Standard-form categorical propositions with the same subject and predicate terms can be aligned in a traditional square of opposition that shows certain immediate inferences that can be drawn from the truth or falsity of one of them, based on such intuitive relationships as contradiction, contrariness, subcontrariness, and subalternation. Statements about empty classes, however, generate problems with this traditional square. All of the legitimate immediate inferences can be used to manipulate the various propositions in an extended argument so as to help put the argument itself in standard form.
Standard-form categorical propositions with the same subject and predicate terms can be aligned in a traditional square of opposition.
Each of the four corners can be obverted (change quality and replace predicate with its complement). The I and E corners can be converted (switch subject and predicate). The A and O corners can be contraposed (replace subject and predicate with their complements).
The square of opposition highlights relationships between standard-form categorical propositions with the same subject and predicate terms that enable certain further immediate inferences to be drawn from the truth or falsity of one of them, as long as none of the classes mentioned is empty (null).
Contradictories (A and O, E and I) differ from each other in both quality and quantity, with the result that contradictories always have opposite truth values. If a statement is true, then its contradictory is false (and vice versa, this being a two-way street).
Contraries (A and E) are universal claims that differ from each other in quality but not in quantity, with the result that while they can both be false, they cannot both be true. However, null classes—those that do not have any members at all, for example, round squares—lead to problems.
Subcontraries (I and O) are particular claims that differ from each other in quality but not in quantity, with the result that while they can both be true, they cannot both be false. Null classes, however, present problems.
Subalternation is the relationship between a universal claim and its dependent or hanging particular claim. Thus, a statement and its subalternate differ from each other in quantity but not in quality, with the result that if a statement is true, its subalternate is also true (but not vice versa, this being a one-way street).
1. However, null classes again present problems.
2. The view of ancient logicians that you could convert a universal proposition “by limitation”—by moving to the subalternate of the universal proposition—does not hold if the class is empty.
3. For example, “All round squares are truly remarkable,” seems true, but “There is at least one truly remarkable round square” is clearly false.
The various immediate inferences can be used to modify the propositions in an argument so as to (a) have each one begin properly with its quantity indicator and (b) reduce the number of terms that occur in it to the three that a syllogism can handle.
A. If one of the statements in an argument amounts to the denial of a standard-form proposition (e.g., “Not all Greeks are Athenians”), one may appeal to the rule of contradiction to replace that denial with the assertion of its contradictory (in this case, “Some Greeks are not Athenians”).
B. If two of the statements in an argument employ complementary terms (e.g., heroes and non-heroes), one may obvert one of them, thus ensuring that both propositions are about the same set.
C. If one of the statements in an argument amounts to the denial of an I proposition (e.g., “It is not the case that some wolves are strict vegetarians”), one may appeal to the rule of subcontraries (or to the rules of contradiction and subalternation) to replace that denial with the assertion of its corresponding O proposition (“Some wolves are not strict vegetarians”). This is highly problematic, however, if the subject term names a null class (as in “It is not the case that some unicorns are carnivores”). We shall examine this problem in Lecture Eight.
D. The same thing can be done with the denial of an O proposition, to replace it with the assertion of its corresponding I, but this is also problematic when null classes are in play. E. Once the statements in an argument are cleaned up, the argument itself can be put in standard form and assessed for validity in terms of formal rules, as we shall see in Lecture Eight.
[Courtesy: Professor James Hall]