Logical Implication Analysis

The core of the problem lies in understanding the given conditional statement and evaluating the logical possibility of other conditional statements based on it.

Given Fact: "If it rains, then the field is wet."

In logic, we represent this as:

• Let $P$ be the statement "It rains".
• Let $Q$ be the statement "The field is wet".
• The fact is written as the implication $P \implies Q$. This means that whenever $P$ is true, $Q$ must also be true.

Representing Statements and Options

The individual statements are:

• (i) It rains: $P$
• (ii) The field is not wet: $\neg Q$
• (iii) The field is wet: $Q$
• (iv) It did not rain: $\neg P$

Now, let's translate the given options into logical notation:

• Option 1: If (iii), then (iv) translates to $Q \implies \neg P$.
• Option 2: If (i), then (iii) translates to $P \implies Q$.
• Option 3: If (i), then (ii) translates to $P \implies \neg Q$.
• Option 4: If (ii), then (iv) translates to $\neg Q \implies \neg P$.

Identifying the Logically Impossible Statement

We need to determine which of these options contradicts the fact $P \implies Q$.

• Option 1 ($Q \implies \neg P$): This is the contrapositive of $P \implies Q$. A statement and its contrapositive are logically equivalent, so this is possible.
• Option 2 ($P \implies Q$): This is the original fact itself, hence it is possible.
• Option 3 ($P \implies \neg Q$): This statement claims that if it rains ($P$), then the field is *not* wet ($\neg Q$). This directly contradicts the given fact that if it rains ($P$), the field *is* wet ($Q$). It's impossible for both $P \implies Q$ and $P \implies \neg Q$ to be true simultaneously when $P$ is true. Therefore, this option is NOT logically possible.
• Option 4 ($\neg Q \implies \neg P$): This is also logically equivalent to $P \implies Q$ (it's the contrapositive of the contrapositive, or simply stated, if the field is not wet, it cannot have rained). This is possible.

Final Conclusion

The statement "If it rains, then the field is not wet" ($P \implies \neg Q$) cannot be true if the fact "If it rains, then the field is wet" ($P \implies Q$) is true. This represents a logical contradiction.