Cover Page



Title Page



Preface to the Revised Edition

Preface to the First Edition


Chapter 1: Concepts of Indiscriminability

1.1 Indiscriminability and Cognition

1.2 Formal Features of Indiscriminability

1.3 The Intentionality of Indiscriminability

1.4 Direct and Indirect Discrimination

1.5 Further Reflections


Chapter 2: Logics of Indiscriminability

2.1 Logical Apparatus

2.2 The Non-Transitivity of Indiscriminability


Chapter 3: Paradoxes of Indiscriminability


Chapter 4: Concepts of Phenomenal Character

4.1 Presentations of Characters

4.2 Presentation-Sensitivity

4.3 The Identity of Characters


Chapter 5: Logics of Phenomenal Character

5.1 Maximal M-Relations

5.2 Ignorance and Indeterminacy

5.3 Matching the Same Experiences


Chapter 6: Paradoxes of Phenomenal Character

6.1 The Paradox of Observational Predicates

6.2 The Paradox of Phenomenal Predicates

6.3 The Failure of Observationality

6.4 Sorites Arguments and Necessary Ignorance


Chapter 7: Generalizations

7.1 Maximal M-Relations as Minimal Revisions

7.2 Examples

7.3 Necessary Conditions for Personal Identity

7.4 Sufficient Conditions

7.5 Close Relations


Chapter 8: Modal and Temporal Paradoxes

8.1 A Modal Paradox

8.2 Two Temporal Paradoxes

8.3 Comparisons


Chapter 9: Criteria of Identity

9.1 Forms

9.2 Functions


Appendix: Maximal M-Relations and the Axiom of Choice



Additional Notes (to the Revised Edition)

Chapter 1  Concepts of Indiscriminability

Chapter 4  Concepts of Phenomenal Character

Chapter 6  Paradoxes of Phenomenal Character

Chapter 7  Generalizations

Chapter 8  Modal and Temporal Paradoxes

Chapter 9  Criteria of Identity

References (to the First Edition)

Additional References (to the Revised Edition)


Title Page

To my mother; for my father

Preface to the Revised Edition

Identity and Discrimination was first published in 1990. It appeared in Basil Blackwell’s Philosophical Theory monograph series, edited by John McDowell, Philip Pettit and Crispin Wright. Like most volumes in that series, it was taken out of print in 1993 as a result of a change in publishing policy. Since then I have been hearing complaints that it is hard to obtain. Although I have subsequently changed my view on some of the topics with which it deals, not least on vagueness, as a whole it still strikes me as useful enough to be worth republishing.

The main text has been left unaltered, even though it contains some things that I would no longer say and many things that I would now say differently. On most of those points, I have had my more recent say elsewhere.

In the case of vagueness, my views were evolving as I wrote Identity and Discrimination. Having started as an orthodox supervaluationist, shortly before I finished the book I came on the idea of margins for error, and its potential use in defence of an independently attractive epistemic interpretation of vagueness. It appears on pp. 104–7. However, I was not yet sure of my ground, and tentatively opted for a hybrid view, epistemicist about some cases, supervaluationist about others. Subsequent reflection and experience convinced me that a uniformly epistemicist view can be robustly defended, and has systematic advantages over a mixed alternative, since much of the cost of backsliding from classical logic and bivalent semantics is incurred if it happens even once. I defended an uncompromisingly uniform epistemicism in Vagueness. Although some details of the account there need refinement, I see no good reason to compromise on the logic and semantics.

Another respect in which the approach of Identity and Discrimination is tentative concerns the treatment of phenomenal character, the topic of chapters 4–6. It is often supposed to be a central feature of consciousness, defined by appeals to ‘what it is like’. I did not (and do not) find appeals to ‘what it is like’ of much explanatory or even descriptive value. Although talk of what it is like may once have served as a useful corrective to behaviourism, it has become a piece of lazy-mind jargon that obstructs serious understanding of how we experience the world. Constraining phenomenal character in terms of indiscriminability promises to impose some much-needed discipline on the discussion. However, I also worried about the hint of verificationism in attempts to capture the supposed psychological reality of phenomenal character in such terms. I therefore adopted an experimental attitude, developing a formal strategy for handling the specific structural problem in which I was interested for such accounts of phenomenal character while warning that the strategy was no panacea (p. 112). I later came to the conclusion that one leading conception of the phenomenal is an artefact of uncritical epistemology (Knowledge and its Limits p. 173). In any case, indiscriminability is a real feature of cognitive life, whose structure needs investigating.

Chapters 1–3 develop an account of discrimination as recognition of distinctness, which enables the logic of indiscriminability to be explained as resulting from the interaction of the logic of epistemic possibility with the logic of identity. The realization that indiscriminability should be understood in terms of knowledge was an early step towards the knowledge-first epistemology of Knowledge and its Limits. The epistemic conception of indiscriminability has subsequently been put to work by various authors in recent discussion of disjunctivist theories of perception, which have some similarities with, but also some differences from, knowledge-first epistemology (Knowledge and its Limits pp. 44–8).

