This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6]. In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a (...) model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3, which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli. We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete. In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest. (shrink)

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. (...) The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here). We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete (whether or not consistent and recursively enumerable). In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show (unlike classical quantum arithmetic) that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest. (shrink)

Our traditional ways of thinking about politics and society are becoming obsolete. We need some new points of reference in order to re-imagine the possible character, growth, and functioning of our private and common life. Such re-imagination would imply doing away with every-man-for-himself individualism as well as consumption-makes-me-happy materialism and the-state-will-take-care-of-it passivity. There is an alternative: Personalism is a forgotten, yet golden perspective on humanity that seeks to describe what a human being is and to then draw the social consequences. (...) Personalism builds upon the thinking of Martin Buber and Emmanuel Levinas, among others, and has been a source of inspiration for Martin Luther King, Desmond Tutu, and other important personalities in recent history. According to personalism, humans are relational and engaged and possess dignity. The person and the relationship amongst persons are the universal point of departure: Human beings have inherent dignity, and good relationships amongst humans are crucial for the good, engaged life and for a good society. Personalism has been greatly neglected in Western political thought. In this book, Jonas Norgaard Mortensen attempts to introduce personalism while simultaneously demonstrating its historical origins, acquainting the reader with its thinkers and those who have practiced it, and showing that personalism has a highly relevant contribution to make in the debate about today's social and political developments. (shrink)

Sociological studies into alcohol use seem to find it difficult to deal with the substance itself. Alcohol tends to be reduced to a symbol of a social process and in this way the sociological research loses sight of effects beyond the social. This article suggests a new theoretical approach to the study of alcohol and teenagers' (romantic) relationships, inspired by actor-network theory (ANT). The central feature of ANT is to search for relationships, or rather networks, between all things relevant to (...) the phenomenon. All material and semantic structures, things, persons, discourses, etc. that influence a given situation are described as actants and are entered into the analysis. The aim of this article is to propose a way of including materiality in sociological analyses of alcohol and to explore ways of using focus group interview material in ANT-inspired analysis. By analysing a girl's development from being a non-drinker at the age of 14, to a heavy drinker at 16, the article investigates how the actants brought out by young people in the focus groups themselves constitute different forms of agency. The analysis is based on a qualitative study consisting of 37 focus groups conducted with teenagers in Denmark over a three-year period. (shrink)

Paraconsistent logic, logic in which inconsistent information does not deliver arbitrary conclusions, is one of the fastest growing areas of logic, with roots in profound philosophical issues, and applications in information processing and philosophy of science. This book contains selected papers presented at the First World Congress on Paraconsistency, held in Ghent in 1997. It contains papers on various aspects of the subject. As such, it should be of interest to all who want to learn what the subject is, and (...) where it is going. (shrink)

Our view is that the folk concept of knowing how is more complicated than many epistemologists assume. We present four studies that go some way towards supporting our view—that the folk concept of knowledge-how is a philosophical hybrid, comprising both intellectualist and anti-intellectualist features. One upshot is, if we are going to award a presumptive status to philosophical theories of know-how that best accord with the folk concept, it ought to go to those that combine intellectualist and anti-intellectualist elements.

First, we consider an argument due to Popper for maximal strength in choice of logic. We dispute this argument, taking a lead from some remarks by Susan Haack; but we defend a set of contrary considerations for minimal strength in logic. Finally, we consider the objection that Popper presupposes the distinctness of logic from science. We conclude from this that all claims to logical truth may be in equal epistemological trouble.

This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal logic, intuitionist logic (...) and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity. (shrink)

In this paper, a survey is made of some of the contributions to the interpretation of Hartle and Hawking’s theory of the wave function of the universe and its beginning. It is argued that there are considerable difficulties with the interpretation of the theory, but that there is at least one interpretation hitherto not found in the literature which survives existing philosophical objections.

Dirac’s treatment of his well known Delta function was apparently inconsistent. We show how to reconstruct his reasoning using the inconsistency-tolerant technique of Chunk and Permeate. In passing we take note of limitations and developments of that technique.

We present experimental results that support the claim that the folk concept of knowledge how is an epistemological hybrid, encompassing both intellectualist and praxist elements.

This is the first book dealing with the reception of Soren Kierkegaard in Japan. It may seem strange that the Danish philosopher, theologian and writer, who is renowned in the western world as individualist and the existentialist, has been read and studied in Japan since the turn of century. The aim of this study is to explain why the Japanese came to read Kierkegaard, how several religious and non-religious lines of reception developed, and why he is still of current interest (...) in postmodern Japan. (shrink)

This paper criticises necessitarianism, the thesis that there is at least one necessary truth; and defends possibilism, the thesis that all propositions are contingent, or that anything is possible. The second section maintains that no good conventionalist account of necessity is available, while the third section criticises model theoretic necessitarianism. The fourth section sketches some recent technical work on nonclassical logic, with the aim of weakening necessitarian intuitions and strengthening possibilist intuitions. The fifth section considers several a prioristic attempts at (...) demonstrating that there is at least one necessary proposition and finds them inadequate. The final section emphasises the epistemic aspect of possibilism. (shrink)

A typical theorem of conaexive logics is Aristotle''s Thesis(A), (AA).A cannot be added to classical logic without producing a trivial (Post-inconsistent) logic, so connexive logics typically give up one or more of the classical properties of conjunction, e.g.(A & B)A, and are thereby able to achieve not only nontriviality, but also (negation) consistency. To date, semantical modellings forA have been unintuitive. One task of this paper is to give a more intuitive modelling forA in consistent logics. In addition, while inconsistent (...) but nontrivial theories, and inconsistent nontrivial logics employing prepositional constants (for which the rule of uniform substitution US fails), have both been studied extensively within the paraconsistent programme, inconsistent nontrivial logics (closed under US) do not seem to have been. This paper gives sufficient conditions for a logic containingA to be inconsistent, and then shows that there is a class of inconsistent nontrivial logics all containingA. A second semantical modelling forA in such logics is given. Finally, some informal remarks about the kind of modellingA seems to require are made. (shrink)

One of Arthur Prior’s constructions of the relational calculus for times within tense logic plus propositional quantifiers is considered using Malcolm Rennie’s multimodal semantics and found wantin...

