What Is Graded Membership?

It has seemed natural to model phenomena related to vagueness in terms of graded membership. However, so far no satisfactory answer has been given to the question of what graded membership is nor has any attempt been made to describe in detail a procedure for determining degrees of membership. We seek to remedy these lacunae by building on recent work on typicality and graded membership in cognitive science and combining some of the results obtained there with a version of the conceptual spaces framework.     

Note on the integration of prototype theory and fuzzy-set theory

Many criticisms of prototype theory and/or fuzzy-set theory are based on the assumption that category representativeness (or typicality) is identical with fuzzy membership. These criticisms also assume that conceptual combination and logical rules (all in the Aristotelian sense) are the appropriate criteria for the adequacy of the above “fuzzy typicality”. The present paper discusses these assumptions following the line of their most explicit and most influential expression by Osheron and Smith (1981). Several arguments are made against the above identification, the most important being that representativeness in prototype theory is exclusively based on element-to-element similarity while fuzzy membership is inherently an element-to-category relationship. Also the above criteria for adequacy are criticized from the viewpoint of both prototype theory and fuzzy-set theory as well as from that of both conceptual and logical combination, and also from that of integration.     

Intuitions,Biases, and Extra-Wide Reflective Equilibrium

It seems that intuitions are indispensable in philosophical theorizing. Yet, there is evidence that our intuitions are heavily influenced by biases. This generates a puzzle: we must use our intuitions, but we seemingly cannot fully trust those very intuitions. This paper develops a methodology for philosophical theorizing that attempts to avoid this puzzle. Specifically, it develops and defends a methodology it calls Extra-Wide Reflective Equilibrium. It argues that this method allows us to use intuitions, while also providing a mechanism to check the influence of bias on our intuitions. In section 1, it defends the claim that intuitions are indispensable in philosophical theorizing. In section 2, it outlines recent arguments against the reliability of intuitions. In section 3, it explains and defends its account of Extra-Wide Reflective Equilibrium.     

Domain differences in absolute judgments of category membership: evidence for an essentialist account of categorization

There has been some debate about the correspondence between typicality gradients and category membership. The present study investigates the relationship between these two measures in the domains of animals and artifacts. Forty-two adults judged the degree of typicality or category membership of 293 animals and artifacts. The subjects’ tendency for animals, but not for artifacts, was to make more absolute ratings on category membership (i.e., judging exemplars as definitely members or definitely not members of their respective category) than on typicality. More importantly, at almost every level of typicality, subjects were more likely to make absolute judgments of category membership for animals than for artifacts. These results indicate that people treat category membership of animals as relatively absolute (which best fits an essentialist model of categorization) and treat category membership of artifacts as relatively graded (which best fits a prototype model of categorization). These domain differences add crucial supporting evidence for claims about the domain-specificity of essentialism.     

Exemplars and prototypes in natural language concepts: A typicality-based evaluation

Are natural language categories represented by instances of the category or by a summary representation? We used an exemplar model and a prototype model, both derived within the framework of the generalized context model (Nosofsky, 1984, 1986), to predict typicality ratings for 12 superordinate natural language concepts. The models were fitted to typicality ratings averaged across participants and to the typicality judgments of individual participants. Both analyses yielded results in favor of the exemplar model. These results suggest that higher-level natural language concepts are represented by their subordinate members, rather than by a summary representation.     

Disjunction of natural concepts

Two experiments investigated the way in which the rated membership of items in disjunctively defined categories, such as FRUITS OR VEGETABLES and PETS OR FARMYARD ANIMALS, varies as a function of membership in individual constituent categories. Items were rated for category membership and typicality in each category separately, and in their disjunction. The results showed non-Boolean effects of both overextension and underextension of the disjunctions. Typicality in the disjunction was highly predictable from constituent typicality values, using regression equations with negative interaction terms. The results are compared with similar effects for concept conjunctions and are discussed in terms of an intensional model of conceptual combination (Hampton, 1987b, 1988).     

Typicality and Graded Membership in Dimensional Adjectives

This paper concerns an investigation of the manner in which typicality constrains graded membership in antonymous dimensional adjectives such as short/tall and cheap/expensive using the conceptual spaces framework. In this framework, items are organized in a space comprised of one or more dimensions along which they can be compared. The items’ graded membership is established by their relative proximity in this space to the prototypical instances of contrasting concepts. Because dimensional adjectives can be applied to an indefinite variety of things and grammatically have no upper bound to serve as cognitive reference point, they have been argued to lack prototypes. We present the results of an empirical study showing that the conceptual spaces framework can nevertheless be extended successfully to dimensional adjectives by complementing them with a comparison class argument (such as short/tall for an adult man and cheap/expensive for a smartphone), allowing participants to retrieve meaningful prototypical instances, which can be used to establish membership degree. Since dimensional adjectives are subjective, we investigate how the framework can accommodate interindividual variability in membership degree judgments. We find that the predictions of the framework significantly improve if prototypical instances themselves are assumed to come with a gradient instead of being considered equally typical, thereby providing a more fine‐grained account of typicality and furthering the development of the conceptual spaces framework.     

