classical and fuzzy logic vtu

1

CLASSICAL LOGIC AND FUZZY LOGIC

Dr S.Natarajan Professor Department of Information Science and Engineering PESIT, Bangalore

Classical Predicate Logic – tautologies, Contradictions, Equivalence, Exclusive Or Exclusive Nor, Logical Proofs, Deductive InferencesFuzzy Logic, Approximate Reasoning, Fuzzy Tautologies, Contradictions, Equivalence and Logical Proofs, Other forms of the Implication Operation, Other forms of the Composition Operation

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Python logic

Tell me what you do with witches?BurnAnd what do you burn apart from witches? More witches! Shh! Wood! So, why do witches burn? [pause] B--... 'cause they're made of... wood? Good! Heh heh. Oh, yeah. Oh. So, how do we tell whether she is made of wood? []. Does wood sink in water? No. No. No, it floats! It floats! Throw her into the pond! The pond! Throw her into the pond! What also floats in water? Bread! Apples! Uh, very small rocks!

ARTHUR: A duck! CROWD: Oooh. BEDEVERE: Exactly. So, logically... VILLAGER #1: If... she... weighs... the same as a duck,... she's made of wood. BEDEVERE:

And therefore? VILLAGER #2: A witch! VILLAGER #1: A witch!

10/14

http://www.montypython.net/cgi-bin/dl2/grail.cgi?aduck.wav

http://www.montypython.net/cgi-bin/dl2/grail.cgi?aduck.wav

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Classical Logic

What is

LOGIC- Small part of Human body to reason

LOGIC- means to compel us to infer correct answers

What is

NOT LOGIC- Not responsible for our creativity or ability to

remember

LOGIC helps in organizing words to form words- not

context dependent

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Fuzzy Logic

FUZZY LOGIC is a method to formalize humancapacity to Imprecise learning called ApproximateReasoning

Such reasoning represents human ability to reason approximately and judge under uncertainty

In Fuzzy Logic --- all truths are partial or approximate Here, the reasoning has been termed as Interpolative reasoning

September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets

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Negation (NOT)Negation (NOT)

Unary Operator, Symbol: Unary Operator, Symbol:

PP PP

truetrue falsefalse

falsefalse truetrue


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Conjunction (AND)Conjunction (AND)

Binary Operator, Symbol: Binary Operator, Symbol:

PP QQ PPQQ

truetrue truetrue truetrue

truetrue falsefalse falsefalse

falsefalse truetrue falsefalse

falsefalse falsefalse falsefalse


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Disjunction (OR)Disjunction (OR)


PP QQ PPQQ


truetrue falsefalse truetrue

falsefalse truetrue truetrue



9

Exclusive Or (XOR)Exclusive Or (XOR)


PP QQ PPQQ

truetrue truetrue falsefalse

truetrue falsefalse truetrue




10

Implication (if - then)Implication (if - then)


PP QQ PPQQ




falsefalse falsefalse truetrue


11

Biconditional (if and only if)Biconditional (if and only if)


PP QQ PPQQ



falsefalse truetrue falsefalse

falsefalse falsefalse truetrue


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Statements and OperatorsStatements and OperatorsStatements and operators can be combined in any Statements and operators can be combined in any

way to form new statements.way to form new statements.

PP QQ PP QQ ((P)P)((Q)Q)

truetrue truetrue falsefalse falsefalse falsefalse

truetrue falsefalse falsefalse truetrue truetrue

falsefalse truetrue truetrue falsefalse truetrue

falsefalse falsefalse truetrue truetrue truetrue


13

Statements and OperationsStatements and OperationsStatements and operators can be combined in any way Statements and operators can be combined in any way

to form new statements.to form new statements.

