Probability and symbolic logic
Item Type text; Thesis-Reproduction (electronic)
Authors Platzman, George William, 1920-
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/553581 PROZ&BILm AID SYiSOUC LOGIC
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Although the Immediate bueineee of the theory of probabilities is with the fro queasy of the ocourrenee of events, and although i t therefore borrows some of its elements from the science of number, yet as the e*» preseion of the occurrence of these events, and also of the relation s, of whatever kind, which connect them, ie the office of language, the common instrument of reason, so the theory of probabilities oust bear some definite relation to logic•
George Boole coK rem
INTRQDUDTIOH
1. 2. 3. 4.
THE CALCULUS OF PROBABILITY 1. Kietoricol • • • 2. Porctzky and Coutur&t 3. «ymee • * * *
PROBABILITY AM) LOSE?
1. Iho Oonoral Ldthod of Boole 2. PoreW s Extension . . 3* Analysis of Examples . , 4. Critiqpe of the Method . • 5. The D lajuitotlve Method « •
CfieSLOSIOM 1. Boole's Method ? ...... 2, Note on Synholle Biltlplleatlon * • • •
3* ProhLezas for Further Study . . • • • . S S S a s s e s s 558(5 5
BIBLIOGRAPHY IKTRODUCTIOH
It iu nearly thrw centuries ago that the theory of probabilities •a a scieaco found its gem in the speculations of Pascal on games of chance # Although attended by obstacles not usually encountered in the more exact sciences, its evolution has been continuous and finds its culmination in relation to the present-day theories of measure and a llie d topics of analysis. It is. & curious fact that we find in the writings of Leibnitz, a near-contemporary of Pascal, the beginnings of symbolic logic. The development of th is subject, however, was somewhat more troubled than that of the science of probabilities and received its im petus only with the work of Boole, roughly a century ago.
The Intent of the present work is to study the relation between symbolic logic and the mathematical theory of probabilities, and to give an account of some of the benefits which accrue when this relation is exploited. To this end, the work is divided into three main sections, the first dealing with the algebra of logic, the second dealing with the calculus of probabilities, making use of the results of the first section, the third treating of certain general problems in classical probability theory, making use of the results of tho first two sections. Tho con cluding chapter is inserted mainly for the purpose of emphasizing the inconclusive nature of tho results of tho preceding section, and for sug gesting tho direction in which further investigations sight profitably be attempted. ii
In respect of Chapter III, I deen it proper to otato that tho paper of ?.S. Po rot sky entitled "Solution of tho General Problem of tho Theory of Probability -with tho Aid of Katheoaticsl Logic", in Russian, was graciously translated for no in its entirety by Dr.'A.3. Boldyreff. Reference to this paper is to be found in Keynes’s A Treat iso on Proba< b ilitv and in G .l. Lewis’s A Survey of Symbolic Lo,:ic. Tho f ir s t of tboso authors implies that be has no knowledge of the contents of this paper, while the latter freely admits that he is familiar with none of Foret sky’s papers of 1881-1887, all in Russian, among which tho present one is included.
References to books and papers are made in the body of the text.
page number is referred to it will bo included within the parentheses, being separated from the reference number by a comma. Thus (3,256) means page 256 of reference number 3. This system eliminates tho use ®f footnotes.
The present study was suggested to me by Hr. A.w. Boldyreff. la Its course I was guided by hie logical acumen and wise counsel, and it 1#
Z5m io a rational animal. ayobolical treatnent of the most gee* oral principles of rational procedure i s the scope of symbolic logic* and it is the aim of the present chapter to acquaint the reader with that
To gain a perspective of the evolution of symbolic logic from the time of loibnits* one can do no bettor than to read Chapter 1 of 0*1* Lewis’s A Survey of Symbolic Logic In cost discussions of logical methods* the notion of "class’* i s Indispensable • Relative to a group of properties P * a class is an assemblage whose members are all objects satisfying P * and no others. In this sense, the extension of a class is the class itself, while the intension of a class is the associated group of properties P * For example, let P be the property of forming a salt by replacing hydrogen in am acid. The membership of the associa ted class w ill then be Lithium* Potassium, Sodium, e tc ., and th is assem blage will constitute, the extension of the class (i«e* the class of met als) § the intension of the class being the property of forming a salt
ed*
Concerning the calculus of classes, suffice it to say that Leibnits (1646-1716) was in possession of the most important properties of the L eitm lis, of C astillos and veBtieations* Auguotuo do J^rgan (l£XJ6-lC7e) lontributlozm to value, such as the theorem ehich s t i l l boars his musej and the idea of a universe of dice cur no* Following do liorgon, ts?o reach the work of George Boolo (1615-1664), who, with the publication in 1654 of the Laws of Thourht (% ), at ono stroke acoiisilated the work of hio predecesaore
a.
Tho a la of ths l^aws of Thought is asnouncod by Boolo in its opon-
Tho design of the following treatise is to Investigate the funda mental laws of those operations of the mind by which reasoning is per formed* to give expression to them is the symbolical language of a Cal culus, and upon th is foundation to establish the science of Logic and construct its method.** V
She purpose of the present section is to outline & few of tho results of this investigation, and this will be done with a minimum of comment. Css of the principal differences which distinguish this algebra from that to be presented in tho next section is that the former is descriptive and the latter postulations!$ for this reason it is possible that many of tho ideas of tho proeont section will assume thoir proper significance only after a reading of section 3, and it is thus hoped that tho follow- the elasa x with inteaaien X is tereM the operation of electing: froa 1 oil cenbers satiofying X , and the aleeenta of the algebra or classoe x,y,s,#te.
Ml the operations of Language, as an instrument of reasoning, m y be conducted by a oyaten of signs composed, of the following olo3ionta,viz. l&t» Literal eysbole, a« x,y,etc», rspreaonting things aa subjects A&* Signs of operation, as + , ~ , standing for those opera tions of tho mind by which the conceptions of things are combined or re solved so aa to form new conceptions involving the same elements. 3rd. The oign of identity, = . And these symbols of logic are in their use subject to definite laws, partly agreeing with and partly differing from the laws of the correspon ding symbols in the science of Algebra. (1,27)
The signs of operation are defined as followst The logical product ef two Momenta x and y , denoted by /xy :, is the operation of elect ing froa 1 all mcBbere satisfying X and also I , and no others; far classes in extension, xy thus is the class ef members common to s and y . The logical sum of two Mesanto, denoted by x + y y is defined onlyjshen xy = 0 and is the operation of electing from 1 all noEbere satisfying X and ell members satisfying Y . The logical difference and y , of addition, being the element x in tho equation y + z = x . The difference is an important one since it is tho class of every thing not contained in x , or simply, not-x.
plots or: strong 4 algebra. ~c uced by Boole it is attended trith many disadvantages, atea-
irg free the fact that the disjunction 2: + x has no place in the light ef lile interpretation. I t has, however, the on® redeeming feature that it is strictly inverse to logical subtraction and this fact proves ef
RULE*- Express sicplo names or (polities by the symbols x,y,s,ete., and their contraries by 1 • x , 1 » y , 1 - c , etc.; classes of things defined by oooron rm es or q ualities, by oonne'eting the corres ponding symbols as in raul tip i faction; oolloctions of things, consisting sr porllsas difforont froa each .ether, by oonhsoting the egressions of thes^j>^o^by^be^ign_+ . In particular, lot the expression, “Either x*c or y*s ", bo expressed by xtl - y) + y(l - x), when the clasoea denoted by x and, y are exclusive, by x +• y(l - x) when they are not exclusive. ... (4 ,57)
Writers on symbolic logic have displayed an inexhaustible variety in representing the operations of disjunction. In the present paper x + y will he used to denote x(l - y) +* y(l - x) which in this section is taken only when xy = 0 following Boole, but which in section 4 will be oxtanded to include cases when xy 4= 0 , no change being made in no tation. Hie weak disjunction x(l - y) + y(l - x) + xy in denoted by Qx + yj to avoid confusion in section 4 where both types of disjunction are discussed. Of course, when xy = u tbo operations of weak and etreag disjunction are eynonoKCuo, so that tho above capanded cur-o could equally well have been written in terms of 4- instead of 4- .
m ( 2) *(r*> = (3) *(y + 2) = xy -5- (#) x(y - s) = zy - xs . JAiltiplicatloa to distributive over sub traction* (5) r. y - y + X . Addition io cocmtativoe . . . , ■ ■ . - (6) x + (jr + t) = (% + y) + s * Addition is associative* la addition to these there is one law tixich can properly be viewed as the cardinal principle of the algebra, nandy (T) xx = x • Tills io knom ns the Index Lav or the Principle of Tautology, and is self- evident for classes ns any oxunplo will show* It is through this law that Boole io led to formulate the fundamental method of his system. V?e have seen that the symbols of Logto are subject to the special lav , x l = x • I:ov of tho symbols of lumber there aro but two, v is. 0 and 1 , which arc subject to the coco formal law* We know th at 0*" = 0 , and that l’-' 1 j aitii the equation x - x , considered as algebraic, has no other roote than 0 and 1 * Hence instead of determining tho measure of formal agreement of tho symbols of Logic with those of generally, it is core immediately suggested to us to compare then with Let ua conceive, then, of an algebra in which the symbols x,y,zfotc, admit indifferently of tho values 0 and 1 , and of these values alone. The laws, the axiom, and the processes of such an Algebra will be identical in their whole extent with the lavs, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established* ( 4 , 37) interpretation. And this is the mode of procedure which will actually be adopted, though i t w ill be deemed unnecessary to state in every in stance tho nature of tho transformation employed. Tho processes to which the symbols x,y ,« , regarded as quantitative and of the spediee above described, aro subject, are not limited by those conditions of thought to ukich they would, i f performed upon purely logical symbols, be subject, and a freedom of operation is given to us in tho use of them, without which the Inquiry after a general method in logic would be a hopeless quest. (4 ,70) Xao guiding principle of Boole*c Kothod is now c lea r, and ooce e f # » conccqticnccc wiiich i t involveo w ill new bo noetic nod. A function of x in the fora Ax 4* B(1 - x) is said to be devel oped, wherein x and 1 - % , toraed constituent*), arc oynbola of quan tity susceptible only of the values 0 or 1 | and wherein a or B , 0 or 1 , or a combination of lo g ic a l symbols and such numbers. Thus, l e t f (x) = Ax 4- B(1 - x) . Then manifestly f(l) = A and f(0) = B , eo that for any developed funetient . . . f(x) = f(l)x 4- f(Q)(l - x) . The extension of the above concepts to many variables is immediate; for two variables x and y % fU»y) = f(l»l)xy 4* f(l,0)x(l » y) * f(0,l)(l - x/y 4r f(0,0)(l - x)(l - y). ,.n important example of this typo of osgiansion is that of the ftoa* ction y ~ xy 4- -5- x(l - y) 4- 0 (1 - xjy 4- & (1 - x)(l - y) . The interpretation of those coefficients is ono of the principle tasks of Boole’s system, and since the most general function in the Algebra of Logie is the “rational" function, i.e . & finite combination of the opera tions of addition and multiplication, it is cow obvious that the oxpan- -7- F * Q + + o s . first three 1 , 0 , and -§• , all obey the Index Law, the latter since it "ada&ta of the numerical values 0 and 1 indifferently" $ i whorcao the fourth, q , does not obey thin law. With this, it is simple to prove that the above expansion is equivalent to P + vR and 0=0, where v replaces % , an indeterminate elcaont. One of the most important processes of the algebra is that of ell#, iaatien. To eliairato any logical symbol x from an equation 0 = 7 , expand the right member in the fora ax -h B(1 • x) , A and B being free of x • The result of the elimination will then bj? 0 - A3 * In Boole’s eyetorn the proof of this is cinple, but since the result is also valid in the Boole-schrtider algebra the proof will not be given hero. The preceding account of sob© of the salient aspects of Boole’s algebra hao obviously not boon intended to servo as even the barest out line of the results of his investigations, but only to indicate the diff erences which underly Boole’s approach with that of the modified Boole an- - SchrMor Algebra, and incidoctly to insure the ^initiated at least a par tia l understanding of sozno of tho at at events which appear in Chapter III. 3. Tho Boolo-achrBdor -Igebra. ®*e algebra of logic, in its generally accepted fans, ... wan foun ded by Bcolo and given i t s present form by BchrSder, who incorporated into i t certain emendations which Sevens ted proposed and certain addi tions - particularly tho relation “is contained in** or “implies" - which -'ir-v: Peirce had mado to Eoolo’c syctoss* I t ic due to SchrSdor’c ocurd judgo- Eont that the result is still on algebra, siapler yet more powerful than Boole*r. calculus» Jevore, in sieplifying Boole’s system, destroyed its mathematical form Peirce, retaining the mathematical form, complicated instead of eisplifying the original calculus* Since the publication of and improved methods have been offered, the most notable of which arc contained in the Studies of Poretsky, and in Uhltchoad*c Universal ^ggSaae (»r,ii8) Ac has boon previously indicated, one of the principle advantages of SchrGdor’o system over that of tho Bars of Thought lion with the fact that the former is postulatioml* Variottc cote of postulates for tho Boolo-SchrMer iVlgobra, or Boolean algebra an i t is commonly called, have been provided, notably those of Huntington Sbeffor (H), and Bernstein (/ ),(* ). although it is possible ( I > to device a sot containing only two primitives, a class aid a single operator*, with as few as four independent postulates, it boat serves the interests of clar ity to employ a system whoso prim itives aro more fully developed and whose postulates are framed with tho view of insuring ease of application* The following sot ic troll suited for this purpose and is that given in Lewis and Langford’s Symbolic horde (1 6 , 20- 9 ). I* Primitive#* 1. ---class of classes &,b,o,etc., in extension* 2. Tho logical product al» of a and b (proviouoly described). 