Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

Which entities should be accepted as part of the furniture of the world, and which not? What are pseudo-objects, if they are not properly objects? This collection explores the answers given to these questions by some key philosophers throughout the 20th century. It brings together essays by leading scholars on a subject of central importance to both metaphysics and the history of philosophy.".

We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem for pairs (...) (RT₂²) splits into a stable version (SRT₂²) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for RT₂² and SRT₂²). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL₀, DNR, and BΣ₂. We show, for instance, that WKL₀ is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π₁¹-conservative over the base system RCA₀. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT₂² (and so does not imply RT₂²). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT₂² on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions. (shrink)

Human control of action in routine situations involves a flexible interplay between (a) task-dependent serial ordering constraints; (b) top-down, or intentional, control processes; and (c) bottom-up, or environmentally triggered, affordances. In addition, the interaction between these influences is modulated by learning mechanisms that, over time, appear to reduce the need for top-down control processes while still allowing those processes to intervene at any point if necessary or if desired. We present a model of the acquisition and control of goal-directed action (...) that goes beyond existing models by operationalizing an interface between two putative systems—a routine and a non-routine system—thereby demonstrating how explicitly represented goals can interact with the emergent task representations that develop through learning in the routine system. The gradual emergence of task representations offers an explanation for the transfer of control with experience from the non-routine goal-based system to the routine system. At the same time it allows action selection to be sensitive both to environmental triggers and to biasing from multiple levels within the goal system. (shrink)

Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.

Hierarchical Bayesian models provide an account of Bayesian inference in a hierarchically structured hypothesis space. Scientific theories are plausibly regarded as organized into hierarchies in many cases, with higher levels sometimes called ‘paradigms’ and lower levels encoding more specific or concrete hypotheses. Therefore, HBMs provide a useful model for scientific theory change, showing how higher-level theory change may be driven by the impact of evidence on lower levels. HBMs capture features described in the Kuhnian tradition, particularly the idea that higher-level (...) theories guide learning at lower levels. In addition, they help resolve certain issues for Bayesians, such as scientific preference for simplicity and the problem of new theories. (shrink)

In this paper we give a positive answer to Julia Robinson's question whether the definability of + and · from S and ∣ that she proved in the case of positive integers is extendible to arbitrary integers (cf. [JR, p. 102]).

Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate $\perp$ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are $\{S, \perp\}$ -definable over Z. Applications to definability over Z and N (...) are stated as corollaries of the main theorem. (shrink)

Originally published in 1958.The history of Morley College provides an illuminating case-history of the growth and spread of adult education in the second half of the nineteenth and early years of the twentieth centuries. Morley College is unique in that it was one of the first of such institutions to proclaim and inculcate absolute sex and class equality. It has always been guided by democratic principles in the sense that the students have been encouraged to play a definite part in (...) the administration of the college – an ethos which continues to this day. (shrink)

Originally published in 1958.The history of Morley College provides an illuminating case-history of the growth and spread of adult education in the second half of the nineteenth and early years of the twentieth centuries. Morley College is unique in that it was one of the first of such institutions to proclaim and inculcate absolute sex and class equality. It has always been guided by democratic principles in the sense that the students have been encouraged to play a definite part in (...) the administration of the college – an ethos which continues to this day. (shrink)

The development of analogical reasoning has traditionally been understood in terms of theories of adult competence. This approach emphasizes structured representations and structure mapping. In contrast, we argue that by taking a developmental perspective, analogical reasoning can be viewed as the product of a substantially different cognitive ability – relational priming. To illustrate this, we present a computational (here connectionist) account where analogy arises gradually as a by-product of pattern completion in a recurrent network. Initial exposure to a situation primes (...) a relation that can then be applied to a novel situation to make an analogy. Relations are represented as transformations between states. The network exhibits behaviors consistent with a broad range of key phenomena from the developmental literature, lending support to the appropriateness of this approach (using low-level cognitive mechanisms) for investigating a domain that has normally been the preserve of high-level models. Furthermore, we present an additional simulation that integrates the relational priming mechanism with deliberative controlled use of inhibition to demonstrate how the framework can be extended to complex analogical reasoning, such as the data from explicit mapping studies in the literature on adults. This account highlights how taking a developmental perspective constrains the theory construction and cognitive modeling processes in a way that differs substantially from that based purely on adult studies, and illustrates how a putative complex cognitive skill can emerge out of a simple mechanism. (shrink)

