# Definition:Consistent/Proof System

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## Definition

Let $\mathcal L$ be a logical language.

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

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

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

That is, some logical formula $\phi$ is **not** a theorem of $\mathscr P$.

### Propositional Logic

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

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

- For every logical formula $\phi$, not
*both*of $\phi$ and $\neg \phi$ are theorems of $\mathscr P$

## Also defined as

**Consistency** is obviously necessary for soundness in the context of a given semantics.

Therefore it is not surprising that some authors obfuscate the boundaries between a **consistent proof system** (in itself) and a sound proof system (in reference to the semantics under discussion).

## Also see

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.4$: Conditions for an Axiom System - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Axiom systems