Bertrand Russell offered an influential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russell's paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some (...) doubt upon some of the uses to which modern descendente of Russell's paradox of propositions have been put in recent literature. (shrink)

Russell's "new contradiction" about "the totality of propositions" has been connected with a number of modal paradoxes. M. Oksanen has recently shown how these modal paradoxes are resolved in the set theory NFU. Russell's paradox of the totality of propositions was left unexplained, however. We reconstruct Russell's argument and explain how it is resolved in two intensional logics that are equiconsistent with NFU. We also show how different notions of possible worlds are represented in (...) these intensional logics. (shrink)

'These propositions may seem mild, yet, if accepted, they would absolutely revolutionize human life.' With these words Bertrand Russell introduces what is indeed a revolutionary book. Taking as his starting-point the irrationality of the world, he offers by contrast something 'wildly paradoxical and subversive' Sceptical Essays has never been out of print since its first publication in 1928. Today, besieged as we are by the numbing onslaught of twenty-first-century capitalism, Russell's defense of scepticism and independence of mind is (...) as timely as ever. In clear, engaging prose, he guides us through the key philosophical issues that affect our daily life. (shrink)

It is fairly well known that Wittgenstein's criticisms of Russell's multiple‐relation theory of judgement had a devastating effect on the latter's philosophical enterprise. The exact nature of those criticisms however, and the explanation for the severity of their consequences, has been a source of confusion and disagreement amongst both Russell and Wittgenstein scholars. In this paper, I offer an interpretation of those criticisms which shows them to be consonant with Wittgenstein's general critique of Russell's conception of logic and (...) which serves to elucidate some of the notoriously enigmatic passages of the Tractatus. In particular, I seek to show the continuity of Wittgenstein's criticisms of the theory of judgement with his remarks on Russell's paradox and the theory of types. In addition, I place these issues in the context of Russell's own philosophical ambitions in order to reveal the deep divisions between the two over the nature of logical form and the analysis of propositional content. (shrink)

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox (...) was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

Russell discovered the classes version of Russell's Paradox in spring 1901, and the predicates version near the same time. There is a problem, however, in dating the discovery of the propositional functions version. In 1906, Russell claimed he discovered it after May 1903, but this conflicts with the widespread belief that the functions version appears in _The Principles of Mathematics_, finished in late 1902. I argue that Russell's dating was accurate, and that the functions version does not (...) appear in the _Principles_. I distinguish the functions and predicates versions, give a novel reading of the _Principles_, section 85, as a paradox dealing with what Russell calls _assertions_, and show that Russell's logical notation in 1902 had no way of even formulating the functions version. The propositional functions version had its origins in the summer of 1903, soon after Russell's notation had changed in such a way as to make a formulation possible. (shrink)

The paradox of propositions, presented in Appendix B of Russell's The Principles of Mathematics (1903), is usually taken as Russell's principal motive, at the time, for moving from a simple to a ramified theory of types. I argue that this view is mistaken. A closer study of Russell's correspondence with Frege reveals that Russell carne to adopt a very different resolution of the paradox, calling into question not the simplicity of his early type theory (...) but the simplicity of his early theory of propositions. (shrink)

