论文标题
直觉强的Löb逻辑的证明理论
Proof Theory for Intuitionistic Strong Löb Logic
论文作者
论文摘要
本文介绍了直觉强的löb逻辑$ {\ sf isl} _ \ box $的两个序列计算，一个基于终止的calculus $ {\ sf g4ip} $ copipigrigitigant in Intuitionistion $ copipimit and for Intuitionistiantif Intuitionistiantif Intuitionistionist $ aigitigrigantigrigitigantim offimitigantimitigantional for Intuitionistionistionistionist $ aigitigantimitigitigantialistiantialist $ { $ {\ sf g3isl} _ \ box $的标准无剪切序列微积分$ {\ sf g3ip} $，没有$ {\ sf ipc} $的结构规则。主要结果之一是$ {\ sf g3isl} _ \ box $的切割定理的句法证明。此外，建立了$ {\ sf isl} _ \ box $的顺序计算和希尔伯特系统之间的等价。从文献中知道$ {\ sf isl} _ \ box $相对于直觉的模态kripke模型，其中模态关系是及时的，相反，有充分的基础和直觉关系的子集。在这里，通过使用基于$ {\ sf g4isl} _ \ box $变体的反模型构造获得建设性证明。因此，该论文包含两个切割的证据，一种语义和句法证明。
This paper introduces two sequent calculi for intuitionistic strong Löb logic ${\sf iSL}_\Box$: a terminating sequent calculus ${\sf G4iSL}_\Box$ based on the terminating sequent calculus ${\sf G4ip}$ for intuitionistic propositional logic ${\sf IPC}$ and an extension ${\sf G3iSL}_\Box$ of the standard cut-free sequent calculus ${\sf G3ip}$ without structural rules for ${\sf IPC}$. One of the main results is a syntactic proof of the cut-elimination theorem for ${\sf G3iSL}_\Box$. In addition, equivalences between the sequent calculi and Hilbert systems for ${\sf iSL}_\Box$ are established. It is known from the literature that ${\sf iSL}_\Box$ is complete with respect to the class of intuitionistic modal Kripke models in which the modal relation is transitive, conversely well-founded and a subset of the intuitionistic relation. Here a constructive proof of this fact is obtained by using a countermodel construction based on a variant of ${\sf G4iSL}_\Box$. The paper thus contains two proofs of cut-elimination, a semantic and a syntactic proof.
