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25.5 Prenex Form and Chain of Equivalences

When a formula has all its quantifiers stacked up out front (widest possible scope), it is in Prenex Normal Form (PNF).

Prenex Normal Form (PNF)
All the quantifiers are widest possible scope.

For example: ExEy(Dog(x)&Dog(y)&~(x=y))

Every sentence of FOL is equivalent to some sentence in PNF. We can use equivalence principles to transform any sentence into its PNF twin.

First, let’s see if you can identify when sentences are in prenex form.

Now let’s take a sentence not in prenex form, and transform it.