Generalizing the depth relevance condition: deep relevant logics not included in R-mingle. (English) Zbl 1408.03017

Robles, Gemma; Méndez, José M.

Notre Dame J. Formal Logic 55, No. 1, 107-127 (2014).

Summary: Brady has shown how to define a class of deep relevant logics from Meyer’s crystal lattice CL. The aim of this paper is to generalize Brady’s result by showing how to define a class of deep relevant logics from each weak relevant matrix (weak relevant matrices only verify logics with the variable-sharing property). A class of deep relevant logics not included in R-Mingle is defined.

MSC:

03B47	Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03F52	Proof-theoretic aspects of linear logic and other substructural logics

Keywords:

deep relevant logics; relevant logics; weak relevant matrices; logical matrices

References:

[1]	Ackermann, W., “ Begründung einer strengen Implikation,” Journal of Symbolic Logic , vol. 21 (1956), pp. 113-28. · Zbl 0072.00106 · doi:10.2307/2268750
[2]	Anderson, A. R., and N. D. Belnap, Jr., Entailment, Vol. 1: The Logic of Relevance and Necessity , Princeton University Press, Princeton, 1975.
[3]	Belnap, N. D., Jr., “Entailment and relevance,” Journal of Symbolic Logic , vol. 25 (1960), pp. 144-46. · Zbl 0107.00802 · doi:10.2307/2964210
[4]	Brady, R. T., “Completeness proofs for the systems \(\mathrm{RM}3\) and \(\mathrm{BN}4\),” Logique et Analyse (New Series) , vol. 25 (1982), pp. 9-32. · Zbl 0498.03012
[5]	Brady, R. T., “Depth relevance of some paraconsistent logics,” Studia Logica , vol. 43 (1984), pp. 63-73. · Zbl 0581.03014 · doi:10.1007/BF00935740
[6]	Brady, R. T., “Hierarchical semantics for relevant logics,” Journal of Philosophical Logic , vol. 21 (1992), pp. 357-74. · Zbl 0770.03008 · doi:10.1007/BF00260741
[7]	Brady, R. T., “Relevant implication and the case for a weaker logic,” Journal of Philosophical Logic , vol. 25 (1996), pp. 151-83. · Zbl 0873.03026 · doi:10.1007/BF00247002
[8]	Brady, R. T., ed., Relevant Logics and Their Rivals, Vol. II , Ashgate Publishing, Aldershot, England, 2003. · Zbl 1044.03014
[9]	Brady, R. T., Universal Logic , vol. 109 of CSLI Lecture Notes , CSLI Publications, Stanford, Calif., 2006.
[10]	Méndez, J. M., G. Robles, and F. Salto, “Depth relevance, containment semantics, and the factor axioms,” in preparation.
[11]	Meyer, R. K., and R. Routley, “Algebraic analysis of entailment, I,” Logique et Analyse (New Series) , vol. 15 (1972), pp. 407-28. · Zbl 0336.02020
[12]	Robles, G., “Negaciones subintuicionistas para lógicas con la conversa de la propiedad Ackermann” (“Subintuitionistic negations for logics with the converse Ackermann property”), Ph.D. dissertation, Ediciones Universidad de Salamanca, Colección Vitor 179, Salamanca, Spain, 2006.
[13]	Robles, G., and J. M. Méndez, “A general characterization of the variable-sharing property by means of logical matrices,” Notre Dame Journal of Formal Logic , vol. 53 (2012), pp. 223-44. · Zbl 1254.03041
[14]	Routley, R., and R. K. Meyer, “The semantics of entailment, III,” Journal of Philosophical Logic , vol. 1(1972), pp. 192-208. · Zbl 0317.02019 · doi:10.1007/BF00650498
[15]	Routley, R., V. Plumwood, R. K. Meyer, and R. T. Brady, Relevant Logics and Their Rivals, Part I : The Basic Philosophical and Semantical Theory , Ridgeview Publishing, Atascadero, Calif., 1982.