THE AXIOMS AND ALGEBRA OF INTUITIVE PROBABILITY

ANNALS or MATHEMATICSVol. 41, No. 2, April, 1940

THE AXIOMS AND ALGEBRA OF INTUITIVE PROBABILITY

By B. O. KOOPMAN

(Received June 1, 1939)

1. The Idea of Probability

The intuitive thesis in probability holds that both in its meaning and in thelaws which i t obeys, probability derives directly from the intuition, and isprior to objective experience ;' it holds that it is for experience to be interpretedin terms of probability and not for probability to be interpreted in terms ofexperience; and it holds that all the so-called objective definitions of probabilitydepend for their effective application to concrete cases upon their translationinto the terms of intuitive probability. I n its purest form i t stems from theaphorism: knowledge is possible, while certainty is not; for it sees in the resolutionof this antinomy—it sees in the varying "probability" in which the articles ofknowledge are held—its very conceptual germ.

Clearly the vindication of a thesis such as this falls outside the domain ofmathematician or physicist, and belongs to the philosopher: the question atissue is one of epistemology.' A n d our function here is to undertake neitherits defense nor its refutation, but solely to attempt the axiomatic formulationof the idea of intuitive probability, and the derivation of the classical mathe-matical theory of probability from this. T h a t such a task is necessary at thepresent time appears from the fact that all the axiomatic treatments of intuitiveprobability current in the literature take as their starting point a number(usually between 0 and 1) corresponding to the "degree of rational belief" or'credibility" of the eventuality in question. N o w we hold that such a numberis in no wise a self-evident concomitant with or expression of the primordialintuition of probability, but rather a mathematical construct derived from thelatter under very special conditions and as the result of a fairly complicatedprocess implicitly based on many of the very intuitive assumptions which weare endeavouring to a)domatize: There is, in short, what appears to us to be a

I "(Objective) experience" is used here practically as the equivalent o f " laboratoryexperiment" i n the narrow objective sense o f the word, and excludes introspective o r"subjective experience." Moreover, i t i s a question here o f rational derivation f romexperience, and the a pr ior i view of probabil i ty is quite consistent w i th the idea thatprobability as well as logic may be derived by race experience through the process ofevolution. I n spite o f terminological similarities, there is no question here of KanteanTranscendentalism.

2 The philosophical controversy here involved is very old and is sti l l unsettled. W i t h -out attempting to give general references to the literature on this point we may cite theIntroduction and First Chapter of J. M. Keynes' A Treatise on Probability (London, 1921),in which the thesis of intuit ive probabil i ty is eloquently urged.

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270 B . O . K O O P M A N

serious rational lacuna between the primal intuition of probability, and thatbranch of the theory of measure which passes conventionally under the nameof probability. Moreover, this assumption commits one to too great precision,leading either to absurdities, o r else to the undue restriction of the field ofapplicability of the idea of probability.3 The fundamental viewpoint of the present work is that the primal intuitionof probability expresses itself in a (partial) ordering of eventualities: A certainindividual at a certain moment considers the propositions a, b, h, k, the meaningsof which he apprehends, and which he considers to be determinate (i.e., eithertrue or false) without (in general) knowing whether they are true or false (forthe sharpening of these ideas, see §2). T h e n the phrase

"a on the presumption that h is true is equally or lessprobable than b on the presumption that k is true"

conveys a precise meaning to his intuition—and this, utterly disregardingwhether he accepts what it says, rejects it, or is non-committal. T h i s is, as wesee it, a first essential in the thesis of intuitive probability, and contains theultimate answer to the question of the meaning of the notion of probability.Let us write the quoted phrase in either of the symbolic forms

alh < blk, b l k > a lhand call such an expression a comparison in probability.4 I t w i l l b e c o n v e n i e n tto call such symbols as alh or b/k eventualities, the propositions a and b, thecontingencies in the eventualities, and the propositions h and k, the presumptionsin the eventualities. A n d we may read alh b / k : a on h infra probable b on k( > being read supraprobable).

The position of acceptance, rejection, or doubt taken with regard to a/h <blk may vary from individual to individual and from moment to moment withthe same individual, depending as it will upon his entire state of mind. Never-

3 We have in mind, on the one hand, events of such elaborate and individual nature asto defy numerical evaluation of probability, yet which can often be definitely comparedas to their likelihood of occurrence; and on the other hand, instances such as the following:Let 0 denote an impossible event, le t A and B consist of a certain normally distributedvariate x being found to have exactly the value a and b respectively, b being much closerto the central value than a, so that for the probabil i ty densities: 99(a) < co(b) T h e n asfar as the numerical probabilities go, p(0) = p(A) = p(B) 0 , but i t is rational to placethem in the order of likelihood: 0 < A < B.

But fundamentally our object is not so much the treatment of such cases, as to securean adequate conceptual framework for the idea of probability.

4 The notation a/h is suggested by that of Keynes (I.e.); yet we are using i t in quite adifferent sense and in a manner much closer to the current notation for the quotient of aring by an ideal (the ideal ( - - -,h) c f . § 2 ) . F o r t o b e g i n w i t h , K e y ne s r e g ar d s a /h as a n u mb e r,

or a t least susceptible of many of the rules of ordinary arithmetic, whereas for us i t isbut an ordered pair of propositions, or, as we shall see later, a remainder class in a Booleanring. A n d secondly, the logical type of proposition which we denote by a, h, etc., is fardifferent from what Keynes appears to understand by such letters.

AXIOMS A N D A L G E B R A O F I N T U I T I V E P R O B A B I L I T Y 2 7 1

theless, i f a certain individual at a certain moment subscribes to both of thefollowing: a/h < b/k, b/k < c/ l , he must at the same moment subscribe toa/h < c/l, and also he must subscribe to each of > >For otherwise he will be in a condition of mental incoherence of the same natureas logical inconsistency. W e thus come to a second essential in the thesis ofintuitive probability, namely, that its comparisons obey fixed laws which, likethe laws of intuitive logic to which they are akin, are axiomatic, and owe theirauthority to man's awareness of his own rational processes.'

The circumstance that notwithstanding the "subjectivity" with which com-parisons in probability may be made, they are none the less fixed both asregards their meaning and their laws, is the very fact which makes a rationalmathematical theory of probability possible. I t is the unique function of sucha theory to develop the rules for the derivation of comparisons in probability fromother comparisons in probability previously given.

When the intuition exerts itself upon a situation involving uncertain events,it is capable of performing two tasks: firstly, i t may assert comparisons inprobability; and secondly, i t may state laws of consistency governing suchcomparisons. There is between these two operations of the intuition an essen-tial cleft; and the failure to realize this has been a most frequent cause of con-fusion. F r o m the laws of consistency a theory may be constructed (and thisis our present object) with the aid of which, comparisons in probability (andindeed, values of numerical probability) can be deduced from others which arehypothesized: but beyond this, the laws of consistency do not allow one to go.This naturally raises the question as to whether there are any general principleswhich determine (non-trivial) comparisons in probability ab ovo,---withouttaking any such comparisons as starting point.

One might consider a concrete case: Two wayfarers are on a road which, asthey know, leads to their destination. A f t e r a while, to their surprize, the roaddivides in two, and as they are ignorant of their orientation, they do not knowwhich way to turn. T h e first wayfarer asserts that the right-hand turn isequally or less probable as the way to their destination than the left ; but thesecond disagrees with him. I s there any general principle which wil l enableeither one rationally to compel the other to accept his position? W e mean a prin-ciple which does not itself implicitly assume a comparison in probability (suchas a statement of "irrelevancy") to have been granted and which can be appliedto the present case without begging the question; a principle which, finally,

5 There is a body of opinion at the present time which regards the laws of probabil ity,and even the laws of logic, as subject to objective experimental proof or disproof, or atleast as requiring possible alteration in the l ight of the evolution of physical theory. T h i sschool strengthens its ease by the claim that certain facts in the quantum theory are atvariance wi th the " laws of probabi l i ty" (or indeed, o f logic). W h i l e we shall treat thisquestion i n detail elsewhere, we may say here tha t we have examined w i th a l l logicalminuteness every alleged ease which we could discover, and have found in every casethat the physical circumstances of the application of the laws of probabil i ty, etc., werethe things that were altered, bu t never the laws themselves.

