hintikka, laudan and newton: an interrogative model of scientific inquiry

J A M E S W. G A R R I S O N

H I N T I K K A , L A U D A N A N D N E W T O N : AN

I N T E R R O G A T I V E M O D E L O F S C I E N T I F I C

I N Q U I R Y *

I N T R O D U C T I O N

This paper has a twofold purpose. First, it will attempt to unite Jaakko Hintikka's recent extension of his logic of questions and answers to an interrogative model of scientific inquiry with Larry Laudan's conception of science as a problem-solving or question-answering activity. This is not as straightforward a task as it might at first seem, since the two erotetic conceptualizations appear to be diametrically opposed. The result might be called the Hintikka-Laudan interrogative interpretation of scientific inquiry. This done, I will then provide a con- crete historical example of a scientific research project that seems to conform to Hintikka's model, as well as illustrate some of the model's more surprising properties. This second task is important since Hin- tikka presents his logic as being, in one straightforward sense, a logic of discovery. This paper then may be seen as an attempt to place Hintikka's so far largely formal work in the logic of science into the broader context of recent philosophy of science, while at the same time providing it with much needed historical evidence.

H I N T 1 K K A , L A U D A N A N D S C I E N T I F I C I N Q U I R Y

Hintikka's model seems to be inspired by Kant 's claim in the preface to the second edition of the Critique of Pure Reason that one of the fundamental insights of the seventeenth century "natural philosophers" was that

They learned that reason has insight only into that which it produces after a plan of its own, and that it must not allow itself to be kept, as it were, in nature's leading-strings, but must itself show the way with principles of judgement based upon fixed laws, constraining nature to give answer to questions of reason's own determining. Accidental observations, made in obedience to no previously thought-out plan, can never be made to yield a necessary law . . . . (Critique of Pure Reason, B xiii, italics added)

Synthese 74 (1988) 145-171. O 1988 by Kluwer Academic Publishers.

146 J A M E S W. G A R R I S O N

Reason in the present case appears in the form of Hintikka's logic of questions and answers. The questions take the form of controlled experiments.

For Laudan, the flow of questions and answers seems to run opposite to that of Hintikka and Kant. For Laudan, the primary role of reason, this time appearing in the guise of theory, is to "provide solutions to problems. If problems constitute the questions of science, it is theories which constitut~ the answers" (PP, p. 13). Thus Laudan concludes, "the counterpoint between challenging problems and adequate theories is the basic dialectic of s c i e n c e . . . " (PP, p. 14). This is certainly correct, as far as it goes, and it is no doubt important to realize that nature does indeed pose questions to the inquirer, but this alone is not enough. Something is missing. What happens, we might ask, when the existing theory is adequate enough to enable us to recognize a problem, but incapable of solving it? Typically the response is for the inquirer to formulate the best question possible within the confines of the existing theory and pose it to nature in the hope of obtaining some new principle that may then, usually in conjunction with the initial theory, be used to answer an initial question posed by nature (or by some third party different from both the inquirer and nature). This of course is precisely what Hintikka and Kant recom- mend.

An entirely satisfactory model of scientific inquiry assigns com- plementary roles to both interrogative directions. Any barriers to this dissolve as soon as we acknowledge that the dialogue between the inquirer and nature runs both ways and usually occurs simultaneously and in interdependent fashion. On the whole this dialogue is one of the most delightful, as well as necessary, aspects of human existence. Nature never lies, but she seldom reveals all of her secrets at once. Nature, like the Oracle at Delphi, will allow us to deceive ourselves by misinterpreting her answers. If we suffer a tragic reversal of fate, we have only ourselves to blame.

Hintikka sees the beginning of the dialogue with nature in a set of theoretical premises T, or theory, from which the inquirer then tries to prove a given conclusion C. This conclusion of course is an answer to a question originally posed to the inquirer by nature and recognizable as such to the inquirer only by means of the presuppositions of the initial theory. The role of theoretical presuppositions in initiating inquiry is crucial to Laudan as well. Laudan writes, "problems of all

M O D E L O F S C I E N T I F I C I N Q U I R Y 147

sor ts . . , arise within a certain context of inquiry and are partly defined by that .context. Our theoretical presuppositions about the natural order tell us what to expect and what seems peculiar or 'problematic' or questionable (in the literal sense of that term)" (PP, p. 15). Laudan then goes on to note that "whether something is regarded as an empirical problem will depend, in part, on the theories we possess" (PP, p. 15). Laudan is quite specific about the character of these presuppositions when he writes, "If we ask, 'How fast do bodies fall near the earth?' we are assuming there are objects akin to ofir conceptions of body and earth which move towards one another according to some regular rule" (PP, p. 15). It is in this context that Laudan speaks of theory-ladenness.

For Hintikka presuppositions are, like those described by Laudan, typically existential presuppositions, that is, existentially quantified propositions. Such presuppositions allow us to see nature's questions as questions in context and thereby initiate inquiry. But presuppositions also have another function on Hintikka's model, that of governing what questions the inquirer may actually pose to nature. One of the most important restrictions placed on question-asking activity on Hintikka's model is that a question cannot be asked unless the presuppositions for formulating it have already been established. In this way theoretical presuppositions not only initiate, but constrain and guide, the directions an inquiry can initially take. Later, as new presuppositions begin to accumulate from the questions that the investigator constrains nature to answer, new avenues of inquiry begin to open up and old ones to close. Problems often dissolve or change their character as presuppositions evolve. Referring to the role of presuppositions in both initiating and constraining inquiry, Hintikka speaks of "the problem-ladenness or question-ladenness of observations". 1

In the hands of investigators actively engaged in research the role of theory is not primarily its explanatory capability, although a theory's past successes are likely to instill confidence; rather it is the ability of the theory to suggest important problems to be solved, as well as to guide and constrain inquiry, that matters most. These functions are carried out at first by theoretical presuppositions that serve to hold down the size and range of questions that can be asked, and, equally important, that it seems at any given stage of the research project reasonable to ask. Laudan must be careful on this point. The basic

148 J A M E S W . G A R R I S O N

dialectic of science might well be the problem solving effectiveness of theories, but by over-emphasizing this aspect Laudan may come dangerously close, in the eyes of those who don't read him carefully, to expressing a somewhat static view of theories, a view that does not agree with his temporal definition of rational inquiry in terms of a theory's progress in solving its problems. It is necessary to say something about how a theory progresses. By emphasizing the active role of theory in guiding individual research projects occurring within larger traditions, we obtain a better perspective on both the development of individual theories and the traditions that contain them.

