In the present paper the well-known Gödels – Churchs argument concerning the undecidability of logic (of the first order functional calculus) is exhibited in a way which seems to be philosophically interestingfi The natural numbers are not used. (Neither Chinese Theorem nor other specifically mathematical tricks are applied.) Only elementary logic and very simple set-theoretical constructions are put into the proof. Instead of the arithmetization I use the theory of concatenation (formalized by Alfred Tarski). This theory proves to be an (...) appropriate tool. The decidability is defined directly as the property of graphical discernibility of formulas. (shrink)
Ad hominem arguments are generally dismissed on the grounds that they are not attempts to engage in rational discourse, but are rather aimed at undermining argument by diverting attention from claims made to assessments of character of persons making claims. The manner of this dismissal however is based upon an unlikely paradigm of rationality: it is based upon the presumption that our intellectual capacities are not as limited as in fact they are, and do not vary as much as they (...) do between rational people. When we understand rationality in terms of intellectual virtues, however, which recognize these limitations and provide for the complexity of our thinking, ad hominem considerations can sometimes be relevant to assessing arguments. (shrink)
With the goal of understanding how Christopher Southgate communicates his in-depth knowledge of both science and theology, we investigated the many roles he assumes as a teacher. We settled upon wide-ranging topics that all intertwine: (1) his roles as author and coordinating editor of a premier textbook on science and theology, now in its third edition; (2) his oral presentations worldwide, including plenaries, workshops, and short courses; and (3) the team teaching approach itself, which is often needed by others (...) because the knowledge of science and theology do not always reside in the same person. Southgate provides, whenever possible, teaching contexts that involve students in experiential learning, where they actively participate with other students.We conclude that Southgate’s ultimate goal is to teach students how to reconcile science and theology in their values and beliefs, so that they can take advantage of both forms of rational thinking in their own personal and professional lives. The co-authors consider several examples of models that have been successfully used by people in various fields to integrate science and religion. (shrink)
The Neo-Aristotelian ethical naturalism of Philippa Foot and Rosalind Hursthouse purports to establish a naturalistic criterion for the virtues. Specifically, by developing a parallel between the natural ends of nonhuman animals and the natural ends of human beings, they argue that character traits are justified as virtues by the extent to which they promote and do not inhibit natural ends such as self-preservation, reproduction, and the well-being of one’s social group. I argue that the approach of Foot and Hursthouse cannot (...) provide a basis for moral universalism, the widely-accepted idea that each human being has moral worth and thus deserves significant moral consideration. Foot and Hursthouse both depict a virtuous agent as implicitly acting in accord with moral universalism. However, with respect to charity, a virtue they both emphasize, their naturalistic criterion at best provides a warrant for a restricted form of charity that extends only to a limited number of persons. There is nothing in the natural ends of human beings, as Foot and Hursthouse understand these, that gives us a reason for having any concern for the well-being of human beings as such. (shrink)
Christopher Peacocke’s A Study of Concepts is a dense and rewarding work. Each chapter raises many issues for discussion. I know three different people who are writing reviews of the volume. It testifies to the depth of Peacocke’s book that each reviewer is focusing on a quite different set of topics.
One of the most noteworthy features of David Gauthier's rational choice, contractarian theory of morality is its appeal to self-interested rationality. This appeal, however, will undoubtedly be the source of much controversy and criticism. For while self-interestedness is characteristic of much human behavior, it is not characteristic of all such behavior, much less of that which is most admirable. Yet contractarian ethics appears to assume that humans are entirely self-interested. It is not usually thought a virtue of a theory that (...) its assumptions are literally false. What may be said on behalf of the contractarian? (shrink)
In this interview, Christopher Norris discusses a wide range of issues having to do with postmodernism, deconstruction and other controversial topics of debate within present-day philosophy and critical theory. More specifically he challenges the view of deconstruction as just another offshoot of the broader postmodernist trend in cultural studies and the social sciences. Norris puts the case for deconstruction as continuing the 'unfinished project of modernity' and—in particular—for Derrida's work as sustaining the values of enlightened critical reason in various (...) spheres of thought from epistemology to ethics, sociology and politics. Along the way he addresses a number of questions that have lately been raised with particular urgency for teachers and educationalists, among them the revival of creationist doctrine and the idea of scientific knowledge as a social, cultural, or discursive construct. In this context he addresses the 'science wars' or the debate between those who uphold t. (shrink)
The sovereignty of the people, it is widely said, is the foundation of modern democracy. The truth of this claim depends on the plausibility of attributing sovereignty to “the people” in the first place, and I shall express skepticism about this possibility. I shall suggest as well that the notion of popular sovereignty is complex, and that appeals to the notion may be best understood as expressing several different ideas and ideals. This essay distinguishes many of these and suggests that (...) greater clarity at least would be obtained by focusing directly on these notions and ideals and eschewing that of sovereignty. My claim, however, will not merely be that the notion is multifaceted and complex. I shall argue as well that the doctrine that the people are, or ought to be, sovereign is misleading in potentially dangerous ways, and is conducive to a misunderstanding of the nature of politics, governance, and social order. It would be well to do without the doctrine, but it may be equally important to understand its errors. Our understandings and justifications of democracy, certainly, should dispense with popular sovereignty. (shrink)
We investigate the “mathematical” strength of the theory I*2. In particular we prove the quadratic reciprocity law and Bertrand's postulate, using fragments of I*2 which employ some well-known number-theoretic functions.
