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40 What i.s this thing called Science?

deflected, the reading on the ammeter may or may not in- crease. \Ye cannot make the oytcomes conform to our theorie~ .. It was because the h sical world is the way i · that the

rimen con ucted by Hertz yie ed no e ection of cath· ode ra sand the modified expenment con ucte Thomson ~- It was the mclterial i erences m e experimental arrangements of the two physicists that led to the differing outcomes, u ot. the differences in the theories held by them. It is the sense in which experimental outcomes are determined by the workings of the world rather than by theoretical views about the world that provides the possibility of testing theo- ries against the world. This ig not to say that significant results arc t'asily achievable and infallible, nor that their significance is always straightforward. But it does help to establish the point that the attempt to tes t the adequacy of scientific theories against experimental results is a meaning- ful quest. What is more , the history of science gives us exam- ples of caRes where the challenge was successfully met.

Further reading

The second half of Hacking (1983) was an important early move in the new interest philosophers of science have taken in experiment. Other explorations of the topic are Franklin (1986), Ji~ranklin (1990), Galison (1987) and Mayo (1996), although th~se detailed treatments will take on their full significance on ly in the light of chapter 13, on the "new experimentalism". The issues raised in this chapter are dis- cussed in a little more detail in Chalmers ( 1984).

CHAPTER 4

Deriving theories from the facts: induction

Introduction

In these early chapters of the book we have been considering the idea that what is characteristic of scientific knowledge is that it is derived from the facts. We have r eached a stage where we have given some detailed attention to the nature of the observational and experimental facts that can be consid- ered as the basis from which scientific knowledge might be derived, although .. we hav~ seen that those facts cannot be established as stntghtforwardly and securely as is commonly supposed. Let us assume, then, that appropriate facts can be established in science. We must now face the question of how s~-ientific knowledge can be derived from those facts.

"Science is derived from the facts" could be interpreted to mean that scientific knowledge is constructed by first esta~ lishing the facts and then subsequently building the theory to fit them. We discussed this view in chapter 1 and rejected it as unreasonable. The is~ue that I wish to explore involvC's interpreting "derive" in some kind of logical rather than temporal sense. No matter which comes fi~~, the facts or the theory, the question to be addressed is the ~t{int to which the theory is borne out by the facts. The strongest possible claim would be that the theory can be lo,brically derived from the facts. That is, given the facts, the theory can be proven as a c~nsequence of them. Thi!'! strong claim cannot be substanti- ated. To see why this is so we must look at some of the basic features of logical reasoning.

Baby logic

Logic is concerned with the deduction of statements from

42 What is this thing called Science?

other, given , statements. It is concerned with what follows from what. )To a ttempt will l.w made to give a detailed account and appraisal of logic or deductive reasoning here. Rather, I will make the points that will be sufficient for our purpose with t he aid of some very simple examples.

Here is an example of a logical argument that is perfectly adequat e or, to use the technical term used by logicians, perfectly valid.

Example 1 1. All books on philosophy are boring. 2. This book is a book on philosophy. 3. This book is boring.

In this a rgument, (1) and (2) ar.e the premi~:>es and (3) is the conclusion. It i,s evident, I take it, that if (1) and (2) are true then (3) is rl~uncf to be true. It i:s not possible for (3) to be false once it is given that (1) and (2) are true. To assert ( 1) and (2) as true and to deny (3) is to contradict onesdf. This is the key feature of a logically valid deduction. If the premises are true then the conclusion must he true. Logic is truth pre- ~~rving.

A slight modification of Example (1) will give us an in- stan ce of an argument t hat is not valid.

Example 2 l. Many books on philosophy arc boring. 2. This book is a book on philosophy. 3 . This book is boring.

In this example, (3) does not follow of necessitv from (1) and (2). Even if (1 i and (2) arc truf:!, then this book-might yet t um out to be one ofthe minority of books on philosophy that a re not boring. Accepting ( lJ and (2) as true and holding (3) to be fa lse does not involve a contradiction. The argument is invalid.

