LOUIS NARENS

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS

ABSTRACT. A probability function P on an algebra of events is assumed. Some of theevents are scientific refutations in the sense that the assumption of their occurrence leadsto a contradiction. It is shown that the scientific refutations form a a boolean sublatticein terms of the subset ordering. In general, the restriction of P to the sublattice is not aprobability function on the sublattice. It does, however, have many interesting properties.In particular, (i) it captures probabilistic ideas inherent in some legal procedures; and (ii) itis used to argue against the commonly held view that behavioral violations of certain basicconditions for qualitative probability are indicative of irrationality. Also discussed are (iii)the relationship between the formal development of scientific refutations presented hereand intuitionistic logic, and (iv) an interpretation of a belief function used in the behavioralsciences to explain empirical results about subjective, probabilistic estimation, includingthe Ellsberg paradox.

1. INTRODUCTION

Since the 1930s, the probability calculus of Kolmogorov (1933) has be-come the standard theory of probability for mathematics and physics. Thisarticle proposes an alternative calculus for empirical science. The alterna-tive is consistent with the standard in the sense that it assigns the sameprobabilities to scientific events as a probability function from the Kol-mogorov theory. It, however, has a different event space and employs adifferent logic to relate events. The difference results from platonic natureof the event space and logic of the standard, which fails account for charac-terizing features of scientific events, like their verifiability and refutability.Such features are accounted for in the alternative, which is based on anon-classical logic of events that intrinsically models key properties ofverifiability and refutability.

An analogous situation occurred in mathematics. Once, euclideangeometry was completely dominate. Manifolds were described and workedout in an euclidean framework. Later, it was found to be far more insight-ful and productive to work completely “inside” the manifold, exploitingits intrinsic characteristics. Thus, for example, it became more produc-tive for mathematicians to describe the geometry of a torus (the surfaceof a doughnut shaped object) intrinsically in terms of its 2-dimensional

Synthese (2005) 145: 397–423 © Springer 2005

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geometrical properties than extrinsically as a 2-dimensional surface in a 3-dimensional euclidean space. In the present case, the Kolmogorov theoryis dominate. The scientific events, i.e., events of the form, “A occurs and is(scientifically) verifiable,” or of the form, “A occurs and is (scientifically)refutable,” of a targeted fragment of science can be viewed as a portion ofa larger set of platonic events. They only form a portion, because the com-plement (or negation) of a scientific event need not be a scientific event;that is, the complement of a verifiable event need be neither verifiablenor refutable, and similarly, the complement of a disjunction of refutableevents need be neither verifiable nor refutable. In short, the complementa-tion operator of a boolean algebra of events does not in general preservescientific events. In this sense, the Kolmogorov theory adds extrinsic struc-ture to the scientific events through its use of (boolean) complementation.This article shows that the scientific events themselves have considerablestructure. This intrinsic structure forms the basis of an alternative proba-bilistic theory that is applied to various examples, including American lawand issues involving the behavioral characterization of rational behavior.

1.1. Probability Functions

Since the 1930s, probability has been defined in accordance of Kol-mogorov (1933) as a finite or σ -additive measure on a boolean algebraof sets. Although σ -additivity adds important mathematical structure, thegeneralizations of probability theory discussed in this article do not em-ploy the additional structure, and thus only finitely additive versions arepresented. Also, to simplify notation and definitions, nonempty sets ofmeasure 0 are excluded. If needed, the development could be extendedto include the σ -additive case and nonempty sets of measure 0.

One productive way for understanding the relationship of standardprobability theory to its generalizations and extensions is through latticetheory. Boolean algebras of sets form a special kind of lattice, calledboolean lattices. The concepts of lattice and boolean lattice are definedas follows:

DEFINITION 1. � is said to be partial ordering on A if and only if A isa nonempty set and the following three conditions hold for each a, b, andc in A:

(i) a � a;(ii) if a � b and b � a, then a = b; and

(iii) if a � b and b � c, then a � c. �DEFINITION 2. 〈A,�, u, z〉 is said to be a lattice (with unit element uand zero element z) if and only if

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(i) � is a partial ordering on A;(ii) for each a and b in A, there exists a unique c in A, called the join of a

and b and denoted by a � b, such that a � a � b, b � a � b, and for alld in A,

if a � d and b � d , then a � b � d;(iii) for each a and b in A, there exists a unique element c in A, called the

meet of a and b and denoted by a � b, such that a � b � a, a � b � b,and for all d in A,

if d � a and d � b , then d � a � b;and

(iv) for all a in A, z � a and a � u. �

Let 〈A,�, u, z〉 be a lattice. Then it easily follows that � and � arecommutative and associative operations on A.

CONVENTION 1. A lattice 〈A,�, u, z〉 is often written as

〈A,�,�,�, u, z〉 . �

DEFINITION 3. Let A = 〈A,�,�,�, u, z〉 be a lattice.A is said to be distributive if and only if for all a, b, and c in A,

a � (b � c) = (a � b) � (a � c) .

A is said to be complemented if and only if for each a in A, there existsan element b in A, called a complement of a (in A), such that a � b = uand a � b = z.

A is said to be boolean if and only if it is complemented anddistributive.

Suppose A is boolean. Then it is easy to show that the complementof each element of A is unique. Thus for each a in A, let – a denote thecomplement of a (in A). Then it follows that – is a well-defined operationon A.

A is said to be a lattice of sets (or a lattice of events) if and only if � is⊆, � is ∪, � is ∩, u is a nonempty set, z is ∅, and each element of A is asubset of u. �

CONVENTION 2. A boolean lattice A = 〈A,�,�,�, u, z〉 is oftenwritten as

〈A,�,�,�, – , u, z〉,where – is the complement operator of A.

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Often when describing two or more boolean lattices, the same sym-bol, – is used to denote the two (and, in general, different) complementoperations of the two lattices. �

It is easy to verify that boolean algebras of events are examples of aboolean lattice. The following is another well-known example.

EXAMPLE 1 (Boolean Lattice of Propositions). The propositional cal-culus from logic is an important example of a boolean lattice. Let →, ↔,∨, ∧, and neg stand for, respectively, the logical connectives of implication(“if . . . then”), logical equivalence, (“if and only if”), disjunction (“or”),conjunction, (“and”), and negation (“not”). Consider two propositions αand β to be equivalent if and only if α ↔ β is a tautology. Equivalentpropositions partition the the set of propositions into equivalence classes.Let A be the set of equivalence classes. The equivalence class containinga tautology is denoted by u, and the equivalence class containing a contra-diction is denoted by z. By definition, equivalence class a ≤ equivalenceclass b if and only if for some elements α in a and β in b, α → β is atautology. It is easy to check that ≤ is a partial ordering on A.