272 B . O . K O O P M A N

genuinely pertains to the general abstract theory of probability, and not tosome specialized branch of experimental science.

While an authoritative answer to this question would go beyond the scopeof the present paper, involving as i t would a critique of the existing state ofthe foundations of probability in toto, we may nevertheless venture an expressionof our personal view: We have never seen formulated nor been able ourselvesto formulate such a principle. A n d we have come to the position that nosuch principle is in the nature of things susceptible of formulation, that we areindeed at the very threshold of the problem of the conditions of the possibilityof knowledge, and that the difficulties here met with are those which must inthe nature of things always be encountered when an attempt is made to give amathematical or physical solution to a metaphysical problem.

2. Elementary Notions of LogicIn the theory of probability one is constantly occupied with what may be

called determinate concrete propositions, that is to say, specific assertions of aphysical or biological character, the truth or falsity of which is regarded asverifiable on the performance of an appropriate experiment. Such propositionsand such only shall in the present paper be symbolized by small Latin lettersor by finite clusters of such letters united with the aid of the logical constants(,,,• V ).8 F o r e xa mp le , the p ro po si ti on "it will rain (at stated time and

place)" would be denoted by a letter such as a, " i t will rain or snow (at statedtime and place," by c or by a V b, etc. N o w all propositions of this sort areconceived as contemplated propositions (cf. Frege and Russell) and not as meresymbols for 0 or 1:7 w e m a y t a l k a b o ut a a nd a V b w i th ou t a u to m at i ca l ly

asserting them. Altogether different shall be our conventions regarding theuse of the logical constants ( C ) ,8 w h i c h s h a l l b e r e g a r d e d a s h a v i ng p r e d ic a -

tive power. T h u s "c = (a V b)" is an asserted proposition and shall neverin the present text be symbolized by a single letter, nor shall assertions like i tbe united by means of logical constants: no formula shall contain more thanone of the signs ( C ) .

To sum up: We are dealing exclusively with intuitive logic, are employingthe logical constants solely as abbreviations for notions which are regarded asintuitively evident and as obeying intuitively evident laws, and we are sym-bolizing by letters only propositions of the lowest "logical type."

6 As usual, d e n o t e s negation, • (or mere juxtaposition) denotes conjunction or logicalproduct, and V denotes disjunction or logical sum.

7 The identically false proposition is denoted by 0 and is regarded as coinciding wi thevery a e'd a , t h e i d en t ic a ll y true p ro po si ti on is denoted by 1 and is regarded as coinciding

with every a V a . T h u s all the Boolean algebras here considered are regarded as havingidentical units.

The symbol for identity = and material implication C (a C b meaning "a implies b")are used here rather than m and D in order to bring out the parallel with the mathematicaltheories of sets and algebras.

AXIOMS AND ALGEBRA OF INTUITIVE PROBABILITY 2 7 3

Parentheses shall be used in the self-evident manner of classical mathematics;but they are economized by the conventions concerning the binding strength ofsymbols, according to which: ( i) i s linked most strongly to the succeedingnearest letter, so that , ,a b o r -,• c t • b e a c h m e a n ( - -,a ) • b , a V b m ea ns ( "—a ) V b,

and --a b and C b mean = b and (, , ,a ) C b r e s p e c t i v e l y ; ( i i ) • ( o r

the absence of symbol) binds the letters on either side of it more strongly than Vso that a•b V c or ab V c means (ab) V c. I n this connection i t may be re-marked that in accordance with our fundamental convention, such expressionsas a V (b C c) or --(a C b) would have no use.

The knowledge of the laws which the symbols (— • V = C ) obey, or formalstatement of the laws of intuitive logic, is here regarded as aquired and thereader is assumed to be familiar with the elementary facts of Boolean algebra.'We shall make explicit mention only of the following definitions and theorems,on account of the prominent role which they are to play in our theory of proba-bility. T h e reader may easily furnish the proofs by analogy with the theoremson homomorphism of groups or rings, and the quotient group or remainderring in classical algebra. Detai led proofs are furnished in the paper of M. H.Stone cited above.

DEFINITION. B y a homomorphism 21 2 I ' from a Boolean ring 21 to a Booleanring 21' is meant a functional correspondence x' = f(x) whose domain is 21 andwhose range is 21' , such that, for all x, y e ?I,

= f ( x Y ) = .f(x)f(Y), . f ( x V Y) = f(x) V AO-

In the case where f(x) = f(y) implies x = y, the correspondence 21 —> 21' can beinverted (the result 21' — l being shown to be a homomorphism) and the corre-spondence is called an isomorphism 21 4-4 2I'.

DEFINITION. A non-empty subset u of elements of a Boolean ring 2t is saidto be an ideal in 21 when it satisfies the following conditions:

1. I f a e u artd b e u, then a V ben2. I faeuandbe21then ab eTHEOREM. I f 21 —+ ?I', then the aggregate u of all elements x of 21 such that

x' = f(x) = 0' (the zero in 21') constitutes an ideal in 21. A n d f(a) = f(b) i f andonly if simultaneously a b e u, e u.

We shall now recall the definition of the quotient ideal ?I/u of 2t with respectto any one of its ideals u.

First, all the elements of ?I are distributed into a system of mutually exclusiveand exhaustive remainder classes of u defined by the convention that any twoelements a and b shall belong to the same remainder class of u if and only if

For general bibliographical references, see E. V. Huntington, Trans. Amer. Math .Soc., Vol. 35 (1933), pp. 274-304. F o r algebraic developments of Boolean rings, part icu-larly in the l ight of their relation with the rings and ideals of classical abstract algebra,see M. H. Stone, The Theory of Representations for Boolean Algebras, Trans. Amer. Math.Soc., Vol . 40 (1936), pp. 37-111.

274 B . O. KOOPMAN

simultaneously a b u , u . L e t the remainder class determined by thefact that it contains the element a be denoted by a/u.

Next, if • V ) applied to classes be used to denote the classes resulting fromthe application of the respective operator to all elements or pairs of elements ofthe classes in question, and if ( = C) be used to denote class identity or inclusion,it is shown that

— (a/u), a b / u = (a/u)(b/u), a V b/u = (a/u) V (a/u).

Finally, 21/u is defined as the class of all remainder classes in 21 of u, the• V ) being defined as before, 21/u is seen to be a Boolean ring (with units

0/u and 1/u) and the correspondence x --> x/u (x E 21) is a homomorphism:21 2 1 / u . A n d the following theorem is an immediate consequence.

THEOREM. I f 21 2 1 ' , and if u is the class of all elements of 21 having 0' as theirimage in 21', then 21' 2 1 / u .

Turning now to the significance of these algebraic notions for the presenttheory of probability, suppose that an aggregate of comparisons in probabilityis being envisaged, involving a set of concrete propositions a, b, h, k, etc. A f t e rthe possible adjunction to this set of all of the finite combinations of its proposi-tions in terms of • V )—in case such combinations are not already includedin the set—it becomes closed, and forms a Boolean ring 21, containing 0, 1.

In 21 the element ( a s s u m i n g —h / 1 i.e., h 0 ) generates a principal ideal(i.e., ideal generated by an unique element / 1), which shall be denoted by(—h), and which is composed of the class of all elements x e 21. I f , now,a and b are two elements of 21 for which simultaneously a — b E ,-,a 1 ) (,- 1 1 ) ,

then a and b are logically equivalent on the assumption that h is true in the followingintuitively self-explanatory sense : since a b = —he(c e 20, to assume h true isto assume a b = (a — b)h = h h e = 0, i.e., to rule out that a is true withoutb being true: a C b. A n d similarly, if = ( d E 20, then the assumptionthat h is true automatically makes —ab = 0, i.e., makes b C a.

I t is convenient in this work to introduce the symbol a/ h as simply an alter-native notation for al (—h): in the light of the above, we become aware of thefact that the intuitive act of passing from the consideration of a on no presump-tion concerning any other of the propositions in 21 (beyond 0, 1) to the considera-tion of a on the presumption that h is true, has its exact algebraic counterpartin the passage from a (which may be replaced by a/1) to a/h = a l (,- h ) . H e n c e -forth i t is in this precise algebraic sense of a remainder class that the eventualitysymbol a lh shall be employed, its intuitive counterpart serving solely as a guideto the formulation of the axioms governing the < sign, as well as for the inter-pretation and application of the theorems developed in the formal mathematicalstructure which we are about to erect.

