Fuzzy and Neutrosophic Systems for Medical Diagnosis: A

Mathematical Review of Uncertainty Representation and Decision

Models

R. Sivasamy1,∗, M. Mohammed Jabarulla1, Broumi Said2

1Department of Mathematics, Jamal Mohamed College, Tiruchirappalli–620020, Bharathidasan University,

Tamil Nadu, India

2Laboratory of Information Processing, Faculty of Science Ben M’Sik, Hassan II University, Casablanca,

Morocco

Abstract

Medical diagnosis is an uncertainty-sensitive decision problem in which a patient state x = (x1, . . . , xm) is

mapped to a candidate disease dk ∈ C using incomplete, gradual, and sometimes conflicting evidence. Fuzzy

systems represent such evidence by a membership grade μ(x) ∈ [0, 1]; intuitionistic fuzzy systems use independently

assigned (μ, ν) with the derived hesitation π = 1 − μ − ν; and single-valued neutrosophic systems

use an independently specified triple (T, I, F ) ∈ [0, 1 ]3. This article provides a structured mathematical review

of these models in medical diagnosis using literature available no later than 31 May 2024. The literature

is synthesized by representation space, constraints, distance and similarity measures, aggregation operators,

temporal structure, and diagnostic decision rules. A common embedding Φ places fuzzy and intuitionistic

fuzzy states inside the neutrosophic cube Ω = [0, 1]3, making the geometric relation among the three model

families explicit. Within this space, a weighted diagnostic distance is written as

D(r)

k =





j=1

(|Tpj − Tkj |r + |Ipj − Ikj |r + |Fpj − Fkj |r)





1/r

and the normalization residual κ = T + I + F − 1 is used to distinguish normalized states from underspecified

or overlapping evidence without interpreting (T, I, F) as probabilities. The review further develops

a synthesis-derived decision layer that combines distance, indeterminacy, and diagnostic margin through

eSk = (1 + eDk)−1, Ck = eSk(1 − ¯Ip)γ, and ΔC = C(1) − C(2). The analysis indicates that fuzzy systems

remain appropriate when uncertainty is predominantly gradual and rule interpretability is central, whereas

neutrosophic systems are better justified when independent support, opposition, and indeterminacy must be

retained. The resulting taxonomy clarifies when additional uncertainty dimensions are mathematically informative

and identifies calibration, metric stability, explainability, temporal modeling, and clinical validation as

the principal unresolved problems.

Keywords: Fuzzy systems; Neutrosophic sets; Medical diagnosis; Intuitionistic fuzzy sets; Clinical decision

support; Similarity measures; Uncertainty modeling

1 Introduction

Clinical diagnosis is fundamentally an inference problem in which a vector of observations x =

(x1, x2, . . . , xm) ∈ Rm must be mapped to a disease class dk ∈ C. In an ideal deterministic model, one would