A Theory of Belief

Journal of Mathematical Psychology 47 (2003) 1–31

A theory of belief Louis Narens Department of Cognitive Sciences, University of California, Irvine, (UCI), 3151 Social Science Plaza, Irvine, CA 92697-5100, USA Received 9 August 1999; revised 8 January 2002

Abstract

A theory of belief is presented in which uncertainty has two dimensions. The two dimensions have a variety of interpretations. The article focusses on two of these interpretations. The first is that one dimension corresponds to probability and the other to ‘‘definiteness,’’ which itself has a variety of interpretations. One interpretation of definiteness is as the ordinal inverse of an aspect of uncertainty called ‘‘ambiguity’’ that is often considered important in the decision theory literature. (Greater ambiguity produces less definiteness and vice versa.) Another interpretation of definiteness is as a factor that measures the distortion of an individual’s probability judgments that is due to specific factors involved in the cognitive processing leading to judgments. This interpretation is used to provide a new foundation for support theories of probability judgments and a new formulation of the ‘‘Unpacking Principle’’ of Tversky and Koehler. The second interpretation of the two dimensions of uncertainty is that one dimension of an event A corresponds to a function that measures the probabilistic strength of A as the focal event in conditional events of the form AjB; and the other dimension corresponds to a function that measures the probabilistic strength of A as the context or conditioning event in conditional events of the form CjA: The second interpretation is used to provide an account of experimental results in which for disjoint events A and B; the judge probabilities of AjðA,BÞ and BjðA,BÞ do not sum to 1. The theory of belief is axiomatized qualitatively in terms of a primitive binary relation h on conditional events. (AjBhCjD is interpreted as ‘‘the degree of belief of AjB is greater than the degree of belief of CjD:’’) It is shown that the axiomatization is a generalization of conditional probability in which a principle of conditional probability that has been repeatedly criticized on normative grounds may fail. Representation and uniqueness theorems for the axiomatization demonstrate that the resulting generalization is comparable in mathematical richness to finitely additive probability theory. r 2003 Elsevier Science (USA). All rights reserved.

1. Introduction Tversky, 1997). Various alternatives to subjective probability for measuring belief have been proposed in 1.1. Introduction the literature. A major drawback to most of them was that they lacked interesting mathematical structure and That beliefs can be compared in terms of strength is effective means of calculation for understanding and intuitively compelling. The validity of such comparisons manipulating degrees of belief. underlie the justiﬁcations of many practical methods of Traditionally the probability calculus, as encom- decision, for example, ‘‘beyond a reasonable doubt’’ passed by the axioms proposed by Kolmogorov decisions by juries, selecting treatments for medical (1933), has been taken as the normative calculus for patients, etc. de Finetti (1931, 1937) proposed that manipulating degrees of uncertainty. While the Kolmo- strengths of belief could be measured and compared gorov calculus is arguably a very good idea for through subjective probabilities. However, many situations like casino gambling in which long random thought this proposal unwarrantably restrictive, because sequences exist and are easily observable, it is much it excluded important belief situations such as normative more controversial for cases of uncertainty in which theories evidence (e.g., Shafer, 1976) or descriptive random sequences are either difﬁcult to observe or are theories of how individuals evaluate uncertain proposi- impossible by the nature of the events involved. tions (e.g., Tversky & Koehler, 1994; Rottenstreich & This article investigates an alternative to the probability calculus that is based on the idea that uncertainty E-mail address: [email protected]. is measured by probability and an additional dimension.

This alternative is viewed as a very modest general- ignorance of c and d: Then, because of the differences in ization of the probability calculus. As such, it allows for the understanding of the nature of the probabilities a focussed discussion about its acceptability as a involve, a lower degree of belief may be assigned to generalization of various concepts based on the prob- ðcjc; dÞ than to ðaja; bÞ; thus invalidating Eq. (2). ability calculus. Suppose Eq. (1) and the judgements of the likelihoods In the literature, there are several different founda- of the occurrences of a and c; given either a or c occurs, tional approaches to probability that produce calculi are based on much information about a and c and the satisfying the Kolmogorov axioms. (Various examples nature of the uncertainty involved, and the judgment the can be found in Fine, 1973; van Lambalgen, 1987.) In likelihoods of the occurrences of b and d given either b particular, de Finetti (1937) and Savage (1954) pio- or d occurs is due to the lack of knowledge of b and d: neered approaches designed to capture personal or This may result in different degrees of belief being subjective probability functions, and the theory of assigned to ðaja; cÞ and ðbjb; dÞ; and such an assignment conditional belief developed here extends these and would invalidate Eq. (3). related approaches to a more general class of personal The primary goal of this article is to examine measuring functions that takes into account not only axiomatic theories of belief that result by making the uncertainty but also possibly information about the above deletion. These theories will be evaluated in terms nature of the uncertainty, and in certain psychological of their mathematical power, philosophical acceptabil- settings information about the processing of uncertain- ity, and applicability. ties. Like de Finetti and Savage, the theories of Such axiomatic theories provide generalizations of conditional belief developed in the article are axiomatic subjective conditional probability. The generalizations, and based on a primitive qualitative preference ordering. which are called theories of subjective conditional belief Several axiomatizations of increasing complexity are or conditional belief for short, will have similar levels of presented. They are all motivated in part by the fact that calculative power and mathematical richness as sub- they can be viewed as specific weakenings of corre- jective conditional (finitely additive) probability. In sponding axiomatic systems for related versions of addition, they will also provide for nonprobabilistic conditional probability. dimensions of belief, for example, a dimension of For events e and f ; let ðeje; f Þ stand for ‘‘likelihood of ‘‘ambiguity.’’ e occurring if either e or f occurs.’’ The key axiom of conditional probability that is deleted in all the general- Convention 1.1. Let A be a subset of the nonempty set izations presented in this article asserts that for all B: In this article, when discussing ðAjBÞ; A will often be distinct states of the world a; b; c; and d; if referred to as the choice set, choice,orfocus (of ðAjBÞ) ðaja; bÞBðbja; bÞ and ðcjc; dÞBðdjc; dÞ; ð1Þ and B as the context (of ðAjBÞ). then In some belief situations, we have for each choice A in ðaja; bÞBðcjc; dÞð2Þ context B very good evidence for the strength of ðAjBÞ— and in other belief situations, we have only poor and evidence for evaluating the strength of ðCjDÞ for choices ðaja; cÞBðbjb; dÞ: ð3Þ C in context D: In cases with good evidence for all The intuition for the desirability of the above deletion choice and context sets, I believe it is reasonable to is the following: Assume measure the strength of beliefs as probabilities; and in cases where there is poor evidence for some choice sets, I ðaja; bÞ¼ðbja; bÞ and ðcjc; dÞ¼ðdjc; dÞ: ð4Þ agree with Shafer (1976) and others that on normative In the context of the other axioms for conditional grounds it is unreasonable to demand that strengths of probability, the above asserts that if a and b have belief be measured as probabilities. In this article, equal likelihood of occurring and c and d have strengths of belief will be measured as ‘‘degrees of equal likelihood of occurring, then the conditional belief,’’ which like probabilities are nonnegative real 1 probabilities of ðaja; bÞ and ðcjc; dÞ are 2; and the numbers, but unlike probabilities, (i) need not be less conditional probabilities of ðaja; cÞ and ðbjb; dÞ are the than or equal to 1, and (ii) need not be additive for same. beliefs of unions of disjoint events. Suppose Eq. (1) and the judgment of equal likelihood of the occurrences of a and b; given either a or b occurs, 1.2. Belief functions is based on much information about a and b and a good understanding of the nature of the uncertainty involved, Definition 1.1. Let D be a nonempty set. Then kD is said and the judgment of equal likelihood of the occurrences to be a belief function on D if and only if kD is a function of c and d; given either b or d occurs, is due to the lack on the set of subsets of D such that for each EDD; (i) of knowledge of b and d; for example, due to complete 0pkDðEÞ; and (ii) kDðEÞ¼0 iff E ¼ |: L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 3

Let kD be a belief function. Then the following ky’s and Koehler’s support theory, Rottenstreich’s and definitions hold: Tversky’s generalization of support theory, and Bren- ner’s and Rottenstreich’s asymmetric support theory are (i) The value of kD on a subset E of D; kDðEÞ; is called accounted for. E’s degree of belief (under kD). The belief theories presented in this article may (ii) kD is additive if and only if and for all subsets E interpreted as either descriptive or normative theories. | and F of D; if E-F ¼ ; then kDðE,FÞ¼ As descriptive theories, they may be useful as alter- kDðEÞþkDðFÞ: natives for existing theories and as a means for (iii) kD is monotonic if and only if for all subsets E and suggesting new experimentation. A separate article will C F of D; if E F then kDðEÞokDðFÞ: argue for their acceptability as normative theories for (iv) kD is modest if and only if it is monotonic and for certain kinds of probabilistic situations. all subsets E of D; 0pkDðEÞp1: 2. Basic axioms for conditional belief (v) kD has norm 1 if and only if kDðDÞ¼1: (vi) kD is a subjective probability function if and only if it has norm 1 and is modest and additive. In this section qualitative characterizations of various systems of conditional belief are presented and quantitative representation and uniqueness theorems are Lemma 1.1. Let kD be a belief function. Then the shown for them. following two statements are true: For purposes of presentation, the proofs of the theorems stated in this section are given in Section 7. 1. If kD is additive, then it is monotonic. 2. If k has norm 1 and is additive, then it is a subjective D Definition 2.1. Throughout this article D will denote the probability function. subset relation, C the proper subset relation, R the real numbers, Rþ the positive real numbers, I the integers, Proof. Immediate from Definition 1.1. & and Iþ the positive integers. Throughout the article, h will denote a binary The values of a belief function are called degrees of relation that is transitive, reflexive, and connected belief, and when the belief function is a subjective (either xhy or yhx for all x; y in the domain of h). probability function, they are often called subjective Such transitive, reflexive, and connected relations on probabilities or simply probabilities. Under one inter- nonempty sets are called weak orderings. The symmetric pretation, degrees of belief are distortions of subjective part of h is denoted by B; and is defined by, probabilities that take into account nonprobabilistic xBy iff xhy and yhx; aspects present in the choice situation, for example, ‘‘ambiguity.’’ Under this interpretation, it may be the for all x and y in the domain of h; and the asymmetric case that some degrees of belief of elements of D distort part of h is denoted by g and is defined by, probability by producing an increase and others by xgy iff xhy and not yhx: producing a decrease so that over D the effects of the distortions cancel and kDðDÞ¼1: Throughout this article X will denote an infinite set of In Sections 2 and 3 axiomatic, qualitative character- objects. By definition, a context is a finite subset of X izations of belief functions of the form that has at least two elements. C will denote the set of contexts. The notation ðajCÞ will denote C is a context k ðAÞ¼P ðAÞvðAÞ B B and aAC: ‘‘ðajCÞ’’ will often be read as ‘‘strength of are presented, where PB is a finitely additive probability a in the context C:’’ When C ¼fa; a1; y; ang; ðajCÞ function on B and v is from a ratio scale of functions will often be written as ðaja; a1; y; anÞ: By convention, into the positive reals. For each family of subsets D on the notation ðaja; a1; y; anÞ assumes the elements which v is constant, kB acts on D like a finitely additive a; a1; y; an are distinct. probability function in the sense that for all E and F in A nonempty set S of functions from a nonempty set þ D such that E-F ¼ |; Y into R is said to be a ratio scale if and only if (i) for each rARþ and each f AS; rf AS; and (ii) for all f and g kBðE,FÞ¼kBðEÞþkBðFÞ: in S there exists s in Rþ such that f ¼ sg: This form of additivity need not hold for events with A nonempty set S of functions from a nonempty set þ different v-values. These features of kB are use in this Y into R is said to be an interval scale if and only if (i) article to account for various empirical phenomena in for each rARþ; each s in R; and each f AS; rf þ sAS; observed in human probability estimation. In particular and (ii) for all f and g in S there exist r in Rþ and s in R phenomena described by the Ellsberg Paradox, Tvers- such that f ¼ rg þ s: 4 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

2.1. Basic belief axioms from the fact that h is a weak ordering that the induced orderings on X and C described above are also The Basic Belief Axioms consist of the following 12 weak orderings. axioms. They provide the mathematical core of the axiom systems for this article. Deﬁnition 2.3. An ordered pair of functions /u; vS is said to be a basic belief representation for h if and only Axiom 1. h is a weak ordering on the set if the following three conditions hold: fða; CÞjCAC and aACg: 1. u and v are functions from X into Rþ: h 2. For all C; D in C; C CD iff X X Axiom 2. Suppose AAC; aAA; and B is a nonempty uðeÞX uðeÞ: ﬁnite subset of X such that B-A ¼ |: Then eAC eAD ðajAÞgðajA,BÞ:

Axiom 3. Suppose A; B are in C and C is a nonempty 3. For all distinct a; a1; y; an and all distinct finite subset of X such that A-C ¼ B-C ¼ |; and b; b1; y; bm; suppose a and b are elements of X. Then ðaja; a1; y; anÞhðbjb; b1; y; bmÞ (i) if aAA and aAB; then if and only if ðajAÞhðajBÞ iff ðajA,CÞhðajB,CÞ; uðaÞ vðaÞ u a u a ? u a (ii) if aAA and bAA; then ð Þþ ð 1Þþ þ ð nÞ uðbÞ ðajAÞhðbjAÞ iff ðajA,CÞhðbjA,CÞ; XvðbÞ : uðbÞþuðb1Þþ? þ uðbmÞ and (iii) if aAA; aAB; bAA; and bAB; then Axioms 2 and 3 are necessary conditions for the ðajAÞhðajBÞ iff ðbjAÞhðbjBÞ: existence of a basic belief representation for h—as is easy to verify directly through Definition 2.3. Similarly, direct verification shows that the following three axioms are also necessary for the existence of a basic belief h Through the use of Axiom 3, induces natural representation for h: orderings on X and C as follows: Axiom 4. For all a; b; c; e; and f in X; if Definition 2.2. Define hX on X by: for all a; b in X; aab; aac; e e; a B e e; b ; and f f ; a B f f ; c ; ahX b if and only if there exists a finite set C such that ð j Þ ð j Þ ð j Þ ð j Þ D h C X Àfa; bg and ðajfa; bg,CÞ ðbjfa; bg,CÞ: then fa; bgBCfa; cg:

Note that it follows from Axiom 3 and the definition 0 0 0 0 h X ah b for all C Axiom 5. For all a; a ; b; b in X and all A; A ; B; B in C; if of X on above that X if and only if 0 0 0 0 aBX a ; bBX b ; and A-A ¼ B-B ¼ |; then the D if C is finite and C X Àfa; bg; following two statements are true: then ðajfa; bg,CÞhðbjfa; bg,CÞ: 1. If ðajAÞhðbjBÞ and ða0jA0Þhðb0jB0Þ; then ðajA, h 0 0 Similarly define C on C by: for all C; D in C; A ÞhðbjB,B Þ: h 0 0 0 0 D CC if and only if 2. If ðajAÞgðbjBÞ and ða jA Þhðb jB Þ; then ðajA, 0 0 ðajfag,CÞhðajfag,DÞ A ÞgðbjB,B Þ: for some a in X such that a is not in C,D: Note that it follows from Axiom 3 and the definition of h on C Axiom 6. Suppose a; b are arbitrary elements of X and above that Dh C if and only if A; B are arbitrary elements of C and aAA and bAB: Then C the following two statements are true: ðajfag,CÞhðajfag,DÞ 1. If AB B then for all a in X such that a is not in C,D: C ahX b iff ðajAÞhðbjBÞ: h Note that Definition 2.2 says that C CD if and only if for some a in X ÀðC,DÞ the strength of a in 2. If aBX b then context fag,C is less than or equivalent to its strength in h h context fag,D: Also note it immediately follows A CB iff ðbjBÞ ðajAÞ: L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 5

Axiom 5 in the presence of the other axioms 2. (Uniqueness theorem) Let corresponds to a well-investigated axiom of measure- U ¼fujthere exists v such that /u; vS is a basic belief ment theory known as ‘‘distributivity.’’ Axiom 6 representation for h corresponds to another well–investigated axiom of g measurement theory known as ‘‘monotonicity.’’ To obtain strong results about basic belief representa- 3. and tions for h; other axioms are needed. The following V ¼fvjthere exists u such that /u; vS is a basic belief ones imply that the situation under consideration is rich h in objects and contexts. representation for g:

A A h Axiom 7. For all a X and B C; if B CA for some A in 4. Then U and V are ratio scales. C such that aAA; then there exist cAX and CAC such that aBX c; BBCC; and cAC: Proof. Similar to Theorem 8.2. & Axiom 8. For all a; b in X and all A in C; if aAA and 2.2. Basic belief axioms with binary symmetry ahX b; then there exists c in X and C in C such that B B b X c; A CC; and cAC: Axiom 14 (Binary symmetry). Let a; b; c; and d be Axiom 9. For all A; BinC and all bAB; there exist cAX arbitrary, distinct elements of X such that and C in C such that bBX c; BBCC; A-C ¼ | and ðaja; bÞBðbja; bÞ and ðcjc; dÞBðdjc; dÞ: ðbjBÞBðcjCÞ: Then B Axiom 10. For each A ¼fa1; y; ang in C there exist ðaja; bÞ ðcjc; dÞ 0 0 y 0 0 | A ¼fa1; ; ang in C and e in X such that A-A ¼ ; and feg-ðA,A0Þ¼|; and for i ¼ 1; y; n; ðaja; cÞBðbjb; dÞ: B 0 ðeje; aiÞ ðeje; aiÞ:

Deﬁnition 2.5. The basic belief axioms with binary g Axiom 11. For all A; BinC; if A CB then there exists C symmetry consists of the basic belief axioms together | g in C such that B-C ¼ and A CB,C: with the axiom of binary symmetry (Axiom 14).

