APPENDIX. ARTICLE I. AN ANALYSIS OF SOME TRAINS OF REASONING. To elucidate and at the same time to test the accuracy of those views of the reasoning process which have been unfolded in the preceding chapters, perhaps the most effectual way will be to examine some specimens of argumentation, not fashioned for the purpose, but taken from productions written without reference to theories or canons of logic. The usual course in logical treatises is to frame syllogisms or enthymemes specially adapted to exemplify the rules and observations brought forward; and this has its advantages; but it ought not to supersede an examination and analysis of the actual reasoning employed by men in their ordinary discourse and writings to convince each other. The latter procedure may be expected to bring out some points which would have otherwise escaped remark, and, at all events, it is likely enough to put to the test the soundness of any theory on the subject. SECTION 1. Analysis of a Demonstration in Euclid. The first instance of reasoning which I shall select for this purpose, is the demonstration of a theorem in Euclid. THEOREM. An exterior angle of a triangle is equal to both its opposite interior angles, and all the interior angles of a triangle are together equal to two right angles. angles CAB, CBA, and BCA, are together equal to two right angles. Through the point c draw the straight line CE parallel to AB. 1. The interior angle BAC is equal to the exterior angle ECD, because AD is a straight line falling upon the parallel lines A B and CE. (book i. prop. 29.*) 2. Again, the alternate angles ABC and BCE are equal, because BC is a straight line falling upon the parallel lines AB and CE. (i. 29.) 3. Wherefore the two interior angles BAC and ABC are together equal to the two angles ECD and BCE or the whole angle BCD. 4. When to each of these equals is added the angle BCA, the angles BCA, BAC, and ABC, which are the three interior angles of the triangle, are together equal to the angles BCA and BCD. 5. But the angles BCA and BCD being made by the straight line BC on the same side of the straight line AD, are together equal to two right angles. (i. 13.) 6. Wherefore the three interior angles of the triangle are also together equal to two right angles. * Simson's Euclid. In this demonstration there are six distinct steps of reasoning. The first and second steps, although in appearance enthymemes, are in reality syllogisms, having the major premises not indeed formally stated nor yet suppressed, but only referred to as propositions formerly proved, viz., “a straight line falling upon two parallel straight lines makes the exterior angle equal to the interior opposite one," and "a straight line falling upon two parallel straight lines makes the alternate angles equal." The general principle or maxim exemplified by these two arguments, is the dictum de omni et nullo. In the latter argument, for example, the equality of the alternate angles ABC and BCE is not self-evident, but proved by the allegation previously demonstrated that all such angles are equal. The third step is an argument not requiring a major premise. The angles BAC and ABC having been shown to be respectively equal to ECD and BCE, the first pair together are intuitively discerned to be equal to the second pair together, or to BCD. To such reasoning, indeed, a major premise is, as we all know, sometimes appended, by citing the maxim (forming the 2nd Axiom in Simpson's Euclid) "if equals are added to equals the wholes are equal," but, as already explained, this can bring no confirmation to the argument, which is in itself perfectly conclusive. The axiom cited is only the general principle exemplified by the reasoning, and when introduced as a major premise is a logical impertinence. The fourth step is also a self-evident argument requiring no major premise, and exemplifies the same axiom, “when equals are added to equals the wholes are equal," or more correctly, "when the same quantity is added to equals, the wholes are equal." The fifth step is again an apparent enthymeme, with the major premise not formally stated but indicated as having been previously proved, viz. " the angles which one straight |