Problem 24 - Thinking Mathematically

Question

Write the converse, inverse, and contrapositive of each statement. "If it doesn't fit, you must acquit." (lawyer Johnnie Cochran)

Step-by-Step Solution

Answer

Converse: 'If you must acquit, it doesn't fit.' Inverse: 'If it fits, you do not need to acquit.' Contrapositive: 'If you do not acquit, then it fits'.

Step 1: Identify the hypothesis and conclusion

Identify the hypothesis and conclusion in the original statement. The hypothesis is 'it doesn't fit' and the conclusion is 'you must acquit'.

Step 2: Formulate the converse

Interchange the hypothesis and the conclusion to form the converse. The converse here would thus be: 'If you must acquit, it doesn't fit'.

Step 3: Formulate the inverse

Negate both the hypothesis and conclusion to form the inverse. So, the inverse would be: 'If it fits, you do not need to acquit'.

Step 4: Formulate the contrapositive

Negate and interchange both the hypothesis and conclusion to form the contrapositive. The contrapositive here would be: 'If you do not acquit, then it fits'.

Key Concepts

Converse Statements

In logical reasoning, a converse statement is created by exchanging the hypothesis and conclusion of the original statement. This transformation plays a key role in determining the relationship between two propositions from different perspectives. By switching their places, you explore if both ideas hold as true or meaningful also in this reversed order.

Take the example from attorney Johnnie Cochran's famous statement: "If it doesn't fit, you must acquit." The hypothesis here is "it doesn't fit," and the conclusion is "you must acquit."

• Original statement: "If it doesn't fit, you must acquit."

For the converse, swap the hypothesis and conclusion:

• Converse: "If you must acquit, it doesn't fit."

This new sentence changes the implied causality but doesn't automatically retain the truth of the original statement. Always make sure to verify if the converse holds true in practice, as it's not universally the case.

Inverse Statements

An inverse statement involves negating both the hypothesis and conclusion of the original statement. This process results in a new statement that can reveal additional insights about the relationship between the propositions.

While the inverse closely relates to the original, it doesn't imply that both will automatically share the same truth value. For instance, in the original statement from Johnnie Cochran: "If it doesn't fit, you must acquit," the inverse is formed by negating the hypothesis, "it doesn't fit," and the conclusion, "you must acquit:"

• Original statement: "If it doesn't fit, you must acquit."

Negate both components to create the inverse:

• Inverse: "If it fits, you do not need to acquit."

This new statement has reversed the initial conditions and outcomes, providing a fresh viewpoint on the logical connection initially suggested.

Contrapositive Statements

Contrapositive statements are both a reversal and negation of the original statement, making them a powerful tool in logical reasoning. Finding the contrapositive involves interchanging and negating both the hypothesis and conclusion, which often results in a statement that is equivalent to the original.

Let’s apply this to Cochran’s legal dictum: the statement "If it doesn't fit, you must acquit," involves the hypothesis "it doesn't fit" and the conclusion "you must acquit."

• Original statement: "If it doesn't fit, you must acquit."

Now, reverse and negate both parts:

• Contrapositive: "If you do not acquit, then it fits."

Unlike the converse and inverse, the contrapositive of a true original statement is always true. In logic, this means that if the original statement holds true, so will the contrapositive, offering a reliable method to test validity.