Christopher G. Timpson provides the first full-length philosophical treatment of quantum information theory and the questions it raises for our understanding of the quantum world. He argues for an ontologically deflationary account of the nature of quantum information, which is grounded in a revisionary analysis of the concepts of information.
Medical analogies are commonly invoked in both Indian Buddhist dharma and Hellenistic philosophy. In the Pāli Canon, nirvana is depicted as a form of health, and the Buddha is portrayed as a doctor who helps us attain it. Much later in the tradition, Śāntideva described the Buddha’s teaching as ‘the sole medicine for the ailments of the world, the mine of all success and happiness.’ Cicero expressed the view of many Hellenistic philosophers when he said that philosophy is ‘a medical (...) science for the mind.’ He thought we should ‘hand ourselves over to philosophy, and let ourselves be healed.’ ‘For as long as these ills [of the mind] remain,’ he wrote, ‘we cannot attain to happiness.’ There are many different forms of medical analogy in these two traditions, but the most general form may be stated as follows: just as medicine cures bodily diseases and brings about physical health, so Buddhist dharma or Hellenistic philosophy cures mental diseases and brings about psychological health—where psychological health is understood as the highest form of happiness or well-being. Insofar as Buddhist dharma involves philosophy, as it does, both renditions of the analogy may be said to declare that philosophy cures mental diseases and brings about psychological health. This feature of the analogy—philosophy as analogous to medical treatment—has attracted considerable attention. (shrink)
Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. In this book Christopher Pincock tackles this perennial question in a new way by asking how mathematics contributes to the success of our best scientific representations. In the first part of the book this question is posed and sharpened using a proposal for how we can determine the content of (...) a scientific representation. Several different sorts of contributions from mathematics are then articulated. Pincock argues that each contribution can be understood as broadly epistemic, so that what mathematics ultimately contributes to science is best connected with our scientific knowledge. In the second part of the book, Pincock critically evaluates alternative approaches to the role of mathematics in science. These include the potential benefits for scientific discovery and scientific explanation. A major focus of this part of the book is the indispensability argument for mathematical platonism. Using the results of part one, Pincock argues that this argument can at best support a weak form of realism about the truth-value of the statements of mathematics. The book concludes with a chapter on pure mathematics and the remaining options for making sense of its interpretation and epistemology. Thoroughly grounded in case studies drawn from scientific practice, this book aims to bring together current debates in both the philosophy of mathematics and the philosophy of science and to demonstrate the philosophical importance of applications of mathematics. (shrink)
This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk :228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation. Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze separation structures and (...) class='Hi'>Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces. (shrink)
We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically (...) interesting, requiring a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here. (shrink)
Whitehead, in his famous book Process and Reality, proposed a definition of point assuming the concepts of "region" and "connection relation" as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper, proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as primitive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.
Let us consider an arbitrary semantical relation. It holds between some linguistic entities and pieces of reality referred to. We may call it meaning. The controversy between psychologism and antipsychologism therefore may be exhibited as an ontological dilemma:Antipsychologism PsychologismThe relation of meaning is independent of human beings The relation of meaning is established by human beingsWhen we describe the meaning of words we do not need to refer to human behavior When we describe the meaning of words we need to (...) refer to human behaviorIt seems to be safe to assume that reality is not contradictory and a good description of reality cannot lead to the contradiction. All presentations of semantical antinomies rest on antipsychologistic description of semantical relations. Thus we can blame the contradictions called semantical antinomies on the antipsychologistic description. (shrink)
The paper proposes a new definition of the conception of effectiveness . A good name for this version of effectiveness is discernibility. The definition is based on the fact that every computation may be reduced to the operation of discerning the fundamental symbols and concatenation of formulas. This approach to effectiveness allows us to formulate the proof of undecidability in such a way that arithmetization of the syntax may be replaced by the use of concatenation in metalogic.
In this excellent introduction, Christopher Shields introduces and assesses the whole of Aristotle’s philosophy, showing how his powerful conception of human nature shaped much of his thinking on the nature of the soul and the mind, ethics, politics and the arts. Beginning with a brief biography, Christopher Shields carefully explains the fundamental elements of Aristotle’s thought: his explanatory framework, his philosophical methodology and his four-causal explanatory scheme. Subsequently he discusses Aristotle’s metaphysics and the theory of categories and logical (...) theory and his conception of the human being and soul and body. In the last part, he concentrates on Aristotle’s value theory as applied to ethics and politics, and assesses his approach to happiness, virtues and the best life for human beings. He concludes with an appraisal of Aristotelianism today. (shrink)