Definition:Gentzen Proof System/Instance 1/Alpha-Rule
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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 $\mathbf A = \neg \neg \mathbf A_1$. |
Sources
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 3.2$: Definition $3.2$