In closing this section we note, firstly, that the symbol a/0 = a/(1) is useless,since its intuitive content is the assertion of 0 : i t shall be implicitly excludedthroughout; and, secondly, that the symbol -< is regarded as having predicativeforce, and hence all symbolic constructs into which i t enters are asserted, not


contemplated, propositions, and are never denoted by letters nor operated uponwith logical constants." T h u s it, like (= C) , has a weaker binding force than

V), so that the parentheses in expressions such as [( )/( )] [ ( )/( )] areomitted. T h e binding strength of ( / ) is greater than ( = , C, < ) and less than

V).Finally, i t might be interesting to consider eventualities a/u in which u is

not a principal ideal (u ( — h ) ) . B u t a careful examination of this matterhas shown us that such an extension is not a useful one at the present stage ofthe theory.

3. The Axioms of ProbabilityFrom the standpoint of intuition,- t h e f o l l o w i n g a x i o m s a r e t h e i n t u i t i ve l y

evident laws of consistency governing all comparisons in probability; in somecases, to be sure, the full understanding of their meaning may offer some diffi-culty: the only claim is that, their meaning once being apprehended, theirtruth will be granted." F r o m the standpoint of pure mathematics, they are theformal postulates governing the < sign, and are accepted by convention. T h eaxioms form a sufficient basis for what we envision as the legitimate role of atheory of intuitive probability. W e offer no discussion of consistency, nor havewe striven systematically to secure independence (Axiom S has indeed beenstated for all integral n, whereas i f assumed for prime n the general case willfollow). I n short, the situation is altogether comparable with the case of thelaws of intuitive logic.

In each of the following axioms a definite Boolean ring 21 (with units 0, 1) ispresupposed in which the remainder classes are formed; and wherever in laterdiscussion a theorem is stated or proved, a definite 21 is again presupposed.But i t has seemed unnecessary in general to make explicit mention of this factin most of the following statements and discussion.

THE AXIOMSV. AXIOM OF VERIFIED CONTINGENCY.

alh < kik.

'° This circumstance shows itself throughout our whole work by the fact tha t we arenever allowed to ta lk about the probabi l i ty of a comparison in probabil i ty. T h i s mayseem to get us unto conflict with Bayes' Rule, which is often regarded as concerning itselfwith " the probabil i ty of a probabi l i ty." A mere change in wording avoids this difficultyfor us, and we regard Bayes' Rule to concern itself only with the probabilities of differentvalues of a physical parameter 0, which, itself, determines the probabil i ty (in particular,by the equation p 0 ) .

" A most useful auxi l iary i n the in tu i t ive comprehension and interpretation o f theaxioms as well as later theorems and proofs is the logical diagram, in which plane regionsrepresent the propositions, and V = c ) are given their set-theoretic rendering. O n emay go further and let a lh < b lk be represented by having the ratio of the area of ah tothat of h t h a t of bk to k.

276 B . O. KOOPMAN

I. AXIOM OF IMPLICATION.I f alh > klk, then h C a.

R. Axiom OF REFLEXIVITY.a/h a l h .

T. AXIOM OF TRANSITIVITY.I f a/h < b/k and b/k < c/1, then alh c / 1 .

A. Axiom OF ANTISYMMETRY.I f alh < blk, then >

C. AXIOMS OF COMPOSITION.Let albihi X 0 and a2b2h2 O.

I f al/hi a2/ h 2 a nd b i /a l hi b 2/a 2h 2 , then albi/h1 < a2b2/h2 •

C2 • I f al/111 < b2/a2h2 and Walk < a2/h2, then alb , / hi < a 2 b 2 / h 2 •

D. Axioms OF DECOMPOSITION (QUASI-CONVERSES OF C).Let albihi X 0, a2b2h2 X 0, and albi/ h1 a 2 b 2 / h 2 . T h e n i f e i t h e r o f t h e

eventualities(i) al/ h1 , bi/ alhi , has the supraprobable relation (>) with either of

(ii) a2h2 , b21 a2h2 , it will follow that the remaining eventuality of (i) will

have the infraprobable relation (< ) with the remaining one of (ii).P. Axiom OF ALTERNATIVE PRESUMPTION.

I f a/bh r l s and a/—bh < rls, then alh r / sS. Ax iom OF SUBDIVISION.

For any integer n let the propositions al , • • • , a . , b , • • • , b , b e s u c h t h a t

a,a, = btb , = 0 ( i j) j = 1, • • • , n; a = al V • • • V a. X 0; b =

bi V • • • V b. X 0; ada • • • • < a,/a; bi/b < • • • < b./b; then alia

be/b.

On fixing the attention upon a given Boolean ring 21 of concrete propositions,in which the < relation is supposed to have been introduced, it is seen that theformulation focuses attention upon the totality a of remainder classes withrespect to all the principal ideals of 21 and that Axioms R and T show that <defines a partial ordering of the elements of .1 2 W e s h a l l a c c o r d i n g l y a p p r o -

priate the conventional definitions and theorems of the theory of partiallyordered sets." I n particular, we shall write in either of the two forms a/h <

12 While i t is shown at once that i f h C k (material implication) then (h) C (k) (setinclusion), and conversely, we may state a t once tha t there is no such simple relationbetween the -< and c when they occur between remainder classes. Examples are mosteasily given after the subject of numerical probabil i ty has been developed.

" For a ful l account of this theory, see H. M. MacNeille, Part ial ly Ordered Sets, Trans.Amer. Math. Soc., Vol. 42 (1937), pp. 416-460.

In most treatments o f part ia l ly ordered sets, the greater par t o f the developmentsfollow from the "Lat t ice Assumption," which in the present connection would take thefollowing form: I f alh e a and b/k e a then a contains c// having the properties

(i) c / / a / h , c/1 b / k and


blk, b/k > a/h, the abbreviation for the combined assertion of alh < b/k anddenial of a/h > b/k, and we shall use the symbol f o r equiprobability: a lhb/k as the abbreviation for the joint assertion of the two following: a lh • b / k ,a/h > blk. (Contrast w i t h — : a/h = b/k signifying identity of remainderclasses).

A process constantly to be employed in all later work is the application of afactor in the presumption to the contingency; for i t is evident in view of theconventions of §2 that, for example alhk = ah/hk ahk/h Ic .

It will be remarked that Axiom I is in sharp contrast with the familiar factthat an eventuality which is not regarded as certain (as alh when h t a) maynevertheless have the same numerical probability (unity) as one held as certain(as k1k). O u r contention is that the truth of Axiom I resides in the very in-tuitions of comparison in probability and certainty, of which the numericalprobability is but an unfaithful representation.

A final remark : I f the contingency a is a determinate arithmetical statement,such as "the 100th decimal in 2 i s 2" our theory asserts that the eventuality a/1eitherc•-z-: 0/1 or 1/1 (cf. Theorem 1 in 0 ) . T h i s is because the truth or falsityof a is determined by the Boolean ring 21 within which all the propositions hereenvisaged find themselves: in this ring a may be computed by the formal lawsof the algebra and found either to = 0 or 1. W e hold that this is as it shouldbe and that if the ideas of intuitive probability are to be applied non-trivially toarithmetic, the contingency must be stated in a logically undetermined formwhich reveals something of the "random process" by which the definition of theinteger was settled upon.

4. Theorems on ComparisonTHEOREM 1. F o r all a / 0, ala 1 / 1 and 0/a 0 / 1 are true.I f ah 0 and ah / h, then 0/1 < alh <111.I t follows from V that a/a -< 1/1 and 1/1 < a/a, hence a/a 1 / 1 . These

with A yield —a/a > ---4/1 and ,• 4 / 1 > , ,a / a ; b u t ,- -,a / a = a a / a = 0 /a

and —1/1 = 0/1; whence the relation 0/a c-,e, 0 / 1 .

From V i t follows that a/h < 1/1; i f a/h > 1/1, I would lead to ah = h,contrary to hypothesis; hence alh <111. N o w if —ah = h i t would follow that0 = a (,-,a h ) = ah , c on tr ar y to hypothes is ; hence we may replace a by ,---,a in the

previous discussion and so obtain - -,a / h < 1 / 1 t o g e t h e r w i t h t h e d e n i a l o f

> 1/1; whereupon A shows that a l h > 0/1 is true and a/h 0 / 1 false;hence 0/1 < a/h. T h u s our theorem is proved.