PP QQ PPQQ (P(PQ)Q) ((P)P)((Q)Q)

truetrue truetrue truetrue falsefalse falsefalse

truetrue falsefalse falsefalse truetrue truetrue

falsefalse truetrue falsefalse truetrue truetrue

falsefalse falsefalse falsefalse truetrue truetrue


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Equivalent StatementsEquivalent Statements

PP QQ (P(PQ)Q) ((P)P)((Q)Q) (P(PQ)Q)((P)P)((Q)Q)

truetrue truetrue falsefalse falsefalse truetrue

truetrue falsefalse truetrue truetrue truetrue

falsefalse truetrue truetrue truetrue truetrue

falsefalse falsefalse truetrue truetrue truetrue

The statements The statements (P(PQ) and (Q) and (P)P)((Q) are Q) are logically equivalentlogically equivalent, ,

because because (P(PQ)Q)((P)P)((Q) is always true.Q) is always true.


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Conditional (Implication)Conditional (Implication)

This one is probably the least intuitive. It’s only This one is probably the least intuitive. It’s only partly akin to the English usage of “if,then” or partly akin to the English usage of “if,then” or “implies”.“implies”.

DEF: DEF: p p q q is true if is true if q q is true, or if is true, or if pp is false. In is false. In the final case (the final case (pp is true while is true while qq is false) is false) p p q q is false.is false.

Semantics: “Semantics: “pp implies implies q q ” is true if one can ” is true if one can mathematically derive mathematically derive q q from from pp..

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Truth Tables

P Q P P Q P Q P Q PQ

False False True False False True True

False True True False True True False

True False False False True False False

True True False True True True True


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Tautologies and ContradictionsTautologies and Contradictions

A tautology is a statement that is always true.A tautology is a statement that is always true.

Examples: Examples: RR((R)R)(P(PQ)Q)((P)P)((Q)Q)

If SIf ST is a tautology, we write ST is a tautology, we write ST.T.If SIf ST is a tautology, we write ST is a tautology, we write ST. This symbol T. This symbol

is also used for logical equivalence.is also used for logical equivalence.


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Tautologies and ContradictionsTautologies and Contradictions

A contradiction is a statement that is alwaysA contradiction is a statement that is always

false.false.

Examples: Examples:

RR((R)R)

(((P(PQ)Q)((P)P)((Q))Q))

The negation of any tautology is a contra-The negation of any tautology is a contra-

diction, and the negation of any contradiction is diction, and the negation of any contradiction is

a tautology.a tautology.

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TAUTOLOGIES

Tautologies – Compound Propositions which are ALWAYS TRUE , irrespective of TRUTH VALUES of INDIVIDUAL SIMPLE PROPOSITIONS

APPLICATIONS- DEDUCTIVE REASONING, THEOREM PROVING , DEDUCTIVE INFERENCING ETC.,Example: A is a set of prime numbers given by (A1 =

1 , A2 = 2, A3 = 3, A4 = 5, A5 = 7, A6 = 11 …) on the real line universe X, then the proposition Ai is not divisible by 6 is A TAUTOLOGY

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Proof by Contradiction

• A method for proving A method for proving p p qq..

• Assume Assume pp, and prove that , and prove that pp ( (qq qq))

• ((qq qq) is a trivial contradiction, equal to ) is a trivial contradiction, equal to FF

• Thus Thus ppFF, which is only true if , which is only true if pp==FF

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Contradiction Proof Example

• Definition:Definition: The real number The real number rr is is rational rational if there if there exist integers exist integers p p and and q q ≠≠ 0, 0, with no common factors with no common factors other than 1 (i.e., gcd(other than 1 (i.e., gcd(pp,,qq)=1), such that )=1), such that r=p/q.r=p/q. A A real number that is not rational is called real number that is not rational is called irrational.irrational.

• Theorem:Theorem: Prove that is irrational. Prove that is irrational.2

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Symbolic logic

• Definition– Language represented by a small set of symbols

reflecting the fundamental structure of reasoning with full precision.

• Propositional logic• Predicate logic

premiseconclusion

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Forms of reasoning

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The structure of propositional logic

• Simple proposition– A proposition that does not contain any other

proposition. (atomic proposition)

• Affirmative proposition– A proposition that contains no negating words or

prefixes.

A dog has four legs and tomorrow is Sunday.

