0N FORMAL AEDS YO MEENHFEC DECISION MAKENG Thais for “10 Degree of M. A. MICHIGAN STATE UNIVERSITY John M. Vickers 1959 ON FORMAL AIDS TL SCIENTIFIC DECISION MAKING By John E. Vickers A THESIS 'ttei to the College of Science end Arts of michigan 'e qniv;rsity 0: Agriculture and Applied Science in partial fulfillment of the requirenents for tea degree of '."/‘1|.1" "I l‘ f" r 4‘ L'u‘XLJJ.:JR CT mils ‘ _/ Department of PhiIOSOphy j.e2,’ ,'/ 1959 , / ACKNOWLEDGEMENT I should like to thank Professors Henry 3. Leonard and Lewis K. Zerby who read a first draft and suggested many improvements. I must acknowledge the con- tinued stiaulation and aid of Professor Richard Rudner; any merit the thesis might have is surelya tribute to his skill and patience as a teacher. He is not, of course, to be held accountable for its shortcomings. II. CONTENTS INTRODUCTION; THE TASK OF Th5 THESIS 1.1 2.1 2.2 2.3 2.4 3.1 3.2 3.3 4.1 5.1 5.2 5.3 3.2 EVALUAIION CF LVIUEhOE; A RULE Introductory . . . . . . . . . Positivism and ontology . . . . The bifurcation of discourse . Internal and external questions Quine's criticisms . . . . . . Emotivisn and the fact/value distinction . . . . . . . . . . Naturalism . . . . . . . . . . SCICHCG 311d lelle o o o o o o o A decision paradigm . . . . . . Discussion of the aradism . . P o Normative and descriptive discourse Scientific goals . . . . . . . Preview . . . . . . . . . . . . The theoretical decision, . . . Higher level hypotheses . . . . Goodman's theory of projection Induction and probability . . . Criteria of confirmation . . . Fl 10 15 17 2O 25 29 33 36 I 30 #1 bl #4 #7 5h 56 III. IV. 3.3 ii A Rule of rejection . . . . . . . h.l Interest as a property of hypotheses} h.2 A method of evaluating evidence . FOUNDATIOHS OF A TALORI OF IMPOdTANCE. 1.1 2.1 2.2 3.1 3.2 h.l Methodological comments . . . . . The scientific decision situation Decision matrices . . . . . . . . Initial definition of importance. Comments on the definition. . . . The utility of hypotheses . . . . -\~ CONCLUSION . . . . . . . . .'. . . 1.1 The cons ructivist method . . . . 1.2 2.1 2.é 3.1 3.2 h.l Confirmation and utility . . . . On the rule of rejection . . . . Certainty and practical certainty Truth and languages . . . . . . . Policies, strategies and decisions. Swmndry . o o o o o - o o o o o o o . 81 . 39 105 .105 108 111 113 l17 120 123 CHAPTER I INTRODUCTION; THE TASK OF THE THESIS 1.1 This is not a thesis in the history of philoso- phy. That is to say, the conclusions of the thesis are not intended to be such as will trace the causal historical development of various philosophical problems and attempts at solutions of those problems. There is a sense, of course, in which any attempt at philosophical discourse in- volves itself with the problems of the history of philoso- phy. That sense, however, is so obvious as to merit no more than recognition here. Except for such involvement as this ineluctable sort the present work does not explicitly concern itself with questions usually treated by historians of philosophy. The problems with which this thesis does concern itself are, perhaps, novel as regards philosophy. Many of these problems -- at least in their present form -- have not historically concerned philosophers, though they have been explicitly dealt with by other disciplines for various purposes and.with various results. Because of this compara- tive novelty, it is to be expected that the thesis would be considered by some not to be a philosophical thesis at all, but rather a step-child properly belonging to any of sev- eral other disciplines. In the light of this possibility ~- a real one, I fear -- it would seem that some argument should be advanced for the philosophical relevance of the thesis. I should hope that the work can argue for itself well enough, as regards the soundness of its conclusions; but I think it not to be expected that it argue for its being philosophical. It would, however, seem reasonable to expect that tne author of the thesis present some argument for this philosophical relevance. One should expect to find, somewhere in the finished work, a body of discourse which persuaded that what was being said was philosophical; No such discourse is included, unless the quite sketchy re- marks of the last chapter along these lines be taken as persuasive, and I fear that the antipathetic reader will find little persuasion here. In short, if you do not think that the problem itself is philosophically relevant, there is probably nothing within the thesis which would incline you to change your attitude. You will go on thinking that my tOpic of concern is not a philosophical one. I Since the attempt at justification is abandoned a priori, perhaps some sort of a causal explanation of the author's attitudes will be of assistance to the reader who wonders just how anyone could consider such a thesis at all philosophical. Since, furthermore, we explicitly claim dispensation from any responsibility as regards historical worth, what we say of the history of philosophy may be taken as indicative of the author's philosophical concerns rather than as historical information. With this in mind we refer to some developments in contemporary philosophical thought. We each -- to some extent —- choose our philosophical ancestry, and the author's choice of ancestry should become evident in the following brief historical comments; Twentieth century philosophy has produced sev- eral seemingly disparate but actually intimately related endeavors. Some of these have enjoyed salutary success, others have fared less well. I should think that one en- deavor which has been -- and continues to be -- largely successful is the formation of the pragmatic tradition in philosophy. The work of Peirce, James, Schiller and Dewey comes to full bloom in the pragmatic philosophy of C. I. Lewis. Lewis' value theory, in turn, lays the foundation for naturalist epistemology and value theory. The notion of value as fel§,value, of value as being inseparable from experience, and of knowledge as being functionally insepar- able from purposes, relate themselves so as to form a coher- ent philosophical position; one which -- to the author -- is quite persuasive. A second important and likewise largely success- ful endeavor of twentieth century philOSOphy centered about the publication of Principig.Mathematigg, by Russell and Whitehead. The new logic -- first conceived in Boole's work of the nineteenth century -— was brought to maturity, exhibiting importance and relevance really heretofore un- dreamed of. Pzincipig was a good indication of the genuine efficacy of the formal sciences; the unity of mathematics and logic was established, and the obligation of philosophy to attend to the new logic became obvious. Closely allied to the advances of Bringipig was the work of Von Neumann in game theory.1 With advances largely attendant to the publication of IE M 91:: gm in l9h4, formal techniques found fertile new fields for application. Conceptually rigorous modes of conceiving conflict of interest in widely varied situations were es- tablished, and psychologists, economists, mathematicians, and even a few philosophers became excited about decision theory, utility theory, and the comprehensive schema of game theory. Interest has blossomed in this direction -- questions of decision and utility theories have shown them- selves involved with topics in all areas of the social, phy- sical and formal sciences. lAlthough The Theory of Games gag Economic fie: havigz (Princeton, 19337 is usuaIIy referred to as the definitive publication of von Neumann on Game Theory, his first publication,"2ur Theorie der Gesellschaftsspiele" (flatnematische Annalen, 100, 1928) contained the conceptual essentials of the later presentation. 5 A very important development of the century was the rise and -- if I may say so -- the fall of logical positivism. Related, no doubt, to the publication of Principig, the efforts of the Vienna Circle to clarify philosophical problems gradually culminated in a purge which threatened to legislate traditional philosophy into meaninglessness. Pragmatism learned from the adventures of positivism, being warned of pitfalls and encouraged to the exploration of new areas of relevance. Principia Mathema- tigg provided the strong skeletal framework of an effica— cious formal apparatus for the expression of positivist doc- trine, and the remarkable advances of science provided a strong set of arguments for the positivistic bifurcation of knowledge -- the factual and the formal. So many of the problems of this century have cen- tered about the positivistic adventure -- its birth in the troubled reactions to Hegelian and British idealistic phil- osophy and the need to account for the importance of scien- tific knowledge, its rise with the appreciation of the im- portance and viability of a linguistic approach to philosoé phical questions; and its eventual demise at the hands of its own techniques and practitioners2 -- that the drama of 2It is noteworthy that the most telling argu- ments against the positivistic theses were those of the positivists and their followers. I should think that one of the most admirable facets of the positivist movement is that it showed itself to be largely self-corrective. 6 positivism becomes, in large part, the drama of twentieth century philosophy. Because of this central position of positivism in the development of the philosophies of mid- century, and, therefore, in the formation of the author's heritage, we commence the thesis with a brief sketch of one tenet of positivistic thought. This tenet, it is felt, is a vital one; upon its success or failure hinged the success or failure of the positivistic constructive endeavor. In our examination of this theme we hope to show how positivism as it became more and more sophisticated finally faced the inadequacies in its own structure. In the resolution of these inadequacies the advances of the century become in- tegrated, and analytic philosophy progresses toward maturity. This thesis is intended to be a tentative step in the direction of that maturity. It is an attempt to utilize some of the formal tools which have become so ex- citingly effective and to make that utilization in self- conscious awareness of pragmatic developments in value theory and epistemology. The tenor of the thesis is prag- matic, insofar as pragmatism recognizes that human values depend upon human purposes. It is empirical where empiri- cism requires that philOSOphy be aware of human experience; and it is formal in that it makes free use of formal techniques. 2.1 Whitehead once remarked that every school of 7 philosophy has two major exponents; the originator of the abstractive scheme which is the mark of the school, and a final exponent who universalizes the scheme. PhilOSOphical positions are usually originated with the elucidation of specific problems in mind, it is at the hands of the con- summator of the scheme that it is stretched to fit the parts of philosophy for which it was not intended: The result is a :eductig gg.gb§ggggm, so to speak, which renders the scheme complete, makes its failings patent, and sometimes makes evident the total inapplicability of the scheme to problems of philosophy. Rudolf Carnap has certainly performed the func- tion of concluding exponent for positivism. It was at the hands of Carnap that the detailed linguistic bias of posi- tivism was extended into the areas of value theory and ontology; the extension into value theory came with the emotivist movement in ethics of which Carnap -- if not the Sophists -- was the foremost early propounder; the exten- sion into ontology will interest us in the next several sec- tions where we shall discuss Carnap's efforts to preserve a united front to varying problems. The inadequacies of positivism became evident in these attempts at universali- zation of what had been a seemingly tenable epistemological thesis; we now praise positivism and the early positivists more for the tenor of their efforts than for their specific conclusions. It is the ontolOgical extension of positivism which first interests us here. The early positivists had maintained that all cognitively meaningful discourse was composed of factually (scientifically) determinate or logically determinate statements. Any expression, the truth value of which could not be ascertained by scientific or logico-mathematical means was declared to be at best omotively meaningful. Such statements conveyed information about no more than the speaker's 'attitudes', and said nothing 'objective' whatever. .As a result of this legisla- tion, all cognitive enterprises were either scientific or logico-mathematical. Science investigated the truth sta- tus of synthetic statements, and logic and mathematics in- vestigated the truth claims of analytic statements. 2.2 This bifurcation was at the heart of the posi- tivistic thesis. Given any statement, one was supposed to be able to assign it to the prOper area for investigation. The business of philosophy was, for the most part, properly to assign statements to mathematics and the physical sci- ences for investigation. As positivism gained more momen- tum, dependence upon the tenability of the analytic/synthe- tic distinction became more and more evident. Attempts to formulate adequate meaning criteria all postulated the di- chotomy, and the attacks on traditional metaphysics and ‘Value theory were largely implementations of it. Such an important axiom merits serious scrutiny, and analytic philosophers commenced an exhaustive examination. The efforts‘were largely directed towards making the distinc- tion clear; attempts were made to define 'analytic' and isynthetic' through extensional logical techniques; These attempts culminated in an article by Quins in 19513, which constituted an indictment of the distinction on the grounds that ...for all its a priori reasonableness, a boundary between analytic and synthe- tic statements simply has not been drawn. That there is such a distinction to be drawn at all is an unempirical dogma of empirifiists, a metaphysical article of faith. Quins, in this article, examines the possibility of found-’ ing the distinction in a notion of synonymy which, he shows, is at least as unclear as the distinction to be justified. He then examines semantical rules as possibly justifying the distinction, and concludes that this foundation too is unsteady. It is Quine's thesis that the status of many statements as 'L-true' depends in large part upon quite arbitrary conventions. Statements which are L true in one language might very well not be so in another language. He maintains -- on this basis -- that the supposed dichotomy is more of a continuum and, as such, inadequate as a basis of division of knowledge. Bwiiiiard V. Quins. "Two Dogmas 01‘ EmpiriCism". Ph' oso hical Review No. 60 (1951), 20-43. ‘ '- Ibid., p. 37. 10 2.3 Carnap's reply to this challenge was not an ' attempt to reinstate the dichotomy, it was rather the for- mulation of a new dichotomy which had its basis in another facet of language than the semantic dimension. The reply was phrased in an important article, Empiricism, Semanticg and Ontologys. ‘ When Quine pointed out -- in.2gg Dogmas Q; Empiricigm -- that statements which are L-true in one language might very well not be so in another language, and that the question.of 'what was logically determinate depended in large part upon what language included the con- sideratum, Carnap was forced to consider at least some of the questions as cognitive which he had formerly relegated to the realm of the non-cognitive. In order to make this consideration possible and not relinquish all of his posi- tivistic tenets he constructed a new realm of discourse. What had formerly been known as cognitively meaningful would henceforth be known as theoreticgl discourse. Those non-theoretical questions which were admissible -- under the new aegis -- to serious discourse, were to be known as practicgl questions. As specifically relevant to ontology, questions of existence which could be considered within a 5Rudolf Carnap, "Emoiricismt_3emantics and Ontology" (Rgvue Internationalede Philgggphie, XI (1950), 20-2}O . I ll given language framework were to be considered interggl theoretical questions; questions about the existence of entities, say, in the universe of discourse of a given language framework, or existential questions about the sub- ject entities of the language which were not answerable within the language, were called gztgrngl practical ques- tions. To ask if unicorns exist is to ask a question which is answerable by empirical scientific means -- one looks about the world for evidence before replying. To ask, on the other hand, if physical objects exist is to ask a ques- tion which is not so answerable -- it is a practical exter- nal question. To ask if there are prime numbers greater than 100 is to ask a question, says Carnap, which is logi- cally determinate as to reply, while to ask if numbers exist is to ask another practical external question. This distinction has an initial intuitive plausibility to it, but upon further examination we see that the difference is not at all as clear as it would at first seem to be. To say that: 1. There are prime numbers greater than 100. is L-true is to say that its truth is ascertainable by re- ference to the semantical rules of some language. In this case by reference to the semantical rules of formalized number theory. But the semantical rules of formalized number theory would never advise us of the truth of (I). To be sure, from the axioms of number theory (1) may be inferred, but then we have shown no more than that 12 2. If there are numbers (i.e. if the Peano axioms are satisfied) then there are prime numbers greater than 100. is L-true, and the L-truth of (1) depends upon the L-truth of 3. There are numbers. which is precisely the external practical statement the truth of which is not theoretically determinable. This being the case, we can see another chance to make the distinction: Une might say that 'Are there numbers?' is a pggg; question to 'Are there prime numbers greater than 100?‘ in the sense that the L-truth of (1) is inferrable from the L-truth of (3). That is, if (3) were L-true, then (1) would be L-true. To be sure, this is nothing like the iron clad distinction originally claimed, but let us nod provisionally to it and grant Carnap the priority (in the sense outlined) of (3) to (l); similarly the priority of ‘ 4. Physical objects exist. 