Chapter 18: Conversion, Obversion, and Squares of Opposition

Conversion, Obversion, and Contraposition (pp. 181-183) Two statement forms are logically equivalent if and only if they are true under exactly the same circumstances You will need to recognize logically equivalent forms of categorical propositions to deal with syllogisms in ordinary English (Chapter 19).

Conversion, Obversion, and Contraposition (pp. 181-183) To convert, switch the positions of the subject and predicate terms. The converse is logically equivalent only for universal negative and particular affirmative propositions. “No S are P” is logically equivalent to “No P are S.” “Some S are P” is logically equivalent to “Some P are S.”

Conversion, Obversion, and Contraposition (pp. 181-183) Complementary classes and complementary terms The complement of a class contains all objects not contained in the original class. Sometimes the complementary class is limited by an assumed context. The complement of the class of senators contains everything that is not a senator. If the assumed context is members of Congress, the complement of senators is representatives. For a term T, it’s complement is non-T. In ordinary English, the complement of a term often, although not always, has a preface such as in- or un-. ‘Flammable’ and ‘inflammable’ are synonyms, for example.

Conversion, Obversion, and Contraposition (pp. 181-183) To obvert, change the quality of a proposition from affirmative to negative or negative to affirmative, and replace the predicate with its complement. Every categorical proposition is equivalent to its obverse. “All S are P” is logically equivalent to “No S are non-P. “No S are P” is logically equivalent to “All S are non-P.” “Some S are P” is logically equivalent to “Some S are not non-P.” “Some S are not P” is logically equivalent to “Some S are non-P.”

Conversion, Obversion, and Contraposition (pp. 181-183) To form the contrapositive, replace the subject and predicate terms with their complements and convert. The contrapositive is logically equivalent only for universal affirmative and particular negative propositions. In those cases in which the contrapositive is logically equivalent to the given, it can be derived by successive uses of conversion and obversion. See p. 183.

The Boolean Square of Opposition (pp. 183-185) Squares of opposition are graphic representations of immediate inferences that can be drawn given the truth or falsehood of a categorical proposition. An immediate inference is made without appealing to another proposition. On the Boolean Square, A and O propositions and I and E propositions are contradictories. Two statements are contradictories if and only if the truth of one entails the falsehood of the other.

The Boolean Square of Opposition (pp. 183-185) The Boolean square looks like this:

The Aristotelian Square of Opposition (pp. 183-185) The Aristotelian interpretation of categorical propositions holds that universal propositions have existential import. A statement has existential import if and only if its truth requires that the subject class has at least one member. A universal proposition in effect also assets its corresponding particular proposition. A: All S are P and some S are P. E: No S are P and some S are not P.

The Aristotelian Square of Opposition (pp. 183-185) A and O propositions and I and E propositions are contradictories. A and E propositions are contraries. Two statements are contraries if it is possible for both to be false, but it is not possible for both to be true. An I proposition is the subaltern of an A proposition, and an O proposition is the subaltern of an E proposition Subalternation is a relation between a universal proposition and its corresponding particular proposition such that if the universal is true, the particular is also true. I and O propositions are subcontraries. Two statements are subcontraries if it is possible for both to be true, but it is not possible for both to be false.

The Aristotelian Square of Opposition (pp. 183-185) The Aristotelian Square looks like this:

The Aristotelian Square of Opposition (pp. 183-185) On the Aristotelian Interpretation, given the truth of a universal proposition, you can determine the truth value of every other statement on the square. Given the falsehood of a universal, you can only determine that its contradictory is true. Given the falsehood of a particular proposition, you can determine the truth value of every other statement on the square. Given the truth of a particular, you can only determine that its contradictory is false.

The Aristotelian Square of Opposition (pp. 183-185) Assume true: False False True

The Aristotelian Square of Opposition (pp. 183-185) False True True Assume false:

The Aristotelian Square of Opposition (pp. 183-185) Assume false True

The Aristotelian Square of Opposition (pp. 183-185) True Assume false

Squares and Equivalences Together You can combine considerations of equivalent forms with the squares “Some S are not non-P” is logically equivalent to “Some S are P. So, if that statement is true, on either square you can infer that “No S are P” is false. “All non-P are non-S” is logically equivalent to “All S are P.” If that statement is true: On the Boolean square you can infer only that “Some S are not P” is false. On the Aristotelian square you can infer that “Some S are not P” is false, that “Some S are P” is true, and that “No S are P” is false.