# Definition:Gentzen Proof System/Instance 1/Alpha-Rule

## Definition

Let $\LL$ be the language of propositional logic.

Let $\mathscr G$ be instance 1 of a Gentzen proof system.

The $\alpha$-rule allows for introducing an $\alpha$-formula in the conclusion.

Its precise formulations is as follows:

$(\alpha)$: For any $\alpha$-formula $\mathbf A$ and associated $\mathbf A_1, \mathbf A_2$ from the table of $\alpha$-formulas:

Given $U_1 \cup \set {\mathbf A_1}$ and $U_2 \cup \set {\mathbf A_2}$, one may infer $U_1 \cup U_2 \cup \set {\mathbf A}$.

### Notation

In a tableau proof, the $\alpha$-rule can be used as follows:

 Pool: Empty. Formula: $U_1 \cup U_2 \cup \set {\mathbf A}$. Description: $\alpha$-Rule. Depends on: The lines containing $U_1 \cup \set {\mathbf A_1}$ and $U_2 \cup \set {\mathbf A_2}$. Abbreviation: $\alpha \circ$, where $\circ$ is the binary logical connective such that $\mathbf A = \mathbf A_1 \circ \mathbf A_2$ or $\mathbf A = \neg \paren {\mathbf A_1 \circ \mathbf A_2}$, or $\neg \neg$ in the case that $\mathbf A = \neg \neg \mathbf A_1$.