The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the (...) realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility. (shrink)

In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ 2 0 (...) -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA. (shrink)

V. Lifschitz defined in 1979 a variant of realizability which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis. In this paper we describe an extension of intuitionistic arithmetic in which the soundness of Lifschitz' realizability can be proved, and we give an axiomatic characterization of the Lifschitz-realizable formulas relative to this extension. By a "q-variant" we obtain a new derived rule. We also show how to extend Lifschitz' realizability to second-order arithmetic. Finally we describe (...) an analogous development for elementary analysis, with partial continuous application replacing partial recursive application. (shrink)

The classical forms of both modified realizability and relative realizability are naturally described in terms of the Sierpinski topos. The paper puts these two observations together and explains abstractly the existence of the geometric morphisms and logical functors connecting the various toposes at issue. This is done by advancing the theory of triposes over internal partial combinatory algebras and by employing a novel notion of elementary map.

Following the book Algebraic Set Theory from André Joyal and leke Moerdijk [8], we give a characterization of the initial ZF-algebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (McCarty [9]).

We study the lattice of local operators in Hylandʼs Effective Topos. We show that this lattice is a free completion under internal sups indexed by the natural numbers object, generated by what we call basic local operators.We produce many new local operators and we employ a new concept, sight, in order to analyze these.We show that a local operator identified by A.M. Pitts in his thesis, gives a subtopos with classical arithmetic.

We employ the notions of "sequential function" and "interrogation" (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using Longley's preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is that every realizability topos is a geometric quotient of a realizability topos on a total combinatory algebra.

Kleene's realizability interpretation for first-order arithmetic was shown by Hyland to fit into the internal logic of an elementary topos, the “Effective topos” . In this paper it is shown, that there is an internal realizability definition in , i.e. a syntactical translation of the internal language of into itself of form “n realizes ” , which extends Kleene's definition, and such that for sentences , the equivalence [harr]n is true in . The internal realizability definition depends on finding separated (...) covers for non-separated objects of . However, for the objects arising in higher-order arithmetic, canonical covers are available, which are definable in higher-order arithmetic. These canonical covers yield “covering axioms”. It is shown that these covering axioms, together with uniformity principles and Extended Church's Thesis, axiomatize a formalized extension of Kleene realizability to a constructive system of higher-order arithmetic. The details are worked out for second and third-order arithmetic, but the method can be extended to any order. As an application, it is shown in the final section that a certain completeness property of the category of “modest sets” can be derived in third-order arithmetic from the realizability axioms. (shrink)

Two straightforward “extensionalisations” of Kleene's realizability are considered; denoted re and e. It is shown that these realizabilities are not equivalent. While the re-notion is a subset of Kleene's realizability, the e-notion is not. The problem of an axiomatization of e-realizability is attacked and one arrives at an axiomatization over a conservative extension of arithmetic, in a language with variables for finite sets. A derived rule for arithmetic is obtained by the use of a q-variant of e-realizability; this rule subsumes (...) the well-known Extended Church's Rule. The second part of the paper focuses on toposes for these realizabilities. By a relaxation of the notion of partial combinatory algebra, a new class of realizability toposes emerges. Relationships between the various realizability toposes are given, and results analogous to Robinson and Rosolini's characterization of the effective topos, are obtained for a topos generalizing e-realizability. (shrink)

An axiomatic treatment of synthetic domain theory is presented, in the framework of the internal logic of an arbitrary topos. We present new proofs of known facts, new equivalences between our axioms and known principles, and proofs of new facts, such as the theorem that the regular complete objects are closed under lifting . In Sections 2–4 we investigate models, and obtain independence results. In Section 2 we look at a model in de Modified realizability Topos, where the Scott Principle (...) fails, and the complete objects are not closed under lifting. Section 3 treats the standard model in the Effective Topos. Theorem 3.2 gives a new characterization of the initial lift-algebra relative to the dominance. We prove that in the standard case it is not the internal colimit of the chain 0 → L → L 2 → ⋯ . The models in Sections 2 and 3 compare via an adjunction. Section 4 discusses a model in a Grothendieck topos. A feature here is that N is not well-complete , whereas 2 is. (shrink)

F. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic $\mathbf{HAH}$: $\forall X\lbrack\forall x(x \in X \vee \neg x \in X) \wedge \forall Y(\forall x(x \in Y \vee \neg x \in Y) \rightarrow \forall x(x \in X \rightarrow x \in Y) \vee \forall x \neg(x \in X \wedge x \in Y)) \rightarrow \exists n\forall x(x \in X \rightarrow x = n)\rbrack$, and if not, whether assuming Church's Thesis CT and (...) Markov's Principle MP would help. Blass and Scedrov gave models of $\mathbf{HAH}$ in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP. In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that $RP$ is derivable in $\mathbf{HAH} + \mathrm{CT} + \mathrm{MP} + \mathrm{ECT}_0$, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos. (shrink)

Theory and Practice of Knowledge Transfer: Studies in School Education in the Ancient Near East and Beyond. Edited by W. S. van Egmond and W. H. van Soldt. Publications de l’Institut histo- rique-archéologique néerlandais de Stamboul, vol. 121. Leiden: Nederlands Instituut voor het NabiJe Oosten, 2012. Pp. 111 + 152. €36.04.

Gödel once offered an argument for the general reflection principle in set theory that took the form of an analogy with Leibniz' Monadology. I discuss the mathematical and philosophical background to Gödel's argument, reconstruct the proposed analogy in detail, and argue that it has no justificatory force.

In this paper I address the current debate on ontic versus epistemic conceptualizations of mechanistic explanation in the mechanisms literature. Illari recently argued that good explanations are subject to both ontic and epistemic constraints: they must describe mechanisms in the world in such fashion that they provide understanding of their workings. Elaborating upon Illari’s ‘integration’ account, I argue that causal role function discovery of mechanisms and their components is an epistemic prerequisite for achieving these two aims. This analysis extends Illari’s (...) account in important ways, putting more pressure on ontic readings of mechanistic explanation and providing an answer to the question how ontic and epistemic constraints on mechanistic explanation are related. I argue these point in terms of cases on memory research drawn from neuroscience and research on extinct neurogenetic mechanisms from early nervous systems biology. (shrink)

While 4E approaches to cognition are increasingly introduced in educational contexts, little has been said about how 4E commitments can inform pedagogy aimed at fostering ethical competencies. Here, we evaluate a 4E-inspired ethics exercise that we developed at a technical university to enliven the moral imagination of engineering students. Our students participated in an interactive tinkering workshop, during which they materially redesigned a healthcare artifact. The aim of the workshop was twofold. Firstly, we wanted students to experience how material choices (...) at the levels of design and functionality can enable morally significant reimaginings of the affordances commonly associated with existing artifacts. We term this type of reimagining world-directed moral imagination. Secondly, through the design process, we wanted students to robustly place themselves in the lived embodied perspectives of (potential) users of their selected artifacts. We term this person-directed moral imagination. While student testimonies about the exercise indicate that both their world-directed and person-directed moral imagination were enlivened, we note that the fostering of robust person-directed moral imagination proved challenging. Using 4E insights, we diagnose this challenge and ask how it might be overcome. To this end, we engage extensively with a recent 4E-informed critique of person-directed moral imagination, raised by Clavel Vázquez and Clavel-Vázquez (2023). They argue that person-directed moral imagination is profoundly limited, if not fundamentally misguided, particularly when exercised in contexts marked by emphatic embodied situated difference between the imaginer and the imagined. Building upon insights from both the 4E field and testimonies from critical disability studies, we argue that, while their critique is valuable, it ultimately goes too far. We conclude that a 4E approach can take on board recent 4E warnings regarding the limits of person-directed moral imagination while contributing positively to the development of moral imagination in engineering ethics education. (shrink)

This paper investigates conceptions of explanation, teleology, and analogy in the works of Immanuel Kant (1724-1804) and Georges Cuvier (1769-1832). Richards (2000, 2002) and Zammito (2006, 2012, 2018) have argued that Kant’s philosophy provided an obstacle for the project of establishing biology as a proper science around 1800. By contrast, Russell (1916), Outram (1986), and Huneman (2006, 2008) have argued, similar to suggestions from Lenoir (1989), that Kant’s philosophy influenced the influential naturalist Georges Cuvier. In this article, I wish to (...) expand on and further the work of Russell, Outram, and Huneman by adopting a novel perspective on Cuvier and considering (a) the similar conceptions of proper science and explanation of Kant and Cuvier, and (b) the similar conceptions of the role of teleology and analogy in the works of Kant and Cuvier. The similarities between Kant and Cuvier show, contrary to the interpretation of Richards and Zammito, that some of Kant’s philosophical ideas, whether they derived from him or not, were fruitfully applied by some life scientists who wished to transform life sciences into proper sciences around 1800. However, I also show that Cuvier, in contrast to Kant, had a workable strategy for transforming the life sciences into proper sciences, and that he departed from Kant’s philosophy of science in crucial respects. (shrink)

