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STUDIES IN PROOF THEORY
Advisory Editors
BIBLIOPOLIS
. PER MARTIN-LOF
Notes 'by Giovanni Sambin of- a series 0/ lecturesgiven in Padua, June i980
INTUITIONISTIC TYPE THEORY
P. MARTIN-LOFG .E . MINCW. POHLERSD. SCOTTW. SIEGC. SMORYNSKIR. STATMANS . TAKASUG . TAKEUTI
D. PRAWITZH. SCHWICHTENBERG
C. CELLUCCIJ..Y. GIRARD
P. AczELC. BOHM .
W. BUCHHOLZE. ENGELER
S . FEFERMANCH . GoAD
W. HOWARDG.HUET
D. LEIVANT
Managing Editors
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STUDIES IN PROOF THEORYLECTURE NOTES
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u .
I . CONTENTS
Natura l numbers . . . . ... . . .. . .. . .. . ..... . . . . . . . .. . . . . . .. . . .. . . 71
Pr-opos Lt i ona l equa l i t y ... ...... . ...... . . : . . . . . .... . . . .. . . . . . . .
Lis t s ; . . . . . . . . . . . . . . . . . . . . . . . 77Wellorder ings ........ . ..... . . . . . . . . . . ......... .. . . . .. .... . . 79Un iverses .... .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
, , '.. " .
Fin i t e se tsConsis tency
Di s j o i n t union of " two s e t s . . .... . .... . . . . ..... . . . . .. . . . . . . . . '. . . 55596569
In t roductory remarks .Proposi t ions and judgements " 3Explanat ions o f th e form s o f judgement ' 7Pr op os i t i ons 11Rule s o f e q ua l i ty , . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 14Hypothe t ica l judgements and s u b s t i t u t i o n ru l e s 16Judgements wi t h more than one as sumption and contex ts 19Sets an d ca te go r i e s _ 2 1General remarks on th e rule s 24Carte s ian product of a family of s e t s 26De f i n i t i o n a l eq ua l i t y : 31App Lf ca t Lcns of the ca r t es ian produc t ; ' ,' 32Disj oin t un ion of a family of s e t s .. . . . .... ..... . . .. . . .. . . . . .. . 39Applic a t ion s o f th e ' di s jo in t un i on 42The axiom of' choice ...... . . ..... . . ... .. . ..... .. . . . . . . . . .. . . 50The n ot io n o f such tha t .. . . . . . . . .. . . . . . .. . . . .... .. ....... . . . .. 53
I\
1l
IitII 1984 by Bibliopolis, edizioni di f i l ~ s o l 1 i i scienze Napoli, via Arangio Ruiz 83 , ';. :,' " " , ,All rights reserved, No part of'ihis>-,1;;qp!i:,ti,ay 'be reproduced , in anyform or by any means without permission' -inwriting from. the publisherP ri nt ed in I taly by Grafitalia Via Censi dell'Arco, 25 Cercola (Napoli)
)
ISBN 88-7088-105-9
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It1,I
I
PrefaceThese lectures were given in Padova a t th a Laboratorio pe r
Ricerche 'd i Dinamica de i Sistemi e di Elettronica Biomedica th e,Cons i g l i o Nazionale delle Ricerche dur in g t he month of June 1980.I am i nd eb te d t o Dr'. , Enrico Pagello of t h a t l abo ra to r y f o r th e op -portunity of s o d oi ng . The audience was made up by phi losophers,mathematicians and compute r s c i e n t i s t s . Accordingly, I t r i e d to sa ysomething which might be of i n t e r e s ~ to each of these three c a t -egor ie s . Es sen t ia l ly th e same l e c t u r e s , a l b e i t in a somewhat im -proved a nd mo re advanced form, were given l a t e r in th e same yearas par t of the meeting on Konstruktive Mengenlehre und Typentheoriewhich was organized in Munich by Prof. Dr. Helmut Schwichtenberg,te whom I am indebted fo r th e i n v i t a t i o n , dur in g t he week 29 Sep-
~ e m b ~ ~ - 3 October 1980.The main improvement of th e Munich .l e c t u r e s , as compared wi t h
those given in Padova, was t o e adop t ion of a sy s tema ti c h ighe r level(Ger. Stufe) notation which allows me to write simply
Fl (A,B), L(A,B), W(A,B) , ),(b),E(c ,d) , D(c ,d,e ) , R(c , d , e ) , T(c;d)
ins tead of
(Tl x EO A)B(x), (L x e A)B(x) , (Wx e iI)B(x) , (AX)b(x),E ( c , ( x , y ) d ( x , y , D ( c , ( x ) d ( x ) , ( y ) e ( y ; R(c,d,(x,y)e(x,y ,T ( c , ( x , y , z ) d ( x , y , z ,
respec t ive ly. Moreover, th e use , of higher l e ve l v a ri ab l es arid con-s t a n t s makes i t possible t o f ormul at e th e elimination and equali tyr ul es fo r th e car tes ian product in such a way tha t they follOw th e
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'. ,.
IT-e l iminat ioncEI : (A ,B )
c = (p(c) , q ( c ) E E (A,B)
c (Ax)Ap(c,x) E n(A,BJC E n (A,B)
F(c ,d ) -= d( (x)Ap(c , x .
s tockho lm, J anuary 1984,
r ~ l e s . Moreover, th e second of these , that i s , th e rule
can be derived by m ea ns of th e ~ - r u l e s in th e sa me wa y as the rule
is .der ived byway of example on p . 62 of th e main text . Conversely ,th e new elimination and equali ty rules can be derived from th e oldones by making th e defini t ion
So , actually , they ar e equivalent.I t only remains fo r me to thank Giovanni Sambin fo r having
undertaken, a t his own s ug g es ti on , t h e c o ns id e ra b le work o f w ri ti ngand t yp in g t he s e n o te s , therebf making th e l e c ture s acces s ib le to awider aUdience.
l!1Idey) 6 C(A(y
d(yl ' 6 cO.(y ,(y Lx ) E. B(x) (x oS A
(y(x) E. B(x) (x E. A
F ( c " d) 6 C( c )
F(A,(b) ,d )
. (x E A)b Cx ) 6 B(x)
C E. n (A ,B )
r-espect.tvely . Here y is a bound function variable, F i s a new nonc a n o ~ i c a l (e l imina to ry ) ope rato r by m ea ns of which th e binary ap plication operation can be defined, putt ing
same ,p a t t e r n as t h e e l imina ti on and e qu al it y r ul es f or a l l th e othertype forming opera t ions . In their new fo rmula ti on, t hese rules read
Ap(c,a) == F(c, (y )y (a , Pel' Martin-Lofand y(x) E. B(x) (x E. A) i s an assumption, i t s e l f hypothe t ica l , whichha s been pu t wi thin pa ren theses to indica te tha t it is being d is -charged. A program of the new form F(c,d) has v a l u e ~ provided c ha svalue A (b ) and deb) h as v al ue e. This rule fo r eva lua t ing F (c ,d )reduces to t he l az y e va lu at io n rule fo r Ap(c,a) when th e above definition i s being made. Choosing C(z) to be B(a), thus independen t ofz, and d(y) to be y(a) , th e new eiimination rule reduces tp t he o ldo ne an d th e new e qu al it y r ul e t o th e f i r s t of th e two ol d equali ty
1jI1
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I',II::1!II
Introductory remarks
,Mat hemat i cal logic and re la t ion between logic and mathematicshave been i n te r pr e te d i n a t l ~ a s t three d i f fe ren t ways:
(1 ) mathematical logic , as symbolic logic, or logic using mathe-matical .ymbolism;(2) mathemat ica l logic 'a s foundations to r philosophy) of mathe-matics;(3 ) mathematical logic as l ogi c s tud ie d by mathematical methods,as a branch of mathematics. '
We sha l l here mainly be interested in mathematical l og ic i n th e secondsense. What sha l l do is also mathematical logic in th e f i r s t sense,bu t certainly no t in th e third .
'The princ ipa l problem tha t remained a f t e rP ' r inc ipia Mathematicawas comple ted was, according to i t s authors , that of justifying th eaxiom of reducibili ty (or , as we would now say, th e impredicative com-prehension axiom) '. The, ramif i ed theo ry of types was predica t ive , bu ti t was no t su f f ic ien t fo r deriving even elementary parts o f a nal y si s .So axiom of reducibili ty was ~ d d e d on th e pragmatic ground that itwas needed, although no . a t is f a c to r y j u s ti f i ca t i o n (explanat ion) of itcould be provided. The whole p oin t of the ramif ica t ion was , then lost ,so that i t might ju s t a s w el l be abol ished. What then remained wast he s impl e theory o f t yp es . I ts o f fi ci al just i f icat ion (Wittgenstein ,
R ~ ~ s e y ) res t s on th e interpretation o f p r oposi ti ons as truth valuesand proposi t ional f un c ti on s ( o f one or s e ver a l v a ri a bl e s; as t ru thfunct ions. The laws of th e classical p r opos it iona l l ogi c a re thenc lear ly valid, and so ar e th e quan t i f ie r laws , as long as quantif ica -tion is res t r ic ted to f ini te domains. However, it does no t seem possible to make sense of quantif ica t ion over in f in i t e domains , l ike th e
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ment: '
separ-a te s teps:
Here th e , d i s ~ i ~ c t i o n b e t w e e n proposi t ion (Ger. Sa tz ) and as s er t ion or judgement '( Ge r . Ur te i l ) i s es s en t i a l . What we combine b y m ea nsof' th e logica l opera t ions ( . l : :> & v V 3 '), , , , " an d hold to be tr ue a reproposi t ions . When we hold a proposi t ion to be t rue , we make a j ~ d g e -
s pe ak , t he y f'orm an open c on ce p t, I n standard textbookpresenta t ions o f f i r s t order l og ic , we d'ca n ~ s t i n g u i s h three qui te
The d i s t inc t ion between proposi t ions and jUdgements was c learfrom Frege to Pr i nc i p i a . These not ions have l a t e r be 1en re p aced by th eformal i s t i c n ot io ns o f formula and theorem (i n a f ormal s y st em ) , re -spec t ive ly. Contrary to formUlas, proposi t ions a re no t def'ined induct ively . So to
(2 ) spec i f ' i ca t ion o f aXioms and ru l e s o f i nf er e nc e ,
Proposi t ions and Judgements
(3 ) semant ica l i n t e rp r e t a t ion .
proposi t ion ~ J U d g e m e n t ,In par t i cu l a r , ' t he premisses and c o nc l us i on o f a l og i c a l i n fe r en c e a r ejudgements.
(1 ) induct ive def in i t i on of terms an d formulas ,
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Fo rmu la s a nd dd t 'e uc ~ o n s ar e given meaning only through semant icswhich i s usual ly done follOWing Tarsk i and ' ,assuming se t theory .What we do here i s t t, mean 0 be c l ose r to ord inary mathematical
p r a c t ~ c e . We wi l l avoid keeping form a nd mea ni ng (con ten t) apa rt In -stead we w il l a t th e same t ime display cer t a in forms o f j U d g e m e n ~ andin ference t ha t a re used in mathemat ica l d ', " proofs an expla in , them seman-t i ca l ly . Thus we make exp l i c i t h tw a i s usual ly impl i c i t l y taken f'or
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l ' C. A. Hoare, An axiomat ic b as is o f computer programming, Communica t ions of th e ACM, Vol. 12 ,196'9, pp . 576-580 and 583.2 E. W. Di jks t r a , A d i s c ip l ine of Programming, P r e nt ic e Ha l l,
Englewood Cl i f f s , N. J ., 1976.3 "P. Mar t in-Lof , Construct ive mathematics an d compu t er p rog r am-
ming, Logic, Methodology and Philosophy of Science VI ; Edited ,b yL. J . Cohen, J . Los, H. Pfe i f f e r and K.-P. Podewski , North-Holland,Amsterdam, 1982, pp . 153-175. '
domain of na t u ra l numbers, on t h is i n te r p re t a ti o n o f th e n oti on s ofproposi t ion an d proposi t iona l f ~ n c t i o n . Fo r th i s reason , afuong othe r s ,what we develop here i s an i n t u i t i o n i s t i c theory o f types ; which i sa ls o p re di ca ti ve ( or ' ramified) . It i s free from th e defic iency ofRusse l l ' s ramif ied theory o f types, as r eg ar ds t he pos s ib i l i t y of de ve lop ing e l emen t ar y part s o f mathematics, l i ke th e theory of ' r e a l numbe rs , because of t h e p re s en c e of the opera t ion which allows us to formth e ca r t e s i an produc t o f ,a ny g i ve n f am i ly o f s e t s , in par t i cu l a r , th ese t of a l l funct ions f rom o ne se t to another .
