D-complete Single Axioms for the Equivalential Calculus with the rules D and R

Marcin Czakon

doi:10.18778/0138-0680.2024.15

Authors

Marcin Czakon John Paul II Catholic University of Lublin, Department of Logic, Poland Author https://orcid.org/0000-0001-6217-6640

DOI:

https://doi.org/10.18778/0138-0680.2024.15

Keywords:

equivalential calculus, D-complete, single axiom, condensed detachment

Abstract

Ulrich showed that most of the known axiomatisations of the classical equivalence calculus (EC) are D-incomplete, that is, they are not complete with the condensed detachment rule (D) as the primary rule of the proof procedure. He proved that the axiomatisation EEpEqrErEqp, EEEpppp by Wajsberg is D-complete and pointed out a number of D-complete single axioms, including one organic single axiom. In this paper we present new single axioms for EC with the condensed detachment and the reversed condensed detachment rules that form D-complete bases and are organic.

References

M. Alizadeh, N. Joharizadeh, Counting weak Heyting algebras on finite distributive lattices, Logic Journal of the IGPL, vol. 23(2) (2015), pp. 247–258, DOI: https://doi.org/10.1093/jigpal/jzu033 DOI: https://doi.org/10.1093/jigpal/jzu033

M. Ardeshir, W. Ruitenburg, Basic propositional calculus I, Mathematical Logic Quarterly, vol. 44(3) (1998), pp. 317–343, DOI: https://doi.org/10.1002/malq.19980440304 DOI: https://doi.org/10.1002/malq.19980440304

G. Birkhoff, Lattice theory, vol. 25, American Mathematical Soc. (1940). DOI: https://doi.org/10.1090/coll/025

S. Celani, R. Jansana, Bounded distributive lattices with strict implication, Mathematical Logic Quarterly, vol. 51(3) (2005), pp. 219–246, DOI: https://doi.org/10.1002/malq.200410022 DOI: https://doi.org/10.1002/malq.200410022

I. Chajda, Weakly regular lattices, Mathematica Slovaca, vol. 35(4) (1985), pp. 387–391.

I. Chajda, Congruence kernels in weakly regular varieties, Southeast Asian Bulletin of Mathematics, vol. 24 (2000), pp. 15–18, DOI: https://doi.org/10.1007/s10012-000-0015-8 DOI: https://doi.org/10.1007/s100120070022

P. Dehornoy, Braids and self-distributivity, vol. 192, Birkhäuser (2012), DOI: https://doi.org/10.1007/978-3-0348-8442-6 DOI: https://doi.org/10.1007/978-3-0348-8442-6

A. Diego, Sur les algebras de Hilbert, Ed. Herman, Collection de Logique Mathématique. Serie A, vol. 21 (1966).

G. Epstein, A. Horn, Logics which are characterized by subresiduated lattices, Mathematical Logic Quarterly, vol. 22(1) (1976), pp. 199–210, DOI: https://doi.org/10.1002/malq.19760220128 DOI: https://doi.org/10.1002/malq.19760220128

S. Ghorbani, MULTIPLIERS IN WEAK HEYTING ALGEBRAS, Journal of Mahani Mathematics Research, vol. 13(3) (2024), pp. 33–46, DOI: https://doi.org/10.22103/jmmr.2024.22758.1563

D. Joyce, A classifying invariant of knots, the knot quandle, Journal of Pure and Applied Algebra, vol. 23(1) (1982), pp. 37–65, DOI: https://doi.org/10.1016/0022-4049(82)90077-9 DOI: https://doi.org/10.1016/0022-4049(82)90077-9

H. Junji, Congruence relations and congruence classes in lattices, Osaka Mathematical Journal, vol. 15(1) (1963), pp. 71–86.