Axiom 12. The following two statements are true: Deﬁnition 2.6. A function u is said to be a basic choice A representation for h if and only if the following four 1. For all a; b in X and all A in C; if a A and bhX a; then there exist c and C such that conditions hold: þ cBX b and ðcjCÞBðajAÞ: 1. u is a function from X into R :

2. For all a; b in X; ahX b iff uðaÞXuðbÞ: 3. For all C; D in ; 2. For all a in X and all A; BinC; if aAA; then there exist XC X h X c and C such that C CD iff uðeÞ uðeÞ: eAC eAD CBCB and ðcjCÞBðajAÞ:

Axiom 13 (Archimedean axiom). For all A; B; B1; y; 4. For all distinct a; a1; y; an and all distinct þ Bi; y in C; if for all distinct i; jinI Bi-Bj ¼ | and b; b1; y; bm; B B B; then for some nAIþ i C ðaja; a1; y; anÞhðbjb; b1; y; bmÞ [n BkhCA: k¼1 5. if and only if uðaÞ uðbÞ X : Definition 2.4. Axioms 1–13 are called the Basic Belief uðaÞþuða1Þþ? þ uðanÞ uðbÞþuðb1Þþ? þ uðbmÞ Axioms. Theorem 2.2. Assume the basic belief axioms with binary Theorem 2.1. Assume the basic belief axioms (Definition symmetry (Definition 2.5). Then the following two 2.4) are true. Then the following two statements hold: statements hold: 1. (Representation theorem) There exists a basic belief 1. (Representation theorem) There exists a basic choice representation for h (Definition 2.3). representation for h (Definition 2.6). 6 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

2. (Uniqueness theorem) The set of basic choice repre- Because in many ways these theories can be viewed as sentations for h forms a ratio scale. minimal generalizations that result from elimination of an obvious questionable principle for conditional belief (binary symmetry), they are ideal candidates for Proof. Similar to Theorem 8.3. & preliminary investigation.

2.3. Comments 2.4. The BTL model of choice

For intuitive purposes, the axioms presented through- The basic belief axioms with binary symmetry out this article may be divided into three rough (Deﬁnition 2.5) provide (via Theorem 2.2) a qualitative categories: (i) substantive axioms that reveal important description of a widely used, important quantitative structural relationships about conditional probability model in the behavioral sciences called the ‘‘BTL and belief; (ii) richness axioms that guarantee that we are Model:’’ dealing with rich probabilistic and belief situations; and Much behavioral science research involves the model- other axioms that belong to neither categories (i) nor (ii). ing of the probabilistic choice of objects from a set of The role of the substantive axioms is to describe what alternatives. A particularly important model is one in conditional probability and belief are in rich settings; which objects are assigned positive numbers by a and the role of the richness and other axioms is to function u so that the probability p that object a is guarantee that such a description can be made easily and chosen from the set of alternatives faja1; y; ang is given will work. Examples of richness axioms are Axioms 7– by the equation, 12 that state the existence of certain kinds of solvability uðaÞ relations in terms of B and g: The Archimedean axiom p ¼ : ð5Þ ? (Axiom 13) is an example of an ‘‘other axiom.’’ uðaÞþuða1Þþ þ uðanÞ Throughout this article, richness axioms are freely In the literature, this model is often called the Bradley– employed to simplify exposition. In some instances this Terry–Luce model, which is often abbreviated to the results in redundancy in the axioms. However, having BTL model.1 nonnecessary and sometimes redundant or unneeded There are many ways in which the ordering h on axioms does not impede the main objectives of the choice strengths of objects in contexts can be established article—to formulate and evaluate generalizations of empirically. In particular, by letting ‘‘ðajAÞhðbjBÞ’’ conditional probability in terms of normativeness, stand for ‘‘The conditional probability of a being chosen mathematical power, and applicability. The substantive from A is at least as great as the conditional probability axioms thus far are Axioms 1–6 and binary symmetry of b being chosen from B;’’ a model of the basic belief (Axiom 14). axioms with binary symmetry results. However, other As Theorem 2.2 shows, the basic belief axioms with interpretations can be given to h that also yield the binary symmetry qualitatively describes a situation of basic belief axioms with binary symmetry. The added conditional probability. Taking the perspective that ﬂexibility of multiple interpretations of primitives is one conditional belief may differ from conditional prob- of the great strengths of the qualitative approach to ability, it then follows that there may be situations in axiomatization. This extends to the basic belief axioms. which one or more of the basic belief axioms with binary For example, the characteristic property of the BTL symmetry may fail to adequately characterize condi- model of choice is that context plays no role in tional belief. The basic belief axioms assume such a determining the odds of alternatives (Luce’s Choice possible failure, namely the failure of binary symmetry Axiom). Smith and Yu (1982) formulate a quantitative (Axiom 14). That binary symmetry should be invalid in generalization where the odds of alternatives may vary certain kinds of belief and choice situations has been with the contexts in which they occur. In this general- repeatedly suggested in the literature. ization, the function The objective of this article is to give and evaluate PðxjCÞ extensions of the basic belief axioms as theoretical ; alternatives to conditional probability. Of course, such ZðxÞ extensions are likely at best to produce only partial theories of conditional belief, because only binary 1 A special case of the BLT model, symmetry (Axiom 14) is deleted. To obtain general uðaÞ theories of belief, other axioms will likely have to be p ¼ : ð6Þ changed as well. Thus, the theories of conditional belief uðaÞþuða1Þ was used by the famous set-theorist E. Zermelo to describe the power of this paper should be viewed and evaluated as of chess players (Zermelo, 1929). The choice models implicit in Eqs. (5) particular generalizations of conditional probability and (6) have been used by Bradley and Terry (1952), Luce (1959), and which apply to a restricted set of belief situations. many others in behavioral applications. L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 7

P which is mathematically equivalent to ðxjCÞvðxÞ; is Axiom 17. For each nonempty finite event A of X there used as a measure of the ‘‘context sensitivity’’ of x in exists e in X À A such that context C: ðPðxjCÞ denotes the probability of x given | C:) Thus, by Theorem 2.1, the basic belief axioms (1) for each finite event B of X, if CACB (and provide a qualitative account of ‘‘context sensitivity.’’ therefore B has at least two elements), then there B Smith and Yu apply their theory to choice situa- exists E in C such that B CE (Definition 2.2) and B tions in which there is a similarity structure on the ðAjBÞ EðejEÞ; and alternatives. (2) there exists f in X such that f ae; f eA; and

3. Belief axioms ðf je; f ÞBEðf jA,ff gÞ:

Definition 3.1. Let F ¼fFjF is a finite subset of Xg: Axiom 18. Suppose |CACB; |CACB0; eAE; e0AE0; B 0B 0 Elements of F are called finite events (of X). Elements of B CE; B CE ; and 0 0 0 F Àf|g are called context events (of X). In the notation ðAjBÞBEðejEÞ and ðAjB ÞBEðe jE Þ: ‘‘ðAjBÞ;’’ where A is in ; B is in Àf|g; and ADB; A F F Then is called the focal event of ðAjBÞ and B is called 0 B 0 0 the context event of ðAjBÞ: By definition, the finite ðeje; e Þ ðe je; e Þ: conditional events (of X) consists of all ðAjBÞ where AAF; B in F Àf|g; and ADB: (In terms of the earlier Axiom 17 associates with each finite conditional event notation, in ‘‘ðaja; bÞ;’’ a is focal event fag and a; b is the ðAjBÞ; with Aa| and B having at least two elements, a context event fa; bg; and thus ðaja; bÞ is the same as finite conditional event of the form ðejEÞ; where eAX ðfagjfa; bgÞ: and EAC; such that ðAjBÞB ðejEÞ: The basic belief axioms and basic belief axioms with E B binary symmetry are concerned with finite conditional Since E is an extension of h; this allows the h events ðAjBÞ; where B has at least two elements and A placement of ðAjBÞ in the E-ordering to be determined has the special form A ¼fbg for some b in B: This by the placement of ðejEÞ in the h-ordering. section extends these axiom systems to all finite Axioms 17 and 18 play a critical role in extending a conditional events. To accomplish this, a new primitive basic belief representation /u; vS for h to elements of h h relation E is introduced and additional axioms the domain of E: (uðAÞ is defined to be uðeÞ and vðAÞ h involving E are assumed. is defined to be vðeÞ; where e is as in condition (1) of Axiom 17.) Axiom 15. The following two statements are true: h Axiom 19. The following two statements are true: (1) E is a weak order on the set of finite conditional events of X. 1. For each a in X there exists E in C such that h h h (2) E is an extension of (where is as in Axiom 1). ðfagjfagÞBEðEjEÞ:

Finite conditional events of the forms ð|jBÞ; ðAjBÞ 2. Let ðAjBÞ and ðCjDÞ be arbitrary ﬁnite conditional with A having at least two elements, and ðBjBÞ are events such that B and D are in C: Then there exist 0 0 0 0 not involved in the h-ordering. The following four ﬁnite events B and D such that B-B ¼ |; BBCB ; 0 0 axioms specify, with respect to the h-ordering, the D-D ¼ |; DBCD ; and placement within the h -ordering of these three kinds 0 0 E ðAjBÞh ðCjDÞ iff ðAjB,B Þh ðCjD,D Þ: of events. E E

Axiom 16. The following two statements are true for all Definition 3.2. The belief axioms consist of the basic finite conditional events ðAjBÞ of X and all context events belief axioms (Definition 2.4) together with Axioms CofX: 15–19. (1) ðAjBÞBEð|jCÞ iff A ¼ |: Definition 3.3. B is said to be a belief representation for h with context function u and definiteness function v if (2) E if Aa| Then ðAjBÞgEð|jCÞ: and only if the following three conditions hold: 8 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

(i) u and v are functions from F into, respectively, that he can always assign relative likelihoods to the Rþ,f0g and Rþ: states of nature. But how does he act in the presence (ii) For each C in F; of his uncertainty? The answer to that may depend on C ¼ | iff uðCÞ¼0; another sort of judgment, about the reliability, credibility, or adequacy of his information (including and X his relevant experience, advice and intuition) as a if Ca|; then uðCÞ¼ uðfcgÞ: whole: not about the relative support it may give to cAC one hypothesis as opposed to another, but about its ability to lend support to any hypothesis at all. (iii) B is a function on the finite conditional events of X If all the information about the events in a set of such that for all finite conditional events ðAjBÞ and gambles were in the form of sample-distributions, the ðCjDÞ of X; ambiguity might be closely related, inversely to the size of the sample. But sample-size is not a universally A B h C D iff B A B XB C D ð j Þ Eð j Þ ð j Þ ð j Þ useful index of this factor. Information about many and events cannot be conveniently described in terms of a uðAÞ sample distribution; moreover, sample-size seems to BðAjBÞ¼vðAÞ : focus mainly on the quantity of information. uðBÞ ‘‘Ambiguity’’ may be high (and the confidence in any particular estimate of probabilities low) even where there is ample quantity of information, when h there questions of reliability and relevance of Belief representations for E with context function u and definiteness function v are useful in applications information, and particularly where there is conflict- where one wants to characterize events in terms of a ing opinion and evidence. probabilistic dimension and another dimension. For the This judgment of the ambiguity of one’s informa- purposes of this article, the other dimension is called tion, of the over-all credibility of one’s composite ‘‘definiteness.’’ Definiteness may split into additional estimates, of one’s confidence in them, cannot be dimensions. In the intended interpretations, u measures expressed in terms of relative likelihoods or events (if the probabilistic dimension and v measures definiteness. it could, it would simply affect the final, compound One naturally encounters various kinds of ‘‘definite- probabilities). Any scrap of evidence bearing on ness,’’ and the intended interpretations of v may vary relative likelihood should already be represented in with the kinds of definiteness. those estimates. But having exploited knowledge, Consider the example of evaluating evidence in guess, rumor, assumption, advice to arrive at a final criminal cases. Here we assume that ‘‘guilty beyond a judgment that one event is more likely than another reasonable doubt’’ is determined by having a sufficiently or that they are equally likely, one can still stand back high ‘‘degree of belief.’’ Consider the following two from this process and ask: ‘‘How much, in the end, is situations: (1) where the evidence is purely circumstan- all this worth? How much do I really know about the tial, and (2) where the evidence is almost entirely problem? How firm a basis for choice, for appro- physical. Suppose the subjective probabilities for the priate decision and action, do I have?’’ The answer, two situations are the same, e.g., 0:998: I believe it is ‘‘I don’t know very much, and I can’t rely on that,’’ reasonable in this case to assign (2) a higher degree of may sound familiar, even in connection with mark- belief than (1). This reflects the idea that degree of belief edly unequal estimates of relative likelihood. If depends not only on probabilities, but also on the kinds ‘‘complete ignorance’’ is rare or nonexistent, ‘‘con- of evidence that the probabilities are based on. A siderable’’ ignorance is surely not. particular version of this idea is presented in the concept of ‘‘belief representation of h with context function u E The next theorem, which extends Theorem 2.1 from and definiteness function v:’’ singleton focal events to finite events, provides existence Definiteness may also be viewed as an ordinal and uniqueness results concerning belief representations opposite of the following ‘‘ambiguity’’ concept of for h : Ellsberg (1961, pp. 659–660): E Theorem 3.1. Assume the belief axioms (Definition 3.2). Then the following two statements are true: Let us assume, for purposes of discussion, that an h individual can always assign relative weights to 1. There exists a belief representation for E with alternative probability distributions reflecting the context function u and definiteness function v. B h relative support given by his information, experience 2. Let be a belief representation for E with context and intuition to these rival hypotheses. This implies function u and definiteness function v. Then the L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 9

following two statements are true: Note that by Definition 4.1, each choice representa- h (i) For all positive reals r and s there exists a belief tion for E is a conditional probability function on the h representation for E with context function ru finite conditional events of X: and definiteness function sv. B h (ii) Let 1 be a belief representation for E with Definition 4.2. The belief axioms with binary symmetry context function u1 and definiteness function v1: consist of the belief axioms together with the axiom of Then for some positive real numbers r and s, binary symmetry (Axiom 14).

