Laurence Goldstein

(1947 - 2014)

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).

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.

Our interpretation of the "parmenides" 132a1 - 132b2 has the following features. (i) it stresses that the third man argument is an infinite regress and (ii) notes its epistemological thrust. (iii) a faithful translation of the last line of the argument reads "and no longer will each of the forms be for you one but each is infinite in multitude." parmenides' point is that each form, which socrates believed to be complete (one), turns out to be an unbounded, incompletable series of subforms useless for comprehending the unity of many particulars. (iv) related problems are seen to occupy plato in the "philebus" and in the "sophist"

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

The views on contradiction and consistency that Wittgenstein expressed in his later writings have met with misunderstanding and almost uniform hositility. In this paper, I trace the roots of these views by attempting to show that, in his early writings, Wittgenstein accorded a ?unique status? to tautologies and contradictions, marking them off logically from genuine propositions. This is integral both to his Tractatus project of furnishing a theory of inference, and to the enterprise of explaining the nature of the Satz (statement, proposition). Wittgenstein mantained that contradictions are not false. In his early writings this surprising thesis is a consequence of his view that contradictions are not statements. In his late writings he continues to advocate the thesis, but for quite different reasons. In these late writings, I contend, Wittgenstein succeeds in making the surprising thesis plausible

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

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