Definition:Consistent (Logic)
Definition
Let $\LL$ be a logical language.
Let $\mathscr P$ be a proof system for $\LL$.
Proof System
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 $\mathscr P$.
Set of Formulas
Let $\FF$ be a collection of logical formulas.
Then $\FF$ is consistent for $\mathscr P$ if and only if:
- There exists a logical formula $\phi$ such that $\FF \nvdash_{\mathscr P} \phi$.
That is, some logical formula $\phi$ is not a provable consequence of $\FF$.
Also defined as
Common confusion arises in the precise interpretation of consistency in logic.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, consistency is placed in the realm of proof systems.
It is also common to place it in the realm of formal semantics.
That is, to define consistent as what is satisfiable on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Because of the variation, any use of these terms in a source work should be treated with the utmost care and precision to determine the exact meaning.
Also see
- Results about logical consistency can be found here.
Sources
- 1944: Eugene P. Northrop: Riddles in Mathematics ... (previous) ... (next): Chapter One: What is a Paradox?
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- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): consistent (in logic)