)', or. if no ambiguity is caused by doing so, we write simply 'f6'. (The quotation marks are not themselves part of the notation. See 0.4.) Also, in the case of a twoplace function, we would write '((hl)O)' instead of 'h(l,O)', and the notation '((hl)O)' can be further simplified to '(hlO)' and even to 'hlO' if no ambiguity results. (It would be understood that the number ten is not referred to in such an expression, because two-digit numbers would be written in some special form, such as '!0'.) The fact that the expression '((hl)O)' can be abbreviated as '(hlO)' is a special instance of the general convention in combinatory logic that omitted parentheses are always understood as inserted as far to the left as possible. Thus, in general, '(abc)' is an abbreviation for '((ab)c)', while '(abed)' is an abbreviation for '(((ab)c)d)', and so on. For example, '(ab(cde)fg)' would be an abbreviation for '((((ab)((cd)e))f)g)'. Also, outermost parentheses may be omitted when no ambiguity results from doing so. 2.2. Since the value of the two-place function h for the pair of applicands (1,0) is written as 'hlO' in combinatory logic, it is seen that there is a double use of application. First h is applied to l to obtain hl, and then
32 hl
is applied to 0 to obtain
ELEMENTS OF COMBINATORY LOGIC
FUNCTIONAL NOTATION IN COMBINATORY LOGIC
hlO.
ha a .•• an when f is defined as being ha 0 and use is 0 1 made of the convention that omitted parentheses are assumed
theses, we would say that first
h
If we retained parenis applied to l to ob-
tain (hl), and then (hl) is applied to 0 to obtain ((hl)O). But, by convention, parentheses may be omitted here. 2.4. In the case of a two-place function
h
it might
seem that 'hl' is a meaningless expression since it denotes the result of applying the two-place function to only a sin-
33
to converge to the left. 2.6. It is worth noticing that an equation of the form, h(x)
= f(g(x)),
would appear as follows in combinatory no-
tation: hx = f(gx). The parentheses around 'gx' cannot be omitted. because then the expression 'fgx' would rightly be
gle applicand. But actually 'hl' can be viewed as denoting
interpreted as '(fg)x', and this is not intended. Similarly
the function
an equation of the form,
f
which, in ordinary functional notation, is
such that
f(y) = h(l,y),
fy
For example, choosing
=
hly.
h
as in paragraph 1.5,
but writing the left side of the defining equation for
h
and
f
satisfy the same equation and are there-
fore the same function. As another example,
hb
would be
the one-place function having the following property (where the left side of the equation is written in combinatory notation): h6y = 36 + 12y + y 2 . 2.5. In general, if
h
is any two-place function and
if (a,b) is an ordered pair of applicands in the domain of definition of b
h,
then
ha
2.7. EXERCISES. (1). Rewrite, omitting parentheses that can be
ha
or in combinatory notation,
because the equation, (ha)b = hab,
hab = hab,
fb
= hab.
f(b)
This is
can also be written as,
owing to the convention about parentheses, so
that if (ha) is called 'f'
derstood as present in combinatory notation: ab(c(defg)) (3).
then
fb = hab.
More generally
nary functional notation• fxy(gxy(hx)) 3.
h
is an n+l-place function and if the ordered
n+l-tuple (a ,a , .•• ,an) is in its domain of definition, 0 1 then ha 0 is an n-place function having the ordered n-tuple (a 1 , ..• ,an) in its domain of definition. Furthermore, if ha 0 is called 'f'. then in ordinary functional notation we have, f(a 1 , ... ,an) = h(a ,a , •.• ,an)' or in combinatory 0 1 notation we have, fa ... a = ha a ..• a . This is because 1 n o l n the latter equation is simply the identity, ha a ••• an = 0 1
.Q.f
~ ~
~-Functions
3.1. We now proceed to construct a general system Q that deals with functions and uses combinatory notation. The system Q can be shown to be consistent. 3.2. The following symbols will designate the primitive (i.e. undefined) functions dealt with by the system Q:
Ia
I '
still, if
Transform from ordinary functional notation.
to combinatory notation: f(g(x, h(z,y,x))) (4). Transform from combinatory notation to ordi-
is cal-
led 'f', then we have in ordinary functional notation,
= h(a,b),
omitted in combinatory notation: (((a(bc))((de)f))g) (2). Rew.t'i~e, .in::>~rt.lng yarentheses that ar·e un-
is a one-place function having
in its domain of definition. Furthermore, if
Here the
sons.
in combinatory notation, we have, hxy = 2 hly = l+2y+y • Since, also, fy = hly = l+2y+y 2 , we see hl
would ap-
hxy = fx(gxy).
parentheses around 'gxy' cannot be omitted for similar rea-
x 2 + 2xy + y 2 • Hence,
that
h(x,y) = f(x, g(x,y)),
pear in combinatory notation as,
or, in combinatory notation,
I
vI '
I
:=II
I '
I
AI
'
IE I
'
I
I
I '
I
BI
'
t
c
I '
I
w
I '
I
KI
'
IN I
•
'-', The
properties of these functions will be explained subsequently. Other primitive functions can be added, if desired, to stand for various scientific or philosophical concepts. 3.3. All the functions with which ~he system Q will deal will be called ~-functions. This class of functions can be defined by induction as follows: (1)
Every one of the primitive functions is a Q-
32 hl
is applied to 0 to obtain
ELEMENTS OF COMBINATORY LOGIC
FUNCTIONAL NOTATION IN COMBINATORY LOGIC
hlO.
ha a .•• an when f is defined as being ha 0 and use is 0 1 made of the convention that omitted parentheses are assumed
theses, we would say that first
h
If we retained parenis applied to l to ob-
tain (hl), and then (hl) is applied to 0 to obtain ((hl)O). But, by convention, parentheses may be omitted here. 2.4. In the case of a two-place function
h
it might
seem that 'hl' is a meaningless expression since it denotes the result of applying the two-place function to only a sin-
33
to converge to the left. 2.6. It is worth noticing that an equation of the form, h(x)
= f(g(x)),
would appear as follows in combinatory no-
tation: hx = f(gx). The parentheses around 'gx' cannot be omitted. because then the expression 'fgx' would rightly be
gle applicand. But actually 'hl' can be viewed as denoting
interpreted as '(fg)x', and this is not intended. Similarly
the function
an equation of the form,
f
which, in ordinary functional notation, is
such that
f(y) = h(l,y),
fy
For example, choosing
=
hly.
h
as in paragraph 1.5,
but writing the left side of the defining equation for
h
and
f
satisfy the same equation and are there-
fore the same function. As another example,
hb
would be
the one-place function having the following property (where the left side of the equation is written in combinatory notation): h6y = 36 + 12y + y 2 . 2.5. In general, if
h
is any two-place function and
if (a,b) is an ordered pair of applicands in the domain of definition of b
h,
then
ha
2.7. EXERCISES. (1). Rewrite, omitting parentheses that can be
ha
or in combinatory notation,
because the equation, (ha)b = hab,
hab = hab,
fb
= hab.
f(b)
This is
can also be written as,
owing to the convention about parentheses, so
that if (ha) is called 'f'
derstood as present in combinatory notation: ab(c(defg)) (3).
then
fb = hab.
More generally
nary functional notation• fxy(gxy(hx)) 3.
h
is an n+l-place function and if the ordered
n+l-tuple (a ,a , .•• ,an) is in its domain of definition, 0 1 then ha 0 is an n-place function having the ordered n-tuple (a 1 , ..• ,an) in its domain of definition. Furthermore, if ha 0 is called 'f'. then in ordinary functional notation we have, f(a 1 , ... ,an) = h(a ,a , •.• ,an)' or in combinatory 0 1 notation we have, fa ... a = ha a ..• a . This is because 1 n o l n the latter equation is simply the identity, ha a ••• an = 0 1
.Q.f
~ ~
~-Functions
3.1. We now proceed to construct a general system Q that deals with functions and uses combinatory notation. The system Q can be shown to be consistent. 3.2. The following symbols will designate the primitive (i.e. undefined) functions dealt with by the system Q:
Ia
I '
still, if
Transform from ordinary functional notation.
to combinatory notation: f(g(x, h(z,y,x))) (4). Transform from combinatory notation to ordi-
is cal-
led 'f', then we have in ordinary functional notation,
= h(a,b),
omitted in combinatory notation: (((a(bc))((de)f))g) (2). Rew.t'i~e, .in::>~rt.lng yarentheses that ar·e un-
is a one-place function having
in its domain of definition. Furthermore, if
Here the
sons.
in combinatory notation, we have, hxy = 2 hly = l+2y+y • Since, also, fy = hly = l+2y+y 2 , we see hl
would ap-
hxy = fx(gxy).
parentheses around 'gxy' cannot be omitted for similar rea-
x 2 + 2xy + y 2 • Hence,
that
h(x,y) = f(x, g(x,y)),
pear in combinatory notation as,
or, in combinatory notation,
I
vI '
I
:=II
I '
I
AI
'
IE I
'
I
I
I '
I
BI
'
t
c
I '
I
w
I '
I
KI
'
IN I
•
'-', The
properties of these functions will be explained subsequently. Other primitive functions can be added, if desired, to stand for various scientific or philosophical concepts. 3.3. All the functions with which ~he system Q will deal will be called ~-functions. This class of functions can be defined by induction as follows: (1)
Every one of the primitive functions is a Q-
ELEMENTS OF COMBINATORY LOGIC
35
THE CLASS OF Q-FUNCTIONS
function.
that
If a and b are Q-functions, then b belongs to the domain of definition of a, and the result of the application of a to b, namely ab, is itself a Q-
identity with
after lower-case (i.e. non-capital) letters will be used, as above, to refer to Q-functions, unless there is some
function.
indication to the contrary.]
(3) The only Q-functions are those functions that can be shown to be Q-functions in virtue of (1) and (2). (Notice that we are assuming that the domain of definition
course is the proposition " I is identical with BI. " This Q-function can also be expressed by the notation, '[I= BI]'
of each Q-function is the whole class of Q-functions, so
in general,
that every Q-function can be applied to every other Q-function and also to itself.)
'bca', that is, for '((bc)a)', and that '[a ... an b c •.• cm]' 1 1 may be treated as an abbreviation for 'b(c 1 ••• cm)(a 1 •.. an)',
3.4, The class of ~ 2£ ~-functions, also called the class of Q-formulas, can be defined by induction in a way
that is, for '((b(c .•• cm))(a 1 ••• an))'. Outer square brack1 ets may be omitted, in this notation, if' no ambiguity re-
that is parallel to the way in which the class of Q-functions itself is defined:
the central expression 'b' cannot be omitted.
(2)
( l)
itive Q-functions are Q-formulas (and indeed may be said to (2)
or simply,
If the expressions 'a' and 'b' are Q-formulas,
is the proposition "a BI "
or " a
has the attribute of
is identical with
The Q-function
BI. "
=(BI)I
[Here-
of'
'I = BI' in accordance with the convention that '[abc)' may be treated as an abbreviation for
sults from doing so, but the blank spaces on each side of'
3.6. Some Q-functicns are two-place relations, and the
'l'he symbols wh1ch are the names o1' the prlm-
be Erimitive 2-formulas).
=(BI)a
values of such Q-functions are attributes. In general, if r is a Q-function that is a two-place relation, then rb, the application of
r
to
b,
is the attribute of bearing the
so is the expression '(ab)' or any abbreviation of it such as 'ab'.
relation r to b. For example, the Q-function denoted by the symbol '=' is the two-place relation identit;r, so that
(3) The only Q-formulas are those expressions that can be shown to be Q-formulas in virtue of (1) and (2), (For example, 'C' and 'K' are Q-formulas by (1), since they
=b is the attribute, "bearing the relation identity to b" or "being identical with b," or simply, "identity with b." Thus '=ba' denotes the proposition, " a
are names of primitive Q-functions, so 'CK' must be a Q-f'or-
identity with
mula by (2). Similarly, mulas.)
proposition can also be written as '[a= b)' or 'a= b' in
'=(CK)' and '=(CK)I' would be Q-for-
b,"
or " a
has the attribute of
is identical with
3.5. Some Q-functions are attributes (i.e. properties),
which asserts that
eral, if
has the attribute of bearing the relation
is a Q-function that is an attribute, then
fa,
bears
r
to
b
the proposition
and the values of such Q-functions are propositions. In genf
a
r,
This
3·5· Similarly, in
accordance with the convention stated in the case of any two-place relation
b."
(that is, that r
to
a
b) can be
the application of' f to a, is the proposition asserting that a has attribute f, For example, the Q-function de-
written as 'rba' or as '[arb)' or (if no ambiguity re-
noted by the symbol '-' is the attribute falsity, so that -a
disjunction may be viewed as two-place relations among Q-
is the proposition " a
functions, so that the conjunction of a with b may be written as '&ba' or as 'a & b', and the disjunction of a
is false."
has the attribute falsity"
As another example, the
~-function
the Q-formula '=(BI)' is the attribute identity
or " a
denoted by with~.
so
sults) merely as ' a r b ' .
with
b
In the system
q, conjunction and
may be written as 'Vba' or as 'a V b'.
ELEMENTS OF COMBINATORY LOGIC
35
THE CLASS OF Q-FUNCTIONS
function.
that
If a and b are Q-functions, then b belongs to the domain of definition of a, and the result of the application of a to b, namely ab, is itself a Q-
identity with
after lower-case (i.e. non-capital) letters will be used, as above, to refer to Q-functions, unless there is some
function.
indication to the contrary.]
(3) The only Q-functions are those functions that can be shown to be Q-functions in virtue of (1) and (2). (Notice that we are assuming that the domain of definition
course is the proposition " I is identical with BI. " This Q-function can also be expressed by the notation, '[I= BI]'
of each Q-function is the whole class of Q-functions, so
in general,
that every Q-function can be applied to every other Q-function and also to itself.)
'bca', that is, for '((bc)a)', and that '[a ... an b c •.• cm]' 1 1 may be treated as an abbreviation for 'b(c 1 ••• cm)(a 1 •.. an)',
3.4, The class of ~ 2£ ~-functions, also called the class of Q-formulas, can be defined by induction in a way
that is, for '((b(c .•• cm))(a 1 ••• an))'. Outer square brack1 ets may be omitted, in this notation, if' no ambiguity re-
that is parallel to the way in which the class of Q-functions itself is defined:
the central expression 'b' cannot be omitted.
(2)
( l)
itive Q-functions are Q-formulas (and indeed may be said to (2)
or simply,
If the expressions 'a' and 'b' are Q-formulas,
is the proposition "a BI "
or " a
has the attribute of
is identical with
The Q-function
BI. "
=(BI)I
[Here-
of'
'I = BI' in accordance with the convention that '[abc)' may be treated as an abbreviation for
sults from doing so, but the blank spaces on each side of'
3.6. Some Q-functicns are two-place relations, and the
'l'he symbols wh1ch are the names o1' the prlm-
be Erimitive 2-formulas).
=(BI)a
values of such Q-functions are attributes. In general, if r is a Q-function that is a two-place relation, then rb, the application of
r
to
b,
is the attribute of bearing the
so is the expression '(ab)' or any abbreviation of it such as 'ab'.
relation r to b. For example, the Q-function denoted by the symbol '=' is the two-place relation identit;r, so that
(3) The only Q-formulas are those expressions that can be shown to be Q-formulas in virtue of (1) and (2), (For example, 'C' and 'K' are Q-formulas by (1), since they
=b is the attribute, "bearing the relation identity to b" or "being identical with b," or simply, "identity with b." Thus '=ba' denotes the proposition, " a
are names of primitive Q-functions, so 'CK' must be a Q-f'or-
identity with
mula by (2). Similarly, mulas.)
proposition can also be written as '[a= b)' or 'a= b' in
'=(CK)' and '=(CK)I' would be Q-for-
b,"
or " a
has the attribute of
is identical with
3.5. Some Q-functions are attributes (i.e. properties),
which asserts that
eral, if
has the attribute of bearing the relation
is a Q-function that is an attribute, then
fa,
bears
r
to
b
the proposition
and the values of such Q-functions are propositions. In genf
a
r,
This
3·5· Similarly, in
accordance with the convention stated in the case of any two-place relation
b."
(that is, that r
to
a
b) can be
the application of' f to a, is the proposition asserting that a has attribute f, For example, the Q-function de-
written as 'rba' or as '[arb)' or (if no ambiguity re-
noted by the symbol '-' is the attribute falsity, so that -a
disjunction may be viewed as two-place relations among Q-
is the proposition " a
functions, so that the conjunction of a with b may be written as '&ba' or as 'a & b', and the disjunction of a
is false."
has the attribute falsity"
As another example, the
~-function
the Q-formula '=(BI)' is the attribute identity
or " a
denoted by with~.
so
sults) merely as ' a r b ' .
with
b
In the system
q, conjunction and
may be written as 'Vba' or as 'a V b'.
36
ELEMENTS OF COMBINATORY LOGIC 3.7. Some Q-functions are three-place relations, others
are four-place relations, and so on. In general, the values of n+l-place relations are n-place relations. (Attributes and classes can be viewed as being one-place relations, and propositions as being zero-place relations.) Thus if an n+l-place relation, then for the applicand
a)
ra
r
is
(which is the value of
is an n-place relation. In case
r
CHAPrER 2
r
is a zero-place relation, that is, a proposition, matters
BOOLEAN CONCEPrS AND IDENTITY
cah be so arranged that ra is also a proposition. The details of this need not concern us here. 3.8. It should also be mentioned that 'ab' can be used to assert that
b
is a member of the class
being used to assert that
b
a,
as well as
has the attribute
a.
4.
This is
because one-place relations can be viewed as being classes th~n
R
rRn
~lso
a is the atbe viewed as being the
class of green things, and 'ab' can mean that
b
is green (i.e. has the attribute green). Similarly, if is a class, then 'aa' can mean that self, and '-(aa)' can mean that
a
a
b
a
is a member of it-
is not a member of it-
of all classes that are not members of themselves. EXERCISES. Suppose that 'L' is an additional prim-
itive Q-formula that denotes the relation ~· and that 'J' and 'M' are additional primitive Q-formulas that denote two persons, Jack and Mary (assuming that in some sense persons can be Q-functions). (1). What Q-formula would denote the proposition, "Jack loves Mary"? (2). What Q-formula would denote the proposition, "Jack does not love Mary, or Mary does not love Jack"? (3).
~
I1isjunctiOI}
4.1. The methods described in the Introduction will now
the role here that is played by proofs there. A
self. We will later (9.11) define in the system Q the class
3.9.
Negation, Conjunction,
that is played by sentences there, and Q-Eroofs will play
is a mem-
ber of the class of green things, as well as meaning that
££2::
be followed, except that Q-formulas will play the role here
as well as being attributes. For example, if
tribute green,
~
What Q-fcrmula would denote the attribute,
"lover of Mary"? What class would this Q-formula also denote'!
~-~
Q·-Eroof is defined to be a sequence of Q-formulas (written as a vertical column in the format described in the Introduction) each of which is a d.c. (direct consequence) of previous Q-formulas of the sequence or is specified as being a hypothesis of the sequence. (The various kinds of d.c. permitted will be described subsequently.) Higher order Q-proofs can be defined by induction in terms of lower order Q-proofs as follows: An
~-££Q~ ~[
is defined to be a se-
quence of Q-formulas, nth-order Q-proofs, and (possibly) Qproofs of order less than
n,
such that each Q-formula of
the sequence is a d.c. of previous members of the sequence, or is specified as being an hypothesis of the sequence, or is reiterated from a previous step of an enclosing Q-proof of higher order. ~.2.
The rules of d.c. for negation will be those of
the Fitch system (0.41). These rules and others will often be presented in schematic form by way of Q-proofs that use the rules.
37
36
ELEMENTS OF COMBINATORY LOGIC 3.7. Some Q-functions are three-place relations, others
are four-place relations, and so on. In general, the values of n+l-place relations are n-place relations. (Attributes and classes can be viewed as being one-place relations, and propositions as being zero-place relations.) Thus if an n+l-place relation, then for the applicand
a)
ra
r
is
(which is the value of
is an n-place relation. In case
r
CHAPrER 2
r
is a zero-place relation, that is, a proposition, matters
BOOLEAN CONCEPrS AND IDENTITY
cah be so arranged that ra is also a proposition. The details of this need not concern us here. 3.8. It should also be mentioned that 'ab' can be used to assert that
b
is a member of the class
being used to assert that
b
a,
as well as
has the attribute
a.
4.
This is
because one-place relations can be viewed as being classes th~n
R
rRn
~lso
a is the atbe viewed as being the
class of green things, and 'ab' can mean that
b
is green (i.e. has the attribute green). Similarly, if is a class, then 'aa' can mean that self, and '-(aa)' can mean that
a
a
b
a
is a member of it-
is not a member of it-
of all classes that are not members of themselves. EXERCISES. Suppose that 'L' is an additional prim-
itive Q-formula that denotes the relation ~· and that 'J' and 'M' are additional primitive Q-formulas that denote two persons, Jack and Mary (assuming that in some sense persons can be Q-functions). (1). What Q-formula would denote the proposition, "Jack loves Mary"? (2). What Q-formula would denote the proposition, "Jack does not love Mary, or Mary does not love Jack"? (3).
~
I1isjunctiOI}
4.1. The methods described in the Introduction will now
the role here that is played by proofs there. A
self. We will later (9.11) define in the system Q the class
3.9.
Negation, Conjunction,
that is played by sentences there, and Q-Eroofs will play
is a mem-
ber of the class of green things, as well as meaning that
££2::
be followed, except that Q-formulas will play the role here
as well as being attributes. For example, if
tribute green,
~
What Q-fcrmula would denote the attribute,
"lover of Mary"? What class would this Q-formula also denote'!
~-~
Q·-Eroof is defined to be a sequence of Q-formulas (written as a vertical column in the format described in the Introduction) each of which is a d.c. (direct consequence) of previous Q-formulas of the sequence or is specified as being a hypothesis of the sequence. (The various kinds of d.c. permitted will be described subsequently.) Higher order Q-proofs can be defined by induction in terms of lower order Q-proofs as follows: An
~-££Q~ ~[
is defined to be a se-
quence of Q-formulas, nth-order Q-proofs, and (possibly) Qproofs of order less than
n,
such that each Q-formula of
the sequence is a d.c. of previous members of the sequence, or is specified as being an hypothesis of the sequence, or is reiterated from a previous step of an enclosing Q-proof of higher order. ~.2.
The rules of d.c. for negation will be those of
the Fitch system (0.41). These rules and others will often be presented in schematic form by way of Q-proofs that use the rules.
37
ELEMENTS OF COMBINATORY LOGIC
38
4.2. Double negation introduction (neg
2
int)
According to this rule, '-(-a)' is a d.c. of 'a'. Notice that it would not be correct to write '--a' in this context since this would be an abbreviation for '((--)a)' in combinatory logic. 4.3. ~negation elimination (neg elim) 2
According to this rule, 'a' is a d.c. of '-(-a)'. 4.4. Negation elimination (neg elim)
RULES FOR NEGATION, CONJUNCTION, AND DISJUNCTION
~;ba
or
~ ~ba
or
39
~: v b
r:
v b
According to this rule, 'Vba' (which can also be written as 'a v b') is a d.c. of 'a', and also it is a d.c. of I b I • 4.8. Disjunction elimination (dis elim) Vba
or
a V b
II II
. c
According to this rule, 'b' is a d.c. jointly of 'a' and '-a'. Rules for conjunction and disjunction now follow. 4.5. Conjunction introduction (conj int) or
a.
b
According to this rule, 'dba' is a d.c. jointly of 'a' and 'b'. The conclusion 'dba' may also be written '[ad b]' or 'ad b' in virtue of the convention stated near the end of paragraph 3.5. 4.6. Conjunction elimination (conj elim) or
a.
b
According to this rule, 'a' is a d.c. of 'dba' (which can also be written as 'ad b'), and so is 'b'. 4.7. Disjunction introduction (dis int)
According to this rule, 'c' is a d.c. jointly of 'Vba' and two Q-proofs, one having 'a' as its only hypothesis and one having 'b' as its only hypothesis, and each having 'c' as one of its steps. ( 'Vba' may also be written as 'a V b' .) Hereafter the notation 'ad b' and 'a V b' will ordinarily be used in place of 'dba' and 'Vba'. Negative rules for conjunction and disjunction now are stated. 4.9. Negative conjunction introduction (neg conj int)
L-a V -b I -[a d b] According to this rule, '-[ad b]' is a d.• c. of '-a V -b'. An alternative notation for '-[a & b]' would be '-(dba)', and an alternative notation for '-a V -b' would be 'V(-b) (-a)'. 4.10. Negative conjunction elimination (neg conj elim)
L-[a db] I -a V -b According to this rule, '-a V -b' is a d.c. of '-[ad b]'.
ELEMENTS OF COMBINATORY LOGIC
38
4.2. Double negation introduction (neg
2
int)
According to this rule, '-(-a)' is a d.c. of 'a'. Notice that it would not be correct to write '--a' in this context since this would be an abbreviation for '((--)a)' in combinatory logic. 4.3. ~negation elimination (neg elim) 2
According to this rule, 'a' is a d.c. of '-(-a)'. 4.4. Negation elimination (neg elim)
RULES FOR NEGATION, CONJUNCTION, AND DISJUNCTION
~;ba
or
~ ~ba
or
39
~: v b
r:
v b
According to this rule, 'Vba' (which can also be written as 'a v b') is a d.c. of 'a', and also it is a d.c. of I b I • 4.8. Disjunction elimination (dis elim) Vba
or
a V b
II II
. c
According to this rule, 'b' is a d.c. jointly of 'a' and '-a'. Rules for conjunction and disjunction now follow. 4.5. Conjunction introduction (conj int) or
a.
b
According to this rule, 'dba' is a d.c. jointly of 'a' and 'b'. The conclusion 'dba' may also be written '[ad b]' or 'ad b' in virtue of the convention stated near the end of paragraph 3.5. 4.6. Conjunction elimination (conj elim) or
a.
b
According to this rule, 'a' is a d.c. of 'dba' (which can also be written as 'ad b'), and so is 'b'. 4.7. Disjunction introduction (dis int)
According to this rule, 'c' is a d.c. jointly of 'Vba' and two Q-proofs, one having 'a' as its only hypothesis and one having 'b' as its only hypothesis, and each having 'c' as one of its steps. ( 'Vba' may also be written as 'a V b' .) Hereafter the notation 'ad b' and 'a V b' will ordinarily be used in place of 'dba' and 'Vba'. Negative rules for conjunction and disjunction now are stated. 4.9. Negative conjunction introduction (neg conj int)
L-a V -b I -[a d b] According to this rule, '-[ad b]' is a d.• c. of '-a V -b'. An alternative notation for '-[a & b]' would be '-(dba)', and an alternative notation for '-a V -b' would be 'V(-b) (-a)'. 4.10. Negative conjunction elimination (neg conj elim)
L-[a db] I -a V -b According to this rule, '-a V -b' is a d.c. of '-[ad b]'.
ELEMENTS OF COMBINATORY LOGIC
'+O
4.11. Negative
disju~
RULES FOR IDENTITY
41
introduction (neg dis int)
L-a li-b
I -[a
V b]
According to this rule, '-[a V b]' is a d. c. of '-a li-b'. 4.12. Negative disjunction elimination (neg dis elim)
L-Ca V b] I -a a -b According to this rule, '-ali -b' is a d.c. of '-[a V b]'. 4.13. ~ of ~Morgan (d m) A derived rule. The four rules given in 4.10-4.12 are actually special cases of this rule (i.e. underived cases of it). All the cases have already been listed on page 25. 4.14. Constructive dilemma (cnst dil) a V b
II II
c
v
d
or
V b
P'+5
1 2 3 4
b V a
5
II II
6
c
v
7 8
9 lO ll d
This rule has already been presented on page 18, and the sense in which it is a derived rule has been explained there. According to this rule, a Q-formula 'c V d' is a consequence of three things jointly: (l) a Q-formula 'a V b' (or 'b V a'), (2) a Q-proof with single hypothesis 'a' and with 'c' as one of its steps, and (3) a Q-proof with single hypothesis 'b' and with 'd' as one of its steps. 4.15. ~ tollendo ponens (m t p) This is a derived rule.
~~a
According to this rule, 'b' is a consequence of any of the following pairs of Q-formulas: 'a' and '-a V b'; 'a' and 'b V -a'; '-a' and 'a V b'; and '-a' and 'b V a'. See pages 26 and 27 for a previous statement of this rule and for a derivation of one of the cases of it. 4.16. An example is now given of a Q-proof that uses several of the rules described above.
12 13
-a -[-a li [[b & c] V [d & e]]] a V [[b li c] V [d lie]] [b li c] V [d & e] &c
1:
1:
&e
b v d e V c [e V c]
& [b v
d]
hyp hyp 2, d m l, 3, m hyp 5, conj 5, conj hyp 8, conj 8, conj 4, 5-7, 4, 5-7. ll, 12,
t p elim elim elim elim 8-10, cnst dil 8-10, cnst dil conj int
4.17. EXERCISES. Construct Q-proofs of the following• (1). 'b 1i e' from 'b li -a' and '[a a c] V [d 1i e]'. (2). I -[-a & [-b v [-ca-d]]]' from 'a V [b li [c v d]] I • (3). 'fVg' from'-(-(-[ali[cad]J))', 'cVf', 'd V g', and '-(-[a & e]))'.
5.
Rules for Identity
5.1. Among the rules for identity that will be employed in the system Q are the rule of identit~ introduction, asserting 'a = a', the rule of identit~ elimination, asserting the substitutability of equals for equals (i.e. identicals for identicals), and the rule of excluded !!~ £££
ELEMENTS OF COMBINATORY LOGIC
'+O
4.11. Negative
disju~
RULES FOR IDENTITY
41
introduction (neg dis int)
L-a li-b
I -[a
V b]
According to this rule, '-[a V b]' is a d. c. of '-a li-b'. 4.12. Negative disjunction elimination (neg dis elim)
L-Ca V b] I -a a -b According to this rule, '-ali -b' is a d.c. of '-[a V b]'. 4.13. ~ of ~Morgan (d m) A derived rule. The four rules given in 4.10-4.12 are actually special cases of this rule (i.e. underived cases of it). All the cases have already been listed on page 25. 4.14. Constructive dilemma (cnst dil) a V b
II II
c
v
d
or
V b
P'+5
1 2 3 4
b V a
5
II II
6
c
v
7 8
9 lO ll d
This rule has already been presented on page 18, and the sense in which it is a derived rule has been explained there. According to this rule, a Q-formula 'c V d' is a consequence of three things jointly: (l) a Q-formula 'a V b' (or 'b V a'), (2) a Q-proof with single hypothesis 'a' and with 'c' as one of its steps, and (3) a Q-proof with single hypothesis 'b' and with 'd' as one of its steps. 4.15. ~ tollendo ponens (m t p) This is a derived rule.
~~a
According to this rule, 'b' is a consequence of any of the following pairs of Q-formulas: 'a' and '-a V b'; 'a' and 'b V -a'; '-a' and 'a V b'; and '-a' and 'b V a'. See pages 26 and 27 for a previous statement of this rule and for a derivation of one of the cases of it. 4.16. An example is now given of a Q-proof that uses several of the rules described above.
12 13
-a -[-a li [[b & c] V [d & e]]] a V [[b li c] V [d lie]] [b li c] V [d & e] &c
1:
1:
&e
b v d e V c [e V c]
& [b v
d]
hyp hyp 2, d m l, 3, m hyp 5, conj 5, conj hyp 8, conj 8, conj 4, 5-7, 4, 5-7. ll, 12,
t p elim elim elim elim 8-10, cnst dil 8-10, cnst dil conj int
4.17. EXERCISES. Construct Q-proofs of the following• (1). 'b 1i e' from 'b li -a' and '[a a c] V [d 1i e]'. (2). I -[-a & [-b v [-ca-d]]]' from 'a V [b li [c v d]] I • (3). 'fVg' from'-(-(-[ali[cad]J))', 'cVf', 'd V g', and '-(-[a & e]))'.
5.
Rules for Identity
5.1. Among the rules for identity that will be employed in the system Q are the rule of identit~ introduction, asserting 'a = a', the rule of identit~ elimination, asserting the substitutability of equals for equals (i.e. identicals for identicals), and the rule of excluded !!~ £££
ELEMENTS OF COMBINATORY LOGIC
42
identitl, asserting '[a= b) V -[a= b)'. (The system Q does not have a general rule of excluded middle, but only certain special ones, such as the one just mentioned. In other words, 'a V -a' is not provable in Q for every Q-formula 'a', but only for certain Q-formulas such as those that express identities.) A special case of the rule of identity elimination will be called the rule of restricted identitl elimination. This is the case of identity elimination when the substitution is made in the left or right side (or both) of an expression which is itself an identity (i.e. of the form 'a= b') or a denial of an identity (i.e. of the form '-[a = b)') • This case of identity elimination may be used under some circumstances when the more general rule of identity elimination is not allowed. Another important rule is called extensional~· According to this rule it is permissible to conclude 'a = b' if 'a' and 'b' are deriv~blc fro~ each other without use of identity elimination (but with use of restricted identity elimination, if needed) and if '-a' and '-b' are similarly derivable from each other. Finally there is the ~ ~ Q£ cancellation. These rules for identity will now be presented in more detail. 5.2. Identi~ ~uction (id int)
Ia
= a
or
I =aa
According to this rule, 'a = a' (which can also be written '=aa') can appear as a step in any Q-proof, for any Q-formula 'a'. Thus, for exawple, •a = a• and '- = -' can, according to this rule, appear as steps in any Q-proof. Steps of this sort may be called ~· since they are allowed to appear without being hypotheses, direct conseauences, or reiterations; and the definition of the concept of (,(-proof should now be extended to allow steps that are axioms. 5.3. Identit~ elimination (id elim)
~
a =
b
( ... a .. •) ( ... b ... )
or
~
b
=
a
( ... a ... ) ( ... b ••• )
RULES FOR IDENTITY
43
If '( .•. a ••• )' is any Q-formula in which a Q-formula 'a' occurs one or more times, and if '( .•• b ••• )' is a Q-formula that results from replacing 'a' by a Q-formula 'b' in one or more places in '( ••• a ••• )', then this rule asserts that '( ... b ... )' is a d.c. of 'a= b' and '( ... a ... )' jointly, and also of 'b =a' and '( •.• a ••• )' jointly. Here '( ..• a .•• )' could of course be 'a' itself, in which case '( ••. b ••• )' would be 'b'. 5.4. Restricted identit~ elimination (res id elim) This rule is exactly like identity elimination except that the expression '( ••• a ••• )' must be an identity or the negation of an identity, and the replacements must be made in its right side or its left side or both. Thus if the replacements are made in both sides and if '( •.. a ••• )' is either '(---a---)= ( , , a , , ) ' or '-[(---a---)= (,,a,,)]' (where ' (---a---)' 1.s an,y (,(-formula in which 'a' occurs one or more times, and similarly for ' ( , , a , , ) ' ) , then we have:
~
a =
b
(---a---) (---b---)
(,,a.,) (,,b.,,)
~
a =
b
-[(---a---) • (,,,a,,,)] - [ (---b---) = (" 'b" ')]
Here '(---a---)' or '(,,,a,,,)' could be 'a' itself. Also, '(---b---)' is the result of replacing 'a' by 'b' in one or places in '(---a---)', and similarly for '(,,,b,,,)', The first hypothesis in each case could be 'b = a' instead of 'a = b'. 5.5. Extensionalit~ (ext) According to the rule of extensionality, 'a = b' is a d.c. of the following four Q-proofs jointly:
ll
II
~-:a
1-=b
-b
-a
These are not permitted to use the general (unrestricted) form of id elim or to contajn reiterations of any step obtained with the help of the general form of id elim. The use of res id elim, however, is permjtted in these Q-proofs,
ELEMENTS OF COMBINATORY LOGIC
42
identitl, asserting '[a= b) V -[a= b)'. (The system Q does not have a general rule of excluded middle, but only certain special ones, such as the one just mentioned. In other words, 'a V -a' is not provable in Q for every Q-formula 'a', but only for certain Q-formulas such as those that express identities.) A special case of the rule of identity elimination will be called the rule of restricted identitl elimination. This is the case of identity elimination when the substitution is made in the left or right side (or both) of an expression which is itself an identity (i.e. of the form 'a= b') or a denial of an identity (i.e. of the form '-[a = b)') • This case of identity elimination may be used under some circumstances when the more general rule of identity elimination is not allowed. Another important rule is called extensional~· According to this rule it is permissible to conclude 'a = b' if 'a' and 'b' are deriv~blc fro~ each other without use of identity elimination (but with use of restricted identity elimination, if needed) and if '-a' and '-b' are similarly derivable from each other. Finally there is the ~ ~ Q£ cancellation. These rules for identity will now be presented in more detail. 5.2. Identi~ ~uction (id int)
Ia
= a
or
I =aa
According to this rule, 'a = a' (which can also be written '=aa') can appear as a step in any Q-proof, for any Q-formula 'a'. Thus, for exawple, •a = a• and '- = -' can, according to this rule, appear as steps in any Q-proof. Steps of this sort may be called ~· since they are allowed to appear without being hypotheses, direct conseauences, or reiterations; and the definition of the concept of (,(-proof should now be extended to allow steps that are axioms. 5.3. Identit~ elimination (id elim)
~
a =
b
( ... a .. •) ( ... b ... )
or
~
b
=
a
( ... a ... ) ( ... b ••• )
RULES FOR IDENTITY
43
If '( .•. a ••• )' is any Q-formula in which a Q-formula 'a' occurs one or more times, and if '( .•• b ••• )' is a Q-formula that results from replacing 'a' by a Q-formula 'b' in one or more places in '( ••• a ••• )', then this rule asserts that '( ... b ... )' is a d.c. of 'a= b' and '( ... a ... )' jointly, and also of 'b =a' and '( •.• a ••• )' jointly. Here '( ..• a .•• )' could of course be 'a' itself, in which case '( ••. b ••• )' would be 'b'. 5.4. Restricted identit~ elimination (res id elim) This rule is exactly like identity elimination except that the expression '( ••• a ••• )' must be an identity or the negation of an identity, and the replacements must be made in its right side or its left side or both. Thus if the replacements are made in both sides and if '( •.. a ••• )' is either '(---a---)= ( , , a , , ) ' or '-[(---a---)= (,,a,,)]' (where ' (---a---)' 1.s an,y (,(-formula in which 'a' occurs one or more times, and similarly for ' ( , , a , , ) ' ) , then we have:
~
a =
b
(---a---) (---b---)
(,,a.,) (,,b.,,)
~
a =
b
-[(---a---) • (,,,a,,,)] - [ (---b---) = (" 'b" ')]
Here '(---a---)' or '(,,,a,,,)' could be 'a' itself. Also, '(---b---)' is the result of replacing 'a' by 'b' in one or places in '(---a---)', and similarly for '(,,,b,,,)', The first hypothesis in each case could be 'b = a' instead of 'a = b'. 5.5. Extensionalit~ (ext) According to the rule of extensionality, 'a = b' is a d.c. of the following four Q-proofs jointly:
ll
II
~-:a
1-=b
-b
-a
These are not permitted to use the general (unrestricted) form of id elim or to contajn reiterations of any step obtained with the help of the general form of id elim. The use of res id elim, however, is permjtted in these Q-proofs,
ELEMENTS OF COMBINATORY LOGIC
44
and in obtaining steps that are reiterated into these Qproofs. An important limitation on the rule of extensionality is that it cannot be used under an hypothesis. In other words, it can be used only for proving steps of categorical proofs (proofs lacking hypotheses) and never for proving steps of hypothetical proofs (proofs having hypotheses). 5.6. Here are some examples of Q-proofs using the various identity rules so far stated: P'+6
~I : :
3r
a = c
~
( .•• b ••• )
~a( ••• = bb ••. ) ( • • •a • . • )
2l~aa
= 3 b =
b a a
5 6 7 8
12
As used in P46, however, it should be thought of as written:
P47
3 4
ll
= a
c ' , and ' ( ••• a ••• ) ' is ' a
l 2
lO
( • . •a • . • )
Here ' ( ... b ••• ) ' is 'b
p48
9
hyp hyp 1, 2, id elim
The above Q-proof expresses the derived rule of transitiv~ Q[ identit~ (trans id). Actually the second case of id elim is being used at step 3 of P46, the one that appears as: b
RULES FOR IDENTITY
c' •
hyp id int 1, 2, id elim
The above Q-proof expresses the derived rule of s~mmet~ Q[ ldentit~ (sym id), according to which 'b = a' follows from 'a= b'. At step 3 the first form of identity elimination is being used. 'a =a' and 'b =a' correspond respectively to '( ..• a •.• )' and '( ... b ... )'. The uses of id elim in P46 and P'+7 can each be refarded, if desired, as uses of res id elim. Hence trans id and sym id can be used in the four Q-proofs required in the rule of extensionality.
r:
a
1:
a
-[a
1
a
a
a
a]
-a V -a -a -a -a V -a
1a
-[a
[a
a a]
a] =
a
45 hyp l, conj elim hyp 3, rep (see 0.29, 0.30) 3, 4, conj int hyp 6, d m 7, dis elim hyp 9, dis int 10, d m l-2, 3-5, 6-8, 9-11, ext
Notice that in connection with the use of dis elim at step 8 two trivial subproofs and have been omitted. r-a r-a -a -a The two subproofs 1-2 and 3-5 may be written as a sort of double subproof which when read downward corresponds to 1-2 (with step 2 repeated) and when read upward corresponds to steps 3-5. The presence of a second horizontal line indicates that the proof is double and can be read both upward and downward. The two other subproofs can also be combined as a double subproof, and P48 is transformed into the following format: (Downward reasons) (Upward reasons) hyp P49 1 a 2, 3, conj int 2 1, conj elim 3. rep 2, rep 3 hyp 4 hyp 5, d m ~-[a a] 4, d m -a V -a 5 6, dis int 6 5, dis elim -a hyp a] = a 1-3, 4-6, ext 7 [a
~:a
a
a
Here is another example of a Q-proof that uses ext. It is assumed that 'a' and 'b' are provable Q-formulas and have been proved in the first part of the proof without using id elim (but possibly by using res id elim), and so can be reiterated into the subproofs used for extensionality:
ELEMENTS OF COMBINATORY LOGIC
44
and in obtaining steps that are reiterated into these Qproofs. An important limitation on the rule of extensionality is that it cannot be used under an hypothesis. In other words, it can be used only for proving steps of categorical proofs (proofs lacking hypotheses) and never for proving steps of hypothetical proofs (proofs having hypotheses). 5.6. Here are some examples of Q-proofs using the various identity rules so far stated: P'+6
~I : :
3r
a = c
~
( .•• b ••• )
~a( ••• = bb ••. ) ( • • •a • . • )
2l~aa
= 3 b =
b a a
5 6 7 8
12
As used in P46, however, it should be thought of as written:
P47
3 4
ll
= a
c ' , and ' ( ••• a ••• ) ' is ' a
l 2
lO
( • . •a • . • )
Here ' ( ... b ••• ) ' is 'b
p48
9
hyp hyp 1, 2, id elim
The above Q-proof expresses the derived rule of transitiv~ Q[ identit~ (trans id). Actually the second case of id elim is being used at step 3 of P46, the one that appears as: b
RULES FOR IDENTITY
c' •
hyp id int 1, 2, id elim
The above Q-proof expresses the derived rule of s~mmet~ Q[ ldentit~ (sym id), according to which 'b = a' follows from 'a= b'. At step 3 the first form of identity elimination is being used. 'a =a' and 'b =a' correspond respectively to '( ..• a •.• )' and '( ... b ... )'. The uses of id elim in P46 and P'+7 can each be refarded, if desired, as uses of res id elim. Hence trans id and sym id can be used in the four Q-proofs required in the rule of extensionality.
r:
a
1:
a
-[a
1
a
a
a
a]
-a V -a -a -a -a V -a
1a
-[a
[a
a a]
a] =
a
45 hyp l, conj elim hyp 3, rep (see 0.29, 0.30) 3, 4, conj int hyp 6, d m 7, dis elim hyp 9, dis int 10, d m l-2, 3-5, 6-8, 9-11, ext
Notice that in connection with the use of dis elim at step 8 two trivial subproofs and have been omitted. r-a r-a -a -a The two subproofs 1-2 and 3-5 may be written as a sort of double subproof which when read downward corresponds to 1-2 (with step 2 repeated) and when read upward corresponds to steps 3-5. The presence of a second horizontal line indicates that the proof is double and can be read both upward and downward. The two other subproofs can also be combined as a double subproof, and P48 is transformed into the following format: (Downward reasons) (Upward reasons) hyp P49 1 a 2, 3, conj int 2 1, conj elim 3. rep 2, rep 3 hyp 4 hyp 5, d m ~-[a a] 4, d m -a V -a 5 6, dis int 6 5, dis elim -a hyp a] = a 1-3, 4-6, ext 7 [a
~:a
a
a
Here is another example of a Q-proof that uses ext. It is assumed that 'a' and 'b' are provable Q-formulas and have been proved in the first part of the proof without using id elim (but possibly by using res id elim), and so can be reiterated into the subproofs used for extensionality:
ELEMENTS OF COMBINATORY LOGIC
'+6 P50
nl n2 n3 n'+ n5 n6 n7 n8 n9
a b hyp n2' reit hyp nl' reit n2' reit
~:
t:·
nl'' reit hyp n8' n7' neg elim nl' reit n2' reit hyp
n5' n6' neg elim n -n4 , n5-n8' ext 3 As in the case of F49, and generally hereafter when there are two columns of reasons, the left column gives reasons for steps of the subproofs when the subproofs are read downward, while the right column gives reasons for steps of the subproofs when the subproofs are read upward. The last two of the primitive rules for identity are now presented. 5.7. Excluded ~ fQ£ identit~ (ex mid id) -b
a
I [a
=
b
= b)
V -[a = b)
According to this rule, '[a b) V -[a= b)' is an axiom, that is, may appear as a step in any q-proof (or subproof of a q-proof). Thus although the principle of excluded middle does not hold in general in the system q, it does hold for those q-formulas that are identity statements. 5.8. ~ ~ Qf cancellation (zeta) According to this rule, 'a = b' is a d.c. of a categorical subproof that has 'ac = be' as a step and is "general with respect to 'c'," that is, remains a correct subproof when 'c' is replaced by any other q-formula. Schematically this rule may be expressed:
RULES FOR IDENTITY
'+7
pect to 'c'. Zeta is a rule of "cancellation" in the sense that the 'c' is, in effect, cancelled. Here is an example of a Q-proof that uses extensionality and zeta. P51 1 b hyp li(-(-a) )b 2, 3, conj int 2 1, conj elim -(-a) '+, neg 2 int b 1, conj elim 5, conj elim 3 '+ a 2, neg 2 elim 5, conj elim liab 3, '+, conj int hyp 5 6 -(li(-(-a))b) hyp 7, d m V(-(-(-a)))(-b) 6, d m 7 10, 9-8, cnst dil 8 hyp L -C-C-a)) 9, neg2 int neg2 elim 1-a 9 hyp 10 V(-a)(-b) 7, 8-9, cnst dil 11, d m 11 -(liab) 11, d m hyp 12 8:(-(-a))b = liab 1-5, 6-11, ext 13 li(-(-a)) = lia 1-12, zeta The full power and usefulness of rule zeta will be more evident after rules for such operators as 'B', 'C', 'K', and 'W' have been presented in Chapter 3· Some further derived rules involving identity will now be presented. 5.9. Monotonic ~ !££ identitr (mon id) This rule is easily derivable by use of id int and res id elim:
t
La
= b
I ( ... a •.• )
= ( ••• b ••• )
5.10. Negation introduction!££ identitr (neg int id) This derived rule is as follows: b
be a
= b
The 'c' to the left of the vertical line of the subproof expresses the fact that the subproof must be general with res-
-[a = b)
This rule asserts that '-[a= b)' is a consequence of a subproof that has 'a = b' as its only hypothesis and has contra-
ELEMENTS OF COMBINATORY LOGIC
'+6 P50
nl n2 n3 n'+ n5 n6 n7 n8 n9
a b hyp n2' reit hyp nl' reit n2' reit
~:
t:·
nl'' reit hyp n8' n7' neg elim nl' reit n2' reit hyp
n5' n6' neg elim n -n4 , n5-n8' ext 3 As in the case of F49, and generally hereafter when there are two columns of reasons, the left column gives reasons for steps of the subproofs when the subproofs are read downward, while the right column gives reasons for steps of the subproofs when the subproofs are read upward. The last two of the primitive rules for identity are now presented. 5.7. Excluded ~ fQ£ identit~ (ex mid id) -b
a
I [a
=
b
= b)
V -[a = b)
According to this rule, '[a b) V -[a= b)' is an axiom, that is, may appear as a step in any q-proof (or subproof of a q-proof). Thus although the principle of excluded middle does not hold in general in the system q, it does hold for those q-formulas that are identity statements. 5.8. ~ ~ Qf cancellation (zeta) According to this rule, 'a = b' is a d.c. of a categorical subproof that has 'ac = be' as a step and is "general with respect to 'c'," that is, remains a correct subproof when 'c' is replaced by any other q-formula. Schematically this rule may be expressed:
RULES FOR IDENTITY
'+7
pect to 'c'. Zeta is a rule of "cancellation" in the sense that the 'c' is, in effect, cancelled. Here is an example of a Q-proof that uses extensionality and zeta. P51 1 b hyp li(-(-a) )b 2, 3, conj int 2 1, conj elim -(-a) '+, neg 2 int b 1, conj elim 5, conj elim 3 '+ a 2, neg 2 elim 5, conj elim liab 3, '+, conj int hyp 5 6 -(li(-(-a))b) hyp 7, d m V(-(-(-a)))(-b) 6, d m 7 10, 9-8, cnst dil 8 hyp L -C-C-a)) 9, neg2 int neg2 elim 1-a 9 hyp 10 V(-a)(-b) 7, 8-9, cnst dil 11, d m 11 -(liab) 11, d m hyp 12 8:(-(-a))b = liab 1-5, 6-11, ext 13 li(-(-a)) = lia 1-12, zeta The full power and usefulness of rule zeta will be more evident after rules for such operators as 'B', 'C', 'K', and 'W' have been presented in Chapter 3· Some further derived rules involving identity will now be presented. 5.9. Monotonic ~ !££ identitr (mon id) This rule is easily derivable by use of id int and res id elim:
t
La
= b
I ( ... a •.• )
= ( ••• b ••• )
5.10. Negation introduction!££ identitr (neg int id) This derived rule is as follows: b
be a
= b
The 'c' to the left of the vertical line of the subproof expresses the fact that the subproof must be general with res-
-[a = b)
This rule asserts that '-[a= b)' is a consequence of a subproof that has 'a = b' as its only hypothesis and has contra-
Ray Nakabayashi ELEMENTS OF COMBINATORY LOGIC
4-8
dictory steps 'c' and '-c'. Here is the derivation• P52
[a = b] V -[a = b]
1
2
ex mid id hyp
ac = b
RULES FOR IDENTITY
4-9
are derived from the original rule as stated in 5.5. In these alternative forms,
as in the original rule, it is re-
quired that no use of id elim be made in the subproofs associated with the rule, and that no step be reiterated into such subproofs if it was obtained by use of id elim. Use of
r
res id elim, however, is permitted in such subproofs and in
-c
-[a = b]
-[a
= b]
5.11. Indirect £roof
1££
n , n , neg elim 2 1 1, 2-n , dis elim 3 identiti (ind pr id)
This is a derived rule, and its derivation is like that of
obtaining steps reiterated into such subproofs. 5.15. The first of these alternative forms of ext may be expressed schematically as follows, assuming that 'a' has been so chosen that 'a V -a'
is provable. The first part of
the proof is a proof of 'a V -a'.
neg int id given in P52 above.
n· I
Ia
5.12.
