Fuzzy truth values

Fuzzy truth values true, false, fairly true, very true,... 

All of these linguistic terms except fuzzy modifiers are represented in each context by appropriate fuzzy sets. Fuzzy predicates are represented by fuzzy sets defined on universal sets of elements to which the predicates apply. Fuzzy truth values and fuzzy probabilities are represented by fuzzy sets defined on the unit interval [0,1]. Fuzzy quantifiers are either absolute or relative they are represented by appropriate fuzzy numbers defined either on the set of natural numbers or on the interval [0,1]. Fuzzy modifiers are operations by which fuzzy sets representing the various other linguistic terms are appropriately modified to capture the meaning of the modified linguistic terms. 

Baldwin recognises that it is possible to interpret the actual philosophical meaning of the fuzzy truth value restrictions in different ways for different problems. For example he argues that we may wish to think in terms of plausibility, of possibility, of importance, or of dependability as our model or interpretation of truth for a particular problem. Recalling the discussions on dependability in Sections 2.11 and 5.8, it is clear that engineers are not so much interested in the truth of a proposition but in its dependability. Whilst the fuzzy logic notation and fuzzy truth values are retained in the rest of this chapter and in Chapter 10, the interpretation should be that fuzzy truth restrictions are fuzzy restrictions on the dependability of a proposition. 

It is perhaps clear from the preceding discussion, that the ideas of fuzzy sets can be used to generalise the binary concepts of true and false in ordinary logic. True and false can be replaced by fuzzy sets which are truth restrictions (Section 6.1.3) defined on the interval [0,1 ], and these fuzzy restrictions are interpreted as fuzzy truth values. This fuzzy logic (FL) is what Zadeh has tentatively suggested [83]. The ideas described in this section are based on the developments of Zadeh s work by Baldwin [84]. 

The traditional deductive syllogism was described in Section 2.5. Where we were dealing with precisely defined statements which can be labelled true or false. Using fuzzy logic we can begin to deal with deductive syllogisms where imprecise statements are labelled with fuzzy truth values. For example,... 

The deductii . . f the previous section, whilst being interesting, are likely to be useful only a )mall number of problems. In order to consider more complex problems it S ob. iously necessary to be able to deal with compound propositions. Let us begin this section by considering a compound proposition such as P Q is T, made up of two propositions P and Q, connected by which is a logical relation such as AND, OR, IMPLIES or EQUIVALENT TO, and r is a fuzzy truth value restriction on. ... 

These are linguistic terms that modify other linguistic terms, for example, very, more or less, fairly, extremely. They may be used to modify fuzzy predicates, fuzzy truth-values, and fuzzy probabilities. The fuzzy set that defines the hedge is called a modifier. 

Fuzzy sets and fuzzy logic. Fuzzy sets differ from the normal crisp sets in the fact that their elements have partial membership (represented by a value between 0 an 1) in the set. Fuzzy logic differs from the binary logic by the fact that the truth values are represented by fuzzy sets. 

Conventional computers initially were not conceived to handle vague data. Human reasoning, however, uses vague information and uncertainty to come to a decision. In the mid-1960 this discrepancy led to the conception of fuzzy theory [14]. In fuzzy logic the strict scheme of Boolean logic, which has only two statements true and false), is extended to handle information about partial truth, i.e., truth values between "absolutely true" and absolutely false". It thus gives a mathematical representation of uncertainty and vagueness and provides a tool to treat them. 

Fuzzy logic extends the Boolean logic so as to handle information about truth values which are between absolutely true and "absolutely false . 

One can see that the truth values in fuzzy logic strongly resemble the stochastic values from the theory of probabilities. However, methods based on the use of statistics are not considered fuzzy by the orthodox fuzzy theory protagonists. Instead of using probability values, fuzzy theory works with possibility values. It is argued that both values are substantially different and that the latter have to be evaluated by methods other than statistical. Our understanding, however, is that at a very fundamental level, both values have essentially the same nature. 

Fuzzy logic is also widely used in process control, because it allows rules to be expressed in a simple linguistic format IF (A) THEN (B), with an associated confidence level that is related to the membership functions. To understand how it is used for control, consider the simple example of a fan heater governed by four rules, summarized in Fig. 8. These rules map onto the four fuzzy sets COLD, COOL, WARM, and HOT also shown in Fig. 8. If the room temperature is 18°C, the heavy line on Fig. 8, then by Rule 3, the fan speed is medium, with truth value 0.7, and by Rule 4, the fan speed is low with truth value 0.3. The process of... 