u1 ¼ ru and v1 ¼ sv: Theorem 4.1. Assume the belief axioms with binary symmetry (Deﬁnition 4.2). Then there exists a choice & h Proof. Similar to Theorem 8.4. representation for E:

/ S h Let u be as in statement 1 of Theorem 3.1, and for Proof. Let u; v be a belief representation for E finite events E and F of X with ECF let with context function u and definiteness function v: It needs to be only shown that for all nonempty finite uðEÞ PF ðEÞ¼ : events A and B of X; vðAÞ¼vðBÞ: Let A and B be uðFÞ arbitrary finite events. If A has at least two elements, Then the following theorem, which is a restatement of then it follows from Axiom 17 that eA in X can be found Theorem 3.1 in terms of belief functions (Definition 1.1), so that is an immediate consequence of Theorem 3.1. uðAÞ¼uðeAÞ and vðAÞ¼vðeAÞ:

Definition 3.4. Suppose B is a belief representation for If A ¼fag; let eA ¼ a and thus again, h with context function u and definiteness function v E uðAÞ¼uðe Þ and vðAÞ¼vðe Þ: (Definition 3.3). Then for all finite conditional events A A ðAjBÞ; Similarly an element eB of X can be found so that uðAÞ BðAjBÞ¼vðAÞP : ð7Þ uðBÞ¼uðeBÞ and vðBÞ¼vðeBÞ: A uðbÞ b B Thus to show vðAÞ¼vðBÞ; it is sufficient to show In the definiteness interpretation of Eq. (7), u is vðeAÞ¼vðeBÞ: But because the belief axioms with binary interpreted as a measure of probabilistic strength, and symmetry include the basic belief axioms with binary v is interpreted as a measure of something that is the symmetry, it follows from Theorem 1.2 that vðxÞ¼vðyÞ opposite of ambiguity or vagueness, which is called for all x and y in X; and thus that vðeAÞ¼vðeBÞ: definiteness. With these interpretations in mind, the right-hand side of Eq. (7) is interpreted as a subjective Definition 4.3. Let B be a belief representation for h P E probability ðAjBÞ of A occurring when B is presented, with context and definiteness functions. Then B is said where to be additive if and only if for all finite conditional uðAÞ events ðAjCÞ and ðBjCÞ of X; if A-B ¼ |; then PðAjBÞ¼P ; u b bAB ð Þ BðA,BjCÞ¼BðAjCÞþBðBjCÞ: weighted by the definiteness factor vðAÞ; i.e., BðAjBÞ¼vðAÞPðAjBÞ: ð8Þ Additive belief representations provide a quantitative Then when Eq. (8) is used to interpret B; B is called a theory of belief that is very close in mathematical form h to that of the probability calculus. However, in several definiteness representation for E with probability function P and definiteness function v: applications involving uncertainty, additivity is often difficult to justify, and often in such applications normative or intuitive arguments can be given for 4. Belief axioms with binary symmetry nonadditivity. There are interesting cases of additive belief repre- Definition 4.1. B is said to be a choice representation for sentations that are not trivial variants of choice h B representations. Examples of these can be constructed E (with support u) if and only if is a belief representation for h with context function u and using the following idea: Let u and w be functions from E Rþ definiteness function v and for all finite events ðAjBÞ X into : Extend u and w to nonempty finite subsets A of X; of X as follows: X X uðAÞ BðAjBÞ¼ : uðAÞ¼ uðaÞ and wðAÞ¼ wðaÞ: uðBÞ aAA aAA 10 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

Then in terms of u and w; the functions v and B are OBðA; BÞ is called the belief support odds (induced by B) defined so that u; v; and B have the algebraic of A over B: By definition, let PB be a function on finite characteristics of a belief representation: For nonempty conditional events of X such that for each finite finite subsets A of X and conditional events ðAjBÞ conditional event ðEjFÞ of X; of X; let BðEjFÞ wðAÞ PBðEjFÞ¼ : vðaÞ¼ ; BðEjFÞþBðF À EjFÞ uðAÞ PB is called the belief support probability function and by the above equation, let (induced by B). PBðEjFÞ is called the belief support wðAÞ uðAÞ probability of ðEjFÞ: BðAjBÞ¼ ¼ vðAÞ : uðBÞ uðBÞ Note by part (ii) of statement 2 of Theorem 3.1, that It is easy to show that B is additive. the definitions of OB and PB in Definition 5.1 are When B is additive, the above process can be independent of the choice of B; i.e., if B0 is another belief reversed: Let B be an additive belief representation for representation with context and definiteness functions, h E with context function u and definiteness function v: then OB ¼ OB0 and PB ¼ PB0 : For each A in F; let Also note that PB behaves like a probability function wðAÞ¼uðAÞvðAÞ: in that for A-B ¼ |; Then for all finite conditional events ðAjBÞ of X; PBðAjA,BÞþPBðBjA,BÞ¼1:

wðAÞ However, unlike a probability function, PB may not be BðAjBÞ¼ ; ð9Þ uðBÞ additive, i.e., situations with finite events C; D; and E can be found such that C D | and and (because B is additive) for all finite conditional - ¼ events ðCjEÞ and ðDjEÞ of X such that C-D ¼ |; PBðC,DjEÞaPBðCjEÞþPBðDjEÞ: wðC,DÞ¼wðCÞþwðDÞ: The notion of a ‘‘fair bet’’ relates the strengths of An intended interpretation of Eq. (9) (when B is beliefs of the events in the bet to the value of the additive or nonadditive) is that for conditional beliefs outcomes of the event. The following is one reasonable ðAjBÞ; w is a measure of the probabilistic strength of the notion of ‘‘fair bet:’’ focal event A; u is a measure of the probabilistic strength B of the context event B; and B is the measure of the Definition 5.2. Let be an individual’s belief represen- strength of belief of ðAjBÞ: This interpretation is tation for h with context and definiteness functions, employed later in the article. and A and B be nonempty finite events such that A-B ¼ |: Consider the gamble of gaining something that has value a40 to the individual if A occurs and 5. Belief support probability losing something that has value b40 to the individual if B occurs. Call this gamble a belief odds fair bet for this Many probabilists and decision analysts believe individual if and only if that the degree of belief of event E is properly mea- b OBðA; BÞ¼ : sured by a probability p; and that the same probability, a p; is the proper weight to assign to E in normative models of utility under uncertainty. In the theory of Note that in terms of the belief support probability belief developed here, the two different kinds of function, the above gamble is a belief odds fair bet if and E measurements of —as (i) a degree of belief and as (ii) only if a weight in a model of utility under uncertainty— are kept separate, and in general, assign different values aPBðAjA,BÞÀbPBðBjA,BÞ¼0: to E: The formulation of ‘‘belief odds fair bet’’ starts with degrees of belief for conditional events. When these Definition 5.1. Let A and B be disjoint finite events of X conditional events occur in evaluation of gambles, other a| B such that B ; and let be a belief representation notions based on degrees of belief are needed to capture h for E with context and definiteness functions. By key concepts inherent in gambling such as a ‘‘belief odds definition, let fair bet.’’ For ‘‘belief odds fair bet’’ this is accomplished very naturally in terms of belief odds. But it is also BðAjA BÞ O , accomplished in a logically equivalent and natural BðA; BÞ¼B : ðBjA,BÞ manner through belief support probabilities. For a task L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 11 where an individual is asked to judge numerically the probability and definiteness assignments: B probabilities of conditional events, both and PB are PðRjUÞ¼PðBjUÞ¼PðYjUÞ¼1; natural candidates for modeling the judgments. Empiri- 3 PðR YjUÞ¼PðB YjUÞ¼PðR BjUÞ¼2; cal results discussed in Section 6 suggest that PB is better , , , 3 in this regard. and Assume B is a belief representation for h with vðBÞ¼vðYÞ vðR YÞ¼vðR BÞ vðB YÞ¼vðRÞ: context function u and definiteness function v: Let A; B; o , , o , C; and D be nonempty finite events such that Then it is easy to verify that the following three statements are true: ðAjA,BÞBEðBjA,BÞ and B 1. BaP; ðCjC,DÞ EðDjC,DÞ: ð10Þ 2. BðBjUÞoBðRjUÞ and BðB,YÞjUÞ4BðR,YjUÞ: Assume that a high definiteness value, say 1, is assigned 3. PBðBjUÞoPBðRjUÞ and PBðB,YÞjUÞo to A and B because much is known about them; and PBðR,YjUÞ: assume a low definiteness value, ao1; is assigned to C and D; because little is known about them. Then by Note that by statement 2, B is not additive (Definition Eq. (10), 4.3). Similarly, by statement 3 PB is not additive. uðAÞ¼uðBÞ and uðCÞ¼uðDÞ: The second inequality of statement 3 yields the following conclusion for belief odds fair bets: Suppose Thus, U is presented. Consider the gamble

B B 1 g1 ¼ða; B,Y; Àb; RÞ ðAjA,BÞ¼ ðBjA,BÞ¼2 and B B 1 ðCjC,DÞ¼ ðDjC,DÞ¼2a: of gaining something that has value a40toan individual if B,Y occurs and losing something that Therefore, has value b40 to the individual if R occurs. Then for

1 this individual this gamble is a belief odds fair bet if and PBðAjA BÞ¼PBðCjC DÞ¼ : , , 2 only if Thus, although the conditional events ðAjBÞ and ðCjDÞ b ¼ 2a: differ in degree of belief and definiteness, they are given Similarly, consider the gamble the same value by the belief support probability function PB: This value is the same as PðAjA,BÞ¼PðCjC,DÞ; g2 ¼ða; R,Y; Àc; BÞ when B is interpreted as a definiteness representation for of gaining something that has value a40 to the h with probability function P and definiteness E individual if R,Y occurs and losing something that function v (Definition 3.4). has value c40 to the individual if B occurs. Then the In general for conditional events ðEjFÞ with vðEÞ¼ latter gamble is a fair bet if and only if vðFÞ; vðR,YÞ P c ¼ 2 a42a: PBðEjFÞ¼ ðEjFÞ: vðBÞ

Thus, to interestingly differentiate PB from P (and This result makes intuitive sense for ‘‘ambiguity therefore, from B), one needs to consider situations adverse’’ individuals: If B had less ambiguity (and where the definitenesses of the focal events differ from therefore more definiteness) to the extent that vðBÞ¼ the definitenesses of their context events. vðR,YÞ; then intuitively, c would equal 2a: Therefore, Such a situation is suggested in Ellsberg (1961). to the extent that vðBÞovðR,YÞ; c is greater than 2a: It Suppose an urn has 90 balls that have been thoroughly is reasonable to suppose that ambiguity adverse mixed. Each ball is of one of the three colors, red, blue, individuals would prefer g1 to g2: or yellow. There are thirty red balls, but the number of The upshot of the above is that even though g1 and g2 blue balls and the number of yellow balls are unknown are belief odds fair bets, g1 is preferable to g2 by except that together they total 60. A ball is to be ambiguity adverse individuals. This implies that in randomly chosen from the urn. Let R be the event that a general, the values of g1 and g2 should not be computed red ball is chosen, B the event a blue ball is chosen, and by the formulas, Y the event a yellow ball is chosen. Let U ¼ R,B,Y: aPBðB,YjUÞÀbPBðRjUÞ and Assume that this situation is part of the domain of a aPBðR,YjUÞÀbPBðBjUÞ; definiteness representation B with probability function P and definiteness function v (Definition 3.4). For this because the first formula would then yield g1 as having situation I consider the following to be reasonable value 0 and the second formula g2 as having value 0, 12 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 thus implying that the individual would be indifferent factors such as memory and attention, and (ii) the between g1 and g2: making of judgments that are mediated by heuristics such as representativeness, availability, and anchoring and adjusting. 6. Support theory Formally, support theory is formulated in terms of ‘‘evaluation frames’’ and ‘‘support functions:’’ 6.1. Tversky’s and Koehler’s theory An evaluation frame ðA; BÞ consists of a pair of exclusive hypotheses: the first element A is the focal Belief support probability provides a theory of hypothesis that the judge evaluates, and the second subjective probability judgment that is similar in many element B is the alternative hypothesis. We assume respects to the ‘‘support theory’’ of Tversky and that when A and B are exclusive the judge perceives Koehler (1994). them as such, but we do not assume that the judge Support theory is a descriptive theory in which can list all the constituents of an implicit disjunction. probability judgments are assigned to descriptions of Thus, the judge recognizes the fact that ‘‘biological events, called hypotheses, instead of to events. It is sciences’’ and ‘‘physical sciences’’ are disjoint cate- assumed that there is a finite set T with at least two gories, but he or she may be unable to list all their elements that generates an event space and a set of disciplines. This is a form of bounded rationality; we hypotheses H such that each hypothesis A in H assume recognition of exclusivity, but not perfect describes an event, called the extension of A and denoted recall. by A0; that is a subset of T: It is allowed that different We interpret a person’s probability judgment as a hypotheses describe the same event, e.g., for a roll of a mapping P from an evaluation frame to the unit pair of dice, the hypotheses ‘‘the sum is 3’’ and ‘‘the interval. To simplify matters we assume that PðA; BÞ product is 2’’ describe the same event, namely one die equals 0 if and only if A is null and it equals 1 if and shows 1 and the other 2. only if B is null; we assume that A and B are not both Hypotheses that describe an event ftg; where tAT; are null. Thus, PðA; BÞ is the judged probability that A called elementary; those that describe | are called null; rather than B holds, assuming that one and only one and nonnull hypotheses A and B in H such that the of them is valid. y conjunction of A and B describe | are called exclusive Support theory assumes that there is a ratio scale s (with respect to H). C in H is said to be an explicit (interpreted as degree of support) that assigns to each disjunction (with respect to H)—or for short, an explicit hypothesis in H a nonnegative real number such that hypotheses (of H)—if and only if there are exclusive A for any pair of exclusive hypotheses A; B in H; and B in H such that C ¼ A3B; where ‘‘3’’ stands for sðAÞ the logical disjunction of A and B; ‘‘A or B:’’ D in H is PðA; BÞ¼ ; ð11Þ said to be implicit (with respect to H) if and only if D is sðAÞþsðBÞ not |; is not elementary, and is not explicit with respect [and] for all hypotheses A; B; and C in H; if B and C to : 0 H are exclusive, A is implicit, and A0 ¼ðB3CÞ ; then H may have implicit and explicit hypotheses that describe the same event, e.g., C: ‘‘Ann majors in a sðAÞpsðB3CÞ¼sðBÞþsðCÞ: ð12Þ natural science,’’ A: ‘‘Ann majors in a biological (Quoted from preprint of Tversky and Koehler, science,’’ and B: ‘‘Ann majors in a physical science.’’ 1994). Then C and A3B describe the same event, i.e., have the same extension, or letting H0 stand for the extension of a Tversky and Koehler show that conditions 11 and 12 hypothesis H; above imply the following four principles for all A; B; C; and D in H: C0 ¼ðA3BÞ0 ¼ A0,B0: 1. Binary complementarity. PðA; BÞþPðB; AÞ¼1: It is assumed that whenever exclusive A and B belong 2. Proportionality.IfA; B; and C are mutually ex- to H; then their disjunction A B also belong to H: 3 clusive and B is not null, then Tversky and Koehler (1994) provide empirical data PðA; BÞ PðA; B3CÞ for many situations in which subjects judge explicit ¼ : hypotheses E to be more likely than implicit ones I with PðB; AÞ PðB; A3CÞ same extensions (E0 ¼ I 0). They suggest that this empirical result reflects a basic principle of human 3. Product rule. Let RðA; BÞ be the odds of A against B; judgment. They explain it in terms of an intuitive theory i.e, let of information processing involving (i) the formation of PðA; BÞ a ‘‘global impression that is based primarily on the most RðA; BÞ¼ : representative or available cases’’ and modulated by PðB; AÞ L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 13

