Appendix 6: Schröder's lecture XI

Appendix 6" Schr6der's Lecture XI

Introduction In this lecture Schr6der primarily works out rules for sum (existential quantifier) and product (universal quantifier) over domains of the first o r d e r [Stufe] (quantifiers over individuals) and of the second o r d e r (quantifiers over all binary relatives on the domain, his version of seco n d - o r d e r logic). This is wholly interpreted; there is no fixed formal system with rules of inference in which he operates. The previous lectures indicate that Schr6der is perfectly happy to regard these operations as operations on propositional functions, rather than operations on formulas. This lecture thus has a highly algebraic flavor, even though it expresses more quantifier rules than one has seen anywhere else in Schr6der (or anywhere else, for that matter). The quantifier rules naturally emphasize interactions with relational operations, a subject probably not taken up since. The most interesting feature of this lecture is that Schr6der's algebraic point of view m e a n t that he regarded a universal existential prefix as a p r o d u c t of sums or a greatest lower b o u n d of least u p p e r bounds. He thus leapt from the finite case to the arbitrary case and simply wrote out the most general distributive law. If x and y range over the integers, he writes (qx)(3y)rh(x, y) as (u where f r a n g e s over all functions on the integers to the integers. From his point of view, this is II,Zj4(i,j) = ZjIIxrh(x,f(x)). Note that the least u p p e r b o u n d Ef on the right is over a c o n t i n u u m of functions. This reflects the complete distributive law (for propositional functions). This device and the formulas it entails is precisely the device used by L6wenheim in his difficult-tofollow p r o o f of the L6wenheim-Skolem theorem. We have used a simpler notation than these authors, since they wrote f as a sequence instead of as a function, giving complex-looking subscripts. This lecture is significant for revealing exactly how far S c h r 6 d e r got 339

SCHRC)DER'S LECTURE XI

34 ~

with q u a n t i f i e r s - - f u r t h e r t h a n the early Peirce; in addition, the algebraic p o i n t of view of the c o m p l e t e distributive law a p p e a r s h e r e for the first time in m a t h e m a t i c a l history, which was the device that L 6 w e n h e i m p i c k e d up for his proof.

Eleventh

Lecture

Studies of Elimination,

P r o d u c t a n d S u m Problems

w 29. O n Peirce's So-called "Development F o r m u l a s " S u m m a t i o n a n d Product Evaluation.

Page 491

On the Inversion Problem.

O n p a g e 190 of "Note B" (Peirce 1883; see also Peirce 1880, p. 55), Peirce r e m a r k s that in the relative algebra t h e r e are a n u m b e r of "curious d e v e l o p m e n t formulas," such as ab ; c = II (a " uc + b" f~c),

(a + b) 3- c = F.,{a d (u + c) }{bo'- (u + c) },

a" bc = II (au" b + ati" c),

a d- (b + c) - E {(a + u) a- b]{(a + zi) a'- c},

~ u

1)

u

w h e r e the II a n d E as identical p r o d u c t a n d identical sum, respectively, e x t e n d over all relatives of the universe o f discourse 12. . . . 1 Page 497

Peirce's p r o p o s i t i o n s 1), a n d o u r e x t e n s i o n o f t h e m , f o r m the "startu p stock" of p r o p o s i t i o n s a n d m e t h o d s which allow us to evaluate s u m s E as well as products II in o u r discipline. It is often useful to be able to give the identity p r o d u c t (the intersection [Gemeinheit]) of all relatives x which fulfill a certain c o n d i t i o n , for e x a m p l e , roots o f a given e q u a t i o n ~ a s we have already shown in the n i n t h lecture. Likewise, the q u e s t i o n r e g a r d i n g the identity sum o f all roots can be of i m p o r t a n c e . For this reason alone, the art o f determ i n i n g the sum a n d p r o d u c t ~ w h i c h is n o t so e a s y - - d e s e r v e s to be cultivated a n d d e v e l o p e d systematically. T h a t this is f u n d a m e n t a l for e l i m i n a t i o n p r o b l e m s , a n d for i n f e r e n c e in g e n e r a l , was shown at the e n d o f w 28. I shall now p r e s e n t a series o f my own findings which aim at i n c r e a s i n g this capital (science); they no l o n g e r relate to Peirce's publications, b u t are nevertheless relevant. We are d e a l i n g with sums E; a n d p r o d u c t s II which have "absolute" e x t e n s i o n , i.e., over the entire d o m a i n o f discourse. D e p e n d i n g o n w h e t h e r the i n d e x is an e l e m e n t symbol i or j, etc., a n d its e x t e n s i o n is the first d o m a i n of discourse 1', or w h e t h e r it a p p e a r s as the s u m or p r o d u c t variable of a binary relative u o f any given type with the s e c o n d ' Schr6der's proofs of Peirce's "curious development formulae" (pp. 521-536) have been omitted here.

341

FROM PEIRCE TO SKOLEM

d o m a i n o f discourse 12 as its e x t e n s i o n , we can distinguish two o r d e r s [Stufe] in s u m a n d p r o d u c t d e t e r m i n a t i o n . Even t h o u g h Peirce's p r o p o s i t i o n s 1) already b e l o n g to the s e c o n d o r d e r [zweite Stufe], we want, first, to c o n s i d e r the p r o b l e m s which b e l o n g to the first o r d e r [erste Stufe]. To b e g i n with, the r e a d e r can easily prove these g r o u p s o f little p r o p ositions by m e a n s of the coefficient evidence:

n i = n ; = n ~ = n ~ = 0,

Ei = E ~ = E i ' = E ~ = 1,

E i i = 1' = l-I(/+ ~) = H(~ + ;),

Ei~ = Ei-/" = 0 ' = II(~+ ~),

~ [ = 1,

II(i + i) = 0.

7)

8)

9,)

T h e i n d e x is a s s u m e d always to be i. For the proof, one only has to add the common coefficient of the suffix hk Page 498 for E or l-I, respectively, and to discuss it. For example, formula 9), on the left, becomes 7

7

/

I

If the domain 1 ~ has more than two elements, there is also an i for h :~ k. For / ! this i, 0;h0;k = 1 because it is different from both elements h and k, and the last Ei equals 1, q.e.d. For the domain 1,~ of only two elements, the right sides in equations 9) would have to be replaced by 1' or 0'. If we take, for the time being, J as the r e p r e s e n t a t i v e o f o n e o f the f o u r e l e m e n t relatives i, i, i, ;, we can s h o r t e n f o r m u l a 7) as follows: EJ=l,

nj=o.

Because II~(i)J~--IIJ, etc., it is clear that m o r e g e n e r a l l y we m u s t have {4~(i) + J} = 1,

II 4~(i)J = o.

Thus, we d o n o t have to write o u t the p r o d u c t f o r m u l a s for e x p r e s s i o n s such as IIi a;i" i, IIii" i;b, b e c a u s e we i m m e d i a t e l y r e c o g n i z e t h e m to be e q u a l to 0. Etc. P r o p o s i t i o n s such as E 0 ij'= 1,

II 0 ( ~ + ] ) =0,

which refer to d o u b l e o r m u l t i p l e sums o r p r o d u c t s , we d o n o t want to take into c o n s i d e r a t i o n for the time being. R e g a r d i n g f o r m u l a 7), the following p r o p o s i t i o n c o m e s f r o m 29) o f w 25:

SCHRODER'S LECTURE XI

342

"

{

- "/

Z i a ; i " i = Z~(aj. i)i

H,(a;i+~)

=Hi(ao ~+~)

IIi(~+ i;a) =II,(~+~j-

Z i i " i, a - Z~i( [j- a)

10) a).

Of course, it is also easy to produce the coefficient evidence for any of the formulas, for example, {if, (a" i + ~)}hk = II;{(a" i)hk + '=kh}= H;(ah; + 0~k) = ahkFinally, these can be i n f e r r e d m i n the form of r.,~a'i, i;l'= a" 1~, for examp l e m f r o m the more general proposition 14) which we will give later. F u r t h e r m o r e , we a r e a b le to e v a l u a t e Z i a n d IIi f o r t h e 16 o p e r a t i o n s c o m b i n i n g a g e n e r a l relative a a n d a relative o f i, w h i c h w e r e d i s c u s s e d Page 499 in f o r m u l a s 21) to 23) o f w 25. T h e results in q u e s t i o n a r e e x p r e s s e d by 32 f o r m u l a s w h i c h , i n c i d e n tally, also p r o v i d e r e p r e s e n t a t i o n s f o r t h e o p e r a t i o n s c o m b i n i n g t h e m o d u l e s a n d a. If we a s s u m e a as c o n s t a n t with r e s p e c t to t h e i n d e x w h i c h is to be a s s u m e d as i, we h a v e 1 = Z(ij- a) = E(~j- a) = Z(a j- i) = Z(a d- ~), 1 1)

O= I I i ; a = H i ; a = H a ; i = H a ; ~ ,

a" 1 = Za" i = Za" ~-= Za" i'= Z a ' ~ = Z:(aj- [), 1 ;a = Ei; a = Z~'a

= Z i ; a = Z ~ ; a = Z ( ~ j - a),

12) a j - 0 = H(aj- i) = II(aj- ~) = 1-I(a 0, i) = I I ( a ~ ~) = H a ; i, 0 0' a = H(i 0~ a) = H(~o~ a) = H ( i o ~ a) = II( ~j- a) = I I i ; a,

{

Ei(a~i)

=(aj-l');1,

I]i (i& a) = 1; (1'0" a),

Hi a ; [ = a ; 0 ' j - 0 , Hi i ; a = 0 0 , 0 " ,

13) a.

For the proof of 11), we should r e m e m b e r m f o r example, on the right s i d e m t h a t i ; a - i " 1 ; a b y 21) o f w 25; therefore, IIi; a = 1 ;a" Hi, which vanishes according to 7). Etc. For 12), we also only have to take 7), according to which we have

F.,a; i = a;F.,i=a;1,

II(a3- i) = a3- Hi = a3-0,

and, finally, a; i = a3- z from 22) of w 25. One part of these formulas we can infer in like m a n n e r (because of i';1 - 1) as Z,a;i=Z;a;i"

i;l=a;1,

II~a;i=II~(a;i+ i';0) = a3-0,

from the more general proposition 14), which will follow.

FROM PEIRCE TO SKOLEM

343

Thus, of these formulas, only the last, 13), needs to be explained.Justification is given for the top right formula by the observation that Lhk- lI,(a; ~)hk is different from Rhk only because of the different designation of the product index (i instead of m)--with respect to the proof given for 28) of w 25. Remarkably simply and i m p o r t a n t , it seems to me, are the following groups of propositions: a ; b = F,,a ; i" i; b = F,, (a j- ~')( ~j- b) ,.,

v

= E~ia; 1 ;bi = Z,{( ~ +a) j- 0}{0 j- (b + ~)},

a j- b = I-I, (a; i + i; b) = II, (a c~ ~ + ~J b) =II~{(~+a) j - 0 j - ( b + ~ ) = I I , ( i a ; 1 + 1 ;bi).

14)

It is a g o o d idea to m e m o r i z e the first formulas on the right of the Page 500 equal sign for each of these expressions. They teach us how to break

down a relative product into an identity sum, and a relative sum into an identity product. The proof is achieved fastest with 32) of w 25 where we have Y~ia;i. i; b = E ~ a ; i b = a ; b E i = a ; b l - a;b according to 7). Etc. The other forms of the proposition are modifications of the one proved from 22) of w 25--and we could state several others as well. T h e dual to 14) is v

II~a;i.

,..

i;b=(aj-O)(O~b),

E~(a;i+i;b)

=a;l+l;b,

because of 12), where, on the left, we must H~a; i 9 II~i'; b, etc. While we now can also easily evaluate

have

II~a;i.

15) i;b=

II~ai ; b = II, a ; ib = (a ~ 0)(0 ~ b), E , { ( a + z) j . b } = E , l a ~ ( i +

b)}=a;1

+ 1;b

15~)

---cf. 32) of w 2 5 - - i t is not at all true for II~a; ~b, and, in general, we still look at the large majority of sum a n d p r o d u c t expressions with confusion. T h e r e f o r e it seems advisable, first, to have at o u r disposal as completely as possible the simplest sum a n d p r o d u c t formulas from the ':start-up stock" of o u r discipline, and, second, to learn m e t h o d s to refer a given s u m m a t i o n p r o b l e m , etc., back to the p r o b l e m s solved with the simple formulas from the "start-up stock." C o n c e r n i n g the first task, we believe we should state, discuss, and

344

S C H R O D E R ' S LECTURE XI

p r o v e at the very least the following set of p r o p o s i t i o n s , as duals a n d s u p p l e m e n t s o f 10)"

{

E ; a ; ~ . i'= a;O',

H;(aj-i+~) = aj-l', 16)

-

I];i" ;; a = 0 ' ; a,

,.,

II; ( i + i'o~ a) = l'j- a,

,.,

Eia; i ; i = ~;(aj- ~ ) i = a ; 0 ' ,

II,(a ; i + ~') = II,(a j- ~ + i) = a j- 1',

E,~" i; a = E,~(~0~ a) = 0 ' ; a,

1-I;(i + i; a) = l-I;(/+ ~

{

a) = 1'~ a,

H , ( a ; ~+ ~) =a;O',

E; (aj- i)i = aj- 1',

H ; ( ~ + ;; a) = 0", a,

E ; i ( i ; a) = 1'~ a,

17)

18)

v

Z;a;['i=a;1,

H , ( a j - i + i) = aj-O,

~;~" i ; a = 1 ;a,

l-I; (i + i'd" a) = 0 d" a,

19.) w h e r e , w h e n w i t h o u t t h e asterisk, r e a d a ; 0 ' ; 0 '

{

E,(a~i)~=(a~l');O', _

,.,

~;i(ij-a) =0'; (l'~a),

Page 501

for a ; 1 . Etc.,

II, (a ; ~ + / ) = a i 0 ' ~ 1', ~,

II;(i+i;a)

= l'~0';a,

20)

In the 32 f o r m u l a s 10) a n d 16) to 20), we will find the ~ a n d II o f all ( b i n a r y identity) p r o d u c t s a n d sums which can be f o r m e d f r o m a ; i o r a;~, as well as f r o m a j - ~ o r a j-i and ; o r ~, e t c . - - a s l o n g as at least t h o s e E, II that can be r e d u c e d to 0 o r 1 at o n e g l a n c e are n o t t a k e n into c o n s i d e r a t i o n . Formulas of the type which, in their general terms a" i or a" ~ appears instead of a" i or a" i, etc., can easily be reduced to something we know because a" i or a'z splits into a" l ' i , respectively, a" 1 9 ~; they do not belong have the same status as the formulas discussed so far and do not deserve to be stated together with them. _

,.,

O f the f o r m u l a s stated, f o r m u l a 18) a p p e a r s as especially r e m a r k a b l e b e c a u s e it t e a c h e s us to r e p r e s e n t c e r t a i n relative p r o d u c t s , such as a ; 0 ' , as identity products, while, in g e n e r a l , this is only c a r r i e d o u t in t h e f o r m of an identity sum. For the proof of the propositions, we refer, for 16), to 30) of w 25 and to 7). Formulas 17) are particular cases which result from our theorem 14), in which. for example, the fight side has to be II,(a" i + z') -- 1-Ii(a" i + f; 1') = aj- 1'. For formula 18), we transform via the identity calculus" a" i+ ~ --a" f" i+ ~= a ' 0 " i + ~ = a ' 0 ' +~ according to 30) of w 25, where we must have ri;(a.0q~) = a" 0' + II;~= a" 0' + 0, according to 7).

