The deductive system in Boole's Laws of Thought (LT) involves both an algebra, which we call proto-Boolean, and a "general method in Logic" making use of that algebra. Our object is to elucidate these two components of Boole's system, to prove his principal results, and to draw some conclusions not explicit in LT. We also discuss some examples of incoherence in LT; these mask the genius of Boole's design and account for much of the puzzled and disparaging commentary LT has (...) received. Our evaluation of Boole's logical system does not differ substantially from that advanced in Hailperin's exhaustive study, Boole's Logic and Probability. Unlike the latter work, however, we make direct use of the polynomials native to LT rather than appealing to formalisms such as multisets and rings. (shrink)

In 1952, Quine showed that the problem of reducing a propositional formula to a simplest normal equivalent can be solved in two steps, viz., (i) express the given formula, Φ, equivalently as the disjunction of all its prime implicants, and (ii) find all non-redundant disjunctions of the latter that are equivalent to Φ (Quine 1952). However, it seems not generally known that an ingenious form of the same two-step process was published by Hugh McColl in 1878.