Understanding Logical Equivalence of Conditional Statements

Logical equivalence means that two statements have the same truth value in every possible situation. For conditional statements, the most common logical equivalent is the contrapositive.

Analyzing the Given Statement

The given statement is: "If Govind is a singer then he will sing a song."

We can represent this statement using symbols:

• Let P be the statement "Govind is a singer."
• Let Q be the statement "He will sing a song."

The original statement is in the form "If P then Q", which is written symbolically as $\text{P} \rightarrow \text{Q}$.

Logical Equivalents Explained

There are several forms related to a conditional statement $\text{P} \rightarrow \text{Q}$:

• Converse: If Q then P ($\text{Q} \rightarrow \text{P}$). This is NOT logically equivalent to the original statement.
• Inverse: If not P then not Q ($\neg\text{P} \rightarrow \neg\text{Q}$). This is also NOT logically equivalent to the original statement.
• Contrapositive: If not Q then not P ($\neg\text{Q} \rightarrow \neg\text{P}$). This IS logically equivalent to the original statement.

The contrapositive statement has the same meaning and truth value as the original conditional statement.

Evaluating the Options

Let's look at each option and determine its logical form relative to our original statement ($\text{P} \rightarrow \text{Q}$).

1. If Govind is not a singer then he will not sing a song.
   • This is "If not P then not Q". This is the Inverse.
   • The inverse is not logically equivalent to the original statement.
2. If Govind is a writer then he will write a story.
   • This statement introduces entirely new conditions ("Govind is a writer") and conclusions ("he will write a story").
   • It is not related logically to the original statement about Govind being a singer and singing a song.
3. If Govind sings a song then he must be a singer.
   • This is "If Q then P". This is the Converse.
   • The converse is not logically equivalent to the original statement.
4. If Govind does not sing a song then he must not be a singer.
   • This is "If not Q then not P". This is the Contrapositive.
   • The contrapositive is logically equivalent to the original statement.

Conclusion

The logical equivalent of the statement "If Govind is a singer then he will sing a song" is its contrapositive, which is "If Govind does not sing a song then he must not be a singer."