Logical operations for reducing the number of terms: “conversion”, “contraposition”, and “obversion”
So far, we have studied how to use Venn diagrams to test categorical syllogisms involving three terms for validity or invalidity. Sometimes, categorical syllogisms involve propositions that are not in a straightforward standard form. Often, we can apply logical operations to the propositions in such syllogisms to yield a standardform categorical proposition that can be expressed in a Venn diagram, and hence tested for validity or invalidity. (Note, however, that we have studied how to do this only for syllogisms that involve three terms.)
On this handout, please place an “x” or shading in the appropriate sectors of the Venn diagrams.
Conversion is the logical operation in which a proposition’s subject and predicate terms are switched. Conversion can be applied to any particular affirmative or universal negative categorical proposition to yield a logically equivalent proposition.
Converse: PA statement converse (PA) UN statement converse (UN)
Sentence logic: S P = S P S P = P S
Predicate logic: (x)(Sx Px) = (x)(Px Sx) (x)(Sx Px) = (x)(Px Sx)
Categorical logic: Some S is P = Some P is S No S are P = No P are S
Some categorical syllogisms contain more than three terms, in which one or more of these terms has the prefix “non”, “un”, or “in”. Such terms are called the “term complement” of the term formed by eliminating this prefix. Changing a term to its term complement is involved in the logical operations of contraposition and obversion. Performing these logical operations often reduces the number of terms in the categorical syllogism, allowing us to represent the argument in a Venn diagram in such a way that it can be tested for validity or invalidity. (Note, however, that we have studied how to do this only for syllogisms that involve three terms.)
Contraposition is the logical operation in which a proposition’s subject and predicate terms are switched, and both terms are replaced by their term complement (the proposition’s affirmative or negative “quality” remains unchanged). Contraposition can be applied to any universal affirmative or particular negative categorical proposition to yield a logically equivalent proposition.
Contrapositive: UA statement = contrapositive (UA) PN statement = contrapositive (PN)
Sentence logic: S P = P S S P = P S
Predicate logic: (x)(Sx Px) = (x)(Px Sx) (x)(Sx Px) = (x)(Px Sx)
Categorical logic: All S are P = All nonP are nonS Some S is not P = Some nonP is not nonS
Obversion is the logical operation in which a proposition’s its “quality” is reversed (from affirmative to negative, or from negative to affirmative), and its predicate term is replaced by its term complement (i.e., adding or subtracting the prefix “non”, “un”, or “in”). Obversion can be applied to any categorical proposition to yield a logically equivalent proposition.
Obverse: UA statement = obverse (UN) UN statement = obverse (UA)
Sentence logic: S P = S P S P = S P
Predicate logic: (x)(Sx Px) = (x)(Sx Px) (x)(Sx Px) = (x)(Sx Px)
Categorical logic: All S are P = No S are nonP No S are P = All S are nonP
Obverse: PA statement = obverse (PN) PN statement = obverse (PA)
Sentence logic: S P = S P S P = S P
Predicate logic: (x)(Sx Px) = (x)(Sx Px) (x)(Sx Px) = (x)(Sx Px)
Categorical logic: Some S is P = Some S is not nonP Some S is not P = Some S is nonP
