We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Rios Insua and Alon and Lehrer is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott, relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.

We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is${\rm{\Delta }}_\alpha ^0 $-complete for someα. To prove this, we extend Montalbán’sη-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinalαand a cone in the Turing degrees such that the exact complexity (...) of computing an isomorphism between the given structure and another copy${\cal B}$in the cone is a c.e. degree in${\rm{\Delta }}_\alpha ^0\left$. In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions. (shrink)

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma (...) }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel. (shrink)

We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{}$-computable Polish space; this answers a question of Nies. (...) We also prove analogous results for compact Polish groups and locally path-connected spaces. (shrink)

Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs (...) where possible, followed by a survey of recent advances. (shrink)

This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.

We define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$, $\Pi _{\alpha }$, or $\mathrm {d-}\Sigma _{\alpha }$. We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit (...) ordinal. This answers a question left open by A. Miller.We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$. There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$, $\Pi _{\omega _1^{CK}+1}$, $\Sigma _{\omega _1^{CK}+1}$. Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$. The existence of such structures was an open question. (shrink)

A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a noncomputable set A and a computable instance of a problem ${\mathsf {P}}$, to find a solution relative to which A (...) is still noncomputable.In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of one cone, of $\omega $ cones, of one hyperimmunity or of one non- $\Sigma ^{0}_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non- $\Sigma ^{0}_1$ definitions coincide. On the other hand, none of these notions coincide in a nonrelativized setting. (shrink)

Universality has been an important concept in computable structure theory. A class C of structures is universal if, informally, for any structure of any kind there is a structure in C with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being (...) universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences, and also answer a question of Alvir, Knight, and McCoy. (shrink)

This article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, (...) we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information. (shrink)

The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation ≥ on events that does not preclude a probabilistic interpretation, in the sense that ≥ has extensions that are probabilistically representable, we characterize the extension ≥+ of ≥ that is exactly the intersection of all probabilistically representable extensions of (...) ≥. This extension ≥+ gives us all the additional comparisons that we are entitled to infer from ≥, based on the assumption that there is some probability measure of which ≥ gives us partial qualitative information. We pay special attention to the problem of extending an order on states to an order on events. In addition to the probabilistic interpretation, this problem has a more general interpretation involving measurement of any additive quantity: e.g., given comparisons between the weights of individual objects, what comparisons between the weights of groups of objects can we infer? (shrink)

A computable structure [Formula: see text] is decidable if, given a formula [Formula: see text] of elementary first-order logic, and a tuple [Formula: see text], we have a decision procedure to decide whether [Formula: see text] holds of [Formula: see text]. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is [Formula: see text]-complete. We also show that for each [Formula: see text] (...) the index set of the computable structures with [Formula: see text]-decidable presentations is [Formula: see text]-complete. (shrink)

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is (...) sufficient to find such an embedding. By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal’cev property, we show that they have computable dimension \. (shrink)

We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\varphi $ (in $\mathcal {L}_{\omega,\omega }$ ) is equivalent to a formula of the infinitary language $\mathcal {L}_{\infty,\omega }$ with n alternations of quantifiers. We prove that $\varphi $ is equivalent to a finitary formula with n alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a (...) simpler way. (shrink)

Our main result is that there exist structures which cannot be computably recovered from their tree of tuples. This implies that there are structures with no computable copies which nevertheless cannot code any information in a natural/functorial way.

An oracle A is low-for-speed if it is unable to speed up the computation of a set which is already computable: if a decidable language can be decided in time $t(n)$ using A as an oracle, then it can be decided without an oracle in time $p(t(n))$ for some polynomial p. The existence of a set which is low-for-speed was first shown by Bayer and Slaman who constructed a non-computable computably enumerable set which is low-for-speed. In this paper we answer (...) a question previously raised by Bienvenu and Downey, who asked whether there is a minimal degree which is low-for-speed. The standard method of constructing a set of minimal degree via forcing is incompatible with making the set low-for-speed; but we are able to use an interesting new combination of forcing and full approximation to construct a set which is both of minimal degree and low-for-speed. (shrink)

We say that a theory [Formula: see text] satisfies arithmetic-is-recursive if any [Formula: see text]-computable model of [Formula: see text] has an [Formula: see text]-computable copy; that is, the models of [Formula: see text] satisfy a sort of jump inversion. We give an example of a theory satisfying arithmetic-is-recursive non-trivially and prove that the theories satisfying arithmetic-is-recursive on a cone are exactly those theories with countably many [Formula: see text]-back-and-forth types.

We say that a theory T satisfies arithmetic-is-recursive if any X′-computable model of T has an X-computable copy; that is, the models of T satisfy a sort of jump inversion. We give an example of a...

