Gregory Landini

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

This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the substitutional theory reveals the unity of Russell's philosophy of logic and offers new avenues for a genuine solution of the paradoxes plaguing Logicism.

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

It is well known that in _Principia Mathematica_ Russell offers a theory of definite descriptions and holds that ‘existence’ is not a property. It is less well known that in “On Denoting” he discusses the version of Anselm’s ontological argument for God formulated by Descartes, accepting the premiss “Existence is a perfection” and assessing the argument as valid but question-begging. This is different from his later comments in _A History of Western Philosophy_ which find the argument invalid. Indeed, given the sanctions of _Principia_, one might have thought he would find the argument logically ungrammatical. This paper shows how Russell might formulate and evaluate Anselm’s ontological argument and the version offered by Descartes in a way that avoids the conflict.1.

Wittgenstein's Tractatus has generated many interpretations since its publication in 1921, but over the years a consensus has developed concerning its criticisms of Russell's philosophy. In Wittgenstein's Apprenticeship with Russell, Gregory Landini draws extensively from his work on Russell's unpublished manuscripts to show that the consensus characterises Russell with positions he did not hold. Using a careful analysis of Wittgenstein's writings he traces the 'Doctrine of Showing' and the 'fundamental idea' of the Tractatus to Russell's logical atomist research program, which dissolves philosophical problems by employing variables with structure. He argues that Russell and his apprentice Wittgenstein were allies in a research program that makes logical analysis and reconstruction the essence of philosophy. His sharp and controversial study will be essential reading for all who are interested in this rich period in the history of analytic philosophy.

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...

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.

Unaware of Frege's 1879 Begriffsschrift, Russell's 1903 The Principles of Mathematics set out a calculus for logic whose foundation was the doctrine that any such calculus must adopt only one style of variables–entity (individual) variables. The idea was that logic is a universal and all-encompassing science, applying alike to whatever there is–propositions, universals, classes, concrete particulars. Unfortunately, Russell's early calculus has appeared archaic if not completely obscure. This paper is an attempt to recover the formal system, showing its philosophical background and its semantic completeness with respect to the tautologies of a modern sentential calculus