Abstract
In this paper, we consider multiplicative-additive fragments of affine propositional classical linear logic extended with n-contraction. To be specific, n-contraction (n ⩾ 2) is a version of the contraction rule where (n+ 1) occurrences of a formula may be contracted to n occurrences. We show that expansions of the linear models for (n + 1)- valued Łukasiewicz logic are models for the multiplicative-additive classical linear logic, its affine version and their extensions with n-contraction. We prove the finite axiomatizability for the classes of finite models, as well as for the class of infinite linear models based on the set of rational numbers in the interval [0, 1]. The axiomatizations obtained in a Gentzen-style formulation are equivalent to finite and infinite-valued Łukasiewicz logics.
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Prijatelj, A. Bounded contraction and Gentzen-style formulation of Łukasiewicz logics. Stud Logica 57, 437–456 (1996). https://doi.org/10.1007/BF00370844
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DOI: https://doi.org/10.1007/BF00370844