The $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $, a universal prefix-free machine U whose halting probability is $\alpha $. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $, one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. (...) nor right-c.e. (shrink)

When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and ${\mathcal {L}}_{\omega _1, \omega }$ -elementary? We find that these are exactly the classes (...) that can be defined by an infinitary formula that has no infinitary disjunctions. (shrink)

Recent advances in wireless data transmission technology have the potential to revolutionize clinical neuroscience. Today sensing-capable electrical stimulators, known as “bidirectional devices”, are used to acquire chronic brain activity from humans in natural environments. However, with wireless transmission come potential failures in data transmission, and not all available devices correctly account for missing data or provide precise timing for when data losses occur. Our inability to precisely reconstruct time-domain neural signals makes it difficult to apply subsequent neural signal processing techniques (...) and analyses. Here, our goal was to accurately reconstruct time-domain neural signals impacted by data loss during wireless transmission. Towards this end, we developed a method termed Periodic Estimation of Lost Packets. PELP leverages the highly periodic nature of stimulation artifacts to precisely determine when data losses occur. Using simulated stimulation waveforms added to human EEG data, we show that PELP is robust to a range of stimulation waveforms and noise characteristics. Then, we applied PELP to local field potential recordings collected using an implantable, bidirectional DBS platform operating at various telemetry bandwidths. By effectively accounting for the timing of missing data, PELP enables the analysis of neural time series data collected via wireless transmission—a prerequisite for better understanding the brain-behavior relationships underlying neurological and psychiatric disorders. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Theologian's Doubts:Natural Philosophy and the Skeptical Games of GhazālīLeor HaleviIn the history of skeptical thought, which normally leaps from the Pyrrhonists to the rediscovery of Sextus Empiricus in the sixteenth century, Abū Ḥāmid Muḥammad al-Ghazālī (1058-1111) figures as a medieval curiosity. Skeptical enough to merit passing acknowledgment, he has proven too baffling to be treated fully alongside pagan, atheist, or materialist philosophers. As a theologian defending certain Muslim (...) dogmas, Ghazālī has not met what historians consider the mark of the true skeptic, a mind doubting the possibility of all systems of knowledge. But what is fascinating about him is that he brought into practical operation the tools of what I call "functional skepticism."1He denied the claims to truth of Aristotelian physics—whose basis he showed to rest on groundless belief—then turned and argued for the possibility of the Resurrection tooth and nail. The scholarly debate on The Incoherence of the Philosophers (Tahāfut al-falāsifa) has concentrated on the extent to which Ghazālī qua Ashcarite theologian was seduced into Aristotelian philosophy despite himself.2 In my view this debate has been misguided in the attempt to distill the [End Page 19] essence of Ghazālī from the book's eclectic theology; I will argue for a different view of Ghazālī on the basis of a close reading of key passages. In the unusual sections where Ghazālī applies Aristotelian language to a world not following the ordinary laws of physics, some have found Ghazālī slipping, unconsciously perhaps, into an Aristotelian frame of mind. I will show that, as a skeptical theologian with a dialogic imagination, he was rather deconstructing Aristotelian discourse while playing a Wittgensteinian sort of language game.Natural Philosopher or Speculative Theologian?The disagreement about the extent to which philosophy infected Ghazālī is ancient. Ghazālī might have studied philosophy only in order to refute it. He himself defended his philosophizing with the claim that one cannot deconstruct a system of thought until one has understood it so deeply as to elaborate upon its fundamental principles.3 His Maqāṣid al-falāsifa was in fact received, especially in trans-Pyrenean Europe, as a philosopher's genuine summary of the object of philosophy.4 The book strikes me as suspiciously creative in its representation of philosophical discourse, but it appears in any case as an expert and surprisingly unbiased treatment.5 Arabic readers knew that Ghazālī had also written a polemical treatise against philosophy, Tahāfut al-falāsifa, but they still wondered about his engagement with the ideas he challenged. Abū Bakr Ibn al-cArabī, for example, commented that Ghazālī had been unable to extricate himself from philosophy.6 Other philosophers pondered whether or not he had been a closeted member of their charmed circle and sought in his writings traces of esoteric philosophy.7Averroës's own sober sense of distance between philosophy and theology was partially a reaction to what he perceived as Ghazālī's dangerous and carefree mixture of the two sciences.8 He attacked Ghazālī's book in The Incoherence [End Page 20] of the Incoherence to restore philosophy's sense of purity, an aim he sought to accomplish by separating religious concerns from the philosopher's field of inquiry.9 Ironically, such a separation is precisely what Ghazālī might have wished to provoke by crisscrossing and blurring the line between religion and philosophy.The modern debate on chapter 17 of Tahāfut al-falāsifa has concentrated on defining Ghazālī as either a natural philosopher or an occasionalist theologian. In his defense of the possibility of miracles Ghazālī presented two theories of causation, one denying the logical basis of Aristotelian notions of natural causality, and the other more or less adopting these notions. Jointly, the two theories have seemed incompatible, and for this reason scholars have attempted to sort Ghazālī out of the apparent confusion. In 1978 L. E. Goodman argued persuasively that Ghazālī exploited rather than denied the philosophers' ideas of causality. In two articles Michael Marmura challenged Goodman, contending... (shrink)