The reader may by now be feeling bored. Experiences of that kind certainly have a bearing on the truth of statements (1) and (3) in Example 1 and Example 2. But a point that

Deriving theories from the facts: induction 43

needs to be stressed here is that logical deduction a lone cannot establish the truth of factual~~tatc~.~~ts-o.f.(he kir~d f~ring .. in oure~ampies. All_ _t~-at logic-can offer in thil:l connection is lhat· £1 the premises are true and the argument is valid then the conclusion must be trw~ .. But whether the ~s ~;;··t~ue ~;-not is .not~ q~e;oon that can be settled by an appeal to logic. 1\n argument can be a perfectly valid deduction even if it involves a fa lse premise. Here is an __ .... -- -- ·. ·· -· .... example.

Example3 1. All cats have five legs. 2. Bugs Pussy is my cat.

3. Bug::i Pussy has five legs. This is a perfectly valid deduction. If (1) and (2) are true

then (3) must be true. It so happens that, in this example (1) and (3) are false. But this does not affect the fact that the ar~:,TUment is valid.

There is a strong ~ense, then, in which lo1,ric alone is not a so~~-ce of new truths. The truth of the factualstat.em~nte.that ·constitute th e pre~~es of arguments cannot be established ·by appeal t o logic. Logic can simply reveal what follows from, or what in a sense is al re-ady containeJ in, the statemer)ts we ·already: have to hand. Agclinst this limitation we have the great st~ength oflogic, namely, its truth-preserving character. I( we .can be sure our premi~es are true then we can be cquaUy ~ure that everything w.e logically derive from them will also be true.

Can scientific laws be derived from the facts?

With thi8 discussion of the nature oflogic behind us, it can be straightforwardly shown that ss_!ent~~c knowledge ~annot be derived from the facts if "derive" is interpreted as "logically deduce".

Some simple examples of scientific knowledge will b~ suf- ficient for the illustration of this bask point. Let us consider

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44 What is this thing called Science?

some low-level scientific laws such as "metals expand when heated" or "acids turn litmus red". These are general state- ments. They are examples of what philosophers refer to as universal statements. They refer to all events of a particular kind, all instances of metals being heated and all instances oflitmus being immersed in acid. Scientific knowledge invari- ably involves general statements of this kind. The situ~~~ is quite otherwise when it comes to the_~~rvation state- ments that con!:ititute the facts that provide the evidence for general scientific laws. Those observable facts'Ot- experimen- tal results are specific claims about a state of affairs that obtains at a particular time. They are what philosophers call singular statements. They include statements such as "the length of the copper bar increased when it was heated" or "the litmus paper turned red when immersed in the beaker of hydrochloric acid". Suppose we have a large nwnber of such facts at our disposal as the basis from which we hope to derive some scientific knowledge (about metals or acids in the case of our examples). What kind of argument can take us from those facts, as premises, to the scientific laws we seek to derive as conclusions? In the case of our example concerning the expansion of metals the argument can be ~chematised as follows:

' Premises 1. Metal x1 expanded when heated on occasion t 1. 2. Metal x2 expanded when heated on occasion t~. n. Metal Xu expanded when heated on occasion tn. Conclusion All metals expand when heated.

Thi~Js..n~t a l~__gic~lly _yalid argument. It lacks the basic features of such an argument. It is simply not the case that if the statements constituting the premises are true then the conclusion must be true. However many observations of ex· pan ding metals we have to work with, that is, however great n might be in our example, there can be no logical guarantee that some sample of metal might on some occasion contract when heated. There is no contradiction involved in claiming

Deriuing theories from the facts: induction 45

both that all known examples of the heating of metals has re~uited in expansfon and that "all metals expand when heated" is false. - ·This straightforward point is illustrated by a somewhat ~oiiie e~ample attributed to Bertrand Russell. It con- cerns a turkey who noted on his first morning at the turkey farm that he was fed at 9 am. After this experience had been repeated daily for several weeks the turkey felt saf~1in draw- ing the conclusion "I am always fed at 9 am". Ala8, this conclusion was shown to be false in no uncertain manner when, on Christmas eve, instead of being fed, the turkey's throat was cut. The turkey's argwnent led it from a nwnber of true observations to a false conclusion, clearly indicating the invalidity of the argument from a logical point of view.

_Ar.gym.ents9f.the k_ind I have illustrated with the example concerning the expansion of metals, which proceed from a finit~-~~mber of specific facts to a general conclusion, are cal1ed i~ductive argwnents, as distinct from logical, deductive argu111.ents. A characteristic of inductive arguments that dis- t~guishes them from deductive ones is that, by proceeding as they do from statements about some to statements about all events of a particular kind, they go beyond what i~ con- tained in the premises. 9-.!!neral_scientific la~sj,nvariably go ~eiQr~~ ~he finite amount of observable evidence that is avail- ~~Je to support them, and that is why they can never be P.!:£>Y.e!l in the sense of being logically deduced irom that evidence.