In two areas , a t l e a s t i ou r language ~ e e m s to have advantages over t r ad i t i ona l foundat ional languages . F i r s t , Zermelo-Fraenkel se ttheory c anno t adequa te l y deal with t h e f o un d a ti o na l problems of c a t egory theory , where th e ca tegory o f a l l s e t s , t he c at eg or y of a l lg ro up s, t he , ca tegory o f f un ct or s fro m o ne such ca tegory t o a no th e re t c . ar e considered . These problems coped with b y m ea ns of th ed i s t inc t ion between s e t s an d c at eg o ri es ( in th e l og ica l o r phi losophi ca l sense , not in th e sense o f c at eg or y t he or y) which i s made in i n tu i t i on i s t i c type theory . Second, present l og ica l symbolisms a re i na dequa te as programming languages, which expla ins why computer sc i ent i s t s have developed t h e i r own languages (FORTRAN, ALGOL, LISP,PASCAL, .. . ) and systems o f proof ru l e s (Hoare1 , Di jks t r a2, . ) . Wehave shown elsewhere 3 how th e add i t i ona l r ichness of ' t y p e theory , ascompared with f i r s t order predica te l og ic , makes it usable a s a programming language .
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A
A v B
4 -
A i s solvable
A is f u l f i l l a bl e. ( rea l izable)
a E A
a i s a method of solving theprob l em (doing t he t as k) A
a is a method of f ~ l f l i l i n g( rea l iz ing) th e intent ion( exp ec ta t ion ) A
A i s a set a i s an 'eLemen t of th e se t . A A is nonempt yA i s a pr opos i t ion a i s a pr oof ( con s t ruc t i on) of A is true
the propos i t i on A
A se t
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( 2 ) A and B equal ~ e t s (A B) ,
(4 ) a and b ar e eq ua l e lemen ts of th e set A (a = b E A) .( 3) a is .an element of th e se t A (a A) ,
. We us e f ou r f orms of j U d g ~ m e n t :(1 ) A is a se t (a bb r . A
i n th e ta.ble:
A i s a problem( task)
A i s an i ntention(expecta t ion)
( I f we read l i t . r a l ly a s ~ a ~ ~ , then we ~ i g h t A E Set ,A = BESe t , a e El (! ) , a = b E El(A ) , re spectively .) Of cou rs e , anysyntact ic variabl e s co u ld be us ed th e f 11se 0 sma l e t t e r s f or e l -ements and ca pi t a l le t ters f or ae'ts i s onl y t o convenience . Not e t ha tin or d i n a r y se t t he or y , a E b and a = b ar e pr oposit i on s , wh i l e theyar e j udgements here. A jUdgement of th e f orm A = B has no mean in gless we already know A and B t o sets . Likewis e , a jUdgement of t hef orm a E A pr esupposes th a t A i s a se t , and a judgement of t he f orma = b E A pr esuppose .s, f i r s t , that A i s t dse , an , se cond, t ha t a andb ar e elemen ts of A .
Each f orm of judgemen t admits f 1severa differe nt read ings , as
ric 'I
A tr-ueprop .
l-AyB
A v B t r ue
A, B prop . .1- A
A prop.
or
where we use, l ike Frege, th e symbol l- to th e le f t of A to s ignifythat A is t rue . In our system of rules , . t hi s wi l l always be exp l ic i t .A r ul e of inference is j us t i f ied by exp la ining the conclusion onth e assumption that th e pr emisses ar e known. H.ence, before a ru le o finferenc e can be just i f ied, i t must be explained wha t i t is tha t wemust know "in ord e r to have th e r igh t to make a judgement of any oneof t he v ar io us f orms tha t th e premisses and cOnclusion c an h av e.
takes f or granted tha t A and B ar e f ormulas, and onl y then doe s i t saythat we can i n f e r A v B to be true when A is true . .I f we ar e t o give af ormai rule , we have to make th i s expllc i t , writing
granted . When one tr e a t s lo gi c as arty ot he r branch of mathematics, asi n th e metamathematical t radi t i on origina ted by Hilber t , such ju dge ments and in f erences are onl y par t ia l ly and f ormal l y re pr ese nted inth e s o- called ob j e c t l an gu ag e, w hi le t hey ar e implic i t ly used; as inany other branch of mathematics, in the s o- ca l le d metalanguage.
Our main aim is to build up a system of f orma l rules representingin th e be s t possible way informal (mathematical ) re asoning. In th eusua l natu r a l deduction style , th e rules given ar e no t quite . f ormal .For i n s t an ce , th e rule .
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4 A. Heyt ing , Die in tu i t ion is t i sche Grundlegung de r Mat hemat ik ,Erkenntnis, Vol. 2, 1931 , pp. 106-115 . .
5 A. N. Kolmogorov , Zur Deutung de r in tu i t ion i s t i schen Logik,Zeitschrif t , Vol. 35 . 1932, pp . ~ 8 - 6 5 .
6 H. B. Curry and R. Feys, Combinatory Logic, Vol. 1, North-Holland ; Amsterdam , 1958 , pp , 312-315.
7 w., A. Howard , The formulae-as-types notio n of cons truc tion , ToH. B. Curry : Essays on Combinatory Logic, Lambda Calculus arid Formalism , Academic Press, London , 1980 , pp. 4 19 -4 90 .
The se cond, logical interpretation is discussed toghether with rulesbelow . The third was suggested by Heytin g4 and t he fourth by Kol mogOrov5 . The las t i s very close to programming. " a is a method . . . " can 'be read as "a i s a program . .. ". Since programming languages have aformal notation fo r th e program a, bu t not fo r A, we complete t he s en t en ce w it h " . . . which meets th e s p e c i f i c a t i on A". In .Kolmogorov 's in - t e r p r e t a t i o n , th e word problem refers to something to be done and th ewor d program to how to do i t . The analogy be tween t he f i r s t and th esecond interpretation i s . implici t in th e Brouwer-Heyting in te rp re t at ion , ~ h e logical c o n s t a n t I t was made more exp l ic i t bY ,Curry ,andFeys6 , b ut o nl y fo r th e impl i ca t iona l fragment , and i t wa s extended t o, . 7in tu i t ion i s t i c f i r s t order ar i thmetic by Howard _ I t is th e only knownway of i n t . r p r e t i n ~ i ~ t u i t i o n i s t i c logic so tha t axiom of 6hoi c ebecomes , valid .
To d is t inguish between proofs. of judgements ( u su a ll y i n t ree - l ikef orm) a nd proofs of propos i t ions (here identif ied with elements , thusto th e l e f t of E . ) we reserve th e wo rd c ons t ruction f or the la t ter andus e it when confusion might occur.
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Explanations of th e forms of judgement
by which i ts elements a re c on st ructed. However" th e we akn es s of th i sdef ini t ion is clear : 1010 , fo r ins tance, though n ot o bt ainable wi tht h e g iven rules, is clearly an element of N, sinc,e we know t ha t we 'can bring it to th e form a ' fo r some a e ' N. W thus ha ve . to di s t i n
~ u i s h th e elements which have a form by which we can di rec t ly se et ha t they ar e th e resu l t of one of the rules , and 'c all th em cano ni ca l , from a l l othe r e lements, wh i ch we wi l l cal l nonc anonica l .
a 'E No E Na N
The f i r s t ,i s th e ontolog i ca l (anc i en t Greek) , t he second th e ep is t e m o l ~ g i c a l (Descartes, Kant, . ) and th e third th e semant ic a l (modern) way of posing essentially th e same question . At f i r s t s igh t , wecould assume that a se t i s defined by prescr ibing how i t s elementsar e f ormed. Th is we do when we say tha t th e se t of na tura l numbersN i s defined by giving th e rules :
What does a judgement of the form "A is a se t" mean?
What i s it t ha t we must know in o rd er t o have th e r ight to judgesomething to be a set?
Wha t is a set?t ions :
For e ac h one of the . four forms of judgement, we. now explain wha ta judgement of t ha t form means . We can explain what a judgement , s a yof th e f i r s t form, means by answering one of th e followin g three ques -
\6 -
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b Eo B
- 9 -
(2 ) two se ts A and Ba r e equal ifa A a E A a . B--- ( tha t i s , and --)a B a B a 6. A
(4 ) two a rb i t ra ry elements a, b of a set A ar e equal if , whenexecuted, a and b yie ld equa l canon ica l elements of A as resul1
This is th e meaning of a judgement of th e form a . A. Note that herewe assume t he n ot io n of method as primit ive . The rules of computatic
of th e present language wil l be such tha t th e computaticof an element a of a se t A terminates with a .va l ue b as soon as th eoutermost form of b t e l l s t hat, it . i s a canonical element of A (norm"order or l a zy eva lua t ion ) . Fo r ins tance , th e computation of 2 + 2 g iv es t he v al ue (2 + 1) ' , which is a canonical element of N since2 + 1 E N.
Final ly :
and
fo r a rb i t ra ry canonical elements a , b.
abE A
(3 ) an element a of a set A is a method (o r progr am) which , wheexecuted, yie lds a canonical element of A as re su l t .
This is th e meaning of a judgement of th e form A= B.When we explain what an element of a se t A i s , we must a s s ~ m e
we know tha t A is a se t , that is , i n p a rt ic u la r , how i t s canonical. elements ar e formed. Then:
b' . N'
b = d . B
b B
and
a
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This is th e meaning of a judgement of the form a = b A. This d e f i -n i t i o n makes good sense since it is ' p a r t of the d e f i n i t i o n ' o f a s e twhat it means fo r two canonical ,elements o f th e s e t to be equal .
Example. I f e, f A x B, then e an d f ar e methods which yieldcanonical elements ( a , b ) , (c ,d) E A x B, r e s p e c t i v e l y , as r e s u l t s ,and e f E A x B i f (a ,b) ( c , d ) E A x B, which in turn ~ o l d s if
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Proposi t ions
8 D. Prawitz, I n t u i t i o n l s t 1 c l o g i c : a phi losophical challenge ,Log ic a nd Ph ilo sophy , Ed i ted by G. H. von W r i g h t Mar ti nu s Ni j h of f ,The Hague , pp . 1-10.
a proposi t ion is def ined by laying down what counts 'a s ' a proofof th e proposi t ion ,
a proposi t ion is t r u e i f it ha s a proof, t h a t is , i f a proof of8it can be given
and t h a t
C l a s s i c a l l y , a proposi t ion is n ot hi ng b ut a t r u t h va l ue , t h a tan element of th e s e t o f t r u t h values, who se two elements ar et rue and th e f a l s e . Because of th e d i f f i c u l t i e s of j u s t i f y i n g
th e r u l e s fo r forming proposi t ions by 'means of q u a n t i f i c a t i o n overi n f i n i t e domains , when a proposi t ion i s understood, as a t r u t h value ,t h i s explanation is r e j e c t e d by th e i n t u i t i o n i s t s and replaced bysaying t h a t
Thus , i n t u i t i o n i s t i c a l l y , t r u t h i s i d e n t i f i e d with p r o v a b i l i t y , thougho f c ou rs e' n o t (because of G ~ d e l incompleteness theorem) wi t h deriva b i l i t y within any p a r t i c u l a r formal system.
The e xp la na ti on s o f t he meanings o f th e l o g i c a l operat ions, whichf i t together with th e i n t u i t i o n i s t i c conception of wha t a proposi t io ni s , ar e given by th e standard t a b l e :
d B;= c E A and b
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' t he f i r s t l ine ' of which should be interpreted a s s ay in g that therei s nothing that counts as a proof .
