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.

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)

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 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)

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)

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.

In der Geschichte des Wiener Kreises lässt sich eine private und eine öffentliche Phase unterscheiden. Die private Phase ist 1929 zu Ende. Im Sommer dieses Jahres beschliessen die Mitglieder mit ihrer Philosophie der wissenschaftlichen Weltauffassung vor die Öffentlichkeit zu treten.

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 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.

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

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)

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)

Abstract.Evolutionary psychology and behavioural genomics are both approaches to explain human behaviour from a genetic point of view. Nonetheless, thus far the development of these disciplines is anything but interdependent. This paper examines the question whether evolutionary psychology can contribute to behavioural genomics. Firstly, a possible inconsistency between the two approaches is reviewed, viz. that evolutionary psychology focuses on the universal human nature and disregards the genetic variation studied by behavioural genomics. Secondly, we will discuss the structure of biological explanations. (...) Some philosophers rightly acknowledge that explanations do not involve laws which are exceptionless and universal. Instead, generalisations that are invariant suffice for successful explanation as long as two other stipulations are recognised: the domain within which the generalisation has no exceptions as well as the distribution of the mechanism described by the generalisation should both be specified. It is argued that evolutionary psychology can contribute to behavioural genomic explanations by accounting for these two specifications. (shrink)

The contributions in this volume focus on the Bayesian interpretation of natural languages, which is widely used in areas of artificial intelligence, cognitive science, and computational linguistics. This is the first volume to take up topics in Bayesian Natural Language Interpretation and make proposals based on information theory, probability theory, and related fields. The methodologies offered here extend to the target semantic and pragmatic analyses of computational natural language interpretation. Bayesian approaches to natural language semantics and pragmatics are based on (...) methods from signal processing and the causal Bayesian models pioneered by especially Pearl. In signal processing, the Bayesian method finds the most probable interpretation by finding the one that maximizes the product of the prior probability and the likelihood of the interpretation. It thus stresses the importance of a production model for interpretation as in Grice’s contributions to pragmatics or in interpretation by abduction. (shrink)

This contribution discusses the expression ‘Image of the Father’ as a case in point to Aquinas's approach of naming Christ: Christology and Trinitarian theology, as well as the discussion of analogical naming in divinis, need to be taken together. ‘Image’ and ‘Father’ are predicated differently of Christ and God the Father than of human beings. Moreover, God is ‘Father’ in a different way towards the Son than regarding human beings. Christ is the unique image of the Father: the invisible image (...) of the invisible Father. (shrink)

Zusammenfassung Sportökonomen empfehlen dem europäischen Profifußball eine Rezeption der Kooperationsdesigns der US-amerikanischen Profiligen, da diese als Prototyp eines langfristig überlebensfähigen Meisterschaftsbetriebs angesehen werden. Der vorliegende Beitrag argumentiert auf der Basis einer Rekonstruktion der Entwicklung der Zentralvermarktung der Fußball-Bundesliga, dass dem deutschen Profifußball diese Option versperrt ist, da die politischen Anspruchsgruppen des Fußballs die Bewahrung des europäischen Sportmodells einfordern. Zugleich hat das Zusammenspiel von politischen Regulierungsbemühungen und Medienentwicklung zur Einführung eines höchst problematischen Wettbewerbsdesigns im europäischen Profifußball geführt, das aber schwer veränderbar (...) erscheint. (shrink)

Zusammenfassung Zu den gegenwärtig am intensivsten diskutierten Themen der Politikwissenschaft gehört die Debatte um den Aufstieg des Regulierungsstaates. Während einige Autoren den Regulierungsstaat für eine neue „intelligente“ Form von Staatlichkeit halten, die ihre Politikziele durch eine geschickte Architektur von Regulierungsregimen erreicht, verweisen andere Autoren auf seine Tendenz zur Kolonialisierung gesellschaftlicher Sektoren und zu utopischen Ambitionen. Der vorliegende Beitrag vertritt die Auffassung, dass das Vordringen staatlicher Regulierung im Sport einen kritischen Fall für die aktuelle Debatte um Konturen und Ambitionen des Regulierungsstaates (...) konstituiert, weil es sich beim Sport um einen hybriden Sektor handelt, der sowohl ein Unterhaltungsgut produziert, als auch sozialintegrative Leistungen bereitstellt. Die Fallstudie zum Aufstieg des Regulierungsstaates im englischen Fußball versucht die Thesen zu unterstützen, dass (a) die Ambitionen des Regulierungsstaates in starkem Maße kontext- und pfadabhängig sind und (b) die Fähigkeit des „intelligenten“ Regulierungsstaates begrenzt ist, Selbstregulierungsregime zu kreieren, die einen Sektor davor bewahren, von starken kommerziellen Interessen „übernommen“ zu werden. (shrink)

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 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)