Hilbert Proof System Instance 2 Independence Results


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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 $(\text A 4)$ is independent from $(\text A 1)$, $(\text A 2)$, $(\text A 3)$.


$RST \, 4$ 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
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