Philosophy, science, and method: essays in honor of Ernest Nagel

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PHILOSOPHY, SCIENCE, AND METHOD

Essays in Honor of Ernest Nagel

PHILOSOPHY,

SCIENCE, AND METHOD Editors:

SIDNEY MORGENBESSER Columbia University

PATRICK SUPPES Stanford University

MORTON WHITE Harvard University

St. Martin s Press, New York

COPYRIGHT © 1969 BY ST. MARTIN’S PRESS, INC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM WITHOUT PERMISSION FROM THE PUBLISHER.

LIBRARY OF CONGRESS CATALOG CARD NUMBER 69-17855

DESIGNED BY CHARLES KAPLAN

MANUFACTURED IN THE UNITED STATES OF AMERICA

PREFACE

Ernest Nagel is one of this generation’s most distinguished philosophers of science and one of its most effective spokesmen for critical naturalism. His philosophy of science and his commitment to naturalism are intimately connected, for he believes that the methods and results of science vindicate the spirit of classical naturalism, and that modern naturalism is intolerably thin when it is not informed by considerable familiarity with those methods and results. Nagel therefore insists that theorists of knowledge should care¬ fully examine the logic of science; and because he doubts the value of detached metaphysical speculation, he holds that philosophers must acquaint themselves with what scientists say and do in order to analyze such concepts as space, time, and causality. In a similar vein he holds that the conclusions and self-corrective methods of science lend support to social and political liberalism. Ernest Nagel is, then, a remarkably wide-ranging analytical philosopher, who has combined logical power and scientific learning in his admirable effort to formulate principles that should guide rational inquiry and rational living. Nagel has conducted this effort by using more than the written word. He has been a most influential teacher and an ideal intellectual companion to many philosophers, scientists, and men of affairs; he has not only ex¬ hibited but also rekindled in others commitments to clarity and concern for truth and for the need to integrate vision and technique. We hope that the essays here collected, all written since 1966 and some finished that year, will bring pleasure to the man they are intended to honor. All of them are concerned with issues that have been of interest to Ernest Nagel—the nature of the scientific enterprise, the conditions for warranted belief and justified action, the tenability of pragmatic theses about logic and scientific theories, and the scope and limitations of scientific approaches to discus¬ sions of human affairs and human action. It is also heartening to note that many of the essays have been contributed by historians, scientists, linguists, and legal theorists who, by exhibiting the philosophical relevance of issues in their field, confirm Ernest Nagel’s conviction that philosophers may learn much from the work of scientists and scholars. Although there is wide agreement among the contributors concerning the method and style of v

vi

PREFACE

philosophy, there is no orthodoxy and there is occasional dissent from some views that Ernest Nagel has defended. We are sure he would not want it any other way.

CONTENTS

Preface

1. SCIENCE AND INQUIRY Nagel’s Lectures on Dewey’s Logic

2

PATRICK SUPPES

Some Difficulties in Knowing

26

STUART HAMPSHIRE

On Objective Intensions and the Law of Inverse Variation

48

R. M. MARTIN

Confirmation of Laws

74

MARY HESSE

Induction and the Aims of Inquiry

92

ISAAC LEVI

Concepts of Statistical Evidence

112

ALLAN BIRNBAUM

Some Half-Baked Thoughts about Induction

144

MAX BLACK

152

G. FEINBERG

165

THEODOSIUS DOBZHANSKY

179

CARL G. HEMPEL

200

SIDNEY MORGENBESSER

STRUCTURE OF SCIENCE On What There May Be in the World On Cartesian and Darwinian Aspects of Biology Reduction: Ontological and Linguistic Facets The Realist-Instrumentalist Controversy

viii

CONTENTS

The Identity Thesis

219

Extensive Measurement When Con¬ catenation Is Restricted and 235 Maximal Elements May Exist

judith jarvis Thomson

r. duncan luce and

A. A. J. MARLEY

Causation and Action

250

MORTON WHITE

Some Empirical Assumptions in Modern Philosophy of Language

260

NOAM CHOMSKY

If Matter Could Talk

286

FRITZ MACHLUP

Functionalism in Social Anthro¬ pology

306

MAURICE MANDELBAUM

Legal Theory: Law, Justice, Ethics, and Social Morality 333

Wolfgang friedmann

3. THE CONSTRUCTION OF THE GOOD Metaphors, Analogies, Models, and All That, in Ethical Theory

364

ABRAHAM EDEL

Absolutism and Human Rights

382

SIDNEY HOOK

Justice and Rationality

400

CHARLES FRANKEL

Beyond Pareto Optimality

415

JAMES S. COLEMAN

Coercion

440

ROBERT NOZICK

Existentialism and Death: A Survey of Some Confusions and Absurdities 473

paul edwards

4. HISTORICAL STUDIES The Quadrature by Lunes in the Later Middle Ages

508

marshall clagett

Isaac Newton’s Principia, the Scrip¬ tures, and the Divine Providence

523

i.

Bernard cohen

CONTENTS

The Law of Inertia: Some Remarks on Its Structure and Significance 549

ARNOLD KOSLOW

Kant’s Philosophy of Arithmetic

568

CHARLES PARSONS

A Soviet Philosopher’s View of Peirce’s Pragmatism

595

PHILIP P. WIENER

John D. Rockefeller and the Historians

602

SIGMUND DIAMOND

ix

■J'

1. Science and Inquiry

NAGEL’S LECTURES ON DEWEY’S LOGIC Patrick Suppes

I first encountered Ernest Nagel in the late winter of 1947 when I entered Columbia as a graduate student. The lectures I attended were those of the course he entitled Types of Logical Theory, and during that term he devoted the lectures to Bradley and Dewey. I listened eagerly and with pleasure to everything that Nagel had to say about Bradley and Dewey, and I marveled at the patient way he went about dissecting their views and stressing the weak points in their arguments. Nagel’s lectures on Dewey have never been published, and because I increasingly see the importance of what Dewey was trying to do, it seemed more than ap¬ propriate to give here an account of Nagel’s lectures. I have shown the date of each lecture as a method of indicating natural breaks in the narrative.

MARCH 10 Nagel finished his lectures on Bradley by making the well-known point that Bradley seems to deny the essential validity of discursive discourse, and he opened his lectures on Dewey by pointing out that this was an aspect of Bradley’s thought that Dewey had absorbed and accepted. Before turn¬ ing to any details of the logic itself, Nagel made some general remarks about Dewey’s motivation and general orientation. He noted four important influences on Dewey’s thought that should be taken account of in ex¬ amining the Logic. The first was the importance that Dewey attached to modern mathematics and experimental science; the second was the in¬ fluence of biological conceptions, particularly those of Darwin; the third was the rejection of psychological empiricism; and the fourth was concern with the social import of thought. Concerning the first point about the importance of modern mathematics and experimental science, Nagel said that Dewey inveighs against the view that science gives us a final grasp of things. Dewey’s arguments are directed against Aristotle and the tradition of British psychological empiricism, which holds that sense gives us a final grasp of things. In this connection as in others, however, one of the difficulties of Dewey’s 2

PATRICK SUPPES

3

analysis is that he is usually directing a specific polemic at various un¬ named schools. His two main foes are traditional rationalists and traditional empiricists, because he wants to eliminate the sharp dichotomy between reason and sense. According to Nagel, Dewey’s attack on classical rational¬ ism has been fortified by the fact that many of the alleged first principles of various sciences have subsequently been replaced by other first principles, as, for example, in geometry. Concerning the influence of biological conceptions in Dewey’s thought, Nagel remarked that it is evident that Dewey emphasizes a genetic ap¬ proach in most of his works. Dewey, Nagel said, has been reproached for confusing genesis with validity, and yet much can be said for Dewey’s approach as a fruitful method. Dewey is dependent on Darwin for his emphasis on change and, in particular, for the thesis that structure is not fixed but functional. Dewey’s conception of the mind as active makes up for some of the deficiencies apparent in associationistic psychologists like Mill. Concerning the social context of Dewey’s thought, Nagel remarked that it is worth remembering that Dewey wrote his Logic to help the social sciences progress at the same pace as the natural sciences. Dewey’s interest in logic, Nagel said, has been controlled by the apparent profound chaos in moral and social thought for which Dewey has sought a solution. One of Dewey’s earliest and most influential papers (1902), “Logical conditions of a scientific morality,” contains the kernel of most of his logical thought. Dewey’s general conception is that an appropriate logic should be an organon for the solution of pressing social problems. After these preliminary and general remarks, Nagel turned to the direct consideration of Dewey’s Logic, The Theory of Inquiry. To begin with, Nagel pointed out, Dewey considers logic as the theory of inquiry and not as a formal science. Dewey distinguishes between the proximate and ultimate subject matter of logic. The proximate subject matter is the tradi¬ tional study of the implicative relations between propositions, and so forth. There is much agreement on this aspect of the subject matter; but there is disagreement over the ultimate subject matter, for example, about the basic units of logic. According to Bradley, the basic units are judgments; others hold that they are terms. For Dewey, the basic unit is a completed act of inference. Dewey continually argues that inference is needed, because any natural immediate experience is incomplete in itself, and its relation to other events involves inference. Dewey’s chief criticism of ancient science is that it mistook immediate qualities as the efficient causes of things. The chief virtue of modern science is in overcoming this and being able to ignore directly experimental qualities as causes. On this view logical theory is regarded as systematic reflection on inquiry. In Dewey’s opinion, and I think also in Nagel’s, too often there has been no systematic investigation of why some inquiries are successful and others are not; the peculiar con-

4

1. SCIENCE AND INQUIRY

ditions that produce success are too seldom understood. These conditions are precisely the subject matter of logic. For Dewey, if there are no problem solvers, there is no logic. Put another way, it may be said that for Dewey logic is equivalent to experimental epistemology. Logic is a positive inquiry into inquiries, having the methods of inquiries as its specific subject matter. In this sense logic is descriptive and not normative, but it is also normative insofar as it sets standards for later inquiries.

MARCH 12 Nagel began by pointing out that a central feature of Dewey’s logical theory is his claim that logical forms arise inside inquiry and do not char¬ acterize things outside of inquiry. Nagel pointed out that while this con¬ textual concept of logical form is a difficult one for those with a back¬ ground in mathematics and formal logic, it is worthwhile to see what Dewey has in mind. According to Nagel, the key to understanding what Dewey means here is to realize that the concept of logical form is not the same in Dewey’s theory and in formal logic. Dewey holds that certain types of functions are to be regarded as the logical forms; for example, the functional relation between evidence and conclusion is not characteristic of things themselves but only in relation to specific inquiries. Those who argue that logical forms have the eternity of Platonic forms hardly under¬ stand Dewey’s point. Another way of putting it, Nagel said, is that Dewey’s logical forms are relations between means and consequences. Nagel said that Dewey’s logical forms are postulates for inquiry. Dewey is essentially saying that the distinctions of the proximate subject matter belong to things only in specific inquiries. In these terms, logic becomes the formulation of the conditions of successful inquiry. Nagel remarked that a functional concept of knowledge is sometimes attributed to Dewey; but unfortunately, the term functional is vague and needs specification. Nagel said that there are two ways of specifying knowl¬ edge. One way is to say that we have knowledge when we have the truth, and to define knowledge in terms of its systematic characteristics, not in terms of how it is acquired. The other way of specifying knowledge is to say that knowledge is acquired through inquiry and that an essential way of characterizing knowledge in this sense is to characterize the context of inquiry. According to the latter view, which is Dewey’s, inquiry begins or arises from doubt when there is a felt tension. What resolves the doubt or tension is knowledge, the resolution of the problem. Nagel remarked that Dewey’s theory of knowledge is functional in two ways. It is functional insofar as knowledge is construed in terms of the resolution of particular problems; it is also functional in the sense that knowledge is identified in terms of a process (the process of inquiry) which can itself be overtly

PATRICK SUPPES

5

located by reference to the behavior of certain organisms. For Dewey, knowledge simply becomes the terminus of inquiry, and this view rules out the question of the possibility of knowledge in general, or knowledge apart from a particular context of inquiry. For Dewey, inquiry is a transformation of indeterminate constituents into a unified, determinate whole. Here Nagel quoted Russell’s famous remark that this characterization would apply to a drill sergeant working with a group of recruits. Nagel pointed out that one difficulty in Dewey’s theory is that of determining to what extent the definition of logic proposed by Dewey is adequate to the traditional problems of philosophy. Dewey seems to intend his definition as an empirical generalization, and he speaks in such broad terms that much of the revolutionary aspect of his thought evaporates when what he says is understood.

MARCH 17 Nagel began by considering Dewey’s conception of the naturalistic char¬ acter of logic. Nagel characterized a theory as naturalistic when a con¬ tinuity is established between it and the biological operations out of which it grew; but Nagel then asked what this emphasis on continuity contributes to a logic interested in forms of warranted assertability. What importance does continuity have in this context? What is the point of raising it? Dewey’s answer has already been stated; namely, hypotheses cannot be introduced independent of a context, and it is precisely the introduction of a context that leads us to the naturalistic analysis of inquiry. Nagel then raised the particular question of the use of symbols as a distinctive case. Nagel stated that he failed to see why Dewey’s emphasis on continuity would change or obscure the usual or traditional account of the role of symbols in inquiry. Continuity with biological operations, Nagel stated, seems irrelevant in terms of the specific functions of symbols in inquiry. Nagel also emphasized that Dewey’s attempt to show the con¬ tinuity between the logical and biological must be regarded as speculative. The present amount of accurate empirical information is insufficient to establish the relation. As an example, Nagel discussed ponendo ponens. Dewey, he said, takes over Peirce’s view of rules of inference as leading principles; for example, ponendo ponens is a habit men have acquired which enables them to go from two premises to a conclusion. It is an effi¬ cient habit that is a generalization from experience; but Nagel argued that whatever biological foundations may underlie the genesis of the principle of ponendo ponens, its validity may be established independently of biological and physical interpretations. As a second example, Nagel considered Dewey’s interpretation of Aristotelian logic. Nagel said Dewey was right in asserting that the

6

1. SCIENCE AND INQUIRY

syllogistic forms are not fruitful, but wrong in denying the validity of the syllogism simply because it was developed at an early stage of science. Nagel then stated that surely there must be some misunderstanding here. Possibly what he was saying was an incorrect interpretation of Dewey. Nagel said he was calling attention to the close connection between genesis and validity for Dewey, and he raised the question of whether a person who ignored the biological evidence would be a supernaturalist. In Nagel’s view, surely not. As a final example, Nagel remarked on Dewey’s treatment of the prin¬ ciple of identity. For Dewey, it is not just the form if p then p, but rather the carrying through of the term so that it has one meaning. It formulates a rule for the handling of terms. Nagel said that for Dewey it then becomes a synthetic principle rather than a tautology.

MARCH 19 Nagel began with some remarks about symbols which, I believe, were meant to reflect Dewey’s views, although my notes are vague on this point. Nagel said that symbols are artificial signs. Between natural signs and their object, there is signification. Between artificial signs or symbols and what they stand for, there is meaning. We can have clusters of symbols or arti¬ ficial signs, but not clusters of natural signs. For Dewey, all inquiry essentially involves the use of symbols or artificial signs. This is a point which much of the criticism of Dewey’s logic seems to have missed. Nagel then considered the important term situation in Chapter IV of the Logic. For Dewey it is not possible to define the term situation, for all definitions are made within situations. There are two points of Dewey’s discussion that Nagel said were important to note. The first is that percep¬ tion itself occurs always within a situation; the second is the sense in which perception is cognitive. The central point here is that Dewey denies that perception per se is cognitive. Isolated acts of perception are for Dewey not knowledge; they are neither true nor false. Validity or invalidity is relevant only if we consider the signification of the perceptions. Through¬ out his many discussions, Dewey criticizes the doctrine of immediate knowl¬ edge, that is, the doctrine that direct or immediate knowledge arises from sensation. Nagel repeated that it is essential to realize that in Dewey’s logic knowledge can never be the case of simple perception alone. The reason for Dewey’s claim is clear. Perception is not the outcome of inquiry, but this is the most important characteristic of knowledge; consequently, perception is not knowledge. Relevant to this discussion, Nagel turned to Dewey’s important distinction between having and knowing. For Dewey, Nagel said, knowing is the terminus of inquiry, but having is essentially an aesthetic experience.

PATRICK SUPPES

7

Knowledge is capable of being formulated in discourse, but that which is had is not. Here we can see how perception fits into Dewey’s scheme of things. Perception is having. Dewey’s criticism of philosophical idealism is that it does not admit the distinction between having and knowing, and thus, does not permit us to break through the egocentric circle. For Dewey, Nagel pointed out, knowing is an instrument of having, but Nagel remarked that it is perhaps impossible to define clearly what having is for Dewey, and it is often hard to distinguish between having and knowing. In fact, Nagel said, he had a certain difficulty in understanding what knowledge is for Dewey. Even if it is agreed that knowledge is the terminus of a situation or an inquiry, it is not clear whether this terminus is a having or a knowing. It seems to be both, but in different respects for Dewey. At this point Nagel digressed in order to consider Dewey’s distinction between scientific and common-sense knowledge. Scientific knowledge has no reference to the immediate situation. Common sense does, but it is also vague. Common sense is interested in qualitative differences for ends of use and for enjoyment. Science is nonqualitative. Science is interested in nonqualitative differences for purposes of knowledge. On the other hand, science originates in common sense. In Dewey’s view, the Greeks’ separation of art from science and their belief in pure reason slowed scientific develop¬ ment tremendously. (Here I suspect that Dewey, like other philosophers, has been seduced by the traditional Platonic tale. It is scarcely possible to claim that the deep and important development of mathematical and observational astronomy by the Greeks represents a theoretical or practical belief in pure reason as a method of learning about the world. On this point, Dewey, and perhaps to some extent Nagel, is simply repeating a common view of Greek science.) For Dewey, the difference between science and common sense is really social and not logical. It is more or less accidental that thus far science has tended to concentrate on different problems from those of common sense. In Dewey’s view, prescientific ideas have held sway too often and for too long in morals and politics. This has made for an essential split in common sense that is reflected in philosophy—the split between pre- and postscientific thinking. Nagel concluded this digres¬ sion by pointing out that Dewey’s central point is to emphasize the funda¬ mental unity of the two kinds of inquiry and not to sanction any absolute difference between them. Nagel concluded this lecture with a brief discussion of Chapter V of the Logic, which deals with the needed reform of logic. Nagel mainly reviewed Dewey’s claims as to why a reform is needed, that is, why classical logic no longer applies or is appropriate. First, classical logic is qualitative only. Second, Greek science, the context in which classical logic arose, asserted heterogeneity of substance and motion, whereas ours asserts homogeneity of the two. Third, in Greek science all quantifications were accidental rather

8

1. SCIENCE AND INQUIRY

than essential. Fourth, relations were in general also accidental, and now they are the prime subject matter of science. Dewey wants modern logic to serve present science and culture in the way that Aristotle served the science and culture of his own time. (It is not clear to me to what extent Dewey realized that Aristotle’s logic was in no sense adequate to Greek mathematics or astronomy. In any case, the thrust of this chapter seems to be one of the weaker parts of the Logic, and, like Nagel, we can quickly dispense with it here.)

MARCH 24 In this lecture Nagel turned to Part II of the Logic, which deals with the structure of inquiry and the construction of judgments. Nagel em¬ phasized that we cannot hope to get the sense of what Dewey is saying by examining minutely any specific argument. Nagel said that Dewey’s def¬ inition of inquiry illustrates this point perfectly. Russell’s criticism of Dewey is well justified if we take strictly the single italicized statement of Dewey in which he defines inquiry (pp. 105-106): “Inquiry is the controlled or directed transformation of an indeterminate situation into one that is so determinate in its constituent distinctions and relations as to convert the elements of the original situation into a unified whole.” Nagel’s point is that it would never do to take seriously for careful analysis this single italicized definition of inquiry. What we have to do is to read the many passages in which Dewey discusses inquiry and put together for ourselves a more or less inductively constructed picture of what Dewey means by inquiry. Nagel then said he would try to give some feeling for how this might be done. He began by saying that the situation with which inquiry begins is indeterminate. The constituents of the situation do not hang together. The organism wants to know in what way they do not hang together, for surely in other ways, they do. Nagel said that apparently the problem is raised in the mind of the inquirer, because he cannot understand the structures of the various parts of the situation. In this sense, the situation is doubtful or indeterminate. It is important to emphasize that for Dewey this sense is not merely a subjective one. Complete determination does not hold for the environment. Dewey says that evidence in the physical sciences about indeterminancy of physical events is evidence that the sense is not purely subjective. Nagel remarked that this argument is questionable, but that the reasons were too complicated to discuss at this point. Here Nagel mentioned Dewey’s lecture on time two years before (1945). Nagel’s version of the lecture was that Dewey held that time and individuality are connected. Individuality is development in time, and Dewey claimed that to some extent his view was derived from statistical mechanics. Nagel asserted that using such evidence from physics to discuss human individuality was, in his

PATRICK SUPPES

9

view, bad philosophy. He said he would agree that the came for doubt can be outside the body of the doubter, but this does not make the situation itself doubtful. Nagel asserted that, as far as he could see, the element of indeterminancy exists wholly, or almost wholly, in the observer. (The sophisticated thing which Dewey might have attempted is a biological interpretation of subjective probability. It seems to me that this is the best line along which to claim that indeterminancy is more than subjective, or at least to take the sting out of the claim that it is subjective.)

MARCH 26 The discussion of Dewey’s conception of inquiry was continued. Nagel said that he would assume that what Dewey meant by an indeterminate situation was not enough. Nagel went on to the second main point, namely, that the existence of the indeterminate situation is not enough; the further step of formulating the problem must be taken. Formulation of a problem involves for Dewey partial transformation of the situation itself. Nagel emphasized that Dewey seems to use the concept of transformation to mean an actual physical change in the situation. Nagel then examined more carefully Dewey’s notion of a problem. He first emphasized that for Dewey a problem is something that must have a possible solution. For Dewey the statement of a probable or indeterminate situation as a problem has meaning only if in the very terms of statement there is a possibility of solution. Nagel said that this may be Dewey’s version of the verifiability theory of meaning. At this point Nagel reviewed briefly some standard versions of the verifiability theory. Nagel felt that, in spite of Dewey’s vagueness, his approach had a certain body and fullness on these matters that the positivistic version of the verifiability theory of meaning lacked. Nagel then commented on a number of related points. For Dewey, he noted, factual conditions determine probable conditions. There need to be settled facts or constituents in a situation. Nagel said that Dewey’s affirmation of the need for settled facts raises the previous question as to what is meant by doubt in the situation as a whole. Nagel asked what Dewey regarded as the facts. The answer seems to be, he said, that the facts are constituents in the situation that can be noted through perception. But he commented that later Dewey says that a subject (in relation to a predicate) always consists of those perceptual elements that help to identify the problem. Nagel remarked that this leads to an ideational element. For Dewey, ideas are anticipated outcomes of possible solutions. The terms of a problem, or the boundary conditions (to use a terminology not used by Dewey), are facts. Ideas, in contrast, are possible solutions of problems. Nagel said that this “in a nutshell” is Dewey’s theory of judgment, which

10

1. SCIENCE AND INQUIRY

he would discuss in greater detail later. To support this analysis from the text, Nagel quoted the following passage from page 111 of the Logic. In logical fact, perceptual and conceptual materials are instituted in functional correlativity with each other, in such a manner that the former locates and describes the problem while the latter represents a possible method of solution. Both are determinations in and by inquiry of the original problematic situation whose pervasive quality controls their institution and their contents. Both are finally checked by their capacity to work together to introduce a resolved unified situation. As distinctions they represent logical divisions of labor.

