I4i EEPORT 1870. 



Here neither s'^=s nor sx=xs. If we say tliat in order to save whole these equa- 

 tions we may employ a different symhol for every application of the adjective small, 

 how can we express the meaning which is common to them all, and in virtue of 

 which the word small exists as an element of language ? 



Diffident as I am with respect to all these remarks on a method in which I find 

 so much to admire, I am yet more so with respect to the following. But it seems 

 to me that we cannot say that 



a;(l-.r)=0 



expresses proprio vie/ore, that is, in virtue of antecedent conventions, what is called 

 the principle of contradiction. 



In ordinary language we have words which, independently of this principle, 

 express negation ; we say red, not red, and the like ; hut in the ' Laws of Thought ' 

 there is no other means of expressing not red than by \—x,x denoting red. 



Now the interpretation of this sjTnhol 1— .r seems to me to bo given by the 

 principle of contradiction, and therefore I should rather say that the equation 

 .r(l—.r) = is interpreted by that principle than that it expresses it. In accord- 

 ance with this view, the equation x- = x would appear to be independent of the 

 principle of conti'adiction. 



On Boole^s ' Laws of TJiour/Jit.' By tJie Ecv. Eobeet Haeley, F.R.S. 



This paper was intended as a supplement to some " Remai-ks on Boole's Mathe- 

 matical Analysis of Logic," which the author submitted to the Section at the 

 Nottingham Meeting, an abstract of which was printed in the Eeport for 186G. 

 (See Transactions of the Sections, pp. 3-G.) 



From the logical equation a:-=.r, the equation a- ■^.■r-=0 is derived by subtracting 

 x"^ from both members, and the result is put under the form .r(l — a-)=Obythelaw 

 of distribution. It is to be observed, however, that at every stfep of the process 

 the principle of identity .v=x is assumed, and in Boole's iuterpretation of the final 

 result the same principle is used, for it is implied that the .r without the brackets is 

 identical with the x within. Further, in the final interpretation not only is the 

 principle of contradiction (or non-contradiction) emploj'ed, as Leslie EUis points 

 out in the latter part of his ' Observations,' but the principle of excluded middle ia 

 also employed. For in interpreting 1— ,r to mean not -.r, it is tacitly assumed 

 that every one of the things of which the universe, represented by xmi^, is made 

 up, is either x or not x. It would thus appear that these three principles, identity, 

 contradiction, and excluded middle, arc incapable of being reduced to more elemen- 

 tary truths. They are axiomatic, and Boole made use of them imconsciously 

 in ii-aming his laws of logical interpretation. ('Laws of Thought,' chap. iii. 

 prop, iv.) 



In chap. iii. § 6, Boole, by three different methods, one of which is partly logical, 

 and the other two are wholly algebraical, deduces the equation 



/(l)/(0)=0 

 from the equation for the expansion or development of any logical functiony(a:), viz. 



/(,r)=/(l)^+/(0)(l-^), 

 where /(.r) may or may not involve other class symbols than x. The latter equa- 

 tion is established in chap. v. § 10, by means of the principle that it is lawftil to 

 treat ,r as a quantative symbol susceptible only of the values and 1. But it is 

 worthy of notice that the fonner equation may be directly established by means 

 of the same principle. For, treating /('.r)=0 as an algebraic equation, of which 

 the root .r has only the values 1 and 0, we have at once, by the theory of equations, 



/(l)/(0)=0. 



The influence of Boole's ideas may be traced in works apparently so diverse as 

 Professor W. Stanley Jevons's ' Substitution of Similars,' Professor P. G. Tait's 

 ' Quaternions,' and Sir Benjnmin Brodie's ' Calculus of Chemical Operations.' The 

 system of logic proposed by Mr. Jevons is closely analogous to, and in some respects 

 identical with, that given by Boole ; but it is distingaiished from the latter by the 

 rejection of the calculus of 1 and 0. In a little work entitled " Pure Logic, or the 