The above table can be made mor e explic i t by saying:I 'III
t
\II,I
a proof of the proposition.L
A & BAV BA :> B
( Vx)B(x) ,
(3 x)B(x)
a pro of of the proposition.i ,
A & B
A v B
A :> B
('o'x)B(x)
(3 x)B(x)
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consis ts of
a proof of A and a proof of Ba proof of A or a p ro of o f Ba method which takes any proofof A into a proof of Ba method which takes an arbi traryindividual a into a proof of B(a)an individual, a and a proof ofB(a)
ha s th e form
(a ,b) , where a is a proof of Aand b is a proof of Bi (a) ; where a i s a proof of A,or j{b), where b i s a proof of B(Ax)b(x), where b(a) i s a proofof , B provided a a proof of A(Ax)b(x), where b(a) i s a proofof B(a) prOVided a is an individual(a ,b) , where a i s an individualand b i s a p ro of o f B(a)
- 13 -
As i t s tands , this table is n o t s t r i c t ly cor-r-ect., since i t, showsproofs of canonical form only. ' An arbitary proof, in anal;ogy with anarbi trary element of a se t, is a method of producing a proof of ca nonical form.
I f we take ser iously , th e idea that a p r o ~ o s i t i o n is defined bylaying down how i t s canon i ca l p roo fs ar e f'or-med (a s in th e secondtable above) and accept that a se t is defined by prescribing how i t scanonical elements ar e formed, then i t is c le ar t ha t i t would onlylead to unnecessary dup li c at ion t o keep t he n ot io ns o f p r oposi ti onand se t (and th e associa ted notions of proof , of a proposition ' and e l-ement of a s et ) a p ar t. Ins tead, we simply identify them, that i s,t r ea t them as o ne an d th e same notion. This i s t h e fo rmu lae -as - types(propositions-as-sets) in terpreta t ion on Which in tu i t ion i s t i c typetheory is based.
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We no w begin to bui ld up a sy stem o f r ul es . F i rs t , we gi ve th efo l lowing rules of equal i ty , which ar e ea s i l y expla ined u si ng t hefa c t t h a t t h e y ho ld , by def in i t i on , fo r can on ic a l e l eme n ts :
Fo r i ns t ance , a de t a i l e d explana t ion o f t r an s it iv i ty i s : a = b E Ameans t ha t a and b yie ld canonica l elem ents d an d e , respec ti ve ly , andt h a t d = e A. S imi la r ly , if c yie lds f , e = f EA . S in c e we assumet rans i t iv it y fo r canoni ca l element s , we obta in d = f E A, which mean st ha t a = c A.
C, we "
Ba = b E B
b e A
b .. A
a E Aa
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(1 ) B(x) se t (x e 'A )
D(x)
(x E A)
(x E A)
D(c)B(x)
B(x) = D(x)
Bealc E. A
a E A
a
(2 ) B(x) = DUe) (x e. 'A )
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. Sub s t i tu t Lon
is correct. We can now d er iv e t he ruleB(a) = D(a)
which says tha t B(x) 'and D(x) ar e equal famil ies of sets over th ese t A; is that B(a) and D(a) a re e qu al sets fo r any element a of A(so , in particular , B (x ) a nd D(x) must be famiiies of sets overTherefore th e rule
B(x) i s a se t under- th e assumption x e .A, which does no t mean ' .t ha t,we mus t h av e a deriva t ion within any particular f o rma l syst em ( l iketh e one tha t we, are in t he p ro ce ss of building up). When an assumptionX E A is discharged by a pp li c at i on o f a rUle, we wri t e it ins idebrackets.
The meaning of a 'h ypot h e t i c a l Judgement of the fo rm
from th e above rules. In fac t , f r o m c ,E A and B(x) set (x E A) ,' we obtain B(a ) '= B(c) by th e second substitution r ul e , and f rom c E A,which is implic i t in a c E: A, B(c) = D(c) by th e t h ir d sub s ti t u ti o nru le . So B(a) = D(c) by , t r ans i t iv i ty .
(x E A)B(x) se t
B(c)(a)C E A
X E: A
a
B(x) s et
( x E A)B(x) se t
B(a) se ta E A
Subst i tut ion
The nota t ion
Hypothe t i ca l Judgements and substitution rules
only reca l l s that we make (have a proof of) the judgement tha t
which says that B(x) is a set under th e assumption x E A, or, better ,that B(x) is a family of se t s over A. A more t rad i t iona l notationi s {B 1 or {B : X E A} The meaning of a judgement of the formx J x E A x .(1 ) is tha t B(a) is a set whenever a i s an element of A, and alsotha t B(a) and B(c) ar e equal sets whenever a and c ar e equa l e lemen tso f A. By virtue of th i s meaning , we LmmedLat.eLy se e that th e follow-in g s u bs t i t u t i o n rules ar e correc t :
The four basic forms of ju dgement a re g en er al iz ed i n .order t oe xp re s i a l so hypothet ical judgements , i .e . judgements which ar e madeunder assumptions . In this sec t ion, we t r ea t th e case of one suchassumption . So assume tha t A is a se t . The f ~ r s t form of judgementi s general ized to t h e hypo the t ica l form
. I
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means t h a t b(a) and d (a ) a re equal elements of the se t B(a) fo r anye lement a of the se t A. We then have
- '9 -
A2(X,) is a family of s e t s ove r A"
(1 ) A(X" . . . xn) s e t (x , E A, . x2 e A2 ( x , ) , .. . .Xn E An ( x " . . . ,x n_,
A, is a se t .
A3(X"x2) i s a family of s e t s with two i nd ic es x , A, andx2 A2 ( X, ) ,
Jud gements with more than one assumption and contexts
An(x, ..,oo. , xn _ , ' i s a family of s e t s with n- ' indices x, . A"X2 E A2 (x,) , . . . , xn_ , E An_,(x, ' . . . xn_2) .
means tha t A(a" . . . ,an) is a s e t whenever a, . A" a2 E A2 ( a , ) , . , . ,an E A ~ ( a " . .. . an_') and that A(a" . . . ,a n) = A(b " .... , bn 'whenever a, = b, e A" . . . , an = bn An(a" .. . ,a n_,) . We sa y thatA(x" . . ,x n) is a family of s e t s with n indices . The rt assumptions ina judgement of th e form ( ,) c o n s t i t u t e what we c a l l th e context . wh i c hplays a role analogous to th e s e t s of f ormula e r . (ex tr a formulae)appearing in Gentzen sequents . Note a lso that any i n i t i a l segment of acontext always a context. Because of th e meaning of a hypothe t ica l
Then a judgement of th e form
We may now fur ther gene ra l iz e judgements t o i nc lu de hypothe t i ca ljudgements with an a r b i t r a r y number n of assumptions. W explain th e i rmeaning by induct ion . that i s , assuming we understand th e meaning ofjudgements with n -' assumptions. So assume we know that
(x E A)b(x) E. B(x)
b(a) = b(c) . B{a)c A
(x . A)b(x) = d(x) e B(x)
(x A)b Cx) . iHx)
a . A
b(a) E. B(a)Final ly, a judgement of th e form
b(a) = d(a) . B(a)
Subst i tut ion
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a A
(4 ) b(x) = d(x) . B(x) (x . A)
Subst i tut ion
A hypothe t ica l j U d ~ e m e n t of th e form(3) b( x) E B(x) (x A)
wQi ch i s th e l a s t s u b s t i t u t i o n rule .
means t h a t we know b(a) to be an element of th e se t B(a) a ssuming weknow a to be an element of th e s e t A, and t h a t b(a) = b I c ) E. B(a)wh ene ve r a and c ar e equal elementa of A. In other wo r ds , b(x) isan extensional function with domain A and .range B(x) depending onth e argument x . Then th e following r ul es a re j u s t i f i e d :
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t he category of f am il ie s o f sets B(x) (x E A)' ove r a given se t A.
t he c a te g ory of elements of a given se t (o r proofs of a proposit ion) .
th e category of sets (o r propositions).
et c .se ts , .
t he c at eg ory of functions c(x.y) E C(x.y) (x E A, y E B(x . whereA is a set . ~ ( x ) (x E A) and C(x.y) . (x "- A. y EB(x families of
- 2 ' -
th e category o f fa mi lie s o f sets C(x .y ) (x EA Y E B(x , whereA se t . B(x) se t (x E A).
Sets and ca tegories
th e category of functions b(x) E a(x) (x E A). whe re A se t .B(x) se t (x E A) ,
In addit ion t o t he se , t he re a re higher ca tegories . l ike th e categoryof b ina ry funct ion s which take two sets into another se t . The functionx . which takes two sets A and B into t he i r cartesian product A x B.
A category. is defined by explaining what an o bj ec t o f th e cat -egory is and when two such ob j ec t s a re equal . A category need no t bea se t , since we can grasp what it means to be an o bj ec t o f a givencategory even without exhaust ive rules fo r forming i t s objec ts . Forins tance . we now grasp what a se t is and when two sets ar e equal , sowe have d ef in ed t he c at eg ory of se t s (and. by th e same token . th ecategory of propositions), bu t it is no t a se t . So fa r . we have de fined several ca tegories :
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(4 ) a(x, . xn) = b(X, . ,x n) E A(x, . xn)(x EA. X E A (x , . .... x ," n n n-(equa l func t ion s with n arguments).
{3 ) a(x" . . . , xn ) E A(x, , xn ) (x , e A, . . . . .xn E An(x , . xn_,(function with n a rguments),
(equal f ami li es o f sets with n ind ices ) ,
and we assume th e corresponding s u bs t it u ti o n r ul es t o be given.
judgement o f the ( ' ) , we se e tha t th e f i rs t two rules of subst i -tu t io n may be extended to t he c as e of n assumptions ,and we understandthese extensions t o be given .
I t is by now clear how t o e xp la in th e meaning of th e remainingforms o f hypot het ic al judgement :
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i s a n e xa mp le of an objec t of t ha t c a te g o r y .We wi l l sa y ob jec t of a category but e le ment of a s e t , wh i c h r e-
f l ec t s th e di ffe rence between ca tegories and s e t s . To d e f i n a ca t -egory it i s no t necessary to prescribe how i t s o b je ct s a re f o r mbu t Jus t to grasp what ( a rb i t r a ry ) objec t of t he c at eg o ry i s . Eachs e t determines a category , namely th e category of e l ements of th e se t ,b ut n ot c on ve rs el y: f or i ns ta n ce , th e c a t eg o r y o f s e t s an d th e ca t -egory of p r opo s it io n s a r e no t s e t s , s ince we cannot descr ibe how a l lt h e i r elements a re formed. We ca n now s a y t h a t a judgement i s a s t a t e -ment to th e e f f e c t t ha t s ome t h i ng i s an obje c t of a c a t e g or y (a 6 A,A s e t , . . . ) o r t ha t tw o objec ts o f a c a te g o r y ar e e qu al (a = b E A,A B, ) .
What about t h e word type in th e l og i c a l s e n s e . given to it byRusse l l with his ramif i ed (resp. s im p le ) t he or y o f type s ? Is type syn-onymous with cate gory or with s e t ? In some cases w ith th e one , its e e ms , and in o th er c as es w it h th e othe r . And it i s t h i s confusion oftwo di f f e r en t concepts which ha s l ed to th e impredica t iv i t y o f th es i mp l e t h eo r y o f types . When a type i s def ined a s th e range o f s ign i -f icance of a propos i t iona l func t ion, so t ha t types ar e wha t th equan t i f i e r s range over , then it seems t ha t a type i s th e same thinga s a s e t . On th e other hand , when one s p ea k s a b ou t th e s imple typeso f p r op o si ti on s , p r o pe rt ie s of ind iViduals , r e l a t ions between i nd i v i d -ua ls e t c . , it seems as if types an d c a te go r ie s a r e th e same . The im -po r t an t d i f fe r en c e between th e ramif ied types of propos i t i ons , prop-e r t i e s , r e l a t ions et c . of some f in i te order and th e simple types ofa l l proposi t ions , prope r t i e s , r e la t io n s e t c . i s prec ise ly t h a t th eramif ied types are (o r ca n be unders tood a s) s e t s , so t ha t i t makessense t o q u an ti fy over them, whereas th e simple types ar e mere ca t -egories .
F o r e xa mp le , BA i s a s e t , th e s e t o f f un c ti on s from th e s e t A to
23 -
t h e s e t B (BA wi l l be in t roduced as an a bb re v ia t ion f o r (n x Ewhen B(x ) i s constant ly equal to B ). In pa r t icu l a r , {O, llAi s a se t ,bu t it i s no t th e s a me thing as ~ ( A ) which i s only a c a t e g o r y. Thereason t ha t BA can be construed t i .
as a ses t ha t we ta ke t he n ot io nof funct ion as primi t ive , ins tead of defining a fu nct ion a s a se t oforde re d pa i r s or a binary r e l a t i o n s a t i sf y i ng th e usual ex i s t ence an duniqueness cond i t i ons , w h ic h w ou ld make it a ca tegory ( l i ke d?(A))ins tead o f a . s e t .
