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)

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)

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.

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)

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)

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.

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)

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.".

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)

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)

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...

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)

All humans have an equal basic moral status. They possess the same fundamental rights, and the comparable interests of each person should count the same in calculations that determine social policy. Neither supposed racial differences, nor skin color, sex, sexual orientation, ethnicity, intelligence, nor any other differences among humans negate their fundamental equal worth and dignity. These platitudes are virtually universally affirmed. A white supremacist racist or an admirer of Adolf Hitler who denies them is rightly regarded as beyond the (...) pale of civilized dialogue. However, a very simple line of argument developed by Peter Singer challenges our understanding of these platitudes and forces us to rethink the basis and nature of the moral equality of all humans. What I call the “Singer Problem” arises if one accepts that the morally significant cognitive capacities that are relevant to the determination of the fundamental moral status of a being vary from individual to individual by degree. The problem is to specify a moral principle determining fundamental moral status that assigns a superior status to humans compared to other animals on the basis of the superior cognitive capacities of humans but also assigns all human persons an equal fundamental moral status regardless of their differing cognitive capacities. More broadly posed, the problem is to specify moral principles that yield intuitively satisfactory implications for the treatment of human individuals and other individual animals given that cognitive capacities differ across species and individuals. A search for a resolution of the Singer problem leads to disappointing results. (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)

When the assertion that some agent is exploiting a person connotes that the exploitation is morally wrong, what is this wrong? Some maintain that exploitation need not involve unfair division of advantages, but instead is essentially domination for self-enrichment. This essay denies this claim and upholds the idea that exploitation claims concern unfair distribution. Some maintain that the hypothetical fully competitive market exchange price can serve, at least in some contexts, as the standard for assessing whether voluntary interaction is exploitative. (...) This essay denies that the idea of the fully competitive market price can serve in this role. Nor should we accept a pure luck egalitarian claim that would identify fair distribution with the outcome of ideally competitive markets proceeding from fair initial distribution. (shrink)

In recent years some moral philosophers and political theorists, who have come to be called “luck egalitarians,” have urged that the essence of social justice is the moral imperative to improve the condition of people who suffer from simple bad luck. Prominent theorists who have attracted the luck egalitarian label include Ronald Dworkin, G. A. Cohen, and John Roemer.1 Larry Temkin should also be included in this group, as should Thomas Nagel at the time that he wrote Equality and Partiality.2 (...) However, each of these theorists asserts a different position. The common ground, if any, is obscure. The idea of luck that is invoked is not transparently clear. Anyway, the term “luck egalitarianism” was coined by a critic of the doctrine, and tendentiously defined to denote an extreme version of the view that looks implausible from the start.3 With some justice Ronald Dworkin, perhaps the chief architect of the luck egalitarian position, has denied that he is a luck egalitarian. (shrink)

There seems to be a lot of opacity in our language. Quotation is opaque. The modal idioms are apparently opaque. Propositional attitude ascriptions seem opaque, as do the environments created by verbs such as ‘seeks’ and ‘fears’. Opacity raises a number of issues — first and foremost, whether there is such a thing. This article concentrates on the question of whether there is any opacity to be found in natural language, examining various reasons one might have for denying that apparent (...) opacity is genuine. (shrink)

