Geodesic
Harmony and Normalisation in Bilateral Logic
Bulletin of the Section of Logic, Tome 52 (2023) no. 3, pp. 377-409.

Voir la notice de l'article provenant de la source Library of Science

In a recent paper del Valle-Inclan and Schlöder argue that bilateral calculi call for their own notion of proof-theoretic harmony, distinct from the usual (or ‘unilateral’) ones. They then put forward a specifically bilateral criterion of harmony, and present a harmonious bilateral calculus for classical logic. In this paper, I show how del Valle-Inclan and Schlöder’s criterion of harmony suggests a notion of normal form for bilateral systems, and prove normalisation for two (harmonious) bilateral calculi for classical logic, HB1 and HB2. The resulting normal derivations have the usual desirable features, like the separation and subformula properties. HB1-normal form turns out to be strictly stronger that the notion of normal form proposed by Nils Kürbis, and HB2-normal form is neither stronger nor weaker than a similar proposal by Marcello D’Agostino, Dov Gabbay, and Sanjay Modgyl.
Keywords: bilateralism, normalisation, harmony 
@article{BSL_2023_52_3_a3,
     author = {del Valle-Inclan, Pedro},
     title = {Harmony and {Normalisation} in {Bilateral} {Logic}},
     journal = {Bulletin of the Section of Logic},
     pages = {377--409},
     publisher = {mathdoc},
     volume = {52},
     number = {3},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BSL_2023_52_3_a3/}
}
TY  - JOUR
AU  - del Valle-Inclan, Pedro
TI  - Harmony and Normalisation in Bilateral Logic
JO  - Bulletin of the Section of Logic
PY  - 2023
SP  - 377
EP  - 409
VL  - 52
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BSL_2023_52_3_a3/
LA  - en
ID  - BSL_2023_52_3_a3
ER  - 
%0 Journal Article
%A del Valle-Inclan, Pedro
%T Harmony and Normalisation in Bilateral Logic
%J Bulletin of the Section of Logic
%D 2023
%P 377-409
%V 52
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BSL_2023_52_3_a3/
%G en
%F BSL_2023_52_3_a3
del Valle-Inclan, Pedro. Harmony and Normalisation in Bilateral Logic. Bulletin of the Section of Logic, Tome 52 (2023) no. 3, pp. 377-409. http://geodesic.mathdoc.fr/item/BSL_2023_52_3_a3/

[1] M. D’Agostino, D. Gabbay, S. Modgil, Normality, Non-Contamination and Logical Depth in Classical Natural Deduction, Studia Logica, vol. 108(2) (2020), pp. 291–357 | DOI

[2] P. del Valle-Inclan, J. J. Schlöder, Coordination and Harmony in Bilateral Logic, Mind, (2022) | DOI

[3] M. Dummett, The Logical Basis of Metaphysics, Harvard University Press (1991).

[4] F. Ferreira, The Co-Ordination Principles: A Problem for Bilateralism, Mind, vol. 117(468) (2008), pp. 1051–1057 | DOI

[5] N. Kürbis, Some Comments on Ian Rumfitt’s Bilateralism, Journal of Philosophical Logic, vol. 45(6) (2016), pp. 623–644 | DOI

[6] N. Kürbis, Normalisation for Bilateral Classical Logic with Some Philosophical Remarks, Journal of Applied Logics, vol. 2(8) (2021), pp. 531–556.

[7] D. Prawitz, Natural Deduction: A Proof-Theoretical Study, Stockholm, Sweden: Dover Publications (1965).

[8] A. N. Prior, The Runabout Inference-Ticket, Analysis, vol. 21(2) (1960), p. 38 | DOI

[9] I. Rumfitt, Yes and No, Mind, vol. 109(436) (2000), pp. 781–823 | DOI

[10] F. Steinberger, On the Equivalence Conjecture for Proof-Theoretic Harmony, Notre Dame Journal of Formal Logic, vol. 54(1) (2013), pp. 79–86 | DOI

[11] N. Tennant, Negation, Absurdity and Contrariety, [in:] D. M. Gabbay, H. Wansing (eds.), What is Negation?, Springer Netherlands, Dordrecht (1999), pp. 199–222 | DOI