The Diagonal Lemma Fails in Aristotelian Logic

TL;DR

This paper explores the limitations of the Diagonal Lemma within Aristotelian logic, demonstrating that self-referential sentences like Gödel's can be considered neither true nor false under Strawson's logic of presuppositions.

Contribution

It extends Strawson's logic of presuppositions to complex sentences and shows the failure of the Diagonal Lemma in Aristotelian logic.

Findings

Self-referential Gödel sentences have empty subjects.

Such sentences are classified as neither true nor false.

The Diagonal Lemma does not hold in this logical framework.

Abstract

In the 1950-ies P. F. Strawson proposed the Logic of Presuppositions. In this system sentences with empty subject are considered neither true nor false. Strawson considered only simple sentences with two predicate letters. But the concept can be extended to arbitrary monadic, even polyadic sentences utilizing truth-relevant logic developed by Richard Diaz. It can be shown that self-referential G\"odel's sentence in fact has an empty subject, and can thus be classified by the Logic of Presuppositions as neither true nor false.