Then captures the important characteristic of the distortion RðA; BÞRðC; DÞ¼RðA; DÞRðC; BÞ; ð13Þ that is due to ‘‘unpacking.’’ Let B be a function from conditional hypotheses to provided A; B; C; D are not null, and the four pairs the positive reals such that for each conditional of hypotheses in Eq. (13) are pairwise exclusive. hypothesis ðAjA3B), 4. Unpacking principle. Suppose B; C; and D are uðAÞ mutually exclusive, A is implicit, and A0 ¼ðB3CÞ0: BðAjA3BÞ¼vðAÞ : ð16Þ uðA BÞ Then 3 Eqs. (14) and (16) give B the same algebraic form as PðA; DÞpPðB3C; DÞ¼PðB; C3DÞþPðC; B3DÞ: the belief representations considered in Section 3. BðAjA3BÞ is intended to be interpreted as a distortion (by a factor of vðAÞ) of the probabilistic strength, Tversky and Koehler show the following theorem: uðAÞ ; Theorem 6.1. Suppose PðA; BÞ is defined for all exclusive uðA3BÞ A; BAH; and that it vanishes if and only if A is null. Then of ðAjA3BÞ: The distortion of interest for the kinds of binary complementarity, proportionality, the product rule, situations covered by Koehler–Tversky theory is due to and the unpacking principle hold if and only if there exists the unpacking principle, which is captured in large part a ratio scale s on that satisfies Eqs. (11) and (12). H by Eq. (15). Let PB be the belief-support probability function Proof. Theorem 1 of Tversky and Koehler (1994). & determined by B; i.e., let Belief-support probabilities can account for phenom- BðAjA3BÞ PBðAjA3BÞ¼ : ena that are the basis of Koehler’s and Tversky’s BðAjA3BÞþBðBjA3BÞ support theory. The key idea for this is to interpret Then it is easy to verify that PB satisfies Binary ‘‘definiteness’’ as a measure of unpackedness. To carry complementarity, proportionality, and the product rule. this out, a few minor modifications are needed. The following theorem is also immediate: Instead of the set T of elementary hypotheses, an infinite set X of elementary hypotheses will be assumed. Theorem 6.2. Suppose A and D are exclusive and B; C; It will also be assume that each hypothesis A in H has 0 0 and D are mutually exclusive, A is implicit, and A ¼ an extension A that is a finite subset of X: To avoid 0 ðB3CÞ : Then the following three statements are true: extra notation and extra conditions, it will be assumed that each element of H is nonnull. Also instead of the 1. (Ordinal unpacking) PBðAjA3DÞpPBðB3Cj evaluation frame notation ðA; BÞ; the conditional B3C3DÞ: hypothesis notation ðAjA3BÞ (where, of course, A3B 2. (Definiteness unpacking) If vðBÞ¼vðCÞ; then is explicit) will be employed to describe the kinds of PBðAjA3DÞpPBðB3CjB3C3DÞ situations that support theory is concerned with. The ¼ PBðBjB C DÞþPBðCjC B DÞ: purpose of these changes is to make the discussion 3 3 3 3 coordinate to the discussions and results given earlier in the article. They are not essential for the points made 3. The following two statements are equivalent: throughout this section. (i) B is additive, i.e., Let u be a function from X into the positive reals. BðB3CjB3C3DÞ Extend u to H as follows: For each nonempty finite ¼ BðBjB C DÞþBðCÞjB C DÞ: subset A of X; let 3 3 3 3 X uðAÞ¼ uðaÞ: ð14Þ (ii) The unpacking principle holds, i.e., aAA PBðAjA DÞ PBðB CjB C DÞ For each H in H; let uðHÞ¼uðH0Þ (where, of course, 3 p 3 3 3 H0 is the extension of H). u is to be interpreted as a ¼ PBðBjB3C3DÞþPBðCjC3B3DÞ: measure of probabilistic strength. Let v be a function from H into the positive reals such that for all A and B in H; if A is implicit, B is explicit, and A0 ¼ B0; then Observe that the inequalities in Theorem 6.2 become strict if the inequality in Eq. (15) becomes strict. vðAÞpvðBÞ: ð15Þ The above shows that belief-support probabilities, v is to be interpreted as a ‘‘distortion factor’’ due to when generated by an additive B; is a form of support specific kinds of cognitive processing, and Eq. (15) theory. This form of support theory generalizes phe- 14 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 nomena outside of support theory when generated by a RðA; BÞRðB; DÞ¼RðA; CÞRðC; DÞ; and nonadditive B: The nonadditive case still retains much of the flavor of support theory, particularly its empirical (ii) if ðA; BÞ; ðB; DÞ; and ðA; DÞ are exclusive, then basis, because ordinal unpacking and definiteness RðA; BÞRðB; DÞ¼RðA; DÞ: unpacking hold. Of possible empirical importance is the consideration that it may be possible to find many natural situations in which definiteness unpacking holds 3. (Odds inequality) Suppose A1; A2; and B are mu- but the unpacking principle fails. tually exclusive, A is implicit, and the judge Let ðAjA3BÞ be a conditional hypothesis. Then in the recognizes A13A2 as a partition of A: That is, 0 0 formula, ðA13A2Þ ¼ A and the judge recognizes that A13A2 has the same extension as A: Then uðAÞ BðAjA3BÞ¼vðAÞ ; uðA3BÞ RðA; BÞpRðA13A2; BÞpRðA1; BÞþRðA2; BÞ: may be viewed as ‘‘distorting’’ the fraction, uðAÞ : uðA3BÞ Rottenstreich and Tversky (1997) show the following By the definition of u; theorem: uðAÞ uðA0Þ uðA0Þ : Theorem 6.3. Suppose P A; B is defined for all exclusive ¼ 0 ¼ 0 0 ð Þ uðA3BÞ uððA3BÞ Þ uðA ,B Þ hypotheses A and B and that it vanishes if and only if A is null. Then the above three assumptions hold if and only if However, by Eq.P (14), 0 there exists a nonnegative function s on the set of uðA Þ 0 uðaÞ P aAA ; 0 0 ¼ hypotheses such that for all exclusive hypotheses C and D, uðA ,B Þ cAA0,B0 uðcÞ sðCÞ may be interpreted as a subjective conditional prob- PðC; DÞ¼ : ability of the conditional, extensional hypothesis sðCÞþsðDÞ 0 0 0 ðA jA ,B Þ: With this interpretation in mind, vðAÞ is Furthermore, if A1 and A2 are exclusive, A is implicit, and the amount that the subjective conditional probability of ðA13A2Þ is recognized as a partition of A, then the extension of ðAjA3BÞ needs to be distorted to sðAÞpsðA13A2ÞpsðA1ÞþsðA2Þ: achieve BðAjA3BÞ: Tversky and Koehler (1994) provided many examples of their theory. However, latter empirical studies Binary complementarity is the same as in Tversky’s showed their theory to be inadequate. To accomodate and Koehler’s theory. Rottenstreich’s and Tversky’s these additional studies, generalizations of Tversky’s product rule is slightly stronger than the product rule of and Koehler’s theory were developed, and two of these Tversky and Koehler, since it contains the additional are discussed next. product condition RðA; BÞRðB; DÞ¼RðA; DÞ: Odds inequality is a replacement for the unpacking condition. 6.2. Rottenstreich’s and Tversky’s support theory Tversky and Rottenstreich (1997) note that in Odds inequality, ‘‘The recognition requirement, which re- Rottenstreich and Tversky (1997) noted empirical stricts the assumption of implicit subadditivity, was not examples in which probabilities for explicit disjunctions explicitly stated in the original (Tversky & Koehler, G3H were subadditive, i.e., empirical situations where 1994) version of the theory, although it was assumed in PðG3HÞpPðGÞþPðHÞ: its applications.’’ In Section 6.1 it is shown that belief support To accommodate such subadditive situations, they probabilities, when generated by an additive B; is a provided the following generalization of Tversky and form of Tversky’s and Koehler’s support theory. When Koehler (1994): For all hypotheses E and F; where F is the belief support probabilities are generated by a nonnull, let subadditive B; this form of support theory naturally PðE; FÞ generalizes to a form of Rottenstreich’s and Tversky’s RðE; FÞ¼ : PðF; EÞ support theory, with the product rule and odds inequality holding. For nonadditive B; much of the flavor of Then the following three assumptions hold for all Tversky and Koehler’s original support theory is still hypotheses A; A ; A ; B; C; and D: 1 2 maintained, particularly its empirical basis, because 1. (Binary complementarity) PðA; BÞþPðB; AÞ¼1: ordinal unpacking and definiteness unpacking hold. Of 2. (Product rule) (i) If ðA; BÞ; ðB; DÞ; ðA; CÞ; and possible empirical importance for Tversky’s and Koeh- ðC; DÞ are exclusive, then ler’s theory is the consideration that it may be possible L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 15 to find many natural situations in which definiteness and B the alternative hypothesis. The theory assumes unpacking holds but the unpacking principle fails. two support functions are used in evaluating PðA; BÞ; sf For subaddtive B; the belief support probabilities add for focal hypotheses s for alternative hypotheses. for hypotheses of the same definiteness, and thus the PðA; BÞ is then determined by the formula support form corresponding to Tversky’s and Koehler’s sf ðAÞ theory apply to hypotheses that have the same v-value. PðA; BÞ¼ : sf ðAÞþsðBÞ This suggests the following for the form of support corresponding to Rottenstreich’s and Tversky’s theory The special case sf ¼ s yields support theory. Accord- (i.e., for subaddtive B): (1) The hypotheses naturally ing to Brenner and Rottenstreich this special case arose partition into families, with the hypotheses in each in the above mentioned studies because ‘‘all earlier family having the same definiteness and hypotheses from studies involved hypotheses that were especially well- different families having different definitenesses. (2) For defined and left no room for variation in their each family, the support form of Tversky’s and representation y:’’ Koehler’s theory holds for hypotheses from the family. Fox et al. (1996), for example, studied hypotheses And (3) the support form of Rottenstreich’s and such as ‘‘the price of Microsoft stock will be above Tversky’s theory holds for hypotheses from different seventy dollars.’’ There is little ambiguity in such families. hypotheses and consequently little room for varia- bility or asymmetry in their representations. We 6.3. Asymmetric support theory suggest that earlier researchers failed to observe [failures of Binary Complementarity] because they Recently, Brenner and Rottenstreich (manuscript) investigated only such especially well-defined hypoth- developed a version of support theory in which binary eses which essentially left on room for representa- complementarity may fail. They call their theory tional asymmetry [i.e., sf as]. (p. 16) asymmetric support theory, and this section presents their model and some of the examples and results Macchi, Osherson, and Krantz (1999) conducted contained in their manuscript. studies involving ultra-difficult general information The empirical bases for support theory are experi- questions. For example subjects were presented one of mental studies of probability judgments that consis- the following: tently find sums of probabilities greater than 1 for The freezing point of gasoline is not equal to that partitions consisting of more than two elements and of alcohol. What is the probability that the sums equal to 1 for binary partitions. For example, Fox, freezing point of gasoline is greater than that of Rogers, and Tversky (1996) asked professional option alcohol? traders to judge the probability that the closing price of Microsoft stock would fall within a particular interval The freezing point of alcohol is not equal to that on a specific future date. When four disjoint intervals of gasoline. What is the probability that the that spanned the set of possible prices were presented for freezing point of alcohol is greater than that of evaluation, the sums of the assigned probabilities were gasoline? typically about 1.50. However, when binary partitions The typical sum of probabilities over all such binary were presented, the sums of the assigned probabilities partition was about 0.90, relatively far from 1, indicat- were very close to 1, e.g., 0.98. Tversky and Fox also ing a failure of binary complementarity. Macchi et al. observed in probability judgments involving future reasoned that given the ultra-difficulty of the questions temperature in San Francisco, the point-spread of there is relatively little evidence in favor of the focal selected NBA and NFL professional sports games, and hypothesis. If the subjects attended relatively more to many other quantities, sums of assigned probabilities the focal rather than the alternative hypothesis, then greater than 1 for partitions consisting of more than two they might not appreciate the fact that the alternative elements and sums nearly equal 1 for binary partitions. hypothesis also has little support, leading to having a This pattern of results were also replicated by Re- sum of judged probabilities less than 1. delmeier, Koehler, Liberman, and Tversky (1995) in a In a follow-up study, Macchi et al. attempted to study of practicing physicians making judgments of equalize the amounts of attention paid to focal and patient longevity. Other researchers, e.g., Wallsten, alternative hypotheses by explicit mention of both Budescu, and Zwick (1992) have also observed binary hypotheses. Subjects were presented one of the follow- complementarity in experimental settings. ing: Asymmetric support theory is based on judgments of probability PðA; BÞ of propositions of the form ‘‘A The freezing point of gasoline is not equal to that of holds rather than B;’’ where A and B are exclusive alcohol. Thus, either the freezing point of gasoline is hypotheses. In the above, A is called the focal hypothesis greater than that of alcohol, or the freezing point of 16 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

alcohol is greater than that of gasoline. What is the AandBinH and it vanishes if and only if A is null. Then probability that the freezing point of gasoline is the following two statements are equivalent: greater than that of alcohol? 1. The asymmetric product rule and the asymmetric triple The freezing point of alcohol is not equal to that of product rule. gasoline. Thus, either the freezing point of alcohol is 2. There exist functions sf and s on H such that for all greater than that of gasoline, or the freezing point of exclusive A and B in H;

