Consider the conditional logical propositions:
- $P$: "The canal gate is opened."
- $Q$: "The tertiary watercourse is pressurized."
- $R$: "The tail-end farmers experience severe moisture stress."
It is known that:
1. $P \implies Q$ (If the canal gate is opened, then the tertiary watercourse is pressurized).
2. $Q \implies \neg R$ (If the tertiary watercourse is pressurized, then the tail-end farmers do not experience severe moisture stress).
If it is observed that tail-end farmers DO experience severe moisture stress ($R$ is True), which of the following statements must logically be TRUE?
Options:
(A) The tertiary watercourse is not pressurized ($\neg Q$).
(B) The canal gate is not opened ($\neg P$).
(C) The canal gate is opened ($P$).
(D) The tertiary watercourse is pressurized ($Q$).
Answer Key:
A, B
Step-by-Step Solution & Explanation
**Method 1: Modus Tollens Chain**
1. From Premise 2: $Q \implies \neg R$.
Its contrapositive is $R \implies \neg Q$.
Given $R$ is True, by Modus Tollens, $\neg Q$ must be True (the tertiary watercourse is NOT pressurized). Hence (A) is True and (D) is False.
2. From Premise 1: $P \implies Q$.
Its contrapositive is $\neg Q \implies \neg P$.
Since $\neg Q$ has been established as True, it follows that $\neg P$ must be True (the canal gate is NOT opened). Hence (B) is True and (C) is False.
Therefore, options **A** and **B** are strictly true.