The Traditional Square of Opposition is a diagram specifying logical relations among four types of categorical propositions. It is useful in the analysis of syllogistic logic, serving to identify the allowed logical conversions from one type to another.

Immediate Arguments

There are four types of immediate arguments: Contradictories, Contraries, Subcontraries, Subalterns.

The Traditional Square of Opposition

1. Contradictories

Two propositions are contradictory  if one of them is denial of another. They cannot both be true, and cannot both be false. A-proposition is contradictory of O-proposition, and E-proposition is contradictory of I-proposition.

2. Contraries

Two propositions are contrary if they cannot both be true, but they might both be false. A-proposition is contrary of E-proposition.

3. Subcontraries

Two propositions are subcontrary if they cannot both be false, bu they might both be true. I-proposition is subcontrary of O-proposition.

4. Subalterns

A proposition is a subaltern of another iff it must be true if its superaltern is true, and the superaltern must be false if the subaltern is false. I-proposition is subaltern of A-proposition, and O-proposition is subaltern of E-proposition.

5. Categorical Propositions & The Traditional Square of Opposition in Venn Diagrams

Square of opposition, set diagrams” by mate2code – Own work. Licensed under Public Domain via Wikimedia Commons.

Immediate Inferences

Immediate inferences are logical operations which can be performed on categorical statements, in order to change it into another. There are three concepts of such operations – Conversion, Obversion, Contraposition.

1. Conversion

Conversion is created by switching the two parts of statement. There are two types of conversion, one is implicational, and second is categorical.

Implicational conversion is where parts of a implication is reversed, may be seen as reverse-implication. Converse of  is .

In categorical conversion, the subject and predicate are interchanged. Example , the converse of “No S are P” is “No P are S“.

Here, only conversion of E and I are valid. The conversion of A-proposition can be valid via limitation. The converse of “All S are P” is “Some P are S“. O-proposition has no valid conversion.

2. Obversion

Obversion changes the quality of proposition, and the predicate term. It acts as complimentary set. But truth value doesn’t change.

Example, “All cats are animals” is changed to “No cats are non-animals” meaning, there is no cat, which is not animal.

3. Contraposition

Contrapositions are created by switching the subject with the predicate while negating both of them.

Example, if all dogs are mammals, then we can say, all non-mammals are non-dogs. It can be easily shown by drawing Venn diagrams.

Contrapositions also apply to implications. The contraposition of  is .