# Scope of Quantifier, bit puzzling

$\forall x(p(x) \rightarrow \exists xq(x))$

$p(x)$ : $x$ is a human. $q(x)$ : $x$ has a job

Help me understand this in english please?

• for all humans there exists a job
– JMP
Jul 17, 2015 at 6:24
• edited. sorry. Please look again. Jul 17, 2015 at 6:39
• The scope of the quantifier is "as little as possible"; deviations are managed with parentheses. In your example, the inner quantifier $\exists x$ acts on $q(x)$ only. Thus, from "the point of view" of the outer quantifier $\forall x$ there is only one variable $x$ to act on : that in $p(x)$. The formula can be rewritten as $\forall x (p(x) \to \exists y q(y))$ and the two have the same meaning. Jul 17, 2015 at 6:40
• About the interplay between the two, you have to note that the formula is true (or satisfied) also in "tricky" cases. Consider as domain of interpretation $\mathbb N$ and as interpretation for $p(x)$ the property : "$x$ is less than $0$" and as interpretation of $q(x)$ : "$x$ is equal to $0$". With tis int the meaning of the formula is : $\forall x [ (x < 0) \to \exists x (x=0)]$ and thus it is true in the said interpretation. Jul 17, 2015 at 6:44

Assume that the variable $x$ represents a creature, then I would interpret your formula as: