It has been sufficiently shown, perhaps, that we can by the Indirect Method of Inference extract the whole truth from a series of propositions, and exhibit it anew in any required form of conclusion. But it may also need to be shown by examples that so long as we follow correctly the almost mechanical rules of the method, we cannot fall into any of the fallacies or paralogisms which are often committed in ordinary discussion. Let us take the example of a fallacious argument, previously treated by the Method of Direct Inference (p. 62), Granite is not a sedimentary rock, (1) Basalt is not a sedimentary rock, (2) and let us ascertain whether any precise conclusion can be drawn concerning the relation of granite and basalt. Taking as before A = granite, B = sedimentary rock, C = basalt, the premises become A = A*b*, (1) C = C*b*. (2) Of the eight conceivable combinations of A, B, C, five agree with these conditions, namely A*b*C *a*B*c* A*bc* *ab*C *abc*. Selecting the combinations which contain A, we find the description of granite to be A = A*b*C ꖌ A*bc* = A*b*(C ꖌ *c*), that is, granite is not a sedimentary rock, and is either basalt or not-basalt. If we want a description of basalt the answer is of like form C = A*b*C ꖌ *ab*C = *b*C(A ꖌ *a*), that is basalt is not a sedimentary rock, and is either granite or not-granite. As it is already perfectly evident that basalt must be either granite or not, and *vice versâ*, the premises fail to give us any information on the point, that is to say the Method of Indirect Inference saves us from falling into any fallacious conclusions. This example sufficiently illustrates both the fallacy of Negative premises and that of Undistributed Middle of the old logic. The fallacy called the Illicit Process of the Major Term is also incapable of commission in following the rules of the method. Our example was (p. 65) All planets are subject to gravity, (1) Fixed stars are not planets. (2) The false conclusion is that “fixed stars are not subject to gravity.” The terms are A = planet B = fixed star C = subject to gravity. And the premises are A = AC, (1) B = *a*B. (2) The combinations which remain uncontradicted on comparison with these premises are A*b*C *a*B*c* *a*BC *ab*C *abc*. For fixed star we have the description B = *a*BC ꖌ *a*B*c*, that is, “a fixed star is not a planet, but is either subject or not, as the case may be, to gravity.” Here we have no conclusion concerning the connection of fixed stars and gravity. *The Logical Abacus.*