How do human beings aggregate subjective categories, and which mathematical models describe this procedure adequately?

—H.‐J. Zimmermann (question from the last chapter of his book Fuzzy Set Theory— and Its Applications)

In Part Two of the book, we study graded evaluation logic and develop logic aggregators using a strictly anthropocentric approach. We try to answer the fundamental question, “How do human beings aggregate subjective categories, and which mathematical models describe this procedure adequately?” [ZIM96]. We investigate this problem in the context of evaluation reasoning. We also believe that all books about logic and/or decision making should ask this fundamental question explicitly and as early as possible (well, not in the last chapter). In our case, the appropriate place for this question is right here. Then, in Section 2.1.8, we offer an explicit answer to this question.

In the area of aggregation, as the point of departure, it is necessary to differentiate two categories of aggregation problems:

1. Mathematical problems of aggregation of anonymous real numbers.
2. Decision engineering problems of aggregation of arguments that have specific semantic identity.

In the case of aggregation of anonymous real numbers, the goal is to study the widest possible class of aggregation functions. In the case of aggregation of arguments that have semantic identity (i.e. the role, meaning, units of measurement, and impact on attaining specific stakeholder's goals) it is necessary to create aggregators that provide appropriate support to the semantic identity of arguments. Following are three most important types of semantic identity:

• Degree of truth of a value statement (a statement that claims the complete satisfaction of justifiable requirements of a specific stakeholder).
• Degree of fuzzy membership (the membership in a fuzzy set of objects that have specific, precisely defined properties, relevant for a given stakeholder).
• Probability (the likelihood of occurrence of a precisely defined event, relevant for specific stakeholder).

Each type of semantic identity affects the necessary properties of basic aggregation operators, and the process of development of compound aggregation structures.

The evaluation methodology, its theoretical background, and all applications presented in this book are based on the following principles of anthropocentric logic aggregation:

1. Logic is a key component of human reasoning. Therefore, logic should be based on observable properties of human reasoning, and it should serve for quantitative modeling of human reasoning.
2. Decisions are the results of human mental activities. Consequently, trustworthy decision models cannot be developed without relating them explicitly to observable patterns of human reasoning.
3. The concepts of suitability, preference, value, importance, simultaneity and substitutability are human percepts. These percepts are primarily used in evaluation decision making.
4. Generally, all human percepts used in evaluation are graded: they have intensity (or degree) that varies in the range from the lowest (quantified as 0) to the highest (quantified as 1).
5. A stakeholder is defined as a decision maker (a single person, or a group, or an organization) who is interested in the ownership and/or use of a specific object. By definition, the stakeholder is capable and authorized to specify goals and requirements that are expected to be satisfied by one or more evaluated objects/alternatives.
6. The assertion that an object or alternative completely satisfies selected stakeholder’s requirements is called the value statement. The degree of truth of a value statement is a human percept that reflects the degree of similarity between the statement and the objective reality. The degree of truth of a value statement can be quantified as a score that is used as an indicator of suitability or preference.
7. Evaluation is a process of creating the percept of overall suitability/preference of an evaluated object. Evaluation must take into account all relevant suitability attributes (i.e., only those attributes that provably affect the suitability of an evaluated object/alternative; various attributes that don’t affect the overall suitability, are irrelevant and should be neglected in the process of evaluation).
8. Evaluation can be intuitive (based on intuitive aggregation of attribute degrees of suitability) or quantitative (based on a mathematical model of the overall degree of satisfaction of stakeholder’s requirements). The result of evaluation creates an intuitive percept (or a quantitative indicator) of the overall suitability, preference, or value of each alternative.
9. The partial truth is a human percept that has intensity from completely false (0) to completely true (1). By definition, the suitability, preference, and value are degrees of truth of corresponding value statements. They can also be interpreted as degrees of fuzzy membership.
10. The number of evaluated objects/alternatives can be one or more. Consequently, evaluation methods must be applicable for evaluation of a single object. Since the evaluation process can be applied to a single alternative, it follows that evaluation must be possible without pairwise comparison of competitive alternatives. Evaluation of complex objects/alternatives is primarily a cardinal process (and much less frequently an ordinal process). Stakeholders are always interested to know the overall cardinal value of each alternative (the degree of satisfaction of their justified requirements), even in cases where evaluation criteria include some pairwise comparisons. The ranking of alternatives can always be a natural consequence of the comparison of cardinal values of alternatives. The goal of evaluation is not to show that object A is better than the object B, but to show how good is each of them (obviously, A can be better than B while both of them are unacceptable).
11. Each evaluation process creates and aggregates degrees of truth. Consequently, evaluation is a logic process based on graded truth values. A graded evaluation logic is a propositional calculus used for processing degrees of partial truth (or fuzzy membership) defined on the interval [0,1].
12. The propositional calculus inside the hypercube must be consistent with the propositional calculus in vertices {0, 1}ⁿ (Boolean logic). Consequently, the graded evaluation logic must be a seamless generalization of classical Boolean logic.

These principles are easily defendable and clearly show in which direction we intend to move and what school of thought we want to promote. Our focal point is the observable human reasoning. It is important to note that the interest in human‐centric approach to logic is clearly visible in the work of the founders of modern logic [DEM47, BOO47, BOO54].