5. Unicorns exist. Carnap thus maintains that the question of the existence of numbers is not one which is asked or answered by mathematics or mathematicians. It is a question, he says, which has historically been a peculiarly philosophical question. It is a question which has been asked and 13 answered differently by realists and idealists throughout the centuries. And, Carnap explains, the roalists and idealists have been confused as to what sort of question they were asking. The question is not a factual question, it is a question of neither logic nor science; this is indicated by the lack of agreement as to what would consti- tute evidence in support of a reply. The question, accord- ing to Carnap, is actually a question of the utility, of the adequacy, of the aptitude of a given language framework relative to our purposes in communicating about a given universe of discourse. One might say that it is a question as to whether or not we decide to discuss the world in such a way that the universe of discourse in question becomes a category of our la1guage. ,When Carnap says that numbers exist, he means that the language of number theory results in a categorization which is largely congruent with his purposes of discourse. When he says that physical objects exist, he is saying that the thing-language affords an ade- quate categorization of the mass of percepta which confront us in common experience. After outlining this foundation of the external- internal dichotomy, Carnap goes on to alleviate the sting of the old positivist legislation: Although only internal questions can have direct cognitive meaning, external ques- tions may be indirectly cognitively meaningful; they are lh pragmatic questions -- questions of the utility of an entire language framework. His conclusions from this are not un- expected: ...the decisive question is not the alleged ontological question of the existence of abstract entities, but rather the question whether the use of abstract linguistic forms, or, in technical terms, the use of variables beyond those of things or phenomenal data, is expedient and fruitful for the purposes for which semantical ana- lyses are made, viz. the analysis, interpretation, clarification, or con- struction of languages of communication, particularly languages of science. and then in more comprehensive criticism of Quine's position; The (nominalistic) critics will have to show that it is possible to con- struct a semantical method which avoids all references to abstract entities and achieves by simpler means essentially the same results as other methods.7 A large part of what Carnap accomplished was to prescribe a different usage for 'exists'. He attempts, fbr perhaps the last time, to sidestep the central ontological question; "What exists?", not by directly denying the mean- ingfulness of the question, but by assigning a new meaning 'to 'exists' and requiring that the question be answered as :if it were "What are the universal categories of languages?" ‘ lfe might agree with much of Carnap's thesis, but as he ‘ 6Ibid., p. 39. 7Ibid., p. 40. 15 evades ontology with his semantical footwork, we still want to ask the same old question, and we find that this is pre- cisely the question that cannot even be asked in Carnap's new vocabulary. A 8 2.4 In a reply to Carnap's article Quins rephrases Carnap's external/internal distinction as a categozysfigpr glggg distinction. He then points out9 that Carnap accepts his standard for judging whether a given theory accepts given alleged entities, i.e. The test is whether the variables of quantification have to include those entities in their rangs in order to make the theory true. Questions of category, says Quine, are questions of what alleged entities are included in the range of the category variables (i.e. the variables of the broadest range) of a language. .Questions of sub-class are questions of what particular entities of a given species or sub-species are included in the range of restricted or 'limited' variables of a language. Assuming this to be congruent with Carnap's treatment, Quine points out that there remains no definite L 8Willard V. Quins, "0n Carnap's Views on Ontology" Philosophicg; Studies, II (1951), 65-72.- - 91bid., p. 67. 101bid., p. 69. U 16 distinction between category and sub-class questions. 'Are there prime numbers greater than lOO?' is a sub-class ques- tion when asked internally to the theory of natural numbers; when asked, in other words, in number theory where 'n’ stands for 'number'. It becomes a category question, how? ever, when asked in a language in which the variable of broadest range is 'p', where 'p', stands for 'prime number greater than 100!.' if we cannot tell from examination of a question alone whether it is a category or a subclass ques- tion, (ergo whether it is an external or an internal ques- tion) then questions are only internal or external relative to a given language framework. But this tells us no more than we already know from the rules of an ordinary quanti- fied logic. When we introduce a limited variable into our language, we do so usually because of its categoricity rela- tive to a given universe of discourse. This saves us the trouble of additional notation and simplifies our deductive procedures considerably. The number theorist has no need for a variable which includes anything in its range other than numbers. It seems that Carnap's attempt to ignore ques- tions of ontology is an abortive one. Though we can sym- pathize with the positivist's desire to avoid the tangled confusions which frequently accompany metaphysical dis- course, we still maintain that he is not exempt from con- sidering such questions if we choose to ask them. We can l7 fairly require that positivism, shown to be possessed of a demon, must either go into quarantine or suffer the rites of exorcism. The legislative act by which traditional meta- physics and value theory were ruled meaningless is now invalidated on the grounds that the legislation made a false and relevant assumption. 3.1 We have noticed that in an attempt to extend the scope of positivism so as adequately to answer ontological questions, a basic flaw became evident in the positivist scheme. The early positivistic legislation became invali- dated when it was shown that the analytic/synthetic dis- tinction was inadequate as an exhaustive and exclusive characterization of meaningful discourse. As a result of this the indifference to metaphysical problems which had marked linguistic philosophy from the time of the Vienna Circle became no longer conscionable. It is not beside the point here to consider the effects of this revelation in another part of the philosophical universe -- to con- sider its effects in particular on theories of value. As the early positivists were indifferent to questions of ontology, so also were they indifferent to questions of value except insofar as predications of value were indicative only of the Speaker's attitudes. Such a 18 value theoretical position might be characterized as emp- Lgxiégpll Positivism and emotivism complemented each other quite well: A positivist could not but be an emotivist, and most emotivists were of an otherwise positivistic per- suasion. Emotivism provided a completely non-cognitive esthetics and ethics, and positivism provided a completely non-emotive logic and epistemology. Because of this con- geniality, the troubles of emotivism are the troubles of positivism and vice versa. If Carnap's external/internal dichotomy is assumed to hold universally, then any expression must be either external or internal. Expressions of value are ob- viously not L-true, and if they were synthetic-internal they would be confirmable through the procedures of some empiri- cal science, and this -- the emotivists assumed -- was not 12 Thus all questionsof value were external the case. questions, and, as pointed out above, all external questions were questions of value. Science, according to this llSee, e.g., C. L. Stevenson, Ethics and Languagg (Yale, 1944) passim, but esp. chap. iv. 12Stevenson, for example, though maintaining that scientific knowledge is in some way contributive to ethical decisions, says "...the task of selecting from the stores of knowledge and bringing together the information that bears on a specific moral issue, is one to which scientists do not address themselves." Ibid, p. 331. l9 viewpoint, patrols only the precincts of the internal, theoretical, non-value decision. The destruction of Carnap's dichotomy disturbs this peaceful picture of delegated authority: If any ques- tion can become external or internal by an extension or restriction of the range of variables, then value questions need not be external in nature -- unless a question changes from a value to a non-value question as the styles of the involved variables change. It would seem fairly obvious that to make such a demand is to play havoc with the way hi which we use logics: When we change the range of a variable, we do it for convenience and ease of deduction. The ques- tion, too, of the 'neutrality' of logic should be considered. Whether we can have metaphysically and ethically neutral logics or not, it is surely desirable to avoid the complete entanglement which would come about if valuings were con- sidered incident upon the ranges of variables. It would be somewhat awkward if one w re forced to pause whenever he changed a variable's range and survey which predications he had made evaluative and which non-evaluative. The only possible support from the value-theoretical side of the issue would include the establishment of a thorough and ex- haustive means/ends distinction, thus building the required categoricity into the value realm instead of the fact realm ~- questions of ultimate ends to be external while questions of means to be internal. But this is surely unsatisfactory, 20 and it is evident that such a dichotomy would run into at least the same difficulties 33; value distinction as it does‘qgg fact distinction. 3.2 If emotivism as an ethical position proves inade- quate, we question its suppositions. Is it feasible to con- ceive of predications of value as indicating no more than the speaker's attitudes? Svidently not, since the assumed dichotomy between the cognitive and the non-cognitive is not at all clearly established. We must look for other ways of interpreting evaluative discourse and hence for a different set of ethical suppositions. If one cannot satisfactorily account for values as arising only from attitudes, then he must account for them as issuing from some portion of what is undergone by the organism without prejudging that they arise from this or that portion of what is undergone to the exclusion of other portions. He are requiring an extension, a broaden- ing of ethical concepts: A factor in this broadening is the discarding of the attitude/belief distinction and the resulting attention to an unbifurcated experience in an attempt to discover those facets of human experience which have most to do with values and valuings. The value theor- ist cannot ignore human purposes and the goal seeking character of human endeavors. He might find more or less obvious ways of describing these purposes, but he cannot 21 ignore them in founding a system of ethics or esthetics. Such vague remarks cannot characterize a position, they can serve at most as a partial criterion of theories of value in general: Such theories as would be congruent with the criterion, I should call naturalistig. But to give a name to a class of ethical sys- tems is not to solve problems of value theory. .It is not a part of the task of this thesis to formulate a theory of value, so we are to some extent free of the difficulties of formulation: It is however a part of the task of this thesis to make specific remarks about values of a restricted sort; so we are required at least to comment on what sorts of theories of value would be needed in order that our re- marks be coherent with respect to some more comprehensive position. Our reference to naturalism is intended as an indication of such theories. We construe naturalism, broadly, as that ethical position which requires that all values be related to felt value. It might be noted that this requirement rules out transcendental schemes of value which assume the origination of values in some extra-ex- periential source. In this ruling out, however, we are anxious not to involve ourselves in the solipsitic pre- dicament of emotivist or extreme relativist ethics. We should also require that naturalistic value theory make 22 possible empirically meaningful predications of value.13 In this requirement is presented the problem of reconciling 'subjective' human experience with 'objective' discourse. ind this problem, we feel, is not far disParate from.the epistemological problems encountered in explicating know- ledge of a less valuative sort. 1 When the problem is phrased in this way, the relations of science and values become obviously important. Naturalistic value theory must make provision for the util- izatibn of scientific itechniques and conclusions. This should lead to an attempt to provide a science of value; which science is not restricted a priori from any meaning- ful investigation. If we can agree that the scientific method -- a method perhaps characterized as well by intent as by procedure -- constitutes the most adequate available mode of prediction and can bring about accurate control, then we should be eager to admit investigation of a scien- tific nature as the ultimate justification for value theo- retical conclusions. I 131 am aware that such a requirement invites accusations of falling into the so-called 'naturalistic fallacy'. I am not at all sure, however, that this pur- ported fallacy has ever been explicated clearly enough that one could tell just what he would be doing were he committ- ing it. It would seem that a careful characterization of ‘the naturalistic position allays any danger of inherent lfallacy. See 0. Lewis, An Anal sis of Knowled e an fiflaluation (Chicago, 19h5) pp. 405 ff; 23 It would be erroneous to interpret the preceding remarks as implying that the task of value theory ig the task of a science of value; to identify the two would be analogous to identifying metaphysics and science in general. To provide a foundation for a science of value is -- as here intended -- to become concerned with the efficacy rather than the efficiency of the science. It is surely the case that con- cern with efficacy cannot be divorced from concern with efficiency, but this is not to say that efficacy cannot be distinguished from efficiency. One might have a very effi- cient science of prediction and control which was only tri- vially efficacious: If the theories of the science in terms of its results have little relevance to the course of ex- perience, then the science is unimportant. An efficient science still has need of justification beyond its effi- ciency, and this justification can only come in indication that the efficiency of the endeavor is relevant to and in- volved with experience. To show this relevancy is to argue for the efficacy of the science, and to neglect it is to encourage the pursuit of nebulous and trivial goals. What is efficacious must be efficient to some extent, and it is a part of the task of this thesis to show that at least some of what is efficient muSt be efficacious in order that it be efficient. But an index of efficiency is not indicative of efficacy, nor does efficacy belie any certain degree of efficiency. 24 Let it be said, then, that naturalistic value theory involves itself with founding a science of value. Another involvement of such theories of value is with decision. The pragmatic requirement that values must be felt values immediately requires that a theory of value pro- vide some methods for choosing to feel one rather than another value. The recognition of the impossibility of un- wavering pursuit of one ultimate goal of activity should carry with it recommendation as to how to direct activities through the manifold of values that are encountered, formed, accepted or rejected. To call x good is not very meaning- ful unless one decides to pursue x or to prefer x to y, It would seem equally obvious that decisions, or the manifes- tations of such preferences, require references to values in their justification. Insofar as the activity of science is decision activity, scientific procedure requires valua- tion and appraisings as well as more directly describable cognitive processes. If the scientist decides to accept a hypothesis, he is deciding, e.g., that the risk of error is not as great as the cost involved in further investigation.14 t is at this point that our thesis commences its investi- gations: We shall attempt partially to analyze this valuational involvement of science, making some recognitions 1“The whole develOpment leading to this conclu- sion is an elaboration of the views expressed by Richard Rudner, "The Scientist qua écientist Makes Value Judgments" !Philosophy of Science, vol. 20, no. 1, 1953) ~ of purposes and the experiential bases of values. 3.3 If science and value theory involve each other a outlined above, then a part of the task of value theory U) is to explicate the valuings of science. Insofar as these valuings remain tacit, the value decision of the individual scientist need not be congruent with the purposes of sci- ence, e.g. as an institution or even with the purposes of the scientist himself. To the extent that the scientist identifies his goals with the goals of science in general, then to that extent will he need to recognize these goals so as to make their joint pursuit feasible in the scientific endeavor. Some attempt must be made to decide if and how the scientific decision-situation differs from the non- scientific decision-situation: we all make decisions con- stantly, of varying importance and accuracy. Are there any features of the scientific decision situation which incline it to result in more or less adequate decisions? And, right on the heels of this question, What is an adequate decision? Such questions raise a plethora of related ques- tions and invite a comprehensive characterization of values and decisions. duch a digression would be a luxury which we cannot afford. Je should like to establish more firmly the direction of our inquiry by the introduction of a deci- sion paradigm and a schematic arrangement. This paradigm is intended to set the “enor of the thesis; it should point to the problems and the methods with which the remainder of the work will be concerned. 