Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-26T04:28:20.651Z Has data issue: false hasContentIssue false
  • English
  • Français

WHAT STANDS BETWEEN GROUNDING RULES AND LOGICAL RULES IS THE EXCLUDED MIDDLE

Part of: Philosophical aspects of logic and foundations Proof theory and constructive mathematics

Published online by Cambridge University Press:  23 December 2024

FRANCESCO A. GENCO
FRANCESCO A. GENCO*
Affiliation:
LUCI GROUP, DEPARTMENT OF PHILOSOPHY “PIERO MARTINETTI” UNIVERSITÀ DEGLI STUDI DI MILANO VIA FESTA DEL PERDONO 7, 20122 MILAN, ITALY CENTER FOR LOGIC, LANGUAGE AND COGNITION (LLC) DEPARTMENT OF PHILOSOPHY AND EDUCATION SCIENCES UNIVERSITY OF TURIN VIA S. OTTAVIO, 20, 10124, TURIN, ITALY
*
E-mail: frgenco@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The distinction between the proofs that only certify the truth of their conclusion and those that also display the reasons why their conclusion holds has a long philosophical history. In the contemporary literature, the grounding relation—an objective, explanatory relation which is tightly connected with the notion of reason—is receiving considerable attention in several fields of philosophy. While much work is being devoted to characterising logical grounding in terms of deduction rules, no in-depth study focusing on the difference between grounding rules and logical rules exists. In this work, we analyse the relation between logical grounding and classical logic by focusing on the technical and conceptual differences that distinguish grounding rules and logical rules. The calculus employed to conduct the analysis provides moreover a strong confirmation of the fact that grounding derivations are logical derivations of a certain kind, without trivialising the distinction between grounding and logical rules, explanatory and non-explanatory parts of a derivation. By a further formal analysis, we negatively answer the question concerning the possible correspondence between grounding rules and intuitionistic logical rules.

Keywords

groundingclassical logicexcluded middle lawintuitionistic logicproof theory

MSC classification

Primary: 03A99: None of the above, but in this section
Secondary: 03F03: Proof theory, general
Type
Research Article
Information
The Review of Symbolic Logic , Volume 18 , Issue 1 , March 2025 , pp. 1 - 27
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

BIBLIOGRAPHY

Aschieri, F., Ciabattoni, A., & Genco, F. A. (2018). Classical proofs as parallel programs . Gandalf 2018, EPTCS , 277, 43–57.Google Scholar
Aschieri, F., Ciabattoni, A., & Genco, F. A. On the concurrent computational content of intermediate logics. Theoretical Computer Science , 813, 375–409.Google Scholar
Barnes, J. (ed.) (1984). The Complete Works of Aristotle . Princeton, New Jersey: Princeton University Press.Google Scholar
Bolzano, B. (2014). Theory of Science , Oxford: Oxford University Press, Translated by Rolf George and Paul Rusnok.Google Scholar
Correia, F. (2010). Grounding and truth-functions. Logique et Analyse , 53(211), 251–279.Google Scholar
Correia, F.(2014). Logical grounds. The Review of Symbolic Logic , 7(1), 31–59.CrossRefGoogle Scholar
d’Agostino, M. (2014). Analytic inference and the informational meaning of the logical operators. Logique et Analyse , 227, 407–437.Google Scholar
D’Agostino, M. (2015). An informational view of classical logic. Theoretical Computer Science , 606, 79–97.CrossRefGoogle Scholar
D’Agostino, M., Gabbay, D., & Modgil, S. (2020). Normality, non-contamination and logical depth in classical natural deduction. Studia Logica , 108(2), 291–357.CrossRefGoogle Scholar
Fine, K. (2012). Guide to Ground. In F. Correia and B. Schnieder (eds.), Metaphysical grounding: understanding the structure of reality. Cambridge: Cambridge University Press, pp. 37–80.Google Scholar
Genco, F. A. (2021). Formal explanations as logical derivations. Journal of Applied Non-Classical Logics , 31(3–4), 279–342.Google Scholar
Genco, F. A., & Poggiolesi, F. (2021). Defining formal explanation in classical logic by substructural derivability. Computability in Europe 2021 . New York: Springer.Google Scholar
Genco, F. A., Poggiolesi, F., & Rossi, L. (2021). Grounding, quantifiers, and paradoxes. Journal of Philosophical Logic , 50, 1417–1448.CrossRefGoogle Scholar
de Groote, P. (1995). A simple calculus of exception handling . In International Conference on Typed Lambda Calculi and Applications, pp. 201–215. Berlin, Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
Hafner, J. (1999). Bolzano’s criticism of indirect proofs. Revue d’histoire des sciences , 52(3/4), 385–398.Google Scholar
Iemhoff, R. (2020). Intuitionism in the philosophy of mathematics . In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy , Metaphysics Research Lab, Fall 2020 ed., Stanford, California: Stanford University.Google Scholar
Lipton, P. (2004). Inference to the Best Explanation . London: Taylor & Francis.Google Scholar
Poggiolesi, F. (2016). On defining the notion of complete and immediate formal grounding. Synthese , 193, 3147–3167.CrossRefGoogle Scholar
Poggiolesi, F. (2018). On constructing a logic for the notion of complete and immediate formal grounding. Synthese , 195, 1231–1254.CrossRefGoogle Scholar
Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study . Stockholm: Almqvist & Wiksell.Google Scholar
Rumberg, A. (2013). Bolzano’s concept of grounding (Abfolge) against the background of normal proofs. Review of Symbolic Logic , 6(3), 424–459.CrossRefGoogle Scholar
Schnieder, B. (2011). A Logic for Because . Review of Symbolic Logic , 4(3), 445–465.CrossRefGoogle Scholar
Szabo, M. E.. The Collected Papers of Gerhard Gentzen . Amsterdam: North-Holland, 1969.Google Scholar
Tennant, N. (1990). Natural Logic , Edinburgh: Edinburgh University Press.Google Scholar
Troelstra, A. S. (1977). Aspects of constructive mathematics . In Studies in Logic and the Foundations of Mathematics , vol. 90, Amsterdam: Elsevier, pp. 973–1052.Google Scholar
van Benthem, J. (1985). The variety of consequence, according to Bolzano. Studia Logica , 44(4), 389–403.CrossRefGoogle Scholar