Theory of Categories: Key Instruments of Human Understanding 1839988134, 9781839988134

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Theory of Categories

Theory of Categories Key Instruments of Human Understanding

Patrick Grim and Nicholas Rescher

Anthem Press An imprint of Wimbledon Publishing Company www.anthempress.com This edition first published in UK and USA 2023 by ANTHEM PRESS 75–76 Blackfriars Road, London SE1 8HA, UK or PO Box 9779, London SW19 7ZG, UK and 244 Madison Ave #116, New York, NY 10016, USA Copyright © Patrick Grim and Nicholas Rescher 2023 The author asserts the moral right to be identified as the author of this work. All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored, or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data A catalog record for this book has been requested. 2023938584 ISBN-13: 978-1-83998-813-4 (Hbk) ISBN-10: 1-83998-813-4 (Hbk) Cover image: Wikipedia This title is also available as an e-book.

For Johan van Benthem Logician and philosopher in a category all his own

CONTENTS Prefaceix 1. The Nature of Categories

1

2. The History of Category Theory

33

3. Empirical Issues in Categorization

59

4. Categories in Science

73

5. Category Mistakes and Philosophical Paradoxes

101

6. Ethical and Social Categories

131

References147 Index161

PREFACE The present book is the third in a series of collaborations, its two predecessors being: •• Beyond Sets: A  Venture in  Collection-Theoretic Revisionism (Frankfurt: Ontos Verlag, 2010). •• Reflexivity: From Paradox to Consciousness (Frankfurt: Ontos Verlag, 2012). In this book, as in the others, a key philosophical concept (collectivity/totality, reflexivity/self-orientation, and in this case categoricity/sortalization) is subject to critical scrutiny and innovative exploration. A great deal of water has flowed under the philosophical bridge since the tract on The Conception of Types in the Light of the Modern Logic (Der Typusbegriff im Lichte der Neuen Logik) by Carl G. Hempel and Paul Oppenheim (Leiden, 1936). This classic venture at  renovating categorization in  the  heyday of logical positivism was predicated on now-outdated perspectives and requires updating in the wake of the wider horizons that subsequent philosophizing has opened up. The present deliberations represent an attempt to accommodate these broadened perspectives. The authors are grateful to Estelle Burris for aid in preparing their text for publication.

Chapter 1 THE NATURE OF CATEGORIES Reality is a manifold of variation; the variety, diversity, and complexity the world affords is virtually endless: individually and collectively, its constituents exhibit an ever-proliferating array of features and characteristics. But the human mind, though also vastly complex, cannot keep in step. Its focus has to be not  in  the  endless detail of real individuality, but  in  its patterns and regularities. The world deals in the variant detail of specifics, while the mind has little choice but to address this via types and in kinds. Reality proliferates, the mind categorizes, in effect endeavoring to deal with an analog Reality by digital means.

Categorization as a Conceptual Necessity To  be intelligibly and usefully managed, information must be systematized. To this  end categories are an indispensable resource. Categories provide  for the conceptual basis of generalization and identification. Categorization is an essential and unavoidable instrumentality for conceptually navigating a world – indeed for being able to conceptualize a world that needs to be navigated. Categories map out the conceptual structure by whose means we map out the domain of our knowledge. They provide for the table of contents and the index by whose means our knowledge is organized and made accessible. They render our otherwise incoherent mass of information cognitively manageable and provide an informatively convenient framework for our cognition about the realms of the actual and the possible. All cognition beyond a dull mental hum demands categorization. All experience beyond passive reception of a perceptual blast demands categorization. All action beyond flailing – and perhaps even that – requires the guidance of categorization. Categories are essential in any cognition, any experience, and any action as we know them, and indeed in any cognition, experience, or action for beings to any degree like us. Successful action in the world demands information about the world, and information as we know it is conceptualized in terms of categories. As creatures that must operate by means of information, we must categorize.

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There is a long philosophical tradition in which the emphasis is on a search for the highest or most general categories. Suppose that someone says, with sublime vagueness, “I am thinking of something.” What would this be? And suppose that we can get answers to just ten questions. What should we ask? We would be well advised to adopt something like the following list: Is it concrete or abstract? Is it unique or does it have many instances? Is it real or imaginary? Is it physical or immaterial? Is it long lasting or short-lived? Is it self-subsistent or man-made? (natural or artificial) Does it age or change over time? Can it be moved in space? It is organic or inorganic? Is it alive or inert? Such questions illustrate very general categories of philosophical taxonomy. If something is going to figure on the agenda of consideration and examination, what are the most basic and fundamental facts about it that we would need and want to know? What are the first – most basic and far-reaching – that we would want to have resolved? Categorization proceeds by distinction and classificatory separation, and the largest and most significant modes of differentiation have hardly changed since classical antiquity. Material/Immaterial Concrete/Abstract Nature-produced/Thought produced Actual (real)/Fictional (imaginary) Organic/Inorganic Human/Nonhuman These fundamental distinctions do and must figure in thought – be it that of the Greeks or ourselves, given the commonality of the human condition and our concern with the basic question: “Where and how does this fit into the framework of our information about things?” Our focus in this book, however, will be not merely on the “highest categories” but on categories and categorization in general. Informational differentiation of the same type is inevitably carried out in the small as well as the large – in the context of particular concerns or particular tasks. It can be carried out with respect to any of a  variety of specific things: clouds, fractions, books,



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buildings, modes of transport, and so forth. In addition to the unrestricted and universal categories of philosophical metaphysics, there will also be the typespecific categories of more narrowly focused ranges of deliberation. Categories provide for the sortalization of objects of consideration. They stand coordinate with the question: What sort of thing is it? Categorization serves multiple cognitive purposes. For the sake of analogy consider the question: What sort of word is it? As it stands this is equivocal and indefinite without further specification. We have to add something on the order of: —in point of —in point of —in point of —in point of

language context. (Is it French or Swahili, etc.) grammar. (Is it a noun or a verb, etc.) pronunciation. (Is it monosyllabic or multisyllabic?) autography. (Does its spelling begin with an A or H, etc.)

The situation with thought-objects is exactly like this. We have to supplement “What  sort of thing is  it?” with specifications along the lines of “in  point of X” when that fill-in is something on the order of creation, composition, function, obligation, and so  forth. There  is  arguably no single unique, omnifunctional, all-predominate thing-classification (see Chapter 5). Without a purpose-specific basis for clarification for categorization, the very idea of categorization becomes a conceptual illusion. In  cognitive as  in  domestic architecture, structure and intent will interact. Our conceptual structure in a particular domain reflects our goals in that domain, though our goals are shaped within  the  structure of our conceptualization as well. Inevitably, as we will emphasize throughout, the world itself plays a role in categorization, since it is in our interaction with the world that  our conceptualization is  tested and revised. As  our body of information expands, cognitive structure changes form to accommodate it. But as Aristotle already saw from the very start it is our questions that give insistence and stability to the origination of knowledge: the answers will vary with the questions that we ask. It  has  long been clear that  the  linguistic management of information demands categories – linguistically grounded sortings of objects, of actions, of the  things true of them and of their relation to each other. If  we are to communicate interpersonally, we must be able to identify the things we are talking about and what we are saying about them – an aspect of categorization crucial for communication. If  we are to address interpersonally acceptable items (“objects”) we must be able to ask and answer questions like “What item is at issue?” “What sort of thing is it?” “How can we get to it (cognitively) from here?” and so forth. Identification requires sortalization. That is, there must be generic categoricity. But  the  story goes far deeper than language,

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and categories go far deeper than linguistic categories. Generic categoricity is also  necessary far beyond the  needs of communication. Even to realize that  something is  in  front of us, and that  it  has  a  particular character, demands an identification of it as “something” of a certain sort (Dummett 1973, Thomasson 2007). To deliberately act on something is to instantiate a conceived category of action on a conceived category of something. Categories are the  tools of identification and generalization, necessary not  only for communication but for perception, action, and cognition quite generally. In  the  end, categorization is  an unavoidable requisite in  the  quest for understanding why things behave as  they do. Successful operation in the world demands information about the world, and information as we know it is written in terms of categories. As creatures that must operate in terms of information, we must categorize. It is important to distinguish the need for some kind of categorization, however, from allegiance to any specific set of categories. Categorization – the systematic organization of information regarding kinds – is an enterprise in which we have an inevitable involvement and a  constant cognitive need. The  fundamentals are unchanging: sortal affiliation, differentiation via distinctions, genus-species subordination, and so forth. The project of categorization itself is a perduring constant. But the specific means of implementing this program and realizing its objectives – the specific kinds in terms of which we categorize – is open to alternatives and can be expected to change. It  is  crucially important to distinguish between process and product, between categories and categorization. The  absolutely crucial observation in this regard was already made by Immanuel Kant (1726–1804). What  Kant sought to show in  his  celebrated “Deduction (i.e., A  priori Validation) of the Categories” is that if there is to be objective knowledge – if existing objects are to be identified and described – then there must be a  framework of categories to accomplish this. He  goes on to argue that this necessary task is contingently achieved by us through the (semantically adjusted) manifold of categories presented by Aristotle. However, the necessity of those particular categories is something he does not maintain. Quite the contrary. For he explicitly tells us that: The feature of our human understanding—namely. that it can integrate information solidly by means of the  categories, and only by certain particular ones, admits of no further accounting. It is a fact of life that is as little capable of deeper explanation on general principles as to why we have just these and no other linguistic resources, and why space and time, and they alone, are our frameworks of perceptual organization. (CPuR. B145–146.)



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The basic point here is clear and correct. Our access to objective experience indispensably demands categorization. But  why we use certain  particular category-schemes rather than others to accomplish this essential task may be a matter of historical fact rather than necessity. Consider an analogy. Physical measurement requites a unit of measure. But what unit we employ to achieve this  – yards or meters, liters or quarts, pounds or stones or kilograms  – is  contingent, a  historical “fact of life” that  lacks any deeper necessitating rationale. In the management of information, the need for some mode of categorization is inescapable, inevitable, and necessary. However, the particular and specific way of realizing this objective is circumstantially negotiable. In the prevailing circumstances, it  is  supplied not  by general principles but  by the  available resources of the time and place. The circumstances and conditions in which we live seem to impose certain  domains of categorization in  us. Whether we are hunters, gatherers, or office clerks, there is a range of life-analyzing issues imposed on us by nature or society with which we must deal by means of categorization. To  say that  categories are essential is  not to say that the particular categories we use, in a particular endeavor or at a particular time, or in a particular culture, are essential. Categorization itself is inevitable in human culture. But the particular form it takes in a particular culture – what categories that culture employs – can be deeply contingent.

How to Misunderstand Categories The familiar spectrum of color appears on the cover (also available at http:// www.pgrim.org/categories/spectrum/). What  is  immediately obvious in that spectrum are clear bands of different color. Green forms a clear central band, moving to bands of yellow and light blue on each side, which in turn move to orange and red at the left extreme and purple and violet on the right. Those are color categories. The fact that you see them as bands marks them as your perceptual color categories. We further examine the empirical work on perceptual and color categories in Chapter 3. As a whole the book presents a revisionist approach to categories. There is a standard philosophical portrayal of categories and categorization that we deem substantially correct in some respects, but markedly incorrect in others. Misunderstanding categories – both our particular categories and the nature of categories in general – is a major source of philosophical problematic and paradox in philosophy of mind, epistemology and metaphysics, philosophy of science, social philosophy, and ethics. Each of the following is a common and tempting conception of categories. Each of these, will argue, involves a  misunderstanding of the  nature of

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categorization and classification as  they actually function in  our cognition and information management: •• Categories are set-like entities. •• Categories are defined by necessary and sufficient conditions. •• Relations of similarity are to be defined in terms of category co-membership. •• We first recognize individuals and then group them into categories. •• Categories are to be judged in terms of whether they cut nature at its joints. •• Categorical classifications form tree-like structures of exclusive and exhaustive sub- categories. The  book seeks to set the  record straight in  these matters, using the philosophical history of categories as  a  background but  relying on contemporary empirical work as  well.1 A  consideration of the  essential cognitive function of categories makes it inescapably clear that they must be understood pragmatically. So  understood, categories will fit  some but  not all aspects of the  traditional picture. Pragmatically understood, moreover, categories will inevitably be topic-, context-, and culture-bound, geared in their nature to the informative tasks in hand. Categories in general can be as variant as our purposes and perspectives. It also follows that categories will be structured in terms of specific cases, tailored to the purposes at hand.

The Foundational Concepts We take the concept of categories to be central in understanding management of information and cognitive navigation of a world. But that concept does not come alone: it comes as part of a cluster: concepts of items or individual things, features or properties, kinds and natural kinds, similarity, classifications, taxonomies, and of course the  concept of concepts themselves. While concentrating on the nature of categories, we have inescapably helped ourselves to other concepts in the cluster. It is time to put the basic concepts in order. An item is  any particular object of discussion or consideration. It  need not be a physical thing, either natural or artificial (the Rock of Gibraltar, the Sphinx) but can be a fiction (Sherlock Holmes) or a quantity (four), a shape (circularity), or a  mood (anger): in  sum, it  could be anything. But  it  must have identity that links it to certain similar things and differentiates it from everything else. Categories are the factors that must be addressed in providing the effective identification of items of possible consideration. Features or properties are descriptive characteristics of items. The concept of similarity is a fellow traveler: items which share a feature will be similar to at least that extent, and items with an identifiable similarity will have an



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identifiable feature in common. In line with our discussion of vagueness and similarity above, however, it should be noted that “descriptive features” need not  be linguistically bound by the  terms of literal description, and neither “identifiable similarities” nor  “identifiable features” need be capable of explicit  description. We often see the  similarities between things, and by that token see that they share features in common, without necessarily being able to eloquently specify the similarity or list features in common. Categories are kinds. They are groupings or collectivities in terms of a shared character, feature, property, or “family resemblance”: groupings under or within a determining characterization. Categories are not sets: not merely sets, in carrying their kind-membership criterion with them, not as profligate as sets, tied to the limited salient groupings relevant to a particular endeavor or context, and unlike sets in often having vague and indeterminate boundaries.2 We take the defining character of a category to be its determining concept, which can also be thought of as a kind membership condition for a particular category. As  noted below, Kindmembership conditions (KMCs) may be either totalistic – demanding for category membership that  an item satisfy all features of a  feature cluster – or qualitative, demanding merely that it display some, most, or at  least n features of that  cluster. But  here as  above it is important not  to construe membership conditions too linguistically. In many cases we can indeed make explicit  an underlying condition of membership: being unmarried as a condition for bachelorhood, for example. But conditions of membership may be inexplicit, and perhaps even incapable of explicit  formulation. Any of us would be hard-pressed to outline the determining concept we use for our everyday category of green things, for example, and any such formulation would be well beyond the capability of three-year-olds fully capable of recognizing membership in  that  category. The role of inexplicit similarity to prototypes or exemplars is quite properly emphasized in  the  empirical literature on categorization, further outlined in Chapter 3 (Rosch 1975, 1983, Lakoff 1990, Hampton 2006, 2017, Rouder and Ratcliff 2006, Reisberg 2015). That  vague and inexplicit  but  crucial similarity must be included as  a  form of category-determining concept or kind membership condition as well. Although concepts are kind-membership-determining, they are not themselves either categories or the  groupings they define. Concepts apply to individual thing in  virtue of determining the  categories to which those things belong, but concepts – unlike both categories and groupings – do not have members. Although the  tradition is  a  rough one, our concepts echo both Mill’s connotation and Frege’s (Mill 1843, Frege 1892). Using a term of philosophical art, concepts are the intensional tool of categorization.3 Here as with items and categories, it is important to note that the range of concepts

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is not merely actuality but possibility as well: they may deal with truth and falsity, with fact and fiction, with reality and irreality alike. Beyond individual categories, a larger architecture of information structures categories within a classification, often instantiated as a tree network descending at  each level from genus to species. Each partition within a classification proceeds by means of a principle of division (principium divisionis), generally either descriptive or functional. The classical ideal of perfect classification is one in which each division is into a set of exhaustive and mutually exclusive sub-categories. That ideal demands that the Xs be divided into a spectrum of subcategories type-i X is. If all goes well, the X is are exclusive (nothing falls into two distinct X is) and exhaustive (every X falls in some X i). A division between true and false is the familiar instance here, with excluded middle and tertium non datur as ruling principles. But just as the categories we actually use need not have sharp edges and carry necessary and sufficient kind membership conditions, so  the  classifications we use need not  always have subdivisions that  are both exhaustive and mutually exclusive. Our purposes are often perfectly well and perhaps even best satisfied by various modes of imperfect classification, in  which exclusiveness and exhaustiveness sometimes fail, with two modes of fuzziness: the  internal (as  between one sub-category and another) and the  external (as between inclusion in one sub-category and “the rest”). Within a category of professional athletes, it is natural to envisage subcategories of professional football players and professional baseball players. But we are quite ready to concede that the division is nonexclusive: Deion Sanders was both, playing in both the Super Bowl and the World Series. We work with a category of criminal enterprises, but  are well-advised not  to assume exhaustiveness for our sub-categories: there will, unfortunately, be types of digital and internet crime that we cannot yet anticipate. A classification is defined by its categories in a specific network structure. The informational function of categories depends crucially on the classification of which they form a  part, and that  classification is  characteristically structured in terms of levels of categories and sub-categories. Alongside  the  substantive matters at  issue with categories, there is also the discourse machinery by which these matters are given expression. These provide the  machinery by which categorical classifications are articulated. The medieval schoolmen characterized these as syncategorematic because they function uniformly across the entire range of categorical deliberation. They include  the  logical and semantic resources that  operate everywhere alike, including operations like negation, induction, generalization, and the like by which various sorts of things – however categorically distinguished – can be correlated, compared, or otherwise connected.



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A  taxonomy is  defined by the  sequential ordering of levels of division within a classification. However approached, categories are bound up with taxonomy. They  provide  the  unity that  holds the  classifying boxes together and coordinates them with what  is  taxonomically subordinate and superordinate. Two items or kinds may be categorically separated or co-characterized depending on the  function that  guides the  categorization at issue. Tomatoes are grouped with fruits in biological but with vegetables in culinary classification. A taxonomy is a sequential division of items into nomically determined kinds, developed functionally under the  guidance of a guiding purpose. The relation between the elements that come into operation here – between items, features, kinds, categories, concepts, and kind membership conditions, classifications, and taxonomies is set out in the glossary of Table 1.1.

Table 1.1  Glossary of basic ideas. •• Items are individual particulars. They have features of varying sorts. •• Features are descriptive characteristics of items. To qualify as descriptive they must be determinable in principle, though perhaps not in practice. Some descriptive features must always be employed in identifying items, though it is not always possible to make relevant features linguistically explicit. •• Categories are kinds, groups (collectivities) of items whose membership is determined by kind membership conditions (KMCs). Every category demands the manifold of failure into the applicable and the inapplicable. (Example: attributing colors to numbers or shapes to odors.) To attribute to an item of category a feature not applicable to its members is to commit a “category mistake.” •• Kind Membership Conditions (KMCs) can be either total or partial. A qualitative KMC takes the format: x ∈ K iff Fx for all of the F ∈ F (comparative) Granted there is room for variation here. A qualitative KMC can take some such format as: x ∈ K iff Fx for some of the F ∈ F (disjunctive) x ∈ K iff Fx for most of the F ∈ F or x ∈ K iff Fx for at least n of the F ∈ F (numerical) And like the categories they define, kind membership conditions can be vague and incapable of being made explicit. Analogy to paradigms or exemplars can qualify as KMCs, determining categories with corresponding vagueness. •• Natural Kinds, though paradigmatically thought of in terms of the categories within a scientific discipline, are best understood pragmatically as those categories salient for a particular endeavor or in a particular purposive or cultural context. •• A classification is a thorough division of items into relevant natural kinds. •• A taxonomy is a hierarchical ordering of the classification of all (issue-relevant) items.

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The  principles used in  the  divisions of a  taxonomy and classification, and in the categorization fundamental to it, maybe of many different forms. Categories then present the kinds of considerations used to distinguish among the types of things. Three familiar ways in which a body of information can be organized are: •• By inferential dependence and derivation (Paradigm: geometry). •• By development and normative sequence (Paradigm: history). •• By subject-order classification and thematic topicality (Paradigm: biology and medicine). Categorization, classification, and taxonomy can employ division by such functions as: •• Mode of origination (growth, prediction). •• Modes of activation (manual, electrical, mechanical). •• Use or function. •• Age or place of origin. •• Descriptive features. Classificatory theses can be true either contingently (“ ‘Yellow’ is a colordescribing term”) or by necessity (“Yellow is  a  color”). It  is  a  contingent fact that  Thomas  Jefferson is  to be classed among the  “Presidents of the U.S.A.” But it is necessary that “Three is to be classed among the odd integers.” Put in  Kantian terminology, categorizations can have either a purely conceptual (a priori) base or they can rest on factual (a posteriori) arrangements. The  main  overall point is  that  categorization provides the  principles (the  procedural guides) for classification (see Table  1.2). Categorization calls for dissemination based on distinctions of pragmatically-sensitive “natural kinds.”

Table 1.2  Illustration of categorization. Classification

Principle of Order (Categorization)

The Aristotelian categories The Linnean system of biology

Objects by questions about them Flora & fauna by mode of production and development Books by theme and subject Expressions by topic or keyword Words by alphabetic spelling

The Dewey decimal system Roget’s Thesaurus A dictionary



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An explicit  categorical taxonomy of a  given domain, to the  extent that is possible, provides a map of its cognitive structure. And the steps by which it accomplishes this is roughly as follows: Step 1: Conceptualization: specifying the  range of objects to be the  subject of considerationand investigation. Explanation of what these objects are to be and how they are to be identified. Step 2: Sortalization: specifying how the objects are to be divided into kinds or types and how such grouping and separation is to be set up. This includes specifying the  definitive features (principles of division) by which sortalization is effected. Step 3: Classificatory Subordination: specifying the definitive (necessary) connection among the different sorts obtaining by virtue of the principles of separation involved. Step 4: Operational Interconnection: determining the  empirical and contingent regularities that interrelate the various sorts of items. Step 5: Application: implementing this  knowledge with respect to the  various (and variable) purposes and objectives of the taxonomy at issue. When you know the categories within a discussion or investigation, you have a general overview of its main issues and problems. Categories indicate both the substantive topics and the conceptual organization of the inquiry, and provide an overview of the issues (problems/questions) even if not as yet the answers. Classification is a prime guide to cognitive importance. A clear indication of this is the list of key ideas associated with the items discussed in reference works. Thus an encyclopedia might augment its article on Buenos Aires with a keywords list including: Argentine metropolis, national capital, major cities in  South America. The  indication of those particular categories indicates that their pertinence to the item at issue represents its most important, most essential, and most informative features.

The Contextual Functionality of Categories As an epistemic resource, categories reflect both our interests (our questions) and relevant beliefs (our answers). Consider an example: people. Our main questions about people are presented in what might be called passport issues: sex, age, appearance, nationalism, profession, and character. Whenever someone is under consideration these are the primary things we want to know about them, followed by such secondary matters as talents, education, cultural relationships, political allegiances. And all these issues have an interest for us, because it is on this basis that we ourselves have to come to terms in our dealings with experience.

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There  is  no question but  that  different cultures and culture-eras  think of the world’s arrangement differently – that  both persons and natural phenomena are differently conceptualized in Chinese, Aztec, Ancient Greek, or contemporary Western civilization. Different peoples at  different times portrays different kinds of things in their thinking about reality and about themselves. For  ancient Romans, the  most crucial categorical distinction among people was whether they were citizens or not. For medievals whether they were fellow believers or not. For those of the old regime whether they were aristocrats or not. For  German Nazis  whether they were Aryans. All depending on time and place, culture, religion, circumstance, or some comparable factor achieved primacy in the categorizations of fellow humans. In matters of legal entitlement and social standing categorization has always been a  paramount factor – and in  these matters naturally enough one of human contrivance and convention. Consider the  illustration of men’s clothing. It  is  clearly subject to a  vast array of classification. •• By  parts of the  body: socks, shoes, underwear, trousers, robes (togas), overcoats, and so forth. •• By type of material: silk, cotton, fur and so forth, •• By age of intended wearer: children, youth, adults, geriatrics, and so forth. •• By size of the individual wearer: large, medium, small, and so forth. •• By season of wear: winter, summer, and so forth. •• By occasion for wearing: leisure, work uniforms, evening dress, operating theater. The range of alternatives indicates that categorization pivots on the pragmatic rationale of classifying. Over and above the  familiar sort of descriptive categories there are also  evaluative categories, such, for example, as: superior, good, adequate, average, poor. These  are often geared to descriptive categories (as  when a student’s work is evaluated as “average/adequate” when the percentage of his correct answers lie in the 55–65 range), but valuation can also proceed on an independently nondescriptive footing. Imagine a  storehouse of information consisting of slips of paper, each marked with a  true statement but  all scrunched together helterskelter. The  information is  all there, but  in  a  form that  does no one any good. There  is  an important lesson here. Truth and fact is  not enough. To  provide  for knowledge, information must be organized and categorization produces the  guiding principles organizing information into knowledge.



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Thus when books are at issue we can sort them (as medieval libraries did) by (1) size and (2) date of acquisition. These features (size and length of possessing) were the book-classifying categories they employed. These categories served an important function in identifying the placement of books in cases, but were not of much use for anything else. But two important considerations emerge even here: (1) categorization is  a  functional process, and (2) categorization changes with the change of interests: it is not set once and for all, definitive and permanent. In medicine, one can taxonomize (classify) ailments by •• Parts of the body affected (headache, toothache, etc.) •• Nature of the symptoms (pain, elevated temperature, malediction dizziness, etc.) •• Nature of the diagnostics (blood tests, x-rays, etc.) •• Nature of the  mode  of treatment (diet and regimen, medicaments and drugs, surgery, etc.) •• Nature of the  casual mechanisms: etiology (injury, infection, systemic imbalances, etc.) Such a taxonomy involves: (1) a unifying theme (here medical malefaction), (2) a ruling principle of categorization (affectively, systemically, diagnostically), and (3) a subordinate cascade of classifications at different levels of taxonomic subordination. The categorical fundamentals are theoretically fixed by the matter at issue. Modern Western medicine, the  practices of traditional Chinese medicine, and African Shamanism all address medical issue on this  categorical basis – conceptually fixed by the fundamentalization at issue. But of course, the taxonomic implementation of the matter is very different, and indeed may well change drastically over time. Like all else, categories can be categorized. This  meta-categorization proceeds principally by role function, these being the distinctive characteristics of categories. There are •• Identifying categories. •• Descriptive categories. •• Explanatory categories. •• Use or function categories. Given our various purposes, we will have categorizations of buildings, machines, books, and so forth. Within any topic area, and even within the

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same classification, we will have any of various forms of categories. With the simple of example of buildings, there will be: Descriptive categories (say of building types in architecture). Aesthetic categories (say of stylistic types in architecture). Purposive categories (in architecture: hotels, private residences, filling stations). Productive/Creative categories (in architecture: bricks, wood, concrete). Locational categories (city-central, remote). Feature categories by special feature (fireproof, earthquake proof, movable). Pricing categories by cost (market value, reproduction cost). The subject to which category theorists have given the most attention over the centuries is the question of what C. S. Peirce called the universal categories, which seek to taxonomize items at the very highest range – the objects of thought and coordination at large. Figure 1.1 illustrates such a totalistic categorization. Philosophers throughout the ages have developed various themes. For the most part, however, the top level of our categorizing taxonomies are substantially more limited and specific than considerability-at-large, with a more definite sphere of application in view. Possible Objects of Consideration Abstracta Types (Kinds of things)

Concreta Hypothetical

Relations among things Laws of operation

Actual Physical Subjects Physical State Processes

Themes (topics) Numbers, quantities, shapes

Figure 1.1  A sampling of philosophical categories.

The crucial role of categories becomes clear in the erotetic dimension: in their function in questions and answers (see Table 1.3). (1) They define the range of questions we pose about the objects that concern us. (2) Our categorizations correspond to the  range of answers we expect. In  the  process, (3) our categories are coordinate with classification. Thus with respect to (say) people, the main categories are: age, sex, activity and/or culture setting, profession and/ or livelihood mode, and so forth. The traditional game of “twenty questions,” or “Animal, Vegetable, Mineral?” proceeds precisely by attempting to identify



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Table 1.3  A register of categorical questions. •• What is the range of objects or items that is to be at issue? •• How are these objects to be identified? How are they to be differentiated and specified? •• How is the overall range of objects divided into different kinds? What considerations are to be at issue here? •• How do these considerations work out? What are the proposed kinds of objects at issue? •• How are the issues/kinds of items related? How do they aid and contrast among themselves?

something by progressively honing in on its categorization within an appropriate classification. In Dickens’ Christmas Carol, the game appears as “Yes and No,” The object to be identified is Scrooge, and “some objected that the reply to ‘Is it a bear?’ ought to have been ‘Yes.’ ” (Dickens 1843). Our knowledge and information about things is  almost unavoidably articulated semantically (Carlson 1977, 1982, Krifka  2004). And the machinery of our symbolic resources shapes the thought frame for representing that information. Whether we proceed in Indo-European or in Chinese or in the mathematical formation will condition our modus operandi in paving the way in which we think about things (Chierchia 1998). No doubt each such thought-mode has its own manifold of categories. Categorization is a functional and purposive procedure, conducted with a view to facilitating the considerations or use of objects of concern, either their explanation or their exploration. Both our informative means and our purposive ends govern the categories whose terms transact our dealings with objects. And both change in the nature of cognitive progression over time. And as  information becomes more fully articulated the  category manifold will expand and change, dictated in part by how well our purposes play out in the world given the categories we employ. We no longer think about meteorological phenomenon in  the  categorical framework that was employed in ancient Greece. The boundaries of classification grow or shrink in  line with the  prevailing state of knowledge. Astrology no longer counts among the sciences; plastic is now among our building materials.4

Categories and Kinds Categories relevant to a particular topic, project, or set of questions are the salient kinds appropriate to that context. They are groupings or collectivities, but  groupings or collectivities of items with a  particular shared character, feature, or property: groupings under a determining characterization. But  as  emphasized, categories are emphatically not sets, in  a  number of regards. Unlike sets or classes, it is important to note that although categories have

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members they must also exhibit a thematic unity, disallowing merely extensioned grouping via  their membership. Thus “North American marsupials” and “opossums” are different categories, even though their members may be the same. Sets are purely extensional, defined by their members. Categories are richer, carrying their characterizations with them. Categories, keyed to particular endeavors, are also  pragmatically salient and hence far less profligate than sets: “American marsupials and coins in my pocket on a Tuesday” constitutes a perfectly respectable set, but in the context of biology does not constitute an acceptable scientific category. A related distinction has  been marked between “sparse” and “abundant” properties, in  which the  natural properties are contrasted with gerrymandered and gruesome sortalizations (Lewis  1986, Armstrong 1978). The  “abundant” properties are those satisfied by any arbitrary set; the  “sparse” are those characteristic of genuine kinds. What has not generally been noticed is that it is categories as commonly conceived, quite distinct from sets, that fill this role. It has also not generally been emphasized that categories are pragmatically determined, variable with distinct human enterprises. Given the unlimited and unrestricted horizon of our potential interests, perspectives, and purposes, the range of categorization and classification seems unlimited. But it is crucial to understand that  within  a  specific context, with an eye to a  specific interest, perspective, or purpose, the range of relevant categories or kinds is typically also limited in a very important way. It  is  tempting to identify genuine categories as  those corresponding to “natural kinds.”5 If the “natural” of “natural kinds” is limited to the scientific realm of chemistry, biology, and the like, this is a serious oversimplification. But identification of genuine categories with natural kinds is  perfectly appropriate as  long as  “natural” is  understood far more generally: as “natural” within a given human project in coordination with its associated reality. Natural kinds in  this  pragmatic sense need not  be authoritatively dictated by science alone. The crux of the matter is rather nomicity – that is, lawful comportment. In  this  sense “natural kindhood” pivots on the  rules, principles, and lawful comportment of the sorts of things at issue relative to the purposive context at issue, however non-scientific or “unnatural” those fundamentals may be in other respects.6 Written exclusively in terms of scientific purposes, the picture that emerges is a vision of the world cut at its natural joints: •• Some sortalizings represent natural kinds whereas others do not; the content of theory becomes determinative here. •• Objects of considerations can always/generally be affiliated to others as being of the same natural kind.



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•• Natural kinds generally divide  into subkinds which continue to exfoliate further divisions in turn. •• Natural kinds are formed by networks of lawful connection. Natural kinds are always bound up with some others in lawful relationships and •• The  nature of such relationships are themselves classifiable into natural kinds (causality, descriptivity, etc.) •• Authentic classification requires differentiation into natural kinds. •• Not all kinds are natural kinds. Precious stones or edible vegetables do not form a natural kind. •• Authentic categorization calls for authentic classifications. A harmonious fabric of lawful interrelating among kinds of things is  required by appropriate classification and categorization. The only thing wrong with this familiar picture is that it is limited to one human endeavor – the scientific – and thus to one concept of “natural kinds.” If broadened to the  various cultures, purposes, and contexts open to us, it holds quite generally. With that breadth of possible human involvement in mind, for example, it is clear that in some contexts it will be precisely the  distinction between precious and semi-precious stones that  is  the salient  category in  play, or a  distinction between edible  and  poisonous vegetation. It  deserves note that  we generally think of kinds within  a  context, and that in  every context these are a  salient few. Within  different projects, different categories will play the role of “natural kinds,” though the  project may be something very different from the  scientific attempt to understand the  natural world and the  kinds may be very far from “natural.” Within  anthropology, “natural kinds” may be cultural types, forms of ritual, or kinship structures. When the topic is farm machinery, the  relevant categories may be those of soil preparation (plowing and fertilizing), planting, and harvesting, with machines sorted into “natural kinds” that are clearly artificial in the sense of man-made. When the project is aesthetic design, “natural kinds” may include colors, shapes, and layout categories. It  is  crucial in  the  cognitive function of categories, within  not only an attempt to understand the natural world but in every human project, that they are restricted to salient and significant kinds rather than including all and every arbitrary grouping. But  that  does not  tell us which the  genuine categories are relative to a  specific endeavor; a  glimpse at  the  history of science makes it clear that we have often been wrong about what the natural kinds were (phlogiston, N-rays, the  brontosaurus) and how to cut nature at the joints (circular planetary orbits, crystal celestial spheres). If scientific

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natural kinds were obvious analytically or on their face, science would neither have nor  need a  history. If  we couldn’t be wrong about which the genuine categories were germane to a particular human endeavor, we wouldn’t be frustrated in  our various endeavors in  many of the  ways we often are. In science nous (theory) tries to model physis (nature) by means of conceptual relationships representing the process of events. Nature’s structure is inherent in the modus operandi of its physical processes. The cognitive structure of our theorizing science seeks to reflect this structure within the informational processing structure of its operations. But  there are crucial differences between processes and the conceptualized laws that seek to represent them. Natural processes either occur or they don’t; they can speed up or slow down: they have placement in space and time. Laws and concepts can do none of these things. Instead, they can be accurate or inaccurate, detailed or vague. Science is thus not a “mirror of nature;” it functions with a complexity far beyond any simple isomorphism between the  reality it  addresses and the conceptual instrument it employs. Its success is not one of descriptive necessity but one of the efficacy of its guidance of our actions within nature’s orbit. And the ultimate standard appraisal of our categories is not looking to see whether they line up with nature’s “joints” but whether they realize pragmatic success in matters of predication and action.

Generalization and Individuation Classification groups categories in  terms of contrastive distinctions. In the historical attempt at classification “from the top,” the starting point would have to be something of maximal content that does not itself issue from a  distinction among yet larger coverage. This  initiating maximum would have to be something on the order of being an object of possible consideration. At this stage there are two standard steps into the next (second) level of philosophical categorization, namely: •• Thing/substance vs. process/occurrence. •• Natural vs. artificial. •• Concrete vs. abstract. From here on the  elaborations of classificatory distinctions has  historically proceeded in  line within  the  setting of the  categorical questions at  issue in the inventory of Aristotelian categories as summarized in Table 1.4. It is clear, however, that they are subject to certain limitations because such general (metaphysical) categories can serve to conceptual wild-goose chases,



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Table 1.4  Categorical Questions (After Aristotle). Question •• What type/kind? •• Where? •• When? —origination? —duration? •• Amount [extent, intensity] •• Composition? —how constituted? —of what constituted? •• Order? [harmony, coherence, contextuality] •• Efficient causality —antecedence? —consequence? •• Final causality? [Fundamentality]

Aristotelian Category Quantity Place – Position or Placement Time

Quantity (Degree) State

Relation Action & Passion [Fundamentality]

Purpose

such as asking for the placement or the direction of numbers or the purpose of triangles. For  example, items that  are abstract and natural (e.g., laws of nature, number) do not admit of temporal characterization (when?, duration?) nor are they matters of composition. •• Only processes (rainstorms, earthquakes) admit  of intensity and degree, not substances (which either exist or don’t—all or nothing). •• Only artifacts admit of purposiveness. •• Abstractions (e.g., numbers) do not exert causality. Aristotle saw categories as the essential initiators of inferential (syllogistic) logic. For the collection of things into categorical grips is the indispensable requisite for relating such groups by the quantitative indices of “all,” “some,” and “more.” They provide for the outcome constitutes “categorial perspectives” whose inferential interrelationships constitute the substance of logic. We follow William James in  emphasizing categories as  an essentially pragmatic instrumentality of cognition matters of both individuation and generalization. Categories are the tools of identification and generalization, and not only speech but perception, action, and cognition require generalization. The act of generalization itself is an act of categorization: a grouping of things as of a general kind. By the same token, any categorization is a generalization: a grouping of these things in general.

