Plithogenic Sets: Contradiction-Aware Aggregation

• Plithogenic sets are generalized uncertainty models that incorporate contradiction degrees among attribute values, extending fuzzy, intuitionistic fuzzy, and neutrosophic frameworks.
• They utilize a contradiction function to modulate aggregation operators, adjusting weights based on the deviation from a dominant attribute value.
• The framework underpins diverse applications—from rental evaluations to graph neural networks—and recovers classical models as special cases.

Plithogenic sets are generalized uncertainty models in which elements are characterized by one or more attributes, each attribute may take many values, each value carries a degree of appurtenance, and aggregation is modulated by a contradiction or dissimilarity degree between attribute values, typically relative to a dominant value. In the hierarchy surveyed in "A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets" (Fujita et al., 12 Mar 2026), they are presented as a contradiction-aware generalization of fuzzy, intuitionistic fuzzy, and neutrosophic sets; in "Plithogeny, Plithogenic Set, Logic, Probability, and Statistics" (Smarandache, 2018), they are introduced together with plithogenic logic, probability, and statistics as a framework intended to refine aggregation, inclusion, and related operators by incorporating the relationships among attribute values themselves.

1. Conceptual position in generalized set theory

Plithogenic sets arise in the progression from fuzzy to intuitionistic fuzzy to neutrosophic modeling. A fuzzy set assigns each element a single membership degree $\mu(x)\in[0,1]$. An intuitionistic fuzzy set assigns a membership/non-membership pair $(\mu(x),\nu(x))$ subject to

$0 \le \mu(x)+\nu(x)\le 1,$

with hesitation $\pi(x)=1-\mu(x)-\nu(x)$. A neutrosophic set assigns the triple $(T(x),I(x),F(x))$, where truth, indeterminacy, and falsity are independent coordinates in $[0,1]$, with no requirement that they sum to $1$ (Fujita et al., 12 Mar 2026).

The plithogenic extension preserves these membership-like structures but adds a new structural ingredient: attribute values may contradict each other to different degrees, and those contradiction degrees affect aggregation. The survey states that plithogenic sets “represent each element through its attribute values, together with the corresponding degrees of appurtenance, and by introducing a contradiction (or dissimilarity) function between distinct attribute values” (Fujita et al., 12 Mar 2026). In the 2018 formulation, a plithogenic set is “a set whose elements are characterized by one or more attributes, and each attribute may have many values,” with each attribute value equipped with a fuzzy, intuitionistic fuzzy, or neutrosophic degree of appurtenance $d(x,v)$ (Smarandache, 2018).

This places plithogenic sets as a higher-level framework rather than a competing replacement for earlier models. The survey repeatedly positions them as a meta-framework in which fuzzy, intuitionistic fuzzy, and neutrosophic sets are recovered as limiting or exact special cases (Fujita et al., 12 Mar 2026). A common misconception is therefore that plithogenic sets merely add another membership coordinate; the defining idea is instead that the semantic relations among attribute values become part of the formalism.

2. Formal structure and notation

The survey gives the basic plithogenic-set structure as

$PS = (P, v, P_v, pdf, pCF),$

where $P$ is the universe, $(\mu(x),\nu(x))$0 is an attribute, $(\mu(x),\nu(x))$1 is the set of possible values of that attribute, $(\mu(x),\nu(x))$2 is the Degree of Appurtenance Function (DAF), and $(\mu(x),\nu(x))$3 is the Degree of Contradiction Function (DCF) (Fujita et al., 12 Mar 2026). The DAF assigns to each pair $(\mu(x),\nu(x))$4 a membership vector of dimension $(\mu(x),\nu(x))$5, with the survey explicitly identifying the principal cases: $(\mu(x),\nu(x))$6 for fuzzy, $(\mu(x),\nu(x))$7 for intuitionistic fuzzy, $(\mu(x),\nu(x))$8 for neutrosophic, and higher $(\mu(x),\nu(x))$9 for richer multipartitioned models (Fujita et al., 12 Mar 2026).

