A note on Lukasiewicz’s three-valued logic
A note on Lukasiewicz’s three-valued logic
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关于卢卡谢维奇三值逻辑的注释
DOI：
10.13128/annali_dip_filos-1969
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
Pierluigi Minari
中科院分区：
文献类型：
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作者：
Pierluigi Minari
众所周知，Lukasiewicz的三值逻辑L3与经典逻辑不同，允许定义两个非平凡的真值泛函模态算子和一个非平凡的真值泛函模态算子。我们解决的问题，找到一个方便的句法表征的“模态内容”的L3。为此，我们考虑了Wajsberg的L3公理化（演算W），并证明了它与模态演算W_的等价性。模态演算W_本质上包括：BCK+双重否定模式，S5（K;T; 4;B）的特征模态模式，盒装公式的完全压缩模式和“部分塌陷”模式α →（α →α）。作为应用，我们得到了一个简单而自然的完整性证明一拉Lindenbaum的W，以及一个相当大的简化Wajsberg的原始，巧妙的完整性证明。
It is well known that Lukasiewicz’s three-valued-logic L3 admits – unlike classical logic – the definition of two non trivial, truth-functional modal operators and ∆. We address the question of finding a convenient syntactic characterization of the “modal content” of L3. To this aim, we consider Wajsberg’s axiomatization of L3 (the calculus W) and prove its equivalence with a modal calculus W_ which, essentially, includes: the BCK+double negation schemas, the characteristic modal schemas of S5 (K;T; 4;B), full contraction for boxed formulas and the “partial collapse” schema α → (α →α). As applications, we obtain a simple and natural completeness proof a la Lindenbaum for W, as well as a considerable simplification of Wajsberg’s original, ingenious completeness proof.