gasoline is greater than that of alcohol. What is the sf ðAÞ probability that the freezing point of alcohol is PðA; BÞ¼ : sf ðAÞþsðBÞ greater than that of gasoline? In this study, the typical sum of judged probabilities was about 1.01. Apparently, mentioning both the focal 6.4. A few comments about support theories and alternate hypotheses made it natural for subjects to evaluate them in similar ways. 1. Judging probabilities degrees of belief: Let u; v; and Brenner and Rottenstreich (manuscript) conducted B be as in Eqs. (14), (15), and (16). In particular, empirical studies where evaluations of category size uðAÞ uðAÞ BðAjA3BÞ¼vðAÞ ¼ vðAÞ : underlay the likelihood judgment and consistently found uðA3BÞ uðAÞþuðBÞ that the sums of percentage judgments for binary Letting wðAÞ¼uðAÞvðAÞ; we then obtain the formula partitions were less than 1. They also conducted wðAÞ empirical studies where evaluations of similarity under- BðAjA3BÞ¼ ; ð17Þ lay the likelihood judgment and consistently found that uðAÞþuðBÞ the sums of probability judgments for binary partitions which is similar in many ways to Brenner and were greater than 1. Rottenstreich formula, Brenner and Rottenstreich presents the following s ðAÞ axiomatization and theorem for the representation PðA; BÞ¼ f : sf ðAÞþsðBÞ sf ðAÞ In Eq. (17), w is interpreted as a measure the probabil- sf ðAÞþsðBÞ istic strength of the focal hypothesis, u as a measure of for partitions ðA; BÞ in terms of the probability or the probabilistic strength of the alternative hypothesis, percentage function P: and BðAjA3BÞ is the probability judgment given by the subject. Using this interpretation, explanations can be Definition 6.1. Let provided for the qualitative shifts in probabilistic PðA; BÞ judgments occurring in the empirical examples of the QðA; BÞ¼ ; 1 À PðA; BÞ previous subsection. Brenner and Rottenstreich (manuscript) also show that their formula quantitatively fits where B is nonnull. Then the following two definitions the data. My guess is that it would be difficult to obtain: distinguish the two formulas in terms of goodness of fit 1. The asymmetric product rule is said to hold if and using the kind of data collected by Brenner and only if Rottenstreich. QðA; BÞQðC; DÞ¼QðA; DÞQðC; BÞ; 2. Dual belief support representations: In the previous subsection it was shown that belief support probabilities whenever the arguments of Q are exclusive. provided an adequate theory for support theory. There, 2. The asymmetric triple product rule is said to hold if the definiteness function vðAÞ was interpreted as a factor and only if that accounted for the amount that a hypothesis A was QðA; BÞQðB; CÞQðC; AÞ¼QðC; BÞQðB; AÞQðA; CÞ; distorted with respect to its extension through cognitive processing. In particular, distortion to due unpacking whenever the arguments of Q are exclusive. (as well as other distortions) could be incorporated in v: However, the kinds of distortions considered in asym- Note that whenever there exists a hypothesis exclusive metrical support theory cannot be included in v; because of A; B; and C; the asymmetric triple product rule vðAÞ depends only on the hypothesis A and thus not on for A; B; and C; follows from the asymmetric product whether A is a appearing as a focal or an alternative rule. hypothesis. Thus to extend the belief support development to the kinds of empirical situations considered by Theorem 6.4. Assume the notation and concepts of Brenner and Rottenstreich, two belief support functions support theory of the previous section. Let H be a set are needed: one, Bf for focal hypotheses, and the other, of hypotheses. Suppose PðA; BÞ is defined for all exclusive B; for alternative hypotheses. The following is a natural L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 17 way of accomplishing this: Let u; uf ; v; and vf be their extensions as Tversky and Koehler (1994), they do functions from nonnull hypotheses into the positive not extend their axiomatization or mathematical model reals and P be such that for each conditional hypothesis to include a version of the unpacking principle. In ðAjA3BÞ; Section 6.1 unpacking was characterized in terms of the belief support probabilities representation. This char- Bf ðAjA3BÞ PðAjA3BÞ¼ acterization generalizes easily to the dual belief support Bf ðAjA3BÞþBðBjA3BÞ representation: uf ðAÞ vf ðAÞvðAÞ uf ðA3BÞ Let A and B be arbitrary elements of H such that A is ¼ : ð18Þ 0 0 v A v A uðAÞ v B uðBÞ implicit, B is explicit, and A ¼ B : Assume f ð Þ ð ÞuðA3BÞ þ ð ÞuðA3BÞ vðAÞpvðBÞð22Þ Note that in Eq. (18) vf is not the definiteness function and for Bf ; rather the product vf ðAÞvðAÞ gives the definite- B B ness of A in for f : vf ðAÞpvf ðBÞ: ð23Þ In the second equality in Eq. (18), u ðAÞ is interpreted f The the following generalizes Theorem 6.2. as the probabilistic strength of the extension of A when A is the focal hypothesis, and similarly, uðAÞ is Theorem 6.5. Suppose the above assumptions, notation, interpreted as the probabilistic strength of the extension and conventions. Suppose A and D are exclusive, B; C; of A when A is the alternative hypothesis. The following and D are mutually exclusive, A is implicit, and A0 ¼ assumption is within the spirit of support theory: ðB3CÞ0: Then the following two statements are true: uf ¼ u: ð19Þ 1. (Ordinal unpacking) PBðAjA3DÞpPBðB3Cj B B Also, because f and are intended to be belief support B3C3DÞ: functions, it is assumed that 2. Suppose Bf is additive, i.e., u A u A0 and u A B u A u B ; 20 ð Þ¼ ð Þ ð 3 Þ¼ ð Þþ ð Þ ð Þ Bf ðB3CjB3C3DÞ¼Bf ðBjB3C3DÞþBf ðCjB3C3DÞ: 0 0 where as usual, A is the extension of A and ðA3BÞ ¼ Then the unpacking principle holds, i.e., A03B0: PBðAjA DÞ PBðB CjB C DÞ¼PBðBjB C DÞþPBðCjC B DÞ: vðAÞ is intended as a factor accounting for the 3 p 3 3 3 3 3 3 3 probabilistic distortion of A with respect to the 0 probabilistic strength, uðA Þð¼uðAÞÞ; of its extension 0 0 0 Proof. By the hypothesis A ¼ðB3CÞ and Eq. (20), A : vf ðAÞ is intended as a factor that accounts for the additional distortion resulting from A being a focal uðAÞ¼uðB3CÞ¼uðBÞþuðCÞ: ð24Þ rather than alternative hypothesis. By Eq. (21), Applying Eqs. (19) and (20) to Eq. (18) then yields, v ðAÞvðAÞuðAÞ uðAÞ PðAjA DÞ¼ f vf ðAÞvðAÞu A u B 3 P A A B ð Þþ ð Þ vf ðAÞvðAÞuðAÞþvðDÞuðDÞ ð j 3 Þ¼ uðAÞ uðBÞ vf ðAÞvðAÞ þ vðBÞ 1 uðAÞþuðBÞ uðAÞþuðBÞ : 25 ¼ vðDÞuðDÞ ð Þ vf ðAÞvðAÞuðAÞ 1 þ ¼ : ð21Þ vf ðAÞvðAÞuðAÞ vf ðAÞvðAÞuðAÞþvðBÞuðBÞ 0 Note that by letting for each hypothesis H in H; Similarly, by Eq. (21), the hypothesis A0 ¼ðB3CÞ ; and Eq. (20), sf ðHÞ¼vf ðHÞvðHÞuðHÞ and sðHÞ¼vðHÞuðHÞ; Eq. (21) becomes PðB3CjB3C3DÞ vf ðB3CÞvðB3CÞuðB3CÞ sf ðAÞ ¼ PðAjA3BÞ¼ ; v ðB CÞvðB CÞuðB CÞþvðDÞuðDÞ sf ðAÞþsðBÞ f 3 3 3 1 which is Brenner’s and Rottenstreich’s asymmetric ¼ 1 þ vðDÞuðDÞ support representation. The representation given by vf ðB3CÞvðB3CÞuðB3CÞ Eq. (21) differs from Brenner’s and Rottenstreich’s in 1 that the correspondents to s and s have an inner ¼ : ð26Þ f 1 þ vðDÞuðDÞ structure to them. This ‘‘inner structure’’ allows for the vf ðB3CÞvðB3CÞuðAÞ formulation of unpacking principles. 3. Extending the dual belief support representation to Then statement 1 follows from Eqs. (22) and (23). include unpacking: Although Brenner and Rottenstreich To show statement 2, suppose Bf is additive. Then the allude to unpacking studies and use the same formal unpacking principle follows from statement 1 and the setup involving implicit and explicit hypotheses and first equality in Eq. (18). 18 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

Observe that all the inequalities in statements 1 and 2 theories on whether UðxÞX0 or whether UðyÞX0or of Theorem 6.5 become strict if the inequality in Eq. (22) whether both UðxÞ and UðyÞ are p0:) becomes strict; and the inequality in statement 1 In Section 6, B was used as a structured support becomes strict if the inequality in Eq. (23) becomes function to produce generalizations of support theories strict. of probability judgment. The structured nature of B allowed for a different kind of formulation of the unpacking principle of Tversky and Koehler (1994) as 7. Conclusions well as for generalizations of it. Then with insights gained from theses formulations to unpacking for The belief axioms (Definition 3.2) yield the existence support theory, it was easy to extend the unpacking of a definiteness representation B with probability principle to the more general and complicated versions function P and definiteness function v (Definition 3.4). of support theory. When the value of v is constant, B may be looked at as a Traditional probability theory has two components to ratio-scaled version of P; i.e., B is essentially a it: A probability function that is a finitely additive conditional probability function. Thus the notion of a measure and an independence relation on events. While definiteness representation generalizes the notion of a the development of the analog of independence for belief probability function. More importantly, in the multi- theory is outside the scope of this article, a few farious uses of probability in mathematical modeling, B observations will be made about it. or other probabilistic concepts generated by it are often In the Kolmogorov theory of probability, indepen- substitutable for probability functions, yielding new and dence is a concept defined in terms of probabilities, i.e., more general models. the events A and B are by definition ‘‘independent’’ if As an example consider utility theory. Let U be an and only if PðA-BÞ¼PðAÞPðBÞ: I and others find this individual’s function from objects of value (both notion of independence problematic: In the most positive and negative) into the real numbers, and let A important applications of probabilistic independence, and B be nonempty, disjoint finite events. Let g ¼ one has a very good idea of which key events are ðx; A; y; BÞ be the gamble of receiving object of value x if independent of one another without resort to calculation A occurs and receiving y if B occurs. In the behavioral or often even without knowing their probabilities and sciences, models of utility have the form the probability of their intersections. Also in most applications where probabilities are used, they are UðgÞ¼UðxÞW1 þ UðyÞW2; admittedly inexact; but the Kolmogorov definition of where W1 and W2 are weights satisfying the conditions, independence require the probabilities be exact. In 0pW1p1; 0pW2p1; and W1 þ W2 ¼ 1: foundations of probability founded on relative frequen- cies, independence of trials are assumed before the Then theories of utility result by specifying W and W : 1 2 probability of events are defined. One of the most important of these theories is Because of these considerations, I find it preferable to subjective expected utility (SEU), which assumes that introduce a primitive binary relation > on events the individual has a conditional probability function P representing probabilistic independence and have it such that linked to probability by the following: W1 ¼ PðAjA,BÞ and W2 ¼ PðBjA,BÞ: if A>B then PðA-BÞ¼PðAÞPðBÞ: Thus, SEU assumes that only one dimension of In intended interpretations, > is often known indepen- uncertainty is relevant for specifying W1 and W2—the dimension that is measured by probability. The belief dently of the underlying probability function, often representation B can be interpreted as the combined through intuition or theory about the nature of reality measurement of two dimensions—a dimension of being modeled, e.g., that ‘‘red’’ coming up on a future probability measured by P; and a dimension of turn of a particular roulette wheel is independent of the sum of points scored being even by the teams involve in definiteness of focal events measured by v: Let FP;v denote a function from the set conditional events into a particular future football game. B the real interval ½0; 1 that is determined by P and v: Let be a belief representation for h with P Then consideration of probability function and definiteness function v (Definition 3.4). Let > be the relation of causal W ¼ FP ðAjA BÞ and W ¼ FP ðBjA BÞ 1 ;v , 2 ;v , independence between two events. Interpreting P as provides a starting point for generating utility theories conditional probability then reasonably yields that generalizes SEU. (Current utility theories would if A>B then PðA-BÞ¼PðAÞPðBÞ also have W1 and W2 depend on features of UðxÞ and UðyÞ; e.g., rank-dependent theories on whether as a desirable condition for belief. However, the UðxÞXUðyÞ or whether UðyÞXUðxÞ; and sign-dependent specification of vðA-BÞ is difficult and may depend L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 19 on the kind of definiteness one is considering. For Definition 8.2. Y ¼ /Y; k0; JS is said to be an example, even though A is causally independent of B; extensive structure if and only if the following seven the definiteness of A need not be causally independent of conditions hold: the definiteness of B: An extreme example of this is 1. Ya|; J is a binary operation on Y; and k0 is a where the definiteness of A and B (A-Ba|) is due to the same kind of lack of information or unreliability of total ordering on Y: information, and thus the definiteness A and B (and 2. J is commutative, i.e., xJy ¼ yJx for all x and y in Y: therefore A-B) are the same and completely causally dependent. The other extreme is when the definiteness of 3. J is associative, i.e., ðxJyÞJz ¼ xJðyJzÞ for all A is causally independent of the definiteness of B: What x; y; and z in Y: g0 is needed to advance further the theories of belief 4. Y is positive, i.e., xJy x for all x and y in Y: considered in this article is a means of sensibly 5. Y is monotonic, i.e., for all x; y; and z in k0 k0 classifying different kinds of definiteness. Notions of Y; x y iff xJz yJz: ‘‘independence’’ may prove to be useful in such an 6. Y is restrictedly solvable, i.e., for all x and y in Y; if g0 g0 effort. x y then for some z in Y; x yJz: The theory of belief developed in this article arose out 7. Y is Archimedean, i.e., for all x and y in Y; there g0 of generalizing probability theory to apply to situations exists a positive integer n such that nx y; where where the axiom of binary symmetry (Axiom 14) may 1x ¼ x; and for all positive integer k; ðk þ 1Þx ¼ fail. These generalizations, which are captured by the ðkxÞJx: belief axioms, use belief representations instead of probability functions. When the belief representations Definition 8.3. Define " on C as follows: For all a; b; g are additive, they provide a theory that is very close in C, a"b ¼ g if and only if there exist AAa; BAb; and to that of probability theory. But for many kinds of CAg such that A-B ¼ | and A,B ¼ C: belief situations additivity is unwarranted and un- wanted. Nevertheless, belief representations without Theorem 8.1. " is an operation on C and C ¼ additivity are still rich in mathematical structure and /C; kC; "S is an extensive structure (Definition 8.2). provide fertile ground for generating probabilistic-like concepts. Proof. Note that if E1; y; Ei; y are elements of C; then by repeated use of axiom 9 elements F1; y; Fi; y of C can be found so that EiBCFi and for all distinct i; j in 8. Additional lemmas, theorems, and proofs þ I ; Ei-Ej ¼ |: From this and the fact that , is commutative and associative, it easily follows that " 8.1. Preliminary lemmas and theorems is a commutative and associative operation on C. From the commutativity of " and Axiom 2 it follows that C is Convention 8.1. Throughout this subsection the Basic positive. From the commutativity and Axioms 3 and 9 it Belief Axioms (Definition 2.4) will be assumed. follows that C is monotonic. From Axiom 11 it follows that C is restrictedly solvable, and from Axioms 9 and 13 Lemma 8.1. B; BC; and BX are equivalence relations. that C is Archimedean. By Lemma 8.2 kC is a total ordering on C: h h h Proof. Since ; C; and X are weak orderings, it B B B easily follows that ; C; and X are equivalence Definition 8.4. Define the binary relation h on X C relations. & % as follows: For all xa; yb in X C; xah%yb if and only if there exist aAx; AAa; bAy; BAb; such that B Definition 8.1. Let C be the set of C-equivalence ðajAÞhðbjBÞ: classes of C and X be the set of BX -equivalence classes k h B k of X: Let C be C= C on C; that is, let C be the Lemma 8.3. The following five statements are true: binary relation on C such that for all a; b in C, akCb if and only if there exist A in a and B in b such that 1. h% is a weak ordering. h k h B A CB: Similarly let X be X = X on X; that is, let 2. For all x; yinX and all a in C, if xa is in the domain of kX be the binary relation on X such that for all x; y in h% and xkXy; then ya is in the domain of h% and X; xkXy if and only if there exist a in x and b in y such xah%ya: that ahX b: 3. For all a; b in C and all x in X, if xa is in the domain of h% and bkCa; then xb is in the domain of h% and Lemma 8.2. kC and kX are total orderings. xah%xb: 4. For all x; yinX, the following two propositions are Proof. Left to reader. & true: (i) if xah%ya for some a in C, then xkXyand 20 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

