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
Showing posts with label ORGs (Organizations of Restricted Generality). Show all posts
Showing posts with label ORGs (Organizations of Restricted Generality). Show all posts
Saturday, August 23, 2008
Large-scale Organizational Computing requires Unstratified Reflection and Strong Paraconsistency
Organizational Computing is a computational model for using the principles, practices, and methods of human organizations. Organizations of Restricted Generality (ORGs) have been proposed as a foundation for Organizational Computing. ORGs are the natural extension of Web Services, which are rapidly becoming the overwhelming standard for distributed computing and application interoperability in Organizational Computing. The thesis of this paper is that large-scale Organizational Computing requires reflection and strong paraconsistency for organizational practices, policies, and norms.
Strong paraconsistency is required because the practices, policies, and norms of large-scale Organizational Computing are pervasively inconsistent. By the standard rules of logic, anything and everything can be inferred from an inconsistency, e.g., “The moon is made of green cheese.” The purpose of strongly paraconsistent logic is to develop principles of reasoning so that irrelevances cannot be inferred from the fact of inconsistency while preserving all natural inferences that do not explode in the face of inconsistency.
Reflection is required in order that the practices, policies, and norms can mutually refer to each other and make inferences. Reflection and strong paraconsistency are important properties of Direct Logic [Hewitt 2007] for large software systems. Gödel first formalized and proved that it is not possible to decide all mathematical questions by inference in his 1st incompleteness theorem. But the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency! This paper proves a generalization of the Gödel/Rosser incompleteness theorem: theories of Direct Logic are 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 theory in Direct Logic has an inconsistency!
The published paper with the above abstract appears in the excellent Springer LNCS volume Coordination, Organizations, Institutions, and Norms in Agent Systems III (edited by Jaime Sichman, Pablo Noriega, Julian Padget and Sascha Ossowski) and can be found at SpringerLink.
A version with some typos corrected is here: Large-scale Organizational Computing requires Unstratified Reflection and Strong Paraconsistency.
Strong paraconsistency is required because the practices, policies, and norms of large-scale Organizational Computing are pervasively inconsistent. By the standard rules of logic, anything and everything can be inferred from an inconsistency, e.g., “The moon is made of green cheese.” The purpose of strongly paraconsistent logic is to develop principles of reasoning so that irrelevances cannot be inferred from the fact of inconsistency while preserving all natural inferences that do not explode in the face of inconsistency.
Reflection is required in order that the practices, policies, and norms can mutually refer to each other and make inferences. Reflection and strong paraconsistency are important properties of Direct Logic [Hewitt 2007] for large software systems. Gödel first formalized and proved that it is not possible to decide all mathematical questions by inference in his 1st incompleteness theorem. But the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency! This paper proves a generalization of the Gödel/Rosser incompleteness theorem: theories of Direct Logic are 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 theory in Direct Logic has an inconsistency!
The published paper with the above abstract appears in the excellent Springer LNCS volume Coordination, Organizations, Institutions, and Norms in Agent Systems III (edited by Jaime Sichman, Pablo Noriega, Julian Padget and Sascha Ossowski) and can be found at SpringerLink.
A version with some typos corrected is here: Large-scale Organizational Computing requires Unstratified Reflection and Strong Paraconsistency.
Subscribe to:
Posts (Atom)