= b]
a V -a
~~
~~
-c
b
=
Non-identit~
~t
introduction (non-id int), first form
This is a derived rule. The derivation, which uses id elim
I
understood that '( •.. a ..• )' is any Q-formula in which a Q-
-b a = b Notice that the first subproof is double but that the second one is not. In order to derive ttis form of extensionality it is sufficient to show that the second subproof could be made double by completing it in the upward direction. This amounts to showing that '-a' can be derived from the hypothe-
formula 'a' occurs and that '( ... b ••• )' is the result of re-
sis '-b' if use is made of 'a V -a' and of the fact that 'b'
placing 'a' by a Q-formula 'b' in one or more places in
is derivable from 'a'.
and neg int id,
~
is omitted.
( ... a •. •) -( ••• b •.• )
-[a = b]
In the above schematic presentation of the rule, it is to be
'( ... a ... )'. 5.13.
5.16. ThP- nc>xt form of ext can be used when 'a V -a'
Non-identit~
introduction (non-id int), second form
and
'b V -b'
are both provable.
It is derivable somewhat
This is a derived rule. The derivation is omitted, but it uses
like the previous form of ext, but by showing that '-b' is
res id elim and neg int id.
obtainable from '-a' as well as that '-a' is obtainable from
L-[( ... a ... ) =( ... b ... )]
'-b'. It can be exDressed schematically as follows, assuming
I
that 'a V -a' and 'b V -b' h3ve been proved (for suitable
-[a
= b)
5.14. We now
~ive
so~e
alt~rnative
forms of ext that
choice of 'a' and 'b') in the first part of the proof.
Ray Nakabayashi
Ray Nakabayashi ELEMENTS OF COMBINATORY LOGIC
4-8
dictory steps 'c' and '-c'. Here is the derivation• P52
[a = b] V -[a = b]
1
2
ex mid id hyp
ac = b
RULES FOR IDENTITY
4-9
are derived from the original rule as stated in 5.5. In these alternative forms,
as in the original rule, it is re-
quired that no use of id elim be made in the subproofs associated with the rule, and that no step be reiterated into such subproofs if it was obtained by use of id elim. Use of
r
res id elim, however, is permitted in such subproofs and in
-c
-[a = b]
-[a
= b]
5.11. Indirect £roof
1££
n , n , neg elim 2 1 1, 2-n , dis elim 3 identiti (ind pr id)
This is a derived rule, and its derivation is like that of
obtaining steps reiterated into such subproofs. 5.15. The first of these alternative forms of ext may be expressed schematically as follows, assuming that 'a' has been so chosen that 'a V -a'
is provable. The first part of
the proof is a proof of 'a V -a'.
neg int id given in P52 above.
n· I
Ia
5.12.
= b]
a V -a
~~
~~
-c
b
=
Non-identit~
~t
introduction (non-id int), first form
This is a derived rule. The derivation, which uses id elim
I
understood that '( •.. a ..• )' is any Q-formula in which a Q-
-b a = b Notice that the first subproof is double but that the second one is not. In order to derive ttis form of extensionality it is sufficient to show that the second subproof could be made double by completing it in the upward direction. This amounts to showing that '-a' can be derived from the hypothe-
formula 'a' occurs and that '( ... b ••• )' is the result of re-
sis '-b' if use is made of 'a V -a' and of the fact that 'b'
placing 'a' by a Q-formula 'b' in one or more places in
is derivable from 'a'.
and neg int id,
~
is omitted.
( ... a •. •) -( ••• b •.• )
-[a = b]
In the above schematic presentation of the rule, it is to be
'( ... a ... )'. 5.13.
5.16. ThP- nc>xt form of ext can be used when 'a V -a'
Non-identit~
introduction (non-id int), second form
and
'b V -b'
are both provable.
It is derivable somewhat
This is a derived rule. The derivation is omitted, but it uses
like the previous form of ext, but by showing that '-b' is
res id elim and neg int id.
obtainable from '-a' as well as that '-a' is obtainable from
L-[( ... a ... ) =( ... b ... )]
'-b'. It can be exDressed schematically as follows, assuming
I
that 'a V -a' and 'b V -b' h3ve been proved (for suitable
-[a
= b)
5.14. We now
~ive
so~e
alt~rnative
forms of ext that
choice of 'a' and 'b') in the first part of the proof.
Ray Nakabayashi
ELEMENTS OF COMBINATORY LOGIC
50
BOOLEAN IDENTITIES 6.
v
-b
-(-a) [a 8. a] [a V a] [a8.b]
a • b
5.17. The last alternative form of ext is as follows:
~:
b
[a V b]
d)
It is derived thus:
t
a
~c
~
b
8. V 8. V
[b [b [b [b
8. V V 8.
a
b) V -[a = b)
[c
d) V
-[c =
[a
b)
[c = d)
d)
= =
ex mid id ex mid id l-n1 , n2 , n , ext (5.16) 3
5.18. EXERCISES. Derive: (1). The monotonic rule for identity (5.9). (2). Indirect proof for identity (5.11). (3)· Non-identity introduction, first form (5.12). (4-). Non-identity introduction, second form (5.13). (5)· 'Vab • Vba'. (6). '=(Vab) = =(Vba)', i.e. '(=(Vab)) = (=(Vba))'. (7). The first alternative form of ext (5.15). (8). The second alternative form of ext (5.16). (g), The third alternative form of ext (5.17) (10). '[[[a= b) V -[a= b)] 8. c] c'. (11). '[(a =b) 8. -[a = bJ] = [[c = d) 8. -[c = d]]'.
= a = a
a a a [b8.aJ [b V a]
c]] c]) c)] c]J
-[a b] -[a V b]
= d
[a
=
([a = [[a [[a [[a [a~ [b V a]] =a [a V [b 8. a]] = a [a [a [a [a
d
[a = b) = [c
P53 l
Boolean Identities
6.1. By use of ext the following identities are easily provable. For example, the second of them can be proved as was done in F48 or F49. After each of these identities the same identity is given in ordinary Boolean notation.
a V -a b
51
8. V 8. V
b] b] b) b)
8. V V 8.
c) c) [a 8. c)] [a V c)]
ab ba a+b b+a a(bc) (ab)c a+(b+c) (a+b)+c a(b+c) ab+ac a+bc = (a+b)(a+c) a(b+a) = a a+ba = a
[-a V -b)
[-a
a -b)
Other similar De Morgan identities can also be proved, for example, '-[-a 8. b) = [a V -b]'. These identities will be called unrestricted Boolean identities (u bid), since they can be proved unrestrictedly in the system Q. 6.2. In addition to the unrestricted Boolean identities there are other Boolean identities which can be proved in the system Q only if certain Q-formulas can be proved to satisfy excluded middle (without use of id elim). More specifically, i f 'a V -a' and 'b v -b' are provable without use of id elim, then the following identities are provable: [a V -a] = [b v -b) [a 8. -a] [b li -b] c = [[a V -a] 8. c) c = [[ali -a] v c) [(a V -a] v c] [a V -a] [[ali-a] 8. c] [a li -a]
1
0 c c 1
0
=
b+b bb lc 0+C l+C Oc
ELEMENTS OF COMBINATORY LOGIC
50
BOOLEAN IDENTITIES 6.
v
-b
-(-a) [a 8. a] [a V a] [a8.b]
a • b
5.17. The last alternative form of ext is as follows:
~:
b
[a V b]
d)
It is derived thus:
t
a
~c
~
b
8. V 8. V
[b [b [b [b
8. V V 8.
a
b) V -[a = b)
[c
d) V
-[c =
[a
b)
[c = d)
d)
= =
ex mid id ex mid id l-n1 , n2 , n , ext (5.16) 3
5.18. EXERCISES. Derive: (1). The monotonic rule for identity (5.9). (2). Indirect proof for identity (5.11). (3)· Non-identity introduction, first form (5.12). (4-). Non-identity introduction, second form (5.13). (5)· 'Vab • Vba'. (6). '=(Vab) = =(Vba)', i.e. '(=(Vab)) = (=(Vba))'. (7). The first alternative form of ext (5.15). (8). The second alternative form of ext (5.16). (g), The third alternative form of ext (5.17) (10). '[[[a= b) V -[a= b)] 8. c] c'. (11). '[(a =b) 8. -[a = bJ] = [[c = d) 8. -[c = d]]'.
= a = a
a a a [b8.aJ [b V a]
c]] c]) c)] c]J
-[a b] -[a V b]
= d
[a
=
([a = [[a [[a [[a [a~ [b V a]] =a [a V [b 8. a]] = a [a [a [a [a
d
[a = b) = [c
P53 l
Boolean Identities
6.1. By use of ext the following identities are easily provable. For example, the second of them can be proved as was done in F48 or F49. After each of these identities the same identity is given in ordinary Boolean notation.
a V -a b
51
8. V 8. V
b] b] b) b)
8. V V 8.
c) c) [a 8. c)] [a V c)]
ab ba a+b b+a a(bc) (ab)c a+(b+c) (a+b)+c a(b+c) ab+ac a+bc = (a+b)(a+c) a(b+a) = a a+ba = a
[-a V -b)
[-a
a -b)
Other similar De Morgan identities can also be proved, for example, '-[-a 8. b) = [a V -b]'. These identities will be called unrestricted Boolean identities (u bid), since they can be proved unrestrictedly in the system Q. 6.2. In addition to the unrestricted Boolean identities there are other Boolean identities which can be proved in the system Q only if certain Q-formulas can be proved to satisfy excluded middle (without use of id elim). More specifically, i f 'a V -a' and 'b v -b' are provable without use of id elim, then the following identities are provable: [a V -a] = [b v -b) [a 8. -a] [b li -b] c = [[a V -a] 8. c) c = [[ali -a] v c) [(a V -a] v c] [a V -a] [[ali-a] 8. c] [a li -a]
1
0 c c 1
0
=
b+b bb lc 0+C l+C Oc
52
ELEMENTS OF COMBINATORY LOGIC
These latter identities will be called restricted Boolean identities (r b id). In ordir.ary Boolean notation at the right (which is not used in the system Q) we can think of '1' as defined as 'a+a' and of '0' as defined as 'aa', for some specific 'a' such that 'a V -a' is provable in Q without use of id elim (but possibly with use of res id elim). Ordinary Boolean algebra does not require any assumptions about satisfying excluded middle in connection with proving these identities. 6.3. EXERCISES. (l). Prove any of the last eight unrestricted Boolean identities. (2). For suitably chosen 'a', prove any of the last four restricted Boolean identities.
CHAPTER 3 COMBINATORS
7.
~
Combinators I
~
K
7.1. The primitive Q-function I, first mentioned in 3.2, is called a combinator, and it has the following simple property: The value of I for each applicand a is a itself. In other words, Ia = a for every Q-function a. This latter identity (or rather the Q-formula 'Ia = a' expressing it) is called the I-ldentitz (I id) and is derivable by use of ext and the rules stated in 7.2 and 7.3 below. 7.2. Rules of I-elimination (I elim) and I-introduction (I int)
Here elimination is downward and introduction is upward. (A similar convention will be used in other cases.) The form of the rule given on the left can be considered to be a special case of the form of the rule given on the right, namely the case where n is zero. The first form of the rule asserts that 'a' is a d.c. of 'Ia' by I elim and that 'Ia' is a d.c. of 'a' by I int. The second form of the rule asserts (for each non-negative integer n) that 'ab 1 ... bn' is a d. c • of 'Iab 1 ••• bn' by I elim and that 'Iab ••• bn' is a d.c. of 1 'ab 1 ••• bn' by I int. For example, if n is 3, then by the second form of the rule, 'IVVVV' is a d.c. of 'VVVV'.
53
52
ELEMENTS OF COMBINATORY LOGIC
These latter identities will be called restricted Boolean identities (r b id). In ordir.ary Boolean notation at the right (which is not used in the system Q) we can think of '1' as defined as 'a+a' and of '0' as defined as 'aa', for some specific 'a' such that 'a V -a' is provable in Q without use of id elim (but possibly with use of res id elim). Ordinary Boolean algebra does not require any assumptions about satisfying excluded middle in connection with proving these identities. 6.3. EXERCISES. (l). Prove any of the last eight unrestricted Boolean identities. (2). For suitably chosen 'a', prove any of the last four restricted Boolean identities.
CHAPTER 3 COMBINATORS
7.
~
Combinators I
~
K
7.1. The primitive Q-function I, first mentioned in 3.2, is called a combinator, and it has the following simple property: The value of I for each applicand a is a itself. In other words, Ia = a for every Q-function a. This latter identity (or rather the Q-formula 'Ia = a' expressing it) is called the I-ldentitz (I id) and is derivable by use of ext and the rules stated in 7.2 and 7.3 below. 7.2. Rules of I-elimination (I elim) and I-introduction (I int)
Here elimination is downward and introduction is upward. (A similar convention will be used in other cases.) The form of the rule given on the left can be considered to be a special case of the form of the rule given on the right, namely the case where n is zero. The first form of the rule asserts that 'a' is a d.c. of 'Ia' by I elim and that 'Ia' is a d.c. of 'a' by I int. The second form of the rule asserts (for each non-negative integer n) that 'ab 1 ... bn' is a d. c • of 'Iab 1 ••• bn' by I elim and that 'Iab ••• bn' is a d.c. of 1 'ab 1 ••• bn' by I int. For example, if n is 3, then by the second form of the rule, 'IVVVV' is a d.c. of 'VVVV'.
53
ELEMENTS OF COMBINATORY LOGIC
7.3. Rules of negative !•elimination (neg I elim) and negative
I-introduct~on
(neg I int)
L -Cia) 1-a 7.4. Hereafter it will be customary to abbreviate two double Q-proofs,
THE COMBINATORS I AND K
55
Observe that Ka is a function which has the value a for for every applicand b. Thus we can think of Ka as a "constant" function which always has the same value a regardless of what applicand b it is applied to. Furthermore, K itself may be viewed as a two-place function which takes as its value the first of its two applicands. In ordinary functional notation this fact would be expressed: K(a,b) ~ a. 7.7. ~ !£! K (K elim, K int, neg K elim, neg K int) These four rules are expressed schematically as follows:
- L Kab
-Ia
as:
For example, the rule neg K int for the case where zero is expressed schematically as follows:
Hence the four rules stated in 7.2 and 7.3 can be stated in a combined form as follows:
- L Iab 1 ..• bn
-I ab 1 •.. bn For
n
7.5. The rule of !-identity may be derived thus:
3
This means that '-(Kab)' is a d.c. of '-a' by neg K int. As another example, the rule of neg K elim for the case where n is 2 is expressed schematically as follows:
L -(Kabcd) I -(acd)
-L Ia -L Ia
is
L-a I -(Kab)
chosen as zero we have:
-Ia P54- l 2
n
This means that '-(acd)' is a d.c. of '-(Kabcd)' by neg K elim. The two negative K-rules may be expressed schematically as follows by a double Q-proof:
hyp
-Ia
1, I elim
Ia = a
1-2, ext
In the quadruple subproof 1-2 above, reasons are given only for steps of the the first of the four combined subproofs. This will be permitted when reasons for the steps of the remaining three subproofs are closely analogous. 7.6. The Q-function K is a combinator and has the property, Kab = a. This latter identity (or rather the Q-formula 'Kab = a' expressing it) is called the K-identitl (K id) and is derivable by ext and the rules stated in 7.7 below.
L -(Kabc 1 ... en) 1-Cac 1 .•• en)
In the case that n is zero, the parentheses in '-(ac ••• cn)' 1 are dropped, and we have simply '-a'. 7.8. Here are examples of Q-proofs using the I- and Krules:
P55
l 2 3 4
hyp II III I(!II)
1, I int (case for n=O) 2, I int (case for n=l)
3. I int (case for n=O)
ELEMENTS OF COMBINATORY LOGIC
7.3. Rules of negative !•elimination (neg I elim) and negative
I-introduct~on
(neg I int)
L -Cia) 1-a 7.4. Hereafter it will be customary to abbreviate two double Q-proofs,
THE COMBINATORS I AND K
55
Observe that Ka is a function which has the value a for for every applicand b. Thus we can think of Ka as a "constant" function which always has the same value a regardless of what applicand b it is applied to. Furthermore, K itself may be viewed as a two-place function which takes as its value the first of its two applicands. In ordinary functional notation this fact would be expressed: K(a,b) ~ a. 7.7. ~ !£! K (K elim, K int, neg K elim, neg K int) These four rules are expressed schematically as follows:
- L Kab
-Ia
as:
For example, the rule neg K int for the case where zero is expressed schematically as follows:
Hence the four rules stated in 7.2 and 7.3 can be stated in a combined form as follows:
- L Iab 1 ..• bn
-I ab 1 •.. bn For
n
7.5. The rule of !-identity may be derived thus:
3
This means that '-(Kab)' is a d.c. of '-a' by neg K int. As another example, the rule of neg K elim for the case where n is 2 is expressed schematically as follows:
L -(Kabcd) I -(acd)
-L Ia -L Ia
is
L-a I -(Kab)
chosen as zero we have:
-Ia P54- l 2
n
This means that '-(acd)' is a d.c. of '-(Kabcd)' by neg K elim. The two negative K-rules may be expressed schematically as follows by a double Q-proof:
hyp
-Ia
1, I elim
Ia = a
1-2, ext
In the quadruple subproof 1-2 above, reasons are given only for steps of the the first of the four combined subproofs. This will be permitted when reasons for the steps of the remaining three subproofs are closely analogous. 7.6. The Q-function K is a combinator and has the property, Kab = a. This latter identity (or rather the Q-formula 'Kab = a' expressing it) is called the K-identitl (K id) and is derivable by ext and the rules stated in 7.7 below.
L -(Kabc 1 ... en) 1-Cac 1 .•• en)
In the case that n is zero, the parentheses in '-(ac ••• cn)' 1 are dropped, and we have simply '-a'. 7.8. Here are examples of Q-proofs using the I- and Krules:
P55
l 2 3 4
hyp II III I(!II)
1, I int (case for n=O) 2, I int (case for n=l)
3. I int (case for n=O)
56 P56
ELEMENTS OF COMBINATORY LOGIC 1 2 3
-FI - II I = II
hyp 1, I int 1-2, ext
8.
1~ K(Ka)bc 2 Kac 3 a
hyp 1, K elim (case for n=l) 2, K elim (case for n~o)
P58
1 2 3 4-
b Kbc KKabc KKKKabc K(KKKK)aabc K(K(KKKK)aab)dc
hyp 1, K 2, K 3, K 4-, K 5, K
c
hyp 1, K elim 2, I int 3. K int 1-4-, ext 1-5, zeta
5 6 P59
1 2 3 45 6
-~ ~c
~ IK - K(IK)c KKc E K(IK)c KK = K(IK)
1 2 3 45
c
-~KKc
IK = K - K(IK)c KKc = K(IK)c KK = K(IK)
(case (case (case (case (case
for for for for for
n=O) n=2) n='+) n=3) n=l)
1 2 3
KK IK KK
KK K K(IK)
~
T
Case for n=O:
- L Cabc
-r acb
8.3. Here are some examples of Q-proofs that use the c-rules: P62
1 2 3 45 6
b
rm
Kbi b - Ib CKib = Ib CKI = I
hyp l, c 2, K 3. I l-4-, l-5.
elim elim int ext zeta
hyp 1, c 2, c l-3, 1-4-, l-5.
elim elim ext zeta zeta
hyp I id 1, 2, id elim 1-3. ext (incorrect) 1-4-. gamma
P63
id int I id 1, 2, (res) id elim
This shows that the converse of the converse of a function is the same as the function itself. Repeated uses of zeta as in P63 above will hereafter be abbreviated as in P64-:
7.9. EXERCISES. Construct proofs of the following: (1). 'KKKKK = KKKIIK' (2). 'TIIII = IIIIII' (3). 'a = KKK(Kab)b'
1 2 3 45 6
The following, however, is correct: P61
£ombinators C
8.1. If two Q-functions f and g are such that gab fba for all a and b, then we say that g is the converse of f. The combinator C when applied to a Q-function f gives the converse of f. Thus Cf is the converse of f, and we have: Cfab = fba. This identity can, of course, also be written Cabc = acb. The Q-formula 'Cabc = acb' is called the c-identity (C id). It is derivable by ext from the following rules for c: 8.2. ~ fQr C (C elim, C int, neg C elim, neg C int)
The following is not a correct Q-proof because step 3 uses id elim in a subproof required for ext. P60
~
=
P57
int int int int int
57
THE COMBINATORS C AND T
P64-
1 2 3 45
a
b
-~C(Cf)ab
Cfba - fab C(Cf)ab = fab C(Cf)a = fa C(Cf) = f
- ~ C(Cf)ab Cfba fab fab C(Cf)ab C(Cf) = f a,b
hyp 1, c 2, c 1-3. 1-4-,
elim elim ext zeta
56 P56
ELEMENTS OF COMBINATORY LOGIC 1 2 3
-FI - II I = II
hyp 1, I int 1-2, ext
8.
1~ K(Ka)bc 2 Kac 3 a
hyp 1, K elim (case for n=l) 2, K elim (case for n~o)
P58
1 2 3 4-
b Kbc KKabc KKKKabc K(KKKK)aabc K(K(KKKK)aab)dc
hyp 1, K 2, K 3, K 4-, K 5, K
c
hyp 1, K elim 2, I int 3. K int 1-4-, ext 1-5, zeta
5 6 P59
1 2 3 45 6
-~ ~c
~ IK - K(IK)c KKc E K(IK)c KK = K(IK)
1 2 3 45
c
-~KKc
IK = K - K(IK)c KKc = K(IK)c KK = K(IK)
(case (case (case (case (case
for for for for for
n=O) n=2) n='+) n=3) n=l)
1 2 3
KK IK KK
KK K K(IK)
~
T
Case for n=O:
- L Cabc
-r acb
8.3. Here are some examples of Q-proofs that use the c-rules: P62
1 2 3 45 6
b
rm
Kbi b - Ib CKib = Ib CKI = I
hyp l, c 2, K 3. I l-4-, l-5.
elim elim int ext zeta
hyp 1, c 2, c l-3, 1-4-, l-5.
elim elim ext zeta zeta
hyp I id 1, 2, id elim 1-3. ext (incorrect) 1-4-. gamma
P63
id int I id 1, 2, (res) id elim
This shows that the converse of the converse of a function is the same as the function itself. Repeated uses of zeta as in P63 above will hereafter be abbreviated as in P64-:
7.9. EXERCISES. Construct proofs of the following: (1). 'KKKKK = KKKIIK' (2). 'TIIII = IIIIII' (3). 'a = KKK(Kab)b'
1 2 3 45 6
The following, however, is correct: P61
£ombinators C
8.1. If two Q-functions f and g are such that gab fba for all a and b, then we say that g is the converse of f. The combinator C when applied to a Q-function f gives the converse of f. Thus Cf is the converse of f, and we have: Cfab = fba. This identity can, of course, also be written Cabc = acb. The Q-formula 'Cabc = acb' is called the c-identity (C id). It is derivable by ext from the following rules for c: 8.2. ~ fQr C (C elim, C int, neg C elim, neg C int)
The following is not a correct Q-proof because step 3 uses id elim in a subproof required for ext. P60
~
=
P57
int int int int int
57
THE COMBINATORS C AND T
P64-
1 2 3 45
a
b
-~C(Cf)ab
Cfba - fab C(Cf)ab = fab C(Cf)a = fa C(Cf) = f
- ~ C(Cf)ab Cfba fab fab C(Cf)ab C(Cf) = f a,b
hyp 1, c 2, c 1-3. 1-4-,
elim elim ext zeta
ELEMENTS OF COMBINATORY LOGIC
58
8.5. We now define 'T' as an abbreviation for 'CI'. The following rules for T are then derivable: 8.6. ~forT (T elim, T int, neg T elim, neg T int)
- LTabc 1 .•. en -I bac 1 •.. en
8.7. From 8.6 by ext we obtain the identity 'Tab= ba'. This will be called the T-identit~ (Tid). 8.8. EXERCISES. Construct proofs of the following: (1). 'I= CT' (2). 'I= CKa' (3). 'KI=TKC' 8.9. It is to be recalled that the names of Q-functions are Q-formulas (3.4). Given a Q-formula of the form 'ba', we will say that 'Tab' is the T-expansion of 'ba'. (Similarly we would say that 'Cabc' is the C-expansion of 'acb', and that 'Kab', 'Kac', and so on, are K-expansions of 'a', and that 'Ia' is the I-exEansion of 'a'.) Not every Q-formula has a T-expansion, because none of the names of the primitive Q-functions is of the form 'ba', and so such names (for example, 'd', 'I', 'K' and 'C') do not have T-expansions. On the other hand, 'T' itself can be considered to be of the form 'ba' because it is an abbreviation for 'CI'. The T-expansion of 'CI' is obviously 'TIC', and so the T-expansion of 'T' is also 'TIC'. Notice that the T-expansion of 'VBV' is 'TV(VB)'. This is because 'VBV' is really an abbreviation for '(VB)V', and because the T-expansion of the latter is 'TV(VB)'. In the following listing we compare various expressions with their T-expansions. The expressions are on the left and the corresponding T-expansions are on the right. fg Tgf fgh Th(fg) f(gh) T(gh)f fghi Ti(fgh) fgh(ij) T( ij) ( fgh) fgh( ijk) T(ijk)(fgh)
THE COMBINATORS C AND T
59
8.10. Every Q-formula other than the names of primitive Q-functions has a ~ section and a right section, so that if a Q-formula is of the form 'ab' we call 'a' the left section and 'b' the right section. More exactly: If 'a' and 'b' are such that 'a' does not occur in 'b' and 'b' does not occur in 'a', then the only occurrence of 'a' in 'ab' is defined as being the left secion of 'ab' and the only occurrence of 'b' in 'ab' is defined as being the right section of 'ab'. If 'a' occurs in 'b', then there is more than one occurrence of 'a' in 'ab', and in this case the leftmost occurrence of 'a' in 'ab' is defined as being the left section of 'ab'. If 'b' occurs in 'a', then there is more than one occurrence of 'b' in 'ab', and in this case the rightmost occurrence of 'b' in 'ab' is defined as being the right section of 'ab'. 8.11. In general, then, the leftmost (or only) occurrence of 'a' in 'ab' is the left section of 'ab', and the rightmost (or only) occurrence of 'b' in 'ab' is the right section of 'ab'. In order to construct the T-expansion of a Q-formula we first write'T' itself, then we write the right section of the Q-formula, and finally the left section of the Q-formula. We now give examples of right and left sections and T-expansions. In all cases except the first, the original expression is written in two different forms. The second form is to show more clearly the separation between left and right sections. Expression
Left Section
Right Section
T-expansion
-=
The only occurrenee of ' '
The only occurrenee of ' = '
T=-
vvv
The third occurrenee of 'V'
TV(VV)
(VV)V
The only occurrenee of '(VV)'
TCC(Iic-) (TCC) (dC-)
The only occurrenee of '(TCC)'
The only occurrenee of '(de-)' .
T(dC-)(TCC)
Ed( Bd( Bit))
The first occurrenee of '(B8:)'
The only occurT(Bd(Bd)) ( Ed) renee of '(Bd(B&))'
(Ed) (BII:(Bd))
-
ELEMENTS OF COMBINATORY LOGIC
58
8.5. We now define 'T' as an abbreviation for 'CI'. The following rules for T are then derivable: 8.6. ~forT (T elim, T int, neg T elim, neg T int)
- LTabc 1 .•. en -I bac 1 •.. en
8.7. From 8.6 by ext we obtain the identity 'Tab= ba'. This will be called the T-identit~ (Tid). 8.8. EXERCISES. Construct proofs of the following: (1). 'I= CT' (2). 'I= CKa' (3). 'KI=TKC' 8.9. It is to be recalled that the names of Q-functions are Q-formulas (3.4). Given a Q-formula of the form 'ba', we will say that 'Tab' is the T-expansion of 'ba'. (Similarly we would say that 'Cabc' is the C-expansion of 'acb', and that 'Kab', 'Kac', and so on, are K-expansions of 'a', and that 'Ia' is the I-exEansion of 'a'.) Not every Q-formula has a T-expansion, because none of the names of the primitive Q-functions is of the form 'ba', and so such names (for example, 'd', 'I', 'K' and 'C') do not have T-expansions. On the other hand, 'T' itself can be considered to be of the form 'ba' because it is an abbreviation for 'CI'. The T-expansion of 'CI' is obviously 'TIC', and so the T-expansion of 'T' is also 'TIC'. Notice that the T-expansion of 'VBV' is 'TV(VB)'. This is because 'VBV' is really an abbreviation for '(VB)V', and because the T-expansion of the latter is 'TV(VB)'. In the following listing we compare various expressions with their T-expansions. The expressions are on the left and the corresponding T-expansions are on the right. fg Tgf fgh Th(fg) f(gh) T(gh)f fghi Ti(fgh) fgh(ij) T( ij) ( fgh) fgh( ijk) T(ijk)(fgh)
THE COMBINATORS C AND T
59
8.10. Every Q-formula other than the names of primitive Q-functions has a ~ section and a right section, so that if a Q-formula is of the form 'ab' we call 'a' the left section and 'b' the right section. More exactly: If 'a' and 'b' are such that 'a' does not occur in 'b' and 'b' does not occur in 'a', then the only occurrence of 'a' in 'ab' is defined as being the left secion of 'ab' and the only occurrence of 'b' in 'ab' is defined as being the right section of 'ab'. If 'a' occurs in 'b', then there is more than one occurrence of 'a' in 'ab', and in this case the leftmost occurrence of 'a' in 'ab' is defined as being the left section of 'ab'. If 'b' occurs in 'a', then there is more than one occurrence of 'b' in 'ab', and in this case the rightmost occurrence of 'b' in 'ab' is defined as being the right section of 'ab'. 8.11. In general, then, the leftmost (or only) occurrence of 'a' in 'ab' is the left section of 'ab', and the rightmost (or only) occurrence of 'b' in 'ab' is the right section of 'ab'. In order to construct the T-expansion of a Q-formula we first write'T' itself, then we write the right section of the Q-formula, and finally the left section of the Q-formula. We now give examples of right and left sections and T-expansions. In all cases except the first, the original expression is written in two different forms. The second form is to show more clearly the separation between left and right sections. Expression
Left Section
Right Section
T-expansion
-=
The only occurrenee of ' '
The only occurrenee of ' = '
T=-
vvv
The third occurrenee of 'V'
TV(VV)
(VV)V
The only occurrenee of '(VV)'
TCC(Iic-) (TCC) (dC-)
The only occurrenee of '(TCC)'
The only occurrenee of '(de-)' .
T(dC-)(TCC)
Ed( Bd( Bit))
The first occurrenee of '(B8:)'
The only occurT(Bd(Bd)) ( Ed) renee of '(Bd(B&))'
(Ed) (BII:(Bd))
-
60
ELEMENTS OF COMBINATORY LOGIC
8.12. We may also speak of the right section of the right section, the left section of the left section, the right section of the left section, and so one, when such sections exist. Thus the second occurrence of '&' in '&&&' is the right section of the left section of '&&&', since •&&&• is really an abbreviation for '(&&)&' and ultimately for '((&&)&)'.The second occurrence of 'BV' (that is, of '(BV)') in 'BV(BVBV)' is the left section of the left section of the right section of 'BV(BVBV)'. This is seen more easily if we write 'BV(BVBV)' in its unabbreviated form '(BV)(((BV)B)V)', or ultimately, '((BV)(((BV)B)V;;'. Similarly we can see that the second occurrence of 'B' in 'BV(BVBV)' is the left section of the left section of the left section of the right section of 'BV(BVBV)'. We will subsequently make important use of the concepts of left and right section. 8.1;. EXERCISES. In terms of right and left sections give the location in 'B&V(B&VK)(KKK)' of each of the following: (1). The first occurrence of 'B' • (2). The second occurrence of 'B&'. (3). The fourth occurrence of 'K'. (4). The only occurrence of 'KKK'.
9·
~
Combinators B
~
W
9.1. The fundamental identity for the combinator B is Babe = a(bc). This identity, known as the B-~~ (B id), is derivable by ext from the following rules for B: 9.2. ~ £££ B (B elim, B int, neg B elim, neg B int) Case for n=O:
- L Babe -la(bc)
q.3. The fundamental identity for the combinator W is Wab = abb. This identity, known as the W-identit~ (W id), is derivable by ext from the following rules for W: 9.4. ~ fQE W (W elim, W int, neg W elim, neg W int) Case for n=O:
-L Wab
-r abb
THE COMBINATORS B AND W
61
9.5. The combinator B may be thought of as a two-place function which operates on a pair of transformations or on a pair of operators to give the Eroduct of those two transformations or operators. By the "product" we mean the single transformation (or single operator) which gives the same effect as the successive application of the two transformations (or operators). For example. if f is a transformation representing a clockwise rotation of 20 degrees, and if g is a transformation representing a counterclockwise rotation of 5 degrees, then Bfg would represent a clockwise rotation of 15 degrees. This is because Bfgx = f(gx), so that if gx is the result of rotating x counterclockwise 5 degrees, and if f(gx) is the result of rotating gx clockwise 20 degrees, then f(gx) is the result of rotating x clockwise 15 degrees, and therefore so is Bfgx. As another example, suppose that 'D' 1s used to stand for differentiation, so that Df would be the function that is the derivative of the function f. Then D(Df) would be the double (i.e. second) derivative of f. Now by the B-identity we have BDDf = D(Df), so that BDD applied to f gives the same effect as a double application of D to f. Thus BDD, which is the product of D with D (i.e. the square of D), is the operation of double differentiation. 9.6. It will be convenient sometimes to abbreviate 'Bab' as '[b.a]', thus emphasizing that B is a kind of multiplication. In fact, in the theory of groups, if the elements of a group are regarded as being transformations, then the fundamental group operation of multiplication is exactly this kind of multiplication. Using this notation, the product of the two rotations g and f (mentioned in 9.5 above) would be simply [g.f], and the operation of double differentiation would be [D.D]. We could also use •n 2 • as a further abbreviation for '[D.D]', but we do not need to do so, because our subse1uent definition of the number 2, together with our subsequent definition of exnonentiation, will enable us to derive the equation, [a.a] = a 2 . Indeed, we can define '2'
60
ELEMENTS OF COMBINATORY LOGIC
8.12. We may also speak of the right section of the right section, the left section of the left section, the right section of the left section, and so one, when such sections exist. Thus the second occurrence of '&' in '&&&' is the right section of the left section of '&&&', since •&&&• is really an abbreviation for '(&&)&' and ultimately for '((&&)&)'.The second occurrence of 'BV' (that is, of '(BV)') in 'BV(BVBV)' is the left section of the left section of the right section of 'BV(BVBV)'. This is seen more easily if we write 'BV(BVBV)' in its unabbreviated form '(BV)(((BV)B)V)', or ultimately, '((BV)(((BV)B)V;;'. Similarly we can see that the second occurrence of 'B' in 'BV(BVBV)' is the left section of the left section of the left section of the right section of 'BV(BVBV)'. We will subsequently make important use of the concepts of left and right section. 8.1;. EXERCISES. In terms of right and left sections give the location in 'B&V(B&VK)(KKK)' of each of the following: (1). The first occurrence of 'B' • (2). The second occurrence of 'B&'. (3). The fourth occurrence of 'K'. (4). The only occurrence of 'KKK'.
9·
~
Combinators B
~
W
9.1. The fundamental identity for the combinator B is Babe = a(bc). This identity, known as the B-~~ (B id), is derivable by ext from the following rules for B: 9.2. ~ £££ B (B elim, B int, neg B elim, neg B int) Case for n=O:
- L Babe -la(bc)
q.3. The fundamental identity for the combinator W is Wab = abb. This identity, known as the W-identit~ (W id), is derivable by ext from the following rules for W: 9.4. ~ fQE W (W elim, W int, neg W elim, neg W int) Case for n=O:
-L Wab
-r abb
THE COMBINATORS B AND W
61
9.5. The combinator B may be thought of as a two-place function which operates on a pair of transformations or on a pair of operators to give the Eroduct of those two transformations or operators. By the "product" we mean the single transformation (or single operator) which gives the same effect as the successive application of the two transformations (or operators). For example. if f is a transformation representing a clockwise rotation of 20 degrees, and if g is a transformation representing a counterclockwise rotation of 5 degrees, then Bfg would represent a clockwise rotation of 15 degrees. This is because Bfgx = f(gx), so that if gx is the result of rotating x counterclockwise 5 degrees, and if f(gx) is the result of rotating gx clockwise 20 degrees, then f(gx) is the result of rotating x clockwise 15 degrees, and therefore so is Bfgx. As another example, suppose that 'D' 1s used to stand for differentiation, so that Df would be the function that is the derivative of the function f. Then D(Df) would be the double (i.e. second) derivative of f. Now by the B-identity we have BDDf = D(Df), so that BDD applied to f gives the same effect as a double application of D to f. Thus BDD, which is the product of D with D (i.e. the square of D), is the operation of double differentiation. 9.6. It will be convenient sometimes to abbreviate 'Bab' as '[b.a]', thus emphasizing that B is a kind of multiplication. In fact, in the theory of groups, if the elements of a group are regarded as being transformations, then the fundamental group operation of multiplication is exactly this kind of multiplication. Using this notation, the product of the two rotations g and f (mentioned in 9.5 above) would be simply [g.f], and the operation of double differentiation would be [D.D]. We could also use •n 2 • as a further abbreviation for '[D.D]', but we do not need to do so, because our subse1uent definition of the number 2, together with our subsequent definition of exnonentiation, will enable us to derive the equation, [a.a] = a 2 . Indeed, we can define '2'
62
ELEMENTS OF COMBINATORY LOGIC
as an abbreviation for 'WB', and we can define 'ab' as 'ba' (this is our definition of exponentiation in the sense of raising to a power), and the proof of the above equation can then be constructed as follows: P65
1
2
3
WBa = Baa aWB = [a. a] [a. a] = a 2
w id 1, rep, def 2, sym id, def
This sort of multiplication will also serve as multiplication between natural numbers (non-negative integers) when we define these numbers. We have already defined '2' as 'WB'. It is also possible to to define '1' as 'I', and to define 'O' as 'CK'. The fundamental identities for 0, 1, and 2 are then the following provable identities: Oab b lab ab 2ab a(ab) Similarly, the fundamental identity for 3 would be 3ab a(a(ab)) and that for 4 would be 4ab a(a(a(ab)) and so on. Definitions of '3', '4', and so on will be given later. 9.7. When we use the notation '[b.a]' in place of 'Bab', we may speak of multiplication instead of speaking of B, and when rules for B are expressec with this dot notation, we may call them rules for multi2lication (mult elim, mult int, neg mult elim. neg mult int), writing them as follows: Case for n=O:
- L [b .a ]c
-r a(bc)
From the above rules the multi2lication identit~ (mult id), [b.a]c = a(bc), follows by ext. 9.8. We now give a proof that [a.I] = a (i.e. a proof of the Q-formula '[a. I] = a' ) .
THE COMBINATORS B AND W P66
[a.I]b
1 2
bl
3
[a. I ] = a
I( a b) ab
63 mult id I id 1-2, zeta
The above proof is an abbreviation for the following proof: P67
1 2 3 4
[a.I]b = I(ab) I(ab) = ab [a. I]b = ab [a. I] = a
b
mult id I id l, 2, trans id l-3. zeta
Notice that this method for abbreviating Q-proofs makes it possible to omit the left-hand side of an identity if it is the same as the right-hand side of a preceding identity. In this connection, reference to use of trans id may also be omitted. This method of abbreviation of Q-proofs will be used frequently from now on. A proof of [I.a] = a could also be given and would be somewhat similar to P67. Since the numeral '1' can be defined to be an abbreviation for 'I', the two equations [a.I] = a and [I.a] = a can also appear as [a.l] =a and [l.a] =a. g.g. A proof is now given of the associative ~ !££ multi2lication (assoc mult). P68
1 2 3 4
5
d
[a. [b.c]]d
[b.c](ad) c(b(ad)) c([a.b]d) [(a.b].c]d [a. [b. c]] = [[a.b].c) =
mult mult mult mult 1-4,
id id id, mon id (5.9) id zeta
Uses of both sym id and trans id will be omitted in abbreviated proofs such as the above. The full proof would be: P69
1 2 3 4
5 6 7
[a. [b.c]]d = [b. c ](ad) [b. c](ad) = c(b(ad)) b(ad) = [a.b]d c(b(ad)) = c([a.b]d) c([a.b]d) = ([a.b].c]d [a. [b.c]]d = [[a.b].c]d [a. [b. c)] = [(a.b].c]
d
mult id mult id mult id, sym id 3. men id mult id, sym id 1, 2, 4, 5. trans id l-6, zeta
62
ELEMENTS OF COMBINATORY LOGIC
as an abbreviation for 'WB', and we can define 'ab' as 'ba' (this is our definition of exponentiation in the sense of raising to a power), and the proof of the above equation can then be constructed as follows: P65
1
2
3
WBa = Baa aWB = [a. a] [a. a] = a 2
w id 1, rep, def 2, sym id, def
This sort of multiplication will also serve as multiplication between natural numbers (non-negative integers) when we define these numbers. We have already defined '2' as 'WB'. It is also possible to to define '1' as 'I', and to define 'O' as 'CK'. The fundamental identities for 0, 1, and 2 are then the following provable identities: Oab b lab ab 2ab a(ab) Similarly, the fundamental identity for 3 would be 3ab a(a(ab)) and that for 4 would be 4ab a(a(a(ab)) and so on. Definitions of '3', '4', and so on will be given later. 9.7. When we use the notation '[b.a]' in place of 'Bab', we may speak of multiplication instead of speaking of B, and when rules for B are expressec with this dot notation, we may call them rules for multi2lication (mult elim, mult int, neg mult elim. neg mult int), writing them as follows: Case for n=O:
- L [b .a ]c
-r a(bc)
From the above rules the multi2lication identit~ (mult id), [b.a]c = a(bc), follows by ext. 9.8. We now give a proof that [a.I] = a (i.e. a proof of the Q-formula '[a. I] = a' ) .
THE COMBINATORS B AND W P66
[a.I]b
1 2
bl
3
[a. I ] = a
I( a b) ab
63 mult id I id 1-2, zeta
The above proof is an abbreviation for the following proof: P67
1 2 3 4
[a.I]b = I(ab) I(ab) = ab [a. I]b = ab [a. I] = a
b
mult id I id l, 2, trans id l-3. zeta
Notice that this method for abbreviating Q-proofs makes it possible to omit the left-hand side of an identity if it is the same as the right-hand side of a preceding identity. In this connection, reference to use of trans id may also be omitted. This method of abbreviation of Q-proofs will be used frequently from now on. A proof of [I.a] = a could also be given and would be somewhat similar to P67. Since the numeral '1' can be defined to be an abbreviation for 'I', the two equations [a.I] = a and [I.a] = a can also appear as [a.l] =a and [l.a] =a. g.g. A proof is now given of the associative ~ !££ multi2lication (assoc mult). P68
1 2 3 4
5
d
[a. [b.c]]d
[b.c](ad) c(b(ad)) c([a.b]d) [(a.b].c]d [a. [b. c]] = [[a.b].c) =
mult mult mult mult 1-4,
id id id, mon id (5.9) id zeta
Uses of both sym id and trans id will be omitted in abbreviated proofs such as the above. The full proof would be: P69
1 2 3 4
5 6 7
[a. [b.c]]d = [b. c ](ad) [b. c](ad) = c(b(ad)) b(ad) = [a.b]d c(b(ad)) = c([a.b]d) c([a.b]d) = ([a.b].c]d [a. [b.c]]d = [[a.b].c]d [a. [b. c)] = [(a.b].c]
d
mult id mult id mult id, sym id 3. men id mult id, sym id 1, 2, 4, 5. trans id l-6, zeta
ELEMENTS OF COMBINATORY LOGIC
9.10. Suppose that, using ordinary mathematical notation, we wish to find a function g satisfying the equation, g(x) = f(x,x), for a given two-place function f. For example, i f f(x,y) = y cos x, we wish a g such that g(x) = f(x,x). The combinator W makes it possible to define the required g in terms of f, because g may be defined simply as Wf, since the W-identity (9.3) gives Wfx = fxx. 9.11. The combinator W plays an important part in various paradoxes. For example, W(B-) can be regarded being "the class of all classes that are not members of themselves," used in the Russell paradox. From the W-identity we get W(B-)a =B-aa, and from the B-identity we get B-aa= -(aa), so that W(B-)a = -(aa) by the transitivity of identity. Now by 3.8 the left side of the last equation means that a is a member of W(B-), while the right side means that a is not a member of itself. Thus the members of W(B-) are exactly those things (in this case, classes) which are not members of themselves. In other words, for a to be a member of W(B-) is equivalent to a not being a member of itself. In particular, if a is W(B-) itself, then the proposition W(B-)(W(B-)), asserting that W(B-) is a member of itself, is identical with the proposition -(W(B-)(W(B-)), asserting that W(B-) is not a member of itself. Thus as a special case of W(B-)a = -(aa), we have the equation, W(B-)(W(B-)) = -(W(B-)(W(B-))), which identifies a proposition with its own denial. Such an identification, however, does not lead to a contradiction in the system Q because neither of these propositions satisfies excluded middle. The following proof shows that if 'W(B-)(W(B-))' satisfied excluded middle, as step l in effect assumes, then a contradiction would be obtainable:
P70
l~W(B-)(W(B-)) V -(W(B-)(W(B-))) 2 W(B-)(W(B-)) = B-(W(B-))(W(B-)) 3 B-(W(B-))(W(B-)) = -(W(B-)(W(B-))) 4 W(B-)(W(B-)) = -(W(B-)(W(B-)))
hyp
65
THE COMBINATORS B AND W
5 6 7 8 9 10 ll
12 13
1'("-)(W(B-)J W(B-)(W(B-))
= -(W(B-)(W(B-))) -(W(B- )(W(B-))) W(B-)(W(B-)) a -(W(B-)(W(B-)))
hyp 4, reit
5. 6, id elim 5. 7, conj int
~-(W("- )(W("-)))
hyp
W(B-)(W(B-)) = -(W(B-)(W(B-))) W(B-)(W(B-)) W(B-)(W(B-)) a -(W(B-)(W(B-))) W(B-)(W(B-)) a -(W(B-)(W(B-)))
4, reit
9. 10, id elim 9. 11, conj int l' 5-8,9-12, diselim
Thus, if step l were a provable Q-formula, the system Q would be inconsistent because step 13 would be provable also. The Russell paradox is avoided by system Q because of the unprovability of step 1. Furthermore, the system Q can be shown to contain no contradictions at all. This can be done by using the method used in my paper, "The System Cf> of Combinatory Logic," Journal 2.£ Symbolic Logic, 23 0958), 13-21. 9.12. In spite of the fact that system Q can be shown to be free from contradiction, it is interesting to observe that a contradiction would arise in Q if the limitation on the use of identity elimination in connection with extensionity is ignored, as is shown in P71. Let 'R' serve as a temporary abbreviation for 'W(B(B-(=[W = W])))'.
P71
l
2 3
I
[RR = [W = W]] V -[RR = [W = W]] RR = [W = W]
w= w
4
RR
5 6
-[RR = [W = W] -(=[W = W](RR)) B-(=[W = W])(RR) B(B-(=[W = W]))RR W(B(B-(=[W = W])))R RR
7 8 9
w id
10
B id 2, 3, trans id
12
ll
RR W(B(B-(=[W = W])))R
ex mid id hyp id int 2, 3. id elim hyp 5. rep, def 6, B int 7. B int 8, W int 9. rep, def 1, 2-4, 5-10, dis elim 11, rep, def
ELEMENTS OF COMBINATORY LOGIC
9.10. Suppose that, using ordinary mathematical notation, we wish to find a function g satisfying the equation, g(x) = f(x,x), for a given two-place function f. For example, i f f(x,y) = y cos x, we wish a g such that g(x) = f(x,x). The combinator W makes it possible to define the required g in terms of f, because g may be defined simply as Wf, since the W-identity (9.3) gives Wfx = fxx. 9.11. The combinator W plays an important part in various paradoxes. For example, W(B-) can be regarded being "the class of all classes that are not members of themselves," used in the Russell paradox. From the W-identity we get W(B-)a =B-aa, and from the B-identity we get B-aa= -(aa), so that W(B-)a = -(aa) by the transitivity of identity. Now by 3.8 the left side of the last equation means that a is a member of W(B-), while the right side means that a is not a member of itself. Thus the members of W(B-) are exactly those things (in this case, classes) which are not members of themselves. In other words, for a to be a member of W(B-) is equivalent to a not being a member of itself. In particular, if a is W(B-) itself, then the proposition W(B-)(W(B-)), asserting that W(B-) is a member of itself, is identical with the proposition -(W(B-)(W(B-)), asserting that W(B-) is not a member of itself. Thus as a special case of W(B-)a = -(aa), we have the equation, W(B-)(W(B-)) = -(W(B-)(W(B-))), which identifies a proposition with its own denial. Such an identification, however, does not lead to a contradiction in the system Q because neither of these propositions satisfies excluded middle. The following proof shows that if 'W(B-)(W(B-))' satisfied excluded middle, as step l in effect assumes, then a contradiction would be obtainable:
P70
l~W(B-)(W(B-)) V -(W(B-)(W(B-))) 2 W(B-)(W(B-)) = B-(W(B-))(W(B-)) 3 B-(W(B-))(W(B-)) = -(W(B-)(W(B-))) 4 W(B-)(W(B-)) = -(W(B-)(W(B-)))
hyp
65
THE COMBINATORS B AND W
5 6 7 8 9 10 ll
12 13
1'("-)(W(B-)J W(B-)(W(B-))
= -(W(B-)(W(B-))) -(W(B- )(W(B-))) W(B-)(W(B-)) a -(W(B-)(W(B-)))
hyp 4, reit
5. 6, id elim 5. 7, conj int
~-(W("- )(W("-)))
hyp
W(B-)(W(B-)) = -(W(B-)(W(B-))) W(B-)(W(B-)) W(B-)(W(B-)) a -(W(B-)(W(B-))) W(B-)(W(B-)) a -(W(B-)(W(B-)))
4, reit
9. 10, id elim 9. 11, conj int l' 5-8,9-12, diselim
Thus, if step l were a provable Q-formula, the system Q would be inconsistent because step 13 would be provable also. The Russell paradox is avoided by system Q because of the unprovability of step 1. Furthermore, the system Q can be shown to contain no contradictions at all. This can be done by using the method used in my paper, "The System Cf> of Combinatory Logic," Journal 2.£ Symbolic Logic, 23 0958), 13-21. 9.12. In spite of the fact that system Q can be shown to be free from contradiction, it is interesting to observe that a contradiction would arise in Q if the limitation on the use of identity elimination in connection with extensionity is ignored, as is shown in P71. Let 'R' serve as a temporary abbreviation for 'W(B(B-(=[W = W])))'.
