Common sense for concurrency and strong paraconsistency using unstratified inference and reflection

http://commonsense.carlhewitt.info/

This article develops a strongly paraconsistent 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 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.

A generalization of the Gödel/Rosser incompleteness theorem is proved, i.e., a strongly paraconsistent theory is self-provably incomplete. 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 self-provably inconsistent!

Alonzo Church, Concurrency, Deduction Theorem, Kurt Gödel, Inconsistency, Logic Programming, Logical Necessity of Inconsistency, Robert Kowalski, John McCarthy, Dana Scott, Strong Paraconsistency
This article is dedicated to John McCarthy.


PDF of full version of this article is available here.


Introduction

“But if the general truths of Logic are of such a nature that when presented to the mind they at once command assent, wherein consists the difficulty of constructing the Science of Logic?” [Boole 1853 pg 3]

Our lives are changing:  soon we will always be online. (If you have doubts, check out the kids and the VPs of major corporations.) Because of this change, common sense must adapt to interacting effectively with large software systems just as we have previously adapted common sense to new technology. Logic should provide foundational principles for common sense reasoning about large software systems.

John McCarthy is the principal founding Logicist of Artificial Intelligence although he might decline the title. (Logicist and Logicism are used in this paper for the general sense pertaining to logic rather than in the restricted technical sense of maintaining that mathematics is in some important sense reducible to logic.) Simply put the Logicist Programme is to express knowledge in logical propositions and to derive information solely by classical logic inferences. The Logicists Bob Kowalski and Pat Hayes extended the Logicist Programme by attempting to encompass programming by using mathematical logic as a programming language to deduce computational steps, which characterizes Logic Programming as it is used in this paper. I.e., Logic Programming is what can be programmed in classical logic.

This paper discusses three challenges to the Logicist Programme:

1.      Inconsistency is the norm and consequently classical logic infers too much, i.e., anything and everything. The experience (e.g. Microsoft, the US government, IBM, etc.) is that inconsistencies (e.g. among implementations, documentation, and use cases) in large software systems are pervasive and despite enormous expense have not been eliminated.

    Standard mathematical logic has the problem that from inconsistent information, any conclusion whatsoever can be drawn, e.g., “The moon is made of green cheese.” However, our society is increasingly dependent on these large-scale software systems and we need to be able to reason about them. In fact professionals in our society reason about these inconsistent systems all the time. So evidently they are not bound by classical mathematical logic.

2.      Unstratified inference and reflection are the norm and consequently logic must be extended to use unstratified inference and reflection for strongly paraconsistent theories. However, the traditional approach (using the Tarskian framework of stratified metatheories) is unsuitable for Software Engineering because unstratified direct and indirect self-reference pervades reasoning about use cases, documentation, and code.

3.      Concurrency is the norm. Logic Programs based on the inference rules of mathematical logic are not computationally universal because the indeterminate computations of concurrent programs in open systems cannot be deduced using mathematical logic.

 

Large software systems are becoming increasingly permeated with inconsistency, unstratified inference and reflection, and concurrency. As these inconsistent reflective concurrent systems become a major part of the environment in which we live, it becomes an issue of common sense how to use them effectively. This paper suggests some principles and practices.

 

Limitations of First Order Logic

“A foolish consistency is the hobgoblin of little minds.”
---Emerson

First Order Logic is woefully lacking for reasoning about large software systems.  For example, a limitation of classical logic for inconsistent theories is that it supports the principle that from an inconsistency anything can be inferred, e.g. “The moon is made of green cheese.”

    For convenience, I have given the above principle the name IGOR for Inconsistency in Garbage Out Redux. IGOR can be formalized as follows in which a contradiction about a proposition Ω infers any proposition Ψ:

                            Ω, ¬Ω ├ Ψ

The IGOR principle of classical logic may not seem very intuitive!  So why is it included in classical logic?

The IGOR principle is readily derived from the following principles of classical logic:

·     Full indirect inference: (Ψ├ Φ, ¬Φ)  →  (├ ¬Ψ) which can be justified in classical logic on the grounds that if Ψ infers a contradiction in a consistent theory then Ψ must be false. In an inconsistent theory, full indirect inference leads to explosion by the following derivation in classical logic by a which a contradiction about P infers any proposition Q:
         P, ¬P ├  ¬Q ├ P, ¬P ├ (¬ ¬Q) ├ Q

·     Disjunction introduction: (Ψ ├ (Ψ∨Φ)) which in classical logic would say that if Ψ is true then (Ψ∨Φ)) is true regardless of whether Φ is true.  In an inconsistent theory, disjunction introduction leads to explosion via the following derivation in classical logic in which a contraction about P infers any proposition Q:
         P,¬ P ├ (P∨Q),¬P ├ Q

First Order Logic has the following additional limitatins:.

