Quantifier shift

A quantifier shift is a logical fallacy in which the quantifiers of a statement are erroneously transposed. The change in the logical nature of the statement may not be obvious when it is stated in a natural language like English.

Definition

The fallacious deduction is that: For every A, there is a B, such that C. Therefore, there is a B, such that for every A, C.

\forall x\,,\exists y\,Rxy\vdash \exists y\,\forall x\,Rxy

However, an inverse switching:

\exists y\,\forall x\,Rxy\vdash \forall x\,\exists y\,Rxy

is logically valid.

Examples

1. Every person has a woman that is their mother. Therefore, there is a woman that is the mother of every person.

∀x∃y(Px → (Wy & M(yx))) therefore ∃y∀x(Wy → (Px & M(yx)))

It is fallacious to conclude that there is one woman who is the mother of all people.

However, if the major premise ("every person has a woman that is their mother") is assumed to be true, then it is valid to conclude that there is some woman who is any given person's mother.

2. Everybody has something to believe in. Therefore, there is something that everybody believes in.

∀x∃y Bxy therefore ∃y∀x Bxy

It is fallacious to conclude that there is some particular concept to which everyone subscribes.

It is valid to conclude that each person believes a given concept. But it is entirely possible that each person believes in a unique concept.

References

Robert Audi (General Editor), The Cambridge Dictionary of Philosophy (Second Edition), 1999, pp. 272–3.
A. R. Lacey, Dictionary of Philosophy (Third Revised Edition) (Barnes & Noble, 1996).
Introduction to Logic, Harry J. Gensler, p. 220
Antony G. Flew, Dictionary of Philosophy: Revised Second Edition
Harry J. Gensler, Historical Dictionary of Logic

Formal fallacies
Masked man fallacy
In propositional logic	Affirming a disjunct Affirming the consequent Denying the antecedent Argument from fallacy False dilemma
In quantificational logic	Existential fallacy Illicit conversion Proof by example Quantifier shift
Syllogistic fallacy	Affirmative conclusion from a negative premise Exclusive premises Existential Necessity Four-term fallacy Illicit major Illicit minor Negative conclusion from affirmative premises Undistributed middle
Other types of formal fallacy List of fallacies

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