B xbh%yb for all b in C such that bkCa; and (ii) if ðajAÞhðbjBÞ and a X b: Therefore by statement 2 g g g h k xa %ya for some a in C, then x X y and xb %yb of Axiom 6, B CA: Thus by Definition 8.1, b Ca: for all b in C such that bkCa: Let y in X be such that xkXy: Then it follows 5. For all a; b in C, the following two propositions are from Definition 8.1 that c and d in y can be found such B true: (i) if xah%xb for some x in X, then bkCa and that ahX c; bhX d; and c X d: Since ðajAÞ and ðbjBÞ yah%yb for all y in X such that xkXy; and (ii) if and ahX c and bhX d; it easily follows that elements h g g g h xa %xb for some x in X, then b Ca and ya %yb and k in X can be found so that A Cfc; hg and k h for all y in X such that x Xy: B Cfd; kg: It then follows from Axiom 7 that e and f in X and E and F in C can be found such that cBX e; dBX f ; ABCE; and BBCF; eAE; and f AF: Since cB e and dB f ; it follows that eAy and f Ay: Proof. 1. It is immediate that h is a transitive X X % From Bh A it follows that Fh E: Therefore by relation on X C: To show reflexivity, let xa be an C C statement 2 of Axiom 6, ðejEÞhðf jFÞ: Thus, by arbitrary element of the domain of h ; and let a be in % Definition 8.4, yah yb: (ii) Proposition (ii) follows X and A be in a: Then by Axiom 7, let cAX and CAC be % by a similar argument. & such that aBX c; ABCC; and cAC: Then from ðcjCÞBðcjCÞ; it follows that xaB%xa: It will next be Lemma 8.4. The following five statements are true: shown that h% is connected. Because h% is reflexive, the domain of h% ¼ the range of h%: Let xa and yb 1. For all x; yinX and all a in C, if ykXx and xa is in the be arbitrary elements of the domain of h%: Let domain of h%; then there exists b in C such that a; A; y; and B be such that aAA; AAC; yAB; BAC: xaB%yb: Then because h is a weak ordering, 2. For all x in X and all a; b in C, if xa is in the domain of h B either ðajAÞhðbjBÞ or ðbjBÞhðajAÞ; %; then there exists y in X such that xa %yb: 3. The following two propositions are true for all x; yinX which by Definition 8.4 yields, 0 0 0 0 and all a; b; a ; b in C: (i) if xah%yb and xa h%yb ; either xah%yb or ybh%xa: then " 0 h " 0 Because h% is transitive, reflexive, and connected, it is xða a Þ %yðb b Þ; and a weak ordering. 0 0 (ii) if xag%yb and xa h yb ; then 2. Suppose x and y are in X, a is in C, xa is in the % 0 0 xða"a Þg%yðb"b Þ: domain of h%; and xkXy: By Axiom 8 and Definitions 8.1 and 8.4, ya is in the domain of h%: Then by Definitions 8.1 and 8.4 and statement 1 of Axiom 6, 4. For all x; y in X and all a; b in C; if xah ya: % xaB yb 3. Suppose a and b are in C, x is in X, and xa is in the % domain of h%; and bkCa: Then by Definitions 8.1 and then 8.4 and Axiom 7, xb is in the domain of h%: Then by xkXy iff akCb: Definitions 8.1 and 8.4 and statement 2 of Axiom 6, xah%xb: 4. (i) Suppose x; y in X and a in C are such that 5. For all x; yinX and all a; b in C, if h xa %ya: It then follows from Definitions 8.4 and 8.1 xaB%yb and gkCb; and Statement 2 of Axiom 12 that aAx; AAa; bAy; 0 and BAa can be found such that ðajAÞhðbjBÞ and then there exists a and such that B 0B A CB: Thus by statement 1 of Axiom 6, ahX b: xa %yg: Therefore by Definition 8.1, xkXy: Let b in C be such that bkCa: Then it follows from Definition 8.1 that C B h and D in b can be found such that C CD; C CA; and Proof. Statements 1 and 2 follow from Definitions 8.1 h D CB: It then follows from Axiom 7 that e and f in X and 8.4 and Axiom 12. and E and F in C can be found such that To show statement 3, note that by Theorem 8.1 and B B B B B " 0 " 0 a X e; b X f ; C CD CE CF; eAE; and f AF: its proof that a a kCa and b b kCb; which by B B " 0 Then since e X ahX b X f ; it follows by statement 1 statement 3 of Lemma 8.3 yields that xða a Þ and " 0 of Axiom 6, that ðejEÞhðf jFÞ: Thus by Definition 8.4, yðb b Þ are in the domain of h%: Statement 3 then xbh%yb: Proposition (ii) follows by a similar argu- follows from Axioms 9 and 5 and Definitions 8.1, 8.3, ment. and 8.4. 5. (i) Suppose a; b in C and x in X are such that To show statement 4, we need only show (i) that xah%xb: Then it follows from Definitions 8.4 and 8.1 xkXy and bgCa leads to a contradiction and (ii) that that aAx; AAa; bAx; and BAb can be found such that ygXx and akCb leads to a contradiction. L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 21

(i) xkXy and bgCa: By statement 2 of Lemma 8.3, Lemma 8.6. Suppose xiai is in the domain of h%; bkCai; gkCai; and dkCai: Then the following xah%ya; three statements are true: and thus by statement 4 of Lemma 8.3, 1. bkCd if and only if Di;gðbÞkCDi;gðdÞ: xbh%yb: 2. Di;gðb"dÞ¼Di;gðbÞ"Di;gðdÞ: 3. For all z in C, if zkCg then there exists y such that Since by hypothesis xaB%yb; it then follows from ykCai and Di;gðyÞ¼z: statement 1 of Lemma 8.3 that Proof. 1. (i) Suppose bk d: Then by statement 3 of xbh%xa; C Lemma 8.3, xid and xib are in the domain of h%; and by statement 3 of Lemma 8.3, which by statement 5 of Lemma 8.3 yields akCb; contradicting the hypothesis akCb: xidh%xib; (ii) ygX x and bkCa: By statement 3 of Lemma 8.3, which by Definition 8.5 yields ybh%ya; h and thus by statement 5 of Lemma 8.3, tiðdÞai %tiðbÞai; xah xa: % which by statement 4 of Lemma 8.3 yields Since by hypothesis, xaB yb; it then follows from % t ðdÞgh t ðbÞg; statement 1 of Lemma 8.3, i % i xbh%yb; which by Definition 8.5 yields, which by statement 4 of Lemma 8.3 yields xk y; X xiDi;gðdÞh%xiDi;gðbÞ; contradicting the hypothesis ygXx: Statement 5 is immediate from Axiom 12 and and which by statement 5 of Lemma 8.3 yields Definition 8.4. & Di;gðbÞkCDi;gðdÞ:

Definition 8.5. Suppose xiai is in the domain of h% and (ii) Suppose Di;gðbÞk Di;gðdÞ: By Definition 8.5, b; and g are arbitrary elements of C; bkCai; and gkCai: C xiDi ðbÞ and xiDi ðdÞ are in the domain of h ; and Then by statement 3 of Lemma 8.3, xib is in the domain ;g ;g % thus by statement 3 of Lemma 8.3, of h%: By statement 2 of Lemma 8.4, let tiðbÞ be such that xiDi;gðdÞh%xiDi;gðbÞ; tiðbÞaiB%xib: ð27Þ which by Definition 8.5 yields By statement 4 of Lemma 8.4, tiðgÞdh%tiðgÞb; tiðbÞ%Xxi: ð28Þ

Since by Eq. (27) tiðbÞai is in the domain of h%; it which by statement 5 of Lemma 8.3 yields follows from statement 3 of Lemma 8.3 that tiðbÞg is in bkCd: the domain of h%: By statement 1 of Lemma 8.4, let Di;gðbÞ be such that 2. Since bk a and dk a ; it follows from Theorem B C i C i xiDi;gðbÞ %tiðbÞg: ð29Þ 8.1 that " is positive (Deﬁnition 8.2), and thus that b"dgCbkCai: By Deﬁnition 8.5, Lemma 8.5. Suppose xi is an arbitrary element of X, and xiDi;gðbÞB%tiðgÞb ai; b; and g are arbitrary elements of C such that xiai is in h k k the domain of % and b Cai; and g Cai: Then the and following two statements are true: xiDi;gðdÞB%tiðgÞd: 1. tiðbÞ%Xxi: k k 2. Di;gðbÞ Cg Cai: Thus by statement 3 of Lemma 8.4,

xiðDi;gðbÞ"Di;gðdÞÞB%tiðgÞðb"dÞ: ð30Þ Proof. Statement 1 follows from Eq. (28). To show k statement 2, note that by hypothesis g Cai and from But by Deﬁnition 8.5, Eq. (29), statement 1 of this lemma, and statement 4 of tiðgÞðb"dÞB%xiDi;gðb"dÞ: Lemma 8.4 that Di;gðbÞkCg: & 22 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

Thus by Eq. (30), Lemma 8.8. There exists an unbounded X–C sequence. x ðD ðbÞ"D ðdÞÞB%x D ðb"dÞ; i i;g i;g i i;g Proof. By Lemma 8.7, let j be an additive representation for which by statement 5 of Lemma 8.3 yields C ¼ /C; kC; "S: Di;gðbÞ"Di;gðdÞÞ ¼ Di;gðb"dÞ:

3. Suppose zk g: Then by statement 3 of Lemma 8.3, It then easily follows from the fact that C is an extensive C structure (Definition 8.2) and j is an additive repre- xigh%xiz: ð31Þ sentation for C that jðCÞ has an infinite sequence of elements, a1; a3; a5; y; such that for all a in C, By statement 2 of Lemma 8.4, let y in X be such that agCa2kÀ1 for some positive integer k: For each positive integer k; let A2kÀ1 be an element of a2kÀ1 and a2kÀ1 be ygB%xiz: ð32Þ an element of A2kÀ1 and x2kÀ1 be the element of X such that a is in x : Then, it follows from Definitions Then by statement 4 of Lemma 8.4, 2kÀ1 2kÀ1 8.1 and 8.4 that x2kÀ1a2kÀ1 is in the domain of h% for xikCy: ð33Þ each positive integer k: It easily follows from the fact that C is a concatena- Thus by statement 2 of Lemma 8.3, yai is in the domain tion structure and j is an additive representation for C of h%: By statement 1 of Lemma 8.4, let y be such that that jðCÞ has an infinite sequence of elements, a2; a4; a6; y; such that for all a in C, a2kgCa for some xiyB%yai: ð34Þ positive integer k: For each positive integer k; let A2k be an element of a : Let a be an element of A : By Axiom Then by Definition 8.5, 2k 2 2 12 let for each positive integer k; b2k and B2k be y ¼ tiðyÞ; ð35Þ elements of respectively X and C such that B B A2k CB2k and ða2jA2Þ ðb2kjB2kÞ: and by Eq. (32), z ¼ Di;gðyÞ: Thus to complete the proof of statement 3, we need to only show that yk a : This We will show by contradiction that for each b in X there C i g follows from Eqs. (33), (34), and statement 4 of Lemma exists a positive integer k such that b2k X b: For h 8.4. & suppose not. Let b in X be such that b X b2k for all positive integers k: By Axiom 12, let B in C and b0 in X B 0 B 0 Definition 8.6. A function L from C into Rþ is said to be be such that b X b and ða2jA2Þ ðb jBÞ: By the choice an additive representation for /C; k ; "S if and only if of a2; a4; a6; let k be a positive integer such that C g g all a and a0 in C, A2k CB: Then B2k CB: Thus since 0 0 0 aka iff LðaÞXLða Þð36Þ ðb2kjB2kÞBða2jA2ÞBðb jBÞ; and g 0 " 0 0 it follows from statement 4 of Lemma 8.4 that b2k X b : Lða a Þ¼LðaÞþLða Þ: ð37Þ 0 Thus, because bBX b ; it follows that b2kgX b; a contradiction. For each positive integer k; let x2k be Lemma 8.7. The following two statements are true: the element of X of which b2k is an element, and let a2k be the element of C of which B is an element. Then for 1. There exists an additive representation for /C; 2k each positive integer k; x a is in the domain of h ; k ; "S: 2k 2k % C and for each x in X, there exists an integer n such that 2. For all additive representations j and j0 of þ 0 x2ngXx: /C; kC; "S; there exists r in R such that j ¼ rj : The sequence defined above by xiai; i a positive integer, is an unbounded X–C sequence. &

Proof. Theorem 2.8.1 of Narens (1985) &. Lemma 8.9. Suppose xiai is an unbounded X–C sequence, j and k are positive integers, ajgCak; and b; g; and d are Deﬁnition 8.7. xiai; i ¼ 1; y; is said to be an unbounded elements of C such that

X–C sequence if and only if the following three bkCak; gkCaj; and dkCaj; statements are true: and D ðdÞ¼D ðdÞ: Then for all z such that h A þ k;b j;g 1. xiai is in the domain of % for all i I : k þ z Caj; Dk;bðzÞ and Dj;gðzÞ are deﬁned and 2. For all a in C, there exists j in I such that agC aj: þ Dk;bðzÞ¼Dj;gðzÞ: 3. For all x in X, there exists k in I such that xkgX x: L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 23

Proof. Suppose z is an arbitrary element of C such that and gkCaj; is said to be a D-set (dependent on the zkCaj: Since ajgCak; it follows that zkCak: Thus by sequence xiai) if and only if the following four statement 2 of Lemma 8.5, Dk;bðzÞ and Dj;gðzÞ are statements are true: defined. Suppose D ðzÞaD ðzÞ: A contradiction will k;b j;g 1. Sa|: be shown. There are two cases to consider. 2. For all Dj;g and Dk;b in S; if Dj;gðdÞ¼Dk;bðdÞ for Case 1: Dj;gðzÞgCDk;bðzÞ: By Lemma 8.7, let L be an some dAC; then Dj;gðsÞ¼Dk;bðsÞ for all s common additive representation for /C; kC; "S: Then to the domains of the functions Dj;g and Dk;b: þ LðDj;gðzÞÞ4LðDj;bðzÞÞ: 3. For all jAI and all gkCaj; if there exist Dk;b in S and d in C such that D d D d ; then D is in By properties of the real number system, let m in Iþ be i;gð Þ¼ k;bð Þ i;g : such that S 4. For all Dj;g and Dk;b in S; there exits d in C such that Àm LðDj;gðzÞÞ À LðDj;bðzÞÞ42 LðDj;gðdÞÞ: Di;gðdÞ¼Dk;bðdÞ: Then, again by using properties of the real number system, let p in Iþ be such that Definition 8.9. f is said to be a C-function if and only if for some D-set S; Àm [ LðDj;gðzÞÞ4p2 LðDj;gðdÞÞZLðDk;bðzÞÞ: ð38Þ f ¼ S: Since L is an additive representation, it follows from a hypothesis of the lemma that LðDj;gðdÞÞ ¼ LðDk;bðdÞÞ; which by Eq. (38) yields Lemma 8.11. Let xiai be an unbounded X–C sequence, m be a positive integer, and lkCam: Then for all d in C, L D z 4p2ÀmL D d 39 ð j;gð ÞÞ ð j;gð ÞÞ ð Þ there exist a positive integer p and an element gp of C such and that B Àm dkCgp and tpðamÞgp %xpl: p2 LðDk;bðdÞÞZLðDk;bðzÞÞ: ð40Þ By statement 2 of Lemma 8.6 and by the property of Proof. Since /C; k ; "S is an extensive structure additive representations expressed in Eq. (37), it easily C ¼ C (Theorem 8.1), it is easy to show that a positive integer k follows from Eqs. (39) and (40) that can be found so that m LðDj;gð2 zÞÞ4LðDj;gðpdÞÞ kamkCl: and Let d be an arbitrary element of C. Since C is an m LðDk;bðpdÞÞZLðDk;bð2 zÞÞ; extensive structure and xiai is an unbounded X–C sequence, it easily follows from Lemma 8.7 and which by the property of additive representations Definition 8.6 that a positive integer p can be found expressed in Eq. (37) and statement 1 of Lemma 8.6 such that a k a and yields m C p k m d Ckap: ð41Þ 2 zgCpd and Since by hypothesis llkCam and by Definition 8.5 m B pdkC2 z; tpðamÞap %xpam; which contradicts that k is a total ordering. C it follows from statement 5 of Lemma 8.4 that g can be Case 2: D z g D z ; follows by a similar argu- p k;bð Þ C j;gð Þ found such that ment. & B tpðamÞgp %xpl: ð42Þ Lemma 8.10. Let xiai be an unbounded X–C sequence. Then By Definition 8.5,

