Paraconsistent weak Kleene (⁠|$\textrm{PWK}$|⁠) logic is the |$3$|-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α|$\textrm{lg} (\textrm{PWK})$|⁠. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, |$\textrm{PWK}$| has only one proper nontrivial extension apart from classical logic: |$\textrm{PWK}_{\textrm{E}}\textrm{,}$| PWK logic plus explosion. This |$6$|-valued logic, unlike |$\textrm{PWK} $|⁠, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.

中文翻译：

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超一致弱Kleene逻辑的扩展

次协调弱克林（⁠| $ \ TEXTRM {PWK} $ |⁠）逻辑是| $ 3 $ | 弱Kleene矩阵并具有两个指定值的逻辑。在本文中，我们研究了广义对合双半格的先验性的波塞特，特别关注Α | $ \ textrm {lg}（\ textrm {PWK}）$ |⁠生成的阶理想。将这个结果应用于Alexej Pynko的一般结果，我们证明，就像Priest的悖论逻辑一样，| $ \ textrm {PWK} $ | 除经典逻辑外，仅具有一个适当的非平凡扩展：| $ \ textrm {PWK} _ {\ textrm {E}} \ textrm {，} $ | PWK逻辑加爆炸。这个| $ 6 $ | 值逻辑，与| $ \ textrm {PWK} $ |⁠不同，无法保持一致。我们通过变量包含准则描述其后果关系，并确定其Suszko简化模型。