da Costa Cn逻辑的限制N矩阵语义的一个简单判定过程
A simple decision procedure for da Costa's Cn logics by Restricted Nmatrix semantics
摘要：
尽管有限非确定性矩阵相当强大，但它无法表征某些形式不一致的逻辑，例如在$\textbf{mbCcl}$和$\textbf{Cila}$之间发现的那些逻辑。为了克服这一限制，我们在这里提出了限制非确定性矩阵（简称RN矩阵），这是非确定性代数连同一个子集的估值。这使我们不仅可以描述$\textbf{mbCcl}$和$\textbf{Cila}$（它在语言上等同于达·科斯塔的逻辑$C_1$），而且可以描述达·科斯塔演算$C_n$的整个层次。这产生了一个新的决策程序，这些逻辑。此外，我们表明，RNmatrix的语义，这里提出的自然诱导每个$C_n$，这构成了这些逻辑的另一个决策过程的标记表系统。这个新的语义允许我们把da Costa的$C$系统的层次结构设想为一个（非确定性的）$(n+2)$值逻辑的家族，其中$n$是“不一致真”真值的个数，2是“经典”或“一致”真值的个数，对于每个$C_n$。
英文摘要：
Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between $\textbf{mbCcl}$ and $\textbf{Cila}$. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only $\textbf{mbCcl}$ and $\textbf{Cila}$ (which is equivalent, up to language, to da Costa's logic $C_1$) but the whole hierarchy of da Costa's calculi $C_n$. This produces a novel decision procedure for these logics. Moreover, we show that the RNmatrix semantics proposed here induces naturally a labelled tableau system for each $C_n$, which constitutes another decision procedure for these logics. This new semantics allows us to conceive da Costa's hierarchy of $C$-systems as a family of (non deterministically) $(n+2)$-valued logics, where $n$ is the number of "inconsistently true" truth-values and 2 is the number of "classical" or "consistent" truth-values, for every $C_n$.
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