论文标题
单变量晶格可价值逻辑的代数语义
Algebraic semantics for one-variable lattice-valued logics
论文作者
论文摘要
任何一阶逻辑的单变量片段都可以视为模态逻辑，其中通用和存在的量词分别被盒子和钻石模态代替。在某些情况下，已经获得了这些逻辑代数语义的公理化：最值得注意的是，对于模态对应物S5和MIPC，分别是一阶古典逻辑和直觉逻辑的单变量片段。但是，在一阶中间逻辑的设置之外，缺乏一般方法。本文为在一阶晶格值逻辑的设置中提供了这种方法的基础，其中公式在具有晶格还原的代数结构中被解释。特别地，通过概括Bezhanishvili的功能表示定理和Monadic Heyting代数，获得了这些逻辑的单变量片段的模态对应物获得的公理。还为一阶次级结构逻辑的单变量片段提供了另一种证明理论证明，该片段具有无剪切的序列微积分，并接受了一定有限的插值属性。
The one-variable fragment of any first-order logic may be considered as a modal logic, where the universal and existential quantifiers are replaced by a box and diamond modality, respectively. In several cases, axiomatizations of algebraic semantics for these logics have been obtained: most notably, for the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic, respectively. Outside the setting of first-order intermediate logics, however, a general approach is lacking. This paper provides the basis for such an approach in the setting of first-order lattice-valued logics, where formulas are interpreted in algebraic structures with a lattice reduct. In particular, axiomatizations are obtained for modal counterparts of one-variable fragments of a broad family of these logics by generalizing a functional representation theorem of Bezhanishvili and Harding for monadic Heyting algebras. An alternative proof-theoretic proof is also provided for one-variable fragments of first-order substructural logics that have a cut-free sequent calculus and admit a certain bounded interpolation property.
