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 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)

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)

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)

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)

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)

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]).

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)

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.

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.

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.

This essay attempts to develop a psychologically informed semantics of perception reports, whose predictions match with the linguistic data. As suggested by the quotation from Miller and Johnson-Laird, we take a hallmark of perception to be its fallible nature; the resulting semantics thus necessarily differs from situation semantics. On the psychological side, our main inspiration is Marr's (1982) theory of vision, which can easily accomodate fallible perception. In Marr's theory, vision is a multi-layered process. The different layers have filters of (...) different gradation, which makes vision at each of them approximate. On the logical side, our task is therefore twofold • to formalise the layers and the ways in which they may refine each other, and • to develop logical means to let description vary with such degrees of refinement. The first task is formalised by means of an inverse systems of first order models, with reality appearing as its inverse limit. The second task is formalised by means of so-called conditional quantifiers, a new form of generalised quantification which can best be described as resource bounded quantification. We show that the logic provides for a semantics and pragmatics of direct perception reports. In particular, direct perception reports have a possibly nonveridical, approximative semantics, which becomes veridical only by virtue of our pragmatic expectation that what is perceived would continue to be the case, were we to perceive more accurately. It is a general feature of resource bounded logics that the underlying logics are weak, but that stronger principles can be obtained pragmatically, by strengthening the resource. For the logic of vision this feature is clarified by showing how changes in the resource capture different notions of partiality, and by studying how the perception verb interacts with connectives and quantifiers in different visual contexts. The inference Veridicality, which is now viewed rather as a nonmonotonic inference, is also studied in depth. We end with an attempt to buttress the proposed model by comparing it with suggestions put forward in Cognitive or Conceptual Semantics, in the literature on evidentials, and in Husserl's philosophy of perception. (shrink)

This volume presents contributions to quantifier theory and its applications and gives a good impression of the depth and diversity of recent work in the field. The book starts with a long introduction aimed at making the individual papers accessible to a wide audience of logicians and linguists.

Theories, Technologies, Instrumentalities of Color is the outcome of a workshop, held in Leuven, Belgium, in May 2000.

Classroom discussion is frequently proposes as an essential part of democratic citizenship education. Literature, however, pays little attention to what kind of discussion is most effective and how teachers can facilitate a discussion. This study aims to contribute to the development of a framework for analysing the characteristics of classroom discussions and the different roles teachers can adopt in guiding a discussion on controversial issues. In addition, we investigated how the way teachers guide the discussion is related to the structure (...) and content features of the discussion. The framework was used to analyse five classroom discussions in secondary education. Our framework appeared to be useful for revealing differences in the structure and content features of the classroom discussions and in the way teachers guide the discussion. The results also indicated that a high degree of teacher regulation was related to high content quality and more participation from students. A high degree of stu... (shrink)

Probably colour is the best worked-out example of allegedly neurophysiologically innate response categories determining percepts and percepts determining concepts, and hence biology fixing the basic categories implicit in the use of language. In this paper I argue against this view and I take C. L. Hardin's Color for Philosophers [1988] as my main target. I start by undermining the view that four unique hues stand apart from all other colour shades (Section 2) and the confidence that the solar spectrum is (...) naturally divided into four categories (Section 3). For such categories to be truly universal, they have to be true for all peoples and in Section 4 I show that Berlin and Kay's [1969] widely quoted theory of basic colour categories is not sufficiently supported to lend it any credibility.Having disposed of the view that inspection of language or ‘pure’ perception unveils the universal colour categories. I turn to neurophysiological and psychophysical theories of colour vision to see whether they provide a more solid basis for deciding what the innate response categories are. In Section 5 I show that Hardin's account of the opponent-process theory neither supports his view that ‘colour-coding’takes place early in the visual neural pathway, nor his view that knowledge of colour vision science will help us solve many philosophical mysteries about colour.In Section 6 I give a more detailed review of what is known today about the neurophysiology of colour vision and I show that there's nothing in the brain which could be called a colour module, let alone a module with homunculi for particular basic colour categories. In Section 7 I show that psychophysical models do not support such rigid constraints on category formation either. Hence (Section 8), at least in the case of colour, current science supports a plasticity in the formation of categories that goes far beyond the requirements of those naturalistic philosophers who would like to ground primitive concepts in biology. (shrink)