Nagel noted that the distinction between perceptual and conceptual materials made in this passage is perhaps the basic distinction for Dewey. The funda¬ mental thing for Dewey is that there is no dualism of perception and conception but rather a functional differentation. Dewey’s theory of inquiry is an attempt to show how two different processes that are widely separated in traditional philosophy and epistemological analysis are but two different functions within a single framework of inquiry. Nagel reviewed the steps in inquiry as Dewey conceived them. We begin with an indeterminate situation. We transform that situation into the fixing of a specific or definite problem. The next step is the introduction of reasoning or symbolic operations in order to determine the acceptability of the possible solutions. What is important about Dewey’s conception of reasoning, which we shall examine in more detail later, is that it is always an intermediate operation. A mathematical investigation or, it would seem, any purely theoretical work is always a phase of inquiry and cannot be the whole of inquiry. The reason for this is that a purely theoretical study does not constitute the formulation of a problem in terms of observation, nor apparently in terms of the explicit and immediate feeling of in¬ determinacy of an actual physical situation. Nagel emphasized that in mak¬ ing these observations he had been supposing that the conception of factual material does not include the observation of symbols as carriers of meaning. It is apparent that from a formal standpoint this would be a way of con¬ ceptualizing a theoretical investigation as a complete inquiry, and in my own judgment, it would be very much in the spirit of Dewey’s general philosophy to give such a radically empirical account of pure mathematics. However, it should be emphasized that Nagel did not support this view, nor is it easy to support it by any direct remarks on Dewey’s part. Nagel concluded the lecture by noting that it seems appropriate to assign mathematics to the immediate operation of reasoning. He noted that Dewey’s use of the word problem is more restricted than the ordinary use and does not include such things as mathematical problems. In this dis¬ cussion of mathematics, Nagel referred to page 405 of the Logic. Several of the passages there do suggest to me that a radically empirical interpretation of mathematical activity would perhaps provide a better reading of Dewey than a more conventional one. I quote here the passage of most interest.

PATRICK SUPPES

11

In its early history, problems of strictly existential subject-matter provided the occasion for mathematical conceptions and processes as means of resolving them. As mathematics developed, the problems were set by mathematical ma¬ terial as that itself stood at the given time. There is no contradiction between the conceptual, non-existential nature of mathematical contents and the existen¬ tial status of mathematical subject-matter at any given time and place. For the latter is an historical product and an historical fact. The subject-matter as it is at a given time is the relatively “given.” Its existing state occasions, when it is investigated, problems whose solution leads to a reconstruction. Were there no inconsistencies or gaps in the constituents of the “given” subject-matter, mathematics would not be a going concern but something finished, ended. As was intimated in an earlier context, material means and procedural means operate conjugately with each other. Now there are material means, having functionally the status of data, in mathematics in spite of their nonexistential character. They constitute the “elements” or “entities” to which rules of operation apply, while the rules have the function of procedural means. For example, in the equation 2 + 3 = 5, 2 and 3 are elements operated upon, while + and = are operations performed. There is no inconsistency in the identity between the logical function of existential data and mathematical elements or entities and the strictly non-existential character of the latter.

APRIL 9 Nagel continued his discussion of the pattern of inquiry. He noted that for Dewey the facts are operational in inquiry in the sense that they are selected in connection with a particular problem, to test particular solutions of that problem. Ideas are also operational in inquiry because it is their function to direct inquiry. For Dewey the facts interact with one another, but Nagel was skeptical that this was an appropriate way to talk about facts. As an illustration he mentioned that from the observation of finger¬ prints we infer the man was present at the scene of the crime. Is it ap¬ propriate to say that the two facts interact? Nagel said he would prefer to talk about our organization of the facts. Referring to pages 118-119, Nagel said that an important distinction for Dewey between content and object of inquiry had been misunderstood by some of his critics, who had taken Dewey as holding that inquiry creates its subject matter. As Nagel conceived it, the distinction Dewey intended was between subject matter undergoing inquiry and the subject matter of a completed inquiry. Dewey wants to use the term subject matter for material undergoing inquiry; he specifically calls this content. The outcome of an inquiry, which he sometimes calls the outcome of subject matter, is what he means by an object. Because this language sounds a little unusual, it may be useful to give the explicit quotations from Dewey on page 119. When it is necessary to refer to subject-matter in the context of either observation or ideation, the name content will be used, and, particularly on account of its representative character, content of propositions.

12

1. SCIENCE AND INQUIRY

The name objects will be reserved for subject matter so far as it has been produced and ordered in settled form by means of inquiry; proleptically, ob¬ jects are the objectives of inquiry. Nagel then turned to the explicit discussion of the nature of judgment for Dewey in Chapter VII of the Logic. Nagel’s first point was that for Dewey judgment is not inquiry itself but the settled outcome of it. Thus, judgment is never found in inquiry except on a partial basis. Secondly, judg¬ ments are to be distinguished from propositions, which do not have direct existential import. For Dewey a judgment issues in existential consequences. It is a decisive directive for future activity, but the judgment itself is not the set of activities that ensue. It has a propositional form, and thus in general, its truth or falsity can be investigated. Nagel indicated that for him this analysis presents a difficulty. Knowledge is defined by Dewey as the terminus of inquiry; but this characterization of judgment indicates that knowledge is, strictly speaking, not the terminus of inquiry, for following the assertion of a judgment a further transformation takes place, namely, the directive for future activity contained in the judgment. (Nagel did not discuss the distinction between judgments as imperatives or directives on the one hand and as true or false indicative sentences on the other, but the reasons for not pursuing this kind of distinction are clear to anyone who has perused, even cursorily, the Logic.) Nagel noted that for Dewey a final judgment is always individual in character. Nagel remarked, in a characteristic vein of his own, “This is a hard saying,” for many judgments have the form of universal proposi¬ tions. Nagel raised the question of how we reconcile Dewey’s assertion that judgments are individual in character with what appear to be the patent facts in many inquiries. Nagel remarked that one way of looking at the matter is that many inquiries by theoretical scientists must be regarded, as previously remarked, only as partial inquiries. Secondly, he said that it was necessary here to make Dewey’s distinction between singular and individual. For Dewey, singulars are named by demonstratives such as this or that. Propositions about singulars occur during the course of inquiry, but the final judgment is about the total situation, which is individual. (It seems clear that Dewey’s views on these matters have been influenced by the way we think of legal judgments of a court, which, almost always, do deal with an individual situation and are not expressed in terms of universals. This particular interpretation was not discussed by Nagel.) In this discussion of judgment Nagel emphasized the overweening im¬ portance for Dewey of practical activity in contrast to theoretical investiga¬ tions. Nagel said he would interpret Dewey as holding that the scientific inquiries of a mathematical physicist would never be regarded as a com¬ pleted inquiry, at least not until they were used in some aspect of practice or at least in direct, experimental investigation. Nagel then turned to Dewey’s view that every judgment has the logical

PATRICK SUPPES

13

form of subject, predicate, and copula. (That Dewey held this view is sub¬ stantiated by many passages in Chapter VII.) Nagel raised the question whether Dewey was ignoring the central fact of modern logic that many propositions are relational in form. Dewey, he said, seemed not to be ignoring the modern logical viewpoint, but to be distinguishing between propositions which are not true or false, but more or less useful, and judgments which are true or false. Dewey seems to take the view that propositions are guides, maps, or blueprints; they are, in other words, means for making more effective judgments. In certain respects, Nagel said, Dewey’s view seems to be close to Wittgenstein’s verifiability theory of meaning. For Dewey only those things are fully warrantable, that is, true or false, which are individual in character. General propositions do not have this property but are useful only as guides. However, it should be emphasized that for Dewey a judgment is not warranted by a simple perception. Nagel pointed out that if Dewey’s view is taken seriously, then it is not only the concept of truth for general propositions that is endangered but the whole concept of confirmation and scientific testing of theories, in the sense of determining the extent to which evidence con¬ firms a theory or, in ordinary terms, the extent to which the theory seems to hold for relevant phenomena. This has the consequence that there is no direct question of induction for judgments because judgments are always particulars. They are not related in functional fashion to general proposi¬ tions. As a result, induction holds only for intermediate stages of inquiry and not for the terminus or final judgment of inquiry.

APRIL 14 In this lecture, on which my notes are rather brief, Nagel turned to a more detailed discussion of Dewey’s analysis of the subject, predicate, and copula, that is, of the constituents of judgments. As already remarked, for Dewey a subject always has a subject-predicate form. Nagel stressed that Dewey’s emphasis on the form of judgments helps to bring out in what sense the logical form accrues to a subject matter. Something is a subject matter not because of its ontological character but because of its logical character, which amounts to a rejection of Aristotelian substance. Contrary to many classical epistemological views, for Dewey a fact functions only as a part of inquiry. To say that something is a subject is to say that a fact has acquired a logical character it did not have previously. In this connection Nagel cited a significant passage from Dewey (pp. 128-129). The condition—and the sole condition that has to be satisfied in order that there may be substantiality, is that certain qualifications hang together as de¬ pendable signs that certain consequences will follow when certain interactions take place. This is what is meant when it is said that substantiality is a logical, not a primary ontological determination.

14

1. SCIENCE AND INQUIRY

Certainly these views on the problem of substance are consistent with Dewey’s general position. Nagel then emphasized Dewey’s view that predicates provide a method of solution and do not themselves constitute a solution. I find my notes on this discussion very thin. I suspect that I probably did not understand very well what Nagel was saying. In a modern vein, it seems appropriate to me to say that when Dewey talks about predicates as methods of solution, he is emphasizing the use of language for purposes of communication in inquiry and here, as always, playing down the use of language for the bare expression of fact. Nagel then turned to Dewey’s view that the copula expresses a functional correspondence of the subject and predicate. In particular, the copula expresses, as Dewey puts it, the act or operation of “subjection”; that is, of constituting the subject. It is a name for the complex of operations by means of which (a) certain existences are restrictively selected to delimit a problem and provide evidential testing material, and by which (b) certain conceptual meanings, ideas, hypotheses, are used as characterizing predicates. It is a name for the functional correspondence between subject and predicate in their relation to each other. The operations which it expresses distinguish and relate at the same time [pp. 132-133]. After completing this discussion of the copula as such, Nagel emphasized Dewey’s view that a judgment is a temporal affair; or, as Dewey puts it, “that judgment is a process of temporal existential reconstitution.” Most philosophical readers will be troubled by this obscure phrase, but the idea that judgments are temporal occurrences does in itself tie in with recent discussions of utterances and statements. In any case, Nagel summarized Dewey’s position as amounting to the view that a judgment consists in overt transformation of the situation or in the actual doing of something, rather than simply in the assertion of a proposition.

APRIL 16 In this lecture, Nagel turned to an analysis of Chapter IX, “Judgments of Practice: Evaluation.” Nagel began by pointing out that declarative propositions are only intermediate and instrumental. They are statements of what conditions do exist. In contrast, judgments of practice contain overt transformations, of the sort to which the classical theory of judgment often denies status. The classical theory asserts that even in the case of a practical judgment there is in the judgment itself no overt transformation. Nagel mentioned that Dewey attempts to separate the linguistic situation from the psychological one. The mere existence of a practical syllogism does not in itself indicate the presence of a judgment of practice. If this syllogism is a matter of habit on the part of the user, then no issue of

PATRICK SUPPES

15

judgment arises. It is only when there is a question of doubt and inquiry that a genuine judgment of practice is made. Nagel pointed out that if we accept this account, then many judgments ordinarily taken as final must be classed as instrumental. A judgment of practice for Dewey always involves a transformation of antecedent con¬ ditions. In this connection, Dewey defends the position (pp. 167-168) that moral evaluations have the character he has ascribed to judgments of practice and are not, as often thought traditionally, predetermined and given ends in themselves. What Dewey has to say about ethics is often better and clearer than his comments on other topics. The following passage puts his view very well. The notion that a moral judgment merely apprehends and enunciates some predetermined end-in-itself is, in fact, but a way of denying the need for and existence of genuine moral judgments. For according to this notion there is no situation which is problematic. There is only a person who is in a state of subjective moral uncertainty or ignorance. His business, in that case, is not to judge the objective situation in order to determine what course of action is re¬ quired in order that it may be transformed into one that is morally satisfactory and right, but simply to come into intellectual possession of a predetermined end-in-itself [p. 168]. At the end of this lecture Nagel remarked that a source of much mis¬ understanding of Dewey in his discussion of practical judgments, and in particular, moral judgments, is Dewey’s distinction between evaluating and valuing. The distinction for Dewey is parallel to that between having and knowing. It is the distinction between having a value experience and evaluating that experience.

APRIL 23 In this lecture Nagel discussed Chapter X, “Affirmation and Negation: Judgment as Requalification.” In this chapter Dewey considers such matters as his view of the traditional A, E, I, and O propositions. Nagel began by emphasizing, once again, that declarative propositions are not simply enunciatory for Dewey, but are means leading to solutions. It is in this light that Dewey’s discussion of affirmative and negative judgments must be viewed. Nagel recalled that the concept of affirmative and negative judgments was introduced by Aristotle in the Prior Analytics, and is ordinarily regarded as a grammatical distinction; but for Dewey the basis of the distinction is not grammatical, but is rather a difference in function. Affirmative propositions represent agreement of subject matters in their evidential capacities. Negative propositions represent subject matters to be eliminated because of their irrelevancy or indifference to the evidential func-

16

1. SCIENCE AND INQUIRY

tion of material in the solution of a given problem. In this connection Nagel remarked that it is rather difficult to find statements that are negative in Dewey’s terms, but affirmative in grammatical form. In any case, Nagel emphasized that for Dewey both forms of proposition have an intrinsic connection with change. Dewey does not mean that every such proposition reports a change, but that affirmative or negative propositions are logically grounded in the exclusion of alternatives. Nagel then turned to Dewey’s interpretation of the traditional square of opposition. As might be expected, Dewey emphasizes that the traditional relations of contrariety, subcontrariety, and contradiction have to be under¬ stood in functional and not in formal or mechanical terms. What is inadmissable for Dewey is the interpretation of propositions as independent sets of objects to be considered by and of themselves and in their relation to each other. For Dewey the traditional opposition of contraries is to be inter¬ preted in terms of setting limits within which specific determinations must fall. His concrete example is that what we know about marine vertebrates must fall between the A proposition “All marine vertebrates are cold¬ blooded,” and the E proposition “No marine vertebrates are cold-blooded.” For Dewey these contrary propositions cannot represent conclusions or the terminus of an inquiry, but are the results of a preliminary survey. In interpreting this view of Dewey’s, it is important always to remember that for him the terminus of inquiry must be an individual judgment, not a general proposition. In terms of this general view it is quite clear and natural to maintain that A and E propositions cannot themselves be the terminus of inquiry. In this connection, Nagel made the point that in experimental inquiries it is usually not sufficient to operate with the bare formality of A and E propositions as setting natural limits of inquiry. Usually much more detailed and more specific information is available. For example, in investigating the temperatures of marine vertebrates we would not ordinarily look at the contraries “All marine vertebrates have a temperature of 60°F.” and “No marine vertebrates have a temperature of 60° F.” We would rather investigate a range of temperatures and would have an entirely different formulation of the relevant propositions to be tested. At this point Nagel said that he really did not know what Dewey would have said to this objection. For Dewey, subcontraries are more determinate than contraries but are still indeterminate compared with the individuality of final judgment. In Dewey’s view, subcontraries are used only if they are determinate, that is, if they have some ground for support. At this point Nagel considered Russell’s famous analysis of the sentence “The present King of France is bald,” and he said that in his judgment it was better to make explicit what you mean by proposition before you consider contraries and subcontraries in relation to factual data or material. Dewey seems to hold that the mean-

PATRICK SUPPES

17

ing of propositions is determined by factual data, and this Nagel felt should be avoided. Nagel next turned to Dewey’s discussion of contradictories. He pointed out that for Dewey a universal or general proposition is negated not by the indeterminate “some” but by the determinate singular. Nagel admitted that Dewey is correct in terms of much actual practice and inquiry. General propositions are denied by the use of a singular proposition and not merely by an existential statement. Nagel made the point, however, that there is a standard, broader usage. It is possible to establish a particular or existential proposition without using singular propositions; insofar as this must be accepted as established doctrine or procedure, Dewey’s position must be qualified. My notes do not show that Nagel discussed this point in detail. Finitistic or constructive positions about the foundations of mathematics would seem to offer specific support of Dewey’s position.

APRIL 28 In this lecture Nagel turned to Chapter XI, “The Function of Proposi¬ tions of Quantity in Judgment.” Nagel began by noting the traditional dis¬ tinction between universal and particular propositions, and emphasized Dewey’s criticism that this classical Aristotelian distinction is too restric¬ tive for modem science. Dewey’s point is that in the context of modern quantitative science, the qualitative distinction between all and some is so crude and general as almost to be irrelevant. Nagel pointed out that Dewey did not have a very thorough understanding of the role of quantifiers in modern logic and of their relation to the construction of the real number system on the foundations of set theory. He gave as an example the way in which quantifiers may be used to express the sentence that there is exactly one mayor of New York City. What Nagel said is certainly correct and sound. It can be said on Dewey’s behalf that his remarks were aimed really at the kind of applications of Aristotelian logic to scientific matters to be found in Aristotle and not to the formal doctrine itself. It is certainly true that much of the discussion in Aristotle and in particular those ap¬ plications that hew as closely as possible to the line of his logic are not scientifically very deep or satisfactory. Nagel noted that for Dewey distinctions between kinds of quantity are for functional purposes, and measurements are instrumental always in relation to certain aims. Throughout this chapter Dewey emphasizes this instrumental character of measurement in contrast to what he likes to call the mistaken cosmological and ontological framework of Aristotle. Nagel then turned to the discussion of some of Dewey’s constructive views about measurement. One of the first things to note in reading the

18

1. SCIENCE AND INQUIRY

chapter is that Dewey generalizes the common conception of measure to include any comparison. In other words, for Dewey it is not necessary to think of measurement as assigning numbers to objects. An example would be the specification that liquid A is denser than B when B will float on A. It is also a characteristic thesis of Dewey’s that there is no fundamental antagonism between quantitative and qualitative distinctions. For Dewey there is an underlying qualitative continuum at the basis of all quantitative measurement. For example, in the measurement of length we assume the quality of spatial extension. What we do in measurement is to ignore cer¬ tain qualities and to take others into account. Against the critics of quantification, Dewey says that we have enhanced the control of the emergence of certain qualities by the introduction of quantitative techniques. The objection to measurement raised by many philosophers is not a sound one. Here, Dewey had in mind the objection that the reduction of a subject matter to numerical measurements made it dead and bare. As Dewey emphasizes throughout the chapter, the develop¬ ment of modern science renders that thesis rather ridiculous. At the end of this lecture Nagel digressed to reject the Kantian view that measurement always involves space. I still have a vivid memory of this discussion. What Nagel had to say about this Kantian view seemed to me to exemplify just the kind of thing a philosopher should be saying and doing. Nagel considered a number of examples from the physical sciences of measurement and asked, in each case, whether or not it was possible to dispense with spatial extension in measurement. After examina¬ tion of a number of cases, he argued that it was not always required that spatial concepts be involved in measurement and he cited as the primary instance the measurement of time. Unfortunately, my notes do not show the detailed remarks Nagel made about the measurement of time, but in any case it would be a digression to enlarge upon this point here.

APRIL 30 This lecture was mainly devoted to Chapter XIII, “The Continuum of Judgment: General Propositions.” Nagel first discussed Dewey’s analysis of the types of existential propositions. There are first particular propositions. These are propositions which qualify a singular. An example would be the proposition “This is sweet.” Singular propositions, on the other hand, deter¬ mine a singular as one of a kind, for example, the proposition “This is a dog.” Nagel pointed out that for Dewey the notion of kind is important and is to a large extent borrowed from earlier logicians, particularly Mill. For Dewey the presence of a kind means the co-occurrence of traits, so that one can serve as the sign for the others. As has already been noted in earlier discussion, in Dewey’s hierarchy of propositions, singular proposi-

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19

tions have a more complicated function and a more central position in inquiry than do particular propositions. Their relation to judgments and determination of inquiry has already been noted. Nagel next turned to Dewey’s concept of generic propositions, which are a species of general proposition. They are primarily propositions about relationships of kinds. What is recurrent in generic propositions is the power of certain qualities to serve as signs. For Dewey it is never the im¬ mediate qualities of things which are general, but the mode of using signs. Nagel pointed out that this is a point of consistency in Dewey. It char¬ acterizes his effort to show distinctions as logical rather than ontological, and to show that logic arises and operates within inquiry. Thus, generality arises from a certain habit that has been established so that an individual acts in a certain way when confronted with signs of a particular kind. Nagel contrasted this view with the traditional assertion that universal arise from certain kinds of signs that stimulate certain responses. Nagel pointed out that instead of saying that a particular color red is a universal because the redness recurs, Dewey says we can claim individuals react a given way to this stimulus because of certain habits. Nagel pointed out, however, that if we say that the recurrence of the habit is due to a common element we have pushed the analysis back only a very small step. For Dewey this would be a matter of generating a new inquiry. Nagel said Dewey could always reply that we need not commit ourselves to a Platonic idealism, but can always say that signs merely function for other things and that we need no Platonic universal for taking care of the facts. Thus, for Dewey, universality consists not inherently in the thing itself but in the mode of response. Pursuing this line of thought, Nagel noted that the inquiry into the response itself requires a response, and thus we have an infinite hierarchy of responses. On the other hand, it is not a vicious regress since for knowledge we do not need to know all levels. As an example of this, Nagel mentioned an animal exposed to an auditory stimulus. The animal is able to respond in appropriate fashion and yet is not able to conduct inquiry at a deeper level.

