A two-stage algorithm for assessing violations of additivity via axiomatic and numerical conjoint analysis

An algorithm for assessing additivity conjunctively via both axiomatic conjoint analysis and numerical conjoint scaling is described. The algorithm first assesses the degree of individual differences among sets of rankings of stimuli, and subsequently examines either individual or averaged data for violations of axioms necessary for an additive model. The axioms are examined at a more detailed level than has been previously done. Violations of the axioms are broken down into different types. Finally, a nonmetric scaling of the data can be done based on either or both of two different badness-of-fit scaling measures. The advantages of combining all of these features into one algorithm for improving the diagnostic value of axiomatic conjoint measurement in evaluating additivity are discussed.     

Empirical evaluation of the axioms of multiplicativity, commutativity, and monotonicity in ratio production of area

Although direct scaling methods have been widely used in the behavioral sciences since the 1950s, theoretical approaches which could clarify the implicit assumptions inherent in Stevens' ratio scaling approach were developed only recently. Today, it is generally accepted that the axioms of commutativity and multiplicativity are fundamental to the subjects' ratio scaling behavior. Therefore, both axioms were evaluated in ratio production of area. Participants were required to adjust the area of a variable circle to prescribed ratio production factors. The results are in accordance with previous empirical findings: commutativity was satisfied, whereas multiplicativity failed to hold. Additionally, the validity of the monotonicity property was analyzed, which postulates that the subjects' adjustments in a ratio production experiment preserve the mathematical order of the ratio production factors. Monotonicity was satisfied empirically, which is consistent with all the current theories of ratio scaling.     

Stevens’ Direct Scaling Methods and the Uniqueness Problem

Stevens postulated that we can use the responses of a participant in a ratio scaling experiment directly to construct a psychophysical function representing the participant's sensations. Although Stevens' methods of constructing measurement scales are widely used in the behavioral sciences, the problem of which scale type is appropriate to describe ratio scaling data is still unresolved. To deal with this problem, we develop a theoretical framework to specify the scale type attained by Stevens' direct scaling methods. It is shown, under fairly mild background assumptions, that the behavioral axioms presented in this paper are necessary and sufficient for the psychophysical functions to be ordinal-, interval-, log-interval-, or ratio-scales. Furthermore, suggestions on how to test these behavioral axioms are provided. Requests for reprints should be sent to thomas.     

Evaluating the Equal-Interval Hypothesis with Test Score Scales

The axioms of additive conjoint measurement provide a means of testing the hypothesis that testing data can be placed onto a scale with equal-interval properties. However, the axioms are difficult to verify given that item responses may be subject to measurement error. A Bayesian method exists for imposing order restrictions from additive conjoint measurement while estimating the probability of a correct response. In this study an improved version of that methodology is evaluated via simulation. The approach is then applied to data from a reading assessment intentionally designed to support an equal-interval scaling.     

Extensions of Narens' theory of ratio magnitude estimation

Stevens postulated that the responses of a participant in a ratio scaling experiment can be used directly to construct a psychophysical function. Today, it is generally accepted that the axioms of commutativity and multiplicativity are crucial for the interpretation of the subjects' ratio scaling behaviour. Empirical findings provide evidence that commutativity holds, whereas multiplicativity fails to hold across different sensory modalities. This shows that, in principle, Stevens' direct scaling methods yield measurements on a ratio scale level, but that the numerals occurring in a ratio scaling experiment cannot be taken at face value. Thus, Narens and others introduced a transformation function f, which converts the numerals used in an experiment into the latent mathematical numbers. The aim of the present paper is to specify the (unknown) shape of the transformation function f, by analysing different extensions of the multiplicative property. The results provide evidence that f is either a power function or a logarithmic function.     