_'These propositions may seem mild, yet, if accepted, they would absolutely revolutionize human life.'_ With these words Bertrand Russell introduces what is indeed a revolutionary book. Taking as his starting-point the irrationality of the world, he offers by contrast something 'wildly paradoxical and subversive' - a belief that reason should determine human actions. Today, besieged as we are by the numbing onslaught of twenty-first-century capitalism, Russell's defence of scepticism and independence of mind is as timely as ever. In (...) clear, engaging prose, he guides us through the key philosophical issues that affect our daily lives - freedom, happiness, emotions, ethics and beliefs - and offers no-nonsense advice. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The humble origins of Russell's paradox by J. Alberto Coffa ON SEVERAL OCCASIONS Russell pointed out that the discovery of his celebrated paradox concerning the class of all classes not belonging to themselves was intimately related to Cantor's proof that there is no greatest cardinal. lOne of the earliest remarks to that effect occurs in The Principles ofMathematics where, referring to the universal class, the class (...) of all classes and the class of all propositions, he notes that when we apply the reasoning of his [Cantor's] proofto the cases in question we find ourselves met by definite contradictions, of which the one discussed in Chapter x is an example. (P. 362) And in a footnote he adds: "It was in this way that I discovered this contradiction". Throughout his writings Russell left a number ofhints concerning the sort of connection he had drawn between Cantor's proof and his own discovery. In fact, his suggestions are so specific that there would seem to be little room left for speculation concerning how the discovery took place.2 The picture that emerges almost immediately from Russell's observations is the following. For reasons which are 1 See, e.g., Bertrand Russell, The Principles of Mathematics, 2nd ed. (London: Allen & Unwin, 1937), §§100, 344-9; G. Frege, Wissenschaftlicher Briefwechsel (Hamburg: Felix Meiner Verlag, 1976), pp. 215-16; B. Russell, Essays in Analysis, ed. D. Lackey (New York: George Braziller, 1973), p. 139; B. Russell, Introducrion to Mathemarical Philosophy (New York: Simon and Schuster, 1971), p. 136; B. Russell, My Philosophical Development (New York: Simon and Schuster, 1959), pp. 75-6; The Autobiography of Bertrand Russell, I (New York: Bantam Books, 1968), 195. 2 See, e.g., Ch. Thiel's "Einleitung des Herausgebers" in Frege, Wissenschaftlicher Briefwechsel, pp. 203 and 216 (footnote), and 1. Grattan-Guinness, "How Bertrand Russell Discovered his Paradox", Historia Marhematica, 5 (1978),127-37. 31 32 Russell, nos. 33-4 (Spring-Summer 1979) never made clear, Russell decided to analyze Cantor's argument applying it to "large" classes such as the universal class V, and the class of all classes. When, for example, we consider V, its power set and the correlationf(x) = {x} if x is not a class, f(x) =. x otherwise, then Cantor's diagonal class D turns out to be the class of all classes not belonging to themselves. Moreover, since the element ofV which is taken toD byfisD itself(i.e., sincef(D) = D), Cantor's reasoning invites us to raise the question whether D belongs tof(D) (i.e., toD) or not; and it establishes that it does precisely if it doesn't.3 The purpose of this note is to present recently uncovered information that complements and corrects our present understanding of Russell's discovery of his paradox. As it turns out, far from originating from his desire to apply the ideas in Cantor's theorem, a version of Russell's paradox first occurred in an argument that Russell had devised, late in 1900, in order to establish the invalidity of that theorem. An appeal to "large" classes such as V or Class was, as we shall see, essential to Russell's attempted refutation. As he displayed the details of his counterexample Russell's paradox emerged, at first unrecognized, as the by-product of a project that the paradox itself would eventually undermine. In a paper written in 1901 and largely devoted to a popular exposition and defence of Cantor's theory of the infinite, Russell expressed what appeared to be a rpinor reservation to Cantor's treatment : There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work.4 3 See Principles, §349. Perhaps I should remind... (shrink)

The paradox of propositions, presented in Appendix B of Russell's The Principles of Mathematics (1903), is usually taken as Russell's principal motive, at the time, for moving from a simple to a ramified theory of types. I argue that this view is mistaken. A closer study of Russell's correspondence with Frege reveals that Russell carne to adopt a very different resolution of the paradox, calling into question not the simplicity of his early type theory (...) but the simplicity of his early theory of propositions. (shrink)

Russell claims in his autobiography and elsewhere that he discovered his 1905 theory of descriptions while attempting to solve the logical and semantic paradoxes plaguing his work on the foundations of mathematics. In this paper, I hope to make the connection between his work on the paradoxes and the theory of descriptions and his theory of incomplete symbols generally clearer. In particular, I argue that the theory of descriptions arose from the realization that not only can a class not be (...) thought of as a single thing, neither can the meaning/intension of any expression capable of singling out one collection (class) of things as opposed to another. If this is right, it shows that Russell’s method of solving the logical paradoxes is wholly incompatible with anything like a Fregean dualism between sense and reference or meaning and denotation. I also discuss how this realization lead to modifications in his understanding of propositions and propositional functions, and suggest that Russell’s confrontation with these issues may be instructive for ongoing research. (shrink)

_'These propositions may seem mild, yet, if accepted, they would absolutely revolutionize human life.'_ With these words Bertrand Russell introduces what is indeed a revolutionary book. Taking as his starting-point the irrationality of the world, he offers by contrast something 'wildly paradoxical and subversive' - a belief that reason should determine human actions. Today, besieged as we are by the numbing onslaught of twenty-first-century capitalism, Russell's defence of scepticism and independence of mind is as timely as ever. In (...) clear, engaging prose, he guides us through the key philosophical issues that affect our daily lives - freedom, happiness, emotions, ethics and beliefs - and offers no-nonsense advice. (shrink)

In Appendix B of Russell's The Principles of Mathematics occurs a paradox, the paradox of propositions, which a simple theory of types is unable to resolve. This fact is frequently taken to be one of the principal reasons for calling ramification onto the Russellian stage. The paper presents a detaiFled exposition of the paradox and its discussion in the correspondence between Frege and Russell. It is argued that Russell finally adopted a very simple solution to (...) the paradox. This solution had nothing to do with ramified types but marked an important shift in his theory of propositions. (shrink)

Solutions to Russell’s paradox of propositions and to Kaplan’s paradox are proposed based on an extension of von Neumann’s method of avoiding paradox. It is shown that Russell’s ‘anti-Cantorian’ mappings can be preserved using this method, but Kaplan’s mapping cannot. In addition, several versions of the Epimenides paradox are discussed in light of von Neumann’s method.

