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21
MeasurementTheoretic Justification of Connectives in Fuzzy Set Theory
, 1995
"... The problem of representing intersection and union in fuzzy set theory is considered. There are various proposals in the literature to model these concepts. The possibility of using continuous triangular norms and conorms (including min and max) are taken up in a measurementtheoretic setting. T ..."
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The problem of representing intersection and union in fuzzy set theory is considered. There are various proposals in the literature to model these concepts. The possibility of using continuous triangular norms and conorms (including min and max) are taken up in a measurementtheoretic setting. The conditions are laid out to arrive at cardinal scales on which addition and multiplication are meaningful and critically discussed. These conditions must either be accepted on normative grounds or must be empirically verified before the modeling process in order to see which operations are meaningful. It is emphasized that the Archimedean axiom and the existence of natural bounds are crucial in arriving at ratio and absolute scale representations. Keywords: Membership functions, measurement theory, operators, relations. 1 Introduction and Preview When Zadeh [45] introduced the concept of a fuzzy set he suggested to use the functions min and max to model set theoretic intersection an...
Genetic Measurement Theory of Epistatic Effects
, 1997
"... Epistasis is defined as the influence of the genotype at one locus on the effect of a mutation at another locus. As such it plays a crucial role in a variety of evolutionary phenomena such as speciation, population bottle necks and the evolution of genetic architecture (i.e. the evolution of dominan ..."
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Epistasis is defined as the influence of the genotype at one locus on the effect of a mutation at another locus. As such it plays a crucial role in a variety of evolutionary phenomena such as speciation, population bottle necks and the evolution of genetic architecture (i.e. the evolution of dominance, canalization and genetic correlations). In mathematical population genetics, however, epistasis is often represented as a mere noise term in an additive model of gene effects. In this paper it is argued that epistasis needs to be scaled in a way that is more directly related to the mechanisms of evolutionary change. A review of general measurement theory shows that the scaling of a quantitative concepts has to reflect the empirical relationships among the objects. To apply these ideas to epistatic mutation effects it is proposed to scale AxA epistatic effects as the change in the magnitude of the additive effect of a mutation at one locus due to a mutation at a second locus. It is shown ...
Normal science, pathological science and psychometrics
 Theory & Psychology
, 2000
"... Abstract. A pathology of science is defined as a twolevel breakdown in processes of critical inquiry: first, a hypothesis is accepted without serious attempts being made to test it; and, second, this firstlevel failure is ignored. Implications of this concept of pathology of science for the Kuhnia ..."
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Abstract. A pathology of science is defined as a twolevel breakdown in processes of critical inquiry: first, a hypothesis is accepted without serious attempts being made to test it; and, second, this firstlevel failure is ignored. Implications of this concept of pathology of science for the Kuhnian concept of normal science are explored. It is then shown that the hypothesis upon which psychometrics stands, the hypothesis that some psychological attributes are quantitative, has never been critically tested. Furthermore, it is shown that psychometrics has avoided investigating this hypothesis through endorsing an anomalous definition of measurement. In this way, the failure to test this key hypothesis is not only ignored but disguised. It is concluded that psychometrics is a pathology of science, and an explanation of this fact is found in the influence of Pythagoreanism upon the development of quantitative psychology. KEY WORDS: measurement, normal science, pathology of science, psychometrics, quantification There is no safety in numbers, or in anything else. (James Thurber) I argued (Michell, 1997a, 1997b) that quantitative psychology manifests methodological thought disorder, eliciting from Lovie (1997) criticisms quite unlike those offered by others invited to comment on my argument (Kline, 1997; Laming, 1997; Luce, 1997; Morgan, 1997). Lovie follows the postpositivist tradition stemming from Kuhn (1970a) and, from that perspective, saw my approach as a ‘hardnosed (and very outdated) positivist and empiricist/realist line ’ (Lovie, 1997, p. 393). The view that positivism is a form of empirical realism remains widespread, despite Passmore’s (1943, 1944, 1948) early critique and recent analyses (e.g. Friedman, 1991). Hence, there may be value in clarifying my argument regarding pathological forms of science and highlighting my reasons for so categorizing psychometrics. My thesis is that psychometricians are not only uncritical of an issue basic to their discipline but that, in addition, they have constructed a conception of quantification that disguises this. If science is a cognitive enterprise, then I
Group actions on onemanifolds, II: Extensions of Hölder’s Theorem
, 2001
"... This selfcontained paper is part of a series [FF1] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on R with restrictions on the fixed point set of each element. One result ..."