What constitutes a good inductive argument?

We have seen that if scientific knowledge is to be understood as being derived from the facts, then "deriv~Il!\:lS~ be under· stood in an inductive rather than a deductive seJ!S.~- But what -are the characteristics of a good inductive argument? The question is of fundamental importance because it is clear that not all generalisations from the observable facts are war- ranted. Some of them we will wish to regard as ~~e~hasty or

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46 What is this thing called Science?

based on insufficient evidence, as when, perhaps, we condemn the attribution of some characteristic to an entire ethnic group based on some unpleasant encounters with just one pair of neighbours. Under precisely what circumstances is it legitimate to assert that a scientific law has been "'derived" from some finite body of observational and experi~ental evidence?

A first att('mpt at an answer to this question involves the demand that, .if an inductive inference from observable f~ct'$ to laws is to be justified, then the following conditions must

c satisfied: The number of observations forming the basis of a gener- alisation must be large.

2. The observations must be repeated under a wide variety of conditions.

3. No accepted observation statement should conflict with ·the.derived law.

Condition 1 is regarded as necessary because it is dearly not legitimate to conclude that all metals expand when heated on the basis of just one observation of an iron bar's expansion, say, any more than it is legitimate to conclude that all Australians are drunkards on the basis of one observation of an intoxicated Australian. ~large number of independent observations would appear to be necessary before eith!:r generalisation can be justified. A good inductive argument does not jump to conclusions. ·

One way of increasing the number of observations in the examples mentioned would be to repeatedly heat a single bar of metal or to continually observe a particular Australian getting drunk night after night, and perhaps morning after morning. Clearly, a list of observation statements acquired in such a way would form a very unsatist~'lctory basis for the respective generalisations. That is why condition 2 is neces- sary. "~!1_ J:!letals expand when heated" will be a legitimate generalisation only if the observations of expansion on which it is based range over a wide variety of conditions. Various kinds of metals should be heated, long bars, short bars, silver

Deriving ilworit~s (rom the facts: induction 47

bars, ~~pp~r bars etc. should be heated at high and low pressures and high and low temperatures and so on. Only if on all such oc(:asions expansion results is it legitimate to generalise by induction to the general law. Further, it is evident that if a particular sample of metal is observed not to expand when heated, then the generalisation to the law will not be justified. Condition 3 is essential.

'l'he above can be summed up by the following statement of the principle of induction.

v .•. ·-· •'.. ...--· ........ ·"···-· -· .•

l . If a lar~e number of A's have been observed under a wide variety ~f conditions, and if all those A's without exception possess the

. !1roperty B, then all A's have the property B.

There are serious problems with this characterisation of induction. Let us consider condition 1, the demand for large numbers of observations. One problem with it is the vague- nc·ss of"large". Are a hundred, a thousand or more observa- tions required? If we do attempt to introduce precision by introducing a number here, then there would surely be a great deal of arbitrariness in the number chosen. The problems do not stop here. There are many instances in which the demand f(Jr a large number of instances seems inappropriate. 'l'o illustrate this, consider the strong public reaction against nuclear warfare that was provoked by the dropping of the first atomic bomb on Hiroshima towards the end of the Second World War. That reaction was based on the realisation of t.he extent to which atomic bombs cause ~idC"~pr~'ad destruction and human suffering. And yet this widespread, and surely reasonable, beH,ef was based on just or~e drarnatk observa- tion. In similar ~ei'n, it would be a very stubb~m investigator who insisted on putting his hand in the fire many times tJef(Jre concluding that fire burns. Let us consider a less f,~~cifui example related to scientific practice. Suppose I reproduced an experiment reported in some recent scientith: journal, and sent my results offfor publication. Surely the editor of the journal would reject my paper, explaining that the

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experiment had already been done! Condition 1 is riddled with problems.

