Let's make a trip to a new world called "Never Never Land".
Regular, ordinary first-order logic has two quantifiers: $\forall$ and $\exists$.
Now, let's imagine we lived in a world in which these quantifiers didn't exist, and instead we only had one quantifier, $\mathrm{N}$. The quantifier $\mathrm{N}$ is the "never" quantifier, and the expression
$$\mathrm{N} x.\; [\text{some formula}]$$
means "[some formula] is never true, regardless of what choice of $x$ we pick." For example, the expression
$\mathrm{Nx}(\mathrm{P}(\mathrm{x}))$ says "There is No element $\mathrm{x}$ in the domain, such that $\mathrm{P}(\mathrm{x})$ is true".
For predicates $\mathrm{A}(\mathrm{x})$ and $\mathrm{B}(\mathrm{x})$, Which of the following is the correct expression for $\text{“All A's are B's" }?$
- $\neg \mathrm{Nx}(\mathrm{A}(\mathrm{x}) \rightarrow \mathrm{B}(\mathrm{x}))$
- $\neg \mathrm{Nx}(\mathrm{A}(\mathrm{x}) \wedge \neg \mathrm{B}(\mathrm{x}))$
- $\mathrm{Nx}(\mathrm{A}(\mathrm{x}) \wedge \neg \mathrm{B}(\mathrm{x}))$
- $\mathrm{Nx}(\neg A(x) \wedge B(x))$
