Which of the following statements are logically equivalent? A. Some animals…

Concept

Logical equivalence means that two statements have the same truth conditions in every interpretation.

In predicate logic, “non-herbivore” is the complement of “herbivore”; therefore “not a non-herbivore” reduces by double negation to “herbivore”. Changing which class is negated changes the statement’s truth conditions.

Application

1. Let A(x) mean “x is an animal” and H(x) mean “x is a herbivore”.

2. Statement A becomes ∃x[A(x) ∧ H(x)]: at least one individual is both an animal and a herbivore.

3. Statement D becomes ∃x[A(x) ∧ ¬(¬H(x))]. Double negation gives ∃x[A(x) ∧ H(x)], exactly the form of A.

4. Statement B becomes ∃x[¬H(x) ∧ ¬A(x)], while C becomes ∃x[A(x) ∧ ¬H(x)]. Neither has the same truth conditions as A or D.

Cross-check and contrast

Replacing “non-herbivore” with “not herbivore” makes D read “Some animals are not not herbivores,” which simplifies to “Some animals are herbivores.”

• “A, B and C only” includes B and C, whose predicates do not reduce to A.

• “A and B only” pairs A with a statement about non-animals.

• “B, C and D only” groups statements with three different predicate patterns.

Result

Therefore, the logically equivalent statements are A and D only.