FROM P E I R C E T O SKOLEM

345

For formulas 19) and 20), we appeal to the coefficient evidence, w h e r e u p o n the left side is, respectively, Lhk=Ei(a'[)hk t^k ~ = EiE t amtu, - -tk^ = E t ahl~,iOliOik , , 7

!

=

(a'O' "O')hk,

=

Lhk=~;IIt(aht+ ilk)Zkh=F.,iIIt(aht+ lt;)01k R^k,

q.e.d.

As d u a l a n d c o m p l e m e n t to f o r m u l a s 14), 15), we likewise have to cite the f o l l o w i n g p r o p o s i t i o n s w h i c h we can see as a g e n e r a l i z a t i o n o f p r o p o s i t i o n s 16) to 20) above: = (aj-l') ; (l'j-b),

E,(aj-i)(ij-b)

...

II,(a;~+~;b)

..,

B,(aj- i)( i ~ b) = r.,(aj, i) 9 i; b

= (a o" 1') ;b,

= a;O'~

...

II,(a ; i+ ;; b) = R ( a ~ ~+ ~; b)

= a ; (l'c~ b),

= aj-O'

E , a ; i" i'; b = E,a; ~. (~j. b) ] -_ _J ~ , a ; i" ~;b = ~,(a ~t i)" i ;b

IIi(ao~i+~j-

b) = I I , ( a j - i +

22)

", b,

= a;O"b,

i';b) }

I-I, (a j- i + i ~ b) = lI, (a ; i + ij- b)

I2,a ; ~. ~; b = a ; 1 ; b,

b)

b,

,1.

{ {

~

II,(a ; ~ + i; b) = II,(a ; i +

~,(a j- ;)( i ~ b) = E,a ; i " ( ij- b)

21)

=a;O'j-O';b,

1' = a j-

j- b,

II,(a j- i + i'0~ b) = a ~ 0 ~ b.

23)

24*)

Page 502

T h e II, o f the g e n e r a l t e r m s o n the left a n d the ~i o f t h o s e o n the r i g h t are easy to state a c c o r d i n g to 11) a n d 13) since the g e n e r a l t e r m s are c o m p o s i t e a n d can be split up. Proof. According to 3) and 4) of w 25, we can write i = 1' at ~,

i=/at 1',

i = 0 '. i,

/' = i ; 0 ' ,

w h e r e u p o n we can then rewrite { a at i = a at l t at i= (aat 1')"i, .

iatb=~at

l'atb=i;(l'atb),

a " ~ = a " O" i , -

.

;;b=i;O"b

25)

_

(as well as a a t i = a ' i ,

;atb=i;b),

in addition to 23) o f w 25, page 418.

T h e r e u p o n , all o f the f o r m u l a s 21) t h r o u g h 24) fall u n d e r t h e s c h e m e ( o f the first e q u a t i o n o n the left a n d the right) o f o u r p r o p o s i t i o n 14). We can see f r o m 25) in c o n n e c t i o n with 21) a n d 22) o f w 25 that, in

34 6

SCHR6DVR'S LECTURE XI

c o m b i n i n g relatives with e l e m e n t r e l a t i o n s , relative a d d i t i o n is always d i s p e n s a b l e , n a m e ly , it c a n be p l a y e d o u t as a relative m u l t i p l i c a t i o n ( e v e n w i t h o u t c o n t r a p o s i t i o n ) , w h e n it d o e s n o t n a t u r a l l y t e r m i n a t e in i d e n t i t y a d d i t i o n . Basically, o n e o n l y n e e d s to l e a r n to c a l c u l a t e well with t h e e x p r e s s i o n s o f th e two f o r m s a ; i a n d i ; b . . . . 2 Page 503

I n s t e a d o f e s t a b l i s h i n g still m o r e f o r m u l a s , we n o w w a n t to d e m o n strate, with a series o f s m a l l e r p r o b l e m s , h o w we c a n d e t e r m i n e , with t h e p r o p o s i t i o n s m e n t i o n e d so far, n u m e r o u s a n d variously s t a t e d p r o d ucts a n d sums. F o r t h a t p u r p o s e , we shall m a i n l y use p r o d u c t s , a n d n o t try to c o m p l e t e t h e g r o u p s o f a s s o c i a t e d f o r m u l a s . F r o m t h e way we t r e a t t h e s e e x a m p l e s , th e r e a d e r will at least b e able to g e t a n a b s t r a c t i d e a o f o u r m e t h o d s , a n d m o r e so in t h e f o l l o w i n g sections. Problem 1. We are looking for x = IIi (a ; i'+ ~; b). We have x = I I ; ( a ; l " i'+~;b) =II;(a;1 +~;b)(i'+~;b) =(a;1 + I I i ; ' , b ) I I , ( ; + f;0';b) =(a;1 + 0 , : t 0 ' ; b ) I I ; i ' ; ( l ' + 0 ' ; b ) by 13), and thus finally x = ( a ; 1 + 0 , r 0'; b)10 0'- (1' + 0'; b)].

Page 504

Problem 2. We are looking for x = H i (a; i'+ f; b). Solution: x = r I i ( a ; 1 9 f + [. 1;b) =(a;1 + 1 ;b)(a;1 + IIih(IIii'+ 1;b)rI,(i+ f) = a ; l " 1 ;b" I I i ( 0 ' ; i + i'; 1')= a; 1 ;b" (0'0~ 1'); therefore x = l " a;1 ;b. Problem 3. We are looking for x = H i (a; i + i; b) = II i (a ~ ~+ i; b).

x = I I i ( a ; i + i" 1 ;b) = ( I I i a ; i + 1 ;b)IIi(a;i+ i) = (a~0 + 1 ;b)IIi(a+ 1 ' ) ; i = ( a ~ 0 + 1;b){(a+ l ' ) ~ 0 } = a ( a + l ' ) ~ 0 + { ( a + l ' ) z t 0 } " 1;b; therefore x=a~ 0 + {(a+ 1') ~0} ;b. In particular, if we let a---0' and thereafter take a for b, we get the second formula on the left of the following group: a" 1 = l-I, (a" i + i; 0') = IIi(a" i+ ~), 1 "a=IIi(0"i+

i'a) =II;(~+ i'a),

a j- 0 = Ei (a j- ~) i', Oj-a=Eii(~o-a),

32)

which is an interesting dual to 18) in as much as it shows how the relative product a; 1, etc., can also be represented as a l-I; (instead of, as usual, Ei). The formulas, incidentally, can also be easily understood directly. Problem 4. We are looking for x = II; (a ~t i + i; b). Considering 25), it comes as x = IIi{(a j- 1') ; i + i; b} u n d e r the previously solved problem, and, therefore, has to be

x = a j - O + {(a~ 1' + l'),:t0l" 1 ;b. We can also determine x by means of a double product as follows. Because of 14), we have Schr6der's first-order identities involving negation and converse (pp. 501-503) have been omitted here.

FROM PEIRCE TO SKOLEM

347

x = II,{IIi(a ;j + j-; i) + i; b} = II u (a ;j + j';i + i; b) = IIj{a ;j + l-I,(j-;i + i; b)}. Now from the scheme of the previous problem, we have

n;(]; i+ i;b)

=j'j-0 + {0"+ 1') ~0}" 1 ; b = 0 + (l'j-j) 9 1 ; b = 0 *

by 3) of w 25, and therefore *x -- Ilja ;j - a3- 0. Problem 5. We are looking for x = I-I; (a j- [ + i; b) = H i (a; i + [; b).

x=IIa(a;i+ i" 1 ;b) = ( I I ~ a ; i + 1 ;b)IIa(a;i+ ~) = (aj-0 + 1 ;b)II;(a + 0 ' ) ; i . Thus x=(aj-O+l;b){(a+O')~O}=a&O+l~a;l'l;b=aj-O+l'a;l'l;b=aj 0+l~t'l" l'b. Page 505

-

Problem 6. We are looking for x=l-I~(aj-i+ i;b) =l-I;{(a~ 1 ' ) ; i + i;b)}. By the scheme of the previous problem, we can immediately state: x = a~ 0 + l'(aj- 1') ;1 ;b. But we can also go t h r o u g h the double product: x = H;Hj(a;j + j'; i + i; b) = Ilfla;j +j-,r + 1)"; 1 "b) = I l f l a ; j + l ' ; j ; b ) =IIj(a;j+j'b) =a~t0 + {(a+ 1')o'-0}" b, according to problems 5 and 3. T h e two results coincide because of the second formula of proposition 30. The II in problems 6 and 3 are the same! Problem 7. We are looking for x = II; (i; a + ~-;b).

x=IIi(i;a+ ~" 1;b) = ( I I ; i ; a + 1;b)IIi(i;a+ ~) = ( 0 + 1 ; b ) I I ; ( 0 ' ; i + i;a); therefore, from p r o b l e m 3: x = I ;a" 1 ;b. As n u m e r o u s as t h e p r o b l e m s a r e w h i c h c a n be s o l v e d in this way, we still c a n n o t discover, for e x a m p l e , Ili a;~b. L e t us t u r n n o w to t h e s u m a n d p r o d u c t p r o b l e m s o f t h e s e c o n d o r d e r [zweiten Stufe]. A n i m p o r t a n t p r o b l e m o f a g e n e r a l c h a r a c t e r is: to determine the "intersection"[ Gemeinheit] H, as well as the ( c o m m o n o r t o t a l ) "domain" [Bereich] ~ of all binary relatives x which fulfill a given condition--for e x a m p l e , t h e e q u a t i o n F(x) = 0 - - a s t h e i r " r o o t s . " It makes sense to d e n o t e these two unknowns (as p r o d u c t and as sum) by P and S. But the p r o d u c t is the Subject and the sum is the Predicate to each of the roots, so that this d e n o t a t i o n could be misleading. I will therefore call them P and Q. By marking the extension condition under the signs H, ~, the m a t h e m a t i c i a n would be inclined to write

P=IIx

Q=2x

x

x

{F(x) =0}

{F(x) =0}.

But our discipline has the advantage that the extension condition can be inc o r p o r a t e d into the H, E-expressions themselves. How this can be d o n e shall be explained shortly with an obvious extension of the problem. T h e p r o b l e m c a n b e c o n s i d e r a b l y g e n e r a l i z e d w h e n we try. to m u l t i p l y

348

SCHRODER'S LECTURE XI

o r a d d a given f u n c t i o n ~(x) o f t h e roots, i n s t e a d o f t h e r o o t x itself. We c o u l d t h e r e f o r e l o o k for P = II r

Page 506

Q =g r

x

x

{F(x) = 0}

{F(x) = 0}.

We give II a n d ~ a b s o l u t e e x t e n s i o n m o v e r all p o s s i b l e relatives x o f t h e s e c o n d d o m a i n o f d i s c o u r s e . N o w we o n l y h a v e to m a k e t h e g e n e r a l t e r m n e u t r a l w h e n x d o e s not fulfill t h e e x t e n s i o n c o n d i t i o n F(x) = O, t h a t is, we o n l y have to w o r r y that, in e a c h s u c h case, t h e g e n e r a l t e r m o f 1-I a n d E is n e u t r a l , n a m e l y e q u a l to 1 if it is a p r o d u c t factor, a n d e q u a l to 0, if it is a s u m m a n d . O n t h e o t h e r h a n d , t h e t e r m ~(x) m u s t a c t u a l l y a p p e a r o r be effective for every x w h i c h fulfills t h e e x t e n s i o n condition. We c a n d o this by w r i t i n g P = 11 { ~ ( x ) +

F(x)

= 0l,

Q = E

x

Depending

~b(x){F(x) =

33)

0}.

x

on whether

x is r o o t o r n o t , t h e f a c t o r p r o p o s i t i o n

F(x) = 0 in Qwill in fact have t h e t r u t h v a l u e 1 o r 0; o n t h e o t h e r h a n d , t h e n e g a t i o n w h i c h a p p e a r s as a s u m m a n d etc.

in P will b e e q u a l 0 o r 1,

In case the polynomial F(x) of our equation would only be able to express the values 0 and 1 as proposition symbols, for example a coefficient function or even as a "distinguished" relative, we can simplify the above as follows: P = 11 {cI,(x) + F(x)], x

Q = ~ ~(x) F(x). x

m

In this case, we would have (F #: 0) - (F= 1) - F a n d ( F = 0 ) -- (F = 1) -- F. In any other case, this would be a grave mistake. In g e n e r a l , we can n o w r e p l a c e , in 33), t h e p r o p o s i t i o n a l t e r m , acc o r d i n g to t h e s c h e m e o f w 11, by a b i n a r y relative, t h a t is to say, a d i s t i n g u i s h e d relative w h i c h t h e n takes o n t h e v a l u e 0 o r 1. T h e n we have m

F(x) = 0 = {F(x) 4: 0} = 1 ;F(x) ; 1,

{F(x) = 0} - 0 ~ F(x) ~ 0.

