An alternative proof is given that to each LLM proof net there corresponds at least one LLM sequent proof. The construction is inspired by the sweepline technique from computational geometry and includes a treatment of the multiplicative constants and of proof boxes.

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: there can be nonmeasurable events that have no probability at all and on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice. We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient to prove (...) the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory. (shrink)

I present solutions to several questions of Paul Bankston [2] by means of another version of the ultracoproduct construction, and explain the relation of ultracoproduct of compact Hausdorff spaces to other constructions combining topology, algebra and logic.

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the (...) Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely (...) universal and purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable. (shrink)

We prove the existence of p-ideals that are nonmeagre subsets of P(ω) under various set-theoretic assumptions.

We give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let X1, . . . , Xn be independent Banach-valued random variables. Let I be a random variable independent of X1, . . . , Xn and uniformly distributed over {1, . . . , n}. Put ˜.

Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets $A \subseteq \mathbf{R}$ is the game determined????? Rules: The two players alternate moves starting with player I. Each move a n is legal iff it is a real number and $0 , and for $n > 1, a_n . The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff ∑ a n (...) exists and ∑ a n ∈ A. We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinancy and the determinancy of other well-known and much-studied games. (shrink)

We investigate Borel reducibility between equivalence relations $E(X;p)=X^{\mathbb{N}}/\ell_{p}(X)'s$ where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(L r ; p)'s and E(c 0 ; p)'s for r, p ∈ [1, +∞). We also answer a problem presented by Kanovei in the affirmative by (...) showing that $\mathrm{C}\left({\mathrm{\mathbb{R}}}^{+}\right)/{\mathrm{C}}_{0}\left({\mathrm{\mathbb{R}}}^{+}\ri ght)$ is Borel bireducible to $\mathbb{R}^{\mathbb{N}}/c_{0}$. (shrink)

R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that $\lambda , where α * is the Σ 1 -projectum of α and λ is the Σ 2 -cofinality of α. The theorem is also established for the metarecursive case, α = ω 1 , where (...) α * = λ = ω. (shrink)

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing Ψ − h, r -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results.

R.I.G. Hughes presents a series of eight philosophical essays on the theoretical practices of physics. The first two essays examine these practices as they appear in physicists' treatises (e.g. Newton's Principia and Opticks ) and journal articles (by Einstein, Bohm and Pines, Aharonov and Bohm). By treating these publications as texts, Hughes casts the philosopher of science in the role of critic. This premise guides the following 6 essays which deal with various concerns of philosophy of physics such as laws, (...) disunities, models and representation, computer simulation, explanation, and the discourse of physics. (shrink)

Sur la décomposition des ensembles de points en parties respectivement congruentes.

R.I.G Hughes offers the first detailed and accessible analysis of the Hilbert-space models used in quantum theory and explains why they are so successful.

This book provides a collection of sources, many of them fragmentary and previously scattered and hard to access, for the development of Peripatetic philosophy in the later Hellenistic period and the early Roman Empire. It also supplies the background against which the first commentator on Aristotle from whom extensive material survives, Alexander of Aphrodisias (fl. c. AD 200), developed his interpretations which continue to be influential even today. Many of the passages are here translated into English for the first time, (...) including the whole of the summary of Peripatetic ethics attributed to 'Arius Didymus'. (shrink)

Reason and Value collects 15 new papers by leading contemporary philosophers on themes from the work of Joseph Raz. Raz has made major contributions in a wide range of areas, including jurisprudence, political philosophy, and the theory of practical reason; but all of his work displays a deep engagement with central themes in moral philosophy. The subtlety and power of Raz's reflections on ethical topics make his writings a fertile source for anyone working in this area. Especially significant are his (...) explorations of the connections between practical reason and the theory of value, which constitute a sustained and penetrating treatment of a set of issues at the very center of moral philosophy as it is practiced today. The contributors to the volume acknowledge the importance of Raz's contributions by engaging critically with his positions and offering independent perspectives on the topics that he has addressed. The volume aims both to honour Raz's accomplishments in the area of ethical theorizing, and to contribute to an enhanced appreciation of the significance of his work for the subject. (shrink)

The Nature of God explores a perennial problem in the philosophy of religion.

In this first paperback edition, a new introduction by Tom Burke establishes the ongoing importance of Sleeper's analysis of the integrity of Dewey's work and ...

This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.

This book is not a study of Plato's philosophy, but a contribution to the literary interpretation of the dialogues, through analysis of their formal structure, ...

The author demonstrates that certain forms of irrationality - incontinent action and self-deception - which many philosophers have rejected as being logically or psychologically impossible, are indeed possible.

Many philosophers believe that there is a causal condition on perception, and that this condition is a conceptual truth about perception. A highly influential argument for this claim is based on intuitive responses to Gricean style thought experiments. Do the folk share the intuitions of philosophers? Roberts et al. (2016) presented participants with two kinds of cases: Blocker cases (similar to Grice’s case involving a mirror and a pillar) and Non-Blocker cases (similar to Grice’s case involving a clock and brain (...) stimulation). They found that a substantial minority agreed that seeing occurs in the Non-Blocker cases, and that in the Blocker cases significantly less agreed that seeing occurs. They thus hypothesized that folk intuitions better align with a no blocker condition than with a causal condition. This paper continues this line of enquiry with two new experiments. The paper investigates the generality and robustness of Roberts et al.’s findings by expanding the sense modalities tested from only vision to audition and olfaction as well. The paper also uses Gricean style thought experiments as a case study for investigating the “reflection defense” against the negative project in experimental philosophy. Our results replicate and extend Roberts et al.’s study and support their hypothesis that folk intuitions better align with a no blocker condition. They also provide an empirical reason to doubt the reflection defense. (shrink)

In this, the first of two ground-breaking volumes on the nature of language in the light of the way it evolved, James Hurford looks at how the world first came ...

The Hellenistic philosophers and schools of philosophy are emerging from the shadow of Plato and Aristotle and are increasingly studied for their intrinsic philosophical value. They are not only interesting in their own right, but also form the intellectual background of the late Roman Republic. This study gives a comprehensive and readable account of the principal doctrines of the Stoics, Epicureans and various sceptical traditions from the death of Alexander the Great in 323 B.C. to around 200 A.D. Discussions are (...) arranged topically in order to address underlying issues and to make clear what the different schools have in common and how they differ. At the same time the coherence of each system as a whole is emphasized. (shrink)

While it might seem prima facie plausible that divine foreknowledge is all that is needed for prophecy, this seems incorrect. To issue a prophecy, God hasto know not just how someone will act, but how someone would act were the prophecy issued. This makes some think that Middle Knowledge is required.I argue that Thomas Flint’s two Middle Knowledge based accounts of prophecy are unsatisfactory, but one of them can be repaired. However the resources needed for repair also yield a sketch (...) of a foreknowledge-only account of prophecy. (shrink)