Oppositional Inference

Contradictory opposition is the opposition of a pair of propositions so related to one another that they cannot be either simultaneously true or simultaneously false. The truth of one excludes the truth of the other, and the falsity of one excludes the falsity of the other. In other words, contradictory propositions are so related to one another that if one is true, the other is false, and vice versa. This exclusion of both simultaneous truth and simultaneous falsity is the essential note of contradictory opposition.

Because contradictory opposition excludes both simultaneous truth and simultaneous falsity, it is the most perfect kind of opposition and is of great importance in controversy and debate. To refute a thesis or destroy the truth of a proposition, it is sufficient and necessary to prove its contradictory: to prove more is superfluous, to prove less is inadequate.

Quantified attributive propositions having the same subject and predicate but differing in BOTH QUALITY AND QUANTITY (that is, A and O, and E and I) are contradictories. For instance, the universal affirmative proposition “Every man is seated” is the contradictory of the particular negative proposition “Some man is not seated,” and vice versa; and the universal negative proposition “No man is seated” is the contradictory of the particular affirmative proposition “Some man is seated,” and vice versa. (See the square above.)

The Rules of Contradictories are based on the very notion of contradiction and may be briefly stated as follows:

1. If one of two contradictory propositions is true, the other is false.

2. If one is false, the other is true.

In case these rules are not already perfectly clear, they will be clarified by an inspection of the following diagrams of the propositions “Every man is seated” and “No man is seated.”

If every man is seated, then it is false that some man is not seated; and if some man is not seated, then it is false that every man is seated.

Similarly, if no man is seated, it is false that some man is seated; and if some man is seated, it is false that no man is seated. Consider the following diagram, in which the quantitative relationship of “man” and “seated ones” is displayed:

If no man is among the seated ones, it is false that some man is among them, and vice versa.

We shall now give some examples of contradictory propositions that are not quantified attributive propositions. Notice that the members of each pair are so related to one another that if one is true the other is false, and vice versa, and that this incompatibility in both truth and falsity is the essential note of contradictory opposition.

1. “Socrates is sick” and “Socrates is not sick.”

2. “Bobby is always eating” and “Bobby is sometimes not eating.”

3. “Johnny is never sick” and “Johnny is sometimes sick.”

4. “If it is raining, the ground is wet” and “It is false that if it is raining, the ground is wet”—which can also be expressed as “If it is raining, it does not follow that the ground is wet,” for the “if ... then” type of proposition asserts a sequence, while “it does not follow” denies a sequence.

5. “Both John and Mary will go” and “Either John or Mary or both will not go”—which can also be expressed as “Either John will not go, or Mary will not go, or neither of them will go.”

Propositions having the same subject but having contradictory terms as predicates are also contradictory propositions. Thus, the propositions “This is a man” and “This is a non-man” are related to one another as contradictories, since if the one is true the other is false, and vice versa. Similarly, propositions having the same subject but having predicates that are immediately opposed contrary terms are contradictories if the subject of the propositions belongs to the genus to which the immediately opposed contrary terms belong—for instance, “Angels are mortal” and “Angels are immortal.”

Later on, when we take up the classification of propositions, we shall give the contradictories of all the types needing special attention.

Contrary opposition is the opposition of a pair of propositions so related to one another that they cannot be simultaneously true but can be simultaneously false (at least as far as their form is concerned). The truth of one excludes the truth of the other, but the falsity of one does not exclude the falsity of the other. In other words, contrary propositions are so related that if one is true the other is false, but if one is false the other is doubtful. This exclusion of simultaneous truth but not of simultaneous falsity is the essential note of contrary opposition.

Universal attributive, or categorical, propositions having the same subject and predicate but differing in quality (that is, A and E) are contraries. For instance, the universal affirmative proposition “Every man is seated” is the contrary of the universal negative proposition “No man is seated,” and vice versa. (See the square above.}

The Rules of Contraries are based on the very notion of contrary opposition and may be stated briefly as follows:

1. If one of two contrary propositions is true, the other is false.

2. If one is false, the other is doubtful.

We shall now give examples of contrary propositions that are not quantified attributive propositions. Notice that the members of each pair are so related to one another that only one of them can be true but both of them can be false (at least as far as their form or structure is concerned).

The propositions “Both men and women are admitted” and “Neither men nor women are admitted” are contraries, as they cannot be simultaneously true but can be simultaneously false—for instance, if men are admitted but not women or if women are admitted but not men.

“This house is entirely white” and “This house is entirely black” illustrate contrary opposition, as both propositions cannot be true but both can be false—as would be the case if the house were partly white and partly black, or of some altogether different color such as red, brown, or green. The following pairs of propositions likewise illustrate contrary opposition:

1. “Johnny is never sick” and “Johnny is always sick.”

2. “Socrates is seated” and “Socrates is standing.”

3. “All his answers are right” and “All his answers are wrong.”

4. “He got 100% in his examination” and “He got 50% in his examination.”

5. “He should go” and “He should not go.”

If you study these pairs of propositions, you will see that the members of each pair are so related to one another that if one is true the other is false, but both of them can be false

Subcontrary opposition is the opposition of two propositions that, cannot be simultaneously false but can be simultaneously true: if one is false, the other must be true; but both of them can be true (at least as far as their form is concerned).

Subcontraries get their name from their position on the square of opposition “under” (sub) the contraries. Usually the name is limited to I and O, that is, to particular propositions having the same subject and predicate but differing in quality; thus, “Some man is seated” and “Some man is not seated” are subcontraries.

The Rules of Subcontraries may be stated briefly as follows:

1. If one is false, the other is true.

2. If one is true, the other is doubtful.

Thus, if “Some man is seated” is false, “Some man is not seated” is obviously true. On the other hand, if “Some man is seated” is true, it is impossible to tell whether or not some man is not seated; maybe some are seated and some are not, and maybe all are seated.

Subalterns are not, strictly speaking, opposites at all because (as far as their form is concerned) neither the truth nor falsity of either of them excludes the truth or falsity of the other. In other words, both of them can be true and both of them can be false. Nevertheless, subalterns (or, more accurately, a subalternant and its subaltern) are frequently called opposites because of their connection with the strict mutually repugnant opposites that we have already studied.

Subaltern opposition is the relationship of attributive, or categorical, propositions having the same subject, predicate, and quality, but differing in quantity. Usually the universal proposition (A or E) is called the subalternant or superior, and the particular proposition (I or O) is called the subalternate or subaltern, but sometimes both are called subalterns.

Rule of Subaltern Opposition. The rules of subaltern opposition may be stated briefly as follows:

1. If the universal is true, the particular is true; but if the universal is false, the particular is doubtful.

2. If the particular is true, the universal is doubtful; but if the particular is false, the universal is false.

A consideration of the examples given on the square will make these rules clear.

Note that induction, which we shall study later, proceeds from I to A; that is, from a limited number of instances to a universal law. Induction, however, is not formal inference but material inference; hence, the rules given here do not govern induction.