Condition .. 2 has serious problems too, stc~ming from difficult ies s"urroullJin.g the question of what counts as a significant variation in circumstances. What counts as a significant. variation in the circumstances under which the expansion of a heated metal is to be investigated? Is it necessacy.to. v.ary thctyp_e of meta_t.th!! Pressure_ and the ti._m~ ~(~ay?_The answer is "'yes" in the first and possibly th~?_$econd. ca.se but "no" in the third. But what are the grounds for that answer? The question is important because unless it can be answered the list of variations can be extended indefinitely by endte'; sfyadding further variations, such as the size of the laborat?.~ .. ~~_the colour ofthe experimenter's socks. Unless such "superfluous" variations can be eliminated, the condi- tion~ under which an inductive inference can be accepted can never be satisfied. What are the grow1ds, then , for regarding a range of possible variations as superfluou~'? The common· sense answcr is straightforward enoug-h.)Vc draw on ourppQr knowledge ofthe situation to distinguish between the factors

.. ihaTmiglitan<fthosid.hat c~~t influence the system we are i'Mi~ig_~t-~: It is our knowledge of meWs and .t he kinds of ways that they can be acted on that leads us to the expectation that their physical behaviour will depend on the type of metal and the surrounding pressure but not on the time of day or the colour ofthe experimenter's socks. We draw on our current stock of knowledge to help judge what is a relevant circum- stance that might need to be varied when investigating the generality of an effect under investigation.

This response to the problem is surely correct. However, it poses a problem for a sufficiently strong version of the claim that scient ific knowledge should be derived from the facts by induction. The problem arises when we pose the question of how the knowledge appealed to when judging the relevance or otherwise of some circumstances to a phenomenon under investigation (such as the expansion of metals) is itself vin- dicated. If we demand that that knowledge itself is to be

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Veri.Fing theories from the facts: induction 49

arr ived at by induction, then our problem will recur, because those further inductive aq,ruments will themselves require t he specification of t he relevant circumstances and so on. Each inductive argument involves an appeal to prior knowl- edge, which needs an inductive argument to justify it, which involves an appeal to further pr ior knowledge and so on in a never-ending chain. The demand that all knowledge be justi- fied by induction becomes a demand that cannot be met.

Even Condition 3 is problematic since little scientific knowledge would survlvelhe demand that there be no known exceptions. This is a point that wiU be discussed in some detail r.i'ciiapter 7.

Further problems with inductivism . .. ---- .. ·- -· ·--- ....

Let us call the position acc.:ording to which scientific knowl- edge is to be derived from the observable facts by some kind of inductive inference in:ductivism and those who subscribe to that view inductivists. We have already pointed to a ser ious problem inherent in that. view, namely, the problem of stating precisely under what conditions a generalisation constitutes a good inductive inference. That is , it is not clear what induction a~ouhi .. ~ to. There are further problems with the inductivist position.

If we take contemporary scientific knowledge at anything like face value, then it has to be admitted that much of that knowledge refers to the unobservable. It refers to such things as protons and electrons, genes and DNA molecules and so on. How can such knowledge be accommodated into the .i~ductivist position? Insofar as inductive reasoning involves so.nle killd-of generalisation from observable facts, it would appear that such reasoning is not capable of yielding know l- edge of the unobservable·. Any generalisation from facts about ~he observable world can yield nothing other than generali- sations about the observable world. Consequently, scientific knowledge of the unobservable world can never be estab- lished by the kind of inductive reasoning we have discussed.

50 What is this thing called Science?

This leav(~S the inductivist in the unoomforb1ble position of having to r<!ject much contemporary science on the grounds that it involves going beyond what can be justified by induc- tive generalisation from the observable.

Another problem stem's from the fact that many scientific laws take the form of exac( mathematically formulated laws. The law of gravitation, which states that the foree between any two Ill<lsses is proportional to the product of those masses divided by the square of the distance that separates them, is a straightforward example. Compared with the exactness of such laws we have the inexactness of any of the measure- ments that constitute the observable evidence for them. It is well appreciated that all obsc•rvations are 1:1ubject to some degree of error, as refle~ted in the practice of scientists when they write the result of a particular measurement as x ± dx, where the dx represents the estimated margin of error. If scientific laws are inductive generalisations from observable facts it is difficult to see how one can escape the inexactness of the measurements that constitute the pn~mises of the inductive arb'Uments. ~tis diffic1.1lt to see how exact laws £::tn ~ver be ind~ctively j.u.sti!!_~d on the basis of inexact evidence . . ~ .!d;hfrd problem for the inductivist is an old philosopliicai

'· chestntifcalied the problem of induction. The problem arises for anyone who subscribes to the view that scientific knowl- edge in all its aspects must be justified either by an appeal to (deductive) logic or by deriving it from experience. David Burne was an eighteenth-century philosopher who did sub- scribe to that view, and it was he who clearly articulated the problem I am about t~ highlight.

