Inference by Opposition of Propositions
Chapter Forty-Two
Syllabus topic 3.2, "inference by Opposition of propositions"
Pages 203 to 206 of 334
In one line
Given the truth or falsity of one of the four propositions, the relations of the square tell you what follows about the other three.
In the wording a student can write in an examination: inference by opposition is the drawing of a conclusion about one proposition from the known truth or falsity of another having the same subject and predicate, by applying the rules of contradiction, contrariety, subcontrariety and subalternation.
The four rules, as instructions
The relations were established at sequence 410. Here they are stated as instructions for drawing a conclusion, which is what an examination question asks for.
Contradictories, A with O and E with I. If one is true, the other is false. If one is false, the other is true. This relation always yields an answer, in both directions.
Contraries, A with E. If one is true, the other is false. If one is false, nothing follows, because both may be false.
Subcontraries, I with O. If one is false, the other is true. If one is true, nothing follows, because both may be true.
Subalterns, A with I and E with O. If the universal is true, the particular is true. If the particular is false, the universal is false. In the other two directions, nothing follows.
Notice the pattern. Only the contradictory relation answers in every case. The other three answer in one direction and are silent in the other, and knowing which direction is silent is what the marks are for.
The sixteen inferences
Take the four propositions with the same subject and predicate. Suppose one of them is known to be true, or known to be false. What follows about each of the other three?
If A is true.
| Follows | By | |
|---|---|---|
| E | False | Contrary |
| I | True | Subaltern |
| O | False | Contradictory |
If A is false.
| Follows | By | |
|---|---|---|
| E | Undetermined | Contraries may both be false |
| I | Undetermined | Falsity does not descend |
| O | True | Contradictory |
If E is true.
| Follows | By | |
|---|---|---|
| A | False | Contrary |
| I | False | Contradictory |
| O | True | Subaltern |
If E is false.
| Follows | By | |
|---|---|---|
| A | Undetermined | Contraries may both be false |
| I | True | Contradictory |
| O | Undetermined | Falsity does not descend |
If I is true.
| Follows | By | |
|---|---|---|
| A | Undetermined | Truth does not ascend |
| E | False | Contradictory |
| O | Undetermined | Subcontraries may both be true |
If I is false.
| Follows | By | |
|---|---|---|
| A | False | Subaltern, falsity ascends |
| E | True | Contradictory |
| O | True | Subcontrary |
If O is true.
| Follows | By | |
|---|---|---|
| A | False | Contradictory |
| E | Undetermined | Truth does not ascend |
| I | Undetermined | Subcontraries may both be true |
If O is false.
| Follows | By | |
|---|---|---|
| A | True | Contradictory |
| E | False | Subaltern, falsity ascends |
| I | True | Subcontrary |
Count the undetermined entries: eight of twenty four. A third of the questions have the answer "nothing follows", and writing "undetermined" with the reason earns the mark exactly as a positive answer does. A student who forces an answer in those eight places loses more than one who says nothing follows.
The rest of this chapter
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The rest of this subject
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