4.1 A woman goes to buy thread to sew a dress. Upon reaching the store she discovers that she has neglected to bring along a sample of the material of the dress with which to match the thread. It is a long trip back home, and she needs tne thread as soon as possible. Upon examining the various colors of thread in the store she finds that tnere are several which approximate to the color of the dress. She must depend upon her memory of the color of the dress to advise her of the proper hue to choose. If she is right, if the dress and the thread are of exactly the same color, she will be rewarded by having the proper materials for her task, with the eventuation that the finished dress will not be marred by a clash of colors. If she should be unsuccessful in her choice to the degree that the dress would be severely marred by utilization of the thread, she is 'punished' by being forced to make another trip to the store -- with the resultant delay in the completion of the dress or by an unhappy eventuation of her labors. If she buys several spools of thread, her chances of obtaining a matching color are increased with every additional Spool she purchases. If she buys a great quantity of Spools, she can be 'practically certain' of obtaining a matching color. Before our distraught shepper selects which and 27 how many spools of thread to buy she weighs the consequences of the alternatives open to her: Consider that there is a quantitative index for any alternative purchase (of a num- ber of spools of thread) which is formed as the product of (l) the probability that she will be successful with the given selection,15 and (2) the 'rewards' of successful choice. Call this index the utility of the choice. 'We might then evaluate the utility for every possible selec- tion. dome simplifications are in order: Let the woman Specify one spool of thread which she feels most strongly is the prOper spool, the n as she increases the cardinality of selection sets, the 'prOper spool' occupies the approxi- mate midpoint of the color set represented by each selec- tion set. As the cardinality of the selection set increases, the probability of mistake decreases, and the negative utility of the spending of the price of selection increases.16 The increased certainty obtained by adding one spool to the selection set is smaller as the selection set is larger, 15For the purposes of this illustration ’proba- bility' is to be taken to mean Simply ’probabilit of occurrence with respect to available evidence'. t is felt that comments upon interpretative procedures with respect to the calculus of probability would be confusing and unne- cessary at this early and explicative stage. These questions will be treated in some detail in chap. II, infra. 16For the purposes of the present illustration 'negative utility' is to be taken in its most obvious intui- tive sense. Once again we refrain from refinements for rea- sons of clarity. The reader who is troubled by such assump- tions may look ahead infra., chap. III, pp. 94-96. while the increment in the negative utility of loss of money which is brought about by the addition of one unit is constant.17 The utility of the money loss may be represen- ted by the number of spools purchased multipled by minus one. We assume that the probability of money loss is al- ways 1. In this representation the probability of a mis- take times the value of a mistake added to the probability of cost (always one) times the value of cost gives the utility of the selection. 0r, where 'gm' abbreviates 'Pro- bability of mistake when set i is chosen'; 'hg' abbreviates 'value of mistake‘. And where 'Pc' is 'probability of cost' and 'gc' is 'value of cost of set i', the utility of a set,. i, is represented by U (i) = (Rm x Em) + (Pp x Vp) 1 1 1 1 Since we have specified that the probability of cost is always 1, we may simplify as follows: (i)= (Rm) (Vm) - (N'i) where "N'i" represents the number of items in 1. Then the 17.It is assumed here that the utility of money is linear with respect to money, i.e. that the numerical quantity of an amount of money is indexical of its value. We are aware that this is not the case, but recent investi— gations Show it to be sufficiently approximate to truth to make the assumption innocuous in illustrations. See Donald Davidson and Patrick Suppes with Sidney Siegel, Decision ggking (Stanford, 1957). 29 choice Situation might be viewed as follows, with appropri- ate probabilities invented fer the example.18 Let the value of mistake be -20, in units corres— ponding to the cost of'a spool of thread. Then utility is maximized19 by maximizing the function U (i), where U (i) = -(§h) (20) -(N'i) The following table then represents the choice situation: number of probability utility utility spools of of of purchased mistake mistake selection 1 .6h - 12.8 - 13.8 2 .32 - 6.A - 8.h 3 .16 - 3.2 - 6.2 4 .08 ~ 1.6 - 5.6 5 .Oh - 0.8 - 5.8 6 .02 - O.h - 6.h 7 .01 - 0.2 - 7.2 5.1 The above schematized choice situation makes available information about several quite relevant facets of any situation in which a choice or decision is to be Inade. We see that there is a state of affairs which dic- tates what might be called the 'inherent logic' of the _ 18The probabilities are invented solely for JLLlustrative purposes. 19Cf. note 16, ggng, 30 situation:20 The color of the dress is just what it is and it is going to be matched by the color of one of the spools of thread or it is not going to be so matched. The decision that the woman makes awaits its ultimate evalua- tion until the color of the dress is compared with the color of the thread selected. Notice, however, that the decision might be capable of appraisal in other respects than this ultimate one of weighing its consequences. We might very well make an evaluation which took into account (1) the purposes of the subject, (2) the information available to the subject, and (3) the probability on the basis of (2) that the chosen activity will bring about (1). If wewere to order alternative decisions in any given situation using the value of (3) associated with each decision as its index (this would be analogous to evaluating selection sets in the paradigm according to the 'utility of mistake' associated with each selection) then for any pair of members of the array we could specify which of the pair was 'more likely'21 to eventuate in the desideratum. This evaluation would be independent of the actual eventuation, and it would make ZOThis phrase is introduced by R. B. Braithwaite, The Theor of Games a Tool for the Moral Philoso he (Cambridge, 1951). 21Actually the relation which would be established 633 ordering the field would be 'is not less likely to even- ?Luate in the desideratum than'.~ For obvious gain in clar- irty'we use a locution which suggests the stronger and, I think, not establishable relation. 31 perfectly good sense to say that, although the decision eventuated unhappily, it was still the best decision in the sense that it was most likely to bring about the desideratum. The attempt to evaluate decisions in this manner, though, would surely end in confusion. Many of the arrays would be either incomplete or infinite, since, for almost any desideratum act and information set, there is some pro- bability that the act will eventuate in the desideratum. Further, we can almost always specify some act such that it is more likely to bring abaut the desideratum than any act included in the array. dotice, in the paradigm, that as we increased the cardinality, we decreased the probability of mistake. It is obvious that the probability would continue to decrease with increases in the cardinality, and it could so be constructed that the probability would approach zero as a limit when the cardinality grew very large. In point of fact this is just the situation, and we are forced to consider alternatives which are less likely to bring about the desideratum than other alternatives as being more worthy than the latter alternatives. The difficulty is alleviated when we consider .probable effects of an act other than the desideratum and assign probabilities and values to these acts as well. Then, in? for each effect there is assignable some function.which idadicates the desirability or value of that effect to the subject, we might conceive of categorizing eventuations in terms of the net values of their results. This, gener- ally, is the program of theories of utility -- to design some format whereby eventuations can be assigned relevant end comparable indices of value. t is obvious that any attempt to evaluate decisions as regards their likelihood of achieving desiderata is going to have to consider some assignment of indices to contemplated eventuations of these decisions, where the contemplated eventuations will include some eventuations other than the desiderata. It is this IOCOQHitiOH of the necessity of considering the undesirable effects of acts which bring about desirable effects that renders utility theorj feasible; increase in info-mation iris giflfi an. proballlity of acting t3 arin; about the host -igujesus results. One might be teupted, at this juncture, to say that a subject on ht to do that act whicn will so eventuate: that, to refer to the paradigm, the woman ought to choose fair spools of thread because by so doing she would maximize ”w“ utility. Dome reflection, however, convinces rs that use of evaluative langwage at such an early stage is pre- ‘ ‘ .-. -. .. 3*. -. w I'. v ‘ .-¢--. 1‘1 4-}\' - ,h . , " - I, -' - v (1. all... .Lxl b-1C lbbue ab regards UUK‘J f5luVGdCO 3:. .uu'Cil iOl‘du' x) k) C_.'. schemata Seneraily: why ought she? Dhe oujht to only if U 2:: accepts that systenatio notions of value, utility, pro- T,c\‘:.21.;«- .4. ,.. 'g'-‘,.‘.,.4.. --,_,. - '. _ "’CAX.).L—L.-EJ', 89C. al‘e 0.1....LJ.C...13'.iler lQJIJOI‘piilc hilt”; 11k]... ,2le- " --’-.- '- I', -' ‘x" -'. -- . ' 1,, -' 3, ' — --“ 7' ,. 1- .~ -' -- ,\ " , ‘J-J LudhmJUfic llb‘uu'J.l:J 0L L‘BCLS—Oll 3.1L- ul‘JJ-li.) LIC‘ JOLLV'L1;C‘C3 11b]... 33 that such Systematic conclusions reveal her real wants. It is at this point that the persuasion tends to become viciously circular. "Why", we Should say, "If you will grant that our definitions are justified, then you must accept our conclusions." find, of course, our argument for acceptance of the definitions is that by utilization of them we arrive at conclusions which are presystematically de- sired. What we are obviously talking around is the dis- tinction between normative and descriptive discourse. 5.2 To offer a normative theory of decision making is to involve oneself in the circularity indicated above unless such a theory is accompanied with empirical evidence which indicates the relevance of the decisions and conclu- sions to actual choice situations. An adequate descriptive theory would doubtless form the best normative theory; it would provide evidence that certain courses of decision activity resulted in specifiable eventuations, it would supply us with data to enable us to forecase utility yields and other types of rewards, and in so doing would instruct us as to maximal utility choices. The instructions of such a.theory would be prOper to the extent that its axioms and definitions were indicated to be applicable. The point here is that any noraative theory of decision making would 13s descriptive to the extent that its advice was well , 22 founded. To offer a descriptive theory of decision making would be to offer -- anon; other things -- a means for pre- dicting how subjects would make decisions when faced with various choice situations. The argument for acceptance of such a theory would have to include empirical evidence that, at least, the axioms were satisfied in some cases. One interesting feature of such a theory is that it would not be drastically different from other theories as regards the sorts of observations which would be admissible as confirma- tory of its hypotheses. Assumedly what psychology will and what it will not admit as evidential behavior on the part of subjects is not going to undergo important modification for the phrasing of a new theory. What will be different, how- ever, is the set of theoretical higher level terms which would characterize such a theory. 5uch theoretical terms would function in Campbellian hypotheses, in statements having to do with the measurement techniques of the theory, and would generally perform the task of knitting the lower level observations into a coherent conceptual whole. If such a descriptive theory is to become possible, it will be possible insofar as there are meaningful theoretical terms 22A very interesting and largely successful attempt to demonstrate the applicability of decision theories is characterized by DaVldSOH et. 81-: 0 - Cit- 35 available which can perform this novel relational function among the established types of relational procedures. In the present thesis some argument is made for the selection of such terms as 'utility', 'importance', etc. as adequate theoretical terms in a descriptive theory of decision mak- ing. The conclusions of this thesis will be directly rele- vant only to the program of a descriptive theory of scien- tific decision, but the indirect relevance to any theory of decision making should become evident in the phrasing of the prOposals. The current thesis, since it does not support its conclusions with direct empirical evidence, cannot pur- port to be a theory of decision making. The point of the preceding remarks is that if there is ever to be an adequate theory of scientific decision making it will require some preliminary explication of concepts and theoretical terms so as to permit the introduction of the novel framework into scientific discourse. There are available empirical theories which utilize such terms as mentioned above, but to my knowe ledge they are almost all in economics or allied disciplines and concern themselves with monetary indices of value and utility. Such empirical endeavors are, indeed, invaluable. 'The salutary success in prediction and explanation resulting :from utilization of these theories is quite sufficient evi- (:0LCG of their worth. What is being contemplated here, how- ‘FVer, is the extension of these concepts of utility and 36 value so as to include other than monetary indices within their scope. Some of the ensuing discourse will be con- cerned with empirical work that has dealt with monetary concepts of utility and value, because most of the empiri- cal work in this area has been so concerned. 5.3 The goals of individual scientists probably include not only such desiderata as fame, fortune, personal satisfaction and conformity, but also such altruistic goals as the making available of knowledge and the increased possibility of control over the environment. The altruis- tic goals are more social in nature; many people agree that a particular scientist should st‘ive to make more knowledge available, while relatively few feel that he -- the particular scientist -- should strive to increase his fame. I should speculate that most scientists would prefer to nake tne personal goals subservient to the social goals, though this is at presen an untestable hypothesis. If, however, tnere were made available to the scientist some method for incorporating social goals into his procedures while diminishing the importance of personal goals, then the hypothesis would become immediately testable. 6.1 The hope that such a method become available in the near future in indeed quixotic, but this does not at all change the desirability of possessing a method of 'value inculcation'. If we realize that the scientific decision is a decision of value, then we want our values influencing 37 the decision to be at least analyzable. Once the scientist becomes aware that his value schema has an appreciable effect on the success or failure of his inquiry, then, if he desires success of his inquiry, he will be eager to make this effect beneficial rather than deterrent. What is con- templated here is not some means of forcing the scientist to investigate what society wants him to investigate, rather it would be some means of permitting examination of the implications of the scientist‘s valuings and the resultant adjustment of decisions so atho conform consistently with the scientist's more encompassing values. That some modi- fication in value schemata will be a result of this adjust- ment is of course true, but whatever a value schema might be it cannot be conceived to be static; our values change at least as frequently and as strongly as do our beliefs. It is very infrequently that we modify a 'big' or central value -- most modifications of value schemata occur with minor and unimportant values and then so as to make minor values accord with more important values. And this is re- markably like the situation with beliefs: Meet changes in minor beliefs come about so as to permit the reconciliation of more central beliefs with the world of 'brute fact}. We very infrequently change our central beliefs.23 The methods of science have traditionally made every provision to J"'230f.lQuine,‘Qp;_g;§. and Quine, M thods f Lo '0, (New York: 1959). ‘ . 