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Inductive reasoning is  essentially projection of a  generalization beyond cases yet considered, quite typically – and crucially – into future cases yet to be observed (see Chapter  4). That  inductive reasoning is  predicated on categorization. An inductive thesis paradigmatically takes the form: In circumstances/conditions of type T things of kind K always behave Xly. (“Water boils at 100° C,” “Elms shed their leaves in springtime.”) To obtain any such generalizations we must categorize kinds, circumstances, and behavioral modes. It is also true, though perhaps less obvious, that categorization is a crucial tool in individuation and identification. It allows not merely for the grouping of things, but for the identification of the individual things that we group and their re-identification over time. For  an item to be identified is  for it  quite generally to be identified as  a  specific thing, distinct from other things, and the  conceptual borders of that  individuation are an implementation of categories (Dummett 1973, Wright 1983, Hale 1987, Lowe 1989, 2009, Thomasson 2007, 2019). The difficulty of attempting to imagine individuating a  re-identifiable “something” without a  “what” – a  purely demonstrative “this” for a green patch in my private visual field, perhaps – simply serves to make clear how central categorization is to individuation and re-identification over time. We use the machinery of categorization in terms of •• Kinds (of things). •• Properties (of things). •• Locations (of things), for example, in order to specify a particular – the one and only item that. •• is of kind K •• is of kind K and has properties P1, P2, … Pn. •• is of kind K and is positioned at location L. •• is of kind K, positioned at L, and possessed of properties P1, … , Pn. •• has properties P1, … , Pn. •• is positioned at L and has properties P2, …, Pn. On  this  outline, categories do not  merely group particulars somehow identified without them. Nor  do they allow us to identify and individuate particulars in terms of groupings somehow conceived without members. In classical terms, we no more form universals as groupings of particulars that we



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have without classification than we form particulars in terms of universals we conceive without members. Particulars and universals are conceived together, as dual aspects of a single process of categorization. It  is  a  tempting mistake to view the  role of categories in  generalization as  a  second-order operation on a  predetermined first-order level of atomic individuals. Steven Harnad, a  leading cognitive science researcher on categories, outlines such a view in terms of Jorge Luis Borges’s story of Ireneo Funes (Harnad 2005). Borges’ Funes has a complete eidetic memory of every experienced detail: He knew the forms of the clouds in the southern sky on the morning of April  30, 1882, and he  could compare them in  his  memory with the veins in the marbled binding of a book he had seen only once, or with the feathers of spray lifted by an oar on the Rio Negro on the eve of the Battle of Quebracho. In  the  seventeenth century, Locke postulated (and condemned) an impossible language in which every individual thing – every stone, every bird, every branch – would have its own name; Funes once contemplated a similar language, but discarded the idea as too general, too ambiguous. The truth was, Funes, remembered not only every leaf of every tree in every patch of forest, but every time he had perceived or imagined that leaf. Funes, we must not  forget, was  virtually incapable of general, platonic ideas. Not only was it difficult for him to see that the generic symbol “dog” took in  all the  dissimilar individuals of all shapes and sizes, it irritated him that the “dog” of three-fourteen in the afternoon, seen in  profile, should be indicated by the  same noun as  the  dog of three-fifteen, seen frontally. (Borges 1944, 1998). What the story of Funes shows, Harnad concludes, is that “living in the world requires the  capacity to forget or at  least ignore what  makes every instant infinitely unique, and hence incapable of exactly recurring.” (Harnad 2005). Categorization is the essential tool of the required generalization. To  the  extent that  Funes’ experience and memory approaches that  of a world of atomic particulars, however, both Funes’ and Borges’ description of that experience will be impossible. At that extreme, neither Funes nor Borges could appropriately characterize his experience as that of each leaf on a tree, nor of a particular dog from a particular angle. “Leaf,” “dog”, and “angle” are all general categories. Nor could Borges characterize Locke’s impossible language as one in which each individual has a proper name. The process of individuation itself, crucial for recognizing two things as distinct or two things rather than one, employs a mechanism to which categories are essential.

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James famously says “[t]he  baby, assailed by eyes, ears, nose, skin  and entrails at once, feels it  all as  one great  blooming, buzzing confusion…” ( James 1890, 488). That passage is quite widely taken to portray the baby’s base experience  – and ours – as  an undifferentiated continuum. James seems sympathetic to that portrayal, speaking repeatedly of experience as a continuum or plenum. Experience is not grounded in an atomic level of sensedata, from which all else arises by composition. “The  ‘simple impression’ of Hume, the  ‘simple idea’ of Locke are both abstractions, never realized in experience.” Experience is  not pointillistic but  smoothly blended, an undiscriminating state. “Such anaesthetics as chloroform, nitrous oxide, etc., sometimes bring about transient lapses … in which numerical discrimination especially seems gone; for one sees light and hears sound, but whether one or many lights and sounds is quite impossible to tell (487).” For  James, both the  general categorization of universals and the individuation of particulars are our imposition on experience, guided by our position and our purposes. “As  I  sit  here, I  think objects, and I  make inferences, which the future is sure to analyze and articulate and riddle with discriminations, showing me many things wherever I now notice one … We all cease analyzing the world at some point, and notice no more differences. The 1st units with which we stop are our objective elements of being. Those of a dog are different from those of Humbolt; those of a practical man from those of a metaphysician (489).” The  ground floor of experience as  an undifferentiated continuum is as  much an abstraction as  Hume’s or Locke’s. We do not  have access to experience in  the  raw, without categories, arguably not  even under the influence of James’ nitrous oxide. We attempt to explain the experience we do have – a developed experience of objects individuated and grouped in terms of kinds  – as  the  result of an active mind meeting an independent world. The  evidence that  what  that  active mind works with is  given as  neither a Hume-Lockean pointillism on the one hand nor a Jamesian continuum on the  other is  simply that  active minds differently placed and with different purposes – a dog or a Humbolt – end up with a world of experience in which different objects and different kinds predominate.

Vagueness and Similarity We have emphasized that categories are not sets. Categories are more finely tuned than sets, carrying a matter of communicative functionality rather than extension. Categories are also more selective than sets, including not every willy-nilly sortalization but genuine kinds germane in the context of a specific pragmatic endeavor.



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A  further crucial distinction is  that  unlike sets, categories and kinds need not  have sharp edges (van Deemter 2010, Hampton 2011). Classical, extensionally defined, obey a law of excluded middle in terms of membership: everything is either a member of a given set S or is not. Membership criteria for sets, therefore, come naturally in  the  form of necessary and sufficient conditions that fully apply or fully fail to apply. Kind membership conditions for categories, in  contrast – their defining concepts – may themselves be vague. The categories they determine will correspondingly be indeterminate. Green and red may be perfectly appropriate categories in certain contexts, even though we are hard-pressed to draw the  precise wavelength at  which either shades to yellow. By the same token, a classification of colors may fail the classical ideal of sharply exclusive and exhaustive sub-categories, but be perfectly useful despite that. We must, in the end, recognize that our categories need reflect no greater precision than is demanded by the purposive contexts they serve. The failure to do so involves a basic misunderstanding of the nature of our own categorization. A rigid classificatory taxonomy is one whose compartments are precise while a loose taxonomy is one whose components fail in this respect and so admit of vague or indefinite boundaries where the placement of items can be indeterminate and uncertain. But ordinary language is imprecise: in  everyday life affairs, most taxonomies are loose in  this  way. Thus consider •• The domain of men’s jacket sizes into: small/medium/large. •• The division of human ages into: child/adolescent/adult. •• The division of watercourses into: rivulets/streams/rivers. •• The division of hillocks/hills/mountains. •• The division of habitation-complexes into villages/towns/cities. It  could be said that  much of the  world as  we view and describe it  is  one of indefinite boundaries and imprecise deliberations. After all, it  makes no sense to endeavor to specify the  age of a  person to six decimal places. We live in  a  world without sharp conceptual boundaries. Our descriptive characterizations and classifications often admit  only an indefinite and indistinct specification with indefinite and imprecise boundaries. (Some answers are clearly IN and other clearly OUT, but  some will always be problematic and discussable.) Our descriptive resources generally resist absolute precision. The world as we view and describe it is pervaded by regions of vague, indefinite, imprecise character. When we ignore this fact and try to improve precision where it is unrealized we fall into misunderstandings and paradox (see Chapter 5).

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The essential vagueness of many of our categories is clear from their tie to similarity, properly understood in terms of its own vagueness. Here again, we have to fight against a tempting picture. Were categories simply sets, similarity between x and y might be analyzable in terms of whether x and y were both members of the same set. Such an approach is broached in the work of both Goodman and Quine (Quine 1969, Goodman 1973). But given the ubiquity of gerrymandered sets, it’s immediately evident that any two things will be similar on such an account, in as many infinite ways as there are sets to which they both can be assigned. Imprecise classification is  not merely something we tolerate – it  is  often precisely what  we need. We employ vague categories for good pragmatic reasons – in order to facilitate the efficient management and communicative transmission of information. In many or most situations a demand for precise counts and exact determinations would seriously hinder rather than help our information processing, both individually and socially. We have to choose between the risk of error and serious complications in communication and information management. The advantage we seek in ordinary life is almost always in  limiting the  latter. The  issue is  ultimately one of a  pragmatic trade-off. Just as there are salient categories within a given endeavor and context, there are salient similarities. Indeed the two travel together: the members of a  category or a  kind can indeed be expected to be similar, precisely because they will both satisfy the classificatory conditions of that category or kind. Categories correspond to context-sensitive natural kinds. And such a kind is not just a set or group of things. On a classical picture, sameness of kind-hood requires feature-commonality: the  items of a  given kind must be affiliated to one another by means of a  shared feature or family of features, a  detectable transit  of similarity between them. If  “detectable features” are construed broadly enough, to include  detectable similarities to a  characteristic prototype or set of standard exemplars of a  category, that picture is both accurate and inevitable (see Chapter 3).7 But it should be remembered that “features” in this sense need not be explicit or capable of linguistic expression. As  regards similarity, we have to distinguish moreover between similarities appropriate to different pragmatic contexts: phenomenal (observational similarity), nomic similarity, and functional similarity. Items that are very disparate on superficial observation can be functionally equivalent. Similarity in  its various forms is  crucial to categorization. And yet it is vastly complex and variable in  respect. No one is  tempted to think that the relation of similarity is other than context-sensitive, nor that it is other than vague. A  question of similarity tout court makes little sense without



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the  “in  what  respects” dictated by project and context. Similarity and categorization travel together as  conceptual partners; it  should not  be surprising that they often reflect the same essentially vague core.

Categorical Mysteries A category must have a  criterion of inclusion providing a  standard of membership. Such a membership criterion may or may not be effective in the sense of determining for any given item whether or not  it  meets the inclusion condition at issue. Thus “divisible by two” is an effective criterion for qualifying as “even integer.” But “cause of cancer” is not an effective criterion for “carcinogen”; it provides no way of telling membership. A mystery category is one whose membership criterion is such as to preclude by its very nature any prospect of ever identifying its members: it  characterizes those members but admits no determination in single cases. Examples include •• Being an ever-ignored grain of sand. •• Being a totally unknown Roman soldier. •• Being a completely forgotten Greek sculptor. All these vagrant properties – that  is, of no known address – are applicable to real individuals and define nonemply groups. But  there is  nothing – no specifiable item – to which such specification can ever be appropriately applied. And when we cannot specify membership, we cannot count it either. We can estimate the  approximate size of such groups but  cannot specify it. Granted, something may be unidentifiable under one description and not  another. But this illustrates rather than undermines the character of categorization. The range of these mystery categories whose membership is by definition cognitively inaccessible is further expanded by another family of categories, the  purely hypothetical, whose membership is  determined by counterfactual conditionals. This includes •• People saved by automobile seat belts. •• People whose life has been shortened by solar-flare radiation. •• People who would not  be alive today had the  Boston strangler not  been caught. Note that the third group includes not only that killer’s potential victims, but their further potential descendants as well. But note too that it excludes those who would have died of other causes in the interval – a circumstance which renders the group still more problematic.

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This sort of speculative grasping is even more problematic with mystery categories such as •• People who would be alive had Hitler died in infancy. Note that  this  class is  clearly nonempty but  nevertheless presents highly speculative category. A further example is: •• People who would have [as opposed to might have] survived the Titanic disaster if the Carpathian had not stood off during the ship’s sinking. In all such cases, we have an intelligible criterion for category membership that cannot possibly be implemented in practice and detail, since in no case do we have an effective (i.e., actually implementable) means for resolving the issue of membership with respect to specific items. It is in this sense that the categories at issue are mystery categories: their (undeniably real) members can be specified but not identified. Just as categories are not limited to the epistemically identifiable, they are not  limited to the  physically actual or real. Consider Figure  1.2’s attempt at an ontological categorization. Note that  the  category of unidentifiable reals constitutes a classificatory abstraction that has identifiable members, as for example, that “ever-ignored grain of sand.” But among the concrete, in addition to the reals, are the fictional entities. In  the  end, the  categorical function of “natural kinds” is  not ultimately limited to the  constraints of nature or actuality. It  is  easy to envisage categorizations of mythical beasts or imaginary beings (Borges 2006). If categories and kinds are relative to pragmatic purposes, not  merely the divisions between kinds but  what  kinds are recognized will be relative to context. In  contexts of literary discussion “fictional detectives” is  a category, and fictional detectives constitute a  kind of fictional personages. Sherlock Holmes and Antoine Lupin are members, though both are entirely entities concrete real/actual predictively identifiable

fictional/possible

predictively unidentifiable

Figure 1.2  Basic ontological classification.

abstract instantiated

uninstantiated



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fictional – and in  fact in  part because both are fictional. For  the  purposes of physics or chemistry, however, we might insist that there is neither a kind nor a category of fictional detectives. Can there be empty categories? Here again, we need to be sensitive to issues of intension and pragmatic context. In light of the point just made, within some contexts, it will make perfect sense to envisage categories all members of which are fictional, or all members of which do not now exist. One of the cardinal achievements of twentieth-century knot theory has been to create a natural classification of knots – even of physically unrealizable ones. In  summary, it  is  clear that  there are some indispensably necessary requisites for classificatory categorization: •• items to be classified. •• descriptive sortalization (of categorized items). •• in-principle (albeit not in practice!) identifiability of categorized items. But it is also clear that the following are not needed for categorization: •• specific identifiability. This is shown by mystery categories and descriptively vagrant characterization. •• concreteness. This is shown by categories of numbers and shapes. •• actual existence. This  is shown by categories of fictional detectives and mythical creatures. Are there contexts in  which we should recognize categories not  only of fictional entities but of impossibilia as well? “Square circles” and “odd numbers divisible by two” are both inconsistent concepts. And because inconsistent, they don’t apply to anything. Categories, we emphasize throughout, are kinds. One might then maintain that “square circles” and “unmarried bachelors” are not kinds, and thus not categories, precisely because the concepts involved are inconsistent. On the other hand, square circles are impossible shapes, as are rectangular triangles. Odd numbers divisible by two and primes that  are multiples of three are impossible numbers. Red shades of green and blue shades of yellow are impossible colors. Might one not maintain, in certain contexts, that these are distinct categories of impossibilia? The  things that  “belong” to such categories could not exist on pain of contradiction, of course, but that would be part of what is required in order for them to belong to categories of such a sort. There is a philosophical alternative here, however: to envisage not categories of impossibilia  but  second-order categories  of inconsistent concepts.8

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Without envisaging a category stocked with square circles, we can recognize that “square circles” is an inconsistent concept, and as such that concept is an element of a category of inconsistent concepts. It is, in particular, an element of a category of inconsistent concepts concerning shapes. There  are moreover at  least two types of phenomena  in  this  netherregion.9 For a purported categorical specification may exhibit either (XU) unintelligibility (falling afoul of semantical meaning) or (XC) conceptual incoherence (falling afoul of logical requirements). “Chartreuse ideas” and “icosahedral emotions” fall into the XU: we cannot get a conceptual grip on what is at issue. Where integers are at issue, “prime squares” fall into XC. In  this  case, we understand the  specification perfectly well, but understand on fundamental principles why there cannot be such things. The application of such a distinction is not always obvious, however.10 We might be urged to put “round squares” in either group, and the charitable power of human imagination is  such that  we strain  to find a  way of intelligibly understanding even “icosahedral emotions.” Here as before, a second-order categorization of variously aberrant forms of specifications themselves is a perfectly acceptable alternative to embracing categories of aberrant “things” so specified.

Categories: Epistemic, Metaphysical, and Linguistic There is an ancient debate (as old at least as Porphyry) as to whether the categories of existences are (mere) human contrivances or ref lect objective differences in the nature of things: genuine natural kinds. The pragmatic perspective seems to provide  for the  answer, BOTH. They  are human contrivances made to enable us to predict, explain, create, understand, and navigate our world(s). But they could not serve this function adequately if they did not genuinely ref lect reality: if the world beyond us did not cooperate with our conceptual divisions. Only by ref lecting our conceptual kinds as  real kinds, including “natural kinds,” can human differentiation be expected to provide a successful basis for human endeavors. Categorization can be successful only if it “cuts nature at its joints.” But from the pragmatic perspective, nature has many joints. The joints relative to our farming endeavors constitute one set, those relative to our medical attempts another, the joints important for understanding genetic relationships a third. All those joints are nature’s joints (Dupré 1993). In each endeavor our attempt is to “cut nature at the joints,” but it is a different skeletal structure of joints that is appropriate to each endeavor. It  is  quite natural to think of categories first and foremost as  mental categories: the mental boxes in which we group things as similar or distinguish



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things as  different. Your category of “green things” is  your mental box for those things characterized by the property you see in that clear central band in the color spectrum (illustrated at  the  cover and at  http://www.pgrim. org/categories/spectrum/). Your category for “purple things” is  a  different category, a different mental box. Kinds, kind membership conditions, features, similarities, classifications, and taxonomies can all be thought of accordingly as mental phenomena. But there are some respects in which this psychological or epistemic take on categories proves incomplete. There are concepts that have never occurred to us, and that may never occur to us; there may be categories that are crucial in how the world works but that we will never successfully grasp. The ideal classification with regard to any of our projects may be one that we will never in fact formulate. With an eye to mathematics, for example, the full realm of constitutive concepts and categories may be one that is impossible of human or even cognitive formulation. Concepts, categories, and classifications in this sense seem to lie beyond the realm of the  purely psychological or epistemically accessible: their philosophical treatment calls for not  merely epistemology or philosophy of mind but  a  full-blown metaphysics. There  are many aspects of our conceptualization regarding our own conceptualization that  function in this way. Our thoughts are clearly aspects of our psychology, but there are thoughts that we will never have and even thoughts we could never have. What we believe are propositions, and yet there are propositions that will never occur to us and indeed propositions that could never occur to us. In the same way, we are led to see categories as crucial to our cognitive lives, and yet to recognize their ontology as  something that  extends beyond us. There  may be similarities – even important similarities – that we may never be able to see. There may be distinctions – even crucial distinctions – that we are never able to draw. In the same sense, there may be categories – even fundamental categories regarding how the world is or how it operates – that turn out to be beyond us. The  fact that  the  concept of categories does have this  double life or this conceptual spread is crucial to their full understanding. At one end, the term “category” designates a  psychological or epistemic tool we actually have at hand, immediately recognizable in the simple example of the color spectrum. At the other end, the term “category” designates something that may lie beyond us: the real “joints” in the world optimal with respect to our particular endeavors, scientific or otherwise. As  a  matter of scientific fact, the  observed properties of chemical compounds appear quite generally to be explainable in  terms of the  periodic table elements of which they are composed. The  periodic table thus appears to represent the fundamental kinds in terms of which nature itself

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operates, in terms of which nature operated long before we were able (or existent) to discover that  fact, and which took a  great  deal of both experimental and theoretical effort to lay bare. For  the  endeavor circumscribed by chemistry, these categories represent nature’s own “joints.” The topic of categories may of course be biological as well as chemical, social as well as physical, and may serve human ends inclusive of explanation and prediction but beyond these as well. To the extent possible, our cognitive goal is to formulate mental categories, organized in  classifications, which match those in  the  world crucial for our specific endeavors. The  philosophical examination of categories and classification thus demands an approach that includes both epistemology and metaphysics within a pragmatic perspective. As  Barbara  C. Malt and her colleagues note, the  road to concepts and categories has  often been taken the  linguistic route via  correlative words (Malt et  al. 2015). It  is  common in  much of the  literature in  linguistics and even psychology to treat  concepts as  either word meanings or mental representations of word meanings (Rosch 1975, Lakoff 1987). Both Mill’s and Frege’s classic treatments of concepts and connotations are explicitly in terms of language. Sellars argues that  category claims are ultimately statements about linguistic expressions (Sellars 1974). It  is  an item of debate whether the  basis  of even Aristotle’s categories is  linguistic or not  (Baumer 1993, Moravcsik 1967, Studtmann 2021). To ask whether categories are mental boxes, metaphysical sortals, or word meanings is to act as if the concept of categories is appropriate to only one of these philosophical topic areas: to act as if one of these can play a role in only one of these contexts; to act as if one of these areas is primary, foundational, or fundamental in its understanding. An argument for fundamentality could admittedly be made on behalf of any of these. One might argue that definitive categories may be beyond our reach, and thus that the ontological understanding of categories is primary. One might argue that  categories play their primary role in  understanding cognition, and thus that  the  epistemic role is  primary. One might argue that our real familiarity and primary scientific access to categories is in terms of word meaning, and thus that categories are best understood as the stuff of language. The  extensive historical debate among realists, conceptualists, and nominalists regarding universals is replicated in closely parallel discussions between Platonic (or ideal) realists, naturalistic realists, and conceptualists in the present era who see categories as nature-made, nature-suggested, and logically conceptualized, respectively. Whether categorization is  properly considered as  fundamentally metaphysical, scientifically empirical, or



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linguistically based may continue in  active debate. But  the  treatment of these as exclusive options may be philosophically misguided. In  the  end, all three perspectives capture something important about concepts and conceptualization; all remain available and contextually appropriate.11 We suggest that  categories form part of a  conceptual cluster of general philosophical importance across all of these: metaphysically, epistemically, and linguistically. There is no priority here, because the philosophical issues salient in each context cannot be separated. Our language is concept-based and it is simply a mistake to detach what we attempt to represent linguistically from what we attempt to represent conceptually – and a further mistake to attempt to separate either of these from the world we attempt to represent. Process and product, endeavor and goal, trajectory, and target are inseparable interlinked here.

Lessons of Category-Deliberations •• Categories are essential for the  formulation of factual information: Cognition requires the  deployment of categories. There  is  no such thing as knowledge without categories. •• There  is  no single all-perspective set of categories. Categorization is intensively diversified. •• Categoricity is  pervasive, essential, and uniform as  a  procedure. But categories vary contingently and contextually with the historical and cultural setting. •• Categories are functional and purposive in nature. They are designed with a view to particular ranges of deliberation and answer different issues. Categories are thus bound to be diversified by division and differentiation along varying lines of purpose functionality. •• Precisely because they function pragmatically, the  world also  plays a role in  categorization, since it  is  in  interaction with the  world that  our conceptualization is tested and revised. •• Even within  a  fixed and determinate purposive range, categories are creations of the  factual situation – determined by the  state of empirical knowledge. •• Owing to functional variation there is bound to be dissension and disagreement about what manifold of categories is paramount in given circumstances. •• But  issues of this  variation and disagreement about the  make-up of the register of categories and about their proper function does not affect the basic fact that some recourse to categories and categorization is indispensable to cognition.

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Notes 1 Disparate aspects of our attack clearly have precedents in various fields. Our attempt is  to build on these toward a  unified philosophical picture of the  pragmatics of categorization. With an eye to linguistics, Lakoff (1987) and Taylor (1989) have been of importance in our work. In cognitive science, prototype and exemplar theory are clear precedents (Rosch 1975, Hampton 2017). For  related work in  developmental psychology we are indebted to Rakison and Oakes (2003). In  philosophy we take Thomasson (2007) and Duprè (1993) as forerunners, and can follow some elements of our approach in a trail backwards through Dummett (1973), Wittgenstein (1953), and Frege (1892), to the Ramist revolt in Antoine Arnauld and Pierre Nicole (1662) (Chapter 2). 2 The conflation of categories with sets is not by any means limited to philosophy. One of the  most recent and most rigorously formal treatment in  sociology and cultural analysis explicitly defines a category as “the set of objects that a person categorizes as instances of a particular concept” (Hannan et al. 2019, p. 98). 3 What form intensions take – description-like specifications, prototypes, or exemplars – may nonetheless remain in dispute. See Abbott (1989) and Hampton (2017). 4 Consistent with the pragmatic emphasis here is an emphasis on categories as flexibly ad hoc with reflect to shifting tasks at hand. See Cassanto and Lupyan (2015). 5 “Natural kinds” have an extended and controversial history. See  Quine (1969), Abbott (1989), Hacking (1991) and Bird and Tobin (2018), and the further discussion in Chapter 4. While the idea of natural kinds may have had its source in biological considerations, Darwin effectively put an end to this connection. For he made it clear that a connection of biological descent is not kind-preserving but can (over time) give rise to entirely new species. Descriptive kind socialization and genetic descendency were prized apart by his work. His masterwork could just as soon have been entitled The End of Species as The Origin of Species. 6 On this point see also Slater and Borghini (2013). 7 Michael Weisberg offers an analysis  of similarity in  terms of weighted elements of a  feature set (Weisberg 2013). What  keeps this  from being “arbitrary and unilluminating” is the supposition that elements and weights of the feature set come from context, conceptualization, and pragmatic goals (p. 148). 8 Such a strategy is at least related to one in Sellars (1974), which Thomasson (2019) traces back to Ockham. 9 Magrador (2013) distinguishes at least four forms of category-mistake type infelicity: candidates which are syntactically ill-formed, meaningless, meaningful but without truth value, and pragmatically inappropriate. 10 XU candidates clearly call to mind Ryle’s “category mistakes” (Ryle 1938, 1949). See  also  Chapter  5. But  the  attempt to clearly circumscribe category mistakes from other absurdities and varieties of radical falsehoods has proven far from easy (Magridor 2013, Goldwater 2018, Thomasson 2019). 11 Rescher (2007) emphasizes that different kinds or universals may have a very different epistemic, metaphysical, or linguistic status. Hampton (2015) makes a related point regarding various understandings of concepts in  terms of a  “semantic triangle” of words, thoughts, and things.

Chapter 2 THE HISTORY OF CATEGORY THEORY An Arc of History Our concern is with categories in general, including chemical categories, biological categories, color categories, social and cultural categories, even mystery categories and categories of fictional entities. But  much of the philosophical history regarding the  nature of categories in  general is buried beneath a history focused on a very specific set of categories: the categories, envisaged in  a  universal classification as  the  highest kinds or genera. A range of alternative and conf licting proposals as to which are the categories in this sense stretches from Aristotle through Kant and Husserl to the present. Which are the highest categories is a topic of controversy, both historical and contemporary. We have also  noted the  ancient debate (as  old at  least as  Porphyry) as  to whether the  categories are (mere) human contrivances or reflect objective differences in  the  nature of things via  natural kinds. The  philosophical history reveals a  range of different approaches to the nature of categories – envisaged alternatively as metaphysical, epistemic, linguistic, or pragmatic. But it also reveals an intriguing trail of development through these approaches. In  broad strokes, the  history of categorytheorizing exhibits an increasing naturalization. We will trace that trail of changing interpretations of the  categories through exemplars in  Aristotle, Avicenna, the Ramist Revolt, Locke, Kant, Peirce, Frege, Husserl, and Ryle to contemporary debates. The  two central areas  of debate, both historically and in  contemporary guise, are debates as  to which the  categories are and to what they are. The  lesson we draw from history is  that  both areas  of controversy reflect deep philosophical mistakes. One mistake is the presupposition that there is some unique and ultimate set that are the categories. Another is the mistaken assumption that  categories must fall to one side  or the  other of a  false metaphysical/epistemic divide.

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Aristotle Aristotle was  equivocal as  to whether categories are about language or about reality.1 But this is understandable, given that our informative talk about things is an endeavor to make manifest what  things actually are, adequatio ad rem: “approximation to reality” usually taken as “correspondence to fact.” All of the Aristotelian categories indicate different aspects of what something is or does. Aristotle’s interpreters are not  unanimous with regard to the  nature and function of his  Categories.2 But  his  main  Anglo-Saxon expositors have approached the matter from an epistemic point of view. For W. D. Ross the categories address the  question: “How is  one to classify the  main  types of entity invoked in the structure of Reality?” (Ross 1923, p. 23). For J. L. Ackrill too the  sorts of questions that  can be raised about things is  pivotal for the Aristotelian categories (Ackrill 1963). With the category of absolute universalization, which is to include everything and anything that is an object of possible consideration, it is (or should be) clear that anything is capable of being identified, described, and classified. And various salient questions can always be raised – and presumably answered – with regard to it: —What is it? —What is it like? —How is it related to other things? —What ensures its existence/being/reality Questions of this kind are always appropriate, and the Aristotelian categories can be viewed as  a  venture in  attempting to understand their range. On this basis, Aristotle was led to consider the following series of categorization questions: Substance:  “What  is  it?”: particular and specific (this  man) or type-constituted and generic (man at large). [With respect to the former we would also ask about its existence: real or fictional. With regard to the latter, we can also ask is it  actually instantiated in  existing exemplifications or not?] And perhaps also: What is its type? Quantity: What is its size, magnitude, amount? What  are its descriptive features? This  could be Quality:  taken to encompass also  its constituent components. What (if anything) is it made of? Relation: How does it connect to other things?



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Time: When—at what date or time—is it (or its instantiation) realized? Place: Where (if anywhere) is  it  located? Where is  it  (or its instantiation) employed? Position or placement: How is it emplaced? Is it prone or sitting or shaking? State: How is it presented, or outfitted, equipped? Is it dressed, shod, armored, masked? What  is  it  doing or undoing? What  changes is it Action/Passion:  producing or undergoing? Approached from such an erotetic standpoint there is no reason to accept Aristotle’s list of categories as  complete and definitive. If, as  it  seems to all appearances, the  factors at  issue are to be issues geared to ways of item-identification and specification, then further additions could certainly be made, for example: Being: What is the item’s existential status: is it real and actual or fictional and merely possible? Origination: How is the item created or produced? Is it material or artificial and man-made? Function: What is the item’s role in Reality’s scheme of things? What (if anything) is its function or purpose? In a synoptic perspective the categories center on the issue of sameness and difference. The crux of the question: How can two objects of consideration manage to be different from one another? And here the available range of possible answers includes: •• In being (existence/nonexistence, reality/irreality) SUBSTANCE. •• In amount (size, measure nature) QUANTITY. •• In description (properties features) QUALITY. •• In relationships (kinship, similarity) RELATION. •• In location (placement, proximity) PLACE. •• In timing (age, curation) TIME. •• In position (prone, upright, quivering) POSTURE. •• In possession (having parts of constituents) POSSESSION. •• In acts, events, and efficacy policies (by pushing, heating, etc.) ACTION. •• In reaction, responses (filling, melting) PASSION. If  the  standard view of the  matter is  correct – if Aristotle’s Categories is intended “to indicate the  most general feature addressed in  our thought

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about the world” – then the Aristotelian categories are not exhaustive even within the overall framework of his own philosophizing, where such matters as existence, origination, and teleological purposiveness also  figure  on the metaphysical agenda. Avicenna The medieval Arabic logicians also construed these categorical matters of identification into matters of question/answer dialectic. 3 It is particularly germane to call attention to the  (relatively brief ) treatment of the theory of questions by the famous Persian-Arabic philosopher Avicenna (980–1037).4 Known principally as  a  metaphysician, Avicenna  also  wrote extensively on issues of logic across the  entire spectrum of this  discipline as  cultivated in medieval Islam. In several of his logical treatises, he sought to provide an analysis  and a  systematic classification of questions.5 The  upshot of these attempts is presented in Table 2.1.

Table 2.1  Avicenna’s classification of questions BASIC QUESTIONS (mutâlib umhât) 1. The is-it question (hal al-shay‹) i. Re existence simply (mawjûd mutlaqan) ii. Re existence in-a-state (mawjûd bi-hal kadhâ) 2. The what-is-it question (mâ al-shay‹) i. Re. essence of the thing (dhât al-shay‹) [a] definition (hadd) [b] description (rasm) ii. Re. meaning-of-the-word (mafhûm al-ism) 3. The what-sort question (ayyu al-shay‹) (Re. the genus, species, and difference of the thing) 4. The why question (limâ al-shay‹) i. Why is: the cause (the four causes: [a] material, [b] formal, [c] efficient, [d] final) ii. Why said: the reason SUBSIDIARY QUESTIONS (mutâlib juz’iyyah) 5. The how question (kayfa al-shay‹) 6. The where question (ayna al-shay‹) 7. The when question (matâ al-shay‹) 8. The how-much question (kammiyyat al-ashyâ‹)6



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The principal distinctions involved in this classification are as follows: 1. Basic questions vs. subsidiary questions. The rationale here appears to be that a basic question is one regarding the existence, the nature, and the causes of a thing; and thus deals with (i) issues concerning substance (rather than “accidents,” in the sense of Aristotelian categories other than substance), together with (ii) issues concerning the traditional Aristotelian causes (which are extra-categorical questions). By contrast, the subsidiary questions deal with the accidental features of things. Apparently, this is the reason why Avicenna designates7 the four basic questions (1–4) as the scientific questions, dealing with matters of essence and existence, and he  characterizes the subsidiary questions (5–8) as  non-scientific precisely because they address accidental matters. For, of course, on the  classical, Aristotelian view of the matter science deals with the essential features of thing, and scientific knowledge of accidents is accordingly impossible. 2. Questions of fact vs. questions of discourse. Avicenna  is  clear and explicit in distinguishing considerations regarding the  nature of things from those regarding the  meanings of words (2i vs. 2ii), and in  distinguishing considerations as to why things are as they are from those regarding why things are spoken of in certain ways (4i vs. 4ii).8 It would seem that Avicenna’s pointed formulation of the matter represents a substantial step towards the later distinction between nominal and real definition, a step indicated by but going beyond the work of the Stoics. 3. The priority of questions. The idea operative here is that it can prove infeasible to raise a question Q1 (e.g., that regarding the purpose of a thing) if a suitable answer to an antecedently presupposed question Q2 (e.g., that regarding the existence of the  thing) is  not forthcoming. Avicenna  consequently maintains that, for example, the why question (4 i) is posterior to the is-it question (1 i). In such a case – when the legitimacy of raising the question Q1 turns on the obtaining of an appropriate (affirmative or negative) answer to Q2 – question Q2 is  said to have (logical) priority over Q1. In  just sense, the question “Is X an accomplished flutist?” would be posterior to the question “Does X play the  flute at  all?”: if the  second is  answered negatively, it would be pointless to raise the first. Avicenna’s treatment of the theory of questions was not altogether original with him. For example, his treatment of the “four scientific questions” agrees in detail with Yahyâ ibn ‘Adî’s discussion presented in his treatise “On the Four Scientific Questions Regarding the Art of Logic.”9 This point of agreement between these two opponents shows that Avicenna’s categorical treatment of the logic of questions altogether reflects earlier Arabic logical tradition.10

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Table 2.2  The correspondence between the Aristotelian categories. The Porphyrean Predicables, and Avicenna’s Questions Category/ Predicable

Greek Name

Arabic Name

Question at Issue

Avicenna’s Question

1. Substance 2. Quantity 3. Quality i. Genus ii. Species iii. Difference iv. Essential Qualities —Definitive —Descriptive** v. accidents 4. Relation 5. Place 6. Time 7. Posture (position) 8. Possession 9. Action 10. Passion [11. Cause***

ousia poson poion genos eidos diaphora

al-jawhar what thing? al-kammiyah how much? ayyu what sort? jins naw‹ fasl

(1i), (1ii)* (8) (3)

idion horos hypographê symbebekota pros ti pou pote keisthai echein poiein paschein aitia

dhât hadd rasm kayfa al-idâfah ayna matâ wad‹ lahu an yaf‹al an yuf‹al al-sabab Annotations

what nature?

(2i)

how functioning? how related? where? when in what attitude? with what ? what doing? what undergoing? why?

(5) (6) (7)

(4)]

* Roughly, the two parts of this question ask regarding the primary and secondary substance at issue, respectively. ** Regarding this entry, see footnote 16 below. *** Regarding this 11th entry see the discussion in the text.

