Difference measurement and simple scalability with restricted solvability

Let A, B be two sets, with B ? A × A, and ≤ a binary relation on B. The problem analyzed here is that of the existence of a mapping u: A → R, satisfying:

$\text{(a,b) ? (}\text{a}\text{?}\text{,}\text{b}\text{?}\text{)}\text{iff}\text{∨∧ μ(b) ? μ(a) ? μ(}\text{b}\text{?}\text{) ? μ(}\text{a}\text{?}\text{)}$

whenever (a, b), (a′, b′) ∈ B. In earlier discussions of this problem, it is usually assumed that B is connected on A. Here, we only assume that B satisfies a certain convexity property. The resulting system provides an appropriate axiomatization of Fechner's scaling procedures. The independence of axioms is discussed. A more general representation is also analyzed:

$\text{(a,b) ? (}\text{a}\text{?}\text{,}\text{b}\text{?}\text{)}\text{iff}\text{∨∧ F[μ(b), μ(a)] ? F[μ}\text{b}\text{?}\text{]}$

, where F is strictly increasing in the first argument, and strictly decreasing in the second. Sufficient conditions are presented, and a proof of the representation theorem is given.     

Some representation problems for semiorders

Suppose we have a number representation of a semiorder 〈A, P〉 such that aPb iff f(a)+δ(a) < f(b), for all a, b ∈ A, where δ is a nonnegative function describing the variable jnd. Such an f (here called a closed representation) may not preserve the simple order relation $\text{R}{}^1$ generated by 〈A, P〉, i.e., $\text{aR}{}^1\text{b}$ but f(a) > f(b) for some f, δ and a, b ∈ A. We show that this “paradox” can be eliminated for closed and closed interval representations. For interval representations it appears to be impossible. That is why we introduce a new type of representation (an R-representation) which is of the most general form for number representations that preserve the linear structure of the represented semiorders. The necessary and sufficient condition for an R-representation is given. We also give some independent results on the semiorder structure. Theorems are proved for semiorders of arbitrary cardinality. The Axiom of Choice is used in the proofs.     

Binary choice probabilities: on the varieties of stochastic transitivity

Let {P_λ} denote the family of decisiveness relations {$\text{P}{}_{\mathrm{\lambda }}\text{:}\text{1}\text{2}\text{≤ λ < 1}$} with aP_λb if and only if P(a,b) > λ, where P is a binary choice probability function. Families in which all decisiveness relations are of the same type, such as all strict partial orders or all semiorders, are characterized by stochastic transitivity conditions. The conditions used for this purpose differ in various ways from the traditional forms of strong, moderate, and weak stochastic transitivity. The family {P_λ} is then examined from the viewpoint of interval representation models, the most general of which is aP_λb if and only if I(a, λ) > I(b, λ), where the I's are real intervals with I(a, λ) > I(b, λ) if and only if the first interval is completely to the right of the second. With I(a, λ) = [f(a, λ), f(a, λ) + σ(a, λ)], the specializations of the interval model that are discussed include those where the location function f (for left end-points) depends only on the set A of alternatives or stimuli and where the length function σ depends only on A or on λ or neither.     

Conditions under which thurstone case III representations for binary choice probabilities are also Fechnerian

By a Thurstone Case III representation for binary symmetric choice probabilities P_x,y we mean that there exist functions F, μ, σ > 0 such that $\text{P}{}_{\mathrm{x,y}}\text{= F[}\text{(μ(x) ? μ(y))}\text{(σ}{}^2\text{(x) + σ}{}^2\text{(y))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}$. We show that the constraint σ = constant, or μ = ασ + β, α ≠ 0, is both necessary and sufficient for a Thurstone Case III representation to be Fechnerian, i.e., to be reexpressable as as P_x,y = G(u(x) ? u(y)) for some suitably chosen functions G, u.     

