The Epistemic Route to Reflection

Gödel’s incompleteness theorems show that any formal theory that contains a certain minimal amount of arithmetic is incomplete: there are statements in the language of the theory that the theory cannot either prove or disprove. Such statements include statements (called reflection principles) that formally express the soundness of the theory, i.e. that everything that is provable from the axioms is true. This seems to contrast with our pre-theoretical understanding of arithmetic in everyday life and in science, and with our readiness to accept that every statement that can be proved from our axioms is true. Many authors have therefore sought to justify the addition to our formal theories of such reflection principles by different means. This paper presents a new way of justifying the addition of reflection principles on the basis of epistemic considerations, by employing certain uncontroversial properties of the notion of informal or absolute provability for arithmetic. S!
tarting from the observation that the fundamental properties of this notion can be formally characterised by using the axioms of the modal logic S4 (along the lines already proposed by Gödel), I will show that the recognition that the axioms and rules of inference of Peano arithmetic (PA) correctly formalise informal arithmetical reasoning is sufficient to formally imply the local and uniform reflection schemes.