  • It lacks reflection so it can’t deal with mutually reflective propositions, e.g., among documentation, uses cases, and implementations of large software systems. Also it is stratified, meaning that different theories cannot mutually refer to each other’s inferences.  In particular a theory cannot directly reason about itself.
  • It doesn’t handle the mathematical induction needed for inferring properties of programs. Nor does it handle reasoning about contention in concurrency.

The plan of this artcle is as follows:

  1. Solve the above problems with First Order Logic by introducing a new system called Direct Logic for large software systems.
  2. Demonstrate that no Logicist system is computationally universal (not even Direct Logic even though it is evidently more powerful than any logic system that has been previously developed).  I.e., there are concurrent programs for which there is no equivalent Logic Program.
  3. Discuss the implications of the above results for common sense.

Inconsistency is the Norm in Large Software Systems

“find bugs faster than developers can fix them and each fix leads to another bug”
--Cusumano & Selby 1995, p. 40

The development of large software systems and the extreme dependence of our society on these systems have introduced new phenomena.  These systems have pervasive inconsistencies among and within the following:

  • Use cases that express how systems can be used and tested in practice
  • Documentation that expresses over-arching justification for systems and their technologies
  • Code that expresses implementations of systems

Adapting a metaphor used by Karl Popper for science, the bold structure of a large software system rises, as it were, above a swamp. It is like a building erected on piles. The piles are driven down from above into the swamp, but not down to any natural or given base; and when we cease our attempts to drive our piles into a deeper layer, it is not because we have reached bedrock. We simply pause when we are satisfied that they are firm enough to carry the structure, at least for the time being.  Or perhaps we do something else more pressing.  Under some piles there is no rock.  Also some rock does not hold.

Different communities are responsible for constructing, evolving, justifying and maintaining documentation, use cases, and code for large, human-interaction, software systems.  In specific cases any one consideration can trump the others.  Sometimes debates over inconsistencies among the parts can become quite heated, e.g., between vendors.  In the long run, after difficult negotiations, in large software systems, use cases, documentation, and code all change to produce systems with new inconsistencies. However, no one knows what they are or where they are located! 

Furthermore there is no evident way to divide up the code, documentation, and use cases into meaningful, consistent microtheories for human-computer interaction.  Organizations such as Microsoft, the US government, and IBM have tens of thousands of employees pouring over hundreds of millions of lines of documentation, code, and use cases attempting to cope.  In the course of time almost all of this code will interoperate using Web Services. A large software system is never done [Rosenberg 2007].

The thinking in almost all scientific and engineering work has been that models (also called theories or microtheories) should be internally consistent, although they could be inconsistent with each other.

Consistency has been the bedrock of mathematics

When we risk no contradiction, it prompts the tongue to deal in fiction. Gay [1727]

Platonic Ideals were to be perfect, unchanging, and eternal. Beginning with the Hellenistic mathematician Euclid [circa 300BC] in Alexandria, theories were intuitively supposed to be both consistent and complete. Wilhelm Leibniz, Giuseppe Peano, George Boole, Augustus De Morgan, Richard Dedekind, Gottlob Frege, Charles Peirce, David Hilbert, etc. developed mathematical logic.  However, a crisis occurred with the discovery of the logical paradoxes based on self-reference by Cesare Burali-Forti [1897], Cantor [1899], Bertrand Russell [1903], etc. In response Russell [1908] stratified types, [Zermelo 1905, Fränkel 1922, Skolem 1922] stratified sets and [Tarski and Vaught 1957] stratified logical theories to limit self-reference. Kurt Gödel [1931] proved that mathematical theories are incomplete, i.e., there are propositions which can neither be proved nor disproved.

 Consequently, although completeness and unrestricted self-reference were discarded for general mathematics, the bedrock of consistency remained.


Paraconsistency has been around for a while. So what’s new?

Within mathematics paraconsistent logic was developed to deal with inconsistent theories. The idea of paraconsistent logic is to be able to make inferences from inconsistent information without being able to derive all propositions, property called “simple paraconsistency” in this paper in contrast to “strong paraconsistency” which is discussed below.

