THE EVOLUTION OF SCIENTIFIC THEORIES 265 



the tightness of its accommodation in the theory? Scholastic science 

 accommodated practically everything known, but in a system of ex- 

 treme looseness. Such looseness is reflected in the j^roduction of the 

 relations through deductive chains flawed by assumptions and ap- 

 proximations for which the theory provides no explicit justification. 

 But this looseness was wholly unapparent (as almost invariably it 

 is) to contemporary eyes. Perhaps we may hope to estimate tightness 

 of correlation by considering the extent to which the theoretic deriva- 

 tion rationalizes the limits in range and accuracy we find in practice 

 to attach to each colligative relation. We may hopefully adopt a 

 modified index ratio of the form: 



2P 



Here t is the factor of tightness ranging from 0, for a relation en- 

 tirely unaccommodated, up to 1 for a relation that has both its 

 strengths and weaknesses perfectly delimited. 



Turning now to the denominator, we face much graver problems. 

 Dealing, as always we do, with incompletely formalized theories, we 

 never have a definite roster of postulated axioms, syntactic rules, 

 semantic rules, auxiliary assumptions, etc. Using common sense and 

 our knowledge of other sciences, as indicated in the preceding chap- 

 ter, often we simply rely on intuition to supply the necessary items, 

 as we need them, without any full awareness of their identity, much 

 less their total number. Even given such a number, we would have 

 yet to render account of the fact that sometimes a simplification of 

 syntactic rules demands an entirely disproportionate complication 

 of semantic rules, and conversely. Attach to each theoretical stipula- 

 tion some factor of complexity, f? If possible, why not? We have 

 then to face "only" this terminal problem: can we even hope to state 

 a correlative index for any single theory when, as we have just seen, 

 in every application it forms a unit with unnumbered others of its 

 kind? 



Noting diese complications, and however improbably, let it be 

 granted that a correlative index can somehow be estimated. Often 

 theories will seem to differ so sharply in index value that even crude 

 estimates are sufficient to make the difference manifest. Do we then 

 e a criterion sufficient for the judgment of competing theories? 

 inly it cannot always be sufficient. Whatever may be the ultimate 



