# What is null quantification?

I want to know the definition of null quantification and when and why we need to use it. In a book I was learning, it says:

"establish rules for null quantification that we can use when a quantified variable does not appear in part of a statement. "

And

" Establish these logical equivalences, where x does not occur as a free variable in A. Assume that the domain is nonempty.

a) $$\forall x ~( P(x) \lor A ) \iff ( \forall x~P(x) ) \lor A$$ "

Now can anyone explain me what is said in this two quotes. And what's the $$A$$ here in the second quote. Is this stands for a proposition without $$x$$ or a free variable or a bound variable.

Please explain this with a proper definition and a clear example.

• A is any wwf that does not have any free occurrences of x. For example, y = t. Nov 1, 2019 at 10:35
• What does it mean by wwf Nov 2, 2019 at 5:34
• Well formed formula. Nov 2, 2019 at 9:35
• Possible duplicate of How do I solve this discrete math/quantifier problem? Nov 7, 2019 at 14:34

$$A$$ is a proposition (a well formed formula) and it explicitly states it is being used "where $$x$$ does not occur as a free variable in $$A$$."

So we have a part of the statement, $$P(x)\vee A$$, where the term $$x$$ does not occur as a free variable; that part being the $$A$$.

We would like to establish a rule if inference where we can 'extract' this part from the quantified statement -- also, conversely, to 'inject' it when we have the disjunction.

That is we wish to show that $$\forall x~(P(x)\lor A)$$ entails, and is entailed by, $$(\forall x~P(x))\lor A$$.   That the two statements are equivalent (well, when the domain is non-empty).

$$\forall x~(P(x)\vee A)$$ claims that: "Every valuation for $$x$$, in this non-empty domain, satisfies the formula $$P(x)\lor A$$."

$$(\forall x~P(x))\vee A$$ claims that: "$$A$$ is satisfied or every valuation for $$x$$, in this non-empty domain, satisfies the formula $$P(x)$$."

In first-order logic (FOL) a well-formed formula (wff) is also called a sentence or a proposition (terminology originated mainly from Aristotle and Greek stoics) if and only if it has no free variable(s) occurrence since all its quantified in-scoped variables are all bounded in their corresponding domain of discourse (using philosophy jargon we say a true proposition must be grounded or corresponded to ontological fact manifested in extensional objects). A null quantification is related to such propositional (zeroth-order) case while a general wff with free variable occurrences is not truth-functional, it can only be satisfied or unsatisfied by some particular value(s) of its bounded variable(s).

Just bear in mind below first-order equivalences (with one exception if noted). So we can mentally imagine the null quantification case as a limiting case of fully bounded case and we can have first-order equivalence (4) in this limit case while we cannot have such nice first-order relationship in the general wff case (2).

(1) $$\forall x ~(P(x) \land A(x)) ≡ \forall x~P(x) \land \forall x~A(x)$$

(2) $$\forall x ~(P(x) \lor A(x)) \not≡ \forall x~P(x) \lor \forall x~A(x)$$

(3) $$\forall x ~(P(x) \land A) ≡ \forall x~P(x) \land A$$

(4) $$\forall x ~(P(x) \lor A) ≡ \forall x~P(x) \lor A$$