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.     

Conjoint Weber laws and additivity

This paper investigates the mathematical consequences of a number of related empirical laws, exemplified by

$\text{P}{}_{\mathrm{ax;by}}\text{= P(ξa)(ξx);(ξb)(ξy)}$

where a, x, b, y, and ξ are real numbers, and P_ax;by is the probability of choosing the two-dimensional object (a, x) in the set {(a, x), (b, y)}. A variety of results is derived showing that, in the presence of such laws, the class of feasible models for choice data is considerably reduced. In particular, it is shown that the above law, together with the “additive conjoint” form

$\text{P}{}_{\mathrm{ax;by}}\text{= F[l(a) + r(x), l(b) + r(y)]}$

(where F, l, and r are unspecified except for continuity and monotonicity properties), requires the choice probabilities to possess one of the following three analytic forms:

$\text{P}{}_{\mathrm{ax;by}}\text{= G}\text{a}{}^{\mathrm{\beta }}\text{+ δx}{}^{\mathrm{\beta }}\text{b}{}^{\mathrm{\beta }}\text{+ δy}{}^{\mathrm{\beta }}\text{, β ≠ 0}$

;

$\text{P}{}_{\mathrm{ax;by}}\text{= G(a}{}^{\mathrm{\beta }}\text{x}{}^{\mathrm{\gamma }}\text{/b}{}^{\mathrm{\beta }}\text{y}{}^{\mathrm{\gamma }}\text{), β + γ ≠ 0}$

;

$\text{P}{}_{\mathrm{ax;by}}\text{= Q}{}_0\text{(a/x, b/y)}$

.     

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.     

Similar threshold functions from contrasting neural signal models

Three neural signal models of increment threshold detection are compared. All assume that the criterion for threshold is the attainment of a critical, minimum neural signal (or difference between two neural signals), and that the signal due to a test flash of intensity λ in the absence of a background light is $\text{λ}\text{(λ + σ)}$ (where σ is the semi-saturation constant). The models differ in the manner in which a background light of intensity θ is assumed to affect the signal. One model (due to Alpern et al., 1970a, Alpern et al., 1970b, Alpern et al., 1970c) assumes that the test flash signal, $\text{λ}\text{(λ + σ)}$, is attenuated by the multiplicative factor $\text{θ}{}_{\mathrm{D}}\text{(θ + θ}{}_{\mathrm{D}}\text{)}$ (where θ_D is a constant interpreted as sensory noise); another model specifies that the test flash signal is simply reduced (by subtraction) by the amount $\text{θ}\text{(θ + K)}$ (K a constant). One main result of this paper is that in the absence of pigment bleaching, these two models imply indistinguishable increment threshold functions. Further, a necessary and sufficient condition for each model guaranteeing the absence of saturation with steady backgrounds is found to be empirically satisfied. A third model is considered where the background field is assumed both to contribute to the neural signal and simultaneously to attenuate it (via a gain change). These assumptions are closely related to theoretical accounts of color induction and color perception. Though this model needs further investigation, it appears to be in better accord with actual increment threshold data than the others.     

Positive-difference structures and bilinear utility functions

Let (M₁, f), (M₂, g) be mixture sets and let ? be a binary preference relation on M₁ × M₂. By using the concept of positive-difference structures, necessary and sufficient conditions are given for the existence of a real-valued utility function u on M₁ × M₂ which represents ? and possesses the bilinearity property

$\text{u(?(α, x}{}_1\text{,x}{}_2\text{),g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{αu(x}{}_1\text{, g(βy}{}_1\text{, y}{}_2\text{))}\text{+}\text{(1 ? α) u(x}{}_2\text{, g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{βu(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_1\text{)}\text{+}\text{(1 ? β) u(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_2\text{)}$

, for all α, β ∈ [0, 1], all x₁, x₂ ∈ M₁ and all y₁, y₂ ∈ M₂. Moreover, uniqueness up to positive linear transformations can be proved for those utility functions. Finally an outline is given of applications of these results in expected utility theory.     

On unpredictability as a causal factor in “learned helplessness”

In two replications, two groups of dogs were exposed to a series of uncontrollable, electric shocks. For one group the shocks were preceded by a tone (i.e., Paired). For the other group the shocks were randomly related to the tones and hence unpredictable (i.e., Random). Each replication also included a third group; in the first it was exposed only to the series of tones (CS-only), while in the second, it was exposed only to a series of shocks (Shocks-only). Then, all dogs were required to learn a discriminative choice escape/avoidance task in which the required response was to lift the correct paw in the presence of each of two visual S^Ds to escape or avoid the shocks [($\text{S}{}_1{}^{\mathrm{D}}\text{?R}{}_1\text{)}\text{(S}{}_2{}^{\mathrm{D}}\text{?R}{}_2$)]. Dogs preexposed to random tones and shocks were least successful in learning the task relative to those groups which experienced either predicted shocks, only the tones, or only the shocks, which in turn did not differ from each other. These results permitted the inference that the proactive interference with choice behavior following random tone CSs and shocks was attributable to a learned irrelevance generalized with respect to CSs.     

An alternative scaling method for priorities in hierarchical structures

The early contributions of Saaty have spawned a multitude of applications of principal right (PR) eigenvector “scaling” of a dominance matrix [R]. Prior to Saaty's work (1977–1984) scaling of dominance matrices received little attention in multidimensional scaling, e.g., see Shepard (1972, pp. 26–27). This eigenvector method (EM) of scaling [R] yields u_i scores (weights) popularly used at each branching of the Analytic Hierarchy Process (AHP) technique that has been increasingly applied in multiple criterion analysis of utility, preference, probability, and performance. In this paper, it is proposed that an alternate least squares method (LSM) scaling technique yielding least squares optimal scores (weights) provides $\text{w}{}_{\mathrm{i}}{}^1$ values having a number of important advantages over u_i scores popularly utilized to date.     

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.     

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.     

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.     

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.     

Developmental changes in conditional reasoning: Linguistic or logical?

This study investigated reasoning with abstract conditional sentences as a function of age. Subjects from third-grade to eleventh-grade were required to evaluate the conclusion of several conditional arguments. The results confirmed the previously established finding that performance improves with age, particularly between 11 and 15 years. This finding could be interpreted to indicate that individuals become more logical as they get older. However, another possible interpretation is that the meaning of a conditional sentence like pthenq for naive Ss may not always be given by the truth function pq true, $\text{p}\text{q}$ false, $\text{p}\text{q}$ true, and $\text{p}\text{q}$ true. Further analysis suggested that at 9 years individuals treat the connective in the sentence, ifpthenq, as if it were either a conjunctive or a biconditional, that the conjunctive meaning disappears with increasing age and after 13 years is gradually superceded by the conditional.     

A family of Gödel hybrid logics

In this paper, we define a family of fuzzy hybrid logics that are based on Gödel logic. It is composed of two infinite-valued versions called ${\mathsf{GH}}_{\infty }$ and ${\mathsf{WGH}}_{\infty }$, and a sequence of finitary valued versions ${({\mathsf{GH}}_n)}_{0<n<\infty }$. We define decision procedures for both ${\mathsf{WGH}}_{\infty }$ and ${({\mathsf{GH}}_n)}_{0<n<\infty }$ that are based on particular sequents and on a set of proof rules dealing with such sequents. As these rules are strongly invertible the procedures naturally allow one to generate countermodels. Therefore we prove the decidability and the finite model property for these logics. Finally, from the decision procedure of ${\mathsf{WGH}}_{\infty }$, we design a sound and complete sequent calculus for this logic.