Immediate inference explained

An immediate inference is an inference which can be made from only one statement or proposition.^[1] For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" (Obverse). There are a number of immediate inferences which can validly be made using logical operations, the result of which is a logically equivalent statement form to the given statement. There are also invalid immediate inferences which are syllogistic fallacies.

Valid immediate inferences

Converse

See main article: Converse (logic).

• Given a type E statement, "No S are P.", one can make the immediate inference that "No P are S" which is the converse of the given statement.
• Given a type I statement, "Some S are P.", one can make the immediate inference that "Some P are S" which is the converse of the given statement.

Obverse

See main article: Obversion.

• Given a type A statement, "All S are P.", one can make the immediate inference that "No S are non-P" which is the obverse of the given statement.
• Given a type E statement, "No S are P.", one can make the immediate inference that "All S are non-P" which is the obverse of the given statement.
• Given a type I statement, "Some S are P.", one can make the immediate inference that "Some S are not non-P" which is the obverse of the given statement.
• Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some S are non-P" which is the obverse of the given statement.

Contrapositive

See main article: Contraposition (traditional logic).

• Given a type A statement, "All S are P.", one can make the immediate inference that "All non-P are non-S" which is the contrapositive of the given statement.
• Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some non-P are not non-S" which is the contrapositive of the given statement.

Invalid immediate inferences

Cases of the incorrect application of the contrary, subcontrary and subalternation relations (these hold in the traditional square of opposition, not the modern square of opposition) are syllogistic fallacies called illicit contrary, illicit subcontrary, and illicit subalternation, respectively. Cases of incorrect application of the contradictory relation (this relation holds in both the traditional and modern squares of opposition) are so infrequent, that an "illicit contradictory" fallacy is usually not recognized. The below shows examples of these cases.

Illicit contrary

• It is false that all A are B, therefore no A are B.
• It is false that no A are B, therefore all A are B.

Illicit subcontrary

• Some A are B, therefore it is false that some A are not B.
• Some A are not B, therefore some A are B.

Illicit subalternation and illicit superalternation

• Some A are not B, therefore no A are B.
• It is false that all A are B, therefore it is false that some A are B.

See also

• Transposition (logic)
• Inverse (logic)

Notes and References

1. Book: Churchill, Robert Paul . Logic: An Introduction . 1990 . St. Martin's Press . New York . 0-312-02353-7 . 21216829 . 2nd . 162 . Immediate inference is the assumption, without intervening—or 'mediating'—premises, that because one categorical statement is true (or false), a logically equivalent categorical statement must also be true (or false)..