This uncommonly fine textbook of the modern theory of functions of a real variable is particularly well-suited for mathematically mature students in the fields of philosophy and foundations of mathematics, philosophy of physics, probability theory, and statistics. Those who wish to achieve first-hand acquaintance with the quantum theory will also need to have a grasp of the material presented in this book. The first chapter presents a capsule survey of topics in abstract set theory and algebra, including a discussion of (...) relations, functions, the axiom of choice, and the well-ordering theorem, and leading up to the construction of the fields of real and complex numbers. The second chapter is an introduction to topology and spaces of topological functions. The approach is set-theoretical. Chapter three deals with the theory of the Lebesgue integral and measure theory. This chapter is quite comprehensive, rigorous and embellished with many exercises for the student. Subsequent sections include the theories of function spaces, Banach spaces, abstract Hilbert spaces, and differentiation. This is clearly one of the more worthwhile texts in its field.—H. P. K. (shrink)

The Jordan-Hahn decomposition and the Lebesgue decomposition, two basic notions of classical measure theory, are generalized for measures on orthomodular posets. The Jordan-Hahn decomposition property (JHDP) and the Lebesgue decomposition property (LDP) are defined for sections Δ of probability measures on an orthomodular poset L. If L is finite, then these properties can be characterized geometrically in terms of two parallelity relations defined on the set of faces of Δ. A section Δ is shown to have the JHDP (...) if and only if every pair of f-parallel faces is p-parallel; it is shown to have the LDP if and only if every pair of disjoint faces is p-parallel. It follows from these results that the LDP is stronger than the JHDP in the setting of finite orthomodular posets. Mielnik's convex scheme of quantum theory provides the frame for a physical interpretation of these results. (shrink)

We use the compactness theorem of continuous logic to give a new proof that $$L^r([0,1]; {\mathbb {R}})$$ isometrically embeds into $$L^p([0,1]; {\mathbb {R}})$$ whenever $$1 \le p \le r \le 2$$. We will also give a proof for the complex case. This will involve a new characterization of complex $$L^p$$ spaces based on Banach lattices.

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f } are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of meager sets via Banach-Mazur games. We (...) study the convergence of Fourier series for Lp-computable functions and show that whereas for every p > 1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager. (shrink)

This paper investigates aspects of measure and randomness in the context of locale theory . We prove that every measure μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X . Furthermore, when μ is a finite measure with μ=M, the σ-locale X has a smallest σ-sublocale of measure M . In particular, when μ is a probability measure, X has a smallest σ-sublocale of measure 1. All (...) σ prefixes can be dropped from these statements whenever X is a strongly Lindelöf locale, as is the case in the following applications. When μ is the Lebesgue measure on the Euclidean space Rn, Theorem 1 produces an isometry-invariant measure that, via the inclusion of the powerset P in the lattice of sublocales, assigns a weight to every subset of Rn. When μ is the uniform probability measure on Cantor space {0,1}ω, the smallest measure-1 sublocale, given by Theorem 2, provides a canonical locale of random sequences, where randomness means that all probabilistic laws are satisfied. (shrink)

We prove that the property "P doesn't make the old reals Lebesgue null" is preserved under countable support iterations of proper forcings, under the additional assumption that the forcings are nep (a generalization of Suslin proper) in an absolute way. We also give some results for general Suslin ccc ideals.

A theory of approximation to measurable sets and measurable functions based on the concepts of recursion theory and discrete complexity theory is developed. The approximation method uses a model of oracle Turing machines, and so the computational complexity may be defined in a natural way. This complexity measure may be viewed as a formulation of the average-case complexity of real functions—in contrast to the more restrictive worst-case complexity. The relationship between these two complexity measures is further studied and compared with (...) the notion of the distribution-free probabilistic computation. The computational complexity of the Lebesgue integral of polynomial-time approximable functions is studied and related to the question “FP = ♯P?”. (shrink)

We introduce the notion of effective Axiom A and use it to show that some popular tree forcings are Suslin+. We introduce transitive nep and present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is a Suslin ccc ideal (e.g. Lebesgue-null or meager) and P is a transitive nep forcing (e.g. P is Suslin+) and P does not make any I-positive Borel set small, then P does not make any I-positive set small.