38 incorporate changes of belief in the structure of inquiry, while there is almost no consideration of valuational changes. So long as it was felt possible to conduct sci- ence without reference to value, this situation was justi- fied; but if we come to the realization that beliefs and values are inextricably related, then we can no longer con- done such indifference to factors which wield so much in- fluence in such a vital direction. The goals of this thesis include, for the most part, a discussion of the relevance of recent work in value theory and utility theory to the philOSOphy of science. More explicitly, the thesis attempts to outline an approach to problems of scientific decision making which would ex~ plicitly recognize the involvement of valuations in such decisions and the efficacy of various formal and quasi-for- mal techniques for analyzing this involvement. An attempt of this nature is almost sure to raise more problems than it solves, and this heritage of puzzles is perhaps what marks the thesis as philosophical rather than scientific; when the philOSOpher answers a question we call him a sci- cntist, when he persists in raisizg uncomfortable problems he is being philosophical. In this first chapter we have attempted to show the involvement of decision with valuation in such a way that no adequate characterization of scientific decisions could omit mention Oi values and valuings. If the reader 39 18 not by now convinced of the ineluctable presence of value in scientific decisions, then the sequel will be not at all persuasive. If he is at least disposed to accept this involvement, then it is hoped that what follows will voke him to pro du tivc thou ht. If one is looking for solutions rather than problems, then he will be disappointed. The second chapter consists of an outline of some contemporary developments in confirmation theor‘. The outline involres itself with some problems of probability heory azxd a tentative position as regards interpretations of probability is accepted, some arguments for this accep- tan e are put forth. A 2.1;tlod of evaluation of evidence is prOposed which depends upon acceptance of the outlined position as regards probability. Some attempt is made to show the relations of various probleu1s in the philosophy of science one to the other and to the problems of character- izing the evaluative cle no: nt in scientific decision. There is disc lesion throughout tie c11apter of difficu ties of measurement in general and of real-valued measurement in particular. The third chapter utilizes the evaluative method prOposed in the second chapter as a tool for ordering hy- potheses according to their values. Some basic vocabulary and concepts of utility theory are introduced and the dis- cussion of value is conducted explicitly in terms of these concepts. In this chapter the goal of the thesis is #0 phrased specifically as acheivement of an ordering of out- comes in a delineated outcome space: After some discussion, of orderings and metricization of functions over the out— come space, such an ordering is proposed and briefly commented upon. The fourth and final chapter of the thesis dis- cusses some of the more significant shortcomings of the ordering preposed in the third chapter. A cursory summary of other work in the thesis area is presented, and some effort is made to comment upon the efficacy of these other approaches. The discussion initiated in the first chapter on the relation of decision and value is resumed in the vocabulary of the second and third chapters, and the con- clusions of the thesis are briefly summarized and discussed. GrinP’i‘Sii II "THE EVALJnTION OF EVIDEI'JCB; A RULE OF LrLEJ"CTION" ‘ ‘ 1.1 it is possible to take the view that an indivi- I G sl wakes a decision when he crystalliz s his attitudes in sich s way that he assumes himself to desire some goal. This crystallization of attitudes -- I think we are aware -- comes about when we rule that the goal seeking activities under consideration eventuate in a net good for us. Such a view involves a 85388 of 'decision' which indicates intro- pective personal activity on the part of the subject. It C) is ineptly described, because it is such a prevalent and acet of hum n experience that it defies F") broadly relevant precise characterization. The point to be made is that be- cause of this ineffability, to use 'decision' in this sense is to render a formalization of decision making at least unfeasible. It would be all but impossible to give any in- tersubjective empirical meaning to this sense of 'decision': One feels that he has decided, and he feels it so strongly. that no amount of evidence will dissuade him from his feel- in; once he has crystallized the relevant attitudes in the manner indicated above. Because of this difficulty with personal decision (a difficulty charactezistic of such subjecti1e states) theories of decision making must usually content themselves #2 with a primitive observable term such as 'exhibits decision behavior'.1 The psychological task then becomes one of attempting to make the systematic meaning of the primitive correspond as fruitfully as possible with the presystematic nearing of 'makes a decision'. The psychologist must test his definition, and if his subjects exhibit symptoms (e.g. telling the psychologist that they did not make decisions under various circumstances) which convince him that the systematic term is violently non-isomorphic with the pre- systematic term, then he so alters his criteria for appli- cation of the systematic term that the extensions become more nearly isomorphic. This is not dissimilar to many other tasks of the scientific endeavor, there is a gradual adjustment of the systematic and presystematic meanings in such a manner that the extensions become as nearly isomor- phic as possible without doing undue violence to either term. In the case of decision making by scientists we shall make an oversimplification, which is justified in the light of the appreciable amount of clarity it pernits, in assuming the adjustment mentioned above to be an easy one. It is surely the case that the scientist makes many hints of decisions; he decides to be a scientist, he le. Davidson et. al., Op. Cit. LL37 decides to investigate a certain field of phenomena, and so on. For our purposes, however, we shall consider only one sort of scientific decision; the decision to accept or re- ject a hypothesis on the basis of specified evidence. We shall see, later, that we cannot consider this decision at all adequately without honoring at least some of the more peripheral sorts of decision, since the hypothesis' acceptance or rejection deperds in large part upon attitudes which are diminished and strengthened by the making of other decisions. This initial simplification, nevertheless, has the decided advantage that clear-cut criteria may be spe- cified for application of the systematic term 'exhibits decision behavior' which seems intuitively suitable for application of the presystematic correlate 'decides'. An index that a subject.X has accepted a hypothesis H is then _X's use of H in prediction. . If the scientist's activity is characterized as behavior in which he accepts or rejects hypotheses, then his behavior is characterized as involving decisions -- de- isions which are in part decisions of value: For when the scientist decides to accept H, he decides that the cost of acceptance is lower than the cost of rejection, or that the rewards of acceptance are higher than the rewards of rejec- tion. The scientist can never be certain that a given hy- pothesis is true or false, but he can and does decide that a probability assignment to the hypothesis is adequate or 44 inadecuate. The task of this chapter and the next is to make explicit tne valuational consideratio s involved in ' this decision: we shall commence by mentioning some impor- tant facets of valuation as directed toward scientific hy— potheses. 1.2 The positive evaluation of a true and accepted hypothesis may be broadly conceived to be a function of its predictive power and as a negative or inverse function of its adverse effect on other hypotheses. Generally pppp and accepted hypotheses tend to disconfirm false theories and confirm true ones, while false and accepted hypotheses confirm and disconfirm inversely. A similar case may be made for true and false rejected hypotheses. Much of the discussion throughout chapter I was intended to indicate the need for eXplicit recognition of evaluational factors unavoidably involved in the acceptance and rejection of hypotheses. Any scheme which aspired to such recognition and which attended only ta the function of hypotheses in direct prediction would surely not meet the needs of sci- ence; indeed, where hypotheses refer directly to states of n tare, we have little or no difficulty in making the de— cision to accept or reject -- we can make it so easily that 1aivc characterizations of science which attend only to this direct relationship of thaery to fact (such as that characterization which was typical of the early logical #5 I positivists)2 can readily persuade us that no complex evaluation is entering into the decision to accept or re- ject hypotn eses. N‘hat is oi inter est to the value theore- tician is the non-direct cas e -- the case, for example, of a Campbellian hypothesis or a law of measurement within a particular theory. when these sorts of a paren ml non-con- fir able statements show themselves necessary for adequate scie nee, the philosopher of science has no real alternative to discovering some way of giving empirical meaning to them. There have, of course, sec n many attempts to stretch the criteria of naive empiricism to provide for the meaningful- ness of theoretical statements of the sorts mentioned above.3 Indeed, the history of the philosophy of science for the past several decades is quite well characterized by reference to the succession of meaning criteria which attempted to egitimize Canpbellian hypotheses and other theoretical statements. whatever the results of this quest, one fact has become increasingly evident: Science is not 2See, for exa: ple, Rudolf Carnap "Logical Founda- tions of the unit*y of Science", International bnc Lclopedia of Unified Science (Chica1o, 1933) T, No. 1. 3See, for e: HOMJle Carnap, "Testability and Mean- in;", Philosophy of :Science, (1936,1937) 3 and a. c. a. Hempel, "Problems and Changes in the Empiricist Criterion of leaning", Revue Interna ionore de PhilOSOphie (l950),h 4.- And Craig, op. cit. *— 46 sing to adapt itself to a naive-empirical view of confir- mation and meaningfulness, theorists of empirical meaning nd their techniques to the procedure u) must adapt themselves 1 o: veriiication WJiCh .re utilized in science. The word ' 1.?4'1 .f‘ Dd Lima '7'1"'r‘-1'.l‘ ° T: ’) '3 *‘oh '7 w ' acapo is use ncic acuiseelj. ne solentist is gdlte sure that he cannot conduct his investigations without using at least some positivistically undesirable statements in his theories; there has been quite some evidence to the effect that no substitute is available for this procedure. rn- , . .F. . _ '1. , ,_. . . ¢ne philosopher 01 sc1ence must either declare his investi- gations irrelevant to the progress or science, or he must be prepared to advise the scientist of ways 01 handling the troublesome statements, instead of merely telling the sci- entist that his higher level hypotheses are not empirically ..1eanin: ul. Jul- . The distinction between higher and lower level hypotheses mipht be characterized in terms of the cost of changing hypotheses. If a very low level statement in a theory (say a direct rep rt of physical phenomena) is varied so as to report difrerentlj, the requirements of modifica- tion higher in the theory are minimal. Frequently no change at all will be reeuired, sin~e the generalizations of the theory will be probabilistic and net prone to disverifica- tion by one particular contr-ry instance. Significant chenres, however, in higher level hypotheses are not so innocuous; the best oi scientific theories display a A7 cohesiveness which prohibits significant change of one Li'he“ level hypothesis without significant change in other higher level hypotheses or sig1i1'ic ant chan5 e in lower level statements. A:1alo:eusly, the ace tance or re ‘ection of a lower level stitcucnt does not require complex and inpor tam t valuations, since the cost 01 error in these cases is small. The diii’ict lty b1: comes evident and vital, h wever, in just these higher level statenents which dis- alay very little direct relevance to evidence: ii are the costs of e1ror are great and the rewards of correctness equally great. what is rurther to be noted is that how the scientist evaluates has an eifect proportional to the level .,. .:. ' r . _ ‘4’. ‘ - ' ._,\ r , '1 " ,_ (1 ,r ('0 _ '1 . ‘W , oi 0J3 geneialization upon ”hat A; declaies to 1e the case, .',.,J-' 4- ‘ ,. ClcdulSo had U) - ~\ ~ 1 -,\.- J— J— - " >2 . ,<. ‘ \_ ,-‘!-. '\ - :' .."\ or, upon what truth claims he Adnpb. Ii tdu . ,.. ,‘ '- Ann? ____'-.°. . _-_ V ,, .1! 1: ,' H '1 ‘ '--fi -1, zo.1e means oi JDULMLULQJ one value 01 ni5hcr leJel hypothe- 0 see, he w ulu nave a tool which would 3reatly 1aeil1tate _. - . “ .. ' ., ..' .!.' . _. ._' r" ‘ ° one picper resolution 01 the {UCSLiOno ine value oe1n3 cen- siae red clai 115 to be henerically inciiaolo, re1usi :13 to be- ‘--z---. ." it" C‘ ,11 ‘1 if v 3"“? '1 ('3-\“‘ 5 '1 .19.." -: ‘fi‘r ._1_ 1.... us.) L, OJ dl-J ...VL..11-- 15;)... “.L ...rlu’vo '_ ' . _ | .0 .-'- " . ,._ o - .0 I _ .JL' ', . _o [‘4 do]. ENC 11.1.5; Lia/l“, O— CL‘LUCl 1C1 OJ. 111133111115 QQQQL"“ v'«‘.,1-a- '- -, . 1- '. - - . .. ~ O-I-' . _ , ,- - '- u;L-_‘.u LO b’JJI‘Cli .LQJ.‘ Jed. ..LC ink. .LCCS OJ. 1‘1ij misses clS f3l'dVeiL-L ul- J—IIA ‘4‘ J-."'\‘ ‘ 3. 4-- \ .,> a ' '1 .\r - 1 -: - r.-.' o0 U11 “ bbclvdiadnu 1.1.1 ..LlpbtlelJ 113 L0 S'JdI‘CU .L.‘i Vain. .1 ' 4‘ r' -- 1. 1' 4‘ J'- ‘ -\ *\-'- F 1'- ~- 1.. o-u ,-. - - u V, I.- ‘ 4‘ ."' - .. UOJoJJLJOlLrJ‘ attaipos up diaper; Dir .'. 11.;3t;-L)QS Oi COllili’ma- ‘" . x‘ ' - - ‘n -: : J- . i . J- ‘. \ W ‘t " n I r ‘, 4“ . . v.‘ ~-. .- u ‘ '1‘ . sion Coilbiudl" one Llnlu 0.- ..L‘JLiilLLl) CO 013 out.) EUCcI‘y wl'lullln -_:' ' , "i . 401. Aempel,"bhan es", op. Cit. ('1 P L) :- O .3 p.) c. in evaluation of meaningful— 5 ness of an entire theory. A. 5. aithwaite describes the which'fiuzlquthesi* Jrocess of theory interpretation and evaluation as a 'zipp- in? up' of the theory -- a orocess in which the lowest level observatian statements 0: the theory aetcr“1*e the inter— wretation to oe pla ced on the hi her level hypot es es. Lraithnaite's view ni3ht he sli3htly amended to maintain that although this is the yrocedure, essentially some ad- J k: Justmant on the meanings o; the lo.1er level ternls is made the interrelations of the hi her level terms. In such a gutually augusting manner is the whole theoretical struc- ture given an interoretation in terms of a broad context of human experience. This attention to the more widespread conception of scientific hoori:s has harhed -- it would seem -- most 01 the success1ul work done in contemporary confirmation theory. Nelson Goodman, in his stimulating book Fact, Fiction, and Foreca§§,5 has outlined an aparoach to UJVOldblua evaliatiOA which eXplicitly invo l as itself 1st only with whole theories, but -- in erfect -- with the H whole hi story 0; ci 1ce. Goodman's results are of suffi- cient relevance ans 1mportance ior our task to merit a summary of his theory of projection: . 5R. B. Braithwaite, Scientific Exolanation (Ca Ibriu“c, 1955), Chap. iii. ’ 6(Harvard, 1955) #9 r1 ‘-* Av ‘\ "l\’\ ' 4". -'. ‘i 5 -. ft ~- '7, - . ~\ uooanan's coac,in with tnc filOJiCfi oi RrOJGCLlOu of inductive logic comma; es Uitn a 9‘szlw1Jnt asaut tnc allaly- r-‘i (V "1‘ ‘ j‘Wl-l Y\r\.C‘f" "l. n r " ”3‘43“! 1“ ‘I‘.‘ y -' ‘3 ' . "21!? 7 yo.” Lxllu 111\3L1.-lil‘i;_) - LA ...LL'VQs) 0.5 ‘uoullUVL J—CsLUL‘u-L CJnKL-LL1-0nbx \J. ll'yl’ _‘--.‘ 3 . .JC' c,,.)~",.a 1.111;”) t ).r;\'l ' «(it C \ n" _, l‘ by baud-ulna,” uv‘vvl u. CAUUV“ UL) 0 LA lUL‘ 8 0L). UCllC‘CUua i.) .. 1‘ J— H F" f1 fl‘lé‘.‘ .‘7 ‘ 1"". .3 _.- 1'] lg " ““ 'fi -",’3 "Y ‘I ‘ '3 ~1r‘ L J- . ....’. UVIJS OJ. ol Luau ... tlzlvb_k.llu.. mauledJS, (1‘10 COLL...) O wile canoluSion Euat tna grolel is -- indeed -- unreasonably than succeed: in Showing that the problem 0f f\ p I t+ F (J H d o (- counterfactuals could be rzsolvcd only if a larger and even more pu mlin; prOslem could be exglicated; namely, the es- tablishnar t of a criterion for distix uisiiin' p J-n L.) 4.. I be - lawl il'e SU acute from non- anlihe statements. In the second wart off Fact, Fiction and Forecast Goodnan faces another V31; closely related problem, that ‘ .fi 3". J. N . 3p7‘-« W,“ : ”w, - a f, - . , ; OJ. 'vr.l1...lC3«L/j11 .nOUI,;J.liL/J’, Jst-Lclt-‘-drljr Ablotidlltf eJ‘CpreSSGCA D“ A use oi tne word 'possihle'. fie renarks that for those who 5‘ o no AJEiOH of gossibility as clear, there Cf . . ‘ . ‘ .~-,»‘ (‘1' '3’: 1 - ’-r" ‘ Cal {.1 UdL-l-UL-L'Ju ”4.461- ‘ {a i MIG Linsclf, nowevor, unable to condone is no problem. 1 such acceptance and is convinced that the notion is one wuicn r quires philosophic clarification. It appears that Lis doutts are well founded, if onl; for heuristic reasons, for he suceeds in showin; tna modality is another of the aulees which would be r-s -lv were a clear criterion of 7Ibido, Chan. 1., pp. 39-4h0 A 50 laulike statements made available.0 What happens, of course, is that the problem of law gains much more importance tnan it night at first have had. We cannot fail to be impressed: Even if we feel that subjunctives are inelininable and that nodal concepts are meaningful without truth functional ex- plication, we must still rJuark sympathetically the impor- ance for Goodman's philosophical position 0 this quite (1‘ central proble: . 'This sympathy entails -— it would 5:. e111 -- a perhaps ;rudgin; admi-sion that a solutian to the now pregnant problem of law would be a quite persuasive argument for th viability of Goodman's whole approach to matters philosophical, or, at least, to matte~s scientific-philoso- phical. The third part of Goodman' prObfalfldulC work is an examination of the problem of induction with an eye to re fo; mulation, if such be permitted. He commences by re- marking on the remarkable persistence and difficulty of the problem of induction; indicates that his own difficulty with law is immediately relevant to the traditional problem, and sets about his task by inquiring 8"Predicates supposedly pertaining to (possible xtiti es} are seen to aiply to actual thin: s, but to have m1 ions related in peculiar ways to, and usually broader han, the extensions of certain manifest predicates. ...