Is Idempotence “More Fundamental” than Non-Contradiction?

We undertake a thorough examination of George Boole’s claim that, as he discovered by means of his algebra, the law of idempotence is “more fundamental” than the law of non-contradiction (The Laws of Thought, Chapter III, Proposition IV). There is a paucity of sources investigating this subject (with a notable exception being (Béziau 2018)). We query Boole’s claim; we examine if and how we can make sense of it; we identify the notable Aristotelian precedent of philosophical reflections on relative fundamentality of logical principles; and we inquire as to what philosophical view of logic is consistent with Boole’s way of thinking about logical principles. Boole’s thinking is apparently burdened by a metaphysically laden view of logic. We argue in detail that it is a radically different way of thinking about logic—a formalist view that regards logic as manipulation of symbolic resources, congenial to logical positivism—which allows us to make some tentative sense of claims about relative fundamentality of logical laws, insofar as we can define such a notion in a meaningful way. However, on the other hand, entanglements in metaphysically laden phantasmagorias fail to support (or perhaps even fail to make sense of) Boole’s claim. In order to substantiate the metalogical and philosophical–logical claims, we advance and construct formal derivations within different Boolean languages with a view to showing how idempotence is primary in some formal systems, but it is derivable (from non-contradiction) in other systems. Hence, Boole’s claim, as we can make sense of it (as relative derivability), is language-dependent, and we argue that this is consistent with a certain philosophical view of what logic is.