A representation theorem for finite random scale systems

This paper investigates necessary and sufficient conditions on choice probabilities P_a,B (of picking an element a in an offered set B), for the existence of random variables U_a, satisfying the equation P_a,B = $\text{P}${U_a = max {U_b | b ∈ B}} for all nonempty finite subsets B in a fixed set A, and all a ∈ B. A complete solution to this representation problem is obtained in the case where A is finite. The proof of the representation theorem provides an algorithm to construct the random variables U_a, up to some uniqueness properties. Investigation of these uniqueness properties show that an important part of the ordinal structure of the underlying random variables can be recovered.     

A locus problem involving the three-dimensional city-block metric

In connection with multidimensional scaling, representations have been considered of the form abDcd ← ?(f(a), f(b)) ≦ ?(f(c), f(d)), for all a, b, c, d ∈ A, where A is a nonvoid finite set, D is a four-place relation on A, f is a function from A into Euclidean n-space, $\text{R}{}^{\mathrm{n}}$, and ? is a metric in $\text{R}{}^{\mathrm{n}}$. For particular metrics there exist finite universal axiomatizations which are necessary and sufficient for the above representation. On the other hand, it is known that no such axiomatizations can be given for either the supremum metric or the ordinary Euclidean metric. Methods for showing this apply easily to the city-block metrics in $\text{R}{}^1$ and $\text{R}{}^2$. This article describes a computer-aided verification of a locus result which shows the impossibility of finite universal axiomatizability for the case of the city-block metric in $\text{R}{}^3$. The result was obtained by dealing with 21,780 cases, each of which involved a set of 10 equations in 12 unknowns along with a related set of inequalities.     

Piaget's logical formalism for formal operations: An interpretation in context

Two interpretations of the equations used by B. Inhelder and J. Piaget in The growth of logical thinking from childhood to adolescence (London: Routledge & Kegan Paul, 1958) are discussed, with implication as the central example. The expression (p ? q) (p implies q) is said to be equal to $\text{(p · q)}\text{V}\text{(}\text{p}\text{· q)}\text{V}\text{(}\text{p}\text{·}\text{q}\text{)}$ (i.e.,p and q, or not p and q, or neither p nor q). According to one, (p ? q) asserts the existence of each case mentioned. According to the other, (p ? q) only asserts that current knowledge allows the possibility of each of the cases. Neither interpretation makes sense of all the relevant passages. Both can be combined in a consistent interpretation when attention is paid to the functional context of the subject's use of logical operations in this book: the “operations” describe the knowledge states which the subject can differentiate and relate, on his way to solving Inhelder's tasks. The logical notation used to represent these states is not a representational format attributed to the subject and manipulated by his cognitive processes, but part of a structuralist account of the subject's reasoning capacities.     

Nontransitive measurable utility

A new theory of preferences under risk is presented that does not use the transitivity and independence axioms of the von Neumann-Morgenstern linear utility theory. Utilities in the new theory are unique up to a similarity transformation (ratio scale measurement). They key to this generalization of the traditional linear theory lies in its representation of binary preferences by a bivariate rather than univariate real valued function. Linear theory obtains a linear function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. The new theory obtains a skew-symmetric bilinear function φ on P × P for which φ(p, q) > 0 if and only if p is preferred to q. Continuity, dominance, and symmetry axioms are shown to be necessary and sufficient for the new representation.     

Two-generator semigroups of binary relations

We study upper bounds on the size of the semigroups generated two randomly chosen n × n Boolean matrices having exactly N one entries. In general, as $\text{N}\text{n}$ increases the semigroups tend to be smaller. If N is unrestricted, or is a function which is at least [$\text{((2 + ?)n}{}^3\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$] the probability tends to one that the size of the semigroup is exactly 3. If N is a function which is at least [(r + 1 + ?)n log n] the probability tends to one that the size of the semigroup will be no more than $\text{2}{}^{\mathrm{<mtext>(</mtext><mtext>(n?1)</mtext><mtext>r</mtext><mtext>)+1</mtext>}}\text{? 1}$. However, if N is any function w(n) such that $\text{w(n)}\text{n}{}^2\text{→ 0}$ and w(n) > n, the average size of the resulting semigroups will be at least . This phenomenon is caused by some semigroups of extremely large size. It is thought that w(n) > n and $\text{w(n)}\text{n}{}^2\text{→ 0}$ hold for those matrices usually encountered in sociology and psychology.     