Dj;gðajÞ¼g: tpðamÞapB%xpam: ð43Þ

Proof. By Deﬁnition 8.5, Thus by applying statement 3 of Lemma 8.4 to Eq. (43) k À 1 times, xjDj;gðajÞB%tjðajÞgB%xjg; tpðamÞðkapÞB%xpðkamÞ; and thus by statement 4 of Lemma 8.4, Dj;gðajÞ¼g: &

which together with kamkCl implies Definition 8.8. Let xiai be an unbounded X–C sequence. þ x lh t ða Þðka Þ; Then a set S of elements of the form Dj;g; where jAI p % p m p 24 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 which by Eq. (42) yields It will first be shown that f is a function on C. Since f is a union of binary relations, f is a binary relation. tpðamÞg h tpðamÞðkapÞ; p % Suppose ðd; aÞ and ðd; bÞ are arbitrary elements of f : By Definitions 8.8 and 8.9 let D and D in be such that which yields j;g k;s S Dj;gðdÞ¼a and Dk;sðdÞ¼b: By statement 2 of Definition kapkCgp; 8.7, a ¼ b: Since ðd; aÞ and ðd; bÞ are arbitrary elements of f ; it follows that f is a function. which by Eq. (41) yields It will now be shown that f is onto C. Let d be an arbitrary element of C. Since by statement 1 of dkCgp: & Definition 8.8 Sa|; let Dm;l be an element of S: By Lemma 8.10, Lemma 8.12. Let xiai be an unbounded X–C sequence. þ Dm;lðamÞ¼l: Then for each m in I and each lkCam; there exists a D- þ set S dependent on xiai such that Dm;l is in S: By Lemma 8.11, let p in I and g in C be such that dk g and t ða Þgh x l: Proof. Let m be an arbitrary element of Iþ; and l be C p m % p k an arbitrary element of C such that l Cam: Let S be Then by Definition 8.5, the largest set of elements of the form Dk;b; where k þ Dp;gðamÞ¼l: is in I and bkCak; such that Dm;l is in D and for all Dk;b in D; if Dk;bðdÞ¼Dm;lðdÞ for some d in C, then Dk;bðsÞ¼ D ðsÞ for all s common to the domains of D Thus by statement 3 of Definition 8.8, Dp;g is in S: Since m;l k;b k and D : S exists by theorems of set theory. It d Cg; it follows from statement 3 of Lemma 8.6 that m;l k will be shown that S satisfies statements 1–4 of Dp;gðyÞ¼d for some y Cap: Thus f is onto C. Definition 8.8. It will next be shown that f is defined on all of C. Let d be an arbitrary element of C. Since f is onto C,it 1. Sa| since Dm;l is in S: follows from Definitions 8.8 and 8.9 that ap; a; and g 2. Suppose Dj;g and Dk;b are in S: Let ap be the largest can be found so that Dp;gðaÞ¼d: Then by Definition 8.5 element of faj; ak; amg and d be an element of C such it follows that dkCap and therefore by Definition 8.5 that dgCap: Then d is in the domains of Dj;g; Dk;b; and that Dp;gðdÞ exists. Thus by Definitions 8.8 and8.9, f ðdÞ Dm;l: By the definition of S; is defined. Dj;gðdÞ¼Dm;lðdÞ¼Dk;bðdÞ: Statements 1 and 2 of Lemma 8.6 together with Thus by Lemma 8.9, Dj;gðsÞ¼Dk;bðsÞ for all s common Definitions 8.8 and 8.9 show that f preserves kC " / to the domains of Dj;g and Dk;b: and : Therefore f is an automorphism of C; þ "S 3. Suppose Dk;b is in S; j is in I ; gkCaj; d is in C, kC; : & and Dj;gðdÞ¼Dk;bðdÞ: Let ap be the largest element of faj; ak; amg and Z be an element of C such that ZgCap: Lemma 8.14. Let xiai be an unbounded X–C sequence, Then Z is in the domains of Dj;g; Dk;b; and Dm;l: By and let j be an additive representation for C ¼ / "S Iþ Lemma 8.9, Dj;gðZÞ¼Dk;bðZÞ: Since Dk;b is in C; kC; : Then for each i in there exists a Rþ S; Dk;bðZÞ¼Dm;lðZÞ: Thus function ci from fxjxAX and xikXxg into such that for all x; y; a; and b; if xikXx; xikXy; akCai; and Dj;gðZÞ¼Dm;lðZÞ: bkCai; then

ciðxÞ ciðyÞ Therefore by Lemma 8.9, Dj;gðsÞ¼Dm;lðsÞ for all s xah yb iff X : % j a j b common to the domains of Dj;g and Dm;l: Thus Dj;g is ð Þ ð Þ in S: 4. Suppose Di;g and Dk;b are in S: Then by the Proof. Since j is an additive representation for C; it argument presented in the numbered 2 paragraph immediately follows that A ¼ /jðCÞ; X; þS is an above, Dj;gðdÞ¼Dk;bðdÞ for some d in C. & extensive structure that is isomorphic to C: It is well- known that each automorphism A is a multiplication by Lemma 8.13. Suppose f is a C-function. Then f is an a positive real number. automorphism of /C; X ; "S: þ C For each i in I and each x in X,ifxikXx; then for some d; xidB%xai and (by statement 4 of Lemma 8.4) Proof. Throughout this proof let, by Deﬁnitions 8.8 and þ dkCai: Therefore, by Deﬁnition 8.5, for each i in I and 8.9, be a D-set dependent on the sequence x a such S i i each x in X such that xikXx; let that [ 1 c ðxÞ¼ : f ¼ S: i À1 jðti ðxÞÞ L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 25

Let i; x; y; a; and b; be arbitrary elements such that i is Thus þ in I ; xikXx; xikXy; akCai; and bkCai: Then by c1ðzkÞ Deﬁnition 8.5, cjðzkÞ¼ cjðzjÞ; c1ðzjÞ xah yb iff t tÀ1ðxÞah t tÀ1ðyÞb % i i % i i which by Eq. (44) yields À1 À1 iff xiDi;a½ti ðxÞh%xiDi;b½ti ðyÞ 0 c1ðzkÞ À1 À1 r ¼ c ðzjÞ¼c ðzkÞ¼r: ð45Þ iff Di;a½t ðxÞ%CDi;b½t ðyÞ j 1 i i cjðzjÞ iff j D tÀ1 x j D tÀ1 y : ð i;a½ i ð ÞÞp ð i;b½ i ð ÞÞ To show c is a function, suppose ðx; sÞ and ðx; s0Þ are Since by Lemmas 8.12 and 8.13 Di;a is part of an elements of c: By the deﬁnition of c; let j and k be such automorphism of C and j is an isomorphism of C onto that A; c1ðzjÞ À1 À1 cjðxÞ¼s jðDi;a½ti ðxÞÞ ¼ j½Di;aðjðti ðxÞÞ; cjðzjÞ where j½Di;a is part of an automorphism of A: Since by and Lemma 8.10 D ða Þ¼a; it then follows that j½D i;a i i;a c ðz Þ must be multiplication by jðaÞ=jða ). Therefore, by the 1 k c ðxÞ¼s0: i c z k above sequence of logical equivalences, kð kÞ À1 À1 Without loss of generality, suppose that zjkXzk: Then xah%yb iff jðDi;a½ti ðxÞÞpjðDi;b½ti ðyÞÞ zk is in the domain of c : By statement 1 of Lemma 8.4, À1 À1 j iff j½Di;aðjðti ðxÞÞpj½Di;bðjðti ðyÞÞ let m and n be such that jðaÞ jðbÞ À1 À1 xmB%z n: iff jðti ðxÞÞp jðti ðyÞÞ k jðaiÞ jðaiÞ Then by Lemma 8.14, jðaÞ 1 jðbÞ 1 iff p j a c x j a c y cjðxÞ jðmÞ ckðxÞ ð iÞ ið Þ ð iÞ ið Þ ¼ ¼ ; c ðxÞ c ðyÞ cjðzkÞ jðnÞ ckðzkÞ iff i X i : & jðaÞ jðbÞ and thus

c ðzkÞ c ðxÞ¼ j c ðxÞ: ð46Þ Lemma 8.15. Let xiai; ji; and ci be as in Lemma 8.14. j k ckðzkÞ For each positive integer i let zi be x1 if xikXx1; and zi be From xi if x1kXxi: (Thus in particular, z1kXzi for all positive integers i:) Let c1ðzjÞ [N s ¼ cjðxÞ; c ðz Þ cjðzjÞ c ¼ 1 i c : c z i i¼1 ið iÞ it follows by Eq. (46) that "# Then c is a function from X into Rþ: c1ðzjÞ cjðzkÞ c1ðzjÞ 1 s ¼ ckðxÞ¼ cjðzkÞ ckðxÞ; c ðzjÞ c ðzkÞ c ðzjÞ c ðzkÞ Proof. Since c is a union of functions, c is a set of j k j k ordered pairs. which by Eqs. (44) and (45) yields Let k be an arbitrary positive integer. Let c ðz Þ s ¼ 1 k c ðxÞ¼s0: c ðzkÞ k r ¼ 1 c ðz Þ¼c ðz Þ: ckðzkÞ c ðz Þ k k 1 k k k To show that c is deﬁned on all of X, let x be Then the ordered pair ðzk; rÞ is in c: Suppose zk is in the an arbitrary element of X. Since xiai is and unboun- 0 domain of cj: Then ðzk; r Þ is in c; where ded X–C sequence, let p be such that xpkCx: Then x is in the domain of c and thus is in the domain c1ðzjÞ 0 p cjðzkÞ¼r : ð44Þ of c: & cjðzjÞ By statement 1 of Lemma 8.4, let d and g be such that Lemma 8.16. Let j be an additive representation for / "S zjdB%zkg: C ¼ C; kC; : Then there exists a function c from X into Rþ such that for all xa and yb in the domain Since z k z and z k z ; it follows by Lemma 8.14 1 X j 1 X k of h ; that % cjðzjÞ jðdÞ c1ðzjÞ cðxÞ cðyÞ ¼ ¼ : xah%yb iff X : cjðzkÞ jðgÞ c1ðzkÞ jðaÞ jðbÞ 26 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

Proof. Let xiai be an unbounded X–C sequence Proof. and c and ci as in Lemmas 8.15 and 8.14, and let xa aBX b iff ðaja; bÞBðbja; bÞ and yb be arbitrary elements of the domain of h%: Since x a is an unbounded X–C sequence, let j be uðaÞvðaÞ uðbÞvðbÞ i i iff ¼ such that uðaÞþuðbÞ uðaÞþuðbÞ iff uðaÞvðaÞ¼uðbÞvðbÞ: & xjkCx and xjkCy:

By use of Lemma 8.4 it is easy to show that we can ﬁnd d Deﬁnition 8.10. For all a and b in X; aEb if and only if and g in C such that there exists e in X such that eaa; eab and dk a ; gk a ; and xdB yg: C j C j % ðeje; aÞBðeje; bÞ:

It will first be shown that Theorem 8.2 (Theorem 2.1). Assume the basic belief xah yb iff xða"dÞh yðb"gÞ: ð47Þ % % axioms (Definition 2.4) are true. Then the following two statements hold: First suppose xah%yb: Since C is an extensive " " " structure, a dkCa and b gkCb: Thus xða dÞ and 1. (Representation theorem) There exists a basic belief " yðb gÞ are in the domain of h%: Therefore, since representation for \ (Definition 2.3). xdB%yg; it follows that from statement 3 of Lemma 8.4 2. (Uniqueness theorem) Let " " that xða dÞh%yðb gÞ: Now suppose not xah%yb: / S \ Then ybg%xa: Thus, since xdB%yg; by statement 3 of U ¼fuj u; v u is a basic belief representation for g; Lemma 8.4, yðb"gÞg%xða"dÞ; and thus not " " and} xða dÞh%yðb gÞ: B Lemma 8.14 applied to xd %yg yields V ¼fvj/u; vS v is a basic belief representation for \g: cjðxÞ cjðyÞ ¼ ; Then U and V are ratio scales. jðdÞ jðgÞ and thus Proof. By Theorem 8.1 let j be and additive representation for C ¼ /C; kC; "S: cjðxÞjðgÞ¼cjðyÞjðdÞ: ð48Þ For each AAC; let aA be the element of C such that Therefore by Eq. (47), Lemma 8.14, Definition 8.6, AAaA: Eq. (48), and Lemma 8.15, For each c in X; let (by Axiom 10 and Definition 8.10) c0 be such that cEc0 and cac0; and let u be the function xah yb iff xða"dÞh yðb"gÞ % % on X defined by

cjðxÞ cjðyÞ iff X jðafc;c0gÞ jða"dÞ jðb"gÞ uðcÞ¼ : 2 cjðxÞ cjðyÞ iff X (To show u is well-deﬁned, let c00 be such that cEc00 and j a j d j b j g ð Þþ ð Þ ð Þþ ð Þ a 00 c c : It is only necessary to show afc;c0g ¼ afc;c00g: It iff c ðxÞjðbÞþc ðxÞjðgÞ 0 00 j j follows from Axiom 4 that fc; c gBCfc; c g; and thus XcjðyÞjðaÞþcjðyÞjðdÞ that afc;c0g ¼ afc;c00g:) 1. Let A ¼fa; a1; y; ang be an arbitrary element of C iff c x j b Xc y j a jð Þ ð Þ jð Þ ð Þ such that a; a1; y; an are distinct. Then by Axiom 10 and Deﬁnition 8.10, let A0 a0; a0 ; y; a0 be such that c ðxÞ c ðyÞ ¼f 1 ng j j 0 0 0 B 0 0 | E 0 iff X a ; a1; y; an are distinct, A CA ; A-A ¼ ; a a ; jðaÞ jðbÞ E 0 B 0 and for i ¼ 1; y; n; ai ai: Because A CA ; it follows cðxÞ cðyÞ iff X : that aA ¼ aA0 ; and thus that jðaÞ jðbÞ jðaAÞ¼jðaA0 Þ:

8.2. Proofs for Section 2 Therefore, because j is an additive representation for C; " Lemma 8.17. Let /u; vS be a basic belief representation 2jðaAÞ¼jðaA aA0 Þ for \ and a and b be in X: Then ¼ jðaA,A0 Þ aB b iff u a v a u b v b : ¼ j½a 0 "a 0 "?"a 0 X ð Þ ð Þ¼ ð Þ ð Þ fa;a g fa1;a1g fan;ang L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 27

Xn The above establishes that ¼ jðafa;a0gÞþ jðafa ;a0gÞ i i U ¼fuj/u; vS u is a basic belief representation for \g 1¼1 Xn is a ratio scale. ¼ 2uðaÞþ 2uðaiÞ; Because /u; vS; a basic belief representation for \; it i¼1 is an immediate verification that for each s in and thus Rþ; /u; svS is a basic belief representation for \: X Let a be an element of X; /t; v0S be an arbitrary basic jða Þ¼ uðeÞ: ð49Þ A h ARþ eAA belief representation for ; and r be such that 0 Because j is an additive representation for C and v ðaÞ¼rvðaÞ: ð50Þ Eq. (49) holds for all A in C; it follows that for all B and Let b be an arbitrary element of X: To complete the C in C, proof of statement 2, it is sufficient to show that 0 B\CC iff aBkCaC v ðbÞ¼rvðbÞ: iff jða ÞXjða Þ XB XC Because iff uðbÞX uðcÞ: U ¼fuj/u; vS u is a basic belief reprsentation for \g bAB cAC is a ratio scale, /ru; vS is a basic belief representation Let c be as in Lemma 8.16. Define the function v on for \: For each C in C; let X as follows: for each e in X; let xe be the element of X X u C u c : such that eAxe and let ð Þ¼ ð Þ cAC cðxeÞ vðeÞ¼ : Case 1: a\ b: Let B in C be such that bAB: Then by uðeÞ X Axiom 12 let c in X and C in C be such that By Lemma 8.16 and Eq. (49) and the above definition cBX a and ðcjCÞBðbjBÞ: ð51Þ of v; Then by Eq. (51) and Lemma 8.17, ðajAÞhðbjBÞ iff xaaAh%xbaB ruðaÞvðaÞ¼ruðcÞvðcÞ and ruðaÞv0ðaÞ¼ruðcÞv0ðcÞ:ð52Þ cðx Þ cðx Þ iff a X b jðaAÞ jðaBÞ By Eq. (51), vðaÞuðaÞ vðbÞuðbÞ ruðcÞvðcÞ ruðbÞvðbÞ iff P XP ¼ ð53Þ eAA uðeÞ eAB uðeÞ ruðCÞ ruðBÞ uðaÞ uðbÞ and iff vðaÞP XvðbÞ P : 0 0 eAA uðeÞ eAB uðeÞ ruðcÞv ðcÞ ruðbÞv ðbÞ ¼ : ð54Þ Let A and B be arbitrary elements of C: By Definition ruðCÞ ruðBÞ 8.1 and Eq. (49), Eqs. (51)–(53) yield

