Restricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃x : A.F (x) is understood as disjoint union, are the projections, and these do not preserve firstorderedness. This note shows, however, that the Curry-Howard theory is conservative over Heyting’s system.

To appear in the Proceedings of Logic Colloquium 2006. (32 pages).

The eminent Veronese physician Gabriele Zerbi held senior positions in three northern Italian cities and was the author of several major books on anatomy, ageing, philosophy and the rol...

There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...) refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper.. (shrink)

These essays present new analyzes of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.

This paper has two central purposes: the first is to survey some of the more important examples of fallacious argument, and the second is to examine the frequent use of these fallacies in support of the psychological construct: Attention Deficit Hyperactivity Disorder. The paper divides 12 familiar fallacies into three different categories—material, psychological and logical—and contends that advocates of ADHD often seem to employ these fallacies to support their position. It is suggested that all researchers, whether into ADHD or otherwise, (...) need to pay much closer attention to the construction of their arguments if they are not to make truth claims unsupported by satisfactory evidence, form or logic. (shrink)

The concept of the Anthropocene draws attention to human activity's impact on the planet at the geological scale. It is tempting to reason that like evolution, a heliocentric solar system or quantum mechanics, climate science compels us to accept as real a radical new ontology, the ‘anthroposphere’, with far-reaching social and political consequences. I wish to argue that this temptation should be resisted. The Anthropocene cannot be understood entirely as a natural scientific phenomenon, although it can be treated as such (...) for certain purposes. It is also an irreducibly social phenomenon. This is not to say that it is a socially constructed concept like nationhood, but that it is constituted by natural causal processes that are irreducibly entangled with social causal processes. Adopting the Anthropocene as a working concept therefore requires that we understand the causal processes involved in bringing it about as social causal processes, while also viewing these processes as objectual, open to public scrutiny and capable of compelling public assent. Social causes are not, however, easily subjected to such a naturalistic treatment. I conclude that the Anthropocene is not currently a suitable candidate for inclusion in a naturalist ontology. (shrink)

The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.

This new book is a wide-ranging, contemporary and accessible analysis of familiar and recurring myths about mass education in the United Kingdom. Looking at a variety of important issues and problems, each chapter begins by dispelling myths and assumptions about the classroom, going beyond class, race and gender, to offer analysis of topics such as discipline, youth cultures, information technology and globalisation. Utilising an interdisciplinary lens, this book offers knowledge from disciplines as diverse as sociology, philosophy, jurisprudence and cultural studies. (...) Gordon Tait examines the strengths and weaknesses of different theoretical approaches to education, from critical theory to postmodernism, and Foucaultian governance to post-colonialism. Analysing the many assumptions about education taken for granted in British public discourse, important conclusions are drawn about which of these assumptions are fair and reasonable, and which we should challenge. This book is an essential resource for advanced undergraduate and postgraduate courses on the sociology of education, culture and education, and the philosophy of education. (shrink)

William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gdel.

Katharine Tait, daughter of Bertrand and Dora Russell, here vividly portrays the extraordinary and stimulating environment she grew up in. In refreshing contrast to the interpretation of Russell as philosopher and public figure, Tait's is a close personal account of her deep love and admiration for her father and its gradual tempering by the imperfections she came to see in him. Touchingly written and beautifully described, the book shows Russell to be a man of great warmth, charm and (...) humour, but is also a testament to the traumas created when he applied his views on marriage and education to his own home. My Father Bertrand Russell is essential reading for those seeking to understand the private background to Russell's ideas, the personal effects of his lifestyle prescriptions, and what it was like to live with a man of such genious. (shrink)

This article examines the role of British translators of European philosophers from 1907 to 1915 in addressing Edwardian political crises. In an age of fragmentation between the main parties as well as crises within political groups, cultural and political magazines were a ‘counter public sphere’ in which intellectuals could discuss the conflicts of public life. This article will show how translators of Friedrich Nietzsche, Georges Sorel, and Henri Bergson opened up debate between intellectuals across the political spectrum in the avant-garde (...) magazine The New Age through the worship of intellectual heroes not bound to any tradition. The translators Oscar Levy, Millicent Murby, and T. E. Hulme all sought to create cultural icons to overcome splits in their different political projects. Whilst Edwardian intellectuals played with tropes of British insularity and philistinism, scholarship has neglected how the geographical concept of insularity was projected onto a reality that was ideological disintegration. (shrink)

There is currently no consensus from the relevant stakeholders regarding the operational and construct definitions of child assent for research. As such, the requirements for assent are often construed in different ways, institutionally disparate, and often conflated with those of parental consent. Development of a standardized operational definition of assent would thus be important to ensure that investigators, institutional review boards, and policy makers consider the assent process in the same way. To this end, we describe a Delphi study that (...) provided consensus from a panel of expert stakeholders regarding the definitions of child assent for research. Based on current guidelines, a preliminary definition of assent was generated and sent out for review to a Delphi panel including pediatric bioethicists and researchers, Institutional Review Board members, parents, and individuals with regulatory/legal expertise. For each subsequent review, the process of summarizing and revising responses was repeated until consensus was achieved. Panelists were also required to rank order elements of assent that they believed were most important in defining the underlying constructs of the assent process. In providing these rankings, panelists were asked to frame their responses in the contexts of younger and adolescents/older children in non-therapeutic and therapeutic trials. Summary rankings of the most important identified elements were then used to generate written construct definitions which were sent out for iterative reviews by the expert panel. Consensus regarding the operational definition was reached by 14/18 of the panel members. Seventeen panelists agreed with the definitions of capacity for assent, elements of disclosure for younger children, and the requirements for meaningful assent, respectively. Fifteen members agreed with the elements of disclosure for adolescents/older children. It is hoped that this study will positively inform and effect change in the way investigators, regulators, and IRBs operationalize the assent process, respect children’s developing autonomy, and in concert with parental permission, ensure the protection of children who participate in research. (shrink)

