Skip to main content
    • Aa
    • Aa

Which structural rules admit cut elimination? An algebraic criterion

  • Kazushige Terui (a1)

Consider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structure-free logic. The aim of this work is to give such a condition by using algebraic semantics.

We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set ℛ of structural rules, the cut elimination theorem holds for FL enriched with ℛ if and only if ℛ satisfies the propagation property.

As an application, we show that any set ℛ of structural rules can be “completed” into another set ℛ*, so that the cut elimination theorem holds for FL enriched with ℛ*. while the provability remains the same.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[9] J.-Y. Girard , Linear logic, Theoretical Computer Science, vol. 50 (1987), pp. 1–102.

[12] R. Hori , H. Ono , and H. Schellinx , Extending intuitionistic linear logic with knotted structural rules, Notre-Dame Journal of Formal Logic, vol. 35 (1994), no. 2, pp. 219–242.

[16] M. Ohnishi and K. Matsumoto , A system for strict implication, Annals of the Japan Association for Philosophy of Science, vol. 2 (1964), pp. 183–188.

[17] M. Okada , Phase semantics for higher order completeness, cut-elimination and normalization proofs (Extended Abstract), Electronic Notes in Theoretical Computer Science, vol.3, a Special Issue on the Linear Logic'96, Tokyo Meeting (J.-Y. Girard, M. Okada, and A. Scedrov, editors), 1996.

[18] M. Okada , Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic, Theoretical Computer Science, vol. 227 (1999), pp. 333–396.

[19] M. Okada , A uniform semantic proof for cut-elimination and completeness of various first and higher order logics, Theoretical Computer Science, vol. 281 (2002), pp. 471–498.

[27] A. Zamansky and A. Avron , Cut-elimination and quantification, Studia Logica, vol. 82 (2006), pp. 157–176.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 49 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th July 2017. This data will be updated every 24 hours.