Relationships between quantifiers

Consider the formula, “no university has no student”, it is equivalent to “all universities have at least one student”, which shows that quantifiers are linked.

$$\begin{array}{c}\neg (\exists x)(\text{University}(x)\wedge \neg (\exists y)\text{Student}(y,x))\\ (\forall x)(\text{University}(x)\wedge \to (\exists y)\text{Student}(y,x))\end{array}$$

Hence we can produce the following the equivalences:

• $\neg \forall x\varphi (x)\equiv \exists x\neg \varphi (x)$
• $\neg \exists x\varphi (x)\equiv \forall x\neg \varphi (x)$