Inference by Opposition of Propositions

How to work one out under pressure

Do not memorise twenty four entries. Work each from the relation.

Step one. Find the contradictory of the given proposition, and write its truth value straight off, since contradiction always answers.

Step two. For the remaining two, name the relation, then apply the rule and ask whether it answers in this direction.

Step three. Where it does not answer, write "undetermined" and give the reason in three words: "contraries may both be false", or "truth does not ascend".

That procedure produces the correct answer every time and takes about a minute per proposition.

A worked example

A written statement asserts: "It is not true that all the goods delivered were defective."

Identify what is given. The proposition denied is A: all the goods delivered were defective. So A is false.

Contradictory first. The contradictory of A is O: some of the goods delivered were not defective. Since A is false, O is true. The defendant has, without saying so, asserted that at least one item was sound.

Now E. The contrary of A is E: none of the goods delivered was defective. Contraries may both be false, so from A being false nothing follows about E. The pleading does not assert that the goods were sound.

Now I. The subaltern of A is I: some of the goods delivered were defective. Falsity does not descend, so nothing follows about I. The pleading is entirely consistent with some of the goods being defective, and indeed with all but one being defective.

What the analysis is worth practically. The defence, as pleaded, admits nothing about the condition of the goods except that not every item was defective. Counsel who reads the denial as an assertion that the goods were sound has misread it, and counsel drafting for the plaintiff should ask for particulars of which goods are said to have been sound, because the defence as it stands is compatible with a case that is almost entirely lost.

And the drafting lesson, from sequence 410. A defendant who really means that the goods were sound must plead E, and denying A does not get there.

Distinctions that carry marks

What this does not mean

"Undetermined" is not a failure to answer. It is the answer, and it is worth the same mark as a truth value, provided the reason is given.

The inferences hold only where the terms are identical. Change the subject or the predicate and none of them applies.

The modern qualification stands. On the modern reading only the contradictory inferences survive, for the reason at sequence 380. An answer should state the traditional table, since that is what is examined, and note the qualification.

Quick revision

Contradiction always answers, in both directions. Start every question with it.

Contrariety answers from truth, not from falsity: if A is true E is false, and if A is false nothing follows.

Subcontrariety answers from falsity, not from truth: if I is false O is true, and if I is true nothing follows.

Subalternation: truth descends, falsity ascends, and neither reverse.

Eight of twenty four entries are undetermined, and saying so with the reason is a full answer.

The four with no undetermined entries are: A true, E true, I false, O false.

Test yourself

1. If A is true, what follows about E, I and O?

E is false, by the contrary relation, since two universals of opposite quality cannot both be true. I is true, by subalternation, since the truth of a universal descends to the particular of the same quality. O is false, by contradiction, since A and O are contradictories and exactly one of them is true. Nothing is undetermined when A is true.

2. If A is false, what follows?

Only that O is true, by contradiction. Nothing follows about E, because contraries may both be false: it may be that some of the class have the predicate and some do not. Nothing follows about I either, because falsity does not descend from a universal to its particular, and the particular may still be true of part of the class.

3. If I is true, what follows?

Only that E is false, by contradiction. Nothing follows about A, because truth does not ascend from a particular to its universal. Nothing follows about O, because I and O are subcontraries and may both be true, which is in fact the commonest situation, some members of a class having the predicate and some not.

4. State, for each relation, the direction in which it yields no conclusion.

Contradiction yields a conclusion in every direction. Contrariety yields nothing from the falsity of one member, since both may be false. Subcontrariety yields nothing from the truth of one member, since both may be true. Subalternation yields nothing from the falsity of the universal and nothing from the truth of the particular.

5. A defence pleads that it is not true that all the goods delivered were defective. What does it assert and what does it leave open?

It asserts that A is false, and therefore, by contradiction, that O is true: at least one item was not defective. It leaves entirely open whether some of the goods were defective, since falsity does not descend to the subaltern, and it says nothing at all in support of the proposition that none was defective, since contraries may both be false. The defence is consistent with almost the whole consignment being defective.

6. Why is "undetermined" a complete answer?

Because the question asks what follows, and in eight of the twenty four cases nothing does. The relations of contrariety, subcontrariety and subalternation each operate in one direction only, so a question that asks about the other direction has "nothing follows" as its correct answer. Stating it with the reason, such as that contraries may both be false, earns the mark; inventing a truth value loses it.