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)

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)

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)

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.

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 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.

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)

Advocates have sought to prove that underdetermination obtains because all theories have empirical equivalents. But algorithms for generating empirical equivalents simply exchange underdetermination for familiar philosophical chestnuts, while the few convincing examples of empirical equivalents will not support the desired sweeping conclusions. Nonetheless, underdetermination does not depend on empirical equivalents: our warrant for current theories is equally undermined by presently unconceived alternatives as well-confirmed merely by the existing evidence, so long as this transient predicament recurs for each theory and body (...) of evidence we consider. The historical record supports the claim that this recurrent, transient underdetermination predicament is our own. (shrink)

It is fortunate for my purposes that English has the two words ‘almighty’ and ‘omnipotent’, and that apart from any stipulation by me the words have rather different associations and suggestions. ‘Almighty’ is the familiar word that comes in the creeds of the Church; ‘omnipotent’ is at home rather in formal theological discussions and controversies, e.g. about miracles and about the problem of evil. ‘Almighty’ derives by way of Latin ‘omnipotens’ from the Greek word ‘pantokratōr’; and both this Greek word, (...) like the more classical ‘pankratēs’, and ‘almighty’ itself suggest God's having power over all things. On the other hand the English word ‘omnipotent’ would ordinarily be taken to imply ability to do everything; the Latin word ‘omnipotens’ also predominantly has this meaning in Scholastic writers, even though in origin it is a Latinization of ‘pantocratōr’. So there already is a tendency to distinguish the two words; and in this paper I shall make the distinction a strict one. I shall use the word ‘almighty’ to express God's power over all things, and I shall take ‘omnipotence’ to mean ability to do everything. (shrink)

By the time of his death in 2006, Sir Peter Strawson was regarded as one of the world's most distinguished philosophers. Unavailable for many years,_ Scepticism and Naturalism_ is a profound reflection on two classic philosophical problems by a philosopher at the pinnacle of his career. Based on his acclaimed Woodbridge lectures delivered at Columbia University in 1983, Strawson begins with a discussion of scepticism, which he defines as questioning the adequacy of our grounds for holding various beliefs. He then (...) draws deftly on Hume and Wittgenstein to argue that we must distinguish between 'hard', scientific naturalism; or 'soft', humanistic naturalism. In the remaining chapters the author takes up several issues in which sceptical doubts play an important role, in particular the nature of transcendental arguments and including the objectivity of moral philosophy, the mental and the physical, and the existence of abstract entities. _Scepticism and Naturalism_ is essential reading for those seeking an introduction to the work of one of the twentieth century’s most important and original philosophers. This reissue includes a substantial new foreword by Quassim Cassam and a fascinating intellectual autobiography by Strawson, which together form an excellent introduction to his life and work. (shrink)

This fourth and final volume of the monumental commentary on Wittgenstein's _Philosophical Investigations_ covers pp 428-693 of the book. Like the previous volumes, it consists of philosophical essays and exegesis.

First published in 1987. Routledge is an imprint of Taylor & Francis, an informa company.

Symmetries in physics are most commonly recognized and discussed in terms of their function in the mathematical formalism of the theories. Discussion of the observation of symmetries in nature is less common. This paper analyses the observation of particular symmetries such as Lorentz and gauge symmetries, distinguishing between direct observation of the symmetry itself and indirect evidence, the latter being the observation of some consequence of the symmetry are, in an important sense, directly observed, while local symmetries such as gauge (...) symmetry are not. (shrink)

James Conant, a proponent of the ‘New American Wittgenstein’, has argued that the standard inter- pretation of Wittgenstein is wholly mistaken in respect of Wittgenstein’s critique of metaphysics and the attendant conception of nonsense. The standard interpretation, Conant holds, misascribes to Wittgenstein Carnapian views on the illegitimacy of metaphysical utterances, on logical syntax and grammar, and on the nature of nonsense. Against this account, I argue that (i) Carnap is misrepresented; (ii) the so-called standard interpretation (in so far as I (...) have contributed to it) is misrepresented; (iii) Wittgenstein’s views, early and late, are misrepresented. I clarify Wittgen- stein’s conception of logical syntax and of the nonsense that results from transgressing it. (shrink)

Hume's 'Natural History of Religion' offers a naturalized account of the causes of religious thought, an investigation into its 'origins' rather than its 'foundation in reason'. Hume thinks that if we consider only the causes of religious belief, we are provided with a reason to suspend the belief. I seek to explain why this is so, and what role the argument plays in Hume's wider campaign against the rational acceptability of religious belief. In particular, I argue that the work threatens (...) a form of fideism which maintains that it is rationally permissible to maintain religious belief in the absence of evidence or of arguments in its favour. I also discuss the 'argument from common consent', and the relative superiority of Hume's account of the origins of religious belief. (shrink)

