Journal of Symbolic Logic 73 (4):1315-1327 (2008)

Abstract
We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski. Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle
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DOI 10.2178/jsl/1230396921
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References found in this work BETA

Incompleteness in Intuitionistic Metamathematics.David Charles McCarty - 1991 - Notre Dame Journal of Formal Logic 32 (3):323-358.
Intuitionistic Completeness and Classical Logic.D. C. McCarty - 2002 - Notre Dame Journal of Formal Logic 43 (4):243-248.
Reflexive Intermediate Propositional Logics.Nathan C. Carter - 2006 - Notre Dame Journal of Formal Logic 47 (1):39-62.

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Citations of this work BETA

What Is an Inconsistent Truth Table?Zach Weber, Guillermo Badia & Patrick Girard - 2016 - Australasian Journal of Philosophy 94 (3):533-548.
Preface.Matteo Pascucci & Adam Tamas Tuboly - 2019 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 26 (3):318-322.
Structuralism and Isomorphism.C. McCarty - 2015 - Philosophia Mathematica 23 (1):1-10.

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