A theoretical framework for testing the additive difference model for dissimilarities data: Representing gambles as multidimensional stimuli

Implicit within the acceptance of most multidimensional scaling models as accurate representations of an individual's cognitive structure for a set of complex stimuli, is the acceptance of the more general Additive Difference Model (ADM). A theoretical framework for testing the ordinal properties of the ADM for dissimilarities data is presented and is illustrated for a set of three-outcome gambles. Paired comparison dissimilarity judgments were obtained for two sets of gambles. Judgments from one set were first analyzed using the ALSCAL individual differences scaling model. Based on four highly interpretable dimensions derived from this analysis, a predicted set of dimensions were obtained for each subject for the second set of gambles. The ordinal properties of the ADM necessary for interdimensional additivity and intradimensional subtractivity were then tested for each subject's second set of data via a new computer-based algorithm, ADDIMOD. The tests indicated that the ADM was rejected. Although violations of the axioms were significantly less than what would be expected by chance, for only one subject was the model clearly supported. It is argued that while multidimensional scaling models may be useful as data reduction techniques, they do not reflect the perceptual processes used by individuals to form judgments of similarity. Implications for further study of multidimensional scaling models are offered and discussed.     

Biased extensive measurement: The homogeneous case

In the homogeneous case of one type of objects, we prove the existence of an additive scale unique up to a positive scaling transformation without transitivity of indifference and with a property of homothetic invariance weaker than monotonicity. The representation, which is a particular case of a semiorder representation, reveals a unique positive factor α?1 that biases extensive structures and explains departures from these standard axioms of extensive measurement (α=1). We interpret α as characterizing the qualitative influence of the underlying measurement process and we show that it induces a proportional indifference threshold.     

Fundamental axioms for preference relations

Summary The basic theory of preference relations contains a trivial part reflected by axioms A1 and A2, which say that preference relations are preorders. The next step is to find other axims which carry the theory beyond the level of the trivial. This paper is to a great part a critical survey of such suggested axioms. The results are much in the negative — many proposed axioms imply too strange theorems to be acceptable as axioms in a general theory of preference. This does not exclude, of course, that they may well be reasonable axioms for special calculi of preference. I believe that many axioms which are rejected here may be plausible if their range of application is restricted by conditions which are possible to formulate only in a language richer than that of the propositional calculus, e.g. in one containing modal operators or probabilistic concepts.     

Monotone mapping of similarities into a general metric space

A method of “maximum variance nondimensional scaling” is described and tested that transforms similarity measures into distances that meet just three conditions: (C1) they exactly satisfy the metric axioms, (C2) they are, as nearly as possible, monotonically related to the similarity measures, (C3) they have maximum variance possible under the two preceding conditions. By achieving an appropriate balance between the last two conditions, one can determine the true underlying distances and the form of the unknown monotone function relating the similarity measures to those distances without assuming that the underlying space has any particular Euclidean, Minkowskian, or even dimensional strucutre. The method appears to have potential applications, e.g., to studies of stimulus generalization and the structure and processing of semantic information.     

Model Existence in Non-Compact Modal Logic

Predicate modal logics based on Kwith non-compact extra axioms are discussed and a sufficient condition for the model existence theorem is presented. We deal with various axioms in a general way by an algebraic method, instead of discussing concrete non-compact axioms one by one.     

On Bayesian Testing of Additive Conjoint Measurement Axioms Using Synthetic Likelihood

This article introduces a Bayesian method for testing the axioms of additive conjoint measurement. The method is based on an importance sampling algorithm that performs likelihood-free, approximate Bayesian inference using a synthetic likelihood to overcome the analytical intractability of this testing problem. This new method improves upon previous methods because it provides an omnibus test of the entire hierarchy of cancellation axioms, beyond double cancellation. It does so while accounting for the posterior uncertainty that is inherent in the empirical orderings that are implied by these axioms, together. The new method is illustrated through a test of the cancellation axioms on a classic survey data set, and through the analysis of simulated data.     

An Elementary System of Axioms for Euclidean Geometry Based on Symmetry Principles

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (1) it supports the thesis that Euclidean geometry is a priori, (2) it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, (3) it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.     

