Mathematical Foundations and Computational Methods for a Digital Logic Machine

Abstract

The manipulations of the propositional calculus (and the analogous Boolean algebra) have recently become of fundamental importance in fields other than set theoretical proofs of pure mathematics. For example, in the analysis and evaluation of intelligence and military reports, preparation and analysis of tactical methods and principles, complex logical nonnumerical problems often arise more frequently than do problems of a numerical nature. Consequently there is the need for a systematic method of analyzing and solving propositional functions and equations. The purpose of this paper is to present the systematic rules of an “arithmetic” for handling propositional functions and equations, formulated in a manner amenable to easy computation and mechanization. Among the important results given are: (a) a simple method of expressing a propositional calculus in terms of the shortest digitalized words possible, (b) a systematic method for generating the simplest and absolute simplest forms for Boolean functions, (c) a method for systematically generating all true and false implications of given propositions, (d) a method for systematically generating all possible changes of variables (i.e., substitution of new variables) that can be made on a propositional calculus while preserving the logical meaning, and (e) the systematic generation of all solutions to any number of simultaneous propositional equations in any number of unknowns. The solution of this latter problem implies the ability to perform manipulations with propositional variables in a manner analogous to manipulations of ordinary algebraic equations. A number of illustrative applications to military situations are given, including an analysis of certain propositions and conclusions of Clausewitz. Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.