classical and fuzzy logic vtu
TRANSCRIPT
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
2
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
3
4
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
5
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
6
Negation (NOT)Negation (NOT)
Unary Operator, Symbol: Unary Operator, Symbol:
PP PP
truetrue falsefalse
falsefalse truetrue
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
7
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
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
8
Disjunction (OR)Disjunction (OR)
Binary Operator, Symbol: Binary Operator, Symbol:
PP QQ PPQQ
truetrue truetrue truetrue
truetrue falsefalse truetrue
falsefalse truetrue truetrue
falsefalse falsefalse falsefalse
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
9
Exclusive Or (XOR)Exclusive Or (XOR)
Binary Operator, Symbol: Binary Operator, Symbol:
PP QQ PPQQ
truetrue truetrue falsefalse
truetrue falsefalse truetrue
falsefalse truetrue truetrue
falsefalse falsefalse falsefalse
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
10
Implication (if - then)Implication (if - then)
Binary Operator, Symbol: Binary Operator, Symbol:
PP QQ PPQQ
truetrue truetrue truetrue
truetrue falsefalse falsefalse
falsefalse truetrue truetrue
falsefalse falsefalse truetrue
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
11
Biconditional (if and only if)Biconditional (if and only if)
Binary Operator, Symbol: Binary Operator, Symbol:
PP QQ PPQQ
truetrue truetrue truetrue
truetrue falsefalse falsefalse
falsefalse truetrue falsefalse
falsefalse falsefalse truetrue
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
12
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
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
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
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
14
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.
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
15
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..
16
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
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
17
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.
September 5, 2006 Applied Discrete MathematicsWeek 1: Logic and Sets
18
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.
19
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
20
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
21
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
22
23
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
24
Forms of reasoning
25
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
26
Logic Operations
27
Negation
• p = 『 a dog has four legs 』• q = 『 Elvis is mortal 』
Truth table
28
Conjunction
29
Disjunction
30
Implication
antecedent consequent
31
Equivalence
32
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
33
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.
34
Classical Logic &Fuzzy Logic
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)
35
Classical Logic &Fuzzy Logic
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.
36
Classical Logic &Fuzzy Logic
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
38
– 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.)
39
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.)
40
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.)
41
Classical Logic &Fuzzy Logic
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
42
Classical Logic &Fuzzy Logic
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.
43
Classical Logic &Fuzzy Logic
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
44
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)
45
Classical Logic &Fuzzy Logic
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
46
Classical Logic &Fuzzy Logic
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.
47
Classical Logic &Fuzzy Logic
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.
48
Classical Logic &Fuzzy Logic
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.
49
Classical Logic &Fuzzy Logic
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
50
51
52
53
54
55
56
57
Truth values of complex propositions
58
Table of a complex proposition
59
Logic functions
60
Valid inference
61
Invalid inferenceerror
62
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
75
76
77
78
79
80
81
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.
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
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
Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs
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
Truth table (approximate modus ponens)
A B AB (A(AB)) (A(AB))B
.4 .1 .6 .4 .6
.4 .9 .9 .4 .9 Quasi tautology
.6 .1 .4 .4 .6
.6 .9 .9 .6 .9
114
Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs
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
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
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
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.
159
Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs
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
Truth table (approximate modus ponens)
A B AB (A(AB)) (A(AB))B
.4 .1 .6 .4 .6
.4 .9 .9 .4 .9 Quasi tautology
.6 .1 .4 .4 .6
.6 .9 .9 .6 .9
160
Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs
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
Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs
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
Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs
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.