Published in ArXiv http://arxiv.org/abs/0812.4852 Common sense for concurrency and inconsistency tolerance using Direct LogicTM Carl Hewitt http://carlhewitt.info This paper is dedicated to John McCarthy. Abstract Transitivity of Argumentation ............................................. 8 Soundness, Faithfulness, and Adequacy .............................. 9 Direct Logic is a minimal fix to classical mathematical logic and statistical probability (fuzzy) inference that meets the requirements Housekeeping ...................................................................... 9 of large-scale Internet applications (including sense making for Self-refutation ...................................................................... 9 natural language) by addressing the following issues: Booleans ............................................................................ 10 inconsistency tolerance, contrapositive inference bug, and direct Two-way Deduction Theorem ........................................... 10 argumentation. Boolean equivalences ........................................................ 11 For example, in classical logic, (not WeekdayAt5PM) can be Splitting ............................................................................. 11 inferred from the premises (not TrafficJam) and Absorption ......................................................................... 12 WeekdayAt5PM infers TrafficJam. However, Direct Logic does Decidability of Inference in Boolean Direct Logic ............ 12 not sanction concluding (not WeekdayAt5PM) because this Inconsistency tolerance facilitates theory development ..... 12 might be an undesirable inference. The same issue affects Reification and Abstraction ............................................... 13 probabilistic (fuzzy) inference. Suppose (as above) the probability Roundtripping.................................................................... 14 of TrafficJam is 0 and the probability of TrafficJam given Diagonal Argument ........................................................... 15 WeekdayAt5PM is 1. Then the probability of WeekdayAt5PM Logical Fixed Point Theorem ............................................ 15 is 0. Varying the probability of TrafficJam doesn‘t change the Liar Paradox ...................................................................... 16 principle involved because the probability of WeekdayAt5PM Absolute Incompleteness Theorem .................................... 16 will always be less than or equal to the probability of TrafficJam. Absolute Inconsistency Theorem ...................................... 18 Also, in the Tarskian framework of classical mathematical Logical necessity of inconsistency .................................... 18 logic, expressing argumentation is indirect and awkward. For Don‘t believe every argument ........................................... 19 example a classical theory cannot directly represent its own Contributions of Direct Logic............................................ 20 inference relationship and consequently cannot directly represent Concurrency is the Norm ........................................................... 20 its rules of inference. Nondeterministic computation .......................................... 20 Gödel and Rosser proved that nontrivial mathematical theories Dijkstra argued against unbounded nondeterminism ......... 22 are incomplete using the assumption of consistency of Peano Arguments for incorporating unbounded nondeterminism 22 Arithmetic. This paper proves a generalization of the Bounded nondeterminism in CSP ...................................... 23 Gödel/Rosser incompleteness theorem: theories in Direct Logic Unbounded nondeterminism in ActorScript ...................... 23 are self-provably incomplete using inconsistency tolerant Bounded Nondeterminism of Direct Logic ....................... 24 reasoning. However, there is a further consequence: Since the Computation is not subsumed by logical deduction .......... 24 Gödelian paradoxical proposition is self-provable, theories in Arrival order indeterminacy .............................................. 25 Direct Logic are self-provably inconsistent! Computational Representation Theorem ........................... 26 This paper also proves that Logic Programming is not Scientific Community Metaphor ....................................... 26 computationally universal in that there are concurrent programs for The admission of logical powerlessness ..................................... 27 which there is no equivalent in Direct Logic. Consequently the Logic Programming paradigm is strictly less general than the Work to be done ......................................................................... 27 Procedural Embedding of Knowledge paradigm. Thus the paper Conclusion ................................................................................. 28 Acknowledgements .................................................................... 30 makes use of a concurrent programming language ActorScript™ Bibliography............................................................................... 30 (suitable for expressing massive concurrency in large software Appendix. Additional Principles of Direct Logic ...................... 39 systems) that is defined meta-circularly in terms of itself. Equality ............................................................................. 39 Nondeterministic λ-calculus .............................................. 39 Contents Set Theory ......................................................................... 39 Introduction .................................................................................. 2 Noncompactness ................................................................ 40 Inconsistency is the Norm in Large Software Systems ................. 3 Direct Logic is based on XML .......................................... 40 Consistency has been the bedrock of mathematics ....................... 4 Limitations of Classical Mathematical Logic ............................... 4 Inconsistency Tolerance ....................................................... 4 Contrapositive Inference Bug .............................................. 5 Direct Argumentation .......................................................... 5 Direct Logic .................................................................................. 6 In the argumentation lies the knowledge .............................. 7 Syntax of Direct Logic ......................................................... 7 March 21, 2010 Page 1 of 41 Introduction software systems. The Tarskian framework of “But if the general truths of Logic are of such a hierarchically stratified theories [Tarski and nature that when presented to the mind they at Vaught 1957] is unsuitable for Software once command assent, wherein consists the Engineering. difficulty of constructing the Science of Logic?” 3. Concurrency is the norm. Logic Programs based [Boole 1853 pg 3] on the inference rules of mathematical logic are not computationally universal because the Our lives are changing: soon we will always be online.1 message order arrival indeterminate computations Because of this change, common sense must adapt to of concurrent programs in open systems cannot be interacting effectively with large software systems just deduced using mathematical logic from sentences as we have previously adapted common sense to new about pre-existing conditions. The fact that technology. Logic should provide foundational computation is not reducible to logical inference principles for common sense reasoning about large has important practical consequences. For software systems. John McCarthy is the principal founding Logicist of example, reasoning used in Semantic Integration Artificial Intelligence although he might decline the cannot be implemented using logical inference title.2 Simply put the Logicist Programme is to express [Hewitt 2008a]. knowledge in logical propositions and to derive information solely by classical logic inferences. As these inconsistent concurrent systems become a Building on the work of many predecessors [Hewitt major part of the environment in which we live, it 2008d], the Logicists Bob Kowalski and Pat Hayes extended the Logicist Programme by attempting to becomes an issue of common sense how to use them encompass programming by using classical effectively. This paper suggests some principles and mathematical logic as a programming language. practices. This paper discusses three challenges to the Logicist The plan of this paper is as follows: Programme: 1. Solve the above problems with First Order 1. Inconsistency is the norm and consequently Logic by introducing a new system called classical logic infers too much, i.e., anything and Direct Logic3 for large software systems. everything. The experience (e.g. Microsoft, the 2. Demonstrate that no Logicist system is US government, IBM, etc.) is that inconsistencies computationally universal (not even Direct (e.g. that can be derived from implementations, Logic even though it is evidently more documentation, and use cases) in large software powerful than any logic system that has been systems are pervasive and despite enormous previously developed). I.e., there are expense have not been eliminated. concurrent programs for which there is no Standard mathematical logic has the problem equivalent Logic Program. that from inconsistent information, any conclusion 3. Discuss the implications of the above results