The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed (...) in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if φ is true, then there exists a proof of φ Building on the work of Artemov (Bull Symb Log 7(1): 1-36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence. (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

Ethics is of vital importance to the Swedish educational system, as in many other educational systems around the world.Yet, it is unclear how ethics should be dealt with in school, and prior research and evaluations have found serious problems regarding ethics in education.The field of moral education lacks clear and widely accepted definitions of key concepts, and these ambiguities negatively impact both research and educational practice. This thesis draws a distinction between three approaches to ethics in school – the descriptive (...) ethics approach, the value transmission approach, and the inquiry ethics approach – and studies in what way (if at all) they are prescribed by the national curriculum for the Swedish compulsory school, how they relate to students’ moral reasoning about technology choices and online behaviour, and what pedagogical merits and disadvantages they have. Hopefully, this both contributes to reducing the ambiguities of the field, and to answering the question of how ethics should be dealt with in education. The descriptive ethics approach asserts that school should teach students empirical facts about ethics, such as what views and opinions people have. The value transmission approach holds that school should mediate some set of predefined values to the students and make sure the students come to accept these values.The inquiry ethics approach is the view that school should teach students to reason and think critically about ethics and to engage in ethical inquiry. The role of ethics in the curriculum has not been studied in light of the above distinction, in prior research,and such an investigation is undertaken here.The results suggest that ethics has a prominent, but complicated, role in the Swedish national curriculum. Although no explicit distinction is drawn or acknowledged in the curriculum, all three approaches are prescribed throughout the curriculum, albeit to different degrees. In the general section of the curriculum, the value transmission and inquiry ethics approaches are more extensively prescribed than the descriptive ethics approach. It was found that most of the syllabi contained explicit references to ethics, while some only contained implicit references to ethics, and two syllabi lacked references to ethics altogether. In the syllabi, the inquiry ethics approach is the most dominant, both in the sense of being present in the most syllabi, and in the sense of being more strongly prescribed in many of the syllabi where several approaches occur.The value transmission approach has the weakest role in the syllabi. In total, the inquiry ethics approach is the approach most strongly prescribed by the curriculum. But prior research has shown that inquiry ethics is very rarely implemented in the classroom. In this thesis, it is found that the inquiry ethics and the value transmission approaches are incompatible, given certain reasonable interpretations, which makes the finding that inquiry ethics is rarely implemented less surprising, since value transmission is practiced in schools. Some possible causes, and some consequences, of this is discussed. The students, in their moral reasoning about technology choices, reasoned in accordance with several classical normative theories – including consequentialism, deontological ethics and virtue ethics – and in doing so, they expressed reasoning that in the discussion is found to be in conflict with the values of the value foundation in the curriculum. These findings complement earlier findings, for example that students in their actions contradict the value foundation, by adding that such conflicts also exist in their reasoning. The existence of these conflicts is found to be problematic for a value transmission approach. Many of the students defended very restrictive views on disclosing personal information online, and prior research as well as the present data has shown that adults typically hold views that are very similar to these, concerning how they think that young people ought to act online. On the other hand, youths’ actual online behaviour, as reported in earlier studies, differs considerably from this. In line with this, the students also seemed to endorse a form of private morals view, according to which moral choices are simply up to one’s own taste, which would yield an escape exit from the restrictive views mentioned above, and permit any behaviour. In the discussion, it is argued that this is the result of an attempt at value transmission from the grown-up community, probably including teachers, which might seem to work, since the students claim to hold certain views, but which likely instead constitutes a false security, since these values are not actually accepted, but only paid lip service to, and the adults are therefore wrong in their belief that the students are protected by a certain set of values (that they think the students are upholding), since the students in fact do not uphold, and therefore do not act based upon, these values. This situation risks making the students more vulnerable than had no value transmission attempt been taken in the first place. Hence, the attempted value transmission runs the risk of counteracting its purpose of helping the students acquire a safe online behaviour. Throughout the moral reasoning mentioned above, extensive variations in the students’ reasoning were found, both interpersonally and intrapersonally, both in the decision method and in the rightness criterion dimensions, as well as in between the dimensions.The existence of such variations is a novel finding, and while possible applications in future research are discussed, it is also noted that this existence constitutes a reason to question the successfulness of both the value transmission and the inquiry ethics endeavours of the educational system. The results and discussions described above highlight the importance of investigating the merits of the different approaches. Several arguments that arise from the material of this thesis are presented, evaluated and discussed. The ability of each approach to fulfil some alleged key aims of ethics education is scrutinised; their abilities to educate for good citizenship, to educate for quality of life of the individual, and to facilitate better educational results in other subjects are all investigated, as well as the ability of each approach to help counteract the influence from online extremist propaganda aimed at young people and to promote safe online behaviour in general. It is concluded that the inquiry ethics approach has the strongest support from the material of this thesis. Some consequences for school practice are discussed, and it is concluded that changing the role of ethics in the curriculum would be beneficial, downplaying the role of value transmission and further increasing, and making more explicit and clear, the role of inquiry ethics. It is also shown that there are strong reasons for the inclusion of a new subject in the Swedish compulsory education with special focus on ethics. 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It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is in fact not even disagreement in any interesting sense, let alone a faultless one, in spite of appearances and claims to the contrary. A position I call classical pluralism is sketched, (...) intended to provide a coherent methodological stance towards the issue. Some reasons to recommend this stance are given, as well as some speculations as to why not everyone might want to follow the recommendation. (shrink)

The dissertation provides an analysis and elaboration of Michael Dummett's proof-theoretic notions of validity. Dummett's notions of validity are contrasted with standard proof-theoretic notions and formally evaluated with respect to their adequacy to propositional intuitionistic logic.