Strict Finitism and the Happy Sorites

Call an argument a ‘happy sorites’ if it is a sorites argument with true premises and a false conclusion. It is a striking fact that although most philosophers working on the sorites paradox find it at prima facie highly compelling that the premises of the sorites paradox are true and its conclusion false, few (if any) of the standard theories on the issue ultimately allow for happy sorites arguments. There is one philosophical view, however, that appears to allow for at least some happy sorites arguments: strict finitism in the philosophy of mathematics. My aim in this paper is to explore to what extent this appearance is accurate. As we shall see, this question is far from trivial. In particular, I will discuss two arguments that threaten to show that strict finitism cannot consistently accept happy sorites arguments, but I will argue that (given reasonable assumptions on strict finitistic logic) these arguments can ultimately be avoided, and the view can indeed allow for happy sorites arguments.     

WHAT WILLIAMSON'S ANTI‐LUMINOSITY ARGUMENT REALLY IS*

Abstract: Williamson argues that when one feels cold, one may not be in a position to know that one feels cold. He thinks this argument can be generalized to show that no mental states are such that when we are in them we are in a position to know that we are in them. I argue that his argument is a sorites argument in disguise because it relies on the implicit premise that warming up is gradual. Williamson claims that his argument is not a sorites argument; I explain why he has not given us any reason to accept the claim.     

论Priest对Sorites悖论的模糊解悖方案

G.Priest在1998-2003年之间有一个对sorites悖论的模糊解悖方案,根据该解悖方案,对含混语言的最佳语意论应该是某种的模糊语意论。在这样的模糊语意论中,边界语句是既真且假的语句,而sorites悖论中所涉及的离断律和等同传递律则不是有效的推论步骤。Priest并试图利用这样的语意论去说明：为什么sorites悖论看起来像是有效的论证,以及为什么切割点的存在尽管必然却违反直觉。我在本文中论证：Priest的模糊解悖方案是一个完全失败的方案,因此最好被放弃。我论证,Priest的模糊解悖方案至少有以下几个主要的问题：它与Priest的真理理论以及双面真理论不兼容、它对于sorites悖论的诊断有特设的嫌疑、它对于sorites悖论为何看起来有效的说明不具有可信度、它对于切割点存在何以违反直觉的说明也不具有可信度、它有将高阶含混性崩塌为一阶含混性之虞。     

V—Vagueness, Realism, Language and Thought

The problem of vagueness and the sorites paradox arise because we try to treat natural language as if it were a unitary formal system. In fact, natural language contains a large variety of representational ontologies that serve different purposes and which cannot be united formally, but which can intuitively be taken as ways of seeing a common basic ontology. Using this framework, we can save classical logic from vagueness and avoid the sorites.     

Inconsistent Languages

The main thesis of this paper is that we sometimes are disposed to accept false and even jointly inconsistent claims by virtue of our semantic competence, and that this comes to light in the sorites and liar paradoxes. Among the subsidiary theses are that this is an important source of indeterminacy in both conditions, that we must revise basic assumptions about semantic competence, and that classical logic and bivalence can be upheld in the face of the sorites paradox.     

The Paradox of Belief Instability and a Revision Theory of Belief

The epistemic paradox of 'belief instability' has recently received notable attention from many philosophers. In this paper I offer a new proposal, which I call a 'revision theory of belief'. This theory is in many respects an application of Gupta's and Belnap's revision theory of truth. They argue that the Liar paradox arises because our notion of truth is circular. I offer a similar proposal for handling the paradox of belief instability. In particular, I argue that our notion involved in the paradox of belief instability is circular, and this circularity of belief is the source of the paradox.     

Chain-Arguments and the Sorites Paradox

A finite chain of valid arguments can never lead from truth to falsehood. Call this the concatenation principle, or CP. Some propose to reject CP in response to the sorites paradox. I offer a justification of a restricted version of CP based on intuitively-appealing principles, all of which are already assumed in mathematics. The restricted version excludes soritical cases, but covers intuitively correct applications of CP in mathematics and elsewhere. The upshot is that the cost of rejecting the unrestricted CP is much lower than is often assumed.     

Paradox and Paul: Catholic and Protestant Theologies of Grace

John Barclay offers a capacious and helpful framework for reflection on large swathes of the history of the theology of grace with his notion of the six ways of ‘perfecting the gift’. In this article, I extend his analysis to a consideration of certain typical differences between Catholic and Protestant conceptions of grace. Where Protestant theology tends toward a juxtaposition of grace with sin, Catholic theology often considers grace in relation to nature, and each side, I suggest, has its own characteristic proclivity towards paradox.     

On Russell's vulnerability to Russell's paradox

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ‘the most important event’ in ‘the most important year in my intellectual life’. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory.     

JUST WHAT IS VAGUENESS?

We argue that standard definitions of ‘vagueness’ prejudice the question of how best to deal with the phenomenon of vagueness. In particular, the usual understanding of ‘vagueness’ in terms of borderline cases, where the latter are thought of as truth‐value gaps, begs the question against the subvaluational approach. According to this latter approach, borderline cases are inconsistent (i.e., glutty not gappy). We suggest that a definition of ‘vagueness’ should be general enough to accommodate any genuine contender in the debate over how to best deal with the sorites paradox. Moreover, a definition of ‘vagueness’ must be able to accommodate the variety of forms sorites arguments can take. These include numerical, total‐ordered sorites arguments, discrete versions, continuous versions, as well as others without any obvious metric structure at all. After considering the shortcomings of various definitions of ‘vagueness’, we propose a very general non‐question‐begging definition.     