The most extreme form of simple paraconsistent mathematics is dialetheism [Priest and Routley 1989] which maintains that there are true inconsistencies in mathematics itself e.g., the Liar Paradox. However, mathematicians (starting with Euclid) have worked very hard to make their theories consistent and inconsistencies have not been an issue for most working mathematicians. As a result:

·   Since inconsistency was not an issue, mathematical logic focused on the issue of truth and a model theory of truth was developed [Dedekind 1888, Löwenheim 1915, Skolem 1920, Gödel 1930, Tarski and Vaught 1957, Hodges 2006]. More recently there has been work on the development of an unstratified logic of truth [Leitgeb 2007, Feferman 2007a].

·   Simple Paraconsistent logic somewhat languished for lack of subject matter. The lack of subject matter resulted in simple paraconsistent proof theories that were for the most part so awkward as to be unused for mathematical practice.

Consequently mainstream logicians and mathematicians have tended to shy away from simple paraconsistency.

One of the achievements of Direct Logic is the development of an unstratified reflective strongly paraconsistent inference system with mathematical induction that does minimal damage to traditional natural deductive logical reasoning.

Previous simple paraconsistent logics have not been satisfactory for the purposes of Software Engineering because of their many seemingly arbitrary variants and their idiosyncratic inference rules and notation. For example (according to Priest [2006]), most simple paraconsistent and relevance logics rule out Disjunctive Syllogism ((Φ∨Ψ), ¬Φ ├ Ψ). However, Disjunctive Syllogism seems entirely natural for use in Software Engineering!


Goals of Direct Logic

The proof of the pudding is the eating. Cervantes [1605] in Don Quixote. Part 2. Chap. 24

Direct Logic is an unstratified strongly paraconsistent reflective formalism for using inference for large software systems with the following goals:

·        Provide a foundation for strongly paraconsistent theories in Software Engineering.

·        Formalize a notion of “direct” inference for strongly paraconsistent theories.

·        Support all “natural” deductive inference [Fitch 1952; Gentzen 1935] in strongly paraconsistent theories with the exception of general Proof by Contradiction and Disjunction Introduction.

·        Support mutual reflection among code, documentation, and use cases of large software systems.

·        Provide increased safety in reasoning about large software systems using strongly paraconsistent theories.

    Direct Logic supports inference for a strongly paraconsistent reflective theory T (├T ). Consequently, ├T does not support either general indirect inference (proof by contradiction) or disjunction introduction. However, ├T does support all other rules of natural deduction [Fitch 1952]. Consequently, Direct Logic is well suited for practical reasoning about large software systems.

    The theories of Direct Logic are “open” in the sense of open-ended schematic axiomatic systems [Feferman 2007b]. The language of a theory can include any vocabulary in which its axioms may be applied, i.e., it is not restricted to a specific vocabulary fixed in advance (or at any other time).  Indeed a theory can be an open system can receive new information at any time [Hewitt 1991, Cellucci 1992].


Direct Logic is based on argument rather than truth

Partly in reaction to Popper, Lakatos [1967, §2]) calls the view below Euclidean (although there is, of course, no claim concerning Euclid’s own orientation):

“Classical epistemology has for two thousand years modeled its ideal of a theory, whether scientific or mathematical, on its conception of Euclidean geometry. The ideal theory is a deductive system with an indubitable truth-injection at the top (a finite conjunction of axioms)—so that truth, flowing down from the top through the safe truth-preserving channels of valid inferences, inundates the whole system.”

Since truth is out the window for inconsistent theories, we have the following reformulation:

Inference in a theory T (├T) carries argument from antecedents to consequents in chains of inference.


The admission of logical powerlessness

Descartes [1644] put forward the thesis that reflection conveys power, specifically the power of existence, as in “I think, therefore I am.”  Reflection conveys ability for large software systems to reason about the possible outcomes of their actions.  However reflection comes with logical limitations including the following

·        Admissibility. It may not be safe to use reflection on propositions (about outcomes) that are not admissible.

·        Incompleteness. It may be impossible to logically prove or disprove outcomes.

·        Undecidability. Outcomes may be recursively undecidable.

·        Strong Paraconsistency. There are typically good arguments for both sides of contradictory conclusions.

·        Necessary Inconsistency. An unstratified reflective strongly paraconsistent theory of Direct Logic is necessarily inconsistent.

·        Concurrency. Other concurrently operating system components may block, interfere with, or revert possible outcomes.

·        Indeterminacy. Because of concurrency, the outcomes may be physically indeterminate.

·        Entanglement. The very process of reflection about possible outcomes can affect the outcomes.

·        Partiality. There might not be sufficient information or resources available to infer outcomes.

·        Nonuniversality. Logic Programs are not computationally universal because they cannot implement some concurrent programs.

