Metalogic, Schopenhauer and Universal Logic

Abstract

Schopenhauer used the word “metalogical” since his first work, On the Fourfold Root of the Principle of Sufficient Reason (1813), being the first to give it a precise meaning and a proper place within a philosophical system. One century later the word “Metalogic” started to be used and promoted in modern logic by the Russian logician Nicolai Vasiliev and the Polish School (Łukasiewicz, Tarski, Wajsberg). The aim of this paper is to examine the relations between the different uses of this word and doing that to try to have a better understanding of what Metalogic is and also logic tout court. In a first section we examine and clarify the meaning of Metalogic in modern logic, comparing Metalogic to Metamathematics and Universal Logic. We make in particular a distinction between two trends in Metalogic that can be crystallized through metatheorem vs. meta-axiom. In a second section we present Schopenhauer’s use of the word, which is essentially through the notion of metalogical truths. We describe their locations within Schopenhauer’s framework, standing side by side with other kinds of truths (metaphysical truths, logical truths, empirical truths), constituting altogether the Principle of Sufficient Reason (PSR) of Knowledge, one of the four roots of the PSR. We explain why Schopenhauer thinks that mathematical truths do not need to have a logical ground and present his view according to which metalogical truths are fundamental laws of thought that cannot be changed. We discuss the feminine nature he attributes to them and establish a parallel with Aristotle’s vision of logic. In a third section we examine how modern logic arose from a double challenge of the fundamental laws of logic: their reformulation and relocation, their relativization and rejection. We emphasize that this dynamic evolution was performed on the basis of some semiotical and conceptual changes at the heart of logic and Metalogic.