Quantifier-Shift Fallacy

Alias: Illicit Quantifier Shift

Type:

Form:

Every P bears the relation R to some Q.
Therefore, some Q bears the inverse of relation R to every P.

Similar Validating Form:

Some Q bears the relation R to every P.
Therefore, every P bears the inverse of relation R to some Q.

Example:

Every event must have a cause, that is, every event bears the effect relation to some other event, which is its cause.
Therefore, there must be a cause of every event, that is, some event bears the inverse of the effect relation—which is the cause relation—to every other event. This first cause is what we call "God".

Counter-Example:

Everybody loves someone.
Therefore, there is somebody whom everyone loves. (Lucky person!)

Exposition:

The phrase "quantifier-shift" refers to the two quantifiers at the beginning of the premiss and conclusion of arguments of this form, namely, "every" and "some". "Shift" refers to the fact that the difference between the premiss and conclusion of this form of argument consists in a shift in the order—or, technically, the scope—of the quantifiers. In the premiss, the universal quantifier, "every", is followed by the existential one, "some", whereas in the conclusion the order is reversed. This means that in the premiss the universal quantifier has widest scope, while in the conclusion the existential quantifier has wider scope.

The fallaciousness of this form of argument is most easily seen by examining some counter-examples that would fool no one. For instance, everyone has a mother—that is, for every person, there is some mother of that person. However, it is false that there is a mother of us all—that is, it is not true that some woman is the mother of everyone.

The converse inference is validating (see the Similar Validating Form); for instance, if there was truly someone who loved everyone, then it would follow that everyone was loved by someone—namely, the all-lover. But it does not follow from the fact that everyone is loved by someone that there is someone who loves everyone—that is, an all-lover. The fact that these two inferences differ only in the direction in which the quantifiers are shifted is probably one psychological reason why this fallacy is so easy to commit.

Sources:

fallacist@fallacyfiles.org