Iductio ad Hypothesis III These difficulties which beset Reichebach's philosophy of iductio are serious, but they still leave us room to hope that it might be possible to costruct a theory of iductio which would justify simple eumeratio without eedig to make ay assumptios about the uiformity of ature. Such a theory, largely free from these difficulties to which Reichebach's view is subject, has bee proposed by Professor Williams.7 His view is ivitig i its simplicity, ad it offers us what he takes to be a good reaso for adoptig ad for trustig the method of iductio by simple eumeratio. I his view, we must commece with the proportioal, or statistical, syllogism. This is a mode of argumet which, he coteds, is basic to the theory of probability, beig the oe ultimate source from which umerical values for probabilities may be obtaied (though of course oce some umerical values have bee itroduced by meas of the statistical syllogism, others may the be derived from these by meas of the usual rules relatig probabilities). The statistical syllogism is a mode of argumet of the followig form: Of ail the thigs that are M, 111 are P. a is a M. Therefore (with a probability of 111) a is P. 6 See Max Black, Problems of Aalysis (Ithaca, N.Y., 19 54), chs. x ad xii; Erest Nagel, Sovereig Reaso (Glecoe, III., 1954), ch. xiv. 7 D. C. Williams, The Groud of Iductio (Cambridge, Mass., 1947). 70
Eumerative Iductio Here m ad are itegers, a is a idividual thig, ad 'M' ad 'P' are empirical predicates. As it stads, the statistical syllogism is ot a iductive mode of argumet; but it ca be brought to bear upo iductio i the followig way. Let us suppose that we have observed raves ad foud m of them to be black, the rest ot black. If is a fairly large umber, the usig oly algebra we ca prove that, whatever the total umber of raves may be (so log as they are fiite i umber), the great majority of the -membered subclasses of the class of raves differ relatively little from the whole class i regard to the fractio of their members which are black. Thus we are provided with major ad mior premises for this statistical syllogism: Of all the -membered subclasses of the class of raves, most differ little from the whole class i regard to the fractio of their members that are black. This sample, whose fractio of black members is m, -membered subclass of the class of raves. IS a Therefore (with a good probability), this sample differg little from the whole class with regard to the fractio of its members that are black. By meas of straightforward algebraic cosideratios, the rough otios of "most," "little," ad "good" could be replaced by exact algebraic formulatios or by defiite umerical values if m ad are specified. Ad o matter how large the class of raves may be (so log as it is fiite), "most" will mea a wholesomely large percetage, "little" a pleasigly small oe, ad "good" a fractio early equal to oe, all provided is fairly large. For istace, if is 2500, "most" ca mea at least 95 per cet, while "little" will the mea ot more tha 2 per cet, ad "high" will mea 95 Furthermore, as icreases
iductio ad Hypothesis 'vithout boud, "most" approaches 100 per cet as its limit, "little" approaches zero as its limit, ad "good" approaches oe as its limit. It is importat to ote the sigificace of the statistical syllogism just discussed. The major premise of that syllogism is ot empirical; rather its truth ca be certified by algebra aloe. The mior premise costitutes eumerative iductive evidece: it sums up a set of observatioal statemets gleaed from ispectio of idividual raves. The coclusio of the syllogism implies a iductive geeralizatio: 'The fractio of all raves that are black differs little from m. ' Thus the statistical syllogism eables us to pass from iductive evidece to a iductive geeralizatio as our coclusio. The fial coclusio is ot so simple i form as are the coclusios of the form 'All Pare Q' to which some other iductive methods might purport to lead us; but for practical purposes this sort of coclusio surely is satisfactory. Here o empirical assumptios about the \ovorld have had to be made; whatever the world may be like, this argumet is impeccably ad ecessarily valid, provided we accept the statistical syllogism itself. \Ve seem to be offered a strog reaso for embracig eumerative iductio as a trustworthy ad fudametal method of odemostrativc iferece. IV Some critics have felt that the statistical syllogism is too good to be true ad that it provides somehow too easy a way out of the difficulties surroudig the logic of iductio. A few of these criticisms ought briefly to be oted, because of the light they may shed upo the issue at stake. For istace, certai critics, takig very seriously the differece betwee past ad future, have objected that this philosophy of iductio is usoud because, though the world may 72