Proposition p Proposition q

Complex proposition

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Logic Operations

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Negation

• p = 『 a dog has four legs 』• q = 『 Elvis is mortal 』

Truth table

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Conjunction

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Disjunction

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Implication

antecedent consequent

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Equivalence

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Classical Logic & Fuzzy Logic

Classical predicate logic

T: uU [0,1]

U: universe of all propositions.

All elements u U are true for proposition P are called the truth set of P: T(P).

Those elements u U are false for P are called falsity set of P: F(P).

T(Y) = 1 T(Ø) = 0

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Classical Logic &Fuzzy Logic

Logic connectives

Disjunction Conjunction Negation –Implication Equivalence

If xA, T(P) =1 otherwise T(P) = 0OrxA(x)={ 1 if x A, otherwise it is 0 }

If T(p)T()=0 implies P true, false, or true P false. P and are mutually exclusive propositions.

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Given a proposition P: xA, P: xA, we have the following logical connectives:

Disjunction PQ: x A or x B hence, T(PQ) = max(T(P),T(Q))Conjunction PQ: xA and xB

hence T(P Q)= min(T(P),T(Q))Negation If T(P) =1, then T(P) = 0 then T(P) =1Implication (P Q): xA or xB Hence , T(P Q)= T(P Q)

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Equivalence

1, for T(P) = T(Q)(P Q): T(PQ)=

0, for T(P) T(Q)

The logical connective implication, i.e.,P Q (P implies

Q) presented here is also known as the classical

implication.

P is referred to as hypothesis or antecedent

Q is referred to as conclusion or consequent.

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T(PQ)=(T(P)T(Q))Or PQ= (AB is true)T(PQ) = T(PQ is true) = max (T(P),T(Q))(AB)= (AB)= ABSo (AB)= ABOr AB false AB

Truth table for various compound propositions

P Q P PQ PQ PQ PQ

T(1) T(1) F(0) T(1) T(1) T(1) T(1)

T(1) F(0) F(0) T(1) F(0) F(0) F(0)

F(0) T(1) T(1) T(1) F(0) T(1) F(0)

F(0) F(0) T(1) F(0) F(0) T(1) T(1)

37General format:

– If x is A then y is B (where A & B are linguistic values defined by fuzzy sets on universes of discourse X & Y).

• “x is A” is called the antecedent or premise• “y is B” is called the consequence or

conclusion– Examples:

• If pressure is high, then volume is small.• If the road is slippery, then driving is dangerous.• If a tomato is red, then it is ripe.• If the speed is high, then apply the brake a little.

Fuzzy if-then rules

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– Meaning of fuzzy if-then-rules (A B)

• It is a relation between two variables x & y; therefore it is a binary fuzzy relation R defined on X * Y

• There are two ways to interpret A B:–A coupled with B–A entails B

if A is coupled with B then:

Fuzzy if-then rules (cont.)

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If A entails B then:

R = A B = A B ( material implication)

R = A B = A (A B) (propositional calculus)

R = A B = ( A B) B (extended propositional calculus)

Fuzzy if-then rules (3.3) (cont.)

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Two ways to interpret “If x is A then y is B”:

A coupled with B

B

A

y

x

Fuzzy if-then rules (cont.)

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PQ: If x A, Then y B, or PQ AB

The shaded regions of the compound Venn diagram in the following figure represent the truth domain of the implication, If A, then B(PQ).

B Y

X

A

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IF A, THEN B, or IF A , THEN CPREDICATE LOGIC (PQ)(PS)Where P: xA, AX

Q: yB, BYS: yC, CY

SET THEORETIC EQUIVALENT (A X B)(A X C) = R = relation ON X Y

Truth domain for the above compound proposition.