There is a growing body of writing, for instance from the nursing profession, espousing an approach to ethics based on care. I suggest that this approach is hopelessly vague and that the vagueness is due to an inadequate analysis of the concept of care. An analysis of 'care' and related terms suggests that care is morally neutral. Caring is not good in itself, but only when it is for the right things and expressed in the right way. 'Caring' ethics assumes (...) wrongly that caring is good, thus it can tell us neither what constitutes those right things, nor what constitutes the right way. (shrink)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras, where * is a continuous t-norm. In this paper we investigate the structure of the (...) variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

Gyenis and Rédei (G&R) have shown that any prior _p_ on a finite algebra _A_, however chosen, significantly restricts the set of posteriors derivable from _p_ by Jeffrey conditioning (JC) on a nontrivial measurable partition (i.e., a partition consisting of members of _A_, at least one of which is not an atom of _A_). They support this claim by proving that the set of potential posteriors _not derivable_ from _p_ in this way, which they call the _Bayes blind spot of (...) p_, is large, having cardinality _c_ and normalized Lebesgue measure 1, as well as being of second Baire category for a natural metrizable topology. In the present paper, we establish results analogous to those of G&R for probability measures on any infinite sigma algebra of subsets of a countably infinite set (which requires distinctly different treatments of the topological and measure-theoretic cases). We also show, in both the finite and infinite cases, that all of the limitative results for a single prior _p_ continue to hold for the intersection of the Bayes blind spots of countably many priors. This leads us to reject the claim of G&R that the large size of blind spots in the single prior case is attributable to the limitations imposed by priors. We argue instead that it is the so-called _rigidity property_ of JC that accounts for the large size of Bayes blind spots. G&R also prove that _any_ potential posterior _q_ can be derived from a prior _p_ by at most two applications of JC on nontrivial partitions. But they remark that their particular two-stage derivation of an \(r \in BS(p)\) from _p_ is barely distinguishable from a single, complete reassessment on a trivial partition. We show, however, that there are two-stage derivations of _r_ from _p_ that require no direct assessment of the probability of any atom at either stage. Finally, in order to situate the aforementioned limitative results in the proper context, we demonstrate that a probability revision that would amount to complete reassessment within a Jeffrey evidentiary framework may arise, in a different evidentiary framework, in a way that involves no direct assessment of the probability of any atom, as, for example, in a generalization of Jeffrey’s solution to the problem of old evidence and new explanation. (shrink)

Let ω be a Kolmogorov–Chaitin random sequence with ω1: n denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω1: n )} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the (...) first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail. (shrink)

The text of Euripides' Troades depends mainly upon the versions preserved in two manuscripts, viz. Palatinus 287 and Vaticanus 909 . Both these manuscripts are well known and have been several times collated. Their importance in relation to one another and to the other existing manuscripts for the study of the text of Euripides in general has been frequently discussed and is no longer a subject of major controversy. The notes which follow are designed to put on record a number (...) of corrections to the readings ascribed to P and V for the Troades in the apparatus criticus of Prinz-Wecklein and later editors; and to correct and amplify the general descriptions of P and to call attention to certain difficulties in the current views of its relationship to Laurentianus xxxii. (shrink)

Psychoanalysis had a profound influence on formation of Merleau-Ponty’s thought. However, at the same time, he rejects Freud’s idea that the unconscious consists of latent mental contents that cause a certain type of behavior. Instead of a hidden experience, Merleau-Ponty argues that the unconscious is an ambiguous consciousness. In The Structure of Behavior and The Phenomenology of Perception, he specifies this ambiguity by means of the concepts of habit, bad faith, bodily expression, affective intentionality and body schema. In this paper, (...) I will present the interconnection of these aspects of the human existence, following Merleau-Ponty’s two early major works. Further, I will show the difference of Merleau-Ponty’s notion of bad faith from that of Sartre, and, finally, I will suggest a limitation of Merleau-Ponty’s approach to the unconscious. (shrink)

