Gregory Landini

In lieu of an abstract, here is a brief excerpt of the content:January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd russell: the Journal of Bertrand Russell Studies n.s. 28 (winter 2008–09): 127–42 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 YABLO’S PARADOX AND RUSSELLIAN PROPOSITIONS Gregory Landini Philosophy / U. of Iowa Iowa City, ia 52242–1408, usa [email protected] Is self-reference necessary for the production of Liar paradoxes? Yablo has given an argument that self-reference is not necessary. He hopes to show that the indexical apparatus of self-reference of the traditional Liar paradox can be avoided by appealing to a list, a consecutive sequence, of sentences correlated one-one withnaturalnumbers.Yabloopens his “Paradox without Self-Reference” (Analysis, 1993) with the assumption that there is a sequence such that: Snz: “(;k)(k > n.!. True Sk)” Each sentence on Yablo’s list is supposed to be correlated one-one with number n. Each sentence is supposed to say that for every natural number kz greater than n, the k-th sentence on the list is not true. By comparing Yablo’s construction to an analogous construction with early Russellian propositions, we show that Yablo has failed to generate a paradox. 1. vicious circularity in yablo’s paradox S elf-reference has been taken to be the source of paradoxes. Poincaré and Russell are commonly associated with this thesis. In an amusing passage, the mathematician Jourdain recalls something of the dialogue between them: Nearly all mathematicians agreed that the way to solve these paradoxes was simply not to mention them; but there was some divergence of opinion as to how they were to be unmentioned. It was clearly unsatisfactory merely not to mention them. Thus Poincaré was apparently of the opinion that the best way of avoiding such awkward subjects was to mention that they were not to be mentioned. But [as Russell put it] “one might as well, in talking to a man with January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd 128 gregory landini 1 Philip Jourdain, The Philosophy of Mr. B*rtr*nd R*ss*llz (London: Allen & Unwin, 1918), p. 77. Russell is quoted from “Mathematical Logic as Based on the Theory of Types” (1908), LK, p. 63. 2 “On ‘Insolubilia’ and Their Solution By Symbolic Logic”, in B. Russell, Essays in Analysis, ed. Douglas Lackey (London: Allen & Unwin, 1973), p. 196. 3 I believe that this is Russell’s view in 1906. See Gregory Landini, “Russell’s Separation of the Logical and Semantic Paradoxes”, Revue internationale de philosophie 58 (2004): 257–94 (issue titled “Russell en héritage / Le centenaire des Principes”, ed. Philippe de Rouilhan). a long nose, say: ‘When I speak of noses, I except such as are inordinately long’, which would not be a very successful eTort to avoid a painful topic.”1 Poincaré maintained that one should exclude the oTending cases and therebyavoid“viciouscircles”ofself-referencewhichgenerateparadoxes. Russell objected: We may illustrate this by what M. Poincaré says concerning Richard’s paradox. Having Wrst put E = “all numbers deWnable in a Wnite number of words” we arrive at a paradox, due, says M. Poincaré, to our having included a number only deWnable in a Wnite number of words by means of E. This vicious circle he proposes to avoid by deWning E as “all numbers deWnable in a Wnite number of words without mentioning Ey”. To the uninitiated, this deWnition looks more circular than ever.2 Russell held that some paradoxes such as those of classes require, in order to exclude oTending cases without mentioning them, a “reconstruction of logical Wrst principles”. Others, such as Richard’s, are to be dismissed because they involve confused and viciously circular notions of “deWnability ”.3 Is self-reference necessary for paradoxes? The question is not well crafted. There are quite diTerent notions ofz “self-reference” involved in paradoxes. The self-reference involved in the Liar “This sentence is false” is provided by an apparatus of indexicals. This apparatus is quite distinct from the self-reference involved in Russell’s early ontology of propositions (as mind- and language-independent states of aTairs). This is an ontological self...

