Unit 3: Immediate Inference & The Square of Opposition

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).

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)

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.

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."

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").