The 2018 formalization gives a closely related single-attribute form. Let $0 \le \mu(x)+\nu(x)\le 1,$0 be a universe of discourse, $0 \le \mu(x)+\nu(x)\le 1,$1 a non-empty set, $0 \le \mu(x)+\nu(x)\le 1,$2 a set of attributes, and $0 \le \mu(x)+\nu(x)\le 1,$3 one attribute. If $0 \le \mu(x)+\nu(x)\le 1,$4 is the spectrum of all possible values of attribute $0 \le \mu(x)+\nu(x)\le 1,$5, and $0 \le \mu(x)+\nu(x)\le 1,$6, then the degree-of-appurtenance function is

$0 \le \mu(x)+\nu(x)\le 1,$7

with $0 \le \mu(x)+\nu(x)\le 1,$8 for fuzzy appurtenance, $0 \le \mu(x)+\nu(x)\le 1,$9 for intuitionistic fuzzy appurtenance, and $\pi(x)=1-\mu(x)-\nu(x)$0 for neutrosophic appurtenance. The tuple

$\pi(x)=1-\mu(x)-\nu(x)$1

is then called a plithogenic set, where $\pi(x)=1-\mu(x)-\nu(x)$2 is the contradiction degree function (Smarandache, 2018).

Both formulations make the same structural point: membership is no longer attached to an element in isolation, but to an element-value pair. The set is therefore attribute-centered. In the survey’s terms, each element $\pi(x)=1-\mu(x)-\nu(x)$3 is evaluated across multiple attribute values $\pi(x)=1-\mu(x)-\nu(x)$4; $\pi(x)=1-\mu(x)-\nu(x)$5 gives appurtenance to each value, and contradiction values govern how those appurtenances interact under aggregation (Fujita et al., 12 Mar 2026).

3. Dominant value and contradiction function

A central notion is the dominant attribute value. In the 2018 treatment, among the values in $\pi(x)=1-\mu(x)-\nu(x)$6 there is usually one dominant value $\pi(x)=1-\mu(x)-\nu(x)$7, chosen by experts as the most important reference value. The same idea appears in the survey, which frequently uses a dominant value $\pi(x)=1-\mu(x)-\nu(x)$8 as the reference context for aggregation (Smarandache, 2018, Fujita et al., 12 Mar 2026).

The contradiction or dissimilarity function is defined in the 2018 paper as

$\pi(x)=1-\mu(x)-\nu(x)$9

with

$(T(x),I(x),F(x))$0

The survey gives the corresponding plithogenic axioms

$(T(x),I(x),F(x))$1

for the scalar-contradiction setting (Smarandache, 2018, Fujita et al., 12 Mar 2026). The interpretation is explicit in the 2018 account: $(T(x),I(x),F(x))$2 means identical values and no contradiction, $(T(x),I(x),F(x))$3 means maximal contradiction, and intermediate values represent partial dissimilarity (Smarandache, 2018).

The contradiction degree is not auxiliary metadata. The survey emphasizes that it is the mechanism by which plithogenic aggregation works: values closer to the dominant context are given more weight, while values more contradictory to the dominant context are penalized more. It repeatedly illustrates this through weights of the form

$(T(x),I(x),F(x))$4

In multipolar settings the weight combines value-level and pole-level compatibilities (Fujita et al., 12 Mar 2026).

The 2018 paper also notes boundary cases that complicate the standard picture. In some applications, no dominant value may exist, or there may be multiple dominant values. In such cases, the contradiction function may be suppressed or replaced by another relationship function (Smarandache, 2018). This suggests that the dominant-value mechanism is fundamental in the canonical theory but not obligatory in every application-specific adaptation.

4. Aggregation operators and derived notions

Plithogenic aggregation is described in the survey as usually a weighted mean whose weights come from contradiction degrees. In the simplest scalar-contradiction case, if $(T(x),I(x),F(x))$5 is the dominant attribute value, then the plithogenic reduction is

$(T(x),I(x),F(x))$6

with

$(T(x),I(x),F(x))$7

In the $(T(x),I(x),F(x))$8-polar setting, the compatibility weight is

$(T(x),I(x),F(x))$9

and the aggregated degree is

$[0,1]$0

with $[0,1]$1 (Fujita et al., 12 Mar 2026). The survey states that this weighted-mean pattern reappears throughout complex, interval-valued, hesitant, cubic, rough, soft, dynamic, probabilistic, triangular, and trapezoidal extensions.

