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}

• 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 values
▫ 0 is absolute falsity
▫ 1 is absolute truth

• 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)

• 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}

• 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

• 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 BL-
tautology if e
*
(φ) = 1 for each evaluation e and each continuous t-norm *.

• 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
BL.

• 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 supremum

• 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 controlled.

• Stanford Encyclopedia of Philosophy:
▫ http://plato.stanford.edu/entries/logic-fuzzy/
• Wikipedia:
▫ http://en.wikipedia.org/wiki/Fuzzy_logic