COLLIGATIVE RELATIONS AND SCIENTIFIC LAWS 117 



template undertaking the extravagant series of studies essential for 

 exact prediction. Demanding such studies, the law of the lever would 

 lose all its value: with but a fraction of the effort required for pre- 

 diction we could simply set up the lever system and observe what 

 happens. Thus, as Duhem remarks, 



A mathematical deduction . . . may therefore be useful or otiose, 

 according to whether or not it permits us to derive a practically 

 definite prediction of the result of an experiment whose conditions are 

 practically given. 



Predictions drawn with difficulty from the more complicated van der 

 Waals relation are usually more reliable than those drawn from 

 Boyle's law. But ordinarily we elect to use Boyle's law— from which 

 we easily derive a host of predictions usually quite well enough 

 borne out in practice. 



Paraphrasing Einstein, may I not insist that insofar as a colligative 

 relation is generally applicable it is not exact, and insofar as it is 

 exact it is not generally applicable? What we seek in a colligative re- 

 lation is efficiency— a reasonable balance of reliability, generality, and 

 convenience— and efficiency is all that is attested by the successful 

 functioning of a colligative relation as such. To be sure, in their cos- 

 mologic speculations, scientists have sometimes claimed discovery 

 of the true laws by which nature is governed. That cosmologic dogma 

 is, however, no present concern of ours— as becomes clear when we 

 consider, for example, the equivalence of inertial and gravitational 

 masses. We may hold it universally and absolutely "true" because we 

 accept as final the relativistic theory in which this equivalence is a 

 deduction entailed by postulates ive accept. But tlien surely we are 

 not treating it simply as a colligative relation. Alternatively, we might 

 hold the equivalence "true" because it has been confirmed to the 

 limit of our experimental measurements. Here we introduce the 

 covert assumption that a law so confirmed must be mathematically 

 exact, and so unite the colligative relation with a superfluous meta- 

 physical assumption. As a colligative relation the "law" is no more 

 than efficient. The generality of this important conclusion is perhaps 

 best illustrated by showing its applicability to a colligative relation 

 apparently wholly different in form from any so far examined. 



The sulfur relation. Sulfur is a yellow crystalline solid with density 

 2.07, melting at 119.0°C and boiling at 444.6°C, a good insulator, 



