Truth value

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 . 

The input variables state now no longer jumps abruptly from one state to the next, but loses value in one membership function while gaining value in the next. At any one time, the truth value of the indoor or outdoor temperature will almost always be m some degree part of two membership functions ... 

Just as there are universal computers that, given a particular input, can simulate any other com-puter, there are NP-complete problems that, with the appropriate input, are effectively equivalent to any NP-hard problem of a given size. For example, Boolean satisfiability -i.e. the problem of determining truth values of the variable s of a Boolean expression so that the expression is true -is known to be an NP-complete problem. See section 12.3.5.2... 

The sign = in propositional equations indicates equality of the truth values of the propositions equated, not necessarily, their identity or equivalence. 

Redefining True, False, and the Value of Connectives. We begin by replacing T with 1, and F with 0, mapping atomic formula onto the range 0,1. Then the truth value of a formula A is either 0 or 1 ... 

In summary, the value of an implication rule is an inverse measure of how often the antecedent has a higher value than the consequent. This is very important in multi-valued logics where the truth value ranges over many numbers rather than just 0 and 1. The more "valuable" the rule, the more often it implies the correct consequence. 

Unlike chemistry, mathematics often deals with infinite domains, and infinite axiom sets. If we allow the fact that two axioms infer the same conclusion to increase the truth value of that conclusion, we must choose some increment that reflects the importance of each individual axiom. If there are an infinite number of such axioms, then each axiom becomes infinitesimally important. Thus LT logic chooses to err on the side of conservatism, assuring that the conclusions will be valid, though perhaps less strong than they could actually be. 

However, the value of an atomic formula is comprised of three parts, the confirmation value, disconfirmation value, and the combined truth value ... 

Incrementally Acquiring Evidence. Unlike LT logic, the IMYL allows successive inferences about a fact to increase the truth value of that fact. One way of viewing the way that the IMYL deals with inferences is to say that an inference in support of a theorem decreases our ignorance about that theorem. Thus, when the theorem is first proposed, the ignorance is maximal, the values for CY, DY, and TY are all 0. The amount of ignorance about the CY (or DY) could be said to be m. 

The first inference of a theorem with value v, (v =< m), then reduces our ignorance by v. If the value v was in confirmation of the theorem, then the values become DY = 0, CY = v, and TY = v. We have reduced our ignorance about the CY to m-v. Further confirmatory evidence for the theorem is applied to the remaining measure of ignorance. The truth value is calculated from CY and DY as normal. 

It can be shown that CV and DV approach the maximal value m asymptotically and that the order of acquiring truth values does not matter. 

From Eq. (G.6) we obtain appropriate probability information via the system operators T = jl T2), while the transformation formulas (G.4) correspond to proper truth-values consistent with Eq. (G.7). The new eigenvectors here are obtained as a superposition of vectors corresponding to legitimate input values for p = 1. For T2 = 0, Eq. (G.6) gives the classical result p =, i.e., no information at all. Consequently r yields a bias to the no information platform. Note that the operator T, or the truth matrix T, is a nonclassical quantity (operator), which will play a crucial role below serving as the square root of the relevant bias" part of the system operator transforming the input information accordingly. 

To keep the rule generally applicable we have to provide alternative rules for the case where one or more arguments are free. In this case we just match A with B without doing any calculations. To realize this alternate rule we need (a) a way to test whether a variable is instantiated and (b) the operator or. For (a) there is a built-in predicate var(A) which evaluates to true if A is a currently uninstantiated variable. The or combination of truth values is symbolized by putting a semicolon ( ) between them. The new subrule is... 

Ordinarily, when someone says something to you, it is immediately and automatically evaluated in relation to the accumulated knowledge that forms part of the CRO. If a salesperson, for example, says, This is the best product on the market, you immediately and automatically evaluate it in terms of your CRO knowledge that salespeople exaggerate, even lie, about the things they have a stake in selling. You take in the statement, but you add the qualification to it that it has questionable truth value. 

True is a very powerful word, with its absolute qualities. It s too powerful. What we actually know is that this Euclidean assumption produces very practical results in a lot of ordinary physical-world situations. This is its truth value. There is also a geometry, though, that... 

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. 

Such an approximate vector representation is depicted in Fig. 1. The vector components have only two meanings, which correspond to the two truth values in classical propositional logic true (1) or false (0). 

Representations of a graph, molecular graph, and their adjacency matrix are shown in Figs. 2(a), (b), and (c), respectively. It is apparent that the elements a, may be treated, again, as Boolean truth values of classical... 

Thus formalized, the structure elucidation problem consists of determining the A matrix elements (0 and 1) from the S vector elements (also 0 and 1). From information about the presence (5, = 1) or absence (5, = 0) of signals in different places within the spectrum range, we shall draw conclusions about the presence (a,y = 1) or absence (a j = 0) of bonds between the atoms i and j within the chemical structure. Note that in relationship (5) we have values that are sharply defined (1 or 0) for both the spectrum parameters and the chemical bonds. Such truth values and the corresponding logic have been designated as crisp ... 

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... 

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