Avicenna’s theory of questions provided the basis of his theory of categories. This  he  based on Aristotle’s duly augmented by the  five predicable of Porphyry. There can be no doubt of this, in the face not only of the parallelism of the concepts at work here, but also the close correspondence of the Arabic terminology at issue in the discussion of categories.11 The relevant data are assembled in Table 2.2. Two aspects of this tabulation should be noted especially: 1. It helps to bring out quite clearly the fact that Avicenna approaches questions from an ontological direction, viewing them all as questions asked about an existing thing (this was already implicit in the Arabic nomenclature), and that this thing is to be considered in isolation (hence the absence of the category of relation as well as its cognates posture and possession), and without regard



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to other things by which it may be affected or upon which it might be acting (hence the absence of the categories of action and passion).12 2. It highlights the prominence – as a category! – of the rubric of causation, which the  important role of why? questions endows with special significance.13 The  introduction of why? questions was  not grounded in  an ontological analysis  of the  Categories but  rooted in  a  deeper analysis  of science: to know a thing is to know its categorical status. Avicenna (in discussing demonstrative syllogism) begins with the analysis of questions in Posterior Analytics, II, i and fills it out by further use of the Categories. One must regard the what-sort? question as  somehow derived from Aristotle’s question hoti: “that  it  is  the  case.”14 For  then the  four Basic Questions coincide  exactly with Aristotle’s four “subjects for inquiry” [loc. cit.]. The process of their transmission can be traced through the Alexandrian Aristotelians to the Arabs in considerable detail.15 The  fact is  that  most of the  departures made  by Avicenna  in  logic from orthodox Aristotelian positions trace back to ultimately Stoic sources. For it is clear that the Stoic logicians also interested themselves in the logic of questions. For example, Diogenes Laertius reports in his register of the logical works on Chrysippus (280–209 B.C ) that this important Stoic logician wrote an entire series of treatises on the logical theory of questions.16 A meager modicum of information about this Stoic theory of questions is provided in sources available to us,17 but this is insufficient to throw much light on the conjecture under discussion. Nonetheless, it seems likely, all things considered, that Avicenna’s treatment of the logic of questions is (ultimately) indebted to the Stoic discussions on the subject. To  be sure, the  reference to “description” is  the  only point in  Avicenna’s classification of questions which, taken in  isolation, is  clearly Stoic and postAristotelian.18 But the tactic of realigning categorial ideas around the organizing theme of questions seems to have the earmarks of a Stoic innovation. In  any case, these initial steps towards a  formal theory of questions (be they of Greek or Islamic origins) mark a  noteworthy advance in the development of logico-semantical theory away from the  emphasis  on assertion that  has  generally characterized the  mainstream of development in this area among Western theoreticians.19 The Ramist Revolt The  Ramist revolt against scholastic Aristotelianism rejected the  basics of Aristotelian logic. As regards the categories, this came to a peak in a classic book by Antoine Arnauld and Pierre Nicole on Logic or the  Art of Thinking (1662, Buroker 1996). After listing Aristotle’s categories, the  authors say:

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“these categories are viewed as based on reason and truth, when in fact they are completely arbitrary, having no foundation but the imagination of one man, who had no authority to prescribe laws to others. Each of us has  as much right to arrange the  objects of our thought in  other ways, according to our own manner of philosophizing” (Buroker 1996, p.  34). No one has more emphatically insisted on the sheer arbitrariness of categorization. But it is hard to reconcile with the usual functional/purposive on the matter. Locke John Locke (1632–1704) also  figures  importantly in  categorical history. His  distinction between nominal and real specifications of the  format  “X obtains whenever Y does” contemplates two prospects: •• The formulation case where this relation just happens to be so. (The moon is the largest physical object to orbit the Earth.) •• The necessary case where this relation obtains as a matter of conceptual necessity. (One, the first integer, is the start of the count process.) The  first sort (nominal specification) proceeds in  terms of unessential categorization (in  the  first example, classification by size). By  contrast the  second (real specification) proceeds by categorization via  essential properties (such as an integer’s position in the count sequence.) In this general context the distinction between the real and the nominal mirrors that between categorization by natural vs. artificial kinds. Kant Immanuel Kant (1726–1804) reorganized the  Aristotelian categories into alignment with different modes of assertion. For  Kant, categories were, like everything else that  is  a  priori, features that the “faculty structure of the human mind” imposed on the machinery of conceptualization. Categories provide for the ways of securing our epistemic (cognitive) fix upon reality. Kant coordinated his categories with modes of assertion (or “judgment”). (see Table  2.3.) In  this  way of proceeding, Kant exploits the  Aristotelian equilibration of language and reality, discourse and fact. However, facticity as he sees it is not a matter of mind-independent (“transcendental”) reality (to which – effectively by definition – our minds do not provide access), but to a mind-conditioned reality-view coordinate with the faculty-structure of the human mind. And it is this inherently mentalesque reality provided by the



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Table 2.3  A Kantian perspective. Modes of Judgement

Categories of Being

• universal (all) • particular (some) • singular (this)

Quantity • unity (this) • plurality (these) • totality (every)

• Affirmative (is) • Negative (is not) • Indifferent (is perhaps)

Quality • reality (is) • negative (is not) • limitation (is perhaps)

• Categorical (definite) • Hypothetical (conditional) • Disjunctive (alternative)

Relation • inherent (subject/attitude) • dependence (grounding/consequence) • reciprocity (against/prohibit)

• problematic (maybe) • assertoric (is) • apodictic (must be)

Modality • possibly (vs. impossibly) • actuality (vs. irreality) • necessity (vs. contingency)

modus operandi of our faculties that is at issue with the Kantian categories. They are features neither of reality as such nor of thought alone, but of objective reality – albeit a reality constituted by the modus operandi of human thought. As  Kant sees it  the  human mind functions via  an inherent logicogrammatical template of four structures with three compartments each •• Quantity [Scope] —Unity [This] —Plausibly [Some] —Totality [All] •• Quality [Measure] —Reality [Is so] —Negation [Is not so] —Limitation Conditionality [May or may not be] •• Relation [Manner] —Inherence [Has by nature] —Dependence [Has by condition] —Reciprocity [Conjoins with] •• Modality [Status] —Possibility [May be] —Actuality/Factuality [Is] —Necessity [Must be]

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Kant’s view of categoricity is thus logico-pragmatic: His categories represent how thought enjoins items of consideration into claims of fact. For instance, the claim at issue in “The cat is on the mat” has the feature of unity (the cat), reality (is  so), dependence (continuality), and actuality/factuality. For  us humans, things can possibly be as  they can possibly be thought-to-be, so that the categories of plausible factuality mirror the actualities of discursive thought. Categories are in effect affirmation-structures: they reflect the modes of formulation of fact in human thought. However, Kant’s thought on categories is  widely misrepresented. Many commentators think that  his  “Deduction of the  Categories” is  an attempt to prove the inevitability of the categorical scheme set out in his tabulation. This  is just plain  false. His  reasoning is  to establish the  need for SOME categories if there is to be objective knowledge. Categoricity is indispensable for the cognition of things, but certainly not the particular categories which reflect our human modus operandi. (Consider an analogy. Having an addressee is essential to reporting, but any possible system of address-provision is one contingent possibility among others.) Kant is very explicit about this: The feature of OUR understanding—that it can produce a priori unity of apprehension only by means of the categories, and only by such and so many—is as little capable of further explanation as why we have just these and no other functions of judgment, or why space and time and the only forms of possible intuition for us. (CPuR, B 144–177). The crux of the Kantian categories is that they afford the means open to us in representing and considering objects. The Kantian categories, that is to say, implement the classical idea of truth as adaequatio ad rem – that the product of our producings reflects their process of production. Peirce Charles Sanders Peirce (1839–1914) figures  among the  principal category theorists after Kant. He  postulated three basic categories as  fundamental in human cognition, which he called Firstness, Secondness, and Thirdness, corresponding roughly to three modes of being: possibility, actuality, and normativity respectively; thinkability, experientiability, and approvability; prospect, fact, and necessity (lawfulness); or might, is, and must-be, respectively. These are, in effect, the three lights in which states of affairs can and must be seen. It is clear that the Peircean categories are linguistic rather than constitutive – that they relate to how things are talked about rather than how they are constituted: how they fashion in thought rather than in their substantive nature.



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Frege Gottlob Frege (1848–1925) also  adopted the  view that  categories are the signposts of differentiation in linguistic expression – that is, in the ways in  which issues are treated in  language, so  that  modes of reference bear the  defining basis  of categorization. Issues of categoricity now  in  effect become features of linguistic practice, and different categorizations simply mirror different modes of designation and reference. The  mission of categorization thus measures the  systemization of our linguistic dealings with the  objects of thoughts. On  this  basis  our mode  of conceptualization becomes communicative (for us at least) and whether categorical distinctions are the same for those who are Arcadian or Mandarin is left as an open issue, with the distinction between the necessary and the contingent left out of sight. Husserl Edmund Husserl (1859–1938) devoted a  monumental effort to clarify and extend the  doctrine of categories. He  distinguished between semantical “categories of meaning” (in  relation to how we think about objects) and ontological categories of being (in  relation to the  realities we mean to consider).20 With respect to ontological categories, we must distinguish between what  is  possible and what  is  not; with respect to semantical categories, we must distinguish between what is meaningful and what is not. Categorization is to serve as a touchstone of meaning – and of meaninglessness, which can occur either formally and generally  [as  with “a  round, non-round thing” which remains absurd, irrespective of what term is substituted for “thing”] or materially and substantially  [as  with “a  round square”] with absurdity dependent on the specifics of the case. On this foundation, Husserl developed an elaborate doctrine of logically diversified categories. He proposed a strictly linguistic approach to categories based on the  idea  of meaningful substantiability. Thus an expression such as “She left in - - -” coordinated with different categorical prospects: —Means (in a sedan chair). —Manner (in great haste). —Mode (in an angry frame of mind). —Time (in time to catch her bus). By contrast “She painted her boudoir in - - -” will admit of: —Means (an oil paint). —Mode (in shades of yellow). —Manner (in great haste).

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With manner, there can thus be substantive identity. But with means, there may not. It will not do to say “She left in oil paint.” So departures and boudoir paintings are categorically differentiated and incomparable proceedings. (This  line of thought was  further developed and extended by Gilbert Ryle in “Categories” and The Concept of Mind (1938, 1949)). Whitehead The early twentieth-century philosopher who more than any other sought to articulate a doctrine of categories was A. N. Whitehead (1861–1947). His aim was  to articulate a  body of principles equitable to the  characterization of human experience at large as reflected in the two subjective, intrauniversal, objective. In line with this idea, he presented a scheme of One “Category of the Ultimate,” namely creativity on becoming (1 = 13) Eight “Categories of Existence” (8 = 23) Twenty-seven “Categories of Explanation” (27 = 33) However, Whitehead’s “categories” are not really tributaries of the mainstream of philosophical tradition on the subject. With Whitehead’s predecessors, categories were designed to present universal features of knowledge; with Whitehead, they became features of universal knowledge: mainstays providing the  supporting foundation of a  Karnak-like structure of philosophizing. They  are building blocks of his particular system of metaphysical thought and do not constitute an effort to represent aspects of the logic of how people generally think about things, but rather address how Whitehead himself thinks about things in general. Whitehead’s “categories” are actually such in name only. Various recents Since the  days of Peirce and Husserl, a  considerable number of other theorists have ventured into the  murky terrain  of philosophically general categorization. The list includes (but is not exhausted by): Samuel Alexander, Space, Time and Deity (London: Macmillan, 1920). Ingvar Johansson: Ontological Investigations (London: Routledge, 1988). Roderick Chisholm: A Realistic Theory of Categories (Cambridge: Cambridge University Press, 1996). Roman Ingarden: Time and Modes of Being, tr. Helen R. Michejda (Springfield, Illinois: Charles C. Thomas, 1964).



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Stephan Körner: Categorical Frameworks (Oxford: Blackwell, 1970). Reinhard Crossman: The Categorical Structure of the World (Bloomington, IN: Indiana University Press, 1983). Joshua Hoffman & Gary Rosenkranz: Substance among other Categories (Cambridge: Cambridge University Press, 1984). E. J. Lowe: Kinds of Being (Oxford: Blackwell, 1989). The  variability of approaches to categorization has  not been lost in  recent discussion. Each such theorist takes somewhat different views of what categories are supposed to do and accordingly arrives at a somewhat different categorical scheme. To  see where we now  are the  matter must be considered in  its historical perspective.

Lessons from History With an emphasis on high points from Aristotle through Kant to Frege, Husserl, and beyond, the  history of category-theorizing can be seen as  exhibiting a certain line of development: a thread of ongoing naturalization. The Greek commentators, and most modern, see the Aristotelian categories as fundamentally ontological – an invoking of the modes of being. Beginning with the Arabs this is given a more pragmatic turning. For them, the categories were instrumentalities of information processing and belonged rather to epistemology (“logic”) than to ontology (metaphysics). In Kant, the categories become a  priori structures of conceptualization imposed by the  “faculty structure of the human mind.” In Frege and Husserl language and its culturecontext become central. Categories are no longer absolute, as they were generally seen before Kant, but increasingly become contextualized to culture-context. A clear theme in much of post-Kantian category theory is that we can no longer ask what  the  categories might ideally to be, but  only what  they actually are in a given setting. And since cultural change is unpredictable, so is categoricity. The history of the doctrine of categories thus exhibits an ongoing transition: •• From the a priori to the a posteriori. •• From necessity to contingency. •• From fixity to variability. •• From absoluteness to functionality. But this holds only for what our categories are seen to be, not for what they are for. The  purpose that  categories serve and the  function they have is a trans-historical constant. The distinction is like that between what travel

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or communication is – which is constant – and how we go about it, which is ever-changing. The functionality of categories is stable, its modus operandi varies. As does our take on their cognitive status. And so  a  certain  absoluteness and necessity remains. It  relates to the difference between categoricity and the  categories, between substance and form, between what is at issue with membership on the list (its idealism), and what the members are that qualify. For while the list of categories grows and changes, the requisite of categoricity remains constant. A  survey of the  history of category theory carries an important lesson. All of the  classical philosophical ventures are predicated on the  shared presupposition that there is indeed a single, uniquely dominant set of general categories indicating the  highest kinds into which objects of considerations can be sorted. But this presupposition is deeply problematic. There really does not seem to be a single, unique, definitive such categorization. The situation is  analogous to seeking for a  unique way of axiomatizing mathematics or of cataloging books or of bridging rivers. In such matters, there is no single unique option but  a  variety of effective procedures variously suited to the specific purposes at issue.21 What thus emerges from the history is not a single schedule of categories but a single set of questions that constructively define the mission of categorization and address how this project is to be implemented. To ask for a unique, allpredominating set of categories is  to ask for too much. Different modes of categorization are required in  addressing different issues and problems. (There just is no single optimal mode to all issues of employment, diet, etc.) Instead, coming up with a list of categorical questions/issues that can operate across the board is the project of categorization. Notwithstanding the  extensive and elaborate discussions of categories in recent philosophizing there is no agreement in the matter and no consensus on the definitive register of categories. And the reason for this is plain and simple: theoreticians simply do not  agree on the  fundamental issue of what it is to be a category: in what is the aim and function of categorization. And even those who agree that it is to be an inventory that classifies everything discussable still fail to agree on what the basic developmental principle is to be. Is the taxonomy at issue to be developed with a view to inquiry, information storage, learning procedure, and so forth. There are, after all, endlessly many ways of sortalizing things, no one of which is definitively and all-purposively correct. Emphasis  on different objectives and pressures opens up room for endless variation of views as to which purpose is to be given priority. The history of category theory reinforces the idea that approach to the topic must include  the  angle of the  fundamentals of epistemology. To  render our knowledge accessible we must organize it. Because the  organization



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of information is  driven by a  functional and pragmatic motivation, form is bound to follow function, so that the manner of classificatory proceedings will reflect the purposive objectives of the particular enterprise at hand. Consider the  analogy generated by the  instruction: “Make a  list of presidents of the  US.” Immediately one will be asked: “But  what  is  to be the guiding principle?” Is it to be: •• By date of entry into the office •• By age of the individual at entry into office •• By order of longevity •• In order of length of service in elective office •• etc. ? And there is no single right answer here. It all depends on the nature of the interest at issue in the particular context of deliberation. And of course, the categorical issue of an investigation of everything whatsoever is in a far worse condition than this. Philosophical development has  given way in  relation to categories in the same way that it has misfired in relation to much else of a conceptual nature – by overlooking the contextualization of the situation. For the question on which philosophers have focused is: “What  is  the  correct register of categories?” whereas the question should properly have been contextualized by asking (1) What  are the  main  alternatives for categorization and (2) What is best suited for the various inherently different areas of application? It is unrealistic to expect a one-size-fits-all recommendation (any more than in matters of social or political theory). Even with the topical range fixed on the issue of thematic perspective – of functionality – this  remains open to variation. The lesson one can carry away from a look at the history of the problem is that the idea of a uniquely comprehensive and correct set of categories is an impossible one. One-size-fits-all categorization is a pipe dream. A given landscape can be regarded from different points of view and accordingly become subject to very different modes of categorization: that  of the mapmaker, the  landscape gardener, the  ordinance engineer, the  social planer, and so forth. And there just is no reason why Reality as a whole will not be in  much the  same position. The  idea  of one supreme overarchingly omnipotent view-set of categories is a philosophical illusion. And the idea that there is a single all-purpose answer to the question of categorization is fixed on a mistaken presumption. It is not that there is no such thing as  an adequate categorization, but  that  the  very conception is  diversified and fluid and embraces a  vast spectrum of purpose contexts.

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It is not that the conception is meaningless but that the presupposition of its unicity is flawed. Categorization like evolution is inherently pluralistic and diversified. Only after you become specific as to the why of categorization can you meaningfully address the issue of its what.

The Duality of It One fundamental point of disagreement pervades the  history of category theory. Are categories fundamentally linguistic/epistemological or are they realistic/ontological: do they address reality itself of rather merely how we think and talk about it? Aristotle goes both ways: ontological in the Categories, erotetically linguistic in  the  Topics. And even some theorists have been divided. Johannsson and Chisholm are both explicitly ontological. Grossman, Westerhoff, Hoffman, and Rosenkranz are linguistico-epistemological, seemingly seeing their task to be “cognitively descriptive” along with Cassirer, Ryle, and Sellars. The  resolution for this  divergence is  clear. In  due rigor, categories should not  be seen as  exclusively epistemic or realistic but  as  representing an interaction of the  two. They  should be seen as  reflecting our best effort at representing whatever sector of reality may be at issue. “Tell me what is, independently of and without reference to what  you think to be so” is  an unfulfillable instruction. Categories do not  and cannot properly be confined to one direction or the other: we cannot deal with reality save via the mediation of thought and language. If categories are to serve as instruments for understanding reality we cannot separate their signification from their realistic dimension: these are two inseparable sides of the  same coin. Like Janus, they look in  two directions at once, and a sensible category theory must relate to reality-aswe-see-it, with neither factor – the seeing and the reality, the thought and its object – separable from the other. Overall, there has  been a  wide  array of responses to the  question of the status of the categories, specifically including: •• Dictated by the inherent logic of reality (Aristotle). •• Inherent in  the  very modus operandi thinking about things (Platonic scholasticism). •• Built into the  contingent make up of language (Nominalistic Scholasticism). •• Purely arbitrary and conventional (Ramism). •• Inherent in the familiar status of the human mind (Kant). •• Contextually purposive and pragmatic resources (Peirce and Pragmatists).



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Historically, then, there are two main  lines of approach. (1) Some see the  matter as  one of logico-semantical fundamentality, with categories dictated by the  first principles of objective thought. (2) Others see them as  imposed by the  contingent nature of empirical reality imposed by our apprehension of the real world as experience reveals it to us. And still, others see them as convenient devices of human artifice embodied in the cultural conventions of language. Our present approach is  one of “splitting the  difference” via  a crucial distinction: that  between categoricity as  generic project and the  categories as its mode of implementation. The difference runs parallel to that between the  human need for communication as  a  generic process and the  specifics of a  given human language as  the  means of its realization. Or between transportation at large, and walking, automobile driving, railway transport, flight, and so forth, as various modes of its realization and implementation. Two basic questions arise in dealing with categories: I. What basic assumptions and presumptions must be satisfied if objectively valued thought about a  thought-independently existing reality is  to be possible? The answer here suggest contentions on the order of 1. There  must actually be an authentically mind interpreted Reality that exists independently of what we think about it. 2. This Reality must have a descriptive nature of some sort in and of itself. 3. This descriptive nature must be of a sort that we can group or approximate in thought. 4. Adequate thought about these facts must have a  format  or structure of some thought-separable sort. Going beyond this basis there is a further crucial issue, that of the question II. Mind-interpreted Reality being what  it  is, and our place within it orbit being what it is, what are the factors which – for us – compose the means by  which we address the  issue of the  descriptive nature at issue in (3) and (4) above? Thus where issue I  deals with categoricity in  general and addresses considerations of the logic of objectivity as such, issue II is something very different. The categoricity of I sets the abstract stage, the categories of II deal with the detail of specific performance.

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Categoricity is  a  necessary resource for the  accomplishment of certain general objectives (object identification and classification above all). Whereas the categories are culturally variable and empirical, variable in different contexts of operation. Thus categoricity as  such is  a  fixture mandated by the nature of various human cognitive and communicative enterprises. Which categories are viable is changeable and contextualized to the diversity of human cognitive and communicative practice. In  Kant’s language, it is  categoricity taken generally and absolutely that  is  necessary and thus a priori for rational beings at  large and in  communication. The  particular categories are but contingent requisites of the human situation, pragmatically dictated by our purposive contexts.

Language and Categorization: Aristotle, Kant, and Wittgenstein Human thought proceeds by symbolic representation: it  indicates things by conceptions and combines ideas  to form assertions, questions, injunctions, evaluations, and the rest of it. It provides verbal representation to informative claims. Thus we humans conduct our conceptual business by means of language. Language is crucial for communication, transmission of information, and thought cooreinagtion between different minds. To convey thought about concepts, ideas, facts, and indeed even about mere possibly requires language. For  us humans, information, concepts, and language are conjoined in an inseparable union. Our conceptualization proceeds via  linguistic characterization. As  it  has  long been noted that  grammatical features have a far-reaching ontological bearing at  the  grand  (gross) level, just the subject/predicate structure of Indo-European languages is reflected in the substance/attribute distinction of Aristotelian logic. Grammar structures Table 2.4 Category Substance Quantity Quality Place Time Posture Cause Relation Possession Action Passion

Greek Name ousia poson poion pou pote keusthai aitia prosti echein poiein pasiskein

Question at Issue What is it? How much? What sort (kind)? Where? When? How structured? How produced? How connected? How endowed? Doing what? Undergoing what?

Theme identification

Description

Connectivity



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our view of reality. As the  Persian philosopher Avicenna  pointed out long ago Aristotle’s categories align with the  questions that  one can rise with regard to objects of consideration, as indicated in Table 2.4: what is at issue?; what is it in itself?; and how is it connected to other items? However, this phenomenon is also found at the small-scale level of particular details. Thus the grammatical distinction between active and passive verbs is reflected in the categorical distinction between agent and patient. And the same holds also  for various other Aristotelian categories. And in  general, it  is  clear that here the grammatical survey of the sorts of questions one would raise about an object of consideration effectively corresponds to the  categorical classification of its ontological status. Accordingly, many grammatical principles accordingly reflect significant categorical distinctions. Consider a few examples: 1. The  human community divides into male and female, a  circumstance reflected in many languages by the division of norms into masculine and feminine. Moreover, the use of diminutives often divides people into adults and children. 2. The  social distinction between elite, middling, and subordinate classes reflected in grammar is the distinction between modes of address that are, respectively, first, normal, and inferior. And this  distinction is  further grammaticized by a distinguishing between formal and informal modes of address. 3. In some languages (e.g., Spanish) there is a systemic difference between persons and impersonal objects, represented by the personal a. 4. In some languages (e.g., Japanese) there are special grammatical devices for dealing with particular classes or types of steps (e.g., birds). Another grammatical distinction with categorical ramifications is that between process verbs and product verbs as in Table 2.5. The question “What are you (engaged in) doing?” can always be answered in the process mode, seeing that  doing is  a  process. But  success cannot be “processized,” Table 2.5 Process Verbs Going to London Running the race Trying to recall a name Aiming at the bull’s eye Learning Greek Buying into the lottery

Achievement Verbs Arriving in London Winning the race Remembering a name Hitting the bull’s eye Knowing Greek Winning the lottery

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since succeeding is  a  terminating achievement that  can only be realized retrospectively and so cannot be something that one is currently doing. Success (in contrast to trying) is only determinable retrospectively. Throughout our doings, the difference between process and product plays an important role. Many modes of grammar serve to represent features that constitute reality as we see it. The grammatical distinction between singular and plural is precisely designed to capture an important feature of reality. The singularizing use of “the” (as in “the sun” or “the moon”) serves a comparable function. Another notable linguistic device harmonizing language with existential reality is the tensing of verbs into past, present, and future. The  division of items into permanent types and transient conceptions as reflected in the Spanish dialectic between ser and estar is a further refinement along similar lines. And an analogous (but vastly more problematic) linguistic device is the classing of nouns into male and female in Romance languages. For language is not ontologically neutral: its devices often reflect the  basic ways in  which we view the structure and modus operandi of the real. The  philosopher who most explicitly develops the  idea  of a  harmonious correspondence between the  grammatology of linguistic presentation and the  categorization of ontological status is  Immanuel Kant. His  theory of categories is based on three fundamental ideas: 1. That with a few mission-correlative modifications, Aristotle’s doctrine of ontological (existence characterizing) categories is correct and adequate. 2. That  the  ontological categories strictly and exactly parallel the  basic rendering of logical-grammatical formalities. 3. And that in consequence, the ontological description-taxonomic categories are subject in  the  modes of discursive procedure, so  that  a  language factually mirrors the realities as our thought rationally presents them. He groups linguistic propositions into four classes: quantity, quality, form (“relation”), and modality. And he aligns these grammatical distinctions with the  four modes of categorization at  issue in  Table  2.6. Kant thus puts the categorical status of ontological characterization into a strict alignment the grammatical modes of linguistic characterization. The Kantian perspective has it that at the most general and basic level, the grammatical principles of discursive procedure provide for the categories of conceptualization that we use in organizing our thought about things. Kant’s approach to the categories is based on his commitment to a theory of isomorphism (conformity of structure) between our reality (reality as we see it) and our grammatology (our means of linguistic representation) – between the  categories and our grammatical modes of expression (the  “linguistic



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Table 2.6 The Kantian Categories I.

Grammatical Counterpart

Quantity

—why (“this”) —plurality (“some”) —totality (“all”) II. Quality

Object of consideration

—reality (“is”) —negation (“is not”) —limitation (“need not be”)

Reality status

III. Relation —inherence (thing and property) —causality (cause and effect) —community [i.e., interaction] (agent and patient) IV. Modality —possibility —actuality —necessity

Relational connection

Possibility status

function of judgement” as he characterized them). As Kant saw it, the-worldas-we-conceive-it is bound to exhibit the grammatical lines of the language we use for its conceptualization, with the result of an isomorphism (a conformity of structure) between the descriptive structure of that  thought-world and the “grammatical structure” of that world-descriptive language. Such a grammatical approach to characterizing reality effectively envisions the following correspondence. On this perspective, the categorical framework provides the  bridging link between language and reality. On  this  view, language embodies salient features of the  categorical framework of our conceptualization which provides the apparatus for our conception of the real: language ~ [categorical framework] ~ reality adequation to fact As  Kant himself emphatically insisted, the  reality at  issue is  phenomenal realty, our reality as we see it, realty as we take it to be in the full light of our preconceptions, presumptions, and presuppositions. The  linguistic – and more generally symbolic – devices for the consideration and discussion of Reality are thus not neutral and indifferent

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about the  nature of that  reality, but  are predicated from the  outset on certain  presumptions about its nature, with various factual aspects presumed as  postulated from the  very outset about the  composition and character of that reality. Ludwig Wittgenstein had it that “The limits of our language are the limits of our word.” But the Kantian perspective went well beyond this, holding in effect that “The structure of our language is the structure of our world – that our thought-world (the  “world as  we address it  in  our thought and discourse) is in  its most fundamental regards characterized and shaped by the  basic grammatical structures of the language that we employ for this purpose. The world’s organization is  thus seen by us as  conformable to the  grammatical principles at work in its characterization.

An Alternative Approach: Replacing Grammatical Isomorphism by Pragmatic Functionalism One of the lessons of history is the consideration of routes that have not been taken. Kant’s isomorphism thesis  is predicated on an a  priori linked to the basic and rudimentary presuppositions built into the grammatical structure of a language. But there is another possible approach, one that does not take such a linguistic route, but rather one of purposive functionalism. It sets out from the idea that the purpose of categories is to seek out the classificatory structure of representation that  provides for efficient, effective, and essential means of reality-descriptive representation, letting communicative efficacy serve as the standard of categorical conceptualization. This is an approach far closer to that we have taken in this book. On  the  mainstream approach to ontological categorization that  runs from Aristotle to Kant to Wittgenstein, we are to look to language, usage, and grammar for guidance regarding the  categorical approach used in the characterizing of the  real. Even where the  focus is  on communication, however, there is an available alternative based not on linguistic structures but on a functionalistic and pragmatic approach. The format of this approach is predicated in the question: •• If  you are to communicate effectively with others about a  shared world equally acceptable to all, what  sort of devices must be at  your common disposal? The question is now not one of linguistic usage but one of means to ends of efficacy in the accomplishment of a postulated task – of efficacy in the goals of communication. The task before us is formulating, conveying, and harmonizing



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information among different individuals for the  sake of coordinating their actions for the  common benefit. And the  means for doing this  will have to involve a  common understanding of the  situation at  hand and the  action it calls for. Specifically, the following issues will have to be resolved. 1. What are the prevailing circumstances? 2. What are the issues that need to be addressed to manage them satisfactorily? 3. What are the means at our disposal for addressing these issues? 4. What are the processes that are required to put these means effectively to work? The line of thought at issue in this issue-resolution ways-and-means approach affords yet another – more analytical than linguistic – approach to the nature of categorization: a procedurally functional rather than descriptively classificatory approach to the categorization. The categorical scheme of such a functional approach to the matter would have something like the following structure: •• Situation Specification —What is the problematic reality at hand? •• Situational Requirements —Whose interests are at stake? —How do these interests arise? —What is required to serve them adequately? •• Recourse Prospects —What resources are at our disposal for meeting these needs? —How can they be effectively and efficiently secured? •• Procedural Possibilities —What steps are at our effective disposal for resolving the problems? This  too is  an erotetic (question-geared) approach to categorization, but geared more to rational action than general description: an approach that is not linguistically descriptive functional, operational, and pragmatic. Of course, the familiar issues of Aristotelian categorization will still into play in this variant approach. All those questions about who, what, how, when, and so forth. will still find places somewhere on the issue-agenda. But they will now do so not in the context of grammatical/descriptive considerations but in that of functional/processual ones. The shift in perspective here mirrors that well-known distinction between the Aristotelian substance-ontolog y of things and properties and the rival processontolog y of process and product.22

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Postscript Regrettably, there is  no modern history of the  theory of categories; all we yet have is Friedrich Trendelenburg’s long-dated Geschichte der Kategorienlehre (1846). However, some recent discussions afford useful material, including the articles on Categories in the Routledge Encyclopedia of Philosophy, the Internet Encyclopedia of Philosophy, and the Stanford Encyclopedia of Philosophy. One must often turn to studies on particular authors (esp. Aristotle, Kant, and Peirce) to find material relevant to their particular contributions to the subject.

Notes 1 Regarding Aristotle’s ambivalence see Ackrill 1963, p. 71. 2 A  range of interpretive options appear in  Paul Studtman, “Aristotle’s Categories,” The  Stanford Encyclopedia  of Philosophy (Spring 2021 Edition), Edward N. Zalta  (ed.), URL = https://plato.stanford.edu/archives/spr2021/entries/aristotle-categories/. 3 In recent years the logic of questions has come into its own as a branch of logic theory which has generated widespread interest and has been extensively cultivated. A pioneer work of the recent discussions is M. and A. Prior, “Erotetic Logic,” The Philosophical Review, vol.  64 (1955), pp.  43–59. Three important monographs are: D. Harrah, Communication: A Logical Model (Cambridge, MA, 1963); N. D. Belnap, Jr., An Analysis of Questions: Preliminary report (Santa Monica, 1963); L. Aqvist, A New Approach to the Logical Theory of Interrogatives, pt. I  (Uppsala, 1965). For  a  brief but  synoptic discussion see the  article “Questions” by C. L. Hamblin  in P. Edwards (ed.), The  Encyclopedia  of Philosophy, vol. VII (New York, 1967), pp. 49–53. 4 On Avicenna as a logician see N. Rescher, The Development of Arabic Logic (Pittsburgh, 1964), especially pp.149–155. 5 Our principal sources are: (1) Dânesh-nâme, anonymously edited in Teheran in 1331 A.H. (=1912); tr. by M. Achena  and H. Massé, Avicenna: Le Livre de  Science, vol.  I, Sections on logic and metaphysics (Paris, 1955), pp. 84–85; (2) Kitâb al ishârât wa-’ltanbîhât, ed. J. Forget (Leiden, 1982); ed. with the commentary of Nâsir al-Dîn al-Tûsî (b.1201) by S. Dunyâ (Cairo, 1960); tr. A.M. Goichon, Livres des Directives et Remarques (Paris, 1951); see pp. 85–86 of the former text and pp. 234–238 of the translation; (3) Kitâb al-najât, ed. M. Kurdî (Cario, 1938); the material on questions is extracted and translated in a series of footnotes on pp. 235–237 of A. M. Goichon, op. cit. 6 Le livre de science, Vol. I, p. 84. 7 This distinction too comes from Aristotle. See Metaphysics, 1030a 27–28. Cf. also ibid., 1029b13 and Posterior Analytics 92b6–8, 26. 8 Listed in the Dânesh-nâme, but omitted in the Ishârât. 9 Translated in  N. Rescher, Studies in  Arabic Philosophy (Pittsburgh: University of Philosophy Press, 1967), pp. 41–47. 10 In fact, the four questions are already to be found in al-Kindî. See A. Altmann and J. M. Stern, Isaac Israeli (London, 1958), p. 11. 11 Cf. D.M. Dunlop, “Al-Fârâbî’s Paraphrase of the Categories of Aristotle,” The Islamic Quarterly, vol. 4 (1958), pp. 168–197; vol. 5 (1959), pp. 21–54. 12 In  view of Avicenna’s occasional dependence on Stoic sources, it  is  worth noting that  his  treatment of questions is  clearly based on the  Aristotelian doctrine of



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categories, in contrast to the simplified Kategorienlehre of the Stoics, for which see E. Zeller, Die Philosophies der Griechen, vol. III, pt. I, 5th ed. (curavit E. Wellman Leipzig, 1923; photoreprinted Hilesheim, 1963), pp. 93–105. Thus the question is-it? what-is-it? why? come straight out of Posterior Analytics, II i: while the Stoic doctrine is entirely different, with no place for what? where? when? how much? etc. 13 The theory of questions has an intimate relationship with the theory of demonstration, which deals with the  establishment of answers. (Note that  Avicenna’s treatment of questions falls into the section on demonstration). Regarding the kinship (particularly with respect to why? questions) see M.E. Marmura, “Ghazali and Demonstrative Science,” Journal of the History of Philosophy, vol. 3 (1965), pp. 183–204 (see especially pp. 190–191). 14 Cf. A. M. Goichon, op. cit., p. 236, n.2. 15 See Altmann and Stern, Isaac Israeli (op. cit.), pp. 13–14. 16 Peri erôtêseôs (“On Questions”: 2 books), Peri peuseôs (“On Queries,” 4 books), Epitomê peri erôtêseôs kai peuseôs (“Epitome on Questions and Queries”), Peri apolriseôs (“On Answers”: 4 books), Epitomê peri apokriseôs (“Epitome on Answers”). See Diogenes Laertius, Lives of the Eminent Philosophers, VII: 191 (ed. D. H. Hicks in the Loeb series, Vol. II, p. 300). A question (erôtêma) can be answered yes of no (e.g., “Is today Monday?”); a query (pysma) is an interrogation that cannot be so answered (e.g., “What day is it?”). (ibid., VII: 66) Compare B. Mates, Stoic Logic (Berkeley, 1953), pp. 18–19. 17 See C. Prantl, Geschichte der Logik im Abendlande, vol. I (Leipzig, 1855; photoreprinted Graz, 1955), p. 441, n. 115. 18 The  term “description” (hypographê) is  not used by Aristotle, but  was  first used by the  Stoics, and came to the  Arabic philosophers via Galen and the  Alexandrian commentators. See Altmann and Stern, Isaac Israeli (op. cit.) pp. 10–11 (n.I). 19 The preceding discussion draws on N. Rescher, Studies in Arabic Philosophy (Pittsburgh: University of Pittsburgh Press, 1967, pp. 48–53). 20 See  David Woodruff Smith, Mind World: Essays in  Phenomenolog y and Ontolog y, Cambridge: Cambridge University Press 2004 and Husserl, London: Routledge, 2007. 21 A  similar skepticism regarding a  unique and absolute set of ontological categories finds voice in  Westerhoff (2005), Bennett (2009), Kriegel (2013) and Thomasson (2015). 22 For  details see N. Rescher, Process Philosophy (Pittsburgh: University of Pittsburgh Press, 2000).

