In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.

Participants were unknowingly exposed to complex regularities in a working memory task. The existence of implicit knowledge was subsequently inferred from a preference for stimuli with similar grammatical regularities. Several affective traits have been shown to influence AGL performance positively, many of which are related to a tendency for automatic responding. We therefore tested whether the mindfulness trait predicted a reduction of grammatically congruent preferences, and used emotional primes to explore the influence of affect. Mindfulness was shown to correlate negatively (...) with grammatically congruent responses. Negative primes were shown to result in faster and more negative evaluations. We conclude that grammatically congruent preference ratings rely on habitual responses, and that our findings provide empirical evidence for the non-reactive disposition of the mindfulness trait. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are (...) closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

A closed λ-term E is called an enumerator if M ε /gL/dg /gTn ε N E/drn/dl = β M. Here Λ° is the set of closed λ-terms, N is the set of natural numbers and the /drn/dl are the Church numerals λfx./tfnx. Such an E is called reducing if moreover M ε /gL/dg /gTn ε N E/drn/dl /a/gb M. In 1983 I conjectured that every enumerator is reducing. An ingenious recursion theoretic proof of this conjecture by Statman is presented in (...) Barendregt . The proof is not intuitionistically valid, however. Dirk van Dalen has encouraged me to find intuitionistic proofs whenever possible. In the lambda calculus this is usually not difficult. In this paper an intuitionistic version of Statmans proof will be given. It took me somewhat longer to find it than in other cases. (shrink)

A λ-theory T is a consistent set of equations between λ-terms closed under derivability. The degree of T is the degree of the set of Godel numbers of its elements. H is the $\lamda$ -theory axiomatized by the set {M = N ∣ M, N unsolvable. A $\lamda$ -theory is sensible $\operatorname{iff} T \supset \mathscr{H}$ , for a motivation see [6] and [4]. In § it is proved that the theory H is ∑ 0 2 -complete. We present Wadsworth's proof (...) that its unique maximal consistent extention $\mathscr{H}^\ast (= \mathrm{T}(D_\infty))$ is Π 0 2 -complete. In $\S2$ it is proved that $\mathscr{H}_\eta(= \lambda_\eta-\text{Calculus} + \mathscr{H})$ is not closed under the ω-rule (see [1]). In $\S3$ arguments are given to conjecture that $\mathscr{H}\omega (= \lambda + \mathscr{H} + omega-rule)$ is Π 1 1 -complete. This is done by representing recursive sets of sequence numbers as λ-terms and by connecting wellfoundedness of trees with provability in Hω. In $\S4$ an infinite set of equations independent over H η will be constructed. From this it follows that there are 2^{ℵ_0 sensible theories T such that $\mathscr{H} \subset T \subset \mathscr{H}^\ast$ and 2 ℵ 0 sensible hard models of arbitrarily high degrees. In $\S5$ some nonprovability results needed in $\S\S1$ and 2 are established. For this purpose one uses the theory H η extended with a reduction relation for which the Church-Rosser theorem holds. The concept of Gross reduction is used in order to show that certain terms have no common reduct. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations (...) are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of $\lambda$ -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which (...) derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both (...) translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. (shrink)

This handbook with exercises reveals the mathematical beauty of formalisms hitherto mostly used for software and hardware design and verification.

When viewing a circular coin rotated in depth, it fills an elliptical region of the distal scene. For some, this appears to generate a two-fold experience, in which one sees the coin as simultaneously circular (in light of its 3D shape) and elliptical (in light of its 2D ‘perspectival shape’ or ‘p-shape’). An energetic philosophical debate asks whether the latter p-shapes are genuinely presented in perceptual experience (as ‘perspectivalists’ argue) or if, instead, this appearance is somehow derived or inferred from (...) experience (as ‘anti-perspectivalists’ argue). This debate, however, has largely turned on introspection. In a recent study, Morales, Bax, and Firestone (2020) aim to provide the first empirical test of this question. They asked subjects to find an elliptical coin seen face-on from a search array that also included a circular coin seen either face-on or at an angle. They found that subjects reacted more slowly when the distracting circle was seen at an angle, such that it’s p-shape matched that of the target ellipse. From this, they concluded that the similar p-shape between the ellipse and circle constituted a phenomenal similarity between the two, and thus that perspectivalism is true. We show that these results can also be explained by pre-attentive guidance by unconscious representations (in what follows, just “unconscious pre-attentive guidance”) and that this explanation is at least as plausible as one from phenomenal similarity. Thus, we conclude that the experiment does not support perspectivalism over anti-perspectivalism. (shrink)

The articles by Henk Bodewitz collected in this volume, published between 1969 and 2013, deal with Vedic cosmology and ethics on basis of a systematic philological study of early Vedic texts, from the Ṛgveda to various Brāhmaṇas, Āraṇyakas and Upaniṣads.

