This study examined the effect of problem-based instruction (PBI) on undergraduate students and prospective secondary mathematics teachers’ perceptions of mathematical proof and pedagogical views. Quantitatively, the Mathematical Proof Survey (MPS) was developed and used to assess the views of mathematical proof held by undergraduates in lecture-based and PBI mathematics courses. Qualitatively, research interviews examined the way teacher candidates’ experiences as mathematics learners in the courses affected their conceptions of mathematical proof and views of learning and teaching mathematics and proof. Findings from quantitative analysis of MPS data and qualitative analysis of interview data are presented, and results from the comparative analysis are discussed for implications. The results of the study suggest that experiences with proof and instruction in such PBI courses provide opportunities for undergraduates and prospective mathematics teachers to develop more humanistic perspectives of proof and process-oriented pedagogical views than do lecture-based courses. / text

published_or_final_version / Education / Master / Master of Education

L'expertise psychiatrique est requise au tribunal dans plusieurs situations juridiques tant en matière criminelle que civile et elle est soumise aux mêmes règles de preuve que n'importe quelle expertise. Pourtant, la psychiatrie et son objet sont tout à fait particuliers. La relation que peuvent entretenir le juge et l'expert-psychiatre est teintée par plusieurs éléments de nature sociale et professionnelle, mais aussi simplement juridique et procédural. Alors que les juristes parlent de cette relation comme d'une usurpation du rôle du juge par l'expert-psychiatre, les psychiatres, au contraire, croient que leur expertise est totalement pervertie dans le processus judiciaire. Mais la réalité n'est pourtant pas univoque: si l'expertise psychiatrique est de façon générale une preuve parmi les autres, elle peut également occuper une place centrale dans le mécanisme décisionnel. / Psychiatric report is required at the court in several legal statements as weIl in criminal matters as in civic matters and is subject to the same codes of proofs as any other expert report. Still, psychiatry and its object are absolutely peculiar. The relation that may be kept between the judge and the expert psychiatrist is tinged with many elements of social and professional nature but also of judicial and quibbling nature, simply. While lawyers and legal experts quote that relationship of an usurpation of the judge part by the expert psychiatrist , psychiatrists on the other hand reckon that the expert report is completely perverted by the judicial process. Yet, reality is not univocal. If the psychiatric report is globally a proof among many others, it can still have a crucial impact in the ruling machinery. / "Mémoire présenté à la Faculté des études supérieures en vue de l'obtention du grade de maître en droit (LL. M.)". Ce mémoire a été accepté à l'unanimité et classé parmi les 10% des mémoires de la discipline. Commentaires du jury : "Excellent mémoire. Bien construit, bien écrit. Encourageons la candidate à poursuivre au doctorat".

Disertacijoje nagrinėjamos žinių logikos su centrinio agento sąveikos aksioma. Tyrimas apima multimodalines logikas Kn, Tn, K4n ir S4n. Disertacijos tikslas – baigtinės išvedimo paieškos sekvenciniai skaičiavimai minėtoms logikoms. Darbe pristatomas naujas išvedimo paieškos baigtinumą užtikrinantis metodas, kuris yra pritaikomas minėtoms logikoms, o taip pat monomodalinėms logikoms K4 ir S4. / In the dissertation epistemic logics with central agent interaction axiom are analysed. The research covers multimodal logics Kn, Tn, K4n and S4n. The aim of the work is finite derivation search sequent calculi for the mentioned logics. A new method to obtain the termination of derivation search is presented in the thesis and this method is applied to the mentioned logics as well as to monomodal logics K4 and S4.

In the dissertation epistemic logics with central agent interaction axiom are analysed. The research covers multimodal logics Kn, Tn, K4n and S4n. The aim of the work is finite derivation search sequent calculi for the mentioned logics. A new method to obtain the termination of derivation search is presented in the thesis and this method is applied to the mentioned logics as well as to monomodal logics K4 and S4. / Disertacijoje nagrinėjamos žinių logikos su centrinio agento sąveikos aksioma. Tyrimas apima multimodalines logikas Kn, Tn, K4n ir S4n. Disertacijos tikslas – baigtinės išvedimo paieškos sekvenciniai skaičiavimai minėtoms logikoms. Darbe pristatomas naujas išvedimo paieškos baigtinumą užtikrinantis metodas, kuris yra pritaikomas minėtoms logikoms, o taip pat monomodalinėms logikoms K4 ir S4.

