In the opening sentence of his Methods of Logic, W. V. O. Quine writes, “Logic is an old subject, and since 1879 it has been a great one.”1 Quine is referring to the year in which Gottlob Frege presented his Begriffschrift, or “concept-script,” one of the first published accounts of a logical system or calculus with quantification and a function-argument analysis of propositions. There can be no doubt as to the importance of these introductions, and, indeed, Frege’s orientation and advances, (...) if not his particular system, have proven to be highly significant for much of mathematical logic and research pertaining to the foundations of mathematics. Nevertheless, Quine’s assessment, often repeated, obscures other.. (shrink)

Henry Sheffer’s 1908 Harvard Ph.D. thesis contains an interesting appendix on a central feature of the logical work of his thesis advisor, Josiah Royce. This is the claim in Royce’s 1905 article “The Relations of the Principles of Logic to the Foundations of Geometry” that an unsymmetric ordering relation can be defined on the single symmetric O-relation for which he gives postulates in that paper. Sheffer criticizes Royce’s specific definition from the point of view of the evolving twentieth century conception (...) of axiomatic method and also shows more generally that an unsymmetric1 order relation cannot be defined in the system of the 1905 paper based on the O-relation. In this article I wish to consider the nature.. (shrink)