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SCIENCE PROGRESS 



RECENT ADVANCES IN SCIENCE 



FTTBiE MATHEMATICS. By Dorothy M, Wrinch, Fellow of Girton 

 College, Cambridge, Member of the Research Staff, University College, 

 London. 



In the algebra of propositions elaborated by Boole and the 

 earlier symbolic logicians, the three operations of disjunction 

 " not p and q," conjunction " p and q " and negation " not 

 p " were taken as undefined. Sheffer {Trans. Amer. Math. Soc, 

 14, 191 3), however, showed that the three operations are not 

 independent, and can be obtained from one operation, which 

 may be written pjq, which turns out to have the properties of 

 " not p or not q." Nicod {Proc. Camb. Phil. Soc, 19, 191 6), 

 using the one fundamental operation pjq, substantially reduced 

 the number of primitive propositions elaborated by Russell and 

 Whitehead for the construction of the logical calculus. 



In the field of ordinary algebra Huntington {Trans. Amer. 

 Math. Soc, 6, 1905) developed a set of thirteen postulates for 

 fields making use of multiplication and addition as undefined 

 notions. Norbert Wiener {Trans. Amer. Math. Soc, 21, 1920) 

 performs the same task for these undefined notions as Sheffer 

 performed in 191 3 for the undefined notions of the algebra of 

 logic. The one undefined operation Wiener uses is represented 

 by the symbol x @ y. In terms of @ multiplication and 

 addition can be defined. This operation turns out to have the 

 formal properties associated with xjy in algebra. 



It was established by L. L. Dines in 191 5 {Bull. Amer. Math. 

 Soc, 21) that Sheffer 's five postulates for Boolean algebras 

 (in terms, of course, of one undefined operation) are independent 

 in the ordinary sense that no one of the postulates is implied by 

 the other four. E. H. Moore, in the New Haven Mathematical 

 Colloquium {Yale University Press, 19 10), gave a new definition 

 for the independence of a set of postulates, as follows : a set of 

 m postulates is completely independent if, and only if, there are 

 no implicational relations existing between the properties 

 defined, either by the postulates or by the negatives of the 

 postulates. For, if the truth or falsity of one postulate implies 

 either that another postulate is true, or that it is false, it would 

 seem either that the two postulates are concerned with two 



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