P71
l
2 3
I
[RR = [W = W]] V -[RR = [W = W]] RR = [W = W]
w= w
4
RR
5 6
-[RR = [W = W] -(=[W = W](RR)) B-(=[W = W])(RR) B(B-(=[W = W]))RR W(B(B-(=[W = W])))R RR
7 8 9
w id
10
B id 2, 3, trans id
12
ll
RR W(B(B-(=[W = W])))R
ex mid id hyp id int 2, 3. id elim hyp 5. rep, def 6, B int 7. B int 8, W int 9. rep, def 1, 2-4, 5-10, dis elim 11, rep, def
66
ELEMENTS OF COMBINATORY LOGIC 13 14 15 16 17 18
B(B-(=[W= W]))RR B-(=[W = W])(RR) -(=[W = W](RR)) -[RR = [W = W]] w = w W] RR = [W
12, w elim 13, B elim 14, B elim 15, rep, def id int
Step 18, which is inconsistent with step 16, is not act~ally a valid step, but it would follow from steps ll and 17 by ext as in P50 were it not for the fact that step ll requires use of id elim in its proof, as is seen at step 4, and cannot be obtained by using res id elim in place of id elim. The method of P71 is due to John R. Myhill. 9.13. EXERCISES. Give Q,-proofs of the following: (l). 'B 2 abcd a(bcd)' (2). 'B[a.b] = [(Ba).(Bb)]'
•c 2 = I' (4). •w2 ab = abbb'
(3).
2 ( 5) . 'K a be = a ' 9.14. Just as the T-expansion of a Q-formula may be formed by writing first 'T', then the right section of the Qformula, and finally the left section of the Q-formula, so also the B-expansion of a Q-formula may be formed according to the following principle: First write 'B', then write the left section of the Q-formula, then write the left section of its right section, and finally write the right section of its right section. Thus the B-expansion of 'a(bc)' is 'Babe'; that of 'ab(cd)' is 'B(ab)cd'; and that of 'abc(def)' is 'B(abc)(de)f'. Notice that in forming a B-expansion it is sometimes necessary to introduce parentheses with respect to the left section of the expression to be B-expanded, and with respect to the left section of its right section. Thus 'abc' and 'de' in 'abc(def)' must be set off with parentheses in forming the B-expansion 'B(abc)(de)f' of 'abc(def)'. Not all Q-formulas have a B-expansion, but only those which have a left section, a left section of a right section, and
THE COMBINATORS B AND W
67
a right section of a right section. Similarly, the only Qformulas that have T-expansions are those that have a left section and a right section. No Q-formula has more than one B-expansion or more than one T-expansion, nor does it have more than one I-expansion, C-expansion, or W-expansion. Every Q-formula 'a' has 'Ia' as its one and only I-expansion. Every Q-formula 'a' has infinitely many K-expansions, 'Kaa', 'Kab', 'Kac', and so on. TheW-expansion of a Q-formula 'abb' is 'Wab'. For a Q-formula to have a W-expansion, the Q-formula must have a left section of a left section, a right section of a left section, and a right section, and the same expression must constitute the right section as constitutes the right section of the left section. For example, •avv• has a w-expansion, namely 'W&V', but '&V&' has no w-expansion. In order to form the W-expansion of a Q-formula that has a w-expansion, we first write 'W', and then write the left section of the left section of the Q-formula, and finally write the right section of the Q-formula (the same as writing the right section of the left section of the Q-formula). As already noted, thew-expansion of 'abb' is 'Wab'; that of 'abcc' is 'W(ab)c'; that of 'abc(def)(def)' is 'W(abc)(def)'; and that of 'a(bcde)(bcde)' is 'Wa(bcde)'. It is interesting to observe that the W-expansion of 'WWW' is 'WWW' itself. Notice that 'abbe' does not have a W-expansion unless 'b' and 'c' are the same Q-formula. If they are the same Q-formula, then 'abbe' is the same as 'abbb' and has 'W(ab)b' as its W-expansion. In one respect W-expansions differ from I-, C-, T-, and B-expansions. Each of the latter expansions is always a longer expression than the expression of which it is an expansion, but this is not always true of W-expansions. For example, 'WWW' is its own W-expansion, as already mentioned, and theW-expansion of 'a(bc)(bc)' is the shorter expression 'Wa(bc)'. 0.15. When two Q-functions can be proved to be identical, we will say that the Q.-formulas that are their names
66
ELEMENTS OF COMBINATORY LOGIC 13 14 15 16 17 18
B(B-(=[W= W]))RR B-(=[W = W])(RR) -(=[W = W](RR)) -[RR = [W = W]] w = w W] RR = [W
12, w elim 13, B elim 14, B elim 15, rep, def id int
Step 18, which is inconsistent with step 16, is not act~ally a valid step, but it would follow from steps ll and 17 by ext as in P50 were it not for the fact that step ll requires use of id elim in its proof, as is seen at step 4, and cannot be obtained by using res id elim in place of id elim. The method of P71 is due to John R. Myhill. 9.13. EXERCISES. Give Q,-proofs of the following: (l). 'B 2 abcd a(bcd)' (2). 'B[a.b] = [(Ba).(Bb)]'
•c 2 = I' (4). •w2 ab = abbb'
(3).
2 ( 5) . 'K a be = a ' 9.14. Just as the T-expansion of a Q-formula may be formed by writing first 'T', then the right section of the Qformula, and finally the left section of the Q-formula, so also the B-expansion of a Q-formula may be formed according to the following principle: First write 'B', then write the left section of the Q-formula, then write the left section of its right section, and finally write the right section of its right section. Thus the B-expansion of 'a(bc)' is 'Babe'; that of 'ab(cd)' is 'B(ab)cd'; and that of 'abc(def)' is 'B(abc)(de)f'. Notice that in forming a B-expansion it is sometimes necessary to introduce parentheses with respect to the left section of the expression to be B-expanded, and with respect to the left section of its right section. Thus 'abc' and 'de' in 'abc(def)' must be set off with parentheses in forming the B-expansion 'B(abc)(de)f' of 'abc(def)'. Not all Q-formulas have a B-expansion, but only those which have a left section, a left section of a right section, and
THE COMBINATORS B AND W
67
a right section of a right section. Similarly, the only Qformulas that have T-expansions are those that have a left section and a right section. No Q-formula has more than one B-expansion or more than one T-expansion, nor does it have more than one I-expansion, C-expansion, or W-expansion. Every Q-formula 'a' has 'Ia' as its one and only I-expansion. Every Q-formula 'a' has infinitely many K-expansions, 'Kaa', 'Kab', 'Kac', and so on. TheW-expansion of a Q-formula 'abb' is 'Wab'. For a Q-formula to have a W-expansion, the Q-formula must have a left section of a left section, a right section of a left section, and a right section, and the same expression must constitute the right section as constitutes the right section of the left section. For example, •avv• has a w-expansion, namely 'W&V', but '&V&' has no w-expansion. In order to form the W-expansion of a Q-formula that has a w-expansion, we first write 'W', and then write the left section of the left section of the Q-formula, and finally write the right section of the Q-formula (the same as writing the right section of the left section of the Q-formula). As already noted, thew-expansion of 'abb' is 'Wab'; that of 'abcc' is 'W(ab)c'; that of 'abc(def)(def)' is 'W(abc)(def)'; and that of 'a(bcde)(bcde)' is 'Wa(bcde)'. It is interesting to observe that the W-expansion of 'WWW' is 'WWW' itself. Notice that 'abbe' does not have a W-expansion unless 'b' and 'c' are the same Q-formula. If they are the same Q-formula, then 'abbe' is the same as 'abbb' and has 'W(ab)b' as its W-expansion. In one respect W-expansions differ from I-, C-, T-, and B-expansions. Each of the latter expansions is always a longer expression than the expression of which it is an expansion, but this is not always true of W-expansions. For example, 'WWW' is its own W-expansion, as already mentioned, and theW-expansion of 'a(bc)(bc)' is the shorter expression 'Wa(bc)'. 0.15. When two Q-functions can be proved to be identical, we will say that the Q.-formulas that are their names
68
ELEMENTS OF COMBINATORY LOGIC
are equal. For example, the Q-functions I and II can be proved to be identical, because we can prove 'II = I' by the I-identity. Hence we are justified in saying that the Q-formulas 'I' and 'II' are equal. They are obviously not the same Q-formulas, because one of them consists of just one occurrence of the symbol 'I', while the other consists of two occurrences of the symbol 'I', but the fact that they are equal Q-formulas indicates that they are names of identical Q-functions, and hence names of one and the same Q-function. In other words, equal Q-formulas are names of the same Qfunction. Every Q-formula that has a T-expansion is equal to that T-expansion, and similarly for I-, K-, C-, B-, and W-expansions. 10.
Abstraction
10.1. Tn constru~ting P71 (thR last step of which is not valid) as given in 9.12, it was necessary to define 'R' in such a way that 'Ra' would be equal (in the sense of 9.15) to '-[aa = [W = W])' for every Q-formula 'a', in other words, in such a way that we would have the theorem, Ra = -[aa : [W = WJJ for every Q-function a, and in particular for the Q-function R itself, giving RR = -[RR = [W = W]]. The process of finding the definition of 'R' that gives these results is a special case of a general process which we will call abstraction. Let us see how this general process enables us to find the desired definition of 'R', namely the definition of 'R' as an abbreviation for the Q-formula 'W(B(B-(=[W = W])))'. We wish 'R' to be such that 'Ra' is equal to '-[aa = [W = W)]', The latter expression may, for convenience, be rewritten as '-(=[W = W](aa))', so that the occurrences of 'a' in it do not fall within square brackets. The process of abstraction consists in setting up a proof that starts with '-(=[W W](aa))' as an hypothesis and that has successive steps in which occurrences of 'a'
ABSTRACTION
69
get moved to the right as far as possible and finally become just a single occurrence of 'a' that constitutes the whole right section of the last step of the proof. The left section of this same last step is then the Q-formula for which 'R' can, by definition, serve as an abbreviation. This proof in which 'a' is gradually moved to the right is as follows: P72
lr-(=[W = W](aa)) 2 B-(=[W = W](aa) 3 B(B-(=[W = W]))aa 4 W(B(B-(=[W = W])))a
hyp l' B int 2, B int 3. w int
Notice that step 2 is the B-expansion of step 1, that step 3 is the B-expansion of step 2, and that step 4 is the W-expansion of step 3. Thus we reached the last step from the first step by a series of expansions. The proof also works in the reverse direction by use of the corresponding elimination rules, and we could form a similar double proof out of the negations of steps 1-4, using the corresponding negative introduction and elimination rules. Hence if we define 'R' as 'W(B(B-(=[W = W])))', ext gives the desired result that 'Ra' is equal to '-(=[W = W](aa))', that is, to '-[aa = [W = W]]'. Steps 1-4 of P72 should be compared with steps 6-9 of P7l. 10.2. The four-step proof constituted by P72 illustrates the following three principles concerned with abstraction: Princi£le l· If there are one or more occurrences of 'a' in the right section of a step in the abstraction process, and if 'a' is not the whole right section, then use B int to get the next step, so that the next step is the B-expansion of the present step. (This is illustrated in going from step l to step 2 in P72, and also in going from step 2 to step 3.) Princi£le ?· If 'a' is the whole right section of a step in the abstraction process, and if 'a' is also the right section of the left section of that step; so that the
68
ELEMENTS OF COMBINATORY LOGIC
are equal. For example, the Q-functions I and II can be proved to be identical, because we can prove 'II = I' by the I-identity. Hence we are justified in saying that the Q-formulas 'I' and 'II' are equal. They are obviously not the same Q-formulas, because one of them consists of just one occurrence of the symbol 'I', while the other consists of two occurrences of the symbol 'I', but the fact that they are equal Q-formulas indicates that they are names of identical Q-functions, and hence names of one and the same Q-function. In other words, equal Q-formulas are names of the same Qfunction. Every Q-formula that has a T-expansion is equal to that T-expansion, and similarly for I-, K-, C-, B-, and W-expansions. 10.
Abstraction
10.1. Tn constru~ting P71 (thR last step of which is not valid) as given in 9.12, it was necessary to define 'R' in such a way that 'Ra' would be equal (in the sense of 9.15) to '-[aa = [W = W])' for every Q-formula 'a', in other words, in such a way that we would have the theorem, Ra = -[aa : [W = WJJ for every Q-function a, and in particular for the Q-function R itself, giving RR = -[RR = [W = W]]. The process of finding the definition of 'R' that gives these results is a special case of a general process which we will call abstraction. Let us see how this general process enables us to find the desired definition of 'R', namely the definition of 'R' as an abbreviation for the Q-formula 'W(B(B-(=[W = W])))'. We wish 'R' to be such that 'Ra' is equal to '-[aa = [W = W)]', The latter expression may, for convenience, be rewritten as '-(=[W = W](aa))', so that the occurrences of 'a' in it do not fall within square brackets. The process of abstraction consists in setting up a proof that starts with '-(=[W W](aa))' as an hypothesis and that has successive steps in which occurrences of 'a'
ABSTRACTION
69
get moved to the right as far as possible and finally become just a single occurrence of 'a' that constitutes the whole right section of the last step of the proof. The left section of this same last step is then the Q-formula for which 'R' can, by definition, serve as an abbreviation. This proof in which 'a' is gradually moved to the right is as follows: P72
lr-(=[W = W](aa)) 2 B-(=[W = W](aa) 3 B(B-(=[W = W]))aa 4 W(B(B-(=[W = W])))a
hyp l' B int 2, B int 3. w int
Notice that step 2 is the B-expansion of step 1, that step 3 is the B-expansion of step 2, and that step 4 is the W-expansion of step 3. Thus we reached the last step from the first step by a series of expansions. The proof also works in the reverse direction by use of the corresponding elimination rules, and we could form a similar double proof out of the negations of steps 1-4, using the corresponding negative introduction and elimination rules. Hence if we define 'R' as 'W(B(B-(=[W = W])))', ext gives the desired result that 'Ra' is equal to '-(=[W = W](aa))', that is, to '-[aa = [W = W]]'. Steps 1-4 of P72 should be compared with steps 6-9 of P7l. 10.2. The four-step proof constituted by P72 illustrates the following three principles concerned with abstraction: Princi£le l· If there are one or more occurrences of 'a' in the right section of a step in the abstraction process, and if 'a' is not the whole right section, then use B int to get the next step, so that the next step is the B-expansion of the present step. (This is illustrated in going from step l to step 2 in P72, and also in going from step 2 to step 3.) Princi£le ?· If 'a' is the whole right section of a step in the abstraction process, and if 'a' is also the right section of the left section of that step; so that the
70
ELEMENTS OF COMBINATORY LOGIC
step has the general form 'faa', then use W int to get the next step, so that the next step is the W-expansion of the present step. (This is illustrated in going from step 3 to step 4 in P72.) Principle 2· If 'a' is the whole right section of a step in the abstraction process, and if 'a' does not occur in the left section of that step, then that step is the last step of the abstraction process, and the left section of it can be used as a definition of the desired Q-formula, that is, as the name of the desired Q-function. (This situation is illustrated by step 4 of P72.) 10.3. We now state further principles of abstraction which are not illustrated in P72. PrinciEle ~· If 'a' is not the right section of a step in the abstraction process, and does not occur in the right section, but does occur in the left section, then use T int to get the next step, so that the next step is the T-expansion of the present step and does contain one or more occurrences of 'a' in its right section. (This procedure obviously serves to move 'a' further to the right. For example, if the given step were '&a(Va)&', the next step would be the Texpansion 'T&(&a(Va))' of '&a(Va)&'.) PrinciEle 2· If 'a' is the whole right section of a step in the abstraction process, and if 'a' occurs in, but is not the whole of, the right section of the left section of that step (so that Principle 2 does not apply), then use B int to get the next step, but do so in such a way that the ~ section of this next step is the B-expansion of the left section of the present step, and its right section simply continues to be 'a'. (For example, if the given step were '&a(Va)a', then the next step would be 'B(&a)Vaa', and the step after that, by Principle 2, would be 'W(B(&a)V)a'. Furthermore, the step after that would be, by the present principle, 'BW(B(&a))Va'.)
ABSTRACTION
71
ErinciEle ~· If 'a' is the whole right section of a step in the abstraction process, and if 'a' occurs in the left section of the left section of that step, but not in the right section of the left section (so that Principle 2 and Principle 5 do not apply), then use T int to get the next step, but do so in such a way that the ~ section of this next step is the T-expansion of the left section of the present step, and its right section simply continues to be 'a'. (For example, if the given step were 'BW(B(&a))Va', which is the last step of the previous example, then the next step would be 'TV(BW(B(&a)))a'.) PrinciEle 2· If 'a' itself is the whole of a step in the abstraction process, then use I int, so that the next step is the I-expansion of 'a', namely 'Ia'. PrinciEle ~· If a step 'b' of the abstraction process has no occurrence of 'a' in it, then use K int, so that the next step is the K-expansion 'Kba' of 'b'. PrinciEle 2· If 'a' is the whole right section of a step in the abstraction process, and if it is also the whole left section so that the step is 'aa', then use I int in such a way that the next step is 'Iaa'. (Then, by Principle 2, the step after that would be 'Wia' and would be the last step.) 10.4. If the abstraction process starts with some Qformula '( ••. a •.• )' involving one or more occurrences of 'a' (or even involving no occurrences of 'a', so that Principle 8 applies), it is eventually bound to end with a Q-formula 'fa' (provided that 'a' itself does not involve 'B', 'T', 'W', 'I', or 'K'), and then we say that 'f' is the 'a'-~ stract of '( ••. a •.• )', and we use the following notation to serve as an abbreviation for the 'a'-abstract of '( ••• a ••• )', that is, for 'f': '[a]( ... a ••• )'. Notice that 'fa' can be the last step of the abstraction process only if Principle 3 applies, and that Principle 3 applies only if 'a' does not occur in 'f' (and, of course, is different from 'f'). Hence,
70
ELEMENTS OF COMBINATORY LOGIC
step has the general form 'faa', then use W int to get the next step, so that the next step is the W-expansion of the present step. (This is illustrated in going from step 3 to step 4 in P72.) Principle 2· If 'a' is the whole right section of a step in the abstraction process, and if 'a' does not occur in the left section of that step, then that step is the last step of the abstraction process, and the left section of it can be used as a definition of the desired Q-formula, that is, as the name of the desired Q-function. (This situation is illustrated by step 4 of P72.) 10.3. We now state further principles of abstraction which are not illustrated in P72. PrinciEle ~· If 'a' is not the right section of a step in the abstraction process, and does not occur in the right section, but does occur in the left section, then use T int to get the next step, so that the next step is the T-expansion of the present step and does contain one or more occurrences of 'a' in its right section. (This procedure obviously serves to move 'a' further to the right. For example, if the given step were '&a(Va)&', the next step would be the Texpansion 'T&(&a(Va))' of '&a(Va)&'.) PrinciEle 2· If 'a' is the whole right section of a step in the abstraction process, and if 'a' occurs in, but is not the whole of, the right section of the left section of that step (so that Principle 2 does not apply), then use B int to get the next step, but do so in such a way that the ~ section of this next step is the B-expansion of the left section of the present step, and its right section simply continues to be 'a'. (For example, if the given step were '&a(Va)a', then the next step would be 'B(&a)Vaa', and the step after that, by Principle 2, would be 'W(B(&a)V)a'. Furthermore, the step after that would be, by the present principle, 'BW(B(&a))Va'.)
ABSTRACTION
71
ErinciEle ~· If 'a' is the whole right section of a step in the abstraction process, and if 'a' occurs in the left section of the left section of that step, but not in the right section of the left section (so that Principle 2 and Principle 5 do not apply), then use T int to get the next step, but do so in such a way that the ~ section of this next step is the T-expansion of the left section of the present step, and its right section simply continues to be 'a'. (For example, if the given step were 'BW(B(&a))Va', which is the last step of the previous example, then the next step would be 'TV(BW(B(&a)))a'.) PrinciEle 2· If 'a' itself is the whole of a step in the abstraction process, then use I int, so that the next step is the I-expansion of 'a', namely 'Ia'. PrinciEle ~· If a step 'b' of the abstraction process has no occurrence of 'a' in it, then use K int, so that the next step is the K-expansion 'Kba' of 'b'. PrinciEle 2· If 'a' is the whole right section of a step in the abstraction process, and if it is also the whole left section so that the step is 'aa', then use I int in such a way that the next step is 'Iaa'. (Then, by Principle 2, the step after that would be 'Wia' and would be the last step.) 10.4. If the abstraction process starts with some Qformula '( ••. a •.• )' involving one or more occurrences of 'a' (or even involving no occurrences of 'a', so that Principle 8 applies), it is eventually bound to end with a Q-formula 'fa' (provided that 'a' itself does not involve 'B', 'T', 'W', 'I', or 'K'), and then we say that 'f' is the 'a'-~ stract of '( ••. a •.• )', and we use the following notation to serve as an abbreviation for the 'a'-abstract of '( ••• a ••• )', that is, for 'f': '[a]( ... a ••• )'. Notice that 'fa' can be the last step of the abstraction process only if Principle 3 applies, and that Principle 3 applies only if 'a' does not occur in 'f' (and, of course, is different from 'f'). Hence,
72
ELEMENTS OF COMBINATORY LOGIC
'a' never occurs in the 'a'-abstract of any Q-formula '( ..• a ••• )', that is, it never occurs in '[a]( ••• a ••• )'. (It might be more accurate to say that 'a' never occurs in the expression for which '[a]( ••. a ••• )' is an abbreviation.) For example, the 'a'-abstract of '-[aa [W = W)]' is, as we have seen, 'W(B(B-(=[W = W])))', and so 'a' does not occur in it. In other words, 'a' does not occur in (the expression for which) '[a](-[aa = [W = W]])' (is an abbreviation), namely 'W(B(B-(=[W = W])))'. 10.5. Whenever we speak of the 'a'-abstract of '( ••• a ••• )', it is to be assumed that 'a' does not involve any of the combinators 'B', 'T', 'W', 'I', or 'K'. This assumption guarantees that no step of the abstraction process introduces additional occurrences of 'a', and so guarantees that all occurrences of 'a' get moved to the right, where they coalesce into a single occurrence, whereupon the abstraction process terminates. 10.6. The abstraction process gives rise to a quadruple proof of the following form, where 'f' is '[a]( ••• a ••• )': P73
Furthermore, the above proof remains valid if 'a' is replaced throughout it by any other Q-formula 'b'. Hence by replacing 'a' by 'b' in P73, and recalling that 'f' is '[a]( ••• a .•• )', we obtain: P74
-~( ... b ... )
-~ [a~( •.. a ••• )b It can also be seen that the above equivalence still holds if a string of Q-forrnulas 'c •. 'c •, ... , 'en' is attached 1 2 on the right. Attaching this string and inverting the equivalence gives:
ABSTRACTION
73
10.7. Abstraction~ (abs elim, abs int, neg abs elim, neg abs int)
_L [a]( .•• a ••• )bc 1 ... en
-t (... b ••• )cl ..• cn
Here '( .•• a ••• )' is any Q-forrnula, and '( .•• b ••• )' is theresult of replacing all occurrences of 'a' in '( •.• a ••• )' by occurrences of 'b'. If there are no occurrences of 'a' in '( ... a ... )', then '( ... b ... )' is the same Q-formula as'( ... a •.• )'. Other letters can be used in the statement of the abstraction rules, for example:
_L [x]( •.• x ••• )ab 1 .•. bn
-t (... a ••• )bl •.. bn
10.8. By ext from 10.7 we obtain the abstraction identi~ (abs id), [x]( ••• x ••• )a = ( •.• a .•• ). This could of course also be written as [a]( •. a ••• )b =( ..• b .•• ) • 10.9. In connection with the abstraction process it should be remarked that if the process is carried out in accordance with the nine principles, a unique Q-formula is bound to result which will be, by definition, the 'a'abstract of '( .•• a ••• )'. If, on the other hand, the occurrences of 'a' are moved to the right by other uses of combinators than those described by the nine principles, then a kind of abstract is also obtained but it will not be the particular Q-formula which is, by definition, the 'a'-abstract of '( ... a ... )', that is, '[a]( ... a ... )', though it rna.y operate essentially like '[a]( ... a ... )' and, indeed, even be equal to '[a]( ••. a ... )' in the sense of 9.15. 10.10. A reformulation of P7l using the abstraction rules is now given. We now stipulate that 'R' is an abbreviation for '[x](-[xx = [W = W]])', that is, for the 'x'-abstract of '-[xx = [W = W))'. It turns out that 'R' is still an abbreviation for the same expression as previously, since the 'x'-abstract of '-[xx = [W = WJ]' is 'W(W(B-(=[W = W])))' · Notice how much the proof is shortened and simplified by the use of the abstraction rules.
72
ELEMENTS OF COMBINATORY LOGIC
'a' never occurs in the 'a'-abstract of any Q-formula '( ..• a ••• )', that is, it never occurs in '[a]( ••• a ••• )'. (It might be more accurate to say that 'a' never occurs in the expression for which '[a]( ••. a ••• )' is an abbreviation.) For example, the 'a'-abstract of '-[aa [W = W)]' is, as we have seen, 'W(B(B-(=[W = W])))', and so 'a' does not occur in it. In other words, 'a' does not occur in (the expression for which) '[a](-[aa = [W = W]])' (is an abbreviation), namely 'W(B(B-(=[W = W])))'. 10.5. Whenever we speak of the 'a'-abstract of '( ••• a ••• )', it is to be assumed that 'a' does not involve any of the combinators 'B', 'T', 'W', 'I', or 'K'. This assumption guarantees that no step of the abstraction process introduces additional occurrences of 'a', and so guarantees that all occurrences of 'a' get moved to the right, where they coalesce into a single occurrence, whereupon the abstraction process terminates. 10.6. The abstraction process gives rise to a quadruple proof of the following form, where 'f' is '[a]( ••• a ••• )': P73
Furthermore, the above proof remains valid if 'a' is replaced throughout it by any other Q-formula 'b'. Hence by replacing 'a' by 'b' in P73, and recalling that 'f' is '[a]( ••• a .•• )', we obtain: P74
-~( ... b ... )
-~ [a~( •.. a ••• )b It can also be seen that the above equivalence still holds if a string of Q-forrnulas 'c •. 'c •, ... , 'en' is attached 1 2 on the right. Attaching this string and inverting the equivalence gives:
ABSTRACTION
73
10.7. Abstraction~ (abs elim, abs int, neg abs elim, neg abs int)
_L [a]( .•• a ••• )bc 1 ... en
-t (... b ••• )cl ..• cn
Here '( .•• a ••• )' is any Q-forrnula, and '( .•• b ••• )' is theresult of replacing all occurrences of 'a' in '( •.• a ••• )' by occurrences of 'b'. If there are no occurrences of 'a' in '( ... a ... )', then '( ... b ... )' is the same Q-formula as'( ... a •.• )'. Other letters can be used in the statement of the abstraction rules, for example:
_L [x]( •.• x ••• )ab 1 .•. bn
-t (... a ••• )bl •.. bn
10.8. By ext from 10.7 we obtain the abstraction identi~ (abs id), [x]( ••• x ••• )a = ( •.• a .•• ). This could of course also be written as [a]( •. a ••• )b =( ..• b .•• ) • 10.9. In connection with the abstraction process it should be remarked that if the process is carried out in accordance with the nine principles, a unique Q-formula is bound to result which will be, by definition, the 'a'abstract of '( .•• a ••• )'. If, on the other hand, the occurrences of 'a' are moved to the right by other uses of combinators than those described by the nine principles, then a kind of abstract is also obtained but it will not be the particular Q-formula which is, by definition, the 'a'-abstract of '( ... a ... )', that is, '[a]( ... a ... )', though it rna.y operate essentially like '[a]( ... a ... )' and, indeed, even be equal to '[a]( ••. a ... )' in the sense of 9.15. 10.10. A reformulation of P7l using the abstraction rules is now given. We now stipulate that 'R' is an abbreviation for '[x](-[xx = [W = W]])', that is, for the 'x'-abstract of '-[xx = [W = W))'. It turns out that 'R' is still an abbreviation for the same expression as previously, since the 'x'-abstract of '-[xx = [W = WJ]' is 'W(W(B-(=[W = W])))' · Notice how much the proof is shortened and simplified by the use of the abstraction rules.
74-
P75
1 2
[RR
>;
IRR ~a W
4-
5 6 7 8 9 10 11
12
=
[W = W] V - [RR [W = W]
=
ELEMENTS OF COMBINATORY LOGIC
ABSTRACTION
[W
10.1'+. The multiple abstraction identity (abs id) is obtainable by ext from the multiple abstraction rules and is as follows: [xl] [x2]. • • [xm] ( • .• xl • .• x2. • • • • .xm. • • )al a2. • • am ( .•• al • •• a2 ••. • •• am ••• ) It is to be understood in the above identity and also in the statement of the multiple abstraction rules that the Q-formula '( ••• a 1 ••• a 2 ••••.• am···)' is the result of replacing • 1y everywh ere ln • I ( x 1 , I x 2 I , ••• , I xm I respec t lve ••• x ••• 1 x2 • . " • • • xm • • •) ' by 'al ' ' 'a2' ' • • • ' 'am' • 10.15. Here is an example of a proof that uses the multiple abstraction identity:
=
WJ]
~-[RR = [W = W]]
[x](- [xx [W = W]])R RR RR [x](-[RR = [W = W] ])R -[RR = [W = W]] w= w RR = [W = W]
ex mid id hyp id int 2, 3. id elim hyp 5, abs int 6, rep, def 1, 2-4-, 5-7, dis elim 8, rep, def 9. abs elim id int (invalid step)
10.11. It is possible to form double abstracts, triple abstracts, quadruple abstracts, and so on. In particular, the 'x'-abstract of the 'y'-abstract of a Q-formula '( ••• x ••• y ••• )' is called the 'x', 'y'-abstract of that Q-formula. It may also be referred to as '[x][y]( ••. x ••• y ••• )'. This is because the 'y'-abstract of '( •.. x •.. y ••• )' is '[y]( ••• x •.• y ••• )', so that the 'x'-abstract of the 'y'-abstract of '( ••• x ••• y ••• )' must be the 'x'-abstract of '[y]( •.• x •.• y .•• )', that is, '[x][y]( ... x ... y ... )'. Similarly the (triple) 'x', 'y','z'-abstract of '( ... x ... y ... z ... )' would be '[x][y][z] ( ••. x ... y ••• z ••• )'. The following rules for multiple abstraction are easily proved by repeated use of the ordinary rule. 10.12. Multiple abstrac~ ~ (abs elim, abs int, neg abs elim, neg abs int)
t
1
P76
a1
[xl][x2](bx2xl)ala2 Cba 1 = [x 1 ][x2 ](bx2 x 1 )a 1 Cb = [x 1 ][x2 J(bx 2 x ) 1
3 4-
t[
x 1 )[x2 Hx,]( ..• x 1 ••• x2 ••• x, ••• )o 1 o 2 o,b [x 2 J[x ]( •.. a .•• x ... x ••• )a a b 1 2 2 3 3 3 [x ]( ..• a ... a 2 ••• x •.• )a b 1 3 3 3 ( ... a 1 ... a 2 ... a ••• )b 3
hyp 1, abs elim 2, abs elim 3, abs e1im
c id 1, abs id 1-2, zeta 1-3, zeta
10.16. EXERCISES. Compute the abstracts presented in (1)-(8) below. In proving the identities in (9)-(11) there is no need to compute the abstracts. (1). The 'a'-abstract of 'aba', assuming that 'a' does not occur in 'b'. (The answer is 'WTb'.) (2). The 'x'-abstract of 'xxb', i.e. '[x](xxb)', assuming that 'x' does not occur in 'b'. (Similarly in (3)(8) below it is assumed that 'x' and 'y' do not occur in 'a' or 'b'.
'[x](xxxb)' '[x](xabx)' ( 5). '[x][y](abxy)' (6). '[x][y](xya)' (7). '[x][y](ayx)' (8). '[x] [y] (xyx)' (9). Prove 'CW = [x][y](yxx)'. (First prove [x][y](yxx)ab', then apply rule zeta twice.) (10). Prove 'C = [x][y][z](xzy)'. (11). Prove 'BW = [x][y][z](xyzz)'. (3). ( 4-).
10.13. As an example, the case of 10.12 for which m=3 and n=l can be proved as follows: 2 3 '+-
a 2 1 Cba 1 a 2
a
-r (... al ••• a2 .••••• am .•• ) b1 ••• bn 1-
= ba 2 ~ 1
1 2
- L [xl] [x2]. ·' [xm] ( • • .xl. • .x2. • · • • ·Xm· • • )al a2.' .ambl.' 'bn
P76
75
'CWab
=
74-
P75
1 2
[RR
>;
IRR ~a W
4-
5 6 7 8 9 10 11
12
=
[W = W] V - [RR [W = W]
=
ELEMENTS OF COMBINATORY LOGIC
ABSTRACTION
[W
10.1'+. The multiple abstraction identity (abs id) is obtainable by ext from the multiple abstraction rules and is as follows: [xl] [x2]. • • [xm] ( • .• xl • .• x2. • • • • .xm. • • )al a2. • • am ( .•• al • •• a2 ••. • •• am ••• ) It is to be understood in the above identity and also in the statement of the multiple abstraction rules that the Q-formula '( ••• a 1 ••• a 2 ••••.• am···)' is the result of replacing • 1y everywh ere ln • I ( x 1 , I x 2 I , ••• , I xm I respec t lve ••• x ••• 1 x2 • . " • • • xm • • •) ' by 'al ' ' 'a2' ' • • • ' 'am' • 10.15. Here is an example of a proof that uses the multiple abstraction identity:
=
WJ]
~-[RR = [W = W]]
[x](- [xx [W = W]])R RR RR [x](-[RR = [W = W] ])R -[RR = [W = W]] w= w RR = [W = W]
ex mid id hyp id int 2, 3. id elim hyp 5, abs int 6, rep, def 1, 2-4-, 5-7, dis elim 8, rep, def 9. abs elim id int (invalid step)
10.11. It is possible to form double abstracts, triple abstracts, quadruple abstracts, and so on. In particular, the 'x'-abstract of the 'y'-abstract of a Q-formula '( ••• x ••• y ••• )' is called the 'x', 'y'-abstract of that Q-formula. It may also be referred to as '[x][y]( ••. x ••• y ••• )'. This is because the 'y'-abstract of '( •.. x •.. y ••• )' is '[y]( ••• x •.• y ••• )', so that the 'x'-abstract of the 'y'-abstract of '( ••• x ••• y ••• )' must be the 'x'-abstract of '[y]( •.• x •.• y .•• )', that is, '[x][y]( ... x ... y ... )'. Similarly the (triple) 'x', 'y','z'-abstract of '( ... x ... y ... z ... )' would be '[x][y][z] ( ••. x ... y ••• z ••• )'. The following rules for multiple abstraction are easily proved by repeated use of the ordinary rule. 10.12. Multiple abstrac~ ~ (abs elim, abs int, neg abs elim, neg abs int)
t
1
P76
a1
[xl][x2](bx2xl)ala2 Cba 1 = [x 1 ][x2 ](bx2 x 1 )a 1 Cb = [x 1 ][x2 J(bx 2 x ) 1
3 4-
t[
x 1 )[x2 Hx,]( ..• x 1 ••• x2 ••• x, ••• )o 1 o 2 o,b [x 2 J[x ]( •.. a .•• x ... x ••• )a a b 1 2 2 3 3 3 [x ]( ..• a ... a 2 ••• x •.• )a b 1 3 3 3 ( ... a 1 ... a 2 ... a ••• )b 3
hyp 1, abs elim 2, abs elim 3, abs e1im
c id 1, abs id 1-2, zeta 1-3, zeta
10.16. EXERCISES. Compute the abstracts presented in (1)-(8) below. In proving the identities in (9)-(11) there is no need to compute the abstracts. (1). The 'a'-abstract of 'aba', assuming that 'a' does not occur in 'b'. (The answer is 'WTb'.) (2). The 'x'-abstract of 'xxb', i.e. '[x](xxb)', assuming that 'x' does not occur in 'b'. (Similarly in (3)(8) below it is assumed that 'x' and 'y' do not occur in 'a' or 'b'.
'[x](xxxb)' '[x](xabx)' ( 5). '[x][y](abxy)' (6). '[x][y](xya)' (7). '[x][y](ayx)' (8). '[x] [y] (xyx)' (9). Prove 'CW = [x][y](yxx)'. (First prove [x][y](yxx)ab', then apply rule zeta twice.) (10). Prove 'C = [x][y][z](xzy)'. (11). Prove 'BW = [x][y][z](xyzz)'. (3). ( 4-).
10.13. As an example, the case of 10.12 for which m=3 and n=l can be proved as follows: 2 3 '+-
a 2 1 Cba 1 a 2
a
-r (... al ••• a2 .••••• am .•• ) b1 ••• bn 1-
= ba 2 ~ 1
1 2
- L [xl] [x2]. ·' [xm] ( • • .xl. • .x2. • · • • ·Xm· • • )al a2.' .ambl.' 'bn
P76
75
'CWab
=
76
ELEMENTS OF COMBINATORY LOGIC
10.17. Some further combinators that are of some interest are s, which is such that Sabc = ac(bc); and t, which is such that tabcd = a(bd)(cd); and, finally, Y, which is such that Yabcd = a(bc)(bd). These combinators can be defined by use of multiple abstracts as follows: We let 'S' be an abbreviation for '[x][y][z](xz(yz))', and let 't' be an abbreviation for '[w][x][y][z](w(xz)(yz))', and let 'Y' be an abbreviation for '[w][x][y][z](w(xy)(xz))'. Then the above identities are obtainable as multiple abstraction identities. 10.18. It is characteristic of combinators as different from other sorts of Q-functions that the Q-formulas which are names of combinators are always equal to multiple abstracts '[x1 Hx2 J •.• [xm]( ••• x 1 ..• x 2 •••••• xm···)' of such a kind that '( ••• x 1 •.• x 2 •••••• x •.• )' is built up solely out 3 of 'x1 ', 'x2 •, ••• , 'xm'• though possibly not out of all of them. It should be clear from the definitions given in 10.1'1 that S, t, and Y are combinators. Other Q-functions that have previously been referred to as combinators can easily be shown to be such. For example, B is a combinator because 'B' is equal to the multiple abstract '[x][y][z](x(yz))'. 10.19. There is a different way of defining the 'a'abstract of '( •.. a .•• )' that makes fundamental use of the combinators S and K. If this method is employed, then the result of computing an abstract (as in 10.16) will usually be different from what it would be under the previous method. This different method can be described very economically by three clauses (1)-(3) as follows: (1). The 'a'-abstract of 'a' itself is 'I', or, if we please, 'SKK', which is equal to 'I'. (2). If 'a' does not occur in 'b', then the 'a'-abstract of 'b' is 'Kb'. (3). If the 'a'-abstract of the left section of 'b' is 'c', and if the 'a'-abstract of the right section of 'b' is 'd', then the 'a'-abstract of 'b' is 'Sed', For example, suppose we wish to compute the 'x'-abstract of
ABS'l'RACT ION
77
'xabx' by this new method. We assume, of course, as in exercise (4) of 10.16, that 'x' occurs in neither 'a' nor 'b'. Since the 'x'-abstract of of 'x' is 'I' and the 'x'-abstract of 'a' is 'Ka', we conclude that the 'x'-abstract of 'xa' is 'SI(Ka)'. Furthermore, the 'x'-abstract of 'b' is 'Kb', so that the 'x'-abstract of 'xab' is 'S(SI(Ka))(Kb)'. Finally, using again the fact that the 'x'-abstract of 'x' is 'I', we conclude that the 'x'-abstract of 'xabx' is 'S(S(SI(Ka))(Kb))I'. It would then be easy to show S(S(SI(Ka))(Kb))Ix = xabx, as required. 10.20. EXERCISES. (1). Compute '[x](xxxb)' by the new method, assuming that 'x' does not occur in 'b'. Show that for '[x](xx xb)' as thus computed, we have '[xJ(xxxb)a = aaab' as provable. (2). Prove 'W CSI' (u,;lng the identity for S given in 10.17). (3). Prove 'W S(CI)'. (4). Prove B(BS)B' (using the identity for given in 10 .17). ( 5). Prove 'S B(B(BW)C)(BB)'. ( 6). Prove 'B S(KS)K'. (7). Prove 'B BSK'. (8). Prove 'W SS(KI)'. (9). Prove 'Y H +(+B) )B(KK)' •
•
'.
76
ELEMENTS OF COMBINATORY LOGIC
10.17. Some further combinators that are of some interest are s, which is such that Sabc = ac(bc); and t, which is such that tabcd = a(bd)(cd); and, finally, Y, which is such that Yabcd = a(bc)(bd). These combinators can be defined by use of multiple abstracts as follows: We let 'S' be an abbreviation for '[x][y][z](xz(yz))', and let 't' be an abbreviation for '[w][x][y][z](w(xz)(yz))', and let 'Y' be an abbreviation for '[w][x][y][z](w(xy)(xz))'. Then the above identities are obtainable as multiple abstraction identities. 10.18. It is characteristic of combinators as different from other sorts of Q-functions that the Q-formulas which are names of combinators are always equal to multiple abstracts '[x1 Hx2 J •.• [xm]( ••• x 1 ..• x 2 •••••• xm···)' of such a kind that '( ••• x 1 •.• x 2 •••••• x •.• )' is built up solely out 3 of 'x1 ', 'x2 •, ••• , 'xm'• though possibly not out of all of them. It should be clear from the definitions given in 10.1'1 that S, t, and Y are combinators. Other Q-functions that have previously been referred to as combinators can easily be shown to be such. For example, B is a combinator because 'B' is equal to the multiple abstract '[x][y][z](x(yz))'. 10.19. There is a different way of defining the 'a'abstract of '( •.. a .•• )' that makes fundamental use of the combinators S and K. If this method is employed, then the result of computing an abstract (as in 10.16) will usually be different from what it would be under the previous method. This different method can be described very economically by three clauses (1)-(3) as follows: (1). The 'a'-abstract of 'a' itself is 'I', or, if we please, 'SKK', which is equal to 'I'. (2). If 'a' does not occur in 'b', then the 'a'-abstract of 'b' is 'Kb'. (3). If the 'a'-abstract of the left section of 'b' is 'c', and if the 'a'-abstract of the right section of 'b' is 'd', then the 'a'-abstract of 'b' is 'Sed', For example, suppose we wish to compute the 'x'-abstract of
ABS'l'RACT ION
77
'xabx' by this new method. We assume, of course, as in exercise (4) of 10.16, that 'x' occurs in neither 'a' nor 'b'. Since the 'x'-abstract of of 'x' is 'I' and the 'x'-abstract of 'a' is 'Ka', we conclude that the 'x'-abstract of 'xa' is 'SI(Ka)'. Furthermore, the 'x'-abstract of 'b' is 'Kb', so that the 'x'-abstract of 'xab' is 'S(SI(Ka))(Kb)'. Finally, using again the fact that the 'x'-abstract of 'x' is 'I', we conclude that the 'x'-abstract of 'xabx' is 'S(S(SI(Ka))(Kb))I'. It would then be easy to show S(S(SI(Ka))(Kb))Ix = xabx, as required. 10.20. EXERCISES. (1). Compute '[x](xxxb)' by the new method, assuming that 'x' does not occur in 'b'. Show that for '[x](xx xb)' as thus computed, we have '[xJ(xxxb)a = aaab' as provable. (2). Prove 'W CSI' (u,;lng the identity for S given in 10.17). (3). Prove 'W S(CI)'. (4). Prove B(BS)B' (using the identity for given in 10 .17). ( 5). Prove 'S B(B(BW)C)(BB)'. ( 6). Prove 'B S(KS)K'. (7). Prove 'B BSK'. (8). Prove 'W SS(KI)'. (9). Prove 'Y H +(+B) )B(KK)' •
•
'.
MULTIPLICATION, ADDITION, AND EXPONENTIATION
CHAPrER 4
FOUNDATIONS OF ARITHMETIC
11. Multiplication, Addition,
~ E~onentiation
11.1. In this section we will present some properties of multiplication, addition, and exponentiation that hold for all ~-functions. In the next section some further properties will be presented that do not hold for all ~-func tions but do hold for those Q-functions which will be called "natural numbers" (non-negative integers). For example, the associative law of multiplication holds for all ~-functions, but the commutative law does not. The latter law, however, does hold for natural numbers and for certain other ~-func tions. We first state some rules for 0 and 1. 11.2. ~ £££ 0 (0 elim, 0 int, neg 0 elim, neg 0 int) -LOabc 1 ••• cn bel ••• en
follows directly from the 1-identity by the monotonic rule for identity (5.9) and may, for convenience, also be referred to as the 1-identity. 11.4. Hereafter an expression of the form, 'abc = de f', will be understood as, '[abc] = [de f]'. Similarly, for example, 'abc de = fg h ij' is to be understood as '[(ab) c (de)]= [(fg) h (ij)]'. Also, 'a.b = c.d' will be understood as '[a.b] = [c.d]'. In general, outermost square brackets can be omitted from '[a.b]' if ambiguity does not result from doing so. 11.5. Associative ~ £££ multiplication (assoc mult) a.[b.c] = [a.b].c This was proved in P68. The above notation, however, uses an abbreviation stated in 11.4. 11.6. gight-~ multiplies~~ l (rt mult l) a.l = a This was proved in P66. 11.7. ~-~multiplication~ l (left mult l) l.a = a
This can be proved by a proof similar to P66. 11.8. &.gh!-~ multiplication~ 0 (rt mult 0) a.o = o This is proved as follows: P77
-r
The above rules are easily proved, assuming that '0' is treated as an abbreviation for 'CK' as in 9.6. The 0-~ ~ (0 id), Oab = b, follows from the above rules by ext. 11.3. ~ £££ l (l elim, l int, neg l elim, neg l int) - L lab 1 ••. bn -t"ab 1 •.. bn These rules are the same as the corresponding rules for I, since 'l' is treated as an abbreviation for 'I'. By ext we get the 1-~~~ (l id), la =a. The identity, lab= ab, 78
79
l 2 3 4
b,c
[a.O]bc
a.o = 0
O(ab)c = c Obc
mult id, mon id 0 id 0 id
l-3, zeta (see P64)
The rule of left-hand multiplication by 0 cannot be proved in F,eneral (unlike the above rule) but can be proved for the case where the multirlied Q-function is a natural number. The proof is ~iven in 12.21. Notice that the above rule holds for every Q-function a, ref>:ardless of whether a is a natural number or not. The same is true of all the rules stated in this section.
MULTIPLICATION, ADDITION, AND EXPONENTIATION
CHAPrER 4
FOUNDATIONS OF ARITHMETIC
11. Multiplication, Addition,
~ E~onentiation
11.1. In this section we will present some properties of multiplication, addition, and exponentiation that hold for all ~-functions. In the next section some further properties will be presented that do not hold for all ~-func tions but do hold for those Q-functions which will be called "natural numbers" (non-negative integers). For example, the associative law of multiplication holds for all ~-functions, but the commutative law does not. The latter law, however, does hold for natural numbers and for certain other ~-func tions. We first state some rules for 0 and 1. 11.2. ~ £££ 0 (0 elim, 0 int, neg 0 elim, neg 0 int) -LOabc 1 ••• cn bel ••• en
follows directly from the 1-identity by the monotonic rule for identity (5.9) and may, for convenience, also be referred to as the 1-identity. 11.4. Hereafter an expression of the form, 'abc = de f', will be understood as, '[abc] = [de f]'. Similarly, for example, 'abc de = fg h ij' is to be understood as '[(ab) c (de)]= [(fg) h (ij)]'. Also, 'a.b = c.d' will be understood as '[a.b] = [c.d]'. In general, outermost square brackets can be omitted from '[a.b]' if ambiguity does not result from doing so. 11.5. Associative ~ £££ multiplication (assoc mult) a.[b.c] = [a.b].c This was proved in P68. The above notation, however, uses an abbreviation stated in 11.4. 11.6. gight-~ multiplies~~ l (rt mult l) a.l = a This was proved in P66. 11.7. ~-~multiplication~ l (left mult l) l.a = a
This can be proved by a proof similar to P66. 11.8. &.gh!-~ multiplication~ 0 (rt mult 0) a.o = o This is proved as follows: P77
-r
The above rules are easily proved, assuming that '0' is treated as an abbreviation for 'CK' as in 9.6. The 0-~ ~ (0 id), Oab = b, follows from the above rules by ext. 11.3. ~ £££ l (l elim, l int, neg l elim, neg l int) - L lab 1 ••. bn -t"ab 1 •.. bn These rules are the same as the corresponding rules for I, since 'l' is treated as an abbreviation for 'I'. By ext we get the 1-~~~ (l id), la =a. The identity, lab= ab, 78
79
l 2 3 4
b,c
[a.O]bc
a.o = 0
O(ab)c = c Obc
mult id, mon id 0 id 0 id
l-3, zeta (see P64)
The rule of left-hand multiplication by 0 cannot be proved in F,eneral (unlike the above rule) but can be proved for the case where the multirlied Q-function is a natural number. The proof is ~iven in 12.21. Notice that the above rule holds for every Q-function a, ref>:ardless of whether a is a natural number or not. The same is true of all the rules stated in this section.
