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)

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)

Charles Sanders Peirce proposed two different solutions to the Liar Paradox. He proposed the first in 1865 and the second in 1869. However, no one has yet noted in the literature that Peirce rejected his 1869 solution in 1903. Peirce never explicitly proposed a third solution to the Liar Paradox. Nonetheless, I shall argue he developed the resources for a third and novel solution to the Liar Paradox.In what follows, I will first explain the Liar Paradox. (...) Second, I will briefly rehearse Peirce's 1865 solution and his reasons for rejecting it.1 Third, I will review his 1869 solution and his reasons for rejecting it in 1903. Lastly, I will propose a novel solution to the Liar Paradox by drawing upon Peirce's later.. (shrink)

In this paper, I try to understand what Buridan means when he suggests that "every proposition, by its very form, signifies or asserts itself to be true." I show how one way of construing this claim - that every proposition is in fact a conjunction one conjunct of which asserts the truth of the whole conjunction - does lead to a resolution of the Liar paradox, as Buridan says, and moreover is not vulnerable to the criticism on the basis (...) of which Buridan came to reject this view. However, I go on to argue that the view causes Truth-Teller worries when applied to non-Liar propositions. (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)

Propositions are more than the bearers of truth and the meanings of sentences: they are also the objects of an array of attitudes including belief, desire, hope, and fear. This variety of roles leads to a variety of paradoxes, most of which have been sorely neglected. Arguing that existing work on these paradoxes is either too heavy-handed or too specific in its focus to be fully satisfactory, I develop a basic intensional logic and pursue and compare three strategies for (...) addressing the paradoxes, one employing truth-value gaps, one restricting propositional quantification, and one restricting our ability to have attitudes like belief and desire. This results in four distinct resolutions of the paradoxes, all but one of which are novel and all of which receive novel and general implementations. While resolving the paradoxes is of course the ultimate goal, I do not here argue that any one of the resolutions is superior. These paradoxes have been so little studied that my primary goal is only to identify the most fundamental costs and benefits of the various approaches one can take to addressing them. Each resolution I develop has significant drawbacks, which I argue highlight tensions between the different roles propositions play. Past researchers have skirted these tensions, and the issues raised by these paradoxes more generally, by focusing on non-propositional paradoxes, such as the most familiar forms of the Liar paradox. At the least, then, I hope this dissertation establishes that the propositional paradoxes deserve attention not only because of their consequences for intensional logic, but also because of their consequences for our understanding of content, truth, quantification, and a host of mental attitudes. (shrink)

This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method B p , which specifies an initial belief state B p (T) that is revised to the new propositional belief state B(E) upon receipt of information E. An acceptance rule tracks Bayesian conditioning when B p (E) = B p|E (T), (...) for every E such that p(E) > 0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The "Lockean" rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1-2):221-262, 1975; Alchourrón et al., J Symb Log 50:510-530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987). (shrink)

ABSTRACT Section 1 discusses properties and propositions, and some of the motivation for an account in which property instantiation and propositional truth behave ‘naively’. Section 2 generalizes a standard Kripke construction for naive properties and propositions, in a language with modal operators but no conditionals. Whereas Kripke uses a 3-valued value space, the generalized account allows for a broad array of value spaces, including the unit interval [0,1]. This is put to use in Section 3, where I add (...) to the language a conditional suitable for restricting quantification. The shift from a value space based on the ‘mini-space’ {0,, 1} to one based on the ‘mini-space’ [0,1] leads to more satisfactory results than I was able to achieve in previous work: a vast variety of paradoxical sentences can now be treated very simply. In Section 4 I make a further addition to the language, a conditional modeled on the ordinary English conditional, paying particular attention to how it interacts with the restricted quantifier conditional. This is all done in the [0,1] framework, and two alternatives are considered for how the ordinary conditional is to be handled; one of them results from adding a tweak to a construction by Ross Brady. Section 5 discusses a further alternative, a standard relevance conditional (for the ordinary conditional, perhaps for use with a different quantifier-restricting conditional), but argues that it is not promising. Section 6 discusses the identity conditions of properties and propositions (again in the setting of a value space based on [0,1]); the issue of achieving naivety for coarse-grained properties is seen to be more complicated than some brief remarks in Field [2010] suggested, but a way to get a fair degree of coarse-grainedness is shown. (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 gregory-landini@uiowa.edu 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)