The psychology of inferring conditionals from disjunctions: A probabilistic study

There is a new probabilistic paradigm in the psychology of reasoning that is, in part, based on results showing that people judge the probability of the natural language conditional, if $A$then $B$, $P\left(ifAthenB\right)$, to be the conditional probability, $P(B\mid A)$. We apply this new approach to the study of a very common inference form in ordinary reasoning: inferring the conditional if not-$A$then $B$ from the disjunction $A$ or $B$. We show how this inference can be strong, with $P$(if not-$A$then $B$) “close to” $P\left(AorB\right)$, when $A$ or $B$ is non-constructively justified. When $A$ or $B$ is constructively justified, the inference can be very weak. We also define suitable measures of “closeness” and “constructivity”, by providing a probabilistic analysis of these notions.     

Lexicographic additive differences

Several authors have identified sets of axioms for a preference relation ? on a two-factor set A × X which imply that ? can be represented by specific types of numerical structures. Perhaps the two best-known of these are the additive representation, for which there are real valued functions f_A on A and f_X on X such that (a, x) ? (b, y) if and only if f_A(a) + f_X(x) > f_A(b) + f_X(y), and the lexicographic representation which, with A as the dominant factor, has (a, x) ? (b, y) if and only if f_A(a) > f_A(b) or {f_A(a) = f_A(b) and f_X(x) > f_X(y)}. Recently, Duncan Luce has combined the additive and lexicographic notions in a model for which A is the dominant factor if the difference between a and b is sufficiently large but which adheres to the additive representation when the difference between a and b lies within what might be referred to as a lexicographic threshold. The present paper specifies axioms for ? which lead to a numerical model which also has a lexicographic component but whose local tradeoff structure is governed by the additive-difference model instead of the additive model. Although the additive-difference model includes the additive model as a special case, the new lexicographic additive-difference model is not more general than Luce's model since the former has a “constant” lexicographic threshold whereas Luce's model has a “variable” lexicographic threshold. Realizations of the new model range from the completely lexicographic representation to the regular additive-difference model with no genuine lexicographic component. Axioms for the latter model are obtained from the general axioms with one slight modification.     

On the basis of early transitivity judgments

This research had two aims. The first was to test three explanations of performance on N-term series tasks by young children: the labeling model of B.DeBoysson-Bardies and K. O'Regan (1973), Nature (London), 246, 531–534, the sequential-contiguity model of L. Breslow (1981, Psychological Bulletin, 89, 325–351), and the ordered array or image model of C. A. Riley and T. Trabasso (1974, Journal of Experimental Child Psychology, 17, 187–202). In the first experiment, 5-year-old children were taught additional premises which would interfere with labeling and sequential-contiguity processes, but not with forming an ordered array. Reasoning performance was essentially comparable to previous results with the paradigm, thus supporting the ordered array model. The second aim was to reexamine children's ability to learn sets of premises which can be assembled into an ordered array, since there was reason to believe that previous studies had created false positives. In the second experiment, 3- to 7-year-old children were taught either overlapping (a > b, b > c, …) or nonoverlapping (a > b, c > d, …) premises. Overlapping premises can be integrated into an ordered array (a, b, c, d, e), but nonoverlapping premises cannot. However, the overlapping condition proved more difficult, and the success rate for preschoolers ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-olds}$) was of zero order. This raises doubts about their ability to learn a set of premises of the kind required for transitive inference. These doubts were strengthened by the third experiment which showed that when premises were not presented in serial order, preschool ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-old}$) children could not learn the premises of an N-term series task.     

