Coming to Terms: Proof Theory Extended to Definite Descriptions and other Terms

Introduction

The project is concerned with two areas which so far have rarely come together: complex terms and proof theory. ExtenDD focuses on definite descriptions as the most important and troublesome singular terms and on sequent calculus and its generalizations as the most important tool of modern proof theory.

Historical Context

Since Russell's 'On Denoting', regarded as a paradigm of analytic philosophy, definite descriptions occupy a central place in philosophical research and many deep and detailed studies have been carried out. The second half of the 20th century saw the development of new approaches to this phenomenon based on non-classical logics, in particular free logic in which, contrary to Frege's and Russell's classical logic, it is not assumed that every term refers.

Research Gap

Yet despite the long history of research into definite descriptions, a paradigm of formal logic has so far rarely been applied to them: proof theory. The methods developed by Gentzen, in particular those relating to his sequent calculus, provide the means for the deepest study of proofs and their properties.

Current Limitations

Yet only a small effort has so far been put into the adequate treatment of definite descriptions in this framework. The same counts for other complex singular terms such as set abstracts and number operators. ExtenDD fills this important gap in research.

Objectives

Applying the methods of proof theory to definite descriptions is profitable to both sides. Competing theories of definite descriptions and complex terms in general, their advantages and shortcomings, are shown in a new light.

Challenges and Solutions

The behaviour of complex terms needs subtle syntactical analysis and requires enriching the toolkit of proof theory. ExtenDD deals with both challenges:

1. It develops formal theories of definite descriptions.
2. It modifies the machinery of proof theory to cover new areas of application.

Impact

The realization of ExtenDD affects significantly the field of proof theory, automated deduction, and philosophy of language.