论文标题
互补的伪树树代数和直觉逻辑，否定为最小的否定
Contrapositionally Complemented Pseudo-Boolean Algebras and Intuitionistic Logic with Minimal Negation
论文作者
论文摘要
该文章是对两个代数结构的研究，即“互补的伪树状代数”（CCPBA）和`codositionally $ \ vee $补充的伪推杆式代数'（c $ \ vee $ cpba）。代数最近是从对粗糙集类别类别的Topos理论研究中获得的。这些代数的显着特征是，在本质上有两种否定，一种是直觉的，另一种是最小的，以及连接两个操作员的条件。我们研究这些代数的属性，举例说明并将其与相关的现有代数进行比较。然后研究了与CCPBAS及其扩展ILM-$ {\ VEE} $相对应的直觉逻辑（ILM）'，用于C $ \ vee $ cpbas。除了与直觉和最小逻辑的关系外，ILM还与Peirce的逻辑有关。侧重于这两种否定的属性，获得了两种用于ILM和ILM-$ {\ vee} $的关系语义，并提供了两种语义之间的跨译。在对代数中的两种否定的特征提取特征，在对否定效果的逻辑研究之后，将进行进一步的研究，该研究独立于二进制含义的二进制操作员而定义操作员。出现了两个逻辑$ k_ {im} $和$ k_ {im- {\ vee}} $的两个逻辑$ k_ {im} $，在该语言中不包含含义。 $ k_ {im} $ - 代数是CCPBA的还原。代数的否定显示以增强的邓恩风筝的否定形式占据不同的位置。 $ k_ {im} $和$ k_ {im- {\ vee}} $的关系语义是基于Dunn的兼容框。最后，在工作中定义的逻辑的不同代数和关系语义之间建立了关系。
The article is a study of two algebraic structures, the `contrapositionally complemented pseudo-Boolean algebra' (ccpBa) and `contrapositionally $\vee$ complemented pseudo-Boolean algebra' (c$\vee$cpBa). The algebras have recently been obtained from a topos-theoretic study of categories of rough sets. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. We study properties of these algebras, give examples, and compare them with relevant existing algebras. `Intuitionistic Logic with Minimal Negation (ILM)' corresponding to ccpBas and its extension ILM-${\vee}$ for c$\vee$cpBas, are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce's logic. With a focus on properties of the two negations, two kinds of relational semantics for ILM and ILM-${\vee}$ are obtained, and an inter-translation between the two semantics is provided. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn's logical framework for the purpose, two logics $K_{im}$ and $K_{im-{\vee}}$ are presented, where the language does not include implication. $K_{im}$-algebras are reducts of ccpBas. The negations in the algebras are shown to occupy distinct positions in an enhanced form of Dunn's Kite of negations. Relational semantics for $K_{im}$ and $K_{im-{\vee}}$ are given, based on Dunn's compatibility frames. Finally, relationships are established between the different algebraic and relational semantics for the logics defined in the work.
