Consider the following logical inferences.
I₁: If it rains then the cricket match will not be played.
The cricket match was played.
Inference: There was no rain.
I₂: If it rains then the cricket match will not be played.
It did not rain
Inference: The cricket match was played.
Which of the following is TRUE?
a)
Both I₁ and I₂ are correct inferences
b)
I₁ is correct but I₂ is not a correct inference
c)
I₁ is not correct but I₂ is a correct inference
d)
Both I₁ and I₂ are not correct inferences
Correct answer is option 'B'. Can you explain this answer?

Question

Consider the following logical inferences.
I₁: If it rains then the cricket match will not be played.
The cricket match was played.
Inference: There was no rain.

I₂: If it rains then the cricket match will not be played.
It did not rain
Inference: The cricket match was played.

Which of the following is TRUE?

a)
Both I₁ and I₂ are correct inferences
b)
I₁ is correct but I₂ is not a correct inference
c)
I₁ is not correct but I₂ is a correct inference
d)
Both I₁ and I₂ are not correct inferences

Correct answer is option 'B'. Can you explain this answer?

Answer

Explanation:

I1:
- The given statement is: 'If it rains then the cricket match will not be played.'
- Inference I1 states: 'The cricket match was played.'
- From the given statement, we can infer that if it rains, the cricket match will not be played. However, the statement does not provide any information about what happens if it does not rain. Therefore, we cannot definitively conclude that the cricket match was played just because it did not rain. There could be other reasons for the match not being played.

I2:
- The given statement is: 'If it rains then the cricket match will not be played.'
- Inference I2 states: 'It did not rain.'
- This inference is correct because the given statement explicitly states that if it rains, the cricket match will not be played. Since it did not rain, we can logically conclude that the cricket match was played.

Therefore, the correct answer is option B. The first inference (I1) is not a correct inference, while the second inference (I2) is a correct inference based on the given statement.

Answer

I₁ states that:
Statement
Consider A = it rains,
B = match played
So, If (it rains) then (match will not be played) that means,
A → (¬ B)
Inference states that “there was no rain” means A = false (F)
Therefore, for any F → (¬ B) i.e. for any "F that implies not B" is true.
So this inference is valid.
Here the only condition is if it rains the match will not be played. That doesn’t mean if it will not rain then the match will be played.
If the match was played then it means it doesn’t rain.
So this inference is valid.

I₂ states that:
Statement
Consider A = it rains,
B = match played
So, If (it rains) then (match will not be played) that means,
A → (¬ B)
Inference states that “the match was played”.
Here the only condition is if it rains the match will not be played. If the match was played then it means it doesn’t rain.
That doesn’t mean if it will not rain then the match will be played.
So this inference is invalid.
Hence, the correct answer is "option b".

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Consider the following two statements.S1: If a candidate is known to be corrupt, then he will not be electedS2: If a candidate is kind, he will be electedWhich one of the following statements follows from S1 and S2 as per sound inference rules of logic?

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