If in an assertion involving < signs, every < be replaced by a new statementis obtained which shall be called the sharpening of the original. I f on the otherhand at least one < is replaced by < the new statement shall be called a strength-

(ii) i f d im e a such that d im < a /h and d im b / k then d im < c l l .The remaining Latt ice Assumption is the ( ) dual o f the above. I t is perhapsnot unworthy o f note tha t these ideas p lay no role i n the present theory o f part ia l lyordered sets.

278 B . O. KOOPMAN

ening of the original statement. I f a deductive statement (axiom, theorem,corollary, etc.) has a < statement for hypothesis and a < statement for con-clusion, the new deductive statement obtained by replacing both hypothesis andconclusion by their sharpening shall be called the sharpening of the originaldeductive statement, while the deductive statement obtained by replacing bothhypothesis and the conclusion by one of their respective strengthenings shall becalled a strengthening of the original deductive statement.

One of our tasks is the establishment of the truth of the sharpening and ofevery strengthening of every one of the axioms which involves the i n bothhypothesis and conclusion, to wit, of all except V, I and R. T h i s is convenientlyaccomplished in two stages announced in Theorems 2 and 11.

THEOREM 2. T h e sharpening and every strengthening of Axioms T, A, C, Dare valid.

The proofs for T and A are obvious and shall be omitted.Turning to CI , l e t t h e n ew h y po t he s i s be c t iblh i X 0 a2b2h2 X 0, and al/hi

c t2/h2 , bl/alhi cz-J b2/a2h2 We are to prove a1b1/1/1 a2b2/h2 Now the hypothe-

sis leads to the original hypothesis of CI , a n d a l s o t o t h i s l a t t e r i n w h i c h t h e

subscripts 1 and 2 have been interchanged; thus CI l e a d s t o t h e t w o c o n c l u s i o n s

a1b1/h1 a2b2/h2 , a1b1/h1 >- a2b2/h2 , which is equivalent to the conclusion

sought.As for the strengthening of CI , l e t t h e n e w h y p o t h e si s b e a l b i hi X 0 , a 2 b2h2 0,

and al/ hi < a2 /h2 , b2/a2h2 ; we are to prove Valhi < a2/h2 In

view of CI a l l t h at i s r eq u ir ed is to show the falsity of ctibl/h1 > a2b2/h2 Let

this relation be assumed true and apply D (with subscripts interchanged) to thetwo relations bi/alhi < b2/ a2h2 , a 1 b 1 /h 1 > a 2 b 2 /h 2 , t h us d e r iv i n g t he f o l lo w in g

contradiction of hypothesis: a2/h2 < al/hiThe remaining possibility (not reducible to the above) is the introduction

into the hypothesis of CI of Valh i < a2h2 T h e t r e a t m e n t o f t h i s c a s e b e i n g

quite similar to the other is left to the reader.The discussion of C2 is omitted, being altogether similar to the case of CI •

And as for D, we shall treat only the following case, leaving the rest to thereader:

Let a lbih i X 0 , a 2b 2h 2 X 0, and a1b1/1 /1 < a2b2/h2 , al/h1 > b2/a2h2 We are to

prove that Valh i < a2/h2 I t is required only to show the falsity of Va lh i >a2/h2 For this purpose we have but to apply C2 (subscripts interchanged)

to the relations a2/h2 bi/ alhi , b 2 /a 2 h 2 < a l /h i , w h e re u p on t he re w il l f ol lo w

the contradiction of hypothesis: a2b2/h2 < ctibl/h1 •In the succeeding theorems, when it is desired to indicate that sharpening and

strengthenings are valid, a statement to this effect is appended to the statementof the theorem. B u t in those cases where the proof of the sharpened or strength-ened theorem is furnished by simply paraphrasing that of the original theorem,with the replacement of axioms, theorems, etc., employed by their sharpening orstrengthenings, the proof is omitted. A n d wherever in the future reference ismade to an axiom or theorem, either the original or its sharpening or strengthen-

ing shall be meant: the reader will have no difficulty in seeing which one isneeded.

THEOREM 3. I f ah C bh then alh < blh. I f in addition, bh c a h , thena/h < blh.

From R and V it follows that b/h < blh, albh < blbh. I t is now possible toapply C, in which we put hi = h2 = h , a1 = a2 = b , b1 = a , b2 = b, a nd so to

derive the following relation, which furnishes the first conclusion desired, ablh <bb/h. H e r e albih , = a b h = a h a nd a2b2h2 = bh, so that if a2b2h2 0 we should

have ah = abh 0 , in which case Theorem 1 would furnish the desired conclusion.If, further, i t is assumed that bh a h , whereas alh < b/h, i.e., alh > b/h,

we could apply D to the following relations blh < alh, b/h > blh, where weput hi = h2 = h, a1 = a2 = b, b1 = b, b2 = a, and so either alblh,(= bh) and

a2b2h2(=abh = ah) are not both / 0 or else the relation blbh < albh is obtained.

But we have b/bh = bh/bh 1 / 1 , hence this would give by Theorem 1 thatbh C a, so that bh C ah, in contradiction with the hypothesis. I f ah 0 thenin view of the relation we are trying to disprove, we should have blh < a/h =0/h 0 / 1 and thus by Theorem 1, bh 0 C ah. F ina l ly, if bh 0 , the samecontradiction with the hypothesis bh a h would obtain. T h u s all parts of thetheorem are proved.

The second part of this theorem may be regarded as a kind of strengtheningof the first part. T h e sharpening in the sense that ah = bh implies ah/hbhlh is a trivial consequence of R. Simi lar remarks apply to the followingtheorem.

THEOREM 4. I f a C k Ch, then a/h < alk. I f in addition h c k and a 0 ,then alh < alk.

From V and R it follows that k/h < k/k,a/hk < alhk. I t is possible to applyC, after setting h, = h, h2 — a , = a2 = k , b1 = b2 = a , f r om w h ic h t he d e si r ed

conclusion follows. He re alblh , = a h k = a , a2b2h2 = a k = a , a nd i f a = 0 t he

conclusion would follow from Theorem 1.If, furthermore, h k and a / 0, we could apply D to the supposed relations

a/k a / h , a/k > alk, where we set hi = k , h2 = h , a1 = a2 = k , b , = b2 = a ,

from which i t would follow,—providing a1b1h1( = a k = a ) a n d a2b2h2( = a k h = a )

are neither = 0, a case excluded by hypothesis,—that 1/1 k l k < k lh; thush C k, and we have a contradiction.

THEOREM 5. I f (i) a1bih1 = a 2 b 2 h 2 = 0 , a n d i f ( i i) a 1 /h 1 a 2/h 2 , ( ii i) <

b2/h2 , then a, V bi/h, < a2 V b2/h2 •

The sharpening and all strengthenings are valid.On applying A to (ii) one obtains

(1)


> , -- - 'a 2 /h 2 •

From (i) it follows that ,- -,a , V ,- - -,b , V ,-,-,h , = 1 , w h e nc e , m u l t ip l y i ng t h ro u gh

by M il: = b1h1 ; thus we have , -- -,a1b1/ h1 h1h1/h1 =

bi/h1 , and similarly for b2/h2 Hence (iii) may be written as

(2) ,• -' a2 b2 /h 2 •

280 B . O. KOOPMAN

It may be assumed that / 0 and ,- -,a 2 b 2 h2 / O . F o r i f = 0 ,

(i) would give us Mil = —alblhi V alblh = 0, and the conclusion would followdirectly from the following relations made evident by (i i) and Theorem 3:al V b1/1/1 = alhi V b1h1/1/1 = a1it1/1/1 = adhl a1/h1 < a2/h2 < a2 V b2/h2 •

While if , , ,a2b2h2 = 0 , w e c ou ld d er iv e b2h2 = 0 as before, whereupon (iii) would

yield with the aid of Theorem 1 the already excluded case = O. W eare, therefore, entitled to apply D to (1) and (2) , -,a2 p l a y i n g t h e r o l e

of "al , a2") and so obtain

(3) < b2/ —a2h2

This with A gives the relation > - —b2/a2h2 • I f ,•,a 1, --,b 1 h 1 X 0

and —a2b2h2 X 0, CI m a y b e a p p l i e d t o t h is r e l a ti o n t o ge t he r w it h ( 1) , ,

—b2 , h2 , , hi playing the role of "al , b1 , h1 , a2 , b2 , h2"), whereupon

,which, on application of A, yields the conclusion ofour theorem.

