Some Useful 16-Valued Logics: How a Computer Network Should Think
Some Useful 16-Valued Logics: How a Computer Network Should Think
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一些有用的 16 值逻辑：计算机网络应该如何思考
DOI：
10.1007/s10992-005-0556-5
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发表时间：
2005
影响因子：
1.5
通讯作者：
H. Wansing
中科院分区：
文献类型：
--
作者：
Y. Shramko;H. Wansing
在Belnap的有用的4值逻辑中，经典真值的集合2={T，F}推广到集合4= p(2)={∅,{T},{F},{T,F}}。在本文中，我们赞成将这一过程扩展到16= p(4)（以及更远）。事实证明，这种推广是有良好动机的，并从具有信息和真假排序的双格FOUR2引出另一个代数结构，即具有信息排序、真排序和（不同）假排序的三格SIXTEEN3。有趣的是，在SIXTEEN3的真阶和假阶代数运算分别生成的逻辑与FOUR2的逻辑一致，即一次蕴涵。这一观察结果可作为一级蕴涵意义的进一步说明。然而，在目前的情况下，将真序和假序的逻辑词汇结合起来而获得的语言中的逻辑系统也就变得相当自然了。我们在语义上定义了扩展语言中这两种顺序的逻辑，并在这两种情况下公理化了一个由三个一元操作组成的片段：否定、对合和它们的组合。我们还提出了完整语言中逻辑的另外两种定义，包括双结果系统。换句话说，除了将一级蕴涵表示为有用的16值逻辑之外，我们还定义了进一步的有用的16值逻辑，用于推理真理和（非）假性。我们期望这些逻辑在信息处理中成为一种有趣而有用的工具，特别是当我们处理由分层互连的计算机组成的网络时。我们还简要讨论了Arieli和Avron关于逻辑双边格的概念，并指出了一些有待未来研究的开放性问题。
In Belnap’s useful 4-valued logic, the set 2={T,F} of classical truth values is generalized to the set 4=℘(2)={∅,{T},{F},{T,F}}. In the present paper, we argue in favor of extending this process to the set 16=℘(4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN3 coincide with the logic of FOUR2, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli’s and Avron’s notion of a logical bilattice and state a number of open problems for future research.