Five-year-olds’ beliefs about the discreteness of category boundaries for animals and artifacts

Research on adult concepts indicates that category structure varies by domain; adults view membership in animal categories as absolute but membership in artifact categories as graded. In this study, we examined domain differences in beliefs about category boundaries among young children (5-year-olds). The results indicated that young children, like adults, were less likely to endorse graded category membership for animal than for artifact categories. These domain differences could not be attributed to domain differences in typicality. Implications for conceptual development and for models of domain specificity in adult cognition are discussed.     

Conceptual combination with prototype concepts

This paper deals with how people combine simple, prototype concepts into complex ones; e.g., how people combine the prototypes for brown and apple so they can determine the typicality of objects in the conjunction brown apple. We first consider a proposal from fuzzy-set theory (Zadeh, 1965), namely, that the typicality of an object in a conjunction is equal to the minimum of that object's typicality in the constituents (e.g., an object's typicality as a brown apple cannot exceed its typicality as a brown or as an apple). We evaluated this “min rule” against the typicality ratings of naive subjects in two experiments. For each of numerous pictured objects, one group of subjects rated its typicality with respect to an adjective concept, a second group rated its typicality vis-à-vis a noun concept, and a third group rated its typicality with respect to the adjective-noun conjunction. In both studies, most objects were rated as more typical of the conjunction than of the noun. These findings violate not only the min rule but also other simple rules for relating typicality in a conjunction to typicalities in the constituents. As an alternative to seeking such rules, we argue for an approach to conceptual combination that starts with the prototype representations themselves. We illustrate one version of this approach in some detail, and show how it accounts for the major findings of the present experiments.     

Moral Learning,Rationality, and the Unreliability of Affect

James Woodward and John Allman [2007, 2008] and Peter Railton [2014, 2016] argue that our moral intuitions are products of sophisticated rational learning systems. I investigate the implications that this discovery has for intuition-based philosophical methodologies. Instead of vindicating the conservative use of intuitions in philosophy, I argue that what I call the rational learning strategy fails to show philosophers are justified in appealing to their moral intuitions in philosophical arguments without giving reasons why those intuitions are trustworthy. Despite the fact that our intuitions are outputs of surprisingly sophisticated learning mechanisms, we do not have reason to unreflectively trust them when offering arguments in moral philosophy.     

Margins for Error: A Reply

I address Peter Mott's 'Margins for Error and the Sorites Paradox' ( The Philosophical Quarterly , 48 (1998), pp. 494–503). Mott criticizes my account of inexact knowledge, on which it satisfies margin for error principles of the form 'If one knows in a given case, one avoids false belief in sufficiently similar cases'. Mott's arguments are shown to be fallacious because they ignore the fact that our knowledge of inexact knowledge is itself inexact. In the examples discussed, the first-level inexact knowledge is perceptual. Since my defence of an epistemicist theory of vagueness explains our ignorance of truth-values in borderline cases as the result of knowledge the inexactness of which has a conceptual source, the paper also contributes to the defence of epistemicism about vagueness.     

Effects of classification context on categorization in natural categories

The patterns of classification of borderline instances of eight common taxonomic categories were examined under three different instructional conditions to test two predictions: first, that lack of a specified context contributes to vagueness in categorization, and second, that altering the purpose of classification can lead to greater or lesser dependence on similarity in classification. The instructional conditions contrasted purely pragmatic with more technical/quasi-legal contexts as purposes for classification, and these were compared with a no-context control. The measures of category vagueness were between-subjects disagreement and within-subjects consistency, and the measures of similarity-based categorization were category breadth and the correlation of instance categorization probability with mean rated typicality, independently measured in a neutral context. Contrary to predictions, none of the measures of vagueness, reliability, category breadth, or correlation with typicality were generally affected by the instructional setting as a function of pragmatic versus technical purposes. Only one subcondition, in which a situational context was implied in addition to a purposive context, produced a significant change in categorization. Further experiments demonstrated that the effect of context was not increased when participants talked their way through the task, and that a technical context did not elicit more all-or-none categorization than did a pragmatic context. These findings place an important boundary condition on the effects of instructional context on conceptual categorization.     

