I was commissioned by Barry Smith, Editor of The Monist , to act as Advisory Editor for issue 88.1, January 2005 on the topic Humor, and we drafted the appended description. The deadline for submissions is January 31, 2004, and you are welcome to submit an article to me for consideration (word limit 7,500 words, including footnotes). What the Editor and I are, hoping for, is some serious and seriously good philosophical writing on this topic.
Is there a key for ‘translating' some set-theoretical paradoxes into counterpart semantical paradoxes and vice-versa? There is, and this encourages the hope of a unified solution. The solution turns not on inventing new axioms that do not entail contradiction, but on imposing a completely intuitive restriction on the comprehension axiom of naive set theory in order to avoid illegitimate (circular) stipulation.
Letting is a common practice in mathematics. For example, we let x be the sum of the first n integers and, after a short proof, conclude that x = n(n+1)/2; we let J be the point where the bisectors of two of the angles of a triangle intersect and prove that this coincides with H, the point at which another pair of bisectors of the angles of that triangle intersect. Karl Weierstrass's colleagues, in an attempt to solve optimization problems, stipulated (...) that the minimum area for a triangle with a given perimeter be a straight line segment conceived as a triangle with zero altitude. (Weierstrass complained that this obscured the insight that some problems have no solutions.) In mathematics applied to physics, we let x be the temperature in Fahrenheit corresponding to 30° Centigrade; we let v be the velocity of the Earth through the luminiferous ether. Before the error was spotted, the official rules for Little League Baseball made an inconsistent stipulation about the dimensions of home plate (Bradley 1996). (shrink)
It is uncontroversial that, on any run through a Sorites series, a subject, at some point, switches from an ‘F’ verdict on one exhibit to a non-‘F’ verdict on the next. (Where this ‘cut-off’ point occurs tend to differ from trial to trial.) It is a fallacy to infer that there must be a cut-off point simpliciter between F items and non-F items. The transition is from firm ground to swamp. In the Sorites reasoning, some conditionals of the form ‘If (...) Item n is F, then Item n + 1 is F’ are not false but nonsensical. This solution respects boundarylessness. (shrink)
This paper focuses on the issue of what to do if a couple who generates embryos chooses to lawfully, and in their (and my) view, ethnically discard those embryos. Specifically, is it appropriate to use the cells that come from “excess” embryos in medical research instead of discarding them when a couple has ceased trying to have any additional children?
Consideration of a paradox originally discovered by John Buridan provides a springboard for a general solution to paradoxes within the Liar family. The solution rests on a philosophical defence of truth-value-gaps and is consistent (non-dialetheist), avoids ‘revenge’ problems, imports no ad hoc assumptions, is not applicable to only a proper subset of the semantic paradoxes and implies no restriction of the expressive capacities of language.
A standard method for refuting a set of claims is to show that it implies a contradiction. Stephen Clark questions this method on the grounds that the Law of Non-Contradiction, together with the other fundamental laws of logic do not accord with everyday reality. He accounts for vagueness by suggesting that, for any vague predicate 'F', an ordinary object is typically to some extent both F and not-F, and that objects do not change abruptly from being F to being not-F. (...) I challenge Clark's 'deconstruction' of logic, and show that, in characterizing vagueness and dealing with the associated Sorites paradox, we can accommodate his observation that change from being F to being not-F is ineradically continuous without tampering with any fundamental logical laws. (shrink)
Wittgenstein discusses speakers exploiting context to inject meaning into the sentences that they use. One facet of situation comedy is context-injected ambiguity, where scriptwriters artfully construct situations such that, because of conflicting contextual clues, a character, though uttering a sentence that contains neither ambiguous words nor amphibolous contruction may plausibly be interpreted in at least two distinct ways. This highlights an important distinction between the (concise) sentence that a speaker uses and what the speaker means, the disclosure of which may (...) require considerable spelling out. Understanding this phenomenon of nonindexical contextualism is the key to solving, inter alia , problems where, puzzlingly, exchanging a singular term in a statement with a co-referential one fails to preserve truth-value. This is a rare case where there is a huge debate in the recent literature that is decisively settled by Wittgenstein’s approach. (shrink)
A syntactically correct number-specification may fail to specify any number due to underspecification. For similar reasons, although each sentence in the Yablo sequence is syntactically perfect, none yields a statement with any truth-value. As is true of all members of the Liar family, the sentences in the Yablo sequence are so constructed that the specification of their truth-conditions is vacuous; the Yablo sentences fail to yield statements. The ‘revenge’ problem is easily defused. The solution to the semantical paradoxes offered here (...) revives the mediaeval cassatio approach, one that largely disappeared due to its incomprehending rejection by influential contemporary writers such as William Shyreswood and Thomas Bradwardine. The diagnosis readily extends to the set-theoretic paradoxes. (shrink)
This paper builds on work done by Graham Priest (1994, 1995, 1998b, 2000) but does not presuppose knowledge of that work. Priest established that many paradoxes, which had been traditionally divided into different families, have a structure in common – which he calls the Inclosure Schema – and, correlatively, that these paradoxes demand a uniform solution. The uniform solution favoured by Priest is a Dialetheist one. I show that, with minor modification, the Inclosure Schema becomes sufficiently embracing to exhibit the (...) underlying structure not just of the logico-semantical paradoxes discussed by Priest, but of some metaphysical paradoxes too. The uniform solution advocated here is a non-Dialetheist one. Although this is not the concern of the present paper, I am persuaded by some recent work (Bromand 2002; Simmons 1993, pp.80-2) that Dialetheism, whatever its other virtues, does not deliver a solution to the semantical paradoxes. (shrink)
Book Information Paradoxes: Their Roots, Range and Resolution. Paradoxes: Their Roots, Range and Resolution Nicholas Rescher , Chicago and La Salle : Open Court , 2001 , xxiii + 293 , US$24.95 ( paper ). By Nicholas Rescher. Open Court. Chicago and La Salle. Pp. xxiii + 293. US$24.95 (paper:).