T h e n it follows P = II {O(x) + 1 ; F(x) ; 1 l, x

Q = 1~O(x){0 e F(x) ~ 0l.

34)

x

H e r e u can be w r i t t e n for x. In t h e l o w e r p o r t i o n o f t h e p r o b l e m , w h i c h i n t e r e s t e d us initially, we have, in p a r t i c u l a r ,

FROM PEIRCE TO SKOLEM

349

II{x + F(x) =0} = H{U + 1 ;F(x) 9 1}, x

u

{F(x) = 0} = g u{0 e F(u) e 0}. x

35)

u

If we could evaluate H, respectively g, taken over u with the absolute e x t e n s i o n (over all binary relatives), for any given f u n c t i o n of u, we would be able, with s c h e m e s 34), 35), to obtain the P a n d Q in quest i o n - - e v e n w i t h o u t knowing or d e t e r m i n i n g the r o o t x [of the e q u a t i o n

F(x) Page 507

= 0].

If, however, we know the g e n e r a l r o o t or the solution of this conditional e q u a t i o n in the f o r m of

x =f(u), we are in an even b e t t e r position to find the solution of o u r p r o b l e m , as we have m o r e i m m e d i a t e l y a n d simply P = H ~{f(u)}, u

Q = ~ ~{f(u)},

36)

as well as for the lower p o r t i o n of 35) of o u r p r o b l e m :

II,, {x + F(x) = 0} = H f(u),

r,x x{F(x) = 0} = ~f(u).

37)

A c c o r d i n g to the idea of the g e n e r a l solution, we have t h e n in fact for e v e r y u:

Fir(u)} = O,

Fl f(u)}

1.

W h e t h e r we chose this way or that, the art consists in determining the II and ~ taken over u with absolute extension from any given relative function "~(u) .

A method of solving this p r o b l e m in its full and u n l i m i t e d generality is n o t known.* But to discover such a m e t h o d is the ideal o f this t h e o r y which will p e r h a p s never be possible. Probably we will only be able to solve it in stages a n d thus a p p r o a c h this faraway goal slowly. For the time being, we can only set a definite goal for ourselves and, in o r d e r to reach it, create a part of the m e t h o d . Such a practical g o a l - - a n d , in fact, the closest o n e f r o m a systematic p o i n t of view--is the d e t e r m i n a t i o n of H a n d E of all roots of o n e of o u r t h r e e e l e m e n t a r y inversion p r o b l e m s . Page 508

Since x= u(aj-~) is the general root of the subsumption x" b 4= a, and, of course, we must have IIu =0, u

* Compare the end of this section.

Yu=l, u

35 ~

SCHRODER'S LECTURE XI

and since u =0 and u = 1 themselves figure u n d e r the values over which u extends, we have to have I I u c = c I I u - - O , I ; u c - - c r . u = c , as well as u

u

u

u

II (x + x" b=6--a) = O,

E (x" b=6--a) = a j- ~.

x

x

quod erat inveniendum.

W h e n e v e r 0 belongs to the roots x of the given condition, then Hx is equal to 0; and when to 1 belongs to them, then Z;x is equal to 1 and will not interest us any l o n g e r here. For example, IIu'a=0,

Eu'a=l'a

u

u

is immediate. T h u s , t h e r e r e m a i n s to b e s o l v e d t h e i n v e r s i o n p r o b l e m f o r t h e (ext e n d e d ) s e c o n d case a n d f o r t h e t h i r d case, a n d h e r e it will b e imp o r t a n t u i f we o n l y d e a l with o n e r e p r e s e n t a t i v e f o r e a c h c a s e m t h a t we l e a r n to e v a l u a t e a p r o d u c t in t h e f o r m o f x = II [u + a{(b + (t) j- c} ; d].

38)

u

In o r d e r to r e a l i z e this g o a l , w h i c h c a n o n l y b e a p p r o a c h e d we n e e d to t a c k l e a s e r i e s o f p r e l i m i n a r y p r o b l e m s .

in s t a g e s ,

Pro b l e m 8. We are looking for: x = II (u + 7i" a). u

According to Peirce's proposition 1) or 4), we can immediately state x = 1 9al', a n d so x = II (u" 1' + ,i" a). u

Because of 1 9i1~= i; 1~= i, we have, in particular, II (u + ti'i) = i, and similarly II (u + fi" i) = [. u

u

As a corollary, we now also have found: II(u+a" u

~'b) =a'l'b=a"

l'bl',

because the general factor can be split into (u + a)(u + (t" b), and therefore the II can be separated into the II of the first factor, I I ( u + a ) = a + I I u = a + O = a, and the II of the second, which falls u n d e r the above schema. P ro b l e m 9. We are looking for: x = II ( u ' a + (t r b). u

According to 14), we can r e p r e s e n t tij-b as a product (over z), by which we have won the game! Because we can now c o n c l u de

35 x

FROM PEIRCE TO SKOLEM

x = I I { u ' a + II;(~i" i + i;b)] u

•II;{II (u" a + ~i'i) + i;b}-II;(1 "ai+ i;b) u

Page 509

a c c o r d i n g to 4), a n d further: x=II~(i;a+ i;b) =I'I;/; ( a + b). By the last f o r m u l a 12), we have thus found: x --0 ~ (a + b) m w h i c h is, curiously e n o u g h , symmetric with respect to a a n d b. If we assume that a -- 1' and t h e n take a for b, t h e n we have f o u n d , in particular, I I ( u + 7i,y a) = 0 ~ ( a + u

1').

[However, if we were to assume b =0', e x c h a n g i n g u for ti, we would get the result of p r o b l e m 8 again, a l t h o u g h in the s o m e w h a t d i f f e r e n t f o r m of 0j-

(a+ 0').] As corollary of this result, we now also know I I { u + a(~i~ b)} =a{Oj- ( b + 1')}, u

which can be i n f e r r e d in a similar way as above. With exactly the same m e t h o d , we can also solve an e x p a n s i o n of the previous problem. P r o b l e m 10. We are looking for x = II {u" a + (~i + b) j- c}. u

x = I I I I i { u ' a + (~i+ b)"i+ i;c} =II~{b" i + i ; c + I I ( u ' a + u

u

fi'i)}

=YI~(b" i + i;c+ 1 "ai) =II;{b" i + i ; ( c + a)} = b,y (a + c) by 14), 4), a n d 14). In particular, we have now f o u n d YI{u+ (7i+ a) o~b} = a,y (1' + b), u

1-[ [u + a{(Ti + b) ,t- c}] = a{b3- (c + 1')}. u

As in 1), by m e a n s of a I~ over u, so we can now also r e p r e s e n t a0,- (b + c)malt h o u g h its f o r m is u n s y m m e t r i c a l m b y a H over u. P r o b l e m 11.* We are l o o k i n g for x -- H{u + mi" b}. u

We will not s u c c e e d in solving this p r o b l e m in the same way as the two previous ones because we c a n n o t r e p r e s e n t the relative p r o d u c t of the s e c o n d t e r m of the g e n e r a l factor as H; but only as Ei, a c c o r d i n g to 14)" however, a I] c a n n o t be t r a n s f e r r e d from b e h i n d a II to the f r o n t of it. In the case of b =0', we can refer to 18). T h e r e f o r e , we must, in factmsolve only the simplest case:

n(u+ a . 0 ' ) : n , l : + n ( , , + a.;)l-n;(~+h =n;~=0 u

u

a c c o r d i n g to 7) a n d p r o b l e m 8 - - w h i c h we have already u n d e r s t o o d on page * Because of u = 1u" 1' its quickest solution comes from 6) p. 496, and is a special case ~

9

35 2

SCHRODER'S LECTURE XI

494 from Peirce's formula 1). Now we also have aft" b0':(= ft'0', and therefore also II (u + aft" b0') :(= II (u + ft" 0') = 0. u

u

If we solve (for x) b =0'b + l'b, and if we bear in m i n d that, according to 24) of w 22, we have a~i" b l ' - a f t " 1" bl', then we can split the general term, aft'bO' + a" 1 "bl',

u+ aft'b=u+

Page 510

suppressing the factor ft in the last term in the presence of the first. We then obtain x = a " l'b,

while the H on the right-hand side vanishes by the previous result. In particular, we have II (u + aft" 1) = a. u

Please observe that, in spite of the p l a c e m e n t of the parentheses which are different from those in the corollary to p r o b l e m 8, the end result of the two is the same. We can also immediately recognize the new value as the lower limit for x because y = II (au + aft" b) = II (au" 1' + aft" b) = a" l'b u

u

according to Peirce's t h e o r e m 1, but, because of au =6- u, we must have y =6= x. T h e same lower limit can also be d e t e r m i n e d by means of Xhk = II [Uhk + E t ahtfthtbtk ] = II E l (Uhk + ahtbtk)(Uhk + fthl) u

u

because we must have EtH :(= HE t according to the propositional s c h e m e o) on page 41. Because we now have Iluhk=0 , we get, as we will show in the next problem, II(uh, + fth,)= l't,, SO follows E, ah, b,,l'~,= (a" l'b)hk=C--Xhk. [It is conceivable to set an u p p e r limit for the lower limit for x. T h e reasoning goes thus: aft;b=6--a;b" ft;b, therefore x=6--a;b" 1 ; b l ' = a ; b ; l ' b , and the proposition a; l'b=6--a;b; l'b has to be valid, which also can easily be proved directly. However, we have already recognized the lower limit as the exact value.] P r o b l e m 12. We a r e l o o k i n g for: x = II {u + (7i 0~ a) ; b}. T h i s is a d i f f i c u l t o n e ! Its s o l u t i o n will o n l y b e p o s s i b l e f o r c e r t a i n s p e c i a l cases, s u c h as I I { u + (7i0~0) ;b} =0, tt

II{u+ u

(720~ a ) ; l ' b } =00~ (a0' + bl'),

f o r w h i c h we g e t t h e r e s u l t easily f r o m w h a t we h a v e l e a r n e d so far: b e c a u s e t h e s e c o n d t e r m o f t h e g e n e r a l f a c t o r splits i n t o (7i0~0) 9 1 ;b

F R O M P E I R C E T O SKOLEM

353

o r (~20~ a) 9 1 ;bl', r e s p e c t i v e l y ~ w h e r e u p o n t h e g e n e r a l f a c t o r itself a n d its II split up. This last r e s u l t is o b t a i n e d by m u l t i p l y i n g t h e c o m b i n a t i o n 0 0 ~ (a + 1') with 1 ;bl', w h i c h is = 0 ~ (b + 0'). An important special case in which the solution is easy is the case b = i. If we let y =II{u + (tij-a)"i} u

and deal first with this subproblem, we have

yhk--~{Uh, + Et(tij-a)mit,} =~{Uhk + (~i3- a)hi} =IIluhk + IIt (liht + at;)} = IItlati + II (u^k + ~iht)} =l-It(at; + l~t) u

u

: (l'j- a)ki = {(l'J- a) "i}kh.

Page 511 That in fact we obtain II (uhk + ~ih~) = l't, u

can be explained as follows: for l = k this II equals 1, but for l ~ k it has to be equal to 0; the latter vanishes as a factor of II because then we also have a m o n g the u (that is, a m o n g all possible ones) some for which u^, =0 and, at the same time, uht= 1; therefore also 7iht=0. We have now found y = i; (d0~ 1'). In t h e g e n e r a l case we can (again) find two limits b e t w e e n w h i c h t h e u n k n o w n ( b u t fully d e t e r m i n e d ) relative x m u s t be. To d e t e r m i n e t h e s e limits b e f o r e h a n d is w o r t h t h e t r o u b l e for two r e a s o n s . First, t h e y give u s m a s in t h e p r e v i o u s special c a s e s m a v a l u a b l e way o f c h e c k i n g t h e e x a c t value o f x w h i c h we shall find later with c o m p l e t e l y n e w m e t h o d s . Also, we can t h e r e b y d i s c o v e r r e m a r k a b l e t h e o r e m s by s h e e r luck. By 14), we have x = IIF,~{u + (~ij-a);i" i;b}, a n d since, by t h e p r o p o s i t i o n a l s c h e m e 0), p a g e 41, we have EII :(= l i E , we m u s t g e t u

E ~ I I { u + (Tiz~ a ) ; i } I I ( u + u

i;b) = ~ i ;

(~izj 1') 9 i;b=F,~i;(gtztl')b:~--x

u

- - - c o m p a r e 26) o f w 25; t h e r e f o r e , by 12), 1 ;(~i# l')b:~--x, w h i c h gives us t h e lower limit. In o r d e r to find t h e u p p e r limit, we write, also a c c o r d i n g to 14)" x = II [u + {II~(~i; i + i;a)};b]. B u t h e r e we are n o t p e r m i t t e d to i g n o r e t h e p a r e n t h e s e s . S u p p r e s s i n g t h e m w o u l d , o n t h e c o n t r a r y , a m o u n t to d i s p l a c i n g t h e m a n d w o u l d set t h e m over, a c c o r d i n g to s c h e m e {IIa} ; blc--IIa ; b, w h i c h = II{a ; b}. T h u s , It

v

,,,

x:~--II~[H(u+ ~ i ; i ; b ) + i;a;b] =II~{1 ; ( i ; b ) l ' + i;a;b}

354

SCHRODER'S LECTURE XI ,..

