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n-Valued Refined Neutrosophic Logic and Its Applications to Physics
Florentin Smarandache
TL;DR
The paper traces extensions from two-valued and three- or four-valued logics to general n-symbol and numerical-valued refined neutrosophic logic. It defines two classes of neutrosophic norms and conorms for physics applications, with analogous generalizations for sets and probability.
Problem
The paper addresses how particular two-, three-, and four-valued logics can be generalized to n-symbol or numerical-valued refined neutrosophic logic.
Method
It refines truth, indeterminacy, and falsity into multiple subcomponents and defines two classes of neutrosophic norms and conorms for combining neutrosophic propositions.
Results
The paper presents n-valued refined neutrosophic logic and lists applications to physics.
Takeaways & Limitations
The same generalization framework can be applied to n-valued refined neutrosophic sets and probabilities.
Takeaways & Limitations
Neutrosophic norms and conorms only approximate interconnectivity between two n-valued neutrosophic propositions, so multiple approximation versions exist.
Abstract
from arXiv · showhide
In this paper we present a short history of logics: from particular cases of 2-symbol or numerical valued logic to the general case of n-symbol or numerical valued logic. We show generalizations of 2-valued Boolean logic to fuzzy logic, also from the Kleene and Lukasiewicz 3-symbol valued logics or Belnap 4-symbol valued logic to the most general n-symbol or numerical valued refined neutrosophic logic. Two classes of neutrosophic norm (n-norm) and neutrosophic conorm (n-conorm) are defined. Examples of applications of neutrosophic logic to physics are listed in the last section. Similar generalizations can be done for n-Valued Refined Neutrosophic Set, and respectively n- Valued Refined Neutrosopjhic Probability.
1. Two-Valued Logic
Two-valued logic represents contraries such as Yin and Yang, or truth and falsity, using either symbols or numerical values. Boolean logic uses T/F or 1/0, while fuzzy logic permits graded values.
- Two-symbol logic represents Yin and Yang as contraries.
- Boolean logic uses two symbol-values: truth T and falsity F.
- Numerical Boolean logic uses truth 1 and falsity 0.
- Fuzzy logic allows truth and falsity to be numbers in [0,1] satisfying T + F = 1.Truth and falsity can also be subsets of [0,1].
2. Three-Valued Logic
Three-valued logics extend true and false with a third status, such as possible, unknown, or neutral. Numerical neutrosophic logic represents truth, falsity, and indeterminacy as independently variable quantities.
- Łukasiewicz, Kleene, and neutrosophic traditions add Possible, Unknown, or Neuter to True and False.
- Neutrosophy studies neutralities between a notion, its opposite, and intermediate neutral ideas.
- Kleene’s numerical logic assigns True 1, False 0, and Unknown 1/2, using min, max, and 1- for its operators.
- Neutrosophic logic lets T, F, and I be any numbers in [0,1], with 0 ≤ T + I + F ≤ 3.These components may also be standard or nonstandard subsets of ]-0, 1+[.
3. Four-Valued Logic
Four-valued logic adds unknown and contradiction to true and false. Its numerical neutrosophic form assigns subsets of [0,1] to all four components and generalizes Belnap’s symbolic logic.
- Belnap’s four symbolic values are True, False, Unknown, and Contradiction.
- Absolute-relative variants combine TA, TR, IA, IR, FA, and FR into logics with 2, 3, 4, 5, or 6 symbols.
- The absolute-relative 4-symbol and 6-symbol cases are identified as particularly interesting variants.
- Four-valued numerical neutrosophic logic refines indeterminacy into Unknown and Contradiction, represented as subsets of [0,1].Contradiction is defined as T/\F, connecting the logic with Extenics.
4. Five-Valued Logic
Refined neutrosophic logic splits truth, indeterminacy, and falsity into multiple subcomponents, yielding n-symbol or n-numerical-valued systems. The framework allows infinite-valued sets, source dependence, and technical simplification to [0,1].
