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I'm reviewing my previous exams for the final, and the only two true or false questions that confuse me are:

  1. If (A⊃B)∨(A⊃C) is true, then A implies either B or C.

  2. (P⊃Q)∨(P⊃~Q) means P⊃(Q∨~Q).

The answer is that both statements are false.

For 1, ((A⊃B)∨(A⊃C))⊃(A⊃(B∨C)), either by assigning truth values or by proof using inference, I find the conditional to be a tautology.

For 2, both (P⊃Q)∨(P⊃~Q) and P⊃(Q∨~Q) are tautologies. So ((P⊃Q)∨(P⊃~Q)) is equivalent to (P⊃(Q∨~Q)).

Then, why are these two statements false?

  • 1
    Perhaps "either B or C" implies exclusive OR, but I do not see what the second issue might be unless you were going over intuitionistic logic. – Conifold Dec 8 '16 at 22:17
  • For the second one, the only thing I can imagine seems utterly pedantic to me, namely that P itself does not entail Q v ~Q; it entails one or the other (since both entailments could never obtain simultaneously). But this seems pedantic and trivial because Q v ~Q is always true in standard sentential logic. and because material implication means 2 = ~P v Q v v ~P v ~Q which is satisfied for any value of Q. – virmaior Dec 9 '16 at 1:46
  • Does the source of the statement explain its answer? Both of these look tautological to me, as well. – Mark Andrews Dec 10 '16 at 2:18

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