Let a and b be arbitrary elements of A, α and β be respectively arbi-trary elements of a and b, j , be the equivalence class of α ∨ β, m be theequivalence class of α ∧ β, and n be the equivalence class of neg α. Thenit is easy to check that j is the join of a and b, m is the meet of a and b, nis the complement of a, and

A = 〈A,�,�,�, – , u, z〉 ,is a boolean lattice. �

Although the assignments of probabilities to propositions are often theitems of interest, they are usually modeled as assignments of probabilitiesto events. This is partially justified by the following theorem of Stone(1936).

THEOREM 1 (Stone’s Representation Theorem). Let

A = 〈A,�,�,�, – , u, z〉be a boolean lattice. Then there exists a boolean algebra of events B =〈B,⊆,∪,∩, – , X,∅〉 and an isomorphism ϕ of the lattice A onto B suchthat ϕ(u) = X and ϕ(z) = ∅. �

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Probability functions have the formal properties of a finitely additivemeasure. The finite additivity comes form the condition,

For all a and b in A, if a � b = z, then P(a � b) = P(a)+ P(b).(1)

For boolean lattices, Equation (1) is equivalent to

For all a and b in A, P(a) + P(b) = P(a � b)+ P(a � b).(2)

Obviously Equation (2) implies Equation (1). The equivalence of Equa-tions (1) and (2) uses the existence of complements. Both Equations (1)and (2) can be used for formulating generalizations of probability func-tions that apply to lattices more general than the boolean ones. BecauseEquation (2) retains more structure of probability functions, it is used inthe generalizations presented in this article.

It is well-known that lattices that have probability functions that satisfyEquation (2) must satisfy the following condition:

Modularity: for all a, b, and c in A, if a ≤ b, then a � (b � c) =(a � b) � (a � c).

Modularity is a generalization of distributivity. Birkhoff and von Neumann(1936) use non-distributive modular lattices to describe the propositionallogic of quantum mechanics. This article considers only the impactof Equation (2) style probability functions on distributive lattices thatgeneralizes boolean lattices in a particular way.

Textbook probability theory defines a and b to be independent, a ⊥ b,if and only if

P(a ∩ b) = P(a)P(b) .(3)

This definition of “independence” strikes me and others as rather odd: (1)In practical applications one usually knows and uses certain events as being“independent” as part of the process of establishing probabilities. In fact,frequency theories of probability (e.g., von Mises 1936) employ a non-numerical concept of independence at the foundational level to guaranteethat the relative frequencies have the desired properties. (2) There are manysituations in which there is not enough information to uniquely deter-mine a probability function, that is, there are situations in which differentprobability functions describe the known facts and hypotheses. In suchsituations, Equation (3) is of little use for deciding independence. And (3),in situations where there is unique probability function, events a and b maybe “independent” for only the accidental reason that Equation (3) holds,

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rather than for some principled reason involving that natures of a and b. Inthis article, we will assume, as is customary in some foundational theoriesof probability, that ⊥ is taken as a primitive concept and require,

if a ⊥ b, then P(a)P(b) .

Some of the above considerations about probability functions aresummarized in the following definition.

DEFINITION 4. P is said to be a probability function on the distributivelattice 〈A,�,�,�, u, z〉 with independence relation ⊥ if and only if thefollowing five conditions hold for all a and b in A:

1. P is a function from A into the closed interval of the reals [0, 1].2. P(u) = 1 and P(z) = 0.3. If a ⊆ b, then P(a) ≤ P(b).4. P(a)+ P(b) = P(a � b)+ P(a � b).5. If a ⊥ b, then P(a � b) = P(a)P(b). �

1.2. Belief Functions

Many researchers consider traditional probability theory to be the onlyrational approach to uncertainty; and many others disagree, with someproposing alternative theories that they consider to be rational. The alter-native theories, leaving aside the issue of whether they are truly “rational,”generally suffer the deficiency of lacking the mathematical power of prob-ability theory. Narens (2003a), however, provides a theory that generalizesprobability theory and has mathematical and calculative power at the samelevel of finitely additive probability theory. He arrives at this theory byfirst providing a qualitative axiomatization of a version of conditionalprobability. This is done in such a way that one of the axioms capturesconsequences of conditional probability that some theorists believe tobe invalid for a general, rational theory of belief. This axiom is theneliminated and the most general quantitative model corresponding to theremaining axioms is found. The result is a belief function B(A|B) havingthe form,

B(A|B) = P(A|B)v(A) ,(4)

where P is a uniquely determined conditional probability function and v isa function into R

+ that is unique up to multiplication by a positive real. Hethen uses this result to give alternative accounts of various findings aboutpsychological judgments of probability.

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The axiom of qualitative conditional probability that Narens (2003a)eliminates is called Binary Symmetry. It is formulated as follows, where“∼” stands for “has the same degree of belief” and “(E |F)” for the condi-tional event of “E occurring given F has occurred:” Suppose A, B, C , andD are mutually disjoint, nonempty events and

(A|A ∪ B) ∼ (B|A ∪ B) and (C |C ∪ D) ∼ (D|C ∪ D) .(5)

Then

(A|A ∪ B) ∼ (C |C ∪ D)(6)

and

(A|A ∪ C) ∼ (B|B ∪ D).(7)

Note that if degrees of belief are measured by a probability functionP and Equation (5) holds, then P(A) = P(B) and P(C) = P(D), fromwhich Equations (6) and (7) follow.

Narens (2003(a)) gives the following rationale for the desirability ofeliminating the axiom of Binary Symmetry:

In the context of the other axioms for conditional probability, Binary Symmetry assertsthat if A and B have equal likelihood of occurring and C and D have equal likelihood ofoccurring, then the conditional probabilities of (A|A ∪ B) and (C |C ∪ D) are 1

2 , and theconditional probabilities of (A|A ∪ C) and (B|B ∪ D) are the same.

Suppose Equation (5) and the judgment of equal likelihood of the occurrences of Aand B, given either A or B occurs, is based on much information about A and B and agood understanding of the nature of the uncertainty involved, and the judgment of equallikelihood of the occurrences of C and D, given either C or D occurs, is due to the lackof knowledge of C and D, for example, due to complete ignorance of C and D. Then,because of the differences in the understanding of the nature of the probabilities involve, alower degree of belief may be assigned to (C |C ∪ D) than to (A|A ∪ B), thus invalidatingEquation (6).

Suppose Equation (5) and the judgments of the likelihoods of the occurrences of A andC , given either A or C occurs, are based on much information about A and C and thenature of the uncertainty involved, and the judgment the likelihoods of the occurrences ofB and D given either B or D occurs is due to the lack of knowledge of B and D. This mayresult in different degrees of belief being assigned to (A|A ∪ C) and (B|B ∪ D), and suchan assignment would invalidate Equation (7).

