Problem 49 - Thinking Mathematically

Question

Express each statement in "if . . then" form. (More than one correct wording in "if ... then" form is possible.) Then write the statement's converse, inverse, contrapositive, and negation. Being neither hedonistic nor ascetic is necessary for following Buddha's "Middle Way."

Step-by-Step Solution

Answer

If-Then Form: 'If one is following the Buddha's Middle Way, then they are neither hedonistic nor ascetic'. Converse: 'If one is neither hedonistic nor ascetic, then they are following the Buddha's Middle Way'. Inverse: 'If one is not following the Buddha's Middle Way, then they are either hedonistic or ascetic'. Contrapositive: 'If one is either hedonistic or ascetic, then they are not following the Buddha's Middle Way'. Negation: 'One is following the Buddha's Middle Way and they are either hedonistic or ascetic'.

Step 1: Conversion to If-Then Form

The statement can be translated as 'If one is following the Buddha's Middle Way, then they are neither hedonistic nor ascetic.'.

Step 2: Converse

The converse of this statement would be 'If one is neither hedonistic nor ascetic, then they are following the Buddha's Middle Way.'.

Step 3: Inverse

The inverse of the statement is 'If one is not following the Buddha's Middle Way, then they are either hedonistic or ascetic.'.

Step 4: Contrapositive

The contrapositive of the statement is 'If one is either hedonistic or ascetic, then they are not following the Buddha's Middle Way.'.

Step 5: Negation

The negation of the statement is 'One is following the Buddha's Middle Way and they are either hedonistic or ascetic.'

Key Concepts

Converse of a Statement

In the realm of logical statements, understanding the converse is essential for building logical reasoning skills. The converse of a statement flips the hypothesis and the conclusion of an original 'if...then' statement.

For example, if the original statement is 'If it rains, then the ground is wet.', the converse would be 'If the ground is wet, then it rains.'. However, it's crucial to note that the truth of the converse is not dependent on the truth of the original statement.

Applying this to our exercise, the original statement, once translated to if-then form, is 'If one is following the Buddha's Middle Way, then they are neither hedonistic nor ascetic.'. Naturally, its converse becomes 'If one is neither hedonistic nor ascetic, then they are following the Buddha's Middle Way.', which is a distinct logical statement that needs separate verification to establish its truth value.

Inverse of a Statement

The inverse of a statement is derived by negating both the hypothesis and the conclusion of the original statement, without changing their order. This process introduces the concept of logical negation, which is the construction of a new proposition that will be true if and only if the original proposition is false.

In simpler words, for an 'if...then' statement, 'If A, then B', the inverse would be 'If not A, then not B'. For our textbook exercise, the original statement gets transformed into 'If one is not following the Buddha's Middle Way, then they are either hedonistic or ascetic.'.

While the inverse relates to the original statement, similar to the converse, the truth of the inverse is not dependent on the truth of the original. It's an independent proposition altogether.

Contrapositive of a Statement

The contrapositive might sound complex, but it's quite a straightforward concept once you get the hang of it. It is obtained by both negating and switching the hypothesis and conclusion of the original 'if...then' statement. Importantly, the contrapositive of any true statement is also always true, making it a valuable tool in mathematical proofs and logic.

To frame the contrapositive, take the statement 'If A, then B'. Its contrapositive is 'If not B, then not A'. From our exercise, 'If one is either hedonistic or ascetic, then they are not following the Buddha's Middle Way.' is the contrapositive of the original if-then statement. Recognizing this logical relationship helps in understanding implications of statements and their deductions.

Negation of a Statement

Negation plays a critical role in logical statements, serving as the foundation for constructing inverses and contrapositives. To negate a statement is to create a new statement that will be true if the original statement is false and vice versa. The negation effectively denies what the original statement asserts.

In the context of 'if...then' statements, the negation doesn't follow the same structural rules as the converse, inverse, and contrapositive. Instead, it directly opposes the assertion as a whole. For instance, the negation of 'If it is hot, then I will swim' is 'It is hot, and I will not swim'. Referring to our example, the negation is 'One is following the Buddha's Middle Way and they are either hedonistic or ascetic.'. This negation presents a scenario that the original statement would exclude, providing a contrasting viewpoint.