By Hugh MacColl
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Extra info for Symbolic Logic and its Applications
F? ); t f but the converse (or inverse) implications are not necessarily true, so that the three formulae would lose their validity if we substituted the sign of equivalence ( The first two formulae for the sign of implication (:). need no proof; the third is less evident, so we will prove = it as denote the above three two being self-evident, to be a certainty, so that we get the Let follows. we assume
P(A) = = ; T € Sup
(A_) We get = A< = e;=(e y = r T , never a certainty, though for a variable is it may turn out true in a particular case. Again, we get ^(AT ) = (AT )e = (^) = e« = e; e T for 6 T means T which is a formal certainty. In this T though we have A = A yet (p(A) is not (0 T ) case, therefore, , , T equivalent to (p(A ).
This shows that the fourth term B'D' cannot be omitted But if we retain as redundant if we omit the third term. term B'D', fourth the third term B'C, we may omit the we then get for B'D'(CD' + CD + B'C7 = B'D'(Ce + C'n + eC')' = B D (C + C ) =B D )' / / , , / / J? = i7. Thus, we may omit either the third term B'C, or else the fourth term B'D', as redundant, but not both. 30. A complex alternative may be said to be in its simplest form* when it contains no redundant terms, and none of its terms (or of the terms left) contains any redundant factor.
Symbolic Logic and its Applications by Hugh MacColl