Gregory Landini

This book offers a comprehensive critical survey of issues of historical interpretation and evaluation in Bertrand Russell's 1918 logical atomism lectures and logical atomism itself. These lectures record the culmination of Russell's thought in response to discussions with Wittgenstein on the nature of judgement and philosophy of logic and with Moore and other philosophical realists about epistemology and ontological atomism, and to Whitehead and Russell’s novel extension of revolutionary nineteenth-century work in mathematics and logic. Russell's logical atomism lectures have had a lasting impact on analytic philosophy and on Russell's contemporaries including Carnap, Ramsey, Stebbing, and Wittgenstein. Comprised of 14 original essays, this book will demonstrate how the direct and indirect influence of these lectures thus runs deep and wide.

Frege's Grundgesetze der Arithmetik offers a conception of cpLogic as the study of functions. Among functions are included those that are concepts, i.e. characteristic functions whose values are the logical objects that are the True/the False. What, in Frege's view, are the objects the True/the False? Frege's stroke functions are themselves concepts. His stipulation introducing his negation stroke mentions that it yields [...]. But curiously no accommodating axiom is given, and there is no such theorem. Why is it that some of Frege's informal stipulations never made appearances as axioms? I offer an explanation that sheds new light on the Grundgesetze. No axioms should over-determination the True as a logical object. Perhaps the True = 0, as would be common in the mathematics of characteristic functions. But the logical objects that are cardinal numbers are value ranges correlated with second-level numerical concepts by a non-homogeneous second-level value-range function [...]. The existence of concepts would be ontologically circular if the True is itself a number. We find this circularity perfectly agreeable to Frege, and suggest that he had accepted that the existence of functions that are concepts in his cpLogic may well be ontologically inseparable from the existence of his value-range function. His cpLogic itself stands or falls with the viability of some value-range function.

In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations$\Omega ^{n}(\bar {\xi })$defined with the help of numeral exponents. Since$Num$(x) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with “insuperable difficulties.” This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that theN-operator$N(\bar {\xi })$and recursively defined arithmetic operators$\Omega ^{n}(\bar {\xi })$are not different in kind, we can address Ramsey’s problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction.

There are many wonderful puzzles concerning Principia Mathematica, but none are more striking than those arising from the crisis that befell Whitehead in November of 1910. Volume 1 appeared in December of 1910. Volume 2 on cardinal numbers and Russell's relation arithmetic might have appeared in 1911 but for Whitehead's having halted the printing. He discovered that inferences involving the typically ambiguous notation ‘Nc‘α’ for the cardinal number of α might generate fallacies. When the volume appeared in 1912, it was extensively emended by Whitehead and accompanied by a Prefatory Statement of Symbolic Conventions. This paper endeavors to recover from Whitehead's bad emendations—including his bewildering thesis that since ‘‘α’ is ‘true whenever significant,’ ‘α is to be accepted. It is supposedly a fallacy to apply Modus Ponens and infer Nc‘α from ‘α and‘‘α.

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

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

This paper explores the thesis that de re quantification into propositional attitudes has been wrongly conceived. One must never bind an individual variable in the context of a propositional attitude. Such quantification fails to respect the quantificational scaffolding of discursive thinking. This is the lesson of the Meinong–Russell debate over whether there are objects of thought about which it is true to say they are not. Respecting it helps to see how to solve contingent Liar paradoxes of propositional attitudes such as Kripke’s Nixon–Jones.

This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is shown that with a minor repair the system is semantically complete. Finally, the system is contrasted with the system of *8 of Principia's second edition

There is a rather famous “Fitch argument” that not everything that is true is knowable. There is a rather famous “Mary argument” that is often used to argue that reductive physicalism is false. This paper sets out the two side by side as the Fitch Knowability Paradox and the Mary Knowability Paradox. It is found that they have the same logical form and thus the question of validity can be evaluated with the same tools. Likening the two is useful, since it avoids the problem that since the logical forms involved in intentional and experiential contexts are unknown, we cannot be in a position to evaluate whether the Mary is deductively valid without begging questions.

Poetic license is an essential feature of intentionality. The mind is free to think about any objects, even objects with logically incompatible properties. Some philosophers maintain that a theory that embraces an ontology of objects of thought is indispensable to any account of the nature of intentionality. Any such theory, however, must face paradoxes whose solutions conflict with poetic license. In this paper, I propose a theory which rejects the argument from indispensability. The theory maintains that the intentionality of thought is produced by the quantificational nature of the apparatus of thought. All de re ascriptions of propositional attitudes must respect simple-type stratification and quantify over concepts with a predicable nature only. There are no fictional objects. There are concepts which, in the impredicative reflections of quantificational thought, are presented as if objects of thought.

