A Syntactic Approach to Maksimova's Principle of Variable Separation for Some Substructural Logics

Notre Dame Journal of Formal Logic 39 (1):94-113 (1998)
Maksimova's principle of variable separation says that if propositional formulas $A_1 \supset A_2$ and $B_1 \supset B_2$ have no propositional variables in common and if a formula $A_1\wedge B_1 \supset A_2\vee B_2$ is provable, then either $A_1 \supset A_2$ or $B_1 \supset B_2$ is provable. Results on Maksimova's principle until now are obtained mostly by using semantical arguments. In the present paper, a proof-theoretic approach to this principle in some substructural logics is given, which analyzes a given cut-free proof of the formula $A_1\wedge B_1 \supset A_2\vee B_2$ and examines how the formula is derived. This analysis will make clear why Maksimova's principle holds for these logics
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