“ihe preble:u of dis sisiens looks suspiciously like one e3 tLe shil osenh r's olde st iriends azd enemies. The pro- blem of induction." lbid., pp. Sof. 51 ...what 3r c1 sely would constitute the justification we seek. If the problem is to explain how we know that certain predictions will turn out to be correct, the sufficient answer is that we don't know any such tuing. If the problem is to find some way of distinguishing ante- cedently between true and false predic- tions, we are asking for prevision rather than for philOSOphical explana- tion. Nor does it help matters much to say that we a~e merely trying to show that or why certain predictions are probable. Often it is said that while we cannot te-: l in advance whether a prediction conccihinU a given throw of a die is tr1e we can decide whether the prediction is a probable one. But if this means determining how the predic- tion is related to actual frequency dis- tributions of future throws of the die, surely there is no way of knowing or proving this in advance. On the other hand, if tee jud nent that the pre dic- tion is probable has nothing to do with subseguent occur ences, then the ques- tion remains in what sense a probable prediction is any getter justified than an improbable one Goodnan gee on to remark that what is involved 0') is not the wholesale justification of all inductions, but rather a q*est for cri aria of evaluation of specific in- ductions. Une justifies deductions by eference to rules and one should eXpeet that the justification of induction; would be by sinilar reference. The rules themselves are justified by reference to accepted inferential practice. The circularity is virtuous; no specific induction justifies its elf, nor does the class of all inductions justify the 9 . Ibid. pp. 05f. 52 process. WA rule is waended if it yields gn.inference we are unwilling to aecep ; an inference is rejected if it violates a rule we are unwilling to amend." \ This characterization sets the theme for the remainder of tne work; Goodman's concern is to characterize what rules have historically operated effectively. The problem is narrowed when put in Goodman's terminology: To make an inductive inference is to utilize a predicate, which applies and has applied to a certain class of things, in such a way that a claim is made that the predicate also applies to another class of things. This stretching of predication Goodman calls 'prgjegtion' of the predicate. A predicate P is projected when the claim is made that some object which in fact does not manifestly exhibit P is 'P— able', or that under given conditions the object would ex- hibit the symptoms associated with the possession of P. If specific inductive problems may be characterized as cases of,"0ught P to be projected in this case?" Then Goodman's problem -- to establish criteria for inductive evaluation -- is phrased by asking, "In what sorts of cases ought pre- dicates to be projected?" This, obviously, is a case it- self of projection; namely of projection of the predicate iprojected': What things, in short, are projectible? lOIbid., p. 67. 53 The reply to this question is.all but obvious: Those predicates are projectible which have been success— fully projected most frequently. Such predicates, says Goodman, are 'well entrenched', and hence deserving of sci- entific attention. The proposed theory of projection then defines 'Hl is a better entrenched hypotheses than H2' as a func- tion of 'P is a well entrenched predicate'. Very generally, those hypotheses which contain better entrenched predicates are better entrenched.ll In conclusion it is maintained that a hypothesis should be projected if and only if it does not disagree with a better entrenched hypothesis which could also be projected. What Goodman accomplishes in his theory of pro- jection is to lay quite solid groundwork for a criterion of scientific interest; Scientists ggght to be interested in well entrenched hypotheses, and —- in this sense e- well entrenched hypotheses are scientifically interesting. The importance of this accomplishment to the present inquiry lies in the fact that it permits us to consider the values of only interesting hypothesis and to assume that a criter- ion is -- if not available -- feasible for the distinction of such interesting hypotheses. 11The establishment of the function which deter- mines degree of entrenchment of a hypothesis from the en- trenchment of contained predicates is, of course, the task of the theory which Goodman outlines. See Ibid., chap. iv. 54 One of the very central points of Goodman's de- velOpment is the phrasing of the generalization; "A rule is amended if it yields an inference we are unwilling to accept; an inference is rejected if it violates a rule we are unwilling to amend." The benevolent circularity Characterized here is the circularity involved in all sys- tematization -- we systematize because we are concerned with what it is we systematize about. We have a set of be- liefs about some matter, and we want to examine the set with an eye to determining, for example, whether the set is con- sistent, whether it is cotenable with other belief sets, what new beliefs are implied by the set, and so on. We modify the systematization when it tells us that two be- liefs are not cotenable and we believe very strongly that they are cotenable, we modify a belief when it disagrees with a systematization in which we believe strongly. The process is one of mutual adjustment between -- to coin a phrase -- method and content, if such an artificial divi- sion may be appealed to. The point, of course, is that the division is not a division at all. 3.1 One might very well characterize the philosophy of science as that branch of philosophy which is concerned with the problem of induction in some one of its many forms. There is a very strong precedent for the attempts of philos- ophers of science to justify induction, but only recently 55 has pause been taken to ask what would constitute the sought after justification. In asking for clarification, it becomes evident that we would be quite content with a suitable set of criteria for confirmation. We would con- sider induction generally justified if we had some means of knowing when every specific induction was prOperly made.12 The problem parallels the problem of the justification of deduction: we consider that deduction is a justified pro- cedure because we have a set of criteria by which we can evaluate specific deductive inferences. If a deductive inference conforms to the rules of some accepted system of logic, then it is a good deduction. Similarly with induc- tive inference, if an inductive inference conforms to the accepted criteria, then it is a good induction. One does not pretend, of course, that the historical problem of in- duction is thus solved by exchanging one unclear term (' justification') for a set of only slightly more clear terms, ('confirmation', ‘properly made induction', 'accep- ted criteria' } but the intuitive clarity thus gained is not entirely specious: The exchange of terms presents more Opportunities for explication, permits.us, perhaps, more definitely to point out where the lack of clarity is most invidious. lzlt is interesting to note that decision pro- cedures for deductive calculi cannot fulfill this require- ment. Perhaps it is too stringent to require such complete decision procedures for im1_1uctive calculi. 56 3.2 To describe the task as one of seeking criteria of confirmation is to say that what is sought is a set of rules which would establish to what degree a given body of evidence confirms a given hypothesis. The decision pro- cedure, however, need not be one which discovers one value among an infinite set of degrees of confirmation: That would be analogous to requiring that decision procedures for a deductive calculus discover all its theorems. What is re- quired is that, given an assignment of a value to the degree to which a given body of evidence confirms a hypothesis, the decision procedure gives us a means for deciding whether the value is authorized.13 It is in this qualified sense that we can say that a set of rules (constituting a decision procedure) establishes degree of confirmation.14 In the sequel 'probability' is to be taken to mean 'degree of confirmation'. The symbol 'p(H,E)' is read 'degree of confirmation of H by the evidence 3'. Those who are uneasy with such an interpretation of the calculus of probability can be assured that no prohibition of other 13This decree of confirmation is abbreviated by Carnap as 'C(h,e)'. See Rudolf Carnap, Lo ical Foundation of P obabilio , (Chicago, 1950), passim. he shall use 'pIfi,E)' throughout to mean 'degree of confirmation of the hypothesis H on the basis of the evidence E.‘ 141bid., pp. 198rr. 57 interpretations (e.g., limiting frequency probability) is intended. The interpretation as degree of confirmation is chosen herein for the following reasons: 1. Provision of an interpretation which permits the assignment of all real values, ospmps $1 to p(H,E). 2. Provision of an interpretation which expli- citly recognizes the relevance of available evidence to determination of a value for p(H,E). 3. Provision of an interpretation which es- tablishes (in the sense outlined.afove) a value of p(H,E) for every H and E. 5 The prOperty described in (1), above, is suffi- ciently important in the sequel that some explicit comment is required on it: It is, to be sure, an assumption of the schemeof analysis proposed in this and the next chapter that degree of confirmation is a real valued function. One might object that this assumption is unwarranted, that no proof is available for adequate demonstration of (1). Whether this is or is not the case is not the concern of the .present work. What is the concern of the present work re- garding this real-valuedness may be summarized in two state- ments: (a) There is evidence that the requirement for proof 15This third requirement renders frequentist probability inept for our purposes. Frequentist assign- ments are to classes of hypotheses, and we require assign- ment to each pair, (H,E). Cf., e.g., Hans Reichenbach: "The Logical oundations of the Concept of Probability' in Feigl and Brodbeck, pp,cit., pp. 456-u7h. . 58 needs clarification before an adequate proof is forthcoming, and (b) None of the transformations on degree of confirma- tion effected in the sequel impugn this supposed real valued feature of the function. In supposing degree of confirma- tion to be real valued we preserve whatever real valuedness might be possessed by functions of it. As regards the need for proof indicated in (a): It has been shown that (a') If p(H,E), p(H',E') are two real numbers such that v - - p(H,E)>p(H',E'), then if R is an adequate comparative notion of probability, R orders p(H,E), p(H',E') so that p(H,E) is more probable than p(H',E').1 This statement is necessarily vague, since it so much de3 pends on what we accept as an 'adequate comparative notion' of probability. This vagueness points out just the diffi- culty involved in asking for the sort of proof seemingly required by (1). If we were confident of the efficacy of some comparative notion of probability, then we could evalu- ate degree of confirmation relative to that notion, by a procedure similar to that alluded to in (a'). If, on the other hand, we were confident of the efficacy of real valued degree of confirmation, then we could evaluate our more intuitive notions of comparative probability in a like manner. As it stands, the sorry shape of our convictions . _ 16L- J. Savage, EQEQdations of etatistics, New fork, chap. iii. 59 prohibits either procedure, and we must look for enlighten- ment on both fronts at once. The search is facilitated by some such requirement of comparison as (a') to at least some extent. Savage purports to show -- with a considerable degree of success -- that his personal probability is an adequate comparative notion,l7 and he further indicates that it can be the foundation for a metricized real valued probability. The problems which surround interpretation of probability are fascinating, but -- perhaps regrettably -- they need not concern us any longer here. On the basis of our two statements, (a) and (b), we shall assume ourselves authorized to treat degree of confirmation as real valued, with the recognition that demonstration to the contrary would require corresponding modification of our thesis. 3.3 It was said above that the task of confirmation theory was in part to establish criteria which in turn es- tablish to what degree a given body of evidence confirms a given hypothesis, Probability, under this interpretation, is a function wnich assigns a real value to each pair (H,d) where d is some hypothesis and E is a set of evidence state- (D nts. V‘ 11'. 1 7Ibid., p. 32. 60 Thus, suppose E is construed as a set of state- E = {81, $2, $3,...’Sn.} then a genuine problem immediately arises. One must Speh cify a criterion for inclusion in E by which evidential statements may be evaluated. Given a statement, how are we to decide if it should be included in E? Carnap’s very de- tailed proposals are adequately defined only for very mea- ger languages, and then with the somewhat dubitable device 18 What about confirmation in richer of state descriptions. languages, and for evidential statements not originating in a state description? The problem may be viewed as one of deciding which evidence is relevant to a given hypothesis. The an- swerin.Carnap's terms is that gll,evidence is relevant to any hypothesis.19 We might agree with this only to point out that stubborn adherence to such a position would pro- hibit utilization of the calculus until an infinite evidence set was accumulated. In view of the difficulty of attain- ing such a set, we should like to declare Carnap's proposal 18My reasons for calling state descriptions "Somewhat dubitable" are based, in part, on the treatments in Nelson Goodman "On Infirmities of Confirmation Theory", Philosoghy and Phenomenological Research (19h?) No. 8. ~ And . . Kemeny with Paul Oppenheim, Degree of Factual Support", Philosoph of Scienc , (1952) 19; 307-32h. 19See Carnap, Foundatio , pp. 164:. 61 unusable, and to maintain -- in the same breath -- that some evidence was surely man: relevant than other evidence. What is needed, granting the worth of.our comment, is some criterion for establishing the degree of relevance of evi- dence. Our proposal is to rule to exclude evidence which does not make a significant difference in degree of confirma- tion. Evidence is to be excluded, in other words, if it does net change p(H,E) by some amount greater than or equal to op. Where mp is an 'interval of accuracy' so to speak, of the confirmation function.20 The question is, to re- phrase again, which si are to be utilized in confirmation of a given hypothesis? The answer is, generally, that in important investigations 5- are to be utilized so that p(H,E) is es- 1 tablished to within very small limits, while in less impor- tant investigations, the establishment need not be so pre- cise. In important investigations B will be a very large 21 . . , . . set, wh1le 1n cases where we are not so anx1ous for accuracy, E will be smaller. We can adjust E to suit the inquiry at hand by letting in statements above a certain 20What is phrased here is analogous to Braithwaite's 'Rule of Reje ction' which is boperative in class-ratio probability. See Braithwaite, Scientific Exp planation, chap. vi. 21E need not, of course, be large in the sense of containing many statene; its. Dee below, chap. iv, pp. 111-113.b 62 degree of relevance. And it would seem quite proper to call a statement relevant if and only if it changes p(H,E) by some specified amount. In short: If E = {31’ 52,...,3n} Then seE if and only if the addition of s to E changes p(H,E) by some amount ap>,cpp. It is evident that mp should be small as the inquiry is im- portant, and large as the inquiry is unimportant. Thus evidence is relevant if and only if utilization of it changes the confirmation function to a degree above the limits of tolerance established by setting op. But now two additional questions present them— (1) What methods should be used to compute p(H,E) from a given H and E? (2) How are values to be set for op? As regards a reply to (l), we shall have something to say later in this chapter. Let us remark right now, however, that a reply to (l) ("The method.M should be used.") is supported by reference to a structuring of belief (not to exclude so called structurings of 'attitudes') which M effects, where the beliefs so structured are relevant to human activity. The reply to (2) is -- though difficult -- not so temptingly philosophical. It is evident that if mp is small, the investigation will be arduous, evidential statements will 63 be included which refer to much evidence. If mp is quite large, the investigation will be more cursory, less evi- dence will be referred to. As we are more anxious to narrow the gap k2 - kl: klfi p(H,EL‘. k2, so do we assign smaller values to mp. Being anxious to narrow the gap is being anxious that utilization of H result in well supported pre- dictions, ergo (if our notions of what is well supported are what we want them to be) true predictions. 