Question

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?

• A. Both $I_1$ and $I_2$ are correct inferences
• B. $I_1$ is correct but $I_2$ is not a correct inference
• C. $I_1$ is not correct but $I_2$ is a correct inference
• D. Both $I_1$ and $I_2$ are not correct inferences

Answer 1

The correct option is A Both $I_1$ and $I_2$ are correct inferences

Let p : It rains
q : cricket match will not be played.
$I_1:p\Rightarrow q$
$\underset{–}{\sim q}$
$\therefore \underset{–}{\sim p}$
Clearly $I_1$ is correct since it is in the form of Modus Tollens (rule of contrapositive)
$I_2:p\Rightarrow q$
$\underset{–}{\sim p}$
$\therefore \underset{–}{\sim q}$
which corresponds$[p\Rightarrow q\wedge \sim P]\Rightarrow \sim q]$
$\equiv \left[\right(p^{\prime }+q\left)p^{\prime }\right]\Rightarrow q^{\prime }$
$\equiv p^{\prime }\Rightarrow q^{\prime }$
$\equiv (p^{\prime }{)}^{\prime }+q^{\prime }\equiv p+q^{\prime }$
which is not a tautology.
So $I_2$ is incorrect inference.