TRANSACTIONS OF THE SECTIONS. 5 



passage intended is that whicli occurs on p. 103, wliere Leibnitz, in a paper entitled 

 "])ifficiiltates quredam logic*, " makes a near approacli to the enunciation of the 

 fundamental law of logic, although he does not, either in that paper or elsewhere, 

 so far as is known, state the law explicitly. He does, however, observe that 

 AB=BA, which is Boole's law of commutation ; and fiu-ther, that from the pro- 

 position, all A is B, we may infer that AB= A,^ — an inference which, applied to the 

 identical proposition A is A, gives us Boole's law of duality, A A = A. One of Leib- 

 nitz's illustrations of this inference is very curious. " Quidam se appellabat Grtjn- 

 BERG, viridis mons. Sodalis ei dicit, sufficeret ut Te appellares Berg, mons. Quid 

 ita ? respondet prior, putasne omnes montes esse virides ? Cui sodalis, ita, inquit, 

 nunc certe, nam testas erat. Ita illi naturalis sensus dictabat luec duo coincidere, 

 omnis mons est viridis et asqiiivalent viridis mons et mons." Boole did not become 

 aware of these anticipations by Leibnitz imtil more than twelve months after the 

 publication of his 'Laws of Thought,' when they were pointed out to him by E. 

 Leslie Ellis. Ellis subsequently addressed to Boole some " Observations,'' which 

 yet remain impublished, on some of the elementary parts of his system. These 

 " Observations," which are chiefly critical, throw much light on the -wi-iter's views 

 respecting the possibility of developing other calculuses of inference besides the 

 mathematical and the logical. 



3. The space at our disposal for this absti-act will not permit us to print Ellis's 

 paper here in e.rtenso, but the following brief extract will give a tolerably clear 

 idea of its general character. 



" It appears to be assumed in Chap. III. Section 8 [' Laws of Thought '], that 

 in deriving one conception from another the mind always moves, so to speak, along 

 the line of predicamentation, always passes from the genus to the species. No 

 doubt everything stands in relation to something else, as the species to the genus, 

 and consequently the symbolical language proposed is in extent perfectly general, 

 that is, it may be applied to all the objects in the universe. But I veutiu-e to 

 doubt whether it can express explicitly all the relations between ideas which really 

 exist, all the threads of connexion which lead the mind from one to the other. It 

 seems to me that the mind passes from idea to idea in accordance with various 

 principles of suggestion, and that in correspondence with the diflerent classes of 

 such principles of suggestion, we ought to recognize ditterent branches of the general 

 theory of inference. This leads me to a further doubt whether logic and the science 

 of quantity can in any way be put in antithesis to one another. From the notion 

 of tan apple we may proceed to that of two apples, and so on in a process of aggi'e- 

 gation, which is the foundation of the science of discrete quantity. Or again, 

 from the notion of an apple we may proceed to that of a red apple, and this move- 

 ment of the mind in lined predicatnentali is the foundation of ordinary logic. But 

 it is plain it priori that there are other principles of suggestion besides these two, 

 and the following considerations lead me to think that there are other exercises of 

 the reasoning faculty than those included in the two sciences here refeiTcd to. In 

 the first place, certain inferences not included in the ordinary processes of conver- 

 sion and syllogism were recognized as exceptional cases by the old logicians. Leib- 

 nitz has mentioned some with the remark that they do not depend on the dictum 

 (le omni et mdio, but on something of equivalent evidence. The only question is 

 whether we should be right in considering these cases as exceptions, and if they are 

 so, to what they owe their existence. One instance is the ini-ersio relationis, e. g. 

 Noah is Shem's father, therefore Shem is Noah's son. Here we pass from the idea 

 of Shem to that of his father, and vice rers(i. The movement of the mind is along 

 a track distinct from that which we follow, either in algebra or what we commonly 

 call logic. The perception of the truth of the inference depends on a recognition 

 of the correlation of the two ideas, father and son." 



The author gave his reasons for believing that, when the " exceptional cases " re- 

 ferred to in the above passage are fully investigated, and a calculus is devised for their 

 symbolical solution, it will be found that the processes involved in such a calculus 

 formally coincide with the processes commonly employed in the solution of func- 

 tional equations. He also pointed out that it was in this direction probably that 

 Boole's method would be found to admit of extension — an extension analogous 

 to that which Boole himself effected for the theory of the solution of differential 



