CHAPTER 6. OPPOSITIONS.
By the ‘opposition’ of two propositions, is meant: the exact logical relation existing between them — whether the truth or falsehood of either affects, or not, the truth or falsehood of the other.
In this context, note, the expression ‘opposition’ is a technical term not necessarily connoting conflict. We commonly say of two statements that they are ‘opposite’, in the sense of incompatible. But here, the meaning is wider; it refers to any mental confrontation, any logical face-off, between distinguishable propositions. In this sense, even forms which imply each other may be viewed as ‘opposed’ by virtue of their contradistinction, though to a much lesser degree than contradictories. Thus, the various relations of opposition make up a continuum.
Now, upon reflection, the logical relations of implication, incompatibility, and exhaustiveness, defined earlier, are found to be incomplete insofar as they leave certain issues open. There is therefore a need to combine them in various ways, to obtain a list of seven fully defining kinds of ‘oppositions’:
a. Mutual Implication (or implicance): is defined as the relation between two propositions which are either both true or both false. Each is called an implicant and is said to implicate the other. P implies Q, and Q implies P; and, nonQ implies nonP, and nonP implies nonQ.
b. Subalternation: is the relation between two propositions which are either both true or both false, or one — called the subalternant — false and the other — called the subaltern — true; the occurrence of ‘subalternant true and subaltern false’ being excluded by definition. The subalternant and subaltern may be referred to jointly as the subalternatives. This relation is, therefore, one-way implication. P implies Q, but Q does not imply P; and, nonQ implies nonP, but nonP does not imply nonQ.
Subalternation, may be counted as two distinct relations, subalternating, and being subalternated, each of whose direction must be specified. This is in contrast to the other five oppositions, which are symmetrical.
c. Contradiction: exists between two propositions which cannot be both true and cannot be both false. If either is true, the other is false; and if either is false, the other is true. They are said to be contradictories. Their affirmations are incompatible and their denials are incompatible. P implies nonQ, and nonP implies Q; and, Q implies nonP, and nonQ implies P.
d. Contrariety: two propositions are contrary if they cannot both be true, but may both be false. If either is true the other is false, but if either is false the truth or falsehood of the other is possible. They are said to be contraries. Their affirmations are incompatible, but not their denials. P implies nonQ, but nonP does not imply Q; and, Q implies nonP, but nonQ does not imply P.
e. Subcontrariety: occurs when two propositions cannot be both false, but may be both true. If either is false, the other is true; but the truth of either leaves that of the other indeterminate. They are said to be subcontraries. Their denials are incompatible, but not their affirmations. nonP implies Q, but P does not imply nonQ; and, nonQ implies P, but Q does not imply nonP.
f. Unconnectedness (or neutrality): two propositions are ‘opposed’ in this way, if neither formally implies the other, and they are not incompatible, and they are not exhaustive. Note that this definition does not exclude that unconnecteds may, under certain conditions, become connected (or remain unconnected under all conditions).
Note that these seven types of opposition define both directions of the relations concerned, in contrast to the basic logical relations. For this reason they may be called ‘full’ relations: they leave no question marks. They are logically exhaustive, allowing us to classify the relation of any pair of propositions.
There are other kinds of compound logical relations, besides the above mentioned seven. These concern paradoxical propositions, which imply even their own contradictory, or some contradiction. For example, ‘X is not X’ formally implies both that ‘something, called X, exists’ (by the law of identity), and that ‘there is no such thing as X’ (by the law of contradiction).
However, paradoxes are very rare in formal logic; rather they occur, only a bit less rarely, with specific contents. Formal logic is mainly interested in the oppositions between normal propositions, which are in principle consistent in form. More will be said about paradoxes later, when we look into the logic of logic.
The official terminology for the various kinds of opposition, here suggested, may not always accord with common usage. Especially note that in practise the word ‘contradiction’ is very often taken as equivalent to ‘incompatibility’, signifying (in official parlance) ‘either contradiction or contrariety’; thus, for instance, with the expression ‘law of contradiction’; we mean incompatibility. Also, the word ‘opposite’ is sometimes used to mean contradictory.
It is curious to note, too, that the words ‘subaltern’ and ‘subcontrary’, though quite old, are rarely used in practise; I have only seen them used by logicians. Such failures of words or meanings to enter the mainstream of language, are sad testimonies to the popular disinterest in studying logic.
The following table summarizes the above through analysis of the possibilities of combination of the affirmations and denials of two propositions, P and Q, which are given as being related by a certain opposition, specified in the left column. ‘Yes‘ indicates possible combinations, ‘no‘ impossible ones.
Table 6.1 Definitions of Full Oppositions.
