LITMIR.BIZ

       Note 9

      Both these reductions are given by Aristotle. Compare 28b14 and 28b20 with 28a24.

       Note 10

      I neglected to refer afterwards to the form of the substitution of some-S and not-P for their definitions. But in Part III, §4 I have reduced such arguments to the first figure.

       Note 11

      Prescision should be spelt with another s as its etymology suggests. But in correcting the proof-sheets Hamilton’s Metaphysics and Chauvin’s Lexicon led me astray.

       Note 12

      It may be doubted whether it was philosophical to rest this matter on empirical psychology. The question is extremely difficult.

       Note 13

      Theorem VII is the associative principle, the difficulty of the demonstration of which is recognized in quaternions. I give here the demonstration in the text in syllogistic form.

      To avoid a very fatiguing prolixity, I employ some abbreviated expressions. If anyone opines that these render the proof nugatory, I shall be happy to go through with him the fullest demonstration. (1). Let (Am,Bn) be any product of the series described in def. 6. By the principle of Contradiction this includes non-identical classes under it or it does not. If it does, let x denote one of these classes not identical with some other (Am,Bn).

      (2) Then, by cond. 3 of def. 6, x is of the form

      By Barbara.

      (3) The term Am occurs in this sum or it does not.

      (4) If it does the B by which it is multiplied is either Bn or it is not.

      (5) If it is, by def. 2, and Barbara,

      which is contradictory of the supposition in (1) that any x is (Am,Bn)

      (6) If this B is not Bn, denote it by Bq. Then by def. 2, and Barbara

      (7) But No Bq is Bn, by definition,

      (8) And, by def. 3 and Barbara,

      (9) Hence, from (7) and (8), by Celarent

      (10) Hence, from (6) and (10), and Felapton

      (11) But, by def. 3, and Barbara,

      (12) Hence, from (10) and (11), by Baroko

      which is contrary to the definition of x.

      (13) This reduces the supposition that Am occurs in the Σ(A,B) defined in (2) to an absurdity. So that Am does not occur in this sum.

      (14) Hence by (2) and Defs. 2 and 3

      (15) Hence, by Def. 7, cond. 1,

      (16) But by (1)

      (17) Whence, by def. 3,

      (18) And from (15) and (17)

      Which is absurd. Hence

      (19) The supposition that Am,Bn includes mutually coexclusive classes is absurd. Or, every thing of the form (Am,Bn) is an individual, as stated on p. 64.

      (20) Every individual X, that is, whatever does not include under it mutually exclusive classes, by condition 3 of def. 7 can be expressed in the form

      (21) Hence, if (Am,Bn) be a term of Σ(A,B) by def. 2

      (22)

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