The problem ari~es when we raise the question of how induction itself is to be justiti,:d. How is the principle of induction to be vindicated'? _Those who take the view under discussion have only two options, to justifY it by an appeal to logic or by an appeal to experience. We have already seen that the first option will not do. Inductive inferences arc not logical (deductive) int~ronces. Thi::; leaves us with the second option, to attempt to justify induct.ion by an appeal to experience.

Deriving theories from the facts: induction 51

\Vhat would such a justification be like? Presumably, it would go something like this. Induction has been observed to work on a lar1:,~ number of occasions. For inr:ltance, the laws of optics, derived by induction from the results of laboratory experiments, have been used on numerous occasions in the de~ign of optical instruments that have operated satisfac- torily. and the laws of planetary motion, inductively derived from the observation of planetary positions, have been suc- cessfully used to predict eclipses and conjunctions._'f.~is list could be greatly extended with accounts of successful predic- tions-and explam\tio.ns that we presume to be made on the ha'sis ofinductively derived. scientific laws and theorie::;. Thus, HO the argument goes, induction is justified by experience. T_P.~.~Q.P ?f !nd.q~::.!J.2!!J§.Yn{-lc.ceml!We. This can he

seen once the form ofthe argument is spelt out schematically as follows:

The principle of induction worked successfully on occasion x1 The principle of induct ion worked SU(:c:es!;fully on occasion :r:! etc.

Tlw principle of induction always worh

A general statement asserting th.e validity of the principle of induction is here inferred from a number of individual instances of its successful application. The argument is there:'".::. fore itself an inductive one. Consequently, the attempt to ju~tify induction by an appeal to experience involves assum- ing what one is tryjngto prove. ltinvolvesj ustifyinginduetion by appealing to induction, and so is totally unsatisfactory.

_.:..:J;.Q~r.-~ttempt to avoid the p~obl~m of induction involves weakerung the demand that scientific knowledge be proven true, and restingt:o!ifcil'twith the claim that scientific claims can be shown to be probably true in the light of the evidence. So the vast number of observations that can be invQkcd to support the claim that materials denser than air fall down~· wards on earth, although it does not pennit us to prove the truth of the claim, does warrant the assertion that the claim is probably true. In line with this suggestion we cal). reformu- late the principle of induction to read, ~if a f~-~ge numhec of

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~shave been observed under a wide vafiety ofconditions,and ~f all these observed A's have the property B, then all Ns probably hav~ the property B". This reformulation does not

x•;·' ove~co~e the p~oblem of induction. The reformulated princi- ple 1s shll a umversal statement. It implies, on the basis of a fmite _number of successes, that all application:;; of the princi- ple wtll lead to general conclusions that are probably true. ConsequontJy, <~tt~mpts to justify the probabilistic version of the principle of induction by an appeal to experience involve ~m ~ppe~ to inductive arguments of the kin4 . .!ha.~ arc being JUstified Just as the principle in its original form did.

There is another basic problem with interpretations of inductive arguments that construe them as leading to prob- able truth rather than truth. This problem arises as soon as ~n~ tries~ be precise about just how probable a law or theory Is m ~he bght of specified evidence. It may seem intuitively ~laus1ble tha~- as the observational support for a general law mcreases tho probability that it is true also increases. But this intuition doe~ not stand up to inspection. Given standard probability theory, it is difficult to avoid the .conclusion that the probability of any generallaw is zero ;~-atev~r the obser- vational evid~nce. To make the point in a non-technical way, any obser.vat10nal evidence will consist of a finite number of observation statements, whereas the general law will make claims a~out an_ infinity of possible cases. The probability of the law m the hght of the evidence is thus a finite number divided by infinity, which remains zero by whatever factor the finite amount of evidence is increased. Looking at it in an- other way, there will always be an infmitc numher of general statements that are compatible with a fmite number of ob- servation statemcnts,just as there is an infinity of curves that c~ be drawn through a finite number of point8. That is, there w~ll always be an infinite numbe_r of hYPotheses comp~tlble Wl~~ a finite amount of evid~nce. Consequently, the prob- a~lht~ of any one of them being true is zero. In chapter 12 we w11l discuss a possible way around this problem.

