Christopher Doble

Almost Connected Orders

One of the standard axioms for semiorders states that no three-point chain is incomparable to a fourth point. We refer to asymmetric relations satisfying this axiom as ‘almost connected orders’ or ‘ac-orders.’ It turns out that any relation lying between two weak orders, one of which covers the other for inclusion, is an ac-order (albeit of a special kind). Every ac-order is bracketed in a natural way by two weak orders, one the maximum in the set of weak orders included in the ac-order, and the other minimal, but not necessarily the minimum, in the set of weak orders that include the ac-order. The family of ac-orders on a finite set with at least five elements is not well graded (in the sense of Doignon and Falmagne, 1997). However, such a family is both ‘upgradable’ and ‘downgradable,’ as every nonempty ac-order contains a pair whose deletion defines an ac-order on the same set, and for every ac-order which is not a chain, there is a pair whose addition gives an ac-order.

A threshold theory account of psychometric functions with response confidence under the balance condition

The study of thresholds for discriminability has been of long-standing interest in psychophysics. While threshold theories embrace the concept of discrete-state thresholds, signal detection theory discounts such a concept. In this paper we concern ourselves with the concept of thresholds from the discrete-state modelling viewpoint. In doing so, we find it necessary to clarify some fundamental issues germane to the psychometric function (PF), which is customarily constructed using psychophysical methods with a binary-response format. We challenge this response format and argue that response confidence also plays an important role in the construction of PFs, and thus should have some impact on threshold estimation. We motivate the discussion by adopting a three-state threshold theory for response confidence proposed by Krantz (1969, Psychol. Rev., 76, 308-324), which is a modification of Luces (1963, Psychol. Rev., 70, 61-79) low-threshold theory. In particular, we discuss the case in which the practice of averaging over order (or position) is enforced in data collection. Finally, we illustrate the fit of the Luce-Krantz model to data from a line-discrimination task with response confidence.

Assessing Mathematical Knowledge in a Learning Space

According to Knowledge Space Theory (KST) (cf. Doignon and Falmagne, 1999; Falmagne and Doignon, 2011), a student’s competence in a mathematics or science subject, such as elementary school mathematics or first year college chemistry, can be described by the student’s ‘knowledge state,’ which is the set of ‘problem types’ that the student is capable of solving. (In what follows, we abbreviate ‘problem type’ as ‘problem’ or ‘item.’) As the student masters new problems, she moves to larger and larger states. Some states are closer to the student’s state than others, though, based on the material she must learn in order to master the problems in those states. Thus, there is a structure to the collection of states, and this structure gives rise to a ‘learning space,’ which is a special kind of knowledge space. These concepts have been discussed at length in Chapter 1 of this volume. We recall here that the collection of states forming a learning space always contains the ‘empty state’ (the student knows nothing at all in the scholarly subject considered) and the ‘full state’ (the student knows everything in the subject). The collection of states must also satisfy two pedagogically cogent principles, which we state below in nonmathematical language.

On Meaningful Scientific Laws

The authors describe systematic methods for uncovering scientific laws a priori, on the basis of intuition, or Gedanken Experiments. Mathematical expressions of scientific laws are, by convention, constrained by the rule that their form must be invariant with changes of the units of their variables. This constraint makes it possible to narrow down the possible forms of the laws. It is closely related to, but different from, dimensional analysis. It is a mathematical book, largely based on solving functional equations. In fact, one chapter is an introduction to the theory of functional equations.

Dimensional Invariance and Dimensional Analysis

The notion of invariance is a fundamental one in mathematics and the sciences. This chapter is a brief introduction to two of its subtopics, dimensional invariance and dimensional analysis, which are relevant to the subject of this book. Specifically, dimensional invariance is closely related to, but technically different from, meaningfulness.

Propagating Axioms via Meaningfulness

The meaningfulness condition introduced in Definition 5.2.1 and Equation (5.6) has a remarkable property. In some cases, a condition imposed on a single code of a meaningful collection may be automatically transported to all the codes in the collection. This applies to many properties, such as solvability, quasi-permutability, symmetry, differentiability, and others.

Order Invariance under Transformations

So far in this book, we have investigated the consequences of some abstract axioms such as transitivity or permutability, combined with the meaningfulness condition, on the mathematical form of the codes. This chapter is in the same spirit, with the abstract axioms replaced by ‘order-invariance’ axioms. The next equation gives an example.

Meaningful Representations of Scientific Codes

In this chapter, which recalls some results of Falmagne (2015), we derive some exemplary consequences of the meaningfulness condition paired with some abstract axioms, in particular: associativity, quasi-permutability, bisymmetry, translatability, and quasi-permutability, the latter in the context of LF-systems.

Abstract Axioms and their Representations

Except for the content of Section 4.7, which is relatively recent, all the results of this chapter are standard parts of the functional equation literature. Specific references are given in due place. These results provide the mathematical foundations on which the meaningfulness axiom will operate.