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This is from "Introduction to Mathmatical Logic" by Elliot Mendelson , forth edition , chapter 2.6 [Rule C]

In the book , The "rule C" deduction in a first-order theory K is shown with the notation $\Gamma \vdash_C \mathscr B$. The confusion for me is that in $\Gamma \vdash_K \mathscr B$ , K is said to be a first order theory , so that would mean that in $\Gamma \vdash_C \mathscr B$ , C is a first order theory , but that seems to be not true. Am I missing something here?

Kripke Platek
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  • Rule C is a new rule. The gist of Prop.2.10 (page 80) is that adding the rule to the predicate calculus (Axioms+MP+Gen) what we get is a conservative extension: if $\Gamma \vdash_C \mathscr B$, then $\Gamma \vdash \mathscr B$ – Mauro ALLEGRANZA Apr 01 '21 at 13:30
  • Can "rule C" be thought of as a rule of inference (like MP or Gen)? – Kripke Platek Apr 01 '21 at 13:33
  • Yes, similar to the Deduction Theorem; having $\Gamma,A \vdash B$ the theorem asserts that we can build a derivation $\Gamma \vdash A \to B$. It is a meta-theory that we can use in a derivation without the necessity of effectively producing the "native" derivation. – Mauro ALLEGRANZA Apr 01 '21 at 13:38
  • See also the post Rule C – Mauro ALLEGRANZA Apr 01 '21 at 13:48

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