Traditionally, in logic, only unary and binary operations are used as basic ones -- e.g., "not", "and", "or" -- while the only ternary (and higher order) operations are the operations which come from a combination of unary and binary ones. For the classical logic, with the binary set of truth values f0; 1g, the possibility to express an arbitrary operation in terms of unary and binary ones is well known: it follows, e.g., from the well known possibility to express an arbitrary operation in DNF form. A similar representation result for [0; 1]-based logic was proven in our previous paper. In this paper, we expand this result to finite logics (more general than classical logic) and to multi-D analogues of the fuzzy logic -- both motivated by interval-valued fuzzy logics. 1.

Abstract. 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 computer-understandable 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 higher-order 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 chapter, we provide results aimed at finding a better (ideally optimal) way of combining these approaches.

. To describe experts' uncertainty in a knowledge-based system, we usually use numbers from the interval [0; 1] (subjective probabilities, degrees of certainty, etc.). The most direct way to get these numbers is to ask the expert; however, the expert may not be 100% certain what exactly number describes his uncertainty; so, we end up with a second-order uncertainty -- a degree of certainty describing to what extent a given number d adequately describes the expert's uncertainty about a given statement A. At first glance, it looks like we should not stop at this second order: the expert is probably as uncertain about his second-order degree as about his first-order one, so we need third order, fourth order descriptions, etc. In this paper, we show that from a realistic (granular) viewpoint, taking into consideration that in reality, an expert would best describe his degrees of certainty by a word from a finite set of words, it is sufficient to have a second-order description; from this v...

. In most knowledge-based systems, the experts' uncertainty is described by a real number from the interval [0; 1] (this number is called subjective probability, degree of certainty, etc.). However, experts usually use a small finite set of words to describe their degree of unecratinty; thus, to adequately describe the expert's optinion, it is desirable to use a finite (granular) logic. If all we know about the expert's opinion on two statements A and B is this expert's degrees of certainty d(A) and d(B) in these two statements, and the user asks a query "A&B?", then we need to estimate the degree d(A&B) based on the given values d(A) and d(B). In this paper, we formalize the natural demand that gradual changes in d(A) and d(B) must lead to gradual changes in our estimate for d(A&B) (we called it continuity). We show that the only continuous &-operation is min(a; b). Likewise, the only continuous -operation is max(a; b), the only continuous "not"\Gammaoperation corresponds to f(a) = 1 ...

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 computer-understandable 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 higher-order 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.

To describe experts' uncertainty in a knowledge-based system, we usually use numbers from the interval [0; 1] (subjective probabilities, degrees of certainty, etc.). The most direct way to get these numbers is to ask an expert; however, the expert may not be 100% certain what exactly number describes his uncertainty; so, we end up with a second-order uncertainty -- a degree of certainty describing to what extent a given number d adequately describes the expert's uncertainty about a given statement A. At first glance, it looks like we should not stop at this second order: the expert is probably as uncertain about his second-order degree as about his first-order one, so we need third order, fourth order descriptions, etc. In this paper, we show that from a realistic (granular) viewpoint, taking into consideration that in reality, an expert would best describe his degrees of certainty by a word from a finite set of words, it is sufficient to have a second-order description; from this vi...

knowledge-based system, we usually use numbers from the interval (subjective probabilities, degrees of certainty, etc.). The most direct way to get these numbers is to ask an expert; however, the expert may not be 100 % certain what exactly number describes his uncertainty; so, we end up with a second-order uncertainty – a degree of certainty describing to what extent § a given number adequately describes the expert’s uncertainty about a given statement At first glance, it looks like we should not stop at this second order: the expert is probably as uncertain about his second-order degree as about his first-order one, so we need third order, fourth order descriptions, etc. In this paper, we show that from a realistic (granular) viewpoint, taking into consideration that in reality, an expert would best describe his degrees of certainty by a word from a finite set of words, it is sufficient to have a second-order description; from this viewpoint, higher order descriptions can be uniquely reconstructed from the second-order one, and in this sense, the secondorder description is sufficient.