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Another paraconsistent algebraic semantics for Lukasiewicz-Pavelka logic

机译:Lukasiewicz-Pavelka逻辑的另一种超协调代数语义

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摘要

As recently proved in a previous work of Turunen, Tsoukias and Oeztuerk, starting from an evidence pair (a, b) on the real unit square and associated with a propositional statement α, we can construct evidence matrices expressed in terms of four values t, f, k, u that respectively represent the logical valuations true, false, contradiction (both true and false) and unknown (neither true nor false) regarding the statement α. The components of the evidence pair (a, b) are to be understood as evidence for and against α, respectively. Moreover, the set of all evidence matrices can be equipped with an injective MV-algebra structure. Thus, the set of evidence matrices can play the role of truth-values of a Lukasiewicz-Pavelka fuzzy logic, a rich and applicable mathematical foundation for fuzzy reasoning, and in such a way that the obtained new logic is paraconsistent. In this paper we show that a similar result can be also obtained when the evidence pair (a, b) is given on the real unit triangle. Since the real unit triangle does not admit a natural MV-structure, we introduce some mathematical results to show how this shortcoming can be overcome, and another injective MV-algebra structure in the corresponding set of evidence matrices is obtained. Also, we derive several formulas to explicitly calculate the evidence matrices for the operations associated to the usual connectives.
机译:正如最近在Turunen,Tsoukias和Oeztuerk的工作中所证明的那样,从真实单位正方形上的证据对(a,b)开始,并与命题陈述α关联,我们可以构造以四个值t表示的证据矩阵, f,k,u分别代表关于陈述α的逻辑评估,正确,错误,矛盾(正确和错误)和未知(既不正确也不错误)。证据对(a,b)的组成部分应分别理解为支持和反对α的证据。此外,所有证据矩阵的集合都可以配备内射MV代数结构。因此,证据矩阵集可以发挥Lukasiewicz-Pavelka模糊逻辑的真值的作用,这是模糊推理的丰富而适用的数学基础,并且以这种方式使得所获得的新逻辑是超一致的。在本文中,我们表明,当在实数单位三角形上给出证据对(a,b)时,也可以获得类似的结果。由于实际单位三角形不允许自然的MV结构,因此我们引入一些数学结果来说明如何克服此缺点,并在相应的证据矩阵集中获得另一个内射MV代数结构。此外,我们导出了几个公式,以明确计算与常规连接词相关的运算的证据矩阵。

著录项

  • 来源
    《Fuzzy sets and systems》 |2014年第1期|132-147|共16页
  • 作者单位

    Faculty of Mathematics, Complutense University of Madrid, Madrid, Spain;

    Department of Mathematics, Tampere University of Technology, Tampere, Finland;

    Belgian Nuclear Research Centre (SCK CEN), Mol, Belgium;

    Faculty of Mathematics, Complutense University of Madrid, Madrid, Spain;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    Mathematical fuzzy logic; Paraconsistency; MV-algebras;

    机译:数学模糊逻辑;超一致性;MV代数;
  • 入库时间 2022-08-18 02:59:07

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