Fact: If it rains, then the field is wet. Read the following statements: (i) It rains (ii) The field is not wet (iii) The field is wet (iv) It did not rain
If (i), then (ii).
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.
Important Questions from Instructions
Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows.
Statement:
I. All film stars are playback singers.
II. All film directors are film stars.
Conclusions:
I. All film directors are playback singers.
II. Some film stars are film directors.
All people in a certain are either ‘Knights’ or ‘Knaves’ and each person knows every other person’s identity. Knights NEVER lie, and knaves ALWAYS lie.
P says “Both of us are knights”, Q says “None of us are knaves”.
Which one of the following can be logically inferred from the above?A smart city integrates all modes of transport, uses clean energy and promotes the sustainable use of resources. It also uses technology to ensure the safety and security of the city, something which critics argue, will lead to a surveillance state.
Which of the following can be logically inferred from the above paragraph?
(i) All smart cities encourage the formation of surveillance states.
(ii) Surveillance is an integral part of a smart city.
(iii) Sustainability and surveillance go hand in hand in a smart city.
(iv) There is a perception that smart cities promote surveillance.The binary operation □ is defined as a □ b = ab + (a + b), where a and b are any two real numbers. The value of the identity element of this operation, defined as the number x such that a □ x = a, for any a, is_______.
Social science disciplines were in existence in an amorphous form until the colonial period when they were intuitionalized. In varying degrees, they were intended to further the colonial interest. In the time of globalization and the economic rise of postcolonial countries like India, conventional ways of knowledge production have become obsolete.
Which of the following can be logically inferred from the above statements?
i) Social science disciplines have become obsolete.
ii) Social science disciplines had a pre – colonial origin
iii) Social science disciplines always promote colonialism
iv) Social science must maintain disciplinary boundaries.
