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.
I
Standard-form categorical propositions with the same
subject and predicate terms can be aligned in a traditional square of opposition.
II
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).
III
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.
IV
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]