Abstract
A calculus for classical propositional sequents is introduced that consists of a restricted version of the cut rule and local variants of the logical rules. Employed in the style of proof search, this calculus explodes a given sequent into its elementary structural sequents—the topmost sequents in a derivation thus constructed—which do not contain any logical constants. Some of the properties exhibited by the collection of elementary structural sequents in relation to the sequent they are derived from, uniqueness and unique representation of formula occurrences, will be discussed in detail. Based on these properties it is suggested that a collection of elementary structural sequents constitutes the purely structural representation of the sequent from which it is obtained.
This is a preview of subscription content, access via your institution.
References
Arndt, M., Eight inference rules for implication, Studia Logica, pp. 28, 2018 https://doi.org/10.1007/s11225-018-9821-9.
Avron, A., Tonk—a full mathematical solution, in A. Biletzki, (eds.), Hues of Philosophy, College Publications, 2010, pp. 17–42.
Došen, K., and Z. Petrić, Graphs of plural cuts, Theoretical Computer Science 484:41–55, 2013.
Gentzen, G., Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen, Mathematische Annalen 107:329–350, 1933.
Gentzen, G., Untersuchungen über das logische Schließen. I–II, Mathematische Zeitschrift 39:176–210, 405–431, 1935.
Gentzen, G., Collected Papers of Gerhard Gentzen, volume 55 of Studies in Logic and the Foundations of Mathematics, North-Holland, 1969.
Girard, J.-Y., Linear logic, Theoretical Computer Science 50:1–102, 1987.
Girard, J.-Y., Transcendental syntax i: deterministic case, Mathematical Structures in Computer Science 27(5):827–849, 2017.
Hertz, P., Über Axiomensysteme für beliebige Satzsysteme. I. Teil. Sätze ersten Grades, Mathematische Annalen 87:246–269, 1922.
Hertz, P., Über Axiomensysteme für beliebige Satzsysteme. II. Teil. Sätze höheren Grades, Mathematische Annalen 89:76–100, 1923.
Hertz, P., Über Axiomensysteme für beliebige Satzsysteme, Mathematische Annalen 101:457–514, 1929.
Hertz, P., Sprache und Logik, Erkenntnis 7(1):309–324, 1937.
Hughes, D. J. D., Proofs without syntax, Annals of Mathematics 164:1065–1076, 2006.
Prawitz, D., Proofs and the meaning and completeness of the logical constants, in J. Hintikka, I. Niiniluoto and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic, Reidel, 1979, pp. 25–40.
Schroeder-Heister, P., Uniform proof-theoretic semantics for logical constants (abstract), Journal of Symbolic Logic 56:1142, 1991.
Schroeder-Heister, P., Rules of definitional reflection, in Proceedings of the 8th Annual IEEE Symposium on Logic in Computer Science, 1993, pp. 222–232.
Schroeder-Heister, P., Generalized definitional reflection and the inversion principle, Logica Universalis 1:355–376, 2007.
von Plato, J., Gentzen’s proof systems: byproducts in a work of genius, Bulletin of Symbolic Logic 18(3):313–367, 2012.
Acknowledgements
Supported by the DFG Project “Paul Hertz and his foundations of structural proof theory” (DFG AR 1010/2-1).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Arndt, M. The Explosion Calculus. Stud Logica 108, 509–547 (2020). https://doi.org/10.1007/s11225-019-09861-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-019-09861-6
Keywords
- Sequent calculus
- Structural reasoning
- Cut rule
- Locality