For events A and B, let A � B stand for “A is more likely than B.” Therelation � can be observed in several ways, for example by asking subjectdirectly, “Is A more likely than B,” or by asking the subject, “Would youprefer the gamble of receiving $100 if A occurs and receiving $0 if A doesnot occur to the gamble of receiving $100 if B occurs and receiving $0 ifB does not occur?” The following has been consistently assumed to be abasic rule of rational probabilistic behavior:

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DEFINITION 5 (de Finetti’s Axiom). If A � B and C ∩ A = C ∩ B = ∅,then A ∪ C � B ∪ C .

Note that de Finetti’s axiom is the qualitative version of,

if P(A) > P(B) and C ∩ A = C ∩ B = ∅,then P(A ∪ C) > P(B ∪ C) .

Ellsberg (1961) presented the following example of human behaviorthat contradicts de Finetti’s Axiom:

EXAMPLE 2 (Ellsberg’s Paradox). Suppose an urn has 90 balls that havebeen throughly mixed. Each ball is of one of the three colors, red, blue, oryellow. There are thirty red balls, but the number of blue balls and thenumber or yellow balls are unknown except that together they total 60. Aball is to be randomly chosen from the urn, and the subject will earn $100if he or she correctly guesses the color of the ball. Let R be the event thata red ball is chosen, B the event a blue ball is chosen, and Y the event ayellow ball is chosen. Let U = R ∪ B ∪ Y . Typically, subjects’ behaviorindicate,

B(R|U ) > B(B|U ) and B(R ∪ Y |U ) < B(B ∪ Y )|U ) ,violating de Finetti’s Axiom. The usual reason given for this result is thatsubjects perceive B as more “ambiguous” than R and R ∪ Y as more“ambiguous” than B ∪ Y and ambiguity is perceived to to increase of theriskiness of the choice. �

Ellsberg’s Paradox is consistent with the belief function B above by theassignments to Equation (4),

P(R|U ) = P(B|U ) , v(R) > v(B) ,

P(R ∪ Y |U ) = P(B ∪ Y ) , and v(R ∪ Y ) < v(B ∪ Y ) .

Ellsberg and others have argued that probability theory is too narrow for ageneral theory of rational belief, with Ellsberg’s Paradox being a demon-stration of its limitation. If one assumes this, then the belief function B inEquation (4) has a normative interpretation. However, Ellsberg’s Paradoxis very controversial as an example of rational behavior.

One of goals of this article is to provide a non-controversial interpreta-tion of B as a rational assignment of degrees of belief. This will be doneby considering a lattice of events where each event can either be verifiedscientifically or refuted scientifically. Such a lattice will be shown to be dis-tributive. The scientific refutations will be shown to be a boolean sublattice

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS 405

of it. A rational assignment of degrees of belief is given by a probabilityfunction P on the scientific events. When P is restricted to the sublattice ofscientific refutations, it can be reinterpreted as a belief function B on thescientific refutations that satisfies Equation (4). Because P was chosen as arational assignment of degrees of belief, it then follows that its restrictionto the sublattice of scientific refutations, reinterpreted as B, must also be arational assignment.

Other issues of rationally are also considered in the article, includingwhat sorts of behavior may be appropriately labeled “irrational.”

2. LATTICES OF SCIENTIFIC EVENTS

2.1. Basic Concepts

In this section we start with a lattice of events S = 〈S,⊆,∪,∩, – ,U,∅〉.In the intended interpretation, S is a set of events about a phenomenonunder investigation. Other events are generated from S. Some of these arethrough a new unary operation, ¬. ¬ applies to events to produce events.It may produce new events; that is, when applied to events in S, it mayproduce an event not in S. It also may apply to events not in S, for example,if Q is in S and ¬ Q is not in S, then ¬ may be applied to ¬ Q toproduce ¬ (¬ Q). In the intended interpretation, some of the new eventsmay be interpretable about the phenomenon under consideration, i.e., assubsets of U that are not in S, while others may be about the science of thephenomenon under consideration, and and as such cannot be interpreted assubsets of U . The new operation is called refutation.

DEFINITION 6. Let Q be an event. Then the following definition holds:¬ Q is the event that the assumption of the occurrence of the event Q leadsto a contradiction. “¬ Q” is read as, “The event that the event Q has beenrefuted.” �

Obviously Definition 6 is incomplete, because “leads to a contradic-tion” has not been specified. More about this will be said later.

CONVENTION 3. As usual, ¬ ¬ A stands for ¬ (¬ A), ¬ A ∪ B for(¬ A) ∪ B, et cetera. �

¬ should not confused with the boolean operation of complementation– . In particular, the Law of the Excluded Middle holds for – , i.e., A ∪ – Ais always the sure event; however, A ∪ ¬ A need not be the sure event. In

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this respect, – behaves like negation in classical logic, whereas ¬ behaveslike negation in intuitionistic logic (see Section 2.2).

Methods that produce scientific contradictions are many and varied. Forexample, an event Q may be refuted through verification of an event Tsuch that Q ∩ T = ∅. Or the assumption of the occurrence of Q maycontradict a fundamental principle of the portion of science under consid-eration, e.g., in a portion of classical physics where the assumption of theoccurrence of Q implies the existence of perpetual motion. For most richfragments of science, we do not have complete descriptions of the methodsof scientific inference. I consider it likely that such complete descriptionsare incapable of formal description. Fortunately, for the kinds of issuesdealt with in this article, a complete description of scientific refutationis not needed; only a few basic properties of ¬ are needed. These arecontained in Axioms 1 to 3 of the following definition.

DEFINITION 7. Let S = 〈S,∪,∩, – ,U,∅〉 be a lattice of events. Then

E = 〈E ,⊆,∪,∩,¬, X,∅〉is said to be the lattice of scientific events (generated by S) if and only ifthe following condition about E and the following three axioms about ¬hold:

E is the smallest set of events such that

(i) S ⊆ E ;(ii) ∅ and X are in E ;

(iii) if A is in E , then A ⊆ X ; and(iv) if A and B are in E , then A ∪ B, A ∩ B, and ¬ A are in E .

AXIOM 1. ¬ X = ∅ and ¬ ∅ = X.

AXIOM 2. For all A and B in E , if A ∩ B = ∅, then B ⊆ ¬ A.

AXIOM 3. For all A in E , A ∩ ¬ A = ∅. �

Note that X is the sure event of E (i.e., is the largest event in E ).

