Fuzzy Logic III

"Another direction of research is fuzzy systems. This will greatly increase the use of mathematics from the inanimate to the animate. In the past, mathematics has been used for the analysis of physical systems. With fuzzy systems and computer simulation we can study many processes in the social sciences." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"In the real world, none of these assumptions are uniformly valid. Often people want to know why mathematics and computers cannot be used to handle the meaningful problems of society, as opposed, let us say, to the moon boondoggle and high energy-high cost physics. The answer lies in the fact that we don't know how to describe the complex systems of society involving people, we don't understand cause and effect, which is to say the consequences of decisions, and we don't even know how to make our objectives reasonably precise. None of the requirements of classical science are met. Gradually, a new methodology for dealing with these 'fuzzy' problems is being developed, but the path is not easy." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"One advantage of the use of fuzzy models is the fact that their complexity can be gradually increased as more information is gathered. This increase in complexity can be done automatically or manually by a careful commission of the new operating point." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)

"Each fuzzy set is uniquely defined by a membership function. […] There are two approaches to determining a membership function. The first approach is to use the knowledge of human experts. Because fuzzy sets are often used to formulate human knowledge, membership functions represent a part of human knowledge. Usually, this approach can only give a rough formula of the membership function and fine-tuning is required. The second approach is to use data collected from various sensors to determine the membership function. Specifically, we first specify the structure of membership function and then fine-tune the parameters of membership function based on the data." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)

"Logic is the study of methods and principles of reasoning, where reasoning means obtaining new propositions from existing propositions. In classical logic, propositions are required to be either true or false; that is, the truth value of a proposition is either 0 or 1. Fuzzy logic generalizes classical two-value logic by allowing the truth values of a proposition to be any numbers in [0, 1]. This generalization allows us to perform fuzzy reasoning, also called approximate reasoning; that is, deducing imprecise conclusions (fuzzy propositions) from a collection of imprecise premises (fuzzy propositions). In this section, we first introduce some basic concepts and principles in classical logic and then study their generalizations to fuzzy logic." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)

"Fuzzy logic is an application area of fuzzy set theory dealing with uncertainty in reasoning. It utilizes concepts, principles, and methods developed within fuzzy set theory for formulating various forms of sound approximate reasoning. Fuzzy logic allows for set membership values to range (inclusively) between 0 and 1, and in its linguistic form, imprecise concepts like 'slightly', 'quite' and 'very'. Specifically, it allows partial membership in a set." (Larbi Esmahi et al,  Adaptive Neuro-Fuzzy Systems, 2009)

"Like classical logic, fuzzy logic uses formulas to formally represent statements about the world. Given an appropriate semantic structure (such as an evaluation of propositional symbols in the case of propositional logic, or a relational structure in the case of predicate logic), a truth degree of formula ϕ is denoted by ||ϕ||. It is significant that the truth degree ||ϕ|| of ϕ may in general be any element of the set of truth degrees. That is, formulas in fuzzy logic are true to degrees , not just true or false as in the case of classical logic." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"Nevertheless, the use of fuzzy logic is supported by at least the following three arguments. First, fuzzy logic is rooted in the intuitively appealing idea that the truths of propositions used by humans are a matter of degree. An important consequence is that the basic principles and concepts of fuzzy logic are easily understood. Second, fuzzy logic has led to many successful applications, including many commercial products, in which the crucial part relies on representing and dealing with statements in natural language that involve vague terms. Third, fuzzy logic is a proper generalization of classical logic, follows an agenda similar to that of classical logic, and has already been highly developed. An important consequence is that fuzzy logic extends the rich realm of applications of classical logic to applications in which the bivalent character of classical logic is a limiting factor." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"The principal idea employed by fuzzy logic is to allow for a partially ordered scale of truth-values, called also truth degrees, which contains the values representing false and true , but also some additional, intermediary truth degrees. That is, the set {0,1} of truth-values of classical logic, where 0 and 1 represent false and true , respectively, is replaced in fuzzy logic by a partially ordered scale of truth degrees with the smallest degree being 0 and the largest one being 1." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"We use the term fuzzy logic to refer to all aspects of representing and manipulating knowledge that employ intermediary truth-values. This general, commonsense meaning of the term fuzzy logic encompasses, in particular, fuzzy sets, fuzzy relations, and formal deductive systems that admit intermediary truth-values, as well as the various methods based on them." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

On Networks XVII (Semantic Networks I)

"In comparison with Predicate Calculus encoding s of factual knowledge, semantic nets seem more natural and understandable. This is due to the one-to-one correspondence between nodes and the concepts they denote, to the clustering about a particular node of propositions about a particular thing, and to the visual immediacy of 'interrelationships' between concepts, i.e., their connections via sequences of propositional links." (Lenhart K Schubert, "Extending the Expressive Power of Semantic Networks", Artificial Intelligence 7, 1976)

"[…] semantic nets [are defined] as graphical analogues of data structures representing "facts" in a computer system for understanding natural language." (Lenhart K Schubert," "Extending the Expressive Power of Semantic Networks", Artificial Intelligence 7, 1976)

"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1 (1), 1980)

"We define a semantic network as 'the collection of all the relationships that concepts have to other concepts, to percepts, to procedures, and to motor mechanisms' of the knowledge." (John F Sowa, "Conceptual Structures", 1984)

"[…] semantic nets fail to be distinctive in the way they (1) represent propositions, (2) cluster information for access, (3) handle property inheritance, and (4) handle general inference; in other words, they lack distinctive representational properties (i.e., 1) and distinctive computational properties (i.e., 2-4). Certain propagation mechanisms, notably 'spreading activation', 'intersection search', or 'inference propagation' have sometimes been regarded as earmarks of semantic nets, but since most extant semantic nets lack such mechanisms, they cannot be considered criterial in current usage." (Lenhart K Schubert, "Semantic Nets are in the Eye of the Beholder", 1990)

"[…] the representational and computational strategies employed in semantic net systems are abstractly equivalent to those employed in virtually all state-of-the-art systems incorporating a substantial propositional knowledge base, whether they are described as logic-based, frame-based, rule-based, or some-thing else." (Lenhart K Schubert, "Semantic Nets are in the Eye of the Beholder", 1990)

"A semantic network or net represents knowledge as a net-like graph. An idea, event, situation or object almost always has a composite structure; this is represented in a semantic network by a corresponding structure of nodes (drawn as circles or boxes) representing conceptual units, and directed links (drawn as arrows between the nodes) representing the relations between the units. […] An abstract (graph-theoretic) network can be diagrammed, defined mathematically, programmed in a computer, or hard-wired electronically. It becomes semantic when you assign a meaning to each node and link. Unlike specialized networks and diagrams, semantic networks aim to represent any kind of knowledge which can be described in natural language. A semantic network system includes not only the explicitly stored net structure but also methods for automatically deriving from that a much larger structure or body of implied knowledge." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The essential idea of semantic networks is that the graph-theoretic structure of relations and. abstractions can be used for inference as well as understanding. […] A semantic network is a discrete structure as is any linguistic description. Representation of the continuous 'outside world' with such a structure is necessarily incomplete, and requires decisions as to which information is kept and which is lost." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The great organizing principle of thought is abstraction. By assigning particular things to abstract categories we are able to dispense with irrelevant detail and yet instantly draw copious conclusions about a thing due to its membership in various categories. Semantic networks specify the structure of interrelated abstract categories and use this structure to draw conclusions." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

On Diagrams (1950-1974)

"The diagrams incorporate a large amount of information. Their use provides extensive savings in space and in mental effort. In the case of many theorems, the setting up of the correct diagram is the major part of the proof. We therefore urge that the reader stop at the end of each theorem and attempt to construct for himself the relevant diagram before examining the one which is given in the text. Once this is done, the subsequent demonstration can be followed more readily; in fact, the reader can usually supply it himself." (Samuel Eilenberg & Norman E. Steenrod, "Foundations of Algebraic Topology", 1952)

