This paper develops a very powerful formalism (called Direct Logic™) that incorporates the mathematics of Computer Science and allows unstratified inference and reflection using mathematical induction for almost all of classical logic to be used. Direct Logic allows mutual reflection among the code, documentation, and use cases of large software systems thereby overcoming the limitations of the traditional Tarskian framework of stratified metatheories.

Gödel first formalized and proved that it is not possible to decide all mathematical questions by inference in his 1st incompleteness theorem. However, the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency! This paper proves a generalization of the

Gödel/Rosser incompleteness theorem: a strongly paraconsistent theory is incomplete. However, there is a further consequence: Although the semi-classical mathematical fragment of Direct Logic is evidently consistent, since the Gödelian paradoxical proposition is self-provable, every reflective strongly paraconsistent theory in Direct Logic is inconsistent!

This paper also proves that Logic Programming is not computationally universal in that there are concurrent programs for which there is no equivalent in Direct Logic. Thus the Logic Programming paradigm is strictly less general than the Procedural Embedding of Knowledge paradigm.

PDF copy of the paper can be downloaded at: Common sense for concurrency and strong paraconsistency using unstratified inference and reflection