In the Netherlands, in 2002, euthanasia became a legitimate medical act, only allowed when the due care criteria and procedural requirements are met. Legally, an Advanced Euthanasia Directive can replace direct communication if a patient can no longer express his own wishes. In the past decade, an exponential number of persons with dementia share a euthanasia request with their physician. The impact this on physicians, and the consequent support needs, remained unknown. Our objective was to gain more insight into the (...) experiences and needs of Dutch general practitioners and elderly care physicians when handling a euthanasia request from a person with dementia. We performed a qualitative interview study. Participants were recruited via purposive sampling. The interviews were transcribed verbatim, and analyzed using the conventional thematic content analysis. Eleven general practitioners and elderly care physicians with a variety of experience and different attitudes towards euthanasia for PWD were included. Euthanasia requests appeared to have a major impact on physicians. Difficulties they experienced were related to timing, workload, pressure from and expectations of relatives, society’s negative view of dementia in combination with the ‘right to die’ view, the interpretation of the law and AEDs, ethical considerations, and communication with PWD and relatives. To deal with these difficulties, participants need support from colleagues and other professionals. Although elderly care physicians appreciated moral deliberation and support by chaplains, this was hardly mentioned by GPs. Euthanasia requests in dementia seem to place an ethically and emotionally heavy burden on Dutch GPs and elderly care physicians. The awareness of, and access to, existing and new support mechanisms needs further exploration. (shrink)

In this paper I present an ordered selection of citations from František Wald's publications and correspondence to give some idea of how he fitted in his times, how his views were received by contemporaries, and to illustrate his epistemological views. I highlight the originality of Wald's aim to develop a theory of chemistry on the basis of the concept of 'phase', instead of the concepts of compound and element. In chemistry there has been a vast increase in substances that are (...) phases, which makes Wald's work still relevant. © 2013 by HYLE and Jaap van Brakel. (shrink)

In this paper we extend Wittgenstein’s notion of family resemblance to translation, interpretation, and comparison across traditions. There is no need for universals. This holds for everyday concepts such as green and qing 青, philosophical concepts such as emotion and qing 情, as well as philosophical categories such as form of life and dao 道. These notions as well as all other concepts from whatever tradition are family resemblance concepts. We introduce the notion of quasi-universal, which connects family resemblance concepts (...) from a limited number of traditions. The possibility and necessity of extending family resemblance concepts across traditions dissolves the false antinomy of universalism versus relativism. (shrink)

This paper is on the update semantics for might of Veltman. Threeconsequence relations are introduced and studied in an abstract setting.Next we present sequent-style systems for each of the consequence relations.We show the logics to be complete and decidable. The paper ends with asyntactic cut elimination result.

In this paper I evaluate the soundness of the prototype paradigm, in particular its basic assumption that there are pan-human psychological essences or core meanings that refer to basic-level natural kinds, explaining why, on the whole, human communication and learning are successful. Instead I argue that there are no particular pan-human basic elements for thought, meaning and cognition, neither prototypes, nor otherwise. To illuminate my view I draw on examples from anthropology. More generally I argue that the prototype paradigm exemplifies (...) two assumptions that dominate cognitive science: (1) If human beings use words they mean something particular and what they mean can be discovered by scientific methods. (2) There is a fixed number of domains of categorization, each made up of a fixed number of basic categories. I suggest that these two assumptions lead to Brave New World. (shrink)

Amidst the progress being made in the various (sub-)disciplines of the behavioural and brain sciences a somewhat neglected subject is the problem of how everything fits into one world and, derivatively, how the relation between different levels of discourse should be understood and to what extent different levels, domains, approaches, or disciplines are autonomous or dependent. In this paper I critically review the most recent proposals to specify the nature of interdiscourse relations, focusing on the concept of supervenience. Ideally supervenience (...) is a relation between different discourses which has all the advantages of reduction, but without its disadvantages. I apply the more abstract considerations to two concrete cases: schizophrenia and colour. Usually an interlevel or interdiscourse relation is seen as asymmetrical: the overlaying discourse depends on the underlying discourse (and not vice versa), where the out- or un-spoken assumption is that the ultimate underlying discourse is physical. Instead I argue that scientific categories referred to in interdiscourse relations are, ultimately, dependent on common sense categories and common sense normative criteria. It is the manifest categories and common sense ideas about what is reasonable and what is right that determine the relevant categorisations at the deeper, underlying levels. I suggest that the implications of this are not merely methodological or epistemological. (shrink)

This paper is on the update semantics for might of Veltman. Threeconsequence relations are introduced and studied in an abstract setting.Next we present sequent-style systems for each of the consequence relations.We show the logics to be complete and decidable. The paper ends with asyntactic cut elimination result.

This study investigated the relationship between the development of students? self-regulated learning and students? perceptions of the learning environment in terms of autonomy support, the emphasis on relevance and collaborative learning. In addition, we compared innovative learning environments that aim to enhance self-regulated learning with traditional learning environments. Questionnaires for measuring self-regulated learning and perceptions of the learning environment were administered by 648 students. Self-regulated learning was measured at the start of secondary education and again half way through the first (...) year. The results point to the importance of how students perceive the learning environment for self-regulated learning. There was a positive relationship between autonomy support and relevance and self-regulated learning. Furthermore, students in innovative environments perceived more autonomy support, more emphasis on relevance and more collaborative learning than those in traditional environments. Students in innovative environments, however, reported no more self-regulated learning than students in traditional environments. (shrink)