4 REPORT 18G6. 



has rigorously established, viz., " that the ultimate laws of logic — those alone upon 

 which it is possible to construct a science of logic — are mathematical in their form 

 and expression, although not belonging to the mathematics of quautit}'." The 

 term mathematics is here used in au enlarged sense, as denoting the science of the 

 laws and combinations of symbols, and in this view there is nothing imphiloso- 

 phical in regai-ding logic as a branch of mathematics, instead of regarding mathe- 

 matics as a branch of logic. The sjTnbols of common algebra are subject to three 

 laws, viz. — 



The law of commutation ccy=yx, (1) 



The law of convertibility of terms , x+i/= +y+.r, (2) 



The law of distribution x (y+~) =xi/+xz (3) 



These laws are fundamental ; the science of algebra is built upon them. And they 

 are axiomatic ; each of them becomes evident in all its generality the moment we 

 clearly apprehend a single instance. Now Boole has shown that the same laws 

 govern the symbols of logic, and that therefore in the logical system the processes of 

 algebra are all Aalid. But at the root of this system there is found to exist a law, 

 derived from the nature of the conception of class, to which the symbols of com- 

 mon algebra are not in general subject. This law is named by Boole " the law of 

 duality,'' and is expressed by the equations 



.r2=.r, y-=y, &c (4) 



Now Aiewing the equation x-=x as algebraic, the only values which will satisfy 

 it are and 1. If therefore an algebra be constructed in which the symbols 

 X, y, z, &c. admit iuditi'erently of the values and 1, and of these alone, it "follows 

 that " the laws, the axioms, and the processes of such an algebra are identical in 

 their whole extent with the laws, the axioms, and the processes of an algebra of 

 logic." Difference of interpretation alone divides them. Upon this principle 

 Boole's logical method is founded. Propositions are represented as equations ; 

 these are dealt with as algebraic, the literal symbols involved being supposed 

 susceptible only of the values and 1 ; all the reqinsite processes of solution are 

 performed; and linally the logical interpretation of the symbols is restored to 

 them. Some illustrations were given of the application of the method. That 

 mg^jtiOd, to use the originator's o^m words, " has for its object the determination 

 of any element in any proposition, however complex, as a logical function of the 

 remaining elements. Instead of confining oiu: attention to the ' subject' and the 

 ' predicate,' regarded as simple terms, we can take any element or any combination 

 of elements entering into either of them, make that element or that combination 

 the ' subject ' of a new proposition, and determine what its ' predicate ' shall be, 

 in accordance with the data afforded to us." In the same way any system of 

 equations whatever, by which propositions or combinations of propositions can be 

 represented, may be analyzed, and all the "conclusion" which those propositions 

 involve be deduced from them. 



2. Bacon, in his 'Novum Organum,' Liber Seeuudus Aphorismorum, A. XXWL, 

 notices incidentally an analogy that exists between a well-known axiom in ma- 

 thematics, and the fundamental canon of syllogism : he says, " Postulatum ma- 

 thematicum, ut qua eidem tertio aqualia sunt etimn inter se sint cequalia, conforme est 

 cum fabrica syllogismi in logica, qui imit ea qua? conveniunt in medio." On this- 

 passage E. Leslie Ellis remarks, " The importance of the parallel here suggested 

 was never imderstood until the present time, because the language of mathematics 

 and of logic has hitherto not been such as to permit the relation between them to be 

 recognized. I\Ir. Boole's ' Laws of Thought' contain the first development of ideas 

 of which the germ is to be found in Bacon and Leibnitz ; to the' latter of whom the 

 fundamental principle, that in logic a- = a, was known (vide Leibnitz, Philos. Works, 

 by Erdmann, p. l-SO). It is not too much to say that Mr. Boole's treatment of the 

 subject is worthy of these great names. Other calculuses of inference (using the 

 word in its •widest sense), besides the mathematical and the logical, yet perhaps 

 remain to be developed." (Bacon's Collected Works, vol. i. footnote on p. 281.) 

 The reference to Leionitz requires some correction, for on p. 130 of the edition 

 cited, there is nothing whatever relating to the logical question. Probably the 



