The high grade of precision predominating in other natural sciences (e.g., mathematics and physics) stands in sharp contrast with the imprecise and often ad hoc definitions which can be found in the medicine. Although medicine undoubtedly is a systematic and empirical scientific discipline, many traditional mathematical techniques fail when they are applied to medical problems. The main reason for this shortcoming may be the inherent uncertainty of medical knowledge and the vagueness of the concepts that are used to denote medical facts. As an example, the following statements about acute bronchitis can be found in medical textbooks: mostly seen in winter; in most cases due to viral infection; onset frequently accompanied by pharyngitis and laryngitis; dry cough for the first 3 days; later production of viscous eventually purulent sputum; frequently pronounced subjective feeling of illness; and; fever up to 39°C.

This typical definition of a disease shows the principal dilemma of the system analyst. Imprecise and vague terms like frequently, in most cases, pronounced, up to 39°C, etc. are common in medical textbooks. Nevertheless, physicians are able to draw reproducible conclusions in this “environment of uncertainty”. So that uncertainty and vagueness play a major role in the problem of medical knowledge representation, while natural languages (e.g., English) are quite suitable to express the uncertainty, present algorithmic languages call for precise recipes, and the translation from the first representation to the second presents a significant challenge. There are several types of vagueness that may appear in clinical knowledge presentation:

1. The Lack of Information: Not every observation of relevance to a medical fact may be available or has been collected, in which case an educated guess sometimes has to be made. Even if collected, the information can be unreliable.
2. Non-Specificity Connected with Sizes of Relevant Sets: Frequently medical facts refer to “other conditions”, “other risk factors”, “other significant co-morbidities”, leaving it up to the doctor to decide what they are. To be translated into an algorithmic language, an explicit list of those conditions is reasonably required.
3. The Probabilistic Nature of Data and Outcomes: There are few clinical signs that unequivocally point to a medical condition, and therefore to a predefined course of actions. Sensitivity and specificity of most clinical tests are far from ideal, and consequently they point to a likelihood, rather than presence or absence of medical condition. The outcome of any non-trivial recommendation is also, in a sense, a gamble. The words usually, likely, commonly, possibly, etc. express this type of uncertainty in natural languages.
4. Vagueness in the Formulation of Recommendations: What is the meaning of such phrases as suggested, recommended, should be strongly considered or not routinely warranted? The medical guidelines, in contrast to precise recipes suitable for direct translation into a computer language, allow for situations in which the recommendation may not be appropriate, without specifying the exact conditions. They urge —not force— doctors to follow the recommendations, and thus do not supplant their decision making.
5. Strife: Strife (or Discord) expresses conflicts among the various sets of alternatives. Often several medical facts may be applicable to the given patient circumstances, each pointing to a specific set of actions. Conflicting facts are not necessarily a feature of poor design or lack of expert agreement. The doctor then has to decide which action or combination of actions is the most appropriate.
6. The Fuzziness in Determination of Clinical Signs: This type of certainty can be subjectivity in the assessment of a patient’s symptoms, or in the interpretation of precise objective data, such as laboratory test results or even a patient’s age. What exactly is the size of an enlarged liver? What exactly do we mean by infants or middle-aged men?

Several mathematical formalisms have been proposed to treat uncertainty. The oldest and best-studied approach is the probabilistic one. It has sound axiomatic foundations laid by Kolmogorov in the 1930s, and allows various interpretations, among which are subjective approaches. In the framework of medical decision making and expert systems, it has been used since the 1970s. The pitfall in using probabilities is that the vast majority of conditional probabilities required for the Bayes rule are not available, and their subjective estimations by medical experts tend to be inconsistent and inaccurate. Directions explored in the past 30 years include belief transfer in semantic networks, Dempster-Shafer evidence theory, possibility theory, and more important, fuzzy set theory. The main advantage of fuzzy set theory is that it allows transparency in knowledge representation. The key concept of fuzzy set theory is that of partial membership of elements in a set. In the next chapters, we show how fuzzy set theory can be used in medical decision support.