Could space consist entirely of extended regions, without any regions shaped like points, lines, or surfaces? Peter Forrest and Frank Arntzenius have independently raised a paradox of size for space like this, drawing on a construction of Cantor’s. I present a new version of this argument and explore possible lines of response.

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call (...) *Countable Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.

'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)

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: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, 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)

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)

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)

Relativized Metaphysical Modality (RMM: Murray and Wilson, 'Relativized metaphysical modality', Oxford Studies in Metaphysics, 2012; Murray, Perspectives on Modal Metaphysics, 2017) exploits 'two-dimensionalist' resources to metaphysical, rather than epistemological, ends: the second dimension offers perspective-dependence without contingency, diverting attacks on 'Classical' analyses of modals (in effect, analyses validating S5 and the Barcan Formulae). Here, we extend the RMM program in two directions. First, we harvest resources for RMM from Lewis's 1980 'Context--Index' (CI) framework: (a) the ban in CI on binding (...) into context-arguments (akin to Kaplan's 'monstrosity' ban) projects a bright line between perspective-dependence and contingency; and (b) CI-postulated connections among meaning, content, truth, argument-structure, context, and modality collectively generate a 'Generalized Humphrey Problem' for any non-Classical analysis (examples covered include appeals to accessibility, contingent domains, and counterpart relations). Second, we sharpen the tools of RMM-based metaphysical analysis, and extend their domain of coverage across familiar anomalies for Classical modals: we revisit earlier RMM-based bulwarks for S5 (against 'Chisholm's Paradox' for moderate flexibility of essence, and nomological necessitarianism); and we now similarly shore up the Barcan Formulae (against the apparent contingency of existence and nonexistence). (shrink)

F. Russell Sullivan analyzes the relationship between faith and reason in Kierkegaard's philosophy. Kierkegaard is widely considered to be an irrationalist. Sullivan argues that he views faith as reasonable in a distinct way that must be uncovered. In some of his pseudonymous works, Kierkegaard speaks of the movement of faith as paradoxical and absurd. There is evidence from his non-pseudonymous works that Kierkegaard does not consider faith irrational. He denigrates reason only in that he wishes to impress upon nominal Christians (...) (who look upon faith only as a body of doctrine) that more and more understanding of the tenets of faith can never yield logical certainty.The doctrines of faith can be argued pro and contra. For Kierkgaard, faith in this context is illogical, but not irrational. (shrink)

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic (...) interest of this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)

Some philosophers have claimed that sentences like all bachelors are unmarried are analytic, where this is to say that they are true in virtue of meaning, and that anyone who understands one can know that it is true. Some have claimed in addition that the notion of analyticity can be used to solve problems in epistemology. However, in the last century the work of Quine and Putnam led many to doubt such claims, and to suspect that there is no analyticity, (...) only an illusion of analyticity to be explained. ;I think that there is something to Quine and Putnams objections. Moreover, I think that they are of help in formulating a better account of analyticity. The core of this dissertation is an account of the analytic/synthetic distinction which uses recent developments in the philosophy of language-including Russellian semantics, distinctions between different kinds of meaning, two-dimensional modal logic and the distinction between analyticity, necessity and a priority---to defend a theory which covers both the traditional necessary analytic truths such as all vixens are female and Kaplans examples of the contingent analytic such as I am here now, dthat □ = and Snow is white iff actually, now, snow is white. ;This account is used to provide solutions to problems in philosophical logic concerning the correct accounts of vagueness, inconsistency and modality. I also show how it can be used to provide a solution to the Liar paradox. (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)

_'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)

This volume covers the period from the beginning of Russell's work on Volume Two of the Principles of Mathematics to the critical discovery of the theory of descriptions in 1905. Foundations of Logic gives a vivid picture of Russell wrestling with the logical paradoxes, often unsuccessfully, as he tries out one foundational scheme after another. This volume provides the key to both Bertrand Russell's philosophy of logic and philosophy of mathematics. It includes unpublished work on the theory of denoting which (...) predates Russell's famous article of 1905 and unpublished manuscripts on the so-called "zig-zag" theory with which Russell attempted to provide a type-free foundation for mathematics. The volume also gathers together for the first time a number of reviews and survey articles, along with two talks on modality and truth. It will be an essential addition to any Bertrand Russell collection. (shrink)

This volume shows Bertrand Russell in transition from a neo-Kantian and neo-Hegelian philosopher to an analytic philosopher of the highest rank. During this period, his research centered on writing The Principles of Mathematics. The volume draws together previously unpublished drafts which shed light on Russell's struggle to accept Cantor's notion of continuum as well as Russell's infinite ordinal and cardinal numbers. It also includes the first version of Russell's Paradox.