DEFINITION 8. Let E = 〈E ,⊆,∪,∩,¬, X,∅〉 be a lattice of scientificevents generated by S. Then S is called the set of initial scientific events(of E) and E is called the set of scientific events (of E). Events of E of theform ¬ A are called refutations.

F is said to be an lattice of scientific events if and only if for some T , Fis the lattice of scientific events generated by T . �

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Although refutations are important in carrying out empirical science,they also have significant roles in other kinds of situations. Because De-finitions 7 and 8 contain only very general properties of contradictions,refutations are also applicable to situations outside of the foundationsof empirical science. The following are two very different examples ofinvolving refutations.

EXAMPLE 3 (Proof Theory). Let � be formal theory of arithmetic usedby Gödel (1931) in his famous incompleteness theorem, and let � � αstand for the first-order sentence α is a formal consequence of �. Let≡ be the equivalence relation defined on the set of first-order arithmeticsentences, F, such that for all φ and ψ in F, φ ≡ ψ if and only if(� � φ ↔ ψ). Let S be the set of ≡-equivalence classes. Define �, ∨,∧, u, and z as follows: For all A and B in S,

(i) A � B if and only if for some α in A and β in B, if � � α, then � � β,(ii) A � B if and only if for some α in A and β in B, � � α or � � β,

(iii) A � B if and only if for some α in A and β in B, � � α and � � β,(iv) u is the set of α in F such that � � α.(v) z is the set of α in F such that � � neg α, where neg is the negation

operator of the language of �.

Then 〈S,�,�,�, u, z〉 is a distributive lattice. For A in S, let ¬ A standfor, “For some α in A, � � α leads to a contradiction.” Let γ be the criticalsentence (“I am not a theorem”) that Gödel used to establish his incom-pleteness theorem, and let G be the element of S to which γ belongs. Gödelshowed that the assumption of � � γ contradicted a fundamental principleof the metatheory of arithmetic, ω-consistency, which in the current setupyields ¬ G.

Note the following analogies between refutation in science and refu-tation in this mathematical example: (1) Verifiability is analogous toprovability from �. (2) Refuting A by verifying a B such that A ∩ B = ∅

is analogous to assuming � ∪ {δ} and showing � ∪ {δ} � κ , where κis a contradiction of F, to conclude ¬ D, where D is the element of Sto which δ belongs. And (3), assuming the occurrence of A to contradicta fundamental principle of the portion of science under consideration isanalogous to assuming � � θ and showing that a fundamental principle ofthe metatheory of arithmetic is contradicted. �

EXAMPLE 4 (Psychology of Judgment). In the psychology of subjec-tive judgments of probability, Narens (2003b) presents a model in whichcognitive representations of linguistic descriptions of events are given atopological structure. It is assumed that when a subject is asked to judge

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the conditional probability of A|B, P(A|B), the subject does this by firstforming cognitive representations A for A and B for B. The subject thenuses cognitive heuristics to determine the support for A, S+(A) (modeledas a non-negative real number), and the support against A, S−(A) (mod-eled a non-negative real number), and produces a probability judgmentconsistent with the formula,

P(A|B) = S+(A)S+(A)+ S−(A)

.

S−(A) is determined as follows: The subject constructs a cognitive com-plement of A, ¬ A (taking B to be the universe in which the comple-mentation takes place), uses cognitive heuristics to find the support for¬ A, S+(¬ A), and takes S−(A) to be S+(¬ A). Narens assumes that thecognitive representations B, A, and ¬ A are open sets in a topology with Bas the universal set in the topology. The presentation of A|B triggers cog-nitively clear and ambiguous instances for judgment. The clear instancestriggered by A form A. The ambiguous instances triggered by A form the(topological) boundary of A. Similarly the cognitive construction of ¬ Aleads to clear and ambiguous instances, with the clear ones forming ¬ Aand the ambiguous ones the boundary of ¬ A. Thus in computing S+(A)and S+(¬ A), the triggered boundaries are ignored. This leads to a failureof the Law of the Excluded Middle, that is, A ∪ ¬ A is a proper subset ofB. Narens assumes that for open sets Q, ¬ Q is the interior of the closureof B – Q, i.e., ¬ Q is the largest open set in B such that Q ∩ (¬ Q) = ∅.With this definition,

〈B,⊆,∪,∩,¬, B,∅〉 ,where B is the set of open subsets of B, is formally a lattice of scien-tific events. Narens (2003b) shows that this topological model accountsfor many puzzling empirical findings in the extensive literature of humanprobability judgments. �

Other examples of lattices of scientific events can be constructed fromapplications of intuitionistic logic for reasons discussed next.

2.2. Relationship to Intuitionistic Logic

Intuitionism was introduced by the mathematician L. L. J. Brouwer as analternative form of mathematics. It followed from Brouwer’s philosophyof mathematics that the methods of derivation used in intuitionistic math-ematical theorems could not be formalized. However, the intuitionistic,

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mathematical theorems that Brouwer produced displayed sufficient regu-larity in their proofs that an axiomatic approach to his methods of proofappeared feasible. Such an axiomatization was accomplished by Heyt-ing (1930). Today logics that are equivalent to Heyting’s axiomatizationare called intuitionistic logic. Although Heyting designed his logic forintuitionistic mathematics, it was shown to have other applications. Forexample, Kolmogorov (1932) showed that Heyting’s logic had the correctformal properties to provide a theory for mathematical constructions; andGödel (1933) showed that it can be interpreted as a foundational logicfor concepts that naturally arise in proof theory of mathematical logic.Kolmogorov and Gödel achieved their results by giving interpretations tothe logical primitives that were different from Heyting’s.

In classical propositional logic, the operation of “if and only if” yieldsan equivalence relation on propositions, and the induced algebra on the re-sulting equivalence classes is a boolean lattice (see Example 1). In a similarmanner, in intuitionistic propositional logic, the intuitionistic “if and onlyif” operation, yields an equivalence relation on intuitionistic propositions,and the induced algebra on its equivalence classes also produces a lat-tice, called a pseudo boolean algebra. Like boolean lattices, each pseudoboolean algebra is isomorphic to a lattice of subsets of the form,

P = 〈P ,⊆,∪,∩, X,∅〉 ,(Stone 1937; McKinsey and Tarski 1946).1 Such lattices of subsets arecalled “pseudo boolean algebras of subsets,” and are defined formally asfollows:

DEFINITION 9. P = 〈P ,⊆,∪,∩,⇒, X,∅〉, where ⇒ is a binary op-eration on P , is said to be a pseudo boolean algebra of subsets if and onlyif the following three conditions hold for all A and B in P :

(1) P = 〈P ,⊆,∪,∩, X,∅〉 is a lattice of subsets.(2) A ∩ (A ⇒ B) ⊆ B.(3) For all C in P , if A ∩ C ⊆ B then C ⊆ (A ⇒ B).