The author argues that the primary lesson of the so-Called logical and semantical paradoxes is that certain entities do not exist, Entities of which we mistakenly but firmly believe that they must exist. In particular, Russell's paradox teaches us that there is no such thing as the property which every property has if and only if it does not have itself. Why should anyone think that such a property must exist and, Hence, Conceive of russell's argument as (...) a paradox rather than a proof for the nonexistence of this property? the author traces this conviction to an uncritical acceptance of the so-Called principle of property abstraction and claims that this principle is false. It is simply not true, As one widely assumes, That propositional forms represent (complex) properties. Then it is argued, On independent grounds, That there are no complex properties, But only complex states of affairs. (shrink)

Sequel to Part I. In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addresses Russell’s own various attempts to solve (...) these paradoxes, including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ‘logical fictions’ or ‘logical constructions’ instead. (shrink)

In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theorem, its proof, how it can be used (...) to manufacture paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. (shrink)

In this article I try to show the philosophical continuity of Russell's ideas from his paradox of classes to Principia mathematica. With this purpose, I display the main results (descriptions, substitutions and types) as moments of the same development, whose principal goal was (as in his The principles) to look for a set of primitive ideas and propositions giving an account of all mathematics in logical terms, but now avoiding paradoxes. The sole way to reconstruct this central (...) period in Russell is to resort to unpublished manuscripts, which show the publications of these years to be the extremities of one same iceberg. Thus the logical problems (doubts about propositions, matrices and functions) are parallel to the ontological (the searching for genuine logical subjects) and methodological ones (the status of eliminative reduction). Paradoxes were the cause that the kind of (constructive) definition already applied by Russell, as an inheritance of Moorean analysis, obtained a new trait: ontological elimination, which, beginning with a ?no classes theory? and the theory of descriptions, led to a new theory of judgment (already present in manuscripts from 1906), ?incomplete symbols, and logical constructions. (shrink)

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also (...) model other axioms of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

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... (shrink)

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. (...) Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process. (shrink)

Prawitz observed that Russell’s paradox in naive set theory yields a derivation of absurdity whose reduction sequence loops. Building on this observation, and based on numerous examples, Tennant claimed that this looping feature, or more generally, the fact that derivations of absurdity do not normalize, is characteristic of the paradoxes. Striking results by Ekman show that looping reduction sequences are already obtained in minimal propositional logic, when certain reduction steps, which are prima facie plausible, are considered in addition to (...) the standard ones. This shows that the notion of reduction is in need of clarification. Referring to the notion of identity of proofs in general proof theory, we argue that reduction steps should not merely remove redundancies, but must respect the identity of proofs. Consequentially, we propose to modify Tennant’s paradoxicality test by basing it on this refined notion of reduction. (shrink)

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. (...) Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

The conventional wisdom has it that between 1905 and 1919 Russell was critical to pragmatism. In particular, in two essays written in 1908–9, he sharply attacked the pragmatist theory of truth, emphasizing that truth is not relative to human practice. In fact, however, Russell was much more indebted to the pragmatists, in particular to William James, as usually believed. For example, he borrowed from James two key concepts of his new epistemology: sense-data, and the distinction between knowledge by acquaintance and (...) knowledge by description. Reasonable explanation of this is that, historically, Russell’s logical realism and James’s pragmatism have the same roots—the German philosopher Rudolph Hermann Lotze (1817–1881). In this paper we are going to explore the fact that in 1905, under Lotze’s influence, Russell married propositions with beliefs. A few years later this step also made Russell prone to embrace the theory of truth-making that has its roots in James. In contrast to the concept of sense-data and to the distinction between knowledge by acquaintance and knowledge by description, however, the understanding that we believe propositions—and not, for example, simply grasp them—was in tension with Russell’s Principle of Extensionality, according to which propositions can be logically connected with other propositions only as truth-functions. The point is that when we judge a mind-relation (for example, a relation of belief) to a proposition, the latter cannot be determined as true or false. The two most talented pupils of Russell, Wittgenstein and Ramsey, severely criticized the central place propositional attitudes play in Russell’s logic. Wittgenstein analyzed “A believes that p” to “ ‘p’ says p” (5.542). Ramsey criticized Russell’s beliefs in propositions the other way round: He stressed that belief is an ambiguous term that can be interpreted for the better in the sense of pragmatism. Prima facie surprisingly, he maintained that his “pragmatism is derived from Mr Russell.” (1927: 51). (shrink)

