Which of the following statements is equivalent to the statement—'Some…

Concept

Negating a complemented class cancels the two negations: ¬(¬P) = P. In categorical logic, ‘Some A are B’ means that at least one individual belongs to both A and B, represented as ∃x(A(x) ∧ B(x)).

Application

1. Let R(x) mean ‘x is a realist’ and P(x) mean ‘x is a philosopher’.

2. ‘non-realist’ becomes ¬R(x), while ‘not non-philosopher’ becomes ¬(¬P(x)) = P(x).

3. The statement therefore becomes ∃x(¬R(x) ∧ P(x)).

4. Because conjunction is commutative, this is ∃x(P(x) ∧ ¬R(x)): ‘Some philosophers are not realists.’

Contrast

• ‘No philosophers are realists’ is universal and excludes every philosopher from the realist class; it does not express an existential claim.

• ‘Some realists are philosophers’ and ‘Some philosophers are realists’ both place the witnessed individual inside the realist class, whereas the given statement places that individual outside it.

Result

‘Some philosophers are not realists’ is the equivalent statement.