For the purpose of deriving the answer , C " to nature's question, the inquirer has available two kinds of 'moves', the choice between which is the inquirer's: deductive moves and interrogative moves. The interchangeability of these two moves is one of the most remarkable features of Hintikka's model.

A deductive move is an application of some suitable set of rules of logical inference. Hintikka assumes that these are E. W. Beth's rules of semantical tableau construction. 2 (They are, of course, but Gent- zen-type rules written in the reverse order.) It is also assumed that only such tableau rules as satisfy the subformula principle of Beth's semantical tableau will be used. The initial entries of the tableau are of course T in the left column and C in the right. In an interrogative move the inquirer addresses a question to nature, assuming~that the presupposition of the question has been established, i.e., occurs in the left column of the relevant subtableau. Interrogative moves; . just like tableau-building rules, are always applied to some one subtableau. If nature can answer, the answer is recorded in the left column as an additional premise that may then be used in deriving C. Obviously the inquirer is trying to close the tableau, with closure defined in the usual way.

How do these two moves compare with each other? The deductive move can always be made, whereas the interrogative move is subject to a number of restrictions, for example, that the presupposition of the question must have already been established in order for the question to be asked. Most of the strategical considerations involved in inquiry concern which presuppositions to instantiate experimentally, and in what order. Another restriction that could be required is that the answer be complete, but this is too restrictive, for it would eliminate one of the most important guides and propellants to inquiry. Partial


answers are a spur to further questioning activity as the inquirer attempts to obtain a complete answer. This is especially the case when the partial answer contains new and perhaps unexpected (anomalous) presuppositions not contained in the original theory. Generally if the question can be asked and answered, the inquirer is better off than if a deductive move had been made instead. For then the researcher will have only one subtableau to worry about whereas a deductive move will leave two. The inquirer's task is thus halved. There is then an extremely close structural connection between deduction and questioning. All the nontrivial steps of inference can be replaced, in favorable circumstances at least, by interrogative steps with potential or actual advantages accruing to the inquirer. So it turns out that question-answer sequences can, at least in favorable circumstances, serve the same purpose as deductions in the logicians' narrow sense, and serve them at least as well and possibly better.

The analogy between deduction and interrogation suggests the remarkable possibility that the , logic of scientific inquiry is, to a surprising degree, just good old-fashioned proof theory - in fine, that many of those things that constrain and guide natural deduction serve, along with some other erotetic components, to constrain and guide scientific research projects. This raises an even more interesting and radical possibility.

For those actively engaged in research, the function of theories is a logical one; to them the raison d'etre of theory is, as we have already said, to sustain and to guide inquiry, and not merely to provide explanations. For Laudan it is progress in answering its empirical problems and overcoming its conceptual ones that is crucial in evalur ating theories and the research traditions that contain them. The examples of Laudan and others provide historical evidence for the constraining role of theories in inquiry. Hintikka's interrogative model displays the logical structure of such constraint. Unlike other pur- ported logics of science, the interrogative model alone exposes the strategic role played by theories in the process of discovery.3~For this reason (among others) it may be regarded as a logic of discovery.

Traditionally many philosophers of science have rejected any notion of a logic of discovery, opting instead for a distinction between the context of discovery and the context of justification. Such a distinction seems to abandon any hope of logically understanding what goes on in most of everyday routine scientific research by declaring it psycho-


logical or even irrational. 4 Small wonder that practicing scientists so rarely find anything of use in the "explications" of the philosophers of science.

Certainly there are no mechanical rules of scientific discovery. This does not mean that there are not systematic, rational and even logical principles that the philosopher may hold up for further study. The notion of logical constraint and guidance we have been outlining illustrates this possibility. All of this is not to say there are not important psychological, or even political or economic considerations present in the context of discovery; there are these and more.

Logically, the nontrivial aspect of research, that which logic cannot decide for us, is which theoretical presuppositions to instantiate, in what order, or how to respond to and use new presuppositions introduced by nature's answers during the course of sustained investigation. The situation is formally analogous to choosing which auxilliary constructions to carry out in what order so as to solve a problem in Euclidian geometry. It can be shown that this is not always a recursively decidable process; hence, there will will be no decision method in the classical sense of the phrase. To acknowledge these considerations is not to reject the possibility of a logic of discovery, but rather to see precisely why there must be one. Were there not a logic of discovery in science we would be very hard pressed to account for how it is that discoveries are made with the frequency and predictability that they so often are. This is just to say once again that theories initiate, constrain and guide inquiry along well established and Understood lines by permitting the inquirer to recognize problems, formulate questions by instantiating theoretical presuppositions, and interpret nature's answers. Without a theory the researcher would face the blooming buzzing confusion of a potentially infinite number of uninterpreted "facts", and with a static theory researchers would not know what to do with those problems that their theory allows them to recognize, but not solve.

There is a very important heuristic role played by logic, interrogative logic at least, in inquiry. Insofar as Hintikka's logic is governed by the traditional laws of proof theory it might even be possible to formulate some very precise suggestions as to what presuppositions to instantiate in what order, even taking fiscal as well as physical constraints into consideration. For example, it might be something of a heuristic axiom to say, on this account, that researchers


ought to ask those experimental questions that instantiate the greatest number of presuppositions given the ever present practical limitations of time, space, energy - and money, providing of course that nature can answer them clearly and unambiguously. It might not be possible to logically deduce why it is that a Newton so often chose precisely the right constructions or questions, but we may be sure that he was guided and constrained in his choices by the presuppositions of the theories originally adhered to and those presuppositions acquired as his inquiry unfolded.

The only general restriction on questions is that the inquirer must establish the presupposition of a question before asking it. There are many other possible restraints. Here we will only mention two that are concerned with the answers to questions. First, it seems reasonable to impose a restriction that allows nature not to answer every admissible question. Some questions are simply ill-conceived or misdirected, as when we misconceive the original problem, i.e., nature's question. Second, there is a whole continuum of restrictions defined by the logical complexity of admissible answers. At one extreme Hintikka has questioning games where questions and answers are limited only by the need of the presuppositions of questions. This is called the u n -

l i m i t e d c a s e . This is the case most resembling deductive proof. The other extreme of the interrogative process is where the only admissible answers are unnegated atomic sentences. This is called the a t o m i c

c a s e . In between there is an entire hierarchy of possible answers. They can be thought of as classified by the quantifier prefixes of the appropriate prenex normal form of admissible answers. If the prefix of the answers is of the form,

(1) ( E x O ( E x 2 ) . . . ( E x k )

the answer is said to be of E-type. If it is

(2 ) ( A x , ) ( m x 2 ) . . . ( A X k )

it is said to be of the A-type. There are also answers of EA-type, but we are most interested in answers of the form,

(3) ( A X l ) ( A x z ) . . . ( A x i ) ( E x , ) ( E x 2 ) . . . ( E x k ) .

