AND ON THE LOGIC OF RELATIONS. 



347 



Y to Z. If I had now produced this principle for the first time, and in the present manner, 

 it would surely have been imputed to me that I had made a fanciful definition of syllogism 

 out of a mathematical analogy. But my second paper will bear witness that I enunciated the 

 identity of inference with combination of relation at a time when I had not noted the extreme 

 closeness of the analogy. For when I in that paper remarked that the generality of the 

 notion of composition (of ratio) prevented the Greek geometers from needing to make separate 

 treatment of decomposition, I made no allusion (having in truth none to make) to the 

 analogous point of syllogism. But if I had generalised the mathematical notion, from the 

 Greek, the process would have been both natural and valid. For ratio is no direct trans- 

 lation of Xoyos: the Greek* word means communication; and the same turn of thought 

 which made Xoyos a technical term of geometry made it stand for any relation in one of its 

 derived meanings; that is, any way in which we talk about one notion in terms of another. 

 Any way of speaking of one notion with respect to a second, joined with a way of speaking of 

 the second notion with respect to a third, must dictate a way of speaking of the first notion 

 with respect to the third. And this is syllogism : it exhibits, in the most general form, the 

 law of thought which connects two notions by their connexions with a third. The character 

 of the connexions belongs as much to the matter of the syllogism as the character of the terms 

 connected. 



: The universal and all containing form of syllogism is seen in the statement of X..LMZ 

 is the necessary consequence of X..LY and Y..MZ. Whether the compound relation LM 

 be capable of presentation to thought under a form in which the components are lost in the 

 compound — in the same manner, to use Hartley's simile, as the odours of the separate in- 

 gredients are not separately perceptible in the smell of the mixture — is entirely a question of 

 matter. 



In the Aristotelian syllogism, figure is a function of the places of the middle term ; and 

 its necessity arises from the nature of the proposition being also a function of the places 

 of its terms : we cannot, in that system, say ' Every X is Y ' without having Y for the pre- 

 dicate. Adopt Hamilton''s expressed quantifications and, as he justly remarks, figure becomes 

 an unessential variation. Introduce the general idea of relation, and figure resumes its 

 importance: but not as connected with the place of the middle term. Whether we say 

 X..LY or LY..X, the figure is the same. Change of figure can be eiFected only by con- 

 version of relation. The^rs^ figure is that of direct transition : X related to Z through X 

 related to Y and Y related to Z. The fourth figure is that of inverted transition: X related 

 to Z through Z related to Y and Y related to X. The second figure is that of reference to 

 (the middle term) : X related to Z through X and Z both related to Y. The third figure 



• Euclid's definition of ralio, most properly when most 

 literally translated, is " Communicating instrument is a habi- 

 tude of two magnitudes of the same kind to one another with 

 respect to quantuplicity." We talk about magnitude in terms 

 of magnitude only by how many times one contains the other. 

 On the meaning of tiiXikotiis see the Penny Cyclopcedia and 



the Supplement, articles Ratio. That the communicating 

 instrument was called communication (Xoyos) was a case of 

 that feature of the Greek under which the excessive curve was 

 called excess (hyperbola), the defective curve the defect 

 (ellipse), the irregular angle the irregularity (anomaly), and 

 so on. Parabola is another instance, of which elsewhere. 



