## Mathematics and Statistics Faculty and Staff Publications

#### Document Type

Book

#### Publication Date

2015

#### Abstract

Symbolic (or Literal) Neutrosophic Theory is referring to the use of abstract symbols (i.e. the letters *T, I, F*, or their refined indexed letters *T _{j}, I_{k}, F_{l}*) in neutrosophics.

In the first chapter we extend the dialectical triad thesis-antithesis-synthesis (dynamics of A and antiA, to get a synthesis) to the neutrosophic tetrad thesis-antithesis-neutrothesis-neutrosynthesis (dynamics of A, antiA, and neutA, 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 (t), partially indeterminate (i), and partially false/nonmembership/improbable (f), where are subsets of the unitary interval .

In the third chapter we introduce for the first time the notions of *Neutrosophic Axiom, Neutrosophic Deducibility, Neutrosophic Axiomatic System, Degree of Contradiction (Dissimilarity) of Two Neutrosophic Axioms, etc.*

The fourth chapter we introduced for the first time a new type of structures, called *(t, i, f)-Neutrosophic Structures*, 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 *space*, and a *set of axioms* (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy of the form *(t, i, f) ≠ (1, 0, 0),* that structure is a *(t, i, f)-Neutrosophic Structure*.

In the fifth chapter we make a short history of: the *neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, etc. *The aim of this chapter is to construct examples of splitting the literal indeterminacy *(I)* into literal sub-indeterminacies *(I _{1},I_{2},…,I_{r})*, and to define a

*multiplication law*of these literal sub-indeterminacies in order to be able to build refined

*I*-

*neutrosophic algebraic structures*.

In the sixth chapter we define for the first time three *neutrosophic actions* and their properties. We then introduce the *prevalence order* on with respect to a given neutrosophic operator , which may be subjective - as defined by the neutrosophic experts. And the *refinement of neutrosophic entities* , , and . Then we extend the classical logical operators to *neutrosophic literal (symbolic) logical operators* and to *refined literal (symbolic) logical operators*, and we define the *refinement neutrosophic literal (symbolic) space*.

In the seventh chapter we introduce for the first time the *neutrosophic quadruple numbers *(of the form ) and the *refined* *neutrosophic quadruple numbers*. Then we define an *absorbance law*, based on a *prevalence order*, both of them in order to multiply the neutrosophic components or their sub-components and thus to construct the *multiplication of neutrosophic quadruple numbers*.

#### Publisher

EuropaNova

#### Language (ISO)

English

#### Keywords

(t, i, f)-Neutrosophic Structures, neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, thesis-antithesis-neutrothesis-neutrosynthesis, Neutrosophic Axiom, Neutrosophic Deducibility, Neutrosophic Axiomatic System.

#### Recommended Citation

Smarandache, Florentin. "Symbolic Neutrosophic Theory." (2015). https://digitalrepository.unm.edu/math_fsp/31

#### Included in

Algebra Commons, Applied Mathematics Commons, Dynamical Systems Commons, Logic and Foundations Commons, Set Theory Commons