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.

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)

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)

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)

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)

The author believes that large‐scale rationality on the part of the interpretant is essential to his interpretability, and therefore, in his view, to her having a mind. How, then are cases of irrationality, such as akrasia or self‐deception, judged by the interpretant's own standards, possible? He proposes that, in order to resolve the apparent paradoxes, one must distinguish between accepting a contradictory proposition and accepting separately each of two contradictory propositions, which are held apart, which in turn requires to (...) conceive of the mind as containing a number of semi‐independent structures of interlocking beliefs, desires, emotions, memories, and other mental states. (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.

Fitch’s paradox of knowability is an apparently valid reasoning from the assumption (typical of semantic anti-realism) that every true proposition is knowable to the unacceptable conclusion that every true proposition is known. The paper develops a critical dialectic wrt one of the best motivated solutions to the paradox which have been proposed on behalf of semantic anti-realism—namely, the intuitionistic solution. The solution consists, on the one hand, in accepting the intuitionistically valid part of Fitch’s reasoning while, on the (...) other hand, exploiting the characteristic weakness of intuitionistic logic in order to preserve the consistency of such acceptance with the denial of omniscience. It is first remarked how the solution still commits one to acceptance of modal claims which are unwarranted even by the lights of standard intuitionistic semantics. A novel form of the paradox is then introduced, which focuses on infallibility rather than omniscience and derives, from semantic anti-realism and a highly plausible constraint on knowledge, that every believed proposition is not untrue. Because of the logical form of this conclusion, an analogue of the intuitionistic solution for the novel form of the paradox would require drawing the characteristic intuitionistic distinctions wrt decidable propositions, which cannot be done. Semantic anti-realism still intuitionistically entails the unacceptable conclusion that every believed (decidable) proposition is true. (shrink)

Given any proposition, is it possible to have rationally acceptable attitudes towards it? Absent reasons to the contrary, one would probably think that this should be possible. In this paper I provide a reason to the contrary. There is a proposition such that, if one has any opinions about it at all, one will have a rationally unacceptable set of propositional attitudes—or if one doesn’t, one will end up being cognitively imperfect in some other manner. The proposition I am concerned (...) with is a self-referential propositional attitude ascription involving the propositional attitude of rejection. Given a basic assumption about what constitutes irrationality, and a few assumptions about the nature of cognitively ideal agents, a paradox results. This paradox is superficially like the Liar, but it is importantly different in that no alethic notions are involved at all. As such, it stands independent of the Liar and is not a ‘revenge’ version of it. After setting out the paradox I discuss possible responses. After considering several I argue that one is best off simply accepting that the paradox shows us something surprising and interesting about rationality: that some cognitive shortfall is unavoidable even for ideal agents. I argue that nothing disastrous follows from accepting this conclusion. (shrink)

We identify a class of paradoxes that is neither set-theoretical nor semantical, but that seems to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we may need to rethink (...) the commonly accepted methods for dealing with the logical paradoxes. (shrink)

Saying that x is ineffable seems to be paradoxical – either I cannot say anything about x, not even that it is ineffable – or I can say that it is ineffable, but then I can say something and it is not ineffable. In this article, I discuss Alston’s version of the paradox and a solution proposed by Hick which employs the concept of formal and substantial predicates. I reject Hick’s proposal and develop a different account based on some (...) passages from Pseudo-Dionysius’ Mystica Theologia. ‘God is ineffable’ is a metalinguistic statement concerning propositions about God: not all propositions about God are expressible in a human language. (shrink)

We construct a a system PLRI which is the classical propositional logic supplied with a ternary construction , interpreted as the intensional identity of statements and in the context . PLRI is a refinement of Roman Suszko’s sentential calculus with identity (SCI) whose identity connective is a binary one. We provide a Hilbert-style axiomatization of this logic and prove its soundness and completeness with respect to some algebraic models. We also show that PLRI can be used to give a partial (...) solution to the paradox of analysis. (shrink)

This paper discusses a new paradox, the paradox of infallibility. Let us define infallibility in the following way: (Def I) t is infallible if and only if (iff) everything t believes is true, where t is any term. (Def I) entails the following proposition: (I) It is necessary that for every individual x, x is infallible iff every proposition x believes is true. However, (I) seems to be inconsistent with the following proposition (P): It is possible that there (...) is some individual who believes exactly one proposition, namely that she is not infallible. So, it seems to be the case that either (I) or (P) must be false. Yet, (I) is simply a consequence of (Def I) and (P) clearly seems to be true. This is the puzzle. I discuss five possible solutions to the problem and mention some arguments for and against these solutions. (shrink)