In ‘Additional Notes (to the Revised Edition)’, I have given a few references to subsequent writings by others where it seemed particularly useful to do so. Naturally, I have made no attempt to provide bibliographies of recent work on large relevant themes such as vagueness, phenomenal consciousness and essentialism. Nor have I indicated the places where I am no longer happy with what I said in 1990; this preface should be sufficient indication. I have added comments on a few points of interest, including some material from my note ‘Fregean Directions’ (Analysis 1991) in a new note to chapter 9. These additions are kept separate from the original text of the book, which remains as it was printed in 1990, except for straightforward minor corrections on pp. 9, 35, 45, 57, 65, 66, 72, 77, 84, 107, and 115.

A note on the origins of the book may amuse a few readers. Early in 1974, as a first year undergraduate reading Mathematics and Philosophy at Balliol College Oxford, I became fascinated by Frege’s idea of identifying obscure entities (such as directions) with equivalence classes of less obscure entities (such as lines) under a comparatively clear equivalence relation (such as parallelism), and especially by Russell’s attempts to extend the method of logical constructions to the world of our experience. I was moved less by the supposed metaphysical or epistemological benefits of the method than by its intellectual elegance. But what particularly intrigued me was an obstacle it faced in some cases: the non-transitivity of the natural candidate for the relevant equivalence relation. Suppose, for example, that we wish to define the perceptible determinates of a given determinable, such as weight. A salient proposal is to equate the identity of the perceptible determinate qualities with indiscriminability in the given respect. But they are not equivalent, because identity is always transitive while indiscriminability in a respect is notoriously not: two differences each too small to be discriminated may add up to a difference large enough to be discriminated.

It occurred to me that by applying Zorn’s Lemma, an equivalent of the Axiom of Choice, one can prove that some partition of the objects to be assigned perceptible qualities lumps together only ones mutually indiscriminable (in the relevant respect), and is maximal in the sense that any coarsening of it lumps together some discriminable objects. Such a maximal partition defines an equivalence relation, with the formal features of a criterion of identity for the relevant perceptible qualities, and in a sense comes maximally close to the natural but incoherent original idea. It struck me as a promising fallback position. However, I also realized that, by an easy extension of the argument, the solution is not unique: whenever indiscriminability is non-transitive, more than one partition is maximal. I experimented with criteria for choosing amongst solutions, although I knew on grounds of symmetry that no formal criterion will always select a unique one: just consider the simplest possible case, a reflexive, symmetric, non-transitive relation of indiscriminability on a domain of three objects. I generalized the applications of the technique, for example to a characterization of meanings in natural language, where the non-transitive relation on linguistic expressions is approximate synonymy, and to the non-Cantorian infinite numbers toyed with in section 7.2. By the summer of 1974 I had most of the material in sections 5.1, 5.3, 7.1 and 7.4 of the present book.

Despite my increasing scepticism about the philosophical presuppositions of a Russellian programme of logical construction, I continued to work intermittently on the ideas, and proved that the lemma about maximal equivalence relations on which the technique depends is itself equivalent to the Axiom of Choice (see pp. 156–7). I wrote up the material in 1975 and submitted it to The Journal of Philosophy. After a long wait, I received what I now know to have been a ‘revise and resubmit’ letter, asking (reasonably enough) for the philosophical significance of the material to be clarified. At the time, ignorant of the conventions of philosophy journals, I read the letter as a patronising rejection and put the paper away in disgust. I abandoned the topic for almost a decade. My doctoral dissertation at Oxford was on ‘The Concept of Approximation to the Truth’, as a topic in the philosophy of science, without reference to vagueness: of two precise but false theories, one may in some sense be closer to the truth than the other is. Its only contribution to Identity and Discrimination was that it became the starting point for an article on the logic of comparative similarity, used in passing in section 7.5. Perhaps it was no coincidence that the two projects shared the theme of approximation. Years later, I rewrote the paper on identity completely, avoiding commitment to metaphysical and epistemological assumptions that I could no longer accept. It finally appeared in The Journal of Philosophy for 1986 as ‘Criteria of Identity and the Axiom of Choice’. The rest of Identity and Discrimination was written without further delay (see the Preface to the first edition). I hope that, through this revised edition, it will encourage others to go further in exploring the fine structure, both logical and epistemological, of the interaction between identity and discrimination.

Preface to the First Edition

This book is not longer than it is. The reader may feel that no apology is needed, but should recall the price of comparative brevity: neglect of alternative views, compromises in formal rigour. In particular, I have assumed without argument that the last thing to give up is a principle of classical logic; even its opponents should agree that no case against it is complete without an understanding of what can be done within its limits; this book aims to contribute to that understanding. Logicians will note that quotation marks have been omitted where ambiguity does not threaten.