If on the other hand , , ,a 2 - -,b2h2 = 0 , i t w o u l d f o l l ow t h at a2 V b2 V - -,h2 = 1,

whence h2 = h 2 (a 2 V b2 V —112) = (a2 V b2)h2 , and thus a2 V b2/h2 = (a2 V b2)

h2/h2 = h2/h2 and our conclusion would be a consequence of V. Finally, if

= 0 we should have (as / 0 is assumed) = ,-,a1- 1 /1h1 V

= —alblhl , and thus by Theorem 1, = 1 / 1 ,which in conjuction with (3) gives b2/a2h2 • 1/1 whence, again by Theorem 1,,-,a2b2h2 = —a2h2 and multiplication of this by gives —a2—b2h 0, the case

previously disposed of. T h u s our proof is complete.THEOREM 6. I f a1tal2h1 = a 2 1 a 22h 2 = 0 f o r a l l i X j , a nd if a1 1/ h1 -•< a2, / h2 for

all i = 1, 2, • • • , n, then an V an V • • • V a l „ / hi -• < a 2 1 V a 22 V • • • V a 2 n/ h 2

The sharpening and all strengthenings are valid.This follows from Theorem 5 by mathematical induction.THEOREM 7. I f (i) hi C a1, b2 C a 2 a n d i f ( i i) al/ h1 < a2/ h2 , ( ii i)

1)21112 then al ,---, b1/h1 a2 b2/h2

The sharpening and all strengthenings are valid.In virtue of (i), a1b1 = b i a n d a2b 2 = b 2 , so t ha t ( ii i) b ec om es a 1b i/h 1 a2b2 /h2 •

Suppose firstly that albihl / 0 , a 2 b 2 h 2 / O . T h e n t h e a p p l ic a t i on of D ( wi th

subscripts interchanged) to this relation together with ( i i ) yieldsb2/a2h2 , which on application of A furnishes the relation

(4) — b 2 / a 2 h 2 •

I t is now permitted to assume that al — b i h i X 0 , a 2 — b 2 h 2 / O . F o r i f a l

= 0, then al - - , b 1 /h 1 = = 0/14 0/1, and the conclusion of

our theorem would be a consequence of Theorem 1. A n d i f a2 — b 2 h 2 = 0 ,relation (4) would give ,- - -,b1/ a1h1 a 2 — b 2 h 2 /h 2 = 0 /h 2 0 /1 , w h en c e T h eo r em 1

leads to the case al ,---, b1h1 = 0 j u s t d i s p o s ed o f . W e m ay , t h en , a pp ly CI to (4)

in conjunction with (ii) (with , —b2 replacing "bi , b2" ) , a n d s o o b t a i n t h e

conclusion of our theorem.Turning lastly to the excluded cases, suppose that a2b2h2 = O. T h i s gives in


view of (i) b2h2 0 , s o t h at ,, ,b2 V —h2 1, = h2 ; thus a2 b2/h2 =

a2(—b2h2)/h2 = a2h2/h2 = a2/ h2 N o w al , -- - , bi C a1, a nd we h av e by T he or em 3

— • al/ hi T he c o nc l u si o n to our t he or em then follows from (ii).

Finally, if alblhl = 0 , ( i ) g i ve s t ha t bihi = 0, and thus b1/h1 = b1111/111 = 0/1/1

0/1; b u t from (iii) i t follows that b2/h2 0 / 1 whence b2h2 = 0,—the case justdisposed of.

THEOREM 8. L e t a C k, b C k, k C h. Then either of the following relationsis a necessary and sufficient condition for the truth of the other:

(i) a/h V i z ; (ii) alk b l k .The sharpening and strengthening are valid.Assume (i). Since ka = a and kb = b, we have on account of (i) and R the

relations kalh < kblh, klh > kilt. N o w kah = a and kbh b , and the sup-position that either or both of these = 0 leads at once to (ii) (Theorem 1). W emay accordingly apply D to the above pair of relations (with k, a, h, k, b, h,replacing al , b1 , h1 , a2 , b2 , h2 , r e sp e ct i ve l y) and so obtain a/kh < b/kh, which,

in view of kh = k, is the desired conclusion (ii).Assume (ii). W e may apply CI t o t h e t w o r e l a t i o n s k / h < k /h , a l k h b /k h ,

(with k, a, h,k,b, h, replacing al , b1 , h1 , a 2 , b 2 , h2 , r e s p e c t i ve l y ) a nd s o o b ta i n a

relation which, since ka = a and kb = b, is identical with (i). H e r e againkah = a and kbh = b, and if either = 0 the conclusion is trivial.

THEOREM 9. L e t ah 0 and bh / O. Then from

(i)it will follow that

(ii)

a/h a / b h

blh < blah.

The theorem remains valid i f the < be replaced by > in both (i) and (ii).The sharpening and strengthening are valid.We may write (i) and a relation evident from R as the pair ab/h a b l h ,

a/bh > a/h, and may then apply D (with b, a, h, a, b, h, replacing al , bi , h1,

a2 , b2 , h2 , respectively), whereupon (ii) is obtained. That this application of D

may be justified, we observe that albihi = a b h = a2b2h2 ; s u p p o s e t h a t a b h 0 ;

then b/ah = abh/ah = 0/ah 0 / 1 , and (ii) becomes a trivial consequence ofTheorem 1.

The proof of the remainder of the theorem follows precisely similar lines andmay be omitted.

THEOREM 10. L e t kh 0 and / O. Then either of the following relationsis a necessary and sufficient condition for the truth of the other:

(i) a/h < a l kh ; ( i i ) a lh > a /The sharpening and strengthening are valid.I t is only necessary for us to show that (i) implies (ii), in view of A. I f ah 0 ,

then a l = ah/ = 0 / ' A t ; 0/1, and (ii) would be a consequence ofTheorem 1. I f ah 0 , Theorem 9 may be applied to (i) (with b = k) and thus a

282 B . O. KOOPMAN

relation obtained which under the action of A yields ,- -,k / h > ,- - / k l a h . A g a i n

applying Theorem 9 (with a , > , replacing a, b, - 0 , we derive (ii).THEOREM 11. T h e sharpening and all strengthenings of Axioms P and S are

valid.For the sharpening of P, the hypothesis a/bh c • - • z . . . . I r l s on the one

hand contains the hypothesis of P and hence its conclusion, and on the otherhand contains the relations a/bh > r l s , a / -,b h > r / s , w h e n c e i t f o l l o w s b y A t h a t

, , ,r /s , whence again by P, < --,r/s, and again

by A, alb > rls, which in conjunction with the former conclusion of P gives thesharpened conclusion.

For the strengthening of P, let it be assumed that a/bh < r/s, r / s .All that is required is to prove the falsity of

(5) alh > rls.

Assume this to be true; our hypothesis yields with the aid of T, a/ - -,bh a l h .Now apply Theorem 10,—the above relation being equivalent to ( i i ) wi thk b , — a n d we have a/h a / b h , which in conjunction with (5) gives thecontradiction of hypothesis a/bh > s . T h e other case is automatically settledby the foregoing.

The sharpening of S is established at once on the double application of S tothe -< and the > hypotheses contained in the sharpened hypothesis.

For the strengthening of S, suppose the strengthened hypothesis to containfor some fixed i(1 i n 1 ) at/ a < a 1+ 1/ a , w e a r e t o p r o v e al/ a < be/ b .

Assume on the contrary that al/ a > b „ / b , i t f o l l o w s a t o n c e t h a t

al/a > bi/b, • • • , a,/a > bi/b,

a1/a > bi+l/b, • • • , an/a b„/b;

whence in view of Theorem 6 (strengthened) the contradiction follows 1/1c/a > b/b 1 / 1 . T h e remaining cases are left to the consideration of thereader.

THEOREM 12. L e t ah 0 , ,-,a h X 0 , b h 0 , - -,b h X O . T h en f r om

(i) a / b h -< a /it follows that

(ii) b / a h

(iii) a / b h a / h .