In this paper a formalized logic of propositions, PA1, is presented. It is proven consistent and its relationships to traditional logic, to PM ([15]), to subjunctive (including contrary-to-fact) implication and to the “paradoxes” of material and strict implication are developed. Apart from any intrinsic merit it possesses, its chief significance lies in demonstrating the feasibility of a general logic containing theprinciple of subjunctive contrariety, i.e., the principle that ‘Ifpwere true thenqwould be true’ and ‘Ifpwere true thenqwould be false’ are (...) incompatible. (shrink)

In this paper it is shown that addition of certain reductions to the standard cut removing reductions of deductions in prepositional logic makes prepositional logic non-normalizable. From this follows that some provable propositions in prepositional logic has no direct proof.

The authors provide an object-theoretic analysis of two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan. After laying out the paradoxes, the authors provide a brief overview of object theory and point out how syntactic restrictions that prevent object-theoretic versions of the classical paradoxes are justified philosophically. The authors then trace the origins of the Russell paradox to a problematic application of set theory in the definition of worlds. Next the authors show (...) that an object-theoretic analysis of the Kaplan paradox reveals that there is no genuine paradox at all, as the central premise of the paradox is simply a logical falsehood and hence can be rejected on the strongest possible grounds—not only in object theory but for the very framework of propositional modal logic in which Kaplan frames his argument. The authors close by fending off a possible objection that object theory avoids the Russell paradox only by refusing to incorporate set theory and, hence, that the object-theoretic solution is only a consequence of the theory’s weakness. (shrink)

Recently, self-fulfilling cases, that is, ones in which an agent's believing a proposition guarantees its truth, have been offered as counterexamples to uniqueness. According to uniqueness, at most one doxastic attitude is epistemically rational given the evidence. I argue that self-fulfilling cases are not counterexamples to uniqueness because belief-formation is not governed by epistemic rationality in such cases. Specifically, this is because epistemic rationality is not just about forming true beliefs, but about tracking mind-independent truths. In support of the latter (...) claim, I offer three arguments, namely that self-fulfilling and non-self-fulfilling cases differ in their phenomenology, in the norms that guide belief formation, and in the way they relate to the evidence. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:“Unauthorized Propositions”The Federalist Papers and Constituent PowerJason Frank (bio)The PEOPLE, who are the sovereigns of the State, possess a power to alter it when and in what way they please. To say otherwise is to make the thing created, greater than the power that created it.—Anonymous, Federal Gazette, March 18, 1789The we of the Constitution’s “We the People” was as much of an artificial construct as the Constitution (...) itself, it was its creature, not its creator.—Sheldon Wolin, The Presence of the PastIn Federalist No. 40, James Madison engaged the vexing question of whether the Philadelphia Convention of 1787 had been properly authorized “to frame and propose” the new Federal Constitution of the United States [83]. Madison explored there “the ground on which the Convention stood” [264], which he identified as the purportedly solid foundation of the people themselves [Morgan 272]. Federalist No. 40 offers a theoretically probing yet avowedly practical reflection on the availability of foundations and on dilemmas of popular authorization in times of constitutional crisis; it may be productively read as an exemplary eighteenth-century American reflection on a state of exception. As such, Federalist No. 40 provides helpful orientation to how the popular constitutionalism of the revolutionary and postrevolutionary years confronted and navigated—if never fully resolved—some of the most insistent paradoxes of modern democratic theory.In No. 40, Madison was preoccupied with the theoretical and historical dilemmas surrounding the eighteenth-century American conception of the people’s constituent power—that is, the ultimate power of the people to alter fundamental law “when and in what way they please.”1 However, in contrast to most contemporary theoretical approaches to constituent power—whether influenced by the political theology of Carl Schmitt or the immanentist optimism of Michael Hardt and Antonio Negri—Madison’s navigation of the attending dilemmas of popular authorization resisted the turn to metaphysical abstraction and the appeal to a transcendent or immanent absolute. In his approach to the problem of constituent power, Madison insisted on the productive irreducibility of institutional mediation and the performative elicitation of a retrospective authorization by the people. While accepting the absence of a firm ground of legal authorization (the unbroken authority of law), Madison also rejected a wholly extralegal or unmediated appeal to the people themselves (the transcendental appeal to spirit beyond the letter of the law). Instead, he endorsed a political navigation of these dilemmas in medias res. Confronting [End Page 103] the dilemma of constituency—how does the people give birth to itself as a self-authorizing subject?—Madison shifted the register of discourse away from the formal and juridical and toward the informally ethicopolitical. Doing so, he was able to affirm a rupture in law while simultaneously enlisting another register of normative continuity. This essay outlines Madison’s productive encounter with these dilemmas in The Federalist, and suggests the relevance of his example for recent debates in democratic theory, too often captivated by the formal rubrics of “norm” and “exception.”1The government which [the Federalists] are so enthusiastically fond of is as yet an ideal phantom, a chimera, a mere theory detested and execrated by every true friend to government.—“One of the People,” Carlisle Gazette, January 9, 1788Madison wrote Federalist No. 40 as a response to widespread and persuasive Anti-Federalist claims that the Philadelphia Convention had exceeded the mandate granted to it by the Annapolis Convention of 1786 and the Congressional Act of 1787, both of which authorized the meeting of the Philadelphia Convention “for the sole and express purpose of revising the Articles of Confederation” [259]. While the Philadelphia Convention was, in Bruce Ackerman’s words, “expressly subordinated to existing institutions and procedures” [Ackerman 43], it had nonetheless produced an entirely new constitutional text and—theoretically more problematic for Madison—an entirely new procedure for its popular ratification [see also Ackerman and Katyal]. Under Article 13 of the Articles of Confederation, constitutional reform could come about only through the unanimous assent of the thirteen state legislatures.2 Under the Philadelphia Convention’s plan, the ratification of the proposed constitution would bypass the constituted power of the state legislatures and appeal instead to the... (shrink)