MAY 5 This lecture was devoted to Chapter XIV, “Generic and Universal Prop¬ ositions.” I find my notes on this lecture to be rather unsatisfactory, and in an effort to bring them into better order, I have read Chapter XIV once again. I must confess that I find Dewey unusually obtuse in this chapter. The whole discussion is at a level of such vague generality that it is difficult to pin down and evaluate his central theses. It seems fair to say that in this lecture Nagel was dealing with very recalcitrant material. Nagel began by pointing out that the distinguishing feature of generic

20

1. SCIENCE AND INQUIRY

propositions for Dewey is that, unlike universal propositions, they deal directly with that which is existential. Universal propositions, on the other hand, connect attributes in a necessary fashion, necessary at least in the context of inquiry. In a characteristic turn of phrase, Dewey says that universal propositions are modes of action or possible ways of acting. The possibility is expressed by the traditional if-then form. A universal proposition tells us the conditions that inquiry aims to introduce. In this respect, universal propositions are in some sense definitional in character. They are definitional in that they constitute an analysis of a concept into its constituent elements. As an example, Nagel gave Newton’s law of gravity. It is a universal proposition; and insofar as it expresses a universal and necessary property of material bodies, it is also a partial definition of material bodies. Here, I think, Nagel was saying something that was clearer as a doctrine in his lectures than is the corresponding set of ideas in Dewey. Nagel mentioned that for Dewey there are two types of universal prop¬ ositions. He referred to Chapter XX, on mathematical discourse, and in particular to page 397. One type of universal proposition is exemplified by physical laws and the other by propositions of mathematics. Nagel then asked what these two types of universal propositions have in common. Dewey’s answer seems to be that both are in some sense definitional in character. On the other hand, Nagel pointed out that Dewey distinguishes different kinds of physical laws, some being generic and some universal. An example of a generic physical law would be the proposition that all whales are mammals. This proposition is generic because it asserts a relation of kinds, more explicitly, an existential connection between kinds. Nagel pointed out that for Dewey the logical status of a generic proposi¬ tion is that of an I or O proposition, an essentially contingent type of proposition. A universal proposition, on the other hand, is a necessary proposition which is not capable of being refuted by experience, but may be abandoned in the light of inquiry. There is no doubt that Dewey intends to deal with some traditional logical distinctions in Chapter XIV, but it is difficult to be very sympathetic with his enterprise, or to believe that he is making distinctions of much contemporary use in this chapter. Even if the distinctions are significant, the maddeningly vague and muddled way in which he discusses them makes it hard to take him seriously.

MAY 7 Nagel continued the discussion of generic and universal propositions. He said that he would like to give his own explanation of how universal propositions are intended to function in Dewey’s system, by referring

PATRICK SUPPES

21

to Peirce’s well-known distinction between premises and leading principles. He made the point that leading principles provide a means of making a transition from premises to conclusion, but also avoid the infinite, regress illustrated by Lewis Carroll’s parable of the tortoise and Achilles. He spent some time in discussing the tale of the tortoise and Achilles in order to bring out the necessity of having some rules of inference. He outlined the situation here with great clarity, but I shall not summarize his presenta¬ tion, because of its general familiarity as an example illustrating the need for rules of inference. Nagel then pointed out that sometimes premises can be converted into leading principles. He gave as an example the following syllogism: All A is B, all C is A, therefore all C is B. We could change this by converting the first premise into a leading principle and then having only the single premise, all C is A. Nagel then remarked that this discussion of Peirce’s conception of leading principles is germane to Dewey’s distinction between generic and universal propositions, because for Dewey universal propositions function primarily as leading principles in inquiry. Universal propositions formulate the kind of operations that are used to determine the sets of traits common to a kind. Nagel then considered some examples, contrasting mathematical and physical propositions in the discussion. Nagel first asserted that it is in one sense patently false to say that Newton’s second law is as necessary as 2 + 2 = 4. For the denial of Newton’s second law will not lead to a contradiction, as in the case of 2 + 2 = 4. On the other hand, he would agree with Dewey that universals are not used for descriptions of matters of fact. Nagel then showed how Newton’s laws could be used as leading principles. He referred to Carnap’s Logical Syntax of Language and said he would follow Carnap’s distinc¬ tion between premises, rules of inference, and conclusions. He pointed out that rules of inference are usually regarded as logical rules, but there was no reason that there could not also be rules of physical inference. Newton’s second law could be taken as such a rule of physical inference. He gave as an example the standard formula for computing how far a body has fallen in t seconds. The formula, s = Vigt2, acts as a principle of physical inference. We are given the premise that the body has fallen for 4 seconds. Our conclusion is to state how far it has fallen. It is easy enough to convert the formula, s = !^gt2, into a rule: Given the numerical value of the time, we calculate the distance by multiplying the time by itself, multiplying that result by the constant g, and dividing by 2. The resulting number is the number of feet that the body has fallen. Nagel pointed out that such physical rules of inference not only deter¬ mine physical consequences, as in the example just discussed, but also in part determine the meanings of terms employed in the premises and in con¬ clusions. According to Nagel, Dewey is suggesting that universal proposi-

22

1. SCIENCE AND INQUIRY

tions of physics, for example, are necessary in the sense that the meanings of terms that we use in investigations are partly fixed by these propositions. If the universal propositions are abandoned, then the meanings of terms necessarily change. Nagel said that he did see a difficulty in the sense that what Dewey calls universal propositions are in fact often used as premises and not as leading principles. Nagel did say that it is possible to look at the matter functionally and to claim that a given proposition can function in some contexts as a premise and in others as a rule of inference. Nagel said that he would put the matter this way: it would be possible to introduce a greater degree of “relativization,” which would permit us to use a prop¬ osition either as a premise or as a leading principle of inference. Nagel concluded the lecture by discussing some of the advantages of regarding universal propositions as leading principles. In the first place we test them only insofar as they are useful. For the pragmatist and especially for Dewey, theories of science are simply the means of getting from one set of singular propositions to another. Nagel admitted that initially this viewpoint does seem to clear the air. We can then assert that physical laws do not necessarily reflect the structure of things in the universe but simply provide tools for getting from one singular proposi¬ tion to another. Nagel did say that he felt there were difficulties in this view as well, but there was not time to pursue them on this occasion.

MAY 12 In this lecture Nagel dealt with that part of Chapter XVII concerned with what are often called the laws of thought or formal canons of logic. Traditionally, this discussion has centered around the principles of identity, contradiction, and excluded middle, and Dewey discusses each of these principles at the end of the chapter. Nagel began by saying that the function of laws of thought or formal canons is to state the ultimate conditions which propositions must satisfy to function properly in inquiry. He turned then to the discussion of the first principle, that of identity. He remarked that this principle is not to be found in Aristotle, and that probably the first explicit formulation is found in Leibniz. Traditionally the principle expresses an ultimate condition on any subject matter, but for Dewey it expresses something different. He here cited Dewey’s own formulation (p. 344) that the prin¬ ciple is “the logical requirement that meanings be stable in the inquirycontinuum.” Nagel remarked that this interpretation of Dewey’s is obviously different from that given by realistic logicians (realistic is here, of course, an ontological term). Nagel next turned to the principle of contradiction. He stated that for

PATRICK SUPPES

23

Dewey this principle sets the ground of complete exclusion. It is “a con¬ dition to be satisfied.” Put another way, Nagel said that for Dewey the principle states a condition that propositions must satisfy to be used in inquiry; thus it says nothing ontologically. Next, Nagel turned to the principle of excluded middle. He formulated the principle, both in the form that everything has either the property A or not A, and also in the form p or not p. He quoted Dewey’s own remark on page 346: The principle “presents the completely generalized formulation of conjunctive-disjunctive functions in their conjugate relation.” Nagel also emphasized that for Dewey the principle is a logical condition to be satisfied. It is a directive for making definitions. Discussing the three principles together, Nagel then emphasized that for Dewey the three principles have no ontological status, but quoting Dewey, page 346, “as formulations of formal conditions (conjunctivedisjunctive) to be satisfied, they are valid as directive principles, as regula¬ tive limiting ideals of inquiry.” In this connection Dewey discusses the classical objection to the law of excluded middle, that the law does not apply to changing relations. Nagel affirmed that he felt Dewey’s answer was wholly sound. The principles are meaningless for changing relations unless they are considered as conditions to be satisfied. Nagel concluded by pointing out that it is essential for Dewey’s logic that these laws are not descriptive of traits that exist outside of inquiry, but play a logical role within the context of inquiry. He mentioned that a predecessor of Dewey in this line of thought was F. C. S. Schiller. It should be mentioned that Nagel spent some time explaining the standard logical formulations of each of the three principles, and I have omitted that material here.

MAY 14 In this lecture, the last one of the term, Nagel began by discussing Dewey’s views on induction. He stated that Dewey’s views on induction and more generally on scientific method were fairly orthodox, but as he would show later in the lecture his views on causality were not. Nagel remarked that Dewey is concerned to show that a given sample or class has representative connections for the whole. This is one formulation of a classical problem of induction. The difficulty, of course, is to know how to state the criterion of representativeness. Nagel pointed out that beyond stating the problem correctly, Dewey says little more. For instance, at no point does he discuss, even at an elementary level, general principles of statistical inference. Nagel remarked that it was disappointing to find so little specific discussion in Dewey, considering the definiteness of much of the literature on induction.

24

1. SCIENCE AND INQUIRY

Nagel then turned to a discussion of Chapter XXII, “Scientific Laws— Causation and Sequences.” He first pointed out that for Dewey causation is taken as having a logical, rather than an ontological, character. Nagel said that he thought that Dewey’s analysis of causation was one of his most successful efforts. His attempt to give causation a logical status, to place it in the context of inquiry, and to deny it a status in nature, as such, represented an attempt to say something important and new. Dewey continually tries to make it clear that we must be wary of asserting anything about order in nature. Causal laws for Dewey are a means of introducing links between events, but he emphasizes that in his view the links do not exist prior to inquiry. In this respect he is very much against Mill’s view of causal laws as necessary and unconditional. According to Dewey, what Mill should have said is that causal laws are means by which a certain kind of uniformity between events can be established. Dewey continually reiterates, in contrast to Mill, that the subject of science is not sequences of events but the establishing of links between traits or characters of events. In summing up Dewey’s analysis of causality, Nagel expressed the view that Dewey’s analysis of causation is one of the most original parts of the instrumentalist-pragmatist position. The particularly distinctive feature is the view that all general propositions have a significance only within inquiry. To ask the question that has repeatedly been asked in the history of philosophy about the representativeness, or even the ground of general propositions, is to raise questions that pull the propositions out of context. What a general proposition is can only be determined by what it is used for. Interpreted this way, we do not take general propositions as representative of the structure of nature as such. It is fair to claim that the adoption of this view permits a wholesale “deontologizing” of a wide range of propositions. This view, which Dewey defends so consistently, provides quite a fresh viewpoint in the history of philosophy. Regarding the further question of whether Dewey can disprove the ontological interpretation against which he argues so vigorously, Nagel said it seems fair to say that Dewey can show only that such an ontological interpretation is not required. The assigning of an ontological status to propositions is in no way necessary for understanding them. However, the final question to be asked of Dewey is whether he can avoid by this move all questions of metaphysical status. Nagel ended the lecture by stating that this question is a matter of considerable debate. The reader should remember that the synopsis of Nagel’s lectures given here is a very much abbreviated version, and, perhaps just as important, is a version based on the notes of a new and rather na'ive student of philosophy. If the reader feels that some points are put too simply or inaccurately, the fault is almost surely mine rather than his. In spite of such faults, the recording of these notes may still be of some service, for

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Dewey has possibly the most impenetrable prose style of any serious philosopher since Hegel. On the other hand, like Hegel, he has important and fundamental things to say. It seems fitting to end with the closing lines of Dewey’s Logic, which describe so well not only the major thrust of his work but the dominating spirit of Ernest Nagel’s philosophy as well. Since scientific methods simply exhibit free intelligence operating in the best manner available at a given time, the cultural waste, confusion, and distortion that results from the failure to use these methods, in all fields in connection with all problems, is incalculable. These considerations reinforce the claim of logical theory, as the theory of inquiry, to assume and to hold a position of primary human importance.

SOME DIFFICULTIES IN KNOWING

Stuart Hampshire

I There are at least three distinct kinds of challenge to, or rebuttals of, a claim to knowledge: The first is the simple rebuttal—“What you claim to know to be true is not true”; the second is a challenge which questions the source of the knowledge or the method by which the alleged knowledge has been obtained. This challenge is.commonly expressed in the words “How do you know?” When such a question is put as a challenge, it is im¬ plied that the claim to genuine knowledge is not acceptable unless a reliable source, or a reliable method, has been used in the particular case. A claim to knowledge is not to be respected unless the knowledge claimed has a respectable origin; the speaker may be required to show that he is an authority on the particular issue, as he has implicitly claimed to be. He must be able to show that he is in a position to know that which he claims to know. Otherwise he is exposed to the rebuttal: “What you say may be true—but you cannot now possibly know that it is.” A third and different challenge, or different kind of challenge, may be phrased in various ways, but is commonly expressed in the words “Are you sure?” Human beings are apt to err: a liability to mistake attends all their performances, not excluding their search for knowledge. They make mere mistakes, as it seems, by chance, or perhaps even inexplicably. They use words care¬ lessly, inaccurately. They forget things. They overlook things. They write down the wrong number for no reason at all, or for no reason that they can give. They often have a reliable source and a reliable method; they have been asked the time; they have a watch; but then they misread the figure on the dial carelessly. A machine may misread the figure on the dial, but not carelessly; for it does not employ, or need to employ, care in reading it correctly. But human beings do. They are not instruments: or, if they are instruments, they are instruments that use themselves, and they may misuse themselves. They need to be careful in order finally to be sure, because they may go wrong without anyone having a reliable method of finding out why they have gone wrong. “Are you sure?” unlike “How do you know?” asks, among other things, whether you have on this occasion been careful, or careful enough, in using a generally reliable 26

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method. Of course you can also make a machine check its results; but this is not the same as asking it whether it is sure that it has not made a mistake. A claim to knowledge is certainly not to be respected unless the knowledge claimed has a respectable origin. But more than this can properly be asked: “Are you sure that you have not just made a mistake?” I am just mentioning well-known facts here. These well-known facts are of the first importance in understanding the notion of knowledge, and per¬ haps also the concept of following a rule. I check the proof to be sure that I have not made a mere mistake, just a slip, somewhere in the derivation. I look at my watch again when the small boy who has asked me the time says, “Are you sure?” This kind of fallibility, or corrigibility, is always in the background, and with some of us, on some topics, it has to be in the foreground when we claim to know. I may be in the best possible position to know that something is the case, and yet I may throw away my ad¬ vantage, inadvertently, carelessly, incompetently, for no reason at all. This is why I can be asked “Are you sure that you are right?” when I have claimed to know something on occasions when it would be absurd to ask me how I know. I may be reminded of one kind of fallibility even when the other kind of fallibility—fallibility in respect of source or method—is not in question. There notoriously are occasions when the question “How do you know?” would be at least absurd, and perhaps unintelligible, as a question. These are the same occasions on which a statement shows, in virtue of its gram¬ matical form and its topic, these two taken together, that the speaker is in the best possible position to claim to know that it is true. For example, he who reports that he is currently experiencing a certain sensation cannot intelligibly be asked how he knows that he is; it is already shown, in the grammar and vocabulary of the statement itself, that he is in the best possible position to claim to know that his statement is true. The grammar and vocabulary show that he is the authority on this matter, that he is in the optimum position for making that statement. But he can intelligibly be asked whether he is sure that his report is correct; doctors often do put just this question, and it is sometimes, and in abnormal conditions, difficult to answer it; on the other hand, it is often, and in normal conditions, very easy indeed to answer it—e.g., when the description is not a very specific one or when the sensation is a familiar one. Of the first person, present tense reports of sensations, one may say that it is evident that the speaker is the authority. But he is also a fallible authority, particularly if he attempts a very specific description, or a description that is in some other way am¬ bitious and highly informative, or when he describes a sensation that is very unfamiliar. We may therefore, as a preliminary, divide statements about states of mind, attitudes, and desires into two classes: first, those statements which admit the challenge “How do you know?” when a claim to knowledge has

28

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been made, as well as the less specific challenge “Are you sure?” In the second class, we have those statements which show, in their grammar and vocabulary, that the speaker is in the best possible position for claiming to know that the statement is true, that he is the authority, and that no ques¬ tion about the source of his knowledge arises; then only the less specific challenge “Are you sure?” is in place, a challenge that requires the speaker to think again about the statement in case he has been careless in making it and has not on this occasion taken due precautions against mistake. This latter challenge is appropriate to any claim to knowledge of any kind, whatever the grammar and the vocabulary of the statement may be. But it will be especially appropriate when the topic is complex, or when for some reason it requires a careful use of words or careful matching of a specific description, or when there is evidence that the speaker has been hasty or careless or that he has been unreliable in the past. We all know, for ex¬ ample, how difficult it may be to describe sensations when they constitute symptoms to a doctor; there is the difficulty of not exaggerating or under¬ stating, of distinguishing between that which is more correctly called pain and that which is more correctly called discomfort. And if we are asked “What kind of pain?” or “What is the pain like?” we know how difficult it may be to give a more specific description and to avoid error in matching the right description to the phenomenon. With this kind of possibility of mistake, I may hesitate, be doubtful, and need to stop to think before answering a question, even though I am clearly in the best position for claiming to know the correct answer; the grammar and vocabulary of the question alone may show that, if I do claim to know the answer, no question would arise of how I know or what the source of my knowledge is. Yet I hesitate, am doubtful, and do not know what the correct answer is. Someone else may think that he knows what the correct answer is, and his belief may prove correct, even though I, the authority, do not yet know. The doctor may know: he has seen lots of cases like mine; perhaps he has had the disease himself, or he has heard lots of reports, and he is thoroughly familiar with the appropriate vocab¬ ulary. He has used a reliable method of inference; and I may subsequently confirm that his conclusion was correct. He had arrived at a conclusion that turns out to be correct by an argument that establishes that so-and-so must be true or is likely to be true. If after careful attention to the phe¬ nomenon and to the description, I confirm his conclusion, I do not present the conclusion of an argument; I report what I claim to know, or what I believe, without benefit of argument. In the old-fashioned, much abused phrase, I now either know, or I think I know, directly. Because of the possibility of carelessness of different kinds and the possibility of inaccuracy in the use of words or symbols, it is never an offense against the proper use of language for someone to argue that he who is in the best possible position for making a certain statement is prob-

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ably, in a particular case, mistaken. It may, in the circumstances of a particular case, be irrational and silly so to argue; but it will not be an offense against the proper use of language, as it would be to ask “How do you know that you are in great pain?” as opposed to “Are you sure that ‘great pain’ is not an overstatement and untrue?” Because of this kind of corrigibility, an inductive argument may be relevant to establishing the truth or falsity of a statement which the speaker himself was in a position to claim to know to be true without any appeal to evidence or argument. From the fact that we may sometimes have good inductive reasons to be¬ lieve that a man is probably mistaken in the report that he gives, for example, of his sensations, we must not conclude that such statements require inductive evidence as their support when a claim to knowledge is made. They do not require inductive support when the speaker is also the designated subject of the statement; but an inductive inference may on occasion lead us to the conclusion that such a statement, made on the best authority, by a man trying to be truthful, is probably false, or even, in some cases, that it must be false, where the “must” is the sign of an inference.

II In the case of sensations, then the subject’s difficulty, if he has one, will typically be one of matching the right description to the phenomenon experienced. If he hesitates and is unsure about the right reply to a question about a present sensation, he may be uncertain about what to call the sensation or how to classify it. Since truth has always been represented as an agreement between a statement and that to which it refers, the subject has a doubt which is a pure case of doubt about the truth of the various possible descriptions that suggest themselves to him. This is one particular kind of doubt among others, and a particular kind of difficulty in achieving a true statement. He needs to be accurate, or exact, to find words that fit perfectly and that are not just approximately right; and it may be difficult to get an exact fit in words. On the one side is the phenomenon, the reality to be described, which is in no way concealed from him, and is, so to speak, transparent when he attends to it; he therefore does not need to investigate further, or to probe or to experiment or to approach the phenomenon from another angle. On the other side, requiring to be matched, is the commonplace vocabulary in use from which he has to make a fitting selection. If he is not sure whether a particular description that suggests itself would be a truthful description, this would not normally be accounted a case of ignorance; it is not exactly that kind of lack of knowl¬ edge from which he suffers, unless it is some ignorance of the standard use of the relevant words. It is an uncertainty of another kind. Let us call it the semantic uncertainty. It is in some respects very like the uncertainty

30

1. SCIENCE AND INQUIRY

that a man may feel when trying to discern whether one pattern exactly matches another pattern in a different medium. It is uncertainty about matching possible descriptions with an independent reality. This kind of uncertainty, which might occur in connection with almost any kind of empirical statement, is often at its most acute when a man is trying to find the most fitting and accurate description of his own sensations. Every kind of statement has its own kind of liability to error, and this matching of descriptions to the phenomena becomes more crucial when other liabilities to error have dropped away. Then one may resolve his uncertainty by an inductive inference. This description is probably the right one. Then he may tell you, by reconstructing the inference, why he believes that this must be the correct description; but he will not tell you how he knows; for his belief evidently falls short of the ideal case and therefore does not constitute knowledge. He was indeed in a position to know, but he was unsure; therefore he had to use an eccentric, and not the standard, method of arriving at his statement. In a parallel case, a man was in a position to know what was said at a meeting because he was present and heard the speeches, but he happens not to be sure. He may fill the gap by inferring what must have been said. If he is asked “Do you know that this is a true account?” he will have to indicate that he is not claiming the authority of a witness. He can tell you why (using what evidence) he believes that this is a true account.

