This document provides an introduction to fuzzy logic, including its history and applications. It discusses classical logic and its limitations in dealing with uncertain propositions. Multi-valued logics are introduced as an approach to handle indeterminate truth values. Fuzzy logic then allows for gradual assessments between true and false by using membership functions and fuzzy set theory. Conditional and quantified fuzzy propositions are defined along with operations on them. The document concludes by mentioning applications of fuzzy logic in areas like controllers for washing machines and computer engineering.

1. FUZZY LOGIC
Theory and
Applications
Dzikra F. – Yosep Dwi K.

2. HISTORICALREMARKS
Charles Sanders
Peirce
Jan Lukasiewicz Lotfi Zadeh

3. INTRODUCTION

4. CLASSICAL LOGIC
Logic 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.

5. 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.

6. ISOMORPHISM
Set Theory Propositional Logic
𝒫 𝑋
⋃
⋂
−
𝑋
∅
⊆
ℒ(𝑉),
⋁
∧
−
1
0
⇒

7. QUANTIFICATION
Existential quantification of a predicate P(𝑥)
is expressed by the form
∃𝑥 𝑃 𝑥 =
𝑥 ∈ 𝑋
𝑃(𝑥)
Universal quantification of a predicate 𝑃(𝑥)
is expressed by the form
∀𝑥 𝑃 𝑥 =
𝑥 𝜖 𝑋
𝑃 𝑥 .
Existential and Universal Quantification

8. FUZZY LOGIC

9. 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

10. Two of three-
valued logics𝒂 𝒃
Lukasiewicz Bochvar
∧ ∨ ⟹ ⟺ ∧ ∨ ⟹ ⟺
0 0 0 0 1 1 0 0 1 1
0
1
2
0
1
2
1
1
2
1
2
1
2
1
2
1
2
0 1 0 1 1 0 0 1 1 0
1
2
0 0
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1 1
1
2
1
2
1
2
1
2
1
2
1
1
2
1 1
1
2
1
2
1
2
1
2
1
2
1 0 0 1 0 0 0 1 0 0
1
1
2
1
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1 1 1 1 1 1 1 1 1 1
Primitives of some three-valued logics

11. 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.

12. TRUTH VALUES
The set 𝑇𝑛 of truth values of an n-valued logic is
thus defined as
𝑇𝑛 = 0 =
0
𝑛 − 1
,
1
𝑛 − 1
,
2
𝑛 − 1
, … ,
𝑛 − 2
𝑛 − 1
,
𝑛 − 1
𝑛 − 1
= 1

13. TRUTHVALUE
The n-value logics (𝑛 ≥ 2) uses truth
values in 𝑇𝑛 and defines the primitives
by the following equations:
𝑎 = 1 − 𝑎
𝑎 ∧ 𝑏 = min 𝑎, 𝑏
𝑎 ∨ 𝑏 = max 𝑎, 𝑏
𝑎 ⟹ 𝑏 = min 𝑎, 1 + 𝑏 − 𝑎
𝑎 ⟺ 𝑏 = 1 − 𝑎 − 𝑏

14. 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 𝐿1.

15. FUZZYPROPOSITIONS

16. UNCONDITIONAL ANDUNQUALIFIED
PROPOSITION
The canonical form of fuzzy propositions of this type,
𝑝, is expressed by sentence
𝑝: 𝒱 is 𝐹
Example:
𝑝: temperatue (𝒱) is high (𝐹).
And the membership grade is
𝑇 𝑝 = 𝐹(𝑣)

17. Components of the fuzzy proposistion

18. UNCONDITIONAL AND QUALIFIED
PROPOSITIONS
𝑝: 𝒱 is 𝐹 is 𝑆
𝑝: Tina (𝒱) is young (𝐹) is very true (𝑆)
𝑇 𝑝 = 𝑆(𝐹 𝑣 )
𝑝: Pro 𝒱 is 𝐹 is 𝑃
Pro {𝒱 is 𝐹} = 𝑣𝜖𝑉 𝑓 𝑣 ∙ 𝐹(𝑣)
𝑇 𝑝 = 𝑃
𝑣𝜖𝑉
𝑓 𝑣 ∙ 𝐹(𝑣)

19. EXAMPLE
𝑝: Pro {temperature t (at given place and time) is
around 75°F} is likely
Pro (t is close to 75°F) = 𝑣𝜖𝑉 𝑓 𝑣 ∙ 𝐹(𝑣) = 0.8
𝑇 𝑝 = 0.95
𝑡 68 69 70 71 72 73 74 75
𝑓(𝑡) .002 .005 .005 .01 .04 .11 .15 .21
𝑡 76 77 78 79 80 81 82 83
𝑓(𝑡) .16 .14 .11 .04 .01 .005 .022 .001

20. CONDITIONAL AND UNQUALIFIEDPROPOSITIONS
Propositions 𝑝 of this type are expressed by the
canonical form
𝑝: if 𝒳 is A, then 𝒴 is B
These propositions may also be viewed as
propositions of the form
(𝒳,𝒴) is R
where
𝑅 𝑥, 𝑦 = 𝒥 𝐴 𝑥 ∙ 𝐵 𝑦
where 𝒥 denotes a binary operation on [0, 1]
representing a suitable fuzzy implication.

