Admissible Hypotheses and Enhanced Learning

This paper discusses s t ra teg ies fo r moving through sequences of hypotheses, each one of which is produced in response to an experimental tes t of the previous member. Previous discussions of t h i s issue have a l l agreed that hypotheses deduct ive ly incompat ible w i t h the evidence at stage ri cannot appear in the sequence beyond n This paper contends that t h i s conclusion is untenable. The use of o v e r s i m p l i f i e d models has led i nves t i ga to r s i n t o over look ing ep is temolog ica l p roper t ies o f more complex hypotheses which al low more soph is t i ca ted methodologies in t e s t i n g and hypothes is generat ion. In p a r t i c u l a r , i t i s shown that t e s t i n g already f a l s i f i e d hypotheses may give more experimental in fo rmat ion than o the r , more t r a d i t i o n a l s t r a t e g i e s . This is shown by cons ider ing a popular board game, but a r e a l i s t i c example is introduced to demonstrate the general importance and usefulness of the s t ra tegy . In t h i s short paper I discuss s t ra teg ies in the t e s t i n g of a sequence of hypotheses. Each of the hypotheses, except perhaps the f i r s t , is proposed a f t e r an experimental tes t of the preceding hypothes is . The aim of the s t ra tegy is to maximise the ra te of convergence of members of the sequence of hypotheses to the cor rec t hypothesis. This problem has been discussed in the l i t e r a t u r e qu i te ex tens ive ly under the general heading of Methodology of Science and to a lesser extent in the f i e l d of h e u r i s t i c s . Two fundamenta l l y d i f f e r e n t views have been taken about the manner of proceeding in such a sequence. The f i r s t , associated w i t h Carnap (Carnap, 1952), Reichenbach (Reichenbach, 1961) and many others f o l l o w i n g them is to put an eva lua t ion on each poss ib le n t n hypothesis and choose tha t hypothesis which has a maximum value fo r t h i s eva lua t i on . Most o f t e n , but by no means always, the evaluat ions were simply the p r o b a b i l i t i e s or con f i rmat ion of the a l t e r n a t i v e hypothesis, al though t h i s is by no means un ive rsa l ( e . g . Reichenbach, op c i t ) . Such proposals are o f t en termed i n d u c t i v i s t . A q u i t e d i f f e r e n t view is taken by Popper (Popper, 1972) and fo l lowers who eschew a p r o b a b i l i s t i c o r conf i rmatory eva luat ion f u n c t i o n . Instead they propose that the func t i on r i ses w i t h the r i s k of f a l s i f i c a t i o n of a hypothes is . By adopt ing as the n t n member of a sequence the hypothesis w i th the minumum r i s k of r e f u t a t i o n , usua l ly i d e n t i f i e d by maximising content of the hypothes is , we maximise the chance of r e f u t a t i o n . Th i s , in t u r n , maximises the expected ra te of progress along the sequence of hypotheses and t h e i r convergence to the t r u t h . The general stance of t h i s l a t t e r group is tha t since there e x i s t no ob jec t i ve measures of the conf i rmat ion of hypothesis by evidence, the appropr iate eva lua t ion is given by a methodolog ica l ly der ived func t ion which r e f l e c t s the a p r i o r i l i k e l i h o o d of r e f u t a t i o n and hence progress along the sequence of hypotheses. Such a thes is concerning the appropriateness of the eva lua t ion is ca l led f a l s i f i c a t i o n i s t . Both approaches to the eva luat ion func t i on share one important common fea tu re . The eva luat ion takes a minimum value fo r any hypothesis which is deduct ive ly incompat ib le w i t h the evidence to date. This assignment is j u s t i f i e d in both approaches by the view tha t each proposed hypothes is must be a poss ib le candidate fo r " the t rue hypothes is " . The i n d u c t i v i s t would res ta te t h i s w i t h term "probable" ra ther than possib le but since "probable" e n t a i l s " poss ib l e " the two views co inc ide . The d iscuss ion of these top ics has been rendered less h e l p f u l to i nves t i ga to rs by the perhaps oversimple cha rac te r i sa t i on of hypothesis . They a re , fo r much of the d i scuss ion , simply s t ruc tu red sets of sentences whose only re levant c h a r a c t e r i s t i c here is tha t they e n t a i l simple observat ion sentences which e i t h e r do or do not accord w i t h ac tua l observat ions. The assessment of any hypothesis given a s ing le piece of evidence is thus a two valued func t i on e i t h e r i ncons i s ten t or cons i s ten t . While such a model of experimental t e s t i n g has the v i r t u e of s i m p l i c i t y i t i s , I contend, so u n r e a l i s t i c as to obscure the rea l and i n t e r e s t i n g problem of s t ra teg ies f o r exper imental t e s t i n g even in simple contex ts . I w i l l argue that when the two major schools of thought agree tha t re fu ted hypotheses should be discarded they are both wrong. In doing so I s h a l l use hypotheses whose observat iona l consequences are min imal ly more s t r u c t u r e d . These hypotheses w i l l assign occupat ion s ta tes to a f i n i t e ordered set o f c e l l s . Accordingly the p o s s i b i l i t y a r ises o f the hypotheses f i t t i n g the