[论文解读] Symbolic Neutrosophic Theory

Florentín Smarandache|arXiv (Cornell University)|Oct 14, 2015

Advanced Mathematical Theories参考文献 10被引用 27

一句话总结

本文提出符号 neutrosophic 理论，通过真理（T）、模糊性（I）和虚假（F）的符号表示，扩展了经典逻辑与代数，并提出了新型结构，如 (t, i, f)-neutrosophic 系统、neutrosophic 公理和 neutrosophic 四元数。它建立了基础运算，如精细逻辑算子、优势顺序和吸收律，以形式化在模糊性下的推理。

ABSTRACT

Symbolic (or Literal) Neutrosophic Theory is referring to the use of abstract symbols (i.e. the letters <em>T, I, F</em>, or their refined indexed letters <em>T_j, I_k, F_l</em>) in neutrosophics. In the first chapter we extend the dialectical triad thesis-antithesis-synthesis (dynamics of &lt;A&gt; and &lt;antiA&gt;, to get a synthesis) to the neutrosophic tetrad thesis-antithesis-neutrothesis-neutrosynthesis (dynamics of &lt;A&gt;, &lt;antiA&gt;, and &lt;neutA&gt;, in order to get a neutrosynthesis). In the second chapter we introduce the neutrosophic system and neutrosophic dynamic system. A neutrosophic system is a quasi- or –classical system, in the sense that the neutrosophic system deals with quasi-terms/concepts/attributes, etc. [or -terms/concepts/attributes], which are approximations of the classical terms/concepts/attributes, i.e. they are partially true/membership/probable ( ), partially indeterminate ( ), and partially false/nonmembership/improbable ), where are subsets of the unitary interval . In the third chapter we introduce for the first time the notions of <em>Neutrosophic Axiom, Neutrosophic Deducibility, Neutrosophic Axiomatic System, Degree of Contradiction (Dissimilarity) of Two Neutrosophic Axioms, etc.</em> The fourth chapter we introduced for the first time a new type of structures, called <em>(t, i, f)-Neutrosophic Structures</em>, presented from a neutrosophic logic perspective, and we showed particular cases of such structures in geometry and in algebra. In any field of knowledge, each structure is composed from two parts: a <em>space</em>, and a <em>set of axioms</em> (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy of the form <em>(t, i, f) ≠ (1, 0, 0),</em> that structure is a <em>(t, i, f)-Neutrosophic Structure</em>. In the fifth chapter we make a short history of: the <em>neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, etc. </em>The aim of this chapter is to construct examples of splitting the literal indeterminacy <em>(I)</em> into literal sub-indeterminacies <em>(I₁,I₂,…,I_r)</em>, and to define a <em>multiplication law</em> of these literal sub-indeterminacies in order to be able to build <em>refined I</em>-<em>neutrosophic algebraic structures</em>. In the sixth chapter we define for the first time three <em>neutrosophic actions</em> and their properties. We then introduce the <em>prevalence order</em> on with respect to a given neutrosophic operator , which may be subjective - as defined by the neutrosophic experts. And the <em>refinement of neutrosophic entities</em> &lt;A&gt;, &lt;neutA&gt;, and &lt;antiA&gt;. Then we extend the classical logical operators to <em>neutrosophic literal (symbolic) logical operators</em> and to <em>refined literal (symbolic) logical operators</em>, and we define the <em>refinement neutrosophic literal (symbolic) space</em>. In the seventh chapter we introduce for the first time the <em>neutrosophic quadruple numbers </em>(of the form ) and the <em>refined</em> <em>neutrosophic quadruple numbers</em>. Then we define an <em>absorbance law</em>, based on a <em>prevalence order</em>, both of them in order to multiply the neutrosophic components or their sub-components and thus to construct the <em>multiplication of neutrosophic quadruple numbers</em>.

研究动机与目标

• 通过使用真理、模糊性和虚假的字面成分 T、I、F 及其精细索引，形式化 neutrosophic 逻辑的符号框架。
• 通过引入 (t, i, f) ≠ (1, 0, 0) 的模糊成分，将经典系统扩展为 neutrosophic 系统。
• 引入 neutrosophic 逻辑的基础构造，如 neutrosophic 公理、可推导性以及矛盾程度。
• 通过将模糊性嵌入空间和公理中，发展几何与代数中的 (t, i, f)-neutrosophic 结构。
• 定义 neutrosophic 四元数，并使用优势顺序定义吸收律，以实现对组件的乘法运算。

提出的方法

• 提出 neutrosophic 四元组：正题-反题-中题-综合，作为辩证三元组的推广。
• 将 neutrosophic 系统定义为具有成分 (t, i, f) ∈ [0,1]³ 的准经典系统，分别表示真理、模糊性和虚假。
• 引入 neutrosophic 公理，并将两个公理之间的矛盾程度作为形式化逻辑构造。
• 通过将模糊性嵌入系统的空间和公理中，发展 (t, i, f)-neutrosophic 结构。
• 定义精细的字面成分 I_j、I_k 等，并引入子模糊性之间的乘法法则。
• 建立优势顺序和吸收律，以定义形如 (a, bI, cF, d) 及其精细变体的 neutrosophic 四元数的乘法。

实验结果

研究问题

• RQ1 如何通过符号 neutrosophic 成分将经典逻辑与代数系统扩展以包含模糊性？
• RQ2 包含部分为真、模糊和为假的成分的 neutrosophic 系统的正式结构是什么？
• RQ3 neutrosophic 公理如何定义？其矛盾程度如何衡量？
• RQ4 几何与代数中 (t, i, f)-neutrosophic 结构的性质与构建规则是什么？
• RQ5 如何使用基于精细优势顺序的吸收律来实现 neutrosophic 四元数的乘法？

主要发现

• 本文引入 neutrosophic 四元组作为辩证三元组的扩展，包含中题与综合。
• 形式化了 neutrosophic 系统，其中成分 (t, i, f) ≠ (1, 0, 0) 表示部分为真、模糊和为假。
• 引入了 neutrosophic 公理的概念，并提出了两个此类公理之间矛盾程度的度量。
• 通过将模糊性嵌入系统的空间和公理中，定义了 (t, i, f)-neutrosophic 结构。
• 通过子模糊性 I_j 及其乘法法则，首次建立了精细 neutrosophic 代数结构的框架。
• 定义了 neutrosophic 四元数和基于优势顺序的吸收律，以实现组件乘法的一致性。