This paper investigates the relationship between James’ radical empiricism and Deleuze’s study of the genesis of sense without a transcendental subject as necessary condition. It shows that James’ concept of pure experience changes the form of relation between mind and world. Considering how to conceptualize experience without a fixed metaphysical or transcendental subject destabilizes ontological identity, leads to a founding conceptual divergence from traditional phenomenology, and motivates Deleuze’s efforts towards transcendental empiricism. The paper reads Deleuze’s work on the genesis of (...) sense in this context, arguing that one important result is an ontological pluralism. Such pluralism is crucial in considering how meaning can be made between and across differences and is in keeping with radical empiricism’s openness to life’s complexity. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2501\REVIEWS.251 : 2005-09-14 19:58  Reviews RUSSELL’S INFLUENCE ON INGEMAR HEDENIUS S A Theology and Religious Studies / U. of Lund  , Lund, Sweden @. Svante Nordin. Ingemar Hedenius. En filosof och hans tid [Ingemar Hedenius. A philosopher and his time]. Stockholm: Natur och Kultur, . Pp. ;  photos.  kr. en years ago I wrote a review article about Gunnar Fredriksson’s book (...) Tabout Bertrand Russell for Russell called “Russell’s Influence in Sweden”. In it I mentioned two Swedish philosophers who introduced and established analytic philosophy in Sweden. The oldest, Ingemar Hedenius (–), was professor in practical philosophy in Uppsala from  to ; the younger, Anders Wedberg (–), was professor in theoretical philosophy in Stockholm from  to . Together they had an enormous influence on the development of Swedish philosophy, since—among other things—they had the  See the review of Bertrand Russell: en intellektuelli i politiken, Russell, n.s.  (): –. _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2501\REVIEWS.251 : 2005-09-14 19:58 Reviews  power to influence most of the new professors in philosophy. And they can both be called disciples of Bertrand Russell. The formal division between theoretical and practical philosophy goes back to Aristotle and Kant and consists in a distinction between the philosophy of thinking and the philosophy of acting. In practice it means that the former subject includes metaphysics, epistemology, philosophy of language and philosophy of mathematics and logic, while the latter deals with ethics, aesthetics, social philosophy and philosophy of religion. It was Wedberg who introduced Russell as a logician and a philosopher of mathematics and for a long time theoretical philosophers primarily wrote about logic and tried to apply the symbols of mathematical logic to classical philosophical problems. Russell’s influence on Hedenius was of a more general character that showed itself in his writings on ethics and religion, but also in his general attitude to life. In  Hedenius published a collection of essays called Tro och Vetande [Faith and Knowledge] that started a public discussion about the truth of Christianity that went on for years. The book was an undisguised attack on Swedish theology and Swedish theologians that caused a lot of hard feelings and undermined the reputation of theology as a legitimate academic subject. It was such a knock out that still today many Swedish theologians suffer from a “Hedenius complex”. (One could call it a “Russell complex”.) There is only one reference to Russell in Hedenius’ book and that is to Sceptical Essays, where Russell writes: “I wish to propose for the reader’s favourable consideration a doctrine which may, I fear, appear wildly paradoxical and subversive. The doctrine in question is this: that it is undesirable to believe a proposition when there is no ground whatever for supposing it true” (SE, p. ). Of course, Russell was not the first philosopher to endorse such a doctrine, which can be traced back to the spirit of Socrates and the old Greek sceptical philosophers, but Hedenius seems to have been impressed by Russell’s humorous and elegant way of formulating it. He took it to his heart and called it “den intellektuella moralens maxim”, which translates into “the maxim of intellectual morality” and sounds like an echo of Kingdon Clifford, although Hedenius might not have heard of him. Anyway, Hedenius thought that it was morally wrong to believe any proposition without having good reasons. With this maxim in mind he examined the major tenets of Christianity and found that very few—if any—passed the test, although he never seems to have doubted that Jesus existed, which he might have done, if he had examined the reasons for believing so more critically. From now on Hedenius was the best known and feared philosopher in Sweden. Another professor in Uppsala with a big sense of humour said: “There is no God and Ingemar Hedenius is his prophet.” Hedenius took an active part _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2501\REVIEWS.251 : 2005-09-14 19:58  Reviews in many public discussions and wrote hundreds of newspaper articles, essays and close to  books that could be read by anyone. The influence of Russell is obvious in many of them, although he is seldom... (shrink)

The Russell-Myhill Antinomy, also known as the Principles of Mathematics Appendix B Paradox, is a contradiction that arises in the logical treatment of classes and "propositions", where "propositions" are understood as mind-independent and language-independent logical objects. If propositions are treated as objectively existing objects, then they can be members of classes. But propositions can also be about classes, including classes of propositions. Indeed, for each class of propositions, there is a proposition stating that (...) all propositions in that class are true. Propositions of this form are said to "assert the logical product" of their associated classes. Some such propositions are themselves in the class whose logical product they assert. For example, the proposition asserting that all-propositions-in-the- class-of-all-propositions-are-true is itself a proposition, and therefore it itself is in the class whose logical product it asserts. However, the proposition stating that all-propositions-in-the-null-class-are-true is not itself in the null class. Now consider the class w, consisting of all propositions that state the logical product of some class m in which they are not included. This w is itself a class of propositions, and so there is a proposition r, stating its logical product. The contradiction arises from asking the question of whether r is in the class w. It seems that r is in w just in case it is not. This antinomy was discovered by Bertrand Russell in 1902, a year after discovering a simpler paradox usually called Russell's paradox ". It was discussed informally in Appendix B of his 1903 Principles of Mathematics . In 1958, the antinomy was independently rediscovered by John Myhill, who found it to plague the "Logic of Sense and Denotation" developed by Alonzo Church. (shrink)

