On Modal Systems with Rosser Modalities

Sufficiently strong axiomatic theories allow for the construction of self-referential sentences, i.e. sentences saying something about themselves. After the Gödel’s paper on incompleteness (Gödel, 1931) the self-reference method found further applications—some are listed below—and became even more important. Around say 1970 it appeared that the self-reference method was not only a useful tool, but also an interesting field of study: it became clear that reasoning about self-referential sentences could be made more transparent and limitations of the self-reference method could be clarified by using modal logic. The connections of meta-mathematics to modal logic, especially after the Solovay’s paper (Solovay, 1976), brought traditional modal logic to the attention of more mathematically oriented logicians and constituted a stimulus in modal logical studies. In this paper we briefly discuss some of the existing modal systems, putting an emphasis on Rosser modalities, also known as witness comparison modalities, and present one new system of that kind. First we fix some symbolism and the way we speak about arithmetic and self-reference.