When on e speaks about data types in compute r sc ience , on e mi ghtju s t as well sa y data s e t s . So h er e ty pe i s always synonymous wi t hs e t and n ot w it h category .
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General remarks 'o n th e ru l e s
We now s t a r t to g ive th e ru l e s fo r th e di f f e r en t symbols we use .We w i ll f ol lo w a common pat te rn in giv ing the m. F or e ac h opera t ionwe have four rules :
s e t format ion,
i n t roduc t i on ,
e l imina t ion,
equa l i t y .
The formation rule says t ha t we ca n form a cer t a in (proposi t ion)from cer t a in other sets (proposi t ions) or f ami l i e s o f s e t s (proposi t i ona l func t i ons ) . The int roduct ion r ~ l e s sa y what a re th e canonicale lements (and e qua l c a non i ca l e leme nt s) of the s e t , , t h us giv ing i t smeaning. The e l imina t ion rule shows how we may d e fi n e f u n ct i on s on th es e t def ined by th e int roduct io n ' r u l e s . The e q u al it y r u le s r e l a t e th eint roduct ion and el iminat ion ru l e s by showing how a funct ion def inedby 'means of th e el iminat ion rule opera tes on t h e c a n on i ca l elementso f the s e t which ar e genera ted by th e int roduct ion ru l es .
In th e i n t e rp r e t a t ion of s e t s as propos i t i ons , th e format ionru l e s a re used to form p r opo si ti o n s, i n tr oduc ti on an d e l imina t ionru l e s ar e l ike th os e o f Gentzen9 , and th e e q u al it y r u le s correspondto t he r ed uc ti on rules o f Prawitz 10 .
9 G. Gentzen, Untersuchungen uber da s log ische Schl iessen,Mathematische Zei t s chr i f t , Vol . 39 , 1934, pp .176-210 an d 405-431.
10 D. Prawitz , Na tu ra l Deduc t ion , A Proof-Theore t ica l Study,Almqvist &Wiksel l , Stockholm, 1965.
' 1I
'.. .;. .'
- 25 -
We remark here a lso t ha t to each ru l e . o f s e t format ion, i n t ro duc t ion and e l im i n a ti on , t h e re cor responds an equal i ty ru l e , whichallows to ~ u b s t i t u t e e qu als f or e qu al s.
The ru l e s should be ru l e s o f immediate inference ; we cannotfu r t he r ana ly se them, b ut o nly expla in them. However , in t h e end , noexplanat ion ca n subs t i t u t e each i nd i v i dua l 's unders tanding .
..1
- - - --- - - ----_=-----_.
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(x E , A)A = CC = A
D(x)x e. A
B(x)
(x E C)
- ' 27 -
b(x} e D(x)
A = CC = A
x E Ab Lx ) e. B(x)
(x E C)
r ec t ion i s s imi l a r .
(AX)b(x) E ( D X E C)D(x)
name of the fo rmer . Since , in g en er al , th er e ar e no exhaust ive ru l e sf o r g e n er a ti n g a l l funct ions f rom o ne s e t to another , it fol lows t ha twe cannot genera te induct ive ly a l l th e elements o f a. s e t of th e form(n x IS, A)B(x} (o r, in par t i cu la r , o f th e 1, form 'B A, l ike NN) .
We ca n now ju s t i fy th e second rule of se t format ion. So l e t(AX}b(x) be a canonical element o f (TIx E A)B(x}. Then b( x} E B(x}(x E A) . Therefore , assuming x E C we get x E A by symmetry an d equali ty of s e t s from th e premiss A C, and h en ce b (x } E B(x}. Now, fromth e premiss B(x} = D(x} (x E A) , aga in by equal i ty of s e t s (which i sassumed to hold a lso fo r f am i li es o f s e t s ) , we obta in b(x} E D(x) ,an d henc e (Ax}b (x ) e: ([ 1 x e C)D(x} by n - i n t r odu c t i on . The other d i-
t ha t these rules i n tr o du c e c a no n ic a l elements an d equal canonie lements , even if b(a} i s no t a canonical element o f B(a) fo rA. Also, we assume t ha t t he u su al variable r es t r i c t ion i s met,
i . e . tha t x does n ot a pp ea r free in an y assumption except ( those ofE A. Note t ha t it i s neccessary to unders tand t ha t
hex) e B(x} (x E A) 'i s a funct ion to be ab le to form th e canonicalelement (AX}b(x) E ( f ix e, A)B(x); we c ou ld s ay t ha t th e l a t t e r i s a
D(x)(x c;; A)
B(x)(Tl x E A)B(x) = (Tix E C)D(x}
(x e A)BOc} s e tse t
( f ix E A)B(x} s e t
(AX}b(x) = (Ax;}d(x) e ( f ix ' e A)B(x}b(x) = d(x} e B(x)
(x e A)(AX)b(x) e (T ] x E A)B(x)
b(x} E. B(x}
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Car tesian product of a family of s e t s
n - i n t roduc t i on
D- fo rma t ion
Given a se t A an d a ,f a mi l y of s e t sB (x} over th e s e t A, we ca n
The second rule says t ha t from equal arguments we ge t equal values.The s am e holds fo r a l l other se t forming opera t ions , an d we wi l l never
i Th conclus ' o n o f the f i r s t rule i s t ha t somethingspe l l it ou t aga n. e i s a s e t . To understand which s e t it i s , we must know how i t s canoni-ca l elements an d i t s equal canonical e lements a re formed. This i s ex pla ined by th e i n tr oduct ion r u l e s:
form th e product :
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(x E A)b(x) E B(:c)
C E (TIx E A)B(x)
a E A
(Ax)Ap(c,x) E ( n x E A)B(x)
ApAX)b(x) ,a ) = bra) E B(a)
n-equal i ty
We have t o e xp la in th e meaning of th e new constant Ap (Ap fo rApplication) ; Ap(c,a) i s a method o f ob ta in ing a canonical element ofB(a) , and we now explain how to execute i t . We know thatc (T l x E A)B(x) , that i s , that c i s a method which yie lds a canoni-ca l element (AX)b(x) of (TIx e A)B(x) as resul t . Now take a E A and
s U b s t i t ~ t e i t fo r x i nb (x ) . Then .b ( a ) E B(a). Calculating b(a ) , weobtain as result a canonical element of B (a ), a s required . Of course,in th is explanation ; no concrete computation is carried out; it ha s th e character of a thought exper imen t (Ger . Gedankenexper iment ) , Weus e Ap(c,a) instead of th e more common bra) to distinguish th e resul to f a pp ly in g t he b in ar y applica t ion function Ap to the two arguments cand a from th e resu l t of applying b to a. Ap(c , a) co rr esponds to th eapplica t ion ope ra t ion (ca) in combinatory logic. But recal l that in
c o ~ b i D a t o r y logic there ar e no type restr ic t ions , s ince one can a l ways form ( ca ), fo r any c and a.
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. .. . .th e pr-ogr-am ~ ( x ) th e. f i r s t rule sa y.s that . applying th e name ofa program to argument yie lds th e same r e s ~ l t as ex ec uti ng t he p ro -gram with that ,rgument a j i n p u t ~ Similar ly, . th e second r ul e is needed
The f i r s t equality rule shows how .the new function Ap operates ' on th ecanonical elements of (F l x E A)B(x): Think of (AX)b(x) as a name of
A) ,
a = b E A
a E A
X . AB[x) = D(x)
(x 6 C)
(Ax)d(x) '" (D x '" C)D(X)= (Ax)d(X) E (TIx E A)B(x)=
C = AA = C
X E A
c e; (nx . A)B(x)Ap(c,a) . B(a)
c = d E (n x . A)B(x)
(Ax)b(x)
Ap(c,a) = AP(d,b) E B(a)
(Ax)b(x) = O.x)d(x) . (TIx E C)D(x)b(X) = d(X) . D(X)
A = C
(x . C)
(Ax)b(x) and (Ax)d( x ) be equalsame assumptions. So le t Then b (X) = d(x) E B(x) (x Ee lement s o f. (n x 6 A)B.,(x) .
n _elimination
1 canonical elements ofand (Ax )d (X) ar e equashows that (Ax)b(x)(f\x e; C)D(x).
and therefore th e derivation
28 -
corresponds directly to th eWe also have to prove that
under th e. canonical
. d . s a formalderiva t ion cannot be considere aWe remark that th e above 0in t yp e t he ory i t se l f S1nce . of the second T1_formation ruleproof 0 l oty between two sets whichformal rule of p ro v1 ng a n e qu a 1th e re is no exp lana ti on o f what such an equali ty means.
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B t ruet rueA & B t rue
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Def i n i t i ona l. equal i ty is i n t ens iona l equa l i t y , o r e q u al it y ofmeaning (synonymy) . We us e th e symbol : : o r =def. (which was f i r s tint roduced by B u r a l i ~ F o r t i ) Def in i t i ona l equal i ty :: i s a r e l a t ionbetween l i ngu i s. t i c expr-e sa t on s ; it s ho ul d n ot be confused with equal i ty between objec ts ( s e t s , e lements .of a s e t e tc . ) which we denoteby =. De f in i ti o n a l e qua li ty is th e equiva lence r e l a t ion genera ted byabbrevia tory def in i t i ons , changes of bound va r i ab l e s and the pr i nc ip l eof sUbst i tut ing equals f or e q ua ls . Therefore it i s decidab le , bu t notin the sense t ha t a == b V .., (a == b) holds, simply because a == b isno t a p r opo s it io n i n th e sense of th e p re se n t t he or y. D e f in i ti o na lequal i ty ca n be used t o r ew r it e e xp re ss io ns , i n which case i t s dec id a b il it y i s es s en t i a l in checking th e formal c o r re c tn e s s o f a proof.In f ac t , to check th e correc tness of an inference l i ke
Def i n i t i ona l equal i ty
fo r i ns t ance , we must in par t i cu la r make sure t ha t t h e o c cu r re n ce s ofth e express ions A an d B above th e l i ne an d th e c o r r e s p o n ~ i n g occur-rences below ar e th e same, t ha t i s , t ha t they ar e def in i t i ona l lyequal . Note tha t th e rewri t ing of an express ion i s no t counted as aformal inference .
rules fo r BA, which i s th eBA . to
(AX)AP(c .x ) B(x) (x E A)
provided aAp(c,a) yi e l ds th e
equal e lements , we needrule er n _int roduct ion fo r ) . B( )By th e (a ) Ap(c,a E a) This means b a =
b(X) -_ Ap(c .x ) E B(x) . (x EA . (' )b(X) and hence. ce c y i e l ds AX .EA . Bu t th is i s t r ue ,same v al ue a s b(a) .
d c ts conta in th eThe rules fo r pr o u t B In fact , we taketh e s e t A to the se .se t o f f un ct io n s from Here th e concept o f d e f ~ -B does no t depend onx .be ( ll .x E A)B, where
n i t i o n a l equa l i t y is use fu l .
f which we k now only( ) fo r a program 0t o o bt ai n a not a t i on , Ap c ,x , . f l lows . .Recal l t ha tl e ca n be e x p l a ~ n e d as 0c The second ru t s as re -th e name y ' e ld equal canonical elemen, . 1 if they )
two elements ar e equa It (Ax)b(x) , where oCx) E B(Xi Id s the resuSu l t s . S o s ~ p p o s e eye t to prove isi s canonica l , what we wan(x A) . Since (Ax)Ap
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I I,11 \ .
