Fuzzy Logic: una introduzione, Dispense di Logica Matematica

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INTRODUCTION TO FUZZY SETS AND

FUZZY LOGIC

Prof.ssa E.B.P. Tiezzi

CONTENTS OF THE COURSE

PART I Introduction Fuzzy sets Operations with fuzzy sets t-norms A theorem about continuous t-norm

INTRODUCTION

There are no whole truths; all

truths are half- truths. It is trying to

treat them as whole truths that

plays the devil.

- Alfred North Whitehead

…THERE ARE TWO MAIN GROUPS..

“…..There two main groups….:

• Logicians: they can see that fuzzy logic is indeed a branch of logic…..
• Researchers involved in fuzzy logic application and soft computing. Most of these are not professional logicians. This fact is not very important but is the opinion of the author that a better understanding of the strictly logical basis of fuzzy logic is very useful for fuzzy logic appliers….. Still more than that: a better mutual understanding between (classical) logician and researchers in fuzzy logic promises to lead to deeper cooperation and new results…” Preface of Mathematics of Fuzzy logic , Petr Hajek

FUZZY LOGIC IN BROAD SENSE Fuzzy logic in broad sense serves mainly as apparatus for fuzzy control, analysis of vagueness in natural language and several other application domains. It is one of the techniques of soft-computing, i.e. computational methods tolerant to suboptimality and impreciseness (vagueness) and giving quick, simple and sufficiently good solutions.

FUZZY LOGIC IN THE NARROW SENSE Fuzzy logic in the narrow sense is symbolic logic with a comparative notion of truth developed fully in the spirit of classical logic (syntax, semantics, axiomatization, truth-preserving deduction, completeness, etc.; both propositional and predicate logic). It is a branch of many-valued logic based on the paradigm of inference under vagueness.

FUZZY LOGIC IS POPULAR The number of papers dealing, in some sense, with fuzzy logic and its applications is immense, and the success in applications is evident, in particular in fuzzy control. There are numerous books written on this subject and numerous papers dealing with fuzzy systems. Naturally, in this immense literature the quality varies; a mathematician (logician) browsing in it is sometimes bothered by papers that are mathematically poor (and he/she may easily overlook those that are mathematically excellent). This should not lead to a quick rejection of the domain. Let us quote Zadeh, the inventor of fuzzy sets "Although some of the earlier controversies regarding the applicability of fuzzy logic have abated, there are still influential voices which are critical and/or skeptical. Some take the position that anything that can be done with fuzzy logic can be done equally well without it. Some are trying to prove that fuzzy logic is wrong. And some are bothered by what they perceive to be exaggerated expectations. That may well be the case but, as Jules Verne had noted at the tum of the century, scientific progress is driven by exaggerated expectations."

LOGIC STUDIES THE NOTION(S) O F CONSEQUENCE. It deals with propositions (sentences), sets of propositions and the relation of consequence among them. The task of formal logic is to represent all this by means of well-defined logical calculi admitting exact investigation. Various calculi differ in their definitions of sentences and notion(s) of consequence (propositional logics, predicate logics, modal propositional/predicate logics, many-valued proposi- tional/predicate logics etc.). Often a logical calculus has two notions of con- sequence: syntactical (based on a notion of proof) and semantical (based on a notion of truth); then the natural questions of soundness (does provability imply truth?) and completeness (does truth imply provability?) pose them- selves.

HAYEK ….

Fuzzy logic is neither a poor man's logic nor poor man's probability. Fuzzy logic (in the narrow sense) is a reasonably deep theory. Fuzzy logic is a logic. It has its syntax and semantics and notion of consequence. It is a study of consequence. There are various systems of fuzzy logic, not just one. We have one basic logic (BL) and three of its most important extensions: Łukasiewicz logic, Gödel logic and the product logic. Fuzzy logic in the narrow sense is a beautiful logic, but is also important for applications: it offers foundations.

SET THEORY REFRESHER A SET IS A MANY THAT ALLOWS ITSELF TO BE THOUGHT OF AS A ONE. GEORG CANTOR. The classical set theory simply designates the branch of mathematics that studies (crisp) sets. Note that in a set the order does not matter: { 7 , 9 } denotes the same set as { 9, 7 }.

Usually, a set is denoted by a capital letter. The empty set is denoted ∅: it contains no

elements. In order to manipulate classical ensembles, we define a set of operations: union of two sets, intersection of two sets and complement.

CRISP SET Example Let X be the set of all real numbers between 0 and 10 and let A = [5, 9] be the subset of X of real numbers between 5 and 9. This results in the following figure:

FUZZY SETS Fuzzy sets generalise this definition, allowing elements to belong to a given set with a certain degree. Instead of considering characteristic functions with value in {0, 1} we consider now functions valued in [0, 1]. A fuzzy subset F of a set X is a function μF (x) assigning to every element x of X the degree of membership of x to F: x ∈ X → μF (x) ∈ [0, 1].

SHOWS THE DIFFERENCE BETWEEN A CONVENTIONAL SET AND A FUZZY SET CORRESPONDING TO A DELICIOUS FOOD.

COMPARE THE TWO MEMBERSHIP FUNCTIONS CORRESPONDING TO THE PREVIOUS SET.