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On Different Intuitionistic Calculi and Embeddings from Int to S4

Uwe Egly
Studia Logica: An International Journal for Symbolic Logic
Vol. 69, No. 2, Analytic Tableaux and Related Methods. Part 2: Non-Classical Logics (Nov., 2001), pp. 249-277
Published by: Springer
Stable URL: http://www.jstor.org/stable/20016348
Page Count: 29
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On Different Intuitionistic Calculi and Embeddings from Int to S4
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Abstract

In this paper, we compare several cut-free sequent systems for propositional intuitionistic logic Int with respect to polynomial simulations. Such calculi can be divided into two classes, namely single-succedent calculi (like Gentzen's LJ) and multi-succedent calculi. We show that the latter allow for more compact proofs than the former. Moreover, for some classes of formulae, the same is true if proofs in single-succedent calculi are directed acyclic graphs (dags) instead of trees. Additionally, we investigate the effect of weakening rules on the structure and length of dag proofs. The second topic of this paper is the effect of different embeddings from Int to S4. We select two different embeddings from the literature and show that translated (propositional) intuitionistic formulae have sometimes exponentially shorter minimal proofs in a cut-free Gentzen system for S4 than the original formula in a cut-free single-succedent Gentzen system for Int. Moreover, the length and the structure of proofs of translated formulae crucially depend on the chosen embedding.

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