We classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for, a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any.

We prove uniformly computable versions of the Implicit Function Theorem in its differentiable and non-differentiable forms. We show that the resulting operators are not computable if information about some of the partial derivatives of the implicitly defining function is omitted. Finally, as a corollary, we obtain a uniformly computable Inverse Function Theorem, first proven by M. Ziegler.

Sufficient conditions are given for the computation of an arc that accesses a point on the boundary of an open subset of the plane from a point within the set. The existence of a not-computably-accessible but computable point on a computably compact arc is also demonstrated.

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.

We show that any partial order with a Σ3 enumeration can be effectively embedded into any partial order obtained by imposing a strong reducibility such as ≤tt on the c. e. sets. As a consequence, we obtain that the partial orders that result from imposing a strong reducibility on the sets in a level of the Ershov hiearchy below ω + 1 are co-embeddable.

Let equation image. We show that for many reducibilities, the requirement that a relation be intrinsically reducible to the α-th jump of a countable mode A has a syntactic equivalent. Furthermore, we show that many reducibilities coincide in such a situation.

We study the following classes: Q* (r 1 A 1 ,..., r kA k ) which is defined to be the collection of all sets that can be computed by a Turing machine that on any input makes a total of r i queries to A i for all i ∈ {1,..., k}. Q(r 1A 1 ,...,r kA k ) which is defined like Q* (r 1A 1 ,..., r kA k ) except that queries to A i must be (...) made before queries to A i+1 for all i ∈ {1,..., k - 1}. QC(r 1A 1 ,..., r kA k ) which is defined like Q(r 1A 1 ,..., r kA k ) except that the Turing machine must halt even if given incorrect answers to some of its queries. We show that if A 1 ,..., A k are jumps that are not too close together, then all three of these classes are identical and are not changed if we permute (r 1A 1 ,..., r kA k ). This extends a result of Beigel's [1]. Since the second class is not affected by permutations, we say that these sets commute with each other. We also show that jumps that are too close together may not commute. We also characterize the commutative sequences of sets obtained by iterating the jump operation through an ordinal notation. (shrink)

Call a set A n-correctable if every set Turing reducible to A via a Turing machine that on any input makes at most n queries is Turing reducible to A via a Turing machine that on any input makes at most n-queries and on any input halts no matter what answers are given to its queries. We show that if a c.e. set A is n-correctable for some n ≥ 2, then it is n-correctable for all n. We show that (...) this is the optimal such result by constructing a c.e. set that is 1-correctable but not 2-correctable. The former result is obtained by examining the logical closure properties of c.e. sets that are 2-correctable. (shrink)

The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u with infinitely many zeros, the Blaschke sum of u provides the exact amount of (...) information necessary to compute the factorization of u. Along the way, we prove some uniform computability results for Blaschke products; these results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation. (shrink)

Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $$\phi ^\mathcal {M}\le r$$, and the open diagram, which encapsulates strict inequalities of the form $$\phi ^\mathcal {M}< r$$. We show (...) that the closed and open $$\Sigma _N$$ diagrams are $$\Pi ^0_{N+1}$$ and $$\Sigma ^0_N$$ respectively, and that the closed and open $$\Pi _N$$ diagrams are $$\Pi ^0_N$$ and $$\Sigma ^0_{N + 1}$$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal. (shrink)

Suppose \ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable \ spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps.

We examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.

We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof.

For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A: $ODD^A_n = \{(x_1, \dots ,x_n): {\tt\#}^A_n(x_1, \dots, x_n) \text{is odd}\}$ and, for m ≥ 2, $\text{MOD}m^A_n = \{(x_1, \dots ,x_n):{\tt\#}^A_n(x_1, \dots ,x_n) \not\equiv 0 (\text{mod} m)\},$ where ${\tt\#}^A_n(x_1, \dots ,x_n) = A(x_1)+\cdots+A(x_n)$ . (We identify A(x) with χ A (x), where χ A is (...) the characteristic function of A.) If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODD A n and $\text{MOD}m^A_n: \bullet\text{ODD}^A_n$ can be decided with n parallel queries to A, but not with n - 1. $\bullet \text{ODD}^A_n$ can be decided with $\lceil log(n + 1)\rceil$ sequential queries to A but not with $\lceil log(n + 1)\rceil - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil n/m\rceil + \lfloor n/m\rfloor$ parallel queries to A but not with $\lceil n/m\rceil + \lfloor n/m\rfloor - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil$ sequential queries to A but not with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil - 1$ . The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.) In particular, every nonzero truth-table degree contains a set A such that ODD A n cannot be decided with n - 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODD B n can be decided with one query to B, a set's query complexity depends more on its structure than on its degree. For a fixed set A, $Q(n,A) = \{S: S \text{can be decided with n sequential queries to} A\},\\Q_\parallel(n, A) = \{S: S \text{can be decided with n parallel queries to} A\}.$ We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies: $\bullet Q(0,A) \subset Q(1,A) \subset Q(2,A) \subset\cdots\\ \bullet Q_\parallel(0, A) \subset Q_\parallel(1, A) \subset Q_\parallel(2,A) \subset\cdots$ The same is true if A is a jump. (shrink)

