Quantifier shift

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A quantifier shift is a formal logical fallacy in which the quantifiers of a statement are improperly transposed. While this cannot always be easily recognized in common language, it can be expressed in both formal and informal terms.

<math>\forall x \exists y Rxy \vdash \exists y \forall x Rxy</math>
For every x there is a y such that R. Therefore, for every y there is an x such that R.

The classical illustration of this fallacy is as follows:

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

Use in Apologetics

This is most commonly illustrated in First Cause arguments for the existence of God.

Every event has a cause. Therefore, there is an event that caused all subsequent events.
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