Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics
Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics
复制标题
超直觉逻辑和模态逻辑的代数、语义和逻辑属性的相互关系
DOI：
10.4064/-46-1-159-168
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
L. Maksimova
中科院分区：
文献类型：
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作者：
L. Maksimova
我们考虑了命题超直观逻辑(S.I.L.)的L家族。和NE(K)的正规模式逻辑(n.m.l)。众所周知，L与伪布尔代数(或Heyting代数)簇的格之间存在对偶，并且NE(K)与模代数簇的格对偶同构。逻辑的许多重要性质，如Craig插值性质(CIP)、析取性质(DP)、Beth性质(BP)、Hallden-完备性(HP)等都具有与其映象相适应的簇性质，并且许多自然代数性质与逻辑的自然性质是一致的。例如，S.I.L。L有CIP当且仅当它的伴随簇V(L)具有合并性(AP)；L是Hallden-完备的当且仅当V(L)是由次直不可约Heyting代数生成的。对于任何n.m.l。L，V(L)的合并性等价于L的插值性的一个较弱形式，超合并性等价于CIP；L是Hallden-完全的当且仅当V(L)满足强形式的联合嵌入性质。众所周知的直觉主义和模态逻辑的关系克里普克语义似乎是比代数逻辑更自然的解释。从某种意义上讲，Klipke框架范畴可以看作是Heyting簇或模代数簇的子范畴。我们讨论了在多大程度上可以将关于逻辑性质的问题归结为考虑其语义模型的问题。
We consider the families L of propositional superintuitionistic logics (s.i.l.) andNE(K) of normal modal logics (n.m.l.). It is well known that there is a duality between L and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also NE(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig’s interpolation property (CIP), the disjunction property (DP), the Beth property (BP), Hallden-completeness (HP) etc. have suitable properties of varieties as their images, and many natural algebraic properties are in accordance with natural properties of logics. For example, a s.i.l. L has CIP iff its associated variety V (L) has the amalgamation property (AP); L is Hallden-complete iff V (L) is generated by a subdirectly irreducible Heyting algebra. For any n.m.l. L, the amalgamation property of V (L) is equivalent to a weaker version of the interpolation property for L, and the superamalgamation property is equivalent to CIP; L is Hallden-complete iff V (L) satisfies a strong version of the joint embedding property. Well-known relational Kripke semantics for the intuitionistic and modal logics seems to be a more natural interpretation than the algebraic one. The categories of Kripke frames may, in a sense, be considered as subcategories of varieties of Heyting or modal algebras. We discuss the question to what extent one may reduce problems on properties of logics to consideration of their semantic models.