3. Tho logical negation a* of a (previously described). 4. Tho null class, unique and represented by 0 * 5. Tho logical equality a = b of a and b , meaning that a and b may be employed interchangeably, having the same extension. II. Definitions. 1* Tho universe class 1 = 0 1 * 2. Tfco logical ouEi a + t = (a*b*)' . (veak disjunction) 3* Tko logical order a C b noans a = ab . In virtue of the first pesfculatos given below, this is soon to no analogous to the arithzotlcal order relation 6 , not to < # III. Pootulatoe. 1. aa =• a . 2. nb = ba • 3. a(te) = {&b)c • 4. oO = 0 . If ab*=. 0 s then aC b • Hero, as elsewhere, ah* mans a.(bs) , not (mb)* * 5 . I f a C b and a < b* , then a - 0 . Tho conaequencos of those postulates arc tko las# which constitute the body of the Boolo-Schi'Scler i-lgobra. The thoorcaa which follow are coco of tlio most useful of those laws and are chosen with tho intent of making intelligible tho discussions of the third chaptor. They are pre sented in tho order of their dependonco, and for this reason the liberty is taken of oaittin their proofs, indications for rainy of which will bo found in Lewis and Langford* <$ book previously referred to (26,30-45). Css is wado of tho triple line S to nbbrsviato "is oquivalont to", naaning "implioo, and convorcdly". IV. Theorems. 1. nb C m . 2. aa* = 0 , This ia tho Law of Contmdiotiea. 3* mb* = 0 .s> a C b . -10- ♦e (a*)* = a . 5» a ~ b a* =■ b* * The definition a + b =■ (a'b*)* constitute* one form of d* Morgan’s Theorem, other useful forms of which are the following three. *• ab = (a* b’) ’ . T. a’b’ = (a+ b)’ . 8. (ab)’ = a’ + b* . Be IZorg&n’s Theorem exhibits a remarkable correspondence between laws ex pressed in terms of the logical product and those expressed in terms of the logical sum. This corfespodance is termed the law of Duality. 9. a 4- a = a . This is the dual of the Index law, o& « a . 10. a 4- 1 - 1 . 11* a 4-a* = 1 • Law of the Excluded Middle. 12. a 4* b = b •=• a C b . 13. ab ( a 4- b * 14. a 4- ab =. a * Law of Absorption. 13. a = ab 4- ab* • Law of Expansion. 16. 1 = (a 4^ a»)(b 4-b’Hc + c')... . 17. a + b =• a 4- a ’b • 18. a 4 b = 0 • = * a - 0 — b • 15. ab - 1 a = 1 = b • 20. a = b -S* 1 =■ ab 4* a ’b’ . H . a = b •=. t =s (ab* 4* a'b)t’ 4- (ab +-a’b*)t • This is called Poetsky’s Law of Forms. By virtue of the Law of Expansion, it is one of the remarkable characteristics of this algebra that if any element does not appear in -11- oppositely it is r,t>i the fact leal function is in which the otituent of each x as a factor, that is, eith er x or x ' • F, form of a function of a eiagle variable x is Ax + fix' The fivo the the canipulation of logical equations, as vrii: Um If f(x) = Ax + Bx» , then f(l) = h: ' and f(0) - B so that f(x) = f(l)x +■ f(0)x* .• S3. AB C Ax 4- Bx* < A + B . 24. I f Ax 4- Bx* = 0 , then AB = 0 . 25- I f Ax 4 Bx* * 1 , then A 4- B = 1 . 26. Ax 4 Bx* = 1 •=* B* C x c A ♦=• x = B* 4- uA (u arb itrary ). This 'will cosplote the present section on the Boole-SohrBder Alge bra, and it is hoped that some indication has been given of its relatlee te the elgebm of the laws of Thought . It seems hardly necessary to point out that the twenty-six theorems listed here barely scratch the Schr8der Algebra will leads their choice has been made solely with the purpose of paving the way for the discussions of the third chapter. -w - Finally, i t should be remarked that the elonents a,b,c,otc» of the Boolo-SchrBdor Algebra nood not bo restricted in th e ir interpretation to representing classes in extension* the algobra my be developed as an abstract rathematieal syateas with a,b,c,otc. as entities satisfying the conditions imposed by tho postulate syotm. a non-logical interpretation would be tho sot of all regions in a plane, or tho sets of points on a lino* Thio treatment of tho algebra as an abstract mathematical system leads directly to the subject matter of tho next section, which deals with the relation of Boolean algebra to tho concepts of group and ring* 4* Subsumption under Ring Theory. It is well to begin this section with a review of the ideas* group, and ring. The idea of a group involves two prim itives, a class K and an operator 4) © io corautativo* a© b =-b©a with a,b in K , tho group (K,@) bocomos abelian* toe idea of a ring involves three primitives, a class K and two operators © and <3 relevant to the meabbre a,b,c,... of K . toe ring postulates are* 1) (K,S) io an abelian group. 8) <3 is K-elssiag. S) , (8 is associative. 4) & ® (b © c) - a Q b © a © c , and (b © c) ® a = b © a © c <£) a . With the additional postulate 5) ® io coasmtativo * , the ring (K,©,®) becomes abelian. For exanple, if I is the set of all integers, positive and nega tiv e , with zero, then (1 ,+ ) io an abelian group and (1, + , *) an abelian ring, where +• and x are the operations of ordinary arith- ao tie. The link between Boolean algebra aM abstract algebra was studied by Bernstein ( l) ,( 3 ),( S") and has culainatod in the work of Stone (3d), (3'),(W and 0. Birkhoff ((,). The results which follow are found in tho principle work of Stone* "Tho Theory of Representations for Boolean Algebras" (3« )> and are taken verbatim from the text of his paper, tho only changes being lm notation, fibers he writes a + b we write &+ b # this is the oporation of strong disjunction and ie the same as that pre viously used except that Boole's restrietion ab - 0 io removed. Whore ho writes a V b wo write a + b j this is weak disjunction and is the operation of tho preceding section. Of tivo terms should not bo meant to imply that ones or that the elements are classes* Boolean algebra is now to od ao an abstract mathematical system, and it is the remltant eoeprehen- sibility that renders an invaluable service to tho.algebra of logic. A. ? is not out of -M- cttittg is a oyston of elements with certain rules of coabimtion oxplicitly defined to provide the relations of one given aLeraent with anothorj or dinary arithmetic i s an example of a calculus, -n algebra, as the term has been coEseriLy employed, is a calculus which deals with general elements and thoir relations in general, the ordinary algebra of arithmetic being an example. It io in this sense that the torn algebra is hero to be tak en. The appelation has, howvere, also been used to mean linear algebra ( i . e . lin ear act over a rin g ), but i t w ill not be here so ct^ployod. Consider now the ring (K, + , • ) where 4- i s the operatiea with respect to which (K, + ) ia a group and i s termed "addition", while the operation • i s termed m ultiplication. We do not assume that multiplication is commutative or that a ring con tains more than one element. It is well known that in any ring the solu tion 0 of the equation x 4- a = a is independent, of a and satisfies the relations a+ 0 — 04-a=a, a0 = 0a=0| that the solution -a of the equation x + a = 0 is unique; and that the solution of the equation x 4- a - b is unique and is given by x = b 4 (-a), commonly written b - a . ue now lay down the following formal definitions Definition 1. A ring in which every dement is idenpotent, satisfying the law aa - a , is called a Boolean ring. Theorem !. A Boolean ring is necessarily commutative; obeys the two equivalent laws a 4- a - 0 , &=-aj and necessarily contains divisors of 0 if it contains moro than two elements. Every Boolean ring A can be embedded in a Boolean ring B which possesses a unit element, in such a manner that B is unique in the following ocnooi if C is a Boolean ring with unit containing A , then C contains also a Boolean ring B* isomorphic to B and containing A . A finite Boolean ring necessarily contains a unit ordinary ring properties of associativity and distributivity, for in a Boolean ring, a 4- b - (a 4- b)(a 4- b) = (a + b) +• (ba 4- ab) , hence ba 4 ab = 0 so that w riting b ~ a , wo have a +• a - 0 , th is imply ing a = -a . Hence from ha 4- ab = 0 we have ha = -(ab) = ah • Shis chows that a Boolean ring ic necessarily abelian and of characteristic 2. The proof of tho reminder of the thooron will be found in the original paper, cited above. It should bo cade dinar at this point that a Boolean algebra of the Boele-SchrSdef type is not of itself a Boolean ring since the operation of ring addition is + and not -h j nor is the algebra of the iawo of Thought a Boolean ring, sines a 4 a has no place in that system. The following theorems show the relation between Boolean ring and Boolean algebra. Theorem 2* If 4 is a Boolean ring with unit # , the introduction of a binary operation and a unary operation * through tho equations (l) a+ b =• a 4- b 4- ab , (2) a* = a 4- o , converts A into an algebraic system B in which (4.3) a + b T%b + a , . (4.4) a4-(b -»r e) — (a +- b) + c , (4.6) (a'+ b')'+ (a* + b)'= a , the old operations being erprossod in tense of the now through tho o ra tions (6) a-4 b = ab'+ a'b = (a' + b ")'+ (a''4k b')* , (7) ab = (a* 4- b*)' . On tho other hand, if B ie an algebraic oyetea obeying tho laws (4.3), (4.4) , and (4.6), then B ie a Boolean algebra} and the Introduction of new operations through the equations (6) and (?) converts 3 into a Bool ean ring A with unit e = a -*■ a' and zero 0 - o1 =- (a + a’j* , tho old operations being expressed in terns of tho new through tho equations (l) m * (S) above. Thie thooroa clearly serves to identify Boolean rings with unit and Boolean algebras, as characterised by Huntington*e postulates. In view of Theorem 1, a Boolean ring without unit con be regarded as embedded in one chick hao a unit. In particular, it ohcras that the operation of addition la a Boolean ring corresponds abstractly to tho operation of forming the oycsotric difforonco or union (mod 2) of classes, as indicated by the re lation (6); and it shows similarly that the operation of multiplication corrocpondo to tho operation of forming tho intersection of classes, as indicated in the relation (7). (31) Those remarks from Stone’s paper will suffice for tho purposes of the present section. In concluding this section, however, brief mention will bo mde of another remarkable approach to the representation of Bool ean algebras, namely that given by tl. Birkhoff in his recent Lattice Theory (6 ). So ehall merely state the definitions loading up to Boolean algebra, and those w ill bo lifte d verbatim from bis tex t. IMMM m .W * By a p artia lly ordered system, is meant a system X in which a relation x & y (road “x includes y") is defined, which satisfies PI* For a ll x , x = x • Ft* If x & y and y & x , then x = y . F3» If x = y and y A s , then x . Definition Jiili A lattice is a partially ordered system any two of whose elements x and y havo a greatest lower bound or “meet" x/> y , and a least upper bound or"join" xw y . L5s I f x = s , then X 'u ty z ^ s ) = (xw y)/-x z . ("Modular identity") Definition 3.1: A lattice will be called "modular", if and only if its elements satisfy the modular identity. Definition 4.1* A (modular) la ttic e is "complemented" if and only i f i t has a 0 and I and 1*7* Every x hao a "complement" x* , ouch that x/-vx* 0 and X'wfX* I . Ms (x/^y) vi (yrx z) v (zr^x) =• ( xvj y)(y v /s ) r \( z v*x) . M’t Xz^(yvyz) =. (xz%y)w (xz-iz) • L6"x xw(yz>z) = (xwy),-\ (xv^z) . Definition 5.1» A lattice will be called “distributive" if and only if it satisfies M, L6», L6" identically. Definition 6.1* A Boolean algebra is a cooplesented dietrilattivo lattice. (&) m s CALCULUS OF PROMBILITT 1* Hiaiorieal. categories« probability in anelyais, and probability in logic. %# first of those categories ms blessed with the goniius of Laplace, whose The second, probability in logic, has experienced a core troubled devel opment, although certainly i t must be admitted that i t has received the earnest attention of many great writers. Some of the preoccupations which th is development has involved are indicated by i&gel (28,15ff)s There are ... both historical and analytic grounds for the view cen tral to empiricism that there is no a* priori knowledge of natters of fa c t, and there are sim ilar grounds for t .e thesis of contemporary empir icism that no amount of empirical evidence can establia& propositions about matters of fact beyond every possibility of doubt or error. On the other hand, the recognition of this state of affairs raises an important problem. Although our beliefs cannot be established with absolute final ity, we, as we must, differentiate between them on the ground of the char acter of the evidence which supports them. «e regard i t is more probable th at lixpoleon was a h isto rical character than that he is a solar myth, tie btitlve that the prognoses of a modern physician are more reliable than those made a century ago. A chemist accepts Lavoisier's theory of combus tion as better founded than S tah l's phlogiston theory, and a physicist will urge that the quantum theory of radiation is today more securely based than it was twenty years ago. There is clearly an obvious need for canons to evaluate the evidence supporting any proposition, and for the formulation of the principles wo employ in deciding that one statement io better grounded than another. Judging by the success of past attempts to supply them, i t may be suspected that every proposed l i s t of such canons and formulations w ill be incomplete and w ill require emendation wil the progress of inquiry. The need, however, is u permanent one; and the a t t empts to satisfy it constitute the broader setting and the larger theme in contccporary discussions of probability. The formal laws in the calculus of probability per ao are disputed fer no apology oinoe it is not the task of the present work to exanino the demarcation between ratiomliea and empiricism in probability; the laws are discussed only with a view to insuring a grasp of the manipula tive procedures employed in Chapter III* The etudy of these law was initiated by Pascal, Fermat, Bernoulli, ete*, and consolidated in their classical fora by Laplace. The following is a summary. Chiefly taken from Laplace, of the principles which have bom applied to the solution of questions of proba bility. They are consequences of its fundamental definitions ... and may be regarded as indicating the degree in which it has been found possible te render those definition available. Principle, 1st. If p be the probability of the occurrence of any event, 1 - p will be the probability of its non-occurrence. 