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This study explores how successful Alvin Plantinga is in his contention that belief in God can be obtained and maintained in a basic way that attains and retains rationality for reflective persons. Plantinga indeed calls into question any confident presumption that theistic belief is epistemically irresponsible. He not only seriously challenges the necessity for propositional evidence to be available for such belief to be justified, he also supplies significant support for the conclusion that it remains legitimate even if it faces a preponderance of contrary considerations. However, Plantinga does not convincingly demonstrate that basic theistic belief merits privileged status by virtue of a character sufficiently analogous to paradigmatic perceptual, memory and ascriptive beliefs. Nor does he adequately argue its independence from the bearing of evidentialist concerns, especially regarding its background moorings. He needs to do more work to show the full warrant for theistic belief.

The research for this thesis has followed two main paths: the one of probability logics and the other of type systems and logical frameworks, bringing them together through interactive theorem proving. With the development of computer technology and the need to capture real-world dynamics, situations, and problems, reasoning under uncertainty has become one of the more important research topics of today, and one of the tools for formalizing this kind of knowledge are probability logics. Given that probability logics, serving as decision-making or decision-support systems, often form a basis for expert systems that find their application in fields such as game theory or medicine, their correct functioning is of great importance, and formal verification of their properties would add an additional level of security to the design process. On the other hand, in the field of logical frameworks and interactive theorem proving, attention has been directed towards a more natural way of encoding formal systems where derivation rules are subject to side conditions which are either rather difficult or impossible to encode naively, in the Edinburgh Logical Framework \LF or any other type-theory based Logical Framework, due to their inherent limitations, or to the fact that the formal systems in question need to access the derivation context, or the structure of the derivation itself, or other structures and mechanisms not available at the object level. The first part of the thesis deals with the development and formal verification of probability logics. First, we introduce a Probability Logic with Conditional Operators - LPCP, its syntax, semantics, and a sound and strongly-complete axiomatic system, featuring an infinitary inference rule. We prove the obtained formalism decidable, and extend it so as to represent evidence, making it the first propositional axiomatization of reasoning about evidence. Next, we show how to encode probability logics LQI and LQnI in the Proof Assistant Coq. Both of these logics extend classical logic with modal-like probability operators, and both feature an infinitary inference rule. LQI allows iterations of probability operators, while LQnI does not. We proceed to formally verify their key properties - soundness, strong completeness, and non-compactness. In this way, we formally justify the use of probabilistic SAT-solvers for the checking of consistency-related questions. In the second part of the thesis, we present LFP - a Logical Framework with External Predicates, by introducing a mechanism for locking and unlocking types and terms into LF, allowing the use of external oracles. We prove that LFP satisfies all of the main meta-theoretical properties (strong normalization, confluence, subject reduction, decidability of type checking). We develop a corresponding canonical framework, allowing for easy proofs of encoding adequacy. We provide a number of encodings - the simple untyped lambda-calculus with a Call-by-Value reduction strategy, the Design-by-Contract paradigm, a small imperative language with Hoare Logic, Modal Logics in Hilbert and Natural Deduction style, and Non-Commutative Linear Logic (encoded for the first time in an LF-like framework), illustrating that in LFP we can encode side-conditions on the application of rules elegantly, and achieve a separation between derivation and computation, resulting in cleaner and more readable proofs. We believe that the results presented in this thesis can serve as a foundation for fruitful future research. On the one hand, the obtained formal correctness proofs add an additional level of security when it comes to the construction of expert systems constructed using the verified logics, and pave way for further formal verification of other probability logics. On the other hand, there is room for further improvement, extensions, and deeper analysis of the LFP framework, as well as the building of a prototype interactive theorem prover based on LFP and discovering its place in the world of proof assistants.