If the answer is of this form it is said to be an AE-type answer. What is established in a controlled experiment is some inter-

dependence between different variables. Depending on the kind of


interdependence sought, answers to experimental questions may yield answers of varying degrees of complexity. The most important and sought for answers are those that yield a functional dependency, i.e., some universal law or principle. This kind of answer requires an AE-sentence for its formulation. Any interrogative logic of science must be at least sophisticated enough to deal with questions resulting in AE-type answers.

We cannot here discuss all the details that follow from the AE- assumption. Nonetheless some of its main features are clear. It leads to a model of scientific inquiry that differs sharply from most theories of the progress of science and scientific method. It is unlike the widely accepted hypothetical-deductive model, for in the AE-case new universal laws, and consequently, new theories can be derived from experiments (answers to questions put to nature) by means of the questioning process, sometimes even without an antecedent theory T. This remarkable possibility only emerges if we first conceive of experiments as questions put to nature, and then inspect the various underlying structures of such questions as they are revealed by the interrogative model. The AE-case is also unlike the inductive model, which perhaps can be thought of as having as its logic the A-case. For instance, the answer to a single question can in the AE-case suffice to establish a general conclusion C. The inquirer does not have to collect a large number of instances and then obtain a conclusion from them by generalization. One crucial experiment suffices to derive the principle. Once a well formulated question is answered it is senseless to ask it again. On the AE-assumption Hintikka's model truly becomes a logic of discovery.

A H I S T O R I C A L P R E C E D E N T F O R Q U E S T I O N I N G

P R O C E D U R E

But are there any real examples of scientists conducting their research in accordance with Hintikka's interrogative model? Where, for instance, do you ever find scientists conducting their lines of research in terms of the kind of mixture between deductive 'moves' and questioning 'moves', i.e., experiments and observations, suggested by Hin- tikka? Or where do we see a researcher drawing perfectly general conclusions from isolated individual experiments?

M O D E L OF S C I E N T I F I C I N Q U I R Y 153

The answer, to state the matter boldly, is Newton's mode of argumentation in his Opticks. There he flaunts all the usual parapher- nalia of an axiomatic and deductive system: axioms, definitions, propositions, theorems, problems, etc. Yet from the very beginning there is something else most unusual that cannot be easily accounted for by inductive and hypothetico-deductive models of science:

My Design in this Book is not to explain the Properties of light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which I shall premise the following Definitions and Axioms. (Opticks p. 1)

There are many barriers to a full and proper understanding of this brief but enigmatic passage as well as the Opticks itself. One of them, proof by experiment, is easily removed by the interrogative model. Newton is prepared to introduce the results of experiments as additional premises of his argument at any stage in its development even though they are not implied by his axioms or definitions. It was argued above that, in applying our interrogative model to actual scientific practice, we must think of experiments as questions addressed to nature. If so, Newton is to all intents and purposes following the interrogative model to a tee.

Normal scientific inquiry is initiated by some theory T assumed by the researcher(s) as true for the purposes of conducting scientific investigation. In the Opticks this initial theory is comprised of eight definitions and eight axioms that make up "the sum of what hath hitherto been treated of in Opticks. For what hath been generally agreed on I content myself to assume under the notion of Principles, in order to what I have further to write" (Opticks, pp. 19-20). I would like to call the reader's attention to Newton's notion of "Principles". Newton's use of this term in the Opticks exactly parallels his usage in the Principia. In both cases the term is used to denote the most general (mathematical) premises used in the deductive demonstration of (empirical) consequences. For Newton the difference between demonstration of and derivation from is immense. This logical isomorphism barely scratches the surface of the broader and deeper logical, methodological and mathematical unity between the two works, a unity that merely expresses the underlying unity and power of their author's thought. The general consensus of 20th century historical opinion has been to assign the "experimental" Opticks a secondary status behind the more "mathematical" Principia. This error arises out


of mistaking a difference in emphasis as a difference in design. Worse still, it overlooks Newton's most enduring achievement: the est- ablishment of a mathematically rigorous "experimental Philosophy".

There are several definitions and axioms that require special attention. Consider for instance the first definition (Def. 1): "By the Rays of Light I understand its least Parts, and those as Well in the same Lines, as Contemporary in several lines". As Henry Guerlac remarks of this definition,

In spirit it is as good an example of Newton 's mathemat ica l way as the Principia: light is treated as a mathemat ica l entity, as rays that can be represented by lines; the axioms with which he begins are the accepted laws of optics; and numbers - the different refrangibilities . . . . W h e n e v e r appropriate, and this is most of the time, his l anguage of exper imental descript ion is the language of number . 5

Two things in this call for additional comment. First, there is Guerlac's recognition that the Opticks is indeed every bit as mathematical as the Principia, i.e., that both works represent "The Mathematical Prin- cipals of Natural Philosophy". That the mathematics of the Opticks is simple Euclidian geometry whereas that of the Principia is, frequently, that of the then newly-emergent analysis, serves only to render the latter more abstruse, but not any more mathematical. Second, Definition I, by establishing a geometrical optics, frames the entire ensuing dialogue between inquirer and nature in the language of mathematics. Both questions and answers must be formulated in this language. The tacit assumption of course is that Galileo was correct, that the book of nature "is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single w o r d . . . ,,.6 There is a sense in which this statement is even truer for Newton than its original author.

Two other definitions, numbers VII and VIII, are deserving of special consideration also, as much for the comments Newton makes concerning them as for what they say. Definition VII reads, "The Light whose Rays are all alike Refrangible, I call Simple, Homogeneal and Similar; that whose Rays some more Refrangible than others, I call compound, Heterogeneal and dissimilar". Of this definition Newton remarks that he calls light homogeneal "not because I affirm it so in all respects, but because the Rays which agree in Refrangibility, agree at least in all those of their other properties which I consider in the


following Discourse". This definition is not complete until Newton examines and defines, experimentally, those "other Properties" that remain to be considered in his discourse. One of the remarkable features of Hintikka's model not dealt with earlier is its treatment Of just such definitory 'moves', where the definition is determined by an experimental question. 7

Definition VIII begins, "The Colours of Homogeneal Lights, I call Primary, Homogeneal and Simple; and those of Hetergeneal Lights, Heterogeneal and Compound". Here too, Newton's definition gets somewhat ahead of itself since it has yet to be shown that light is "Heterogeneal and Compound". That light even has the property "heterogeneity" is yet to be derived from phenomena. Here the definition not only initiates inquiry but anticipates some of the results.