The idea that temporal propositions are vague predicates is examined with attention to the nature of the objects over which the predicates range. These objects should not, it is argued, be identified once and for all with points or intervals in the real line (or any fixed linear order). Context has an important role to play not only in sidestepping the Sorites paradox (Gaifman 2002) but also in shaping temporal moments/extent (Landman 1991). The Russell-Wiener construction of time from (...) events (Kamp 1979) is related to a notion of context given by a string of observations, the vagueness in which is brought out by grounding the observations in the real line. With this notion of context, the context dependency functions in Gaifman 2002 are adapted to interpret temporal propositions. (shrink)

The notion of a proposition is central to philosophy. But it is subject to paradoxes. A natural response is a hierarchical account and, ever since Russell proposed his theory of types in 1908, this has been the strategy of choice. But in this paper I raise a problem for such accounts. While this does not seem to have been recognized before, it would seem to render existing such accounts inadequate. The main purpose of the paper, however, is to provide a (...) new hierarchical account that solves the problem. (shrink)

This paper consists of an investigation of three debates concerning propositional identity: the tension between structured propositions and higher-order logic, the principle Only Logical Circles, and Kaplan’s Paradox. The literature at large has mistaken the consequences of each of these debates. Structuralists are not committed to the claim that identical properties have different extensions; rather, they are committed to existence monism. Only Logical Circles does not preclude the identification of green in terms of grue; some further (and, as (...) of yet, undisclosed) principle is required for that result. And Kaplan’s Paradox does not count against the possible-worlds conception of propositions; it applies to every account of propositions whatsoever. (shrink)

As Alasdair Urquhart has noted, Bertrand Russell asserted that developing the theory of definite descriptions from 1905 was the first step towards solving the paradoxes that were finally resolved after 1908 in Principia Mathematica with the theory of types. I extend Urquhart’s suggestion that Russell was referring to the use of the notion of incomplete symbol in his solution to the paradoxes in his doomed theory “substitutional theory” of “Russellian propositions” in 1906. The Introduction to PM states that expressions (...) for propositions are incomplete symbols. This paper assesses the status of propositions in PM and connects the theory of types with the theory of descriptions. (shrink)