Aristotelian Logic Axioms in Propositional Logic: The Pouch Method

A new theoretical approach to Aristotelian Logic (AL) based on three axioms has been recently introduced. This formalization of the theory allowed for the unification of its uncommunicated traditional branches, thus restoring the theoretical unity of AL. In this brief paper, the applicability of the three AL axioms to Propositional Logic (PL) is explored. First, it is shown how the AL axioms can be applied to some simple PL arguments in a straightforward manner. Second, the development of a proof method for PL inspired by the AL axioms is presented. This method mimics the underlying mechanics of the proof method from AL, and offers a complementary alternative to proof methods such as truth trees.     

On extremal axioms

In the paper translated here, Carnap and Bachmann shows that the apparently metalinguistic ‘extremal' axioms that are added to some axiom systems to the effect that the foregoing axioms are to apply as broadly, or as narrowly, as possible may be formulated directly as proper axioms. They analyze such axioms into four fundamental types, with the help of a concept of ‘complete’ isomorphism.     

Routes to Triviality

It is known that a number of inference principles can be used to trivialise the axioms of naïve comprehension – the axioms underlying the naïve theory of sets. In this paper we systematise and extend these known results, to provide a number of general classes of axioms responsible for trivialising naïve comprehension.     

A method of weighing qualitative preference axioms

A method is developed for determining the absolute and relative strengths of qualitative preference axioms in normative Bayesian decision theory. These strengths are calculated for the three most common qualitative axioms; transitivity, the sure-thing principle, and dominance. The relative strength of the latter two axioms with respect to transitivity is calculated for special cases, and a bound is derived which is applicable to a larger class of decision problems. Possible implications of this theoretical work for decision heuristics are discussed.     

Linking social axioms with indicators of positive interpersonal, social and environmental functioning in Iran: an exploratory study

Social axioms are people's general beliefs about how the world functions and always involve the relationship between two conceptual entities. Social axioms have been proposed as a construct that can be useful in helping researchers interpret cultures and explain people's behaviors in different cultural contexts. Despite the growth of studies on social axioms in various countries, no effort has been made so far to investigate specifically the relation between social axioms and indicators of interpersonal, social, and environmental functioning. To fill this gap, this exploratory study sought to examine the relation between social axioms and a set of variables indicating positive interpersonal, social, and environmental functioning (namely, gratitude, connectedness to nature, social participation, perspective-taking, and empathic concern) in a sample of 303 Iranian university students. Findings showed that reward for application, religiosity, and social complexity significantly predicted gratitude when sex was controlled for. Social complexity and reward for application significantly contributed to explaining the variance in connectedness to nature over and above sex. Social cynicism and social complexity also predicted perspective-taking significantly after controlling for sex. Social axioms were not successful in predicting social participation and empathic concern. Overall, it is possible to conclude that the findings support the utility of social axioms in predicting interpersonal, social, and environmental functioning. That is, generalized beliefs about oneself, the social and physical environment, or the spiritual world are associated with individuals' interpersonal, social, and environmental functioning in this Iranian sample.     

Empirical evaluation of axioms fundamental to Stevens's ratio-scaling approach: I. Loudness production

Stevens's direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens's central axioms of commutativity and multiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens's (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.     

What I do and what I think they would do: Social axioms and behaviour

The social axioms system uniquely predicted a large variety of behaviours and preferences. It is suggested that (a) the assistance social axioms provide in predicting the behaviour of others, and (b) the self‐characteristics embedded in the axioms account for this unique prediction ability. Three studies, each pertaining to a different axiom, tested the prediction power of the social axiom regarding two types of behaviours: One that is directly impacted by how others are expected to behave, and another that is more self‐directed. Results consistently revealed a unique contribution of the social axioms over personal characteristics in prediction of behaviours directed by how others are expected to behave, whereas behaviours that are more self‐directed were largely explained by relevant personal characteristics. Copyright © 2010 John Wiley & Sons, Ltd.     

Axiomatizing Relativistic Dynamics without Conservation Postulates

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc ². The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.