How the truth or falsity of one proposition necessitates the truth or falsity of another.
1. Immediate vs. Mediate Inference
In logic, Inference is the process by which a conclusion is derived from one or more premises.
- Immediate Inference: The conclusion is drawn from one single premise (e.g., Square of Opposition, Conversion).
- Mediate Inference: The conclusion is drawn from two or more premises (e.g., Syllogisms).
2. The Aristotelian Square of Opposition
The Square of Opposition maps the logical relationship between A, E, I, and O propositions when they have the same subject and predicate.
(Diagram representing Contradictories, Contraries, Subcontraries, and Subalternation)
3. The Four Logical Relations
| Relation | Propositions | Logical Rule |
|---|---|---|
| Contradictories | A-O and E-I | They have opposite truth values. If one is true, the other must be false. They cannot both be true and cannot both be false. |
| Contraries | A-E | They cannot both be true, but they can both be false. If one is true, the other is false. If one is false, the other is undetermined. |
| Subcontraries | I-O | They cannot both be false, but they can both be true. If one is false, the other is true. If one is true, the other is undetermined. |
| Subalternation | A-I and E-O | If the Universal (A/E) is true, the Particular (I/O) is true. If the Particular is false, the Universal is false. |
4. Eduction: Conversion and Obversion
Eduction is a form of immediate inference where we rephrase a proposition to see what else it implies.
- Conversion: Interchanging the subject and predicate.
Example (E): "No cats are dogs" converts to "No dogs are cats." - Obversion: Changing the quality and replacing the predicate with its complement.
Example (A): "All men are mortal" obverts to "No men are non-mortal."
Exam Essentials
- The Contradictory Rule: This is the strongest relation. If an exam question asks "If A is True, what is O?", the answer is always "False."
- Undetermined Values: In Contraries (A-E), if A is false, E is undetermined (it could be true or false). Many students wrongly assume it must be true.
- Conversion Limits: Remember that "A" propositions cannot be converted simply (e.g., "All dogs are animals" does NOT mean "All animals are dogs").