Three experiments examined whether the mere priming of potential action effects enhances people’s feeling of causing these effects when they occur. In a computer task, participants and the computer independently moved a rapidly moving square on a display. Participants had to press a key, thereby stopping the movement. However, the participant or the computer could have caused the square to stop on the observed position, and accordingly, the stopped position of the square could be conceived of as the potential effect (...) resulting from participants’ action of pressing the stop key. The location of this position was primed or not just before participants had to stop the movement. Results showed that priming of the position enhanced experienced authorship of stopping the square. Additional experimentation demonstrated that this priming of agency was not mediated by the goal or intention to produce the effect. (shrink)

Ubuntu is een levensfilosofie. Spirituele levenskracht verbindt de mens met zijn samenleving en de natuur. De werkelijkheid is een dynamiek van op elkaar inwerkende krachten. Het streven naar herstel van harmonie en vrede in een versplinterde wereld is daarbij een cruciaal doel. Ubuntu is verzoeningsgezind. Nelson Mandela was als leider van de bevrijdingsbeweging ANC, na zijn langdurige gevangenschap op Robbeneiland, sterk door ubuntu geïnspireerd. Het gaf hem moed om met de witte minderheidsregering gesprekken aan te gaan om het discriminerende systeem (...) van Apartheid te overwinnen. Tussen zijn vrijlating in 1990 en de eerste non-raciale verkiezingen in 1994 balanceerde Zuid-Afrika op de rand van een bloedige burgeroorlog. Met geduld en vasthoudendheid wist Mandela echter zijn verzoeningsstrategie door te zetten en een catastrofe te voorkomen. Zijn verzoeningspolitiek is tot op de dag van vandaag een inspiratiebron voor het oplossen van nationale en internationale conflicten. (shrink)

Analyse van de positie van de vrouw in gezins- en arbeidssituaties en de rol van de staat daarbij.

Understanding is a central aim of science and highly important in present-day society. But what precisely is scientific understanding and how can it be achieved? This book answers these questions, through philosophical analysis and historical case studies, and presents a philosophical theory of scientific understanding that highlights its contextual nature.

Robots are increasingly being studied for use in education. It is expected that robots will have the potential to facilitate children’s learning and function autonomously within real classrooms in the near future. Previous research has raised the importance of designing acceptable robots for different practices. In parallel, scholars have raised ethical concerns surrounding children interacting with robots. Drawing on a Responsible Research and Innovation perspective, our goal is to move away from research concerned with designing features that will render robots (...) more socially acceptable by end users toward a reflective dialogue whose goal is to consider the key ethical issues and long-term consequences of implementing classroom robots for teachers and children in primary education. This paper presents the results from several focus groups conducted with teachers in three European countries. Through a thematic analysis, we provide a theoretical account of teachers’ perspectives on classroom robots pertaining to privacy, robot role, effects on children and responsibility. Implications for the field of educational robotics are discussed. (shrink)

The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.

This study is an archeological investigation into the historically changing relationship between words and images. The result is an encyclopedia of interpretative techniques in which language functions as a model of thought. Three periods come to the fore. In the classical one, grammatical structures are responsible for the dominance of describing and identifying activities. Thought about art departs from the idea, that classificatory systems represent images. _Art criticism_ is the form of interpretation in this period. In the modern period time (...) moves to the foreground. Now attention is focused on changes in grammatical forms instead of changes in constant structures. A hermeneutic form of interpretation comes into being: _Art History_. In the postmodern period one realizes that words fail to establish a consistent relationship with visuality. New, semiological theories of interpretation now explain what images can signify. This volume is of interest for philosophers of art and art theoreticians, as well as for students and professionals in both fields. (shrink)

The paper is replaced by a new version (12-2019): DOI: 10.5281/zenodo.3572846 -/- Physical phenomena emerge from the quantum fields everywhere in space. However, not only the phenomena emerge from the quantum fields, the law of the conservation of energy must have its origin from the same spatial structure. This paper describes the relations between the main law of physics and the mathematical structure of the “aggregated” quantum fields.

