5U 



SCIENCE. 



[Vol. I., No. 18. 



STUDIES IN LOGIC. 

 Studies in logic. By members of the Johns Hopkins 



university. Boston, Little, Brown, Sf Co., 1883. 



7 + 203 p., 2 pi. 16°. 



Mr. C. S. Peirce and four of his students, 

 present or recent members of his logic classes 

 at Baltimore, offer us in this worli sis distinct 

 essays on topics of recent logical theory, 

 besides three shorter contributions classed as 

 notes. The volume is throughout studiously 

 unpretentious and very solid work, that might 

 have made much greater claims with perfect 

 safety. The style is extremelj' compact, and 

 the purchaser of the book will pay for no 

 padding. 



Four of the longer studies appeal only to 

 very special students. The two others, Mr. 

 Marquand's essa}- on the ' Logic of the Epi- 

 cureans ' and Mr. Peirce's very important 

 study of the logic of induction, entitled ' A 

 theory of probable inference,' will interest the 

 general student either of philosophj' or of 

 scientific method. 



Mr. Marquand's essay on the Epicurean 

 logic opens the book, and gives us an account 

 of the Epicurean theor}' of induction as it is 

 stated in the work of Philodemus, that has 

 been preserved in fragments in a Herculaneum 

 papyrus. One could wish that this essaj- had 

 been fuller upon some points ; but as a whole 

 we must accept it with thankfulness, as con- 

 taining useful and not otherwise so easily 

 accessible information. Mr. Marquand then 

 discusses a ' Machine for producing syllogistic 

 variations,' and adds a ' Note on an eight-term 

 logical machine.' 



Then follow two 'Algebras of logic,' by Miss 

 Christine Ladd (now Mrs. Fabian Franklin) 

 and Mr. O. H. Mitchell respectively. These 

 are new structures on Boole's foundation. 

 Miss Ladd uses two copulas, expressed by 

 the symbols v and v. With these she is able 

 to write algebraically all the old forms of 

 statement, and to perform the customary op- 

 erations of symbolic logic with great brevity 

 and facility. The copula v, a wedge, is used 

 to signify exclusion. A v B means that A is 

 wholly excluded from B ; i.e., that no A is B. 

 This copula is not to imply the existence of 

 the terras of the statement. The copula v, an 

 incomplete wedge, is the symbol of imperfect 

 exclusion. A v B means that some A is B. 

 And this copula is taken to imply the existence 

 of the terms of the statement. The symbol 

 00 is used for the universe of discourse. The 

 symbol finds no use in this algebra, a; v oo 

 expresses the non-existence of the class x; 

 and this is written more briefiy a; v. The 



notation thus established has the convenience 

 that avb = ab\, a&cv = a v 6c, etc., and, 

 with a corresponding notation for the other 

 copula, abc v = av be, etc. ; so that the fac- 

 tors of an excluded or not excluded combina- 

 tion ma3- be written in txny order, and the 

 copula may be inserted at any point or writ- 

 ten at either end. The notation is further 

 applied to combinations of propositions, and 

 to the processes of elimination ; and the rela- 

 tive simplicity of expression is preserved 

 throughout. 



Mr. Mitchell expresses propositions as logi- 

 cal polynomials, consisting of sums of terms, 

 formed after Boole's fasliion. The classes 

 indicated by the poh'nomials are stated in the 

 propositions to form either the whole or some 

 part of the universe of discourse. Thus, the 

 proposition that the universe U = a -|- 6 

 would mean that no a \s b. Such a proposi- 

 tion Mr. Mitchell expresses by the notation 

 {a + b)i; or, in general, if F be any logical 

 pol3'nomial, Fj means that F precisely fills up 

 the universe. F„ would express that F forms 



some part of the universe. F„ means that P 

 forms part of the universe. Propositions thus 

 formed are used for the purposes of inference 

 in a simple way, expressed in Mr. Mitchell's 

 words by the rule, " Take the logical product 

 of the premises, and erase the terms to be 

 eliminated." 



The foregoing may serve to suggest to any 

 one acquainted with Boole's notation the drift 

 of the innovations proposed in these two 

 algebras. Psychological importance, as Mr. 

 Peirce himself suggests', these two notations 

 can hardly claim. Thej- tell us nothing new 

 about the nature of the thinking process, but 

 are interesting onlj' as ingenious and possiblj' 

 useful methods for expressing very briefly 

 complex facts and elaborate logical calcula- 

 tions. As such expressions, they will hold 

 their own, and maj' even be noticed in that 

 not verj" distant time when the whole earth 

 shall be filled with logical algebras, whereof 

 there shall be, for all we can now see, as many 

 as there are tiles on the roofs of the houses. 



Mr. B. I. Oilman's verj' special studj- fol- 

 lows, on ' Operations in relative number, with 

 application to the theorj- of probabilities.' 

 Then comes the strong piece of the book, Mr. 

 Peirce's before-mentioned discussion of the 

 logic of induction. Tlais we have read, not 

 with entire conviction, but certainly with no 

 little admiration. Readers of Mr. Peirce's 

 fine papers called ' Illustrations of the logic 

 of science,' in the Popular science monthly of 