80
ELEMENTS OF COMBINATORY LOGIC 11.9. EeEonentiation
£z
-L [a+
0 (exp 0)
=~a
11.11. Multiplication£[ ab.c = (ab)c This is proved as follows: P80 l ab.c = a[b.c] 2 3 4-
[b.c]a c(ba) (ab)c
exponent~
(mult exp)
id int, def (11.4-) id int, def (9. 6) mult id id int, def (q,6)
We are to understand 'Cab)' to be an abbreviation for 'ba' in cases where the omission of the parentheses might lead to ambiguity. Thus in step 4- above we use 'Cab)' rather than 'ab' as an abbreviation for 'ba' (i.e, for '(ba)' ). 11.12. ~~. denoted by '+', will now be defined in such a way that the law of exponents (11.16) will be derivable, and will hold not merely for natural numbers but for Q-functions in general. We let '+' serve as an abbreviation for '[x][y][z][yz.xz]'. Here '[yz.xz]' abbreviates '[(yz).(xz)]'. 11.13. ~les fQr elim, neg add int)
~ion
"l [• •
11.14-. The derivation of the above rules is as follows:
id int, def (9. 6) 0 id 1 id l-3, zeta
Recall that we treat 'ab' as an abbreviation for 'ba', as stated in 9. 6. 11.10. Exponentiation £z 1 (exp 1) a1 s a This is proved as follows: 1 id int, def P79 a l id
~~
b]cd 1 ... dn
-1 [ac.bc]d1 ••• dn
0
= 1 This is proved as follows: P78 1 b a 0 b Oab 2 b lb 3 4- ao = 1 a
81
MULTIPLICATION, ADDITION, AND EXPONENTIATION
(add elim, add int, neg add
P81
b)oo 1 ... ' " +bacd 1 ••• dn [x)[y][z][yz.xz]bacd 1 ••• dn [ac.bc]d 1 ..• dn
2 3 4--
hyp 1, rep, def. (3.6) 2, rep, def ( 11. 12) 3. abs elim (10 .12)
11.15. From the rules for addition by ext we obtain the addition identity (add id): [a+ b]c = [ac.bc] 11.16. Addition Qf exponent~ (add exp) c[a + b) = ca.cb This is the same as the addition identity, but it is expressed in a way that uses the definition of exponentiation. 11.17. Right-~ addition £[ 0 Crt add 0) a + 0 = a Proof is left to reader as an exercise. 11.18. ~-~ addition of 0 (left add 0) 0 + a = a Proof is left to reader as an exercise. 11.19. Associative~ £2£ addition (assoc add) a + [b + c) = [a + b) + c This is proved as follows: P82
1 2 3 45 6
ad. [b + c]d ad.[bd.cd] [ad.bd].cd [a + b]d.cd [[a + b) + c]d a + [b + c) = [a + b) + c
d
[a + [b + c]]d
add id add id, mon id assoc mult add id, mon id add id 1-5. zeta
11.20. ~-~ distribution Q[ multiplication ~ addition (left dist mult add) a.[b +c) = a.b + a.c Here 'a.b + a,c' is an abhreviation for '[a.b] + [a.c]'.
80
ELEMENTS OF COMBINATORY LOGIC 11.9. EeEonentiation
£z
-L [a+
0 (exp 0)
=~a
11.11. Multiplication£[ ab.c = (ab)c This is proved as follows: P80 l ab.c = a[b.c] 2 3 4-
[b.c]a c(ba) (ab)c
exponent~
(mult exp)
id int, def (11.4-) id int, def (9. 6) mult id id int, def (q,6)
We are to understand 'Cab)' to be an abbreviation for 'ba' in cases where the omission of the parentheses might lead to ambiguity. Thus in step 4- above we use 'Cab)' rather than 'ab' as an abbreviation for 'ba' (i.e, for '(ba)' ). 11.12. ~~. denoted by '+', will now be defined in such a way that the law of exponents (11.16) will be derivable, and will hold not merely for natural numbers but for Q-functions in general. We let '+' serve as an abbreviation for '[x][y][z][yz.xz]'. Here '[yz.xz]' abbreviates '[(yz).(xz)]'. 11.13. ~les fQr elim, neg add int)
~ion
"l [• •
11.14-. The derivation of the above rules is as follows:
id int, def (9. 6) 0 id 1 id l-3, zeta
Recall that we treat 'ab' as an abbreviation for 'ba', as stated in 9. 6. 11.10. Exponentiation £z 1 (exp 1) a1 s a This is proved as follows: 1 id int, def P79 a l id
~~
b]cd 1 ... dn
-1 [ac.bc]d1 ••• dn
0
= 1 This is proved as follows: P78 1 b a 0 b Oab 2 b lb 3 4- ao = 1 a
81
MULTIPLICATION, ADDITION, AND EXPONENTIATION
(add elim, add int, neg add
P81
b)oo 1 ... ' " +bacd 1 ••• dn [x)[y][z][yz.xz]bacd 1 ••• dn [ac.bc]d 1 ..• dn
2 3 4--
hyp 1, rep, def. (3.6) 2, rep, def ( 11. 12) 3. abs elim (10 .12)
11.15. From the rules for addition by ext we obtain the addition identity (add id): [a+ b]c = [ac.bc] 11.16. Addition Qf exponent~ (add exp) c[a + b) = ca.cb This is the same as the addition identity, but it is expressed in a way that uses the definition of exponentiation. 11.17. Right-~ addition £[ 0 Crt add 0) a + 0 = a Proof is left to reader as an exercise. 11.18. ~-~ addition of 0 (left add 0) 0 + a = a Proof is left to reader as an exercise. 11.19. Associative~ £2£ addition (assoc add) a + [b + c) = [a + b) + c This is proved as follows: P82
1 2 3 45 6
ad. [b + c]d ad.[bd.cd] [ad.bd].cd [a + b]d.cd [[a + b) + c]d a + [b + c) = [a + b) + c
d
[a + [b + c]]d
add id add id, mon id assoc mult add id, mon id add id 1-5. zeta
11.20. ~-~ distribution Q[ multiplication ~ addition (left dist mult add) a.[b +c) = a.b + a.c Here 'a.b + a,c' is an abhreviation for '[a.b] + [a.c]'.
82
ELEMENTS OF COMBINATORY LOGIC
The proof of the above distributive law is as follows: P83
l 2 3 4 5 6
d
[a. [b + c]]d
[b + c](ad) b(ad).c(ad) [a.bJd.c(ad) [a.bJd.[a.cJd [a.b + a.cJd a. [b + cJ = a.b + a.c
mult id add id mult id, mon id mult id, mon id add id l-5. zeta
The corresponding right-hand distributive law can be proved for natural numbers, but not for Q-functions in general. See 12.28. 11.21. ~-successorshi£ of 0 (non-sue 0) -[0 = a + lJ This non-identity states that 0 is not the successor of any Q-function. The proof is as follows: P84
l 2 3 4 5 6
7 8 9 lO
0 = a + l -[0 OJ
00-[0 = OJ = [a + l]O- [O = OJ = [aO.lOJ-[0 = OJ [aO.OJ-[0 = OJ 0-[0 = OJ [O = OJ
0 = 0 -[0 = OJ -[0 a + lJ
hyp 0 id hyp, mon id add id, mon id l id, mon id rt mult o, mon id 0 id id int 7. 8, id elim 1-9, neg int id (5.10)
Notice that in giving the reason for step 7, we treat step 7 as being '0-[0 =OJ = [0 =OJ', but in referring to step 7 in step 9 we treat step 7 as being '-[0 =OJ = [O =OJ'. This same convention will be followed elsewhere when a series of identity symbols is used as in steps 2-7. 11.22. An alternative way for defining addition is to treat '+' as an abbreviation for '[wJ[xJ[yJ[zJ(wy(xyz))'. This definition gives the same effect as the other definition, but shows that '+' is a combinator in the sense of 10.18.
THE NATURAL NUMBERS
83
11.23. Just as B may be regarded as multiplication, so . 'T a b = a b, . t·1on, s1nce also T may be regarde d as exponen t 1a is easily proved. Thus the Q-functions 0, +, B, and T are the four mathematical concepts zero, addition, multiplication, and exponentiation. These four Q-functions are in fact combinators in the sense of 10.18, and they will be called the basic mathematical combinators. It can be shown that they ~l~e been chosen to serve as the primitive (i.e. undefined) combinators in place of I, K, C, B, and w. Definitions of expressions denoting the latter combinators (other than B) could then be stated as follows: , I' for '00'. 'C, for 'TBBTBBTBB, (i.e. 'BBT(BBT) (BBT)' ) , 'K' for 'CO' 'W' for 'C(B(+II)T)'. It is easy to verify that the rules for I, C, K, and W would then be derivable from the rules for O, +, B, and T. ll. 21+. EXERCISES. (l).
Prove 'a+ 0 =a'.
(2). Prove '0 +a= a'. (3). Show how an attempt to prove '[b + cJ.a b.a + c.a', using methods like those of P83, would fail. ( 4) • Prove ' - [ 0 = l J ' • (5). Let '+' be '+' as defined in 11.12, and let '+' be '+' as defined in 11.22. Prove '+ = + ' • (6). Prove 'C TBBTBBTBB ••
.
(7). 12.
~
Prove 'W = C(B(+II)T)'.
Natural Numbers
12.1. By the ~~numbers we mean the non-negative finite integers 0, 1, 2, 3, 4, and so on. We assume that the Q-formulas '0', '1', '1+1', '[l+lJ +1', '[[l+lJ+lJ + l', and so on, stand as names of them in the system Q. In other words, we regard the natural numbers as being Q-functions, and we use the following abbreviations:
82
ELEMENTS OF COMBINATORY LOGIC
The proof of the above distributive law is as follows: P83
l 2 3 4 5 6
d
[a. [b + c]]d
[b + c](ad) b(ad).c(ad) [a.bJd.c(ad) [a.bJd.[a.cJd [a.b + a.cJd a. [b + cJ = a.b + a.c
mult id add id mult id, mon id mult id, mon id add id l-5. zeta
The corresponding right-hand distributive law can be proved for natural numbers, but not for Q-functions in general. See 12.28. 11.21. ~-successorshi£ of 0 (non-sue 0) -[0 = a + lJ This non-identity states that 0 is not the successor of any Q-function. The proof is as follows: P84
l 2 3 4 5 6
7 8 9 lO
0 = a + l -[0 OJ
00-[0 = OJ = [a + l]O- [O = OJ = [aO.lOJ-[0 = OJ [aO.OJ-[0 = OJ 0-[0 = OJ [O = OJ
0 = 0 -[0 = OJ -[0 a + lJ
hyp 0 id hyp, mon id add id, mon id l id, mon id rt mult o, mon id 0 id id int 7. 8, id elim 1-9, neg int id (5.10)
Notice that in giving the reason for step 7, we treat step 7 as being '0-[0 =OJ = [0 =OJ', but in referring to step 7 in step 9 we treat step 7 as being '-[0 =OJ = [O =OJ'. This same convention will be followed elsewhere when a series of identity symbols is used as in steps 2-7. 11.22. An alternative way for defining addition is to treat '+' as an abbreviation for '[wJ[xJ[yJ[zJ(wy(xyz))'. This definition gives the same effect as the other definition, but shows that '+' is a combinator in the sense of 10.18.
THE NATURAL NUMBERS
83
11.23. Just as B may be regarded as multiplication, so . 'T a b = a b, . t·1on, s1nce also T may be regarde d as exponen t 1a is easily proved. Thus the Q-functions 0, +, B, and T are the four mathematical concepts zero, addition, multiplication, and exponentiation. These four Q-functions are in fact combinators in the sense of 10.18, and they will be called the basic mathematical combinators. It can be shown that they ~l~e been chosen to serve as the primitive (i.e. undefined) combinators in place of I, K, C, B, and w. Definitions of expressions denoting the latter combinators (other than B) could then be stated as follows: , I' for '00'. 'C, for 'TBBTBBTBB, (i.e. 'BBT(BBT) (BBT)' ) , 'K' for 'CO' 'W' for 'C(B(+II)T)'. It is easy to verify that the rules for I, C, K, and W would then be derivable from the rules for O, +, B, and T. ll. 21+. EXERCISES. (l).
Prove 'a+ 0 =a'.
(2). Prove '0 +a= a'. (3). Show how an attempt to prove '[b + cJ.a b.a + c.a', using methods like those of P83, would fail. ( 4) • Prove ' - [ 0 = l J ' • (5). Let '+' be '+' as defined in 11.12, and let '+' be '+' as defined in 11.22. Prove '+ = + ' • (6). Prove 'C TBBTBBTBB ••
.
(7). 12.
~
Prove 'W = C(B(+II)T)'.
Natural Numbers
12.1. By the ~~numbers we mean the non-negative finite integers 0, 1, 2, 3, 4, and so on. We assume that the Q-formulas '0', '1', '1+1', '[l+lJ +1', '[[l+lJ+lJ + l', and so on, stand as names of them in the system Q. In other words, we regard the natural numbers as being Q-functions, and we use the following abbreviations:
ELEMENTS OF COMBINATORY LOGIC
84
'0'
for 'CK',
'l'
for
'2' • 3'
for
'4'
for
for
I
I
85
THE NATURAL NUMBERS
12.4. From the above rules by ext the following identities can be obtained: the
I '
'l + l' '2 + l ' ' i.e. for '[l + l] + l. ' '3 + l ' ' i.e. for '[[l + l] + l] +l].'
0-~,
and so on. Previously we defined '2' as 'WB', but now we let
lab
the present definition replace the earlier one. Either defi-
2ab
a( a b)
3ab
a(a(ab))
4ab
a(a(a(ab)))
nition could be used because we can prove
WB
~
l + 1.
In
the system Q a two-digit number such as twenty-five would have to be written in some such way as this,
'25', where 'ab'
12.2. The Q-formulas '0',
'1',
'2', and so on, as de-
fined above, will be called ~-numerals. For example, 'I',
'CK',
'I+ I', and '[l + l] + l' are Q-numerals, but '1 + [l +
specifically to the Q-numeral '[1 + 1] + 1', since we can 1 + [1 + l]
~
[l + l ] + 1.
Except for
'0' and ' l ' , every Q-numeral is of the form 'a+ 1', where
a 1b 2 a b a3b 4 a b
ab
12. 5. Observe that 'Oab'
ab a( a b) a(a(ab)) a(a(a(ab)))
(or 'a 0 b') means, in effect,
the result of applying a to b zero times, and that 'lab' 1 (or •a b') means the result of applying a to b one time, 2 and that '2ab' (or •a b') means the result of applying a to
l]' is not a Q-numeral. although it is equal to a Q-numeral, prove the identity,
the 2-
and so on.
would be an abbreviation for '[9.a +a]+ b'. Numbers involving three or more digits can be handled in a similar way.
1-identit~,
the
!dentitl, the 3-identit~, the 4-!dentitl, and so on (0 id, l id, 2 id, 3 id, 4 id, and so on), specifically: Oab b or a 0b b
b
twice, and so on. 12.7. We will regard the Q-formula 'N' as being the
name of
~ ~
of
~ ~
natural
numb~
well, as being the name of the attribute ~).
Q£
(or, equally being
~
natural
There will be an introduction rule for the Q-func-
'a' is a Q-numeral. Also, if 'a' is a Q-numeral other than
tion N, that is, for the class of natural numbers, and also
'0', then 'a+ l' is a Q-numeral. Notice that '0 + l' is not
an elimination rule for N. The elimination rule has the pe-
a Q-numeral, but it is equal to the Q-numeral '1'. Any Q-for-
culiarity that it makes a step in a proof a direct conse-
mula equal to a Q-numeral is a name of the same natural num-
quence of infinitely many previous steps, thus requiring
ber as that Q-numeral. Hence '0 + l' is the name of l, just
proofs to be infinite in length wherever the rule is used.
as '1' is the name of 1. The following rules are easily de-
This may seem ob,iectional from the standpoint of those lo-
rivable. (Some have already been stated.) 12.3. ~for 0, 1, 2, 3, 4, ~ ~ QQ (0 elim, 0 int, neg 0 elim, neg 0 int, 1 elim, l int, neg l elim, neg l int, 2 elim, 2 int, neg 2 elim, neg 2 int, 3 elim, and so on) -~labc 1 ... cn
-, abc 1 ••• en -L3abc 1 •.• c
-raCa(ab))c~ ••• cn
And so on.
gicians who wish to keep all logical processes as finite as possible. The elimination rule for N, however, will be used only twice--both times to derive rules which themselves are usually regarded as perfectly acceptable and sufficiently finite. The presence of the infinitary elimination rule for N does have one especially important advantage: it guarantPes t~at
the system Q will not suffer from incompleteness with
r •·c;pect to ( elPmentary) arithmetic,
the kind of incomplete-
ness that GHdel showed must exist for all systems that do not possess
~uch
an infinitary rule. Not onl.Y does Q contain
all of elementary arithmetic, but Q CH.n be shown to be free
ELEMENTS OF COMBINATORY LOGIC
84
'0'
for 'CK',
'l'
for
'2' • 3'
for
'4'
for
for
I
I
85
THE NATURAL NUMBERS
12.4. From the above rules by ext the following identities can be obtained: the
I '
'l + l' '2 + l ' ' i.e. for '[l + l] + l. ' '3 + l ' ' i.e. for '[[l + l] + l] +l].'
0-~,
and so on. Previously we defined '2' as 'WB', but now we let
lab
the present definition replace the earlier one. Either defi-
2ab
a( a b)
3ab
a(a(ab))
4ab
a(a(a(ab)))
nition could be used because we can prove
WB
~
l + 1.
In
the system Q a two-digit number such as twenty-five would have to be written in some such way as this,
'25', where 'ab'
12.2. The Q-formulas '0',
'1',
'2', and so on, as de-
fined above, will be called ~-numerals. For example, 'I',
'CK',
'I+ I', and '[l + l] + l' are Q-numerals, but '1 + [l +
specifically to the Q-numeral '[1 + 1] + 1', since we can 1 + [1 + l]
~
[l + l ] + 1.
Except for
'0' and ' l ' , every Q-numeral is of the form 'a+ 1', where
a 1b 2 a b a3b 4 a b
ab
12. 5. Observe that 'Oab'
ab a( a b) a(a(ab)) a(a(a(ab)))
(or 'a 0 b') means, in effect,
the result of applying a to b zero times, and that 'lab' 1 (or •a b') means the result of applying a to b one time, 2 and that '2ab' (or •a b') means the result of applying a to
l]' is not a Q-numeral. although it is equal to a Q-numeral, prove the identity,
the 2-
and so on.
would be an abbreviation for '[9.a +a]+ b'. Numbers involving three or more digits can be handled in a similar way.
1-identit~,
the
!dentitl, the 3-identit~, the 4-!dentitl, and so on (0 id, l id, 2 id, 3 id, 4 id, and so on), specifically: Oab b or a 0b b
b
twice, and so on. 12.7. We will regard the Q-formula 'N' as being the
name of
~ ~
of
~ ~
natural
numb~
well, as being the name of the attribute ~).
Q£
(or, equally being
~
natural
There will be an introduction rule for the Q-func-
'a' is a Q-numeral. Also, if 'a' is a Q-numeral other than
tion N, that is, for the class of natural numbers, and also
'0', then 'a+ l' is a Q-numeral. Notice that '0 + l' is not
an elimination rule for N. The elimination rule has the pe-
a Q-numeral, but it is equal to the Q-numeral '1'. Any Q-for-
culiarity that it makes a step in a proof a direct conse-
mula equal to a Q-numeral is a name of the same natural num-
quence of infinitely many previous steps, thus requiring
ber as that Q-numeral. Hence '0 + l' is the name of l, just
proofs to be infinite in length wherever the rule is used.
as '1' is the name of 1. The following rules are easily de-
This may seem ob,iectional from the standpoint of those lo-
rivable. (Some have already been stated.) 12.3. ~for 0, 1, 2, 3, 4, ~ ~ QQ (0 elim, 0 int, neg 0 elim, neg 0 int, 1 elim, l int, neg l elim, neg l int, 2 elim, 2 int, neg 2 elim, neg 2 int, 3 elim, and so on) -~labc 1 ... cn
-, abc 1 ••• en -L3abc 1 •.• c
-raCa(ab))c~ ••• cn
And so on.
gicians who wish to keep all logical processes as finite as possible. The elimination rule for N, however, will be used only twice--both times to derive rules which themselves are usually regarded as perfectly acceptable and sufficiently finite. The presence of the infinitary elimination rule for N does have one especially important advantage: it guarantPes t~at
the system Q will not suffer from incompleteness with
r •·c;pect to ( elPmentary) arithmetic,
the kind of incomplete-
ness that GHdel showed must exist for all systems that do not possess
~uch
an infinitary rule. Not onl.Y does Q contain
all of elementary arithmetic, but Q CH.n be shown to be free
86
ELEMENTS OF COMBINATORY LOGIC
from contradiction, and this is more than can be said for many well-known systems which use only finite rules. 12.8. Introduction ~ f£r N (N int) The rule states that any Q-formula of the form 'Na', where 'a' is a Q-numeral, may appear as a step in a proof. Such a step would be interpreted as meaning that a is a member of N, that is, that a is a natural number. This introduction rule enables us to assert 'NO', 'Nl', 'N2', 'N3', and so on, as steps in proofs. 12.9. Elimination ~~ f£r N (N elim) This rule states that '( ••• a •.• )' is a direct consequence of the infinitely many Q-formulas 'Na', '( •.. o ... )', '( ... 1 ... )', '( ... 2 ... )', '( ••• 3 ... )', and so on, where '( ... a ... )' is any Q-formula in which a Q-formula 'a' occurs, and the results of replacing 'a' respectively by '0', '1', '2', '3', and so on, in '( ... a ... )' are '( •.• o... )', '( ... 1 ... )', '( ... 2 ••• )', '( ... 3 ... )', and so on. This rule may be expressed schematically thus: Na
THE NATURAL
~
( ... 1. .. )
~
b
N( ... a ... ) N( ... b ... )
a
=
b
-(N( ••. a ••• )) -(N( ••• b ••• ))
Here '( ••• a.,.)' could be 'a' itself. The first hypothesis in each case could be 'b =a' instead of 'a= b'. 12.12. ~ of closure of N ~ res2ect 1Q succession (clos N sue)
~:~a
+ 1]
12.13. The above rule is derived as follows: P85
1 2 3 4 5 6
7 8 9 10
( ... o... )
a =
87
NUMBE~S
Na Nl 0 + N[O N2 N[l N3 N[2 N4 N[3
hyp N int left add 0 2, 3, res id elim (l2.ll) N int 5. rep, def N int 7. rep, def N int 9. rep, def
l = l + 1] + 1] + 1] + 1]
( ... 2 ... )
"'
( ... a ... ) 12.10. We assume a rule of N), as follows:
~~
middle
f£r
N (ex mid
I Na V -(Na) According to this rule, 'Na V -(Na)' may appear as a step in any Q-proof. 12.11. The rule of :;:estricted .i£fntit;y: elimination (res id elim) (5.4) is to be extended in the following way: The expression in which the replacement of 'a' by 'b' is made can be of the form 'N( ..• a .•. )' or of the form '-(N( ••. a ••• ))'. In this case we have:
N[a + 1]
l, 4, 6, 8, 10, ••• , N el im
The above Q-proof illustrates the fact that although we cannot ever write down the infinity of premisses of the rule N elim, it is sufficient that for each premiss we have a way of writing it down and, if necessary, of proving it absolutely (as by N int in the above proof) or from previous steps. 12.14. ~ Q£ N-induction (N indue) Na
( ... o... ) b~( ... b ... )
I c... ~b + ( ... a .. •)
1 J ...
)
86
ELEMENTS OF COMBINATORY LOGIC
from contradiction, and this is more than can be said for many well-known systems which use only finite rules. 12.8. Introduction ~ f£r N (N int) The rule states that any Q-formula of the form 'Na', where 'a' is a Q-numeral, may appear as a step in a proof. Such a step would be interpreted as meaning that a is a member of N, that is, that a is a natural number. This introduction rule enables us to assert 'NO', 'Nl', 'N2', 'N3', and so on, as steps in proofs. 12.9. Elimination ~~ f£r N (N elim) This rule states that '( ••• a •.• )' is a direct consequence of the infinitely many Q-formulas 'Na', '( •.. o ... )', '( ... 1 ... )', '( ... 2 ... )', '( ••• 3 ... )', and so on, where '( ... a ... )' is any Q-formula in which a Q-formula 'a' occurs, and the results of replacing 'a' respectively by '0', '1', '2', '3', and so on, in '( ... a ... )' are '( •.• o... )', '( ... 1 ... )', '( ... 2 ••• )', '( ... 3 ... )', and so on. This rule may be expressed schematically thus: Na
THE NATURAL
~
( ... 1. .. )
~
b
N( ... a ... ) N( ... b ... )
a
=
b
-(N( ••. a ••• )) -(N( ••• b ••• ))
Here '( ••• a.,.)' could be 'a' itself. The first hypothesis in each case could be 'b =a' instead of 'a= b'. 12.12. ~ of closure of N ~ res2ect 1Q succession (clos N sue)
~:~a
+ 1]
12.13. The above rule is derived as follows: P85
1 2 3 4 5 6
7 8 9 10
( ... o... )
a =
87
NUMBE~S
Na Nl 0 + N[O N2 N[l N3 N[2 N4 N[3
hyp N int left add 0 2, 3, res id elim (l2.ll) N int 5. rep, def N int 7. rep, def N int 9. rep, def
l = l + 1] + 1] + 1] + 1]
( ... 2 ... )
"'
( ... a ... ) 12.10. We assume a rule of N), as follows:
~~
middle
f£r
N (ex mid
I Na V -(Na) According to this rule, 'Na V -(Na)' may appear as a step in any Q-proof. 12.11. The rule of :;:estricted .i£fntit;y: elimination (res id elim) (5.4) is to be extended in the following way: The expression in which the replacement of 'a' by 'b' is made can be of the form 'N( ..• a .•. )' or of the form '-(N( ••. a ••• ))'. In this case we have:
N[a + 1]
l, 4, 6, 8, 10, ••• , N el im
The above Q-proof illustrates the fact that although we cannot ever write down the infinity of premisses of the rule N elim, it is sufficient that for each premiss we have a way of writing it down and, if necessary, of proving it absolutely (as by N int in the above proof) or from previous steps. 12.14. ~ Q£ N-induction (N indue) Na
( ... o... ) b~( ... b ... )
I c... ~b + ( ... a .. •)
1 J ...
)
ELEMENTS OF COMBINATORY LOGIC
88
Here it is to be understood that '( ... a •.. )' is any Q-formu-
Na
la in which 'a' occurs, and the Q-formulas '( ••• o ...
c---o---)
)',
'( ... b ..• )', and '( ..• [b + 1] ..• )' result from replacing 'a' respectively by '0',
'b', and 'b+ l' throughout '( •.• a ••• )'.
According to this derived rule, the three things:
'Na',
b
'( •.. a ... )' is a consequence
'( .•. o ...
)',
and a subproof that is
l ;~--~---)
j c---[b (---a---)
general with respect to 'b' (i.e. is correct regardless of which Q-formula 'b' happens to be) and that has '( ••• b ••• )'
+
lJ---)
This can be derived by using the previous form of N induct
as its hypothesis and '( ... [b + l] •.• ) as its conclusion.
and choosing '( ... a ••. )' in that form as
duction, perhaps in slightly unfamilar form. It asserts, in
Also, this can be derived, ,just as the previous form was,
effect, that if
by a direct use of N elim.
a
is a natural number, if 0 has a certain
property, and if whenever b + l,
then
a
b
12.17. The symbol
has that property so does
symbol '=' will be extended to apply to the symbol
Na
hyp
2
c ..• o..• )
h,yp
n+l
( •• • [O + l] ... )
2, as in subproof of 12.14
n+2
0 + l = l ( ... l. .. )
left add 0
example, 'a + b
*
2n+3
( •.• [ l + l] ••• ) ( •.• 2 ••• )
* c'
P87
n+l. n+2, res id elim
l
Na
2
0 + l
3
n+3, as in subproof of 12.14
5
2n+2, rep, def
6
3n+3
( .•• [2 + l] ..• ) ( •.• 3 ••• )
£!
addition
Q£
l (com add l)
left add 0
= l = l
rt add 0
+ 0
blb+l=l+b [b + l] +l [l + b] +l
= l
+ [b + l]
hyp 4, mon id as soc add l, 3, 4-6, N indue
2n+3, as in subproof of 12.14
Jn applying the rule of N indue at step 7,
3n+2, rep def
'a + l
=
' ( ••• a ••• )' is
l + a', so ' ( ••• [b + l] •.• )' is '[b + l] + l =
' l + [b + l] ' •
w
( .•. a ••• )
l, 2, n+3, 2n+3, 3n+3, ••• , N elim
12.16. An alternative form of the ~ (N indue) is as follows:
£!
N-induction
i
For
hyp
7 a+l=l+a 3n+2
'*'·
will be understood as an abbreviation
for '[a+ b] c'. 12.18. Qommutativit~
4
2n+2
is then eas-
sumed where needed. Many of the conventions applying to the
steps:
l
n+3
[a j, b] = -[a = b],
ily established. The use of this identity can be tacitly as-
12.15. The rule N indue is derivable as follows, assum-
P86
will be defined as an abbreviation
for 'B(B-)'. The identity,
has that property.
ing that the subproof in 12.1 1+ requires n
'*'
'Na
& (---a---)'.
This rule is the familar rule of ordinary mathematical in-
12.19. Commutativity of addition (com add)
I I
j The proof of this is left as an exercise.
ELEMENTS OF COMBINATORY LOGIC
88
Here it is to be understood that '( ... a •.. )' is any Q-formu-
Na
la in which 'a' occurs, and the Q-formulas '( ••• o ...
c---o---)
)',
'( ... b ..• )', and '( ..• [b + 1] ..• )' result from replacing 'a' respectively by '0',
'b', and 'b+ l' throughout '( •.• a ••• )'.
According to this derived rule, the three things:
'Na',
b
'( •.. a ... )' is a consequence
'( .•. o ...
)',
and a subproof that is
l ;~--~---)
j c---[b (---a---)
general with respect to 'b' (i.e. is correct regardless of which Q-formula 'b' happens to be) and that has '( ••• b ••• )'
+
lJ---)
This can be derived by using the previous form of N induct
as its hypothesis and '( ... [b + l] •.• ) as its conclusion.
and choosing '( ... a ••. )' in that form as
duction, perhaps in slightly unfamilar form. It asserts, in
Also, this can be derived, ,just as the previous form was,
effect, that if
by a direct use of N elim.
a
is a natural number, if 0 has a certain
property, and if whenever b + l,
then
a
b
12.17. The symbol
has that property so does
symbol '=' will be extended to apply to the symbol
Na
hyp
2
c ..• o..• )
h,yp
n+l
( •• • [O + l] ... )
2, as in subproof of 12.14
n+2
0 + l = l ( ... l. .. )
left add 0
example, 'a + b
*
2n+3
( •.• [ l + l] ••• ) ( •.• 2 ••• )
* c'
P87
n+l. n+2, res id elim
l
Na
2
0 + l
3
n+3, as in subproof of 12.14
5
2n+2, rep, def
6
3n+3
( .•• [2 + l] ..• ) ( •.• 3 ••• )
£!
addition
Q£
l (com add l)
left add 0
= l = l
rt add 0
+ 0
blb+l=l+b [b + l] +l [l + b] +l
= l
+ [b + l]
hyp 4, mon id as soc add l, 3, 4-6, N indue
2n+3, as in subproof of 12.14
Jn applying the rule of N indue at step 7,
3n+2, rep def
'a + l
=
' ( ••• a ••• )' is
l + a', so ' ( ••• [b + l] •.• )' is '[b + l] + l =
' l + [b + l] ' •
w
( .•. a ••• )
l, 2, n+3, 2n+3, 3n+3, ••• , N elim
12.16. An alternative form of the ~ (N indue) is as follows:
£!
N-induction
i
For
hyp
7 a+l=l+a 3n+2
'*'·
will be understood as an abbreviation
for '[a+ b] c'. 12.18. Qommutativit~
4
2n+2
is then eas-
sumed where needed. Many of the conventions applying to the
steps:
l
n+3
[a j, b] = -[a = b],
ily established. The use of this identity can be tacitly as-
12.15. The rule N indue is derivable as follows, assum-
P86
will be defined as an abbreviation
for 'B(B-)'. The identity,
has that property.
ing that the subproof in 12.1 1+ requires n
'*'
'Na
& (---a---)'.
This rule is the familar rule of ordinary mathematical in-
12.19. Commutativity of addition (com add)
I I
j The proof of this is left as an exercise.
ELEMENTS OF COMBINATORY LOGIC
90
12.20. Commutativity Qf multiplicati£~ £z l (com mult l) l.a s a.l This follows from left mult 1 and rt mult 1. 12.21. ~-~ multi£llcation £z 0 (left mult 0) P88
Na o.o
l
2
=
0
3
0
brO.b o. [b + l]
'+ 5 6
7
o.b + O.l o.b + 0 O.b 0
8
o.a
=
12.22. Closure Qf N P89
3
b + 0
=
~
N[b
0]
5
c~N[b
6
+
b
+ c)
N [ [b + c] + l]
b + [c + l] = [b + c) +l N[b + [c + l)]
7 8
N[b + a]
9
hyp hyp rt add 0 2, 3, res id elim hyp 5, clos N sue (12.12) assoc add 6, 7, res id elim l, ~. 5-8, N indue
11 2
a.[b + l] = a.b + a.l =a.b+a
12.24.
~
Ql9~
left dist mult add rt mult 1, man id
Qf N fQ£ multiplication (clos N mult)
5 6
7 12.26.
~-~
[[a+ b)+ c] + d [a+[b+c])+d [a+ [c + b]] + d [[a+c]+b]+d [a+ c] + [b + d]
hyp hyp assoc add as soc add, man id 1, 2, com add, monadd as soc add, man add as soc add
multi£lication £l successor (left mult
sue)
II =:
[a+ l].b = a.b + b
• I
,I
The proof of this is left as an exercise. Use com assoc add and N indue. In using the latter, show that the required result is true for 'b' chosen as '0', and that if it is true for 'b' chosen as 'c', then it is true for 'b' chosen as 'c + 1'. Also, use the second form of N indue, so that the subproof has two hypotheses, one of which is 'Nc'. 12.27. Commutativitl £[ multi£licatio~ (com mult) PC)2
l 2 3
Na Nb b.O
'+ 5 6 7 8
0
o.b c
Nc b.c c.b Nb b.[c + l]
9
Na
lO
Nb N[b.a]
ll
The proof of this is left as an exercise. 12.25. ~~ associativiJx £[ addition (com assoc add)
Nb Nc [a+ b) + [c +d)
~
hyp rt mult 0 (11.8) hyp left dist mu1t add rt mult 1, mon id rt add 0
12.23. Right-hand multi£lication £l successor (rt mult sue) PC)O
l 2
3
fQ£ addition (clos N add)
Na Nb
l
2
P91
3 l, 2, 3-7, N indue
o
91
THE NATURAL NUMBERS
b.a = a.b
b.c + b c.b + b [c + 1]. b
hyp hyp rt mult 0 2, left mult 0 hyp hyp 2, reit rt mult sue 6, man id 5, 7, left mult sue l, ~. 5-10, N indue
12.28. Right-~ ~~91! £[ multi£lication addition (rt dist mult add)
i£!2
ELEMENTS OF COMBINATORY LOGIC
90
12.20. Commutativity Qf multiplicati£~ £z l (com mult l) l.a s a.l This follows from left mult 1 and rt mult 1. 12.21. ~-~ multi£llcation £z 0 (left mult 0) P88
Na o.o
l
2
=
0
3
0
brO.b o. [b + l]
'+ 5 6
7
o.b + O.l o.b + 0 O.b 0
8
o.a
=
12.22. Closure Qf N P89
3
b + 0
=
~
N[b
0]
5
c~N[b
6
+
b
+ c)
N [ [b + c] + l]
b + [c + l] = [b + c) +l N[b + [c + l)]
7 8
N[b + a]
9
hyp hyp rt add 0 2, 3, res id elim hyp 5, clos N sue (12.12) assoc add 6, 7, res id elim l, ~. 5-8, N indue
11 2
a.[b + l] = a.b + a.l =a.b+a
12.24.
~
Ql9~
left dist mult add rt mult 1, man id
Qf N fQ£ multiplication (clos N mult)
5 6
7 12.26.
~-~
[[a+ b)+ c] + d [a+[b+c])+d [a+ [c + b]] + d [[a+c]+b]+d [a+ c] + [b + d]
hyp hyp assoc add as soc add, man id 1, 2, com add, monadd as soc add, man add as soc add
multi£lication £l successor (left mult
sue)
II =:
[a+ l].b = a.b + b
• I
,I
The proof of this is left as an exercise. Use com assoc add and N indue. In using the latter, show that the required result is true for 'b' chosen as '0', and that if it is true for 'b' chosen as 'c', then it is true for 'b' chosen as 'c + 1'. Also, use the second form of N indue, so that the subproof has two hypotheses, one of which is 'Nc'. 12.27. Commutativitl £[ multi£licatio~ (com mult) PC)2
l 2 3
Na Nb b.O
'+ 5 6 7 8
0
o.b c
Nc b.c c.b Nb b.[c + l]
9
Na
lO
Nb N[b.a]
ll
The proof of this is left as an exercise. 12.25. ~~ associativiJx £[ addition (com assoc add)
Nb Nc [a+ b) + [c +d)
~
hyp rt mult 0 (11.8) hyp left dist mu1t add rt mult 1, mon id rt add 0
12.23. Right-hand multi£lication £l successor (rt mult sue) PC)O
l 2
3
fQ£ addition (clos N add)
Na Nb
l
2
P91
3 l, 2, 3-7, N indue
o
91
THE NATURAL NUMBERS
b.a = a.b
b.c + b c.b + b [c + 1]. b
hyp hyp rt mult 0 2, left mult 0 hyp hyp 2, reit rt mult sue 6, man id 5, 7, left mult sue l, ~. 5-10, N indue
12.28. Right-~ ~~91! £[ multi£lication addition (rt dist mult add)
i£!2
92
ELEMENTS OF COMBINATORY LOGIC
~
THE NATURAL NUMBERS
:: Nc [a+ b].c
~
12.33.
=
11 BCTabK 2
a.c + b.c
=
Kab
=
a
BCT-identit~
fQ£ K (BCT id K) BCT id K id
In virtue of this rule, the first term of the couple BCTab is the result of applying the couple to K. 12.34. ~ Qf BCT-identit~ 12£ 0 (BCT id 0)
The proof of this is left as an exercise. Use clos N add, com mult, and left dist mult add. 12.29. Closure Qf N £Q£ exponentiation (clos N exp)
The proof of this is left as an exercise. 12.30. EXERCISES. (1). Prove '2.2 = ~·. (2). Prove '2 + 3 = 5'. (?). Prove '23 = 8'. (~). Derive the alternative form of N indue (12.16) from the original form of it (12.14). (5). Derive the original form of N indue (12.1~) from the alternative form of it (12.16). (6). Prove '[a t b) = -[a = b)'. (7). Derive com add (12.19). (8). Derive clos N mult (12. 21+). (9). Derive left mult sue (12.26). ( 10). Derive rt dist mult add (12.28). ( 11). Derive clos N exp (12.29). 12.31. Jn order to derive some further results about natural numbers, it is desirable to introduce the concept of ordered courle in the following way: BCTab will be said to be an ordered courle of which a is the ~ ~ and b is the ~£QQQ ~· For brevity, ordered couples will simply be called ~· 12.32. ~ Qf BCT identit~ (BCT id) BCTabc = cab The above equation is very easily proved.
Qf
93
11 BCTabO
=
2
= b
Oab
BCT id 0 id
In virtue of this rule, the second term of the couple BCTab is the result of applying the couple to o. 12.35. ~ Qf identit~ 12£ courles (id coup) BCTab = BCTcd a = c
r
I
I i
I
I
i
!'
b = d
Proof omitted. According to this rule, two couples are identical if and only if their first terms are the same and their second terms are the same. 12.36. We now define a Q-function F which is such that if it is applied to a couple BCTab, the result is the couple BCT[a + l]a. We define F by letting 'F' be an abbreviation for '[x](BCT[xK + l](xK))' 12:37. ~of F-identit~ (F id), first form
P93
11 Fa 2
= =
[x](BCT[xK + l](xK))a BCT[aK + l](aK)
id int, def abs id
12.38. ~ Qf F-identit~ (F id), second form P94
~I F(BCTab)
BCT[BCTabK + l](BCTabK) BCT[a + l]a
D id (first form) BCT id K, mon id
12.39. We now consider the effect of multiple applications of F to the couple BCTOO. One application of F to BCTOO gives BCTlO. A second application of F to BCTOO, that is, an application of F to BCTlO, gives BCT2l. A third application of F to BCTOO gives BCT32. And so on. In fact we can prove:
92
ELEMENTS OF COMBINATORY LOGIC
~
THE NATURAL NUMBERS
:: Nc [a+ b].c
~
12.33.
=
11 BCTabK 2
a.c + b.c
=
Kab
=
a
BCT-identit~
fQ£ K (BCT id K) BCT id K id
In virtue of this rule, the first term of the couple BCTab is the result of applying the couple to K. 12.34. ~ Qf BCT-identit~ 12£ 0 (BCT id 0)
The proof of this is left as an exercise. Use clos N add, com mult, and left dist mult add. 12.29. Closure Qf N £Q£ exponentiation (clos N exp)
The proof of this is left as an exercise. 12.30. EXERCISES. (1). Prove '2.2 = ~·. (2). Prove '2 + 3 = 5'. (?). Prove '23 = 8'. (~). Derive the alternative form of N indue (12.16) from the original form of it (12.14). (5). Derive the original form of N indue (12.1~) from the alternative form of it (12.16). (6). Prove '[a t b) = -[a = b)'. (7). Derive com add (12.19). (8). Derive clos N mult (12. 21+). (9). Derive left mult sue (12.26). ( 10). Derive rt dist mult add (12.28). ( 11). Derive clos N exp (12.29). 12.31. Jn order to derive some further results about natural numbers, it is desirable to introduce the concept of ordered courle in the following way: BCTab will be said to be an ordered courle of which a is the ~ ~ and b is the ~£QQQ ~· For brevity, ordered couples will simply be called ~· 12.32. ~ Qf BCT identit~ (BCT id) BCTabc = cab The above equation is very easily proved.
Qf
93
11 BCTabO
=
2
= b
Oab
BCT id 0 id
In virtue of this rule, the second term of the couple BCTab is the result of applying the couple to o. 12.35. ~ Qf identit~ 12£ courles (id coup) BCTab = BCTcd a = c
r
I
I i
I
I
i
!'
b = d
Proof omitted. According to this rule, two couples are identical if and only if their first terms are the same and their second terms are the same. 12.36. We now define a Q-function F which is such that if it is applied to a couple BCTab, the result is the couple BCT[a + l]a. We define F by letting 'F' be an abbreviation for '[x](BCT[xK + l](xK))' 12:37. ~of F-identit~ (F id), first form
P93
11 Fa 2
= =
[x](BCT[xK + l](xK))a BCT[aK + l](aK)
id int, def abs id
12.38. ~ Qf F-identit~ (F id), second form P94
~I F(BCTab)
BCT[BCTabK + l](BCTabK) BCT[a + l]a
D id (first form) BCT id K, mon id
12.39. We now consider the effect of multiple applications of F to the couple BCTOO. One application of F to BCTOO gives BCTlO. A second application of F to BCTOO, that is, an application of F to BCTlO, gives BCT2l. A third application of F to BCTOO gives BCT32. And so on. In fact we can prove:
ELEMENTS OF COMBINATORY LOGIC
THE NATURAL NUMBERS
OF(BCTOO) = BCTOO. lF(BCTOO) = BCTlO, 2F(BCTOO) = BCT21, 3F(BCTOO) = BCT32. and so on. In general, n+l applications of F to BCTOO give a couple that has n as its second term and n+l as its first term. 12.40. ~ Q! multiple applications £[ F (m F)
4
hyp left add 0, mon id 1 id, mon id F id hyp
9
P95
1 2 3 4 5 6 7 8 q
10 11
Na [O + l]F(BCTOO)
lF(BCTOO) F(BCTCO) BCT[O + 1]0 b [b + l]F(BCTOO) = BCT[b + l]b [[b + 1] + l]F(BCTOO) [lF.[b + l]F](BCTOO) lF([b + l]F(BCTOO)) F([b + l]F(BCTOO)) F(BCT[b + llb) BCT[[b + 1] + lJ[b + 1] [a + l]F(BCTOO) = BCT[a + l]a
5 6 7 8
P98
1 2 3 4 5 6 7 8
hyp hyp hyp BCT id 0 1, m F, mon id 3, mon id 2, m F, mon id BCT id 0
Na Nb a + 1 = b + 1 a = BCT[a + l]aO [a + l]F(BCTOO)O [b + l]F(BCTOO)O BCT[b + l]bO b
1 2 3 4
7 8 9 lO
11 12
P99
1 2 3 4
themselves. 12.42. Monotogy £[ successor tit~ (mon sue non-id)
5 6
P97
llNa 2 Nb 3 a +b
respect !£ hyp hyp hyp
~-~
b
a +b a + 1 +b + 1
Na Nb Nc a • b a+ 0 + b+O d Nd a+d • b + d N[a + d) N[b + d) [a+d]+l + [b + d) + 1 a+ [d + l] + b+[d+l] a+C.b+C
12.44, Cancellation
The rule in effect asserts that if the successors of two natural numbers are the same, so are the natural numbers ~
hyp 1, reit 2, reit 5. 6, 7. can sue 3. reit 4-8, neg int id (5.10)
12.43. Monotogy £[ addition ~ respect !2 ~-~ (mon add non-id)
5 6
12,41. ~ Q[ cancellation £Qr successor (can sue) P96
1 = b + 1
In the above proof step 8 is treated as though it were '-[a= b)', to which it is equal. A similar procedure will sometimes be followed elsewhere. tit~
add id mult id 1 id 4, mon id F id 1, 4, 5-10, N indue
,.. Na Nb a
95
7 8
Na Nb Nc a + c
1:
a
=
•
b + c + c b
b + c
+b
+ c b + c
hyp hyp hyp hyp 4, rt add 0, res id elim hyp hyp 1, reit, 6, clos N add 2, reit, 6, clos N add 7. 8, 9. mon sue non-id 10, assoc add, res id elim 3. 5. 6-11, N indue
f£r addition (can add) hyp hyp hyp hyp hyp 1, 2, 3, reit, 5, mon add non-id 4, reit 5-7. ind pr id (5.11)
ELEMENTS OF COMBINATORY LOGIC
THE NATURAL NUMBERS
OF(BCTOO) = BCTOO. lF(BCTOO) = BCTlO, 2F(BCTOO) = BCT21, 3F(BCTOO) = BCT32. and so on. In general, n+l applications of F to BCTOO give a couple that has n as its second term and n+l as its first term. 12.40. ~ Q! multiple applications £[ F (m F)
4
hyp left add 0, mon id 1 id, mon id F id hyp
9
P95
1 2 3 4 5 6 7 8 q
10 11
Na [O + l]F(BCTOO)
lF(BCTOO) F(BCTCO) BCT[O + 1]0 b [b + l]F(BCTOO) = BCT[b + l]b [[b + 1] + l]F(BCTOO) [lF.[b + l]F](BCTOO) lF([b + l]F(BCTOO)) F([b + l]F(BCTOO)) F(BCT[b + llb) BCT[[b + 1] + lJ[b + 1] [a + l]F(BCTOO) = BCT[a + l]a
5 6 7 8
P98
1 2 3 4 5 6 7 8
hyp hyp hyp BCT id 0 1, m F, mon id 3, mon id 2, m F, mon id BCT id 0
Na Nb a + 1 = b + 1 a = BCT[a + l]aO [a + l]F(BCTOO)O [b + l]F(BCTOO)O BCT[b + l]bO b
1 2 3 4
7 8 9 lO
11 12
P99
1 2 3 4
themselves. 12.42. Monotogy £[ successor tit~ (mon sue non-id)
5 6
P97
llNa 2 Nb 3 a +b
respect !£ hyp hyp hyp
~-~
b
a +b a + 1 +b + 1
Na Nb Nc a • b a+ 0 + b+O d Nd a+d • b + d N[a + d) N[b + d) [a+d]+l + [b + d) + 1 a+ [d + l] + b+[d+l] a+C.b+C
12.44, Cancellation
The rule in effect asserts that if the successors of two natural numbers are the same, so are the natural numbers ~
hyp 1, reit 2, reit 5. 6, 7. can sue 3. reit 4-8, neg int id (5.10)
12.43. Monotogy £[ addition ~ respect !2 ~-~ (mon add non-id)
5 6
12,41. ~ Q[ cancellation £Qr successor (can sue) P96
1 = b + 1
In the above proof step 8 is treated as though it were '-[a= b)', to which it is equal. A similar procedure will sometimes be followed elsewhere. tit~
add id mult id 1 id 4, mon id F id 1, 4, 5-10, N indue
,.. Na Nb a
95
7 8
Na Nb Nc a + c
1:
a
=
•
b + c + c b
b + c
+b
+ c b + c
hyp hyp hyp hyp 4, rt add 0, res id elim hyp hyp 1, reit, 6, clos N add 2, reit, 6, clos N add 7. 8, 9. mon sue non-id 10, assoc add, res id elim 3. 5. 6-11, N indue
f£r addition (can add) hyp hyp hyp hyp hyp 1, 2, 3, reit, 5, mon add non-id 4, reit 5-7. ind pr id (5.11)
ELEMENTS OF COMBINATORY LOGIC
96 12.4-5. :!!2!!-.!,!!.~n.th:!~;y: PlOO l 2
fN:
successor (non-id sue)
Na
hyp
Nb
hyp b + [a + l]
b Na
l, reit
5 6
Nb N[a + l]
2, reit
0 + b
7
[a + l] + b
8
9
NO
lO
0
ll
0
12
b
t
a + l
t a + l b + [a + l]
o,
bl
ab
Aa According to this rule, the Q-formula 'Aa'
subproof that is general with respect to 'b' and has 'ab' as
tion the transitivity and symmetry of identity, the symmetry laws for addition and multiplication. Also,
'a+ b + c' will
'[a+ b)+ c', and 'a+ b + c + d' will abbrevi-
ate '[[a+ b) +c)+ d', and so on. A similar convention will apply to '&' and 'V' as to '+'· Also,
'a.b.c' will ab-
breviate '[a.b].c', and 'a.b.c.d' will abbreviate '[[a.b].c] . d', and so on. 12.4-7. EXERCISES. (1). Prove BCT id (12.32). (2). Derive id coup (12.35). (3). Given two Q-functions G and G2 , define a 1 Q-funotion G such that G (BCTab) = BCT(G a)(G 2 b). 1 3 3 Universalltl
13.3. Universalit;y: introduction (univ int)
3-ll, neg int id
of non-identity (which is easily proved) and the associative
13.
that is
5. 6, 8, 9. can add non-sue 0 (see 12.17)
12.4-6. Hereafter in proofs we will assume without men-
abbreviate
~-emptiness,
LAa I ab
3. trans id 5. 6, com add N int rt add
regard E as existence in the sense of
class. 13.2. Universalit;y: elimination (univ elim)
4-, clos N sue b + [a + l]
97
as the attribute possessed by every non-empty attribute or
hyp
3 4-
UNIVERSALITY AND EXISTENCE
~££ ~~
13.1. Introduction and elimination rules, and negative introduction gnd negative elimination rules are stated below for the primitive Q-functions A and E. We may regard A as universalit.Y, that is, as the attribute possessed by every all-inclusive cuuniversal") attribute or class, and we may
is a d.c. of a
a step (ordinarily the last step). In saying that the subproof is general with respect to 'b' we mean that it remains a valid proof if 'b' Q-formula.
is replaced throughout it by any other
13.4-. The expression '(x)( ... x ••• )' will serve as an abbreviation for 'A[x] ( ... x •. ,) '. Here ' ( ... x •.• )' is any Q-formula, possibly one in which 'x' occurs, but not necessarily one in which 'x' occurs. The expression '(x)' in '(x)( ... x ... )' is called a ~ivers~ ~uantifier and is read, "for every Q-function x." Thus, for example, the Q-formula '(x)[x = x]' would be read as (and would express the proposition) "for every Q-function x,
xis identical with x."