By the time of his death in 2006, Sir Peter Strawson was regarded as one of the world's most distinguished philosophers. Unavailable for many years,_ Scepticism and Naturalism_ is a profound reflection on two classic philosophical problems by a philosopher at the pinnacle of his career. Based on his acclaimed Woodbridge lectures delivered at Columbia University in 1983, Strawson begins with a discussion of scepticism, which he defines as questioning the adequacy of our grounds for holding various beliefs. He then (...) draws deftly on Hume and Wittgenstein to argue that we must distinguish between 'hard', scientific naturalism; or 'soft', humanistic naturalism. In the remaining chapters the author takes up several issues in which sceptical doubts play an important role, in particular the nature of transcendental arguments and including the objectivity of moral philosophy, the mental and the physical, and the existence of abstract entities. _Scepticism and Naturalism_ is essential reading for those seeking an introduction to the work of one of the twentieth century’s most important and original philosophers. This reissue includes a substantial new foreword by Quassim Cassam and a fascinating intellectual autobiography by Strawson, which together form an excellent introduction to his life and work. (shrink)

According to many philosophers, psychological explanation canlegitimately be given in terms of belief and desire, but not in termsof knowledge. To explain why someone does what they do (so the common wisdom holds) you can appeal to what they think or what they want, but not what they know. Timothy Williamson has recently argued against this view. Knowledge, Williamson insists, plays an essential role in ordinary psychological explanation.Williamson's argument works on two fronts.First, he argues against the claim that, unlike knowledge, (...) belief is``composite'' (representable as a conjunction of a narrow and a broadcondition). Belief's failure to be composite, Williamson thinks, undermines the usual motivations for psychological explanation in terms of belief rather than knowledge.Unfortunately, we claim, the motivations Williamson argues against donot depend on the claim that belief is composite, so what he saysleaves the case for a psychology of belief unscathed.Second, Williamson argues that knowledge can sometimes provide abetter explanation of action than belief can.We argue that, in the cases considered, explanations that cite beliefs(but not knowledge) are no less successful than explanations that citeknowledge. Thus, we conclude that Williamson's arguments fail both coming andgoing: they fail to undermine a psychology of belief, and they fail tomotivate a psychology of knowledge. (shrink)

We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set. We give a simplified account of varied revision sequences-as a generalised algorithmic theory of truth. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.

This paper offers a general characterization of underdetermination and gives a prima facie case for the underdetermination of the topology of the universe. A survey of several philosophical approaches to the problem fails to resolve the issue: the case involves the possibility of massive reduplication, but Strawson on massive reduplication provides no help here; it is not obvious that any of the rival theories are to be preferred on grounds of simplicity; and the usual talk of empirically equivalent theories misses (...) the point entirely. (If the choice is underdetermined, then the theories are not empirically equivalent!) Yet the thought experiment is analogous to a live scientific possibility, and actual astronomy faces underdetermination of this kind. This paper concludes by suggesting how the matter can be resolved, either by localizing the underdetermination or by defeating it entirely. Introduction A brief preliminary Around the universe in 80 days Some attempts at resolving the problem 4.1 Indexicality 4.2 Simplicity 4.3 Empirical equivalence 4.4 Is this just a philosophers' fantasy? Move along... ...nothing to see here 6.1 Rules of repetition 6.2 Some possible replies Conclusion. (shrink)

We develop and apply a multi-dimensional conception of explanatory depth towards a comparative analysis of inflationary and bouncing paradigms in primordial cosmology. Our analysis builds on earlier work due to Azhar and Loeb (2021) that establishes initial condition fine-tuning as a dimension of explanatory depth relevant to debates in contemporary cosmology. We propose dynamical fine-tuning and autonomy as two further dimensions of depth in the context of problems with instability and trans-Planckian modes that afflict bouncing and inflationary approaches respectively. In (...) the context of the latter issue, we argue that the recently formulated trans-Planckian censorship conjecture leads to a trade-off for inflationary models between dynamical fine-tuning and autonomy. We conclude with the suggestion that explanatory preference with regard to the different dimensions of depth is best understood in terms of differing attitudes towards heuristics for future model building. (shrink)

We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise the revision theoretically definable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta’s theory of truth over arithmetic using \emph{fully varied revision} sequences yields a complete \Pi13 set of integers; (iii) the set of \emph{stably categorical} sentences using their revision operator ψ is similarly \Pi13 and which (...) is complete in Gödel’s universe of constructible sets L; (iv) give an alternative account of a theory of truth—realistic variance that simplifies full variance, whilst at the same time arriving at Kripkean fixed points. (shrink)

This paper discusses the metaphor of projection in relation to Hume’s treatment of causal necessity. I argue that the best understanding of projection shows it to be compatible with taking Hume to be a ‘sceptical realist’ about causal necessity, albeit an agnostic one.

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated (...) with the jump operator satisfy a Spector criterion, and correspond to the L ζ -stables. It also implies that the machines devised are "Σ 2 Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. (shrink)