POSSIBILITY OF: | P+Q | P+nonQ | nonP+Q | nonP+nonQ |
Implicance | yes | no | no | yes |
Subalternating | yes | no | yes | yes |
Being Subalternated | yes | yes | no | yes |
Contradiction | no | yes | yes | no |
Contrariety | no | yes | yes | yes |
Subcontrariety | yes | yes | yes | no |
Unconnectedness | yes | yes | yes | yes |
Note that incompatibles are either contradictory or contrary, while exhaustives are either contradictory or subcontrary. Also worth noting, compatibles may be either implicant, or subalternative (in one or the other direction), or subcontrary, or unconnected. The seven definite oppositional relations are mutually exclusive (i.e. contrary, to be exact), but one of the seven must hold.
The doctrine of opposition arose out of the need to apply the laws of thought to propositions more complex than the initial forms ‘S is P’ and ‘S is not P’. The concepts of equality, conflict, and limitation, had to be expanded upon, to reflect the more qualified relations found to exist between forms once they are quantified.
We know that two singular propositions differing only in polarity (viz. R, G) are contradictory, for at any given instant This-S cannot both be P and not-be P, and must be one or the other. But what of the plural versions of these forms? The following diagram shows their interrelationships.
Diagram 6.1 Rectangle of Opposition.
We note, to begin with, that for each polarity, the universal (all) subalternates the singular (any specific individual), which in turn subalternates the particular (some is an indefinite quantity, meaning one or more). Next, A universal and particular of opposite polarity (A and O, or E and I) are contradictory, just as two singulars (R and G) concerning one and the same individual are contradictory. Lastly, universals are contrary to universals or singulars of opposite polarity (A and E, A and G, R and E), and particulars are subcontrary to particulars or singulars of opposite polarity (I and O, I and G, R and O).
We may summarize these findings in the form of a ‘truth-table’. This tells us which other propositions must be true (T) or false (F), or may be either (.), in the context of each form given on the left under heading T being true, or each form given on the right under F being false.
Table 6.2 Truth-table.
T | A | R | I | E | G | O | F |
A | T | T | T | F | F | F | O |
R | . | T | T | F | F | . | G |
I | . | . | T | F | . | . | E |
E | F | F | F | T | T | T | I |
G | F | F | . | . | T | T | R |
O | F | . | . | . | . | T | A |
The conjunction of I and O may be viewed as a form of proposition in its own right, though composite. If we oppose this to the above standard forms, we obtain the following. Since ‘I + O‘ subalternates I and O (considered separately), it is contrary to A and E. It is unconnected to R and G, since either may be true or false without affecting it.
Also note in passing the position of forms quantified by ‘most’ or ‘few’, which we mentioned earlier… See Appendix 2 for remarks on this topic.
Note that two propositions with the same subject, but with different predicates, may be considered opposites, if the predicates are well known to be antithetical. Thus, ‘S is P’ and ‘S is Q’ may implicitly intend ‘S is P (but not Q)’ and ‘S is Q (but not P)’, respectively. In such case, the forms may of course be treated as effective contradictories.
These oppositions are proved as follows. Remember that each of the plural propositions can be defined by a series of singular propositions of the same polarity. Thus, A and I are reducible to a series S1 is P, S2 is P, S3 is P, etc., differing in that All-S covers the whole class of S, whereas Some-S covers only part of the same class. Likewise in the case of negatives, E and O. Thus the subalternation of singular or particular, to a generality of like polarity, is simply the inclusion by the whole of the class of any part thereof. This relation is unidirectional in that if the whole is affirmed or denied so is every part of it, whereas if some part is affirmed or denied it does not follow that other parts are.
Similarly, the contradictions of A and O, or E and I, are proven by consideration of their subsumptions. If all the members of a class are included in a predication, then any which is declared excluded would be found to be both P and nonP, an impossibility. The same can be argued in the negative case: if all are excluded, then none can be included without inconsistency.
With regard to I and O (or I and G, or R and O), they are subcontrary insofar as conflicting predicates can consistently be applied to different parts of the same subject-class, although it is impossible to evade either affirming or denying any predicate of a subject, i.e. one must be true. The contrariety of A and E (or A and G, or R and E) is due to the observation that, while they cannot be both true without implying some singular case(s) of inconsistency, they could be both false without antinomy, as occurs in the case of I and O being both true.
The concepts of inclusion and exclusion are geometrically evident. They were implicit in the original formulation of the laws of thought, when we referred to the whole or part of a singular phenomenon. In this logical discipline, we broaden the laws of thought, by treating individual instances as parts of a larger phenomenon we call a class or universal, and then applying our laws to this new whole. Essentially, no information has been added, we have merely in fact elucidated inherent data.
To conclude, let us point out that ‘opposition’ can be viewed as a kind of immediate inference, like eduction. This is especially obvious when we draw out an implicant or subaltern, but can also be said about affirming a proposition on the basis of another’s falsehood, or denying one on the basis of another’s truth or falsehood. Opposition is not a mere theoretical construct for logicians, but of practical value to the layman.