These limitations lead to an admission of logical powerlessness:

In general, a component of a large software system is logically powerless over the outcome of its actions.

This admission of powerlessness needs to become part of the common sense of large software systems.

Work to be done

There is much work to be done to further develop Direct Logic:

·        The consistency of the semi-classical fragment of Direct Logic needs to be proved relative to the consistency of classical mathematics.

·        The decidability of the Variable-free Fragment of Direct Logic needs to be settled. As remarked above, the Boolean Fragment is very close to R-Mingle (which is decidable).

·        Strong Paraconsistency of reflective theories of Direct Logic needs to be formally defined and proved.
     Church remarked as follows concerning a Foundation of Logic that he was developing:

Our present project is to develop the consequences of the foregoing set of postulates until a contradiction is obtained from them, or until the development has been carried so far consistently as to make it empirically probable that no contradiction can be obtained from them. And in this connection it is to be remembered that just such empirical evidence, although admittedly inconclusive, is the only existing evidence of the freedom from contradiction of any system of mathematical logic which has a claim to adequacy. [Church 1933]

Direct Logic is in a similar position except that the task is to demonstrate strong paraconsistency instead of consistency. Also Direct Logic has overcome many of the problems of Church’s Foundation of Logic.

·         Inconsistencies such as the one about ├T ParadoxT are relatively benign in the sense that they lack significant consequences to software engineering. Other propositions such as ├T 1=0 are more malignant because it can be used to paraconsistently infer that all integers are equal to 0. To address malignant propositions, deeper investigations of provability using ╟T must be undertaken.

·        Tooling for Direct Logic needs to be developed to support large software systems.


Conclusion

We are now approaching the half century mark of the Logicist Programme for Artificial Intelligence that was initiated by McCarthy.  It has been a fascinating adventure full of twists and turns!

Logicists are now challenged as to whether they agree that

·        Strong Paraconsistency is the norm.

·        Unstratified inference and reflection are the norm.

·        Logic Programming is not computationally universal.

A number of Logicists feel threatened by the results in this paper.

·        Some would like to stick with just classical logic and not consider strong paraconsistency.

·        Some would like to stick with the Tarskian stratified theories and not consider unstratified inference and reflection.

·        Some would like to stick with just Logic Programming (e.g. nondeterministic Turing Machines and λ calculus) and not consider concurrency.

And some would like to have nothing to do with any of the above!  However, the results in this paper (and the driving technological and economic forces behind them) tend to push towards strong paraconsistency, unstratified inference and reflection, and concurrency. The requirements of large software systems are pushing towards strong paraconsistency and unstratified inference and reflection while Web Services and many-core architectures are pushing towards concurrency. [Hewitt 2005]

Software engineers for large software systems often have good arguments (proofs) for some proposition P and also good arguments (proofs) for the negation of P, which is troubling. So what do large software manufacturers do? If the problem is serious, they bring it before a committee of stakeholders to try and sort it out. In many particularly difficult cases the resulting decision has been to simply live with the problem for a while. Consequently, large software systems are shipped to customers with thousands of known inconsistencies of varying severity. The challenge is to try to keep the situation from getting worse as systems continue to increase in complexity.

 A big advantage of strongly paraconsistent logic is that it makes fewer mistakes than classical logic when dealing with inconsistent theories. Since software engineers have to deal with theories chock full of inconsistencies, strong paraconsistency should be attractive. However, to make it relevant we need to provide them with tools that are cost effective.

This paper develops a very powerful formalism (called Direct Logic) that incorporates the mathematics of Computer Science and allows unstratified inference and reflection for almost all of classical logic to be used in strongly paraconsistent theories in a way that is suitable for Software Engineering. Direct Logic allows unstratified direct and indirect mutual reference among use cases, documentation, and code thereby overcoming the limitations of the traditional assumption of hierarchical 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 theory in Direct Logic 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 theory in Direct Logic is inconsistent! The mathematical exploration of diagonalization and reflection has been through Eubulides [4th century BC], Cantor [1890], Zermelo [1908], Russell [1908], Gödel [1931], Rosser [1936], Turing [1936], Curry [1942], Löb [1955], etc. leading ultimately to logically necessary inconsistency.

The concept of TRUTH has already been hard hit by the pervasive inconsistencies of large software systems. Accepting the necessary logical inconsistency of reflective strongly paraconsistent theories would be another nail in its coffin. Ludwig Wittgenstein (ca. 1939) said “No one has ever yet got into trouble from a contradiction in logic.” to which Alan Turing responded “The real harm will not come in unless there is an application, in which case a bridge may fall down.”[Holt 2006] It seems that we may now have arrived at the remarkable circumstance that we can’t keep our systems from crashing without allowing contradictions into our logic!