Chapter 3 EMPIRICAL ISSUES IN CATEGORIZATION Preliminaries Categories, categorization, and the  organization of information are crucial topics throughout the  broad realm of cognitive studies, not  merely in philosophy but  in  both theoretical and empirical work in  linguistics, psychology, and the brain sciences. Each of these disciplines offers intriguing findings regarding the ways we categorize—findings of importance for the understanding of human knowledge. The central philosophical point that categories are crucial for cognition, in  general, is  fully recognized in  the  empirical work of other disciplines. As the linguist George Lakoff emphasizes, Categorization is not a matter to be taken lightly. There is nothing more basic than categorization to our thought, perception, action and speech. Every time we see something as a kind of thing, for example, a tree, we are categorizing. Whenever we reason about kinds of things—chairs, nations, illnesses, emotions, any kind of thing at all—we are employing categories.[…] Without the ability to categorize, we could not function at all, either in the physical world or in our social and intellectual lives. (Lakoff 1987, 5–6) A  point that  is  clear philosophically, and essentially a  priori, is that categorization and discrimination are linked. To recognize things as belonging to different categories is to be able to discriminate, at least conceptually, between items in those different categories. The “at least conceptually” allows us to say that we can discriminate between prime numbers greater than a googol and non-primes greater than a googol, though actually being able to name any of the former may be beyond our abilities. With that proviso, indeed, to recognize things as belonging to different categories is to discriminate between them. This in no way forces us to say that all discrimination entails categorization: that if we recognize that this thing is different than that, it must be because

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we have assigned the  two things to different categories. Categories are kinds, with the kinship of kinds determined by the pragmatic context of our purposes. Two things x and y will always belong to different sets or different arbitrary collections: we need to merely think of (1) the  set to which the coins in  my pocket and x belong as  members and (2) the  set to which the coins in my pocket and y belong as members. But with an eye to kinds and categories as basic groupings important for real distinctions in the context of our purposes, mere differentiation between x and y need not entail assigning them to different kinds or categories. The  importance of this  philosophical point is  emphasized in  a  range of empirical results. Many of the most striking empirical findings are the result of a  methodology that  contrasts discrimination with something more: a grouping into kinds that qualifies as categorization.

Perceptual Discrimination and Categorization The  empirical work on color perception offers particularly clear cases in which categorization is taken to be a mechanism of association that goes beyond simple differentiation. Perceptual categories are recognized as elements of information management that  are importantly different from mere transitions of discrimination. Color is in fact a very complicated subject, physically and physiologically as well as philosophically.1 We outline a somewhat simplified story. The familiar spectrum of color appears on the cover. We ask you to actually look at that spectrum of wavelengths, either in the cover form or at http://www. pgrim.org/categories/spectrum/. That experience is crucial for the point at issue here, which is not a remotely conceptual but an immediate empirical one. What  is  immediately obvious in  looking at  that  spectrum is that it is composed of clear bands of different color. Green forms a clear central band, bleeding to other distinct bands of yellow and light blue on each side, which in turn move to orange and red at the left extreme and purple and violet on the right. The fact that you do see these as bands indicates that these function as categories in your perception. Those clearly distinct bands represent your color categories. A spectrum of gray, in contrast, shows no such categorical bands:



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Gray is  not a  wavelength of light: there is  no gray light. What  the  gray spectrum tracks instead is  the  reflective property of a  surface, from approximately 3% reflective (black) to 80% or 90% (white). What is clear from the display above, in contrast with the color spectrum, is that we do not see bands of gray. We do not have perceptual categories of gray in the same sense that  we have perceptual categories of color. When exposed to a  spectrum consisting merely of brightness between white and black, we see a  smooth smudge from one side to the other. It is that distinction that lies at the core of empirical psychological work on perceptual categories. The “quality spaces” characteristic in the use of color categories cluster colors in ways that smooth discrimination along a spectrum would not. Humans are not alone in treating the color and gray spectrums differently. If  a  pigeon is  trained to peck black keys and avoid white keys, and if we subsequently expose that pigeon to a spectrum of gray keys in between, we find a gray point at which its response rate becomes essentially random. Its response rate at either side of that point, moreover, varies smoothly: the blacker a key, the higher the response rate, the whiter, the lower, in a smooth probability distribution. The  same does not  hold for training in  animals with normal blue-green color vision. If trained between blue and green stimuli, and then exposed to frequencies in between, there will again be a midpoint at which the animal’s response rate becomes virtually random. But on either side of that midpoint, there is not a smooth transition between the two different responses, as there is in the case of black and white. Here, there is a much more abrupt change in  the  rate of response on either side  of the  boundary: the  quality spaces of blue and green are separated much more along the lines of categories than are the  quality spaces of black and white (Berlin  and Kay 1969, Harnad 2003). The  same is  true for us. Exposed to a  spectrum between black and white, we see only a smoothly graduated smudge. But exposed to a spectrum from infrared to ultraviolet, we see not something smooth and “shades of gray”-like, but something with clear qualitative color bands. Those are our color categories. The really basic quality spaces of our color perception appear to be innate, for complicated physiological reasons having to do with mechanisms of visual perception (three detectors in  different frequencies with inhibitory links). Other categories, of course—including more sophisticated color categories— are not  innate. Consider taupe and puce, for example. But  the  basic characteristics of categorization that  appear in  the  innate case appear in learned categorization as well. The  abrupt change in  response around midpoints noted for color vision is spoken of as “compression” within color ranges, and “expansion” between them.

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In effect, samples within categorical color ranges are seen as more similar than can be read from mere wavelength, with greater differences between them and neighboring ranges than pure wavelength would dictate. In at least some cases, supervised learning of other perceptual categories in people produces the same compression and expansion in terms of discriminatory similarity and dissimilarity (Pevtzow and Harnad 1997). There  is  another though related distinction between discrimination and categorization that  appears in  the  empirical literature. We present two colors side by side and ask whether they are the same color. This tests our discrimination, and people are very good at it. There is indeed a “just noticeable difference” (relative to observer and observation conditions) below which our errors increase radically, but that just noticeable difference is very small in terms of wavelengths. At and above that just noticeable difference, we make relatively few errors in discriminating sameness and difference. This is simultaneous pairwise or relative discrimination. But the situation is different if we are shown a color, interrupted in our concentration, and are then shown another sample and asked whether it is the same color as the one previously seen. Here we do much worse. This second task is  spoken of as  absolute discrimination and is  interpreted in  terms of categorization. The idea is that it involves categorizing the first sample, later categorizing the second, and asking ourselves whether the two categories are the same. In relative discrimination, the size of just noticeable differences is small, with the  number of discriminable differences very large. In  the  case of absolute discrimination, the number of distinguishable regions, interpreted as categories, shrinks enormously. In judgments of length, for example, we end up with approximately seven categories of absolute discrimination. If we attempt to divide any of those categories into smaller ones, errors increase (Miller 1956, Harnad 2003).

Categorization in Animals and Machines The fact that at least some animals have color categories, with abrupt rather than smooth changes in  response rates at  their borders, has  already been mentioned (see also  Caves 2018). Measurable perceptual categorization of this type, across sensory modalities, is in fact quite widespread in the animal kingdom, extending even to insects (Avarguès-Weber et al. 2011, AvarguèsWeber and Giurfa 2015, Dyer and Chittka 2004, Dyer 2012). A  classic case of perceptual categorization has  been shown in  humans in terms of elements of speech. “Voice onset time” is a continuous variable important for distinguishing the sound /b/ from the sound /p/: “boo” from



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“pooh” and “bat” from “pat.” If voice onset time is below a certain boundary, we hear the  sound as  /b/; above that, we hear the  sound as  /p/. But the boundary is  categorical: our ability to distinguish between onset time between variants on either side of the boundary is greater than our ability to distinguish between onset time variations for samples both of which are on the same side  of the  boundary. Similar categorization holds for perceptual distinctions between /d/ and /t/ and between /g/ and /k/ (Lisker and Abramson 1964, Liberman et al. 1957, 1961, Studdert-Kennedy et  al. 1970). It turns out, intriguingly, that  both chinchillas  and rhesus macaques not only categorize along a voice onset time continuum but do so at the same point as we do (Kuhl and Miller 1975, Morse and Snowdon 1975). The suggestion is that perceptual categorization in this regard is not created in the learning of human speech but is recruited for purposes of human communication from a  broader evolutionary background (Green et al. 2020). Perceptual categorization of different aspects of vocalization has  been shown for Japanese macaques in distinguishing juvenile from female “coos” and for swamp sparrows in the elements of their songs (Green 1975, Nelson and Marler 1989). Categorization in  the  latter case shows the  influence of learning, and categorization of Barbary macaque calls varies by localized populations (Fischer 1998, Prather et al. 2009, Lachlan and Nowicki 2015). It has been suggested that perceptual categorization is particularly prominent in cases of communicative signals and where a quick decision is demanded (Green et al. 2020). Machine learning, like any learning, requires an ability to generalize from past experience to new situations: from an initial or training data set to a wider application within an expanded wealth of data. On the basis of repeated exposure using a  training set, a  neural network is  trained to associate property x with property y, for example, a specific set of symptoms with a particular disease, or a financial history with a reliability of paying back a  loan. But  generalization in  machine learning faces the  same difficulties as generalization in any kind of learning. There are not merely an indefinite but  an infinite number of properties that  distinguish one thing from another, and one set of things from another. By the same token, there are not merely an indefinite but an infinite number of properties or configurations or properties that distinguish a subset of the training sample with property y from the  subset which does not  have property y. Which of these properties or configurations of properties offers the  appropriate inductive generalization? Truly unbiased learning would have no basis  on which to distinguish one configuration of subset properties from any other. All property

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configurations that  sorted y’s from non-y’s would have the  same status. Watanabe (1985) characterizes this  in  terms of the  “Ugly Duckling Theorem.” Considered without property bias, and given all possible properties and relations, the number of features shared by the  ugly duckling—the  cygnet in  the  Hans Christian Anderson tale—with any of the other ducklings will be the same number shared by any of the ducklings with each other. It  is  the  weighting of some features in  our perception that makes the cygnet stand out. A similar weighting is required in machine learning. In  “The  Need for Biases in Learning Generalizations,” Tom Mitchell noted long ago that in practice, machine learning quite standardly builds biases into generalization by the simple fact that it employs a limited language in  which subsets or their properties are characterized (Mitchell 1980). Potential generalizations are radically reduced in  number by the simple fact that only a few properties or combinations of properties are expressible in the formal language employed and thus only a few properties or combinations of properties are candidates for algorithmic generalization. The data itself is standardly characterized in terms of a small handful of parameters, with combinations of these the only elements in terms of which generalization can be made. The danger, of course, is that parameters and properties that are in fact relevant are omitted—environmental conditions in  specific residential areas  for predicting susceptibility to a  disease, for example. In  such a  case, the  generalization is  bound to be wrong simply because it lacks the necessary parameters to work with. In some cases, on the other hand, parameters that might in fact increase predictive accuracy are quite deliberately omitted for ethical reasons. The  race of applicants is  excluded by law in  the  calculation of credit  scores, for example. Given the past history of redlining in home loans, calculations in terms of home address and zip code may be suspicious for the same reason (Érdi 2020; see also Chapter 6). Even within the limits of the formal language employed, machine language algorithms can show categorization in terms outlined above for perceptual categorization in  humans and animals: “compression” within  some ranges and “expansion” between them. Tijsseling and Harnad 1997 trained a neural network with three hidden nodes on the sample shapes shown in the upper image of Figure 3.1. On the left, the lower image shows the three-dimensional space of node values after pairwise “distinction” training but before training on categories. On the right, the lower image shows the space after supervised learning of the three categories. How similar items were within each group and how dissimilar from other groups, as measured by hidden unit weights, increases in machine learning to reflect the same perceptual categorization patterns seen in humans and animals.



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Figure  3.1  Upper: three sets of stimuli to a  neural network. Lower left: hidden node  distances after auto-association but  before training. Lower right: compression within categories and expansion between categories after training. From Tijsselling and Harnad (1997).

Beyond Perceptual Categorization The  empirical work we have briefly surveyed focuses on perceptual categorization. For categorization more generally it offers tantalizing hints, but these are admittedly only hints. Perceptual categorization is  tested and indeed defined in terms of differentiation on an objective scale. Color categorization, for example, is tested and defined in terms of the ability to distinguish samples at some points on the wavelength spectrum with greater or lesser accuracy than samples at  other points on the  spectrum, despite the fact that wavelength differences in the two cases are objectively the same. Our categories do not  always, and in  fact do not  generally, have an objective scale of this type against which they can be measured. We categorize animals, machines, goals and daily tasks, friends and enemies, threats and opportunities, and good acts and bad acts, and in none of these cases is there the simple objective scale used in studies regarding perceptual categorization. Intuitively and qualitatively, however, our categorizations, in  general, do seem to reflect the  character evident in  these simpler cases. Categories

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in  general feel something like conceptual spaces or gravitational clusters, analogous to quality spaces in the color spectrum. Items within a category cluster in  terms of similarity, though that  similarity may be difficult or impossible to define: the mark of “compression.” Items in different categories repel each other in terms of dissimilarity: the mark of “expansion.” Although relative discrimination between items within a common category is generally possible, absolute discrimination between items in  different common categories is generally easy. We have noted that  aspects of categorization in  sound recognition hold across species. The  basis  of at  least much of human color categorization appears to be innate, tied to the  specific mechanisms of color processing. But the evidence indicates that categorization in a wider sense—even human perceptual categorization—involves multiple systems (Maddox and Ashby 2004). Categorical development of various sorts, with models of a variety of mechanisms, has been extensively studied in infants and children (Rakison and Oakes 2003, Mareschal et al. 2010). It is obvious that much of our more sophisticated categorization is  learning-, environment-, language-, and culture-relative. We have no innate categorizations regarding races, religions, or political systems, let  alone prime numbers or varieties of open-source software. Even the case of color categorizations remains an area of dispute (Taylor 1989). The classical Whorfian line emphasized differences in color terminology between languages (Whorf 1956). The  Bassa  language of Liberia  has  only two terms for classifying colors, corresponding to the  cool and warm ends of the  spectrum. In  the  Bambara  language of the  Congo, there are three basic color terms, distinguishing (1) white or beige, (2) reddish and brownish colors, and (3) green, indigo, and black (McNeill 1972). The  Whorfian hypothesis was that conceptual distinctions and even abilities to distinguish samples must follow suit. The  fact that  there is  no distinction between red and orange in  the  Shona  language in  Zimbabwe and the  Bassa  language in Liberia, for example, might be offered as indicating that the speakers of these languages will be unable to distinguish between those colors. As  early as  1910, however, contrary linguistic evidence was  offered supporting hypotheses of color universals across cultures. In “The Puzzle of Color Vocabularies,” R. S. Woodworth concluded that there is an invariable sequence of basic color terms: “Color nomenclature begins, almost always, with red, and spreads to the other colors in spectral order, usually, however, skipping such transitional colors as orange, blue-green and violet” (Woodworth 1910, p. 327). That universalist tradition became firmly implanted in the work of Berlin and Kay (1969), though with later modifications (Kay and Regier 2003, Correia and Ocelák 2019). Berlin and Kay propose a universal inventory



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Table 3.1  The essentials of the Berlin and Kay sequence (McNeill 1972). I White, black II Red III Green, yellow IV Yellow, green (whichever did not appear at stage III appears at stage IV) V Blue VI Brown VII Purple, pink, orange, gray.

of eleven color terms from which the color terms in any language are drawn. Those terms, moreover, form a specific series: languages will have terms of one step in the series only if they have color terms in the steps that proceed it. The Berlin and Kay sequence is essentially as in Table 3.1. Both Woodworth, and Berlin and Kay, however, admit variations in the patterns they propose. Color relativists in  the  Whorfian tradition emphasize cultural and specifically linguistic influences. Universalists have tended to emphasize perceptual and neurological constraints in color perception. Contemporary work indicates that  neither of these alone is  sufficient to explain  the  data: that  both biological and cultural forces are in  play. The  wider cultural significance of colors may also  be of importance—the  ritual role of colors among the  Navaho, for example (McNeil 1972). Environmental conditions seem to play a  role as  well—the  prevalence of particular colors in  deserts as opposed to tropical environments, for example (Correia and Ocelák 2019).

Prototypes and Exemplars How do people learn and employ categories? In cognitive science, prototype and exemplar theories have played a  major role in  trying to answer that question. In broad outline, at least, each of these is entirely consistent with and supported by the  empirical results surveyed above. In  line with the discussion of categories and similarity in Chapter 1, both prototype and exemplar theories function in terms of a primitive concept of similarity. In  prototype theory, a  single abstracted prototype forms the  core of a category, with the application of that category learned and applied in terms of salient similarity to the  prototype (Posner and Keele 1968, 1970, Reed 1972, Rosch 1975, 1983, Lakoff 1999, Smith and Minda 2001, Taylor 2004, Gärdenfors 2004, 2017, Hampton 2006, 2017). In  exemplar theory, which we will emphasize here, it  is  not a  single abstracted prototype but  a  set of very concrete exemplars that are taken as the core of a category, stored with

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all details in  place. Categories are learned and applied in  terms of salient similarity to those exemplars (Medin and Schaffer 1978, Nosofsky 1986, 2014, Estes 1994, Storms et al. 2000, Rouder and Ratcliff 2006, Reisberg, 2015). Often categories are formed and applied in terms of salient similarity to sets of both exemplars and counter-exemplars: core exemplars of things to which the category applies and of things to which it does not. On exemplar theory, a category is learned first and foremost by acquiring a set of exemplars. It is often learned as a set of contrasts, each of which will have exemplars. To use the classical example from the Sorites (Chapter 5), “bald” may have a  set of exemplars, and “not bald” another set. “Green,” “blue,” and “black” may all have sets of exemplars. From there on, category use is dictated by the core concept of similarity: It is elements that are saliently similar to the exemplars for which the same term is used. It is when a new example is  similar enough to the  exemplars of “bald” that  is  categorized as bald, when it is similar to “not bald” exemplars instead that it is categorized as  “not bald.” Hidden beneath the  apparent monadic “bald” on such an account, therefore, is a buried relation: “bald” functions as “is saliently similar to exemplars a, b, and c,” where a, b, and c are the exemplars for the term: 2

Bx ≡ Sx{a,b,c}

In exemplar theory, it should be emphasized, a, b, and c are not predicate defined. Exemplars are stored lock stock and barrel, with all properties in place. They are property multidimensional, but those properties need not be named or even conceptualized. It is full-blooded full-property representatives that moor the end of a concept. Similarity judgments will be multidimensional as well, and one may decide later that someone really is “bald” on the basis of an element of similarity to exemplars that  did not  originally occur to one: it is  the  pattern of hair distribution among the  exemplars that  matters, perhaps, rather than the  number of hairs. In  the  psychological literature, it is  put forward as  a  virtue of exemplar accounts that  relevant properties may not  themselves be conceptualized or distinguished, and may not  even be “linear” (Ashby and Maddox 2005). But the line between exemplar and prototype theories in  this  respect is  far from clear: it  can be hypothesized that  a  prototype is  learned from exemplars, with properties no more explicit than those from which it is formed. The  central emphasis  on similarity in  exemplar and prototype theories accords both with a range of empirical evidence and with the philosophical approach to categories we have outlined above. An essentially vague relation of similarity, operating beneath our monadic categories, tied for each to unnamed exemplars, accords both with the  available psychological and



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linguistic literature and explains many of the  familiar features of vague predicates. Vague categories can be multidimensional—as  are “nice” and “nasty,” “wise” and “reckless”—just as is the underlying concept of similarity. Vague categories can be salient-sensitive, just as  the  underlying concept of similarity can be salient-sensitive. A  further dimension that  is  also  clearly operative is  the  pragmatic dimension: categories are often context- and purpose-relative, just as is the underlying concept of similarity.

The Pragmatics of Basic Categories The  work of Eleanor Rosch is  rightly regarded as  a  foundational turning point in  empirical work regarding categories, largely setting the  agenda  for psychological work on the topic since the 1970s. At the core of that work is empirical support for a hypothesis that not all categories are of a kind: that some categories are “basic” categories, on a central level correlated with faster recognition and ease of childhood learning (Rosch and Mervis 1975, Rosch et al. 1976, Rosch 1978). That hypothesis is anticipated in Brown (1958): The  dime in  my pocket is  not only a  dime. It  is  also  money, a  metal object, a thing, and, moving to subordinates, it is a 1952 dime, in fact a 1952 dime with a unique pattern of scratches, discolorations, an smooth places. When such an object is  named for a  very young child how is it called? It may be named money or dime but probably not metal object, thing, 1952 dime, or particular 1952 dime. The dog out on the lawn is  not only a  dog but  is  also  a  boxer, a  quadruped, an animate being; it is the landlord’s dog, named Prince. How will it be identified for a child? Sometimes it will be called a dog, sometimes Prince, less often a boxer, and almost never a quadruped, or animate being. (Brown 1958, p. 17; Hajibayova 2013) The empirical support for such a hypothesis is widespread, from work on childhood learning to color terminology, picture categorization, face and emotion recognition, and data and library organization (Blewitt 1983, Greco and Daehler 1985, Mervis and Crisafi 1982, Berlin and Kay 1991, Murphy and Smith 1982, Jorgensen 2003, Schmidt and Stock 2009, Bates 1998, Lancaster 1970). The explanations offered for a distinction between basic and non- (or less) basic categories are uniformly pragmatic, though the form and emphasis of those explanations can vary. Lakoff’s explanation is in terms of embodiment: basic-level categories are “human-sized,” categories at  the  level of mediumsized dry goods reflecting the ways that embodied humans interact with things

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(Lakoff 1987, Lakoff and Johnson 1999). Basic-level categories “depend not on the objects themselves, independent of people, but on the way people interact with objects: the way they perceive them, image them, organize information about them and behave toward them with their bodies” (Lakoff 1987, p. 51). Rosch’s outline of “basic” categories leads directly to a  pragmatic information-theoretic explanation. The level of basic categories is distinguished from superordinate and subordinate levels in that “basic” categories have a greater similarity of members within categories and a greater dissimilarity of members from those of contrasting categories. “Fish” and “birds” are offered as  basic categories. “Animals” form a  superordinate category because its members are far too various—being informed that something is an animal is thus likely to convey too little. “Goldfish” form a subordinate category because they are insufficiently different from other fish, and thus may provide superfluous information. Brown’s “dime,” halfway between an insufficiently informative “metal object” and an over-specific “1952  dime with scratches over the motto” fits precisely this pattern. The level of “basic” categories can also be defined as that level on which categorical “compression” and “expansion” are maximized, outlined above in  terms of perceptual categories and machine learning. The  identification of an individual in  terms of a  “basic” category thus embodies a  compromise between too little information in  context and too much. “The basic level in a taxonomy is the level at which categories carry the most information” (Rosch et al. 1976, p. 383). Although we are not aware of it  being noted in  the  psychological literature, the  use of basic categories rather than the  “too little” information of superordinate categories or the “too much” of subordinate categories is  very much in  the  spirit  of Grice’s principles and maxims governing conversational implicature, which he himself characterizes in terms of communicative information (Grice 1998). Despite speculations in  Rosch to the  contrary (1976), early work often seemed to assume that the “basic” level of categories would be universal across individuals, contexts, and cultures. Later work indicates that this is not the case: that what level of information is “basic” will itself be pragmatically fixed. For the worker storing personal belongings, “a painting” may be on the basic level. For the museum-goer, it may be artist-individuations like “a Picasso” that will fill that role ( Jolicoeur et al. 1984, Belke et al. 2010). Novices and experts with regard to dogs and birds use quite different categorization levels as  “basic” (Tanaka  and Taylor 1991). All of this  is  precisely in  line with the  general approach we have taken throughout: categories are fixed by the informationprocessing needs of particular tasks. What  the  empirical work adds to that picture is the distinction between basic, subordinate, and superordinate category levels particular to the pragmatics of particular goals and projects.



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Empirical Lessons Regarding Categories We have used perceptual categorization with regard to color as a guide to lessons from the empirical literature. From that literature, we draw several lessons that we propose hold for categories and categorization in general: •• First is  the  simple fact that  we do categorize and operate conceptually in  terms of categories. Categorization is  essential to our management of information in the ways necessary to navigate and operate in a world. •• Categorization goes beyond the simple ability to make distinctions between cases, defining gravitational conceptual spaces of “compressed” similarity and “expanded” dissimilarity. •• Our categories are variable but pragmatically constrained by our contact with reality. Those contacts with reality are in  turn circumscribed by perceptual, cognitive, cultural, and historical limitations. •• Prototype and exemplar theories from cognitive science and the empirical evidence in linguistics and psychology converge with the broadly pragmatic philosophical approach outlined in  previous chapters and in  those that follow.

Notes 1 Not to mention linguistic complications which we put aside  here. Leaves “green” in having not yet turned as opposed to “green” in having been painted, among other senses, form primary examples in radical contextualist approaches. See Travis (1997) and Hansen (2011). 2 See also Grim (forthcoming).

Chapter 4 CATEGORIES IN SCIENCE In the Phaedrus, Socrates outlines two essential procedures of classification. One is to bring a plurality under the appropriate form. Phaedrus And what is the other principle, Socrates? Socrates That of dividing things again by classes, where the natural joints are, and not trying to break any part, after the manner of a bad carver.1 The  aim of our science, it  is  often said, is  to “cut nature at  its joints” (Campbell, O’Rourke, and Slater 2011). But  does nature have joints? And what determines where those joints are? Particular categories exfoliate from others via  principles of division which both separate (from taxonomic predecessors) and unite (among their taxonomic successors). So  in  the  end we have a  unified grouping united by specification commonalities. And just herein lies the difficulty. For in committing ourselves to a claim of the format “All items of category C are united in possessing feature F ” we undertake risky factual commitments when definitions are not at issue. “Swans are white” – but unfortunately there are those antipodean exceptions. “Species breed true by type” – but if they did so invariably then there could be no evolution. The history of science is a litany of unraveled taxonomies as  new phenomena  fail to conform to established patterns (Oppenheim 1936, Hempel and Oppenheim 1936).

Scientific Categorization Categorization is  crucial to all cognition: without categories, we have the possibility of neither generalization nor  individuation: an appreciation for kinds of things and for identification of things, essentially as  of a  kind. Navigation in our world, or any world, requires its conceptualization in terms

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of categories. The task of science is of a piece: an extended cultural tradition of attempting to navigate our world in terms of understanding and prediction. The goal is to represent a world well enough to accommodate ourselves to it and, where possible, to accommodate the world to us. As in all conceptual navigation, as in all cognition, categorization is crucial. It is possible to write the history of science as the history of an attempt to get the right representation of the structure and mechanisms of the world around us. That representation is inevitably in terms of categories – categories of both things and mechanisms. In that sense, the history of science is the history of an attempt to conceptualize the world in terms of the right categories. What  makes them the  right categories? Here there are two answers. The first is that the right categories are nature’s own: the categories of entities and mechanisms in terms of which nature itself operates. The second answer is  that  the  right categories are those that  are right for us: those that  serve our purposes. Here as elsewhere, it is both answers in combination that offer the  best picture: the  right categories are those that  serve our purposes in interacting with nature, where nature is a full partner in that interaction. Some choices of categories, like some theories they constitute, will face a  recalcitrant and uncooperative nature. Nature will not  “answer to” just any theory, or any set of categories, in terms of offering us successful options for prediction and intervention. That is what lies behind the idea that nature has categories independent of us: “joints” that our categorization attempts to mirror, but does not create. But the act of categorization is ultimately our own, and is ultimately to be judged in terms or our purposes. Understanding and prediction of the world are our goals, as much as justice in our social organizations and beauty in our aesthetic productions are our goals. Only intellectual hubris  would make us think that  we are gods enough to seek a  god’s-eye view. The  particular questions we pursue in science at a particular time will be dictated by what we take to be the important questions within reach, and that importance will be a reflection of our pragmatic interests in context. The particular categories we use will be a reflection of those we think will best offer us the contextualized understanding and intervention we seek. The  history of science is  the  history of discarded theories, and often of the categories in which they were phrased as well. There have been attempts to understand the  world around us using categories of phlogiston, witches, crystal spheres, epicycles, ectoplasm, auras, unicorns, psychokinesis, an elixir of life, the philosopher’s stone, black bile, humors, caloric, coronium, luminiferous aether, nebulium, and N-rays. Intelligence was  rarely lacking in the pursuits characterized by those categories. Often those categorizations were worked out in great and clever detail: Ptolemy’s Almagest is a primary



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example. Always those categorizations were worked out with an eye to precisely the  pragmatic purposes of explanation, prediction, and control that guide our own theories and our own categorizations. Things didn’t work out as expected. Nature did not cooperate in facilitating our cognitive purposes with the  use of those categories. That  is  the  sense in which those categories failed to cut nature at its joints. A full understanding of the scientific attempt to “cut nature at its joints” demands a  pragmatic understanding that  incorporates both nature and our own purposes. There are thus two potential sources of misunderstanding here. One potential mistake is to think that it is only our purposes that determine the success of our theories. That way lies anti-realism in the worst possible sense: a portrayal of scientific procedure as nothing more than a cultural or broadly political dance of power, dominance, and social authority. This  is the  path that  has  been taken in  some aspects of “sociology of science” – in  the  work of Bruno Latour, for example, though even he  has  explicitly stepped back from the hyperbolic extremes (LaTour 2007). Another potential mistake, however, is to think of our theories as somehow dictated by nature alone, independently of our purposes and the conceptual tools we devise to serve those purposes. Theory bears the marks of the theorymaker as well as that which it is a theory of: theory is inevitably is a product of our conceptual interaction with nature. Our theories, from the specific level of individual hypothesis  to the  general level of categorical structure, are inevitably our cognitive tools. Nature is both the testing ground as to which theories will prove most adequate to our purposes and the ultimate ground of explanation for why they prove adequate when they do. Theories are our tools for understanding, explanation, prediction, and intervention – all with regard to the  things that  are of importance to us in a particular context and at a particular time. As tempting is the idea that our theories should simply mirror, match, or be isomorphic to an independent reality, that is certainly not the whole truth, and in some ways distorts the basic character of scientific exploration (Cartwright 1983). The world itself doesn’t predict its future, doesn’t explore counterfactuals, doesn’t formulate its basic mechanisms, and doesn’t compute their outcomes. All of these are on our side: essentially cognitive attempts in our interaction with nature. Numbers are not quantities and equations are not forces. Our theoretical symbolisms characterize the world itself only to the extent that they play a role in successful attempts to navigate within a natural world independent of us. Our theories are cognitive tools, analogous to the  compass and sextant with which we attempt to track magnetic field and solar elevation. Given its inevitably pragmatic setting, even theoretical success doesn’t guarantee a simple “match” with reality. A discrete mathematics is adequate

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for a  wide  range of relatively successful scientific theory, but  that  does not  entail that  nature itself is  discrete. A  continuous mathematics is the primary tool for understanding a wide range of natural phenomena. Calculus is  a  primary tool for everyday physics. But  our continuous mathematics may simply be a conceptual approach that proves successful for us, with our purposes: it need not entail that nature itself is fully continuous. The  lessons of quantum physics seem to indicate that  nature is  ultimately chunky rather than smooth. There are other lessons here regarding our scientific categorizations. One is that theories can be more or less accurate with regard to the phenomenon at  issue, and that  in  context increased pragmatic accuracy may come at a pragmatic cost that is too high. If we could have a vaccine that is 99.0% effective at the end of a single month of development, with the possibility of a  superior vaccine that  is  99.9% effective, but  only with another ten years of development, it  would be ridiculous to defer vaccine development and use until we could get the more perfect form. The aim will often be to find scientific theories that are good enough for our purposes. Newtonian dynamics was good enough to get us to the moon and back; to do the same calculations in full Einsteinian form would have been simply otiose. Given the crucial pragmatics of all theory, it is to be expected that there will often or even quite generally be alternatives to a  given categorization, and that there may well be equally good categorizations – or categorizations that are equally good for a given set of purposes. In that sense, there would be alternative true descriptions of the universe. In that sense, nature might well not have single set of categorical joints. Multiple categorizations might “cut nature at the joints,” though at different joints.

Categories in Dispute, Categorization Gone Awry We have noted categorizations that  litter the  graveyard of scientific theories: phlogiston, humors, and N-rays among them. There  are categories the introduction of which was hotly debated even in the history of mathematics. A category of irrational numbers faced resistance among the  Pythagoreans. Negative numbers have a  history that  extends back to China in 200 BC and India in the 600s AD, but as late as 1758 the British mathematicians Frances Masares and William Friend claimed that negative numbers did not  exist; that  they “darken the  very whole doctrines of the  equations  and make dark of the  things which are in  their nature excessively obvious and simple” (Rogers 2009). The square roots of negative numbers were dismissed derisively as “imaginary” in Descartes’ Géométrie, a category that did not gain full acceptance until the work of Euler, Cauchy,



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and Gauss (Giaquinta  and Modica  2004, p.  121). Controversy regarding Cantor’s categories of infinity continued into the twentieth century and fed much of his personal depression. There  are categorizations that  remain  the  stuff of scientific debate and scientific controversy. In  paleoanthropology, in  biology, in  chemistry, and in  physics the  search and debate continues regarding the  appropriate scientific categories. Are Neanderthals properly categorized as  Homo Sapiens or as  distinct species Homo Neanderthalensis? Do the  range of specimens categorized as  Homo Habilis  represent individual variations or species differences? Should this find be categorized as Homo or as Australopithecus? Continuing debates regarding categorization in paleoanthropology are nicely represented in Figure 4.1. Biological classification is  in  terms of species. But  there are rival approaches to what the primary category – that of species – is or should represent, and thus rival approaches as  to what  species there are. In Linnaeus the criterion of species is morphological, an approach revived using computer analysis of similarities in numerical taxonomy (Sokal and Sneath 1973). Ernst Mayr’s legacy, tied to theories of isolation in divergent

Figure 4.1 An illustration of hominin species distributions, emphasizing Australopithecus, Paranthropus, and Homo. Overlaps indicate debates regarding the genus of particular samples. Adapted from Wood 2010 and McNulty 2016.

