Fuzzy Logic

Fuzzy Logic
Frank Costanzo – MAT 7670 Spring 2012
• Fuzzy logic began with the introduction of Fuzzy
Set Theory by Lotfi Zadeh in 1965.
• Fuzzy Set
▫ Sets whose elements have degrees of membership.
▫ A fuzzy subset A of a set X is characterized by
assigning to each element of x in X the degree of
membership of x in A.
▫ Example let X={x|x is a person} and A={x|x is an
old person}
What is Fuzzy Logic?
• In Propositional Logic, truth values are either
True or False
• Fuzzy logic is a type of Many-Valued Logic
▫ There are more than two truth values
• The interval [0,1] represents the possible truth
▫ 0 is absolute falsity
▫ 1 is absolute truth
Fuzzy Connectives
• t-norms (triangular norms) are truth functions
of conjunction in Fuzzy Logic
▫ A binary operation, *, is a t-norm if
It is Commutative
It is Associative
It is Non-Decreasing
1 is the unit element
▫ Example of a possible t-norm: x*y=min(x, y)
Fuzzy Connectives Continued
• t-conorms are truth functions of disjunction
▫ Example: max(x, y)
• Negation – This function must be nonincreasing and assign 0 to 1 and vice versa
▫ 1-x
• R-implication – The residuum of a t-norm;
denoting the residuum as → and t-norm, *
▫ x → y = max{z|x*z≤y}
Basic Fuzzy Propositional Logic
• The logic of continuous t-norms (developed in
Hajek 1998)
• Formulas are built from proposition variables
using the following connectives
Conjunction: &
Implication: →
Truth constant 0 denoting falsity
Negation ¬ φ is defined as φ → 0
Basic Fuzzy Propositional Logic cont….
• Given a continuous t-norm * (and hence its
residuum →) each evaluation e of propositional
variables by truth degrees for [0,1] extends
uniquely to the evaluation e*(φ) of each formula
φ using * and → as truth functions of & and →
• A formula φ is a t-tautology or standard BLtautology if e*(φ) = 1 for each evaluation e and
each continuous t-norm *.
Basic Fuzzy Propositional Logic cont….
• The following t-tautologies are taken as axioms of
the logic BL:
(A1) (φ → ψ) → ((ψ → χ) → (φ → χ))
(A2) (φ & ψ) → φ
(A3) (φ & ψ) → (ψ & φ)
(A4) (φ & (φ → ψ)) → (ψ & (ψ → φ))
(A5a) (φ → (ψ → χ)) → ((φ & ψ) → χ)
(A5b) ((φ & ψ) → χ) → (φ → (ψ → χ))
(A6) ((φ → ψ) → χ) → (((ψ → φ) → χ) → χ)
(A7) 0 → φ
• Modus ponens is the only deduction rule; this gives
the usual notion of proof and provability of the logic
Basic Fuzzy Predicate Logic:
• Basic fuzzy predicate logic has the same
formulas as classical predicate logic (they are
built from predicates of arbitrary arity using
object variables, connectives &, →, truth
constant 0 and quantifiers ∀, ∃.
• The truth degree of an universally quantified
formula ∀xφ is defined as the infimum of truth
degrees of instances of φ
• Similarly ∃xφ has its truth degree defined by the
Various types of Fuzzy Logic
• Monoidal t-norm based propositional fuzzy logic
▫ MTL is an axiomatization of logic where conjunction is
defined by a left continuous t-norm
• Łukasiewicz fuzzy logic
▫ Extension of BL where the conjunction is the Łukasiewicz tnorm
• Gödel fuzzy logic
▫ the extension of basic fuzzy logic BL where conjunction is
the Gödel t-norm: min(x, y)
• Product fuzzy logic
▫ the extension of basic fuzzy logic BL where conjunction is
product t-norm
• Fuzzy Control
▫ Example: For instance, a temperature measurement
for anti-lock breaks might have several separate
membership functions defining particular temperature
ranges needed to control the brakes properly.
▫ Each function maps the same temperature value to a
truth value in the 0 to 1 range. These truth values can
then be used to determine how the brakes should be
• Stanford Encyclopedia of Philosophy:
▫ http://plato.stanford.edu/entries/logic-fuzzy/
• Wikipedia:
▫ http://en.wikipedia.org/wiki/Fuzzy_logic

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