The article addresses the emerging framework of ‘positioning theory’, which has become influential in educational theory. We will present the central concepts and ideas of ‘positioning theory’ and subsequently how they are being used to contribute to educational theory. We will focus on an article by Svend Brinkmann where he uses ‘positioning theory’ as the foundation for moral education. Seemingly, Brinkmann uses ‘positioning theory’ rather uncritically in an attempt to ‘strengthen’ the educational process by introducing fixed categorisations and concepts to (...) it. Contrary to this approach, we will use the educational theories of Knud Grue-Sørensen and Gert Biesta to emphasise how we instead should recognise what is here defi ned as the ‘weakness’ of educational theory, and how we have to consider both the adaptive element of education as well as the emancipatory element of it. (shrink)

This paper explores the model theory of relevant arithmetic, emphasizing the structure of nonstandard natural numbers in the relevant arithmetic R#. In particular, the authors prove the “Alien Intruder Theorem” guaranteeing the existence of a model of R# including the rational numbers in which each rational acts as a nonstandard natural number. The authors conclude by considering some consequences of and open questions about the construction used in the theorem.

In the mathematics education research literature, there is a growing body of scholarship on how mathematicians practice their craft. The purpose of this chapter is to survey some of this literature and explain how it can contribute to the philosophy of mathematical practice. We first describe how mathematics educators use empirical methodologies to investigate the behaviors of mathematicians and argue that findings from these studies can inform the philosophy of mathematical practice. We then illustrate this by summarizing research on mathematicians’ (...) understandings and mathematicians’ proof reading. (shrink)

On several levels, there is now a debate whether the concept of God can be made compatible with modern science. In an attempt to elucidate this debate, I give an account of my own experiences from writing a book on the foundation of quantum mechanics. In my opinion, one can give two independent arguments for the existence of God by taking as departure an epistemic (knowledge-based) interpretation of quantum theory. However, I also argue that any religious belief should be the (...) result of an existential choice governed by each person’s own context (history and culture). In particular, I give my account of Christianity as a religion connected to our western culture, how one as a scientist can relate to the relevant dogmas. (shrink)

The well known Bell experiment with two actors Alice and Bob is considered. First the simple deduction leading to the CHSH inequality under local realism is reviewed, and some arguments from the literature are recapitulated. Then I take up certain background themes before I enter a discussion of Alice’s analysis of the situation. An important point is that her mind is limited by the fact that her Hilbert space in this context is two-dimensional. General statements about a mind’s limitation during (...) a decision process are derived from recent results on the reconstruction of quantum theory from conceptual variables. These results apply to any decision situation. Let all the data from the Bell experiment be handed over to a new actor Charlie, who performs a data analysis. But his mind is also limited: He has a four-dimensional Hilbert space in the context determined by the experiment. I show that this implies that neither Alice nor Charlie can have the argument leading to the CHSH inequality as a background for making decisions related to the experiment. Charlie may be any data analyst, and he may communicate with any person. It is argued that no rational person can be convinced by the CHSH argument when making empirical decisions on the Bell situation. (shrink)

This document contains the appendices, which provides the stimulus materials, for the four studies reported in: Gonnerman, Mortensen, & Robbins (forthcoming). KNOWING HOW as a philosophical hybrid. Synthese. -/- .

Nudging is a concept in behavioural science, political theory and economics that proposes indirect suggestions to try to achieve non-forced compliance and to influence the decision making and behaviour of groups and individuals. Researchers in medical ethics are currently discussing whether nudging is ethically permissible in healthcare. In this article, we examine current knowledge about how different decisions are made and how this decision-making process pertains to patients. We view this knowledge in light of the nursing project and the ongoing (...) debate regarding the ethical legitimacy of nudging in healthcare. We argue that it is insufficient to discuss nudging in nursing and healthcare in light of free will and patient autonomy alone. Sometimes, nurses must take charge and exhibit leadership in the nurse–patient relationship. From the perspective of nursing as leadership, nudging becomes a useful tool for directing and guiding patients towards the shared goals of health, recovery and independence and away from suffering. The use of nudging in nursing to influence patients’ decisions and actions must be in alignment with the nursing project and in accordance with patients’ own values and goals. (shrink)

Sven Erik Nordenbo; Bildung and the Thinking of Bildung, Journal of Philosophy of Education, Volume 36, Issue 3, 16 December 2002, Pages 341–352, https://doi.or.

Popper's definition looked initially promising provided that the restriction of classical logic was removed. As we have seen, this promise is not fulfilled. The search for a satisfactory verisimilitude ordering must therefore be pursued along more mainstream lines. The present exercise ought, however, to make us aware of the possibility that breakdowns of proposed definitions might only occur because of strictly classical assumptions.