Material from my ‘Criteria of Identity and the Axiom of Choice’, The Journal of Philosophy 83 (1986), pp. 380–94, appears, completely rewritten, in chapters 5 and 7 and the appendix; I thank the editors of The Journal of Philosophy for permission to print it here. I have been greatly helped by responses to oral presentations of ideas in this book. Early versions of chapter 8 were given as talks at Brandeis University and Williams College in 1987 (I thank Peter Lipton in particular). Chapter 9 stems from a paper read to a conference on Identity at Dubrovnik in 1987; David Charles replied. Much of chapters 1 and 6 evolved in a class I gave at Oxford University in Michaelmas 1988. It developed further when I made parts of chapters 1, 2 and 4 the basis of two talks at the University of California at Los Angeles in 1989. I should also like to thank Graeme Forbes, Eli Hirsch, Michael Morris, Stig Alstrup Rasmussen, Nathan Salmon and Crispin Wright for written comments on various parts of the material. Robin Gandy, Dan Isaacson, Bill Newton-Smith and Andrew Pigdon may have forgotten older debts. A first draft was completed at Trinity College, Dublin, and efficiently typed by Anne Burke. Later drafts have been written at University College, Oxford, of which I am not the first Fellow to have written a book about identity; I take the work of my colleague David Wiggins to justify certain assumptions which I have tacitly made. The contrasting influences of the work of Michael Dummett and Saul Kripke are active in many parts of the book. Most important of all, my wife Elisabetta gave me encouragement when it was most needed.


Intelligent life requires the ability to discriminate, but not with unlimited precision. A way of discriminating is usually ineffective below a certain threshold. When two things differ by just more than the threshold they can be discriminated, but an intermediate thing may differ from both by less than the threshold, and therefore be discriminable from neither. Discriminability is a rough guide to distinctness; discriminable things are always distinct, distinct things are often but not always discriminable. By the same token, indiscriminability is a rough guide to identity: in longer words, an approximate criterion. If indiscriminability is a shadow of identity, the shape of the latter is distorted but recognizable in the shape of the former. The logic of identity generates a logic of approximate criteria of identity, in some ways similar and in some different (identity is reflexive, symmetric and transitive; indiscriminability is reflexive, symmetric and non-transitive). It is the theme of this book. In particular, techniques are developed for working approximate criteria of identity into exact ones.

Chapter 1 analyses discrimination between things as activation of the knowledge that they are distinct, and indiscriminability as the impossibility of activating such knowledge. The analysis permits an explanation of the reflexivity, symmetry and non-transitivity of indiscriminability in terms of the reflexivity, symmetry and transitivity of identity. However, that is to treat indiscriminability as a relation, and there is a sense in which it is not one, because it is as intentional as any other cognitive phenomenon: things may be discriminable when presented in one way and not when presented in another. The intentional and non-intentional senses are compared. Chapter 2 is more technical; it uses an epistemic interpretation of modal logic to formalize the two senses, and to make precise connections between their formal properties and more general claims about the logic of knowledge. Chapter 3 uses this apparatus to formulate conditions in which an observational predicate applies to both or neither of two indiscriminable things, thereby giving rise to a sorites (slippery slope) paradox.

Can indiscriminability ever be an exact criterion of identity? The phenomenal character of an experience is supposed to be wholly given to its subject; if such characters exist, they are subjective qualities in the sense that they form a kind for which identity and indiscriminability coincide. According to a well-known argument there are no subjective qualities, for a non-transitive relation of indiscriminability cannot coincide with the transitive relation of identity. Chapter 4 defends phenomenal characters against that argument, by showing it to run foul of the intentionality of indiscriminability already discussed. Discrimination between phenomenal characters depends on which experiences present them. A positive condition is derived on the relation in which two experiences stand when they have the same phenomenal character. Chapter 5 shows that the condition is satisfied, but by more than one relation. The question is raised whether the concept of sameness in phenomenal character is indeterminate between those relations, or refers to just one of them even if we must be ignorant of which. The account of phenomenal character is contrasted with the view that experiences are the same in phenomenal character just in case they match the same experiences. Chapter 6 uses the plurality of candidate relations of sameness in phenomenal character to explore and defeat sorites paradoxes like those of chapter 3 as threats to phenomenal character. Generalizations to other sorites paradoxes are discussed; as before, they can be understood in terms of either indeterminacy or ignorance.

A necessary and sufficient criterion of identity is an equivalence relation: it is reflexive, symmetric and transitive. The treatment of phenomenal character can be seen as a matter of finding equivalence relations which best approximate a given necessary criterion that is insufficient because non-transitive; various equally good approximations are candidate necessary and sufficient criteria. Chapter 7 generalizes the technique to other cases, including the identity of species and persons, and makes the appropriate concept of approximation precise. A comparison is made with the problem of finding equivalence relations which best approximate a given sufficient condition that is unnecessary because non-transitive. Chapter 8 applies the same technique to sorites paradoxes about the identity of artifacts as their constituent materials vary from possible world to possible world or from time to time. In the light of the foregoing, chapter 9 begins to clarify the confused notion of a criterion of identity. A criterion of identity for things of some kind may be a relation that is not a species of identity and relates things of some other kind, as having the same phenomenal character is not a species of identity and relates experiences, not their phenomenal characters. It is suggested that a criterion of identity is a metaphysical principle apt to explain certain epistemic phenomena.