⇒ is called the operation of relative pseudo complementation. �

⇒ is the pseudo boolean algebra of subsets version of the implicationoperation of intuitionistic logic.

Let B = 〈B,⊆,∪,∩, – , X,∅〉 be a boolean algebra of subsets. Thenit is not difficult to show that

〈B,⊆,∪,∩,−→ , X,∅〉

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is pseudo boolean algebra of subsets, where for all A and B in B,

A −→ B = – A ∪ B,

i.e., −→ is the boolean algebra of subsets version of the implication op-eration of classical logic. Thus each boolean algebra of subsets is a pseudoboolean algebra with – A ∪ B = −→ = ⇒.

The following theorem show the relationship between the refutationoperator ¬ of an empirical algebra of events and the relative pseudocomplementation operator ⇒ of a pseudo boolean algebras of subsets.

THEOREM 2. Suppose P = 〈P ,⊆,∪,∩,⇒, X,∅〉 is a pseudo booleanalgebra of subsets. For each A in P , let

¬ A = A ⇒ ∅.

Then 〈P ,⊆,∪,∩,¬, X,∅〉 is an lattice of scientific events.Proof. Axioms 1 to 3 are immediate consequences of Definition 9. �

In intuitionistic logic, ¬ in Theorem 2 is called intuitionistic negation.Lattices of scientific events are lattices of subsets with a weakened form

of complementation that has the fundamental properties of intuitionisticnegation. They differ from pseudo boolean algebras in that the operationof relative pseudo complementation, ⇒, is not used. While in some appli-cations of lattices of scientific events ⇒ has cogent interpretations, this isnot always the case. For all the applications of Intuitionistic logic of whichI am aware, ¬ has cogent interpretations.

2.3. Properties of Lattices of Scientific Events

THEOREM 3. Let E = 〈E ,⊆,∪,∩,¬, X,∅〉 be a lattice of scientificevents and A and B be arbitrary events in E . Then the following fivestatements are true:

1. If B ⊆ A, then ¬ A ⊆ ¬ B.2. A ⊆ ¬ ¬ A.3. ¬ A = ¬ ¬ ¬ A.4. ¬ (A ∪ B) = ¬ A ∩ ¬ B.5. ¬ A ∪ ¬ B ⊆ ¬ (A ∩ B).

Proof. 1. Suppose B ⊆ A. By Axiom 3, A∩¬ A = ∅. Thus B∩¬ A =∅. Therefore, by Axiom 2, ¬ A ⊆ ¬ B.

2. By Axiom 3, ¬ A∩ A = ∅. Thus by Axiom 2, A ⊆ ¬ ¬ A, showingStatement 2.

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS 411

3. By Statement 2, A ⊆ ¬ ¬ A. Thus by Statement 1,

¬ ¬ ¬ A ⊆ ¬ A.

However, by Statement 2,

¬ A ⊆ ¬ ¬ (¬ A) = ¬ ¬ ¬ A.

Therefore, ¬ A = ¬ ¬ ¬ A.4. By Axiom 3, (i) ¬ A ⊆ – A and (ii) ¬ B ⊆ – B. Thus

¬ A ∩ ¬ B ⊆ – A ∩ – B = – (A ∪ B).

Therefore (A ∪ B) ∩ (¬ A ∩ ¬ B) = ∅, and thus by Axiom 2,

¬ A ∩ ¬ B ⊆ ¬ (A ∪ B).(8)

Because A ⊆ A ∪ B and B ⊆ A ∪ B, it follows from Statement 1 that

¬ (A ∪ B) ⊆ ¬ A and ¬ (A ∪ B) ⊆ ¬ B.

Therefore

¬ (A ∪ B) ⊆ ¬ A ∩ ¬ B.(9)

Equations (8) and (9) show that

¬ (A ∪ B) = ¬ A ∩ ¬ B.

5. From

A ∩ B ⊆ A and A ∩ B ⊆ B,

it follows from Statement 1 that

¬ A ⊆ ¬ (A ∩ B) and ¬ B ⊆ ¬ (A ∩ B),

and thus

¬ A ∪ ¬ B ⊆ ¬ (A ∩ B). �

2.4. Boolean Lattice of Refutations

DEFINITION 10. Let E = 〈E ,⊆,∪,∩,¬, X,∅〉 be a lattice of scientificevents. By definition, let

R = {¬ B|B ∈ E} = {A|A is a refutation in E}.

By definition, for each A and B in R, let

A � B = ¬ ¬ (A ∪ B).

Then R is said to be the boolean lattice of refutations (of E) if and only if

R = 〈R,⊆,�,∩,¬, X,∅〉. �

412 LOUIS NARENS

Let E be a lattice of scientific events and R be the boolean lattice ofrefutations in E. The following lemmas and theorem establish facts aboutR, including that R is a boolean lattice.

Hypotheses for Lemmas 1 to 6: Let E = 〈E ,⊆,∪,∩,¬, X,∅〉 be alattice of scientific events and R = 〈R,⊆,�,∩,¬, X,∅〉 be the booleanlattice of refutations of E. Let A, B, and C be arbitrary elements of R,and A′ and B ′ be elements of E such that A = ¬ A′ and B = ¬ B ′.

LEMMA 1. A = ¬ ¬ A.Proof. Because A = ¬ A′, it follows from Statement 3 of Theorem 3

that

¬ ¬ A = ¬ ¬ ¬ A′ = ¬ A′ = A. �

LEMMA 2. A ∩ B is in R.Proof. By Statement 4 of Theorem 3,

A ∩ B = ¬ A′ ∩ ¬ B ′ = ¬ (A′ ∪ B ′). �

LEMMA 3. If B ⊆ A, then ¬ ¬ B ⊆ A.Proof. By Lemma 1, B = ¬ ¬ B. �

LEMMA 4. L = 〈R,⊆, X,∅〉 is a lattice with A ∩ B as the meet of Aand B, and A � B as the join of A and B.

Proof. By Lemma 2, A ∩ B is in R. Because A ∩ B is the meet in Eand R ⊆ E , it follows that A ∩ B is the meet in L.

Let D in R be such that

A ⊆ D and B ⊆ D.

(Such a D exists, because the above expression holds for D = X .) ThenA ∪ B ⊆ D. By two applications of Statement 1 of Theorem 3,

¬ ¬ (A ∪ B) ⊆ ¬ ¬ D.

By Statement 2 of Theorem 3, A ∪ B ⊆ ¬ ¬ (A ∪ B), and by Lemma 1,¬ ¬ D = D. Thus,

A ∪ B ⊆ ¬ ¬ (A ∪ B) = A � B ⊆ D.

