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Question 1008187: Assume that the following 4 logical propositions are all true :
1. (A → B) ∧ (A → ¬B)
2. ¬A → B
3. ¬(B ∧ D)
4. C ∨ ¬A
What can you say about the truth value of propositions A, B, C and D. You
can answer true, false, or uncertain. Justify your answers.
Truth value of proposition A :
Truth value of proposition B :
Truth value of proposition C :
Truth value of proposition D :
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Assumption: All four propositions given are true.
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Proposition 1: ( A --> B) & (A --> ~B)
( A --> B) & (A --> ~B) being true only happens when BOTH
(A --> B) is true
AND
(A --> ~B) is true
If B is true, then ~B is false. Or vice versa. What does this mean? It means that one of those two, (A --> B) or (A --> ~B), is going to be false if A is true. Consider B to be false. That means (A --> B) is false if A is true. If B was true, then the issue occurs with (A --> ~B)
So to summarize, A must be false for each piece of proposition 1 to be true. A must be false for proposition 1 as a whole to be true. B is unknown at this point
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Proposition 2: ~A ---> B
A is false (found earlier)
~A is true
In order for ~A ---> B to be true, B must be true as well. If ~A were true and B were false, then ~A --> B would be false.
So B is true.
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Proposition 3: ~(B & D)
~(B & D) is only true if either B or D is false. B was found to be true, so D has to be false.
D is false
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Proposition 4: C v ~A
A is false
~A is true
C could be true or it could be false. It's impossible to determine. The outcome of C v ~A is going to be true because ~A is true.
Because C doesn't pop up in any other proposition, and because of the issue discussed above, this means that C's truth value is unknown.
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Truth value of proposition A : false
Truth value of proposition B : true
Truth value of proposition C : uncertain
Truth value of proposition D : false
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