Applicat ions of the cartes ia n product
Fi r s t , us .ing def ini t ional equal i ty, we ca n now de f ine BA byput t i ng
V - int roduct ion(x E A)
B(x) t rue
prov ided B does not depend on x , We n ex t c on si de r t he n - ru l e s inth e i n te r pre t a t i o n of p ropos i t ions as sets . I f , i n th e . f i r s t rUle ,n - fo rmat ion, we th in k of B(x) as a p ro p os it io n i ns te ad o f a se t ,then , a f te r th e defini t ion
(V x 6 A)B(x) _ (Tl x G A)B(.x) ,it becomes th e rule
a( x 1 , ,x ) .0 E i\{x 1 , ,x ) (XE Am 1 1 ' " ., X E A (xx E A ( m m 1"" , Xm_ 1 ) ,m+l m+ l x 1 , .. ,xm) , .. . , x E A ( x.0 .0 l' . . . , xmV - format ion
('r;fx E A)B(x) prop .
t rue ) .t rue (x E A1 1 ' . ' X E A (xA m m l' , xm 1)'m+l(x 1 , ,x ) t rue A ( -m ' . . . , .0 x1 ' , xm)
prop . (x E "i1 1 , .. , X G A ( xA m m l ""'x 1)'m+l(x 1 , . . ,x ) t rue A ( m-m ' . . , .0 x 1 " " ' xm) t rue)
A(X1 , ,x j. m
A(x 1 , . .. ,x )m
Sim i l a r l y , we wri te
tha t A(a s , x1 ' . . . , x ) i s a px . m roposi t ion provided e Am e Am (x 1" , x ) and ' A xl 1 " ,m-l ' m+l(X1" " 'x ) A (m ' . . . , .0 x1" " ' x ): m
. namely, ' suppose tha t A. . . 1 up to A and AThen ,. if we ' a .' m+ .0 depend only on xre merely i n te r es te d i n th e l' . , . x .inesser i t'; a 'l t ruth of A(x ) mto wri t .e expli c i t l " " 'x , i t i ssymbols fo th mA . s . r e elements of A. n' 0 we abbrev ia te i t with m.l ' .. . ,(x 6 A)
B(x) prop . A se t
whic h says t ha t un i versa l quant i f ica t ion forms propos i t i ons . A s e tmer e l y says that th e domain over which th e un iv e rs a l qua n t if i e r rangesi s a se t an d th i s is why we do no t change it into A prop. Note t ha tth e r ul e of V - f o r ma t i o n i s j us t an instance of n- forma t ion. Wes im i l a r ly have
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ar e a l l t rue , as an a b br e vi a ti o n o fx E Am(X" . . . ,xm_,) ,(x x ) prop. (x , e A" .. . , m1 ' , m . x E A (x" . . , xm) x EO A , (x" .. . , xm) , .. . , n nm+l m+
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::> a re :
::> ~ i n t r o d u c t i o n
wh ic h i s a g e n e ra l iz a t io n o f th e usual rule of forming A::> B, sincewe may a ls o us e th e assumpt ion A t rue to prove B .prop. This genera li z a t i o n i s pe r h a p s "more e vid en t i n th e Kolmogorov i n t e rp re t a t i on ,where we might be in th e posi t ion to jUdge B to be a problem only un .de r th e assumption tha t th e problem A ca n be so lved , wh i c h i s c learlysu f f i c i en t fo r th e ,problem A::> B, t ha t i s , th e problem o f s o lv in g Bprovided tha t A ca n be so lved , t o make sen se . The in ference rules fo r
(V x eA)B(x) t rueE AV -e l iminat ion
th e ru le of n - e l im ina t i on ,back t o th e V - ru l es , fromTurningwe have in pa r t i cu l a r
B(a) t rue (A t rue)f o f (V x A)B(x),we -se e t ha t , if c i s a prooR e s to r in g p r oo f s , (V it E. A)B(x) i sPr oo f o f B (a ); so a proof oft he n Ap (c ,a ) i s a A . t a proof o f B (a ),
"t y element o f 0a method which t akes an a r b ~ r a r " of th e universa li n tu i t i on i s t i c i n t e r p r e t a t ~ o nin agreement w it h t hequan t i f i e r .
I f we now define
B t rueA ::> B t rue
wh i ch comes from th e ru le of O- in t r oduc t ion by suppressing proofs ,an d
::> -e l iminat ion ( n x E. A)B,
A :::> B prop .
::> - format ion
we obta in from th e I1 - ru les th e . r u l e swhere B does no t depend on x , tru le of n - f o r ma t i o n , assuming. B does nof o r i m p li c at i on . From th e
depend on x , we obta in which i s obtained from th e ru le o f n - e l im ina t i on by th e same process .Example (the combinator I ) . Assume A s e t an d x EA . Then , by
n- i n t roduc t i on , we o b t ~ i n (AX)x E. A --. A, an d therefore , fo r an y .prop o s i t io n A, A ::> A t rue . This expresses th e fac t t ha t a proof o f A ::> Ai s th e method: t ake th e same proof (const ruc t ion) . We ca n def ine th ecombina tor I put t ing I =(AX)X. No t e t h a ~ th e same I belongs to an yse t o f th e form A -+ A, since we do no t have di f fe ren t va r i ab l e s fo rd i f f e ren t types.
A t rueB t rue
A ::> B t rue
B p rop .(A t r u e )
A prop. i
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of B.
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S =: (Ag)(Af)(Ax)Ap(Ap(g,x) ,Ap(f,x
(Y x E A)(Vy E B(xC(x,y)::>(Yf e (n x e A)B(x(Y x E A)C(x,Ap(f,x t rue
(A ::> (B ::> C ::> A::> B) ::> (A ::> 15 t rue .
(V x E A)(B(x) ::> C(x ::> Y x E A)B(x) ::> (Y x E A).C(x true.
S E (Tl x E A)(S(x) - C ( x - (f Tlx E A)B(x) _ (Tl x E A)C(x .
which is t he u su al combinator S, denoted by Agfx.gX(fx) i i i combinatory logic. In th is way , we have assigned a type ( se t ) to th e combinator S. Now t hi nk o f C(x,y) as a proposi t ional function . Then wehave proved
(V x e A)(Vy e: B(xC(x,y)::> (Vf e fT B ) (Vx A)C(x,f(x.xA 'xIf.we assume that C(x,y) does no t depend on y , then(n y e B(xC(x,y) =B(x) -C ( x ) and therefore
.whi ch is t radi t ional ly written
So , .i f we think of B(x ) and C(X) as proposi t ions , we have
Now assume that B(x ) doe s no t depend on x and that C(x;y) does notdepend on x and : y. Then we obtain
that i s , in th e logical in terpreta t ion,
This i s just th e secone axiom of th e Hilbert style proposi t ional calcUlus. In th is las t ~ a s e , t he p ro of above, when writ ten in treeform ,
So, again by
Assume A se t , B(x) se t (x E A),le t x E A, f E (Tl x E A)B(x) andThen Ap(f,x) E B (x ) an d
We may now pu t
) E (Tl x E A)(ny E B(icC(x,y)Ag) (.Ar) (Ax)Ap(Ap (g,x) ,Ap ( f ,x )- (T I fE (nX EA)B(x(nx E A)C(x,Ap(f,x).
Since th e se t to th e r igh t does no t depend on g, abstracting on g, weobtain
(Af)(Ax)Ap(Ap(g,x),Ap(f,x ( n f E ( n x eA )B( x ( nx eA)C(x,Ap(f,x.
and, by A -abstraction on f,(Ax)Ap(Ap(g,x) ,Ap(f .x ) e (n x E A)C(x ,Ap(f , x ,
Now, by A -abstraction on x, we obtainAp(Ap(g,x),Ap(f,x E C(x,Ap(f,x.
n-elimination,
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,c
Example (the combinator S) .C(x,y) set (x E A, y E B(x andg e (Tl x E A)(ny e B(xC(x,y) .Ap (g ,x) ( n y B(xC(x,y) by n-e l imina t ion .
Example (the combinator K). Assume A se t , B(x) se t (x E A) andA B(x) Then byA -abstraction on y, we obtainet x . . ,y . ' .
(Ay)X E B(x) - A, and, by A -abstraction on x,d fi e th e combinator KAX)(AY)X E (n x E A)(B(x) - - A). We can e n
fAd B as proposi t ions , whereutting K == (Ax) (Ay)X. If we think 0 anB does no t depend on x, K a pp ea rs a s a proof of A ? (B ? A); soA=> (B ? A) is true. K.expresses th e me:hod: given any proof x of A,take th e function from B to A which is constantly x fo r any proof y
I.
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(x E A)
b EO: B(a)
B(x) s e t
(x E A)B(xi = D(x)
A s e t
a E A(a ,b ) E (L x A)B(x)
A = C(Lx E A)B(x) = ( L X E C)D(x)
L - in t roduct io n
W "c a n now j us t i f y th e equal i ty rule associa ted wit h L - forma -
(L x E A)B(x) s e t
L - fo rmat i on
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union o f a fa mily of se t s
The s ec on d g ro u p o f rules i s abou t th e di s jo in t un i on o f afamily of se t's ,
A mo r e t r ad i t i ona l n o ta ti on f or (E x E A)B(x)" wou l d be L BxEA x"( U B orU B ) . We now expla in wha t s e t (L x E A)B( x ) i s byx"A x xe:A x
prescr ib ing how i t s canonica l elements ar e fo rmed . Thi s we do withth e ru l e :
t i on :
In "f a c t , any canonica l element of (L x E A)B(x) i s of th e f orm (a ,b )" "" With a EO: A an d b EO: B(a) by 2: ; - in t roduct ion . Bub then we a lso have
( g E A -(B-CAp ( g , x ) E B -- C
(x E A)
Ap(Ap(g ,x ) ,Ap(f ,x C
(f A - B )
" EA-C(AX)Ap(Ap( g ,x ) , Ap (f , x ,
Ap(f ,x ) E B
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\ \
( x A)
( E (A -+ B) - (A -- C)O,f) (Ax)Ap(Ap(g , X) ,Ap f , x """ E A _ (B _C _A -B )- (A -C (Ag ) (Af } (Ax ) Ap(Ap (g , x ) , Ap ( f . x ) (
becomes:
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- 40 - 41 -
a E C and b E D(a) by equali ty of sets and subst i tut ion. Hence(a j b ) E: (Ex E C)D(x) by L-in t roduc t ion . The other direction issimilar .
E -e l iminat ion
L-equality
a E; A b e fiCa)(x E A, yeB (xd(x,y) E CUx,y
E(c,(x,y)d(x,y E C(C)
e; Ca ,b , andE CU a , b . The
d(a ,b) E Ca,b
A second ru le o f L -equali ty , analogous to th e second rule ofn-equal i ty , is now derivable, as we shal l se e la ter .
is -correc t .
(Here, as in L -e l iminat ion,. C( ). ( ...z se t z e (L.X E A)B(x is an im-pl ic i t premiss.) Assuming that we know th e p remis se s, t he c on cl us io nj us t i f i e d by imagining Ea ,b ) , (x ,y)d(x,y to be executed . In fact ,we f i r s t execute (a,b), which yields (a,b) i t se I f as . resul t ; then we
subst i tute a, b fo r x, y in d(X,y) , obta iningd(a ,b)execute d(a,b) un t i l .we obtain a canonical element esame canonical element is produced by d(a,b), and t hu s t he c on cl us io n
d(x,y) E C(x,y(x A, y B(x
C E (L x E A)B(x)
where we presuppose th e premiss C(z) se t (z E (Lx E A)B(x, althoughit is no t writ ten ou t explic i t ly . (To be precise, we should alsow ri te o ut t h e p remi sse s A se t and B(x) set (x E A).) We explain th e. rule of L -e l iminat ion by showing how th e new constant E operateson i t s arguments. So. assume we know t he p r em i ss es . Then we e x e ~ u t eE(c , (x,y)d(x ,y as follows. Fi r s t execute c, which yie lds a canonicalelement of th e form (a,b) with a E A and b E B(a). Now. substitute aand b fo r x and y, respectively, in th e r ight premiss, obtaining. d(a ,b) e: Ca ,b . Executing d(a, l we .obt a i n a canonical element e ofCa , b . We now want to show that e is also a canonical element ofC(C) . I t i s a general fact that , i f a E A ahd a ha s value b, thena = b E A (note, however, that t h i ~ does no t mean that a = b E A isnecessarily formally derivable by some part icular se t of formal rules) .In our case , c = (a,b) E (L x E A)B(x) and hence, by subs t i tu t ion ,C(C) = Ca ,b . Remembering what it means fo r two sets to be equal,we conclude from th e f ac t t ha t e is a canonical element o f C a ,b that e i s also a canonical element of C(c) .
Another no ta ti on f o r E(Ci(X,y)d(x,y could be (Ex,y)(c ,d(x,y,bu t we prefer th e f i r s t since it shows more clear ly tha t x and y be
c o m ~ bound only i n d (x ,y ).
III
t
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C t rue
B prop.(A t rue)
C t rue
A prop.