Ill Consider next another type of case in which a man is asked a question about himself, in respect of which he is in the best position to know the answer; and yet he does not know and admits that he is uncertain about it. He is asked whether he wants, or would like, to go to Italy with a certain group of people or not; let us suppose that it is certain that his emotions are strongly engaged by the prospect, and that he is far from being indiffer¬ ent about it. He is not being asked whether he will go; that is, he is not being asked to decide an immediate practical question. Let us suppose that the inquirer is intensely curious about the subject’s state of mind, as it concerns this question, and that there is no practical possibility immediately in view at the time of the inquiry. The questioner wants the subject to re¬ flect and to tell him what his desires are; the questioner might even ask how he feels about going to Italy. The request is a request for information. The subject reflects and replies: “I do not know whether I want to go or not. I can’t tell you yet; you must allow me to think about it further.” We are very often uncertain about what we want to do in a specific matter which is very far from indifferent to us. “I don’t know what I want

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to do about it: I am not sure whether I want to or not” is a form of words often used when, for example, a man has a divided mind about the project; perhaps there are features of it that are attractive to him and features that are repugnant. Or perhaps he feels that he has not thought about it enough, or perhaps he is confused about it. Or again there is a wide range of different types of cases where “I am not sure what I want” may express an uncertainty in specifying precisely the nature of the object wanted rather than an uncertainty about the balance of desirability in an already fully specified object. Or the uncertainty may be an unsureness in focusing my desire on its proper object and in eliminating alternatives. Or the subject may be vacillating and in a state of turmoil about the matter. Anyone who (in delineating the notion of knowledge) respects the actual uses of language will admit that uncertainties of this general kind are as genuine as any other cases of uncertainty. They are genuine cases of not knowing. As I may be uncertain about the real properties of some object before me, and as I may be uncertain both about the truth of some highly specific statement about my sensations, so I may be uncertain again when asked to make some statement about my desires or my aims, ambitions, hopes, attitudes, sentiments. When I am not sure what I want to do or want to have, because my desires are confused or inchoate or because they conflict or because they are not clearly formed, I may try to end the uncertainty; for someone re¬ quires the information about myself which only I can give him author¬ itatively. When in this situation I stop to think, my problem is not gen¬ erally, typically, that of matching words to an independently recognized reality; my uncertainty is not usually a semantic uncertainty, although it could be that, or it could be that as well. In the journey-to-Italy situation, I would naturally consider the merits of the proposed courses of action. In determining what I want to do when I am in a state of uncertainty and am not sure and do not know, I attend to the features of the possible courses of action. If after careful reflection, I announce my conclusion in the words “I now know the answer to your question: I now know what I want,” I would normally be able to give my reasons, the considerations that have led me to the conclusion; now I can say “I now know what I want.” The reasons would be reasons for wanting, and they would not naturally be counted as evidence that a certain account of my already ex¬ isting wants is true. For these reasons some typical cases of “I do not know what I want” may be assimilated to some typical cases of “I do not know what I shall do,” in respect of the kind of knowledge involved. And it is not surprising that making up one’s mind, or coming to know, what one wants to do is often very like the formation of an intention: very like, in that the required precautions against error are very often precautions against misguidedness

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or incorrectness in the desire and the intention rather than precautions against incorrectness in the statement of the desire or intention, taken as something already independently formed. There are contrasting cases when a man notices and reports his desires, cravings, and impulses as phenomena of experience, exactly as he may report his sensations; he notices that he wants a drink or finds that, being hungry, he wants to eat. In such a situation a man may be uncertain about the correct characterization (or the name) of the thing that he wants, although there is an acceptable sense in which he does know what he wants; for he may know that he would unhesitatingly and unerringly recognize the particular thing, or the kind of thing, that he wants if it were produced before him. In such cases his knowledge of what he wants, when attained, is knowledge of a fact about himself, closely parallel with the fact that he has such-and-such a sensation in his leg. He may often be surprised to discover, or to notice, that at this moment he has an impulse, or desire, which he had not expected that he would have. In such cases he would not be said to have formed the desire, even less to have formed it as the con¬ clusion of a process of thought; rather he has come across it, as a fact of his consciousness. The desire occurred and emerged in his consciousness, independently of any calculations. The word “lust,” for example, par¬ ticularly if it has a sexual connotation, is almost, but not quite, the name of a sensation. There is a whole spectrum of cases between the felt bodily craving, which approximates to a sensation in respect of the kind of knowl¬ edge that we have of it, and the reflective desire or interest that is formed as the outcome of a process of thought. A man may, for example, be led to recognize that he has at this time (already) a desire which he had not known that he had and which a friend had inferred from the evidence of his behavior. The desire was not something that he had felt; nor was it a desire that he had formed; he had not come to know of its existence in either of these ways. Prompted by his friend, he had originally inferred its existence from his behavior—“This must be what I have been wanting to do.” But here I want to concentrate on the case of the man who does not know whether he wants to go to Italy and who has to stop to think whether he does. If from this initial state of uncertainty he moves to a conclusion which amounts to his now knowing what he wants, or to his now knowing what his attitude is, his process of thought is properly characterized as delib¬ eration. Deliberation is a process of thought that begins with uncertainty and that is aimed at some conclusion, accepted by the subject, of the form “This is to be true of me.” The uncertainty from which the process of thought started was not an uncertainty about the matching of a statement with an independent reality. The uncertainty that leads to deliberation is always an uncertainty about what is to be the case and not about what is or was the case. It is an uncertainty about the future, conceived as alter¬ able, one way or the other, by removing the uncertainty.

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Formation of a present belief, desire, intention, attitude, or sentiment is a case of coming to know what my belief, desire, interest, attitude, or sentiment is to be, starting from now, and not of coming to know what it already is. Coming to know or to be sure what my attitudes and senti¬ ments about something are counts as a decision, insofar as the subject has aimed, in reaching his conclusion, at some kind of appropriateness in attitudes or sentiments taken in relation with their object. Perhaps one can distinguish this intentional self-knowledge and intentional uncertainty from semantic uncertainty in this way: if the subject is still uncertain and is wondering whether he has a certain aim or ambition or not, then an ob¬ server cannot know either that he does have the aim or ambition or that he does not, and any knowledge that an observer possesses on this topic must be knowledge of the future. An observer may believe that the sub¬ ject has had an aim or ambition, perhaps an unconscious one, which can be inferred from his past behavior. But if the subject is still doubtful about his present aims, the observer will not claim to know that the subject still has this aim, definitely and without qualification, unless and until the sub¬ ject’s uncertainty ends. A doctor may know what I am now feeling and that a statement about my present sensations is true, even when I am my¬ self still uncertain what the correct account of my sensations is. In respect of aims and ambitions, and that which I want to achieve, an uncertainty in giving a correct account, whether for my own benefit or for the benefit of others, usually amounts to an uncertainty of aim. The opposite case would be the situation in which I am uncertain of the right words to describe my objective. The verb “believe,” being (like “try”) a strongly intentional verb, scarcely admits of this opposite case. But there are many situations in which I do not know which of two things I believe about something, and therefore no one else can know what I believe until I make up my mind and thereby come to know what I believe. An observer can only predict that I will make up my mind in a particular way. And he, no less than I, may infer from my behavior and from other evidence that I must have believed so-and-so before now, but not that I must now believe so-and-so when I myself am doubtful. So I may say of my friend— “He cannot know what I want to do, because I do not yet know myself.” He can only use his psychological knowledge to predict what I will want to do.

IV It seems that if I am still uncertain about what I want to do, I do not, as my friend might, use the scientific knowledge of causes that I may happen to possess to settle the question of what I now want to do by some kind of inductive inference. There does seem to be a real difference between on

34

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the one hand self-knowledge, in the sense of contemporary knowledge of one’s own mind at the present time, and, on the other, knowledge of the desires, beliefs, and sentiments of others, or knowledge of one’s own past, of the history of one’s desires, beliefs, and sentiments, which one may regard exactly as one regards the history of another person. Referring to one’s own past, one may try to explain why the sequence of desires and attitudes was what it was. This might be a standard causal type of ex¬ planation, because the facts to be explained can be established prior to, and independently of, the explanation of them. Consider two different questions: (1) “What do I now want to do?” and (2) “What will I want to do?” A man who has an adequate psy¬ chological theory may in many cases use his knowledge of causal con¬ nections to predict what he will want to do, if and when certain sufficient conditions are satisfied. He knows that he will want to eat later in the day. He knows from experience that he will want to laugh when he meets his old friend, or that he will want to run away when he meets his enemy. But it does not follow that he can use his knowledge of causes to discover what his wants are in cases where he is doubtful and does not already know what he wants to do. If he finds that he does actually feel certain desires, he will then use his psychological knowledge to explain why his present desires are as he finds them to be. Suppose a man believes that if he knew the relevant covering laws, and if he knew that the initial condi¬ tions stated were satisfied, he would know that it must be true that he will want to do so-and-so. It must be true that he will want to eat before six o’clock. But could he settle his uncertainty about what he now wants to do by a similar inductive inference? If he could, then the conclusion of the inference could be expressed in the words “Given that such-and-such sufficient conditions have occurred, I must want so-and-so”: or, more plausibly, “So-and-so must be what I want.” But then it seems that he will naturally ask—“But is it?” He will look for an endorsement of this conclusion. He will ask himself whether he finally does want what he has inferred that he must want. An observer might have said to him, “I know all about you and about people in your condition: so-and-so must be what you want.” But he will wait for the subject’s confirmation; for while the subject is still doubtful whether the so-and-so is what he wants, his desire is still inchoate and unformed. There is a typical use of the word “must” which will, I think, serve to mark the distinction between the two kinds of knowledge which I am pursuing; this is the “must” that marks an inference from evidence that compels one to draw a certain conclusion. This is the use of “must” in such statements as “This must be my hat” or “Your hat must be upstairs since it is not here” or “He must have been very unhappy when he heard the news.” It may be useful to interpolate something about this use of “must” in claims to knowledge of matters of fact.

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It has sometimes been said, mistakenly, that this use of the modal form makes a stronger statement of fact than the corresponding plain indicative statements—“This is your hat” and “Your hat is upstairs” and “He is very unhappy.” If by a “stronger statement” is meant a statement which claims a greater degree of certainty, this is surely not true. The flat assertion “This is your hat” would commonly be taken to imply a greater degree of cer¬ tainty than the more tentative “This must be your hat,” if any implications at all about certainty can be drawn from the wording of these statements alone. The function of “must” in such cases as these is to imply that the assertion about the hat is the conclusion of an inference, when the statement might have been known to be true directly and without the aid of inference. In a well-known short story by Saki, a woman, descending from a train at a country station, is accosted by a stranger who says,—“You must be the new governess.” The woman replies, “Well, if I must, I must,” and the story proceeds from there, from this willful confusion of two kinds of necessity. Had the stranger simply said, “You are the new governess,” she might simply have denied it. The necessity in “You must be the new governess” is contrasted with the possibility represented by the weaker form of modal statement “You may be the new governess”; both show something of the weight of evidence for the conclusion, in the one case compelling evidence, in the other not. Because “It must have been X” and “It must be X” indicate the conclusion of an inference, they often do not make such a strong claim, in respect of certainty, as the unmodified “It was X” and “It is X.” The man who says, “It must have rained” makes no stronger claim, in respect of certainty, than the man who says, “It was raining”; the former is typically the man who has seen the wet pave¬ ments, the latter the man who remembers the rain coming down. So much is true of putative statements of fact about the present and the past. But the situation in respect of statements about the future course of events is quite different. Here the contrast between statements which are the conclusions of an inference and statements known to be true noninferentially is at least far from obvious. If I am considering the future course of events, nothing corresponds to my just seeing my hat and recognizing it, or my having seen the rain and having remembered it. For this reason we may be inclined to say that all statements about the future course of events, other than statements of intention, are known to be true, if known at all, by some kind of inference. “Fascism will not happen here,” no less than “Fascism cannot happen here,” will be supported by some inference; “It will happen” no less than “It must happen” will be sup¬ ported, in face of the challenge “How do you know that it will?” by an appeal to some evidence that it will, or to some method of inference, and not by any claim to the authority of direct knowledge, parallel to “I can see it” or “I remember it.” For this reason there is a definite sense in which “It must happen” is normally stronger than “It will happen.” “It must hap-

36

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pen” here has a greater logical force, because it says both that it will happen and that there is compelling evidence that it will. “This horse, Ajax, must win the race” says more than “This horse, Ajax, will win”: namely, that there is no possibility of its losing; and this cannot be a correct thing to say if the evidence does not exclude the possibility of its losing. The pos¬ sibility is typically excluded, and the “must” is justified, when a contrary outcome is incompatible with some well confirmed general proposition. Let us then try to transfer this idiom of “must,” marking the conclusion of an inference that compels assent, into the present tense. A man may say “I must be in love” or “I must be jealous” because he has noticed features of his own behavior that amount to compelling evidence that he is. In respect of these passions, revealed by their typical symptoms, he here speaks of himself as he might speak of another or of his own past. These are the facts that he has discovered about himself, and he can tell you how he knows and of the evidence that has led him to this conclusion. Such an inference is not unusual. On the other hand, only in very peculiar circumstances would a man intelligibly say “I must be in pain,” or “I think I must be in pain”; he scarcely ever infers that he is in pain, and there is not normally any question of him telling you how he has discovered that he is in pain, or of his giving you the source of his knowledge. Least of all would he say “I must be in pain because such-and-such has happened, which, as you know, causes pain.” There are exceptional circumstances in which the modal idiom has a use: a man might reasonably say to a doctor—“I suppose you are right when you infer, on the basis of your experience and medical knowledge, that I cannot be in great pain, only in some acute discomfort; perhaps I am exaggerating.” He infers that the doctor’s description is likely to be, or even must be, the correct one and that he must be misdescribing; what he is calling great pain is not what is ordinarily called great pain. He resolves the semantic doubt (exceptional for the unspecific description “pain,” less so for “great pain”) by an in¬ ductive inference. But one cannot naturally say of oneself “I must believe p,” although one may say “I must have believed p at that time.” “I must believe p,” representing an inference from the evidence of behavior, is a way of saying “I must have believed p up to now,” the moment of realization. I may infer from signs and behavioral indications what my past beliefs were—“I must have believed him”—as I may infer the present beliefs of another. But one cannot infer what one’s beliefs are to be, starting from now. Either one already knows, or one has to answer a normative question, to form a belief on the evidences of truth, as one takes them to be. I may in exceptional circumstances infer what my beliefs will be in the future. For example, I might say “I know from experience that when I see him I shall fall under his influence, and that I will believe what he tells me, even in defiance of the evidence.” The implication here is that I shall no longer know, re-

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member, or realize, when I am in his presence, that my beliefs are formed under his influence and only under his influence. I will not have in mind the correct explanation, as I now suppose, of my having the belief. I can¬ not transfer this melancholy self-knowledge into the present without in¬ coherence-—“I only believe this story because I am under his influence.” This is not the expression of a contemporary belief but a kind of irony; for a declared belief is necessarily endorsed by the subject as to be explained, at least in part, by his reasons, and as not to be explained wholly by external causes which are not taken to be evidence of truth. He who can truly claim to believe p to be true necessarily intends his favorable attitude to p to be alterable by evidences of error. For this is how believing that p is true is distinguished from cognate propositional attitudes—e.g., hoping that p is true, liking to imagine that p is true, wishing p to be true, and many others. So it would be absurd to arrange an experiment to determine the causes of beliefs of a certain kind, and to ask the cooperation of the sub¬ jects: “The experimenter will apply certain stimuli and change your en¬ vironment in various ways, and you are to report the change in your beliefs that are the effects of these external changes.” There would be a conceptual impossibility in carrying out these instructions. Some apparent psychological effects could be reported—e.g., a changing inclination to believe a certain proposition. But the subjects, in the circumstance of the experiment, would not call these noticeable psychological effects changes in belief. Nor can a man under posthypnotic suggestion say “I only believe this because I have been hypnotized to believe it.” My declared beliefs are not facts about myself that I may discover after a preliminary uncertainty, as I may dis¬ cover, after a preliminary uncertainty, that I must be in the grip of some passion because I exhibit the symptoms. For the same reason I cannot intelligibly say “I must intend to do so-and-so,” although I can intelligibly say “I must have intended to post the letter and then forgotten it.” On the other hand, I could intelligibly say, using the concept of purpose, and the present tense, “I see now: this must be my ultimate aim or purpose in doing this although I had not known that it was.” I may infer what my more or less unconscious aim or purpose has been up to now from signs in my overt behavior. But now I have to endorse or repudiate the purpose; what is to be my purpose, starting from now? Perhaps I do not know the answer to this question; perhaps I do not know, in respect of some activity in which I am engaged, what my aim or purpose really is. In removing this uncertainty, I fix my aim. Similarily with the two-faced concept of desire: While I make up my mind what I now want, there are no knowable facts to be expressed in the words “I want to do so-and-so.” If I infer that I must already really want so-and-so, or that I must already have suchand-such an unconscious desire, it is still an open question whether I dissociate myself from this desire, now brought to consciousness. The “openness” resides in the reference to the immediate future, to “now” as

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1. SCIENCE AND INQUIRY

meaning “starting from now” as opposed to “up to now.” Given that I have discovered, perhaps as the conclusion of an inference from my previous behavior, that I do have the desire, do I want to get rid of this desire, if I can, or do I now endorse it? Does it now persist as a conscious desire? The fact that I have learned is a fact about the immediate past, leading into the present. Starting from now, and from this fact, I may think again about what is to be desired. In this uncertainty there is room for delibera¬ tion, that is, for determining what is to be true of me. Suppose that I am doubtful about what I want, and suppose that, under the influence of an adequate and tested psychological theory, I think that it must be possible to infer what I now want, just as I infer what another man now wants on the basis of a theory—“Given so-and-so that has hap¬ pened, he must want so-and-so”; for the relevant initial conditions, and causal factors, are known; so I conclude that I must now want so-and-so, given that so-and-so has occurred. Is this odd and unusual? I think it is: but is it odd and unusual for the same reason that it would be odd for a doctor to conclude by a parallel argument that he, the doctor, must be in pain because he has a disease which is always painful? Surely not— the oddity has a different source. Of the doctor and his pain, one would merely say that if he already has a pain, he must know directly that he has. “Am I in pain?” is an unusual question for which a particular context of semantic uncertainty must be imagined: the context, already men¬ tioned, in which the speaker is doubtful whether he would be exaggerat¬ ing if he described his condition as pain rather than, e.g., discomfort. But the question “Do I want so-and-so?” is far from unusual and is intelligible in a very ordinary context. That I should not be sure what I want is entirely normal. One would not appeal to the principle that if a man has a desire he must generally, and except in odd circumstances, know directly that he has; for the very range of cases under consideration show that no such principle operates. It is an intrinsic feature of desires, aims, ambitions, purposes, attitudes, that they may be uncertain, unfocused, confused, inchoate. It was one of the strengths of Aristotle’s Ethics to have recognized this, as against those empiricists who have taken desires as in¬ variably given facts of consciousness. The oddity of saying “Applying a reliable psychological theory, I have come to the conclusion that I must now want this” resides in the kind of knowledge claimed, which does not match the original uncertainty, ex¬ pressed as “I do not know what I want.” Admittedly there are cases in which a man may infer that he must really want to do so-and-so, although he had not realized this until his attention was drawn to the evidence. But in coming to know this, he has not removed the kind of uncertainty which he might have expressed in the words “I do not know whether I now want to do X or not.” If he had been asked “Do you want X?” he might have answered “I do not know,” and then be convinced that he had been wanting

STUART HAMPSHIRE

39

X. But he still has the immediate future, starting from now, to determine. The man who does not know what he now wants in a matter that deeply concerns him, and who also does not know whether he admires an action or not, normally has a question for decision. Just because he has a question for decision, it seems that he cannot settle his doubt by a factual inference, which could only lead him to a conclusion about what must have already been true before this doubt arose. In removing this present uncertainty, he has to consider the features of X that make it desirable for him and the features of Y that make it admirable. In confessing his uncertainty, he has confessed that his desire for X and his admiration of Y are still mere pos¬ sibilities, to be endorsed or eliminated by him. An observer may infer, applying a theory, that he will make up his mind in a particular way and that he will emerge from his deliberations with a particular desire and a particular attitude. But the subject will not now remove his uncertainty in this way. If therefore someone were to say to him, in this situation of uncertainty, “You must want X” and “You must admire Y,” he would normally take this “must” as prescriptive, as an imperative of rationality or perhaps even of morality—perhaps as meaning “You have compelling reasons for wanting X and for admiring Y.” There would be an implied allusion to some standard of correctness—for example, “You must, on pain of being inconsistent” or “You must, on pain of being utterly mis¬ guided in your desires and admirations.” The “must” would not naturally be taken as “It must be the case that you do in fact want X and admire Y.” Suppose I say to you: “You must want to save your friend.” I may intend this to be taken as “It must be true that you do. No human being could be so unfeeling.” But you would not express agreement with this proposition in the words “All right, I must.” This would be Saki’s governess again. You would say “Strangely enough, I don’t” or “I do.” Either you know whether you do or do not, or it is not yet true either that you do or you do not, just because you are still uncertain about it, after the question has been put to you. Even if the inference to the desire had been correct, and the desire was an unconscious desire unrecognized by the subject, it still would not be true that you simply want now to help your friend if, at the conscious level, you are now doubtful. However, “You must have wanted X and admired Y” would be construed as “It must be the case that you did.” “I must have wanted X and admired Y” would in similar fashion be construed as meaning “It must be the case that I did.” On the other hand “I suppose I must still want X” would often be interpreted as “I suppose I must if I am to be consistent and rational.” I might be committed, logically or otherwise, to wanting X by my other desires and admirations. If the “must” is expanded as “I must, on the pain of being inconsistent or ir¬ rational,” then it is not misleading, I think, to characterize the statement as normative. It is normative in the same way as “I must accept the conclusion of this argument” or “This cannot be doubted,” which does not normally

40

1. SCIENCE AND INQUIRY

represent a psychological impossibility. But “I suppose I must still want X might represent the discovery and acknowledgment of a desire, particularly a long-range, or so-called dispositional, desire, which has existed up to now; so a man might say “I suppose I must still want to please him,” acknowledg¬ ing this fact about himself. And then the question arises—“Given that it is an inferred fact that I have this desire, do I want to get rid of it or not?” He has discovered and acknowledged that he has wanted to please up to now; but does this desire, once recognized, disappear, and if it does not, but rather lingers, does he wish that it would disappear? In any case his state of mind and its explanation have become more complex. I view the history of my past states of mind and attitudes from the standpoint of an observer; that I am writing an autobiography rather than a biography imposes no limits upon the inferences that I may make about my desires and attitudes, as they have existed up to now, or upon the explanations that I may give of them. The settlement of any doubts that I may have about what these states of mind actually were does not modify the states of mind themselves; in recognizing them for what they were, I do not endorse them. But while I do not yet know what I want and am still uncertain what to aim at, the adequate psychological theory cannot be applied, just because uncertainty exists in my mind, and uncertainty of a particular kind, the kind of uncertainty which is neither ignorance, nor a semantic uncertainty, but rather an intentional uncertainty, which raises, at least in part, a normative question. Imagine a man who believed that he had an adequate psychological theory which would never leave him un¬ certain about his future conscious desires, aims, ambitions, and attitudes. He would expect to find himself, as it were, saddled with desires, aims, ambitions, with states and attitudes, which he had all along expected to occur in the foreseen conditions of his particular case. He would expect all his own desires, aims, ambitions, and attitudes to occur, just as the ordinary man now expects that, after his long fast, he will want to eat, or that under certain conditions he will have a craving for sugar, or for alcohol, or a longing to fall asleep. Such a psychological theorist would expect to find himself, under foreseeable conditions, acquiring a certain ambition or falling into an attitude of admiration, as the ordinary man expects to find himself, under foreseeable conditions, becoming angry or frightened or jealous, or having an impulse to run away, or becoming depressed about something. There are indeed desires, emotions, moods, which I know will descend upon me under certain conditions; I can often infer and then observe their occurrence in myself no less than in other people. There is nothing in the characterization of such desires, emotions, moods which excludes the pos¬ sibility of my being the passive and helpless observer of them, and my knowing of their existence in this way. I may, without contradiction, be said to observe myself “falling into” these states or being subject to these

