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)

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)

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]).

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.

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.

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)

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)

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.

In recent years, there has been a heated debate about how to interpret findings that seem to show that humans rapidly and automatically calculate the visual perspectives of others. In the current study, we investigated the question of whether automatic interference effects found in the dot-perspective task (Samson, Apperly, Braithwaite, Andrews, & Bodley Scott, 2010) are the product of domain-specific perspective-taking processes or of domain-general “submentalizing” processes (Heyes, 2014). Previous attempts to address this question have done so by implementing inanimate (...) controls, such as arrows, as stimuli. The rationale for this is that submentalizing processes that respond to directionality should be engaged by such stimuli, whereas domain-specific perspective-taking mechanisms, if they exist, should not. These previous attempts have been limited, however, by the implied intentionality of the stimuli they have used (e.g. arrows), which may have invited participants to imbue them with perspectival agency. Drawing inspiration from “novel entity” paradigms from infant gaze-following research, we designed a version of the dot-perspective task that allowed us to precisely control whether a central stimulus was viewed as animate or inanimate. Across four experiments, we found no evidence that automatic “perspective-taking” effects in the dot-perspective task are modulated by beliefs about the animacy of the central stimulus. Our results also suggest that these effects may be due to the task-switching elements of the dot-perspective paradigm, rather than automatic directional orienting. Together, these results indicate that neither the perspective-taking nor the standard submentalizing interpretations of the dot-perspective task are fully correct. (shrink)

What if the Big Bang was an exodus of non-existing reality into existence? In this book a theory of the reality is started from the principle that 'existing takes time' but in states of the universe there are pre-things in moments, thus non-existing, which realize in strings over specific time intervals as existing phenomena. Causality must be reviewed now and new paradigms for reality follow. The existing and observed universe are discontinuous and "limits" in mathematical models do not correspond to (...) reality. New interpretations in evolution theory and the human condition are being investigated. (shrink)

In this paper we consider the algebra of differential operators with polynomial coefficients acting on Clifford algebra-valued functions from both sides. We characterize the subalgebra of SO(m)-invariant differential operators, which itself contains the subalgebra of GL(m)-invariant differential operators.

Libertarians often bill their theory as an alternative to both the traditional Left and Right. _The Routledge Handbook of Libertarianism_ helps readers fully examine this alternative, without preaching it to them, exploring the contours of libertarian thinking on justice, institutions, interpersonal ethics, government, and political economy. The 31 chapters--all written specifically for this volume--are organized into five parts. Part I asks, what should libertarianism learn from other theories of justice, and what should defenders of other theories of justice learn from (...) libertarianism? Part II asks, what are some of the deepest problems facing libertarian theories? Part III asks, what is the right way to think about property rights and the market? Part IV asks, how should we think about the state? Finally, part V asks, how well can libertarianism deal with some of the major policy challenges of our day, such as immigration, trade, religion in politics, and paternalism in a free market. Among the _Handbook_’s chapters are those from critics who write about what they believe libertarians get right as well as others from leading libertarian theorists who identify what they think libertarians get wrong. As a whole, the _Handbook_ provides a comprehensive, clear-eyed look at what libertarianism has been and could be, and why it matters. (shrink)

Millions of human biological samples are stored worldwide for medical research or treatment purposes. These biospecimens are of enormous potential value to law enforcement as DNA profiles can be obtained from these samples. However, forensic use of such biospecimens raises a number of ethical questions. This article aims to explore ethical issues of using human bodily material in medical biobanks for crime investigation and prosecution purposes. Concerns about confidentiality, trust, autonomy and justice will be discussed. We explore how to balance (...) these concerns against the importance of crime solving. Relevant case examples of forensic use of medical biobanks show that requests by law enforcement to access biobanks are handled in disparate ways. We identify some core ethical issues and conclude that further research on these issues is needed to provide ethical guidance. (shrink)

Over 30 million people worldwide have taken a commercial at-home DNA test, because they were interested in their genetic ancestry, disease predisposition or inherited traits. Yet, these consumer DNA data are also increasingly used for a very different purpose: to identify suspects in criminal investigations. By matching a suspect’s DNA with DNA from a suspect’s distant relatives who have taken a commercial at-home DNA test, law enforcement can zero in on a perpetrator. Such forensic use of consumer DNA data has (...) been performed in over 200 criminal investigations. However, this practice of so-called investigative genetic genealogy raises ethical concerns. In this paper, we aim to broaden the bioethical analysis on IGG by showing the limitations of an individual-based model. We discuss two concerns central in the debate: privacy and informed consent. However, we argue that IGG raises pressing ethical concerns that extend beyond these individual-focused issues. The very nature of the genetic information entails that relatives may also be affected by the individual customer’s choices. In this respect, we explore to what extent the ethical approach in the biomedical genetic context on consent and consequences for relatives can be helpful for the debate on IGG. We argue that an individual-based model has significant limitations in an IGG context. The ethical debate is further complicated by the international, transgenerational and commercial nature of IGG. We conclude that IGG should not only be approached as an individual but also—and perhaps primarily—as a collective issue. There are no data in this work. (shrink)

