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Obviously, "$\vdash$" defines a relation on $L$. This relation is called the called the derivability property.

Definition: Derivability Property

Let \(L\) be a logical calculus. We say that a string $\phi\in L$ is derivable in $L$ if there is is proof $\phi_1,\ldots,\phi_n\vdash \phi$. In this case, we write $\vdash_L\phi$, or just $\vdash \phi$. Otherwise, we write $\not\vdash\phi,$ denoting that $\phi$ has not derivable in $L$.

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References

Bibliography

  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Beierle, C.; Kern-Isberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000