Some propositions are structurally unknowable for certain agents. Let me call them ‘Moorean propositions’. The structural unknowability of Moorean propositions is normally taken to pave the way towards proving a familiar paradox from epistemic logic—the so-called ‘Knowability Paradox’, or ‘Fitch’s Paradox’—which purports to show that if all truths are knowable, then all truths are in fact known. The present paper explores how to translate Moorean statements into a probabilistic language. A successful translation should enable (...) us to derive a version of Fitch’s Paradox in a probabilistic setting. I offer a suitable schematic form for probabilistic Moorean propositions, as well as a concomitant proof of a probabilistic Knowability Paradox. Moreover, I argue that traditional candidates to play the role of probabilistic Moorean propositions will not do. In particular, we can show that violations of the so-called ‘Reflection Principle’ in probability need not yield structurally unknowable propositions. Among other things, this should lead us to question whether violating the Reflection Principle actually amounts to a clear case of epistemic irrationality, as it is often assumed. This result challenges the importance of the principle as a tool to assess both synchronic and diachronic rationality—a topic which is largely independent of Fitch’s Paradox—from a somewhat unexpected source. (shrink)

Frege famously argued that truth is not a property or relation. In the “Notes on Logic” Wittgenstein emphasised the bi-polarity of propositions which he called their sense. He argued that “propositions by virtue of sense cannot have predicates or relations.” This led to his fundamental thought that the logical constants do not represent predicates or relations. The idea, however, has wider ramifications than that. It is not just that propositions cannot have relations to other propositions but (...) also that they cannot have relations to anything at all. The paper explores the consequences of this insight for the way in which we should read the Tractatus. In the “Notes on Logic” the insight led to Wittgenstein's emphasis on “facts” in any attempt to understand the nature of symbolism. This emphasis is continued in the Tractatus. It is central to his view that propositions are facts which picture facts which prevent us from construing such picturing as a relation between what pictures and what is pictured. It illuminates the importance of context principle with regard to the distinction between showing and saying to which Wittgenstein attached so much importance and it underlies the non-relational view of psychological propositions which he advocates. Finally, if propositions by virtue of sense cannot have predicates or relations the paradox at the end of a work which consist largely of propositions about propositions becomes intelligible. (shrink)

Metaphysically possible worlds have many uses. Epistemically possible worlds promise to be similarly useful, especially in connection with propositions and propositional attitudes. However, I argue that there is a serious threat to the natural accounts of epistemically possible worlds, from a version of Russell’s paradox. I contrast this threat with David Kaplan’s problem for metaphysical possible world semantics: Kaplan’s problem can be straightforwardly rebutted, the problems I raise cannot. I argue that although there may be coherent accounts of (...) epistemically possible worlds with fruitful applications, any such an account must fundamentally compromise the basic idea behind epistemic possibility. (shrink)

A novel normal form for propositional theories underlies the logic pdl, which captures some essential features of natural discourse, independent from any particular subject matter and related only to its referential structure. In particular, pdlallows to distinguish vicious circularity from the innocent one, and to reason in the presence of inconsistency using a minimal number of extraneous assumptions, beyond the classical ones. Several, formally equivalent decision problems are identified as potential applications: non-paradoxical character of discourses, admissibility of arguments in argumentation (...) networks, propositional satisfiability, and the existence of kernels of directed graphs. Directed graphs provide the basis for the semantics of pdl and the paper concludes by an overview of relevant graph-theoretical results and their applications in diagnosing paradoxical character of natural discourses. (shrink)

Are there ‘self-referential’ propositions? That is, propositions that say of themselves that they have a certain property, such as that of being false. There can seem reason to doubt that there are. At the same time, there are a number of reasons why it matters. For suppose that there are indeed no such propositions. One might then hope that while paradoxes such as the Liar show that many plausible principles about sentences must be given up, no such (...) fate will befall principles about propositions. But the existence of self-referential propositions would dash such hopes. Further, the existence of such propositions would also seem to challenge the widespread claim that Liar sentences fail to express propositions. The aim of this paper is thus to settle the question–at least given an assumption. In particular, I argue that if propositions are structured, then self-referential propositions exist. (shrink)

