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24.2 Null Quantification (Null)

If a quantifier doesn’t bind any variables in a formula, it is said to be null on it.

AxGuilty(pia) looks weird, but it is well-formed. The quantifier Ax is not conflicting in scope with any other quantifier binding x, so it is okay.

But since Guilty(pia) has no x in it, Ax is null. It is idle, not doing anything. But that doesn’t make it ungrammatical.

Null quantification looks a lot less weird when we have multiple quantifiers. For example:

ExEy(Dog(x)&Dog(y)&~(x=y))

Says that there are two dogs: x is a dog, and y is a dog, and they are not the same dog.

But notice that Dog(y) is inside the scope of Ex, so Ex is null on Dog(y).

Basically, if a quantifier is null on a formula (if there are no variables in the formula that the quantifier binds), then the quantifier can always be put wide scope around it.

So:

P&AxQ(x) ⇔ Ax(P&Q(x))

Null Quantification (Null)
P&AxQ(x) ⇔ Ax(P&Q(x))

We use P to represent some formula, possibly complex, that has no xs in it. Since Ax is can’t bind anything in P, we can expand the scope of the quantifier to cover it.

That’s now null quantification works.

25.2 Null Quantification (Null)