Review of Symbolic Logic 6 (3):394-423 (2013)

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics. Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.
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DOI 10.1017/S1755020313000099
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References found in this work BETA

Entailment: The Logic of Relevance and Necessity.[author unknown] - 1975 - Studia Logica 54 (2):261-266.
Substructural Fuzzy Logics.George Metcalfe & Franco Montagna - 2007 - Journal of Symbolic Logic 72 (3):834 - 864.

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Implicational Logics III: Completeness Properties.Petr Cintula & Carles Noguera - 2018 - Archive for Mathematical Logic 57 (3-4):391-420.
Axiomatization of Non-Associative Generalisations of Hájek's BL and psBL.Yaroslav Petrukhin - 2020 - Journal of Applied Non-Classical Logics 30 (1):1-15.

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