2nd. The probability of the concurrence of two independent events is the product of the probabilities of these events. The probability of the concurrence of two dependent events is equal to the product of the probability of one of them by the probability that i f that event .occur, the other w ill happen alee. ith* The probability that if an event, S , take place, an event, F , will also take place, is equal to the probability of the concurrence of the events £ and F , divided by the probability of the occurrence of E . The probability of the occurrence of one or the other of two events which cannot concur is equal to the sum of their separate prebaM- lit le e . 6tb. I f an observed event can only resu lt from some one of n d iff erent causes which are a’priori equally probable, the probability of any one of the causes is a fraction whose numerator i# the probability of the event, on the hypothesis of the existence of that cause, and whose denom inator is the sum of the similar probabilties relative to all the causes. 7-th* The probability of a furore event is the sum of the produote formed by multiplying the probability of each cause by the probability that if that cause exists, the said future event will take place* (9,248-9) This was essentially the state of affairs at the time of Boole; the fundamental laws of the probability calculus were held mainly as working principles stoning from the classical ration definition of probability, no attoapt being made to systematize thorn into the body of a sound deduc tive system which would earn tho designation of a strict mathematical science. Koynooj in his (ISa) otat'ec that Several modern w riters have made some attempt a t a symbolic tre a t- mont ef probability. But with the exception of Boolo ••• no one has work ed out anything very elaborate. (20,155) Tho most notable of these attempts io perhaps that of I’acGoil in his sev en papers on "Tho Calculus of Equivalent Statements" (27)j his treatment, i f elaborated, would have parallelod that of Keynes. The above statement ef Keynes is likely to convoy tho impression that Boole developed a sym bolic treatment of the calculus of probability, but this is not actually LacColl’s papers appeared over a span of years, 1677-1898$ prior te th is , however, Peirce had w ritten in 1867 ( 29)* (we have changed hi® notation; numerical operations being enclosed in circles) Let every expression fo r a class have a second meaning, which is i t s cleaning in an equation, iiamely, le t i t denote the proportion of in dividuals of that class to be found among all the individuals examined in tho long run* Then wo have a If a = b a f»b a© b - (a + b )© ab Let b ex. denote the frequency of the b*s amongst the a a • Then eeosidered as a class, if & and b are events b*, denotes the fast that i f a happens b happens. a <9 bo. €) * properties of the function . a ® b® ag. (b^))*. (1 *b)a © 1 (3 bo. ba © 4©1>„^U -5.) («pB)flu © I Characterietically obscure, this passage of Poirco'o nevortholoas indicates a treatment of probability which wculld differ from that of Eoolo, as has boon summod up by Lewis (25,106) as follows* (1) "boro Boole puts p,q,otc» for tbo “probability of a , of b , etc.", in paeoing from the logical to tho arithmetical interpretation of hie-equations, Poirco oinply changes tho rolations involved,• from logical relations to the corresponding arithmetical relations, .** and lets the terms a,b,etc. stand for the frequency of the a*o , b1s * etc.* in thoi-system under discussion. (2) Boole hao no symbol for the frequency of tho a*o amongst tho b*s , which Poirco represents by a& . As a result, Boole is led to treat the probabilities of all unconditioned events as independent - a procedure which involved him in cany difficulties and some errors. (3) Peirce has a complete set of four logical operations, and four analagous operations of arithmetic. Ibis greatly facilitates the passage from tho purely logical expression of relations of classes or events to the arithmetical expression of their relative frequencies or probabilities. Lewis then continues with a passage pertinent to tho present discusooions Probably there is no one pj^Lce of work which would so immediately reward and investigator in symbolic logic as would)/the development ©f this calculus of probabilities in such a shape ao'to make it simple and prcacticable. Sxcopt for a monograph by Porotxky andv,tho studies of li. iZacColl, tho subject above in 1867. (25,106) tion, and it should bo rocarkod that Lewis indicates in a footnote the poothunzo publication of a paper by Couturat, too la te to include in h is text; this paper is also included in tho next section, 2, Perotaky and Couturat. The subject ratter of the present section centers about the idea cff tee r.unerisation of classes in extension. If x represent a class ta ken in extension, then n(x) will bo used to represent the cardinal cumber of the class Essbarship.' according to Russell*s definition of cardinal number, n(x) is merely the intensive class whose extensive members aro similar to x , but this point is of no particular interest .hero,. Ono of tho earliest explicit uses of nusorization is to bo found ia the chapter entitled "Of Statistical Conditions”, in Boole’s haws As dorivod from actual experience, the probability of any event la the result of a process of approximation. It is the limit of the ratio of tho number of cases in which the event is observed to occur, to the whole number of equally possible cases which observation records,- a limit to which we approach tho more nearly as the number of observa tions is increased. How let tho symbol n , prefixed to the expression of any class, represent the number of individuals contained in that olasa. ... In accordance with this notation n(l) will represent tho number of individuals contained in tho universe of discourse, and bCx)/ b(1) will represent tho probability that any individual ... , se lected out of ... n(l) , (is an x). If observation has not made us acquainted with tho total values of a(x) and n(l) , then tho prob ability in question is tho limit to which n(x)/n(l) approaches ao the number of individual observations is increased. In like manner ... if x represent an event of a particular kind observed, n(x) will represent the number of occurrences of that •went, »(1) tho number of events (equally probable) of all kinds, and b(x)/b(1) , or its limit, the probability of tho occurrence of the event x . spec ting tho ratios of the quantities n(x) , n(y) , n(l) , otc. cay bo converted into conclusions respecting tho probabilities of the events represented by x , y * etc. ...g e n e ra lly any such s ta tis tical relations ... will be converted into relations connecting the probabilities of tho events concerned, by changing n(l) to 1 , and any other symbol a(x) into Frob. x . (»,815-6) The use to which this mcerisation was put by . Boole is host ex- ecpiifiod b; tho following proposition which ho sots about t® eti.ve* Given the respective numbers of individuale comprised in any •lassos, a , t t e tc ., logically defined, to deduce a system of nu merical limits of any other class w , also logically defined. (9,304) Perhaps he considered it as trivial, but it books remarkable that he cade no comment of how tho concept of nucorization night be employed is tho way that it me later done by Poretzky and Geuturat. Hevorthe- l##e it le clear that ho was in poosossion of tho fhndamentsl priaoi- ploo of the idea. next to the paper "Solution of the General Problem of tho Theory of Probability eith tho Aid of hathosatical Logic" by Poretzky (31). Sittoe this paper has never received publication in the English Ian- part of tho paper, the Appendix, which is pertinent to the present ties. . . . to each logical equality (l) f(a,b,6,...) — ♦ Dividing both sides of tho equality by H(l) , denoting the number of objects in tho universe of discourse, v;o got another nunori- cal equality* ...... (&) ^ [^f (a,b,c,«**) J = # eapreocing the equality of probabilities of tho logical classes f and d> • «o shall call tho passage fron (l) to (3) tho "probabilization" of tho logical equality (l)$ tho passage from (1) to (2) - "nuaerisa- tion" of (l)e ••• if wo establish rules for passing from (l) to (2) and deter mine the properties of the symbol U independently of the Theory of Probability, then, in view of the simple connection between (2) and (3), in those properties of II wo shall at tho saco tiao discover corns of tbo proportion of tho symbol P * It should bo observed that (2) has meaning not only as an inter ned into stop from (1) to (3), but also in its e lf , sinco i t may have applications to other fields, o.g. Stetistiee. W© now consider tho construetlen of rules for tho nuKorimtion of logical oquiditieo. To “nunoriee" a logical equality It suffices to rairaeriz© each mesbor separately and thon sot tbo two results equal to each other. Thee nunorization of logical equalities reduces to nuneriration of logical function* The determination of tho number of objects contained in a given logical class a , i.o. tho determination of 1!(a) , cay of course bo accomplished by direct counting. But if wo know tho relation be tween cuch symbols os K(a) , 11(b) , H(a -t- b) , e tc ., then we could empress tho magnitude of ono of those in teres of tho others. The object of tbo theory of numoriration is to establish relations between tho various symbols II • we shall f ir s t seek a relation between ll [f*(a,b,ct*..)} and H [f(a,b,c,...)3 , where f* is tho logical negation of f . From the logical identity* f + f * = 1 we have: H(f +' f*) ^ 11(1) . Since tho product ff* = 0 , the ©lascoo f and f * are disjunct, •and therefore K(f + f ) - H(f) +• H(f*) and K(f) + K(f') = M(l) , 1.0. H(f*) = N(l) - 11(f) . Dividing both ncnbors by IJ(l) we got P(f') -1 - P(f) » ono of tho basic truths of the Theory of Probability* lloxt tro seek an expression fo r H(a +• b) • If ab = 0 , i.e. a and b disjunct, then clearly K(at-b) -N(a) t-H(b) . But cuppoco n and b aro conjunct, i.o. ab * o . Froa tho identity a - ab +.ab' , where the right member is disjunct, we derive N(a) - !Kab) 4- * W ) . Similarly from b - ab +- a'b wo derive N(b) = K(db) + K(a’b) * Adding* . i«1a) + n(b) - Sir(db) + l(db») 4- l:(a*b) . On tho other Hand, oinco for a logical class m , a 4- a * a f at- b - = ab 4- ab* 4- a*b , where the right member is disjunct. Therefore* fl(a+b) « B(eb)+H(ab*) 4-K(a*b) ♦ Comparing with tho previous result, wo got K(a4- b) - K(a) + K(b) - K(&b) , a general result from which, in p articu lar, who,- ab = 0 we deduce K(a 4- b) - Ii(a) 4- 11(b) . _ n*rthor, in view of previous observations and of the logical law cm # n , wo generalise as follows* N(a t b +■ c) — n\ifa + b) c] 4- II(a + b) 4- li(c) - II[(a + b)@] - = *(&) + K(b) + K(c) - N(ab) - K(ae) - H(bc) 4- H(abc) . ... Tho general result is now obvious. In particular, for a sum 2 0*. in which ava^ - 0 for a ll l ^ j * K(X a j - SL (K (aJ) whenco dividing by 11(1) P( 5. a.) =- 2-P (aJ , He shall new establish another formula for *( 2 a*} • la formal logic wo have the identity a, -t- a x= a , 4- (a, )*a^. where the cum on the rig h t i s disjunct* Honce N(a,+- &J =- H(a.) *- *((&,)ea*3 . Similarly from a, * a*.+ a, + (a,)*ax+ (a,)'(a J'a, » wo get K(a% +- a%+ - H(a,) + N[(a,)*ax]+ H[(a,)*Cat)*a»] * In gonoral* N(SaO ~ K(a,) +» M[^a,) *a J + K [(a,)'(aj'a J + ... The third method of determining N( 2. a J consists in decomposing tbo sum 3E nw into olemento of total disjunction. If such a decom position is 5L a,, a s, 4- ox4- s» +* . . . , then ^ 11(8.) • The fourth method is this. Since the negation of the sum a, +- a* a > 4-... is the product (a,)*(aJ‘(aO***• • then N(a, 4- a x4- a-^4- ••• > ~ H(l) - » I ( a ,) #( a J ’ ( a ,)*...][ . He shall new consider U for the product of logics! classes, vie already have H(a 4-b) - E(a) 4- K(b) - H(ab) . Thoroforo N(ab) - K(a) -w K(b) - K(& > b) . Also (S) N(ab) - K(l) - K[(ab)'l - M(l) - H(a' * b») . #e shallanot consider generalizations of these formulae. Instead we consider the following. The next to the last formula shews that knowing K(a) and N(b) we s tib l camtot determine N(ab) • However, it is eapy to find the bounds inside which this magnitude is contained. Karnely, ll(ab) in not lo s s than zero, and not core than the sm aller of the numbers R(a) and H(b) . Boole has shown that the lower limit can bo improved on. ihmely* H(ab) - K(a) 4- N(b) - H(l) . In fact from fonmla (£) it follows that H(ab) — M(l) - H(a' + b’) =■ 11( 1 ) - [H(a') 4- *(he) - M(a'b')] ~ - K(l) - [H(l) - 11(a) +- M(l) - H(b) - H(a'b’>l = - K(a) 4- N(b) - 11(1) + *<»•*') # This Is a ja«ar resu lt fo r N(ab) , from which in fact H(ah) - M(a) + N(b) - B(l) . • •• Geasrslly* ., . H (ria .) ^ i H(aO - (n - l)K (l) . As far as tho upper bound Is concerned, the magnitude of «(□ a j is equal to or leas than tho smlleat of the tf(&J * This io n il that can bo done in the nay of rules for 1-iimorization. If, however, wo odd tho hypothesis of uniform distribution of tho objects of oach class over the whole universe of discourse, then H(ab) * !i(a) = K(b) s ii(l) whence = m i * ® P(ab) = F(a) » P(b) . This concludes the Appendix to Poretzky’a paper (31), It is clear that Porotzky boxrowB the idea of numerization from Boti.e, but utilizes it in a new and ingenious fashion. In 1917 was published posthumously a paper by Coutunti entitled "La Logique Algorithmique et lo Calcul dee Prebabilitee" (15), Since Couturat cakes no mention of tho above paper of Poetzky, i t is assumed that ho worked Independently of tho rem its of the latter; his paper is a development of numorization along much tho sano linos that Porotzky bad followed. He denotes by p(A) the probability "du jugeaont (simple) A", an# hf n(4) "le nombre des cas ou la proposition A est vraie". Then it follows that ‘ P ( a ) = * La fornulo la probability do A* euimnte ,(A ,) = Tollo oat la probabilito pour gie A* aolt vraio, ou A soli feusao. 51 I'on addltlonne cmbro a on trouvo* pW + pUM.