 # Fuzzy logic Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. Compared to traditional binary sets (where variables may take on true or false values) fuzzy logic variables may have a truth value that ranges in degree between 0 and 1. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false. Furthermore, when linguistic variables are used, these degrees may be managed by specific functions. The term "fuzzy logic" was introduced with the 1965 proposal of fuzzy set theory by Lotfi A. Overview Classical logic only permits propositions having a value of truth or falsity. Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. Applying truth values Fuzzy logic temperature In this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. Related:  Fuzzy

Fuzzy cognitive map A collection of papers with applications in various disciplines is presented in the book "Fuzzy Cognitive Maps: Advances in Theory, Methodologies, Applications and Tools edited by Michael Glykas and prefaced by Bart Kosko. A simple application of FCMs is described in a book of William R. Taylor, where the war in Afghanistan and Iraq is analyzed. And in Bart Kosko's book Fuzzy Thinking, several Hasse diagrams illustrate the use of FCMs. As an example, one FCM quoted from Rod Taber describes 11 factors of the American cocaine market and the relations between these factors. While applications in social sciences introduced FCMs to the public, they are used in a much wider range of applications, which all have to deal with creating and using models of uncertainty and complex processes and systems.

Completeness In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields. Logical completeness Mathematical completeness Computing Economics, finance, and industry Botany A complete flower is a flower with both male and female reproductive structures as well as petals and sepals. References Three-valued logic In logic, a three-valued logic (also trivalent, ternary, trinary logic, or trilean,[citation needed] sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Jan Łukasiewicz and C. Representation of values As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the ternary numeral system. Inside a ternary computer, ternary values are represented by ternary signals. This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, and true}, and extends conventional Boolean connectives to a trivalent context. Logics Kleene logic , and its truth table is

Model Theory 1. Basic notions of model theory Sometimes we write or speak a sentence S that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that S comes to express a true or false statement, we are said to interpret S, and the added information is called an interpretation of S. For example I might say He is killing all of them, and offer the interpretation that ‘he’ is Alfonso Arblaster of 35 The Crescent, Beetleford, and that ‘them’ are the pigeons in his loft. The structure I in the previous paragraph involves one fixed object and one fixed class. Note that the objects and classes in a structure carry labels that steer them to the right expressions in the sentence. If the same class is used to interpret all quantifiers, the class is called the domain or universe of the structure. One of those thingummy diseases is killing all the birds. 2. Take for example the following set of first-order sentences:

Fuzzy set It has been suggested by Thayer Watkins that Zadeh's ethnicity is an example of a fuzzy set because "His father was Turkish-Iranian (Azerbaijani) and his mother was Russian. His father was a journalist working in Baku, Azerbaijan in the Soviet Union...Lotfi was born in Baku in 1921 and lived there until his family moved to Tehran in 1931." Definition A fuzzy set is a pair where is a set and For each the value is called the grade of membership of in For a finite set the fuzzy set is often denoted by Let Then is called not included in the fuzzy set if is called fully included if , and is called a fuzzy member if . The set is called the support of and the set is called its kernel. is called the membership function of the fuzzy set of a given kind; usually it is required that Fuzzy logic As an extension of the case of multi-valued logic, valuations ( ) of propositional variables ( ) into a set of membership degrees ( Fuzzy number A fuzzy number is a convex, normalized fuzzy set . Entropy

Predicate logic In informal usage, the term "predicate logic" occasionally refers to first-order logic. Some authors consider the predicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from an informal, more intuitive development. Predicate logics also include logics mixing modal operators and quantifiers. See also  Jump up ^ Eric M. References A.

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. Background First incompleteness theorem Diagonalization B.

Infinitary Logic First published Sun Jan 23, 2000; substantive revision Wed Feb 22, 2012 Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via “Gödel numbering”) and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning “languages” some of whose formulas would be naturally identified as infinite sets. 1. Given a pair κ, λ of infinite cardinals such that λ ≤ κ, we define a class of infinitary languages in each of which we may form conjunctions and disjunctions of sets of formulas of cardinality < κ, and quantifications over sequences of variables of length < λ. Let L — the (finitary) base language — be an arbitrary but fixed first-order language with any number of extralogical symbols. ∧i∈I φ φ0 ∧ φ1 ∧ … 2.

Lotfi A. Zadeh Lotfali Askar Zadeh (born February 4, 1921), better known as Lotfi A. Zadeh, is a mathematician, electrical engineer, computer scientist, artificial intelligence researcher and professor emeritus of computer science at the University of California, Berkeley. Life and career Zadeh was born in Baku, Azerbaijan SSR, as Lotfi Aliaskerzadeh, to an Iranian Azerbaijanis father from Ardabil, Rahim Aleskerzade, who was a journalist on assignment from Iran, and a Russian Jewish mother, Fanya Koriman, who was a pediatrician. The Soviet government at this time courted foreign correspondents, and the family lived well while in Baku. Zadeh attended elementary school for three years there, which he has said "had a significant and long-lasting influence on my thinking and my way of looking at things. In 1931, when Zadeh was ten years old, his family moved to Tehran in Iran, his father's homeland. Personal life and beliefs Work Fuzzy sets and systems 1965. Notes

Paraconsistent Logic A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or "inconsistency-tolerant") systems of logic. Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term paraconsistent ("beside the consistent") was not coined until 1976, by the Peruvian philosopher Francisco Miró Quesada. Definition In classical logic (as well as intuitionistic logic and most other logics), contradictions entail everything. Which means: if P and its negation ¬P are both assumed to be true, then P is assumed to be true, from which it follows that at least one of the claims P and some other (arbitrary) claim A is true. Paraconsistent logics and classical logic Motivation Philosophy Tradeoff Example means that

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