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Possibility theory

Definition

Possibility theory is a mathematical framework for representing uncertainty through degrees of possibility and necessity. It can describe which alternatives remain compatible with incomplete or imprecise information, without requiring one precise probability distribution. Introduced by Lotfi Zadeh and developed further by other researchers, it is closely connected with fuzzy sets, while distinguishing uncertainty about a fact from the graded meaning of a vague description.

What it means

Suppose a supplier says that a delivery will arrive 'around Tuesday.' That statement may support several dates to different degrees without providing observed frequencies for those dates. A possibility model can retain the alternatives and their relative compatibility with the statement. Possibility asks how far an event remains open; necessity asks how strongly competing alternatives are excluded. Neither quantity should automatically be read as the percentage of deliveries that will satisfy the event.

How it works

For a finite set of alternatives, a normalized possibility distribution assigns each alternative a score pi between 0 and 1, with at least one score equal to 1. The possibility of an event A is the largest score among the alternatives in A. The empty event has possibility 0.
Necessity is defined by N(A) = 1 - Possibility(not A). Thus, an event becomes necessary to the extent that its complement is ruled out. Possibility uses the maximum when combining events by union, rather than adding their scores. Necessity uses the minimum for intersections. This entry uses the numerical scale and ordinary crisp events; qualitative versions can instead emphasize ordering. The choice of scale and interpretation must be specified before calculations are used operationally.

Three practical examples

• A planning team represents an imprecise delivery estimate across several days. It distinguishes a date that remains entirely possible from a deadline strongly supported by the available information.
• An equipment-diagnosis system records which faults remain compatible with a technician's incomplete observations. It avoids presenting these compatibility levels as measured fault probabilities unless a separate probabilistic justification exists.
• A knowledge system encodes statements with different levels of certainty. When sources disagree, it examines the conflict and source assumptions rather than averaging numbers whose meanings have not been aligned.

Think of it this way

Imagine a set of doors representing possible outcomes. Possibility describes how open an event's most open door remains. Necessity depends on how closed the doors outside that event are. One door being fully open does not mean you must pass through it if other doors also remain open.

Worked example

A fictional delivery has four mutually exclusive, exhaustive arrival alternatives: day 1, day 2, day 3 and day 4. An analyst explicitly chooses possibility scores 0.2, 1.0, 0.7 and 0.1 to represent the available imprecise report. These are illustrative judgments, not probabilities estimated from shipment data.
Let A mean 'arrives by day 2.' Its possibility is max(0.2, 1.0) = 1. Its complement contains days 3 and 4, with possibility max(0.7, 0.1) = 0.7. Therefore N(A) = 1 - 0.7 = 0.3. Arrival by day 2 is fully possible but only necessary to degree 0.3 in this model.
Let B mean 'arrives by day 3.' Its possibility is 1, and its complement contains only day 4, with score 0.1. Thus N(B) = 0.9. The later deadline has stronger necessity because fewer compatible alternatives lie beyond it.
For 'arrives on day 3,' possibility is 0.7 and necessity is 0: day 2 remains outside that event with score 1. Adding the scores for days 1 and 2 would give 1.2, which is not their union's possibility. The correct operation gives 1.
The four scores sum to 2.0. Dividing by that total would create numbers summing to 1, but that arithmetic would not establish a justified probability model. If every day instead had possibility 1, every nonempty proper subset would have possibility 1 and necessity 0. That represents no exclusion among the listed alternatives, rather than a claim that each occurs with probability one quarter.

Illustrative case study

In a fictional events business, a coordinator uses the delivery example to explain why an unqualified 'possible by day 2' label is inadequate for scheduling installation. Both deadlines have possibility 1, but their necessity values differ substantially.
The team documents how the scores were elicited, what evidence would change them and why day 4 is the final alternative in this simplified model. Before using the model for real shipments, it considers an additional 'later than day 4' alternative. Excluding that possibility without evidence could create false confidence. The team also separates the uncertainty description from the decision rule about when to book installers; the scores alone do not encode disruption costs.

Common misconceptions and mistakes

A possibility of 1 is not automatically certainty, and a necessity of 0 does not automatically make an event impossible. Possibility scores need not sum to 1. Nor is a fuzzy membership value automatically a probability: the interpretation of a vague property differs from uncertainty about whether a precise statement is true.
Information fusion also requires care. The appropriate combination rule depends on assumptions about source reliability, consistency and dependence. Choosing minimum, maximum or another operation merely because it produces a convenient answer can change what the result means.

Business use

Possibility theory can make incomplete judgments explicit when forcing a single precise distribution would conceal important assumptions. Record the alternatives, score meanings, evidence sources and interpretation of necessity. Explain what zero and one mean to users. Keep decision preferences separate from uncertainty: a costly failure may justify a different action even when the underlying possibility distribution is unchanged. Where reliable statistical information is available, compare with an appropriate probabilistic model rather than assuming possibility theory must replace it.

People also ask

Can two exclusive outcomes both have possibility 1?

Yes; neither may be excluded by the available information.

Is necessity the same as probability?

No. It is defined through the possibility of the complement.

Does possibility theory automatically resolve conflicting experts?

No. Conflict handling still requires explicit modeling and combination assumptions.

Check your understanding

What is the necessity of arrival by day 3?

Answer: It is 0.9 under the stated scores.

Why is day 3 not necessary despite possibility 0.7?

Answer: Another day outside that event has possibility 1.

Does dividing scores by their sum validate probabilities?

Answer: No; normalization alone supplies no probabilistic justification.

Related terms

• Degree of belief
• Possibility distribution
• necessity measure
• fuzzy set
• imprecise probability
• epistemic uncertainty
• possibilistic logic
• information fusion

Learn next

Add a fifth arrival alternative with possibility 0.4. Recalculate the necessity of arrival by day 3 and explain why the additional alternative changes the conclusion.

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Sources

Dubois and Prade, Modeling uncertain and vague knowledge in possibility and evidence theories: https://arxiv.org/pdf/1304.2349 Dubois, Liu, Ma and Prade, The basic principles of uncertain information fusion: https://research-information.bris.ac.uk/ws/files/104165943/IF2016.pdf AIMA glossary, possibility theory: https://github.com/aimacode/aima-glossary/blob/master/glossary.md

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