Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. (...) The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists. (shrink)

Scholars of Plato are divided between those who emphasize the literature of the dialogues and those who emphasize the argument of the dialogues, and between those who see a development in the thought of the dialogues and those who do not. In this important book Russell Dancy focuses on the arguments and defends a developmental picture. He explains the Theory of Forms of the Phaedo and Symposium as an outgrowth of the quest for definitions canvassed in the Socratic dialogues, by (...) constructing a Theory of Definition for the Socratic dialogues based on the refutations of definitions in those dialogues, and showing how that theory is mirrored in the Theory of Forms. His discussion, notable for both its clarity and its meticulous scholarship, ranges in detail over a number of Plato's early and middle dialogues, and will be of interest to readers in Plato studies and in ancient philosophy more generally. (shrink)

Through consideration of another practical case, this chapter opens the way to a generalization of the method of argument outlined previously. Multilateral cases raise the question of how the interests of all parties can be resolved into a determinate moral conclusion, which brings the discussion to a standpoint that has affinities with classical utilitarianism. Like the principle of universalizability, the form of the utilitarian principle espoused is purely logical. In both cases, the moral substance comes from fleshing out the parties’ (...) actual inclinations and interests. The chapter embarks on a short survey of the challenges faced by a utilitarian system of this kind, before considering the concept of happiness, act‐utilitarianism, and rule‐utilitarianism. (shrink)

MY AIM in writing this paper is to ask help in the laying of a pair of ghosts who continue to haunt moral philosophy. Their names are ‘objectivism’ and ’subjectivism'; but the second sometimes assumes the name of ‘relativism'. I have long been convinced that the supposed battles between these two have no real substance; but for all that, they go on confusing many moral philosophers. It would be a great benefit to our profession if somebody could put an end (...) to the confusion once for all; but it has proved harder than might have been hoped, and that is my excuse for trying again. (shrink)

ARCHAEOLOGYTHE Classical Spartans were noted for their austerity, which seemed already ancient to writers of the fifth century B.C. The early poetry and art of their country show a considerable aesthetic sense. This apparent contradiction has caused some students to conclude that the strict Lycurgan regimen was not introduced till the middle or even the end of the sixth century and that before that date Sparta had culturally been developing in much the same way as other important Greek states. The (...) argument is unsound. (shrink)

Arrian's enthusiasm for Ptolemy's account of Alexander has often been echoed in modern times. With much justification it is generally agreed that Arrian's account of Alexander, through its reliance on the works of Ptolemy and Aristobulus, is our best and, on the whole, most reliable account of Alexander. Recent work, however, has illuminated Ptolemy's weaknesses, and we can no longer regard Ptolemy as utterly reliable in every important respect. His version of the Alexander story is centred on Alexander, therefore Alexander (...) is depicted out of the close context of the Macedonian court. It is only through the information preserved in other writers—traditionally, but undiscriminatingly, considered unreliable—that, for instance, the picture of Alexander's struggle with his Macedonian nobles has begun to emerge. And in matters of this kind Ptolemy's version is so much the court ‘official’ version that it cannot be regarded as trustworthy. (shrink)

Gasping respiration in the dying patient is the last respiratory pattern prior to terminal apnoea. The duration of the gasping respiration phase varies; it may be as brief as one or two breaths to a prolonged period of gasping lasting minutes or even hours. Gasping respiration is very abnormal, easy to recognise and distinguish from other respiratory patterns and, in the dying patient who has elected to not be resuscitated, will always result in terminal apnoea.Gasping respiration is also referred to (...) as agonal respiration and the name is appropriate because the gasping breaths appear uncomfortable and raise concern that the patient is suffering and in agony. Enough uncertainty exists about the influence of gasping respiration on patient wellbeing, that it is appropriate to assume that the gasping breaths are burdensome to patients. Therefore, gasping respiration at the end of life should be treated.We propose that there is an ethical basis, in rare circumstances, for the use of neuromuscular blockade to suppress prolonged episodes of agonal respiration in the well-sedated patient in order to allow a peaceful and comfortable death. (shrink)

To make out in what way Einstein’s 1905 ‘annus mirabilis’ writings hang together one has to hang on Einstein’s strive for unity evinced in his stubborn attempts to coordinate with one another the basic research traditions of classical physics. Light quanta hypothesis and special theory of relativity turn out to be mere milestones of maxwellian electrodynamics and statistical thermodynamics reconciliation programme. The conception of luminiferous ether was an insurmountable stumbling block for Einstein’s statistical thermodynamics programme in which the leading role (...) was played by the light quanta paper . (shrink)

The issue on which I intend to focus is whether there is anything else, anything more than ontological economy, which, in Russell's mature account of the constituents of propositions, is gained by his rejection of denoting concepts. I will argue that in order to answer this question, it is necessary to appreciate that by the time of "On Denoting," Russell was not merely advancing a claim of philosophical logic or a theory of the logical form of the descriptive phrases of (...) English. (edited). (shrink)

Wittgenstein's Tractatus contains a wide range of profound insights into the nature of logic and language – insights which will survive the particular theories of the Tractatus and seem to me to mark definitive and unassailable landmarks in our understanding of some of the deepest questions of philosophy. And yet alongside these insights there is a theory of the nature of the relation between language and reality which appears both to be impossible to work out in detail in a way (...) which is completely satisfactory, and to be bizarre and incredible. I am referring to the so-called logical atomism of the Tractatus . The main outlines of this theory at least are clear and familiar: there are elementary propositions which gain their sense from being models of possible states of affairs; such propositions are configurations of names of simple objects, signifying that those simples are analogously configured; every proposition has its sense through being analysable as a truth-functional compound of elementary propositions, thus deriving its sense from the sense of the elementary propositions when this view is taken in conjunction with the idea that the sense of a proposition is completely specified by specifying its truth-conditions. In this way the Tractatus incorporates in its working out a philosophical system analogous to the classical philosophical systems of Leibniz or Spinoza which are regarded by many people, in a sense rightly, as the prehistoric monsters of philosophy which are not to be studied as living organisms, but studied as the curiosities of human thought. And we may here agree that in the end we must simply reject a philosophy which incorporates such features as its postulation of simple eternal objects, or of a possibility of an analysis of a proposition which was presented as a pre-condition for the propositions that we ordinarily utter to make sense, and yet the specific form of which we are unaware of, and so on. (shrink)