Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

All-encompassing text examines every aspect of coma from neurochemistry, monitoring, and treatments to prognostic factors.

Summary A study of the metallurgy and manufacturing techniques of a group of eight astrolabes (seven from Lahore, one attributed to India) using non-destructive methods has produced the earliest evidence for systematic use of high-zinc (α?+??) brass. To produce this alloy, the brass industry supplying the Lahore instrument makers must have co-melted metallic copper and zinc. This brass-making technology was previously believed to have been developed on an industrial scale in the nineteenth century in Europe. This work hypothesizes that this (...) technology was used in Lahore on an industrial scale as early as ad 1601. In addition, this work hypothesizes that the α?+?? brass alloy was used specifically for its ease in manufacturing the thin sheet brass required for astrolabe-component manufacture. (shrink)

This book offers an introduction to the Sophists of fifth-century Athens and a new overall interpretation of their thought. Since Plato first animadverted on their activities, the Sophists have commonly been presented as little better than intellectual mountebanks - a picture which Professor Kerferd forcefully challenges here. Interpreting the evidence with care, he shows them to have been part of an exciting and historically crucial intellectual movement. At the centre of their teaching was a form of relativism, most famously expressed (...) by Protagoras as 'Man is the measure of all things', and which they developed in a wide range of views - on knowledge and argument, virtue, government, society, and the gods. On all these subjects the Sophists did far more than simply provoke Plato to thought. Their contributions were substantial and serious; they inaugurated the debate on many central philosophical questions and decisively shifted the focus of philosophical attention from the cosmos to man. (shrink)

"It is the purpose of this article to attempt to re-examine the account of Thrasymachus' doctrine in Plato's Republic, and to show how it can form a self-consistent whole. [...] In this paper it is maintained that Thrasymachus is holding a form of [natural right]." Note: Volume 40 = new series 9.

The practice of paternalism has changed along with developments in medicine, philosophy, law, sociology and psychology. Physicians have learned that a patient's values are a factor in determining what is best for that patient. Modern paternalism continues to be guided by the principle that the physician decides what is best for the patient and pursues that course of action, taking into account the values and interests of the patient. In the autonomy model of the doctor-patient relationship, patient values are decisive. (...) In the paternalistic model, they are but one among several factors the physician must consider in making a medical decision. Although difficult to practise because of limitations in empathising with another person, modern paternalism remains a way to achieve maximum patient benefit. (shrink)

In this paper, the author attempts to explore some of the problems connected with the formulation and application of international biomedical ethical guidelines, with particular reference to Africa. Recent attempts at revising and updating some international medical ethical guidelines have been bedevilled by intractable controversies and wrangling regarding both the content and formulation. From the vantage position of relative familiarity with both African and Western contexts, and the privilege of having been involved in the revision and updating of one of (...) the international ethical guidelines, the author reflects broadly on these issues and attempts prescribing an approach from both the theoretical and practical angles liable to mitigate, if not completely eliminate, some of the problems and difficulties. (shrink)

John Rawls’ use of a contractarian strategy for justifying basic principles of justice has raised the hope that a similar strategy might work for a theory of right and moral principles generally. I want to show that this hope cannot be fulfilled.In what follows I interpret contractarianism in a Rawlsian way on the grounds that his is the most plausible version of the doctrine we are likely to get. I am not however concerned with the details of Rawls’ argument for (...) justice but instead with an idea that appears to underlie the contractarian strategy. In order to avoid the complications of Rawlsian exegesis, I choose to discuss the doctrine as it might be used to justify a moral principle of mutual assistance and not as Rawls in fact uses it to justify principles of justice. (shrink)

This paper attempts to delineate some of the principal features and tasks of a politics of postmodernity. An attempt is made in the first part of the paper to reflect on the democratic revolutions of 1989 in Eastern Europe and to discern what lessons they might have to offer. What is called for, it is maintained, is a renewed theory of democracy and, more particularly, a reformulation of traditional liberalism. In the second part of the paper the author seeks to (...) suggest how such traditional liberal concepts as the individual, reason, universality, and freedom can be reformulated in a genuinely postmetaphysical, postmodern fashion. (shrink)

Among classical Latin poets, Juvenal is unusually richly provided with ancient scholia; at the same time, the scholia exhibit an unusual degree of ignorance and sheer stupidity. What is perhaps most surprising, however, is the extent to which these commentators appear to have been worse informed than we are today concerning the identity of individuals who appear in the satires.

In an earlier study I argued that the appearance of the name of Memmius in the first, second, and fifth books alone of Lucretius de Rerum Natura is only the most striking indication of a fundamental change in the poet's attitude towards his reader which is already well established quite a short way through book 5, and which makes it almost incontestable that Lucretius wrote books 3, 4, and 6 after he had lost all hope of converting Memmius to Epicureanism.

Plato's Sophist begins with an attempt to arrive by division at a definition of a Sophist. In the course of the attempt six different descriptions are discussed and the results summarized at 231 c-e. A seventh and final account may be said to occupy the whole of the rest of the dialogue, including the long digression on negative statements. The first five divisions characterize with a considerable amount of satire different types of sophist, or more probably different aspects of the (...) sophistic art. The sixth division is very different. To quote Cornford's words, ‘satire is dropped. The tone is serious and sympathetic, towards the close it becomes eloquent’. (shrink)

Anselm: I agreed that Pegasus is a flying horse according to the stories people tell, the paintings painters paint and so on . That is, Pegasus is a flying horse in the understanding of storytellers, their readers and the artists who depict Pegasus. You asked whether flying is not an unmediated causal power . Well, it could be an unmediated causal power if you or I had it, but not if a being with only mediated powers had it. And so (...) poor Pegasus, a being whose powers are only those given him by storytellers and artists, has only the mediated power of flying that they have given him. He flies in the thoughts and depictions of beings with unmediated powers. Still, Pegasus has not only the mediated power to fly but also the mediated power to entertain us, and perhaps even inspire us. Fool: But people could tell stories about you, too, about how you sprouted wings and flew onto the roof of Canterbury Cathedral. Anselm: Alas, if it were only in stories that I could do that, then flying would not be an unmediated causal power of mine. I would still have unmediated causal powers all right, and so would be greater than Pegasus. But flying is not one of my unmediated causal powers. However, something than which nothing greater can be conceived would, and does, as I have proved in my Proslogion, have unmediated causal powers – indeed, much greater unmediated causal powers than any poor mortals like you and me.1 •↵1 For …. (shrink)