AND ON LOGIC IN GENERAL. 



219 



F, affirmative; N, negative; V, universal; P, particular. The compartment NFN contains 

 all the syllogisms in which the first (minor) premise negative and the second premise affirmative 

 give a negative conclusion. The columns (PVP), (VPP) contain all in which the one pre- 

 mise is particular and the other universal. The middle column contains all the universal 

 syllogisms. It is flanked by four compartments of six each : and each one compartment con- 

 tains all the syllogisms of one particular conclusion. Thus all in the upper left flank give the 

 conclusion (). The canon* of validity is as follows : — Every pair of universals gives a con- 

 clusion : and every universal and particular in which the middle terms are of different quantities. 

 The canon of inference is : — Erase the symbols of the middle term, and what is left shews the 

 conclusion. Thus () )'( gives (•(, by which I signify that the copartient of an external is 

 exient : or reading metaphysically, [] ]'[ gives [•[, the irrepugnant of a repugnant is inde- 

 pendent. Supplying the terms, we have X () Y)-( Z gives X (■( Z ; or, on the abacus, 'Some 

 Xs are Ys, no Y is Z ; therefore some Xs are not Zs.' 



XXXVIII. Thirty-two combinations give valid syllogisms; and as many are invalid. 

 Sixteen of these invalid combinations, of which eight repeated twice, in conjunction with eight 

 of the valid forms, thirty-two in all, have a meaning of their own, as follows. The form of 

 our syllogism is. A, B, C being relations : — 



Every A of B is a C. 

 Now there aref also thirty-two truths of this form, derivable as follows : — 

 Every A is a C of every (converse of B). 



Thus every complement of every species is a complement : therefore every complement is 

 a complement of every genus. Again, every genus of every partient is a partient : therefore 

 every genus is a partient of every partient. The symbolic rule is as follows : — Choose any 

 one of the thirty-two combinations in which the middle spiculse are of different quantities. 

 Reject a universal followed by a particular. In any other case, strike out the middle spiculae, 

 and if the result be a universal, either let it stand, or change the second spicula : but if the 

 result be a particular, there is no choice but to change the second spicula. 



Thus (■) () is inoperative: there is no relation A of which we can say Every A is a 



Let P + Q = R express that P and Q, coexisting, give R : let 

 — P represent the contrary of P (or contradictory); let be 

 the symbol of impossibility of coexistence. If then P, Q, R be 

 three propositions which cannot coexist, so that P + Q + R = 0, 

 we have three modes of inference P+Q = — R, Q + R = -P, 

 R + P = - Q. Now Barbara may be expressed thus 



x))Y + y))z+x(-(z = o 



whence X))Y + Y))Z = X))Z Barbara I. 



y ))Z + X(-(Z = X(-(Y BarolcoU. 



X ))Y + X(-(Z = Y(-(Z BokardoUl. 



This process is carried through all the syllogisms of the first 

 figure. 



• There are various ways in which the symbols may be 

 put togetlier so as to give all the syllogistic forms by consecu- 

 tive pairs. Thus the following set 



)) )) )•( (•) 

 taken cyclically, that is, the first and last being considered as 



neighbours, give all the eight universal syllogisms in consecu- 

 tive pairs, if we read both backwards and forwards. And 

 under the same rule, eight particular syllogisms are seen in 

 each of the two following cycles : 



)) )•) )•( 



)•) )) )( (•) 



The following conceit gives a kind of zodiac of syllogism. 

 Put round a circle the twelve symbols here consecutively writ- 

 ten, distinguishing the universals by the thicker parentheses ; 



)) )•) )■) )) )( )•) )•( X (•) )•) 



Any two consecutive universals give a universal syllogism : 

 any universal with a contiguous particular gives a particular 

 syllogism. And these whether we read forwards or back- 

 wards. 



t If cj)'j/x<xx for all values of x, which is the proper 

 analogy for the composition of relations in the syllogism, then 

 (px < x'P'^^i '>'*' "* must not say \ltx< <t>'^x^- 



28 — 2 