"A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. […] A logic diagram is a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement. These spatial relations are usually of a topological character, which is not surprising in view of the fact that logic relations are the primitive relations underlying all deductive reasoning and topological properties are, in a sense, the most fundamental properties of spatial structures. Logic diagrams stand in the same relation to logical algebras as the graphs of curves stand in relation to their algebraic formulas; they are simply other ways of symbolizing the same basic structure." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"The diagrams and circles aid the understanding by making it easy to visualize the elements of a given argument. They have considerable mnemonic value […] They have rhetorical value, not only arousing interest by their picturesque, cabalistic character, but also aiding in the demonstration of proofs and the teaching of doctrines. It is an investigative and inventive art. When ideas are combined in all possible ways, the new combinations start the mind thinking along novel channels and one is led to discover fresh truths and arguments, or to make new inventions. Finally, the Art possesses a kind of deductive power." (Martin Gardner, "Logic Machines and Diagrams", 1958) 

"An information retrieval system is therefore defined here as any device which aids access to documents specified by subject, and the operations associated with it. The documents can be books, journals, reports, atlases, or other records of thought, or any parts of such records - articles, chapters, sections, tables, diagrams, or even particular words. The retrieval devices can range from a bare list of contents to a large digital computer and its accessories. The operations can range from simple visual scanning to the most detailed programming." (Brian C Vickery, "The Structure of Information Retrieval Systems", 1959)

"Diagrams are sometimes used, not merely to convey several pieces of information such as several time series on one chart, but also to provide visual evidence of relationships between the series." (Alfred R Ilersic, "Statistics", 1959)

"It is extremely helpful to the imagination to have a geometric picture available in terms of which we can visualize sets and operations on sets. […] Diagrammatic thought of this kind is admittedly loose and imprecise; nevertheless, the reader will find it invaluable. No mathematics, however abstract it may appear, is ever carried on without the help of mental images of some kind, and these are often nebulous, personal, and difficult to describe." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"A model is a qualitative or quantitative representation of a process or endeavor that shows the effects of those factors which are significant for the purposes being considered. A model may be pictorial, descriptive, qualitative, or generally approximate in nature; or it may be mathematical and quantitative in nature and reasonably precise. It is important that effective means for modeling be understood such as analog, stochastic, procedural, scheduling, flow chart, schematic, and block diagrams." (Harold Chestnut, "Systems Engineering Tools", 1965) 

"A diagram thus enables us to discover the internal categorization which characterizes the information being processed in a much shorter time than does a map. […] A diagram permits the rapid and precise internal processing of information having three components, but it does not permit introducing the information into a universal system of visual memorization and geographic comparison. It is a closed graphic system, limited solely to the information being processed. […] In a diagram, one begins by attributing a meaning to the planar dimensions, then one plots the correspondences." (Jacques Bertin, "Semiology of graphics", 1967) 

"A graphic is a diagram when correspondences on the plane can be established among all elements of another component." (Jacques Bertin, "Semiology of graphics", 1967) 

"To analyse graphic representation precisely, it is helpful to distinguish it from musical, verbal and mathematical notations, all of which are perceived in a linear or temporal sequence. The graphic image also differs from figurative representation essentially polysemic, and from the animated image, governed by the laws of cinematographic time. Within the boundaries of graphics fall the fields of networks, diagrams and maps. The domain of graphic imagery ranges from the depiction of atomic structures to the representation of galaxies and extends into the spheres of topography and cartography."  (Jacques Bertin, "Semiology of graphics", 1967) 

"When the correspondences on the plane can be established between: - all the divisions of one component - and all the divisions of another component, the construction is a DIAGRAM." (Jacques Bertin, "Semiology of graphics", 1967) 

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"A diagram is worth a thousand proofs." (Carl E Linderholm, "Mathematics Made Difficult", 1971)

"Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows." (Saunders Mac Lane, "Categories for the Working Mathematician", 1971)

"The result of the implementation, the logical design, is traditionally shown as a series of block diagrams. These blocks represent in effect a series of statements, Actually, a direct presentation of these statements is more suitable and, although less familiar, more easily understood. The Harvard Mark IV was to large degree designed and described by such statements, as has been the case with several subsequent developments." (Gerrit Blaauw, "Computer Architecture", 1972) 

