The Square of Opposition (Boolean Interpretation) Keith ...

The Square of Opposition (Boolean Interpretation)
Keith Burgess-Jackson
19 September 2015

There are four standard-form categorical propositions (SFCPs):

• Universal affirmative (“A”): All S are P.
• Universal negative (“E”): No S are P.
• Particular affirmative (“I”): Some S are P.
• Particular negative (“O”): Some S are not P.

On the modern or Boolean interpretation of SFCPs, which we shall adopt in this course, only the
particular propositions—“I” and “O”—have existential import.1 This means that only “I” and “O”
propositions make existence claims. The “I” proposition says that there is (“exists”) at least one
object that is both an S and a P, i.e., that the SP area is not empty (SP ≠ 0). The “O” proposition
says that there is (“exists”) at least one object that is an S but not a P, i.e., that the SP� area is not
empty (SP� ≠ 0). The universal SFCPs—“A” and “E”—lack existential import. This means that “A”
and “E” propositions make no existence claims. The “A” proposition says that if there are any S’s,
then they are P’s, i.e., that the SP� area is empty (S�P = 0). The “E” proposition says that if there are
any S’s, then they are not P’s, i.e., that the SP area is empty (SP = 0). Here is a summary:

• All S are P: S�P = 0.
• No S are P: SP = 0.
• Some S are P: SP ≠ 0.
• Some S are not P: S�P ≠ 0.

To say that two SFCPs are opposed to one another is to say that they differ in quantity or quality or
both. The various forms of opposition between “A,” “E,” “I,” and “O” propositions are depicted in
the Square of Opposition. Here is a complete list of the relations between SFCPs:

• The “A” proposition and its corresponding “E” proposition have the same quantity
(namely, universal), but are opposed in quality, with the “A” proposition being affirmative
and the “E” proposition negative.

• The “I” proposition and its corresponding “O” proposition have the same quantity (namely,
particular), but are opposed in quality, with the “I” proposition being affirmative and the
“O” proposition negative.

• The “A” proposition and its corresponding “I” proposition have the same quality (namely,
affirmative), but are opposed in quantity, with the “A” proposition being universal and the
“I” proposition particular.

• The “E” proposition and its corresponding “O” proposition have the same quality (namely,

1 There is also a traditional or Aristotelian interpretation, according to which all four SFCPs have existential
import.

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negative), but are opposed in quantity, with the “E” proposition being universal and the
“O” proposition particular.
• The “A” proposition and its corresponding “O” proposition are opposed in both quantity
and quality, with the “A” proposition being universal affirmative and the “O” proposition
being particular negative.
• The “E” proposition and its corresponding “I” proposition are opposed in both quantity and
quality, with the “E” proposition being universal negative and the “I” proposition being
particular affirmative.

The following generic Square of Opposition shows the four SFCPs (“A,” “E,” “I,” and “O”) in their
proper positions (as depicted in Venn diagrams):

Before proceeding, let us pause to define two logical relations between (among) propositions (or,
more accurately, propositional forms):2

• Contradictoriness: X is the contradictory of Y (i.e., X and Y are contradictories) iff3 (1) it is
possible for X to be true while Y is false; (2) it is possible for Y to be true while X is false; (3)
it is impossible for both X and Y to be true; and (4) it is impossible for both X and Y to be
false. In other words, X and Y necessarily have different truth values.4

• Independence: X is independent of Y (i.e., X and Y are independent) iff (1) it is possible for
X to be true while Y is false; (2) it is possible for Y to be true while X is false; (3) it is possible
for both X and Y to be true; and (4) it is possible for both X and Y to be false.

