塔斯基经典相关逻辑
Tarskian classical relevant logic
摘要：
塔斯基经典相关逻辑TR起源于塔斯基关于关系演算的基础和限制于多个变量的一阶逻辑的工作，由塔斯基和吉万特在他们的书《无变量的集合论的形式化》中提出，并在前九节中总结。TR与著名的逻辑KR密切相关。在塔斯基的二元关系扩展一阶逻辑中，关联逻辑的每一个公式都有一个对应的语句，关系符号上有运算符。公式在TR中（根据定义），或在KR中（根据定理），当且仅当其相应的句子可以在一阶逻辑中证明，使用最多四个变量，假设所有二元关系都是稠密的，并且对于TR，复合可换，或者对于KR，是对称的。TR的词汇表与Meyer和Quilley提出的经典相关逻辑CR $^*$相同，但TR适当地包含CR $^*$。TR的特征帧是CR $^*$的特征帧并且满足额外帧条件。在TR中有一些公式（但在CR $^*$中没有）与这个框架条件相对应，并为T的定理提供了一个反例。科瓦尔斯基TR或KR的特征框架是其复代数分别是交换或对称的整数稠密关系代数的框架。对于这两个类，同构类型的数量像三元关系的同构类型的数量一样增长。这两个类的渐近公式。类似的结果也适用于由相应句子的一阶证明中使用的变量数量定义的逻辑层次。
英文摘要：
The Tarskian classical relevant logic TR arises from Tarski's work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of Set Theory without Variables, and summarized in first nine sections. TR is closely related to the well-known logic KR. Every formula of relevance logic has a corresponding sentence in Tarski's extended first-order logic of binary relations with operators on the relation symbols. A formula is in TR (by definition), or in KR (by a theorem), if and only if its corresponding sentence can be proved in first-order logic, using at most four variables, from the assumptions that all binary relations are dense and, for TR, commute under composition, or, for KR, are symmetric. The vocabulary of TR is the same as the classical relevant logic CR$^*$ proposed by Meyer and Routley but TR properly contains CR$^*$. The frames characteristic for TR are the ones that are characteristic for CR$^*$ and satisfy an extra frame condition. There are formulas in TR (but not in CR$^*$) that correspond to this frame condition and provide a counterexample to a theorem of T. Kowalski. The frames characteristic for TR, or KR, are the ones whose complex algebras are integral dense relation algebras that are commutative, or symmetric, respectively. For both classes, the number of isomorphism types grows like the number of isomorphism types of ternary relations. Asymptotic formulas are obtained for both classes. Similar results apply to a hierarchy of logics defined by the number of variables used in the first-order proofs of their corresponding sentences.
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