In the first part of the paper, factual information is given about developments in European business ethics since it started on a more or less institutionalized basis, five or six years ago. In the second part some comments are presented on the meaning of the developments and the possible causes. Attention is given to resemblances and differences between American and European business ethics. In the short last part some suggestions are proposed about tasks business ethics will face in the next (...) decade. (shrink)

In this paper we have two aims: first, to draw attention to the close connexion between interpretation and scientific understanding; second, to give a detailed account of how theories without a spacetime can be interpreted, and so of how they can be understood. In order to do so, we of course need an account of what is meant by a theory ‘without a spacetime’: which we also provide in this paper. We describe three tools, used by physicists, aimed at constructing (...) interpretations which are adequate for the goal of understanding. We analyse examples from high-energy physics illustrating how physicists use these tools to construct interpretations and thereby attain understanding. The examples are: the ’t Hooft approximation of gauge theories, random matrix models, causal sets, loop quantum gravity, and group field theory. (shrink)

The rise of the mechanistic worldview in the seventeenth century had a major impact on views of biological generation. Many seventeenth century naturalists rejected the old animist thesis. However, the alternative view of gradual mechanistic formation in embryology didn’t convince either. How to articulate the peculiarity of life? Researchers in the seventeenth century proposed both “animist” and mechanistic theories of life. In the eighteenth century again a controversy in biology arose regarding the explanation of generation. Some adhered to the view (...) that life is a physical property of matter (e.g. Buffon), others saw living entities as the result of the development of pre-existing germs (e.g. Bonnet). Naturalists, lacked a convincing account that could guide their research. In interaction with leading naturalists of his time Immanuel Kant articulated an approach to explaining generation. Kant’s account, delineated in his Kritik der Urteilskraft (Critique of the power of judgment) (1790), is a combination of Newtonian non-reductionist mechanism in explanation, and a concept of natural end comparable to Stahl’s formal conception of organic bodies. It consists of two claims: a) in biology only mechanical explanation is explanatory, and b) living entities contain some original organisation, which is mechanically unexplainable. In the nineteenth century this approach influenced naturalists as Müller, Virchow, and Von Baer, in their physiological research. Dissatisfied with a sheer mechanistic or, on the other hand, a sheer teleological approach, they appreciated the Kantian account of mechanical explanation of natural ends. In Germany, in the second halve of the nineteenth century, Ernst Haeckel reopened the debate about abiogenesis, which still continuous. (shrink)

Global society is facing formidable current and future problems that threaten the prospects for justice and peace, sustainability, and the well-being of humanity both now and in the future. Many of these problems are related to science and technology and to how they function in the world. If the social responsibility of scientists and engineers implies a duty to safeguard or promote a peaceful, just and sustainable world society, then science and engineering education should empower students to fulfil this responsibility. (...) The contributions to this special issue present European examples of teaching social responsibility to students in science and engineering, and provide examples and discussion of how this teaching can be promoted, and of obstacles that are encountered. Speaking generally, education aimed at preparing future scientists and engineers for social responsibility is presently very limited and seemingly insufficient in view of the enormous ethical and social problems that are associated with current science and technology. Although many social, political and professional organisations have expressed the need for the provision of teaching for social responsibility, important and persistent barriers stand in the way of its sustained development. What is needed are both bottom-up teaching initiatives from individuals or groups of academic teachers, and top-down support to secure appropriate embedding in the university. Often the latter is lacking or inadequate. Educational policies at the national or international level, such as the Bologna agreements in Europe, can be an opportunity for introducing teaching for social responsibility. However, frequently no or only limited positive effect of such policies can be discerned. Existing accreditation and evaluation mechanisms do not guarantee appropriate attention to teaching for social responsibility, because, in their current form, they provide no guarantee that the curricula pay sufficient attention to teaching goals that are desirable for society as a whole. (shrink)