Which statement is logically equivalent to ‘All atheists are pessimists’?

Concept

A universal statement ‘All P are Q’ can be written as the implication P → Q.

An implication is logically equivalent to its contrapositive, ¬Q → ¬P. Its converse Q → P and inverse ¬P → ¬Q are not generally equivalent to it.

Application

1. Let P mean ‘is an atheist’ and Q mean ‘is a pessimist’. The given statement is P → Q.

2. Form the contrapositive by reversing the implication and negating both parts: ¬Q → ¬P.

3. In words, ¬Q → ¬P is ‘All non-pessimists are non-atheists.’

Contrast

• ‘All non-atheists are non-pessimists’ is the inverse ¬P → ¬Q.

• ‘All pessimists are atheists’ is the converse Q → P.

• ‘None of the above’ is excluded because the contrapositive is listed.

Cross-check

If a person is not a pessimist, that person cannot be an atheist under the original rule; otherwise the original implication would be violated.

Therefore, the equivalent statement is ‘All non-pessimists are non-atheists.’