The theorem is valid if < be replaced by > in every one of (i), (ii), (iii).The sharpening and strengthening are valid.We have in view of R that a/—bh < a /,- -,b h . A p p l y P t o t h i s i n c o n j u n c t io n

with (i) (with a, —bh, replacing r, ,$); we obtain a/h a l N o w we applyTheorem 10 (with k = a n d so have (iii). T h e relation (ii) now follows by T.

The following extension of P may be given; we omit the proof which is im-mediate.

AXIOMS A N D A L G E B R A O P I N T U I T I V E P R O B A B I L I T Y 2 8 3

THEOREM 13. Under the hypothesis that

hihi = 0: (i j) j = 1, • • • , m;

kik = 0: (i j) j = 1, • • • , n;

alhi blki ; i 1, • • • , m; 1, • • • , n.

I t follows thata/h, V • • • V h„, b / k , V • • • V k„

The sharpening and strengthenings are valid.

5. Numerical Probability

The object of this section is to lay the foundations of the theory of numericalprobability, or mathematical theory of probability in the conventional sense,upon the basis of the theory expounded in the preceding sections so that theclassical developments can be made to follow from the present work. I n addi-tion to the axioms of §3 it is necessary to make a further assumption, the precisephilosophical character of which we shall not undertake to examine, concerningthe existence of what we shall call n-scales.

DEFINITION 1. B y an n-scale shall be meant any set of n propositions Oh , • • • ,ILO for which the following is posited:

1. u = ul V • • • V un O.

2. utui = 0 (i j) for all i, j 1, • • • , n.

3. ut/ u udu for all i, j = 1, • • • , n.

ASSUMPTION. I f n is any given positive integer, at least one n-scale may beregarded as existing.

THEOREM 14. I f (14 , • • • , u„) is an n-scale and (vi , • • • , ) a n m - s c a l e ,

and if p and cr are integers with 0 c r :5 n, 0 p f l m, thenui, V • • • V ui,lu <, , > v„ V • • • V vip/v

according as crIn <, = , > plm.Here it is understood that i1 , • • • , icr a r e e a c h b e t w ee n 1 a nd n , a nd t ha t no

two are equal; similarly for j, , • • • , jp and m. A n d , finally, when a = 0 thefirst member is replaced by 0/u, and when p 0 , the second by 0/v; the casewhere either or both of these are true is trivial, and shall be left out of con-sideration in our proof.

Since by hypothesis

u i ,/ u uilu, • • • , ui„/u

v„/v vi/ v , • • • , vav vidv,

we have by Theorem 6 (sharpened) that

ui, V • • • V ui, /u u l V • • • V ucr/u,

vi, V • • • V v i , / v v i V • • • V vp/v,

284 B . O . KOOPMAN

and thus it suffices to prove that

ul V • • • V %du <, ,› vl V • • • V v,/v

according as oin < , , › p/m.In virtue of our underlying assumption, we are permitted posit the existence

of an mn-scale( wj, • • • , w . ) . L e t Ui (i = 1, • • • , n) and Vi (j = 1, • • • , m)

be defined as follows

Now

U i = Wm(i-1)-1-1 V W m ( i — I ) + 2 V • " V W m i

V i = W n ( 1 - 1 ) + 1 V W n ( i - 1 ) + 2 V • • • V W O •

Clearly we have

w = wi V • • • V Winn = (11 V • • • V Un = VI V • • • V VM

= 0 , V i V i = 0 , /

U i / W U i / W , V i / W V i / W ;

the last, again by Theorem 6. T h u s we may apply Axiom S to ul , • • • , u nand Ul , • • • , Un and so we obtain udu U,/w, (i, j 1, • • • , n. Similarly,

for vi , • • • , vm and VI , • • • , Vm , we have vdvcz•-•--; Vdw, i, j = 1, • • • ,

Applying Theorem 6 again, we derive

v • • • V u,/u U l V • • • V U, /w

vi V • • • V v,,/v VI V • • • V Vp/w•

(11 v • • • v U, = Wi v • • • V w. ,

v • • • V V, = wi V • • • V wn, ,and thus by Theorem 3

v • • • v u,/u < , , › vi V • • • V vo/ v

according as

wi V • • • V Wm G and 1), = , D and l t wi V • • • V wn,that is, as mu < , = , n p , from which the theorem follows.

Fixing the attention upon a given eventuality a/h, let n be an arbitrarypositive integer, and NI , • • • , un) a c e r t a in a -s c a l e. T he r e l at i o n ul V • • • V

udu a / h will be true for at least one value of t (0 t n ) , i f we agree toinclude the case 0/u -< a/h for t = O. Furthermore, the maximum value of tfor which this relation holds is independent of the particular n-scale (ul , • • • , un)

chosen, as an obvious application of Theorem 14 will show: This value of tis thus determined when n is given (for fixed alh) and shall be denoted by On).

Similarly, the minimum value of T for which a/h -<,ul V • • • V

ur/u is

AXIOMS AND ALGEBRA OF INTUITIVE PROBABILITY

true is determined when n is given (for fixed alh), and shall be denoted by T(n).Evidently we have 0 t ( n ) T ( n ) n .

THEOREM 15. T h e following limits always exist:

p*(a,h) = limt(n) 13* (a, h) — lim

n —0 co n n -*co n

and they satisfy the relation

0 p*( a, h) 71*(a, h) 1.

Let n, m be any positive integers, and (241, • • • , u . ) a n d O h , • • •

n-scale and m-scale. B y definition of t(n), we can assertul V • • • V utoolu < a/h

and deny

Hence we must have

Similarly we prove that

On combining these we have

2,4 V • • • V u t ( * )+1 I u a l h .

t(m) t ( n ) +m n

for otherwise, by Theorem 14, we should have

al V • • • V atoo-Fau v • • • V vt(m)/v,whereupon a contradiction is produced in view of the known relation

th V • • • V vt(,..)/v < a/h.

On) < gm) + IM

1t ( m ) t ( n ) 1— — — ,m m n n

285

, v„,) any

and thus t(m)/m — t(n)ln 0 as n, m o o , and the existence of p*( a , h ) i sestablished. Simi lar ly for p*(a, h). T h e relation 0 p* p * 1 i s a nimmediate consequence of 0 t ( n ) T ( n ) n .

THEOREM 16. T h e relation al/hi a2 /h2 implies both ( a l , hi) P*( a 2 h 2 )

and p*(al , hi) P * ( a 2 , h2). A n d p*(al , hi) < , h2) implies the relational/hi 0,2/h2 •

The first part of the theorem is obvious. T o provd the second, we observethat since

712(n)

— 1 )*( a l , < 11* (a2 h2) — lim I2(n)

lim ,n n n

28(3 O. KOOPMAN'

we may take n such that Ti( n ) / n < t2( n ) / n , w h e r e u po n

al/hi < /LI V • • • V ur,(n)/o < ol V • • • V o f2( n ) / o a 2 / h 2 ,

in virtue of Theorem 14.THEOREM 17. p*( a , h ) + h) = 1.

Let (RI, • • • , u„) be an n-scale, and observe that for any s (0 s n),

—(al v • • v u8)/u = a8+1 V • • • V %du Uj v • • • v 24,_s/u.

Thus with the aid of Axiom A and the relation

v • • • V ut(t)/u a / h u l V • • • V u/.(„)/ u

we derive the formula

RI V • • • V un_T(,,)/u — a / h u l V • • • V un_g(„)/u.

Comparing this with the relation defining t'(ñ), T'(n) (for - -,a / h ) : v • • • V a t

,( n ) / a • •< u1 v

, • • • v ur,(n) / u ,

we have t' (a) n - T(n) and T ' (a) n - On). D iv id ing by n and lettingn 0 0 we have

p*(-a, h) + p*(a, h) 1, p*(a, h) + p*(a, h) 1.

Now the corresponding relations with a and —a interchanged are evidentlyvalid. T h u s 1 p*( a , h ) + h ) 1, and our t he or em is p ro ve d.

THEOREM 18. ( i ) p*(ai V a2 , h ) p * ( a l , h ) + p* (a2 , h ) . ( i i ) I fa

1a2h = 0, then p*(al v a2 , h) p*(al , h) + p*(a2 , h).

Let t(n), T(n) correspond to al V a 2 / h , a n d ti( n ) , Ti( n ) c o r r e sp o n d t o a dh ,

and, finally, t2(n), T2(n) correspond to a2/ 11 .

