Carnap's Early Metatheory: Scope and Limits

TL;DR

This paper reevaluates Carnap's early attempts at formulating a unified, interpreted type-theoretic metatheory, highlighting its relative coherence and contributions despite initial failures.

Contribution

It offers a detailed reassessment of Carnap's early metatheoretical work, clarifying its significance and limitations in the development of modern logic.

Findings

Carnap's approach was less confused than previously thought

His framework provided valuable insights into meta-logical notions

The reassessment clarifies Carnap's role in logic's development

Abstract

In his Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Rudolf Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap's attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap's contributions to the development of modern logic.