Encoding many-valued logic in {\lambda}-calculus

We extend the well-known Church encoding of two-valued Boolean Logic in $\lambda$-calculus to encodings of $n$-valued propositional logic (for $3\leq n\leq 5$) in well-chosen infinitary extensions in $\lambda$-calculus. In case of three-valued logic we use the infinitary extension of the finite $\lambda$-calculus in which all terms have their B\"ohm tree as their unique normal form. We refine this construction for $n\in\{4,5\}$. These $n$-valued logics are all variants of McCarthy's left-sequential, three-valued propositional calculus. The four- and five-valued logic have been given complete axiomatisations by Bergstra and Van de Pol. The encodings of these $n$-valued logics are of interest because they can be used to calculate the truth values of infinitary propositions. With a novel application of McCarthy's three-valued logic we can now resolve Russell's paradox. Since B\"ohm trees are always finite in Church's original $\lambda{\mathbf I}$-calculus, we believe their construction to be within the technical means of Church. Arguably he could have found this encoding of three-valued logic and used it to resolve Russell's paradox.Comment: 15 page

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An In Vitro Comparison of the Rake Angles Between K3 and ProFile Endodontic File Systems

The purpose of this study was to compare rake angles of the ProFile and K3 file systems. Twenty-five 40/0.06 taper files were obtained for each system. Five files from the same manufacturer were placed perpendicularly into a vial of Epoxicure Resin and left to set for 24 h. The set-ups were removed from the vials and each were sectioned 5 mm from the tip of the files and polished. A photomicrograph was taken of each file with 100× magnification. Five sets of ProFile and five sets of K3 files were processed in this manner. Images were captured digitally, and rake angles of each file were measured. Multivariate ANOVA found a significant difference (p \u3c 0.001) among the three negative rake angles of the ProFile system compared with the K3 system

The effects of the design and development of a chemistry curriculum reform on teachers’ professional growth: a case study

A curriculum innovation requires new learning material for students and a preparation program for teachers, in which teacher learning is a key ingredient. In this paper we describe how three experienced teachers, involved in the development and subsequent classroom enactment of student learning material for context-based chemistry education, professionalized. For data collection a questionnaire, three interviews and discussion transcripts were used. Our results show that: (a) teachers, cooperating in a network under supervision of an expert, can develop innovative learning material; (b) the development of learning material can be seen as a powerful program to prepare teachers for an innovation; and (c) teachers’ knowledge increased in all five pedagogical content knowledge domains during the development and class enactment phases

Salut personnel et socialisation religieuse dans les assemblées de Dieu de Polynésie française

Un des paradoxes du pentecôtisme semble reposer sur sa capacité à promouvoir à la fois individualisme et communautarisation. Cet article, fondé sur une étude extensive de la socialisation opérée par les Assemblées de Dieu polynésiennes, s’efforce de dénouer cette contradiction. Il montre comment un travail institutionnel « invisible » permet aux croyants d’établir une continuité symbolique entre l’autonomisation vis-à-vis des structures traditionnelles d’encadrement qui a précédé leur conversion et leur intégration à une communauté religieuse exigeante au sein de laquelle les contrôles communautaires et institutionnels sont subjectivement vécus comme des relations personnelles « enchantées » avec Dieu.An apparent paradox of Pentecostalism lies in its ability to promote both individualism and community. This article, based on an extended study of socialization within the Polynesian Assemblies of God, attempts to unravel this contradiction. It shows how an « unseen » institutional work enables believers to establish a symbolic continuity between the distanciation from traditional structures of control that preceded their conversion, and their integration into a religious constraining community where institutional and community controls are subjectively experienced as personal « enchanted » relations with God. In this religious paradigm, « brothers and sisters in Christ » do not replace the traditional family, but rather form a new space of possible « familial » ties based upon personal affinities.Una de las paradojas del pentecostismo parece apoyarse en su capacidad de promover al mismo tiempo individualismo y comunitarización. Este articulo, basado en un estudio extensivo de la socialización que se realiza en las Asambleas de Dios polinesias, trata de desentrañar esta contradicción. Muestra cómo el trabajo institucional ‘invisible’ permite a los creyentes establecer una continuidad simbólica entre la autonomización de las estructuras tradicionales de control que precedió las conversiones y la integración en una comunidad religiosa exigente al interior de la cual los controles comunitarios e institucionales se viven subjetivamente en tanto que relaciones personales ‘encantadas’ con Dio

Assessing teachers’ beliefs to facilitate the transition to a new chemistry curriculum: what do the teachers want?

In this article, we describe the results of a study of chemistry high school teachers’ beliefs (N = 7) of the chemistry curriculum and their roles, their beliefs on the teacher as developer of materials, and their beliefs about professional development. Teachers’ beliefs influence the implementation of a curriculum. We view the use of a new curriculum as a learning process, which should start at teachers’ prior knowledge and beliefs. The results reveal that it is possible to develop a new curriculum in which teachers’ beliefs are taken as a starting point. Promising approaches to prepare teachers for a new curriculum is to let them (co)develop and use curriculum materials: It creates ownership, and strengthens and develops teachers’ pedagogical content knowledge (PCK)

Qualitative Evaluation of Emotional Intelligence In-Service Program for Secondary School Teachers

This paper is an attempt to evaluate the Emotional Intelligence (EQ) In-Service Program on the basis of experiences of 20 secondary school teachers who attended the program in a private school in Turkey. A phenomenological approach, with a focus group method was used. The first objective of this study was to evaluate EQ program on the basis of teachers’ experiences. The second was to explore the teachers’ perceptions about the implications of EQ skills in classroom situations. The results were discussed in terms of the evaluation of the EQ program and its implications for learning and teaching activities in the classroom

Unitary expansion of the time evolution operator

We propose an expansion of the unitary evolution operator, associated to a given Schr\"odinger equation, in terms of a finite product of explicit unitary operators. In this manner, this unitary expansion can be truncated at the desired level of approximation, as shown in the given examples.Comment: 6 pages, 7 figures. Updated version, minor final change