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Boolean Differences between Two Hexagonal Extensions of the Logical Square of Oppositions

机译：两个六角形延伸的反对派的两个六边形扩展之间的布尔差异

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The classical Aristotelian Square characterizes four formulae in terms of four relations of Opposition: contradiction, contrariety, subcontrariety, and subalternation. This square has been extended into a hexagon by two different strategies of inserting intermediate formulae: (1) the horizontal SB-insertion of Sesmat-Blanche and (2) the vertical SC-insertion of Sherwood-Czezowski. The resulting visual constellations of opposition relations are radically different, however. The central claim of this paper is that these differences are due to the fact that the SB hexagon is closed under the Boolean operations of meet, join and complement, whereas the SC hexagon is not. Therefore we define the Boolean closure of the SC hexagon by characterizing the remaining 8 (non-trivial) formulae, and demonstrate how the resulting 14 formulae generate 6 SB hexagons. These can be embedded into a much richer 3D Aristotelian structure, namely a rhombic dodecahedron, which also underlies the modal system S5 and the propositional connectives.
机译：古典亚里士敦方形在四个反对关系方面表征了四种公式：矛盾，对照，分包令和亚总和。通过插入中间体公式的两种不同的策略延伸到六边形中：（1）水平SB插入的SESMAT-BLANCHE和（2）SHERWOM-CZEZOWSKI的垂直SC-INSTERTION。然而，由此产生的反对关系的视觉星座是完全不同的。本文的核心索赔是，这些差异是由于SB六边形在满足，加入和补充的布尔操作下关闭，而SC六边形则不是。因此，我们通过表征剩余的8（非平凡）公式来定义SC六边形的布尔闭合，并证明所得到的14个公式产生6 SB六边形。这些可以嵌入到一个更丰富的3D亚里士典结构中，即菱形十二锭，其也为模态系统S5和命题连接而下潜。

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