---cf. p r o b l e m 8. But b e c a u s e o f i ; b = i . 1 ; b a n d 1 ; i 1 ' - i ; first t e r m can be c h a n g e d into i" 1" b a n d we obtain:

,,,

1'-i,

the

x ~: II~i; (1' + a;b) 9 (1 ; b + fl~i;a;b) ={0ff(l'+a;b)}(1;b+0j-a;b) =0ea;b+{0~(a;b+

]~)}. 1;b.

H e r e the first term, which is c o n t a i n e d in the s e c o n d o n e , can be o m i t t e d b e c a u s e we have 0 0~ a ; b:(=0 0~ (a ; b + 1'), as well as 0 o~ a ; b=(c--a; b=~-1 ; b. T h e r e f o r e , the s e c o n d t e r m r e m a i n s as the upper limit that we s o u g h t a n d is c o n f i r m e d by the whole expression, which m u s t be

1;(dj-l')b~x~{O~.(a;b+l')}.

1;b=0j-(a;b+l;b.

1').

In the last e x p r e s s i o n for x [Translator's note: x = II [u + {IIi(~; i + i; a)} ;b]], we c a n n o t pull the /I~ o u t o f the braces by an e q u i v a l e n t t r a n s f o r m a t i o n , and, by the s a m e token, we c a n n o t p u s h the E; in f r o n t Page 512 o f the II, in the first e x p r e s s i o n for x [Translator's note: x = HE~{u + (~ 0~ a) ; i" i; b}], as was asserted above, in o r d e r to find the lower limit. This c o u l d only s u c c e e d with a c o u r a g e o u s p r o c e d u r e : T h e m e t h o d would be to o p e r a t e with infinite (or u n l i m i t e d ) multiple p r o d u c t s H; even with o n e whose H-sign c o u l d possibly f o r m a continu u m (in case we would write it d o w n in detail); for e x a m p l e , if we assign to each point of the linea lI c o r r e s p o n d i n g to a p r o d u c t variable specifically chosen. For such p r o d u c t s a n d sums we may also w i t h o u t hesitation transfer a n d apply the i n f e r e n c e rules which are g u a r a n t e e d by the p r o p o sitional s c h e m e based o n the dictum de omni. This is p r o b a b l y the first time in m a t h e m a t i c s that this has b e e n d o n e . I will t h e r e f o r e g u i d e the s t u d e n t heuristically a l o n g the p a t h o n which the m e t h o d first o c c u r r e d to me. I first tried to e x t e n d the p a r t i c u l a r case y o f o u r p r o b l e m - - w h e r e the solution was f o u n d - - b y trying to o b t a i n (as a s u b p r o b l e m ) : "

u

u

"

u

z - H{u + (~j- a) ;i + (~o" a) ;j}. u

We have Zhk = II{Uhk + ({tJ" a)h , + (~J- a)hj} u

"-

I-I {Uhk + u

tim( Uhm "~ a,,,,) +

1-I,,(~h,, + a,q)}

= II,,,,{ami + a,q + II (Uhk + ftnm + ~h,,)}" u

And

now

II (Uh, + Uhm + Uh,,) = l'mk + l',,k, u

FROM PEIRCE TO SKOLEM

355

namely, equal to 0 for (m :/: k)(n :/: k) because, a m o n g o t h e r things, we will have a factor with Uhk = O, Uhm = 1, Uh,, = 1, a n d equal to 1 for (m = k) + (n = k). It follows that:

Z,,k = H,,H,,(I',~ + am, + 1~,,, + a,,j) Hm( l/kin + a,,,) + H,(ak, + a,j) = (l'j- a)k, + (l'J- a)ki = {(1' J- a) ;i + (1' ~ a) ;j}kh, =

w h e r e b y we have f o u n d ;

As we now f o u n d a solution to P r o b l e m 12 easily in the case w h e n b = i is an e l e m e n t , as well as w h e n b = i + j r e p r e s e n t s a system of two e l e m e n t s , we c a n n o t foresee why it s h o u l d n o t also work in the case w h e r e b = b ; 1 is a system a n d t h e r e f o r e a sum of any n u m b e r of e l e m e n t s that could possibly as points c o n t i n u o u s l y fill a line. We i m m e d i a t e l y observe that the investigation only has to be g e n e r a l i z e quantitatively, and, i n d e e d , we will find

H { u + (ftj-a) ;b;1} = l ;f~; (Sj- l'), Page 513

u

H{u + ((tj-a) ;1} = l ; (Sj- l'). u

L o o k i n g back at Zhk, we observe that o u r i n f e r e n c e s would n o t have b e e n possible if w e - - w h i c h s e e m e d feasible at f i r s t - - h a d used the s a m e letter m for the i n d e x n of the last H as for the previous H. If, however, all terms are p r o d u c t s II in a sum with m u t u a l l y indep e n d e n t indices, n a m e d i n d e p e n d e n t l y , t h e n it is possible to advance all of the H, each affixed with its index as suffix, to the left; it is possible to use this insight also for a sum r e p r e s e n t e d symbolically by g. We now i n t e n d to d e d u c e the f o r m u l a on the left, o n p a g e 512 (bott o m ) (of which the o n e on the right is m e r e l y a special c a s e ) - - b y calling s the p r o d u c t H over u we are looking f o r - - b e c a u s e this will give us g r o u n d s to formalize o u r p r o c e d u r e . For that p u r p o s e it is c o n v e n i e n t to use for the system b;1 = b the f o r m u l a o b t a i n e d f r o m w 27 as b = E~b~i or, shorter, ~ii; w h e r e we only have to k e e p in m i n d that the sum over i, the ~ which does n o t have the i n d e x a d d e d as d e p e n d e n t suffix, but w h e r e it a p p e a r s (ad hoc) written u n d e r n e a t h , that this s u m does n o t have the full, b u t a s o m e h o w given (limited or u n l i m i t e d ) e x t e n s i o n f r o m the d o m a i n of discourse 11 of the e l e m e n t s . For s = I I { u + (~2o~a);E;}, t h e r e f o r e Shk=II{u + E;(~20~ a);i}hk u

=HlUhk + E,({tJ- a)h,} = IIluhk + Z,flm(5h,, + am,)}. u

u

W i t h o u t a new idea, we c a n n o t c o n t i n u e b e c a u s e we c a n n o t b r i n g the II m b e f o r e the ~i a n d t h e r e b y b e f o r e the II. T h e idea (already a l l u d e d to) which will get us f u r t h e r is the o n e which~we now generally f o r m u l a t e

35 6

Page 5 1 4

SCHRC)DER'S

LECTURE

XI

in the m a i n text a n d c o n f r o n t it with its dual c o u n t e r p a r t - - w i t h o u t m e n t i o n i n g the latter very m u c h . If we have a E i of a lI m of a g e n e r a l t e r m f(i,m), a n d we wish for s o m e r e a s o n to push the E b e h i n d the II in an equivalent transformation, this is n o t i m m e d i a t e l y possible. Because of ~II :(= II~, we could only do so by drawing weakened conc l u s i o n s - - i f we want to be satisfied with such a p r o c e d u r e . Otherwise, n o t h i n g h i n d e r s us f r o m r e n a m i n g the i n d e x of the IIm in all the o t h e r t e r m s of the ~i, that is, "to differentiate" all these indices as m, (m with the suffix i o t a ) m w h e r e we only have to r e m e m b e r that L c h a n g e s in "parallel" with i. It a p p e a r s suggestive to take for ~ the letter i itself as suffix for m. D i s r e g a r d i n g the fact that m~ already has a fixed m e a n i n g as relative coefficient of the e l e m e n t m in w 27, it still would n o t be correct. As we will soon s e e - - i n case we s u c c e e d m w e may not c h o o s e for L a symbol which contains the n a m e /-----such as 4~(i). This will have the advantage that we now can push every single II to be taken over a certain m, to the front, in f r o n t of o u r Z. We can now justify the i m p o r t a n t f o r m u l a

{ ~,Ilmf(i,m ) = E,IIm,f(i,m,) = II, (IIm,)]],f(i,m,), II,~.f(i,m) = II,~m,f(i,m,) = l-I, (~ml)II,f(i,m,),

39)

by which we have a t t a i n e d o u r goal of having p u s h e d all II in front of

the ~. To facilitate the printing, we have set the indices i a n d m as if they, as e l e m e n t s , would have full e x t e n s i o n over 11. I a m sure this is permissible. But it is not at all necessary for the p r o o f of o u r s c h e m e . T h e indices i a n d m may have any given e x t e n s i o n s in l ~ - - p r e s u p p o s i n g of course that the e x t e n s i o n of m is i n d e p e n d e n t of i, the s a m e for e a c h i a n d is also t r a n s f e r r e d to each m,, that is, associated to e a c h of these (a limitation of which even the last p a r t of each of the d o u b l e s c h e m a t a is i n d e p e n d e n t ) . In o t h e r words, we may also write Ei for Zi or 1-Im for I-I m .

O u r s c h e m a would r e m a i n in force even if the two indices, or o n e of t h e m , were not e l e m e n t letters but would have their e x t e n s i o n in 12 as a u or v. But we do n o t want to discuss this possibility here. T h e last part of o u r s c h e m e n e e d s a f u r t h e r e x p l a n a t i o n , but has to be understood first. If L (in parallel with i) would have to run t h r o u g h a series of values 1,2,..., we could explain the m e a n i n g of the mysterious o p e r a t o r in f r o n t of the last ~ in 39) by writing it in the o r d i n a r y way, explicitly--by n o t m e n t i o n i n g the g e n e r a l t e r m or factor, in the f o r m of

FROM

PEIRCE

TO SKOLEM

357

II,(II,,,,) =IIm II,, IIm.~'''

or

Hmlm2m3.." =

Hiltm

t

a n d d e f i n e it as p r o d u c t symbol for a (possibly u n l i m i t e d ) "multiple p r o d u c t . " And t h e n II,(E,,,) = EmtEm2E,,~...

or

E,,~m2m3... = En, m,

would be n o t h i n g but the s u m m a t i o n symbol to indicate a " multiple s u m . " Since the latter c o r r e s p o n d s , dually, to the m u l t i p l e p r o d u c t , we can Page 515 see that in the new symbol (which is shown to be i n d i s p e n s a b l e for abbreviation h e r e ) the II, may not be transcribed, dually, to E,, but will r e m a i n as II, in the dual c o u n t e r p a r t to the s c h e m e . It r e m a i n s o p e n w h e t h e r the t h e o r y will ever m a k e use of symbols such as E,(IIm, ), E,(Em,), by which only some, any, but at least o n e of the II or E over m, could be set. O f the given r e p r e s e n t a t i o n s or m e t h o d s of expressions, the ones on the right are less good, even misleading, for the r e a s o n that the composite suffix m 1m.2m3"" of a II or E s h o u l d not be a real p r o d u c t ( n e i t h e r an identity n o r a relative o n e ) , but stands conventionally for the "series" m l , m 2 , m 3, "" (cf. p. 24). It is true that o u r II, does not p o i n t to a real p r o d u c t either, but to a succession of signs (of the type i n d i c a t e d after it in p a r e n t h e s e s ) which may also b e c o m e a c o n t i n u u m . If the L in parallel with i has to run t h r o u g h a c o n t i n u u m of values, such as all the points of a line, we can no l o n g e r write the m e a n i n g of II,(IIm,) explicitly. A r i t h m e t i c allows us, however, to n a m e t h e m all a n d differently by assigning each of those points to a real n u m b e r f r o m an interval. For e x a m p l e , we could let m, be the n u m b e r c o r r e s p o n d i n g to p o i n t L. Thus, we can only say with respect to the e x p l a n a t i o n o f o u r symbols: we have to a s s u m e f o r each p o i n t L of the line a II m t 9 T h e o r d e r in which such II over d i f f e r e n t indices are taken (if we want to a s s u m e t h e m in a definite s e q u e n c e , which is n o t always necessary) is of no c o n s e q u e n c e , as is well known. B e c a u s e ~ a c c o r d i n g to the broadly used dictum de omni: what applies at e a c h m for e a c h n necessarily has to apply at each n for each m, etc. To prove o u r s c h e m e 39) on the left-hand side, we want to think, for didactic reasons, of a discrete series of i a n d L, w h e r e we c h o o s e the n a m e s m l, m2, m3, for the index m of IIm r e f e r r i n g to the values A, B, C, ..., or also il, i2, i3 . . . . . of i. T h e n the left side of o u r s c h e m e is . . .

L = I l m l f ( i l , m x ) + Ilmzf(iz,m,2) + I I , , 3 f ( i ~ , m 3 ) +

-

rlm,n rlm,

...

{f(i~,m,) + f ( i z , m 2 ) + f ( i ~ , m 3 ) + ...} = R.

For its proof, we only have to say that the sum of the t e r m s of L,

358

SCHRODER'S LECTURE XI

which, for l-I, p r e c e d e or s u c c e e d a definite m• do n o t c o n t a i n this i n d e x m• and, as a c o n s t a n t with respect to it, can be d e s i g n a t e d as a Page 516 o r b, ad hoc. T h e n it is only necessary for the t r a n s f o r m a t i o n o f L to R for e a c h o f the d i f f e r e n t i a t e d (that is, differently n a m e d ) rn to use the scheme" a + IImf(m) + b = rl m{a + f ( m ) + b}, w h e r e the II m can be e x t e n d e d over the p r e c e d i n g o r the s u c c e e d i n g c o n s t a n t a d d e n d . This very scheme---cf. 26), p a g e 1 0 0 ) - - c o u l d easily be p r o v e d by the initial p r o p o s i t i o n a l s c h e m e 3,), p a g e 40. In a similar way, we would have for the s c h e m e 39), the r i g h t - h a n d side" L = r,m,f(i,,ml)E,,.~f(iz,mz)~,m~f(i3,m 3) ... = E,,Em, E,,:~ . . . f ( i ~ , m ~ ) f ( i z , m z ) f ( i ~ , m 3 )

....