"Whether or not a given conceptual model or representation of a physical system happens to be picturable, is irrelevant to the semantics of the theory to which it eventually becomes attached. Picturability is a fortunate psychological occurrence, not a scientific necessity. Few of the models that pass for visual representations are picturable anyhow. For one thing, the model may be and usually is constituted by imperceptible items such as unextended particles and invisible fields. True, a model can be given a graphic representation - but so can any idea as long as symbolic or conventional diagrams are allowed. Diagrams, whether representational or symbolic, are meaningless unless attached to some body of theory. On the other hand theories are in no need of diagrams save for psychological purposes. Let us then keep theoretical models apart from visual analogues."  (Mario Bunge, "Philosophy of Physics", 1973)

Anatol Rapoport - Collected Quotes

"The first attempts to consider the behavior of so-called ‘random neural nets’ in a systematic way have led to a series of problems concerned with relations between the 'structure' and the ‘function’ of such nets. The ‘structure’ of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of "this" neuron synapsing on ‘that’ one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport. "Cycle distributions in random nets." The Bulletin of Mathematical Biophysics 10 (3), 1948)

"A fundamental value in the scientific outlook is concern with the best available map of reality. The scientist will always seek a description of events which enables him to predict most by assuming least. He thus already prefers a particular form of behavior. If moralities are systems of preferences, here is at least one point at which science cannot be said to be completely without preferences. Science prefers good maps." (Anatol Rapoport, "Science and the goals of man: a study in semantic orientation", 1950)

"No map contains all the information about the territory it represents. The road map we get at the gasoline station may show all the roads in the state, but it will not as a rule show latitude and longitude. A physical map goes into details about the topography of a country but is indifferent to political boundaries. Furthermore, the scale of the map makes a big difference. The smaller the scale the less features will be shown." (Anatol Rapoport, "Science and the goals of man: a study in semantic orientation", 1950)

"[…] theoretical science is essentially disciplined exploitation of metaphor." (Anatol Rapoport, "Operational Philosophy", 1953)

"Scientific metaphors are called models. They are made with the full knowledge that the connection between the metaphor and the real thing is primarily in the mind of the scientist. And they are made with a clearly definable purpose - as starting points of a deductive process. […] Like every other aspect of scientific procedure, the scientific metaphor is a pragmatic device, to be used freely as long as it serves its purpose, to be discarded without regrets when it fails to do so." (Anatol Rapoport, "Operational Philosophy", 1954) 

"The predictions of physical theories for the most part concern situations where initial conditions can be precisely specified. If such initial conditions are not found in nature, they can be arranged. Such arrangements are considerably easier to realize with inanimate than with animate matter, because the properties of animate matter are much more sensitive to being tampered with than inanimate matter. In particular, living tissue in vitro may behave quite differently than in situ. Controlled biological experiments are, of course, possible, but they are more difficult and their scope is more limited than that of physical experiments. For this reason, biology has had to depend to a greater extent than physics on theories of larger speculative scope, in which reasoning by imaginative analogy plays a more important role." (Anatol Rapoport, "The Search for Simplicity", 1956)

"A theorem is a proposition which is a strict logical consequence of certain definitions and other propositions" (Anatol Rapoport, "Various meanings of theory", American Political Science Review 52, 1958)

"A thorough understanding of game theory, should dim these greedy hopes. Knowledge of game theory does not make one a better card player, businessman or military strategist." (Anatol Rapoport, "The Use and Misuse of Game Theory", 1962)

"Although the drama of games of strategy is strongly linked with the psychological aspects of the conflict, game theory is not concerned with these aspects. Game theory, so to speak, plays the board. It is concerned only with the logical aspects of strategy." (Anatol Rapoport, "The Use and Misuse of Game Theory", 1962)

"The outstanding feature of behavior is that it is often quite easy to recognize but extremely difficult or impossible to describe with precision." (Anatol Rapoport. "An Essay on Mind", General Systems, 1962)

"[Game theory is] essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not." (Anatol Rapoport, "Prisoner's Dilemma: A study in conflict and cooperation", 1965)

"The usefulness of the models in constructing a testable theory of the process is severely limited by the quickly increasing number of parameters which must be estimated in order to compare the predictions of the models with empirical results" (Anatol Rapoport, "Prisoner's Dilemma: A study in conflict and cooperation", 1965)