Note, however, that the universal quantifier is not a Q-formula and does not denote a Q-function. The following introduction and elimination rules for the universal quantifer are easily derived by use of the abstraction rules and the universality rules. 13.5. ~1:§1 ~t.~ elimination (u q elim)
LCx)( ... x ... )
I c ... b ... )
ELEMENTS OF COMBINATORY LOGIC
96 12.4-5. :!!2!!-.!,!!.~n.th:!~;y: PlOO l 2
fN:
successor (non-id sue)
Na
hyp
Nb
hyp b + [a + l]
b Na
l, reit
5 6
Nb N[a + l]
2, reit
0 + b
7
[a + l] + b
8
9
NO
lO
0
ll
0
12
b
t
a + l
t a + l b + [a + l]
o,
bl
ab
Aa According to this rule, the Q-formula 'Aa'
subproof that is general with respect to 'b' and has 'ab' as
tion the transitivity and symmetry of identity, the symmetry laws for addition and multiplication. Also,
'a+ b + c' will
'[a+ b)+ c', and 'a+ b + c + d' will abbrevi-
ate '[[a+ b) +c)+ d', and so on. A similar convention will apply to '&' and 'V' as to '+'· Also,
'a.b.c' will ab-
breviate '[a.b].c', and 'a.b.c.d' will abbreviate '[[a.b].c] . d', and so on. 12.4-7. EXERCISES. (1). Prove BCT id (12.32). (2). Derive id coup (12.35). (3). Given two Q-functions G and G2 , define a 1 Q-funotion G such that G (BCTab) = BCT(G a)(G 2 b). 1 3 3 Universalltl
13.3. Universalit;y: introduction (univ int)
3-ll, neg int id
of non-identity (which is easily proved) and the associative
13.
that is
5. 6, 8, 9. can add non-sue 0 (see 12.17)
12.4-6. Hereafter in proofs we will assume without men-
abbreviate
~-emptiness,
LAa I ab
3. trans id 5. 6, com add N int rt add
regard E as existence in the sense of
class. 13.2. Universalit;y: elimination (univ elim)
4-, clos N sue b + [a + l]
97
as the attribute possessed by every non-empty attribute or
hyp
3 4-
UNIVERSALITY AND EXISTENCE
~££ ~~
13.1. Introduction and elimination rules, and negative introduction gnd negative elimination rules are stated below for the primitive Q-functions A and E. We may regard A as universalit.Y, that is, as the attribute possessed by every all-inclusive cuuniversal") attribute or class, and we may
is a d.c. of a
a step (ordinarily the last step). In saying that the subproof is general with respect to 'b' we mean that it remains a valid proof if 'b' Q-formula.
is replaced throughout it by any other
13.4-. The expression '(x)( ... x ••• )' will serve as an abbreviation for 'A[x] ( ... x •. ,) '. Here ' ( ... x •.• )' is any Q-formula, possibly one in which 'x' occurs, but not necessarily one in which 'x' occurs. The expression '(x)' in '(x)( ... x ... )' is called a ~ivers~ ~uantifier and is read, "for every Q-function x." Thus, for example, the Q-formula '(x)[x = x]' would be read as (and would express the proposition) "for every Q-function x,
xis identical with x."
Note, however, that the universal quantifier is not a Q-formula and does not denote a Q-function. The following introduction and elimination rules for the universal quantifer are easily derived by use of the abstraction rules and the universality rules. 13.5. ~1:§1 ~t.~ elimination (u q elim)
LCx)( ... x ... )
I c ... b ... )
98
ELEMENTS OF COMBINATORY LOGIC
bl
( •.• b ... )
(x)( ... x ... )
13.7. Existence
~~
(exist elim)
Ea
UNIVERSALITY AND EXISTENCE
99
According to this rule, the Q-formula 'c' is a d.c. of the Q-formula '(:;rx)( ••• x •.• )' together with a subproof that is general with respect to 'b', has '( ••• b ••• )' as its only hypothesis. and has 'c' as a step. 13.11. Existence quantifier introduction (e q int)
Lc••• b ••• )
I (~x) ( •.• x ... ) c
According to this rule, the Q-formula 'c' is a d.c. of 'Ea' together with a subproof that is general with respect to 'b', has 'ab' as its only hypothesis, and has 'c' as a step. 13.8. Existence introduction (exist int) Lab
I Ea
13.9. The expression '(~x)( ••• x ••• )' will serve as an abbreviation for 'E[x]( .•. x .•• )'. The expression '(~x)' in '(~x)( ••. x ..• )' is called an existence guantifier and is read, "for some Q-function x," or "there is a Q-function x such that." Thus, for example, the Q-formula ' ( 1Ix)[x = a]' would be read as "for some Q-function x, x is identical with a," or as "there is a Q-function x such that x is identical with a." The existence quantifier, like the universal quantifier, is not a Q-formula and does not denote a Q-function. The following introduction and elimination rules for the existence quantifier are easily derived by use of the abstraction rules and the existence rules. 13.10. Existence quan~~ elimlg~ (e q elim)
13.12. We now state negative introduction and elimination rules for A and E, and corresponding derived rules for quantifiers. 13.13. Negative universalit~ elimination (neg univ elim) and negative universalitl introduction (neg univ int)
L-(Aa) t"E(B-a) The elimination rule is downward in the above schema, and the introduction rule is upward. Similarly in other cases below. 13.14. Negative existence elimination (neg exist elim) and ne~tive existence introduction (neg exist int)
L-(Ea)
t" A(B-a)
13.15. Negative universal s.uantifier elimination (neg u q elim) and negative universal quantifier introduction (neg u q int) L-Cx)( ... x ... )
r (~x)(-( ••• x ••• )) 13.16. Negative existence quantifier elimination (neg e q elim) and negative existence quantifier introduction (neg e q int) L-C~x)( ... x ... )
t"Cx)(-( ••. x ••• )) 13.17. ~Morgan~ fQ£ quantifiers (d m q) These derived rules include the rules in 13.15 and 13.16 to-
98
ELEMENTS OF COMBINATORY LOGIC
bl
( •.• b ... )
(x)( ... x ... )
13.7. Existence
~~
(exist elim)
Ea
UNIVERSALITY AND EXISTENCE
99
According to this rule, the Q-formula 'c' is a d.c. of the Q-formula '(:;rx)( ••• x •.• )' together with a subproof that is general with respect to 'b', has '( ••• b ••• )' as its only hypothesis. and has 'c' as a step. 13.11. Existence quantifier introduction (e q int)
Lc••• b ••• )
I (~x) ( •.• x ... ) c
According to this rule, the Q-formula 'c' is a d.c. of 'Ea' together with a subproof that is general with respect to 'b', has 'ab' as its only hypothesis, and has 'c' as a step. 13.8. Existence introduction (exist int) Lab
I Ea
13.9. The expression '(~x)( ••• x ••• )' will serve as an abbreviation for 'E[x]( .•. x .•• )'. The expression '(~x)' in '(~x)( ••. x ..• )' is called an existence guantifier and is read, "for some Q-function x," or "there is a Q-function x such that." Thus, for example, the Q-formula ' ( 1Ix)[x = a]' would be read as "for some Q-function x, x is identical with a," or as "there is a Q-function x such that x is identical with a." The existence quantifier, like the universal quantifier, is not a Q-formula and does not denote a Q-function. The following introduction and elimination rules for the existence quantifier are easily derived by use of the abstraction rules and the existence rules. 13.10. Existence quan~~ elimlg~ (e q elim)
13.12. We now state negative introduction and elimination rules for A and E, and corresponding derived rules for quantifiers. 13.13. Negative universalit~ elimination (neg univ elim) and negative universalitl introduction (neg univ int)
L-(Aa) t"E(B-a) The elimination rule is downward in the above schema, and the introduction rule is upward. Similarly in other cases below. 13.14. Negative existence elimination (neg exist elim) and ne~tive existence introduction (neg exist int)
L-(Ea)
t" A(B-a)
13.15. Negative universal s.uantifier elimination (neg u q elim) and negative universal quantifier introduction (neg u q int) L-Cx)( ... x ... )
r (~x)(-( ••• x ••• )) 13.16. Negative existence quantifier elimination (neg e q elim) and negative existence quantifier introduction (neg e q int) L-C~x)( ... x ... )
t"Cx)(-( ••. x ••• )) 13.17. ~Morgan~ fQ£ quantifiers (d m q) These derived rules include the rules in 13.15 and 13.16 to-
ELEMENTS OF COMBINATORY LOGIC
100
UNIVERSALITY AND
gether with the following additional derived rules: L-Cx)C-c ••• x ••• )) l(~x)( ••• x ••• )
101
in a way similar to the way in which N indue is derived from N elim. Jt is not possible to derive str univ int from univ int.
L -(:>!x)(-( ••• x ••• )) l (x)( •.. x •.• )
13.22. We may also assume a strong ~ Q£ exist~ ~ limination (str exist elim) as follows, where we assume that a ll th e n~- f' ormu l as are enumera t e d as I q I , 1 q I , I q I • ••• ·• 1 2 3 Ea
13.18. It is desirable to assume the following~~ ~ universalit;y: and disjunction (univ dis): LA[a.Vc] Aa V c
I ~
~XIST~NCE
13.19. From the above rule we can derive the following relating universal _guantificatiog .enQ disjunction (u q
dis): L (x)[b V ( •.. x ••• )]
I
b V (x)( ••• x ••• )
13.20. The converses of univ dis and u q dis, namely, LAa V c A[a.Vc]
I
L b V (x)( ••• x ••• ) I (x)[b V ( ... x .•• )]
are derivable without use of univ dis or u q dis, and may be referred to as cnv univ dis and cnv u q dis. The rule of univ dis is not as arbitrary as it may seem. It is derivable from some "strong» rules which will now be stated. 13.21. Instead of assuming the ordinary rule of universality introduction (13.3) it is possible to assume a stronger rule which we will call the strong~ of universalit;y: introduction (str univ int). This rule, likeN elim, requires infinitely many premisses. Jf we suppose that all the Q-formu 1 as are enumera t e d as I q I , I q I , I q I , ... , th en th e ru le 2 1 3 may be stated schematically as follows:
b
Here it is to be understood that the rule is asserting that 'b' is a d.c. of 'Ea' and the infinity of subproofs which begin respectively with 'aq •, 'aq •, 'aq •, ..• as hypotheses 1 2 3 and end with 'b'. The ordinary rule exist elim may be derived from str exist elim, but the strong rule cannot be derived from the ordinary rule. The two strong rules for universality and existence are derivable from each other by appropriate use of negative rules and negation rules. 13.23. The following strQgg ~
Q£
universal guantif~
~ introduction (str u q int) is derivable from str univ int
by use of the abstraction rules and the definition of '(x) ( •.• x ..• )'. We suppose that all Q-formulas are enumerated as 'q ', I
Aa The ordinary rule univ int may be derived from str univ int
q2
I '
I
I
q
1
3
I '
'
• •
•
(... ql •.• ) ( ••• q2 ... )
ELEMENTS OF COMBINATORY LOGIC
100
UNIVERSALITY AND
gether with the following additional derived rules: L-Cx)C-c ••• x ••• )) l(~x)( ••• x ••• )
101
in a way similar to the way in which N indue is derived from N elim. Jt is not possible to derive str univ int from univ int.
L -(:>!x)(-( ••• x ••• )) l (x)( •.. x •.• )
13.22. We may also assume a strong ~ Q£ exist~ ~ limination (str exist elim) as follows, where we assume that a ll th e n~- f' ormu l as are enumera t e d as I q I , 1 q I , I q I • ••• ·• 1 2 3 Ea
13.18. It is desirable to assume the following~~ ~ universalit;y: and disjunction (univ dis): LA[a.Vc] Aa V c
I ~
~XIST~NCE
13.19. From the above rule we can derive the following relating universal _guantificatiog .enQ disjunction (u q
dis): L (x)[b V ( •.. x ••• )]
I
b V (x)( ••• x ••• )
13.20. The converses of univ dis and u q dis, namely, LAa V c A[a.Vc]
I
L b V (x)( ••• x ••• ) I (x)[b V ( ... x .•• )]
are derivable without use of univ dis or u q dis, and may be referred to as cnv univ dis and cnv u q dis. The rule of univ dis is not as arbitrary as it may seem. It is derivable from some "strong» rules which will now be stated. 13.21. Instead of assuming the ordinary rule of universality introduction (13.3) it is possible to assume a stronger rule which we will call the strong~ of universalit;y: introduction (str univ int). This rule, likeN elim, requires infinitely many premisses. Jf we suppose that all the Q-formu 1 as are enumera t e d as I q I , I q I , I q I , ... , th en th e ru le 2 1 3 may be stated schematically as follows:
b
Here it is to be understood that the rule is asserting that 'b' is a d.c. of 'Ea' and the infinity of subproofs which begin respectively with 'aq •, 'aq •, 'aq •, ..• as hypotheses 1 2 3 and end with 'b'. The ordinary rule exist elim may be derived from str exist elim, but the strong rule cannot be derived from the ordinary rule. The two strong rules for universality and existence are derivable from each other by appropriate use of negative rules and negation rules. 13.23. The following strQgg ~
Q£
universal guantif~
~ introduction (str u q int) is derivable from str univ int
by use of the abstraction rules and the definition of '(x) ( •.• x ..• )'. We suppose that all Q-formulas are enumerated as 'q ', I
Aa The ordinary rule univ int may be derived from str univ int
q2
I '
I
I
q
1
3
I '
'
• •
•
(... ql •.• ) ( ••• q2 ... )
ELEMENTS OF COMBINATORY LOGIC
102
UNIVERSALITY AND EXISTENCE
103
d d
13.24. The following strong ~ Q[ existence guantifi~ elimination (str e q elim) is derivable from str exist elim. We suppose that all Q-formulas are enumerated as 'q ', 1 I
q2 I
'
I
q
3
I '
" •
"
"
13.26. The above rule can be used to derive univ dis: PlOl 1 2 3 w W+l w+2 w+3
2w 2w+l 2w+2
b
13.25. Another strong rule that can be assumed is the strong ~ Q[ disjJ
A[a.Vc] Vc(aq 1 ) Vc(aq2 )
r~c(Aa) aql aq2
Aa Vc(Aa) Vc(Aa)
hyp l, univ elim, mult elim 1, univ elim, mult elim hyp w, dis int hyp hyp
W+2, W+3,
...
2w, dis int l' 2' ... '
tu-
.
str univ int
W+l, w+2- 2w+l,
str dis elim In the above proof there are understood to be infinitely many steps, and w is understood to be the smallest infinite ordinal number. It is assumed, also, that all the Q-formulas are en urn e rae t d as I q I , I q I , I q I , •••. 1 2 3 13.27. EXERCISES. Give derivations of the following: (1). u q elim and u q int. (2). e q elim and e q int. ( 3). neg u q elim and neg u q int. ( 4). neg e q elim and neg e ~ int. ( 5). d m q as in 13.17. ( 6). u q dis from univ dis. ( 7). cnv univ dis (13.20). ( 8). univ int from str univ int. ( 0). str u q int from str univ int. ( 10). str e q elim from str exist elim.
ELEMENTS OF COMBINATORY LOGIC
102
UNIVERSALITY AND EXISTENCE
103
d d
13.24. The following strong ~ Q[ existence guantifi~ elimination (str e q elim) is derivable from str exist elim. We suppose that all Q-formulas are enumerated as 'q ', 1 I
q2 I
'
I
q
3
I '
" •
"
"
13.26. The above rule can be used to derive univ dis: PlOl 1 2 3 w W+l w+2 w+3
2w 2w+l 2w+2
b
13.25. Another strong rule that can be assumed is the strong ~ Q[ disjJ
A[a.Vc] Vc(aq 1 ) Vc(aq2 )
r~c(Aa) aql aq2
Aa Vc(Aa) Vc(Aa)
hyp l, univ elim, mult elim 1, univ elim, mult elim hyp w, dis int hyp hyp
W+2, W+3,
...
2w, dis int l' 2' ... '
tu-
.
str univ int
W+l, w+2- 2w+l,
str dis elim In the above proof there are understood to be infinitely many steps, and w is understood to be the smallest infinite ordinal number. It is assumed, also, that all the Q-formulas are en urn e rae t d as I q I , I q I , I q I , •••. 1 2 3 13.27. EXERCISES. Give derivations of the following: (1). u q elim and u q int. (2). e q elim and e q int. ( 3). neg u q elim and neg u q int. ( 4). neg e q elim and neg e ~ int. ( 5). d m q as in 13.17. ( 6). u q dis from univ dis. ( 7). cnv univ dis (13.20). ( 8). univ int from str univ int. ( 0). str u q int from str univ int. ( 10). str e q elim from str exist elim.
ELEMENTS OF COMBINATORY LOGIC
104
~-QE-~ ~ ~
14. ~·
14.1. The relation "less than" will be called simply and the relation "less than or equal to" will be called
si~ply ~-Q£-equal. Less will be denoted by the symbol '<',
and less-or-equal will be denoted by the symbol ·~·· The latter symbol will be treated as an abbreviation for [z][x](~y)[Ny & [x + y = z]] The symbol '<' will be treated as an abbreviation for rz][x][x + 1 ~ z] The symbols ·~· and '<' will be subject to some of the same conventions as '='·For example, 'x + 1 ~ z' is regarded as an abbreviation for '[x + 1] ~ z'. The following rules for less-or-equal and for less are easily derivable. As elsewhere, the downward direction expresses the elimination rules, and the upward direction the introduction rules. 14.2. ~ fQ! ~-QE-~ (~ elim, ~ int, neg ~ elim, neg ~ int)
-La
< b
-rc~y)lNy 14.3.
~
& ca fQ!
+ Y 1~
=
(< elim, < int, neg< elim, neg<
int)
-La
-ra+l~b
14.4. The following sEecial ~ Q[ lt~-Q£-eaual ~ (sp::; elim) can be easily derived from~ elim:
~
b
~
l' a
+ y
~ :c; ~
b
d
d
Here 'y' cannot be mentioned in 'd' because otherwise the
b
14.6, It is also easy to derive the following identity, [a~ b] = (~y)[Ny & [a+ y = b]] which will he called~ ~-Q£-egual identi~ (~ id), and the following identity, [a < b] = [a + 1 ~ b] which will be called~~ ide~tit~ (< id). 14.7. Transitivity Q[ 1ess-Q£-~ (trans ~) Pl02 1 2 3 4 6
7 8 9
< b
a
subproof would not be general with respect to 'y' and still end with 'd'. 14.5. The following sEeci~ ~ Q[ ~-Q£-eaual ~ duction (sp ~ int) can be easily derived from ~ int:
5
bJ
105
LESS-OR-EQUAL AND LESS
10 11
12 13 14
a b
~
b c X Nx a + X =b b :::: c y Ny b + y ~ c Nx N[x + y] a + X =b a + X+ y a :::; c a :::; c a :::; c ~
In accordance with 12.46 in going from step 11 to as transformed into 'a+ also that 'a+ x + y' is by 12.46.
=
c
hyp hyp hyp hyp 2, reit hyp hyp 3. reit 6, 8, clos N add 4, reit 7. 10, res id elim (see 12 .46) 9. 11, sp :::; int 5. 6-12, sp :::: elim 1, 3-13, sp ~ elim
there is tacit use of assoc add made step 12, since step 11 can be viewed [x + y] = c' by assoc add. Notice an abbreviation for '[a+ x] + y'
ELEMENTS OF COMBINATORY LOGIC
104
~-QE-~ ~ ~
14. ~·
14.1. The relation "less than" will be called simply and the relation "less than or equal to" will be called
si~ply ~-Q£-equal. Less will be denoted by the symbol '<',
and less-or-equal will be denoted by the symbol ·~·· The latter symbol will be treated as an abbreviation for [z][x](~y)[Ny & [x + y = z]] The symbol '<' will be treated as an abbreviation for rz][x][x + 1 ~ z] The symbols ·~· and '<' will be subject to some of the same conventions as '='·For example, 'x + 1 ~ z' is regarded as an abbreviation for '[x + 1] ~ z'. The following rules for less-or-equal and for less are easily derivable. As elsewhere, the downward direction expresses the elimination rules, and the upward direction the introduction rules. 14.2. ~ fQ! ~-QE-~ (~ elim, ~ int, neg ~ elim, neg ~ int)
-La
< b
-rc~y)lNy 14.3.
~
& ca fQ!
+ Y 1~
=
(< elim, < int, neg< elim, neg<
int)
-La
-ra+l~b
14.4. The following sEecial ~ Q[ lt~-Q£-eaual ~ (sp::; elim) can be easily derived from~ elim:
~
b
~
l' a
+ y
~ :c; ~
b
d
d
Here 'y' cannot be mentioned in 'd' because otherwise the
b
14.6, It is also easy to derive the following identity, [a~ b] = (~y)[Ny & [a+ y = b]] which will he called~ ~-Q£-egual identi~ (~ id), and the following identity, [a < b] = [a + 1 ~ b] which will be called~~ ide~tit~ (< id). 14.7. Transitivity Q[ 1ess-Q£-~ (trans ~) Pl02 1 2 3 4 6
7 8 9
< b
a
subproof would not be general with respect to 'y' and still end with 'd'. 14.5. The following sEeci~ ~ Q[ ~-Q£-eaual ~ duction (sp ~ int) can be easily derived from ~ int:
5
bJ
105
LESS-OR-EQUAL AND LESS
10 11
12 13 14
a b
~
b c X Nx a + X =b b :::: c y Ny b + y ~ c Nx N[x + y] a + X =b a + X+ y a :::; c a :::; c a :::; c ~
In accordance with 12.46 in going from step 11 to as transformed into 'a+ also that 'a+ x + y' is by 12.46.
=
c
hyp hyp hyp hyp 2, reit hyp hyp 3. reit 6, 8, clos N add 4, reit 7. 10, res id elim (see 12 .46) 9. 11, sp :::; int 5. 6-12, sp :::: elim 1, 3-13, sp ~ elim
there is tacit use of assoc add made step 12, since step 11 can be viewed [x + y] = c' by assoc add. Notice an abbreviation for '[a+ x] + y'
106 14.8. Reflexivitl Qf Pl03 1 2 3
NO a + 0 a ~ a
=
14.9. ~ Pl04
LESS-OR-EQUAL AND LESS
ELEMENTS OF COMBINATORY LOGIC
l~Na 2 0 + 3
0
~
a a
=
less-Q£-~
N int rt add 0 1, 2, sp
a ~-Q£-e,gual
~
b a
~
b
=
Pl07 1 2 ~
int
3 I+
~)
5
hyp left add 0 1, 2, sp ~ int
a
14.10. Identit;r entails Pl05 l~a 2 a 3 a
(0
14.14. ~ ~~
~)
(refl
~-_Q£-e_gual
6 e~ual
(id ent ~) ~'
hyp refl ~ 1, 2, id elim
Although the reason given for step 3 is id elim, it is always possible to regard such a substitution into a less-orequal formula, or into a less formula, as a case of res id elim. This is because alternative steps can always be used in which matters are so arranged that the substitution is made in a Q-formula which is an identity.
r.
fl06 2
3 4 5
b a + 0 a + 0 NO a ~ b
14.11.
=a
=b
~-_Q£-.!:.9.!!JU
~Nb
(~ add)
(Proof omitted)
a < a + b
14,12.
~
&!h res{!ect !Q syccessQ! (< sue) (Proof omitted)
1'+.13. !>es£ &!h !.w2m !Q addition !Q suocessor (< add sue)
~Nb
a < a + 1 + b
Pl08
(Proof omitted)
Na a + 0 + 0 + 0 + 0 <
1 + 1 1 + a = 1 + a = 1 ::; a + a + 1
a 1 + a a + 1 1
suc~e.essor
(0 < sue)
hyp 1, com add 1 left add 0 2, 3. res id elim '+. sp ~ int 5, < int
14.15. Transitivit;r £f le~s &!h res{!ect !Q les~-Q£ (trans < ~), first form (see 14.30 for second form)
l"'
hyp b hyp 2 b ~ c 1, < elim 3 a + 1 :: b 2, 3. trans < 4 a + 1 ::; c 4, < int 5 a < c 14.16. ~ entails less-Q£-egual (< ent !f)
Pl09 1 2 3 4
hyp rt add 0 1, 2, res id elim N int 3. 4, sp ~ int
&!h resi!ect !Q addition
107
5 6 7 8
a < b a + 1
rx
~
b
a + 1 + X Nl N[l + x] a ~ b a ~ b
b
hyp 1, < elim hyp hyp N int 3. 5. clos N add 4, 6, sp ::; int 2, 3-7. sp !:: elim
14.17. Transi t_i vi tl Q! les!! (trans <)
PllO
r· 2
3 4
~
b b < c b ~ c a < c
hyp hyp 2, < ent ::; 1, 3. trans < <
14.18. Monoto~ Qf addition &!h £BS{!ect !Q (mon add ~)
ILa~b
~-Q£
106 14.8. Reflexivitl Qf Pl03 1 2 3
NO a + 0 a ~ a
=
14.9. ~ Pl04
LESS-OR-EQUAL AND LESS
ELEMENTS OF COMBINATORY LOGIC
l~Na 2 0 + 3
0
~
a a
=
less-Q£-~
N int rt add 0 1, 2, sp
a ~-Q£-e,gual
~
b a
~
b
=
Pl07 1 2 ~
int
3 I+
~)
5
hyp left add 0 1, 2, sp ~ int
a
14.10. Identit;r entails Pl05 l~a 2 a 3 a
(0
14.14. ~ ~~
~)
(refl
~-_Q£-e_gual
6 e~ual
(id ent ~) ~'
hyp refl ~ 1, 2, id elim
Although the reason given for step 3 is id elim, it is always possible to regard such a substitution into a less-orequal formula, or into a less formula, as a case of res id elim. This is because alternative steps can always be used in which matters are so arranged that the substitution is made in a Q-formula which is an identity.
r.
fl06 2
3 4 5
b a + 0 a + 0 NO a ~ b
14.11.
=a
=b
~-_Q£-.!:.9.!!JU
~Nb
(~ add)
(Proof omitted)
a < a + b
14,12.
~
&!h res{!ect !Q syccessQ! (< sue) (Proof omitted)
1'+.13. !>es£ &!h !.w2m !Q addition !Q suocessor (< add sue)
~Nb
a < a + 1 + b
Pl08
(Proof omitted)
Na a + 0 + 0 + 0 + 0 <
1 + 1 1 + a = 1 + a = 1 ::; a + a + 1
a 1 + a a + 1 1
suc~e.essor
(0 < sue)
hyp 1, com add 1 left add 0 2, 3. res id elim '+. sp ~ int 5, < int
14.15. Transitivit;r £f le~s &!h res{!ect !Q les~-Q£ (trans < ~), first form (see 14.30 for second form)
l"'
hyp b hyp 2 b ~ c 1, < elim 3 a + 1 :: b 2, 3. trans < 4 a + 1 ::; c 4, < int 5 a < c 14.16. ~ entails less-Q£-egual (< ent !f)
Pl09 1 2 3 4
hyp rt add 0 1, 2, res id elim N int 3. 4, sp ~ int
&!h resi!ect !Q addition
107
5 6 7 8
a < b a + 1
rx
~
b
a + 1 + X Nl N[l + x] a ~ b a ~ b
b
hyp 1, < elim hyp hyp N int 3. 5. clos N add 4, 6, sp ::; int 2, 3-7. sp !:: elim
14.17. Transi t_i vi tl Q! les!! (trans <)
PllO
r· 2
3 4
~
b b < c b ~ c a < c
hyp hyp 2, < ent ::; 1, 3. trans < <
14.18. Monoto~ Qf addition &!h £BS{!ect !Q (mon add ~)
ILa~b
~-Q£
108
ELEMENTS OF COMBINATORY LOGIC
The above rule is derivable by use of sp 5 elim and sp 5 int. addition~
14.19. Monotonl Qf add <)
La Ia
res£ect
!£ less (mon
LESS-OR-EQUAL AND LESS
109
In the above proof it is understood that < int and neg < int are used in connection with subproofs required for cnst dil in step 2. These subproofs were omitted for economy, and so
(Proof omitted)
< b + c < b + c
were the corresponding subproofs in Plll. 14.24. Negation introduction
~.£2tiot!
14.20. Monotonl Qf (mon mult 5)
J!ill re:mect !£ :toess-
5) and
~
ra
~-egua:to
(Proof omitted)
I :c< b
f£E
-[a
Use special rules for 5 and rt mult sue in doing this proof.
5
~-Q£-egual
(neg int
(Proofs omitted)
5 b
'-~
fc.a5c.b
£££
(neg int <)
b)
14.21. ~ Qf closure Qf e~cluded m1dd:be (clos ex mid)
~ l
: ~ =:
(Proofs omitted)
[a a b) v -[a a b) [a V b) V -[a V b)
La V -a I -a V -(-a)
(x)[( ••. x •.. ) V -( ... x ••• )]
The above rules are easily derived since less-or-equal and less satisfy excluded middle.
(x)( ••• x ••• ) V -(x)( •.• x ••. )
Pll3 14.25.
(lx)( ••• x •.. ) V -(lx)( ..• x ... )
14.22. Excluded ~ [Q£ 1~-Q£-~ (ex mid 5) Plll 1
x
Nx V -(Nx)
ex mid N
2
[a + x = b) V -[a + x = b)
3
[Nx
& [a+
x =b)] V -[Nx
3 4
ex mid id
a
[a+ X= b))
5 6
1, 2, clos ex mid 4
(x)[[Nx & [a + x
5
(lx)[Nx
& [a+
b)] V -[Nx & [a+ x 1-3, u q int
x = b]) V -(3x)[Nx
& [a+
b)]]
x
7 b)]
4, clos ex mid 6
[a
5
b) V
-[a
5
b)
5, cnst dil (using 5 int and neg 5 int in the subproofs)
14.23. Excluded ~ [Q£ les_e (ex mid <) Pll2
~I
ex mid 5 [a + l 5 b] V -[a + l 5 b) [a < b) V -[a < b] d.il l' cnst
1 2
~
Na [O = 0]
r
identical witg Q£
v
~
hyp id int, dis int hyp
[0 < 0]
[O = b) V [0 < b) 0 < b + 1 [0
[0 = a]
hyp
3. 0 < sue [0 < b + 1) 5. dis int [O < a] 1' 2, 3-6, N indue
b + 1]
v
tgan (0 id <)
v
14.26. We will let '[a
1
b]' serve as an abbreviation
for '-[a 5 b)', and we will let '[a { b]' serve as an abbreviation for '-[a< b]'.
I.
14.27. ~ ~ t~ ~~ Pll4 1 2 3 '+
({
0) hyp
a < + 01 5 0 x Nx la+l+x
l' < elim hyp
0
hyp
108
ELEMENTS OF COMBINATORY LOGIC
The above rule is derivable by use of sp 5 elim and sp 5 int. addition~
14.19. Monotonl Qf add <)
La Ia
res£ect
!£ less (mon
LESS-OR-EQUAL AND LESS
109
In the above proof it is understood that < int and neg < int are used in connection with subproofs required for cnst dil in step 2. These subproofs were omitted for economy, and so
(Proof omitted)
< b + c < b + c
were the corresponding subproofs in Plll. 14.24. Negation introduction
~.£2tiot!
14.20. Monotonl Qf (mon mult 5)
J!ill re:mect !£ :toess-
5) and
~
ra
~-egua:to
(Proof omitted)
I :c< b
f£E
-[a
Use special rules for 5 and rt mult sue in doing this proof.
5
~-Q£-egual
(neg int
(Proofs omitted)
5 b
'-~
fc.a5c.b
£££
(neg int <)
b)
14.21. ~ Qf closure Qf e~cluded m1dd:be (clos ex mid)
~ l
: ~ =:
(Proofs omitted)
[a a b) v -[a a b) [a V b) V -[a V b)
La V -a I -a V -(-a)
(x)[( ••. x •.. ) V -( ... x ••• )]
The above rules are easily derived since less-or-equal and less satisfy excluded middle.
(x)( ••• x ••• ) V -(x)( •.• x ••. )
Pll3 14.25.
(lx)( ••• x •.. ) V -(lx)( ..• x ... )
14.22. Excluded ~ [Q£ 1~-Q£-~ (ex mid 5) Plll 1
x
Nx V -(Nx)
ex mid N
2
[a + x = b) V -[a + x = b)
3
[Nx
& [a+
x =b)] V -[Nx
3 4
ex mid id
a
[a+ X= b))
5 6
1, 2, clos ex mid 4
(x)[[Nx & [a + x
5
(lx)[Nx
& [a+
b)] V -[Nx & [a+ x 1-3, u q int
x = b]) V -(3x)[Nx
& [a+
b)]]
x
7 b)]
4, clos ex mid 6
[a
5
b) V
-[a
5
b)
5, cnst dil (using 5 int and neg 5 int in the subproofs)
14.23. Excluded ~ [Q£ les_e (ex mid <) Pll2
~I
ex mid 5 [a + l 5 b] V -[a + l 5 b) [a < b) V -[a < b] d.il l' cnst
1 2
~
Na [O = 0]
r
identical witg Q£
v
~
hyp id int, dis int hyp
[0 < 0]
[O = b) V [0 < b) 0 < b + 1 [0
[0 = a]
hyp
3. 0 < sue [0 < b + 1) 5. dis int [O < a] 1' 2, 3-6, N indue
b + 1]
v
tgan (0 id <)
v
14.26. We will let '[a
1
b]' serve as an abbreviation
for '-[a 5 b)', and we will let '[a { b]' serve as an abbreviation for '-[a< b]'.
I.
14.27. ~ ~ t~ ~~ Pll4 1 2 3 '+
({
0) hyp
a < + 01 5 0 x Nx la+l+x
l' < elim hyp
0
hyp
ELEMENTS OF COMBINATORY LOGIC
110
5
X + l
6
7
a + x + l 0 -[a + x + l 0]
8
a {: 0
+ X
a {: 0
9 lO
l
a{:O
14.28.
Irreflexivit~
£[
3, com add l 4, 5, res id elim non-sue 0 (11.21) 6, 7, neg elim 2, 3-7, sp ~ elim l-9, neg int < ~
(irrefl <)
(Proof omitted)
LESS-OR-EQUAL AND LESS
~
14.30. Transitivity Qf less~ respect 1Q ~ess-Q£ (trans < ~), second form (see 14.15 for first form)
r:
(Proof omitted)
I: ~ :
r
a < c
14.31. Converse of Pll6 l
2
3 This rule can be derived similarly to 14.27, but using 12.45 in place of 11.21. The hypothesis 'Na' is required in connection with 12.45. 14.29. ~-££-~~ ~tai~s !dept~t~ Q£ ~ (~ ent id dis <) hyp Pll5 l a ~ b hyp 2 x Nx hyp 3 a+x=b 4 [O = x] V [O < x] 2, 0 id < (14.25) hyp X 5 3, reit, 5, res id elim 6 + 0 b rt add o, 6, res id elim b 7 hyp 8 0 < X 8, < elim 0 + l ~ X 9 left add 0, 9, res id elim 10 l ~ X hyp ll y Ny 12 hyp l + y X a + x b 3, reit 13 14 12, mon id, 13, res id elim a + l + y b ll, 14, sp ~ int 15 a + l ~ b 16 a < b 15, sp < int a < b 10, ll-16, sp ~ elim 17 18 [a = bl V [a < b] 4, 5-7, 8-17, cnst dil l, 2-18, sp ~ elim 19 [a = b] V [a < b]
lll
4
~
Q£
~-Q£-~
Na Nb
hyp hyp
0 !: b [b < 0] V [O ~ b]
2, 0
5 c
[b < c] V [c ~ b]
6
r~:~:~ I
7 8
~
hyp
c
ll
[c = b] V [c < b]
12 13 14 15 16 17 18
~~:;:~
~ b]
b
< b + 1 ~ b [b < c + l] V [c + l ~ b]
[b < c + l] V [c + l
~
b]
[b < a] V [a ~ b]
14.32. Converse £[ ~ id dis <)
££
identit~
< sue 12, 13, res id elim hyp 15, < elim 11,12-14,15-l&,cnst dil 5, 6-9,10-17, diselim l, 4, 5-18, N indue Q£ le€s (cnv
I :: [b < a]
6. 7, trans < 8, dis int hyp 10, ~ ent id dis < hyp
L cc I
r
~
< sue
lO
19
v
[a = b]
v
[a < b]
14.33. ~-££-~ epta!~ negative Qf converse Q[ ent neg cnv <)
~ (~
~)
3, dis int hyp
[b < c + l] V [c + l
9
(cnv < dis
ELEMENTS OF COMBINATORY LOGIC
110
5
X + l
6
7
a + x + l 0 -[a + x + l 0]
8
a {: 0
+ X
a {: 0
9 lO
l
a{:O
14.28.
Irreflexivit~
£[
3, com add l 4, 5, res id elim non-sue 0 (11.21) 6, 7, neg elim 2, 3-7, sp ~ elim l-9, neg int < ~
(irrefl <)
(Proof omitted)
LESS-OR-EQUAL AND LESS
~
14.30. Transitivity Qf less~ respect 1Q ~ess-Q£ (trans < ~), second form (see 14.15 for first form)
r:
(Proof omitted)
I: ~ :
r
a < c
14.31. Converse of Pll6 l
2
3 This rule can be derived similarly to 14.27, but using 12.45 in place of 11.21. The hypothesis 'Na' is required in connection with 12.45. 14.29. ~-££-~~ ~tai~s !dept~t~ Q£ ~ (~ ent id dis <) hyp Pll5 l a ~ b hyp 2 x Nx hyp 3 a+x=b 4 [O = x] V [O < x] 2, 0 id < (14.25) hyp X 5 3, reit, 5, res id elim 6 + 0 b rt add o, 6, res id elim b 7 hyp 8 0 < X 8, < elim 0 + l ~ X 9 left add 0, 9, res id elim 10 l ~ X hyp ll y Ny 12 hyp l + y X a + x b 3, reit 13 14 12, mon id, 13, res id elim a + l + y b ll, 14, sp ~ int 15 a + l ~ b 16 a < b 15, sp < int a < b 10, ll-16, sp ~ elim 17 18 [a = bl V [a < b] 4, 5-7, 8-17, cnst dil l, 2-18, sp ~ elim 19 [a = b] V [a < b]
lll
4
~
Q£
~-Q£-~
Na Nb
hyp hyp
0 !: b [b < 0] V [O ~ b]
2, 0
5 c
[b < c] V [c ~ b]
6
r~:~:~ I
7 8
~
hyp
c
ll
[c = b] V [c < b]
12 13 14 15 16 17 18
~~:;:~
~ b]
b
< b + 1 ~ b [b < c + l] V [c + l ~ b]
[b < c + l] V [c + l
~
b]
[b < a] V [a ~ b]
14.32. Converse £[ ~ id dis <)
££
identit~
< sue 12, 13, res id elim hyp 15, < elim 11,12-14,15-l&,cnst dil 5, 6-9,10-17, diselim l, 4, 5-18, N indue Q£ le€s (cnv
I :: [b < a]
6. 7, trans < 8, dis int hyp 10, ~ ent id dis < hyp
L cc I
r
~
< sue
lO
19
v
[a = b]
v
[a < b]
14.33. ~-££-~ epta!~ negative Qf converse Q[ ent neg cnv <)
~ (~
~)
3, dis int hyp
[b < c + l] V [c + l
9
(cnv < dis
112
ELEMENTS OF COMBINATORY LOGIC
Pll7 l 2 3 4 5 6 7
Na a ::: b
~~
hyp hyp hyp 2, reit 3, 4, trans < ::: l' reit, irrefl < 3-6, neg int <
< a
::: b < a {: a b {: a
14.34. ~ ent~ negative Q.[ QQ:Q.verse Q.[ l_ess-m::~ (< ent neg cnv ::;) Pll8 l 2 3 4 5 6 7
Na a < b
hyp hyp hyp 2, reit 3, 4, trans < < 1' reit, irref1 < 3-6, neg int ::;
< a
n
< b < a
b
1
{ a a
14. 35. ~ entails Pll9 l Na 2 a < b b 3 4 < b < a 5 6 { a 7 a t b
t!Qt!-identit~
hyp hyp hyp 2, reit 3. 4, res id elim l, reit, irref1 < 3-6, neg int id
~~
14. 36. Mono tom: Q.[ (man mult <) Pl20 l 2 3 4 5 6 7 8
Nc a < b a + l ::: b N[c + l] [c + l]. [a + l] c + l = l + c [c + l]. [a + l]
~ication
:s =
(< ent non-id)
&.!1! J;:eS:QeCt
.tQ~
hyp hyp 2, < elim l, clos N sue [c + l]. b 3. 4, mon mult ::: l, com add l [c + l]. a + c + l rt rnult sue [c + l]. a + l + c 6, mon id
LESS-OR-EQUAL AND LESS 9 lO
11
[c [c [c
+ + +
l].a + l ::: [c + l].[a + l] l].a < [c + l]. [a + 1] l].a < [c + 1]. b
14.37. Cancellation Pl2l 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
113
£2£
l , 8, sp ::; int 9. < int 5. 10, trans <
:::
multi£lication (can mult)
Na hyp Nb hyp Nc hyp [c + 1]. a = [c + l]. b hyp [b < a] v [a b) v [a < b) 1' 2' cnv < dis id dis < b < a hyp N[c + l] 3. rt>it, clos N sue [c + 1]. b < [c + 1] .a 6, 7. rnon mult < N[ [ c + l).b] 2, reit, 7. clos N mult [c + 1] .a [c + l].b 8, 9. < ent non-id [c + l] .a "' [c + l]. b '+' reit a 10, 11, neg elim b b hyp b 13. rep b hyp < b similarly to 12 a = b 5, 6-12, 13-14, 15-16, dis elim
r: r:
14.38. EXERCISES. Derive the rules stated in 14.4, 14.5, 14.12, 14.13, 14.18, 14.19, 14.20, 14.21, 14.24, 14.28, 14.30, 14.32.
112
ELEMENTS OF COMBINATORY LOGIC
Pll7 l 2 3 4 5 6 7
Na a ::: b
~~
hyp hyp hyp 2, reit 3, 4, trans < ::: l' reit, irrefl < 3-6, neg int <
< a
::: b < a {: a b {: a
14.34. ~ ent~ negative Q.[ QQ:Q.verse Q.[ l_ess-m::~ (< ent neg cnv ::;) Pll8 l 2 3 4 5 6 7
Na a < b
hyp hyp hyp 2, reit 3, 4, trans < < 1' reit, irref1 < 3-6, neg int ::;
< a
n
< b < a
b
1
{ a a
14. 35. ~ entails Pll9 l Na 2 a < b b 3 4 < b < a 5 6 { a 7 a t b
t!Qt!-identit~
hyp hyp hyp 2, reit 3. 4, res id elim l, reit, irref1 < 3-6, neg int id
~~
14. 36. Mono tom: Q.[ (man mult <) Pl20 l 2 3 4 5 6 7 8
Nc a < b a + l ::: b N[c + l] [c + l]. [a + l] c + l = l + c [c + l]. [a + l]
~ication
:s =
(< ent non-id)
&.!1! J;:eS:QeCt
.tQ~
hyp hyp 2, < elim l, clos N sue [c + l]. b 3. 4, mon mult ::: l, com add l [c + l]. a + c + l rt rnult sue [c + l]. a + l + c 6, mon id
LESS-OR-EQUAL AND LESS 9 lO
11
[c [c [c
+ + +
l].a + l ::: [c + l].[a + l] l].a < [c + l]. [a + 1] l].a < [c + 1]. b
14.37. Cancellation Pl2l 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
113
£2£
l , 8, sp ::; int 9. < int 5. 10, trans <
:::
multi£lication (can mult)
Na hyp Nb hyp Nc hyp [c + 1]. a = [c + l]. b hyp [b < a] v [a b) v [a < b) 1' 2' cnv < dis id dis < b < a hyp N[c + l] 3. rt>it, clos N sue [c + 1]. b < [c + 1] .a 6, 7. rnon mult < N[ [ c + l).b] 2, reit, 7. clos N mult [c + 1] .a [c + l].b 8, 9. < ent non-id [c + l] .a "' [c + l]. b '+' reit a 10, 11, neg elim b b hyp b 13. rep b hyp < b similarly to 12 a = b 5, 6-12, 13-14, 15-16, dis elim
r: r:
14.38. EXERCISES. Derive the rules stated in 14.4, 14.5, 14.12, 14.13, 14.18, 14.19, 14.20, 14.21, 14.24, 14.28, 14.30, 14.32.
115
SOME SPECIAL FUNCTIONS pio = 0
(since
PO = 0)
15.3. The function expressed by ·~· is such that if i is greater than j, then i ~ j is the difference between i
and
j, while if
CHAPrER 5
~ o 0 ~ j
P0 i
i
i ~
j, then
i
~
j
be an abbreviation for 'TP'. Then the
following identities can be proved: i ~ j ~ji TPji = jPi = Pji
RECURSION AND CIRCUITS
15.
is not greater than
i
·~·
is zero. we let
i
= PjO
0
i
~
i~[i+j]
SEecial Eunctio~s
[i+j]~j
15.1. Various functions of natural numbers will now be
i
15.4. The function expressed by
·~·
is such that if
described, and it will be shown how they can be defined with-
j
in the system Q. From now on the procedure will be somewhat
two, while if they are equal, it is each of them. Let ·~· be an abbreviation for '[x][y][y ~ [y ~ x]]'. Then the fol-
more informal in the sense that many identities will be presented that are provable, but the proofs of them will be left for the reader to do. The lowercase letters 'h',
and
i
are unequal, then
~ji
is the lesser of the
lowing identities can be proved:
'i',
~ji
~
= i
~
[i
~igOi
j]
= ~jO
= 0
'j', and 'k' will stand for arbitrary Q-numerals (except possibly when being used as subscritpts, or later on, when 'h'
j
will denote an arbitrary function). Thus they may be thought of as denoting natural numbers. The notation 'a b = c' will
two, while if they are equal. it is each of them. Let ·~· be an abbreviation for '[x][y][[x + y] ~ ~xy]'. Then the
stand for '[a= b] stand for '[a= b]
a a
[b = c]', while 'a= b = c = d' will
a
[b = c]
15.5. The function expressed by ·~· is such that if and i are unequal, then ~ji is the greater of the
following identities can be proved:
[c = d]', and so on.
15.2. The first function to be considered is the predecessor function, denoted by •p•. This function always has a
= [j + i] ~ ~ji ~Oj = ~jO = j 15.6. The function expressed by ·~· is such that if
~ji
i
value one less than its applicand for an applicand greater
is O, then ~i is also o, while if i is not 0, then ~i is l. Let·~· be an abbreviation for '[x][l ~ [l ~ x]]'.
than zero, so that
Then the following identities can be proved:
P[i + l]
i, but for applicand zero we
have PO = o. In the system Q we let 'P' be an abbreviation for '[x](xF(BC'l'uO)u', that is, for '[x](Fx(BCTOO)O)'. (See
~i
l
~
[l
~
il
~
l
~
[l
~
0]
12.36-12.41.) The following identities can easily be proved:
~[ i
Pi = Fi(BCTOO)O PO = F 0 (BCTOO)O = BCTOOO P[i + l] = Fi + 1 (BCTOO)O
0 BCT[i + l] iO
i i
=
0
+ l] = l ~ [ l ~ [ i + l J J = l ~ 0 = l 15.7. The function expressed by ·~.e.g· is such that if
is not o, then
is 1. Let
·~·
~i
is 0, while i f
be an abbreviation for
i
is 0, then
'G~l',
i.e. for
'(C~)l'. Then the following identities can be proved:
114
~i
115
SOME SPECIAL FUNCTIONS pio = 0
(since
PO = 0)
15.3. The function expressed by ·~· is such that if i is greater than j, then i ~ j is the difference between i
and
j, while if
CHAPrER 5
~ o 0 ~ j
P0 i
i
i ~
j, then
i
~
j
be an abbreviation for 'TP'. Then the
following identities can be proved: i ~ j ~ji TPji = jPi = Pji
RECURSION AND CIRCUITS
15.
is not greater than
i
·~·
is zero. we let
i
= PjO
0
i
~
i~[i+j]
SEecial Eunctio~s
[i+j]~j
15.1. Various functions of natural numbers will now be
i
15.4. The function expressed by
·~·
is such that if
described, and it will be shown how they can be defined with-
j
in the system Q. From now on the procedure will be somewhat
two, while if they are equal, it is each of them. Let ·~· be an abbreviation for '[x][y][y ~ [y ~ x]]'. Then the fol-
more informal in the sense that many identities will be presented that are provable, but the proofs of them will be left for the reader to do. The lowercase letters 'h',
and
i
are unequal, then
~ji
is the lesser of the
lowing identities can be proved:
'i',
~ji
~
= i
~
[i
~igOi
j]
= ~jO
= 0
'j', and 'k' will stand for arbitrary Q-numerals (except possibly when being used as subscritpts, or later on, when 'h'
j
will denote an arbitrary function). Thus they may be thought of as denoting natural numbers. The notation 'a b = c' will
two, while if they are equal. it is each of them. Let ·~· be an abbreviation for '[x][y][[x + y] ~ ~xy]'. Then the
stand for '[a= b] stand for '[a= b]
a a
[b = c]', while 'a= b = c = d' will
a
[b = c]
15.5. The function expressed by ·~· is such that if and i are unequal, then ~ji is the greater of the
following identities can be proved:
[c = d]', and so on.