(: I x E A)B(x) t r-ue :
3-e i iminat ion
This rule is an instance Of ~ - f o r m a t i o n and a genera l iza t ion of th eusual rule of forming proposi t ions of th e form A & B, since we mayknow that B i s a proposi t ion onl y und er th e assumption that A i s t rue .
A & .B vp r -op ,
&-forma t ion
(x E B(x) t rue)
A & B = A x B == (L.X E A)B,
- 113 -
where B does no t depend on x. We derive here only th e rules of. conjunc t ion.
Here, as usual , no assumptions, e xc ep t t ho se explic i t ly writ ten ou t ,may depend on th e variable x. The rule of I : -e l iminat ion is s trongerthan th e :3 -elimina t ion ru le , which i s obtained from it by suppressingp roo f s, s in c e we take into conside ra t ion a l so proofs (construc t ions) ,which i s no t po ss ib le w it hi n t he language of f i r s t order predica te
1 > logic . This addi tiona l s t r ength wil l be visible when treating th e l e f tand r igh t projections below.
The r ul es o f d is jo in t union deliver a lso t he u su al ru les o f con.j u nc t i o n and th e usual p r ope r ti es o f th e cartesian product of twosets i f we define
(x E A)
B(a) t rue
B(x) prop .
( 3x E A)B(x) t ruea : A
.A se t
3-introduction
(3 x A)B(x) prop.
Appl ic at io ns o f t he d is jo in t union
3 -formation
(3 X ."E A)B(x) :: (E x E A)B(x),
In accordance w it h t he in tu i t ion i s t i c i n t ~ r p r e t a t i o n of th e exis tent i a l quan t i f ie r , th e rule of 'L- introduction may be inte rpre ted assaying tha t a (canonical) proof of (3 x E A)B(x) is a pa i r (a ,b ) ,where b is a p ro of o f th e fac t that a sa t i s f i e s B. Suppressingproofs , we o bt ai n t he r ul e of 3- in t roduc t ion , in w h ~ c h , however,th e f i r s t premiss a E A i s usual ly no t made explic i t .
then, from th e E-ru le s , interpreting B(x) as a proposi t ional ' funct ion over A; we obtain as par t icu la r cases:
As we have already done w ith t he cartesian product,_we sha l lnow se e what ar e the logica l in te rp re ta t ions of the dLs j oLn t una'on .I f we pu t
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- 44 -
b E B(a)pa,b = a E A
' a E Ap(c) e A
Left projection
C E (3 x E A)B(x)p f c ) e A
I f we now turn to th e logical in terpreta t ion, we see that
(3 x)B(x) true(Ex)B(x) individual
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a counte rpa rt of which we have jus t proved, and a rule of the form(3 x)B(x) true
Bex)B(x true
holds, which means that from a proof of . (:3 x . A)B(x) we can obtain anelement of A fo r which t he p roper ty B holds. So we have no need of th edescription operator (1x)B(x) (the x such that B(x) holds) or thechoice operator (Ex )B(x ) ( an x such that B(x) ho ld s ), s i nc e , from th eintuitionist!c 'poi nt of view, (3 x E A)B(x) ' i s true when we have aproof of it . The dif 'f iculty witQ an epsilon term (Ex)B(x) is that i tis cons trued as a function of t he p roper ty B(x) i t se l f ,n d no t of th eproof of (3 x)B(x). This is why Hilbert had to postulate both a ruleof th e 'form
fore, taking C(z) to be A and d(x,y) to be x in th e ru le s of E -e lim Ln a t Lon and r:-equali.ty, we o bt ai n a s d er iv ed rules:
C true
B true
C true
A & B trueA true
'A & B true
A & B true (A true) A & B true (B true)A true B true
Example ( lef t projection). We definep(c) - E(c, (x ;y)x)
&-introduction
(A t rue , B true)&-elimination
and cal l i t th e l e f t p roj e ct i on o f c since i t i s a method of obtaining the value of the f i r s t ( lef t ) c oo r di nat e o f .the pair produced byan arbi trary element c of (E x E A)B(x). In fact , i f we take th e termd(x,y) in t he e xp la na ti on o f r : -e l im in at i on to be x, then we ,s ee that
obta1"n th e pair (a,b) with a E A and b E B(a)t o e xe cu te p (c ) we f i r s twhich is th e value of c, and tlien subst i tute . a, b fo r x, y in x, ob taining a, which i s executed to yield a canonical element of A. There-
From t hi s r ul e of &-eliminat ion, we o b ta in t he standard &-eliminationi C to be A and B themselves:rules by choos ng
Restoring proofs, we see that a ( ca non ic al ) p roof o f A & B is pairb a.re given Proofs of Aand B respectively.(a,b) , 'where a and
Ie
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I ;
- 46 -
which ha s a counterpart in th e f i r s t _of th e ru le s o f r ight proje c t ionthat we sha l l se e in t he n ex t example .
Example (r ight projec t ion) . We defineq Cc ) =-E(c,(x ,y)y) .
Take d (x ,y ) t o be y in -the r u le of L -e l im i nation. Fr om x E A,Y E B(x) we obtain px ,y = x E A by l e f t pro jec t ion , and therefore B(x) = B(px , y ) . So , by th e rule equali ty of se ts ,y E B(p( ( x ,y ). Now cho os e C(z) -set -( z E (L.x E A)B(x to be th efa mily B( p ( z set (z E (L .X E A)B(x . Then th e rule of L. - e l im na t ion gives q(c) E B(p(c . More f or ma l l y :
- 47 -
( 3 x E A)B(x) , then q Cc) i s a const ruct ion of B(p(c , where , by l e f tpro je c t ion , p(c) E A. Thus , supp r es s i ng t he con s t r uc t i on i n th e con c lu s io n , B( p ( c ) is true. Note , howeve r , t h at , in ca s e B(x ) dependson x, i t is impossible t o su pp ress th e con s tr u c ti on in th e premiss ,s i n c e th e con c l u s i o n depends on it .Fi nally, when B(x) does no t depend on x, so t ha t - we may wri tei t s imply as B, and both A and -B ar e thought of as proposi t ions , th ef i r s t ru le _of ri gh t proje c t ion reduces to
&-el i minati onA & B true
(x E A) (YEB(x B true
So we have :
px , y = x E. A
qt c ) == sre , ( x ,y)y) E. B(p(c
( ' Ix E A)( B(x):::> (3 x E A)B(x true,
A ::> (B => (A & B t rue .
by s up press ing t he co n s t r uc t i o n s in bo th t he premis s and t he conclu s ion.
f rom whi ch , in particular, when B does no t depend on x ,
Example (axioms of co n j u n c t i o n ) . We f i r s t deriveA (B (A & B true, which i s th e ax iom corre sponding to th e r ul eof &- i n t roduction . i ssume A se t ; B(x) se t (x E A) and le t x E A,Y E B( x ) . Then (x,y) E (L x E A)B(x) by L.-in t roduc t ion , an d , byI l - i n t r o d u c t i on , O,y )( x ,y ) E B(x) ~ (L x E A)B(x) (note tha t(L x E A)B(x) doe s no t de pe nd on y ). and (Ax) O..y)(x,y)E(T ] x ( x ) ~ (L x E. A)B(x.The l o g i c a l reading is th en
We now us e th e l e f t and -right projec t i ons to de r iv e A & B :::> A trueand A & B::> B t r ue . To ob tai n t he f i r s t , as s ume z E. (Ex E. A)B(x) .
bE . B(a)b E Bf a )
B(x) = B( p( l x , y )
a E A
- x = px,y E A
qa,b
Y E _B(p( (x, y )(y E B(x
C E (L x E. A)B(x )q Co ) E. B'(p Lc)
Ri g h t projec t io n
C E_( L X E A)B(x)
The second o f t he se r ule s is derived by L -e quali ty in much the sameway as th e f i rs t was derived by- L -e l i min a t io n .
When B(x) i s th ought -of a s -a pr opos i t i ona l f un c t i on , th e f i rs trule of r igh t pr oj e c t ion s ay s t ha t , i f c i s a c on s t ruc ti o n of
I
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Then V(z) A by le f t projectiori, and, by ~ - a b s t r a c t i o n on z,(Az)p(z) E: (L.X E A)B(x ) - A,
In par t icu la r , when B (x ) d oe s no t depend on x, we obtainA & B :::> A t rue .
,' To obtain th e second , from 'z E. (I : x e. A)B(x), we have q (z ) E: B(p (z by right projection , and , hence, by A-abstrac t ion,
(Az)q(Z) E: (Tl e E. (L x E. A)B(xB(p(z(note that B (p (z d ep en ds on z ) . In part icular , when B(x) does notdepend on x , we obtain
A & B :::> B true.Example (another a ppl ic at ion o f th e disjoint union). The rule of
L-el i ; inat ion says tha t any function ,d(x,y) with arguments in A andB(x) gives also a func t ion (with th e same values, by L -equali ty) witha p ai r in (L x E: A)B(x) as single argument. What we now prove is anaxiom corresponding to th is rule. So , assume A se t , B(x) se t (x E A),C(z) se t (z E (L x E. A)B(x and le t f E: (TIx E A)(Dy E: B(xCqx ,y .We want to find an element of
(TIx E A)(TIy E. B(xCx,y - ( O z E (L x e A)B(lCC(Z).We define Ap(f ,x,y) = Ap(Ap(f,x),y) fo r convenience. Then Ap(f,x,y)is a ternary functiori , and Ap(f,x,y) E Cx,y (x E A, y B(x . So,assuming z E (L x EO 'A) B( x ) , by L:-el imination, we obtainE(z , (x,y)Ap(i,x,y C(z) (discharging x A and y e. B(x , and, byA-abstraction on z , we obtain th e function
....
- 49 -
(Az)E(z,(x,y)Ap(f,x,y EO (TIz E (L x eA)B(xC(z)with argument f . So we ,s t i l l have th e assumption
f E (D x e A)( Ilv e B(xC(x,y) ,which we discharge by A-abstraction , obtaining
(Af) (Az)E(z ,(x,y)Ap(f ,x,y (O x e A)(Oy e B(xCx ,y -+ (TIz (L x E A)B(xC(z) .
In the logica l reading, we have(Y x e A)(Vy B (xC x , y ::> (Y z ( Lx A)B(xC(z) t rue ,
which reduces to th e common
(Y x e A)(B(x)::J C) ::J 3 x e: A)B(x) ::> C) t ruewhen C does no t depend on z, and to
(A :::> (B::>' C:::> A & B) '::> C) t ruewhen, i n a ddi ti on, B Is independent of x.
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The axiom of choi ce
We now show t ha t , with th e ru le s in troduced so far , we ca n givea proof of th e axiom of choice , which in o u r s ym bo li sm read s:
( \ I x E A)(3 y E B(xC(x,y)=:>(3f e ( n x e A)B(x) ( ' Ix E A)C(x ,Ap(f ,x t rue .The usual argument in ma t h e ma t i c s , based on th e i n tu i t i on i s t i c i n t e rp r e t a t i on of th e l og i ca l constants , i s roughly asfollows : to prove ('II x ) (3 y)C(X,y) ::> ( 3 f ) (v x)C(x , f (x , assume thatwe have a .p r o o f of th e antecedent. This means that we have a methodwhich, a pp li ed t o an arb i t r a ry x , y ie lds a proof of (3 y)C(x ,y ) , thati s , a pa i r cons i s t ing of an element y and a proof o f C ( x, y) . Le t fbe th e method wh i c h , to an a rb i t ra r il y g iven x , a ssi gn s t he f i r s tcomponent of t h i s pa i r . Then C (x ,f(x h old s fo r an arb i t r a ry x, andh enc e so do es th e consequent . The s ame idea ca n be pu t in to symbols,ge t t i ng a fo rma l proof in i n tu i t i o n i s t i c type theory . Le t A s e t ,B(x) se t (x e A) , C(x ;y ) s e t (x E A, Y E B(x , a nd a ss um eze (Tl x E, A)(Ly e B(xPC(x,y) . I f x is an arb i t r a ry elemen t ofA, L e . x e A, th en , by n-e l imina t ion , we obta in
Ap(z,x) : (E y e B(x))C(x,y) .We now apply l e f t p r o j ~ c t i o n t o o bt ai n
p(Ap(z ,x E B(x)
an d r i gh t p ro j e c ti on to obta inq(Ap(z,x e C(x ,p (Ap(z ,x ) .
