Abstract
The system GLS, which is a modal sequent calculus system for the provability logic GL, was introduced by G. Sambin and S. Valentini in Journal of Philosophical Logic, 11(3), 311–342, (1982), and in 12(4), 471–476, (1983), the second author presented a syntactic cut-elimination proof for GLS. In this paper, we will use regress trees (which are related to search trees) in order to present a simpler and more intuitive syntactic cut derivability proof for GLS1, which is a (more connectively and inferentially economical) variant of GLS without the cut rule.
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Notes
Stemming from one of the meanings of the word “regress,” which is “The reasoning involved when one assumes the conclusion is true and reasons backward to the evidence.”
One reason we use this notation is to avoid confusion with the expression Γ ⊢ Δ which sometimes is a short for “the sequent Γ ⊢ Δ is provable,” which might not be true since Γ,Δ can be any sets of formulas whatsoever. This notation also serves to indicate that the context is of regress trees and not of the (corresponding) Gentzen proof system.
For example, if the root is
we can use the GLR regress rule to regress it to
or to regress it to
(and therefore
has at least two associated regress trees).Technically, S 1 and S 3 are not in the right form to allow a legal application of the (→ -left) rule; however, they can easily be weakened and strengthened to the right form, thus allowing us to obtain S 4. This remark will apply to all the following instances involving a similar “illegal” use of the (→ -left) rule.
we can use the GLR regress rule to regress it to
or to regress it to
(and therefore
has at least two associated regress trees).References
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Acknowledgements
I would like to thank David Schwartz for his guidance and great insights.
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Brighton, J. Cut Elimination for GLS Using the Terminability of its Regress Process. J Philos Logic 45, 147–153 (2016). https://doi.org/10.1007/s10992-015-9368-4
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DOI: https://doi.org/10.1007/s10992-015-9368-4