Classification of concatenation measurement structures according to scale type

A relational structure is said to be of scale type (M,N) iff M is the largest degree of homogeneity and N the least degree of uniqueness (Narens, 1981a, Narens, 1981b) of its automorphism group.Roberts (in Proceedings of the first Hoboken Symposium on graph theory, New York: Wiley, 1984; in Proceedings of the fifth international conference on graph theory and its applications, New York: Wiley, 1984) has shown that such a structure on the reals is either ordinal or M is less than the order of at least one defining relation (Theorem 1.2). A scheme for characterizing N is outlined in Theorem 1.3. The remainder of the paper studies the scale type of concatenation structures 〈X, ?, ° 〉, where ? is a total ordering and ° is a monotonic operation. Section 2 establishes that for concatenation structures with M>0 and N<∞ the only scale types are (1,1), (1,2), and (2,2), and the structures for the last two are always idempotent. Section 3 is concerned with such structures on the real numbers (i.e., candidates for representations), and it uses general results of Narens for real relational structures of scale type (M, M) (Theorem 3.1) and of Alper (Journal of Mathematical Psychology, 1985, 29, 73–81) for scale type (1, 2) (Theorem 3.2). For M>0, concatenation structures are all isomorphic to numerical ones for which the operation can be written $\text{x°y = yf(}\text{x}\text{y}\text{)}$, where f is strictly increasing and $\text{f(x)}\text{x}$ is strictly decreasing (unit structures). The equation f(x^?)=f(x)^? is satisfied for all x as follows: for and only for ? = 1 in the (1,1) case; for and only for ?=kⁿ, k > 0 fixed, and n ranging over the integers, in the (1, 2) case; and for all ?>0 in the (2, 2) case (Theorems 3.9, 3.12, and 3.13). Section 4 examines relations between concatenation catenation and conjoint structures, including the operation induced on one component by the ordering of a conjoint structure and the concept of an operation on one component being distributive in a conjoint structure. The results, which are mainly of interest in proving other results, are mostly formulated in terms of the set of right translations of the induced operation. In Section 5 we consider the existence of representations of concatenation structures. The case of positive ones was dealt with earlier (Narens & Luce (Journal of Pure & Applied Algebra27, 1983, 197–233). For idempotent ones, closure, density, solvability, and Archimedean are shown to be sufficient (Theorem 5.1). The rest of the section is concerned with incomplete results having to do with the representation of cases with M>0. A variety of special conditions, many suggested by the conjoint equivalent of a concatenation structure, are studied in Section 6. The major result (Theorem 6.4) is that most of these concepts are equivalent to bisymmetry for idempotent structures that are closed, dense, solvable, and Dedekind complete. This result is important in Section 7, which is devoted to a general theory of scale type (2, 2) for the utility of gambles. The representation is a generalization of the usual SEU model which embodies a distinctly bounded form of rationality; by the results of Section 6 it reduces to the fully rational SEU model when rationality is extended beyond the simplest equivalences. Theorem 7.3 establishes that under plausible smoothness conditions, the ratio scale case does not introduce anything different from the (2, 2) case. It is shown that this theory is closely related to, but somewhat more general, than Kahneman and Tversky's (Econometrica47, 1979, 263–291) prospect theory.     

A theory of belief

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 A|B, 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 C|A. The second interpretation is used to provide an account of experimental results in which for disjoint events A and B, the judge probabilities of A|(A∪B) and B|(A∪B) do not sum to 1.The theory of belief is axiomatized qualitatively in terms of a primitive binary relation ? on conditional events. (A|B?C|D is interpreted as “the degree of belief of A|B is greater than the degree of belief of C|D.”) 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.     

