In this memoir I examine the principles of the logical calculus of Boole, as laid down in his celebrated treatise on the “Laws of Thought,” and also the criticisms which have been published concerning these principles. I bring forward a new theory of the operation of the mind, founded upon an analysis of language and the nature of mathematical reasoning, which enables me to correct these principles, to place them on a clear, rational, and generalised basis, and to show that there is a logical algebra which coincides with the ordinary algebra when its symbols are integral, but is a generalised form of the ordinary algebra when its symbols are fractional.