In 1988 Fodor and Pylyshyn issued a challenge to the newly-popular connectionism: explain the systematicity of cognition without merely implementing a so-called classical architecture. Since that time quite a number of connectionist models have been put forward, either by their designers or by others, as in some measure demonstrating that the challenge can be met (e.g., Pollack, 1988, 1990; Smolensky, 1990; Chalmers, 1990; Niklasson and Sharkey, 1992; Brousse, 1993). Unfortu- nately, it has generally been unclear whether these models actually do (...) have this implication (see, for instance, the extensive philosophical debate in Smolensky, 1988; Fodor and McLaughlin, 1990; van Gelder, 1990, 1991; McLaughlin, 1993a, 1993b; Clark, 1993). Indeed, we know of no major supporter of classical orthodoxy who has felt compelled, by connectionist models and argu- ments, to concede in print that connectionists have in fact delivered a non-classical explanation of systematicity. (shrink)

An innovative, multifaceted approach to scientific experiments as designed by and shaped through interaction with the modeling process The role of scientific modeling in mediation between theories and phenomena is a critical topic within the philosophy of science, touching on issues from climate modeling to synthetic models in biology, high energy particle physics, and cognitive sciences. Offering a radically new conception of the role of data in the scientific modeling process as well as a new awareness of the problematic aspects (...) of data, this cutting-edge volume offers a multifaceted view on experiments as designed and shaped in interaction with the modeling process. Contributors address such issues as the construction of models in conjunction with scientific experimentation; the status of measurement and the function of experiment in the identification of relevant parameters; how the phenomena under study are reconceived when accounted for by a model; and the interplay between experimenting, modeling, and simulation when results do not mesh. Highlighting the mediating role of models and the model-dependence of data measurement, this volume proposes a normative and conceptual innovation in scientific modeling--that the phenomena to be investigated and modeled must not be precisely identified at the start but specified during the course of the interactions arising between experimental and modeling activities. Contributors: Nancy D. Cartwright, U of California, San Diego; Anthony Chemero, U of Cincinnati; Ronald N. Giere, U of Minnesota; Jenann Ismael, U of Arizona; Tarja Knuuttila, U of South Carolina; Andrea Loettgers, U of Bern, Switzerland; Deborah Mayo, Virginia Tech; Joseph Rouse, Wesleyan U; Paul Teller, U of California, Davis; Michael Weisberg, U of Pennsylvania; Eric Winsberg, U of South Florida. (shrink)

Although most consumers are positive about socially responsible companies, in order to benefit from CSR efforts, effective and clear CSR communication is important. However, due to the constantly rising profusion of eco-labels, based on either own claims from the organization or claims made by an external third party, consumers may encounter difficulties in identifying truly responsible firms, which could result in less effective CSR initiatives, even for those responsible firms. Therefore, building on attribution theory, this study seeks to identify how (...) uncertified internal CSR claims and external third-party CSR labels should be used in order to deter greenwashing and increase positive consumer evaluations. Within a 3 × 2 design, respondents are exposed to different coffee product packages measuring their attitude toward the brand, corporate credibility, purchase intention, and scent perception, as well as perceived attributional CSR motives. Overall, findings indicate that especially an external CSR label affects consumer responses toward the firm. Moreover, perceived CSR motives serve as a mediator between an external CSR label and corporate credibility and brand attitude, respectively. These findings warrant further consideration of introducing an external multilevel rating systems by governmental law. (shrink)

Experimental modeling is the construction of theoretical models hand in hand with experimental activity. As explained in Section 1, experimental modeling starts with claims about phenomena that use abstract concepts, concepts whose conditions of realization are not yet specified; and it ends with a concrete model of the phenomenon, a model that can be tested against data. This paper argues that this process from abstract concepts to concrete models involves judgments of relevance, which are irreducibly normative. In Section 2, we (...) show, on the basis of several case studies, how these judgments contribute to the determination of the conditions of realization of the abstract concepts and, at the same time, of the quantities that characterize the phenomenon under study. Then, in Section 3, we compare this view on modeling with other approaches that also have acknowledged the role of relevance judgments in science. To conclude, in Section 4, we discuss the possibility of a plurality of relevance judgments and introduce a distinction between locally and generally relevant factors. (shrink)

After having defined the KARO logic for specifying intelligent agents in earlier work we now turn to the question how to realise agents specified in the KARO framework. To this end we look at agent programming languages that we have defined, and investigate how programs in these languages can be linked to the KARO logic.