By (iii), Everything is a or be or Cd or cD ;
and by (iv). Everything is AB or AE or CD or cd ;
therefore, Everything is ABCd or ABcD or ACdE or AcDE or aCD or acd or bCDe or bcde ;
therefore by (i), Everything is ABCd or ABcD or Abcde or ACdE or AcDE or aBcdE or aCD or bCDe ;
therefore by (ii), Everything is ABCd or Abcde or ACdE or AcDE or aBcdE or aCD. (v)
Hence, All B is ACd or AcDE or acdE or aCD ;
All acd is BE ;
All AC is Bd or dE ;
All ac is BdE.
Eliminating E from each of the above we have the results arrived at by Boole.
Eliminating both A and E from (v) we have
Everything is BCd or bcd or Cd or cD or Bcd or CD ;
that is Everything is C or D or cd, which is an identity. This is equivalent to Boole’s conclusion that “there is no independent relation among the properties B, C, and D” (Laws of Thought, p. 148).
Any further results that may be desired are obtainable immediately from (v).
505. Given XY = A, YZ = C, find XZ in terms of A and C.
[Venn, Symbolic Logic, pp. 279, 310–312. Johns Hopkins Studies in Logic, pp. 53, 54.]
The premisses may be written as follows:
Everything is AXY or ax or ay ; (1)
Everything is CYZ or cy or cx. (2)
By (1), All XZ is AY or ay, and by (2), All XZ is CY or cy ; therefore, All XZ is ACY or acy. Hence, eliminating Y, All XZ is AC or ac.
This solves the problem as set. But in order to get a complete solution equivalent to that which would be obtained by Boole, the following may be added: Solving as above for x or z, and eliminating Y, we have All that is either x or z is AcXz or aCxZ or ac. Whence, by contraposition, Whatever is AC or Ax or AZ or CX or Cz is XZ. In other words, Whatever is AC or AZ or CX is XZ ; and Nothing is Ax or Cz.
518 506. Shew the equivalence between the three following systems of propositions: (1) All Ab is cd ; All aB is Ce ; All D is E ; (2) All A is B or c or D ; All BE is A ; All Be is Ad or Cd ; All bD is aE ; (3) Whatever is A or e is B or d ; All a is bE or bd or BCe ; All bC is a ; All D is E. [K.]