Epistemicism,Parasites, and Vague Names

John Burgess has recently argued that Timothy Williamson's attempts to avoid the objection that his theory of vagueness is based on an untenable metaphysics of content are unsuccessful. Burgess's arguments are important, and largely correct, but there is a mistake in the discussion of one of the key examples. In this note I provide some alternative examples and use them to repair the mistaken section of the argument.     

Category-based induction: An effect of conclusion typicality

Category-based induction involves the willingness of a thinker to project some newly learned property of one or more classes of objects to another class on the basis of their shared membership in a common superordinate category. Previous research has established that the perceived strength of arguments of the form "Class A has Property P; therefore, Class B has Property P" is influenced by the similarity of A to B and by the typicality or representativeness of A in a shared category, superordinate to both A and B. (The nature of P is also crucial, but we do not examine it in this study.) There is, however, no prior evidence that the relation between B and the category is influential. Three experiments were designed to test whether the typicality of B in the superordinate category also has an effect on inductive argument strength. By using multiple regression (Experiment 1) and an experimental design (Experiment 3), an effect of conclusion typicality was found, so that people are more willing to project properties to more typical conclusions. Experiment 2 ruled out conclusion familiarity as a potential confounding variable. The results are interpreted in the light of current models of category-based induction.     

Vagueness as closeness

This paper presents and defends a definition of vagueness, compares it favourably with alternative definitions, and draws out some consequences of accepting this definition for the project of offering a substantive theory of vagueness. The definition is roughly this: a predicate ‘F’ is vague just in case for any objects a and b, if a and b are very close in respects relevant to the possession of F, then ‘Fa’ and ‘Fb’ are very close in respect of truth. The definition is extended to cover vagueness of many-place predicates, of properties and relations, and of objects. Some of the most important advantages of the definition are that it captures the intuitions which motivate the thought that vague predicates are tolerant, without leading to contradiction, and that it yields a clear understanding of the relationships between higher-order vagueness, sorites susceptibility, blurred boundaries, and borderline cases. The most notable consequence of the definition is that the correct theory of vagueness must countenance degrees of truth.     

A formal ideal-based account of typicality

Inspired by Barsalou’s (Journal of Experimental Psychology: Learning, Memory, and Cognition, 11, 629–654, 1985) proposal that categories can be represented by ideals, we develop and test a computational model, the ideal dimension model (IDM). The IDM is tested in its account of the typicality gradient for 11 superordinate natural language concepts and, using Bayesian model evaluation, contrasted with a standard exemplar model and a central prototype model. The IDM is found to capture typicality better than do the exemplar model and the central tendency prototype model, in terms of both goodness of fit and generalizability. The present findings challenge the dominant view that exemplar representations are most successful and present compelling evidence that superordinate natural language categories can be represented using an abstract summary, in the form of ideal representations. Supplemental appendices for this article can be downloaded from .     

Mass nouns,vagueness and semantic variation

The mass/count distinction attracts a lot of attention among cognitive scientists, possibly because it involves in fundamental ways the relation between language (i.e. grammar), thought (i.e. extralinguistic conceptual systems) and reality (i.e. the physical world). In the present paper, I explore the view that the mass/count distinction is a matter of vagueness. While every noun/concept may in a sense be vague, mass nouns/concepts are vague in a way that systematically impairs their use in counting. This idea has never been systematically pursued, to the best of my knowledge. I make it precise relying on supervaluations (more specifically, ‘data semantics’) to model it. I identify a number of universals pertaining to how the mass/count contrast is encoded in the languages of the world, along with some of the major dimensions along which languages may vary on this score. I argue that the vagueness based model developed here provides a useful perspective on both. The outcome (besides shedding light on semantic variation) seems to suggest that vagueness is not just an interface phenomenon that arises in the interaction of Universal Grammar (UG) with the Conceptual/Intentional System (to adopt Chomsky’s terminology), but it is actually part of the architecture of UG.     

VAGUE PARTS AND VAGUE IDENTITY

We discuss arguments against the thesis that the world itself can be vague. The first section of the paper distinguishes dialectically effective from ineffective arguments against metaphysical vagueness. The second section constructs an argument against metaphysical vagueness that promises to be of the dialectically effective sort: an argument against objects with vague parts. Firstly, cases of vague parthood commit one to cases of vague identity. But we argue that Evans' famous argument against will not on its own enable one to complete the reductio in the present context. We provide a metaphysical premise that would complete the reductio, but note that it seems deniable. We conclude by drawing general morals from our case study.