Reductio ad absurdum, but just once

Question

Can we prove that $\exists x \lnot F$ follows from $\lnot \forall x F$ by natural deduction using the intuitionistic rules, and RAA (reductio ad absurdum) just once?

It is usually done by using RAA twice on the assumptions $\lnot F$ and $\lnot \exists x \lnot F$. I know that $\forall x F$ doesn't follow from $\lnot \exists x \lnot F$ in intuitionistic logic, so we cannot derive $\exists x \lnot F$ from $\forall x F, \lnot \forall x F$. Moreover, if we introduced $\exists x \lnot F$ by RAA, we would have to derive $\bot$ from $\lnot \exists x \lnot F, \lnot \forall x F$; if we introduced $\exists x \lnot F$ by $\exists$-intro, we would have to derive $\lnot F$ from $\lnot \forall x F$ using RAA just once, which seems not quite likely... In both cases I don't know how to proceed.

Answer 1

You cannot derive $\lnot\forall x F(x) \rightarrow \exists x\lnot F(x)$ in an intuitionistic system without RAA. This is because there is a Soundness Theorem for intuitionistic logic with respect to Kripke Semantics (https://en.wikipedia.org/wiki/Kripke_semantics#Semantics_of_intuitionistic_logic; I spell them out a bit below): if we could derive that implication using the rules (without RAA), then the formula would be valid in all Kripke Models. However, we can construct a counter-example in which $\lnot\forall x F(x) \rightarrow \exists x\lnot F(x)$ does not hold, so it cannot be derived.

Consider the following Kripke Structure $\mathcal{K} = (K, \leq, D, \Vdash)$:

Since $0\leq1$, $1\not\Vdash F(b)$ and $b\in D(1)$, we have $0\not\Vdash\forall x F(x)$, and also $1\not\Vdash\forall x F(x)$, so $0\Vdash \lnot\forall x F(x)$. But at 0, there is no $d\in D(0)$ satisfying $\lnot F(d)$, so $0\not\Vdash\exists x \lnot F(x)$. Hence, $0\not\Vdash\lnot\forall x F(x) \rightarrow \exists x\lnot F(x)$.

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In case you're unfamiliar with Kripke Structures. $(K, \leq)$ is a partially ordered set where you can consider the elements of $K$ to be "states of knowledge": at each stage $k\in K$ we "know" some theorems but perhaps not others, which may be proved or refuted at a later stage $k' \geq k$. Each state has a domain of objects, $D(k)$, and $\Vdash$ tells us which sentences hold at each node.

Two observations:

1. Objects do not "disappear" once constructed: $k \leq k' \implies D(k) \subset D(k')$.
2. Proved theorems remain proved: $k \Vdash\varphi \implies k' \Vdash\varphi$ for all $k' \geq k$.

(You can easily see that the example above satisfies these conditions.)

We can now stipulate the conditions under which $k \Vdash\varphi$:

• $k\Vdash F(d) \iff$ $F(d)$ holds at $k$ and $d\in D(k)$;
• $k\Vdash \lnot\varphi \iff$ for all $k'\geq k$, $k'\not\Vdash\varphi$;
• $k\Vdash \varphi\land\psi \iff k\Vdash\varphi \text{ and } k\Vdash\psi$;
• $k\Vdash \varphi\lor\psi \iff k\Vdash\varphi \text{ or } k\Vdash\psi$;
• $k \Vdash \varphi\rightarrow\psi \iff $ for all $k' \geq k$, $k'\Vdash\varphi$ implies $k'\Vdash\psi$;
• $k\Vdash\forall x F(x) \iff $ for all $k' \geq k$ and all $d\in D(k')$, $k'\Vdash F(d)$;
• $k\Vdash\exists x F(x) \iff $ there is a $d\in D(k)$ such that $k\Vdash F(d)$;

We say that a formula $\varphi$ is Kripke valid if, for all Kripke Structures $\mathcal{K} = (K, D, \Vdash)$ and all $k\in K$, $k\Vdash\varphi$, and we write "$\Vdash\varphi$". Writing "$\vdash_i\varphi$" for "$\varphi$ is intuitionistically provable", we have

Soundness Theorem If $\vdash_i\varphi$, then $\Vdash\varphi$.

Hence, since we showed that $\not\Vdash\lnot\forall x F(x) \rightarrow \exists x\lnot F(x)$ (by providing a Kripke Structure as a counter-example), it follows from the Soundness Theorem that $\not\vdash_i\lnot\forall x F(x) \rightarrow \exists x \lnot F(x)$.