Now as we have noted earlier, fuzzy logic (FL) is an extension of MVL with truth values as fuzzy sets or more accurately as fuzzy restrictions on the truth. We will still call the truth space U defined on the interval [0, 1], but in FL a truth value will be a fuzzy subset t CU and Xr U- [0, 1]. We will still require rules to define the implication relation 1 as well as negation, conjunction C, disjunction D and equivalence. Zadeh suggested the use of the Lukasiewicz rules given above and they will be used in the rest of this book. Baldwin, Pilsworth and Guild [85, 86] have examined various alternative rules for implication. 

In fuzzy logic the process of obtaining a modified truth value for a proposition, given data. 

Another similar way is to define the various goals by calculating truth values. For example, imagine a structure is to be designed so that it is safe and economic where these fuzzy goals are defined by Fig. 6.10, and the elements of safe are n e N where pf = 10 " the probability of failure, and the elemenis of economic aiehe H the utility measure [0,1]. 

In fact Baldwin gives special labels to certain truth value fuzzy sets or truth value restrictions such as true, false, unrestricted, impossible, absolutely true and absolutely false. In Fig. 6.12 a set of definitions are illustrated which were those adopted in Baldwin s earlier work. Later these definitions were slightly amended (Fig. 6.20). It is most important to note that we are now dealing with truth value restrictions. Thus the membership of any element of a given truth value restriction will be the maximum possible or least restrictive value, given the available information. 

The method for calculating tb is exactly that described in the introduction to this chapter for binary and multi-valued logic. The process is one of calculating fuzzy truth restrictions for the first and second lines of the deduction on the space Ux X Uy, intersecting them to produce an equivalent restriction and then projecting the result on to Uy Thus... 

The method of comparing alternative design solutions is as follows. Truth values for the propositions NP, RH, HE, DC, MC, FC, B, C, WL, that is Ti, T2. . . T9, are calculated for a particular design solution. For example Ti may be calculated by ITFM as in Fig. 6.19. A fuzzy set value for the notional probability of a particular design solution is calculated or estimated by other means and the truth value restriction Ti is then the truth of NP given the design solution fuzzy... 

The definition can be extended to give a fuzzy possibility by composing v(A/A ) not just with true but with a range of truth values either side of true from undecided to absolutely true. If a membersliip level of one is given to the possibility value calculated above, then the other possibility values can be given... 

Fuzzy propositions are assigned to fuzzy sets. Suppose proposition P is assigned to the fuzzy set A, then the truth value of a proposition, (P), is given by T P) = P/ x) where 0 < pj < 1. 

The fundamental difference between classical propositions and fuzzy propositions is in the range of their truth values. While each classical proposition is required to be either true or false, the truth or falsity of fuzzy propositions is a matter of degree (Klir and Yuan 1995). Assuming that tmth and falsity are expressed by values 1 and 0, respectively, the degree of tmth of each fuzzy proposition is expressed by a number in the unit interval [0,1]. 

Fuzzy rule sets usually have several antecedents that are combined using fuzzy operators. The combination is called a premise, and it generates a single truth value that determines the rule s outcome. In general, one rule by itself is not sufficient, but two or more rules that can play off one another are needed. The output of each rule is a fuzzy set, but in general, the output of an entire collection of rules should be a single number. 

An assertion or belief is something understood or felt to be understood by the domain expert. There is no "truth" value or certainty check. It would be possible to indicate the certainty value of an assertion using a form of fuzzy reasoning or simply attaching a probability value to all assertions. 

It can be verified that the relations above reduce to their usual counterparts when applied to binary logic. The standard Lukasiewicz logic Li is isomorphic to fuzzy set theory based on the standard fuzzy operations in the same way the two-valued logic is isomorphic to the crisp set theory. The membership degree A(x) for x e X may be interpreted as the truth value of the proposition x is a member of the set A . The reciprocal is also valid. 

Human processing of information is not based on two-valued, off-on, either-or logic. It is based on fuzzy perceptions, fuzzy truths, fuzzy inferences, etc., all resulting in an averaged, summarized, normalized output, which is given by the human a precise number or decision value which he/she verbalizes, writes down or acts on. It is the goal of fuzzy logic control systems to also do this. 

The truth value of a rule is determined from the conjunction (i.e. minimum degree of membership of the rule antecedents) (Zadeh (1973)). Thus the trath-value of the rule is taken to be the smallest degree of tmth of the rule antecedents. This tmth-value is then applied to all consequences of the rule. If any fuzzy output is a consequent of more than one rule, that output is set to the highest (maximum) tmth-value of all the mles that include it as a consequent. The result of the mle evaluation is a set of fiizzy conclusions that reflect the effects of all the mles whose tmth-values are greater than zero. 

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