4.1 From our remarks in the preceding section, we should like to conclude that 'the' problem of induction is really several problems. A schema was presented which indi- cated that at least two questions could be considered rele- vant to fruitful solution of the problem. What is most im- portant is that there are known methods to commence answer- ing the pertinent questions raised by the two approaches mentioned. It is important to realize that the translation of the problem presented in section 3.2 is by no means the only adequate translation. Instead of searching for criteria of confirmation, one might -- as did Goodman -- ask for cri- teria for distinguishing interesting hypotheses from Efllfl“ teresting hypotheses, and thus point the way to justifica- tions which were relevant to the status of interesting hy- potheses. It is the writer's opinion that any such attempt would involve itself ultimately with all the problems of confirmation criteria, but then it is readily evident that 64 a confirmation criteria approach is very soon going to be- come involved.with distinguishing interesting.from unin- teresting hypotheses. We have characterized Goodman's 'distinction approach' to the problem of induction as one which attempted to distinguish interesting from uninteresting hypotheses. The importance of his criteria for the remainder of this thesis requires that we discuss these terms, 'interesting' and 'uninteresting': One might intend by the phrase 'inter- esting hypothesis' to indicate those hypotheses in which scientists show interest; but this would not indicate an intuitively acceptable set, for scientists have shown interest in some quite unfruitful hypotheses and wasted -- by their own admission -- much time in so doing. The old adage that a negative demonstration is just as valuable as a positive one applies only to negative demonstration rele- vant to interesting hypotheses, and not to those confused quests which result from inept phrasing or show themselves unrelated to other investigations. The sociologist of sci- ence -- a rare bird -- is of course quite justified in in- vestigating what hypotheses actually interest scientists. But the philOSOpher of science would make insignificant con- tribution were he thus to limit his work. No, what must result from the philosophy of science is a set of recommen- dations as to what sorts of hypotheses scientists ought to show interest in. The temptation is strong to involve 65 oneself in the sort of circularity mentioned in 1.3.2 and 1.3.3, above, and such circularity can only be avoided by indicating areas where empirical evidence is needed to show relevance of whatever recommendations are pr0posed. The net result of this thesis should be some such recommenda- tions, and it is hoped that the indication of need for evi- dence is not herein neglected. Much of the more interesting recent work in the philosophy of science has been concerned with what we have called the problem of distinction; how does one tell an in- teresting hypothesis from an uninteresting hypothesis? Or, to paraphrase, by what criterion may lawlike statements be distinguished? To search for such a criterion is to search for some property which might serve as an index of distinc- ion, or perhaps for some property present in all hypotheses which is maximally (minimally) manifested in interesting hypotheses and minimally (maximally) in uninteresting ones. Goodman finds such a prOperty as a function of the histories of the terms involved in hypotheses. Hypotheses which con-. tain terms with interesting histories are worth bothering about, hypotheses which contain only terms with insignifi- cant scientific histories are not worth detailed and exhaus- tive investigation. Goodman's recommendation appears at first to be ultra conservative -- since only terms which have historically interested scientists (or terms coextensive with them) can be called interesting. But this is not the 66 case; what is required is that the investigation must commence in terms of extant theories, extending itself to virgin areas as the associated terms gain meaning through relation to the rest of science. Such a requirement implies no more than that what knowledge is available be made rele— vant to the investigation at hand; and this amounts to say- ing that science should be recognized in scientific investi- gation. Goodman's criteria of interest grew out of (\ several attempts to make the distinction of lawlike state- ments from non-lawlige. One of the sorts of attempts which will interest us here was made in terms of determining the effects of the hypothesis in question upon extant theories; not really far removed from determination of entrenchment. Any hypothesis, it was said, distinguishes two notable classes among extant hypotheses 22 those which agree with the hypothesis, and those which disagree with it. If the value of the first class was greater than the value of the second class, then -- went the recommendation -- the hypo- thesis should be accepted, if the comparison was weighted the other way, then the hypothesis should be rejected. There are, of course, many variations on this theme -- as the conscientious vagueness and ambiguity of the term ’value' 22The distinction is actually into three classes, the simplification is made possible by defining two ex- haustive and exclusive classes, see infra. Chapt. III, pp. 70-79. 67 permits: One might say that the more evidence relevant, the more interesting the hypothesis, etc. Goodman's comments on this procedure are most enlightening, and -; to one who would try to formulate a criterion in terms of the proce- dure; maddening. He shows that by clever definition of pre- dicates logically equivalent hypotheses can be evaluated quite disparately, and that as much evidence can be garnered for the acceptance of trivial hypotheses as for the accep- tance of important ones.23 In short, the condition is not sufficient to guarantee distinction. To say, however, that the primitive distinction criterion is inadequate as a mode of distinction is not to say that it is completely useless. Indeed, the causes for its development qua criterion probably lie in the fact that it approaches the status of a necessary condition for inter- esting hypotheses. Most of the interesting hypotheses of science have schismatized scientific knowledge to some de- gree. Given a new hypothesis, if it be interesting, then there are interesting hypotheses which are in agreement with it and interesting hypotheses which are in disagreement with it. This follows from a definition of 'interesting' in terms of entrenchment. If the new candidate agrees with important scientific knowledge and disagrees only with un- important scientific knowledge, then the cost of accepting 23See Goodman, Férecafit, chap. iii. 68 it when it is false is not as large as the cost of rejecting it when it is true. Let it be emphasized that such a com- parison is justified only in case the hypothesis in question has been established as interesting by some prior criterion. It is a simple task to invent uninteresting hypotheses which appear as interesting under the specified conditions; but given an interesting hypothesis we are prepared to consider the agreement condition as specifying part of the cost in- volved in acceptance or rejection. h.2 In this section we shall prOpose a method of analysing evidence relevant to a given hypothesis. It is intended throughout that when evidence is determined rele- vant, it must be relevant according to some mp; Where mp is not specified, it does not enter into the immediate cal- culation and mention of it is omitted for the sake of sim- plification. Another simplification of the present section is the highly elliptical phrasing, 'H is interesting.' This is lot intended to assume that a criterion of interest is available which absolutely delineates interesting from un- interesting hypothoses: It‘gg intended to assume that some mode is available for deciding whether H is sufficiently interesting to merit consideration in the inquiry at hand. Given, then, a hypothesis h which is being con- sidered; assume that H is interesting. Consider ng...,L& a set of laws such that (i)p(Li,H)>-O where all the Li are .3 . ' . "P r h ‘v 4. accepted and interesting. lne Li, tnen, are all the 69 accepted laws which H confirms, where this confirmation is established within limits set by some pp. b‘imilarly define {L'1,...,L'% so that (i)p(L'i,H)=0 1 9:2, where all the L'i are accepted and interesting. The L'i are all the accepted laws which H disconfirms within limits established by some op. Similarly‘define {hl,...,h;% and {h'l,...,h'% so that all hi, h' are accepted and interesting and l (i)p(‘d,hi)>0 and-(i)p(H,h'i) = o. All the hi confirm H and all the h'i disconfirm h. Then lét T = {h1,...,hr,h,Ll,...,L,;k and T'= {h'l,...,h's,L'l,...,LH§ T will be called H's including theo , and T' will be called h's.;;yg; theory. . ‘ ‘ Let E be the set of evidence statements such that p T E)>O pET‘,E)>O That is to say, 3 is the set of evidence statements which confirms both T and T', both the including and the rival theory. ‘ Let F be the set of evidence statements such that p(T F))>O P(T",F) = O F is the set of evidence statements which confirm T (the including theory) and do not confirm T' (the rival theory). Similarly define F' so that F' is the set of evidence 70 statements where T F') = o 5(T",F')>o Then vH‘E)=p(TLE) + T' E ‘ 2 VH(F)=p(T,EuF) _ VH(E) VH(F' )=p(T',EUF') _ VH(E) That is to say, the value of evidence supporting only the including (rival) theory is the difference in the degree of confirmation of the including (rival) theory and the value of evidence which supports both the rival and the including theory. The three functions listed above are then de- fined OVer the same range as is degree of confirmation. (An implicit assumption has been that theories are weighed in accordance with the evidence which confirms them.) The definitions assume that degree of confirmation is real valued, that it is an extensive measure over the range of all pairs (H,E). Thus the proposed formulations constitute some attempt to weigh Egg relative importance of the rival and including theory. We shall consider that the task of this chapter has been accomplished; some means has been proposed for the evalutation of evidence sets and theories which relate to a given interesting hypothesis. In the next chapter we shall attempt to utilize this evaluative method for weighing I different hypotheses. CHAPTER III FOUNDAIIONS OF A THEORI OF IMPORTANCE 1.1 In the previous chapter a method for appraisal of evidence relevant to a given hypothesis was outlined. Such an outline is certainly required if even a programma- tic preposal for the evaluation1 of hypotheses is to be made. The present chapter comprises an attempt to utilize the out- line as a basis for some comments on the evaluation of hy- potheses: These comments will explicitly presuppose that there has been antecedently provided, first, some effica- cious method of evidential assessment, and second, some criterion for distinguishing scientifically interesting hy- potheses. Without such criteria, the evaluation to be under- taken below of hypotheses in terms of importance becomes meaningless; for uninteresting hypotheses can show high importance values relative to similar interesting hypotheses. The method could thus -- in the absence of a criterion of interest -- be construed as indicating that uninteresting hypotheses ought to be accepted by scientists. In view of 1In the present and ensuing chapters we shall use 'value’ and its cognates only where reference to value in the ethical or esthetical sense is intended. Locutions such as 'evaluation of evidence' will be avoided in favor of such expressions as 'appraisal of evidence'. This convention is adhered to not because of any misgivin 3 about uses of 'value', but for clarity of exposition. here has been no attempt to use the word thus univocally in the preceding chapters since there is little danger of confusion. 72 this, any prOposal for the weighing of hypotheses which makes use of the evidential criterion outlined in chapter II will be restricted by the presupposal of a criterion of interest. The restriction to interesting hypotheses, how- ever, is not as stringent as might at first appear. Good- man's theory of projection provides an informal scheme for the partial ordering of hypotheses as regards interest. If, to be sure, the criterion of evidential assessment were to be metricized, it would be necessary to metricize the theory of projebtion; but the lack of a metricized theory of pro- jection should not prevent enlightened speculation as to what such a theory would be like were it available, and should not, in turn, prevent speculation as to what a theory of evidential assessment would be like if a metricized theory of projection were available. It is hoped that Goodman's theory of projection and the outline of evidential assess- ment procedures presented in chapter II are, in turn, ade- quate bases for speculation about a method of evaluation of hypotheses. In the sequel we shall attempt just such specu- lation, bearing in mind that such comments as eventuate are justified only in case both the relevant presupposed theories are shown to be empirically and formally sufficient. Some comment should be made about the utilization of formal and quasi-formal modes of exposition in this and the preceding chapter. The exposition of evidential 73 assessment on the basis of degree of confirmation assumed the additivity and multiplicativity of various functions. It is obvious that such assumptions are only to be justi- fied by a quite thoroughgoing formal investigation accom— panied by an interpreted theory dealing with the behavior of the values of the functions. It might be objected that what has been given is an irreSponsible display of arithme- tic sleight of hand, at best confusing and at worst erroneous. Such an objection is, I think, quite tenable if one takes the presented formalism to be a fully interpreted theory about the phenomena at hand. The formalism, however, is not intended to be such a theory: Speculation in formal language is no more noxious than speculation in any other language ~- just as long as one clearly labels it speculation and does not purport to be offering mathematical 'proofs' for demon- strated' conclusions. The nature of the concepts dealt with and the author's ineptitude of expression make it advisable in the interests of simplicity and precision that what is said be said —- at least in part -- formally. This formalism may be taken as a heuristic device to aid in the explication attempted discursively if the reader's conscience balks at speculative formal discourse; but the intention of the writer is to make speculative comments in a quasi-formal idiom. Thus, the treatment of a concept as metricizable or linearly orderable or whatever is not intended to imply that the concept has been shown to be or is of such a nature; 7h when such implications are intended they will be made expli~ cit. The primary goal of this chapter indeed, of this thesis -- is to ShOW’the feasibility of treating certain concepts as at least partially orderable. The 239g; that these concepts are so orderable would require that other concepts be metricizable or completely orderable, and fur- ther.that such an ordering had empirical meaning of a quite specific nature. No such proof will be offered, but it should at least become evident where such proofs are needed and where empirical evidence is required to render a partial theory of scientific decision making. If such indication is clearly and correctly made, then the thesis has fulfilled one of its important functions. 2.1 The decision of a scientist to accept or reject a hypothesis is a product of many factors. Though it is true that science is the long arm of common sense, the arm is uncommonly long in that complex evidential rules and pro- cedures of observation and inference are made an explicit part of scientific procedure, whereas in common sense judgments such components of decision making machinery are left more or less to function without explicit utilization or checks on their operation. This checking on the opera- tion of decision making apparatus has become a carefully incorporated part of scientific procedure. The scientific method is replete with checks and counter checks -- observa- tions are contrasted with other observations, inferential 75 patterns are compared with other inferential patterns, hy- potheses are checked against related hypotheses, and the whole machinery is made self-corrective insofar as that is possible. The methodologist must attend to this feature of science with scrupulous care. Advances in the efficacy of science's capacity for self-correction are significant im- provements in scientific method. Historically much attention has been directed at observational and inferential procedures. Science itself has invented observational equipment as the need for in- creasingly accurate observation became felt. Logicians and mathematicians have traditionally been concerned with the inferential patterns used by scientists. It is regrettable, however, that little attention has been directed at the influence of valuings or appraisingg -- in Dewey's sense -- 'n science. It would seen pretty obvious that valuings in- fluence the scientific decision, and that the interests of science would be best served by making valuative procedures explicitly self-corrective insofar as possible. In the en- suing sections some attempt is made to outline a method of such self correction. The method is quite narrow in sc0pe, dealing only with decisions to accept or reject hypotheses, and at that only with decisions to accept or reject hypo- theses previously judged interesting. But it affords some Opportunity for ordering and is not -- it is hoped -- com- pletely quixotic in its aspirations. The relevant alternatives for scientists when 76 faced with interesting hypotheses may be characterized as being in one of three general classes. After contemplating an interesting hypothesis the scientist either rejects it, accepts it, or holds it in abeyance. We shall assume that his initial attitude is one of abeyance -- that is to say that he neither accepts nor rejects the hypothesis, but decides to subject it to further testing. After a period of testing he assimilates his data and again chooses one of the three alternatives. If at some juncture he decides to accept the hypothesis, then he utilizes the hypothesis in prediction, assumes that it supports higher level generali- zations and verifies lower level hypotheses,2 and assumes that the probability associated with the hypothesis by the relevant theory is justified. If he decides to reject the hypothesis, then he assumes that higher level generaliza- tions are disverified, that lower level hypotheses dependent upon it are disconfirmed. If at some juncture the scientist decides that the hypothesis in question is not as interes- tins as it at first appeared, or that it lacks some other Q quality necessary to legitimately testably hypotheses, then he may assign it to a Limbo, so to speak, of always-to-be- held-in-abeyance hypotheses. To hold a hypothesis in abeyance 2We say that a hypothesis verifies lower level statements and confirms higher level hypotheses. Reversing the implication, hypotheses disconfirm lower level state- ments and disverify higher level hypotheses. The barbarism 'disverif' seems condoned in view of its intuitive aptness. 77 is to refuse to accept or reject pending further evidence, the decision in no way requires that the scientist show any degree of diligence in accumulating such evidence. The deci- sion to hold in abeyance might result in feverish investiga- tive activity, or it might result in completely ignoring the hypothesis as uninteresting or otherwise unsuitable. In characterizing this situation, one might be tempted to speak of a linear or quasi-linear ordering of pre- ferences; we might ask scientists to specify under varying conditions in what order they prefer abeyance, rejection and acceptance for a given hypothesis. There are available ingenious and empirically tested methods for establishing such n-membered preference arrays3 so the project is not at all chimerical. If however, we conducted such an investiga- tion, in all probability we should find that certain kinds of arrays are never presented}P For instance, in any array where the scientist was indifferent between acceptance and rejection, he would prefer abeyance to both of them, and in 3Savage records one such device: "...