The calculus of boolean structures

Abstract

In this chapter we develop the calculus of boolean structures in a rather algebraic fashion. We do so for a variety of reasons. Firstly, we have to introduce the reader to the repertoire of general formulae that will be used throughout the remainder of this booklet. Secondly, by proving all formulae that have not been postulated, we give the reader the opportunity of gently familiarizing himself with our style of conducting such calculational proofs. Thirdly, we wish to present this material in a way that does justice to how we are going to use it. Since value-preserving transformations are at the heart of our calculus, so are the notions of equality and function application; hence our desire to develop this material with the equality relation in the central rôle. (It is here that our treatment radically departs from almost all introductions to formal logic: it is not uncommon to see the equality —in the form of “if and only if”— being introduced much later as a shorthand, almost as an afterthought.)