STUART HAMPSHIRE

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impulses; I can see myself in these respects as a specimen. I know that I will want this or fear that or be disappointed by the other. But then further uncertainty occurs at a higher level, which by itself introduces a new complexity into my state of mind. For there is always the question of the attitude that I am now to adopt to these presumed facts or presumed probabilities about myself: do I want them to be otherwise or not? The “doubling” of my references to myself is unavoidable, as long as I have the means to infer and to notice these facts about myself. As soon as we develop this hypothesis of the man who applies a psy¬ chological theory in self-prediction, we see that he might be led to discard many of the idioms that a man now uses to talk about himself, about his aims, ambitions, interests, hopes, and attitudes. The state into which the “psychological theorist” might expect himself to fall, a state in some respects like the attitude of admiration, still might not be admiration. The state into which the man falls might be like admiration in, for example, the behavior that accompanies it. It would not be admiration unless the pre¬ dictor was ready to ask himself whether he thought that the action or person admired was in some respects admirable; having “fallen into” the state that resembles admiration, he would not be said to admire unless he thought that his state was to some degree appropriate to its object. Even if he admires only reluctantly or unwillingly, he still must think of the object as in some way admirable. If this thought that the object is in some way admirable is predicted by him, the prediction will be justified, at least in part, by reference to the predicted features of the object that will explain his thinking the object admirable. Will this thought be uncontrollable? I labor these points because, through the empiricist tradition, we have received an idea of knowledge, and therefore of self-knowledge, as essen¬ tially and in all cases, knowledge of something that has an independent existence, independent, that is, of the knowing. The tradition therefore has difficulty in admitting that the very same process of thought may be both a coming to be sure or to know, that something is to be the case and a process of making it the case. The reasoning that makes me sure that some¬ thing is true of me is sometimes also the reasoning that makes it true of me—e.g., I have to admit that I want X, or that I am trying to do X. The empiricist tradition freely admits that a man has genuine knowledge of his own given sensations and impressions, which are kinds of transparent objects that we encounter in our experience. Sensations are taken within the tradi¬ tion to differ from external physical objects in that we see right through them, and that we do not need to investigate them from different points of view in order to know what they are; they are presented all at once and in their entirety, and there is no distance between the observer and the object, which needs to be judged in determining what the object really is. There is no difference between their surface and their substance. In recent years it has been widely admitted that the more complex

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1. SCIENCE AND INQUIRY

states of mind, passions, and emotions do often need to be investigated by the subject, and the elements and aspects seen as a pattern, before a man can be sure what they are, even though they are his own states of mind. It is sometimes also admitted that the subject’s knowledge of their proper characterization may be essential to the states being characterized in a particular way, and therefore that self-knowledge makes the states of mind different, and more complex, from what they would otherwise be. It is recognized that there are states of mind—for example, the state of being indignant about something or of being embarrassed by it—which are of their nature fully intentional states, just because this kind of selfknowledge is constitutive of them. Someone—for example, a very young child—who does not possess any of the concepts which would enable him to discriminate his state as one of indignation, as opposed, for example, to anger, could not in fact be indignant, although he could be angry. This is the feature of self-knowledge on which I am insisting because I think that, following this path, one can come to see what is peculiar to inten¬ tional knowledge, merely as a kind of knowledge. One difference between being made to feel uncomfortable by something that someone has just said and being embarrassed by his remark is that in the latter case the subject believes or knows that the remark is part of the explanation of his feeling; his knowledge of the explanation of his feeling distinctly modifies the state of mind. A remark may have the effect, or consequence, that he feels uncomfortable, without his recognizing that his feeling is caused by, or is the consequence of, just that remark. If he does come to believe, or to know, that his feeling is the effect of that remark, then a further question arises for him—“Is there anything about the remark that is embarrassing? Was it an embarrassing remark?” That is, was there ground or good reason for embarrassment? Certainly he may recognize that he is embarrassed by it, while admitting that this is to some degree an inappropriate, or even a ridiculous, feeling for him to have about it. But he must believe that his feeling has some minimum appropriateness, if it is to count as a case of embarrassment. Of a child, it may be true that he is feeling uncomfortable and ill at ease, and that this is the effect of a cause, namely, a remark made which touched off certain associations in his mind. But it might be false, and in some cases absurd, to say that he was embarrassed at or by the remark, and, even more absurd, to say that he found the remark em¬ barrassing. The reflexive self-knowledge is essential in converting a case of being made uncomfortable by it, where this is an instance of correlation between stimulus and effect, into a case of being embarrassed by it or ashamed of it or guilty about it, where these are typical intentional relations, constituted and distinguished from each other by the accompanying knowledge of, or belief about, the partial explanation of the feeling. The knowledge is arrived at by considering the properties of the object which constitute reasons for being embarrassed, ashamed, or guilty. I am em-

STUART HAMPSHIRE

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barrassed, ashamed, or guilty because the object seems to me to have certain properties, and I would mention my noticing of these properties, or my belief that the object has these properties, in giving a partial ex¬ planation of why I am embarrassed, ashamed, or guilty. Each of these states has a standard target, or paradigm case. There is something that it is normal to be ashamed of, embarrassed by, guilty about. This is part of the sense, I think, of Aristotle’s doctrine of the mean in respect of the passions: that there is, built into the concept of any one of the passions, a norm of appropriateness in the object of the passion. If I have not been sure whether I am embarrassed by something or not, then my becoming sure is often the outcome of my being sure that there is something normally embarrassing about it, that there is something in the object to explain the embarrassment as not altogether inappropriate. I can usually say why I am embarrassed by it, even if I admit that the reasons are insufficient; giving my reasons for the attitude or feeling is in such cases more like giv¬ ing my reasons for a present or future action than it is like finding a causal explanation of a state of mind which I have independently identified. If the state of mind is the outcome of reflection or deliberation, I have not identified it as existing independently of the reasons that I now take to be the partial explanation of it. In the case imagined, when I ask myself the question “Is this uneasy feeling that I have embarrassment about so-andso?” I look for an explanation of the feeling, and, as a result of this search for an explanation, my state of mind may cease to be what it was before. Originally it was only true that I felt uneasy; and then, with my recog¬ nition of the source of this feeling as a normal one, it became true that I was embarrassed by the object, e.g., a remark made. The situation is very like that in which a man’s attention is drawn to something that he is doing without his being distinctly aware that he is doing it. His action acquires a new character in virtue of his knowledge that he is doing it; new descriptions are applicable to his performances in virtue of the fact that they are no longer unintentional in these respects. Previously there was a cause: now there is room for a reason, and for an explanation of his doing it, known to him, which, in virtue of this knowl¬ edge, changes the character of the action. The difference between “He is breaking the box,” where this is the causal judgment that he is the cause of its breaking, and “He is breaking the box” in a stronger sense, including the intention to break it, is very like the difference between “He is made uncomfortable by it” and “He is embarrassed by it.” In virtue of his knowing what he is doing, the subject will normally be in a position to explain his action by some reason that he has for doing it, or at least authoritatively to reject other explanations that are suggested. He can give some explanation of why he is doing it, however rough and vague and incomplete. To summarize: “I do not know what I shall now do,” “I do not know

44

1. SCIENCE AND INQUIRY

what I want,” “I do not know what I feel about it,” “I do not know what my attitude to it is,” typically confess uncertainty, not about what is already true, but about what is to be true of me, starting from now. This requires a process of reflection, akin to deliberation, which is aimed at the kind of knowledge or certainty which has as its outcome that I do now intend to do so-and-so, want so-and-so, feel so-and-so, have such-and-such an attitude, or sentiment. If I start from an uncertainty about myself of this kind— about what is to be true of me in these respects—I will not infer from a statement of initial conditions together with some general psychological proposition what must be true of me, starting from now, as I may infer what must have been true of me up to now. If deliberation ends with the conclusion that I must, starting now, want so-and-so, feel so-and-so, have such-and-such an attitude, the “must” will allude to some standard of rationality or morality: so to a kind of correctness, which is not truth. A present tense question put to me as a request for information—“Do you want?” “What is your attitude to this?”—places me in an ambiguous position, a position of facing both ways; I am confronted with the need to report the facts, as they already are, and with the need to make a decision; for I may dissociate myself, with a second-level desire or attitude, from the desire or attitude which I find that I already have. The request for information forces me into this position of active self-consciousness; and the asking of the question may change the facts just by engendering this selfconsciousness. In response to the question, I may for the first time recognize that I in fact have a desire or attitude that I now wish that I did not have. The movement from knowledge of the facts about oneself, about one’s desires and attitudes up to this moment, to a knowledge of that which is to be true of me is a continuous movement of consciousness; it is so pervasive that it is easily overlooked. Present tense avowals of desires and attitudes have been found baffling in their logic, partly because they mark this movement from the one kind of knowledge to the other. “I want X” is open to a two-faced challenge, when the question “Are you sure that you want X?” is asked; the first face of the challenge calls on the autobiographer not to disregard any evidence, perhaps of behavior, that suggests that the facts are not yet as he reports; the second challenge calls on him not to disregard any reasons there may be for not desiring X. The contrast between evidence and reasons here is a contrast between steps that lead to, and are taken to justify, a conclusion which is a belief about already wanting X, and steps that lead to, and are taken to justify, wanting X; justification of both kinds might be needed because the belief that X is already wanted might be a mistake, and wanting X might be a mistake. The observer, considering the desires of the subject, would hold these possibilities of mistake apart; he would know from which point of view he was considering the desires. But the subject would normally not separate the factual from the deliberative question. This is part of the asymmetry,

STUART HAMPSHIRE

45

stressed in the controversy about knowledge of other minds, between first-person present tense statements and other statements about states of mind, or at least about desires and attitudes. It is plain that this distinction between two kinds of uncertainty and of knowledge touches the issue of freedom and determinism—but at what point and with what consequences? 1 will state the minimum consequence first. Whatever facts a man may learn about his present desires and at¬ titudes, he always has the higher order question of his desires and attitudes in the face of these facts—does he want them to be otherwise? He may sometimes find that he is powerless to change his passionate attitude toward something, although he wishes to, and that he cannot control or suppress a desire that he already has. He is then left in a state of conflict. He might know the explanation of his original desire and understand its causes; he might have been in a position to predict, accurately and with confidence, that he would have just this desire; and he might have known that he would be powerless to prevent himself feeling this desire, although he had wanted not to feel it. Experimental knowledge of the explanation of his states, the kind of knowledge that is stressed by a determinist, is knowledge of the conditions under which he would be able to prevent something being true of him, if he wanted to prevent it. The “if he wanted to” states a con¬ dition which always arises no matter what scientific explanation of desires is available. If a man had known that under stated conditions he would want X and also that he would want not to want X, he has before him the further question, which arises in virtue of this knowledge, of whether he wishes this conflict not to occur. Even if he foresees accurately the sequence of his desires and moods, and also his conflicts of desire, he would still adopt an attitude and have wishes in respect of this sequence and of the various elements in it. My argument is that any knowledge which a man acquires from experiment and observation about his own present and future states presents him with another potential uncertainty and with the need of knowledge of another kind, and that this is a feature of knowl¬ edge itself. This feature of knowledge could as well be illustrated by reference to perceptual knowledge as by knowledge of mental states and attitudes. Suppose that a man comes to know, for the first time, about some physiological mechanism of perception; using this knowledge, he is able to predict what his perceptions will be when a stimulus of a very specific type is provided; the stimulus is a sufficient condition of a certain physical state, which is in turn a sufficient condition of his perceiving something— for example, a certain pattern of colors on the ceiling; he may still be uncertain, when the perception occurs, whether the perception is veridical or not, whether his inclination to believe that there is a pattern of colors on the ceiling is to be endorsed or not. He must then find reasons to accept or reject the belief, and must actively inquire whether the explanation of his perception is of the kind that allows it to be a veridical one. As one

46

1. SCIENCE AND INQUIRY

learns about the causes and conditions upon which one’s perceptions de¬ pend, one is in a position to apply this knowledge of causes in distinguish¬ ing veridical from deceiving perceptions; and applying the knowledge in¬ cludes actively performing tests—e.g., by changing the perspective or angle of vision or by performing some simple interference experiments. I can legitimately regard myself, with my sensory equipment, as an instru¬ ment that records the presence of objects in the environment; but I am an instrument which deliberately employs itself to find the answers to ques¬ tions which I have raised. When I find myself inclined to believe that there is a certain pattern of colors on the ceiling, and not as the conclusion of an inference, I may still suspend belief until I have performed tests and found good reasons for believing. I am normally free, and not powerless, to question any of my beliefs, whether they arise directly from perception or not. In the setting of questioning an inclination to believe, or of resolving uncertainty, that which causes me to believe is called a reason for believing; in a setting in which a perceptual belief has formed without any active inquiry, or even the shadow of an uncertainty, there must still be an ascertainable cause of the belief, but it may not be called a reason. If I am altogether ignorant of the psychology of perception, I will not know which perceptual clues make me form certain beliefs, e.g., about the distance be¬ tween visible objects. I will not know how I know, and I will say that I know directly, in cases where there has been no uncertainty resolved by inquiry and where my belief is true. When I am uncertain and actively investigate, my final belief has an explanation which includes a reason, the evidence from which I have inferred. Sometimes my reason may only be the belief that my first inclinations to believe (my first impressions of distance) almost always are confirmed, that is, that I am a reliable instrument in this matter; this is a reason for endorsing my first impression of distance on a particular occasion. But the further question—“Am I sure that there is nothing abnormal in the circumstances in which this perceptual belief was formed?”—is never an absurd question. I do not wish to deny, but rather to stress, the many differences between knowledge of the external world and the knowledge that a man may have of his own states of mind, desires, and attitudes. But the corrigibility in principle of any empirical knowledge claim, apparent in perceptual claims, has often neglected consequence in claims to self-knowledge. Suppose that a man knows what he will want to do when certain sufficient conditions obtain, which he knows will occur very soon. When the anticipated desire occurs under the foreseen conditions, he can still raise the question “But am I sure that I do want to do this? Are there good reasons why I should want this?” Of course he knows that he does now want it, that this primary desire exists, and, in the case supposed, he knows why it exists. He is so far like the man who sees the pattern of colors on the ceiling, and who knows that this is a possible correct

STUART HAMPSHIRE

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description of what he sees, but is still not sure whether there is in fact a pattern of colors on the ceiling. The mere raising of the reflective question has already changed his state of mind; and this distinguishes self-knowledge of this kind from knowledge of the external world. For there is now a sense in which he knows what he wants to do and a sense in which he does not know what he wants to do. Recognizing this primary desire and knowing the conditions (perhaps a physical state) that have produced it, he may still be hesitant and uncertain, and any conclusion of the uncertainty will have a reason as at least part of its explanation. The uncertainty in this case of desire amounts to a kind of conflict, and will be plainly a conflict of desire if the outcome is that he does not want, on reflection, to do that which he also feels a desire to do. Similarly with attitudes of mind: I may find myself predictably feeling admiration for someone and being inclined to act correspondingly, while on reflection I am uncertain about my attitude to him, because I am uncertain whether there is anything admirable about him. I have therefore a divided and uncertain attitude, which may be continuously modified as I continue to reflect upon what my attitude is. The transition, in self-knowledge, from knowledge of psychological fact, and of the causes that explain my desires and attitudes, to a reflection on these facts, which introduces causes that are reasons, depends, firstly, on a feature of knowledge itself: that a claim to empirical knowledge is always challengeable by the question “Am I sure that this is so?” This challenge to reflect upon desires and states of mind, when it is contemporary with these states, leads to the consideration of reasons. Insofar as a desire that is to be in part explained by reasons differs from a previous desire that was independent of these reasons, the reflection by itself brings about a change. This conclusion obviously leaves many of the traditional issues as¬ sociated with a thesis of determinism untouched. But it is incompatible with a particular picture that is often associated with a thesis of determin¬ ism: namely, the picture of advancing psychological knowledge gradually displacing the kind of uncertainty that is expressed by the questions “Do I want to do this?” and “What is my attitude to this?” when these questions call for a decision. Any additions to our systematic knowledge in psychology, and thus to our power of calculating exactly what we will or would feel under specifiable conditions, will always leave a place open for this kind of uncertainty.

ON OBJECTIVE INTENSIONS AND THE LAW OF INVERSE VARIATION1 R. M. Martin

I One of the most fascinating but neglected branches of modern logic is the semantical theory of intensions. By the intension of a term is meant vari¬ ously the meaning or connotation or Sinn of the term as over and against its designation or denotation or Bedeutung. Intensions in one form or an¬ other have played a small, but persistent, role in the history of logic. In modern mathematical logic they have played scarcely any role at all. In¬ tensions in fact are the kind of entity in which mathematicians have shown little interest. But surely the theory of intensions must play a fundamental role in philosophical, as contrasted with purely mathematical, logic. A fundamental “principle” concerning intensions is the so-called law of inverse variation of extension and intension. There was much controversy over this law in the latter half of the nineteenth century, and its exact logical status was in doubt. The controversy seems never to have been set¬ tled and the “law,” if such it be, seems never to have been stated ade¬ quately. Although occasionally mentioned by more recent writers, no one seems to have probed the logic, if there be such, underlying it. In par¬ ticular, the law seems not to have been examined by such proponents of intensions as Carnap, Church, Fitch, or Frege. And yet it has often been regarded as one of the fundamental laws concerning intensions. A clear and unobjectionable statement of it is necessary if the theory of intensions is to be given a secure modern footing. Let us examine critically a little of the fascinating history of this law, preparatory to attempting to give it a more exact form. We shall not present a complete account but will merely select a few formulations of outstanding interest. And if we start in the center, this will at least (in Bradley’s phrase) get us into the heart of the matter. Bradley in fact called the law “that preposterous article of orthodox logic [that] turned the course of our reasoning into senseless miracle.”2 But Bradley never formulated the law very clearly. “Extension and in¬ tension,” he says almost mockingly, “. . . are related and must be related in a certain way. The less you happen to have of the one, the more there48

R. M. MARTIN

49

fore you must have of the other [italics added]. This statement has often passed itself off as both true and important. I confess that to me it has always seemed false or frivolous.”3 Frivolous, indeed, lacking clear and defensible definitions of ‘intension,’ ‘less,’ ‘happen to have,’ ‘must have,’ and no doubt other terms. Such definitions are not, it is to be feared, to be found anywhere in the annals of idealist logic. For Bradley, as for others in the Hegelian tradition, the notion of intension is intimately bound up with the doctrine of the “concrete uni¬ versal.” There, however, ‘intension’ takes on a quite different meaning in which extension and intension seem to coalesce. Hence naturally, on such a meaning, a law of inverse variation does not hold.4

II As good a preliminary statement of the law as any—and better than most—is to be found in (the fourth edition, 1906, of) Neville Keynes’ Formal Logic.5 (L)

In a series of common terms standing to one another in a relation of subordination the extension and intension vary inversely.

Keynes does not himself accept the law in this form but uses it merely as a basis for discussion. We note first that this preliminary formulation of the law concerns terms, not what the terms designate or denote. Thus straightaway we see that it is to be a law of syntax or semantics, not of some object-language. But the terms are common, i.e., are denotative and denote objects in common, so that the law is clearly a semantical law, not a syntactical one and not one of logic in the narrow sense (comprising just the theories of truth-functions, quantifiers, and identity). Further, the terms must be ar¬ ranged in a series, although strictly this is an elaboration which is not needed. The relation of subordination between terms is presumably a semantical relation, analogous to class-inclusion. A term a is subordinate to a term b if and only if every object denoted by a is also denoted by b.6 The extension of a term a is presumably the class (or virtual class) of all objects denoted by a. So far so good, barring niceties. The extension of a term, Keynes says (p. 22), “consists of objects of which the name can be predicated,” whereas “its intension consists of properties which can be predicated of it,” i.e., of the extension. Intensions are of three kinds, conventional, subjective, and objective. The conventional intension, or connotation, of a class-name consists of “those qualities which are essential to the class in the sense that the name implies them in its definition. Were any of this set of qualities absent the name would not be applicable. . . .” (How about primitive class-names, not defined?) Perhaps

50

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we may make Keynes’ definition a little more precise by regarding the connotation of a defined class-name ‘P’ as the class of all “qualities” Q such that necessarily for all x, if x has P then x has Q. The subjective intension of a term depends upon the individual user of the term and is “less important from the logical standpoint. 7 Therefore we need say nothing more about it here. Finally, there “is the sum-total of qualities actually possessed in com¬ mon by all members of the class. These will include all the qualities included under the two preceding heads, and usually many others in addition.” This sum-total comprises the objective intension or comprehension of the classname. Perhaps a “sum-total” is a logical sum of classes. Or is it rather a set? Keynes is not too explicit on this point, although, as we have seen, the word ‘set’ creeps surreptitiously into his explanation of the meaning of ‘connotation.’ But a logical sum is not a set, and it is most important to know which is intended if we are to have here a clear meaning for ‘connotation’ and ‘objective intension.’ Keynes speaks of “classes” as, presumably, the objects designated by class-names. On the other hand, intensions are sum-totals or sets of “qualities.” His ontology is thus at best rather mixed. Keynes distinguishes three laws of variation which “must together be substituted for the law of inverse variation between extension and intension in its usual form (L) if full precision of statement is desired.” The first of these laws is (p. 37): (K)

If the connotation of a term is arbitrarily enlarged or restricted, the denotation in an assigned universe of discourse will either remain un¬ altered or will change in the opposite direction.

The statement of (K) is surely clearer than that of (L). In particular, note that ‘vary inversely’ has been dropped. What is meant anyhow by ‘inverse variation’ in connection with the extension and intension of terms? We have no quantitative measure of terms, so that inverse variation cannot mean here what it means in physics, for example, in the statement of Boyle’s law. Let ‘M’ be a class-term. Let con(‘M’) be the conventional intension or connotation of ‘M’. What does it mean now to say that the connotation of ‘M’ is “arbitrarily enlarged or restricted”? Presumably by this—in accord with (L)—we are invited to consider the connotation of another class P in which M is properly included or which properly includes M. Let con(‘P’) be the connotation of ‘P’. In saying that the connotation of ‘M’ is enlarged, we are intended perhaps to consider that con(‘M’) is properly included in or subordinate to con(‘P’). Of course we can say this only where con(‘M’) and con(‘P’) are themselves sets (or classes of some sort, perhaps classes of classes or of “qualities”), so that by ‘inclusion’ we mean set-inclusion of the type appropriate to sets of “qualities.” If, on the

R. M. MARTIN

51

other hand, conventional intensions (or connotations) are to be construed as logical sums of classes, inclusion must be of the type appropriate to classes of individuals. We must distinguish then two statements correspond¬ ing to (K), depending upon whether con(‘P’) and con(‘M’) are taken as logical sums or as classes. Both of these statements, however, will have the form (K')

If con(‘P’) is properly included in con(‘M’), then M is properly included in or identical with P.