(The sub- ject) is instructed to rank the three acts in order, subject to the consideration that two of them will be drawn at ran- dom..., and that he is then to have whichever of these two acts ne has assigned (more preference)." 'Savage, op.cit., p. 29. hThis, of course, is an empirical thesis. But in view of our stipulation the assumption seems safe. It is noteworthy that procedures such as that mentioned in note 3, above, are inapplicable to such situations. 78 any array where he greatly preferred acceptance to rejection or rejection to acceptance, he would also prefer that more preferred member to abeyance. In short, the preference value of abeyance is at least in part a function of the relative status of acceptance and rejection. Because of this relationship among the three options, the construction of a scheme of appraisal which recognizes all three of them awaits the construction of an scheme of appraisal which recognizes two of them -- acceptance and rejection. If a fully articulated method of appraising acceptance and rejection were available, the problem of intro- ducing the third option to explicit consideration would be quite manageable: There are techniques of sequential analy- sis in statistical inference and sampling theory which would be of great assistance here.5 It is obvious that appraisal of acceptance and rejection is the prior and major task. Since we are considering that the initial atti- tude toward a hypothesis is always abeyance, and that later decisions to acuept or reject follow a decision not to hold in abeyance, we may informally characterize the entire deci- sion situation as being a decision between two alternatives (abeyance or not abeyance) one of which (not abeyance) results in a two alternative decision itself, (accept or reject) and the other of which (abeyance) can lead back to a reposing of 5See for example, Savage, Qp. Qit., pp. 142ff. 79 the original decision. we shall attempt a partial formaliza- tion of appraising procedures which rank the two alternatives of acceptance and rejection for a large though quite restric- ted class of hypotheses. 2.2 If the scientist is depicted as facing two al- ternatives, then we can appraise his choice in one of two ways -- as mentioned in chapter I. we can decide if he hose rightly on the basis of available evidence and goals, 0 no matter what the outcome, or we can play 'Monday morning quarterback' and distinguish right from wrong choices in terms of their eventuatiens. Our task here is to prescribe a method of appraisal in the first sense -- on the basis of goals and evidence -- such that specific appraisings by the use of this method coincide, with greater regularity than any alternative method provides, with ultimate appraisals in terms of eventuations. In short, we wish to prescribe a method for appraising alternative courses of action by which a course of action is appraised independently of whether it is undertaken or not and, if undertaken, independently of its specific results. In order to weigh alternatives and thus appraise decisions, we shall consider that the appraisal is to be made in terms of a probability distribution over the space of outcomes of the decision. We have described the scien- tist's decision as schematically representable by two alternatives (abstracting from the decision to not decide). 80 We shall consider that the hypothesis in question must be either true or false. The decision situation may then be schematized by reference to a 2X2 matrix: H is true T_ H is false , scientist H is accepted and true H is accepted and false accepts H scientist H is rejected and true H is rejected and false rejects H FIGURE 16 It is generally agreed that we ought to accept true hypotheses and reject false ones,7 so a generic assign— :ient of possible evaluations may be made as follows: H is true H is false accept good bad reject bad good FIGURE 2 6This format, in an abbreviated form, will be used throughout the remainder of the work. Figure l is in- tended as a guide to reading the matrix format. The matrix presentation invites such interpretations as "2-person zero- sum game" in which the scientist is depicted as playing against nature. Such anthropomorphic references are inaccep- table -- it would seem -- in that it is difficult to associ- ate the notion of a strategy with the activities of nature, even if the universe is case in a purposive role. cf chap. V pp. 113-117. 7This, of course, is an oversimplification. It frequently comes about that rejection of a true hypothesis or acceptance of a false one results in a utility maximisa- tion. lhis shortcoming of the present scheme is noted and briefly discussed in chapter IV pp. 117-120. 81 The matrix depicts an evaluation of points in the outcome space, and not of procedures of appraisal.8 We shall even- tually make some comments about assessment of decisions according to methods, but our concern here is to make a general prediction as to what we shall judge about the out— comes of the decisions, regardless of the procedures used in arriving at those decisions. We are in truth playing 'Monday morning quarterback‘, but we are doing it on Friday night; we know that if we should decide to accept a hypothe- sis which turns out to be true it will be good for us,9 that f‘we should decide to accept one which is false it will be H. bad, and so on. What methods we use to reach these deci— sions to accept or reject the hypothesis in question has no bearing whatsoever on the W 21; the W. Good Inethods, it is true, usually result in happy eventuations, but good eventuations need not arise from good methods. 3.1 Before the assignment of values to points in the outcome space can be made more specific, some additional properties of hypotheses must be noted which will make such assignment possible. The first of these properties which we shall note is importance: A hypothesis is important insofar as its acceptance results in the rejection of formerly accepted hypotheses. Importance is distinct from interest in 80f. chap., I, pp. 24, f. Except as noted in note 7, above. 82 that very interesting hypotheses (i.e. well entrenched hy- potheses) might be acceptable at very little expense of re- linquishing other accepted hypotheses. In the extreme case of a highly interesting ‘ut not important hypothesis, the lmrvotzma sis may be considered to be a re-statement of extant theories. Such a hypothesis might offer a gain in systema- tic clarity and economy by way of reformulation. To say, then, that the only hypotheses considered for evaluation are interesting hypotheses is not to have filled the need for assessing the impprtance of considered hypotheses. Importance is defined as follows: Let the de- gggg g: importance of H be wH, where ¢H=VH(F')=p(T ,ELJF') - VH(E) and where VH(E)=Q(T,E] + 2(T'.E1 2 . so that "VH(E)" indicates the worth, or weight, so to speak, of evidence which confirms both T and T'; where "T" and "T'" indicate, respectively, the comprehensive theory which in- cludes H and the comprehensive theory which rivals H. (We shall speak of these as "H's including theory" and "H's rival theory.") Thus, “¢H" may be interpreted as referring to the increment in confirmation bestowed on the rival theory by evidence which does not confirm the including theory. The definition of “pH" guarantees that for highly important H, the rival theory will in general be highly 83 confirmed, and moreover that the rival theory and the inclu- ding theory will have little supporting evidence in common. notice the fact that highly important H might be included by theories which are highly confirmed or not. VH(F) is irrele- vant to the determination of the importance function, wH. Perhaps it will now become a bit more clear what was intended by saying that H was important insofar as its acceptance teQQgQ'EQ bring about the rejection of previously accepted theories. It is evident that the formal definition of "degree of importance" is an incomplete definition, but it is noteworthy that counter-examples to the definition (i.e. hypotheses which are formally important but intuitively unim- portant) are -- in most cases -- either excluded by criteria of interest or are also counter exemplary to criteria of interest.10 It is for this reason that such emphasis is placed on the priority of a criterion of interest to a cri- terion of appraisal of importance. Insofar as this develop- ment is taken to be a theory, it fails in the face of counter examples not excluded by criteria of interest: Insofar, how- ever, as the deveIOpment is explicative and speculative, it xiight be said that such counter examples show its success -- the notion must be understood before it can be counter exem- plified. In the preceding section we noted that a generic lch. p. 33, this chapter. 81+ assignment of values to each of four points in the decision outcome space was at least intuitively acceptable. We also noted that insofar as such an assignment was innocuous, it was trivial: One doesn't need a 2X2 matrix to learn that it is good to accept true hypotheses. In order to utilize a system of evaluation by matrix inspection, some further re- finement of value assignments must be accomplished. The first step in this refinement is the establishment of an 'importance index' to be determinable for any interesting hypothesis. 'Importance index' is so defined above as to be metricizable. This definition is, of course, quite Optimis- tic. To render such a concept amenable to metrical measure- ment would require a much more thoroughgoing formalism than is feasible within the confines of the present work. What -- it is hoped -- is accomplished by such formal sketching as that presented here, is some idea of how important indices would (functionally) behave if the concept were made thoroughly metrical. An assumption which is essential to what theori- zing does go on in the sequel is that degree of importance is at least partially orderable; it is assumed that given any two interesting hypotheses, it is ascertainable which if either brings about in the event of its acceptance more ex- tensive MOdifiCptiOH of extant theories. It is for this rea- son that we introduce the notion of weighing theories in accordance with the support lent them by evidence statements, 85 and then bring about comparison of evidence statement sets among themselves. If the explicating theory is rejected, then the evidence statements are left unexplained, so to speak, and one must re-tnearize to bring them under the rubric of explicative theory. If the acceptance of a hypo- thesis brings about such rejection -- so that some evidence statements are left unexplicated -- then that hypothesis is important to an extent which agrees with the amount of ob- served phenomena left unexplicated. 'Amount of phenomena' is defined in terms of the degree of confirmation (or the in- crement to degree of confirmation) lent to some theory by the evidence statements in question. dome reflection will assure us that this measurement of evidence is a difficult matter;11 to count the evidence statements won't do, since any amount of evidence can be expressed in any number of statements: One would like to speak of the gzea of phenomena 'covered' by the theory, but this would seem to require cardinal comparison of very large sets,12 and.where such comparison is not required, eventuate in drastically counter- intui ive conclusions. For these reasons the notion of importance is introduced. Although, as mentioned above, we treat importance 11See, for example, the v ry interesting attempt in J. G. Kemeny with Paul Oppenheim, Degree of Factual Support", loc, cit. The reader will note similarities be- tween our notion of importance and the Keaeny-Oppenheim 'F'. t is hoped that the importance function does away with necessity of reference to state descriptions. . lzCimpare for instance, the number of stars with the number of e e trohs. 86 as if it were extensively measurable -- i.e. metricizable -- for explicative purposes, our concern is primarily that we indicate it to be intensively measurable. Je want to intro- duce a relative notion of importance ('H1 is more important than H2.') as a relation which orders the field of interes- ting hypotheses. We do this by supposing importance to be metricizable and then, to render this supposition innocuous, only utilize 'importance' in comparative locutions, e.g. 'The importance of H1 is g‘cater than the importance of H2.’ in the two culminating formulas of the development, T.III.lI and T.III.2, we compare importance with another evidential function of H, vH(F). This comparison is permissible even for partially ordered importance because VH(F)=¢H'. H' is the rival hypothesis of H, so to speak, and we need know no more about it than that - 1. VH(F)=$H' 2. vH'=¢A. 3.2 Consider hypotheses of the highest generality -- axioms of articulated and developed theories. Such hypothe- ses have no corresponding set of laws {l1,...LQ5 ‘WhiCh they confirm, and the including theory for such a hypothesis con- sists of H, the highest level hypothesis in question, and the set {hl,...,h£5 of lower level hypotheses which it verifies. Ther might, of course, be more than one highest level hypo- thesis, but this in no way effects the absence of a set 87 ‘{Ll,...L;5 for any highest level H.13 Similarly, there will be few laws disverified by H, because of its high generality, but many lower level hypotheses which it discon- firms. In this extreme case, the evidence for the including theory will be the evidence for H and the evidence for the rival theory will be the evidence against H. B will be -- if not empty -- of very small weight,14 and v(F') will be simply the degree of confirmation of the rival theory on the basis of available evidence evaluated with respect to some pp. In such a case the information afforded by the computa- tion of importance functions is at best trivial, and, possi- bly, erroneous as regards intuitive evaluations. If the importance function of a highest level hypothesis is nothing more nor less than the degree of confirmation of the rival theory of the hypothesis, then it is of no more use than a straightforward examination of relevant probabilities without consideration of the utility of acceptance or rejection. 13The distinction might be made in terms of factual suppggg as op osed to degree of confirmation -- see Keneny and Oppenheim ref. note 11, above) -- but there seems to be no need to appeal to a distinction this fine. 14Incompatible statements can, of course be confirmed by the same evidence. See Goodman, Fact, riction and 5 recast pp. 69ff. If one makes pp small enough, how- ever, to permit such support, in the case of the statement which supports both incompatible statements, the contrary of the supporting statement, which will disconfirm both state— ments, will also be admitted. 88 In view of such considerations as the above, we shall exclude highest level hypotheses from our subject matter. The exclusion, perhaps, is not as indicative of narrowness of conclusion as might at first seem. If such a highest level hypothesis is a reformulation of already accepted and con- firmed theorizing, then its empirical relevance is indirect -- though not by any means insignificant -- and the utility of accepting it results from its function in simplifying and clarifying extant theories. We shall consider such hypo- theses (reformulations of extent theories) as limit cases of highly interesting and unimportant hypotheses. If, on the other hand, the highest level hypothesis in question is not a reformulation, but a novel assertion in the form of a very inclusive generalization, then its degree of interest would not be nearly so high -- such a hypothesis would contain novel terms and exhibit a low degree of entrenchment. It's importance, however, might be very high. One is minded of current theorizing in extra-sensory perception; the terms concerned have very little significant scientific history, (I. hence arr not well entrenched; but the acceptance of the hypothesis would require not only modification of laws of psychology and biology, but also of physics and chemistry.15 151 fail to find a clear presentation of these hypotheses. One gets some idea, however, of what an articu- lated theory of extrasensory perception would be like from J. B. Rhine, New Frontiers of the Mind (New York, 1937). 89 The question as regards such hypotheses is Goodman's inquiry; ought they be projected? If only interesting hypotheses are to be projected, then they ought not. If one considers some other property than interest, then perhaps they ought. The situation seems counter exemplary to the theory of projection, and provides an interesting case for study of that theory and its limitations: What is of most interest to us here is that a case which proves difficult for the theory of impor- tance to deal with, shows itself to be difficult for the theory of projection to deal with. As we remarked before, this would seem to be indicative of the degree to which an adequate theory of importance must depend upon an adequate theory of projection, or interest. The above comments indicate that our caveat about hypotheses of highest level generality need not perhaps be as stringent as it is. There are cases in which importance func- tions of highest level generalizations are indicative and relevant. There are also, however, cases in which such func- tions are neither indicative nor relevant. This leads us to remark that where the theory works on highest level generali— zations, it works; and hence to consider highest level generali— zations properly outside the subject matter of our comments. We shall make reference to limit cases in our illustrative develOpment, but we do not intend to offer conclusive infor- nation as regards these limit cases. 