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Embedding Intuitionistic into Classical Logic

21 pagesPublished: June 3, 2023


The famous double negation translation [16, 17] establishes an embedding of classical into intuitionistic logic. Curiously, the reverse direction has not been covered in literature. Utilizing a normal form for intuitionistic logic [20], we establish a small model property for intuitionistic propositional logic. We use this property for a direct encoding of the Kripke semantics into classical propositional logic and quantified Boolean formulae. Next, we transfer the developed techniques to the first order case and provide an embedding of intuitionistic first-order logic into classical first-order-logic. Our goal here is an encoding that facilitates the use of state-of-the-art provers for classical first-order logic for deter- mining intuitionistic validity. In an experimental evaluation, we show that our approach can compete with state-of-the-art provers for certain classes of benchmarks, in particular when the intuitionistic content is low. We further note that our constructions support the transfer of counter-models to validity, which is a desired feature in model checking applications.

Keyphrases: automated reasoning, intuitionistic logic, model theory, proof theory

In: Ruzica Piskac and Andrei Voronkov (editors). Proceedings of 24th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 94, pages 329--349

BibTeX entry
  author    = {Alexander Pluska and Florian Zuleger},
  title     = {Embedding Intuitionistic into Classical Logic},
  booktitle = {Proceedings of 24th International Conference on Logic for Programming, Artificial Intelligence and Reasoning},
  editor    = {Ruzica Piskac and Andrei Voronkov},
  series    = {EPiC Series in Computing},
  volume    = {94},
  pages     = {329--349},
  year      = {2023},
  publisher = {EasyChair},
  bibsource = {EasyChair,},
  issn      = {2398-7340},
  url       = {},
  doi       = {10.29007/b294}}
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