Therefore,

A ⊆ A � B ⊆ D and B ⊆ A � B ⊆ D.

Thus A � B is the join of A and B in L. �

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS 413

LEMMA 5. (A ∩ B) � (A ∩ C) = A ∩ (B � C).Proof.

(A ∩ B) � (A ∩ C) = ¬ ¬ [(A ∩ B) ∪ (A ∩ C)]= ¬ ¬ [A ∩ (B ∪ C)]⊇ ¬ [¬ A ∪ ¬ (B ∪ C)]

(Statement 5 of Theorem 3)

= [¬ ¬ A ∩ ¬ ¬ (B ∪ C)](Statement 4 of Theorem 3)

= [¬ ¬ A ∩ (B � C)]= A ∩ (B � C)] (Lemma 1),

that is,

(A ∩ B) � (A ∩ C) ⊇ A ∩ (B � C) .(10)

Because,

(A ∩ B) ∪ (A ∩ C) = A ∩ (B ∪ C)

⊆ A ∩ ¬ ¬ (B ∪ C)

(Statement 2 of Theorem 3)

= A ∩ (B � C),

it follows that

(A ∩ B) ∪ (A ∩ C) ⊆ A ∩ (B � C) .

Then

A ∩ B ⊆ A ∩ (B � C) and A ∩ C ⊆ A ∩ (B � C) .

Thus, because by Lemma 4 (A ∩ B) � (A ∩ C) is the ⊆-smallest elementin R such that

A ∩ B ⊆ A ∩ (B � C) and A ∩ C ⊆ A ∩ (B � C) ,

it then follows that

(A ∩ B) � (A ∩ C) ⊆ A ∩ (B � C).(11)

Equations (10) and (11) show

(A ∩ B) � (A ∩ C) = A ∩ (B � C) . �

414 LOUIS NARENS

LEMMA 6. A � ¬ A = X.Proof.

A � ¬ A = ¬ ¬ (A ∪ ¬ A)

= ¬ (¬ A ∩ ¬ ¬ A) (Statement 4 of Theorem 3)

= ¬ (¬ A ∩ A) (Lemma 1)

= ¬ ∅ (Axiom 3)

= X. (Axiom 1) �

THEOREM 4. R = 〈R,⊆,�,∩,¬, X,∅〉 is a boolean lattice.Proof. ⊆ partially orders R, and X and ∅ are respectively the maximal

and minimal elements of R with respect to ⊆. By Lemma 4, ∩ and � arerespectively the meet and join operations of R. By Lemma 5, ∩ distributesover �, and by Lemma 6 and Axiom 3, ¬ is the complement operator onR. Thus R is a boolean lattice. �

3. PROBABILITY THEORY FOR SCIENTIFIC EVENTS

One way to produce probability functions on an lattice of scientific eventsE is to find a probability function P on a boolean lattice of events thatcontains E and restrict P to the domain of E.

DEFINITION 11. Let E = 〈E ,⊆,∪,∩,¬, X,∅〉 be an lattice of scien-tific events. Let B be the smallest set such that (i) E ⊆ B, and (ii) for allA and B in E , A ∪ B, A ∩ B, and – A are in B. Then

B = 〈B,⊆,∪,∩, – , X,∅〉is called the outer boolean lattice generated by E. �

Let E be an lattice of scientific events and B be the outer boolean latticegenerated by E. Then under usual assumptions about rationality and prob-ability, subjective probabilities can be assigned to the events in B througha probability function P. Then the restriction of P to the domain of E, PE

is a probability function (in the sense of Definition 4) on the lattice E.Let R = 〈R,⊆,�,∩,¬, X,∅〉 be the boolean lattice of refutations of

E (Definition 10). Let B be the restriction of PE to R. Then B is, in general,not a probability function of R. In fact, B is superadditive; that is, for allA and B in R such that A ∩ B = ∅,

B(A)+ B(B) ≤ B(A � B) ,

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS 415

and there may be elements C and D in R such that C ∩ D = ∅ and

B(C)+ B(D) < B(C � D) .

The supperadditivity of B results from the fact that R may have elementsA and B such that A ∩ B = ∅ and A ∪ B is not a refutation, and therefore,

A � B = ¬ ¬ (A ∪ B) ⊃ A ∪ B.

Let L = 〈A,≤,�,�, u, z〉 be a lattice and � be a binary relation onA. The intended interpretation of “a � b” is “the degree of belief for a isless than or equal to the degree of belief for b.” For boolean lattices, thefollowing condition has been considered as necessary for rationality:

DEFINITION 12. Let L and � be as above. Then � is said to be quali-tatively additive if and only if � is a total ordering on A and de Finetti’sAxiom (Definition 5) holds, that is, for all a, b, and c in A, if a � c = zand b � c = z, then

a ≺ b iff a � c ≺ b � c. �

Suppose L is boolean. In the behavioral sciences, particularly in eco-nomics and experimental psychology, it is often considered appropriateto formulate “rationality” in terms of an individual’s behavior rather thanin terms of the individual’s interpretation of what is giving rise to his orher behavior. Thus it is said that a person’s behavior is consistent withrationality if it satisfies certain rationality conditions. It is recognized thata person can display rational behavior by employing nonrational subjectivemethods of evaluation. This is part of the reason of why the word “consis-tent” is used in the phrase “consistent with rationality.” In the literature,the qualitative additivity of � has been repeatedly taken as a conditionused in showing behavior is consistent with rationality. However, as dis-cussed below, I find it problematic to assert that the failure of qualitativeadditivity is sufficient to establish that a person’s behavior is inconsistentwith rationality.

Consider the case where P is a probability function on the outer booleanlattice B of an scientific lattice of events, E. Let

R = 〈R,⊆,�,∩,¬, X,∅〉be the boolean lattice of refutations of E, and let B be the restriction of P

to R. Define ≺ on R as follows: For all S and T in R,

S ≺ T iff B(S) < B(T ).

416 LOUIS NARENS

Because there may exist A, B, and C in R such that A ∩ C = B ∩ C = ∅,B ≺ A, and

A � C = A ∪ C and B � C = ¬ ¬ (B ∪ C) ⊃ B ∪ C,

it may be the case that

P(A � C) < P(B � C),

and thus, because P(A � C) = B(A � C) and P(B � C) = B(B � C),

A � C ≺ B � C,

i.e., de Finetti’s Axiom, and therefore qualitative additivity, may fail in R.Such a failure should not be considered to be inconsistent with rationality,for if the probability assignments via P to the events in the outer booleanlattice B of E are to be considered to be “consistent with rationality,” thenso should assignments via P with respect to the boolean lattice R (whichis a substructure of B).