Throughout this paper my objective will be to establish and clarify Hume's original intentions in his discussion of causation in Book I of the Treatise. I will show that Hume's views on ontology, presented in Part IV of that book, shed light on his views on causation as presented in Part III. Further, I will argue that Hume's views on ontology account for the original motivation behind his two definitions of 2 cause. This relationship between Hume's ontology and his account (...) of causation explains something which has baffled Hume scholars for some time,- namely, why does Hume's discussion of causation in I, iii, 14 have such a paradoxical air about it? I will show that Hume's views on causation have a paradoxical air about them because they rest on an ontology of "double existence" - an ontology which Hume describes as the monstrous offspring of two principles, which are contrary to each other, which are both at once embrac'd by the mind, and which are unable mutually to destroy each other (T 215) My interpretation will centre on the following two claims: (i) When Hume wrote Section 14, Of the idea of necessary connexion, he was primarily concerned to attack the view that the origin of our idea of necessity was to be discovered in the operations of matter or bodies. Of the suggested sources from which our idea of necessity could be thought to originate this is the source which, initially, interested Hume the most. It is, therefore, of great importance that we interpret Hume's remarks in light of this fact. (ii) Hume offers the first definition of cause as an account of causation as it exists in the material world independent of our thought and reasoning.He offers the second definition as an account of causation as we find it in our perceptions. It will also be argued, in this context, that necessity constitutes "an essential part" of both of Hume's two definitions of cause. (shrink)

Could it ever be right to say that a language---as opposed to a speaker of the language---makes, or presupposes or somehow commits itself to certain claims? Such as that certain kinds of objects exist, or that things are a certain way? It can be tempting to think not, to think that languages are just the neutral media through which speakers make claims. Yet certain, surprisingly diverse, phenomena---analyticity, racial epithets, object-involving direct reference, arithmetic, and semantic paradoxes like the Liar---have pushed philosophers (...) towards views according to which languages can have presuppositions or commitments of their own---to things like the existence of numbers, the marital status of bachelors, the existence of water, and even to contradictions or morally abhorrent views. In this paper I want to present some recent data from linguistics that supports a less commonly discussed, and rather surprising version of this idea: namely that English presupposes the existence of locations or places. In section one I do some work to clarify what this claim could possibly mean by identifying some central ways in which languages have been thought to presuppose various things about the world. In section two I present the core of the linguistic data and theory from the work of Susan Rothstein. In section three I compare it to some older work by David Kaplan, arguing that the significance of the new results is greater for the issue at hand, and then in the final section I examine the philosophical significance of this work. One might attempt to draw quite impressive conclusions: such as that the existence of space is analytic, and hence a priori. I will argue that such a conclusion here would be over hasty, and that what we really have is just a surprising fact about our not-so-neutral natural language. (shrink)

Der Autor beginnt mit einer Erörterung der Leibnizschen These, man müsse substantielle Formen annehmen, wenn man die inertia erklären wolle, und fragt dann, in welcher Weise man den Begriff der substantiellen Form zu dem in Beziehung zu setzen habe, was Leibniz über materielle Substanzen sagt. Der zentrale Begriff ist hier der Begriff der Repräsentation von einem bestimmten Gesichtspunkt oder einer bestimmten Lage aus, welche die Substanz von ihrer materia prima gewinnt. Bei der Erörterung der Frage, wie Leibniz' Substanzbegriff mit den (...) anderen Teilen seiner Philosophie zusammenstimmt, merkt der Verfasser an, Leibniz sei versucht zu sagen, materielle Dinge seien nur Phänomene. Aber diese Auffassung würde die konkreten Teile der Leibnizschen Philosophie zerstören. Davon abgesehen macht Leibniz es aufgrund seines Bestehens auf dem vollständigen In-sich-selbst-Enthaltensein (self-containedness) der Substanz schwer, die Einheit einer zusammengesetzten Substanz zu erklären. Anschließend behandelt der Autor einige Auffassungen, die er für Mißverständnisse Leibnizens hält. Wenn man behauptet, Raum und Zeit seien Ordnungen, so besagt das nicht, daß Monaden keine raum-zeitlichen Beziehungen zukommen. In gleicher Weise folgt aus der Behauptung, materielle Dinge seien phaenomena bene fundata, nicht, daß sie nur Repräsentationen im Geiste seien. Zum Schluß legt der Verfasser dar, daß Leibniz' scheinbar paradoxe Behauptung, Substanzen handelten nicht aufeinander, mit den'Lehren der modernen Wissenschaft übereinstimmt. (shrink)

