Electronic Theses and Dissertations

Award Winner

Recipient of the 1995 Outstanding Master's Thesis Award - Second Place.

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Open Access Thesis


The problem to be studied for this thesis was that of whether the usual statement calculus is a suitable formal system for every many-valued logic in a particular collection of logics. The logics in question are those that fall between the usual two-valued logic and a modified form of the Lukasiewicz-Tarski three-valued logic.

Since this betweenness relationship was an original concept and appeared nowhere in the literature, the first goal in the research plan was to define this relationship precisely. Preliminary concepts included truth value mapping and forgivingness of logics, concepts that, like betweenness, are original to this paper and that facilitate the comparison of many-valued logics. After betweenness was defined, the next stage of the research would be to investigate the logics between the classical logic and the modified Lg and to see for which of these logics the statement calculus is suitable. This would involve direct calculations with truth tables as well as the use of any published results on the axiomatization and also the comparison of many-valued logics.

Because of the scarcity of work or literature on the problem of comparing many-valued logics, direct calculation turned out to be the most effective method of research. The introduction of a device called a truth class table proved to be invaluable. Such a table allows the logician to work with sets of truth values instead of with individual truth values themselves. Truth class tables were calculated that are characteristic of the logics under consideration.

It was discovered that the usual statement calculus is not suitable for every logic between the two-valued logic and the modified L3. The main result of the thesis is a theorem relating sufficient conditions under which a many-valued logic will have the usual statement calculus for a suitable formal system. It is not yet known whether these conditions are necessary as well as sufficient. Concluding remarks demonstrate, as a corollary to the main result, that for any integer n there exist n-valued logics for which the usual statement calculus is suitable.

Year of Submission


Year of Award

1995 Award


Department of Mathematics

First Advisor

Michael Millar


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Date Original


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