Plithogenic DNMA

Four things change.

Cells. In crisp DNMA every cell is a single number. Here every cell is a truth (T), indeterminacy (I), falsity (F) triple. As on the plithogenic data-type card, one criterion is taken as dominant with a contradiction degree of zero; how far each other criterion opposes the dominant one is given by a contradiction degree between 0 and 1. For a cost criterion the triple is first complemented (T,I,F becomes F,I,T), every cell is then adjusted by its own criterion's contradiction degree, and it descends to a single number with the neutrosophic score function ((1+T-2I-F)/2). Crisp DNMA already expects a number; here this score simply takes the place of crisp DNMA's input number.

Two separate normalisations. Crisp DNMA equalises this score in two different ways: linear-maximum (dividing by the column maximum) and vector norm (dividing by the square root of the sum of the squared column values). The plithogenic extension applies the same two normalisations, but to the contradiction-adjusted score rather than the raw number. Linear normalisation always falls between 0 and 1; the vector norm need not.

Combining three measures, and an important limit. This extension's engine computes three measures: u1, the weighted sum of the linear norm; u2, a weighted sum built from the linear norm's complement; and u3, a ratio built with the vector norm. Algebraically, u2 is identical to u1 as long as the weights sum to 1: u2 = 1 − Σw(1−r) = Σw·r = u1. This is a property that comes from pure algebra, independent of the data; it has been verified by this card's author by running the kernel directly (see the verification notes). The consequence is that only two independent signals, not three, actually enter the final score. Furthermore, the rank-consistency penalty that is crisp DNMA's signature step (comparing the three models' rankings and penalising the inconsistent alternative) is absent from this extension's engine; the three measures are combined by a simple average alone (U = (u1+u2+u3)/3).

The contradiction degree's effect is partial and measured. The linear norm (u1, and hence u2) is entirely independent of the contradiction degree, exactly as in P-COCOSO and P-VIKOR; min-max scaling fully absorbs the linear transformation of the form c+(1−c)·s(0) produced by the contradiction adjustment. But the vector norm does not absorb this transformation, because it is not a scaling rule that accommodates an additive shift (the c term). As a result, u3 changes measurably as the contradiction degree rises: in this card's illustrative example, raising the contradiction degrees from 0 to 0.90 shrinks u3's spread across the three alternatives from 0.362 to 0.011, that is, to almost a tenth (see Case 1). The final U score is therefore also affected in a measured way; the finding that the contradiction degree has zero effect, which holds for P-COCOSO and P-VIKOR, does NOT hold in exactly the same way here.

DecisionMind holds the contradiction adjustment, the two normalisations, and the simple average of the three fixed in this extension. Weights come from outside; the method does not generate weights.

As in crisp DNMA, the U score is a number summarising how far several measures agree; it can be negative, and it cannot be compared with a different alternative set. But in this extension the claim that "three models confirm one another" is not as strong as in crisp DNMA, because u1 and u2 are already the same number and the rank-consistency penalty is never computed. Furthermore, u3's discriminating power narrows as the contradiction degree rises; in an analysis run with high contradiction degrees, most of the score effectively comes from a single signal (u1 = u2).

Thus instead of writing:

"P-DNMA offers an assurance arising from the consistency of three independent aggregation models, so the result is robust"

the report should read:

"In this engine u1 and u2 are algebraically the same number, and the rank-consistency penalty is not computed; the real discriminating contribution comes from u3, and that weakens as the contradiction degree rises"

Consider this extension when your criterion scores are given as truth-indeterminacy-falsity triples and some criteria are a more independent, or more contradictory, source of information than others. But because of the finding above, entering high contradiction degrees only leans further on u3, whose discriminating power is already weak; using the contradiction degrees as a "fine-tuning" tool is misleading. Turning a measured criterion into a T-I-F triple is producing uncertainty, not modelling it; this principle holds here too. The exit condition of crisp DNMA holds here as well: this extension is not suitable if a criterion column is constant or if no compromise is acceptable on one criterion.

Presenting the result as "the consensus of three independent models." u1 and u2 are always the same number; this engine effectively has two signals (u1=u2 and u3), not three. The report should say so.

Assuming the rank-consistency penalty is running. The crisp DNMA card describes this penalty as DNMA's signature; there is no such step in this extension's engine, and U is a simple average.

Treating the contradiction degree as a tool for "pulling the result the way I want." The contradiction degree affects the result only through u3, and only indirectly; it is not a direct, predictable lever.

Choosing the dominant criterion arbitrarily. Even though the contradiction degree's effect is partial, which criterion is treated as dominant (zero contradiction) must be stated openly and justified in the report.

Skipping the cost complementation. If the calculation proceeds without rewriting (T,I,F) as (F,I,T) for a cost criterion, the highest-cost alternative appears to have been drawn towards the ideal.

The governing principle is this:

In P-DNMA the number of independent signals feeding the final score is smaller than it appears, and the contradiction degree's effect lies, in a measured way, in only one of them; the report must show both limits together.