Foundations of a General Theory of Manifolds [Cantor, 1883], which I will refer to as the Grundlagen, is Cantor’s first work on the general theory of sets. It was a separate printing, with a preface and some footnotes added, of the fifth in a series of six papers under the title of “On infinite linear point manifolds”. I want to briefly describe some of the achievements of this great work. But at the same time, I want to discuss its connection (...) with the so-called paradoxes in set theory. There seems to be some agreement now that Cantor’s own understanding of the theory of transfinite numbers in that monograph did not contain an implicit contradiction; but there is less agreement about exactly why this is so and about the content of the theory itself. For various reasons, both historical and internal, the Grundlagen seems not to have been widely read compared to later works of Cantor, and to have been even less well understood. But even some of the more recent discussions of the work, while recognizing to some degree its unique character, misunderstand it on crucial points and fail to convey its true worth. (shrink)

Of course, I do not mean by the title of this paper to deny the existence of something called.

Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in (...) so far as its aim was to give constructive content to intuitionism) is superfluous. (shrink)

For consciousness to exist, an entity must have prerequisite characteristics and attributes to give rise to it. We explore these “building blocks” of consciousness in detail in this paper, which range from perceptive to computational to meta-representational characteristics of an entity’s cognitive architecture. We show how each cognitive attribute is strictly necessary for the emergence of consciousness, and how the building blocks may be used for any entity to be classified as being conscious. The list of building blocks is not (...) limited to human or organic consciousness and may be used to classify artificial and organisational conscious entities. We further explore a list of attributes that seem intuitively necessary for consciousness, but on further investigation, are neither required nor sufficient. The building blocks do not represent a theory of consciousness but rather a meta-theory on the emergence and classification of consciousness. (shrink)

The “self” does not exist within a vacuum. For an entity to be considered to have a sense of self, it requires certain characteristics and attributes. This paper investigates these “structures” of the sense of self in detail, which range from a unified consciousness to self-awareness to personal identity. The paper details how each attribute and characteristic is strictly necessary for an entity to be classified as having a self, and how the five structures detailed within may be used as (...) a guide for categorising and classifying entities as having selfhood or not (or any point along the spectrum between these). The five structures do not represent a theory of selfhood, but rather a meta-theory on the potential emergence and classification of the self. (shrink)

Conscious sentient AI seems to be all but a certainty in our future, whether in fifty years’ time or only five years. When that time comes, we will be faced with entities with the potential to experience more pain and suffering than any other living entity on Earth. In this paper, we look at this potential for suffering and the reasons why we would need to create a framework for protecting artificial entities. We look to current animal welfare laws and (...) regulations to investigate why certain animals are given legal protections, and how this can be applied to AI. We use a meta-theory of consciousness to determine what developments in AI technology are needed to bring AI to the level of animal sentience where legal arguments for their protection can be made. We finally speculate on what a future conscious AI could look like based on current technology. (shrink)

History and Philosophy of Logic, Volume 32, Issue 2, Page 177-183, May 2011.

The reduction of the lambda calculus to the theory of combinators in [Sch¨ onfinkel, 1924] applies to positive implicational logic, i.e. to the typed lambda calculus, where the types are built up from atomic types by means of the operation A −→ B, to show that the lambda operator can be eliminated in favor of combinators K and S of each type A −→ (B −→ A) and (A −→ (B −→ C)) −→ ((A −→ B) −→ (A −→ C)), (...) respectively.1 I will extend that result to the case in which the types are built up by means of the general function type ∀x : A.B(x) as well as the disjoint union type ∃x : A.B(x)– essentially to the theory of [Howard, 1980]. To extend the treatment of −→ to ∀ we shall need a generalized form of the combinators K and S, and to deal with ∃ we will need to introduce a new form of the combinator S.. (shrink)

There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...) refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper.. (shrink)

The established theories and frameworks on consciousness in the academic literature as related to artificial intelligence (AI), are rooted in anthropocentricism. Even those theories created intentionally for AI are based on the levels of consciousness as it is understood in humans primarily, and in other animals secondarily. This paper will discuss why such anthropocentric frameworks are built on unsecure foundations. We will do this by comparing the capacities and functions of human and AI cognitive architectures, discussing the ramifications and consequences (...) of the behaviors that stem from these, and looking at the neurological conditions in humans that can give the most promising hints as to what a potential conscious AI entity would look like. The paper ends with a proposed solution for building a nonanthropocentric foundation of cognition that could lead toward a truly AI-focused framework of consciousness. (shrink)