The thesis of this book is that Löwenheim’s and Skolem’s work on what is now known as the downward Löwenheim–Skolem theorem developed directly from Schröder’s Algebra der Logik, which was itself an avowed elaboration of the work of the American logician Charles S. Peirce and his student O. H. Mitchell. We have been unable to detect any direct influence of Frege, Russell, or Hilbert on the devel...

We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of minimal logic plus Peirce’s law into minimal logic, which we refer to as the Peirce translation, as it eliminates uses of Peirce’s law. When combined with modified realizability this translation applies to full classical...

Isaac Levi (1980) targets an implicit tension in C.S. Peirce’s epistemology, one that exists between the need to always be open-minded and aware of our propensity to make mistakes so that we do not “block the road of inquiry,” and the need to treat certain beliefs as infallible and to doubt only in a genuine way so that inquiry can proceed in the first place. Attempts at alleviating this tensio...