2 Let “X” and “Y” be propositional forms.
3 “iff” is an abbreviation for “if and only if.” To say “Ø iff Ψ” is to say two things: (1) Ø if Ψ; and (2) Ø only if Ψ.
For example, if I say that my dog Autie gets a treat iff Autie is good, I am saying two things: (1) Autie gets a treat if Autie
is good; and (2) Autie gets a treat only if Autie is good. The former says that Autie’s being good is a sufficient condition
for Autie’s getting a treat. The latter says that Autie’s being good is a necessary condition for Autie’s getting a treat.
Another example: x receives an A in this course iff x’s final score in this course is at least 90. This says that receiving a
final score in this course of at least 90 is both a necessary and a sufficient condition for receiving an A in this course.
4 Examples: (1) It’s raining (here and now); it’s not raining (here and now); (2) 2 + 2 = 4; 2 + 2 ≠ 4; (3) Barack
Obama is the 44th president of the United States; Barack Obama is not the 44th president of the United States.

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Here is a summary:

Contradictoriness (X Is it possible for Is it possible for Is it possible for Is it possible for
is the contradictory X to be true Y to be true both X and Y to both X and Y to

of Y) while Y is false? while X is false? be true? be false?
Yes Yes No No
Independence (X is
Logical Relation independent of Y) Yes Yes

Yes Yes

Note that these logical relations are general in nature. They are logical relations between (among)
propositions (or propositional forms), not merely categorical propositions. Each logical relation
gives rise to one or more inferences. For example, suppose (1) that X and Y are contradictories
and (2) that X is true. It follows that Y is false. Here is a complete list of inferences based on these
logical relations:

Supposition

X is true X is false Y is true Y is false
Y is false X is true
X and Y are Y is true X is false
contradictories Y is X is
Supposition undetermined5 undetermined
X and Y are
independent Y is X is
undetermined undetermined

We are now in a position to ask how the four SFCPs are logically related to one another. Let us
take them in the same order as above:

• “A” and “E”: The Venn diagrams show four things: (1) it is possible for the “A” proposition
to be true while its corresponding “E” proposition is false; (2) it is possible for the “E”
proposition to be true while its corresponding “A” proposition is false; (3) it is possible for
both the “A” proposition and its corresponding “E” proposition to be true;6 and (4) it is

5 To say that the truth value of a given proposition (or propositional form) is undetermined is not to say that it
has no truth value; it is to say that there is not enough information to determine its truth value.

6 This may seem strange. How can it be true both that All S are P and that No S are P? The answer is that the
Boolean interpretation of “A” and “E” propositions makes it the case. According to Boole, the “A” proposition says that
if there are S’s, then they are P’s, and the “E” proposition says that if there are S’s, then they are not P’s. These are
conditional (if-then) propositions. As we will see later in the course (when we get to propositional logic), a conditional
proposition is true whenever (1) it has a false antecedent or (2) it has a true consequent. (The antecedent of a
conditional proposition is the proposition that follows the word “if”; the consequent is the proposition that follows the
word “then.”) Suppose the subject class is unicorns. The “A” proposition says that if there are unicorns, then they are
P’s. The “E” proposition says that if there are unicorns, then they are not P’s. The antecedent of these conditional
propositions is “there are unicorns.” Since there are (in fact) no unicorns, the antecedent is false in both cases, and
since the antecedent is false in both cases, the conditional propositions are true. Thus, as strange as it may seem, it is
true both that all unicorns are S’s and that no unicorns are S’s. This result is generalizable. Whenever the subject class
is empty, both the “A” proposition and its corresponding “E” proposition will be true. Since the “A” proposition is the
contradictory of the “O” proposition, and since the “E” proposition is the contradictory of the “I” proposition, it follows
that the corresponding “I” and “O” propositions will be false.