Axiom V states that "The Sine of Incidence is either accurately or very nearly in a given Ratio to the Sine of Refraction". This is of course merely the Snell-Descartes law. The other axioms serve similarly to sum up the developments in optics up to Newton's time.

The first six propositions of Book One, Part I of the Opticks are all proved either entirely or in part by experiments. None of these propositions could be proven on the basis of the original theory T alone, although these proofs do make frequent use of T and/or other propositions already established by their own "experimental Proof". It is the very fact that these propositions cannot be proven from the already existing "generally agreed on" theory that marks them as a significant research advance in the development of optics, The "proofs" of these propositions depend upon nature providing answers to experimental questions posed by the inquirer. Newton skillfully employs the analogy between deductive and interrogative moves in the logic of questions and answers to bring about what represented at least something very close to a revolution in the optical theory at the time. We must take a closer look a-t some of Newton's experimental proofs and how they interrelate.

Experiments three through ten prove "Prop. II. Theor. II."; i.e., that "The light of the Sun consists of Rays differently Refrangible". Here we will only be concerned with experiments three and five. Before moving on, it is worth noting that all of these experiments serve to complete definitions VII and VIII while simultaneously requiring that those definitions be already well enough understood to allow the proof to pass.


Ernst Mach describes the experimental arrangement for question three:

The rays which transverse a prism at minimum deviation subtend equal angles with incident and emergent faces of the prism, and small variations in the angle of incidence do not cause appreciable variation in the deviation by reason of the property of the minimum [see Figure 1]. For this reason similar rays of sunlight, which subtend an angle of 31' with each other at incidence, emerge from the prism inclined at the same angle. The length of the spectrum is thus explicable only by unequal refraction of the different CO|OUrS. 8

M

C

N Fig. l.

E X

This proof is not especially mathematical, although it is important to note that it does have an underlying geometricality that may be revealed by invoking Definition I. Here the inquirer is placed into direct perceptual acquaintance with the "celebrated Phaenomena of Colours". Hintikka refers to such perceptual answers as demonstrative or ostensive answers. Newton himself stresses the perceptual character of this proof: "And therefore seeing by Experience it is found that the image is not round, but about five times longer than broad, the Rays which going to the upper end P of the Image suffer the greatest Refraction, must be more refrangible than those which go to the lower end T . . . " (Opticks, p. 32). No measurement or other mathematical determination need be made to see "by Experience" that the proposition is indeed true.

The most noteworthy and remarkable consequence of experiment three is that it proves Axiom V of the original theory T does not hold for all rays apart as was "vulgarly supposed". Newton's response to this anomalous result is typical. First he checks to see if the question has been correctly formulated and properly asked. Newton writes, "because it is easy to commit a Mistake in placing the Prism in due


Posture, I repeated the Experiment four or five Times and always found the Length of the Image that which is set down above". Newton next tries the experiment "With another Prism of clearer Glass and better Polish", and "with different thickness of the Prism where the Rays passed through it, and different inclinations of the Prism to the Horizon, [all of which] made no sensible changes in the length of the Image". Newton continues on in this vein before concluding that "according to the Laws of Opticks vulgarly received, they could not possibly be so much inclined to one another" (Opticks, pp. 30-31). Note that Newton does not repeat the experiment in order to expand his inductive base, but only to be sure the experimental question was properly asked. Experiment three and" its companion experiment four reopen a question assumed t 9 have already been answered by the initial theory T. Not only are the answers of experimental questions capable in some instances of introducing general principles in- dependently of the initial theory, they are even logically capable of calling a general principle - in this case a theoretical axiom - of the initial theory back into question. It is not Axiom V alone that is called into question, but every proposition whose demonstration depends upon it. Newton has a problem.

Experiments three and four not only ~reopen a question already assumed answered, but give rise to a number of new questions as well. Newton notes some of these questions at the end of his discussion of experiment four:

So then, by these two Experiments it appears, that in Equal Incidences there is a considerable inequality of Refraction. But whence this inequality arise, whethei- it be that some of the incident Rays are refracted more, or other less, constantly, or by chance, or that one and the same Ray is by Refraction disturbed, shatter'd dilated, and as it were split and spread into many diverging Rays, as Grimaldo supposes, does not yet appear by these Experiments, but will appear by those that follow (Opticks, p. 34).

Experiments three and four provide the necessary presuppositions for a series of questions that would literally be inconceivable without them. These experiments~'open up one line of inquiry and reopen another. Either one of these conditions is enough to initiate what Kuhn has called abnormal or revolutionary science. Perhaps, but if so there is nothing patently irrational about it. True, the inquirer has a newfound logical freedom, but this is not radical freedom of the kind likely to induce psychological nausea or angst. The inquirer has some new presuppositions and must discard some old ones; the world-


picture will change, but not entirely beyond recognition; the subsequent inquiry will be constrained and guided by the presuppositions that continue to be accepted, along with any new ones that may emerge in the subsequent interrogative "chase of Pan", to borrow a phrase from Francis Bacon.

The configuration of Newton's next experimental proof is readily constructed from that of experiment three by adding, in Newton's words, "a second Prism immediately after the first in a cross Position to it .... " (Opticks, p. 34). See Figure 2 below. Experiment five logically as well as physically presupposes experiment three. The introduction of a second prism produces a second spectrum which may then be compared to the first to yield a mathematically more precise and complete answer. Newton describes what "is found by Experience" in this experiment: "by the Refraction of the second Prism, the breadth of the Image was not increased, but its superior part, which in the first Prism suffered the greater Refraction, and appeared violet and blue, did again in the second Prism suffer a greater Refraction than its inferior part, which appeared red and yellow, and this without any Dilation of the Image in breadth" (Opticks, pp. 34-35). This result is enough to answer Grimaldi's supposition in the negative; the spectrum cannot be due to diffusion.

By comparing the two prisms it is possible to determine, mathematically determine, that the ray "which in the first Prism is more or less refracted, is exactly in the same proportion more or less refracted in the second" (Opticks, p. 42). Thus experiment five provides a more exact and mathematically determined answer and proof of proposition VI than does experiment three. Newton concludes that every ray

Q[-'-I K q i..., ..... ! k

R U L r ~; . . . . . . . : l S M o i; . . . . . . . : ' :: ira v N

T H

Fig. 2.

S

7


considered apart does indeed "differ in degree of Refrangibility, and that in some certain and constant Proportion" (Opticks, p. 42). Exactly what that constant of proportion is, is not determined by the experiment; that is another question.

