Existential Import

EXERCISE 6.2

For each pair of propositions, given that the first one is true, determine whether the second is true, false, or undetermined, in accordance with the traditional square of opposition.

❋ 1. All S are P. Some S are P.

2. All S are P. No S is P.

3. No S is P. All S are P.

❋ 4. Some S are P. All S are P.

5. Some S are not P. All S are P.

6. No S is P. Some S are P.

❋ 7. Some S are not P. No S is P.

8. All S are P. Some S are not P.

9. Some S are P. Some S are not P.

❋10. Some S are P. No S is P.

11. Some S are not P. Some S are P.

12. No S is P. Some S are not P.

E : No S is P A : All S are P

O : Some S are not P I : Some S are P

subalternate subalternate

subcontraries I

A

O contraries E

contr adictories

contr a

dictories

SUMMARY

The Traditional Square of Opposition

6.3 Existential Import

In discussing the relationships that make up the square of opposition, we have been assuming that subject and predicate terms stand for categories of things in reality. But some terms, such as “unicorn” and names of other mythological beings, are vacuous:

They do not have any referents in reality. How does this affect the truth or falsity of categorical statements involving these terms? It seems natural to say that the A propo-sition “All unicorns have horns” is true, at least as a statement about mythology. In the same way, when a teacher issues the warning, “All students who miss three or more

classes will fail the course,” the statement may be true even if there aren’t any students who miss three or more classes. Indeed, the whole point of the warning is to discourage anyone from becoming an instance of the subject term.

These statements seem to lack what logicians call existential import, because their truth doesn’t depend on the existence of unicorns or students who miss three or more classes. A statement has existential import only when its truth depends on the existence of things in a certain category—in the case of a categorical proposition, the existence of things in the category signified by its subject term. A statement with existential im-port implies that things in that category exist, so if they do not exist, the statement is false.

In modern logic, existential import is a function of a statement’s logical form.

According to this view, universal categorical statements (A and E) always lack existential import, just because they are universal, whereas particular statements (I and O) always have existential import. Propositions of the form All S are P and No S is P do not imply that there are any Ss, but propositions of the form Some S are P and Some S are not P do have that implication.

Affirmative Negative

Universal statements lack existential import: Their truth does not depend on existence of Ss

A: All S are P E: No S is P

Particular statement have existential import: Their truth does depend on exis-tence of Ss

I: Some S are P O: Some S are not P

Because this distinction is based on logical form, it doesn’t matter what the subject term is. The principle that universal propositions lack existential import is not limited to statements like “All unicorns have horns.” It also applies to statements like “All dogs are animals,” which is clearly about a class of existing things. According to the modern view, however, the statement does not logically imply their existence; it would be true even if there were no dogs. In contrast, the statements “Some dogs are animals” and

“Some unicorns have horns” do have existential import because of their logical form.

Since dogs exist but unicorns don’t, the first statement is true but the second is false.

The same distinction applies to negative propositions. The universal E proposition

“No perpetual motion machine has been patented” is true, even though there are no perpetual motion machines. But the particular O statement “Some perpetual motion machines have not been patented” is false.

The issue of existential import has major implications for the square of opposition.

The traditional square presupposes that the terms of any categorical proposition do have referents. If we adopt the modern doctrine about existential import, however, then some of the relations in the traditional square no longer hold. Let’s start with subal-ternation. On the traditional view, an A proposition of the form All S are P entails the corresponding I proposition Some S are P. Now suppose there are no Ss. On the modern view, the I proposition would be false, but the A proposition could still be true. So the

6.3 Existential Import 157

truth of the A proposition does not imply the truth of the corresponding I. The same is true for E and O. Subalternation must therefore be removed from the square.

The next casualty is the relation between A and E. On the traditional view, these propositions are contraries: They cannot both be true. If all Ss are P, then it cannot also be true that no S is P. On the modern doctrine, however, neither of these universal state-ments has existential import, so they could both be true in the case where there are no Ss. For example, “All unicorns have horns” and “No unicorns have horns” are both true by default because there are no unicorns.

Finally, the I and O propositions no longer fit the definition of subcontraries. Two statements are subcontraries if, in virtue of their logical form, they could both be true but could not both be false. If no Ss exist, however, then both particular statements are false. To stay with our mythological example, the absence of unicorns means that both “Some unicorns have horns” and “Some unicorns do not have horns” are false statements.

The only relationship that survives in the modern square of opposition is that of contradictories. If there exists a single thing that is both S and P, then the I proposition is true and the E is false. But if nothing is both S and P, then the I proposition is false and the E is true—even if the absence of things that are both S and P is due to the fact that there aren’t any Ss at all. The same reasoning applies to the A and O propositions.

So E is true if and only if I is false, and A is true if and only if O is false.

The modern square of opposition, then, is quite different from the traditional square. All of the horizontal and vertical relationships in the traditional square are re-moved, leaving only the diagonal relationship between contradictory statements.

Which version of the square of opposition—the traditional or the mod ern—should we adopt? Our answer depends on which is the correct view of existential import, and that remains a controversial issue. Something can be said on both sides. Against the modern doctrine, it can be argued that it’s unfair to deprive all universal statements of existential import just because of a few unusual cases. Normally, we do take such state-ments to be about a class with real members. For example, if I said that everyone con-victed of terrorism last year was sentenced to life in prison, you would feel cheated if I

SUMMARY

The Modern Square of Opposition

contr

a dictories contr

adictories

O: Some S are not P I: Some S are P

E: No S is P A: All S are P

went on to claim that the statement is true only because no one was convicted of terror-ism last year. However, the exceptions do exist—there are statements that lack existen-tial import—and we want our principles of logical form to be true without exception.

In light of this unresolved theoretical issue, it is useful for you to be familiar with both the traditional and the modern square of opposition. If you are dealing with a universal statement that clearly refers to class of actual things, it is safe to use the tradi-tional version of the square in deriving implications. But if there is any question about the existence of Ss or Ps, then you should rely on the modern version of the square.

EXERCISE 6.3

For each pair of propositions, given that the first one is true, determine whether the sec-ond is true, false, or undetermined, in accordance with the modern square of opposition.

❋ 1. All S are P. Some S are P.

2. All S are P. No S is P.

3. Some S are not P. All S are P.

❋ 4. Some S are P. All S are P.

5. No S is P. Some S are P.

6. Some S are not P. Some S are P.

❋ 7. All S are P. Some S are not P.

8. Some S are P. Some S are not P.

9. Some S are P. No S is P.

❋10. No S is P. Some S are not P.

11. Some S are not P. No S is P.

12. No S is P. All S are P.

In document Art of Reasoning, The - Kelley, David [SRG] (Page 173-176)