(K'), on the set-meaning, although an improvement on (K), is still far from satisfactory. In particular there is implicit reference to “attributes” or “characteristics.” It is doubtful that we need recognize a separate realm of attributes as distinguished from classes, for they are a dubious entity at best. A clear condition under which two attributes may be said to be identi¬ cal is lacking, as Quine and others have repeatedly pointed out.8 Also the notion of connotation is not too clear. What are “those qualities which are essential to the class in the sense that the name implies them in its definition”? What is meant by ‘essential’ here? By ‘implies’? Without acceptable answers to these questions, we cannot claim to have put forward a clear account of connotation at all. It has been said that “clarity is not enough,” but still it is a good deal and surely a prerequisite for any adequate logical theory of connotation. It should be noted, incidentally, that Keynes is usually careful to dis¬ tinguish between the use and mention of expressions, a care not shown by all writers on the subject. But he vacillates between speaking of class-names and of “properties” or “qualities” or “attributes” or “characteristics,” as we have noted. Many of his statements concerning class-names are ac¬ ceptable on the basis of modern syntax and semantics. When he speaks of “attributes” or “characteristics,” in the wake no doubt of Aristotelian essentialism, he is less clear and not always defensible. Keynes goes on to distinguish two other laws of variation, which, how¬ ever, concern more special notions and are merely supplementary to (K).

Ill

Two further, but very different, formulations of the “law” are to be found in Cohen and Nagel’s influential An Introduction to Logic and Scientific Method.9 First: (A)

When a series of terms is arranged in order of subordination, the extension and intension vary inversely.

And secondly: (B)

If a series of terms is arranged in order of increasing intension, the denotation of the terms will either remain the same or diminish.

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1. SCIENCE AND INQUIRY

Cohen and Nagel criticize (A) but seem to condone (B) as a satisfactory law. Although these two formulations appear similar to those of Neville Keynes, actually they are very different. In the first place, by a teia. Cohen and Nagel do not understand an expression, but rather what th~ expression stands for. Propositions, according to them and following Aristotle, either assert or deny something of something else. That about which the assertion is made is called the subject, and that which is asserted about the subject is called the predicate. The subject and the predicate [i.e., presumably the things, not the expressions] are called the terms of the proposition. ... A term may be viewed in two ways either as a class of objects (which may have only one member), or as a set of attributes or characteristics which determine the ob¬ jects. The first phase or aspect is called the denotation or extension of the term, while the second is called the connotation or intension. Thus the extension of the term “philosopher” is “Socrates,” “Plato,” “Thales,” and the like; its in¬ tension is “lover of wisdom,” “intelligent,” and so on. It is interesting that Cohen and Nagel do regard the intension as a set of attributes or characteristics. They are perhaps the first to recognize explicitly that intensions are sets in some fashion, and this recognition constitutes an advance in the subject. By the conventional intension or connotation of a term, Cohen and Nagel mean, more precisely, “the set of attributes which are essential to it.” Thus the intension of the attribute P is presumably the set of all attributes A essential to P. And by “essential,” they say, “we mean the necessary and sufficient condition [in the singular, notice] for regarding any object as an element of the term.” This definition is not too clear. Perhaps, according to it, we may say that attribute Q is essential to attribute P if and only if necessarily every object which has P has Q and conversely. The intension of P then becomes the set of all attributes A such that necessarily every object which has P has A and conversely. But this is not the set intended, for presumably it would be a set with only one member, namely, P itself. Perhaps the intension of P here is to be only (i) the set of all attributes A such that necessarily every object which has P has A. Or perhaps (ii) the finite set of attributes (Ch, . . . , Q(l} such that necessarily every x which has P has Qi, has Q2, . . . , has Q„. But all this is a little murky. For the subordination of intensions Cohen and Nagel have a clear meaning, for intensions are certain kinds of sets and hence subordination is merely class- or set-inclusion. Let for the moment ‘int(P)’ designate the in¬ tension of P in whatever sense they intend. If we disregard for the moment the unnecessary complication of bringing in the notion of a series, their formulation (A) seems to be: (A')

If P is subordinate to Q, then int(Q) is subordinate to int(P).

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53

By ‘subordinate’ here is meant for the moment properly subordinate or is properly included in. (B), on the other hand, would read: (By)

If int(Q) is subordinate to int(P), then P is subordinate to or identical with Q.

We do not claim that (A') and (B') are precisely what Cohen and Nagel intend by (A) and (B), but only that they are reasonably close approxima¬ tions to what they perhaps intend. It should be noted that Cohen and Nagel’s formulations are given as principles within an object-language. For Keynes, however, as we have observed, these principles are meta-linguistic, more precisely, semantical, principles. It is not altogether clear what is thought to be gained by their statement within an object-language. Usually by a term one under¬ stands an expression of such and such a kind. Keynes is explicit on this point. According to him, “. . . it seems better to start from the names. . . . Neglect to consider names in . . . connexion [with extension and intension] has been responsible for much confusion.” Also Carnap and Frege have emphasized the importance of regarding intensions as intensions of expres¬ sions. In fact, it is now almost universally recognized that the theory of extension, denotation, connotation, intension, and the like, belongs to semantics, and hence is to be formulated within a semantical meta¬ language. There is also the difficulty concerning ‘necessary’ in Cohen and Nagel’s formulations. Is this to be construed in the sense of some modal operator? If so, we must then face squarely the need for mixing modal operators and quantifiers. We do not wish to contend that this cannot be done satisfactorily, but only that there are grave difficulties here which must not be glossed over. Quine and others have pointed these out repeatedly. Surely we can¬ not expect a satisfactory formulation in the manner of Cohen and Nagel of the law or laws of inverse variation until these difficulties are overcome. Also closely connected with these is the dubious ontology of attributes, etc., which we have already met in the formulations of Keynes.10 The reader may object that we are being excessively laborious as to logi¬ cal detail. But, to borrow a phrase which Cohen and Nagel themselves use in another context, “it is necessary to . . . [be] so if we wish to avoid elementary confusions.” In trying to avoid such we must occasionally be allowed to refer to more advanced matters in modern logic, syntax, and semantics, as well as to pay strict attention to what might appear as minutiae. To refuse to allow this is often to refuse to let a subject grow or develop beyond its incipient (or textbook) stages. As logic itself, including now the theory of intensions, has developed enormously within recent years, it must carry its history along with it, subjecting it to fresh critical scrutiny in the light of present knowledge.

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IV Let us reflect now upon what is needed if we are to gain a more adequate formulation of the law or laws of inverse variation. In the first place, we must have a well-articulated semantics including a syntax. Nothing less than this surely is now acceptable. Because the law or laws are so intimately connected with denotation or designation, the most natural logical underpinning is no doubt to be found in denotational or designational semantics. Next we must have a clear and acceptable notion of what intensions are. We must have a clear ontic description of them. One searches far and wide in the literature for this. In the author’s Intension and Decision, a theory of many kinds of intension was suggested. Certain defects, however, marred the theory there, but these we shall try to correct. Intensions should emerge in a natural way, so we should contend, as certain kinds of entities already available in the underlying denotational semantics. They are not, in other words, to be regarded as sui generis. The notion of analytic truth should play a fundamental role, supplanting the somewhat vague and unsatisfactory traditional use of ‘necessary’ or ‘essential.’ Also intensions must be dissected into parts (or members) in some fashion, and are not to be regarded as indivisible wholes. As regards this, there seem to be two historical lines of research. On the one hand, there is the German tradition, stemming no doubt from Kant, through Frege to Carnap. And, on the other, there is the English tradition to which Stuart Mill and Neville Keynes have contributed notably. The key difference seems to be that in the German tradition intensions are regarded as sui generis and as indivisible wholes, whereas in the other they are dissected into components. On this crucial matter we side with the English. We shall try to be as clear as possible as to precisely what these components are. Heretofore this has been left rather vague, as we have seen. Let us turn now to recent semantical theory, in order to pave the way for a more sophisticated and (hopefully) adequate formulation of the law or laws of inverse variation. To designate is to be a proper name of, whether of an individual, a class, or a relation. To denote is to be a common name of an individual and of an individual only. Semantical rules concerning designation and denotation reflect both actual usage and convention, but are no doubt to be laid down in part by meta-linguistic fiat. There is a third notion, also of great semantical interest, that of L-designation. It is perhaps a more “natural” notion than either designation or denotation, simpler, and more closely reflecting in some sense actual usage. L-designation has occasionally

R. M. MARTIN

55

been mentioned by Carnap, although very rarely used even by him.11 Also it is definable, with certain desirable restrictions, in terms of denotation. Hence an intensional semantics based on it shares the advantages enjoyed by the semantical meta-languages based upon denotation. We shall also need to introduce a notion of virtual-class L-designation. Virtual classes, it will be recalled, are classes manquees, very like real classes, but lacking the crucial feature of being values for variables in the formalism at hand.12 The theory that emerges, with virtual classes used in place of real classes and L-designation in place of designation, represents an important improvement over the theory put forward in Intension and Decision. Classes are treacherous entities. They lead us astray in mathematics, and hence there is little reason to regard them as suitable tools in philosophy. In their favor is the circumstance that they are extensional entities and hence relatively clear. If we reject them, it would not do to reinstate intensional entities in their place, entities such as properties or class-con¬ cepts, relation-concepts, individual-concepts, or propositions.13 If we are to reject classes, we must do so toto caelo and reject intensional entities as well. Rejection here of course consists only of refusal to admit as values for variables. If we can succeed in defining a suitable notation for such entities, or somehow succeed in gaining the effect of such, upon an ac¬ ceptable foundation, this is all to the good. In fact, this is just what we hope to do for the various kinds of intensions. The philosophic importance attaching to the rejection of intensions as values for a special kind of variable (intensions sui generis, so to call them) is multifold. First, there is the distrust of pseudoentities. At best intensional entities are suspect and should not be multiplied beyond neces¬ sity. Then there is the complexity of the laws governing them. These are notoriously sticky and more complex than ordinary laws governing ex¬ tensions. Also there is the further complexity of the whole meta-language in which intensions sui generis can be accommodated. Extensional meta¬ languages are always considerably simpler, both in structure and in what is assumed. There is also the relevant thesis of the unity of science. Why should only extensional entities be required in certain parts of science and then all of a sudden abandoned when we turn to others? It is as though we were to condone here a fundamental severance similar to the alleged one between the Natur- and Geisteswissenschaften. Also there is the very considerable historical confusion that surrounds the subject of intensions. When properly viewed and analyzed, so we contend, the assumption of intensions sui generis is seen not only not to have been needed but to have been positively harmful. The role of intensions, to a large extent at least, can be played instead by certain kinds of virtual constructs, as we hope to show.

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V As is frequently done, let us presuppose some suitable first-order lan¬ guage-system L as object-language, rich or expressive enough to be of some philosophic interest. Our task is to construct a semantics for L in as simple a way as possible, in which a theory of objective intensions for L may be accommodated. There are no doubt many ways of formulating such a semantics for L. Let us choose one here, as elsewhere, based on denotation. The primitive is ‘Den’, it will be recalled, significant in contexts of the form ‘a Den x’, read ‘the expression a denotes the individual x\ We recall also that, in the underlying syntax, every expression of L is given a structuraldescriptive name, 7p' for ‘(’, ‘tilde’ for and so on. Abundant use will be made of virtual classes, which, as we have noted, differ from real ones primarily in not being values for variables. Although the terminology and notation of Intension and Decision are used here for the most part, famili¬ arity with them, or it, is not presupposed. Let ‘(—x—)’ be some sentential function of the object-language L containing ‘x' as its only free variable. The virtual class of all individuals x such that (—x—) may then be expressed by ‘x3(—x—)’, the ‘3’ being read ‘such that.’ Any expression of the form ‘x3(—x—)’ will be, then, a one-place abstract. Let ‘PredConOne o’ express that a is a primitive oneplace predicate constant of L or a one-place abstract containing no free variables. We shall concern ourselves primarily with one-place predicate constants, but we can pass on to two-place, three-place, etc., predicate constants easily as desired. Governing the primitive ‘Den’ for multiple denotation, we have the fol¬ lowing two rules. First, that for all x, m Den x if and only if (—x—), where (i) ‘m’ is taken as the structural description of the abstract ‘*3(—x—)’, ‘(—x—)’ being any sentential function of L containing ‘jc’ as its only free variable, or (ii) ‘m’ is taken as the structural description of a primitive predicate constant and ‘(—x—)’ consists of that predicate con¬ stant concatenated with ‘x’. And, second, that for all a and for all x, if a Den x then PredConOne a. These two rules fix the properties of Den. According to the first, just certain PredConOne’s denote certain objects, and according to the second, no expressions other than PredConOne’s can denote.14 In terms of ‘Den’ we may define now virtual-class designation. Thus an expression a may be said to designate a virtual class X3(—x—) provided a is a PredConOne and for all x, a Den * if and only if (—x—). Thus, in symbols, (Dl)

‘a Des X3(—x—)’ abbreviates ‘(PredConOne a* (x) (a Den x= (—x—)))’, if (etc.)

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Let L contain ‘M’ (for men) and ‘R’ (for rational beings) as PredConOne’s. According to this definition-schema we may say that ‘M’ designates the virtual class of individuals who are men, ‘r3(Mr • Rx)’ designates the virtual class of individuals who are both men and rational, and so on. Also a here designates the virtual class X3(—x—), no matter what notation we use to refer to that class. Thus if X3(—x—) = *3(- .x. .), then also a Des X3(.. x . .). After all, a rose by any other name. . . . We may go on to virtual-class L-designation as follows. An expression a is said to L-designate a virtual class xs(—x—) if and only if a designates X3(—x—) and a = m, where in place of ‘m’ we put in the structuraldescriptive name of the abstract ‘x3(—x—)’. As a matter of fact, however, the clause concerning designation here may be dropped, for it follows from the other clause by one of the Rules of Denotation. Hence we are left with the especially simple definition, (D2)

‘a LDes X3(—x—)’ for ‘a = m’, where (etc.)

This definition, or rather definition-schema, defines ‘LDes’ only in con¬ texts in which ‘a’ is the only free syntactical or expressional variable. The same is true, note, of the definition-schema for ‘Des’. We have no variables for virtual classes, only abstracts, and hence we do not define ‘Des’ and ‘LDes’ in contexts wherein the second argument is a variable. Nor would it be desirable to do such where we are dealing with only virtual classes. When we say that a LDes X3(—x—), the notation we use to refer to X3(—x—) is all-important. Here a rose by any other name differs remark¬ ably, because the names themselves differ. L-designation is a matter pri¬ marily of the names, as it were, and only secondarily of what the names stand for. Thus ‘M’ L-designates X3Mx, but not X3(Mx • Rx), although the two virtual classes are the same. We see then that L-designation, strictly speaking, is a notion of syntax. Hence the virtual classes referred to in the definiendum of (D2) may be called syntactic or nominal virtual classes. But to say this is misleading. It suggests that we can subdivide the virtual classes into those which are nominal and those which are not, and this is not correct. To be nominal is rather a property of the occurrence of a virtualclass expression. Although we gain the effect of speaking of a virtual class in the definiendum of (D2), this is merely an illusion, for strictly we are speaking only of a certain expression which designates, more properly, Ldesignates, it. Although a virtual-class expression occurs in the definiendum of (D2), no such expression occurs in the definiens. The definiens merely stipulates that a be of such and such a shape. For a to L-designate is then merely for it to be of such and such a shape. But, as we have noted, any expression of such and such a shape does in fact designate such and such a virtual class, in view of the Rules of Denotation. Hence one and the same virtual

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class is, in a roundabout way, implicitly involved in the definiens as well as in the definiendum. We should note that the relation of L-designation here differs remarkably from that of Carnap. That of Carnap is of course a semantical relation, not one of syntax—and further, within the theory of meaning or intension sui generis rather than within the theory of extension or reference. Accord¬ ing to him, and in essentially his own words, an expression a L-designates an entity in L if and only if it can be shown that a designates that entity merely by using the semantical rules of L without any reference to facts.15 For Carnap, the entity L-designated may be a real class, whereas here it can be a virtual one only. On the other hand, suppose we know that a — m, where in place of ‘m’ we put in the structural description of lX3(—x—)’. It then follows, merely by the Rules of Denotation, that a designates X3(—x—), as we have already remarked. This circumstance seems sufficient to justify referring to the relation here as one of L-designa¬ tion. Nominal virtual classes might also be referred to as thus-designated virtual classes. A natural reading of the definiendum of (D2) is ‘the expression a stands for the virtual class X3(—x—) as thus designated’. We may then, if we wish, speak of thus-designated virtual classes, although the terminology is perhaps a bit awkward. Nominal virtual classes might also be called virtual classes in intension. They are not strictly intensions but are merely “taken in intension,” as the traditional phrase has it. We shall see in a moment that they are members of intensions, entering into their inner constitution, as it were, but are not to be confused with intensions themselves. Nominal virtual classes may also be thought of as virtual-class concepts. Also we may speak of occurrences of expressions for nominal virtual classes as being opaque or indirect rather than direct. At any event, there is surely enough kinship here to justify these various terminologies.

VI

Let us turn now to the various kinds of objective intensions, essentially as in Intension and Decision, mutatis mutandis. An expression a is said to be analytically included in an expression b, in symbols ‘a Anlytclnc b\ provided the sentence which consists of ‘(x) (’ fol¬ lowed by (or concatenated with) a followed by ‘xD ’ followed by b followed by ‘x)’, where a and b are one-place predicate constants, is an analytic sentence of L. Note that here we are merely spelling out the structural description of the given sentence. Suppose a is the predicate ‘P’ and b ‘Q’. Then a is analytically included in b if and only if ‘(x) (PxD Qx)’ is an analytic sentence of L. Given a one-place predicate constant a, the members of its objective

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analytic intension are to be the nominal virtual classes L-designated by oneplace predicate constants b such that a is analytically included in b. We cannot introduce the notion in just this fashion, but instead say that a given nominal virtual class X3(—x—) is a member of the objective analytic intension of an expression a, if and only if a is a one-place predicate constant of L and there exists a one-place predicate constant b which designates X3(—x—) and such that a is analytically included in b. In symbols,

re¬

(D3)

‘x3(—x—) e ObjAnlytclnt(a)’ abbreviates ‘(PredConOne a • (Eh) (b LDes X3(—x—) • a Anlytclnc b))’.

And of course the whole definiendum must be regarded as an indivisible unit. In other words, (D3) provides a contextual definition of the entire phrase ‘x3(—x—) is a member of the objective analytic intension of and not of any part of this phrase in isolation. Thus ‘member of is used here only by proxy, but there is enough similarity with class-membership, or rather virtual-class-membership, to justify the use. Likewise ‘the analytic intension of is defined only as embedded in the given kind of context. Further contexts can be introduced as we go on. Let us now turn to the classic example, homo animal rationalis. In accord with this, let ‘M’ be regarded as short for ‘x3(Rx‘Ax)’, where ‘R’ (for rationals) and ‘A’ (for animals) are likewise PredConOne’s. Let us assume that ‘R’ and ‘A’ are primitive. As members of the analytic intension of ‘M’ we have then the nominal virtual classes of rationals, of animals, and of men, together with any other nominal virtual classes in the L-designators for which ‘M’ is analytically included. Let ‘F’ (featherless) and ‘B’ (bi¬ peds) be further primitive PredConOne’s. Although ‘(x)(MxD (Fx• Bx))’, which states roughly that every man is a featherless biped, is true in L, it is not analytically so. Hence the nominal virtual class of featherless bipeds, as well as that of bipeds and that of featherless objects, cannot be members of the analytic intension of ‘M’. This is of course as we wish it and as it should be. Let us say that a is veridically included in b, in symbols ‘a Verlnc b\ pro¬ vided a and b are both PredConOne’s and the sentence consisting of ‘(x) (’ followed by a followed by ‘xD’ followed by b followed by ‘x)’ is true. Veridical inclusion is like analytic inclusion, with ‘Tr’ for truth replacing ‘Anlytc’ in the appropriate definientia. Similarly we go on to theoremic and synthetic (or factual) inclusion, ‘a Thmlnc b’ expresses that a is theoremically included in b, and la Synthclnc b' that a is synthetically so. These relations prepare the way for other kinds of intensions. We may introduce now objective synthetic, veridical, and theoremic in¬ tensions by replacing ‘Anlytclnc’ in the definiens of (D3) by ‘Synthclnc’,

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‘Verlnc’, and ‘Thmlnc’ respectively. It is natural to think that these, to¬ gether with analytic intensions, would constitute the four fundamental types, one corresponding to the semantical notion of being analytic, one to that of being synthetic, one to that of being true, and one to that of being a therorem. We let ‘ObjSynthcInt’, ‘ObjVerlnt’, and ‘ObjThmlnt’ respec¬ tively symbolize these notions. It should be noted that analytic intensions do intimately depend of course upon the analytic truths of L. These in turn are merely the logical truths of L together with the results of abbreviating them by using the definitions. Nothing more elaborate is involved, so that the notion of being analytic here is as well-founded as that of being logically true.