As a consequence of experiment five and its successors Newton is prepared to open his discussion of Prop. VI by confidently declaring "That every Ray considered apart, is to itself in some degree of Refrangibility, is sufficiently manifest out of what has been said" (Opticks, p. 75). Newton specifically mentions experiments five through nine and twelve through fourteen. Newton then closes the opening paragraph of his discussion by asserting that "The Refraction therefore of every Ray apart is regular, and what Rule that Refraction observes we are now to shew" (Opticks, p. 76). Clearly Newton seeks to prove some comprehensive principle with which to unite the vague, inexact and mathematically undetermined but suggestive experimental appearances, comparisons and measurements that have gone before.

The enunciation of "Prop. VI. Theor. V." of Book One, Part I of the Opticks reads "The Sine of Incidence of every Ray considered apart, is to its Sine of Refraction in a given Ratio". In the words of E. W. Strong, "A principle in physics is 'mathematical' if its enunciation states a ratio or proportion, or as we would now say, a formula or functional relationship" (Strong, p. 418). Prop. VI is the first universal principle proven in the Opticks. Needless to say it is intended to reestablish a more general version of Axiom V. Prop. VI is proven by means of experiment fifteen.

However useful the intervening experimental proofs, propositions, theorems and problems may have been in answering the questions posed at the end of experiment four, they nonetheless give rise to many new questions of their own that require answering, while failing to derive a replacement for Axiom V. In particular they establish that there does indeed exist a constant proportion between the angle of incidence and the angle of refraction, but they fail to specify exactly what that constant of proportionality might be, nor whether or not it /,aries, as well it might, with the angle of incidence. The determination of the latter question is important for the continued development of Newton's inquiry in the Opticks. What is lacking in these earlier experiments is brought out by the following statement of E. W. Strong: "Methodologically, there is a 'determining mathematically' in observation and experiment in making measurements; but such


measurements do not of themselves yield laws" (p. 419). Earlier Strong had declared that "quantity is only traced out if there are measures for it is measurement alone which provides quantified data for calculation" (p. 418). Proposition VI draws together, both logically and experimentally, into one unifying law the mathematical deter- minations that precede it. This type of convergence is typical of the interrogative model wherein inquiry is directed toward establishing some conclusion, i.e., solving some problem. Strong clarifies the nature of such a unifying question: "To compute a ratio or proportion of quantified data is to institute a rule of measure: and such a rule of measure is the comprehension of what he has measured" (p. 419). Strong refers specifically to Newton's "mathematical demonstration" of Prop. VI. of the Opticks as an instance of such a comprehensive rule.

Strong's remarks are suggestive, although precisely what the difference is between those observations and experiments, such as experiments five through nine and twelve through fourteen, that contain a "determining mathematically" (the phrase is Newton's) but do not express a rule of measure, and those that do, is left unclear. Nor is Strong especially clear on what their relation might be. He seems to suggest that the former provide "da ta" for the latter, but as we shall see this cannot be the relation between experiments three, five and fifteen, in spite of the fact that they bear a most intimate, almost familial relationship to one another.

Newton's approach is, once again, methodologically and logically conservative, not to mention atypical. Experiments three and four only prove that Axiom V is not a general principle applicable, as it stands, to every ray apart; it leaves open the question whether or not it may hold for some rays. The error of the "vulgar" optics of the day was that the homogeneity of the sun's light was erroneously presup- posed. Actually it is not so much Axiom V per se that is rejected; rather it is one of its tacit presuppositions. One of the valuable uses of questions, whether put to oneself or nature, is to activate such tacit knowledge. This is a valuable process even when the "knowledge" proves false. Experiments three and four expose the error, yet Newton remains quite willing to accept the measurements of his predecessors, " the late Writers in Opticks", as holding for "Rays which have a mean Degree of R e f r a n g i b i l i t y . . . " (Opticks, p. 76). Newton's ploy is to reinterpret Axiom V as an answer to a more restricted question


regarding rays in the middle of the spectrum by convert ing an un- restricted, universally quantified answer of A E structure into a restricted universally quantified answer that may then serve a s a much needed principle to be applied in the experimental proof of Prop. VI. What Newton is arguing is that nature answered the earlier question as honestly as she could, but that her answer was misinterpreted due to overgeneralization, as a consequence not of faulty inductive tech- niques, but rather of a tacit and mistaken presupposition regarding the homogenei ty of sunlight. Newton's move is conservative in that it conserves Axiom V as a special or limiting case useful in formulating more general questions that, hopefully, may result in more universal answers. Newton's strategical ploy, if that is what we should call it, makes perfectly good sense in terms of the interrogative model.

Newton's maneuver also seems to accord well with his own methodological remarks near the end of the Opticks. There Newton writes that,

although the arguing from Experiments and Observations by Induction be no Demon- stration of general Conclusions; yet is the best way of arguing which the Nature of Things admits of, and may be looked upon as so much the stronger, by how much the Induction is more general. And if no Exception occur from Phaenomena, the Con- clusion may be pronounced generally. But if at any time afterwards any Exception shall occur from Experiments, it may then begin to be pronounced with such Exceptions as occur. (Opticks, p. 404).

Clearly, in the case of Axiom V, an exception has occurred from experiments three and four, and as a consequence Newton begins to pronounce the weaker and less general conclusion. Note how different this interpretation is from that offered for the same methodological passage by Imre Lakatos. In Proofs and Refutations (pp. 26-27) Lakatos refers to Newton's ploy as an "exception-barring method" , and so it may appear as an isolated methodological statement. Placed, as it is here, in the broader context of Newton's actual methodological practice Newton's maneuver looks quite different. Newton's ploy is not to block exceptions so much as to arrive at a restricted universal that may then be used in the experimental proof of a stronger, more general principle that expands the domain of application including, among other things, the earlier more restricted principle. This boot- strapping technique will be further discussed in the conclusion.

Newton's maneuver is very much in the spirit, if not the letter, of some of Larry Laudan's comments concerning 'Convert ing Anomalies


to Solved Problems', as he titles one section of his book. There the topic is competing research traditions. One way of co-opting a competing research tradition, or what is logically equivalent, earlier research, is to absorb its results into your own tradition as a limiting or special case; this is precisely what Newton does. Laudan writes

One of the most cognitively significant activities in which any scientist can engage is the successful transformation of a presumed empirical anomaly for a theory into a confirmation instance for that t h e o r y . . , conversion of anomalies into problem-solving successes does double service: it not only exhibits the problem solving capacities of a t h e o r y . . , but it simultaneously eliminates one of the major cognitive liabilities con- fronting the theory• This process of converting anomal ies • . , into solved problems [answered questions] is as old as science i t se l f . • . (PP, p. 307).