It would be an understatement to say that Russell was interested in Cantorian diagonal paradoxes. His discovery of the various versions of Russell’s paradox—the classes version, the predicates version, the propositional functions version—had a lasting effect on his views in philosophical logic. Similar Cantorian paradoxes regarding propositions—such as that discussed in §500 of The Principles of Mathematics—were surely among the reasons Russell eventually abandoned his ontology of propositions.1 However, Russell’s reasons for abandoning what he called “denoting concepts”, (...) and his rejection of similar “semantic dualisms” such as Frege’s theory of sense and reference—at least in “On Denoting”—made no explicit mention of any Cantorian paradox. My aim in this paper is to argue that such paradoxes do pose a problem for certain theories such as Frege’s, and early Russell’s, about how definite descriptions are meaningful. My first aim is simply to lay out the problem I have in mind. Next, I shall turn to arguing that the theories of descriptions endorsed by Frege and by Russell prior to “On Denoting” are susceptible to the problem. Finally, I explore what responses a.. (shrink)

Russell takes his paper ?On denoting? to have achieved the repudiation of the theory of denoting concepts and Frege?s theory of sense, and the invention of the notion of incomplete symbols.This means that Russell attempts to solve the set theoretic and semantic paradoxes without making use of a theory of sense.Instead, his strategy is to revise his logical ontology by arguing that certain symbols should be treated as incomplete.In constructing such arguments Russell, at various points, makes use of epistemological and (...) metaphysical considerations.These arguments do not form themselves into a systemic set of considerations to be used in appraising a logical system.Finally, the vicious circle principle is argued for on the basis of considerations, which are presumed evident, about the nature of propositional functions.The stringency of this principle is a basic problem for the system of Principia mathematica.However, even given the terms of the argument, ?On denoting? does not offer a complete repudiation of the notion of sense.This allows the possibility of retaining some of the insights of Principia mathematica whilst rejecting the stringency of the vicious circle principle.The basis of such a system is the theory of sense. (shrink)

This note presents a transcription of Russell's letter to Hawtrey of 22 January 1907 accompanied by some proposed emendations. In that letter Russell describes the paradox that he says "pilled" the "substitutional theory" developed just before he turned to the theory of types. A close paraphrase of the derivation of the paradox in a contemporary Lemmon-style natural deduction system shows which axioms the theory must assume to govern its characteristic notion of substituting individuals and propositions for (...) each other in other propositions. Other discussions of this paradox in the literature are mentioned. I conclude with remarks about the significance of the paradox for Russell. (shrink)

This note presents a transcription of Russell's letter to Hawtrey of 22 January 1907 accompanied by some proposed emendations. In that letter Russell describes the paradox that he says "pilled" the "substitutional theory" developed just before he turned to the theory of types. A close paraphrase of the derivation of the paradox in a contemporary Lemmon-style natural deduction system shows which axioms the theory must assume to govern its characteristic notion of substituting individuals and propositions for (...) each other in other propositions. Other discussions of this paradox in the literature are mentioned. I conclude with remarks about the significance of the paradox for Russell. (shrink)

Edmund Husserl’s engagement with Bertrand Russell’s paradox stands in a continuum of reciprocal reception and discussions about impossible objects in the School of Brentano. Against this broader context, we will focus on Husserl’s discussion of Russell’s paradox in his manuscript A I 35α from 1912. This highly interesting and revealing manuscript has unfortunately remained unpublished, which probably explains the scant attention it has received. I will examine Husserl’s approach in A I 35α by relating it to earlier discussions (...) of relevant topics in his manuscripts and the broader historical context of the School of Brentano and early phenomenology. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:russell: the Journal of Bertrand Russell Studies n.s. 33 (summer 2013): 59–94 The Bertrand Russell Research Centre, McMaster U. issn 0036–01631; online 1913–8032 oeviews PRINCIPIA’S SECOND EDITION Russell Wahl English and Philosophy / Idaho State U. Pocatello, id 83209, usa [email protected] Bernard Linsky. The Evolution of Principia Mathematica: Bertrand Russell’s Manuscripts and Notes for the Second Edition. Cambridge: Cambridge U. P., 2011. Pp. vii, 407; 2 plates. isbn: 978-1-10700-327-9. (...) £96; us$159. ussell and Whitehead’s Principia Mathematica was a pioneering work which was in many ways overtaken by the success of the new discipline it helped found. There is little doubt that Principia Mathematica was one of the most important works in the development of symbolic logic, as even Quine, one of the critics of the foundations proposed in it, said: “This is the book that has meant the most to me.”Yet few mathematicians or logicians follow the theory of types proposed in Principia as the foundation of mathematics, nor the logicist project of which it is a variant. While much of Principia ’s notation has been incorporated into modern formal logic and set theory, much of it also is dated and not easy for contemporary mathematicians and logicians to read. Among those who were dissatisfied with the philosophical foundations of the first edition were the authors themselves.1 During 1923 and 1924 Russell revised and updated Principia, and the second edition of the first volume appeared in 1925 with the other two volumes appearing in 1927. What resulted was a new edition with the philosophical underpinning modi- fied, but with the superstructure, that is, the resulting theorems of the three volumes, unchanged. Russell wrote a new Introduction and added three Appendices, but other than that made only minor changes.The new philosophy of logic had as its core idea a principle of extensionality which Russell expressed as the view that “functions of propositions are always truth-functions and a function can only occur in a proposition through its values” (PM2 1: ______ 1 Russell wrote the new material for the second edition, but it is clear from a letter to him byWhitehead on 24 May 1923, quoted by Linsky, that he, too, thought the foundational system needed tweaking: “I don’t think that ‘types’ are quite right” (p. 16). o= 60 Reviews xiv). It is interesting that at times Russell seems to have been hesitant about even endorsing this change.2 The second edition of Principia has not been well received; by and large it has either been ignored or scorned. Linsky cites especially negative remarks by Monk, but others have dismissed the second edition additions and it has generally been thought a failure since 1944, when Gödel pointed out an error in the Appendix B proof that mathematical induction can be “rectified” even without the axiom of reducibility. The Evolution of Principia Mathematica makes available to scholars Russell’s notes and manuscripts of the new material added to the second edition. Linsky also seeks to redress the indifference to the second edition, to set the record straight about several misunderstandings, and to give an account of the new material.The book he has produced will be very valuable to scholars of Russell and to anyone interested in the development of type theory and indeed of logic as a whole during this time. This is also a difficult book, both because the material can get quite technical at times and also because the new material included in the introduction to the second edition is somewhat sketchy, and there are philosophical and technical points about which Russell himself is less than clear. As a result there have been disagreements about how the second edition is best to be interpreted. Linsky recognizes that the interpretive issues are under-determined by the text and wants to stay relatively neutral with respect to the different alternatives about the various controversies. At times it can be difficult to keep the various positions straight.The difficulty lies with the material, and Linsky does a very good job of explaining the issues to the layman. The work is divided into... (shrink)

