Quantifier incompatibility

Quantifier incompatibility describes the property of first-order predicate logic that the universal quantifier does not “understand” itself with the disjunction and the existential quantifier does not “understand” itself with the conjunction (there is no distributive law for these quantifiers and operators): ${\displaystyle \forall }$ ${\displaystyle \exists }$

${\displaystyle (\forall x\colon F(x))\lor (\forall x\colon G(x))\,\not \equiv \,\forall x\colon (F(x)\lor G(x))}$

${\displaystyle (\exists x\colon F(x))\land (\exists x\colon G(x))\,\not \equiv \,\exists x\colon (F(x)\land G(x))}$

example

The statement

The following applies to every car: it drives or it stops

is not synonymous with the statement

Each car is stationary or moving any car ,

because in the latter case either no car is driving or all cars are driving, but no intermediate state is possible, such as

Some cars drive, some don't .