The purpose of the paper is to show that by cleaning Classical Logic (CL) from redundancies (irrelevances) and uninformative complexities in the consequence class and from too strong assumptions (of CL) one can avoid most of the paradoxes coming up when CL is applied to empirical sciences including physics. This kind of cleaning of CL has been done successfully by distinguishing two types of theorems of CL by two criteria. One criterion (RC) forbids such theorems in which parts of the (...) consequent (conclusion) can be replaced by arbitrary parts salva validitate of the theorem. The other (RD) reduces the consequences to simplest conjunctive consequence elements. Since the application of RC and RD to CL leads to a logic without the usual closure conditions, an approximation to RC and RD has been constructed by a basic logic with the help of finite (6-valued) matrices. This basic logic called RMQ (relevance, matrix, Quantum Physics) is consistent and decidable. It distinguishes two types of validity (strict validity) and classical or material validity. All theorems of CL (here: classical propositional calculus CPC) are classically or materially valid in RMQ. But those theorems of CPC which obey RC and RD and avoid the difficulties in the application to empirical sciences and to Quantum Physics are separated as strictly valid in RMQ. In the application to empirical sciences in general the proposed logic avoids the well known paradoxes in the area of explanation, confirmation, versimilitude and Deontic Logic. Concerning the application to physics it avoids also the difficulties with distributivity, commensurability and with Bell's inequalities. (shrink)

The paper shows that the paradox of the totality of propositions rest on assumptions characteristic of some theories of structured contents (like Jeffrey King's "new account of structured propositions").

I argue that one ought not be epistemically akratic. Although this position may look self-evident, it is hard to pin down exactly what’s wrong with the akratic subject. Indeed, some philosophers argue that epistemic akrasia is permissible. The standard anti-akratic response focuses on the weird downstream implications of this state for action and assertion. This approach, however, is unsatisfactory, since it fails to explain the epistemic impermissibility of epistemic akrasia. Here, I argue that epistemic akrasia is impermissible on a purely (...) epistemic basis. Doing so brings out the connection between positive epistemic statuses and truth: it is that connection that explains the fundamental impermissibility of epistemic akrasia. (shrink)

Drawing on the so-called “doctrinal paradox”, List and Pettit (2002) have shown that, given an unrestricted domain condition, there exists no procedure for aggregating individual sets of judgments over multiple interconnected propositions into corresponding collective ones, where the procedure satisfies some minimal conditions similar to the conditions of Arrow’s theorem. I prove that we can avoid the paradox and the associated impossibility result by introducing an appropriate domain restriction: a structure condition, called unidimensional alignment, is shown to (...) open up a possibility result, similar in spirit to Black’s median voter theorem. Specifically, I prove that, given unidimensional alignment, propositionwise majority voting is the unique procedure for aggregating individual sets of judgments into collective ones in accordance with the above mentioned minimal conditions. (shrink)

There is an apparent dilemma for hierarchical accounts of propositions, raised by Bruno Whittle : either such accounts do not offer adequate treatment of connectives and quantifiers, or they eviscerate the logic. I discuss what a plausible hierarchical conception of propositions might amount to, and show that on that conception, Whittle’s dilemma is not compelling. Thus, there are good reasons why proponents of hierarchical accounts of propositions did not see the difficulty Whittle raises.

The semantic paradoxes, whose paradigm is the Liar, played a crucial role at a crucial juncture in the development of modern logic. In his 1908 seminal paper, Russell outlined a system, soon to become that of the Principia Mathematicae, whose main goal was the solution of the logical paradoxes, both semantic and settheoretic. Russell did not distinguish between the two and his theory of types was designed to solve both kinds in the same uniform way. Set theoreticians, however, were content (...) to treat only the set-theoretic paradoxes, putting aside the semantic ones as a non-mathematical concern. This separation was explicitly proposed, eighteen years after Russell’s paper, by Ramsey, though he, like Russell, advocated a system that addresses both kinds. Since then, the semantic paradoxes have been viewed within the perspective of the theory of truth, where they have occupied a respectable niche, but one of rather specialized interest. (shrink)

