The given statement is at once resolvable into the four following propositions:
| ⎧ ⎨ ⎩ | All Ab is Cd or cD, | (i) | |
| All bC is Cd or cD, | (ii) | ||
| All Cd is Ab or bC, | (iii) | ||
| All cD is Ab or bC. | (iv) | ||
| (i) may be resolvedinto | ⎰ | All Abc is D, | (v) |
| ⎱ | All AbD is c. | (vi) |
But (vi) is inferable from (ii); and observing some other obvious simplifications we obtain immediately the following solution:
| (1) All Abc is D ; |
| (2) All bC is d ; |
| (3) All Cd is b ; |
| (4) All cD is Ab. |
543. Shew the equivalence between the two sets of propositions given in section [541]. [K.]
544. Find which of the following propositions may be omitted without affecting the information given by the propositions as a whole: All Ab is cDE ; All Ac is bDE ; All Ad is BCe ; All Ae is BCd ; No aE is B or C ; No B is c ; All Bd is ACe ; No bD is C or e ; No bE is Ad or C ; All C is B ; All Cd is ABe ; All cD is bE ; All cE is AbD or ab ; All de is ABC or abc. [K.]
545. Resolve each of the following complex propositions into a conjunction of propositions not containing any alternative combination of terms:
(1) Everything is ABCD or AbCd or aBcD or abcd ;
(2) Everything is AbCD or AbCd or Abcd or aBcd or abCD or abCd or abcd ;
(3) Everything is AbcDE or aBCd or aBCE or aBcd or aBde or abCe or abce or abDe or abde or BcdE or bCDe ;
(4) Everything is ABCE or ABcd or ABcE or ABde or Abcd or abCE or abcE or abdE or abde or BCde ;
(5) Everything is ABCDE or ABCdE or ABcDE or ABcDe or ABcde or AbCdE or Abcde or aBCDE or aBCde or abCDE or abcDe ;
538 (6) Everything is ABDe or ABDF or AcDe or Acef or aBDe or aBDF or abCD or abCd or abcD or abcd or aCDE or aCDe or aCdE or aCde or acDe or aDEF or aDEf or aDeF or aDef or BcDF or bceF or bcef ;
(7) Everything is AbdE or Abef or AbF or Acdef or aBDF or abCF or aCdE or ade or bCDe or bCdf or bDEF ;
(8) Everything is ABCEf or Abe or aBCdf or aBcdE or aBcdeF or abef or bceF. [K.]
546. Express the following proposition in as small a number as you can of propositions in which no alternative combination of terms occurs: Everything is ABCDe or ABCdE or ABcDe or AbCdE or AbCde or aBCdE or aBcDE or aBcde or aBcdE or abCde or abCdE. [J.]