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evolution, is  a  species criterion based on links of actual or possible reproduction (Mayr 1963). A necessary condition for species within cladistics is shared and distinct characteristics inherited from a common ancestor, but proponents vary regarding sufficient conditions and the  choice of defining characteristics (Dupré 1993). Chemical categories seem some of the most secure, philosophically cited as  paradigms of natural kinds. But  even the  familiar categorical claim that water is H2O may require qualification. Hydrogen occurs in the form of several isotopes – not only with one atom and one electron (H), but with one neutron, one proton, and one electron (D) and with two neutrons, one proton, and one electron (T). Oxygen has 16O, 17O, and 18O as isotopes, all of which have 8 protons and 9 electrons, but with 8, 9, and 10 neutrons respectively. “If we look at enough samples of enough water, we will find H217O, H218O, HD16O, D217O, T218O, etc., in addition to H216O. In fact, natural samples of water almost always contain  a  mixture of these other isomers” (Weisberg 2006). Physics has  long faced its own categorical difficulties. The  appropriate category scheme appears to be one not  in  which space and time are each distinct categories but in which both should be treated in terms of a unified category of space-time. On the other hand, the phenomena of light seems to fit  neither a  category of particle behavior nor  a  category of wave behavior exclusively. The  demand seems to be for an appropriate category that incorporates both but is not simply either of these. Controversy can of course lead to change in  scientific categories. In the wake of scientific discovery and innovation, we change the  terms of reference by whose means we conceptualize reality. And this  brings new distinctions and new categories to the  fore. After all, we no longer classify people’s dispositions with reference to the  humors of Greek medicine. We no longer grade substances in terms of their proportional content of Joseph Priestley’s phlogiston. We no longer classify people’s psychological division in terms of their head configuration via phrenological theory. When scientific progress yields innovations in our understanding of nature’s modus operandi we readjust our taxonomies and categorization accordingly. The categories in  vogue at  a  given juncture reflect the  substance of its scientific state of the art. It is easy to construct illustrations of ways in which categories can go awry. In any context of discussion or deliberation the broadest category is going to be the “domain of discourse,” the totality of items we are going to be dealing with and addressing. This may well produce a dissonance from the realities at  work in  our concerns. If  we are trying to understand how weather conditions (rainstorms or snowfalls) come about, our domain  of discourse



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should not  be shapes and numbers. Our categories can be represented in  terms of variables that  range over the  elements of specific categories. If we are Quineans for whom “to be is to be the value of a variable” then the domain of a variable had best fit the realities at issue. If our quantifier blinders are the real numbers, then we might as well give up on trying to understand the origin of species. Mathematical examples offer particularly clear illustrations of ways in  which categories can misfit  the  reality at  issue. We have variables for individual positive integers, allowing us to solve for 3x = 9 and for truths such as ∀y∃x(x = y+1). But suppose our variables could range over only even numbers. Expressed in  terms of that  variable range, it  would not  be true that  ∀y∃x(x = y+1), and our “math” would be unable to express relations between quantities that exist in the real world. In such a case we would be attempting to work with suboptimal variables, or suboptimal categories. At the extreme, suppose our numerical categories were just “many” and “few.” The pragmatics of counting sheep would become difficult and unreliable, calculating acreage yield virtually impossible. On  the  other hand, we might attempt to work with an arithmetic in which there is  no simple category of “even number,” but  rather a  series “number divisible by 2 and 1 less than 3,” “number divisible by 2 and 1 less than 5,”… and into which we could not  quantify in  the  last position. Now  certain  generalizations regarding all even numbers would become difficult or impossible. Or consider an arithmetic written in terms of categories such as “number larger than the first one considered last Tuesday…” It is clear that our categories, represented by the range of our variables, can be either apt or awry, pragmatically clumsy or point on. Our theories can be false, of course, even if we have the “right” categories. But they may not even reach into the  right realm of being capable of being right or wrong if our categories are from the wrong domain, too coarse-grained, inappropriately fine-grained, or skewed in some other way.2

Counterfactuals, Natural Laws, and Natural Kinds There  are deep disagreements, with an extensive philosophical history, regarding the nature and status of natural laws, natural kinds, and essences (Whewell 1847, Mill 1884, Quine 1969, Armstrong 1983, Dupré 1993, Carroll 1994, 2020, Lange 2000, Hacking 2007, Bird 2018, Bird and Tobin 2022, Chakravartty 2023). We will not attempt in these few pages to settle all such disagreements. Indeed it  is  not clear that  these don’t represent a cluster of different disagreements, regarding different concepts of natural laws, natural kinds, and essences. Ian Hacking has proposed that “there are

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so many radically incompatible theories of natural kinds now in circulation that the concept itself has self-destructed” (Hacking 2007, p. 205). That verdict may be overstated. Our attempt in this section is to focus on one clear and basic distinction that  occupies an arguable center in the conceptual space of natural laws and natural kinds. That distinction is between two different kinds of categories, functioning differently in terms of counterfactuals, and their relation to two different kinds of generalizations. What we would suggest is that it is this distinction that is at the core of concepts both of “natural kinds” and “natural laws,” or at least should be. It is a central conceptual difference that really does make a difference. The  concepts of natural kinds and natural laws are fellow travelers, both in general and in our treatment here. As a first pass, natural kinds are the kinds in terms of which natural laws are written. Given this link, natural kinds have sometimes been defined as  sortalizations that  are scientifically fundamental or explanatorily basic, that  have some natural properties in common or that permit inductive inferences (Mill 1884, Quine 1969, Bird and Tobin 2023). To the extent that natural laws are laws of natural processes, moreover, natural kindhood will be fundamentally a  process-based conception. The members of a natural kind will have to exhibit a common fundamentality of comportment: there must be a manifold of shared modes of process – shared mechanisms or regularities of modus operandi – that link the members of a kind together in mode of processual uniformity. Natural kinds so conceived are groups of items linked into a commonality by lawful regularities. With natural kinds “birds of a  feather must flock together” in  point of shared functionality of lawful comportment. The concepts of natural kinds and natural laws are fellow travelers. But it is clear that any account of natural kinds in terms of natural laws begs an important question – the question of the nature and status of “lawfulness” or of natural laws. What makes a natural law a natural law as opposed to simply a  universal but  accidental generalization? Without an answer to that  we don’t yet have an answer as to what makes a sortalization a “natural kind” as opposed to merely a kind. The  basic distinction we want to emphasize as  an answer to these core questions is  a  distinction in  terms of counterfactuals, which tracks at  least in  large part concepts and standard examples of both natural kinds and natural laws. The  scientific context makes it  clear that  not all of our sortlizations are the same. “Copper” is one sortalization: some things are copper and some things are not. “Is a coin in my pocket today” is another sortalization: some



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things are coins in  my pocket and some are not. And we may have the following true universal statements: 1. All copper things conduct electricity. 2. All the coins in my pocket today are pennies. 3. All the coins in my pocket today conduct electricity. Given these universal statements, the following follow immediately: 1′. If x is copper, it conducts electricity. 2′. If x is a coin in my pocket today, it is a penny. 3′. If x is a coin in my pocket today, it conducts electricity. All of these are true, and all three cases are parallel in terms of linguistic syntax and in  forms captured in  elementary logic. But  there is  a  crucial difference nonetheless. In considering a plastic coin of the sort characteristic of children’s play money, we take the first of the following to be true but the others to be false: 1″. (Showing a plastic coin) Were this copper, it would conduct electricity. 2″. (Showing a plastic coin) Were this in my pocket, it would be a penny. 3″. (Showing a  plastic coin) Were this  in  my pocket, it  would conduct electricity. We hold (1″) to be true in  accord with the  universal generalization in  (1), which we maintain in full force. But instead of holding (2″) and (3″) to be true we backpedal on the generalizations in (2) and (3). What we hold instead are these: 2′″. Were this in my pocket, not all the coins in my pocket would be pennies. 3′″. Were this in my pocket, not all the coins in my pocket would conduct electricity. The only thing that can explain the different treatment of (1) and (1″) on the one hand and (2) and (2″) and (3) and (3″) on the other is how we treat the categories and the  generalizations involved. Beyond those categories the  language and logic are the  same. Thus “copper” and “coin  in my pocket today” are radically different sortalizations. One of these supports counterfactuals – the case of 1″ – while the other does not – the case of 2″ and 3″. One of these maintains a generalization (1) in full force under counterfactual considerations, whereas the others qualify generalizations in the manner of (2′″) and (3′″). The  concepts of natural law and natural kinds, we’ve noted, are fellow travelers. (1) in this example is treated as a natural law, and it is quite natural

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to take as a mark of natural law or “law-likeness” whether a generalization supports counterfactuals in  the  way that  (1) does. (2) and (3), on the  other hand, are treated as accidental generalizations which do not carry through counterfactual supposition in the same way. But all that differs in the two cases is the sortalization involved and the generalization in terms of it: “copper” as  opposed to “a  coin  in my pocket today.” Here again, it  is  quite natural to take “copper” as a natural kind, “a coin in my pocket today” as what we might term an accidental kind. We know that  all native North American marsupials are opossums. But the two sortalizations function quite differently with regard to supporting counterfactuals. We accept (5) rather than (4): 4. Had the kangaroo been native to North America, it would have been an opossum. 5. Had the kangaroo been native to North America, not all North American marsupials would have been opossums. To  the  extent that  the  distinction between natural laws and accidental generalizations is genuine, we would suggest, what underwrites that distinction is whether the  generalizations at  issue support counterfactuals in  the  way these examples indicate. To the extent that the distinction between natural and accidental kinds is genuine, what underwrites that distinction is whether the  categories at  issue support counterfactuals in  the  way these examples indicate. Whatever has been meant by “natural law” and “natural kind,” this is what we suggest should be meant. Copper and kangaroos are in fact the kinds of examples that are offered for “natural kinds;” “the coins in my pocket” and “North American marsupials” are the kinds of examples offered as accidental generalizations. In the case of copper, at least, “copper conduct electricity” is the type of generalization standardly offered as a law of nature. Other characteristics associated with natural kinds and natural laws also accord with this outline in terms of counterfactuals. Natural laws and natural kinds are thought to permit inductive inferences, a projection to futurefactuals which is  very much of a  kind with a  projection in  counterfactuals (Mill 1884, Quine 1969). They  are thought of as  offering fundamental explanations in ways in which accidental kinds and accidental generalizations do not, which again  accords with the  fact that  they seem to explain  why certain counterfactuals are true and others are not. As noted, natural laws are thought of as laws of natural processes, giving us natural kinds that  are process based. Here again, the  course of natural processes seems to be what we are accessing in thinking about what would be the case in counterfactual instances.



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If the distinction between counterfactual-supporting and counterfactualnon-supporting categories and generalizations is  accepted as  what  is  really at  issue, however, there are important implications that  are not  generally accepted. First of all there are generalizations and categories that  support counterfactuals but which are not standardly thought of as laws of nature or natural kinds. In  line with the  general approach we have been proposing, moreover, both of these may be pragmatically contextual in ways not always recognized. Although the distinction at issue applies to natural science, we should not be misled into thinking that “natural kinds” designate only natural as opposed to artificial kinds. An easy case is the social sciences: within anthropology, “natural kinds” may be cultural types, forms of ritual, or kinship structures. But  the  same kind of distinction applies with regard to farm machinery, where relevant categories may be those of soil preparation, planting, and harvesting, with machines sorted into “natural kinds” that  are clearly artificial in the sense of being man-made. We employ counterfactuals, categories, and generalizations in  much the  same way when it  comes to human inventions as  we do in  the  case of natural science. In  examining an exclusively gas-powered tractor at  an agricultural exhibit, we might plausibly propose: 6. Were this machine designed to be hybrid, it would be on the cutting edge of invention. But even if we know that all the machines manufactured by Tesla happen to be designed to be hybrid, we might hesitate to endorse (7): 7. Were this machine manufactured by Tesla, it would be on the cutting edge of invention. In that case, we might plausibly offer (8) as a countersuggestion instead: 8. Were this machine manufactured by Tesla, not everything produced by Tesla would be designed to be hybrid. Here “designed to be hybrid” functions more like “copper” in  supporting counterfactuals. “Manufactured by Tesla” functions more like “is a coin in my pocket.” Both, however, reflect aspects of human artifacts rather than elements of the  natural sciences. Counterfactual and non-counterfactual kinds – “natural” and “accidental” kinds – thus appear in  the  context of human artifacts as well as our descriptions of a nonhuman nature.

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They  appear in  the  context of purely human social arrangements and ethical judgments as well. Both of the following might hold of a particular case:   9. This is a case of attempted murder. 10. This case involves a banana hidden in a back room. But counterfactual (11) might well be more plausible than counterfactual (12): 11. Were this not a case of attempted murder, the prisoner would not deserve punishment. 12. Were this  not a  case involving a  banana  hidden in  a  back room, the prisoner would not deserve punishment. The  crucial distinction, we propose, is  thus between categories that  support counterfactuals, sortalizations that  don’t, and the  “law-like” and accidental generalizations that  are formulated in  terms of them. Natural kinds like “copper” and natural laws like “copper conducts electricity” fit  the  bill, but they are not the only things that do so. Counterfactuals seem to be a common currency across information organization and projection in  general, with a distinction between kinds of categories and generalizations that goes with them. “Natural kinds” in the sense outlined here thus extend beyond the realm of natural science, or even science more generally. They can also be pragmatically and contextually sensitive. In all of the examples outlined our categories or sortalizations have been explicitly announced. But that need not be the case. Quine’s conflicting counterfactuals hide different background categorizations and different background generalizations (Quine 1960): 13. If Caesar had been in charge, he would have used nuclear weapons. 14. If Caesar had been in charge, he would have used catapults. Implicit  in  (13) is  a  categorization of Caesar as  an aggressive military strategist who used the most advanced weapons available. Implicit in (14) is a categorization of Caesar as a man of his military times, the first century BC. Both categories might qualify as “natural kinds” in the service of different “natural laws.” Both support counterfactuals. But here, depending on context, it  is  different categories that  are salient and thus different counterfactuals that become plausible in context. Goodman emphasizes this point in arguing that “natural kinds” should be replaced with “relevant kinds,” where what is relevant is quite clearly a matter of context and purpose (Goodman 1985). “Obviously, what  makes a  class



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relevant to a person or a community may be facts about nature in the wild, on the farm, in the stars, in the lab, in the human psyche, or in the nucleus of an atom” (Hacking 2007, p. 204). Still further, what gives a class the salience and relevance of a “natural kind” may be a matter of different cloud formations variously important as ultraviolet blocks, as indicators of calm sailing ahead, of needed rain  in the  near future, of a  prospectively beautiful sunrise or much needed bombing cover, let alone those cloud categories important for distinguishing different periods of English landscape painting. We return to the simple fact is that we don’t treat all sortalizations as equal. Some we treat as genuine categories or natural kinds, with counter-factual supporting “laws” as conceptual fellow travelers. Some we treat as arbitrary collections. Why do we treat these sortalizations differently? What justifies these distinctions? Although the  first question is  of a  descriptive nature and the second normative, the answer to both is the same. It is a pragmatic answer: we treat them differently, and are justified in doing so, because these distinctions work. When they don’t, the pressure is on both “descriptively” and “normatively” to adopt a different categorization, with different “laws” and dictating differently supported counterfactuals. Here lies one of the deep mysteries. We live in a universe that is in some sense fully actual. Nothing physically happens, ever happened, or ever will that  doesn’t actually happen. The  merely possible is  never physically actual. And yet our cognitive lives are deeply enmeshed in considerations of what is and isn’t possible, both for us and for our environment. By the same token, counterfactuals are by definition counterfactuals. We cannot simply observe what would have happened if something that didn’t hold had held instead. Our evidence is limited to the actual and what actually does occur. But  we crucially use that  evidence in  the  service of cognition regarding the possible and what might have occurred. Although actual occurrences are obviously limited to the  actual, our conceptual system and our concept of truth clearly extend to both the non-actual possible and the counterfactual: what might have happened but didn’t, and what would have happened if… In the end, what we really want to know is what will happen contingent on certain  events, including importantly what  will happen contingent on particular interventions on our part. Our tool for formulating that  kind of information – and guiding our own actions – is a conceptual system written in  terms of categories, laws, possibilities, and counterfactuals. These  are crucial aspects of our major conceptual tool for dealing with the universe. So  are natural laws something genuinely operative in  the  world? Are natural kinds where nature itself cuts the  “joints”? Or are these artifacts of our own information management? Here as  elsewhere, the  answer we want to give is  “both.” Our categorical distinctions are our own, and are

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framed in terms of our purposes – often contextually very different purposes. As such, they are aspects of our information management. But they would not work, and could not be maintained, if nature didn’t cooperate. Nature will support some attempts at  cuts, some generalizations, and some of the projections reflected in counterfactuals. What we discover is those aspects of conceptual organization that  nature will genuinely support. Nature’s side of the bargain might be characterized in terms of Dupré’s “promiscuous realism;” she  offers a  variety of “joints” consistent with the  variety of our potential purposes (Dupré 1993). All of this, we want to suggest, applies to scientific “essences” as well.

Categories and Essences Much has  been made  of scientific essentialism: that  water is  not merely contingently, but necessarily and essentially composed of two hydrogen and one oxygen atom (Putnam 1973, 1975, Kripke 1980). Anything which didn’t have that  composition would not  be water. Gold has  an atomic weight of 196.96657u. Were a  lump to change its atomic weight, it  would no longer be gold. In the case of definitional categories, essentialism is far from surprising. Bachelors are unmarried; when they cease to be unmarried they cease to be bachelors. Were Michael married, he  would not  be a  bachelor. Here “essence” is built into the definitional criteria of the category at issue; it can effectively be read off explicit kind membership conditions for the category. This form of essentialism is transparent to use a priori, or at least as a priori as the membership conditions for our categories are explicitly available to us. But  it  was  not always known that  water was  H2O, nor  that  gold has  an atomic weight of 196.96657u. Indeed the claims of scientific essentialism are more carefully phrased as “if water is H2O, it is essentially H2O.” and “if gold has an atomic weight of 196.96657u it has that weight essentially.” The fact that we take these substances to have these properties essentially, but that we discovered they have these properties, offers a picture of essences discovered a  posteriori. Nature so  portrayed not  only has  joints but  has  true essences. The suggestion is that science is the search for those true essences in nature. The relation between counterfactual-supporting natural laws and natural kinds outlined in the previous section has an explanatory place for a posteriori essences as  well, fully in  accord with the  intuitions that  drive Putnam and Kripke’s arguments. At  the  core of those arguments are intuitions that  identity statements, if true, must be necessarily true. This  holds for both identity statements regarding individuals and for identity statements regarding substances such



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as water and gold. With an eye to individuals, the intuition is that Samuel Clemens isn’t Mark Twain by happenstance: these are names for the same man, and thus it is necessary that Samuel Clemens is Mark Twain. A similar intuition is that Hesperus doesn’t merely happen to be Phosphorus. These are both names for Venus, and thus it is necessary that Hesperus is Phosphorus. It must be admitted that necessity-supporting intuitions in these cases demand that we read designations transparently – as rigid designators (Kripke 1980). If “Mark Twain” is read as a character description on the order of “a turn of the  last century author signing himself ‘Mark Twain,’ ” the  necessity of identity and all that  follows from it  disappears. The  same is  true if we use “the  morning star” in  place of “Phosphorus,” reading it  as  a  mere description. Identity statements are statements regarding individuals, and it is when designators are read “rigidly” as pegging individuals rather than independently describing them that we have genuine identity statements and the intuitions of necessity that Kripke and Putnam mine. Identity does not merely apply in the case of individuals, of course. It can be seen as the source of a posteriori necessity in the case of scientific categories as well. The classic examples are precisely such identity statements: Water is H2O Gold is the element with atomic weight of 196.96657u These function on the general level precisely as “Samuel Clemens is Mark Twain” function on the  individual level. Indeed these may function even more clearly as identity statements, with less temptation to confuse them with mere descriptions. Here again, our intuitions are that if these two stuffs are identical – if water is H2O – then water could not have been anything else any more than any stuff could be non-self-identical. It is interesting to note that the core examples generally given of scientific identities grounding essentialism often involve designators from different periods of scientific development. “Water” was  a  common term far before chemistry developed to the point that it could be decomposed into hydrogen and oxygen. Gold was  commonly recognized long before a  contemporary notion of elements, let alone identifying atomic weights. “Water is H2O” and “Gold is the element with atomic weight of 196.96657u” are thus something

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like bridge principles between rigid designators at  different periods of scientific progress: bridge principles in the form of identity statements. Essentialism in general can be seen as grounded in identity, and scientific essentialism in particular can be seen as grounded in the identity of scientific categories (Fine 2002). In all the cases above, we have full identities that seem to qualify as  full essences: the  full essence of water is  H2O. More limited essential properties, however, will be a  clear accompaniment: entailments from one or the other side of an identity statement. In the case of gold, for example, both Gold is an element and Gold has an atomic weight of 196.96657u will designate essential properties short of full essences. The same will hold for Water is composed of hydrogen and oxygen. If natural laws and natural kinds are to be understood in terms of counter­ factuals, as  argued in  the  preceding section, we should expect essentialist claims regarding them to be expressible in terms of counterfactuals as well. And indeed they are, of a particularly strong modal sort. Were this sample not H2O it would not – could not – be water. Were this sample gold rather than pyrite, it  would have to have the  atomic weight 196.96657u. Were this water, it would have to be composed of hydrogen and oxygen. To return to earlier examples: were this  copper, it  would have to conduct electricity. Were this a kangaroo, it would necessarily be a marsupial. So do essences and essential properties characterize the world itself – its ultimate joints, as it were? Natural laws and natural kinds, we have argued, are both crucial tools for navigating a world independently of us. They are inevitably aspects of our own cognition, with a  specific character in  terms of supporting counterfactuals. That is the conceptual character that makes them natural laws and natural kinds rather than accidental generalizations and sortalizations. But while they are inevitably aspects of our cognition, they are intended to answer to a world, to be revised or rejected if they do not. Do we discover essences? Indeed we do, precisely in  the  sense that  we discover identities. We have discovered that Samuel Clemens is Mark Twain, that  Venus is  both the  morning star and the  evening star, and that  water is H2O. But  it  is  inescapable and perhaps deflationary that  what  we have discovered is  that  several of our own terms are co-referring. This  is, in fact, a good example of the double-sided nature of what is at issue, and the  impropriety of a  false dilemma  asking whether what  is  at  issue lies



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exclusively “with us” or “with the world.” The terms in identity statements are inevitably ours, as are all statements, on our side of an attempt to navigate nature. The fact that those terms are co-referential, however, must often be decided by the nature we navigate. Precisely because they are discovered a posteriori, we can be as wrong about identities and essences as we can be about anything else. The claim that water is  necessarily or essentially H2O is  no more epistemically secure than the simple claim that water is H2O. We can thus no more rest scientifically when we think we have discovered an essence or essential property of this  kind than we can rest when we think we have discovered some other kind of fact. The identity-based a posteriori necessity of “water is H2O” is thus a very different and epistemically far more vulnerable kind of necessity than is the definitional security of “bachelors are unmarried.” Consider some of the  ways that  our history might have turned out differently, and the demands that would have put on our categories. It might have turned out that  samples of what  we took to be “the  same stuff” had quite different properties if they originated east and west of the Greenwich meridian, or if they came from an artesian spring as opposed to the surface of a lake. In that case, we might well have decided that these weren’t really “the  same stuff,” distinguishing two natural kinds, for each of which we sought microstructural identities and differing essential properties. We might on the other hand have continued to insist on a single category and a single “stuff,” which just happened to have different accidental properties in  artesian wells, on the  surface of lakes, east and west of the  Greenwich meridian. Whether one of these courses proved preferable to the other would depend on our wider purposes and our further interactions with nature. The point is simply that categories and categorical change are pragmatically constrained by nature, but are by no means simply and directly dictated by nature. We cannot simply peer into nature and see her essences any more than we can peer into nature and see natural kinds or identities. All of these we find through a great deal of conceptual development in an extended and ongoing interaction with nature. In this regard, it must be remembered that our discarded categories were often thought to come with defining characteristics, conclusive indicators, or essential properties as  well. Phlogiston is  released in  the  process of combustion: anything not  so  released would not  have been phlogiston. Witches are in league with the devil, or have superhuman abilities: someone who has neither would no longer qualify as a witch. Auras are energy fields. Ectoplasm is  a  physical manifestation. N-rays increase the  glow of sparks (Figure 4.2). When we abandon these categories, of course, we abandon these associated aspects of their “identity” as well.

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Figure 4.2  Critical phenomena of N-rays.

There is another role that categories play with regard to essentialism, most clearly illustrated in the case of necessities regarding individuals rather than kinds, though both might be designated de  re. The  inventor of the  bifocals was our first postmaster general. But he might have been neither. Had history been different, Benjamin Franklin might have refused that political position. Had history been different, the idea of bifocals might never have occurred to him. It is much harder to accept that Benjamin Franklin might have had different parents, or have come from a different sperm and egg. Surely any individual born from different parents or a different sperm and egg would have been a different person.



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Even in the case of individual essentialism categories are very much in play. Individuation, we have argued, quite generally requires categorization: a thing must generally be approached as  a  thing of a  particular kind. “Benjamin Franklin” is individuated as a man or a person, and that category – complete with its essential and independent of its accidental properties – is lurking just beneath the surface. This becomes clear when the defender of de re essentialism makes the ultimate (and very plausible) appeal: “Surely this man (indicated by gesture or the  laying of hands) couldn’t have come from a  different sperm and egg.” It is the individuating criteria of “the same person” or “the same man,” consequent on appeal to categories of person and man, that are doing the work of de re essentialism. That  essence is  at  least often a  matter of reference under a  category in this  way is  clear when we think of individuals that  can have no real essential properties because they do not  exist. Sherlock Holmes must have had particular parents  – Siger Holmes and Violet Sherrinford in the reconstruction of W. S. Baring-Gould 1962. If those were his parents, anyone with different parents would have been a  different person. All of this holds in applying our individuating criteria for “the same person” that comes as  part of the  package of a  category of people, exhibited even in a fictional context where we are perfectly aware that  Sherlock Holmes is a purely fictional character.

The Problem of Induction In An Enquiry Concerning Human Understanding, Hume begins with the observation that I have found that such an object has always been attended with such an effect, and I foresee, that other objects, which are, in appearance, similar, will be attended with similar effects. (Hume 1748, Section IV, Part II, [16]) But  what  justifies that  inductive leap? The  inference cannot be deductive, because it not deductively guaranteed. Deductive inferences are on pain of contradiction, but  there is  nothing contradictory in  the  supposition that  an object similar to today’s might not be attended by similar effects tomorrow. In  the  A  Treatise of Human Nature Hume phrases the  issue in  terms of a background principle: If  Reason determin’d us, it  would proceed upon that  principle that instances of which we have had no experience, must resemble those, of which we have had experience, and that the course of nature continues always uniformly the same. (Hume 1739–1740, Book I, Part III, Section II)

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But what can support that background principle? It is not deductively certain by any means. And if its justification is inductive – “that principle has always worked for us in the past” – that seems to offer merely circular inductive support for our crucial inductive premise. Such is the traditional problem of induction. An emphasis on the pragmatics of categories and category use offers a very different approach to the problem. To  begin  with, there is  no clear principle of uniformity independent of a  particular choice of categories. The  future will quite definitely be unlike the past in infinitely many ways. Tomorrow will have a sunrise, as today did, but that sunrise will be different in many ways. The fact that it has rained for a week is, if anything, a counterindication that tomorrow’s weather will be the same. I have been alive every day since 1950. But that is clearly no indication that I will be alive every day into the future. If there were a matter of principle at stake, it would not be anything as unbridled as Hume’s principle “that  the  course of nature continues always uniformly the  same” (Salmon 1953, Sober 1988, Norton 2003). Any assumption regarding uniformity would have to be tied to an assumption regarding our categories: that  we are substantially right regarding prioritized categories or natural kinds, and that  unobserved or not  yet observed instances of those natural kinds will retain the characteristic properties of those categories. Any basic assumption would thus be that  our categories are substantially correct. Were we to be surprised by the inconstancy of things as we have categorized them, we would take that  as  a  strong indication that  our categorization was  not adequate: that we have failed to cut nature at its joints. But there is a very basic mistake in thinking that any principle is at work in  our use of induction, and thus in  charging the  use of such a  principle as  “circular.” Here it  is  important to recognize a  categorical distinction between discursive and procedural contexts. The objection of circularity and self-invocation functions very differently in these two contexts. It is certainly improper and unprofitable to use a premise in its own deduction or to use an assertion as its own justification. But to use a procedure in its own justification is  another matter (van Cleve 1984, Papineau 1992, Lange 2011). To  use reason in justifying reason, discourse in justifying discourse, and deduction in understanding deduction is not only appropriate but effectively inevitable. To appeal to a principle of the uniformity of nature is again to mistake inductive reasoning for something like deductive reasoning: that we deduce a  conclusion from certain  premises, one of which is  a  crucial universal regarding the  uniformity of nature. This  doesn’t work even in  the  case of deductive reasoning, as is evident from Lewis Carroll’s “What the Tortoise said to Achilles” (Carroll 1895). Were every inference from any p to q to demand an additional premise p → q, then even a modus ponens from p → q



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and p to q would demand an additional premise: ((p → q and p) → q), except of course that an inference regarding that premise would require another, ad infinitum. But  induction is  not a  deduction from premises at  all, and thus not a deduction from a  premise of the  uniformity of nature. It  is  not deduction, not even probabilistic or hidden-principle deduction. Charges of circularity are appropriate to derivations, but induction is not a derivation. It is a cognitive mechanism, and is ultimately to be understood and evaluated in terms of its pragmatic fruitfulness. Hume’s initial question can be taken in either a descriptive or a prescriptive sense: either “why do we believe that tomorrow’s bread will nourish as today’s bread has,” or “what  justifies us in  believing that  tomorrow’s bread will nourish s today’s bread has”? His  answer – “habit” – is  often taken as  an adequate response to the first question, but not to the second. But it is not too far off as an answer to both the descriptive and normative question, for there are aspects of habit that are fully rational. Induction is  entirely a  matter of pragmatics. For  pragmatic reasons, we need to shape our behavior beyond observed samples: in particular, beyond observed samples from the  past into the  future. We rely on procedures that have worked in the past. What is the alternative? To rely on procedures that  have failed in  the  past? To  rely on procedures for which we have no track record at  all? It  is  that  pragmatic procedure that  is  quasi-formalized as induction. But the formalization can mislead us into thinking it is more like deduction than it is. Our inductive inferences are indeed habitual, but aspects of behavior that have become habitual precisely because they have worked in the past. In induction, as Quine quips, “nothing succeeds like success.” The projection of that which has worked in the past is the basis of induction. In this sense, yes, induction gets an inductive justification. But it is not “circular”: the charge of circularity applies to deductive justification in particular, not to cognitive procedures in general. There is no circularity when we document the benefits of careful attention to detail through careful attention to detail. There  is  no circularity when we use rational reflection in  thinking about the  value of rational reflection. And there is  no circularity in  choosing pragmatically successful procedures on pragmatic grounds. Induction in terms of a grounded categorization that has shown its worth is such a procedure.

The New Riddle Nelson Goodman’s “new riddle of induction” is  phrased in  terms of the introduced predicate “grue” (Goodman 1955). Something is grue just in case it is first examined before 2050 and is green or is not so examined and is blue.

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Correlate to “grue” is  a  predicate “bleen.” Something is  bleen just in  case it is examined before 2050 and is blue or is not so examined and is green.3 All emeralds we have seen to date have been green. In  philosophical caricature, it  is  on that  basis  that  we can hypothesize that  any emerald newly discovered in 2051 will also be green. But all emeralds we have seen to date have also been grue. If we hypothesize that any emerald discovered in 2051 will be grue, we will be hypothesizing that  any such emerald will be not green but blue. With the right gerrymandered predicates, Goodman notes, it appears that any observation might support any conclusion outside of the sample. Goodman posed the  “new riddle,” among other things, as  a  challenge to any appeal to a  “principle of uniformity” offered in  answer to Hume’s problem. Even were we to concede that “that the future will be like the past,” we would be forced to specify “like the past with respect to what predicate.” If emeralds are uniformly green, emeralds newly discovered in 2051 will be green. If  emeralds are uniformly grue, emeralds newly discovered in  2051 will be blue. What  Goodman’s riddle underscores is  the  point that  has  guided our exploration throughout: that  not all sortalizations are on an equal footing. There is something acceptable – or we think acceptable – about thinking about emeralds in terms of color that is not equally acceptable about thinking about emeralds in terms of “golor.” The question here as elsewhere is a question of prioritized categorization, categories beyond mere sets, and natural kinds. Why do we favor hypotheses regarding “grue” as  opposed to “green”? The  short answer is  that  “grue” is  a  gerrymandered predicate in  terms of time. Those haven’t worked well for us in the past, and so haven’t become habitual. But there is also a longer and richer answer, perhaps. Our investigations of the world quite generally assume that our point of investigation, in space or time, is not privileged: that we are trying to figure out something that is not contingent on the way we attempt to figure it out. Our assumption is that we are investigating something independent of us, and independent of our position of investigation. We may ultimately find out otherwise, but  our default is that our investigatory position is nothing special. What justifies us in an assumption that the world doesn’t bend to the point at which we attempt to investigate it? The  fact that  that  assumption has  succeeded in the past. “Grue” is  not defined merely in  terms of color and time. It  is  crucially defined in  terms of our own observation. A  predicate like “grue” builds in  the  fact of observation: something is  “grue” just in  case it  is  green and examined before t or blue and examined afterward. That quite explicitly builds



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in universe-relativity contingent on our points of observation. Past results don’t indicate much confidence in  that  kind of predicate. Were those predicates favored, “nature’s joints” would depend fundamentally on our perspective of observation. Our evidence has quite generally been that the world doesn’t function in terms of those categories. To  a  number of points here there is  a  skeptical Goodmanian response. The claim is that “grue” is defined in terms of “green,” time, and examination. But as Goodman notes, we can also define the extension of “green” in terms of “grue,” time, and examination: Something is green just in case it is grue and examined before t or bleen and examined after t. The extension of “green” can thus be given a syntactically parallel definition, also using “examined” and the suspicious time t. Thus, the  reasoning goes, only with a  prejudicial assumption in  favor of “green” as  basic is  “grue” to be condemned on this  basis. For  a  “grue” speaker, the  same points would count against “green.” Goodman’s own response to the  “new riddle” parallels Hume’s appeal to “habit” in  many ways, though it  has  a  sociological rather than psychological air. “Green” is “projectible” in ways that “grue” is not, and predicates are “projectible” when they are entrenched: ultimately, when they have a past track record of success. Goodman ultimately holds a form of “irrealism,” in which the entire rationality of our approach to the world is language-relative, contingent on the predicates we have inherited from the past (Goodman 1985). One of Goodman’s strongest points is that the extension of “green” can be specified in terms of “grue,” “bleen,” and an examination time t precisely as the extension of “grue” can be defined in terms of “green,” “blue,” and an examination time t.4 That  symmetry clearly vitiates attempts to solve the new riddle by syntax and extension alone, within either traditional logics or Bayesian approaches (Sober 1994). But  the  asymmetry of “green” and “grue” is  ultimately not  a  matter of syntax or definitional semantics. It  is  a  deeply empirical and pragmatic difference in their status as categories. A first way to see this is to consider the fact that “grue” defined in terms of a time t of 2050 is only one of a vast range of grue-like predicates, the behavior of many of which we have already observed. Consider for example: Grue1921 = df g reen and examined before 1921 or blue and examined after Grue1922 = df g reen and examined before 1922 or blue and examined after Grue1923 = df g reen and examined before 1923 or blue and examined after

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The hypothesis that emeralds are grue1921 entails that all emeralds examined since 1921 are blue. That  hypothesis  is a  clear empirical failure. The hypotheses that  emeralds are grue1922 and grue1923, let  alone grue1865 and grueOct.29,1950 are all empirical failures as  well. We thus have an extensive track record for predicates of the “grue” family, and it is a track record of repeated failure. By meta-induction over our categories themselves, “grue”like categories seem to be an empirically bad bet. Here we are tracking not a syntactic, semantic, or even logical difference between “green” and “grue” but a genuinely empirical difference. This is clear if we attempt Goodman’s symmetry move in terms of the family of “green” hypotheses. But in order for these definitions to genuinely specify appropriate extensions for dated versions of “green” we need to use not  merely “grue” but “grue1921” and the like in each definition: Green1921 = df g rue1921 and examined before 1921 or bleen1921 and examined after Green1922 = df g rue1921 and examined before 1922 or bleen1921 and examined after Green1923 = df g rue1921 and examined before 1923 or bleen1921 and examined after These are indeed the syntactical “inverses” of those considered above, and follow the  same pattern of entailments. The  hypothesis  that  emeralds are green1921 entails that  all emeralds examined since 1921 are bleen1921, for example. But in fact, all emeralds examined since 1921 have been bleen1921, just as all emeralds examined before 1921 were grue1921. In each case, that simply means that they were green. “Grue”-like hypotheses are of a date-specified family for which we have repeated empirical disconfirmation. A Goodmanian symmetrical family of date-specified “green” hypotheses would be one for which we have repeated empirical confirmation. Although it may be true that every emerald we have seen to date qualifies as  grue2050 as  well as  green2050, simply because 2050 is yet in the future, that does not hold for the differently dated representatives of each category. The  category of “grue”-like predicates has  a  history of empirical failure. “Green”-like predicates have an empirical history of success. Meta-induction on predicate categories thus seems to vindicate “green” and vitiate “grue.” Is that inductive response to the new riddle “circular”? No more so here than in the inductive response to Hume’s problem offered above. What is at issue here is simply a specific instance of the same general cognitive procedure at issue there. Our selection of categories is justified by past success. But that justification is not “circular”: as noted, the charge of



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circularity applies to deductive justification in  particular, not  to cognitive procedures in general. There  is  also  another way to see the  fundamentally empirical and pragmatic distinction between “green” and “grue.” Goodman’s approach leaves out something beyond language and beyond the syntactical symmetry of definitions regarding “grue” and “green.” The  simple fact is  that  we can immediately see color and not golor: we can see the green character of things simply by looking at them, without knowing either what time it is or the history of past examination. Because we can only see green, we cannot tell whether something is “grue” unless we know the temporal history of its examination. Without knowing whether the  date today is  before or after 2050, I am unable to know whether a newly found emerald is grue or not. Even if I know the time is after 2050, I don’t know whether the emerald before me is grue or not without knowing its history of examination. It’s a simple fact about us that we can see “green” but cannot see “grue.” Should we be prejudiced in  terms of things we can see? Not  always, but as a pragmatic starting point and a defeasible presumption, the answer is arguably “yes.” Here there are two considerations: 1. We cannot help it. We’re built that way. 2. If  we look for an explanation for why we’re built that  way, we have a background explanation for inductively successful properties built into our own evolution. We weren’t evolved to see grue-like characteristics. Borrowing from Dickens, “the wisdom of our ancestors” is in the categorization, “and my unhallowed hands shall not disturb it” (Dickens 1843). If  there is  to be a  Goodmanian symmetry between “grue” and “green,” therefore, it must be a symmetry not merely between interdefinability between alternative terms but  between alternative forms of perception. What  of the  “grue”-seer? If  there is  reason for a  defeasible presumption in  favor of perceivable properties, then were there such a  thing as  a  grue-seer, he  or she would be justified in projecting “grue” rather than “green.” Could there be a “grue”-seer? It is interesting to note that a “grue”-seer wouldn’t plausibly also be a green-seer, any more than we are could plausibly see both. If he could see both green and grue directly, he would also know just by looking at something whether it had been examined before 2050 or not. If it’s grue and green, it was so examined. If it’s grue and blue, it was not. Here as before the ability to see that something had been previously examined just by looking at it would be a radically unfamiliar form of perception.5 That  radically unfamiliar form of perception would seem to demand that on a basic level, the world wasn’t independent of us in the ways we assume

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in our investigation. Either form of biperceptive agent would always be seeing the fingerprints of history of examination. If we take that as reason to believe that grue/green biperception is not possible, it becomes clear not only that we as green-seers cannot see grue, but that a grue-seer could not see green. And yet we know that the ability to see green is a common one: we have it. We have no evolutionary conjecture as to how a perceptual ability to “see” grue could ever arise. The upshot is this: If nature has joints independent of us, we have strong reasons to believe that  they aren’t grue-like. And the  answer to the  new riddle of induction is the same as the older one: Yes, particular inductions are inductively justified – justified by the general success of induction. How do we know that induction in general is successful? Its instances have generally been successful, at  least in  terms of the  particular predicates we use. And when it hasn’t been, we are dubious of the predicates at issue. One reason to reject them is that they illegitimately build in a particular unjustified position of observation. Induction is justified, as our moral principles are justified, by a process of reflective equilibrium. The general procedure is justified by the success of its individual instances. Our use in new instances is justified by the justification of the general procedure. That is also built into our psychology, but it doesn’t follow that  it  is  “merely psychological”: habit  can be rational, and given the  methods in  which habits are established – both individually and culturally – it may even quite generally be rational.

Lessons from Scientific Categories •• A full understanding of the scientific attempt to “cut nature at its joints” demands a  pragmatic understanding that  incorporates both nature and our own purposes in categorization. •• Clear examples of categories in  dispute appear in  paleoanthropology, in biology, chemistry, physics, and across the history of science. •• Multiple categorizations might “cut nature at its joints” at different joints in the context of different purposes. •• We take some generalizations to support counterfactuals, whereas we take others to lack this  feature, distinguishing natural laws and natural kinds from accidental generalizations and sortalizations. The  explanation and justification of these distinctions are ultimately pragmatic. •• Identities between our categories ground a posteriori necessities, essences, and essential properties, but by the same token these are no more epistemically secure than is our categorization of nature in general.



