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Consider the following logical inferences.

$I_{1}$: If it rains then the cricket match will not be played.
The cricket match was played.
Inference:  There was no rain.

$I_{2}$: 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?

  1. Both $I_{1}$ and $I_{2}$ are correct inferences
  2. $I_{1}$ is correct but $I_{2}$ is not a correct inference
  3. $I_{1}$ is not correct but $I_{2}$ is a correct inference
  4. Both $I_{1}$ and $I_{2}$ are not correct inferences
in Mathematical Logic by
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0
Given: $R\rightarrow \ \sim C$

Hence: $C\rightarrow \ \sim R$

 

When it rains, we can imply there's no cricket match.

When it doesn't rain, we can' say anything. (because $\sim R$ doesn't imply anything)

4 Answers

+40 votes
Best answer
$I_1$ is a correct inference. $I_2$ is not a correct inference as it was not mentioned what would have happened if it hadn't rained- They might have played or they might not have played.
by
selected by
+6
This is an example of Modus tollen. Isn't it ?
+1
yes it is example of modus tollens
0

Only I1 is example of Modus Tollen. I2 does not look like Modus Tollen. Pooja Palod mam. please confirm. 

+1

I2 is an example of Logical Fallacy.Usually fallacy occurs in two forms:

Form 1)The Fallacy of affirming the consequence

If p then q.  AND
q

Then we cannot conclude anything.

Form 2) The fallacy of denying the antecedent.

If p then q.  AND

NOT P

Then we cannot conclude anything.

 

+21 votes

Let us assume p=It rains, the q=cricket match will not be played.

I1: If it rains then the cricket match will not be played   (p->q)

The cricket match was played. (~q)
Inference:  There was no rain.(~p)

p->q

~q


~p    (Modus tollens )

Thus I1 is a correct inference.

I2: can be written as 

p->q   
 ~p


Given inference - ~q 
Not a correct inference. When p is false and q is true, both the premises becomes true but conclusion becomes false

Hence answer is B 

by
edited by
0
Nice detailed answer .....
+2

I2 is a famous Fallacy.
This type of incorrect reasoning is known as Fallacy of denying the hypothesis.

Another one is -
p->q
q


p
It is known as Fallacy of affirming the conclusion.  


 

+18 votes

B) I1 is correct but I2 is not a correct inference.

by
+7 votes

No, the inference 'I' is not Correct.

Let P = it rains.

Q = Cricket match will not be played.

For P $\rightarrow$ Q to be true.

If P is true then Q has to be true. (There is no other choice for Q).

But if P is false then Q can be anything (True or False). Still P $\rightarrow$ Q is true.

This is also known as "The fallacy of denying the antecedent".

P $\rightarrow$ Q
~P

$\therefore$ (Nothing can be concluded).

by
0
but ,

P -> Q

 ~Q

-----------------

This is possible right ?

~P
+2

P $\rightarrow$ Q
~Q

$\therefore$   ~P

This is also known as Modus Tollen

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