15.2. The first function to be considered is the predecessor function, denoted by •p•. This function always has a
= [j + i] ~ ~ji ~Oj = ~jO = j 15.6. The function expressed by ·~· is such that if
~ji
i
value one less than its applicand for an applicand greater
is O, then ~i is also o, while if i is not 0, then ~i is l. Let·~· be an abbreviation for '[x][l ~ [l ~ x]]'.
than zero, so that
Then the following identities can be proved:
P[i + l]
i, but for applicand zero we
have PO = o. In the system Q we let 'P' be an abbreviation for '[x](xF(BC'l'uO)u', that is, for '[x](Fx(BCTOO)O)'. (See
~i
l
~
[l
~
il
~
l
~
[l
~
0]
12.36-12.41.) The following identities can easily be proved:
~[ i
Pi = Fi(BCTOO)O PO = F 0 (BCTOO)O = BCTOOO P[i + l] = Fi + 1 (BCTOO)O
0 BCT[i + l] iO
i i
=
0
+ l] = l ~ [ l ~ [ i + l J J = l ~ 0 = l 15.7. The function expressed by ·~.e.g· is such that if
is not o, then
is 1. Let
·~·
~i
is 0, while i f
be an abbreviation for
i
is 0, then
'G~l',
i.e. for
'(C~)l'. Then the following identities can be proved:
114
~i
ELEMENTS OF COMBINATORY LOGIC
116 ~i
C~li
= ~il =
l
~
i
nsgo l ~ 0 = l nsg[i + l] = l ~ [i + l] = 0 15.8. There is no trouble, incidentally, in obtaining such results as the following:
I ~[a +
l] = 1
This is because N indue could be used for the purpose, or N elim could be used as follows: Na !!.ej[O + 1] !!£;[ 1 + 1] .eg[2 + l]
l l 1
w .eg[a + 1]
=l
3 4
hyp 15.6 15.6 15.6 l,
2, 3. 4,
...
•
N elim
"urthermore, '(x) [Nx ::o [~[x + l] = 1]]' can be obtained as follows, where 'p ::0 q' is treated as an abbreviation for '-p v q': Pl23 1 2 3 4
5 6
a
117
Jfa = BCT[aK + .eg(aO)][~(aO).f[aK + 1]] Jf(BCTiO) = BCT[i + .egOJ[.egO.f[i + l] = BCTiO Jf(BCTi[j + 1)) = BCT[i+.eg[j + l]][.eg[j + l].f[i+ l]J = BCT[i + l](f[i + 1]) Also, if k 0 is the least k such that fk o, then: (Jf)h(BCTO(fO)) (Jf)h(BCTO(fO))
~Na
Pl22 1 2
SOME SPECIAL FUNCTIONS
Na V -(Na)
ex mid N hyp ~Na .e.g[ a + l] = l 2, as in Pl22 -(Na) V [.eg[a + l] = l] 1, 2-3, cnst dil /~. rep, def Na ::o [.eg[a + 1] = l] (x)[Nx :::> [.eg[x + l] = 1] l-5. u q int
Step 6 could also be obtained by res imp int. See 0.52. 15.9. The function to be expressed by 'J' is such that Jf(BCTij) is simply BCTij for j = O, but is BCT[i + l](f[i + 1]) for j > o. (Recall that BCTi,j is the ordered couple consisting of i and j in that order, as stated in 12.31.) Let 'J' be an abbreviation for '[x] [y] (BCT[yK +.e.g( yO) J Le,e;(yO) .x[yK + 1]])'. Then using 12.32, 12.33, and 12.34, the following identities can be proved:
BCTh(fh) for h < k 0 BCTh(fh) BCTk 0 0 for h = k 0 (Jf)h(BCTO(fO)) BCTk 0 0 for h > k 0 Thus if h is greater than or equal to the least k such that fk = 0, then (Jf)h(BCTO(fO))K is that least k, and otherwise is h. In other words, if there is an i such that fi = 0 and i ~ h, then (Jf)h(BCTO(fO))K is the least such i, and is h if there is no such i. 15.10. Let the notation e:Hfi = OJ i~h
represent the least i ~ h such that fi = 0 i f there is such an i, and represent 0 if there is no such i. The the above notation could be treated as an abbreviation for:
((Jf)h(BCTO(fO))K).(~((Jf)h(BCTO(fO))O)) If there is an i (and therefore a least i) such that i ~ h
and fi = O, then the left-hand factor of the above product equals the least such i by 15.9, and the right-hand factor equals 1, so that the whole product equals the least such i. On the other hand if there is no i such that i < h and fi = o, then it is not hard to see that (Jf)h(BCTO(fO))O must be greater than O, so that the right-hand factor equals 0 by 15.7 and the whole product therefore equals o. Thus the above product has the requisite properties. 15.11. Consider next a slightly different situation. Suppose that f is such that there is some i such that fi = o. Then there must be a least such i. Let the expression £i[fi = 0]
ELEMENTS OF COMBINATORY LOGIC
116 ~i
C~li
= ~il =
l
~
i
nsgo l ~ 0 = l nsg[i + l] = l ~ [i + l] = 0 15.8. There is no trouble, incidentally, in obtaining such results as the following:
I ~[a +
l] = 1
This is because N indue could be used for the purpose, or N elim could be used as follows: Na !!.ej[O + 1] !!£;[ 1 + 1] .eg[2 + l]
l l 1
w .eg[a + 1]
=l
3 4
hyp 15.6 15.6 15.6 l,
2, 3. 4,
...
•
N elim
"urthermore, '(x) [Nx ::o [~[x + l] = 1]]' can be obtained as follows, where 'p ::0 q' is treated as an abbreviation for '-p v q': Pl23 1 2 3 4
5 6
a
117
Jfa = BCT[aK + .eg(aO)][~(aO).f[aK + 1]] Jf(BCTiO) = BCT[i + .egOJ[.egO.f[i + l] = BCTiO Jf(BCTi[j + 1)) = BCT[i+.eg[j + l]][.eg[j + l].f[i+ l]J = BCT[i + l](f[i + 1]) Also, if k 0 is the least k such that fk o, then: (Jf)h(BCTO(fO)) (Jf)h(BCTO(fO))
~Na
Pl22 1 2
SOME SPECIAL FUNCTIONS
Na V -(Na)
ex mid N hyp ~Na .e.g[ a + l] = l 2, as in Pl22 -(Na) V [.eg[a + l] = l] 1, 2-3, cnst dil /~. rep, def Na ::o [.eg[a + 1] = l] (x)[Nx :::> [.eg[x + l] = 1] l-5. u q int
Step 6 could also be obtained by res imp int. See 0.52. 15.9. The function to be expressed by 'J' is such that Jf(BCTij) is simply BCTij for j = O, but is BCT[i + l](f[i + 1]) for j > o. (Recall that BCTi,j is the ordered couple consisting of i and j in that order, as stated in 12.31.) Let 'J' be an abbreviation for '[x] [y] (BCT[yK +.e.g( yO) J Le,e;(yO) .x[yK + 1]])'. Then using 12.32, 12.33, and 12.34, the following identities can be proved:
BCTh(fh) for h < k 0 BCTh(fh) BCTk 0 0 for h = k 0 (Jf)h(BCTO(fO)) BCTk 0 0 for h > k 0 Thus if h is greater than or equal to the least k such that fk = 0, then (Jf)h(BCTO(fO))K is that least k, and otherwise is h. In other words, if there is an i such that fi = 0 and i ~ h, then (Jf)h(BCTO(fO))K is the least such i, and is h if there is no such i. 15.10. Let the notation e:Hfi = OJ i~h
represent the least i ~ h such that fi = 0 i f there is such an i, and represent 0 if there is no such i. The the above notation could be treated as an abbreviation for:
((Jf)h(BCTO(fO))K).(~((Jf)h(BCTO(fO))O)) If there is an i (and therefore a least i) such that i ~ h
and fi = O, then the left-hand factor of the above product equals the least such i by 15.9, and the right-hand factor equals 1, so that the whole product equals the least such i. On the other hand if there is no i such that i < h and fi = o, then it is not hard to see that (Jf)h(BCTO(fO))O must be greater than O, so that the right-hand factor equals 0 by 15.7 and the whole product therefore equals o. Thus the above product has the requisite properties. 15.11. Consider next a slightly different situation. Suppose that f is such that there is some i such that fi = o. Then there must be a least such i. Let the expression £i[fi = 0]
ELEMENTS OF COMBINATORY LOGIC
118
represent the smallest number i such that fi = Q. The problem is now to find some way of defining ·~i[fi = 0]' for an f of this kind, that is, for an f such that there is an i for which fi = o. (Notice that in this case, unlike the case in 15.10, there is no upper bound h.) This problem apparently could be solved if we could define a Q-function H satisfying the two equations: fj = o, for Hfj = j Hfj = Hf[j + 1] for fj > o. This would solve the problem because if i 0 is the least i such that fi = O, then by the above equations we would have HfO = Hfl = Hf2 = •.• = Hfi 0 = i 0 , so that '~i[fi = 0]' could be defined as 'HfO'. In order for H to have the properties expressed by the above two equations, it would be sufficient for H to satisfy the following equation for every natural number j: (H-identity) Hfj = [nsg(fj).j] + [~(fj).Hf[j + 1)] That this would be sufficient is seen as follows: If the above equation holds and if fj = 0, then Hfj [~.j] + [~.Hf[j + 1]] = [l.j] + [O.Hf[j + 1]) = j
On the other hand, if fj > O, then Hfj = [O.j] + [l.Hf[j + 1]] = Hf[j + 1] In order to define 'H' in such a way that the H-identity is satisfied and the above results ensue, it suffices to let 'H' be an abbreviation for 'LL', where 'L' is a temporary abbreviation for [x][y]rzJ[[nsg(yz).z] + [~(yz).xxy[z + l]]J Then Lafj = [nsg(fj).j] + [~(fj).aaf[j + 1)], and LLfj = [~(fj).j] + [~(fj).LLf[j + 1]]. This last identity is the required H-identity when 'H' is treated as an abbreviation for 'LL'. Hence, ·~i[fi =OJ' can be viewed as an abbreviation for 'HfO'.
RECURSIVE FUNCTIONS
119
15.12. EXERCISEs. Prove the equations stated in 15.215.7, 15.9, and 15.11. 16.
Recursive
~~~
16.1. There is a class of functions of natural numbers known as recursive f2gctkogs. The values of these functions can always be effectively computed in a finite number of steps and are always natural numbers. Conversely it appears to be the case that every function of natural numbers (taking natural numbers as values) which is effectively computable in a finite number of steps is recursive ("Church's thesis"), but this has never been proved, and it is hard to see how it could. Every recursive function, it will be shown, coincides with a Q-function for the cases where the Q-function is applied to natural numbers. (A Q-function f may, in effect, be limited to natural numbers as applicands by using '[x] [Nx li fx]' in place of 'f'. This because if a is a natural number, then it is possible to prove [x][Nx li fx]a = fa~ while if a is not a natural number this equation would ordinarily not be provable, and nothing of any importance could be proved concerning the left side of the equation.) We begin by considering a subclass of the class of recursive functions, namely the class of £rimitjve recur€~~ functiops. 16.2. The succ~ fu~ is the function f such that, in ordinary functional notation, f(i) = i + 1. This function is the same as the Q-function ~ (as applied to natural numbers), where ·~· is an abbreviation for '+1'. We have: ~i = +li = i + 1 The successor function is a primitive recursive function. 16.3. The n-place constant functiop which has value i is the function f such that, in ordinary functional notation, f(j 1 , j2' ·•• , jn) = i, and in combinatory notation, fj 1 j 2 ••• jn = i. This function is the same as the Q-function constni (as applied to natural numbers), where 'const ' is ~ -n an abbreviation for '[y)[x J[x ] ••• [xn]y'. We then have: 1 2
ELEMENTS OF COMBINATORY LOGIC
118
represent the smallest number i such that fi = Q. The problem is now to find some way of defining ·~i[fi = 0]' for an f of this kind, that is, for an f such that there is an i for which fi = o. (Notice that in this case, unlike the case in 15.10, there is no upper bound h.) This problem apparently could be solved if we could define a Q-function H satisfying the two equations: fj = o, for Hfj = j Hfj = Hf[j + 1] for fj > o. This would solve the problem because if i 0 is the least i such that fi = O, then by the above equations we would have HfO = Hfl = Hf2 = •.• = Hfi 0 = i 0 , so that '~i[fi = 0]' could be defined as 'HfO'. In order for H to have the properties expressed by the above two equations, it would be sufficient for H to satisfy the following equation for every natural number j: (H-identity) Hfj = [nsg(fj).j] + [~(fj).Hf[j + 1)] That this would be sufficient is seen as follows: If the above equation holds and if fj = 0, then Hfj [~.j] + [~.Hf[j + 1]] = [l.j] + [O.Hf[j + 1]) = j
On the other hand, if fj > O, then Hfj = [O.j] + [l.Hf[j + 1]] = Hf[j + 1] In order to define 'H' in such a way that the H-identity is satisfied and the above results ensue, it suffices to let 'H' be an abbreviation for 'LL', where 'L' is a temporary abbreviation for [x][y]rzJ[[nsg(yz).z] + [~(yz).xxy[z + l]]J Then Lafj = [nsg(fj).j] + [~(fj).aaf[j + 1)], and LLfj = [~(fj).j] + [~(fj).LLf[j + 1]]. This last identity is the required H-identity when 'H' is treated as an abbreviation for 'LL'. Hence, ·~i[fi =OJ' can be viewed as an abbreviation for 'HfO'.
RECURSIVE FUNCTIONS
119
15.12. EXERCISEs. Prove the equations stated in 15.215.7, 15.9, and 15.11. 16.
Recursive
~~~
16.1. There is a class of functions of natural numbers known as recursive f2gctkogs. The values of these functions can always be effectively computed in a finite number of steps and are always natural numbers. Conversely it appears to be the case that every function of natural numbers (taking natural numbers as values) which is effectively computable in a finite number of steps is recursive ("Church's thesis"), but this has never been proved, and it is hard to see how it could. Every recursive function, it will be shown, coincides with a Q-function for the cases where the Q-function is applied to natural numbers. (A Q-function f may, in effect, be limited to natural numbers as applicands by using '[x] [Nx li fx]' in place of 'f'. This because if a is a natural number, then it is possible to prove [x][Nx li fx]a = fa~ while if a is not a natural number this equation would ordinarily not be provable, and nothing of any importance could be proved concerning the left side of the equation.) We begin by considering a subclass of the class of recursive functions, namely the class of £rimitjve recur€~~ functiops. 16.2. The succ~ fu~ is the function f such that, in ordinary functional notation, f(i) = i + 1. This function is the same as the Q-function ~ (as applied to natural numbers), where ·~· is an abbreviation for '+1'. We have: ~i = +li = i + 1 The successor function is a primitive recursive function. 16.3. The n-place constant functiop which has value i is the function f such that, in ordinary functional notation, f(j 1 , j2' ·•• , jn) = i, and in combinatory notation, fj 1 j 2 ••• jn = i. This function is the same as the Q-function constni (as applied to natural numbers), where 'const ' is ~ -n an abbreviation for '[y)[x J[x ] ••• [xn]y'. We then have: 1 2
120
ELEMENTS OF COMBINATORY LOGIC ~nijlj2 .••
[y][xl)[x2] •.• [xn]yijlj2 ••• jn i (by the abstraction identity) In applying the abstraction identity we replace 'y', 'x •, •x 2 •, ... ,
jn
=
'xn' respectively by 'i',
'j ',
1
'j ', ... 2
,
1
'jn'
in 'y', but since only 'y' occurs in 'y', the result is simply 'i'. Every constant function is a primitive recursive function. 16.4. EXERCISE. Show that an alternative way to define 'constn' is to treat it as an abbreviation for 'Kn'. 16.5. The n-place identity function which has a value identical with that of its ith argument is the function f such that, in ordinary functional notation, f(jl' j2' ••• • ji' ··• • jn) ji This function is the same as the Q-function ident. l,n (as applied to natural numbers), where 'ident. ' is an abbrevia----1,n tion for '[x 1 Hx2 J ••• [xi] •.. [xn]xi'. We have:
~i,n'jlj2 ..• ji .•• jn
[xl][x2] ... [xi] .•• [xn]xijlj2 .•• ji .•• jn ji (by the abstraction identity) Every identity function is a primitive recursive function. 16.6. EXERCISE. Show that an alternative way to define 'ident. ------l.,n ' is to treat it as an abbreviation for BiKn.:. i(Bi .:.1 0 1) =
=
16.7. Given an n-place function f and also n m-place functions g1 , g2 , ..• , gn' the function h such that, in ordinary functional notation, h(jl, ... ,jm) = f(gl(jl, ... ,jm), ••• ,gn(jl'"""'jm)) is said to be the compos~ fupctio~ off with g , g , ..• , 2 1 gn' and it is primitive recursive iff, g , g , .•. gn are 2 1 primitive recursive. This function is the same as the Q-function com:2n,mfg 1 g 2 .•• gm (as applied to natural numbers), where 'com:2 n,m ' is an abbreviation for [u][v 1 ] .•• [vn][x 1 ] ... [xml(u(v 1 x ... xm) .•• (vnx •.• xm)) 1 1 We then have: comEn,mfgl ... gnjl ..• jm = [u)[vl] .•. [vnl[xl] .•. [xm](u(vlxl ... xm) ... (vnxl ... xm)fgl .•. gn'il•••,jm)
121
RECURSIVE FUNCTIONS
= f(g j ••• jm) ..• (gnj ••• jm) (by the abstraction identity) 1 1 1 as required. (In this paragraph and hereafter 'h' refers to an arbitrary function rather than to an arbitrary natural number.) 16.8. Given a natural number i and a two-place function g, the one-place function h such that, in ordinary functional notation, h(O) = i h(j + l) = g(j, h(j)), is said to be the function defined £ecursive!l ln ~ Q[ i and g. It is primitive recursive if g is pr~mitive recursive. This function is the same as the Q-function ~ig (as applied to natural numbers, and assuming g to be a Q-function), where ·~· is an abbreviation for [x][u][y]((Mu)Y(BCTxO)K) and 'M' is a temporary abbreviation for [v][z](BCT(v(zO)(zK))[zO + l]) It will be shown that we have ~igO = i ~ig[j + 1]
= gj(~igj)
thus establishing that ~ig has the properties required for the function h described above. From the definition of 'M' (using the abstraction identity) we get Mgc = BCT(g(cO)(cK))[cO + 1], Mg(BCTab) = BCT(gba)[b + 1] by the BCT rules. and so Then (Mg) 0 (BCTiO) = BCTiO, and by use of the previous equation we have (Mg) 1 (BCTiO) Mg(BCTiO) BCT( gOi) 1 2 Mg(Mg(BCTiO)) (Mg) (BCTiO) " Mg(BCT(gOi)l) BCT(gl(gOi)) Mg((Mg) 2 (BCTi0)) "Mg(RCT(gl(g0i;)2) BC T ( g2! gl ( gC i) ) ) 3 and so on. Now from the definition of 'Ef._
120
ELEMENTS OF COMBINATORY LOGIC ~nijlj2 .••
[y][xl)[x2] •.• [xn]yijlj2 ••• jn i (by the abstraction identity) In applying the abstraction identity we replace 'y', 'x •, •x 2 •, ... ,
jn
=
'xn' respectively by 'i',
'j ',
1
'j ', ... 2
,
1
'jn'
in 'y', but since only 'y' occurs in 'y', the result is simply 'i'. Every constant function is a primitive recursive function. 16.4. EXERCISE. Show that an alternative way to define 'constn' is to treat it as an abbreviation for 'Kn'. 16.5. The n-place identity function which has a value identical with that of its ith argument is the function f such that, in ordinary functional notation, f(jl' j2' ••• • ji' ··• • jn) ji This function is the same as the Q-function ident. l,n (as applied to natural numbers), where 'ident. ' is an abbrevia----1,n tion for '[x 1 Hx2 J ••• [xi] •.. [xn]xi'. We have:
~i,n'jlj2 ..• ji .•• jn
[xl][x2] ... [xi] .•• [xn]xijlj2 .•• ji .•• jn ji (by the abstraction identity) Every identity function is a primitive recursive function. 16.6. EXERCISE. Show that an alternative way to define 'ident. ------l.,n ' is to treat it as an abbreviation for BiKn.:. i(Bi .:.1 0 1) =
=
16.7. Given an n-place function f and also n m-place functions g1 , g2 , ..• , gn' the function h such that, in ordinary functional notation, h(jl, ... ,jm) = f(gl(jl, ... ,jm), ••• ,gn(jl'"""'jm)) is said to be the compos~ fupctio~ off with g , g , ..• , 2 1 gn' and it is primitive recursive iff, g , g , .•. gn are 2 1 primitive recursive. This function is the same as the Q-function com:2n,mfg 1 g 2 .•• gm (as applied to natural numbers), where 'com:2 n,m ' is an abbreviation for [u][v 1 ] .•• [vn][x 1 ] ... [xml(u(v 1 x ... xm) .•• (vnx •.• xm)) 1 1 We then have: comEn,mfgl ... gnjl ..• jm = [u)[vl] .•. [vnl[xl] .•. [xm](u(vlxl ... xm) ... (vnxl ... xm)fgl .•. gn'il•••,jm)
121
RECURSIVE FUNCTIONS
= f(g j ••• jm) ..• (gnj ••• jm) (by the abstraction identity) 1 1 1 as required. (In this paragraph and hereafter 'h' refers to an arbitrary function rather than to an arbitrary natural number.) 16.8. Given a natural number i and a two-place function g, the one-place function h such that, in ordinary functional notation, h(O) = i h(j + l) = g(j, h(j)), is said to be the function defined £ecursive!l ln ~ Q[ i and g. It is primitive recursive if g is pr~mitive recursive. This function is the same as the Q-function ~ig (as applied to natural numbers, and assuming g to be a Q-function), where ·~· is an abbreviation for [x][u][y]((Mu)Y(BCTxO)K) and 'M' is a temporary abbreviation for [v][z](BCT(v(zO)(zK))[zO + l]) It will be shown that we have ~igO = i ~ig[j + 1]
= gj(~igj)
thus establishing that ~ig has the properties required for the function h described above. From the definition of 'M' (using the abstraction identity) we get Mgc = BCT(g(cO)(cK))[cO + 1], Mg(BCTab) = BCT(gba)[b + 1] by the BCT rules. and so Then (Mg) 0 (BCTiO) = BCTiO, and by use of the previous equation we have (Mg) 1 (BCTiO) Mg(BCTiO) BCT( gOi) 1 2 Mg(Mg(BCTiO)) (Mg) (BCTiO) " Mg(BCT(gOi)l) BCT(gl(gOi)) Mg((Mg) 2 (BCTi0)) "Mg(RCT(gl(g0i;)2) BC T ( g2! gl ( gC i) ) ) 3 and so on. Now from the definition of 'Ef._
122
ELEMENTS OF COMBINATORY LOGIC
tion identity we have, ~igj = (Mg)j(BCTiO)K and so (Mg) 0 (BCTiO)K ~igO BCTiOK
= (~
identity)
(Mg) 1 (BCTiO)K BCT(gOi)lK gOi ~ig2 = (Mg) 2 (BCTiO)K BCT(gl(gOi))2K = gl(gOi) ~ig3 (Mg)3(BCTiO)K BCT(g2(gl(gOi)))3K g2 ( gl ( gOi)) and so on, so that, in general, ~ig[j + 1] = gj(~igj) These results can also be written as the pair of equations ~igO = i E!£ig[j + 1] = gj(~igj) which were the two equations to be proved. 16.9. EXERCISE. Show that if i is 1 and g is such that gjk = [j + l].k, then ~ig is the factorial function. 16.10. Given an n-1-place function f and an n+l-place function g, the n-place function h such that, in ordinary functional notation, for n > 1, =
h(jl'''''jn-l'O) = f(jl'''''jn-1) h(jl'''''jn-l'jn+ l) = g(jl'''''jn-l'jn,h(jl''''jn-l'jn)), is said to be the function de£ined rec~rsivelz ~ ~ Q£ f ~ g. It is primitive recursive if f and g are primitive recursive. This function is the same as the Q-function recnfg (as applied to natural numbers, and assuming f and g to be Q-functions), where 'rec • is an abbreviation for -n
[u][v][xl] ... [xn-l](~(uxl'''xn-l)(vxl'''xn-1)) Then we have: ~nfgjl ..•
jn-ljn
~(fjl'''jn-l)(gjl'''jn-l)jn
(~n identity)
Consequently, ~nfgjl ••• jn-1° = ~(fjl'''jn-l)(gjl'''jn-l)O
fjl' .. jn-1 as required. Also:
i
~igl
123
RECURSIVE FUNCTIONS
' J ,v]Lxl] r r = LU ..• ,xn-ll(~(uxl'''xn-l)(vxl ••. xn-l)fgjl'''jn-ljn
recnfgjl'''jn-l[jn + ll = !!£(fjl'''jn-l)(gjl'''jn-l)[jn + l] = gjl'''jn-ljn(~(fjl'''jn-l)(gjl'''jn-l)jn) = gjl'''jn-ljn(~nfgjl'''jn-ljn) as required. Thus re~nfg can play the role of h. 16.11. EXERCISE. If h is the Q-function ~ 2 ~ 1 , 1 (com:e1 , 1 .!!E£ ident , ) 3 3 show that hj 0 = jl and hj 1 [j 2 + 1] = hj 1 j 2 + 1, and show 1 also that h as applied to natural numbers is primitive recursive. Since h, as applied to natural numbers, is the same as addition, this would show that addition is primitive recursive. 16.12. The class of primitive recursive functions may be defined as the class of all functions which can be shown to be primitive recursive in virtue of 16.2, 16.3, 16.5, 16.7, 16.8, and 16.10. 16.13. EXERCISES. Show that the following functions are primitive recursive when restricted to natural numbers• multiplication, exponentiation, the functions P, ~. ~· ~· .!!! ' :MIS. 16.14. The class of recursive £~ct!On@ may be defined as the class consisting of all primitive recursive functions together with all functions g such that, in ordinary functional notation, g(j , ••• ,jn) = ~i[f(j , .•• ,jn,i) = 0], where 1 1 ·~· is used as in 15.10, and where f itself is primitive recursive or recursive and such that, for all j 1 , •.• , jn' there is ani such that f(j , ••• ,jn,i) = o. The Q-function 1 corresponding tog would then be [x1 ] ••• [xn](H(fx 1 ••• xn)O), where 'H' is as defined in 15.10. Thus every recursive function is some Q-function restricted to natural numbers.
122
ELEMENTS OF COMBINATORY LOGIC
tion identity we have, ~igj = (Mg)j(BCTiO)K and so (Mg) 0 (BCTiO)K ~igO BCTiOK
= (~
identity)
(Mg) 1 (BCTiO)K BCT(gOi)lK gOi ~ig2 = (Mg) 2 (BCTiO)K BCT(gl(gOi))2K = gl(gOi) ~ig3 (Mg)3(BCTiO)K BCT(g2(gl(gOi)))3K g2 ( gl ( gOi)) and so on, so that, in general, ~ig[j + 1] = gj(~igj) These results can also be written as the pair of equations ~igO = i E!£ig[j + 1] = gj(~igj) which were the two equations to be proved. 16.9. EXERCISE. Show that if i is 1 and g is such that gjk = [j + l].k, then ~ig is the factorial function. 16.10. Given an n-1-place function f and an n+l-place function g, the n-place function h such that, in ordinary functional notation, for n > 1, =
h(jl'''''jn-l'O) = f(jl'''''jn-1) h(jl'''''jn-l'jn+ l) = g(jl'''''jn-l'jn,h(jl''''jn-l'jn)), is said to be the function de£ined rec~rsivelz ~ ~ Q£ f ~ g. It is primitive recursive if f and g are primitive recursive. This function is the same as the Q-function recnfg (as applied to natural numbers, and assuming f and g to be Q-functions), where 'rec • is an abbreviation for -n
[u][v][xl] ... [xn-l](~(uxl'''xn-l)(vxl'''xn-1)) Then we have: ~nfgjl ..•
jn-ljn
~(fjl'''jn-l)(gjl'''jn-l)jn
(~n identity)
Consequently, ~nfgjl ••• jn-1° = ~(fjl'''jn-l)(gjl'''jn-l)O
fjl' .. jn-1 as required. Also:
i
~igl
123
RECURSIVE FUNCTIONS
' J ,v]Lxl] r r = LU ..• ,xn-ll(~(uxl'''xn-l)(vxl ••. xn-l)fgjl'''jn-ljn
recnfgjl'''jn-l[jn + ll = !!£(fjl'''jn-l)(gjl'''jn-l)[jn + l] = gjl'''jn-ljn(~(fjl'''jn-l)(gjl'''jn-l)jn) = gjl'''jn-ljn(~nfgjl'''jn-ljn) as required. Thus re~nfg can play the role of h. 16.11. EXERCISE. If h is the Q-function ~ 2 ~ 1 , 1 (com:e1 , 1 .!!E£ ident , ) 3 3 show that hj 0 = jl and hj 1 [j 2 + 1] = hj 1 j 2 + 1, and show 1 also that h as applied to natural numbers is primitive recursive. Since h, as applied to natural numbers, is the same as addition, this would show that addition is primitive recursive. 16.12. The class of primitive recursive functions may be defined as the class of all functions which can be shown to be primitive recursive in virtue of 16.2, 16.3, 16.5, 16.7, 16.8, and 16.10. 16.13. EXERCISES. Show that the following functions are primitive recursive when restricted to natural numbers• multiplication, exponentiation, the functions P, ~. ~· ~· .!!! ' :MIS. 16.14. The class of recursive £~ct!On@ may be defined as the class consisting of all primitive recursive functions together with all functions g such that, in ordinary functional notation, g(j , ••• ,jn) = ~i[f(j , .•• ,jn,i) = 0], where 1 1 ·~· is used as in 15.10, and where f itself is primitive recursive or recursive and such that, for all j 1 , •.• , jn' there is ani such that f(j , ••• ,jn,i) = o. The Q-function 1 corresponding tog would then be [x1 ] ••• [xn](H(fx 1 ••• xn)O), where 'H' is as defined in 15.10. Thus every recursive function is some Q-function restricted to natural numbers.
124
ELEMENTS OF COMBINATORY LOGIC 17.
~-Referentialll ~efined Eunct!on~
17.1. The method used to define H in 15.11 is a special case of a general method for defining a Q-function M such that Mb 1 b 2 ... bn = ( ... M... b 1 ... b 2 ...... bn ... ), where ' ( ... M ••• b 1 ••• b 2 •••••• bn···)' represents any context involving 'M' itself as well as 'b 1 ', 'b 2 ', ••• , 'bn' and standing for the value of M when Misapplied to b , b , ••• , bn as an 2 1 n-place function. Such a Q-function M is ~-refereptia~~l defined in the sense that the value specified for the function involves reference to the function M itself, as well as reference to the entities to which the function is applied. The general method for defining such a function is as follows: First let 'L' be a temporary abbreviation for [x][yl J[y2] · '· [yn] ( · '.xx.' •Y1• · ·Y2• · ·'' ·Yn· ")' Then Lab 1 b 2 ... bn = ( ... aa ... b 1 ... b 2 ...... bn'") and LLb 1 b 2 ... bn = ( ... LL ... b 1 ... b ...... bn ... ), 2 so that if 'M' is a temporary abbreviation for 'LL', then M has the required properties, that is Mblb2'''bn = ( ••• M.•• bl'''b2''''''bn···)· For example, suppose that we wish M to be such that Mb = [M =b). We let 'L' be a temporary abbreviation for '[x][y] [xx = y]', so that Lab= [aa =b) and LLb = [LL =b), and we let 'M' be a temporary abbreviation for 'LL', so that Mb = [M =b). Notice, in this example, that 'M' denotes a class that has itself as its only member. This is because 'Mb' asserts that b is a member of M, while 'M = b' asserts that M is identical with b. Since Mb = [M = b), it follows that to be a member of M is to be identical with M, so that the one and only member of M must be M itself. As another example, M can be defined in such a way that M = -M. Here 'L' would serve as a temporary abbreviation for '[x](-(xx))'. Notice that in this case there are no 'b ', ••• , 'bn' and 1 no 'y 1 ', .•• , 'yn'• since n = o. (In the previous example, n = 1.) From the way 'L' is defined, we get La= -(aa) and so LL = -(LL), and since 'M' is a temporary abbreviation
SELF-REFERENTIALLY DEFINED FUNCTIONS
125
for 'LL', we have M = -M, as required. In this example, 'L' denotes the class of classes that are not members of themselves, and so gives rise to the famous Russell Paradox according to which L is a member of itself if and only if it is not a member of itself. No contradiction arises from this situation in the system Q, since M as defined here does not satisfy excluded middle. The Russell Paradox, in a slightly different formulation, has already been discussed in 9.11, and something very much like it appears in 9.12 and again in 10.10. Notice that 'R', as defined in 9.12 and 10.10, illustrates, as another example, the general method for defining self-referentially defined functions. 17.2. It is also possible to define two functions M 1 and M2 such that: Ml bl b2 .•• bn = ( ••• Ml •. •142 •.• bl ••• b2 •••••• bn • • • ) 142 bl b2. • • bn = ( ' • 'Ml • ' '142' ' 'bl' ' 'b2' ' • • ' 'bn' ' ' ) In this case we let 'L 1 ' be a temporary abbreviation for [xl] [x2] [yl ][y2]' • • [y n](' · 'xl xl x2. · .x2xl x2 • · •Y1•' •Y2' • • ... yn ... ) and we let 'L 2 ' be a temporary abbreviation for [xl)[x2][yl)[y2] ••• [yn](,,,xlxlx2'''x2xlx2'''yl'''y2''' Then:
•••Yn''')
Llala2blb2'''bn ~ ( •.• al al a2 .• • a2al a2. • . bl • • • b2 • • • • • • bn • • • ) L2ala2blb2'''bn = (,,,alala2'''a2ala2'''bl'''b2''''''bn''') so that Ll Ll L2bl b2. • • bn ( .•• Ll Ll L2' • ·L2LlL2 •. • bl •. • b2' • • • • • bn • • •) L2LlL2blb2'''bn (,,,LlLlL2'''L2LlL2'''bl'''b2''''''bn''') Then, by treating 'M ' as a temporary abbreviation for 1 'L 1L1L2 ' and 'M 2 ' as a temporary abbreviation for 'L L L ', 2 1 2 we see that M1 and M2 have the required properties. As an example, it is possible to define two classes each of which is the only member of the other. In this case n = 1, and we let 'Ll' be an abbreviation for '[x JCx J[y][x x x y] ', 1 2 2 1 2 and 'L 2 ' be an abbreviation for '[x J[x J[y][x x x = y]'. 1 2 1 1 2
124
ELEMENTS OF COMBINATORY LOGIC 17.
~-Referentialll ~efined Eunct!on~
17.1. The method used to define H in 15.11 is a special case of a general method for defining a Q-function M such that Mb 1 b 2 ... bn = ( ... M... b 1 ... b 2 ...... bn ... ), where ' ( ... M ••• b 1 ••• b 2 •••••• bn···)' represents any context involving 'M' itself as well as 'b 1 ', 'b 2 ', ••• , 'bn' and standing for the value of M when Misapplied to b , b , ••• , bn as an 2 1 n-place function. Such a Q-function M is ~-refereptia~~l defined in the sense that the value specified for the function involves reference to the function M itself, as well as reference to the entities to which the function is applied. The general method for defining such a function is as follows: First let 'L' be a temporary abbreviation for [x][yl J[y2] · '· [yn] ( · '.xx.' •Y1• · ·Y2• · ·'' ·Yn· ")' Then Lab 1 b 2 ... bn = ( ... aa ... b 1 ... b 2 ...... bn'") and LLb 1 b 2 ... bn = ( ... LL ... b 1 ... b ...... bn ... ), 2 so that if 'M' is a temporary abbreviation for 'LL', then M has the required properties, that is Mblb2'''bn = ( ••• M.•• bl'''b2''''''bn···)· For example, suppose that we wish M to be such that Mb = [M =b). We let 'L' be a temporary abbreviation for '[x][y] [xx = y]', so that Lab= [aa =b) and LLb = [LL =b), and we let 'M' be a temporary abbreviation for 'LL', so that Mb = [M =b). Notice, in this example, that 'M' denotes a class that has itself as its only member. This is because 'Mb' asserts that b is a member of M, while 'M = b' asserts that M is identical with b. Since Mb = [M = b), it follows that to be a member of M is to be identical with M, so that the one and only member of M must be M itself. As another example, M can be defined in such a way that M = -M. Here 'L' would serve as a temporary abbreviation for '[x](-(xx))'. Notice that in this case there are no 'b ', ••• , 'bn' and 1 no 'y 1 ', .•• , 'yn'• since n = o. (In the previous example, n = 1.) From the way 'L' is defined, we get La= -(aa) and so LL = -(LL), and since 'M' is a temporary abbreviation
SELF-REFERENTIALLY DEFINED FUNCTIONS
125
for 'LL', we have M = -M, as required. In this example, 'L' denotes the class of classes that are not members of themselves, and so gives rise to the famous Russell Paradox according to which L is a member of itself if and only if it is not a member of itself. No contradiction arises from this situation in the system Q, since M as defined here does not satisfy excluded middle. The Russell Paradox, in a slightly different formulation, has already been discussed in 9.11, and something very much like it appears in 9.12 and again in 10.10. Notice that 'R', as defined in 9.12 and 10.10, illustrates, as another example, the general method for defining self-referentially defined functions. 17.2. It is also possible to define two functions M 1 and M2 such that: Ml bl b2 .•• bn = ( ••• Ml •. •142 •.• bl ••• b2 •••••• bn • • • ) 142 bl b2. • • bn = ( ' • 'Ml • ' '142' ' 'bl' ' 'b2' ' • • ' 'bn' ' ' ) In this case we let 'L 1 ' be a temporary abbreviation for [xl] [x2] [yl ][y2]' • • [y n](' · 'xl xl x2. · .x2xl x2 • · •Y1•' •Y2' • • ... yn ... ) and we let 'L 2 ' be a temporary abbreviation for [xl)[x2][yl)[y2] ••• [yn](,,,xlxlx2'''x2xlx2'''yl'''y2''' Then:
•••Yn''')
Llala2blb2'''bn ~ ( •.• al al a2 .• • a2al a2. • . bl • • • b2 • • • • • • bn • • • ) L2ala2blb2'''bn = (,,,alala2'''a2ala2'''bl'''b2''''''bn''') so that Ll Ll L2bl b2. • • bn ( .•• Ll Ll L2' • ·L2LlL2 •. • bl •. • b2' • • • • • bn • • •) L2LlL2blb2'''bn (,,,LlLlL2'''L2LlL2'''bl'''b2''''''bn''') Then, by treating 'M ' as a temporary abbreviation for 1 'L 1L1L2 ' and 'M 2 ' as a temporary abbreviation for 'L L L ', 2 1 2 we see that M1 and M2 have the required properties. As an example, it is possible to define two classes each of which is the only member of the other. In this case n = 1, and we let 'Ll' be an abbreviation for '[x JCx J[y][x x x y] ', 1 2 2 1 2 and 'L 2 ' be an abbreviation for '[x J[x J[y][x x x = y]'. 1 2 1 1 2
126
ELEMENTS OF COMBINATORY LOGIC
Notice that 'x 1 x 1x 2 • is not mentioned in the definition of 'L 1 ' and that 'x 2x 1 x 2 • is not mentioned in the definition of of 'L 2 '. Such omissions are of course allowed. We then have L1 a 1 a 2 b = [a 2 a 1 a 2 =b) and L2 a 1 a 2 b = [a a a 2 =b), and so 1 1 L1L1L2 b = [L 2L1L2 =b) and 1 2L1L2 b = [L 1L L2 =b). Letting 1 'M1 ' be an abbreviation for 'L 11 1 2 • and letting •u • be an 1 2 abbreviation for 'L L L ' we get M b = [M2 =b) and u b 2 1 2 2 1 = [M 1 =b). Thus, to be a member of u is to be identical 1 with M2 , and to be a member of M2 is to be identical with u . 1 17.3. EXERCISES. (1). Given a Q-formula 'a', define 'M' in such a way that M = aM (2). Define Q-formulas 'M 1 ' and •u 2 • such that M 1 is the only member of u2 , while M1 and u2 are the only members of u2 • (3). Define 'M 1 ' and •u • in such a way that 2 'a M1 b' is provable if and only if 'a= M ' and 'b = u2 • 1 are both provable, while 'a M2 b' is provable if and only if 'a = M2 ' and 'b = M ' are both provable. 1 17.4. In ~eneralizing the method of 17.2 to three functions, M1 , M2 , and u , the following three equations would 3 have to be satisfied: M1 b 1 b 2 .•• bn ( ••• u1 •.. M2 •.. M ••• b ••• b 2 •••..• bn .•• ) 1 3 M2blb2' • • bn = ( , ' ,Ml'' ,M2', ,M3'"bl' ',b2'','' ,bn' , ) M3blb2'''bn = (---Ml---M2---M3---bl---b2------bn---) This can be done by treating 'Mi' (for i = 1,2,3) as a tempoary abbreviation for 'LiL 1L2L ', where 'L ', 'L 2 ', and '1 • 1 3 3 are themselves temporary abbreviations respectively for [xl] [x2J[ x3] [yl J[y2] ... [yn l( .. • xl xl x2x3 • •. x2xl x2x3 •. • x3xlx2x3. • ·Yl•. •Y2· • •. • ·Yn·. •) [xl] [x2] [x3] [y 1] (y 2] .. • [y n J (' ' 'xl xl x2x3' ' 'x2xl x2x3' '' x3xlx2x3'''Yl'''Y2''''''Yn''') [xl][x2][x3][yl][y2] ... [yn](---xlxlx2x3---x2xlx2x3--x3xlx2x3---yl---y2------yn---) The cases for four or more (but finitely many) self-referen-
SELF-REFERENTIALLY DEFINED FUNCTIONS
127
tially defined functions are easily constructed by analogy with the cases for three and less. 17.5. EXERCISES. (1). Show that M1 , M2 , and u in 17.4 have the 3 intended properties. (2). Define three classes such that the first is the only member of the second. the second is the only member of the third, and the first and third are the only members of the first. (3). Define four classes such that the first and second are the only members of the third, the second and third are the only members of the fourth, the third and fourth are the only members of the first, and the fourth and first are the only members of the second. 17.6. Using the method of definition described in 17.1, an operator 'Z' will now be defined such that Zf = f(Zf) This identity will be called the Z-identitx (Z id). We let 'L' be a temporary abbreviation for '[x)[y](y(xxy))', so that Laf = f(aaf) and LLf = f(LLf). Then if 'Z' is viewed as an abbreviation for '1L', the Z-identity is obtained. If 'f' is '-', we have Z- = -(Z-) by the Z-identity. This gives an effect very much like that of the Russell Paradox since the Q-formula 'Z-' is equal to its own denial. 17.7. EXERCISES. (1). Another method for defining the 'M' of 17.1 is by letting 'M' be an abbreviation for 'Zf', where 'f' itself i< a temporary abbreviation for [u][yl] [y2] .. · [yn]( · · · u. • •Y1 · · ·Y2" • • • •Y n' • ·) Show that this method gives correct results. (2). Define •z 1 , 2 • and •z , • such that 2 2 z1 , 2 fg = f(Z 1 , 2 fg)(z 2 , 2 fg) z2,2fg = g(Zl,2fg)(Z2,2fg) using methods of 17.2. (3). Define 'M 1 ' and •u2 • of 17.2 in a way that makes use of the properties of •z , • and •z , • of ex. (2). 1 2 2 2
126
ELEMENTS OF COMBINATORY LOGIC
Notice that 'x 1 x 1x 2 • is not mentioned in the definition of 'L 1 ' and that 'x 2x 1 x 2 • is not mentioned in the definition of of 'L 2 '. Such omissions are of course allowed. We then have L1 a 1 a 2 b = [a 2 a 1 a 2 =b) and L2 a 1 a 2 b = [a a a 2 =b), and so 1 1 L1L1L2 b = [L 2L1L2 =b) and 1 2L1L2 b = [L 1L L2 =b). Letting 1 'M1 ' be an abbreviation for 'L 11 1 2 • and letting •u • be an 1 2 abbreviation for 'L L L ' we get M b = [M2 =b) and u b 2 1 2 2 1 = [M 1 =b). Thus, to be a member of u is to be identical 1 with M2 , and to be a member of M2 is to be identical with u . 1 17.3. EXERCISES. (1). Given a Q-formula 'a', define 'M' in such a way that M = aM (2). Define Q-formulas 'M 1 ' and •u 2 • such that M 1 is the only member of u2 , while M1 and u2 are the only members of u2 • (3). Define 'M 1 ' and •u • in such a way that 2 'a M1 b' is provable if and only if 'a= M ' and 'b = u2 • 1 are both provable, while 'a M2 b' is provable if and only if 'a = M2 ' and 'b = M ' are both provable. 1 17.4. In ~eneralizing the method of 17.2 to three functions, M1 , M2 , and u , the following three equations would 3 have to be satisfied: M1 b 1 b 2 .•• bn ( ••• u1 •.. M2 •.. M ••• b ••• b 2 •••..• bn .•• ) 1 3 M2blb2' • • bn = ( , ' ,Ml'' ,M2', ,M3'"bl' ',b2'','' ,bn' , ) M3blb2'''bn = (---Ml---M2---M3---bl---b2------bn---) This can be done by treating 'Mi' (for i = 1,2,3) as a tempoary abbreviation for 'LiL 1L2L ', where 'L ', 'L 2 ', and '1 • 1 3 3 are themselves temporary abbreviations respectively for [xl] [x2J[ x3] [yl J[y2] ... [yn l( .. • xl xl x2x3 • •. x2xl x2x3 •. • x3xlx2x3. • ·Yl•. •Y2· • •. • ·Yn·. •) [xl] [x2] [x3] [y 1] (y 2] .. • [y n J (' ' 'xl xl x2x3' ' 'x2xl x2x3' '' x3xlx2x3'''Yl'''Y2''''''Yn''') [xl][x2][x3][yl][y2] ... [yn](---xlxlx2x3---x2xlx2x3--x3xlx2x3---yl---y2------yn---) The cases for four or more (but finitely many) self-referen-
SELF-REFERENTIALLY DEFINED FUNCTIONS
127
tially defined functions are easily constructed by analogy with the cases for three and less. 17.5. EXERCISES. (1). Show that M1 , M2 , and u in 17.4 have the 3 intended properties. (2). Define three classes such that the first is the only member of the second. the second is the only member of the third, and the first and third are the only members of the first. (3). Define four classes such that the first and second are the only members of the third, the second and third are the only members of the fourth, the third and fourth are the only members of the first, and the fourth and first are the only members of the second. 17.6. Using the method of definition described in 17.1, an operator 'Z' will now be defined such that Zf = f(Zf) This identity will be called the Z-identitx (Z id). We let 'L' be a temporary abbreviation for '[x)[y](y(xxy))', so that Laf = f(aaf) and LLf = f(LLf). Then if 'Z' is viewed as an abbreviation for '1L', the Z-identity is obtained. If 'f' is '-', we have Z- = -(Z-) by the Z-identity. This gives an effect very much like that of the Russell Paradox since the Q-formula 'Z-' is equal to its own denial. 17.7. EXERCISES. (1). Another method for defining the 'M' of 17.1 is by letting 'M' be an abbreviation for 'Zf', where 'f' itself i< a temporary abbreviation for [u][yl] [y2] .. · [yn]( · · · u. • •Y1 · · ·Y2" • • • •Y n' • ·) Show that this method gives correct results. (2). Define •z 1 , 2 • and •z , • such that 2 2 z1 , 2 fg = f(Z 1 , 2 fg)(z 2 , 2 fg) z2,2fg = g(Zl,2fg)(Z2,2fg) using methods of 17.2. (3). Define 'M 1 ' and •u2 • of 17.2 in a way that makes use of the properties of •z , • and •z , • of ex. (2). 1 2 2 2
ELEMENTS OF COMBINATORY LOGIC
128
(4), Using the method of 17.4 and extensions of it to four or more functions, show that it is possible to define operators 'Z.J.,n 'for i=l, 2, .•• , n such that zi,nflf2'''fn= fi(zl,nflf2'''fn)(Z2,nflf2'''fn) .• , (Zn,nflf2'''fn) (5) • Sh ow h ow th e opera t ors 'Z l,~ ' , 'Z ,~ ' , and 2 'Z ' defined in ex. (4) can be used to define 'M ', •y •, ~.~ 1 2 and 'M ' of 17.4. ~
18.
