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 In Wittgenstein's Apprenticeship with Russell, GregoryLandini draws extensively from his work on Russell's unpublished manuscripts to show that the consensus characterises Russell with positions he did not hold.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. (shrink)
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 The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory.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. (shrink)
Do the rich descriptions and narrative shapings of literature provide a valuable resource for readers, writers, philosophers, and everyday people to imagine and confront the ultimate questions of life? Do the human activities of storytelling and complex moral decision-making have a deep connection? What are the moral responsibilities of the artist, critic, and reader? What can religious perspectives—from Catholic to Protestant to Mormon—contribute to literary criticism? Thirty well known contributors reflect on these questions, including iterary theorists Marshall Gregory, James (...) Phelan, and Wayne Booth; philosophers Martha Nussbaum, Richard Hart, and Nina Rosenstand; and authors John Updike, Charles Johnson, Flannery O'Connor, and Bernard Malamud. Divided into four sections, with introductory matter and questions for discussion, this accessible anthology represents the most crucial work today exploring the interdisciplinary connections between literature, religion and philosophy. (shrink)
Considerable variation exists not only in the kinds of transposable elements (TEs) occurring within the genomes of different species, but also in their abundance and distribution. Noting a similarity to the assortment of organisms among ecosystems, some researchers have called for an ecological approach to the study of transposon dynamics. However, there are several ways to adopt such an approach, and it is sometimes unclear what an ecological perspective will add to the existing co-evolutionary framework for explaining transposon-host interactions. This (...) review aims to clarify the conceptual foundations of transposon ecology in order to evaluate its explanatory prospects. We begin by identifying three unanswered questions regarding the abundance and distribution of TEs that potentially call for an ecological explanation. We then offer an operational distinction between evolutionary and ecological approaches to these questions. By determining the amount of variance in transposon abundance and distribution that is explained by ecological and evolutionary factors, respectively, it is possible empirically to assess the prospects for each of these explanatory frameworks. To illustrate how this methodology applies to a concrete example, we analyzed whole-genome data for one set of distantly related mammals and another more closely related group of arthropods. Our expectation was that ecological factors are most informative for explaining differences among individual TE lineages, rather than TE families, and for explaining their distribution among closely related as opposed to distantly related host genomes. We found that, in these data sets, ecological factors do in fact explain most of the variation in TE abundance and distribution among TE lineages across less distantly related host organisms. Evolutionary factors were not significant at these levels. However, the explanatory roles of evolution and ecology become inverted at the level of TE families or among more distantly related genomes. Not only does this example demonstrate the utility of our distinction between ecological and evolutionary perspectives, it further suggests an appropriate explanatory domain for the burgeoning discipline of transposon ecology. The fact that ecological processes appear to be impacting TE lineages over relatively short time scales further raises the possibility that transposons might serve as useful model systems for testing more general hypotheses in ecology. (shrink)
Bertrand Russell: Logic For Russell, Aristotelian syllogistic inference does not do justice to the subject of logic. This is surely not surprising. It may well be something of a surprise, however, to learn that in Russell’s view neither Boolean algebra nor modern quantification theory do justice to the subject. For Russell, logic is a synthetic … Continue reading Russell: Logic →.
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. (shrink)
Principia Mathematic goes to great lengths to hide its order/type indices and to make it appear as if its incomplete symbols behave as if they are singular terms. But well-hidden as they are, we cannot understand the proofs in Principia unless we bring them into focus. When we do, some rather surprising results emerge ? which is the subject of this paper.
This paper offers an interpretation of Russell's multiple-relation theory of judgment which characterizes it as direct application of the 1905 theory of definite descriptions. The paper maintains that it was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes, that Russell generated the philosophical explanation of the hierarchy of orders and the ramified theory of types of _Principia mathematica (1910). The interpretation provides a new understanding of Russell's abandoned book _Theory of Knowledge (1913), the 'direction (...) problems' and Wittgenstein's criticisms. (shrink)
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. (shrink)
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. (shrink)
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. (shrink)
On investigating a theorem that Russell used in discussing paradoxes of classes, Graham Priest distills a schema and then extends it to form an Inclosure Schema, which he argues is the common structure underlying both class-theoretical paradoxes (such as that of Russell, Cantor, Burali-Forti) and the paradoxes of ?definability? (offered by Richard, König-Dixon and Berry). This article shows that Russell's theorem is not Priest's schema and questions the application of Priest's Inclosure Schema to the paradoxes of ?definability?.1 1?Special thanks to (...) Francesco Orilia for criticisms of an early draft of this article. (shrink)
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. (shrink)
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. (shrink)
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. (shrink)
Confronted with Russell's Paradox, Frege wrote an appendix to volume II of his _Grundgesetze der Arithmetik_. In it he offered a revision to Basic Law V, and proclaimed with confidence that the major theorems for arithmetic are recoverable. This paper shows that Frege's revised system has been seriously undermined by interpretations that transcribe his system into a predicate logic that is inattentive to important details of his concept-script. By examining the revised system as a concept-script, we see how Frege imagined (...) that minor modifications of his former proofs would recover arithmetic. (shrink)