Two-valued logic

Thus, propanol, C3H70H, has a membership of 1 in the three-carbon molecule class, while ethanol, C2H5OH, has a membership of 0 in the same class. As the membership in a crisp set must take one of only two possible values, Boolean (two-valued) logic can be used to manipulate crisp sets. If all the knowledge that we have can be described by placing objects in sets that are separated by crisp divisions, the sort of rule-based approach to the development of an expert system described in the previous chapter is appropriate. 

In version 1, the definition is qualified by the statement In the last case of this definition, let logically equivalent mean equivalent according to the rules of classical two-valued propositional calculus. This qualification means, in effect, that Definition 1 defines the system of classical, two-valued logic. The definition is employed in the following theorem ... 

Fuzzy logic must violate, by definition, some laws of classical logic or, alternatively, it must violate the truth functionally.Hence, Elkan s theorem (however trivial) does not say anything about fuzzy logic at all. The technical limitations implied by the theorem do not apply to fuzzy logic, but to classical two-valued logic. 

To conclude it is natural to abandon two-valued logic in science because, when one deals with the phenomena of nature, one enters a stage in logic in which we recognize the utility of imprecision . The vagueness of chiral and achiral, when used with reference to a chemical system, should therefore be regarded as a virtue rather than as a defect. 

Using a variety of parallel architectures. Because the digital circuit adopts two value logic, as long... 

The reason for including a discussion of the philosophical foundations of mathematics and science is to demonstrate that traditional two-valued logic is but one way of setting up a deductive system for communicating scientific ideas there are alternative logics, and fuzzy logic as presented in Chapter 6 is one of them. 

Fuzzy logic maintains that all things are a matter of degree and challenges traditional two-valued logic which holds... 

The basic assumption upon which two-valued logic is based, that every proposition is either true or false, has been questioned since Aristotle. For example, propositions about future events are neither actually true, nor actually false, but potentially either. Consequently, their truth-value is undetermined, at least prior to the event. Propositions whose truth-value is problematic are not limited to future events. For example, propositions that imply physical or chemical measurements depend on the limitations of mea-surement. While several types of multivalued logic have been proposed, here we will discuss the infinite-valued logic, whose truth-values are represented by all the real numbers in the interval [0,1]. This is also called standard Lukasiewicz logic L. The primitives of this logic are defined as... 

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. 

The outcome of agglomerative hierarchical cluster analysis is a crisp cluster membership function, which can take only the values 0 (no membership) or 1 (membership). Other non-hierarchical clustering techniques such as k-means cluster (KMC) analysis still follow this concept, whereas fuzzy C-means (FCM) clustering returns fuzzy class memberships. The latter method thus departs from the classical (0 or 1) two-valued logic and uses soft linguistic system variables, i.e. degrees of class membership values varying between 0 and 1. 

Two-valued logic is assumed, so all signals (and possibly variables) will only occupy Ihe states 0 or 1. Therefore each element must be of type BIT. 

As with the example in section 4.2, all signals are of type BIT (two-value logic). 

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