VII

It is to be observed that the theory here does not allow that one and the same nominal virtual class can be a member of both the ObjAnlytcInt(a) and of the ObjSynthcInt (a). The way in which the member is specified or referred to makes all the difference. (Hence our concern with nominal virtual classes.) We may see this as follows. Let (i)

X3(—x—) e ObjSynthcInt (a)

and let a L-designate some nominal virtual class N. Then N is a subclass of X3(—x—), and this synthetically. More precisely, we should say here that there is a b such that a Synthclnc b where b LDes X3(—x—). But also N is a subclass of the logical sum of N and X3(—x—), i.e., of X3(Njc v —x—), and this analytically. More precisely, there is a c such that a Anlytclnc c where c LDes *3(N;c v —x—). Hence we have that (ii)

X3(N;c v —*—) e ObjAnlytcInt(n),

according to (D3). Although xs(—*—) and X3(Nx v —*—) are in fact the same virtual class, N being a subclass of X3(—x—), they are not one and the same nominal virtual class. Hence we are not allowed to inter¬ change salva veritate the abstracts for them in either (i) or (ii). The way in which the member of an objective intension is referred to is crucial. Strictly the members of an objective intension are merely nominal virtual classes, not virtual classes simpliciter, as we have noted. If we regard objective intensions as real classes of real classes, in the manner of Intension and Decision, the situation just described raises diffi¬ culty. That every member of an analytic intension, in the sense of Intension and Decision, is also a member of the corresponding veridical one is im¬ mediate. The converse can be established by an argument similar to that of the preceding paragraph.10 Hence, within that theory there seems to

R. M. MARTIN

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be no way of distinguishing properly the two kinds of intension. In the theory just sketched, however, in terms of Den and nominal virtual classes, there is no such difficulty. Although the form of the definiens of (D3) is essentially the same as in the corresponding definition in Intension and Decision, with ‘LDes’ replac¬ ing ‘Des’, the definienda differ, as well as the whole ambiente. In particular, objective intensions, as now conceived, are merely virtual classes of nominal virtual classes, no longer real classes of real classes. Hence expressions for them, as well as for their members, can occur only in suitably defined contexts. Hence also, strictly speaking, there are no such things as objective intensions at all. We cannot quantify over them nor over their members. But it is not clear that we ever need to do this anyhow, as in the case of virtual classes generally. We can, however, specify an objective inten¬ sion as having such and such specified nominal virtual classes as members. And this is what, after all, we wish. A meaning or intension is nothing if not specified. And if it is properly specified in some sense, nothing more is needed. Thus, although it might seem that something important is sacrificed in the virtual treatment of intensions, it is by no means clear that this is the case. It might be thought that in the theory here extensionality is in some fashion abandoned. But this is not the case, as we have already in effect noted. Suitable extensionality laws hold for L-designation. To say that a LDes X3(—x—) is merely to say that a is of such and such a shape. Just because X3(—*—) might be the same virtual class as X3(. . x . .) by no means gives grounds for thinking that a LDes X3(. . x . .) also. That a has one shape in fact rules out that it can have another. But clearly if a = m and such and such holds of a, then such and such holds of m likewise (where in place of ‘m’ we put in the structural description of a PredConOne). Also if a = b and a LDes X3(—x—), then b LDes X3(—x—) also. Finally, if a LDes X3(—x—) and a LDes X3(. .*..), then not only is X3(—x—) the same virtual class as X3(. . x . but also m = n (where in place of ‘m’ and ‘n’ we put in respectively the structural de¬ scriptions of ‘*3(—x—)’ and ‘X3(. . x . .)’). Thus we see that suitable extensionality laws hold for LDes without restriction. Nominal virtual classes differ from ordinary virtual classes only in a maniere de parler. There is no difference in ontology—in fact there is no ontology here at all, for virtual classes strictly do not exist in the sense of being values for variables. We speak, however, as though there were such things as nominal virtual classes and thus give the theory of inten¬ sions a kind of ontological flavor. Distinctions in intension thus seem to reflect an ontological difference, but actually are reducible merely to differences in the manieres de parler. Strictly there are no such things as virtual classes, nominal virtual classes, or intensions. They are mere forms of nonbeing.

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1. SCIENCE AND INQUIRY

vm We note that the only contexts thus far introduced in which ‘ObjAnlytcInt (a)’ may occur significantly are of the form of the definiendum of (D3), in which a nominal virtual class is said to be a member of the ObjAnlytcInt (a). And similarly for ‘ObjVerlnt(a)’, ‘ObjSynthcInt(a)’, and ‘ObjThmlnt (a)\ But further contexts may be introduced definitionally, in particular, contexts in which we may say that one intension is included in another or that an intension is identical with an intension. Various relations of inclusion as between one-place predicate constants have already been introduced. The following definitions introduce various relations of inclusion as between objective intensions. ‘ObjAnlytcInt (a) C ObjAnlytcInt (ft)’ abbreviates ‘(PredConOne a • PredConOne b • (c) (a Anlytclnc c D b Anlytclnc c))’, ‘ObjAnlytcInt(a) C ObjVerlnt(b)’ abbreviates ‘(PredConOne a • PredConOne b • (c) (a Anlytclnc c D b Verlnc c))’, and so on, for all possible cases. There are sixteen in all. And identity for each case is merely mutual inclusion. Thus ‘ObjAnlytcInt(a) = ObjAnlytcInt(fi)’ abbreviates ‘(ObjAnlytcInt(a) C ObjAnlytcInt(b) • ObjAnlytcInt(b) C ObjAnlytcInt (a))’, and so on. Also we can introduce an existential operator for objective intensions. We let ‘E!ObjAnlytcInt(a)’ abbreviate ‘(Eb)a Anlytclnc b’, and so on. We have immediately existence theorems, to the effect that if a is a PredConOne, then ElObjAnlytcInt(a), E!ObjVerInt(a), and EIObjThmInt(a). For synthetic or factual intensions much depends upon the axioms of the object-language, which in turn depend upon what the facts are, so to speak. Ordinarily we should be able to show that the objective synthetic intension of a PredConOne exists also, but it is not clear that this can always be done. We have some interesting inclusional laws for intensions as follows. Tla.

i- PredConOne a D ObjAnlytcInt (a)

C ObjVerlnt(a).

(The V’ is to be read ‘is a theorem’ and is to absorb the quotation marks around the string of symbols following it.) Tib.

h-PredConOne a D ObjThmlnt(a) C ObjVerlnt(a).

R. M. MARTIN

Tic.

h-PredConOne a D ObjAnlytcInt (a) C ObjThmlnt(a).

Tld.

b-PredConOne a D ObjSynthcInt(a) C ObjVerlnt(a).

63

Let lF\ ‘G’, and ‘H’ now be any PredConOne’s. That analytic and synthetic intensions are properly distinguished is shown by the following law. T2.

hf e ObjSynthcInt(a) D — F e ObjAnlytcInt (a).

And that they are properly combinable, by the following. T3.

\-F e ObjVerlnt(a) = (F e ObjAnlytcInt (a) v F e ObjSynthcInt(a)).

We are now in a position to state several laws of inverse variation in an exact and (hopefully) adequate form. T4a.

b- (PredConOne a • PredConOne b) ObjAnlytcInt(b) = b Anlytclnc a).

D

(ObjAnlytcInt(a)

C

T4b.

b- (PredConOne a • PredConOne b) D (ObjVerlnt(a) C ObjVerlnt(h) = b Verlnc a).

T4c.

(— (PredConOne a • PredConOne b) D (ObjThmlnt(a) C ObjThmlnt(h) = b Thmlnc a).

Also we have several mixed laws of inverse variation as follows. T5a.

h (PredConOne a • PredConOne b • (ObjVerlnt(a) C ObjAnlytcInt(h) v ObjThmlnt(a) C ObjAnlytcInt{b)) D b Anlytclnc a.

T5b.

i-(PredConOne a • PredConOne b) D ((ObjAnlytcInt(a) C ObjVerlnt(h) v ObjThmlnt(a) C ObjVerlnt(h)) = b Verlnc a).

T5c.

i- (PredConOne a • PredConOne b • ((ObjAnlytcInt(a) C ObjThmlnt(h) v ObjVerlnt(a) C ObjThmlnt(h)) D b Thmlnc a.

Various relations of equivalence as between PredConOne’s may now be introduced as mutual inclusions. Let .‘a AnlytcEquiv b’ abbreviate ‘(a Anlytclnc b • b Anlytclnc a)\ The definiendum reads ‘a is analytically equivalent with b’. Similarly let ‘a SynthcEquiv b’ abbreviate ‘(a Synthclnc b • b Synthclnc a)\ and so on. As corollaries of T4a-T4c we have the following laws concerning the various kinds of equivalence. T6a.

i-(PredConOne a • PredConOne b) ObjAnlytcInt(b) = a AnlytcEquiv b).

D

(ObjAnlytcInt(a)

=

64

1. SCIENCE AND INQUIRY

T6b.

I- (PredConOne a • PredConOne b) D (ObjVerlnt(a) = ObjVerlnt(b) == a VerEquiv b).

T6c.

i- (PredConOne a • PredConOne b) D (ObjThmlnt(a) = ObjThmlnt(6) = a ThmEquiv b).

Let us say that one nominal virtual class is analytically included in another if the expression which LDes the first is analytically included in the expres¬ sion which LDes the second. Thus ‘F Anlytclnc G’ abbreviates ‘(Ea) (Eb) (a LDes F • b LDes G • a Anlytclnc b)\ Also we may introduce the relation of analytical equivalence as between nominal virtual classes. We let ‘F AnlytcEquiv G’ abbreviate ‘(F Anlytclnc G • G Anlytclnc F)’. We now have some laws of inverse variation involving L-designation. T7a.

b- (PredConOne a • PredConOne b • a LDes F • b LDes G) (ObjAnlytcInt(a) C ObjAnlytcInt(6) = G Anlytclnc F),

D

and hence T7b.

\- (PredConOne a • PredConOne b • a LDes F • b LDes G) (ObjAnlytclnt(a) = ObjAnlytcInt(6) = F AnlytcEquiv G).

T8a.

i- (PredConOne a • PredConOne b • a LDes F • b LDes G) D (ObjVerlnt(a) C ObjVerlnt(fe) = F Verlnc G),

D

and hence T8b.

h- (PredConOne a • PredConOne b -a LDes F* b LDes G) D (ObjVerlnt(a) = ObjVerInt(6) = F VerEquiv G).

We need not tarry with these technicalia further, the main line of de¬ velopment being clear enough.

IX We go on now to further kinds of intension, as suggested in Intension and Decision, refashioning them as needed. In particular, the so-called Whiteheadian intensions must be handled rather differently. Here we have no type-theory by way of logical scaffolding and hence must make the virtual theory suffice. Whiteheadian intensions will have as members nomi¬ nal virtual classes of virtual classes! The general definition for Whiteheadian analytic intensions is as follows. Here as above we introduce only contexts in which such and such a nominal virtual class (of higher type) is said to be a member of a Whiteheadian

R. M. MARTIN

65

analytic intension of a PredConOne. We presuppose that abstracts of higher type such as ‘G3(—G—)’ have been introduced contextually, and that ‘d LDes G3(—G—)’ has been suitably defined. Then, without more ado, (D4) ‘G3(—G—) e WhtdAnlytclnt(a)’ is to abbreviate ‘(PredConOne a• (Eb)(b LDes G3(—G—) • Anlytc (The here is the sign of concatenation, so that (d^a) is the string of symbols b followed by a). And similarly for the other kinds of Whiteheadian intension, synthetic, veridical, and theoremic. As an example, consider again the defined PredConOne ‘M\ We note that ‘(x)(Mx D Mx)’, ‘(x)(Mx D (Rx-Ax))’, ‘(x)(Mx D Rx)\ ‘(x) (Mx D Ax)’, etc., etc., are a few Anlytc’s containing ‘M’. As members of the Whtd Anly tclnt (a), where a LDes M, we have then the nominal virtual classes of virtual classes G3(x)(Mx D Gx), G3(x)(Gx D Mx), G3(x) (Gx D Gx), G3(x)(Gx D (Rx-Ax)), G3(x)(Gx D Rx), G3(x)(Gx D Ax), etc., etc. An important point to note is that the various kinds of Whiteheadian in¬ tension may also be introduced for individual constants and Russellian descriptions of individuals, and this in contrast to the other types of objective intension mentioned. First let us consider primitive individual constants and then Russellian descriptions. Let ‘InCon a’ express that a is a primitive individual constant of L. Then (D5) ‘x3(—x—) e WhtdAnlytcInt(fl)’ may abbreviate ‘(InCon a(Ed) (d LDes x3(—x—) • Anlytc (d^a)))’. And similarly for the other kinds, synthetic, veridical, and theoremic. Now for Russellian descriptions, taken as primitives of L. Let ‘inviota’ be the structural-descriptive name of the inverted iota V, and ‘ex’ of ‘x’. If c is the structural-descriptive name of ‘(—x—)’ then i{inviota'~'ex'^cy is the structural description of ‘(-)X.(—x—) )’. Likewise let ‘invep’ be the structural description of the inverted epsilon ‘3\ And finally let ‘SentFuncOne n,d’ express that a is a sentential function of L containing d as its only free variable. Then we may let (D6) ‘x3(—x—) e WhtdAnlytcInt(a)’ abbreviate ‘(Ed) (Ec) (d LDes x3(—x—) • SentFuncOne c,ex • a = (inviota^ ex^ c) • (ex'~'invep 0, Pij = cpiy for each iefi. For example in the model of the genetical example above,

£2 = (Pa)

(9/16

6/16

1/16

11/4

1/2

1/4 _

= 1/16 •

=

no two possible outcomes would, under (Si), represent equivalent evidence, since no two columns of the model of E2 are proportional.

ALLAN BIRNBAUM

119

But in the model

£ = = 1/16

{94

l

l

\}

the second and third columns are proportional (in fact identical); the condition of (Si) is met, with c = 1: pt2 = pi3, for i = 1 and 2. Thus (Sx) implies Ev(E,2) = Ev(E,3). To exemplify the extramathematical content of this concept, consider the genetical experiment described above which is represented by E2. It is quite possible in practice to complement such an experiment, in case its outcome is / = 2, by tossing a fair coin and observing whether it falls heads or tails. Precisely because this addition to the genetical experimental procedure seems so clearly trivial, and irrelevant to the statistical and genetical evi¬ dence in each possible outcome, it is interesting and useful to consider formally the model of the augmented experiment and to compare it with the original model: The new sample space has four points, which we may label by

1, if / = 1, „ _ -

2, if j — 2 and heads is observed, 3, if j = 2 and tails is observed, 4, if j = 3.

It is readily verified that the new model has the form E = (p\y) given above. The indicated relations between probabilities in the respective models are Pa = p'ii, Pi2 = p'i2 + p'i3, and Pi3 = p'i4, for i = 1 and 2. Thus E is seen to be another model of the genetical experiment repre¬ sented by E2, including an addition which is trivial but harmless. The addi¬ tion cannot hinder us in recognizing that the two outcomes /' = 2 and 3 of E represent the same genetical and statistical evidence, so long as we think of those outcomes as differing only in a way that is irrelevant to evidential meaning. This equivalence is conveniently suggested by the respective labels: (/ = 2, heads) and (/ = 2, tails). However since we consider the model E as valid (even if unparsimonious), we may also represent it in our usual generic notation by E — (pV). What formal aspects of the two models of statistical evidence (E,2) and {E,3) correspond to their (agreed or postulated) equivalence of evidential meaning? The answer is just the condition (Si) given above: In our ex¬ ample, the use of a fair coin corresponds to the value c = 1 in (Si); use of a coin with probability c/(c+ 1) of heads corresponds to the general case. Thus (Si) may be described heuristically as the concept that recognizable pure “noise” is irrelevant to evidential meaning. We have illustrated the extramathematical content of concept (Si) by

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1. SCIENCE AND INQUIRY

an example of the kind usually used to illustrate its plausibility and to sup¬ port its adoption. In the example, a given model was made more com¬ plicated, in a way considered trivial and irrelevant to evidential meaning, as expressed by (Si). Conversely, adoption of (SO supports certain con¬ venient commonly-made simplifications of models used in practice: If just the evidential meaning of outcomes is of interest, there is no need to dis¬ tinguish between outcomes which are equivalent under (Si), and they may be treated as equivalent alternative indications of one outcome. Such treatment is naturally represented by a simpler new model, with a smaller sample space, determined from the original model. An example is provided once more by the genetical experiment: Instead of the model E2 above, we may obtain a different, more complete model of the same experimental situation as follows: If the progeny are labeled by u = 1, 2, respectively, then the experiment has the four possible outcomes (yi,y2) = (0, 0), (o, i), (i, o), or (l, i), where yu = 1 or 0 according as progeny u has trait A or does not. Denoting these sample points respectively by j — 1, 2, 3, 4, the more complete model is seen to have the form E = (p' iy) given above. Thus this more com¬ plete and accurate model E of the genetical experiment is identical with the model we obtained earlier by augmenting E2 in a way considered trivial and irrelevant for evidential meaning. Thus simplification of E to the form E2 is parsimonious and without effect on evidential meanings. The indicated correspondence between the sample spaces and the probabilities in the models E2 = (ptj) and E = (p' ir) is summarized by

i = /(/')

T, for j' = 1, 2, for j' = 2 or 3, 3, for f = 4,

and by the equations given above relating

Pi/s

and

p'i/s.

The concept of evidence denoted by (Si) above, together with its conse¬ quences for simplification of models just illustrated,2 constitute the suf¬ ficiency concept of statistical evidence. This is conveniently formulated as follows: (S):

The sufficiency axiom: If E = (py) is determined from E' = (p^ ) by any function j=j(j') which takes distinct values on any two values of j' which do not label proportional columns of E', then for each (E', j') we have Ev(E',j')=Ev(E,j) where j=j(JD-

The simplification of models by use of sufficient statistics is sometimes justified by appeal to operational considerations which may be judged more or less distinct from the appeal to concepts of evidence (or of evidential irrelevance) just illustrated. This is particularly the case in decision-theoretic

ALLAN BIRNBAUM

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discussions, where concepts of evidence play no specific role. In terms of the genetical examples above, such an operational justification of replacing E by the simpler E2 is the following: The genetical experiment, when represented just by the simplified model E2, can be complemented if desired in an operationally realizable way by the toss of a fair coin, as described above, so as to yield an experiment whose model is identical with the original E. Here the two perspectives, operational and evidence-conceptual, seem not only compatible but even complementary and mutually support¬ ing. Unfortunately this is not the case more generally, as we shall see. The sufficiency concept was formulated by R. A. Fisher (1920) in con¬ nection with relatively specific problems of estimation, namely comparisons of alternative estimators of precision of measurements under the normal error model. (It played a distinct though implicit role earlier, in certain simplifications of models like the one just illustrated.) Like most of the other precise general concepts of evidence to be discussed, the main con¬ tent of the sufficiency concept is linked rather delicately with the forms of experimental models. An investigator’s judgment that a given model is approximately valid and useful often entails no very precise guidance in terms of “approximately sufficient” statistics. This is another important kind of consideration which tends to support eclectic and informal practice in applied statistics. (For some historical and applied-statistical methodo¬ logical comments, and further references, cf. Fisher, 1950, p. 2.757a, and Tukey, 1960, pp. 471-473.) It is convenient to introduce here some definitions required below: Any function which depends only upon a sample point is called a statistic. Examples are each of the functions j — /(/') considered above. Any statistic meeting the condition in (5) is called a sufficient statistic.3 Each of the statistics /(/') we have considered thus far is sufficient, and so is the trivial function /(/') = /'. A statistic which is not sufficient, in

is /(/') = 1 for /' =1,2. (This function fails to take distinct values on values /' labeling nonproportional columns.) Thus the sufficiency axiom (5) may be stated thus: If /(/') is a sufficient statistic in E, and if j(j\) = /(A), then Ev(E',j'o) = Ev(E,j) where / = j(j\). 2.2 The Confidence Concept We shall refer to the concept of statistical evidence by which estimates having the confidence region form are usually interpreted, as the confidence concept. The formal theory of confidence region estimation does not include reference to any concepts of statistical evidence, as has been emphasized

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consistently by the theory’s inaugurator, Neyman (1937, 1938, 1962). Thus when a confidence region estimate is interpreted as representing statistical evidence about a parameter point of interest, an investigator or expositor has adjoined to an application of the formal theory his concept of statistical evidence, which we refer to as the confidence concept. We illustrate in terms of the important binomial case represented by our examples. Such examples are often treated by reference to charts which appear in many texts and manuals of statistical methods, given originally by Clopper and Pearson (1934). Our brief discussion of details may be complemented by reference to such a chart (or to equivalent tables of the binomial distributions or of the approximating normal distributions; cf. e.g. Walker and Lev, 1953, p. 461). A lower 99 percent confidence limit estimator of the binomial parameter 6 is by definition any function of the observed outcome x, denoted con¬ veniently by 0{.99,x), which has the property of providing a correct lower bound for the unknown value of 6 with probability at least .99, under each possible 6; that is, any function 6(.99,x) of x satisfying Prob(0(.99,x) ^ 6\8) ^ .99, for each 0, 05=02^1. Application of such an estimator to the statistical evidence (£,40), for example, might give the result 0(.99,4O) = .63 (as found below). Such a lower confidence limit estimate is usually interpreted as a lower bound on the unknown true value 6, whose correctness is supported by fairly strong statistical evidence, indexed by .99. (An estimate is a fixed number, ob¬ tained by use of an estimator, a function of jc. ) Such an interpretation is represented conveniently by writing Conf (6 ^ .63) ^ .99. (Such notation has been used, for example, in the text of Walker and Lev, l.c. p. 54.) The number .99 appears above as the lower bound of probabilities of correctness (of bounds on 0, given by an estimator). Such a number is called the confidence level, or the confidence coefficient,4 of the estimator 9(.99,x) and of an estimate such as 0(.99,4O) = .63. This probability property is cited as basic in the usual explanations of the confidence con¬ cept, and in justifications of the use of confidence region estimators. It is referred to as an operational property of the method. (The term “operating characteristics” is used more or less generally to refer to the error-probabili¬ ties which are the basic terms of the theory of estimation due to Neyman and the related theory of tests due to Neyman and Pearson. “Operational” here refers to the realizability, in long sequences of comparable independent applications, of observable relative frequencies approximating the prob¬ abilities mentioned.) These explanations usually include implicit or explicit reference to the confidence concept, without which such estimates would receive no evidential interpretation. In our binomial examples, a convenient commonly used approximate formula for such an estimator is

ALLAN BIRNBAUM

123

6(.99,x) =— - ^(.99) a/—(1 - — )/n = n Vn n

jo-23W^-io)/50For (£,40) this gives #(.99,40) a .67. ( denotes the standard normal X

distribution function.) Here — is of course the usual point estimator of a binomial parameter, and the second term is a multiple of the usual estimator of its standard error. A precise formula for such an estimator is #(.99,*) = the smallest value # for which ^ f(v,6) 5= .99, where /(*,#) is the vrgx

binomial pdf of our examples,

#*(1 — #)50'L (The verification that

this estimator has the required basic property is immediate, and is described conveniently in many texts by reference to the Clopper-Pearson charts. Because of the discreteness of binomial distributions, no estimator (func¬ tion of x) can meet exactly the indicated bound on error-probabilities.) Antecedents of both the method of confidence region estimation and the confidence concept can be traced back as far as the earliest systematic consideration of statistical evidence, that of Bernoulli, l.c. (Cf. also e.g. Dempster, 1964, pp. 56-7). As the example illustrates, the general method of confidence region estimation is often applied by use of techniques provided by the theory of point estimation, a self-contained branch of mathematical statistics. Point estimates, accompanied by distribution theory adequate to indicate their precision, yield confidence region estimates (often approximate ones) for a wide variety of problems. Indeed, use of the confidence concept is cur¬ rently the favored mode of application and interpretation of point estima¬ tion methods. Thus confidence region estimation techniques share important features with the long-standing practice of reporting measurements (esti¬ mates) of quantities accompanied by their standard errors (or other indices of precision); and the confidence concept may be regarded as in¬ cluding usual interpretations of such reports of measurements. Further ex¬ pository remarks on these relations are given e.g. in Birnbaum (1961a). The paper cited also describes briefly a mode of treating tests of statistical hypotheses which is currently favored when feasable, namely, embedding testing problems in confidence region estimation problems. To illustrate in terms of our genetical example where # has just the two possible values .75 and Vi, the confidence interval estimation result Conf(.63 ^ # rg .94) fS .98 incorporates conveniently an application of a test (between the hypotheses .5 and .75) and its interpretation (“reject # = .5, at significance level stronger than 2%”).5

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1. SCIENCE AND INQUIRY

In the modern general theory of estimation by confidence regions, at¬ tention is directed toward minimizing probabilities that confidence regions will include values of 6 other than the true value, subject to the basic defining condition (inclusion of the true value of 6 with specified prob¬ ability). Such minimization also tends to give small confidence regions, or sharp confidence limits. Precise definitions of optimality or efficiency, in terms of minimization of various error probabilities, play a key role in this theory as the basis for choosing among the numerous possible esti¬ mators meeting the basic condition. In our example the estimator 0,40)). The various concepts and techniques of evidential interpretation discussed above, and others, are applicable here in their usual general ways. For example an optimal 99 percent lower con¬ fidence limit estimator of 6 can be given as before (not withstanding greater complexity of the definition of optimality and the calculations). However it has seemed to many theoretical and applied statisticians that a model of evidence such as (£,(£50,40)) here, comprehensive and ac¬ curate though it is, contains parts which are clearly irrelevant to evidential meaning: given the bad luck of getting just 50 rather than 200 ob¬ servations, the evidential meaning of the outcome of the fifty observations is fully represented by (£50,40); and the hypothetical possibility that an additional 150 observations might have become available but in fact did not, seems irrelevant to the evidential meaning of the result. Formally, this is represented by adopting (£50,40) in place of (£,(£50,40)) as the ap¬ propriate model of the statistical evidence obtained (or as a more par¬ simonious, though equivalent model). Within this “conditional” model any chosen mode of evidential interpretation could be applied. For example the confidence limit estimate found for (£50,40) in section 2.2 above might be determined and interpreted in the usual way. The general form of the concept just illustrated may be stated generally by reference to any model £ = (pi;) which is a mixture of two other models £' = (p'ij-) and E" = (p"ir) in the sense illustrated, namely: (i)

The three models have common parameter space.

