The use of dialogues in logic goes back at least to Lorenzen's 1958 paper "Logik und Agon", and has its roots in much older traditions, going back to the pre-Socratics. Our project aims to (1) describe the foundational value of Lorenzen's dialogical logic, and (2) embed it into a modern scientific context taking into account its historical roots.

(1) The foundational investigations consist in (i) discussing and clarifying technical points of dialogue semantics, and (ii) evaluating its philosophical background claims as well as its potential to lay the foundations for logical reasoning in mathematics, computer science and linguistics. These foundational investigations are the heart of the project and will be carried out in the form of two Master Projects, which are not assigned to a particular site, but are central projects which all three principal investigators are jointly committed to.

(2) The embedding into a modern context and the historical roots are subjects of several collaborative projects between two sites each, as well as of individual local projects. They investigate the role of negation and of definitional reasoning as paradigms for reasoning in general, the use of dialogues in informal and semi-formal mathematical proofs, the use of zero-knowledge proofs and extended logic programs in computer science, and of dialogical versions of the propositions-as-types approach within linguistics. Concerning the historical roots we concentrate, besides studying the modern history of dialogical logic, on medieval theories of obligationes.