1. (P • ∼Q) ≡ (R $ \lor $ S)
2. ∼(R ≡ ∼T)
3. (T $ \lor $ ∼S) • (T $ \lor $ P)
4. ∼T
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) ∼P
B) Q
C) R
D) S
E) T

Question

Quizplus · Accepted Answer

Given ∼T from premise 4, we can use it in premise 3, (T ∨ ∼S) • (T ∨ P), to deduce that ∼S must be true for the first part of the conjunction to hold, since T is false. With ∼S, looking at premise 1, (P • ∼Q) ≡ (R ∨ S), and knowing S is false, for the equivalence to hold, P • ∼Q must also be false. Since we don't have information directly about P, we focus on the other side of the equivalence, R ∨ S, which now depends solely on R being true or false for the equivalence to hold. However, from premise 2, ∼(R ≡ ∼T), knowing T is false, R cannot be equivalent to true (since ∼T is true), suggesting R has to be true for the statement to be not equivalent (since if R were false, it would be equivalent to ∼T, which is true). However, this does not directly lead to any of the options being derivable without further assumptions. The mistake in the initial explanation is assuming specific truth values for R and the direct derivability of other options without considering all implications of the premises. The correct approach involves analyzing the logical structure and implications of the premises more carefully, particularly how they interact with each other under the rules of propositional logic. Given the premises, without a direct path to deduce the truth value of Q, R, S, or P from the given information and the mistake in the initial reasoning, the correct derivation from the premises is not straightforwardly provided in the explanation. The correct answer relies on a detailed analysis of the premises and the application of propositional logic rules, which was not accurately captured in the initial explanation.

1 (P • ∼Q) ≡ (R ∨ \lor ∨ S) 2 ∼(R ≡ ∼T) 3

1 (P • ∼Q) ≡ (R $\lor$ S)
2 ∼(R ≡ ∼T)
3