Fuzzy Logic

1
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 Russells 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.
2
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
3
Fuzzy Logic
Negation Disjunction Conjunction Implication
Zadeh, 1973
4
Fuzzy Logic
5
Fuzzy Logic
6
Fuzzy Logic
7
Fuzzy Logic
8
Fuzzy Logic
9
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 A B A ? R A ? R
0.4/1 0.4/2 1/3 0.8/4 0.4/5 0.4/6
? B
10
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.
11
Fuzzy Tautologies, Contradictions, Equivalence,
and Logical Proofs
Truth table (approximate modus ponens)
Truth table (approximate modus ponens)
12
Fuzzy Tautologies, Contradictions, Equivalence,
and Logical Proofs
13
Fuzzy Tautologies, Contradictions, Equivalence,
and Logical Proofs
14
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.
15
Fuzzy Rule-based systems
16
Fuzzy Rule-based systems
17
Fuzzy Rule-based systems
Composite
18
Linguistic Hedges
19
Precedence of the Operations
20
Example (contd)
Then we construct a phrase, or a composite
term ? not very small and not very, very
large which involves the following set-theoretic
operations
Suppose we want to construct a linguistic
variable intensely small (extremely small) we
will make use of the equation defined before to
modify small as follows
21
Example (contd)
22
Rule-based Systems

• IF-THEN rule based form
• Canonical Rule Forms
• Assignment statements
• x large, x ? y
• 2. Conditional statements
• If A then B,
• If A then B, else C
• 3. Unconditional statements
• stop
• go to 5
• unconditional can be
• If any conditions, then stop
• If condition Ci, then restrict Ri

23
Decomposition of Compound Rule
24
Multiple Disjunctive Antecedents
IF THEN
25
Condition Statements
26
Condition Statements
likely very likely highly likely true
fairly true very true false fairly false
very false
27
Aggregation of fuzzy rule
The process of obtaining the overall consequent
(conclusion) from the individual consequent
contributed by each rule in the rule-base is
known as aggregation of rules. Conjunctive System
of Rules
Disjunctive System of Rules
28
Graphical Technique of Inference
29
Graphical Technique of Inference
For r disjunctive rules A11 refers to the
first fuzzy antecedent of the first rule. A12
refers to the second fuzzy antecedent of the
first rule.