University of Texas at El Paso DigitalCommons@UTEP Departmental Technical Reports (CS) Department of Computer Science 3-1-2001 Why Unary and Binary Operations in Logic: General Result Motivated by Interval-Valued Logics Hung T. Nguyen Vladik Kreinovich University of Texas at El Paso, [email protected] I. R. Goodman Follow this and additional works at: http://digitalcommons.utep.edu/cs_techrep Part of the Computer Engineering Commons Comments: UTEP-CS-01-07. Published in the Proceedings of the Joint 9th World Congress of the International Fuzzy Systems Association and 20th International Conference of the North American Fuzzy Information Processing Society IFSA/NAFIPS 2001, Vancouver, Canada, July 25-28, 2001, pp. 1991-1996. Recommended Citation Nguyen, Hung T.; Kreinovich, Vladik; and Goodman, I. R., "Why Unary and Binary Operations in Logic: General Result Motivated by Interval-Valued Logics" (2001). Departmental Technical Reports (CS). Paper 371. http://digitalcommons.utep.edu/cs_techrep/371 This Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports (CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Why Unary and Binary Operations in Logic: General Result Motivated by Interval-Valued Logics Hung T. Nguyen Vladik Kreinovich Mathem. Sciences, New Mexico State Univ. Comp. Science, Univ. of Texas at El Paso Las Cruces, NM 88003, USA El Paso, TX 79968, USA [email protected] [email protected] I.R. Goodman Code D42215, Space & Naval Warfare Systems Center San Diego, CA 92152, USA [email protected] Abstract We are interested in explaining why unary and binary logical operations are the only basic ones. If we assume Traditionally, in logic, only unary and binary operations that the logic of human reasoning is the two-valued are used as basic ones – e.g., “not”, “and”, “or” – while (classical) logic, then the possibility to transform every the only ternary (and higher order) operations are the logical function into a DNF form explains this empirical operations which come from a combination of unary fact. and binary ones. For the classical logic, with the binary set of truth values ¢¡¤£¦¥¨§ , the possibility to express an However, classical logic is not a perfect description of arbitrary operation in terms of unary and binary ones is human reasoning: for example, it does not take into con- well known: it follows, e.g., from the well known pos- sideration fuzziness and uncertainty of human reason- sibility to express an arbitrary operation in DNF form. ing. This uncertainty is taken into consideration in fuzzy ¡¤£¦¥ A similar representation result for © -based logic was logic [9, 25, 29]. In the traditional fuzzy logic, the set ¡¤£¥ ( proven in our previous paper. In this paper, we expand of truth values is the entire interval © . This this result to finite logics (more general than classical interval has a natural notion of continuity, so it is natu- logic) and to multi-D analogues of the fuzzy logic – both ral to restrict ourselves to continuous unary and binary motivated by interval-valued fuzzy logics. operations. With this restriction in place, a natural question is: can ¡£¥ ¡¤£¥ #+% © an arbitrary continuous function )!*© be represented as a composition of continuous unary and 1. Introduction binary operations? The positive answer to this question was obtained in our papers [19, 22]. Traditionally, in logic, only unary and binary operations ¡£¥ In © -based fuzzy logic, an arbitrary logical operation are used as basic ones – e.g., “not”, “and”, “or” – while can be represented as a composition of unary and binary the only ternary (and higher order) operations are the ¡¤£¦¥ ones. However, the © -based fuzzy logic is, by itself, operations which come from a combination of unary only an approximation to the actual human reasoning and binary ones. about uncertainty. A natural question is: are such combinations sufficient? Indeed, how can we describe the expert’s degree of con- I.e., to be more precise, can an arbitrary logical op- / fidence ,.-0/21 in a certain statement ? A natural way eration be represented as a combination of unary and to determine this degree is, e.g., to ask an expert to es- binary ones? timate his degree of confidence on a scale from 0 to 10. ¥¦¡ 6587 For the classical logic, with the binary set of truth values If he selects 8, then we take ,3-4/21 . ¨¨£¤¨§ ¡£¥§ (= ), the positive answer to this To get a more accurate result, we can then ask the same question is well known. Indeed, it is known that an arbi- expert to estimate his degree of confidence on a finer $#&%' trary logical operation "! can be represented, scale, e.g., from 0 to 100, etc. For example, if an expert ¥ ¥¦¡9¡ ¡¤: ¥ 7 5 e.g., in DNF form and thus, it can indeed be represented 5 selects 81, we will take ,.