4.1 In previous sections we have spoken of "the four 90 points in the outcome space" of a decision situation. This location, convenient initially to describe a way of viewing decision situations, is sufficiently misleading to require Xplicative comments: The outcome space is perhaps best considered as a d§n§g_set of points, this sense set may be partitioned into four subsets, mapping every point in each subset to an 'ideal' representative point, or, mapping every point in the outcome Space to one and only one point in another Space of four points. The generic ordering illus- trated in Figure 2 imperfectly ordered the four points in this second space. A more nearly complete ordering of out- comes must attend to two tasks: (1) An ordering of points (or of some significant subset of points) in the partitions of the original dense outcome space, and (2) An ordering of the partitions.g£ the dense outcome space, or a more complete ordering of the four discrete points in the simpli- fied space. In this section we shall attempt this more com- prehensive ordering, first within each partition, then amen? the partitions. Our first attempt at ordering will be made, as was said, in terms of value. It is issumed that, in order 93 to have value or disvalue, an outcome must make a felt ex- periential difference which is mainly to be exemplified in chang- in; what is accepted by science. If an outcome requires that science reject certain theories which it once accepted, then this is an outcome which makes a difference; similarly 91 for outcomes which require the acceptance of what was once rejected. If an outcome requires no such variation in acceptance, then it is to be considered generally better than bad outcomes and worse than good ones. Outcomes are to be considered good if they consist in changing policy so that true statements are accepted and false ones re- jected, they are to be called bad if they bring about the converse. In order to make this evaluation of outcomes relevant to appraisal of decisions irrespective of specific outcomes, some means is obviously required to functionally relate the two values. He shall call that decision X0.§§§E in a given situation if for all x, u(xo)33.u(x). Where u(x) is the sum of all products, p(y)v(y), where p(Y) is the probability that y will occur given that x occurs, and v(y) is the value of ylé (in this case the variation in scientific policy). Svmbelically u(x)= : [p(yi,x)v(yi)] i: The set of y's can be, of course, infinite; and one faces an analogous problem to that of deciding which statements are to be adaitted to an evidence set for the computation of degree of confirmation. much the same sort of qualifi- cation can be made as regards itility (for such is the léBy 'value' is meant, e.g., 'felt good', not nathematical or logical value. See note 1, this chapter. 92 conventional name for the function u(x) )3 the number of y's included in the set is directly preportional to the in- quirer's need to ascertain exactly what the results will be. 1 a We shall not define this formally here, the reader is re- ferred to the contemporary literature for various formal treatments.l7 4.2 Consider two hypotheses Ho and H1 such that H1 has a high importance index and H0 has a low importance index. If no and H1 are both true, then they both ought to be acceeted, but the obligation is stronger in one case than the other. Acceptance of Ho results in little modifi- d . , . lo .. .. . cation of truth claims, while acceptance of the highly important H results in extensive modification. Theories l which were heretofore accepted are rejected and, what is decisive, true (interesting) hypotheses tend to confirm true (interesting) theories and disconfirm false (interesting) theories. Thus the acceptance of a true H1 tends to dis- confirm a large body of false and relevant theory, while the acceptance of a true He brings about little such modification. }7See 4. D. Luce and h. Raiffa, Games and Decisions (Jew York, 1957), pp. 19-23. Savage, op. cit., pp. 70-75, Davidson, et. al., op, cit., pp. 9-12. 18The expression 'truth claim' is used as equi- vale.t to 'what is aecepted"in the preceding section. The change is made to avoid such confusing locutions as "Accep- tance results in modification of what is accepted.’ «There is no intention to imply that science 'claims' anything. 93 In the limit case where He is a reformulation of extant theorizing, what results from its acceptance is a more econ- omical and facile expression of what has previously been expressed. Reformulations are, so to Speak, accepted with little regard for direct empirical evidence; the evidence has been regarded in the acceptance of the lower level gen- eralizations, and the acceptance of the minimally important but highly interesting hypothesis is dictated by considera- tions of systematic economy. This is not to say that sys- tematic economy is an unimportant end, far from it;19 but it I”). s to say that the benefits which accrue to science from such economical moves is a product of advances in formal techniques, and that no important empirical scientific de- isio; is needed to adopt unimportant reformulations. (J we might say, then that all other factors being equal, if H1 is more important than Ho, then the consequences of accepting the true Hl are more valuable than those of accep ing the true HO. Utilizing a conventional and sugges- tive notation, we shall let "v[A,T](Hi)" abbreviate "the when it is true". The comparison may A value of accepting Hi then be phrased in a formula III.l.l If (le> mic) then v.[A,T](Hl) > VLA,TJ(HO) “0 If Ho’ as above, is low in importance and true, then to reject it is to tend to reject its supporting and Wfi H 19See 0.3. Goednan, ‘tructure of Appearance, C1153? . Ill. supported hypotheses. Since in the case of a true and -- the inclu‘inr tLao ry -- would tend ' 4-, .. .. .L: ‘, ' llluCL-‘VSULIL ll _, to be true, rejection in this case would be tantamount to comma rid .1 a true theory disconfirmbd. If H1 is high in infert 11cc, true and rejec 3d, then the evil is dou ble edred in that a true theorr 's eensidered disconfinned and a false one confirmed. here the phrase "all other things equal" " ‘ - ,. a - VJ ‘f‘ I 7 ~ r ~~~~~~r~ ' ‘ -‘ \r‘ o . ~\( ‘I USCG-19:) efasellbldl, Hr: dSO'J...\J ln tile CUhlfurl-gorl 451th tile J.- V ’| “"J. ‘1' n. "N; 4“: 4" V' '3 \r 'l'.‘: -u., ‘ . n 44-. ,«r a . ‘ L .- wr CLOLJ v 01. illbaf} u.’.C\_/TJ boilllidv’u lb c.10 same In 00th. Cdbd U) \o fer, therefore, only in the amount of tr e theory disconfirned. All other factors equal, the rejection of a true 31, is of greater disvalue than the re- jection of a true Ho, when hl is more important than Ho One mi ght choose to think that the realms of value and disvalue present two discrete distributive realms. ihiS aaeants to claiming that the relation 'better than' is not connected in its field. Such, it would seen, was the assnnption behind our initial HM ric ordering. is are now prepared to recent anv such assum :tion that we might have made, and explicitly to assume that good and bad outcomes may be repres- ented on a lizear scale, unique up to the assigl- ment of a zero and a unit. On this assumption -- which, it might be remarned, the author has found nor e intuitively satisfactory with closer acruaintance —- to speak of disvalue is to speak of less value. The advantages of this for a lane of ordering such as that we contemplate are so obvious as not to need mention, cut one ni.j ht obj ct that sucn a scheme of ordering does injustice to the natu~e of good and evil, or the customarj notions of the distinction between the two disjunctive realms. lo the pi ota gonist of disjunction we remark that he is free, if he chooses, to represent the states of good and evil by positive and negative numoers respectively. If he does this, however, it would seem that ne at least im- plicitly assames a point of staztus quo —~ that the point of zero inc1e11ent or loss is the neutial point of value; and to assume this is to assume that the ordering has a great many no: e pro operties to it than would seem justified. It is to assume, most importantly, that the scale is completely or- deraL -le, that values are mecninéfully additive, and that the intervals eetween value assignments signify 'real' units 01 UlllCretce oetwaen values. Our preposal that a scale of partial comparative ordering be utilized carries none of these disagreeable results with it. If the protagonist of disjunction could meaninjfully explicate his disjunction, then there would be reason to heed his admonitions. Until such explication is forthcoming, however, we shall assume that the assignment of value indices to points in the out- come space indicates no mo1e tlan a partial comparative ordering. The arguments put forth here are the traditional 96 and conventional arguments for the use of utility theory.20 It should be no secret, by now, that we intend to implement our measurement of values by reference to the vocabulary and the ems of utilit' and decision theory. The vocabulary we have introduce d so far is obviously kindred to that of tility theory, and the introduction of concepts has been unashamedly pointed toward an eventual association through formulations of these theories. It might be objected that though utility theory and decision theory are good things, the‘ are not philosophical sorts of things. he mention this objection only because it seems to be somewhat prevalent. f the philosophical relevance of these theories is recog- nized, then there is no need to argue for this relevance. If it is not recognized, than this is not the place to mak such argument. The doubtful reader is referred to the literature.21 We shall henceforth consider 'x has more disvalue than y' as as Mival nt to 'x has less value tr an y' Text- ially we shall Speak of value and disvalue for verbal 20Luce and Raiffa, Games and Decisions, chaps. l and 2 contains a quite favorable and persuasive introduction to utility theory. Savage, Foundations of Statistics, chap. 5,1,10313 1ore Ior111all‘r with the introduction of util- :t;,'1nile Savage in chapts. l throu h L presents an interes- tin n1d clear general exposition. The philosophical pro- alems, as wel , become evident in such works as Braithwaite, ‘hoerr o; ”axes The stron;est presentation and argument, 0”“V”r vreuld still seem to be the classic treatment in Von aeunann and horje nstern, Theory of dames and Economic Behavior chaps. I and II. '-1 21 See note 2 , above. ~ 97 facility and intuitive comprehension, but in formal notation we utilize onl, expressions of positive value. To return, at last, to the comparison with re- soect to true and rejected hypotheses: We said that if A H1 is more important m11n filo, then the rejection of a true H1 -- all other things being equal -- is of greater disvalue (of less value) than the rejection of a true Ho' Hence the formula: III.l.2 If $Hli>¢HO Then v[R,T](Hl)<.v[R,T](HO) If H0 is accepted and false; if it is of low im- portance its acc;pt ance does not call for the rejection of J- a larre body of hue. ml 3a, thus does not necessarily tend to disconfirm a large body of true extant theory. Since it is false however, its acceptance does tend to confirm a body oi accept1d false tie ry. The case is analogous to accept- in: true and uni 1:120 rta nt hypotheses -- if tne hy Joshcsis is very interesting and not at all importent, it may be con- sidered as a reformulation. In the case of accepting true and an -101o1nt hyootheses, the benefits accrued were larsely systematic benefits. So, in this case of accepting the false and unimportant hypothesis, the sadvantabbs are largely those of nakin; iteasier for ourselves to persist in some mistake. This sort o1 disadvanta; e to be sure, is insidious as can be: One only infrequently tests such refor- nulations, and just incorporates then into accepted doctrine. "r ' - r1 A' L. -‘D " A-41 ‘tf‘ r. w . n ‘ » - . ' we should s1r that 11 tne1e 1s a disad var tags then it lS 98 insidious. It might very well be, however, that such sim- plifie' systematization would make the errors of false hy- potheses more readily discoverable, and than we should re- mark h v fortunate was the acceptance of this false hypo- thesis. But I think that the fact that we should call our- selves fortunate in such a case is sufficient to persuade against the assignnent of a generic positive value to the acceptance of false ny10on Ms, ho.:ever unimportant. The argumen seems to indicate that some low index of disvalue she Id be assigned to the acceptance of unimportant false hypotheses. If, on the other hand, the hi hly in1portant H1 4. th accrued disvalue is accordingly (L) (1' (D Q Cu :5 {L H) C.) H U) (D u is acc arre. T’(Hl) would tend to be true. The more important is H1 the more disvalue is accrued. If, in short, H0 is less important than H1, then -- all else equal -- more disvalue (less value) accrues by the 1coeuoon of H1 if f ls e than by acceptance of Ho if false. 111.1.3 If eHl> 1,1110 then v[A,F](Hl) < vLA,F](HO) The remaining case is similar to the preceding three cases; If H1 is more important than Ho, then to reject the false H1 is of more value than to reject the false HO, incc by rejecting the false H1 we make more of a difference in tru h clain1s the we do by rejecting the false HO. This diffe re 11cc, furthermore, is a good difference, since false hypothesest end to con1ir1 false (interesting) theories and to sconiir1 tru>(1nte-g..11;) t series. in an; linit r -. .-. r org—\J'. -.'r ”.‘V-‘l .‘J- . ,- * As“. 3-1- r‘vxt ..' '. ’.‘ . ~‘ _/‘I’\fi '<‘ ‘ l Chu M b v-LIO&--“LJ.~.\UlOn UCL LIA (.-L '14- tile J-‘\‘.I “ ‘ ‘ q Q Q ‘ a. _ C. . .‘ .. .. ._... . . A a 4- -.I ,\ I. I ‘m. fl ’1‘ s a- . .u q ‘ - u“:— Iv ' -- 4‘ :‘J ’ sadly". dolwtubi UAAM; ~-'L\c--IV «v--E.|iu34. wations £101; :43 2131 3 L 31-3- -. a, . .7,.-.,,.. 1.1. ,. .11.... A: .. Axbu‘ J—ol Ci.—“Jl~a~~J-LLL-) Vial») L;-—-_‘ - l-§—L.§L\-l U— KAV'1 ‘-'-"‘ ~ '4‘ -“»“’ :‘*"r‘ . ~r‘ '.-‘ ‘11"? ’~ ‘I‘S “ I‘ "1 J'1 A . .ni~,ortanc “,Aotneses. All other factors eeaal, there 15 ' ‘ "I‘ "I \v ~ ' ‘ fi' 4- , \~\ i.\ . > r J,‘ ‘. \»‘1 I‘ "- . .v - 1..-, ; J<.-‘..'tse £33-}..Laallt %;JUIA tlte rQJ'QCULK/il UJ. H1 Ullall 0.1- do, ' ~. 1' .~-~ ' n “'r\ ~~~~ - - t '3 . - - ~:v-— ~ 4 .\ 43",- w ‘.;.(.‘1’;‘ do Lulu. h]. are 4.93:1..-w3, .xIICL 1.1 15 111038 JiuijOltun'b Vilad III.l.Lr If 4:111 > tho then v[11,F](Hl) > vLR,F](HO) L.3 For conveaignce of r fe1;nce he rep: odz1ce the ..L: fornulas from the or :edinx section; If 111)}11311 III. .1 Ov[A,T](l-Il)_> v[A,T](HO) 1%,T](Hl)4 vER,TJ(HO) .3 VIA,FJ(H1)< VEA,E](HO) 1.4 vER,FJ(Hl) > viR.FJ(Ho) nfl examin ition of III. 1. l revealst hat the value H l—i ¢\) El ;1 aJJOptilg a true nypct133is varies proportionately with th. imjortance of the hypothesis -- the more import at is the hypothesis, the better it be accepted if true. Suppose nae-egoose to express this _s III.2.1 vLA,T](H)=(¢H)(fH) for some fH The functional expression 'fH' is innocuousl3r va :ue. It would seem unjustifiedl; restrictive here to assume that A III.l.l permitted reglacenent of 'fH' with a constant, we have been cautiousl, avsiding assuming any additivity or ‘ lOO multiglicativi ty of inoo1c;; e arrays, and reglacement with e constant VIould violate Iust these caveats. The utiliza- U “3 tion of fH'rather~than a constant corresponds to our use 0 chrcse 'other factors beinfi eoual’ in the discursive tne _ A \_ presentatiOn of III.l.lt r*eu3h III. 1. 4. This qualification was omitted from the for ulas because no adequate way was found to express it which would not render the formalism ex- cessively cumbe some and, perhaps, defeat the explicative end of tlie previous seccion. No such omission need be con- 1 donod nere, however, for the utilisation of functional no- (I, totie1 is sufficient qualification. lne analogues of 111.1.2, III.l.3, 111.1.4 are written similarly; III.2.2 vLR,T](H)=jH/vfl for some in III.2.3 v[A,F](H):5H/$H for some 5H III.2.4 v[R,F](H):(¢H)(kH) for some kH he use of differe-nt functional notation in each case assures that the: multiplicative and divisible formats are hsfi;lggg, The point, let us repeat, is an explicative one. do shall make no further denonst rative use of III.2.l through III.2.4 but shall develog another formula set which will be the foundation of a final partially ordered matrix set. Let us compare III.2.l and III.2.4. mhe first e: these ezzsr esszs the value of accepting true Hi as a func- tion of importance, while the latter expresses the value of rejecting false Hi as a i auction of importance. The guestion 101 to which we new address ourselves is this; For a given Hi how does vLA,T](Hi) compare with VLR,F](H1)? It is obvious that, abstracting from variations in fH and k“, vLA,T](H) and vLR,F](H) vary proportionately with variation in wH. Some unpacking of the functional notations must be effected. It is noteworthy that 'tH' ex- presses nothing about the degree of confirmation of the in- cluding theory, but only permits consideration of H's rival theory as a function of H's acceptance or rejection. Let us now explicitly consider a function of H's degree of con- firmation, namely vH(F). If the inCluding theory explains more than the rival theory, that is to say if uh <.vH(F), then a more significant change in truth claims is effected by accepting the theory if true than by rejecting it if false. In both cases the changes in truth claims are for the good, but in one case more good is brought about than in the other. The converse variation can also be seen to hold: If the rival theory explains more than the including theory, then the change in truth claims is greater when the false theory is rejected than it is when the true one is accepted. These considerations give rise to two formulas. 111.4.1 [w (mm) 3 v[R,F](H) < vL'A,T](H) 111.4.2 LVHU“) < “WI! 3 VEA,T](H) < vLR,F](H) The analogous deve10pment of the formulas for comparison of true rejection and false acceptance is obvious: 102 111.4.15 HrH < vH(F)] D viR,T](H)< v[A,F](H) 111.4.25 [mm < en] 3 v[A,F](H) < v[R,T](H) We shall now repeat formally the assumption.made in the first generic value assignments; III.3.l v[R,T](H) < vLR,F](H) for all H 111.3.2 v[R,T](H)< vagina) for all H III.3.3 v[A,F](H) < vLR,F](H) for all H 111.3.1. vU-,F])H) < vL’A,T](H) for all H Two orderings then become evident, as consequences of the two distinct conditionals.22 T.III.1 W} < v1-1(F)]D viR T](H)< vLA F](H)4v[R,F](H) < v[A,T](H) Lfrom 111.1.1, 111.4.15,111.3.3] T-III-Z [v (PKWP vLA F](H)< vLR T](H)< v[A,T](H) 5§§H