   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.

Of course the results of this paper do not diminish the importance of logic. There is much work to be done!

Our everyday life is becoming increasingly dependent on large software systems. And these systems are becoming increasingly permeated with inconsistency, reflection and concurrency. As these strongly paraconsistent reflective concurrent systems become a major part of the environment in which we live, it becomes an issue of common sense how to use them effectively. We will need sophisticated software systems to help people understand and apply the principles and practices suggested in this paper. Creating this software is not a trivial undertaking!


Acknowledgments

Sol Feferman, Mike Genesereth, David Israel, Bill Jarrold, Ben Kuipers, Pat Langley, Vladimir Lifschitz, Frank McCabe, John McCarthy, Fanya S. Montalvo, Peter Neumann, Ray Perrault, Natarajan Shankar, Mark Stickel, Richard Waldinger, and others provided valuable feedback at seminars at Stanford, SRI, and UT Austin to an earlier version of the material in this paper. For the AAAI Spring Symposium’06, Ed Feigenbaum, Mehmet Göker, David Lavery, Doug Lenat, Dan Shapiro, and others provided valuable feedback.  At MIT Henry Lieberman, Ted Selker, Gerry Sussman and the members of Common Sense Research Group made valuable comments.  Reviewers for AAMAS ’06 and ‘07, KR’06, COIN@AAMAS’06 and IJCAR’06 made suggestions for improvement.

In the logic community, Mike Dunn, Sol Feferman, Mike Genesereth, Tim Hinrichs, Mike Kassoff, John McCarthy, Chris Mortensen, Graham Priest, Dana Scott, Richard Weyhrauch and Ed Zalta provided valuable feedback

Dana Scott made helpful suggestions on reflection and incompleteness. Richard Waldinger provided extensive suggestions that resulted in better focusing a previous version of this paper and increasing its readability. Sol Feferman reminded me of the connection between Admissibility and P1. Discussion with Pat Hayes and Bob Kowalski provided insight into the early history of Prolog. Communications from John McCarthy and Marvin Minsky suggested making common sense a focus. Mike Dunn collaborated on looking at the relationship of the Boolean Fragment of Direct Logic to R-Mingle. Greg Restall pointed out that Direct Logic does not satisfy some Relevantist principles. Gerry Allwein and Jeremy Forth made detailed comments and suggestions for improvement. Bob Kowalski and Erik Sandewall provided helpful pointers and discussion of the relationship with their work. Discussions with Ian Mason and Tim Hinrichs helped me develop Löb’s theorem for Direct Logic. Scott Fahlman suggested introducing the roadmap in the introduction of the paper. At CMU, Wilfried Sieg introduced me to his very interesting work with Clinton Field on automating the search for proofs of the Gödel incompleteness theorems. Also at CMU, I had productive discussions with Jeremy Avigad, Randy Bryant, John Reynolds, Katia Sycara, and Jeannette Wing. At my MIT seminar and afterwards, Marvin Minsky, Ted Selker, Gerry Sussman, and Pete Szolovits made helpful comments. Les Gasser, Mike Huhns, Victor Lesser, Pablo Noriega, Sascha Ossowski, Jaime Sichman, Munindar Singh, etc. provided valuable suggestions at AAMAS’07. I had a very pleasant dinner with Harvey Friedman at Chez Panisse after his 2nd Tarski lecture.

Jeremy Forth, Tim Hinrichs, Fanya S. Montalvo, and Richard Waldinger provided helpful comments and suggestions on the logically necessary inconsistencies in theories of Direct Logic. Rineke Verbrugge provided valuable comments and suggestions at MALLOW’07. Mike Genesereth and Gordon Plotkin kindly hosted my lectures at Stanford and Edinburgh, respectively, on “The Logical Necessity of Inconsistency”. Inclusion of Cantor’s diagonal argument as motivation as well as significant improvements in the presentation of the incompleteness and inconsistency theorems were suggested by Jeremy Forth. John McCarthy pointed to the distinction between Logic Programming and the Logicist Programme for Artificial Intelligence. Reviewers at JAIR made useful suggestions. Mark S. Miller made important suggestions for improving the meta-circular definition of ActorScript. Comments by Michael Beeson helped make the presentation of Direct Logic more rigorous. Conversations with Jim Larson helped clarify the relationship between classical logic and the logic of paraconsistent theories.


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