Information and preference in partial orders: A bimatrix representation

This paper presents a bimatrix structure for examining ordinal partial rankings. A set of axioms is given similar to those of Kemeny and Snell (1962) and Bogart (1973), which uniquely determines the distance between any pair of such rankings. Thel ¹ norm is shown to satisfy this set of axioms, and to be equivalent to the Kemeny and Snell distance on their subspace of weak orderings. Consensus formation is discussed.This research was supported by a NSERC grant A8966.     

Machine short-cuts in the computation of chisquare and the contingency coefficient

Rapid computational routines are presented for calculatingx ² from frequency data in the following cases: (1) test of goodness of fit between an observed and a theoretical distribution; (2) test of independence of distributions displayed in anr ×c table; (3) test of independence of distributions displayed in anr × 2 table. A rapid method of computing the contingency coefficient also follows from the procedure used in the second of these cases.     

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.     

Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function

Understanding the psychological interpretation of numerals is of both practical and theoretical interest. In classical magnitude estimation, respondents match numerals to sensations and in magnitude production they select sensations that stand in a prescribed numerical ratio to a given standard. The present work focusses on evaluating several possible, and related, forms for the function W formulating the distortion of numerals. The main form, of which a power function is a special case, is the Prelec exponential/power representation. Behavioral equivalents to power and to Prelec functions are formulated, tested, and rejected. It is argued that either the mathematical form or the assumption W(1)=1 is wrong. Whereas, the axiomatic literature has focussed exclusively on the former inference, we explore the alternate that W(1)≠1. Behavioral axioms are formulated in each case and experimentally tested. We conclude that most respondents satisfy a general power function and that those who do not, satisfy the general Prelec function.     

Multidimensional unfolding: Determining the dimensionality of ranked preference data

A model is proposed which treats rankings given by a group of judges as representing regions in an isotonic space of dimensionalityr. Three possible criteria for estimating lower bound dimensionality are discussed: mutual boundary, cardinality, and the occurrence of transposition groups. Problems associated with each criterion are mentioned.Deceased.     

An axiomatization of the ratio/difference representation

If ≥_r and ≥_d are two quaternary relations on an arbitrary set A, a ratio/difference representation for ≥_r and ≥_d is defined to be a function f that represents ≥_r as an ordering of numerical ratios and ≥_d as an ordering of numerical differences. Krantz, Luce, Suppes and Tversky (1971, Foundations of Measurement. New York, Academic Press) proposed an axiomatization of the ratio/difference representation, but their axiomatization contains an error. After describing a counterexample to their axiomatization, Theorem 1 of the present article shows that it actually implies a weaker result: if ≥_r and ≥_d are two quaternary retations satisfying the axiomatization proposed by Krantz et al. (1971), and if ≥_r′ and ≥_d′ are the relations that are inverse to ≥_r and ≥_d, respectively, then either there exists a ratio/difference representation for ≥_r and ≥_d, or there exists a ratio/difference representation for ≥_r′ and ≥_d′, but not both. Theorem 2 identifies a new condition which, when added to the axioms of Krantz et al. (1971), yields the existence of a ratio/difference representation for relations ≥_r and ≥_d.     

On the creative role of axiomatics. The discovery of lattices by Schröder,Dedekind, Birkhoff,and others

Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various ways in which lattices have been introduced in mathematics by Schröder et al. These historical episodes reveal that the axiomatic method is not only a way of systematizing our knowledge, but that it can also be used as a fruitful tool for discovering and introducing new mathematical notions. Looked at it from this perspective, the creative aspect of axiomatics for mathematical practice is brought to the fore.     

Notes on Mally’s deontic logic and the collapse of {\varvec{Seinsollen}} and {\varvec{Sein}}

This paper analyzes Mally’s system of deontic logic, introduced in his The Basic Laws of Ought: Elements of the Logic of Willing (1926). We discuss Mally’s text against the background of some contributions in the literature which show that Mally’s axiomatic system for deontic logic is flawed, in so far as it derives, for an arbitrary A, the theorem “A ought to be the case if and only if A is the case”, which represents a collapse of obligation. We then try to sort out and understand which axioms are responsible for the collapse and consider two ways of amending Mally’s system: (i) by changing its original underlying logical basis, that is classical logic, and (ii) by modifying Mally’s axioms.     

