rnel: Refined Neutrosophic Evidential Logic

A toolkit for neutrosophic evidence

Documentation: https://mleyvaz.github.io/rnel/

rnel computes with typed evidence. A probability of truth cannot say why a system is unsure: two sources that contradict each other, a single source with balanced evidence, and no evidence at all can all give 0.5. RNEL (Smarandache and Leyva-Vázquez) keeps them apart:

With only T and F it reduces exactly to Subjective Logic (Jøsang) and to evidential deep learning (Sensoy et al.).

Install

pip install rnel                                    # core, no dependencies
pip install "rnel[credal]"                          # with numpy, scikit-learn and scipy
pip install "rnel[nn]"                              # with PyTorch models
pip install git+https://github.com/mleyvaz/rnel     # development version

The core has no dependencies. The modules credal, neutro_credal, existential, mvc, active and cluster use numpy, scipy or scikit-learn (install rnel[credal]), and rnel.nn uses PyTorch (install rnel[nn]).

Betting triplet (experimental)

from rnel.betting import gaussian_evalues, log_ratio_from_ci, betting_triplet

# IPASS: progression-free survival hazard ratios with and without EGFR mutation
est = [log_ratio_from_ci(0.48, 0.36, 0.64), log_ratio_from_ci(2.85, 2.05, 3.98)]
lp, lm = gaussian_evalues([e for e, _ in est], [s for _, s in est], delta=0.2877, direction="less")
betting_triplet(lp, lm, lam=0.5)["verdict"]   # 'conflict'      (benefit in one stratum, harm in the other)
betting_triplet(lp, lm, lam=1.0)["verdict"]   # 'insufficient'  (pooling cancels them)

Refined truth (contributed by F. Smarandache): relative truth ("true in at least one source") and absolute truth ("true in every source"), each with its own error bound.

import math
from rnel.betting import refined_triplet
lp = [math.log(30), math.log(8), math.log(0.5)]          # e-values for A from three sources
r = refined_triplet(lp, [-x for x in lp], lam=0.5)
r["T_rel"], r["T_abs"], r["verdict"]                      # (0.981, 0.0, 'supported in some source')

Typed plithogenic statistics (experimental)

Contradiction degree between attribute values computed from data, with a guarantee (contributed by F. Smarandache).

from rnel.plithogeny import typed_plithogenic_report
r = typed_plithogenic_report({"lab1": (62, 30, 8), "lab2": (55, 38, 7), "lab3": (20, 72, 8)})
r["c_hat"]   # {('lab1','lab2'): 0.0, ('lab1','lab3'): 0.208, ('lab2','lab3'): 0.125}

Quick start

from rnel import Reports, rnel_tuple, fused_contradiction
from rnel.decide import Policy

# two confident sources that disagree, and a fair coin: SL fuses both to the same opinion
x = fused_contradiction([Reports(t=10), Reports(f=10)])
coin = fused_contradiction([Reports(t=10, f=10)])
print(x.C, coin.C)            # 0.579 vs 0.0 (Proposition 8.3 / Definition 8.4)
print(Policy().decide(x))     # ('C', 'present both positions and flag the dispute')

# the evidence tuple of Definition 8.1
print(rnel_tuple(Reports(t=3, f=1, c=2, v=1, n=1)).as_dict())

From text, with a public NLI model and no training (pip install transformers torch):

from rnel.text import NLIReader, rnel_from_text
t, readings = rnel_from_text(NLIReader(), "Coffee increases the risk of heart disease.",
                             ["A cohort found higher coffee intake linked to more heart disease.",
                              "A meta-analysis found moderate coffee lowers the risk of heart disease."])

NLI models often label unrelated text as "contradiction", so give only evidence that is about the claim.

Demo app: streamlit run examples/app.py.

Credal tools for neutrosophic triples

rnel.neutro_credal (experimental; pip install "rnel[credal]", needs numpy and scipy) reads a triple (T, I, F) of an event as the bounds T <= P(A) <= 1 - F of a credal set (Walley's betting reading). Through this reading the neutrosophic triple inherits Walley's quality checks and tools, applied directly to triples:

from rnel import neutro_credal as nc

members = [(0.7, 0.2, 0.1), (0.6, 0.3, 0.1), (0.1, 0.8, 0.1)]   # three classifiers, classes A, B, C
rep = nc.credal_diagnosis(members, labels=["A", "B", "C"], events={"A or C": ["A", "C"]})
rep["triples"]         # A (0.1, 0.6, 0.3), B (0.2, 0.6, 0.2), C (0.1, 0.0, 0.9): intervals [0.1,0.7], [0.2,0.8], [0.1,0.1]
rep["status"]          # 'coherent'
rep["events"]["A or C"]  # (0.2, 0.6, 0.2): P(A or C) in [0.2, 0.8]
rep["decision"]        # ['A', 'B'] (maximality; E-admissibility and interval dominance agree here)
rep["rnel"]            # why: per class a (support), b (rejection), u (absence), c (conflict between sources); C*

nc.sure_loss_degree((0.7, 0.1, 0.6))       # 0.15 = (T + F - 1) / 2
nc.to_glut_frame((0.8, 0.3, 0.7)).represents()   # True: the glut frame represents T + F > 1

The same call accepts a fitted rnel.credal.CredalEnsemble (credal_diagnosis(ensemble=ens, X=X)), using the leaf counts of each tree as evidence per source. A longer walk-through is in examples/neutro_credal_demo.py. Theorem numbers refer to Leyva-Vázquez and Smarandache, Neutrosophic probability and credal sets: exact reduction theorems and a glut lifting (manuscript, 2026); the tests check the closed forms of Theorem 2 against linear programming on 3000 random assessments (16 812 events, maximum error 3.3e-16) and reproduce Table 2 and counterexamples C1-C4.

Modules

Verified

pytest runs the results of the papers and book as automated tests. Numbered results carry @pytest.mark.theorem(...); the generated theorem-to-test map records the current exact counts.

• the worked example of Proposition 8.3 and Definition 8.4;
• Theorem 8.2;
• Theorem 5.2 on random tuples: involution, De Morgan, multiplicative mass, and projection onto SL multiplication and comultiplication;
• Theorem 6.2 for n = 3;
• De Morgan and SL projection for the RNEL operators;
• the neural head and the per-source network.

Version 0.4.0 adds rnel.betting (with refined truth) and typed plithogenic statistics in rnel.plithogeny. Version 0.3.0 consolidates the credal, conflict, refined-statistics, multi-view learning, DSmT, copula, plithogenic and score modules used by Neutrosophic Evidence, and adds rnel.existential. It is archived at https://doi.org/10.5281/zenodo.23178344 (version 0.2.0: https://doi.org/10.5281/zenodo.23040628; all versions: https://doi.org/10.5281/zenodo.23040627).

Evidence on real data

The preregistered evaluation of the neural head on AVeriTeC, CLIMATE-FEVER and ChaosNLI is at https://github.com/mleyvaz/rnel-neural-factcheck.

Cite

F. Smarandache, M. Y. Leyva-Vázquez, The Neutrosophic (T, I, N, F), Multi-Uncertainty, n-Valued Refined, and Refined Evidential Logics with Neutrosophic Probability as Extensions of Subjective Logic, manuscript, 2026.

MIT licence.