Date of Award

1965

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

Konrad Suprunowicz

Abstract

From an intuitive point of view the notion of effective procedure consists of a set of rules or instructions that enables one, in a finite number of steps and in a purely mechanical way, to answer yes or no to any one of a given class of questions. This procedure requires no intelligence to carry out the instructions and, in fact, it is con­ceivable that some mechanical contrivance may be constructed to carry out these instructions. Should such an effective procedure exist, that answers either yes or no, then the group of problems in question is said to be effectively decidable; otherwise not decidable. In this paper, some of the more important properties of the pro­positional and predicate calculi will be established with the thought in mind of considering the notion of effective procedure relative to these properties. In achieving this end, the propositional and predicate calculi will be considered in a purely formal context. Formal in the sense that on the outset the symbols employed within the theories will be devoid of any interpretation. Later, however, an interpretation will be placed on these symbols in order to answer certain questions concern­ing decidability. In considering the propositional and predicate calculi as formal theories, a distinction must be drawn between those symbols used in the particular theory and the language used to describe this theory. The former use will be referred to as the object language and the latter as the syntax or metalanguage. The object language, for the formal theory under consideration, will

be given explicitely, whereas the metalanguage will consist of, only that portion of the English language needed to clearly describe the formal system. In some instances, since no confusion will result, certain symbols may appear not only in the object language but also in the metalanguage. This will be evident by the two-fold use of the symbols: ~, >,, (, ), and '. Should specific reference be made to some one of the symbols of the theory, this symbol will be enclosed in single quotes. Furthermore, the reasoning employed in establishing results about the formal systems will consist only of those notions which have great intuitive appeal. Included among these will be mathematical induction.

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