It is shown that the paradoxes of confirmation are closely linked to the paradoxes of material implication and that they can be avoided by formulating natural laws in terms of a genuine if-Connective rather than the material conditional. However, Natural laws so expressed are not confirmed by simple conjunctions. The question then is whether the common assumption that simple conjunctions do confirm universal generalizations is correct. The answer given is that it is not. In particular, A confirming proposition of the (...) form 'this is a black raven' is not equivalent to 'this is a raven and this is black'. (shrink)

Though we may be competent at using many technologies, most of what we think we know about technology in general is false. Our error stems from the everyday conception of things as separate from each other and from us. In reality technologies belong to an interconnected network the nodes of which cannot exist independently qua technologies. What is more we tend to see technologies as quasi-natural objects, but they are just as much social as natural, just as much determined by (...) the meanings we give them as by the causal laws that rule over their powers. The errors of common sense have political consequences in domains such as, development, medicine and environmental policy. In this paper I summarize many of the conclusions philosophy of technology has reached reflecting on the reality of our technological world. These conclusions appear as paradoxes judged from our everyday perspective.This paper presents a philosophy of technology. It draws on what we have learnt in the last 30 years as we abandoned old Heideggerian and positivist notions and faced the real world of technology. It turns out that most of our common sense ideas about technology are wrong. This is why I have put my ten propositions in the form of paradoxes, although I use the word loosely here to refer to the counter-intuitive nature of much of what we know about technology. (shrink)

Syntactical treatments of propositional attitudes are attractive to artificial intelligence researchers. But results of Montague (1974) and Thomason (1980) seem to show that syntactical treatments are not viable. They show that if representation languages are sufficiently expressive, then axiom schemes characterizing knowledge and belief give rise to paradox. Des Rivières and Levesque (1988) characterize a class of sentences within which these schemes can safely be instantiated. These sentences do not quantify over the propositional objects of knowledge and belief. We (...) argue that their solution is incomplete, and extend it by characterizing a more inclusive class of sentences over which the axiom schemes can safely range. Our sentences do quantify over propositional objects. (shrink)

This paper discusses a new paradox, the paradox of infallibility. Let us define infallibility in the following way: (Def I) t is infallible if and only if (iff) everything t believes is true, where t is any term. (Def I) entails the following proposition: (I) It is necessary that for every individual x, x is infallible iff every proposition x believes is true. However, (I) seems to be inconsistent with the following proposition (P): It is possible that there (...) is some individual who believes exactly one proposition, namely that she is not infallible. So, it seems to be the case that either (I) or (P) must be false. Yet, (I) is simply a consequence of (Def I) and (P) clearly seems to be true. This is the puzzle. I discuss five possible solutions to the problem and mention some arguments for and against these solutions. (shrink)

This paper argues that we should solve paradoxes for propositions (such as the Russell–Myhill paradox) in essentially the same way that we solve Russellian paradoxes for sets. That is, the standard, iterative approach to sets is extended to include properties, and then the resulting hierarchy of sets and properties is used to construct propositions. Propositions on this account are structured in the sense of mirroring the sentences that express them, and they would seem to serve the (...) needs of philosophers of language and metaphysicians who rely on such propositions. The resulting account has limitations, comparable to those faced by the iterative approach to sets, but it is argued to be in important ways preferable to others that have been proposed. (shrink)

Abstract This essay discusses the paradox of the N?g?rjunian negation as presented in his Vigrahavy?vartani. In Part One it is argued that as the Naiy?yika remarks, N?g?rjuna's speech act ?No proposition has its own intrinsic thesis? seemingly contradicts his famous claim that he has no negation whatsoever. In Parts Two and Three I consider the traditional as well as modem responses to this paradox and offer my own. I argue that N?g?rjuna's speech act does not generate a (...) class='Hi'>paradox for two reasons: (a) the equivalence thesis of the kind??P = ?P is obviously false; and (b) since N?g?rjuna's speech act is situated in the dialogical/conversational universe of discourse as opposed to the argumentative/systematic universe of discourse, the teaching of the non?intrinsic thesis of all statements that it purports, holds for all statements in its class, including itself. Lastly, it is argued that even though the N?g?rjunian speech act is not a negation situated in the argumentative universe of discourse, it serves both philosophical and soteriological purposes. (shrink)