Higher-Order Sorites Paradox

The naive theory of vagueness holds that the vagueness of an expression consists in its failure to draw a sharp boundary between positive and negative cases. The naive theory is contrasted with the nowadays dominant approach to vagueness, holding that the vagueness of an expression consists in its presenting borderline cases of application. The two approaches are briefly compared in their respective explanations of a paramount phenomenon of vagueness: our ignorance of any sharp boundary between positive and negative cases. These explanations clearly do not provide any ground for choosing the dominant approach against the naive theory. The decisive advantage of the former over the latter is rather supposed to consist in its immunity to any form of sorites paradox. But another paramount phenomenon of vagueness is higher-order vagueness: the expressions (such as ‘borderline’ and ‘definitely’) introduced in order to express in the object language the vagueness of the object language are themselves vague. Two highly plausible claims about higher-order vagueness are articulated and defended: the existence of “definitely ^ω ” positive and negative cases and the “radical” character of higher-order vagueness itself. Using very weak logical principles concerning vague expressions and the ‘definitely’-operator, it is then shown that, in the presence of higher-order vagueness as just described, the dominant approach is subject to higher-order sorites paradoxes analogous to the original ones besetting the naive theory, and therefore that, against the communis opinio, it does not fare substantially better with respect to immunity to any form of sorites paradox.     

Vagueness and the Laws of Metaphysics

This is a paper about the nature of metaphysical laws and their relation to the phenomenon of vagueness. Metaphysical laws are introduced as analogous to natural laws, and metaphysical indeterminism is modeled on causal indeterminacy. This kind of indeterminacy is then put to work in developing a novel theory of vagueness and a solution to the sorites paradox.     

The Paradox of Wuwei? Yes (and No)

This essay considers P. J. Ivanhoe's critical challenge to Slingerland's analysis of wuwei(‘effortless action’). While I agree with Ivanhoe that we should do more work to embody and understand the concept of wuwei, I will defend Slingerland's notion that wuwei involves paradox—particularly in the cases of Zhuangi and Laozi. The present essay is not a defense of the specifics of Slingerland's analysis. Nonetheless, this essay focuses on defending the notion of paradox. Ivanhoe offers an alternative view of wuwei, one that sees the paradox as a riddle. I argue that this kind of formulation would frame the problem of wuwei in an unhelpful manner. I offer several novel ways of overcoming, or at least qualifying, the experience of paradox that seems to be at play in nondoing.     

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.     

Tolerant, Classical, Strict

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases.     

K ⊈ E

In a series of very influential works, Tim Williamson has advanced and defended a much discussed theory of evidence containing, among other claims, the thesis that, if one knows P, P is part of one's evidence (K ? E). I argue that K ? E is false, and indeed that it is so for a reason that Williamson himself essentially provides in arguing against the thesis that, if one has a justified true belief in P, P is part of one's evidence: together with a very plausible principle governing the acquisition of knowledge by non‐deductive inference based on evidence, K ? E leads, in a sorites‐like fashion, to what would seem a series of unacceptably bootstrapping expansions of one's evidence. I then develop some considerations about the functions of and conditions for evidence which are suggested by the argument against K ? E. I close by discussing the relationship of the argument with anti‐closure arguments of the style exemplified by the preface paradox: I contend that, if closure is assumed, it is extremely plausible to expect that the diagnosis of what goes wrong in the preface‐paradox‐style argument cannot be used to block my own argument.     

Measuring Inconsistency

I provide a method of measuring the inconsistency of a set of sentences – from 1-consistency, corresponding to complete consistency, to 0-consistency, corresponding to the explicit presence of a contradiction. Using this notion to analyze the lottery paradox, one can see that the set of sentences capturing the paradox has a high degree of consistency (assuming, of course, a sufficiently large lottery). The measure of consistency, however, is not limited to paradoxes. I also provide results for general sets of sentences.     

Who cares about winning?

Why do we so often care about the outcomes of games when nothing is at stake? There is a paradox here, much like the paradox of fiction, which concerns why we care about the fates and threats of merely fictional beings. I argue that the paradox threatens to overturn a great deal of what philosophers have thought about caring, severing its connection to value and undermining its moral weight. I defend a solution to the paradox that draws on Kendall Walton's solution to the paradox of fiction, developing his idea that it be extended to games. The solution takes games to involve make-believe: in particular, players and spectators make-believe that the outcome of the game matters. I also explore how the phenomenon extends beyond games. And I explore some moral implications: in particular, my view preserves the idea that we have reason not to impede others in their pursuit of what they care about.     

Soritic thinking, vagueness, and weakness of will

Soritic thinking based on reasoning that is involved in the sorites paradox plays a crucial role in some forms of weakness of will. Such soritic reasoning leads to failures of behavior, but cannot be shown to be irrational by standard means. Thus weakness of will appears to be rational, whereas strength of will is irrational when viewed soritically. The puzzle is how to undermine weakness of will and expose it as irrational. Even though such weakness of will is not moral, moral-type reasoning involving the principle of equality can be brought to bear. Weakness of will can also be seen to be analogous to free-rider problems and the prisoner's dilemma.