Degrees of inconsistency. Carefully combining classical and paraconsistent negation
Degrees of inconsistency. Carefully combining classical and paraconsistent negation
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不一致的程度。
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发表时间：
2013
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通讯作者：
P. Verdée
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作者：
P. Verdée
本文致力于将经典逻辑的否定与格雷厄姆·普里斯特的LP的否定以一种忠实于组合逻辑的中心性质的方式结合起来。我们给出了一个逻辑L，它结合了两个否定的一些必要条件。这些必要条件包括：（a）L应该是真值泛函，（B）L对于次协调否定应该是严格非爆炸性的（即如果A和A都有一个非平凡的后果集，那么这对于包含两者的集合也应该是这种情况）和（c）L应该是CL和LP的保守扩张。这种渴望是由一种特殊的财产理论视角对次协调的动机。接下来，我们设计逻辑CLP。我们提出了一个公理化的逻辑和三个语义特征（非确定性语义，一个无限值集理论语义和一个无限值语义与整数值）。我们证明，CLP是唯一的逻辑满足所有假设的desiderata。CLP的无限值语义可以被看作是产生了一种解释，在这种解释中，不一致和不一致的属性是不同程度的：不是每个涉及不一致的句子都是同样不一致的。
This paper is devoted to combining the negation of Classical Logic (CL) and the negation of Graham Priest’s LP in a way that is faithful to central properties of the combined logics. We give a number of desiderata for a logic L which combines both negations. These desiderata include the following: (a) L should be truth functional, (b) L should be strictly non-explosive for the paraconsisent negation ∼ (i.e. if A and ∼A both have a non-trivial set of consequences, then this should also be the case for the set containing both) and (c) L should be a conservative extension of CL and of LP. The desiderata are motivated by a particular property-theoretic perspective on paraconsistency. Next we devise the logic CLP. We present an axiomatization of this logic and three semantical characterizations (a non-deterministic semantics, an infinitely valued set-theoretic semantics and an infinitely valued semantics with integer numbers as values). We prove that CLP is the only logic satisfying all postulated desiderata. The infinitely valued semantics of CLP can be seen as giving rise to an interpretation in which inconsistencies and inconsistent properties come in degrees: not every sentence which involves inconsistencies is equally inconsistent.