By A-abs t r a c t ion on x (o r n . - in t roduc t ion) , discharging x E A, we
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have
(Ax) p (Ap (z .x ) E (Il x E A)B(x ) ,and, by TI-equality,
ApAx)p(Ap(z,x . x ) - p(Ap(z,x E Bf x ) ,By sUbs t i tu t ion , we ge t
C(x,ApAx)p(Ap(z,x .x ) = C( x ,p(Ap(z , x )and hence , by e q ua l it y o f se t s ,
q(Ap(z ,x}) e C ( x , A p ~ x ) p ( A p ( z , x ) , x where (Ax) P( Ap(z , x ) i s independent of x .. By abstract ion on x ,
(Ax)q(Ap(z ,x E (TIx EA)C(x,ApAx)p(Ap(z,x,x .We now .use th e rule o f pa i r ing ( t ha t E ~ i n t r o d u c t i o n ) to ge t
Ax)p(Ap(z ,x , (Ax)q(Ap(z ,x)( L . f E. (TIx E A)B(x)(TIx e A)C(x,Ap(f,x
(note t ha t , in th e l a s t step, th e new va r iab le f i s int roduced an dsUbsti tuted fo r (Ax)p(Ap(z,x in th e r i gh t member) . Fina lly , byabstract ion on z, we ~ b t a i n
(Az)Ax)p(Ap (z ,x, (AX)q(Ap(Z ,X )E (TIx E A)(E y E. B(xC( x , y )=:> (E f E (TIx E A)B(x(nx e A) C(x ,Ap( f ,x .
In Zermelo-Fraenkel se t theory , there i s no proof of th e axiomo f c ho ic e, so it must be t ak en a s an ax iom, fo r which , however, itseems to be d i f f icu l t to c l a i m se lf-evidence . Here a de t a i led
II
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is clear when developing in tu i t ion i s t i c
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Cauchyf a ) _ (V e E Q)(e > 0 :=> (3 in E N)(Vn E N)(\am+n-aml e) ,
The notion of such that
R =: (LxEN-Q)Cauchy(x )
- 53 -
i s t h e d e f in i ti o n of reals as th e se t of sequences of r a t i o n a lnumbers satisfying th e Cauchy condition ,
11 S . Feferman, Constructive theories of functions and classes ,Logic Colloquium .78 , Edited by M. Boffa, D. van Dalen and K. McAloon ,North-Holland, Amsterdam , 1919, pp . 159-224 .
where a i s th e sequence a O' a 1 , In th is way, a real number i s asequence of rational numbers t o g h e ~ h e r with a proof tha t i t sa t i s fie sth e Cauch1 condit ion. So , assuming c E R, e E Q and d 'E (e > 0) (i n
In a d di ti on t o disjoint union ; exis tent ia l quan t.Lf'Lca t.Lon ,cartesian product A X B and . conjunction A & B, th e operation L ha sa fi .fth Lnt.er-pr-e t at.Lon: th e set of a ll a EO A such that B(a ) holds.Le t A be a se t and B{x) a p r opo si ti o n f o r x EO A. We wan t to defineih e se t of a l l a& i ~ u c h that B(a) holds (which i s usually . wr i t t e n[x E A: B(x . ll> . 'ha ve an ' element a EO A such that B(a) holds meansto have an element a EO A together with a proof of B(a), namely anelement b E B(a). So t h e e lemen ts of th e se t of a l l e lemen ts o f Asatisfying B(x) ar e. pairs (a ,b) .wi t h b E B(a) , Le elemen ts of(E x E A)B(x) . Then th e L-rules p la y th e role of th e comprehensionaxiom ( or the separation principle in ZF). The in format ion given byb B(a) i s called th e witnessing information by Feferman 11 A typ i c a l a ppl ic a ti o n is th e following.
,Bxampl e (the reals as Cauchy sequences).
of a sequence 'choicefo r i n s t an ce, i n f ind ing t h e l imi tfunct ion.
th e axiom of choice has been provided in th e formjus t i fi ca t ion of f In mani sorted l anguages , t he axiom of ,choice isof the ,above pr oo " .. t there is no mechanism to prove 1 t. Fo r ins tance, 1nexpress ib le bU f f ini te type , it must be t ak en a s .a n axiom . TheHeyting arithmet ic 0'" i o m ofneed for the a
mathematiCs ' a t dep t h ,t i 1 inverse of a surjectiveof reals or pa r a
II1ii I,I. ' \
\
'I \II\
=
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b = d E B
b E B
j ( b ) = j (d) E A + B
j ( b ) E A + B
B s e tA + B s e t
A se t
a Ai ( a ) E A + B
i (a ) = i ( c ) E A + B
+-format ion
. +-int roduct ion
where i and . j ar e two new pr imiti ve cons t an t s ; the i r us e i s t o g iv et h e i n fo rmat io n t ha t an element of A + B comes from A or B , a nd whi chof th e two i s t he c as e. I t goes w it ho u t s ay in g t ha t we a lso have th er u le s o f + -i ntr od uc ti on f or equal elements:
Disjoint union of tw o s e t a
The canonical elements o f A + B ar e formed using :
We now g iv e th e r ule s f or th e sum (d i s jo in t union o r c opr oduc t)of two s e t s .
Since an arb i t r a ry element c o f A + B yi e l ds a canonical element o fth e form i (a ) or j (b ) , knowing c e A + B means .t h a t we a lso ca n de t e r mi n e f rom wh ic h of th e two s e t s A an d B the element c comes.
Only b y m ea ns of t he p ro o f q( c) do we know how f a r to go th eapproximation desi red.
Ap(p(c) ,p (Ap(Ap(q(c) ,e ) , d ) E Q.
Ap(Ap(q(c) ,e ) ,d) E (3 m E N)(Vn E N)(lam+n-aml e ).
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p ( A p ( ~ P ( q ( c ) , e ) , d N,
an d we now obta in am by a ppl y in g p ( c) to it ,
Applying l e f t pro j e c t i on , we obta in th e m we need , i . e .
and
. d d' p ro of o f the pr oposi t ion e > 0 ), then , b y m ea nsother wo r s ' . 1S ao f the pro j e c t i ons , we obta in p Cc ) and q(c) E Cauchy(p(c .Then
I\II
f.I
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I ,
+-el iminat ion
C E A + B(x E A)
d(x) C(i(x(y e B)
e(y) E C(j(y
, ,bec ome ' e vi d en t .T h ~ d i s j u n c t i o n of two proposi t ions i s now inte rpre ted as ' th e
sum ,of two se t s . We t h er ef o re pu t:
C true(B t rue)
B t rueA V B true
B prop.
C t rue(A t rue)
C t rue
A V B prop.A prop.
A t rueA V B t rue
A v B t rue
V -formation
V - introduction
v -elimina t ion
A V B == A + B.From th e formation and i nt ro du ct io n r ul es f or +, we then o bt ai n t hecorresponding rules ' f or V :
Note tha t , i f a i s a proof of A, then i (a) i s a (canon ica l) p roo f ofA v B, and s imilar ly fo r B.
follows from th e rule o f + -e l imina ti on by choosing a familyC == C(Z) (z E A + B) which does no t depend on z and suppressingproofs (constructions) both in t he p rem is se s, i nc lu di ng t he assumpt ions, and t h e conc lus ion .
(y E B)
(y E B)e I y ) E C( j (y
e(y) E ' C(j ( y
e I b ) E C(j(b
d f a ) E C(i(a(x E A)
d Ix ) E C(i(x
D(c,(x)d(x),(y)e(y E C(c)
D(j(b) , (x)d(x) ,(y)e(yb E B
i (a) , then c = i ta ) E A + B and hencevalue j (b) , then c = j(b) E A + Bandth i s explanation of the meaning of D,
+-equali ty(x E A)
a E A d(x) E C(i(xD(i(a) , (x)d(x) , (y)e(y
where th e premisses A 's e t , B se t and C(z) se t (z E A + B) ar e presupposed, although no t exp l ic i t ly w r it te n o u t. We must now explainhow to exeaute a program of th e new form D(c , (x )d (x ) , (y )e (y . 'As sume we know c E A + B. Then c wil l yie ld a canonical element i ta )with a E A or j(b) with b E B. In th e f i r s t case , subs t i tu te a fo r xin d (x ), obtaining d(a) , and execute i t . By th e second premiss,d(a) e C( i ( a , so d ( ~ ) yields a canonical element of C( i ( a . Simil a r ly , in th e second c as e, e (y ) i ns te ad o f d(x) must be used to ob t ai n e (b ), which produces a canonical element o fC ( j ( b . In ei thercase , we obtain a canonical element of C(C), s ince , i f c has value
C(c) = C( i ( a , and, i f c ha shence C(C) = C( j (b . Fromth e e qual i ty r u l es :
,I
L
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(4 ) I (A ,a ,b ) .b e A,B,
(3 ) a(2 ) A(1 ) == or =def. '
P ropos i ti o n al e qua l it y
We now tu rn to th e axioms f or e q ua li ty . I t i s a t r ad i t i on(de r i ving i t s origin from Pr i nc i p i a ' Mathematica) to ca l l equa l i t yin predica te log ic i den t i ty . However, th e word i den t i ty i s moreproper ly used fo r def in i t i ona l equa l i t y , =or =d f ' discussede. 2a bo ve . I n f ac t , an . equal i ty sta tement , fo r ins tance , 2 = 2+2 ina r i t hme t i c , does no t mean t ha t th e two members ar e th e same, bu tmerely t ha t they have th e same value . Equal i ty i n p re di ca te l og ic ,however, i s a lso di f f e r en t f r ~ m ou r equa l i t y a = b E A, because th eformer i s a proposi t ion, while th e l a t t e r is a judgement . A form o fp r opo si t io n a l e qua li ty i s neve r the l e s s i nd i spensab le : we w an t ane q u al it y I (A ,a ,b ) , which a s se rt s t ha t a and b ar e equal elements oft he s e t A, .bu t on which we ca n opera te w ith th e l og i c a l opera t ions( r eca l l t ha t e . g . th e n eg atio n o r quan t i f i c a t i on of a judgement doesnot make sense) . In a cer t a in sen se , I (A, a , b f i s an i n t e rna l formo f =. We then have f ou r k in ds o f equa l i t y :
Equal i ty between objec ts is expressed in a judgement an d must be de -f ined s e p ar a te l y f o r each category , l i ke t he c at eg or y se ts , as in (2 ) ,o r the ca tegory of elements of a s e t , as in (3); (4 ) i s a proposi t ion,whe re as ( 1) i s a mere s t ipu la t ion , a r e l a t ion between l i ngu i s t i cexpress ions . Note howeve r t ha t I (A ,a ,b ) t ru e i s a judgement, which
. wi l l tu rn ou t to be equivalen t to a = b A (which i s no t to sa y
. i
(A ? C) B :::> C) :::> (A Y B ::> C t r ue .
I f , moreover, C(z) does no t d ep en d o n z and A, B ar e proposi t ions aswel l , we have
This, when C(z) i s thought as a proposi t ion, gives
(Ar ) (Ag) (Az)D(z, (x )Ap(f ,x ) , (y)Ap(g,yE (n x E A) C( i (x ) ) - ( (n y B) C( j (y ) ) - (n z E A + B) C( z .
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("Ix E A)C( iex: :> "Iy B)C( j(y : :> ( 'Vz A + B)C(z t r ue .
Example ( int roductory axioms of di s junc t ion) . Assume A s e t ,B s e t and l e t x A. Then i (x ) A + B by +- i n t roduc t i on , an d henceO.x) iex) e A -A + B by A-abst rac t ion on x.lf A an d B ar e proposi t i ons , we have A ::> A V B t r ue . In th e same way, (Ay) j (y ) e B - - A + B,and hence B A V B t r ue .
