Picture Fuzzy

A picture fuzzy data structure expresses an alternative's assessment on a criterion through three degrees: support for the judgement (μ, yes), abstention (η, neither yes nor no), and rejection (ν, no). All three lie between 0 and 1, and their sum cannot exceed 1. The remainder is the "no opinion given" share: those who did not vote, left the question blank, or did not engage with it.

This structure is the natural counterpart of a vote or a survey. For a board member, "55 per cent yes, 20 per cent abstain, 25 per cent no" is directly the cell (0.55, 0.20, 0.25). Unlike the intuitionistic fuzzy structure, the abstention share is not left over from support and rejection; it is a separate degree. Unlike the neutrosophic structure, the three degrees are bound together: the sum cannot exceed 1, because the same person cannot say both a full yes and a full no at once.

Use it wherever the assessment is gathered as a vote, a survey, or a distribution of opinion. A board's votes on a proposal, a survey's support/opposition/undecided proportions, an expert panel's spread of opinion: these are the type's natural sources. It suits problems where the undecided share matters to the decision itself, where "wherever the undecided go, the result follows."

By contrast, where the assessment is a single expert's single numerical judgement, or the abstention share was never separately measured, there is no need for the picture fuzzy structure; inventing that share afterwards removes its whole contribution.

Yes, but each of the three degrees must come from separately counted shares of the same measurement. Two steps are required.

First, turn the criterion into a judgement: the proposition "the alternative meets this criterion." A price or a measurement is not itself true or false; but the judgement "this price is acceptable" can be counted as yes, abstain or no.

Second, count each of the three degrees separately from the same source: μ as the proportion saying yes, η as the proportion abstaining, ν as the proportion saying no. Those who gave no opinion are written into none of the three degrees; the remaining share is theirs.

What must not be done is deriving the other two degrees from a single support figure: "55 per cent support, so 45 per cent must be against" sets the abstention share to zero and reduces the structure to an ordinary proportion.

The abstention degree (η) is the share of those who heard the question and said "neither yes nor no." The no-opinion share is the share of those who did not answer at all, measured by what is left over from the sum of the three degrees. Someone who ticks "undecided" is not the same as someone who leaves the survey blank.

For that reason:

"Everyone who is neither yes nor no is an abstainer"

is better replaced by:

"Abstention is counted separately; non-response is kept as the share left outside the three degrees"

If the abstention share and the non-response share are put in the same basket, the audience that could be won over with more information gets confused with the audience that cannot be reached at all.

The chief advantage of the picture fuzzy structure is that it carries a distribution of opinion into the calculation without reducing it to a single proportion. Where two alternatives share the same support rate but one has many opponents and the other many undecided voters, this difference is preserved and shown in the ranking.

Furthermore, keeping abstention as a separate component shows how "settled" the decision is: the result for an alternative with a large abstention share can still shift with further information.

Because the three degrees are counted from the same source, they are bound together; missing information and opposing opinion cannot be represented separately. "I am undecided" and "there is no information on this" fall into the same component. Where the unknown share must be carried independently of the opposing view, a neutrosophic structure is required.

For ranking, the three degrees must be reduced to one value, and how abstention is weighted in that reduction is itself a choice; different score functions can give different rankings. The structure is designed for a distribution of opinion; it does not suit a single expert's approximate judgement (fuzzy) or several plausible values (hesitant).

The most frequent mistake is generating a triple from a single proportion: writing (0.55, 0, 0.45) from "55 per cent support" ignores the abstention share, and erases the structure's one real contribution.

The second is filling in the abstention share afterwards by guesswork: if a survey offered no "undecided" option, η is simply unknown, not "presumably around 10 per cent." A third is forcing the triple's sum to equal 1, or writing one whose sum exceeds 1; the first erases the non-response share, the second is invalid for this type.

The governing principle is this:

Every degree in a picture fuzzy triple must be a share counted separately in the same measurement; no degree may be derived from the others, and their sum must not exceed 1.

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