Let B, E, R, P, and B be as above. As discussed previously, it is oftenthe case that a person believes in the independence of two events A and Bwithout recourse to checking whether P(A ∩ B) = P(A)P(B). Let A ⊥ Bstand for A and B exhibiting such “belief independence.” I will assume—although many would insist that rationality demands—that if A and B areelements of R and when considered as elements of B, A ⊥ B, then whenconsidered as elements of R, it is also the case that A ⊥ B. That is, A ⊥ Bdoes not depend on the boolean lattice to which they belong. Consistencybetween ⊥ and P requires that for all elements C and D of B such thatC ⊥ D, P(C ∩ D) = P(C)P(D). The following condition, stated forB and called union independence, is often taken as a rational qualitativeaxiom about probability with an independence relation ⊥: For all A, B,and C in B, if A ⊥ B, A ⊥ C , and B ∩ C = ∅, then A ⊥ (B ∪ C).Union independence does not hold for R, because it may be the case thatfor some A, B, and C in R,

A ⊥ B, A ⊥ C, B ∩ C = ∅, B � C = ¬ ¬ (B ∪ C) ⊃ B ∪ C,

and not A ⊥ B � C .

DEFINITION 13. Let R = 〈R,⊆,�,∩,¬, X,∅〉 be the boolean latticeof refutations of an lattice of scientific events.

B′ is said to be a belief function on R if and only if B

′ is a function fromR into the closed interval of reals [0, 1] such that B

′(∅) = 0, B′(X) = 1,

and for all A in R, if A �= ∅ and A �= X , then 0 < B′(A) < 1. Similar

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS 417

definitions hold for belief function on lattices of scientific events and onboolean lattices of events.

Suppose B′ is a belief function on R. Then (i) B

′ is said to be monotonicif and only if for all A and B in R,

if A ⊂ B then B′(A) < B

′(B),

and (ii) B′ is said to be superadditive if and only if for all C and D in R

such that C ∩ D = ∅,

B′(C)+ B

′(D) ≤ B′(C � D). �

Let R = 〈R,⊆,�,∩,¬, X,∅〉 be the boolean lattice of refutations ofan lattice of scientific events E, and let B be the outer boolean lattice of E.Then not every monotonic, superadditive belief function B

′ on R can beextended to a probability function on B. A necessary condition for suchextensions is that for all A, B, C , and D in R, if

A � B ⊆ C � D ,

then

B′(A)+ B

′(B)− B′(A ∩ B) ≤ B

′(C)+ B′(D)− B

′(C ∩ D) .

This condition is behaviorally testable in probability estimation experi-ments. It appears to me to be sensible to test this and related conditions onsuperadditive belief functions before asserting such functions exhibit “ir-rational behavior,” even for researchers committed to textbook probabilitytheory as a criterion for rationality.

4. CONDITIONAL EVENTS

Let B = 〈B,⊆,∪,∩, – , X,∅〉 be a boolean algebra of events, B be abelief function on B, and P be any probability function on B. For eachnonempty A in B, let

v(A) = B(A)

P(A).

Then B(A) = P(A)v(A). Thus a belief function having the form“P(A)v(A),” puts no restriction on the belief function; that is, given aprobability function P on B, each belief function is obtainable throughan appropriate choice of v . However, such is not the case for conditionalbelief representations of the form,

B(C |B) = P(C)v(C)

P(B)= P(C |B)v(C),

418 LOUIS NARENS

where C and B are arbitrary events in B such that C ⊆ B and B �= ∅.This is because v(C) does not vary with the conditioning event B in suchrepresentations—a powerful restriction. In this section v(C) is viewed asa factor that describes the amount the probability P(C) is distorted. Notethat this way of viewing v requires that the distortion of P(C) be the sameas the distortion of P(C |B), for all B such that C ⊆ B.

Let E = 〈E ,⊆,∪,∩,¬, X,∅〉 be a lattice of scientific events,

R = 〈R,⊆,�,∩,¬, X,∅〉be the boolean lattice of refutations of E, and P be a probability functionon E. For each nonempty A in E , let

v(A) = P(¬ ¬ A)

P(A),

and for each B and C in E such that B is nonempty and C ⊆ B, let

B(C |B) = P(C)v(C)

P(B)= P(C |B)v(C).

The above equation may be interpreted as an individual’s belief in C ishis or her subjective probability that given an scientific demonstration ofthe event B, a scientific refutation can be given to the claim that C is sci-entifically refutable. Although complicated, such conditional beliefs andtheir associate degrees of belief make sense. They are especially useful insituations where the demonstration of the double refutation of an event A(i.e., ¬ ¬ A) is sufficient to take the same action, or have the same effect,as the event A.

EXAMPLE 5 (American Law). In American law there is a presumptionof innocence; that is, in a criminal proceeding, a lawyer needs only toaccomplish the double refutation demonstrating innocence in order to freehis or her client; that is, the lawyer needs only to refute the claim that itcan be demonstrated that the innocence of the client is refutable. For legalpurposes, this has the same effect as the demonstration of innocence. Letpi be the probability of demonstrating innocence, I , and p the probabilityof demonstrating the double refutation of innocence, ¬ ¬ I . Then pi ≤ p,because I ⊆ ¬ ¬ I . For various legal decisions and actions, for example,plea bargaining, it is p rather than pi that is used. �

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS 419

5. DISCUSSION

Narens (2003a) produced a qualitative axiomatization of conditional prob-ability in which one axiom – Binary Symmetry – contains principles thathave been challenged in the literature (e.g., Ellsberg 1961) as necessaryfor the rational measurement of uncertainty. He explored the mathemati-cal consequences of deleting this axiom and found that the most generalrepresentation of the remaining axioms measured uncertainty in terms of aconditional belief function B of the form,

B(A|B) = P(A|B)v(A),where P is a uniquely determined conditional probability function and v isa function that is unique determined up to multiplication by positive reals.

Narens (2003a) investigates the applicability of the conditional beliefrepresentation as a descriptive theory for various empirical results in theliterature involving subjective probability. Except for the Ellsberg Para-dox, these empirical results are about “irrational” probabilistic phenomena.And the Ellsberg Paradox is highly controversial as a proper application ofrationality. Thus Narens (2003a) does not provide an example where theconditional belief function has a clear, rational application. Such exam-ples are provided here by considering the restriction of rational probabilityassignments to boolean lattices of refutations.

The approach to rationality in this article is a limited one: Only observ-able rational behavior is considered. Such an approach deliberately ignoresthe motivations and internal decision processes resulting in behavior. Inexperimental psychology and economics this version of rationality hasproven to be very productive.

