# Hilbert Proof System Instance 2 Independence Results

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## Contents

## Theorem

Let $\mathscr H_2$ be Instance 2 of the Hilbert proof systems.

Then the following independence results hold:

### Independence of $(A1)$

Axiom $(A1)$ is independent from $(A2)$, $(A3)$, $(A4)$.

### Independence of $(A2)$

Axiom $(A2)$ is independent from $(A1)$, $(A3)$, $(A4)$.

### Independence of $(A3)$

Axiom $(A3)$ is independent from $(A1)$, $(A2)$, $(A4)$.

### Independence of $(A4)$

Axiom $(A4)$ is independent from $(A1)$, $(A2)$, $(A3)$.

### $RST \, 4$ is Derivable

Rule of inference $RST \, 4$ is derivable from $RST \, 1, RST \, 2, RST \, 3$ and the axioms $(A1)$ through $(A4)$.

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.6$: Independence