In this paper I consider two paradoxes that arise in connection with the concept of demonstrability, or absolute provability. I assume—for the sake of the argument—that there is an intuitive notion of demonstrability, which should not be conflated with the concept of formal deducibility in a (formal) system or the relativized concept of provability from certain axioms. Demonstrability is an epistemic concept: the rough idea is that a sentence is demonstrable if it is provable from knowable basic (“self-evident”) premises by (...) means of simple logical steps. A statement that is demonstrable is also knowable and a statement that is actually demonstrated is known to be true. By casting doubt upon apparently central principles governing the concept of demonstrability, the paradoxes of demonstrability presented here tend to undermine the concept itself—or at least our understanding of it. As long as we cannot find a diagnosis and a cure for the paradoxes, it seems that the coherence of the concepts of demonstrability and demonstrable knowledge are put in question. There are of course ways of putting the paradoxes in quarantine, for example by imposing a hierarchy of languages a` la Tarski, or a ramified hierarchy of propositions and propositional functions a` la Russell. These measures, however, helpful as they may be in avoiding contradictions, do not seem to solve the underlying conceptual problems. Although structurally similar to the semantic paradoxes, the paradoxes discussed in this paper involve epistemic notions: “demonstrability”, “knowability”, “knowledge”... These notions are “factive” (e.g., if A is demonstrable, then A is true), but similar paradoxes arise in connection with “nonfactive” notions like “believes”, “says”, “asserts”.3 There is no consensus in the literature concerning the analysis of the notions involved—often referred to as “propositional attitudes”—or concerning the treatment of the paradoxes they give rise to. (shrink)

In this paper, I motivate a modal account of propositions on the basis of an iterative conception of propositions. As an application, I suggest that the account provides a satisfying solution to the Russell-Myhill paradox. The account is in the spirit of recently developed modal accounts of sets motivated on the basis of the iterative conception of sets.

I argue that on very weak assumptions about truth (in particular, that there are coherent norms governing the use of "true"), there is a proposition absolutely inexpressible with conventional language, or something very close. I argue for this claim "constructively": I use a variant of the Berry Paradox to reveal a particular thought for my readership to entertain that very strongly resists conventional expression. I gauge the severity of this expressive limitation within a taxonomy of expressive failures, and argue (...) that despite its strength there is nothing incoherent about admitting its existence. The argument forms part of a project of clarifying precisely what trade-offs are required to secure the kinds of expressive power truth theorists typically want, in the process showing that the admission of very strong expressive limitations may ultimately prove to be the lesser of two evils. (shrink)

The paradox of material implication has remained unresolved since antiquity because it was believed that the nature of implication was entailment. The article shows that this nature is opposition and therefore the name "implication" should be replaced with the name "competition". A solution to the paradox is provided along with appropriate changes in nomenclature, the addition of connectives and the postulate that the biconditional take over the role of the previous implication. In addition, changes to the nomenclature of (...) logic gates reflecting the competition in electronics are proposed. (shrink)

A probability function is non-conglomerable just in case there is some proposition E and partition \ of the space of possible outcomes such that the probability of E conditional on any member of \ is bounded by two values yet the unconditional probability of E is not bounded by those values. The paradox of non-conglomerability is the counterintuitive—and controversial—claim that a rational agent’s subjective probability function can be non-conglomerable. In this paper, I present a qualitative analogue of the (...) class='Hi'>paradox. I show that, under antecedently plausible assumptions, an analogue of the paradox arises for rational comparative confidence. As I show, the qualitative paradox raises its own distinctive set of philosophical issues. (shrink)

Zeno’s paradoxes of motion, allegedly denying motion, have been conceived to reinforce the Parmenidean vision of an immutable world. The aim of this article is to demonstrate that these famous logical paradoxes should be seen instead as paradoxes of immobility. From this new point of view, motion is therefore no longer logically problematic, while immobility is. This is convenient since it is easy to conceive that immobility can actually conceal motion, and thus the proposition “immobility is mere illusion of the (...) senses” is much more credible than the reverse thesis supported by Parmenides. Moreover, this proposition is also supported by modern depiction of material bodies: the existence of a ceaseless random motion of atoms—the ‘thermal agitation’—in the scope of contemporary atomic theory, can offer a rational explanation of this ‘illusion of immobility’. Our new approach to Zeno’s paradoxes therefore leads to presenting the novel concept of ‘impermobility’, which we think is a more adequate description of physical reality. (shrink)