Knowledge and its Limits presents a systematic new conception of knowledge as a kind of mental stage sensitive to the knower's environment. It makes a major contribution to the debate between externalist and internalist philosophies of mind, and breaks radically with the epistemological tradition of analyzing knowledge in terms of true belief. The theory casts new light on such philosophical problems as scepticism, evidence, probability and assertion, realism and anti-realism, and the limits of what can be known. The arguments are (...) illustrated by rigorous models based on epistemic logic and probability theory. The result is a new way of doing epistemology and a notable contribution to the philosophy of mind. (shrink)

This is not your psychoanalyst's unconscious.

Knepper criticizes existing efforts in the philosophy of religion for being out of step with, and therefore useless to, the academic study of religion, then forwards a new program for philosophy of religion that is in step with, and therefore useful to, the academic study of religion.

According to the Fragmentation Analysis, epistemic akrasia is a state of conflict between beliefs formed by the linguistic and non-linguistic belief-formation systems, and epistemic akrasia is irrational because it is a state of conflict between beliefs so formed. I argue that there are cases of higher-order epistemic akrasia, where both beliefs are formed by the linguistic belief-formation system. Because the Fragmentation Analysis cannot accommodate this possibility, the Fragmentation Analysis is incorrect. I consider three objections to the possibility of higher-order epistemic (...) akrasia. Along the way, I offer a revision of the Fragmentation Analysis that can allow for the possibility of higher-order akrasia while avoiding the problems I point out for the original view. (shrink)

Second-order logic and modal logic are both, separately, major topics of philosophical discussion. Although both have been criticized by Quine and others, increasingly many philosophers find their strictures uncompelling, and regard both branches of logic as valuable resources for the articulation and investigation of significant issues in logical metaphysics and elsewhere. One might therefore expect some combination of the two sorts of logic to constitute a natural and more comprehensive background logic for metaphysics. So it is somewhat surprising to find (...) that philosophical discussion of secondorder modal logic is almost totally absent, despite the pioneering contribution of Barcan. (shrink)

In this global introduction to philosophy of religion you begin not with a single tradition, but with religious philosophies from East Asia, South Asia, West Africa, and Native North America, alongside the classical Abrahamic and modern European traditions. Matching this diversity of traditions, chapters are organized around questions that acknowledge there is no single understanding of any god or ultimate reality. Instead you approach six different traditions of philosophizing about religion by asking questions about the journeys of both the self (...) and the cosmos such as "What is my path?" and "Where did the cosmos come from?" Accompanied by introductory materials and an extensive glossary, each chapter includes learning objectives, questions for discussion, and suggested primary and secondary sources. The categories of religion and philosophy are interrogated throughout. Equipped with study tools and universal questions about the self and the cosmos, Philosophies of Religion: A Global and Critical Introduction shows you how to philosophize about religions around the world. (shrink)

On method, discursive logics, and epistemology -- Questions of medieval discursive practice -- From the middle ages to the (w)hole of Utopia -- Kepler, his Dream, and the analysis and pattern of thought -- Campanella and Bacon: concerning structures of mind -- The masculine birth of time -- Cyrano and the experimental discourse -- The myth of sun and moon -- The difficulty of writing -- Crusoe rights his story -- Gulliver's critique of Euclid -- Emergence, consolidation, and dominance of (...) a discourse. (shrink)

