Thesis--Köln. / Vita. Includes bibliographical references (p. 253-256).

5 Abstract Diploma thesis Parallel between the development of the logical thinking of pupils and the history of logic deals with aplications of method of genetic parallel in logic. On the selected logical problems are shown their fylogenetic (historical) and ontogenetic development. In the historical part of the thesis is discussed especialy the problem of implication, which develops mainly in megaric-stoic logic. Following chapter deal with so-called alternative fhylogeny, ethnographic research. Overview summaries several research articles and monographs exploring the issue of the logical development of children in past decades. Theory of Jean Piaget is mentioned. Next short chapter deals with definition of subject matter of logic at the primary and secondary school (RVP). Also some textbooks are mentioned. The last chapter discusses author's own research.

This thesis addresses an issue in the philosophy of Mathematics which is little discussed, and indeed little recognised. This issue is the phenomenon of a ‘change of setting’. Changes of setting are events which involve a change in a scientific framework which is fruitful for answering questions which were, under an old framework, intractable. The formulation of the new setting usually involves a conceptual re-orientation to the subject matter. In the natural sciences, such re-orientations are arguably unremarkable, inasmuch as it is possible that within the former setting for one’s thinking one was merely in error, and under the new orientation one is merely getting closer to the truth of the matter. However, when the subject matter is pure mathematics, a problem arises in that mathematical truth is (in appearance) timelessly immutable. The conceptions that had been settled upon previously seem not the sort of thing that could be vitiated. Yet within a change of setting that is just what seems to happen. Changes of setting, in particular in their effects on the truth of individual propositions, pose a problem for how to understand mathematical truth. Thus this thesis aims to give a philosophical analysis of the phenomenon of changes of setting, in the spirit of the investigations performed in Wilson (1992) and Manders (1987) and (1989). It does so in three stages, each of which occupies a chapter of the thesis: 1. An analysis of the relationship between mathematical truth and settingchanges, in terms of how the former must be viewed to allow for the latter. This results in a conception of truth in the mathematical sciences which gives a large role to the notion that a mathematical setting must ‘explain itself’ in terms of the problems it is intended to address. 2. In light of (1), I begin an analysis of the change of setting engendered in mathematical logic by Gottlob Frege. In particular, this chapter will address the question of whether Frege’s innovation constitutes a change of setting, by asking the question of whether he is seeking to answer questions which were present in the frameworks which preceded his innovations. I argue that the answer is yes, in that he is addressing the Kantian question of whether alternative systems of arithmetic are possible. This question arises because it had been shown earlier in the 19th century that Kant’s conclusion, that Euclid’s is the only possible description of space, was incorrect. 3. I conclude with an in-depth look at a specific aspect of the logical system constructed in Frege’s Grundgesetze der Arithmetik. The purpose of this chapter is to find evidence for the conclusions of chapter two in Frege’s technical work (as opposed to the philosophical). This is necessitated by chapter one’s conclusions regarding the epistemic interdependence of formal systems and informal views of those frameworks. The overall goal is to give a contemporary account of the possibility of setting-changes; it will turn out that an epistemic grasp of a mathematical system requires that one understand it within a broader, somewhat historical context.

In broadest terms, this thesis is concerned to answer the question of whether the view that arithmetic is analytic can be maintained consistently. Lest there be much suspense, I will conclude that it can. Those who disagree claim that accounts which defend the analyticity of arithmetic are either unable to give a satisfactory account of the foundations of mathematics due to the incompleteness theorems, or, if steps are taken to mitigate incompleteness, then the view loses the ability to account for the applicability of mathematics in the sciences. I will show that this criticism is not successful against every view whereby arithmetic is analytic by showing that the brand of "conventionalism" about mathematics that Rudolf Carnap advocated in the 1930s, especially in Logical Syntax of Language, does not suffer from these difficulties. There, Carnap develops an account of logic and mathematics that ensures the analyticity of both. It is based on his famous "Principle of Tolerance", and so the major focus of this thesis will to defend this principle from certain criticisms that have arisen in the 80 years since the book was published. I claim that these criticisms all share certain misunderstandings of the principle, and, because my diagnosis of the critiques is that they misunderstand Carnap, the defense I will give is of a primarily historical and exegetical nature. Again speaking broadly, the defense will be split into two parts: one primarily historical and the other argumentative. The historical section concerns the development of Carnap's views on logic and mathematics, from their beginnings in Frege's lectures up through the publication of Logical Syntax. Though this material is well-trod ground, it is necessary background for the second part. In part two we shift gears, and leave aside the historical development of Carnap's views to examine a certain family of critiques of it. We focus on the version due to Kurt Gödel, but also explore four others found in the literature. In the final chapter, I develop a reading of Carnap's Principle - the `wide' reading. It is one whereby there are no antecedent constraints on the construction of linguistic frameworks. I argue that this reading of the principle resolves the purported problems. Though this thesis is not a vindication of Carnap's view of logic and mathematics tout court, it does show that the view has more plausibility than is commonly thought.

[pt] Não obstante o reconhecimento de que a lógica desenvolvida por Guilherme de Ockham é consideravelmente interessante tanto do ponto de vista histórico quanto filosófico, pouca atenção foi dada àquelas doutrinas lógicas que envolvem aspectos temporais. Este fato, por sua vez, constitui um obstáculo à compreensão integral da lógica ockhamista, já que acarreta uma série de controvérsias motivadas basicamente por interpretações parciais que menosprezam o papel desempenhado pelo tempo na lógica medieval. Com efeito, o presente estudo analisa o caráter temporal da lógica de Ockham a fim de contextualizá-lo junto às teorias lógicas do referido autor. Para tanto, reservou-se uma parte da investigação ao esclarecimento das noções fundamentais da lógica ockhamista e, em seguida, destinou-se outra parte à determinação da interação entre o tempo e tais noções. Neste contexto, evidenciou-se que a lógica concebida por Ockham é essencialmente temporal, pois o fato dela concentrar-se na análise da estrutura da língua latina, aliado ao reconhecimento de que a doutrina das proposições temporalmente flexionadas e a silogística temporal desenvolvida pelo referido autor apóiam-se numa teoria da suposição capaz de lidar com uma concepção ampla de significação, cujo domínio dos objetos significados encerra o que é ou poderia ser tanto no presente, quanto no passado ou no futuro, indica que até as noções fundamentais da lógica ockhamista presumem o caráter temporal da linguagem ordinária. / [en] Although the logic developed by William of Ockham is regarded as having considerable interest, both from a historical and from a philosophical point of view, little attention has been paid to the temporal aspects of his doctrines. This creates a barrier to the full understanding of Ockham´s logic because it leads to many controversies that are due to partial interpretations which underestimate the role of time in the logic of the Middle Ages. In the present study, the temporal character of Ockham´s logic is analyzed in order to contextualize it within his general theories. The first part of our investigation is concerned with the basic notions of Ockham´s logic, and the second part studies their interaction with temporal notions. It becomes clear that Ockham´s logic is essentially temporal. This is mainly due to the fact that it concentrates on the analysis of the latin language, and that the doctrine of temporal propositions and of temporal syllogism are based on a conception of supposition that must be able to deal with a broad conception of signification. The domain of objects signified includes what is, or what could be, in the present as well as in the past and in the future. This shows that fundamental notions of Ockham´s logic presuppose the temporal character of ordinary language.