Definition:Consistent (Logic)/Proof System/Propositional Logic

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Definition

Let $\LL_0$ be the language of propositional logic.

Let $\mathscr P$ be a proof system for $\LL_0$.


Definition 1

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 $\mathscr P$.


Definition 2

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 see