Let M be a first-order structure; we denote by DEF(M) the set of all first-order definable relations and functions within M. Let π be any one-to-one function from N into the set of prime integers. Let ∣ and $\bullet$ be respectively the divisibility relation and multiplication as function. We show that the sets DEF(N,π,∣) and $\mathrm{DEF}(\mathbb{N},\pi,\bullet)$ are equal. However there exists function π such that the set DEF(N,π,∣), or, equivalently, $\mathrm{DEF}(\mathbb{N},\pi,\bullet)$ is not equal to $\mathrm{DEF}(\mathbb{N},+,\bullet)$ . Nevertheless, in all cases (...) there is an $\{\pi,\bullet\}$ -definable and hence also {π,∣}-definable structure over N which is isomorphic to $\langle\mathbb{N},+,\bullet\rangle$ . Hence theories TH(N,π,∣) and $\mathrm{TH}(\mathbb{N},\pi,\bullet)$ are undecidable. The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice. The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables. (shrink)

In this response, we reiterate the importance of development (both ontogenetic and phylogenetic) in the understanding of a complex cognitive skill – analogical reasoning. Four key questions structure the response: Does relational priming exist, and is it sufficient for analogy? What do we mean by relations as transformations? Could all or any relations be represented as transformations? And what about the challenge of more complex analogies? In addressing these questions we bring together a number of supportive commentaries, strengthening our emergentist (...) case for analogy (in particular with insights from comparative psychology), and review new supportive evidence. We also rebut those commentaries that ignore development at their peril. Along the way, we revisit the main assumptions underlying the analogy as relational priming (ARP) account of analogy, clarifying and elaborating as necessary. (shrink)

Let $ be the restriction of usual order relation to integers which are primes or squares of primes, and let ⊥ denote the coprimeness predicate. The elementary theory of $\langle\mathbb{N};\bot, , is undecidable. Now denote by $ the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure $\langle\mathbb{N};\bot, . Furthermore, the structures $\langle\mathbb{N};\mid, and $\langle\mathbb{N};=,+,x\rangle$ are interdefinable.

We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1-types.

This book builds on the idea that pragmatics and philosophy are strictly interconnected and that advances in one area will generate consequential advantages in the other area. The first part of the book, entitled ‘Theoretical Approaches to Philosophy of Language’, contains contributions by philosophers of language on connectives, intensional contexts, demonstratives, subsententials, and implicit indirect reports. The second part, ‘Pragmatics in Discourse’, presents contributions that are more empirically based or of a more applicative nature and that deal with the pragmatics (...) of discourse, argumentation, pragmatics and law, and context. The book presents perspectives which, generally, make most of the Gricean idea of the centrality of a speaker’s intention in attribution of meaning to utterances, whether one is interested in the level of sentence-like units or larger chunks of discourse. (shrink)

Embodied approaches to cognition conceive of mental life as emerging from the ongoing relationship between neural and extra-neural resources. The latter include, first and foremost, our entire body, but also the activity patterns enacted within a contingent milieu, cultural norms, social factors, and the features of the environment that can be used to enhance our cognitive capacities (e.g., tools, devices, etc.). Recent work in music education and sport psychology has applied general principles of embodiment to a number of social contexts (...) relevant to their respective fields. In particular, both disciplines have contributed fascinating perspectives to our understanding of how skills are acquired and developed in groups; how musicians, athletes, teachers, and coaches experience their interactions; and how empathy and social action participate in shaping effective performance. In this paper, we aim to provide additional grounding for this research by comparing and further developing original themes emerging from this cross-disciplinary literature and empirical works on how performative skills are acquired and optimized. In doing so, our discussion will focus on: (1) the feeling of being together, as meaningfully enacted in collective musical and sport events; (2) the capacity to skillfully adapt to the contextual demands arising from the social environment; and (3) the development of distributed forms of bodily memory. These categories will be discussed from the perspective of embodied cognitive science and with regard to their relevance for music education and sport psychology. It is argued that because they play a key role in the acquisition and development of relevant skills, they can offer important tools to help teachers and coaches develop novel strategies to enhance learning and foster new conceptual and practical research in the domains of music and sport. (shrink)

This paper consists of my responses to the comments by nine commentators on my book Are we Bodies or Souls? It makes twelve separate points, each one relevant to the comments of one or more of the commentators, as follows: I defend my understanding of “knowing the essence” of an object as knowing a set of logically necessary and sufficient conditions for an object to be that object; I claim that there cannot be thoughts without a thinker; I argue that (...) my distinction of “mental” from “physical” events in terms of whether anyone has privileged access to whether or not they occur, is a clear one; and I defend my account of metaphysical modality and its role in defending my account of personal identity; I defend my view that Descartes’s argument in favor of the view that humans are essentially souls fails, but that my amended version of that argument succeeds; I claim that my theory acknowledges the closeness of the connection in an earthly life between a human soul and its body; I argue that my Cartesian theory of the soul-body relation is preferable to Aquinas’s theory of that; I argue that a material thing cannot have mental properties; I argue that any set of logically necessary conditions for an object to be the object it is, which together form a logically sufficient condition for this, mutually entails any other such set; I deny that a dualist needs to provide an explanation of how the soul has the capacities that it has; and finally I defend my view that souls have thisness, and claim that that is not a difficulty for the view that God determines which persons will exist. (shrink)

