Moral Dilemmas and Vagueness

In this paper we point out some interesting structural similarities between vagueness and moral dilemmas as well as between some of the proposed solutions to both problems. Moral dilemma involves a situation with opposed obligations that cannot all be satisfied. Transvaluationism as an approach to vagueness makes three claims concerning the nature of vagueness: (1) it involves incompatibility between mutually unsatisfiable requirements, (2) the underlying requirements retain their normative power even when they happen to be overruled, and (3) this incompatibility turns out to be rather benign in practice. Given that transvaluationism is inspired by moral dilemmas, these claims are assessed in respect to them. Transvaluationism provides a smooth account of the mentioned claims concerning vagueness. Following a brief discussion of Sorensen’s views on moral dilemmas and conflict vagueness, we offer a model of moral pluralism accommodating structurally similar claims about the nature of moral conflict and moral dilemmas.     

Vagueness as Semantic Indecision: Metaphysical Vagueness vs Indeterminate Reference

After presenting a negative characterization of metaphysical vagueness and the main tenets of the view of vagueness as semantic indecision, the paper critically discusses the objection that such a view requires that at least some vagueness not be just constituted by semantic indecision—but rather by the metaphysical vagueness of some semantic relations themselves submitted by Trenton Merricks and, more recently, Nathan Salmon.     

Introduction: Vagueness and Ontology

The article introduces a special issue of the journal Metaphysica on vagueness and ontology. The conventional view has it that all vagueness is semantic or representational. Russell, Dummett, Evans and Lewis, inter alia, have argued that the notion of “ontic” or “metaphysical” vagueness is not even intelligible. In recent years, a growing minority of philosophers have tried to make sense of the notion and have spelled it out in various ways. The article gives an overview and relates the idea of ontic vagueness to the unquestioned phenomenon of fuzzy spatiotemporal boundaries and to the associated “problem of the many”. It briefly discusses the question of whether ontic vagueness can be spelled out in terms of “vague identity”, emphasizes the often neglected role of the difference between sortal and non-sortal ontologies and suggests a deflationary answer to the ill-conceived question of whether the “ultimate source” of vagueness lies either in language or in the world.     

Metaphysical Vagueness and Metaphysical Indeterminacy

The topic of this paper is whether there is metaphysical vagueness. It is shown that it is important to distinguish between the general phenomenon of indeterminacy and the more narrow phenomenon of vagueness (the phenomenon that paradigmatically rears its head in sorites reasoning). Relatedly, it is important to distinguish between metaphysical indeterminacy and metaphysical vagueness. One can wish to allow metaphysical indeterminacy but rule out metaphysical vagueness. As is discussed in the paper, central argument against metaphysical vagueness, like those of Gareth Evans and Mark Sainsbury, would if successful rule out metaphysical indeterminacy. One way to argue specifically against the possibility of metaphysical vagueness might be thought to be to argue for a specific theory of the nature of vagueness according to which vagueness is a semantic phenomenon. But it is shown that there are complications also pertaining to arguments with that structure. Toward the end of the paper, I discuss Trenton Merricks’ well-known argument against a semantic view on vagueness and for a metaphysical view.     

Naturalness, Vagueness, and Sortals

In the past few years, deflationary positions in the debate on the nature of composite material objects have become prominent. According to Ted Sider these include the thesis of quantifier variance, against which he has defended ontological realism. Recently, Sider has considered the possibility of rejecting his arguments against the vagueness of the unrestricted quantifiers in terms of translation functions. Against this strategy, he has presented an intuitive complaint and has argued that it can only be resisted if quantifier variance is accepted. But this is false. In this paper I argue, against Sider, that there is a coherent way to combine the rejection of quantifier variance with the vagueness of the unrestricted quantifiers. I sketch a model to show this, and then I consider, on the basis of it, several versions of the indeterminacy argument against the vagueness of the unrestricted quantifiers that Sider has formulated over the years.     

Moral Ambivalence,Relativism, and Pluralism

David Wong (2006) has introduced the notion of moral ambivalence in the philosophical debate. In this paper, we focus on the nature of moral ambivalence and on its interpretation. We hold that moral ambivalence is not a phenomenon that provides evidence for relativism, as Wong claims, and as relativism is usually understood. Rather, ambivalence denotes a pluralist attitude, an attitude characterized by the thought that two different, even incompatible, courses of action can both be permissible when considered from a single perspective. We distinguish pluralism from relativism and argue that it is more appropriate to consider Wong’s position as an objectivist pluralism (as pluralism is characterized by Berlin and others). We also highlight the similarity between Wong and Hume’s accounts.     