Skeptical theism is a family of responses to the evidential problem of evil. What unifies this family is two general claims. First, that even if God were to exist, we shouldn’t expect to see God’s reasons for permitting the suffering we observe. Second, the previous claim entails the failure of a variety of arguments from evil against the existence of God. In this essay, we identify three particular articulations of skeptical theism—three different ways of “filling in” those two claims—and describes (...) their role in responding to evidential arguments of evil due to William Rowe and Paul Draper. But skeptical theism has been subject to a variety of criticisms, several of which raise interesting issues and puzzles not just in philosophy of religion but other areas of philosophy as well. Consequently, we discuss some of these criticisms, partly with an eye to bringing out the connections between skeptical theism and current topics in mainstream philosophy. Finally, we conclude by situating skeptical theism within our own distinctive methodology for evaluating world views, what we call “worldview theory versioning.”. (shrink)

The paper explores the main competing interpretations of Aristotle's view of the relation between happiness and external goods in the Nicomachean Ethics. On the basis of a careful analysis of what Aristotle says in the Nicomachean Ethics (and other works such as the Eudemian Ethics, Politics, Rhetoric, etc.) it is argued that it is likely that Aristotle takes at least some external goods to be actual constituents of happiness provided that (1) they are accompanied by virtuous activity and (2) the (...) agent enjoying and using those external goods does so in ways compatible with the continuous aim of acting virtuously. (shrink)

This is the first single volume to offer a comprehensive and systematic account of British and American aesthetics from the early eighteenth century to the late twentieth century.

The chapter argues that a model‐building methodology like that widespread in contemporary natural and social science already plays a significant role in philosophy. One neglected form of progress in philosophy over the past fifty years has been the development of better and better formal models of significant phenomena. Examples are given from both philosophy of language and epistemology. Philosophy can do still better in the future by applying model‐building methods more systematically and self‐consciously, with consequent readjustments to its methodology. Although (...) it is neither likely nor desirable for model‐building to become the sole or even main philosophical method, its use enhances the power and reliability of philosophical thinking. (shrink)

Jecker and colleagues open (2024) a critical and needed dialogue about the ethics of international conferencing. In particular, they focus on proposing a set of principles in selecting the location...

Things in common: an introduction -- Life -- Specters -- Subscendence -- Species -- Kindness.

This volume offers readers a unique and comprehensive overview of theoretical perspectives on "the sublime," the singular aesthetic response elicited by phenomena that move viewers by transcending and overwhelming them. The book consists of an editor's introduction and fifteen chapters written from a variety of disciplinary perspectives. Part One examines philosophical approaches advanced historically to account for the phenomenon, beginning with Longinus, moving through eighteenth and nineteenth century writers in Britain, France, and Germany, and concluding with developments in contemporary continental (...) philosophy. Part Two explores the sublime with respect to particular disciplines and areas of study, including Dutch literature, early modern America, the environment, religion, British Romanticism, the fine arts, and architecture. Each chapter is both accessible for nonspecialists and offers an original contribution to its respective field of inquiry. (shrink)

The philosophical relevance of experimental psychology is hard to dispute. Much more controversial is the so‐called negative program's critique of armchair philosophical methodology, in particular the reliance on ‘intuitions’ about thought experiments. This chapter responds to that critique. It argues that, since the negative program has been forced to extend the category of intuition to ordinary judgments about real‐life cases, the critique is in immediate danger of generating into global scepticism, because all human judgments turn out to depend on intuitions. (...) Recently, some proponents of the negative program have tried to demarcate the target of their critique more narrowly. However, their attempts are still far too indiscriminate, and over‐generate scepticism. Nevertheless, once experimental philosophy has refined its own methodology, it may contribute to the refinement of the methodology of mainstream philosophy, by filtering out the effects of cognitive bias, although it offers no prospect of doing without judgments on real or imaginary cases. (shrink)

Lewisian reference magnetism about linguistic content determination [Lewis 1983 has been defended in recent work by Weatherson [2003] and Sider [2009], among others. Two advantages claimed for the view are its capacity to make sense of systematic error in speakers' use of their words, and its capacity to distinguish between verbal and substantive disagreements. Our understanding of both error and disagreement is linked to the role of usage and first order intuitions in semantics and in linguistic theory more generally. I (...) argue, partially on the basis of these more general considerations, that reference magnetism delivers implausible results. Specifically, I argue that the proponent of reference magnetism maintains her analysis of genuinely systematic error at the cost of an empirically unjustifiable error theory regarding ordinary usage. In response, I describe an alternative view of content determination—MUMPS, or Meaning is Use Minus Pragmatics—which is not committed to such error theories. Despite this advantage, MUMPS has high prima facie costs. On such a view, there is a great deal of variation in linguistic meaning across speakers and times. As a result, a large number of seemingly mistaken claims are analysed as expressing true propositions. Correspondingly, a large number of seemingly substantive disagreements are analysed as terminological. However, I argue that these consequences are not as costly as they seem. Despite appearances, MUMPS is consistent with objective, metaphysically realist adjudication of disagreements, even in cases where meanings are not shared and where both parties to a dispute speak truly. MUMPS thus allows for a more nuanced understanding of linguistic usage, change, and variation, without imposing a commitment to any form of metaphysical anti-realism. (shrink)