So it is. The connections between the earlier experiments, three and four,

may be seen, literally, by looking at Figure 3. As in experiment 3, S represents the sun's "round white Image painted on the opposite wall b y his [the sun's] direct Light", and as in the experiment five "PT [represents] his oblong coloured Image made by refracting that Light with a prism placed at the Window". Finally, "pt, or 2p2t, 3p3t, [represents] his oblong colour'd Image made by refracting again the same Light side-ways with a second Prism placed immediately after the first in a cross Position to it, as was explained in the fifth

3t 2t t T

iiiiiiiiiiiii iiiiiiii iiiiiiiiiii 3p. 2p p P

• ".. ~ "'.. ..,• -

•-.-5..\i "...-..::.. "~.

~-.t s ~m q!n

Fig. 3.


Experiment; that is to say, pt when the Refraction of the second Prism is small, 2p2t when its Refraction is greater, and 3p3t when it is greatest" (Opticks, pp. 7-8). Newton himself suggests angles of "fifteen or twenty Degrees . . . of thirty or fo r ty . . , and of sixty", to provide the three additional spectrums generated by the second 'prism.

The ancestral relation between experiment fifteen and its predecessors may be viewed in two seemingly different and yet compatible ways. One way of conceiving what Newton does in experiment fifteen, or any of its predecessors is to take an instantial (or configurational) interpretation. On this interpretation we see Newton or any other inquirer as analyzing the experimental configuration as an instantiated instance of the physical theorem to be proven in precisely the same way a geometrician might analyze a geometrical instantiation of a geometrical configuration or figure. What he studies is the interdependence between known and unknown geometrical entities. If necessary the analyst introduces additional auxiliary constructions into the figure in order to arrive at a more mathematically determined configuration, one that will lead him, hopefully, at least closer to the proof sought. As we have already indicated, which constructions to carry out and in what order is the nontrivial part of any proof, and is what gives rise to strategical difficulties that are not always easily decided - similarly in the physical case.

We may see Newton making precisely analogous moves in the Opticks. There he carries out an analysis of the experiment configuration. Just as the spectrum pt was first added to experiment three in order to obtain a configuration sufficiently determined mathematically to prove a "certain and constant Proportion", in experiment five, so too were spectrums 2p, 2t and 3p, 3t added as optical auxiliary constructions to configuration five in order to obtain a configuration sufficiently determined mathematically to prove a rule of measure. The analogy between the geometrical and 01btical cases actually collapses if we invoke Definition I and recall that Newton's Opticks is indeed a geometrical optics.

Conceived geometrically and analytically the difference between those experiments and observations that determine a rule of measure and those that don't comes down to this. Those experiments that do represent completed analyses, whereas those that don't represent only incomplete or partial analyses. A complete analysis reduces what is sought, but assumed known by hypothesis, to that which is given by


some fixed and determinate ratio. A partial analysis, for example experiment five, may only yield some more or less indeterminate, vague and inexact relation. The configuration of experiment fifteen depends geometrically on experiments three and five for its existence. It is constructed out of them by adding auxiliary constructions to the original figure, i.e., experiment three. Prop. VI is related to experi-

m e n t fifteen, and the five propositions and fourteen experiments that preceded, as conclusion is to proof.

The dependence of experiment fifteen upon its predecessors, especially experiments three and five, may also be straightforwardly gras- ped using the interrogative model. Experiment three raises the logical possibility of even asking a question like experiment fifteen by revok- ing a presupposition that would otherwise block its formulation while simultaneously establishing a presupposition, i.e., the heterogeneity of light, in the left hand column of the tableaux such that the question may be asked. Experiment five provides a partial answer to the question reopened by experiments three and four. Experiments three and five in conjunction with the other experiments preceding fifteen furnish the presuppositions and principles necessary to formulate the experimental question clearly and unambiguously. Stated differently. experiments one through fourteen collectively prove five propositions - four theorems and a solved problem (Prob. IV) - t h a t may then serve as additional premises in the proof of proposition VI.

Surprisingly the two in/erpretations, instantial and interrogative, are virtually identical. The individual experimental questions we have discussed are all formulated in the language of nature as Newton saw it - geometry. The answers were similarly geometrical or quasi- geometrical, that is they yielded some mathematical interdependency more or less exact, or more or less general depending on the quantifier prenex of the answer. Experiment fifteen, to state the matter bluntly, is the first experimental question posed by Newton in the Opticks whose answer has the internal logical complexity characteristic of the AE- case. ° Only a question so formulated as to receive such an answer is logically capable of proving an entirely general, functional rule of measure. An answer of any lesser degree of complexity, e.g., atomis- tic, A, E, or even EA cases, is simply logically incapable of establishing a general conclusion.

Geometrically, the best the earlier experiments, for example


experiment five, could do was to provide a vague, inexact, only partially mathematically determined relation: what was sought was a general rule of measure; only experiment fifteen is capable logically or mathematically (analytically) of doing this.

The analogy between the two interpretations seems even deeper once we recognize that geometrical auxiliary constructions and the presuppositions of questions have exactly the same logical structure, are in fact the same thing. Both are existentially quantified propositions.

From this vantage point it is easy to see what Strong's distinction amounts to in terms of the interrogative model. Logically, on Hin- tikka's model, experiment fifteen is the first experiment of the Opticks whose answer is an AE-type sentence. This means that it alone is capable, logically capable, of )Tielding a universal generalization in the form of a functional rule of measure. This not only allows us to draw Strong's distinction sharply and on logical rather than exclusively mathematical grounds, but also to see exactly what the relation between experiment fifteen and its predecessors amounts to. The earlier experiments establish the presuppositions necessary to formulate the concluding question. Without all of the correct presuppositions the experimental questions would be more or less under- determined and subject to misinterpretation, a potentially tragic situation.

The last observation leads directly to another. Why doesn't Newton attempt to jump directly to the desired conclusion by asking the "big" question immediately, rather than asking a series of "smaller" questions? The answer might well be that Newton does not wish to commit the fallacy of begging the question. This fallacy is given rigorous characterization by Hintikka's model where a question may not be asked by the inquirer before its presuppositions have been established. Of course, the presuppositions requisite to asking the principle, question are not usually immediately available to the inquirer. Con- sequently, to avoid the "~allacy" of petitioprineipii Newton petitions nature with a protracted series of smaller questions first, before asking the principle question, that is the question that is intended to establish the universal principle. Even so, it is necessary at the end of this long series of questions, answers, and deductions to convert Axiom V into an additional principle in order to proceed to the concluding question.


C O N C L U S I O N S A N D D I S C U R S I V E P O S T S C R I P T

We have seen how Hintikka's interrogative model of inquiry may be extended to include scientific inquiry by c?nceiving experiments as questions put to nature; and how, by acknowledging that the dialogue between the inquirer and nature runs both ways, Hintikka's model may be drawn into agreement with Larry Laudan's notion of science as a proble m solving activity. Next we saw how this combined interpretation might be illustrated by part of a scientific research project carried out by Isaac Newton in the context of one of his most famous discoveries.

The latter task went smoothly enough, although there is much that requires further and more detailed study. Newton's methodology has alWays been an anomaly ~ to historians and philosophers of science, especially to philosophers who are committed (as, e.g., Karl Popper is) to the hypothetico-deductive conception of science. I suspect that the reason for this is that they have, almost without exception, assumed the hypothetical-deductive method, in one form or another, as their normative and descriptive standard. 9 Frustrated, some, such as J. M. Keynes, have given up and declared Newton "that last of the magi- cians", arguing that for Newton "experiments were always . . . . a means, not of discovery, but always of verifying-what he knew already". 1° Others, such as Hempel, seemto think that Newton was simply methodologically confused. ~ Current Newtonian scholarship, on the other hand, mostly seems bent on saving Newton as a hypothetico-deductivist.

All of this seems rather remarkable given that Newton himself makes it perfectly clear what his method actually was; it is the ancient and venerated method of analysis. Newton's longish methodological statement made near the end of the Opticks is ample testimony to this fact:

As in Mathematics, So in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and in admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy . . . . By this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis; And

MODEL OF SCIENTIFIC INQUIRY 167

the Synthesis consists in assuming the Causes discover'd and establish'd as Principles, and by them explaining the Phaenomena proceeding from them, and proving the Explanations. (Opticks, pp. 404-05) 12

It seems at least reasonable to have a go at evaluating Newton by the methodological standards he sets for himself. This Hintikka's m e t h o d makes it possible for us to do.

It is the second sentence in the above passage that prompts the instantial or analysis of configurations interpretation of experimental questions taken in this paper. Unlike others I propose taking Newton's analogy between experimental and geometrical analysis, especially in the context of a geometrical optics, quite seriously. By so doing it becomes possible to make a start toward unraveling Newton's use of such methodological terms and phrases as induction, hypotheses, crucial experiments and deriving principles from phenomena.

We are further aided in understanding Newton's method by dis- tinguishing a second sense of analysis not unrelated to the first. This second sense might be referred to as propositional analysis. In propositional analysis the investigator seeks to deductively demonstrate some propositional conclusion C from a given set of theoretical propositions T. Sounds familiar. The heuristic secret of propositional analysis is that the principles and presuppositions of the conclusion, as well as those of the premises, may be used in seeking the desired proof in the hope that if some connection may be made the process may then be reversed in synthesis to obtain a deductive demonstration - or explanation. This is precisely what E. W. Beth had in mind when he declares of his method of semantical tableau that it "realises to a considerable extent the conception of a purely analytical method, which has played such an important role in t h e history of logic and philosophy". 13 This means tha t Newton's method is built directly into Hintikka's model. We may see Newton making use of the heuristic power of this method when he converts Axiom V into what amounts to a special limiting case of the general conclusion he is trying to prove and then using it as a principle in asking the experimental quegtion that proves that conclusion.

Experimental (configurational) analyses are but small questioning steps used to prove some principle, i.e., establish answers, that may then be used in the deductive demonstration of the sought for conclusion: "the big" question originally posed by nature. The two methods are thus used simultaneously. The heuristic power of


configurational analysis lies in its assuming the desired construction (experimental entity or variable) as complete or carried out (existent) and then working backwards to the original figure. In both senses of analysis the heuristic element consists in the ability to work backwards as well as forwards, or, in the words of Pappus "we suppose the thing sought as being and as being true, and then we pass through its concommitants [consequences] in order, as though they were true and existent by hypothesis, to something admitted . . . . 14 Hypotheses as they might be used in both senses of analysis are, I believe, precisely what Newton had in mind when he wrote "For hypotheses should be subservient only in explaining the properties of things, but not assumed in determining them; unless so far as they may furnish experiments". 15 What Newton means by hypotheses when he is con- demning them are explanatory principles not derived in one way or another from an experimental analysis. This is the case in the conversion of Axiom V into a special case of the desired conclusion. There Newton "presumed" that "the late Writers in Opticks" must have "adapted their Measures only to the middle of the refracted L i g h t . . . " (Opticks, p. 76). Newton's target when he speaks nega- tively of hypotheses is above all Descartes, who by his own admission does not hesitate to use explanatory functionsJ 6

In the present context it is easy to understand what Newton means by such notions as deriving principles from phenomena, induction and crucial experiments. The individual experimental question formulated in the mathematical language of nature contains a "determining mathematicality", perhaps even a rule of measure, within the experimental configuration. Newton's "derivation" amounts to little more than recognizing the universal in the particular. For Newton induction was neither elimination or enumeration. Newton explicitly defends his experimentum crucis against the "many hundreds of trials" of Hooke. All induction meant for Newton was the generalizing step, pronounc- ing the universal contained in the experimental particular generally; or, what's the same thing, recording the answer to the experimental question on the left side of the Beth tableau as a premise to be used in subsequent deductions. A well conceived experimental question, if properly asked and answered, need only be asked once; it is in this sense crucial. It is senseless, even ill-mannered, to repeat a question already answered. Just as a geometer would see no reason to construct more geometrical figures to verify his analysis and corresponding proof of a geometrical theorem or problem, Newton likewise felt no compulsion to repeat a well executed experiment unless it was to make

MODEL OF S C I E N T I F I C INQUIRY 169

sure tha t the ques t ion was a sked co r rec t ly , o r unless some " E x c e p t i o n shall o c c u r f rom E x p e r i m e n t s " , in which ins tance , as in the case of A x i o m V and e x p e r i m e n t three , " i t may then beg in to be p r o n o u n c e d with such E x c e p t i o n s as o c c u r " . It has been a puzzle to m a n y in- duc t iv i s t s that thei r m e t h o d o l o g i c a l qua lms and c o n c e r n s go so l a rge ly u n n o t i c e d by the m e t h o d o l o g i c a l l y " n a i v e " in the m o r e m a t h e m a t i c - al ly a d v a n c e d sc iences . C o u l d it be that those sc iences that have t r ad i t iona l ly re l i ed m o r e on the ana ly t ica l than the h y p o t h e t i c o - d e d u c t i v e m e t h o d con t inue to do so?