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•• The  answer to Hume’s problem is  induction’s pragmatic success, not deductive appeal to a  principle of uniformity. Circularity would be an appropriate charge in  the  case of a  deductive justification, but  not with regard to a cognitive mechanism evaluated pragmatically. •• The asymmetry of “green” and Goodman’s “grue” is not a matter of syntax or semantics but a deeply empirical and pragmatic difference in their status as categories. It is meta-induction on categories of predicates that justifies our preference, no more circular here than in the case of Hume’s problem.

Notes 1 Phaedrus 265e. 2 Glymour and Spirtes (1994) considers issues of both unfortunate choices of variables and choices inadequate to decide causal structure. 3 Different understandings of “grue” appear in the literature. Prime candidates would include: 1. x is grue =df x is green before t and blue after t. 2. x is grue =df x is green and examined before t or blue and examined after t. 3. x is grue =df x is green and first examined before t or blue and first examined after t. On definition (1), used in Barker and Achinstein (1960), emeralds would need to change color at t in order to remain grue, an interpretation Goodman resists. On (2), the dating of “examined” remains ambiguous and with it the status of emeralds examined, but not first examined, after t: must they have changed color in order to remain  grue? The  formulation that  seems cleanest, that  captures all relevant aspects of the issue, and is explicit in Goodman (Goodman and Elgin 1988) is (3), which is used here. It is worthy of note that even Cohnitz and Rossberg (2020) uses (2) and (3) as if they were interchangeable. 4 Goodman repeatedly claims that  “green” can be defined in  terms of “grue” and “bleen” just as “grue” can be defined in terms of “green” and “blue.” What is clear is that symmetrical specifications of extension are possible along these lines. Whether specification of extension is enough for genuine definition is another matter, but one we put aside here. 5 Here author G has benefitted from communications with Beppe Brivec. See also Brivec (2021).

Chapter 5 CATEGORY MISTAKES AND PHILOSOPHICAL PARADOXES A  number of the  lessons regarding categories that  we have emphasized throughout also  play a  role in  classical puzzles and paradoxes. We have attempted to counter a  range of tempting philosophical conceptions of categories, arguing among other things that: •• Categories are not set-like entities. •• Categories typically carry the intentionality of their membership conditions rather than being defined purely extentionally in terms of membership. •• Both categories and the similarity relations that travel with them can be essentially vague. •• Categories function pragmatically in terms of salient purposes and interests, and their application can thus vary with different purposes and interests. •• Categorical classifications need not  form tree-like structures of exclusive and exhaustive sub-categories. Each of these points appears again in a consideration of category mistakes and a range of paradoxes, in either classical or contemporary form: •• The Sorites and its ancient kin: the Phalakros and the Millet Seed. •• The Ship of Theseus, the Statue and the Clay, Dion and Theon and the Problem of 1,001 Cats. •• The Sancho Panza Hanging Paradox and The Contract of Protagoras. •• The Liar. Paradox is often presented in terms of a set up followed with a final “gotcha” question: •• Does a trio of sand grains constitute a heap or not? •• Once we have replaced each plank, do we have the ship of Theseus or not? •• Is this lump of clay a statue or not? •• Is the Liar sentence true or not?

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But  the  appropriate response is  sometimes not  an attempted “solution” in the terms in which the “gotcha” question is phrased but a rejection of the question and perhaps the  entire setup. For  the  paradoxes we will survey, the proper is  often “The  question, and perhaps the  entire set up, is  based on a misunderstanding of the nature of the categories in terms of which it is phrased.” For  a  range of classical paradoxes, such an approach allows us to characterize fairly precisely what that categorical misunderstanding amounts to, even if it doesn’t always offer the kind of solution that the “gotcha” question seems to demand. Our aim is to point out the crucial role that categories play – and that our assumptions regarding the  nature of the  categories play – in  creating these puzzles in the first place. Although our analysis may at times implicitly suggest a, what we have to offer need not stand or fall with so ambitious a goal. Attention to the particular way categories are involved in a paradox may also simply allow us a better understanding of the source and structure of the problem. Indeed our point will often be that a search for a “solution” in the terms that are usually demanded – a final semantic category for the Liar, for example, or a discrete logic adequate to vagueness – may be a futile search. The demand for a solution of these standard types may itself involve a misunderstanding of the nature of our categories and the pragmatics of categorization.

Classification Errors and Category Mistakes There  are a  range of errors possible in  the  handling and management of categories, classifications, and taxonomies. Primary among these are mis-categorization and mis-classification as  fundamental errors in the organization of information. Misclassification can extend even to meaninglessness and incoherence, in the form of classical “category mistakes.” Categories serve to set limits to cognitively viable thoughts and discourse. Features that can only hold for one sort of thing cannot meaningfully be applied elsewhere. To position an item in one category is to exclude it from others. And what is characteristic of some cannot be attributed to another. Only written words can be palindromic, only shapes can be oblong. So one cannot speak literally of palindromic shapes or oblong words. To  do so  is  to become entangled in  the  categorical mistake of attributing to one sort of things features that can only hold for another, different sort. Accordingly, there are two main forms of category mistake: 1. Ascribing to (or presupposing for) items in one category features that can only appertain to members of another (e.g., attributing age to number.)



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2. Denying to (or rejecting for) items in  a  category features that  have to characterize items of this sort. (e.g., treating stationery as a natural kind rather than an artifact.) The  fact is  that  categorization issues from principles of distinction (principia divisionis). But if the X’s are to be Y’s that are Z, then – more or less by definition – there cannot possibly be a  Y-type X that  lacks Z. It  would be a  fundamental mistake in  view of the  very nature of the  categorization process to claim or consider such a  thing, which would, in  effect, become a contradiction in terms. Thus if a pauper is to be someone who lacks funds, we cannot have a rich pauper. Or if a youth is to be someone of few years we cannot have an ancient youth. These  are paradigmatic category mistakes: imputing to something of a putative categorization a feature which items of this category cannot possibly have. Misunderstanding is the inevitable consequence of mis-categorization and mis-classification, caught up in a cycle of error as per the following: misunderstand oversimplify Mis-classify

Assimilate different kinds Separate identical kinds

lose detail no longer distinguish distinct versions

misunderstand

If  we do not  classify things correctly, we cannot generalize about them correctly and so cannot do proper inductive reasoning with regard to them. Our understanding will not just be incomplete but incorrect. One route to paradox is  the  use of flawed and improper standards of category membership (principia divisionis). Thus consider: (T) People’s age is  to be determined by the  number of birthday anniversaries they have had. But now consider the paradox: •• Henry has lived for sixty-six years •• Henry has had sixteen birthday anniversaries (since he was born on leap day). The first indicates his age to be sixty-six; the second – via thesis (T) – an age of only sixteen years. The air of paradox is obviously due to the impropriety of (T).

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A category specification is also flawed and improper when the sort of thing it specifies is something that cannot possibly exist – that is, when that grouping is empty on grounds of general logical or semantic principles. The fallacy of purported categories of this sort are well known for philosophical discussion: •• Round squares. •• Maximal integers. •• Small multitudes. •• False truths. There cannot be any meaningfully specified X-type Y’s when being X-type is incompatible with being a Y. Similar but subtler problems arise with regard to •• A list of precisely those lists that don’t list themselves. If such a list listed itself, it would follow that it didn’t, since this is specified to be a list of precisely those lists that don’t list themselves. If it does not list itself, however, it must: this is a list of precisely those lists that don’t list themselves. Ironically, however, there could be a  precise list of, say, audio recordings which don’t mention themselves. Improper standards of category membership offer a particularly plausible analysis of paradox and its avoidance when paradoxes such as these involve setlike categories – or explicitly involve sets. On a Russellian analysis, for example, given the law of excluded middle in the form that anything must either belong to a given set or not, any viable criterion of set-membership would demand that: 1. The  claim that  an item meets this  condition must be meaningful. (Coherence) and 2. Any items meeting this condition must not presuppose that circumstance has already (previously) been satisfied. (Non-circularity) Condition 1 is violated by Russell’s specified •• The set of all sets that do not contain themselves Condition 2 is violated by. •• The set of all sets meeting condition C. The first of these is as inconsistent as a round square: just as a round square would have to be both round and not-round, Russell’s “set” would have



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to both contain  itself and not. On  condition 2 and a  Russellian analysis, the second of these inappropriately presupposes the collectability of all sets meeting condition C prior to the question of whether C itself is a member.

Vague Categories On the classical picture, categories are set-like collections sharply determined by necessary and sufficient conditions. Issues of misclassification, improper standards of category membership, and Ryle-like category mistakes all pose problems and threaten paradox even to the  extent that  categories do fit that classical picture. We have argued throughout, however, that  our standard categories do not fit that classical conception. Our categories are often vague, for example, with membership indeterminate at the edges. Just as misapplication of our categories can lead to paradox, misunderstanding of the nature of our own categories can lead to paradox. Failing to recognize their common and essential vagueness is one such source of misunderstanding. We have emphasized that categories are not sets and are importantly nonset-like. Categories are more finely tuned than sets, including an intensional determining element rather than being defined purely extensionally. Categories are also more selective than sets, including not every willy-nilly sortalization but genuine kinds germane in the context of a specific pragmatic endeavor. Classical sets must also  have sharp edges: everything must either be a member of a given set or not, with no “in between.” Membership criteria for sets thus come naturally in  the  form of necessary and sufficient conditions capable of “yes or no” application. Kind-membership conditions for categories, in contrast – their defining concepts – may themselves be vague. The categories they determine will correspondingly have an element of indeterminacy at the edges. Dangerous animals and tranquil colors, let alone fairly angry expressions and pleasing shades of green, may qualify as perfectly appropriate categories in certain contexts, despite the fact that all of these are vague. Membership in classical sets is all or nothing; everything must qualify as a full member or not a member at all. But it is worth noting that it is the definiteness of membership that categories often violate, rather than merely the  “all or none.” It  is  for that  reason that  fuzzy logic, though it  admits degrees of set membership, will still fall short of capturing the  vagueness of categories (Zadeh 1965a, 1965b, Rescher 1969, Bellman and Zadeh 1977, Haack 1978, Klir and Folger 1988). Within  fuzzy logic, everything has a precise degree of membership in every set. The vagueness inherent in

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categories of dangerous animals, tranquil colors, or fairly angry expressions comes with no precise degrees. We must, in the end, distinguish between rough and precise classifications and recognize that  our own categories are often of the  former character. When we ignore this fact and try to impose precision where it is unrealized we fall into misunderstandings and paradox. Why do we tolerate imprecise classification and almost always use loose taxonomies in ordinary life? For good pragmatic reasons, as emphasized in Chapter 1. A tolerance of categorical imprecision often eases the transmission of information and facilitates communication. In many situations a demand for precise counts and exact determinations would impose burdens on cognition, communication, and action, greatly hindering the ready processing, exchange, and exploitation of information. Categorical imprecision can be not  only enormously convenient but  pragmatically necessary. We have to choose between inexactitude on the one hand and awkward and inefficient information-management and complication of communication on the other, with the  advantage in  ordinary life almost always in  reducing the  latter. The issue is ultimately one of pragmatic trade-off. Even with a recognition of essential vagueness in many of our categories and classifications, it should be noted, a number of the lessons of a classical conception will carry over in a modified form. Although category membership may not  have sharp edges, there are still errors of mis-categorization: something may be inappropriately grouped with the wrong kind, it may be inappropriately excluded from a genuine kind to which it belongs, and may be inconsistently included in conflicting kinds. And although vague categories may be appropriate in some endeavors, it is also possible for categories to be improperly vague in others.

Paradoxes of Vagueness Vague terms will generally have a  more-or-less well-defined central core of application surrounded by a  large penumbra  of indefiniteness and uncertainty. And so when a term T is vague, non-T will automatically also be so. There will accordingly be a fuzzy region of ambivalent overlap between T-situations and non-T situations where our inclination is  to see the  issue both ways. The aporetic inconsistency of paradox then becomes a tempting possibility. Overlooking or misunderstanding the essential vagueness of our everyday categories lies at the root of a number of related paradoxes with an extended ancient pedigree.



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The  eristic (from hê eristikê as  wrangling or dialectic) that  the  Megarian school of Greek antiquity took over from the Sophists in the fourth century BC was  an early version of dialectic—or of spurious dialectic according to Aristotle,1 its principal opponent, who dealt with the  matter extensively in  his  Sophistical Refutations.2 It  was  thematically oriented towards puzzles, paradoxes, and sophisms. Aporetics is  the  descendant of this  enterprise as it continued in the medieval study of sophisms and insolubilia. Eubulides of Megara  (b. ca. 400 B. C.) was  the  most prominent and influential member of the  Megarian school of dialecticians, succeeding its founder Euclid of Megara, a pupil of Socrates, as its head.3 The Megarians derived their concern with dialectics from the ancient Sophists, who reveled in reason-baffling paradoxes because they brought grist to the mill of their teaching that  reason is  inadequate for enabling us to grasp the  reality of things. (How—they asked—can there be a point in rational inquiry if one must already know what is true in order to recognize it when one finds it?4) Like the Sceptics later on, the Sophists were not concerned with paradoxes for their own sake but because they saw them as making manifest a fundamental aspect of the human condition. The Megarians took the view that what was pivotal for dialectic was not so much logic, the theory of correct inference, as eristics, the theory of error avoidance.5 As  they saw it, conflict is  the  avenue to health. Knowing how to deal with sophisms—how to manage thought in situations of conflict and competition—is  as  important for tuning the  mind as  sporting competition is for tuning the body. Eubulides did more to promote concern for paradoxes than any other single thinker in  the  history of the  subject with the  exception of Zeno of Elea. He is credited with seven important paradoxes: The Liar (pseudomenos), The  Overlooked Man (dialanthanôn), Electra  and her Brother, The  Masked Man (egkekalummenos), The  Horns (keratinês), The  Bald Man (phalakros) and The Heap (sôritês). The  last of these – the  Sorites paradox – is  posed in  the  following account: A  single grain  of sand is  certainly not  a  heap. Nor  is  the  addition of a single grain of sand enough to transform a non-heap into a heap: when we have a collection of grains of sand that is not a heap, then adding but one single grain will not create a heap. And so by adding successive grains, moving from 1 to 2 to 3, and so on, we will never arrive at a heap. And yet we know full well that a collection of 1,000,000 grains of sand is a heap, even if not an enormous one.6

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To achieve a more perspicuous formalization of the argument, it is helpful to use some abbreviated symbolism. Let g i represent a collection of i grains of sand and let us adopt H(g) to abbreviate the statement: “The group g of sandgrains is a heap.” We then have: (1) (2) (3)

~H(g1) H(g1,000,000) (i )[~ H ( g i ) ~ H ( g i 1 )]

an observable fact an observable fact a seemingly evident general principle

(4)

~ ( ∃i )H ( g i )

from (1), (3) by mathematical induction

(5) (6)

~H(g1,000,000) (5) contradicts (2)

from (4)

Here (1) and (3) together logically entail (4). And (4) yields (5) which contradicts (2). The  triad {(1), (2), (3)} accordingly constitutes an aporetic cluster and one of its members must be abandoned if the  paradox is  to be resolved. But  what  of the  acceptability-status of these propositions? What  sorts of considerations of priority and precedence are operative here? As  the  above indications make clear, (1) and (2) are observable facts relative to our understanding of the  category of “a  heap.” On  the  other hand, thesis  (3) is  cast at  a  high level of abstract generality and is  really no more (though also no less) than an eminently plausible generalization – a theory. The theses of our inconsistent triad accordingly have the following priority ranking: 1 ,  2     3  .

We have three retention/abandonment alternatives here: Retain (1) and (2), abandoning (3)Retain (1) and (3), abandoning (2) Retain (2) and (3), abandoning (1) The first alternative alone enables us to retain all of the top priority theses; the optimal option here is accordingly to abandon (3). Plausible though it may seem, this thesis is less so than its rivals: it is the weakest link in this chain of inconsistency. And this  is  perfectly reasonable. For  (3) is  actually rather problematic. When dealing with vague concepts that  have a  slippery slope, as  it  were, it is clear that we are in a more confidence-inspiring situation with particular claims like (1) and (2) and (5) than we are with an unrestricted generalization like (3). (After all “we know a heap when we see one.”) With factual issues of



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this  sort we stand on firmer ground with particular and concrete matters than we do with matters of abstract generality. And yet despite the comparative frailty of (3) we would not (and should not) simply reject it as flat-out false. For its negation entails ( i ) ~ H  gi  & H  gi  1, and we would find it very difficult indeed to conceive of such an i. But we do not (and need not) set up (3)’s negation as true; instead, we have the option of seeing (3) as  plausible even though we propose to abandon it  pro tem as “contextually untenable” in the present context. The course of traditional paradox resolution via the dismissal of premises as false is  here made  impracticable by disassembling premiss (3) into the following multitude: (3.1) ~ H ( g1 ) → ~ H ( g 2 ) (3.2) ~ H ( g 2 ) → ~ H ( g 3 ) —— —— ——

3.999999  ~ H ( g 999,999 )  ~ H ( g1,000,000 ) What  makes this  Heap perplex a  “paradox of vagueness” is that the imprecision of what a “heap” is undermines the acceptability of a generalized principle like (i )[~ H ( g i ) ~ H ( g i 1 )], which rides roughshod over the  fact that a slippery slope is at issue. This sort of generalization seems to rest on a mistaken impression that the category at issue is more sharply defined than is  the  case. This  can be seen graphically by looking at  the  situation from a variant angle. Thus consider the (overly brief) series: ~H ~H ~H ? ? ?H H H

Note here that the insertion of a group of indeterminate (neither ~H nor H ) cases between the ~H’s and the H’s enables us to retain the rule: Whenever gi is a non-heap, then gi+1 is not a heap that is, a non-heap (~H ) is never succeeded by a heap (H ). On this simple an approach, of course, there are now two ways of failing to be a heap, namely to be a be non-heap (~H ) or to be an indecisive or borderline heap (as indicated by ?). It is now clear that (1) a single step forward from ~H will never carry us over to H, while nevertheless (2) a  series of steps along the  series will eventually carry us from ~H’s to H’s. But  the  basic problem

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inevitably appears with a question of higher-order vagueness: at what point does something shift from a full-fledged heap H or a full-fledged non-heap ~H to a borderline case such as that marked by “?.” The vagueness of our categories quite characteristically includes not merely a vagueness between H and ~H but also the higher-order vagueness between H and a borderline case. Multiplication of any number of sharp intermediates provides no remedy, as the problem of sharp transitions from H to a first borderline case, or from any of those to any other, will remain. An epistemicist treatment of the Sorites maintains both classical logic and set-like categories, maintaining that things really do shift from heaps to nonheaps with a  single grain  of sand. Any indeterminacy is  purely epistemic: we just don’t know where the  points of sharp division are (Sorensen 1988, Williamson 1992, 1994). Indeed, since we couldn’t then recognize any proposed division as  the  right one, it  appears that  our ignorance must be irremediable (Raffman 2014). It  is  widely recognized, however, in  accord with our approach here, that the puzzlement of the Sorites relies on a genuine semantic phenomenon of categorical vagueness (Hyde and Raffman 2018, Abalerzhad and Bueno, forthcoming). Our categories themselves are vague. Premise (3) of the Sorites argument – (i )[~ H ( g i ) ~ H ( g i 1 )] – is  indeed the  thing we must abandon. What is not agreed is what the proper analysis of vagueness is, how we can best model it, and why premise (3) seems so tempting in the first place. There have been many marvelously clever attempts to address the Sorites by reshaping a logic and semantics in ways that will do some kind of justice to our conceptual categories (among the  most clever, perhaps, are Zadeh 1965a, 1965b, van Fraassen 1966, Fine 1975, and Field 2003). All have points of plausibility, but  all attempt to substitute something far more definitive for vagueness itself. This is true of the values assigned not merely by fuzzy logics but by gapped, glutted, intuitionistic, intentional, and various forms of ascending semantic hierarchies. None of these are entirely successful, and we would suggest that none can be. The simple truth, amply illustrated by the Sorites, is that our categories don’t work in  terms of discrete semantic values. All that  our categories require in practice are cases of clear applicability and clear inapplicability. Vagueness in between, and the vagueness of that vagueness, is quite plausibly an inherent, essential, and unavoidable characteristic of many or most of our categories. An alternative is thus to recognize essential vagueness by explicitly building it  into our logic, either in  terms a  vague quantifier (“balder than the great majority of…”) or a fundamental vague relation (“is saliently similar to exemplars a, b, and c…”) (Grim 2005, forthcoming). The latter, it should



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be noted, carries explicit ties to empirical work on categories and category learning noted in Chapter 3. Vagueness is in many cases a feature rather than a bug of our categorization. For a broad range of our purposes, we don’t need the precision that discrete values would give us. Our categories often require are cases of clear applicability and clear inapplicability – H and ~H – but  that  requirement carries no commitment to a  decisive value appropriate to every case. Our categories indeed carry no commitment to “every case” at  all. We don’t need the precision of discrete value for every case, and we couldn’t handle it. As  noted, demands that  we replace vagueness with exactitude  in  every case or in general would impose burdens on cognition, communication, and action, greatly hindering the  ready processing, exchange, and exploitation of information. Vagueness is  often not  a  curse but  a  blessing, facilitating rather than impeding the pragmatic purposes of information management necessary for navigating a world. One of the points that we have emphasized throughout is that categories are structured and function pragmatically in  terms of the  salient purposes and interests of different contexts. That pragmatic and contextual sensitivity of categories is a feature that has been noticed in the literature on the Sorites, even where allied with a variety of formal treatments. Fara and Stanley offer forms of epistemicism in which there is indeed a one-grain transition point for heaps, but maintain that the location of that hidden point hidden point can shift with speaker’s interests of even focus of attention (Graff 2000, Fara 2008, Stanley 2003). Contextual sensitivity is  often appealed to in  a  variety of semantic and alternative logic accounts, often with emphasis  on the  point in  a  series considered (Raffman 1994, Soames 1999, 2002, Shapiro 2006, Åkerman 2009, Hyde and Raffman 2018). Although formal accounts may often fail to capture vagueness precisely because they attempt to legislate it into something else, they can nonetheless serve as models of important aspects of the phenomenon at issue. This can be illustrated by the example of fuzzy logic, with its over-definitive legislation in terms of precise degrees of “heapness.” One of the  questions that  the  Sorites poses is  not merely what  goes wrong with the reasoning in (1) to (5) above, but why that reasoning seems so  plausible. If  the  universal generalization in  (3) is  the  culprit, as  we and many other have proposed, the question is why (3) is nonetheless so plausible. Taken as a model, fuzzy logic provides a plausible answer to such a question. In such a logic, for each step of a “forced march” Sorites from case g n to case g n+1, there will be a conditional of the form ~ H ( g n ) ~ H ( g n 1 ), for example. If  the  disparity in  truth values of the  two cases is  small, the  truth value

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of that  conditional in  a  fuzzy logic will be very high in  each case, as  will the truth-value of the universal (i )[~ H ( g i ) ~ H ( g i 1 )]. It will nonetheless be true that  at each step of the  forced march, the  H or ~H value of the  case at  issue will “leak” a  little “heapness” or truth-value, such that  at the  end we will have gone from very true H cases to very false ones. Precise values aside, all of this seems very true to our thinking about a Sorites forced march and very true to the nature of vagueness. There does seem little difference in “heapness” or “truth of heapness” between two neighboring cases. And yet discrepancy from clear cases builds systematically in multiple steps. Despite insufficiency a full account of vagueness, a range of formal treatments can offer important insights if taken as models (Grim 1997, Hyde 2008). An approach to the Sorites which leaves us with vagueness as an essential and unavoidable aspect of our categories, as we have urged here, may have an ancient parallel as well. The Greek philosopher Chrysippus (c. 280–208 B. C.), was one of the most prolific authors of the Stoic school and its ablest logician. He devoted entire tracts to several of the  most important paradoxes—five of them dealt with the Liar paradox alone.7 With the Liar in particular, Chrysippus offered the resolution (characterized as cassatio, that is, “null and void,” by the medievals) that the proposition at issue is simply meaningless.8 But limitation to a trichotomy of true/false/meaningless led the  Stoics into difficulty with the  analysis  of the  Sorites where this  machinery is  clearly insufficient. Thus it  is  greatly to his  credit  that  Chrysippus made  the  important point that  the  Sorites cannot be resolved on the basis of classing the premises as true, false, or meaningless but requires a more complete suspension of judgment.9 The  ancient Phalakros (or Bald Man) paradox is  a  close relative to the Sorites in which the movement proceeds not by adding grains to a nonheap but by removing hairs from a head. The key to understanding paradox is the search for the weak spots – those chinks in the armor of paradox where its defenses are the weakest. In the case of the Phalakros, this will once more be the iteration thesis which lays down that general rule: “loss of a single hair does not a bald man make.”10 The Millet Seed paradox of Zeno of Elea is an ancestor to these paradoxes. It  too roots in  the  effect of unnoticeably small (“subliminal”) differences, arising from the question of how it is that while dropping as single millet seed makes no sound, dropping a  bushel of them makes a  loud thud.11 Leibniz used this example as a case in point for his theory of “minute,” unnoticeably small, perceptions beneath the  threshold of consciousness. In  any case, in this instance, it is also the iteration rule akin to mathematical induction that is the weakest link in an otherwise plausible chain. Here as in the other cases, a recognition of the vagueness of the concept at issue affords a clear recognition of the weakness of that link.



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Paradoxes of Identity A  salient question is  whether certain  sorts of item-modifications are kind preserving. Consider the  category “play by Shakespeare.” One editor modernizes the  language. Another Shakespearean translates it  into Urdu. Is the  result still a  “play by Shakespeare”? A  Roman villa  is  restored and reconstructed to the  point where 90% of its stonework is  new. Are we still dealing with “a Roman villa?” At what point in such matters is the thread of kind-identity broken? When the boundaries of kind-identity are blurred, categorical sortalization becomes problematic and kind-identity issues become an invitation to paradox. Because individuation is  tied to categorization, it  isn’t surprising that  a  misunderstanding of our categories will lead to a  misunderstanding of individuation. Our categories are often vague, and paradoxes of identity and reidentification may well turn on a misunderstanding of that vagueness. But  there are also  other features of our categories – and other sources of potential misunderstandings regarding categorization – that appear in these paradoxes as well. The  classic presentation of the  Ship of Theseus problem appears in Plutarch’s Lives. On returning to Athens after defeating the Minotaur, The  ship on which Theseus sailed with the  youths and returned in safety, the thirty-oared galley, was preserved by the Athenians down to the time of Demetrius Phalereus. They took away the old timbers from time to time, and put new and sound ones in their places, so that the vessel became a standing illustration for the philosophers in the mooted question of growth, some declaring that it remained the same, others that it was not the same vessel. (Plutarch circa 100 CE, 1914). The problem is posed in a still more pointed way in Hobbes’ retelling (De Corpore 1655). The  Ship of Theseus is  enshrined as  a  museum exhibit, with rotting planks occasionally replaced. It appears we have simply maintained the original ship. The  supervising carpenter, however, takes the  replaced planks home and reassembles them. If something cannot be in two places at once, and two distinct ships cannot be the same, which of these is the Ship of Theseus? The  problem will clearly apply to anything that  is  composed of parts. In a different work, Hobbes applies it to people: If  one asks: “Is a  man, when old and young, the  same being, or matter, in  number?” it  is  clear that, because of the  continual casting of [existing] body-tissue and the acquisition of new, it is not the same

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material [that endures], and hence not the same body; yet, because of the unbroken nature of the flux by which matter decays and is replaced, he is always the same man. (Hobbes, Thomas White’s De Mundo Examined, 1642, 1976, 12.4) This  too has  an ancient precedent. In  a  comic scene, the  ancient playwright Epicharmus has a debtor argue that he should not be responsible for what he owes. Our bodies and hence our selves change from moment to moment, and thus he is no longer the person who contracted the debt. The exasperated lender strikes the debtor, who objects to the abuse. In response, the lender pleads the same case: that he is no longer the person who struck him (Wasserman 2018). These puzzles all concern identity through change over time: under what changes in parts is something the same ship or the same man? A closely related class of puzzles, again with ancient pedigree, concerns identity, number, and the “boundaries” of individual things. Chrysippus, as represented in Philo, poses a puzzle concerning Dion and Theon, that part of Dion that excludes his left foot. Do we have two people, or one? And if Dion’s left foot is amputated, do they both survive? In contemporary guise, with an emphasis on number, this is the “Problem of 1,001 Cats” (Geach 1962, 1967, Lowe 1983, 1985, Unger 1980, Wiggins 1980, 2001, Lewis 1993, 1999). Tibbles lays on the mat on my feet, with all of her 1000 hairs. But were we to draw a boundary that included all of Tibbles except for one specific hair, the  thing within  the  boundary would still be a cat. Similarly for boundaries with other specific hairs. These things cannot be identical, because they differ in specific hairs. Yet all are cats, apparently forcing the conclusion that there is not one cat on the mat but at least 1,000. As is typical of philosophical problems, the puzzles of identity constitute a conceptual briar patch. Our everyday operating assumptions, and our assumptions regarding those assumptions, don’t always play well together. The philosophical analysis of paradox embodies precisely the recognition of those conflicts. As illustrated above with regard to the Sorites, navigating the briar patch often calls for sacrificing some operating assumption, or some assumption regarding those assumptions, in order to maintain coherence or consistency (Rescher 2001), as illustrated above with regard to the Heap. The philosophical history of the  paradoxes of identity is  a  history of contrasting attempts to make such sacrifices. Here perhaps we should sacrifice or temper concepts of identity (Lewis  1993). Perhaps we should recognize an unparsimonious ontology of multiple objects in the same space (Wiggins 1980, 2001). Perhaps we should deny the existence of standard objects altogether (Unger 1979).



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Much of the air of paradox disappears, however, with minimal sacrifices in  our assumptions regarding our assumptions, if we recognize the  fact that categories lie at the core of the paradoxes and that the assumptions to be abandoned often involve a  misunderstanding of the  nature of our own categories. Paradoxes of identity rely on two primary sources of misunderstanding: (a)  a  failure to recognize the  vagueness and pragmatic relativity of catego­ rization, thinking incorrectly of categories on the model of sets, and (b)  conflating concepts of sameness and identity as  they normally function in terms of categories with a  far more technical ideal that  we’ll term “Leibnizian identity.” Categories are not  sets, and concepts of sameness and identity normally operative in  terms of categories are not  Leibnizian identity. We will use “identity” in an ordinary and nontechnical sense of sameness throughout this section, turning to its contrast with Leibnizian identity only toward the end. We have emphasized in  Chapter  1 that  categories are crucial for both cognitive processes of both generalization and individuation. Categorization in terms of kinds is necessary not merely for giving us groupings of things, allowing for induction across the group, but often for the basic identification of the individual things that we group. For an item to be identified is quite generally for it to be identified as a specific element of a  category, distinct from other elements of that  category, and the conceptual borders of that individuation are inevitably a consequence of the category at issue. Phrased in classical terms, particulars and universals are conceived together, dual aspects of a single process of categorization.12 There are three aspects of individuation that are of particular importance with regard to the paradoxes at issue: identity, counting, and edges. All of these are tied to individuation. Sameness and identity are inseparable from individuation. The fact that our categorization individuates what is before us as a statue dictates at the same time the conditions under which something is  the  same statue, a  statue identical to this  one. That, in  turn, tells us when two statues are not  the  same, and thus are two: the  basis  of counting. And individuation defines edges: spatial, temporal, and even modal. A  statue may be distinct from its constitutive material on the  basis  of our categorization, and may exist through time in  spite of changes to its basic structure. Whether this would be the same statue under certain transformation, or would have been had it had a different history, will be dictated by the principles of individuation consequent on the category under which we consider it.

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Identity in a common sense is simply the question of whether two things are the same thing: whether this is the same statue that perched on your bookcase yesterday, or whether this lump of clay is the same one I manipulated yesterday. Identity in that sense – the ordinary sense of “the same” that is ubiquitous in our conceptualization – is at least often category-bound. Because salient categories in use are often pragmatically determined by the purposes at hand, whether two things are “the same” in the relevant sense is often determined by the categories pragmatically in use. There was a precious Ming vase on the shelf yesterday. A pile of shards lies on the floor today. Are these the same thing? Without a contextual specification for the relevant category at issue, the question may simply not be well-defined: the only practicable answer may be “yes-and-no.” Context relativity can occur even when a single term is at issue, used in different contexts with a different significance. “Is Henry drunk the same person as Henry sober?” The answer may well be “yes” in  relation to biological kinship or property ownership, but “no” in relation to personality, amicability, or moral responsibility.13 At  the  core of the  paradoxes we’ve outlined lies the  concept of identity. But  when viewed in  terms of this  ordinary and category-bound sense of identity as “the same X,” many of the paradoxes disappear. And even where the conceptual fog is not entirely dispersed, it is often lifted: an understanding in terms of categories offers a more satisfactory even if not total philosophical understanding of what is at issue. The  category of “statue” is  importantly different than the  category of “lump,” distinct in the individuation conditions it brings with it as it is distinct in  the  pragmatic interests that  it  serves. The  identity conditions of statues are clearly tied to the  spatial form that  they take, and often to the  artistic conditions under which they were formed. Interestingly, a statue might lose an arm, later replaced, and in that respect not remain the same lump of clay. Even more tellingly, the multiple casts made from Remington’s original might all be “The Bronco Buster,” despite the fact that they are clearly composed of different pools of bronze and now reside in different museums and personal collections. Statues are aesthetic objects, with the  individuation conditions appropriate thereto. Lumps of clay, on the  other hand, though malleable in form, have an identity far more finely tied to material. Halve a lump of clay and add an equal portion to what was lost and in terms of the individuation appropriate for many purposes you will have a different lump. If we attend to the categories at issue and restrict ourselves to categories of “statue” and “lump of clay,” it becomes almost impossible even to formulate the paradox at issue in terms of sameness. On reflection, “is this statue the same statue as this lump of clay?” makes little sense. “Is this statue the same lump of clay as  this  lump of clay?” doesn’t do much better. In  order to even get



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the problem off the ground we need what appears to be a third category – “thing” or “object.” Is this  statue the  same thing as  this  lump of clay? Are this statue and this lump of clay the same object? The problem of the statue and the lump of clay thus concerns not two categories but what appear to be three. Here there appear to be several options. It is perfectly possible for different categories to have different conditions of individuation, evident already in  the  categories of “statue” and “lump of clay.” The  same lump of clay may take the  form of different statues on different days, and Remington’s “Bronco Buster” may appear as  a  different parcel of bronze in  separate cities. In context, a term like “physical object” can simply add a further set of conditions of individuation. A statue sitting before me and a lump of clay sitting before me may qualify as  the  same physical object despite the  fact that that physical object would remain the same lump of clay but not the same statue if deformed. One complication here is that in context, “thing” and “object” are often themselves category-vague: they often appear to be less categories in  their own right than place-holders for explicit or assumed categories. In a context in which garden plants are salient, my question as to whether what appear in two photographs are the same thing may carry the force of “is this the same plant?” or alternatively “is this the same species of plant?” Where books are at  issue, the  question of whether this thing is  the  same as  that  thing may be a  question regarding a  single physical copy, different editions, or the  same literary work. “Thing” and “object” thus appear to have something like the anaphoric function of pronouns rather than higher-level nouns, inheriting a contextual categorization rather than rising above it. A  recognition of an ordinary sense of category-relative identity, then, does much to dispel the  air of paradox regarding the  statue and the  clay. Much the same holds for the counting versions of the paradoxes of identity.14 Just as identity in the sense of “the same thing” can be bound to a specific category at issue, and precisely because identity can be so bound, counting can be bound to a category as well. Where lumps of bronze are the category as issue, what you see in a museum in Ogdensburg New York, and in Forth Worth Texas are clearly not the same: these are two distinct lumps of bronze. Where statues or works of art are the relevant category, the same Remington work is exhibited in the Remington Museum in Ogdensburg New York, and the Amon Carter Museum in Fort Worth. Remington didn’t return to the subject: there is only one “Bronco Buster,” which you can see in either place. Here a final element in the mix is the vagueness of edges. Many of our categories, we have noted, are circumscribed not  in  terms of necessary and sufficient conditions but  in  terms of similarity to a  paradigm or set of exemplars. Spatial, temporal, and modal edges can often be far from specific.