Se~uential
Circuits
18.1. Use will now be made of combinatory logic in dealing with a type of circuit that has some importance in computer theory, the seguential circuit· The essential properties of sequential circuits will be described in the following paragraphs. Hereafter sequential circuits will be referred to simply as "circuits." Every sequential circuit has finitely many inputs and one output. It will be shown that BiC represents the process of interchanging the i+lst and the i+2nd inputs of a circuit, that BiW represents the process of fusing thA i+lst with the i+2nd input to form a single input, that BiZ represents the process of feeding a branch off the output of an i-input circuit into its own last input, and that Bi represents the process of forming a circuit by feeding the output of an i-input circuit into the first input of another circuit. Also, BiK represents the trivial process of introducing an inoperative input between the ith and the i+ist inputs. Thus BiC, BiW, BiZ, and BiK represent transformations of circuits into circuits, and Bi represents a transformation of a pair of circuits into a circuit. All sequential circuits may be viewed as constructed from certain elementary sequential circuits (to be described below) by means of these transformations. 18.2. The class of elfmenta~ seguent~al citcyit€ consists of the and-elements, or-elements, nand-elements, nor-elements, not-elements, and delay elements described in 18.3-18.18. The class of ~egyential circuits consists of all
SEQUENTIAL CIRCUITS
129
elementary sequential circuits together with circuits that result from joining together or transforming sequential circuits by the processes described in 18.1. It is to be understood that every sequential circuit has exactly one output and zero or more inputs (always finite in number). Branches from the one output, however, may be fed into inputs of the circuit and into inputs of other circuits. Strictly speaking, the output, or a branch from the output, may be fed only into the last input of the circuit itself or into the first input of another circuit (according to the processes allowed by 18.1), but this limitation is indirectly overcome by the fact that inputs may be permuted into a different order by repeated uses of the transformations of the kind represented by B1c. Thus, in effect, feeding the output, or a branch from the output of any sequential circuit into any input of another sequential circuit produces a new sequential circuit, and feeding the output, or a branch from the output of any sequential circuit into an input of that same sequential circuit produces a new sequential circuit. 18.~. An !E£-eleme~t is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if all its inputs are activated at that time. If there are n inputs and if a class ai of natural numbers represents the times (in seconds or other specified units of time) at which the ith input is activated (i = 1, 2, ••• , n), then the intersection of the classes a • a 2 , ••• , an (that is, the class of those num1 bers each of which is a member of all of a 1 , a 2 , ••• , ah) represents the times at which the output is activated. This intersection is expressible in the system Q by a 1 0 a 2 0 ••• o an where 'a 0 b n c' is an abbreviation for '[an b) 0 c' and where 'O' is an abbreviation for '[x)[y)[z][xz d yz)', The equation [a n b]c = ac & be is easily proved and may be called the intersection identit~ (intersect id). It is also possible to define an operator 'n n ' such that
ELEMENTS OF COMBINATORY LOGIC
128
(4), Using the method of 17.4 and extensions of it to four or more functions, show that it is possible to define operators 'Z.J.,n 'for i=l, 2, .•• , n such that zi,nflf2'''fn= fi(zl,nflf2'''fn)(Z2,nflf2'''fn) .• , (Zn,nflf2'''fn) (5) • Sh ow h ow th e opera t ors 'Z l,~ ' , 'Z ,~ ' , and 2 'Z ' defined in ex. (4) can be used to define 'M ', •y •, ~.~ 1 2 and 'M ' of 17.4. ~
18.
Se~uential
Circuits
18.1. Use will now be made of combinatory logic in dealing with a type of circuit that has some importance in computer theory, the seguential circuit· The essential properties of sequential circuits will be described in the following paragraphs. Hereafter sequential circuits will be referred to simply as "circuits." Every sequential circuit has finitely many inputs and one output. It will be shown that BiC represents the process of interchanging the i+lst and the i+2nd inputs of a circuit, that BiW represents the process of fusing thA i+lst with the i+2nd input to form a single input, that BiZ represents the process of feeding a branch off the output of an i-input circuit into its own last input, and that Bi represents the process of forming a circuit by feeding the output of an i-input circuit into the first input of another circuit. Also, BiK represents the trivial process of introducing an inoperative input between the ith and the i+ist inputs. Thus BiC, BiW, BiZ, and BiK represent transformations of circuits into circuits, and Bi represents a transformation of a pair of circuits into a circuit. All sequential circuits may be viewed as constructed from certain elementary sequential circuits (to be described below) by means of these transformations. 18.2. The class of elfmenta~ seguent~al citcyit€ consists of the and-elements, or-elements, nand-elements, nor-elements, not-elements, and delay elements described in 18.3-18.18. The class of ~egyential circuits consists of all
SEQUENTIAL CIRCUITS
129
elementary sequential circuits together with circuits that result from joining together or transforming sequential circuits by the processes described in 18.1. It is to be understood that every sequential circuit has exactly one output and zero or more inputs (always finite in number). Branches from the one output, however, may be fed into inputs of the circuit and into inputs of other circuits. Strictly speaking, the output, or a branch from the output, may be fed only into the last input of the circuit itself or into the first input of another circuit (according to the processes allowed by 18.1), but this limitation is indirectly overcome by the fact that inputs may be permuted into a different order by repeated uses of the transformations of the kind represented by B1c. Thus, in effect, feeding the output, or a branch from the output of any sequential circuit into any input of another sequential circuit produces a new sequential circuit, and feeding the output, or a branch from the output of any sequential circuit into an input of that same sequential circuit produces a new sequential circuit. 18.~. An !E£-eleme~t is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if all its inputs are activated at that time. If there are n inputs and if a class ai of natural numbers represents the times (in seconds or other specified units of time) at which the ith input is activated (i = 1, 2, ••• , n), then the intersection of the classes a • a 2 , ••• , an (that is, the class of those num1 bers each of which is a member of all of a 1 , a 2 , ••• , ah) represents the times at which the output is activated. This intersection is expressible in the system Q by a 1 0 a 2 0 ••• o an where 'a 0 b n c' is an abbreviation for '[an b) 0 c' and where 'O' is an abbreviation for '[x)[y)[z][xz d yz)', The equation [a n b]c = ac & be is easily proved and may be called the intersection identit~ (intersect id). It is also possible to define an operator 'n n ' such that
130
ELEMENTS OF COMBINATORY LOGIC
nna1 a 2 ••• an = a 1 n a 2 n ••• nan In fact, the functional dependence of output on input in the case of an n-input and-element is expressed in the system Q by the Q-function nn. For example, if n = 3 and if a , a , 2 1 and a are respectively [1,2,3}, [1,3,4-}, and [1,2,3,5} 3 (that is, a 1 has just 1, 2, and 3 as members, a has just 2 1, 3, and 4- as members, and a has just 1, 2, 3, and 5 as 3 members), then n a 1a 2a (or, equivalently, a n a n a ) 3 3 1 2 3 would be (1,3} (that is, would have just 1 and 3 as members). Here, in general, '[c 1 ,c 2 , ... ,cn}' may be regarded as an abbreviation for '[x][[x = c 1 ] V [x = c 2 ] V ••. V [x =en]]', so that (c 1 ,c 2 , ••• ,cn} is the class having exactly c , c , 1 2 , en as its members. The and-element of the above example can be diagrammed thus:
--------u ------+ (
((1,2.3} 1' 3,4-} -------(1,2,3,5} ------
1, 3}
n3
The inputs are shown on the left and the output is shown on the right. In an actual and-element the output is activated very slightly later than the inputs rather than at exactly the same time. This difference is so small, however, especially as compared with the delay caused by a delay element, that it can be ignored for theoretical purposes in most cases. The above diagram would have the same meaning if presented as follows: 001110 011010 101110
=========n-------+
001010
n3
Here, reading the digits from right to left, '011010' means activation at times 1, 3, and 4- (and nonactivation at times O, 2 and times ?: 5), just as '[1, 3,4-}' does in the other diagram.
131
SEQUENTIAL CIRCUITS
18.4-. An £E-element is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if at least one of its inputs is activated at that time. If there are n inputs and if a class a 1 of natural numbers represents the times at which the ith input is activated (i = 1, 2, .•. , n), then the union of a 1 , a 2 , . . . . an (that is, the class of those numbers each of which is a member of at least one of a , a 2 , ••• , 1 an) represents the times at which the output is activated. This union is expressible in the system Q by a 1 U a 2 U .•• U an where 'au b U c' is an abbreviation for '[aU b] u c' and where 'U' is an abbreviation for '[x][y)[z][xz V yz]'. The equation [a U b]c = ac V be is easily proved and may be called the ~ ident;j, t:~: (union id). It is also possible to define an operator 'Un' such that una 1 a 2 •.• an = a 1 u a 2 u •.• u an The functional dependence of output on input in the case of an n-input or-element is expressed in the system Q by the Qfunction un. For example we would have u4 (1}(1,3}(2}(6,5} (1,2,3,5,6}. The corresponding diagram would be:
-----------n --------+ (
(1] 3} --------{1 (2} -----------
'
1 '2' 3' 5' 6}
(6,5} --------u418.5. A ~-element is an elementary sequential circuit having just one input and such that the output is activated at a given time if and only if the input is not activated at that time. If a class a of natural numbers represents the times at which the input is activated, then the complement of a within the natural numbers (that is, the class of all natural numbers not in a ) represents the times at which the output is activated. This class is expressible in the
130
ELEMENTS OF COMBINATORY LOGIC
nna1 a 2 ••• an = a 1 n a 2 n ••• nan In fact, the functional dependence of output on input in the case of an n-input and-element is expressed in the system Q by the Q-function nn. For example, if n = 3 and if a , a , 2 1 and a are respectively [1,2,3}, [1,3,4-}, and [1,2,3,5} 3 (that is, a 1 has just 1, 2, and 3 as members, a has just 2 1, 3, and 4- as members, and a has just 1, 2, 3, and 5 as 3 members), then n a 1a 2a (or, equivalently, a n a n a ) 3 3 1 2 3 would be (1,3} (that is, would have just 1 and 3 as members). Here, in general, '[c 1 ,c 2 , ... ,cn}' may be regarded as an abbreviation for '[x][[x = c 1 ] V [x = c 2 ] V ••. V [x =en]]', so that (c 1 ,c 2 , ••• ,cn} is the class having exactly c , c , 1 2 , en as its members. The and-element of the above example can be diagrammed thus:
--------u ------+ (
((1,2.3} 1' 3,4-} -------(1,2,3,5} ------
1, 3}
n3
The inputs are shown on the left and the output is shown on the right. In an actual and-element the output is activated very slightly later than the inputs rather than at exactly the same time. This difference is so small, however, especially as compared with the delay caused by a delay element, that it can be ignored for theoretical purposes in most cases. The above diagram would have the same meaning if presented as follows: 001110 011010 101110
=========n-------+
001010
n3
Here, reading the digits from right to left, '011010' means activation at times 1, 3, and 4- (and nonactivation at times O, 2 and times ?: 5), just as '[1, 3,4-}' does in the other diagram.
131
SEQUENTIAL CIRCUITS
18.4-. An £E-element is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if at least one of its inputs is activated at that time. If there are n inputs and if a class a 1 of natural numbers represents the times at which the ith input is activated (i = 1, 2, .•. , n), then the union of a 1 , a 2 , . . . . an (that is, the class of those numbers each of which is a member of at least one of a , a 2 , ••• , 1 an) represents the times at which the output is activated. This union is expressible in the system Q by a 1 U a 2 U .•• U an where 'au b U c' is an abbreviation for '[aU b] u c' and where 'U' is an abbreviation for '[x][y)[z][xz V yz]'. The equation [a U b]c = ac V be is easily proved and may be called the ~ ident;j, t:~: (union id). It is also possible to define an operator 'Un' such that una 1 a 2 •.• an = a 1 u a 2 u •.• u an The functional dependence of output on input in the case of an n-input or-element is expressed in the system Q by the Qfunction un. For example we would have u4 (1}(1,3}(2}(6,5} (1,2,3,5,6}. The corresponding diagram would be:
-----------n --------+ (
(1] 3} --------{1 (2} -----------
'
1 '2' 3' 5' 6}
(6,5} --------u418.5. A ~-element is an elementary sequential circuit having just one input and such that the output is activated at a given time if and only if the input is not activated at that time. If a class a of natural numbers represents the times at which the input is activated, then the complement of a within the natural numbers (that is, the class of all natural numbers not in a ) represents the times at which the output is activated. This class is expressible in the
132
ELEMENTS OF COMBINATORY LOGIC
system Q by '..,a', where '..,• is an abbreviation for the Q-formula '[x][Nx &. -(ax)]'. The identity .., ab = [Nb &. -Cab)] is then provable. The functional dependence of output on input in the case of a not-element is expressed in the system Q by the Q-function .., • For example, i f the input of a notelement is activated at times 1 and 3, then the output is activated at all times other than 1 and 3, that is, at times 0, 2, 4, 5, 6, •••• The diagram would be: [ 1, 3}
--------0 ---------~ . , (
SEQUENTIAL CIRCUITS
are the input classes. It is possible to define an operator 'u n ' such that Una 1 a 2 •.• an = -,[a 1 U a 2 u ••• u an] The functional dependence of output on input in the case of an n-input nor-element is expressed in the system Q by the Qfunction Un. For example we would have u3 n.3H2H5,6l = ..,£1,2,3,5,6l The corresponding diagram would be
(1.3}
1, 3}
133
------n
[2} --------
------~
.., [1,2,3,5,6}
[5,6) -----18.6. A ~-element is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if not all of its inputs are activated at that time. The output class is expressible in the system Q by •.., [a 1 n a 2 n ••• n an]', where al' a , 2 ••• , an are the input classes. It is possible to define an operator 'lin' such that lina 1a 2 ••• an = -,[a 1 n a 2 n ••• nan] The functional dependence of output on input in the case of an n-input nand-element is expressed in the system Q by the Q-function nn. For example we would have l'i3(1,2.3](1,3,4}(1,2,3,5l
------n
2
..,
(1.3}
The corresponding diagram would be:
{1.2.3} (1,3,4} -----(1,2,3,5} ----
------~..,
[1,3}
lin
18.7. A ~-elem~ is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if none of its input is activated at that time. The output class is expressible in the system Q by •.., [a 1 u a 2 u •.. u an]', where a , a , .•• , an 2 1
18.8. Tt should be clear that the effect of a nand-element can be produced by feeding the output of an and-element into the input of a not-element, and that the effect of a nor-element can be produced by feeding the output of an orelement into the input of a not-element.
18.9. EXERCISES. (l). Formulate introduction and elimination rules for n and nn· (2). Formulate introduction and elimination rules for u and un. (3). Prove a n [b u c] = [a n bJ u [a n c] (4). Prove -,a = -,-,-,a -,a u -,b ( 5). Prove -,[a n b] ( 6). Prove -,[a u b] = -,a n -,b 18.10. A ~e,y ili~.n.:t is an elementary sequential circuit having ,just one input and such that the output is activated at a given time if and only if the input is activated at one unit of time earlier. where some arbitrarily chosen interval of time has been designated as a unit. Thus if a is a class of natural numbers that represents the times at which the input is activated, then a class of natural numt:ers obtained by adding 1 to each member of a r~presents
132
ELEMENTS OF COMBINATORY LOGIC
system Q by '..,a', where '..,• is an abbreviation for the Q-formula '[x][Nx &. -(ax)]'. The identity .., ab = [Nb &. -Cab)] is then provable. The functional dependence of output on input in the case of a not-element is expressed in the system Q by the Q-function .., • For example, i f the input of a notelement is activated at times 1 and 3, then the output is activated at all times other than 1 and 3, that is, at times 0, 2, 4, 5, 6, •••• The diagram would be: [ 1, 3}
--------0 ---------~ . , (
SEQUENTIAL CIRCUITS
are the input classes. It is possible to define an operator 'u n ' such that Una 1 a 2 •.• an = -,[a 1 U a 2 u ••• u an] The functional dependence of output on input in the case of an n-input nor-element is expressed in the system Q by the Qfunction Un. For example we would have u3 n.3H2H5,6l = ..,£1,2,3,5,6l The corresponding diagram would be
(1.3}
1, 3}
133
------n
[2} --------
------~
.., [1,2,3,5,6}
[5,6) -----18.6. A ~-element is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if not all of its inputs are activated at that time. The output class is expressible in the system Q by •.., [a 1 n a 2 n ••• n an]', where al' a , 2 ••• , an are the input classes. It is possible to define an operator 'lin' such that lina 1a 2 ••• an = -,[a 1 n a 2 n ••• nan] The functional dependence of output on input in the case of an n-input nand-element is expressed in the system Q by the Q-function nn. For example we would have l'i3(1,2.3](1,3,4}(1,2,3,5l
------n
2
..,
(1.3}
The corresponding diagram would be:
{1.2.3} (1,3,4} -----(1,2,3,5} ----
------~..,
[1,3}
lin
18.7. A ~-elem~ is an elementary sequential circuit having two or more inputs and such that its output is activated at a given time if and only if none of its input is activated at that time. The output class is expressible in the system Q by •.., [a 1 u a 2 u •.. u an]', where a , a , .•• , an 2 1
18.8. Tt should be clear that the effect of a nand-element can be produced by feeding the output of an and-element into the input of a not-element, and that the effect of a nor-element can be produced by feeding the output of an orelement into the input of a not-element.
18.9. EXERCISES. (l). Formulate introduction and elimination rules for n and nn· (2). Formulate introduction and elimination rules for u and un. (3). Prove a n [b u c] = [a n bJ u [a n c] (4). Prove -,a = -,-,-,a -,a u -,b ( 5). Prove -,[a n b] ( 6). Prove -,[a u b] = -,a n -,b 18.10. A ~e,y ili~.n.:t is an elementary sequential circuit having ,just one input and such that the output is activated at a given time if and only if the input is activated at one unit of time earlier. where some arbitrarily chosen interval of time has been designated as a unit. Thus if a is a class of natural numbers that represents the times at which the input is activated, then a class of natural numt:ers obtained by adding 1 to each member of a r~presents
ELEMENTS OF COMBINATORY LOGIC
134-
the times at which the output is activated. This latter class may be expressed in the system Q by '6a', where '6' is an abbreviation for '[x][y](~z)[xz & [y = z + 1]]'. Thus the functional dependence of output on input in the case of a delay element is expressed in the system Q by the Q-function 6· For example, we would have ~[2 5,8} = [3,6,9}. The correspending diagram would be: [2.5.8}
------n-------1>
SEQUENTIAL CIRCUITS
135
In the following diagram the inner rectangle stands for the original circuit and the outer rectangle stands for the transformed circuit. Here i = 2, and each circuit has five inputs.
--~>
(3.6.9}
6
It should be remarked that in dealing with sequential circuits attention need be paid to what is happening only at times O, 1, 2, 3, 4-, and so on, and not at intermediate times between these. At least this is the case for the theory of so-called slnchronous sequential circuits, the kind being considered here, and it is made possible because of antecedent clockwork control built into such circuits. The structure of such clockwork control is not treated here as part of the abstract or logical theory of synchronous sequential circuits. There are also aslnchronous sequential circuits which lack such clockwork control. The theory of these circuits is slightly different and more complicated. 18.11. Four ways of transforming sequential circuits will now be presented, corresponding to the Q-functions BiC, BiW, BiZ, and BiK, and one way of combining two sequential circuits to form a new sequential circuit, corresponding to the Q-function Bi. By repeated use of these ways of transforming and combining, all sequential circuits can be built from elementary sequential circuits. 18.12. The first of the four basic ways of transforming a sequential circuit is to interchange the i+lst and the i+2nd inputs (i=O, 1, 2, 3, ... ) in the case of circuits having at least i+2 inputs. If the functional dependence of output on input is expressed by f for the original circuit, it will be expressed by B1cf for the transformed circuit.
=
B2 Cfa a 2 a a 4 a 1 3 5 B(BC)fa a 2 a a 4 a 1 3 5 BC(fa1 )a2 a a 4 a 3 5 C( fa 1 a 2 ) a a 4 a 3 5 fa a 2 a 4 a a 1 3 5
By repeated use of transformations of this kind, inputs may be rearranged in any desired order. 18.13. The second of the four basic ways of transforming a sequential circuit is to fuse into a single input the l+lst and the i+2nd inputs in the case of a sequential circuit having at least i+2 inputs. If the functional dependence of output on input is expressed by f for the original circuit, it will be expressed by Biwf for the transformed circuit. Here is a diagram of an example:
f
-+ B2wfa a 2 a a4 1 3 B(BW)fa1 a 2 a a 4 3 BW(fa )a a a 1 2 3 4 W(fa 1 a 2 )a a4 3 = fa 1 a 2 a a a4
3 3
Here the original circuit is a 5-input circuit and the trans-
ELEMENTS OF COMBINATORY LOGIC
134-
the times at which the output is activated. This latter class may be expressed in the system Q by '6a', where '6' is an abbreviation for '[x][y](~z)[xz & [y = z + 1]]'. Thus the functional dependence of output on input in the case of a delay element is expressed in the system Q by the Q-function 6· For example, we would have ~[2 5,8} = [3,6,9}. The correspending diagram would be: [2.5.8}
------n-------1>
SEQUENTIAL CIRCUITS
135
In the following diagram the inner rectangle stands for the original circuit and the outer rectangle stands for the transformed circuit. Here i = 2, and each circuit has five inputs.
--~>
(3.6.9}
6
It should be remarked that in dealing with sequential circuits attention need be paid to what is happening only at times O, 1, 2, 3, 4-, and so on, and not at intermediate times between these. At least this is the case for the theory of so-called slnchronous sequential circuits, the kind being considered here, and it is made possible because of antecedent clockwork control built into such circuits. The structure of such clockwork control is not treated here as part of the abstract or logical theory of synchronous sequential circuits. There are also aslnchronous sequential circuits which lack such clockwork control. The theory of these circuits is slightly different and more complicated. 18.11. Four ways of transforming sequential circuits will now be presented, corresponding to the Q-functions BiC, BiW, BiZ, and BiK, and one way of combining two sequential circuits to form a new sequential circuit, corresponding to the Q-function Bi. By repeated use of these ways of transforming and combining, all sequential circuits can be built from elementary sequential circuits. 18.12. The first of the four basic ways of transforming a sequential circuit is to interchange the i+lst and the i+2nd inputs (i=O, 1, 2, 3, ... ) in the case of circuits having at least i+2 inputs. If the functional dependence of output on input is expressed by f for the original circuit, it will be expressed by B1cf for the transformed circuit.
=
B2 Cfa a 2 a a 4 a 1 3 5 B(BC)fa a 2 a a 4 a 1 3 5 BC(fa1 )a2 a a 4 a 3 5 C( fa 1 a 2 ) a a 4 a 3 5 fa a 2 a 4 a a 1 3 5
By repeated use of transformations of this kind, inputs may be rearranged in any desired order. 18.13. The second of the four basic ways of transforming a sequential circuit is to fuse into a single input the l+lst and the i+2nd inputs in the case of a sequential circuit having at least i+2 inputs. If the functional dependence of output on input is expressed by f for the original circuit, it will be expressed by Biwf for the transformed circuit. Here is a diagram of an example:
f
-+ B2wfa a 2 a a4 1 3 B(BW)fa1 a 2 a a 4 3 BW(fa )a a a 1 2 3 4 W(fa 1 a 2 )a a4 3 = fa 1 a 2 a a a4
3 3
Here the original circuit is a 5-input circuit and the trans-
136
ELEMENTS OF COMBINATORY LOGIC
formed circuit in a 4-input circuit. As before, the inner rectangle stands for the original circuit and the outer rectangle stands for the transformed circuit. By first rearranging the order of inputs by use of the BiC transformations so that two inputs to be fused are adjacent to each other, any two inputs of a circuit may be fused. Also, by repeated use of the BiW transformations, three or more inputs may be fused. 18.14. The third of the four basic ways of transforming a sequential circuit is to connect a branch from the output into the last input of the circuit, thus producing a "feedback" effect. The resulting circuit has one less input than the original circuit, since what was the last input of the original circuit now serves merely to receive content from the output. This kind of transformation is ordinarily appled only to a circuit that contains a delay element in an appropriate way so that the output will not have an instantaneous effect on itself, but only a delayed effect. Such an instantaneous effect, in any case, would be physically impossible since every real circuit involves at least an extremely small amount of delay which can usually be ignored for theoretical purposes. If the original circuit has i+l inputs and if the functional dependence of output on input for it is expressed by f, then the dependence of output on input for the i-input transformed circuit will be expressed by BiZf. Here is the diagram of an example:
al-~ ------al----n --fa
a 2 -~ ------a2 ----
r-Z(fala2)- I f
a (Z(fa a
1 2
1 2
»-r I
-+B2 Zfa a 1 2 = B(BZ)fa a 1 2 I = BZ(fa 1 )a2 = Z(fa.l a2)
1+---------------------------------·
1 2 (Z(fa 1 ~))
= fa a
Here the original circuit has three inputs and the trans-
SEQUENTIAL CIRCUITS·-----------------------------------1_3__7 formed circuit has two inputs, so i = 2. As before, the inner rectangle stands for the original circuit and the outer rectangle stands for the transformed circuit. If it is desired to feed the output back into some input ot~er than the last input, it is necessary merely to use the Bl.C trans f orma t.J.ons to move the designated input into last position and then perform the present transformation so as to feed the output into that input. 18.15. The fourth of the four basic ways of transforming a sequential circuit is to introduce a non-operative input immediately after the ith input. The activity or non-activity of this input has no effect on the output, but the transformed circuit has one more input than the original circuit. If the functional dependence of output on input is expressed by f for the original circu~t, then for the transformed circuit it is expressed by B1 Kf. In the example diagramed below, the original circuit has four inputs, and an inoperative input is introduced after the third input, so i
=
3.
--+ B3Kfa a a a 4 a 1 2 3 5
f
B(B(BK))fa 1 a 2 a a 4 a 3 5 K(fa 1a 2 a )a4 a 3 5 fa 1a 2 a a 3 5
18.16. The basic way considered here for combining two circuits to form a third circuit is to connect the output of the first circuit with the first input of the second circuit. The inputs of the resulting circuit consist of the inputs of the first circuit in their given order followed, in their given order, by all but the first of the inputs of the sec-
136
ELEMENTS OF COMBINATORY LOGIC
formed circuit in a 4-input circuit. As before, the inner rectangle stands for the original circuit and the outer rectangle stands for the transformed circuit. By first rearranging the order of inputs by use of the BiC transformations so that two inputs to be fused are adjacent to each other, any two inputs of a circuit may be fused. Also, by repeated use of the BiW transformations, three or more inputs may be fused. 18.14. The third of the four basic ways of transforming a sequential circuit is to connect a branch from the output into the last input of the circuit, thus producing a "feedback" effect. The resulting circuit has one less input than the original circuit, since what was the last input of the original circuit now serves merely to receive content from the output. This kind of transformation is ordinarily appled only to a circuit that contains a delay element in an appropriate way so that the output will not have an instantaneous effect on itself, but only a delayed effect. Such an instantaneous effect, in any case, would be physically impossible since every real circuit involves at least an extremely small amount of delay which can usually be ignored for theoretical purposes. If the original circuit has i+l inputs and if the functional dependence of output on input for it is expressed by f, then the dependence of output on input for the i-input transformed circuit will be expressed by BiZf. Here is the diagram of an example:
al-~ ------al----n --fa
a 2 -~ ------a2 ----
r-Z(fala2)- I f
a (Z(fa a
1 2
1 2
»-r I
-+B2 Zfa a 1 2 = B(BZ)fa a 1 2 I = BZ(fa 1 )a2 = Z(fa.l a2)
1+---------------------------------·
1 2 (Z(fa 1 ~))
= fa a
Here the original circuit has three inputs and the trans-
SEQUENTIAL CIRCUITS·-----------------------------------1_3__7 formed circuit has two inputs, so i = 2. As before, the inner rectangle stands for the original circuit and the outer rectangle stands for the transformed circuit. If it is desired to feed the output back into some input ot~er than the last input, it is necessary merely to use the Bl.C trans f orma t.J.ons to move the designated input into last position and then perform the present transformation so as to feed the output into that input. 18.15. The fourth of the four basic ways of transforming a sequential circuit is to introduce a non-operative input immediately after the ith input. The activity or non-activity of this input has no effect on the output, but the transformed circuit has one more input than the original circuit. If the functional dependence of output on input is expressed by f for the original circu~t, then for the transformed circuit it is expressed by B1 Kf. In the example diagramed below, the original circuit has four inputs, and an inoperative input is introduced after the third input, so i
=
3.
--+ B3Kfa a a a 4 a 1 2 3 5
f
B(B(BK))fa 1 a 2 a a 4 a 3 5 K(fa 1a 2 a )a4 a 3 5 fa 1a 2 a a 3 5
18.16. The basic way considered here for combining two circuits to form a third circuit is to connect the output of the first circuit with the first input of the second circuit. The inputs of the resulting circuit consist of the inputs of the first circuit in their given order followed, in their given order, by all but the first of the inputs of the sec-
~LE~TS
OF COMBINATORY LOGIC
ond circuit. If the functional dependence of output on input is expressed by f for the first circuit and by g for the second circuit, then it will be expressed by Bigf for the combined circuit, where i is the number of inputs of the first circuit. In the example given in the following diagram, the first circuit has two inputs and the second circuit has three inputs. The combined circuit has four inputs.
-+B~fa 1 a 2 a 3 a4
~----------------------------~
B(Bg)fa a 2 a a 4 1 3 Bg(fa )a 2 a a4 1 3 • g(fa a 2 )a a 4 1 3
If it is desired to connect the output of the first circuit with some input of the second circuit other than its first input, the procedure is to move the specified input of the second circuit into first position by repeated use of the BiC transformations, and then combine the two circuits in the present way. Notice that if the first of the two circuits is an and-element and the second is a not-element, then the combined circuit has the same properties as a nandelement. Similarly, combining an or-element with a not-element gives the effect of a not-element. Also, combining a nand-element with a not-element gives the effect of an andelement, and combining a nor-element with a not-element gives the effect of an or-element. 18.17. Se'l_uential circuits with more than one output may be included in the account given here provided that each of the outputs is treated as though it was the output of a oneoutput sequential circuit embedded in the multi-output sequential circuit.
SEQUENTIAL CIRCUITS
139
18.18. The remarks made here about sequential circuits have been only a partial introduction to a much more extensive subject. The aim has been to show that this subject does bear a close relationship to combinatory logic. 18.19· EXERCISES. If a circuit has three inputs and if a Q-function f expresses the functional dependence of its output on its inputs, give a Q-formula for a function that expresses the functional dependence of output on input for the circuit that results when the original circuit has been transformed in each of the following ways. (In each case go back to the original circuit as a starting point. Notice that in some cases a succession of transformations may be needed.) (1). Interchange the first two inputs. (2). Interchange the first and third inputs. (3). Fuse the first and second inputs into a single input. (4). Fuse the first and third inputs into a single input. (5). Feed a branch from the output into the third input. (6). Feed a branch from the output into the second input. (7). Introduce a non-operative input between the second and third inputs. (8). Feed the output into the first input of a 2-input circuit for which a Q-function g expresses the functional dependence of output on input. (g). Feed the output into the second input of a 2-input circuit for which a Q-function g expresses the functional dependence of output on input.
~LE~TS
OF COMBINATORY LOGIC
ond circuit. If the functional dependence of output on input is expressed by f for the first circuit and by g for the second circuit, then it will be expressed by Bigf for the combined circuit, where i is the number of inputs of the first circuit. In the example given in the following diagram, the first circuit has two inputs and the second circuit has three inputs. The combined circuit has four inputs.
-+B~fa 1 a 2 a 3 a4
~----------------------------~
B(Bg)fa a 2 a a 4 1 3 Bg(fa )a 2 a a4 1 3 • g(fa a 2 )a a 4 1 3
If it is desired to connect the output of the first circuit with some input of the second circuit other than its first input, the procedure is to move the specified input of the second circuit into first position by repeated use of the BiC transformations, and then combine the two circuits in the present way. Notice that if the first of the two circuits is an and-element and the second is a not-element, then the combined circuit has the same properties as a nandelement. Similarly, combining an or-element with a not-element gives the effect of a not-element. Also, combining a nand-element with a not-element gives the effect of an andelement, and combining a nor-element with a not-element gives the effect of an or-element. 18.17. Se'l_uential circuits with more than one output may be included in the account given here provided that each of the outputs is treated as though it was the output of a oneoutput sequential circuit embedded in the multi-output sequential circuit.
SEQUENTIAL CIRCUITS
139
18.18. The remarks made here about sequential circuits have been only a partial introduction to a much more extensive subject. The aim has been to show that this subject does bear a close relationship to combinatory logic. 18.19· EXERCISES. If a circuit has three inputs and if a Q-function f expresses the functional dependence of its output on its inputs, give a Q-formula for a function that expresses the functional dependence of output on input for the circuit that results when the original circuit has been transformed in each of the following ways. (In each case go back to the original circuit as a starting point. Notice that in some cases a succession of transformations may be needed.) (1). Interchange the first two inputs. (2). Interchange the first and third inputs. (3). Fuse the first and second inputs into a single input. (4). Fuse the first and third inputs into a single input. (5). Feed a branch from the output into the third input. (6). Feed a branch from the output into the second input. (7). Introduce a non-operative input between the second and third inputs. (8). Feed the output into the first input of a 2-input circuit for which a Q-function g expresses the functional dependence of output on input. (g). Feed the output into the second input of a 2-input circuit for which a Q-function g expresses the functional dependence of output on input.
141
THE SYSTEM QD
The effect of this rule is to assert that every proposition satisfies the principle of excluded middle, so that every proposition is true or false. 19.4. D-D-introduction (DD int)
I D(Da)
CHAPTER 6
The effect of this rule is to assert that Da (the proposition that a is a proposition) is a proposition. 19.5. D-identit;z introduction (D id int)
A THEORY OF PHOPOSITIONS
19.
~
I D[a
S;zstem QD
The effect of this rule is to assert that every identity is a proposition. 19.6. D-N-introduction (DN int)
19.1. In order to develop a theory of propositions in as explicit a way as possible, the system QD will now be constructed and presented. This system is an extension of the system Q, that is, it contains all the theorems of Q together with further theorems. In constructing the system QD a new primitive Q-formula 'D' is to be added to the list of primitive Q-formulas (3.4). This new primitive Q-formula denotes the Q-function which is the class of propositions (or the property of being a proposition). In other words, D is the Q-function which is the class of propositions (or the property of being a proposition). Thus the Q-formula 'Da' expresses the proposition that a is a proposition, and the Q-formula 'D(Da)' expresses· the (true) proposition that Da is a proposition, since Da is the proposition that (truly or falsely) asserts that a is a proposition. 19.2. The rules employed in the system QD consist of all the rules employed in the system Q together with additional rules concerned with n. These additional rules may be expressed schematically as follows: 19.3. D-elimination (D elim) LDa a V -a
I D(Na) The effect of this rule is to assert that Na (the proposition that a is a natural number) is a proposition. 19.7. D-£egation introduction (D neg int) LDa D(-a)
I
The effect of this rule is to assert that the negation of a proposition is a proposition. 19.8. D-conjunction introd~gtiQU (D conj int)
I I l
I ~:
l D[a &
14-0
b)
The effect of this rule is to assert that conjunctions of propositions are propositions. 19.9. D-disjunction introduction (D dis int)
~
I
= b)
Da
Db D[a V b)
141
THE SYSTEM QD
The effect of this rule is to assert that every proposition satisfies the principle of excluded middle, so that every proposition is true or false. 19.4. D-D-introduction (DD int)
I D(Da)
CHAPTER 6
The effect of this rule is to assert that Da (the proposition that a is a proposition) is a proposition. 19.5. D-identit;z introduction (D id int)
A THEORY OF PHOPOSITIONS
19.
~
I D[a
S;zstem QD
The effect of this rule is to assert that every identity is a proposition. 19.6. D-N-introduction (DN int)
19.1. In order to develop a theory of propositions in as explicit a way as possible, the system QD will now be constructed and presented. This system is an extension of the system Q, that is, it contains all the theorems of Q together with further theorems. In constructing the system QD a new primitive Q-formula 'D' is to be added to the list of primitive Q-formulas (3.4). This new primitive Q-formula denotes the Q-function which is the class of propositions (or the property of being a proposition). In other words, D is the Q-function which is the class of propositions (or the property of being a proposition). Thus the Q-formula 'Da' expresses the proposition that a is a proposition, and the Q-formula 'D(Da)' expresses· the (true) proposition that Da is a proposition, since Da is the proposition that (truly or falsely) asserts that a is a proposition. 19.2. The rules employed in the system QD consist of all the rules employed in the system Q together with additional rules concerned with n. These additional rules may be expressed schematically as follows: 19.3. D-elimination (D elim) LDa a V -a
I D(Na) The effect of this rule is to assert that Na (the proposition that a is a natural number) is a proposition. 19.7. D-£egation introduction (D neg int) LDa D(-a)
I
The effect of this rule is to assert that the negation of a proposition is a proposition. 19.8. D-conjunction introd~gtiQU (D conj int)
I I l
I ~:
l D[a &
14-0
b)
The effect of this rule is to assert that conjunctions of propositions are propositions. 19.9. D-disjunction introduction (D dis int)
~
I
= b)
Da
Db D[a V b)
1 1+2
ELEMENTS OF COMBINATORY LOGIC
The effect of this rule is to assert that disjunctions of propositions are propositions. This rule is derivable from D neg int and D conj int since a V b =-[-a a -b]. Similarly, D conj int is derivable from this rule and D neg int since a b =-[-a v -b]. 19.10. D-A-introduction (DA int)
THE SYS'l'EM QD 19.13. D-intersection introductiou (D intersect int) and D-£Dion introductioQ (D union int)
a
LCx)(D(ax)) D(Aa)
I
19.11. D-E-introduction (DE int) L (x)(D(ax)) D(Ea)
I
This rule and the previous one have the effect that if ab is a proposition for every Q-function b, then Aa (asserting that a is a property of every Q-function) is a proposition, and so is Ea (asserting that a is a property of some Qfunction). 19.12. Let 'a ~ b' be an abbreviation for '-a V b'. (This definition was previously introduced in 15.8.) The following rule, called~ £Onens (m p), is then easily derived:
The following two rules are also easily derived. They will be called restricted im£~iqatiQU ~QtrQ~YQt~QQ (res imp int), first form and second form. (See also 0.52.)
Some further rules for D will now be presented. They are concerned with intersections and unions. The intersection of a class of relations holds if all the relations hold, while the union holds if some of the relations hold.
D intersect int (first form) D union int (first form) In order to show how the above rules refer implicitly to intersections and unions, the concepts of intersection of a class of n-place relations and union of a class of n-place relations can be respectively denoted by Q-formulas 'Tid and '[ •, where 'Tin' is an abbreviation for '[z][y 1 ][y2] ... [yn] (x)n[zx ~ xy y •.• y ] ' and where '[d is an abbreviation for 1 2 '[z][y ][y ] .•• [ynJ(lx)[zx xy y 2 ••• yn]'. The following Tin1 2 1 and l..n \ -identities CTI n id, [ n id) are easily proveds
a
TI 0 ab 1 b 2 ••• b 0
=
(x)[ax
~
2:nablb2' •• bn = (lx)[ax
xb 1b 2 •.• bn]
a xblb2 •.• bn]
By use of the above identities and res id elim (19.14-), these forms of D intersect int and D union int become derivable:
~
(x)(D(ax)) (x)[ax ~ D(xb 1 b 2 ..• bn)] DCTI ab b ••• b) D intersect int (second form) n 1 2 n D([nab 1 b 2 ••• bn) D union int (second form)
It will be said that c is a definite relatiop. of degree n (or a definite n-£lace re1at~Qn, or, for n = 1, a definite ~) if (y )(y ) .•• (yn)(D(cy y ••• yn)), Thus the rules of 2 1 1 2 D intersect int and D union int, in either their first or second forms, are seen to imply the result that if all the members of a definite class a are definite relations of degree n, then the intersection Tina of the class a is a definite relation of degree n, and so is the union l:na of the class a. Propositions are definite relations of degree zero. Therefore it is the case that i f a is a definite class of propositions, then TI a and [ca. which equal (x)[ax ~ x] and 0
1 1+2
ELEMENTS OF COMBINATORY LOGIC
The effect of this rule is to assert that disjunctions of propositions are propositions. This rule is derivable from D neg int and D conj int since a V b =-[-a a -b]. Similarly, D conj int is derivable from this rule and D neg int since a b =-[-a v -b]. 19.10. D-A-introduction (DA int)
THE SYS'l'EM QD 19.13. D-intersection introductiou (D intersect int) and D-£Dion introductioQ (D union int)
a
LCx)(D(ax)) D(Aa)
I
19.11. D-E-introduction (DE int) L (x)(D(ax)) D(Ea)
I
This rule and the previous one have the effect that if ab is a proposition for every Q-function b, then Aa (asserting that a is a property of every Q-function) is a proposition, and so is Ea (asserting that a is a property of some Qfunction). 19.12. Let 'a ~ b' be an abbreviation for '-a V b'. (This definition was previously introduced in 15.8.) The following rule, called~ £Onens (m p), is then easily derived:
The following two rules are also easily derived. They will be called restricted im£~iqatiQU ~QtrQ~YQt~QQ (res imp int), first form and second form. (See also 0.52.)
Some further rules for D will now be presented. They are concerned with intersections and unions. The intersection of a class of relations holds if all the relations hold, while the union holds if some of the relations hold.
D intersect int (first form) D union int (first form) In order to show how the above rules refer implicitly to intersections and unions, the concepts of intersection of a class of n-place relations and union of a class of n-place relations can be respectively denoted by Q-formulas 'Tid and '[ •, where 'Tin' is an abbreviation for '[z][y 1 ][y2] ... [yn] (x)n[zx ~ xy y •.• y ] ' and where '[d is an abbreviation for 1 2 '[z][y ][y ] .•• [ynJ(lx)[zx xy y 2 ••• yn]'. The following Tin1 2 1 and l..n \ -identities CTI n id, [ n id) are easily proveds
a
TI 0 ab 1 b 2 ••• b 0
=
(x)[ax
~
2:nablb2' •• bn = (lx)[ax
xb 1b 2 •.• bn]
a xblb2 •.• bn]
By use of the above identities and res id elim (19.14-), these forms of D intersect int and D union int become derivable:
~
(x)(D(ax)) (x)[ax ~ D(xb 1 b 2 ..• bn)] DCTI ab b ••• b) D intersect int (second form) n 1 2 n D([nab 1 b 2 ••• bn) D union int (second form)
It will be said that c is a definite relatiop. of degree n (or a definite n-£lace re1at~Qn, or, for n = 1, a definite ~) if (y )(y ) .•• (yn)(D(cy y ••• yn)), Thus the rules of 2 1 1 2 D intersect int and D union int, in either their first or second forms, are seen to imply the result that if all the members of a definite class a are definite relations of degree n, then the intersection Tina of the class a is a definite relation of degree n, and so is the union l:na of the class a. Propositions are definite relations of degree zero. Therefore it is the case that i f a is a definite class of propositions, then TI a and [ca. which equal (x)[ax ~ x] and 0
144
ELEMENTS OF COMBINATORY LOGIC
(:>:x) [ax &. x] respectively, are, in effect, the conjunction of the propositions that are in the class a and the disjunction of those same propositions. The above rules guarantee that this conjunction and this disjunction are both of them propositions, so that the conjunction of a definite class of propositions is a proposition, and the disjunction of a definite class of propositions is also a proposition. 19.14. The rule of restricted identitz el~mknat~og (res id elim) (5.4, 12.11) is to be extended in the following way: The expression in which the replacement of 'a' by 'b' is made can be of the form 'D( ... a ••. )' or of the form '-(D( .•• a ..• ))'. In this case we have:
~
a
=b
D( ••• a ••• ) D( ••• b ••• )
~
a
=b
-(D( ••• a ••• )) -(D( ••• b ••• ))
Here '( .•• a ••• )' could be 'a' itself. The first hypothesis in each case could be 'b = a' instead of 'a = b'. 19.15. This completes the list of rules that must be added to the rules of system Q in order to generate the system QD. Although the consistency of Q appears to be beyond question, the same cannot at present be said about QD. It seems likely, however, that QD is free from contradiction. Some derived results concerning QD will now be presented. 10.16. The derived rule of D-universal gQantifier int£2duction (D u q int) asserts that 'D(x)( ... x ••. )' is a consequence of '(x)(D( ... x ... ))', and it is derived as follows: Pl24 l 2 3 4-
5 6
7 8
(x)(D( ... x ... )) a (x)(D( ... x •.. )) D( ••• a ..• )
[x]( ... x ... )a ~( ... a ... ) D( [x] ( ... x ... )a) (x)(D[x]( ... x ..• )x) D(Alx]( •.. x ••. )) D(x)( ... x ... )
THE SYSTEM QD
19.17. The derived rule of D-existence guantif!er introduction (De q int) asserts that 'D(~x)( •.• x ••• )' is a consequence of '(x)(D( .•• x ••• ))'. It is derived similarly to D u q int. 19.18. It has been pointed out (0.51) that not all the rules presented in the Introduction hold for the system Q. In particular the rules ex mid, neg int, ind pr, and imp int do not hold in ~eneral in Q, nor do they hold in general in QD. These rules, however, do all hold in QD if we assume them to be formulated as in the Introduction (using the letters 'p', 'q', 'r', and so on) but under the hypotheses 'Dp', 'Dq', 'Dr', and so on (or, equally well, under the hypotheses 'p V -p', 'q V -q', and so on). In other words, these rules do hold in QD if 'p', 'q', 'r', and so on, denote propositions in the sense of being members of D, or at least in the sense of being Q-functions that satisfy excluded middle. (In the latter case, the rules would even hold in Q itself.) When used in this way, these rules will be said to be restricted ~ of the original rules. Thus, for example, restricted ~ndirect £roof (res ind pr) can appear in either of the two forms that can be represented schematically thus: P V -P
q v -q
2, u q elim
abs id 3, 1+, res id elim 2-5, u q int b, DA int 7, rep, def
[can be omitted]
Dp Dq
[can be omitted]
It
II"
p
p
-q
hyp J, reit
145
-q
The second hypothesis in each of these forms of the rule is actually superfluous and can be omitted. The derivation of the second form (omitting the superfluous hypothesis) is as follows: Pl25
lpp 21 p V -P 'I ~ -P
hyp 1, D elim
hyp
144
ELEMENTS OF COMBINATORY LOGIC
(:>:x) [ax &. x] respectively, are, in effect, the conjunction of the propositions that are in the class a and the disjunction of those same propositions. The above rules guarantee that this conjunction and this disjunction are both of them propositions, so that the conjunction of a definite class of propositions is a proposition, and the disjunction of a definite class of propositions is also a proposition. 19.14. The rule of restricted identitz el~mknat~og (res id elim) (5.4, 12.11) is to be extended in the following way: The expression in which the replacement of 'a' by 'b' is made can be of the form 'D( ... a ••. )' or of the form '-(D( .•• a ..• ))'. In this case we have:
~
a
=b
D( ••• a ••• ) D( ••• b ••• )
~
a
=b
-(D( ••• a ••• )) -(D( ••• b ••• ))
Here '( .•• a ••• )' could be 'a' itself. The first hypothesis in each case could be 'b = a' instead of 'a = b'. 19.15. This completes the list of rules that must be added to the rules of system Q in order to generate the system QD. Although the consistency of Q appears to be beyond question, the same cannot at present be said about QD. It seems likely, however, that QD is free from contradiction. Some derived results concerning QD will now be presented. 10.16. The derived rule of D-universal gQantifier int£2duction (D u q int) asserts that 'D(x)( ... x ••. )' is a consequence of '(x)(D( ... x ... ))', and it is derived as follows: Pl24 l 2 3 4-
5 6
7 8
(x)(D( ... x ... )) a (x)(D( ... x •.. )) D( ••• a ..• )
[x]( ... x ... )a ~( ... a ... ) D( [x] ( ... x ... )a) (x)(D[x]( ... x ..• )x) D(Alx]( •.. x ••. )) D(x)( ... x ... )
THE SYSTEM QD
19.17. The derived rule of D-existence guantif!er introduction (De q int) asserts that 'D(~x)( •.• x ••• )' is a consequence of '(x)(D( .•• x ••• ))'. It is derived similarly to D u q int. 19.18. It has been pointed out (0.51) that not all the rules presented in the Introduction hold for the system Q. In particular the rules ex mid, neg int, ind pr, and imp int do not hold in ~eneral in Q, nor do they hold in general in QD. These rules, however, do all hold in QD if we assume them to be formulated as in the Introduction (using the letters 'p', 'q', 'r', and so on) but under the hypotheses 'Dp', 'Dq', 'Dr', and so on (or, equally well, under the hypotheses 'p V -p', 'q V -q', and so on). In other words, these rules do hold in QD if 'p', 'q', 'r', and so on, denote propositions in the sense of being members of D, or at least in the sense of being Q-functions that satisfy excluded middle. (In the latter case, the rules would even hold in Q itself.) When used in this way, these rules will be said to be restricted ~ of the original rules. Thus, for example, restricted ~ndirect £roof (res ind pr) can appear in either of the two forms that can be represented schematically thus: P V -P
q v -q
2, u q elim
abs id 3, 1+, res id elim 2-5, u q int b, DA int 7, rep, def
[can be omitted]
Dp Dq
[can be omitted]
It
II"
p
p
-q
hyp J, reit
145
-q
The second hypothesis in each of these forms of the rule is actually superfluous and can be omitted. The derivation of the second form (omitting the superfluous hypothesis) is as follows: Pl25
lpp 21 p V -P 'I ~ -P
hyp 1, D elim
hyp
ELEMENTS OF COMBINATORY LOGIC
n n+l n+2 n+3
q -q p p
6 7 8 n, n+l, neg elim 2, 3- n+2, dis elim
Statements of res imp int have already been given in 0.52 and 19.12. Notice that there was no need there to have 'Db' or 'b V -b' as an hypothesis, just as there is no need to have 'Dq' or 'q V -q' as an hypothesis in res ind pr. 19.19. EXERCISES. (1). Derive D dis int from D neg int, D conj int, and rules of system Q· (2). Derive an equation that shows that 'E' could be defined in terms of 'A' and other Q-formulas. Using this equation derive DE int from DA int, D neg int, and rules of system Q. (3). Derive the rule D e q int. (4). State the two forms of res neg int (analogous to the two forms of res ind pr) and derive both forms. 20.
Definite Qlasses
~ ~elatio~s
20.1. It is seen from 19.13 that a definite class is a Q-function a such that the Q-function ax is a proposition for every Q-function x. The class of definite classes will be denoted by 'D ', and the latter expression will serve as 1 an abbreviation for '[y](x)(D(yx))'. Consequently from the abstraction identity the following identity is obtained: D1 a = (x)(D(ax)) This latter identity will be called the D 1 -identit~ (D 1 id). It will next be shown that the class of definite classes is itself a definite class. This is the rule of D -definiteness 1 (D 1 dft). Pl26 l 2 3 I+
5
a
bl D(D(ab)) (x)(D(D(ax)) D( x)(D( ax)) D1 a = (x)(D(ax)) D(D 1 a)
DEFINITE CLASSES AND RELATIONS
DD int l- 1, u q int 2, D u q int Dl id 3. 4, res id elim
(x) (D(D x))
1 [y](x)(D(yx))D 1 D1D1
l- 5, u q int 6, abs int 7, rep, def
20.2. It is seen from 19.13 that a definite relation of degree n (also called a definite n-place relation) is a Qfunction a such that the Q-function ax 1x 2 .•. xn is a proposition for all Q-functions x 1 , x 2 , ••. , xn. The class of definite relations of degree n will be denoted by 'Dn'• and the latter expression will serve as an abbreviation for '[y] (x 1 )(x2 ) .•• (xn)(D(yx1 x 2 ••• xn))'. Consequently from the abstraction identity the following identity is obtained: Dna= (x1 )(x2 ) .•• (xn)(D(ax 1 x 2 •.• xn)) This latter identity will be called the Dn-identitl (Dn id). It is easy to prove that Dn' the class of definite n-place relations, is a definite class, that is, D Dn· This is done 1 analogously to Pl26 above. This result may be called the rule of Dn-definiteness (Dn dft). Recall not only that definite one-place relations are classes, but that definite zero-place relations are propositions, so that D is the 0 same as D, and we can prove that the class of propositions (namely D or D ) is a definite class, since this is merely 0 the case of D1Dn for which n = o. There also can be derived the following rules for Dn (Dn elim, Dn int, neg Dn elim, neg Dn int): - LDna
- f' (x 1 )(x2 ) ... (xn)(D(ax 1x 2 ... xn)) These rules follow directly from the abstraction rules and the definition of 'Dn'• and they do not require use of the Dn-identity and identity elimination. 20.3. It will now be useful to refer again to the combinator ~ that has already been referred to in 10.17, and to recall the identity, ~abed= a(bd)(cd), which will now be known as the t-identity (t id). There are also the derived rules for ~ (t elim, t int, neg t elim, neg t int):
ELEMENTS OF COMBINATORY LOGIC
n n+l n+2 n+3
q -q p p
6 7 8 n, n+l, neg elim 2, 3- n+2, dis elim
Statements of res imp int have already been given in 0.52 and 19.12. Notice that there was no need there to have 'Db' or 'b V -b' as an hypothesis, just as there is no need to have 'Dq' or 'q V -q' as an hypothesis in res ind pr. 19.19. EXERCISES. (1). Derive D dis int from D neg int, D conj int, and rules of system Q· (2). Derive an equation that shows that 'E' could be defined in terms of 'A' and other Q-formulas. Using this equation derive DE int from DA int, D neg int, and rules of system Q. (3). Derive the rule D e q int. (4). State the two forms of res neg int (analogous to the two forms of res ind pr) and derive both forms. 20.
