Inference by Opposition of Proposition

Inference by opposition of proposition involves drawing conclusions based on the logical relationships between opposing propositions. This method leverages the types of opposition (contradictory, contrary, subcontrary, and subaltern) to infer the truth or falsehood of a proposition based on the truth or falsehood of another, related proposition.

Here's how inferences work with each type of opposition:

1. Contradictory Opposition

Rule: In contradictory opposition, if one proposition is true, the other must be false, and vice versa.

Inference:

If "All swans are white" (universal affirmative) is true, then "Some swans are not white" (particular negative) must be false.

Conversely, if "Some swans are not white" is true, then "All swans are white" must be false.

This is useful in debates and proofs where establishing the truth of one statement immediately disproves the other.

2. Contrary Opposition

Rule: In contrary opposition, if one proposition is true, the other must be false, but if one is false, the other could be true or false.

Inference:

If "All swans are white" (universal affirmative) is true, then "All swans are black" (universal negative) must be false.

However, if "All swans are white" is false, "All swans are black" could still be false as well (some swans could be gray or a mix of colors).

This type of inference helps when disproving one universal statement but doesn’t necessarily confirm the truth of the opposing statement.

3. Subcontrary Opposition

Rule: In subcontrary opposition, if one proposition is false, the other must be true; if one is true, the other may also be true.

Inference:

If "Some swans are white" (particular affirmative) is false, then "Some swans are not white" (particular negative) must be true

Conversely, if "Some swans are not white" is false, then "Some swans are white" must be true.

Subcontraries allow us to infer truth from the falsehood of particular statements, though both can also be true simultaneously.

4. Subaltern Opposition

Rule: In subaltern opposition, if the universal statement is true, the particular statement must also be true; if the particular statement is false, the universal statement must also be false.

Inference:

If "All swans are white" (universal affirmative) is true, then "Some swans are white" (particular affirmative) must also be true.

If "Some swans are white" is false, then "All swans are white" must also be false.

Subaltern inferences are helpful in validating particular statements from universal ones and disqualifying universal statements based on particular falsehoods.

Summary of Inference by Opposition

Contradictory: One true, other false.

Contrary: One true, other false; both false possible.

Subcontrary: One false, other true; both true possible.

Subaltern: Universal true implies particular true; particular false implies universal false.

These inferences allow logical deduction based on the structure of opposition, strengthening reasoning and argument analysis.