Structural rules for multi-valued logics Nissim Francez and Michael Kaminski Abstract. We study structural rules in the context of multi-valued log- ics with finitely-many truth-values. We first extend Gentzen’s traditional structural rules to a multi-valued logic context; in addition, we propos some novel structural rules, fitting only multi-valued logics. Then, we propose a novel definition, namely, structural rules completeness of a collection of structural rules, requiring derivability of the restriction of consequence to atomic formulas by structural rules only. The restric- tion to atomic formulas relieves the need to concern logical rules in the derivation. Mathematics Subject Classification (2010). Primary 03F03 Secondary 03B22, 03A02. Keywords. structural rules, multi-valued logics, structural rules com- pleteness. 1. Introduction Usually, under the term structural rules, one refers to the following rules, devised by Gentzen for incorporation in his sequent calculi LJ/LK for intu- itionistic/classical logics (see [3]). Γ 1 ,ψ,ϕ, Γ 2 →χ Γ 1 ,ϕ,ψ, Γ 2 →χ (PL) Γ→χ Γ,ϕ→χ (WL) Γ,ϕ,ϕ→χ Γ,ϕ→χ (CL) (1.1) Here Γ is a meta-variable over sequences of object language formulas, and ϕ,ψ and χ are meta-variables ranging over object language formulas. In the case of multi-conclusions sequents Γ→Δ, there are also analogous rules (PR), (WR), (CR) for modifying Δ, the r.h.s. of a sequent. A talk based on this paper was presented at the second Substructural logics: semantics, proof theory, and applications (SYSMICS) conference, Vienna, February 2018.