(ii)

The sample joints / of £ are labeled alternatively, in any fixed order, by the symbols (£'/), for ]’ = 1,2, ..., and (£"/'), for

f = 1,2,....

ALLAN BIRNBAUM

131

(iii)

Pii

(

J(l/2 )p'ij;

if j=(E',n,

1(1/2)p"ir,

if/=. (£"/')•

(Mixtures are usually defined more generally, allowing more than two component experiments, and possibly unequal weights; but this simple restricted definition suffices for our discussion.) The general form of the concept illustrated is11 (C):

The conditionality axiom: If £ is a mixture of the experiments E'^fp'jj.) and E"=(p"ir), then Ev(E,(E',j'))=Ev(E',jO.

The conditionality concept emerged in Fisher’s theory of estimation, a little later than the sufficiency and likelihood concepts. In his last writings on statistical inference, he emphasized both the importance he attached to the concept and the incompleteness of knowledge of its mathematical and extramathematical aspects. (“The most important step which has been taken so far to complete the theory of estimation is the recognition of Ancillary statistics” [1956, pp. 157-8].) Other theoretical and applied statisticians have seen the conditionality concept as an appropriate and even essential complement to the oper¬ ationally-based confidence concept. They have supported its application, in cases illustrated by the example above, as supplying necessary content of an intrinsically evidence-conceptual kind, and guidance for choices among confidence region estimators. (Cox, 1958, pp. 359-63, Wallace, 1958, p. 864, and further references therein.) Against this background, the non¬ uniqueness of conditional models in certain problems (Basu, 1959, Birnbaum, 19616) appeared initially as a significant but not insuperable problem, to be met by further investigation of the scopes of the conditionality and confidence concepts. 2.7 Incompatibility Between the Conditionality and Confidence Concepts The general approach described in the preceding paragraph is one of reliance on the confidence concept, with restrictions to accommodate the conditionality as well as the sufficiency concept. Although this perspective remains a preferred one among many applied and theoretical statisticians, it has become clear that it must remain at best a rather eclectic one, be¬ cause it embraces conceptual ingredients fully as disparate as the likelihood and confidence concepts, whose incompatibility was discussed above: It has been seen that the conditionality and sufficiency axioms together are equivalent to the likelihood axiom. This has suprised and disappointed some, including this writer, who remain without an adequate precise general concept of statistical evidence, and without even consistent criteria for adequacy of such a concept. The proof that (C) and (5) imply (L) is elementary: For any E' =

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1. SCIENCE AND INQUIRY

(p'ij-) and E" = (p"ir) with common parameter space, let (E',j') and (£",/") be any models of evidence determining a common likelihood func¬ tion: For some c > o, p"ir = c p'iy for each i. By applying (C) to the mixture E of E' and E" we obtain Ev(E'j') = Ev(E,(E',j') ) and Ev(E",j") = Ev(E,(E",j”)) Now the likelihood functions determined by (£,(£',/)) and (£,(£",/")) are, respectively, V2 p'ir and V2p"ir. These are proportional columns of E. Therefore an application of (5) gives £v(£,(£'/)) = £v(£, (£",/"))• Hence

Ev(E',n = Ev(E",j"), which is the conclusion of (L), completing the proof (given by this writer, 1962). (For another concept which many have found highly plausible, and which along with (5) implies (L), see Pratt, 1961, 1962.)

3. STATISTICAL THEORIES OF DECISION-MAKING, RATIONAL BEHAVIOR, AND INDUCTIVE BEHAVIOR 3.1 General Remarks

The historical roots of probability concepts were linked with those of concepts of utility and of reasonable decision-making under uncertainty, in the problems concerning betting odds in games of chance, which led to early probability theory, and also in the earliest work concerned critically with the interpretation of probabilities, that of James Bernoulli. Many writers have since affirmed, and many denied, that other interpretations of probabilities can and should be given, interpretations independent of utility and decision concepts. In any case, the probabilities Pij appearing in standard models of experiments have usually been given other inter¬ pretations, generally referred to as frequency interpretations. It was in another role that utility and decision concepts re-entered mathematical statistics in recent decades, notably with Neyman (e.g. 1957) and Wald (1950), and played a major role. (The concepts had also ap¬ peared clearly in this role in the introductions by Laplace and Gauss of the estimation criteria of mean-absolute-error and mean-square-error. But as the normal error model came to receive almost exclusive attention in estima¬ tion theory, that model’s precision parameter assimilated the technical role of such criteria, and their specific conceptual content was neglected. New technical and conceptual problems appeared with increasing consideration

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of other models, and also of the normal model with precise attention to the imperfect estimate of its precision parameter.) This role of utility and decision concepts might be described as connected with interpretations and applications of outcomes j of given experiments E = (Pij). But more precisely, this role is not related to interpretations of cases of statistical evidence (E,j), nor to concepts of evidence, in the gen¬ eral sense of this paper. (This point has not always been clearly appreciated. Some sources of misunderstanding are indicated below.) Rather, this role is related primarily to another aspect of the experimental situation, represented by another mathematical ingredient which complements the model E of an experiment, when statistical decision problems are formu¬ lated. This is the decision space (illustrated briefly in section 1 above), a set D = {d} of points (labels) d of the respective alternative decisions or actions, one of which is to be adopted on the basis of the observed out¬ come j of E. An artificially simplified but relevant illustration can be devised in connection with the genetical example above: Let d1 denote the decision to launch a commercial stock-breeding program, or alternatively to launch a program of further genetical research, in a way which would be ad¬ vantageous and appropriate if and only if two specified genes lie on different chromosomes. Suppose d2 denotes the sole alternative possible decision, for example to launch a different program, or to do nothing. The problematic situation represented in part by the model (E,D) is resolved when a decision, d-i or d2, is adopted. In statistical decision theory, each possible policy, or rule for choice of decisions, is represented by a decision function d(j) with values in D (pos¬ sibly randomized, i.e., depending also upon an auxiliary randomization variable). The theory develops concepts and techniques of advantageous choice, based on the utility concepts of loss and risk or utility, and on criteria for optimal choice (admissibility, minimax, etc.). Statistical decision theory has played significant roles in several devel¬ opments in modern mathematical statistics. One of these developments is a new approach to prior probability concepts, due to Savage (1954): The admissibility criterion for reasonable choice is a weak and basic, noncontroversial one in the context of statistical decision problems. It is of some intuitive as well as technical interest that this criterion admits accurate formal interpretation in terms of hypothetical implicitly-held prior prob¬ abilities: Each admissible decision function would be also an optimal one for a person holding a suitably corresponding prior probability distribution. (Here an extended technical sense of prior pdf is sometimes required, a sense generally favored also by those holding prior probability concepts.) Savage showed that certain richer criteria, characterizing also consistency among all preferences in such contexts, entail prior probabilities and linked utilities, at least hypothetical implicitly-held ones, which would account for

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any specified consistent preference pattern. (Savage is also a notable modern exponent of the view that probabilities in general, not only prior prob¬ abilities, are essentially linked with utilities. But this issue is rather in¬ dependent of the result mentioned.) Another important role of statistical decision theory has been based on its introduction of new mathematical approaches by which previously established branches of mathematical statistics have been appreciably ex¬ tended, notably the theories of testing hypotheses and estimation due to Neyman and Pearson, and of point estimation. In this connection the loss and utility concepts of decision theory play no distinctive role, since errorprobabilities remain the basic terms of the theories thus extended. Here the specification of the decision space is chosen to represent one of the standard statistical problems mentioned, and not to represent more concrete or practical alternative decisions faced by an investigator using a test or estimate. As an example of testing, in the genetics experiment, “rejection or non-rejection of the hypothesis” that the genes lie on different chro¬ mosomes would be called the alternative “decisions.” (The loss and utility or risk functions for testing, and for confidence interval estimation which is linked technically with testing, are here simply indicators of errors and of probabilities of errors, not measures of more specific practical losses and utilities which might be entailed by errors. [Cf. e.g. Birnbaum, 1961a.] For point estimation the loss function may be squared error, linked to the traditional variance criterion and precision concept, or absolute error, or any other function specified as an appropriate measure of the loss entailed by a given error of estimation. The case of squared error has been most fully treated, for reasons partly technical and no doubt partly intuitive or traditional. The role of general loss functions in such formal point estimation theory is correlated with the limited application of point estimates except when accompanied by an estimate of precision [not loss].) Now concepts of statistical evidence, including those discussed above, played essential roles at various stages in the development of theories and applications of estimation and testing hypotheses, notwithstanding the perennially problematic character of such concepts. But in recent decades statistical decision theory has proved eminently successful in providing in¬ creased mathematical scope, and also one kind of conceptual unity, for these theories. These two circumstances have been interpreted by many (but far from all) as showing that the concepts of decision and utility introduced by decision theory have provided appropriate or successful ex¬ plications of the previously troublesome and obscure concepts of statistical evidence; and/or that they have provided an appropriate way of circum¬ venting questions concerning such concepts. Such interpretations are illusory and unfortunate. Concepts of statistical evidence can hardly be circumvented in general, since they are involved

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explicitly, inextricably and irreducibly, in the actual structures and processes of scientific disciplines such as genetics where the essential decision-theoretic concepts, utility and decision, have little or no clearly relevant role. The point concerning decisions in particular has been expressed notably by Cox (1958, p. 354): “It might be argued that in making an inference we are ‘deciding’ to make a statement of a certain type about the popula¬ tions and that, therefore, provided the word decision is not interpreted too narrowly, the study of statistical decisions embraces that of inferences. The point here is that one of the main general problems of statistical inference consists in deciding what types of statement can usefully be made and exactly what they mean. In statistical decision theory, on the other hand, the possible decisions are considered as already specified.” 3.2 The Compounding Approach Important extensions of decision theory are provided by the compound¬ ing approach, in which decision problems which might be treated separately are treated jointly in ways which yield appreciable gains in terms of error frequencies or utilities. (Neyman, 1962, and Robbins, 1963, give non¬ technical accounts.) A particularly striking feature of the compounding approach is that these gains are available even when the respective prob¬ lems lie in arbitrarily related or unrelated empirical research fields. This feature has been appreciated and endorsed by the originator of the approach (Robbins, l.c., p. 112). (The problems might lie respectively in agronomy and astronomy; and/or in eugenics, euthenics, eumetrics, and hermetics. Thus the approach may be described as opening wider domains over which utility is to be gained, by the compounding of errors.) Now the notion is widely and firmly held that errors or apparent errors in one research field are irrelevant to evidence in another field of in¬ vestigation judged unrelated to the first.12 For those who see significant scope for this notion, the compounding approach provides an inadvertant but striking reductio ad absurdum of the view that decision theory provides the appropriate ways of interpreting and using statistical evidence in general.

4. PRIOR PROBABILITY CONCEPTS 4.1 Technical Introduction A prior probability model of an experiment, (G,E), is constituted by a model of an experiment E = (p^) as above, complemented by a mathe¬ matical probability distribution over the parameter space O, denoted by G and represented by a pdf gi = Prob(i), called the prior (or a priori) dis¬ tribution. Correspondingly, a prior probability model of statistical evidence,

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(G,E,j), is a model (£,/') of evidence as above, complemented by a prior distribution G. Now (G,E) is a probability model, as distinct from a statistical model, of an experiment, in our usage introduced above. That is, it represents just one, not two or more, probability distributions on a given sample space. Its sample space is not S = {/} (the sample space of E), but the set of points (/,/'), for ( e fi and j e S. (We may denote that space by ClxS = {(U)}). Correspondingly, (G,E,j) is not a statistical model of an experiment (although it includes such a model (£,/) as an ingredient), and the con¬ cepts of statistical evidence discussed above are not even formally appli¬ cable. Moreover, the observed outcome j here is not an elementary outcome (sample point) of the probability model (G,E) but represents the set of sample points (/,/), /eO. Finally, the model (G,E) is given in a form which specifies some conditional distributions: Pn,Pi2, ■ ■ Pa, • ■ ■ Pu, is now the conditional pdf of j, given i. From (G,E,j), a standard elementary formula (Bayes’ theorem, not Bayes’ principle or postulate) gives the conditional probability of i, given •

g*i = Prob(i|/) = kgipy, for each i e Q,

where k is a normalizing constant (1/k = 2 giPa) • This represents a new i’eO mathematical probability distribution G* over O, called the posterior (or a posteriori) distribution. In our genetical example, a mathematically possible prior distribution G is the uniform one, which assigns equal probabilities gi — g2 = Vi to the two parameter points i = 1,2. The latter labels indicate respectively the param¬ eter values 6 = Vi and 14 in the binomial distributions in that model. In a more general genetical model, allowing for breaking and rearranging of chromosomes, the parameter space becomes Va =4 0 =4 Vi, and the model of our experiment with 50 observations is once again constituted by the cor¬ responding set of binomial distributions. On this parameter space also, various possible mathematical probability distributions G may be defined. 4.2 Relation to the Likelihood Concept We may note that in this context of prior probability models, a precise version of the likelihood principle is evident: The preceding formula shows that the posterior probabilities determined by (G,E,j) depend upon the statistical evidence (£,/') only through its likelihood function cp^ (since the latter, multiplied by gi, gives g*i up to a constant, and the constant is de¬ termined as above); and relative posterior probabilities

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Q.

differ from corresponding relative prior probabilities — only by the factor Si'

L{i,i') = —the likelihood ratio. Pi'}

Of course this observation does not provide interpretations or justifica¬ tions of the likelihood concept outside of prior probability contexts, except by way of broad and loose analogy. On the other hand, any support for the likelihood principle on grounds independent of prior probability con¬ cepts may be regarded as also partial support for the latter concepts (since it diminishes the scope of questions of extramathematical interpretation and justification of prior probability concepts). Such possible independent grounds include self-evidence of the likelihood concept; or its entailment by other concepts which may be considered plausible, such as (5) and (C) as above. 4.3 Interpretations of Prior Probability Concepts Of course prior opinions and judgments, based on previously available observations and theory, play important broad roles in investigations in all fields. What many investigators and theoretical statisticians have not found is an adequate account of the appropriateness, the meaning, and the use¬ fulness of the specified forms of prior probability concepts and methods, in relation to the background of a specific investigation. (Cf. e.g. Nagel, 1939.) (Of course prior probability concepts are not directly supported by the difficulties which may be met in other approaches to explication of probabilities such as the py’s; not, in particular, by the objections which have been raised to proposed “relative frequency interpretations” of such prob¬ abilities, as in Keynes’ discussion of the structure of Darwin’s theory [1921, pp. 108-9].) Poincare’s view of the empirical sciences as “but unconscious applications of the calculus of probability,” without which “Science would be impos¬ sible”, has been endorsed by one of the foremost modern exponents of prior probability concepts, de Finetti (1937; pp. 99, 156 of the translation cited below). Here “Science” evidently refers to the actual structures and developments of the sciences, as these might be characterized by general principles amounting to a “logic of science” (including, in the case of de Finetti’s work, a kind of solution of Hume’s general problem of induc¬ tion). The rich literature on prior probability concepts attests abundantly to this broad observation: If there is a “logic of science,” in any one of many proposed precise senses, then that “logic” will include formal prior prob¬ abilities. It is the actual or potential existence of science, or of a logic of science, in such a sense which has been proposed by some and questioned by others for two hundred years. During recent years, prior distributions have been introduced and used systematically by some investigators of genetical linkage (Haldane and

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Smith, 1947, pp. 12-13; Morton, 1955, pp. 281-4, and 1962, pp. 38-43; Smith, 1959, pp. 298-9, and 1963, and references therein; and Renwick and Lawler, 1963, pp. 69-71, 84). In another area, a problem in at¬ tribution of authorship has been investigated by both Bayesian and nonBayesian methods, described in detail, by Mosteller and Wallace (1964). It is interesting to compare and contrast such applications of prior prob¬ ability concepts with examples which figured in earlier stages of discussions of such concepts. The latter include Karl Pearson’s and Fisher’s discussions of anthropometric correlation measurements (Soper et al., 1916; Fisher, 1921). A discussion of such comparisions will be presented elsewhere, together with fuller discussion of other points touched on briefly in this paper.

NOTES 1. The writer is indebted for helpful comments on earlier versions to John Pratt, Charles Fisher, and Valerie Mike, and to Churchill Eisenhart for guidance on histori¬ cal material. This work was done in part while the author was a Fellow of the John Simon Guggenheim Foundation. It was also supported in part by the Office of Naval Research and the National Science Foundation. 2. The general form of such determination of one model E from another E', by any given function /'(/') (not necessarily having the property considered above), is: E has as its sample space S — {/} the range of values of /(/'), for f in S’; without loss of generality for our purposes, we assume this range to have (after possible relabeling) the form /= 1,2, . . . J. The probabilities in E — (pu) are given by the usual basic probability rule, p(j = 2p'(j' where the summation is over all j' such that ;■(/') = /. 3. The condition given above is well known to be equivalent to the usual de¬ fining condition for a sufficient statistic. (Cf. e.g. Cramer, 1946, p. 488, Lehmann, 1959, Lindgren, 1962.) Since we shall not use the latter condition, it is convenient to regard the condition in (S) as defining sufficient statistics in our discussion. 4. It is of course common nontechnical usage to call any proposition probable or likely if it is supported by strong evidence of some kind. Thus it would not be unnatural here to call the confidence coefficient .99 the probability or the likelihood of the proposition that 8 S: .63. However such usage is to be avoided as misleading in this problem-area, because each of the terms probability, likelihood, and con¬ fidence coefficient is given a distinct mathematical and extramathematical usage here. That of likelihood is introduced in the next section. Probabilities are specified in our models E and (E,x) only for points and subsets of the sample space S = {*}. For points 8 or sets (like 8 2: .63) in the parameter space Q = (0), no probabilities are defined, except under specific additional formulations (such as the Bayesian, in which the parameter space is made the sample space of specified prior and posterior distributions). 5. The concept of statistical evidence which has been associated with tests since their appearance in 1710, which may be termed the significance-probability concept, has proved an elusive one to explicate (cf. Anscombe, 1963). The paper in which the Neyman-Pearson theory of tests originated was an approach to the problem of explicating and justifying this concept in connection with a systematic mathematical theory of tests. The paper introduced a new definition of a test as a two-decision procedure, whose operational error-probability properties allow application of the confidence concept as it applies to results of such tests. (Cf. especially p. 336 of Neyman and Pearson, 1933.) 6. When convenient the last form may be used, or the symbol c may be replaced by any positive number. Thus the only essential aspect of a likelihood function is the set of likelihood ratios it determines; these are the respective values of the likelihood ratio function: L(i,i’) = Pn/pvi, for i,i’ e 0.

7. We observe that the definition of a sufficient statistic and the statements of

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(Si) and (S) above can be stated conveniently in terms of the likelihood function. In fact, the likelihood function is readily seen to be the sufficient statistic which gives the greatest possible simplification of models in the sense illustrated above; the latter property makes it a “minimal sufficient” (or “necessary and sufficient”) statistic. 8. Fisher’s program for a theory of fiducial probability shows that he considered that a fully adequate concept of statistical evidence would be one having the form of probability distributions over parameter spaces, determined however in such a way as to be incompatible with the likelihood axiom (cf. Anscombe, 1957). The fiducial concept is not discussed further here because the present writer (among others) has found insufficient clear substance in the mathematical and extra mathe¬ matical aspects of that concept. For some recent criticisms, reinterpretations, and/or revisions, see Fraser, 1961, Dempster, 1964, 1966, and Hacking, 1965. 9. But a semblance of a “likelihood school” has been suggested in recent years by, for example, Barnard et al., 1962, Anscombe, 1964, D. G. Kendall, 1965, Sprott and Kalbfleisch, 1965, Zweifel, 1966, and some tentative discussion by this writer, 1961, 1962, pp. 286-98, 325-6. 10. We indicate here in further detail the incompatibility referred to: The given likelihood function could have arisen alternatively from a different experimental procedure, E*, in which the newly-bent coin is tossed until just 40 heads have been observed. If the fortieth head is observed after just 10 tails have appeared, the likeli¬ hood function is the same. But the model of this experiment is quite different from E. The method of confidence limit estimation is applicable in E* in the same general way as in E. But in general the (optimal nonrandomized) .99 lower confidence limit determined from (E*,x*) and (E,x) are different, even when the latter deter¬ mine the same likelihood function. Furthermore, there are still other experiments in which the same likelihood function may appear, but where the method of confidence limit estimation is inapplicable, or at best applicable only awkwardly and formalistically. An example is represented by the model with just two sample points, j — 1,2, and the pdf /(1,