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This selfcontained paper is part of a series [FF1] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on R with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in Diff 2 (R) as those groups whose elements have at most one fixed point. 1
Nilpotent pseudogroups of functions on an interval
, 2008
"... A nearidentity nilpotent pseudogroup of order m ≥ 1 is a family f1,...,fn: (−1,1) → R of C 2 functions for which: fi − id  C 1 < ǫ for some small positive real number ǫ < 1/10 m+1 and commutators of the functions fi of order at least m equal the identity. We present a classification of neariden ..."
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A nearidentity nilpotent pseudogroup of order m ≥ 1 is a family f1,...,fn: (−1,1) → R of C 2 functions for which: fi − id  C 1 < ǫ for some small positive real number ǫ < 1/10 m+1 and commutators of the functions fi of order at least m equal the identity. We present a classification of nearidentity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks. As an application, we classify certain foliations of nilpotent manifolds.
An Irrational Construction of R from Z
"... This paper reports on a construction of the real numbers in the ProofPower implementation of the HOL logic. Since the original construction was implemented, some major improvements to the ideas have been discovered. The improvements involve some entertaining mathematics: it turns out that the De ..."
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This paper reports on a construction of the real numbers in the ProofPower implementation of the HOL logic. Since the original construction was implemented, some major improvements to the ideas have been discovered. The improvements involve some entertaining mathematics: it turns out that the Dedekind cuts provide many routes one can travel to get from the ring of integers, Z, to the eld of real numbers, R.
These Properties Go a Long Way Toward Implying
"... Since the time of Savage (1954) it has been accepted that subjective expected utility (SEU) embodies the concept of rational individual behavior under uncertainty. If, however, one alters the domain formulation in two ways, by distinguishing gains from losses and by adding a binary operation of join ..."
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Since the time of Savage (1954) it has been accepted that subjective expected utility (SEU) embodies the concept of rational individual behavior under uncertainty. If, however, one alters the domain formulation in two ways, by distinguishing gains from losses and by adding a binary operation of joint receipt, then equally rational arguments lead in the case of binary mixed gambles to predictions quite different from those of SEU. A question, raised but not really answered, is whether there is a rational argument for choosing
FILTRATIONS IN SEMISIMPLE LIE ALGEBRAS, I
"... Abstract. In this paper, we study the maximal bounded Zfiltrations of a complex semisimple Lie algebra L. Specifically, we show that if L is simple of classical type An, Bn, Cn or Dn, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain part ..."
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Abstract. In this paper, we study the maximal bounded Zfiltrations of a complex semisimple Lie algebra L. Specifically, we show that if L is simple of classical type An, Bn, Cn or Dn, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine KacMoody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups. 1.
Some Topics in System Theory
, 1970
"... The foundations of measurement theory were invesUgated by modeltheoreUc methods. The purpose was to establish a firm ba~is for <;leneral system theory. One major result was the formulatton of the concepts of scale, scale transformation and the proof of the existence, for an arbitrary scale, of the g ..."
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The foundations of measurement theory were invesUgated by modeltheoreUc methods. The purpose was to establish a firm ba~is for <;leneral system theory. One major result was the formulatton of the concepts of scale, scale transformation and the proof of the existence, for an arbitrary scale, of the group of scale transformatiqns which "leave the scale form invariantII. As an illustration of the appUcabiUty of these concepts and because of its intrinsic interest an expo~ition was <;liven of the theory of measurement for extensive quantities. A novel formulation of the usual axioms was developed which made them ~lementary formulae in the first order predicate calculus. Thus it was P9ssible to show that this theory is modelcomplete 0 Then A. Robinson I f! tlleorems were applied to show that this theory is negation complete. In the formulation of the existence theorem for scale transformation 1 groups no restriction was placed on the empirical 'Emodel Qr the numerical 'Emodel and in the other illustrative examples it wa $ shown how different choices of these 'Emodels lead to different scale transformation groups. An example of a theory of a nonextensive quantity was presented but the methods used before would not Yield modelcompleteness for this theory and thus an interesting unsolved problem remains.