Example (e l iminatory axiom of di s junc t ion) . Assume' A s e t , B s e t ,C(z) s e t (z Eo A + B) and l e t f (Tl x e A)C( i ex , g ( D y E B)C(j(yan d z e A + B. Then, by n-e l imina t ion , from x e A, we haveAp(f ,x ) EC ( i e x , and, . from y e B, we have Ap(g,y) e C( j ( y . So ,using z E A + B, we c an a pp ly +-e l imina t ion to obta inD(z, (x)Ap(f ,x ) , (y )Ap(g .y Eo C(z) , thereby discharg ing .x e A andy e B. By A-abst rac t ion on z , g , f in t ha t order , we get
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a=be:A
(x A)
Finally,.mi t 's th e
c = r I(A , a , b )C E I (A ,a ,b )
C I (A ,a ,b )
I-e l imina t ion
We would then d ie r ve th e fOllowing r u l e s wh i has primi t ive : ' c we h ere t ak e ins tead
I - equa l i ty
note t ha t I - formation i s th e only rUleformation o f f am i li es up to now wh i c h per-+ , NNW of s e t s . I f only thn ' , were allowed e opera t ions n , L:, we would only .Example ( in troducto r ge t constant se ts .x Y aXiom of i den t i ty )A. Then x = x A Assume A s e t and l e tabs tract ion on x (A) ' and, by I - in t roduc t ion , r E l (A,x ,x ) . By, x r ( \ {XE .A ) I (Ac anon ic a l p r oo f of th 1 . , x , x ) . Therefore (Ax)r i s ae aw of i den t i ty on A.
.v'
b e: AE. AI(A,a ,b) s e t
A s e t
I-format ion
I - introduct ion
r e: I(A,a ,b)
We now have to expla in how to form canonical elements of I (A, a , b ) .The s tandard way to ~ n o w t ha t I(A,a ,b) is t rue is to have a =b E. A.Thus th e int roduct ion rule i s s imply: if a = b e: A, then there i s acanonical proof r of I (A,a,b) . Here r does no t depend on a , b or A;it does no t matter what canonica l element I(A,a ,b} ha s when a = b E. A,as long as it ha s o ne .
t ha t it ha s th e same sense) . (1 ) i s i n t e n s i o n ~ l (sameness . of meani ng ) , while (2) , (3 ) and (4 ) ar e e x te n s iona l ( e qual i ty between obj e c t s ) . As fo r F re g e, e lem en ts a, b may have di f f e r en t meanings, orbe di f f e r en t methods, but have th e same value . Fo r i ns t ance , wecer t a in ly have 22 = 2+2 N, b ut n ot 22 _ 2+2.
o f e q u al it yhOlds, namely
. (Ax) r E (V x E A) I (A , x, x)r E I(A,x,x)
Example (e l imina tory aXiom o f i den t i ty )property Sex) prop (x E A) Given a se t A and a. over A we c l .correspOnding to L . ' a ~ m t ha t th e lawe l bn i z 's p r in c i p letha t eq u 1 I . of ind isce rn ib i l i tya e ements sa t i s fy th e same proper t i es ,We could now adopt e l imina t ion and e q u al it y r u le s fo r I in th e sames ty le a s fo r Tl , L. . + , namely int roducing a new e l imina to ry ope ra t o r .
Also, note t ha t th e rule f o r i n tr oduc ing equal elements of I (A,a ,b )i s th e t r i v i a l one:
a = b E Ar = r E. I(A,a ,b)
I.
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(AZ)(AW)w E I(A,x,y) ? (B(x) ? B(y
( y e B(xx e A)qx,y = y E B(x)
(x ,y ) (1 : x E A)B(x)
(x,y) E (E x e. A)B(x) .
(y e B(xpx,y = X " A(px , y , q x ,y )
(x e: A)
r E IL : x e. A)B(x) , (px ,y ,q x ,y ) , (x ,y
(p ( (x ,y ,qx , y )
- 63 -
(p(c) , q (c = c e (1: x e A)B(x)E(c,(x,y)r) e IL : x E A)B(x) , (p(c) , q ( c . c )
r- e ILX e A)B(x) , ( px , y ,qx ,y ) , ( x ,y .
E (c , (x , y ) r ) e I( ( L x E A) B( x) , ( p ( c ) , q ( c ) ) , c )
This example i s typical. The I-rules ar e used systematic al ly to showth e uniqueness of a function , whose existence is given by an elimin ation rule , and whose p r ope r ti e s a re expressed by t he a ss oc iatedequali ty rules.
C E (L x. e A) B ( x )
and hence, by I -e l iminat ion , (p (c ) ,q (c = c e. (L x E A)B(x) .
By I-introduction,
is derivable . I t is analogue of th e second n-equa l i ty r ul e, wh i c hcould also be derived, provided th e TI-rules were formulated -f o l l owi ngth e same patte rn as th e other rules . Assume E: A, Y e B(x). By th eprOjection laws , px ,y = x 6 A and qx,y = y e B(i) . Then , byE-introduction (equal elements form equal pairs) ,
Fow ta ke th e family C ( ~ ) in th e r ul e o f I:-ellmination t o be1(0:' x E A)B(x), (p(z) ,q (\z ,z ) . Then we obtain
Y EO A and
(x e. A)B(x) se t
B(y)(x)x = Y 6 A
(z e. I(A,x,y
w 6 'B(y){ ~ w ) w EO B(x) ? B(y)
(w 6 B(x
( Vx E A)( v s e A)( I (A, x , y) .::> (B (x)::> B(y ) true.
. " . .propertie s is no t possible , and hence th e meaning of identi:ty . ha s: t. obe de fined in anothe r way , wi t h out invalidating Leibniz 's p ~ i n C l ~ l e ,
Example (proof of th e converse of the projec t ion l aws ) ". We cannow prove that t he i nf e re n ce rule
C E: (L x e A)B(x)c = (p(c) ,q (c e (E x e A)B(x) .
(a = b) == (V X)(X(a) ? X(bfrom wh i c h Leibniz 's princ iple is obvious, s ince it i s taken to definet he meaning of identity . In the present language , quantif ica t ion over
The same problem (o f justifying Leibniz 's principle) was solvedin Principia by th e use o f impred i ca t ive second order quantification .T he re o ne defines
(AX)(Ay)(Az)(AW)We. ("Ix E A)(Vy 6 A)(I(A,x ,y ) ::> (B(x) ? B(y)
hence B(x) = B(y) by substitution. S o, a ss um in g w E B(x), by equalit yof s e ts , we obtain w EO B(y) . Now, by abstraction on w, z , y , x in th a torder , we obtain a p ro of of t he c l a i m:
To prove i t , assume x E A, Y e. A and z 6 .I(A ,x,y) . Then x
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c(mn) and continue byd e C(C) . since c ba s
(m = 0 ,1 . . n- i. t )
se tn
cm E C(mn) , (m = 0 , ' 1 . . n-1)
N -formationn
.N - introductionn
N -e l iminat ionn
Finite sets
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been . seen to be equal to mn and cm e C(mni 'i s a p r e m i s s Rn i s recursion over th e f in it e s et Nn; it i s a kind of d e f i n i t i o n cases .
Here, as usual, th e famfly of sets C(z) se t (z eN ) may be interpretednknow th e premisses , Rn i s explained .resul t i s m fo r some m be tween 0
n 'and n-1. Selec t th e cor-r-espond t ng element c of. ' . . mexecuting i t . The resu l t i s a .canonLce L element
as a property over N : Assuming we. n .as fol lows: f i rSt execute c, whose
So we "have th e sets ' NO with no elements, N,. with th e s ingle canonicalelement 01, with canonical elements O2, '12 , et c .
Note th a.t , up to 'n ow, we have no operations to build up setsfrom nothing, bu t only o pe ra ti on s t o o bt ai n new sets from given ones(and from families ' of se t s ) . We now introduce f in it e s e ts , wh i ch ar egiven outright; hence t he ir s et formation rules will ~ a v e no premisses.Actually, we have inf ini te ly 'many ru les , one group of rules fo r eachn = O. 1,
t he e qu iv al en ce o f t he se tworules. we can prove(2 ) t h e co rre spond ing p rop -indexed familY as in ,
(3 x E A)I(B,b(X) ,y ) (y E B) ,
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1 , f elements) . ~ h e r e(proper t ies and indexed f a m ~ 1e s 0o f l oo k in g at s ub se ts o f a se t B:
p Lx) e B (x E (E y E. B)C(y,
an indexed fami ly of elements(2 ) a subset ofb Lx ) E B (x e A).
roposi t ional f u ~ c t i o n (property)(1 ) a subset of B is a P .C(y) (y EO B) ;
ty as in (1 ) t h e co rre spond ingand, conversely , given a proper indexed family is
Using th e ident i tyconcepts. Given anerty i s
Examplear e two 'ways
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.1. t rue '(B t rue )
A t rueA V B t rue
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.1. -el1mina t .Lon
C t rue.1. t rue
Then 0 , is a (canonical) proof of ,l r , s ince 0, E N, by N, - i n t r o du ct i on . So T' i s t r ue . We now want to prove t ha t 0, i s .i n f ac t th e onlyelement of N" t ha t i s , t ha t th e ru l e ,
i s derivable . In fact" , from 0, 6 N" we ge t 0, : 0, 6 N" and hencer I (N"O"O;>. Now app l t 'N,-e l imina t ion with I (N"z ,O , ) (z E N, )fo r th e family of s e t s C(z) (z E N, >. ~ s i n g th e assumption c e N"we get R, (c , r ) E I (N"c ,O, ) , an d hence c : 0, EN , .
When C (z ) d oe s no t depend on z , it i s possible to suppress th eproof (const ruc t ion) n ot o nly in t h e conc l us io n but a lso in th epremiss . We then arr ive a t th e l og ica l inference rule
which i s eas i ly seen to be e qu iv a ie n t t o th e form above .Example (about N,). We define
' t r a d i t i o n a l l y ca l led ex falso quodl ibe t . This rule i s often used inord inary mathematics, bu t in th e form
we have th e
(m: 0, ', . . . , n -1 )
by th e above explana t ion,N _in t roduc t i on) :n
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R (m ,c o , , c n _ , )n n
N _equal i tyn
The ~ l i m i n a t i o n rule becomes simply :
N _el imina t ionR (c ) E: ct c)
t ha t we 'u n de r s t a n d t ha t we sha l l neverf th e rule isTh exp ' lanation 0 ' t R (c )e we sha l l never have to execu eO , element C6 NO' so t ha tget execut ing program of th e formThus th e se t of ins truct ions fo r b ti ' i mi la r to th e programmi'ng s tatement ~R (c ) i s vacuous . It s s , '2in troduced by Dijkst ra'2 Se e note 2 .
, in th e conclu, i of m - " ..., n-fo r each ch o ce ' - ,(one such rule Id b to pos t u l a t e th e r ule s fo r ns ion) . An a l t e rna t ive approach wou ' e == N + N e t c . , an d ,equal to a n d ' on iy , ',de f i ne N2 :: 'N, + N" N3 - , 2then det ive a l l other ru l es . no i n tr oduc ti on r u le hence noExample (about NO) NO ha s
na tu r a l to pu te lements ; it is thus
From th e meaning of Rn , givenn rules (note t ha t mn 6 Nn by
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a l l th e usual rules o f n e ga ti o n.we can e a s i l y derive
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Consistency
What can .we s ay about th e consistency of ou r system o f r ul es ?We can understand consistency i n two d i f f e r e n t ways :
( , ) M e t a m a t ~ ~ m a t i c a l consistency. Then , to prove mathematicallyth e consistency of theory T, we cons ide r ano the r t h ~ o r y T ', wh i c hcontains codes fo r p r oposi ti on s o f th e o r i g i n a l theory T and a predicate Der such that Der('A') e xp re s se s t he fac t t h a t th e propos i t i o nA with code ' A' i s derivable in T. Then we define Cons ==- 'Der( ' l . ' ) =. Der( '.L ' ) ::>.l. and ( t r y to ) prove that 'Cons i s t rue inT' . This method i s t he o nl y one applicable when, l i k e Hilber t , wegive up thehope 'o f a semant i c a l j u s t i f i c a t i o ~ of th e axioms and rulesof inference; it could be followed , wi th success , also f or i n t u i t i o n i s t i c type th eo ry , b ut , s in ce we have been as met icu ious abou t i t ss eman ti cs a s a bo ut i ts syntax , we have no need of i t . Instead, weconvince ourselves .d i r e c t l y of i t s consistency in th e following simpleminded way.
(2 ) Simple minded consistency . This means simply t h a t JL cannotbe proved, or that we ah aLl, never have th e r i g h t to judge .L t rue(which, un li ke t he p rop o si ti o n Cons above , i s no t a mathematicalproposi t ion) . To convince ourselves of th is , we a rg ue a s fol lows : i fc e 1- woul