ABSTRACT In our response Field's ‘Properties, Propositions and Conditionals’, we explore the methodology of Field's program. We begin by contrasting it with a proof-theoretic approach and then commenting on some of the particular choices made in the development of Field's theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field's response to Restall’s [2010] ‘What Are We to Accept, and What Are We to (...) Reject, While Saving Truth from Paradox?’. (shrink)

Fitch showed that not every true proposition can be known in due time; in other words, that not every proposition is knowable. Moore showed that certain propositions cannot be consistently believed. A more recent dynamic phrasing of Moore-sentences is that not all propositions are known after their announcement, i.e., not every proposition is successful. Fitch's and Moore's results are related, as they equally apply to standard notions of knowledge and belief (S 5 and KD45, respectively). If we interpret (...) ‘successful’ as ‘known after its announcement’ and ‘knowable’ as ‘known after some announcement’, successful implies knowable. Knowable does not imply successful: there is a proposition ϕ that is not known after its announcement but there is another announcement after which ϕ is known. We show that all propositions are knowable in the more general sense that for each proposition, it can become known or its negation can become known. We can get to know whether it is true: ◊(Kϕ ∨ K¬ϕ). This result comes at a price. We cannot get to know whether the proposition was true. This restricts the philosophical relevance of interpreting ‘knowable’ as ‘known after an announcement’. (shrink)

This article distinguishes between two historical ways of presenting the catchphrase “Once true, always true” (semel verum, semper verum), associated with the twelfth-century logical school of the nominales. Within the Time-Jumping Model, a hypothetical tenseless propositional content (enuntiabile) is treated as the common significate of differently tensed statements, such as “Socrates will die” and “Socrates died,” uttered before and after Socrates’s death. This hypothetical enuntiabile is “always true” thanks to its tenseless nature. By contrast, the Fixed-Present Model preserves the tensed (...) character of enuntiabilia. By revealing an implicit temporal index, the nominales could apply “Once true, always true” to enuntiabilia such as Christum venisse (that Christ has come), which not only is true and will always be true but also – paradoxically – has always been true. It has always been true because it amounts to the propositional content that Christ came before this instant, where “this instant” rigidly denotes the present moment. The Fixed-Present Model alleviates the worry that the nominales confused truth-bearers with truthmakers. (shrink)

The author puts forth an approach to propositional attitude contexts based upon the view that one does not have beliefs of ordinary extensional entitiessimpliciter. Rather, one has beliefs of such entities as presented in various manners. Roughly, these are treated as beliefs of ordered pairs — the first member of which is the ordinary extensional entity and the second member of which is a predicate that it satisfies. Such an approach has no difficulties with problems involving identity, such as of (...) The Morning Star and The Evening Star (section 1). Given the second members of the pairs, the modes of presentation, it is quite natural to allow exportation everywhere. There is no need for essentialism. (One also can have non-essentialistic modal logic if one grants analyticity or the like.) (section 2). Given that the second member of the pair need only be one that is satisfied by the entity that is the first member (and need not be specificative), the method has no difficulties when one is concerned only with discriminations (and not specifications) (section 3). When this method is combined with the Frege-Carnap method of descriptions, fictional entities can be accommodated; Goodman''s unicorn-picture and the like can be brought within a Tarskian semantics; and Geach''s difficulties with intentional identity appear to be handled (section 4). Given the author''s ordered pair construals, there appears to be no additional need for notional construals; i.e., the author''s one unified method appears satisfactory for dealing with both traditionalde re (relational) andde dicto (notional) construals. The Paradox of the Knower and the like do not appear formulatable against the author''s approach. (section 5). The author also argues against the basic principles behind the Church-Langford translation argument (section 6). (shrink)

The present work outlines a logical and philosophical conception of propositions in relation to a group of puzzles that arise by quantifying over them: the Russell-Myhill paradox, the Prior-Kaplan paradox, and Prior's Theorem. I begin by motivating an interpretation of Russell-Myhill as depending on aboutness, which constrains the notion of propositional identity. I discuss two formalizations of of the paradox, showing that it does not depend on the syntax of propositional variables. I then extend to (...) class='Hi'>propositions a modal predicative response to the paradoxes articulated by an abstraction principle for propositions. On this conception, propositions are “shadows” of the sentences that express them. Modal operators are used to uncover the implicit relation of dependence that characterizes propositions that are about propositions. The benefits of this approach are shown by application to other intensional puzzles. The resulting view is an alternative to the plenitudinous metaphysics of impredicative comprehension principles. (shrink)