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Some common tautologies follow:

BB X AX; A X X

AB (A(AB))B (modeus ponens)(B(AB))A (modus tollens)Proof:(A(AB)) B(A(AB)) B Implication((AA) (AB))B Distributivity((AB))B Excluded middle laws(AB)B Identity(AB)B Implication(AB)B Demorgans lawA(BB) AssociativityAX Excluded middle lawsX T(X) =1 Identity; QED

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Classical Logic & Fuzzy Logic

Proof(B(AB))A(B(AB))A((BA)(BB)) A((BA))A(BA)A

(BA)A

(BA)AB(AA)BX = X T(X) =1 A B AB (A(AB) (A(AB)B

O 0 1 0 1

O 1 1 0 1

1 0 0 0 1

1 1 1 1 1

Truth table (modus ponens)

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Contradictions

BBA; AEquivalencePQ is true only when both P and Q are true or when both P and q are false.Example

Suppose we consider the universe positive integers X={1 n8}. Let P = “n is an even number “ and let Q =“(3n7)(n6).” then T(P)={2,4,6,8} and T(Q) ={3,4,5,7}. The equivalence PQ has the truth set T(P Q)=(T(P)T(Q)) (T(P) (T(Q)) ={4} {1} ={1,4}

T(A)

T(B)Venn diagram for equivalence

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Exclusive orExclusive NorExclusive or P “” Q(AB) (AB)Exclusive Nor(P “” Q)(PQ)Logical proofsLogic involves the use of inference in everyday life.

In natural language if we are given some hypothesis it is often useful to make certain conclusions from them the so called process of inference (P1P2….Pn) Q is true.

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Hypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers.Conclusion : Engineers do not believe in magic.Let us decompose this information into individual propositionsP: a person is an engineerQ: a person is a mathematicianR: a person is a logical thinkerS: a person believes in magicThe statements can now be expressed as algebraic propositions as((PQ)(RS)(QR))(PS)It can be shown that the proposition is a tautology.ALTERNATIVE: proof by contradiction.

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Deductive inferences

The modus ponens deduction is used as a tool for making inferences in rule based systems. This rule can be translated into a relation between sets A and B.

R = (AB)(AY)

Now suppose a new antecedent say A’ is known, since A implies B is defined on the cartesian space X Y, B can be found through the following set theoretic formulation B= AR= A((AB)(AY))

Denotes the composition operation. Modus ponens deduction can also be used for compound rule.

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Whether A is contained only in the complement of A or whether A’ and A overlap to some extent as described next:

IF AA, THEN y=B

IF AA THEN y =C

IF AA , AA, THEN y= BC

57

Truth values of complex propositions

58

Table of a complex proposition

59

Logic functions

60

Valid inference

61

Invalid inferenceerror

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Basic Inference forms

63

Rules of Replacement

64

Predicate Logic

Singular Proposition

General Proposition

Subject term Predicate term

65

Singular Propositions

Lassie is a dog

DlIndividual constant

lPredicate variable

D

Fido is a dog DfBuster is a dog DbGinger is a dog Dg

66

Generalization

Lassie is a dog

DlIndividual constant

lPredicate variable

D

Fido is a dog DfBuster is a dog DbGinger is a dog Dg

Dx x: Individual variableDx: propositional functionDf: substitution instances

Dx

Dl

instantiation generalization

67

General Propositions

(∃x)Dx : There exists at least one x, such that the x is a dog

(∃x)( Dx Q∧ x) : There exists at least one thing, such that it is both a dog and a quadruped.

( ∀ x) Dx : For any x, x is a dog

( ∀ x) Dx Qx : for any x, if x is a dog, then x is a quadruped

Existential generalization ∃x : Existential quantifier

universal generalization ∀x : universal quantifier

68

Relations represented by predicate logic

• John loves Mary --- LjmL ： relation j,m ： individual constant

• Everything is attracted by something --- ( ∀ x )(∃y)Ayx

x y

69

Quantifier Negation

• It is false that everything is square --- ¬( ∀ x )Sx

• There is something which is not square --- ( ∃ x) ¬Sx

Quantifier negation equivalences

70

The Square of Opposition

Spring 2003 CMSC 203 - Discrete Structures 71

Rules of InferenceRules of Inference

Rules of inferenceRules of inference provide the justification of provide the justification of the steps used in a proof.the steps used in a proof.