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possible for both the “A” proposition and its corresponding “E” proposition to be false.
Therefore, the “A” proposition and its corresponding “E” proposition are independent.
• “I” and “O”: The Venn diagrams show four things: (1) it is possible for the “I” proposition to
be true while its corresponding “O” proposition is false; (2) it is possible for the “O”
proposition to be true while its corresponding “I” proposition is false; (3) it is possible for
both the “I” proposition and its corresponding “O” proposition to be true; and (4) it is
possible for both the “I” proposition and its corresponding “O” proposition to be false.7
Therefore, the “I” proposition and its corresponding “O” proposition are independent.
• “A” and “I”: The Venn diagrams show four things: (1) it is possible for the “A” proposition
to be true while its corresponding “I” proposition is false; (2) it is possible for the “I”
proposition to be true while its corresponding “A” proposition is false; (3) it is possible for
both the “A” proposition and its corresponding “I” proposition to be true; and (4) it is
possible for both the “A” proposition and its corresponding “I” proposition to be false.
Therefore, the “A” proposition and its corresponding “I” proposition are independent.
• “E” and “O”: The Venn diagrams show four things: (1) it is possible for the “E” proposition
to be true while its corresponding “O” proposition is false; (2) it is possible for the “O”
proposition to be true while its corresponding “E” proposition is false; (3) it is possible for
both the “E” proposition and its corresponding “O” proposition to be true; and (4) it is
possible for both the “E” proposition and its corresponding “O” proposition to be false.
Therefore, the “E” proposition and its corresponding “O” proposition are independent.
• “A” and “O”: The Venn diagrams show four things: (1) it is possible for the “A” proposition
to be true while its corresponding “O” proposition is false; (2) it is possible for the “O”
proposition to be true while its corresponding “A” proposition is false; (3) it is impossible
for both the “A” proposition and its corresponding “O” proposition to be true; and (4) it is
impossible for both the “A” proposition and its corresponding “O” proposition to be false.
Therefore, the “A” proposition and its corresponding “O” proposition are contradictories.
• “E” and “I”: The Venn diagrams show four things: (1) it is possible for the “E” proposition to
be true while its corresponding “I” proposition is false; (2) it is possible for the “I”
proposition to be true while its corresponding “E” proposition is false; (3) it is impossible
for both the “E” proposition and its corresponding “I” proposition to be true; and (4) it is
impossible for both the “E” proposition and its corresponding “I” proposition to be false.
Therefore, the “E” proposition and its corresponding “I” proposition are contradictories.

These logical relations are depicted on the following Square of Opposition, which is known as the
Modern (Boolean) Square of Opposition:

7 This may seem strange. How can it be false both that Some S are P and that Some S are not P? The answer
is that the subject class may be empty. Suppose the subject class is unicorns. It is false that some unicorns are P’s, for
this says that there is at least one thing that is both a unicorn and a P (which is false, there being no unicorns), and it is
false that some unicorns are not P’s, for this says that there is at least one thing that is both a unicorn and not a P
(which is false, there being no unicorns). This result is generalizable. Whenever the subject class is empty, both the “I”
proposition and its corresponding “O” proposition will be false. Since the “I” proposition is the contradictory of the “E”
proposition, and since the “O” proposition is the contradictory of the “A” proposition, it follows that the corresponding
“A” and “E” propositions will be true. See the previous note.

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The Modern (Boolean) Square of Opposition provides the basis for a number of immediate
inferences:8

• If you know that a given “A” proposition is true, then you may infer that the corresponding
“O” proposition is false (since they are contradictories).

• If you know that a given “E” proposition is true, then you may infer that the corresponding
“I” proposition is false (since they are contradictories).

• If you know that a given “I” proposition is true, then you may infer that the corresponding
“E” proposition is false (since they are contradictories).

• If you know that a given “O” proposition is true, then you may infer that the corresponding
“A” proposition is false (since they are contradictories).

• If you know that a given “A” proposition is false, then you may infer that the corresponding
“O” proposition is true (since they are contradictories).

• If you know that a given “E” proposition is false, then you may infer that the corresponding
“I” proposition is true (since they are contradictories).

• If you know that a given “I” proposition is false, then you may infer that the corresponding
“E” proposition is true (since they are contradictories).

• If you know that a given “O” proposition is false, then you may infer that the corresponding
“A” proposition is true (since they are contradictories).

8 An immediate inference is an inference in which the conclusion is drawn from a single premise, without the
mediation of a second premise. A mediate inference is an inference in which a conclusion is drawn from two (or more)
premises.

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