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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.

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When you quantify using $\exists x$ and $\forall x$, the quantifiaction (some/all) extends over all instances of the quantified variable $x$.
So if you instantiate $\exists x(Px \wedge Rx)$ with the constant $a$, we get $Pa \wedge Ra)$.
Similarly, instantiating $\forall x(Rx \to Wx)$ with the constant $b$, we get $Rb \to Wb$.
In short, a quantifier applies to the entire statement as described above.

When it does not, the quantifier's range is only part of the statement, we have a null quantifier e.g. $\forall x(Wx \vee F)$. The $\forall x$ applies only to $Wx$ and not to $F$. This quantification is pointless because $\forall x(Wx \vee F) \iff \forall x(Wx) \vee F$

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$A$ 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 $P(x)$."

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

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