One important rule is called One important rule is called modus ponensmodus ponens or the or the law of detachmentlaw of detachment. It is based on the . It is based on the tautology tautology (p (p (p (p q)) q)) q. We write it in the following q. We write it in the following way:way:

ppp p q q________ qq

The two The two hypotheseshypotheses p and p p and p q q are are written in a column, and the written in a column, and the conclusionconclusionbelow a bar, where below a bar, where means means “therefore”.“therefore”.

Spring 2003 CMSC 203 - Discrete Structures 72

Rules of InferenceRules of Inference

The general form of a rule of inference is:The general form of a rule of inference is:

pp11

pp22 .. .. .. ppnn________ qq

The rule states that if pThe rule states that if p11 andand p p22 andand … … andand p pnn are all true, then q is true as are all true, then q is true as well.well.

Each rule is an established tautology Each rule is an established tautology ofof pp11 p p22 … … p pnn q q

These rules of inference can be used These rules of inference can be used in any mathematical argument and do in any mathematical argument and do not not require any proof.require any proof.

73

CS 173 Proofs - Modus Ponens

I am Mila.If I am Mila, then I am a great swimmer.

I am a great swimmer!

p

p q

q

Tautology:

(p (p q)) q

Inference Rule:

Modus Ponens

74

CS 173 Proofs - Modus Tollens

I am not a great skater.If I am Erik, then I am a great skater.

I am not Erik!

q

p q

p

Tautology:

(q (p q)) p

Inference Rule:

Modus Tollens

3.1-82

Rules of Inference

• Many (implication) tautologies are rules of inference, and have the form:

H1 H2 … Hn C

where Hi are the hypotheses, and C is the conclusion. • They can be represented by the symbolic form:

H1

H2

.

.

Hn

C

3.1-83

Fallacies

• Fallacies are incorrect inferences.– Based on contingencies, NOT tautologies.

• Some common fallacies are:– Affirming the conclusion (or the consequent) – Denying the hypothesis (the antecedent)– Begging the question (or circular reasoning)

3.1-84

The Fallacy of Affirming the Conclusion

• This invalid argument has the form:

p q

q

p• It is based on the implication:

[(p q) q] p,which is NOT a tautology.

3.1-85

Example

• Is the following argument valid:If you do every problem in the ‘Rosen’ textbook, then you will learn discrete mathematics.

You learned discrete mathematics.

Therefore, you did every problem in the textbook.

3.1-86

Example - Solution

No. Let p and q be the following propositionsp: “You did every problem in the ‘Rosen’ textbook,”q: “You learned discrete mathematics.”The argument used is of the form:

p qq p

It is based on the implication:

[(p q) q] p,which is NOT a tautology.

103

Fuzzy Logic

The restriction of classical propositional calculus to a two-valued logic has created many interesting paradoxes over the ages. For example, the barber of Seville is a classic paradox (also termed as Russell’s barber). In the small Spanish town of Seville, there is a rule that all and only those men who do not shave themselves are shaved by a barber. Who shaves the barber?

Another example comes from ancient Greece. Does the liar from Crete lie when he claims, “All Cretians are liars”? If he is telling the truth, then the statement is false. If the statement is false, he is not telling the truth.

104

Fuzzy Logic

Let S: the barber shaves himself

S’: he does not

S S’ and S’ S

T(S) = T(S’) = 1 – T(S)

T(S) = 1/2

But for binary logic T(S) = 1 or 0

Fuzzy propositions are assigned for fuzzy sets:

10

~

~~

A

A xPT

105

Fuzzy Logic

~~

1 PTPT

~~~~

~~~~

,max

:

QTPTQPT

BorAxQP

~~~~

~~~~

,min

:

