Understanding Logical Possibility

The core of this question lies in understanding a conditional statement and its logical implications. We are given a fact: If it rains, then the field is wet. This can be represented using propositional logic.

• Let P represent the statement "It rains".
• Let Q represent the statement "The field is wet".

So, the given fact is a conditional statement: \(P \rightarrow Q\) (If P, then Q).

Analyzing Each Statement

Let's break down each numbered statement provided in the question:

• (i) It rains: This corresponds to P.
• (ii) The field is not wet: This is the negation of Q, represented as \(\neg Q\).
• (iii) The field is wet: This corresponds to Q.
• (iv) It did not rain: This is the negation of P, represented as \(\neg P\).

Evaluating Each Option for Logical Possibility

We need to determine which of the given options is NOT logically possible based on the initial fact \(P \rightarrow Q\).

Option 1: If (iii), then (iv).

• This translates to: If the field is wet (Q), then it did not rain (\(\neg P\)).
• Logically: \(Q \rightarrow \neg P\).
• Analysis: Is it possible for the field to be wet (Q) even if it did not rain (\(\neg P\))? Yes, absolutely. The field could be wet for other reasons, such as dew, sprinklers, or a flood, without rain being the cause. The original fact only states that if it rains, it gets wet; it doesn't say rain is the only way for it to get wet. Therefore, this statement is logically possible.

Option 2: If (i), then (iii).

• This translates to: If it rains (P), then the field is wet (Q).
• Logically: \(P \rightarrow Q\).
• Analysis: This statement is precisely the given fact. Since it is the established fact, it is by definition logically possible and true within the context of the problem.

Option 3: If (i), then (ii).

• This translates to: If it rains (P), then the field is not wet (\(\neg Q\)).
• Logically: \(P \rightarrow \neg Q\).
• Analysis: Let's compare this with our given fact: \(P \rightarrow Q\).
  • The fact states: If it rains (P), then the field is wet (Q).
  • This option states: If it rains (P), then the field is not wet (\(\neg Q\)).
• These two statements are direct contradictions if P is true. If P (it rains) is true, then according to the fact \(P \rightarrow Q\), Q (the field is wet) must be true. However, this option suggests that if P is true, then \(\neg Q\) (the field is not wet) must be true. It's impossible for both Q and \(\neg Q\) to be true at the same time. Therefore, this statement directly contradicts the given fact and is NOT logically possible.

Option 4: If (ii), then (iv).

• This translates to: If the field is not wet (\(\neg Q\)), then it did not rain (\(\neg P\)).
• Logically: \(\neg Q \rightarrow \neg P\).
• Analysis: This statement is the contrapositive of the original conditional statement \(P \rightarrow Q\). In logic, a conditional statement and its contrapositive are logically equivalent. This means if the original statement is true, its contrapositive must also be true, and vice-versa. Since the original fact (\(P \rightarrow Q\)) is given as true, its contrapositive (\(\neg Q \rightarrow \neg P\)) must also be true. Therefore, this statement is logically possible.

Conclusion

Based on the detailed analysis, the only option that is NOT logically possible, given the fact "If it rains, then the field is wet," is "If (i), then (ii)" because it states that if it rains, the field is not wet, which directly contradicts the established fact.