This paper describes and defends the “virtues of ingenuity”: detachment, lucidity, thoroughness. Philosophers traditionally praise these virtues for their role in the practice of using reasoning to solve problems and gather information. Yet, reasoning has other, no less important uses. Conviction is one of them. A recent revival of rhetoric and argumentative approaches to reasoning (in psychology, philosophy and science studies) has highlighted the virtues of persuasiveness and cast a new light on some of its apparent vices—bad faith, deluded confidence, (...) confirmation and myside biases. Those traits, it is often argued, will no longer look so detrimental once we grasp their proper function: arguing in order to persuade, rather than thinking in order to solve problems. Some of these biases may even have a positive impact on intellectual life. Seen in this light, the virtues of ingenuity may well seem redundant. Defending them, I argue that the vices of conviction are not innocuous. If generalized, they would destabilize argumentative practices. Argumentation is a common good that is threatened when every arguer pursues conviction at the expense of ingenuity. Bad faith, myside biases and delusions of all sorts are neither called for nor explained by argumentative practices. To avoid a collapse of argumentation, mere civil virtues (respect, humility or honesty) do not suffice: we need virtues that specifically attach to the practice of making conscious inferences. (shrink)

In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theorem, its proof, how it can be used to manufacture (...) paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. (shrink)

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ?the most important event? in ?the most important year in my intellectual life?. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus (...) vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory. (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)