The 2018 monograph presents a complementary operator-based formulation. If a $[0,1]$2-norm is applied to the dominant value, then on another value $[0,1]$3 one applies the linear combination

$[0,1]$4

If the dominant value uses a $[0,1]$5-conorm, then the other value uses

$[0,1]$6

For one-attribute fuzzy values $[0,1]$7 and $[0,1]$8, the plithogenic intersection and union are

$[0,1]$9

$1$0

When $1$1, these reduce to ordinary $1$2-norm and $1$3-conorm behavior; when $1$4, the roles are reversed; and when $1$5, the operators coincide (Smarandache, 2018).

The same source extends contradiction-sensitive behavior to complement, inclusion, equality, and partial order. For one-attribute fuzzy values, the plithogenic complement may be written as $1$6. For intuitionistic fuzzy sets, one form is

$1$7

and for neutrosophic sets one form is

$1$8

Inclusion is also modified by contradiction. For intuitionistic fuzzy values, the plithogenic inclusion relation is

$1$9

while for the chosen neutrosophic model,

$d(x,v)$0

Equality is then defined through mutual inclusion (Smarandache, 2018).

These two presentations are compatible in emphasis. The survey foregrounds contradiction-weighted means; the 2018 work foregrounds contradiction-weighted blends of $d(x,v)$1-norm and $d(x,v)$2-conorm. In both cases, the decisive mathematical principle is that aggregation depends on how far each value is from the dominant one.

5. Special cases, reductions, and extended families

A major theme in the survey is that plithogenic sets subsume several existing uncertainty theories. It explicitly states the following reductions: if $d(x,v)$3 and $d(x,v)$4, one gets the classical fuzzy set; if $d(x,v)$5 and $d(x,v)$6, one gets the intuitionistic fuzzy set; if $d(x,v)$7 and $d(x,v)$8, one gets the neutrosophic set. The corresponding scalar-contradiction variants are listed as plithogenic fuzzy ($d(x,v)$9), plithogenic intuitionistic fuzzy ($PS = (P, v, P_v, pdf, pCF),$0), and plithogenic neutrosophic ($PS = (P, v, P_v, pdf, pCF),$1) (Fujita et al., 12 Mar 2026).

The 2018 book presents the same inclusion hierarchy through explicit value sets. For a fuzzy set, $PS = (P, v, P_v, pdf, pCF),$2 with contradiction $PS = (P, v, P_v, pdf, pCF),$3. For an intuitionistic fuzzy set, $PS = (P, v, P_v, pdf, pCF),$4 with

$PS = (P, v, P_v, pdf, pCF),$5

and full contradiction between membership and nonmembership. For a neutrosophic set,

$PS = (P, v, P_v, pdf, pCF),$6

with $PS = (P, v, P_v, pdf, pCF),$7 and $PS = (P, v, P_v, pdf, pCF),$8 (Smarandache, 2018). In that formulation, indeterminacy is treated as “half-opposite” to truth and falsehood.

The survey further extends the reduction pattern to a large family of enriched plithogenic models. The variants it enumerates include m-polar plithogenic sets, complex plithogenic sets, superhyperplithogenic sets, linguistic plithogenic sets, q-rung orthopair plithogenic sets, type-$PS = (P, v, P_v, pdf, pCF),$9 plithogenic sets, iterative multi-plithogenic sets, interval-valued plithogenic sets, offset plithogenic sets, cubic plithogenic sets, hesitant plithogenic sets, spherical and T-spherical plithogenic sets, plithogenic rough and soft rough sets, linear Diophantine plithogenic sets, tree/forest plithogenic sets, plithogenic soft expert sets, dynamic plithogenic sets, probabilistic plithogenic sets, triangular/trapezoidal plithogenic sets, nonstandard plithogenic sets, refined plithogenic sets, subset-valued plithogenic sets, and picture plithogenic sets (Fujita et al., 12 Mar 2026).

A plausible implication is that the theory’s unifying role derives from two orthogonal parameters: the internal shape of appurtenance, indexed by the dimension $P$0, and the contradiction structure, indexed by the contradiction function. The survey explicitly supports the first part of this reading through the $P$1 interpretation (Fujita et al., 12 Mar 2026).