R.

W h a t we have said shall n o t j u s t be justified a n d stated for an arbitrary discrete value series of i a n d L (for which we illustrated it above, so to s p e a k ) - - f o r e x a m p l e , by p r o o f by i n d u c t i o n , b u t we w a n t to use it for all i, L a c c o r d i n g to the i. If the i, L would have to r u n t h r o u g h a c o n t i n u u m o f values, we can a s s u m e for each i x, X of their value, as we have p r o v e d f r o m / 3 ) , p a g e 37, that a c c o r d i n g to 18), p a g e 98, each t e r m o f a "E" can also be r e p r e s e n t e d as a real term o f a (binary) "sum" (in the n a r r o w e s t sense), the o t h e r t e r m o f which, d e s i g n a t e d as a ( i n d e p e n d e n t o f the m app e a r i n g in this term) m u s t to be subject to the s c h e m e Ilmf(m) + a = IIm{f(m ) + a}. Etc. q.e.d. F u r t h e r m o r e , we can see that o u r s c h e m e would be false a n d illusionary if we would replace the i n d e x n a m e m, by m i or by 4)(i). Because, in its last part, F,if(i,m ) would a p p e a r as the g e n e r a l factor o f II, a n d this w o u l d have to have a value totally i n d e p e n d e n t of i since the letter i only f u n c t i o n s as p l a c e h o l d e r for the values which are assigned to it f r o m the e x t e n s i o n o f i. T h e r e f o r e , we can also n o l o n g e r use m i in its e v a l u a t e d expression. (In analogy to a definite integral which is indep e n d e n t o f its i n t e g r a t i o n variables!) T h e r e f o r e , we can totally o m i t the o p e r a t o r Ili(IIm;) p r e c e d i n g the term, a c c o r d i n g to the law of tautology H a = a, a n d o u r s c h e m e would t h e n be very m u c h simplified. T h a t such simplifications are generally n o t admissible c o u l d be illustrated with examples. T h e application o f o u r s c h e m a to o u r p r o b l e m is now: Page 517

Shk = I'Iluhk + II,(IIm,)~-,i(Uhm ' + am, i)} = nn,(nm,)lu,,, + r,(a,,~, + am, i) } = II,(IIm,) H (Uhk + ~,~2,,~, + Z;,am,,) tt

=

am,, + n

+

FROM PEIRCE TO SKOLEM

359

Again, we can easily see that we have to have 40)

IIu (Uhk + I]iuhm,) = I]il'kmt

which, in case even only o n e of the m, is equal to k, m a k e s sense in the f o r m of the e q u a t i o n 1 = 1; in the case, however, that all m, of the sum over i are u n e q u a l to k, to be r e c o g n i z e d in the f o r m of the e q u a t i o n 0 = 0, t h e n a m o n g the admissible values of u there will be o n e such for which uh, = 0, as well as i (and the L c h a n g i n g in parallel with it) a n d each Uhm' = 1, that is, each Uhm, = 0, and the factor of 1-I thus vanishes. T h e n we have

Shk = II,(1-Im,)E,(a'k,,, + am,,) = E,II,,(lkm + am,) = E,(I'~ a)k,, w h e r e we used o u r s c h e m e 39) again in reverse. T h a t is,

S,,k = E,{(I' ~ a) ; ilk,, = 1(1' ~ a) ;F,,iik, , = 1(1' J- a) ;bi,h = 1/~; ( ~

1')1,,,,

w h e r e b y it is f o u n d consistent with the above p r o b l e m s = 1 ;/~; (5~f 1'). Now that we have the method for the solution, we want to tackle and solve the g e n e r a l p r o b l e m , r a t h e r than the special p r o b l e m 12, which we c h a r a c t e r i z e d in p r o b l e m 8. Before we start, we have to say that by specializing the result of the g e n e r a l investigation, the solution to p r o b l e m 12 can be easily f o u n d as

x=l:(d0

~l')b.

T h a t is, the lower limit for x f o u n d on page 511 r e p r e s e n t s in this case the exact value of this u n k n o w n . T h e r e f o r e , the c h e c k given by the previous d e t e r m i n a t i o n of this limit c o r r e s p o n d s to o u r result. This result also gives us an answer for the p a r t i c u l a r cases of the p r o b l e m settled previously. Thus, it is i m m e d i a t e for a - 0. If, on the o t h e r h a n d , we have bl' for b, we have to u n d e r s t a n d 1 ; (~-

Page 518

l ' ) b l ' = 0 0~ (a0' + bl').

To that end, we can (and I anticipate a little), a c c o r d i n g to the already stated propositions 47), 46), separate the left-hand side into 1 ; ( ~ 0 ~ 1 ' ) 1 ' - 1 ; (1'0~ a ) l ' = 0 cr (a + 1') a n d 1 ; b l ' - 0 el- (b + 0'), t h e n the product of these two expressions 0 el- (a + l')(b + 0') can be e x p r e s s e d as the r i g h t - h a n d side, q.e.d. If, finally, b ; 1 stands for b, we i m m e d i a t e l y have 1 ; (~0~ l')(b ; 1) 1 ;b; ( ~ 1), q.e.d. At times, o n e obtains interesting propositions even if o n e m a k e s a mistake! I have b e e n led to wrongly c h a n g e (i0~ a) ;b into i'cl- a ; b in the d e d u c t i o n of x by inaccurately r e m e m b e r i n g p r o p o s i t i o n s 27) of w 29,

SCHRODER'S LECTURE XI

36o

and thus obtained the value 0 ~t (a;b + 1') for x with which all of the four checkings, except the last, are correct. Strangely enough, this incorrect result lies correctly between the previous determined limits, and by checking it, one obtains remarkable propositions! Indeed, we must have

1;(dd-l')b~O~(a;b+l')=(=O~(a;b+

1 " 1;b).

Because the major has already been split into {0~ ( a ; b + 1')}{0 (a;b+l;b)}=x" ( 0 c t l ; b ) = x " 1;b, we only have to represent x=(= x. 1;bofx~=l;bby0~(a;b+l'):(=0j-(1;b+l')=0j-l'+l;b=l;b. More valuable is what the minor, the first partial subsumption of our double summation, teaches. According to the first inversion theorem, it can be transcribed equivalently into 1 ;1 ; ( ~ t l')b:~--a;b + 1', or into the first proposition of the following pair: 0' 9 1 ; (g~t l ' ) b ~ : a ; b ,

0'

a(l'j-/~)" 1 :,~--a;b,

aj-b~--l'+ Oj. ( d ; 0 ' + b), a~b:~--l' + (a + 0' ;/~) j-0,

41)

of which the conjugated propositions, combined with each other and with what is already known, enclose the relative product and the relative sum between the following limits:

{

0'{a(l'ct/~) ;1 + l'(dj-l')b}:~--a;b:~--a;l"

1;b,

actO + Oj-b~--a~b::~--l' + {(a+ 0' ;/~) cr 0}{0 ~t ( ~ ; 0 ' + b)}.

42)

To prove the first proposition 41) from the coefficient evidence, we have to show, bringing the right side to 0, that 0 ' ' 1 ; ( a c t l')b" (dot /~) = 0; t h e r e f o r e 0~j~hI-lk(akh + l'kj)bhjII l (di, + btj) = 0, or that we have ~hI-IktOtij(akh + l'k~)(d,t + bo)bhj = O. Since j ~e i, k g: j in the effective terms and factors, the value k = i is represented, and for each h a factor of IIkt with k = i, l = h as O0(a~h + 10)(d~h + bhj)bhi equals 0, and therefore each term of I; h vanishes, q.e.d. Afortiori, we also have I

0'{a(l'ct/~) + ( ~ t l')bl~--a;b;

therefore, e.g.,

0'(a~t l')b(d~/~) =0,

whereby certain identity products are proved to be those which are at Page 519 least contained in the relative product. Etc. Problem 13. We are looking for x = II [u + a{(~2 + b) ~ c} ;d] as in 38), page 508. It comprises the previous Problems 8 to 12 as special cases--but Problems 9 and 10 not fully, that is, only the m i n o r cases. We have--because of 39) u

FROM

PEIRCE

TO

36x

SKOLEM

x,, k = II {Uhk + I]~II,,ah, (~ih. , + b,,m + C,,~)d~k] tt

= I],(I'Im,)[r, iahi(bhm ' + Cm,i)dik -k- ~ (Uhk -t- F.,iahidik~thm,)] a n d w o n d e r w h i c h v a l u e t h e last II has. At this p o i n t we h a v e to o b s e r v e t h a t m, is n o t c o n s t a n t with r e s p e c t to i, b u t c h a n g e s in p a r a l l e l with i in E;, t e r m by t e r m . If o v e r L all m, are u n e q u a l to k, Uhk = 0 will o c c u r n e x t to all Uhm' = 1 o v e r L, a n d o u r II vanishes. If, h o w e v e r , o v e r L s o m e m, is e q u a l to k, t h e last I2~ o f t h e f a c t o r 7ibm' (= ~ihk) in all a c c o m p a n y i n g t e r m s will be s u p p r e s s e d in t h e p r e s e n c e o f t h e s u m m a n d Uhk, a n d r, iahidi, l',m ' o c c u r s as a n o n e x p r e s s i v e c o m p o n e n t o f t h e g e n e r a l f a c t o r in o u r lI, to w h i c h t h e w h o l e II is also r e d u c e d , b e c a u s e n e x t to Uhk = 0 t h e ot"her ~ihk' (in w h i c h m, is ~ i f f e r e n t f r o m k) = 0 will also o c c u r - - s i n c e all possible values f r o m 12 h a v e to be w r i t t e n for u. We m u s t t h e r e f o r e have

II (Uhk + )2iahidik~ih.,, ) u

=

' }] iahidik 1kin,,

43)

w h i c h also gives t h e c o r r e c t value, 0, for t h e p r e v i o u s case. T h e s u m o n t h e r i g h t - h a n d side, o f c o u r s e , c a n n o t be r e d u c e d to a single t e r m acc o r d i n g to s c h e m e 12), p a g e 121, b e c a u s e in it m, is n o t c o n s t a n t with r e s p e c t to i b u t its m e a n i n g c h a n g e s in p a r a l l e l with i; this s u m c a n h a v e a n y n u m b e r o f effective terms. We t h u s o b t a i n

x,,~ = II,(II.,,)~,a,,,(b,,..,

+ Cm,, + l',,,k)d,k = F,,ah,IIm(bhm + Cm, + l ' k ) d , k

if we use o u r s c h e m e 39) in r e v e r s e ( a b o v e it was u s e d in t h e f o r w a r d direction). N o w we c a n write c,,,~ = i~,,, = (i; c)hm = (c;i)mk in a n y way we w a n t a n d also a s s u m e t h e t e r m as t a u t o l o g i c a l l y d o u b l e d , a n d c h o o s e for t h e o n e t h e f o r m e r , for t h e o t h e r t h e l a t t e r f o r m . A f t e r this, we o b t a i n /

/

II ,, ( b,, ,,, + c,,, + l mk ) = {(b + i; c') e 1 }hk : {b j- ( c ; i + l')}hk = {(b + i; c') ~ ( c ; i + l')}hk, as well as ahi = (a ; i)hk, dik = ( i ; d)hk, a n d we have ,.,

,.,

xj, k = F,~[a;i" {(b + i; ~) o~ ( c ; i + 1')} 9 i; d]h k or therefore v

..,

x = F,~a;i" {(b + i; c") j- ( c ; i + 1')} 9 i ; d , w h e r e o f t h e two t e r m s i; ~ a n d c;i, t h e o n e o r t h e o t h e r ( b u t n o t b o t h ) c a n be s u p p r e s s e d . We m a y write o u r r e s u l t m o r e s i m p l y as

SCHRI3DER'S LECTURE XI

362

x = ~ a ; i " {brt (c;i+ 1')} 9 i;d.

44)

T h e x, however, is not completely r e p r e s e n t e d in closed form, but the II over u has been reduced from the second o r d e r to a I2 over i of Page 520 the first level or order. T h e latter can also be given in a simpler form:

x = E j . a{b~ (c + i)};d

45)

- - w h e r e again the s u m m a n d i can be s e p a r a t e d from c a n d a d d e d as i" to b as a s u m m a n d . This can be easily verified by establishing the g e n e r a l coefficient Xhk for the last ~, as a result, we have in fact i m m e d i a t e l y the f o r m e r expression of Xhk----only l is taken for k. O n the o t h e r hand, we can also derive systematically the last expression of x, 45), from the previous, 4 4 ) r u b y passing t h r o u g h a d o u b l e sum. To that end, we write the m i d d l e factor in 44) as (b + i;c) ~t 1' in the f o r m of e~t 1' and choose f r o m the four r e p r e s e n t a t i o n s for e~ 1' which we have a c c o r d i n g to 14) or 17), 16) or 22), a n d 18): e0~ 1'= IIj(e ;j + ]) = II~(ectj+ ]) = IIj(ertj+]) = ~j(e,j)], the last because the two s u m m a t i o n signs can be e x c h a n g e d . T h e n we get (b + i; i) ~ j = be (c; i + j) = b e ( c + j ) ; i = { b ~ t ( c + j ) } ; i because j =j;i, etc., a n d we have

x = ~ j ] " ~ , a ; i ' { b r t ( c + j ) } ; i " i;d = F~i]" E,a{brt (c + j ) } ; i " i; d = Zj]" a{6e (c + j ) } ; d , which is the r e p r e s e n t a t i o n 45) of x, except for the d e s i g n a t i o n of the s u m m a t i o n variable. [We had to consider the propositions (because i = i ; 1 ) 10) of w 27, t h e n 27) and 26) of w 25, and, finally 14).] T h e r e are m a n y ways to check o u r result. In particular, we want to derive the solution of P r o b l e m 12 already checked. For that purpose, we let a = 1, b = 0 in 45), a n d t h e r e f o r e write a a n d b for c and d. Thus, we have now x = I],i" {0ct (a + i)};b = E,i. (i0~ a) ;b =l~,i" i; (1'~ a) ;b; cf. 32) of w 25, and 25). A c c o r d i n g to 26), x = 1 ;{(1'0~ a) ; b}l' = 1 ; 1'{/~; (gj- 1')}, Page 521

because 1~ = l'c'~ Now this r e m a r k a b l e proposition is valid:

FROM PEIRCE TO SKOLEM

363

l'(a; b) ; 1 = a/~; 1,

l'(a j- b) ; 1 = (a =/~) 0~ 0,

1 ; (a; b)l' = 1 ; ~b,

1 ; (aj- b ) l ' - 0 j- ( ~ / + b),

46)

of which the s e c o n d f o r m u l a on the left (necessary here) can be p r o v e d with the coefficient evidence by m e a n s of L o = l~t 1;t~ h au, bhjl ~ = E h ajhbhj = I~h 1 ~h(~/b)h~ = R o 9

This p r o p o s i t i o n belongs to a g r o u p of propositions which refer to relatives of the f o r m of l~z; 1, etc., some of which we have studied u n d e r 24), 25) of w 22 (p. 335), a special case in the f o r m of 30). T h a t requires a l s o - - a s obvious from aiibii = (ab)ii, l h ; 1 9 l'b;1 = lhb; 1,

l h ; 1 + l'b;1 = l'(a + b) ; 1,

1;al''

etc.