In this review of Milan Ćirković’s The Great Silence: Science and Philosophy of Fermi’s Paradox, we attempt to reconstruct the logic of Fermi’s paradox as understood by the author, and we critically examine the reasoning that leads to the paradox. We show that there is no plausible solution to Fermi’s paradox that can satisfy all of Ćirković’s proposed desiderata, which in turn suggests that the author’s standards for hypothesis adjudication need to be revised.

In recent years a number of general studies of Hume’s philosophy have appeared. It is in this rather crowded traffic that Professor Antony Flew’s David Hume must make its way.... Flew claims that “no previous study of Hume’s philosophy has made nearly enough of the fact that almost all his conclusions are, for better or for worse, conditioned and sometimes determined by an interlocking set of Cartesian assumptions”(p. 2). In this way, Flew suggests that earlier interpreters have rarely recognised the (...) “grotesquely paradoxical” nature of Hume’s Cartesian assumptions and have, therefore, “failed to point the strangeness, the often self-frustrating perversity, of some of the positions which Hume was misled to adopt later” (p. 3).... ..... In a short introductory work it is, of course, very difficult to defend particular interpretations and criticisms in much detail. Nevertheless, I find that Flew’s discussion frequently moves at an excessive speed and that too often his claims are not given the care and attention that they require. Flew certainly does have some interesting points to make and, at his best, Flew is an insightful and stimulating commentator. On the whole, however, his claims come across as thinly defended, poorly articulated and, as a result, rather unconvincing.... (shrink)

A formula ϕ in the one non-logical symbol ϵ with one free variable x is Russell if the sentence Vχ) is logically valid. This note describes a pattern common to the classical examples of Russell formulas, adds a couple of new ones, and constructs many formulas that are near-Russell.

I shall use upper case "Ki" to form the abstract singular term which stands to lower case "ki" as " fi-ness" stands to " fi." In other words, I shall drop the use of the suffix "-kind" in favor of this new device. Thus the platonistic counterpart of.

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 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)

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)

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)

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.

Cette édition numérique a été réalisée à partir d'un support physique, parfois ancien, conservé au sein du dépôt légal de la Bibliothèque nationale de France, conformément à la loi n° 2012-287 du 1er mars 2012 relative à l'exploitation des Livres indisponibles du XXe siècle. Pages de début Avertissement Les paradoxes logiques Première partie - Signification et dénotation Les expressions simplement significatives et une catégorie d'expressions dénotatives : les descriptions définies Autres catégories d'expressions dénotatives L'élimination des expressions dénotatives Seconde partie - (...) Prédicativité et universalité Le principe du cercle vicieux La théorie substitutionnelle La théorie des types Unecertaine idée de la logique Références bibliographiques Pages de fin. (shrink)

Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (...) (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the property of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell, who discovered it in 1901. (shrink)

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)

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)

I attempt to rescue Frege's naive conception of a set according to which there is a set for every property by redefining the technical concept of degree of an open sentence. Instead of making degree a function of the number of free variables, I make it a function of free variable occurrences. What Russell proved, then, is that there is not a relation-in-extension for every relation-in-intension. In a brief paper it is not possible to discuss how redefining the function-argument correlation (...) affects Frege's system. (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)

Though the phrase 'x is true of x' is well formed grammatically, it does not express any predicate in the logical sense, because it does not satisfy the principle of reduction for statements containing 'x is true of'. recognition of this allows for solution of russell's paradox without his restrictive theory of types.