A Gap Between Fuzzy . . . What We Do in This . . . Source of Multi- . . . We Have, in Effect, at . . . Isn’t Every Sufficiently How Many Truth . . . Complex Logic Multi-Valued Need to Consider . . . How’s This Applicable . . . Already: Lindenbaum-Tarski What Happens in the . . . Relation to Fuzzy Algebra and Fuzzy Logic Home Page Are Both Particular Cases Title Page of the Same Idea ◭◭ ◮◮ ◭ ◮ Andrzej Pownuk and Vladik Kreinovich Page 1 of 17 Computational Science Program, University of Texas at El Paso El Paso, Texas 79968, USA, ampownuk@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 1. A Gap Between Fuzzy Logic and the Tradi- Source of Multi- . . . tional 2-Valued Fuzzy Logic We Have, in Effect, at . . . • One of the main ideas behind fuzzy logic is that: How Many Truth . . . Need to Consider . . . – in contrast to the traditional 2-valued logic, in How’s This Applicable . . . which every statement is either true or false, What Happens in the . . . – in fuzzy logic, we allow intermediate degrees. Relation to Fuzzy • In other words, fuzzy logic is an example of a multi- Home Page valued logic. Title Page • This led to a misunderstanding between researchers in ◭◭ ◮◮ fuzzy and traditional logics. ◭ ◮ • Fuzzy logic books claim that the 2-valued logic cannot Page 2 of 17 describe intermediate degrees. Go Back • On the other hand, 2-valued logicians criticize fuzzy Full Screen logic for using “weird” intermediate degrees. Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 2. What We Do in This Paper Source of Multi- . . . • We show that the mutual criticism is largely based on We Have, in Effect, at . . . a misunderstanding. How Many Truth . . . Need to Consider . . . • It is possible to describe intermediate degrees in the How’s This Applicable . . . traditional 2-valued logic. What Happens in the . . . • However, such a representation is complicated. Relation to Fuzzy • The main advantage of fuzzy techniques is that they Home Page provide a simply way of doing this. Title Page • And simplicity is important for applications. ◭◭ ◮◮ • We also show that the main ideas of fuzzy logic are ◭ ◮ consistent with the 2-valued foundations. Page 3 of 17 • Moreover, they naturally appear in these foundations Go Back if we try to adequately describe expert knowledge. Full Screen • We hope to help researchers from both communities to better understand each other. Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 3. Source of Multi-Valuedness in Traditional Source of Multi- . . . Logic: G¨ odel’s Theorem We Have, in Effect, at . . . • A naive understanding of the 2-valued logic assumes How Many Truth . . . that every statement S is either true or false. Need to Consider . . . How’s This Applicable . . . • This is possible in simple situations. What Happens in the . . . • However, G¨ odel’s showed that this not possible for Relation to Fuzzy complex theories. Home Page • G¨ odel analyzed arithmetic – statements obtained Title Page – from basic equalities and inequalities between poly- ◭◭ ◮◮ nomial expressions ◭ ◮ – by propositional connectives &, ∨ , ¬ , and quanti- Page 4 of 17 fiers over natural numbers. Go Back • He showed that it is not possible to have a theory T in Full Screen which for every statement S , either T � S or T � ¬ S . Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 4. We Have, in Effect, at Least Three Different Source of Multi- . . . Truth Values We Have, in Effect, at . . . • Due to G¨ odel’s theorem, there exist statements S for How Many Truth . . . which T � � S and T � � ¬ S . So: Need to Consider . . . How’s This Applicable . . . – while, legally speaking, the corresponding logic is What Happens in the . . . 2-valued, Relation to Fuzzy – in reality, such a statement S is neither true nor Home Page false. Title Page • Thus, we have more than 2 possible truth values. ◭◭ ◮◮ • At first glance, we have 3 truth values: “true”, “false”, ◭ ◮ and “unknown”. Page 5 of 17 • However, different “unknown” statements are not nec- Go Back essarily provably equivalent to each other. Full Screen • So, we may have more than 3 truth values. Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 5. How Many Truth Values Do We Actually Have Source of Multi- . . . • It is reasonable to consider the following equivalence We Have, in Effect, at . . . relation between statements A and B : How Many Truth . . . Need to Consider . . . � ( A ⇔ B ) How’s This Applicable . . . • Equivalence classes with respect to this relation can be What Happens in the . . . viewed as the actual truth values. Relation to Fuzzy Home Page • The set of all such equivalence classes is known as the Title Page Lindenbaum-Tarski algebra . ◭◭ ◮◮ • Lindenbaum-Tarski algebra shows that any sufficiently complex logic is, in effect, multi-valued. ◭ ◮ Page 6 of 17 • However, this multi-valuedness is different from the multi-valuedness of fuzzy logic. Go Back • We show that there is another close-to-fuzzy aspect of Full Screen multi-valuedness of the traditional logic. Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 6. Need to Consider Several Theories Source of Multi- . . . • In the previous section, we considered the case when We Have, in Effect, at . . . we have a single theory T . How Many Truth . . . Need to Consider . . . • G¨ odel’s theorem states that: How’s This Applicable . . . – for every given theory T that includes formal arith- What Happens in the . . . metic, Relation to Fuzzy – there is a statement S that can neither be proven Home Page nor disproven in this theory. Title Page • This statement S can neither be proven not disproven ◭◭ ◮◮ based on the axioms of theory T . ◭ ◮ • So, a natural idea is to consider additional reasonable Page 7 of 17 axioms that we can add to T . Go Back Full Screen Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 7. Need to Consider Several Theories (cont-d) Source of Multi- . . . • This is what happened in geometry with the V-th pos- We Have, in Effect, at . . . tulate P – that How Many Truth . . . Need to Consider . . . – for every line ℓ in a plane and for every point P How’s This Applicable . . . outside this line, What Happens in the . . . – there exists only one line ℓ ′ which passes through Relation to Fuzzy P and is parallel to ℓ . Home Page • It turned out that neither P not ¬ P can be derived Title Page from all other (more intuitive) axioms of geometry. ◭◭ ◮◮ • So, a natural solution is to explicitly add this statement ◭ ◮ as a new axiom. Page 8 of 17 • If we add its negation, we get Lobachevsky geometry Go Back – historically the first non-Euclidean geometry. Full Screen Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 8. Need to Consider Several Theories (cont-d) Source of Multi- . . . • A similar thing happened in set theory, with the Axiom We Have, in Effect, at . . . of Choice and Continuum Hypothesis. How Many Truth . . . Need to Consider . . . • They cannot be derived or rejected based on the other How’s This Applicable . . . (more intuitive) axioms of set theory. What Happens in the . . . • Thus, they (or their negations) have to be explicitly Relation to Fuzzy added to the original theory. Home Page • The new – extended – theory covers more statements Title Page that the original theory T . ◭◭ ◮◮ • However, the same G¨ odel’s theory still applies to the ◭ ◮ new theory: Page 9 of 17 – there are statements that Go Back – can neither be deduced nor rejected based on this new theory. Full Screen • Thus, we need to add one more axiom, etc. Close Quit
A Gap Between Fuzzy . . . What We Do in This . . . 9. We Have a Family of Theories Source of Multi- . . . • So, instead of a single theory, it makes sense to consider We Have, in Effect, at . . . a family of theories { T α } α . How Many Truth . . . Need to Consider . . . • In the above description, we end up with a family which How’s This Applicable . . . is linearly ordered in the sense that: What Happens in the . . . – for every two theories T α and T β , Relation to Fuzzy – either T α � T β or T β � T α . Home Page • However, it is possible that on some stage, different Title Page groups of researchers select two different axioms. ◭◭ ◮◮ • In this case, we will have two theories which are not ◭ ◮ derivable from each other. Page 10 of 17 • Thus, we have a family of theories which is not linearly Go Back ordered. Full Screen Close Quit
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