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2006 Reflexive Intermediate Propositional Logics
Nathan C. Carter
Notre Dame J. Formal Logic 47(1): 39-62 (2006). DOI: 10.1305/ndjfl/1143468310


Which intermediate propositional logics can prove their own completeness? I call a logic reflexive if a second-order metatheory of arithmetic created from the logic is sufficient to prove the completeness of the original logic. Given the collection of intermediate propositional logics, I prove that the reflexive logics are exactly those that are at least as strong as testability logic, that is, intuitionistic logic plus the scheme $\neg φ ∨ \neg\neg φ. I show that this result holds regardless of whether Tarskian or Kripke semantics is used in the definition of completeness. I also show that the operation of creating a second-order metatheory is injective, thereby insuring that I am actually considering each logic independently.


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Nathan C. Carter. "Reflexive Intermediate Propositional Logics." Notre Dame J. Formal Logic 47 (1) 39 - 62, 2006.


Published: 2006
First available in Project Euclid: 27 March 2006

zbMATH: 1105.03025
MathSciNet: MR2211181
Digital Object Identifier: 10.1305/ndjfl/1143468310

Primary: 03F50
Secondary: 03F55

Rights: Copyright © 2006 University of Notre Dame


Vol.47 • No. 1 • 2006
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