The philosophy of logical atomism.--Logical atomism.

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)

In lieu of an abstract, here is a brief excerpt of the content:Scientific Method in PhilosophyAuthor's note: Thanks to Gregory Landini for helpful clarifications.Gregory Landini. Repairing Bertrand Russell's 1913 Theory of Knowledge. (History of Analytic Philosophy.) London: Palgrave Macmillan, 2022. Pp. x, 397. isbn: 978-3-030-66355-1, us$139 (hb); 978-3-030-66356-8, us$109 (ebook).The title of this book might suggest a rather narrow study of a problem with Russell's Theory of Knowledge and a proposed solution. But as with Landini's first book, (...) class='Hi'>Russell's Hidden Substitutional Theory, the title does not capture the full scope of the work. That work did not just point out the lingering substitution theory in Principia, but gave a sweeping view of Russell's overall development from Principles to Principia and one of the most detailed accounts of the formal system in Principia in any work. This work gives a sweeping overview of Russell's philosophy up to 1918, including not just Russell's multiple-relation theory of judgment, but the whole project of what Russell called scientific method in philosophy, which was the subtitle to his 1914 Our Knowledge of the ExternalWorld. Landini thinks this is the height of Russell's philosophy and proposes a fix to the problems which led Russell to abandon the project.Landini does not just present Russell's views during this period, but vigorously defends them, and not just the multiple-relation theory of judgment mostly associated with Theory of Knowledge, but also the views that logic should be understood as synthetic a priori, Russell's doctrine of acquaintance, including the acquaintance with universals as the grounding of synthetic a priori knowledge, and Russell's view that logic is the essence of philosophy. The book contains a wide range of discussions, including issues in philosophy of mind, theories of representation, theories of universals, evolution in philosophy, and the metaphysics of time and space.The book opens with an overview of some main themes, including Landini's view of the three main phases of Russell's post-idealist philosophy and his discussion of what he calls the revolutions in logic and mathematics. The three phases Landini has in mind are the Principles phase, the Principia phase [End Page 81] (which includes The Problems of Philosophy and Theory of Knowledge) and the neutral monist phase, which Landini sees as a mistake Russell made when he didn't see how to repair his 1913 work. According to Landini, the Principles era ends with the failure of the substitution theory from the po /ao paradox, the Principia era ends by 1918 with the abandonment of the attempt to finish the project of the 1913 Theory of Knowledge, and the neutral monist era begins in 1919 and continues through 1948. Landini wants to concentrate on repairing the view in the Principia era. Any remarks of Russell's from later rejecting, for example, his view of the subject, of acquaintance, of neutral monism, and remarks concerning logic as consisting of tautologies are to be rejected. Landini firmly believes that after abandoning the 1913 project, Russell took a wrong turn. Landini's discussion of the multiple-relation theory of judgment is therefore quite different from the other discussion in the literature, in that most commentators have thought Russell was correct in abandoning the theory and most think there is something to Wittgenstein's criticism. Landini thinks Wittgenstein's criticism was based on his new vision of logic, and that Russell should have rejected it and proceeded with the project of Theory of Knowledge.Given the controversies involved in these claims, I think the best way forward is an explication of Landini's new emphasis on the revolutions in logic and mathematics and some of the key claims Landini makes concerning the Principia era. Then we can look at the details of Landini's repairing of the project.the two revolutionsWhat is striking about Landini's more recent work is his emphasis on these two revolutions. One is the revolution within logic, which originates with Frege; the other a revolution in mathematics which originates with Weierstrass, von Staudt, Pieri and Cantor. While most historians of this period are aware of the important shifts... (shrink)