The ‘Pascalian’ tradition in French thought is a moral rigorism that demands practical embodiment while denying that any embodiment of its demands can ever be complete. The power of this tradition may be seen even in French political moralists of the 20th century. It is revealed in Bergson’s view that the open morality must seek practical expression through the closed society, while constantly subverting it. It is revealed in Levinas’s claim that the ‘saying’ requires to be ‘said’ but always undermines (...) the said, so that what is explicit is ‘always wanting’. Directly, and also through the influence of Levinas, the Pascalian tradition exerts a complex influence on Derrida, in whose thoughts about the topic of justice moral rigorism transforms into an idea of radical epistemic incompleteness, or infinite deferral. ‘Pascalians’ such as Bergson, Levinas and Derrida are convinced, consequently, that the ethical can be given only sporadic or uninstitutionalized political expression; in this regard, they may be contrasted with ‘Lockeans’ who believe that political theory is at one remove from our deepest ethical or religious conclusions. In light of that contrast, the differences between Continental and Anglophone political theory may be traced to post-Reformation circumstances rather than, as is often supposed, to the Enlightenment. (shrink)

Michael A. PetersBeijing Normal UniversityMy deepest condolences to Pepe, Dom and Marcus and to Jim’s grandchildren. Tina and I spent a lot of time at the Marshall family home, often attending dinn...

One way to think about capitalism-versus-socialism is to examine the extent to which capitalist economic institutions are compatible with the fulfillment of socialist ideals. The late G. A. Cohen has urged that the two are strongly incompatible. He imagines how it would make sense for friends to organize a camping trip, distills the socialist moral principles that he sees fulfilled in the camping trip model, and observes that these principles conflict with a capitalist organization of the economy. He adds that (...) these principles are ethically attractive, so if it is feasible to organize the economy on the camping trip model, we ought to do so. This essay argues to the contrary that for all that has been said, capitalist economic arrangements might be in the set of institutional arrangements that overall would best fulfill the camping trip principles, and anyway, the principles themselves ought to be rejected, so the question whether or not a capitalist set-up might satisfy these principles should not interest us. The grounds for rejecting the camping trip principles support a form of welfarist consequentialism that denies that equality of distribution of any sort is per se ethically desirable and also denies that liberal freedoms to live as one chooses are per se morally desirable. Equality and freedom should rather be regarded as in the light of possible means (or hindrances) to advancing good for people, fairly distributed. (shrink)

The conceptual and technical difficulties involved in creating a quantum theory of gravity have led some physicists to question, and even in some cases to deny, the reality of time. More surprisingly, this denial has found a sympathetic audience among certain philosophers of physics. What should we make of these wild ideas? Does it even make sense to deny the reality of time? In fact physical science has been chipping away at common sense aspects of time ever since its inception. (...) Section 1 offers a brief survey of the demolition process. Section 2 distinguishes a tempered from an extremely radical form that a denial of time might take, and argues that extreme radicalism is empirically self-refuting. Section 3 begins an investigation of the prospects for tempered radicalism in a timeless theory of quantum gravity. (shrink)

Few philosophers have left a legacy like that of Gottfried Wilhelm Leibniz. He has been credited not only with inventing the differential calculus, but also with anticipating the basic ideas of modern logic, information science, and fractal geometry. He made important contributions to such diverse fields as jurisprudence, geology and etymology, while sketching designs for calculating machines, wind pumps, and submarines. But the common presentation of his philosophy as a kind of unworldly idealism is at odds with all this bustling (...) practical activity. In this book Richard. T. W. Arthur offers a fresh reading of Leibniz’s philosophy, clearly situating it in its scientific, political and theological contexts. He argues that Leibniz aimed to provide an improved foundation for the mechanical philosophy based on a new kind of universal language. His contributions to natural philosophy are an integral part of this programme, which his metaphysics, dynamics and organic philosophy were designed to support. Rather than denying that substances really exist in space and time, as the idealist reading proposes, Leibniz sought to provide a deeper understanding of substance and body, and a correct understanding of space as an order of situations and time as an order of successive things. This lively and approachable book will appeal to students of philosophy, as well as anyone seeking a stimulating introduction to Leibniz's thought and its continuing relevance. (shrink)

