Immediate inference

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." we can make the immediate inference that "No toads are not green." 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

Given a type E statement, from the traditional square of opposition, "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

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

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 are syllogistic fallacies called illicit contrary, illicit subcontrary, and illicit subalternation. Cases of incorrect application of the contradictory relation are so infrequent, that an "illicit contradictory" fallacy is usually not recognized.

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 (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.

References

Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 162. ISBN 0-312-02353-7. OCLC 21216829. 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).

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 162. ISBN 0-312-02353-7. OCLC 21216829. 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).