Trespassers will be eaten

Existential Fallacy

Alias: The Fallacy of Existential Assumption

Type: Quantificational Fallacy

Form:

Any argument whose conclusion implies that a class has at least one member, but whose premisses do not so imply.

Example Counter-Example
All trespassers will be prosecuted.
Therefore, some of those prosecuted will have trespassed.
All unicorns are animals.
Therefore, some animals are unicorns.
Venn diagram

Venn Diagram:

This diagram represents both the Example and Counter-Example, which it shows to be invalid, since the area with the question mark would be empty if the class S were empty.

Exposition:

A proposition has existential import if it implies that some class is not empty, that is, that there is at least one member of the class. For example:

Existential Import No Existential Import
There are black swans. There are no Sasquatch.

To reason from premisses that lack existential import for a certain class to a conclusion that has it is to commit the Existential Fallacy.

History:

In the traditional formal logic developed by Aristotle and subsequent logicians through the Middle Ages, it was implicitly assumed that the classes of things referred to by the subject and predicate terms of categorical propositions were non-empty. For this reason, certain arguments were considered valid which would not be valid if some class were empty. For example, an A-type proposition implies an I-type, and an E-type implies an O-type:

Subalternation
All Catholics are christians.
Therefore, some christians are Catholics.
No atheists are christians.
Therefore, some christians are not atheists.

This type of inference is called "subalternation". Unfortunately, subalternation is an invalid form of argument if one of the terms refers to an empty class, such as "unicorns"; see the Counter-Example.

For reasons explained in the Exposure, logicians of the nineteenth century dropped the traditional assumption of non-emptiness, and adopted what is called the "Boolean interpretation"—after logician George Boole—of universal quantifiers. Under the Boolean interpretation, A- and E-type propositions lack existential import, while both I- and O-type have it. This has the consequence that some immediate inferences—such as subalternation—and categorical syllogisms which were valid under the traditional interpretation become instances of the Existential Fallacy. Of course, as long as the relevant classes are known to be non-empty, an argument should be considered to be an enthymeme instead of an instance of this fallacy.

Exposure:

The traditional theory makes it impossible to reason about empty classes, which might seem to be a small price to pay if all that we had to give up were classes such as unicorns. However, some classes may be empty for all we know, yet we manage to reason about them all the same. For instance, there may be no extraterrestrial aliens, but we cannot even say this meaningfully in the traditional theory, let alone use the class in an argument. Shoplifters Will Be Prosecuted

Also, consider a shopkeeper who puts up the following sign:

The shopkeeper hopes that potential thieves will reason as follows:

According to the sign, if I shoplift, I'll be prosecuted. I don't want to be prosecuted. Therefore, I'd better not shoplift in this store.

According to the traditional theory, if the sign succeeds in deterring shoplifters, then they cannot reason this way! Yet, it is partly because people reason this way that there are no shoplifters.

Source:

Copi and Cohen, Introduction to Logic (Tenth Edition) (Prentice Hall, 1998), pp. 181-184.


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