21. 𝒥 𝑎, 𝑏 = min 1, 1 − 𝑎 + 𝑏
Let 𝐴 = 0.1
𝑥1 + 0.8
𝑥2 + 1
𝑥3 and B = 0.5
𝑦1 + 1
𝑦2.
Then
𝑅 = 1 𝑥1, 𝑦1 + 1 𝑥1, 𝑦2 + 0.7 𝑥2, 𝑦1 + 1 𝑥2, 𝑦2 + 0.5 𝑥3, 𝑦1 + 1 𝑥3, 𝑦2

22. CONDITIONALAND QUALIFIEDPROPOSITION
Propositions of this type can be characterized by
either the canonical form
𝑝: If 𝒳 is 𝐴, then 𝒴 is 𝐵 is S
or the canonical form
𝑝: Prop {𝒳 is 𝐴 𝒴 is 𝐵 } is 𝑃

23. FUZZY QUANTIFIERS
Fuzzy quantifiers of the first kind are defined on ℝ
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.

24. There are two basic forms of propositions that
contain fuzzy quantifiers of the first kind. One of
them is the form
𝑝: There are 𝑄 i’s in 𝐼 such that 𝒱 𝑖 is 𝐹
Example:
There are about 10 students in a given class whose
fluency in English is high.

25. Alternatively, we can use
𝑝′: There are 𝑄 E’s
where,
𝐸 𝑖 = 𝐹 𝒱 𝑖
Example:
There are about 10 high-fluency English-speaking
students in a given class.

26. Also, a proposition before can be rewritten
as,
𝑝′: 𝒲 is 𝑄
where, 𝒲 = |𝐸|
𝐸 =
𝑖∈𝐼
𝐸 𝑖 =
𝑖∈𝐼
𝐹 𝒱 𝑖
and,
𝑇 𝑝 = 𝑇 𝑝′ = 𝑄 𝐸

27. EXAMPLE
𝑝: There are about three students in 𝑰 whose
fluency in English, 𝒱 𝑖 , is high.
Assume that 𝐼 = {Adam, Bob, Cathy, David, Eve}, and
𝑉 is a variable with values in the interval [0, 100] that
express degrees of fluency in English. And following
scores are given: 𝒱(Adam) = 35, 𝒱(Bob) = 20,
𝒱(Cathy) = 80, 𝒱(David) = 95, 𝒱(Eve) = 70. Determine
the truth value of the proposition 𝑝.

29. From the graph, we get
𝐸 = 0/Adam + 0/Bob + 0,75/Cathy + 1/David +
0,5/Eve
Then,
𝐸 =
𝑖∈𝐼
𝐸 𝑖 = 2,25
Finally,
𝑇 𝑝 = 𝑄 2,25 = 0,625

30. The second basic form of the first kind of fuzzy
quantifiers can be expressed as
𝑝: There are 𝑖's in 𝐼 such that 𝒱1 𝑖 is 𝐹1 and
𝒱2 𝑖 is 𝐹2
Example:
There are about 10 students in a given class whose
fluency in English is high and who are young.

31. The proposition before also can be expressed as,
𝑝′: 𝑄𝐸1’s 𝐸2’s
where,
𝐸1 𝑖 = 𝐹1 𝒱1 𝑖
𝐸2 𝑖 = 𝐹2 𝒱2 𝑖
or,
𝑝′: There are 𝑄(𝐸1 and 𝐸2)’s.
or
𝑝′: 𝒲 is 𝑄

32. The value 𝒲 and 𝑇 𝑝 can be determined as
𝒲 =
𝑖∈𝐼
min 𝐹1 𝒱1 𝑖 , 𝐹2 𝒱2 𝑖
and,
𝑇 𝑝 = 𝑇 𝑝′ = 𝑄 𝒲

33. FUZZY PROPOSITIONS WITH QUANTIFIERSOF SECOND
KIND
𝑝: Among 𝑖's in 𝐼 such that 𝒱1 1 is 𝐹1 there are 𝑄 𝑖's
in 𝐼 such that 𝒱2 𝑖 is 𝐹2
Or,
𝑝′: 𝑄𝐸1’s are 𝐸2’s
Where,
𝐸1 = 𝐹1 𝒱1 𝑖
𝐸2 = 𝐹2 𝒱2 𝑖
Example: Almost all young students in a given class
are students whose fluency in English is high.

34. Proposition before can be written as,
𝑝′: 𝒲 is 𝑄
where,
𝒲 =
𝐸1 ∩ 𝐸2
𝐸1
And we obtain,
𝒲 =
𝑖∈𝐼 min 𝐹1 𝒱1 𝑖 , 𝐹2 𝒱2 𝑖
𝑖∈𝐼 𝐹1 𝒱1 𝑖

35. APPLICATIONS

36. Hardware implementation of a fuzzy controller
COMPUTER ENGINEERING

37. WASHING MACHINE
Type_of_dirt
Dirtness_of_clothes
Linguistic input
Fuzzy controller
Output
Fuzzyfication
Fuzzy arithmetic
& applying criterion
Defuzzyfication
Wash_time

39. THANK YOU.