outrageous remarks about contradictions. Perhaps the most striking remark he makes is that they are not false. This claim first appears in his early notebooks (Wittgenstein 1960, p.108). In the Tractatus, Wittgenstein argued that contradictions (like tautologies) are not statements (Sätze) and hence are not false (or true). This is a consequence of his theory that genuine statements are pictures.
Wittgenstein's Tractatus is widely regarded as a masterpiece, a brilliant, if flawed attempt to achieve an ‘unassailable and definitive … final solution’ to a wide range of philosophical problems. Yet, in a 1931 notebook, Wittgenstein confesses: ‘I think there is some truth in my idea that I am really only reproductive in my thinking. I think I have never invented a line of thinking but that it was always provided for me by someone else’. This disarming self-assessment is, I believe (...) accurate. The Tractatus, despite making significant advances on the logical doctrines of Frege and Russell, is essentially a derivative work—Wittgenstein, as he elsewhere acknowledges, provided a fertile soil in which the original seeds of other peoples' thought grew in a unique way. In a play of mine, published in Philosophy (1999), Wittgenstein fails a tough viva on the Tractatus because he fails to properly support some of the weak arguments in the work and because of his inadequate acknowledgment of sources. The present paper further explores some of the antecedents of Wittgenstein's early views and answers some criticisms of the play. (shrink)
Logicism is one of the great reductionist projects. Numbers and the relationships in which they stand may seem to possess suspect ontological credentials - to be entia non grata - and, further, to be beyond the reach of knowledge. In seeking to reduce mathematics to a small set of principles that form the logical basis of all reasoning, logicism holds out the prospect of ontological economy and epistemological security. This paper attempts to show that a fundamental logicist project, that of (...) defining the number one in terms drawn only from logic and set theory, is a doomed enterprise. The starting point is Russell's Theory of Descriptions, which purports to supply a quantificational analysis of definite descriptions by adjoining a 'uniqueness clause' to the formal rendering of indefinite descriptions. That theory fails on at least two counts. First, the senses of statements containing indefinite descriptions are typically not preserved under the Russellian translation. Second (and independently), the 'uniqueness clause' fails to trim 'some' to 'one'. The Russell-Whitehead account in Principia Mathematica fares no better. Other attempts to define 'one' are covertly circular. An ontologically non-embarrassing alternative account of the number words is briefly sketched. (shrink)
Logicism is one of the great reductionist projects. Numbers and the relationships in which they stand may seem to possess suspect ontological credentials â to be entia non grata â and, further, to be beyond the reach of knowledge. In seeking to reduce mathematics to a small set of principles that form the logical basis of all reasoning, logicism holds out the prospect of ontological economy and epistemological security. This paper attempts to show that a fundamental logicist project, that of (...) defining the number one in terms drawn only from logic and set theory, is a doomed enterprise. The starting point is Russell's Theory of Descriptions, which purports to supply a quantificational analysis of definite descriptions by adjoining a 'uniqueness clause' to the formal rendering of indefinite descriptions. That theory fails on at least two counts. First, the senses of statements containing indefinite descriptions are typically not preserved under the Russellian translation. Second (and independently), the 'uniqueness clause' fails to trim 'some' to 'one'. The RussellâWhitehead account in Principia Mathematica fares no better. Other attempts to define 'one' are covertly circular. An ontologically non-embarrassing alternative account of the number words is briefly sketched. (shrink)
The Russell class does not exist because the conditions purporting to specify that class are contradictory, and hence fail to specify any class. Equally, the conditions purporting to specify the Liar statement are contradictory and hence, although the Liar sentence is grammatically in order, it fails to yield a statement. Thus the common source of these and related paradoxes is contradictory (or tautologous) specifying conditions-for such conditions fail to specify. This is the diagnosis. The cure consists of seeking and destroying (...) the deep-seated preconceptions that make almost irresistible our belief in the existence of items which provably do not exist. (shrink)
(1996). Centrifugal states and Centripetal Courts: Early state reaction to European Court of Justice (1958–1994) and U.S. Supreme Court (1789–1860) The European Legacy: Vol. 1, Fourth International Conference of the International Society for the study of European Ideas, pp. 703-709.