To resolve putative liar paradoxes it is sufficient to attend to the distinction between liar-sentences and the propositions they would express, and to exercise the option of turning would-be deductions of paradox (of contradictions) into reductions of the existence of those propositions. Defending the coherence of particular resolutions along these lines, leads to recognition of the non-extensionality of some liar-sentences. In particular, it turns out that exchanges of terms for identicals in the open-sentence '- does not expression (...) a true proposition' are not invariably truth-preserving because they are not invariably proposition-expression preserving. All of this recommends propositions as fruitful subjects of interesting renewed research. (shrink)

The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all infinite sets. (...) However, only Henkin's proposition equivalent to an internal position to infinity is consistent . This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or " observer'' , or "user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint. (shrink)

This note is an analysis of the paradox given by Myers. It is shown, assuming that the resources available in paraconsistent logic may be applied, how the conclusion of the paradox may be perfectly acceptable, but that the argument is, nonetheless, invalid. This provides a dialethic solution to the paradox.

The ancient semantic paradoxes were thought to undermine the rationalist metaphysics of Plato, and their modern relatives have been used by Russell and others to administer some severe logical and epistemological shocks. These are not just tricks or puzzles, but are intimately connected with some of the liveliest and most basic philosophical disputes about logical form, universals, reference and predication. Dr Cargile offers here an original and sustained treatment of this range of issues, and in fact presents an unfashionable defence (...) of a platonistic ontology. He argues that the paradoxes arise not from mistakes in classical assumptions about truth or from an ontology that includes propositions and properties, but from mistakes in describing what propositions and properties are conveyed by particular linguistic expressions. The book should interest, and may well surprise, philosophers and others concerned with semantics and the foundations of logic. (shrink)

Understandably absorbed in technical details, discussion of the semantic paradoxes risks losing sight of broad methodological principles. This chapter sketches a general approach to the comparison of rival logics, and applies it to argue that revision of classical propositional logic has much higher costs than its proponents typically recognize.

I present a paradoxical combination of desires. I show why it's paradoxical, and consider ways of responding. The paradox saddles us with an unappealing trilemma: either we reject the possibility of the case by placing surprising restrictions on what we can desire, or we deny plausibly constitutive principles linking desires to the conditions under which they are satisfied, or we revise some bit of classical logic. I argue that denying the possibility of the case is unmotivated on any reasonable (...) way of thinking about mental content, and rejecting those desire-satisfaction principles leads to revenge paradoxes. So the best response is a non-classical one, according to which certain desires are neither determinately satisfied nor determinately not satisfied. Thus, theorizing about paradoxical propositional attitudes helps constrain the space of possibilities for adequate solutions to semantic paradoxes more generally. (shrink)

In this paper, I present a novel paradox that pertains to a variety of representational states and activities. I begin by proving that there are certain contingently true propositions that no one can occurrently believe. Then, I use this to develop a further proof by which I derive a contradiction, thus giving us the paradox. Next, I differentiate the paradox from the Liar Paradox, and I show how a common response to the different variations of (...) the Liar Paradox fails to avoid the type of paradox provided in this paper. Finally, I demonstrate how the general ideas behind the paradox regarding occurrent belief can be extended to a wide range of other representational states and activities. (shrink)

Thomas Kroedel argues that the lottery paradox can be solved by identifying epistemic justification with epistemic permissibility rather than epistemic obligation. According to his permissibility solution, we are permitted to believe of each lottery ticket that it will lose, but since permissions do not agglomerate, it does not follow that we are permitted to have all of these beliefs together, and therefore it also does not follow that we are permitted to believe that all tickets will lose. I present (...) two objections to this solution. First, even if justification itself amounts to no more than epistemic permissibility, the lottery paradox recurs at the level of doxastic obligations unless one adopts an extremely permissive view about suspension of belief that is in tension with our practice of doxastic criticism. Second, even if there are no obligations to believe lottery propositions, the permissibility solution fails because epistemic permissions typically agglomerate, and the lottery case provides no exception to this rule. (shrink)