Coherent Odds and Subjective Probability

A well-known Dutch book, due to de Finetti, shows how violations of the additivity law of probability theory (Pr(A∪B)=Pr(A)+Pr(B)−Pr(A∩B)) open the door to a sequence of bets leading to a sure loss. In this paper we show that in a market environment, when bookies act strategically, it may well be optimal for them to post incoherent (nonadditive) odds. This is true even when the bookie's own preferences are expected utility, provided she plays against at least two nonexpected utility bettors.     

Decision making in a case of mixed uncertainty: A normative model

We consider a case of uncertainty which is frequently met in various fields, e.g., in parametric statistics: Events {θ}, θ ∈ ∵, are members of family $\text{E}$ on which the decision maker possesses no information at all; however, conditionally on the realization of {θ}, he is able to affix probabilities to all members of another family of events, $\text{F}$. We assume that the decision maker: (1) has a rational behavior under complete ignorance, for decisions whose results only depend on events of $\text{E}$; (2) with {θ} known, maximizes his conditional expected utility for decisions whose results only depend on events of $\text{F}$; (3) has (unconditional) preferences which are consistent with his conditional ones. These assumptions are shown to be sufficient to ensure an approximate representation of the decision maker's preference by a real-valued function W which has the form $\text{W(f) = v[}\text{Inf}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f),}\text{Sup}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f)]}$, where u and v, respectively, characterize the decision maker's attitudes toward risk and toward complete ignorance.     

Against Probabilistic Measures of Coherence

It is shown that the probabilistic theories of coherence proposed up to now produce a number of counter-intuitive results. The last section provides some reasons for believing that no probabilistic measure will ever be able to adequately capture coherence. First, there can be no function whose arguments are nothing but tuples of probabilities, and which assigns different values to pairs of propositions {A, B} and {A, C} if A implies both B and C, or their negations, and if P(B)=P(C). But such sets may indeed differ in their degree of coherence. Second, coherence is sensitive to explanatory relations between the propositions in question. Explanation, however, can hardly be captured solely in terms of probability.     

On the functional form of partial masking functions in psychoacoustics

Partial masking of pure tones is often investigated in terms of matching functions which record the intensity φ(x, n) of an unmasked tone which matches, in perceived loudness, a tone of intensity x embedded in a flat, broadband noise of intensity n. Empirically, the following property of “shift invariance” is observed to hold for these matching functions: φ(λx, λ^θn) = λφ(x, n), for any λ > 0, and some θ < 1. In the context of the model $\text{φ(x, n) = F}{}_0\text{[}\text{g}{}_0\text{(x)}\text{(h}{}_0\text{(x) + k}{}_0\text{(n))}\text{]}$ which expresses the idea that the effect of a noise mask is to control the gain of a pure tone signal in a multiplicative fashion, shift invariance proves to be an extremely powerful theoretical constraint. Specifically, we show that only two parametric families of functions (F₀, g₀, h₀, k₀) are possible candidates for interpolating empirical data. These two parametric families are given by the following expressions: $\text{φ}{}_1\text{(x, n) = A(}\text{x}{}^{\mathrm{\alpha }}\text{(x}{}^{\mathrm{\alpha \prime }}\text{+ Kn}{}^{\mathrm{<mtext>\alpha \prime </mtext><mtext>\theta </mtext>}}\text{)}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>(\alpha \; ?\; \alpha \prime )</mtext>}}$, $\text{φ}{}_2\text{(x, n) = A[x}{}^{\mathrm{\alpha }}\text{(x}{}^{\mathrm{\alpha \prime }}\text{? Kn}{}^{\mathrm{<mtext>\alpha \prime </mtext><mtext>\theta </mtext>}}\text{)]}{}^{\mathrm{<mtext>1</mtext><mtext>(\alpha \; +\; \alpha \prime )</mtext>}}$. Both of these expressions are in good agreement with a large array of partial masking data.     

Subjunctive Conditional Probability

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P(B//A) be the probability of B on the subjunctive supposition that A. Is P(B//A) equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. On the other hand, I argue that even if we can equate P(B//A) with P(A □→B), we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped.