Question

Given $p$ : A man is a judge, $q$ : A man is honest.
If $S_1$ : If a man is a judge, then he is honest
$S_2$ : If a man is a judge, then he is not honest
$S_3$ : A man is not a judge or he is honest
$S_4$ : A man is a judge and he is honest
Then

• A. $S_2 \equiv S_3$
• B. $S_1 \equiv S_2$
• C. $S_2 \equiv S_4$
• D. $S_1 \equiv S_3$

Hint:

Memorize the fundamental rule: "If P then Q" ($\text{P} \rightarrow \text{Q}$) can always be rewritten as "Not P or Q" ($\sim\text{P} \vee \text{Q}$). Recognizing this identity allows you to match statements $S_1$ and $S_3$ instantly without building truth tables!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given two simple logical propositions, $p$ and $q$. We are also given four compound statements, $S_1$, $S_2$, $S_3$, and $S_4$, expressed in regular text prose. We need to identify which pair of statements are logically equivalent ($\equiv$).

Step 2: Key Formula or Approach:
We must convert each statement from natural language into formal symbolic mathematical logic notation: "If $A$, then $B$" translates to the conditional connective: $A \rightarrow B$ "not $A$" translates to negation: $\sim A$ "or" translates to disjunction: $\vee$ "and" translates to conjunction: $\wedge$ We will then use the conditional equivalence identity: $p \rightarrow q \equiv \sim p \vee q$.

Step 3: Detailed Explanation:
Let's systematically translate each sentence into symbolic logic notation: 1. $S_1$: "If a man is a judge, then he is honest" $\rightarrow$ $p \rightarrow q$ 2. $S_2$: "If a man is a judge, then he is not honest" $\rightarrow$ $p \rightarrow \sim q$ 3. $S_3$: "A man is not a judge or he is honest" $\rightarrow$ $\sim p \vee q$ 4. $S_4$: "A man is a judge and he is honest" $\rightarrow$ $p \wedge q$ Now, let's look at the standard conditional identity in mathematical logic: $$p \rightarrow q \equiv \sim p \vee q$$ By inspection, the symbolic expression for $S_1$ is $p \rightarrow q$, and the symbolic expression for $S_3$ is $\sim p \vee q$. Since these two expressions are logically identical, we have: $$S_1 \equiv S_3$$

Step 4: Final Answer:
The logical equivalence relation is $S_1 \equiv S_3$, which corresponds to option (D).