- d* oa l|ee eeaelut Izraodlatononti p(A') = 1 - »(A) . Coutuzmt then obtains the expression for n(A + B) in precisely the sane canner that it ws obtained by Poretzky (above). T* continue with his diccuD*l*B« . . . Ia preWbilito do B relutive a A n*eet pas autre chose quo la probabilito do la proposition* "Si A ost vraio, B cot vraio**, c'oot-a-diro do liinferenco ou inclusion* A < B . En effect, c'est la proMbilite pour p U < B) - 1 ^ 1 ©t I'on pourra repreeenter par p(A< B) la probabilite do B rela tive a a . . . . on diviso loa deux torces do la fraction par n (l) , on trouvo* ••« Gonsiderone d’abord le cub de deux propositions, A et B . On a lea ogalitos aritbrntiques szivantee P(AB) = ^ - ,U ) *• *{ l< • (IS) Relative to the present discussion, this is all that Ceutorsi has to Bay that is of Interest. The fifteenth topic of his paper is en title d "Problems do Boole", but consists only of a number of blank lines. A a baa been previously noted, ibis paper ms published post humously (Ceuturat met an early death, struck by a military truck in 1 9 1 6 ) and it is indeed a misfortune that hie observations on Boole’s goaoral method did not find thoir nay Into print. 3. Keynes. fe J.M. Keynes goes the credit fo r undertaking the f ir s t system atic investigation into the foundations of tho concept of probability. His treatment, embodied in A. Treatise on Probability , although it leaned more to the side of logic and philosophy, included chat is per- hapo the f ir s t completely deductive and symbolic organization of tho probability calculus. It ie to this latter aspect of his work that the attention of the present section is directed, Keynes viewed the subject of probability from the standpoint of pure logic; he wan thus a non-freqaontiot in tho highest sense of that tera, and in fact devotes an entire chapter of hie work to an examina tion of may logical difficulties inherent in the frequency theory, of which ho cays (20,109-10)s . . . I an cure that the objections which I have raised cannot bo not without a great complication of tho theory, and without robbing it of the eisp lialty wfcioh is i t s greatest preliminary recommendation. Until the theory has boon given now foundations, its logical basis is not no secure as to permit controversial applications of it now in practice. As has been previously remarked, it is not the office of this puper to out or into an examination of the controversy between tho frequency and otato Keynes*e position on tho matter, oinco it io in the nature of th is position that lio a tho key to his subsequent symbolic development of the subjeet# With a vioo to clarifying Koynos'o own attitude, thon, wo pre sent herewith a portion of his own summary of it (20,111-12)$ That part of our knowledge which wo obtain directly, supplies the promisees of that part which we obtain by argument* From these pre misses no cook to ju stify soma dogroo of rational b d io f about a ll sorts of conclusions. Wo do this by perceiving certain logical rolan tions between tho premisses and tho conclusions. The kind of rational belief which we infor in this manner is termed nrobablo (or in the IWLt cort&ij)), and the logical relatione, by the perceptions of which The probability, of a conclusion a derived from premisses h wo write a/h j and th is symbol is of fundamental importance* The object of the Theory or Logic of Probability is to systema tise such processes of inference* 2m particular it aims at elucida ting rules by means of which tho probabilities of different arguments can be compared. I t la of great practical importance to determine which of two conclusions in on tho evidence core probable. The most important of those rules is tho principle of Indiffer ence. According to th is principle wo must roly upon direct judgement for discriminating between the relevant nd the irrelevant parts of the evidence* i>o con only discard those parts of the evidence which are Irrelevant by seeing that they have no logical bearing on the con- elusion* The Irrelevant evidence being thus discarded, the principle lays it down that if the evidence for either conclusion is tho same (i.e* symmetrical), then their probabilities also are the same (i.o. OtJJOl)* If, on the other bead, there is additional evidence (i.e* in add itio n to tho symmetrical evidence) for one of tho conclusions, and this evidence is favourably reievent, then that conclusion is the more probable* ••• There arc, however, many eases in which these rules furnish no means of comparison; and in which i t is certain that i t is net actually within our power to cake the comparison. •*• in these cases the prob abilities are, in fact, not comp arable. 4s in the example of similar ity, where there aro different orders of increasing and diminishing similarity, but where it is not possible to say of every pair of ob jects which of then is on the whole core like a third object, so there are different orders of probability, and probabilities, which are not of the sane order, cannot bo conpared. I t is eoGetlciee of.practical importance, when, for example, we wish to evaluate a chance or to determine the amount of cur expecta tion, to say not only that one probability is greater than another, but by how much i t is greater* We wish, that is to cay, to have a nu merical measure of degrees of probability. This is only occasionally possible. A rule can bo given for nu merical acacurcrnent when the conclusion is one of a number of equi- prebable, exclusive, and exhaustive, alternatives, but not otherwise. The fact, as thus stated, that in this theory probability has only occasionally a numerical measure, serves to emphasize its radical departure from frequency theoretic methods. A complete theory of the passage from abstract to “moasarouble" probabilities has been given by Koopaan (22), (24)g see also (23). Coming now to the symbolic part of Keynes's treatment, ho has thio to cay relative to notation (20,119); ...o u r f ir s t tack is to establish the axioms and definitions which aro to make operative ear symbolical processes. These processes are al most entirely a development of the idea of representing a probability by the symbol a/h , where h is the promise ef an argument and & its eoneltieieiu It might have boon a notation more in accordance with our fundamental id eas,'to have employed the symbol a/h to designate the argument from h to a , and to have represented the probability ef the argument, or rather the degree of rational belief about a which the argument authorizes, by the symbol P(a/Jh) • . . . But in a section where we aro only concerned with probabilities, the use of ?{a/k) would have boon unnecessarily cumbrous, and i t is , therefore, convenient to drop the prefix P and to denote th# probability Itself by c/h . The discovery of a convenient symbol, lik e that of an essential word, has often proved of more than verbal importance. Clear thinking on the subject of probability is not possible without a symbol which takes explicit account of the premiss of the argument as well as of it# conclusion; and endless confusion has arisen through discussions about the probability ef a conclusion without reference to the argument as a whole. I claim, therefore, the Introduction of the symbol a/h, as an essential etep towards any progress in the subject. ~o.» cent regarding-tho use of an appropriate oynbol of probability! the la tte r ' g representation a/h appears ccdnently seccaoeaful in oscap* ing the nany p itfa lls to ehich a careless symbol w ill lead, and hence it will bo adopted throughout tho resaindor of this paper. I lio t cow tho definitions and axioms out of which Keynes io able to dwfaiee the accepted principles of the probability calculus. (20,134-8) Definitions* I . If there exists a relatio n of probability P between a propo sition a and premies h a/h - P . II. If P is a relation of certainty P = 1 . III. If ? is the relation of impossibility P = 0 . IV. If P io a relation of probability, but not the relation of certainty P < 1 . Y. I f P is a relation of probability, but tho relation of impossibility P > 0 . Tho class of propositions a such that a/h =. 1 Is the or for short. fill. If b/ah — 1 and a/bh — 1 , (a Eb)/h - 1 • This may bo regarded ae the definition of equivalence. Tim* we 800 that equivalence io relative to a procias h . & is relative to given h , if b fellows from ah , and & from bh . H. ab/h 4- ab/h - a/h . (Tho supra-bar denotes negation) X. ab/h — a/bh . b/h =■ b/ah . a/h . XI. I f PQ - R i ? ~ q • (P»Q»R are prebaMlltiee) XII. If o/bh = a/h and b/aja - b/h , the probabilities n/h aad b/h are independent. m . If a/bh - a/h b is or, Axioms i (i) Provided that a and h arc propositions or conjunctions or disjunctions of propositions, and that h is not an inconsistent con junction, there exists ono and only one relation of probability P be tween a as conclusion and h as proalso. Thus any conclusion a boars to any consistent premiss h ono and only ono relation of prob a b ility . (ii) If (a s b)/h 1 , and x is a proposition , x/ah = x/bh . This is the Axiom of Squivalencc. ( i l l ) (oT~b = a b)/h =-1 (aa a a)/h = 1 (a s a)/i - 1 (V) ( db Pdr Q) 4- ( ARdhd) - ( ±PdLR) - ( TQ Vs) = ( 6 Pzb R) + (db Qd= S) =(db PdLQ) - ( ? R ^S) in every caso in nhich the probabilities ( ± P t Q) , (± R ±.5) , ( i. P±. R) , etc., exist, i.e., in which these suns satisfy the eeaAl- tiono necessary in order that a nooning bo given to then in terns of our definition. (vl) P(RdLti) - PR±. ps , ifbthe sun R ^ ii and the products PR and PS exist as p robabilities. This . Yet, however -shat restric tiv e mature enables i t s to probability will be held in the a contrast to the polished and 1 Unity of the opoento which are to be outlined in the next section. two of the cost comprehensible of these methods, that of Koloogoroff ef 0. Birkhoff in his recent Lattice Theory (6). the following axioms* (21,*) * welehe S I le II* t? enthfilt die Mango IqrKrplr.nlenoa...A .* I f . P(E) = 1 . ?• uonn A und B dlajt-iikt slnd, so gilt P(A +• B) = P(A) + P(B) • %ln l^engonoystaa ^ s i t e lm r host lent en zuordmog dor "ahlen P(A) , trelche don Axloaen I - V genfigt, nannt mn eln tiahroehoin- listitiudMaM* To indicate tha my in %hioh the standard principles follow from these eii^lo assas^tloBs, observe * (21,6) Aua A+ A = S und dm Axiosen IV und V folgt P(A) 4* P(A) - 1 , P(X) ^ 1 - P(A) . Da E — 0 1st, erM lt man ittsbeaondere P(O) = 0 . «6 tm AjB^**.,H unvorelnbar slM , so folgt bus Axiom IV d ie formal P(A + B •«■ **• H) - P(a) +, P(B) 4- ••• + p ( ij) (dor Additloncsats)* , »enn P(A) > 0 l e t , so nennt man den Quotienten die .dee Si^ignisses B unter der Bedingung fhie will suffice to indicate in a very general way the procedure of Kelesogoreff. The followie® of bis remarks should however be included. la connection tilth Axiom It (21,2) Kin I'engenaystwi holeat eln Ktfrper, tterai Suma, Durehschnitt, und Difforonz von xw@l Kangen dee Systwa tsioder dm Systm angehBrest. Jo- dor nicht loore HengenkSrper enth&llt die IMLI-mnge 0 • tlir besoich- non ••• den Durchaohniit von A und 3 nit A3 , die Vereinigunge- oenge von A und B is Falle AB = 0 sit A +* B , allgemein aber mit A 4- 3 , und die Dlfforenx iron A ung B a it A - B • Dae Kon~ plment 3 - A dor Kongo A wird durch A bezelchnet. And not all events can be associated with a definite probability* (21,15) Bin ifccgensystem ^ 1st eln Borolschor KSrper, vena a lle abzlhl- baren Succxsn 2 Am dor Itmgm Am awe Sp zu 9r gtiiflren* Van nennt Borolschor liJrpor uuch vollatSndig additive Eengensyatom. ... Sin wahrschoinlichkeitefeld . .. let eln Borelechoe a'ohrschein- liehkeitsfold, wenn der entepreebende HengenKSzper ^Sr eln Borelecher 1st. Fur ih Bordlecher ^ahrsoheinlichkoitBfeldorn erhUllt die Snhr- ebhoinlichkoitsrochnung eine volletUndige Handlungsfreiheit, die s it keiner Gefahr verbunden 1st, au Eraignieoen xu gelangen, v^Lohe koine Wahreohoinlichkeit besitees. "O turn now to mention Chapter IX, "Applications to Probability", « of 0. Birkhoff *o Li-ttioo Theory. In eo doing we shall make use of the definitionB appearing on page 16 ef the present paper, and in addition of (6,40) Definltlon.3.4* A functional m[x) defined on a lattice is called "modular" i f and only i f his m(x] + m[yl - 4- m jx ^ y l identically. It la colled "pooitive", if and only if IS* x ^ y implies a[x}- *[/} • And now in one f e ll swoop we have (6,132) The general theory of probability can be based on the following Definition 9.1* By a "distribution” or "probability functional" on a Boolean Algebra, ft , la eoent a positive, modular functional p fi] ehich oatiefioo p[p] = 0 , p[l] - 1 . I t ahould perhaps bo rmarked that a "fumstioiml" is takes to W a red-rallied function. Chapter III Theory of Probabilitiaa", & utateaent of the gwiearel problem eteich he Sivea the probabilities of any system of events* to determine by a general method the consequent or derived probability of any other event. • &) The present chapter is the principle part of this paper, and its task will consist in surveying the efforts which were made by Boole in the direction of the problem above stated . Boole’s studies of th is problem are not confined to the Laws of appearance of the Lam of Thought (1854). Ibis section, which will be a review of Boole's studies (of the general problem), will thus inelude the most important of these pipers in addition to the Lam of Thought. The next section will describe an salification of Boole's method by the Russian logician Foret sky (31), which forms the subject matter of his paper discussed in Chapter II of the present work. The method of the first two sections will be illustrated in Beotian three by the analysis of certain examples * while section four will concern its e lf chapter dlosoB with aection fire, & sketch of a cethod for precisely statiue the genei'al problem, and an Indication cf the direction in thick It# solution Blight properly he attempted. the subject of probability mas never very precise, and in fact rare often than not was actually fallacious. That this should he so is a remarkable thing, in view of the fact that he was a professor of ~ath- eeatlco (at Queen’s College, Cork); yet it can alee be viewed as only another disclosure of the unsettled state of the subject, and can pro perly he ascribed to the fact that no truly deductive traatEmt me known. The earliest of Boole’s considered reflections on our topic appear ed in 1851 (7)i . . . Although the immediate business of the theory of probabilities is with the frequency of the occurrence of events, and although it there fore borrows some of i t