Recipient of the 1998 Outstanding Master's Thesis Award - Third Place.
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Open Access Thesis
The focus of this thesis is to determine exactly which functions serve as appropriate fuzzy negation, conjunction and disjunction functions. To this end, the first chapter serves as motivation for why fuzzy logic is needed, and includes an original demonstration of the inadequacy of many valued logics to resolve the sorites paradox. Chapter 2 serves as an introduction to fuzzy sets and logic. The canonical fuzzy set of tall men is examined as a motivating example, and the chapter concludes with a discussion of membership functions.
Four desirable conditions of the negation function are given in Chapter 3, but it is shown that they are not independent. It suffices to take two of these conditions, monotonicity and involutiveness, as negation axioms. Two characterization proofs are given, one with an increasing generator and the other with a decreasing generator. An example of a general class of negation functions is studied, along with their corresponding increasing and decreasing generators.
Chapters 4 and 5 provide an analysis of fuzzy conjunction and disjunction functions, respectively. Five axioms for each are given: boundary conditions, commutativity, associativity, monotone non-decreasing, and continuity. Yager's class of conjunction and disjunction functions are each shown to satisfy all five of these axioms. The additional assumption of strict monotonicity is added to obtain pseudo-characterizations analogous to the characterizations of the negation function. Finally, it is shown that although the min function is a conjunction function, it does not have a decreasing or an increasing generator. Similar results are obtained in Chapter 5 for disjunction functions, with a concluding theorem that the max function has no generators.
The interactions of these three connectives is the content of Chapter 6. In this chapter, negation, conjunction, and disjunction triples are considered that satisfy both of DeMorgan's laws. Distributivity of conjunction and disjunction over each other is examined. It is then shown that the only conjunction and disjunction pair that satisfies the distributivity axiom is the min, max pair.
In conclusion, Chapter 7 discusses why having unique functions serve as conjunction and disjunction is desirable. It also contains a brief discussion of the implication connective and some areas for further investigation.
Year of Submission
Year of Award
Master of Arts
Department of Mathematics
Joel Haack, Chair, Thesis Committee
1 PDF file (v, 66 pages)
©1997 John Hamman
Hamman, John F., "Fuzzy logic: An analysis of logical connectives and their characterizations" (1997). Dissertations and Theses @ UNI. 659.