超模态逻辑K[T_nK]和K[TK_n](英文)

本文的工作是在D.M.Gabbay的一篇论文《超模态逻辑理论:在模态逻辑中的模转换》基础上所做的,主要是将他的两类满足关系扩充到n+1种满足关系,然后在此基础上得到两类一般性的逻辑类KT_nK]和KTK_n],其中n≥1。我们得到了一些更为一般性的结论:（1）逻辑类KT_nK]的定理模式是:对任意n≥1,□^j+（n+1）kp→□^i+（n+1）kp,其中0≤in]的定理模式是:□^1+（n+1）kp→□^（n+1）kp,其中b≥1。不过,□^j+（n+1）kp→□^i+（n+1）kp,其中0≤in]的定理模式,因此,（3）每一个逻辑KT_nK]都是相应的逻辑KTK_n]的真扩张,其中n≥1;（4）必然化规则在两类逻辑KT_nK]和KTK_n]中都不成立,但是,这样的规则成立,即如果A分别是两类逻辑KT_nK]和KTK_n]的定理,那么对于任意n≥1,□ⁿ⁺¹A也分别是逻辑类KT_nK]和KTK_n]的定理;（5）等值替换规则在逻辑类KT_nK]和KTK_n]下都不封闭;此外,（6）我们将D.M.Gabbay的从超模态逻辑到正规模态逻辑K的两类翻译τ₀和τ₁扩充到n+1类翻译τ₀,τ₁,…,τ_n。在超模态逻辑KT_nK]和KTK_n]与正规模态逻辑K之间,我们找到了点模型满足对应理论,即对任意的超模态逻辑公式α,在某个世界ω上为真,当且仅当,在正规模态逻辑K中τ_i（α）在世界ω上也为真。其中τ_i（α）是公式α从超模态逻辑到正规模态逻辑K的翻译。

Hypermodal Logics K[TnK]and K[TKn]

The work of this paper is based on one paper of D. M. Gabbay, which is ＇A theory of hypermodal logics： Mode shifting in modal logic＇. Specially, based on his two satisfaction relations, I define n ＋ 1 satisfaction relations, and then gain two kinds of more general logics KTnK]and KTKn], whereas n 〉 1. There we will get more general results： （1） For any n ≥ 1,口j＋（n＋l）kp→ 口i＋（n＋l）kp are theorems of KTnK], whereas 0 ≤i 〈 j ≤ n. （2） For any n ≥1, every logic of ICTKn] has theorems 口j＋（n＋l）kp→ 口i＋（n＋l）kp, whereas n 〉 1. However, 口j＋（n＋l）kp→ 口i＋（n＋l）kp are not theorems of ICTK@ whereas 0 ≤i 〈 j ≤ n., and j ≠ 1, or i ≠0. So, （3） every logic of 1CTnK] is a proper extension of logic KTnK], for each n 〉 1. （4） The rule of Necessitation is not available to logics of ICTnK] and K.TKn], well, for any n ≥1, Yqn＋lA is a theorem of KTnK]and KTKn], whereas A is a theorem of ICKTnK]and KTKn], respectively. （5） Logics of KTnK]and KTKn] are not closed under substitution of provably equivalent formulas. Besides, based on his two translations T0, T1 from hypermodal logic KTK] to standard K, we define n ＋ 1 translations TO, T1.....Tn from hypermodal logics KTnK] and KTKn] to standard K, respectively. Then we get a correspondence of pointed model satisfaction with logics ICTnK], 1CTKn] and standard K, respectively, i.e. a hypermodal formula is true in a world by some satisfaction relation iff the corresponding translation of the hypermodal formula is true in that world by the standard K satisfaction relation.