s elements from the science of number, yet as la tio n s, of whatever kind, which eomeot them, is the office of lan guage, the common Instrument of reason, co the theory of probabilities eaet boar some definite relation to logic. The events of shieh it tak es account are expressed by propositions. Regarded in this light, the object of the theory of probabilities cay thus be statedt~ Given the separate probabilities of any propositions to find the probability of another proposition. By the probability of a proposition, I here mean ••• the probability that in any particular instance, arbitrarily chosen, the event or condition which it affirms will come to pass. In confirmation of this view, let it be remarked, that as simple events are expressed by simple propositions, so combinations of events are expressed by combinations of propositions, i.e. by propositions ex pressing some logical connection among the simple propositions which they involve. Upon the nature of that connection depends the mode in which the probability of the compound event represented is derived from the probabilities of the single events. The relation of cause and seek to determine is aleo the probability of a proposition. Kew, every proposition cay be considered either with ref to its matter to its fora. \ may be thus state*t- The probability sought is a function of iho pro- logioal connection of the proposition whose probability is sought with those propositions whose probabilities are given. There are then two conditions necessary toward the construction of a perfect method for the calculus of probabilities•» 1. The prior construction of a general method for determining the logical dependence of any proposition upon another given proposition, or set of propositions. S. The deduction from that expression of the corresponding rela tion among their probabilities. This is quite a clear etatmeni of the ends to which all of Boole’s efforts along th is lin e were subsequently to bo directed, and h is in vestigations always centered principally around the conditions 1. and 2. above. "fe have next a paper appearing in the year (1851), in which is discussed a very elementary problem (8)« **. Given p the probability of an event x , and q the probabi lity of the joint occurrence of the events x and y « required the probability of the event y » The solution of this problem afforded by the general method des cribed in my last letter is ?rob. of y = q + c(l - p} * ttbmre o represents the unknown probability, that if the event x does not take place the event y will take place. Hence it appears that the Halting probabilities of the event y are q end 1 4- q - p • This result is easily verified. The only published solution of this problem with which 1 an acquainted is Prob. of y t b result which involves tho supposition that tho events x and y aro independent. This supposition is, however, only legitimate when the distinct probabilities of x and y aro afforded in tho data of the (The eolation of th is problem refers appears in Wm of Thought pp. 276-7 .) the above extract reveals to a Boole’s peculiar concept of independence. 1# ascribed to two events the other or both are withheld. By holding thus, one can easily eee the sane set of circumstances be dependent in another, and this would cause the independence of events to reel solely upon the nature ef the data concerning them. This same point will appear later* As it happens, Boole is led to the correct solution of tho above problem because the probability of y is act provided in the data. That his solution is actually correct is easily seen* j/h - xy/h + %& = xy/h 4- y/3i « x /h . Writing a/b - P , xy/h = q , y/xh - o , we have f A ~ q 4- *(i - p) » u® turn now to the Lag a of Thought* Chaptw XVI of th is trea tise i s entitled "Of the Theory of P robabilities", and consists i a an ex* tom being applied to each events as consist in a eoebimtios ox simple events. In this manor wo might define it as the practical end of the witii which, by of its definition, it (9,84#) ... it will be advantageous to notice, that there win bo another fora under which all questions in the theory of probabilities may be citiono which assort t h a t or w ill te the truth of those propositions, not to the occurrence of the eventa concerning which they cako assertion. (9,247) I t has boon defined that "the measure of the probability of an event is the ration of the number of eases favourable to that event, to the total number of eaeas favourable or unfavourable, and all equally possible”. In the following investigations the tens probability will bo used in the above asaoo of "measure of probability". (9,253) BSF1SCK0M.- Too events are said to be independent chon tho prob ability of the happening of either of them is unaffected by our expecta tion of the occurrence or failure of the other. (9,255) "a perfectly correct definition", making a point of indicating the im portance of the word expectation, which appears. Interpreted in the proper light it my will be that this definition is compatible with hero it could possibly ba used as a basis for calculating probabilities» Let us repeat hare Eoynas's definition of the independence of the • : . ■ • : ■■■■■■ probabilities z/h and y/h. s If x/jh = x/h and y/zh - y/h , the probabilities z/h and y/h axe iBkaam teA. (9,238) the distinction inrolvod is not merely a verbal one, that is to say, the synfcol r/h boars to the symbol y/h the relation of indcpondonco W definition, whenever xy/h - V h • yA * 2n th is aonao, independence is the mips fo r a relatio n in the calculus, ju st as, fo r example, great- S L M m or lcq3,thft|t are names for certain relations in arlthootie. By thus distinguishing the tons independent as the name of a relation be tween tho elements of our probability algebra, rather than the relation it calf» t?e arc enabled to circuiarcnt all the delicate questions of the subjective causal conuectiono between events, quest ions uhich are ess entially alien to a mathematical theory, and which have preoccupied the bulk of classical writings on probability to a deplorable extent, Y.hy Boole's definition of independence is condoned as "perfectly correct” by Keynes, she himself the term properly, is a systeryi not that Boole’s definition is perfectly fallacious, but rather that, as far aa it# utility is concerned, it is perfectly espty, I should like to rake it perfectly clear, however, that by holding thus, we do theory la to provide '*significance testa** for different eerie# of with a tnathogiatioal theory, we trea t Bind does not dwell* Of course, and it is quite natural so to argue, when we com to utilise the cathecatieel theory by substituting events and propositions for symbols, wo must know how to fit the objective data into an abstract framework, and this colls into play our capacity fo r making subjective judgements. Undeniably true. The important judgements with the operations of the theory* it is a failure to main tain this distinction that accounts for many of the ill# of the eub- jeet* To return to Boole*# definition stated above, suffice i t to say that further comment is not necessary since it is not this definition which ho actually uses, as seems obvious that he cannot. Bis second definition will appear shortly. X would remark that the distinction between simple and co&qiound events is not one founded in the nature of the events themseiLvss, but upon the mode or connection in which they are presented to the mind. Hoar many separate particulars, for instance, are implied in the terms "To be in health," "To prosper," e tc ., each of which might s t i l l be regarded as expressing a "single event" t The prescriptive usages of language which have assigned to particular combinations of events single and definite appellations, and have le f t unnumbered other combination* to be expressed by corresponding combinations of distinct terms or phrases, is essentially arbitrary. Sfhen, them, we designate as sispla Hotr i f is not it cannot Effect the fact, vis«, that the probability of the of one of i» P S ? then I is M * But hut that the data afford of events are given, bet aH information independent. And this are staple or a single vwb or a (9,255-6) This is the i t , net have it ant I t -■ >• with Se fast that one is a and B 3 b. . Thie tion wao roundly criticised, as ms Boole’s it, by a certain Hr. mbrahas (33), (27), and of I t is elm of an y t the event since we are totally i t lie s between tap© awl unity. Thus, Bo©1«*b stateeent should reads faet, viz, that the probability of the occurrence of ©ae of them is § and that of the other ia q $ then 1 am totally ignorant as to the independence of tho events, and coaeeqpeatly can affirm aoithsr th at the probability of their joint ocourranco ia pq aor that it is net m . Let ue continue with Boole s of certain events, we poeeese some definite information respecting their respective combinations. For example, lot it be known that certain combinations are excluded from happening, and therefore that the remain ing combinations alone are possible# Then still is the same general principle applicable. The mode in which we avail ourselves of this information in tho calculation of the probability of any conceivable issue of event® depends not upon the nature of the evenst whose proba bilities and whose limits of possible connection are given, it masters net whether they are simple or compound# I t is indifferent from what of their connecting relatione have been derived, »• must events as information. And this leads us primeipl# in question, vis.* arily implied, in the data; and the mode is which our knowledge of that The practical importance of the above principle consists in the abilities are given, - whatever tho natal# of the event whose probabi lity is sought, we are always able, by an application of the Cel cuius of Logic, to determine the expression of the latter event as a definite combination of the former events, and definitley to assign the wfcole of the implied relatione connecting the former events with each other. In other words, we earn determine what that combination of the given went® those probability is r e tir e d , and «dmt corabimtions of them are alone possible. It follows then that from the above principle, we can upon those events as if thoy v;oro oirnlo events, who no condi- poeeibility of a upon th is (9,256-f) not true* and the above ‘principle** innocent as i t saay seem, plays m role is falsifying this judgement. V.'o aro non prepared to turn our attention tor. tho actual nothod itself, as prooentod by Boole* In doing so, wo shall csko use of tho laro of lboueht and also of a subsequent publication (13) + The oiiqple events x,y,x will be eaid te be “conditioned" when they aro not free to occur in every possible combination; in other words, when some compound event depending upon them Is precluded from occurring... • Simple unconditioned events are by definition Indepen dent. (slot) Proposition I let. If p,q,r are the respective probabilities of any uncondi tioned simple events x*y,s , the probability of any eoapound event V will be tyl , this function (yl being forced by changing in the function V , the sycbbls x,y,s into p,q,r etc. 2ndly. Under the sense circumstances, the probability that if the event V occur, another event V* will .Iso occur, will be [W]/£f3 whsrin W* denotes the result obtained multiplying together the logical functions V and V* , and changing in the result x,y,*,ete. into p,q,r,#ts. (9,858) Concerning th is proposition no ccrrsont noed bs cado, except to observe that it is true only when "unconditioned cirplo events" is re placed by "indepondont events"j #ien this is done, its truth is evi dent. f. are P,q,r,«*« resneGiively when a certain condition 1 is catiefiodi V being in expression a function of x,y,s,».» . Re quired the absolute probabilities of the events x,y,$,,•», that is, the probabilitios of their respective occurrence independently of the condition V . Let p',q*,r*,... be tho probabilities retired, i.o. the prob abilities of the events x,y,z,... re^rded not only as simile, but as ; independent evonts* *iho by Prop. I tho probabilitioo that thoso events will occur when the condition 7 * represented by the logical equation f 1 , is satisfied, are in which [x7] denotes the result obtained by multiplying . f by x e according to the rules of the Calculus of Logic, and changing im the » result x,y,r,*.. into p',q',r*,... re^eetirely. Hence we have, from which oyetea of stations equal in number to tho quantities the values of these quantities may be determined. (9,261) the events are independent, i.e* x?t/h - p'q'r* , etc. tilth these ties (13), more explicit in its #*# nm**" lee s events to a now sot of logical symbols s,t,etc ale any ability is sought in terms of all the given, and let the result be A 4- OB + ^ C + 3 D . % — » * * «■ v m CD If tho bo a r w l U ) w ill in (II), will Probe w * and i t indicatee the new ejgierienee requisite to eeag>lete the solution of the problem* If the system (I) dees net furnish a oitssle system of positive fractional value* of **t,#te.