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A Realistic (NonAssociative) Logic And a Possible Explanations of 7±2 Law
, 2000
"... When we know the subjective probabilities (degrees of belief) p1 and p2 of two statements S1 and S2 , and we have no information about the relationship between these statements, then the probability of S1 &S2 can take any value from the interval [max(p1 + p2 \Gamma 1; 0); min(p1 ; p2 )]. If we must ..."
Abstract

Cited by 9 (7 self)
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When we know the subjective probabilities (degrees of belief) p1 and p2 of two statements S1 and S2 , and we have no information about the relationship between these statements, then the probability of S1 &S2 can take any value from the interval [max(p1 + p2 \Gamma 1; 0); min(p1 ; p2 )]. If we must select a single number from this interval, the natural idea is to take its midpoint. The corresponding "and" operation p1 & p2 def = (1=2) \Delta (max(p1 +p2 \Gamma 1; 0)+min(p1 ; p2)) is not associative. However, since the largest possible nonassociativity degree j(a & b) & c \Gamma a & (b & c)j is equal to 1/9, this nonassociativity is negligible if the realistic "granular" degree of belief have granules of width 1=9. This may explain why humans are most comfortable with 9 items to choose from (the famous "7 plus minus 2" law). We also show that the use of interval computations can simplify the (rather complicated) proofs. 1 1 In Expert Systems, We Need Estimates for the Degree of...
A Semantics for Fuzzy Logic
 Soft Computing
, 1997
"... We present a semantics for certain Fuzzy Logics of vagueness by identifying the fuzzy truth value an agent gives to a proposition with the number of independent arguments that the agent can muster in favour of that proposition. Introduction In the literature the expression `Fuzzy Logic' is used in ..."
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Cited by 7 (0 self)
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We present a semantics for certain Fuzzy Logics of vagueness by identifying the fuzzy truth value an agent gives to a proposition with the number of independent arguments that the agent can muster in favour of that proposition. Introduction In the literature the expression `Fuzzy Logic' is used in two separate ways (at least). One is where `truth values' are intended to stand for measures of belief (or condence, or certainty of some sort) and the expression `Fuzzy Logic' is taken as a synonym for the assumption that belief values are truth functional. That is, if w() denotes an agent's belief value (on the usual scale [0; 1]) for 2 SL, where SL is the set of sentences from a nite propositional language L built up using the connectives :; ^; _ (we shall consider implication later), then w satises w(:) = F: (w()); w( ^ ) = F^ (w(); w()); w( _ ) = F_ (w(); w()); (1) for some xed functions F: : [0; 1] ! [0; 1] and F^ ; F_ : [0; 1] 2 ! [0; 1]; where ; 2 SL. Two p...
NonAssociative Operations
 PROCEEDINGS OF THE SECOND INTERNATIONAL CONFERENCE ON INTELLIGENT TECHNOLOGIES INTECH'2001
, 2001
"... How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A&B and B&A are the same, the corresponding andoperation should be commutative. Similarly, since A&A means the same as A, we should expect that the andoperation shou ..."
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Cited by 3 (2 self)
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How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A&B and B&A are the same, the corresponding andoperation should be commutative. Similarly, since A&A means the same as A, we should expect that the andoperation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict to the picked ones. This idea leads to a neat mathematical theory, but the analysis of reallife expert reasoning shows that all the requirements are only approximately satisfied. we should require all of these requirements to be satisfied to some extent. In this paper, we show the preliminary results of analyzing such operations. In particular, we show that nonassociative operations explain the empirical 7 \Sigma 2 law in psychology according to which a person can normally distinguish between no more than 7 plus minus 2 classes.
A Realistic (NonAssociative) Interval Logic And How Interval Computations Help In Proving Results About It
, 2000
"... Experts' uncertainty about their statements S i is described by probabilities p i . The conclusion C of an expert system normally depends on several statements S i , so to estimate the reliability p(C), we must estimate the probability of Boolean combinations like S1&S2 . We cannot ask experts abou ..."
