If the proposition ‘domestic animals are hardly ferocious’ is taken to be…

Concept — contradictories and subalternation.

In the traditional square of opposition the four standard-form categorical propositions are A (All S are P), E (No S is P), I (Some S are P) and O (Some S are not P). A and O are contradictories, and so are E and I: a pair of contradictories can never be true together and never false together, so declaring one of them false makes the other certainly true. Subalternation adds a second rule — a true universal carries every weaker claim about the same class with it, so if every member of a class has a property, then most members and some members have it too. Because the square applies only to these four standard forms, an ordinary-language sentence must first be rewritten in standard form before its contradictory can be read off.

Application to this question

1. Put the given proposition into standard form. “Domestic animals are hardly ferocious” is a denial: it says that ferocity does not hold of domestic animals. The paper supplies the standard form in its own list — proposition (d), “Some domestic animals are non-ferocious”, is that denial written as the particular negative O. So stipulating that the given proposition is false is stipulating that this O proposition is false.

2. Apply contradiction. The contradictory of O is A: “All domestic animals are ferocious.” Because contradictories cannot both be false, A is certainly true.

3. Apply subalternation. With “All domestic animals are ferocious” true, the weaker quantified claim “Most of the domestic animals are ferocious” must also be true, since a property held by every member of a class is held by more than half of its members.

Cross-check against the remaining propositions

• “No domestic animal is ferocious” is the contrary of A. Two contraries cannot both be true, so with A established this universal negative is false, not certainly true.

• “Some domestic animals are non-ferocious” is precisely the O proposition the question declares false, so the premise itself rules it out.

• Reverse check: suppose “All domestic animals are ferocious” were false. Its contradictory “Some domestic animals are not ferocious” would then be true — which is exactly what the question forbids. So A cannot be anything but true.

Hence the propositions that can be claimed certainly true are “All domestic animals are ferocious” and “Most of the domestic animals are ferocious”, i.e. the code selecting propositions (a) and (b).