* the problem is not & real one and does not in its statesont represent a possible experience. <13) Of th is p rin cip le, se much nhould be said. I t is ceme#en@e of 13oclo*o algebra that in any such expansion no tho above for v , tho sum of the ccmstituente, A + B + C + B 1* equal to unity. Ibrthor- more (coo page 7), such an expansion 1* equivalent to w - A f eC , 1=0, where e is an indoibmiaat* elase. Bynerlting 7 = A + B + C , it is thus soon that V = 1 represent* the condition to t^ich the fun ction B - A + cC , vhose prqMbility is sought, is subject. Beale’s conclusion, then, and it is ono consistent with his system, is that Prob. u - (A + cC)/Vh = rhere this latter oxprocsion involves the probabilities */h -p* , t/h - q’ , etc. all obtainablo from the system (I) by an application of Proposition 21 above. its involved to I t of then in terns of Boole*o Calculus of Logic with all its troubling artificialities, ic not a «i in the following section, an exposition of the above procedure uliich St not only ito equivalent (in that it loads to tho ccno results), but a die- cuooicr. of th is ’Rulo* u n til Porotxky*^ paper has been given consider- Prior to that of ItacColl in 1807 (27), of direct and unequivocal censures of Ecolo’c nothed I know only one, that of a certain I'r. v;il= lie atio n of Boole’s book. He eyas (38)« ... The object of this paper le to show that Professor Boole does in leal equations, over tho problem, and to chow how those aacumod conditions cay be algebra- ically expressed* ... the aseueption of the independence of «•* eia^le events •*» la ... tacitly mdo in the logical solutions given in Professor Boole*# Book* In Proposition I of Cluster XVII the events represented by z,y,ete. are by hypetheeie independent. In other nerds, the equations of condition implied by that Independence (in number, 1 if ther be 2 ovonte, 4 if there be 3 events, Z* - 1 - a if there be n events) ef the simple events x,y,ete* The theorem is proved and provable only on this asaui$|>tion* This Proposition is assumed in Prepeeition 11, and will bolons to a certain group of event a reprooontod by V 9 tho chance of x happening ie p $ of y » q # etc; reqiiroi tho ab solute probabilities of whw we have no such previous know ledge* As in the solution in tho book, the conditions of Prop* 1 are assumed, and one of these is th a t x»y»***, are "einple unconditioned evenin'*, which (page 258) ieplioo th at they aro independent* Conno- quently x,ys**» in Prop. II are eumuned to be independent. ... of events the ^olo mnber of possible combinations, in t^ich of the simple events which ie ineeneietent with such a supposition, instead of considering V as a condition which, if it obtain, the #** form method of one than we should ot XVIII, might be mere shortly solved without the logical equations. In '• method determinate or requires further assumptions,- whether, in fact, the not determinable by ordinary algebra, bis system is this; he takes a definitely stated in hie book, but which cay be shewn, as I have done. * problem which was solved by Boole (the eo-eelled Challenge problem) * Bcole’e own solution, differing from the true one, ia tacitly based. his logical calculus and fearful lest mine is false, there meet surely be some case in iMtii the two would lead to different results, and in which, from the comparison of these course I refer only to proMroe of the kind discussed, in my work, vis. events. Should any method, even of lim ited application* be discovered which should lead to solutions satisfying the conditions to which I have referred, and yet different from those furnished by my own method, which io not of lim ited anolication. and which alwavn causes those con ditions to bo satisfied, I should regard it as a very interesting and #% 1 e _ jL gm 2 . P4nretstar’ s /. <- I t ^ * 1 SchrGdcr type) j but mkflQ ko rcfcrnse to tho cork of othor vrritom. £ioao lirht on this JNnointf Wl \7Quld 'Wm v^WWi 1 ikclv vWP ho chod hv Hl^g» cn I^NPa®invcatirition w w#"##" pP"* of hio '^1four othor nanorn In 1284 I published a paper "On tho IZothodo of Solution of Logical Equelitlea", giving there a complete thwry of much equalitiee. Hero I propose to apply this theory to the followign problcn in the Theory of Probability t te ieteraiae the probability of & coa^ound event, deponeing on given simple evente, from the prehabilltiee of all or a number of (arb itrarily oheeen) eisple events, m sell as the prob abilities of certain other compound events, assuming that the given events are subject to an arbitrary number of arbitrary condidiona. in its general form* Thus its solution vith tiie aid of Hathecatlcal Koto: od on an arbitrary and purely empirical theory of logical oqusl ate wy* to ••• I f two or more jo in t F(abc.**) = rCalFCWW*** PU + A.B + A.B.O+...) =■ P(A) + P(A.)P(B)+ P(^)?(B„)P(C) + ... tCP(»)^(6),P(e),...) In the first oxpreoaion f rqprcsonto a totnlity of certain Ion- leal operations on *ialitative eyehole a,b,o,... $ in the second case the same f represent* a totality of algebraic operations on the quantitative syabole P(a)t P(h), P(c), ... . Here is found the first breach in his system. It ic «f f (a>b#...) - 1 =. f q> -t- f „ (p0 verse of ell the evento under discussion. Likewise, it is well known that the simultaneous conditions giv- 1 = c r v +- the la tte r my be condensed to .. . 'y% = . -- iJ(aBbiCeeee) I™# * • ..c-. I4»t p,q#r**»* . . _ je ct to the condition 1 = M s and lute probabilities. M litie s that i f li . _ 1 . « P = » p%q*»r**« * • ibose in 1 U , follese as & coa- AB = 1 is equivalent to A = 1 = B • Of course the Let i t other classes b,c,df... # nith a by tho equalities i f - * , f = ^ 9 " , 1 — &K&,b*o*d$*,.) On tho e - aaci,b,e»d,.s.J =- m i ) & -t-MU) . *.&) X = M(l,b,c,d,»*«) H(0»b,c,d„..) - m ) + M(0) ♦ Hero «(1) io tho rooult of roplaciog; In tho function %(o,b,c,...) the claso a by 1 $ and lie negation *„ by 0 ; MCO) 1# Mie re sult of replasing in %(**&,#*...) the slam# a by 0 and by 1 I Me(0) is the negation of H(Q) « Tho first of trio last throe oquolitioo ohowo that a is contain ed in M(l) , the aeeead shpws that a eeatalaa U0(Q)M{1) . lime the last two equalities say be replaeed by the inequalityee a 6 m(1) , a ^ 11.(0) . 1(1) • Finally, the third equality 1 = M(l) 4- 1(0) , depending on the classes b,c,d,... , but net containing a , represents tho condition to shioh the functions 1(1} and 1,(0) 1(1) determining a are sub ject In view of the In itia l condition# of the problem* ■ M,(0)K1) * #1) the two inequalities above reduce to the equality« a ~ K l) * If i t Is desired to determine a from the equation 1 — iz(&,b,**«) not through a ll, but through 9om» ef the elaeaea b,e,d,*«« § then the classes not used must bs ensluded from the equality 1 - l(a,b,.*$) * For this purpose it will suffiee to replass in this equality all emelu- dod clacBOfl*-m~wm •wr'wm'w* mpy wmmmclq uoXl mm vnmi## nn jpm thmmwmvm o ir nocrn-tlona* mmm&mwmmmm y bv ™unltn* w-mtim WmmmW mw*m Lot 1* wvefflP tbo # vnnuTt bo a can be determined from the equality 1-1* in the same way as it m s deteralm d from 1 - M • Here a few remarks are pertinent. In the first place, that 1 = 1(a) implies 1 «• 1(1) 4- 1(0) was given by Poire# in 1867 (29). farther, Poretsky's lower lim it for a can be improved cn, namely by using 1.(0) instead of Mo(0)M(l) * This ie a conooqicnco of tho theorem 1 - U(l)a + M(0)ae implies Me(o) c a C 1(1) , which shows also that his upper lim it, 1(1) , ie as it should bs. As will appear shortly, this ohango in tha.lonor linlt.of a loads.to no actual al te r olaseos from 1 = s j were aeeiroa t-o eiim naie d irs « s a into normal fora, obtaining . .1 = M(a#l >a,*.*)b+K(o,0,c . 0»t this l e M tA-ji, c,, •••) > ft ( a . o , c ^ ) , #loh is seen to be obtainable from the normal fora by writing b = 1 and be = 1 I t can be In fact, let there be give* n simple elaeeee a,b,e,*.$ net connected by any equations, and a confound class A denoting a defin ite function of theee elweee. Let a = w , ot what is the eaae, & - a* + A.*, . Then #e have a 4-1 elaeeee« v,a,b,c,... ebieh 1 -=• Aw -► 4>w. - From this condition the elmple class w (i.e. the function ^ ) ») in te no condition to one subject to a definite condition. If simultaneously with a set of a Independent simple classes a,b,o,... , we consider m functions of these classes u,v,w,.«. , then, letting W » » , V V , * - W , ... the problem reduces to one in n + a simple elaaseet a,b,c,...,w,v,w, ... , subject to the conditions 1 - (uO + H.y»)(vfv»¥e)..s - li(a,b,c,...,u,T,wt...) , and from th is any of the class## u ,v ,w ,... can be logically dotor- md-ned with the aid of the remaining classes, am shown above, i .e . any of the functions u ,v ,w ,... can be expressed in terms of all or acme of the given simple classes a,b,e,... and all er seme of the functions. Finally, if n aicplo cloncco a,b,c,... arc dopondcnt, through g, eonditiom* "*•*»•, .... r , .... where sure funotioas of «»%,«,•*• then to detemirto one ef a eet of , ftinetiene «,▼,«»••• ia to oolvo a problcn involv ing n + a aiople olaseees a,bfe*...,u,v,w,... euhjeet to p +- a eeaditiene ’ ^ ^ - •<*", ; . r ■, . * , A* = B* , A** — B* , **#,« B , ▼ = ?§*•» redueing to a single condition 1 = ( A 'B 'f A ^ ) . . . ( a y -t- u M (tf vefe}..» 1 — M(a,h,e, •• • ,a,v,w, •••) This reduces to tho proviouoly diBCueood caoo, except that all the ini tial conditions are accounted for* .7o now liavo all tho data for tho solution of tho general problcn of determining the probability of one function (or a cocpound event) by mean# of the probabilities of all or eoao of the rain in g functions .. "ocuno th a t, ao before, wo have arrived a t tho equality 1 — h(atb,o,***u,v,w,.**) a,b,e,****v,#***# e ■ - H(l) . « ^ X.(O)MU) —ere «(1) aad B(0) ere the reeulte ef nplulag is the Hue ties M the claea u by 1 and 0 respectively (and its nsgatlsa by # % * 11(1) +- K(0) = K . It remains to pass to the deteraiaation ef the probability of tu Let the probabilities of the classes a,b,s,•••▼,«,*•• (found in accor dance with the initial conditions ef the problem, and consequently sub ject to the same condition 1 = K) be p»q*r,*.*,*,^,..* . In such ? 'Vi »— - 1i ....“ W f - % £:?::r’iru:&;™ =^= sjs: itis s «= M(l) , u^lW'OjUU) giving us nil) 024 %(0)M(1) » 1,©, ction u • i ^. 1 K . t m and (0;H(1)K^ n tlm i a*b;a***#v*w#*** probabilities Hat K = Ii(l) + u(Q) » and 2i(l)K = ,M a )[m ) +- M(0)] - H(l) %.(0)%(1)K =- %.(0)K(1)[%(1) k M(0)] = Mo(0)M(l) • l!(l) and CMiD ~ k>(o)u(ii3 *** m U this is so, •ties u by P(u) , we get P(u,.tsLuL . Hu)5L tism i I .# a,b,e,*#.,v*w»... were replaeed by p*#% # r \ f ***# wbatitution the latter symbol# were * expressed in term# of P»S*r,.**,ot*/3*..« P — * % - ^T rT *****...... <<'” # » + * But i f in (1) emd (2) are replaced by p**q*, out this tiiaination. It is euffisieat to reinterpret in (1) and (2) the symbols a,b,e,.and trea t them as i f they were symbol# # quantity and could be treated in ascendance with the roles of Algebra. %us the fin al form of the solution of the problem of finding P(u) is as fcOlowes by means oft . where, after all polynomials ars put into disjunct form, tho symbols a,b,@* are treated algebraically, those latter symbols are tiiminatod from; likerico in disjunct form* r buch ic tho general method of solution of tho problem, formulated at the betinning of the paper. In general, we obtain only the bounds for the probability P(u) , and only in the ease when M,(0)K{1) = M(l) - i U l e Here, then, is the complete theory of Boole’s method, more aeess- ibis to critical inspection than is the fora presented by Boole him self. so shall reserve it for section 4 to examine the theory as to validity* being content here with the following remarks. The trans formation from the logical inequality a c u c b to the numerical in equality k/h 6 u/h - B/k can he justified in this manner. A C u is equivalent to A = Au , hence A/h = Au/h - A/uh • u/k - u/k since A/ub - 1 $ in the same way u/h B/h • As was previously remarked, the lower limit for u mould, strictly speaking, by M#(0) instead of %(0)M(1) , but it makes no real differ#** hero cine* it is not Halt Eultipilea by & r mil +■ moi . would be UjO)M(l) f and th is ie the mma m M.(0)M(1)K . ^ ' V - : " ; . .. - certain eii^le problewi. » - * r ^ . % , : solutions to nine eiaple probleas# In this section we tiisll solve five using, however, Foret sky* s etatewent of it rather than Boole's* The V s 1 Poretsky*# probability, i*e* to on inocjiolU » 1 , end Kciw this is . be proven. A- =-ho(0)K ^ M.(0}M(1) i C =zh(l)K =M(1| e ility of c being w/oh # as Boole’s , th at th is is so *.(e) C w C M(l) , but this rslstion out be shown to imply w = M.(O) + #*(1) nhere « is an arbitrary class# Renee Prcb. w = , but l£e(0)K ^/l and H(1)K = 0 » so M,(0)K + c^(l)K — A. •+ eC t so Pw b#1B - , tdiioh is ju st Boole’s solution, since Port sky’s & is the saso as 'a V . of the ler of application* lead into error, to sake & to the «$ oral nothod*, by a Swffij&Q 1. (Boole, p* 275) Data* x/h - p , xy/h « q « Qu&ositum* y/h • 1. First Solution. r Let sy» u . Then 1= axy -t- i*e(a6.-i-y.)•=- S M(l)= ux + U oX . l K(0) = «. | H o(0) = » I 1^(0)M <1) % ux , mid K - H{l) + H(0) =• ux + a.x0+ u. «• a* + tt.it *+ u0x. * Honco ° ~ 1 “a4 X lC - UX"*- U»x = X , CO p =. - Hones lux.t«.W = _ , + i - p , IK] IH LK3 eo that q ^ JA ^ 1 * P % • II. Second Solution. y/h ~ yx/h 4- yx A = 7*/*+ y/% h . x A = % + e(l - p) # Ihia agrees with Boole*e solution, and alee eoineMes with the aboves o being and arbitrary class aseumee the lim iting value# of © and 1 , thus giving y/h the limiting value# q and q4* 1 - p . Tho propriety of this method of obtaining tho limiting valueo of yA eight reasonably bp questioned. In tho prooont caoo it happens that these limits are oorreot, a* can be shown by other methods, but in general we will more likely than not be kn error by merely assigning limiting values 0 and 1 to the arbitrary classes appearing in the solution, the general problem of establishing the closest limits of the qinoaitua in toras of the data of the problem He of It is valid I tm it closoly enough* but I # and in a Section 5 9 on 2. - 2 ', Data* (s y)/h = p , (x»-t- y»)/h. = q * (xy»> x»y)A * X First Solutions Lot x+y-u $ *•+ y. * t j xy0 ^ x.y = w . 1 ^ B(w) ^ ww(w»+ x®lf> "+ u„w„xeye-i- uv.*f0xy • laicinato x and y » 1 ~M*(w) = u t» + udw e uvww0 » «» M»(l) - uv , H*{0) - UV04- UgV , U*(0) — UV + U.Y0 » K. -= M*(1) -t- ii'(O) - U + V e M%0)K = , H'(i)K = uv . Honco (xy»-*- x0y)/^ ^ . But M ± i $ 2 . H us (ay0 + x»y)A * P 4- q * 1 I I . Second Solution. P 4- q - xy/tx-t- xy0/h4- x.y/k + x»yoA + 3Qr«A + - 1 4- (xy04- XoFJA so that (x/o-t- %»y)/h - p + q - 1 * : f Data, y/h^q, ■ ' - ■ - " ' ■■■ - ...... ■ * . - ; Quaositaffl, xy/h ♦ I, iirct SoltttlSBe Let xy. +- x.y - u $ xy- ▼ • %en 1 - K(t) - u.ray + ar^Cxr* + a^y) +- a .% x 0yo » ^U ) = u0agr * 2Ko> - + x»y) + u»x^r» * H.(O) = u(xy 4- at,/,,) 4- « .(* + /) t K - M(l) + M(0) tt(xy0 +- %=y) + .Kisz + Klo) * ^(1)K ^ ttaasy , #»(0)K ^ «.xy » XK - tucy, 4- i^xy » yK = uxof 4- u0sy , uK = u(xye 4- x$y) . Tims p =. ^ gr<>£:gjWaiqr- * ^ ,g , co that » 4-q - r = .SJk|g^ , xyA = 1L+J.SJE „ II. tselution* p + q - 2 * x/A + *y»A + **tA = 2 • *arA +*•*■* - r v A ^ # Sxneple 4. (Boole p. 281, PereteJqr Sx, 3) . Data* s A = P • 3rA e t » W . # , A * f^u&esituias #A . r /. - -j- I. First Solution* Let Xoy0Xo •= a , then 1 - 8(s) - ux»yes„ 4- u*(x 4-y + s) , H(l) - u0 , 1(0) = «x*y04. ue(x 4- y) , MJO) - «(*-*-y)* *<,*#* # m «(x + y) + UaXaj’0 C s C «» 1 —K - aa^y.-t- e, e H(1)K = *, «#(0)K - UoXa/o t xK - a^x , yK^Lttjr » «K = x x ^ , , p = e , r^IaMsl , 6^, &W . Nm 1 - (p + r) = UI c I m ] r I g r y ? = I l t f i ^ = w . Ic to Cbtai,. [«.*.*.] - L«.)L«.lLr.l . n»n LmyW » 1«?]LWW, » IL q » r) 1 ” r .tt-- j -c.jr]a C.ii - ri. t 6 1 _ , . U* Second Solution. * ' C*+-f-^*}A + *»y.