If the statement ‘No saints are materialists’ is given as false, which of the…

Concept

Categorical propositions in the traditional square of opposition have fixed logical relations.

An E-proposition, ‘No S are P,’ and its I-proposition, ‘Some S are P,’ are contradictories: exactly one is true and the other is false.

Application

1. The given proposition is the E-form ‘No saints are materialists,’ and it is stipulated to be false.

2. The contradictory I-form ‘Some saints are materialists’ must therefore be true.

3. An A-form is contrary to an E-form and an O-form is subaltern to an E-form, so falsity of the E-form does not by itself fix either one. ‘All materialists are saints’ reverses the terms and changes the quality from negative to affirmative while keeping the quantity universal; the result is a universal affirmative proposition. It is not the converse of the given E-form, and its truth is also undetermined.

Contrast

• ‘All saints are materialists’ is the A-form, the contrary of the given E-form.

• ‘Some saints are materialists’ is the I-form, the contradictory of the given E-form.

• ‘Some saints are not materialists’ is the O-form, the subaltern of the given E-form.

• ‘All materialists are saints’ reverses the terms and changes the quality from negative to affirmative while keeping the quantity universal; the result is a universal affirmative proposition. It is neither the converse nor the contradictory of the given E-form.

Cross-check

Set-theoretically, saying ‘No saints are materialists’ is false means that the intersection of saints and materialists is not empty. Hence at least one saint is a materialist, which is exactly the I-proposition.

Result

Therefore, the statement ‘Some saints are materialists’ alone is immediately inferred.