Implication of vacuous truth on the space of possible statements

Question

Considering the concept of vacuous truth, I thought of the following argument:

Say the following proposition is true: $$ \forall{y}\left[\forall{x}\left[x\in{y}\rightarrow \varphi(x)\right]\rightarrow\psi(y)\right] $$ Then there exists no predicate $\tau$ such that: $$ \forall{y}\left[\forall{x}\left[x\in{y}\rightarrow \tau(x)\right]\rightarrow\neg\,\psi(y)\right] $$ For otherwise, we would have: $ \neg\,\psi(\emptyset) \land \psi(\emptyset)$.

Is the above argument correct? If so, does it have a name?

Answer 1

Long comment

A similar issue we have using the definition of the empty set :

$\lnot \exists x (x \in \emptyset)$

we have, by Ex Falso Quodlibet tautological schema : $\lnot \mathcal P \to (\mathcal P \to \mathcal Q)$, that :

$\mathsf {ZF} \vdash \exists x (x \in \emptyset) \to \varphi$,

with a formula $\varphi$ whatever.

So, in conclusion, if $\mathsf {ZF}$ is consistent, we cannot have :

$\mathsf {ZF} \vdash ∀y[\exists x(x∈y) → φ] → ψ$

and :

$\mathsf {ZF} \vdash ∀y[\exists x(x∈y) → \tau] → \lnot ψ.$

But this is true in general for every theorem $\text {T}$ of $\mathsf {ZF}$ : if the theory is consistent, we cannot have both : $\mathsf {ZF} \vdash \text {T} \to ψ$ and $\mathsf {ZF} \vdash \text {T} \to \lnot ψ$.