Showing papers in "History and Philosophy of Logic in 2001"

Russell's Paradox in Appendix B of the Principles of Mathematics : Was Frege's response adequate?

Abstract: In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell's Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege's extant logical system, if Frege's system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege's logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies wh...

Gottlob Frege, August Bebel, and the Return of Alsace-Lorraine: on the dating of the distinction between 'Sinn' and 'Bedeutung'

Abstract: A detailed chronology is offered for the writing of Frege’s central philosophical essays from the early 1890s. Particular attention is given to (the distinction between) Sinn and Bedeutung. Suggestions are made as to the origin of the examples concerning the Morning Star/Evening Star and August Bebel’s views on the return of Alsace-Lorraine. Likely sources are offered for Frege’s use of the terms Bestimmungsweise, Art des Gegebenseins and Sinn und Bedeutung.

A conceptualist interpretation of Lesniewski's ontology

Abstract: A first-order formulation of Leśniewski's ontology is formulated and shown to be interpretable within a free first-order logic of identity extended to include nominal quantification over proper and common-name concepts. The latter theory is then shown to be interpretable in monadic second-order predicate logic, which shows that the first-order part of Leśniewski's ontology is decidable.

An Athenaeum curiosity : De Morgan's reviews of Boole and Jevons

Abstract: In this note we reproduce the book reviews that De Morgan wrote on Boole's and Jevons's first logical works. The most notable property of these documents is the mere fact of their existence and the absence of any reference to them in the specialized literature.

A survey of the life of Hugh MacColl (1837-1909)

Abstract: The Scottish logician Hugh MacColl is well known for his innovative contributions to modal and nonclassical logics. However, until now little biographical information has been available about his academic and cultural background, his personal and professional situation, and his position in the scientific community of the Victorian era. The present article reports on a number of recent findings.

Restoration: S. A. Yanovskaya's path in logic

Abstract: This article presents the story of S. A. Yanovskaya's epiphany—particularly, her shift from hard-line communist orthodoxy and hostility towards ‘bourgeois minded’ Soviet-Russian mathematicians to vigorous support of mathematical logic. In light of this evidence, S. A. Yanovskaya (1896–1966) may be considered as a spiritual leader and administrative founder of modern mathematical research and education in the USSR/Russia.

Extensions of the Finitist Point of View

Abstract: Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ω -rule in his 1931 paper ‘Die Grundlegung der elementaren Zahlenlehre’. The main question we discuss here is whether the finitist (meta-)mathematician would be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ‘Die Widerspruchsfreiheit der reinen Zahlentheorie’ (1936). As to Hilbert and Bernays 1939, we end on a ‘critical’ no...

On Russell's vulnerability to Russell's paradox

Abstract: 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 som...

Chrysippus and the destruction of propositions: a defence of the standard interpretation

Abstract: One of the most intriguing claims of Stoic logic is Chrysippus's denial of the modal principle that the impossible does not follow from the possible. Chrysippus's argument against this principle involves the idea that some propositions are ‘destroyed’ or ‘perish’. According to the standard interpretation of Chrysippus's argument, propositions cease to exist when they are destroyed. Ide has presented an alternative interpretation according to which destroyed propositions persist after destruction and are false. I argue that Ide's alternative interpretation as well as some versions of the standard interpretation conflict with Stoic doctrines about the nature of propositions. I propose another version of the standard interpretation based on Frede's account of the Stoic theory of the proposition. I hold that this version of the standard interpretation both escapes Ide's objections and is consistent with Stoic logic and philosophy of language.

Testing Hypotheses on fouriercoefficients in non parametricregression model

Abstract: We consider the nonlinear nonparametnc regression model with regression function ƒ We construct tests for hypotheses on Fourier coefficients of ƒ in a given band of frequencies. Such tests can be used in particular to compare two noisy signals in a frequency band. The test statistics, we use, are based on the empirical Fourier coefficients of the regression function ƒ.

Richard (Routley) Sylvan: Writings on Logic and Metaphysics

Abstract: Richard Sylvan (ne Routley) was one of Australasia's most prolific and systematic philosophers. Though known for his innovative work in logic and metaphysics, the astonishing breadth of his philosophical endeavours included almost all reaches of philosophy. Taking the view that very basic assumptions of mainstream philosophy were fundamentally mistaken, he sought radical change across a wide range of theories. However, his view of the centrality of logic and recognition of the possibilities opened up by logical innovation in the fundamental areas of metaphysics resulted in his working primarily in these two, closely connected fields. It is this work in logic and metaphysics that is the main focus of what follows.

Three Twentieth-century Commonplaces about 'If'

Abstract: The commonplaces, all grammatically confused, are that ‘conditionals’ are ternary in structure, have ‘antecedents’ and conform to the traditional taxonomy. It is maintained en route that ‘The bough will not break’ is consistent with ‘If the bough breaks …’, that there is no logical difference between “future indicatives” and “subjunctives”, and that there is a difference between the logic of propositions (e.g. ‘The bough broke’) and that of judgments (‘The bough will/might/could/should/must/needn't break’).

A lemma of logarithmic derivative for some δ-subharmonic functions

Abstract: We prove the lemma of logarithmic derivative for some δ-subharmonic functions using some mathematical tools from probability theory. Our work generalizes the lemma of logarithmic derivative for meromorphie functions due to Nevanlinna.