Definition:Theorem/Formal System
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Definition
Let $\LL$ be a formal language.
Let $\mathscr P$ be a proof system for $\LL$.
A theorem of $\mathscr P$ is a well-formed formula of $\LL$ which can be deduced from the axioms and axiom schemata of $\mathscr P$ by means of its rules of inference.
That a WFF $\phi$ is a theorem of $\mathscr P$ may be denoted as:
- $\vdash_{\mathscr P} \phi$
Also known as
A theorem $\phi$ of $\mathscr P$ is also called provable from $\mathscr P$.
Also see
- Definition:Theorem of Logic is a specific case of this concept, if one defines the apparatus of logic as a formal system.
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.3$: Derivable Formulae
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Axiom systems
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous): $\S 3.1$: Definition $3.1$