Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 2
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Definition
Let $\LL$ be the language of propositional logic.
Let $\mathscr P$ be a proof system for $\LL_0$.
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 $\mathscr P$
Sources
- 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
- 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