The so-called Paradox of Serious Possibility is usually regarded as showing that the standard axioms of belief revision do not apply to belief sets that are introspectively closed. In this article we argue to the contrary: we suggest a way of dissolving the Paradox of Serious Possibility so that introspective statements are taken to express propositions in the standard sense, which may thus be proper members of belief sets, and accordingly the normal axioms of belief revision apply (...) to them. Instead the paradox is avoided by making explicit, for any occurrence of an introspective modality in the object language, the belief state to which this occurrence refers; this will make it impossible for any doxastic modality to refer to two distinct belief sets within one and the same context of doxastic appraisal. By this move the standard derivation of a contradiction from the theory of belief revision in the presence of introspectively closed belief sets does not go through any more, and indeed the premisses of the Paradox of Serious Possibility become jointly consistent once they are reformulated with our amended introspective modalities only. Additionally, we present a probabilistic version of the Paradox of Serious Possibility which can be avoided in a perfectly analogous manner. (shrink)

Professor Barbara Stanosz was a years-long lecturer at the Institute of Philosophy, University of Warsaw. In her work she mainly – but not exclusively – focused on the theory of language, particularly semantics and the issues of logical description of phrases in language. She was an author of renowned textbooks, including the famous Ćwiczenia z logiki [Exercises in logic], a vastly popular exercise book helping students to acquire the material on propositional logic, predicate logic and set theory. It is worth (...) mentioning that besides her academic and teaching activities, she was also a social activist and a great supporter of the state’s worldview neutrality. She died on 7th June 2014. (shrink)

We are familiar with various set-theoretical paradoxes such as Cantor's paradox, Burali-Forti's paradox, Russell's paradox, Russell-Myhill paradox and Kaplan's paradox. In fact, there is another new possible set-theoretical paradox hiding itself in Wittgenstein’s Tractatus. From the Tractatus’s Picture theory of language we can strictly infer the two contradictory propositions simultaneously: the world and the language are equinumerous; the world and the language are not equinumerous. I call this antinomy the world-language paradox. Based (...) on a rigorous analysis of the Tractatus, with the help of the technical resources of Cantor’s naive set theory and Zermelo-Fraenkel set theory with the axiom of choice, I outline the world-language paradox and assess the unique possible solution plan, i.e., the mathematical plan utilizing ‘infinity’. I conclude that Wittgenstein cannot solve the hidden set-theoretical paradox of the Tractatus successfully unless he gives up his finitism. (shrink)

This paper deals with the simple paradoxes of validity and with the possibility of solving them in terms of Bradwardinian-Buridanian semantics. The paradoxes of validity as conceived here are cases of semantic pathology, which result due to the use of terms signifying the validity of inference. Semantic paradoxes are a semantico-epistemological phenomenon which is a symptom of the need to revise several apparently acceptable semantic assumptions. The analysis of possible solutions to the paradoxes focuses on Bradwardinian-Buridanian semantics and as a (...) result on the closed, token-based semantic theories that assume the existence of an implicit meaning of propositions. The key theses, as far as the solution to the paradoxes is concerned, are the principle of truth-implication which claims that every proposition expresses or implies its own truth and the closure principle which claims that every proposition asserts or expresses everything that follows from it logically. The present paper advances on recent research in claiming that (with certain reservations) the application of these principles can effectively solve inconsistency-paradoxes but not indeterminacy-paradoxes of validity. (shrink)

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic (...) logic and semantic information theory, and show how removing conceptual omniscience helps resolve Wittgenstein's paradoxes and explain the puzzle of deduction, its ability to generate new knowledge and meaning. (shrink)

This is a critical discussion of Nino B. Cocchiarella’s book “Formal Ontology and Conceptual Realism.” It focuses on paradoxes of hyperintensionality that may arise in formal systems of intensional logic.

