Every Berman’s variety $$\mathbb {K}_p^q$$ K p q which is the subvariety of Ockham algebras defined by the equation $${\sim ^{2p+q}}a = {\sim ^q}a$$ ∼ 2 p + q a = ∼ q a ( $$p\ge 1$$ p ≥ 1 and $$q\ge 0$$ q ≥ 0 ) determines a finitary substitution invariant consequence relation $$\vdash _p^q$$ ⊢ p q . A sequent system $$\mathsf {S}_p^q$$ S p q is introduced as an axiomatization of the consequence relation $$\vdash _p^q$$ ⊢ p q . The system $$\mathsf {S}_p^q$$ S p q is characterized by a single finite frame $$\mathfrak {F}_p^q$$ F p q under the frame semantics given for the formal language. By the duality between frames and algebras, $$\mathsf {S}_p^q$$ S p q can be viewed as a $$4^{2p+q}$$ 4 2 p + q -valued logic as it is characterized by a distributive lattice of $$4^{2p+q}$$ 4 2 p + q elements with a unary operator. Moreover, a structural-rule-free, cut-free and terminating sequent system $$\mathsf {G}_p^q$$ G p q is established for $$\vdash _p^q$$ ⊢ p q . The Craig interpolation property of $$\vdash _p^q$$ ⊢ p q is shown proof-theoretically utilizing $$\mathsf {G}_p^q$$ G p q .

中文翻译：

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Belnap-Dunn Logic 的许多弱点

每个伯曼变体 $$\mathbb {K}_p^q$$ K pq 是由方程 $${\sim ^{2p+q}}a = {\sim ^q}a$ 定义的奥卡姆代数的子变体$ ∼ 2 p + qa = ∼ qa ( $$p\ge 1$$ p ≥ 1 and $$q\ge 0$$ q ≥ 0 ) 确定一个有限代换不变结果关系 $$\vdash _p^q$$ ⊢ pq 。一个后续系统 $$\mathsf {S}_p^q$$ S pq 被引入作为推论关系 $$\vdash _p^q$$ ⊢ pq 的公理化。系统 $$\mathsf {S}_p^q$$ S pq 的特征在于在为形式语言给出的框架语义下的单个有限框架 $$\mathfrak {F}_p^q$$ F pq。通过框架和代数之间的对偶性，$$\mathsf {S}_p^q$$ S pq 可以被视为 $$4^{2p+q}$$ 4 2 p + q 值逻辑，因为它的特点是$$4^{2p+q}$$ 4 2 p + q 个元素的分配格，带有一元运算符。而且，为 $$\vdash _p^q$$ ⊢ pq 建立了无结构规则、无切割和终止的序列系统 $$\mathsf {G}_p^q$$ G pq 。$$\vdash _p^q$$ ⊢ pq 的 Craig 插值属性被证明是使用 $$\mathsf {G}_p^q$$ G pq 证明的。