CONVENTION 4. Throughout the rest of this section let

E = 〈E ,⊆,∪,∩,¬, X,∅〉be a lattice of scientific events,

B = 〈B,⊆,∪,∩, – , X,∅〉the outer boolean algebra of E, and

R = 〈R,⊆,�,∩,¬, X,∅〉the boolean lattice of refutations of E. Also let P be a probability functionon B, and PR be the belief function that is the restriction of P to R. �

420 LOUIS NARENS

EXAMPLE 6 (Behavioral Irrationality/Rationality). Suppose that P is aperson P’s subjective probability function on B. In Section 3, it was shownthat PR fails both standard quantitative and qualitative properties generallyattributed to rational belief. In particular, PR is superadditive and failsqualitative additivity. It was argued that the assignments of degrees ofbelief to elements of R by PR are consistent with rationality, because theassignments of probabilities to elements of B by P are consistent withrationality. Note that in this situation, we have a complete view of therelationship of R to B without entering into P’s mind. Because of this, itmight well be argued that an eccentric interpretation is being given to thejoin operator � in R—that is, for P, � is not the set-theoretic correlate of“or,” (the latter being, of course, ∪). However, other examples can be givenwhere this latter argument does not hold without entering into a person’smind.2

Consider the case where the Experimenter E presents events to per-son Q for evaluation of subjective probability. E can check behaviorallythat Q is treating the events as if they were events from a boolean alge-bra of events. For example, E can give (from E’s perspective) the eventsA∩(B∪C) and (A∩B)∪(A∩C) and ask Q if they are “identical.” SupposeQ passes this test by treating the events “rationally;” that is, treating themfrom the Experimenter’s perspective like they were from a boolean algebraof events. Of course Q could have an idiosyncratic perspective about theevents. For example, through odd socialization Q may be what E wouldconsider to be a rabid empiricist who interprets the stimuli different thanE: Whereas E considers events to be platonic – either they occur or theydo not occur, Q interprets them as empirical and identifies an occurrenceof each event A with its double empirical refutation, ¬ ¬ A. Suppose foreach event A given by E to Q for “probabilistic evaluation,” PR(¬ ¬ A)is Q’s response. Based on this behavior, should Q’s behavior be consid-ered as “inconsistent with rationality,” because PR is superadditive on Rand violates qualitative additivity? I think not. I believe a different ver-sion of “consistent with rationality” is called for: Q’s behavior should beconsidered as consistent with rationality as long as we can conceive ofan isomorphic situation of events in which, under isomorphism, anotherperson could give the same probability estimates to event stimuli as Qand still be considered “rational.” Under this version, the demonstration ofthe non-rationality of Q’s behavior would consist of showing that no suchisomorphism exists. �

For all A and B in E such that A ⊆ B and B �= ∅, let

P(A|B) = P(A)

P(B),

A THEORY OF BELIEF FOR SCIENTIFIC REFUTATIONS 421

and

v(A) = P(¬ ¬ A)

P(A),

and

B(¬ ¬ A|B) = P(¬ ¬ A)

P(B)= P(A|B)v(A).

Let S be the set of initial events that generates E (Definitions 8 and 7).

EXAMPLE 7 (Updating Due to New Evidence). In the following, ele-ments of S are considered as possible new pieces of evidence (e.g., theoutcome of a particular and ongoing DNA test; a new, and very reliable,witness is discovered who provides testimony asserting the proposition ;et cetera). Suppose, like in law, the success (= the occurrence) a doublerefutation depends on evidence. For the sake of argument, assume that (i)the evidence are indisputable facts, (ii) only elements of S can be presentedas additional evidence, and (iii) the success of a double refutation dependsonly on the evidence presented. Suppose P is P’s probability functionon B. P describes P’s current degree of beliefs of events in R. As newevidence comes in, P will update his or her event space and probabilityfunction. Because the success of a double refutation depends only on evi-dence, P’s probability function will only be updated when new evidence Ein S is presented and the updating of an event D in E will be of the form

P(D|E) = P(E ∩ D)

P(E).

Suppose ¬ ¬ A and ¬ ¬ B are in R and

∅ ⊂ ¬ ¬ A ⊂ ¬ ¬ B ⊂ X,

where X is the sure event.Given the above, When is

P(¬ ¬ A)

P(¬ ¬ B)

a proper updating of P? The answer is “rarely.” The reason is that for¬ ¬ B to occur, new evidence needs to be presented, and by hypothesis,this evidence, E , is in S. Because, by assumption, the occurrence of ¬ ¬ Bcompletely depends on the occurrence of E , E ⊆ ¬ ¬ B. The probabilitytheory updating of P for ¬ ¬ A in this circumstance is given by

P(E ∩ ¬ ¬ A)

P(E)= P((E ∩ ¬ ¬ A)|E) ,

422 LOUIS NARENS

and only rarely is this same as

P(¬ ¬ A)

P(¬ ¬ B).

However, E ∩ ¬ ¬ A need not be in R, and, as discussed in Section 4,it is double refutations that are relevant. Therefore, unlike in traditionalprobability theory, the double refutation event space R cannot in generalbe properly updated to a different double refutation event space, becausefrom the point of view of B, R should be updated to {E ∩ F |F ∈ R},which generally contains non-refutations. I do not see this presenting seri-ous difficulties, because for taking actions only updated degrees of beliefare needed, and these can always be computed for events in the event spaceR, conditionalized on conjunctions of events from S. �

The last two examples establish that conditional belief representationsof the form

B(A|B) = P(A|B)v(A)are in some circumstances a basis for a rational assignment of degreesof belief. The argument for rationality rests on conceiving a circumstancewhere for all relevant events A and B, the conditional degrees of beliefB(A|B) agree with the rational assignments of conditional probabilities ina structurally isomorphic belief situation. The argument for the existenceof appropriate applications of such belief representations rests on the ef-fectiveness and correctness of such representations for boolean lattices ofrefutations of lattices of scientific events.

NOTES

1 See Rasiowa and Sikorski 1968 for a complete development of intuitionistic logics andpseudo boolean algebras.2 Of course, a person can enter his own mind and provide an explanation of his behavior.However, experimental psychologists have found that such verbal reports of subjects tobe highly suspect and often unreliable as veridical accounts of how the behaviors wereproduced – see, for example, Nisbet and Wilson (1977). For this and related reasons,behavioral scientists choose to model directly the behavior of subjects rather than theirverbal reports of it or a combination of the two.

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Cognitive SciencesUniversity of California, IrvineSocial Sciences, UClIrvine, CA 92697U.S.A.E-mail: lnarens@uci.edu