The Existential Fallacy

Alias: The Fallacy of Existential Assumption

Taxonomy: Logical Fallacy > Formal Fallacy > Quantificational Fallacy > The Existential Fallacy

History:

In the traditional formal logic of categorical syllogisms developed by Aristotle and subsequent logicians through the Middle Ages and up to the middle of the nineteenth century, it was 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 that would not be valid if some class were empty: in particular, it was thought that an A-type proposition implied an I-type with the same subject and predicate terms, and an E-type implied an O-type, again with the same subject and predicate terms:

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

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

For reasons explained in the Exposure, below, logicians of the later 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, I- and O-type propositions have existential import―see the Exposition, below―whereas both A- and E-types lack 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.

Form:

Any argument whose conclusion implies that a class has at least one member, but whose premisses do not so imply. Usually, this involves arguing from a universal premiss or premisses to a particular conclusion.

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

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 ghosts.

"There are black swans" implies that the class of black swans is not empty, whereas "There are no ghosts" implies that the class of ghosts is empty. To reason from premisses that lack existential import for a certain class to a conclusion that has it is to commit the Existential Fallacy.

Exposure:

• As long as the relevant classes are known to be non-empty, an argument should be considered to be an enthymeme with an implicit existential premiss―that is, a suppressed premiss asserting that there are things of a relevant type―instead of an instance of this fallacy. For instance, in the examples of subalternation, above, since we know that there are christians, both arguments are valid with this additional information.
• People sometimes assume that because there is a term for a type of thing that type of thing must exist. Of course, it's generally true that meaningful words stand for existing things, since the usual reason for creating a new word is to be able to refer to actual things. Nonetheless, there are exceptions, and we all know many words that refer only to imaginary or fictional objects, such as "vampire", "werewolf", "Martian", and so on. Moreover, the traditional logic supported the notion that nouns must have reference in order to be meaningful, thus encouraging a kind of superstitious word magic according to which naming something brings it into existence.

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.

Also, consider a shopkeeper who puts up a sign saying: "All shoplifters will be prosecuted." 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.

• A general principle of deductive reasoning is what I call "the no-free-lunch (NFL) principle", which is that you get in the conclusion what you pay for in the premisses. If you don't pay for something in the premisses, you don't get something in the conclusion. Another, more fancy, way of putting this is that deduction is non-ampliative, that is, the information contained in the conclusion does not go beyond that contained in the premisses. This is why in a valid deductive argument the premisses necessitate the conclusion: if the information contained in the premisses is true then the information contained in the conclusion must also be true, since the latter information is contained in the former. One correlary of the NFL principle is that in deduction you can't get something for nothing, which is why the existential fallacy is fallacious. To conclude that something exists you must assume in the premisses that it exists.

*Note: Irving Copi & Carl Cohen, Introduction to Logic (Tenth Edition) (Prentice Hall, 1998), pp. 278-9