Distributed Knowability and Fitch’s Paradox

Recently predominant forms of anti-realism claim that all truths are knowable. We argue that in a logical explanation of the notion of knowability more attention should be paid to its epistemic part. Especially very useful in such explanation are notions of group knowledge. In this paper we examine mainly the notion of distributed knowability and show its effectiveness in the case of Fitch’s paradox. Proposed approach raised some philosophical questions to which we try to find responses. We also show how we can combine our point of view on Fitch’s paradox with the others. Next we give an answer to the question: is distributed knowability factive? At the end, we present some details concerning a construction of anti-realist modal epistemic logic.     

Fitch’s Paradox and Probabilistic Antirealism

Fitch’s paradox shows, from fairly innocent-looking assumptions, that if there are any unknown truths, then there are unknowable truths. This is generally thought to deliver a blow to antirealist positions that imply that all truths are knowable. The present paper argues that a probabilistic version of antirealism escapes Fitch’s result while still offering all that antirealists should care for.     

Intuitionistic Epistemic Logic,Kripke Models and Fitch’s Paradox

The present work is motivated by two questions. (1) What should an intuitionistic epistemic logic look like? (2) How should one interpret the knowledge operator in a Kripke-model for it? In what follows we outline an answer to (2) and give a model-theoretic definition of the operator K. This will shed some light also on (1), since it turns out that K, defined as we do, fulfills the properties of a necessity operator for a normal modal logic. The interest of our construction also lies in a better insight into the intuitionistic solution to Fitch’s paradox, which is discussed in the third section. In particular we examine, in the light of our definition, DeVidi and Solomon’s proposal of formulating the verification thesis as \(\phi \rightarrow \neg \neg K\phi\). We show, as our main result, that this definition excapes the paradox, though it is validated only under restrictive conditions on the models.     

Note on Florensky’s Solution to Carroll’s ‘Barbershop’ Paradox: Reverse Implication for Russell?

Pavel Florensky solves Lewis Carroll??s ??Barbershop?? paradox to support his reasoning in a previous chapter. Our discussion includes a) the problem (which we also refer to as the p paradox), b) Carroll??s solution, c) Bertrand Russell??s solution, d) Florensky??s solution and then e) a material example proffered by Florensky. Both Russell and Florensky disagree with Carroll??s solution, yet, (ostensibly) unbeknownst to themselves they offer the same solution, which is ??p implies not-q??. Given Florensky??s material example, the solution seems to tell us something about the logic of belief. We ask whether Florensky??s example has reverse implications for Russell??s solution.     

Grelling’s Paradox

Grelling’s Paradox is the paradox which results from considering whether heterologicality, the word-property which a designator has when and only when the designator does not bear the word-property it designates, is had by ‘ ȁ8heterologicality’. Although there has been some philosophical debate over its solution, Grelling’s Paradox is nearly uniformly treated as a variant of either the Liar Paradox or Russell’s Paradox, a paradox which does not present any philosophical challenges not already presented by the two better known paradoxes. The aims of this paper are, first, to offer a precise formulation of Grelling’s Paradox which is clearly distinguished from both the Liar Paradox and Russell’s Paradox; second, to offer a solution to Grelling’s Paradox which both resolves the paradoxical reasoning and accounts for unproblematic predications of heterologicality; and, third, to argue that there are two lessons to be drawn from Grelling’s Paradox which have not yet been drawn from the Liar or Russell’s Paradox. The first lesson is that it is possible for the semantic content of a predicate to be sensitive to the semantic context; i.e., it is possible for a predicate to be an indexical expression. The second lesson is that the semantic content of an indexical predicate, though unproblematic for many cases, can nevertheless be problematic in some cases.     

Kant’s Solution to the Euthyphro Dilemma

Are our actions morally good because we approve of them or are they good independently of our approval? Are we projecting moral values onto the world or do we detect values that are already there? For many these questions don’t state a real alternative but a secular variant of the Euthyphro dilemma: If our actions are good because we approve of them moral goodness appears to be arbitrary. If they are good independently of our approval, it is unclear how we come to know their moral quality and how moral knowledge can be motivating. None of these options seems attractive; the source of moral goodness unclear. Despite the growing literature on Kant’s moral epistemology and moral epistemology the question remains open what Kant’s answer to this apparent dilemma is. The Kantian view I attempt to lay out in this paper is supposed to dissolve the secular version of the Euthyphro dilemma. In responding to this dilemma we need to get clear about the source or the origin of our moral knowledge: Voluntary approval or mind-independent moral facts? Projectivism or detectivism? Construction or given? I believe that all these ways of articulating the problem turn out, on closer inspection, to be false alternatives.     