The paradox of fiction presents an inconsistent triad of propositions, all of which are purported to be plausible or difficult to abandon. Here is an instance of the paradox: (1) Sally pities Anna (where Anna is the character Anna Karenina). (2) To pity someone, one must believe that they exist and are suffering. (3) Sally does not believe that Anna exists. Here is the problem. The paradox was formulated during the heyday of the cognitive theory of (...) the emotions when there was a lot of theoretical commitment to (2) or a variant of it. But now virtually no one accepts (2). To solve the paradox, we just have to find a way to reject one of the inconsistent statements. It appears easy to reject (2). So why do even some of those who do reject (2) not leave matters there? I argue that there is still good reason to consider other solutions to the paradox. It is only by doing so that we fully understand the affective states with which we respond to fiction, in particular, the distinctive functional role that they have in this context. (shrink)

This book attacks an assortment of tendencies and assumptions that the author believes endemic to traditional epistemology. Perhaps the main target is what she sees as a tendency to sublimate the concepts of knowledge and belief, whose roles in everyday life are mundane and unsystematic, into rigid abstractions. This tendency is said to show itself in the allegedly false assumptions that propositions are the objects of knowledge and belief, and that there is a definite set of propositions that (...) one knows and another that one believes. One or the other of these assumptions is said to underlie Chisholm’s attempt to state necessary and sufficient conditions for knowledge, Prichard’s to distinguish knowledge from belief, Malcolm’s to distinguish a ’strong’ from a ‘weak’ sense of ‘to know’, Moore’s to rebut skepticism by giving examples of things he knows, and Descartes’ to imagine that all his beliefs are false. All make the mistake, Wolgast thinks, of speaking of knowledge or belief in the abstract, without reference to the actual use of language. In particular, Chisholm, Prichard, and Moore fail to realize that the force of ‘I know’ varies from context to context. (shrink)

The aggregation of individual judgments on logically interconnected propositions into a collective decision on the same propositions is called judgment aggregation. Literature in social choice and political theory has claimed that judgment aggregation raises serious concerns. For example, consider a set of premises and a conclusion where the latter is logically equivalent to the former. When majority voting is applied to some propositions (the premises) it may give a different outcome than majority voting applied to another set (...) of propositions (the conclusion). This problem is known as the discursive dilemma (or paradox). The discursive dilemma is a serious problem since it is not clear whether a collective outcome exists in these cases, and if it does, what it is like. Moreover, the two suggested escape-routes from the paradox—the so-called premise-based procedure and the conclusion-based procedure—are not, as I will show, satisfactory methods for group decision-making. In this paper I introduce a new aggregation procedure inspired by an operator defined in artificial intelligence in order to merge belief bases. The result is that we do not need to worry about paradoxical outcomes, since these arise only when inconsistent collective judgments are not ruled out from the set of possible solutions. (shrink)

The object of the present discussion is to show that the analysis of indirect discourse obtained when the concept of assertion is construed as a relationship that obtains between the asserting person and the asserted proposition—along the familiar lines proposed by Church [3, 4]—is entirely adequate of itself to circumvent the semantical paradoxes in which indirect discourse is involved.

Propositions are central to at least most theorizing about the connection between our mental lives and the world: we use them in our theories of an array of attitudes including belief, desire, hope, fear, knowledge, and understanding. Unfortunately, when we press on these theories, we encounter a relatively neglected family of paradoxes first studied by Arthur Prior. I argue that these paradoxes present a fatal problem for most familiar resolutions of paradoxes. In particular, I argue that truth-value gap, contextualist, (...) situation theoretic, revision theoretic, ramified, and dialetheist approaches to the paradoxes must deny us the conceptual resources that they themselves make use of, on pain of contradiction. I then detail the costs of the extant strategies that avoid this issue: Hartry Field’s paracomplete approach; Andrew Bacon’s classical treatment of indeterminacy; a generalization of ideas from Prior, Nicholas J.J. Smith, and Hartley Slater; and free logics as recently explored by Bacon, John Hawthorne, and Gabriel Uzquiano. I argue that none of these is perfect, and that each restricts the theories we can endorse in a variety of areas of philosophy. I spell these restrictions out, showing, I hope, that the further investigation of these paradoxes must be a part of future research on propositional attitudes. (shrink)

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. (...) If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes. (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)

Key issues in epistemology for the most part have to do with epistemic values such as justification, truth, and knowledge—that is, values related to the epistemic status of our propositional attitudes, mental events, and states. However, another important issue that is worth examining is the extent to which a subject is in a position to evaluate the strength of her epistemic position. In this paper, I wish to emphasize two properties of our mental states that play a decisive part in (...) that respect: their opacity and transparency. In the following, I will assume that a mental state is opaque whenever it presents itself with no underlying reason, whereas states that are supported by apparent reasons are transparent. One main argument that I will defend is that even when the opacity and transparency of our mental states are not reliable cues, still they remain highly informative. Notably, I draw some implications relative to the externalist/internalist debate in epistemology. I examine the claim that only mental states that are justified are well grounded or justification conferring and the idea that often stands behind, that is, that only states whose reasons are accessible are justified—namely, the Accessibility Requirement. I also examine the source of a recent debate in epistemology: the epistemic status of our intuitive states, as these are perfect instances of opaque mental states. I conclude that intuitive states in some respect are less misleading than states that are supported by apparent reasons. (shrink)

A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.

