A Brief History of Validity #3

We read in my last post on this subject about the nineteen traditional valid forms of syllogisms named by medieval scholars, plus the five forms which can be deduced by subimplication of those with a universal conclusion. These comprise the twenty-four forms of syllogisms identified as valid in my Introductory Logic text.

Statements with Existential Import

However, since the time of George Boole, a 19th-century mathematician, only fifteen of those twenty-four forms are recognized as valid. Why is this? Boole argued that the truth of a particular statement cannot be inferred from the truth of its corresponding universal, because a particular statement asserts the existence of its subject, but a universal statement does not. That is, to say, “Some athletes are dedicated people” is to assert that at least one athlete exists, but to say that “All athletes are dedicated people” is only to say that if one is an athlete then one is a dedicated person. According to Boole, the four categorical statements should be interpreted this way:

All S is P = If S exists then it is P
No S is P = If S exists then it is not P
Some S is P = There exists at least one S that is P
Some S is not P = There exists at least one S that is not P

Particular statements are said to have existential import; they claim that the terms in the statement exist. Universal statements, however, do not have existential import; they are considered as material conditionals.

The Existential Fallacy

In this interpretation, no particular statement can be inferred from a universal statement, or from universal premises. One could not validly argue, for instance,

All grandfathers are fathers.
All fathers are men.
∴ Some men are grandfathers.

This AAI-4 (Bramantip) syllogism is said to make the existential fallacy, which is based on this sixth rule of validity: “A valid syllogism cannot have universal premises and a particular conclusion.” By the modern interpretation, the premises only say this:

If grandfathers exist then they are fathers.
If fathers exist then they are men.
∴ There exists at least one man who is a grandfather.

These premises do not claim that grandfathers or fathers or men exist, but the conclusion does. Thus the conclusion claims more than is contained within the premises, which means the syllogism is invalid.

There is much to commend the modern view of interpreting categorical statements. For example, in the traditional view, this is a valid chain of reasoning:

No athletes are people that breathe underwater ⇒ (by converse)
No people that breathe underwater are athletes ⇒ (by obverse)
All people that breathe underwater are non-athletes ⇒ (by subimplication)
Some people that breathe underwater are non-athletes.

Everyone would agree that the first statement is true, but most people would say that the last statement is false, because it seems to imply that there are people that breathe underwater.

Rethinking the Modern Interpretation

However, I think that the modern interpretation of categorical statements is potentially flawed. Does not the statement “No people that breathe underwater are athletes” seem to imply that there are people that breathe underwater? And why insist that particular statements have existential import? Consider these particular statements:

Some hobbits are not Shire dwellers.
Some black holes are members of binary stars.
Some of your sons will be the king’s horsemen.

Most people would argue that, in the sub-created world of Tolkien, the first statement is true, even though (in our world) hobbits do not exist. Most astronomers would argue that the second is almost certainly true, even though the existence of black holes is still in doubt. The last statement was uttered by Samuel to the people in the hope that it would not be true, that such sons would not exist.

Much more could be said, but the logic student should at least be aware that this debate exists. I would appreciate your thoughts.

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