[Paper Review] About Nonstandard Neutrosophic Logic (Answers to Imamura 'Note on the Definition of Neutrosophic Logic')

Florentín Smarandache|arXiv (Cornell University)|Nov 24, 2018

Advanced Mathematical Theories8 references4 citations

TL;DR

This paper refines neutrosophic logic by embedding it within nonstandard analysis, introducing neutrosophic inequalities, equality, infimum, supremum, and standard intervals to handle truth, indeterminacy, and falsity values beyond the classical [0,1] unit interval. It addresses Imamura’s critique by formalizing the nonstandard components T, I, F, and provides a rigorous foundation for neutrosophic logic despite its lack of practical applications to date.

ABSTRACT

In order to more accurately situate and fit the neutrosophic logic into the framework of nonstandard analysis, we present the neutrosophic inequalities, neutrosophic equality, neutrosophic infimum and supremum, neutrosophic standard intervals, including the cases when the neutrosophic logic standard and nonstandard components T, I, F get values outside of the classical real unit interval [0, 1], and a brief evolution of neutrosophic operators. The paper intends to answer Imamura criticism that we found benefic in better understanding the nonstandard neutrosophic logic, although the nonstandard neutrosophic logic was never used in practical applications.

Motivation & Objective

• To resolve ambiguities in neutrosophic logic by situating it within nonstandard analysis.
• To define neutrosophic equality, inequalities, infimum, and supremum for nonstandard truth, indeterminacy, and falsity values.
• To extend the neutrosophic logic framework to include values outside the classical [0,1] interval.
• To respond to Imamura’s critique by clarifying the theoretical foundations of nonstandard neutrosophic logic.
• To provide a formal evolution of neutrosophic operators within the nonstandard framework.

Proposed method

• Introduces neutrosophic inequalities to compare nonstandard truth, indeterminacy, and falsity values.
• Defines neutrosophic equality based on infinitesimal differences in nonstandard components T, I, F.
• Proposes neutrosophic infimum and supremum for sets of nonstandard neutrosophic values.
• Introduces neutrosophic standard intervals to represent ranges of nonstandard values.
• Applies tools from nonstandard analysis to extend classical neutrosophic logic components T, I, F beyond [0,1].
• Traces the evolution of neutrosophic operators under the nonstandard framework to ensure consistency.

Experimental results

Research questions

• RQ1 How can neutrosophic logic be formally grounded within nonstandard analysis to handle values outside the classical [0,1] interval?
• RQ2 What are the appropriate definitions of neutrosophic equality and inequalities in the nonstandard setting?
• RQ3 How can neutrosophic infimum and supremum be defined for nonstandard neutrosophic values?
• RQ4 What role do neutrosophic standard intervals play in representing nonstandard truth, indeterminacy, and falsity?
• RQ5 How does the formalization of neutrosophic operators under nonstandard analysis resolve Imamura’s critique?

Key findings

• The paper successfully extends neutrosophic logic to include truth, indeterminacy, and falsity values beyond the classical [0,1] interval using nonstandard analysis.
• Neutrosophic equality is defined via infinitesimal equivalence, allowing precise comparison of nonstandard values.
• Neutrosophic inequalities are introduced to order nonstandard neutrosophic components T, I, F.
• Neutrosophic infimum and supremum are formally defined for sets of nonstandard neutrosophic values.
• Neutrosophic standard intervals are established as a tool to represent ranges of nonstandard values.
• The framework provides a coherent response to Imamura’s critique, strengthening the theoretical consistency of nonstandard neutrosophic logic.