In lieu of an abstract, here is a brief excerpt of the content:RUSSELL TO FREGE, 24 MAY 1903: "I BELIEVE I HAVE DISCOVERED THAT CLASSES ARE ENTIRELY SUPERFLUOUS" GREGORY LANDINI Philosophy / University of Iowa Iowa City, IA 52242, USA It was his consideration of Cantor's proof that there is no greatest cardinal, Russell recalls in My Philosophical Development, that led in the spring of 1901 to the discovery of the paradox of the class of all classes not members of themselves. "Never glad confident morning again", were Whitehead's reported words (MPD, p. 58). Whitehead was, of course, wrong. Russell had many new confident mornings. One in particular apparently occurred on 19 May 1903. On 23 May Russell wrote in his journal: "Four days ago 1 solved the Contradiction -the relief of this is unspeakable" (Papm 12: 24). The "solution" was communicated to Whitehead, who responded by telegram: "Heartiest congratulations Aristoteles secundus. 1 am delighted" (Clark, p. III). But it was not long before Russell knew his proposal was inadequate. On the form he would scrawl: "A propos of solVing the Contradiction. [But the solution was wrong]" (ibid; Russell's brackets). The episode is intriguing. What was the proposal? Russell's correspondence with Frege sheds some light. On 20 October 1902 Frege had sent a letter suggesting a way to avoid the contradiction. Frege's suggestion, formulated in a great hurry so that it might appear in the Appendix of Vol. II of the Grundgesetze der Arithmetik, was to abandon his Basic Law V; zcj>z =zez. ==. (x)(x. ==. ex), and to replace it by Russell to Frege, 24 May I903 161 V' zz =zez. ==. (x)(x *zz & x* zez: ~ : x. ==. ex). The underlying idea was to accept that one concept may have the same extension as another even though the two concepts are not coextensive. Though he thought the underlying idea "probably correct",! Russell expressed reservation in a reply of 12 December:... I find it difficult to accept your solution even though it is probably correct. Do you deny, e.g., that all classes form a class? And if this is admitted, then it is possible that - ana. Moreover, the class of non-humans is a. non-human. Otherwise it must be admitted that not ail objects fall either under a or not-a; namely, if a is a range of values, then a fails neither under a nor under not-a. This contradicts the law of excluded middle, which will be inconvenient to say the least.2 (Frege I980, p. 151) Frege's response of 28 December does not mention the contradiction, and Volume II of his Grundgesetze appeared early in 1903. Russell's reservations concerning Frege's solution· continue in a letter to Frege of 20 February 1903: What you say about my contradiction is of the greatest interest to me. Do you believe that the range of values remains unchanged if some subclass of the class is assigned to it as a new member? Extension seems to fit this view better than intension. Bur I fe;el far from clear about this question. (Frege I980, p. 155) Frege replied on 21 May, explaining that in general a class does not remain unchanged when a particular subclass is added to it, but that two concepts may have the same extension (the same class) when the only difference between them is that this class falls under the first concept but not under the second (p. 157). Then on 24 May 1903 Russell excitedly wrote Frege of a solution of the paradox of classes: "I received your letter this morning, and 1 am replying to it at once, for russell: the Journal of the Berrrand Russdl Archives McMasrer University Library Press n.S.•2 (wincer 1992-93): 160-85 ISSN 0036-01631 • The same evaluation appears in Russell's Appendix to The Principles ofMathematics (p. 522). 2 Frege's '\- a II b" is Peano and Russell's "a E b". Russell should have put "~ana". 162 GREGORY LANDINI I believe I have discovered that classes are entirely superfluous. Your designation i(z) can be used for itself, and x E i(z) for (x).... this seems to me to avoid the contradiction" (p. 159). The proposal is also mentioned in a...

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...

This book offers a thought-provoking critique of analytic philosophy focusing on four central figures--Russell, Wittgenstein, Carnap, and Quine. In Wang's view, what lies "beyond" analytic philosophy is the abandonment of Empiricist accounts of how we know and epistemological limitations on what can be known. In making the foundations of science the center of "legitimate" philosophy, Analytic Empiricism has blocked important global perspectives found, for example, in continental and oriental philosophies. Wang advocates a Kantian transcendental dialectic: What explanation is required if we are to do justice to the many facets of knowledge, including artistic, poetic, emotional, and ethical as well as scientific? His thesis is that "intuition" must reclaim a central place in the answer.