In this and the preceding section we have revealed two

Deriving theories from the facts: induction 53

kinds of problem with the idea that sdentific knowledge is derived from the facts by some kind of inductive inference. The first concerned the issue of specifying just what an ~d~quate inductive argument is. The sec~nd involved the ·circularity involved i~ attempts to justify ~duction. I regard the ~~~Lproblem as more "severe'ih~ln the latter. The re-ason that I do not take the problem of induction too seriously is that any attempt to provide an account of sc~ence is bound to confront a problem of a similar kind. We are \lound to run into trouble if we seek rational justifications of every principle we use, for we cannot provide a rational argument for rational

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argument itself without assuming what we are arguing for. .Not even logic can he argu.ed for in a way that is not question

~·begging. H~\Vever, what constitutes a valid deductive argu- ment can be specified with a high degree of precision, where·~ what c.onstitutes a good inductive argument has not been . made at all clear.

The appeal of inductivism

A concise expression of the inductivist view of science, the view that scientific knowledge is derived from the facts by inductive inference which we have discussed in the opening chapters of this book, is contained in the following passage written by a twentieth-century economist.

If we try to imagine how a mind ofsupcrhwnan power and reach, but normal so far as the logical processes of its thought are concerned , , . would use the scientific method, the process would be as follows: First, all .facts would be observed and recorded, w_ithout sele_ction or a priori guess as to their relative import.uncc. Secondly, the ohservcd and recorded faet;; would be analysed, compared and cla~sitied, without hypothesis or postulates. otht·r than those necessarily involved in the logic of thought. Third, from this analysis of the facts, gen.eralizations would be induc- tively drawn as to the relations, classificatory or causal, between .the~;. Fourth, further research would be deductive as well as inductive, employing inferences from prev1ously established generalizations. t

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We have seen that the idea that the collection offacts can and should take place prior. to the acquisition and acceptance of any knowledge does not bear analysis. To suggest otherwise is to believe that my observations of the flora in the Austra- lian bush will be of more value than those of a trained botanist precisely because I know little botany. Let us reject this part of our economist's characterisation of scict1ce. What remains is an account that has a certain appt!al. It is summaris ed in figure 2. The laws and theories t hat make up scientific knowl- edge are derived by induction from a factual basis supplied by observation and experiment. Once such general know ledge is available, it can be drawn on to make predictions and offer

explanatio ... :ns~. -----------

Facts acquired through observation

Laws and theories

Figure2

Consider the following argument:

Predictions and

1. Fairly pure water freezes at about O"C (if b>iven sufficient time).

2. :Vfy car radiator contains fairly pure water.

3. If the temperature falls well below 0°C, the water in my car radiator will freeze (if given sufficient time).

Here we have an example of a valid logical argument to

Deriuing theories from the facts: induction 55

deduce the prediction 3 from the scientific knowledge con- tained in premise 1. If 1 and 2 are true, 3 must be true. However, the truth of 1, :2 or 3 arc not established by this or any other deduction. For the indurtivist the source. of !;t\en- tific truth is experience not logic. O.Q this view, . 1 will be ascertained by direct observation of various instan.ces of freezing water: Once J and 2 have been established by obscr· v~ti~!l. and 'induction, then the prediction 3 can be d~duced . frotn them. ---r:eg; trivial examples will be more complicated, but the roles played by observation, induction and deduction remain essentially the same. As a final example, I will consider the inductivist account of how physical science is able to explain the rainbow.

The simple premise 1 of th0 previous example is here replaced by a number of laws governing the behaviour of light, namely the laws of r eflection and refraction of light. and assertions about the dependence of t he amount ofrefraetion on the colour of the light. These general laws are to be derived from. experience by induction. A large number of laboratory experiments are performed, reflecting rays of light from mir- rors and water surfaces, measuring angles of r efraction for rays of light passing from air to water, water to air and so on, ~nder a wide variety of circumstances, until whatever condi- tions are presumed to be necessary to warrant the inductive derivation of the laws of optics from the experimental results are satisfied.

