Logical fallacy

A **quantifier shift** is a logical fallacy in which the quantifiers of a statement are erroneously transposed during the rewriting process. 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.

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Examples

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

- $\forall x\,\exists y\,(Px\to (Wy\land M(yx)))\vdash \exists y\,\forall x\,(Px\to (Wy\land M(yx)))$

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.

- $\forall x\,\exists y\,Bxy\vdash \exists y\,\forall x\,Bxy$

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.

3. Every natural number $n$ has a successor $m=n+1$, the smallest of all natural numbers that are greater than $n$. Therefore, there is a natural number ${m))$ that is a successor to all natural numbers.

- $\forall n\,\exists m\,Snm\vdash \exists m\,\forall n\,Snm$

It is fallacious to conclude that there is a single natural number that is the successor of every natural number.