Carnap and Beth on the Limits of Tolerance

Benjamin Marschall

doi:10.1017/can.2021.16

Abstract

AbstractRudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance does not suffice to remove all obstacles to the employment of infinitary rules.

Highlights

In his Logical Syntax of Language, Rudolf Carnap develops an account of the nature of logic and mathematics that differs radically from the views of his predecessors and contemporaries (1937a)
At the heart of Logical Syntax is the principle of tolerance, according to which we can freely adopt any system of logic we like without further philosophical justification
This principle plays a crucial role in Carnap’s philosophy of mathematics. Since it licences the use of strong metalanguages with infinitary inference rules, Carnap thinks that he is able to capture the idea that mathematical truth is determined by linguistic rules despite the limitative results of Gödel’s incompleteness theorems

Summary

Introduction

In his Logical Syntax of Language, Rudolf Carnap develops an account of the nature of logic and mathematics that differs radically from the views of his predecessors and contemporaries (1937a). At the heart of Logical Syntax is the principle of tolerance, according to which we can freely adopt any system of logic we like without further philosophical justification. This principle plays a crucial role in Carnap’s philosophy of mathematics. Since it licences the use of strong metalanguages with infinitary inference rules, Carnap thinks that he is able to capture the idea that mathematical truth is determined by linguistic rules despite the limitative results of Gödel’s incompleteness theorems. I will outline Carnap’s philosophy of mathematics with an emphasis on the role of tolerance in dealing with Gödel’s incompleteness theorems. Section five concludes the paper by comparing my reading of Beth to other well-known model-theoretic arguments

Carnap’s philosophy of mathematics

Beth and model-theoretic arguments

Findings

Conclusion