Conversion, Obversion, And Contraposition

When you apply the logical operations - conversion, obversion, and contraposition - to categorical propositions, you yield new categorical propositions that are sometimes valid (logically equivalent) and sometimes not valid (not logically equivalent).

Definitions of the logical operations:
1.)    Conversion consists of simply switching the subject term and the predicate term of a categorical proposition while leaving the quality and the quantity of the proposition the same. The result of applying conversion to a categorical proposition is called the converse of the proposition. Thus, for example, the converse of "All cats are mammals" is "All mammals are cats" – for we dog lovers and cat haters, a depressing thought. The converses of E and I propositions are logically equivalent, for example some cats are mammals and some mammals are cats. If the propositions are logically equivalent then they also have the same truth-values). The converses of A and O propositions are not logically equivalent to them. Thus, if we form an argument whose premise is a categorical proposition and whose conclusion is the converse of it,  the argument is valid if the premise is an E or an I proposition and is invalid if the premise is an A or an O proposition.
2.)    Obversion consists of two changes: (1) changing the quality of the proposition (leaving the quantity the same), and (2) changing the predicate term to its complement. To change the predicate term to its complement, one typically attaches the prefix "non-" to it. The result of obversion is called the obverse of the original proposition to which it is applied. The obverse of any categorical proposition, A, E, I, or O, is logically equivalent to it. Any argument whose premise is a categorical proposition and whose conclusion is its obverse is a valid argument.

                  The obverse of the A proposition "All S is P" is the E proposition "No S is non-P."
                    The obverse of the E proposition "No S is P" is the A proposition "All S is non-P."
                    The obverse of the O proposition "Some S is not P" is the I proposition "Some S is non-P."
                    The obverse of the I proposition "Some S is P" is the O proposition "Some S is not non-P."

3.)    Contraposition consists of two changes: (1) switching the subject term with the predicate term (while leaving the quality and quantity of the proposition unaltered), and (2) complementing both the new subject and predicate terms. Thus, for example, the contrapositive of "All cats are mammals" is "All non-mammals are non-cats." The result of contraposition is called the contrapositive of the original proposition. The contrapositives of A propositions and O propositions are logically equivalent to the originals, while the contrapositives of E and I propositions are not logically equivalent to the originals. If we form an argument whose premise is a categorical proposition and whose conclusion is the contrapositive of it, then the argument is valid if the premise is an A or an O proposition and is invalid if the premise is an E or an I proposition. 

Immediate inference ( Robinson's power-point presentation)

INCONSISTENCY
Consider discourse as a series of statements joined together by the word "and". Thus all the statements in a book or essay would be one long statement. Are the individual statements that make up the long statement compatible with each other? If two statements in the long statement are incompatible with each other (can not be true at the same time or if one is true then the other must be false) then the discourse is inconsistent. Discourse that affirms and denies the same statement (by converse of E or I, or obverse, or contrapostive of A and O). The discourse is self-contradictory.

The logical relationship between statements: we are going to explore the logical relationship between statements in order to discover whether the truth and falsity of one statement is logically connected (depends on the logical form of the two statements) to the truth and falsity of another statement. When comparing two statements they must be or we must get them in standard logical form ( A, E, I, O). Then we examine the relationship between the logical forms of the two statements.

Summary of Equivalent statements:
Conversion (valid for E and I)   Use hammers  and tools as subject and predicate to test A and O.
                    All ( hammers) are (tools) by conversion gives us the statement "All (tools) are (hammers).
                    Some (tools) are not (hammers) by conversion becomes Some (hammers) are not (tools).
Obversion (valid for A, E, I, and O) Changes the quality of the prediate and the quality of the statement as a whole.
Contraposition (valid for A and O) Switch the subject and predicate and make the new subject and predicate the complement or negative of the original subject and predicate: the contrapositive of All (S) are (P) is "All (non-P) are (non-S)."