FUZZY LOGICTheory and ApplicationsDzikra F. – Yosep Dwi K.

HISTORICAL REMARKS

Charles Sanders Peirce

Jan Lukasiewicz Lotfi Zadeh

INTRODUCTION

CLASSICAL LOGICLogic is the study of the methods and principles of reasoning in all its possible forms. Classical logic deals with propositions that are required to be either true or false. Each proposition has its opposite, which is usually called an negation of the proposition. A proposition and its negation are required to assume opposite thruth values.

One area of logic, referred to as propositional logic, deals with combinations of variables that stand for arbitrary propositions.

Two of the many complete sets of primitives have been predominant in propotional logic: (i) negation, conjuction, and disjunction; and (ii) negation and implication.

When the variable represented by a logic formula is always true regardless of the truth values assigned to the variables participating in the formulas, it is called a tautology; when it is always false, it is called a contradiction.

ISOMORPHISM

Set Theory Propositional Logic

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Existential quantification of a predicate P() is expressed by the form

Universal quantification of a predicate is expressed by the form

Existential and Universal Quantification

FUZZY LOGIC

Propositions about future events are neither actually true not actually false, but potentially either; hence, their truth value is undetermined, at least prior to the event.

In order to deal with such propositions, we must relax the true/false dichotomy of classical two-valued logic by allowing a third truth value, which may be called indeterminate.

MULTIVALUED LOGICS

Partly cloudy

Two of three-valued logics

Lukasiewicz Bochvar

0 0 0 0 1 1 0 0 1 1

0 0 1

0 1 0 1 1 0 0 1 1 0

0 0

1 1

1 1 1

1 0 0 1 0 0 0 1 0 0

1 1

1 1 1 1 1 1 1 1 1 1

Primitives of some three-valued logics

QUASI-TAUTOLOGY AND QUASI-CONTRADICTION

We say that a logic formula in a three-valued logic which does not assume the truth value 0 (falsity) regardless of the truth values assigned to its proposition variables is a quasi-tautology.

Similarly, we say that a logic formula which does not assume the truth value 1 (truth) is a quasi-contradiction.

TRUTH VALUESThe set of truth values of an n-valued logic is thus defined as

TRUTH VALUE

The n-value logics () uses truth values in and defines the primitives by the following equations:

INFINITE-VALUE LOGIC

Lukasiewicz used only negation and implication as primitives

Generally, the term infinite-valued logic is usually used in the literature to indicate the logic whose truth values are represented by all the real numbers in the interval [0, 1]. This is also quite often called the standard Lukasiewicz logic .

FUZZY PROPOSITIONS

UNCONDITIONAL AND UNQUALIFIED PROPOSITION

The canonical form of fuzzy propositions of this type, , is expressed by sentence

is Example:

temperatue is high ().And the membership grade is

Components of the fuzzy proposistion

UNCONDITIONAL AND QUALIFIED PROPOSITIONS

: Tina () is young () is very true ()

Pro { is 𝒱

EXAMPLE

: Pro {temperature t (at given place and time) is around 75°F} is likely

Pro (t is close to 75°F)

68 69 70 71 72 73 74 75

.002 .005 .005 .01 .04 .11 .15 .21

76 77 78 79 80 81 82 83

.16 .14 .11 .04 .01 .005 .022 .001

CONDITIONAL AND UNQUALIFIED PROPOSITIONS

Propositions of this type are expressed by the canonical form

These propositions may also be viewed as propositions of the form

( , ) is R𝒳 𝒴where

where denotes a binary operation on [0, 1] representing a suitable fuzzy implication.

Let and B . Then

CONDITIONAL AND QUALIFIED PROPOSITION

Propositions of this type can be characterized by either the canonical form

or the canonical form

FUZZY QUANTIFIERS

Fuzzy quantifiers of the first kind are defined on and characterize linguistic terms such as about 10, much more than 100, at least about 5, and so on.

Fuzzy quantifiers of the second kind are defined on [0, 1] and characterize linguistic terms such as almost all, about half, most, and so on.

There are two basic forms of propositions that contain fuzzy quantifiers of the first kind. One of them is the form

: There are i’s in such that is

Example:There are about 10 students in a given class whose fluency in English is high.

Alternatively, we can use: There are E’s

where,

Example:There are about 10 high-fluency English-speaking students in a given class.

Also, a proposition before can be rewritten as,

: is where,

and,

EXAMPLE

: There are about three students in whose fluency in English, , is high.

Assume that = {Adam, Bob, Cathy, David, Eve}, and is a variable with values in the interval [0, 100] that express degrees of fluency in English. And following scores are given: (Adam) = 35, (Bob) = 20, (Cathy) = 80, (David) = 95, (Eve) = 70. Determine the truth value of the proposition .

From the graph, we get = 0/Adam + 0/Bob + 0,75/Cathy + 1/David +

0,5/EveThen,

Finally,

The second basic form of the first kind of fuzzy quantifiers can be expressed as

: There are 's in such that is and is Example:There are about 10 students in a given class whose fluency in English is high and who are young.

The proposition before also can be expressed as,: ’s ’s

where,

or,: There are ( and )’s.

or: is

The value and can be determined as

and,

FUZZY PROPOSITIONS WITH QUANTIFIERS OF SECOND KIND

: Among 's in such that is there are 's in such that is Or,: ’s are ’sWhere,

Example: Almost all young students in a given class are students whose fluency in English is high.

Proposition before can be written as,: is

where,

And we obtain,

APPLICATIONS

Hardware implementation of a fuzzy controllerCOMPUTER ENGINEERING

WASHING MACHINE

Type_of_dirt

Dirtness_of_clothes

Linguistic input

Fuzzy controller

Output

Fuzzyfication

Fuzzy arithmetic& applying criterion

Defuzzyfication

Wash_time

THANK YOU.