The general process view of learning, which guided research into learning for the first half of this century, has come under attack in recent years from several quarters. One form of criticism has come from proponents of the so-called biological boundaries approach to learning. These theorists have presented a variety of data showing that supposedly general laws of learning may in fact be limited in their applicability to different species and learning tasks, and they argue that the limitations are drawn (...) by the nature of each species' adaptation to the particular requirements of its natural environment. The biological boundaries approach has served an important critical function in the move away from general process learning theory, but it is limited in its ability to provide an alternative to the general process approach. In particular, the biological boundaries approach lacks generality, it is in some respects subservient to the general process tradition, and its ecological content is in too many cases limited toex post factoadaptive explanations of learning skills. A contrasting, ecological approach to learning, which can provide a true alternative to general process theory, is presented. The ecological approach begins by providing an ecological task description for naturally occurring instances of learning; this step answers the question:Whatdoes this animal learn to do? The next step is an analysis of the means by which learning occurs in the course of development, answering the question:Howdoes the animal learn to do this? On the basis of such analyses, local principles of adaptation are formulated to account for the learning abilities of individual species. More global principles are sought by generalization among these local principles and may form the basis for a general ecological theory of learning. (shrink)

Operational epistemology is, to a first approximation, the attempt to provide cognitive rules such that one is in principle always in a position to know whether one is complying with them. In Knowledge and its Limits, I argue that the only such rules are trivial ones. In this paper, I generalize the argument in several ways to more thoroughly probabilistic settings, in order to show that it does not merely demonstrate some oddity of the folk epistemological conception of knowledge. Some (...) of the generalizations involve a formal semantic framework for treating epistemic probabilities of epistemic probabilities and expectations of expectations. The upshot is that operational epistemology cannot work, and that knowledge-based epistemology has the right characteristics to avoid its problems. (shrink)

This book explores the relation between agape (or Christian charity) and social justice. Timothy Jackson defines agape as the central virtue in Christian ethical thought and action and applies his insights to three concrete issues: political violence, forgiveness, and abortion. Taking his primary cue from the New Testament while drawing extensively from contemporary theology and philosophy, Jackson identifies three features of Christian charity: unconditional commitment to the good of others, equal regard for others' well-being, and passionate service open to (...) self-sacrifice for the sake of others. Charity, prescribed by Jesus for his disciples and named by Saint Paul as the "greatest" theological virtue, is contrasted with various accounts of justice. Jackson argues that agape is not trumped by justice or other goods. Rather, agape precedes justice: without the work of love, society would not produce persons capable of merit, demerit, and contract, the elements of most modern conceptions of justice. Jackson then considers the implications of his ideas for several questions: the nature of God, the relation between Christian love and political violence, the place of forgiveness, and the morality of abortion. Arguing that agapic love is to be construed as a gift of grace as well as a divine commandment, Jackson concludes that love is the "eternal life" that makes temporal existence possible and thus the "first" Christian virtue. Though foremost a contribution to Christian ethics, Jackson's arguments and the issues he takes up will find a broader readership. (shrink)

The range of topics in this volume covers a multitude of historical periods and topics, which in turn figure in the new media environments of contemporary life. These include discussions of the Aristotelian and classical models of a "good life" that inform animated fairy tales today, 1930s French and Hollywood films which respond to the dire need for productive human relationships in a turbulent decade, the polemical positions of black film criticism through the lens of James Baldwin's work, a discussion (...) of contemporary filmic quests for happiness, the challenges for women filmmakers today in mapping the values of their own world, landscapes of austerity and poverty in the cinematic homelands today, the scientific, psychological, and philosophical base for human value, and the shifting media frames of modern society and selves. (shrink)