Eumerative Iductio have exhibited uiformities i the past, there remais a possibility that it may ot cotiue to do so i the future, ad thus iductive reasoig may go wrog. This objectio is wide of the mark, of course, for the argumet metioed i the precedig sectio cotais ad eed cotai o factual presuppositio whatever, ad certaily oe about the uiformity of ature through time or otherwise. It is just this freedom from such presuppositios which is its pricipal merit. To be sure, the coclusio of a statistical syllogism may be false eve though its premises are true; but to poit this out is merely to poit out that the iferece is a odemostrative, ot a demostrative oe. I the same vei, critics have objected also that this kid of argumet fails o accout of its refusig to employ ay empirical evidece to show that the observed sample is a "fair" oe. The sample could be biased, they argue, i which case this mode of argumet would yield misleadig coclusios. If all the black raves were i the top of the ur, so to speak, it might be the case that early all raves are ot black eve though all those observed are. Such a objectio as this, however, likewise eglects the fact that the statistical syllogism does ot purport to guaratee the correctess of its coclusio; if it did, it would be a demostrative, ot a odemostrative mode of argumet. \Vhat the statistical syllogism does claim to show is that its coclusio is supported by its evidece. No more should be demaded; for to demad that the coclusio be ecessitated by the evidece is i effect to claim that there caot be ay odemostrative argumets at all. It is eough, surely, that there should be o positive reaso for supposig the sample to be misleadig; give the evidece that we have, we must draw whatever odemostrative coclusios we ca from it. To coted that give evidece caot be employed i odemostrative argumet uless there is further positive evidece 73
Iductio ad Hypothesis that the give evidece is ot misleadig is to embark o a vicious ifiite regress, a regress which would destroy the possibility of there beig ay valid odemostrative argumets at all. There is a further poit about the statistical syllogism which deserves otice, because it may seem to militate agaist the usefuless of this mode of argumet. This ivolves the matter of ifiity. \Ve oted that the statistical syllogism is applicable oly >vhere the populatio cocered may be assumed to be fiite i size. If the populatio were to cosist of a actually ifiite umber of thigs, the the hyperpopulatio cosistig of all the subclasses havig the same size as the give sample would likewise be a ifiite class. I that case, it would be impossible to say that most of the possible samples closely resemble the populatio as regards compositio; for while a ifiite umber of them might so resemble it, a ifiite umber of others would fail to do so, ad o defiite ratio could the be said to exist betwee the umber of those that do ad the umber of those that do ot. Thus the statistical syllogism caot legitimately be employed except where we are etitled to assume the populatio to be fiite. It may be thought that this is a serious shortcomig. Ideed, some writers o iductio have blithely asserted (alog with Reichebach) that populatios of empirical thigs i the world may be, are, or must be actua1ly ifiite i umber. Some like to facy that the class of all swas that ever exist must embrace a ifiite umber of birds, that there are a limitless umber of ihabitats of Africa, ad so o. All who regard the spatiatemporal world as this full of thigs will have to regard the statistical syllogism as yieldig a iadequate rule of iductio, sice it caot serve to make probable ay geeralizatios about such prodigious classes. But is this prodigality justified? Need we seriously suppose 74
Eumerative Iductio that classes of thigs i the world may be so big? Surely this is a eedless metaphysical assumptio. For to say of ay class of empirical thigs that it cotais a actually ifiite umber of members is to make a assertio utterly devoid of empirical sigificace. This follows from the fact that o possible tests or observatios could coceivably establish the statemet; we might have empirical evidece that there exist at least a millio swas, or at least a billio; but i the ature of the case, o evidece could establish that the umber of swas is greater tha every fiite umber. The suppositio that the umber of actual members of some class of empirical idividuals is ifiite is a utestable metaphysical suppositio, ad eed ot be take seriously, so far as empirical kowledge is cocered. There are less thigs i heave ad earth tha are dreamed of i some philosophies. I./