Overcoming Skolem's Criticism of Formal Language Using Robinson's Tools

Abstract

This paper presents Robinson's diagram as a tool to address Skolem's criticism of formal language, which argues that there is no formal way to uniquely define any set of objects. Robinson's diagram symbolically represents information and captures the full "reality" of any given mathematical structure while making the formal language sufficiently comprehensive to fully express mathematical structure. The paper argues how Robinson's empirical and logical tools connect semantics and syntax, and epistemology, formal language, and existence. Robinson believed that we understand a concept only when we can describe it by a set of axioms that brings out the essence of that concept. A complete set of axioms fully describes a concept and has semantic as well as ontological significance. The paper also discusses the transfer principle, which asserts that any statement of a specified type which is true for one structure of class, is true also for some other structure or class of structures. Robinson's diagram has rightfully earned the title ‘diagram’ since it symbolically represents the syntactic as well as the semantic information of a complete set of axioms K and its intended model M. This paper shows how the diagram serves as a useful tool for locating unique models described by a set of axioms while preserving the classical notion of truth and reference without postulating non-natural mental powers.