1;bl'=l;abl',

47)

which can be i m m e d i a t e l y e x t e n d e d to m o r e than two terms. By 46), we have f o u n d the solution of P r o b l e m 12: x = 1 ; (~/~t l')b, as given on page 517. A n d t h e r e f o r e we have also c h e c k e d o u r investigation's m a i n result r I - . . = E~... consisting of setting the values of x f r o m 38), 44), a n d 45) equal to each other: u

,.,

II [u + a{(b + ~i) ;t c} ; d] u

=E~a;i.{bj-(c;i+

1')}" i; d

,.,

= E~i" a { b e (c + i)} ; d.

48)

As a f u r t h e r check, it is left to the s t u d e n t to derive the r e m a i n i n g p r o d u c t values, which fall u n d e r the s c h e m e of o u r P r o b l e m 13, and from this s c h e m e the results that solve the p r o b l e m . . . . ~ P r o b l e m 19. We may now also decide the question which a p p e a r e d on page 268, namely w h e t h e r the general solution of e q u a t i o n x ; 0 ' = a ; 0 ' given u n d e r 22) c o r r e s p o n d s in essence with that in the first line of 26). T h e answer to this question is positive. This also brings us to propositions which are of s o m e interest. First, we notice that the solution 25), page 269, to x ; 0 ' = a is by using a :t 0 = (a 0~ l ' ) a a n d is f u r t h e r simplified to x = (a# l'){d + u + (z/# 1') ;1},

94)

w h e r e it now consists of seven instead of nine terms w i t h o u t having lost a n y t h i n g in clarity. Instead of the (last) relative factor 1, we could also write 0'--cf. 15), page 229. We can also bring the expression of x in 26), page 269, closer to the o n e in 22) by writing "~Problems 14-18 (pp. 521-538) have been omitted here.

364

SCHR()DER'S

a ; 0 ' o ` 0 = (a;0'o` 1') 9 a ; 0 '

LECTURE

XI

(do` l')a = (a'0'o` l')(do` 1')

T h e r e f o r e we can also set apart the factor a;0'o` 1' in 26) and obtain x = (a;0'o` l'){do` 1' + u + (6o` 1') ;0'1

95)

as a simpler expression for the general root of e q u a t i o n x" 0 ' = a ; 0 ' , which now consists of nine instead of 10 terms. Page 539 It has the advantage that only a appears in the c o m b i n a t i o n a ; 0 ' = c, do` 1'= t? in the two expressions of x which have to be proved together. If we again take a for this c and b for u, we can i n d e e d prove as a universally valid formula that (ao` l')[b + dl(b+ d ; 0 ' ) O` 1'} ;0'] = (ao` l'){d + b + (go` 1');0'1.

96)

Proof. The two partial s u b s u m p t i o n s of this e q u a t i o n L = R divide because their predicate is a product, and they are therefore obviously valid as partial conditions of L ~ a o ` 1' and R ~ a o ` 1'. We thus only have to show that R=6-b+ a{(/~+ d ; O ' ) o ` l ' } ; O '

and

L:~--d+ b + (/~o`l');0',

that is, /~R and aDL relative to =(= of the last term on the right. T h e latter, b r o u g h t completely to 0, results in a/~(b;0'o` l')(ao ~ 1') 9 a{(/~+ d ; 0 ' ) 0~ 1'} ; 0 ' = 0, a n d the factor/~ proves to be irrelevant. If we replace (ao` l ' ) a by a o` 0 we can use the proposition (ao`O) 9 ab; c = (ao`O) 9 b; c,

97)

which is easily proved based on 24), page 255, a n d (a o` 0)a = (a o` 0). We now only have to show (a o` 0)(b ; 0' o` 1') 9 {(d;0' +/~) O` 1'} ; 0 ' = 0. Also without the above proposition, the p r o o f would follow a fortiori from the latter. F u r t h e r m o r e , we can suppress the term d ; 0 ' , and the last factor now appears as the negation of the second but last and makes the p r o o f clear a n d understandable. This is because we must categorically have (ao`0) 9 {(d" b + c) o` d } ; e - (ao`0) 9 (co" d) ;e. If we multiply with (a o` 0) o, namely Hha~h in F,klIt (F,,,,di,,bml + cu + dlk)ekj,

98)

365

FROM PEIRCE TO SKOLEM

we eliminate the whole Em because each dim coincides with a factor aim of II h. T h e s u b s u m p t i o n for a/~L is t h e r e f o r e done, a n d we only have to prove, for/~R, /~R= (a0~ I')M~ + (aft 1')/~ 9 (/~0~ 1');0'=(= a{(/~ + 4 ; 0 ' ) ~t 1'} ;0', which divides into two parts because we a d d e d the subjects. This is clear even without factor/~--after we transcribe the last two terms a n d multiply the a j-1' with the negative on the right side as a 9 b(a ~ 1') ; 0'0 ~ 1' =(==b ; 0'0 ~ 1' because the relative p r e s u m m a t i o n on the left a p p e a r s as =~= to b ; 0 '

Page 540 on the right. If we suppress identical factors, we m u s t necessarily get c o n t a i n m e n t [ Obergeordnete]. T h e o t h e r part, b r o u g h t to 0 on the right, likewise requires dbla" b(a~ 1 ' ) ; 0 ' # 1'} = 0, w h e r e the factor a can also be omitted, as we can see, a n d this brings b, a for d, /~ to a m o r e accessible proposition: ab :~-{(a + b ; 0 ' ) j- 1'}0',

a(b j- 1') ; 0 ' 3 1' :~: a + b,

99)

etc. In o r d e r to prove it, we have to use the coefficient evidence since n e i t h e r a~--(aj- 1') ;0' n o r b:~-(b ;0'j- 1') ; 0 ' - b ; 0 ' is i n d e p e n d e n t . If we state S:(c--P for the first s u b s u m p t i o n , we have to show S0 -aob!i:~--PO, where ' P0 = ~J, 0hjIIk (aik + F'lbilOtk' + 1 '

kh) .

To p r o c e e d in a completely analytical way, we adjoin to the universal factor o f l I k the t e r m 0'k~l'kjwhich is = 0 , and we analyze f r o m the s c h e m e a + bc = (a + b)(a + c), evaluating also IIk of the first factor a c c o r d i n g to 12+), page 121, and we have" PO = F.,,,O,'o(a 0 + F.,t bitO'O)IIk (aik + F.,t bitOtk' + lk, ' , + lkj)}. ' If we f u r t h e r multiply the general term of the last I~t with 1'0 + 0~i, which is =1, a n d we evaluate the I] t which comes from the first term, a c c o r d i n g to 12x), page 121, we get the s u m m a n d boO~k which can be simplified to b o because of the following l~)k, a n d we can write this term, i n d e p e n d e n t of k, in front of II k. We now have Po = F"J,Ojo(ao'

+ F'tb,Oo){bo'

+ IIk(aik + Et bitO'tk...O~3'+ lkht + lkj) }, .

' = ( a b ) it 9 1 = S O, a n d eveBy multiplying we easily get the terms aobuF, hO~o rything is thus proved. In 99) we could, of course, also add b' 0' left of the subject.

366

S C H R O D E R ' S LECTURE XI

Better than the formula for x given on page 272, this formula,

x=(a;b&{~)[(dj-[O ;/~;b + u +

(u,,.'l"/~) ;/~1,

comprises empirically the results which we obtained for b = for one of the four module values on page 268, etc., for the general root of x;b = a;b. However, I have not succeeded in simplifying the solution on page 266 of the third inversion problem with an arbitrary b in a similar way. Exercise 20. Also in regard to partial solutions of the general (third) inversion problem, I can add a few observations to the result obtained i n w 19. In o r d e r to prove that substituting u = a" a ; b ; b in the general solution 64) of our problems must also give x = u, and that therefore this u represents a particular solution, root, we had to prove the following, almost monstrous equation as a universally valid formula:

(a;bj-{))[a " a ; b ; / ~ + (a;b)({d + do~/~+/~;/) +(dj-/)) ;/~}~/~) ;/~] = a " a;b In it, the underlined term must vanish, according to the formula d + (d0~/~) ;/~=d m e n t i o n e d on page 267 and given from a;b=~c--a;b. Since we then have a ( a ; b ; b ) ' b = a ; b b y 9 ) o f w 19, we now have

and therefore the second term in parentheses equals

(a;b)(d&f)) ;/~= 0 ;/~=0, and we only have to show that

(a;bj-{))a" a ; b ; b = a "

a ; b ; / ~ o r a" a ; b ; D ~ - a ; b j - { ) .

Page 541

This can be done with a(a; b; D)b=~--a; b from 5) of w 6, q.e.d. However, we also get involved with the curious circumstance, troublesome for our discipline, that the t h e o r e m 9) of w 19 which we had to discover as a particular case of the general third inversion theorem, had already to be used in the process, so that we cannot vouch for its i n d e p e n d e n t discovery and justification (as stated earlier)! In this connection, we would like to draw attention to the fact as well that x = a" a;b; 1 represents an i n d e p e n d e n t solution, so that in addition to the formula already m e n t i o n e d , we also have the pair of formulas,

FROM

PEIRCE

TO

367

SKOLEM

a(a;b;1);b=a;b, a;(1;a;b)b=a;b,

(a+ a~b~O) j-b=aj-b, aj-(Oj-aztb+ b) =a~b,

100)

b e c a u s e we have i m m e d i a t e l y

a(a;b;1) ;b=a;b;1 9 a;b=a;b,

q.e.d.

B o t h g r o u p s o f f o r m u l a s can be c o l l e c t e d in t h e g e n e r a l p r o p o s i t i o n t h a t x = a 9 a ; b ; (/~ + w) r e p r e s e n t s a class o f p a r t i c u l a r s o l u t i o n s o f t h e p r o b l e m x; b = a ; b for any a r b i t r a r y w, etc., a class o f c o n s i d e r a b l e generality, yet s i m p l e in its e x p r e s s i o n o f t h e root. P r o b l e m 21. Finally, we also w o u l d like to m a k e a c o n t r i b u t i o n to t h e o f o u r g e n e r a l p r o b l e m 64). T h e m o s t i m p o r t a n t q u e s t i o n s are: W h e n ( t h a t is, u n d e r w h i c h cond i t i o n s for a a n d b) d o e s t h e x, d e t e r m i n e d by t h e r e q u i r e m e n t x; b = a;b, r e m a i n C o m p l e t e l y arbitrary? A n d w h e n is x p e r f e c t l y d e t e r m i n e d by this r e q u i r e m e n t ? T h e first q u e s t i o n can be easily a n s w e r e d by saying b m u s t be e q u a l to 0, if x = u m u s t r e m a i n u n d e t e r m i n e d . B e c a u s e if we h a v e to have u ; b -- a; b for e a c h u, we m u s t have a; b = 0 - - a s t h e a s s u m p t i o n u = 0 i n d i c a t e s - - t h e r e f o r e also u ; b = 0 for e a c h u, t h u s 1 ; b = 0 o r b = 0, q.e.d. To affirm the s e c o n d q u e s t i o n , we a c c e p t e d b = 1, w h e r e we h a d to have x = a, as sufficient c o n d i t i o n , b u t it is n o t at all necessary. To find t h e n e c e s s a r y a n d sufficient c o n d i t i o n for t h e fact t h a t t h e r e is o n l y o n e r o o t o f t h e e q u a t i o n x; b = a;b, o r t h a t t h e s o l u t i o n 64) is c o n s t a n t in r e l a t i o n to u, we have to go d e e p e r . T h e sufficient o r a d e q u a t e a n d n e c e s s a r y o r r e q u i r e d c o n d i t i o n for t h e fact t h a t a f u n c t i o n f(u) o f a ( r e s t r i c t e d o r u n r e s t r i c t e d ) variable relative u is c o n s t a n t with r e s p e c t to it is

"determination"

Page 5 4 2

~f(u)

= Hf(u)

101)

( w h e r e in t h e first case in p a r e n t h e s e s , t h e E a n d II a r e e x t e n d e d o n l y o v e r t h e d o m a i n o f variability o f u, in t h e s e c o n d - - p r e s e n t - - - c a s e , they have a b s o l u t e e x t e n s i o n ) . If we have f(u) - e for all u o f t h e s a m e value, t h e a b o v e e q u a t i o n is c e r t a i n l y valid b e c a u s e o f ~ e a n d IIe. By t h e s a m e t o k e n , if this is valid, tt u we c a n call t h e c o r r e s p o n d i n g value o f t h e i r two-sided e x p r e s s i o n e ( a n d it b e c o m e s i n d e p e n d e n t o f u b e c a u s e u only a p p e a r s as i n d e x in t h e m ) . B e c a u s e o f f ( u ) ~ E f ( u ) a n d 1-If(u)::(c-f(u) we also g e t f o r e a c h u, f(u) ::~--ea n d e::~--f(u), a n d thus f(u) = e, as we have s h o w n . By a p p l y i n g this s c h e m e ' s 101), we n o w find by c o l l a t i n g 65) with 66): c = cE~..., o r c :~: E~ .... t h a t is, c :~- Z , i . (c; b)lizt (/~ + i)}; 1

102)

368

SCHRI~DER'S

LECTURE

XI

as the necessary a n d sufficient c o n d i t i o n for the fact that r o o t x o f the e q u a t i o n x; b = a ; b is c o m p l e t e l y d e t e r m i n e d by a n d (= a). With this r e q u i r e m e n t , the c d e f i n e d as a;b:t/~ has to be e q u a l to a, which results--infinitely m o r e easily than f r o m i t s e l f - - f r o m the observation that also these have to c o i n c i d e b e c a u s e c a n d a are always roots. In any case, the general, k n o w n particular solutions o f the e q u a t i o n have to c o r r e s p o n d , which gives us--cf, p a g e 2 6 0 - - c = a = a" a ; b ; b = c" a ; b ; b a n d leads to the d o u b l e s u b s u m p t i o n :

a; b j- ~:/~--a:~--a; b; b,

103)

o f which the first part has the s t r e n g t h of an e q u a t i o n . If we i g n o r e the m i d d l e term, we get

(a ; b j- ~) ( d j- [~j- ~) = (a ; b ) ( d j- [~) 0~ /~ = Oj-/~=(=O, which converts into /~ct 0 = 0 a n d thus supplies a result for b: b; 1 = 1.