Dual-ranking act-consequentialism

Dual-ranking act-consequentialism (DRAC) is a rather peculiar version of act-consequentialism. Unlike more traditional forms of act-consequentialism, DRAC doesn’t take the deontic status of an action to be a function of some evaluative ranking of outcomes. Rather, it takes the deontic status of an action to be a function of some non-evaluative ranking that is in turn a function of two auxiliary rankings that are evaluative. I argue that DRAC is promising in that it can accommodate certain features of commonsense morality that no single-ranking version of act-consequentialism can: supererogation, agent-centered options, and the self-other asymmetry. I also defend DRAC against three objections: (1) that its dual-ranking structure is ad hoc, (2) that it denies (putatively implausibly) that it is always permissible to make self-sacrifices that don’t make things worse for others, and (3) that it violates certain axioms of expected utility theory, viz., transitivity and independence.

Consumer preferences for color combinations: An empirical analysis of similarity-based color relationships

In this paper, we examine aesthetic color combinations in a realistic product self-design task using the NIKEiD online configurator. We develop a similarity-based model of color relationships and empirically model the choice likelihoods of color pairs as a function of the distances between colors in the CIELAB color space. Our empirical analysis reveals three key findings. First, people de-emphasize lightness and focus on hue and saturation. Second, given this shift in emphasis, people generally like to combine colors that are relatively close or exactly match, with the exception that some people highlight one signature product component by using contrastive color. This result is more consistent with the visual coherence perspective than the optimal arousal perspective on aesthetic preference. Third, a small palette principle is supported such that the total number of colors used in the average design was smaller than would be expected under statistical independence.     

A characterization of the concept of independence in knowledge structures

In knowledge space theory a knowledge structure provides a deterministic representation of the implications among the items in a given set Q. Concrete procedures for the efficient assessment of knowledge by means of a knowledge structure have been proposed by Doignon and Falmagne [Falmagne, J.-C., & Doignon, J.-P. (1988a). A class of stochastic procedures for the assessment of knowledge. British Journal of Mathematical and Statistical Psychology, 41, 1-23; Falmagne, J.-C., & Doignon, J.-P. (1988b). A markovian procedure for assessing the state of a system. Journal of Mathematical Psychology, 32, 232-258]. The primitive idea at the core of such procedures is that the (correct or wrong) answers of a student to a subset A⊆Q of items could be inferred from the answers to a subset B⊆Q of items that were previously presented to that student. Since B provides information about A, from the viewpoint of the teacher these two subsets are not independent. This idea of dependence vs. independence is formalized in this paper in terms of an independence relation on the power set of Q. A nice characterization of this relation allows to express an arbitrary knowledge structure as the combination of a number of substructures each of which is independent of each other. An algorithm is then proposed which checks for independence in a knowledge structure and decomposes this last into a collection of independent substructures.     

Probability weights in rank-dependent utility with binary even-chance independence

We examine the effects of a weak version of expected utility's independence axiom on the probability weighting function in rank-dependent utility. Our weak independence axiom says that a 50-50 lottery between a two-outcome gamble and its certainty equivalent is indifferent to the certainty equivalent. A variety of nonlinear probability weighting functions satisfy this axiom, but most weighting functions proposed by others do not. Nevertheless, the axiom accommodates weighting functions that are quite similar to the inverse S-shaped concave-convex functions of others that overvalue small probabilities and undervalue large probabilities.     

