Russell. N.s. Vol.
31, no. 1.
“PRINCIPIA MATHEMATICA @ 100”
Edited by Nicholas Griffin, Bernard Linsky and Kenneth Blackwell
TABLE OF CONTENTS
|Preface by Nicholas Griffin and Bernard Linsky|
|Graham Stevens||“Logical Form in Principia
Mathematica and English”|
ABSTRACT: The theory of descriptions, presented informally in “On Denoting” and more formally in Principia Mathematica, has been endorsed by many linguists and philosophers of language as a contribution to natural-language semantics. However, the syntax of Principia’s formal language is far from ideal as a tool for the analysis of natural language. Stephen Neale has proposed a reconstruction of the theory of descriptions in a language of restricted quantification that gives a better approximation of the syntax of English (and, arguably, of other natural languages). This has led to resistance from some Russell scholars who object to the identification of descriptions with quantifiers at the level of logical form in this new language on the grounds that the identification fails to respect the Russellian conception of descriptions as incomplete symbols. I defend Neale’s reconstruction of the theory and argue that he has preserved everything essential to the theory, including the notion of an incomplete symbol. However, I then go on to argue, contrary to Neale and his objectors as well as Russell himself, that the doctrine of incomplete symbols is a superfluous and undesirable element of the theory that is best jettisoned from the theory.
|Ray Perkins, Jr.||“Incomplete Symbols in
Mathematica and Russell’s ‘Definite
ABSTRACT: Early in Principia Mathematica Russell presents an argument that “‘the author of Waverley’ means nothing”, an argument that he calls a “definite proof”. He generalizes it to claim that definite descriptions are incomplete symbols having meaning only in sentential context. This Principia “proof” went largely unnoticed until Russell reaffirmed a near-identical “proof” in his philosophical autobiography nearly 50 years later. The “proof” is important, not only because it grounds our understanding of incomplete symbols in the Principia programme, but also because failure to understand it fully has been a source of much unjustified criticism of Russell to the effect that he was wedded to a naive theory of meaning and prone to carelessness and confusion in his philosophy of logic and language generally. In my paper, I (1) defend Russell’s “proof” against attacks from several sources over the last half century, (2) examine the implications of the “proof” for understanding Russell’s treatment of class symbols in Principia, and (3) see how the Principia notion of incomplete symbol was carried forward into Russell’s conception of philosophical analysis as it developed in his logical atomist period after 1910.
|Russell Wahl||“The Axiom of
ABSTRACT: The axiom of reducibility plays an important role in the logic of Principia Mathematica, but has generally been condemned as an ad hoc non-logical axiom which was added simply because the ramified type theory without it would not yield all the required theorems. In this paper I examine the status of the axiom of reducibility. Whether the axiom can plausibly be included as a logical axiom will depend in no small part on the understanding of propositional functions. If we understand propositional functions as constructions of the mind, it is clear that the axiom is clearly not a logical axiom and in fact makes an implausible claim. I look at two other ways of understanding propositional functions, a nominalist interpretation along the lines of Landini and a realist interpretation along the lines of Linsky and Mares. I argue that while on either of these interpretations it is not easy to see the axiom as a non-logical claim about the world, there are also appear to be difficulties in accepting it as a purely logical axiom.
|Conor Mayo-Wilson||“Russell on Logicism and
ABSTRACT: According to Quine, Charles Parsons, Mark Steiner, and others, Russell’s logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as aprioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell’s explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent work by Andrew Irvine and Martin Godwyn, I argue that Russell thought a systematic reduction of mathematics increases the certainty of known mathematical theorems (even basic arithmetical facts) by showing mathematical knowledge to be coherently organized. The paper outlines Russell’s theory of coherence, and discusses its relevance to logicism and the certainty attributed to mathematics.
|Brice Halimi||“Generality of Logical
ABSTRACT: My aim is to examine logical types in Principia Mathematica from two (partly independent) perspectives. The first one pertains to the ambiguity of the notion of logical type as introduced in the Introduction (to the first edition). I claim that a distinction has to be made between types as called for in the context of paradoxes, and types as logical prototypes. The second perspective bears on typical ambiguity as described in Russell and Whitehead’s “Prefatory Statement of Symbolic Conventions”, inasmuch as it lends itself to a comparison with specific systems of modern typed λ-calculus. In particular, a recent paper shows that the theory of logical types can be formalized in the way of a λ-calculus. This opens the way to an interesting reconciliation between type theories in the Russellian sense of the word, and type theories in the modern sense. But typical ambiguity is left aside in the paper. I would like to extend the suggestion by taking up the question of typical ambiguity, still in the realm of typed λ-calculus.
ABSTRACT: Ramsey defined truth in the following way: x is true if and only if ∃p(x = [p] & p). This definition is ill-formed in standard first-order logic, so it is normally interpreted using substitutional or some kind of higher-order quantifier. I argue that these quantifiers fail to provide an adequate reading of the definition, but that, given certain adjustments, standard objectual quantification does provide an adequate reading.
|Roman Murawski||“On Chwistek’s
ABSTRACT: The paper is devoted to the presentation of Chwistek’s philosophical ideas concerning logic and mathematics. The main feature of his philosophy was nominalism, which found full expression in his philosophy of mathematics. He claimed that the object of the deductive sciences, hence in particular of mathematics, is the expression being constructed in them according to accepted rules of construction. He treated geometry, arithmetic, mathematical analysis and other mathematical theories as experimental disciplines, and obtained in this way a nominalistic interpretation of them. The fate of Chwistek’s philosophical conceptions was similar to the fate of his logical conceptions. The system of rational meta-mathematics was not developed by him in detail. He worked on his own ideas without any collaboration with other logicians, mathematicians or philosophers. His investigations were not in the mainstream of the development of logic and philosophy of mathematics.
|Irving H. Anellis||“Did Principia
Precipitate a ‘Fregean Revolution’?”|
ABSTRACT: I begin by asking whether there was a Fregean revolution in logic, and, if so, in what did it consist. I then ask whether, and if so, to what extent, Russell played a decisive role in carrying through the Fregean revolution, and, if so, how. A subsidiary question is whether it was primarily the influence of The Principles of Mathematics or Principia Mathematica, or perhaps both, that stimulated and helped consummate the Fregean revolution. Finally, I examine cases in which logicians sought to integrate traditional logic into the Fregean paradigm, focusing on the case of Henry Bradford Smith. My proposed conclusion is that there were different means adopted for rewriting the syllogism, in terms of the logic of relations, in terms of the propositional calculus, or as formulas of the monadic predicate calculus. This suggests that the changes implemented as a result of the adoption of the Russell–Fregean conception of logic could more accurately be called by Grattan-Guinness’s term convolution, rather than revolution.
|Kenneth Blackwell||“The Wit and Humour of
ABSTRACT: Except for belatedly proving that “1 + 1 = 2”, Principia Mathematica doesn’t feature in studies of mathematical humour. Yet there is humour in that work, despite the inauspicious conditions under which it was written. Russell, to take one of the authors, had an irrepressible talent for enlivening any subject matter. This paper reports the results of exploring even the “obscure corners” of PM to uncover its humour and wit.