In his "Grundgesetze", Frege hints that prior to his theory that cardinal numbers are objects he had an "almost completed" manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege's cardinal numbers is a theory of concept-correlates. Frege held that, where n > 2, there is a one—one correlation between each n-level function and an n—1 level function, and a one—one correlation between each first-level function and an object. Applied to cardinals, the correlation offers new answers to some perplexing features of Frege's philosophy. It is shown that within Frege's concept-script, a generalized form of Hume's Principle is equivalent to Russell's Principle ofion — a principle Russell employed to demonstrate the inadequacy of definition by abstraction. Accordingly, Frege's rejection of definition of cardinal number by Hume's Principle parallels Russell's objection to definition by abstraction. Frege's correlation thesis reveals that he has a way of meeting the structuralist challenge that it is arithmetic, and not a privileged progression of objects, that matters to the finite cardinals.

This paper examines Russell's substitutional theory of classes and relations, and its influence on the development of the theory of logical types between the years 1906 and the publication of Principia Mathematica (volume I) in 1910. The substitutional theory proves to have been much more influential on Russell's writings than has been hitherto thought. After a brief introduction, the paper traces Russell's published works on type-theory up to Principia. Each is interpreted as presenting a version or modification of the substitutional theory. New motivations for Russell's 1908 axiom of infinity and axiom of reducibility are revealed

This article compares the theory of Meinongian objects proposed by Edward Zalta with a theory of fiction formulated within an early Russellian framework. The Russellian framework is the second-order intensional logic proposed by Nino B. Cocchiarelly as a reconstruction of the form of Logicism Russell was examining shortly after writing The Principles of Mathematics. A Russellian theory of denoting concepts is developed in this intensional logic and applied as a theory of the "objects' of fiction. The framework retains the Orthodox early Russellian ontology of existents, possible non-existents, and properties and relations in intension. This avoids the assumption, found in Meinongian theories, of impossible and incomplete objects. It also obviates the need to preserve consistency by distinguishing a new "mode of predication", or a "distinction in kinds of predicates". Thus, it is argued that an early Russellian theory forms a powerful rival to a Meinongian theory of objects.

The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows that Russell successfully antedates Quine's system of quantification theory without free variables. It is shown as well, that as with Quine's system, a slight modification yields a quantification theory inclusive of the empty domain

Zermelo once wrote that he had anticipated Russell's contradiction of the set of all sets that are not members of themselves. Is this sufficient for having anticipated Russell's Paradox — the paradox that revealed the untenability of the logical notion of a set as an extension? This paper argues that it is not sufficient and offers criteria that are necessary and sufficient for having discovered Russell's Paradox. It is shown that there is ample evidence that Russell satisfied the criteria and that Zermelo did not

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle.

Frege seems to hold two incompatible theses:(i) that sentences differing in structure can yet express the same sense; and (ii) that the senses of the meaningful parts of a complex term are determinate parts of the sense of the term. Dummett offered a solution, distinguishing analysis from decomposition. The present paper offers an embellishment of Dummett?s distinction by providing a way of depicting the internal structures of complex senses?determinate structures that yield distinct decompositions. Decomposition is then shown to be adequate as a foundation for the informativity and analyticity of logic

This article compares the theory of Meinongian objects proposed by Edward Zalta with a theory of fiction formulated within an early Russellian framework. The Russellian framework is the second-order intensional logic proposed by Nino B. Cocchiarelly as a reconstruction of the form of Logicism Russell was examining shortly after writing The Principles of Mathematics. A Russellian theory of denoting concepts is developed in this intensional logic and applied as a theory of the "objects' of fiction. The framework retains the Orthodox early Russellian ontology of existents, possible non-existents, and properties and relations in intension. This avoids the assumption, found in Meinongian theories, of impossible and incomplete objects. It also obviates the need to preserve consistency by distinguishing a new "mode of predication", or a "distinction in kinds of predicates". Thus, it is argued that an early Russellian theory forms a powerful rival to a Meinongian theory of objects.

In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Godel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered