# Intuitionistic completeness of first-order logic

@article{Constable2014IntuitionisticCO,
title={Intuitionistic completeness of first-order logic},
author={Robert L. Constable and Mark Bickford},
journal={Ann. Pure Appl. Log.},
year={2014},
volume={165},
pages={164-198}
}
• Published 7 October 2011
• Mathematics, Computer Science
• Ann. Pure Appl. Log.
Abstract We constructively prove completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable in iFOL if and only if it is uniformly valid in intuitionistic evidence semantics as defined in intuitionistic type theory extended with an intersection operator. Our completeness proof provides an effective procedure that converts any uniform evidence into a formal iFOL proof. Uniform evidence can involve arbitrary concepts from type theory such as ordinals, topological…
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## References

SHOWING 1-10 OF 101 REFERENCES
Completeness and incompleteness for intuitionistic logic
• C. McCarty
• Mathematics, Computer Science
Journal of Symbolic Logic
• 2008
A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle, which is corollaries of a single theorem.
An Abstract Notion of Realizability for Which Intuitionistic Predicate Calculus is Complete
Publisher Summary This chapter presents an abstract notion of realizabillty for which intuitionistic predicate calculus is complete. It is shown that A is derivable in Heyting's predicate calculus if
Uniform provability realization of intuitionistic logic, modality and lambda-terms
This tutorial talk presents a provability realization of Int and S4 that solves both the problem of finding an adequate formalization of the provability semantics and establishing the completeness of the intuitionistic logic Int.
Continuation-passing style models complete for intuitionistic logic
• Danko Ilik
• Mathematics, Computer Science
Ann. Pure Appl. Log.
• 2013
A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic, and a $\beta$-normalisation-by-evaluation program for simply typed lambda calculus with sum types.
Some Intuitions Behind Realizability Semantics for Constructive Logic: Tableaux and Läuchli Countermodels
• Mathematics, Computer Science
Ann. Pure Appl. Log.
• 1996
It is argued in some detail that, in spite of a certain inherent inexactness of the analogy, every intuitively constructive truth is valid in Lauchli semantics, and therefore the Heyting Calculus is powerful enough to prove all constructive truths.
Undecidability and intuitionistic incompleteness
• D. McCarty
• Mathematics, Computer Science
J. Philos. Log.
• 1996
These results give extensions of the theorem of Gödel and Kreisel that completeness for pure intuitionistic predicate logic requires MP.
An Intuitionistically Plausible Interpretation of Intuitionistic Logic
• H. D. Swart
• Mathematics, Computer Science
J. Symb. Log.
• 1977
To be able to settle the converse question: “if A is intuitively true, then ⊦ IPC A ”, one should make the notion of intuitionistic truth more easily amenable to mathematical treatment.
A framework for defining logics
• Computer Science
JACM
• 1993
The Edinburgh Logical Framework provides a means to define (or present) logics through a general treatment of syntax, rules, and proofs by means of a typed λ-calculus with dependent types, whereby each judgment is identified with the type of its proofs.
Semantical Analysis of Intuitionistic Logic I
Publisher Summary The chapter discusses a semantical analysis of intuitionistic logic I. The chapter presents a semantical model theory for Heyting's intuitionist predicate logic and proves the
An Intuitionistic Theory of Types: Predicative Part
Publisher Summary The theory of types is intended to be a full-scale system for formalizing intuitionistic mathematics as developed. The language of the theory is richer than the languages of