If for infinitely many values of n, Ti( n ) + T2( n ) n , d i v i d i n g b y n a n d

allowing n —> 00 through such values would show that p*(al , h) + p*(a2 , h ) 1 ,and conclusion (i) becomes obvious. O n the contrary assumption, for all suffi-ciently large n, Ti( n ) + T2( n ) < n , a nd we s ho u ld have

adh R I V • • • V ur,(n)/u,

a2/h v • • • V UT2(n) UT1(n)-1-1 V • • • V UT1(n)+T2(n)/U,

and hence, by Theorem 5

al V a2/h u l V • • • V uT1(n)+7,2(n)/u•(In this application of Theorem 5, the requirement that a

1a2h = 0 i s d i s p e n s e d

with, inasmuch as we replace a2 by a2 — a1, a n d c a n t h e n r e p l a c e a2/ h i n t h e

above relation by a2 — al/h which, by Theorem 3, - < a2/ 11 . ) T h u s w e o b t a i n

T (n) T 2(n) + T 2 ( n ), w he n ce the c on c lu s io n (i).


Turning now to (ii), suppose that for some n, ti( n ) + t2( n ) n . W e s h o u l d

have/4 V • • • V ati(n)/u al/ h ,

ut,(„)+1 V • • • V un/u ul V • • • V ut2(*)/u < a21h,

and thus by Theorem 5, u /u < al V a21 h , h e n c e b y T h e o r e ms 1 a n d 1 6 ,

P*(al V a2 , h) 1 , and (i i) is established. O n the contrary assumption,ti(n) + t2(n) < n, and we have

U t i ( n ) + 1 V • • • V U t l ( n ) + g 2 ( . ) / U V • • • V U t 2 ( n ) / U a2l hwhich, with the other relation, again by Theorem 5, yields

v • • • V uti(n)-F12(a)/u < al V a2/ h .

Hence ti( n ) + t2( n ) l (n) , and the conclusion (ii) follows.

THEOREM 19. I f a, b, c are any propositions for which a C b C c (b, c 0 ) ,then

(ii)

p*(a, b)p*(b, c) p*(a, c) p*(a, b)p*(13, c)

13* (a, b ) p*( b , c ) 6 7 )* ( a, c) p* (a, Op* (b, c).

We shall assume in the proof that a r 0 , the contrary case being trivial.Let ti( n ) , Ti( n) cor respond to a/c, let I2(n), T2(n) correspond to b/c, and

let t3( n ), 7 '3( n) correspond to a/b.

From a C b it follows by Theorem 3 that a/c < b/c, whence by Theorem 16that

(1) p*( a, c) 6 p*( b, c), 13*(a, c) p*(b, c).

We shall first establish the inequality

(2) p*( a, b )p*( b , c) p*(a, c).

We may evidently assume that t2( n ) 0 0 a s n — ) 0 0 , a s o t h e r w is e t h e c o n c lu s i o n

p*(b, c) 0 could be drawn from Theorem 15, and (2) would be trivial. Clearly,

for I2( n ) = 0 (

, , " • , u t20 0 ) f o r m s a t2( n ) -s c a l e : F or if ul V • • • V ut 2( * ) = 0

it would follow that ul = 0 ; h e n c e t h a t u , / u ui/ u 0 /u ; w he nc e by T he or em 6

that u/u 0 / u , contrary to Theorem 1. A n d the equiprobability of all ex-pressions

It dill v • • • V ut,(n) , 1 i t 2 ( n ) ,

is a consequence of that of all the ui/ u ( T h e o r e m 8 , w i t h a , b , k , h r e p l a ce d

by ui , u, , /11 V • • • V ut2(„) , u, respectively). Consequently we may write

v • • • V ut3(e2(.du1 V • • • V ut 2( „ ) < a l b .

This may be combined withv • • • V ut 2 (, ,)/ u b /c ,

288

the relation

B. O . K O O P M A N

the application of Axiom CI g i v i n g , u p o n s e t t i n g

, bi , h1 = ul V • • • V ut2(n) Ili V • • • V ut,[12(n)] , u;

, b2 , h2 = b , a, c,

ul V • • • v u tar t ,(.) J/ u < a /c ;

from which by definition of ti( n ) w e c o n c l u d e t h a t

t2[t2(n)]

Since I2(n) -> 0 0 w i t h n , we h av e for s u ff i ci e nt l y large n

t2(n) 13[I2(n)] < ti(n)n 12( n) — n

and on letting n --> 00 the inequality (2) is obtained.In a precisely similar manner, we establish the inequality

(3) p* (a, c) p * (a, b)p* (b, c).

In this case, the failure of the assumption T 2(n) 0 0 w i t h n w o u l d l e a d t o

73* (b, c) = 0 and hence by inequality (1), to /1*(a, c) 0 , and thus triviallyto (3). T h e remainder of the proof of (3) follows that of (2).

We turn i now to the proof of the inequality

(4) 1 3 * ( a , b ) p*( b , c ) p * ( c t , c ) .

Suppose that for infinitely many values of n, 12( n ) TI( n ) : i t w i l l f o l l o w a t

once that p*(b, c) 1 , * ( a , c), whence (4) results immediately, inasmuch asp*(a , b) 1 . Suppose on the contrary that Ti( n ) < t2( n ) f o r a l l s u f f i c i e n t l y

large n. W e invoke Axiom D in which we set

b, = a, hi c , h2 u ,

= ul V • • • V Ut2(n) b2 = Ul V • • • V Uri (n) •

In virtue of a C b C c and the definition of 12( n ) , Ti( n ) , w e h a v e

albi/h1 < a2b2/h2 , al/hi a2/h2

furthermore, we have effectively excluded the possibility of either of the equa-tions (since Ti( n ) = 0 w o ul d i mp l y a/c OA hence that a = 0)

( = a) = 0, a2b2h2 ( = U1 v • • • V /I T, ( n) ) = 0 ;

and thus the conclusion of Axiom D permits us to write

a/b u l V • • • V u ri( n ) / u t V • • • V U S2( n ) •


From this we conclude as in the proof of (2) that T3[t2(71)] T i ( n ) . I f t2( n )fails to become infinite with n, then p*( b , c ) 0 , a n d ( 4 ) i s i m m e d i a t e. A n d

when t2(n) 0 0 with n, we may write

12(n) T3[t2(n)] < Ti( n )

n 1 2 ( n ) = nwhence on letting n 0 0 the inequality (4) is established.

Turning lastly to the inequality

(5) p*(a, c) p*(a, b)p*(b, c),

we may always assume ti( n ) T2( n ) . F o r i f f o r s o me n 0 t i (n ) > 7 '2( n )

we should have by Theorem 14

< U1 v • • • V u2,2(n)iu < ul V • • • v ug,(n)/u < a/c

and b/c < a/c contradicts a C b, in view of Theorem 3. W e may then write

= ul V • • • V UT2(n)

ui V • • • V Uti (70 u,

a2 = b, b2 = a, h2 = c,

and exclude albihi = 0 as leading to ti( n ) = 0 a n d t h u s t o a t r i v i a l i t y , a n d

also observe that a2b2h2 = a X O. N o w by definition of VII) , T2(n), etc., wehave the following relations

alth/h1 a2b2/h2 , al/h1 a2/h2,

and thus by Axiom D, a relation obtains which takes the form

v • • • V t t e1( n ) / u 1 V • • • V u r2( , , ) a /b ;

thus, as before, ti( n ) t2[ T2( n ) ] . I f 7 '2( n ) f a il e d to b ec om e i nf in it e with n,

11*(1,, c) 0 , whence by (3) 2)*(a, c) 0 , so that p*( a , c ) 0 , g i v i n g ( 5 )

trivially. F ina l ly, in the case where 7 '2( n ) 0 0 w i t h n , w e h a v e

ti(n) < T2(n) 12[12(n)]

n = n T 2 ( n )

whence the inequality (5) when n 0 0 . T h u s the proof of our theorem iscomplete.

DEFINITION 2. T h e eventuality alh shall be said to be appraisable in the casewhere p*( a , h ) = 2 3* (a , h). When a/h is appraisable, the common limit

p(a, h) = p*( a , h ) = I ) * ( a, h)

shall be called the (numerical) probability of a/h (or of the contingency a on thepresumption h).