Through both an historical and philosophical analysis of the concept of possibility, we show how including both potentiality and actuality as part of the real is both compatible with experience and contributes to solving key problems of fundamental process and emergence. The book is organized into four main sections that incorporate our routes to potentiality: potentiality in modern science [history and philosophy; quantum physics and complexity]; Relational Realism [ontological interpretation of quantum physics; philosophy and logic]; Process Physics [ontological interpretation of (...) relativity theory; physics and philosophy]; on speculative philosophy and physics [limitations and approximations; process philosophy]. We conclude that certain fundamental problems in modern physics require complementary analyses of certain philosophical and metaphysical issues, and that such scholarship reveals intrinsic features and limits of determinism, potentiality and emergence that enable, among others, important progress on the quantum theory of measurement problem and new understandings of emergence. (shrink)

For millennia self has been conjectured to be necessary for consciousness. But scant empirical evidence has been adduced to support this hypothesis. Inconsistent explications of “self” and failure to design apt experiments have impeded progress. Advocates of phenomenological psychiatry, however, have helped explicate “self,” and employed it to explain some psychopathological symptoms. In those studies, “self” is understood in a minimalist sense, sheer “for-me-ness.” Unfortunately, explication of the “minimal self” (MS) has relied on conceptual analysis, and applications to psychopathology have (...) been hermeneutic, allowing for many degrees of interpretive latitude. The result is that MS’s current scientific status is analogous to that of the “atom,” at the time when “atom” was just beginning to undergo transformation from a philosophical to a scientific concept. Fortunately, there is now an opportunity to promote a similar transformation for “MS.” Discovery of the brain’s Default Mode Network (DMN) opened the door to neuroimaging investigations of self. Taking the DMN and other forms of intrinsic activity as a starting point, an empirical foothold can be established, one that spurs experimental research and that enables extension of research into multiple phenomena. New experimental protocols that posit “MS” can help explain phenomena hitherto not thought to be related to self, thereby hastening development of a mature science of self. In particular, targeting phenomena wherein consciousness is lost and recovered, as in some cases of Unresponsive Wakefulness Syndrome (UWS), allow for design of neuroimaging probes that enable detection of MS during non-conscious states. These probes, as well as other experimental protocols applied to NREM Sleep, General Anesthesia (GA), and the waking state, provide some evidence to suggest that not only can self and consciousness dissociate, MS might be a necessary precondition for conscious experience. Finally, these findings have implications for the science of consciousness: it has been suggested that “levels of consciousness” (LoC) is not a legitimate concept for the science of consciousness. But because we have the conceptual and methodological tools with which to refine investigations of MS, we have the means to identify a possible foundation—a bifurcation point—for consciousness, as well as the means by which to measure degrees of distance from that foundation. These neuroimaging investigations of MS position us to better assess whether LoC has a role to play in a mature science of consciousness. (shrink)

I. An argument is presented for the conclusion that the hypothesis that no one will ever decide a given proposition is intuitionistically inconsistent. II. A distinction between sentences and statements blocks a similar argument for the stronger conclusion that the hypothesis that I have not yet decided a given proposition is intuitionistically inconsistent, but does not block the original argument. III. A distinction between empirical and mathematical negation might block the original argument, and empirical negation might be modelled on Nelson''s (...) strong negation, but only on intuitionistically unacceptable assumptions. IV. Intuitionists may have to accept the original argument, and therefore be committed to a dubious view of time on which there cannot be merely inductive evidence for statements about the infinite future. (shrink)

This paper introduces a referential reading of Kant’s practical project, according to which maxims are made morally permissible by their correspondence to objects, though not the ontic objects of Kant’s theoretical project but deontic objects (what ought to be). It illustrates this model by showing how the content of the Formula of Universal Law might be determined by what our capacity of practical reason can stand in a referential relation to, rather than by facts about what kind of beings we (...) are (viz., uncaused causes). This solves the neglected puzzle of why there are passages in Kant’s works suggesting robust analogies between mathematics and ethics, since to universalize a maxim is to test a priori whether a practical object with that particular content can be constructed. An apparent problem with this hypothesis is that the medium of practical sensibility (feeling) does not play a role analogous to the medium of theoretical sensibility (intuition). In response I distinguish two separate Kantian accounts of mathematical apriority. The thesis that maxim universalization is a species of construction, and thus a priori, turns out to be consistent with the account of apriority that informs Kant’s understanding of actual mathematical practice. (shrink)