The paradox of mass voting is not, generally speaking, matched by a paradoxical mass attempt to be politically well informed. As Converse underscored, most people are grossly politically ignorant—just as they would be if, as rational‐ignorance theory holds, they realized that their votes don't matter. Yet many millions of them contradict the theory by voting. This contradiction, and the illogical reasons people offer for voting, suggest that the logic of collective action does not come naturally to people. To equate (...) public ignorance with “rational ignorance,” then, attributes too much theoretical self‐consciousness to people. Even if their ignorance is not based on a grasp of the collective‐action problem, however, people's political ignorance is “rational”: but it is less a matter of calculating the low benefit likely to flow from one vote in a sea of millions, than of encountering the much higher cost of being well informed as compared to voting. As the high cost of becoming well informed appears likely to grow, it may be advisable to move off of the political agenda issues that have no clear continuum toward which a mostly ignorant median voter may gravitate. (shrink)

This article examines the development of Russell's treatment of propositions, in relation to the topic of psychologism. In the first section, we outline the concept of psychologism, and show how it can arise in relation to theories of the nature of propositions. Following this, we note the anti-psychologistic elements of Russell's thought dating back to his idealist roots. From there, we sketch the development of Russell's theory of the proposition through a number of its key (...) transitions. We show that Russell, in responding to a variety of different problems relating to the proposition, chose to resolve these problems in ways that continually made concessions to psychologism. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:September 27, 2008 (1:09 pm) G:\WPData\TYPE2801\russell 28,1 048RED.wpd russell: the Journal of Bertrand Russell Studies n.s. 28 (summer 2008): 69–93 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 eviews WITTGENSTEIN, CRITIC OF RUSSELL Russell Wahl English and Philosophy / Idaho State U Pocatello, id 83209, usa [email protected] Jérôme Sackur. Formes et faits: Analyse et théorie de la connaissance dans l’atomisme logique. Paris: Vrin, 2005. Pp. 313. (...) ¤31.00. isbn 978-2-7116-1722-7. Sackur joins several other recent French authors in writing penetrating analyses of Russell’s and Wittgenstein’s work. Forms and Factsz is an explication of the Tractatus, conducted by means of a contrast between Russell’s work and Wittgenstein’s. Sackur examines the interplay between Russell’s and Wittgenstein ’s versions of logical atomism, particularly, as Sackur puts it, between Russell’s original thought on the one hand, which provides the material for Wittgenstein’s position in the Tractatus, and Wittgenstein’s critique and development on the other, which, he says, provides a unity to Russell’s position (p. 12). While recognizing the inXuence of Russell on Wittgenstein’s project, Sackur in the end sees Wittgenstein as undermining and ultimately rejecting Russell’s logicism. He does not just repeat old claims about Wittgenstein, Russell and logical form. His analysis is very subtle and, perhaps unusually for a work on Wittgenstein which defends Wittgenstein over Russell, Formes et faitsz displays a surprisingly detailed knowledge of Russell’s work, before, during and after the period in question, 1913–19. Anyone who is interested in Russell’s and Wittgenstein ’s work during this period, and in particular Wittgenstein’s criticisms of Russell, will beneWt greatly from this book. There are detailed, well informed discussions of important points of disagreement, particularly Russell and Whitehead ’s account of succession and the ancestor relation, Wittgenstein’s remarks on operators, in particular the Nz operator, various remarks about the theory of types, and Wittgenstein’s now well known criticism of Russell’s theory of judgment. The subtitle of the work, “Analysis and the theory of knowledge in logical atomism”, reXects the points where Sackur sees the greatest disagreement between Russell and Wittgenstein. In the Wrst place, he sees them diTering in their attitude toward analysis, with Russell oTering translations of sentences into a September 27, 2008 (1:09 pm) G:\WPData\TYPE2801\russell 28,1 048RED.wpd 70 Reviews 1 For the Wrst point, see (among others) P.yM.yS. Hacker, Wittgenstein’s Place in Twentieth Century Philosophy (Oxford: Blackwell, 1996), pp. 69–70. For the second, see (again among others), David Pears, The False Prison (Oxford: Clarendon P., 1987), i: 63, where Pears says that Russell’s atomism was founded on the doctrine of “forced acquaintance”. 2 P.yJ. Hager, Continuity and Change in the Development of Russell’s Philosophy (Dordrecht: Klewer, 1994). perspicuous language and Wittgenstein oTering explications, or elucidations, of senses already in the sentences. Secondly, they diTer on the role that epistemology should play in logical atomism, with Wittgenstein criticizing the role of acquaintance in the development of logic. While both of these points have been made before,1 Sackur’s discussion is more thorough than previous studies and leads to some new interpretations of passages in the Tractatus. Sackur’s discussion of analysis is important to his argument. He is interested in a tension between what he sees as two aims of analysis in Wittgenstein’s work, one found in Tractatusz 3.2–3.3 and the other in Tractatusz 4.21–4.2211. He sees analysis in the Wrst group as revealing or elucidating the sense a proposition expresses, a sense understood and already there in the ordinary unanalyzed proposition, and analysis in the second group, driven by the theses of necessity and logical independence, as focusing on truth-functional analysis in terms of elementary propositions that consist of names referring to simple objects. He argues that these two aims don’t automatically go together and in fact appear to be at cross purposes. In the course of the Wrst chapter, the major distinction Sackur makes is between analysis... (shrink)

