Common sense for concurrency and strong paraconsistency using unstratified inference and reflection
Carl E. Hewitt · arXiv (Cornell University) · 2008
This paper develops a strongly paraconsistent formalism called Direct Logic(TM) 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 mutually chock full of inconsistencies code, documentation, and use cases of large software systems thereby overcoming the limitations of the traditional Tarskian framework of stratified metatheories. Goedel first formalized and proved that it is not possible to decide all mathematical questions by inference in his first incompleteness theorem. However, the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency! This paper proves a generalization of the Goedel/Rosser incompleteness theorem: a strongly paraconsistent theory is self-provably incomplete. However, there is a further consequence: Although the semi-classical mathematical fragment of Direct Logic is evidently consistent, since the Goedelian paradoxical proposition is self-provable, every reflective strongly paraconsistent theory in Direct Logic is self-provably 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. Consequently the Logic Programming paradigm is strictly less general than the Procedural Embedding of Knowledge paradigm. Thus the paper defines a concurrent programming language ActorScript(TM) that is suitable for expressing massive concurrency in large software systems meta-circularly in terms of itself.