M. Nourany, S. Ghorbani, A. B. Saeid, On self-distributive weak Heyting algebras, Mathematical Logic Quarterly, vol. 69(2) (2023), pp. 192–206, DOI: https://doi.org/10.1002/malq.202200073 DOI: https://doi.org/10.1002/malq.202200073

H. J. San Martín, Compatible operations on commutative weak residuated lattices, Algebra universalis, vol. 73 (2015), pp. 143–155, DOI: https://doi.org/10.1007/s00012-015-0317-4 DOI: https://doi.org/10.1007/s00012-015-0317-4

H. J. San Martín, Principal congruences in weak Heyting algebras, Algebra universalis, vol. 75 (2016), pp. 405–418, DOI: https://doi.org/10.1007/s00012-016-0381-4 DOI: https://doi.org/10.1007/s00012-016-0381-4

H. J. San Martín, On congruences in weak implicative semi-lattices, Soft Computing, vol. 21 (2017), pp. 3167–3176, DOI: https://doi.org/10.1007/s00500-016-2188-9 DOI: https://doi.org/10.1007/s00500-016-2188-9

A. Visser, A propositional logic with explicit fixed points, Studia Logica, (1981), pp. 155–175, DOI: https://doi.org/10.1007/BF01874706 DOI: https://doi.org/10.1007/BF01874706

J. R. Hindley, BCK and BCI logics, condensed detachment and the 2-property, Notre Dame Journal of Formal Logic, vol. 34(2) (1993), pp. 231–250, DOI: https://doi.org/10.1305/ndjfl/1093634655 DOI: https://doi.org/10.1305/ndjfl/1093634655

J. R. Hindley, D. Meredith, Principal Type-Schemes and Condensed Detachment, The Journal of Symbolic Logic, vol. 55(1) (1990), pp. 90–105, URL: http://www.jstor.org/stable/2274956 DOI: https://doi.org/10.2307/2274956

K. Hodgson, Shortest Single Axioms for the Equivalential Calculus with CD and RCD, Journal of Automated Reasoning, (20) (1998), p. 283–316, DOI: https://doi.org/10.1023/A:1005731217123 DOI: https://doi.org/10.1023/A:1005731217123

J. A. Kalman, Condensed Detachment as a Rule of Inference, Studia Logica, vol. 42(4) (1983), pp. 443–451, DOI: https://doi.org/10.1007/bf01371632 DOI: https://doi.org/10.1007/BF01371632

S. Leśniewski, Grundzüge eines neuen Systems der Grundlagen der Mathematik, Fundamenta Mathematicae, vol. 14(1) (1929), pp. 1–81. DOI: https://doi.org/10.4064/fm-14-1-1-81

C. A. Meredith, A. N. Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, vol. 4(3) (1963), pp. 171–187, DOI: https://doi.org/10.1305/ndjfl/1093957574 DOI: https://doi.org/10.1305/ndjfl/1093957574

J. G. Peterson, Shortest single axioms for the classical equivalential calculus, Notre Dame Journal of Formal Logic, vol. 17(2) (1976), pp. 267–271, DOI: https://doi.org/10.1305/ndjfl/1093887534 DOI: https://doi.org/10.1305/ndjfl/1093887534

J. A. Robinson, A Machine-Oriented Logic Based on the Resolution Principle, Journal of the ACM, vol. 12(1) (1965), pp. 23–41, DOI: https://doi.org/10.1145/321250.321253 DOI: https://doi.org/10.1145/321250.321253

D. Ulrich, D-complete axioms for the classical equivalential calculus, Bulletin of the Section of Logic, vol. 34 (2005), pp. 135–142.

M. Wajsberg, Metalogische Beiträge, Wiadomości Matematyczne, vol. 43 (1937), pp. 131–168.

L. Wos, D. Ulrich, B. Fitelson, XCB, The last of the shortest single axioms for the classical equivalential calculus, Bulletin of the Section of Logic, vol. 3(32) (2003), pp. 131–136.

J. Łukasiewicz, Równoważnościowy rachunek zdań, [in:] J. Łukasiewicz (1961) (ed.), Z zagadnień Logiki i Filozofii, Państwowe Wydawnictwo Naukowe, Warszawa (1939), pp. 234–235.