I t may be remarked that i f every CT were completely ordered by < , everyeventuality in i t would be appraisable.

If al/ h and a2/h are appraisable, then the inequality p(ai , hi) < p(a2 , h2)

290 B . O . K O O P M A N

implies the relation adhl < c t2/ h2 b y T h e o r em 1 6 . B ut t he e q ua t i on p (al , hi) =

P(a2 , h2) is perfectly consistant with any one of the relations al/h2 < , › , '------i,

a2/h2 Thus relations of numerical probabilities furnish but a blurred picture

of the more fundamental comparisons in probability; a circumstance which isillustrated by the familiar fact that we may have h(a, h) 0 even when ah O .

THEOREM 20. I f , • • • , a„) is a p-scale (p = 1, 2, • • • ) and i f b = sumof T distinct elements ai b = a „ v • • • V ( b 0 for r = 0),

then b/a is appraisable, and p(b, a) = r /v• I n particular (whenever a 0 ) ,0/a and ala are appraisable, and p(0, a) = 0, p(a, a) = 1.

In virtue of Theorem 14, to say that t(n) is the maximum t for whichv • • • v ug/ u b /a i s to say t ha t it is the maximum t such that t/n r/v,

and for such t(n),lim t(n)/n = T/p. A n d likewise lim T(n)/n = x/p, and ourtheorem is evident.

THEOREM 21. I n order that alh be appraisable i t is necessary and sufficientthat, corresponding to any e > 0, there should exist appraisable a' /h' and a" fit"such that

a' /h! < a/h < a"/h" ,p(a", h") — p(a', h') < e•

The necessity of the condition appears at once on setting a' = a " = a,h' = h" h .

To prove the sufficiency, i t is only necessary to observe that p(a' , h')p ,o( ct , h): otherwise, by Theorem 16, a' fit' > a/h; and similarly, 13*(a, h)

p(a", h"). Hence 13*(a, h) — p **( a , h ) < e a n d t h i s e b e i n g a r b i t r a ri l y s m a ll ,

1)*(a, h) = h ) follows.An obvious theorem is the following:THEOREM 22. I f 2)*(a, h) 0 , a/h will be appraisable. I f h ) = 1,

a/h will be appraisable.THEOREM 23. I f alh is appraisable, then ,- -,a / h w i l l b e a p p r a i s a b l e , a n d

p(a, h) + p(—a, h) 1 wi l l hold.For on applying Theorem 17, first directly, then with a and i n t e r -

changed, we obtain

13,1,(a, h) + h) = 1,

13*(a, h) + h ) = 1

which in conjunction with the hypothesized relation p,p(a, h) = 1,*(a, h), makethe theorem evident.

THEOREM 24. I f a1a2h = 0 a n d al/ h a nd a2/ h a re b ot h a p p ra i s ab l e , then

al V ct2/h will be appraisable, and

/ * I V a2 , h) = P(ai , h) + p(a2 , h ) .

This is the principle of total probability; i t may be extended to any finitenumber of disjoint summands. I t s truth is an immediate consequence o fTheorem 18.

THEOREM 25. I f a C b C c and if alb and blc are appraisable, then a/c willbe appraisable and p(a, c) = p(a, b)p(b, c) will hold.

This follows immediately from Theorem 19.THEOREM 26. I f a C b C c and a/c and b/c are appraisable, and lastly i f

p(b, c) 0 , then a/b will be appraisable, and p(a, c) p ( a , b)p(b, c) will hold.Setting p*( b , c ) 2 ) *( b , c) p(b, c) and p*(a, c) p*(a, c) p(a, c) in

Theorem 19, we havep*(a, b)p(b, c) 6 p(a, c) p*(a, b)p(b, c),

p)*(a, b)p(b, c) 6 p(a, c) 6 13*(a, b)p(b, c)whence 23*(a, b)p(b, c) 6 p*( a , b ) p ( b , c ) , a n d t h e p r o of i s c o m p le t e d by d i v id i ng

by p(b, c).THEOREM 27. 1 1 a C b C c and i f alb and blc are appraisable, then a ,-,-, blc

is appraisable, and p(a b , c) p ( a , c) — p(b, c).First suppose that p(b, c) O . T h e above proof of (6) may be applied,

whence i t follows that p(a, c) O . O n the other hand b a C b; henceb a / c < b lc (Theorem 3), so that by Theorem 16 0 p*( b a , c )23'1'(b a, c) 6 p(b, c) O. Thus b ,---, a/c is appraisable, and p(b a, c) 0 =

p(a, c) — p(b, c).Next, let p(b, c) O . T h e n by Theorem 26, alb is appraisable, and hence

by Theorem 23 —,a l b i s a p p r a i sa b l e a nd p ( - -,a , b) 1 — p(a, b). T he or em 25

with b a C b C c replacing a C b C c gives the appraisability of b a / cand we have

(6)

AXIOMS A N D ALGEBRA O F I N T U I T I V E PROBABILITY 2 9 1

p(b --, a, c) p ( b --, a, b)p(b, c) = b ) p ( b , c) = [1 — p(a, b)]p(b, c)p(b, c) — p(a, b)p(b, c) = p(b, c) — p(a,

THEOREM 28. L e t al , a 2 , h b e a n y p r o p o si t i o n s s uc h t ha t al C h, a2 C h,

and that adh and (12Ih a r e a p p r a i s ab l e . T h en al V a 2/ h w il l be a p pr a is a bl e

and only i f ala2/ h i s a p p r ai s a b le , and

P(al h ) P ( a 2 h ) = P(al V a2 , h) P ( a l a 2 , h).

Firstly, let ala2/ h b e a p p ra i s a bl e . If p(a l , h) = 0 and p(a2 , h) = 0, it

would follow by Theorem 18 that p*(al V a2 , h) = 0, whence the appraisabilityof al v a2/h (Theorem 22). In the contrary case, we should. have, let us say,

P(al h ) > O. T h e n by Theorem 27 applied with a = a1a2 , b = al , i t w o u l d

follow (since al --' (ala2) = a 2 ) that al , - - - , a2l h i s a p p r a i s a b l e , a n d t h a t

P(al a 2 , h) = P(al h ) P ( a l a 2 , h). N o w we have al v a2/h a 2 Va2/h, (al a2) a2h 0 ; a nd h en ce by Theo rem 24, a, v a2lh is appraisable, and

P(al V a , h) = P(al a 2 , h) + P(a2 = P(al h ) — 1)(a1a2 , h) + p(a2, h);and the first half of our theorem is established.

292 B . O . K O O P M A N

Secondly, let al v a2/ h b e a p p ra i s a bl e . T he or em 23 then e st ab li sh es the

appraisability of V a2)/h = a 2 / h , a n d ,a2/ h , a n d g i v e s t h e

relationsa2 , h) = 1 — P(al V a2 , h),

= 1 — P(ai 1 ) ( ' ' a 2 , h) = 1 — p(a2 , h ) .Now the established half of the present theorem yields the appraisability of

a21h, and the equation+ , h) = p(--'al V , + a 2 , h).

A second application of Theorem 23 gives us the appraisability of V--'a2)/h = ala2/h, and also that p(aia2 , h) = 1 — p(--aal V , h). O n com-bining these equations, the desired conclusion is established.

We thus reach the conclusion of this section, and with it our final goal, havingdefined the numerical propability and established its conventional properties:In virtue of the Theorem of Representation of Boolean Algebras of GarrettBirkhoff and M. H. Stone' our underlying Boolean ring 21 corresponds with aclosed system of subclasses of a certain E, in a subsystem of which a (restrictedly)additive set function pH( A ) = p ( a , h ) i s d e f i ne d w h ic h s a t is f i es the A xi om s of

Kolmogoroff,15 f r o m w h i ch t he c l as s ic a l t he or y of p ro ba bi l it y may be derived.

Our proposed task is thus accomplished.The relation with frequency in sequences will be studied in a forthcoming

paper.

COLUMBIA UNIVERSITY.

14 For ful l references and discussion, see M. H. Stone, The Representation of BooleanAlgebras, Bul l . Amer. Math. Soc., Vol . 44, Dec. 1938, pp. 807-816.

16 A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeits Rechnung. Ergebnisse derMathematik, B2. ( B e r l i n 1933.)

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THE AXIOMS AND ALGEBRA OF INTUITIVE PROBABILITY