Ah B iff aAkCaB iff jðaAÞXjðaBÞ uðaÞvðaÞ uðbÞvðbÞ XC X ¼ ; ð55Þ iff uðeÞX uðeÞ: uðCÞ uðBÞ eAA eAB and Eqs. (51), (52), and (54) yield 2. Let /u; vS be a basic belief representation for \: It uðaÞv0ðaÞ uðbÞv0ðbÞ ¼ : ð56Þ is an immediate verification that for each s in uðCÞ uðBÞ Rþ; /su; vS is a basic belief representation for \: Suppose /t; vS is also a basic belief representation for Then by Eqs. (50), (55), and (56), \: Then it follows from Eq. (49) that v0ðbÞ¼rvðbÞ: X X uðAÞ¼ uðeÞ and tðAÞ¼ tðeÞ; Case 2: b\X a: Let A in C be such that aAA: Then by eAA eAA Axiom 12 let c in X and C in C be such that for A in C: It is easy to verify that uˆ and tˆ defined on C cBX b and ðcjCÞBðajAÞ: ð57Þ by Then by Eq. (57) and Lemma 8.17, uˆða Þ¼uðAÞ and tˆða Þ¼tðAÞ A A ruðbÞvðbÞ¼ruðcÞvðcÞ and ruðbÞv0ðbÞ¼ruðcÞv0ðcÞ:ð58Þ are additive representations for /C; k ; "S: Since C ¼ C By Eq. (57), by Lemma 8.7 the additive representations for C form a ruðcÞvðcÞ ruðaÞvðaÞ ratio scale, let r be a positive real such that ruˆ ¼ tˆ: Then ¼ ð59Þ ru ¼ t: ruðCÞ ruðAÞ 28 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 and and applying Axiom 14 and Theorem 8.2 to Eq. (63) ruðcÞv0ðcÞ ruðaÞv0ðaÞ yields> ¼ : ð60Þ ruðCÞ ruðAÞ uðaÞ uðbÞ vðaÞ ¼ vðbÞ ð65Þ uðaÞþuða0Þ uðbÞþuðb0Þ Eqs. (57)–(59) yield uðbÞvðbÞ uðaÞvðaÞ and ¼ ; ð61Þ uðCÞ uðAÞ vðaÞuðaÞ vða0Þuða0Þ ¼ : ð66Þ uðaÞþuðbÞ uða0Þþuðb0Þ and Eqs. (57), (58), and (60) yield uðbÞv0ðbÞ uðaÞv0ðaÞ Another application of Axiom 14 and Theorem 8.2 to ¼ : ð62Þ uðCÞ uðAÞ Eq. (63) yields vðaÞuðaÞ vða0Þuða0Þ ¼ : ð67Þ Then by Eqs. (50), (61), and (62), uðaÞþuðb0Þ uða0ÞþuðbÞ v0ðbÞ¼rvðbÞ: & Eq. (64) implies vðaÞuðaÞ¼vða0Þuða0Þ: Thus Eq. (66) implies Theorem 8.3 (Theorem 2.2). Assume the basic belief axioms with binary symmetry (Definition 2.5). Then the uðaÞþuðbÞ¼uða0Þþuðb0Þ; ð68Þ following two statements hold: and Eq. (67) implies 1. (Representation theorem) There exists a basic choice uðaÞþuðb0Þ¼uða0ÞþuðbÞ: ð69Þ representation for \ (Definition 2.6). 2. (Uniqueness theorem) The set of basic choice repre- Adding Eqs. (68) and (69) and reducing then yields, sentations for \ forms a ratio scale. uðaÞ¼uða0Þ;

Proof. 1. Because the basic belief axioms with binary which by Eq. (69) yields, symmetry imply the basic belief axioms, by Theorem 8.2 0 let /u; vS be a basic belief representation for \: Then uðbÞ¼uðb Þ; by Theorem 8.2, (i) u is a function from X into Rþ; (ii) A B and thus by Eq. (65), for all andXin C; X v a v b : 70 A\CB iff uðaÞX uðbÞ; ð Þ¼ ð Þ ð Þ aAA bAB and (iii) for all finite conditional events ðajAÞ and ðbjBÞ Thus to complete the proof of statement 1 it needs to of X; only be shown that uðaÞ uðbÞ a\ b iff uðaÞXuðbÞ: ðajAÞ\ðbjBÞ iff vðaÞ P XvðbÞP : X eAA uðeÞ eAB uðeÞ By Definition 2.2, Theorem 8.2, and Eq. (70), Thus to show statement 1, it is sufficient to show that for all a and b in X; a\X b iff ðaja; bÞ\ðbja; bÞ vðaÞ¼vðbÞ and ½a\ b iff uðaÞXuðbÞ: uðaÞ uðbÞ X iff vðaÞ XvðbÞ uðaÞþuðbÞ uðaÞþuðbÞ Let a and b be arbitrary elements of X and A and B be iff uðaÞXuðbÞ: arbitrary elements of C. Suppose aAA and bAB: By 0 0 0 0 0 Axiom 9, let a and b be such that aaa ; bab ; aBX a ; 0 2. By Theorem 8.2, and bBX b : Then by Definition 2.2, ðaja; a0ÞBða0ja; a0Þ and ðbjb; b0ÞBðb0jb; b0Þ: ð63Þ U ¼fuj/u; vS is a basic belief representation for \g;

Applying Theorem 8.2 to Eq. (63) yields, forms a ratio scale. Thus statement 2 is true. & vðaÞuðaÞ vða0Þuða0Þ ¼ and 8.3. Proofs for Section 3 uðaÞþuða0Þ uðaÞþuða0Þ vðbÞuðbÞ vðb0Þuðb0Þ ¼ ; ð64Þ Lemma 8.18. Assume the belief axioms. By Theorem 2.1, uðbÞþuðb0Þ uðbÞþuðb0Þ let /u; vS be a basic belief representation for \: Define u% L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 29 and von% F as follows: Let A be an arbitrary element of and thus F: uðf Þ uðf Þ vðf Þ ¼ vðf Þ ; (i) Suppose A ¼ |: Define u%ðAÞ¼0 and v%ðAÞ¼1: uðeÞþuðf Þ uðAÞþuðf Þ a| (ii) Suppose A : Let B be an element ofF such that yielding ACB: By Axiom 17, let e; E; and f be such that uðeÞ¼uðAÞ: ð72Þ BBCE; B Similarly, the hypothesis ðAjBÞ EðejEÞ; 0 0 0 B 0 0 f ae; f eA; and ðf je ; f Þ Eðf jA,ff gÞ B yields ðf je; f Þ Eðf jA,ff gÞ: 0 Then define u%ðAÞ¼uðeÞ and v%ðAÞ¼vðeÞ: uðe Þ¼uðAÞ: ð73Þ Then the following three statements are true: Thus by Eqs. (72) and (73), u%ðAÞ¼uðeÞ¼uðe0Þ: ð74Þ 1. u%ð|Þ¼0 and v%ð|Þ¼1: Because by hypothesis, 2. If for some a; A ¼fag; then u%ðfagÞ ¼ uðaÞ and 0 0 0 ðAjBÞBEðejEÞ; ðAjB ÞBEðe jE Þ; BBCE; v%ðfagÞ ¼ vðaÞ: and B0B E0; 3. u% and v% are well-defined on F: That is, if Aa| and C B0; e0; E0 and f 0 are such that B0 is in it follows by Axiom 18 that 0 0 0 F; ACB ; B BCE ; 0 0 0 ðeje; e ÞBEðe je; e Þ: ðAjB0ÞB ðe0jE0Þ; E Thus by Theorem 2.1, 0a 0 0 f e ; f eA; and uðeÞvðeÞ uðe0Þvðe0Þ ; 0 0 0 0 0 0 ¼ 0 ðf je ; f ÞBEðf jA,ff gÞ; uðeÞþuðe Þ uðeÞþuðe Þ then u%ðAÞ¼uðe0Þ and v%ðAÞ¼vðe0Þ: which by Eq. (74) yields v%ðeÞ¼vðeÞ¼vðe0Þ: & Proof. To simplify notation, for each nonempty C in C, let X Lemma 8.19. Assume the hypotheses and notation of uðCÞ¼ uðcÞ: Lemma 8.18. Let A be a nonempty finite event of X; and cAC let, by definition, X 1. Statement 1 immediately follows from condition (i). uðAÞ¼ uðaÞ: 2. Suppose A ¼fag: Then By condition (ii), aAA f e f B f a f Then ð j ; Þ Eð j ; Þ; X X and thus u%ðAÞ¼ u%ðfagÞ ¼ uðaÞ¼uðAÞ: A A uðf Þ uðf Þ a A a A vðf Þ ¼ vðf Þ ; uðeÞþuðf Þ uðaÞþuðf Þ Proof. Since /u; vS is a basic belief representation for yielding \; it follows by statement 2 of Lemma 8.18 that for u%ðfagÞ ¼ uðeÞ¼uðaÞ: ð71Þ each x in X; % Also by condition (ii), uðfxgÞ ¼ uðxÞ: B B Thus, B CE and ðfagjBÞ EðejEÞ; X X and thus u%ðfagÞ ¼ uðaÞ¼uðAÞ; aAA aAA uðfagÞ uðeÞ uðBÞ¼uðEÞ and vðfagÞ ¼ vðeÞ; uðBÞ uðEÞ and by Eqs. (73) and (74), u%ðAÞ¼uðAÞ: & which together with Eq. (71) yields, v%ðfagÞ ¼ vðeÞ¼vðaÞ: Theorem 8.4 (Theorem 3.1). Assume the belief axioms 3. Assume the hypotheses of statement 3. Suppose (Definition 3.2). Then the following two statements are true: Aa|: By condition (ii), 1. There exists a belief representation for \ with B E ðf je; f Þ Eðf jA,ff gÞ; context function u and definiteness function v: 30 L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31

B B / S 2. Let be a belief representation for \E with context Because B CE and u; v is a basic belief representa- function u and definiteness function v: Then the tion for \; uðBÞ¼uðEÞ: By Lemma 8.19, following two statements are true: u%ðBÞ¼uðBÞ: (i) For all positive reals r and s there exists a belief Thus, representation for \E with context function ru % and definiteness function sv: B uðAÞ uðAÞ uðeÞ B ðAjBÞ¼v%ðAÞ ¼ v%ðAÞ ¼ vðeÞ : (ii) Let 1 be a belief representation for \E with u%ðBÞ uðBÞ uðEÞ context function u and definiteness function v 1 1 Similarly, Then for some positive real numbers r and s; u%ðCÞ uðgÞ u ¼ ru and v ¼ sv: BðCjDÞ¼v%ðCÞ ¼ vðgÞ : 1 1 u%ðDÞ uðGÞ Thus, because /u; vS is a basic belief representation for Proof. 1. Let u; v; u%; and v% be as in Lemma 8.18. For \ and \E is an extension of \; each nonempty finite event of X; let, by definition, X ðAjBÞ\EðCjDÞ iff ðejEÞ\ðgjGÞ uðCÞ¼ uðcÞ: uðeÞ uðgÞ A iff vðeÞ XvðgÞ c C uðEÞ uðGÞ And for each finite event ðAjBÞ of X; let iff BðAjBÞXBðCjDÞ: u%ðAÞ BðAjBÞ¼v%ðAÞ : ð75Þ Case 3: A ¼ BorC¼ D: Because of statement 1 of u%ðBÞ Axiom 19, we may assume that B and D are in C. Then Then it will be shown that B is a belief representation for by statement 2 of Axiom 12, let B0 and D0 be such that \ % % 0 0 0 0 E with context function u and definiteness function v BBCB ; B-B a|; DBCD ; and D-D a|; and (Definition 3.3). It is immediate that conditions (i), (ii), ðAjBÞ\ ðCjDÞ iff ðAjB,B0Þ\ ðCjD,D0Þ: ð76Þ and (iv) of Definition 3.3 hold for B; u%; and v%: E E Condition (iii) follows from the definition of u% and Thus by Case 2 (which has already been shown), Lemma 8.19. To show condition (v), let A B and 0 0 ð j Þ ðAjB,B Þ\EðCjD,D Þ ðCjDÞ be arbitrary conditional events. iff BðAjB,B0ÞXBðCjD,D0Þ: ð77Þ Case 1: Either A ¼ | or C ¼ |: Suppose | ðAjBÞ\EðCjDÞ: (i) If A ¼ ; then by Axioms 15 and But by conditions (iii) and (iv) of Definition 3.3, 16, C ¼ |: Thus by Eq. (75) and statement 1 of Lemma B 0 1B B 0 1B ðAjB,B Þ¼2 ðAjBÞ and ðCjD,D Þ¼2 ðCjDÞ: 8.18, ð78Þ BðAjBÞ¼0X0 ¼ BðCjDÞ: Thus by Eqs. (76)–(78). | B B (ii) If C ¼ ; then by Eq. (75) and statement 1 of ðAjBÞ\EðCjDÞ iff ðAjBÞX ðCjDÞ: Lemma 8.18, 2. Part (i), statement 2 of the theorem follows by % | a| % BðAjBÞX0 ¼ BðCjDÞ: direct verification. Because uð Þ¼0 and for A ; uðAÞ and v%ðAÞ are defined to be, respectively, uðeÞ and vðeÞ for B B | B Suppose ðAjBÞX ðCjDÞ: If A ¼ ; then ðCjDÞ¼0; an appropriately chosen element e of X; part (ii) follows which by Eq. (75) and the definition of u% in Lemma 8.18 from statement 2 of Theorem 2.1, statement 2 of Lemma | yields C ¼ ; which by Axioms 15 and 16 yields 8.18, and the definitions of u% and v%: & | ðAjBÞ\EðCjDÞ: If C ¼ ; then ðAjBÞ\EðCjDÞ by Axiom 12. a| a| Case 2: A ; C ; and both B and D have at least References two elements: By Axiom 17, let e; E; f and g; G; h be such that the following two conditions hold: Bradley, R. A., & Terry, M. E. (1952). The rank analysis of incomplete B B block designs. Biometrika, 39, 324–345. (1) B CE; D CG; eeA; geC; and Brenner, L., & Rottenstreich. Asymmetric support theory: Focus- dependence and context-independence in likelihood judgment. ðAjBÞBEðejEÞ and ðCjDÞBEðgjGÞ: Manuscript. de Finetti, B. (1931). Sul Significato Soggettivo della Probabilita`. (2) f ae; f eA; hag; heC; and Fundamenta Mathematica, 17, 298–329. B B de Finetti, B. (1937). La pree´vision: ses lois logiques, ses sources ðf je; f Þ Eðf jA,ff gÞ and ðhjg; hÞ EðhjC,fhgÞ: subjectives. Annales de Institut Henri Poincare´ , 7, 1–68 (Translated into English. In H. E. Kyburg jr., & H. E. Smokler (Eds.), Studies Then by Lemma 8.18, in subjective probability (pp. 93–158). New York: Wiley, 1964). % % Ellsberg, D. (1961). Risk, ambiguity and the Savage axioms. Quarterly uðAÞ¼uðeÞ and vðAÞ¼vðeÞ: Journal of Economics, 75, 643–649. L. Narens / Journal of Mathematical Psychology 47 (2003) 1–31 31

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