Historians of modern design and sociology have shown little interest in the leaders of the ever resourceful and influential British Positivist Society. One of the aims of this essay is to show that the Positivist polymath Frederic Harrison (1831–1923) cultivated ideas and practices that are compatible with modernists’ aspirations to improve the lives of the masses. It is accordingly shown that Harrison was an ardent supporter of working-class causes and that on this basis he developed sociological survey methods (...) and produced social programmes to initiate the comprehensive reorganisation of cities. Harrison intended to realise a modern utopia called the “Occidental Republic”, created by Auguste Comte, the Positivist Philosopher who – during the 1830 and 1840s – introduced both sociology and the Religion of Humanity. While many studies pore over the Positivists’ “church” rituals, this essay is the first to argue that the Religion of Humanity and sociology were of equal consequence – and that together they formed the basis of their controversial spatial theory and practice, which was considered the means to realise Comte’s utopia. This vision, it is argued, is central to appreciating how Positivism percolated into the “modern movements” of architecture and planning. (shrink)

The book draws heavily on unparalleled access to the archives of Astrid Kirchherr and includes photographs of John Lennon, Paul McCartney, George Harrison and Kirchherr's former fiance, Stuart Sutcliffe, as well as other key protagonists in ...

This paper examines the apology for the life of the mind Francis Bacon gives in Book I of his 1605 text The Advancement of Learning. Like recent work on Bacon led by the ground-breaking studies of Corneanu, Harrison and Gaukroger, I argue that Bacon’s conception and defence of intellectual inquiry in this extraordinary text is framed by reference to the classical model, which had conceived and justified philosophising as a way of life or means to the care of the (...) inquirer’s soul or psyche. In particular, Bacon’s proximities and debts to the Platonic Apology and Cicero’s defence of intellectual pursuits in Rome are stressed, alongside the acuity and eloquence of Bacon’s descriptions of the intellectual virtues and their advertised contributions to the theologically and civically virtuous life. (shrink)

"In this study, James R. Harrison compares the modern cult of celebrity to the quest for glory in late republican and early imperial society. He shows how Paul's ethic of humility, based upon the crucified Christ, stands out in a world obsessed with mutual comparison, boasting, and self-sufficiency." --.

The notion of presentational phenomenology has powerful epistemological implications. According to Elijah Chudnoff, an experience has presentational phenomenology with respect to p insofar as that experience makes it seem to you that p, and makes it seem as if you are aware of a truth-maker for p. Chudnoff argues that only experiences that have presentational phenomenology with respect to p provide immediate prima facie justification to the belief that p. That is, my visual experience of the orange provides me with (...) defeasible justificatory grounds for believing that there is an orange by virtue of that experience possessing presentational phenomenology. Call this epistemological thesis presentationalism. The central purpose of this paper is to provide a novel analysis of presentational phenomenology, and determine whether it is capable of delivering the epistemic yield detailed above. Here, I endeavour to show the following: presentational phenomenology is neither sufficient nor necessary for immediate prima facie justification, and ambiguity in Chudnoff’s analysis of presentational phenomenology leads the account to a troubling dilemma. Seemings and seeming awareness of truth-makers are not plausible candidates for justification-conferring states. (shrink)

The first edition of The Scientific Background to Modern Philosophy took the dialogue of science and philosophy from Aristotle through to Newton. This second edition adds eight chapters, taking the dialogue through the Enlightenment and up to Darwin. This anthology is an attempt to help bridge the gap between the history of science and the history of philosophy.

Imagine I hold up a Granny Smith apple for all to see. You would thereby gain justified beliefs that it was green, that it was apple, and that it is a Granny Smith apple. Under classical foundationalism, such simple visual beliefs are mediately justified on the basis of reasons concerning your experience. Under dogmatism, some or all of these beliefs are justified immediately by your experience and not by reasons you possess. This paper argues for what I call the looks (...) view of the justification of simple visual beliefs. According to the looks view, such beliefs are mediately justified on the basis of reasons concerning how the relevant things look. Unlike under classical foundationalism, under the looks view as I develop it, these reasons are public. They are public with respect to both their content and possession: with respect to content, they are not about ourselves and our experiences, and with respect to their possession, many people can have the very same looks-related reasons. (shrink)

PREFATORY NOTE T may be well at the outset to say clearly what is the aim of the present volume. The title is Ancient Art and Ritual, but the reader will ...

ASHÉ: Ritual Poetics in African Diasporic is a collection of interdisciplinary essays contributed by international scholars and practitioners. Having distinguished themselves across such disciplines as Anthropology, Art, Music, Literature, Dance, Philosophy, Religion, and Theology and conjoined to construct a defining approach to the study of Aesthetics throughout the African Diaspora with the Humanities at the core, this collection of essays will break new ground in the study of Black Aesthetics. This book will be of great interest to scholars, practitioners, and (...) students interested in tracing African heritage identities throughout the African Diaspora through close examination of a variety of discourses directly connected to expressive elements of cultural production and religious rituals. (shrink)