Definition:Gentzen Proof System/Instance 1

Definition

This instance of a Gentzen proof system is used in:

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.)

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

The Gentzen system applies to sets of propositional formulae.

The intuition behind the system is that a set $U$ represents its disjunction.

$\mathscr G$ has the following axioms and rules of inference:

Axioms

A set $U$ of propositional formulae is an axiom of $\mathscr G$ if and only if $U$ contains a complementary pair of literals.

The invocation of an axiom may be denoted by:

$\vdash U$

Rules of Inference

Let $U_1, U_2$ be sets of propositional formulae.

$\mathscr G$ has two rules of inference, the $\alpha$-rule and the $\beta$-rule:

$\alpha$-Rule

$(\alpha)$: 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 $U_1 \cup U_2 \cup \set {\mathbf A}$.

$\beta$-Rule

$(\beta)$: For any $\beta$-formula $\mathbf B$ and associated $\mathbf B_1, \mathbf B_2$ from the table of $\beta$-formulas:

Given $U_1 \cup \set {\mathbf B_1, \mathbf B_2}$, one may infer $U_1 \cup \set {\mathbf B}$.

Invocations of these rules in a proof can be denoted as:

$(\alpha) \dfrac {\vdash U_1, \mathbf A_1 \hspace{3em} \vdash U_2, \mathbf A_2} {\vdash U_1, U_2, \mathbf A} \hspace{3em}

(\beta) \dfrac {\vdash U_1, \mathbf B_1, \mathbf B_2} {\vdash U_1, \mathbf B}$

This notation suppresses the set notation as a matter of convenience.

Source of Name

This entry was named for Gerhard Karl Erich Gentzen.

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 3.2$: Definition $3.2$