On Three Levels of Abstractness in Peirce’s Beta Graphs

Abstract

Peirce’s beta graphs are roughly equivalent to our first-order predicate logic. However, Bellucci and Pietarinen have recently argued that the beta graphs are not well-equipped to handle asymmetric relative terms. I survey four proposed solutions to the problem and find them all wanting. I offer a fifth solution according to which Peirce’s beta graphs function at three different levels of abstractness from natural language. I diagnose the problem of asymmetric relative terms as arising when we transition from the first to the second level of abstractness. Making grammatical information encoded in natural language explicit at the first level of abstractness and interpreting graphs at the second level of abstractness as shorn of grammatical information resolves the problem. The solution is both well-motivated by Peirce’s own commitments and increases the expressiveness of the beta graphs.