Premise 2 of our previous example will also be replaced by a more complex array of statements. These wili include asser- tions to the effect that the sun is situated in some specified position in the sky relative to an observer on earth, and that raindrops are falling from a cloud situated in some specified region relative to the observer. Sets of stat~ments like these, which describe the s et-up under investigation, will be re- ferred to as initial conditions. Descriptions of experimental set-ups will be typical initial conditions.

Given the laws of optics and the initial conditions, it is now

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56 What is this thing called Science?

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possible to perform deductions yielding an explanation of the forma tion of a rainbow vis ible to the observer. These deduc· tions wi 11 no longer be as self-evident as in our previous examples, and will involve mathematical. as well as verbal at:g)J.ment.s. The derivation will run nfti~i~ as follows. If we

• ~,, . L ' • assume a ramdrop .to be roughly spherical, then the path of a ray of light through a raindrop will be roughly as depicted in figure 3. For a ray of white light from the sun incident on a raindrop at a, the red light will travel along ab and the blue light along ab' according to the law of refraction. The law of reflection requires that ab be reflected along be and ab' along b'c'. Refraction at c and c' will again be determined by the law of refraction, so that an observer viewing the raindro.P will see the red and blue components of the white light separated (and also all the other colors ofthe spedrum). The same separation of colours will be visible to our observer for any raindrop that is situated in a region of the sky such that the line joining the raindrop to the sun makes an angleD with the line joining the raindrop to the observer. Geometrical considerations yield the conc;l~sion that a coloured arc will be visible to the observ~r pfo~i;led the rain cloud is sufficiently extended.

I have only sketched an explanation of the rainbow here, but it should suffice to illustrate the general form of the

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Deriving theories from the facts: induction 57

reasoning involved. 'Given that the laws of optics are true (and for the unqualified inductivist this can be esta~l~s~ed fro~ observation by induction), and given that the mtbal condt· tions are correctly described, then the explanation of the rainbow necessarily follows. The general form of all scientific explanations and predictions can be summarised thus:

Laws and theories

Initial conditions

Predictions and explanations

This is the step depicted on the right-hand side of Figure 2. The basic inductivist account of science does h ave some

immediate appeal . Its attraction lies in the fact that it does seem to capture in a form al way some ()f th~ common~y h~ld intuiticms about the special charac..t eristks of sc1ent1fic knowledge, namely its objectivity, its reliability and its use- fulness. We have discussed the inductivist account of the usefulness of science insofar as it can facilitate predictions and explanations already in this section .

The objectivity of science as construed .by t?e ind~ctivist derives from the extent to which observat10n, mductlon and deduction arc themselves seen as objective. Observable facts are understood to be established by an unprejudiced use of the senses in a way that leaves no room for subjective opinion to intrude. As far as inductive and deductive reasoning arc concerned, these are adequate to the extent that they confo~m to publicly formulated criteria of adequacy, so, o~ce agam, there is no room for personal opinion. Inferences e1ther con- form to t~ q.bje:ctive standards or the~ don't. . . . .. ,

The reliibility of science follows trom the mducth'l~t" claims about observation and both inductive and deducttve reasoning. According to the unqualified inductivist, observa· tion statements that form the factual basis for science can be securely established directly by careful use of the senses. Further this security will be transmitted !A? the laws and theorie~ inductively derived from those fac~.provided the conditions for adequate inductive generalisa tions are met.

II'

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58 What is this thing called Science?

This is guaranteed by the principle of induction which is presumed to form the basis of science. - · -· ··· ···· · .. .

Attractive as it may have appeared, we have seen that the inductivist posit~on is1 at best, in need of severe qualification and, at worst, ttlorougbly inadequate. We have seen that facts adequate for science are by no means stra1ghtforw'3:rdly given but have to be practically constructed, are in some important senses dependent on the knowledge that they presuppose, a complication overlooked in the schematisation in figure 2, and are subject t•J improvement and replacement. More seri- ously, we have been unable to give a precise specification of induction in a way that will help distipguish a justifiable generalisation from the facts from a il.'asfy or ~ash ·o~e, a

. f~rmj9able task given nature's cap~pty to surprise, epito- . mised in the discovery that superc'ooled liquids can flow uphill~~.t.~cr.

In ch apter 12 we will discuss some recent attempts to rescue the inductivist account of science from its diftkult.ies. Meanwhile, we will turn in the next two chapters to a philosopher who attempts to sidestep problems with induc- tion by putting forward a view of science that does not involve induction.