Logic aggregation is the central component of our methodology, and consequently, this book complements other books on aggregation. At this time the most general books on aggregation are [BEL07] and [GRA09]. These books promote a purely mathematical approach to aggregation as a formal theory: They define aggregation functions (aggregators) in an ultimately permissive way as nondecreasing (in each variable) functions , that satisfy boundary conditions and . The result of this definition is an extremely wide family of functions that are not focused on any particular application area. More significantly, this definition also eliminates the need for semantics of aggregation, i.e., it is neither necessary nor interesting to know what is being aggregated and for what reason. The general purpose of aggregation is to aggregate anonymous real numbers. Of course, this is a legitimate approach that perfectly fits in mathematics, but it does not fit in decision engineering. In decision engineering, we don’t aggregate anonymous real numbers, but meaningful degrees of truth. Each degree of truth has precisely defined semantic identity and a justifiable degree of importance. In addition, such degrees of truth are used to make decisions that have both organizational and financial consequences. Therefore, in decision engineering, the purpose of aggregation is insight, not numbers. That yields the need for a modified definition of aggregators.

A more restrictive approach to aggregation is presented in [TOR07] and [BEL16], where the focus is mostly on averaging aggregators and applicability in the area of modeling decisions. This approach increases the practical value of the studied material and interacts with the mathematical literature devoted to means [GIN58, MIT69, MIT77, MIT89, BUL88, BUL03].

In the areas of averaging aggregators and means, the most fundamental properties are internality and idempotency. However, the internality is most frequently interpreted statistically as averaging and not logically as a continuous transition from conjunction to disjunction. In this book, we interpret means as logic functions, and our approach to aggregators will be strictly human‐centric and logic oriented. We will define and use logic aggregators as models of human aggregation of degrees of truth, or (what is equivalent) degrees of fuzzy membership.

Human evaluation reasoning can be investigated from many different standpoints. In particular, in this book we do not study evaluation reasoning from the standpoint of philosophy, psychology, sociology, and other social sciences. We are only interested in those aspects of intuitive evaluation processes that are observable and can be quantitatively modeled with intention to develop computerized decision‐support systems used in decision engineering. Therefore, our primary area of interest is restricted to evaluation and its modeling; it is intentionally kept sufficiently narrow to secure the depth of our study and to avoid conflicting interference with other disciplines.

Observations and analyses of evaluation in the context of human intuitive and professional reasoning show that this logic process has both semantic and formal logic components (see Section 2.2.1). Semantic components reflect the goals, interests, and justified requirements of the stakeholder (decision maker) and affect the selection of relevant attributes of evaluated objects and specification of their overall importance. The formal logic components primarily define the aggregation of subjective categories, where “subjective categories” denote the percepts of value of selected components of evaluated objects. Thus, when we use the term evaluation reasoning in the context of logic, we primarily think about the process of aggregation of various percepts of value.

A natural way to answer the fundamental question about the aggregation of subjective categories is based on careful observation of human intuitive evaluation reasoning (primarily the evaluation reasoning in the context of complex professional decision problems, but also various frequently visible processes of intuitive evaluation in the context of personal decision making). The observations show that the human aggregation of subjective categories is most frequently characterized by the continuity, monotonicity, and grading of all variables and parameters (e.g., the degrees of truth, importance, simultaneity, and substitutability), as well as the internality (idempotency), noncommutativity, and compensativeness of aggregators. All these properties are clearly visible in the process of creating human evaluation decisions (i.e., in the intuitive assessment of worth). Their systematic and detailed analysis is presented in Chapter 2.2.

Mathematical models of logic aggregation of various percepts of value form an infinite‐valued propositional calculus that can be identified as the evaluation logic. The main property of evaluation logic is that it operates with graded values. Such logic is based on degrees of truth, degrees of suitability, degrees of importance, degrees of conjunction, and degrees of disjunction: all values are graded. So, it is reasonable to call such logic a graded logic (GL).

GL is a necessary theoretical background for evaluation methodology presented in Part Three and Part Four of this book. As a propositional calculus, GL has the goal to determine the degree of truth of a compound statement from the degrees of truth of component statements and the logic connectives between them. All degrees of truth can also be interpreted as degrees of fuzzy membership, because fuzziness and partial truth are closely related interpretations of the same perceptual reality [DUJ17].

In extreme cases, the degrees of truth can be bivalent (0 or 1), and in such cases GL becomes the classical Boolean propositional logic. Thus, GL must be developed as a seamless generalization of the classical bivalent propositional calculus. That is the objective of Part Two of this book.

The search for the most appropriate name of GL produced multiple alternatives. Initially, in [DUJ73b] GL was interpreted as a “generalization of some functions in continuous mathematical logic.” Then, in [DUJ07c], the use of GL aggregators for computing preferences resulted in the name “preference logic for system evaluation.” Contrary to classical logic, GL uses weights to express importance of statements and has strong compensative properties that resulted in the name “weighted compensative logic” in [DUJ15a]. As a generalization of classical bivalent logic, GL might also be called “the generalized Boolean logic” and, because of its applicability in soft computing evaluation models, GL can also be called “the soft computing evaluation logic” or shortly “evaluation logic.” The name “graded logic” that we use in this book has an obvious advantage: it is short and it is not limiting GL only to the area of evaluation. Indeed, GL models can be useful in many areas and in applications that are different from decision models.

The goal of Part Two of this book is to present the development of GL and its main properties. Initially, we study the necessary generalizations of Boolean logic, and the properties of fundamental functions used in GL models. Then, we present and compare various mathematical models of aggregators. Finally, we study aggregation structures used for modeling complex criteria and aggregation methods that are necessary in professional evaluation of complex alternatives.