*d* = AB*cd* ꖌ A*b*C*d* ꖌ *abcd*, or *d* = A*d* (B*c* ꖌ *b*C) ꖌ *abcd*. In ordinary language, what is not-opaque is either metal which is gold, and then not-silver, or silver and then not-gold, or else it is not-metal and neither gold nor silver. *Fourth Example.* A good example for the illustration of the Indirect Method is to be found in De Morgan’s *Formal Logic* (p. 123), the premises being substantially as follows:-- From A follows B, and from C follows D; but B and D are inconsistent with each other; therefore A and C are inconsistent. The meaning no doubt is that where A is, B will be found, or that every A is a B, and similarly every C is a D; but B and D cannot occur together. The premises therefore appear to be of the forms A = AB, (1) C = CD, (2) B = B*d*. (3) On examining the series of sixteen combinations, only five are found to be consistent with the above conditions, namely, AB*cd* *a*B*cd* *ab*CD *abc*D *abcd*. In these combinations the only A which appears is joined to *c*, and similarly C is joined to *a*, or A is inconsistent with C. *Fifth Example.* A more complex argument, also given by De Morgan,[78] contains five terms, and is as stated below, except that the letters are altered. Every A is one only of the two B or C; D is both B and C, except when B is E, and then it is neither; therefore no A is D. [78] *Formal Logic*, p. 124. As Professor Croom Robertson has pointed out to me, the second and third premises may be thrown into a single proposition, D = D*e*BC ꖌ DE*bc*. The meaning of the above premises is difficult to interpret, but seems to be capable of expression in the following symbolic forms-- A = AB*c* ꖌ A*b*C, (1) De = D*e*BC, (2) DE = DE*bc*. (3) As five terms enter into these premises it is requisite to treat their thirty-two combinations, and it will be found that fourteen of them remain consistent with the premises, namely AB*cd*E *a*BCD*e* *ab*C*d*E AB*cde* *a*BC*d*E *ab*C*de* A*b*C*d*E *a*BC*de* *abc*DE A*b*C*de* *a*B*cd*E *abcd*E *a*B*cde* *abcde*. If we examine the first four combinations, all of which contain A, we find that they none of them contain D; or again, if we select those which contain D, we have only two, thus-- D = *a*BCD*e* ꖌ *abc*DE. Hence it is clear that no A is D, and *vice versâ* no D is A. We might draw many other conclusions from the same premises; for instance-- DE = *abc*DE, or D and E never meet but in the absence of A, B, and C. *Fallacies analysed by the Indirect Method.*