# Hilbert Proof System Instance 2 is Consistent

## Contents

## Theorem

Instance 2 of the Hilbert proof systems $\mathscr H_2$ is consistent.

## Proof

Consider Instance 1 of a constructed semantics, denoted $\mathscr C_1$.

Note that $\neg p$ is not a tautology for $\mathscr C_1$.

We will establish that every $\mathscr H_2$-theorem is a $\mathscr C_1$-tautology.

That is, that $\mathscr H_2$ is sound for $\mathscr C_1$.

Starting with the axioms:

\((A1)\) | $:$ | Rule of Idempotence | \(\displaystyle (p \lor p) \implies p \) | Proof of Tautology | ||||

\((A2)\) | $:$ | Rule of Addition | \(\displaystyle q \implies (p \lor q) \) | Proof of Tautology | ||||

\((A3)\) | $:$ | Rule of Commutation | \(\displaystyle (p \lor q) \implies (q \lor p) \) | Proof of Tautology | ||||

\((A4)\) | $:$ | Factor Principle | \(\displaystyle (q \implies r) \implies \left({ (p \lor q) \implies (p \lor r)}\right) \) | Proof of Tautology |

Next it needs to be shown that the rules of inference of $\mathscr H_2$ preserve $\mathscr C_1$-tautologies.

### Rule $RST \, 1$: Rule of Uniform Substitution

By definition, any WFF is assigned a value $1$ or $2$.

Thus, in applying Rule $RST \, 1$, we are introducing $1$ or $2$ in the position of a propositional variable.

But all possibilities of assignments of $1$s and $2$s to such propositional variables were shown not to affect the resulting value $2$ of the axioms.

Hence Rule $RST \, 1$ preserves $\mathscr C_1$-tautologies.

### Rule $RST \, 2$: Rule of Substitution by Definition

Because the definition of $\mathscr C_1$ was given in terms of Rule $RST \, 2$, it cannot affect any of its results.

### Rule $RST \, 3$: Rule of Detachment

Suppose $\mathbf A$ and $\mathbf A \implies \mathbf B$ both take value $2$.

Then using Rule $RST \, 2$, definition $(2)$, we get:

- $\neg \mathbf A \lor \mathbf B$

taking value $2$ by assumption.

But $\neg \mathbf A$ takes value $1$ by definition of $\neg$.

So from the definition of $\lor$ it must be that $\mathbf B$ takes value $2$.

Hence Rule $RST \, 3$ also produces only WFFs of value $2$.

### Rule $RST \, 4$: Rule of Adjunction

Suppose $\mathbf A$ and $\mathbf B$ take value $2$.

Then:

\(\displaystyle \mathbf A \land \mathbf B\) | \(=\) | \(\displaystyle 2 \land 2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \neg ( \neg 2 \lor \neg 2 )\) | Rule $RST \, 2 \, (1)$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \neg ( 1 \lor 1 )\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \neg 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2\) |

proving that Rule $RST \, 4$ also produces only $2$s from $2$s.

Hence $\mathscr H_2$ is sound for $\mathscr C_1$.

In particular:

- $\not\vdash_{\mathscr H_2} \neg p$

Hence $\mathscr H_2$ is consistent.

$\blacksquare$

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.5$: Consistency