Relation between efq, DS, DNE, LEM

Question

I am contemplating on the relation between the above formulae schema in the setting of minimal logic (which is just classical logic stripped of efq and DNE; thus, $\bot$ is just some constant formula) using natural deduction. Note that DS stands for Disjunctive Syllogism.

Firstly, I present the relations I believe I have proven (I will gladly present my proofs if needed¹):

1. DNE $\implies$ LEM $\implies$ DS $\iff$ efq
2. LEM + DS $\implies$ DNE

But then, this means that LEM $\iff$ DNE, without any "explicit" need of efq, which refutes a convincing negative stance that DNE requires LEM + efq, leading to obvious confusion.

Question: I am not at all experienced in logic, and my proofs might be erroneous. Thus, if someone can confirm if the above relations are indeed correct in minimal logic, I will be able to sleep alright. I'd also appreciate if you can say something about the correctness of the other implications.

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¹Let me show how I seem to have proven LEM $\implies$ DS:

Assume $A\vee B$ and $\neg A$. Goal is $B$. Since we have LEM, the goal becomes $\neg\neg B$. Thus, assume $\neg B$ with the goal of $\bot$. Now that we have $\neg A$ and $\neg B$, we may conclude $\neg(A\vee B)$ (this version of De Margan is easily proven in minimal logic). Denoting $A\vee B$ by $P$, we thus have $P\wedge\neg P$, and since we have $\neg(P\wedge\neg P)$ (again easily proven in minimal logic), we can conclude $\bot$.

Answer 1

In minimal logic, $A \vee B, \neg A \vdash \neg \neg B$, but also $A \vee B, \neg A \nvdash B$.

Adding the LEM doesn't help here, because you also need EFQ for DNE with the LEM.

$\begin{array}{|c|lr|}1. & A\vee B & Prem. \\ 2. & \neg A & Prem. \\ & \begin{array}{|c|lr|}3. & \neg B & SM\neg I \\ & \begin{array}{|c|lr|}4. & A & SM\to I \\ 5. & \bot & \bot I,4,2\end{array}\\ 6. & A \to \bot & \to I,4-5 \\ & \begin{array}{|c|lr|}7. & B & SM\to I \\ 8. & \bot & \bot I,7,3\end{array}\\ 9. & B \to \bot & \to I,7-8 \\ 10. & \bot & \vee E,1,6,9 \end{array}\\ 11. & \neg \neg B & \neg I,3-10\\ \end{array}$

For DNE to work given this minimal result, you'll need to find a way to derive $\neg \neg B \vdash B$. The principal way to do that is to add an axiom or rule to intuitionistic logic, which will have explosion (or an EFQ rule), as well.

The LEM, alone, does not guarantee DNE, so the following reasoning was flawed:

Assume $A \vee B$ and $\neg A$. Goal is B. Since we have LEM, the goal becomes $\neg \neg B$.

Instead, the only attainable minimal-logic derivation is $\neg \neg B$, and the LEM plays no part in it.