Logic, Metaphysics, and the Natural Sociability of Mankind. Francis Hutcheson

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      CHAPTER 5

      The concluding modes in the four figures are six.

      1. AAA, EAE, AII, EIO, *AAI, *EAO.

      2. EAE, AEE, EIO, AOO, *EAO, *AEO.

      3. AAI, EAO, IAI, AII, OAO, EIO.

      4. AAI, AEE, IAI, EAO, EIO, *AEO.⁹

      Thus there are two [modes] in the first [figure], two likewise in the second, and one in the fourth, which are useless and have no names, because they make a particular inference where the valid conclusion would be universal.

      The named modes are contained in these verses:

      Barbara, Celarent, Darii, and Ferio are of the First;

      Cesare, Camestres, Festino, Baroko are of the Second;

      The Third claims Darapti and Felapton,

      And includes Disamis, Datisi, Bocardo, Ferison.

      Bramantip, Camenes, Dimaris, Fresapo, Fresison,

      Are of the Fourth. But the five which arise from the five universal [modes]

      Are unnamed, and have no use in good reasoning.

      Here are examples of the modes according to the vowels which are contained in the words [of the mnemonic], A, E, I, O.

      CHAPTER 6

      From axioms 1 and 2 (p. 32) the force of the inference in all of these modes will be clear, since both of the extremes are compared with the middle, and one of them with the distributed middle; and either both agree with it, or one only does not agree.

      The Aristotelians neatly demonstrate the force of the inference, and perfect the syllogisms, by means of reduction, since the validity of all [the syllogisms] in figure 1 is evident from the dictum de omni et nullo (see p. 26); they also give, in their technical language, the rules of conversion and opposition, by means of which all the other modes can be reduced to the four modes of the first figure, which Aristotle calls the perfect [modes].¹⁰

      There are two kinds of reduction, ostensive and ad absurdum. The initial letters in each of the modes (B, C, D, and F) indicate the modes of the first figure to which the modes of the other [figures] are to be reduced, i.e., those of which the initial letter is the same.¹¹ S and P following a vowel show that that proposition is to be converted, S simpliciter, P per accidens. M shows that the propositions are to be transposed, K that the reduction is made per impossibile, of which more later. When this is done, the conclusion reached will be either the same as in reducing Cesare, Festino, etc., or [a conclusion] which implies the same conclusion, or the contradictory to the conceded premiss. The validity of an ostensive reduction is known from the rules of conversion and subalternation.

      Reduction to the impossible is as follows. If it is denied that a given conclusion follows from true premisses, let the contradictory of the conclusion be substituted for the premiss whose symbol includes a K, like the major in Bokardo and the minor in Baroko; these premisses will then show in Barbara the truth of the contradictory of the premiss which was claimed to be true. If therefore the given premisses had been true, the conclusion would also have been true; for if it was not, its contradictory would have been true, and if that had been true, it will show (in Barbara) that the other premiss is false, contrary to the hypothesis.