Q.Consider the following statements.
If D is dog, then D is very good.
If D is very good, then D is dog.
If D is not very good, then D is not a dog.
If D is not a dog, then D is not very good.
Identify the pairs of statements having the same meaning. Justify.

Symbolise the four statements; a conditional equals its contrapositive ($p\to q \equiv\ \sim q\to\sim p$), so pair (1)&(3) match and pair (2)&(4) match.

Let $p$: "$D$ is a dog" and $q$: "$D$ is very good." Then the four statements are:

• Statement (1): "If $D$ is a dog, then $D$ is very good" $\;\equiv\; p \rightarrow q$.
• Statement (2): "If $D$ is very good, then $D$ is a dog" $\;\equiv\; q \rightarrow p$.
• Statement (3): "If $D$ is not very good, then $D$ is not a dog" $\;\equiv\; \sim q \rightarrow \sim p$.
• Statement (4): "If $D$ is not a dog, then $D$ is not very good" $\;\equiv\; \sim p \rightarrow \sim q$.

The key logical law is that a conditional is logically equivalent to its contrapositive:

$p \rightarrow q \;\equiv\; \sim q \rightarrow \sim p.$

Applying it:

• Statement (3) $\sim q \rightarrow \sim p$ is exactly the contrapositive of Statement (1) $p \rightarrow q$, so $(1) \equiv (3)$. …