Addressing circular definitions via systems of proofs

Definitions are important tools for our attempt to construct an intelligible imagine of reality. Regarded as such, there are interesting epistemological clues about them to consider: What does a legitimate definition look like? Or: Is there a privileged form that definitions should have? How do we find good definitions? Or: How do we know that a given definition is a good one? Traces of the debate on these, and similar other questions are ubiquitous in the history of philosophy. The aim of this note, however, is not to give a historical account on this matter. Rather, on the basis of some very recent work of a proof–theoretical nature, I plan to address those issues in the context of a discussion concerning circular definitions. A definition is circular in the sense of this paper if the very concept that one is defining is used in the condition defining it. Despite their peculiar character, circular definitions are neither rare, nor easy to dispense with: it turns out that they affect in a significant way the ordinary life, as well as the philosophically interesting level of speech. That is the very reason why they have slowly gathered consideration from scholars in recent times, and have become matter of debate. Circular definitions break the traditional schema that can be used to give an account for ordinary definitions, therefore they raise the problem whether they are legitimate or should be avoided. The goal of this paper is to discuss the issue of circular definitions, illustrate the problematic features connected to it, and present recent developments in the logical research on the topic that help providing them with an arguably plausible justification.