Contemporary conversations about religion and culture are framed by two reductive definitions of secularity. In one, multiple faiths and nonfaiths coexist free from a dominant belief in God. In the other, we deny the sacred altogether and exclude religion from rational thought and behavior. But is there a third way for those who wish to rediscover the sacred in a skeptical society? What kind of faith, if any, can be proclaimed after the ravages of the Holocaust and the many religion-based (...) terrors since? Richard Kearney explores these questions with a host of philosophers known for their inclusive, forward-thinking work on the intersection of secularism, politics, and religion. An interreligious dialogue that refuses to paper over religious difference, these conversations locate the sacred within secular society and affirm a positive role for religion in human reflection and action. Drawing on his own philosophical formulations, literary analysis, and personal interreligious experiences, Kearney develops through these engagements a basic gesture of hospitality for approaching the question of God. His work facilitates a fresh encounter with our best-known voices in continental philosophy and their views on issues of importance to all spiritually minded individuals and skeptics: how to reconcile God's goodness with human evil, how to believe in both God and natural science, how to talk about God without indulging in fundamentalist rhetoric, and how to balance God's sovereignty with God's love. (shrink)

This paper examines the relationship between the KK principle and the epistemological theses of externalism and internalism. In particular we examine arguments from Okasha (Analysis 73(1):80–86, 2013) and Greco (J Philos 111(4):169–197, 2014) which deny that we can derive the denial of the KK principle from externalism.

In lieu of an abstract, here is a brief excerpt of the content:Against Ethical CriticismRichard A. PosnerOscar Wilde famously remarked that “there is no such thing as a moral or an immoral book. Books are well written, or badly written. That is all.” He was echoed by Auden, who said in his poem in memory of William Butler Yeats that poetry makes nothing happen (though the poem as a whole qualifies this overstatement), by Croce, and by formalist critics such as (...) Cleanth Brooks, a doyen of the New Critics, who insisted that edification was the function of religion but not of poetry. 1 George Orwell, though himself a didactic novelist, was of this view as well, 2 and it is one that most critics adopt when the issue is censorship. I accept Wilde’s thesis—the creed of aestheticism, of art for art’s sake—if understood to mean that the moral content and consequences of a work of literature are irrelevant to its value as literature; or as the critic Helen Vendler has put it, that “treating fictions as moral pep-pills or moral emetics is repugnant to anyone who realizes the complex psychological and moral motives of a work of art.” 3 This is not to deny that reading can have consequences, including moral and political ones. Information affects people’s outlook and behavior, and reading is a source of information. Imaginative literature conveys ideas, opinions, and information, often with great power. Think of the role that novels by Turgenev (Fathers and Sons), Dostoevsky (The Possessed), Conrad (The Secret Agent, Under Western Eyes), Koestler (Darkness at Noon), Orwell (Animal Farm, Nineteen Eighty-Four), and, of course, Solzhenitsyn played in exposing the horrors of anarchism and communism. Of A Passage to India it has been said that “as an account of the social conditions of British India it was powerful enough to have influenced events.” 4 Upton Sinclair’s novel The Jungle is thought to have incited federal regulation of food processing; and few doubt the effect of Uncle Tom’s Cabin (1852) on the abolitionist cause. [End Page 1]At the core of the aesthetic tradition, which I shall be defending in this essay particularly against its opponents in the “law and literature” movement, are three theses. First, immersion in literature does not make us better citizens or better people. One might be able to pick out some works of literature that would have such an effect because of the information they convey or the emotional state they induce, but they would constitute a skewed sample of literary works. Second, we should not be put off by morally offensive views encountered in literature even when the author appears to share them. A work of literature is not to be considered maimed or even marred by expressing unacceptable moral views; by the same token, a mediocre work of literature is not redeemed by expressing moral views of which we approve. The proper criteria for evaluating literature are aesthetic rather than ethical. Third, authors’ moral qualities or opinions should not affect our valuations of their works.The insistence, in short, is on the separation of the moral from the aesthetic—but with two qualifications. Some literature has little interest or value apart from the didactic and for it the proper criticism is didactic. And the separation of moral from aesthetic values is not a rejection of the former. The aesthetic outlook is a moral outlook, one that stresses the values of openness, detachment, hedonism, curiosity, tolerance, the cultivation of the self, and the preservation of a private sphere—in short, the values of liberal individualism.The counter tradition in literary criticism to the aesthetic originates with Plato and insists upon the importance, in some versions to the near or even total exclusion of anything else, of the ethical or political content and effects of works of literature, and less commonly of the author’s own morality. Martha Nussbaum, for example, deems Greek tragedies and Anglo-American realistic novels a part of moral philosophy. 5 Reading novels, she argues, “develops moral capacities without which citizens will not succeed in making reality out of the normative conclusions of any moral or political theory, however excellent.” 6 Wayne Booth, disagreeing with his illustrious... (shrink)