104)

Page 543 It also tells us that b may n o t have an e m p t y row. O b t a i n i n g this result

directly with the e l i m i n a t i o n o f a f r o m 102) may have its difficulties. In 102), we can write a for c. If we do this, after c o n s i d e r i n g all four ways o f writing from 66) a n d r e d u c i n g the right to O, we get f o u r forms:

{aII,{(~/~);i + a ; O ' ( b ; i ) + i';/~} =0, an,{i+(~t~+ a;~b)o~/~} = 0

105)

- - w h e r e the u n d e r l i n e d t e r m can be e l i m i n a t e d - - a s e x p r e s s i o n o f the r e q u i r e d c o n d i t i o n . However, it can n o l o n g e r be called "sufficient," only in c o n n e c t i o n with the first s u b s u m p t i o n 103) which has p r o v e d c = a. T h e sufficient or c o m p l e t e c o n d i t i o n was still e x p r e s s e d in 105), if we r e i n t r o d u c e d c for a a n d t h e n wrote for c its value a; b0~/~). As coefficient, this last r e q u i r e m e n t 105) is m o s t simply written as

ahkIIm{( t~ ~" b)ltm de_ ~,anlOtk~lm}

=0.

105 ~

We can draw m a n y i n f e r e n c e s f r o m 105). T h e r e q u i r e m e n t has to exist a fortiori if we suppress any terms after the Hi or, possibly, a d d s o m e factors. We have, for example" aII~{(50~/~);i + i;/~} =0, i.e., a c c o r d i n g to 14) a(53-/~0~/~)=0

or

a:~--a;b'b

as a c o n f i r m a t i o n of the s e c o n d s u b s u m p t i o_ n 103). It also i m m e d i a t e l y follows o u t o f the last form o f 105) as a(d0 ~ b ~ 0 ) = 0 or a:~--a; b; 1 which can be c o n f i r m e d with 104). Etc.

FROM PEIRCE TO SKOLEM

369

Further, we must have aII~( ~+a ; ~b# 0) - 0. According to 7) of w 6, we have, however, a ; (~)b 0~0) :(= a ; ~b~ 0, and here the subject equals a; (~:t0)(bo~0) = a; ~(bc~0) = a(Oj-b) ;~, thus a f o r t i o r i aIIi{a(Oj-b) ; i + [) =0, which results in, according to 18), a" {a(0 j-/~) ;0' or a 9 a ; 0'(b o~0) - 0

106)

as a further, necessary condition. It is possible to satisfy the first subsumption 103) i n d e p e n d e n t l y in the most general way with a = oL;b~/~, as already, shown in 4 ~ of w 19, and the second subsumption with a = ol'o~;b;b. As I have found in a more painful procedure, both requirements 103), involving also 104), can be satisfied in the most general way with i n d e p e n d e n t parameters c~,/3 with the following formula which is easily verifiable, a=ol;(/~j-O) + c ~ ; / 3 j - O + (c~;f3j-/~) 9 1;/3,

b=/3o~0+/3,

107)

and it may be possible to continue to gradually satisfy further partial requirements or subrequirements of the p r o b l e m - - a s 1 0 6 ) - - b y determining the parameters. But as long as we cannot evaluate the products II~ in 105) in closed form by transforming them equivalently as functions of a and b, which are only construed by means of the six species of these arguments and perhaps also by the modules, there is little hope that we can completely solve our difficult "determination problem" (for the third inversion problem) in this way. We therefore have to leave the problem here as it is. Page 544 To evaluate II~ hardly promises success as long as it did not succeed with m u c h simpler products, such as

x = II~a; ~b = Ilia;~b.

108)

C o n c e r n i n g this last problem of the first o r d e r [erster Stufe], with which we return to the main theme of this section and w h i c h - - a l r e a d y for b = 0'---cannot yet be solved, m u c h could be said that is of interest. But we have to abstain and r e c o m m e n d it as a problem, next in difficulty a m o n g the still unsolved problems, for further investigation.

In view of the importance, the fundamental significance for the completion of elimination, for inference in general, which the p r o b l e m of sum and p r o d u c t on the second o r d e r [zweiter Stufe] gained at the end of w 28, we will devote some time to this problem. It will be shown by modifying (in a practically insignificant way) what was said on page 468 that our algebra has also methods for the solution of this p r o b l e m which

37 ~

SCHRODER'S LECTURE XI

are theoretically usable in a g e n e r a l way, but which s e e m to be e n t a n g l e d in u n s u r m o u n t a b l e technical difficulties or m a t h e m a t i c a l c o m p l i c a t i o n s w h e n we try to use t h e m practically. W h e n the e x t e n s i o n of H, E is absolute (over the e n t i r e universe o f discourse 12), w e can assume as self-evident that H f(u) = H f(5)

= H f((z) - H f ( u )

109)

and, similarly for E - - w h e r e the p r o d u c t variable is r e p l a c e a b l e by e a c h o f its relatives. If u can take on any value, t h e n also z~, fi, etc. If, f u r t h e r m o r e , the e x p r e s s i o n o f f ( u ) only a p p e a r s as u a n d fi, b u t n o t z~, u (or reverse), after its p r o p e r r e d u c t i o n by e x e c u t i n g all negations o n c o m p o u n d s u b e x p r e s s i o n s (next to any p a r a m e t e r relatives or constants relating to u), t h e n we k n o w

nf(u,(,)

=tip,o) = nf(~,u),

which we can write in a s i m p l e r way as Hf(u)

- f(O).

110)

T h e n we have f ( 0 ) in o u r rl as the m i n i m a l factor, c o n t a i n e d in all others, the p r o o f of which c o u l d be given in detail by using the t h e o r e m s u0 :(= uv, u + 0 :(= u + v, u; 0 :(= u ;v, u 0~ 0 :(= u 0~v a n d its c o n j u g a t i o n s in c o m b i n a t i o n , or mainly a c c o r d i n g to t h e o r e m 1) of w 6 by c o n s i d e r i n g 0 :~:v. We thus have, for e x a m p l e , i m m e d i a t e l y rl [a{(u + b) j. c} ; d + e; (tf] = a(b j- c) ; d.

Page 545

We leave it to the r e a d e r to state the dually c o r r e s p o n d i n g p r o p o s i t i o n for I2. As a p r o b l e m , the discovery of only such H, E is o f i n t e r e s t in which in the g e n e r a l term f ( u ) n o t only u or ~ b u t also z~o r u a p p e a r essentially. We have solved p r o b l e m s o f this kind in 1) a n d 6). T h e s e can be m a d e into the m o r e g e n e r a l t h e o r e m by m e t h o d s which we will c o n s i d e r later:

u

111)

w h e r e the sums over K a n d X e x t e n d over any series or systems o f suffix values, such as 1,2, 3 . . . . .

371

FROM PEIRCE TO SKOLEM

B e f o r e discussing the m e t h o d s , we want to state s o m e c o n c r e t e exa m p l e s which are i n t e r e s t i n g for their derivations a n d results. P r o b l e m 22. W h e n we look at 5), we are inclined to i n q u i r e after the value o f the following p r o d u c t l-I, after which we i m m e d i a t e l y state

II(au;b+

c;gtd) = a d " 1%;1;b1',

u

II(a;ub+oi'd)

=bc" 1 ~ ; 1 ; d 1 ' ,

u

112) which gives the a n s w e r m w h e r e the s e c o n d result can be o b t a i n e d by e x c h a n g i n g letters in view o f 109). For proof, we call the x the first, s o u g h t for II a n d Uits g e n e r a l factor, a n d we have

x-nU.

.,;- n v,j.

u

U 0 = r,h a~hbhiu~h + Ek qkdkj~ikj" We now "develop" this e x p r e s s i o n for u 0 by m a k i n g the latter ( a n d its n e g a t i o n ) "prominent" or by "bringing it into evidence." F o r that p u r p o s e it is necessary a n d sufficient to let the terms w h e r e u o r ~i are i n d e x e d with this p a r t i c u l a r suffix ij to e m e r g e or to let c o m e o u t w h e r e v e r they can be f o u n d . This can be d o n e by p u r e calculation, in o u r case: by m u l t i p l y i n g the g e n e r a l e l e m e n t o f ~j, with (1=) (l'hj + 0,'0), the ~k with (1=) (l'h, + 0~,,), t h e n resolve t h e m , t h e n apply to the first terms the s c h e m e 12x) f r o m p a g e 121. Now we have i n d e e d :

U0 = aobiiuii + c,d~gt 0 + ~,,O,',~a~hbjou,h + ~kO~kc~kdk~(%, w h e r e we only have in the last two sums u-coefficients which are d i f f e r e n t f r o m e a c h o t h e r a n d f r o m u o. T h e s e may also a p p e a r in H with 0-values u~h = 0 (at h r j), ~ikj = 0 (at k r i), so that only the first two terms can c o n t r i b u t e s o m e t h i n g to the value o f x. T h e actually o c c u r r i n g m i n i m a l value, which is c o n t a i n e d in all values, o f a h o m o g e n e o u s linear function" Tt

~u + ~i--c~

Page 546

+ c~ju + o ~ i

m u s t be the p r o d u c t o f its coefficients, t h e r e f o r e be ol/3, since the last two terms vanish i n d e e d w h e n we assume u = o~/3. T h e r e f o r e , the U# f o r m e d for all possible u c o n t a i n s at least the t e r m aobjjc~d 0 a n d will, for certain values o f u, no l o n g e r c o m p r i s e its terms, so that we have f o u n d

U!i = a ijd ~jc.bjj. We now have

SCHRt~DER'S LECTURE XI

372

x o= (ad)o(l~c;1)~i(1 ;bl') 0 a n d

x = ad" 1~; 1 ;bl',

q.e.d.

T h e p r o c e d u r e is certainly impeccable; however, the conditions for developing it so smoothly and simply are almost never as favorable. We can gain a d e e p e r insight into the p r o d u c t m e t h o d which promises general success, with the derivation (of the first) of the following results: P r o b l e m 23. We have to discover that

II {au ; b + c( ~t j- d) ; e} = (arid)c; ( [t j- b)e, I~I{a ; ub + c; (de 7i)e} = (a j- it)c; (d j- b)e, ~ { ( a + u) rtb}{(c+ ~i; d)0~ e} = ( a ; d + c) e ( d j - b + ~{a~(u+b)IlcJ-(d;fz+e)}=(a;d+

c)~(d;b+

e),

113)

e),

tt

- - w h e r e the ones standing below each o t h e r are immediately o m i t t e d by e x c h a n g i n g d with d in o n e proposition. Derivation. By formulating the first p r o b l e m again as x = IIU we get

x 0 = II U0 a n d

U0 = Eta,boU,l + ~kc, kek~IIt(~i, + dj, l).

T h e p r o b l e m is simple insofar as the u consistently appears with the first index i. We now make uih p r o m i n e n t for any o n e definite h. We have shown previously how to use Et to that e n d (we multiply the general term by l'u, + O~h). In dual c o r r e s p o n d e n c e , we only have to add (0=) Ou, l u ;' to Page 547 the general factor of II~, break it into O'u, and l'u, by the dual equivalent of the distributive law in (72. + dtk + O~h)(~i. + dtk + l'th), a n d take IIt individually of these two factors, where for the first factor the s c h e m e 12 • ) of page 121 can be used. We thus obtain

U 0 = a,,,b,ou,, , + ElO~,,a,,bljU . + EkC,keki(fZ,,, + d,,k)IIt(l'u, + 7i,t + dtk). This has the (linear) form which we have already developed: U, i = c~ u ~, + 3 czar, + 7 ,

which we are better to leave in the n o n h o m o g e n o u s state. Prior to writing the specific values of or, 3, "t' which are here not visible, we want to i n t r o d u c e a s o m e w h a t m o r e accessible symbolism that we r e c o m m e n d for all similar problems. A sum of the form Et01h4)(/) represents n o t h i n g m o r e than the sum of all 4~(1) without do(h) and can also be expressed completely by I2~-h4~(l). By analogy, I2~-"-k4)(/) = I; t O~,,Oikch(l)l b e c o m e s the sum o v e r / o f all 4)(1) without 4)(h) and 4)(k), and so forth.

