Published in ArXiv http://arxiv.org/abs/0812.4852 http://commonsense.carlhewitt.info Common sense for concurrency and strong paraconsistency using unstratified inference and reflection Carl Hewitt http://carlhewitt.info This paper is dedicated to John McCarthy. Abstract Unstratified Reflection is the Norm....................................... 11 Abstraction and Reification .............................................. 11 This paper develops a strongly paraconsistent formalism (called Direct Logic™) that incorporates the mathematics of Diagonal Argument .......................................................... 12 Computer Science and allows unstratified inference and Logical Fixed Point Theorem ........................................... 12 reflection using mathematical induction for almost all of Disadvantages of stratified metatheories ........................... 12 classical logic to be used. Direct Logic allows mutual Reification Reflection ....................................................... 13 reflection among the mutually chock full of inconsistencies Incompleteness Theorem for Theories of Direct Logic ..... 14 code, documentation, and use cases of large software systems Inconsistency Theorem for Theories of Direct Logic ........ 15 thereby overcoming the limitations of the traditional Tarskian Consequences of Logically Necessary Inconsistency ........ 16 framework of stratified metatheories. Concurrency is the Norm ...................................................... 16 Gödel first formalized and proved that it is not possible to Nondeterministic computation .......................................... 16 decide all mathematical questions by inference in his 1st Computation is not subsumed by logical deduction .......... 18 incompleteness theorem. However, the incompleteness Arrival order indeterminacy .............................................. 19 theorem (as generalized by Rosser) relies on the assumption of Computational Representation Theorem ........................... 19 consistency! This paper proves a generalization of the Concurrency requires unbounded nondeterminism ........... 20 Gödel/Rosser incompleteness theorem: a strongly Unbounded nondeterminism in an Actor programming paraconsistent theory is self-provably incomplete. However, language ........................................................................... 21 there is a further consequence: Although the semi-classical Scientific Community Metaphor ....................................... 22 mathematical fragment of Direct Logic is evidently consistent, The admission of logical powerlessness ................................ 22 since the Gödelian paradoxical proposition is self-provable, Work to be done ................................................................... 23 every reflective strongly paraconsistent theory in Direct Logic Conclusion ............................................................................ 23 is self-provably inconsistent! Acknowledgements............................................................... 25 This paper also proves that Logic Programming is not References ............................................................................ 26 computationally universal in that there are concurrent programs Appendix. Additional Principles of Direct Logic ................. 33 for which there is no equivalent in Direct Logic. Consequently Relevance Logic ............................................................... 33 the Logic Programming paradigm is strictly less general than Equality ............................................................................ 33 the Procedural Embedding of Knowledge paradigm. Thus the Nondeterministic λ-calculus ............................................. 33 paper defines a concurrent programming language Set Theory ........................................................................ 34 ActorScript™ (that is suitable for expressing massive Noncompactness............................................................... 34 concurrency in large software systems) meta-circularly in Direct Logic is based on XML .......................................... 34 terms of itself. Provably Inference Reflected Propositions in Theories of Direct Logic ..................................................................... 35 Contents Introduction ............................................................................ 2 Limitations of First Order Logic ............................................. 2 Inconsistency is the Norm in Large Software Systems ............ 3 Consistency has been the bedrock of mathematics .................. 4 Paraconsistency has been around for a while. So what‟s new? 4 Direct Logic ........................................................................... 5 Direct Logic is based on argument rather than truth ............ 6 Syntax of Direct Logic ....................................................... 6 Soundness, Faithfulness, and Adequacy ............................. 7 Self-refutation (Direct Indirect Inference) ........................... 8 Booleans ............................................................................ 8 Two-way Deduction Theorem ............................................ 9 Boolean equivalences ....................................................... 10 Disjunction Introduction by Negation ............................... 10 Additional principles ........................................................ 10 Direct Logic uses strong paraconsistency to facilitate theory development ..................................................................... 10 September 26, 2009 Page 1 of 36 Introduction indirect mutual reference pervades reasoning “But if the general truths of Logic are of such a about use cases, documentation, and code. nature that when presented to the mind they at 3. Concurrency is the norm. Logic Programs based once command assent, wherein consists the on the inference rules of mathematical logic are difficulty of constructing the Science of Logic?” not computationally universal because the [Boole 1853 pg 3] message order arrival indeterminate computations of concurrent programs in open systems cannot be Our lives are changing: soon we will always be online. deduced using mathematical logic. The fact that (If you have doubts, check out the kids and the VPs of computation is not reducible to logical inference major corporations.) Because of this change, common has important practical consequences. For sense must adapt to interacting effectively with large example, reasoning used in Semantic Integration software systems just as we have previously adapted cannot be implemented using logical inference common sense to new technology. Logic should provide [Hewitt 2008a]. foundational principles for common sense reasoning about large software systems. John McCarthy is the principal founding Logicist of Large software systems are becoming increasingly Artificial Intelligence although he might decline the permeated with inconsistency, unstratified inference and title.1 Simply put the Logicist Programme is to express reflection, and concurrency. As these inconsistent knowledge in logical propositions and to derive reflective concurrent systems become a major part of information solely by classical logic inferences. the environment in which we live, it becomes an issue Building on the work of many predecessors [Hewitt 2008d], the Logicists Bob Kowalski and Pat Hayes of common sense how to use them effectively. This extended the Logicist Programme by attempting to paper suggests some principles and practices. encompass programming by using classical mathematical logic as a programming language. This paper discusses three challenges to the Logicist Limitations of First Order Logic Programme: 1. Inconsistency is the norm and consequently “A foolish consistency is the hobgoblin of little minds.” classical logic infers too much, i.e., anything and ---Emerson [1841] everything. The experience (e.g. Microsoft, the First Order Logic is woefully lacking for reasoning about US government, IBM, etc.) is that inconsistencies large software systems. (e.g. among implementations, documentation, and For example, a limitation of classical logic for use cases) in large software systems are pervasive inconsistent theories is that it supports the principle that and despite enormous expense have not been from an inconsistency anything can be inferred, e.g. “The eliminated. moon is made of green cheese.” Standard mathematical logic has the problem For convenience, I have given the above principle the that from inconsistent information, any conclusion name IGOR for Inconsistency in Garbage Out Redux.2 whatsoever can be drawn, e.g., “The moon is IGOR can be formalized as follows in which a made of green cheese.” However, our society is contradiction about a proposition Ω infers any increasingly dependent on these large-scale proposition:3 software systems and we need to be able to reason about them. In fact professionals in our society Ω, ¬ Ω ├ reason about these inconsistent systems all the The IGOR principle of classical logic may not seem very time. So evidently they are not bound by classical intuitive! So why is it included in classical logic? mathematical logic. 2. Unstratified inference and reflection are the The IGOR principle is readily derived from the following norm and consequently logic must be extended to principles of classical logic: use unstratified inference and reflection for strongly paraconsistent theories. However, the Full