Logical necessity is one of the perennial problems of philosophy. Statements like “g = 9.81 m/sec2” or “Radioactivity causes cancer” may be physically necessary, but they are contingent: they could be false, and be refuted by new experience. By contrast, it seems that statements like “¬(p & ¬p),” 2 + 2 = 4,” and “All material objects are located in space” are logically necessary. They do not just happen to be true, since their being false is not merely extremely improbable, but inconceivable. By the same token, disciplines like logic, mathematics, and metaphysics, which seek to discover such truths, seem to be a priori, completely independent of experience.

At the turn of the century there were three accounts of this special type of truth. According to psychologistic logicians like Boole, the laws of logic describe how human beings (by and large) think, their basic mental operations, and are determined by the nature of the human mind. Against this Platonists like Frege protested that logical truths are strictly necessary and objective, and that this objectivity can only be secured by assuming that their subject matter - thoughts and their structures - are not private ideas in the minds of individuals, but abstract entities inhabiting a “third realm” beyond space and time. Finally, according to Russell, logical propositions are completely general truths about the most pervasive traits of reality, a view which is in some ways reminiscent of Mill's claim that mathematical propositions are well-corroborated empirical generalizations.