A proof-theoretic test for paradoxicality was famously proposed by Tennant: a paradox must yield a closed derivation of absurdity with no normal form. Drawing on the remark that all derivations of a given proposition can be transformed into derivations in normal form of a logically equivalent proposition, we investigate the possibility of paradoxes in normal form. We compare paradoxes à la Tennant and paradoxes in normal form from the viewpoint of the computational interpretation of proofs and from the viewpoint (...) of proof-theoretic semantics. (shrink)

Andrew Bacon, John Hawthorne, and Gabriel Uzquiano have recently argued that free logics—logics that reject or restrict Universal Instantiation—are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior. These logics encompass ramified and contextualist approaches to paradoxes, and broadly speaking, there are two kinds of criticism they face. First, they fail to address every version of the Priorean paradoxes. Second, the theoretical considerations behind the logics make absolutely general statements about all propositions, properties (...) of propositions, etc., but because this sort of intensional quantification is always restricted in the logics, they cannot even express those considerations. I present a novel sort of free logic, which rejects the standard Universal Instantiation but validates a restricted form of the rule, and which addresses both kinds of criticism by allowing some propositions to be unrestricted in their quantification. (shrink)

ABSTRACTTwo-dimensional accounts of speech and thought make use of so-called ‘diagonal’ propositions. If diagonals are indeed propositions, they can be negated: an ‘anti-diagonal’ is the negation o...

Tarski’s assessment that natural language is inconsistent on account of the Liar Paradox is shown to be incorrect: what Tarski’s theorem in fact shows is that Truth is not a property of sentences but of propositions. By using propositions rather than sentences as the bearers of Truth, semantic closure within the same language is easily obtained. Tarski’s contrary assessment was partly based on confusions about propositions and their grammatical expression. But more centrally it arose through blindness (...) to pragmatic factors in language — a blindness that was common in his time, and it has continued to the present day, in discussions of ‘Open Pairs’, and Yablo-type paradoxes, for instance. For completeness, it is also shown that the Fixed Point Theorem does not apply to propositions, because of categorical differences between sentences and propositions — also predicates and properties. (shrink)

It is commonly believed that philosophica l comparison depends on having some common measure or standard between and above the compared parts. The paper is to show that the foregoing common belief is incorrect and therewith to inquire into the possibility of cross-cultural philosophical comparison. First, the ‘comparison paradox’ will be expounded. It is a theoretical difficulty for the philosophical tendency represented by Plato’s theory of Ideas to justify comparative activities. Further, the connection of the comparative paradox with (...) the obstacles met by cross-cultural philosophical comparisons will be demonstrated. It will be shown that to attribute the difficulty of cross-cultural comparisons to incommensurability of traditions is irrelevant and misleading. It is to be argued that the original possibility of comparison depends on the ‘comparative situation’, i.e., the mechanism of meaning-production that functions in a non-universalistic and anonymous way. A philosophical paradigm does facilitate the attendance of such a situation, but it is also possible for the situation to emerge between paradigms in a gamesome way. Accordingly, the genuine comparison at issue will not originate primarily and merely on the level of concepts and propositions, but can only be achieved through inter-paradigmatic conditions, where we have the sharp awareness of a paradigm’s boundary from which we can attempt to achieve situational communication with another paradigm. In light of this, the perspective of a philosophical comparison differs not only from the traditional or universalistic one, but also from Gadamer’s hermeneutics, such as the doctrine of ‘fusion of horizons’. The new perspective finds an illustration in Heidegger’s relations with Daoism. (shrink)

If I were to say, “Agnes does not know that it is raining, but it is,” this seems like a perfectly coherent way of describing Agnes’s epistemic position. If I were to add, “And I don’t know if it is, either,” this seems quite strange. In this chapter, we shall look at some statements that seem, in some sense, contradictory, even though it seems that these statements can express propositions that are contingently true or false. Moore thought it was (...) paradoxical that statements that can express true propositions or contingently false propositions should nevertheless seem absurd like this. If we can account for the absurdity, we shall solve Moore’s Paradox. In this chapter, we shall look at Moore’s proposals and more recent discussions of Moorean absurd thought and speech. (shrink)