FROM PEIRCE TO SKOLEM

373

In dual c o r r e s p o n d e n c e , we can also write H,{I',,, + r

=

H-f" r

'

H,{I'lh + l',k + r

= /it-n-*

r

'

a n d so forth, w h e r e the expressions o n the left r e p r e s e n t n o t h i n g m o r e t h a n the p r o d u c t o f all 4)(1) w i t h o u t ~(h) a n d 4)(k), respectively, etc. This small m o d i f i c a t i o n of the symbolism which is legitimate in o u r discipline has the a d v a n t a g e that the g e n e r a l t e r m o f E a n d o f 1-I always has the s a m e (a constantly simple) e x p r e s s i o n (whereas in the n o r m a l way of r e p r e s e n t a t i o n , it would grow a n d increase in bulk), as m o r e a n d m o r e terms are o m i t t e d f r o m the sum, factors o f lI; t h e r e f o r e , also the g e n e r a l term, which o f course r e m a i n s the old o n e , d o e s n o t n e e d to be m e n t i o n e d a n d r e p e a t e d . W h e n finally all terms o f Et, all factors o f / I t are e x c l u d e d , that is, e l i m i n a t e d , the f o r m e r will be equal to 0 a n d the latter e q u a l to 1. If we use these results, we now have

a = aihb#,

13-

-h

~ks

,

3" = ~-~h q_ ~kCikekjdhkI-i1 h '

w h e r e the g e n e r a l t e r m o f E t is now a,~Ou ., a n d the g e n e r a l t e r m o f lI t is u~h + dhk, jUSt as in the first expressions o f o u r U0. In w h a t e v e r f o r m the o t h e r u , (without uih) are given, we now can d e t e r m i n e , choose, u~h in such a way that the above linear f u n c t i o n s o f it take U0 as its m i n i m a l value which m u s t be

which can be r e d u c e d slightly a n d results in 3/' + OL~ = ~27 h + ~,, Cikekj(aihbhj + d,,k)/ii -h .

Page 548

We now m a k e a n o t h e r u~h, p r o m i n e n t in this 3' + c~/3, w h e r e we have to a s s u m e h' 4: h b e c a u s e h does n o t o c c u r in it. We n o w have 3" + ol/3 = E~-h-h' + a~h,bh,jU~h, + Ekc~kekj(a~hbhj+ dhk)(6~h, + dh,k)II~-h-h' I

= O~ u ih, +

i~l

-

I

uih, + 3" ,

and, again, the m i n i m a l value o f this e x p r e s s i o n in view o f the variables (or f u n c t i o n s of) u~ z is 3" + ol'{3'=

F.,[h-# +

F.,kcikekj(aihbhj + dhk)(aih, b~lj +

dn,,)/i-/h-n'.

This c o u l d have b e e n written i m m e d i a t e l y w i t h o u t the i n t e r m e d i a r y calculation o n the basis o f the o b s e r v a t i o n that 3' + o~j3 with r e s p e c t to u , again gives the same f o r m as u o - - e x c e p t for the a b s e n c e o f o n e t e r m (that is, u p to the e x t e n s i o n ) in E a n d /i over l, a n d e x c e p t for the c i r c u m s t a n c e that the p a r a m e t e r p r e c e d i n g the iIt as a factor---or expression o f the c o n s t a n t s % % in U 0 ~ h a s now b e c o m e m o r e complicated in 3' + c~/3 (as we can see it t h e r e now).

374

SCHRODER'S LECTURE XI

This observation is also valid for "y' + a'{3', a n d the law of f o r m a t i o n has b e c o m e clear. If we assume these conclusions to be u n l i m i t e d until all terms of the E t a r e e l i m i n a t e d , thereby, at the same time, all factors of IIt have b e c o m e ineffective, = 1, we have f o u n d x 0 as the last e x p r e s s i o n of the m i n i m a l value, 3,~~176 + a~~176176

Xij = ~kCikekjI-Ih (aihbh~ + dhk) = E~c,JL(a,,, +

~)rI,,(~,,~

+ bhj)ek) = {(a0~ d)c; (dj- b)e} o,

a n d t h e r e f o r e , x = (a ~ d)c ; (d j- b)e, q.e.d. As we can see, the p r o c e d u r e leads us to a " d e t e r m i n a t i o n of the limiting value" a n d is c h a r a c t e r i z e d as a kind of "method of exhaustion": the E a n d H over I were gradually "exhausted"--as we would e l i m i n a t e in a p r o b l e m of e l i m i n a t i o n of x the u n l i m i t e d d o u b l e series which c o u l d possibly f o r m a c o n t i n u u m of its coefficient Xhk by progressive e l i m i n a t i o n of o n e after the other. In the same way, we have taken o u t o f c o n s i d e r a t i o n or "annihilated" o n e U~h after the o t h e r by k e e p i n g only what it c a n n o t avoid c o n t r i b u t i n g U0 to H; in o t h e r words, we were l o o k i n g for the m i n i m a l value of the m i n i m a l value o f the m i n i m a l value, etc., of UO with respect to o n e of its a r g u m e n t s u~h (of the propositional f u n c t i o n ) after the o t h e r - - b y a "minimal value)' of a f u n c t i o n 4~, "with r e s p e c t to an a r g u m e n t u," we u n d e r s t a n d , now briefly speaking, a value which is actually o b t a i n e d for s o m e u, while it is c o n t a i n e d in all values which the f u n c t i o n can take on for any u - - n a m e l y we have to c o n t i n u e this until the last a r g u m e n t uih, if it exists; m o r e generally, t h e n w h e n this has b e e n d o n e with respect to each a r g u m e n t uih. T h u s Page 549 the resulting expression for the m i n i m a l value of UO has b e c o m e ind e p e n d e n t of all a r g u m e n t s u~h, in s h o r t of u; a n d t h e r e f o r e we could e l i m i n a t e the sign H in f r o n t of this c o n s t a n t by the law of tautology. Incidentally, for a = 1, b = 1', by c h a n g i n g s o m e letters, o u r first result 113) b e c o m e s 5 0 ) - - 4 h a t f o r m u l a which previously we could only find in an artful but c o n v o l u t e d way t h r o u g h an infinite n u m b e r of m u l t i p l e II. O u r result for d = 0' also includes o u r t h e o r e m 6) a n d a p p e a r s as o n e of the m o s t g e n e r a l ones of H which could so far be written in closed form. W h a t we have l e a r n e d in the last p r o b l e m can now easily be generalized "th eo re tically." If we have to know a x = II U with the "absolute extension," w h e r e U =f(u) is a given relative func"tion, we have to lookforthe general coefficient x,,k = II U,,k. We will first have no difficulties, following the stipulations (10) to (13) of w 3, r e p r e s e n t i n g the g e n e r a l factor Uhk of the last II explicitly as a "propositional function," which is built f r o m the coefficients of the u

VROM PEZRCE TO SKOLEM

375

a r g u m e n t u a n d its n e g a t i o n 6, as well as of all the possible p a r a m e t e r s a,b,c . . . . o f f ( u ) using the t h r e e species o f the p r o p o s i t i o n a l calculus and, possibly, the ~ a n d H signs in a certain way. We can "develop" this e x p r e s s i o n in the f o r m o f U~,k = a u o + 3 a u + %

for the u-coefficient with any given suffix ij. This d e v e l o p m e n t is linear a n d also h o m o g e n o u s , b u t to the h o m o g e n o u s f o r m

(~ + v)u 0 + (3 + ~,)a 0, the o n e which is n o t yet h o m o g e n o u s has to be p r e f e r r e d , as we will s o o n see. T h e p o l y n o m i a l coefficients a,/3, -y are n o t yet i n d e p e n d e n t Page 550 o f u (they carry possibly the o t h e r u-coefficients), b u t they are indep e n d e n t o f u O. To " m a k e p r o m i n e n t , " to "bring into evidencd' u 0 - - w h e r e the cases i = or r j as well as i,j = o r r h,k are to be t r e a t e d s e p a r a t e l y - - w e only have to observe: w h e n e v e r the first i n d e x 1 o f u or ~ is d o m i n a t e d by a Zt, we can multiply the g e n e r a l t e r m o f this E by l~u + 01t ( = l ) - - f o r a s e c o n d index, however, we multiply with 1~ + 0~.. If, o n the o t h e r h a n d , they are d o m i n a t e d by a Ht, we can a d d 0al' ', (=0), o r 00'1'0, respectively, to the g e n e r a l t e r m - - i n b o t h cases, b r e a k it into 0 ~th a n d 1~th a c c o r d i n g to the distributive law a(b + c) = ab + ac, o r a + bc = (a + b)(a + c), respectively. T h u s Z or H falls respectively into a s u m o r a p r o d u c t o f two o f t h e m , and, for o n e part, the s c h e m a 12) o f p a g e 121 is applicable w h e r e b y we ultimately o b t a i n u 0 or 5 o as explicit factor o r sum. In the o t h e r part, a t e r m previously effective is ineffective, e x h a u s t e d o r exc l u d e d by the factor 0~t or 0~, respectively, by the a d d e n d 1', o r 1'tj, respectively, f r o m the r e m a i n i n g Z t with r e s p e c t to H t. Now o u r Uhk actually takes o n its " m i n i m a l value" for u 0 = d3, which has to be c o n t a i n e d in all values of the Uhk for all u a n d is

w h e r e ij n o l o n g e r occurs as a suffix o f u a n d ~. If m n is any new, arbitrary suffix, we can also d e v e l o p I-

!

Vhk = ~ 'U m,, + 3 Urn,, + ~/ ,

o f which as a factor o f Xhk only the m i n i m a l value c o n t a i n e d in all Vhk for all u a n d realized for s o m e Urn,, n a m e l y w,,~ = v , , : ~ "

=

vh-,i-m.= "v' + ~'~',

in which now n e i t h e r ij n o r m n can o c c u r as an i n d e x o f u o r ~, a n d in s o m e E, H even two terms can be e x c l u d e d a n d a p p e a r e x h a u s t e d . A n d so on.

376

S C H R O D E R ' S LECTURE XI

It is now only a matter of studying the law of constructing minimal values r e p e a t e d l y m a n d until all terms have been excluded, exhausted from ~ and II referring to indices of u or 4. This is theoretically possiblemin practice, the complications can quickly astonish us. T h e completely exhausted ~ vanishes, becomes =0, the 11 is equal to 1. T h e p r o c e d u r e leaves further possibilities o p e n for ingenuity: we can try to eliminate the u-coefficient in series (by rows or columns), or also the uii along the main diagonal or also the pairs conversely to each other, etc. If this succeeds, we have Xhk as a "propositional function" which no longer appears in any u-coefficient, in front of which the II can be eliminated and which consists merely of coefficients of p a r a m e t e r s a, b, c. . . . of f(u). It remains to "compress" (as I would like to say), to "condense" the propositional function, that is, to represent it as a coefficient with suffix hk i n d e p e n d e n t of h and k, a relative built out of a, b, c, ... by the six species including ~, l-I, a "relative function" X, and we will have f o u n d u

Page 551

x=X. Practically, it will always succeed, at least by using sum a n d p r o d u c t forms of the first order [Stufe] which only extend over the universe of discourse 12~as we will show later in a last problem. However, in most cases, the a t t e m p t misfires in p r a c t i c e ~ a n d remains o p e n for f u r t h e r research and study. Problem 26. We are looking for

x=II[a(bu;c) ; d + e{(f+ 4) ~g} ;h], u

114)

where h is not valid as a e l e m e n t l e t t e r - - s o that our 11 has eight different relatives as parameters. With the m e t h o d of exhaustion, we easily find

x O = F"keikhkjHt(buF',,,a~,,,ctmdmj+ ft + gtk),

115)

a n d we only have to "condense" the expression, that is, to express the relative x with the eight parameters, based on the coefficient relation. We can do this when we replace the index k,l with j,i in the form of

x=2je{(b+ f ) j - g } ; j ' j ; h "

II,{a;([;i)d+ f ; i + i;g;j},

116)

a result which allows us to derive correctly most previous results as special cases, for example, for a = 1, d = 1' by e x c h a n g i n g letters of 113). By using l l 6 ) m a b s t r a c t i n g part of its m e t h o d - - w e transform % = [t~ = ([;/),,f---of. page 423---and ~j is transformed into {a; ([; l)d},,j. We, f u r t h e r m o r e , divide the 11t in IIt (b, + f t + gtk) = {(b + f ) J- g}~k, on the one hand, which can be r e d u c e d with the factor % to [e{(b + f ) j- g}]~k~Which may be called for the time being r~k, and, on the

FROM PEIRCE TO SKOLEM

377

o t h e r hand, 1-It{a; ([; l)d + f ; l + l; g; k}0. If r~khkg is written equivalently as (r; k" k; h)ij, we can completely relieve the suffix ij on the right of 115), and then omit it on both sides according to (14), page 33, since the equation 115) was to be assumed u n d e r the d o m i n a n c e of the sign II 0. We now have result 116) from 115). By using the propositions 34) to 36) of w 25 effectively, we can always achieve such "condensation," reduction. It is thus only the elimination or exhaustion p r o c e d u r e which contains u n s u r m o u n t a b l e difficulties. The researcher will immediately become aware of them, when he tries, for example, to find

II {a(u j- b) ; c + d(~t j- e) ;f}. u

Page 552

We let the problem stand as it is at this point and only observe that it shows certain analogies with the mathematical p r o b l e m of "rationalizing" algebraic equations, the elimination of roots which a p p e a r in it. Even if each individual root can be eliminated by isolating it on one side of the equation and then empowering the equation on both sides with its root exponents, we do not succeed in this way to eliminate all roots. More refined methods are necessary. We u n d e r s t a n d this because when eliminating (with the m e n t i o n e d m e t h o d ) a certain root, the n u m b e r of the other, still to be eliminated roots, is increased. With respect to the relative coefficients, this is not the case when exhausting, eliminating a certain one of them; however, the difficulty is increased because the other coefficients appear.