If the proposition “Vegetarians are not meat eaters” is false, then which of…

Concept

In the square of opposition, an E-proposition (No S are P) and the corresponding I-proposition (Some S are P) are contradictories. Exactly one proposition in a contradictory pair is true.

The falsity of E alone does not make the universal affirmative A or the particular negative O determinate; each of those truth values remains undetermined.

Application

1. Let E be No vegetarians are meat eaters. The stem states that E is false.

2. Because E and I are contradictories, I—Some vegetarians are meat eaters—is true. This establishes statement 1.

3. A—All vegetarians are meat eaters—may be true or false when E is false. Therefore statement 2 appropriately treats A as doubtful.

4. O—Some vegetarians are not meat eaters—is not fixed by the falsity of E. It may be true, but it need not be true; therefore statement 4 treats it as doubtful, whereas statement 3 overstates it as true.

Contrast

• 1, 2 and 3 includes statement 3, which treats O as definitely true.

• 2, 3 and 4 omits the contradictory conclusion in statement 1 and includes both status claims about O.

• 1, 3 and 4 includes both O is true and O is doubtful, which cannot be the intended single status.

• 1, 2 and 4 combines the forced truth of I with the two undetermined judgements.

Cross-check

If E is false, at least one vegetarian is a meat eater, which proves I. That fact alone does not say whether every vegetarian is a meat eater or whether any vegetarian remains outside the class of meat eaters, so A and O remain undetermined.

Therefore, statements 1, 2 and 4 are correct.