Introduction: Paraconsistent logics
Introduction: Paraconsistent logics
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简介：次一致逻辑
DOI：
10.1007/bf00935736
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发表时间：
1984
期刊：
影响因子：
0.7
通讯作者：
R. Routley
中科院分区：
文献类型：
--
作者：
G. Priest;R. Routley
设Ih是一个逻辑推论关系。Ih可以被定义为语义（27 lh A保持当且仅当对于某个指定的赋值集，只要E中的所有公式在一个赋值下都为真，那么A也是）或证明理论（27 lh A保持当且仅当对于某个指定的赋值集，只要E中的所有公式在一个赋值下都为真，那么A也是）。cally（E Ih A成立当且仅当对于某个特定的规则集，存在A的导出，其所有（未解除）前提都在E中），或者以其他方式。lh是爆炸性的当且仅当对所有A和B {A，~ A} ih B.当它不是爆炸性的时，它是次协调的。一个逻辑是次协调的当且仅当它的逻辑结果关系为0。E是不相容的当且仅当，对某个A，{A，~ A} c E. E是平凡的当且仅当对所有B，B ∈ E.关于爸爸的重要事实？一致逻辑的一个重要特征是，它们为不一致但非平凡的理论提供了基础。换句话说，存在逻辑推论下封闭的句子集合，它们是不一致的，但不是平凡的。这一事实有时被认为是“次协调”的另一种定义，并且由于逻辑结果是传递的，它等价于我们的定义。[1]因此，我们称不一致但非平凡的理论为次协调的。等价性表明了次协调逻辑值得研究的一个原因。因为有一些重要的不一致的理论并不是微不足道的。因此，对它们的逻辑结构的任何分析都必须使用次协调逻辑。显然，采用弗雷格/尤塞尔这样的爆炸性逻辑或直觉主义逻辑会使它们变得微不足道。
Let Ih be a relation of logical consequence. Ih may be defined either semantically (27 lh A holds iff for some specified set of valuations, wherever all the formulas in E are true under an evaluation, so is A) or proof theoreti? cally (E Ih A holds iff for some specified set of rules, there is a derivation of A, all of whose (undischarged) premises are in E), or in some other way. lh is explosive iff for all A and B {A,~ A} ih B. It is paraconsistent iff it is not explosive. A logic is paraconsistent iff its logical consequence relation is0.Let E be a set of sentences. E is inconsistent iff, for some A,{A,~ A} c E. E is trivial iff for all B, B e E. The important fact about pa? aconsis tent logics is that they provide the basis for inconsistent but non-trivial theories. In other words, there are sets of sentences closed under logical consequence which are inconsistent but non-trivial. This fact is sometimes taken as an alternative definition of'paraconsistent'and, given that logical consequence is transitive, it is equivalent to our definition. 1 For this reason we call inconsistent but non-trivial theories paraconsistent The equivalence indicates one reason why paraconsistent logics are worthy of study. For there are important inconsistent theories which are not trivial. Any analysis of their logical structure must therefore be done using a paraconsistent logic. Clearly, to adopt an explosive logic such as Frege/Eussell or intuitionist logic would trivialise them.