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How many cats lie on the  mat  at my feet? Just one. “Cat” and indeed “Tibbles” are categories indeterminate in terms of numbers of hairs. Were Tibbles to have had one fewer hair or one more she would still be Tibbles and would be the  same cat; such are the  vague edges under individuation consequent on these categories. Some presentations of this  paradox introduce new categories as  well: categories cat1, cat2, and so  forth, envisaged as  applying only to a  region of Tibbles exclusive of specific hairs. Each of these is a cat, and yet cannot by definition be identical, leaving us with many too many cats on the mat. But once we have recognized that different categories have different conditions of individuation, and moreover that  common categories such as  “cat” are vague, it  becomes clear that  Tibbles considered exclusive of specific hairs is still Tibbles, and is still the same cat, although perhaps not the same catn given those artificial demarcations. What, finally, of the Ship of Theseus? Here recognition of the vagueness and pragmatic contextualization of categories is the key to lifting the philosophical fog. That recognition will not tell us which is “the” Ship of Theseus. But it does reveal the pragmatic questions that are genuinely at issue. Which is  the  true Ship of Theseus? That  very much depends on the purposes at  issue, and we need to recognize that  these purposes may themselves be the  subject of debate. One set of interests may demand a history of functional continuity with the  vessel that  Theseus brought from Crete. There the museum’s restoration has pride of place. Do we want to walk the planks that Theseus himself trod, feeling rails and ropes that his hands touched? If so, it is the carpenter’s reconstructed ship that is what we want. Which is  the  true Ship of Theseus can depend on what  categorization is grounded in our purposes in posing the question. The sense of sameness and identity that we have leaned on in this section has  explicitly been the  common notion of sameness and identity that  can be category-relative: to ask whether what  is  shown in  two photographs is the same is to ask whether it is the same X for a category either explicit or assumed in context. There  is, however, another sense of identity, explicitly formulated by Leibniz: two things are Leibniz identical only if they have all properties in common. This  sense of identity undoubtedly has  classical precedents, and may linger as part of our conceptual toolkit as well. The paradoxes of identity would not have even an initial air of perplexity, and would not have had such in antiquity, were Leibniz identity or something like it not a tempting concept beside the category-relative sense outlined. Nor is Leibniz’s a sense of identity entirely distinct from that in common use. If I have a statue before me, or



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a lump of clay, or my favorite artwork, everything true of that statue, lump of clay, or artwork must be true of it: none of these can both have a property and lack that property. The Leibniz criterion must be handled with extreme care, however, particularly with regard to what is admitted among the properties over which the Leibnizian criterion quantifies. Sam Clemens was the same person as Mark Twain, and yet I might cite Mark Twain as my favorite author and deny that honor to Sam Clemens. Does Twain then have a property – being cited as my favorite author – that Clemens lacks? What  is  clear is  that  the  paradoxes of identity reappear if we insist on formulating them in  terms of Leibniz identity: full Leibnizian identity is a major tangle in the conceptual briar patch with which we began. In the Tibbles case, a given cat1 cannot be Leibniz identical to any cat2, since they differ in the property of number of hairs. If each is a cat, we do indeed have too many Leibniz-distinct cats. Precisely because of the different category-individuation conditions we’ve noted, a statue and the clay of which it is composed must be Leibniz distinct. Indeed the statue, the lump of clay, and the physical object before us must be Leibniz distinct. We end up gazing at three rather than one Leibniz object. In each of these cases, then, a major portion of our philosophical perplexity may trace to a bait-and-switch between two senses of identity: categories of “identity,” appropriately enough, which carry different conceptual demands. If  phrased in  terms of everyday and category-relative identity we get one answer to the problems. If phrased in terms of Leibniz identity we get a very different answer. It is unsurprising that conceptual conflict arises when we conflate the two. It  is  not clear that  the  perplexity of the  Ship of Theseus can be laid at Leibniz’s door.15 But to the extent that the Ship of Theseus poses a pragmatic question of which category and conditions of individual we should care about, Leibniz’s criterion of identity is at any rate of little help. The most it would assure us is  that  the  museum’s remodeling is  distinct from the  carpenter’s reconstruction, a  matter obvious and the  grounds of the  problem even in  terms of a  far more familiar concept of category-relative sameness and identity that we have emphasized throughout. Puzzles of ordinary identity and reidentification seem resolvable, or at least understandable, in terms of principles of categorization. Items can be viewed in  different classificatory perspectives based on different functional characterizations, and when seen in one perspective alone may not be subject to considerations applicable in  the  other case. Thus the  same mass of clay can be the bust of a friend and the only gift from a beloved wife. A different assemblage of wood can be the same charted vessel.

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Is There a Highest Category? Although it  does not  strictly constitute a  classical paradox, the  question of a  highest category raises many of the  questions we’ve addressed. Is there a universal or highest category, containing all things? The  question has  several aspects or angles. The  first question is what the status of a  “highest” would be. As  with other questions regarding categories, the question can be approached from two directions: metaphysics and epistemology. From a metaphysical angle, we would get started on the question of objects, entities, or things in terms of the actual or real (ens realis). From an epistemic angle of approach, we would get started in  terms of thought-objects via conceptualization (ens rationis). Metaphysically we might propose that the highest genus would be that of possible existence, admitting of different modes of being. From the epistemic angle, the highest genus would be that of thought objects, admitting of different modes of conceptualization. Throughout, we have emphasized that  categories occupy a  space that  is  both metaphysical and epistemic. By  tradition, moreover, correct thought is a matter of adequation to reality. Thus in both cases (metaphysical and epistemic), we get basically the same trio of objects, entities, and things. The philosophical quarrel has been whether actual being or conception-ofbeing is to be taken as fundamental. From either approach, questions of the possibility of a universal or highest category remain, with similar difficulties on each side. There are two major difficulties at  issue: whether a  universal collectivity of any kind exists, and whether any such collectivity – even if possible – would qualify as a category. In  the  Metaphysics, Aristotle argues against a  highest genus inclusive of all objects, things, or beings of any sort whatsoever (998b23, 1059b31). His argument is  based on a  principle of classification: that  every genus must be definable in terms of differentia that fall outside that genus. The only differentia applicable to all things, however, would be being or thinghood: hardly a  differentia  at  all, and presumably not  something falling outside a  category of everything. With such a  principle there can be no highest genus. There are other arguments to much the same effect. Michael Dummett and Aimee Thomasson both emphasize that individuation, even in naming, requires an implicit  identification in  terms of a  category (Dummett 1973, Thomasson 2007). To  individuate, identify, or reidentify any “thing” is to do so  in  terms of one or more categories under which it  falls, and by implication under which it  fails to fall. In  line with Aristotle’s comments, inclusion in  a  universal category would fail to differentiate its members as individuals, failing to individuate them from anything or from each other.



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Membership in such a category would thus be metaphysically empty, failing to determine a field of identifiable “things” at all. From an epistemic direction, Dummett also  argues that  since “things” would have to include things of any category, speaking of all things would seem to require speaking of all things of any category whatsoever, whether or not those categories have yet been discovered, imagined, or invented, or indeed whether they could ever be. If we can have no grasp of all possible categorial concepts, it  seems that  we can “form no definite conception of the  totality of all objects which could be spoken of” (Dummett 1973, 566–567). Were there either a highest universal category or collectivity, it would seem plausible to ask purely generic questions in  terms of number: without any specification of “thinghood,” how many things are there? Hilary Putnam argues that  purely generic existence questions or claims of this  sort are ill-formed. There are alternative categorizations, giving us alternative catalogs of “things” and thus alternative numerical accountings of “things.” What things there are, Putnam argues – and how many – can only be answered within a given categorical system or “version” (Putnam 1987, Thomasson 2007). Any universal category would entail a  universal collectivity, to which unrestricted quantification over “everything” would seem a natural correlate. But the logical grounds for rejecting any universal collectivity are well-known. By an argument structured along the lines of Russell’s paradox, within any universal collectivity would have to be a  collectivity of all and only those collectivities that did not include themselves. We are then forced to a violation of the law of excluded middle: on pain of contradiction, such a collectivity could neither contain itself nor fail to do so. By an argument structured along Cantorian lines, a universal collectivity would have to include each of its sub-collectivities as members. But then by various versions akin to the argument for Cantor’s theorem, it would have to contain more members than it contains. In a further reflection of the problems of identity, its elements could not even be put into one-to-one correspondence with themselves (Grim 1991). We have explored elsewhere the  radical (though breathtaking) revisions of logic that would be required to envisage plena, collectivities adequate for a genuine totality of facts (Rescher and Grim 2008, 2011). The crucial point here is simply that basic and familiar logical principles are firmly lined up against the possibility of any such totalities. It should be admitted, however, that  a  simple denial – “There  is  no totality of everything” – has  at  least the appearance of using the very concept it denies (Lewis 1991). This doesn’t by any means rule out rejection of a “highest genus,” but it does call for some subtlety in the handling and interpretation of negative claims (Grim 1991).

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In  the  face of clear logical obstacles to any unrestricted quantification, Timothy Williamson attempts a  reconstruction in  terms of quantification first over all expressions, and secondarily over the  things to which they apply (Williamson 2003). As Thomasson argues, however, such an approach threatens either to be restricted to the  limits of a  language or to demand quantification over all possible expressions or categorical concepts in  ways that leave open once again the threat of Russellian and Cantorian paradoxes (Thomasson 2007). Both independently and collectively these points offer reason for suspicion regarding any supposed highest genus or universal category. To  these, we can add reasons from the  pragmatic approach to categories emphasized throughout. If  categorization is  a  purposive device in  the  management of thought, then what is to be said of a hyper-categorization that encompasses all categories at large? It seems implausible. If categories are constituted in terms of salience with regard to contexts and purposes, a universal category would seem to demand some purpose with regard to which everything is  salient. But any such “universal purpose” would render membership in a “universal category” an indeterminate and empty characterization. Membership in a universal category would arguably fall short of the  distinctions and differentia metaphysically required for individuation and thinghood at all. In line with Dummett’s arguments above, it is not clear that we have any grasp of an entirely unlimited field of purpose. It is therefore unclear that we have any grasp of what “things” a category of “all things” would include. Although categories and collectivities seem to attract the  same kinds of logical difficulties in  terms of totalities, we should finally note a  major distinction between them that  we have relied on throughout. With an eye to the  particular contexts and purposes that  define categories as  salient groupings, categories are neither sets nor any other kind of crisp and purely extensional grouping. A grouping including all living American marsupials and the coins currently in  my pocket inevitably forms a  sharply defined and extensional set. But  for neither natural history nor numismatics does it form a category. A vague grouping of potentially useful designs for cutting tools may fail with regard to both extensionality and crisp edges. But for purposes of woodworking and tool design, it may be quite prominent as an important category. The logical difficulties of collective totalities arise because on some intuitive level, it seems there must be such totalities, and that they must be sharp-edged in terms of membership (Rescher and Grim 2011). The pragmatic understanding of categories we have offered throughout relieves us of any correlate pressure to think either that categories must obey the sharp-edged logic of sets or that they must be ordered in terms of some highest genus or overarching category of everything.



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Sancho Panza and Protagoras A  puzzle invariably arises when we face a  situation involving categories X and non-X that takes the form of: (a) A  purported categorical dichotomy of all relevant items into X’s and non-X’s and (b) A  relevant item x that  cannot coherently (consistently, meaningfully) be classed neither as an X nor as a non-X. This structure is manifested in the case of the Liar paradox, the subject of the next section. But it also appears in lesser puzzles as well. For some of these, at least, a satisfactory resolution is to reject the supposed grounds for (a). An historic example is  afforded by the  Sancho Panza  Hanging paradox of Cervantes’ novel Don Quixote (Bk. II, Chap. 51). Here Sancho Panza  is  made  governor of an island where he  must uphold a  curious law that  stipulates that  on arrival visitors are to be questioned about their plans and to be hung if they answer falsely. One day Sancho had to resolve the problem posed by a visitor who, when asked what he would be doing on the island, responded “I am here to be hung.” Now if he is hung, the answer will be true and the penalty inappropriate, while if he is not hung, the answer will be false and the law would require his hanging.16 The situation that results answers to the following summary: The traveler is

the traveler speaks

the law therefore requires

Hung

Truly

Not hung

Not hung

Falsely

Hung

In no case can the traveler’s fate be brought into agreement with the law's requirements. This  situation gives rise to the  following aporetic cluster of individually plausible but ultimately inconsistent theses: (1) The law should always be obeyed. Requirements of law should always be met. People ought (always) to do as the laws require. (2) Cannot implies need not: what people cannot possibly do is never actually obligatory for them. (Ultra posse nemo obligatur as the Roman legal maxim maintained.) (3) Ought implies can: what people legally should do must be possible for them. From (2) by contraposition. (4) In  some circumstances – such as  those indicated by the  example of the Hanging story – people cannot possibly do what the laws require.

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(5) In some circumstances people need not do as the laws require. (From (3) and (4).) (6) (5) contradicts (1). Here {(1), (2), (4)} constitute an inconsistent triad. Now (1) specifies a plausible legal ideal, while (2) is a basic axiom of juridico-ethical rationality. And (4) is an aspect of the defining hypothesis of the situation. The overall rating of priorities is thus:

 4   2  1 And in consequence, maintaining (2) and (4) while sacrificing (1) seems the appropriate resolution for the paradox. Sancho’s sagacious response took exactly the  line “Whichever way I decide, the law is going to be disobeyed, so I might as well be merciful and let this wretch go free.” And this makes perfectly good sense. The premise that “The law is to be obeyed: what is done should be in accordance with the law” needs to be abandoned in the case at hand precisely because this case is so constructed that there just is no possible way for the law to be obeyed. In  violation of (a) above, the  law does not  in  fact prescribe a  categorical dichotomy (answering truly and not being hung / answering falsely and being hung) that can consistently apply to all cases. Another juridical example of a similar structure is afforded by the ancient paradox of the Contract of Protagoras. This paradox, widely disseminated in classical antiquity, arises from the following story: The Greek sophist and teacher Protagoras contracted with some students for a tuition fee to be paid if, but only if, the student won his first case. One of these, the  clever Euathlos, sued in  court for free tuition with the argument: “If I win this case then by the court’s judgment I will owe nothing. And if I lose, then I will owe nothing thanks to the contract, since this is my first case.” Protagoras, being no less clever, of course, argued exactly the reverse.17 The paradox can be represented by the following aporetic pair: (1) The court’s finding should be in accord with what the contract stipulates. However – (2) In  the  specified circumstances, that  contract and the  court’s ruling are bound to disagree on the following basis:



CATEGORY MISTAKES AND PHILOSOPHICAL PARADOXES The Court’s Finding

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Winner of the Case

The Contract Specifies

A fee is due

P

No fee is due

No fee is due

E

A fee is due

Thus in no case can the court’s finding be brought into line with the contract’s specification. Now since thesis  (2) represents the  defining hypothesis  of the  problem, there is  no arguing with it. The  only available option is  to sacrifice (1), invoking the  legal axiom that  an absurd contract is  no contract because it  invalidates itself (conventum absurdum non  est conventum). In  violation of (a) above, the contract does not in fact prescribe a categorial dichotomy (winning the case and fee is due / losing the case and no fee is due) that can consistently apply to the case at hand. To  be sure, the  general considerations of aporetics indicating that  (1) should be abandoned do not tell us what – if anything – should be put on its place. This requires a closer look at the substantive detail of the particular paradoxical situation that  is  before us. And on this  basis, the  court’s only sensible option is to make a ruling and to prioritize it over what the contract specifies in  the  circumstances (thereby abandoning (1)). Here the  salient feature of this  case is  that  the  situation between the  contending parties is entirely symmetric, exactly as the response of Protagoras indicates. Thus by deploying the  basic principle of justice “Treat  like cases alike” the  Court’s sensible ruling would be to divide  the  amount in  contention (the  student’s fee) equally between the parties, awarding just half of the agreed amount to Protagoras. To be sure, this requires the explicit – and deliberate – sacrifice of thesis (1), but this is entirely appropriate in the circumstances. In any event, however, this paradox is decisively resolvable since it is clear which aporetic premise must be abandoned. The  only real problem here is that of why – that is, of establishing just exactly how the justifying grounds for this abandonment are to be articulated – and what is to be put in its place. It deserves stress that this sort of thing is often the most challenging and difficult aspect of paradox management, a point illustrated in both our discussion of the Sorites in a previous section and our discussion of the Liar in the next. In comparing the Sancho Panza paradox and the Contract of Protagoras paradox one observes that  they have exactly the  same generic structure, illustrated in  our tabulations. On  this  basis  the  Stoic paradoxologists of classical antiquity characterized such paradoxes as based on “table-turning” (antistrophê),18 in  which a  fundamental symmetry obtains so  that  whatever can be said on one side  can be answered by an equivalent counterpart on the other. In each of these cases not just equally cogent but in fact structurally identical arguments can be given both for a thesis p and for its negation not-p.

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Exclusiveness, Exhaustiveness and the Liar Some of the lessons of Sancho Panza and Protagoras carry over to the much more heavily debated and logically more serious case of the  Liar. We will not  to “solve” the  paradox in  these few pages. Here as  in  other cases, our primary goal is to understand the structure of the paradox, a structure clear in terms of the categories involved. Just as the classic image of categories is of set-like entities satisfying sharp necessary and sufficient conditions, the classical ideal of classification is one of exclusive and exhaustive sub-categories under any division. Just as paradox can easily arise when ordinary categories are assumed to fit the classic picture but don’t, paradox can easily arise where classificatory conditions of exclusivity and exhaustiveness are assumed to hold but do not, or indeed cannot. Classifications that  allow an entity to be in  each of two categories, though these are taken to be dichotomous and exclusive, will clearly lead us to inconsistency. Classificatory gaps that  leave some items out, resulting in  a  situation of classificatory homelessness, can lead to paradox as  well. This will be the case when a division is taken to be exhaustive and it is not: some further category has been left out, inclusion of which will often be appealed to in an attempt to resolve paradox, as it is in both the Sorites and the Liar. What we will term the simplest Liar is this: (1) This sentence is false. In  the  standard argument, we assume truth and falsity of sentences to be exhaustive sub-categories of semantic value. Thus (1) must be either true or false. But if (1) is true, it is false. If it is false, it is true. In either case, we violate an equally strong assumption that truth and falsity are exclusive – a genuine dichotomy applicable in all cases. This  is recognizably the  structure of both semantic paradoxes such as the Liar and set-theoretical paradoxes on the order of Russell’s paradox. The  paradoxes arise precisely because truth and falsity or set-membership and non-set-membership are taken to be both exclusive and exhaustive, and yet we are faced with items such as (1) or a set of all non-self-membered sets which seem to violate at least one half of that assumption. The  main  point here is  simply that  this  class of paradoxes is  fueled by classificatory issues of exclusiveness and exhaustiveness. Given linguistic resources of propositional self-reference, if we categorize truth and falsity as both exclusive and exhaustive the paradox of the Liar is simply inevitable. It  is  clear what  has  gone wrong: despite our understanding of our understanding of truth and falsity, enshrined in basic logical principles, our common categories of truth and falsity cannot be treated as both exclusive



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and exhaustive. But  here as  in  the  case of Protagoras, though it  is  clear what has gone wrong it is not clear what to do about it.19 In this regard, there are at least two general routes to resolution that have been attempted: gluts and gaps. One route is to deny that the distinctions at issue – truth and falsity, for example – should properly be treated as exclusive. Perhaps something such as (1) can be both T and F, instantiating a semantic “glut.” This, of course, amounts to a violation of the Law of Contradiction. Some sentences can be both true and false. Such is the route of dialetheism (Priest 2006a, 2006b, 2008, a different approach appears in Rescher and Brandom 1979.) A  far more traveled route is  to deny that  the  distinctions at  issue are exhaustive: to reject the  dichotomy at  issue as  improper, introducing some further alternatives. This amounts to a denial of the Law of Excluded Middle. Not every sentence, we can propose, is either true or false. (1) is an exception, demonstrating that there are “gaps” between truth and falsity. With a middle value – perhaps “meaninglessness” (M) – we can categorize (1) as neither true nor false but M. This  route can boast an ancient historical pedigree. We’ve noted that Chrysippus (c. 280–208 BC) wrote entire tracts on several of the most important paradoxes, with five devoted to the  Liar Paradox in  particular.20 With the  Liar, Chrysippus offered the  resolution that  the  sentence at  issue has a middle value of meaninglessness.21 A simple Chrysippean strategy may be sufficient against the simplest Liar. But it fails as a strategy against Liar-like sentences in general. Given a middle value M, we immediately face the strengthened Liar (2) This sentence is either M or false. If true, (2) is something other than true. If either M or false, it appears it must simply be true. In more sophisticated forms, the denial that truth and falsity should be treated as exhaustive categories appears in various contemporary forms in multi-valued logics, supervaluations, and other gapped approaches (Rescher 1969, van Fraassen 1968, 1970, Kripke 1975). We have argued elsewhere that given linguistic capacities broad enough, even these more sophisticated forms often face forms of the  strengthened Liar (Grim 1991, Rescher and Grim 2011). We have framed such an argument in  terms of a  Paradox of Paradox Analysis: Could there be a  consistent semantic category for the  Liar and all its relatives, definable in  terms of necessary and sufficient conditions for Liar-like paradox? That does indeed seem to be the goal of a full range of attempts at “solution,” carrying the promise of an avoidance of paradox by

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recognition, classification, isolation, and avoidance of sentences that  meet those conditions. But  there can be no necessary and sufficient conditions for Liar-like paradox, and in  that  sense no semantic category adequate to the  purpose. For suppose we have a set of conditions C proposed as necessary and sufficient for paradox on the general model of the Liar. And then consider (3): (3) Either this sentence meets conditions C, or it is false. Sentence (3), like any other, must either meet conditions c or not. If it does meet conditions C, it  will be simply true by virtue of its first disjunct – and thus not  paradoxical at  all. In  that  case, our conditions C will not  be sufficient for paradox. If  it  fails to meet conditions C, on the  other hand, the first disjunct will be false and the truth of the sentence as a whole will rest on the second disjunct alone. Under those conditions, the sentence will selfattribute falsehood precisely as the simple Liar does, and will be paradoxical for precisely the same reasons. It fails to meet C, but is paradoxical nonetheless: in that case, our conditions C will not be necessary for paradox. No set of conditions C can be both necessary and sufficient for Liar-like paradox (Rescher and Grim 2011). In the case of the Sorites, we suggested that our categories are essentially and unavoidably vague. All that  our categories require are cases of clear applicability and clear inapplicability. That requirement carries no commitment to a decisive value appropriate to every intermediate case. A similar position is possible here. We do indeed require clear cases of truth and falsity. But not only may we deny exhaustiveness by recognizing the phenomenon of gaps, we need not be able to assign a semantic value in  every case of gaps. As  we have argued elsewhere, what  is  required is  a  strong denial of the  Law of Excluded Middle: It  is  not merely that sentences may be neither true nor false, but they may be neither true, false, nor x for any x or plurality of x’s proposed (Rescher and Grim 2011). Like vagueness, we would propose, the strong failure of a Law of Excluded Middle may be an inherent and unavoidable characteristic of some of our ordinary categories.

Lessons from Category Mistakes and Mistakes Regarding Our Own Categories •• There are two basic forms of category mistake: —Misclassification under a category. —Applying improper standards of category membership.



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•• There are further ways in which paradox arises through a misunderstanding of the nature of our own categorization: —Our categories may be essentially vague. —Even sameness and identity may be category- and context-relative. —Our categories may violate exhaustiveness and exclusivity. •• The vagueness of our categories is clear from the Sorites and its kin. Formal attempts to analyze or replace that vagueness in terms of something more discrete typically underestimate the essential and unavoidable vagueness of our categories, but may nonetheless serve as important models of some of the phenomena at issue. •• Paradoxes of identity and counting – the Ship of Theseus, the Statue and the Clay, the 1,000 cats – rest on a failure to recognize the category- and context- relativity of individuation. Our common category-tied concepts of identity are importantly non-Leibnizian. •• Paradoxes involving reflexivity such as Sancho Panza and the Liar clearly reveal the  limits of assumptions of exclusiveness and exhaustiveness in classification. Attempts at  “solution” for the  Liar – in  particular, in  terms of gaps – may again underestimate the depths of indeterminacy within our semantic categories. •• The most fundamental error is to suppose that categories of one type are actually quite different, raising paradox by treating categories of one type according to the ground rules of another.

Notes 1 Aristotle, Soph. Elen., II, 6. 2 On the Megarians in general see Zeller, Philosophie der Griechen, Vol. II/1, pp. 244–275. For  Eubulides see Diogenes Laertius, Lives of the  Philosophers, II  x 108–110; Prantl, Geschichte, I, pp. 21, 50–58; Aristotle, Soph. Elen., 179a32ff. 3 Pretty well all that is known about Eubulides derives from Diogenes Laertius, Lives of the Philosophers, bk. II, sect.’s 106–20. See Zeller, Philosophie der Griechen, Vol. II/1, pp. 246. 4 Zeller, Philosophie der Griechen, Vol.  I/2, p.  1380. On  the  Sophists in  general see the larger context of this passage on pp. 1371–1384. 5 On this aspect of the Megarian teaching, see Prantl, Geschichte, Vol. I, pp. 41–58. 6 On this paradox and its ramifications see Chapter 2 of R. M. Sainsbury, Paradoxes (2nd. ed., Cambridge: Cambridge University Press, 1995), pp.  23–51. Originally the  paradox also  had a  somewhat  different form, as  follows: Clearly 1 is  a  small number. And if n is a small number so is n + 1. But this leads straightway to having to say that an obviously large number (say a zillion billion) is a small number. (See Prantl, Geschichte, Vol. I, p. 54.) 7 Zeller, Philosophie der Griechen. Vol. III/1, p. 116. For this paradox see Chap. 10 below. 8 See  Rüstow (1908), p.  115: Cassantes autem dicunt, quod dicens se dicere falsum nihil dicit, or again  non est verum nec falsum, quia  nullam tale est propositio. See also Prantl, Geschichte, Vol. IV, p. 41.

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9 See Prantl I, 489. Sextus Empiricus, Adv. Math, VII, 416. Cicero, Academica, II, 29 and 93. See Zeller, Philosophie der Griechen, Vol. III/1, p. 116. 10 Again, breaking up the  general rule into a  large series of particulars will make this  paradox more baffling. For  now while we can identify the  region in  which the  remedy of premiss rejection must fall, we cannot pinpoint the  exact location at which this must be effected. 11 The paradox also took a variant form in antiquity: A single drop of water effects no change on the form of a stone, while a long-continued succession of drops wears a hole into it. (See Aristotle, Physics, 253b14.) 12 The role of categories in individuation is also stressed in Hale (1987), Lowe (1989 and (2009), and in treatments of Frege in Dummett (1973) and Wright (1983). A treatment of paradoxes of identity largely consonant with ours appears in Thomasson (2007 and 2017). 13 Hampton (2017) traces a  similar relativity to domain  of discourse with respect to “fish” (p. 100). 14 Thomasson (2007) uses the success of categorical-relativity as a response to identity and counting problems as an argument that all category-neutral formulations of these are ill-formed. 15 It  may be that  the  Leibniz criterion can be handled in  such a  way as  to allow for vagueness, one of the aspects emphasized above. The requirement that two things must share all properties in  order to be Leibniz identical need not  exclude  vague properties; though hardly part of the tradition, two things might be Leibniz identical though vaguely defined in all the same regards. It may also be that the Leibniz criterion can incorporate pragmatic concerns, as  long as  those are built into the  properties at issue. 16 This  classic paradox also  has  the  ancient variant of the  Nasty Crocodile paradox mentioned twice by Diogenes Laertius (Lives of the  Philosophers, 44 and 82). Having snatched her baby the nasty crocodile turned to the father: “Answer carefully,” he said, “for your baby’s life depends on your truthfulness: Will I eat your baby?” After thinking for a moment the cagey father replied: “Yes, I do believe you will.” For this paradox see Prantl, Geschichte, Vol.  I, p.  493. (Ashworth 1974, p.  103, discusses the  variant Bridgekeeper Paradox featuring the functionary who threw liars into the water.) 17 On ancient discussions of this paradox see Prantl, Geschichte, Vol. I, pp. 493–494. 18 See Prantl, Geschichte, I 493–494. 19 An approach that we don’t belabor here may seem to more closely parallel that suggested in the case of Sancho Panza and Parmenides: to deny that (1) is a sentence, or to insist that it is propositions that are true or false and that (1) fails to express a proposition. On the obstacles to a propositionalist approach see Grim 1991, pp. 18–25. 20 Zeller (1919), Vol. III/1, p. 116. 21 See Rüstow (1910), p. 115: Cassantes autem dicunt, quod dicens se dicere falsum nihil dicit, or again non est verum nec falsum, quia nullam tale est propositio. See also Prantl (1855), Vol. IV, p. 41.

Chapter 6 ETHICAL AND SOCIAL CATEGORIES It is not merely in terms of logic, metaphysics, and the natural world around us that we use categories in an attempt to predict, explain, manipulate, and understand. We employ and apply categories in ethical and social cognition as well. Categorical misapplications with regard to the natural world can result in failed attempts to predict, explain, understand, or intervene in the course of nature. Categorical misapplications in  the  social realm can carry an additional burden of ethical mistake and negative social consequences. There are a range of lessons to be learned here, both regarding the nature of categories – studied here in a realm independent of “nature’s joints” – and with regard to social construction and ethical action.

Categorization Errors in Social Cognition: Oversimplification A  major purpose of categorization is  simplification. The  world around is  a  complex place, with no event exactly repeated in  all particulars and no process repeatable without some variation. Effective cognition and effective action demand that we work with a simplified representation of such a world, in  which items, events, and processes are conceptually grouped in  terms of categories. One clear danger in  any such categorization is  oversimplification: a categorization that is too simple to do justice to the genuine complexity with which we are attempting to deal. That oversimplification can be particularly troublesome when it is complex social realities which we are trying to address. Consider for example categories very often used in thinking about groups of people in  our society, and the  ethical emotions we find appropriate for different groups: •• The homeless. •• The mentally ill. •• The drug-addicted. •• Petty criminals.

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We sympathize with the homeless and want to relieve their plight. A similar sympathy seems appropriate with the mentally ill, though the measures to be taken would seem to be very different. With regard to the drug-addicted we may have mixed emotions, particularly in recognizing lives that have been characterized by repeated bad but  voluntary choices. In  thinking of petty criminals our sympathy may waver or disappear. What is wrong with this exercise is the illusion that these categories are genuinely distinct. Many people fall into several or all of them. The  most extensive survey of the  homeless, taken on a  single night in  2014, reached the consensus that 45% of the homeless had some form of mental illness, with serious mental illness in an estimated 25% (Torrey 2021). Didenko and Pankratz (2007) found that two-thirds of the homeless report that drugs and/or alcohol were a major reason for their becoming homeless, though substance abuse can be a  result of homelessness as  well as  a  cause (National Coalition for the Homeless 2009). It is estimated that not merely petty criminals but  at  least half of American prisoners have some form of mental health concern, and 10%to 25% have some form of serious mental illness: major affective disorders or schizophrenia. Dependence on drugs, alcohol, or both is common in the criminal population (Collier 2014). Taken singly, the  social categories above and the  ethical sentiments with which we often approach them appear to be far too simple. A genuine understanding, as well as a socially effective and ethically acceptable approach to the issues represented by these categories, would have to recognize common dynamics that drive them and the interrelationships between those dynamics. If our categories are inappropriately simple they can blind us to social complexities in a way that leads to pervasive ethical and social error. Oversimplification of a  very similar kind can occur in  categorical generalizations – even true generalizations – and the inferred linkages they can fallaciously be taken to suggest. Each of the following factors correlates with having a higher risk of lung cancer: •• Having quicker access to matches or lighters. •• Buying more tobacco products. •• Having close family members diagnosed with lung cancer.1 •• Living in a household with more ashtrays. Given that better medical care can be expected to produce earlier diagnoses, even having more frequent medical checkups and better medical care can be expected to correlate with earlier diagnoses of lung cancer. In each case we may thus have a true generalization: that those with better medical care are more likely to face an early diagnosis  of lung cancer, that  those living



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in households with more ashtrays have higher lung cancer risks, that having a  close family member diagnosed with lung cancer increases one’s own probability of such a diagnosis, and the like. But precisely because of the range of correlated factors, it  would obviously be wrong to infer explanatory connections from those statistical generalizations. Despite the truth of each of these generalizations, the causes of lung cancer are not to be sought in pure genetics, in the purchase of tobacco products, or in the presence of ashtrays. What we have is a conglomerate of tangled correlations between categories. But among those categories it is only smoking that is genuinely explanatory. The  same issue of misidentifying the  genuinely explanatory factors among a conglomerate of correlations arises with regard to social categories. In a society with the yawning inequities of our own, each of these shows a statistical correlation with membership in certain minority ethnic or racial groups: •• Living below the poverty line. •• Having poorer access to quality education. •• Having poor health care. •• Having reduced occupational opportunity. •• Having a higher unemployment rate. •• Living in higher crime areas. •• Being more likely to be the victim of crime. Given the facts of correlation, there will thus be a range of generalizations – true generalizations – linking minority status with other factors. All too commonly, however, those statistical generalizations are taken to support an inference regarding the explanatory role of minority status itself – precisely as if we inferred that ashtrays cause cancer. The  social causes and consequences of inequality in  our society are almost undoubtedly complex, with tangled conglomerates of correlations and categories. Poverty, its causes and effects, unequal education, unemployment, criminal as opposed to legitimate working opportunities, a cycle of imprisonment, and unsupported release – all of these are likely part of the dynamics. Search for a  single simple explanatory category is  bound to fail in  our attempts to understand and address that complexity.2 All too often the error of oversimplification involves latching on to a very specific category – minority status, ethnicity, or race – as “the” explanation, with “true generalizations” offered as  if they supported that  fallacious inference. At  that  point, the  consequences of oversimplification extend in  egregious ways beyond the  epistemic into the  ethical. Latching onto not  merely a  single explanatory category but  a  fundamentally wrong one

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can reinforce social injustice. Faulty analysis  using wrong and oversimple social categories can perpetuate and exacerbate precisely the  inequities it is fallaciously thought to explain.

Stereotypes Categories are necessary and inevitable in order to navigate a complex world, a  truth that  holds in  navigating a  social world as  much as  a  natural one. But  with the  inevitability of social categories come the  dangers of social stereotypes. Stereotypes represent four intertwined dangers: •• The danger of oversimplification. •• The danger of the wrong category. •• The danger of false attribution. •• The danger of overgeneralization. The first two of these dangers are those noted in the previous section. It is typical of stereotypes to oversimplify a  complex phenomenon in  inappropriately simple terms, particularly in terms of categories of people – simply in terms of the homeless, the addicted, or petty criminals, for example. It is also typical of stereotypes to focus on what may well be the wrong category: to associate high crime areas  with race, for example, rather than with socio-economic status, education, or demographic opportunity level. Stereotypes are a  form of categorization, and as  such they make a significant contribution to the issue of cognitive economy. We often lack the information that would be necessary to deal with the multitude and variety of individual cases in detail. In that all too familiar kind of situation we are forced to treat people in terms of groups: Spanish-speaking children in the case of elementary education, for example, or the  unvaccinated in the  midst of a  pandemic. But  in  treating people via  group classifications it is  crucial to get the  relevantly presupposed facts right. Thus if we are to operate an eligibility criterion for driving licenses we had best ensure that  nineteenyear-olds are safer drivers than seventeen-year-olds. If  military service is  to deny combat  roles for women, the  decision had better be based on and be prepared to provide validating data. The functional relevancy of classificatory differentiations is (and should be) crucial. Traditional stereotypes have carried not only oversimple characterization but  quite generally false attribution. This  has  often been in  terms of race or ethnic background. Racial and ethnic stereotypes were tracked from 1933 to 1969 by a  series of studies known as  the  “Princeton trilogy” (Katz and Braly 1933, Gilbert 1951, Karlins, Coffman and Walters 1969).



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All three studies asked undergraduates to assign the 5 adjectives from a list most characteristic of given racial and national categories. In 1933 Germans were rated scientifically-minded and industrious but  also  extremely nationalistic, for example; Irish as  pugnacious, quick-tempered, and extremely religious; Italians as artistic, impulsive, and passionate. It is dubious to what extent any of these characterizations held for any sizeable proportion of the designated populations. It  is  notable that  the  later studies of the  trilogy found that characterizations became less monolithic, often changed significantly in content, and tilted more positive – at least those explicitly held and admitted to in such a survey. In addition to the other dangers, stereotypes inevitably carry the danger of overgeneralization. That danger may be of particular concern in social matters. Overgeneralizations regarding the calorie count of cereals may involve thinking of things incorrectly. Overgeneralizations regarding people on the  basis  of stereotypes can be expected to result in thinking of people not merely incorrectly but unjustly. We take it as a matter of principle that people have a right to be thought of in terms of their individual characters and capabilities, rather than in terms of a social category within which they fall. Insofar as  people have a  right to be thought of in  terms of their individuality, categorization must be thought of in  ethical beyond merely epistemic terms: in terms not merely of general accuracy but dignity, respect, and rights. There is thus admittedly a tension between the epistemic goals of categorization in  general – a  pragmatically successful simplification for dealing with the  world around us – and ethical principles regarding how people have a right to be thought of. These  issues, sometimes treated under the  term “moral encroachment,” simply extend rather than contradict our general thesis regarding the broadly pragmatic context of proper categorization (Moss 2018). Just as that context may be one of specific goals in  specific endeavors, it  may be ethical and social with regard to certain  visions of humanity and justice. That  can apply, moreover, not  merely with regard to the  specific categories at  issue but the fineness or granularity with which categorization is made. The fourth danger of stereotypes is  that  they tend to overgeneralize by cutting things far too roughly, ignoring the  complexities of individuals much as  they oversimplify the complexities of social phenomena. Although it  is  negative stereotypes that  come first to mind, there are also  stereotypes that  involve arguably positive features: that  Asians are intellectually diligent and good at math, for example, or that women are warmer and more caring than men. In terms of false attribution, however, even these positive stereotypes often come with negative accompaniments: that  Asians are good at  math but  emotionally cold, for example, and that  women are

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warm but weak (Siy and Cheryan, 2016). All the other dangers of stereotypes would apply to positive as  well as  negative stereotypes: oversimplification, overgeneralization, and the use of the wrong category. We should not  be surprised that  epistemology and ethics are entangled where social categories are in question: this is just one of a variety of levels on which social questions and controversies arise regarding the ethical and political handling of information. If nearsighted and color-blind people are marginally less safe as  drivers, would that  justify charging them different insurance rates? Is there a reason to investigate inherent gender differences, or not  to? When do natural inequities call for “leveling the  playing field,” when not, and in what way? How much evidence of how much harm (from smoking cigarettes, drinking alcohol, or texting while driving) is required to warrant social intervention? Different information will entail different ethical conclusions, but  different ethical approaches will dictate that  we should handle information in different ways as well.