Definite Qlasses
~ ~elatio~s
20.1. It is seen from 19.13 that a definite class is a Q-function a such that the Q-function ax is a proposition for every Q-function x. The class of definite classes will be denoted by 'D ', and the latter expression will serve as 1 an abbreviation for '[y](x)(D(yx))'. Consequently from the abstraction identity the following identity is obtained: D1 a = (x)(D(ax)) This latter identity will be called the D 1 -identit~ (D 1 id). It will next be shown that the class of definite classes is itself a definite class. This is the rule of D -definiteness 1 (D 1 dft). Pl26 l 2 3 I+
5
a
bl D(D(ab)) (x)(D(D(ax)) D( x)(D( ax)) D1 a = (x)(D(ax)) D(D 1 a)
DEFINITE CLASSES AND RELATIONS
DD int l- 1, u q int 2, D u q int Dl id 3. 4, res id elim
(x) (D(D x))
1 [y](x)(D(yx))D 1 D1D1
l- 5, u q int 6, abs int 7, rep, def
20.2. It is seen from 19.13 that a definite relation of degree n (also called a definite n-place relation) is a Qfunction a such that the Q-function ax 1x 2 .•. xn is a proposition for all Q-functions x 1 , x 2 , ••. , xn. The class of definite relations of degree n will be denoted by 'Dn'• and the latter expression will serve as an abbreviation for '[y] (x 1 )(x2 ) .•• (xn)(D(yx1 x 2 ••• xn))'. Consequently from the abstraction identity the following identity is obtained: Dna= (x1 )(x2 ) .•• (xn)(D(ax 1 x 2 •.• xn)) This latter identity will be called the Dn-identitl (Dn id). It is easy to prove that Dn' the class of definite n-place relations, is a definite class, that is, D Dn· This is done 1 analogously to Pl26 above. This result may be called the rule of Dn-definiteness (Dn dft). Recall not only that definite one-place relations are classes, but that definite zero-place relations are propositions, so that D is the 0 same as D, and we can prove that the class of propositions (namely D or D ) is a definite class, since this is merely 0 the case of D1Dn for which n = o. There also can be derived the following rules for Dn (Dn elim, Dn int, neg Dn elim, neg Dn int): - LDna
- f' (x 1 )(x2 ) ... (xn)(D(ax 1x 2 ... xn)) These rules follow directly from the abstraction rules and the definition of 'Dn'• and they do not require use of the Dn-identity and identity elimination. 20.3. It will now be useful to refer again to the combinator ~ that has already been referred to in 10.17, and to recall the identity, ~abed= a(bd)(cd), which will now be known as the t-identity (t id). There are also the derived rules for ~ (t elim, t int, neg t elim, neg t int):
ELEMSNTS OF COMBINATORY LOGIC
1 1+8
- L tabcd -fa(bd)(cd)
or
- L[c
-r cd
~a
b]d ~&
-r
Also the derived rules for tV (tV elim, tV int, etc.): - L tVbcd - fV(bd)(cd)
or
-LCc
-r
tv b]d cd v bd
- L [c tna b]d 1 ... d -fcd 1 ... dn a bd .~.dn
1
From these follow the derived rules for tn& (tn& int, tn& elim, etc.): - L tn&bcd 1 ••• d or -f&(bd 1 ... dn)Ccd 1 ... dn)
- L [c tn& b]d 1 ... d -fcd 1 ... dn &. bd .~.dn
1
Also the derived rules for tOy (tOy elim, tnv int, etc.): - L tnvbcd 1 •.• d or -IV(bd 1 ..• dn)Ccd 1 ..• dn)
- L [c tnv b]d 1 ... d -fcd 1 ..• dn V bd .~.dn
1
We also have the tn-identitl (tn id), n
n
-LB -bd 1 ... d -(bd 1 ... ds
-r
They are special cases of the derived rules for Bn (Bn elim, Bn int, etc.): - L Bnabd 1 ••• d -racbd 1 ... ds we also have the Bn-identitl (Bn id),
20.4. The rules for t may be generalized into the following derived rules for tn (tn elim, tn int, etc.): - L tnabcd 1 ••• d or -fa(bd 1 ... dn)Ccd 1 .•• dn)
149
junction. The following rules for Bn- (Bn- elim, Bn- int, etc.) are easily established:
a bd
From the above follow the derived rules for ~& (~& elim, int, etc.): or - L t&bcd -L[c t&b]d &(bd) ( cd) cd &. bd
-r
DEFINITE CLASSES AND RELATIONS
t abcd 1 ••• dn = a(bd 1 .•• d 0 )(cd 1 ... dn) and the snecial cases of the latter, tn& id and tOy id, when 'a' is taken to be '&' or 'V'. We may call tn&bc the n-Elace intersection of b with c, and tnvbc the n-;Elace ~ of b with c. Here we are to think of b and c as n-place relations (classes for n = l, propositions for n = O). Notice that 1 t & (or t&) and t 1v (or tV) are respectively the same as n and U referred to in 18.3 and 18.4, and that t 0 & and t 0 v are respectively the same as &. and v. 20.5. The combinator B may be used to play a role for negation like the role played by t for conjunction and dis-
Bnabd 1 •.. dn
=
a(bd 1 ..• dn)
and the special case of it which is the Bn--identitl (Bn- id), Bn-bd 1 ..• dn
=
-(bd 1 ... dn)
We may call Bn-b the n-place _£omplement of b, where b is thought of as an n-place relation (a class for n = 1, a proposition for n = O). The comnlement of a relation relates exactly those things (in the specified order) that the relation itEelf does not relate, and the complement of a class has as members exactly those things that the class itself does not have as members. (At least these statements are true for definite relations and definite classes; otherwise a slight qualification would be in order.) 20.6. The concepts of n-place intersection, union, and complement were introduced in as much detail as they were as a preliminary to stating the following derived rules (Dn tn& int, Dn tOy int, Dn Bn- int):
r
Dnc D"b Dn(tn&.bc) nnetnvbc) Dn(Bn-b)
Dn tn&. int Dn ;Oy int Dn Bn- int
These rules are easily derived. They assert that the n-place intersection of two definite n-place relations is a definite
ELEMSNTS OF COMBINATORY LOGIC
1 1+8
- L tabcd -fa(bd)(cd)
or
- L[c
-r cd
~a
b]d ~&
-r
Also the derived rules for tV (tV elim, tV int, etc.): - L tVbcd - fV(bd)(cd)
or
-LCc
-r
tv b]d cd v bd
- L [c tna b]d 1 ... d -fcd 1 ... dn a bd .~.dn
1
From these follow the derived rules for tn& (tn& int, tn& elim, etc.): - L tn&bcd 1 ••• d or -f&(bd 1 ... dn)Ccd 1 ... dn)
- L [c tn& b]d 1 ... d -fcd 1 ... dn &. bd .~.dn
1
Also the derived rules for tOy (tOy elim, tnv int, etc.): - L tnvbcd 1 •.• d or -IV(bd 1 ..• dn)Ccd 1 ..• dn)
- L [c tnv b]d 1 ... d -fcd 1 ..• dn V bd .~.dn
1
We also have the tn-identitl (tn id), n
n
-LB -bd 1 ... d -(bd 1 ... ds
-r
They are special cases of the derived rules for Bn (Bn elim, Bn int, etc.): - L Bnabd 1 ••• d -racbd 1 ... ds we also have the Bn-identitl (Bn id),
20.4. The rules for t may be generalized into the following derived rules for tn (tn elim, tn int, etc.): - L tnabcd 1 ••• d or -fa(bd 1 ... dn)Ccd 1 .•• dn)
149
junction. The following rules for Bn- (Bn- elim, Bn- int, etc.) are easily established:
a bd
From the above follow the derived rules for ~& (~& elim, int, etc.): or - L t&bcd -L[c t&b]d &(bd) ( cd) cd &. bd
-r
DEFINITE CLASSES AND RELATIONS
t abcd 1 ••• dn = a(bd 1 .•• d 0 )(cd 1 ... dn) and the snecial cases of the latter, tn& id and tOy id, when 'a' is taken to be '&' or 'V'. We may call tn&bc the n-Elace intersection of b with c, and tnvbc the n-;Elace ~ of b with c. Here we are to think of b and c as n-place relations (classes for n = l, propositions for n = O). Notice that 1 t & (or t&) and t 1v (or tV) are respectively the same as n and U referred to in 18.3 and 18.4, and that t 0 & and t 0 v are respectively the same as &. and v. 20.5. The combinator B may be used to play a role for negation like the role played by t for conjunction and dis-
Bnabd 1 •.. dn
=
a(bd 1 ..• dn)
and the special case of it which is the Bn--identitl (Bn- id), Bn-bd 1 ..• dn
=
-(bd 1 ... dn)
We may call Bn-b the n-place _£omplement of b, where b is thought of as an n-place relation (a class for n = 1, a proposition for n = O). The comnlement of a relation relates exactly those things (in the specified order) that the relation itEelf does not relate, and the complement of a class has as members exactly those things that the class itself does not have as members. (At least these statements are true for definite relations and definite classes; otherwise a slight qualification would be in order.) 20.6. The concepts of n-place intersection, union, and complement were introduced in as much detail as they were as a preliminary to stating the following derived rules (Dn tn& int, Dn tOy int, Dn Bn- int):
r
Dnc D"b Dn(tn&.bc) nnetnvbc) Dn(Bn-b)
Dn tn&. int Dn ;Oy int Dn Bn- int
These rules are easily derived. They assert that the n-place intersection of two definite n-place relations is a definite
150
ELEMENTS OF COMBINATORY LOGIC
n-place relation, that the n-place union of two definite nplace relations is a definite n-place relation, and that the n-place complement of a definite n-place relation is a definite n-place relation. 20.7. The Boolean identities stated in 6.1 also hold if •a•, 'V', and '-' are respectively replaced by ·~na•, •tlly•, and 'Bn_,, as can be easily shown. The other Boolean identities stated in 6.2 will hold if this same replacement is made, but sub,ject to the condition that a and b are definite n-place relations, or even to the weaker condition that 'Cx 1 ) •.• (xn)[ax 1 ••• xn V -(ax1 ... xn)]' and '(x ) ••• (xn) 1 [bx 1 .•• xn V -(bx1 ••• xn)J' are provable without use of id elim (but possibly with use of res id elim). 20.8. The relative Erodu~ of a two-place relation r with a two-place relation s is a two-place relation t such that a bears t to b .iust in case that a bears r to something that bears s to b. For example, the two-place relation aunt-of is the relative product of the two-place relation sister-of with the two-place relation parent-of. This is because an aunt of a person is a si~ter of a parent of that person. If we let 'r I s' (that is, ' I sr') denote the relative product of r with s, then we may treat '1' as an abbreviation for '[u][w][y][x](3z)[[x w z] a [z u y]]'. By use of the abstraction identity, we then obtain the Eflative £reduct identit~ (rel prod id):
or
[a [r
lsrba
(3z)[[a r z]
s] bl
(3z)[[a r z]
& [z & [z
s b]] s b]]
20.9. If r and s are definite two-place relations, so is the r(!lative product of r with s. This is the rule of D2 -relative £~~ introQuction (D 2 rel prod int):
f"
?127 2
3
~
5
D2 s a; b' c
hyp r.yp D2 r D2 s (x)(y) (D [y r
'
),
D2 elim
2. reit X
J)
7 8
9 10 11
12 13 1~
15 16
20.10. The relative square of a two-place relation r is the relative product of r with r itself, that is, r I r, the ~!tY~ ~ is [r I r] I r, the relative fourth~ is [[r I r] I r] I r, and so on. Of course r is its own relative ~ £Ower. The successive powers can also be said tobe r, lrr, lr(lrr), lr(lr(lrr)), andsoon,thatis, ( lr) 0 r, ( lr) 1 r, ( lr) 2 r. ( lr)3r, and so on. Thus then+ lst power of r is ( lr)n+ 1 , where n is a natural number. As examples of the powers of a relation, the relation of a to b such that a is parent of b has the following sequence of powers: parent of, grandparent of, great-grandparent of, great-great-grandparent of, and so on. It will now be shown that if r is a definite two-place relation, so is ( lr)nr for each natural number n. This result guarantees that every power of a definite two-place relation is a definite twoplace relation. (wor present purposes we disregard so-called zeroth powers of relations, which are so~etimes treated as bein!" the identity relation, or a re;:tricted form or the latter.) We first prove a lemma. 2
reit
I+, D2 elim (x)(y)(D[y s x]) 5, u q elim (twice) D[a r c] 6, u q elim (twice) D[c s b] D[[a r c] & [c s b]] 7, 8, D conj int (z)(D[[a r zl a [z s b]] 3-9, u q int D(3z)[[a r z] & [z s b]] 10, D e q int 11, rel prod id, res id elim D( I srba) 3- 12, u q int (y)(D( lsrby)) 3- 13, u q int (x)(y)(D( lsrxy)) 14, D2 int D (1sr) 2 15, rep, def D [r I s] 2
6
1"8 1
"'
151
DEFINITE CLASSES AND RELATIONS
r'z"
3
~
':>!
r = s a b D2 r I (x) (y)(D( rxy))
I
I
1 D(rab)
hyp hyp reit D2 elim I+' u q elim (twice) l,
),
150
ELEMENTS OF COMBINATORY LOGIC
n-place relation, that the n-place union of two definite nplace relations is a definite n-place relation, and that the n-place complement of a definite n-place relation is a definite n-place relation. 20.7. The Boolean identities stated in 6.1 also hold if •a•, 'V', and '-' are respectively replaced by ·~na•, •tlly•, and 'Bn_,, as can be easily shown. The other Boolean identities stated in 6.2 will hold if this same replacement is made, but sub,ject to the condition that a and b are definite n-place relations, or even to the weaker condition that 'Cx 1 ) •.• (xn)[ax 1 ••• xn V -(ax1 ... xn)]' and '(x ) ••• (xn) 1 [bx 1 .•• xn V -(bx1 ••• xn)J' are provable without use of id elim (but possibly with use of res id elim). 20.8. The relative Erodu~ of a two-place relation r with a two-place relation s is a two-place relation t such that a bears t to b .iust in case that a bears r to something that bears s to b. For example, the two-place relation aunt-of is the relative product of the two-place relation sister-of with the two-place relation parent-of. This is because an aunt of a person is a si~ter of a parent of that person. If we let 'r I s' (that is, ' I sr') denote the relative product of r with s, then we may treat '1' as an abbreviation for '[u][w][y][x](3z)[[x w z] a [z u y]]'. By use of the abstraction identity, we then obtain the Eflative £reduct identit~ (rel prod id):
or
[a [r
lsrba
(3z)[[a r z]
s] bl
(3z)[[a r z]
& [z & [z
s b]] s b]]
20.9. If r and s are definite two-place relations, so is the r(!lative product of r with s. This is the rule of D2 -relative £~~ introQuction (D 2 rel prod int):
f"
?127 2
3
~
5
D2 s a; b' c
hyp r.yp D2 r D2 s (x)(y) (D [y r
'
),
D2 elim
2. reit X
J)
7 8
9 10 11
12 13 1~
15 16
20.10. The relative square of a two-place relation r is the relative product of r with r itself, that is, r I r, the ~!tY~ ~ is [r I r] I r, the relative fourth~ is [[r I r] I r] I r, and so on. Of course r is its own relative ~ £Ower. The successive powers can also be said tobe r, lrr, lr(lrr), lr(lr(lrr)), andsoon,thatis, ( lr) 0 r, ( lr) 1 r, ( lr) 2 r. ( lr)3r, and so on. Thus then+ lst power of r is ( lr)n+ 1 , where n is a natural number. As examples of the powers of a relation, the relation of a to b such that a is parent of b has the following sequence of powers: parent of, grandparent of, great-grandparent of, great-great-grandparent of, and so on. It will now be shown that if r is a definite two-place relation, so is ( lr)nr for each natural number n. This result guarantees that every power of a definite two-place relation is a definite twoplace relation. (wor present purposes we disregard so-called zeroth powers of relations, which are so~etimes treated as bein!" the identity relation, or a re;:tricted form or the latter.) We first prove a lemma. 2
reit
I+, D2 elim (x)(y)(D[y s x]) 5, u q elim (twice) D[a r c] 6, u q elim (twice) D[c s b] D[[a r c] & [c s b]] 7, 8, D conj int (z)(D[[a r zl a [z s b]] 3-9, u q int D(3z)[[a r z] & [z s b]] 10, D e q int 11, rel prod id, res id elim D( I srba) 3- 12, u q int (y)(D( lsrby)) 3- 13, u q int (x)(y)(D( lsrxy)) 14, D2 int D (1sr) 2 15, rep, def D [r I s] 2
6
1"8 1
"'
151
DEFINITE CLASSES AND RELATIONS
r'z"
3
~
':>!
r = s a b D2 r I (x) (y)(D( rxy))
I
I
1 D(rab)
hyp hyp reit D2 elim I+' u q elim (twice) l,
),
,,,
152
ELEMENTS OF COMBINATORY LOGIC 6 7 8
9 10
I ~(:a:) (y) (D( say)) (x) (y)(D( sxy)) D2 s
2, reit 5. 6, res id elim 3- 7. u q int 3- 8, u q int 9, D2 int
The above lemma shows that we can go from 'D r• and 'r = s' 2 to 'D 2 s' without use of id elim and by using res id elim instead. Such a transition will hereafter be considered a case of the mere use of res id elim. Other analogous cases that can be similarly proved will be considered in the same way. Pl29 l 2 3 45 6 7 8 g 10 ll 12 13 1415
D2 r a Na V -(Na) Na D2 r 0( lr)r = r D2 (o( lr)r) b D2 (b( lr)r) D2 r D2 ( lr(b( lr)r)) D2 (l( lr)(b( lr)r)) D2 (+lb(lr)r) D2 ([b + l]( lr)r) D2 (a( lr)r) Na :o D2 (a( lr)r) (x)[Nx :J D2 ((1r)xr)]
hyp ex mid N hyp 1, reit 0 id 4-, 5. res id elim hyp 4-, reit 8, 9. D2 rel prod int l id, 9, res id elim + id, 10, res id elim ll, rep, def 3. 6, 7-12, N indue 2, 3-13. res imp int 2- 14-, u q int, def
The above rule may be referred to as D -relative-~ower ~ 2 duction (D 2 rel po int). According to this rule, if r is a definite two-place relation. so is every power of it. 20.11. The :ero12er ancestra.Jo of a two-place relation r is the relation that a bears to b exactly in case that a bears some (positive) power of r to b. It should therefore be clear that a bears to b the proper ancestral of r exactly in case (:.x) [Nx & ( lr)xrbal, that is, (:.x) [Nx & [a ( lr)xr b]], so that if we designate the proper ancestral of r by '*r',we may treat'*' as an abbreviation for '[w][y][z]
I
DEFINITE CLASSES AND RELATIONS
153
(~x)[Nx ~
& ( lw)xwyz]'. We can then derive the :eroper ~ identitl (prop anc id): (~x)[Nx
& ( lr)xrba]
& [a & [a
or
[an b]
(:.x)[Nx
or
[a (n) b]
(~x)[Nx
( lr)xr b]] (( lr)xr) b]]
These three equations are simply different notations for the same equation. The following rules for the proper ancestral are also derivable (prop anc elim, prop anc int, etc.). The same alternative notations as above would of course be available.
20.12. It will now be shown that i f r is a definite two-place relation, so is the proper ancestral of r. This rule will be called D 2 -:ero~er-ancestral introduction CD 2 prop anc int): Pl30 l 2
3 45 6
7 8
9 10 ll 12 13 1415 16 17
hyp DN int 2-2, u q int ex mid N hyp Dr l, reit 2 X (x)[Nx :o D2 (( lr) r)] 6, D2 rel po int Nd :o D2 ((1r)dr) 7, u q elim D2 (C lr)dr) 5, 8, m p (u)(w)(D(( lr)druw)) 9, n2 elim D(( lr)drba) u q elim (twice) D(d( lr)rba) rep, def Nd :o D(d( lr)rba) 4-, 5- 12, res imp int (x)[Nx :o D(x( lr)rba) 4- - 13, u q in t D(~x)[Nx & ( lr)xrba] 2, 14-, Dunion int, def D(•rba) 15, prop anc id, res id elim (x)(y) (D(nxy)) 2- 16, u q int (twice) 17, D2 int cl D(Nc) (x) (D(Nx)) d Nd V -(Nd) Nd
,,,
152
ELEMENTS OF COMBINATORY LOGIC 6 7 8
9 10
I ~(:a:) (y) (D( say)) (x) (y)(D( sxy)) D2 s
2, reit 5. 6, res id elim 3- 7. u q int 3- 8, u q int 9, D2 int
The above lemma shows that we can go from 'D r• and 'r = s' 2 to 'D 2 s' without use of id elim and by using res id elim instead. Such a transition will hereafter be considered a case of the mere use of res id elim. Other analogous cases that can be similarly proved will be considered in the same way. Pl29 l 2 3 45 6 7 8 g 10 ll 12 13 1415
D2 r a Na V -(Na) Na D2 r 0( lr)r = r D2 (o( lr)r) b D2 (b( lr)r) D2 r D2 ( lr(b( lr)r)) D2 (l( lr)(b( lr)r)) D2 (+lb(lr)r) D2 ([b + l]( lr)r) D2 (a( lr)r) Na :o D2 (a( lr)r) (x)[Nx :J D2 ((1r)xr)]
hyp ex mid N hyp 1, reit 0 id 4-, 5. res id elim hyp 4-, reit 8, 9. D2 rel prod int l id, 9, res id elim + id, 10, res id elim ll, rep, def 3. 6, 7-12, N indue 2, 3-13. res imp int 2- 14-, u q int, def
The above rule may be referred to as D -relative-~ower ~ 2 duction (D 2 rel po int). According to this rule, if r is a definite two-place relation. so is every power of it. 20.11. The :ero12er ancestra.Jo of a two-place relation r is the relation that a bears to b exactly in case that a bears some (positive) power of r to b. It should therefore be clear that a bears to b the proper ancestral of r exactly in case (:.x) [Nx & ( lr)xrbal, that is, (:.x) [Nx & [a ( lr)xr b]], so that if we designate the proper ancestral of r by '*r',we may treat'*' as an abbreviation for '[w][y][z]
I
DEFINITE CLASSES AND RELATIONS
153
(~x)[Nx ~
& ( lw)xwyz]'. We can then derive the :eroper ~ identitl (prop anc id): (~x)[Nx
& ( lr)xrba]
& [a & [a
or
[an b]
(:.x)[Nx
or
[a (n) b]
(~x)[Nx
( lr)xr b]] (( lr)xr) b]]
These three equations are simply different notations for the same equation. The following rules for the proper ancestral are also derivable (prop anc elim, prop anc int, etc.). The same alternative notations as above would of course be available.
20.12. It will now be shown that i f r is a definite two-place relation, so is the proper ancestral of r. This rule will be called D 2 -:ero~er-ancestral introduction CD 2 prop anc int): Pl30 l 2
3 45 6
7 8
9 10 ll 12 13 1415 16 17
hyp DN int 2-2, u q int ex mid N hyp Dr l, reit 2 X (x)[Nx :o D2 (( lr) r)] 6, D2 rel po int Nd :o D2 ((1r)dr) 7, u q elim D2 (C lr)dr) 5, 8, m p (u)(w)(D(( lr)druw)) 9, n2 elim D(( lr)drba) u q elim (twice) D(d( lr)rba) rep, def Nd :o D(d( lr)rba) 4-, 5- 12, res imp int (x)[Nx :o D(x( lr)rba) 4- - 13, u q in t D(~x)[Nx & ( lr)xrba] 2, 14-, Dunion int, def D(•rba) 15, prop anc id, res id elim (x)(y) (D(nxy)) 2- 16, u q int (twice) 17, D2 int cl D(Nc) (x) (D(Nx)) d Nd V -(Nd) Nd
17'+
ELEMENTS OF COMBINATORY LOGIC
' t -\"
20.13. We will say that a is strongl;r n-included in b a = Tn&ba, that is, a = [a Tn& b]. For n = 1, this would be the same as a = [an b]. The relation strong n-inclusion will be denoted by ·~n'• where ·~n' is an abbreviation for '[x] [y] [x = tn&yx], or in other notation, '[x] [y] [x = [x Tn& y]' (with spaces on each side of 'tn&'). There is no difficulty in proving n2;::n, that is, in showing the relation ~n of strong n-inclusion to be a definite two-place relation. The class -n c a of n-place relations strongly n-included in an nplace relation a can be shown to be a definite class, so that n 1 (~na) is provable. In particular, t h e c 1 ass ~ a of 1 classes strongly 1-included in a class a can be shown to be a definite class. The theory of propositions presented in this chapter is thus seen to give rise to a powerful theory of definite classes and definite n•place relations. 20.14. In conclusion, it is worth mentioning that although excluded middle does not hold in general in the systems Q and QD, still a ErinciEle Qf excluded foMrth can be shown to hold in general in the system QD, namely the following principle:
'
I I
if
a V -a V -(Da) 20.15. EXERCISES. (1). Prove in QD the principle of excluded fourth stated immediately above. (2). Prove the following rules: Dn elim, t elim, 4 T int, n2 T2V int. (3). Prove n 2~ • 3 (4). Prove D 1 (~ 2 a).
BIBLIOGRAPHY Church, A. Cogan, E.
Curry, H.
Curry, H. Curry, H. Fitch, F.
The Calculi o£ Lambga-CQgvers~on. Princeton: Princeton University Press, 1941. J, "A formalization of the Theory of Sets from the Point of View of Combinatory Logic." b.-..£:. Math. Logik und ~r. der Matb. 1 (1955): 198240. B. "Grundlagen der Kombinatorischen Logik." ~ J, Math. 52 (1930): 509-36, 789-834. "The Paradox of Kleene and Rosser." Trans. Am. Math. Soc. 50 (1941): 454- 516. B., and Feys, R. Combinator;r LQg~~· vol. 1. Amster~ dam: North-Holland, 1958. B., Hindley, J, R., and Seldin, J, P. Combinator;r Logic, vol. 2. Amsterdam: North-Holland, 1972. B. "A system of Formal Logic without an Analogue to the Curry W Operator."~ l (1936): 92100. "A Basic Logic." JSL 7 (1942): 105-ll4. "Representations ~Calculi."~ 9 (1944): 5762. "A Minimum Calculus for Logic."~ 9 (1944): 89- 94. "Self-Reference in Philosophy." ~ 55 (1946): 64- 73· "An Extension of Basic Logic."~ 13 (1948): 95-106. "The Reine-Borel Theorem in Extended Basic Logic."~ 14 (1949): 9-15.
155
17'+
ELEMENTS OF COMBINATORY LOGIC
' t -\"
20.13. We will say that a is strongl;r n-included in b a = Tn&ba, that is, a = [a Tn& b]. For n = 1, this would be the same as a = [an b]. The relation strong n-inclusion will be denoted by ·~n'• where ·~n' is an abbreviation for '[x] [y] [x = tn&yx], or in other notation, '[x] [y] [x = [x Tn& y]' (with spaces on each side of 'tn&'). There is no difficulty in proving n2;::n, that is, in showing the relation ~n of strong n-inclusion to be a definite two-place relation. The class -n c a of n-place relations strongly n-included in an nplace relation a can be shown to be a definite class, so that n 1 (~na) is provable. In particular, t h e c 1 ass ~ a of 1 classes strongly 1-included in a class a can be shown to be a definite class. The theory of propositions presented in this chapter is thus seen to give rise to a powerful theory of definite classes and definite n•place relations. 20.14. In conclusion, it is worth mentioning that although excluded middle does not hold in general in the systems Q and QD, still a ErinciEle Qf excluded foMrth can be shown to hold in general in the system QD, namely the following principle:
'
I I
if
a V -a V -(Da) 20.15. EXERCISES. (1). Prove in QD the principle of excluded fourth stated immediately above. (2). Prove the following rules: Dn elim, t elim, 4 T int, n2 T2V int. (3). Prove n 2~ • 3 (4). Prove D 1 (~ 2 a).
BIBLIOGRAPHY Church, A. Cogan, E.
Curry, H.
Curry, H. Curry, H. Fitch, F.
The Calculi o£ Lambga-CQgvers~on. Princeton: Princeton University Press, 1941. J, "A formalization of the Theory of Sets from the Point of View of Combinatory Logic." b.-..£:. Math. Logik und ~r. der Matb. 1 (1955): 198240. B. "Grundlagen der Kombinatorischen Logik." ~ J, Math. 52 (1930): 509-36, 789-834. "The Paradox of Kleene and Rosser." Trans. Am. Math. Soc. 50 (1941): 454- 516. B., and Feys, R. Combinator;r LQg~~· vol. 1. Amster~ dam: North-Holland, 1958. B., Hindley, J, R., and Seldin, J, P. Combinator;r Logic, vol. 2. Amsterdam: North-Holland, 1972. B. "A system of Formal Logic without an Analogue to the Curry W Operator."~ l (1936): 92100. "A Basic Logic." JSL 7 (1942): 105-ll4. "Representations ~Calculi."~ 9 (1944): 5762. "A Minimum Calculus for Logic."~ 9 (1944): 89- 94. "Self-Reference in Philosophy." ~ 55 (1946): 64- 73· "An Extension of Basic Logic."~ 13 (1948): 95-106. "The Reine-Borel Theorem in Extended Basic Logic."~ 14 (1949): 9-15.
155
156
ELEMENTS OF
COMBINA~ORY
LOGIC
"On Natural Numbers, Integers, and Rationals.• ill 14 (1949): 81- 84. "A Further Consistent Extension of Basic Logic.• l'+ 1949 l : 209 - 18. "A Demonstrably Consistent Mathematics.• ill 15 ( 1950): 17 - 24. §lmbolic Logic. ~n lntrQguctiQ£· New York: Ronald Press, 1952. "A Simplii'ication of Basic Logic.» JSL 18 (1953): 317-25. "Self-Referential Relations.• Actes du XI~me Con' Internatiopal de .El:!;i.loso"Qhie 14: 121-27. gres Amsterdam: North-Holland, 1053; Louvain: Editions E. Nauwelaerts, 1953. "A Definition of Negation in Extended Basic Logic.• 19 (1°54): 29-36. "The Reality of Propositions.• :Bcev. of MetaJ:l_hysJc.se 9 (1955): 3-13. "Recursive Functions in Basic Logic.• llili 21 (1956): 337-'+6. "A Definition of Existence in Terms of Abstraction and Dis,junction." ~ 22 (1957): 343-44. "Combinatory Lop;ic and Whitehead's Theory of Prehensions.» ~~ciepce 24 ( 1957): 331- 35"An Extensional Variety of Extended Basic Logic." 23 0958): 13- 21 "Representation of Sequential Circuits in Combinatory Logic." Phi los. of Science 25 (19)8): 263-79. "Quasi-Constructive Foundations for Mathematics." In A. Heyting, ed., Constructivitz_;i.p_M~ema !i~· Amsterdam: North-Holland, 1059. "A System of Combinatory Logic." ~·echnical report, Office of Naval Research, Group Psychology Branch. Contract No. c.~R/Nonr-609( 16),
m
c
m
m
1)7
BIBLIOGRAPHY
New Haven, Conn., November, 1960 "The System C6 of Combinatory Logic."~ 28 (1963): 87-97· "A Logical Analysis of Some Value Concepts." .w128 (1963): 135-142. "Universal Metalanguages for Philosophy.• ~ £f Meta"Qhysi.-s 17 (1964): 396-402. "A Note on Recursive Relations.• ~ 33 (1968): 107. "A Method for Avoiding the Curry Paradox." In N. Rescher, ed., Essals in Honor of Qarl ~· Hempel. Dordrecht-Holland: D. Reidel, 1969 "Combinatory Logic and Negative Numbers." In K. Lambert, ed., !Qe Logical Way of ~oing ~higgs, in honor of Henry s. Leonard. New Haven: Yale University Press, 1969. "Natural Deduction Rules for English."~ ~QJl.hical Studies 24 (1973): 89- 104. •i:iber formal unentscheidbare Satze der Principia Godel, K. Mathematica und verwandter System, I." Mon. f. Math. und .Ehysik 38 (1931): 173-98. Grosky. w. I. "A Mathematical Theory of Constructible Automata.• Ph. D. Thesis, Yale University, New Haven, 1071. ~numerability~ ~ecidab;i.l~tl, Computabilit~. New Hermes H. York: Springer-Verlag, 196). f. c. "A Theor,y of Positive Intep;ers in Formal Logic." KleP.ne, Amer. J. Math. 57 (lD35): 153- 73· urr:a~:s,
H. J., and Fitch, F. B. "A Theory of Computing r~1a
chines." Studium Gener,?.k 22 (196')): 83- 101+. "!< Theory of Programming Languar;es." §.!;~.hll!!l (i<:.Q~E§l.f. 22 (l96o) : 113 - 36. ~.uine,
w. v.
"Variables ~~xplained Away." P.EQ£::~. 0M.e. . ___ .QL--~-
Amer. Philos. §oc. 101+ (1960): 3'+3- 47. I
156
ELEMENTS OF
COMBINA~ORY
LOGIC
"On Natural Numbers, Integers, and Rationals.• ill 14 (1949): 81- 84. "A Further Consistent Extension of Basic Logic.• l'+ 1949 l : 209 - 18. "A Demonstrably Consistent Mathematics.• ill 15 ( 1950): 17 - 24. §lmbolic Logic. ~n lntrQguctiQ£· New York: Ronald Press, 1952. "A Simplii'ication of Basic Logic.» JSL 18 (1953): 317-25. "Self-Referential Relations.• Actes du XI~me Con' Internatiopal de .El:!;i.loso"Qhie 14: 121-27. gres Amsterdam: North-Holland, 1053; Louvain: Editions E. Nauwelaerts, 1953. "A Definition of Negation in Extended Basic Logic.• 19 (1°54): 29-36. "The Reality of Propositions.• :Bcev. of MetaJ:l_hysJc.se 9 (1955): 3-13. "Recursive Functions in Basic Logic.• llili 21 (1956): 337-'+6. "A Definition of Existence in Terms of Abstraction and Dis,junction." ~ 22 (1957): 343-44. "Combinatory Lop;ic and Whitehead's Theory of Prehensions.» ~~ciepce 24 ( 1957): 331- 35"An Extensional Variety of Extended Basic Logic." 23 0958): 13- 21 "Representation of Sequential Circuits in Combinatory Logic." Phi los. of Science 25 (19)8): 263-79. "Quasi-Constructive Foundations for Mathematics." In A. Heyting, ed., Constructivitz_;i.p_M~ema !i~· Amsterdam: North-Holland, 1059. "A System of Combinatory Logic." ~·echnical report, Office of Naval Research, Group Psychology Branch. Contract No. c.~R/Nonr-609( 16),
m
c
m
m
1)7
BIBLIOGRAPHY
New Haven, Conn., November, 1960 "The System C6 of Combinatory Logic."~ 28 (1963): 87-97· "A Logical Analysis of Some Value Concepts." .w128 (1963): 135-142. "Universal Metalanguages for Philosophy.• ~ £f Meta"Qhysi.-s 17 (1964): 396-402. "A Note on Recursive Relations.• ~ 33 (1968): 107. "A Method for Avoiding the Curry Paradox." In N. Rescher, ed., Essals in Honor of Qarl ~· Hempel. Dordrecht-Holland: D. Reidel, 1969 "Combinatory Logic and Negative Numbers." In K. Lambert, ed., !Qe Logical Way of ~oing ~higgs, in honor of Henry s. Leonard. New Haven: Yale University Press, 1969. "Natural Deduction Rules for English."~ ~QJl.hical Studies 24 (1973): 89- 104. •i:iber formal unentscheidbare Satze der Principia Godel, K. Mathematica und verwandter System, I." Mon. f. Math. und .Ehysik 38 (1931): 173-98. Grosky. w. I. "A Mathematical Theory of Constructible Automata.• Ph. D. Thesis, Yale University, New Haven, 1071. ~numerability~ ~ecidab;i.l~tl, Computabilit~. New Hermes H. York: Springer-Verlag, 196). f. c. "A Theor,y of Positive Intep;ers in Formal Logic." KleP.ne, Amer. J. Math. 57 (lD35): 153- 73· urr:a~:s,
H. J., and Fitch, F. B. "A Theory of Computing r~1a
chines." Studium Gener,?.k 22 (196')): 83- 101+. "!< Theory of Programming Languar;es." §.!;~.hll!!l (i<:.Q~E§l.f. 22 (l96o) : 113 - 36. ~.uine,
w. v.
"Variables ~~xplained Away." P.EQ£::~. 0M.e. . ___ .QL--~-
Amer. Philos. §oc. 101+ (1960): 3'+3- 47. I
158
ELEMENTS OF COMBINATORY LOGIC
Rosser, J. B. "A Mathematical Logic without Variables." A.E£.:._ of Math. 36 0935): 127- 50, and Duke Math. J. l (1935): 328- 55. "New Sets of Postulates for Combinatory Logic." ~ 7 ( 1942): 18- 27 Deux Esguisse€ ge LQgigye. Collection de logique math~matique, S~rie A, 7. Paris: Gauthier-Villars, 1955; Louvain: E. Nauwelaerts, 1955. Schonfinkel, M. "Uber die Bausteine der mathematischen Logik." Math. Ann. 92 ( 1924): 305- 16. Stenlund, s. Introduction to QQmb~Qato;y LQg~Q· Mimeographed. Filosofiska Studier, nr 11, Uppsala University, 1971. Tarski, A. "Der Wahrheitsbegriff in den formalisierten Sprachen." Studia J::hilosJ2his;;a l 0936): 261405. ~ogic, Semagt~cs, Metam?themat~£€• New York: Oxford University Press, 1956. Titgemeyer, R. "Doer einen Widerspruch in Cogans Darstellung der Mengenlehre." Z. f. Math. Logik und Qr. der ~ 7 (1961): 161-63.
INDEX OF ABBREVIATIONS FOR RULES
clos N add, 90 clos N exp, 92 clos N mult, 90 clos N sue, 87 cnst dil, 18, 40 cnv u q dis, 100 cnv univ dis, 100 cnv < dis :;:, lll cnv < dis id dis <, lll co imp elim, 20 co imp int, 19 com add, 89 com add 1, 89 com assoc add, 90 com mult, 91 com mult l, 90 conj elim, 2, 38 conj int, 2, 38 D conj int, 141 D dis int, 141 D e q int, 145 D elim, 140 D intersect int, d m, 25, 40 d m q, 99 D neg int, 11-1-l
abs elim, 73, 75 abs id, 73, 76 abs int, 73, 75 add cond, 15 add elim, 80 add exp, 81 add id, 81 add int, 80 assoc add, 81 assoc mult, 63, 79 B elim, 60 B id, 60 B int, 60 BCT id, 92 BCT id K, 93 BCT id O, 93 Bn- elim, 149 Bn- id, 149 Bn- int, 149 C elim, 57 c id, 57 C int, 57 can add, 95 can mult, 113 can sue, 9'+ clos ex mid, 108 159
158
ELEMENTS OF COMBINATORY LOGIC
Rosser, J. B. "A Mathematical Logic without Variables." A.E£.:._ of Math. 36 0935): 127- 50, and Duke Math. J. l (1935): 328- 55. "New Sets of Postulates for Combinatory Logic." ~ 7 ( 1942): 18- 27 Deux Esguisse€ ge LQgigye. Collection de logique math~matique, S~rie A, 7. Paris: Gauthier-Villars, 1955; Louvain: E. Nauwelaerts, 1955. Schonfinkel, M. "Uber die Bausteine der mathematischen Logik." Math. Ann. 92 ( 1924): 305- 16. Stenlund, s. Introduction to QQmb~Qato;y LQg~Q· Mimeographed. Filosofiska Studier, nr 11, Uppsala University, 1971. Tarski, A. "Der Wahrheitsbegriff in den formalisierten Sprachen." Studia J::hilosJ2his;;a l 0936): 261405. ~ogic, Semagt~cs, Metam?themat~£€• New York: Oxford University Press, 1956. Titgemeyer, R. "Doer einen Widerspruch in Cogans Darstellung der Mengenlehre." Z. f. Math. Logik und Qr. der ~ 7 (1961): 161-63.
INDEX OF ABBREVIATIONS FOR RULES
clos N add, 90 clos N exp, 92 clos N mult, 90 clos N sue, 87 cnst dil, 18, 40 cnv u q dis, 100 cnv univ dis, 100 cnv < dis :;:, lll cnv < dis id dis <, lll co imp elim, 20 co imp int, 19 com add, 89 com add 1, 89 com assoc add, 90 com mult, 91 com mult l, 90 conj elim, 2, 38 conj int, 2, 38 D conj int, 141 D dis int, 141 D e q int, 145 D elim, 140 D intersect int, d m, 25, 40 d m q, 99 D neg int, 11-1-l
abs elim, 73, 75 abs id, 73, 76 abs int, 73, 75 add cond, 15 add elim, 80 add exp, 81 add id, 81 add int, 80 assoc add, 81 assoc mult, 63, 79 B elim, 60 B id, 60 B int, 60 BCT id, 92 BCT id K, 93 BCT id O, 93 Bn- elim, 149 Bn- id, 149 Bn- int, 149 C elim, 57 c id, 57 C int, 57 can add, 95 can mult, 113 can sue, 9'+ clos ex mid, 108 159
160 D union int, 143 DA int, 142 DD int, 141 DE int, 11+2 dis elim, 8, 39 dis int, 2, 38 Dn Bn- int, 149 Dn dft, 147 Dn id, 147 DN int, 141 Dn tfly int, 149 Dn tn& int, 149 n1 id, 146 n1 dft, 146 D2 prop anc int, 153 n2 rel po int, 152 n2 rel prod int, 150 e q elim, 98 e q int, 99 ex mid, 26 ex mid id, 41, 46 ex mid N, 87 ex mid ~· 108 ex mid <, 108 exist elim, 98 exist int, 98 exp o, 80 exp 1, 80 ext, 42, 43, 49, 50 F id, 93 I elim, 53 I id, 53 I int, 53 id coup, 93 id elim, 41, 42 id ent ~· 106
ELEMENTS OF COMBINATORY LOGIC id int, 41, 1+2 imp elim, 2 imp int, 5 ind pr, 22 ind pr id, 48 intersect id, 129 irrefl <, 110 K int, 55 K id, 54 K elim, 55 left add 0, 81 left dist mult add, 81 left mul t sue, 91 left mult 0, 90 left mult 1, 79 m F, 94 m p, 2, 142 m p c, 19 m t p, 26, 40 man add non-id, 95 mon add ~· 107 mon add <, 112 man id, 47 mon mul t ~· 108 mon mul t <, 112 mon sue non-id, 94 mult elim, 62 mult exp, 80 mult id, 62 mult int, 62 N elim, 87 N indue, 87, 88 N int, 87 neg abs elim, 73, 75 neg abs int, 73, 75 neg add elim, 80
INDEX OF ABBREVIATIONS FOR RULES
161
---------------------------------
neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg
add int, 80 B elim, 60 B int, 60 C elim, 57 C int, 57 conj elim, 22, 39 canj int, 22, 39 dis elim, 22, 40 dis int, 22, 40 e q elim, 99 e q int, 99 elim, 22, 38 exist elim, 99 exist int, 99 I elim, 54 I int, 54 int, 22 int id, 47
neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg
int ~· 109 int <, 109 K elim, 55 K int, 55 mult elim, 62 mult int, 62 T elim, 58 T int, 58 u q elim, 99 u q int, 99 univ elim, 99 univ int, 99 W elim, 60 W int, 60 t elim, 147 t int, 147 0 elim, 78 0 int, 78
neg 1 e1im, 78, 84 neg 1 int, 78, 84 neg 2 elim, 84 neg 2 int, 84 neg 3 elim, 84 neg 3 int, 84 neg ~ elim, 104 neg ~ int, 104 neg < elim, 104 neg < int, 104 neg 2 elim 22, 38 neg2 int, 22, 38 non-id int, 48 non-id sue, 96 non-sue o, 82 prop anc e1im, 153 prop anc id, 153 prop anc int, 153 r b id, 21 refl coimp, 21 refl imp, 16 refl ~· 106 reit, 2 rep, 16 res id elim, 42, 43, 87, 144 res imp int, 28, 142 res ind pr, 145 rt add 0, 81 rt dist mult add, 91 rt mult sue, 90 rt mult o, 79 rt mult 1, 79 sp ~ elim, 104 sp ~ int, 105 str dis elim, 102 str e q elim, 102
160 D union int, 143 DA int, 142 DD int, 141 DE int, 11+2 dis elim, 8, 39 dis int, 2, 38 Dn Bn- int, 149 Dn dft, 147 Dn id, 147 DN int, 141 Dn tfly int, 149 Dn tn& int, 149 n1 id, 146 n1 dft, 146 D2 prop anc int, 153 n2 rel po int, 152 n2 rel prod int, 150 e q elim, 98 e q int, 99 ex mid, 26 ex mid id, 41, 46 ex mid N, 87 ex mid ~· 108 ex mid <, 108 exist elim, 98 exist int, 98 exp o, 80 exp 1, 80 ext, 42, 43, 49, 50 F id, 93 I elim, 53 I id, 53 I int, 53 id coup, 93 id elim, 41, 42 id ent ~· 106
ELEMENTS OF COMBINATORY LOGIC id int, 41, 1+2 imp elim, 2 imp int, 5 ind pr, 22 ind pr id, 48 intersect id, 129 irrefl <, 110 K int, 55 K id, 54 K elim, 55 left add 0, 81 left dist mult add, 81 left mul t sue, 91 left mult 0, 90 left mult 1, 79 m F, 94 m p, 2, 142 m p c, 19 m t p, 26, 40 man add non-id, 95 mon add ~· 107 mon add <, 112 man id, 47 mon mul t ~· 108 mon mul t <, 112 mon sue non-id, 94 mult elim, 62 mult exp, 80 mult id, 62 mult int, 62 N elim, 87 N indue, 87, 88 N int, 87 neg abs elim, 73, 75 neg abs int, 73, 75 neg add elim, 80
INDEX OF ABBREVIATIONS FOR RULES
161
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neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg
add int, 80 B elim, 60 B int, 60 C elim, 57 C int, 57 conj elim, 22, 39 canj int, 22, 39 dis elim, 22, 40 dis int, 22, 40 e q elim, 99 e q int, 99 elim, 22, 38 exist elim, 99 exist int, 99 I elim, 54 I int, 54 int, 22 int id, 47
neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg neg
int ~· 109 int <, 109 K elim, 55 K int, 55 mult elim, 62 mult int, 62 T elim, 58 T int, 58 u q elim, 99 u q int, 99 univ elim, 99 univ int, 99 W elim, 60 W int, 60 t elim, 147 t int, 147 0 elim, 78 0 int, 78
neg 1 e1im, 78, 84 neg 1 int, 78, 84 neg 2 elim, 84 neg 2 int, 84 neg 3 elim, 84 neg 3 int, 84 neg ~ elim, 104 neg ~ int, 104 neg < elim, 104 neg < int, 104 neg 2 elim 22, 38 neg2 int, 22, 38 non-id int, 48 non-id sue, 96 non-sue o, 82 prop anc e1im, 153 prop anc id, 153 prop anc int, 153 r b id, 21 refl coimp, 21 refl imp, 16 refl ~· 106 reit, 2 rep, 16 res id elim, 42, 43, 87, 144 res imp int, 28, 142 res ind pr, 145 rt add 0, 81 rt dist mult add, 91 rt mult sue, 90 rt mult o, 79 rt mult 1, 79 sp ~ elim, 104 sp ~ int, 105 str dis elim, 102 str e q elim, 102
162
str exist e1im, 101 str u q int, 101 str univ int, 100 sym eoimp, 21 sym id, 4-4T elim, 58 T id, 58 T int, 58 trans eo imp, 21 trans id, 4-4trans imp 8, 14trans ::;. 105 trans <, 107 trans < ::::;, 107, lll u b id, 51 u q dis, 100 u q e1im, 97 u q int, 97 union id, 131 univ dis, 100 univ elim, 97 univ int, 97 W e1im, 60 w id, 60 W int. 60 zeta, 4-6 ~ int. 14-7 i id, 14-7 ~ elim, 14-7 ~n e1im, 11+8 tn id, 14-8 ~n int, 14-8 ~nv elim, 14-8 ~~ id, 14-8 t~ int, 14-8 tnli e1im. 11+8
ELEMENTS OF COMBINATORY LOGIC ~nli
e1im, 14-8 14-8 ~nli int, 14-8 0 elim, 78 0 id, 78, 85 0 id <, 109 0 int, 78 0 ::;. 106 0 < sue, 107 1 elim, 78 1 id, 78, 85 1 int, 78 2 elim, 842 id, 85 2 int, 843 elim, 8'+ 3 id, 85 3 int, 844- elim, 844- id, 85 4- int, 84::: add, 106 :: elim, 104::: ent id dis, 110 :: ent neg env <, 111 :: id, 105 :: int, 104< ae1d sue, 106 < e1im, 1.01+ < ent neg env ::;. 112 < ent non-id, lV < ent; ::;. 107 < id, 10 ') < int, 101+ < sue, 106 ~nli id,
{ o. 109