Empirical studies among small-scale societies show that participation in national development programs impact traditional norms of community cooperation. We explore the extent to which varying levels of village and individual involvement in development policies relate to voluntary cooperation within community settings. We used a field experiment conducted in seven villages from an indigenous society in Indonesia known for their strong traditional cooperative norms, the Punan Tubu. We framed the experiment in terms of an ongoing government house-building program. The results indicate (...) that there were synergistic and antagonistic interactions between existing cooperative norms and government development policies. Participants’ cooperation in the experimental setting was low, probably because the Punan Tubu are used to cooperating and sharing both under demand and in a context in which uncooperative behavior is largely unpunished. Variation in experimental behavior was related to both village- and individual-level variables, with participants living in resettlement villages and participants living in a house constructed under the government program displaying more cooperative behavior. The cooperation evident in resettled villages may indicate that people in these villages are more comfortable interacting in anonymous settings and less committed to the demand-sharing norms still prevalent in the upstream villages. The more cooperative behavior among villagers who have previously received a house might indicate that they recognize that they are now better off than others and feel more obliged to cooperate. Policies aiming to capitalize on existing cooperative behavior to stimulate community collective action should consider the specific conditions under which cooperation occurs in real settings since traditional norms that regulate cooperative behavior might not translate well to cooperation in government-led programs. (shrink)

The Russian invasion of Ukraine on February 24, 2022, has already caused large amounts of greenhouse gas (GHG) emissions and will continue to do so for manyyears after hostilities have ceased mainly because of the emissions linked to the rebuilding of destroyed or damaged housing, public buildings, infrastructure, factories, and the like. My aim in this paper is to discuss how in a time of climate emergency such emissions of war should impact the political morality of states initiating, continuing, and (...) ending war (through a just and enduring peace) as understood by just war theory (JWT). My point of departure is a study by Lennard de Klerk and six co-authors detailing the emissions of the first year of the Russo-Ukrainian war. (shrink)

Dynamic Epistemic Logic This article tells the story of the rise of dynamic epistemic logic, which began with epistemic logic, the logic of knowledge, in the 1960s. Then, in the late 1980s, came dynamic epistemic logic, the logic of change of knowledge. Much of it was motivated by puzzles and paradoxes. The number … Continue reading Dynamic Epistemic Logic →.

Decentralized clinical trials (DCTs) have the potential to advance the conduct of clinical trials, but raise several ethical issues, including obtaining valid informed consent. The debate on the ethical issues resulting from digitalization is predominantly focused on direct risks relating to for example data protection, safety, and data quality. We submit however, that a broader view on ethical aspects of DCTs is needed to touch upon the new challenges that come with the DCT practice. Digitalization has impacts that go beyond (...) its direct purposes, by shaping behaviors, experiences, social relations, and values. We examine four elements of the informed consent procedure that are affected by DCTs, while taking these soft impacts of technologies into account: (i) informing participants and testing understanding, (ii) freedoms in relation to responsibilities and burdens, (iii) trust in participant-researcher relations, and (iv) impacts on the concept of privacy. Our analysis reveals that a broad view is key for optimal conduct of DCTs. In addition, it provides insight into the ethical impacts of DCTs on informed consent. Technologies such as DCTs potentially have profound impacts which are not immediately addressed by the existing regulatory frameworks, but nonetheless important to recognize. These findings can guide future practices of DCTs to foster the important values of clinical research in this novel approach for conducting clinical trials. (shrink)

This book is about the interrelationship between nature, semiosis, metarepresentation and (self-)consciousness, and the role played by metarepresentation in ...

Dynamic Epistemic Logic This article tells the story of the rise of dynamic epistemic logic, which began with epistemic logic, the logic of knowledge, in the 1960s. Then, in the late 1980s, came dynamic epistemic logic, the logic of change of knowledge. Much of it was motivated by puzzles and paradoxes. The number … Continue reading Dynamic Epistemic Logic →.