A questionnaire on the attitudes towards the functions of research ethical committees was sent to members of selected research ethical committees in Wessex and some controls. Almost all respondents felt there was a need for ethical review of research projects; 42 per cent thought there was a need for some training before joining a committee; 67 per cent thought the system could be improved and 47 per cent thought that monitoring or follow-up procedures should be adopted. Ethical committees were thought (...) to be purely advisory, as opposed to mandatory, by 33 per cent, and 63 per cent thought they should restrict their review to ethical problems as opposed to scientific or design problems. Views about the function of non-medical members ranged from 'none at all' to 'very important'. Of the 10 controls who were asked whether they would become a member of an ethical committee if asked, seven said that on balance they would and the reasons stated varied from the view that it was a 'very important committee' to the feeling that it was 'a necessary but irksome job'. (shrink)

My original paper suggested that an ethics of care which failed to specify how, and about what, to care would be devoid of normative and descriptive content. Bradshaw's approach provides such a specification and is, therefore, not devoid of such content. However, as all ethical approaches suggest something about the 'what' and 'how' of care, they are all 'ethics of care' in this broader sense. This reinforces rather than undermines my original conclusion. Furthermore, Bradshaw's 'ethics of care' has philosophical and (...) historical problems which I outline. (shrink)

We describe and analyse how large herbivores strongly diminished a woody vegetation, dominated by the unpalatable shrub Sambucus nigra L. and changed it into grassland. Density of woody species and cover of vegetation were measured in 1996, 2002 and 2012 in the grazed Oostvaardersplassen. In 2002 and 2012 we also measured density and cover in an ungrazed control site. In 2002 we measured intensity of browsing and bark loss of Sambucus shrubs in the grazed and control sites. In the grazed (...) site the density of Sambucus and Salix spp. declined significantly between 1996 and 2012, and large areas changed into grassland. In the control site the density of Sambucus increased significantly during this period, the density of Salix spp. did not change, and the vegetation consisted of a mixture of woody species and a field layer dominated by tall herbs. In 2002 and 2012 the percentages of dead Sambucus shrubs were significantly higher in the grazed site than in the control site. In 2002 the percentages of twigs browsed and ring barked stems of Sambucus shrubs were significantly higher in the grazed site than in the control site. Our results show that debarking caused mature Sambucus shrubs to die, but that heavy browsing may have helped this process. Our results also point to a significant neighbour effect on the break down of Sambucus, suggesting that Aggregational Resistance and Associational Palatability were both active. Essential conditions for the break down of this woody vegetation were the presence of large herbivores, the low ratio between the areas of summer and winter feeding habitats and the competition amongst herbivores. Browsing may have been responsible for seedling death, as seedlings were found only in the control site and not on the old and newly established grasslands in the grazed site. (shrink)

Husserl's investigations of internal time-consciousness take sound as the primary temporal object. However, in these investigations, the structure of the flux of temporal subjectivity is established to the detriment of the rich tonal content of sound. Just as Husserl has enlarged the significance of the spatial object of mathematical physics to include the historically-sedimented layers of its appearance, so the temporal object will receive additional intelligibility if the rich texture of musical sound is taken into consideration. Particularly useful for this (...) task is Bergson's philosophy of the listening experience. (shrink)

It is consistent that there exists a set mapping $F: \lbrack\omega_2\rbrack^2 \rightarrow \lbrack\omega_2\rbrack^{<\omega}$ such that $F(\alpha, \beta) \subseteq \alpha$ for $\alpha < \beta < \omega_2$ and there is no infinite free subset for F. This solves a problem of A. Hajnal and A. Mate.

Originally published in 1960, The Future of Man is a chronicle of Professor Medwar's Reith lectures of 1959. The book outlines his predictions about the future estate of man, with the 'process of foretelling, rather than with what is actually foretold'. He asks, can we predict the future size of populations? What is the evidence and theoretical background for the belief that human intelligence is declining? Could human beings become uniformly excellent or is inborn diversity and inequality a necessary part (...) of the texture of human populations? The lectures tried to answer these questions and attempts to end with a definition of the biological standing of man. This book will be of interest to anthropologists, biologists and natural historians. (shrink)