Physicalism and phenomenal concepts

Frank Jackson’s famous Knowledge Argument moves from the premise that complete physical knowledge is not complete knowledge about experiences to the falsity of physicalism. In recent years, a consensus has emerged that the credibility of this and other well-known anti-physicalist arguments can be undermined by allowing that we possess a special category of concepts of experiences, phenomenal concepts, which are conceptually independent from physical/functional concepts. It is held by a large number of philosophers that since the conceptual independence of phenomenal concepts does not imply the metaphysical independence of phenomenal properties, physicalism is safe. This paper distinguishes between two versions of this novel physicalist strategy—Phenomenal Concept Strategy (PCS)—depending on how it cashes out “conceptual independence,” and argues that neither helps the physicalist cause. A dilemma for PCS arises: cashing out “conceptual independence” in a way compatible with physicalism requires abandoning some manifest phenomenological intuitions, and cashing it out in a way compatible with those intuitions requires dropping physicalism. The upshot is that contra Brian Loar and others, one cannot “have it both ways.”     

S5 knowledge without partitions

We study set algebras with an operator (SAO) that satisfy the axioms of S5 knowledge. A necessary and sufficient condition is given for such SAOs that the knowledge operator is defined by a partition of the state space. SAOs are constructed for which the condition fails to hold. We conclude that no logic singles out the partitional SAOs among all SAOs.     

An Independence Relation for Sets of Secrets

A relation between two secrets, known in the literature as nondeducibility, was originally introduced by Sutherland. We extend it to a relation between sets of secrets that we call independence. This paper proposes a formal logical system for the independence relation, proves the completeness of the system with respect to a semantics of secrets, and shows that all axioms of the system are logically independent.     

A Unified Nonparametric IRT Model for <Emphasis Type="Italic">d</Emphasis>-Dimensional Psychological Test Data (<Emphasis Type="Italic">d</Emphasis>-ISOP)

The (univariate) isotonic psychometric (ISOP) model (Scheiblechner, 1995) is a nonparametric IRT model for dichotomous and polytomous (rating scale) psychological test data. A weak subject independence axiom W1 postulates that the subjects are ordered in the same way except for ties (i.e., similarly or isotonically) by all items of a psychological test. A weak item independence axiom W2 postulates that the order of the items is similar for all subjects. Local independence (LI or W3) is assumed in all models. With these axioms, sample-free unidimensional ordinal measurements of items and subjects become feasible. A cancellation axiom (Co) gives, as a result, the additive isotonic psychometric (ADISOP) model and interval scales for subjects and items, and an independence axiom (W4) gives the completely additive isotonic psychometric (CADISOP) model with an interval scale for the response variable (Scheiblechner, 1999). The d-ISOP, d-ADISOP, and d-CADISOP models are generalizations to d-dimensional dependent variables (e.g., speed and accuracy of response). The author would like to thank an Associate Editor and two anonymous referees and also Professor H.H. Schulze for their very valuable suggestions and corrections.     

Generalized thurstone models for ranking: Equivalence and reversibility

Previous work has determined the conditions under which generalized versions of Thurstone's theory of comparative judgment are formally equivalent (i.e., empirically indistinguishable) for choice experiments. This note solves the analogous problem for ranking experiments: It is shown that if two “Generalized Thurstone Models” are equivalent for choice experiments with n alternatives they are also equivalent for ranking experiments with n alternatives, despite the fact that ranking generates many more preference probabilities. This result in turn allows one to determine which Generalized Thurstone Models are “reversible,” i.e., satisfy the requirement that regardless of whether the subject ranks from best to worst or from worst to best, rankings that express the same preference order will occur with the same probability.     

A free IPC is a natural logic: Strong completeness for some intuitionistic free logics

IPC, the intuitionistic predicate calculus, has the property

1. Vc(Γ?A ^c /x) ? Γ??xA.Furthermore, for certain important Γ, IPC has the converse property
2. Γ??xA ? Vc(Γ?A ^c /x).
3. may be given up in various ways, corresponding to different philosophic intuitions and yielding different systems of intuitionistic free logic. The present paper proves the strong completeness of several of these with respect to Kripke style semantics. It also shows that giving up (i) need not force us to abandon the analogue of (ii).