Injective Models and Disjunctive Relations
Injective Models and Disjunctive Relations
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DOI：
10.1093/logcom/3.3.231
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发表时间：
1993-06
期刊：
影响因子：
--
通讯作者：
M. Freund
中科院分区：
文献类型：
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作者：
M. Freund
在本文中，我们证明了在语言 £ 上定义的所有优先推理关系集合与该语言元素之间的传递关系族之间存在一对一映射的存在。给定优先推理关系，我们使用相应的传递关系来定义与 £ 相关的所有世界的集合之间的顺序。我们刻画了这个有序集可以表示的所有优先推理关系。作为一种特殊情况，我们研究析取关系族，即从前提析取得出的结论可以从至少一个单独采用的前提中得出的关系。我们证明，对于这种类型的关系，关联模型是单射和过滤的：如果两个世界 m 和 n 满足命题 α 并且对于该属性不是最小的，则存在一个小于它们的世界 p 满足 α。相反，当我们证明任何过滤模型都定义析取推理关系时，我们获得了这些关系的表示定理。最后将这些结果应用于有理关系族，我们通过排序单射模型得到了这些关系表示的新证明。
In this paper, we prove the existence of a one-to-one mapping between the set of all preferential inference relations defined on a language £ and a family of transitive relations among the elements of this language. Given a preferential inference relation, we use the corresponding transitive relation to define an order among the set of all worlds associated to £. We characterize all the preferential inference relations that can be represented by this ordered set. As a particular case, we study the family of disjunctive relations, that is relations where a conclusion drawn from a disjunction of premisses can be drawn from one at least of those premisses taken alone. We show that for this type of relation, the associated model is injective and filtered: if two worlds m and n satisfy a proposition α and are not minimal for that property, then there exists a world p less than both of them that satisfies α. As we prove, conversely, that any filtered model defines a disjunctive inference relation, we obtain a representation theorem for these relations. Applying finally these results to the family of rational relations, we get a new proof of the representation of these relations by means of ranked injective models.