QTPTQPT

BandAxQP

~~~~~~

~~

,max QTPTQPTQPT

QP

Negation

Disjunction

Conjunction

Implication [Zadeh, 1973]

106

Fuzzy Logic

xyxyx

YABAR

ABAR~~~~

1,max,~~~~

Example:

= medium uniqueness =

= medium market size =

Then…

4

2.0

3

1

2

6.0

5

3.0

4

8.0

3

1

2

4.0

~A

~B

107

Fuzzy Logic

108

Fuzzy Logic

When the logical conditional implication is of the compound form,

IF x is , THEN y is , ELSE y is

Then fuzzy relation is:

whose membership function can be expressed as:

~A

~B

~C

~~~~~CABAR

yxyxyx CABAR

~~~~~

1,max,

109

Fuzzy Logic

Rule-based format to represent fuzzy information.

Rule 1: IF x is , THEN y is , where and represent fuzzy propositions (sets)

Now suppose we introduce a new antecedent, say, and we consider the following rule

Rule 2: IF x is , THEN y is

~A

~B

~B

~A

'~A '

~B

RAB ''~~

110

Fuzzy Logic

111

Fuzzy Logic

Suppose we use A in fuzzy composition, can we get

The answer is: NO

Example:

For the problem in pg 127, let

A’ = AB’ = A’ R = A R = {0.4/1 + 0.4/2 + 1/3 + 0.8/4 + 0.4/5 + 0.4/6} ≠ B

RBB ~~

112

Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs

The extension of truth operations for tautologies, contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow identically those in classical logic. However, the use of partially true (or partially false) simple propositions in compound propositional statements results in new ideas termed quasi tautologies, quasi contradictions, and quasi equivalence. Moreover, the idea of a logical proof is altered because now a proof can be shown only to a “matter of degree”. Some examples of these will be useful.

113


Truth table (approximate modus ponens)

A B AB (A(AB)) (A(AB))B

.3 .2 .7 .3 .7

.3 .8 .8 .3 .8 Quasi tautology

.7 .2 .3 .3 .7

.7 .8 .8 .7 .8



.4 .1 .6 .4 .6


.6 .1 .4 .4 .6

.6 .9 .9 .6 .9

114


The following form of the implication operator show different techniques for obtaining the membership function values of fuzzy relation defined on the Cartesian product space X × Y:

~R

155

Fuzzy Logic

Rule-based format to represent fuzzy information.

Rule 1: IF x is , THEN y is , where and represent fuzzy propositions (sets)

Now suppose we introduce a new antecedent, say, and we consider the following rule

Rule 2: IF x is , THEN y is

~A

~B

~B

~A

'~A '

~B

RAB ''~~

156

Fuzzy Logic

157

Fuzzy Logic

Suppose we use A in fuzzy composition, can we get

The answer is: NO

Example:

For the problem in pg 127, let

A’ = AB’ = A’ R = A R = {0.4/1 + 0.4/2 + 1/3 + 0.8/4 + 0.4/5 + 0.4/6} ≠ B

RBB ~~

158


The extension of truth operations for tautologies, contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow identically those in classical logic. However, the use of partially true (or partially false) simple propositions in compound propositional statements results in new ideas termed quasi tautologies, quasi contradictions, and quasi equivalence. Moreover, the idea of a logical proof is altered because now a proof can be shown only to a “matter of degree”. Some examples of these will be useful.

159




.3 .2 .7 .3 .7


.7 .2 .3 .3 .7

.7 .8 .8 .7 .8



.4 .1 .6 .4 .6


.6 .1 .4 .4 .6

.6 .9 .9 .6 .9

160


The following form of the implication operator show different techniques for obtaining the membership function values of fuzzy relation defined on the Cartesian product space X × Y:

~R

161


The following common methods are among those proposed in the literature for the composition operation , where is the input, or antecedent defined on the universe X, is the output, or consequent defined on the universe Y, and is a fuzzy relation characterizing the relationship between specific inputs (x) and specific outputs (y):

Refer fig on next slide…

~~~RAB

~A

~B

~R

162


where f(.) is a logistic function (like a sigmoid or step function) that limits the value of the function within the interval [0,1]

Commonly used in Artificial Neural Networks for mapping between parallel layers of a multi-layer network.

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