# Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations

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## Theorem

Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations.

That is, for every WFF $\mathbf A$:

- $\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathscr H_2} \mathbf A$

## Proof

## Also see

The Soundness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations in which it is proved that:

- If $\vdash \mathbf A$ then $\models \mathbf A$.

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous): $\S 4.8$: Completeness: Theorem $14$