On présente une Logique Probabiliste avec des Operateurs Conditionnels - LPCP, sa syntaxe, sémantique, axiomatisation correcte et fortement complète, comprenant une règle de déduction infinitaire. On prouve que LPCP est décidable, et on l'étend pour qu'il puisse représenter l'évidence, en créant ainsi la première axiomatisation propositionnelle du raisonnement basé sur l'évidence. On codifie les Logiques Probabilistes LPP1Q et LPPQ2 dans l'Assistant de Preuve Coq, et on vérifie formellement leurs propriétés principales: correction, complétude fort et non-compacité. Les deux logiques étendent la Logique Classique avec des opérateurs de probabilité, et présentent une règle de déduction infinitaire. LPPQ1 permet des itérations des opérateurs de probabilité, lorsque LPPQ2 ne le permet pas. On a formellement justifié l'utilisation des solveurs SAT probabilistes pour vérifier les questions liées à la cohérence. On présente LFP, un Cadre Logique avec Prédicats Externes, en introduisant un mécanisme pour bloquer et débloquer types et termes dans LF, en permettant l'utilisation d'oracles externes. On démontre que LFP satisfait tous les principales propriétés et on développe un cadre canonique correspondant, qui permet de prouver l'adéquation. On fournit diverses encodages - le λ-calcul non-typé avec la stratégie de réduction CBV, Programmation-par-Contrats, un langage impératif avec la Logique de Hoare, des Logiques Modales et la Logique Linéaire Non-Commutative, en montrant que en LFP on peut codifier aisément des side-conditions dans l'application des règles de typage et atteindre une séparation entre vérification et computation, en obtenant des preuves plus claires et lisibles.

The aim of this thesis is to analyze the role that environmental technology plays in the solution of environmental problems in cities, and discuss models and conditions that can facilitate the processes of selection, implementation and use of environmental technologies in and by cities. The technological component is perhaps one of the most important characteristics of modern cities. The dependence of humans on technology is in most cases a given, something that is not ignored in the sustainability debate. The development and implementation of new, “better” technologies is however hindered by the inertia that modern societies have and the influence of the dominant systems (e.g. economic systems based on growth, extraction of natural resources and environmental disturbance). So-called environmental technologies are not always able to efficiently compete against other technologies that are embedded in societies by lock-in mechanisms, e.g. learning by doing and using, scale economies, subsidies, and network externalities. Even with the “right” technologies, an exclusively techno-centered approach to sustainability can result in other problems, and it might reduce the sustainability debate and the cities’ role in it to discussions of an administrative nature. The actual role of local actors and their agency must be also considered in the models and frameworks directed at understanding sustainability transition processes. It is thus important to analyze the dynamics of technology selection, implementation, use and diffusion in cities from a stakeholders’ perspective as well. Not only is the availability of technology of interest for understanding the impact it has on the environment, but also the intensity of its use. This has resulted in increased attention from politicians and scholars on the so-called global cities (e.g. London, New York, Tokyo), which are characterized by their intense use of e.g. transport, security and surveillance, and information and communication. Paradigmatic models of sustainability can however be contested when the role of local actors, power and agency are considered in detail and not isolated from the context. Some authors recognize the need to address what they call “ordinary cities”, since focusing on the cities’ comparative level of development (be it political, economic or technological) hinders the possibility of bidirectional learning. In the end, sustainability is a “collective good,” which means that it is in everyone’s interest to coordinate efforts and learn from the best practices, regardless of where they come from. This thesis focuses on “ordinary cities,” and promises to offer conclusions that can contribute to a better understanding of how societies can learn from each other and how environmental technologies can have deeper and better results when implemented in different contexts than the ones where they were developed. Two questions related to the process of environmental-technology adaptation are addressed in this thesis: How do technology adaptation processes for the solution of urban environmental problems take place in cities? And how do cities benefit from environmental technologies? It is found that environmental technology is not only seen as a solution to environmental problems in cities, but every day more as a component of strategies to attract attention and compete for resources in national and international markets. Cities have different adaptation and learning strategies. This means that technological solutions have to be flexible and adaptive to local conditions, and allow for vernacular knowledge and past experiences to enrich their performance by facilitating their connection to existing systems. Learning between cities is important and necessary for global sustainability transitions. When it comes to environmental technology, this process is facilitated by strong proof-of-concept projects. Such projects are not only expected to be able to show their technical ability to solve a problem, but must also offer contextual connections to the problems faced by interested cities or potential implementers.