6. Examples, application patterns, and graph-theoretic extensions

The survey is application-heavy and uses concrete examples to show how contradiction-aware modeling operates. The apartment rental example uses attribute values such as low_rent, near_station, and large_space, with contradiction assignments

$P$2

showing that “low rent” is more compatible with “near station” than with “large space.” The dominant value then determines how appurtenance degrees are weighted in aggregation (Fujita et al., 12 Mar 2026).

The same survey lists a broad range of application settings: apartment rental criteria, diet planning, sustainable car choice, hiring under multi-polar evaluation, vendor selection, smart-city intersection design, medical triage, shipping and logistics, customer satisfaction, supplier evaluation, supply chain sustainability, disease-risk assessment, and air-quality risk (Fujita et al., 12 Mar 2026). The recurring pattern is not domain-specific machinery but the reuse of contradiction-aware aggregation in situations where labels are semantically non-equivalent and partially conflicting.

The 2018 book provides additional illustrative patterns. It includes a size example with values small, medium, big, and very big, together with contradiction degrees relative to the dominant value small; a multi-attribute student example combining altitude, weight, and hair color; a color example showing discrete, countable, and continuous value spaces; an expert evaluation example with attributes color and height; a plithogenic logic example for the proposition “John is a knowledgeable person”; and a plithogenic probability example concerning Jenifer graduating (Smarandache, 2018). These examples demonstrate that the framework is intended to operate across set theory, logic, probability, and statistics rather than only in membership-based classification.

A later theoretical extension appears in "Superhypergraph Neural Networks and Plithogenic Graph Neural Networks: Theoretical Foundations" (Fujita, 2024). That paper does not restate the full foundational definition of a plithogenic set in a standalone block, but it builds graph-theoretic structures on top of the plithogenic framework. A Plithogenic Graph is defined as

$P$3

with vertex component

$P$4

and edge component

$P$5

where appurtenance and contradiction functions are defined separately for vertices and edges (Fujita, 2024). The edge appurtenance compatibility condition is

$P$6

and contradiction is constrained by

$P$7

The paper treats plithogenic graphs as generalizations of both fuzzy and neutrosophic graphs and uses them as the basis for Plithogenic Graph Neural Networks, in which message aggregation coefficients incorporate appurtenance and contradiction directly (Fujita, 2024).

This extension is significant because it shows that plithogenic sets are not confined to static generalized membership spaces. They also provide the semantic basis for structured learning on uncertain relational data, where attributes, appurtenance, and contradiction jointly control propagation and aggregation.

7. Interpretation, scope, and common misunderstandings

The central insight of plithogenic theory is that uncertainty should be quantified not only by membership-like degrees but also by the relationships among the labels themselves. The survey’s five-point summary states that elements are evaluated by attribute values, each attribute value carries a degree of appurtenance, attribute values are related by a degree of contradiction, aggregation is performed by contradiction-weighted means, and many classical fuzzy-type models appear as exact special cases (Fujita et al., 12 Mar 2026). This directly distinguishes plithogenic sets from frameworks in which labels are merely coordinates without explicit semantic interaction.

Compared with intuitionistic fuzzy sets, plithogenic sets retain the membership/non-membership structure but add contradiction among attribute values; hesitation is therefore only one part of the formal picture. Compared with neutrosophic sets, they retain truth-indeterminacy-falsity style coordinates but aggregate them in a context-sensitive way according to attribute contradictions (Fujita et al., 12 Mar 2026). The repeated message in the survey is that plithogenic sets do not replace neutrosophic or intuitionistic fuzzy sets; they recover them as limiting cases within a broader framework.

Another common misunderstanding is to identify plithogenicity solely with multi-valued membership. The graph-theoretic extension makes clear that plithogenic structures are defined by the joint presence of attribute values, appurtenance functions, and contradiction functions, together with compatibility constraints (Fujita, 2024). Likewise, the 2018 operator theory makes clear that the contradiction degree affects intersection, union, complement, inclusion, equality, and partial order, rather than merely annotating values after the fact (Smarandache, 2018).

Taken together, the 2018 foundational monograph, the 2026 survey, and the 2024 graph-theoretic extension depict plithogenic sets as a general, contradiction-aware formalism with two defining commitments: attribute-centered representation and contradiction-sensitive aggregation. Their mathematical purpose is to refine generalized set operations when the semantics of attribute values are themselves graded, heterogeneous, and partially opposed (Smarandache, 2018, Fujita et al., 12 Mar 2026, Fujita, 2024).