T h e p r e c e d i n g is on ly a s tar t , an a t t e m p t to p r o v i d e a nascen t me tasc ien t i f i c r e s e a r c h t r ad i t ion with some p laus ib i l i ty by c o n v e r t i n g one of the ou t s t and ing r e sea rch anoma l i e s of the cu r r en t ly d o m i n a n t t r ad i t ion into a p r o b l e m solv ing success . If the resul ts a re not to ta l ly conv inc ing , e spec ia l ly to those who owe a l l eg i ance to the be t t e r e s t ab l i shed t r ad i t ion , it mus t be c o n c e d e d , I think, tha t it wou ld be at leas t r e a s o n a b l e to pursue the inves t iga t ion fur ther .

NOTES

* Research for this paper was supported by NSF Grant IST-8310936. An earlier version was read at the Conference on Knowledge-Seeking by Questioning, April 1985, at The Florida State University, Tallahassee, Florida.

Jaakko Hintikka and Merrill B. Hintikka: 1982, 'Sherlock Holmes Confronts Modern Logic: Toward a Theory of Information-Seeking Through Questioning'; in E. M. Barth and J. L. Martins (eds.), Argumentation Approaches to Theory Formation. 2 E. W. Beth: 1969, 'Semantic Entailment and Formal Derivability', reprinted in Jaakko Hintikka (ed.), The Philosophy of Mathematics. 3 One exception would seem to be the structuralist approach of Sneed and Stegmuller whose semantical treatment of propositions as class models appears on the surface at least to resemble that of Hintikka, but this resemblance is more apparent than real. As V. N. Sadovsky has observed, all that the structural formulations offer is a "static dynamics of theories". Sneed and Stegmuller, not unlike the earlier explicators of completed scientific theories, seem only able to identify the logical difference between successive theories alter the changes have already taken place. See V. N. Sadovsky: 1981, 'Logic and the Theory of Scientific Change', in J. Hintikka, D. Gruender, and E. Agazzi (eds.), Theory Change, Ancient Aiomatis and Galileo's Methodology. 4 See Reichenbach, Experience and Prediction, p. 6. 5 Henry Guerlac, 'Newton and the Method of Analysis' in the Dictionary of the History of Ideas, Vol. III, p. 389. 6 The Assayer (el Saggiotore), translated by Stillman Drake in (1956) Discoveries and Opinions of Galileo, Doubleday and Company, New York, p. 238. 7 See Opticks, p. 125 for such an experimental proof.

Ernst Mach: 1925, The Principles of Physical Opticks, E. P. Dutton Publishers, New York, p. 88.


9 The first principles of the Opticks are induced, but not in a way that the modern inductivist would recognize or even acknowledge. Inquiry must begin somewhere and in a somewhat arbitrary manner lest we fall into an infinite regress or the dilemma of the Meno. For Newton the dialogue with nature begins with "what hath been generally agreed on" in order to proceed to that which he has "farther to write". Remarkably this is precisely the foundation laid down by Aristotle for dialectical arguments. In Topics (B.1, 100a pp. 25ff.) Aristotle writes:

Now reasoning is an argument in which, certain things being laid down, something other than these necessarily comes about through them. (a) It is a 'demonstration,' when the premisses from which the reasoning starts are true and primary, or are such that our knowledge of them has originally come through premisses are primary and true: (b) reasoning, on the other hand, is 'dialectical', if it reasons from opinions that are generally accep ted . . , these opinions are 'generally accepted' which are accepted by every one or by the majority or by the [natural?] philosophers - i.e., by all or by the majority, or by the most notable and illustrious of them.

In initiating inquiry Newton simply departed from that which was "vulgarly supposed". This and the fact that experiments three and four clearly establish that Axiom V is false is enough to make it clear that Newton is engaged in dialectical rather than demonstrative inquiry. The preceding passage seems to indicate that the principles, or primary premises, of a science need not be induced from sense-perception at all, but from generally accepted opinions, endoxa. But as G. E. L. Owen has documented, well founded opinions, endoxa, were among the phenomena a scientific explanation was supposed to account for, according to Aristotle. Ultimately, Owen suggests, and I would include Newton in the remark as well, "endoxa also rest on experience, even if they misrepresent it" (cf. Parva Naturalia 462b pp. 14-18). 1o j . M. Keynes, 'Newton the Man', in Essays in Biography, p. 313. tl Carl G: Hempell: 1966, Philosophy of Natural Science, Prentice-Hall, Englewood Cliffs, N.J., p. 15. The mention of "rules of Induction" is a rather clear reference to Newton's five rules in the Principia. t2 The middle of this passage was cited on page 22 and bears rereading in the present context. 13 E. W. Beth, 'Semantic Entailment and Formal Derivability', p. 19. 14 This translation, is borrowed from Jaakk0 Hintikka and Unto Remes: 1974, The Method of Analysis, D. Reidel, Dordrecht, p. 9. My paper owes a great deal to chapter IX of this work. 15 See I. B. Cohen (ed.): 1978, Isaac Newton's Papers and Letters on Natural Philoso- phy, Harvard University Press, Cambridge, Massachusetts, p. 196. 16 See Alexander Koyre: 1965, Newtonian Studies, Harvard University Press, Cam- bridge, Massachusetts, especially p. 34.

R E F E R E N C E S

Hintikka, Jaakko: 1976, 'The Semantics of Questions and the Questions of Semantics, Acta Philosophica Fennica 28, no. 4, Helsinki.

MODEL OF SCIENTIFIC INQUIRY 171

Hintikka, Jaakko: 1984a, 'The Logic of Science As A Model-Oriented Logic', in P. D. Asquith and P. Kitcher (eds.), PSA 1984, Philosophy of Science Association, East Lansing, MI, pp. 117-85.

Hintikka, Jaakko: 1984b, 'Rules, Utilities, and Strategies in Dialogical Games', in Lucia Vaina and Jaakko Hintikka (eds.), Cognitive Constraints on Communication, D. Reidel, Dordrecht, pp. 227-94.

Hintikka, Jaakko: forthcoming, 'Some Central Concepts of Metascience In An Inter- rogative Model of Inquiry'.

Hintikka, Jaakko: forthcoming, 'A Spectrum Of Logics Of Questioning'. Lakatos, Imre: 1976, Proof and RefutationS, Cambridge University Press, Cambridge. Laudan, Larry: 1977, Progress and its Problems, University of California Press, Ber-

keley. Newton, Isaac: 1730, Opticks, fourth covered edition reprinted by Whittlesey House,"

McGraw-Hill, New York, 1931. Strong, E. W.: 1957, 'Newton's Mathematical Way', reprinted in Phillip P. Wiener and

Aaron Noland (eds.), Roots of Scientific Thought, Basic Books, New York.

College of Education Division of Curriculum and Instruction Virginia Polytechnic Institute and State University Blacksburg, VA 24061-8498 U.S.A.

hintikka, laudan and newton: an interrogative model of scientific inquiry

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