Deleuze’s “transcendental empiricism” and the “empirical side” of Whitehead’s metaphysics are paradoxical unless placed in the context of their unorthodox readings of empiricism. I explore this context focusing on their engagements with Hume. Both subvert presumptions of a categorical gap between external nature and internal human experience and open possibilities for a speculative empiricism that is non-reductive while still affirming experience as source for philosophical thinking. Deleuze and Whitehead follow Hume in beginning with events of sensation as primary but do (...) not presume the logic of (human) subjects and objects (of nature) as necessary structuring polarities for their interpretation. This challenges a basic distinction (between inner and outer or between self and world) that seems inherent in the ordinary concept of experience, thus earning the moniker speculative. The speculative empiricist studies how these abstractions arise from events of experience prior to their consolidation in representation. This includes a critical component: to what extent do unexamined assumptions about conceptual abstraction hinder, block, or prefigure experiential attention? This critical component has existential implications for how we attend to the affective, intuitive, and preconceptual. (shrink)

The objective of this paper is to show that Russell's paradox cannot be solved just by defining a class as what is classified, as Balzer thinks. It can be solved not by defining a class, as he does, but by rejecting the assumption on which the validity of argument is based, that is, not conceding the truth of the disjunctive premise that a class is either an instance of itself or not an instance of itself.

While it has perhaps always accompanied philosophical thought – one immediately thinks of Plato’s Dialogues – the problem of the communication of that thought, and therefore of its capacity to be taught, has acquired a new insistence in the work of post-Kantian thinkers. As evidence of this one could cite Fichte’s repeated efforts to formulate a definitive version of his Wissenschaftslehre, the model of the Bildungsroman that Hegel adopts for his Phenomenology of Spirit, Kierkegaard’s pseudonymous works, Nietzsche’s Thus Spoke Zarathustra, (...) and Heidegger’s uncompleted Being and Time. Certainly each of these thinkers, as well as their projects, are quite distinct from the others, but their works are also signposts. It is not a question here of reducing the puzzles and difficulties of these works to some single element, some trace of a philosophical zeitgeist. Rather, it is enough to indicate this preoccupation with the problem of the communicability of thinking in order to raise the question of the modality of such a communication. How has thinking come to be so fundamentally troubled by its own expression? That Kant’s Critical philosophy has precipitated this preoccupation ought to come as no surprise, for it was Kant, most famously in the Critique of Pure Reason, who sought to definitively set the limits of philosophical thinking. In doing so, however, Kant is unable to remain utterly silent about what he himself sets outside the bounds of meaningful thought and speech: things-in-themselves. The mere fact that this meaningless element is an essential component of Kant’s philosophical exposition is an indication of the paradoxical logic of the limit: to postulate a limit is simultaneously to set out its obverse side. In Kant’s case, then, the demarcation of what can be thought carries with it the implicit thinking of what is ruled out of bounds for thought. Beginning with Fichte, the philosophical project of Kant’s successors and inheritors is then largely determined by an effort to incorporate or at least accommodate this alien element within thinking. The initial phase of this effort is marked by a tragic motif, a conjoining of thought with the experience of loss and suffering that is characteristic of Romanticism, that culminates in the dialectic of Hegel’s Absolute Knowing. In Absolute Knowing, thought has not vanquished its limitations, it has instead discovered the resources with which it can perpetually surmount them. This first phase is followed by a reaction or response to this seeming completion and overcoming of the Kantian limitation. Now, the tragic movement of thought comes to be understood as itself limited, and necessarily so according to the logic of the limit that imposes itself upon every gesture of closure. Tragedy thereby gives way to comedy insofar as the limitation that is incorporated as an obstacle to be overcome by tragic thought is shown to be the external condition of that very thought. Finally, if the comic is the fate of the limitations of tragedy, the thought of comedy itself demands a third mode of thinking, one capable of something more than the mere reiteration of the limit between comedy and tragedy that troubles them both. This third form of expression is parody, an expression appropriate to a thinking that repeats and completes the threefold movement of tragic thought while, in this very repetition, creating a limitless domain of thoughtful expression. (shrink)

Bertrand Russell's 1903 masterpiece "The Principles of Mathematics" places great emphasis on the need to separate propositions from psychological items such as thoughts. In 1919 Russell explicitly retracts this view, however, and defines propositions as "psychological occurrences". These psychological occurrences are held by Russell to be mental images. In this paper, I seek to explain this radical change of heart. I argue that Russell's re-psychologising of the proposition in 1919 can only be understood against the background (...) of his struggle with the problem of the unity of the proposition in earlier work. Once this is recognized, and the solution to the problem offered by the 1919 theory is appreciated, new light is also shed on Russell's naturalism. I go on to compare Russell's psychological "picture theory" with the vehemently anti-psychological picture theory of Wittgenstein's "Tractatus Logico-Philosophicus" and suggest that, once the background of the dispute is brought into clearer focus, Russell's position can be seen to have many advantages over its more celebrated rival. (shrink)