Relative Properties of Propositions

Hitherto we have considered propositions in themselves. Now we consider them in relation to other propositions made up of the same terms. Three relative properties will be studied:

1. Opposition — existing between propositions that have the same subject and predicate but differ in quantity or quality (or both).
2. Equipollence — existing between two propositions that have the same subject and predicate and the same force of meaning, yet differ in the number of negations they contain.
3. Conversion — existing between a proposition and itself transposed (subject and predicate having changed places), the truth of the proposition being conserved.

By means of these properties we may draw immediate inferences — direct conclusions to other propositions, or to knowledge of the truth or falsity of other propositions.

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a) Opposition of Propositions

Opposition is a relative property existing between two propositions that have the same subject and the same predicate but differ in quantity or quality or both. (Without that common ground — same subject, same predicate — there can be no opposition, only independent unrelated propositions.)

There are two kinds of opposition properly so called (contradiction and contrariety) and two kinds improperly so called (subcontrariety and subalternity).

1. Contradiction

Between an A- and an O-proposition, or between an E- and an I-proposition (same subject and predicate). One is necessarily true, the other necessarily false; they exhaust the possibilities.

Of contradictories, one is necessarily true, the other necessarily false.

Example: “Every man is wise” — “Some man is not wise.”

2. Contrariety

Between a universal affirmative (A) and a universal negative (E) proposition (same subject and predicate). Contraries are sweeping denials of each other, but they leave a middle ground; they do not exhaust the possibilities.

Of contraries, both cannot be simultaneously true, although both may be false.

Example: “Every man is wise” — “No man is wise.”

3. Subcontrariety

Between a particular affirmative (I) and a particular negative (O) proposition (same subject and predicate). The “some man” in each need not be the same individual; there is no proper opposition.

Of subcontraries, both may be true; both cannot be false.

Example: “Some man is wise” — “Some man is not wise.”

4. Subalternity

Between a universal and a particular proposition of the same quality (A and I; or E and O) with the same subject and predicate. The universal is called the subalternant; the particular is the subalternate.

Of subalterns:

• If the subalternant is true → the subalternate must be true.
• If the subalternate is true → the subalternant may be true or false.
• If the subalternate is false → the subalternant is false.
• If the subalternant is false → the subalternate may be true or false.

The Logical Square

The four types of opposition are displayed in the Logical Square:

         A ————— contraries ————— E
         |  "All men are wise"    |
         |  "No man is wise"      |
  sub-   |                        |  sub-
  alter- |                        |  alter-
  nants  |                        |  nants
         |                        |
         I —— subcontraries —— O
   "Some man is wise"    "Some man is not wise"

     A and O are contradictories.
     E and I are contradictories.

Practical value of opposition in argument:

1. Avoid sweeping A- or E-propositions in debate, lest the entire argument be overthrown by proof of the contradictory — a single contrary instance.
2. To defeat a general statement, prove its contradictory, not its contrary — proving the contrary leaves the argument unsettled (“both contraries may be false”).
3. The contradictory is the most powerful and invincible argument.
4. Guard against an opponent trying to disprove your position by establishing only the subcontrary (“both subcontraries may be true”).
5. Avoid concluding to the truth of a subalternant from the truth of its subalternate.

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b) Equipollence of Propositions

Equipollence (or equivalence) is the relative property existing between two propositions that have the same subject and predicate and mean the same thing, but differ in one or more negations. Example: “All men are animals” — “No man is not an animal.”

Practical value: Equipollence makes for accuracy of thought and expression. Sometimes a seeming denial may be shown by equipollence to be an affirmation. It also affords a means of expressing an obscure or vague proposition in clear and distinct form.

Rules for forming equivalents by equipollence:

1. To form the equivalent of the contradictory of any simple proposition, place a negative particle before the subject.

“All men are wise” → “Not all men are wise” = “Some men are not wise” (the O-proposition, contradictory of the original A).

2. To form the equivalent of the contrary of any simple proposition, place a negative particle before the predicate.

“All men are wise” → “All men are not-wise” = “No man is wise” (the E-proposition, contrary of the original A).

3. To form the equivalent of the subaltern (subalternant if original is I or O; subalternate if original is A or E), place a negative particle before both subject and predicate.

“All men are wise” → “Not all men are not-wise” = “Some man is wise” (the I-proposition, subalternate of the original A).

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c) Conversion of Propositions

Conversion is the process by which one proposition is immediately inferred from another by transposing the subject and predicate, the resultant proposition being as true as the original. The original proposition is the convertend; the resultant proposition is the converse.

Rules of Conversion

Rule 1. The converse must be of the same quality as the convertend. (Affirmative converts to affirmative; negative to negative.)

Rule 2. No term in the converse can have a wider Extension than it had in the convertend. (The conclusion may not state more than the premises warrant.)

Rule 3 (Special rules by proposition type):

Type	Converts to	Name
A	I	Accidental conversion
E	E	Simple conversion
I	I	Simple conversion
O	cannot be converted	—

Why A converts to I: In “All men are wise,” all men is universal and wise beings is particular (predicate of affirmative). In the converse, wise beings becomes the subject — we cannot make it universal, but we may use men in a narrower Extension. Hence: “Some wise beings are men” (I-proposition). This also follows from the subalternate principle: what is true of all men is true of some men.

Why E converts to E: In “No man is wise,” both subject and predicate are universal (subject qualified as universal; predicate universal as predicate of negative). Hence: “No wise being is a man” (E-proposition).

Why I converts to I: In “Some men are wise,” both subject and predicate are particular (subject limited by some; predicate as predicate of affirmative). Hence: “Some wise beings are men” (I-proposition).

Why O cannot be converted: In “Some men are not wise,” the subject some men is particular but the predicate wise is universal (predicate of negative). The converse must be negative (same quality). As predicate of a negative proposition in the converse, some men would become universal — but this conflicts with Rule 2, which forbids expanding Extension. Direct conversion is therefore impossible.

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Summary of the Article

We have learned Opposition, Equipollence, and Conversion of propositions as relative properties and as processes of immediate inference. These equip the dialectician to see the illogical nature of unwarranted statements and to form clear and valid inferences.

A practical example: a lecturer observes that French Canadian Catholics are intensely religious yet materially poor, then concludes “The Catholic religion stands against the progress of science.” The move is: from “French Canadian Catholics are unprogressive” to “All Catholics are unprogressive” — an inference of the truth of a subalternant from the truth of its subalternate, which is unwarranted by the rules of subalternacy.

The student should apply knowledge of these relative properties immediately in daily reading of books and newspapers, reducing arguments to logical form and criticising them accordingly.