Ethical Joints In metaphysics and science, the search for proper categories is traditionally thought of as an attempt to “cut nature at its joints.” Can the same view be held with regard to ethical and social categories? Is it plausible that the world has ethical or social joints in the same way that it has chemical or biological joints? The view that there are genuine ethical categories that are embedded in the nature of reality would demand a particularly strong ethical realism and absolutism. The tradition of “natural rights,” ethical entitlements that people have simply in  virtue of being people, would suggest precisely that  view. A Kantian notion that there are some categories of behavior or treatment of fellow agents that are everywhere demanded simply because they are rational agents and thus members of a  kingdom of ends would reflect such a  view as well. We don’t merely have ethical categories, but  have conceptions of the character of those categories. On some realist views, those categories are indeed dictated by the nature of the world or the beings within it. But even a realist view of ethical joints does not entail that those joints will be empirically revealed. We don’t have an empirical test for the existence of rights in the sense that we have a litmus test for the presence of an acid. We don’t have a moral microscope that lets us examine an interaction in search of a perceptual property of wrongness or impermissibility. Here there are alternative routes in  our attempt to understand ethical categories. One route is  to abandon realism in  favor of a  broadly social constructivist view: that in terms of ethics, at least, “man is the measure of all



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things” and “morality was made for man, not man for morality.” On a view of this  kind, ethics is  a  code  of social behavior arrived at, grounded, and constituted socially. It is clear that legal rights are grounded in legal institutions and legal traditions, socially constituted throughout. On one approach, moral rights in general should be thought of in something of the same light, though far less formal in structure. It is we in social interaction, rather than a nature independent of us, that grounds our ethical categories. Another route is  to push back on the  notion that  ethics is  not open to empirical test, or at  least that  ethical evaluation is  not open to empirical input. Here a  major step is  to recognize that  our ethical concepts are far more complex than is often thought. Our ethical concepts include not merely “right” and “wrong,” not  merely “rights” and “obligations,” but  a  range of evaluations of people and the  things they do, the  impulses and motives from which they do them, the  opportunities, alternatives, constraints, and limitations under which they operate. Our ethical concepts include concepts of generosity, cruelty, selfishness, betrayal, self-sacrifice, love, friendship, weakness of will, promises, lies, reciprocity, deception, modesty, courage, integrity, obedience, humility, coercion, hypocrisy, manipulation, loyalty, abuse, diligence, respect, kindness and much more. We certainly do consider the  empirically observable details of an interaction in  deciding that it  was  a  genuine act of self-sacrifice or an obvious instance of cruelty, in deciding whether a  promise was  actually given or deliberately violated, for example. If our ethical concepts are not autonomous, hermetically sealed from the  rest of our cognition – if our ethical concepts are informed by, blended with, or supervenient on concepts much closer to the  empirical – then at least a form of realism might be maintained that insists that we can see ethical joints in the structure of human interactions. There is a sense in which we don’t have to decide between these two routes if we take them as complementary rather than as opposed. Ethical codes are characteristically written in  the  language of realism: rights grounded in  rationality or mere humanity, for example, and justice grounded in terms of characteristics that genuinely are of moral significance and characteristics that are not. In that regard, the understanding “internal” to an ethical code speaks in language akin to the Declaration of Independence: self-evident truths regarding natural rights, for example. There  is  a  sense in which ethical codes cannot be understood without recognizing that as lived and acted within they carry that aura of realism. But there is also a truth regarding ethical codes that takes the very different perspective that characterizes the Constitution: “We the people, in order to form a more perfect union…” This is a socially constituted and socially contracted code. In the Constitution that character is explicit, but in other ethical codes

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it  will not  be: codes socially evolved, socially arrived at, socially agreed and accommodated to, but codes that are ultimately the product of social interaction in the governance of social interaction. Game-theoretic portraits of cooperative principles in ethics as an evolved social equilibrium offer contemporary versions of such an approach (Skyrms 1996, Alexander 2007). Here as elsewhere we may be dealing with a phenomenon accessible from alternative perspectives, roughly and admittedly inadequately characterized in  terms of an “inside” and an “outside.” Other phenomena  of such a  character may include  observations characterized in  terms of indexicals, semantic meaning, and consciousness (Nagel 1986, Grim and Rescher 2012). Sunrises may offer an analogy: though fully real, they can be seen only from a specific spatial position at a specific time, but can only be fully understood in  terms of a  “view from nowhere” conception of the  revolution of planets independent of a  specific place and date. In  much the  same way, realism and social construction both apply to ethical codes and ethical concepts. On such a view, the difference is the perspective from which the phenomenon is  being considered: a  perspective of realism built into the  code  itself, or the constructivist perspective that sees the development of the code as a social product.

Ethical Histories and Ethical Contrasts Viewed as a social construct, ethical codes and the categories that characterize them clearly have a shifting context and a shifting history. The categories of agents to which an ethical code is taken to apply is itself historically variant. In the code of ancient Romans, the crucial categorical distinction was between citizens and noncitizens. In many medieval codes, and even many today, the  crucial distinction is  between believers and nonbelievers. Codes have marked as  crucial distinctions between those owning property and those without, between aristocrats and commoners, between Aryans and non-Aryans. Depending on the  cultural context of time and place citizenship, religion, status, race, or some comparable factor have each been given primacy in the categorizations of fellow humans. Such distinctions will be reflected in the philosophical works of a time. Aristotle’s ethics distinguishes between fully rational agents and slaves, whose souls lack a governing rational component, as well as between men and women, thought of on faulty biological grounds as incomplete males. Kant’s ethics is an ethics in which it is rational agents that form the major category. Hume’s is geared to those capable of moral sentiments. A range of contemporary treatments expand those of moral concern to all sentient agents, human or animal, rational or not (Singer 1975).



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Just as  the  relevant categories of agents within  an ethical code  vary historically, so do the grounds on which action is judged. Although precursors can be read into earlier work from Aristotle to Aquinas, a  full doctrine of moral human rights in our contemporary sense wasn’t fully articulated until as recently as the end of the seventeenth-century, in the work of John Locke, Voltaire, and others. Rights have arguably become increasingly central to our ethical code. Rare is the contemporary debate that isn’t at some point phrased in terms of rights, whether negative rights of liberty and choice or positive rights of entitlement to a decent wage or health care, for example. Both the  central categories of an ethical code, therefore (rights, duties, virtues, and vices, for example), and who it is about (rational agents, sentient agents, or citizens) are historical variables. So, of course, is  the  content of an ethical code  in  terms of what  actions are prescribed and proscribed, and the basis on which actions are taken to be ethically obligatory, neutral, or impermissible. The  “experiment of Darius” is  a  classical example from Herodotus, retold in Montaigne: Darius asked some Greeks what it would take to persuade them to adopt the Indian custom of eating their dead fathers (for that was the ritual among Indians who reckoned that the most auspicious burial they could give their fathers was within themselves): they replied that nothing on earth would make them do it. Then he made an assay at persuading those Indians to abandon their way and adopt that of the Greeks (which was  to cremate their fathers’ corpses): he  horrified them even more. (Montaigne, 1580, “On Habit,” in Screech 2003) This example is used in Herodotus, Montaigne, and elsewhere as an example of the radical variance of ethical codes. But it might equally be taken as an example of how intertwined ethical codes are with broader metaphysical views regarding the  nature of the  spiritual universe. Darius’s Greeks and Indians agree that there is a fundamental value in honoring one’s dead. Their disagreement is more superficial than that, tied to views of the appropriate way of instantiating that central value. An example of both ethical code  difference and ethical code  change is the category of usury, or the  charging of interest – any interest – on money loaned. Usury was  widely condemned in  Medieval Europe, classified explicitly as a sin by the Church. Anselm equates usury to theft (Kirschenbaum 1985). But interest on money loaned is both fundamental to and definitional of capitalism, which is not merely characteristic of our modern world but aspects of which are often applauded in ethical terms, such as the self-determination and self-advancement of entrepreneurship.

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Just as  Darius’s differences between Greeks and Indians may ref lect a  background beyond central values, however, the  same may be true of usury. It  can be argued that  usury was  indeed a  serious threat  to the  stability of the  feudal economy characteristic of medieval Europe  – as  much as  prohibition of interest on loans would be to the  stability of the  contemporary capitalism characteristic of our world. A  conservative value of avoiding serious threats to an existing economic order may thus ground both the  medieval prohibition against usury and the  fact that  it  survives in  our ethical codes today only in  an extenuated form as a prohibition against loan sharking. It can be extremely difficult to tell whether apparent disagreements between ethical codes are fundamental disagreements or not  – independent of or dependent on non-ethical spiritual beliefs or economic contexts, for example. An instructive example is that of two notable attempts to determine whether the ethical code of another culture differed from our own in fundamental or largely superficial ways. In the 1950s, two philosophers worked with two different Native American cultures in an attempt to answer such a question. The philosopher John Ladd worked among the  Dine or Navajo (Ladd 1957). The  philosopher Richard Brandt worked among the Hopi (Brandt 1954). They came to radically different conclusions. Traditional Hopi beliefs are very different from those of most Americans. But  Brandt concluded that  many of these differences did not  qualify as fundamental differences. The Hopi had a sharper disapproval of alcohol and intoxication than is characteristic of most Americans, but they also have a long history of difficulties with alcoholism, which may lie in both genetic predispositions and social impact. The  Hopi had a  strong disapproval of sexual relations between cousins of the  same clan, despite biological relationships distant enough that they would not be frowned upon by most Americans. Given the importance of social structure of Hopi clans, however, and the consequent threat posed by inter-clan sexual contact, this too might be explained by wider facts of social context rather than fundamental ethical differences. Indeed the one fundamental ethical difference that Brandt found was  something very different: a  greater Hopi tolerance of what  would be regarded elsewhere as cruelty to animals. John Ladd’s conclusions regarding the  Dine were radically different. On  his  reconstruction, the  Navajo code  is  essentially egoistic. Time and again his informant Bidaga would tell him that the goal of action is to promote the personal welfare of the agent. According to Ladd’s analysis, the welfare of others, the common good, and general altruism did not generally appear within Navajo ethics.



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Why did two philosophers examining Native American cultures in the same period come to such radically different conclusions regarding fundamental differences? One possibility, of course, is  that  Dine and Hopi culture are very different, both historically and today. But another possibility is  that  the  difference between Ladd and Brandt’s conclusions – or at least the  extent of that  difference – reflects their methodologies as  much as characteristics of the cultures themselves. Brandt’s attempt was to extract an “average” moral view. Toward that end, he worked with a range of different informants. Ladd picked out a single individual as a Dine “moralist”: Bidaga, son of Many Beads. Both philosophers were intent on disentangling ethical beliefs from religious beliefs, factual beliefs, or aspects of social context. That disentanglement is of course crucial to any attempt to answer questions of fundamental ethical differences. But in that attempt at disentanglement, Ladd and Brandt used very different criteria  for what  was  to count as  part of the  ethical code. Ladd followed Durkheim in taking morality to be a system of conduct with “priority” and “legitimacy”: ethical considerations are those that are taken to have priority over other considerations and to have a legitimacy intended to bind both listener and speaker. Bidaga’s dictums were delivered in precisely that spirit. Brandt used very different criteria, and took himself to be studying “ethically affective reactions” rather than merely explicit  statements. First and foremost, Brandt took it that in order to count as ethical a reaction must be disinterested “…in  the  sense of not  being causally dependent on any of the  individual’s personal desires or attachments for particular persons…” On  Brandt’s criteria, Bidaga’s dictums would not  count as  ethical precisely because they were explicitly egoistic. On Brandt’s criteria, what Ladd gave us may have been Bidaga’s egoistic worldview, but was not what his title promised: the structure of a moral code. Such an example points up a lesson implicit in Herodotus and Montaigne’s “experiment of Darius” and the  example of usury as  well, reinforcing a  point made  with regard to ethics and empirical content: we would have great difficulty in disentangling a “pure” ethical code from the rest of our beliefs and motivations regarding the world, both natural and social. Ladd and Brandt both had great difficulty in separating ethics from other actionguiding beliefs, and came to different conclusions at  least in  part because of very different criteria  for what  counted as  the  ethical code  they were trying to disentangle. At  least one lesson is  that  we should not  expect our ethical categories to be hermetically sealed from the  rest of our cognition, to answer to a  distinct pragmatics rather than being part of a  network of belief, individual action, and social interaction that  answers to pragmatic considerations as a whole.

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One can categorize people either individually by their personal dispositions and nature (as Galenic physicians categorized them as choleric, melancholic, etc.), or collectively by their mode  of collective socio-political organization (as  Plato categorized modes of communal organization as  democratic, oligarchic, etc.). An interesting and seldom-investigated question arises on this  connection, namely: “In  which mode  of social organization do people of a given psycho-physical category flourish best?” It is commonly supposed that a liberal democratic capitalist mode of socio-political organization best enables individuals of all sorts to live most satisfactorily. But social scientists rarely examine this  particular question of cross-categorical relationship head-on. In theory, at least, it is perfectly possible that some of us are asked to pay a price so that most of us can flourish.

Ethical Alternatives Might ethical codes, though very different in  content, serve the  same pragmatic ends – meaning that  there might be no pragmatic reason for preferring one over another? This certainly does seem to hold in some cases. According to Malinowski’s classical portrayal of the Trobriand Islanders (Malinowski 1922), the bonds of obligation and child-raising responsibilities between biological fathers and their children are relatively weak. The  bonds of obligation and childraising responsibilities between a  mother’s brother and her children, on the other hand, are taken to be extremely strong. Understanding this pattern is somewhat complicated by Malinowski’s report that the Trobrianders don’t recognize facts of biological conception. But it is clear that the codes of male paternal responsibility are almost directly the reverse of our own. We take a father’s responsibilities for his biological children to be extensive, serious, and deep. We take an uncle’s responsibilities for nephews and nieces to be variable and far less binding. It  is  Trobriand uncles, in  contrast, that  play the role and carry the responsibilities of American fathers. Is one of those ethical codes superior to the other? If it is the proper care of children and the equal distribution of male responsibility that are at issue, there seems little reason to think so. As far as we know, children might be equally well cultivated and provided for on either system, with a comparable distribution of responsibility across the male population in general. In this case, interestingly enough, we might find two ethical codes equally acceptable even though only one of them is our own. We might do so both from the perspective on ethical codes as socially developed to serve general social ends, and from the perspective of things we value within our own code, such as a nurturing environment for children.



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There are other cases, however, in which these perspectives will inevitably come apart. There will be cases in which, though we might recognize the social stability and historical evolution of an alternative ethical code, we cannot find it acceptable. The historical origins and social impact of a Nazi code embodying antisemitic genocide must indeed be intellectually understandable. The same must be true for the  cultural and economic centrality of human sacrifice among the  Aztecs and Mayans (Kubler 1990). But  in  neither case can we recognize these as acceptable alternative ethical codes. This shouldn’t be surprising. We cannot simply shed our ethical concepts or categories any more than we can shed our scientific concepts and categories. Even in recognizing that ethical codes arise in historical context, we must operate in a specific historical context, and that context is our own. To understand that codes may have historical alternatives is not to embrace them as ethically acceptable alternatives. Every ethical code carries the force of ethical realism when lived and acted upon “from the  inside.” Despite the fact that we hold tolerance and open-mindedness to be virtues, there will be limits to what we can see as ethically acceptable alternatives. Among those limits are any ethical acceptance of genocide and human sacrifice.

Ethical Change Our scientific theories, structured in  part in  terms of categories we take to correspond to natural kinds, change directly in  the  face of empirical experience. In  that  sense, our empirical contact with the  world offers validation for our scientific claims and scientific categories. But  it  does not  seem plausible that  ethical claims change or are validated in  precisely the same way in terms of empirical experience. The separation between “is” and “ought,” though perhaps not absolute, seems too great a gap for that. How then do ethical codes change? In what way are they validated? Short of divine revelation, perhaps, independent validation for an autonomous ethics does not  seem to exist. But  as  we’ve argued above, ethics is  not autonomous in many ways. As previous examples demonstrate, our attitudes and beliefs as  to how we should act and interact are inevitably entangled with other beliefs regarding the nature of ourselves and of the world, physical and spiritual. Those other beliefs may be empirically impacted. Even if not directly impacted by empirical experience of the physical world, precisely because ethics is not autonomous, ethical beliefs and ethical categories may change in virtue of what we might think of as ethical experience. Such is the case with many people’s experience of war. Many a soldier has gone off to war with beliefs about his  country, himself, the  enemy, and the  honor and glory of combat only to come back with very different views of all of these.

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The physical realities of war, empirically experienced, can change ethical views and ethical categories. Reduction of prejudice offers a further example. One of the hypotheses regarded as  best confirmed within  social psychology as  a  whole is  “the  contact hypothesis.” Members of one’s own group and of others are quite standardly schematized in  terms of categories. According to the contact hypothesis, categorical prejudice against members of one group by members of another will be reduced with increasing social contact between the  two (Alport 1954). This  hypothesis  underlies a  number of social policies, its most famous association being the  desegregation of U.S. public schools (Patchen 1982, Schofield and Sagar 1977, Stephan 1978). Support for the hypothesis comes from laboratory, field, and survey methods alike (Cook 1985, Desforges et al. 1991, Robinson 1980, Sigelman and Welsh 1993, Stephan and Rosenfield 1978). Although the  reality of the phenomenon seems established, there are different explanations offered for why it occurs (Grim et al. 2004). What is clear is that it is the experience of contact with others in  a  category that  can break down prejudicial stereotypes, resulting in ethical change. Examples in a previous section also suggest that change in other aspects of social context can produce ethical change. Change from a feudal to a free market economy may well follow a historical pattern of forces beyond ethical belief, and yet will result in a change in ethical codes regarding the category of usury. But  even if ethical change can be effected by context and empirical experience “at one remove,” there certainly is a sense in which ethics is not an empirical science and a degree to which it is autonomous. What can lead to ethical change in that regard? A  primary motivator for ethical change is  consistency, in  a  broad sense including the force of analogy. If a code pays lip service to one set of principles but carries positive parables or valued historical examples of action contrary to those principles, consistency will demand that  something has  to give. In  reflective equilibrium, ethical evaluation of specific cases may change. But principles on higher levels can also be changed, abandoned, or replaced as well. Harriet Beecher Stowe’s Uncle Tom’s Cabin offered an easily grasped narrative that  contradicted stereotypical categories of the  enslaved and the  principles that  those categories seemed to support. As  such it  fueled the ethical changes demanded by abolitionism. In an anecdote that may be apocryphal, Lincoln is said to have greeted Stowe as “the little woman who wrote the book that made this great war” (Vollaro 2009). Contact between cultures can result in confrontations between ethical codes that result in changes in one or both. In part, this may be a change in background context. But if bits of each code are adopted, consistency can be a motivator for



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ethical change as well. From the perspective of cultural evolution, it may also be that patterns of interaction within one social group can advantage that group in  competition with another. Evolution of cooperation may well be a  case in point (Sober 1992, Boyd and Richerson 1985, Lewens 2015). Alasdair MacIntyre argues that  our own situation is  one in  which we have inherited bits and pieces of very different ethical codes, each of which was at home in a specific context but which are fundamentally inconsistent. “What we possess, if this view is true, are the fragments of a conceptual scheme, parts of which now lack those contexts from which their significance derived” (MacIntyre 1981, p. 2). If our own principles and parables are inconsistent – embodying conflicts between appeals to autonomy and liberty and appeals to justice and equality, for example, using appeals to rights on both sides – we can expect ours to be a time of ethical conflict driving ethical change.3 In attempting to understand how ethical codes change, and to what extent there are internal mechanisms even in  the  absence of external empirical validation, the law can again serve as a useful example. Ethics is not the law, nor law ethics. But the ways in which formal law changes can offer a model of some of the ways informal ethical codes can change as well. Ronald Dworkin’s view of the action of the Supreme Court in carrying forward an inevitably reinterpreted history applied to new cases is perhaps even more vivid if taken as a picture of our attempts to apply and adapt past ethical categories and principles to new issues. The Court is writing on each occasion or ought to see itself as writing a  new chapter in  a  story that’s been going on for a  very long time. And that means that there’s a constraint on the exercise of its creative imagination. The constraint is just this – it’s got to see itself as part of history, as the story it’s telling is a development of a story that’s been told before. (Dworkin, Moyers interview 1987) In  light of new cases, other principles, and “fresh moral insight,” we can similarly decide that a principle regarding liberty, long held though inherently vague, is best interpreted in one way rather than another. We can decide that our categories regarding agents, actions, or motivations were inadequate or even unethical. We can find ethical motivation from within our ethical codes to change those codes themselves. That  is  quite plausibly a  sense in  which there is moral progress. In the case of law, Dworkin also emphasizes that the process of adaptation and reinterpretation in  light of past practice and principle doesn’t dictate a single route forward. There are always alternative options. In the ethical parallel, options for ethical change always involve ethical alternatives.

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But that fact is not unique to ethical change. It holds for scientific change and changes in our mathematics as well. Taking “pragmatics” broadly enough, and despite clear differences in different areas, change in categorization in all areas of cognition is fundamentally motivated by pragmatics.

Lessons of Social and Ethical Categories A number of points emerge as salient in our consideration: •• Oversimplification is  a  common danger: attempting to deal with a complex social phenomenon as if one category using one characterization was sufficient. •• Stereotypes instantiate the danger of simplification, together with dangers of applying the wrong category, of false attribution, and of overgeneralization. The appropriate granularity of categories used in social context may itself be an issue of ethical concern. •• Although there will be differences between ethical codes, the question of which differences are genuinely fundamental – independent of context and non-ethical beliefs – is a delicate and difficult one. •• Our ethical beliefs and categories are quite generally entangled with other beliefs and categories regarding people, social roles, and the  nonhuman world. In many ways, ethics is not as “autonomous” as is often thought. •• There will be ethical alternatives that can be recognized as equally legitimate. But judgment is always from within an ethical code as it is within a scientific milieu, and there will be elements of other ethical codes that we inevitably reject as unacceptable. •• Understanding both the process of ethical change and the grounds of ethical validation requires a broadly pragmatic perspective that recognizes the roles of “empirical experience,” consistency, and reinterpretation of the code itself.

Notes 1 Not because of a genetic link but because family members tend to live together and thus even non-smoking members will be exposed to secondhand smoke. 2 In a recent work Cailin O’Connor shows how easily the introduction of social categories in the attempt to solve coordination problems can result in a stable but inequitable equilibria, even without relevant differences or prior bias (O’Connor 2019). 3 Though we appeal to MacIntyre’s “disquieting suggestion” of inherited bits and pieces within our ethics, we don’t see that as motivating his conclusion that “we have – very largely, if not  entirely – lost our comprehension, both theoretical and practical, of morality” (Macintyre 1981, p. 2).

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Sober, Elliott, “No Model, No Inference: A  Bayesian Primer on the  Grue Problem,” in Douglas Stalker (ed.), Grue! The New Riddle of Induction (Chicago, IL: Open Court, 1994), pp. 225–240. Sorensen, Roy A., Blindspots (Oxford: Clarendon Press, 1988). Sorensen, Roy A., Vagueness and Contradiction (New York: Oxford University Press, 2001). Stanley, Jason, “Context, Interest-Relativity, and the  Sorites,” Analysis 63 (2003), pp. 269–280. Stephan, Walter G., “School Desegregation: An Evaluation of Predictions Made in Brown v. Board of Education,” Psychological Bulletin 85 (1978), pp. 217–238. Stephan, Walter. G. and David Rosenfield, “Effects of Desegregation on Racial Attitudes,” Journal of Personality and Social Psycholog y 36 (1978), pp. 795–804. Storms, Gert, Paul De Boeck and Wim Ruts, “Prototype and Exemplar-based Information in Natural Language Categories,” Journal of Memory and Language 42 (2000), pp. 51–73. Studdert-Kennedy, Michael, Alvin M. Liberman, Katherine S. Harris and Franklin S. Cooper, “Motor Theory of Speech Perception: A Reply to Lane’s Critical Review,” Psychological Review 77 (1970), pp. 234–249. Studtman, Paul, “Aristotle’s Categories,” The  Stanford Encyclopedia  of Philosophy (Spring 2021 Edition), Edward N. Zalta  (ed.), URL = https://plato.stanford.edu/archives/ spr2021/entries/aristotle-categories/. Tanaka, James W. and Marjorie Taylor, “Object Categories and Expertise: Is the Basic Level in the Eye of the Beholder?,” Cognitive Psycholog y 23 (1991), pp. 457–482. Taylor, John R., Linguistic Categorization (New York: Oxford University Press, 2004). Thomasson, Amie, Ordinary Objects (New York: Oxford University Press, 2007). Thomasson, Amie, Ontolog y Made Easy (New York: Oxford University Press, 2015). Thomasson, Amie, “Categories,” The  Stanford Encyclopedia  of Philosophy (Summer 2019 Edition), Edward N. Zalta  (ed.), URL = . Tijsseling, Adriaan and Stevan Harnad, “Warping Similarity Space in Category Learning by Backprop Nets,” in M. Ramscar, U. Hahn, E. Cambouropolos and H. Pain (eds.), Proceedings of Simcat  1997: Interdisciplinary Workshop on Similarity and Categorization (Edinburgh: Department of Artificial Intelligence, Edinburgh University, 1997), pp. 263–269. Torrey, E. Fuller, “Homeless Mentally Ill Facts and Figures” Mental Illness Policy Org, https://mentalillnesspolicy.org/consequences/homeless-mentally-ill.html retrieved 2021. Travis, Charles, “Pragmatics,” in  Bob Hale and Crispin  Wright (eds.), A  Companion to the Philosophy of Language (Oxford: Oxford University Press, 1997). Trendlenburg, Friedrich, Geschichte der Kategorienlehre (Berlin: G. Bethge, 1846; photoreprinted London: Wentworth Press, 2018). Unger, Peter, “There Are No Ordinary Things,” Synthese 41 (1979), pp. 117–154. Vollaro, Daniel R., “Lincoln, Stowe, and the  ‘Little Woman/Great  War’ Story: The Making, and Breaking, of a Great American Anecdote,” Journal of the Abraham Lincoln Association 30 (2009), pp. 18–34. Wasserman, Ryan, “Material Constitution,” The  Stanford Encyclopedia  of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.), URL = . Watanabe, Satosi, “Theorem of the  Ugly Duckling,” Pattern Recognition: Human and Mechanical (Hoboken, NJ: Wiley, 1985) http:///www.kamalnigam.com/papers/ thesis-­nigam.pdf.

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INDEX

Bates, Marcia J. 69 Baumer, Michael 30 Belke, Benno 70 Bellman, R. E. 105 Berlin, Brent 61, 66–67, 69 Bird, Alexander 79 Blewitt, Pamela 69 Borges, Jorge Luis 21–22 Braly, K. W. 134 Brandom, Robert 127 Brandt, Richard 140–141 Brown, Roger 69 Bueno, Otavio 110 Buroker, Jill Vance 39

Carlson, Gregory N. 15 Carroll, John W. 79 Carroll, Lewis 92 Cartwright, Nancy 75 carving nature at its joints 73, 75, 76, 98 ethical analog 136–138 category mistake 102–105 Cauchy, Augustin-Louis 76 Cervantes, Miguel de 123 Chakravartty, Anjan 79 Cheryan, Sapna 136 Chierchia, Gennaro 15 Chisholm, Roderick 44 Chrysippus 39, 112, 114, 127 classification 8, 9, 11, 23 Cleve, James van 92 Collier, Lorna 132 color 5, 28–29, 62, 65 concepts 7, 9 Constitution vs. Declaration of Independence, ethical concepts 137 contact hypothesis, and prejudice 144 context 11 Cook, Stuart W. 144 Correia, Jose Pedro 66, 67 counterfacutals 79–86 Crisafi, M. 69 Crossman, Reinhard 45 cutting nature at the joints 28, 85

Campbell, Joseph K. 73 Cantor, Georg 121

Daehler, Marvin W. 69 Darius, experiment of 139

1,001 cats paradox 101, 118, 129 Abasenzhad, Ali 110 Abramson, Arthur S. 63 Ackrill, J. L. 34 Åkerman, Jonas 111 Alexander, J. McKenzie 138 Alexander, Samuel 44 Alport, Gordon W. 144 animals, color perception in 61 Anselm 139 Aristotle 3, 4, 10, 19, 33–36, 45, 48, 107, 120 Armstrong, David M. 16, 79 Arnauld, Antoine 39 Ashby, F. Gregory 66, 68 Avarguès-Weber, Aurore 62 Avicenna 33, 36–39, 50

162

THEORY OF CATEGORIES

Declaration of Independence vs. Constitution, ethical concepts 137 Deemter, Kees van 23 Descartes, René de 76 Desforges, D. M. 144 Dickens, Charles 15 Didenko, Eugenia 132 Diogenes Laertius 39 Dion and Theon paradox 101, 114 discrimination 62 Dummett, Michael 4, 20, 121 Dupré, John 28, 78, 79, 86 Dworkin, Ronald 145–146 Einstein, Albert 76 Epicharmus 114 Érdi, Peter 64 essentialism 91 Estes, William K. 68 ethical relativism 138–143 Eubulides of Megara 107 Euler, Leonhard 76 exemplar theory 67–69, 71 Fara, Delia Graff 111 fictions 26, 91 Field, Hartry 110 Fine, Kit 110 Fischer, Julia 63 Folger, Tina A. 105 Fraassen, Bas van 110, 127 Frege, Gottlob 7, 30, 33, 43, 45 Friend, William 76 Gärdenfors, Peter 67 Gauss, Carl Friedrich 77 Geach, Peter 114 generalization 18, 18–22, 115 Giaquinta, Mariano 77 Gilbert, G. M. 134 Goodman, Nelson 24, 84, 93–99 Graff, Delia 111 gray 61 Greco, Carolyn 69 Green, Patrick A. 63 Green, Steven 63

Grice, Paul 70 Grim, Patrick 112, 121, 122, 127, 128, 138, 144 grue 93–99 Haack, Susan 105 Hacking, Ian 79, 85 Hale, Bob 20 Hampton, James A. 7, 23, 67 Harnad, Steven 21, 61, 62, 64 Hempel, Carl G. ix, 73 Herodotus 139 highest category 120–123 Hobbes, Thomas 113–114 Hoffman, Joshua 45 Hume, David 22, 91–93, 99 Husserl, Edmund 33, 43–45 Hyde, Dominic 110–112 identification 20, 73 identity 129 paradoxes of identity 113–120 impossibilia 27 individualization 6 individuation 18–22, 115–117 induction 20 new riddle of 93–98 problem of 91–98 Ingarden, Roman 44 James, William 19, 22 Johansson, Ingvar 44 Johnson, Mark 70 Jolicoeur, Pierre 70 Jorgensen, Connie 69 Kant, Immanuel 4, 33, 40–42, 45, 48, 136 language and categorization 52–54 Karlins, M. T. 134 Katz, D. 134 Kay, Paul 61, 66–67, 69 Keele, Steven W. 67 kinds 7, 9, 15–18, 27, 73 natural kinds 9, 16, 17, 29, 74, 79–86 Klir, George J. 105 knot theory 27

INDEX 163 Körner, Stephan 45 Krifka, Manfred 15 Kripke, Saul 86, 127 Kubler, George 143 Kuhl, Patricia 63 Lachlan, Robert F. 63 Ladd, John 140–141 Lakoff, George 7, 30, 59, 67, 70 Lancaster, F. Wilfrid 69 Lange, Marc 79, 92 language 50–54 Latour, Bruno 75 Law of Excluded Middle 127, 128 Leibniz, Gottfried Wilhelm 115 Leibnizian identity 115–116, 118–119, 129 Lewis, David 16, 114, 121 Liar paradox 102, 107, 112, 126–129 Liberman, A. M. 63 Linnaeus, Carl 10, 77 Lisker, Leigh 63 Locke, John 21, 22, 33, 40, 139 Lowe, Edward Jonathan 20, 45, 114 machine learning 63–65 MacIntyre, Alasdair 145 Maddox, W. Todd 66, 68 Malinowski, Bronislaw 142 Malt, Barbara C. 30 Mareschal, Denis 66 Marler, Peter 63 Masares, Frances 76 Mayr, Ernst 77 McNeil, N. B. 67 Medin, Douglas L. 68 Megarians 107 Mervis, C. 69 Mill, John Stuart 7, 30, 79, 82 Miller, George 62 Miller, Janes D. 63 millet seed paradox 112 Minda, John Paul 67 Mitchell, Tom M. 64 Modica, Guiseppe 77

Montaigne, Miguel de 139 Moravcsik, Julius M. E. 30 Morse, Philip A. 63 Moss, Sarah 135 Murphy, Gregory 69 mystery categories 25–26 Nagel, Thomas 138 National Coalition for the Homeless 132 natural laws 79–86 necessary and sufficient conditions 6 Nelson, Douglas A. 63 Norton, John D. 92 Nosofsky, Robert M. 68 Nowicki, Stephen 63 N-rays 17, 74, 89 O’Rourke, Michael 73 Oakes, Lisa M. 66 Ocelák, Radek 66, 67 Oppenheim Paul ix, 73 oversimplification, in ethical categories 131–134 paleoanthropology, continuing debates in 77 Pankratz, Nicole 132 Papineau, David 92 paradox of paradox analysis 127–128 Patchen, Martin 144 Peirce, Charles Sanders 14, 33, 44, 48 Pevtzow, Rachel 62 Phalakros (bald man) paradox 112 Philo 114 phlogiston 17, 74, 78, 89 phrenology 78 Pierre Nicole, Pierre 39 Plutarch 113 Porphyry 28 Posner, Michael I. 67 pragmatic functionalism 54–56 Prather, Johathan F. 63 Priest, Graham 127 Priestley, Joseph 78 Princeton trilogy, racial and ethnic stereotypes 134

164

THEORY OF CATEGORIES

properties sparse and abundant 16 vagrant properties 25 Protagoras contract paradox 124–125 prototype theory 67–69, 71 Ptolemy 74 Putnam, Hilary 86, 121 Pythagoras 76 Quine, W. V. O. 24, 79, 82, 84, 93 Raffman, Diana 110, 111 Rakison, David H. 66 Ramist revolt 33, 39, 48 Ratcliff, Roger 7, 68 Reed 67 Regier, Terry 66 Reisberg, Daniel 7, 68 Rescher, Nicholas 105, 114, 121, 122, 127, 128, 138 Robinson, James Lee 144 Rogers, Leo 76 Rosch, Eleanor 7, 30, 67, 69–71 Rosenfield, David 144 Rosenkranz, Gary 45 Ross, W. D. 34 Rouder, Jeffrey N. 7, 68 Russell, Bertrand 105 Russell’s paradox 121 Ryle, Gilbert 33, 105 Sagar, H. Andrew 144 Salmon, Wesley C. 92 Sancho Panza paradox 123–126, 129 Schaffer, Marguerite M. 68 Schmidt, Stefanie 69 Schofield, Janet W. 144 scientific laws 16, 18 Sellars, Wilfrid 30 sets, categories distinct from 6, 7, 15, 22, 101, 105 Shapiro, Stewart 111 Ship of Theseus 101 Ship of Theseus paradox 113–114, 118, 129 Sigelman, Lee 144 similarity 6, 22–25, 68–69, 101

Singer, Peter 138 Siy, John Over 136 Skyrms, Brian 138 Slater, Matthew H. 73 Smith, Edward 69 Smith, J. David 67 Sneath, Peter H. A. 77 Snowdon, Charles T. 63 Soames, Scott 111 Sober, Elliott 92 Socrates 73, 107 Sokal, Robert H. 77 Sorensen, Roy 110 sorites paradox 68, 101, 107–112, 129 species 77 Stanley, Jason 111 statue and the clay 101 statue and the clay paradox 115–117, 129 Stephan, Walter G. 144 stereotypes 134–136 Stock, Wolfgang 69 Storms, Gert 68 Stowe, Harriet Beecher 144 Studdert-Kennedy, Michael 63 Studtmann, Paul 30 Tanaka, James W. 70 taxonomies 9, 13, 23 Taylor, John R. 66, 67 Taylor, Marjorie 70 Thomasson, Amie 4, 20, 120–122 Tijselling, Adriaan 64 Tobin, Emma 79 Torrey, E. Fuller 132 Trendelenburg, Friedrich 56 Unger, Peter 114 universals 30 usury 139 vagueness 22–25, 105–113 vagueness, paradoxes of 106–112 voice onset time 62–63 Vollaro, Daniel R. 144 Voltaire, F. M. A. de 139 Wasserman, Ryan 114 Watanabe, Satosi 64

INDEX 165 Weisberg, Michael 78 Welsh, Susan 144 Whewell, William 79 Whitehead, Aflred North 44 Whorf, Benjamin Lee 66 Wiggins, David 114 Williamson, Timothy 110, 122

Wittgenstein, Ludwig 50, 54 Woodworth, Robert Simpson 66 Wright, Crispin 20 Yahyâ ibn ‘Adî’ 37 Zadeh, Lotfi 105, 110 Zeno of Elea 107