Some theorists hold that there is no serious, significant issue concerning cosmopolitanism. They hold that cosmopolitanism is either the anodyne doctrine that we have some duties to distant strangers merely on the ground of shared humanity or the absurd doctrine that we have no special moral duties based on special-ties such as those of friendship, family, and national community. This essay argues against this deflationary position by defending (1) a very extreme cosmopolitan doctrine that denies special-tie moral duties altogether and (...) (2) a slightly milder but still extreme form of cosmopolitanism that allows that there might be special-tie moral duties to intimate associates such as friends and family members but denies that broader social group associations such as those of subjection to a national state or national or ethnic community memberships ever constitute special ties that ground special moral duties. The defense proceeds by rebutting bad arguments leveled against extreme cosmopolitanisms. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Reconstructing Pragmatism: Richard Rorty and the Classical Pragmatists by Chris VoparilRichard Kenneth AtkinsChris Voparil. Reconstructing Pragmatism: Richard Rorty and the Classical Pragmatists. Oxford: Oxford University Press, 2022. Pp. xiv + 377. Hardback, $74.00.A house divided cannot stand, or so Jesus tells us. As far as I can ascertain, Jesus was right about many things (his followers perhaps less so). Accordingly, that the house early pragmatists built, (...) with all of its subsequent divisions and internecine controversies, still stands is something of an anomaly. Perhaps it is explained by the fact that pragmatists have also been blessed by peacemakers, among whom Chris Voparil is to be numbered. In this carefully researched and clearly written book, Voparil examines the ties that bind Richard Rorty to his pragmatist predecessors: C. S. Peirce, William James, John Dewey, Josiah Royce, and Jane Addams. The hope animating Voparil's book is that with a closer look at these ties he "can unsettle the current map of opposing pragmatist camps by uncovering previously obscured or overlooked alignments and common aims" (259). He argues that Rorty did not misunderstand his predecessors but "consciously radicalized" (280) their positions.The book is five chapters long, with each of the chapters devoted to Rorty's relationship to one of the five aforementioned classical pragmatists, in that order. There are also substantial introductory and concluding chapters. The more controversy-laden chapters are the first (on Peirce) and third (on Dewey). As Voparil explains in chapter 1, Peirceans are wary of Rorty's apparent antirealism. But to soothe the worries of wary Peirceans, Voparil argues that commitments to realism can be found in Rorty's early and later works and that some of the more outrageous statements from the middle period are exaggerations that are ultimately qualified, if not disavowed. The Peirce scholar may wish that Voparil had done more to distinguish different varieties of realism found in Peirce's writings, as Robert Lane has done in his recent Peirce on Realism and Idealism (Cambridge: Cambridge University Press, 2018), but the crux of Voparil's argument is persuasive enough. Rorty did not deny that there is a world that affects us and that there is such a thing as "getting it right," even if he thought that varieties of representationalism about mental states and the correspondence theory of truth are not topics worthy of further discussion. Voparil does not [End Page 530] cite the footnote, but Rorty states the matter sufficiently clearly when he writes, "I think that [Nelson] Goodman's trope of 'many worlds' is misleading and that we need not go beyond the more straightforward 'many descriptions of the same world' (provided one does not ask 'And what world is that?'). But his point that there is no way to compare descriptions of the world in respect of adequacy seems to me crucial, and in the first two chapters of this book [Ways of Worldmaking] he makes it very vividly" (Consequences of Pragmatism [Minneapolis: University of Minnesota Press, 1982], xlvii). Rorty's point is that talk of "the world" can play no interesting explanatory role in our accounts of truth, representation, and the like just because we cannot now (or at any given future time) claim to have the final vocabulary to talk about the world. There is an interesting parallelism to Peirce's theory of induction here that Voparil does not explore. On Peirce's theory of statistical induction, we cannot claim at any given time that we have ascertained the real frequencies in nature, but we can be assured that in the long run we will eventually approximate them. Similarly, for Rorty, we cannot claim at any given time that we have the final vocabulary for describing the world, but that does not prohibit us from claiming that we can improve our descriptions of the world, where improvement is relative to some purpose. As Rorty concedes, the descriptions of nature that emerge in the modern period are better descriptions than those of the Aristotelian physics, given various purposes that we have (Consequences of Pragmatism, 191).Turning to chapter 3, Rorty's misappropriations and misrepresentations of Dewey's views have peeved Dewey scholars. In particular, Rorty... (shrink)

SummaryThis paper addresses the question of what scientific realism implies and what it does not when it is articulated so as to provide the best defense against plausible philosophical alternatives. A summary is presented of “abductive” arguments for scientific realism, and of the epistemological and semantic conceptions upon which they depend. Taking these arguments to be the best current defense of realism, it is inquired what, in the sense just mentioned, realism implies and what it does not. It is concluded (...) that realism implies the strong rejection of epistemological foundationalism, a non‐Humean conception of causation and of explanation, and a causal rather than conceptual account of the unity of natural definitions. It is denied that realism implies bivalence or the existence of one true theory, one preferred vocabulary or one distinctly privileged science. It is further denied that realism implies that there are no unrecognized conventional aspects to scientific theorizing and it is denied that realism implies that scientists routinely do good experimental metaphysics. (shrink)