The former gives by being multiplied by s
The second gives
Hence,
This symbol is of the utmost importance.
Let ψa denote the case in which a does not exist or in which
Then hypothetical propositions are expressed thus:—
Particulars thus:—
Let us now take the premises
From the latter
Note 6
An advantageous method of working Boole’s calculus is given by Jevons in his Pure Logic, or the Logic of Quality.
Note 6
I have discussed this question in the last paper contained in this volume.
Note 7.
Aristotle uses this form to prove the validity of the simple conversion of the universal negative. He says: “If No B is A, no A is B. For if not let Γ be the A which is B. Then it is false that no B is A, for Γ is some B.”
Note 8
The Aristotelean method of writing these would be
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)