*.A-1 * *» l»T-p+% + t/U ~ 9 inhere e - aqr/H +• jt/h + %x/h - xy*A • Beede s /h - 1 - (p 4- q + r ) 4- ■ # I have not yet found the U nite « t i / b 9 eltheugh I believe them te be 0 and 1 - r • leede to error. I diall reveel the aaeiaptlon vhioh is taoitly invol ved in Boole's solution. He hem obtained (p. 299) t/h - i l , 4- g(p 4- q - —29w) shore e - t / ( x + y)b • How e /( x 4- y)h a. s/h - 1 - (p+q + r)+ -s-l - (p + q + r) 4- xy/b 4- e(p + q - jy/b) * But %„!)„ = 1 - (p 4. q-vr) + >-6SL , so it is x j / k = I Exraplo 5. (Boole p. 284, Pelree (S3) Sau 1) ' 'V, i Data* V l* ~ P t j /A = 4 • « a ^ • ..'v-J .v % " I# First Solution# . . • Let ay - u , y% - v , ex «-w • tk m 1 « M(er) * uwaqrs + #0v0W<,* +• u»wexjr* +• ttv#vo3qrs0 + m0v.we (*jr* 4- y#**+ *.*e) t m a im tln g x i 1 - M*(tr> - uvny* 4- *.%*&* + «bV%y* + Wo*#?*® + ttoT„w»(y0 4- *,) I M*(l) = wry* + a»v.ye* , %"(0) - a.irys + wr0f * 4- a.Vo(y* 4- *„) $ lie#(®) =L u(v + y« 4- s) 4- m.[v(y. 4* s.) 4- %y*] # 1 ■= K — M* (1) 4- M*(0) •=■ uyCve *■ t „*0) + ae[vy* 4- ▼„ (ye 4- y„s 4- e*) # M!(1)K e tnrys 4- u.v.y.e , M*(0)K - my* , yK = y(vs 4 ▼,*0) , uK - ny(v* + Vo*,) , ^ - m f * 4- «0yy* = vy* # tu A sK = s(my 4* ' ' - ...... 4 ^ -'' """ ' , f 4- tt«ey 4- a,Toy.). Now write e - y/k , e* =- t/k # Hen biKj-LuyK}, LuyKl - Lul[yK} A:: and in addition, [uvy*3 - [ul[vy*l # mca = . , a » , e'pq ^ W* ~ # M 4- e* - *'* Aiob are Boole’s limit*. .1- SI# Second Uethod# 1 - Xvjoto/h 4. v ^ > y A 4-*A • aqrA * y*A - *V^ + xy^A * *301 = 30* . (1 • p) + wfh » (1 « t) * 301 + 3^r<,s»A + «yiA - 1 * I have not yet determined the lim it, for tx/k / hut I have . ... ■ ...... , , ... ■ ■ ■., ■ shorn that Boole's solution, «30i - */h . pq + */h • */y,d& . Cl • q) , : ■ ■ involves the taoit assumption s/yh .=-■ z/^fh , as the reader can V' ": For other ; :v i -G9- h(xs proviouoly boon pointed out tho dfo3jLacioo of tho ^gonorul m n t. to bo indopondactj th. oothod I otoI tm onuopU.. w » 1^. That thi. is «, 1. brought out 1 — lr b , t h . N H « 1 W «n.ot yield a oolution Moopt by a « a b « r o f -w ^tU n. of th. t y p . [ « ] - L u 3 M , and thos. a.Mmption., a. 1 . to b . « P « = t.d . l « d t o an erroneous solution. Hot;, this solution, in tho coco of czacplo 4 the typo xy/h - %/h . y/h , fo r, as w have tiiowa, Boole*s eolution aro thus led to put our finger upon aocucptions of the t>-po £uvj WB8 # no tlio primiplo broach in tho’gcnorolcothod', and this Judgeciont is ouppotted by the evidence of eameplee 1,2, and 3. In these oxacplco, as tho reader trill note upon inspection, tho 'genoral netted* u ill lead to c. solution without any ancucptiono of tho indica- ted typo cuid clo it turno out thoso solutions urc tho correct onco This illustrates too points* first, the fast that the ‘general ■ '■ ■ ;...... saa. Under certain cinuwtancw, Imd U .« r« t rw lt. would seem to indicate that there la poeoibly socothinc that is inher ently sound in tho cothodj and second, if any revision of the csthod ie attempted, it met include an elimination of all ussusptieas ef the gonoral typo above indicated. Of course, front our knorlodgo of tho laws of probability, wo could nako tho la tte r judgeDont without rover- A z 4q ^jhw wa wwmgf GUOrts jf W m w Wof # QBut^ W v 4 W * w m*m mwmi ft yt\ Vf&TTig # ^ Wlm0 V m f A ^ V ^ Wmma* Let ub teat it will lead to a solution* ineeo aa pooGlblo, while yet retaining tho core of the method* Dat&* 1) Bwents ' 2) Condition 1 - F{S) 3) Functions tg(S) aM S(K) 4) Probabilities ^ Tg(E)/SF(B)h SmzdtuB* Probability e = s(E)/y(s)b poooiblo vhich can cuboiot botweon thoea events, for any ono condition the single relation 1. - Q ^ f k(3)f;(S) -h f k (i)* Let Tg = Tg(E) , S =.S(B) , so that oqtti*mlontly with l^F (E ), 1 = n { T 3Tg(S) + T gTg(S)}{s3(L) + BalS) ^F (S ) . From relation amo%% the T’s and 5 , namely 1 = 1(1,T)8 -t M(0»T)5 = B(S,T) * From this we obtain directly 1 =*(1,T) + - K(T) , *’-71*- Ke,T) CSC 2!{1 ,T) , S = M(0 ,T) -v , etewe la the last, » roprocents an arbitrary claao. Thio la s t relation io an sltom n tivo fom of tho preceding one* and doponding upon nhich form t?o ugo, do no SchrBdor algebra and uo shall >rccor.tly turn our attention to it (coo oootlon 5 )| suffice to cay here that tho cotkoda involved arc rigor ous and the results obtained valid. «. » and by virtue of the coxsetratni 1 - F(S), as a nee set of events vkieh* when 1 - K(T), have probabilities t 3 = T3/K(T)h. (ill) • 'wo now rcaeon ao follona: consider 1 - K(T> ao tho relation under vhioh 8 - H(0 ,T) +• uM(l,T) is valid* and bear in Mad that the constraint 1 =- F(E) la Included in tho condition 1 = K(T) . Then e - 5/Fh ^ ^ll(0 ,T) + uM(l*T) } /M h ~ the arbitrary class u the Uniting values 0 and 1 • Lot uo take etoek of shat has been attained. Writing S(1 ,T) = 0 (f)* y(0 *T)Lf(l,T) =^(T) * TaK(T) Sg(T) , we have tho equations * » ^ U Z L ± ^ 1 A . what we seek is to eapreea # in term of the . Hew ia this aeeenpliehod by tlio ’General csethod’ ? Siaply (taking all functions in diojunct fora) by writing *J' ^.(T)A ” ^(TA) * * «- = , f l t t i A ^MTA).±.n0i.,O(TA.l . I t ehould be clear if if is possible to solve ;f of this typo. X dcubt. le t us k l» s&ich is its final fora is 1 ^ K(T)» But by chat token arc ro Just ified in writing tg =' Ta/K(T)h ? In this ormoction it would bo coll m t y It will to rocollcd that the oonco in which toynoa uccd the 1 bol (see Chapter II, cooties 3), tlio loiters a and h stand to » I » I , • With the term (9,177)* of approprlAtias the cycbols z,y,= ruthe of propesltieos, we eight with ply thm to represent the oeearrenee of event#. In fact, 1 •nee of an event both layllee and 1# Implied by, tiie truth eltlon, vis. of the proposition *ieh asserts 1 •rent. The one signification ©f the symbol x the other.*. • Let ua w rite & end B fi>r 1= A and 1 -B them to bo the propositions mA oeeure" and "B occurs”. I t from th is th at the proposition 0 -A w ill be *a fa ils " , by the rules of logic It is the asms as 1= A "A fails"; likewise for 0—3. with that the event A by th (1 —A)/(l —B)h # A and B oified" or is a of h A=-l -B ie to 1=AB , it - 4 f # {U= A) + (1 »B)}A - (l^A)A + (1= 8)A - (1 —AB)/h • i t w ill =74- W obocrvcd that thoir forn iu not oltorod rhtan tko propositions 1-A and 1= B are replaced #imply by A and 3 ♦ Appreciating that thooo tvro la%c arc the cord of the whole probability calculus, it now appears tlonc, ic identical with tho.calculus of probabilities in which theso propositions arc inierproiod as the events which they acscrt. It ic in th is scnco that the nynbolo A/n and B/li are to bo understood in the present eea$#%t* How let us return to tho qi Cation which is under inspootiom by s&a-t token are we justified in writing t, = jh t Tho probabi lity tg was given is the data ae I^(S)/r<£)h . If wo regard T - tAM) as a proposition, then it is porhapo not unreasonable to write t ^ - Ta/{ la ^ T3(S)]F(S)h $ but why is tho proposition T ,= T^(E) (3*4 j ) relevant to the probability of Tg t th is is oocentially vfeat is involved in saying that has the probability in virtue of the relation 1 - K(T) , and ny efforts to clarify this point have been eminently in vain. that involved in step (ill). The relations 1 =• K(T) and 3 - igO,l) -t- 4- uM(l,T) both follow from 1 — M(ltT)3 ■+ H(OtT)S , but is tho first of them necessary to the validity ef the eeooa) T It sees* to me un likely that it is, for the reason that the relation of a function l 1 Ax -HE to its oliainant 1 = A -*■ B is a maoy-to-one relation, whereas x — A + uE is onuivalmt to 1 - Ax + Dx ; this moans noroly that 1 ~ A + B does not necessarily imply x = A uB • -7 5 — !?ot? I do not discount the possibility that elope (ii) and (ill) in ?, i t caiot never bo contingont upon the validity of a logical argu- sent. Blue, In any e«iM recasting of stpe (11) and (111)$ tho princi ple that a - b iaplieo n/k - b/h met of a surety be Obeyed. In summing up the writer* e attitude toward the fertility of the •general taothod*, it is perhaps not out of place to include a final . Illustration, introduced by Boole, tie eyas (9,262)t * part ocular colour, "white*, or of a particular conp coition, "marble those eased inz.uhich a ball that was neither white nor of garble me drawn. Let i t then have been found, that whenever the supposed condi tion was satisfied, there was a probability p that a white ball would be drawn, and a probability q th at a aarblo ball wtmld be drawn: and from these data alone let it be required to find the probability that in tho next drawing, without reference at all to the condition above mentioned, a white ball will be drawn; also the probability that a mar ble ball w ill be drawn. Here if x represent the drawing of a white ball, y that of a marble ball, the condition V will be represented by tho logical func tion *y + x (l - y) + (1 . x)y . Hence wo have x7 — xy + xCl * y) - x , fT - xy 4- (1 x)y = y i [xV] - P . Lyy3 = a » and the final oquatione of the problem are ______P* _ ' jV ______- t p*q* 4-p*(1 - q*) 4- q*(1 - p*) * p'q'-t- p*(l q'y+q'd - P'j from which wo find p 4- q — 1 p + 4 - 1 # q 'e -* p ~ s P In term of the eyiaboliea of 1 that Eoolo*o V is our F(U) , m i *iA » q* = yA • correct solution to this nlrplo prohlcc; is obtained ao follows: »' - ^ - x(x-t- y)A - VCx + y)k . (%4-y)A -P " (x-+ y)A • But (x y) A - -A +yA - %yA ~p* -t- q* - d , shore d = aqrA » eo that p* = p(p* 4- P»^ — 5e_ . , q* z, — 53—- (4 =- ayA) p - 1 p-i- q - 1 os tho solution. Boole*e ressuit involves only the assumption *y/h =30i . y/h , es can easily bo verified by writing this into the, true solution. It is nn interesting coraentary on Boole's way of, thinking, that he re~ nerko (9,262)* fo moot a poosiblo objection, not should bo solution, but it colour, structure, etc. It is our i I t is statementa in and then on the heals of to that norar.t of "those things" (33) says is"at a loss to understand" thie ted to this question. We lorn is author l i f t s the problem #* Hove that the sol*tle* not to bo taken to task ibr leading the reader to place confidcnco in the efficacy of Boole's method ? For ho says (25,71)« With the d etail of th is method, and with the theoretical d iffi culties of its application and interpretation, we need not her® concern oureolves. Suffice it to say that, with certain modifications, it is m entirely workable method and scone to possess certain narked advan tages over those note generally in use* It is a matter of surprise that this immediately useful application of symbolic logic has been so generally overlooked. (25,212). Ac Boole correctly claimed, the most powerful application of th is algebra ic to problems of probability. With due respect, I suet confess that I can find no justification for Professor Lewis's statement that nit is an entirely workable method*• To my own mind, at least, thle has never bean doaonotratodj and if this paper has accomplished nothing eloo, it oocms to no that it euroly indicatco that "the thro rotleal difficultico" of the "application and interpretation" of the 'general method' are not sc definitely resolvable as Professor Lewis would have us believe. Whether the'generaleethod* will admit of a sufficiently workable rejuvenation, and whether it will provide a solution of tho general typo which Boole imagined ho had given, I do not know and so perhaps should net judge. But although I have no proof of it, my own belief io that tho method leans too heavily upon the side of symbolic log ic, ani that in order to mako any real progrooc in tho direction of n gennral solution, insofar as such is attainable, emphasis exist be placed :. : upon the noro ary algebra* Hits view will '.ST section of the preeeai 5, The Disjunetiee men AB = 0 » cum % A,. Is A ., Ac (i+i») then the SB! * Let us not Idea of total disjunction. Consider a sot of c clacaes, and denote by H(l) sot consisting; of the vT 1- 0(2. + !.) | lot uo oall 2(1) the nrinainle act of the set {2^ . . ■ . , ■ * ■■■ - sot has several in^ortaat properties. To study them, it will bo lent to set up a one-to-one correepondanee between the el principle set and the first 2 element of the principle set by # # k is an bdongii^ to the (1,2*3,...*2'*). ..l") .T.-f In the first place, it is A.k , A k' (k+k») of elements of 2(# elecumis will be a involving only the si as a sun of 2(1) The truth of this latter 8tat©cent can be aade evident as fellows* The most general form which i t is possible fo r i(E) to assume is the sum of terms which are products of the elements £*. or their negations* How if any of these elements or its negation does net appear in one of these terms of F(E) , it can bo introduced into it by the Law of Ex pansion, 1 - Ev-i- B,, , so that by a continued application of this pro cess, we arrive a t F(S) in a form which is the sum of terms each of which is tho product of n factors Sw or (1-1 to n); but any term of this type is of necessity an eleemt of 2(1) , as can be seen by inspecting the universe expansien, and this is the desired result. Shea a function is so expanded, we shall say that i t is in totally disjunct fom* %'* are now in position to introduce am important symbol, similar la i t s properties to the Kroneoker delta in th at i t assumes only the values 0 or 1 * In view of the fact that, as we have just seen, any function F(2) of the elements of the set alone, can bo oxproo- it is apparent that any such function earn be represented as a to ta l sum Z A* taken ever a l l the elements A k of 2(1) (2 M in number), provided the coefficients to* use of 1 or 0 ^oerdlng as the term /\* the expansion of F(2) • Sfe have then a convenient and compact sunbol- Im to represent any function of the given type in totally disjunct form, and if we introduce the operator & which when applied to any function F(S) will bring it into totally disjunct form, our results consider the product of two functions F(£) and 0(E) • In view of the formal oquivalencoo F(E) = I F(S) and 0(2) = F(E)G(2) = Z Ak / In the same manner, since A»« + A k ~ A k *• 2(2) + 0(2) = 2 W ^A k F- o F + <- ehere CVk + 6o,t has been abbreviated to W* . Concerning tho operator <££-<£» it io apparent that $ [f (2)0(E)] - [S F(£)] [A 0(2)] . Also, <5 [r(E) -h 0(2)3 - & F(2) + (T 0(2) . Finally, by definition of A k • have 1 - J (1) ■= 2 Ak • la me preceding section wo have boon concerned only with functions involving elements of the set -[s ^ and no othere, and have defined our oumboliom in terms of this typo of function. It will bo apprecia- of a mucker of sets {sv^ « f {b ^ > etc. * In particular, we cay epmk of ouch a general function as being - > ' ' - - • when i t i s given the form Z Fk(A ,B ,.e.) A k _ ^ - .. where the Ak are again elemente of the principle set 2(1) and the > ■ ’ sum extends overvthe 2 members of this set. In tho m m way, the operator j tiiich is such that In order to exemplify tho results of tho preceding parographo, we shall apply them to the solution of a problem which has been termed ono of the most general in the algebra of logic (9,140), (25,162). This is the problem of step (1) of the 'general method* of the preceding section, and i t s solution is p rio r to the application of that method| i t is the problem which occupies the attention of Foretsky in the first part of hie paper (31). Data* 1) Tho set {st^ of dasees S» • 2) The set of functions of those claoooo. 