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)

This paper concerns two paradoxes involving propositions. The first is Russell's paradox from Appendix B of The Principles of Mathematics, a version of which was later given by Myhill. The second is a paradox in the framework of possible worlds, given by Kaplan. This paper shows a number of things about these paradoxes. First, we will see that, though the Russell/myhill paradox and the Kaplan paradox might appear somewhat different, they are really just variants of (...) the same phenomenon. Though they do this in different ways, the core of each paradox is to use the notion of a proposition to construct a function, f, from the power set of some set into the set itself. Next we will see how this paradox fits into the Inclosure Schema. Finally, I will provide a model of the paradox in question, showing its results to be non‐trivial, though inconsistent. (shrink)

G. E. Moore opined that the paradox of analysis might be avoided if it could be shown that sentences expressing conceptual analyses convey information not only about concepts, but also about the expressions they use. If so, “to be a brother is to be a male sibling” and “to be a brother is to be a brother” might express the same proposition, and yet not be identical in information value as the paradox suggests. How sentences might do this, (...) Moore could not see. Many philosophers have pointed out an obvious way in which sentences might be said to convey information about the expressions they use. Some have suggested this information might be used to develop Moore's intuition and resolve the paradox of analysis. I argue that this approach fails. I present a version of the paradox of analysis that resists this sort of solution. (shrink)

In normative reasoning one typically refers to intervals of time across which norms are intended to hold, as well as to alternative possibilities representing hypothetical developments of a given scenario. Thus, deontic modalities are naturally intertwined with temporal and metaphysical ones. Furthermore, contemporary debates in philosophy suggest that a proper understanding of fundamental ethical principles, such as the Ought-Implies-Can thesis, requires a simultaneous analysis of these three families of concepts. In the present article we propose a general formal framework which (...) allows for fine-grained multimodal reasoning in the normative domain. We provide an axiomatization for a novel system of propositional logic encoding the way in which possibilities and norms arising from different sources change over intervals of time. The usefulness of our framework is illustrated by analysing an ancient and particularly challenging `cold case', the Paradox of the Court. (shrink)

The famous dispute between Protagoras and Euathlus concerning Protagoras’s tuition fee reportedly owed to him by Euathlus is solved on the basis of practical argumentation concerning actions. The dispute is widely viewed as a kind of a logical paradox, and I show that such treating arises due to the double confusion in the dispute narrative. The linguistic expressions used to refer to Protagoras’s, Euathlus’s and the jurors’ actions are confused with these actions themselves. The other confusion is the collision (...) between the pairs of incompatible actions ambiguously expressed by two different pairs of sentences, one of which is a propositionally consistent pair whereas the other is an inconsistent one. The actional paradox solution aims to clear up these confusions by means of two core borderlines, propositional and expressive, drawn between the actions and the propositions. The propositional distinction says that actions are empirical facts and they lack truth values unlike propositions, which are mental entities and are often employed for referring to the actions. This distinction helps to avoid the confusion between the empirical incompatibility of actions and the truth-functional inconsistency of propositions. The expressive distinction claims that although the same linguistic sentences can be used to refer both to actions and propositions, two empirically incompatible actions can be expressed both by a pair of inconsistent propositions as well as by a pair of consistent ones. Therefore, the action of Protagoras’s being paid may be linguistically symbolized in four different ways: Protagoras gets paid due to the verdict, Protagoras gets paid due to the contract that amount to Protagoras does not get paid by the contract and Protagoras does not get paid by the verdict respectively, and likewise for Euathlus’s actions. The two distinctions are used for classifying the two groups of paradox solutions, legal and logical, proposed so far depending on which of the two confusions they purport to escape from. The actional reconstruction of the paradox suggests that there is only one single agent in the dispute, Protagoras, while the other named Euathlus is a ‘phantom,’ which most probably was invented by Protagoras himself for the sake of creating this challenging sophism. (shrink)