The Problem of Elimination in the Algebra of Logic
Judy Green · Birkhäuser Boston eBooks · 2008
A central objective in any system of logic is to determine what conclusions follow from given premises. A special case is the recognition of valid syllogisms. Intermediate in generality is the problem of elimination, i.e., eliminating a logical variable from an equation or set of equations. The use of the word elimination in this context appears as early as 1854 in George Boole’s treatise, An Investigation of the Laws of Thought, which contains the following description of the problem: As the conclusion must express a relation among the whole or among a part of the elements involved in the premises, it is requisite that we should possess the means of eliminating those elements which we desire not to appear in the conclusion, and of determining the whole amount of relation implied by the premises among the elements which we wish to retain. Those elements which do not present themselves in the conclusion are, in the language of the common Logic, called middle terms; and the species of elimination exemplified in treatises on Logic consists in deducing from two propositions, containing a common element or middle term, a conclusion connecting the two remaining terms. But the problem of elimination, as contemplated in this work, possesses a much wider scope. It proposes not merely the elimination of one middle term from two propositions, but the elimination generally of middle terms from propositions, without regard to the number of either of them, or to the nature of their connexion. To this object neither the processes of Logic nor those of Algebra, in their actual state, present any strict parallel.