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Experts' uncertainty about their statements S i is described by probabilities p i . The conclusion C of an expert system normally depends on several statements S i , so to estimate the reliability p(C), we must estimate the probability of Boolean combinations like S1&S2 . We cannot ask experts about all 2 n (? 10 10 ) such combinations, so we must estimate p(S1&S2) based on p1 = p(S1) and p2 = p(S2 ). One can use the interval p = [max(p1 +p2 \Gamma 1; 0); min(p1 ; p2 )] of possible values of p(S1&S2 ), but this often leads to p(C) = [0; 1]. A natural idea is to use a midpoint of p instead; this midpoint is a mathematical expectation of p(S1&S2) over a uniform (second order) distribution on all possible probability distributions. This midpoint operation & is not associative (which fits well with human reasoning). We show that some properties of this operation, like semiassociativity and the upper bound (1/9) on the difference a&(b&c) \Gamma (a&b)&c, can be derived by using inter...
On Efficient Representation of Expert Knowledge by Fuzzy Logic: Towards an Optimal Combination of Granularity and HigherOrder Approaches
, 2001
"... A natural approach to designing an intelligent system is to incorporate expert knowledge into this system. One of the main approaches to translating this knowledge into computerunderstandable terms is the approach of fuzzy logic. It has led to many successful applications, but in several aspects, t ..."
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A natural approach to designing an intelligent system is to incorporate expert knowledge into this system. One of the main approaches to translating this knowledge into computerunderstandable terms is the approach of fuzzy logic. It has led to many successful applications, but in several aspects, the resulting computer representation is somewhat different from the original expert meaning. Two related approaches have been used to make fuzzy logic more adequate in representing expert reasoning: granularity and higherorder approaches. Each approach is successful in some applications where the other approach did not succeed so well; it is therefore desirable to combine these two approaches. This idea of combining the two approaches is very natural, but so far, it has led to few successful practical applications. In this paper, we provide results aimed at finding a better (ideally optimal) way of combining these approaches.
A Realistic (NonAssociative) Logic And a Possible Explanations of 7
, 2000
"... When we know the subjective probabilities (degrees of belief) p1 and p2 of two statements S1 and S2 , and we have no information about the relationship between these statements, then the probability of S1 &S2 can take any value from the interval [max(p1 + p2 \Gamma 1; 0); min(p1 ; p2 )]. If we must ..."
Abstract
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When we know the subjective probabilities (degrees of belief) p1 and p2 of two statements S1 and S2 , and we have no information about the relationship between these statements, then the probability of S1 &S2 can take any value from the interval [max(p1 + p2 \Gamma 1; 0); min(p1 ; p2 )]. If we must select a single number from this interval, the natural idea is to take its midpoint. The corresponding "and" operation p1 & p2 def = (1=2) \Delta (max(p1 +p2 \Gamma 1; 0)+min(p1 ; p2)) is not associative. However, since the largest possible nonassociativity degree j(a & b) & c \Gamma a & (b & c)j is equal to 1/9, this nonassociativity is negligible if the realistic "granular" degree of belief have granules of width 1=9. This may explain why humans are most comfortable with 9 items to choose from (the famous "7 plus minus 2" law). We also show that the use of interval computations can simplify the (rather complicated) proofs. 1 1 In Expert Systems, We Need Estimates for the Degree of Certainty of S 1 &S 2 and S 1 S 2 In many areas (medicine, geophysics, military decisionmaking, etc.), top quality experts make good decisions, but they cannot handle all situations. It is therefore desirable to incorporate their knowledge into a decisionmaking computer system. Experts describe their knowledge by statements S 1 ; : : : ; Sn (e.g., by ifthen rules). Experts are often not 100% sure about these statements S i ; this uncertainty is described by the subjective probabilities p i (degrees of belief, etc.) which experts assign to their statements. The conclusion C of an expert system normally depends on several statements S i . For example, if we can deduce C either from S 2 and S 3 , or from S 4 , then the validity of C is equivalent to the validity of a Boolean combination...
Bridges Between Contextual Linguistic Models of Vagueness and Tnorm Based Fuzzy Logic
, 2009
"... Linguistic models of vagueness usually record contexts of possible precisifications. A link between such models and fuzzy logic is established by extracting fuzzy sets from context based word meanings and analyzing standard logical connectives in this setting. In a further step Lawry’s voting semant ..."
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Linguistic models of vagueness usually record contexts of possible precisifications. A link between such models and fuzzy logic is established by extracting fuzzy sets from context based word meanings and analyzing standard logical connectives in this setting. In a further step Lawry’s voting semantics for fuzzy logics is used to reinterpret standard tnorm based truth functions from the point of view of context update semantics. 1