Vagueness, ignorance, and margins for error

We argue that the epistemic theory of vagueness cannot adequately justify its key tenet-that vague predicates have precisely bounded extensions, of which we are necessarily ignorant. Nor can the theory adequately account for our ignorance of the truth values of borderline cases. Furthermore, we argue that Williamson’s promising attempt to explicate our understanding of vague language on the model of a certain sort of “inexact knowledge” is at best incomplete, since certain forms of vagueness do not fit Williamson’s model, and in fact fit an alternative model. Finally, we point out that a certain kind of irremediable inexactitude postulated by physics need not be-and is not commonly-interpreted as epistemic. Thus, there are aspects of contemporary science that do not accord well with the epistemicist outlook.     

Ontological Vagueness, Existence Monism and Metaphysical Realism

Recently, Terry Horgan and Matja? Potr? have defended the thesis of ‘existence monism’, according to which the whole cosmos is the only concrete object. Their arguments appeal largely to considerations concerning vagueness. Crucially, they claim that ontological vagueness is impossible, and one key assumption in their defence of this claim is that vagueness always involves ‘sorites-susceptibility’. I aim to challenge both the claim and this assumption. As a consequence, I seek to undermine their defence of existence monism and support a common-sense pluralist ontology of ‘ordinary objects’ as being fully consistent with a thoroughgoing metaphysical realism.     

Lessons for Vagueness from Scrambled Sorites

Vagueness demands many boundaries. Each is permissible, in that a thinker may without error use it to distinguish objects, though none is mandatory. This is revealed by a thought experiment—scrambled sorites—in which objects from a sorites series are presented in a random order, and subjects are required to make their judgments without access to any previous objects or their judgments concerning them.     

Interrelations and Dissimilarities Between Distinct Approaches to Ontic Vagueness

This paper outlines the often striking parallels of various approaches to ontic vagueness, as well as their even more striking differences. Though circling around the same idea, some of these approaches were developed to solve quite diverse theoretical problems and encounter different challenges. In addition to these difficulties, the frequently disregarded epistemological problems of all theories of ontic vagueness turn out to be even more serious under critical scrutiny. The same holds for the difficulties of deciding, for every case of vagueness, whether the vagueness involved is semantic or ontic.     

Remarks on the Interest-relative Theory of Vagueness

I discuss the interest-relative account of vagueness and argue for a distinction between relational vague predicates and non-relational vague predicates depending on the kind of properties expressed by them. The strategy rests on three arguments arising from the existence of clear cases of a vague predicate, from contexts in which a different answer is required for questions about whether a vague predicate applies to an item, and whether such an item satisfies the interest of an agent, and from cases where an object changes up to the point of becoming P, where P is a vague predicate. In the second part of the paper, I distinguish between relational properties and non-relational properties, and I argue for the view that some vague predicates can express non-relational properties, comparative relational properties and interest-relative properties. On the basis of these arguments, I conclude that vagueness cannot be reduced to interest-relativity.     

Persistence, Ontic Vagueness and Identity: Towards a Substantialist Four–dimensionalism

Four-dimensionalism, the stage theory version in particular, has been defended as the best solution for avoiding vagueness in regards to composition, persistence and identity. Stage theory is highly problematic by itself, and the two views usually packed with it, unrestricted composition and counterpart theory, are a heavy burden. However, dispensing with these two views, four-dimensionalism could avoid vague persistence by issuing a criterion that would establish sharp temporal boundaries for the existence of genuine entities (simples, molecules and living organisms). This would avoid vague existence and vague identity, but in a way that is still compatible with endurantism. Nevertheless, a minimal (substantialist) four-dimensionalism, a worm perdurantist ontology, would fit better with the unique way in which organisms persist: by retaining both identity and intrinsic change.     

Linear birth and death models and associated Laguerre and Meixner polynomials

We study birth and death processes with linear rates λ_n = n + α + c + 1, μ_{n + 1} = n + c, n 0 and μ₀ is either zero or c. The spectral measures of both processes are found using generating functions and the integral transforms of Laplace and Stieltjes. The corresponding orthogonal polynomials generalize Laguerre polynomials and the choice μ₀ = c generates the associated Laguerre polynomials of Askey and Wimp. We investigate the orthogonal polynomials in both cases and give alternate proofs of some of the results of Askey and Wimp on the associated Laguerre polynomials. We also identify the spectra of the associated Charlier and Meixner polynomials as zeros of certain transcendental equations.     

Distributive and Multiplication Modules and Rings

We study rings in which every ideal is a finitely generated multiplication right ideal.     

Affine-inscribed and affine-circumscribed polygons and polytopes

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ³, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ³ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298. Translated by B. M. Bekker.     

Partitions and pairwise sums and products

A seven cell partition of N is constructed with the property that no infinite set has all of its pairwise sums and products in any one cell. A related Ramsey Theory question is shown to have different answers for two and three cell partitions.