- Five-Valued Logic: Five-symbol neutrosophic logic refines indeterminacy into Unknown, Contradiction, and Ignorance.It defines U as neither T nor F, C as T/\F, and G as T\/F.
- Five-Valued Logic: The n-symbol-valued refinement splits T, I, and F into p truth, r indeterminacy, and s falsity types, with p + r + s = n.
- Five-Valued Logic: Each refined numerical subcomponent can be a subset of ]-0, 1+[, and T, I, or F may be countable or uncountable infinite sets.
- Five-Valued Logic: Independent information sources yield independently treated subcomponents, whereas dependent subcomponents are grouped together.For example, dependent T2 and I3 are combined as -0 ≤ T2 + I3 ≤ 1+.
- Five-Valued Logic: The standard interval [0,1] replaces ]-0, 1+[ for technical applications, while contradiction remains linked to Extenics.
7. n-Valued Neutrosophic Logic Connectors
The paper defines neutrosophic conjunction and disjunction operators by combining t-norms and t-conorms across refined truth, indeterminacy, and falsity components. Alternative priority orderings and approximations support different applications.
- The n-norm is the neutrosophic AND operator, while the n-conorm is the neutrosophic OR operator.
- The operators can be defined using t-norm and t-conorm operators from fuzzy logic, with normalization applied if needed.
- The n-norm and n-conorm approximate interconnectivity between two n-valued neutrosophic propositions, so multiple versions are possible.
- Indeterminate components can use opposite t-conorm or t-norm choices for pessimistic and optimistic bounds in the n-norm and n-conorm.
- For a lower-bound n-norm, priorities are T < I < F; for an upper-bound n-norm, priorities are I < T < F.
- For an upper-bound n-conorm, priorities are T > I > F; for a lower-bound n-conorm, priorities are T > F > I.
8. Particular Cases
The refined neutrosophic framework contains particular fuzzy-logic cases determined by the treatment of indeterminacy. Eliminating indeterminacy yields refined fuzzy logic, while retaining one indeterminacy type yields refined intuitionistic fuzzy logic.
- When all indeterminacy components I_k equal 0, the framework becomes n-valued refined fuzzy logic.
- When there is one positive indeterminacy type, I_1 = I > 0, the framework becomes n-valued refined intuitionistic fuzzy logic.
9. Distinction between Neutrosophic Physics and Paradoxist Physics
Neutrosophic Physics combines a physical entity, its opposite, and their neutral counterpart, whereas Paradoxist Physics combines only contradictory entities. Both domains are situated within the neutrosophical framework established in 1995.
- Neutrosophic Physics studies mixtures of A, antiA, and neutA that hold together, producing neutrosophic fields, objects, and states.
- Paradoxist Physics combines contradictory physical entities A and antiA without referring to their neutral counterpart neutA.
- Neutrosophic Physics is described as an extension of Paradoxist Physics because it adds neutrality to combinations of physical contradictories.
- These research domains were established in 1995 within neutrosophy and its logic, set, probability, and statistics frameworks.
10. n-Valued Refined Neutrosophic Logic Applied to Physics
The paper applies refined neutrosophic logic to physical cases in which entities or states can coexist with contradictory, intermediate, or neutral characteristics. Examples span particles, materials, quantum states, fields, and cosmological models.
- Scientific cases include entities that simultaneously exhibit A, antiA, or neutA characteristics.
- Applications include neutrosophic methods in general relativity, cosmological models, and gravitation.
- Examples include qubit superposition, semiconductors, semi-transparent optical components, and metastable quantum states.
- Other examples include a neutrino-photon doublet and an elementary-particle multiplet characterized as a neutrosophic field with two or more values.
- A neutrosophic field can be generalized to selective operators, with effects described as somehow equivalent to tunneling or spontaneous symmetry breaking.
- The paper presents two classes of neutrosophic operators for combining neutrosophic-valued propositions in physics.
- Similar generalizations are stated for n-valued refined neutrosophic sets and n-valued refined neutrosophic probability.