Mathematical concepts for analyzing ambiguous situations

Abstract

We examine some mathematical tools for dealing with ambiguous situations. The main tool is the use of non-standard logic with truth-values in what is called a locale. This approach is related to fuzzy set theory, which we briefly discuss. We also consider probabilistic concepts. We include specific examples and describe the way a researcher can set up a suitable locale to analyse a concrete situation.     

Categoricity Theorems and Conceptions of Set

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to the pure sets of the other.This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine.     

幼儿在集合比较中运用数数的实验研究

本研究中运用了两个实验探讨数数干预和测查条件对儿童在集合比较中运用数数的影响。干预对3岁儿童(M=3：9)没有影响。在平均年龄为4岁4个月时．干预组儿童比控制组儿童更倾向于用数数比较集合．自然组儿童也比传统组更倾向于用数数。许多4岁儿童在无干预时不用数数可能是因为,1)不知数数比视觉性比较更有效,或2)他们在集合比较中的数数极易受测查情景因素的影响。儿童在集合比较中的数数运用与他们的数数水平密切关联。     

The Undecidability of Grisin's Set Theory

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.     

Elementary Extensions of External Classes in a Nonstandard Universe

In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" class L[S(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in S(R). We prove that under some mild saturation-like requirements for R the class L[S(R)] models a certain -version of HST including the principle of +-saturation; moreover, in this case L[S(R)] is an elementary extension of L[S(R)] in the st--language whenever sets R R satisfy the requirements.     

A Very Strong Set Theory?

Studia Logica - Using two distinct membership symbols makes possible to base set theory on one general axiom schema of comprehension. Is the resulting system consistent? Can set theory and...     

Multi‐set factor analysis by means of Parafac2

We consider multi‐set data consisting of observations, k = 1,…, K (e.g., subject scores), on J variables in K different samples. We introduce a factor model for the J × J covariance matrices , k = 1,…, K, where the common part is modelled by Parafac2 and the unique variances , k = 1,…, K, are diagonal. The Parafac2 model implies a common loadings matrix that is rescaled for each k, and a common factor correlation matrix. We estimate the unique variances by minimum rank factor analysis on for each k. The factors can be chosen orthogonal or oblique. We present a novel algorithm to estimate the Parafac2 part and demonstrate its performance in a simulation study. Also, we fit our model to a data set in the literature. Our model is easy to estimate and interpret. The unique variances, the factor correlation matrix and the communalities are guaranteed to be proper, and a percentage of explained common variance can be computed for each k. Also, the Parafac2 part is rotationally unique under mild conditions.     

Extending Standard Models of ZFC to Models of Nonstandard Set Theories

We study those models of ZFCwhich are embeddable, as the class of all standard sets, in a model of internal set theory >ISTor models of some other nonstandard set theories. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fourier‐Motzkin elimination method in MOLP problems

This paper introduces a method for finding the non‐dominated set in criterion space of an MOLP problem based on Fourier–Motzkin Elimination. The method incorporates a procedure that eliminates decision variables from constraints and transforms a feasible region from decision space to criterion space. While avoiding characterization of the efficient set in decision space, it finds the non‐dominated set in criterion space. After the decision maker's most preferred criterion vector has been identified, the corresponding efficient solutions can be found by backward substitution. The method was implemented in MATLAB using a tabular form and computational experiments were conducted. The results indicate that although the method performs well for relatively small problems, it can be computationally intensive for larger problems. Nonetheless, the method is intuitively appealing and it provides useful insights into the geometry and theory of MOLP. As such it makes it a valuable educational tool for students of multi‐criteria decision analysis. Copyright © 2003 John Wiley & Sons, Ltd.