-0/21 . If we as a combination of unary (“not”) and binary (“and” and want an even more accurate estimate, we can ask the “or”) operations. expert to estimate his degree of confidence on an even Uncertainty of expert estimates is only one reason why ¡¤£¥ finer scale, etc. we may want to go beyond the traditional © -valued logic; there are also other reasons: The problem with this approach is that experts cannot describe their degrees of too fine scales. For example, an expert can point to 8 on a scale from 0 to 10, but this A 1-D value is a reqsonable way of describing the same expert will hardly be able to pinpoint a value on a uncertainty of a single expert. However, the con- ¯ scale from 0 to 100. fidence strongly depends on the it consensus be- tween different experts. We may want to use ad- So, to attain a more adequate description of human ditional dimensions to describe how many expert ¡£¥ reasoning, we must modify the traditional © -based share the original expert’s opinion, and to what de- fuzzy logic. Two types of modifications have been pro- gree; see, e.g., [13, 23]. posed. Different experts may strongly disagree. To de- One possibility is to take the finest (finite) scale which scribe the dgeree of this disagreement, we also an expert can still use, and take the values on this scale need additional numerical characteristics, which as the desired degrees of confidence. This approach make the resulting logic multi-D; see, e.g., [21]. leads to a finite-valued fuzzy logic, in which the set of truth values is finite. In all these cases, we need a multi-D logic to adequately This approach has been successfully used in practice; describe expert’s degree of confidence. see, e.g., [1, 5, 20, 26]. It is therefore desirable to check In this paper, we show that both for finite-valued logics whether in a finite logic, every operation can be repre- and for multi-D logics, every logical operation can be sented as a composition of unary and binary operations. represented as a composition of unary and binary op- The problem with finite-valued logics is that the set erations. Thus, we give a general explanation for the of resulting truth values depends on which scale we use. above empirical fact. Instead of fixing a finite set, we can describe the expert’s ¡¤£¦¥ degree of confidence by an interval from © . For ex- ample, if an expert estimates his degree of confidence 2. Finite-Valued Logics by a value 8 on a 0 to 10 scale, then the only thing that we know about the expert’s degree of confidence is that 2.1. What Was Known Before it is closer to 0.8 (8/10) than to 0.7 or to 0.9, i.e., that it ¡¤: £¡¤: ¢¡ 5£¡ belongs to the interval © . In the Introduction, we have already mentioned that for ¡£¥¨§ So, a natural way of describing degrees of confidence the 2-element set of truth values , an ar- # % £ ! © ¥§¦ ¥©¨ more adequately is to use intervals ¤ instead bitrary logical operation can be repre- of real numbers. In this representation, real numbers can sented as a composition of unary and binary operations. be viewed as particular – degenerate – cases of intervals Specifically, in this case, an arbitrary logical operation % £ ¥ ¥ ¥ ¥ © . The idea of using intervals have been originally can be represented as a composition of negation , proposed by Zadeh himself and further developed by conjunction , and disjunction . Bandler and Kohout [2], Turks¨ ¸en [27], and others; for a In [6], we proved that the same is true for the case when recent survey, see, e.g., [24]. is a finite Boolean algebra. Specifically, we prove In interval-valued fuzzy approach, to describe each de- that for such sets , an arbitrary logical operation can be gree of confidence, we must describe two real numbers: represented as a composition of negation, conjunction the lower endpoint and the upper endpoint of the corre- (“intersection”), disjunction (“union”), constants, and a sponding “confidence interval”. special unary operation called absolute truth: We can go one step further and take into consideration Definition 1. For an arbitrary Boolean algebra , -¥ 1 that the endpoints of the corresponding interval are also we define an absolute truth operation as follows: - 1 -¥ 1 ¥ not precisely known. Thus, each of these endpoints is, and for all . in actuality, an interval itself. So, to describe a degree of -31 The function is similar to the delta-function (see, confidence, we now need four real numbers: two to de- e.g., [30]), which is defined, crudely speaking, as a scribe the lower endpoint, and two to describe the upper function which is different from 0 only at one point ¡ one.