3) An additional function S(E) • Cuaooltum* 1) The relation of the T^(E) among themeelvee. S) Tho moot general relation afforded by tho data, of 5(2) to the Ta(S) . In our solution to this problem, we shall use tho operator $ as relative (see above) to the set {s<^ , and shell represent by A k the elements of the principle set 2(1) . In addition, wo shall use A + B = A3 + A B . Finally, wo ohall employ tho Conoral!tod theorem of inclination, carcely* the reraltant over {e j of 1 - F(£eA,B,...) = — $ F(E^,B,..*) = Fk(A;B**..)/\% is given by 1 -2 • lb© only proof of tlxie latter theorem known to no is that given by lew is (25,153-5), based on an Induction. With these facts in mind, let us proceed as follows, let Tj = T^E) . Then for a ll j i — T^(E) = 1 = T^(E) 1= H T^(E)] = P(S,T) . Eon Tj(S) = J T^ E) = 2cu5 Ak so ^P(E,T) = JO [T, + T (2)> g Ta(B)] . But ^ LTg + Ta HenCe for its resultant over {e ^ • I"" is quaeaitum 1 ). . « To obtain quaesitua 2), write S =: S(2) . then 1 = S 5 S(£) , and since 1 = P(E,T) , we have the e#ivalent 1 = P (B ,T )[s ^ 8(2)] , 1 ^ [P{E,T)S(E)}S + [P(E,T)S(1)]S . we desire now the resultant ever {s^ of this equation. To obtain it, wo have SF(E,T) = £-Qi< A* , so that 5 [P(2,T)3(3)] - a [ip(*,T)] [Js(*)l - [2 ilkAk][2 wlAk] = fi2 K^ Ak , and in tho same way, $ [p (E,T)s(e ) ] = 2 jf2k COk Ak • Horce I = 2 i 2h w t SA k + 2 i 2K S A k for ohich tbs resultant over \ e is obvieuslf ,= ( ^ n Kool)S+ (zJ2*c3l)s This is thoequation we need, its solution for S being 2 1 2 ~ t i t C 5 C K k The lower licit can bo sisplifiedi z a * w i = n l i M = m that the final remit takes tiie fora s c z j X wk whore 1 2 * = f | ( l j + w j ) and _f2K = ('^.4- £o5) . mechanical procedure in finding the desired eolation or resultant of a - ' Ta £ given equation, i/e have only to rake a table of the 6J* and of CJ* and apply the terra of this table to the above formulas directly. With this conowhat extended treatcont of tho idea of a totally diejunoi function, we turn mir attention now onee mere to the •gonorol1 probloa in prebabilitiee,. The preblea will be stated in a fora which io nrobablv sonev/hat noro ronoral than that ccnnidorod oithor bv Boolo or by Porotzky, and we riaall show how its solution can be made to de pend upon the laws of ordinary algebra, rather than upon those of the Booie-Schxtider Alegebra. 8) Ikaetiona Tg(E) , T*(2) , S(E) , aid S'(E) » 3) Probabilities t g « Tg(E)/T* (E)b . O m n iu m Probability o - • ; Conaaming this otatement of the problem, a £&a remarks are in order. In the first place, by writing t*(S) - S*(£) - i'(E) , the statnont reduces to that given on page 70, and banco is coon to bo core general than this latter, further, the given probabilities tg aare ##t4#atly the most general which it is possible to provide, at least in teras of tho given universe of events Eu , By writing ?’(£) id en ti- saa&l 1 for any particular j , the corresponding reduces to the more common fora Tg(E)/h , and this applies also to e , the Kow lot Afc represent the dements of the priraiple set 1(1). Then, in accordance with the previous notation, for any function F(E), 8(1) = ^r(S) = 5 coj Ak • Recalling that the Ak are essentially compound events, le t us write fik /h - XK , so that using the thooroa of total probability and bearing In mind that A* is a disjunct sum, we have ( 5 U)k Ak)J~ IE OJtc Xk . Thus F(l) = ^ Ak for any function 8(1)* I t is evident that the gi\ probabilities say bo written* n r / A, S = Xh and that sought. _ ss* i ^ CV/c Ait a * # ~ z 7 Z T * Writing u > y i C^AU,l'-u>? oystan of equations s c }k xK = o ^ Ck Xfc - o from nhieh, together with 0 - Xk ^ 1 and 2. = 1 , the value of a is to be determined whether exactly or within lim its, in terns ef the tp . This is the reduction which we set about to effect. Of course, and i t need hardly be remarked, th is procedure docs net in any way ob viate the solution of the problem, it merely recasts it into a fora which may perhaps prove to be more fru itfu l as a starting point for a solution, i f such to any measure ex ists. In concluding this section on the disjunctive method, it ie per- Kote on oarly uses Ih® earliest explicit four contingoncie® AB, AB, AB, A I be 0 , X SyA-, To determine these we have the equations - 9 + \ + q> - i > s + j j l =l a, , 0-t-X = & Another equation is given in Professor Boole*s aseacption th at & Is m liktily to happed i f B happen as i t is i f B f a il, v iz. 6NA = A+ ™ "9 Ihe same elation is given by the condition that B is as likely to happen if A happen as i t is i f i t f a il. These four ematione deter mine the values of (33) wilbrahan eonstrueted this example for the purpose of criticising Boole’s contention th at the data x/h - P i JfA * q imply xy/k = pq 9 but Boole in his reply to Wilbznhaa (10), (11) seems to have missed the import of this sisplo argument. Be that as it aey, it is intended here only to point out that Wilbrsham put to good uoo this typo of ana ly sis in laying open Boole’s errors. Boole, too, made use of the idea of total disjunction, althougi not quite in the sense of the above. Be applied the method to tho question of determining what he called the "conditions of possible e%- nciricnco". Tho meaning of this nhm.se in best oxnlninod in Boolo • n own words; th is i s tho opening sentence of Chapter XIX of tho laws of entitled "Of Statistical Gesditioas"; By the tom :t tho i 1 ■V subject of probability (< a gonorol rule for tho "< # into the mtuelly oxcluaire altermtione Which they involve... . Rq> resent the probabilities of these alternations by A , , V # etc.* and express the probabilities given or sought by the correspon ding m m of those quantities, this will furnish a series of equations which we w ill suppose to be a in am ber. Dotenaino from these equations any of the a quantities A » A» *V § •te.* in terms of the others. / Substitute A ^ o , o , v ± o t e t c . (1) A +* A" t V 4- • • • = 1 |g | Eliminate in succession each of the quantities A * u. t V 9 etc. as are left in the above inequations after eubeiituUen. The elimination of any quantity such as t from the inequations is effected by reducing each inequation to the form ^ ^ or to the form t a G , and ohaerviag th a t two such forms as toe above give a i b . If the "alternations* ... extend to all possible eestbinationo of to# eicple events* the inequation (2) must replaced by the equation X+ > + v -t- . •. = 1 . The rest of toe process will be the ease ^ (13) Tho siiailarity of needs the critical office is lik e ly to that the theory ehieti X * l c h tho as a scientific Bat I have - •. , ' ^ f - ' . .T- :- --: t 1 Boole*a Method I ■ ' ■ . ., logic. lh@ Galeulo* of Logie, or Symbolic Logic as it is connonly cal led, had its origin in the xrork of George Boole, who in 1654 published cpiration to this work. bilitioo (intho classical sense), at least insofar as it concerned the happening of events, ms, aside frost logic itself, a field which bore the most intimate relation to the results and methods of the new logical calculus. That this contention van fundamentally Bound and of pregnant import we are well able to apprehend today in the light of modern which are elements of the ring of xml umbers mapped on a certain ab stract fora of Boole* s calculus, the Boolean ring, and subject to a trell-chooon defining condition. (30) Tot, although Boole’s theless rigorous of /V/ ^ " r--' --:;%*** '^: been shod on the a b ilitie s . Although i t Is not the intent of the preoont paper to carry the icplication that, with regard to probabilitioo, an% logical approach w typified by the •general method* of Boole met be diocardod in ito ovory aspect, tho view is expressed that tho procedure of the •general cothod* in valuoloeo, and that the qiootion nust ooriously bo enter tained of whether a problem of so broad a scope as proposed by Boole can bo brought under tbo domain of a general r.othod of tho typo which ho Finally, it would soon that in tho calculus of probabilities* tho in a determinate way, they rust at prooent be restricted by whatever limitations are inherent a* A Koto on Symbolic KultipUcation. In Chapter III, section 5, it ©as shorn hou any function F(A) • f the elements of the set { aJJ could be given the tram ^ cuk A k # as a conooquonco of tho Inn of Expansion, A = AD + A5 . In tho pro of the La© ofi Expansion is p(A) - p(AB) + p(AB), or p(AB) = pU ) • p(AS). With a little reflection it will now be appreciated that any tern p{/U) nay bo reduced to a linear expression In terns of tho type P(Aji>eeAx), following device* A a eorical quontitios, writ© (1 - A) for each A that appears and ox- a linear expression in terns of tho typo , with numerical coefficients. To each of those term prefix tho symbol p( ), retain ing the coefficients, and then restore the A. to thoir logical mean ing, thus obtaining the desired reduction. Tho (14) in # e n title d " from tho conoid, Mono of tho ThoSEELl* ^ C + Zc(ot...X)^(Aa... Aa) shore tho field of summation extends over all combinations **'*.£ of the integers 1 , and e is 1 or 0 according as 1*...AX belongs to ry or not. -9 2 - Conconiins tho notation, hio io cioroly any function such as F(A} e and io tho sot xhooo dlozcnts aro tho ooparato tores of Tim oro2_ 2. If (K^) = 2 C. 4 a) and I>(Gq) = 6 E i/H E S9b^E isB ?E 5SE iESEt. plied” and then tho probability token. Fot* tlic nroof Af 4 1 aiir ■f.1, anrtiri, tHn vr-n/l at* 4 #:#„##»%#fomuTzi ####%#*%#& withoutww m* WmM>W W ####y ansus^in^ ##%%mw# # # ##%#»####indG^ondn^oo* *%# %# w #Vov futHthor ##-# iroatcortn “* liP^ » # # w WF of this and allied topics, see a recent paper by Frochot (16). 3* Protoena for Further Study. In cootion 5 of Cbaptor III it rao shoun hor/ tho general problem C^k^k = o 2 C k X k - 0 ' with tho mixilliary conditions 0 ^ 1 JEX* = * * K ’ With regard to these equations. It io evident that tho following 1) la it possible to discover somo cot of conditions undor which . tho above equations will yield a determinate solution for tho unknown in question (e) f : : •' : 2) Is it possible to formulate a general procedure which when applied to those equations will givo on expression for n o^lioit in all tho to these equations (together with the auxilliary conditions) will givo a ccnplcto sot of "conditions of possible experience" for tho data ? 4) Finally and cost important, is it possible to fon.ulato a general procedure which will provide the closest licito, oxproosod in terms of the data, within which the required probability s must lio in ordor to satisfy tho given equations and auxilliary conditions t by a reading of Chapter XIX of tho laws of Thought. Ref- cordacce \rith the eysten; eirolalned la the latroduetlea. p. m . 5) . . . "Soto of postulates fo r Boolean groupo", Annala of rath., vole 40, 1939, p. 420. 6) BIRKKOFF, G., ^ttice.J%pq.n » Acorlcaa Hxthoptical Society Colloquium Publis&tioae, volume XK¥, Mew fork, 1940. 7) BOOLE, G., “On the theory of probablliUee, and in particular on Eitoholl's problem of the distribution of the fixed stars", P h il. ser. iv , vol. 1, 16S1, p. 523. 6) ... "Furthor observations on the theory of probabilities", Phil, ms., ser. iv, vol. 2, 1651, p. 96. Gears, 19#. Volume I contains all hie papers on tho theory of probab i l i t i e s j volumo I I is tho Lssrs of Thought. 10) BOOLE, G., “Reply to coco observations published by Kr. wilbroham in tho Phil. Lhg. vii p. 465, on Boole’s ’Law# of Thought* ", %%., ser. iv, vol. 8, 1854, p. 87. 11) BOOLS, 0 ., "Furthor oboorvatlono in reply to Vr* Uilbrahaa", ir, vol. 8, 1854, p. If5. Phil *. * « "On tho application of the theory of probabilties to Trgns.JTloy. Ooe. Cdinburnh. vol. 21, 1857, p. 597 14) mmmJCK, "On some eyx&olle foraalac in prebebillty theory". Prop# Roy. I ll eh vol. 44, 1937, p . 19. 15) COtFItRAT, L., "la logique algorlthcique et 1# eelcul des probab- Hites", I f f i a . » vol. 24, 1917, p. 291. 14) yfiSCHST, ii,. If) FBIHIC, 0. "On the existenoe of linear algebras in Boolean Algebras", Bill. See., vol. 34, 1926, p. 329. IS) HOMn.J«toH, E.V., 'Sets of indepondont postulates for tho Algebra of Logic", Tram. Am., ISatfe. .See., vo. 5, 1904, p. 288, if) *ltev sets of independent postulates for the e|ra of Logie, with special reference to Shitchcad anc •Principia J&theaatica* ", Trans. Aa. Hath. Soe«. vti„. 35, 1933, p. 274. so) Ksnes, j j u . s Loudon, llacsiill&n, 1921. a) KOL’XGCKOFF, A., "Grundbogriffo dor •-ahrschoinlichkeitorochnung", Ergobnlsse der LMh.. vol. 2, Berlin, Julius Springer, 1933* 22) KDCPEXH, B., "The axioms arri algebra of Intuitive probability". Annals of Lath., vol. 41, 1940, p. 269. 23) . . . "The bases of probability", vol* 46, 1940, p. 763. 24) . . . Hath. . vol• 42, 1941, p. 169• 25) Utitts, C .I., A Survey of _5ygbolic Lorric. Berkeley, U.Gal.Press, 1918. 26) LEWIS, C .I. and LMIGFORD, C.H., Symbolic Logic. lion York, Century, *32 , ®f) iSicCOLL, H., "Tho calculus ot equivalent statements. Sixth paper". Prog. London Math. Sec., vol. 28, 1897, p. 555. 26) MXGBL, Bneyo. of tMlflod Science, vo. I , no. 6, U. Chicago Prase, 1939. 2S> SHOTS!, H.i!., “a sot of five independent postulates for Boolean Algobrao uith applications to logical constants", Trans. An. I’ath» See. , vol. 14, 1913, p. 481. .* See Errata following Bibliography for papers of 0.3. Peirce. 30) STCtK, L'.H., "oubsunption of the theory of Boolean Algobras under tho theory of ringc", Proc. ?b.t. read. 3ci.« vol. 21, 1935, p.103 31) . . . "Theory of roproeontationo for Boolean Algebras", Trsnn. Path. Soc». vol. 40, 1936, p« 37. SS) ... "Tho representation of Boolean Algebras", Bull. An. Math. See.* vo. 44, 1938, p. 807. 33) WILBRiUiAM, K., "On the theory of chances developed in Professor Boole*o ‘Laws of Thought* ", *er. iv , vol. 7, 1854, p . 465. The twotpapera of C.3. Peirce lis te d as (29) and (30) in the text *ra# ...... 29) PEIRCE, C.S., "On an ieproveaent in Boole*# calculus of logic". 30) "On the algebra of logic", Aa. Jour. Math., vol. 3» I860, p. 15. These two papers appear also la S3 ce / f S // Ooy- ^