LITMIR.BIZ

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or Elijah, b that he is a man, c that he is mortal, and d that the Bible errs. Then a:c means that if any person is Enoch or Elijah he is mortal, or what is the same thing, that Enoch and Elijah are mortal. Then, (a:c):d means that if Enoch and Elijah are mortal the Bible errs, which is the first premise. The second premise is b:c, or if any person is a man he is mortal. Multiplying the two premises together, we have [(a:c):d](b:c). Now a:c = ā + c and (ā + c):d is to be converted into the regular form by negativing the antecedent and putting a + instead of the colon. We have, therefore, to find the negative of ā + c. The rule for finding the negative of any expression is this: put a line over every letter that has no line over it, and take a line off every letter that has a line over it, and everywhere substitute multiplication for addition and addition for multiplication. Applying this rule, the negative of ā + c is . That is to say, to deny that anything is either not Enoch or Elijah, or else is mortal, is equivalent to asserting that something is Enoch or Elijah and at the same time is not mortal. To prove that this is so it will be sufficient to show that these two expressions satisfy the formulae of contradiction and excluded middle. By the principle of contradiction their product ought to vanish. Now their product is . By the distributive principle this is the same as , but aā = 0 and , so that the whole is . Now and 0a = 0, so that it comes to 0 + 0 which is 0.

      Thus, the principle of contradiction is satisfied. According to the principle of excluded middle, the sum of any expression and its negative gives $. Adding the two expressions we have

. By the distributive principle of addition with respect to multiplication this is the same as . Now ā + a = $ and , so that the whole becomes ($ + c)(ā + $).

      But $ + c = $ and $ + ā = $, so that it reduces to $ + $ which is $; and thus, the principle of excluded middle is also satisfied. Our first premise, then, is

, and the product of the two premises is . We may arrange this by the associative principle in the following order, . We now put in a new parenthesis, which we are entitled to do by the associative principle, so as to write . We now break down the inner parenthesis and thus have , and since this is , or (a:b):d, which means that if Enoch or Elijah is a man the Bible errs.

      I will now give an example of another variety of indirect syllogism. Take the premises, Translated persons are not mortal and All men are mortal. Let a mean that any person is translated, b that he is a man, c that he is mortal. The first premise, no translated persons are mortal, might be written ac = 0. But if we take the negative of both members of this conclusion we have

, or simply . The other premise is . The product of the two premises is this, . Treating this precisely as in the case of a direct syllogism we get the conclusion , or no translated persons are men.

      We now go on to a slightly more complicated kind of reasoning, the dilemma. The dilemma is a reasoning in which you show that there are two (or more) possible alternatives, and then show that in either case a consequence follows. This kind of reasoning, although treated in books on Rhetoric, was first introduced into the treatises on logic about the year 1500 by Laurentius Valla. In point of fact the dilemma was very little used during the middle ages and it forms the most elementary example of the falsity of the traditional and Aristotelian notion that all reasoning is syllogistic. In the present system of signs, however, we fail to see anything peculiar about the dilemma, for the reason that we have in this system arbitrarily twisted every syllogism into a dilemmatic form, by writing all a is b in the form of either non-a or b. The truth is that this system of signs is altogether framed to meet the case of the dilemma. Syllogistic reasoning is so easy that it is got rid of by a little artifice. The old stock example of a dilemma is as follows: “It is not good to marry a wife, for if she be fair she will be common, if foul then loathesome” (Blundeville, Art of Logic, 1599). Let a mean that any object is a wife, b that she is fair, c that she is foul, d that she is eligible. Then, the first premise is ā + b + c, or any object is either a wife, or is fair, or is foul. The other two premises are ab:d and ac:d, that is, if any object is a fair wife it is not eligible, and if any object is a foul wife it is not eligible. These two propositions may be otherwise written,

and . Now, multiplying the three premises we have . By the application of the distributive principle of addition with respect to multiplication, this is the same as and by the application of the distributive principle of multiplication with respect to addition, this is the same as . Again, applying the same principle, and striking out bb and , we have , and by the striking out of factors this reduces to ā + d + d, or ā + d, or if any object be a wife she is not eligible, or no wife is eligible.

      Without the introduction of any further signs, the above kind of reasoning is all that this system can comprehend, but it is useful in two ways. First, the practice of the method gives us great facility in imagining facts in other logical relations, so that we reason much more easily than we did before, even without the use of the method. Secondly, the algebra is itself very useful whenever we meet with a very complicated state of things, provided it is not so excessively complicated as to be unmanageable even with this aid. I shall now add a number of examples sufficient to thoroughly exercise the student and give him a mastery of this really very simple system.

      Correspondence Course on the Art of Reasoning

      

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