# Equivalence of Definitions of Consistent Set of Formulas

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## Contents

## Theorem

The following definitions of the concept of **Consistent Proof System for Propositional Logic** are equivalent:

Let $\mathcal L_0$ be the language of propositional logic.

Let $\mathscr P$ be a proof system for $\mathcal L_0$.

Let $\mathcal F$ be a collection of logical formulas.

### Definition 1

Then $\mathcal F$ is **consistent for $\mathscr P$** if and only if:

- There exists a logical formula $\phi$ such that $\mathcal F \not \vdash_{\mathscr P} \phi$

That is, some logical formula $\phi$ is **not** a $\mathscr P$-provable consequence of $\mathcal F$.

### Definition 2

Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds.

Then $\mathcal F$ is **consistent for $\mathscr P$** if and only if:

- For every logical formula $\phi$, not
*both*of $\phi$ and $\neg \phi$ are $\mathscr P$-provable consequences of $\mathcal F$