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Converse Statement – Definition and Examples
A converse statement is a conditional statement with the antecedent and consequence reversed.
A converse statement will itself be a conditional statement. It is only a converse insofar as it references an initial statement.
Before moving on with this section, make sure to review conditional statements.
This section covers:
- What is a Converse Statement?
- Truth Value of a Converse Statement
- How to Write a Converse Statement
- Converse Statement Definition
- Other Types of Statements
What Is a Converse Statement?
A converse statement is a conditional statement in which the antecedent and consequence of a given conditional statement are reversed.
Recall that a conditional statement is one that is in the format “if…, then…” They can be true or false.
The antecedent of a conditional statement is the part that follows the word “if.” Similarly, the consequence is the part that follows the word “then.”
Truth Value of a Converse Statement
The truth value of the original statement implies nothing about the truth value of the converse. Likewise, the truth value of the converse implies nothing about the truth value of the original statement.
Indeed, it is possible to have a false statement with a true converse and a false statement with a false converse. Likewise, a true statement can have a false converse or a true converse.
When a statement and its converse are both true, it is called a “biconditional statement.”
How to Write a Converse Statement
To write a converse statement, first, identify the antecedent and the consequence of the original statement.
Then, simply reverse the antecedent and the consequence. That is, whatever initially followed “if” should follow “then.” Likewise, whatever followed “then” should follow “if.”
Converse Statement Definition
A converse statement is a conditional statement in which the antecedent and consequence are reversed relative to a given conditional statement.
Using logic symbols, the converse of a conditional statement “$P \rightarrow Q$ is “$Q \rightarrow P$.
Other Types of Statements
In addition to converses are two other conditional statements derived from a given conditional statement. They are inverses and contrapositives.
The inverse negates both the antecedent and the consequence. The contrapositive is the converse of the inverse. That is it both reverses and negates the antecedent and the consequence of a given conditional statement.
Using logic symbols, for a given statement $P \rightarrow Q$, the inverse is $\neg P \rightarrow \neg Q$,” and the contrapositive is $\neg Q \rightarrow \neg P$.”
For the converse, inverse, and contrapositive, a conditional reference statement is necessary.
For example, note that the converse of the contrapositive results in the inverse. That is, swapping the antecedent and consequence of the contrapositive results in the inverse.
Note that the contrapositive and the original statement will always have the same truth value. The converse and the inverse will always have the same truth value. On the other hand, the truth values of the original statement and the converse, the original statement and the inverse, the contrapositive and the converse, and the contrapositive and the inverse are independent of each other.
It is also important to note that making any logical statement done twice results in the original statement.
That is, the converse of the converse is the original statement.
Consider $P \rightarrow Q$. The converse is $Q \rightarrow P$. Then, the converse of this statement is $P \rightarrow Q$, the original conditional statement.
Examples
This section covers common examples of problems involving converse statements and their step-by-step solutions.
Example 1
Think of a conditional statement that is false whose converse is also false.
Solution
When trying to think of conditional statements and truth values, it is helpful to think of the animal kingdom. Easy options for antecedents and consequences include cats, dogs, mammals, and fish.
Even selecting the first two from this list results in a conditional statement that is false with a false converse.
Conditional statement: If it is a cat, then it is a dog.
Converse statement: If it is a dog, then it is a cat.
These statements are both false, with a tabby cat being a counterexample for the first and a Golden Retriever being a counterexample for the second.
Example 2
“Franco’s car is red.”
Convert this sentence to a conditional statement and then find the converse. What are the truth values of the two statements?
Solution
The sentence converts to the conditional statement, “If it is Franco’s car, then it is red.”
Here, the antecedent is “it is Franco’s car.” The consequence is, “it is red.”
Therefore, the converse is “If it is red, it is Franco’s car.”
Clearly, this is false. An apple is a counterexample.
The true value of the original statement, however, is unknown. A picture or a view of Franco’s car is necessary to determine this.
Example 3
If it is raining, then it is not sunny.
Determine whether the converse of the given statement is true or false.
Solution
First, it is required to identify the converse. In this case, it is “If it is not sunny, then it is raining.”
Is this true? No. A dry but overcast day is a counterexample that shows this is false.
Example 4
“If it is basketball, then it is a sport.”
Identify the inverse, converse, and contrapositive of a given statement. Then, identify the truth value of each statement.
Solution
First, the converse reverses the antecedent and the consequence. That makes the converse:
“If it is a sport, then it is basketball.”
This is false. Ping pong is a counterexample.
The contrapositive reverses and negates the antecedent and consequence. Therefore, it is:
“If it is not a sport, then it is not basketball.”
This is true since basketball is a sport.
Finally, the inverse simply negates the antecedent and consequence. In this case, it is:
“If it is not basketball, then it is not a sport.”
As with the converse, this statement is false. Ping pong is a counterexample.
Note, as expected, the statement and the contrapositive have the same truth value. The converse and the inverse also have the same truth value.
Example 5
Find the converse of the inverse of the converse of the contrapositive of a statement.
Solution
It is best to work on this problem beginning at the end.
Consider a given conditional statement $P \rightarrow Q.$
The contrapositive of this statement is $\neg Q \rightarrow \neg P$.
Then, the converse of this statement is $\neg P \rightarrow \neg Q$.
Next, the inverse of this statement is $\neg \neg P \rightarrow \neg \neg Q$. Since two negatives make a positive, this is just $P \rightarrow Q$.
Finally, the converse of this statement is $Q \rightarrow P$.
Therefore, the converse of the inverse of the converse of the contrapositive is just the converse. That’s because the inverse of the converse of the contrapositive is the initial statement.
Practice Problems
- Think of a statement that is false with a true converse. Use cat, dog, mammal, and fish to help if needed.
- “Mammals do not lay eggs.” Write this statement as a conditional statement. Then, find the converse of this statement. Is the original statement true? How about the converse?
- If it is a country, then it is Germany. Determine whether the converse of this statement is true or false.
- “If it is a 90-degree angle, then it is a right angle.” Identify the converse, inverse, and contrapositive of this statement. Then, determine the truth value of each.
- What is the contrapositive of the inverse of the contrapositive of the converse of the inverse of a statement?
Answer Key
- There are many such examples. One is, “If it is a mammal, then it is a cat.” This is false because dogs are also mammals. The converse, “If it is a cat, then it is a mammal, is true.”
- “If it is a mammal, it does not lay eggs.” The converse is, “If it does not lay eggs, then it is a mammal.” These statements are both false. The counterexample for the first is a platypus, and the counterexample for the second is a Great White Shark. (Note that there are other valid counterexamples).
- This statement is false, and a counterexample is Peru. But its converse, “If it is Germany, then it is a country,” is true.
- Converse: “If it is a right angle, then it is a 90-degree angle.” This is true.
Inverse: “If it is not a 90-degree angle, then it is not a right angle.” This is also true.
Contrapositive: “If it is not a right angle, then it is not a 90-degree angle.” Like the others, this is true. - This is the contrapositive, $\neg Q \rightarrow \neg P$.