Hume’s Solution of the Goodman Paradox and the Reliability Riddle (Mill’s problem)

Many solutions of the Goodman paradox have been proposed but so far no agreement has been reached about which is the correct solution. However, I will not contribute here to the discussion with a new solution. Rather, I will argue that a solution has been in front of us for more than two hundred years because a careful reading of Hume’s account of inductive inferences shows that, contrary to Goodman’s opinion, it embodies a correct solution of the paradox. Moreover, the account even includes a correct answer to Mill’s question of why in some cases a single instance is sufficient for a complete induction, since Hume gives a well-supported explanation of this reliability phenomenon. The discussion also suggests that Bayesian theory by itself cannot explain this phenomenon. Finally, we will see that Hume’s explanation of the reliability phenomenon is surprisingly similar to the explanation given lately by a number of naturalistic philosophers in their discussion of the Goodman paradox.     

McTaggart’s Paradox and Crisp’s Presentism

In his review of The Ontology of Time, Thomas Crisp (Notre Dame Philosophical Reviews, 2005a) argues that Oaklander's version of McTaggart's paradox does not make any trouble for his version of presentism. The aim of this paper is to refute that claim by demonstrating that Crisp's version of presentism does indeed succumb to a version of McTaggart's argument. I shall proceed as follows. In Part I I shall explain Crisp's view and then argue in Part II that his analysis of temporal becoming, temporal properties and temporal relations is inadequate. Finally, in Part III, I shall demonstrate that his presentist ontology of time is susceptible to the paradox he so assiduously sought to avoid.     

Mengzi’s Externalist Solution to the Role Dilemma

The role dilemma raises a problem for role ethic interpretations of Confucianism. The dilemma arises from the conflict between the demands and obligations of Humaneness and the demands and obligations of roles one occupies. Favoring the demands of Humaneness (i.e. externalism) undermines a role ethic because roles and role-obligations no longer ground the ethic. However, favoring social role-obligations (i.e. internalism) permits immoral and unjust role-obligations and allows for uncharitable readings of Confucianism.

This paper examines how Mengzi resolves the dilemma. I argue that Mengzi’s account of human nature privileges the demands of Humaneness; social roles are central but defeasible in light of Humaneness. I briefly discuss a prominent articulation of Confucian role ethics as well as the role dilemma. Then, after considering the technical resources within the Mengzi, I argue that Mengzi espouses an externalism about roles. Finally, I explore the relationship between Mengzi’s externalism and role ethics.     

An axiomatic version of Fitch’s paradox

A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the paradox.     

Moore’s Paradox for God

Philosophia - I argue that ‘Moore’s paradox for God’. I do not believe this proposition shows that nobody can be both omniscient and rational in all her beliefs. I then anticipate...     

Katz’s Revisability Paradox Dissolved

Quine's holistic empiricist account of scientific inquiry can be characterized by three constitutive principles: noncontradiction, universal revisability and pragmatic ordering. We show that these constitutive principles cannot be regarded as statements within a holistic empiricist's scientific theory of the world. This claim is a corollary of our refutation of Katz's [1998, 2002] argument that holistic empiricism suffers from what he calls the Revisability Paradox. According to Katz, Quine's empiricism is incoherent because its constitutive principles cannot themselves be rationally revised. Using Gärdenfors and Makinson's logic of belief revision based on epistemic entrenchment, we argue that Katz wrongly assumes that the constitutive principles are statements within a holistic empiricist's theory of the world. Instead, we show that constitutive principles are best seen as properties of a holistic empiricist's theory of scientific inquiry and we submit that, without Katz's mistaken assumption, the paradox cannot be formulated. We argue that our perspective on the status of constitutive principles is perfectly in line with Quinean orthodoxy. In conclusion, we compare our findings with van Fraassen's [2002] argument that we should think of empiricism as a stance, rather than as a doctrine.     

