By obversion, the first set of propositions become No Ab is C or D ; No aB is c or E ; No D is e ; and these propositions are combined in the statement, Nothing is either AbC or AbD or aBc or aBE or De. (1)
By obverting and combining the second set of propositions, we have Nothing is AbCd or aBE or aBce or BDe or AbD or bDe. (2)
But AbCd or AbD is equivalent to AbC or AbD ; aBE or aBce to aBE or aBc ; BDe or bDe to De. Hence (1) and (2) are equivalent.
Again, by obverting and combining the third set of propositions, we have Nothing is AbD or bDe or aBc or aBE or abDe or acDe or AbC or De. (3)
But since bDe, abDe, acDe are all subdivisions of De, (3) immediately resolves itself into (1).
507. From the premisses (1) No Ax is cd or cy, (2) No BX is cde or cey, (3) No ab is cdx or cEx, (4) No A or B or C is xy, deduce a proposition containing neither X nor Y. [Johns Hopkins Studies, p. 53.]
By (2), No X is Bcde, and by (1) and (3), No x is Acd or abcd or abcE ; therefore, by section [491], No Acd or abcd or abcE is Bcde ; therefore, No Acd is Be.
It will be observed that since Y does not appear in the premisses, y can be eliminated only by omitting all the terms containing it.
508. The members of a scientific society are divided into three sections, which are denoted by A, B, C. Every member must join one, at least, of these sections, subject to the following conditions: (1) any one who is a member of A but not of B, of B but not of C, or of C but not of A, may deliver a lecture to the members if he has paid his subscription, but otherwise not; (2) one who is a member of A but not of C, of C but not of A, or of B but not of A, may exhibit an experiment to the members if he has paid his subscription, but otherwise not; but (3) every member must either deliver a lecture or perform an experiment annually before the other members. Find the least addition to these rules which will compel every member to pay his subscription or forfeit his membership. [Johns Hopkins Studies, p. 54.]
Let A = member of section A, &c.; X = one who gives a lecture; 519 Y = one who performs an experiment; Z = one who has paid his subscription.
The premisses are
(1) All Ab or aC or Bc is x or Z ;
(2) All Ac or aB or aC is y or Z ;
(3) Every member is X or Y ;
(4) Every member is A or B or C.
The problem is to find what is the least addition to these rules which will result in the conclusion that Every member is Z.
By (1), All z is either x or else (a or B) (A or c) (b or C);
therefore, All z is x or ABC or abc.
Similarly, by (2), All z is y or AC or abc ;
therefore, All z is xy or xAC or ABC or abc.
By (3), All z is X or Y ;
therefore,All z is XABC or Xabc or xYAC or YABC or Yabc.
By (4), All z is A or B or C ;
therefore, All z is XABC or xYAC or YABC ;
but All YABC is either XYABC or xYABC ;
therefore, All z is XABC or xYAC.
Hence, we gain the desired result if we add to the premisses, No z is XABC or xYAC. The required rule is therefore as follows: No one who has not paid his subscription may join all three sections and deliver a lecture, nor may he join A and C and exhibit an experiment without delivering a lecture.
509. What may be inferred independently of X and Y from the premisses: (1) Either some A that is X is not Y, or all D is both X and Y ; (2) Either some Y is both B and X, or all X is either not Y or C and not B? [Johns Hopkins Studies, p. 85.]
The premisses may be written as follows: (1) Either something is AXy, or everything is XY or d ; (2) Either something is BXY, or everything is x or y or bC.
By combining these premisses as in [chapter 4], Either something is AXy and something is BXY, or something is AXy and everything is x or y or bC, or something is BXY and everything is XY or d, or everything is bCXY or bCd or dx or dy.[518]
Therefore, eliminating X and Y (see sections [490] and [492]), Either something is A and something is B, or something is A, or 520 something is B, or everything is bC or d ; and by combining the first three alternants as in section [481], this becomes