| Let | V = voter, | v = not voter; |
| H = householder, | h = not householder; | |
| L = lodger, | l = not lodger. |
495 The first of the given statements is No citizen is VHL or vhl ; therefore (by obversion), Every citizen is either v or h or l and is also either V or H or L ; therefore (combining these possibilities), Every citizen is either Hv or Lv or Vh or Lh or Vl or Hl.
But (by the law of excluded middle), Hv is either HLv or Hlv ; therefore, Hv is Lv or Hl. Similarly, Lh is Vh or Lv ; and Vl is Hl or Vh.
Therefore, Every citizen is Vh or Hl or Lv, which is the second of the given statements.
Again, starting from this second statement, it follows (by obversion) that No citizen is at the same time v or H, h or L, l or V ; therefore, No citizen is vh or vL or HL, and at the same time l or V ; therefore, No citizen is vhl or VHL, which brings us back to the first of the given statements.
459. Given “All D that is either B or C is A,” shew that “Everything that is not-A is either not-B and not-C or else it is not-D.” [De Morgan.]
This example and those given in section [466] are adapted from De Morgan, Syllabus, p. 42. They are also given by Jevons, Studies, p. 241, in connexion with his Equational Logic. They are all simple exercises in contraposition.
We have What is either BD or CD is A ; therefore, All a is (b or d) and (c or d); therefore, All a is bc or d.
460. Infer all that you possibly can by way of contraposition or otherwise, from the assertion, All A that is neither B nor C is X. [R.]
The given proposition may be thrown into the form
Everything is either a or B or C or X ;
and it is seen to be symmetrical with regard to the terms a, B, C, X, and therefore with regard to the terms A, b, c, x. We are sure then that anything that is true of A is true mutatis mutandis of b, c, and x, that anything that is true of Ab is true mutatis mutandis of any pair of the terms, and similarly for combinations three and three together.
We have at once the four symmetrical propositions:
| All A is B or C or X ; | (1) |
| All b is a or C or X ; | (2) |
| All c is a or B or X ; | (3) |
| All X is a or B or C. | (4) |