On Kripke’s Dogmatism Paradox: A Logical Dynamical Analysis

As a byproduct of solving the surprise-exam paradox, Saul Kripke formulates a “dogmatism paradox” which seems to show that knowledge entails dogmatism. In this paper, the author analyzes the nature of the dogmatism paradox from a logical dynamical perspective. The author suggests that the dogmatism paradox is better understood as a paradox of knowledge attribution rather than of knowledge. Therefore, the dogmatism paradox could be solved without sacrificing the principle of epistemic closure. Based on a famous version of relevant alternatives theory, the author formalizes a logic of knowledge attribution in the style of logical dynamics, namely, public retraction logic, and analyzes how knowledge attributions are retracted with the expansion of relevant alternatives.     

Curry’s Paradox and ω -Inconsistency

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes¹. In this paper I show that a number of logics are susceptible to a strengthened version of Curry’s paradox. This can be adapted to provide a proof theoretic analysis of the ω-inconsistency in ?ukasiewicz’s continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of ukasiewicz logic which individually, but not jointly, lack the problematic feature.     

Mereological Dominance and Simpson’s Paradox

Philosophia - Numerous papers have investigated the transitivity principle of ‘better-than.’ A recent argument appeals to the principle of mereological dominance for transitivity....     

Yablo’s Paradox and ω-Inconsistency

It is argued that Yablo’s Paradox is not strictly paradoxical, but rather ‘ω-paradoxical’. Under a natural formalization, the list of Yablo sentences may be constructed using a diagonalization argument and can be shown to be ω-inconsistent, but nonetheless consistent. The derivation of an inconsistency requires a uniform fixed-point construction. Moreover, the truth-theoretic disquotational principle required is also uniform, rather than the local disquotational T-scheme. The theory with the local disquotation T-scheme applied to individual sentences from the Yablo list is also consistent.     

Moore’s Paradox and Moral Motivation

Assertions of statements such as ‘it’s raining, but I don’t believe it’ are standard examples of what is known as Moore’s paradox. Here I consider moral equivalents of such statements, statements wherein individuals affirm moral judgments while also expressing motivational indifference to those judgments (such as ‘hurting animals for fun is wrong, but I don’t care’). I argue for four main conclusions concerning such statements: 1. Such statements are genuinely paradoxical, even if not contradictory. 2. This paradoxicality can be traced to a form of epistemic self-defeat that also explains the paradoxicality of ordinary Moore-paradoxical statements. 3. Although a simple form of internalism about moral judgment and motivation can explain the paradoxicality of these moral equivalents, a more plausible explanation can be provided that does not rely on this simple form of internalism. 4. The paradoxicality of such statements suggests a more credible understanding of the thesis that those who are not motivated by their moral judgments are irrational.     

Recollecting the Religious: Augustine in Answer to Meno’s Paradox

Studies in Philosophy and Education - Philosophers of education often view the role of religion in education with suspicion, claiming it to be impossible, indoctrinatory or controversial unless...     

Rescuing Poincaré from Richard’s Paradox

Poincaré in a 1909 lecture in Göttingen proposed a solution to the apparent incompatibility of two results as viewed from a definitionist perspective: on the one hand, Richard’s proof that the definitions of real numbers form a countable set and, on the other, Cantor’s proof that the real numbers make up an uncountable class. Poincaré argues that, Richard’s result notwithstanding, there is no enumeration of all definable real numbers. We apply previous research by Luna and Taylor on Richard’s paradox, indefinite extensibility and unrestricted quantification to evaluate Poincaré’s proposal. We emphasize that Poincaré’s solution involves an early recourse to indefinite extensibility and argue that his proposal, if it is to completely avoid Richard’s paradox, requires rejecting absolutely unrestricted quantification: Richard’s paradox provides a context in which paradox seems inescapable if unrestricted quantification is possible. In proposing his solution to the apparent conflict between Richard’s and Cantor’s results, Poincaré employs temporal expressions whose exact meaning he does not clarify. We suggest an interpretation of these expressions in terms of order of availability and briefly discuss its explanatory power in topics like paradoxes, limitation theorems and indefinite extensibility.